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0704.0002 | Sparsity-certifying Graph Decompositions | Sparsity-certifying Graph Decompositions
Ileana Streinu1∗, Louis Theran2
1 Department of Computer Science, Smith College, Northampton, MA. e-mail: [email protected]
2 Department of Computer Science, University of Massachusetts Amherst. e-mail: [email protected]
Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac-
terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern-
ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have
received increased attention in recent years. In particular, our colored pebbles generalize and strengthen
the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri-
zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble
game with colors. Our work also exposes connections between pebble game algorithms and previous
sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].
1. Introduction and preliminaries
The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs
that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a
graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a
(k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper
range of sparse graphs and 0≤ `≤ k the lower range.
In this paper, we present efficient algorithms for finding decompositions that certify sparsity
in the upper range of `. Our algorithms also apply in the lower range, which was already ad-
dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs
and graphs admitting the decomposition coincide.
Our algorithms are based on a new characterization of sparse graphs, which we call the
pebble game with colors. The pebble game with colors is a simple graph construction rule that
produces a sparse graph along with a sparsity-certifying decomposition.
We define and study a canonical class of pebble game constructions, which correspond to
previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide
a unifying framework for all the previously known special cases, including Nash-Williams-
Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the
properties of the augmenting paths used in matroid union and intersection algorithms[5, 6].
Since the sparse graphs in the upper range are not known to be unions or intersections of the
matroids for which there are efficient augmenting path algorithms, these do not easily apply in
∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO
CCR-0310661 to the first author.
2 Ileana Streinu, Louis Theran
Term Meaning
Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges
Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− `
Block H in G G is sparse, and H is a tight subgraph
Component H of G G is sparse and H is a maximal block
Map-graph Graph that admits an out-degree-exactly-one orientation
(k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs
`Tk Union of ` trees, each vertex is in exactly k of them
Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′)
Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk
Table 1. Sparse graph and decomposition terminology used in this paper.
the upper range. Pebble game with colors constructions may thus be considered a strengthening
of augmenting paths to the upper range of matroidal sparse graphs.
1.1. Sparse graphs
A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤
kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this
paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges
is called tight.
For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges
in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail
in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge.
There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of
a sparse graph. A component is a maximal block.
Table 1 summarizes the sparse graph terminology used in this paper.
1.2. Sparsity-certifying decompositions
A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees.
Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described
by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight
graphs.
A map-graph is a graph that admits an orientation such that the out-degree of each vertex is
exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map-
graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible
configuration certifying that each color forms a map-graph. Map-graphs may be equivalently
defined (see, e.g., [18]) as having exactly one cycle per connected component.1
A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint
map-graphs and ` spanning trees.
Another characterization of map-graphs, which we will use extensively in this paper, is as
the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that
the converse holds as well.
1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases
of the bicycle matroid or spanning pseudoforests.
Sparsity-certifying Graph Decompositions 3
Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a
(2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is
shown with a certifying orientation.
A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each
vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2.
Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the
components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute
multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come
from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition
of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of
tree-pieces in any subgraph G′ has size at least `.
Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an
isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree-
pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges.
These count as three tree-pieces, even though they come from the same back tree when the
whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three
gray tree-pieces and one black one.
Table 1 contains the decomposition terminology used in this paper.
The decomposition problem. We define the decomposition problem for sparse graphs as tak-
ing a graph as its input and producing as output, a decomposition that can be used to certify spar-
sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions;
and the pebble-game-with-colors decomposition, which is defined in the next section.
2. Historical background
The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to
the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint,
4 Ileana Streinu, Louis Theran
Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right
corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray
tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a
single vertex) and one black tree-piece.
Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps-
and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and
matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19].
In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman)
graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay
[21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight
graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the
equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a
direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1.
Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and
proper `Tk graphs for the general upper range. We observe that aside from our new pebble-
game-with-colors decomposition, all the combinatorial characterizations of the upper range of
sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24].
A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick-
son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the
pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and
Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and
left as an open problem using the pebble game to find sparsity certifying decompositions.
3. The pebble game with colors
Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative
integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm
for the decomposition problem later in this paper. Since the phrase “with colors” is necessary
only for comparison to [12], we will omit it in the rest of the paper when the context is clear.
Sparsity-certifying Graph Decompositions 5
We now present the pebble game with colors. The game is played by a single player on a
fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the
addition and/or orientation of an edge. At any moment of time, the state of the game is captured
by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored
by the pebbles on them. While playing the pebble game all edges are directed, and we use the
notation vw to indicate a directed edge from v to w.
We describe the pebble game with colors in terms of its initial configuration and the allowed
moves.
Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices
are shown as black or gray dots. Edges are colored with the color of the pebble on them.
Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start
by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k.
Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume
(w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw
to E(H) and put the pebble picked up from v on the new edge.
Figure 3(a) shows examples of the add-edge move.
Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace
vw with wv in E(H); put the pebble that was on vw on v; and put p on wv.
Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows
examples. The convention in these figures, and throughout this paper, is that pebbles on vertices
are represented as colored dots, and that edges are shown in the color of the pebble on them.
From the definition of the pebble-slide move, it is easy to see that a particular pebble is
always either on the vertex where it started or on an edge that has this vertex as the tail. However,
when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is
sometimes convenient to think of this path reversal sequence as bringing a pebble from the end
of the path to the beginning.
The output of playing the pebble game is its complete configuration.
Output: At the end of the game, we obtain the directed graph H, along with the location
and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble
game configuration colors the edges.
We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game
or that H is a pebble-game graph.
Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions
the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble-
game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a
pebble-game decomposition.
Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges,
and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con-
6 Ileana Streinu, Louis Theran
(a) (b) (c)
Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to
show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an
empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two
black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph.
(c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges
contain a cycle and do not contribute a piece of tree to the subgraph.
Notation Meaning
span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)|
peb(V ′) Number of pebbles on V ′ ⊂V
out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′
pebi(v) Number of pebbles of color ci on v ∈V
outi(v) Number of edges vw colored ci for v ∈V
Table 2. Pebble game notation used in this paper.
nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic
subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′
otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with
the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub-
graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned
by G′.
The properties of pebble-game decompositions are studied in Section 6, and Theorem 2
shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows
this.
For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom-
position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated
vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black
tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not
contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees.
In the following discussion, we use the notation peb(v) for the number of pebbles on v and
pebi(v) to indicate the number of pebbles of colors i on v.
Table 2 lists the pebble game notation used in this paper.
4. Our Results
We describe our results in this section. The rest of the paper provides the proofs.
Sparsity-certifying Graph Decompositions 7
Our first result is a strengthening of the pebble games of [12] to include colors. It says
that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games
discussed in this paper are our pebble game with colors unless noted explicitly.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse
with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph.
Next we consider pebble-game decompositions, showing that they are a generalization of
proper `Tk decompositions that extend to the entire matroidal range of sparse graphs.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse
graphs in the decomposition.
The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus
Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained
by playing the pebble game defined in the previous section. Notice the similarity between the
requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a
proper `Tk .
Our next results show that for any pebble-game graph, we can specialize its pebble game
construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these
specialized pebble game constructions canonical, and using canonical pebble game construc-
tions, we obtain new direct proofs of existing arboricity results.
We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo-
sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning
trees contributes at least one piece of tree to every subgraph.
The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition
is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper
`Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is
sufficient to certify sparsity.
We now state the main theorem for the upper and lower range.
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)-
maps-and-trees.
Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game
graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper
`Tk with kn− ` edges.
As corollaries, we obtain the existing decomposition results for sparse graphs.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph
G is tight if and only if has a (k, `)-maps-and-trees decomposition.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a
proper `Tk .
Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo-
rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem.
Our last result improves on this, showing that a canonical pebble game construction, and thus
8 Ileana Streinu, Louis Theran
a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in
O(n2) time and space.
These time and space bounds mean that our algorithm can be combined with those of [12]
without any change in complexity.
5. Pebble game graphs
In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game
with colors. Since many of the relevant properties of the pebble game with colors carry over
directly from the pebble games of [12], we refer the reader there for the proofs.
We begin by establishing some invariants that hold during the execution of the pebble game.
Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following
invariants are maintained in H:
(I1) There are at least ` pebbles on V . [12]
(I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12]
(I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12]
(I4) For every vertex v ∈V , outi(v)+pebi(v) = 1.
(I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with
a pebble of color ci or a cycle.
Proof. (I1), (I2), and (I3) come directly from [12].
(I4) This invariant clearly holds at the initialization phase of the pebble game with colors.
That add-edge and pebble-slide moves preserve (I4) is clear from inspection.
(I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of
the same color on it. If there is no pebble of that color reachable, then the path must eventually
visit some vertex twice.
From these invariants, we can show that the pebble game constructible graphs are sparse.
Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the
pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight.
The main step in proving that every sparse graph is a pebble-game graph is the following.
Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce
the out degree of v by one.
Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If
peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w.
It follows that any sparse graph has a pebble game construction.
Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse
with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph.
6. The pebble-game-with-colors decomposition
In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We
start with the following lemmas about the structure of monochromatic connected components
in H, the directed graph maintained during the pebble game.
Sparsity-certifying Graph Decompositions 9
Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub-
graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for
i = 1, . . . ,k.
Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex.
Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H
in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these
is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge.
Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′.
Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality
that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and
out-edge tail is the root of a tree-piece.
Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed
path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle.
Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the
monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′.
Since this argument works for any vertex in any color, for each color there is a partitioning
of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each
pebble and out-edge tail is the root of a monochromatic tree, as desired.
Applied to the whole graph Lemma 11 gives us the following.
Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of
color ci is the root of a (possibly empty) monochromatic tree-piece of color ci.
Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′
edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it
to the lower range and giving a construction that finds the tree-pieces, showing the connection
between the `+1 pebble condition and the hereditary condition on proper `Tk .
We conclude our investigation of arbitrary pebble game constructions with a description of
the decomposition induced by the pebble game with colors.
Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game
graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each
is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse
graphs in the decomposition.
Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub-
graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs.
For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can
span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game
decomposition must be sparse. Apply Theorem 1 to complete the proof.
Remark: We observe that a pebble-game decomposition for a Laman graph may be read out
of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed,
pebble game orientations have a natural correspondence with the bipartite matchings used in
10 Ileana Streinu, Louis Theran
Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there
are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be
spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range
pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de-
composition is a proper `Tk .
In the next section, we show that the pebble game can be specialized to correspond to maps-
and-trees and proper `Tk decompositions.
7. Canonical Pebble Game Constructions
In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves-
tigation of decompositions induced by pebble game constructions by studying the case where a
minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15
and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that
this is always possible, implying that monochromatic map-graphs are created only when we
add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that
every color is a forest. Every decomposition characterization of tight graphs discussed above
follows immediately from the main theorem, giving new proofs of the previous results in a
unified framework.
In the proof, we will use two specializations of the pebble game moves. The first is a modi-
fication of the add-edge move.
Canonical add-edge: When performing an add-edge move, cover the new edge with a color
that is on both vertices if possible. If not, then take the highest numbered color present.
The second is a restriction on which pebble-slide moves we allow.
Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a
monochromatic cycle.
We call a pebble game construction that uses only these moves canonical. In this section
we will show that every pebble-game graph has a canonical pebble game construction (Lemma
14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and
maps-and-trees decompositions (Theorem 3 and Theorem 4).
We begin with a technical lemma that motivates the definition of canonical pebble game
constructions. It shows that the situations disallowed by the canonical moves are all the ways
for cycles to form in the lowest ` colors.
Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and
let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created
in exactly one of the following ways:
(M1) The edge vw is added with an add-edge move.
(M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse
edge vw.
Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7.
By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a
connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble
game construction, since the color of an edge only changes when it is inserted the first time or
a new pebble is put on it by a pebble-slide move.
Sparsity-certifying Graph Decompositions 11
vw vw
Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by
adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are
labeled according to their role in the definition of the moves.
Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves,
respectively, in a (2,0)-pebble game construction.
We next show that if a graph has a pebble game construction, then it has a canonical peb-
ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa-
rately. The proof gives two constructions that implement the canonical add-edge and canonical
pebble-slide moves.
Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc-
tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where
`′ = min{k, `}.
Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If
this is not possible, then there are `+1 distinct colors present. Use the highest numbered color
to cover the new edge.
Remark: We note that in the upper range, there is always a repeated color, so no canonical
add-edge moves create cycles in the upper range.
The canonical pebble-slide move is defined by a global condition. To prove that we obtain
the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma
9 to only canonical moves. The main step is to show that if there is any sequence of moves that
reorients a path from v to w, then there is a sequence of canonical moves that does the same
thing.
Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading
to an add-edge move can be replaced with one that has no (M2) steps and allows the same
add-edge move.
In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added,
then it is possible to do this without creating any monochromatic cycles.
12 Ileana Streinu, Louis Theran
Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this
the shortcut construction by analogy to matroid union and intersection augmenting paths used
in previous work on the lower range.
Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step
at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one
application of the shortcut construction reorients a simple path from a vertex w′ to w, and a
path from v to w′ is preserved, the shortcut construction can be applied inductively to find the
sequence of moves we want.
Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines
indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle,
shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part
of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray
tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is
simple, and the shortcut construction can be applied inductively to it.
Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple
path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble
of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v
and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe
that it is rooted at w.
Now consider the edges reversed in our sequence of moves. As noted above, before we make
any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on
this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every
move before the one that reverses some edge yz; prepend onto what is left a sequence of moves
that moves the pebble on w to z in H ′i .
Sparsity-certifying Graph Decompositions 13
Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path.
The path where the pebbles move is indicated by doubled lines.
Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is
(M2); (b) avoiding the (M2) and simplifying the path.
Since no edges change color in the beginning of the new sequence, we have eliminated
the (M2) move. Because our construction does not change any of the edges involved in the
remaining tail of the original sequence, the part of the original path that is left in the new
sequence will still be a simple path in H, meeting our initial hypothesis.
The rest of the lemma follows by induction.
Together Lemma 14 and Lemma 15 prove the following.
Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction.
Using canonical pebble game constructions, we can identify the tight pebble-game graphs
with maps-and-trees and `Tk graphs.
14 Ileana Streinu, Louis Theran
Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game
graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)-
maps-and-trees.
Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game
decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game
graph.
For the reverse direction, consider a canonical pebble game construction of a tight graph.
From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The
definition of the canonical add-edge move implies that there must be at least one pebble of
each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma
12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding
the required ` edge-disjoint spanning trees.
Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph
G is tight if and only if has a (k, `)-maps-and-trees decomposition.
We next consider the decompositions induced by canonical pebble game constructions when
`≥ k +1.
Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb-
ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it
is a proper `Tk with kn− ` edges.
Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show
is that a canonical pebble game construction of a tight graph produces a proper `Tk .
By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom-
position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must
in in exactly k of the trees, as required.
Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a
proper `Tk .
8. Pebble game algorithms for finding decompositions
A naı̈ve implementation of the constructions in the previous section leads to an algorithm re-
quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n)
applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running
time of Θ(n3) for the decomposition problem.
In this section, we describe algorithms for the decomposition problem that run in time
O(n2). We begin with the overall structure of the algorithm.
Algorithm 17 (The canonical pebble game with colors).
Input: A graph G.
Output: A pebble-game graph H.
Method:
– Set V (H) = V (G) and place one pebble of each color on the vertices of H.
– For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide
moves as described by Lemma 15.
Sparsity-certifying Graph Decompositions 15
– If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma
14, otherwise discard vw.
– Finally, return H, and the locations of the pebbles.
Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent
sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction
found is canonical, the main theorem shows that the coloring of the edges in H gives a maps-
and-trees or proper `Tk decomposition.
Complexity. We start by observing that the running time of Algorithm 17 is the time taken to
process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an
edge of G that is added to H.
Each of the pebble game moves can be implemented in constant time. What remains is to
describe an efficient way to find and move the pebbles. We use the following algorithm as a
subroutine of Algorithm 17 to do this.
Algorithm 18 (Finding a canonical path to a pebble.).
Input: Vertices v and w, and a pebble game configuration on a directed graph H.
Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated.
Method:
– Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and
return ‘no.’
– Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi
are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use
the array c[] to keep track of the colors of pebbles on vertices and edges after we move them
and the array s[] to sketch out a canonical path from v to u by finding a successor for each
edge.
– Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in
reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to
the next i. Otherwise, check to see if c[vi+1] = c[ei].
– If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge.
– Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If
a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x
that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i =
1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro-
matic c[x] path from xq−1 to x using this same procedure.
– Finally, slide pebbles along the path from the original endpoints v to u specified by the
successor array s[v], s[s[v]], . . .
The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut
construction. Efficiency comes from the fact that instead of potentially moving the pebble back
and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three
times: once in the initial depth-first search, and twice while converting the initial path to a
canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time
spent processing edges in H.
Although we have not discussed this explicity, for the algorithm to be efficient we need to
maintain components as in [12]. After each accepted edge, the components of H can be updated
in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1)
time each.
16 Ileana Streinu, Louis Theran
Summarizing, we have shown that the canonical pebble game with colors solves the decom-
position problem in time O(n2).
9. An important special case: Rigidity in dimension 2 and slider-pinning
In this short section we present a new application for the special case of practical importance,
k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally
rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the
current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com-
binatorially, we model the bar-slider frameworks as simple graphs together with some loops
placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each
color.
We characterize the minimally rigid bar-slider graphs [20] as graphs that are:
1. (2,3)-sparse for subgraphs containing no loops.
2. (2,0)-tight when loops are included.
We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse
graphs studied in our paper [14].
The connection with the pebble games in this paper is the following.
Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we
replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph.
Proof. Follows from invariant (I3) of Lemma 7.
In [15], we study a special case of slider pinning where every slider is either vertical or
horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction.
For this axis parallel slider case, the minimally rigid graphs are characterized by:
1. (2,3)-sparse for subgraphs containing no loops.
2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each
monochromatic tree spans exactly one loop of its color.
This also has an interpretation in terms of colored pebble games.
Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)-
pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the
graph of a minimally pinned axis-parallel bar-slider framework.
Proof. Follows from Theorem 4, and Lemma 12.
10. Conclusions and open problems
We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and
used it to give an efficient algorithm for finding decompositions of sparse graphs into edge-
disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range
and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the
upper range from [12].
We also used the pebble game with colors to describe a new sparsity-certifying decomposi-
tion that applies to the entire matroidal range of sparse graphs.
Sparsity-certifying Graph Decompositions 17
We defined and studied a class of canonical pebble game constructions that correspond to
either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash-
Williams arboricity theorem and a unified proof of the previously studied decomposition cer-
tificates of sparsity. Canonical pebble game constructions also show the relationship between
the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting
paths, which do not apply in the upper range.
Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2)
algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from
dense ones. Their technique is based on efficiently finding matroid union augmenting paths,
which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to
find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch
scanning, which finds groups of disjoint augmenting paths.
We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester-
mann’s algorithm without changing the running time. The data structures used in the implemen-
tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those
used to support cyclic scanning.
The two major open algorithmic problems related to the pebble game are then:
Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain
an implementable O(n3/2) algorithm for the lower range.
Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble
game algorithm for the upper range.
In particular, it would be of practical importance to find an implementable O(n3/2) algorithm
for decompositions into edge-disjoint spanning trees.
References
1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th
European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003)
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and applications. Algorithmica 7(1), 465–497 (1992)
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sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11
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9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing
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18 Ileana Streinu, Louis Theran
10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the
pebble game. Journal of Computational Physics 137, 346–365 (1997)
11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering
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308(8), 1425–1437 (2008)
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dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg.
cs.uwindsor.ca/papers/72.pdf
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Computer Science 13(10) (2007)
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Canadian Conference on Computational Geometry (CCCG’07) (2007)
16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland,
Amsterdam (1979)
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Mathematical Society 39, 12 (1964)
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(1992)
19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees.
Mathematics of Operations Research 10(4), 701–708 (1985)
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http://cccg.cs.uwindsor.ca/papers/72.pdf
http://cccg.cs.uwindsor.ca/papers/72.pdf
Introduction and preliminaries
Historical background
The pebble game with colors
Our Results
Pebble game graphs
The pebble-game-with-colors decomposition
Canonical Pebble Game Constructions
Pebble game algorithms for finding decompositions
An important special case: Rigidity in dimension 2 and slider-pinning
Conclusions and open problems
|
0704.0003 | The evolution of the Earth-Moon system based on the dark matter field
fluid model | The evolution of the Earth-Moon system based on the dark fluid model
The evolution of the Earth-Moon system based on
the dark matter field fluid model
Hongjun Pan
Department of Chemistry
University of North Texas, Denton, Texas 76203, U. S. A.
Abstract
The evolution of Earth-Moon system is described by the dark matter field fluid
model with a non-Newtonian approach proposed in the Meeting of Division of Particle
and Field 2004, American Physical Society. The current behavior of the Earth-Moon
system agrees with this model very well and the general pattern of the evolution of the
Moon-Earth system described by this model agrees with geological and fossil evidence.
The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago,
which is far beyond the Roche’s limit. The result suggests that the tidal friction may not
be the primary cause for the evolution of the Earth-Moon system. The average dark
matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1.
This model predicts that the Mars’s rotation is also slowing with the angular acceleration
rate about -4.38 × 10-22 rad s-2.
Key Words. dark matter, fluid, evolution, Earth, Moon, Mars
1. Introduction
The popularly accepted theory for the formation of the Earth-Moon system is that
the Moon was formed from debris of a strong impact by a giant planetesimal with the
Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the
formation of the Earth-Moon system, it has been evolving at all time scale. It is well
known that the Moon is receding from us and both the Earth’s rotation and Moon’s
rotation are slowing. The popular theory is that the tidal friction causes all those changes
based on the conservation of the angular momentum of the Earth-Moon system. The
situation becomes complicated in describing the past evolution of the Earth-Moon
system. Because the Moon is moving away from us and the Earth rotation is slowing, this
means that the Moon was closer and the Earth rotation was faster in the past. Creationists
argue that based on the tidal friction theory, the tidal friction should be stronger and the
recessional rate of the Moon should be greater in the past, the distance of the Moon
would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon
would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence
indicates that the recession of the Moon in the past was slower than the present rate, i. e.,
the recession has been accelerating with time. Therefore, it must be concluded that tidal
friction was very much less in the remote past than we would deduce on the basis of
present-day observations (Stacey 1977). This was called “geological time scale
difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the
tidal friction theory (Brush 1983).
But we have to consider the case carefully in various aspects. One possible
scenario is that the Earth has been undergoing dynamic evolution at all time scale since
its inception, the geological and physical conditions (such as the continent positions and
drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at
remote past could be substantially different from currently, in which the tidal friction
could be much less; therefore, the receding rate of the Moon could be slower. Various
tidal friction models were proposed in the past to describe the evolution of the Earth-
Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable
distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray
et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present
rate of tidal dissipation is anomalously high because the tidal force is close to a resonance
in the response function of ocean (Brush 1983). Kagan gave a detailed review about those
tidal friction models (Kagan 1997). Those models are based on many assumptions about
geological (continental position and drifting) and physical conditions in the past, and
many parameters (such as phase lag angle, multi-mode approximation with time
dependent frequencies of the resonance modes, etc.) have to be introduced and carefully
adjusted to make their predictions close to the geological evidence. However, those
assumptions and parameters are still challenged, to certain extent, as concoction.
The second possible scenario is that another mechanism could dominate the
evolution of the Earth-Moon system and the role of the tidal friction is not significant. In
the Meeting of Division of Particle and Field 2004, American Physical Society,
University of California at Riverside, the author proposed a dark matter field fluid model
(Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with
this model very well. This paper will demonstrate that the past evolution of Moon-Earth
system can be described by the dark matter field fluid model without any assumptions
about past geological and physical conditions. Although the subject of the evolution of
the Earth-Moon system has been extensively studied analytically or numerically, to the
author’s knowledge, there are no theories similar or equivalent to this model.
2. Invisible matter
In modern cosmology, it was proposed that the visible matter in the universe is
about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently
invisible which is called dark matter and dark energy, such invisible matter has an anti-
gravity property to make the universe expanding faster and faster.
If the ratio of the matter components of the universe is close to this hypothesis,
then, the evolution of the universe should be dominated by the physical mechanism of
such invisible matter, such physical mechanism could be far beyond the current
Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian
physics could reflect only a corner of the iceberg of the greater physics.
If the ratio of the matter components of the universe is close to this hypothesis,
then, it should be more reasonable to think that such dominant invisible matter spreads in
everywhere of the universe (the density of the invisible matter may vary from place to
place); in other words, all visible matter objects should be surrounded by such invisible
matter and the motion of the visible matter objects should be affected by the invisible
matter if there are interactions between the visible matter and the invisible matter.
If the ratio of the matter components of the universe is close to this hypothesis,
then, the size of the particles of the invisible matter should be very small and below the
detection limit of the current technology; otherwise, it would be detected long time ago
with such dominant amount.
With such invisible matter in mind, we move to the next section to develop the
dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible
matter (dark matter, dark energy and possible other terms) is called dark matter here.
3. The dark matter field fluid model
In this proposed model, it is assumed that:
1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly
filled with the dark matter which is in quiescent state relative to the motion of the
celestial body. The dark matter possesses a field property and a fluid property; it can
interact with the celestial body with its fluid and field properties; therefore, it can have
energy exchange with the celestial body, and affect the motion of the celestial body.
2. The fluid property follows the general principle of fluid mechanics. The dark matter
field fluid particles may be so small that they can easily permeate into ordinary
“baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter
field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the
manner of a sponge moving thru water. The nature of the field property of the dark matter
field fluid is unknown. It is here assumed that the interaction of the field associated with
the dark matter field fluid with the celestial body is proportional to the mass of the
celestial body. The dark matter field fluid is assumed to have a repulsive force against the
gravitational force towards baryonic matter. The nature and mechanism of such repulsive
force is unknown.
With the assumptions above, one can study how the dark matter field fluid may
influence the motion of a celestial body and compare the results with observations. The
common shape of celestial bodies is spherical. According to Stokes's law, a rigid non-
permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds
number experiences a resistance force F
rvF πμ6−= (1)
where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity
constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the
velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual
properties of the dark matter field fluid and its permeation into the sphere, the force F
may not be proportional to the radius of the sphere. Also, F may be proportional to the
mass of the sphere due to the field interaction. Therefore, with the combined effects of
both fluid and field, the force exerted on the sphere by the dark matter field fluid is
assumed to be of the scaled form
(2) mvrF n−−= 16πη
where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be
viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η
is the dark matter field fluid constant, which is equivalent to μ. The direction of the
resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force
described by Eq. 2 is velocity-dependent and causes negative acceleration. According to
Newton's second law of motion, the equation of motion for the sphere is
mvr
m n−−= 16πη (3)
Then
(4) )6exp( 10 vtrvv
n−−= πη
where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a
massive gravitational center, there are three forces in the line between the sphere and the
gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and
(3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the
orbital velocity and causes the sphere to move inward to the gravitational center.
However, if the sum of the centripetal acceleration force and the repulsive force is
stronger than the gravitational force, then, the sphere will move outward and recede from
the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is
sufficiently slow and the repulsive force is small compared to the gravitational force and
centripetal acceleration force, then the rate of receding will be accordingly relatively
slow. Therefore, the gravitational force and the centripetal acceleration force can be
approximately treated in equilibrium at any time. The pseudo equilibrium equation is
GMm 2
2 = (5)
where G is the gravitational constant, M is the mass of the gravitational center, and R is
the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields
)12exp( 1
R n−= πη (6)
(7) )12exp( 10 trRR
n−= πη
where
R = (8)
R0 is the initial distance to the gravitational center. Note that R exponentially increases
with time. The increase of orbital energy with the receding comes from the repulsive
force of dark matter field fluid. The recessional rate of the sphere is
dR n−= 112πη (9)
The acceleration of the recession is
( Rr
Rd n 21
12 −= πη ) . (10)
The recessional acceleration is positive and proportional to its distance to the
gravitational center, so the recession is faster and faster.
According to the mechanics of fluids, for a rigid non-permeable sphere rotating
about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere
ωπμ 38 rT −= (11)
where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is
opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent
dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted
on the sphere is
( ) ωπη mrT n 318 −−= (12)
The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The
torque causes the negative angular acceleration
= (13)
where I is the moment of inertia of the sphere in the dark matter field fluid
( )21
2 nrmI −= (14)
Therefore, the equation of rotation for the sphere in the dark matter field fluid is
ωπη
d −−= 120 (15)
Solving this equation yields
(16) )20exp( 10 tr
n−−= πηωω
where ω0 is the initial angular velocity. One can see that the angular velocity of the
sphere exponentially decreases with time and the angular deceleration is proportional to
its angular velocity.
For the same celestial sphere, combining Eq. 9 and Eq. 15 yields
(17)
The significance of Eq. 17 is that it contains only observed data without assumptions and
undetermined parameters; therefore, it is a critical test for this model.
For two different celestial spheres in the same system, combining Eq. 9 and Eq.
15 yields
67.1
1 −=−=⎟⎟
(18)
This is another critical test for this model.
4. The current behavior of the Earth-Moon system agrees with the model
The Moon-Earth system is the simplest gravitational system. The solar system is
complex, the Earth and the Moon experience not only the interaction of the Sun but also
interactions of other planets. Let us consider the local Earth-Moon gravitational system as
an isolated local gravitational system, i.e., the influence from the Sun and other planets
on the rotation and orbital motion of the Moon and on the rotation of the Earth is
assumed negligible compared to the forces exerted by the moon and earth on each other.
In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data
about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are
listed below for the readers' convenience to verify the calculation because the data may
vary slightly with different data sources.
Moon:
Mean radius: r = 1738.0 km
Mass: m = 7.3483 × 1025 gram
Rotation period = 27.321661 days
Angular velocity of Moon = 2.6617 × 10-6 rad s-1
Mean distance to Earth Rm= 384400 km
Mean orbital velocity v = 1.023 km s-1
Orbit eccentricity e = 0.0549
Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2
= (-1.255 ± 0.024) × 10-4 rad century-2
= (-1.260 ± 0.024) × 10-23 rad s-2
Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1
Earth:
Mean radius: r = 6371.0 km
Mass: m = 5.9742 × 1027 gram
Rotation period = 23 h 56m 04.098904s = 86164.098904s
Angular velocity of rotation = 7.292115 × 10-5 rad s-1
Mean distance to the Sun Rm= 149,597,870.61 km
Mean orbital velocity v = 29.78 km s-1
Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2
The Moon's angular rotation acceleration rate and increase in mean distance to the Earth
(receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey
et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the
result is
039.054.1
10662.21021.1
1092509.31026.1
(19)
The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number
384400 km is assumed to be the distance from the Moon's surface to the Earth's surface.
Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord
with the model used here. The difference (about 7.8%) between the values of -1.54 and -
1.67 may come from several sources:
1. Moon's orbital is not a perfect circle
2. Moon is not a perfect rigid sphere.
3. The effect from Sun and other planets.
4. Errors in data.
5. Possible other unknown reasons.
The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data
sets. The third data set can be used to further test the model. If this model correctly
describes the situation at hand, it should give consistent results for different motions. The
values of n and η calculated from three different data sets are listed below (Note, the
mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are
used in the calculation).
The value of n: n = 0.64
From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1
From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1
From the Moon's recession: η = 4.64 × 10-22 s-1 m-1
One can see that the three values of η are consistent within the range of error in the data.
The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1
By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into
Eq. 18, the result is
14.053.1
6371000
1738000
1021.11029.7
1092509.3105.5
)64.01(
(20)
This is also in accord with the model used here.
The dragging force exerted on the Moon's orbital motion by the dark matter field
fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between
the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field
fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm,
respectively.
5. The evolution of Earth-Moon system
Sonett et al. found that the length of the terrestrial day 900 million years ago was
about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al.,
1996). According to the model presented here, back in that time, the length of the day
was about 19.2 hours, this agrees very well with Sonett et al.'s result.
Another critical aspect of modeling the evolution of the Earth-Moon system is to
give a reasonable estimate of the closest distance of the Moon to the Earth when the
system was established at 4.5 billion years ago. Based on the dark matter field fluid
model, and the above result, the closest distance of the Moon to the Earth was about
259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago,
this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and
McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion
years ago was 250000 km, this is probably the most reasonable number that most
astronomers believe and it agrees excellently with the result of this model. The closest
distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or
242000 km (Hansen, 1982).
According to this model, the length of day of the Earth was about 8 hours at 4.5
billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the
length of day of the Earth with the age of the Earth-Moon system described by this model
along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One
can see that those data fit this model very well in their time range.
Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from
Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model
for past 900 million years. One can see that the model agrees with the geological and
fossil data beautifully.
The important difference of this model with early models in describing the early
evolution of the Earth-Moon system is that this model is based only on current data of the
Moon-Earth system and there are no assumptions about the conditions of earlier Earth
rotation and continental drifting. Based on this model, the Earth-Moon system has been
smoothly evolving to the current position since it was established and the recessional rate
of the Moon has been gradually increasing, however, this description does not take it into
account that there might be special events happened in the past to cause the suddenly
significant changes in the motions of the Earth and the Moon, such as strong impacts by
giant asteroids and comets, etc, because those impacts are very common in the universe.
The general pattern of the evolution of the Moon-Earth system described by this model
agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially
increases with time. One may then imagine that the recessional rate will quickly become
very large. The increase is in fact extremely slow. The Moon's recessional rate will be
3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years.
However, whether the Moon's recession will continue or at some time later another
mechanism will take over is not known. It should be understood that the tidal friction
does affect the evolution of the Earth itself such as the surface crust structure, continental
drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s
rotation, however, such role is not a dominant mechanism.
Unfortunately, there is no data available for the changes of the Earth's orbital
motion and all other members of solar system. According to this model and above results,
the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km
century-1, the length of a year increases about 6.8 ms and the change of the temperature is
-1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on
the Earth. The length of a year at 1 billion years ago would be 80% of the current length
of the year. However, much evidence (growth-bands of corals and shellfish as well as
some other evidences) suggest that there has been no apparent change in the length of the
year over the billion years and the Earth's orbital motion is more stable than its rotation.
This suggests that dark matter field fluid is circulating around Sun with the same
direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the
Earth's orbital motion experiences very little or no dragging force from the dark matter
field fluid. However, this is a conjecture, extensive research has to be conducted to verify
if this is the case.
6. Skeptical description of the evolution of the Mars
The Moon does not have liquid fluid on its surface, even there is no air, therefore,
there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the
Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees
with the model very well. Based on this, one may reasonably think that the Mars’s
rotation should be slowing also.
The Mars is our nearest neighbor which has attracted human’s great attention
since ancient time. The exploration of the Mars has been heating up in recent decades.
NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect
data and study this mysterious planet. So far there is still not enough data about the
history of this planet to describe its evolution. Same as the Earth, the Mars rotates about
its central axis and revolves around the Sun, however, the Mars does not have a massive
moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no
liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to
slow its rotation by tidal friction theories. Based on the above result and current the
Mars's data, this model predicts that the angular acceleration of the Mars should be about
-4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the
solar days/Mars year, the vertical dash line marks the current age of the Mars with
assumption that the Mars was formed in a similar time period of the Earth formation.
Such description was not given before according to the author’s knowledge and is
completely skeptical due to lack of reliable data. However, with further expansion of the
research and exploration on the Mars, we shall feel confident that the reliable data about
the angular rotation acceleration of the Mars will be available in the near future which
will provide a vital test for the prediction of this model. There are also other factors
which may affect the Mars’s rotation rate such as mass redistribution due to season
change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be
carefully analyzed.
7. Discussion about the model
From the above results, one can see that the current Earth-Moon data and the
geological and fossil data agree with the model very well and the past evolution of the
Earth-Moon system can be described by the model without introducing any additional
parameters; this model reveals the interesting relationship between the rotation and
receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in
the same gravitational system, such relationship is not known before. Such success can
not be explained by “coincidence” or “luck” because of so many data involved (current
Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a
“ad hoc” or a wrong model, although the chance for the natural happening of such
“coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s
data will clarify this; otherwise, a new theory from different approach can be developed
to give the same or better description as this model does. It is certain that this model is
not perfect and may have defects, further development may be conducted.
James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar
regions will no longer be regarded as waste places in the universe, which the Creator has
not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find
them to be already full of this wonderful medium; so full, that no human power can
remove it from the smallest portion of space, or produce the slightest flaw in its infinite
continuity. It extends unbroken from star to star ….” The medium that Maxwell talked
about is the aether which was proposed as the carrier of light wave propagation. The
Michelson-Morley experiment only proved that the light wave propagation does not
depend on such medium and did not reject the existence of the medium in the interstellar
space. In fact, the concept of the interstellar medium has been developed dramatically
recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field
fluid is just a part of such wonderful medium and “precisely” described by Maxwell.
7. Conclusion
The evolution of the Earth-Moon system can be described by the dark matter field
fluid model with non-Newtonian approach and the current data of the Earth and the Moon
fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the
Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of
day was about 8 hours. The general pattern of the evolution of the Moon-Earth system
described by this model agrees with geological and fossil evidence. The tidal friction may
not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation
is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2.
References
S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century:
Creationist Arguments for a young Earth. Scientists confront creationism. W. W.
Norton & Company, New York, London, pp 49.
E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood
Cliffs, NJ 07632.
J. O. Dickey, et al., 1994. Science, 265, 482.
D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109.
K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457.
W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504.
B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173.
B. A. Kagan, 1997. Prog. Oceanog. 40, 109.
E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J.
Sediment. Res. 69(6), 1154.
K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York.
H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135.
R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653.
C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the
Earth’s Rotation. Springer-Verlag, Berlin, pp. 154.
L. B. Slichter, 1963. J. Geophys. Res. 68, 14.
C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100.
F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons.
J. W. Wells, 1963. Nature, 197, 948.
Caption
Figure 1, the evolution of Moon’s distance and the length of day of the earth with
the age of the Earth-Moon system. Solid lines are calculated according to the dark matter
field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the
length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al.
The dash line marks the current age of the Earth-Moon system.
Figure 2, the evolution of Solar days of year with the age of the Earth-Moon
system. The solid line is calculated according to dark matter field fluid model. The data
are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current
age (4.5 billion years).
Figure 3, the skeptical description of the evolution of Mars’s length of day and the
solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5
billion years). The vertical dash line marks the current age of Mars.
Figure 1, Moon's distance and the length of day of Earth
change with the age of Earth-Moon system
The age of Earth-Moon system (109 years)
0 1 2 3 4 5
Distance
Length of day
Roche's limit
Hansen's result
Figure 2, the solar days / year vs. the age of the Earth
The age of the Earth (109 years)
3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
|
0704.0004 | A determinant of Stirling cycle numbers counts unlabeled acyclic
single-source automata | A Determinant of Stirling Cycle Numbers Counts Unlabeled
Acyclic Single-Source Automata
DAVID CALLAN
Department of Statistics
University of Wisconsin-Madison
1300 University Ave
Madison, WI 53706-1532
[email protected]
March 30, 2007
Abstract
We show that a determinant of Stirling cycle numbers counts unlabeled acyclic
single-source automata. The proof involves a bijection from these automata to
certain marked lattice paths and a sign-reversing involution to evaluate the deter-
minant.
1 Introduction The chief purpose of this paper is to show bijectively that
a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata.
Specifically, let Ak(n) denote the kn × kn matrix with (i, j) entry
[ ⌊ i−1
⌊ i−1
⌋+1+i−j
, where
is the Stirling cycle number, the number of permutations on [i] with j cycles. For
example,
A2(5) =
1 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0
0 1 3 2 0 0 0 0 0 0
0 0 1 3 2 0 0 0 0 0
0 0 0 1 6 11 6 0 0 0
0 0 0 0 1 6 11 6 0 0
0 0 0 0 0 1 10 35 50 24
0 0 0 0 0 0 1 10 35 50
0 0 0 0 0 0 0 1 15 85
0 0 0 0 0 0 0 0 1 15
http://arxiv.org/abs/0704.0004v1
As evident in the example, Ak(n) is formed from k copies of each of rows 2 through n+1
of the Stirling cycle triangle, arranged so that the first nonzero entry in each row is a 1
and, after the first row, this 1 occurs just before the main diagonal; in other words, Ak(n)
is a Hessenberg matrix with 1s on the infra-diagonal. We will show
Main Theorem. The determinant of Ak(n) is the number of unlabeled acyclic single-
source automata with n transient states on a (k + 1)-letter input alphabet.
Section 2 reviews basic terminology for automata and recurrence relations to count
finite acyclic automata. Section 3 introduces column-marked subdiagonal paths, which
play an intermediate role, and a way to code them. Section 4 presents a bijection from
these column-marked subdiagonal paths to unlabeled acyclic single-source automata. Fi-
nally, Section 5 evaluates detAk(n) using a sign-reversing involution and shows that the
determinant counts the codes for column-marked subdiagonal paths.
2 Automata
A (complete, deterministic) automaton consists of a set of states and an input alphabet
whose letters transform the states among themselves: a letter and a state produce another
state (possibly the same one). A finite automaton (finite set of states, finite input alphabet
of, say, k letters) can be represented as a k-regular directed multigraph with ordered edges:
the vertices represent the states and the first, second, . . . edge from a vertex give the effect
of the first, second, . . . alphabet letter on that state. A finite automaton cannot be acyclic
in the usual sense of no cycles: pick a vertex and follow any path from it. This path must
ultimately hit a previously encountered vertex, thereby creating a cycle. So the term
acyclic is used in the looser sense that only one vertex, called the sink, is involved in
cycles. This means that all edges from the sink loop back to itself (and may safely be
omitted) and all other paths feed into the sink.
A non-sink state is called transient. The size of an acyclic automaton is the number of
transient states. An acyclic automaton of size n thus has transient states which we label
1, 2, . . . , n and a sink, labeled n + 1. Liskovets [1] uses the inclusion-exclusion principle
(more about this below) to obtain the following recurrence relation for the number ak(n)
of acyclic automata of size n on a k-letter input alphabet (k ≥ 1):
ak(0) = 1; ak(n) =
(−1)n−j−1
(j + 1)k(n−j)ak(j), n ≥ 1.
A source is a vertex with no incoming edges. A finite acyclic automaton has at least
one source because a path traversed backward v1 ← v2 ← v3 ← . . . must have distinct
vertices and so cannot continue indefinitely. An automaton is single-source (or initially
connected) if it has only one source. Let Bk(n) denote the set of single-source acyclic
finite (SAF) automata on a k-letter input alphabet with vertices 1, 2, . . . , n + 1 where 1
is the source and n + 1 is the sink, and set bk(n) = | Bk(n) |. The two-line representation
of an automaton in Bk(n) is the 2× kn matrix whose columns list the edges in order. For
example,
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
2 4 6 6 6 6 6 6 6 3 5 3 2 2 6
is in B3(5) and the source-to-sink paths in B include 1
→ 6, 1
→ 6, 1
→ 6, where the alphabet is {a, b, c}.
Proposition 1. The number bk(n) of SAF automata of size n on a k-letter input alphabet
(n, k ≥ 1) is given by
bk(n) =
(−1)n−i
(i+ 1)k(n−i)ak(i)
Remark This formula is a bit more succinct than the the recurrence in [1, Theorem
3.2].
Proof Consider the setA of acyclic automata with transient vertices [n] = {1, 2, . . . , n}
in which 1 is a source. Call 2, 3, . . . , n the interior vertices. For X ⊆ [2, n], let
f(X) = # automata in A whose set of interior vertices includes X,
g(X) = # automata in A whose set of interior vertices is precisely X.
Then f(X) =
Y :X⊆Y⊆[2,n] g(Y ) and by Möbius inversion [2] on the lattice of subsets of
[2, n], g(X) =
Y :X⊆Y⊆[2,n] µ(X, Y )f(Y ) where µ(X, Y ) is the Möbius function for this
lattice. Since µ(X, Y ) = (−1)|Y |−|X| if X ⊆ Y , we have in particular that
g(∅) =
Y⊆[2,n]
(−1)| Y |f(Y ). (1)
Let | Y | = n − i so that 1 ≤ i ≤ n. When Y consists entirely of sources, the vertices
in [n+ 1]\Y and their incident edges form a subautomaton with i transient states; there
are ak(i) such. Also, all edges from the n − i vertices comprising Y go directly into
[n + 1]\Y : (i + 1)k(n−i) choices. Thus f(Y ) = (i + 1)k(n−i)ak(i). By definition, g(∅) is
the number of automata in A for which 1 is the only source, that is, g(∅) = bk(n) and the
Proposition now follows from (1).
An unlabeled SAF automaton is an equivalence class of SAF automata under relabeling
of the interior vertices. Liskovets notes [1] (and we prove below) that Bk(n) has no
nontrivial automorphisms, that is, each of the (n− 1)! relabelings of the interior vertices
of B ∈ Bk(n) produces a different automaton. So unlabeled SAF automata of size n on
a k-letter alphabet are counted by 1
(n−1)!
bk(n). The next result establishes a canonical
representative in each relabeling class.
Proposition 2. Each equivalence class in Bk(n) under relabeling of interior vertices has
size (n− 1)! and contains exactly one SAF automaton with the “last occurrences increas-
ing” property: the last occurrences of the interior vertices—2, 3, . . . , n—in the bottom row
of its two-line representation occur in that order.
Proof The first assertion follows from the fact that the interior vertices of an au-
tomatonB ∈ bk(n) can be distinguished intrinsically, that is, independent of their labeling.
To see this, first mark the source, namely 1, with a mark (new label) v1 and observe that
there exists at least one interior vertex whose only incoming edge(s) are from the source
(the only currently marked vertex) for otherwise a cycle would be present. For each such
interior vertex v, choose the last edge from the marked vertex to v using the built-in
ordering of these edges. This determines an order on these vertices; mark them in order
v2, v3, . . . , vj (j ≥ 2). If there still remain unmarked interior vertices, at least one of them
has incoming edges only from a marked vertex or again a cycle would be present. For
each such vertex, use the last incoming edge from a marked vertex, where now edges are
arranged in order of initial vertex vi with the built-in order breaking ties, to order and
mark these vertices vj+1, vj+2, . . .. Proceed similarly until all interior vertices are marked.
For example, for
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
2 4 6 6 6 6 6 6 6 3 5 3 2 2 6
v1 = 1 and there is just one interior vertex, namely 4, whose only incoming edge is from
the source, and so v2 = 4 and 4 becomes a marked vertex. Now all incoming edges to
both 3 and 5 are from marked vertices and the last such edges (built-in order comes into
play) are 4
→ 5 and 4
→ 3 putting vertices 3, 5 in the order 5, 3. So v3 = 5 and v4 = 3.
Finally, v5 = 2. This proves the first assertion. By construction of the vs, relabeling each
interior vertex i with the subscript of its corresponding v produces an automaton in Bk(n)
with the “last occurrences increasing” property and is the only relabeling that does so.
The example yields
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5
5 2 6 4 3 4 5 5 6 6 6 6 6 6 6
Now let Ck(n) denote the set of canonical SAF automata in Bk(n) representing un-
labeled automata; thus | Ck(n) | =
(n−1)!
bk(n). Henceforth, we identify an unlabeled au-
tomaton with its canonical representative.
3 Column-Marked Subdiagonal Paths
A subdiagonal (k, n, p)-path is a lattice path of steps E = (1, 0) and N = (0, 1), E for
east and N for north, from (0, 0) to (kn, p) that never rise above the line y = 1
x. Let
Ck(n, p) denote the set of such paths.For k ≥ 1, it is clear that Ck(n, p) is nonempty only
for 0 ≤ p ≤ n and it is known (generalized ballot theorem) that
|Ck(n, p) | =
kn− kp+ 1
kn+ p+ 1
kn+ p + 1
A path P in Ck(n, n) can be coded by the heights of its E steps above the line y = −1;
this gives a a sequence (bi)
i=1 subject to the restrictions 1 ≤ b1 ≤ b2 ≤ . . . ≤ bkn and
bi ≤ ⌈i/k⌉ for all i.
A column-marked subdiagonal (k, n, p)-path is one in which, for each i ∈ [1, kn], one of
the lattice squares below the ith E step and above the horizontal line y = −1 is marked,
say with a ‘ ∗ ’. Let C
k(n, p) denote the set of such marked paths.
b b b
b b b b
b b b b
∗ ∗ ∗
(0,0)
(8,4)
y = −1
y = 1
A path in C
2(4, 3)
A marked path P ∗ in C
k(n, n) can be coded by a sequence of pairs
(ai, bi)
where
i=1 is the code for the underlying path P and ai ∈ [1, bi] gives the position of the ∗ in the
ith column. The example is coded by (1, 1), (1, 1), (1, 2), (2, 2), (1, 2), (3, 3), (1, 3), (2, 3).
An explicit sum for |C
k(n, n) | is
k(n, n) | =
1≤b1≤b2≤...≤bkn,
bi ≤ ⌈i/k⌉ for all i
b1b2 . . . bkn,
because the summand b1b2 . . . bkn is the number of ways to insert the ‘ ∗ ’s in the underlying
path coded by (bi)
It is also possible to obtain a recurrence for |C
k(n, p) |, and then, using Prop. 1, to
show analytically that |C
k(n, n) | = | Ck+1(n) |. However, it is much more pleasant to
give a bijection and in the next section we will do so. In particular, the number of SAF
automata on a 2-letter alphabet is
| C2(n) | = |C
1(n, n) | =
1≤b1≤b2≤...≤bn
bi ≤ i for all i
b1b2 . . . bn = (1, 3, 16, 127, 1363, . . .)n≥1,
sequence A082161 in [3].
4 Bijection from Paths to Automata
In this section we exhibit a bijection from C
k(n, n) to Ck+1(n). Using the illustrated
path as a working example with k = 2 and n = 4,
b b b
b b b b
b b b b
∗ ∗ ∗
(0,0)
(8,4)
y = −1
y = 1
first construct the top row of a two-line representation consisting of k + 1 each 1s, 2s,
. . . ,n s and number them left to right:
The last step in the path is necessarily anN step. For the second last, third last,. . .N steps
in the path, count the number of steps following it. This gives a sequence i1, i2, . . . , in−1
satisfying 1 ≤ i1 < i2 < . . . < in−1 and ij ≤ (k + 1)j for all j. Circle the positions
i1, i2, . . . , in−1 in the two-line representation and then insert (in boldface) 2, 3, . . . , n in
the second row in the circled positions:
2 3 4
These will be the last occurrences of 2, 3, . . . , n in the second row. Working from the last
column in the path back to the first, fill in the blanks in the second row left to right as
follows. Count the number of squares from the ∗ up to the path (including the ∗ square)
http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082161
and add this number to the nearest boldface number to the left of the current blank entry
(if there are no boldface numbers to the left, add this number to 1) and insert the result
in the current blank square. In the example the numbers of squares are 2,3,1,2,1,2,1,1
yielding
2 4 5 3 3 5 4 5 4 5 5
This will fill all blank entries except the last. Note that ∗ s in the bottom row correspond
to sink (that is, n+1) labels in the second row. Finally, insert n+1 into the last remaining
blank space to give the image automaton:
1 1 1 2 2 2 3 3 3 4 4 4
2 4 5 3 3 5 4 5 4 5 5 5
This process is fully reversible and the map is a bijection.
5 Evaluation of detAk(n)
For simplicity, we treat the case k = 1, leaving the generalization to arbitrary k
as a not-too-difficult exercise for the interested reader. Write A(n) for A1(n). Thus
A(n) =
1≤i,j≤n
. From the definition of detA(n) as a sum of signed products, we
show that detA(n) is the total weight of certain lists of permutations, each list carrying
weight ±1. Then a weight-reversing involution cancels all −1 weights and reduces the
problem to counting the surviving lists. These surviving lists are essentially the codes for
paths in C
1(n, p), and the Main Theorem follows from §4.
To describe the permutations giving a nonzero contribution to detA(n) =
σ sgn σ×
i=1 ai,σ(i), define the code of a permutation σ on [n] to be the list c = (ci)
i=1 with
ci = σ(i)−(i−1). Since the (i, j) entry of A(n),
, is 0 unless j ≥ i−1, we must have
σ(i) ≥ i−1 for all i. It is well known that there are 2n−1 such permutations, corresponding
to compositions of n, with codes characterized by the following four conditions: (i) ci ≥ 0
for all i, (ii) c1 ≥ 1, (iii) each ci ≥ 1 is immediately followed by ci − 1 zeros in the list,
i=1 ci = n. Let us call such a list a padded composition of n: deleting the zeros
is a bijection to ordinary compositions of n. For example, (3, 0, 0, 1, 2, 0) is a padded
composition of 6. For a permutation σ with padded composition code c, the nonzero
entries in c give the cycle lengths of σ. Hence sgnσ, which is the parity of “n−#cycles
in σ”, is given by (−1)#0s in c.
We have detA(n) =
σ sgn σ
i=1 ai,σ(i) =
σ sgn σ
2i−σ(i)
, and so
detA(n) =
(−1)#0s in c
i+ 1− ci
where the sum is restricted to padded compositions c of n with ci ≤ i for all i (A002083)
because
i+1−ci
= 0 unless ci ≤ i.
Henceforth, let us write all permutations in standard cycle form whereby the smallest
entry occurs first in each cycle and these smallest entries increase left to right. Thus,
with dashes separating cycles, 154-2-36 is the standard cycle form of the permutation
( 1 2 3 4 5 65 2 6 1 4 3 ). We define a nonfirst entry to be one that does not start a cycle. Thus the
preceding permutation has 3 nonfirst entries: 5,4,6. Note that the number of nonfirst
entries is 0 only for the identity permutation. We denote an identity permutation (of any
size) by ǫ.
By definition of Stirling cycle number, the product in (2) counts lists (πi)
i=1 of permu-
tations where πi is a permutation on [i+1] with i+1− ci cycles, equivalently, with ci ≤ i
nonfirst entries. So define Ln to be the set all lists of permutations π = (πi)
i=1 where πi
is a permutation on [i + 1], #nonfirst entries in πi is ≤ i, π1 is the transposition (1,2),
each nonidentity permutation πi is immediately followed by ci − 1 ǫ’s where ci ≥ 1 is the
number of nonfirst entries in πi (so the total number of nonfirst entries is n). Assign a
weight to π ∈ Ln by wt(π) = (−1)
# ǫ’s in π. Then
detA(n) =
wt(π).
We now define a weight-reversing involution on (most of) Ln. Given π ∈ Ln, scan the
list of its component permutations π1 = (1, 2), π2, π3, . . . left to right. Stop at the first
one that either (i) has more than one nonfirst entry, or (ii) has only one nonfirst entry, b
say, and b > maximum nonfirst entry m of the next permutation in the list. Say πk is the
permutation where we stop.
http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002083
In case (i) decrement (i.e. decrease by 1) the number of ǫ’s in the list by splitting πk
into two nonidentity permutations as follows. Let m be the largest nonfirst entry of πk
and let ℓ be its predecessor. Replace πk and its successor in the list (necessarily an ǫ) by
the following two permutations: first the transposition (ℓ,m) and second the permutation
obtained from πk by erasing m from its cycle and turning it into a singleton. Here are
two examples of this case (recall permutations are in standard cycle form and, for clarity,
singleton cycles are not shown).
i 1 2 3 4 5 6
πi 12 13 23 14-253 ǫ ǫ
i 1 2 3 4 5 6
πi 12 13 23 25 14-23 ǫ
i 1 2 3 4 5 6
πi 12 23 14 13-24 ǫ 23
i 1 2 3 4 5 6
πi 12 23 14 24 13 23
The reader may readily check that this sends case (i) to case (ii).
In case (ii), πk is a transposition (a, b) with b > maximum nonfirst entry m of πk+1. In
this case, increment the number of ǫ’s in the list by combining πk and πk+1 into a single
permutation followed by an ǫ: in πk+1, b is a singleton; delete this singleton and insert b
immediately after a in πk+1 (in the same cycle). The reader may check that this reverses
the result in the two examples above and, in general, sends case (ii) to case (i). Since the
map alters the number of ǫ’s in the list by 1, it is clearly weight-reversing. The map fails
only for lists that both consist entirely of transpositions and have the form
(a1, b1), (a2, b2), . . . , (an, bn) with b1 ≤ b2 ≤ . . . ≤ bn.
Such lists have weight 1. Hence detA(n) is the number of lists
(ai, bi)
satisfying
1 ≤ ai < bi ≤ i+ 1 for 1 ≤ i ≤ n, and b1 ≤ b2 ≤ . . . ≤ bn. After subtracting 1 from each
bi, these lists code the paths in C
1(n, n) and, using §4, detA(n) = |C
1(n, n) | = | C2(n) |.
References
[1] Valery A. Liskovets, Exact enumeration of acyclic deterministic au-
tomata, Disc. Appl. Math., in press, 2006. Earlier version available at
http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html
http://www.i3s.unice.fr/fpsac/FPSAC03/articles.html
[2] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, 2nd ed., Cambridge
University Press, NY, 2001.
[3] Neil J. Sloane (founder and maintainer), The On-Line Encyclopedia of Integer Se-
quences http://www.research.att.com:80/ njas/sequences/index.html?blank=1
http://www.research.att.com:80/~njas/sequences/index.html?blank=1
|
0704.0005 | From dyadic $\Lambda_{\alpha}$ to $\Lambda_{\alpha}$ | FROM DYADIC Λα TO Λα
WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Abstract. In this paper we show how to compute the Λα norm , α ≥ 0,
using the dyadic grid. This result is a consequence of the description of
the Hardy spaces Hp(RN ) in terms of dyadic and special atoms.
Recently, several novel methods for computing the BMO norm of a function
f in two dimensions were discussed in [9]. Given its importance, it is also of
interest to explore the possibility of computing the norm of a BMO function,
or more generally a function in the Lipschitz class Λα, using the dyadic grid
in RN . It turns out that the BMO question is closely related to that of
approximating functions in the Hardy space H1(RN ) by the Haar system.
The approximation in H1(RN ) by affine systems was proved in [2], but this
result does not apply to the Haar system. Now, if HA(R) denotes the closure
of the Haar system in H1(R), it is not hard to see that the distance d(f,HA)
of f ∈ H1(R) to HA is ∼
f(x) dx
∣, see [1]. Thus, neither dyadic atoms
suffice to describe the Hardy spaces, nor the evaluation of the norm in BMO
can be reduced to a straightforward computation using the dyadic intervals.
In this paper we address both of these issues. First, we give a characterization
of the Hardy spaces Hp(RN ) in terms of dyadic and special atoms, and then,
by a duality argument, we show how to compute the norm in Λα(R
N ), α ≥ 0,
using the dyadic grid.
We begin by introducing some notations. Let J denote a family of cubes
Q in RN , and Pd the collection of polynomials in R
N of degree less than or
equal to d. Given α ≥ 0, Q ∈ J , and a locally integrable function g, let pQ(g)
denote the unique polynomial in P[α] such that [g − pQ(g)]χQ has vanishing
moments up to order [α].
For a locally square-integrable function g, we consider the maximal function
α,J g(x) given by
α,J g(x) = sup
x∈Q,Q∈J
|Q|α/N
|g(y)− pQ(g)(y)|
1991 Mathematics Subject Classification. 42B30,42B35.
http://arxiv.org/abs/0704.0005v1
2 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The Lipschitz space Λα,J consists of those functions g such that M
α,J g is
in L∞, ‖g‖Λα,J = ‖M
α,J g‖∞; when the family in question contains all cubes
in RN , we simply omit the subscript J . Of course, Λ0 = BMO.
Two other families, of dyadic nature, are of interest to us. Intervals in R of
the form In,k = [ (k−1)2
n, k2n], where k and n are arbitrary integers, positive,
negative or 0, are said to be dyadic. In RN , cubes which are the product of
dyadic intervals of the same length, i.e., of the form Qn,k = In,k1 ×· · ·×In,kN ,
are called dyadic, and the collection of all such cubes is denoted D.
There is also the family D0. Let I
n,k = [(k− 1)2
n, (k+ 1)2n], where k and
n are arbitrary integers. Clearly I ′n,k is dyadic if k is odd, but not if k is even.
Now, the collection {I ′n,k : n, k integers} contains all dyadic intervals as well
as the shifts [(k − 1)2n + 2n−1, k 2n + 2n−1] of the dyadic intervals by their
half length. In RN , put D0 = {Q
n,k : Q
n,k = I
× · · · × I ′n,kN }; Q
n,k is
called a special cube. Note that D0 contains D properly.
Finally, given I ′n,k, let I
n,k = [(k − 1)2
n, k2n], and I
n,k = [k2
n, (k + 1)2n].
The 2N subcubes of Q′n,k = I
× · · · × I ′n,kN of the form I
× · · · × I
Sj = L or R, 1 ≤ j ≤ N , are called the dyadic subcubes of Q
Let Q0 denote the special cube [−1, 1]
N . Given α ≥ 0, we construct a
family Sα of piecewise polynomial splines in L
2(Q0) that will be useful in
characterizing Λα. Let A be the subspace of L
2(Q0) consisting of all functions
with vanishing moments up to order [α] which coincide with a polynomial
in P[α] on each of the 2
N dyadic subcubes of Q0. A is a finite dimensional
subspace of L2(Q0), and, therefore, by the Graham-Schmidt orthogonalization
process, say, A has an orthonormal basis in L2(Q0) consisting of functions
p1, . . . , pM with vanishing moments up to order [α], which coincide with a
polynomial in P[α] on each dyadic subinterval of Q0. Together with each p
we also consider all dyadic dilations and integer translations given by
pLn,k,α(x) = 2
n(N+α)pL(2nx1 + k1, . . . , 2
nxN + kN ) , 1 ≤ L ≤ M ,
and let
Sα = {p
n,k,α : n, k integers, 1 ≤ L ≤ M} .
Our first result shows how the dyadic grid can be used to compute the
norm in Λα.
Theorem A. Let g be a locally square-integrable function and α ≥ 0. Then,
g ∈ Λα if, and only if, g ∈ Λα,D and Aα(g) = supp∈Sα
∣〈g, p〉
∣ < ∞. Moreover,
‖g‖Λα ∼ ‖g‖Λα,D +Aα(g) .
Furthermore, it is also true, and the proof is given in Proposition 2.1 be-
low, that ‖g‖Λα ∼ ‖g‖Λα,D0 . However, in this simpler formulation, the tree
structure of the cubes in D has been lost.
FROM DYADIC Λα TO Λα 3
The proof of Theorem A relies on a close investigation of the predual of
Λα, namely, the Hardy space H
p(RN ) with 0 < p = (α + N)/N ≤ 1. In the
process we characterize Hp in terms of simpler subspaces: H
, or dyadic Hp,
and H
, the space generated by the special atoms in Sα. Specifically, we
Theorem B. Let 0 < p ≤ 1, and α = N(1/p− 1). We then have
Hp = H
where the sum is understood in the sense of quasinormed Banach spaces.
The paper is organized as follows. In Section 1 we show that individual
Hp atoms can be written as a superposition of dyadic and special atoms;
this fact may be thought of as an extension of the one-dimensional result of
Fridli concerning L∞ 1- atoms, see [5] and [1]. Then, we prove Theorem B.
In Section 2 we discuss how to pass from Λα,D, and Λα,D0 , to the Lipschitz
space Λα.
1. Characterization of the Hardy spaces Hp
We adopt the atomic definition of the Hardy spaces Hp, 0 < p ≤ 1, see
[6] and [10]. Recall that a compactly supported function a with [N(1/p− 1)]
vanishing moments is an L2 p -atom with defining cube Q if supp(a) ⊆ Q, and
|Q|1/p
| a(x) |2dx
≤ 1 .
The Hardy space Hp(RN ) = Hp consists of those distributions f that can be
written as f =
λjaj , where the aj ’s are H
p atoms,
|λj |
p < ∞, and the
convergence is in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
|λj |
where the infimum is taken over all possible atomic decompositions of f . This
last expression has traditionally been called the atomic Hp norm of f .
Collections of atoms with special properties can be used to gain a better
understanding of the Hardy spaces. Formally, let A be a non-empty subset
of L2 p -atoms in the unit ball of Hp. The atomic space H
spanned by A
consists of those ϕ in Hp of the form
λjaj , aj ∈ A ,
|λj |
p < ∞ .
It is readily seen that, endowed with the atomic norm
‖ϕ‖Hp
= inf
|λj |
: ϕ =
λj aj , aj ∈ A
becomes a complete quasinormed space. Clearly, H
⊆ Hp, and, for
f ∈ H
, ‖f‖Hp ≤ ‖f‖Hp
4 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
Two families are of particular interest to us. When A is the collection
of all L2 p -atoms whose defining cube is dyadic, the resulting space is H
or dyadic Hp. Now, although ‖f‖Hp ≤ ‖f‖Hp
, the two quasinorms are not
equivalent on H
. Indeed, for p = 1 and N = 1, the functions
fn(x) = 2
n[χ[1−2−n,1](x) − χ[1,1+2−n](x)] ,
satisfy ‖fn‖H1 = 1, but ‖fn‖H1
∼ |n| tends to infinity with n.
Next, when Sα is the family of piecewise polynomial splines constructed
above with α = N(1/p − 1), in analogy with the one-dimensional results in
[4] and [1], H
is referred to as the space generated by special atoms.
We are now ready to describe Hp atoms as a superposition of dyadic and
special atoms.
Lemma 1.1. Let a be an L2 p -atom with defining cube Q, 0 < p ≤ 1,
and α = N(1/p − 1). Then a can be written as a linear combination of 2N
dyadic atoms ai, each supported in one of the dyadic subcubes of the smallest
special cube Qn,k containing Q, and a special atom b in Sα. More precisely,
a(x) =
i=1 di ai(x) +
L=1 cL p
−n,−k,α(x), with |di| , |cL| ≤ c.
Proof. Suppose first that the defining cube of a is Q0, and let Q1, . . . , Q2N
denote the dyadic subcubes of Q0. Furthermore, let {e
i , . . . , e
i } denote an
orthonormal basis of the subspace Ai of L
2(Qi) consisting of polynomials in
P[α], 1 ≤ i ≤ 2
N . Put
αi(x) = a(x)χQi (x)−
〈aχQi , e
j(x) , 1 ≤ i ≤ 2
and observe that 〈αi, e
j〉 = 0 for 1 ≤ j ≤ M . Therefore, αi has [α] vanishing
moments, is supported in Qi, and
‖αi‖2 ≤ ‖aχQi‖2 +
‖aχQi‖2 ≤ (M + 1) ‖aχQi‖2 .
ai(x) =
2N(1/2−1/p)
M + 1
αi(x) , 1 ≤ i ≤ N ,
is an L2 p - dyadic atom. Finally, put
b(x) = a(x) −
M + 1
2N(1/2−1/p)
ai(x) .
FROM DYADIC Λα TO Λα 5
Clearly b has [α] vanishing moments, is supported in Q0, coincides with a
polynomial in P[α] on each dyadic subcube of Q0, and
‖b‖22 ≤
|〈aχQi , e
2 ≤ M ‖a‖22 .
So, b ∈ A, and, consequently, b(x) =
L=1 cL p
L(x), where
|cL| = |〈b, p
L〉| ≤ c , 1 ≤ L ≤ M .
In the general case, let Q be the defining cube of a, side-length Q = ℓ, and
let n and k = (k1, . . . , kN ) be chosen so that 2
n−1 ≤ ℓ < 2n, and
Q ⊂ [(k1 − 1)2
n, (k1 + 1)2
n]× · · · × [(kN − 1)2
n, (kN + 1)2
Then, (1/2)N ≤ |Q|/2nN < 1.
Now, given x ∈ Q0, let a
′ be the translation and dilation of a given by
a′(x) = 2nN/pa(2nx1 − k1, . . . , 2
nxN − kN ) .
Clearly, [α] moments of a′ vanish, and
‖a′‖2 = 2
nN/p 2−nN/2‖a‖2 ≤ c |Q|
1/p|Q|−1/2‖a‖2 ≤ c .
Thus, a′ is a multiple of an atom with defining cube Q0. By the first part of
the proof,
a′(x) =
i(x) +
L(x) , x ∈ Q0 .
The support of each a′i is contained in one of the dyadic subcubes of Q0, and,
consequently, there is a k such that
ai(x) = 2
−nN/pa′i(2
−nx1 − k1, . . . , 2
−nxN − kN )
ai is an L
2p -atom supported in one of the dyadic subcubes of Q. Similarly
for the pL’s. Thus,
a(x) =
di ai(x) +
−n,−k,N(1/p−1)(x) ,
and we have finished. �
Theorem B follows readily from Lemma 1.1. Clearly, H
→֒ Hp.
Conversely, let f =
j λj aj be in H
p. By Lemma 1.1 each aj can be written
as a sum of dyadic and special atoms, and, by distributing the sum, we can
write f = fd + fs, with fd in H
, fs in H
, and
‖fd‖Hp
, ‖fs‖Hp
|λj |
Taking the infimum over the decompositions of f we get ‖f‖Hp
c ‖f‖Hp , and H
p →֒ H
. This completes the proof.
6 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
The meaning of this decomposition is the following. Cubes in D are con-
tained in one of the 2N non-overlapping quadrants of RN . To allow for the
information carried by a dyadic cube to be transmitted to an adjacent dyadic
cube, they must be connected. The pLn,k,α’s channel information across ad-
jacent dyadic cubes which would otherwise remain disconnected. The reader
will have no difficulty in proving the quantitative version of this observation:
Let T be a linear mapping defined on Hp, 0 < p ≤ 1, that assumes values in
a quasinormed Banach space X . Then, T is continuous if, and only if, the
restrictions of T to H
and H
are continuous.
2. Characterizations of Λα
Theorem A describes how to pass from Λα,D to Λα, and we prove it next.
Since (Hp)∗ = Λα and (H
)∗ = Λα,D, from Theorem B it follows readily that
Λα = Λα,D ∩ (H
)∗, so it only remains to show that (H
)∗ is characterized
by the condition Aα(g) < ∞.
First note that if g is a locally square-integrable function with Aα(g) < ∞
and f =
j,L cj,L p
nj ,kj ,α
, since 0 < p ≤ 1,
|〈g, f〉| ≤
|cj,L| |〈g, p
nj ,kj ,α
≤ Aα(g)
|cj,L|
and, consequently, taking the infimum over all atomic decompositions of f in
, we get g ∈ (H
)∗ and ‖g‖(Hp
)∗ ≤ Aα(g).
To prove the converse we proceed as in [3]. Let Qn = [−2
n, 2n]N . We begin
by observing that functions f in L2(Qn) that have vanishing moments up to
order [α] and coincide with polynomials of degree [α] on the dyadic subcubes
of Qn belong to H
‖f‖Hp
≤ |Qn|
1/p−1/2‖f‖2 .
Given ℓ ∈ (H
)∗, for a fixed n let us consider the restriction of ℓ to the space
of L2 functions f with [α] vanishing moments that are supported in Qn. Since
|ℓ(f)| ≤ ‖ℓ‖ ‖f‖Hp
≤ ‖ℓ‖ |Qn|
1/p−1/2‖f‖2 ,
this restriction is continuous with respect to the norm in L2 and, consequently,
it can be extended to a continuous linear functional in L2 and represented as
ℓ(f) =
f(x) gn(x) dx ,
FROM DYADIC Λα TO Λα 7
where gn ∈ L
2(Qn) and satisfies ‖gn‖2 ≤ ‖ℓ‖ |Qn|
1/p−1/2. Clearly, gn is
uniquely determined in Qn up to a polynomial pn in P[α]. Therefore,
gn(x) − pn(x) = gm(x)− pm(x) , a.e. x ∈ Qmin(n,m) .
Consequently, if
g(x) = gn(x)− pn(x) , x ∈ Qn ,
g(x) is well defined a.e. and, if f ∈ L2 has [α] vanishing moments and is
supported in Qn, we have
ℓ(f) =
f(x) gn(x) dx
f(x) [gn(x)− pn(x)] dx
f(x) g(x) dx .
Moreover, since each 2nN/ppL(2n ·+k) is an L2 p-atom, 1 ≤ L ≤ M , it readily
follows that
Aα(g) = sup
1≤L≤M
n,k∈Z
|〈g, 2−n/ppL(2n ·+k)〉|
≤ ‖ℓ‖ sup
‖pL‖Hp ≤ ‖ℓ‖ ,
and, consequently, Aα(g) ≤ ‖ℓ‖ , and (H
)∗ is the desired space. �
The reader will have no difficulty in showing that this result implies the
following: Let T be a bounded linear operator from a quasinormed space X
into Λα,D. Then, T is bounded from X into Λα if, and only if, Aα(Tx) ≤
c ‖x‖X for every x ∈ X .
The process of averaging the translates of dyadic BMO functions leads to
BMO, and is an important tool in obtaining results in BMO once they are
known to be true in its dyadic counterpart, BMOd, see [7]. It is also known
that BMO can be obtained as the intersection of BMOd and one of its shifted
counterparts, see [8]. These results motivate our next proposition, which
essentially says that g ∈ Λα if, and only if, g ∈ Λα,D and g is in the Lipschitz
class obtained from the shifted dyadic grid. Note that the shifts involved in
this class are in all directions parallel to the coordinate axis and depend on
the side-length of the cube.
Proposition 2.1. Λα = Λα,D0 , and ‖g‖Λα ∼ ‖g‖Λα,D0 .
Proof. It is obvious that ‖g‖Λα,D0 ≤ ‖g‖Λα . To show the other inequality we
invoke Theorem A. Since D ⊂ D0, it suffices to estimate Aα(g), or, equiva-
lently, |〈g, p〉| for p ∈ Sα, α = N(1/p − 1). So, pick p = p
n,k,α in Sα. The
defining cube Q of pLn,k,α is in D0, and, since p
n,k,α has [α] vanishing moments,
8 WAEL ABU-SHAMMALA AND ALBERTO TORCHINSKY
〈pLn,k,α, pQ(g)〉 = 0. Therefore,
|〈g, pLn,k,α〉| = |〈g − pQ(g), p
n,k,α〉|
≤ ‖pLn,k,α‖2 ‖g − pQ(g)‖L2(Q)
≤ |Q|α/N |Q|1/2‖pLn,k,α‖2 ‖g‖Λα,D0 .
Now, a simple change of variables gives |Q|α/N |Q|1/2‖pLn,k,α‖2 ≤ 1, and, con-
sequently, also Aα(g) ≤ ‖g‖Λα,D0 . �
References
[1] W. Abu-Shammala, J.-L. Shiu, and A. Torchinsky, Characterizations of the Hardy
space H1 and BMO, preprint.
[2] H.-Q. Bui and R. S. Laugesen, Approximation and spanning in the Hardy space, by
affine systems, Constr. Approx., to appear.
[3] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a
distibution, II, Advances in Math., 24 (1977), 101–171.
[4] G. S. de Souza, Spaces formed by special atoms, I, Rocky Mountain J. Math. 14 (1984),
no. 2, 423–431.
[5] S. Fridli, Transition from the dyadic to the real nonperiodic Hardy space, Acta Math.
Acad. Paedagog. Niházi (N.S.) 16 (2000), 1–8, (electronic).
[6] J. Garćıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related
topics, Notas de Matemática 116, North Holland, Amsterdam, 1985.
[7] J. Garnett and P. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2,
351–371.
[8] T. Mei, BMO is the intersection of two translates of dyadic BMO, C. R. Math. Acad.
Sci. Paris 336 (2003), no. 12, 1003–1006.
[9] T. M. Le and L. A. Vese, Image decomposition using total variation and div( BMO)∗,
Multiscale Model. Simul. 4, (2005), no. 2, 390–423.
[10] A. Torchinsky, Real-variable methods in harmonic analysis, Dover Publications, Inc.,
Mineola, NY, 2004.
Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: [email protected]
Department of Mathematics, Indiana University, Bloomington IN 47405
E-mail address: [email protected]
1. Characterization of the Hardy spaces Hp
2. Characterizations of
References
|
0704.0007 | Polymer Quantum Mechanics and its Continuum Limit | Polymer Quantum Mechanics and its Continuum Limit
Alejandro Corichi,1, 2, 3, ∗ Tatjana Vukašinac,4, † and José A. Zapata1, ‡
Instituto de Matemáticas, Unidad Morelia, Universidad Nacional Autónoma de México,
UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
A. Postal 70-543, México D.F. 04510, Mexico
Institute for Gravitational Physics and Geometry, Physics Department,
Pennsylvania State University, University Park PA 16802, USA
Facultad de Ingenieŕıa Civil, Universidad Michoacana de San Nicolas de Hidalgo,
Morelia, Michoacán 58000, Mexico
A rather non-standard quantum representation of the canonical commutation relations of quan-
tum mechanics systems, known as the polymer representation has gained some attention in recent
years, due to its possible relation with Planck scale physics. In particular, this approach has been
followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we
explore different aspects of the relation between the ordinary Schrödinger theory and the polymer
description. The paper has two parts. In the first one, we derive the polymer quantum mechanics
starting from the ordinary Schrödinger theory and show that the polymer description arises as an
appropriate limit. In the second part we consider the continuum limit of this theory, namely, the
reverse process in which one starts from the discrete theory and tries to recover back the ordinary
Schrödinger quantum mechanics. We consider several examples of interest, including the harmonic
oscillator, the free particle and a simple cosmological model.
PACS numbers: 04.60.Pp, 04.60.Ds, 04.60.Nc 11.10.Gh.
I. INTRODUCTION
The so-called polymer quantum mechanics, a non-
regular and somewhat ‘exotic’ representation of the
canonical commutation relations (CCR) [1], has been
used to explore both mathematical and physical issues in
background independent theories such as quantum grav-
ity [2, 3]. A notable example of this type of quantization,
when applied to minisuperspace models has given way to
what is known as loop quantum cosmology [4, 5]. As in
any toy model situation, one hopes to learn about the
subtle technical and conceptual issues that are present
in full quantum gravity by means of simple, finite di-
mensional examples. This formalism is not an exception
in this regard. Apart from this motivation coming from
physics at the Planck scale, one can independently ask
for the relation between the standard continuous repre-
sentations and their polymer cousins at the level of math-
ematical physics. A deeper understanding of this relation
becomes important on its own.
The polymer quantization is made of several steps.
The first one is to build a representation of the
Heisenberg-Weyl algebra on a Kinematical Hilbert space
that is “background independent”, and that is sometimes
referred to as the polymeric Hilbert space Hpoly. The
second and most important part, the implementation of
dynamics, deals with the definition of a Hamiltonian (or
Hamiltonian constraint) on this space. In the examples
∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
studied so far, the first part is fairly well understood,
yielding the kinematical Hilbert space Hpoly that is, how-
ever, non-separable. For the second step, a natural im-
plementation of the dynamics has proved to be a bit more
difficult, given that a direct definition of the Hamiltonian
Ĥ of, say, a particle on a potential on the space Hpoly is
not possible since one of the main features of this repre-
sentation is that the operators q̂ and p̂ cannot be both
simultaneously defined (nor their analogues in theories
involving more elaborate variables). Thus, any operator
that involves (powers of) the not defined variable has to
be regulated by a well defined operator which normally
involves introducing some extra structure on the configu-
ration (or momentum) space, namely a lattice. However,
this new structure that plays the role of a regulator can
not be removed when working in Hpoly and one is left
with the ambiguity that is present in any regularization.
The freedom in choosing it can be sometimes associated
with a length scale (the lattice spacing). For ordinary
quantum systems such as a simple harmonic oscillator,
that has been studied in detail from the polymer view-
point, it has been argued that if this length scale is taken
to be ‘sufficiently small’, one can arbitrarily approximate
standard Schrödinger quantum mechanics [2, 3]. In the
case of loop quantum cosmology, the minimum area gap
A0 of the full quantum gravity theory imposes such a
scale, that is then taken to be fundamental [4].
A natural question is to ask what happens when we
change this scale and go to even smaller ‘distances’, that
is, when we refine the lattice on which the dynamics of
the theory is defined. Can we define consistency con-
ditions between these scales? Or even better, can we
take the limit and find thus a continuum limit? As it
http://arxiv.org/abs/0704.0007v2
mailto:[email protected]
mailto:[email protected]
mailto:[email protected]
has been shown recently in detail, the answer to both
questions is in the affirmative [6]. There, an appropriate
notion of scale was defined in such a way that one could
define refinements of the theory and pose in a precise
fashion the question of the continuum limit of the theory.
These results could also be seen as handing a procedure
to remove the regulator when working on the appropri-
ate space. The purpose of this paper is to further explore
different aspects of the relation between the continuum
and the polymer representation. In particular in the first
part we put forward a novel way of deriving the polymer
representation from the ordinary Schrödinger represen-
tation as an appropriate limit. In Sec. II we derive two
versions of the polymer representation as different lim-
its of the Schrödinger theory. In Sec. III we show that
these two versions can be seen as different polarizations
of the ‘abstract’ polymer representation. These results,
to the best of our knowledge, are new and have not been
reported elsewhere. In Sec. IV we pose the problem of
implementing the dynamics on the polymer representa-
tion. In Sec. V we motivate further the question of the
continuum limit (i.e. the proper removal of the regulator)
and recall the basic constructions of [6]. Several exam-
ples are considered in Sec. VI. In particular a simple
harmonic oscillator, the polymer free particle and a sim-
ple quantum cosmology model are considered. The free
particle and the cosmological model represent a general-
ization of the results obtained in [6] where only systems
with a discrete and non-degenerate spectrum where con-
sidered. We end the paper with a discussion in Sec. VII.
In order to make the paper self-contained, we will keep
the level of rigor in the presentation to that found in the
standard theoretical physics literature.
II. QUANTIZATION AND POLYMER
REPRESENTATION
In this section we derive the so called polymer repre-
sentation of quantum mechanics starting from a specific
reformulation of the ordinary Schrödinger representation.
Our starting point will be the simplest of all possible
phase spaces, namely Γ = R2 corresponding to a particle
living on the real line R. Let us choose coordinates (q, p)
thereon. As a first step we shall consider the quantization
of this system that leads to the standard quantum theory
in the Schrödinger description. A convenient route is to
introduce the necessary structure to define the Fock rep-
resentation of such system. From this perspective, the
passage to the polymeric case becomes clearest. Roughly
speaking by a quantization one means a passage from the
classical algebraic bracket, the Poisson bracket,
{q, p} = 1 (1)
to a quantum bracket given by the commutator of the
corresponding operators,
[ q̂, p̂] = i~ 1̂ (2)
These relations, known as the canonical commutation re-
lation (CCR) become the most common corner stone of
the (kinematics of the) quantum theory; they should be
satisfied by the quantum system, when represented on a
Hilbert space H.
There are alternative points of departure for quantum
kinematics. Here we consider the algebra generated by
the exponentiated versions of q̂ and p̂ that are denoted
U(α) = ei(α q̂)/~ ; V (β) = ei(β p̂)/~
where α and β have dimensions of momentum and length,
respectively. The CCR now become
U(α) · V (β) = e(−iα β)/~V (β) · U(α) (3)
and the rest of the product is
U(α1)·U(α2) = U(α1+α2) ; V (β1)·V (β2) = V (β1+β2)
The Weyl algebra W is generated by taking finite linear
combinations of the generators U(αi) and V (βi) where
the product (3) is extended by linearity,
(Ai U(αi) +Bi V (βi))
From this perspective, quantization means finding an
unitary representation of the Weyl algebra W on a
Hilbert space H′ (that could be different from the ordi-
nary Schrödinger representation). At first it might look
weird to attempt this approach given that we know how
to quantize such a simple system; what do we need such
a complicated object as W for? It is infinite dimensional,
whereas the set S = {1̂, q̂, p̂}, the starting point of the
ordinary Dirac quantization, is rather simple. It is in
the quantization of field systems that the advantages of
the Weyl approach can be fully appreciated, but it is
also useful for introducing the polymer quantization and
comparing it to the standard quantization. This is the
strategy that we follow.
A question that one can ask is whether there is any
freedom in quantizing the system to obtain the ordinary
Schrödinger representation. On a first sight it might seem
that there is none given the Stone-Von Neumann unique-
ness theorem. Let us review what would be the argument
for the standard construction. Let us ask that the repre-
sentation we want to build up is of the Schrödinger type,
namely, where states are wave functions of configuration
space ψ(q). There are two ingredients to the construction
of the representation, namely the specification of how the
basic operators (q̂, p̂) will act, and the nature of the space
of functions that ψ belongs to, that is normally fixed by
the choice of inner product on H, or measure µ on R.
The standard choice is to select the Hilbert space to be,
H = L2(R, dq)
the space of square-integrable functions with respect to
the Lebesgue measure dq (invariant under constant trans-
lations) on R. The operators are then represented as,
q̂ · ψ(q) = (q ψ)(q) and p̂ · ψ(q) = −i ~ ∂
ψ(q) (4)
Is it possible to find other representations? In order to
appreciate this freedom we go to the Weyl algebra and
build the quantum theory thereon. The representation
of the Weyl algebra that can be called of the ‘Fock type’
involves the definition of an extra structure on the phase
space Γ: a complex structure J . That is, a linear map-
ping from Γ to itself such that J2 = −1. In 2 dimen-
sions, all the freedom in the choice of J is contained in
the choice of a parameter d with dimensions of length. It
is also convenient to define: k = p/~ that has dimensions
of 1/L. We have then,
Jd : (q, k) 7→ (−d2 k, q/d2)
This object together with the symplectic structure:
Ω((q, p); (q′, p′)) = q p′ − p q′ define an inner product on
Γ by the formula gd(· ; ·) = Ω(· ; Jd ·) such that:
gd((q, p); (q
′, p′)) =
q q′ +
which is dimension-less and positive definite. Note that
with this quantities one can define complex coordinates
(ζ, ζ̄) as usual:
q + i
p ; ζ̄ =
q − i d
from which one can build the standard Fock representa-
tion. Thus, one can alternatively view the introduction
of the length parameter d as the quantity needed to de-
fine (dimensionless) complex coordinates on the phase
space. But what is the relevance of this object (J or
d)? The definition of complex coordinates is useful for
the construction of the Fock space since from them one
can define, in a natural way, creation and annihilation
operators. But for the Schrödinger representation we are
interested here, it is a bit more subtle. The subtlety is
that within this approach one uses the algebraic prop-
erties of W to construct the Hilbert space via what is
known as the Gel’fand-Naimark-Segal (GNS) construc-
tion. This implies that the measure in the Schrödinger
representation becomes non trivial and thus the momen-
tum operator acquires an extra term in order to render
the operator self-adjoint. The representation of the Weyl
algebra is then, when acting on functions φ(q) [7]:
Û(α) · φ(q) := (eiα q/~ φ)(q)
V̂ (β) · φ(q) := e
(q−β/2)
φ(q − β)
The Hilbert space structure is introduced by the defini-
tion of an algebraic state (a positive linear functional)
ωd : W → C, that must coincide with the expectation
value in the Hilbert space taken on a special state ref-
ered to as the vacuum: ωd(a) = 〈â〉vac, for all a ∈ W .
In our case this specification of J induces such a unique
state ωd that yields,
〈Û(α)〉vac = e−
d2 α2
~2 (5)
〈V̂ (β)〉vac = e−
d2 (6)
Note that the exponents in the vacuum expectation
values correspond to the metric constructed out of J :
d2 α2
= gd((0, α); (0, α)) and
= gd((β, 0); (β, 0)).
Wave functions belong to the space L2(R, dµd), where
the measure that dictates the inner product in this rep-
resentation is given by,
dµd =
d2 dq
In this representation, the vacuum is given by the iden-
tity function φ0(q) = 1 that is, just as any plane wave,
normalized. Note that for each value of d > 0, the rep-
resentation is well defined and continuous in α and β.
Note also that there is an equivalence between the q-
representation defined by d and the k-representation de-
fined by 1/d.
How can we recover then the standard representation
in which the measure is given by the Lebesgue measure
and the operators are represented as in (4)? It is easy to
see that there is an isometric isomorphism K that maps
the d-representation in Hd to the standard Schrödinger
representation in Hschr by:
ψ(q) = K · φ(q) = e
d1/2π1/4
φ(q) ∈ Hschr = L2(R, dq)
Thus we see that all d-representations are unitarily equiv-
alent. This was to be expected in view of the Stone-Von
Neumann uniqueness result. Note also that the vacuum
now becomes
ψ0(q) =
d1/2π1/4
2 d2 ,
so even when there is no information about the param-
eter d in the representation itself, it is contained in the
vacuum state. This procedure for constructing the GNS-
Schrödinger representation for quantum mechanics has
also been generalized to scalar fields on arbitrary curved
space in [8]. Note, however that so far the treatment has
all been kinematical, without any knowledge of a Hamil-
tonian. For the Simple Harmonic Oscillator of mass m
and frequency ω, there is a natural choice compatible
with the dynamics given by d =
, in which some
calculations simplify (for instance for coherent states),
but in principle one can use any value of d.
Our study will be simplified by focusing on the funda-
mental entities in the Hilbert Space Hd , namely those
states generated by acting with Û(α) on the vacuum
φ0(q) = 1. Let us denote those states by,
φα(q) = Û(α) · φ0(q) = ei
The inner product between two such states is given by
〈φα, φλ〉d =
dµd e
~ = e−
(λ−α)2 d2
4 ~2 (7)
Note incidentally that, contrary to some common belief,
the ‘plane waves’ in this GNS Hilbert space are indeed
normalizable.
Let us now consider the polymer representation. For
that, it is important to note that there are two possible
limiting cases for the parameter d: i) The limit 1/d 7→ 0
and ii) The case d 7→ 0. In both cases, we have ex-
pressions that become ill defined in the representation or
measure, so one needs to be careful.
A. The 1/d 7→ 0 case.
The first observation is that from the expressions (5) and
(6) for the algebraic state ωd, we see that the limiting
cases are indeed well defined. In our case we get, ωA :=
lim1/d→0 ωd such that,
ωA(Û(α)) = δα,0 and ωA(V̂ (β)) = 1 (8)
From this, we can indeed construct the representation
by means of the GNS construction. In order to do that
and to show how this is obtained we shall consider several
expressions. One has to be careful though, since the limit
has to be taken with care. Let us consider the measure
on the representation that behaves as:
dµd =
d2 dq 7→ 1
so the measures tends to an homogeneous measure but
whose ‘normalization constant’ goes to zero, so the limit
becomes somewhat subtle. We shall return to this point
later.
Let us now see what happens to the inner product
between the fundamental entities in the Hilbert Space Hd
given by (7). It is immediate to see that in the 1/d 7→ 0
limit the inner product becomes,
〈φα, φλ〉d 7→ δα,λ (9)
with δα,λ being Kronecker’s delta. We see then that the
plane waves φα(q) become an orthonormal basis for the
new Hilbert space. Therefore, there is a delicate interplay
between the two terms that contribute to the measure in
order to maintain the normalizability of these functions;
we need the measure to become damped (by 1/d) in order
to avoid that the plane waves acquire an infinite norm
(as happens with the standard Lebesgue measure), but
on the other hand the measure, that for any finite value
of d is a Gaussian, becomes more and more spread.
It is important to note that, in this limit, the operators
Û(α) become discontinuous with respect to α, given that
for any given α1 and α2 (different), its action on a given
basis vector ψλ(q) yields orthogonal vectors. Since the
continuity of these operators is one of the hypotesis of
the Stone-Von Neumann theorem, the uniqueness result
does not apply here. The representation is inequivalent
to the standard one.
Let us now analyze the other operator, namely the
action of the operator V̂ (β) on the basis φα(q):
V̂ (β) · φα(q) = e−
~ e(β/d
2+iα/~)q
which in the limit 1/d 7→ 0 goes to,
V̂ (β) · φα(q) 7→ ei
~ φα(q)
that is continuous on β. Thus, in the limit, the operator
p̂ = −i~∂q is well defined. Also, note that in this limit
the operator p̂ has φα(q) as its eigenstate with eigenvalue
given by α:
p̂ · φα(q) 7→ αφα(q)
To summarize, the resulting theory obtained by taking
the limit 1/d 7→ 0 of the ordinary Schrödinger descrip-
tion, that we shall call the ‘polymer representation of
type A’, has the following features: the operators U(α)
are well defined but not continuous in α, so there is no
generator (no operator associated to q). The basis vec-
tors φα are orthonormal (for α taking values on a contin-
uous set) and are eigenvectors of the operator p̂ that is
well defined. The resulting Hilbert space HA will be the
(A-version of the) polymer representation. Let us now
consider the other case, namely, the limit when d 7→ 0.
B. The d 7→ 0 case
Let us now explore the other limiting case of the
Schrödinger/Fock representations labelled by the param-
eter d. Just as in the previous case, the limiting algebraic
state becomes, ωB := limd→0 ωd such that,
ωB(Û(α)) = 1 and ωB(V̂ (β)) = δβ,0 (10)
From this positive linear function, one can indeed con-
struct the representation using the GNS construction.
First let us note that the measure, even when the limit
has to be taken with due care, behaves as:
dµd =
d2 dq 7→ δ(q) dq
That is, as Dirac’s delta distribution. It is immediate to
see that, in the d 7→ 0 limit, the inner product between
the fundamental states φα(q) becomes,
〈φα, φλ〉d 7→ 1 (11)
This in fact means that the vector ξ = φα − φλ belongs
to the Kernel of the limiting inner product, so one has to
mod out by these (and all) zero norm states in order to
get the Hilbert space.
Let us now analyze the other operator, namely the
action of the operator V̂ (β) on the vacuum φ0(q) = 1,
which for arbitrary d has the form,
φ̃β := V̂ (β) · φ0(q) = e
(q−β/2)
The inner product between two such states is given by
〈φ̃α, φ̃β〉d = e−
(α−β)2
In the limit d → 0, 〈φ̃α, φ̃β〉d → δα,β. We can see then
that it is these functions that become the orthonormal,
‘discrete basis’ in the theory. However, the function φ̃β(q)
in this limit becomes ill defined. For example, for β > 0,
it grows unboundedly for q > β/2, is equal to one if
q = β/2 and zero otherwise. In order to overcome these
difficulties and make more transparent the resulting the-
ory, we shall consider the other form of the representation
in which the measure is incorporated into the states (and
the resulting Hilbert space is L2(R, dq)). Thus the new
state
ψβ(q) := K · (V̂ (β) · φ0(q)) =
(q−β)2
We can now take the limit and what we get is
d 7→0
ψβ(q) := δ
1/2(q, β)
where by δ1/2(q, β) we mean something like ‘the square
root of the Dirac distribution’. What we really mean is
an object that satisfies the following property:
δ1/2(q, β) · δ1/2(q, α) = δ(q, β) δβ,α
That is, if α = β then it is just the ordinary delta, other-
wise it is zero. In a sense these object can be regarded as
half-densities that can not be integrated by themselves,
but whose product can. We conclude then that the inner
product is,
〈ψβ , ψα〉 =
dq ψβ(q)ψα(q) =
dq δ(q, α) δβ,α = δβ,α
which is just what we expected. Note that in this repre-
sentation, the vacuum state becomes ψ0(q) := δ
1/2(q, 0),
namely, the half-delta with support in the origin. It is
important to note that we are arriving in a natural way to
states as half-densities, whose squares can be integrated
without the need of a nontrivial measure on the configu-
ration space. Diffeomorphism invariance arises then in a
natural but subtle manner.
Note that as the end result we recover the Kronecker
delta inner product for the new fundamental states:
χβ(q) := δ
1/2(q, β).
Thus, in this new B-polymer representation, the Hilbert
space HB is the completion with respect to the inner
product (13) of the states generated by taking (finite)
linear combinations of basis elements of the form χβ :
Ψ(q) =
bi χβi(q) (14)
Let us now introduce an equivalent description of this
Hilbert space. Instead of having the basis elements be
half-deltas as elements of the Hilbert space where the
inner product is given by the ordinary Lebesgue measure
dq, we redefine both the basis and the measure. We
could consider, instead of a half-delta with support β, a
Kronecker delta or characteristic function with support
on β:
χ′β(q) := δq,β
These functions have a similar behavior with respect to
the product as the half-deltas, namely: χ′β(q) · χ′α(q) =
δβ,α. The main difference is that neither χ
′ nor their
squares are integrable with respect to the Lebesgue mea-
sure (having zero norm). In order to fix that problem we
have to change the measure so that we recover the basic
inner product (13) with our new basis. The needed mea-
sure turns out to be the discrete counting measure on R.
Thus any state in the ‘half density basis’ can be written
(using the same expression) in terms of the ‘Kronecker
basis’. For more details and further motivation see the
next section.
Note that in this B-polymer representation, both Û
and V̂ have their roles interchanged with that of the
A-polymer representation: while U(α) is discontinuous
and thus q̂ is not defined in the A-representation, we
have that it is V (β) in the B-representation that has this
property. In this case, it is the operator p̂ that can not
be defined. We see then that given a physical system for
which the configuration space has a well defined physi-
cal meaning, within the possible representation in which
wave-functions are functions of the configuration variable
q, the A and B polymer representations are radically dif-
ferent and inequivalent.
Having said this, it is also true that the A and B
representations are equivalent in a different sense, by
means of the duality between q and p representations
and the d↔ 1/d duality: The A-polymer representation
in the “q-representation” is equivalent to the B-polymer
representation in the “p-representation”, and conversely.
When studying a problem, it is important to decide from
the beginning which polymer representation (if any) one
should be using (for instance in the q-polarization). This
has as a consequence an implication on which variable is
naturally “quantized” (even if continuous): p for A and q
for B. There could be for instance a physical criteria for
this choice. For example a fundamental symmetry could
suggest that one representation is more natural than an-
other one. This indeed has been recently noted by Chiou
in [10], where the Galileo group is investigated and where
it is shown that the B representation is better behaved.
In the other polarization, namely for wavefunctions
of p, the picture gets reversed: q is discrete for the A-
representation, while p is for the B-case. Let us end this
section by noting that the procedure of obtaining the
polymer quantization by means of an appropriate limit
of Fock-Schrödinger representations might prove useful in
more general settings in field theory or quantum gravity.
III. POLYMER QUANTUM MECHANICS:
KINEMATICS
In previous sections we have derived what we have
called the A and B polymer representations (in the q-
polarization) as limiting cases of ordinary Fock repre-
sentations. In this section, we shall describe, without
any reference to the Schrödinger representation, the ‘ab-
stract’ polymer representation and then make contact
with its two possible realizations, closely related to the A
and B cases studied before. What we will see is that one
of them (the A case) will correspond to the p-polarization
while the other one corresponds to the q−representation,
when a choice is made about the physical significance of
the variables.
We can start by defining abstract kets |µ〉 labelled by
a real number µ. These shall belong to the Hilbert space
Hpoly. From these states, we define a generic ‘cylinder
states’ that correspond to a choice of a finite collection of
numbers µi ∈ R with i = 1, 2, . . . , N . Associated to this
choice, there are N vectors |µi〉, so we can take a linear
combination of them
|ψ〉 =
ai |µi〉 (15)
The polymer inner product between the fundamental kets
is given by,
〈ν|µ〉 = δν,µ (16)
That is, the kets are orthogonal to each other (when ν 6=
µ) and they are normalized (〈µ|µ〉 = 1). Immediately,
this implies that, given any two vectors |φ〉 =
j=1 bj |νj〉
and |ψ〉 =
i=1 ai |µi〉, the inner product between them
is given by,
〈φ|ψ〉 =
b̄j ai 〈νj |µi〉 =
b̄k ak
where the sum is over k that labels the intersection points
between the set of labels {νj} and {µi}. The Hilbert
space Hpoly is the Cauchy completion of finite linear com-
bination of the form (15) with respect to the inner prod-
uct (16). Hpoly is non-separable. There are two basic
operators on this Hilbert space: the ‘label operator’ ε̂:
ε̂ |µ〉 := µ |µ〉
and the displacement operator ŝ (λ),
ŝ (λ) |µ〉 := |µ+ λ〉
The operator ε̂ is symmetric and the operator(s) ŝ(λ)
defines a one-parameter family of unitary operators on
Hpoly, where its adjoint is given by ŝ† (λ) = ŝ (−λ). This
action is however, discontinuous with respect to λ given
that |µ〉 and |µ + λ〉 are always orthogonal, no matter
how small is λ. Thus, there is no (Hermitian) operator
that could generate ŝ (λ) by exponentiation.
So far we have given the abstract characterization of
the Hilbert space, but one would like to make contact
with concrete realizations as wave functions, or by iden-
tifying the abstract operators ε̂ and ŝ with physical op-
erators.
Suppose we have a system with a configuration space
with coordinate given by q, and p denotes its canonical
conjugate momenta. Suppose also that for physical rea-
sons we decide that the configuration coordinate q will
have some “discrete character” (for instance, if it is to
be identified with position, one could say that there is
an underlying discreteness in position at a small scale).
How can we implement such requirements by means of
the polymer representation? There are two possibilities,
depending on the choice of ‘polarizations’ for the wave-
functions, namely whether they will be functions of con-
figuration q or momenta p. Let us the divide the discus-
sion into two parts.
A. Momentum polarization
In this polarization, states will be denoted by,
ψ(p) = 〈p|ψ〉
where
ψµ(p) = 〈p|µ〉 = ei
How are then the operators ε̂ and ŝ represented? Note
that if we associate the multiplicative operator
V̂ (λ) · ψµ(p) = ei
~ = ei
(µ+λ)
p = ψ(µ+λ)(p)
we see then that the operator V̂ (λ) corresponds precisely
to the shift operator ŝ (λ). Thus we can also conclude
that the operator p̂ does not exist. It is now easy to
identify the operator q̂ with:
q̂ · ψµ(p) = −i~
ψµ(p) = µ e
~ = µψµ(p)
namely, with the abstract operator ε̂. The reason we
say that q̂ is discrete is because this operator has as its
eigenvalue the label µ of the elementary state ψµ(p), and
this label, even when it can take value in a continuum
of possible values, is to be understood as a discrete set,
given that the states are orthonormal for all values of
µ. Given that states are now functions of p, the inner
product (16) should be defined by a measure µ on the
space on which the wave-functions are defined. In order
to know what these two objects are, namely, the quan-
tum “configuration” space C and the measure thereon1,
we have to make use of the tools available to us from
the theory of C∗-algebras. If we consider the operators
V̂ (λ), together with their natural product and ∗-relation
given by V̂ ∗(λ) = V̂ (−λ), they have the structure of
an Abelian C∗-algebra (with unit) A. We know from
the representation theory of such objects that A is iso-
morphic to the space of continuous functions C0(∆) on a
compact space ∆, the spectrum of A. Any representation
of A on a Hilbert space as multiplication operator will be
on spaces of the form L2(∆, dµ). That is, our quantum
configuration space is the spectrum of the algebra, which
in our case corresponds to the Bohr compactification Rb
of the real line [11]. This space is a compact group and
there is a natural probability measure defined on it, the
Haar measure µH. Thus, our Hilbert space Hpoly will be
isomorphic to the space,
Hpoly,p = L2(Rb, dµH) (17)
In terms of ‘quasi periodic functions’ generated by ψµ(p),
the inner product takes the form
〈ψµ|ψλ〉 :=
dµH ψµ(p)ψλ(p) :=
= lim
L 7→∞
dpψµ(p)ψλ(p) = δµ,λ (18)
note that in the p-polarization, this characterization cor-
responds to the ‘A-version’ of the polymer representation
of Sec. II (where p and q are interchanged).
B. q-polarization
Let us now consider the other polarization in which wave
functions will depend on the configuration coordinate q:
ψ(q) = 〈q|ψ〉
The basic functions, that now will be called ψ̃µ(q), should
be, in a sense, the dual of the functions ψµ(p) of the
previous subsection. We can try to define them via a
‘Fourier transform’:
ψ̃µ(q) := 〈q|µ〉 = 〈q|
dµH|p〉〈p|µ〉
which is given by
ψ̃µ(q) :=
dµH〈q|p〉ψµ(p) =
dµH e
−i p q
~ = δq,µ (19)
1 here we use the standard terminology of ‘configuration space’ to
denote the domain of the wave function even when, in this case,
it corresponds to the physical momenta p.
That is, the basic objects in this representation are Kro-
necker deltas. This is precisely what we had found in
Sec. II for the B-type representation. How are now the
basic operators represented and what is the form of the
inner product? Regarding the operators, we expect that
they are represented in the opposite manner as in the
previous p-polarization case, but that they preserve the
same features: p̂ does not exist (the derivative of the Kro-
necker delta is ill defined), but its exponentiated version
V̂ (λ) does:
V̂ (λ) · ψ(q) = ψ(q + λ)
and the operator q̂ that now acts as multiplication has
as its eigenstates, the functions ψ̃ν(q) = δν,q:
q̂ · ψ̃µ(q) := µ ψ̃µ(q)
What is now the nature of the quantum configurations
space Q? And what is the measure thereon dµq? that
defines the inner product we should have:
〈ψ̃µ(q), ψ̃λ(q)〉 = δµ,λ
The answer comes from one of the characterizations of
the Bohr compactification: we know that it is, in a precise
sense, dual to the real line but when equipped with the
discrete topology Rd. Furthermore, the measure on Rd
will be the ‘counting measure’. In this way we recover the
same properties we had for the previous characterization
of the polymer Hilbert space. We can thus write:
Hpoly,x := L2(Rd, dµc) (20)
This completes a precise construction of the B-type poly-
mer representation sketched in the previous section. Note
that if we had chosen the opposite physical situation,
namely that q, the configuration observable, be the quan-
tity that does not have a corresponding operator, then
we would have had the opposite realization: In the q-
polarization we would have had the type-A polymer rep-
resentation and the type-B for the p-polarization. As
we shall see both scenarios have been considered in the
literature.
Up to now we have only focused our discussion on the
kinematical aspects of the quantization process. Let us
now consider in the following section the issue of dynam-
ics and recall the approach that had been adopted in the
literature, before the issue of the removal of the regulator
was reexamined in [6].
IV. POLYMER QUANTUM MECHANICS:
DYNAMICS
As we have seen the construction of the polymer
representation is rather natural and leads to a quan-
tum theory with different properties than the usual
Schrödinger counterpart such as its non-separability, the
non-existence of certain operators and the existence of
normalized eigen-vectors that yield a precise value for
one of the phase space coordinates. This has been done
without any regard for a Hamiltonian that endows the
system with a dynamics, energy and so on.
First let us consider the simplest case of a particle of
mass m in a potential V (q), in which the Hamiltonian H
takes the form,
p2 + V (q)
Suppose furthermore that the potential is given by a non-
periodic function, such as a polynomial or a rational func-
tion. We can immediately see that a direct implementa-
tion of the Hamiltonian is out of our reach, for the simple
reason that, as we have seen, in the polymer representa-
tion we can either represent q or p, but not both! What
has been done so far in the literature? The simplest
thing possible: approximate the non-existing term by a
well defined function that can be quantized and hope for
the best. As we shall see in next sections, there is indeed
more that one can do.
At this point there is also an important decision to be
made: which variable q or p should be regarded as “dis-
crete”? Once this choice is made, then it implies that
the other variable will not exist: if q is regarded as dis-
crete, then p will not exist and we need to approximate
the kinetic term p2/2m by something else; if p is to be
the discrete quantity, then q will not be defined and then
we need to approximate the potential V (q). What hap-
pens with a periodic potential? In this case one would
be modelling, for instance, a particle on a regular lattice
such as a phonon living on a crystal, and then the natural
choice is to have q not well defined. Furthermore, the po-
tential will be well defined and there is no approximation
needed.
In the literature both scenarios have been considered.
For instance, when considering a quantum mechanical
system in [2], the position was chosen to be discrete,
so p does not exist, and one is then in the A type for
the momentum polarization (or the type B for the q-
polarization). With this choice, it is the kinetic term the
one that has to be approximated, so once one has done
this, then it is immediate to consider any potential that
will thus be well defined. On the other hand, when con-
sidering loop quantum cosmology (LQC), the standard
choice is that the configuration variable is not defined
[4]. This choice is made given that LQC is regarded as
the symmetric sector of full loop quantum gravity where
the connection (that is regarded as the configuration vari-
able) can not be promoted to an operator and one can
only define its exponentiated version, namely, the holon-
omy. In that case, the canonically conjugate variable,
closely related to the volume, becomes ‘discrete’, just as
in the full theory. This case is however, different from the
particle in a potential example. First we could mention
that the functional form of the Hamiltonian constraint
that implements dynamics has a different structure, but
the more important difference lies in that the system is
constrained.
Let us return to the case of the particle in a po-
tential and for definiteness, let us start with the aux-
iliary kinematical framework in which: q is discrete, p
can not be promoted and thus we have to approximate
the kinetic term p̂2/2m. How is this done? The stan-
dard prescription is to define, on the configuration space
C, a regular ‘graph’ γµ0 . This consists of a numerable
set of points, equidistant, and characterized by a pa-
rameter µ0 that is the (constant) separation between
points. The simplest example would be to consider the
set γµ0 = {q ∈ R | q = nµ0 , ∀ n ∈ Z}.
This means that the basic kets that will be considered
|µn〉 will correspond precisely to labels µn belonging to
the graph γµ0 , that is, µn = nµ0. Thus, we shall only
consider states of the form,
|ψ〉 =
bn |µn〉 . (21)
This ‘small’ Hilbert space Hγµ0 , the graph Hilbert space,
is a subspace of the ‘large’ polymer Hilbert space Hpoly
but it is separable. The condition for a state of the form
(21) to belong to the Hilbert space Hγµ0 is that the co-
efficients bn satisfy:
n |bn|2 <∞.
Let us now consider the kinetic term p̂2/2m. We have
to approximate it by means of trigonometric functions,
that can be built out of the functions of the form eiλ p/~.
As we have seen in previous sections, these functions can
indeed be promoted to operators and act as translation
operators on the kets |µ〉. If we want to remain in the
graph γ, and not create ‘new points’, then one is con-
strained to considering operators that displace the kets
by just the right amount. That is, we want the basic
shift operator V̂ (λ) to be such that it maps the ket with
label |µn〉 to the next ket, namely |µn+1〉. This can in-
deed achieved by fixing, once and for all, the value of the
allowed parameter λ to be λ = µ0. We have then,
V̂ (µ0) · |µn〉 = |µn + µ0〉 = |µn+1〉
which is what we wanted. This basic ‘shift operator’ will
be the building block for approximating any (polynomial)
function of p. In order to do that we notice that the
function p can be approximated by,
p ≈ ~
(µ0 p
~ − e−i
where the approximation is good for p << ~/µ0. Thus,
one can define a regulated operator p̂µ0 that depends on
the ‘scale’ µ0 as:
p̂µ0 · |µn〉 :=
[V (µ0) − V (−µ0)] · |µn〉 =
(|µn+1〉 − |µn−1〉) (22)
In order to regulate the operator p̂2, there are (at least)
two possibilities, namely to compose the operator p̂µ0
with itself or to define a new approximation. The oper-
ator p̂µ0 · p̂µ0 has the feature that shifts the states two
steps in the graph to both sides. There is however an-
other operator that only involves shifting once:
p̂2µ0 · |νn〉 :=
[2 − V̂ (µ0) − V̂ (−µ0)] · |νn〉 =
(2|νn〉 − |νn+1〉 − |νn−1〉) (23)
which corresponds to the approximation p2 ≈ 2~
cos(µ0 p/~)), valid also in the regime p << ~/µ0. With
these considerations, one can define the operator Ĥµ0 ,
the Hamiltonian at scale µ0, that in practice ‘lives’ on
the space Hγµ0 as,
Ĥµ0 :=
p̂2µ0 + V̂ (q) , (24)
that is a well defined, symmetric operator on Hγµ0 . No-
tice that the operator is also defined on Hpoly, but there
its physical interpretation is problematic. For example,
it turns out that the expectation value of the kinetic term
calculated on most states (states which are not tailored
to the exact value of the parameter µ0) is zero. Even
if one takes a state that gives “reasonable“ expectation
values of the µ0-kinetic term and uses it to calculate the
expectation value of the kinetic term corresponding to
a slight perturbation of the parameter µ0 one would get
zero. This problem, and others that arise when working
on Hpoly, forces one to assign a physical interpretation
to the Hamiltonian Ĥµ0 only when its action is restricted
to the subspace Hγµ0 .
Let us now explore the form that the Hamiltonian takes
in the two possible polarizations. In the q-polarization,
the basis, labelled by n is given by the functions χn(q) =
δq,µn . That is, the wave functions will only have sup-
port on the set γµ0 . Alternatively, one can think of a
state as completely characterized by the ‘Fourier coeffi-
cients’ an: ψ(q) ↔ an, which is the value that the wave
function ψ(q) takes at the point q = µn = nµ0. Thus,
the Hamiltonian takes the form of a difference equation
when acting on a general state ψ(q). Solving the time
independent Schrödinger equation Ĥ · ψ = E ψ amounts
to solving the difference equation for the coefficients an.
The momentum polarization has a different structure.
In this case, the operator p̂2µ0 acts as a multiplication
operator,
p̂2µ0 · ψ(p) =
1 − cos
(µ0 p
ψ(p) (25)
The operator corresponding to q will be represented as a
derivative operator
q̂ · ψ(p) := i~ ∂p ψ(p).
For a generic potential V (q), it has to be defined by
means of spectral theory defined now on a circle. Why
on a circle? For the simple reason that by restricting
ourselves to a regular graph γµ0 , the functions of p that
preserve it (when acting as shift operators) are of the
form e(i m µ0 p/~) for m integer. That is, what we have
are Fourier modes, labelled by m, of period 2π ~/µ0 in p.
Can we pretend then that the phase space variable p is
now compactified? The answer is in the affirmative. The
inner product on periodic functions ψµ0(p) of p coming
from the full Hilbert space Hpoly and given by
〈φ(p)|ψ(p)〉poly = lim
L 7→∞
dp φ(p)ψ(p)
is precisely equivalent to the inner product on the circle
given by the uniform measure
〈φ(p)|ψ(p)〉µ0 =
∫ π~/µ0
−π~/µ0
dp φ(p)ψ(p)
with p ∈ (−π~/µ0, π~/µ0). As long as one restricts at-
tention to the graph γµ0 , one can work in this separable
Hilbert space Hγµ0 of square integrable functions on S
Immediately, one can see the limitations of this descrip-
tion. If the mechanical system to be quantized is such
that its orbits have values of the momenta p that are
not small compared with π~/µ0 then the approximation
taken will be very poor, and we don’t expect neither the
effective classical description nor its quantization to be
close to the standard one. If, on the other hand, one is al-
ways within the region in which the approximation can be
regarded as reliable, then both classical and quantum de-
scriptions should approximate the standard description.
What does ‘close to the standard description’ exactly
mean needs, of course, some further clarification. In
particular one is assuming the existence of the usual
Schrödinger representation in which the system has a be-
havior that is also consistent with observations. If this is
the case, the natural question is: How can we approxi-
mate such description from the polymer picture? Is there
a fine enough graph γµ0 that will approximate the system
in such a way that all observations are indistinguishable?
Or even better, can we define a procedure, that involves
a refinement of the graph γµ0 such that one recovers the
standard picture?
It could also happen that a continuum limit can be de-
fined but does not coincide with the ‘expected one’. But
there might be also physical systems for which there is
no standard description, or it just does not make sense.
Can in those cases the polymer representation, if it ex-
ists, provide the correct physical description of the sys-
tem under consideration? For instance, if there exists a
physical limitation to the minimum scale set by µ0, as
could be the case for a quantum theory of gravity, then
the polymer description would provide a true physical
bound on the value of certain quantities, such as p in
our example. This could be the case for loop quantum
cosmology, where there is a minimum value for physical
volume (coming from the full theory), and phase space
points near the ‘singularity’ lie at the region where the
approximation induced by the scale µ0 departs from the
standard classical description. If in that case the poly-
mer quantum system is regarded as more fundamental
than the classical system (or its standard Wheeler-De
Witt quantization), then one would interpret this dis-
crepancies in the behavior as a signal of the breakdown
of classical description (or its ‘naive’ quantization).
In the next section we present a method to remove
the regulator µ0 which was introduced as an intermedi-
ate step to construct the dynamics. More precisely, we
shall consider the construction of a continuum limit of
the polymer description by means of a renormalization
procedure.
V. THE CONTINUUM LIMIT
This section has two parts. In the first one we motivate
the need for a precise notion of the continuum limit of
the polymeric representation, explaining why the most
direct, and naive approach does not work. In the sec-
ond part, we shall present the main ideas and results of
the paper [6], where the Hamiltonian and the physical
Hilbert space in polymer quantum mechanics are con-
structed as a continuum limit of effective theories, follow-
ing Wilson’s renormalization group ideas. The resulting
physical Hilbert space turns out to be unitarily isomor-
phic to the ordinary Hs = L2(R, dq) of the Schrödinger
theory.
Before describing the results of [6] we should discuss
the precise meaning of reaching a theory in the contin-
uum. Let us for concreteness consider the B-type repre-
sentation in the q-polarization. That is, states are func-
tions of q and the orthonormal basis χµ(q) is given by
characteristic functions with support on q = µ. Let us
now suppose we have a Schrödinger state Ψ(q) ∈ Hs =
L2(R, dq). What is the relation between Ψ(q) and a state
in Hpoly,x? We are also interested in the opposite ques-
tion, that is, we would like to know if there is a preferred
state in Hs that is approximated by an arbitrary state
ψ(q) in Hpoly,x. The first obvious observation is that a
Schödinger state Ψ(q) does not belong to Hpoly,x since it
would have an infinite norm. To see that note that even
when the would-be state can be formally expanded in the
χµ basis as,
Ψ(q) =
Ψ(µ) χµ(q)
where the sum is over the parameter µ ∈ R. Its associ-
ated norm in Hpoly,x would be:
|Ψ(q)|2poly =
|Ψ(µ)|2 → ∞
which blows up. Note that in order to define a mapping
P : Hs → Hpoly,x, there is a huge ambiguity since the
values of the function Ψ(q) are needed in order to expand
the polymer wave function. Thus we can only define a
mapping in a dense subset D of Hs where the values of the
functions are well defined (recall that in Hs the value of
functions at a given point has no meaning since states are
equivalence classes of functions). We could for instance
ask that the mapping be defined for representatives of the
equivalence classes in Hs that are piecewise continuous.
From now on, when we refer to an element of the space
Hs we shall be refereeing to one of those representatives.
Notice then that an element of Hs does define an element
of Cyl∗γ , the dual to the space Cylγ , that is, the space
of cylinder functions with support on the (finite) lattice
γ = {µ1, µ2, . . . , µN}, in the following way:
Ψ(q) : Cylγ −→ C
such that
Ψ(q)[ψ(q)] = (Ψ|ψ〉 :=
Ψ(µ) 〈χµ|
ψi χµi〉polyγ
Ψ(µi)ψi < ∞ (26)
Note that this mapping could be seen as consisting of two
parts: First, a projection Pγ : Cyl
∗ → Cylγ such that
Pγ(Ψ) = Ψγ(q) :=
i Ψ(µi)χµi(q) ∈ Cylγ . The state
Ψγ is sometimes refereed to as the ‘shadow of Ψ(q) on
the lattice γ’. The second step is then to take the inner
product between the shadow Ψγ(q) and the state ψ(q)
with respect to the polymer inner product 〈Ψγ |ψ〉polyγ .
Now this inner product is well defined. Notice that for
any given lattice γ the corresponding projector Pγ can be
intuitively interpreted as some kind of ‘coarse graining
map’ from the continuum to the lattice γ. In terms of
functions of q the projection is replacing a continuous
function defined on R with a function over the lattice
γ ⊂ R which is a discrete set simply by restricting Ψ to
γ. The finer the lattice the more points that we have
on the curve. As we shall see in the second part of this
section, there is indeed a precise notion of coarse graining
that implements this intuitive idea in a concrete fashion.
In particular, we shall need to replace the lattice γ with
a decomposition of the real line in intervals (having the
lattice points as end points).
Let us now consider a system in the polymer represen-
tation in which a particular lattice γ0 was chosen, say
with points of the form {qk ∈ R |qk = ka0 , ∀ k ∈ Z},
namely a uniform lattice with spacing equal to a0. In this
case, any Schrödinger wave function (of the type that we
consider) will have a unique shadow on the lattice γ0. If
we refine the lattice γ 7→ γn by dividing each interval in
2n new intervals of length an = a0/2
n we have new shad-
ows that have more and more points on the curve. Intu-
itively, by refining infinitely the graph we would recover
the original function Ψ(q). Even when at each finite step
the corresponding shadow has a finite norm in the poly-
mer Hilbert space, the norm grows unboundedly and the
limit can not be taken, precisely because we can not em-
bed Hs into Hpoly. Suppose now that we are interested
in the reverse process, namely starting from a polymer
theory on a lattice and asking for the ‘continuum wave
function’ that is best approximated by a wave function
over a graph. Suppose furthermore that we want to con-
sider the limit of the graph becoming finer. In order
to give precise answers to these (and other) questions we
need to introduce some new technology that will allow us
to overcome these apparent difficulties. In the remaining
of this section we shall recall these constructions for the
benefit of the reader. Details can be found in [6] (which
is an application of the general formalism discussed in
[9]).
The starting point in this construction is the concept
of a scale C, which allows us to define the effective the-
ories and the concept of continuum limit. In our case a
scale is a decomposition of the real line in the union of
closed-open intervals, that cover the whole line and do
not intersect. Intuitively, we are shifting the emphasis
from the lattice points to the intervals defined by the
same points with the objective of approximating con-
tinuous functions defined on R with functions that are
constant on the intervals defined by the lattice. To be
precise, we define an embedding, for each scale Cn from
Hpoly to Hs by means of a step function:
Ψ(man) χman(q) →
Ψ(man) χαm(q) ∈ Hs
with χαn(q) a characteristic function on the interval
αm = [man, (m + 1)an). Thus, the shadows (living on
the lattice) were just an intermediate step in the con-
struction of the approximating function; this function is
piece-wise constant and can be written as a linear com-
bination of step functions with the coefficients provided
by the shadows.
The challenge now is to define in an appropriate sense
how one can approximate all the aspects of the theory
by means of this constant by pieces functions. Then the
strategy is that, for any given scale, one can define an
effective theory by approximating the kinetic operator
by a combination of the translation operators that shift
between the vertices of the given decomposition, in other
words by a periodic function in p. As a result one has a
set of effective theories at given scales which are mutually
related by coarse graining maps. This framework was
developed in [6]. For the convenience of the reader we
briefly recall part of that framework.
Let us denote the kinematic polymer Hilbert space at
the scale Cn as HCn , and its basis elements as eαi,Cn ,
where αi = [ian, (i + 1)an) ∈ Cn. By construction this
basis is orthonormal. The basis elements in the dual
Hilbert space H∗Cn are denoted by ωαi,Cn ; they are also
orthonormal. The states ωαi,Cn have a simple action on
Cyl, ωαi,Cn(δx0,q) = χαi,Cn(x0). That is, if x0 is in the
interval αi of Cn the result is one and it is zero if it is
not there.
Given any m ≤ n, we define d∗m,n : H∗Cn → H
as the ‘coarse graining’ map between the dual Hilbert
spaces, that sends the part of the elements of the dual
basis to zero while keeping the information of the rest:
d∗m,n(ωαi,Cn) = ωβj ,Cm if i = j2
n−m, in the opposite case
d∗m,n(ωαi,Cn) = 0.
At every scale the corresponding effective theory is
given by the hamiltonian Hn. These Hamiltonians will
be treated as quadratic forms, hn : HCn → R, given by
hn(ψ) = λ
(ψ,Hnψ) , (27)
where λ2Cn is a normalizaton factor. We will see later
that this rescaling of the inner product is necessary in
order to guarantee the convergence of the renormalized
theory. The completely renormalized theory at this scale
is obtained as
hrenm := lim
d⋆m,nhn. (28)
and the renormalized Hamiltonians are compatible with
each other, in the sense that
d⋆m,nh
n = h
In order to analyze the conditions for the convergence
in (28) let us express the Hamiltonian in terms of its
eigen-covectors end eigenvalues. We will work with effec-
tive Hamiltonians that have a purely discrete spectrum
(labelled by ν) Hn · Ψν,Cn = Eν,Cn Ψν,Cn . We shall also
introduce, as an intermediate step, a cut-off in the energy
levels. The origin of this cut-off is in the approximation
of the Hamiltonian of our system at a given scale with
a Hamiltonian of a periodic system in a regime of small
energies, as we explained earlier. Thus, we can write
hνcut−offm =
νcut−off
Eν,CmΨν,Cm ⊗ Ψν,Cm , (29)
where the eigen covectors Ψν,Cm are normalized accord-
ing to the inner product rescaled by 1
, and the cut-
off can vary up to a scale dependent bound, νcut−off ≤
νmax(Cm). The Hilbert space of covectors together with
such inner product will be called H⋆renCm .
In the presence of a cut-off, the convergence of the
microscopically corrected Hamiltonians, equation (28) is
equivalent to the existence of the following two limits.
The first one is the convergence of the energy levels,
Eν,Cn = E
ν . (30)
Second is the existence of the completely renormalized
eigen covectors,
d⋆m,n Ψν,Cn = Ψ
∈ H⋆renCm ⊂ Cyl
⋆ . (31)
We clarify that the existence of the above limit means
that Ψrenν,Cm(δx0,q) is well defined for any δx0,q ∈ Cyl. No-
tice that this point-wise convergence, if it can take place
at all, will require the tuning of the normalization factors
λ2Cn .
Now we turn to the question of the continuum limit
of the renormalized covectors. First we can ask for the
existence of the limit
Ψrenν,Cn(δx0,q) (32)
for any δx0,q ∈ Cyl. When this limits exists there is
a natural action of the eigen covectors in the continuum
limit. Below we consider another notion of the continuum
limit of the renormalized eigen covectors.
When the completely renormalized eigen covectors
exist, they form a collection that is d⋆-compatible,
d⋆m,nΨ
= Ψrenν,Cm . A sequence of d
⋆-compatible nor-
malizable covectors define an element of
, which is
the projective limit of the renormalized spaces of covec-
H⋆renCn . (33)
The inner product in this space is defined by
({ΨCn}, {ΦCn})renR := lim
(ΨCn ,ΦCn)
The natural inclusion of C∞0 in
is by an antilinear
map which assigns to any Ψ ∈ C∞0 the d⋆-compatible
collection ΨshadCn :=
ωαiΨ̄(L(αi)) ∈ H⋆renCn ⊂ Cyl
ΨshadCn will be called the shadow of Ψ at scale Cn and acts
in Cyl as a piecewise constant function. Clearly other
types of test functions like Schwartz functions are also
naturally included in
. In this context a shadow is
a state of the effective theory that approximates a state
in the continuum theory.
Since the inner product in
is degenerate, the
physical Hilbert space is defined as
H⋆phys :=
/ ker(·, ·)ren
Hphys := H⋆⋆phys
The nature of the physical Hilbert space, whether it is
isomorphic to the Schrödinger Hilber space, Hs, or not, is
determined by the normalization factors λ2Cn which can
be obtained from the conditions asking for compatibil-
ity of the dynamics of the effective theories at different
scales. The dynamics of the system under consideration
selects the continuum limit.
Let us now return to the definition of the Hamilto-
nian in the continuum limit. First consider the contin-
uum limit of the Hamiltonian (with cut-off) in the sense
of its point-wise convergence as a quadratic form. It
turns out that if the limit of equation (32) exists for
all the eigencovectors allowed by the cut-off, we have
νcut−off ren
: Hpoly,x → R defined by
νcut−off ren
(δx0,q) := lim
hνcut−off renn ([δx0,q]Cn). (34)
This Hamiltonian quadratic form in the continuum can
be coarse grained to any scale and, as can be ex-
pected, it yields the completely renormalized Hamilto-
nian quadratic forms at that scale. However, this is not
a completely satisfactory continuum limit because we can
not remove the auxiliary cut-off νcut−off . If we tried, as
we include more and more eigencovectors in the Hamilto-
nian the calculations done at a given scale would diverge
and doing them in the continuum is just as divergent.
Below we explore a more successful path.
We can use the renormalized inner product to induce
an action of the cut–off Hamiltonians on
νcut−off ren
({ΨCn}) := lim
hνcut−off renn ((ΨCn , ·)renCn ),
where we have used the fact that (ΨCn , ·)renCn ∈ HCn . The
existence of this limit is trivial because the renormalized
Hamiltonians are finite sums and the limit exists term by
term.
These cut-off Hamiltonians descend to the physical
Hilbert space
νcut−off ren
([{ΨCn}]) := h
νcut−off ren
({ΨCn})
for any representative {ΨCn} ∈ [{ΨCn}] ∈ H⋆phys.
Finally we can address the issue of removal of the cut-
off. The Hamiltonian hren
→ R is defined by the
limit
:= lim
νcut−off→∞
νcut−off ren
when the limit exists. Its corresponding Hermitian form
in Hphys is defined whenever the above limit exists. This
concludes our presentation of the main results of [6]. Let
us now consider several examples of systems for which
the continuum limit can be investigated.
VI. EXAMPLES
In this section we shall develop several examples of
systems that have been treated with the polymer quanti-
zation. These examples are simple quantum mechanical
systems, such as the simple harmonic oscillator and the
free particle, as well as a quantum cosmological model
known as loop quantum cosmology.
A. The Simple Harmonic Oscillator
In this part, let us consider the example of a Simple Har-
monic Oscillator (SHO) with parameters m and ω, clas-
sically described by the following Hamiltonian
mω2 x2.
Recall that from these parameters one can define a length
scale D =
~/mω. In the standard treatment one uses
this scale to define a complex structure JD (and an in-
ner product from it), as we have described in detail that
uniquely selects the standard Schrödinger representation.
At scale Cn we have an effective Hamiltonian for the
Simple Harmonic Oscillator (SHO) given by
HCn =
1 − cos anp
mω2x2 . (35)
If we interchange position and momentum, this Hamilto-
nian is exactly that of a pendulum of mass m, length l
and subject to a constant gravitational field g:
ĤCn = −
+mgl(1 − cos θ)
where those quantities are related to our system by,
mω an
, g =
, θ =
That is, we are approximating, for each scale Cn the
SHO by a pendulum. There is, however, an important
difference. From our knowledge of the pendulum system,
we know that the quantum system will have a spectrum
for the energy that has two different asymptotic behav-
iors, the SHO for low energies and the planar rotor in
the higher end, corresponding to oscillating and rotating
solutions respectively2. As we refine our scale and both
the length of the pendulum and the height of the periodic
potential increase, we expect to have an increasing num-
ber of oscillating states (for a given pendulum system,
there is only a finite number of such states). Thus, it
is justified to consider the cut-off in the energy eigenval-
ues, as discussed in the last section, given that we only
expect a finite number of states of the pendulum to ap-
proximate SHO eigenstates. With these consideration in
mind, the relevant question is whether the conditions for
the continuum limit to exist are satisfied. This question
has been answered in the affirmative in [6]. What was
shown there was that the eigen-values and eigen func-
tions of the discrete systems, which represent a discrete
and non-degenerate set, approximate those of the contin-
uum, namely, of the standard harmonic oscillator when
the inner product is renormalized by a factor λ2Cn = 1/2
This convergence implies that the continuum limit exists
as we understand it. Let us now consider the simplest
possible system, a free particle, that has nevertheless the
particular feature that the spectrum of the energy is con-
tinuous.
2 Note that both types of solutions are, in the phase space, closed.
This is the reason behind the purely discrete spectrum. The
distinction we are making is between those solutions inside the
separatrix, that we call oscillating, and those that are above it
that we call rotating.
B. Free Polymer Particle
In the limit ω → 0, the Hamiltonian of the Simple
Harmonic oscillator (35) goes to the Hamiltonian of a
free particle and the corresponding time independent
Schrödinger equation, in the p−polarization, is given by
(1 − cos anp
) − ECn
ψ̃(p) = 0
where we now have that p ∈ S1, with p ∈ (−π~
Thus, we have
ECn =
1 − cos
≤ ECn,max ≡ 2
. (36)
At each scale the energy of the particle we can describe
is bounded from above and the bound depends on the
scale. Note that in this case the spectrum is continu-
ous, which implies that the ordinary eigenfunctions of
the Hilbert are not normalizable. This imposes an upper
bound in the value that the energy of the particle can
have, in addition to the bound in the momentum due to
its “compactification”.
Let us first look for eigen-solutions to the time inde-
pendent Schrödinger equation, that is, for energy eigen-
states. In the case of the ordinary free particle, these
correspond to constant momentum plane waves of the
form e±(
) and such that the ordinary dispersion re-
lation p2/2m = E is satisfied. These plane waves are
not square integrable and do not belong to the ordinary
Hilbert space of the Schrödinger theory but they are still
useful for extracting information about the system. For
the polymer free particle we have,
ψ̃Cn(p) = c1δ(p− PCn) + c2δ(p+ PCn)
where PCn is a solution of the previous equation consid-
ering a fixed value of ECn . That is,
PCn = P (ECn) =
arccos
1 − ma
The inverse Fourier transform yields, in the ‘x represen-
tation’,
ψCn(xj) =
∫ π~/an
−π~/an
ψ̃(p) e
p j dp =
ixjPCn /~ + c2e
−ixjPCn /~
.(37)
with xj = an j for j ∈ Z. Note that the eigenfunctions
are still delta functions (in the p representation) and thus
not (square) normalizable with respect to the polymer
inner product, that in the p polarization is just given
by the ordinary Haar measure on S1, and there is no
quantization of the momentum (its spectrum is still truly
continuous).
Let us now consider the time dependent Schrödinger
equation,
i~ ∂t Ψ̃(p, t) = Ĥ · Ψ̃(p, t).
Which now takes the form,
Ψ̃(p, t) =
(1 − cos (an p/~)) Ψ̃(p, t)
that has as its solution,
Ψ̃(p, t) = e−
(1−cos (an p/~)) t ψ̃(p) = e(−iECn /~) t ψ̃(p)
for any initial function ψ̃(p), where ECn satisfy the dis-
persion relation (36). The wave function Ψ(xj , t), the
xj-representation of the wave function, can be obtained
for any given time t by Fourier transforming with (37)
the wave function Ψ̃(p, t).
In order to check out the convergence of the micro-
scopically corrected Hamiltonians we should analyze the
convergence of the energy levels and of the proper cov-
ectors. In the limit n → ∞, ECn → E = p2/2m so
we can be certain that the eigen-values for the energy
converge (when fixing the value of p). Let us write the
proper covector as ΨCn = (ψCn , ·)renCn ∈ H
. Then we
can bring microscopic corrections to scale Cm and look
for convergence of such corrections
ΨrenCm
= lim
d⋆m,nΨCn .
It is easy to see that given any basis vector eαi ∈ HCm
the following limit
ΨrenCm(eαi,Cm) = limCn→∞
ΨCn(dn,m(eαi,Cm))
exists and is equal to
ΨshadCm (eαi,Cm) = [d
⋆ΨSchr](eαi,Cm) = Ψ
Schr(iam)
where ΨshadCm is calculated using the free particle Hamilto-
nian in the Schrödinger representation. This expression
defines the completely renormalized proper covector at
the scale Cm.
C. Polymer Quantum Cosmology
In this section we shall present a version of quantum
cosmology that we call polymer quantum cosmology. The
idea behind this name is that the main input in the quan-
tization of the corresponding mini-superspace model is
the use of a polymer representation as here understood.
Another important input is the choice of fundamental
variables to be used and the definition of the Hamiltonian
constraint. Different research groups have made differ-
ent choices. We shall take here a simple model that has
received much attention recently, namely an isotropic,
homogeneous FRW cosmology with k = 0 and coupled
to a massless scalar field ϕ. As we shall see, a proper
treatment of the continuum limit of this system requires
new tools under development that are beyond the scope
of this work. We will thus restrict ourselves to the intro-
duction of the system and the problems that need to be
solved.
The system to be quantized corresponds to the phase
space of cosmological spacetimes that are homogeneous
and isotropic and for which the homogeneous spatial
slices have a flat intrinsic geometry (k = 0 condition).
The only matter content is a mass-less scalar field ϕ. In
this case the spacetime geometry is given by metrics of
the form:
ds2 = −dt2 + a2(t) (dx2 + dy2 + dz2)
where the function a(t) carries all the information and
degrees of freedom of the gravity part. In terms of the
coordinates (a, pa, ϕ, pϕ) for the phase space Γ of the the-
ory, all the dynamics is captured in the Hamiltonian con-
straint
C := −3
+ 8πG
2|a|3
The first step is to define the constraint on the kine-
matical Hilbert space to find physical states and then a
physical inner product to construct the physical Hilbert
space. First note that one can rewrite the equation as:
p2a a
2 = 8πG
If, as is normally done, one chooses ϕ to act as an in-
ternal time, the right hand side would be promoted, in
the quantum theory, to a second derivative. The left
hand side is, furthermore, symmetric in a and pa. At
this point we have the freedom in choosing the variable
that will be quantized and the variable that will not be
well defined in the polymer representation. The standard
choice is that pa is not well defined and thus, a and any
geometrical quantity derived from it, is quantized. Fur-
thermore, we have the choice of polarization on the wave
function. In this respect the standard choice is to select
the a-polarization, in which a acts as multiplication and
the approximation of pa, namely sin(λ pa)/λ acts as a
difference operator on wave functions of a. For details of
this particular choice see [5]. Here we shall adopt the op-
posite polarization, that is, we shall have wave functions
Ψ(pa, ϕ).
Just as we did in the previous cases, in order to gain
intuition about the behavior of the polymer quantized
theory, it is convenient to look at the equivalent prob-
lem in the classical theory, namely the classical system
we would get be approximating the non-well defined ob-
servable (pa in our present case) by a well defined object
(made of trigonometric functions). Let us for simplicity
choose to replace pa 7→ sin(λ pa)/λ. With this choice
we get an effective classical Hamiltonian constraint that
depends on λ:
Cλ := −
sin(λ pa)
λ2|a|
+ 8πG
2|a|3
We can now compute effective equations of motion by
means of the equations: Ḟ := {F, Cλ}, for any observable
F ∈ C∞(Γ), and where we are using the effective (first
order) action:
dτ(pa ȧ+ pϕ ϕ̇−N Cλ)
with the choice N = 1. The first thing to notice is that
the quantity pϕ is a constant of the motion, given that
the variable ϕ is cyclic. The second observation is that
ϕ̇ = 8πG
has the same sign as pϕ and never vanishes.
Thus ϕ can be used as a (n internal) time variable. The
next observation is that the equation for
, namely
the effective Friedman equation, will have a zero for a
non-zero value of a given by
λ2p2ϕ.
This is the value at which there will be bounce if the
trajectory started with a large value of a and was con-
tracting. Note that the ‘size’ of the universe when the
bounce occurs depends on both the constant pϕ (that
dictates the matter density) and the value of the lattice
size λ. Here it is important to stress that for any value
of pϕ (that uniquely fixes the trajectory in the (a, pa)
plane), there will be a bounce. In the original description
in terms of Einstein’s equations (without the approxima-
tion that depends on λ), there in no such bounce. If
ȧ < 0 initially, it will remain negative and the universe
collapses, reaching the singularity in a finite proper time.
What happens within the effective description if we re-
fine the lattice and go from λ to λn := λ/2
n? The only
thing that changes, for the same classical orbit labelled
by pϕ, is that the bounce occurs at a ‘later time’ and for
a smaller value of a∗ but the qualitative picture remains
the same.
This is the main difference with the systems considered
before. In those cases, one could have classical trajecto-
ries that remained, for a given choice of parameter λ,
within the region where sin(λp)/λ is a good approxima-
tion to p. Of course there were also classical trajectories
that were outside this region but we could then refine the
lattice and find a new value λ′ for which the new clas-
sical trajectory is well approximated. In the case of the
polymer cosmology, this is never the case: Every classical
trajectory will pass from a region where the approxima-
tion is good to a region where it is not; this is precisely
where the ‘quantum corrections’ kick in and the universes
bounces.
Given that in the classical description, the ‘original’
and the ‘corrected’ descriptions are so different we expect
that, upon quantization, the corresponding quantum the-
ories, namely the polymeric and the Wheeler-DeWitt will
be related in a non-trivial way (if at all).
In this case, with the choice of polarization and for a
particular factor ordering we have,
sin(λpa)
· Ψ(pa, ϕ) = 0
as the Polymer Wheeler-DeWitt equation.
In order to approach the problem of the continuum
limit of this quantum theory, we have to realize that the
task is now somewhat different than before. This is so
given that the system is now a constrained system with
a constraint operator rather than a regular non-singular
system with an ordinary Hamiltonian evolution. Fortu-
nately for the system under consideration, the fact that
the variable ϕ can be regarded as an internal time allows
us to interpret the quantum constraint as a generalized
Klein-Gordon equation of the form
Ψ = Θλ · Ψ
where the operator Θλ is ‘time independent’. This al-
lows us to split the space of solutions into ‘positive and
negative frequency’, introduce a physical inner product
on the positive frequency solutions of this equation and
a set of physical observables in terms of which to de-
scribe the system. That is, one reduces in practice the
system to one very similar to the Schrödinger case by
taking the positive square root of the previous equation:
Θλ · Ψ. The question we are interested is
whether the continuum limit of these theories (labelled
by λ) exists and whether it corresponds to the Wheeler-
DeWitt theory. A complete treatment of this problem
lies, unfortunately, outside the scope of this work and
will be reported elsewhere [12].
VII. DISCUSSION
Let us summarize our results. In the first part of the
article we showed that the polymer representation of the
canonical commutation relations can be obtained as the
limiting case of the ordinary Fock-Schrödinger represen-
tation in terms of the algebraic state that defines the
representation. These limiting cases can also be inter-
preted in terms of the naturally defined coherent states
associated to each representation labelled by the param-
eter d, when they become infinitely ‘squeezed’. The two
possible limits of squeezing lead to two different polymer
descriptions that can nevertheless be identified, as we
have also shown, with the two possible polarizations for
an abstract polymer representation. This resulting the-
ory has, however, very different behavior as the standard
one: The Hilbert space is non-separable, the representa-
tion is unitarily inequivalent to the Schrödinger one, and
natural operators such as p̂ are no longer well defined.
This particular limiting construction of the polymer the-
ory can shed some light for more complicated systems
such as field theories and gravity.
In the regular treatments of dynamics within the poly-
mer representation, one needs to introduce some extra
structure, such as a lattice on configuration space, to con-
struct a Hamiltonian and implement the dynamics for the
system via a regularization procedure. How does this re-
sulting theory compare to the original continuum theory
one had from the beginning? Can one hope to remove
the regulator in the polymer description? As they stand
there is no direct relation or mapping from the polymer
to a continuum theory (in case there is one defined). As
we have shown, one can indeed construct in a systematic
fashion such relation by means of some appropriate no-
tions related to the definition of a scale, closely related
to the lattice one had to introduce in the regularization.
With this important shift in perspective, and an appro-
priate renormalization of the polymer inner product at
each scale one can, subject to some consistency condi-
tions, define a procedure to remove the regulator, and
arrive to a Hamiltonian and a Hilbert space.
As we have seen, for some simple examples such as
a free particle and the harmonic oscillator one indeed
recovers the Schrödinger description back. For other sys-
tems, such as quantum cosmological models, the answer
is not as clear, since the structure of the space of classi-
cal solutions is such that the ‘effective description’ intro-
duced by the polymer regularization at different scales
is qualitatively different from the original dynamics. A
proper treatment of these class of systems is underway
and will be reported elsewhere [12].
Perhaps the most important lesson that we have
learned here is that there indeed exists a rich inter-
play between the polymer description and the ordinary
Schrödinger representation. The full structure of such re-
lation still needs to be unravelled. We can only hope that
a full understanding of these issues will shed some light
in the ultimate goal of treating the quantum dynamics
of background independent field systems such as general
relativity.
Acknowledgments
We thank A. Ashtekar, G. Hossain, T. Pawlowski and P.
Singh for discussions. This work was in part supported
by CONACyT U47857-F and 40035-F grants, by NSF
PHY04-56913, by the Eberly Research Funds of Penn
State, by the AMC-FUMEC exchange program and by
funds of the CIC-Universidad Michoacana de San Nicolás
de Hidalgo.
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|
0704.0008 | Numerical solution of shock and ramp compression for general material
properties | Numerical solution of shock and ramp compression
for general material properties
Damian C. Swift∗
Materials Science and Technology Division,
Lawrence Livermore National Laboratory,
7000, East Avenue, Livermore, CA 94550, U.S.A.
(Dated: March 7, 2007; revised April 8, 2008 and July 1, 2008 – LA-UR-07-2051)
Abstract
A general formulation was developed to represent material models for applications in dynamic
loading. Numerical methods were devised to calculate response to shock and ramp compression, and
ramp decompression, generalizing previous solutions for scalar equations of state. The numerical
methods were found to be flexible and robust, and matched analytic results to a high accuracy.
The basic ramp and shock solution methods were coupled to solve for composite deformation
paths, such as shock-induced impacts, and shock interactions with a planar interface between
different materials. These calculations capture much of the physics of typical material dynamics
experiments, without requiring spatially-resolving simulations. Example calculations were made of
loading histories in metals, illustrating the effects of plastic work on the temperatures induced in
quasi-isentropic and shock-release experiments, and the effect of a phase transition.
PACS numbers: 62.50.+p, 47.40.-x, 62.20.-x, 46.35.+z
Keywords: material dynamics, shock, isentrope, adiabat, numerical solution, constitutive behavior
∗Electronic address: [email protected]
http://arxiv.org/abs/0704.0008v3
mailto:[email protected]
I. INTRODUCTION
The continuum representation of matter is widely used for material dynamics in sci-
ence and engineering. Spatially-resolved continuum dynamics simulations are the most
widespread and familiar, solving the initial value problem by discretizing the spatial domain
and integrating the dynamical equations forward in time to predict the motion and defor-
mation of components of the system. This type of simulation is used, for instance, to study
hypervelocity impact problems such as the vulnerability of armor to projectiles [1, 2], the
performance of satellite debris shields [3], and the impact of meteorites with planets, notably
the formation of the moon [4]. The problem can be divided into the dynamical equations
of the continuum, the state field of the components s(~r), and the inherent properties of
the materials. Given the local material state s, the material properties allow the stress τ
to be determined. Given the stress field τ(~r) and mass density field ρ(~r), the dynamical
equations describe the fields of acceleration, compression, and thermodynamic work done
on the materials.
The equations of continuum dynamics describe the behavior of a dynamically deforming
system of arbitrary complexity. Particular, simpler deformation paths can be described more
compactly by different sets of equations, and solved by different techniques than those used
for continuum dynamics in general. Simpler deformation paths occur often in experiments
designed to develop and calibrate models of material properties. These paths can be regarded
as different ways of interrogating the material properties. The principal examples in material
dynamics are shock and ramp compression [5, 6]. Typical experiments are designed to induce
such loading histories and measure or infer the properties of the material in these states
before they are destroyed by release from the edges or by reflected waves.
The development of the field of material dynamics was driven by applications in the
physics of hypervelocity impact and high explosive systems, including nuclear weapons [7].
In the regimes of interest, typically components with dimensions ranging from millime-
ters to meters and pressures from 1GPa to 1TPa, material behavior is dominated by the
scalar equation of state (EOS): the relationship between pressure, compression (or mass
density), and internal energy. Other components of stress (specifically shear stresses) are
much smaller, and chemical explosives react promptly so can be treated by simple mod-
els of complete detonation. EOS were developed as fits to experimental data, particularly
to series of shock states and to isothermal compression measurements [8]. It is relatively
straightforward to construct shock and ramp compression states from an EOS algebraically
or numerically depending on the EOS, and to fit an EOS to these measurements. More
recently, applications and scientific interest have grown to include a wider range of pressures
and time scales, such as laser-driven inertial confinement fusion [9], and experiments are
designed to measure other aspects than the EOS, such as the kinetics of phase changes, con-
stitutive behavior describing shear stresses, incomplete chemical reactions, and the effects of
microstructure, including grain orientation and porosity. Theoretical techniques have also
evolved to predict the EOS with ∼1% accuracy [10] and elastic contributions to shear stress
with slightly poorer accuracy [11].
A general convention for representing material states is described, and numerical methods
are reported for calculating shock and ramp compression states from general representations
of material properties.
II. CONCEPTUAL STRUCTURE FOR MATERIAL PROPERTIES
The desired structure for the description of the material state and properties under dy-
namic loading was developed to be as general as possible with respect to the types of material
or models to be represented in the same framework, and designed to give the greatest amount
of commonality between spatially-resolved simulations and calculations of shock and ramp
compressions.
In condensed matter on sub-microsecond time scales, heat conduction is often too slow to
have a significant effect on the response of the material, and is ignored here. The equations
of non-relativistic continuum dynamics are, in Lagrangian form, i.e. along characteristics
moving with the local material velocity ~u(~r),
Dρ(~r, t)
= −ρ(~r, t)div~u(~r, t) (1)
D~u(~r, t)
ρ(~r, t)
div τ(~r, t) (2)
De(~r, t)
= ||τ(~r, t)grad~u(~r, t)|| (3)
where ρ is the mass density and e the specific internal energy. Changes in e can be related
to changes in the temperature T through the heat capacity. The inherent properties of
each material in the problem are described by its constitutive relation or equation of state
τ(s). As well as experiencing compression and work from mechanical deformation, the local
material state s(~r, t) can evolve through internal processes such as plastic flow. In general,
Ds(~r, t)
≡ ṡ[s(~r, t), U(~r, t)] : U ≡ grad ~u(~r, t) (4)
which can also include the equations for ∂ρ/∂t and ∂e/∂t. Thus the material properties must
describe at a minimum τ(s) and ṡ[s(~r, t), U(~r, t)] for each material. If they also describe T (s),
the conductivity, and ṡ(ė), then heat conduction can be treated. Other functions may be
needed for particular numerical methods in continuum dynamics, such as the need for wave
speeds (e.g. the longitudinal sound speed), which are needed for time step control in explicit
time integration. Internally, within the material properties models, it is desirable to re-use
software as much as possible, and other functions of the state are therefore desirable to allow
models to be constructed in a modular and hierarchical way. Arithmetic manipulations must
be performed on the state during numerical integration, and these can be encoded neatly
using operator overloading, so the operator of the appropriate type is invoked automatically
without having to include ‘if-then-else’ structures for each operator as is the case in non-
object-oriented programming languages such as Fortran-77. For instance, if ṡ is calculated
in a forward-time numerical method then changes of state are calculated using numerical
evolution equations such as
s(t+ δt) = s(t) + δtṡ. (5)
Thus for a general state s and its time derivative ṡ, which has an equivalent set of compo-
nents, it is necessary to multiply a state by a real number and to add two states together.
For a specific software implementation, other operations may be needed, for example to
create, copy, or destroy a new instance of a state.
The attraction of this approach is that, by choosing a reasonably general form for the
constitutive relation and associated operations, it is possible to separate the continuum
dynamics part of the problem from the inherent behavior of the material. The relations
describing the properties of different types of material can be encapsulated in a library form
where the continuum dynamics program need know nothing about the relations for any spe-
cific type of material, and vice versa. The continuum dynamics programs and the material
properties relations can be developed and maintained independently of each other, provided
that the interface remains the same (Table I). This is an efficient way to make complicated
material models available for simulations of different types, including Lagrangian and Eule-
rian hydrocodes operating on different numbers of dimensions, and calculations of specific
loading or heating histories such as shock and ramp loading discussed below. Software in-
terfaces have been developed in the past for scalar EOS with a single structure for the state
[12], but object-oriented techniques make it practical to extend the concept to much more
complicated states, to combinations of models, and to alternative types of model selected
when the program is run, without having to find a single super-set state encompassing all
possible states as special cases.
A very wide range of types of material behavior can be represented with this formalism.
At the highest level, different types of behavior are characterized by different structures for
the state s (Table II). For each type of state, different specific models can be defined, such
as perfect gas, polytropic and Grüneisen EOS. For each specific model, different materials
are represented by choosing different values for the parameters in the model, and different
local material states are represented through different values for the components of s. In the
jargon of object-oriented programming, the ability to define an object whose precise type
is undetermined until the program is run is known as polymorphism. For our application,
polymorphism is used at several levels in the hierarchy of objects, from the overall type of a
material (such as ‘one represented by a pressure-density-energy EOS’ or ‘one represented by
a deviatoric stress model’) through the type of relation used to describe the properties of that
material type (such as perfect gas, polytropic, or Grüneisen for a pressure-density-energy
EOS, or Steinberg-Guinan [13] or Preston-Tonks-Wallace [14] for a deviatoric stress model),
to the type of general mathematical function used to represent some of these relations (such
as a polynomial or a tabular representation of γ(ρ) in a polytropic EOS) (Table III). States
or models may be defined by extending or combining other states or models – this can be
implemented using the object-oriented programming concept of inheritance. Thus deviatoric
stress models can be defined as an extension to any pressure-density-energy EOS (they are
usually written assuming a specific type, such as Steinberg’s cubic Grüneisen form), homo-
geneous mixtures can be defined as combinations of any pressure-density-temperature EOS,
and heterogeneous mixtures can be defined as combinations of materials each represented
by any type of material model.
Trial implementations have been made as libraries in the C++ and Java programming
languages [15]. The external interface to the material properties was general at the level
of representing a generic material type and state. The type of state and model were then
selected when programs using the material properties library were run. In C++, objects
which were polymorphic at run time had to be represented as pointers, requiring additional
software constructions to allocate and free up physical memory associated with each object.
It was possible to include general re-usable functions as polymorphic objects when defining
models: real functions of one real parameter could be polynomials, transcendentals, tabular
with different interpolation schemes, piecewise definitions over different regions of the one
dimensional line, sums, products, etc; again defined specifically at run time. Object-oriented
polymorphism and inheritance were thus very powerful techniques for increasing software
re-use, making the software more compact and more reliable through the greater use of
functions which had already been tested.
Given conceptual and software structures designed to represent general material proper-
ties suitable for use in spatially-resolved continuum dynamics simulations, we now consider
the use of these generic material models for calculating idealized loading paths.
III. IDEALIZED ONE-DIMENSIONAL LOADING
Experiments to investigate the response of materials to dynamic loading, and to calibrate
parameters in models of their behavior, are usually designed to apply as simple a loading
history as is consistent with the transient state of interest. The simplest canonical types of
loading history are shock and ramp [5, 6]. Methods of solution are presented for calculating
the result of shock and ramp loading for materials described by generalized material models
discussed in the previous section. Such direct solution removes the need to use a time-
and space-resolved continuum dynamics simulation, allowing states to be calculated with
far greater efficiency and without the need to consider and make allowance for attributes of
resolved simulations such as the finite numerical resolution and the effect of numerical and
artificial viscosities.
A. Ramp compression
Ramp compression is taken here to mean compression or decompression. If the material
is represented by an inviscid scalar EOS, i.e. ignoring dissipative processes and non-scalar
effects from elastic strain, ramp compression follows an isentrope. This is no longer true
when dissipative processes such as plastic heating occur. The term ‘quasi-isentropic’ is
sometimes used in this context, particularly for shockless compression; here we prefer to
refer to the thermodynamic trajectories as adiabats since this is a more appropriate term:
no heat is exchanged with the surroundings on the time scales of interest.
For adiabatic compression, the state evolves according to the second law of thermody-
namics,
de = T dS − p dv (6)
where T is the temperature and S the specific entropy. Thus
ė = T Ṡ − p v̇ = T Ṡ −
pdiv~u
, (7)
or for a more general material whose stress tensor is more complicated than a scalar pressure,
de = T dS + τn dv ⇒ ė = T Ṡ +
τndiv~u
where τn is the component of stress normal to the direction of deformation. The velocity
gradient was expressed through a compression factor η ≡ ρ′/ρ and a strain rate ǫ̇. In all
ramp experiments used in the development and calibration of accurate material models,
the strain has been applied uniaxially. More general strain paths, for instance isotropic or
including a shear component, can be treated by the same formalism, and that the working
rate is then a full inner product of the stress and strain tensors.
The acceleration or deceleration of the material normal to the wave as it is compressed
or expanded adiabatically is
, (9)
from which it can be deduced that
where cl is the longitudinal wave speed.
As with continuum dynamics, internal evolution of the material state can be calculated
simultaneously with the continuum equations, or operator split and calculated periodically
at constant compression [16]. The results are the same to second order in the compression
increment. Operator-splitting allows calculations to be performed without an explicit en-
tropy, if the continuum equations are integrated isentropically and dissipative processes are
captured by internal evolution at constant compression.
Operator-splitting is desirable when internal evolution can produce highly nonlinear
changes, such as reaction from solid to gas: rapid changes in state and properties can
make numerical schemes unstable. Operator-splitting is also desirable when the integration
time step for internal evolution is much shorter than the continuum dynamics time step.
Neither of these considerations is very important for ramp compression without spatial res-
olution, but operator-splitting was used as an option in the ramp compression calculations
for consistency with continuum dynamics simulations.
The ramp compression equations were integrated using forward-time Runge-Kutta nu-
merical schemes of second order. The fourth order scheme is a trivial extension. The
sequence of operations to calculate an increment of ramp compression is as follows:
1. Time increment:
δt = −
| ln η|
2. Predictor:
s(t + δt/2) = s(t) +
ṡm(s(t), ǫ̇) (12)
3. Corrector:
s(t+ δt) = s(t) + δtṡm(s(t+ δt/2), ǫ̇) (13)
4. Internal evolution:
s(t+ δt) → s(t+ δt) +
∫ t+δt
ṡi(s(t
′), ǫ̇) dt′ (14)
where ṡm is the model-dependent state evolution from applied strain, and ṡi is internal
evolution at constant compression.
The independent variable for integration is specific volume v or mass density ρ; for
numerical integration finite steps are taken in ρ and v. The step size ∆ρ can be controlled so
that the numerical error during integration remains within chosen limits. A tabular adiabat
can be calculated by integrating over a range of v or ρ, but when simulating experimental
scenarios the upper limit for integration is usually that one of the other thermodynamic
quantities reaches a certain value, for example that the normal component of stress reaches
zero, which is the case on release from a high pressure state at a free surface. Specific
end conditions were found by monitoring the quantity of interest until bracketed by a finite
integration step, then bisecting until the stop condition was satisfied to a chosen accuracy.
During bisection, each trial calculation was performed as an integration from the first side
of the bracket by the trial compression.
B. Shock compression
Shock compression is the solution of a Riemann problem for the dynamics of a jump
in compression moving with constant speed and with a constant thickness. The Rankine-
Hugoniot (RH) equations [5] describing the shock compression of matter are derived in
the continuum approximation, where the shock is a formal discontinuity in the continuum
fields. In reality, matter is composed of atoms, and shocks have a finite width governed by
the kinetics of dissipative processes – at a fundamental level, matter does not distinguish
between shock compression and ramp compression with a high strain rate – but the RH
equations apply as long as the width of the region of matter where unresolved processes
occur is constant. Compared with the isentropic states induced by ramp compression in
a material represented by an EOS, a shock always increases the entropy and hence the
temperature. With dissipative processes included, the distinction between a ramp and a
shock may become blurred.
The RH equations express the conservation of mass, momentum, and energy across a
moving discontinuity in state. They are usually expressed in terms of the pressure, but are
readily generalized for materials supporting shear stresses by using the component of stress
normal to the shock (i.e., parallel with the direction of propagation of the shock), τn:
u2s = −v
τn − τn0
v0 − v
, (15)
∆up =
−(τn − τn0)(v0 − v), (16)
e = e0 −
(τn + τn0)(v0 − v), (17)
where us is the speed of the shock wave with respect to the material, ∆up is the change in
material speed normal to the shock wave (i.e., parallel to its direction of propagation), and
subscript 0 refers to the initial state.
The RH relations can be applied to general material models if a time scale or strain rate
is imposed, and an orientation chosen for the material with respect to the shock. Shock
compression in continuum dynamics is almost always uniaxial.
The RH equations involve only the initial and final states in the material. If a material
has properties that depend on the deformation path – such as plastic flow or viscosity –
then physically the detailed shock structure may make a difference [17]. This is a limitation
of discontinuous shocks in continuum dynamics: it may be addressed as discussed above
by including dissipative processes and considering ramp compression, if the dissipative pro-
cesses can be represented adequately in the continuum approximation. Spatially-resolved
simulations with numerical differentiation to obtain spatial derivatives and forward time
differencing are usually not capable of representing shock discontinuities directly, and an
artificial viscosity is used to smear shock compression over a few spatial cells [18]. The
trajectory followed by the material in thermodynamic space is a smooth adiabat with dissi-
pative heating supplied by the artificial viscosity. If plastic work is also included during this
adiabatic compression, the overall heating for a given compression is greater than from the
RH equations. To be consistent, plastic flow should be neglected while the artificial viscosity
is non-zero. This localized disabling of physical processes, particularly time-dependent ones,
during the passage of the unphysically smeared shock was previously found necessary for
numerically stable simulations of detonation waves by reactive flow [19].
Detonation waves are reactive shock waves. Steady planar detonation (the Chapman-
Jouguet state [20]) may be calculated using the RH relations, by imposing the condition
that the material state behind the shock is fully reacted.
Several numerical methods have been used to solve the RH equations for materials repre-
sented by an EOS only [21, 22]. The general RH equations may be solved numerically for a
given shock compression ∆ρ by varying the specific internal energy e until the normal stress
from the material model equals that from the RH energy equation, Eq. 17. The shock and
particle speeds are then calculated from Eqs 15 and 16. This numerical method is particu-
larly convenient for EOS of the form p(ρ, e), as e may be varied directly. Solutions may still
be found for general material models using ṡ(ė), by which the energy may be varied until
the solution is found.
Numerically, the solution was found by bracketing and bisection:
1. For given compression ∆ρ, take the low-energy end for bracketing as a nearby state
s− (e.g. the previous state, of lower compression, on the Hugoniot), compressed adia-
batically (to state s̃), and cooled so the specific internal energy is e(s−).
2. Bracket the desired state: apply successively larger heating increments ∆e to s̃, evolv-
ing each trial state internally, until τn(s) from the material model exceeds τn(e − e0)
from Eq. 17.
3. Bisect in ∆e, evolving each trial state internally, until τn(s) equals τn(e − e0) to the
desired accuracy.
As with ramp compression, the independent variable for solution was mass density ρ,
and finite steps ∆ρ were taken. Each shock state was calculated independently of the rest,
so numerical errors did not accumulate along the shock Hugoniot. The accuracy of the
solution was independent of ∆ρ. A tabular Hugoniot can be calculated by solving over a
range of ρ, but again when simulating experimental scenarios it is usually more useful to
calculate the shock state where one of the other thermodynamic quantities reaches a certain
value, often that up and τn match the values from another, simultaneous shock calculation
for another material – the situation in impact and shock transmission problems, discussed
below. Specific stop conditions were found by monitoring the quantity of interest until
bracketed by a finite solution step, then bisecting until the stop condition was satisfied to a
chosen accuracy. During bisection, each trial calculation was performed as a shock from the
initial conditions to the trial shock compression.
C. Accuracy: application to air
The accuracy of these numerical schemes was tested by comparing with shock and ramp
compression of a material represented by a perfect gas EOS,
p = (γ − 1)ρe. (18)
The numerical solution requires a value to be chosen for every parameter in the material
model, here γ. Air was chosen as an example material, with γ = 1.4. Air at standard tem-
perature and pressure has approximately ρ = 10−3 g/cm3 and e = 0.25MJ/kg. Isentropes
for the perfect gas EOS have the form
pρ−γ = constant, (19)
and shock Hugoniots have the form
p = (γ − 1)
2e0ρ0ρ+ p0(ρ− ρ0)
(γ + 1)ρ0 − (γ − 1)ρ
. (20)
The numerical solutions reproduced the principal isentrope and Hugoniot to 10−3% and 0.1%
respectively, for a compression increment of 1% along the isentrope and a solution tolerance
of 10−6GPa for each shock state (Fig. 1). Over most of the range, the error in the Hugoniot
was 0.02% or less, only approaching 0.1% near the maximum shock compression.
IV. COMPLEX BEHAVIOR OF CONDENSED MATTER
The ability to calculate shock and ramp loci in state space, i.e. as a function of vary-
ing loading conditions, is particularly convenient for investigating complex aspects of the
response of condensed matter to dynamic loading. Each locus can be obtained by a single
series of shock or ramp solutions, rather than having to perform a series of time- and space-
resolved continuum dynamics simulations, varying the initial or boundary conditions and
reducing the solution. We consider the calculation of temperature in the scalar EOS, the
effect of material strength and the effect of phase changes.
A. Temperature
The continuum dynamics equations can be closed using a mechanical EOS relating stress
to mass density, strain, and internal energy. For a scalar EOS, the ideal form to close the
continuum equations is p(ρ, e), with s = {ρ, e} the natural choice for the primitive state
fields. However, the temperature is needed as a parameter in physical descriptions of many
contributions to the constitutive response, including plastic flow, phase transitions, and
chemical reactions. Here, we discuss the calculation of temperature in different forms of the
scalar EOS.
1. Density-temperature equations of state
If the scalar EOS is constructed from its underlying physical contributions for continuum
dynamics, it may take the form e(ρ, T ), from which p(ρ, T ) can be calculated using the
second law of thermodynamics [10]. An example is the ‘SESAME’ form of EOS, based on
interpolated tabular relations for {p, e}(ρ, T ) [23]. A pair of relations {p, e}(ρ, T ) can be
used as a mechanical EOS by eliminating T , which is equivalent to inverting e(ρ, T ) to find
T (ρ, e), then substituting in p(ρ, T ). For a general e(ρ, T ) relation, for example for the
SESAME EOS, the inverse can be calculated numerically as required, along an isochore. In
this way, a {p, e}(ρ, T ) can be used as a p(ρ, e) EOS.
Alternatively, the same p(ρ, T ) relation can be used directly with a primitive state field
including temperature instead of energy: s = {ρ, T}. The evolution of the state under
mechanical work then involves the calculation of Ṫ (ė), i.e. the reciprocal of the specific heat
capacity, which is a derivative of e(ρ, T ). As this calculation does not require e(ρ, T ) to be
inverted, it is computationally more efficient to use {p, e}(ρ, T ) EOS with a temperature-
based, rather than energy-based, state. The main disadvantage is that it is more difficult
to ensure exact energy conservation as the continuum dynamics equations are integrated in
time, but any departure from exact conservation is at the level of accuracy of the algorithm
used to integrate the heat capacity.
Both structures of EOS have been implemented for material property calculations. Taking
a SESAME type EOS, thermodynamic loci were calculated with {ρ, e} or {ρ, T} primitive
states, for comparison (Fig. 2). For a monotonic EOS, the results were indistinguishable
within differences from forward or reverse interpolation of the tabular relations. When
the EOS, or the effective surface using a given order of interpolating function, was non-
monotonic, the results varied greatly because of non-uniqueness when eliminating T for the
{ρ, e} primitive state.
2. Temperature model for mechanical equations of state
Mechanical EOS are often available as empirical, algebraic relations p(ρ, e), derived from
shock data. Temperature can be calculated without altering the mechanical EOS by adding
a relation T (ρ, e). While this relation could take any form in principle, one can also follow
the logic of the Grüneisen EOS, in which the pressure is defined in terms of its deviation
∆p(ρ, e − er) from a reference curve {pr, er}(ρ). Thus temperatures can be calculated by
reference to a compression curve along which the temperature and specific internal energy
are known, {Tr, er}(ρ), and a specific heat capacity defined as a function of density cv(ρ).
In the calculations, this augmented EOS was represented as a ‘mechanical-thermal’ form
comprising any p(ρ, e) EOS plus the reference curves – an example of software inheritance
and polymorphism.
One natural reference curve for temperature is the cold curve, Tr = 0K. The cold curve
can be estimated from the principal isentrope e(ρ)|s0 using the estimated density variation
of the Grüneisen parameter:
er(ρ) = e(ρ)|s0 − T0cpe
a(1−ρ0/ρ)
)γ0−a
[24]. In this work, the principal isentrope was calculated in tabular form from the mechanical
EOS, using the ramp compression algorithm described above.
Empirical EOS are calibrated using experimental data. Shock and adiabatic compression
measurements on strong materials inevitably include elastic-plastic contributions as well as
the scalar EOS itself. If the elastic-plastic contributions are not taken into account self-
consistently, the EOS may implicitly include contributions from the strength. A unique
scalar EOS can be constructed to reproduce the normal stress as a function of compression
for any unique loading path: shock or adiabat, for a constant or smoothly-varying strain
rate. Such an EOS would not generally predict the response to other loading histories. The
EOS and constitutive properties for the materials considered here were constructed self-
consistently from shock data – this does not mean the models are accurate for other loading
paths, as neither the EOS nor the strength model includes all the physical terms that real
materials exhibit. This does not in any case matter for the purposes of demonstrating the
properties of the numerical schemes.
This mechanical-thermal procedure was applied to Al using a Grüneisen EOS fitted to the
same shock data used to calculate the {p, e}(ρ, T ) EOS discussed above [24]. Temperatures
were in good agreement (Fig. 2). The mechanical-thermal calculations required a similar
computational effort to the tabular {p, e}(ρ, T ) EOS with a {ρ, T} primitive states (and
were thus much more efficient than the tabular EOS with {ρ, e} states), and described the
EOS far more compactly.
B. Strength
For dynamic compressions to o(10GPa) and above, on microsecond time scales, the flow
stress of solids is often treated as a correction or small perturbation to the scalar EOS.
However, the flow stress has been observed to be much higher on nanosecond time scales
[25], and interactions between elastic and plastic waves may have a significant effect on
the compression and wave propagation. The Rankine-Hugoniot equations should be solved
self-consistently with strength included.
1. Preferred representation of isotropic strength
There is an inconsistency in the standard continuum dynamics treatment of scalar (pres-
sure) and tensor (stress) response. The scalar EOS expresses the pressure p(ρ, e) as the
dependent quantity, which is the most convenient form for use in the continuum equations.
Standard practice is to use sub-Hookean elasticity (hypoelastic form) [16] (Table II), in
which the state parameters include the stress deviator σ, evolved by integration
σ̇ = G(s)ǫ̇ (22)
where G is the shear modulus and ǫ̇ the strain rate deviator. Thus the isotropic and devia-
toric contributions to stress are not treated in an equivalent way: the pressure is calculated
from a local state involving a strain-like parameter (mass density), whereas the stress de-
viator evolves with the time-derivative of strain. This inconsistency causes problems along
complicated loading paths because G varies strongly with compression: if a material is sub-
jected to a shear strain ǫ, then isotropic compression (increasing the shear modulus from
G to G′, leaving ǫ unchanged), then shear unloading to isotropic stress, the true unloading
strain is −ǫ, whereas the hypoelastic calculation would require a strain of −ǫG/G′. Using
Be and the Steinberg-Guinan strength model as an example of the difference between hy-
poelastic and hyperelastic calculations, consider an initial strain to a flow stress of 0.3GPa
followed by isothermal, isotropic compression to 100GPa,. the strain to unload to a state
of isotropic stress is 0.20% (hyperelastic) and 0.09% (hypoelastic). The discrepancy arises
because the hypoelastic model does not increase the deviatoric stress under compression at
constant deviatoric strain.
The stress can be considered as a direct response of the material to the instantaneous state
of elastic strain: σ(ǫ, T ). This relation can be predicted directly with electronic structure
calculations of the stress tensor in a solid for a given compression and elastic strain state [11],
and is a direct generalization of the scalar equation of state. A more consistent representation
of the state parameters is to use the strain deviator ǫ rather than σ, and to calculate σ from
scratch when required using
σ = G(s)ǫ (23)
– a hyperelastic formulation. The state parameters are then {ρ, e, ǫ, ǫ̃p}.
The different formulations give different answers when deviatoric strain is accumulated
at different compressions, in which case the hyperelastic formulation is correct. If the shear
modulus varies with strain deviator – i.e., for nonlinear elasticity – then the definition of
G(ǫ) must be adjusted to give the same stress for a given strain.
Many isotropic strength models use scalar measures of the strain and stress to parame-
terize work hardening and to apply a yield model of flow stress:
fǫ||ǫ2||, σ̃ =
fσ||σ2||. (24)
Inconsistent conventions for equivalent scalar measures have been used by different workers.
In the present work, the common shock physics convention was used that the flow stress
component of τn is
Y where Y is the flow stress. For consistency with published speeds
and amplitudes for elastic waves, fǫ = fσ =
, in contrast to other values previously used
for lower-rate deformation [26]. In principle, the values of fǫ and fσ do not matter as long as
the strength parameters were calibrated using the same values then used in any simulations.
2. Beryllium
The flow stress measured from laser-driven shock experiments on Be crystals a few tens
of micrometers thick is, at around 5-9GPa [25], much greater than the 0.3-1.3GPa mea-
sured on microsecond time scales. A time-dependent crystal plasticity model for Be is being
developed, and the behavior under dynamic loading depends on the detailed time depen-
dence of plasticity. Calculations were performed with the Steinberg-Guinan strength model
developed for microsecond scale data [24], and, for the purposes of rough comparison, with
elastic-perfectly plastic response with a flow stress of 10GPa. The elastic-perfectly plastic
model neglected pressure- and work- hardening.
Calculations were made of the principal adiabat and shock Hugoniot, and of a release
adiabat from a state on the principal Hugoniot. Calculations were made with and without
strength. Considering the state trajectories in stress-volume space, it is interesting to note
that heating from plastic flow may push the adiabat above the Hugoniot, because of the
greater heating obtained by integrating along the adiabat compared with jumping from
the initial to the final state on the Hugoniot (Fig. 3). Even with an elastic-perfectly plastic
strength model, the with-strength curves do not lie exactly 2
Y above the strengthless curves,
because heating from plastic flow contributes an increasing amount of internal energy to the
EOS as compression increases.
An important characteristic for the seeding of instabilities by microstructural variations
in shock response is the shock stress at which an elastic wave does not run ahead of the
shock. In Be with the high flow stress of nanosecond response, the relation between shock
and particle speeds is significantly different from the relation for low flow stress (Fig. 4). For
low flow stress, the elastic wave travels at 13.2 km/s. A plastic shock travels faster than this
for pressures greater than 110GPa, independent of the constitutive model. The speed of a
plastic shock following the initial elastic wave is similar to the low strength case, because the
material is already at its flow stress, but the speed of a single plastic shock is appreciably
higher.
For compression to a given normal stress, the temperature is significantly higher with
plastic flow included. The additional heating is particularly striking on the principal adi-
abat: the temperature departs significantly from the principal isentrope. Thus ramp-wave
compression of strong materials may lead to significant levels of heating, contrary to com-
mon assumptions of small temperature increases [27]. Plastic flow is largely irreversible, so
heating occurs on unloading as well as loading. Thus, on adiabatic release from a shock-
compressed state, additional heating occurs compared with the no-strength case. These
levels of heating are important as shock or release melting may occur at a significantly lower
shock pressure than would be expected ignoring the effect of strength. (Fig. 5.)
C. Phase changes
An important property of condensed matter is phase changes, including solid-solid poly-
morphism and solid-liquid. An equilibrium phase diagram can be represented as a single
overall EOS surface as before. Multiple, competing phases with kinetics for each phase trans-
formation can be represented conveniently using the structure described above for general
material properties, for example by describing the local state as a set of volume fractions
fi of each possible simple-EOS phase, with transition rates and equilibration among them.
This model is described in more detail elsewhere [19]. However, it is interesting to investi-
gate the robustness of the numerical scheme for calculating shock Hugoniots when the EOS
has the discontinuities in value and gradient associated with phase changes.
The EOS of molten metal, and the solid-liquid phase transition, can be represented to a
reasonable approximation as an adjustment to the EOS of the solid:
ptwo-phase(ρ, e) = psolid(ρ, ẽ) (25)
where
e : T (ρ, e) < Tm(ρ)
e−∆h̃m : ∆h̃m ≡ cv(ρ, e) [T (ρ, e)− Tm(ρ)] < ∆hm
e−∆hm : otherwise
and ∆hm is the specific latent heat of fusion. Taking the EOS and a modified Lindemann
melting curve for Al [24], and using ∆hm = 0.397MJ/kg, the shock Hugoniot algorithm was
found to operate stably across the phase transition (Fig. 6).
V. COMPOSITE LOADING PATHS
Given methods to calculate shock and adiabatic loading paths from arbitrary initial
states, a considerable variety of experimental scenarios can be treated from the interaction
of loading or unloading waves with interfaces between different materials, in planar geometry
for uniaxial compression. The key physical constraint is that, if two dissimilar materials are
to remain in contact after an interaction such as an impact or the passage of a shock, the
normal stress τn and particle speed up in both materials must be equal on either side of the
interface. The change in particle speed and stress normal to the waves were calculated above
for compression waves running in the direction of increasing spatial ordinate (left to right).
Across an interface, the sense is reversed for the material at the left. Thus a projectile
impacting a stationary target to the right is decelerated from its initial speed by the shock
induced by impact.
The general problem at an interface can be analyzed by considering the states at the
instant of first contact – on impact, or when a shock traveling through a sandwich of ma-
terials first reaches the interface. The initial states are {ul, sl; ur, sr}. The final states are
{uj, s
l; uj, r
r} where uj is the joint particle speed, τn(s
l) = τn(s
r), and s
i is connected to si
by either a shock or an adiabat, starting at the appropriate initial velocity and stress, and
with orientation given by the side of the system each material occurs on. Each type of wave
is considered in turn, looking for an intersection in the up − τn plane. Examples of these
wave interactions are the impact of a projectile with a stationary target (Fig. 7), release of a
shock state at a free surface or a material (e.g. a window) of lower shock impedance (hence
reflecting a release wave into the shocked material – Fig. 8), reshocking at a surface with a
material of higher shock impedance (Fig. 8), or tension induced as materials try to separate
in opposite directions when joined by a bonded interface (Fig. 9). Each of these scenarios
may occur in turn following the impact of a projectile with a target: if the target is layered
then a shock is transmitted across each interface with a release or a reshock reflected back,
depending on the materials; release ultimately occurs at the rear of the projectile and the
far end of the target, and the oppositely-moving release waves subject the projectile and
target to tensile stresses when they interact (Fig. 10).
As an illustration of combining shock and ramp loading calculations, consider the problem
of an Al projectile, initially traveling at 3.6 km/s, impacting a stationary, composite target
comprising a Mo sample and a LiF release window [28, 29]. The shock and release states were
calculated using published material properties [24]. The initial shock state was calculated to
have a normal stress of 63.9GPa. On reaching the LiF, the shock was calculated to transmit
at 27.1GPa, reflecting as a release in the Mo. These stresses match the continuum dynamics
simulation to within 0.1GPa in the Mo and 0.3GPa in the LiF, using the same material
properties (Fig. 11). The associated wave and particle speeds match to a similar accuracy;
wave speeds are much more difficult to extract from the continuum dynamics simulation.
An extension of this analysis can be used to calculate the interaction of oblique shocks
with an interface [30].
VI. CONCLUSIONS
A general formulation was developed to represent material models for applications in
dynamic loading, suitable for software implementation in object-oriented programming lan-
guages. Numerical methods were devised to calculate the response of matter represented
by the general material models to shock and ramp compression, and ramp decompression,
by direct evaluation of the thermodynamic pathways for these compressions rather than
spatially-resolved simulations. This approach is a generalization of earlier work on solutions
for materials represented by a scalar equation of state. The numerical methods were found
to be flexible and robust: capable of application to materials with very different properties.
The numerical solutions matched analytic results to a high accuracy.
Care was needed with the interpretation of some types of physical response, such as plas-
tic flow, when applied to deformation at high strain rates. The underlying time-dependence
of processes occurring during deformation should be taken into account. The actual history
of loading and heating experienced by material during the passage of a shock may influence
the final state – this history is not captured in the continuum approximation to material
dynamics, where shocks are treated as discontinuities. Thus care is also needed in spa-
tially resolved simulations when shocks are modeled using artificial viscosity to smear them
unphysically over a finite thickness.
Calculations were shown to demonstrate the operation of the algorithms for shock and
ramp compression with material models representative of complex solids including strength
and phase transformations.
The basic ramp and shock solution methods were coupled to solve for composite defor-
mation paths, such as shock-induced impacts, and shock interactions with a planar interface
between different materials. Such calculations capture much of the physics of typical ma-
terial dynamics experiments, without requiring spatially-resolving simulations. The results
of direct solution of the relevant shock and ramp loading conditions were compared with
hydrocode simulations, showing complete consistency.
Acknowledgments
Ian Gray introduced the author to the concept of multi-model material properties soft-
ware. Lee Markland developed a prototype Hugoniot-calculating computer program for
equations of state while working for the author as an undergraduate summer student.
Evolutionary work on material properties libraries was supported by the U.K. Atomic
Weapons Establishment, Fluid Gravity Engineering Ltd, andWessex Scientific and Technical
Services Ltd. Refinements to the technique and applications to the problems described were
undertaken at Los Alamos National Laboratory (LANL) and Lawrence Livermore National
Laboratory (LLNL).
The work was performed partially in support of, and funded by, the National Nuclear Se-
curity Agency’s Inertial Confinement Fusion program at LANL (managed by Steven Batha),
and LLNL’s Laboratory-Directed Research and Development project 06-SI-004 (Principal
Investigator: Hector Lorenzana). The work was performed under the auspices of the U.S.
Department of Energy under contracts W-7405-ENG-36, DE-AC52-06NA25396, and DE-
AC52-07NA27344.
References
[1] J.K. Dienes, J.M. Walsh, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic
Press, New York, 1970).
[2] D.J. Benson, Comp. Mech. 15, 6, pp 558-571 (1995).
[3] J.W. Gehring, Jr, in R. Kinslow (Ed), “High-Velocity Impact Phenomena” (Academic Press,
New York, 1970).
[4] R.M. Canup, E. Asphaug, Nature 412, pp 708-712 (2001).
[5] For a recent review and introduction, see e.g. M.R. Boslough and J.R. Asay, in J.R. Asay,
M. Shahinpoor (Eds), “High-Pressure Shock Compression of Solids” (Springer-Verlag, New
York, 1992).
[6] For example, C.A. Hall, J.R. Asay, M.D. Knudson, W.A. Stygar, R.B. Spielman, T.D. Pointon,
D.B. Reisman, A. Toor, and R.C. Cauble, Rev. Sci. Instrum. 72, 3587 (2001).
[7] M.A. Meyers, “Dynamic Behavior of Materials” (Wiley, New York, 1994).
[8] G. McQueen, S.P. March, J.W. Taylor, J.N. Fritz, W.J. Carter, in R. Kinslow (Ed), “High-
Velocity Impact Phenomena” (Academic Press, New York, 1970).
[9] J.D. Lindl, “Inertial Confinement Fusion” (Springer-Verlag, New York, 1998).
[10] D.C. Swift, G.J. Ackland, A. Hauer, G.A. Kyrala, Phys. Rev. B 64, 214107 (2001).
[11] J.P. Poirier, G.D. Price, Phys. of the Earth and Planetary Interiors 110, pp 147-56 (1999).
[12] I.N. Gray, P.C. Thompson, B.J. Parker, D.C. Swift, J.R. Maw, A. Giles and others (AWE
Aldermaston), unpublished.
[13] D.J. Steinberg, S.G. Cochran, M.W. Guinan, J. Appl. Phys. 51, 1498 (1980).
[14] D.L. Preston, D.L. Tonks, and D.C. Wallace, J. Appl. Phys. 93, 211 (2003).
[15] A version of the software, including representative parts of the material model library and the
algorithms for calculating the ramp adiabat and shock Hugoniot, is available as a supplemen-
tary file provided with the preprint of this manuscript, arXiv:0704.0008. Software support,
and versions with additional models, are available commercially from Wessex Scientific and
Technical Services Ltd (http://wxres.com).
[16] D. Benson, Computer Methods in Appl. Mechanics and Eng. 99, 235 (1992).
http://arxiv.org/abs/0704.0008
http://wxres.com
[17] J.L. Ding, J. Mech. and Phys. of Solids 54, pp 237-265 (2006).
[18] J. von Neumann, R.D. Richtmyer, J. Appl. Phys. 21, 3, pp 232-237 (1950).
[19] R.M. Mulford, D.C. Swift, in preparation.
[20] W. Fickett, W.C. Davis, “Detonation” (University of California Press, Berkeley, 1979).
[21] R. Menikoff, B.J. Plohr, Rev. Mod. Phys. 61, pp 75-130 (1989).
[22] A. Majda, Mem. Amer. Math. Soc., 41, 275 (1983).
[23] K.S. Holian (Ed.), T-4 Handbook of Material Property Data Bases, Vol 1c: Equations of State,
Los Alamos National Laboratory report LA-10160-MS (1984).
[24] D.J. Steinberg, Equation of State and Strength Properties of Selected Materials, Lawrence
Livermore National Laboratory report UCRL-MA-106439 change 1 (1996).
[25] D.C. Swift, T.E. Tierney, S.-N. Luo, D.L. Paisley, G.A. Kyrala, A. Hauer, S.R. Greenfield,
A.C. Koskelo, K.J. McClellan, H.E. Lorenzana, D. Kalantar, B.A. Remington, P. Peralta,
E. Loomis, Phys.Plasmas 12, 056308 (2005).
[26] R. Hill, “The Mathematical Theory of Plasticity” (Clarendon Press, Oxford, 1950).
[27] C.A. Hall, Phys. Plasmas 7, 5, pp 2069-2075 (2000).
[28] D.C. Swift, A. Seifter, D.B. Holtkamp, and D.A. Clark, Phys. Rev. B 76, 054122 (2007).
[29] A. Seifter and D.C. Swift, Phys. Rev. B 77, 134104 (2008).
[30] E. Loomis, D.C. Swift, J. Appl. Phys. 103, 023518 (2008).
TABLE I: Interface to material models required for explicit forward-time continuum dynamics
simulations.
purpose interface calls
program set-up read/write material data
continuum dynamics equations stress(state)
time step control sound speed(state)
evolution of state (deformation) d(state)/dt(state,grad ~u)
evolution of state (heating) d(state)/dt(state,ė)
internal evolution of state d(state)/dt
manipulation of states create and delete
add states
multiply state by a scalar
check for self-consistency
Parentheses in the interface calls denote functions, e.g. “stress(state)” for “stress as a function of
the instantaneous, local state.” The evolution functions are shown in the operator-split structure
that is most robust for explicit, forward-time numerical solutions and can also be used for
calculations of the shock Hugoniot and ramp compression. Checks for self-consistency include
that mass density is positive, volume or mass fractions of components of a mixture add up to one,
TABLE II: Examples of types of material model, distinguished by different structures in the state
vector.
model state vector effect of mechanical strain
s ṡm(s, gradu)
mechanical equation of state ρ, e −ρdiv~u,−pdiv~u/ρ
thermal equation of state ρ, T −ρdiv~u,−pdiv~u/ρcv
heterogeneous mixture {ρ, e, fv}i {−ρdiv~u,−pdiv~u/ρ, 0}i
homogeneous mixture ρ, T, {fm}i {−ρdiv~u,−pdiv~u/ρcv , 0i
traditional deviatoric strength ρ, e, σ, ǫ̃p −ρdiv~u,
−pdiv~u+fp||σǫ̇p||
, Gǫ̇e,
fǫ||ǫ̇
The symbols are ρ: mass density; e: specific internal energy, T : temperature, fv: volume fraction,
fm: mass fraction, σ: stress deviator, fp: fraction of plastic work converted to heat, gradup:
plastic part of velocity gradient, G: shear modulus, ǫ̇e,p: elastic and plastic parts of strain rate
deviator, ǫ̃p: scalar equivalent plastic strain, fǫ: factor in effective strain magnitude. Reacting
solid explosives can be represented as heterogeneous mixtures, one component being the reacted
products; reaction, a process of internal evolution, transfers material from unreacted to reacted
components. Gas-phase reaction can be represented as a homogeneous mixture, reactions
transferring mass between components representing different types of molecule. Symmetric
tensors such as the stress deviator are represented more compactly by their 6 unique upper
triangular components, e.g. using Voigt notation.
TABLE III: Outline hierarchy of material models, illustrating the use of polymorphism (in the
object-oriented programming sense).
material (or state) type model type
mechanical equation of state polytropic, Grüneisen, energy-based
Jones-Wilkins-Lee, (ρ, T ) table, etc
thermal equation of state temperature-based Jones-Wilkins-
Lee, quasiharmonic, (ρ, T ) table,
reactive equation of state modified polytropic, reactive Jones-
Wilkins-Lee
spall Cochran-Banner
deviatoric stress elastic-plastic, Steinberg-Guinan,
Steinberg-Lund, Preston-Tonks-
Wallace, etc
homogeneous mixture mixing and reaction models
heterogeneous mixture equilibration and reaction models
Continuum dynamics programs can refer to material properties as an abstract ‘material type’
with an abstract material state. The actual type of a material (e.g. mechanical equation of
state), the specific model type (e.g. polytropic), and the state of material of that type are all
handled transparently by the object-oriented software structure.
The reactive equation of state has an additional state parameter λ, and the software operations
are defined by extending those of the mechanical equation of state. Spalling materials can be
represented by a solid state plus a void fraction fv, with operations defined by extending those of
the solid material. Homogeneous mixtures are defined as a set of thermal equations of state, and
the state is the set of states and mass fractions for each. Heterogeneous mixtures are defined as a
set of ‘pure’ material properties of any type, and the state is the set of states for each component
plus its volume fraction.
0.0001
0.001
0.01
0.001 0.01
mass density (g/cm3)
isentrope
Hugoniot
0.0001
0.001
0.01
0.001 0.01
mass density (g/cm3)
isentrope
Hugoniot
FIG. 1: Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations for
general material models, compared with analytic solutions.
0 1000 2000 3000 4000 5000
temperature (K)
solid: Grueneisen
dashed: SESAME 3716
FIG. 2: Shock Hugoniot for Al in pressure-temperature space, for different representations of the
equation of state.
0.7 0.75 0.8 0.85 0.9 0.95 1
volume compression
each pair of lines:
upper is Hugoniot,
lower is adiabat
FIG. 3: Principal adiabat and shock Hugoniot for Be in normal stress-compression space, neglecting
strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with
Y = 10GPa (dotted).
0 20 40 60 80 100 120 140
normal stress (GPa)
elastic wave
plastic shock
FIG. 4: Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space, neglecting
strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly plastic with
Y = 10GPa (dotted).
0 1000 2000 3000 4000 5000
temperature (K)
principal
adiabat
principal
Hugoniot
release
adiabat
FIG. 5: Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-temperature
space, neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-perfectly
plastic with Y = 10GPa (dotted).
0 1000 2000 3000 4000 5000
temperature (K)
melt locus
solid Hugoniot
FIG. 6: Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of Al,
for different initial porosities.
initial state
particle speed
initial state
of projectile
principal Hugoniot
of target
principal
Hugoniot
of projectile
shock state:
intersection
of target
FIG. 7: Wave interactions for the impact of a flat projectile moving from left to right with a
stationary target. Dashed arrows are a guide to the sequence of states. For a projectile moving
from right to left, the construction is the mirror image reflected in the normal stress axis.
states
particle speed
secondary
Hugoniot
of target
initial shock state
in target
principal Hugoniot:
high impedance window
low impedance
window
target release isentrope
target release at free surface
window
release
FIG. 8: Wave interactions for the release of a shocked state (shock moving from left to right) into
a stationary ‘window’ material to its right. The release state depends whether the window has
a higher or lower shock impedance than the shocked material. Dashed arrows are a guide to the
sequence of states. For a shock moving from right to left, the construction is the mirror image
reflected in the normal stress axis.
projectile release
in projectile and target
final tensile state
in projectile and target
particle speed
target release
target release
projectile release
initial shock state
FIG. 9: Wave interactions for the release of a shocked state by tension induced as materials try
to separate in opposite directions when joined by a bonded interface. Material damage, spall, and
separation are neglected: the construction shows the maximum tensile stress possible. For general
material properties, e.g. if plastic flow is included, the state of maximum tensile stress is not just
the negative of the initial shock state. Dashed arrows are a guide to the sequence of states. The
graph shows the initial state after an impact by a projectile moving from right to left; for a shock
moving from right to left, the construction is the mirror image reflected in the normal stress axis.
tension
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target
impact shocks
transmitted shock;
reflected wave
free surface
release
release interactions:
FIG. 10: Schematic of uniaxial wave interactions induced by the impact of a flat projectile with a
composite target.
0 5 10 15 20
position (mm)
LiFAl Mo
reflected
transmitted
release
shock
original
shock state
FIG. 11: Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF
release window, 1.1µs after impact. Structures on the waves are elastic precursors.
List of figures
1. Principal isentrope and shock Hugoniot for air (perfect gas): numerical calculations
for general material models, compared with analytic solutions.
2. Shock Hugoniot for Al in pressure-temperature space, for different representations of
the equation of state.
3. Principal adiabat and shock Hugoniot for Be in normal stress-compression space,
neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-
perfectly plastic with Y = 10GPa (dotted).
4. Principal adiabat and shock Hugoniot for Be in shock speed-normal stress space,
neglecting strength (dashed), for Steinberg-Guinan strength (solid), and for elastic-
perfectly plastic with Y = 10GPa (dotted).
5. Principal adiabat, shock Hugoniot, and release adiabat for Be in normal stress-
temperature space, neglecting strength (dashed), for Steinberg-Guinan strength
(solid), and for elastic-perfectly plastic with Y = 10GPa (dotted).
6. Demonstration of shock Hugoniot solution across a phase boundary: shock-melting of
Al, for different initial porosities.
7. Wave interactions for the impact of a flat projectile moving from left to right with a
stationary target. Dashed arrows are a guide to the sequence of states. For a projectile
moving from right to left, the construction is the mirror image reflected in the normal
stress axis.
8. Wave interactions for the release of a shocked state (shock moving from left to right)
into a stationary ‘window’ material to its right. The release state depends whether
the window has a higher or lower shock impedance than the shocked material. Dashed
arrows are a guide to the sequence of states. For a shock moving from right to left,
the construction is the mirror image reflected in the normal stress axis.
9. Wave interactions for the release of a shocked state by tension induced as materials
try to separate in opposite directions when joined by a bonded interface. Material
damage, spall, and separation are neglected: the construction shows the maximum
tensile stress possible. For general material properties, e.g. if plastic flow is included,
the state of maximum tensile stress is not just the negative of the initial shock state.
Dashed arrows are a guide to the sequence of states. The graph shows the initial state
after an impact by a projectile moving from right to left; for a shock moving from
right to left, the construction is the mirror image reflected in the normal stress axis.
10. Schematic of uniaxial wave interactions induced by the impact of a flat projectile with
a composite target.
11. Hydrocode simulation of Al projectile at 3.6 km/s impacting a Mo target with a LiF
release window, 1.1µs after impact. Structures on the waves are elastic precursors.
Introduction
Conceptual structure for material properties
Idealized one-dimensional loading
Ramp compression
Shock compression
Accuracy: application to air
Complex behavior of condensed matter
Temperature
Density-temperature equations of state
Temperature model for mechanical equations of state
Strength
Preferred representation of isotropic strength
Beryllium
Phase changes
Composite loading paths
Conclusions
Acknowledgments
References
References
List of figures
|
0704.0010 | Partial cubes: structures, characterizations, and constructions | Partial cubes: structures, characterizations, and
constructions
Sergei Ovchinnikov
Mathematics Department
San Francisco State University
San Francisco, CA 94132
[email protected]
May 8, 2006
Abstract
Partial cubes are isometric subgraphs of hypercubes. Structures on a
graph defined by means of semicubes, and Djoković’s and Winkler’s rela-
tions play an important role in the theory of partial cubes. These struc-
tures are employed in the paper to characterize bipartite graphs and par-
tial cubes of arbitrary dimension. New characterizations are established
and new proofs of some known results are given.
The operations of Cartesian product and pasting, and expansion and
contraction processes are utilized in the paper to construct new partial
cubes from old ones. In particular, the isometric and lattice dimensions of
finite partial cubes obtained by means of these operations are calculated.
Key words: Hypercube, partial cube, semicube
1 Introduction
A hypercube H(X) on a set X is a graph which vertices are the finite subsets
of X ; two vertices are joined by an edge if they differ by a singleton. A partial
cube is a graph that can be isometrically embedded into a hypercube.
There are three general graph-theoretical structures that play a prominent
role in the theory of partial cubes; namely, semicubes, Djoković’s relation θ, and
Winkler’s relation Θ. We use these structures, in particular, to characterize bi-
partite graphs and partial cubes. The characterization problem for partial cubes
was considered as an important one and many characterizations are known.
We list contributions in the chronological order: Djoković [9] (1973), Avis [2]
(1981), Winkler [20] (1984), Roth and Winkler [18] (1986), Chepoi [6, 7] (1988
and 1994). In the paper, we present new proofs for the results of Djoković [9],
Winkler [20], and Chepoi [6], and obtain two more characterizations of partial
cubes.
http://arxiv.org/abs/0704.0010v1
The paper is also concerned with some ways of constructing new partial
cubes from old ones. Properties of subcubes, the Cartesian product of partial
cubes, and expansion and contraction of a partial cube are investigated. We
introduce a construction based on pasting two graphs together and show how
new partial cubes can be obtained from old ones by pasting them together.
The paper is organized as follows.
Hypercubes and partial cubes are introduced in Section 2 together with
two basic examples of infinite partial cubes. Vertex sets of partial cubes are
described in terms of well graded families of finite sets.
In Section 3 we introduce the concepts of a semicube, Djoković’s θ and Win-
kler’s Θ relations, and establish some of their properties. Bipartite graphs and
partial cubes are characterized by means of these structures. One more charac-
terization of partial cubes is obtained in Section 4, where so-called fundamental
sets in a graph are introduced.
The rest of the paper is devoted to constructions: subcubes and the Carte-
sian product (Section 6), pasting (Section 7), and expansions and contractions
(Section 8). We show that these constructions produce new partial cubes from
old ones. Isometric and lattice dimensions of new partial cubes are calculated.
These dimensions are introduced in Section 5.
Few words about conventions used in the paper are in order. The sum
(disjoint union) A+B of two sets A and B is the union
({1} ×A) ∪ ({2} ×B).
All graphs in the paper are simple undirected graphs. In the notation G =
(V,E), the symbol V stands for the set of vertices of the graph G and E stands
for its set of edges. By abuse of language, we often write ab for an edge in a
graph; if this is the case, ab is an unordered pair of distinct vertices. We denote
〈U〉 the graph induced by the set of vertices U ⊆ V . If G is a connected graph,
then dG(a, b) stands for the distance between two vertices a and b of the graph
G. Wherever it is clear from the context which graph is under consideration, we
drop the subscript G in dG(a, b). A subgraph H ⊆ G is an isometric subgraph
if dH(a, b) = dG(a, b) for all vertices a and b of H ; it is convex if any shortest
path in G between vertices of H belongs to H .
2 Hypercubes and partial cubes
Let X be a set. We denote Pf (X) the set of all finite subsets of X .
Definition 2.1. A graph H(X) has the set Pf (X) as the set of its vertices; a
pair of vertices PQ is an edge of H(X) if the symmetric difference P∆Q is a
singleton. The graph H(X) is called the hypercube on X [9]. If X is a finite
set of cardinality n, then the graph H(X) is the n-cube Qn. The dimension of
the hypercube H(X) is the cardinality of the set X .
The shortest path distance d(P,Q) on the hypercube H(X) is the Hamming
distance between sets P and Q:
d(P,Q) = |P∆Q| for P,Q ∈ Pf . (2.1)
The set Pf (X) is a metric space with the metric d.
Definition 2.2. A graph G is a partial cube if it can be isometrically embedded
into a hypercube H(X) for some set X . We often identify G with its isometric
image in the hypercube H(X), and say that G is a partial cube on the set X .
Figure 2.1: A graph and its isometric embedding into Q3.
An example of a partial cube and its isometric embedding into the cube Q3
is shown in Figure 2.1.
Clearly, a family F of finite subsets of X induces a partial cube on X if and
only if for any two distinct subsets P,Q ∈ F there is a sequence
R0 = P,R1, . . . , Rn = Q
of sets in F such that
d(Ri, Ri+1) = 1 for all 0 ≤ i < n, and d(P,Q) = n. (2.2)
The families of sets satisfying condition (2.2) are known as well graded fam-
ilies of sets [10]. Note that a sequence (Ri) satisfying (2.2) is a shortest path
from P to Q in H(X) (and in the subgraph induced by F).
Definition 2.3. A family F of arbitrary subsets ofX is a wg-family (well graded
family of sets) if, for any two distinct subsets P,Q ∈ F, the set P∆Q is finite
and there is a sequence
R0 = P,R1, . . . , Rn = Q
of sets in F such that |Ri∆Ri+1| = 1 for all 0 ≤ i < n and |P∆Q| = n.
Example 2.1. The induced graph can be a partial cube on a different set if
the family F is not well graded. Consider, for instance, the family
F = {∅, {a}, {a, b}, {a, b, c}, {b, c}}
of subsets of X = {a, b, c}. The graph induced by this family is a path of length
4 in the cube Q3 (cf. Figure 2.2). Clearly, F is not well graded. On the other
hand, as it can be easily seen, any path is a partial cube.
Figure 2.2: A nonisometric path in the cube Q3.
Any family F of subsets of X defines a graph GF = (F, EF), where
EF = {{P,Q} ⊆ F : |P∆Q| = 1}.
Theorem 2.1. The graph GF defined by a family F of subsets of a set X is
isomorphic to a partial cube on X if and only if the family F is well graded.
Proof. We need to prove sufficiency only. Let S be a fixed set in F. We define
a mapping f : F → Pf (X) by f(R) = R∆S for R ∈ F. Then
d(f(R), f(T )) = |(R∆S)∆(T∆S)| = |R∆T |.
Thus f is an isometric embedding of F into Pf (X). Let (Ri) be a sequence of
sets in F such that R0 = P , Rn = Q, |P∆Q| = n, and |Ri∆Ri+1| = 1 for all
0 ≤ i < n. Then the sequence (f(Ri)) satisfies conditions (2.2). The result
follows.
A set R ∈ Pf (X) is said to be lattice between sets P,Q ∈ Pf (X) if
P ∩Q ⊆ R ⊆ P ∪Q.
It is metrically between P and Q if
d(P,R) + d(R,Q) = d(P,Q).
The following theorem is a well-known result about these two betweenness re-
lations on Pf (X) (see, for instance, [3]).
Theorem 2.2. Lattice and metric betweenness relations coincide on Pf (X).
Let F be a family of finite subsets of X . The set of all R ∈ F that are
between P,Q ∈ F is the interval I(P,Q) between P and Q in F. Thus,
I(P,Q) = F ∩ [P ∩Q,P ∪Q],
where [P ∩Q,P ∪Q] is the usual interval in the lattice Pf .
Two distinct sets P,Q ∈ F are adjacent in F if J(P,Q) = {P,Q}. If sets P
and Q form an edge in the graph induced by F, then P and Q are adjacent in
F, but, generally speaking, not vice versa. For instance, in Example 2.1, the
vertices ∅ and {b, c} are adjacent in F but do not define an edge in the induced
graph (cf. Figure 2.2).
The following theorem is a ‘local’ characterization of wg-families of sets.
Theorem 2.3. A family F ⊆ Pf (X) is well graded if and only if d(P,Q) = 1
for any two sets P and Q that are adjacent in F.
Proof. (Necessity.) Let F be a wg-family of sets. Suppose that P and Q are
adjacent in F. There is a sequence R0 = P,R1, . . . , Rn = Q that satisfies
conditions (2.2). Since the sequence (Ri) is a shortest path in F, we have
d(P, Pi) + d(Pi, Q) = d(P,Q) for all 0 ≤ i ≤ n.
Thus, Pi ∈ I(P,Q) = {P,Q}. It follows that d(P,Q) = n = 1.
(Sufficiency.) Let P and Q be two distinct sets in F. We prove by induction
on n = d(P,Q) that there is a sequence (Ri) ∈ F satisfying conditions (2.2).
The statement is trivial for n = 1. Suppose that n > 1 and that the
statement is true for all k < n. Let P and Q be two sets in F such that
d(P,Q) = n. Since d(P,Q) > 1, the sets P and Q are not adjacent in F.
Therefore there exists R ∈ F that lies between P and Q and is distinct from
these two sets. Then d(P,R) + d(R,Q) = d(P,Q) and both distances d(P,R)
and d(R,Q) are less than n. By the induction hypothesis, there is a sequence
(Ri) ∈ F such that
P = R0, R = Rj , Q = Rn for some 0 < j < n,
satisfying conditions (2.2) for 0 ≤ i < j and j ≤ i < n. It follows that F is a
wg-family of sets.
We conclude this section with two examples of infinite partial cubes (more
examples are found in [17]).
Example 2.2. Let Z be the graph on the set Z of integers with edges defined
by pairs of consecutive integers. This graph is a partial cube since its vertex set
is isometric to the wg-family of intervals {(−∞,m) : m ∈ Z} in Z.
Example 2.3. Let us consider Zn as a metric space with respect to the ℓ1-
metric. The graph Zn has Zn as the vertex set; two vertices in Zn are connected
if they are on the unit distance from each other. We will show in Section 6
(Corollary 6.1) that Zn is a partial cube.
3 Characterizations
Only connected graphs are considered in this section.
Definition 3.1. Let G = (V,E) be a graph and d be its distance function. For
any two adjacent vertices a, b ∈ V let Wab be the set of vertices that are closer
to a than to b:
Wab = {w ∈ V : d(w, a) < d(w, b)}.
Following [11], we call the sets Wab and induced subgraphs 〈Wab〉 semicubes of
the graph G. The semicubes Wab and Wba are called opposite semicubes.
Remark 3.1. The subscript ab in Wab stands for an ordered pair of vertices,
not for an edge of G. In his original paper [9], Djoković uses notation G(a, b)
(cf. [8]). We use the notation from [15].
Clearly, two opposite semicubes are disjoint. They can be used to charac-
terize bipartite graphs as follows.
Theorem 3.1. A graph G = (V,E) is bipartite if and only if the semicubes Wab
and Wba form a partition of V for any edge ab ∈ E.
Proof. Let us recall that a connected graph G is bipartite if and only if for every
vertex x there is no edge ab with d(x, a) = d(x, b) (see, for instance, [1]). For
any edge ab ∈ E and vertex x ∈ V we clearly have
d(x, a) = d(x, b) ⇔ x /∈ Wab ∪Wba.
The result follows.
The following lemma is instrumental and will be used frequently in the rest
of the paper.
Lemma 3.1. Let G = (V,E) be a graph and w ∈ Wab for some edge ab ∈ E.
d(w, b) = d(w, a) + 1.
Accordingly,
Wab = {w ∈ V : d(w, b) = d(w, a) + 1}.
Proof. By the triangle inequality, we have
d(w, a) < d(w, b) ≤ d(w, a) + d(a, b) = d(w, a) + 1.
The result follows, since d takes values in N.
There are two binary relations on the set of edges of a graph that play a
central role in characterizing partial cubes.
Definition 3.2. Let G = (V,E) be a graph and e = xy and f = uv be two
edges of G.
(i) (Djoković [9]) The relation θ on E is defined by
e θf ⇔ f joins a vertex in Wxy with a vertex in Wyx.
The notation can be chosen such that u ∈Wxy and v ∈ Wyx.
(ii) (Winkler [20]) The relation Θ on E is defined by
eΘf ⇔ d(x, u) + d(y, v) 6= d(x, v) + d(y, u).
It is clear that both relations θ and Θ are reflexive and Θ is symmetric.
Lemma 3.2. The relation θ is a symmetric relation on E.
Proof. Suppose that xy θ uv with u ∈ Wxy and v ∈ Wyx. By Lemma 3.1 and
the triangle inequality, we have
d(u, x) = d(u, y)− 1 ≤ d(u, v) + d(v, y)− 1 = d(v, y) =
= d(v, x)− 1 ≤ d(v, u) + d(u, x) − 1 = d(u, x).
Hence, d(u, x) = d(v, x) − 1 and d(v, y) = d(u, y)− 1. Therefore, x ∈ Wuv and
y ∈ Wvu. It follows that uv θ xy.
Lemma 3.3. θ ⊆ Θ.
Proof. Suppose that xy θ uv with u ∈Wxy, v ∈ Wyx. By Lemma 3.1,
d(x, u) + d(y, v) = d(x, v) − 1 + d(y, u)− 1 6= d(x, v) + d(y, u).
Hence, xyΘ uv.
Example 3.1. It is easy to verify that θ is the identity relation on the set of
edges of the cycle C3. On the other hand, any two edges of C3 stand in the
relation Θ. Thus, θ 6= Θ in this case.
Bipartite graphs can be characterized in terms of relations θ and Θ as follows.
Theorem 3.2. A graph G = (V,E) is bipartite if and only if θ = Θ.
Proof. (Necessity.) Suppose that G is a bipartite graph, two edges xy and uv
stand in the relation Θ, that is,
d(x, u) + d(y, v) 6= d(x, v) + d(y, u),
and that edges xy and uv do not stand in the relation θ. By Theorem 3.1, we
may assume that u, v ∈ Wxy. By Lemma 3.1, we have
d(x, u) + d(y, v) = d(y, u)− 1 + d(x, v) + 1 = d(x, v) + d(y, u),
a contradiction. It follows that Θ ⊆ θ. By Lemma 3.3, θ = Θ.
(Sufficiency.) Suppose that G is not bipartite. By Theorem 3.1, there is an
edge xy such that Wxy ∪Wyx is a proper subset of V . Since G is connected,
there is an edge uv with u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Clearly, uv does
not stand in the relation θ to xy. On the other hand,
d(x, u) + d(y, v) 6= d(x, v) + d(y, u),
since u /∈ Wxy ∪Wyx and v ∈ Wxy ∪Wyx. Thus, xyΘ uv, a contradiction, since
we assumed that θ = Θ.
By Theorem 3.2, the relations θ and Θ coincide on bipartite graphs. For
this reason we use the relation θ in the rest of the paper.
Lemma 3.4. Let G = (V,E) be a bipartite graph such that all its semicubes are
convex sets. Then two edges xy and uv stand in the relation θ if and only if the
corresponding pairs of mutually opposite semicubes form equal partitions of V :
xy θ uv ⇔ {Wxy,Wyx} = {Wuv,Wvu}.
Proof. (Necessity) We assume that the notation is chosen such that u ∈ Wxy
and v ∈ Wyx. Let z ∈ Wxy ∩Wvu. By Lemma 3.1, d(z, u) = d(z, v) + d(v, u).
Since z, u ∈ Wxy and Wxy is convex, we have v ∈ Wxy, a contradiction to the
assumption that v ∈Wyx. Thus Wxy ∩Wvu = ∅. Since two opposite semicubes
in a bipartite graph form a partition of V , we haveWuv =Wxy andWvu =Wyx.
A similar argument shows that Wuv = Wyx and Wvu = Wxy, if u ∈ Wyx
and v ∈ Wxy.
(Sufficiency.) Follows from the definition of the relation θ.
We need another general property of the relation θ (cf. Lemma 2.2 in [15]).
Lemma 3.5. Let P be a shortest path in a graph G. Then no two distinct edges
of P stand in the relation θ.
Proof. Let i < j and xixi+1 and xjxj+1 be two edges in a shortest path P from
x0 to xn. Then
d(xi, xj) < d(xi, xj+1) and d(xi+1, xj) < d(xi+1, xj+1),
so xi, xi+1 ∈ Wxjxj+1 . It follows that edges xixi+1 and xjxj+1 do not stand in
the relation θ.
The converse statement is true for bipartite graphs (we omit the proof); a
counterexample is the cycle C5 which is not bipartite.
Lemma 3.6. Let G = (V,E) be a bipartite graph. The following statements are
equivalent
(i) All semicubes of G are convex.
(ii) The relation θ is an equivalence relation on E.
Proof. (i) ⇒ (ii). Follows from Lemma 3.4.
(ii) ⇒ (i). Suppose that θ is transitive and there is a nonconvex semicube
Wab. Then there are two vertices u, v ∈ Wab and a shortest path P from u to
v that intersects Wba. This path contains two distinct edges e and f joining
vertices of semicubes Wab and Wba. The edges e and f stand in the relation θ
to the edge ab. By transitivity of θ, we have e θf . This contradicts the result
of Lemma 3.5. Thus all semicubes of G are convex.
We now establish some basic properties of partial cubes.
Theorem 3.3. Let G = (V,E) be a partial cube. Then
(i) G is a bipartite graph.
(ii) Each pair of opposite semicubes form a partition of V .
(iii) All semicubes are convex subsets of V .
(iv) θ is an equivalence relation on E.
Proof. We may assume that G is an isometric subgraph of some hypercube
H(X), that is, G = (F, EF) for a wg-family F of finite subsets of X .
(i) It suffices to note that if two sets in H(X) are connected by an edge then
they have different parity. Thus, H(X) is a bipartite graph and so is G.
(ii) Follows from (i) and Theorem 3.1.
(iii) LetWAB be a semicube of G. By Lemma 3.1 and Theorem 2.2, we have
WAB = {S ∈ F : S ∩B ⊆ A ⊆ S ∪B}.
Let Q,R ∈WAB and P be a vertex of G such that
d(Q,P ) + d(P,R) = d(Q,R).
By Theorem 2.2,
Q ∩R ⊆ P ⊆ Q ∪R.
Since Q,R ∈WAB , we have
Q ∩B ⊆ A ⊆ Q ∪B and R ∩B ⊆ A ⊆ R ∪B,
which implies
P ∩B ⊆ (Q ∪R) ∩B ⊆ A ⊆ (Q ∩R) ∪B ⊆ S ∪B.
Hence, P ∈ WAB, and the result follows.
(iv) Follows from (iii) and Lemma 3.6.
Remark 3.2. Since semicubes of a partial cube G = (V,E) are convex subsets
of the metric space V , they are half-spaces in V [19]. This terminology is used
in [6, 7].
The following theorem presents four characterizations of partial cubes. The
first two are due to Djoković [9] and Winkler [20] (cf. Theorem 2.10 in [15]).
Theorem 3.4. Let G = (V,E) be a connected graph. The following statements
are equivalent:
(i) G is a partial cube.
(ii) G is bipartite and all semicubes of G are convex.
(iii) G is bipartite and θ is an equivalence relation.
(iv) G is bipartite and, for all xy, uv ∈ E,
xy θ uv ⇒ {Wxy,Wyx} = {Wuv,Wvu}. (3.1)
(v) G is bipartite and, for any pair of adjacent vertices of G, there is a unique
pair of opposite semicubes separating these two vertices.
Proof. By Lemma 3.6, the statements (ii) and (iii) are equivalent and, by The-
orem 3.3, (i) implies both (ii) and (iii).
(iii) ⇒ (i). By Theorem 3.1, each pair {Wab,Wba} of opposite semicubes of
G form a partition of V . We orient these partitions by calling, in an arbitrary
way, one of the two opposite semicubes in each partition a positive semicube.
Let us assign to each x ∈ V the set W+(x) of all positive semicubes containing
x. In the next paragraph we prove that the family F = {W+(x)}x∈V is well
graded and that the assignment x 7→ W+(x) is an isometry between V and F.
Let x and y be two distinct vertices of G. We say that a positive semicube
Wab separates x and y if either x ∈ Wab, y ∈ Wba or x ∈ Wba, y ∈ Wab. It is
clear that Wab separates x and Y if and only if Wab ∈ W
+(x)∆W+(y). Let P
be a shortest path x0 = x, x1, . . . , xn = y from x to y. By Lemma 3.5, no two
distinct edges of P stand in the relation θ. By Lemma 3.4, distinct edges of P
define distinct positive semicubes; clearly, these semicubes separate x and y. Let
Wab be a positive semicube separating x and y, and, say, x ∈Wab and y ∈Wba.
There is an edge f ∈ P that joins vertices in Wab and Wba. Hence, f stands in
the relation θ to ab and, by Lemma 3.4, Wab is defined by f . It follows that any
semicube inW+(x)∆W+(y) is defined by a unique edge in P and any edge in P
defines a semicube in W+(x)∆W+(y). Therefore, d(W+(x),W+(y)) = d(x, y),
that is x 7→W+(x) is an isometry. Clearly, F is a wg-family of sets.
By Theorem 2.1, the family F is isometric to a wg-family of finite sets.
Hence, G is a partial cube.
(iv) ⇒ (ii). Suppose that there exist an edge ab such that semicube Wba is
not convex. Let p and q be two vertices in Wba such that there is a shortest
path P from p to q that intersects Wab. There are two distinct edges xy and uv
in P such that x, u ∈ Wab and y, v ∈ Wba. Since ab θ xy and ab θ uv, we have,
by (3.1),
Wab =Wxy =Wuv.
Hence, u ∈ Wxy and v ∈Wyx. By Lemma 3.1,
d(x, u) = d(x, v) − 1 = 1 + d(v, y)− 1 = d(v, y),
a contradiction, since P is a shortest path from p to q.
(ii) ⇒ (iv). Follows from Lemma 3.4.
It is clear that (iv) and (v) are equivalent.
4 Fundamental sets in partial cubes
Semicubes played an important role in the previous section. In this section we
introduce three more classes of useful subsets of graphs. We also establish one
more characterization of partial cubes.
Let G = (V,E) be a connected graph. For a given edge e = ab ∈ E, we
define the following sets (cf. [15, 16]):
Fab = {f ∈ E : e θf} = {uv ∈ E : u ∈Wab, v ∈Wba},
Uab = {w ∈Wab : w is adjacent to a vertex in Wba},
Uba = {w ∈Wba : w is adjacent to a vertex in Wab}.
The five sets are schematically shown in Figure 4.1.
Figure 4.1: Fundamental sets in a partial cube.
Remark 4.1. In the case of a partial cube G = (V,E), the semicubes Wab and
Wba are complementary half-spaces in the metric space V (cf. Remark 3.2).
Then the set Fab can be regarded as a ‘hyperplane’ separating these half-spaces
(see [17] where this analogy is formalized in the context of hyperplane arrange-
ments).
The following theorem generalizes the result obtained in [16] for median
graphs (see also [15]).
Theorem 4.1. Let ab be an edge of a connected bipartite graph G. If the
semicubes Wab and Wba are convex, then the set Fab is a matching and induces
an isomorphism between the graphs 〈Uab〉 and 〈Uba〉.
Proof. Suppose that Fab is not a matching. Then there are distinct edges xu
and xv with, say, x ∈ Uab and u, v ∈ Uba. By the triangle inequality, d(u, v) ≤ 2.
Since G does not have triangles, d(u, v) 6= 1. Hence, d(u, v) = 2, which implies
that x lies between u and v. This contradicts convexity of Wba, since x ∈ Wab.
Therefore Fab is a matching.
To show that Fab induces an isomorphism, let xy, uv ∈ Fab and xu ∈ E,
where x, u ∈ Uab and y, v ∈ Uba. Since G does not have odd cycles, d(v, y) 6= 2.
By the triangle inequality,
d(v, y) ≤ d(v, u) + d(u, x) + d(x, y) = 3.
Since Wba is convex, d(v, y) 6= 3. Thus d(v, y) = 1, that is, vy is an edge. The
result follows by symmetry.
By Theorem 3.4(ii), we have the following corollary.
Corollary 4.1. Let G = (V,E) be a partial cube. For any edge ab the set Fab
is a matching and induces an isomorphism between induced graphs 〈Uab〉 and
〈Uba〉.
Figure 4.2: Graph G.
Example 4.1. Let G be the graph depicted in Figure 4.2. The set
Fab = {ab, xu, yv}
is a matching and defines an isomorphism between the graphs induced by subsets
Uab = {a, x, y} and Uba = {b, u, v}. The set Wba is not convex, so G is not a
partial cube. Thus the converse of Corollary 4.1 does not hold.
We now establish another characterization of partial cubes that utilizes a
geometric property of families Fab.
Theorem 4.2. For a connected graph G the following statements are equivalent:
(i) G is a partial cube.
(ii) G is bipartite and
d(x, u) = d(y, v) and d(x, v) = d(y, u), (4.1)
for any ab ∈ E and xy, uv ∈ Fab.
Proof. (i)⇒(ii). We may assume that x, u ∈ Wab and y, v ∈ Wba. Since θ is an
equivalence relation, we have xy θ uv θab. By Lemma 3.4, Wuv = Wxy = Wab.
By Lemma 3.1,
d(x, u) = d(x, v) − 1 = d(v, y) + 1− 1 = d(y, v).
We also have
d(x, v) = d(y, v) + 1 = d(y, u),
by the same lemma.
(ii)⇒(i). Suppose that G is not a partial cube. Then, by Theorem 3.4, there
exist an edge ab such that, say, semicube Wba is not convex. Let p and q be two
vertices in Wba such that there is a shortest path P from p to q that intersects
Wab. Let uv be the first edge in P which belongs to Fab and xy be the last edge
in P with the same property (see Figure 4.3).
Figure 4.3: An illustration to the proof of theorem 4.2.
Since P is a shortest path, we have
d(v, y) = d(v, u) + d(u, x) + d(x, y) 6= d(x, u),
which contradicts condition (4.1). Thus all semicubes of G are convex. By
Theorem 3.4, G is a partial cube.
Remark 4.2. One can say that four vertices satisfying conditions (4.1) define
a rectangle in G. Then Theorem 4.2 states that a connected graph is a partial
cube if and only if it is bipartite and for any edge ab pairs of edges in Fab define
rectangles in G.
5 Dimensions of partial cubes
There are many different ways in which a given partial cube can be isometrically
embedded into a hypercube. For instance, the graph K2 can be isometrically
embedded in different ways into any hypercube H(X) with |X | > 2.
Following Djoković [9] (see also [8]), we define the isometric dimension,
dimI(G), of a partial cube G as the minimum possible dimension of a hypercube
H(X) in which G is isometrically embeddable. Recall (see Section 2) that the
dimension of H(X) is the cardinality of the set X .
Theorem 5.1. (Theorem 2 in [9].) Let G = (V,E) be a partial cube. Then
dimI(G) = |E/θ|, (5.1)
where θ is Djoković’s equivalence relation on E and E/θ is the set of its equiv-
alence classes (the quotient-set).
The quotient-set E/θ can be identified with the family of all distinct sets Fab
(see Section 4). If G is a finite partial cube, we may consider it as an isometric
subgraph of some hypercube Qn. Then the edges in each family Fab are parallel
edges in Qn (cf. Theorem 4.2). This observation essentially proves (5.1) in the
finite case.
Let G be a partial cube on a set X . The vertex set of G is a wg-family F of
finite subsets of X (see Section 2). We define the retraction of F as a family F′
of subsets of X ′ = ∪F \ ∩F consisting of the intersections of sets in F with X ′.
It is clear that F′ satisfies conditions
∩ F′ = ∅ and ∪ F′ = X ′. (5.2)
Proposition 5.1. The partial cubes induced by a wg-family F and its retraction
F′ are isomorphic.
Proof. It suffices to prove that metric spaces F and F′ are isometric. Clearly,
α : P 7→ P ∩X ′ is a mapping from F onto F′. For P,Q ∈ F, we have
(P ∩X ′)∆(Q ∩X ′) = (P∆Q) ∩X ′ = (P∆Q) ∩ (∪F \ ∩F) = P∆Q.
Thus, d(α(P ), α(Q)) = d(P,Q). Consequently, α is an isometry.
Let G be a partial cube on some set X induced by a wg-family F satisfying
conditions (5.2), and let PQ be an edge of G. By definition, there is x ∈ X such
that P∆Q = {x}. The following two lemmas are instrumental.
Lemma 5.1. Let PQ be an edge of a partial cube G on X and let P∆Q = {x}.
The two sets
{R ∈ F : x ∈ R} and {R ∈ F : x /∈ R}
form the same bipartition of the family F as semicubes WPQ and WQP .
Proof. We may assume that Q = P + {x}. Then, for any R ∈ F,
R∆Q = R∆(P + {x}) =
(R∆P ) + {x}, if x ∈ R,
R∆P, if x /∈ R.
Hence, |R∆P | < |R∆Q| if and only if x ∈ R. It follows that
WPQ = {R ∈ F : x ∈ R}.
A similar argument shows that WQP = {R ∈ F : x /∈ R}.
Lemma 5.2. If F is a wg-family of sets satisfying conditions (5.2), then for
any x ∈ X there are sets P,Q ∈ F such that P∆Q = {x}.
Proof. By conditions 5.2, for a given x ∈ X there are sets S and T in F such
that x ∈ S and x /∈ T . Let R0 = S,R1, . . . , Rn = T be a sequence of sets in
F satisfying conditions (2.2). It is clear that there is i such that x ∈ Ri and
x /∈ Ri+1. Hence, Ri∆Ri+1 = {x}, so we can choose P = Ri and Q = Ri+1.
By Lemmas 5.1 and 5.2, there is one-to-one correspondence between the set
X and the quotient-set E/θ. From Theorem 5.1 we obtain the following result.
Theorem 5.2. Let F be a wg-family of finite subsets of a set X such that
∩F = ∅ and ∪F = X, and let G be a partial cube on X induced by F. Then
dimI(G) = |X |.
Clearly, a graph which is isometrically embeddable into a partial cube is a
partial cube itself. We will show in Section 6 (Corollary 6.1) that the integer
lattice Zn is a partial cube. Thus a graph which is isometrically embeddable
into an integer lattice is a partial cube. It follows that a finite graph is a partial
cube if and only if it is embeddable in some integer lattice. Examples of infinite
partial cubes isometrically embeddable into a finite dimensional integer lattice
are found in [17].
We call the minimum possible dimension n of an integer lattice Zn, in which
a given graph G is isometrically embeddable, its lattice dimension and denote
it dimZ(G). The lattice dimension of a partial cube can be expressed in terms
of maximum matchings in so-called semicube graphs [11].
Definition 5.1. The semicube graph Sc(G) has all semicubes in G as the set
of its vertices. Two vertices Wab and Wcd are connected in Sc(G) if
Wab ∪Wcd = V and Wab ∩Wcd 6= ∅. (5.3)
If G is a partial cube, then condition (5.3) is equivalent to each of the two
equivalent conditions:
Wba ⊂Wcd ⇔ Wdc ⊂Wab, (5.4)
where ⊂ stands for the proper inclusion.
Theorem 5.3. (Theorem 1 in [11].) Let G be a finite partial cube. Then
dimZ(G) = dimI(G) − |M |,
where M is a maximum matching in the semicube graph Sc(G).
Example 5.1. Let G be the graph shown in Figure 2.1. It is easy to see that
dimI(G) = 3 and dimZ(G) = 2.
Example 5.2. Let T be a tree with n edges and m leaves. Then
dimI(T ) = n and dimZ(T ) = ⌈m/2⌉
(cf. [8] and [14], respectively).
Example 5.3. For the cycle C6 we have (see Figure 8.2)
dimI(C6) = dimZ(C6) = 3.
6 Subcubes and Cartesian products
Let G be a partial cube. We say that G′ is a subcube of G if it is an isometric
subgraph of G.
Clearly, a subcube is itself a partial cube. The converse does not hold; a
subgraph of a graph G can be a partial cube but not an isometric subgraph of
G (cf. Example 2.1).
If G′ is a subcube of a partial cube G, then dimI(G
′) ≤ dimI(G) and
dimZ(G
′) ≤ dimZ(G). In general, the two inequalities are not strict. For
instance, the cycle C6 is an isometric subgraph of the cube Q3 (see Figure 8.2)
dimI(C6) = dimZ(C6) = dimI(Q3) = dimZ(Q3) = 3.
Semicubes of a partial cube are examples of subcubes. Indeed, by Theo-
rem 3.4, semicubes are convex subgraphs and therefore isometric. In general,
the converse is not true; a path connecting two opposite vertices in C6 is an
isometric subgraph but not a convex one.
Another common way of constructing new partial cubes from old ones is by
forming their Cartesian products (see [15] for details and proofs).
Definition 6.1. Given two graphs G1 = (V1, E1) and G2 = (V2, E2), their
Cartesian product
G = G1�G2
has vertex set V = V1 × V2; a vertex u = (u1, u2) is adjacent to a vertex
v = (v1, v2) if and only if u1v1 ∈ E1 and u2 = v2, or u1 = v1 and u2v2 ∈ E2.
The operation � is associative, so we can write
G = G1� · · ·�Gn =
for the Cartesian product of graphs G1, . . . , Gn. A Cartesian product
i=1Gi
is connected if and only if the factors are connected. Then we have
dG(u, v) =
dGi(ui, vi). (6.1)
Example 6.1. Let {Xi}
i=1 be a family of sets and Y =
i=1 be their sum.
Then the Cartesian product of the hypercubes H(Xi) is isomorphic to the hy-
percube H(Y ). The isomorphism is established by the mapping
f : (P1, . . . , Pn) 7→
Formula (6.1) yields immediately the following results.
Proposition 6.1. Let Hi be isometric subgraphs of graphs Gi for all 1 ≤ i ≤ n.
Then the Cartesian product
i=1Hi is an isometric subgraph of the Cartesian
product
i=1Gi.
Corollary 6.1. The Cartesian product of a finite family of partial cubes is a
partial cube. In particular, the integer lattice Zn (cf. Examples 2.2 and 2.3) is
a partial cube.
The results of the next two theorems can be easily extended to arbitrary
finite products of finite partial cubes.
Theorem 6.1. Let G = G1�G2 be the Cartesian product of two finite partial
cubes. Then
dimI(G) = dimI(G1) + dimI(G2).
Proof. We may assume that G1 (resp. G2) is induced by a wg-family F1 (resp.
F2) of subsets of a finite set X1 (resp. X2) such that ∩F1 = ∅ and ∪F1 = X1
(resp. ∩F2 = ∅ and ∪F2 = X1) (see Section 5). By Theorem 5.2,
dimI(G1) = |X1| and dimI(G2) = |X2|.
It is clear that the graph G is induced by the wg-family F = F1 +F2 of subsets
of the set X = X1 + X2 (cf. Example 6.1) with ∩F = ∅, ∪F = X . By
Theorem 5.2,
dimI(G) = |X | = |X1|+ |X2| = dimI(G1) + dimI(G2).
Theorem 6.2. Let G = (V,E) be the Cartesian product of two finite partial
cubes G1 = (V1, E1) and G2 = (V2, E2). Then
dimZ(G) = dimZ(G1) + dimZ(G2).
Proof. Let W(a,b)(c,d) be a semicube of the graph G. There are two possible
cases:
(i) c = a, bd ∈ E2. Let (x, y) be a vertex of G. Then, by (6.1),
dG((x, y), (a, b)) = dG1(x, a) + dG2(y, b)
dG((x, y), (c, d)) = dG1(x, c) + dG2(y, d).
Hence,
dG((x, y), (a, b)) < dG((x, y), (c, d)) ⇔ dG2(y, b) < dG2(y, d).
It follows that
W(a,b)(c,d) = V1 ×Wbd. (6.2)
(ii) d = b, ac ∈ E1. Like in (i), we have
W(a,b)(c,d) =Wac × V2. (6.3)
Clearly, two semicubes given by (6.2) form an edge in the semicube graph
Sc(G) if and only if their second factors form an edge in the semicube graph
Sc(G2). The same is true for semicubes in the form (6.3) with respect to their
first factors. It is also clear that semicubes in the form (6.2) and in the form (6.3)
are not connected by an edge in Sc(G). Therefore the semicube graph Sc(G) is
isomorphic to the disjoint union of semicube graphs Sc(G1) and Sc(G2). If M1
is a maximum matching in Sc(G1) and M2 is a maximum matching in Sc(G2),
then M =M1 ∪M2 is a maximum matching in Sc(G). The result follows from
theorems 5.3 and 6.1.
Remark 6.1. The result of Corollary 6.1 does not hold for infinite Cartesian
products of partial cubes, as these products are disconnected. On the other
hand, it can be shown that arbitrary weak Cartesian products (connected com-
ponents of Cartesian products [15]) of partial cubes are partial cubes.
7 Pasting partial cubes
In this section we use the set pasting technique [5, ch.I, §2.5] to build new partial
cubes from old ones.
Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs, H1 = (U1, F1) and
H2 = (U2, F2) be two isomorphic subgraphs of G1 and G2, respectively, and
ψ : U1 → U2 be a bijection defining an isomorphism between H1 and H2. The
bijection ψ defines an equivalence relation R on the sum V1+V2 as follows: any
element in (V1 \U1)∪ (V2 \U2) is equivalent to itself only and elements u1 ∈ U1
and u2 ∈ U2 are equivalent if and only if u2 = ψ(u1). We say that the quotient
set V = (V1 + V2)/R is obtained by pasting together the sets V1 and V2 along
the subsets U1 and U2. Since the graphs H1 and H2 are isomorphic, the pasting
of the sets V1 and V2 can be naturally extended to a pasting of sets of edges
E1 and E2 resulting in the set E of edges joining vertices in V . We say that
the graph G = (E, V ) is obtained by pasting together the graphs G1 and G2
along the isomorphic subgraphs H1 and H2. The pasting construction allows
for identifying in a natural way the graphs G1 and G2 with subgraphs of G, and
the isomorphic graphs H1 and H2 with a common subgraph H of both graphs
G1 and G2. We often follow this convention below.
Remark 7.1. Note that in the above construction the resulting graph G de-
pends not only on graphs G1 and G2 and their isomorphic subgraphs H1 and
H2 but also on the bijection ψ defining an isomorphism from H1 onto H2 (see
the drawings in Figures 7.1 and 7.2).
Figure 7.1: Pasting of two trees.
Figure 7.2: Another pasting of the same trees.
In general, pasting of two partial cubes G1 and G2 along two isomorphic
subgraphs H1 and H2 does not produce a partial cube even under strong as-
sumptions about these subgraphs as the next example illustrates.
Figure 7.3: Pasting partial cubes G1 and G2.
Example 7.1. Pasting of two partial cubes G1 = C6 and G2 = C6 along
subgraphs H1 and H2 is shown in Figure 7.3. The resulting graph G is not a
partial cube. Indeed, the semicubeWab is not a convex set. Note that subgraphs
H1 and H2 are convex subgraphs of the respective partial cubes.
In this section we study two simple pastings of connected graphs together,
the vertex-pasting and the edge-pasting, and show that these pastings produce
partial cubes from partial cubes. We also compute the isometric and lattice
dimensions of the resulting graphs.
Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1 ∈ V1,
a2 ∈ V2, and H1 = ({a1},∅), H2 = ({a2},∅). Let G be the graph obtained
by pasting G1 and G2 along subgraphs H1 and H2. In this case we say that
the graph G is obtained from graphs G1 and G2 by vertex-pasting. We also say
that G is obtained from G1 and G2 by identifying vertices a1 and a2. Figure 7.4
illustrates this construction. Note that the vertex a = {a1, a2} is a cut vertex
of G, since G1 ∪ G2 = G and G1 ∩ G2 = {a}. (We follow our convention and
identify graphs G1 and G2 with subgraphs of G.)
Figure 7.4: An example of vertex-pasting.
In what follows we use superscripts to distinguish subgraphs of the graphs
G1 and G2. For instance, W
stands for the semicube of G2 defined by two
adjacent vertices a, b ∈ V2.
Theorem 7.1. A graph G = (V,E) obtained by vertex-pasting from partial
cubes G1 = (V1, E1) and G2 = (V2, E2) is a partial cube.
Proof. We denote a = {a1, a2} the vertex of G obtained by identifying vertices
a1 ∈ V1 and a2 ∈ V2. Clearly, G is a bipartite graph. Let xy be an edge of G.
Without loss of generality we may assume that xy ∈ E1 and a ∈ Wxy. Note
that any path between vertices in V1 and V2 must go through a. Since a ∈Wxy,
we have, for any v ∈ V2,
d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y),
which implies V2 ⊆ Wxy and Wyx ⊆ V1. It follows that Wxy = W
xy ∪ V2 and
Wyx = W
yx . The sets W
xy , W
yx and V2 are convex subsets of V . Since
xy ∩ V2 = {a}, the set Wxy = W
xy ∪ V2 is also convex. By Theorem 3.4(ii),
the graph G is a partial cube.
The vertex-pasting construction introduced above can be generalized as
follows. Let G = {Gi = (Vi, Ei)}i∈J be a family of connected graphs and
A = {ai ∈ Gi}i∈J be a family of distinguished vertices of these graphs. Let G
be the graph obtained from the graphs Gi by identifying vertices in the set A.
We say that G is obtained by vertex-pasting together the graphs Gi (along the
set A).
Example 7.2. Let J = {1, . . . , n} with n ≥ 2,
G = {Gi = ({ai, bi}, {aibi})}i∈J , and A = {ai}i∈J .
Clearly, each Gi is K2. By vertex-pasting these graphs along A, we obtain the
n-star graph K1,n.
Since the star K1,n is a tree it can be also obtained from K1 by successive
vertex-pasting as in Example 7.3.
Example 7.3. Let G1 be a tree and G2 = K2. By vertex-pasting these graphs
we obtain a new tree. Conversely, let G be a tree and v be its leaf. Let G1 be
a tree obtained from G by deleting the leaf v. Clearly, G can be obtained by
vertex-pasting G1 and K2. It follows that any tree can obtained from the graph
K1 by successive vertex-pasting of copies of K2 (cf. Theorem 2.3(e) in [12]).
Any connected graph G can be constructed by successive vertex-pasting of
its blocks using its block cut-vertex tree [4] structure. Let G1 be an endblock of
G with a cut vertex v and G2 be the union of the remaining blocks of G. Then
G can be obtained from G1 and G2 by vertex-pasting along the vertex v. It
follows that any connected graph can be obtained from its blocks by successive
vertex-pastings.
Let G = (V,E) be a partial cube. We recall that the isometric dimension
dimI(G) of G is the cardinality of the quotient set E/θ, where θ is Djoković’s
equivalence relation on the set E (cf. formula (5.1)).
Theorem 7.2. Let G = (V,E) be a partial cube obtained by vertex-pasting
together partial cubes G1 = (V1, E1) and G2 = (V2, E2). Then
dimI(G) = dimI(G1) + dimI(G2).
Proof. It suffices to prove that there are no edges xy ∈ E1 and uv ∈ E2 which
are in Djoković’s relation θ with each other. Suppose that G1 and G2 are
vertex-pasted along vertices a1 ∈ E1 and a2 ∈ E2 and let a = {a1, a2} ∈ E. Let
xy ∈ E1 and uv ∈ E2 be two edges in E. We may assume that u ∈ Wxy. Since
a is a cut-vertex of G and u ∈Wxy, we have
d(u, a) + d(a, x) = d(u, x) < d(u, y) = d(u, a) + d(a, y).
Hence, d(a, x) < d(a, y), which implies
d(v, x) = d(v, a) + d(a, x) < d(v, a) + d(a, y) = d(v, y).
It follows that v ∈ Wxy. Therefore the edge xy does not stand in the relation θ
to the vertex uv.
The next result follows immediately from the previous theorem. Note that
blocks of a partial cube are partial cubes themselves.
Corollary 7.1. Let G be a partial cube and {G1, . . . , Gn} be the family of its
blocks. Then
dimI(G) =
dimI(Gi).
In the case of the lattice dimension of a partial cube we can claim only much
weaker result than one stated in Theorem 7.2 for the isometric dimension. We
omit the proof.
Theorem 7.3. Let G be a partial cube obtained by vertex-pasting together partial
cubes G1 and G2. Then
max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2).
The following example illustrate possible cases for inequalities in Theo-
rem 7.3. Let us recall that the lattice dimension of a tree with m leaves is
⌈m/2⌉ (cf. [14]).
Example 7.4. The star K1,6 can be obtained from the stars K1,2 and K1,4 by
vertex-pasting these two stars along their centers. Clearly,
max{dimZ(K1,2), dimZ(K1,4)} < dimZ(K1,6) = dimZ(K1,2) + dimZ(K1,4).
The same star K1,6 is obtained from two copies of the star K1,3 by vertex-
pasting along their centers. We have dimZ(K1,3) = 2, dimZ(K1,6) = 3, so
max{dimZ(K1,3), dimZ(K1,3)} < dimZ(K1,6) < dimZ(K1,3) + dimZ(K1,3).
Let us vertex-paste two stars K1,3 along their two leaves. The resulting
graph T is a tree with four vertices. Therefore,
max{dimZ(K1,3), dimZ(K1,3)} = dimZ(T ) < dimZ(K1,3) + dimZ(K1,3).
We now consider another simple way of pasting two graphs together.
Let G1 = (V1, E1) and G2 = (V2, E2) be two connected graphs, a1b1 ∈ E1,
a2b2 ∈ E2, and H1 = ({a1, b1}, {a1b1}), H2 = ({a2, b2}, {a2b2}). Let G be the
graph obtained by pasting G1 and G2 along subgraphs H1 and H2. In this case
we say that the graph G is obtained from graphs G1 and G2 by edge-pasting.
Figures 7.1, 7.2, and 7.5 illustrate this construction.
Figure 7.5: An example of edge-pasting.
As before, we identify the graphs G1 and G2 with subgraphs of the graph
G and denote a = {a1, a2}, b = {b1, b2} the two vertices obtained by pasting
together vertices a1 and a2 and, respectively, b1 and b2. The edge ab ∈ E is
obtained by pasting together edges a1b1 ∈ E1 and a2b2 ∈ E2 (cf. Figure 7.5).
Then G = G1∪G2, V1∩V2 = {a, b} and E1∩E2 = {ab}. We use these notations
in the rest of this section.
Proposition 7.1. A graph G obtained by edge-pasting together bipartite graphs
G1 and G2 is bipartite.
Proof. Let C be a cycle in G. If C ⊆ G1 or C ⊆ G2, then the length of C is
even, since the graphs G1 and G2 are bipartite. Otherwise, the vertices a and
b separate C into two paths each of odd length. Therefore C is a cycle of even
length. The result follows.
The following lemma is instrumental; it describes the semicubes of the graph
G in terms of semicubes of graphs G1 and G2.
Lemma 7.1. Let uv be an edge of G. Then
(i) For uv ∈ E1, a, b ∈ Wuv ⇒ Wuv =W
uv ∪ V2, Wvu =W
(ii) For uv ∈ E2, a, b ∈ Wuv ⇒ Wuv =W
uv ∪ V1, Wvu =W
(iii) a ∈ Wuv, b ∈Wvu ⇒ Wuv =Wab.
Figure 7.6: Edge-pasting of graphs G1 and G2.
Proof. We prove parts (i) and (iii) (see Figure 7.6).
(i) Since any path from w ∈ V2 to u or v contains a or b and a, b ∈Wuv, we
have w ∈Wuv. Hence, Wuv =W
uv ∪ V2 and Wvu =W
(iii) Since ab θ uv in G1, we have W
uv = W
, by Theorem 3.4(iv). Let w
be a vertex in W
uv . Then, by the triangle inequality,
d(w, u) < d(w, v) ≤ d(w, b) + d(b, v) < d(w, b) + d(b, u).
Since any shortest path from w to u contains a or b, we have
d(w, a) + d(a, u) = d(w, u).
Therefore,
d(w, a) + d(a, u) < d(w, b) + d(b, u).
Since ab θ uv in G1, we have d(a, u) = d(b, v), by Theorem 4.2. It follows that
d(w, a) < d(w, b), that is, w ∈ W
. We proved that W
uv ⊆ W
symmetry, W
vu ⊆ W
. Since two opposite semicubes form a partition of V2,
we have W
uv =W
. The result follows.
Theorem 7.4. A graph G obtained by edge-pasting together partial cubes G1
and G2 is a partial cube.
Proof. By Theorem 3.4(ii) and Proposition 7.1, we need to show that for any
edge uv of G the semicube Wuv is a convex subset of V . There are two possible
cases.
(i) uv = ab. The semicube Wab is the union of semicubes W
and W
which are convex subsets of V1 and V2, respectively. It is clear that any shortest
path connecting a vertex in W
with a vertex in W
contains vertex a and
therefore is contained in Wab. Hence, Wab is a convex set. A similar argument
proves that the set Wba is convex.
(ii) uv 6= ab. We may assume that uv ∈ E1. To prove that the semicube
Wuv is a convex set, we consider two cases.
(a) a, b ∈ Wuv. (The case when a, b ∈ Wvu is treated similarly.) By
Lemma 7.1(i), the semicube Wuv is the union of the semicube W
uv and the
set V2 which are both convex sets. Any shortest path P from a vertex in V2 to
a vertex in W
uv contains either a or b. It follows that P ⊆ W
uv ∪ V2 = Wuv.
Therefore the semicube Wuv is convex.
(b) a ∈ Wuv, b ∈ Wvu. (The case when b ∈ Wuv , a ∈ Wvu is treated
similarly.) By Lemma 7.1(ii), Wuv = Wab. The result follows from part (i) of
the proof.
Theorem 7.5. Let G be a graph obtained by edge-pasting together finite partial
cubes G1 and G2. Then
dimI(G) = dimI(G1) + dimI(G2)− 1.
Proof. Let θ, θ1, and θ2 be Djoković’s relations on E, E1, and E2, respectively.
By Lemma 7.1, for uv, xy ∈ E1 (resp. uv, xy ∈ E2) we have
uv θ xy ⇔ uv θ1xy (resp. uv θ xy ⇔ uv θ2xy).
Let uv ∈ E1, xy ∈ E2, and uv θ xy. Suppose that (uv, ab) /∈ θ. We may
assume that a, b ∈ Wuv . By Lemma 7.1(i), V2 ⊂ Wuv, a contradiction, since
xy ∈ E2. Hence, uv θ xy θ ab. It follows that each equivalence class of the
relation θ is either an equivalence class of θ1, an equivalence class of θ2 or the
class containing the edge ab. Therefore
|E/θ| = |E1/θ1|+ |E2/θ2| − 1.
The result follows, since the isometric dimension of a partial cube is equal to the
cardinality of the set of equivalence classes of Djoković’s relation (formula (5.1)).
We need some results about semicube graphs in order to prove an analog of
Theorem 7.3 for a partial cube obtained by edge-pasting of two partial cubes.
Lemma 7.2. Let G be a partial cube and WpqWuv , WqpWxy be two edges in
the graph Sc(G). Then WxyWuv is an edge in Sc(G).
Proof. By condition (5.4), Wqp ⊂ Wuv and Wyx ⊂ Wqp. Hence, Wyx ⊂ Wuv.
By the same condition, WxyWuv ∈ Sc(G).
As before, we identify partial cubes G1 and G2 with subgraphs of the partial
cube G. Then G1 ∪G2 = G and G1 ∩G2 = ({a, b}, {ab}) = K2 (cf. Figure 7.6).
Lemma 7.3. Let G be a partial cube obtained by edge-pasting together partial
cubes G1 and G2. Let W
xy (resp. W
xy ) be an edge in the semicube
Sc(G1) (resp. Sc(G2)). Then WuvWxy is an edge in Sc(G).
Figure 7.7: Semicubes forming an edge in Sc(G1).
Proof. It suffices to consider the case of Sc(G1) (see Figure 7.7). By condi-
tion (5.4),W
vu ⊂W
xy andW
yx ⊂W
uv . Suppose that a ∈ W
vu and b ∈W
(the case when b ∈ W
vu and a ∈ W
yx is treated similarly). Then ab θ1xy and
ab θ1uv. By transitivity of θ1, we have uv θ1xy, a contradiction, since semicubes
uv and W
xy are distinct. Therefore we may assume that, say, a, b ∈ W
Then, by Lemma 7.1, Wvu = W
vu ⊂ V1. Since W
vu ⊂ W
xy ⊆ Wxy, we have
Wvu ⊂Wxy. By condition (5.4), WuvWxy is an edge in Sc(G).
Lemma 7.4. LetM1 andM2 be matchings in graphs Sc(G1) and Sc(G2). There
is a matching M in Sc(G) such that
|M | ≥ |M1|+ |M2| − 1.
Proof. By Lemma 7.3, M1 and M2 induce matchings in Sc(G) which we denote
by the same symbols. The intersection M1 ∩M2 is either empty or a subgraph
of the empty graph with vertices Wab and Wba.
If M1 ∩M2 is empty, then M = M1 ∪M2 is a matching in Sc(G) and the
result follows.
If M1 ∩M2 is an empty graph with a single vertex, say, in M1, we remove
fromM1 the edge that has this vertex as its end vertex, resulting in the matching
M ′1. Clearly, M =M
1 ∪M2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1.
Suppose now that M1 ∩M2 is the empty graph with vertices Wab and Wba.
Let WabWuv, WbaWpq (resp. WabWxy, WbaWrs) be edges in M1 (resp. M2).
By Lemma 7.2, WxyWrs is an edge in Sc(G2). Let us replace edgesWabWxy and
WbaWrs in M2 by a single edge WxyWrs, resulting in the matching M
2. Then
M =M1 ∪M
2 is a matching in Sc(G) and |M | = |M1|+ |M2| − 1.
Corollary 7.2. Let M1 and M2 be maximum matchings in Sc(G1) and Sc(G2),
respectively, and M be a maximum matching in Sc(G). Then
|M | ≥ |M1|+ |M2| − 1. (7.1)
By Theorem 5.3, we have
dimI(G1) = dimZ(G1) + |M1|, dimI(G2) = dimZ(G2) + |M2|,
dimI(G) = dimZ(G) + |M |,
where M1 and M2 are maximum matchings in Sc(G1) and Sc(G2), respectively,
and M is a maximum matching in Sc(G). Therefore, by Theorem 7.5 and (7.1),
we have the following result (cf. Theorem 7.3).
Theorem 7.6. Let G be a partial cube obtained by edge-pasting from partial
cubes G1 and G2. Then
max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) ≤ dimZ(G1) + dimZ(G2).
Example 7.5. Let us consider two edge-pastings of the stars G1 = K1,3 and
G2 = K1,3 of lattice dimension 2 shown in figures 7.1 and 7.2. In the first case
the resulting graph is the star G = K1,5 of lattice dimension 3. Then we have
max{dimZ(G1), dimZ(G2)} < dimZ(G) < dimZ(G1) + dimZ(G2).
In the second case the resulting graph is a tree with 4 leaves. Therefore,
max{dimZ(G1), dimZ(G2)} = dimZ(G) < dimZ(G1) + dimZ(G2).
Let c1a1 and c2a2 be edges of stars G1 = K1,4 and G2 = K1,4 (each of
which has lattice dimension 2), where c1 and c2 are centers of the respective
stars. Let us edge-paste these two graphs by identifying c1 with c2 and a1 with
a2, respectively. The resulting graph G is the star K1,7 of lattice dimension 4.
Thus,
max{dimZ(G1), dimZ(G2)} ≤ dimZ(G) = dimZ(G1) + dimZ(G2).
8 Expansions and contractions of partial cubes
The graph expansion procedure was introduced by Mulder in [16], where it is
shown that a graph is a median graph if and only if it can be obtained from
K1 by a sequence of convex expansions (see also [15]). A similar result for
partial cubes was established in [6] (see also [7]) as a corollary to a more general
result concerning isometric embeddability into Hamming graphs; it was also
established in [13] in the framework of oriented matroids theory.
In this section we investigate properties of (isometric) expansion and con-
traction operations and, in particular, prove in two different ways that a graph is
a partial cube if and only if it can be obtained from the graph K1 by a sequence
of expansions.
A remark about notations is in order. In the product {1, 2} × (V1 ∪ V2), we
denote V ′i = {i} × Vi and x
i = (i, x) for x ∈ Vi, where i, j = 1, 2.
Definition 8.1. Let G = (V,E) be a connected graph, and let G1 = (V1, E1)
and G2 = (V2, E2) be two isometric subgraphs of G such that G = G1 ∪ G2.
The expansion of G with respect to G1 and G2 is the graph G
′ = (V ′, E′)
constructed as follows from G (see Figure 8.1):
(i) V ′ = V1 + V2 = V
1 ∪ V
(ii) E′ = E1 + E2 +M , where M is the matching
x∈V1∩V2
{x1x2}.
In this case, we also say that G is a contraction of G′.
Figure 8.1: Expansion/contraction processes.
It is clear that the graphs G1 and 〈V
1〉 are isomorphic, as well as the graphs
G2 and 〈V
We define a projection p : V ′ → V by p(xi) = x for x ∈ V . Clearly, the
restriction of p to V ′1 is a bijection p1 : V
1 → V1 and its restriction to V
2 is a
bijection p2 : V
2 → V2. These bijections define isomorphisms 〈V
1〉 → G1 and
〈V ′2〉 → G2.
Let P ′ be a path in G′. The vertices of G obtained from the vertices in P ′
under the projection p define a walk P in G; we call this walk P the projection
of the path P ′. It is clear that
ℓ(P ) = ℓ(P ′), if P ′ ⊆ 〈V ′1〉 or P
′ ⊆ 〈V ′2〉. (8.1)
In this case, P is a path in G and either P = p1(P
′) or P = p2(P
′). On the
other hand,
ℓ(P ) < ℓ(P ′), if P ′ ∩ 〈V ′1〉 6= ∅ and P
′ ∩ 〈V ′2 〉 6= ∅, (8.2)
and P is not necessarily a path.
We will frequently use the results of the following lemma in this section.
Lemma 8.1. (i) For u1, v1 ∈ V ′1 , any shortest path Pu1v1 in G
′ belongs to 〈V ′1 〉
and its projection Puv = p1(Pu1v1) is a shortest path in G. Accordingly,
dG′(u
1, v1) = dG(u, v)
and 〈V ′1〉 is a convex subgraph of G
′. A similar statement holds for u2, v2 ∈ V ′2 .
(ii) For u1 ∈ V ′1 and v
2 ∈ V ′2 ,
dG′(u
1, v2) = dG(u, v) + 1.
Let Pu1v2 be a shortest path in G
′. There is a unique edge x1x2 ∈M such that
x1, x2 ∈ Pu1v2 and the sections Pu1x1 and Px2v2 of the path Pu1v2 are shortest
paths in 〈V ′1 〉 and 〈V
2 〉, respectively. The projection Puv of Pu1v2 in G
′ is a
shortest path in G.
Proof. (i) Let Pu1v1 be a path in G
′ that intersects V ′2 . Since 〈V1〉 is an isometric
subgraph of G, there is a path Puv in G that belongs to 〈V1〉. Then p
1 (Puv) is
a path in 〈V ′1 〉 of the same length as Puv. By (8.1) and (8.2),
ℓ(p−11 (Puv)) < ℓ(Pu1v1).
Therefore any shortest path Pu1v1 in G
′ belongs to 〈V ′1 〉. The result follows.
(ii) Let Pu1v2 be a shortest path in G
′ and Puv be its projection to V .
By (8.2),
dG′(u
1, v2) = ℓ(Pu1v2) > ℓ(Puv) ≥ dG(u, v).
Since there is no edge of G joining vertices in V1 \ V2 and V2 \ V1, a shortest
path in G from u to v must contain a vertex x ∈ V1 ∩ V2. Since G1 and G2
are isometric subgraphs, there are shortest paths Pux in G1 and Pxv in G2 such
that their union is a shortest path from u to v. Then, by the triangle inequality
and part (i) of the proof, we have (cf. Figure 8.1)
dG′(u
1, v2) ≤ dG′(u
1, x1) + dG′(x
1, x2) + dG′(x
2, v2) = dG(u, v) + 1.
The last two displayed formulas imply dG′(u
1, v2) = dG(u, v) + 1.
Since u1 ∈ V ′1 and v
2 ∈ V ′2 the path Pu1v2 must contain an edge, say x
1x2, in
M . Since this path is a shortest path in G′, this edge is unique. Then the sec-
tions Pu1x1 and Px2v2 of Pu1v2 are shortest paths in 〈V
1 〉 and 〈V
2〉, respectively.
Clearly, Puv is a shortest path in G.
Let a1a2 be an edge in the matchingM = ∪x∈V1∩V2{x
1x2}. This edge defines
five fundamental sets (cf. Section 4): the semicubes Wa1a2 and Wa2a1 , the sets
of vertices Ua1a2 and Ua2a1 , and the set of edges Fa1a2 . The next theorem
follows immediately from Lemma 8.1. It gives a hint to a connection between
the expansion process and partial cubes.
Theorem 8.1. Let G′ be an expansion of a connected graph G and notations
are chosen as above. Then
(i) Wa1a2 = V
1 and Wa2a1 = V
2 are convex semicubes of G
(ii) Fa1a2 =M defines an isomorphism between induced subgraphs 〈Ua1a2〉 and
〈Ua2a1〉, which are isomorphic to the subgraph G1 ∩G2.
The result of Theorem 8.1 justifies the following constructive definition of
the contraction process.
Definition 8.2. Let ab be an edge of a connected graph G′ = (V ′, E′) such
(i) semicubes Wab and Wba are convex and form a partition of V
(ii) the set Fab is a matching and defines an isomorphism between subgraphs
〈Uab〉 and 〈Uba〉.
A graph G obtained from the graphs 〈Wab〉 and 〈Wba〉 by pasting them along
subgraphs 〈Uab〉 and 〈Uba〉 is said to be a contraction of the graph G
Remark 8.1. If G′ is bipartite, then semicubesWab andWba form a partition of
its vertex set. Then, by Theorem 4.1, condition (i) implies condition (ii). Thus
any pair of opposite convex semicubes in a connected bipartite graph defines a
contraction of this graph.
By Theorem 8.1, a graph is a contraction of its expansion. It is not difficult
to see that any connected graph is also an expansion of its contraction.
The following three examples give geometric illustrations for the expansion
and contraction procedures.
Example 8.1. Let a and b be two opposite vertices in the graph G = C4.
Clearly, the two distinct paths P1 and P2 from a to b are isometric subgraphs
of G defining an expansion G′ = C6 of G (see Figure 8.2). Note that P1 and P2
are not convex subsets of V .
Example 8.2. Another isometric expansion of the graph G = C4 is shown
in Figure 8.3. Here, the path P1 is the same as in the previous example and
G2 = G.
Example 8.3. Lemma 8.1 claims, in particular, that the projection of a shortest
path in an extension G′ of a graphG is a shortest path in G. Generally speaking,
Figure 8.2: An expansion of the cycle C4.
Figure 8.3: Another isometric expansion of the cycle C4.
the converse is not true. Consider the graph G shown in Figure 8.4 and two
paths in G:
V1 = abcef and V2 = bde.
The graph G′ in Figure 8.4 is the convex expansion of G with respect to V1 and
V2. The path abdef is a shortest path in G; it is not a projection of a shortest
path in G′.
Figure 8.4: A shortest path which is not a projection of a shortest path.
One can say that, in the case of finite partial cubes, the contraction procedure
is defined by an orthogonal projection of a hypercube onto one of its facets.
By Theorem 8.1, the sets V ′1 and V
2 are opposite semicubes of the graph G
defined by edges in M . Their projections are the sets V1 and V2 which are not
necessarily semicubes of G. For other semicubes in G′ we have the following
result.
Lemma 8.2. For any two adjacent vertices u, v ∈ V ,
Wuivi = p
−1(Wuv) for u, v ∈ Vi and i = 1, 2.
Proof. By Lemma 8.1,
dG′(x
j , ui) < dG′(x
j , vi) ⇔ dG(x, u) < dG(x, v)
for x ∈ V and i, j = 1, 2. The result follows.
Corollary 8.1. If uv is an edge of G1 ∩G2, then Wu1v1 =Wu2v2 .
The following lemma is an immediate consequence of Lemma 8.1. We shall
use it implicitly in our arguments later.
Lemma 8.3. Let u, v ∈ V1 and x ∈ V1 ∩ V2. Then
x1 ∈Wu1v1 ⇔ x
2 ∈Wu1v1 .
The same result holds for semicubes in the form Wu2v2 .
Generally speaking, the projection of a convex subgraph of G′ is not a con-
vex subgraph of G. For instance, the projection of the convex path b2d2e2 in
Figure 8.4 is the path bde which is not a convex subgraph of G. On the other
hand, we have the following result.
Theorem 8.2. Let G′ = (V ′, E′) be an expansion of a graph G = (V,E) with
respect to subgraphs G1 = (V1, E1) and G2 = (V2, E2). The projection of a
convex semicube of G′ different from 〈V ′1〉 and 〈V
2 〉 is a convex semicube of G.
Proof. It suffices to consider the case when Wuv = p(Wu1v1) for u, v ∈ V1 (cf.
Theorem 8.2). Let x, y ∈Wuv and z ∈ V be a vertex such that
dG(x, z) + dG(z, y) = dG(x, y).
We need to show that z ∈Wuv.
Figure 8.5: A shortest path from x to y.
(i) x, y ∈ V1 (the case when x, y ∈ V2 is treated similarly). Suppose that
z ∈ V1. Then x
1, y1, z1 ∈ V ′1 and, by Lemma 8.1,
dG′(x
1, z1) + dG′(z
1, y1) = dG′(z
1, y1).
Since x1, y1 ∈ Wu1v1 and Wu1v1 is convex, z
1 ∈ Wu1v1 . Hence, z ∈Wuv.
Suppose now that z ∈ V2 \ V1. Consider a shortest path Pxy in G from x
to y containing z. This path contains vertices x′, y′ ∈ V1 ∩ V2 such that (see
Figure 8.5)
dG(x, x
′) + dG(x
′, z) = dG(x, z) and dG(y, y
′) + dG(y
′, z) = dG(y, z).
Since Pxy is a shortest path in G, we have
dG(x, x
′) + dG(x
′, y) = dG(x, y), dG(x, y
′) + dG(y
′, y) = dG(x, y),
′, z) + dG(z, y
′) = dG(x
′, y′).
Since x, x′, y ∈ V1, we have x
1, x′1, y1 ∈ V ′1 . Because x
1, y1 ∈ Wu1v1 and Wu1v1
is convex, x′1 ∈ Wu1v1 . Hence, x
′ ∈ Wuv and, similarly, y
′ ∈ Wuv. Since
x′2, y′2, z2 ∈ V ′2 and Wu1v1 is convex, z
2 ∈Wu1v1 . Hence, z ∈Wuv.
(ii) x ∈ V1 \V2 and y ∈ V2 \V1. We may assume that z ∈ V1. By Lemma 8.1,
dG′(x
1, y2) = dG(x, y) + 1 = dG(x, z) + dG(z, y) + 1
= dG′(x
1, z1) + dG′(z
1, y2).
Since x1, y2 ∈ Wu1v1 and Wu1v1 is convex, z
1 ∈ Wu1v1 . Hence, z ∈Wuv.
By using the results of Lemma 8.1, it is not difficult to show that the class
of connected bipartite graphs is closed under the expansion and contraction
operations. The next theorem establishes this result for the class of partial
cubes.
Theorem 8.3. (i) An expansion G′ of a partial cube G is a partial cube.
(ii) A contraction G of a partial cube G′ is a partial cube.
Proof. (i) Let G = (V,E) be a partial cube and G′ = (V ′, E′) be its expansion
with respect to isometric subgraphs G1 = (V1, E1) and G2 = (V2, E2). By
Theorem 3.4(ii), it suffices to show that the semicubes of G′ are convex.
By Lemma 8.1, the semicubes 〈V ′1〉 and 〈V
2〉 are convex, so we consider a
semicube in the formWu1v1 where uv ∈ E1 (the other case is treated similarly).
Let Px′y′ be a shortest path connecting two vertices in Wu1v1 and Pxy be its
projection to G. By Theorem 8.2, x, y ∈ Wuv and, by Lemma 8.1, Pxy is a
shortest path in G. Since Wuv is convex, Pxy belongs to Wuv. Let z
′ be a
vertex in Px′y′ and z = p(z
′) ∈ Pxy. By Lemma 8.1,
dG(z, u) < dG(z, v) ⇒ dG′(z
′, u1) ≤ dG′(z
′, v1).
Since G′ is a bipartite graph, dG′(z
′, u1) < dG′(z
′, v1). Hence, Px′y′ ⊆ Wu1v1 ,
so Wu1v1 is convex.
(ii) Let G = (V,E) be a contraction of a partial cube G′ = (V ′, E′). By
Theorem 3.4, we need to show that the semicubes of G are convex. By The-
orem 8.2, all semicubes of G are projections of semicubes of G′ distinct from
〈V ′1〉 and 〈V
2〉. By Theorem 8.2, the semicubes of G are convex.
Corollary 8.2. (i) A finite connected graph is a partial cube if and only if it
can be obtained from K1 by a sequence of expansions.
(ii) The number of expansions needed to produce a partial cube G from K1
is dimI(G).
Proof. (i) Follows immediately from Theorem 8.3.
(ii) Follows from theorems 8.2 and 5.1 (see the discussion in Section 5 just
before Theorem 5.2 ).
The processes of expansion and contraction admit useful descriptions in the
case of partial cubes on a set. Let G = (V,E) be a partial cube on a set X ,
that is an isometric subgraph of the hypercube H(X). Then it is induced by
some wg-family F of finite subsets of X (cf. Theorem 2.1). We may assume (see
Section 5) that ∩F = ∅ and ∪F = X .
In what follows we present proofs of the results of Theorem 8.3 and Corol-
lary 8.2 given in terms of wg-families of sets.
The expansion process for a partial cube G on X can be described as follows:
Let F1 and F2 be wg-families of finite subsets of X such that F1 ∩ F2 6= ∅,
F1∪F2 = F, and the distance between any two sets P ∈ F1 \F2 and Q ∈ F2 \F1
is greater than one. Note that 〈F1〉 and 〈F2〉 are partial cubes, 〈F1〉∩ 〈F2〉 6= ∅,
and 〈F1〉 ∪ 〈F2〉 = 〈F〉 = G. Let X
′ = X + {p}, where p /∈ X , and
2 = {Q+ {p} : Q ∈ F2}, F
′ = F1 ∪ F
It is quite clear that the graphs 〈F′2〉 and 〈F2〉 are isomorphic and the graph
G′ = 〈F′〉 is an isometric expansion of the graph G.
Theorem 8.4. An expansion of a partial cube is a partial cube.
Proof. We need to verify that F′ is a wg-family of finite subsets of X ′. By
Theorem 2.3, it suffices to show that the distance between any two adjacent
sets in F′ is 1. It is obvious if each of these two sets belong to one of the families
F1 or F
2. Suppose that P ∈ F1 and Q+ {p} ∈ F
2 are adjacent, that is, for any
S ∈ F′ we have
P ∩ (Q+ {p}) ⊆ S ⊆ P ∪ (Q+ {p}) ⇒ S = P or S = Q+ {p}. (8.3)
If Q ∈ F1, then
P ∩ (Q + {p}) ⊆ Q ⊆ P ∪ (Q+ {p}),
since p /∈ P . By (8.3), Q = P implying d(P,Q + {p}) = 1.
If Q ∈ F2 \ F1, there is R ∈ F1 ∩ F2 such that
d(P,R) + d(R,Q) = d(P,Q),
since F is well graded. By Theorem 2.2,
P ∩Q ⊆ R ⊆ P ∪Q,
which implies
P ∩ (Q + {p}) ⊆ R+ {p} ⊆ P ∪ (Q+ {p}).
By (8.3), R + {p} = Q+ {p}, a contradiction.
It is easy to recognize the fundamental sets (cf. Section 4) in an isometric
expansion G′ of a partial cube G = 〈F〉. Let P ∈ F1∩F2 and Q = P +{p} ∈ F
be two vertices defining an edge in G′ according to Definition 8.1(ii). Clearly,
the families F1 and F
2 are the semicubes WPQ and WQP of the graph G
′ (cf.
Lemma 5.1) and therefore are convex subsets of F′. The set FPQ is the set
of edges defined by p as in Lemma 5.1. In addition, UPQ = F1 ∩ F2 and
UQP = {R+ {p} : R ∈ F1 ∩ F2}.
Let G be a partial cube induced by a wg-family F of finite subsets of a set
X . As before, we assume that ∩F = ∅ and ∪F = X . Let PQ be an edge of G.
We may assume that Q = P + {p} for some p /∈ P . Then (see Lemma 5.1)
WPQ = {R ∈ F : p /∈ R} and WQP = {R ∈ F : p ∈ R}.
Let X ′ = X \ {p} and F′ = {R \ {p} : R ∈ F}. It is clear that the graph G′
induced by the family F′ is isomorphic to the contraction of G defined by the
edge PQ. Geometrically, the graph G′ is the orthogonal projection of the graph
G along the edge PQ (cf. figures 8.2 and 8.3).
Theorem 8.5. (i) A contraction G′ of a partial cube G is a partial cube.
(ii) If G is finite, then dimI(G
′) = dimI(G)− 1.
Proof. (i) For p ∈ X we define F1 = {R ∈ F : p /∈ R}, F2 = {R ∈ F : p ∈ R},
and F′2 = {R \ {p} ∈ F : p ∈ R}. Note that F1 and F2 are semicubes of G
and F′2 is isometric to F2. Hence, F1 and F
2 are wg-families of finite subsets
of X ′. We need to prove that F′ = F1 ∪ F
2 is a wg-family. By Theorem 2.3,
it suffices to show that d(P,Q) = 1 for any two adjacent sets P,Q ∈ F′. This
is true if P,Q ∈ F1 or P,Q ∈ F
2, since these two families are well graded. For
P ∈ F1 \ F
2 and Q ∈ F
2 \ F1, the sets P and Q + {p} are not adjacent in F,
since F is well graded and Q /∈ F. Hence there is R ∈ F1 such that
P ∩ (Q+ {p}) ⊆ R ⊆ P ∪ (Q + {p})
and R 6= P . Since p /∈ R, we have
P ∩Q ⊆ R ⊆ P ∪Q.
Since R 6= P and R 6= Q, the sets P and Q are not adjacent in F′. The result
follows.
(ii) If G is a finite partial cube, then, by Theorem 5.2,
dimI(G
′) = |X ′| = |X | − 1 = dimI(G)− 1.
9 Conclusion
The paper focuses on two themes of a rather general mathematical nature.
1. The characterization problem. It is a common practice in mathematics
to characterize a particular class of object in different terms. We present new
characterizations of the classes of bipartite graphs and partial cubes, and give
new proofs for known characterization results.
2. Constructions. The problem of constructing new objects from old ones
is a standard topic in many branches of mathematics. For the class of partial
cubes, we discuss operations of forming the Cartesian product, expansion and
contraction, and pasting. It is shown that the class of partial cubes is closed
under these operations.
Because partial cubes are defined as graphs isometrically embeddable into
hypercubes, the theory of partial cubes has a distinctive geometric flavor. The
three main structures on a graph—semicubes and Djoković’s and Winkler’s
relations—are defined in terms of the metric structure on a graph. One can say
that this theory is a branch of discrete metric geometry. Not surprisingly, geo-
metric structures play an important role in our treatment of the characterization
and construction problems.
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[14] F. Hadlock and F. Hoffman, Manhattan trees, Util. Math. 13 (1978) 55–67.
[15] W. Imrich and S. Klavžar, Product Graphs, John Wiley & Sons, 2000.
[16] H.M. Mulder, The Interval Function of a Graph, Mathematical Centre
Tracts 132, Mathematisch Centrum, Amsterdam, 1980.
[17] S. Ovchinnikov, Media theory: representations and examples, Discrete Ap-
plied Mathematics, (in review, e-print available at
http://arxiv.org/abs/math.CO/0512282).
[18] R.I. Roth and P.M. Winkler, Collapse of the metric hierarchy for bipartite
graphs, European Journal of Combinatorics 7 (1986) 371–375.
[19] M.L.J. van de Vel, Theory of Convex Structures, Elsevier, The Netherlands,
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[20] P.M. Winkler, Isometric embedding in products of complete graphs, Dis-
crete Appl. Math. 8 (1984) 209–212.
http://arxiv.org/abs/math.CO/0512282
Introduction
Hypercubes and partial cubes
Characterizations
Fundamental sets in partial cubes
Dimensions of partial cubes
Subcubes and Cartesian products
Pasting partial cubes
Expansions and contractions of partial cubes
Conclusion
|
0704.0011 | Computing genus 2 Hilbert-Siegel modular forms over $\Q(\sqrt{5})$ via
the Jacquet-Langlands correspondence | COMPUTING GENUS 2 HILBERT-SIEGEL MODULAR
FORMS OVER Q(
5) VIA THE JACQUET-LANGLANDS
CORRESPONDENCE
CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
Abstract. In this paper we present an algorithm for computing Hecke
eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of
narrow class number one. We give some illustrative examples using the
quadratic field Q(
5). In those examples, we identify Hilbert-Siegel
eigenforms that are possible lifts from Hilbert eigenforms.
Introduction
Let F be a real quadratic field of narrow class number one and let B be the
unique (up to isomorphism) quaternion algebra over F which is ramified at
both archimedean places of F and unramified everywhere else. Let GU2(B)
be the unitary similitude group of B⊕2. This is the set of Q-rational points
of an algebraic group GB defined over Q. The group GB is an inner form of
G := ResF/Q(GSp4) such that G
B(R) is compact modulo its centre. (These
notions are reviewed at the beginning of Section 1.)
In this paper we develop an algorithm which computes automorphic forms
on GB in the following sense: given an idealN inOF and an integer k greater
than 2, the algorithm returns the Hecke eigensystems of all automorphic
forms f of level N and parallel weight k. More precisely, given a prime p in
OF , the algorithm returns the Hecke eigenvalues of f at p, and hence the
Euler factor Lp(f, s), for each eigenform f of level N and parallel weight k.
The algorithm is a generalization of the one developed in [D1 2005] to the
genus 2 case. Although we have only described the algorithm in the case of
a real quadratic field in this paper, it should be clear from our presentation
that it can be adapted to any totally real number field of narrow class
number one.
The Jacquet-Langlands Correspondence of the title refers to the conjec-
tural map JL : Π(GB) → Π(G) from automorphic representations of GB to
automorphic representations of G, which is injective, matches L-functions
and enjoys other properties compatible with the principle of functoriality;
Date: October 29, 2018.
1991 Mathematics Subject Classification. Primary: 11F41 (Hilbert and Hilbert-Siegel
modular forms).
Key words and phrases. Hilbert-Siegel modular forms, Jacquet-Langlands Correspon-
dence, Brandt matrices, Satake parameters.
http://arxiv.org/abs/0704.0011v3
2 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
in particular, the image of the Jacquet-Langlands Correspondence is to be
contained in the space of holomorphic automorphic representations. If we
admit this conjecture, then the algorithm above provides a way to produce
examples of cuspidal Hilbert-Siegel modular forms of genus 2 over F and
allows us to compute the L-factors of the corresponding automorphic repre-
sentations for arbitrary finite primes p of F .
In fact, we are also able to use these calculations to provide evidence for
the Jacquet-Langlands Correspondence itself by comparing the Euler factors
we find with those of known Hilbert-Siegel modular forms obtained by lifting.
This we do in the final section of the paper where we observe that some of
the Euler factors we compute match those of lifts of Hilbert modular forms,
for the primes we computed. Although this does not definitively establish
that these Hilbert-Siegel modular forms are indeed lifts, in principle one can
establish equality in this way, using an analogue of the Sturm bound.
The first systematic approach to Siegel modular forms from a computa-
tional viewpoint is due to Skoruppa [Sk 1992] who used Jacobi symbols to
generate spaces of such forms. His algorithm, which has been extensively
exploited by Ryan [R 2006], applies only to the case of full level structure.
More recently, Faber and van der Geer [FvdG1 2004] and [FvdG2 2004]
also produced examples of Siegel modular forms by counting points on hy-
perelliptic curves of genus 2; again their results are available only in the
full level structure case. The most substantial progress toward the com-
putation of Siegel modular forms for proper level structure is by Gunnells
[Gu 2000] who extended the theory of modular symbols to the symplectic
group Sp4/Q. However, this work does not see the cuspidal cohomology,
which is the only part of the cohomology which is relevant to arithmetic
geometric applications. To the best of our knowledge, there are no numer-
ical examples of Hilbert-Siegel modular forms for proper level structure in
the literature, with the exception of those produced from liftings of Hilbert
modular forms.
The outline of the paper is as follows. In Section 1 we recall the basic
properties of Hilbert-Siegel modular forms and algebraic automorphic forms
together with the Jacquet-Langlands Correspondence. In Section 2 we give
a detailed description of our algorithm. Finally, in Section 3 we present
numerical results for the quadratic field Q(
Acknowledgements. During the course of the preparation of this paper,
the second author had helpful email exchanges with several people includ-
ing Alexandru Ghitza, David Helm, Marc-Hubert Nicole, David Pollack,
Jacques Tilouine and Eric Urban. The authors wish to thank them all.
Also, we would like to thank William Stein for allowing us to use the SAGE
computer cluster at the University of Washington. And finally, the sec-
ond author would like to thank the PIMS institute for their postdoctoral
fellowship support, and the University of Calgary for its hospitality.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 3
1. Hilbert-Siegel modular forms and the Jacquet-Langlands
correspondence
Throughout this paper, F denotes a real quadratic field of narrow class
number one. The two archimedean places of F and the real embeddings of
F will both be denoted v0 and v1. For every a ∈ F , we write a0 (resp. a1)
for the image of a under v0 (resp. v1). The ring of integers of F is denoted
by OF . For every prime ideal p in OF , the completion of F and OF at p
will be denoted by Fp and OFp , respectively.
Let B be the unique (up to isomorphism) totally definite quaternion al-
gebra over F which is unramified at all finite primes of F . We fix a maximal
order OB of B. Also, we choose a splitting field K/F of B that is Ga-
lois over Q and such that there exists an isomorphism j : OB ⊗Z OK ∼=
M2(OK)⊕M2(OK), where M2(A) denotes the ring of 2× 2-matrices with
entries from a ring A. For every finite prime p in F , we fix an isomorphism
Bp ∼= M2(Fp) which restricts to an isomorphism from OB, p onto M2(OFp ).
The algebraic group G = ResF/Q(GSp4) is defined as follows. For any
Q-algebra A, the set of A-rational points of G is given by
G(A) =
γ ∈ GL4(A⊗Q F )
t = νG(γ)J2
νG(γ) ∈ (A⊗Q F )×
where
−12 0
This group admits an integral model with A-rational points for every Z-
algebra A given by
GZ(A) =
γ ∈ GL4(A⊗Z OF )
t = νG(γ)J2
νG(γ) ∈ (A⊗Z OF )×
For any Q-algebra A, the conjugation on B extends in a natural way to the
matrix algebra M2(B ⊗Q A).
The algebraic group GB/Q is defined as follows. For any Q-algebra A,
the set of A-rational points of GB is given by
GB(A) =
γ ∈ M2(B ⊗Q A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Q F )×
This group also admits an integral model with A-rational points for every
Z-algebra given by
GBZ (A) =
γ ∈ M2(OB ⊗Z A)
γγ̄t = νGB(γ)12
νGB (γ) ∈ (A⊗Z OF )×
The group GB/Q is an inner form of G/Q such that GB(R) is compact
modulo its center. Combining the isomorphism j (see above) with con-
jugation by a permutation matrix, we obtain an isomorphism GBZ (OK) ∼=
4 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
GZ(OK), which we fix from now on. For every prime ideal p in F , the split-
ting of GB at p amounts to the splitting of the quaternion algebra B at p;
we refer to [D1 2005] for further details.
By the choice of the quaternion algebra B, we have GB(Q̂) ∼= G(Q̂). (We
denote the finite adèles of Q (resp. Z) by Q̂ (resp. Ẑ)).
1.1. Hilbert-Siegel modular forms. We fix an integer k ≥ 3 and, for
simplicity, we restrict ourselves to Hilbert-Siegel modular forms of parallel
weight k. The real embeddings v0 and v1 of F extend to G(Q) = GSp4(F )
in a natural way. We denote by GSp+4 (F ) the subgroup of elements γ with
totally positive similitude factor νG(γ). We recall that the Siegel upper-half
plane of genus 2 is defined by
H2 = {γ ∈ GL2(C)
∣ γt = γ and Im(γ) is positive definite }.
We also recall that GSp+4 (F ) acts on H
(τ0, τ1) :=
(a0τ0 + b0)(c0τ0 + d0)
−1, (a1τ1 + b1)(c1τ1 + d1)
This induces an action on the space of functions f : H22 → C by
, f |kγ(τ) =
νG(γi)
det(ciτi + di)k
f(τ).
Let N be an ideal in OF and set
Γ0(N) =
∈ GSp+4 (OF )
∣ c ≡ 0(N)
A Hilbert-Siegel modular form of level N and parallel weight k is a
holomorphic function f : H22 → C such that
∀γ ∈ Γ0(N), f |kγ = f.
The space of Hilbert-Siegel modular forms of parallel weight k and level N
is denoted Mk(N). Each f ∈Mk(N) admits a Fourier expansion, which by
the Koecher principle takes the form
∀τ ∈ H22, f(τ) =
{Q}∪{0}
2πiTr(Qτ),
where Q ∈ M2(F ) runs over all symmetric totally positive and semi-definite
matrices. A Hilbert-Siegel modular forms f is a cusp form if, for all γ ∈
4 (F ), the constant term in the Fourier expansion of f |kγ is zero. The
space of Hilbert-Siegel cusp forms is denoted Sk(N).
COMPUTING HILBERT-SIEGEL MODULAR FORMS 5
1.2. The Hecke algebra. The space Sk(N) comes equipped with a Hecke
action, which we now recall. Take u ∈ GSp+4 (F ) ∩M4(OF ), and write the
finite disjoint union
Γ0(N)uΓ0(N) =
Γ0(N)ui.
Then the Hecke operator [Γ0(N)uΓ0(N)] on Sk(N) is given by
[Γ0(N)uΓ0(N)]f =
f |kui.
Let p be a prime ideal in OF and let πp be a totally positive generator of
p; let T1(p) and T2(p) be the Hecke operators corresponding to the double
Γ0(N)-cosets of the symplectic similitude matrices
1 0 0 0
0 1 0 0
0 0 πp 0
0 0 0 πp
1 0 0 0
0 πp 0 0
0 0 π2p 0
0 0 0 πp
respectively. (We remind the reader of the symplectic form J2 fixed at
the beginning of Section 1.) The Hecke algebra Tk(N) is the Z-algebra
generated by the operators T1(p) and T2(p), where p runs over all primes
not dividing N .
1.3. Algebraic Hilbert-Siegel autormorphic forms. We only consider
level structure of Siegel type. Namely, we define the compact open subgroup
U0(N) of G(Q̂) by
U0(N) =
GSp4(OFp )×
ep ),
where N =
p|N p
ep and
ep ) :=
∈ GSp4(OFp )
∣ c ≡ 0 mod pep
The weight representation is defined as follows. Let Lk be the repre-
sentation of GSp4(C) of highest weight (k− 3, k− 3). We let Vk = Lk ⊗Lk
and define the complex representation (ρk, Vk) by
ρk : G
B(R) −→ GL(Vk),
where the action on the first factor is via v0, and the action on the second
one is via v1.
The space of algebraic Hilbert-Siegel modular forms of weight k and
level N is given by
MBk (N) :=
f : GB(Q̂)/U0(N) → Vk
∣ ∀γ ∈ GB(Q), f |kγ = f
6 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
where f |kγ(x) = f(γx)γ, for all x ∈ GB(Q̂)/U0(N). When k = 3, we let
IBk (N) :=
f : GB(Q)\GB(Q̂)/U0(N) → C
∣ f is constant
Then, the space of algebraic Hilbert-Siegel cusp forms of weight k and
level N is defined by
SBk (N) :=
MBk (N) if k > 3,
MBk (N)/I
k (N) if k = 3.
The action of the Hecke algebra on SBk (N) is given as follows. For any
u ∈ G(Q̂), write the finite disjoint union
U0(N)uU0(N) =
uiU0(N),
and define
[U0(N)uU0(N)] : S
k (N) → SBk (N)
f 7→ f |k[U0(N)uU0(N)],
f |k[U0(N)uU0(N)](x) =
f(xui), x ∈ G(Q̂).
For any prime p ∤ N , let ̟p be a local uniformizer at p. The local Hecke alge-
bra at p is generated by the Hecke operators T1(p) and T2(p) corresponding
to the double U0(N)-cosets ∆1(p) and ∆2(p) of the matrices
1 0 0 0
0 1 0 0
0 0 ̟p 0
0 0 0 ̟p
1 0 0 0
0 ̟p 0 0
0 0 ̟2
0 0 0 ̟p
respectively. We let TBk (N) be the Hecke algebra generated by T1(p) and
T2(p) for all primes p ∤ N .
1.4. The Jacquet-Langlands Correspondence. The Hecke modules Sk(N)
and SBk (N) are related by the following conjecture known as the Jacquet-
Langlands Correspondence for symplectic similitude groups.
Conjecture 1. The Hecke algebras Tk(N) and T
k (N) are isomorphic and
there is a compatible isomorphism of Hecke modules
Sk(N)
∼−→ SBk (N).
It is common, but perhaps not entirely accurate, to attribute this con-
jecture to Jacquet-Langlands. To the best of our knowledge, the correspon-
dence in this form was first discussed by Ihara [Ih 1964] in the case F = Q. In
[Ib 1984], Ibukiyama provided some numerical evidence. On the other hand,
it is appropriate to refer to Conjecture 1 as the Jacquet-Langlands Corre-
spondence (for GSp(4)) since it is an analogue of the Jacquet-Langlands
COMPUTING HILBERT-SIEGEL MODULAR FORMS 7
Correspondence (for GL(2)) which relates automorphic representations of
the multiplicative group of a quaternion algebra with certain automorphic
representations of GL(2) (see [JL 1970]). Both correspondences are, in turn,
special consequences of the principle of functoriality, as expounded by Lang-
lands. Finally, it appears that Conjecture 1 may soon be a theorem due to
the work of [So 2008] and the forthcoming book by James Arthur on auto-
morphic representations of classical groups.
2. The Algorithm
In this section, we present the algorithm we used in order to compute
the Hecke module of (algebraic) Hilbert-Siegel modular forms. The main
assumption in this section is that the class number of the principal genus
of GB is 1. (We refer to [D3 2007] to see how one can relax this condition
on the class number.) We recall that since B is totally definite, GB satis-
fies Proposition 1.4 in Gross [Gr 1999]. Thus the group GB(R) is compact
modulo its centre, and Γ = GB(Z)/O×F is finite.
For any prime p in F , let Fp = OF /p be the residue field at p and define
the reduction map
M2(OB, p) → M4(Fp)
g 7→ g̃,
where we use the splitting of OB,p that was fixed at the beginning of Sec-
tion 1. Now, choose a totally positive generator πp of p and put
Θ1(p) := Γ\
u ∈ M2(OB)
∣ uūt = πp12and rank(g̃) = 2
Θ2(p) := Γ\
u ∈ M2(OB)
∣ uūt = π2
12 and rank(g̃) = 1
We let H20(N) = G(Ẑ)/U0(N). Then the group Γ acts on H20(N), thus on
the space of functions f : H20(N) → Vk by
∀x ∈ H20(N),∀γ ∈ Γ, f |kγ(x) := f(γx)γ.
Theorem 2. There is an isomorphism of Hecke modules
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
where the Hecke action on the right hand side is given by
f |kT1(p) =
u∈Θ1(p)
f |ku,
f |kT2(p) =
u∈Θ2(p)
f |ku.
Proof. The canonical map
φ : GB(Z)\GB(Ẑ)/U0(N) → GB(Q)\GB(Q̂)/U0(N)
is an injection. Making use of the fact that the class number in the principal
genus of GB is one (GB(Q̂) = GB(Q)GBZ (Ẑ)), we see that φ is in fact a
8 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
bijection. Since each element f ∈ MBk (N) is determined by its values on a
set of coset representatives of GB(Q)\GB(Q̂)/U0(N), the map φ induces an
isomorphism of complex vector spaces
MBk (N)
f : H20(N) → Vk
∣ f |kγ = f, γ ∈ Γ
f 7−→ f ◦ φ.
We make this into a Hecke module isomorphism by defining the Hecke action
on the right hand side as indicated in the statement of the theorem. �
In the rest of this section, we explain the main steps of the algorithm
provided by Theorem 2.
2.1. The quotient H20(N). Keeping the notations of the previous section,
we recall that N =
p|N p
ep . Let p be a prime dividing N and consider the
rank 4 free
OFp/pep
-module L =
OFp/pep
endowed with the symplectic
pairing 〈 , 〉 given by the matrix
−12 0
where 12 is the identity matrix in M2(OFp/pep ). Let M be a rank 2
OFp/pep
-submodule which is a direct factor in L. We say that M is
isotropic if 〈u, v〉 = 0 for all u, v ∈ M . We recall that GSp4(OFp ) acts
transitively on the set of rank 2, isotropic
OFp/pep
-submodules of L and
that the stabilizer of the submodule generated by e1 = (1, 0, 0, 0)
T and
e2 = (0, 1, 0, 0)
T is U0(p
ep ). The quotient H20(pep ) = GSp4(OFp )/U0(pep )
is the set of rank 2, isotropic
OFp/pep
-submodules of L. Via the reduction
map ÔF → OF /N , the quotient GZ(Ẑ)/U0(N) can be identified with the
product
H20(N) =
H20(pep ).
The cardinality of H20(N) is extremely useful and is determined using the
following lemma.
Lemma 1. Let p be a prime in F and ep ≥ 1 an integer. Then, the cardi-
nality of the set H20(pep ) is given by
#H20(pep ) = N(p)3(ep−1)(N(p) + 1)(N(p)2 + 1).
Proof. For ep = 1, the cardinality of the Lagrange variety over the finite
field Fp = OF /p is given by (N(p) + 1)(N(p)2 + 1). Proceed by induction
on ep. �
We have more to say about elements of H20(pep ) in Subsection 2.5.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 9
2.2. Brandt matrices. Let F = {x1, . . . , xh} be a fundamental domain
for the action of Γ on H20(N) and, for each i, let Γi be the stabilizer of xi.
Then, every element in MBk (N) is completely determined by its values on
F . Thus, there is an isomorphism of complex spaces
MBk (N) →
f 7→ (f(xi)),
where V
is the subspace of Γi-invariants in Vk.
For any x, y ∈ H20(N), we let
Θ1(x, y, p) :=
u ∈ Θ1(p)
∣ ∃γ ∈ Γ, ux = γy
Θ2(x, y, p) :=
u ∈ Θ2(p)
∣ ∃γ ∈ Γ, ux = γy
Proposition 3. The actions of the Hecke operators Ts(p), s = 1, 2, are
given by the Brandt matrices Bs(p) = (bsij(p)), where
bsji(p) : V
k → V
v 7→ v ·
u∈Θs(xi, xj ,p)
γ−1u u
Proof. The proof of Proposition 3 follows the lines of [D1 2005, §3]. �
2.3. Computing the group GB(Z). It is enough to compute the subgroup
Γ consisting of the elements in GB(Z) with similitude factor 1. But it is easy
to see that
u, v ∈ O1B
u, v ∈ O1B
where O1B is the group of norm 1 elements.
2.4. Computing the sets Θ1(p) and Θ2(p). Let us consider the quadratic
form on the vector space V = B2 given by
V → F
(a, b) 7→ ||(a, b)|| := nr(a) + nr(b),
where nr is the reduced norm on B. This determines an inner form
V × V → F
(u, v) 7→ 〈u, v〉.
An element of Θ1(p) (resp. Θ2(p)) is a unitary matrix γ ∈ M2(OB) with
respect to this inner form such that the norm of each row is πp (resp. π
and the rank of the reduced matrix is 1). So we first start by computing
all the vectors u = (a, b) ∈ O2B such that ||u|| = πp (resp. ||u|| = π2p). And
for each such vector u, we compute the vectors v = (c, d) ∈ O2B of the same
10 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
norm such that 〈u, v〉 = 0. The corresponding matrix γ =
belongs
to Θ1(p) (resp. Θ2(p)) when its reduction mod p has the appropriate rank.
We list all these matrices up to equivalence and stop when we reach the
right cardinality.
2.5. The implementation of the algorithm. The implementation of the
algorithm is similar to that of [D1 2005]. However, it is important to note
how we represent elements in H20(N) so that we can retrieve them easily
once stored. As in [D1 2005] we choose to work with the product
H20(N) =
H20(pep ).
Using Plucker’s coordinates, we can view H20(pep ) as a closed subspace of
P5(OFp/pep ). We then represent each element in H20(pep ) by choosing a
point x = (a0 : · · · : a5) = [u ∧ v] ∈ P5(OFp/pep ) such that the submodule
M generated by u and v is a Lagrange submodule, and the first invertible
coordinate is scaled to 1.
Remark 1. In [LP 2002], Lansky and Pollack describe an algorithm which
computes algebraic modular forms on the same inner form of GSp4/Q that
we use. We would like to note that there are some differences between the
two algorithms. Although [LP 2002] also uses the flag variety H20(N) in
order to determine the double coset space GB(Q)\GB(Q̂)/U0(N), it later
returns to the adelic setting in order to compute the Brandt matrices. In
contrast, Theorem 2 and Proposition 3 allow us to avoid that unnecessary
step by describing the Hecke action on the flag variety H20(N) directly. As
a result, we get an algorithm that is more efficient.
3. Numerical examples: F = Q(
5) and B =
−1,−1
In this section, we provide some numerical examples using the quadratic
field F = Q(
5). It is proven in K. Hashimoto and T. Ibukiyama [HI 1980]
that, for the Hamilton quaternion algebra B over F , the class number of
the principal genus of GB is one. We use our algorithm to compute all the
systems of Hecke eigenvalues of Hilbert-Siegel cusp forms of weight 3 and
level N that are defined over real quadratic fields, where N runs over all
prime ideals of norm less than 50. We then determine which of the forms
we obtained are possible lifts of Hilbert cusp forms by comparing the Hecke
eigenvalues for those primes.
3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3. In
Table 1 we list all the systems of eigenvalues of Hilbert-Siegel cusp forms of
weight 3 and level N that are defined over real quadratic fields, where N
runs over all prime ideals in F of norm less than 50. Here are the conventions
we use in the tables.
COMPUTING HILBERT-SIEGEL MODULAR FORMS 11
(1) For a quadratic field K of discriminant D, we let ωD be a generator
of the ring of integers OK of K.
(2) The first row contains the level N , given in the format (Norm(N), α)
for some generator α ∈ F of N , and the dimensions of the relevant
spaces.
(3) The second row lists the Hecke operators that have been computed.
(4) For each eigenform f , the Hecke eigenvalues are given in a row, and
the last entry of that row indicates if the form f is a probable lift.
(5) The levels and the eigenforms are both listed up to Galois conjuga-
tion.
For an eigenform f and a given prime p ∤ N , let a1(p, f) and a2(p, f) be the
eigenvalues of the Hecke operators T1(p) and T2(p), respectively. Then the
Euler factor Lp(f, s) is given (for example, in [AS 2001, §3.4]) by
Lp(f, s) = Qp(q
−s)−1,
where
Qp(x) = 1− a1(p, f)x+ b1(p, f)x2 − a1(p, f)q2k−3x3 + q4k−6x4,
b1(p, f) = a1(p, f)
2 − a2(p, f)− q2k−4,
q = N(p).
3.2. Tables of Hilbert cusp forms of parellel weight 4. In Table 2, we
list all the Hilbert cusp forms of parallel weight 4 and level N that are defined
over real quadratic fields, with N running over all prime ideals of norm less
than 50. (They are computed by using the algorithm in [D1 2005]). We use
this data in order to determine the forms in Table 1 that are possible lifts
from GL2.
3.3. Lifts. There are two types of lifts from GL2 to GSp4. The first one
corresponds to the homomorphism of L-groups determined by the long root
embedding into GSp4, and the second one by the short root embedding.
(See [LP 2002] for more details). Let f be a Hilbert cusp form of parallel
weight k and level N with Hecke eigenvalues a(p, f), where p is a prime not
dividing N . Let φ be the lift of f to GSp4 via the long root, and ψ the one
via the short root. Then the Hecke eigenvalues of φ are given by
a1(p, φ) = a(p, f) N(p)
2 +N(p)2 +N(p)
a2(p, φ) = a(p, f) N(p)
2 (N(p) + 1) +N(p)2 − 1,
and the Hecke eigenvalues of ψ are given by
a1(p, ψ) = a(p, f)
2 − 2 a(p, f) N(p)
a2(p, ψ) = a(p, f)
N(p)4−2k − 3 a(p, f)2 N(p)3−k +N(p)2 − 1.
The second lift ψ is the so-called symmetric cube lifting.
12 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
N = (4, 2) : dimMB
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 0 20 −36 140 580 yes
N = (5, 2 + ω5) : dimM
(N) = 2, dimSB
(N) = 1
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 20 15 −5 0 40 −420 yes
N = (9, 3) : dimMB
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 25− 3ω41 40− 15ω41 30 + 6ω41 24 + 36ω41 −9 0 yes
N = (11, 3 + ω5) : dimM
(N) = 3, dimSB
(N) = 2
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 24 35 34 48 88 60 yes
f2 −20 35 −10 4 0 60 no
N = (19, 4 + ω5) : dimM
(N) = 5, dimSB
(N) = 4
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 4 11 −20 28 6 76 no
f2 7 −50 15 −66 73 −90 yes
f3 24 + ω161 35 + 5ω161 36− ω161 60− 6ω161 98− 3ω161 160− 30ω161 yes
N = (29, 5 + ω5) : dimM
(N) = 9, dimSB
(N) = 8
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 −4 11 10 20 30 60 no
f2 8 −45 30 24 50 −320 yes
f3 17 0 9 −102 86 40 yes
N = (31, 5 + 2ω5) : dimM
(N) = 12, dimSB
(N) = 11
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 13 −20 20 −36 76 −60 yes
N = (41, 6 + ω5) : dimM
(N) = 19, dimSB
(N) = 18
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 10 20 −10 29 30 −20 no
f2 −1 1 5 14 −2 −56 no
f3 27 50 40 84 124 420 yes
f4 −12 19 30 65 0 0 no
f5 16− 2ω21 −5− 10ω21 21 + 4ω21 −30 + 24ω21 72− 2ω21 −100− 20ω21 yes
f6 2− 6ω5 11− 2ω5 8 + 4ω5 11− 4ω5 −12 + 54ω5 160 + 40ω5 no
N = (49, 7) : dimMB
(N) = 26, dimSB
(N) = 25
T1(2) T2(2) T1(
5) T2(
5) T1(3) T2(3) Lift?
f1 5 −60 46 120 40 −420 yes
f2 4 + 4ω65 32 + 3ω65 12− 4ω65 44− 4ω65 −6− 12ω65 145 + 8ω65 no
Table 1. Hilbert-Siegel eigenforms of weight 3
COMPUTING HILBERT-SIEGEL MODULAR FORMS 13
N (4, 2) (5, 2 + ω5) (9, 3) (11, 3 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f1) a(p, f1)
4 2 −4 0 5− 3ω41 4
5 2 + ω5 −10 −5 6ω41 4
9 3 50 −50 −9 −2
11 3 + 2ω5 −28 32 −18− 6ω41 −10
11 3 + ω5 −28 32 −18− 6ω41 −11
19 4 + 3ω5 60 100 −40 + 24ω41 −94
19 4 + ω5 60 100 −40 + 24ω41 28
N (19, 4 + ω5) (29, 5 + ω5)
N(p) p a(p, f1) a(p, f2) a(p, f1) a(p, f2)
4 2 −13 5− ω161 −12 −3
5 2 + ω5 −15 5 + ω161 0 −21
9 3 −17 5 + 3ω161 −40 −4
11 3 + 2ω5 −6 2 + 8ω161 −68 37
11 3 + ω5 33 7− 7ω161 30 −66
19 4 + 3ω5 −139 −15− 9ω161 −28 −40
19 4 + ω5 19 −19 84 −9
N (31, 5 + 2ω5) (41, 6 + ω5)
N(p) p a(p, f1) a(p, f1) a(p, f2)
4 2 −7 7 −4− 2ω21
5 2 + ω5 −10 10 −9 + 4ω21
9 3 −14 34 −18− 2ω21
11 3 + 2ω5 −20 −60 −19
11 3 + ω5 −28 −2 −24− 4ω21
19 4 + 3ω5 −12 74 4− 50ω21
19 4 + ω5 28 16 −29 + 44ω21
N (49, 7)
N(p) p a(p, f1) a(p, f2)
4 2 −15 −2
5 2 + ω5 16 −10
9 3 −50 −11
11 3 + 2ω5 −8 −7− 28ω13
11 3 + ω5 −8 −35 + 28ω13
19 4 + 3ω5 −110 −26 + 14ω13
19 4 + ω5 −110 −12− 14ω13
Table 2. Hilbert eigenforms of weight 4
Remark 2. So far, our algorithm has been implemented only for congruence
subgroups of Siegel type. We intend to improve the implementation in the
near future so as to include more additional level structures such as the
Klingen type. Indeed, Ramakrishnan and Shahidi [RS 2007] recently showed
the existence of symmetric cube lifts for non-CM elliptic curves E/Q to
GSp4/Q. And their result should hold for other totally real number fields,
with the level structures of the lifts being of Klingen type. Unfortunately,
14 CLIFTON CUNNINGHAM AND LASSINA DEMBÉLÉ
those lifts cannot be seen in our current tables. For example, there are
modular elliptic curves over Q(
5) whose conductors have norm 31, 41 and
49, but the corresponding symmetric cubic lifts do not appear in Table 1.
We would like to remedy that in our next implementation.
References
[D1 2005] L. Dembélé, Explicit computations of Hilbert modular forms on Q(
5). Exper-
iment. Math. 14 (2005), no. 4, 457–466.
[D2 2007] L. Dembélé, Quaternionic M -symbols, Brandt matrices and Hilbert modular
forms. Math. Comp. 76, no 258, (2007), 1039-1057. Also available electronically.
[D3 2007] L. Dembélé, On the computation of algebraic modular forms (submitted).
[AS 2001] Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations,
Manuscripta Math. 104 (2001), 173–200.
[FvdG1 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
I, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 381–384.
[FvdG2 2004] Carel Faber and Gerard van der Geer, Sur la cohomologie des systèmes
locaux sur les espaces de modules des courbes de genre 2 et des surfaces abéliennes.
II, C. R. Math. Acad. Sci. Paris 338 (2004), no. 6, 467–470.
[JL 1970] Hervé Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture
notes in mathematics 114 and 278, 1970.
[Gr 1999] Benedict H. Gross, Algebraic modular forms. Israel J. Math. 113 (1999), 61–93.
[Gu 2000] P. Gunnells, Symplectic modular symbols, Duke Math. J. 102 (2000), no. 2,
329-350.
[HI 1980] K. Hashimoto and T. Ibukiyama, On the class numbers of positive definite
binary quaternion hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980),
549-601.
[Ib 1984] T. Ibukiyama, On symplectic Euler factors of genus 2. J. Fac. Sci. Univ. Tokyo
30 (1984), 587614.
[Ih 1964] Y. Ihara, On certain Dirichlet series, J. Math. Soc. Japan 16 (1964), 214-225.
[LP 2002] J. Lansky and D. Pollack, Hecke algebras and automorphic forms. Compositio
Math. 130 (2002), no. 1, 21–48.
[RS 2007] Dinakar Ramakrishnan and Freydoon Shahidi, Siegel modular forms of genus
2 attached to elliptic curves (preprint). Available at www.math.arxiv.
[R 2006] N. C. Ryan, Computing the Satake p-parameters of Siegel modular forms. (sub-
mitted).
[Sk 1992] Nils-Peter Skoruppa, Computations of Siegel modular forms of genus two. Math.
Comp. 58 (1992), no. 197, 381–398.
[So 2008] Claus M. Sorensen, Potential level-lowering for GSp(4), arXive:0804.0588v1.
Department of Mathematics, University of Calgary
E-mail address: [email protected]
Institut für Experimentelle Mathematik, Universität Duisburg-Essen
E-mail address: [email protected]
Introduction
1. Hilbert-Siegel modular forms and the Jacquet-Langlands correspondence
1.1. Hilbert-Siegel modular forms
1.2. The Hecke algebra
1.3. Algebraic Hilbert-Siegel autormorphic forms
1.4. The Jacquet-Langlands Correspondence
2. The Algorithm
2.1. The quotient H02(N)
2.2. Brandt matrices
2.3. Computing the group GB(Z)
2.4. Computing the sets 1(p) and 2(p)
2.5. The implementation of the algorithm
3. Numerical examples: F=Q(5) and B=(-1,-1F)
3.1. Tables of Hilbert-Siegel cusp forms of parallel weight 3
3.2. Tables of Hilbert cusp forms of parellel weight 4
3.3. Lifts
References
|
0704.0012 | Distribution of integral Fourier Coefficients of a Modular Form of Half
Integral Weight Modulo Primes | DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS OF A
MODULAR FORM OF HALF INTEGRAL WEIGHT MODULO
PRIMES
D. CHOI
Abstract. Recently, Bruinier and Ono classified cusp forms f(z) :=
af (n)q
Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]] that does not satisfy a certain distribution property for modulo
odd primes p. In this paper, using Rankin-Cohen Bracket, we extend this result to
modular forms of half integral weight for primes p ≥ 5. As applications of our main
theorem we derive distribution properties, for modulo primes p ≥ 5, of traces of singular
moduli and Hurwitz class number. We also study an analogue of Newman’s conjecture
for overpartitions.
1. Introduction and Results
Let Mλ+ 1
(Γ0(N), χ) and Sλ+ 1
(Γ0(N), χ) be the spaces, respectively, of modular forms
and cusp forms of weight λ + 1
on Γ0(N) with a Dirichlet character χ whose conductor
divides N . If f(z) ∈Mλ+ 1
(Γ0(N), χ), then f(z) has the form
f(z) =
a(n)qn,
where q := e2πiz. It is well-known that the coefficients of f are related to interesting
objects in number theory such as the special values of L-function, class number, traces of
singular moduli and so on. In this paper, we study congruence properties of the Fourier
coefficients of f(z) ∈Mλ+ 1
(Γ0(N), χ) ∩ Z[[q]] and their applications.
Recently, Bruinier and Ono proved in [3] that g(z) ∈ Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]] has a
special form (see (2.1)) by modulo p when p is an odd prime and the coefficients of f(z)
do not satisfy the following property for p:
Property A. IfM is a positive integer, we say that a sequence α(n) ∈ Z satisfies Property
A for M if for every integer r
♯{ 1 ≤ n ≤ X | α(n) ≡ r (mod M) and gcd(M,n) = 1}
if r 6≡ 0 (mod M),
X if r ≡ 0 (mod M).
2000 Mathematics Subject Classification. 11F11,11F33.
Key words and phrases. Modular forms, Congruences.
http://arxiv.org/abs/0704.0012v1
2 D. CHOI
θ(f(z)) :=
f(z) =
n · a(n)qn.
Using Rankin-Cohen Bracket (see (2.3)), we prove that there exists
f̃(z) ∈ Sλ+p+1+ 1
(Γ0(4N), χ) ∩ Z[[q]]
such that θ(f(z)) ≡ f̃(z) (mod p). We extend the results in [3] to modular forms of half
integral weight.
Theorem 1. Let λ be a non-negative integer. We assume that f(z) =
n=0 a(n)q
Mλ+ 1
(Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p ≥ 5 is a prime and
there exists a positive integer n for which gcd(a(n), p) = 1 and gcd(n, p) = 1, then at least
one of the following is true:
(1) The coefficients of θp−1(f(z)) satisfies Property A for p.
(2) There are finitely many square-free integers n1, n2, · · · , nt for which
(1.1) θp−1(f(z)) ≡
a(nim
2)qnim
(mod p).
Moreover, if gcd(4N, p) = 1 and an odd prime ℓ divides some ni, then
p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N.
Remark 1.1. Note that for every odd prime p ≥ 5,
θp−1(f(z)) ≡
a(n)qn (mod p).
As an applications of Theorem 1, we study the distribution of traces of singular moduli
modulo primes p ≥ 5. Let j(z) be the usual j-invariant function. We denote by Fd the
set of positive definite binary quadratic forms
F (x, y) = ax2 + bxy + cy2 = [a, b, c]
with discriminant −d = b2−4ac. For each F (x, y), let αF be the unique complex number
in the complex upper half plane, which is a root of F (x, 1). We define ωF ∈ {1, 2, 3} as
ωF :=
2 if F ∼Γ [a, 0, a],
3 if F ∼Γ [a, a, a],
1 otherwise,
where Γ := SL2(Z). Here, F ∼Γ [a, b, c] denotes that F (x, y) is equivalent to [a, b, c].
From these notations, we define the Hecke trace of singular moduli.
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 3
Definition 1.2. If m ≥ 1, then we define the mth Hecke trace of the singular moduli of
discriminant −d as
tm(d) :=
F∈Fd/Γ
jm(αF )
where Fd/Γ denotes a set of Γ−equivalence classes of Fd and
jm(z) := j(z)|T0(m) =
az + b
Here, T0(m) denotes the normalized mth weight zero Hecke operator.
Note that t1(d) = t(d), where
t(d) :=
F∈Fd/Γ
j(αF )− 744
is the usual trace of singular moduli. Let
h(z) :=
η(z)2
η(2z)
· E4(4z)
η(4z)6
and Bm(1, d) denote the coefficient of q
d in h(z)|T (m2, 1, χ0), where
E4(z) := 1 + 240
d3qn, η(z) := q
(1− qn) ,
and χ0 is a trivial character. Here, T (m
2, λ, χ) denotes the mth Hecke operator of weight
λ + 1
with a Dirichlet chracter χ (see VI. §3. in [5] or (2.5)). Zagier proved in [11] that
for all m and d
(1.2) tm(d) = −Bm(1, d).
Using these generating functions, Ahlgren and Ono studied the divisibility properties
of traces and Hecke traces of singular moduli in terms of the factorization of primes in
imaginary quadratic fields (see [2]). For example, they proved that a positive proportion
of the primes ℓ has the property that tm(ℓ
3n) ≡ 0 (mod ps) for every positive integer n
coprime to ℓ such that p is inert or ramified in Q
. Here, p is an odd prime, and
s and m are integers with p ∤ m. In the following theorem, we give the distribution of
traces and Hecke traces of singular moduli modulo primes p.
4 D. CHOI
Theorem 2. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3).
(1) Then, for every integer r, p ∤ r,
♯{ 1 ≤ n ≤ X | t1(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p)
X if r ≡ 0 (mod p).
(2) Then, a positive proportion of the primes ℓ has the property that
♯{ 1 ≤ n ≤ X | tℓ(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p)
X if r ≡ 0 (mod p).
for every integer r, p ∤ r.
As another application we study the distribution of Hurwitz class number modulo
primes p ≥ 5. The Hurwitz class number H(−N) is defined as follows: the class number
of quadratic forms of the discriminant −N where each class C is counted with multiplicity
Aut(C)
. The following theorem gives the distribution of Hurwitz class number modulo
primes p ≥ 5.
Theorem 3. Suppose that p ≥ 5 is a prime. Then, for every integer r
♯{ 1 ≤ n ≤ X | H(n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
We also use the main theorem to study an analogue of Newman’s conjecture for overpar-
titions. Newman’s conjecture concerns the distribution of the ordinary partition function
modulo primes p.
Newman’s Conjecture. Let P (n) be an ordinary partition function. If M is a positive
integer, then for every integer r there are infinitely many nonnegative integer n for which
P (n) ≡ r (mod M).
This conjecture was already studied by many mathematicians (see Chapter 5. in [8]).
The overpartition of a natural number n is a partition of n in which the first occurrence
of a number may be overlined. Let P̄ (n) be the number of the overpartition of an integer
n. As an analogue of Newman’s conjecture, the following theorem gives a distribution
property of P̄ (n) modulo odd primes p.
Theorem 4. Suppose that p ≥ 5 is a prime such that p ≡ 2 (mod 3). Then, for every
integer r,
♯{ 1 ≤ n ≤ X | P̄ (n) ≡ r (mod p)} ≫r,p
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
Remark 1.3. When r ≡ 0 (mod p), Theorem 2, 3 and 4 were proved in [2] and [10].
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 5
Next sections are detailed proofs of theorems: Section 2 gives a proof of Theorem 1. In
Section 3, we give the proofs of Theorem 2, 3, and 4.
2. Proof of Theorem 1
We begin by stating the following theorem proved in [3].
Theorem 2.1 ([3]). Let λ be a non-negative integer. Suppose that g(z) =
n=0 ag(n)q
Sλ+ 1
(Γ0(4N), χ) ∩ Z[[q]], where χ is a real Dirichlet character. If p is an odd prime and
a positive integer n exists for which gcd(ag(n), p) = 1, then at least one of the following
is true:
(1) If 0 ≤ r < p, then
♯{ 1 ≤ n ≤ X | ag(n) ≡ r (mod p)} ≫r,M
if r 6≡ 0 (mod p),
X if r ≡ 0 (mod p).
(2) There are finitely many square-free integers n1, n2, · · · , nt for which
(2.1) g(z) ≡
ag(nim
2)qnim
(mod p).
Moreover, if gcd(p, 4N) = 1, ǫ ∈ {±1}, and ℓ ∤ 4Np is a prime with
∈ {0, ǫ}
for 1 ≤ i ≤ t, then (ℓ−1)g(z) is an eigenform modulo p of the half-integral weight
Hecke operator T (ℓ2, λ, χ). In particular, we have
(2.2) (ℓ− 1)g(z)|T (ℓ2, λ, χ) ≡ ǫχ(p)
(−1)λ
ℓλ + ℓλ−1
(ℓ− 1)g(z) (mod p).
Recall that f(z) =
a(n)qn ∈ Mλ+ 1
(Γ0(4N), χ) ∩ Z[[q]]. Thus, to apply Theorem
2.1, we show that there exists a cusp form f̃(z) such that f̃(z) ≡ θp−1(f(z)) (mod p) for
a prime p ≥ 5.
Lemma 2.2. Suppose that p ≥ 5 is a prime and
f(z) =
a(n)qn ∈Mλ+ 1
(Γ0(N), χ) ∩ Z[[q]].
Then, there exists a cusp form f̃(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(N), χ) ∩ Z[[q]] such that
f̃(z) ≡ θp−1(f(z)) (mod p).
Proof of Lemma 2.2. For F (z) ∈Mk1
(Γ0(N), χ1) and G(z) ∈Mk2
(Γ0(N), χ2), let
(2.3) [F (z), G(z)]1 :=
θ(F (z)) ·G(z)−
F (z) · θ(G(z)).
This operator is referred to as a Rankin-Cohen 1-bracket, and it was proved in [4] that
[F (z), G(z)]1 ∈ S k1+k2
(Γ0(N), χ1χ2χ
6 D. CHOI
where χ′ = 1 if k1
and k2
∈ Z, χ′(d) =
2 if ki
∈ Z and k3−i
+ Z, and
χ′(d) =
) k1+k2
2 if k1
and k2
For even k ≥ 4, let
Ek(z) := 1−
dk−1qn
be the usual normalized Eisenstein series of weight k. Here, the number Bk denotes the
kth Bernoulli number. The function Ek(z) is a modular form of weight k on SL2(Z), and
(2.4) Ep−1(z) ≡ 1 (mod p)
(see [6]). From (2.3) and (2.4), we have
[Ep−1(z), f(z)]1 ≡ θ(f(z)) (mod p)
and [Ep−1(z), f(z)]1 ∈ Sλ+p+1+ 1
(Γ0(N), χ). Repeating this method p− 1 times, we com-
plete the proof. �
Using the following lemma, we can deal with the divisibility of ag(n) for positive integers
n, p ∤ n, where g(z) =
n=1 ag(n)q
n ∈ Sλ+ 1
(Γ0(N), χ) ∩ Z[[q]].
Lemma 2.3 (see Chapter 3 in [8]). Suppose that g(z) =
n=1 ag(n)q
n ∈ Sλ+ 1
(Γ0(N), χ)
has coefficients in OK , the algebraic integers of some number field K. Furthermore,
suppose that λ ≥ 1 and that m ⊂ OK is an ideal norm M .
(1) Then, a positive proportion of the primes Q ≡ −1 (mod 4MN) has the property
g(z)|T (Q2, λ, χ) ≡ 0 (mod m).
(2) Then a positive proportion of the primes Q ≡ 1 (mod 4MN) has the property that
g(z)|T (Q2, λ, χ) ≡ 2g(z) (mod m).
We can now prove Theorem 1.
Proof of Theorem 1. From Lemma 2.2, there exists a cusp form
f̃(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(N), χ) ∩ Z[[q]]
such that
f̃(z) ≡ θp−1(f(z)) (mod p).
Note that, for F (z) =
n=0 aF (n)q
n ∈ Mk+ 1
(Γ0(N), χ) and each prime Q ∤ N , the
half-integral weight Hecke operator T (Q2, λ, χ) is defined as
(2.5)
F (z)|T (Q2, k, χ)
aF (Q
2n) + χ∗(Q)
Qk−1aF (n) + χ
∗(Q2)Q2k−1aF (n/Q
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 7
where χ∗(n) := χ∗(n)
(−1)k
and aF (n/Q
2) = 0 if Q2 ∤ n. If F (z)|T (Q2, k, χ) ≡ 0
(mod p) for a prime Q ∤ N , then we have
aF (Q
2 ·Qn) + χ∗(Q)
Qk−1aF (Qn) + χ
∗(Q2)Q2k−1aF
Qn/Q2
≡ aF (Q3n) ≡ 0 (mod p)
for every positive integer n such that gcd(Q, n) = 1. Thus, we have the following by
Lemma 2.3-(1):
♯{ 1 ≤ n ≤ X | a(n) ≡ 0 (mod p) and gcd(p, n) = 1} ≫ X.
We apply Theorem 2.1 with f̃(z). Then the purpose of the remaining part of the proof
is to show the following: if gcd(p, 4N) = 1, an odd prime ℓ divides some ni, and
(2.6) θp−1(f(z)) ≡
a(nim
2)qnim
(mod p),
then p|(ℓ− 1)ℓ(ℓ+ 1)N or ℓ | N . We assume that there exists a prime ℓ1 such that ℓ1|n1,
p ∤ (ℓ1 − 1)ℓ1(ℓ1 + 1)N and ℓ | N . We also assume that nt = 1 and that ni ∤ n1 for every
i, 2 ≤ i ≤ t − 1. Then, we can take a prime ℓi for each i, 2 ≤ i ≤ t − 1, such that ℓi|ni
and ℓi ∤ n1. For convention, we define
(−1)(n−1)2/8 if n is odd,
0 otherwise,
and χQ(d) :=
for a prime Q. Let ψ(d) :=
i=2 χℓi(d). We take a prime β such that
ψ(n1)χβ(n1) = −1. If we denote the ψ-twist of f̃(z) by f̃ψ(z) and the ψχβ-twist of f̃(z)
by f̃ψχβ(z), then
f̃ψχ2
(z)− f̃ψχβ(z) ≡ 2
gcd(m,β
ℓj)=1
a(n1m
2)qn1m
(mod p)
and f̃ψχβ(z) ∈ Sλ+(p+1)(p−1)+ 1
(Γ0(Nα
2β2), χ) ∩ Z[[q]] (see Chapter 3 in [8]). Note that
gcd(Nα2β2, p) = gcd(Nα2β2, ℓ1) = 1.
Thus, (f̃ψ(z)− f̃ψχβ(z))|T (ℓ21, λ+ (p+ 1)(p− 1), χ) satisfies the formula (2.2) of Theorem
2.1 for both of ǫ = 1 and ǫ = −1. This results in a contradiction since
(f̃ψ(z)− f̃ψχβ(z))|T (ℓ
1, λ+ (p+ 1)(p− 1), χ) 6≡ 0 (mod p)
and p ≥ 5. Thus, we complete the proof. �
8 D. CHOI
3. Proofs of Theorem 2, 3, and 4
3.1. Proof of Theorem 2. Note that h(z) =
η(z)2
η(2z)
·E4(4z)
η(4z)6
is a meromorphic modular form.
In [2] it was obtained a holomorphic modular form on Γ0(4p
2) whose Fourier coefficients
generate traces of singular moduli modulo p (see the formula (3.1) and (3.2)). Since the
level of this modular form is not relatively prime to p, we need the following proposition.
Proposition 3.1 ([1]). Suppose that p ≥ 5 is a prime. Also, suppose that p ∤ N , j ≥ 1 is
an integer, and
g(z) =
a(n)qn ∈ Sλ+ 1
(Γ0(Np
j)) ∩ Z[[q]].
Then, there exists a cusp form G(z) ∈ Sλ′+ 1
(Γ0(N)) ∩ Z[[q]] such that
G(z) ≡ g(z) (mod p),
where λ′ + 1
= (λ+ 1
)pj + pe(p− 1) for a sufficiently large e ∈ N.
Using Theorem 1 and Proposition 3.1, we give the proof of Theorem 2.
Proof of Theorem 2. Let
(3.1) h1,p(z) := h(z)−
hχp(z),
where hχp(z) is the χp-twist of h(z). From (1.2), we have
h1,p(z) := −2 −
0<d≡0,3 (mod 4)
t1(d)q
d − 2
0<d≡0,3 (mod 4)
(−dp )=−1
t1(d)q
hm,p(z) := h1,p(z)|T (m2, 1, χ0)
= −2 −
0<d≡0,3 (mod 4)
tm(d)q
d − 2
0<d≡0,3 (mod 4)
(−dp )=−1
tm(d)q
for every positive integer m. Let
Fp(z) :=
η(4z)p
η(4pz)
It was proved in [2] that if α is a sufficiently large positive integer, then h1,p(z)Fp(z)
(Γ0(4p
2)) and
(3.2) h1,p(z)Fp(z)
α ≡ h1,p(z) (mod p),
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 9
where k0 = α · p
. Lemma 2.2 and Proposition 3.1 imply that there exists f1,p(z) ∈
Sλ′+ 1
(Γ0(4)) ∩ Z[[q]] such that
f1,p(z) ≡ −2
0<d≡0,3 (mod 4)
(−dp )=−1
tm(d)q
d (mod p),
where λ′ = (k0 + 1 + (p− 1)(p+ 1) + 12)p
2 + pe(p− 1) for a sufficiently large e ∈ N.
We assume that the coefficients of f1,p(z) do not satisfy Property A for an odd prime
p ≡ 2 (mod 3). Note that
= −1 and that p ∤ (3−1)3(3+1). So, Theorem 1 implies
2t1(3) ≡ 0 (mod p).
This results in a contradiction since 2t1(3) = 2
4 ·31. Thus, we obtain a proof when m = 1.
For every odd prime ℓ, we have
f1,p(z)|T (ℓ2, λ′, χ0) ≡ θp−1(h1,p(z))|T (ℓ2, λ′, χ0)
≡ θp−1(h1,p(z)|T (ℓ2, 1, χ0)) ≡ θp−1(hℓ,p(z)) (mod p).
Moreover, Lemma 2.3 implies that a positive proportion of the primes ℓ satisfies the
property
f1,p(z)|T (ℓ2, λ′, χ0) ≡ 2f1,p (mod p).
This completes the proof. �
3.2. Proofs of Theorem 3. The following theorem gives the formula for the Hurwitz
class number in terms of the Fourier coefficients of a modular form of half integral weight.
Theorem 3.2. Let T (z) := 1 + 2
n=1 q
n2. If integers r3(n) are defined as
r3(n)q
n := T (z)3,
r(n) =
12H(−4n) if n ≡ 1, 2 (mod 4),
24H(−n) if n ≡ 3 (mod 8),
r(n/4) if n ≡ 0 (mod 4),
0 if n ≡ 7 (mod 8).
Note that T (z) is a half integral weight modular form of weight 1
on Γ0(4). Combining
Theorem 1 and Theorem 3.2, we derive the proof of Theorem 3.
Proof of Theorem 3. Let G(z) be the
-twist of T (z)3. Then, from Theorem 3.2, we
G(z) = 1 +
n≡1 (mod 4)
12H(−4n)qn +
n≡3 (mod 8)
24H(−n)qn
10 D. CHOI
and G(z) ∈ M 3
(Γ0(16)). Note that 24H(−3) = 8. This gives the complete proof by
Theorem 1. �
3.3. Proofs of Theorem 4. In the following, we prove Theorem 4.
Proof of Theorem 4. Let
W (z) :=
η(2z)
η(z)2
It is known that
W (z) =
P̄ (n)qn
and that W (z) is a weakly holomorphic modular form on Γ0(16). Let
G(z) :=
W (z)−
Wχp(z)
Fp(z)
where Fp(z) =
η(4z)p
η(4p2z)
and β are positive integers. Then we have
G(z) ≡ 2
(−np )=−1
P̄ (n)qn +
P̄ (n)qn (mod p).
We claim that there exists a positive integer β such that G(z) is a holomorphic modular
form of half integral weight on Γ0(16p
2). To prove our claim, we follow the arguments
of Ahlgren and Ono ([1], Lemma 4.2). Note that, by a well-known criterion, Fp(z) is a
holomorphic modular form on Γ0(4p
2) that vanishes at each cusp a
∈ Q for which p2 ∤ c
(see [7]). This implies that G(z) is a weakly holomorphic modular form on Γ0(16p
2). If β
is sufficiently large, then G(z) is holomorphic except at each cusp a
for which p2|c′.
Thus, we prove that G(z) is holomorphic at 1
for 0 ≤ m ≤ 3. Let, for odd d,
ǫd :=
1 if d ≡ 1 (mod 4),
i if d ≡ 3 (mod 4).
If f(z) is a function on the complex upper half plane, λ ∈ Z, and γ = ( a bc d ) ∈ Γ0(4), then
we define the usual slash operator by
f(z) |λ+ 1
)2λ+1
ǫ−1−2λd (cz + d)
−λ− 1
az + b
cz + d
Let g :=
e2πiv/p be the usual Gauss sum. Note that
Wχp(z) =
W (z)|− 1
1 −v/p
Choose an integer kv satisfying
16kv ≡ 15v (mod p).
DISTRIBUTION OF INTEGRAL FOURIER COEFFICIENTS MODULO PRIMES 11
Then, we have
(3.3)
2mp2 1
= γv,m
2mp2 1
1 −16v
+ 16kv
where
γv,m =
1− 2m+4p(v + kv + 2mv2p− 2mvkvp) 1p(15v − 16kv − 2
m+4(v2p+ vkvp))
22mp2(−16vp+ 16kvp) 2m+4vp− 2m+4kvp+ 1
Note that W (z) has its only pole at z ∼ 0 up to Γ0(16). Since γv,m ∈ Γ0(16), the
formula (3.3) implies that Wχp(z) is holomorphic at 2
mp2 for 1 ≤ m ≤ 3. Thus, G(z) is
holomorphic at 2mp2 for 1 ≤ m ≤ 3.
If m = 0, then we have
W (z)|− 1
γv,0 =
−16vp3 + 16kvp3
16vp− 16kvp+ 1
W (z) =
p2(−vp+ kvp)
16vp− 16kvp + 1
W (z) = W (z).
Note that
(3.4) W (z)|− 1
= α · q−
16 +O(1),
where α is a nonzero complex number. The q-expansion of Wχp(z) at
is given by
(3.5) Wχp(z)|− 1
Using (3.3) and (3.4), the only term in (3.5) with a negative exponent on q is the term
(v−kv).
If N is defined by 16N ≡ 1 (mod p), then we have
(v−kv) =
Thus, we have that
(W (z)−Wχp(z))|− 1
= O(1).
This implies that G(z) is a holomorphic modular form of half integral weight on Γ0(16p
Noting that
P̄ (3) = 8,
the remaining part of the proof is similar to that in Theorem 3. Thus, it is omitted. �
12 D. CHOI
References
[1] S. Ahlgren and M. Boylan, Central Critical Values of Modular L-functions and Coeffients of Half
Integral Weight Modular Forms Modulo ℓ, to appear in Amer. J. Math.
[2] S. Ahlgren and K. Ono, Arithmetic of singular moduli and class polynomials, Compos. Math. 141
(2005), no. 2, 293–312.
[3] J. H. Bruinier and K. Ono, Coefficients of half-integral weight modular forms, J. Number Theory 99
(2003), no. 1, 164–179.
[4] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters,
Math. Ann. 217 (1975), no. 3, 271–285.
[5] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag New York, GTM 97,
1993.
[6] S. Lang, Introduction to Modular Forms, Grundl. d. Math. Wiss. no. 222, Springer: Berlin Heidelberg
New York, 1976 Berlin, 1995.
[7] B. Gordon and K. Hughes, Multiplicative properties of eta-product, Cont. Math. 143 (1993), 415-430.
[8] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, Amer.
Math. Soc., CBMS Regional Conf. Series in Math., vol. 102, 2004.
[9] J.-P. Serre, Divisibilite de certaines fonctions arithmetiques, Enseignement Math. (2) 22 (1976), no.
3-4, 227–260.
[10] S. Treneer, Congruences for the Coefficients of Weakly Holomorphic Modular Forms, to appear in
the Proceedings of the London Mathematical Society.
[11] D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I, Int. Press
Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002, pp.211-244.
School of Mathematics, KIAS, 207-43 Cheongnyangni 2-dong 130-722, Korea
E-mail address : [email protected]
1. Introduction and Results
2. Proof of Theorem ??
3. Proofs of Theorem ??, ??, and ??
3.1. Proof of Theorem ??
3.2. Proofs of Theorem ??
3.3. Proofs of Theorem ??
References
|
0704.0013 | $p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral
Weight | p-ADIC LIMIT OF THE FOURIER COEFFICIENTS OF WEAKLY
HOLOMORPHIC MODULAR FORMS OF HALF INTEGRAL WEIGHT
D. CHOI AND Y. CHOIE
Abstract. Serre obtained the p-adic limit of the integral Fourier coefficients of modular
forms on SL2(Z) for p = 2, 3, 5, 7. In this paper, we extend the result of Serre to weakly
holomorphic modular forms of half integral weight on Γ0(4N) for N = 1, 2, 4. The proof
is based on linear relations among Fourier coefficients of modular forms of half integral
weight. As applications of our main result, we obtain congruences on various modular
objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of
Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.
November 4, 2018
1. Introduction and Statement of Main Results
Serre obtained the p-adic limits of the integral Fourier coefficients of modular forms on
SL2(Z) for p = 2, 3, 5, 7 (see Théorème 7 and Lemma 8 in [20]). In this paper, we extend
the result of Serre to weakly holomorphic modular forms of half integral weight on Γ0(4N)
forN = 1, 2, 4. The proof is based on linear relations among Fourier coefficients of modular
forms of half integral weight. As applications of our main result, we obtain congruences
for various modular objects, such as those for Borcherds exponents, for Fourier coefficients
of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the
Maass Space.
For odd d, let
:= γtΓ0(4N)tγ
where γt = (
c d ) ∈ Γ(1) and γt(t) = ∞. We denote the q-expansion of a modular form
f ∈Mλ+ 1
(Γ0(4N)) at each cusp t of Γ0(4N) by
(1.1) (f |λ+ 1
γt)(z) = (cz + d)
−λ− 1
az + b
cz + d
atf (n)q
t , qt := q
where
(1.2) r(t) ∈
2000 Mathematics Subject Classification. 11F11,11F33.
Key words and phrases. modular forms, p-adic limit, Borcherds exponents, Maass space .
This work was partially supported by KOSEF R01-2003-00011596-0 , ITRC and BRSI-POSTECH.
http://arxiv.org/abs/0704.0013v2
2 D. CHOI AND Y. CHOIE
When t ∼ ∞, we denote atf (n) by af (n). Note that the number r(t) is independent of the
choice of f ∈Mλ+ 1
(Γ0(4N)) and λ. We call t a regular cusp if r(t) = 0 (see Chapter IV.
§1. of [15] for a more general definition of a λ-regular cusp ).
Remark 1.1. Our definition of a regular cusp is different from the usual one.
Let U4N := {t1, · · · , tν(4N)} be the set of all inequivalent regular cusps of Γ0(4N). Note
that the genus of Γ0(4N) is zero if and only if 1 ≤ N ≤ 4. LetMλ+ 1
(Γ0(4N)) be the space
of weakly holomorphic modular forms of weight λ + 1
on Γ0(4N) and let M0λ+ 1
(Γ0(N))
denote the set of f(z) ∈ Mλ+ 1
(Γ0(N)) such that the constant term of its q-expansion at
each cusp is zero. Let Up be the operator defined by
(f |Up)(z) :=
af(pn)q
Let OL be the ring of integers of a number field L with a prime ideal p ⊂ OL. For
f(z) :=
af(n)q
n and g(z) :=
ag(n)q
n ∈ L[[q−1, q]] we write
f(z) ≡ g(z) (mod p)
if and only if af (n)− ag(n) ∈ p for every integer n.
With these notations we state the following theorem.
Theorem 1. For N = 1, 2, 4 consider
f(z) :=
af (n)q
n ∈ M0
(Γ0(4N)) ∩ L[[q−1, q]].
Suppose that p ⊂ OL is any prime ideal such that p|p, p prime, and that af(n) is p-integral
for every integer n ≥ n0.
(1) If p = 2 and af (0) = 0, then there exists a positive integer b such that
(f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N.
(2) If p ≥ 3 and f(z) ∈ M0
(Γ0(4N)) with λ ≡ 2 or 2+
(mod p−1
), then there
exists a positive integer b such that
(f |(Up)b)(z) ≡ 0 (mod pj) for each j ∈ N.
Remark 1.2. The p-adic limit of a sum of Fourier coefficients of f ∈ M 3
(Γ0(4N)) was
studied in [13].
Our method only allows to prove a weaker result if f(z) 6∈ M0
(Γ0(4N)).
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 3
Theorem 2. For N = 1, 2 or 4, let
f(z) :=
af (n)q
n ∈ Mλ+ 1
(Γ0(4N)) ∩ L[[q−1, q]].
Suppose that p ⊂ OL is any prime ideal with p|p, p prime, p ≥ 5, and that af (n) is
p-integral for every integer n ≥ n0. If λ ≡ 2 or 2 +
(mod p−1
), then there exists a
positive integer b0 such that
p2b−m(p:λ)
t∈U4N
∆4N,3−α(p:λ)(z)
R4N (z)
e·ω(4N)
(0)atf (0) (mod p)
for every positive integer b > b0 (see Section 3 for detailed notation ).
Example 1.3. Recall that the generating function of the overpartition P̄ (n) of n(see [11])
P̄ (n)qn =
η(2z)
η(z)2
is in M− 1
(Γ0(16)), where η(z) := q
n=1(1− qn). Therefore, theorem 2 implies that
P̄ (52b) ≡ 1 (mod 5), ∀b ∈ N.
2. Applications: More Congruences
In this section, we study congruences for various modular objects such as those for
Borcherds exponents and for quotients of Eisenstein series.
2.1. p-adic Limits of Borcherds Exponents. Let MH denote the set of meromorphic
modular forms of integral weight on SL2(Z) with Heegner divisor, integer coefficients and
leading coefficient 1. Let
(Γ0(4)) := {f(z) =
af(n)q
n ∈ M 1
(Γ0(4)) | a(n) = 0 for n ≡ 2, 3 (mod 4)}.
If f(z) =
af(n)q
n ∈ M+1
(Γ0(4)), then define Ψ(f(z)) by
Ψ(f(z)) := q−h
(1− qn)af (n2),
where h = − 1
af(0) +
1<n≡0,1 (mod 4) af (−n)H(−n). Here H(−n) denotes the usual
Hurwitz class number of discriminant −n. The following was proved by Borcherds.
Theorem 2.1 ([4]). The map Ψ is an isomorphism from M+1
(Γ0(4)) to MH , and the
weight of Ψ(f(z)) is af (0).
4 D. CHOI AND Y. CHOIE
Let j(z) be the usual j-invariant function with the product expansion
j(z) = q−1
(1− qn)A(n).
Let F (z) := q−h
n=1(1 − qn)c(n) be a meromorphic modular form of weight k in MH .
The p-adic limit of
d|n d · c(d) was studied in [5] for p = 2, 3, 5, 7. Here we obtain the
p-adic limit of c(d) for p = 2, 3, 5, 7.
Theorem 3. Let F (z) := q−h
n=1(1− qn)c(n) be a meromorphic modular form of weight
k in MH .
(1) If p = 2, then for each j ∈ N there exists a positive integer b such that
c(mpb) ≡ 2k (mod pj)
for every positive integer m.
(2) If p ∈ {3, 5, 7}, then, for each j ∈ N there exists a positive integer b such that
5c(mpb)−̟(F )A(mpb) ≡ 10k (mod pj)
for every positive integer m. Here, ̟(F ) is a constant determined by the constant
term of the q-expansion of Ψ−1(F ) at 0.
2.2. Sums of n-Squares. For u ∈ Z>0, let
rn(u) := ♯{(s1, · · · , sn) ∈ Zn : s21 + · · ·+ s2n = u}.
Theorem 4. Suppose that p ≥ 5 is a prime. If λ ≡ 2 or 3 (mod p−1
), then there exists
a positive integer C0 such that
r2λ+1
p2b−m(p:λ)
≡ − (14− 4α (p : λ)) + 16
)[ λp−1 ]+α(p:λ)m(p:λ)
(mod p),
for every b > C0.
Remark 2.2. As for an example, if λ ≡ 2 (mod p− 1) and p is an odd prime, then there
exists a positive integer C0 such that
r2λ+1
≡ 10 (mod p), ∀b > C0
2.3. Quotients of Eisenstein Series. Congruences for the coefficients of quotients of
elliptic Eisenstein series have been studied in [3]. Let us consider the Cohen Eisenstein
series Hr+ 1
(z) :=
N=0H(r,N)q
n of weight r+ 1
, r ≥ 2 (see [7]). We derive congruences
for the coefficients of quotients of Hr+ 1
(z) and Eisenstein series.
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 5
Theorem 5. Let
F (z) :=
E4(z)
aF (n)q
G(z) :=
E6(z)
aG(n)q
W (z) :=
E6(z)
aW (n)q
Then there exists a positive integer C0 such that
aF (11
2b+1) ≡ 1 (mod 11),
aG(11
2b+1) ≡ 6 (mod 11),
aW (11
2b+1) ≡ 2 (mod 11)
for every integer b > C0.
2.4. The Maass Space. Next we deal with congruences for the Fourier coefficients of
a Siegel modular form in the Maass space. To define the Maass space, let us introduce
notations given in [17]: let T ∈ M2g(Q) be a rational, half-integral, symmetric, non-
degenerate matrix of size 2g with discriminant
DT := (−1)g det(2T ).
Let DT = DT,0f
T , where DT,0 is the corresponding fundamental discriminant. Further-
more, let
G8 :=
2 0 −1 0 0 0 0 0
0 2 0 −1 0 0 0 0
−1 0 2 −1 0 0 0 0
0 −1 −1 2 −1 0 0 0
0 0 0 −1 2 −1 0 0
0 0 0 0 −1 2 −1 0
0 0 0 0 0 −1 2 −1
0 0 0 0 0 0 −1 2
and G7 be the upper (7, 7)-submatrix of G8. Define
Sg :=
(g−1)/8
2, if g ≡ 1 (mod 8),
(g−7)/8
G7, if g ≡ −1 (mod 8).
6 D. CHOI AND Y. CHOIE
For each m ∈ N such that (−1)gm ≡ 0, 1 (mod 4), define a rational, half-integral, sym-
metric, positive definite matrix Tm of size 2g by
Tm :=
0 m/4
, if m ≡ 0 (mod 4),
e2g−1
e′2g−1 [m+ 2 + (−1)n]/4
, if m ≡ (−1)g (mod 4)
Here e2g−1 ∈ Z(2n−1,1) is the standard column vector and e′2g−1 is its transpose.
Definition 2.3. (The Maass Space) Take g, k ∈ N such that g ≡ 0, 1 (mod 4) and
g ≡ k (mod 2). Let
SMaassk+g (Γ2g)
F (Z) =
A(T )qtr(TZ) ∈ Sk+g(Γ2g)
∣∣∣∣∣∣
A(T ) =
ak−1φ(a;T )A(T|DT |/a2)
(see (6.2) for details). This space is called the Maass space of genus 2g and weight g + k.
In [17] it was proved that the Maass space is the same as the image of the Ikeda lifting
when g ≡ 0, 1 (mod 4). Using this fact together with Theorem 1, we derive the following
congruences for the Fourier coefficients of F (Z) in SMaassk+g (Γ2g).
Theorem 6. For g ≡ 0, 1 (mod 4), let
F (Z) :=
A(T )qtr(TZ) ∈ SMaassk+g (Γ2g)
with integral coefficients A(T ), T > 0. If k ≡ 2 or 3 (mod p−1
) for some prime p, then,
for each j ∈ N, there exists a positive integer b for which
A(T ) ≡ 0 (mod pj)
for every T > 0, det(2T ) ≡ 0 (mod pb).
This paper is organized as follows. Section 3 gives a linear relation among Fourier
coefficients of modular forms of half integral weight. The remaining sections contain
detailed proofs of the main theorems.
3. Linear Relation among Fourier Coefficients of modular forms of Half
Integral Weight
Let V (N ; k, n) be the subspace of Cn generated by the first n coefficients of the q-
expansion of f at ∞ for f ∈ Sk(Γ0(N)), where Sk(Γ0(N)) denotes the space of cusp forms
of weight k ∈ Z on Γ0(N). Let L(N ; k, n) be the orthogonal complement of V (N ; k, n)
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 7
in Cn with the usual inner product of Cn. The vector space L(1; k, d(k) + 1), d(k) =
dim(Sk(Γ(1))), was studied by Siegel to evaluate the value of the Dedekind zeta function
at a certain point. The vector space L(1; k, n) is explicitly described in terms of the
principal part of negative weight modular forms in [9]. These results were extended in [8]
to the groups Γ0(N) of genus zero. For 1 ≤ N ≤ 4, let
4N, λ+
at1f (0), · · · , a
tν(4N)
f (0), af(1), · · · , af(n)
∈ Cn+ν(4n)
∣∣∣ f ∈Mλ+ 1
(Γ0(4N))
where U4N := {t1, · · · , tν(4N)} is the set of all inequivalent regular cusps of Γ0(4N). We
define EL(4N, λ+ 1
;n) to be the orthogonal complement of EV (4N, λ+ 1
;n) in Cn+ν(4N).
Let ∆4N,λ := q
δλ(4N)+O(qδλ(4N)+1) be inMλ+ 1
(Γ0(4N) with the maximum order at ∞,
that is, its order at ∞ is bigger than that of any other modular form of the same level
and weight. Furthermore, let
R4(z) :=
η(4z)8
η(2z)4
, R8(z) :=
η(8z)8
η(4z)4
R12(z) :=
η(12z)12η(2z)2
η(6z)6η(4z)4
and R16(z) :=
η(16z)8
η(8z)4
For ℓ, n ∈ N, define
m(ℓ : n) :=
≡ 0 (mod 2)
≡ 1 (mod 2)
α(ℓ : n) := n− ℓ− 1
Let ω(4N) be the order of zero of R4N (z) at ∞. Note that R4N (z) ∈ M2(Γ0(4N)) has
its only zero at ∞. So, using the definition of η(z) = q 124
n=1(1− qn), we find that
(3.1) ω(4) = 1, ω(8) = 2, ω(12) = 4, ω(16) = 4.
For each g ∈Mr+ 1
(Γ0(4N)) and e ∈ N, let
(3.2)
R4N (z)e
e·ω(4N)∑
b(4N, e, g; ν)q−ν +O(1) at ∞.
With these notations we state the following theorem:
Theorem 3.1. Suppose that λ ≥ 0 is an integer and 1 ≤ N ≤ 4. For each e ∈ N
such that e ≥ λ
− 1, take r = 2e − λ + 1. The linear map Φr,e(4N) : Mr+ 1
(Γ0(4N)) →
8 D. CHOI AND Y. CHOIE
EL(4N, λ+ 1
; e · ω(4N)), defined by
Φr,e(4N)(g)
R4N (z)
(0), · · · , htν(4N)a
tν(4N)
R4N (z)
(0), b(4N, e, g; 1), · · · , b(4N, e, g; e · ω(4N))
is an isomorphism.
Proof of Theorem 3.1. Suppose that G(z) is a meromorphic modular form of weight 2 on
Γ0(4N). For τ ∈ H∪C4N , let Dτ be the image of τ under the canonical map from H∪C4N
to a compact Riemann surface X0(4N). Here H is the usual complex upper half plane,
and C4N denotes the set of all inequivalent cusps of Γ0(4N). The residue ResDτGdz of
G(z) at Dτ ∈ X0(4N) is well-defined since we have a canonical correspondence between a
meromorphic modular form of weight 2 on Γ0(4N) and a meromorphic 1-form of X0(4N).
If ResτG denotes the residue of G at τ on H, then
ResDτGdz =
ResτG.
Here lτ is the order of the isotropy group at τ . The residue of G at each cusp t ∈ C4N is
(3.3) ResDtGdz = ht ·
atG(0)
Now we give a proof of Theorem 3.1.
To prove Theorem 3.1, take
G(z) =
R4N (z)e
f(z),
where g ∈Mr+ 1
(Γ0(4N)) and f(z) =
n=1 af(n)q
n ∈Mλ+ 1
(Γ0(4N)). Note that G(z) is
holomorphic on H. Since g(z), R4N (z) and f(z) are holomorphic and R4N (z) has no zero
on H, it is enough to compute the residues of G(z) only at all inequivalent cusps to apply
the Residue Theorem. The q-expansion of
R4N (z)
ef(z) at ∞ is
R4N(z)e
f(z) =
e·ω(4N)∑
b(4N, e, g; ν)q−ν + a g(z)
R4N (z)
(0) +O(q)
af(n)q
Since R4N (z) has no zero at t ≁ ∞, we have
R4N (z)e
γt = a
R4N (z)
(0)af(0) +O(qt).
Further note that, for an irregular cusp t,
at g(z)
R4N (z)
(0)af(0) = 0.
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 9
So the Residue Theorem and (3.3) imply that
(3.4)
t∈U4N
e·ω(4N)
(0)atf(0) +
e·ω(4N)∑
b(4N, e, g; ν)af(ν) = 0.
This shows that Φr,e(4N) is well-defined. The linearity of the map Φr,e(4N) is clear.
It remains to check that Φr,e(4N) is an isomorphism. Since there exists no holomorphic
modular form of negative weight except the zero function, we obtain the injectivity of
Φr,e(4N). Note that for e ≥ λ−12 ,
4N ;λ+
, e · ω(4N)
= e · ω(4N) + ν(4N)− dimC
Mλ+ 1
(Γ0(4N))
However, the set C4N , 1 ≤ N ≤ 4, of all inequivalent cusps of Γ0(4N) are
∞, 0, 1
∞, 0, 1
C12 =
∞, 0, 1
C16 =
∞, 0, 1
and it can be checked that
(3.5) ν(4) = 2, ν(8) = 3, ν(12) = 4, ν(16) = 6
(see §1 of Chapter 4. in [15] for details). The dimension formula of Mλ+ 1
(Γ0(4N)) (see
Table 1) together with the results in (3.1) and (3.5), implies that
4N, λ+
; e · ω(N)
= dimC(Mr+ 1
(Γ0(4N)))
since r = 2e− λ+ 1.
Table 1. Dimension Formula for Mk(Γ0(4N))
N k = 2n + 1
k = 2n+ 3
k = 2n
N = 1 n + 1 n + 1 n + 1
N = 2 2n+ 1 2n+ 2 2n+ 1
N = 3 4n+ 1 4n+ 3 4n+ 1
N = 4 4n+ 2 4n+ 4 4n+ 1
So Φr,e(4N) is surjective since the map Φr,e(4N) is injective. This completes our claim.
10 D. CHOI AND Y. CHOIE
4. Proofs of Theorem 1 and 2
4.1. Proof of Theorem 1. First, we obtain linear relations among Fourier coefficients
of modular forms of half integral weight modulo p. Let
Op := {α ∈ L | α is p-integral}.
M̃λ+ 1
, p(Γ0(4N)) := {H(z) =
aH(n)q
n ∈ Op/pOp[[q−1, q]] |
H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩Mλ+ 1
(Γ0(4N))}.
S̃λ+ 1
, p(Γ0(4N)) := {H(z) =
aH(n)q
n ∈ Op/pOp[[q−1, q]] |
H ≡ h (mod p) for some h ∈ Op[[q−1, q]] ∩ Sλ+ 1
(Γ0(4N))}.
The following lemma gives the dimension of M̃λ+ 1
, p(Γ0(4N)).
Lemma 4.1. Take λ ∈ N, 1 ≤ N ≤ 4 and a prime p such that
p ≥ 3 if N = 1, 2, 4,
p ≥ 5 if N = 3.
Now take any prime ideal p ⊂ OL, p|p. Then
dim M̃λ+ 1
, p(Γ0(4N)) = dimMλ+ 1
(Γ0(4N))
dim S̃λ+ 1
, p(Γ0(4N)) = dimSλ+ 1
(Γ0(4N)).
Proof. Let
j4N (z) = q
−1 +O(q)
be a meromorphic modular function with a pole only at ∞. Explicitly, these functions
j4(z) =
η(z)8
η(4z)8
+ 8, j8(z) =
η(4z)12
η(2z)4η(8z)8
j12(z) =
η(4z)4η(6z)2
η(2z)2η(12z)4
, j16(z) =
η2(z)η(8z)
η(2z)η2(16z)
Since the Fourier coefficients of η(z) and 1
are integral, the q-expansion of j4N (z) has
integral coefficients.
Recall that ∆4N,λ = q
δλ(4N) + O(qδλ(4N)+1) is the modular form of weight λ + 1
Γ0(4N) such that the order of its zero at ∞ is higher than that of any other modular form
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 11
of the same level and weight. Denote the order of zero of ∆4N,λ at ∞ by δλ(4N). Then
the basis of Mλ+ 1
(Γ0(4N)) can be chosen as
(4.1) {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} .
If ∆4N,λ(z) is p-integral, then {∆4N,λ(z)j4N (z)e | 0 ≤ e ≤ δλ(4N)} also forms a basis of
M̃λ+ 1
,p(Γ0(4N)). Note that δλ(4N) = dimMλ+ 1
(Γ0(4N))− 1. So from Table 1 we have
(4.2) ∆4N,λ(z) = ∆4N,j(z)R4N (z)
where λ ≡ j (mod 2), j ∈ {0, 1}. More precisely, one can choose ∆4N,j(z) as followings:
∆4,0(z) = θ(z), ∆4,1(z) = θ(z)
∆8,0(z) = θ(z), ∆8,1(z) =
(θ(z)3 − θ(z)θ(2z)2) ,
∆12,0(z) = θ(z), ∆12,1(z) =
x,y,z∈Z q
3x2+2(y2+z2+yz) −
x,y,z∈Z q
3x2+4y2+4z2+4yz
∆16,0(z) =
(θ(z)− θ(4z)) , ∆16,1(z) = 18 (θ(z)
3 − 3θ(z)2θ(4z) + 3θ(z)θ(4z)2 − θ(4z)3) .
Since θ(z) = 1+ 2
n=1 q
n, the coefficients of the q-expansion of ∆4N,j(z), j ∈ {0, 1}, are
p-integral. This completes the proof. �
Remark 4.2. The proof of Lemma 4.1 implies that the spaces of Mλ+ 1
(Γ0(4N)) for
N = 1, 2, 4 are generated by eta-quotients since θ(z) =
η(2z)5
η(z)2η(4z)2
For 1 ≤ N ≤ 4 set
4N, λ+
(af(1), · · · , af(n)) ∈ Fnp | f ∈ S̃λ+ 1
(Γ0(4N))
,Fp := Op/pOp.
We define L̃S(4N, λ +
;n) to be the orthogonal complement of ṼS(4N, λ +
;n) in Fn
Using Lemma 4.1, we obtain the following proposition.
Proposition 4.3. Suppose that λ is a positive integer and 1 ≤ N ≤ 4. For each e ∈ N,
e ≥ λ
− 1, take r = 2e−λ+1. The linear map ψ̃r,e(4N) : M̃r+ 1
,p(Γ0(4N)) → L̃S(4N, λ+
; e · ω(4N)), defined by
ψ̃r,e(4N)(g) = (b(4N, e, g; 1), · · · , b(N, e, g; e · ω(4N))) ,
is an isomorphism. Here b(4N, e, g; ν) is defined in (3.2).
Proof. Note that dimS 3
(4N) = 0 and that
dimSλ+ 1
(4N) +N + 1 +
= dimMλ+ 1
(see [10]). So, from Lemma 4.1 and Table 1, it is enough to show that ψr,e(4N) is injective.
If g is in the kernel of ψr,e(4N), then
R4N (z)
e · R4N (z)e ≡ 0 (mod p) by Sturm’s formula
(see [21]). So we have g(z) ≡ 0 (mod p) since R4N(z)e 6≡ 0 (mod p). This completes the
proof. �
12 D. CHOI AND Y. CHOIE
Theorem 4.4. Take a prime p,N = 1, 2, 4 and
f(z) :=
af(n)q
n ∈ Sλ+ 1
(Γ0(4N)) ∩ L[[q]].
Suppose that p ⊂ OL is any prime ideal with p|p and that af (n) is p-integral for every
integer n ≥ n0. If λ ≡ 2 or 2 +
(mod p−1
) or p = 2, then there exists a positive
integer b such that
≡ 0 (mod p), ∀n ∈ N.
Proof of Theorem 4.4. i) First, suppose that p ≥ 3: Take positive integers ℓ and b such
(4.3)
3− 2α(p : λ)
p2b +
pm(p:λ) + ℓ(p− 1) = 2.
Note that if b is large enough, that is, b > logp
3−2α(p:λ)
pm(p:λ) − 2
, then there
exists a positive integer ℓ satisfying (4.3). Also note that atf(0) = 0 for every cusp t of
Γ0(4N) since f(z) is a cusp form. So, if r = 2e− α(p : λ) + 1, then Theorem 3.1 implies
that, for g(z) ∈ M̃r+ 1
(Γ0(4N)),
e·ω(4N)∑
b(4N, e, g; ν)af(νp
2b−m(p:λ)) ≡ 0 (mod p),
since
R4N (z)e
f(z)p
m(p:λ)
Eℓp−1(z)
e·ω(4N)∑
b(4N, e, g; ν)q−νp
+ a g(z)
R4N (z)
(0) +
a g(z)
R4N (z)
(n)qnp
af(n)q
npm(p:λ)
(mod p).
So Proposition 4.3 implies that
p2b−m(p:λ)
2p2b−m(p:λ)
, · · · , a
e · ω(4N)p2b−m(p:λ)
∈ ṼS
4N,α(p : λ) + 1
If α(p : λ) = 2 or 2 +
, then
dimSα(p:λ)+ 1
(Γ0(4N)) = dim ṼS
4N,α(p : λ) +
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 13
ii) p = 2: Note that
∆4N,1(z)
R4N (z)
= q−1+O(1) for N = 1, 2, 4. So, there exists a polynomial
F (X) ∈ Z[X ] such that
F (j4N(z))
∆4N,1(z)
R4N (z)
= q−n +O(1).
For an integer b, 22
> λ+ 2, let
G(z) :=
F (j4N(z))
∆4N,1(z)
R4N(z)
f(z)θ(z)2
1+2b−2λ+3.
Since θ(z) ≡ 1 (mod 2), Theorem 3.1 implies that af(2b · n) ≡ 0 (mod p). �
To apply Theorem 4.4, we need the following two propositions.
Proposition 4.5 (Proposition 3.2 in [22]). Suppose that p is an odd prime, k and N are
integers with (N, p) = 1. Let
f(z) =
a(n)qn ∈ Mλ+ 1
(Γ0(4N)).
Suppose that ξ :=
cp2 d
, with ac > 0. Then there exist n0, h0 ∈ N with h0|N, a sequence
{a0(n)}n≥n0 and r0 ∈ {0, 1, 2, 3} such that
(f |Upm|λ+ 1
ξ)(z) =
4n+r0≡0 (mod p
a0(n)q
4n+r0
m , ∀m ≥ 1.
Proposition 4.6 (Proposition 5.1 in [1]). Suppose that p is an odd prime such that p ∤ N
and consider
g(z) =
a(n)qn ∈ Sλ+ 1
(Γ0(4Np
j)) ∩ L[[q]], for each j ∈ N.
Suppose further that p ⊂ OL is any prime ideal with p|p and that a(n) is p-integral for
every integer n ≥ 1. Then there exists G(z) ∈ Sλ′+ 1
(Γ0(4N)) ∩OL[[q]] such that
G(z) ≡ g(z) (mod p),
where λ′ + 1
= (λ+ 1
)pj + pe(p− 1) with eN large.
Remark 4.7. Proposition 4.6 was proved for p ≥ 5 in [1]. One can check that this holds
also for p = 3.
Now we prove Theorem 1.
Proof of Theorem 1. Take
Gp(z) :=
η(8z)48
η(16z)24
∈M12(Γ0(16)) if p = 2,
η(z)27
η(9z)3
∈M12(Γ0(9)) if p = 3,
η(4z)p
η(4p2z)
∈M p2−1
(Γ0(p
2)) if p ≥ 5.
14 D. CHOI AND Y. CHOIE
Using properties of eta-quotients (see [12]), note that Gp(z) vanishes at every cusp of
Γ0(16) except ∞ if p = 2, and vanishes at every cusp ac of Γ0(4Np
2) with p2 ∤ N if p ≥ 3.
Thus, Proposition 4.5 implies that there exist positive integers ℓ,m, k such that
(f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1
(Γ0(16)) if p = 2,
(f |Upm)(z)Gp(z)ℓ ∈ Sk+ 1
(Γ0(4p
2N)) if p ≥ 3.
Note that k ≡ λ (mod p− 1). Using Proposition 4.6, we can find
F (z) ∈ Sk′+ 1
(Γ0(4N))
such that F (z) ≡ (f(z)|Upm)Gp(z)ℓ ≡ (f |Upm)(z) (mod p) and k′ ≡ k (mod p − 1).
Theorem 4.4 implies that there exists a positive integer b such that (F |Up2b)(z) ≡ 0
(mod p). Thus, we have shown so far that if ρ ∈ p \ p2, all the Fourier coefficients of
· F (z)|Upm+2b are p-integral. Repeat this argument to complete our claim. �
4.2. Proof of Theorem 2. Theorem 2 can be derived from Theorem 3.1 by taking a
special modular form.
Proof of Theorem 2. Take a positive integer ℓ and a positive even integer u such that
3− 2α(p : λ)
pm(p:λ) + ℓ(p− 1) = 2.
Let F (z) :=
∆4N,3−α(p:λ)(z)
R4N (z)
and G(z) := Ep−1(z)
ℓf(z)p
m(p:λ)
. Since Ep−1(z) ≡ 1
(mod p), we have
F (z)G(z) ≡
a∆4N,3−α(p:λ)(z)
R4N (z)
(n)qnp
af(n)q
nm(p:λ)
(mod p).
If Fourier coefficients of f(z) at each cusp are p-integral, then
((F ·G)|2γt) (z) ≡
atF (n)q
atG(n)q
atf (n)q
at∆4N,3−α(p:λ)(z)
R4N (z)
(mod p)
for t ≁ ∞. Since
aF (z)G(z)(0) ≡ a∆4N,3−α(p:λ)(z)
R4N (z)
(0)af (0) + af (p
u−m(p:λ)) (mod p) ,
F (z)G(z)
(0) ≡ at∆4N,3−α(p:λ)(z)
R4N (z)
(0)atf (0) (mod p) for t ≁ ∞,
for large u, the Residue Theorem implies Theorem 2 by letting u = 2b. Therefore it
is enough to check a p-integral property of Fourier coefficients of f(z) at each cusp:
take a positive integer e such that ∆(z)ef(z) is a holomorphic modular form, where
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 15
∆(z) := q
n=1(1− qn)24. Note that the q-expansions of j4N (z) and ∆4N,12e+λ(z) at each
cusp are p-integral. Thus (4.1) implies that
∆(z)ef(z) =
δ12e+λ(4N)∑
cnj4N (z)
n∆4N,12e+λ(z).
Moreover, cn is p-integral since
j4N (z)
n∆4N,12e+λ(z) = q
δ12e+λ(4N)−n +O
qδ12e+λ(4N)−n+1
and f(z) ∈ OL[[q, q−1]]. Note that p ∤ 4N since 1 ≤ N ≤ 4 and p ≥ 5 is a prime.
So Fourier coefficients of j4N (z), ∆N,12e+λ(z) and
at each cusp are p-integral. This
completes our claim. �
5. Proof of Theorem 3
Theorem 3 follows from Theorem 1 and Theorem 2.1.
Proof of Theorem 3. Note that j(z) ∈ MH . Let
g(z) := Ψ−1(j(z)) and f(z) := Ψ−1(F (z)) =
af (n)q
It is known (see §14 in [4]) that
g(z) =
(θ(z))E10(4z)
4πi∆(4z)
θ(z) d
(E10(4z))
80πi∆(4z)
θ(z).
Since the constant terms of the q-expansions at ∞ of f(z), θ(z) and g(z) are 0, a0
(0) =
and a0g(0) =
· 456
, respectively, we have
f(z)− kθ(z)−
a0f(0) + k(1− i)/2
a0g(0)
g(z) ∈ M01
(Γ0(4)).
Applying Theorem 1, one obtains the result. �
6. Proofs of Theorem 4 and 5
We begin with the following proposition.
Proposition 6.1. Let p be an odd prime and
f(z) :=
af (n)q
n ∈Mλ+ 1
(Γ0(4)) ∩ Zp[[q]].
If λ ≡ 2 or 3 (mod p−1
), then
p2b−m(p:λ)
≡ −(14 − 4α(p : λ))af(0) + 28
2−1 − 2−1i
)pb(7−2α(p:λ))
a0f (0) (mod p)
16 D. CHOI AND Y. CHOIE
for every integer b > logp
2α(p:λ)−3
pm(p:λ) + 2
Proof of Proposition 6.1. For ν ∈ Z≥0,
pm(p:λ) := ν · (p− 1) + α(p : λ) + 1
For an integer b with
3− 2α(p : λ)
pm(p:λ) − 2
there exists an ℓ ∈ N such that
3− 2α(p : λ)
p2b +
pm(p:λ) + ℓ(p− 1) = 2,
since
3− 2α(p : λ)
p2b +
pm(p:λ) − 2 = 3− 2α(p : λ)
(p2b − 1) + ν(p− 1).
We have
F (z) ≡
n=0 af (n)q
npm(p:λ) (mod p),
G(z) ≡ q−pb + 14− 4α(p : λ) + aG(1)q + · · · (mod p).
Note that aG(n) is p-integral for every integer n. Moreover, we obtain
F (z)G(z)|2 ( 0 −11 0 ) ≡
a0f (0) + · · ·
−26pb
)pb(7−2α(p:λ))
+ · · ·
(mod p),
where a0f (0) is given in (1.1). Note that
∞, 0, 1
is the set of cusps of Γ0(4), so Theorem
2 implies that
af (p
2b−m(p:n)) + (14− 4α(p : λ))af(0)− 28a0f (0)
)pb(7−2α(p:λ))
≡ 0 (mod p).
This proves Proposition 6.1. �
6.1. Proof of Theorem 4. Now we prove Theorem 4.
Proof of Theorem 4. Take
f(z) := θ2λ+1(z) = 1 +
r2λ+1(ℓ)q
af(n)q
Note that f(z) ∈Mλ+ 1
(Γ0(4)). Since (θ| 1
( 0 −11 0 ))(z) =
, we obtain
af(0) = 1 and a
f (0) =
)2λ+1
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 17
Since λ ≡ 2, 3 (mod p−1
) and
, we have
)p2u(7−2α(p:λ))
a0f (0)
pm(p:λ)
)p2u(7−2α(p:λ)) (
)pm(p:λ)(2α(p:λ)+(p−1)(2[ λp−1 ]+m(p:λ))+1)
)(7−2α(p:λ))(p2u−1)(
)8+2(p−1)[ λp−1 ]+m(p:λ)pm(p:λ)(p−1)+(pm(p:λ)−1)(1+2α(p:λ))
)8+2[ λp−1 ](p−1)+2α(p:λ)(pm(p:λ)−1)
)[ λp−1 ]+α(p:λ)m(p:λ)
(mod p),
for some u ∈ N. Applying Proposition 6.1, we obtain the result. �
6.2. Proof of Theorem 5. Consider the Cohen Eisenstein seriesHr+ 1
(z) :=
N=0H(r,N)q
of weight r + 1
, where r ≥ 2 is an integer. If (−1)rN ≡ 0, 1 (mod 4), then H(r,N) = 0.
If N = 0, then H(r, 0) = −B2r
. If N is a positive integer and Df 2 = (−1)rN , where D is
a fundamental discriminant, then
(6.1) H(r,N) = L(1− r, χD)
µ(d)χD(d)d
r−1σ2r−1(f/d).
Here µ(d) is the Möbius function. The following theorem implies that the Fourier coeffi-
cients of Hr+ 1
(z) are p-integral if p−1
Theorem 6.2 ([6]). Let D be a fundamental discriminant. If D is divisible by at least two
different primes, then L(1−n, χD) is an integer for every positive integer n. If D = p, p >
2, then L(1−n, χD) is an integer for every positive integer n unless gcd(p, 1−χD(g)gn) 6= 1,
where g is a primitive root (mod p).
Proof of Theorem 5. Note that E10(z) = E4(z)E6(z). So, E10(z)F (z), E10(z)G(z) and
E10(z)W (z) are modular forms of weights, 8 · 12 , 7 ·
and 8 · 1
respectively. Moreover, the
Fourier coefficients of those modular forms are 11-integral, since the Fourier coefficients
of H 5
(z), H 7
(z) and H 9
(z) are 11-integral by Theorem 6.2. We have
E10(z)F (z) =
+O(q),
E10(z)F (z)| 17
( 0 −11 0 ) =
(1 + i)(2i)−5 +O
E10(z)G(z) =
+O(q),
E10(z)G(z)| 15
( 0 −11 0 ) =
(1− i)(2i)−7 +O
E10(z)W (z) =
+O(q),
E10(z)W (z)| 17
( 0 −11 0 ) =
(1 + i)(2i)−9 +O
18 D. CHOI AND Y. CHOIE
where B2r is the 2rth Bernoulli number. The conclusion now follows from Proposition
6.1. �
6.3. Proof of Theorem 6. We begin by introducing some notations (see [17]). Let
V := (F2np , Q) be the quadratic space over Fp, where Q is the quadratic form obtained
from a quadratic form x 7→ T [x](x ∈ Z2np ) by reducing modulo p. We denote by < x, y >:=
Q(x, y)−Q(x)−Q(y), x, y ∈ F2np , the associated bilinear form and let
R(V ) := {x ∈ F2np : < x, y >= 0, ∀y ∈ F2np , Q(x) = 0}
be the radical of R(V ). Following [14], define a polynomial
Hn,p(T ;X) :=
1 if sp = 0,∏[(sp−1)/2]
j=1 (1− p2j−1X2) if sp > 0, sp odd,
(1 + λp(T )p
(sp−1)/2X)
∏[(sp−1)/2]
j=1 (1− p2j−1X2) if sp > 0, sp even,
where for even sp we denote
λp(T ) :=
1 if W is a hyperbolic space or sp = 2n,
−1 otherwise.
Following [16], for a nonnegative integer µ, define ρT (p
µ) by
ρT (p
µ)Xµ :=
(1−X2)Hn,p(T ;X), if p|fT ,
1 otherwise.
We extend the functions ρT multiplicatively to natural numbers N by defining
ρT (p
µ)X−µ :=
((1−X2)Hn,p(T ;X)).
D(T ) := GL2n(Z) \ {G ∈M2n(Z) ∩GL2n(Q) : T [G−1] half-integral},
where GL2n(Z) operates by left-multiplication and T [G
−1] = T ′G−1T . Then D(T ) is
finite. For a ∈ N with a|fT , let
(6.2) φ(a;T ) :=
G∈D(T ),|det(G)|=d
ρT [G−1](a/d
Note that φ(a;T ) ∈ Z for all a. With these notations we state the following theorem:
Theorem 6.3 ([17]). Suppose that g ≡ 0, 1 (mod 4) and let k ∈ N with g ≡ k (mod 2).
A Siegel modular form F is in SMaassk+n (Γ2g) if and only if there exists a modular form
f(z) =
c(n)qn ∈ Sk+ 1
(Γ0(4))
THE p-ADIC LIMIT OF WEAKLY HOLOMORPHIC MODULAR FORMS 19
such that A(T ) =
ak−1φ(a;T )c
|DT |
for all T . Here,
DT := (−1)g · det(2T )
and DT = DT,0f
T with DT,0 the corresponding fundamental discriminant and fT ∈ N.
Remark 6.4. A proof of Theorem 6.3 given in [17] implies that if A(T ) ∈ Z for all T ,
then c(m) ∈ Z for all m ∈ N.
Proof of Theorem 6. From Theorem 6.3 we can take
f(z) =
c(n)qn ∈ Sk+ 1
(Γ0(4)) ∩ Zp[[q]]
such that
F (Z) =
A(T )qtr(TZ) =
ak−1φ(a;T )c
|DT |
qtr(TZ).
By Theorem 1, there exists a positive integer b such that, for every positive integer m,
c(pbm) ≡ 0 (mod pj),
since k ≡ 2 or 3 (mod p−1
). Suppose that pb+2j ||DT |. If pj|a and a|fT , then
ak−1φ(a;T )c
|DT |
≡ 0 (mod pj).
If pj ∤ a and a|fT , then pb
∣∣∣ |DT |a2 and a
k−1φ(a;T )c
|DT |
≡ 0 (mod pj). �
Acknowledgement
We thank the referee for many helpful comments which have improved our exposition.
References
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School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-
dong, Goyang, Gyeonggi, 412-791, Korea
E-mail address : [email protected]
Department of Mathematics and Pohang Mathematical Institute, POSTECH, Pohang,
790–784, Korea
E-mail address : [email protected]
1. Introduction and Statement of Main Results
2. Applications: More Congruences
2.1. p-adic Limits of Borcherds Exponents
2.2. Sums of n-Squares
2.3. Quotients of Eisenstein Series
2.4. The Maass Space
3. Linear Relation among Fourier Coefficients of modular forms of Half Integral Weight
4. Proofs of Theorem ?? and ??
4.1. Proof of Theorem ??
4.2. Proof of Theorem ??
5. Proof of Theorem ??
6. Proofs of Theorem ?? and ??
6.1. Proof of Theorem ??
6.2. Proof of Theorem ??
6.3. Proof of Theorem ??
Acknowledgement
References
|
0704.0014 | Iterated integral and the loop product | Iterated integrals and the loop product
Koichi Fujii
1 Introduction
The purpose of this paper is to describe string topology from the viewpoint of
Chen’s iterated integrals. Let M be a compact closed oriented d-manifold and
LM be the free loop space ofM , the set of unbased smooth maps from S1 toM .
Let H∗(LM) be the homology of the free loop space shifted by the dimension of
the manifold i.e. H∗(LM) = H∗+d(LM). Chas and Sullivan found the product
on H∗(LM) which they called loop product [1]:
Hp(LM)⊗Hq(LM)→ Hp+q(LM).
They showed that this product makes H∗(LM) an associative, commutative
algebra.
Merkulov constructed a model for this product based on the theory of iter-
ated integrals, especially of the formal power series connection [10]. He showed
that there is an isomorphism of algebras
H∗(LM) ∼= H∗(ΛM ⊗ R
where ΛM is the de Rham differential graded algebra of M and R
the formal completion of the free graded associative algebra generated by some
noncommutative indeterminates.
On the other hand, Chen showed that the cohomology of the free loop space
of the simply-connected manifold is isomorphic to the cohomology of the cyclic
bar complex of differential forms via Chen’s iterated integrals (see [5] or [8]):
H∗(LM) ∼= H
∗(C(ΛM)).
In this paper, we construct a model for the loop product based on the the-
ory of the cyclic bar complex. We define a complex Hom(B(ΛM),ΛM) and
its subcomplex Hom(B(ΛM),ΛM) so that the Poincaré duality induces the
isomorphism of vector spaces
H∗(Hom(C(ΛM),R)) ∼= H∗−d(Hom(B(ΛM),ΛM)).
We can define a product on Hom(B(ΛM),ΛM) which realizes the loop product.
http://arxiv.org/abs/0704.0014v1
Theorem 1.1. Let M be a compact closed oriented simply-connected manifold.
Assume that H∗(M) is of finite type. Let A be a differential graded subalge-
bra of ΛM such that H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an
isomorphism of associative, commutative algebras
H∗(LM) ∼= H∗(Hom(B(A), A)).
The product defined on H∗(Hom(B(A), A)) corresponds to the loop product un-
der the isomorphism.
The paper is organized in the following way. In section 2, we briefly review
Chen’s iterated integrals. In section 3, we give a construction of a complex
Hom(B(A), A), and discuss its properties. In section 4, we give a proof of
theorem 1.1. In section 5, we study the iterated integrals on the free loop space
of the non-simply-connected manifolds. In section 6, we describe a relation
between the product on Hom(B(A), A) and the Goldman bracket. In this paper,
all the homologies have their coefficients in the field of real numbers.
Acknowledgement: The author would like to thank Professor Toshitake Kohno
much for helpful comments and gentle support.
2 Chen’s iterated integrals
We briefly review Chen’s iterated integrals (see [5], or [8]). Let M be a finite
dimensional smooth manifold and let LM be the free loop space of M , that is
the space of all smooth maps from S1 to M . Let ∆k be the k-simplex
{(t1, · · · , tk) ∈ R
k | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}.
We have an evaluation map
Φk : ∆k × LM →M
defined by
Φk(t1, · · · , tk; γ) = (γ(t1), · · · , γ(tk)).
Then define Pk to be the composition
(Λ∗M)⊗k → Λ∗Mk
→ Λ∗(∆k × LM)
→ Λ∗−kLM
where p∗ is the integration along the fiber of the projection p : ∆k×LM → LM .
Given ω1, · · ·ωk ∈ Λ
∗M , the iterated integral
ω1 · · ·ωk
is a differential form on LM of total degree |ω1| + · · · |ωk| − k, defined by the
formula
ω1 · · ·ωk = (−1)
(k−1)|ω1|+(k−2)|ω2|+···+|ωk−1|+k(k−1)/2Pk(ω1, · · · , ωk).
3 Preliminaries
In this section, we give a construction of some complexes. Let A be an arbitrary
differential graded algebra in this section. Let A∨ denote the dual of A. The
bar complex of A, (B(A), dB), is defined by
B(A) = ⊕r≥0 ⊗
r sA,
dB(ω1, · · · , ωr) = −(−1)
(ω1, · · · , ωi−1, dωi, ωi+1, · · · , ωr)
−(−1)εi
(ω1, · · · , ωi−1, ωi ∧ ωi+1, ωi+2, · · · , ωr).
Here (sA)q = Aq+1 or Aq according as 0 ≤ q or 0 < q, and εi = deg(ω1, · · · , ωi).
We denote the totality of degree n elements by B(A)n. The coproductH
∗(B(A))
→ H∗(B(A)) ⊗H∗(B(A)) is defined by
(ω1, · · · , ωn) 7→
(ω1, · · · , ωi)⊗ (ωi+1, · · · , ωn).
Chen proved the following theorem.
Theorem 3.1 (Chen [5]). Let M be a simply-connected manifold and H∗(M)
be of finite type. Let A be a differential graded algebra of ΛM such that A0 =
R and H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of
coalgebras
H∗(B(A)) ∼= H
∗(ΩM)
given by
(ω1, · · · , ωn) 7→
ω1 · · ·ωn.
Let F pB(A) be a filtration of B(A) such that
F pB(A) = ⊕0≤r≤p ⊗
r sA.
Let Hom(B(A), A∨)n =
p+q=n Hom(B(A)p, A
q∨) and Hom(B(A), A∨) =
n Hom(B(A), A
∨)n. Its boundary is defined by
δϕ(ω1, · · · , ωr)(ω)
= ϕ(ω1, · · · , ωr)(dω) + (−1)
|ω|ϕ(dB(ω1, · · · , ωr))(ω)
− (−1)|ω|ϕ(ω2, · · · , ωr)(ω ∧ ω1)
+(−1)|ω|+εr−1(|ωr|+1)ϕ(ω1, · · · , ωr−1)(ω ∧ ωr).
Let us define the subcomplex of Hom(B(A), A∨), Hom(B(A), A∨), according
to the Chen’s normalization of the cyclic bar complex (see [4] or [8]). We define
Hom(B(A), A∨) to be the set of elements in Hom(B(A), A∨) which satisfy the
following equations for any ω, ωi ∈ A
>0 and f ∈ A0:
−ϕ(· · ·ωi−2, fωi−1, ωi, · · · )(ω) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )(ω)
+ϕ(· · · , ωi−1, df, ωi, · · · )(ω) = 0, 1 ≤ i ≤ r − 1,
−ϕ(ω1, · · · , ωr)(fω) + ϕ(fω1, · · · , ωr)(ω) + ϕ(df, ω1, · · · , ωr)(ω) = 0,
−ϕ(ω1, · · · , fwr)(ω) + ϕ(ω1, · · · , ωr)(fω) + ϕ(ω1, · · · , ωr, df)(ω) = 0.
It can be easily seen that it is isomorphic to the dual of the normalized cyclic
bar complex of A:
Hom(B(A), A∨) ∼= C(A)
Similarly, let Hom(B(A), A)n =
p−q=n Hom(B(A)p, A
q) and Hom(B(A), A)
n Hom(B(A), A)n. Its boundary is defined by
δϕ(ω1, · · · , ωr)
= (−1)|ϕ|−εrdϕ(ω1, · · · , ωr)− (−1)
|ϕ|−εrϕ(dB(ω1, · · · , ωr))
+(−1)|ϕ|−εrω1 ∧ ϕ(ω2, · · · , ωr)
−(−1)(|ωr|+1)(|ϕ|+1)ϕ(ω1 · · · , ωr−1) ∧ ωr.
We define Hom(B(A), A) to be the set of elements in Hom(B(A), A) which
satisfy the following equations for any ω, ωi ∈ A
>0 and f ∈ A0:
−ϕ(· · ·ωi−2, fωi−1, ωi, · · · ) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )
+ϕ(· · · , ωi−1, df, ωi, · · · ) = 0, 1 ≤ i ≤ r − 1,
−f ∧ ϕ(ω1, · · · , ωr) + ϕ(fω1, · · · , ωr) + ϕ(df, ω1, · · · , ωr) = 0,
−ϕ(ω1, · · · , fwr) + ϕ(ω1, · · · , ωr) ∧ f + ϕ(ω1, · · · , ωr, df) = 0.
The cup product on Hom(B(A), A) is defined by
ϕ1 ∪ ϕ2(ω1, · · · , ωr)
0≤i≤r
(−1)|ϕ1|(|ϕ2|+εr−εi)ϕ1(ω1, · · · , ωi) ∧ ϕ2(ωi+1, · · · , ωr).
Since δ(ϕ1 ∪ ϕ2) = δϕ1 ∪ ϕ2 + (−1)
|ϕ1|ϕ1 ∪ δϕ2, H∗(Hom(B(A), A)) becomes
an algebra. This product can be induced on H∗(Hom(B(A), A)).
The E1-term of their spectral sequences associated with the filtration F
pB(A)
can be calculated from the cohomology of A.
Proposition 3.2. There is an isomorphism of vector spaces
H∗(Hom(F
pB(A)/F p−1B(A), A∨)) ∼= Hom(⊗
psH(A), H(A)∨)
Proof. Let A be a differential graded subalgebra of A such that A
= Ap for p
> 1, A
= R and
A1 = dA0 ⊕A
There is an isomorphism of vector spaces
Hom(F qB(A)/F q−1B(A), A∨) ∼= Hom(F
qB(A)/F q−1B(A), A
Since A
= R, there is an isomorphism
H0(Hom(F
qB(A)/F q−1B(A), A
)) ∼= Hom(⊗sH(A), H(A)
Therefore we obtain the proposition.
4 Proof of Theorem 1.1
We give the proof of theorem 1.1 in this section. There is a differential graded
subalgebra of A, A, such that A
= R and H(A) ∼= H(A) by the inclusion. Then
we obtain the isomorphism of algebras
H∗(Hom(B(A), A)) ∼= H∗(Hom(B(A), A))
by proposition 3.2. Therefore it suffices to verify the theorem in the case A0 = R.
The following result is due to Chen.
Theorem 4.1 (Chen [5]). H∗(LM) ∼= H∗(Hom(B(A), A
Proof. We define ψ : C∗(LM)→ Hom(B(A), A
∨) by
ψ(σ)(ω1, · · · , ωn)(ω) =
π∗ω ∧
ω1 · · ·ωn.
Let FpC∗(LM) be a filtration of C∗(LM) such that
FpCr(LM) = { σ : ∆
r → LM | π ◦ σ = σ′ ◦ π′ for some σ′ ∈ Cq(M),
q ≤ p, π′ : ∆r → ∆q } .
Let {Erp,q} be the associated spectral sequence. Define a filtration of Hom(B(A), A
FpHom(B(A), A) = {f ∈ Hom(B(A), A
∨) | f(ω1, · · · , ωn)(ω) = 0, ∀ω ∈ A
≥p+1}.
It can be easily shown that ψ preserves the filtrations of C∗(LM) and Hom(B(A), A
On E2-level, the map
ψ : Hp(M)⊗Hq(ΩM)→ Hp(A
∨)⊗Hq(B(A)
is given by
σ1 ⊗ σ2 7−→
(ω1, · · · , ωn 7→
ω1 · · ·ωn)
Theorem 3.1 asserts that this is an isomorphism. Therefore we obtain the
theorem.
Lemma 4.2. H∗(Hom(B(A), A)) ∼= H∗−d(Hom(B(A), A
Proof. We define a chain map P : Hom(B(A), A)→ Hom(B(A), A∨) by
P (ϕ)(ω1, · · · , ωn)(ω) =
ω ∧ ϕ(ω1, · · · , ωn).
Define a filtration of Hom(B(A), A) by
FpHom(B(A), A) = {ϕ ∈ Hom(B(A), A) | ϕ(ω1, · · · , ωn) ∈ A
≥d−p}.
The map P preserves those filtrations. On E2-level, the map
P : Hd−p(A)⊗Hq(B(A)
∨)→ Hp(A
∨)⊗Hq(B(A)
is given by
ω ⊗ ϕ 7−→
ω ∧ τ
This is isomorphic and we obtain the lemma.
Proof of theorem 1.1. We can verify that H∗(LM) is isomorphic to
H∗(Hom(B(A), A)) as vector spaces by composing the maps in theorem 4.1 and
lemma 4.2. We can also verify that there is an isomorphism of associative,
commutative algebras. Indeed, the cup product of Hom(B(A), A) on E2-level
Hd−p(A)⊗Hq(B(A)
∨)⊗Hd−s(A)⊗Ht(B(A)
∨)→ H2d−p−s(A)⊗Hq+t(B(A)
is given by
a⊗ g ⊗ b⊗ h 7→ (−1)(d−p+q)(d−s)a ∧ b⊗ g · h,
where g · h satisfies
g · h(ω1, · · · , ωn) =
g(ω1, · · · , ωi)h(ωi+1, · · · , ωn).
Then the following theorem asserts that the loop product and the cup product
coincide on E2-level.
Theorem 4.3 (Cohen-Jones-Yan [6]). Let M be a simply-connected manifold.
Then {Erp,q} becomes an algebra and converges to H∗(LM) as algebras. On
E2-level, the product
µ : Hp(M ;Hq(LM))⊗Hs(M ;Ht(LM))→ Hp+q−d(M ;Hs+t(LM))
is given by
µ((a⊗ g)⊗ (b ⊗ h)) = (−1)(d−s)(p+q−d)(a · b)⊗ (gh)
where a ∈ Hp(M), b ∈ Hs(M), g ∈ Hq(ΩM), h ∈ Ht(ΩM), a · b is the intersec-
tion product and gh is the Pontryagin product.
Therefore we obtain the theorem.
5 The conjugacy classes of fundamental groups
Let π denote a fundamental group of a smooth manifold M and J denote an
augmentation ideal of the group ring of π, Rπ. Chen showed that the completion
of the fundamental group with respect to the powers of its augmentation ideal is
isomorphic to the dual of the 0-th cohomology of the bar complex of differential
forms via iterated integrals [3]:
Rπ/Jp ∼= H
0(B(A))∨
where A is a differential graded subalgebra of ΛM such that A0 = R and
H∗(A) ∼= H∗(M).
Based on this work, we study iterated integrals on the free loop space of the
non-simply-connected manifold. Let π̃ denote the set of conjugacy classes of π
and J̃p denote pr(Jp) where pr is the projection of Rπ onto Rπ̃.
Theorem 5.1. Let M be a smooth manifold and H∗(M) is of finite type. Let A
be a differential graded subalgebra of ΛM such that the map Hq(A)→ Hq(ΛM)
induced by the inclusion is isomorphic if q = 0, 1 and injective if q = 2. Then
there is an isomorphism of vector spaces
Rπ̃/J̃p ∼= H0(Hom(B(A), A
We give the proof of this theorem in this section. Let ∗ be a fixed point
in S1. In this section, let LM be a set of smooth maps from S1 to M which
are constant maps near ∗. Let ΩxM be a subspace of LM whose elements
send ∗ to x ∈ M . Let Diff(S1, ∗) denote diffeomorphisms of S1 which coincide
with identity map near ∗. We define α, β : ∆q → LM to be equivalent by a
reparameterization iff there is a smooth map τ : ∆q → Diff(S1, ∗) such that
β(ξ)(t) = α(ξ)(τ(t, ξ)), ∀(t, ξ) ∈ S1 ×∆q.
Let C∗(LM) be a chain complex having as a basis the totality of equiva-
lence classes of smooth simplexes of LM . Let C∗(ΩxM) be a chain complex
having as a basis the totality of equivalence classes of smooth simplexes of ΩxM .
C∗(ΩxM) becomes a noncommutative associative algebra as follows. The prod-
uct of σ1 and σ2 in C∗(ΩxM) is defined to be the path product or 0 according
as degσ1+degσ2 ≤ 1 or > 1. The augmentation ε : C∗(ΩxM) → R is given by
εσ = 1 or 0 according as degσ = 0 or > 0.
Let σ be a smooth simplex of M . Define for each σ
Cq(LM)(σ) = {
niτi ∈ Cq(LM) | π♯τi = σ}.
Cq(LM)(σ) becomes a noncommutative associative algebra. Let ε(σ) denote
the augmentation of Cq(LM)(σ), given by
niτi 7→
ni. Define a filtration
of Cq(LM)(σ) by
FpCq(LM) = (kerε)
p ⊕ (⊕σ:∆q→M (kerε(σ))
Proposition 5.2. The map ψp : FpCq(LM) → Hom(F
p−1B(A), A∨) given by
(ω1, · · · , ωp) 7→
π∗ω ∧
ω1 · · ·ωp
is well-defined, chain map and FpCq(LM) ⊂ kerψp.
Proof. The well-definedness can be verified by the following lemma which can
be verified as in proposition 1.5, proposition 4.1.1 [2], and in proposition 1.5.3
Lemma 5.3 (Chen). (1) If α and β ∈ C∗(LM) are equivalent by a reparame-
terization, then
ω1 · · ·ωn = β
ω1 · · ·ωn.
(2) If τ1, τ2 ∈ Cq(LM)(σ), then
(τ1 · τ2)
ω1 · · ·ωn =
ω1 · · ·ωi ∧ τ
ωi+1 · · ·ωn.
(3) If f ∈ Λ0M , then for any i
ω1 · · · fωi−1 · · ·ωn +
ω1 · · · fωi · · ·ωn +
ω1 · · ·ωi−1df ωi · · ·ωn = 0.
To verify FpCq(LM) ⊂ kerψp, it suffices to show (kerε(σ))
p ⊂ kerψp. Let
s denote the section of π, which sends points of M to the constant map. Take
(σ1 − s♯σ) · (σ2 − s♯σ) · · · · ·(σp − s♯σ) ∈ (kerε(σ))
p, where σ ∈ Cq(M) and
σi ∈ Cq(LM)(σ). Then
(σ1 − s♯σ) · (σ2 − sσ) · · · · · (σp − s♯σ)
π∗ω ∧
ω1 · · ·ωp−1
σ∗ω ∧ (σ1 − s♯σ)
ω1 · · · (σk − s♯σ)
∗1 · · · ∧ (σp − s♯σ)
Therefore we obtain the proposition.
Let C∗(M,x) denote a set of smooth simplexes ofM neighborhood of whose
vertices are at x in M . We define
C ⊗ sC⊗p = C∗(M,x)⊗ sC∗(M,x)
Here (sC∗(M,x))q = Cq+1(M,x) or 0 according as q > 0 or q ≤ 0. Its boundary
is given by the sum of the boundary on each complex. Let us construct a chain
map Φ : C ⊗ sC⊗p → FpC∗(LM)/Fp+1C∗(LM) considering the following three
cases:
case 1: If (σ1, · · · , σp) ∈
sC(M,x)⊗p
, then
Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · · · · (σp − x)
where x is regarded as a constant map.
case 2: If (σ1, · · · , σp) ∈
sC(M,x)⊗p
, then
Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · ·σi · · · (σp − x)
where σi : ∆
1 ∋ ξ 7→ σi(ξ)(t) ∈ ΩxM is
σi(ξ)(t)
σi((1 − ξ)((1 − t)v0 + tv2) + ξ(1 − 2t)v0 + 2ξtv1), if 0 ≤ t ≤ 1/2
σi((1 − ξ)((1 − t)v0 + tv2) + ξ(2 − 2t)v1 + ξ(2t− 1)v2), if 1/2 ≤ t ≤ 1
Here v0, v1, v2 are the vertices of the standard simplex ∆
case 3: If (γ, σ1, · · · , σp) ∈ C1(M,x)⊗
sC(M,x)⊗p
, then
Φ : (γ, σ1, · · · , σp) 7−→ γ
t (σ1 − x)γt · · · γ
t (σp − x)γt
where γt : [0, 1] ∋ s 7→ γ(st) ∈ M , t ∈ ∆
Lemma 5.4. The following diagram commutes:
C ⊗ sC⊗p
−−−−→ FpC1(LM)/Fp+1C1(LM)
C ⊗ sC⊗p
−−−−→ FpC0(LM)/Fp+1C0(LM)
Proof. For case 2,
∂′Φ(σ1, · · · , σp)− Φ∂(σ1, · · · , σp)
= (σ1 − x) · · · (σ
i · σ
i − σ
i − σ
i + σ
i − σ
i + x) · · · (σp − x)
= (σ1 − x) · · · (σ
i − x) · (σ
i − x) · · · (σp − x) ∈ Fp+1C0(LM)
where σ
i , σ
i , σ
i are the faces of σi.
For case 3,
∂′Φ(γ, σ1, · · · , σp)− Φ∂
′(γ, σ1, · · · , σp)
= γ−1 · (σ1 − x) · γ · · · γ
−1 · (σp − x) · γ − (σ1 − x) · · · (σp − x)
∈ Fp+1C0(LM).
Therefore we obtain the lemma.
Proposition 5.2 gives the map
Hq(FpC(LM)/Fp−1C(LM))→ Hq(Hom(F
pB(A)/F p−1B(A), A∨)).
Lemma 5.5. For q = 0, the following map is isomorphic:
H0(FpC(LM)/Fp+1C(LM)) ∼= H0(Hom(F
pB(A)/F p−1B(A), A∨)).
Proof. We obtain the following surjection by lemma 5.4.
Φ : H0(C ⊗ sC
⊗p) ։ H0(FpC(LM)/Fp+1C(LM)).
Composing with the isomorphism ⊗pH1(M) ∼= H0(C ⊗ sC
⊗p), the map
⊗pH1(M) ։ H0(FpC(LM)/Fp+1C(LM))→ Hom(⊗
pH1(A),R)
is given by
(σ1, · · · , σn) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
This is isomorphic and we obtain the lemma.
Lemma 5.6. For q = 1, the following map surjective:
H1(FpC(LM)/Fp+1C(LM)) ։ H1(Hom(F
pB(A)/F p−1B(A), A∨)).
Proof. It suffices to show that the following map obtained by lemma 5.4 is
surjective.
ker∂ → H1(FpC(LM)/Fp+1C(LM))→ Hom(⊗
psH(A), H(A)∨)1
If (γ, σ1, · · · , σp) ∈ ker∂ ∩
C0(M,x)⊗
sC(M,x)⊗p
, then
(γ, σ1, · · · , σp) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
ωp, if deg ω = 0
0, otherwise
through the above map.
If (γ, σ1, · · · , σp) ∈ ker∂ ∩
C1(M,x)⊗
sC(M,x)⊗p
, then
(γ, σ1, · · · , σp) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
when deg ω = 1. Then we can verify the surjectivity and obtain the lemma.
Proof of theorem 1.1. Consider the spectral sequences ofC(LM)/FpC(LM) and
Hom(F p−1B(A), A∨) associated with FqC(LM) and Hom(F
qB(A), A∨), re-
spectively. Lemma 5.5 asserts that ψp is isomorphic on E1-level at degree 0:
H0(FqC(LM)/Fq+1C(LM)) ∼= H0(Hom(F
qB(A)/F q−1B(A), A∨)).
Lemma 5.6 asserts that ψp is surjective on E1-level at degree 1:
H1(FqC(LM)/Fq+1C(LM)) ։ H1(Hom(F
qB(A)/F q−1B(A), A∨)).
Then there is an isomorphism on Er-level at degree 0 for r ≥ 1. We have
Rπ̃/J̃p ∼= H0(C(LM)/FpC(LM)) ∼= H0(Hom(F
pB(A), A∨)).
Therefore we obtain the theorem.
6 The Goldman bracket
This section is devoted to the proof of the following theorem.
Theorem 6.1. Let M be a compact closed oriented surface with genus g. Then
the Goldman bracket induces a Lie algebra structure on lim
Rπ̃/J̃pand there is
an isomorphism of Lie algebras
Rπ̃/J̃p ∼= H0(Hom(B(H
∗(M)), H∗(M)∨)).
Goldman showed that the vector space spanned by the free homotopy classes
of closed curves on a closed oriented surface has a Lie algebra structure [9]. This
work led Chas and Sullivan to the string topology. We would verify that this
structure makes lim
Rπ̃/J̃p a Lie algebra. On the other hand, we can construct
a bracket on H0(Hom(B(H
∗(M)), H∗(M)∨)) by the cup product defined in
section 3 and the Connes’s operator. Here we regard H∗(M) as a differential
graded algebra with a trivial differential. Theorem 6.1 asserts that those two
Lie algebras are isomorphic.
First we describe a relation between this bracket and the augmentation ideal
of the group ring of the surface group to induce a Lie algebra structure on
Rπ̃/J̃p. Then we construct a bracket on H0(Hom(B(A), A
∨)) and verify
the isomorphism of Lie algebras
Rπ̃/J̃p ∼= H0(Hom(B(A), A
Finally we verify the isomorphism
H0(Hom(B(A), A
∨)) ∼= H0(Hom(B(H
∗(M)), H∗(M)∨).
The following proposition makes lim
Rπ̃/J̃p a Lie algebra.
Proposition 6.2. (1) If p ≥ 1 and q ≥ 2, then [J̃p, J̃q] ⊂ J̃p+q−2.
(2) If p ≥ 2 , then [J̃p,Rπ̃] ⊂ J̃p−1.
Proof. We give a proof of (1). Take (σ1−x) · · · (σp−x) ∈ J̃p, (τ1−y) · · · (τq−y)
∈ J̃q, where σi ∈ ΩxM and τi ∈ ΩyM . Assume that all curves are immersions
and σi τj intersect transversally for any i, j. Let {σi♯τj} denote the set of
intersection points of σi and τj . Also assume that all the intersection points are
distinct i.e. {σi♯τj} ∩ {σk♯τl} = φ if i 6= k or j 6= l. Then,
[σ, τ ] =
s∈σi♯τj
{ε(s;σi, τj)γs,x · (σi − x) · · · (σp − x)(σ1 − x) · · ·
·(σi−1 − x) · γ
s,x · ·γs,y · (τj − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ
−γs,x · (σi+1 − x) · · · (σp − x)(σ1 − x) · · · (σi−1 − x) · γ
s,x ·
·γs,y · (τj+1 − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ
∈ J̃p+q−2.
Here γs,x is a path from s to x along σi and γs,y is a path from s to y along τj .
The proof of (2) can be verified in the same way.
Let A be a differential graded subalgebra of ΛM such thatH∗(A) ∼= H∗(ΛM)
by the inclusion.
Proposition 6.3. There is an isomorphism of vector spaces
H∗(Hom(F
pB(A), A)) ∼= H∗−2(Hom(F
pB(A), A∨)).
Proof. We define P : H∗−2(Hom(F
pB(A), A))→ H∗(Hom(F
pB(A), A∨)) by
P (ϕ)(ω1, · · · , ωp)(ω) =
ω ∧ ϕ(ω1, · · · , ωp).
This map preserves the filtrations. On E1-level, the map
Hom(⊗qH(A), H(A))→ Hom(⊗qH(A), H(A)∨)
is isomorphic. Therefore we obtain the proposition.
Now we construct a bracket on H0(Hom(B(A), A
∨)). First, we define the
Connes’s operator B : H∗(Hom(F
pB(A), A∨)) → H∗+1(Hom(F
p−1B(A), A∨))
B(ϕ)(ω1, · · · , ωp−1)(ω)
0≤k≤p−1
(−1)(εk+1)(εp−1−εk)ϕ(ωk+1, · · · , ωp−1, ω, ω1, · · ·ωk)(1).
Composing these maps and the cup product, we can define a bracket on
H0(Hom(F
pB(A), A∨)) by
[ϕ1, ϕ2] = −P (P
−1Bϕ1 ∪ P
−1Bϕ2) ∈ H0(Hom(F
p−1B(A), A∨)).
Take 2g closed 1-forms on M , α1, · · · , αg, β1, · · ·βg, such that
αi ∧ βj =
δij . Let {E
p.q} denote the spectral sequence of Hom(B(A), A
∨) associated with
F pB(A). Notice that the cyclic group Z/pZ acts on E
∼= Hom(⊗pH1(A),R)
ιϕ(ω1, · · · , ωp) = ϕ(ω2, · · · , ωp, ω1)
where ι is a generator of Z/pZ. The bracket [ , ] : E
p,−p⊗E
q,−q → E
p+q−2,−p−q+2
[ϕ1, ϕ2](ω1, · · · , ωp+q−2)
i,m,n
ιmϕ1(αi, ω1, · · · , ωp−1)̺
nϕ2(βi, ωp, · · · , ωp+q−2)
−ιmϕ1(βi, ω1, · · · , ωp−1)̺
nϕ2(αi, ωp, · · · , ωp+q−2)
where ι and ̺ are generators of Z/pZ and Z/qZ, respectively.
Proposition 6.4. The following diagram commutes for p, q ≥ 1:
J̃p/ ˜Jp+1 ⊗ J̃q/ ˜Jq+1 −−−−→ E
∞ ⊗ E
[ , ]
[ , ]
J̃p+q−2/J̃p+q−1 −−−−→ E
p+q−2,−p−q+2
Proof. Take σ = (σ1 − x) · · · (σp − x) ∈ FpC0(LM), τ = (τ1 − y) · · · (τq − y) ∈
FqC0(LM). Take 2g curves in M , ai, bi, as in Figure 1. Assume that σi and τj ,
ak, or bk, intersect transversally for any i, j, k. Also assume that τj and ak, or
bk, intersect transversally for any j, k.
Assume that all the intersection points are distinct. Then for any i, j, k, we
can take each tubular neighborhoods of ai and bi so that it does not include some
neighborhoods of intersection points of σj and τk. We fix such neighborhoods
of intersection points and denote them by Up for each p. We can also take
a tubular neighborhood of the diagonal map from M to M×M outside those
neighborhoods of intersection points of σi and τj for any i, j i.e.
S1 \ ∪pσ
i (Up)
S1 \ ∪pτ
j (Up)
= φ, ∀i, j.
Here N∆ denotes the tubular neighborhood of the diagonal map. Thom class Φ
of this tubular neighborhood satisfies
Φ = −ε(p;σi, τj),
where ε(p;σi, τj) is the intersecion number of σi and τj at p.
Fig. 1
・ ・ ・
Define e♯ : C0(LM) → C1(LM) by e♯γ(ξ)(t) = γ(ξ + t). Let ωk, 1 ≤
k ≤ n, be differential forms on M which has its support inside the tubular
neighborhoods of ai and bi. Then
[σi,τj]
ω1 · · ·ωn
p∈σi♯τj,k
ε(p;σi, τj)
(σi)p
ω1 · · ·ωk
(τj)p
ωk+1 · · ·ωn
p∈σi♯τj ,k
×τj |
(σi)p
ω1 · · ·ωk
(τj)p
ωk+1 · · ·ωn
e♯σi×e♯τj
π∗Φ ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωn.
Here p1, p2 : LM×LM → LM are the projections. The last equality is obtained
by the following lemma.
Lemma 6.5. If p ∈ σi♯τj and p
′ ∈ Up ∩ σi([0, 1]), then
(σi)p
ω1 · · ·ωn =
(σi)p′
ω1 · · ·ωn.
Proof. F Let γ be the curve from p to p′ along σi inside Up. If γ and σ are in
the same direction, then
(σi)p′
ω1 · · ·ωn =
γ·(σi)p′
ω1 · · ·ωn =
(σ)p·γ
ω1 · · ·ωn
ω1 · · ·ωn.
We can also verify the case where γ is in the direction opposite to σ in the same
We have the equality
e♯σ×e♯τ
π∗Φ ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
e♯σ×e♯τ
− p∗1(α1 ∧ β1)− p
2(α1 ∧ β1) + p
1αj ∧ p
2βj − p
1βj ∧ p
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
In fact, if η ∈ Λ(M ×M) then
(−1)|η|+1
e♯σ×e♯τ
π∗dη ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
e♯σ×e♯τ
π∗η ∧ d
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
+(e♯σ)
ω1 · · ·ωk
(e♯τ)
ωk+1 · · ·ωj ∧ ωj+1 · · ·ωp+q−2
The last equality is obtained by the following lemma.
Lemma 6.6. If σ ∈ FpC0(LM), then
(e♯σ)
ω1 · · ·ωp−2 = 0.
Proof. It suffices to show the case σ = (τ1 − x) · · · (τp − x) where x ∈ M and τi
∈ ΩxM . We define τ̄i ∈ ΩxM by
τ̄i(t) =
τi(pt), if (i − 1)/p ≤ t ≤ i/p
0, otherwise.
Let σ̄ denote (τ̄1 − x) · · · (τ̄p − x). It can be shown that e♯σ̄ restricted on [(i −
1)/p, i/p] is contained in Fp−1C1(LM) for any i. Therefore
(e♯σ)
ω1 · · ·ωp−2 = (e♯σ̄)
ω1 · · ·ωp−2 = 0.
Jones, Geztler, and Petrack describes the map e♯ in terms of iterated inte-
grals by the following theorem.
Theorem 6.7 (Geztler-Jones-Petrack [8]). If σ ∈ C0(LM) and ω, ωi ∈ Λ
1 ≤ i ≤ p, then
π∗ω ∧
ω1 · · ·ωp =
ωk · · ·ωpωω1 · · ·ωk−1.
This theorem asserts the equality
e♯σ×e♯τ
− p∗1(α1 ∧ β1)− p
2(α1 ∧ β1) + p
1αj ∧ p
2βj − p
1βj ∧ p
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωn
j,k,l
ωk+1 · · ·ωp−1αjω1 · · ·ωk
ωl+1 · · ·ωp+q−2βjωp · · ·ωl
ωk+1 · · ·ωp−1βjω1 · · ·ωk
ωl+1 · · ·ωp+q−2αjωp · · ·ωl
Finally we obtain the equality
[σ,τ ]
ω1 · · ·ωp+q−2
j,k,l
ωk+1 · · ·ωp−1αjω1 · · ·ωk
ωl+1 · · ·ωp+q−2βjωp · · ·ωl
ωk+1 · · ·ωp−1βjω1 · · ·ωk
ωl+1 · · ·ωp+q−2αjωp · · ·ωl
Since we can take ωi ∈ H
1(M), 1 ≤ i ≤ p + q − 2, so that their support are
inside the tubular neighborhoods of aj and bj, we obtain the proposition.
Proof of theorem 6.1. We obtain the following isomorphism of Lie algebras by
proposition 6.4.
Rπ̃/J̃p ∼= H0(Hom(B(A), A
To obtain the isomorphism of Lie algebras
H0(Hom(B(A), A
∨) ∼= H0(Hom(B(H
∗(M)), H∗(M)∨),
we introduce the following lemma, which asserts the formality of the compact
Kähler manifolds.
Lemma 6.8 (ddcLemma, Deligne-Griffiths-Morgan-Sullivan [7]). Let X be a
compact Kähler manifold and dc = J−1dJ where J gives the complex structure
in the cotangent bundle. If α is a differential form on X such that dα = 0 and
dcα = 0, and such that α = dγ, then α = ddcβ for some β.
Cor. There are quasi-isomorphisms of differential graded algebras
(ΛX, d)← (kerdc, d)→ (H∗dc(X), 0).
Notice that a closed oriented surface endowed with a complex structure become
a Kähler manifolds for the dimensional reason. Therefore the following lemma
completes the proof of the theorem.
Lemma 6.9. If f : A1 → A2 is a quasi-isomorphism of differential graded
algebras, then the map induced by f
H0(Hom(B(A1), A
1 )→ H0(Hom(B(A2), A
is an isomorphism.
Proof. It suffices to verify that the map induced by f
f : H0(Hom(F
pB(A1), A
1 )→ H0(Hom(F
pB(A2), A
is an isomorphism for any p. On E1-level, the map induced by f
Hom(⊗sH(A1), H(A1)
∨)→ Hom(⊗sH(A2), H(A2)
is an isomorphism because f is quasi-isomorphism. Therefore we obtain the
lemma.
Therefore we obtain the theorem.
References
[1] M. Chas and D. Sullivan, String topology, preprint, 1999, http://arXiv.org
/abs/math.GT/9911159.
[2] K.T. Chen, Iterated integrals of differential forms and loop space homology,
Ann. of Math. (2) 97(1973), 217-246.
[3] K.T. Chen, Iterated integrals, fundamental groups and covering spaces,
Trans. Amer. Math. Soc. 206 (1975), 83-98.
[4] K.T. Chen, Reduced bar constructions on de Rham complexes, in:A.Haller
and M.Tierney (eds), (Algebra, topology and category theory, 1977, pp. 19-
[5] K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no.5,
831-879.
[6] R.L. Cohen, J.D.S. Jones and J. Yan, The loop homology algebra of spheres
and projective spaces, Categorical Decomposition Techniques in Algebraic
Topology (Isle of Skye, 2001), Progr. Math., vol. 215. Birkhäuser, Basel,
2004, pp.77-92.
[7] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory
of Kähler manifolds, Invent. Math. 29 (1975), 245-274.
[8] E. Getzler, J.D.S. Jones and S. Petrack Differential forms on loop spaces
and the cyclic bar complex, Topology 30 (1991), no.3, 339-371.
http://arXiv.org
[9] W.M. Goldman, Invariant functions on Lie groups and Hamlitonian flows
of surface group representation, Invent. Math. 85 (1986), no.2, 263-302.
[10] S.A. Merkulov, De Rham Model for String Topology, International Mathe-
matics Research Notices 55 (2004), 2955-2981.
Introduction
Chen's iterated integrals
Preliminaries
Proof of Theorem 1.1
The conjugacy classes of fundamental groups
The Goldman bracket
|
0704.0015 | Fermionic superstring loop amplitudes in the pure spinor formalism | arXiv:0704.0015v2 [hep-th] 10 Mar 2008
Preprint typeset in JHEP style - HYPER VERSION
Fermionic superstring loop amplitudes in the
pure spinor formalism
Christian Stahn
Department of Physics, University of North Carolina
Chapel Hill, NC 27599–3255, USA
E-mail: [email protected]
Abstract: The pure spinor formulation of the ten-dimensional superstring leads to
manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor super-
space. This paper explores different methods to evaluate these integrals and then uses
them to calculate the kinematic factors of the one-loop and two-loop massless four-point
amplitudes involving two and four Ramond states.
Keywords: Superstrings, Pure Spinors.
http://arxiv.org/abs/0704.0015v2
mailto:[email protected]
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Zero mode integration 2
2.1 Symmetry considerations and tensorial formulae 3
2.2 A spinorial formula 5
2.3 Component-based approach 7
3. One-loop amplitudes 7
3.1 Review: four bosons 8
3.2 Four fermions 10
3.3 Two bosons, two fermions 10
4. Two-loop amplitudes 12
4.1 Review: four bosons 13
4.2 Four fermions 14
4.3 Two bosons, two fermions 15
5. Discussion 16
A. Reduction to kinematic bases 17
A.1 Four bosons 17
A.2 Four fermions 18
A.3 Two bosons, two fermions 20
B. A gamma matrix representation 21
1. Introduction
The quantisation of the ten-dimensional superstring using pure spinors as world-sheet
ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and
Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest space-
time supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multi-
loop amplitudes, leading to new insights into perturbative finiteness of superstring theory
[2, 3].
Counting fermionic zero modes is a powerful technique in the computation of loop
amplitudes in the pure spinor formalism and has for example been used to show that
at least four external states are needed for a non-vanishing massless loop amplitude [2].
Furthermore, the structure of massless four-point amplitudes is relatively simple because all
– 1 –
fermionic worldsheet variables contribute only through their zero modes. In the expressions
derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest
by expressing the kinematic factors as integrals over pure spinor superspace [5] involving
three pure spinors λ and five fermionic superspace coordinates θ,
K1-loop =
(λA)(λγmW )(λγnW )Fmn
K2-loop =
(λγmnpqrλ)(λγsW )FmnFpqFrs
(1.1)
where the pure spinor superspace integration is denoted by 〈. . . 〉, and Aα(x, θ), Wα(x, θ)
and Fmn(x, θ) are the superfields of ten-dimensional Yang-Mills theory.
The kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states
at two loops [6] and one loop [7], and were found to match the amplitudes derived in
the RNS formalism [8]. This provided important consistency checks in establishing the
validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had
been reported in [9].)
In this paper, it will be shown how to compute the kinematic factors in (1.1) when
the superfields are allowed to contribute fermionic fields, as is relevant for the scattering
of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that
the calculation of fermionic amplitudes presents no additional difficulties, making (1.1)
a good practical starting point for the computation of four-point loop amplitudes in a
unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was
also emphasised in [10], where the tree-level amplitudes were used to construct the fermion
and Ramond/Ramond form contributions to the four-point effective action of the type II
theories.
This paper is organised as follows. In section 2, different methods to compute pure
spinor superspace integrals are explored. These methods are then applied to the explicit
evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level
in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calcu-
lations are briefly reviewed before separately considering the cases of two and four Ramond
states. Particular attention will be paid to the constraints imposed by simple exchange
symmetries. An appendix contains algorithms which were used to reduce intermediate
expressions encountered in the amplitude calculations to a canonical form.
2. Zero mode integration
The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over
zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component
Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure
spinor constraint (λγmλ) = 0. The amplitude prescriptions [1, 2] require three zero modes
of λ and five zero modes of θ to be present, and a Lorentz covariant object
T̄ αβγ,δ1...δ5 ≡
λαλβλγθδ1 . . . θδ5
= T̄ (αβγ),[δ1...δ5] (2.1)
was constructed such that the Yang-Mills antighost vertex operator
V = (λγmθ)(λγnθ)(λγpθ)(θγmnpθ) has
= 1 . (2.2)
– 2 –
In this section, different methods of computing such “pure superspace integrals” are ex-
plored. As an example, a typical correlator encountered in the two-loop calculations of
section 4 is considered:
F (ki, ui) = k
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
(2.3)
Here, ki and ui are the momenta and spinor wavefunctions of the four external particles.
2.1 Symmetry considerations and tensorial formulae
One systematic approach to evaluate the zero mode integrals is to find expressions for all
tensors that can be formed from (2.1). By Fierz transformations, one can always write the
product of two θ spinors as (θγ[3]θ), where γ[k] denotes the antisymmetrised product of k
gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ[5]λ), and
it is thus sufficient to consider the three cases
(λγ[5]λ)(λ{γ[1] or γ[3] or γ[5]}θ)(θγ[3]θ)(θγ[3]θ)
. (2.4)
Lorentz invariance then implies that it must be possible to express these tensors as sums of
suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus
a parity-odd part made up from terms which in addition contain an epsilon tensor. The
parity-even parts may be constructed [6] starting from the most general ansatz compatible
with the symmetries of the correlator and then using spinor identities along with the
normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the
spinor bilinears can be used to determine the parity-odd part [7]. An extensive (and almost
exhaustive) list of correlators is found in [11], including the (λγ[1]θ) and (λγ[3]θ) cases of
the above list:
(λγmnpqrλ)(λγuθ)(θγfghθ)(θγjklθ)
= − 4
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
δm̄n̄fg δ
(δr̄l δ
u + δ
u − δr̄uδhl )
[fgh][jkl]
(2.5)
(λγmnpqrλ)(λγstuθ)(θγfghθ)(θγjklθ)
= −24
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
δm̄j δ
δq̄sδ
u − δkhδr̄u)
[fgh][jkl](fgh↔jkl)
(2.6)
(Here, the brackets (fgh↔ jkl) denote symmetrisation under simultaneous interchange of
fgh with ijk, with weight one.) The remaining correlator with the (λγ[5]θ) factor can be
derived in the same way, using an ansatz consisting of six parity-even structures. Taking
a trace between the two γ[5] factors and noting that
(λγmnpqrλ)(λγabcdeθ) . . .
(λγmnpq[bλ)(λγcde]θ) . . .
one finds a relation to (2.6). This is sufficient to determine all coefficients in the ansatz,
and the result is
(λγmnpqrλ)(λγabcdeθ)(θγfghθ)(θγjklθ)
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
m̄n̄p̄
(−δehδr̄l + 2δel δr̄h) + δm̄n̄ab δcdfgδ
(δehδ
l − 3δel δr̄h)
[abcde][fgh][jkl](fgh↔jkl)
(2.7)
– 3 –
One may find it surprising that the derivation of these tensorial expressions only made
use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it
can be seen from representation theory that the correlator (2.1) is uniquely characterised,
up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ3
and θ5 transform in
λ(αλβλγ) : Sym3 S+ = [00003] ⊕ [10001]
θ[δ1 . . . θδ5] : Alt5 S+ = [00030] ⊕ [11010] .
(2.8)
(Here, λ and θ are taken to be in the S+ irrep of SO(1,9), with Dynkin label [00001].) The
tensor product of these contains only one copy of the trivial representation. This applies
to any spinors λ, which means that the pure spinor property cannot be essential to the
derivation of the tensorial identities. The use of the pure spinor constraint merely allows
for simpler derivations of the same identities.
As an illustration of this approach, consider the correlator of eq. (2.3). Leaving the
momenta aside for the moment by setting F = k2ak
r F̃ , the task is to compute
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
After applying two Fierz transformations,
(λγmnpq[r|λ)(λγcθ)(θγn
abθ)(θγjklθ)
|s]γcγbu2)
3!·16
(λγmnpq[r|λ)(λγcdeθ)(θγn
abθ)(θγjklθ)
|s]γcdeγbu2)
2·5!·16
(λγmnpq[r|λ)(λγcdefgθ)(θγn
abθ)(θγjklθ)
|s]γcdefgγbu2)
3!·16(u3γqγjklγsu4) ,
one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7).
A reliable evaluation of the numerous index symmetrisations is made possible by the use of
a computer algebra program. In doing these calculations with Mathematica, an essential
tool is the GAMMA package [13], expanding the products of gamma matrices in a γ[k] basis.
The result consists of two parts, F̃ = F̃ (δ) + F̃ (ε), with
F̃ (δ) = 1
mpru2)(u3γ
au4) +
r (u1γ
iu2)(u3γiu4) + . . .
ai1i2u2)(u3γ
i1i2u4) (92 terms) (2.9)
F̃ (ε) = − 1
1209600
εi1...i7
mpr(u1γ
i1...i7u2)(u3γ
au4) + . . .
604800
εampri1...i6(u1γ
i3...i9u2)(u3γ
i7i8i9u4) (34 terms) (2.10)
The epsilon tensors in the second part can be eliminated using the fact that the ui are
chiral spinors: If all the indices on γ[k]ui are contracted into an epsilon tensor, one uses
εi1...ik′j1...jkγ
j1...jkγ11 = (−)
k(k+1)
k! γi1...ik′ , (2.11)
where γ11 = 1
εi0...i9γ
i0...i9 . More generally, if all but r indices of γ[k]ui are contracted,
εi1...ik′ j1...jkγ
p1...prj1...jkγ11 = (−)
k(k+1)
(k′ − r)!
pr ...p1
[i1...ir
γir+1...i′k]
. (2.12)
– 4 –
The result of these manipulations is
F̃ (ε) =− 1
mpru2)(u3γ
au4)− 1280δ
r (u1γ
amiu2)(u3γiu4) + . . .
11200
i1i2i3u2)(u3γ
i1i2i3u4) (53 terms) (2.13)
(Note that while the epsilon terms in the basic correlator formulae were easily obtained from
the delta terms by using Poincaré duality, this cannot be done here in any obvious way.)
The last step in the evaluation of (2.3) is to contract with the momenta, F = k2ak
r F̃ ,
and to simplify the expressions using the on-shell identities
i ki = 0, k
i = 0, /kiui = 0. It
is shown in appendix A.2 that there are only ten independent scalars, denoted by B1 . . . B10,
that can be formed from four momenta and the four spinors u1 . . . u4. With respect to this
basis, the result is
F (δ) = 1
48·10080
695s12(u1/k3u2)(u3/k1u4) + · · ·+ 233s213(u1γau2)(u3γau4)
(7 terms)
48·10080 (695, 775, 0,−80, 356, 356, 0, 233, 233, 0)B1 ...B10 ,
F (ε) = 1
48·10080 (−23,−7, 0,−16, 28, 28, 0, 7, 7, 0)B1 ...B10 ,
F = 1
10080
(14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.14)
where sij = ki · kj .
2.2 A spinorial formula
While the derivation of tensorial identities for correlators of the form (2.4) is relatively
straightforward and elegant, it may be a tedious task to transform the expressions encoun-
tered in amplitude calculations to match this pattern. As seen in the example calculated
above, this is particularly true if additional spinors are involved, making it necessary to ap-
ply Fierz transformations. It is therefore desirable to use a covariant correlator expression
with open spinor indices. Such an expression was given in [1, 2]:
T̄αβγ,δ1...δ5 = N−1
(γm)αδ1(γn)βδ2(γp)γδ3(γmnp)
(αβγ)[δ1...δ5]
, (2.15)
where N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation
with weight one. (Note that the right hand side is automatically gamma-matrix traceless:
any gamma-trace
(γr)αβ × (γm)α[δ1|(γn)β|δ2|(γp)γ|δ3(γmnp)δ4δ5] = −(γmnr)[δ1δ2(γmnp)δ3δ4(γp)δ5]γ = 0
vanishes due to the double-trace identity (γabθ)
α(θγabcθ) = 0, which follows from the
fact that the tensor product (Alt3 S+)⊗ S− does not contain a vector representation and
therefore the vector (ψγabθ)(θγ
abcθ) has to vanish for all spinors ψ, and can also be shown
by applying a Fierz transformation.) This prescription was originally motivated [2] by the
fermionic expansion of the Yang-Mills antighost vertex operator V ,
V = Tαβγ,δ1...δ5λ
αλβλγθδ1 . . . θδ5 (2.16)
Tαβγ,δ1...δ5 =
(γm)αδ1(γ
n)βδ2(γ
p)γδ3(γmnp)δ4δ5
(αβγ)[δ1...δ5]
– 5 –
where T is related to T̄ by a parity transformation, up to the overall constant N . (Since
T̄ is uniquely determined by its symmetries, any covariant expression will be proportional
to T̄ , after symmetrisation of the spinor indices, and this is merely the simplest choice.)
Equation (2.15) immediately yields an algorithm to convert any correlator into traces
of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It
is, however, already very tiresome to determine the normalisation constant N by hand.
The main advantage of this approach is that it clearly lends itself to implementation on
a computer algebra system, which can easily carry out the spinor index symmetrisations,
simplify the gamma products (again using the GAMMA package), and compute the traces.
For example,
N〈V 〉 =
(γm)αδ1(γn)βδ2(γp)γδ3(γmnp)
(αβγ)[δ1...δ5]
(γx)αδ1(γy)βδ2(γz)γδ3(γ
xyz)δ4δ5
= − 1
Tr(γxγ
m)Tr(γyγ
n)Tr(γzγ
p)Tr(γxyzγpnm) + . . .
Tr(γzγpnmγ
zyxγnγxγ
p) (60 terms)
= 5160960 .
The correct normalisation is therefore obtained by setting N = 5160960.
Returning to the example correlator (2.3), one finds that the calculation is by far
simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δi],
one obtains
NF̃ = 1
Tr(γxγ
mnpq[r|)(u3γqγ
xyzγsu4)(u1γ
|s]γzγbu2) + . . .
(u2γbγ
xyzγqu3)(u1γsγyγ
mnpq[rγzγ
s]u4) , (24 terms)
where elementary index re-sorting has reduced the number of terms from 60 to 24. Ex-
panding the gamma products leads to
NF̃ = 476
δpr (u1γ
mu4)(u2γ
au3) + · · ·+ 815(u1γ
ai1i2i3i4u2)(u3γ
i1i2i3i4u4) , (294 terms)
which, in contrast to (2.10), contains no epsilon terms as there are not enough free indices
present. Note that this intermediate result contains terms with with u1 paired with u3
or u4, so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after
contracting with the momenta k2ak
r and decomposing the result in the basis B1 . . . B10,
one again obtains
F = 1
10080
(14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.17)
in agreement with (2.14).
The algorithm just outlined will be the method of choice for all correlator calculations
in the later sections of this paper and can easily be applied to a wider range of problems.
The only limitation is that the larger the number of gamma matrices and open indices
of the correlator, the slower the computer evaluation will be. For example, the correlator
considered in eq. (5.2) of [11],
mnm1n1...m4n4
(λγpγm1n1θ)(λγqγm2n2θ)(λγrγm3n3θ)(θγmγnγpqrγ
m4n4θ)
= − 2
m1n1...m4n4
εmnm1n1...m4n4
, (2.18)
can still be verified with this method but this already requires substantial runtime.
– 6 –
2.3 Component-based approach
A third method to evaluate the zero mode integrals consists of choosing a gamma matrix
representation, expanding the integrand as a polynomial in spinor components, and then
applying (2.15) to the individual monomials. This procedure seems particularly appealing
if at some stage of the calculation one works with a matrix representation anyhow, in
order to reduce the results to a canonical form (e.g. as outlined in appendix A). An
efficient decomposition algorithm (of k4u1u2u3u4 scalars, say) only needs a few non-zero
momentum and spinor wavefunction components to distinguish all independent scalars, and
therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation
allows for a much quicker numeric evaluation of correlator components than a naive use
of (2.15): In view of (2.16), one can equivalently compute the components of the parity-
transformed expression V̄ = (λ̄γmθ̄)(λ̄γnθ̄)(λ̄γpθ̄)(θ̄γmnpθ̄), where λ̄ and θ̄ are spinors of
chirality opposite to that of λ, θ. In the representation given in appendix B, V̄ coincides
with V |λ→λ̄,θ→θ̄, and
V = 192λ9λ9λ9θ1θ2θ3θ4θ9 + · · ·+ 480λ1λ2λ3θ1θ9θ10θ13θ15 + . . . (100352 terms)
The monomials in the fermionic expansion of V̄ then correspond to the arguments of
non-zero correlators, and the coefficients of those monomials are, up to normalisation and
symmetry factors, the correlator values.
Unfortunately, it turns out that the complexity of typical correlators (e.g. the one
given in (2.3)) makes it difficult to carry out the expansion in fermionic components in
any straightforward way and limits this method to special applications. For example, the
coefficients in (2.18) can be checked relatively easily by choosing particular index values,
such as
(λγpγ12θ)(λγqγ21θ)(λγrγ34θ)(θγ0γ0γpqrγ
12λ1λ1λ1θ1θ9θ10θ11θ12 + · · ·+ 12λ16λ16λ16θ5θ6θ7θ8θ16
(For fixed values of pqr, one gets no more than about 105 monomials of the form λ3θ5).
This approach may thus still be helpful in situations where the result has been narrowed
down to a simple ansatz.
3. One-loop amplitudes
The amplitude for the scattering of four massless states of the type IIB superstring was
computed [2] in the pure spinor formalism as
A = KK̄
(Im τ)5
G(zi, zj)
ki·kj , (3.1)
where G(zi, zj) is the scalar Green’s function, and the kinematic factor is given by the
product KK̄ of left- and right-moving open superstring expressions,
K1-loop =
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
cycl(234)
. (3.2)
– 7 –
Here the indices 1 . . . 4 label the external states and “· · ·+
cycl(234)
” denotes the addition
of two other terms obtained by cyclic permutation of the indices 234. The spinor super-
field Aα and its supercovariant derivatives, the vector gauge superfield Am =
m DαAβ
as well as the spinor and vector field strengths Wα = 1
(γm)αβ(DβAm − ∂mAβ) and
Fmn = 18(γmn)
β = 2∂[mAn], describe ten-dimensional super-Yang-Mills theory.
The physical fields of this theory, a gauge boson and a gaugino, are found in the leading
components Am| = ζm and Wα| = ûα and correspond to the Neveu-Schwarz and Ramond
superstring states.
The superfields Aα and W
α as well as the gaugino field ûα are anticommuting.1 To
facilitate computer calculations involving polynomials in the spinor components, and for
easier comparison with the literature, it will be more convenient to work with commuting
fermion wavefunctions uα. Fortunately, as the kinematic factors with fermionic external
states are multilinear functions of the distinctly labelled spinors ûi, it is straightforward to
translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly
fermionic coordinates θ) corresponds to the same expression in u1 . . . u4, multiplied by the
signature of the permutation sorting the ûi (and any θ variables) into some fixed order,
such as (θ · · · θ)ûα11 û
Choosing a gauge where θαAα = 0, the on-shell identities
2D(αAβ) = γ
αβAm , DαW
β = 1
(γmn)α
have been used to derive recursive relations [10, 14, 15] for the fermionic expansion
A(n)α =
(γmθ)αA
(n−1)
m , A
(θγmW
(n−1)) , Wα(n) = − 1
(γmnθ)α∂mA
(n−1)
where f (n) = 1
θαn · · · θα1(Dα1 · · ·Dαnf)|. These recursion relations were explicitly solved
in [10], reducing the fermionic expansion to a simple repeated application of the derivative
operator Omq = 12 (θγm
qpθ)∂p:
A(2k)m =
(2k)!
[Ok]mqζq ,
A(2k+1)m =
(2k+1)!
[Ok]mq(θγqû) .
(3.3)
With this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2)
for the three cases of zero, two, or four fermionic states.
3.1 Review: four bosons
The kinematic factor involving four bosons was considered in [7] and this calculation will
now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result
must be gauge invariant [2] and therefore expressible in terms of the field strengths F1 . . . F4.
The cyclic symmetrisation in (3.2) yields expressions symmetric in F2, F3, F4, and acting
on scalars constructed from the Fi only, the (234) symmetrisation is equivalent to complete
symmetrisation in all labels (1234). Thus the result must be a linear combination of the
1Thanks to Carlos Mafra for pointing this out.
– 8 –
two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F(1F2F3F4)) and
double trace Tr(F(1F2)Tr(F3F4)), leaving one relative coefficient to be determined.
Since all four states are of the same kind, one may first evaluate the correlator for one
labelling and then carry out the cyclic symmetrisation:
1-loop =
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
cycl (234)
The different ways to saturate θ5 result in a sum of terms of the form
XABCD =
1 )(λγ
2 )(λγ
(3.4)
with A+B +C +D = 5 and A, B, C odd, D even:
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
= X3110 +X1310 +X1130 +X1112 .
Note that X1310 and X1130 are related by exchange of the labels 2 and 3. This exchange
can be carried out after computing the correlator, an operation which will in the following
be denoted by π23. Using (3.3) for the superfield expansions and replacing ∂m → ikm, one
obtains
X3110 = − 1512F
tuX̃3110 , X̃3110 =
(λγ[t|γpqθ)(λγ|u]γrsθ)(λγaθ)(θγ
amnθ)
X1112 = − 1128 ik
tuX̃1112 , X̃1112 =
(λγ[m|γpqθ)(λγ|a]γrsθ)(λγnθ)(θγa
X1310 = − 1384 ik
tuX̃1310 , X̃1310 =
(λγ[t|γmaθ)(λγ|u]γrsθ)(λγnθ)(θγa
The method outlined in section 2.2 is readily applicable to these correlators. For example,
for X3111, the trace evaluation yields
X̃3110 = N
Tr(γaγ
z)Tr(γxyzγ
anm)Tr(γxγqpγ
[t|)Tr(γyγsrγ
|u]) + · · ·
· · ·+ 1
Tr(γ[u|γrsγzyxγqpγ
|t]γxγaγ
yγmnaγz)
(60 terms)
δmprs δ
tu − 1315δ
rs − 145δ
δmnpr δ
[mn][pq][rs][tu](pq↔rs)
Upon contracting with the field strengths, momenta and polarisations, and symmetrising
over the cyclic permutations (234) (with weight 3), one finds that all three contributions
are separately gauge invariant:
X3110 +
cycl(234)
= − 11
13440
Tr(F(1F2F3F4)) +
Tr(F(1F2)Tr(F3F4))
X1112 +
cycl(234)
= − 19
53760
Tr(F(1F2F3F4)) +
215040
Tr(F(1F2)Tr(F3F4))
(1 + π23)X1310 +
cycl(234)
= − 1
10240
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
The sum X3110 +X1112 has the right ratio of single- and double-trace terms to be propor-
tional to the well-known result t8F
4, and the last line exhibits the right ratio by itself. The
overall kinematic factor is therefore
K4B1-loop = − 12560
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
= − 1
15360
4 , (3.5)
in agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] for-
malisms.
– 9 –
3.2 Four fermions
The four-fermion kinematic factor could be evaluated in the same way as in the four-boson
case by summing up all terms XABCD, A + B + C + D = 5, now with A, B, C even
and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic
symmetrisation in (3.2) leads to a completely symmetric dependence on û2, û3, û4, and
therefore to a completely antisymmetric dependence on u2, u3, u4. Acting on scalars of
the form k2u1u2u3u4, antisymmetrising over [234] is equivalent to antisymmetrising over
[1234], and there is only one completely antisymmetric k2u1u2u3u4 scalar. Without further
calculation, one can infer that the kinematic factor is proportional to that scalar,
K4F1-loop = const ·
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
which of course agrees with the RNS amplitude (see e.g. [16], eq. (3.67)).
3.3 Two bosons, two fermions
In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect
whether the W and F superfields contribute bosons or fermions. Only the label of the Aα
superfield stays unaffected, and one has to choose whether it should contribute a boson
or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector,
A1,α ∼ (/ζ1θ)α, the calculation can be simplified by choosing a labelling where particle 1 is
a fermion. (Of course, the final result must be independent of this choice.) The assignment
of the other three labels is then irrelevant and will be chosen as f1f2b3b4. Writing out
the cyclic permutations, two of the three terms are essentially the same because they are
related by interchange of the labels 3 and 4. The kinematic factor is then
K2B2F1-loop(f1f2b3b4) = (1 + π34)
(even)
1 )(λγ
(even)
2 )(λγ
(odd)
(even)
(even)
1 )(λγ
(odd)
3 )(λγ
(odd)
(odd)
Unlike in the four-fermion calculation, the result is not fixed by symmetry. There are five
independent ku1u2F3F4 scalars (see appendix A, eq. (A.6)), denoted by C1 . . . C5, and there
are two independent combinations of these scalars with the required [12](34) symmetry.
Expanding the superfields and collecting terms with θ5, the first line yields a combination
of terms XABCD with A, B, D odd and C even. There is only one θ
5 combination coming
from the second line, which will be denoted by X ′2111 ≡ (−π24)X2111:
K2B2F1-loop = (1 + π34) (X4010 +X2210 +X2030 +X2012) +X
2111 ,
with the correlators
X4010 =
ζ3c k
nX̃4010 , X̃4010 =
(λγaθ)(θγa
pqθ)(θγpu1)(λγ
[mu2)(λγ
n]γbcθ)
X2210 = − i12k
nX̃2210 , X̃2210 =
(λγaθ)(θγau1)(λγ
[m|γbcθ)(θγcu2)(λγ
|n]γdeθ)
X2030 = − i36k
nX̃2030 , X̃2030 =
(λγaθ)(θγau1)(λγ
[mu2)(λγ
n]γbcθ)(θγc
X2012 = − i12k
ζ3c k
ζ4e X̃2012 , X̃2012 =
(λγaθ)(θγau1)(λγ
[mu2)(λγ
n]γbcθ)(θγn
X ′2111 =
ζ3c k
2111 , X̃
2111 =
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγnu2)
– 10 –
(The numerical coefficient in X ′2111 includes a sign coming from the θ, û ordering: there
is an odd number of θs between u1 and u2.) Evaluating these expressions as outlined in
section 2.2, the spinor wavefunctions ui present no complication. The last part takes the
simplest form: One finds
(λγaθ)(θγau1)(λγ
mγbcθ)(λγnγdeθ)(θγnu2)
= − 1
(2δbcm[d(u1γe]u2) + δ
m(u1γ
c]deu2))
and therefore
X̃ ′2111 = − 1480
δ[bm(u1γ
c]γdeu2) + δ
m(u1γ
e]γbcu2)
The result for X̃4010 is
X̃4010 =
δbqmn(u1γ
cu2)− 190δ
mq(u1γ
nu2) +
δbcmn(u1γ
qu2)− 12520δ
q (u1γ
bcnu2)
δbq(u1γ
cmnu2) +
δbm(u1γ
cnqu2) +
bcmnqu2)
[bc][mn]
For the evaluation of X̃2210, it is useful to consider the more general correlator
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγxu2)
mn(u1γ
cu2) + . . .
201600
δmx (u1γ
bcdenu2) + · · · − 11403200 (u1γ
bcdemnxu2)
[mn][bc][de]
(27 terms)
9676800
εbcdemni1i2i3i4(u1γ
i1i2i3i4xu2)− 12419200εbcdemnxi1i2i3(u1γ
i1i2i3u2) .
This time, even using the method of section 2.2, there are sufficiently many open indices
and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they
can be re-written into γ[5,7] terms:
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγxu2)
mn(u1γ
cu2) + . . .
16800
δmx (u1γ
bcdenu2) + · · · − 133600 (u1γ
bcdemnxu2)
[mn][bc][de]
(27 terms)
A good check on the sign of the epsilon contributions is that X̃ ′2111 is recovered when
contracting with ηnx, involving a cancellation of all γ
[5] terms. To obtain X̃2210, one
multiplies by −ηcx:
X̃2210 =
δdemn(u1γ
bu2) +
δbdmn(u1γ
eu2) +
δbmde (u1γ
nu2) +
20160
δdm(u1γ
benu2)
δbm(u1γ
denu2) +
20160
δbd(u1γ
emnu2) +
bdemnu2)
[de][mn]
For the calculation of X2030 and X2012, one may first evaluate a more general correlator
〈(λγaθ)(θγau1)(λγ[mu2)(λγn]γbcθ)(θγxγdeθ)〉 and then contract with ηcx and ηnx, respec-
tively. The results are
X̃2030 =
δdemn(u1γ
bu2) +
δbdmn(u1γ
eu2)− 11440δ
de (u1γ
nu2)− 1710080δ
m(u1γ
benu2)
10080
δbm(u1γ
denu2)− 11440δ
d(u1γ
emnu2) +
bdemnu2)
[mn][de]
X̃2012 =
δdebm(u1γ
cu2) +
δbcdm(u1γ
eu2)− 11440δ
de(u1γ
mu2) +
δdm(u1γ
bceu2)
10080
δbm(u1γ
cdeu2) +
10080
δbd(u1γ
cemu2)− 13360 (u1γ
bcdemu2)
[bc][de]
– 11 –
After multiplication with the momenta and polarisations, all individual contributions are
gauge invariant and can be expanded in the basis C1 . . . C5 listed in (A.6):
(1 + π34)X4010 =
483840
(−6,−16,−40, 6, 0)C1 ...C5
(1 + π34)X2210 =
483840
(−18,−104,−176, 18, 0)C1 ...C5
(1 + π34)X2030 =
483840
(−21, 42,−42, 21, 0)C1 ...C5
(1 + π34)X2012 =
483840
(−39, 78,−78, 39, 0)C1 ...C5
X ′2111 = − i11520 (1, 0, 4,−1, 0)C1 ...C5
The sum can be written as
K2B2F1-loop = X
2111 = − i3840 (1, 0, 4,−1, 0)C1 ...C5
= − i
s13(u2/ζ3(/k2 + /k3)/ζ4u1) + s23(u2/ζ4(/k2 + /k4)/ζ3u1)
(3.6)
and again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37).
4. Two-loop amplitudes
The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude
involving four massless states,
d2Ω11d
2Ω12d
i,j ki · kj G(zi, zj)
(det ImΩ)5
K2-loop(ki, zi) ,
where Ω is the genus-two period matrix, and the integration over fermionic zero modes is
encapsulated in
K2-loop = ∆12∆34
(λγmnpqrλ)(λγsW1)F2,mnF3,pqF4,rs
perm(1234)
(4.1)
≡ ∆12∆34K12 +∆13∆24K13 +∆14∆23K14 . (4.2)
The kinematic factors K12, K13, K14 are accompanied by the basic antisymmetric biholo-
morphic 1-form ∆, which is related to a canonical basis ω1, ω2 of holomorphic differentials
via ∆ij = ∆(zi, zj) = ω1(zi)ω2(zj) − ω2(zi)ω1(zj). The superfields Wαi and Fi,mn are the
spinor and vector field strengths of the i-th external state, as in section 3. One encounters
superspace integrals of the form
Y (abcd) =
(λγmnpqrλ)(λγsWa)Fb,mnFc,pqFd,rs
. (4.3)
The symmetries of the λ3 combination [4] in this correlator include the obvious symmetry
under mn↔ pq, and also (λγ[mnpqrλ)(λγs])α = 0 (this holds for pure spinors λ and can be
seen by dualising, and holds for unconstrained spinors λ as part of a λ3θ5 scalar, as seen
from the representation content (2.8)), and allow one to shuffle the F factors:
Y (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4)
– 12 –
4.1 Review: four bosons
The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed
here. As all three kinematic factors K12, K13 and K14 are equivalent, it is sufficient to
consider K12 in detail. With all external states being identical, the symmetrisations of
(4.1) can be carried out at the end of the calculation:
K4B12 = 4
W[1F2]F[3F4]
W[3F4]F[1F2]
= (1− π12)(1− π34)(1 + π13π24)
W1F2F3F4
Expanding the superfields and adopting the notation
YABCD(abcd) =
(λγmnpqrλ)(λγsW (A)a )F
F (C)c,pqF
the Neveu-Schwarz states come from terms of the form YABCD ≡ YABCD(1234) with A odd
and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains
W1F2F3F4
= Y5000 + Y1400 + Y1040 + Y1004 + Y3200 + Y3020 + Y3002 + Y1220 + Y1202 + Y1022
= (1 + π23)(1− π24)
Y5000 + Y1400 + Y3200 + Y1022
and therefore K4B12 can be written as the image of a symmetrisation operator S4B:
K4B12 = S4B
Y5000 + Y1400 + Y3200 + Y1022
S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24)
It is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars
built from the four field strengths Fi and two momenta, the symmetriser S4B has rank four.
The correlators were computed in [6], using the method outlined in section 2.1. Two are
zero, Y5000 = Y1400 = 0, and the remaining ones are
Y3200 =
(λγmnpqrλ)(λγsγabθ)(θγb
cdθ)(θγn
Y1022 =
F 1abF
(λγmnpq[rλ)(λγs]γabθ)(θγq
cdθ)(θγs
In reducing those two contributions to a set of independent scalars, one finds that they
both are not just sums of (k · k)F 4 terms but also contain terms of the form k · F terms.
The latter are projected out by the symmetriser S4B, and the result is
K4B12 = S4B(Y3200 + Y1022) = 1120 (s13 − s23)
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
(s13 − s23)t8F 4 .
By trivial index exchange, one obtains K13 and K14, and the total is
K4B2-loop =
(s13 − s23)∆12∆34 + (s12 − s23)∆13∆24 + (s12 − s13)∆14∆23
4 , (4.5)
a product of the completely symmetric one-loop kinematic factor t8F
4 and a completely
symmetric combination of the momenta and the ∆ij.
– 13 –
4.2 Four fermions
The calculation involving four Ramond states is very similar to the bosonic one. Focussing
on the K12 part, the symmetrisations in (4.1) can again be rewritten as action of sym-
metrisation operators on the correlator of superfields with one particular labelling:
K4F12 (ûi) = (1− π12)(1 − π34)(1 + π13π24)
W1F2F3F4
û1û2û3û4
= 4(1− π12)
W1F2F3F4
û1û2û3û4
The last step follows from the fact that all scalars of the form k4u4 (see appendix A.2),
and therefore all k4û4 scalars, are invariant under π13π24 and have π12 = π34. This time,
on expanding the superfields, one collects the terms YABCD with A even and B, C, D odd.
After using (4.4) to simplify,
W1F2F3F4
û1û2û3û4
= Y2111 + Y0311 + Y0131 + Y0113
= (1 + π23)(1− π24)
Y2111 + Y0311
and after translating to commuting wavefunctions ui, which multiplies every permutation
operator with its signature, one obtains
K4F12 (ui) = S4F
Y2111(ui) + Y0311(ui)
, S4F = 4(1 + π12)(1− π23)(1 + π24) .
This symmetriser has rank three, and the result is again not determined by symmetry.
Two correlators have to be computed:
Y2111(ui) = (−2)k1ak2mk3pk4r
(λγmnpq[rλ)(λγs]γabθ)(θγbu1)(θγnu2)(θγqu3)(θγsu4)
Y0311(ui) = (−23)k
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
With four fermions present, the method of section 2.2 is preferred as it does not involve re-
arranging the fermions using Fierz identities. The first correlator was covered as an example
in that section, and the second one can be evaluated in the same fashion. Expressed in the
basis listed in (A.5), the results are
Y2111(ui) =
(−19,−21, 21, 19,−17,−17, 0, 0, 0, 0)B1 ...B10 ,
Y0311(ui) =
15120
(−14,−16, 0, 2,−8,−8, 0,−5,−5, 0)B1 ...B10 .
After acting with the symmetriser S4F, one obtains the same u4 scalar encountered in the
one-loop amplitude,
K4F12 (ui) = S4F(13Y2111(ui) + Y0311(ui)) =
(−1,−2, 1, 2,−1,−2, 0, 0, 0, 0)B1 ...B10
(s23 − s13)
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
The K13 and K14 parts again follow by index exchange, and the total result
K4F2-loop(ui) =
(s23 − s13)∆12∆34 + (s23 − s12)∆13∆24 + (s13 − s12)∆14∆23
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
(4.6)
is again a simple product of the one-loop kinematic factor and a combination of the ∆ij
and momenta.
– 14 –
4.3 Two bosons, two fermions
As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention
to the permutations in (4.1) since they affect which superfields contribute fermionic fields.
The complete symmetrisation makes it irrelevant which labels are assigned to the two
fermions, and the convention f1f2b3b4 will be used here. The kinematic factor K
12 is then
distinguished from the other two, K2B2F13 and K
14 . Carrying out the symmetrisations in
(4.1) and using the identities (4.4), one finds
K12(û1, û2, ζ3, ζ4) = (1− π12)(1− π34)K̃ ,
K13(û1, û2, ζ3, ζ4) = (2 · 1+ π12 + π34 + 2π12π34)K̃ ,
K14(û1, û2, ζ3, ζ4) = (1+ 2π12 + 2π34 + π12π34)K̃ ,
where, schematically,
(even)
(odd)
(even)
(even)
(odd)
(even)
(odd)
(odd)
. (4.7)
In translating to commuting variables u1 and u2, the permutation operator π12 changes
sign, and therefore2
K12(u1, u2, ζ3, ζ4) = (1+ π12)(1− π34)K̃ ,
K13(u1, u2, ζ3, ζ4) = (2 · 1− π12 + π34 − 2π12π34)K̃ ,
K14(u1, u2, ζ3, ζ4) = (1− 2π12 + 2π34 − π12π34)K̃ .
Expanding the superfields, the contributions to K̃ are:
Y4100 = − i48k
(λγmnpqrλ)(λγsγabθ)(θγbγ
cdθ)(θγcu1)(θγnu2)
Y0500 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγb
cdθ)(θγdu2)
Y0140 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγq
abθ)(θγb
Y0104 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγ|s]
abθ)(θγb
Y2300 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγn
cdθ)(θγeu2)
Y2120 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγq
Y2102 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγ|s]
Y0320 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγbu2)(θγq
Y0302 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγbu2)(θγ|s]
Y0122 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγq
abθ)(θγs]
Y3011 =
(λγmnpqrλ)(λγsγabθ)(θγb
cdθ)(θγcu1)(θγnu2)
Y1211 =
F 3abk
(λγmnpqrλ)(λγsγabθ)(θγn
cdθ)(θγqu1)(θγ|s]u2)
Y1031 =
F 3abF
(λγmnpqrλ)(λγsγabθ)(θγq
cdθ)(θγdu1)(θγ|s]u2)
Y1013 =
F 3abF
(λγmnpqrλ)(λγsγabθ)(θγqu1)(θγ|s]
cdθ)(θγdu2)
2This sign change is crucial to avoid the erroneous conclusion that the two-boson, two-fermion kinematic
factor cannot be of the same product form as in the four-boson or four-fermion cases, which would be in
contradiction to the supersymmetric identities derived in [18].
– 15 –
These correlators can be evaluated exactly as described in section 3.3. One finds that
Y0500 = Y0140 = Y0104 = 0, and the sum of the remaining terms reduces to
K̃ = Y4100 + Y2300 + Y2120 + Y2102 + Y0320 + Y0302 + Y0122 + Y3011 + Y1211 + Y1031 + Y1013
(s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 .
After applying the symmetrisation operators,
(1+ π12)(1− π34)K̃ = i180 (s12 + 2s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
(2 · 1− π12 + π34 − 2π12π34)K̃ = i180 (2s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
(1− 2π12 + 2π34 − π12π34)K̃ = i180 (s12 − s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
the total kinematic factor is seen to be
K2-loop(u1, u2, ζ3, ζ4) = − i180
(s23−s13)∆12∆34+(s23−s12)∆12∆34+(s13−s12)∆12∆34
× (1, 0, 4,−1, 0)C1 ...C5 (4.8)
and displays the same simple product form as in the four-boson and four-fermion case.
5. Discussion
In this paper, different methods were discussed to efficiently evaluate the superspace inte-
grals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending
previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the
treatment of Ramond states poses no additional difficulties.
While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry,
already established the equivalence of the massless four-point amplitudes derived in the
pure spinor and RNS formalisms, it would be interesting to make contact between the
results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed
in the RNS formalism (see for example [19]).
The assistance of a computer algebra system seems indispensible in explicitly evaluat-
ing pure spinor superspace integrals. To avoid excessive use of custom-made algorithms,
it would be desirable to implement these calculations in a wider computational framework
particular adapted to field theory calculations [20].
The methods outlined in this paper should be easily applicable to future higher-loop
amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other
superspace integrals.
Acknowledgements
The author would like to thank Louise Dolan for discussions, and Carlos Mafra for valuable
correspondence. This work is supported by the U.S. Department of Energy, grant no. DE-
FG01-06ER06-01, Task A.
– 16 –
A. Reduction to kinematic bases
In calculating scattering amplitudes one encounters kinematic factors which are Lorentz
invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the
scattered particles. It can be a non-trivial task to simplify such expressions, taking into
account the on-shell identities
i ki = 0, k
i = 0, ki · ζi = 0, /kiui = 0, and, in the case of
fermions, re-arrangements stemming from Fierz identities.
More generally, one would like to know how many independent combinations of some
given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expres-
sion with respect to some chosen basis. This appendix outlines methods to address these
problems, with an emphasis on algorithms which can easily be transferred to a computer
algebra system. These methods are not limited to dealing with pure spinor calculations
but the scope will be restricted to amplitudes of four massless vector or spinor particles in
ten dimensions.
A.1 Four bosons
It is not difficult to reduce polynomials in the momenta and polarisations to a canonical
form. The momentum conservation constraint
i ki = 0 is solved by eliminating one
momentum (for example k4), all k
i are set to zero, and one of the two remaining quadratic
combinations of momenta is eliminated (for example s23 → −s12− s13, where sij ≡ ki ·kj).
Then all products ki · ζi are set to zero, and one extra k · ζ product is replaced (when
eliminating k4, the replacement is k3 · ζ4 → (−k1 − k2) · ζ4). The remaining monomials are
then independent. (This is at least the case with the low powers of momenta encountered
in the calculations of sections 3 and 4, where there are enough spatial directions for all
momenta/polarisations to be linearly independent.)
The implementation of these reduction rules on a computer is straightforward. The
easiest way to obtain scalars which are also invariant under the gauge symmetry ki → ζi
is to start with expressions constructed from the field strengths F abi = 2∂
i . For the
one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars
containing only the four field strengths F1 . . . F4. One finds six independent combinations,
Tr(F1F2F3F4)
Tr(F1F2F4F3)
Tr(F1F3F2F4)
Tr(F1F2)Tr(F3F4)
Tr(F1F3)Tr(F2F4)
Tr(F1F4)Tr(F2F3)
In the two-loop calculations of section 4.1, all monomials have two more momenta. There
are sixteen independent gauge invariant scalars of the form kkF1F2F3F4, and twelve of
them may be constructed from the previous basis by multiplication with s12 and s13:
A1 = s12 Tr(F1F2F3F4), A2 = s13 Tr(F1F2F3F4), etc. One choice for the additional four is
A13 = k3 · F1 · F2 · k3 Tr(F3F4) A15 = k3 · F1 · F4 · k2 Tr(F2F3)
A14 = k4 · F1 · F3 · k2 Tr(F2F4) A16 = k4 · F2 · F3 · k4 Tr(F1F4) .
– 17 –
As an example application of the computer algorithms, one may check that the symmetri-
sation operator of section 4.1,
S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) ,
acts as
S4BA1 = 8A1 + 4A2 − 4A3 + 4A4 + 8A5 + 16A6
. . .
S4BA16 = −6A1 + 6A3 − 6A5 − 12A6 + 32A7 + 3A8 +
A9 + 3A10 +
A11 + 3A12
and has rank four.
A.2 Four fermions
In dealing with the spinor wavefunctions ui one has to face two issues: Fierz identities, and
the Dirac equation. Fierz identities not only allow one to change the order of the spinors
but also give rise to relations between different expressions in one spinor order. The Dirac
equation often simplifies terms with momenta contracted into (uiγ
[n]uj) bilinears.
In this section it is shown how to construct bases for terms of the form (k2 or k4) ×
u1u2u3u4. A significant simplification comes from noting that the Dirac equation allows
one to rewrite (uiγ
[n]uj) bilinears into terms with lower n if more than one momentum is
contracted into the γ[n]. A good first step is therefore to disregard the momenta temporarily
and find all independent scalars and two-index tensors built from u1, . . . , u4. From the
SO(10) representation content,
(S+)⊗4 = 2 · 1+ 6 · + 3 ·˜+ (tensors with rank > 2) ,
one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as
in [21],
T1(1234) = (u1γ
au2)(u3γau4) ,
T3(1234) = (u1γ
abcu2)(u3γabcu4) .
and similarly for the other two inequivalent orders of the four spinors. (Note there is no T5
because of self-duality of the γ[5].) From Fierz transformations, one learns that all T3 terms
can be reduced to T1 by T3(1234) = −12T1(1234)− 24T1(1324) and permutations, and the
identity (γa)(αβ(γ
a)γ)δ = 0 implies that T1(1234) + T1(1324) + T1(1423) = 0, leaving for
example T1(1234) and T1(1324) as independent scalars.
Generalising this approach to two-index tensors, it turns out that it is sufficient to
start with
T11(1234) = (u1γ
mu2)(u3γ
nu4) ,
T31(1234) = (u1γ
aγmγnu2)(u3γau4) ,
T33(1234) = (u1γ
abγmu2)(u3γabγ
nu4) ,
– 18 –
and permutations of the spinor labels. It would be very tiresome to systematically apply
a variety of Fierz transformations by hand and to find an independent set. Fortunately,
by choosing a gamma matrix representation (such as the one listed in appendix B) and
reducing all expressions to polynomials in the independent spinor components u1i , . . . , u
this problem can be solved with computer help. As expected, one finds that the Tij(abcd)
span a nine-dimensional space, and a basis can be chosen as
T11(1234), T11(1324), T11(1423), T11(3412), T11(2413), T11(2314),
T31(1234), T31(1324), T31(2314) . (A.1)
A typical relation reducing the other Tij(abcd) to this basis is
T31(3412) = 2T11(1234) − 2T11(3412) + T31(1324) + T31(2314) + 2ηmnT1(1234) . (A.2)
Having solved the first step, it is now easy to include the two or four momenta, taking
the Dirac equation into account. Consider first the case of two momenta. Starting from
the two-tensors in (A.1), one gets the three independent scalars
(u1/k3u2)(u3/k1u4) , (u1/k2u3)(u2/k1u4) , (u1/k2u4)(u2/k1u3) .
In addition, there are four products of the two independent scalars T1(1234) and T1(1324)
with the two independent momentum invariants s12 and s13. By contracting (A.2) with
momenta, one can show that
s12T1(1324) − s13T1(1234)
= −(u1/k3u2)(u3/k1u4) + (u1/k2u3)(u2/k1u4)− (u1/k2u4)(u2/k1u3) , (A.3)
and this relation can be used to eliminate s12T1(1324). (It will become clear later that
there are no independent other relations like this one.) There are thus six independent
k2u1 · · · u4 scalars:
(u1/k3u2)(u3/k1u4) s12 T1(1234)
(u1/k2u3)(u2/k1u4) s13 T1(1234) (A.4)
(u1/k2u4)(u2/k1u3) s13 T1(1324)
Note that there is only one completely antisymmetric combination of those, given by the
right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent
k4u1 · · · u4 scalars:
B1 = s12 (u1/k3u2)(u3/k1u4) B2 = s13 (u1/k3u2)(u3/k1u4)
B3 = s12 (u1/k2u3)(u2/k1u4) B4 = s13 (u1/k2u3)(u2/k1u4)
B5 = s12 (u1/k2u4)(u2/k1u3) B6 = s13 (u1/k2u4)(u2/k1u3) (A.5)
B7 = s
12 T1(1234) B8 = s12s13 T1(1234)
B9 = s
13 T1(1234) B10 = s
13 T1(1324)
– 19 –
Working in a gamma matrix representation, it is again simple to construct a computer
algorithm which reduces any given k2u1 · · · u4 or k4u1 · · · u4 scalar into polynomials of the
spinor and momentum components. The Dirac equation can then be solved by breaking
up the sixteen-component spinors ui into eight-dimensional chiral spinors u
i and u
i , as in
eq. (B.1). One obtains polynomials in the momentum components kai and the independent
spinor components (uci )
1...8. However, a great disadvantage of this procedure is that it
breaks manifest Lorentz invariance. For example, one encounters expressions which contain
subsets of terms proportional to the square of a single momentum and are therefore equal
to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution
is to choose several sets of particular vectors ki satisfying k
i = 0 and
i ki = 0 and to
evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids
issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases
(A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly
independent.
Equipped with a computer algorithm for these basis decompositions, one finds, for
example, that the symmetriser S4F of section 4.2,
S4F = 4(1 + π12)(1 − π23)(1 + π24) ,
acts on the B1 . . . B10 basis as
S4FB1 = −12B4 + 12B5 + 12B6 ,
. . .
S4FB10 = 8B1 + 16B2 − 8B3 − 16B4 + 8B5 + 16B6 − 24B7 − 24B8 − 24B9
and has rank three.
A.3 Two bosons, two fermions
The combined methods of the last two sections can easily be extended to the mixed case
of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters
scalars of the form ku1u2F3F4. A basis of such objects is given by
C1 = (u1γ
au2)k
C2 = (u1γ
au2)F
C3 = (u1γ
au2)F
c (A.6)
C4 = (u1γ
abcu2)F
C5 = (u1γ
abcu2)F
There are two combinations antisymmetric in [12] and symmetric in (34):
−C1 + 4C2 +C4 and C2 + C3 .
Finally, there are ten independent scalars of the form k3u1u2F3F4 (relevant to the two-loop
calculation of section 4.3), and they can all be obtained by multiplication of C1 . . . C5 with
the two momentum invariants s12 and s13.
– 20 –
B. A gamma matrix representation
A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices
0 (γa)αβ
(γa)αβ 0
where
(γ0)αβ = 116 = (γ
0)αβ ,
(γ9)αβ =
−18 0
= −(γ9)αβ ,
and (γa)αβ = −(γa)αβ, a = 1 . . . 8, is a real, symmetric 16×16 representation for the SO(8)
Clifford algebra,
(γa)αβ =
(σa)T 0
, a = 1 . . . 8 ,
as given in appendix 5.B of [21]. The matrices Γa satisfy the SO(1,9) Clifford algebra
relations,
{Γa,Γb} = 2ηab132 , ηab = (+−− · · · −) ,
and bilinears of chiral spinors (with, say, positive chirality) are constructed as
(uΓ[a1...ak]v) = (uγ[a1...ak ]v) = uα(γ[a1)αβ(γ
a2)βγ . . . (γak ])γδv
This representation is particularly suitable for the calculations outlined in appendix A
because it allows a simple decomposition of SO(1,9) spinors into SO(8) spinors due to its
block structure:
Γ0 · · ·Γ9 =
116 0
0 −116
, Γ1 · · ·Γ8 =
18 0 0 0
0 −18 0 0
0 0 18 0
0 0 0 −18
Therefore, the Dirac equation for a chiral 16-component spinor u,
(γa)αβ∂au
α = 0 ,
can be solved by splitting u into two chiral eight-component spinors of SO(8),
with γ1...8
One obtains the coupled equations
(∂0 + ∂9)u
s − (σ · ∂)uc = 0
(∂0 − ∂9)uc − (σT · ∂)us = 0
(with eight-dimensional dot products). These can be solved for us in terms of uc:
(σ · ∂)uc =
(σ · k)uc , (B.1)
where k+ = −i∂+ = −i√
(∂0 + ∂9).
– 21 –
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|
0704.0016 | Lifetime of doubly charmed baryons | Lifetime of doubly charmed baryons
Chao-Hsi Chang1,2 ∗, Tong Li3†, Xue-Qian Li3‡ and Yu-Ming Wang4§
1CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China
2Institute of Theoretical Physics, Chinese Academy of Sciences,
P.O.Box 2735, Beijing 100080, P.R. China
3 Department of Physics, Nankai University, Tianjin, 300071, P.R. China
4Institute of High Energy Physics, Chinese Academy of Sciences,
P.O.Box 918, Beijing 100049, P.R. China
Abstract
In this work, we evaluate the lifetimes of the doubly charmed baryons Ξ+cc, Ξ
cc and Ω
cc. We care-
fully calculate the non-spectator contributions at the quark level where the Cabibbo-suppressed di-
agrams are also included. The hadronic matrix elements are evaluated in the simple non-relativistic
harmonic oscillator model. Our numerical results are generally consistent with that obtained by
other authors who used the diquark model. However, all the theoretical predictions on the lifetimes
are one order larger than the upper limit set by the recent SELEX measurement. This discrepancy
would be clarified by the future experiment, if more accurate experiment still confirms the value
of the SELEX collaboration, there must be some unknown mechanism to be explored.
∗ email: [email protected]
† email: [email protected]
‡ email: [email protected]
§ email: [email protected]
http://arxiv.org/abs/0704.0016v1
I. INTRODUCTION
The quite large difference of the lifetimes between D± and D0 and the lifetimes close
to each other for B± and B0 are well explained by taking into account the non-spectator
effects[1]. This success implies that the mechanism which governs the reactions at quark
level is well understood. When we apply the mechanism to the heavy baryon case, some
problems emerge. The famous puzzle in the heavy-flavor field that the lifetime of Λb is
remarkably shorter than that of B meson is much alleviated recently when the operators
of higher dimensions are taken into account[2, 3]. The more recent experimental value of
the ratio τ(Λb)/τ(B
0) = 1.041± 0.057[4] is close to the theoretical evaluation[3]. However,
in the theoretical works, one can notice that the evaluation of hadronic matrix elements is
still very rough and based on some approximations. The possible errors brought up by the
uncertainties in the hadronic matrix elements are still uncontrollable. In our recent work[5],
we find that the short-distance contributions to the branching ratio of Λb → Λγ which is
evaluated in the PQCD approach, are much smaller than that from long-distance effects.
Therefore, even though one has a full reason to believe that the low-energy QCD should
solve the discrepancy if it exists, he must find a proper way to deal with the hadronic matrix
elements.
The observation of doubly charmed baryon Ξ+cc by the SELEX Collaboration at
FERMILAB[6] provides an opportunity to investigate the hidden problems. Hopefully the
study may shed some lights on the unknown non-perturbative QCD effects which result in
obvious difference between baryons and mesons. Because Ξ+cc contains two heavy quarks,
by the heavy quark effective theory (HQET) the situation may become relatively simple
and clear compared to the case of Λb or Λc which possesses only one heavy quark. Thus a
careful study on the Ξ+cc is necessary and interesting. Several groups already investigated
the two-heavy-flavor baryons a long time ago[7, 8]. In their work, the evaluation of the
hadronic matrix elements is based on the quark-diquark structure of the baryons. This is
definitely reasonable, it is believed that two heavy quarks can constitute a more stable and
compact color-anti-triplet diquark[9]. However, since charm quark, even b-quark, is not so
heavy that the degree of freedom of the light flavor can be ignored, the diquark scenario
may bring up certain errors, especially when evaluating lifetimes of baryons, because only
inclusive processes are concerned. In this work, we do not use the diquark picture, but
instead, adopt a simpler non-relativistic model for the baryon and re-evaluate the hadronic
matrix elements. As a by-product, one can compare the results by the diquark picture with
that by the three valence-quark picture. It may help us to better understand the diquark
picture and its application range.
The advantage is obvious, that we only concern the inclusive processes in terms of the
optical theorem when calculating the lifetime. Therefore, we do not need to deal with
the hadronization to light hadrons. The only non-perturbative effects come from the wave
function of the heavy baryon. Moreover, since there are two heavy quarks in the baryon,
the relativistic effects are not so significant and the framework of non-relativistic harmonic
oscillator model might lead to a reasonable result.
Moreover, at the quark level, we carry out similar calculations as that in the literature,
but we keep some new operators which are CKM suppressed and contribute to the lifetime.
They appear at the non-spectator scattering at order of 1
in heavy quark expansion(HQE).
Later, our numerical results show that their contributions are indeed very tiny to make any
substantial contributions.
All the concerned parameters in the model are obtained by fitting data, therefore we
avoid some theoretical uncertainties and obtain reasonable results. Comparing these results
with data, we may gain information about the the whole picture.
Our paper is organized as follows. In Section.II we derive the formulation for the lifetimes
of Ξ+cc, Ξ
cc and Ω
cc which include the non-spectator effects. In Section.III, we use a simple
model, i.e. the harmonic oscillator, to estimate the hadronic matrix elements. In Section.IV
we present our numerical results along with the values of all the input parameters. The last
section is devoted to our conclusion and discussion.
II. FORMULATION FOR LIFETIMES OF Ξ+cc, Ξ
cc , Ω
A. Spectator Contribution to Lifetimes of Ξ+cc, Ξ
cc , Ω
The lifetime is determined by the inclusive decays. Thus one can use the optical theorem
to obtain the total width (lifetime) of the heavy hadron by calculating the absorptive part
of the forward-scattering amplitude.
The total width is then written as
Γ(HQ → X) =
d4x〈HQ|T̂ |HQ〉 =
〈HQ|Γ̂|HQ〉, (1)
where
T̂ = T{iLeff(x),Leff(0)} (2)
and Leff is the relevant effective Lagrangian. 1/mQ is the expansion parameter, and the
non-local operator T̂ is expanded as a sum of local operators and the corresponding Wilson
coefficients include terms with increasing powers of 1/mQ. Definitely, the lowest dimen-
sional term dominates in the limit mQ → ∞ and it is the dimension-three operator c̄c.
The total width of a charmed hadron Hc is determined by Im〈Hc|T̂ |Hc〉[10] with a proper
normalization[11].
Γ(Hc → f) =
192π3
|VCKM |2{c3(f)〈Hc|c̄c|Hc〉
+c5(f)
〈Hc|c̄iσµνGµνc|Hc〉
6 (f)
〈Hc|(c̄Γiq)(q̄Γic)|Hc〉
)}, (3)
where the coefficients ci(f) depend on the masses of the internal quarks in the loop. The
coefficient c3(f) has been calculated to one-loop order[12, 13, 14] whereas the coefficient
c5(f) is evaluated at the tree level[15, 16]. VCKM is the Cabibbo-Kabayashi-Maskawa mixing
matrix elements and Gµυ is the gluonic field strength tensor. Since the third term involves
light quarks, it can be different for charmed hadrons with various light flavors. Thus, the
difference appears at the 1/m3c order and in the hadronic matrix elements of four-quark
operators. The contributions at orders higher than 1/m3c are neglected.
To the lowest order, the main contribution comes from the heavy quark(charm quark)
decays, while the light flavors are treated as spectators. The contributions are due to the
semileptonic and the nonleptonic decays as follows:
Γ(c→ s) =
l=e,µ
Γc→sl̄υ +
q(q′)=u,d,s
Γc→sq̄q′ (4)
The semileptonic and nonleptonic decay rates of the c quark up to order 1/m2c has been
evaluated by many authors[17], and here we would directly use their results.
B. Non-spectator Contributions to Inclusive Decays of Ξ+cc, Ξ
cc , Ω
The total width of hadrons which involve at least one charm quark c can be decomposed
into two parts
Γ(HQ → f) = Γspectator + Γnonspectator. (5)
For the spectator scenario, the contribution to the total width of the (ccd)-baryon ground
state Ξ+cc, the (ccu)-baryon ground state Ξ
cc and the (ccs)-baryon ground state Ω
cc should
be a sum of decays rates of two c−quarks individuallynamely
Γspecccq ≃ 2Γspecc , q = u, d, s. (6)
To derive the non-spectator contributions for decays of Ξ+cc, Ξ
cc and Ω
cc, we need the
relevant effective Lagrangian:[18]
L(∆c=1)eff (µ = mc) = −
{VcsV ∗ud[C1(µ)s̄γµLcūγµLd+ C2(µ)ūγµLcs̄γµLd]
+VcdV
ud[C1(µ)d̄γ
µLcūγµLd+ C2(µ)ūγ
µLcd̄γµLd]
+VcsV
us[C1(µ)s̄γ
µLcūγµLs+ C2(µ)ūγ
µLcs̄γµLs]
l=e,µ
s̄γµLcν̄lγ
µLl}+ h.c. (7)
where L denotes 1−γ5
(i) The inclusive decays of Ξ+cc:
There are four diagrams which contribute to the the width of Ξ+cc, as shown in Fig.1. Here
we also include the Feynman diagrams which are CKM suppressed. Fig 1.(a),(c) are the
W-exchange diagrams (WE), while Fig 1.(b),(d) are the pauli-interference diagrams (PI).
Here Fig 1.(d) is arisen from the semi-leptonic decay of the charm quarks with the d−quark
in Ξ+cc. For the WE-type diagrams, we derive the contribution to the width as
(|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))P 2+
{[C21 (µ) + C22 (µ)]c̄γµLcd̄γµLd+ 2C1(µ)C2(µ)c̄γµLdd̄γµLc}, (8)
where P+ = pc + pd, zq+ =
(q = u, d, s). The definition of the function C(z1, z2) is
C(z1, z2) = −[−2(x32 − x31)− (x22 − x21)(3 + 2z1 − 2z2) + 4z1(x2 − x1)], (9)
where x1,2 =
(1+z1−z2)∓
(1+z1−z2)2−4z1
. In the expressions q and q̄ are free field opearotors of
quark and antiquark, and we will show in next section that all the non-perturbative QCD
effects are included in the wavefunctions. Their explicit expressions are given as
(2π)3
α=1,2
bqα(k)u
q (k)e
−ikx + d+qα(k)υ
q (k)e
(2π)3
α=1,2
b+qα(k)ū
q (k)e
ikx + dqα(k)ῡ
q (k)e
. (11)
For Ξ+cc, q=c, u.
The contributions from the Pauli-interference(PI) non-spectator diagrams to the width
of Ξ+cc are:
PI = −
{|Vud|2|Vcd|2Fµν(zu−, zd−)[NC21 (µ)c̄γµLdd̄γνLc+ C22 (µ)c̄iγµLdj d̄jγνLci
+2C1(µ)C2(µ)c̄γ
µLdd̄γνLc] + 2|Vcd|2Fµν(0, zl−)c̄γµLdd̄γνLc}, (12)
where zq− =
(q = u, d, e, µ) and P− = pc−pd. The definition of the function Fµν(z1, z2) is
Fµν(z1, z2) = −[2(x32 − x31)−
(2 + z1 − z2)(x22 − x21) + 3(x2 − x1)]P 2−gµν
+[2(x32 − x31)− 3(x22 − x21)]P−µP−ν , (13)
where the definitions of z1 and z2 are the same as before.
(ii) The inclusive decays of Ξ++cc :
The non-spectator contribution to the width of Ξ++cc come from the diagrams shown in Fig.2.
That is caused by an interference of the produced u−quark from decay of one of the charm
quarks with the u−quark in Ξ++cc . Here we also include the CKM suppressed Feynman
diagrams. The contribution is
PI = −
{|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−)
+|Vcd|2|Vud|2Fµν(zd−, zd−)}
{C21(µ)c̄iγµLuj ūjγνLci +NC22 (µ)ūγµLcc̄γνLu+ 2C1(µ)C2(µ)ūγµLcc̄Lνu},
where z− =
(q = s, d), P− = pc − pu.
(iii) For the inclusive decays of Ω+cc:
The non-spectator contributions for Ω+cc not only come from the Pauli interference of the
s− quark produced in the non-leptonic, but also from the semi-leptonic decay of the charm
quarks with the s−quark in Ω+cc, the later one is suggested by Voloshin et al.[19]. As above,
here we include the CKM suppressed WE non-spectator diagrams. The WE non-spectator
contribution to the width Ω+cc is
|Vus|2|Vcs|2C(zu+, zs+)P 2+
{[C21 (µ) + C22 (µ)]c̄γµLcs̄γµLs+ 2C1(µ)C2(µ)c̄γµLss̄γµLc},
where zq+ =
, q = u, s and P+ = pc + ps.
The PI non-spectator contribution to the width of Ω+cc is
PI = −
{|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)}
{NC21(µ)c̄γµLss̄γνLc + C22(µ)c̄iγµLsj s̄jγνLci + 2C1(µ)C2(µ)c̄γµLss̄γνLc}
|Vcs|2Fµν(0, zl−)c̄γµLss̄γνLc, (16)
where zq− =
, q = u, d, s, e, µ and P− = pc − ps. Sandwiching the operators between
initial and final Ξ+cc, Ξ
cc , Ω
cc states, we obtain the hadronic matrix elements:
WE/PI
= 〈Ξ+cc(P = 0, s)|Γ̂
WE/PI
|Ξ+cc(P = 0, s)〉
PI = 〈Ξ
cc (P = 0, s)|Γ̂
PI |Ξ
cc (P = 0, s)〉
WE/PI
= 〈Ω+cc(P = 0, s)|Γ̂
WE/PI
|Ω+cc(P = 0, s)〉. (17)
III. THE HADRONIC MATRIX ELEMENTS
Because the hadronic matrix elements are fully determined by the non-perturbative
QCD effects which cannot be reliably evaluated at present yet, we need to invoke concrete
phenomenological models to carry out the computations. In this work, we adopt a simple
non-relativistic model, i.e. the harmonic oscillator[20]. This model has been widely em-
ployed in similar researches[21, 22, 23, 24, 25, 26]. In fact, an advantage of the calculations
of the lifetimes of heavy hadrons is that one does not need to deal with the hadronization
process of lighter products (quarks or even gluons) and the heavy hadrons can be well
described by such simple non-relativistic models, and the results are relatively reliable than
for light hadron decays.
(i)The inclusive decays of Ξ+cc:
In the harmonic oscillator model, the wavefunction of Ξ+cc is expressed as |Ξ+cc〉 and
|Ξ+cc(P, s)〉 = AB
color,spin
χspin,flavorϕcolor
d3pρd
3pλΨΞ+cc(pρ,pλ)|ci(pq1 , sq1), cj(pq2, sq2), dk(pq3, sq3)〉.
The normalization condition for |Ξ+cc(P, s)〉 is
〈Ξ+cc(P, s)|Ξ+cc(P′, s′)〉 = (2π)3
δ3(P−P′)δs,s′, (19)
where χspin,flavorϕcolor are the spin-flavor and color wavefunctions respectively. Their explicit
expressions are
χs= 1
,flavor =
(2|c↑c↑d↓〉 − |c↑c↓d↑〉 − |c↓c↑d↑〉) (20)
ϕcolor =
ǫijk. (21)
AB is the normalization constant. The spatial wavefunction ΨΞ+cc is a three-body harmonic
oscillator wavefunction and expressed as
ΨΞ+cc = exp(−
). (22)
Here aρ and aλ parameters reflecting the non-perturbative effects. In the above expressions,
the Jacobi transformations of p1, p2, p3 which are the momenta of the three valence quarks
ccd, and variables pρ, pλ, P are
p1 − p2√
,pλ =
p1 + p2 − 2mcmd p3
22mc+md
,P = p1 + p2 + p3. (23)
We choose the center-of-mass frame of Ξ+cc, i.e. (P=0) to calculate the hadronic matrix
elements. Substituting the four-quark operators into the expressions, we obtain the non-
spectator WE contributions to the width of Ξ+cc as
WE = 64π
2G2FP
+(|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))(C1(µ)− C2(µ))2
|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγµLucūdγ
µLud,
and the PI contribution is
PI = −
π2G2F{|Vcd|2|Vud|2Fµν(zu−, zd−)[−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)]
−2|Vcd|2Fµν(0, zl−)}|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγ
µLud ūdγ
νLuc,
where the sum over spin means a sum over the polarizations of the three valence quarks of
Ξ+cc with their corresponding C-G coefficients in the spin-flavor wavefunction. uq, ūq denote
the Dirac spinors of free quarks q and the expression is
Eq +mq
Eq+mq
χ (26)
ūq =
Eq +mq
1 − σ·p
Eq+mq
in our case q denotes c and d quarks.
(ii)The inclusive decays of Ξ++cc :
The contribution from the PI non-spectator diagrams to the width of Ξ++cc is
Ξ++cc
PI = −
π2G2F{|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−)
+|Vcd|2|Vud|2Fµν(zd−, zd−)}(C21(µ)−NC22 (µ)− 2C1(µ)C2(µ))|AB|2[2(1 +
)]3/2
d3pρd
ρexp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
µLuu ūuγ
νLuc. (28)
Similar to the case of Ξ+cc, the sum over spin means a sum of the polarizations of the three
valence quarks of Ξ++cc with their C-G coefficients. One only needs to replace u by d in pρ,
pλ and other expressions are similar to that for Ξ
(iii)The inclusive decays of Ω+cc:
The contribution from the W-boson exchange(WE) non-spectator diagrams to the width of
Ω+cc is
WE = 64π
2G2FP
+|Vus|2|Vcs|2C(zu+, zs+)(C1(µ)− C2(µ))2
|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγµLucūsγ
µLus,
whereas that from the Pauli-interference(PI) non-spectator diagrams is
PI = −
π2G2F{[|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)]
[−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)]− 2|Vcs|2Fµν(0, zl−)}|AB|2[2(1 +
)]3/2
d3pρd
ρexp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
µLusūsγ
νLuc. (30)
The sum over polarizations is similar to that for Ξ+cc and Ξ
IV. INPUT PARAMETERS AND NUMERICAL RESULTS
To obtain the decay amplitudes, we adopt the input parameters as follows[7, 27]: GF =
1.166×10−5GeV−2, |Vcs| = 0.9737, |Vud| = 0.9745, C1(mc) = 1.3, C2(mc) = −0.57,mc = 1.60
GeV, ms = 0.45 GeV, mu = md = 0.3 GeV, m
s = 0.2GeV, m
u = m
d = 0, MΞ+cc = MΞ++cc =
3.519 GeV, MΩ+cc = 3.578 GeV, MΞ+∗cc −MΞ+cc = MΞ++∗cc −MΞ++cc = MΩ+∗cc −MΩ+cc = 0.132
GeV. Here mq∗ denotes the current quark mass of flavor q.
The non-perturbative parameters aρ, aλ in the harmonic oscillator wavefunctions are
selected as follows: for J/ψ, in ref.[20], a2ρ = 0.33GeV
2, for D−mesons, a2ρ = 0.25GeV2. For
TABLE I: The numerical results about the contributions from the different components and the
evaluated lifetime for the doubly charmed baryons. For a comparison, in the following table, we
list the corresponding lifetimes predicted by the authors of ref.[7] where the diquark picture was
employed. It is noted that in ref.[7], the authors used various input parameters and obtained
slightly diverse results, we take average values of the numbers in the table. There is only one
datum for the lifetimes on τ
given by the SELEX collaboration which is also listed the table.
Ξ+cc Γspec(10
−12GeV) ΓWEnon (10
−13GeV) ΓPInon(10
−15GeV) τ
(ps) τ
(ps) in ref.[7] exp(ps)
2.01 6.43 -3.36 0.25 0.19 0.033
Ξ++cc Γspec(10
−12GeV) ΓPInon(10
−12GeV) τ
(ps) τ
(ps) in ref.[7]
2.01 -1.02 0.67 0.52 −
Ω+cc Γspec(10
−12GeV) ΓWEnon (10
−14GeV) ΓPInon(10
−12GeV) τ
(ps) τ
(ps) in ref.[7]
2.01 4.25 1.10 0.21 0.22 −
the doubly charmed baryons, because aρ reflects the coupling between two charm quarks,
we set it to be the same as that for J/ψ. aλ reflects the coupling of the light quark with
these two charm quark, thus we can reasonably set it to be the same as aρ in D-mesons.
With these parameters as input, the lifetimes of the doubly charmed baryons can be
evaluated out (see TABLE.I), if the non-spectator effects are taken into account.
V. CONCLUSION AND DISCUSSION
In this work, we evaluate the lifetimes of doubly charmed baryons with the non-spectator
effects being properly taken into account. As argued in the introduction, to evaluate the
lifetimes (the total widths), only the inclusive processes are concerned, and then the non-
perturbative effects are all from the wavefunctions of the doubly charmed baryons. Due
to existence of the two heavy charm quarks, the non-relativistic harmonic oscillator model
should apply in this case. Mainly, we carefully calculate the contribution of non-perturbative
effects to the lifetimes in the model, which are closely related to the bound states of the
baryons.
Our numerical results indicate that the non-spectator contributions to the lifetimes of
Ξ+cc, Ξ
cc and Ω
cc are substantial. The non-spectator contributions to the width of Ξ
cc are
mainly from the WE diagrams (the PI diagrams which contribute are CKM suppressed),
since the WE contribution is constructive, therefore the lifetime of Ξ+cc is much suppressed.
By contraries, for Ξ++c and Ω
cc, the non-spectator contributions are mainly from the PI
diagrams and the net effect is destructive. It is noted that for Ω+cc there still are Cabibbo-
suppressed WE diagrams, but for Ξ++cc there are only PI diagrams. Therefore the predicted
lifetime of Ξ++cc is larger than that of other two baryons. We also employ other values for
parameters aρ, aλ and find that the resultant values can vary within 20% uncertainty.
Our results are
τ(Ξ+cc) = 0.25 ps τ(Ξ
cc ) = 0.67 ps and τ(Ω
cc) = 0.21 ps.
These are generally consistent with the results obtained by Kiselev et al.[7] and Guberina
et al.[8], even though they used different models for calculating the hadronic matrix elements.
Concretely, they used the diquark picture and attributed the non-perturbative effects into
the wavefunction of the diaquark at origin. Kiselev et al. gave τ(Ξ+cc) ∼ 0.16 − 0.22 ps
τ(Ξ++cc ) ∼ 0.40− 0.65 ps and τ(Ω+cc) ∼ 0.24− 0.28.
Although all the theoretical predictions based on different models agree with each other,
they are obviously one order larger than the upper limit of the measured value on the
lifetime of Ξ+cc (0.033 ps) by the SELEX collaboration[6]. This deviation, as suggested by
some authors, may come from experiments[28]. So far the difference between theoretical
predictions and experimental data may imply some unknown physics mechanisms which
drastically change the value, if the future experiment, say at LHCb, confirms the measure-
ment of the SELEX. Recently, several groups have studied the possibility of doubly heavy
baryon production at hadron collider LHC and future linear collider ILC[29, 30] and the
effective field theories for two heavy quarks system are also further investigated[31]. We are
expecting the new data from more accurate experiments at LHC and ILC to improve our
theoretical framework and determine if there are contributions from new physics beyond
the standard model.
Acknowledgement: This work is supported by the National Natural Science Foundation of
China.
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FIG. 1: non-spectator effects contribution to lifetime of Ξccd
FIG. 2: non-spectator effects contribution to lifetime of Ξccu
FIG. 3: non-spectator effects contribution to lifetime of Ωccs
Introduction
Formulation for Lifetimes of cc+, cc++, cc+
Spectator Contribution to Lifetimes of cc+, cc++, cc+
Non-spectator Contributions to Inclusive Decays of cc+, cc++, cc+
The hadronic matrix elements
Input parameters and Numerical results
Conclusion and Discussion
References
|
0704.0017 | Spectroscopic Observations of the Intermediate Polar EX Hydrae in
Quiescence | Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 30 October 2018 (MN LATEX style file v2.2)
Spectroscopic Observations of the Intermediate Polar EX Hydrae in
Quiescence
N. Mhlahlo1,2⋆, D.A.H. Buckley2, V.S. Dhillon3, S.B. Potter2, B. Warner1 and
P.A. Woudt1
1Astronomy Department, University of Cape Town, Rondebosch 7700, Cape Town, South Africa
2South African Astronomical Observatory, Observatory 7935, Cape Town, South Africa
3Physics and Astronomy Department, University of Sheffield, Sheffield, S3 7RH, UK
30 October 2018
ABSTRACT
Results from spectroscopic observations of the Intermediate Polar (IP) EX Hya in quiescence
during 1991 and 2001 are presented. Spin-modulated radial velocities consistent with an outer
disc origin were detected for the first time in an IP. The spin pulsation was modulated with
velocities near ∼ 500− 600 km s−1. These velocities are consistent with those of material
circulating at the outer edge of the accretion disc, suggesting corotation of the accretion cur-
tain with material near the Roche lobe radius. Furthermore, spin Doppler tomograms have
revealed evidence of the accretion curtain emission extending from velocities of ∼ 500 km
s−1 to ∼ 1000 km s−1. These findings have confirmed the theoretical model predictions of
King & Wynn (1999), Belle et al. (2002) and Norton et al. (2004) for EX Hya, which predict
large accretion curtains that extend to a distance close to the Roche lobe radius in this system.
Evidence for overflow stream of material falling onto the magnetosphere was observed,
confirming the result of Belle et al. (2005) that disc overflow in EX Hya is present during
quiescence as well as outburst.
It appears that the Hβ and Hγ spin radial velocities originated from the rotation of the
funnel at the outer disc edge, while those of Hα were produced due to the flow of material
along the field lines far from the white dwarf (narrow component) and close to the white dwarf
(broad-base component), in agreement with the accretion curtain model.
Key words: accretion discs, binary - stars: cataclysmic variables.
1 INTRODUCTION
EX Hya is an Intermediate Polar (IP), a sub-class of magnetic Cat-
aclysmic Variable Stars (mCVs) where a late-type main sequence
star transfers material to the magnetic white dwarf star as the two
stars orbit each other under the influence of their mutual grav-
itation. Unlike in Polars, another subclass of mCVs, where the
white dwarf is in synchronous rotation with the binary rotation
(Pspin = Porb), the white dwarf in an IP is in asynchronous rotation
with the orbital motion of the system. EX Hya, however, is nearer
synchronism than the majority of IPs as it has a spin period (∼67.03
min) which is about 2/3 its orbital period (98.26 min) (Mumford
1967; Hellier et al. 1987), and is one of only six out of thirty nine
confirmed IPs with its orbital period below the 2-3 h CV period gap
(Norton et al. 2004). It has an inclination i = 78◦±1◦.
Recent studies have shown that EX Hya does not conform
to the traditional IP model (King & Wynn 1999; Wynn 2000;
Belle et al. 2002; Norton et al. 2004; Belle et al. 2005). This sys-
⋆ E-mail: [email protected]
tem has a large Pspin/Porb ratio (∼ 0.68) implying that it can-
not be in the usual spin equilibrium rotation since most IPs
have been shown to attain spin equilibrium near Pspin/Porb ∼
0.1 (King & Wynn 1999; Wynn 2000). This further implies that
the corotation radius is far greater than the circularisation radius
(Rco ≫ Rcir) and that EX Hya cannot possess a Keplerian disc.
Systems with Keplerian discs are expected to have Rco < Rcir and
thus a smaller Pspin/Porb. These factors have prompted theorists to
suggest that the spin equilibrium state in EX Hya is determined
by Rco ∼ b, where b is the distance to the inner Lagrangian point,
L1 (King & Wynn 1999; Wynn 2000; Norton et al. 2004). In this
model the accretion curtains extend to near the L1 point, and EX
Hya resembles an asynchronous Polar where most of the material
accretes via the stream (King & Wynn 1999; Wynn 2000) and via
both the ring of material near the Roche lobe radius of the primary
and the stream (Norton et al. 2004), depending on the orbital and
spin periods of the system, and the magnetic field strength. In the
later publication it was shown that material in EX Hya is fed from
a ring of material at the outer edge of the Roche lobe, and that for
c© 0000 RAS
http://arxiv.org/abs/0704.0017v1
2 N. Mhlahlo, D.A.H. Buckley, V.S. Dhillon, S.B. Potter, B. Warner and P.A. Woudt
Date HJD (start) Time Spectra
24-04-91 2448371.3884097 3.28 90
25-04-91 2448372.3603850 2.93 72
29-04-91 2448376.2402973 7.00 100
24-03-01 2451993.5427099 2.79 42
25-03-01 2451994.3581177 3.77 56
25-03-01 2451994.5147196 3.96 66
26-03-01 2451995.4360089 1.94 48
26-03-01 2451995.5412087 2.92 50
Table 1. Table of spectroscopic observations during quiescence in 1991 and
2001. The column Date denotes the date at the beginning of the observing
night (before midnight), the column Time denotes the number of observing
hours and Spectra the number of spectra obtained.
the Pspin/Porb of EX Hya, this mode of accretion is preferred over
stream-fed accretion.
In this work we present spectroscopic data of EX Hya in qui-
escence obtained from the SAAO in 1991 (just before EX Hya went
into outburst, and a day or two after outburst) and in 2001. Outburst
data of 1991 will be discussed in a later publication.
2 OBSERVATIONS AND DATA REDUCTION
2.1 1991 Observations
EX Hya was observed in April of 1991 by Buckley et al. (1991)
using the SAAO 1.9-m telescope with the Reticon photon counting
system (RPCS) detector on the Cassegrain spectrograph. A grating
with a resolution of 1200 mm−1 was used and a wavelength range
of 4000 - 5080 Å was covered at a spectral resolution of ∆λ ∼1.2
Å and at a time resolution of 100 - 120 s. The spectrograph slit
width was 250 µm (∼1.5 arcsecs). Wavelength calibration expo-
sures were taken using a CuAr arc lamp. Three nights of observa-
tions (24, 25 and 29 April 1991) were covered in quiescence and,
in total, 262 spectra were obtained. The observing log is given in
Table 1 together with the starting times of the observations.
Following wavelength-calibration and sky-subtraction, the
data were flux calibrated using the spectra of the standard star
LTT3864.
2.2 2001 Observations
The 2001 observations were obtained using the SITe CCD detector
(266× 1798 pixels) on the Cassegrain spectrograph of the SAAO
1.9-m telescope. A grating with a resolution of 1200 mm−1 was
used over the wavelength range 4200 - 5100 Å on the nights of
the 25th and 26th April. Another grating with a resolution of 1200
mm−1 was used on the 24th, 25th and 26th April over the range
6300 - 7050 Å. The spectral resolution was ∼1.0 Å and a 1×2
binning scheme was employed (i.e. binning by 2× in the spatial
direction). The exposure times during observations were 60 s. The
observations covered the period 24 April - 26 April 2001 and, in
total, 262 spectra were obtained (Table 1). The extraction and re-
duction of the data were performed the standard way using the
Image Reduction and Analysis Facility (IRAF)1 package and the
1 IRAF is a software package for the reduction and analysis of astronom-
ical data distributed by National Optical Astronomy Observatory (NOAO)
Figure 1. Radial-velocity Fourier amplitude spectra from the 1991 com-
bined data are shown for α = 3000 km s−1 and 900 km s−1 for the Hβ line
(top left panel and second left panel from the top) and for 3000 km s−1 and
1200 km s−1 for the Hγ line (top right panel and second right panel from the
top). Ω denotes the orbital frequency of the system, 2Ω its first harmonic
and ω+Ω the upper orbital side band where ω is the spin frequency. The
data were prewhitened by the orbital frequency and are displayed in the
third panels from the top. Window spectra are plotted below the amplitude
spectra (bottom panels).
spectra were flux calibrated using observations of the standard star
LTT3218.
3 THE RADIAL VELOCITIES
It is widely accepted that in a canonical CV, the high velocity emis-
sion line wings are formed in the inner parts of the accretion disc
orbiting close to the white dwarf and thus should reflect its orbital
motion (Shafter 1983; Shafter & Szkody 1984; Shafter 1985).
In IPs, however, high velocity emission line wings are formed
in the gas streaming towards the white dwarf at high velocities
(Hellier et al. 1987; Ferrario & Wickramasinghe 1993). The ra-
dial velocities were determined by measuring the wings of the Hβ
and Hγ emission lines from the 1991 data (24 and 25/04/91 - the
data obtained on the 29th was not added since EX Hya had not
fully recovered from outburst); and the wings of Hα, Hβ and Hγ
emission lines from the 2001 data using the Gaussian Convolution
which is operated by the Association of Universities for Research in As-
tronomy (AURA)
c© 0000 RAS, MNRAS 000, 000–000
Spectroscopic Study of the IP EX Hydrae in Quiescence 3
Figure 2. Radial velocity amplitude spectra shown for the Hα line from
2001, for α = 3500 km s−1 and 1200 km s−1. The vertical dashed line
shows the expected position of the orbital and spin period peaks. The data
were prewhitened by Ω and are shown in the third panel from the top. A
window spectrum is shown at the bottom.
Scheme (GCS, Schneider & Young 1980; Shafter & Szkody 1984;
Shafter 1985). The GCS method convolves each spectrum with two
identical Gaussian, one in the red wing and one in the blue wing.
The separation between the two Gaussians is 2α. Care was taken
not to include regions far out in the wings where the continuum be-
gins to dominate by choosing reasonable values of the width (σ) of
the Gaussians and α. Twelve standard Gaussian band-passes were
used with α values ranging from 3500 to 100 km s−1 and corre-
sponding width values from 1200 to 100 km s−1.
3.1 Period Searches
The radial velocities were Fourier transformed using the Discrete
Fourier Transform (DFT) algorithm to search for any periods in
the data (Deeming 1975; Kurtz 1985). The Hβ and Hγ amplitude
spectra from 1991 (24 and 25 April) are shown in Figure 1 and the
Hα amplitude spectra from 2001 are shown in Figure 2.
A prominent peak at a frequency corresponding to the 98-
minute orbital frequency, Ω, is observed in all the emission lines.
Second in strength to the orbital frequency is the spin frequency,
ω, of the narrow s-wave component (NSC) (α = 900,1200 km
s−1) (Figures 1 and 2). The third panels from the top in Figures 1
and 2 show the data after prewhitening by Ω. Power at ω is clearly
present.
The spin frequency was not detected at high values of α
(α = 3000,3500 km s−1) where it is most expected (since at these
velocities the material is quite close to the white dwarf and its emis-
sion is expected to be modulated at the white dwarf spin period).
It was also not present in Hβ and Hγ radial velocities of 2001. The
amplitude of ω relative to Ω was found to be ∼81% for Hβ, ∼64%
for Hγ and ∼18% for Hα.
4 ORBITAL VARIATIONS OF THE EMISSION LINES
The data were phase-binned on the orbital ephemeris of Hellier &
Sprouts (1992),
Teclipse = 2437699.94179+0.068233846(4)E, (1)
where E is the number of orbital cycles and T is the time of mid-
eclipse. This ephemeris is defined by the zero point of mid-eclipse,
where minimum intensity is at phase 0.0. This means that for the
radial velocities, maximum blueshift is perpendicular to the line of
centres, at phase 0.75, meaning that spectroscopic phase zero oc-
curs at the blue-to-red crossing of the emission line radial velocity
curve. 40 phase bins were used to produce the radial velocities.
4.1 Orbital Tomograms and Trailed Spectra
The Hα, Hβ and Hγ Doppler tomograms were computed using
the Back Projection Method (BPM) with the application of a fil-
ter, and the Maximum Entropy Method (MEM) (Marsh & Horne
1988; Marsh 1988; Horne 1991; Spruit 1998). The BPM Doppler
maps are shown in Figure 3 and those constructed using MEM are
shown in Figure 4. Error in ephemerides (both orbital and spin) is
sufficiently small to phase all of our data accurately on the orbital
and spin cycles.
A velocity amplitude of the primary, K1 = 74 ± 2 km s
from our radial velocity measurements and that of the secondary,
K2 = 360± 35 km s
−1, taken from Vande Putte et al. (2003) and
Beuermann et al. (2003), were used to fix the positions of the Roche
lobe and the stream trajectories on the tomograms. A secondary
mass, M2 = 0.10±0.01 M⊙ for EX Hya was derived from the up-
to-date secondary mass-period relation of Smith & Dhillon (1998).
The mass of the primary, M1, was then determined from the above
values using K1K2 =
, and was found to be 0.50±0.05 M⊙.
The Hβ, Hγ and HeI λ4471 Doppler tomograms (Figure 3
and 4) show strong emission at the bright spot, some at the Roche
lobe and the stream, and some from the disc. Those of Hα also
show strong bright spot emission but less or no emission from the
stream. Disc emission is diminished in Hα when compared to other
emission lines, especially at higher velocities. This is more obvious
in the BPM tomogram. The bright spot emission falls near the re-
gion (-100, 350) km s−1. Average-subtracted trailed spectra (Fig-
ure 4) show the corresponding NSC. ∼ 1× 10−11ergs cm−2 s−1
(∼60-70%) of the original line fluxes is contained in the average-
subtracted profiles of Hβ and Hγ, and ∼ 1× 10−12ergs cm−2 s−1
(∼80-90%) is contained in Hα and HeI λ4471. This flux is mainly
due to the bright spot and the stream. The trailed spectra have been
repeated over 2 cycles for clarity.
The trailed spectra of 2001 have revealed two interesting fea-
tures. The first one is the asymmetry in the intensity of the s-wave
(Figure 4). In Hα, the red wing of the NSC is brighter at φ98 ∼
0.1−0.3 and seems to reach maximum brightness near φ98 ∼ 0.25,
whereas the blue wing is dimmer in the range φ98 ∼ 0.7−0.9 and
seems to reach minimum brightness near φ98 ∼ 0.75. A similar ef-
fect is seen in the Hβ and the Hγ lines, and to a lesser extent in HeI
λ4471. The second feature is redshifted emission extending from
the NSC to high velocities (∼ 1000 km s−1) at early binary phases
(φ98 ∼ 0.0−0.2).
c© 0000 RAS, MNRAS 000, 000–000
4 N. Mhlahlo, D.A.H. Buckley, V.S. Dhillon, S.B. Potter, B. Warner and P.A. Woudt
Figure 3. The panels show the Hα, Hβ, Hγ and HeI λ4471 orbital Doppler maps from 2001, and Hβ tomograms of 1991, constructed using the Back-Projection
Method with the application of a filter (top panels) and after subtracting the average of the line profile (bottom panels). The positions of the Roche lobe and
stream trajectories are shown (velocity amplitudes of K1 = 74 km s
−1 and K2 = 360 km s
−1 for the primary and secondary stars, respectively, were used). The
two curves with marked intervals represent the gas stream velocity (upper curve) and the Keplerian velocity along the stream (lower curve). The circles on all
tomograms represent 0.1 of the distance from the L1 point to the primary. The three crosses are centres of mass of the secondary, system and primary, from
top-to-bottom. The asterisk represents the velocity of closest approach. All the maps are plotted on the same velocity scale. The lookup table of this figure is
such that the brightest emission features appear with decreasing intensity from yellow/green to light blue in the online edition, or white to grey in the printed
edition.
The reconstructed trailed spectra suggest that this latter fea-
ture is another s-wave, which we shall refer to as the high velocity
component (HVC), crossing the NSC near φ98 ∼ 0.2 − 0.3. The
Doppler tomograms show emission extending from the bright spot
position, passing along the stream path, to the bottom-left quadrant
at high velocities near 1000 km s−1 which is responsible for the
HVC; most of this emission does not fall within the disc and gas
stream velocities on the map, suggesting that there was small or no
overlap of the stream component with the disc.
The 1991 tomograms showed similar results. It is worth noting
that most of the disc emission in 1991 came from the outer disc than
in 2001.
Even though the MEM and BPM pick out the same features,
in the BP tomograms some features are more prominent than in
the MEM tomograms while the reconstruction obtained using the
MEM reproduces the observed data well.
The advantage of BPM over MEM is that it is faster and it
is easier to get a consistent set of maps of different emission lines
(in terms of the apparent noise in the images). For this reason both
methods have been used.
5 SPIN VARIATIONS OF THE EMISSION LINES
5.1 The Spin Radial Velocity Curve
The radial velocities were phase-folded using 30 bins on the
quadratic spin ephemeris of Hellier & Sproats (1992), where spin
maximum was defined as φ67 = 0. Figure 5 shows the variation of
the Hβ, Hγ and Hα narrow components with ω. Maximum blueshift
is seen at φ67 = 0.79 for Hβ and at φ67 = 0.77 for Hγ. Whereas for
Hα, maximum blueshift is seen at φ67 = 0.90. It should be noted
that Hα and Hβ / Hγ have not been observed simultaneously and so
both data sets probably sample the spin phases at different orbital
phases.
Figure 6 shows the Hα narrow component (α = 1200 km s−1)
and the broad-base component (α= 3500 km s−1) overplotted. The
two components are in phase. The radial velocity variation with the
spin period of the Hβ and Hγ broad-base component could not be
detected, possibly due to velocity cancellation We discuss this in
Section 6.
5.2 Spin Tomograms and Trailed Spectra
Spin tomograms of EX Hya were constructed by Hellier (1999)
but revealed little information. Also, Belle et al. (2005) observed
no coherent emission site/s on their tomograms folded on the spin
phase.
The Hβ and the Hγ BPM and MEM spin tomograms from
2001, however, have revealed a coherent emission site between
Vx ∼ 500 km s
−1 and ∼ 1000 km s−1 which is evidence of emis-
sion from the accretion curtains (Figures 7 and 8). But it is a well
known fact that since the spin period is ∼ 23 of the orbital period in
EX Hya, orbital cycle variations do not smear out when folded on
the spin phase but repeat every 3 spin cycles (Hellier et al. 1987).
This is thought to be the origin of most of the structure in the emis-
sion lines at velocities < 1000 km s−1 (Hellier 1999). To address
this problem, phase-invariant subtraction is performed where emis-
sion that does not vary with the spin cycle is subtracted from the
data. This is achieved by measuring minimum flux at each wave-
length and subtracting this value. The results are shown in the sec-
ond panels from the top of Figure 8. It should be mentioned though,
that even subtracting the invariant part of the line profiles does not
guarantee that the influence of the orbital period variations has been
completely removed.
c© 0000 RAS, MNRAS 000, 000–000
Spectroscopic Study of the IP EX Hydrae in Quiescence 5
Figure 4. 2001 Hα, Hβ, Hγ and HeI λ4471 trailed spectra (top row of pan-
els) and MEM orbital Doppler maps (second row of panels from the top)
as well as the average-subtracted trailed spectra (third row of panels) are
shown plotted on the same scale except for Hα panels. The HVC and the
NSC are indicated. The fourth row shows the average-subtracted Doppler
maps and the models plotted for q = 0.21, i = 78◦ and M1 = 0.50M⊙ . The
bottom panels are the reconstruction of the average-subtracted data. The
fourth and bottom panels are also plotted on the same scale except for Hα
panels. The lookup table of this figure is such that the brightest emission
features appear with decreasing intensity from black to light grey.
A spin-wave (to differentiate it from the s-wave which is nor-
mally caused by the bright spot) in the Hα trailed spectra (Figure 8)
was detected from the data after the phase-invariant subtraction was
performed. This is the first detection of modulation over the spin
cycle in the optical emission line data of EX Hya. This spin-wave
can be seen in the trailed spectra before (but hard to see) and af-
ter subtraction of the phase-invariant line profile. The narrow peak
component is responsible for this spin-wave which is shown ex-
Ηγα =1200
α = 900
α =1200 αΗ
SPIN PHASE
Figure 5. The Hβ (top panel), Hγ (middle panel) and Hα (bottom panel)
spin radial velocities of the narrow component from the 1991 combined data
(Hβ and Hγ) and 2001 data (Hα). The radial velocities were prewhitened by
the orbital frequency and phase-folded on the spin frequency using 30 bins
and are shown plotted as a function of the spin phase.
Figure 6. The spin radial velocity curves of the Hα narrow (crosses) and
broad (dots) components from 2001 (30 bins) plotted as a function of the
spin phase. The solid line represents a fit to the data.
panded in the second column of panels in Figure 8 (the narrow peak
component was selected by hand over a velocity range of ±500 km
s−1). The spin-wave shows maximum blueshift near phase 1.0 and
maximum redshift near phase 0.5, and has an amplitude of ∼ 500
km s−1. The Hα tomogram shows corresponding emission near the
“3 o’clock” position (blob of emission right at the edge of the map),
around ∼ 500 km s−1. DFTs show lower amplitude (∼ 40 km s−1
for Hα and ∼ 130−140 km s−1 for Hβ and Hγ) probably due to di-
lution by stationary material. Also, the Hα MEM tomogram shows
stronger emission that peaks in a broad structure at lower velocities
(Vx ∼ -200 km s
−1 – ∼ +200 km s−1 – around the “5-6 o’clock”
position). Circular motion gives rise to low or zero radial velocities
when the motion is perpendicular to the line of sight, and the emis-
sion seen around the “5-6 o’clock” position could not be from such
velocities since it shows maximum blueshift at φ67 ∼ 0.2− 0.25.
Similar emission was observed in a Polar and was thought to be
due to material that has just been decelerated after having attached
to the magnetic field lines (Schwarz et al. 2005) (we discuss an
alternative explanation in Section 6). The emission near the edge
c© 0000 RAS, MNRAS 000, 000–000
6 N. Mhlahlo, D.A.H. Buckley, V.S. Dhillon, S.B. Potter, B. Warner and P.A. Woudt
Figure 7. The Hα, Hβ, Hγ and HeI λ4471 trailed spectra from 2001 folded
on the spin period are shown at the top panels and the average-subtracted
spectra are shown at the second panels. Doppler maps constructed from
the phase-invariant subtracted spectra are shown in the bottom panels. The
Doppler maps were constructed using the BPM and are shown on the same
velocity scale with the trailed spectra. The lookup table is as in Figure 3.
of the tomogram (at ∼ 500 km s−1) shows maximum blueshift at
φ67 ∼ 1.0 and therefore cannot be due to motion perpendicular to
the line of sight either.
The Hα trailed spectra also show emission coupled to the spin-
wave near φ67 ∼ 0.1 − 0.6 that extends to high velocities in the
red, a similar situation to that seen the in orbital tomograms due
to the HVC (Section 4.1). The corresponding emission in the Hα
tomograms extending to higher velocities in the red spectral region
is not clear.
Both the Hβ and Hγ phase-invariant subtracted trailed spectra
show three weak-intensity spin-waves. The most clearly visible of
the three is phased with maximum redshift near φ67 ∼ 0.3− 0.4,
with an estimated velocity amplitude of ∼ 900 km s−1 and corre-
sponds to the emission near the “3 o’clock” position in the tomo-
gram (Figure 8). The reconstructed trailed spectra reproduce the
observed data.
The Hβ and Hγ trailed spectra in Figure 7 seem to support
these results. The emission observed near the “3 o’clock position”
in the tomograms has also been seen in other IPs such as AO Psc
and FO Aqr (Hellier 1999) and was interpreted as emanating from
the upper accretion curtain.
The spin-waves are weak in intensity though, and more data
are needed to support these results.
6 DISCUSSION OF THE ORBITAL AND SPIN DATA
The generally accepted model of EX Hya has the material leaving
the secondary star through the L1 point, passing via a stream of
material which orbits about the white dwarf, to form an accretion
Figure 8. Hα, Hβ and Hγ trailed spectra of 2001 folded on the spin period
are shown in the top panels and the phase-invariant subtracted trailed spectra
are shown in the second panels from the top. MEM spin Doppler tomograms
constructed from the phase-invariant subtracted spectra are shown in the
third panels with the reconstructed spectra in the bottom panels. The spin
wave observed in the Hα phase-invariant subtracted trailed spectra, which
was caused by the Hα narrow component, is shown expanded on a smaller
velocity scale. The first column of panels are plotted between -1500 km s−1
and 1500 km s−1 and the last two column are plotted between -2000 km
s−1 and +2000 km s−1. The lookup table is as in Figure 4.
disc. The magnetic field lines of the white dwarf which form accre-
tion curtains above and below the orbital plane channel the material
from the disc, starting from the co-rotation radius (Rco) where the
disc is truncated by the field lines, to the surface of the white dwarf
(Hellier et al. 1987; Rosen et al. 1991).
King & Wynn (1999) challenged this model by arguing that
systems with Pspin/Porb > 0.1 cannot possess Keplerian discs since
this implies Rco ≫Rcir. They showed that the spin equilibrium state
in EX Hya is determined by Rco ∼ b, where b is the distance to the
L1 point. In this model the accretion curtains extend to near the L1
point, and EX Hya resembles an asynchronous polar where most of
the material accretes via the stream (King & Wynn 1999; Wynn
2000).
Belle et al. (2002) revised the model of EX Hya after they
showed that their EUV data support the model of King & Wynn
c© 0000 RAS, MNRAS 000, 000–000
Spectroscopic Study of the IP EX Hydrae in Quiescence 7
(1999). Their revised model suggested that the magnetic field in
EX Hya forms a large accretion curtain extending to the outer edge
of the Roche lobe causing:
• part or all of the non-Keplerian disc (hereafter the ring of ma-
terial or the ring) to rotate with the white dwarf,
• an extended bulge (later, Belle et al. (2005) showed that there
was Vertically Extended Material (VEM) obscuring the s-wave
emission during φ98 = 0.57− 0.87, and evidence for overflowing
stream accretion in EX Hya), and
• the ring of material to feel magnetic force at the regions of the
ring close to the poles, causing the ring material at these locations
to be controlled by the magnetic field, forming two chunks along
the accretion ring that rotate with the white dwarf.
Recently, Norton et al. (2004, 2004a) have shown that for sys-
tems with Pspin/Porb ∼ 0.72, when the mass ratio is smaller at
q = 0.2, the material forms a ring near the edge of the primary
Roche lobe, from where accretion curtains funnel down to the white
dwarf surface, in agreement with King & Wynn (1999) and Belle
et al. (2002). The material is fed from the ring (ring-fed accretion)
and channeled along the magnetic field lines (when the angle be-
tween the white dwarf spin axis and magnetic dipole axis is small
i.e < 30◦, which is true for EX Hya).
The discussion by Eisenbart et al. (2002) on the IR-UV flux
distribution in EX Hya implies a disc (isobaric and isothermal) with
an outer radius of 1.6×1010 cm and a thickness of 2×108 cm, and
an assumed central hole of 6×109 cm, but Eisenbart et al. (2002)
suggested that the structure could also be a ring with a larger in-
ner radius, in line with the suggestion of King & Wynn (1999);
Belle et al. (2002) and Norton et al. (2004). They found that the
disc component contains about 1/6 of the total flux which is a bit
more than expected from gravitational energy release at the inner
radius, Rin > 6×10
9 cm.
Our spectroscopic data support both the model of Belle et al.
(2002) and Norton et al. (2004) in which material from a ring, cir-
cling the white dwarf and co-rotating with the magnetic field lines
at the outer edge of the Roche lobe, is accreted by the white dwarf.
The presence of the bright spot revealed by the trailed spectra,
the DFTs of the radial velocities and the Doppler maps (Figures 3
and 4) suggest the presence of a disc or ring of material extending
to near the Roche lobe radius, around the white dwarf. When com-
paring the 1991 and 2001 tomograms for the Hβ they appear to be
in the same state or similar, given the fact that they are ten years
apart. It is reassuring that the fact that the two groups of lines have
not been measured simultaneously is not a significant problem in
the analysis.
More importantly, a spin pulse modulated at velocities con-
sistent with those of the material circulating at the outer edge of
the disc (∼ 500−600 km s−1) (Figures 1 and 2) was detected and
provides evidence for co-rotation of the extended accretion curtains
with the ring material. As discussed in Section 5.2, these low radial
velocities mentioned above were not caused by motion perpendic-
ular to the line of sight near the white dwarf, neither were they
caused by velocity cancellation as will be shown later.
A spin wave was detected in the spin-folded trailed spectra
of Hα (Figure 8) with a velocity semi-amplitude of ∼ 500-600
km s−1. The spin wave shows maximum blue-shift near phase
φ67 ∼ 1.0 (when the upper magnetic pole is pointed away from
the observer) and maximum redshift near phase φ67 ∼ 0.5. The Hα
equivalent widths show maximum flux near φ67 ∼ 1.0. This picture
is consistent with the accretion curtain model of IPs and is possible
if accretion occurs via a disc/ring. The spin tomograms (Figures 7
and 8) show evidence of the accretion curtain emission extending
from ∼ 500 km s−1 to high velocities (∼ 1000 km s−1), suggesting
that material is channeled along the field lines from the outer ring.
The Hα narrow and broad base components show similar phase
variation, suggesting same position of maximum radial velocity as
shown in Figures 6 and 9 (line OA). This indicates that material is
channeled from the ring (at low velocities) to high velocities along
the field lines.
A mass ratio of q ∼ 0.2 was measured from our data, and so
the period ratio Pspin/Porb ∼ 0.68 is consistent with the ring accre-
tion model of Norton et al. (2004).
Decreased prominence of the narrow s-wave component
around φ98 = 0.57−0.87 (Figure 3 and 4) was observed and sug-
gests the presence of VEM at the outer edge of the ring of ma-
terial obscuring the emission at these phases. The presence of the
overflow stream may be infered from this observation (Belle et al.
2005). But direct evidence comes from orbital Doppler tomograms
which show an asymmetry in the emission, where more emission
is observed from the secondary Roche lobe to the lower left quad-
rant than from the opposite side. Average-subtracted orbital tomo-
grams show this emission at higher velocities (∼900-1000 km s−1)
(Figures 3 and 4), and it corresponds to the HVC observed in the
trailed spectra, which is modulated with a velocity semi-amplitude
of ∼ 1000 km s−1. This HVC is reminiscent of that detected by
Rosen et al. (1987) in the trailed spectra of the AM Her system
V834 Cen. Their HVC was blueshifted with a velocity of 900 km
s−1and was said to be produced in the stream close to the white
dwarf. The only difference is that there was no evidence of the
HVC emission when it was expected to be seen redward of another
component (medium-velocity component) in their data, whereas in
EX Hya the evidence of the HVC emission is missing between
φ98 ∼ 0.3− 0.85. The HVC emission is maximally blueshifted at
φ98 ∼ 0.3−0.4. This phasing is consistent with the expected phase
of impact of a stream of material from the secondary with the disc
or of the overflow stream material free-falling onto the magneto-
sphere of the primary (Hellier et al. 1989).
Support for overflow stream is also provided by spin tomo-
grams where emission is observed on the upper accretion curtain
with velocities consistent with stream velocities. This suggests that
this emission site may also have resulted due to impact of overflow
stream with the magnetosphere. The resulting emission is receding
from the observer at maximum redshift near φ67 ∼ 0.4 (Figure 8),
in agreement with the accretion curtain model.
The model of King & Wynn (1999) is not fully supported by
our observations since it predicts direct accretion via a stream. Our
observations, however, fit the models of Norton et al. (2004) and
Belle et al. (2002, 2005). There is evidence for strong Hα emis-
sion of the narrow s-wave component in the spin tomograms, cen-
tred around ∼ 100 km s−1 (Figure 8), that is not accounted for by
these models. This emission shows maximum blueshift at phase
φ67 ∼ 0.2, suggesting that these are rotational velocities (or a com-
bination of streaming and rotational velocities) of the antiphased
motion of a source locked to the white dwarf. One possible expla-
nation is that this emission comes from the opposite pole of the
white dwarf, at a radial distance of 6×109 cm (∼8Rwd). Siegel
(1989) found that the eclipsed optical source in EX Hya is centred
at a radial distance of 1.5×109 cm (∼2Rwd), which is about four
times closer to the white dwarf compared to our result. This could
be the same emission region, but in our observations the emission is
spread out, possibly due to the quality of the data, and this could ac-
count for the difference in the radial distance values quoted above.
But we cannot imagine a geometry where such low rotational ve-
c© 0000 RAS, MNRAS 000, 000–000
8 N. Mhlahlo, D.A.H. Buckley, V.S. Dhillon, S.B. Potter, B. Warner and P.A. Woudt
Observer
ring of material
through upper pole
magnetic axis
phases of maximum blueshift of the
phase of maximum blueshift
phases of maximum redshift
chunk of material’s
overflowing stream
of the s−wave
narrow s−wave
chunk of
material
with field
corotating
phase of maximum
blueshift of the HVC
Observer
0.75 0.7
outer disc
corotating with field
part of disc
maximum blueshift
Ηα BBC
phase of
maximum intensity
phase of
magnetic axis
through
upper pole
phase of
minimum intensity
Ηα BBC
maximum redshift
phase of
Figure 10. A model of EX Hya in quiescence. The figures are drawn over the orbital cycle (left) and spin cycle (right) and show the magnetosphere extending
to the outer edge of the ring, and the chunk of material corotating with the field lines. A verticaly extended material (VEM) is irradiated by the white dwarf in
its inner regions (left).
Observer
Figure 9. A depiction of the regions where Hα was formed. Both the nar-
row and broad base components fall along the same radial direction, OA,
resulting in similar phase variation.
locities can dominate over streaming velocities along the field lines
near the white dwarf. We therefore suggest that this is evidence for
material that is diverted out of the orbital plane. Since one of the
assumptions of Doppler tomography is that everything lies on the
plane, it is not possible to locate the exact position of this emission
relative to the white dwarf.
6.1 White dwarf and secondary masses
Hellier et al. (1987) showed that maximum line widths of ±3500
km s−1 constrain the mass of the white dwarf, and a free-fall
velocity of this magnitude could be achieved for white dwarf
masses greater than 0.48 M⊙. We found M1 = 0.50 ± 0.05M⊙ ,
in good agreement with the results obtained from recent stud-
ies by Hoogerwerf et al. (2004); Beuermann et al. (2003) and
Vande Putte et al. (2003).
For the secondary, we derived M2 = 0.10 ± 0.01 M⊙ from
the secondary mass-period relation of Smith & Dhillon (1988), and
this value agrees with that obtained by Vande Putte et al. (2003).
Beuermann et al. (2003) and Hoogerwerf et al. (2004) find lower
values for M2 consistent with 0.09 M⊙. Eisenbart et al. (2002) ar-
gues that for a secondary mass as low as 0.1 M⊙ the secondary
would have to be substantially expanded by ∼10%.
6.2 The revised model of EX Hya
We propose a model where one of the two chunks alluded to by
Belle et al. (2002), which are formed by the magnetic pull along
the accretion ring, co-rotates with the accretion curtains at the outer
edge of the Roche lobe at ∼ 500-600 km s−1, giving rise to the pul-
sation of emission at the spin period which we observe in our data,
while the other is hidden by the accretion curtain below the ring
of material. The resulting emission is maximally blueshifted near
φ67 ∼ 0.8 (Figure 5). In the accretion curtain model, at φ67 ∼ 0.5
in the spin cycle, minimum flux (due to higher opacity) is observed
when the upper accretion pole of the white dwarf is pointed to-
wards the observer (Hellier et al. 1987), and so the phasing men-
tioned above is compatible with the motion of a rotating accretion
funnel. This is illustrated in Figure 10, where the position of the
observer at pulse maximum is indicated, and the axis of the mag-
netic pole is shown. The disruption of the disc by the magnetic
field at the outer disc is illustrated and part of the disc co-rotating
with the magnetosphere is shown. At a corotation radius, Rc ∼
b = a(0.500 − 0.227log M2M1 ) (∼ 3× 10
10 cm), the material is ro-
tating at a velocity of v2 = GMb ∼ 500 km s
−1, in good agreement
with the observations. Also, a rotation velocity of ∼ 600 km s−1
was measured from the spectra and the radial distance from the star
to the ring of material was found to be ∼ 3× 1010 cm, which is
similar to b, for a white dwarf mass of 0.5 M⊙ (Keplerian motion
about the white dwarf had to be assumed in these calculations).
At this radius, the accretion curtain is also rotating at a velocity of
2πRco/Pspin ∼ 500 km s−1.
∼ 6× 10−12 ergs cm−2 s−1 (64% - integrated over one spin
cycle) of the original line fluxes that is contained in the average-
c© 0000 RAS, MNRAS 000, 000–000
Spectroscopic Study of the IP EX Hydrae in Quiescence 9
subtracted profile of Hα shows radial velocity variations with the
spin period. Assuming that Hβ and Hγ also show a similar flux
variation (Hβ and Hγ spin tomograms also show a low-velocity s-
wave but this result is not secured due to poor quality of data),
the total line fluxes showing radial velocity variations with the
spin period can be estimated to be ∼ 2× 10−11 ergs cm−2 s−1
for the three emission lines. This is ∼2/10 of the total disc flux
(Eisenbart et al. 2002), suggesting that only part of the ring coro-
tates with the white dwarf while the rest of the material may be in-
volved in a near Keplerian motion (this is a rough comparison since
the flux is integrated over one spin cycle for Hα, Hβ and Hγ in our
data whereas Eisenbart et al. (2002) derived their total flux values
from one spectrum over the wavelength range λ = 912−24000 Å).
While some of the ring material co-rotates with the accre-
tion curtains (i.e. remains in the disc rather than being immedi-
ately channeled along the field lines), some is channeled along the
field lines at ∼ 500 km s−1 towards the white dwarf. There is also
some material that overflows the ring and attaches onto the mag-
netic field lines. The overflow stream hits the magnetosphere, prob-
ably causing a second bright spot on the slowly rotating magneto-
sphere (Figure 10). The overflow stream is irradiated by the white
dwarf in its inner regions close to the white dwarf (the regions fac-
ing the white dwarf). This results in the HVC emission being ob-
scured at φ98 ∼ 0.4− 0.9, which are phases where the stream is
viewed from behind-opposite the side facing the white dwarf, hid-
ing the irradiated inner regions. HVC emission from the stream is
blueshifted when that from the narrow s-wave component shows
maximum redshift. Near φ98 ∼ 0.25 the two s-waves intersect, ex-
plaining the asymmetry in the brightness of the s-wave seen near
φ98 ∼ 0.25 (Figure 4). The overflow stream curls nearly behind the
white dwarf and it is truncated by the field when the upper magnetic
pole is facing the stream.
Ferrario & Wickramasinghe (1993) and Ferrario, Wickramas-
inghe & King (1993) showed that in IPs the accretion curtain below
the orbital plane can contribute in the radial velocities of a system
if it can be seen either through the central hole of the truncated
disc, or from below the disc, or both. This effect will result in ve-
locity cancellation due to nearly equal quantities of material that
are blueshifted and redshifted on the accretion curtains (Ferrario,
Wickramasinghe & King 1993).
In EX Hya where the inclination is high (78◦) and the disrup-
tion radius is large (∼ 40 RW D, for a white dwarf mass of 0.5 M⊙)
as proposed in Figure 10, it is clear that we see spin-varying emis-
sion from two opposite magnetic poles, producing a fairly symmet-
ric structure in the spin-folded line profiles (Hellier et al. 1987;
Rosen et al. 1991). If emission from these opposite poles is can-
celling out then the sum will have a much lower velocity. This could
explain the near zero and low amplitude of the radial velocity vari-
ation at the spin period of the Hβ and Hγ, and Hα (6 40 km s−1)
broad-base component, respectively (see also Hellier et al. (1987)
and Ferrario, Wickramasinghe & King (1993)).
One could take this argument further by suggesting that the
spin modulation we observe in our data at velocities near ∼ 500
km s−1 (Figures 1 and 2) is just the slight asymmetries between
the two poles. The resulting velocity could just be a measure of
the degree to which the poles cancel their velocities near ±3500
km s−1(Coel Hellier; private communication). This, however, can-
not be the case for Hβ and Hγ since these two emission lines show
motion that is consistent with that of a rotating object, suggesting
that the line profiles are not dominated by the infall velocities at
the two opposite accretion poles. If they were produced close to
the white dwarf then maximum rotational velocity near ±3500 km
s−1 would be 2πR/Pspin ∼ 30 km s−1, which is much smaller than
∼ 500 km s−1. However, rotional velocities close to the ring are
∼ 500 km s−1. For Hα, however, we observe maximum blueshift
at φ67 ∼ 1.0, and so velocity due to cancellation anywhere between
0 and ±3500 km s−1are expected, depending on how much the
two poles cancel. If both accretion curtains are still visible and
symmetric at large radii (which is possible as suggested by Fer-
rario, Wickramasinghe & King (1993) and our model), velocity
cancellation will still result in smaller amplitudes than those of
∼ 500 km s−1observed in our data. This would then count against
the argument above. Furthermore, Hα orbital Doppler tomograms
show strong emission at the bright spot. If our model is correct, the
field lines should also attract this Hα dominated material, which is
chanelled along the field lines, as already shown above. The veloc-
ity of this material due to streaming motion near the outer ring is
less than that of the Hα broad-base component close to the white
dwarf, as expected. A strong constraint on our model is that the
disruption radius of EX Hya has been shown to be at 5-9×109 cm
(Hellier et al. 1987; Beuermann et al. 2003) which implies a white
dwarf magnetic moment of µ ∼ 7× 1031 G cm3. For our model
this would imply that the accretion curtains do not extend to near
the Roche lobe radius. The theoretical analysis of King & Wynn
(1999) and Wynn (2000), however, has shown that equilibrium ro-
tation is possible if the magnetic moment in EX Hya falls within the
range of 1033 6 µ6 1034 G cm3. These are comparable to weakest
field AM Hers below the period gap, and that EX Hya could pos-
sess such magnetic moments is supported to a certain extent by the
average-subtracted trailed spectra of EX Hya that are reminiscent
of emission lines seen in some Polars, e.g. V834 Cen (as discussed
above), EF Eri (Crampton et al. 1981; Cowley et al. 1982), QS Tel
(Romero-Colmenero et al. 2003) and VV Pup (Diaz 1994). Fur-
thermore, Cumming (2002) raised the possibility that the magnetic
fields in IPs are buried by the material due to high accretion rates
and so are not really as low as they appear. The ring structure in EX
Hya could imply higher accretion rates in EX Hya than previously
thought since the capacity of the ring of material to store matter
may be low when compared to that of a classical disc, resulting in
the accretion of more material than in a classical disc case.
7 SUMMARY
Optical observations of EX Hya and the analysis have suggested
that large accretion curtains extending to a distance close to the
L1 point exist in this system. The DFTs and spin tomograms have
for the first time provided evidence for corotation of the field lines
with the ring material near the Roche lobe. Also, tomography and
the phasing of the spin waves have suggested that feeding by the
accretion curtains of the material from the ring (ring-fed accretion)
takes place. These findings support the models of Belle et al. (2002)
and Norton et al. (2004) for EX Hya and the simulations done by
Norton et al. (2004a) which have shown that for systems with the
parameters of EX Hya, the accreting material forms a ring at the
outer edge of the primary Roche lobe, from where accretion cur-
tains funnel down to the white dwarf surface.
Evidence for stream overflow accretion has been observed.
The HVC caused by the overflow stream disappeared at φ98 ∼
0.4−0.9 due to obscuration by the stream. Obscuration of the NSC
at φ98 ∼ 0.57−0.87 suggested the presence of the VEM which was
irradiated by the white dwarf in its inner regions.
The Hα broad-base component shows a radial velocity varia-
tion with the spin period whereas that of Hβ and Hγ could not be
c© 0000 RAS, MNRAS 000, 000–000
10 N. Mhlahlo, D.A.H. Buckley, V.S. Dhillon, S.B. Potter, B. Warner and P.A. Woudt
detected. The low-amplitude velocity variations modulated at the
spin period for Hα and for Hβ and Hγ is explained in terms of ve-
locity cancellation effects.
We have provided an explanation for the asymmetry in the
intensity of the narrow s-wave component seen in EX Hya trailed
spectra in the optical. The narrow s-wave component and the HVC
cross at φ98 ∼ 0.25, resulting in the asymmetry in brightness that
we observe at these phases.
The spin-folded trailed spectra are not of good quality and
more data are needed to confirm these results.
ACKNOWLEDGMENTS
NM would like to acknowledge financial support from the Sains-
bury/Linsbury Fellowship Trust and the University of Cape Town.
We would like to thank Kunegunda Belle, Coel Hellier and Andrew
Norton for invaluable discussions and for their constructive com-
ments. We acknowledge use of D. O’Donoghue’s and Tom Marsh’s
programs Eagle and Molly, respectively.
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http://arxiv.org/abs/astro-ph/9806141
Introduction
Observations And Data Reduction
1991 Observations
2001 Observations
The Radial Velocities
Period Searches
Orbital Variations of the Emission Lines
Orbital Tomograms and Trailed Spectra
Spin Variations of the Emission lines
The Spin Radial Velocity Curve
Spin Tomograms and Trailed Spectra
Discussion of the Orbital and Spin Data
White dwarf and secondary masses
The revised model of EX Hya
Summary
References
|
0704.0018 | In quest of a generalized Callias index theorem | arXiv:0704.0018v2 [hep-th] 21 Apr 2007
In quest of a generalized Callias index theorem
Andreas Gustavsson1
Förstamajgatan 24,
S-415 10 Göteborg, Sweden
Abstract
We give a prescription for how to compute the Callias index, using as regulator an
exponential function. We find agreement with old results in all odd dimensions. We
show that the problem of computing the dimension of the moduli space of self-dual
strings can be formulated as an index problem in even-dimensional (loop-)space. We
think that the regulator used in this Letter can be applied to this index problem.
[email protected]
http://arxiv.org/abs/0704.0018v2
1 Introduction
We do not know what six-dimensional (2, 0) theory really is. It is believed
that it can sustain solitonic self-dual strings [1], although no one today knows
what a (non-Abelian) self-dual string really is. But if we break the gauge group
maximally to U(1)r, then we should be able to define the charges of these
mysterious self-dual strings by the asymptotic behaviour of the U(1) gauge
fields. One should expect these asymptotic U(1) fields to be (at least isomorphic
with) a copy of the familiar abelian two-form gauge potentials (with self-dual
field strengths).
It now seems to make sense to ask a question like, what is the dimension of
the moduli space of self-dual strings of a given charge?
If the gauge group is SU(2) and is broken to U(1) by the Higgs vacuum
expectation value (that should also determine the tension of the string), then the
intuitive answer to this question is 4N where N is the U(1) charge in a suitable
normalization, such that N = 1 corresponds to one self-dual string. One may
argue that half the supersymmetry is broken by the string. Therefore one string
should sustain 4 fermionic zero modes. Since some (half) of the supersymmety
is unbroken there should also be 4 corresponding bosonic zero modes. These
are naturally identified with the translational zero modes associated with the
four transverse directions to the string. Furthermore, the strings being BPS,
should be possible to separate at no cost of energy (thus staying in the moduli
space approximation). If we take them far from each other, one may suspect
that we can just add 4 bosonic zero modes from each string, to get 4N bosonic
zero modes in total in a configuration of N strings [2].
It would of course be nice to have a proof of this conjecture. Could it be
proven if one had some index theorem? We will not provide a full solution to
this problem in this Letter. But we will make it plausible that the problem can
indeed be solved by computing the index of a certain Dirac operator in loop
space.
To address our index problem, we think that one can lend the methods
that Callias [3] used to prove his index theorem in odd-dimensional spaces. In
our case we have an even number of dimensions (namely the four transverse
direction) so it is apparent that we would have to construct a new type of
index. This we do in section 3.
In section 2 we recall the Callias method [3] to address index problems
in open spaces, though we will modify Callias’ regularization, using the more
convergent exponential function to obtain the index, as the limit
γe−sD
, (1)
(here D2 > 0 and γ =diag (1,−1)) rather than
D2 +M2
, (2)
which is the regularization that Callias used. We think that using the more
convergent regularization of an exponential function is interesting in itself, as
it could possibly extend the Callias index theorem to a wider class of index
problems. Therefore we will devote the first part of this Letter on this subject.
But let us at once say that our regulator probably has no advantages when
attacking these old problems. It does not provide us with a solution for how to
count the number of zero modes in a multimonopole configuration with a non-
maximally broken gauge group, where the index can not be reliable computed
due to a contribution from the continuum portion of the spectrum. What we
hope though, is that our regulatization can be useful when attacking our new
index problem associated with the moduli space of self-dual strings.
In section 2 we obtain the index in one and three dimensions. In three
dimensions we apply this on the multimonopole moduli space and re-derive the
result in [4]. A recent review article on monopoles and supersymmetry is [5].
The one and three-dimensional index problems have also been studied in [6].
We then indicate how our method manages to reproduce the correct results in
any odd dimensions. In section 3 we show how one at least in principle should
be able to compute the dimension of the moduli space of N self-dual strings by
computing a certain index.
2 Computing the Callias index in odd-dimensional
spaces
For Dirac operators on open n− 1-dimensional space where n− 1 is odd, there
is an index theorem by Callias [3]. This applies to Dirac equations of the form
Dψ = 0 (3)
where the Dirac operator D is of the form
D = γiiDi + γnφ. (4)
Here i = 1, ..., n− 1 and γµ ≡ (γi, γn) denote the Dirac gamma matrices,
{γµ, γν} = 2δµν . (5)
We define the gauge covariant derivative as iDis = i∂is +Ais and all our fields
are hermitian. If n− 1 is odd, the gamma matrices can be represented as
One may use the n-dimesional notation Aµ = (Ai, φ), D = γµiDµ, but one must
then remember that space is really n− 1 dimensional.
If n − 1 is even there is no Weyl representation of the gamma matrices
(because of the inclusion of the ‘gamma-five’), and no index theorem of this
form exists.
We define the ‘gamma-five’ for even n as
γ ≡ −i−
2 γ1···n (7)
which then is hermitian, and we define the projectors
(1∓ γ) . (8)
In odd dimensions n− 1, the Dirac operator splits into two Weyl operators
D ≡ P+DP−
D† ≡ P−DP+ (9)
Because P± andD are all hermitian, it follows thatD† is the hermitian conjugate
of D. Also, because D is already of an off-block diagonal form, it suffices to
include just one of the projectors, so we can just as well write this as
D = P+D = DP−
D† = P−D = DP+ (10)
The index can now be defined as
dimkerD − dimkerD† (11)
Since kerD = ker
and kerD† = ker
we can express this as2
dimker
− dimker
= dimker
. (12)
where we have noted that γ = P− − P+.
Callias, Weinberg and others used the regulator
I(M2) = Tr
D2 +M2
to obtain the index as the limit M2 → 0. In this Letter we will be slightly more
general. We define
Ji(x, y) ≡ tr 〈x |γγif(D)| y〉 , (14)
for any function f (and of course D is not dimensionless, so D has to be accom-
panied by M in a suitable way). Then we notice that
W (x, y) ≡ (iγi∂xi + γµAµ(x) +M) 〈x |f (D)| y〉
= 〈x |f (D)| y〉
−iγi∂yi + γµAµ(y) +M
where (manifestly)
W (x, y) = 〈x |(D +M)f (D)| y〉 . (16)
From this, we obtain the following identity
∂xi + ∂yi
Ji(x, y) = 2tr 〈x |γDf(D)| y〉
+tr (Aµ(y)−Aµ(x)) 〈x |γγµf(D)| y〉 (17)
In odd dimensions, the second term in the right hand side vanishes as x ap-
proaches y. This can be seen as being equivalent to the statement that there is
no chiral anomaly in odd dimensions (by using point-splitting and inserting a
Wilson line). So we get
i∂iJi(x, x) = 2tr 〈x |γDf(D)|x〉 (18)
2To see this that kerD = kerD†D we apply the definition of hermitian conjugate
with respect to the inner product (ψ, χ) =
dxψ†χ and the property of the norm, to
0 = (ψ,D†Dψ) = (Dψ,Dψ).
If we wish to compute the index as in Eq (13), then we can take
f(D) =
D2 +M2
(however there is no unique choice of Ji). We then get
Ji(x, y) = tr
D2 +M2
−D2 +D2 +M2
D2 +M2
= −tr
D2 +M2
. (20)
provided
= 0 (21)
We will see in the next few paragraphs how one can achieve this by using a
principal value prescription.
The virtue of expressing Eq (13) as a total divergence, is that we then can
compute the index as a boundary integral over an (n− 2)-sphere at infinity as
I(M2) =
dΩn−2r
n−2x̂iJi(x, x). (22)
where r is the radius of the sphere and dΩn−2 denotes the volume element of
the unit sphere.
If instead we wish to compute the index as the limit of
I(s) = Tr
γe−sD
. (23)
as s→ ∞, then we get
Ji(x, y) = tr
. (24)
It might seem confusing that we can have a plus sign here, when we have a
minus sign in Eq (20). These peculiar signs seem to be correct though. Why we
can have opposite signs should be a reflection of the fact that these expressions
can not be continuously connected with each other, at least not in any obvious
way (like taking M to zero and s to zero. In fact s should be taken to plus
infinity as M goes to zero). We will now illustrate how one can use this Ji to
compute the index in odd dimensions.
One dimension
We choose our gamma matrices as
, γ2 =
and we have
γ = iγ1γ2 =
. (26)
The Dirac operator reads
D = iγ1∂ + γ2φ (27)
We need the square of the Dirac operator,
D2 = −∂2 + φ2 + γ∂φ. (28)
We make the choice
J1(x, y) = −tr
D2 +M2
We assume that φ(x) converges towards some constant values at x = −∞ and
x = +∞. That means that we may ignore ∂φ(x) for sufficiently large |x|, where
we then get
J1(x, x) = −tr (γγ1γ2)
k2 + φ2 +M2
φ2 +M2
The index is now given by
(J1(+∞)− J1(−∞)) = ±1 (31)
if φ flips the sign an odd number of times when going from −∞ to +∞, and 0
otherwise.
If instead we choose
J(x, y) = tr
then we get
J(x, x) = tr (γγ1γ2)
k2 + φ2
e−s(k
2+φ2) (33)
If we compute the integral over k in the most natural way, then we get a result
that vanishes in the limit s → ∞. Could there be another way of defining this
integral, such that we do not get zero as the result? We notice that the integral
A(s) ≡
e−s(k
k2 + 1
for s > 0 is convergent only if we integrate k along a line in the complex plane
which is such that it asymptotically is such that −π
< θ < π
where k = |k|eiθ.
Integrating along any such line in the complex plane, we get the same value of
this integral. If on the other hand we integrate over a line that asymptotically
lies outside this cone, then we get a divergent integral for s > 0. But we get
a convergent integral for s < 0. We then define the value of the integral for
s > 0 as the analytic continuation of the same integral for s < 0. It remains to
compute this convergent integral. Replacing k by ik and s by −s, we get the
integral
A(−s) = −i
e−s(k
k2 − 1 (35)
We can compute its derivative
′(−s) = −i
−s(k2−1) = −i
s (36)
The right-hand side can obviously be analytically continued to −s, and that is
how we will define A(s) where the integral representation does not converge.
We can then integrate up A′(s),
A(∞) = A(0)−
e−s = A(0)−
= A(0)− π (37)
and we then need to compute
A(0) = i
k2 − 1
We define this as the principal value. This is ad hoc – we have no argument
why one should define it like this. But if we accept this, then we get A(0) = 0.
We conclude that we could just as well define the integral that we had, as
e−s(k
k2 + 1
= −π. (39)
But this requires us to perform the integration of k in the cone where it diverges
for s > 0, and then define this integral by analytic continuation. This seem to
be rather ad hoc. We have three rather week arguments why one should Wick
rotate. First, if we keep x − y as a small number, then we get the factor
eik(x−y) and this can act as a convergence factor only if we Wick rotate. (We
illustrate this in the Appendix where we compute the corresponding integral in
any complex number of dimensions.) Second, it seems to be the only way that
we could produce a non-trivial answer. Third, with this prescription we will
manage to reproduce the right answer in any odd number of dimensions, where
we can check our result against the safer regularization used by Callias.
If we compute the integral by this prescription, then we get
J(x, x) = tr (γγ1γ2) lim
k2 + φ2
e−s(k
2+φ2) = i
and we see that we indeed get the right answer.
Three dimensions and magnetic monopoles
The physics problem that we will consider in three dimensions, is to compute
number of zero modes of the Bogomolnyi equation
Fij = ǫijkDkφ (41)
We choose the convention that our fields are hermitian. It is convenient to group
the fields into ‘gauge potential’
Aµ = (Ai, φ) (42)
We define Dµ = (Di, φ) such that iDµ = i∂µ + Aµ and we let Gµν = i[Dµ, Dν ]
be the associated ‘field strength’. Then the Bogomolnyi equation reads
Gµν =
ǫµνρσGρσ . (43)
Linearizing this, we get
DµδAν =
ǫµνρσDρδAσ (44)
Contracting with γµν , we get
(1 + γ)γµνDµδAν = 0 (45)
and if we impose the background gauge condition
DµδAµ = 0 (46)
which is to say that zero modes are orthogonal to gauge variations with respect
to the moduli space metric, then we can write this linearized equation as a Dirac
equation
Dψ ≡ γµDµψ = 0 (47)
where
ψ := (1 + γ)γµδAµ. (48)
We compute
D2 = −D2i + φ2 +
iγµνGµν (49)
Inserting the Bogomolnyi configuration we can write this, thus using the fact
that Gµν is selfdual,
2 = −D2i + φ2 +
(1 + γ)iγµνGµν . (50)
and get a vanishing theorem. Namely, dimkerDD† = 0 as DD† > 0 is strictly
postive. Hence we can compute the dimension of the moduli space dim kerD ≡
dimkerD†D just by computing the index of D. To compute the index, we now
wish to compute
Ji(x, x) = tr
γγiγkDk
We assume that asymptotically φ approaches a constant value at infinity. This
corresponds to a gauge choice where we have a Dirac string singularity. Some
further examination reveals that we get a non-negligible contribution to Ji, for
a sufficiently large two-sphere, only from the term
Ji(x, x) = tr
γγiγ4φ
(2π)3
k2 + φ2 + 1
iγµνGµν
−s(k2+φ2+ 1
iγµνGµν)
We thus need to perform an integral of the form
A(s) =
k2 + 1
e−s(k
2+1) (53)
If we choose the same prescription as we did in one dimension, then we get the
result
A(+∞) = π. (54)
For details of such a computation we refer to appendix A.
If we apply this result to the integral that we had, we get
Ji(x, x) =
γγiγ4φ
iγµνGµν
We expand the square root,
iγµνGµν = φ+
iγµνGµν + ... (56)
In the far distance, in a charge Q monopole configuration, we find that
γµνGµν = 2γkγ4(1− γ)
Q (57)
and so when we trace over the gamma matrices, we get
Ji(x, x) =
. (58)
If we now for instance assume SU(2) gauge group, broken to U(1), then if we
integrate i
Ji over S
2, we get the index 2Q. The number of bosonic zero modes
is twice the index, i.e. −4Q in our conventions [4, 5].
(2m+ 1) dimensions
In 2m+ 1 dimensions we get the integral
A(µ) ≡ lim
k2 + µ2
e−s(k
2+µ2) (59)
if we use our regulator. Here
µ2 ≡ v2 +G (60)
(and G is an abbreviation for 1
iγµνGµν .) This should be compared to the
integral
B(µ) ≡ − lim
(−1)m
(k2 + v2 +M2)
Gm (61)
that we get using the Callias regulator. 3 In order to compare these integrals,
we rewrite them as
A(µ) = µ2m−1a
B(µ) = v−1bGm (63)
where
a = lim
ξ2 + 1
−s̃(ξ2+1)
b = − lim
(−1)m
ξ2 + 1 + M̃2
We compute a according the prescription introduced above in one and three
dimensions, that is by Wick rotating ξ and continue analytically in s. (Details
are in appendix A.) We can compute b using residue calculus (introducing a
regulator so that we can close the contour on a semi-circle at infinity). The
result is
a = −(−1)mπ
b = (−1)m 1
) (65)
We next expand
vA(µ) = v
v2 +G
)m− 1
= v2ma+ ...+
m + ...
vB(µ) = bGm (66)
and we find that the coefficient of Gm becomes equal to
−(−1)m
) π (67)
if one uses our regularization, and equal to
(−1)m 1
) π (68)
3This integral comes from expanding
k2 + v2 +G+M2
k2 + v2 +M2
+ ... (62)
in powers of G as a geometric series [4].
if one uses the Callias regularization. We see that the two expressions coincide
for all m.
We have now showed that if we use our prescription of Wick rotating k to
compute the integrals over the exponential, then we get the right answer for all
cases that can be safely computed using a regulator that is less convergent. We
are inclined to think that our prescription for how to compute the integral, will
also work for index problems where the Callias regulator diverges. But we have
no proof. It is perhaps not so obvious that more general index problems can be
formulated. In the next section we will give one example of a more general type
of index problem.
3 Four dimensions and self-dual strings
To introduce the notation, we first consider the free Abelian tensor multiplet
theory in 1 + 5 dimensions. The on-shell field content is a two-form gauge
potential Bµν , five scalar fields φ
A and corresponding Weyl fermions ψ. The
field strength Hµνρ = ∂µBνρ + ∂ρBµν + ∂νBρµ is selfdual. The supersymmetry
variation of the Weyl fermions is
ΓµνρHµνρ + Γ
µΓA∂µφ
ǫ (69)
where we use eleven-dimensional gamma matrices splitted into SO(1, 5)×SO(5),
so that in particular
{Γµ,ΓA} = 0. (70)
In a static and x5 independent field configuration, in which only φ5 =: φ is
non-zero, we find the SUSY variation
Γ0i5H0i5 + Γ
iΓA=5∂iφ
ǫ (71)
If we assume that the classical bosonic field configuration is such that
∂iφ = H0i5 (72)
then the SUSY variation reduces to
δψ = ∂iφΓ
Γ05 + ΓA=5
ǫ (73)
and we find the condition for unbroken SUSY as
1 + Γ05ΓA=5
ǫ = 0 (74)
If we use the Weyl condition
Γǫ = −ǫ (75)
of the (2, 0) supersymmetry parameter ǫ, then we can also write this as
1 + Γ1234ΓA=5
ǫ = 0. (76)
We may represent the gamma matrices as
Γµ = (Γ0,Γi,Γ5) =
1⊗ iσ2 ⊗ 1, γi ⊗ σ1 ⊗ 1, γ ⊗ σ1 ⊗ 1
ΓA = 1⊗ iσ2 ⊗ σA (77)
where σ1,2,3 are the Pauli sigma matrices, γ = γ1234. Then the condition for
unbroken SUSY is
(1 + γ ⊗ σ) ǫ = 0 (78)
where σ = σ1234 = σA=5.
We have found that if
Hijk = ǫijkl∂lφ (79)
then half SUSY is unbroken. This equation is the Bogomolnyi equation for self-
dual strings [1]. We are interested in finding the number of parameters needed
to describe solutions of this equation. We can linearize it and get the equation
γi∂iχ = 0 (80)
for the bosonic zero modes, that we have gathered into a matrix
χ ≡ γijδBij + γδφ. (81)
For this to work we must also assume the background gauge condition
∂iBij = 0. (82)
Now this linearized equation Eq (80) does not make any reference to the gauge
field. So there is no way that we could count the number of parameters of a
multi-string configuration just using this equation. This should of course not be
a surprise. The strings that we have in the Abelian theory are not solutions of
the field equations. They have to be inserted by hand, that is we need to insert
delta function sources by hand, in the same spirit as for Dirac monopoles.
To be able to count the number of zero modes, we must consider some
interacting theory which (at the classical level) has solitonic string solutions.
To pass to non-Abelian theory we begin by rewriting the Abelian theory
in loop space. Loop space consists of parametrized loops C: s 7→ Cµ(s). We
introduce the Abelian ‘loop fields’ [7]
Aµs = Bµν(C(s))Ċ
ν (s)
φµs = φ(C(s))Ċµ(s)
µs = ψ(C(s))Ċµ(s) (83)
With these definitions, a short computation reveals that Aµs transforms as a vec-
tor and φµs a contra-variant vector under diffeomorphisms in loop space induced
by diffeomorphisms in space-time. One may then extend these transformation
properties to any diffeomorphism in loop space. Space-time diffeomorphism and
reparametrizations of the loops then get unified and are both diffemorphisms
in loop space. The only thing to remember is what is kept fixed under the
variation. If it is the parameter of the loop, or the loop itself.
The field strength becomes
Fµs,νt = Hµνρ(C(s))Ċ
ρ(s)δ(s− t) (84)
In terms of these fields, the Bogomolnyi equation will read4
Fis,jt = ǫijkl∂k(sφlt). (85)
We pass to the non-Abelian theory by letting these loop fields become non-
Abelian, in the sense that Aµs = A
a(s) where λa(s) are generators of a loop
algebra associated to the gauge group [7]. We introduce a covariant derivative
Dµs = ∂µs +Aµs. (86)
Local gauge transformations act as
δΛAµs = DµsΛ
µs = [φµs,Λ]. (87)
Given a loop C, we automatically get a tangent vector Ċµ(s) that makes no
reference to space-time. We can therefore impose the loop space constraints
Ċµ(s)Aµs = 0 (88)
for each s, and also
φµs = Ċµ(s)φ(s;C) (89)
for some subtle field φ(s;C) on loop space. As a consequence, we find that
µs = 0. (90)
These constraints are covariant under diffeomorphisms of space-time and reparametriza-
tions of loops. They are invariant also under local gauge transformations, pro-
vided that the gauge parameter is subject to the condition
µ(s)∂µsΛ = 0 (91)
which is the condition of reparametrization invariance. With the assumption
made that λa(s) are generators of a loop algebra, we find that the constraint
can also be written as
[Aµs, φ
µt] = 0 (92)
A local gauge variation of this constraint is
[DµsΛ, φ
µt] + [Aµs, [φ
µt,Λ]] = [∂µsΛ, φ
µt] + [[Aµs,Λ], φ
µt] + [Aµs, [φ
µt,Λ]]
= [∂µsΛ, φ
µt] + [Λ, [φµt, Aµs]] (93)
The last term vanishes by the constraint. The first term gives us the constraint
Eq (92) that we must impose on the gauge parameter
dsΛa(s, C)λa(s). (94)
4We denote by ∂is the usual functional derivative with respect to C
µ(s).
We have now introduced non-local non-Abelian fields with infinitely many
components. It is also likely that consisteny of the theory requires an infinite
set of constraints on these fields. Maybe then, it could be that we may in the
end descend to a finite degrees of freedom. But this is just a speculation. The
problem appears to be difficult and ill-defined – How should one define a degree
of freedom in a strongly coupled non-local theory?
The non-Abelian generalization of the Bogomolnyi equation should be given
by [7]
Fis,jt = ±ǫijklDk(sφlt). (95)
This equation is gauge invariant and invariant under the residual SO(4) Lorentz
group that is preserved by the strings. We can not think of any reasonable
modification of this equation that would preserve these symmetries, so on this
grounds alone one could suspect this equation to be correct. Of course this is not
the only requirement that the BPS condition imposes. We also get conditions
on the 0s and the 5s components. But these BPS equations will be of no interest
to us right now.
We will show below that the linearized Bogomolnyi equation can be written
Di(s + σφi(s
χt) = 0 (96)
We will also see below that we (presumably) can actually drop the symmetriza-
tion in s and t in this equation. The fields transform in the adjoint represen-
tation of the loop algebra, by which we mean that φisχt = [φis, χt]. We define
the Dirac operator
Ds = γi (Dis + σφis) (97)
and the projectors
(1∓ γσ) , (98)
We can now formulate an index problem, in an even-dimensional (loop-)space.
The even-dimensional space in this case is given by the 4-dimensional transverse
space to the strings, and the index is given by
dimkerDs − dimkerD†s (99)
where
Ds = P+Ds = DsP−
D†s = P−Ds = DsP+. (100)
Since Ds and P± are hermitian, it is manifest that D†s defined this way will be
the hermitian conjugate of Ds, thus justifying the notation.
Computing the index alone is not sufficient in order to obtain the dimension
of the moduli space of self-dual strings. We also need a vanishing theorem that
says that dimkerD†s = 0.
Linearizing the Bogomolnyi equation, we get
2D[isδAjt] = ±ǫijkl (Dksδφlt + φksδAlt) (101)
Contracting by γij , we get
γijD̃isχjt = 0 (102)
where we have defined
D̃is ≡ Dis ∓ γφis
χis ≡ δAis ∓ γδφis (103)
To see that the linearized BPS equation can be written like this, one must use
the constraint
γijφisδφjt = 0. (104)
We can avoid having explicit ± signs by introducing the other chiraly matrix
at our disposal, namely σ that lives in a different vector space than γ. We can
then hide the ± signs in the tensor product
γ ⊗ σ = ±1 (105)
which amounts to
D̃is ≡ Dis + σφis
χis ≡ δAis + σδφis (106)
without any ±.5 If we define
χs ≡ γiχis (108)
then we can write the zero mode equation as
γiD̃isχt + D̃
sχit = 0. (109)
Let us analyze the second term in this equation. It is given by
DisδAit + φ
sδφit
φisδAit +D
sδφit
(110)
We should not count variations that are gauge variations as bosonic zero modes.
We can insure this by demanding the zero modes to be orthogonal to gauge
variations, with respect to the metric on the moduli space,
(δΛAis, δAit) + (δΛφis, δφjt) = 0 (111)
This leads to the background gauge condition
sδAit + φ
sδφit = 0. (112)
5To really understand what is going on, one should apply (1± γσ) on everything, on ψs
and on Ds. Then one notices that
∓γ (1∓ γσ) = σ (1∓ γσ) . (107)
That is, we can trade ∓γ for σ, once we apply (1± γσ) on everything. This is what we really
should do, but to keep the notation simple, we do not spell this out.
This condition implies that the gauge variation of the zero modes vanishes,
δΛδAis = 0 = δΛδφis (113)
To see this, we make a gauge variation δΛδAis = DisΛ, δΛφis = φisΛ, and
ask which gauge parameters Λ will respect the background gauge condition.
Inserting this gauge variation into the background gauge condition, we get
sDit + φ
Λ = 0. (114)
For this to work nicely, it seems that we must constrain the non-locality of our
loop field such that ∂i
∂it) < 0. Then the only solution to this equation is Λ = 0.
In other words all gauge variations of the zero modes have to vanish.
Furthermore we want the variation to preserve the orthogonality between
Ais and φis,
(Ais, δφit) + (δAis, φit) = 0 (115)
If we make a gauge variation of this, then we get the condition
(δΛAis, δφit) + (δAis, δΛφit) = 0 (116)
which amounts to
φisδAit +D
sδφit = 0. (117)
We conclude that the zero mode equation can be written as
Dsχt = 0 (118)
where
Ds = γi (Dis + σφis) (119)
We are interested in counting the number of such modes in a background of k
BPS strings. We compute
D2 = (Dis)
2 + (φis)
γij (Fis,js + γσǫijklDksφls) (120)
(Here D2 ≡ DsDs ≡
DsDs, and analogously for the other fields or opera-
tors.) In a BPS configuration, we get is
2 = (Dis)
2 + (φis)
ij (1 + γσ)Fis,js (121)
Furthermore, in the subspace where 1 + γσ = 0, we find that
D2 = (Dis)
2 + (φis)
2 (122)
is a strictly negative operator, hence has no zero modes. This means that we
have a vanishing theorem, dimkerD† = 0.
A small comment
The zero mode equation was really
D(sχt) = 0 (123)
where we should symmetrize in s and t. That means that we should rather
consider
DsD(sχt) =
(DsDsχt +DsDtχs)
(DsDsχt +DtDsχs + [Ds, Dt]χs) . (124)
If now D[sDt] = 0 and Dsχs = 0, then we get
DsDsχt = 0 (125)
The latter condition, Dsχs = 0 is of course a consequence of D(sχt) = 0 with
s = t. The former condition reads
0 = D[sDt]
= Di[sDit] + φi[sφit] + σDi[sφit] (126)
which we would like to impose as a constraint. Restricting to the abelian case
this is condition is of course true as 0 ≡ ∂i[s∂|i|t]. If we can impose this as a
constraint on the non-abelian fields, then we have now seen that the zero mode
equation Eq (123) implies that
dsD†sDsχt = 0 (127)
because Ds is anti-self-adjoint with respect to the inner product
(ψs, χt) =
ψ†s(C)χt(C)
(128)
on loop space. We can also go in the opposite direction. Assuming that Eq
(127) holds, we get
χt, D
sDsχt
= (Dsχt, Dsχt) (129)
and we conclude that (123) implies
Dsχt = 0 (130)
with no symmetrization in s, t.
How to compute the index
We should now be able to compute an index associated to self-dual strings, as
the limit
I(s) = Tr
(131)
when s→ ∞. We define the quantity
Jis(C,C
′) = tr
γσγiγk (Dks + σφks)
(132)
(it should be clear that the two s’s involved in this formula are totally unrelated)
and find that
I(s) =
DC∂isJ is(C,C) (133)
We can separate the functional integral over parametrized loops C into several
pieces. We can keep a point on the loops C(s) = x fixed, and separate it as
DxC (134)
Then we can write I(s) as an integral over a large three-sphere at spatial infinity,
∂Jis(C)
∂Ci(s)
dΩ3x̂
DxCJis(C,C) (135)
where thus x = C(s).
If we assume that the gauge group is maximally broken to a product of
U(1)’s by the Higgs vacuum expectation values, then we should have U(1) loop
fields at spatial infinity.
If we assume that the gauge group is SU(2) and that it is broken to U(1),
then we need only the asymptotic form of the U(1) fields at spatial infinity,
Fis,jt = Hijl(x)Ċ
l(s)δ(s− t)
φks = vĊk(s) (136)
Without doing any computations, we can guess what the outcome of the
index calculation should be. A term like
ǫijkl
DxCtr (Fis,jt(C)Fks,jt(C)) (137)
could certainly arise somewhere (in odd dimensions a corresponding term van-
ished since there is no chiral anomaly in odd dimensions). In our case this term
vanishes identically by the Bogomolnyi equation and the constraint6
Fis,jtDisφjt = 0. (139)
Then there can be a term
ǫijkl
DxCtr
Fis,jtφks
(140)
6For U(1) fields this would read
Fis,jt∂isφjt ∼ Hijk(C(s))∂iφ(C(s))Ċ
k(s)Ċj(s)δ(s − t)2 ≡ 0. (138)
that should arise in a very similar way as the corresponding term arose for
monopoles. If we insert the asymptotic U(1) fields, this term becomes propor-
tional to
ǫijklHijk(x) (141)
That means that the index should be given by some numerical constant, times
the magnetic charge
H. (142)
A Integrals over the exponential
The integral we will analyze here is
a(s) =
k2 + 1
−s(k2+1)eiǫk (143)
for any complex number ζ. (The ǫ > 0, say, will be taken towards zero. It
arose from ǫ = x− y and we keep it here just as a convergence factor.) We first
compute
a(0) =
k2 + 1
eiǫk (144)
In order to make this integral converge for any ζ, we should Wick rotate k to
ik, and henceforth we will always mean by i the branch eiπ/2, and by −1 we
mean eiπ . Then we get
a(0) = −i2ζ+1
k2 − 1
e−ǫk (145)
and this integral we evaluate as a principal value. That means to evaluate the
residues along the real axis and multiply them not by 2πi, but by half of it, that
is, by πi. We get
a(0) = (−1)ζπ 1− (−1)
. (146)
Next we turn to our integral a(s). It is easier to first compute the derivative.
We should still work with the Wick rotated integral. Making the substitution
ξ = k2 we can put it on the form of two gamma functions. The result is that
′(−s) = −eiπ(ζ+
1 + (−1)2ζ
−ζ− 1
s (147)
which we can trivially continue analytically to +s, and then integrate up. The
result is
a(+∞) = −π 1 + (−1)
cos(πζ)
+ π(−1)ζ 1− (−1)
. (148)
References
[1] P. S. Howe, N. D. Lambert and P. C. West, “The self-dual string soliton,”
Nucl. Phys. B 515, 203 (1998) [arXiv:hep-th/9709014].
[2] D. S. Berman and J. A. Harvey, “The self-dual string and anomalies in
the M5-brane,” JHEP 0411, 015 (2004) [arXiv:hep-th/0408198].
[3] C. Callias, “Index Theorems On Open Spaces,” Commun. Math. Phys.
62, 213 (1978).
[4] E. J. Weinberg, “Parameter Counting For Multi - Monopole Solutions,”
Phys. Rev. D 20, 936 (1979).
[5] E. J. Weinberg and P. Yi, “Magnetic monopole dynamics, supersymmetry,
and duality,” Phys. Rept. 43, 65 (2007) [arXiv:hep-th/0609055].
[6] M. Hirayama, “Supersymmetric Quantum Mechanics And Index Theo-
rem,” Prog. Theor. Phys. 70, 1444 (1983).
[7] A. Gustavsson, “A reparametrization invariant surface ordering,” JHEP
0511, 035 (2005) [arXiv:hep-th/0508243].
A. Gustavsson, “The non-Abelian tensor multiplet in loop space,” JHEP
0601, 165 (2006) [arXiv:hep-th/0512341].
|
0704.0019 | Approximation for extinction probability of the contact process based on
the Gr\"obner basis | Approximation for extinction probability of
the contact process based on the Gröbner basis
Norio Konno
Department of Applied Mathematics
Yokohama National University
Abstract. In this note we give a new method for getting a series of approxi-
mations for the extinction probability of the one-dimensional contact process by
using the Gröbner basis.
1 Introduction
Let X = {0, 1}Zd denote a configuration space, where Zd is the d-dimensional
integer lattices. The contact process {ηt : t ≥ 0} is an X-valued continuous-
time Markov process. The model was introduced by Harris in 1974 [1] and
is considered as a simple model for the spread of a disease with the infection
rate λ. In this setting, an individual at x ∈ Zd for a configuration η ∈ X is
infected if η(x) = 1 and healthy if η(x) = 0. The formal generator is given
Ωf(η) =
c(x, η)[f(ηx)− f(η)],
where ηx ∈ X is defined by ηx(y) = η(y) (y 6= x), and ηx(x) = 1−η(x). Here
for each x ∈ Zd and η ∈ X, the transition rate is
c(x, η) = (1− η(x))× λ
y:|y−x|=1
η(y) + η(x),
http://arxiv.org/abs/0704.0019v2
with |x| = |x1|+ · · ·+ |xd|. In particular, the one-dimensional contact process
001 → 011 at rate λ,
100 → 110 at rate λ,
101 → 111 at rate 2λ,
1 → 0 at rate 1.
Let Y = {A ⊂ Zd : |A| < ∞}, where |A| is the number of elements in A.
Let ξAt (⊂ Zd) denote the state at time t of the contact process with ξA0 = A.
There is a one-to-one correspondence between ξAt (⊂ Zd) and ηt ∈ X such
that x ∈ ξAt if and only if ηt(x) = 1. For any A ∈ Y , we define the extinction
probability of A by limt→∞ P (ξ
t = ∅). Define νλ(A) = νλ{η : η(x) = 0
for any x ∈ A}, where νλ is an invariant measure of the process starting
from a configuration: η(x) = 1 (x ∈ Zd) and is called the upper invariant
measure. In other words, let δ1S(t) denote the probability measure at time t
for initial probability measure δi which is the pointmass η ≡ i(i = 0, 1). Then
νλ = limt→∞ δ1S(t). Then self-duality of the process implies that νλ(A) =
limt→∞ P (ξ
t = ∅). The correlation identities for νλ(A) can be obtained as
follows:
Theorem 1.1 For any A ∈ Y ,
y:|y−x|=1
νλ(A ∪ {y})− νλ(A)
νλ(A \ {x})− νλ(A)
From now on we consider the one-dimensional case. We introduce the fol-
lowing notation:
νλ(◦) = νλ({0}), νλ(◦◦) = νλ({0, 1}), νλ(◦ × ◦) = νλ({0, 2}), . . . .
By Theorem 1.1, we obtain
Corollary 1.2
2λνλ(◦◦)− (2λ+ 1)νλ(◦) + 1 = 0,(1)
λνλ(◦ ◦ ◦)− (λ+ 1)νλ(◦◦) + νλ(◦) = 0,(2)
2λνλ(◦ ◦ ◦◦) + νλ(◦ × ◦)− (2λ+ 3)νλ(◦ ◦ ◦) + 2νλ(◦◦) = 0,(3)
λνλ(◦ ◦ ×◦)− (2λ+ 1)νλ(◦ × ◦) + λνλ(◦ ◦ ◦) + νλ(◦) = 0.(4)
The detailed discussion concerning results in this section can be seen in
Konno [2, 3]. If we regard λ, νλ(◦), νλ(◦◦), νλ(◦ ◦ ◦), . . . as variables, then
the left hand sides of the correlation identities by Theorem 1.1 are polyno-
mials of degree at most two. In the next section, we give a new procedure
for getting a series of approximations for extinction probabilities based on
the Gröbner basis by using Corollary 1.2. As for the Gröbner basis, see [4],
for example.
2 Our results
Put x = νλ(◦), y = νλ(◦◦), z = νλ(◦ ◦ ◦), w = νλ(◦ × ◦), s = νλ(◦ ◦
◦◦), u = νλ(◦ ◦ ×◦). Let ≺ denote the lexicographic order with λ ≺ x ≺
y ≺ w ≺ z ≺ u ≺ s. For m = 1, 2, 3, let Im be the ideals of a polynomial
ring R[x1, x2, . . . , xn(m)] over R as defined below. Here x1 = λ, x2 = x, x3 =
y, x4 = z, x5 = w, x6 = s, x7 = u and n(1) = 3, n(2) = 4, n(3) = 7.
2.1 First approximation
We consider the following ideal based on Corollary 1.2 (1):
I1 = 〈 2λy − 2λx− x+ 1, y − x2 〉 ⊂ R[λ, x, y].(5)
Here y−x2 corresponds to the first (or mean-field) approximation: ν(1)
(◦◦) =
λ (◦))2. Then
G1 = {(x− 1)(2λx− 1), y − x2}(6)
is the reduced Gröbner basis for I1 with respect to ≺. Therefore the solution
except a trivial one x(= y) = 1 is x = ν
(◦) = 1/(2λ). Remark that the
trivial solution means that the invariant measure is δ0. From this, we obtain
the first approximation of the density of the particle, ρλ = Eνλ(η(x)), as
follows:
= 1− ν(1)
(◦) = 2λ− 1
for any λ ≥ 1/2. This result gives the first lower bound λ(1)c of the critical
value λc of the one-dimensional contact process, that is, λ
c = 1/2 ≤ λc.
However it should be noted that the inequality is not proved in our approach.
The estimated value of λc is about 1.649.
2.2 Second approximation
Consider the following ideal based on Corollary 1.2 (1) and (2):
I2 = 〈 2λy − 2λx− x+ 1, λz − λy − y + x, xz − y2 〉 ⊂ R[λ, x, y, z].
Here xz−y2 corresponds to the second (or pair) approximation: ν(2)
(◦)ν(2)
◦) = (ν(2)λ (◦◦))2. Then
G2 = {(x− 1)((2λ− 1)x− 1), 1 + 2λ(y − x)− x,
−y − yx+ 2x2,−z − y(2 + y) + 4x2}
is the reduced Gröbner basis for I2 with respect to ≺. Therefore the solution
except a trivial one x(= y = z) = 1 is x = ν
(◦) = 1/(2λ − 1). As in a
similar way of the first approxaimation, we get the second approximation of
the density of the particle:
2(λ− 1)
2λ− 1 ,
for any λ ≥ 1. This result implies the second lower bound λ(2)c = 1. We
should remark that if we take
I ′2 = 〈 2λy − 2λx− x+ 1, λz − λy − y + x, y − x2, z − x3 〉 ⊂ R[λ, x, y, z],
then we have
G′2 = {z − 1, y − 1, x− 1}
is the reduced Gröbner basis for I ′2 with respect to ≺. Here y−x2 and z−x3
correspond to an approximation: ν
(◦◦) = (ν(2
(◦))2 and ν(2
(◦ ◦ ◦) =
(◦))3, respectively. Then we have only trivial solution: x = y = z = 1.
2.3 Third approximation
Consider the following ideal based on Corollary 1.2 (1)–(4):
I3 = 〈 2λy − 2λx− x+ 1, λz − λy − y + x,
2λs+ w − (2λ+ 3)z + 2y, λu− (2λ+ 1)w + λz + x,
ys− z2, xu− yw 〉 ⊂ R[λ, x, y, z, w, s, u].
Here ys−z2 and xu−yw correspond to the third approximation: ν(3)
(◦◦)ν(3)
◦◦) = (ν(3)
(◦ ◦ ◦))2 and ν(3)
(◦)ν(3)
(◦ ◦×◦) = ν(3)
(◦◦)ν(3)
(◦× ◦), respectively.
G3 = {(x− 1)((12λ3 − 5λ− 1)x2 − 2λ(2λ+ 3)x− λ+ 1), . . .}
is the reduced Gröbner basis for I3 with respect to ≺. Therefore the solution
except a trivial one x = 1 is x = ν
λ (◦) = (λ(2λ+3)+
D)/(12λ3−5λ−1),
where D = 16λ4 + 4λ2 + 4λ+ 1. Then we obtain the third approximation of
the density of the particle:
4λ(3λ2 − λ− 3)
12λ3 − 2λ2 − 8λ− 1 +
for any λ ≥ (1 +
37)/6. This result corresponds to the third lower bound
c = (1 +
37)/6 ≈ 1.180.
3 Summary
We obtain the first, second, and third approximations for the extinction
probability, the density of the particle, and the lower bound of the one-
dimensional contact process by using the Gröbner basis with respect to a
suitable term order. These results coincide with results given by the Harris
lemma (more precisely, the Katori-Konno method, see [3]) or the BFKL
inequality [5] (see also [3]). As we saw, the generators of Im in Section 2
have degree at most two in x1, x2, . . ., such as 2λy − 2λx− x+ 1, ys− z2 in
the case of I3. We expect that this property will lead to get the higher order
approximations of the process (and other interacting particle systems having
a similar property) effectively.
Acknowledgment. The author thanks Takeshi Kajiwara for valuable dis-
cussions and comments.
References
[1] T. E. Harris, Contact interactions on a lattice, Ann. Probab. 2: 969–988
(1974).
[2] N. Konno, Phase Transitions on Interacting Particle Systems, World
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[3] N. Konno, Lecture Notes on Interacting Particle Systems,
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http://www.math.kobe-u.ac.jp/publications/rlm03.pdf.
[4] D. A. Cox, J. B. Little, and D. O’Shea, Ideals, Varieties, And Al-
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http://www.math.kobe-u.ac.jp/publications/rlm03.pdf
Introduction
Our results
First approximation
Second approximation
Third approximation
Summary
|
0704.0020 | Measurement of the Hadronic Form Factor in D0 --> K- e+ nue Decays | BABAR-PUB-07/015
SLAC-PUB-12417
Measurement of the Hadronic Form Factor in D0 → K−e+νe Decays.
B. Aubert, M. Bona, D. Boutigny, Y. Karyotakis, J. P. Lees, V. Poireau, X. Prudent, V. Tisserand, and A. Zghiche
Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France
J. Garra Tico and E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
L. Lopez and A. Palano
Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy
G. Eigen, B. Stugu, and L. Sun
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
G. S. Abrams, M. Battaglia, D. N. Brown, J. Button-Shafer, R. N. Cahn, Y. Groysman,
R. G. Jacobsen, J. A. Kadyk, L. T. Kerth, Yu. G. Kolomensky, G. Kukartsev, D. Lopes Pegna,
G. Lynch, L. M. Mir, T. J. Orimoto, M. T. Ronan,∗ K. Tackmann, and W. A. Wenzel
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
P. del Amo Sanchez, C. M. Hawkes, and A. T. Watson
University of Birmingham, Birmingham, B15 2TT, United Kingdom
T. Held, H. Koch, B. Lewandowski, M. Pelizaeus, T. Schroeder, and M. Steinke
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
D. Walker
University of Bristol, Bristol BS8 1TL, United Kingdom
D. J. Asgeirsson, T. Cuhadar-Donszelmann, B. G. Fulsom,
C. Hearty, N. S. Knecht, T. S. Mattison, and J. A. McKenna
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan, M. Saleem, and L. Teodorescu
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov, A. D. Bukin, V. P. Druzhinin, V. B. Golubev, A. P. Onuchin,
S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, and K. Yu Todyshev
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
M. Bondioli, S. Curry, I. Eschrich, D. Kirkby, A. J. Lankford, P. Lund, M. Mandelkern, E. C. Martin, and D. P. Stoker
University of California at Irvine, Irvine, California 92697, USA
S. Abachi and C. Buchanan
University of California at Los Angeles, Los Angeles, California 90024, USA
S. D. Foulkes, J. W. Gary, F. Liu, O. Long, B. C. Shen, and L. Zhang
University of California at Riverside, Riverside, California 92521, USA
H. P. Paar, S. Rahatlou, and V. Sharma
University of California at San Diego, La Jolla, California 92093, USA
J. W. Berryhill, C. Campagnari, A. Cunha, B. Dahmes, T. M. Hong, D. Kovalskyi, and J. D. Richman
University of California at Santa Barbara, Santa Barbara, California 93106, USA
http://arxiv.org/abs/0704.0020v1
T. W. Beck, A. M. Eisner, C. J. Flacco, C. A. Heusch, J. Kroseberg, W. S. Lockman,
T. Schalk, B. A. Schumm, A. Seiden, D. C. Williams, M. G. Wilson, and L. O. Winstrom
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
E. Chen, C. H. Cheng, F. Fang, D. G. Hitlin, I. Narsky, T. Piatenko, and F. C. Porter
California Institute of Technology, Pasadena, California 91125, USA
G. Mancinelli, B. T. Meadows, K. Mishra, and M. D. Sokoloff
University of Cincinnati, Cincinnati, Ohio 45221, USA
F. Blanc, P. C. Bloom, S. Chen, W. T. Ford, J. F. Hirschauer, A. Kreisel, M. Nagel,
U. Nauenberg, A. Olivas, J. G. Smith, K. A. Ulmer, S. R. Wagner, and J. Zhang
University of Colorado, Boulder, Colorado 80309, USA
A. M. Gabareen, A. Soffer, W. H. Toki, R. J. Wilson, F. Winklmeier, and Q. Zeng
Colorado State University, Fort Collins, Colorado 80523, USA
D. D. Altenburg, E. Feltresi, A. Hauke, H. Jasper, J. Merkel, A. Petzold, B. Spaan, and K. Wacker
Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany
T. Brandt, V. Klose, M. J. Kobel, H. M. Lacker, W. F. Mader, R. Nogowski,
J. Schubert, K. R. Schubert, R. Schwierz, J. E. Sundermann, and A. Volk
Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany
D. Bernard, G. R. Bonneaud, E. Latour, V. Lombardo, Ch. Thiebaux, and M. Verderi
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
P. J. Clark, W. Gradl, F. Muheim, S. Playfer, A. I. Robertson, and Y. Xie
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
M. Andreotti, D. Bettoni, C. Bozzi, R. Calabrese, A. Cecchi, G. Cibinetto, P. Franchini,
E. Luppi, M. Negrini, A. Petrella, L. Piemontese, E. Prencipe, and V. Santoro
Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy
F. Anulli, R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro,
S. Pacetti, P. Patteri, I. M. Peruzzi,† M. Piccolo, M. Rama, and A. Zallo
Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy
A. Buzzo, R. Contri, M. Lo Vetere, M. M. Macri, M. R. Monge,
S. Passaggio, C. Patrignani, E. Robutti, A. Santroni, and S. Tosi
Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy
K. S. Chaisanguanthum, M. Morii, and J. Wu
Harvard University, Cambridge, Massachusetts 02138, USA
R. S. Dubitzky, J. Marks, S. Schenk, and U. Uwer
Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
D. J. Bard, P. D. Dauncey, R. L. Flack, J. A. Nash, M. B. Nikolich, and W. Panduro Vazquez
Imperial College London, London, SW7 2AZ, United Kingdom
P. K. Behera, X. Chai, M. J. Charles, U. Mallik, N. T. Meyer, and V. Ziegler
University of Iowa, Iowa City, Iowa 52242, USA
J. Cochran, H. B. Crawley, L. Dong, V. Eyges, W. T. Meyer, S. Prell, E. I. Rosenberg, and A. E. Rubin
Iowa State University, Ames, Iowa 50011-3160, USA
A. V. Gritsan, Z. J. Guo, and C. K. Lae
Johns Hopkins University, Baltimore, Maryland 21218, USA
A. G. Denig, M. Fritsch, and G. Schott
Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany
N. Arnaud, J. Béquilleux, M. Davier, G. Grosdidier, A. Höcker, V. Lepeltier, F. Le Diberder, A. M. Lutz, S. Pruvot,
S. Rodier, P. Roudeau, M. H. Schune, J. Serrano, V. Sordini, A. Stocchi, W. F. Wang, and G. Wormser
Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,
Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
C. A. Chavez, I. J. Forster, J. R. Fry, E. Gabathuler, R. Gamet,
D. E. Hutchcroft, D. J. Payne, K. C. Schofield, and C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan, K. A. George, F. Di Lodovico, W. Menges, and R. Sacco
Queen Mary, University of London, E1 4NS, United Kingdom
G. Cowan, H. U. Flaecher, D. A. Hopkins, P. S. Jackson, T. R. McMahon, F. Salvatore, and A. C. Wren
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
D. N. Brown and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
J. Allison, N. R. Barlow, R. J. Barlow, Y. M. Chia, C. L. Edgar, G. D. Lafferty, T. J. West, and J. I. Yi
University of Manchester, Manchester M13 9PL, United Kingdom
J. Anderson, C. Chen, A. Jawahery, D. A. Roberts, G. Simi, and J. M. Tuggle
University of Maryland, College Park, Maryland 20742, USA
G. Blaylock, C. Dallapiccola, S. S. Hertzbach, X. Li, T. B. Moore, E. Salvati, and S. Saremi
University of Massachusetts, Amherst, Massachusetts 01003, USA
R. Cowan, P. H. Fisher, G. Sciolla, S. J. Sekula, M. Spitznagel, F. Taylor, and R. K. Yamamoto
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
S. E. Mclachlin, P. M. Patel, and S. H. Robertson
McGill University, Montréal, Québec, Canada H3A 2T8
A. Lazzaro and F. Palombo
Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy
J. M. Bauer, L. Cremaldi, V. Eschenburg, R. Godang, R. Kroeger, D. A. Sanders, D. J. Summers, and H. W. Zhao
University of Mississippi, University, Mississippi 38677, USA
S. Brunet, D. Côté, M. Simard, P. Taras, and F. B. Viaud
Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7
H. Nicholson
Mount Holyoke College, South Hadley, Massachusetts 01075, USA
G. De Nardo, F. Fabozzi,‡ L. Lista, D. Monorchio, and C. Sciacca
Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy
M. A. Baak, G. Raven, and H. L. Snoek
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop and J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
G. Benelli, L. A. Corwin, K. K. Gan, K. Honscheid, D. Hufnagel, H. Kagan, R. Kass,
J. P. Morris, A. M. Rahimi, J. J. Regensburger, R. Ter-Antonyan, and Q. K. Wong
Ohio State University, Columbus, Ohio 43210, USA
N. L. Blount, J. Brau, R. Frey, O. Igonkina, J. A. Kolb, M. Lu,
R. Rahmat, N. B. Sinev, D. Strom, J. Strube, and E. Torrence
University of Oregon, Eugene, Oregon 97403, USA
N. Gagliardi, A. Gaz, M. Margoni, M. Morandin, A. Pompili,
M. Posocco, M. Rotondo, F. Simonetto, R. Stroili, and C. Voci
Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy
E. Ben-Haim, H. Briand, G. Calderini, J. Chauveau, P. David, L. Del Buono,
Ch. de la Vaissière, O. Hamon, Ph. Leruste, J. Malclès, J. Ocariz, and A. Perez
Laboratoire de Physique Nucléaire et de Hautes Energies,
IN2P3/CNRS, Université Pierre et Marie Curie-Paris6,
Université Denis Diderot-Paris7, F-75252 Paris, France
L. Gladney
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
M. Biasini, R. Covarelli, and E. Manoni
Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy
C. Angelini, G. Batignani, S. Bettarini, M. Carpinelli, R. Cenci, A. Cervelli, F. Forti, M. A. Giorgi,
A. Lusiani, G. Marchiori, M. A. Mazur, M. Morganti, N. Neri, E. Paoloni, G. Rizzo, and J. J. Walsh
Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy
M. Haire
Prairie View A&M University, Prairie View, Texas 77446, USA
J. Biesiada, P. Elmer, Y. P. Lau, C. Lu, J. Olsen, A. J. S. Smith, and A. V. Telnov
Princeton University, Princeton, New Jersey 08544, USA
E. Baracchini, F. Bellini, G. Cavoto, A. D’Orazio, D. del Re, E. Di Marco, R. Faccini, F. Ferrarotto, F. Ferroni,
M. Gaspero, P. D. Jackson, L. Li Gioi, M. A. Mazzoni, S. Morganti, G. Piredda, F. Polci, F. Renga, and C. Voena
Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy
M. Ebert, H. Schröder, and R. Waldi
Universität Rostock, D-18051 Rostock, Germany
T. Adye, G. Castelli, B. Franek, E. O. Olaiya, S. Ricciardi, W. Roethel, and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
R. Aleksan, S. Emery, M. Escalier, A. Gaidot, S. F. Ganzhur, G. Hamel de Monchenault,
W. Kozanecki, M. Legendre, G. Vasseur, Ch. Yèche, and M. Zito
DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France
X. R. Chen, H. Liu, W. Park, M. V. Purohit, and J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
M. T. Allen, D. Aston, R. Bartoldus, P. Bechtle, N. Berger, R. Claus, J. P. Coleman, M. R. Convery,
J. C. Dingfelder, J. Dorfan, G. P. Dubois-Felsmann, D. Dujmic, W. Dunwoodie, R. C. Field, T. Glanzman,
S. J. Gowdy, M. T. Graham, P. Grenier, C. Hast, T. Hryn’ova, W. R. Innes, J. Kaminski, M. H. Kelsey,
H. Kim, P. Kim, M. L. Kocian, D. W. G. S. Leith, S. Li, S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane,
H. Marsiske, R. Messner, D. R. Muller, C. P. O’Grady, I. Ofte, A. Perazzo, M. Perl, T. Pulliam,
B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, J. Schwiening, A. Snyder, J. Stelzer, D. Su,
M. K. Sullivan, K. Suzuki, S. K. Swain, J. M. Thompson, J. Va’vra, N. van Bakel, A. P. Wagner,
M. Weaver, W. J. Wisniewski, M. Wittgen, D. H. Wright, A. K. Yarritu, K. Yi, and C. C. Young
Stanford Linear Accelerator Center, Stanford, California 94309, USA
P. R. Burchat, A. J. Edwards, S. A. Majewski, B. A. Petersen, and L. Wilden
Stanford University, Stanford, California 94305-4060, USA
S. Ahmed, M. S. Alam, R. Bula, J. A. Ernst, V. Jain, B. Pan, M. A. Saeed, F. R. Wappler, and S. B. Zain
State University of New York, Albany, New York 12222, USA
W. Bugg, M. Krishnamurthy, and S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
R. Eckmann, J. L. Ritchie, A. M. Ruland, C. J. Schilling, and R. F. Schwitters
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen, X. C. Lou, and S. Ye
University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchi, F. Gallo, D. Gamba, and M. Pelliccioni
Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy
M. Bomben, L. Bosisio, C. Cartaro, F. Cossutti, G. Della Ricca, L. Lanceri, and L. Vitale
Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy
V. Azzolini, N. Lopez-March, F. Martinez-Vidal, D. A. Milanes, and A. Oyanguren
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
J. Albert, Sw. Banerjee, B. Bhuyan, K. Hamano, R. Kowalewski, I. M. Nugent, J. M. Roney, and R. J. Sobie
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
J. J. Back, P. F. Harrison, T. E. Latham, G. B. Mohanty, and M. Pappagallo§
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band, X. Chen, S. Dasu, K. T. Flood, J. J. Hollar,
P. E. Kutter, Y. Pan, M. Pierini, R. Prepost, S. L. Wu, and Z. Yu
University of Wisconsin, Madison, Wisconsin 53706, USA
H. Neal
Yale University, New Haven, Connecticut 06511, USA
(Dated: October 25, 2018)
The shape of the hadronic form factor f+(q
2) in the decay D0 → K−e+νe has been measured
in a model independent analysis and compared with theoretical calculations. We use 75 fb−1 of
data recorded by the BABAR detector at the PEPII electron-positron collider. The corresponding
decay branching fraction, relative to the decay D0 → K−π+, has also been measured to be RD =
BR(D0 → K−e+νe)/BR(D
→ K−π+) = 0.927± 0.007± 0.012. From these results, and using the
present world average value for BR(D0 → K−π+), the normalization of the form factor at q2 = 0
is determined to be f+(0) = 0.727 ± 0.007 ± 0.005 ± 0.007 where the uncertainties are statistical,
systematic, and from external inputs, respectively.
PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
∗Deceased †Also with Università di Perugia, Dipartimento di Fisica, Perugia,
I. INTRODUCTION
Measurements of exclusive semileptonic D decays pro-
vide an accurate determination of the hadronic form fac-
tors entering in these decays. Assuming that the CKM
matrix is unitary, the elements |Vcs| and |Vcd| can be de-
termined:
|Vcs| = |Vud| −
|Vcb|2
+O(λ6) = 0.9729± 0.0003, (1)
using the measured values [1] of |Vud| and |Vcb|, and the
sine of the Cabibbo angle λ = sin(θc) ≃ 0.227. Theo-
retical predictions give estimates of the form factors in
exclusive semileptonic B and D meson decays. Precise
measurements of the hadronic form factors in D decays
can help to validate predictions from QCD calculations in
both D and B decays. Better understanding of the form
factors in B decays is necessary to improve the precision
on the determination of |Vcb| and |Vub|.
In D0 → K−e+νe decays [2], with a pseudoscalar
hadron emitted in the final state, and neglecting the elec-
tron mass, the differential decay rate depends only on one
form factor f+(q
|Vcs|2
∣~pK(q
∣f+(q
, (2)
where GF is the Fermi constant, q
2 is the invariant mass
squared of the two leptons, e+ and νe, and ~pK(q
2) is
the kaon three-momentum in the D0 rest frame [3]. In
this paper we present measurements of the q2 variation
and absolute value of the hadronic form factor at q2 = 0
for the decay D0 → K−e+νe(γ). The data consist of
D mesons produced in e+e− → cc̄ continuum events at
a center of mass energy near the Υ (4S) mass, and were
recorded by the BABAR detector at the Stanford Linear
Accelerator Center’s PEP-II collider. A semi-inclusive
reconstruction technique is used to select charm semilep-
tonic decays with high efficiency. As a result of this ap-
proach, events with a photon radiated during the D0 de-
cay are included in the signal. The systematic uncertain-
ties are kept as low as possible by using control samples
extracted from data where possible.
Measurements of D → Kℓ̄νℓ, based on smaller signal
events samples, have been published by the CLEO [4],
FOCUS [5] and Belle [6] collaborations.
This paper is organized as follows. A general descrip-
tion of the hadronic form factor, f+(q
2), is given in Sec-
tion II, where the different parameterizations considered
Italy
‡Also with Università della Basilicata, Potenza, Italy
§Also with IPPP, Physics Department, Durham University,
Durham DH1 3LE, United Kingdom
in this analysis are explained. In Section III a short de-
scription of the detector components that are important
to this measurement is given. The selection of signal
events and the rejection of background are considered in
Section IV. In Section V, the measured q2 variation of
the hadronic form factor is discussed and compared with
previous measurements. In Section VI the measured de-
cay rate is given and in Section VII these measurements
are combined to obtain the value of f+(0).
II. THE F+(Q
2) HADRONIC FORM FACTOR
The amplitude for the decay D0 → K−ℓ+νℓ depends
on two hadronic form factors:
< K(p′)|Vµ|D(p) > =
pµ + p
µ − qµ
m2D −m2K
m2D −m2K
qµf0(q
2) (3)
where Vµ = s̄γµc. The constraint f+(0) = f0(0) en-
sures that there is no singularity at q2 = 0. When the
charged lepton is an electron, the contribution from f0
is proportional to m2e and can be neglected in decay rate
measurements.
The parameterizations of f+(q
2) which have been com-
pared with present measurements and a few examples of
theoretical approaches, proposed to determine the val-
ues of corresponding parameters, are considered in the
following.
A. Form factor parameterizations
The most general expressions of the form factor f+(q
are analytic functions satisfying the dispersion relation:
Res(f+)q2=m2
ℑf+(t)
t− q2 − iǫ
. (4)
The only singularities in the complex t ≡ q2 plane orig-
inate from the interaction of the charm and the strange
quarks in vector states. They are a pole, situated at the
D∗s mass squared and a cut, along the positive real axis,
starting at threshold (t+ = (mD+mK)
2) for D0K− pro-
duction.
1. Taylor expansion
This cut t-plane can be mapped onto the open unit
disk with center at t = t0 using the variable:
z(t, t0) =
t+ − t−
t+ − t0√
t+ − t+
t+ − t0
. (5)
In this variable, the physical region for the semileptonic
decay (0 < t < t− = q
max = (mD−mK)2) corresponds to
a real segment extending between ±zmax = ±0.051. This
value of zmax is obtained for t0 = t+
1− t−/t+
The z expansion of f+ is thus expected to converge
quickly. The most general parameterization [7], consis-
tent with constraints from QCD,
f+(t) =
P (t)Φ(t, t0)
ak(t0) z
k(t, t0), (6)
is based on earlier considerations [8]. The function
P (t) = z(t,m2D∗
) has a zero at the D∗s pole mass and
|P | = 1 along the unit circle; Φ is given by:
Φ(t, t0) =
24πχV
t+ − t
t+ − t0
t+ − t+
t+ − t+
t+ − t0
t+ − t+
t+ − t−
(t+ − t)
where χV can be obtained from dispersion relations using
perturbative QCD and depends on u = ms/mc [9]. At
leading order, with u = 0 [10],
32π2m2c
. (8)
The choice of P and Φ is such that:
a2k(t0) ≤ 1. (9)
Having measured the first coefficients of this expansion,
Eq. (9) can constrain the others. This constraint, which
depends on χV , may have to be abandoned in the case of
charm decays as the charm-quark mass may not be large
enough to prevent the previous evaluation of χV from
receiving large 1/mc and QCD corrections. However the
parameterization given in Eq. (6) remains valid and it
has been compared [7] with available measurements. The
first two terms in the expansion were sufficient to describe
the data.
2. Model-dependent parameterizations
A less general approach assumes that the q2 variation
of f+(q
2) is governed mainly by the D∗s pole and that
the other contributions can be accounted for by adding
another effective pole at a higher mass [11]:
f+(0)
1− αpole
1− q2
αpole
1− q2
γpolem
= f+(0)
1− δpole q
1− q2
1− βpole q
) (10)
with δpole = (1/γpole − αpole)/(1 − αpole) and βpole =
1/γpole.
If in addition, the form factors f+ and f0 must obey a
relation, valid at large recoil and in the heavy quark limit,
then αpole = 1/γpole [11] (βpole = αpole and δpole = 0 in
this case). Equation (10) becomes:
f+(0)
1− q2
1− αpole q
) , (11)
known as the modified pole ansatz. Initially an even
simpler expression, the simple pole ansatz, was proposed
which considered only the contribution from the D∗s pole.
In the following, the pole mass entering in
f+(0)
1− q2
is fitted. Note that such an effective pole mass value has
no clear physical interpretation and that the proposed q2
variation does not comply with constraints from QCD.
The obtained value may nonetheless be useful for com-
parison with results from different experiments.
B. Quantitative expectations
Values of the parameters that determine f+(q
2) were
obtained initially from constituent quark models and
from QCD sum rules. These two approaches have an in-
trinsically limited accuracy. In this respect, results from
lattice QCD computations are more promising because
their accuracy is mainly limited by available computing
resources.
1. Quark Models
Quark model calculations estimate meson wave func-
tions and use them to compute the matrix elements that
appear in the hadronic current. There are a large variety
of theoretical calculations [12]. Among these models we
have selected the ISGW model [13], simply because it is
widely used to simulate heavy hadron semileptonic de-
cays. This model was expected to be valid in the vicinity
of q2max, a region of maximum overlap between the ini-
tial and final meson wave functions. In ISGW2 [14] the
exponential q2 dependence of the form factor has been
replaced by another parameterization, with a dipole be-
havior, expected to be valid over a larger q2 range:
f ISGW2+ (q
(1 + αI(q2max − q2))
, αI =
r2. (13)
The predicted values of the parameters are f+(q
max) =
1.23 and r = 1.12 GeV−1 [14].
TABLE I: Parameterizations of f+(q
modeling parameters expected values
z expansion[8] a0, rk = ak/a0 no prediction
general 2-poles [11] f+(0), βpole, δpole no prediction
modified pole [11] f+(0), αpole δpole = 0
simple pole f+(0), mpole mpole = mD∗
ISGW2[14] f+(t−), αI f+(t−) = 1.23
αI = 0.104 GeV
2. QCD sum rules
QCD sum rules [15] and their extension on the light
cone [16], are expected to be valid at low q2. Using a
value of 150 MeV for the strange quark mass, one obtains
[16]:
f+(0) = 0.78± 0.11 and αpole = −0.07+0.15−0.07, (14)
using the modified pole ansatz. The uncertainty of f+(0)
is estimated to be of order 15%, and the q2 dependence
is expected to be dominated by a single pole at the D∗s
mass because the value of αpole is compatible with zero.
3. Lattice QCD
Lattice QCD computation is the only approach able
to compute f+(q
2) from first principles. Current results
must be extrapolated to physical values of light quark
masses and corrected for finite lattice size and discriti-
zation effects. There have been several evaluations of
2) for different values of the momentum transfer in
the quenched approximation [17, 18]. These results have
been combined [17], giving f+(0) = 0.73 ± 0.07. The
first unquenched calculation has been published recently
[19]: f+(0) = 0.73± 0.03± 0.07 and αpole = 0.50± 0.04,
using the modified pole ansatz to parameterized the q2
dependence of the form factor.
C. Analyzed parameterizations
The different parameterizations of f+(q
2) considered in
this analysis are summarized in Table I, along with their
corresponding parameters and expected values, where
available.
III. THE BABAR DETECTOR AND DATASET
A detailed description of the BABAR detector and of the
algorithms used for charged and neutral particle recon-
struction and identification is provided elsewhere [20, 21].
Charged particles are reconstructed by matching hits
in the 5-layer double-sided silicon vertex tracker (SVT)
with track elements in the 40-layer drift chamber (DCH),
which is filled with a gas mixture of helium and isobu-
tane. Slow particles which do not leave enough hits in the
DCH due to the bending in the 1.5-T magnetic field, are
reconstructed in the SVT. Charged hadron identification
is performed combining the measurements of the energy
deposition in the SVT and in the DCH with the informa-
tion from the Cherenkov detector (DIRC). Photons are
detected and measured in the CsI(Tl) electro-magnetic
calorimeter (EMC). Electrons are identified by the ra-
tio of the track momentum to the associated energy de-
posited in the EMC, the transverse profile of the shower,
the energy loss in the DCH, and the Cherenkov angle in
the DIRC. Muons are identified in the instrumented flux
return, composed of resistive plate chambers interleaved
with layers of steel and brass.
The results presented here are obtained using a to-
tal integrated luminosity of 75 fb−1 registered by the
BABAR detector during the years 2000-2002. Monte Carlo
(MC) simulation samples of Υ (4S) decays, charm and
other light quark pairs from continuum equivalent, re-
spectively, to 2.8, 1.2 and 0.7 times the data statistics,
respectively, have been generated using GEANT4 [22].
These are used mainly to evaluate background compo-
nents. Quark fragmentation, in continuum events, is de-
scribed using the JETSET package [23]. The MC distri-
butions have been rescaled to the data sample luminosity,
using the expected cross sections of the different compo-
nents (1.3nb for cc, 0.525 nb for B+B− and B0B̄0, 2.09nb
for light uū, dd̄ and ss̄ quark events). Dedicated sam-
ples of pure signal events, equivalent to seven times the
data statistics, are used to correct measurements for effi-
ciency and finite resolution effects. They have been gen-
erated using the modified pole parameterization ansatz
for f+(q
2) with αpole = 0.50. Radiative decays (D
K−e+νeγ) are modeled by PHOTOS [24]. To account for
one of the most important sources of background, a spe-
cial event sample with, in each event, at least one cascade
decay D∗+ → D0π+, D0 → K−π0e+νe (or its charge
conjugate) has been generated with a parameterization
of the form factors in agreement with measurements from
the FOCUS collaboration [25]. Events with a D∗+ and
a D0 decaying into K−π+ or K−π+π0 have been recon-
structed in data and simulation. These control samples
have been used to adjust the c-quark fragmentation dis-
tribution and the kinematic characteristics of particles
accompanying the D meson in order to better match the
data. They have been used also to measure the recon-
struction accuracy on the missing neutrino momentum.
IV. SIGNAL RECONSTRUCTION
We reconstruct D0 → K−e+νe(γ) decays in e+e− →
cc̄ events where theD0 originates from theD∗+ → D0π+.
The main sources of background arise from events with
a kaon and electron candidate. Such events come from
Υ (4S) decays and the continuum production of charmed
hadrons. Their contribution is reduced using variables
sensitive to the particle production characteristics that
are different for signal and background events.
A. Signal selection
Charged and neutral particles are boosted to the cen-
ter of mass system (c.m.) and the event thrust axis is
determined. The direction of this axis is required to be
in the interval | cos(θthrust)| < 0.6 to minimize the loss of
particles in regions close to the beam axis. A plane per-
pendicular to the thrust axis is used to define two hemi-
spheres, equivalent to the two jets produced by quark
fragmentation. In each hemisphere, we search for pairs
of oppositely charged leptons and kaons. For the charged
lepton candidates we consider only electrons or positrons
with c.m. momentum greater than 0.5 GeV/c.
Since the νe momentum is unmeasured, a kinematic
fit is performed, constraining the invariant mass of the
candidate e+K−νe system to the D
0 mass. In this fit,
theD0 momentum and the neutrino energy are estimated
from the other particles measured in the event. The D0
direction is taken as the direction opposite to the sum of
the momenta of all reconstructed particles in the event,
except for the kaon and the positron associated with the
signal candidate. The energy of the jet is determined
from the total c.m. energy and from the measured masses
of the two jets. The neutrino energy is estimated as the
difference between the total energy of the jet and the
sum of the energies of all reconstructed particles in the
hemisphere. A correction, which depends on the value
of the missing energy measured in the opposite jet, is
applied to account for the presence of missing energy due
to particles escaping detection, even in the absence of a
neutrino from the D0 decay.
The D0 candidate is retained if the χ2 probability of
the kinematic fit exceeds 10−3. Detector performance for
the reconstruction of the D0 direction and for the missing
energy are measured using events in which the D0 decays
into K−π+. Corrections are applied to account for ob-
served differences between data and simulation. Each
D0 candidate is combined with a charged pion, with the
same charge as the lepton, and situated in the same hemi-
sphere. The mass difference δ(m) = m(D0π+)−m(D0)
is evaluated and is shown in Fig. 1. This distribution con-
tains events which in addition pass the requirements on
the Fisher discriminant FBB̄ suppressingBB̄ background
and also give a satisfactory kinematic fit constraining the
invariant mass . This last requirement is the reason of
the slow decrease of the δ(m) distribution. At large δ(m)
values, a small excess of background is measured in data
and the simulation is rescaled accordingly. Only events
with δ(m) < 0.16 GeV/c2 are used in the analysis.
10000
15000
20000
0.15 0.2 0.25 0.3
Signal
Peaking cc bkg
Non-peaking cc bkg
0.15 0.2 0.25 0.3
δ(m) (GeV/c2)
FIG. 1: Comparison of the δ(m) distributions from data and
simulated events. MC events have been normalized to the
sample luminosity according to the different cross sections.
An excess of background events of the order of 5% is observed
for large values of δ(m). The arrow indicates the additional
selection applied for the q2 distribution measurement.
B. Background rejection
Background events arise from Υ (4S) decays and
hadronic events from the continuum. Three variables
are used to reduce the contribution from BB̄ events:
R2 (the ratio between the second and zeroth order Fox-
Wolfram moments [26]), the total charged and neutral
multiplicity and the momentum of the soft pion (πs) from
the D∗+. These variables exploit the topological differ-
ences between events with B decays and events with cc̄
fragmentation. The particle distribution in Υ (4S) decay
events tends to be isotropic as the B mesons are pro-
duced near threshold, while the distribution in cc̄ events
is jet-like as the c.m. energy is well above the charm
threshold. This also results in a softer D∗+ momentum
spectrum in Υ (4S) decays compared to cc̄ events.
Corresponding distributions of these variables for sig-
nal and background events are given in Fig. 2. These
variables have been combined linearly in a Fisher dis-
criminant. The requirement FBB̄ > 0.5 retains 65% of
signal and 6% of BB̄-background events.
0 0.5 1
B events
c events
Charged + neutral multiplicity
0 0.25 0.5
pπs (GeV/c)
-5 0 5
FIG. 2: MC simulations of distributions of the variables used
in the Fisher discriminant analysis to reduce the BB̄ event
background: a) the normalized second Fox-Wolfram moment
(R2), b) the event particle multiplicity, c) the slow-pion mo-
mentum distribution, in the c.m. frame, d) the Fisher variable
for BB̄ and for charm signal events.
Background events from the continuum arise mainly
from charm particles as requiring an electron and a kaon
reduces the contribution from light-quark flavors to a low
level. Because charm hadrons take a large fraction of the
charm quark energy charm decay products have higher
average energies and different angular distributions (rel-
ative to the thrust axis or to the D direction) compared
with other particles in the hemisphere emitted from the
hadronization of the c and c quarks. These other parti-
cles are referred to as “spectator” in the following; the
“leading” particle is the one with the largest momentum.
To reduce background from cc̄ events, the following vari-
ables are used:
• the D0 momentum;
• the spectator system mass, msp., which has lower
values for signal events;
• the direction of the spectator system momentum
relative to the thrust axis cos θsp.−thrust;
• the momentum of the leading spectator track;
• the direction of the leading spectator track relative
to the D0 direction;
0 2.5 5
0 2.5
msp. (GeV/c
c bkg.
signal
cosθsp.-thrust
-1 0 1
cosθe
FIG. 3: MC simulations of some of the variables used in
the Fisher discriminant analysis to reduce the cc̄-event back-
ground: a) the D0 momentum after the kinematic fit, b) the
mass of the spectator system (peaks, at low mass values cor-
respond to events with a single charged pion or photon recon-
structed in the spectator system), c) the cosine of the angle
between the spectator system momentum and the thrust di-
rection, d) the cosine of the angle of the positron direction,
relative to the kaon direction, in the eνe c.m. frame.
• the direction of the leading spectator track relative
to the thrust axis;
• the direction of the lepton relative to the kaon di-
rection, in the dilepton rest frame, cos θe;
• the charged lepton momentum, pe, in the c.m.
frame.
The first six variables depend on the properties of c-quark
hadronization whereas the last two are related to de-
cay characteristics of the signal. Distributions for four
of the most discriminating variables are given in Fig. 3.
D0 → K−π+ events have been used to tune the simu-
lation parameters so that distributions of the variables
used to reject background agree with those measured
with data events. These eight variables have been com-
bined linearly into a Fisher discriminant variable (Fcc)
and events have been kept for values above 0. This selec-
tion retains 77% of signal events that were kept by the
previous selection requirement and rejecting 66% of the
background (Fig. 4).
The remaining background from cc̄-events can be di-
vided into peaking (60%) and non-peaking (40%) candi-
-1 0 1 2 3
signal
c bkg.
B+ bkg.
B0 bkg.
uds bkg.
-1 0 1 2 3
FIG. 4: Distribution of the values of the Fisher variable in
the signal region (δ(m) < 0.16 GeV/c2 in a), and for masses
above the signal region (δ(m) > 0.16 GeV/c2 in b).
dates. Peaking events are those background events whose
distribution is peaked around the signal region. These
are mainly events with a real D∗+ in which the slow π+
is included in the candidate track combination. Back-
grounds from e+e− annihilations into light uū, dd̄, ss̄
quarks and BB̄ events are non-peaking. These compo-
nents, from simulation, are displayed in Fig. 1.
C. q2 measurement
To improve the accuracy of the reconstructed D0 mo-
mentum, the nominal D∗+ mass is added as a constraint
in the previous fit and only events with a χ2 probabil-
ity higher than 1% are kept (Fig. 1 is obtained requiring
only that the fit has converged). The measured q2r distri-
bution, where q2r = (pD − pK)
, is given in Fig. 5. There
are 85260 selectedD0 candidates containing an estimated
number of 11280 background events. The non-peaking
component comprises 54% of the background.
To obtain the true q2 distribution, the measured one
has to be corrected for selection efficiency and detector
resolution effects. This is done using an unfolding algo-
rithm based on MC simulation of these effects.
The variation of the selection efficiency as a func-
tion of q2 is given in Fig. 6. The resolution of the
q2 measurement for signal events is obtained from MC
10000
15000
0 0.5 1 1.5 2
Signal
Background
q2r (GeV
FIG. 5: The measured q2r distribution (data points) compared
to the sum of the estimated background and of the fitted
signal components.
simulation. The resolution function can fitted by the
sum of two Gaussian functions, with standard deviations
σ1 = 0.066 GeV
2 and σ2 = 0.219 GeV
2, respectively.
The narrow component corresponds to 40% of the events.
To obtain the unfolded q2 distribution for signal events,
corrected for resolution and acceptance effects, the Singu-
lar Value Decomposition (SVD) [27] of the resolution ma-
trix has been used. This method uses a two-dimensional
matrix which relates the generated q2 distribution to the
detected distribution, q2r , as input. After subtracting the
estimated background contribution, the measured binned
q2r distribution is linearly transformed into the unfolded
q2 distribution. This approach provides the full covari-
ance matrix for the bin contents of the unfolded distri-
bution. Singular values (SV) are ordered by decreasing
values. These values contain the information needed to
transform the measured distribution into the unfolded
spectrum, along with statistical uncertainties from fluc-
tuations. Not all SV are relevant; non-significant values
have zero mean and standard deviation equal to unity
[27]. Using toy simulations, we find that seven SV have
to be kept with events distributed over ten bins. Because
the measurement of the form-factor parameters relies on
the measured q2r distribution, it does not require unfold-
ing, and is independent of this particular choice.
0 0.5 1 1.5 2
(GeV
FIG. 6: The efficiency as a function of q2, measured with
simulated signal events, after all selection criteria applied.
V. RESULTS ON THE Q2 DEPENDENCE OF
THE HADRONIC FORM FACTOR
The unfolded q2 distribution, normalized to unity, is
presented in Fig. 7 and in Table II. Also given in this
table are the statistical and total uncertainties and the
correlations of the data in the ten bins. Figure 7 shows
the result of fits to the data for two parameterizations of
the form factor with a single free parameter, the simple
pole and the modified pole ansatz. Both fitted distribu-
tions agree well with the data.
A summary of these and other form factor parame-
terizations is given in Table III. These results will be
discussed in detail in Section VB.
The fit to a model is done by comparing the number
of events measured in a given bin of q2 with the expecta-
tion from the exact analytic integration of the expression
∣~pK(q
∣f+(q
over the bin range, with the overall
normalization left free. The result of the fit correspond-
ing to the parameterization of the form factor using two
parameters (see Eq. 10) is given in Fig. 8.
A. Systematic Uncertainties
Systematic uncertainties of the form factor parame-
ters are likely to originate from imperfect simulation of
c-quark fragmentation and the detector response, from
uncertainties in the background composition and the in-
dividual contributions for the selected signal sample, the
0 0.5 1 1.5 2
Data unfolded
2 ) Pole mass
Modif. pole mass
-0.01
0 0.5 1 1.5 2
(GeV
FIG. 7: Comparison between the normalized unfolded q2 dis-
tribution obtained from this analysis, and those corresponding
to two fitted models. Lower plot gives the difference between
measured and fitted distributions. The error bars represent
statistical errors only.
0 0.5 1 1.5
βpole
FIG. 8: Contours at 70% and 90% CL resulting from the fit of
the parameterization of the form factor q2 dependence with
two parameters as given in Eq. (10). The value δpole = 0
corresponds to the modified pole ansatz.
uncertainty in the modeling of the signal decay and the
TABLE II: Statistical and total uncertainty matrices for the normalized decay distribution (corrected for acceptance and finite
resolution effects) in ten bins of q2 from 0 to 2 GeV2, and for the ratio RD (see Section VI). The total decay distribution
has been normalized to unity for q2 varying over ten 0.2 GeV2 intervals. The uncertainty matrices are provided for both the
statistical (upper half) and total (lower half) uncertainties. The uncertainty on each measured value is given along the diagonal.
Off-diagonal terms correspond to the correlation coefficients.
q2 bin (GeV2) [0, 0.2] [0.2, 0.4] [0.4, 0.6] [0.6, 0.8] [0.8, 1.0] [1.0, 1.2] [1.2, 1.4] [1.4, 1.6] [1.6, 1.8] [1.8, 2.0]
RD and 0.9269
fractions 0.2008 0.1840 0.1632 0.1402 0.1122 0.0874 0.0602 0.0367 0.0146 0.0007
statistical 0.0072 0.166 0.122 0.111 0.107 0.107 0.102 0.101 0.116 0.081 0.060
uncertainties 0.0031 -0.451 -0.155 0.117 0.005 0.023 0.002 0.002 -0.002 -0.002
and 0.0037 -0.225 -0.304 0.095 0.041 -0.025 -0.005 0.005 -0.005
correlations 0.0033 -0.155 -0.345 0.079 0.058 -0.018 -0.010 -0.006
0.0029 -0.113 0.352 0.058 0.066 -0.013 -0.024
0.0025 0.073 -0.345 0.018 0.075 0.067
0.0020 -0.029 -0.329 -0.004 0.060
0.0016 0.110 -0.339 -0.347
0.0011 0.217 0.012
0.00075 0.965
0.000057
total 0.0139 -0.154 0.072 0.034 0.093 0.046 0.084 0.231 0.278 0.210 0.184
uncertainties 0.0041 -0.462 -0.257 0.022 0.099 0.089 -0.232 -0.092 -0.056 -0.048
and 0.0040 -0.102 -0.247 0.005 0.020 0.015 -0.035 -0.055 -0.048
correlations 0.0035 -0.122 -0.377 0.050 0.074 -0.043 -0.033 -0.027
0.0030 -0.127 0.320 0.092 0.056 -0.038 -0.048
0.0026 0.041 -0.331 0.033 0.089 0.079
0.0022 0.084 -0.264 0.010 0.057
0.0018 0.159 -0.235 -0.254
0.0012 0.369 0.194
0.0009 0.973
0.000065
TABLE III: Fitted values of the parameters corresponding to different parameterizations of f+(q
2). The last column gives the
χ2/NDF of the fit when using the value expected for the parameter.
Theoretical Unit Parameters χ2/NDF Expectations
ansatz [χ2/NDF]
z expansion r1 = −2.5± 0.2 ± 0.2 5.9/7
r2 = 0.6± 6.± 5.
Modified pole αpole = 0.377 ± 0.023 ± 0.029 6.0/8
Simple pole GeV/c2 mpole = 1.884 ± 0.012 ± 0.015 7.4/8 2.112 [243/9]
ISGW2 GeV−2 αI = 0.226 ± 0.005 ± 0.006 6.4/8 0.104 [800/9]
measurement of the q2 distribution. We study the origin
and size of various systematic effects, correct the MC sim-
ulation, if possible, assess the impact of the uncertainty
of the size of correction on the fit results, and adopt the
observed change as a contribution to the systematic un-
certainty on the fitted parameters for the different param-
eterizations under study. Some of these studies make use
of standard BABAR evaluations of detection efficiencies,
others rely on special data control samples, for instance
hadronic decays D0 → K−π+ or K−π+π0.
1. c-quark hadronization tuning
The signal selection is based on variables related to
c-quark fragmentation and decay properties of signal
events. Simulated events have been weighted to agree
with the distributions observed in data. Weights have
been obtained using events with a reconstructed D0 de-
caying into K−π+. After applying these corrections,
the distribution of the Fisher discriminant that contains
these variables is compared for data and simulation. The
remaining difference is used to evaluate the correspond-
ing systematic uncertainty. It corresponds to the varia-
tions on fitted quantities obtained by correcting or not
for this difference, which is below 5% over the range of
this variable.
2. Reconstruction algorithm
It is important to verify that the q2 variation of the
selection efficiency is well described by the simulation.
This is done by analyzing D0 → K−π+π0 as if they were
K−e+νe events. The two photons from the π
0 are re-
moved and events are reconstructed using the algorithm
applied to the semileptonic D0 decay. The “missing” π0
and the charged pion play, respectively, the roles of the
neutrino and the electron. To preserve the correct kine-
matic limits, it is necessary to take into account that
the “fake” neutrino has the π0 mass and that the “fake”
electron has the π+ mass.
Data and simulated events, which satisfy the same
analysis selection criteria as for Keνe, have been com-
pared. For this test, the cos(θe) and pe are removed from
the Fisher discriminant, because distributions for these
two variables are different from the signal events.
The ratio of efficiencies measured in data and simula-
tion is fit with a linear expression in q2. The correspond-
ing slope, (0.71±0.68)%, indicates that there is no signif-
icant bias when the event selection criteria are applied.
The measured slope is used to define a correction and to
estimate the corresponding systematic uncertainty.
3. Resolution on q2
To measure possible differences between data and
simulation on the q2 reconstruction accuracy, D0 →
K−π+π0 events are used again. Distributions of the dif-
ference q2r − q2, obtained by selecting events in a given
bin of q2 are compared. These distributions are system-
atically slightly narrower for simulated events and the
fraction of events in the distant tails are higher for data
(see Fig. 9).
With the D0 → K−π+ sample we study, in data and
simulation, the accuracy of the D0 direction and missing
energy reconstruction for the D0 → K−e+νe analysis.
This information is used in the mass-constrained fits and
thus influences the q2 reconstruction. Once the simula-
tion is tuned to reproduce the results obtained on data
for these parameters, the q2 resolution distributions agree
very well, as shown in Fig. 9. One half of the measured
variation on the fitted parameters from these corrections
has been taken as a systematic uncertainty.
4. Particle identification
Effects from a momentum-dependent difference be-
tween data and simulated events on the charged lepton
and on the kaon identification have been evaluated. Such
-0.4 -0.2 0 0.2 0.4
-0.4 -0.2 0 0.2 0.4
q2r-q
2 (GeV2)
-0.4 -0.2 0 0.2 0.4
-0.4 -0.2 0 0.2 0.4
q2r-q
2 (GeV2)
FIG. 9: Distribution of the difference between the true and
the reconstructed value of q2. D0 → K−π+π0 data events
correspond to dark squares and open circles are used for
simulated ones. Ratio (Data/MC) of the two distributions
are displayed. The distributions in a) compare data and
simulated events before applying corrections measured with
D0 → K−π+ events, whereas these corrections have been ap-
plied for plots in b).
differences, which are typically below 2%, have been mea-
sured for selected, high purity samples of electrons and
kaons. These corrections have been applied and the ob-
served variation has been taken as the estimate of the
systematic uncertainty.
5. Background estimate
The background under the D∗+ signal has two compo-
nents that have, respectively, non-peaking and peaking
behavior.
The non-peaking background originates from non-cc
events and from continuum charm events in which the
πs candidate does not come from a cascading D
∗+. By
comparing data and simulated event rates for δ(m) >
0.18 GeV/c2 (see Fig. 1), a correction of 1.05 is deter-
mined from simulation for the non-peaking background.
This correction is applied and an uncertainty of ±0.05 is
used as the corresponding systematic uncertainty.
Events which include a slow pion originating from D∗+
decay contribute in several ways to the peaking back-
ground. The production rate of D∗+ mesons in the sim-
ulation is in agreement with expectations based on mea-
surements from CLEO [28]. The uncertainty of ±0.06 on
this comparison is dominated by the systematic uncer-
tainty from the CLEO result.
To study the remaining effects, the peaking back-
ground components have been divided according to the
process from which they originate and have been ordered
by decreasing level of importance:
• the K− and the electron originate from a D0 de-
cay (54%). The main source comes from D0 →
K−π0e+νe. We have corrected the decay branch-
ing fraction used for this channel in the MC (2.02%)
using recent measurements (2.17± 0.16% [1]). The
uncertainty on this value has been used to evaluate
the corresponding systematic uncertainty.
• the electron comes from a converted photon or a
Dalitz decay (24%). It has been assumed that the
simulation correctly accounts for this component;
• the K− does not originate from a charm hadron
(14%). This happens usually when there is another
negative charged kaon accompanying the D∗+. We
have studied the production of charged kaon accom-
panying a D∗+ using D0 → K−π+ events and mea-
sure a correction factor of 0.87±0.02 and 0.53±0.02,
respectively, for same sign and opposite sign K-
D∗ pairs. The simulation is modified accordingly
and the remaining systematic uncertainty from this
source becomes negligible;
• fake kaon candidate (mainly pions) (6%) or fake
electrons (1%). Differences between data and MC
on the evaluation of fake rates have been studied
in BABAR. As this affects small components of the
total peaking background rate, the effect of these
differences has been neglected.
6. Fitting procedure and radiative events
To fit form factor parameters we compare the num-
ber of expected events in each bin with the measured
one after all corrections. In this approach it is always
assumed that the q2 variation of f+(q
2) is given ex-
actly by the form factor parameterization. This hy-
pothesis is not correct, a priori, for radiative decays as
q2 = (pD − pK)2 = (pe+ pν + pγ)2 is not (perhaps) equal
to the variable that enters in f+ for such decays. PHO-
TOS is used to generate decays with additional photons
and the modified pole ansatz is taken to parameterize
the hadronic form factor in signal events. To quantify
possible distortion of the fit we compare the fitted value
of a form factor parameter with the one obtained from a
fit to the generated q2 distribution (see Table IV).
TABLE IV: Measured differences between the nominal and
fitted values of parameters. Quoted uncertainties correspond
to MC statistics. The last column gives the impact of the
radiative effects on the form factor measurements as predicted
by PHOTOS.
Parameter Measured difference Bias from
(true-fitted) radiation
δ(r1) (×0.01) 1.2± 13.0 −0.4± 2.7
δ(r2) +1.7± 3.6 +1.9± 0.7
δ(αpole) (×0.01) −1.2± 1.4 −1.1± 0.3
δ(mpole) (MeV/c
2) +4.5± 6.3 +4.6± 1.4
δ(αI) (×0.001GeV
−2) −2.7± 3.1 −2.4± 0.7
Corresponding corrections, given in the second column
of Table IV, have been applied and quoted uncertainties
enter in the systematic uncertainty evaluation.
To evaluate the importance of corrections induced by
radiative effects, we have compared also the fitted value
of a parameter on q2 distributions generated with and
without using PHOTOS. Measured differences are given
in the last column of Table IV. They have not been
applied to the values quoted in Table III for the different
parameters. We measure also that radiative effects affect
mainly the fraction of the decay spectrum in the first bin
in Table II which has to be increased by 0.0012 to correct
for this effect.
7. Control of the statistical accuracy in the SVD approach
Once the number of SV is fixed, one must verify that
the statistical precision obtained for each binned un-
folded value is correct and if biases generated by remov-
ing information are under control. These studies are done
with toy simulations. One observes that the uncertainty
TABLE V: Summary of systematic uncertainties on the fitted parameters.
Source δ (mpole) δαpole δ (αI) δ (r1) δ (r2)
(MeV/c2) (×0.01) (×0.001 GeV−2) (×0.01)
c-hadronization tuning 3.0 0.6 1.1 6.7 1.4
Reconstruction algorithm 7.8 1.6 3.1 6.5 0.2
Resolution on q2 3.4 0.7 1.4 3.0 2.1
Particle ID 5.5 1.1 2.3 3.8 0.4
Background estimate 7.7 1.4 3.0 12.7 2.0
Fitting procedure 6.3 1.4 3.1 13.0 3.6
Total 15.0 2.9 6.0 21.0 4.9
obtained from a fit of the unfolded distribution is un-
derestimated by a factor which depends on the statistics
of simulated events and is ∼ 1.06 in the present anal-
ysis. Pull distributions indicate also that the unfolded
values, in each bin, have biases which are below 10% of
the statistical uncertainty. Similar studies are done for
the determination of form factor parameters.
8. Summary of systematic errors
The systematic uncertainties for determining form fac-
tor parameters are summarized in Table V.
The systematic error matrix for the ten unfolded val-
ues is computed by considering, in turn, each source of
uncertainty and by measuring the variation, δi, of the cor-
responding unfolded value in each bin (i). The elements
of the uncertainty matrix are the sum, over all sources
of systematic uncertainty, of the quantities δi · δj . The
total error matrix is evaluated as the sum of the matrices
corresponding, respectively, to statistical and systematic
uncertainties.
B. Comparison with expectations and with other
measurements
The summary of the fits to the normalized q2 distri-
butions are presented in Table III. As long as we allow
the form factor parameters to be free in the fit, the fit-
ted distributions agree well with the data and it is not
possible to reject any of the parameterizations.
However, if the form factor parameters are constrained
to specific predicted values, the agreement is not good.
For the ISGW2 model, the predicted dependence of the
form factor on q2 disagrees with the data and the fit-
ted value of the parameter αI differs from the predicted
value, αI = 0.104 GeV
−2 by more than a factor two.
As observed by previous experiments, the simple pole
model ansatz, with mpole = mD∗
= 2.112 GeV/c2 does
not reproduce the measurements. This means that the
contribution from the continuum DK interaction can-
not be neglected. If one introduces a second parame-
ter δpole to account for contributions from an effective
pole at higher mass (see Eq. 10) the two parameters are
fully correlated and there is no unique solution, as illus-
trated in Fig. 8. The modified pole ansatz corresponds
to δpole = 0.
In Table VI the fitted parameters for the simple pole
ansatz and the modified pole [11] ansatz are compared
for different experiments. The fitted pole masses are all
well below the mass of the D∗s meson. The results pre-
sented here are consistent within the stated uncertainties
with earlier measurements. Except for the BELLE mea-
surement, all other measurements appear to favor a value
of αpole that is lower than the value predicted by lattice
QCD, namely αpole = 0.50± 0.04.
TABLE VI: Fitted values for the parameters corresponding
respectively to a pole mass and a modified pole mass model
for the form factor.
Experiment mpole (GeV/c
2) αpole
CLEO [4] 1.89 ± 0.05+0.04−0.03 0.36 ± 0.10
+0.03
−0.07
FOCUS [5] 1.93 ± 0.05 ± 0.03 0.28 ± 0.08 ± 0.07
BELLE [6] 1.82 ± 0.04 ± 0.03 0.52 ± 0.08 ± 0.06
This analysis 1.884 ± 0.012 ± 0.015 0.38 ± 0.02 ± 0.03
In Fig. 10, the dependence of the form factor on q2 is
presented. The data are compared to earlier measure-
ments by the FOCUS experiment, as well as with pre-
dictions from lattice QCD calculations [19]. As stated
above, the data favor a somewhat lower value for αpole.
The data have also been mapped into the variable z.
Figure 11 shows the product P ×Φ× f+ as a function of
z. By convention, this quantity is constrained to unity
at z = zmax, which corresponds to q
2 = 0. We perform a
fit to a polynomial, P × Φ × f+ ∼ 1 + r1z + r2z2. The
data are compatible with a linear dependence, which is
fully consistent with the modified pole ansatz for f+(q
as illustrated in Fig. 11.
VI. BRANCHING FRACTION MEASUREMENT
The D0 → K−e+νe branching fraction is measured
relative to the reference decay channel, D0 → K−π+.
0 0.5 1 1.5 2
q2(GeV2)
BABAR
FOCUS
Lattice-QCD (αpole = 0.50(4))
FIG. 10: Comparison of the measured variation of
2)/f+(0) obtained in the present analysis and in the FO-
CUS experiment [5]. The band corresponds to lattice QCD
[19] with the estimated uncertainty.
Specifically, we compare the ratio of rates for the decay
chains D∗+ → D0π+, D0 → K−e+νe, and D0 → K−π+
in data and simulated events, this way, many systematic
uncertainties cancel,
BR(D0 → K−e+νe)data
BR(D0 → K−π+)data
BR(D0 → K−e+νe)MC
BR(D0 → K−π+)MC
N(cc̄)Keν
N(cc̄)Kπ
L(data)Kπ
L(data)Keν
N(D0 → K−e+νe)data
N(D0 → K−e+νe)MC
N(D0 → K−π+)MC
N(D0 → K−π+)data
ǫ(D0 → K−e+νe)MC
ǫ(D0 → K−e+νe)data
ǫ(D0 → K−π+)data
ǫ(D0 → K−π+)MC
The first line in this expression is the ratio of the
branching fraction for the two channels used in the sim-
ulation:
BR(D0 → K−e+νe)MC
BR(D0 → K−π+)MC
0.0364
0.0383
. (16)
The second line is the ratio of the number of cc̄ simu-
lated events and the integrated luminosities for the two
channels:
N(cc̄)Keν
N(cc̄)Kπ
L(data)Kπ
L(data)Keν
117.0× 106
117.3× 106 ×
73.43 fb−1
74.27 fb−1
-0.05 -0.025 0 0.025 0.05
) Modif. pole fit
BaBar
FIG. 11: Measured values for P×Φ×f+ are plotted versus−z
and requiring that P ×Φ× f+ = 1 for z = zmax. The straight
lines represent the result for the modified pole ansatz, the fit
in the center and the statistical and total uncertainty.
The third line corresponds to the ratios of measured
numbers of signal events in data and in simulation, and
the last line gives the ratios of the efficiencies to data and
simulation.
A. Selection of candidate signal events
The selection ofD0 → K−e+νe candidates is explained
in section IVA. For the rate measurement, the con-
straint on the D∗+ mass is not applied and also the
momentum of the soft pion candidate is not included
in the Fisher discriminant variable designed to suppress
BB̄-background. Since generic simulated signal events
used in this measurement have been generated with the
ISGW2 model, they have been weighted so that their q2
distribution agrees with the measurement presented in
this paper. Furthermore, we require for the Fisher dis-
criminant FBB̄ > 0 and restrict δ(m) < 0.16 GeV/c
After background subtraction, there remain 76283± 323
and 95302 ± 309 events in data and simulation, respec-
tively. This gives:
N(D0 → K−e+νe)data
N(D0 → K−e+νe)MC
= 0.8004± 0.0043. (18)
To select D0 → K−π+ candidates, the same data sam-
ples are used and particles, in each event, are selected in
the same way. The same selection criteria on the Fisher
1.75 1.8 1.85 1.9 1.95
D0→K-π+
D0→K-π+γ
D0→K-K+
D0→π-π+
Other
m(Kπ) (GeV/c2)
0.14 0.1425 0.145 0.1475 0.15
D0→K-π+
D0→K-π+γ
Other bkg
m(D0π)-m(D0) (GeV/c2)
0.14 0.16 0.18 0.2
MC bkg
m(D0π)-m(D0) (GeV/c2)
0.14 0.1425 0.145 0.1475 0.15
m(D0π)-m(D0) (GeV/c2)
FIG. 12: Events selected for the reference channel D∗+ → D0π+, D0 → K−π+. a) Kπ mass distribution for events selected
in the range δ(m) ∈ [0.143, 0.148] GeV/c2. b) δ(m) distribution for events selected in the range m(Kπ) ∈ [1.83, 1.89] GeV/c2.
c) same distribution as in b), displayed on a larger mass range with the non-peaking background indicated (shaded area). d)
δ(m) distribution after non-peaking background subtraction, data (points with statistical errors) and simulated events (shaded
histogram).
discriminant to suppress BB̄-events, on the thrust axis
direction and on other common variables are applied.
Events are also analyzed in the same way, with two hemi-
spheres defined from the thrust axis. In each hemisphere
a D0 candidate is reconstructed by combining a charged
K with a pion of opposite sign. These tracks have to
form a vertex and the Kπ mass must be within the range
[1.77, 1.95] GeV/c2. Another charged pion of appropri-
ate sign is added to form a D∗+ candidate.
In addition, the following selection criteria are used:
• the fraction of the beam momentum, in the c.m.
frame, taken by the D∗ candidate must exceed 0.48
to remove contributions from BB̄ events;
• the measured D0 mass must be in the range be-
tween 1.83 and 1.89 GeV/c2. This requirement
eliminates possible contributions from remaining
D0 → K−K+ or π+π− decays (see Fig. 12-a);
• the vertex fit for the D0 and D∗ have to converge.
The δ(m) distribution for candidate events is shown in
Fig. 12-c. The following components contribute to the
D∗+ signal (see Fig. 12-b):
• D0 → K−π+ with no extra photon;
• D0 → K−π+ with at least one extra photon;
• D0 → K−π+(γ) where the π+, mainly from the
D∗+, decays into a muon.
The δ(m) distribution corresponding to other event
categories does not show a peaking component in the
D∗+ signal region. The total background level, is nor-
malized to data using events in the δ(m) interval between
0.165 and 0.200 GeV/c2 (see Fig. 12-c). This global scal-
ing factor is equal to 1.069±0.011. After background sub-
traction, the δ(m) distributions obtained in data and sim-
ulation can be compared in Fig. 12-d. Since the D∗+ sig-
nal is narrower in the simulation, we use a mass window
such that this difference has a small effect on the mea-
sured number of D∗+ events in data and in the simula-
tion. There are 166960±409 and 134537±374 candidates
selected in the interval δ(m) ∈ [0.142, 0.149] GeV/c2 for
simulated and data events respectively. This means:
N(D0 → K−π+)MC
N(D0 → K−π+)data
= 1.230± 0.0046. (19)
B. Efficiency corrections
The impact of the selection requirement on the recon-
structed Kπ mass, has been studied. The Kπ mass dis-
tribution signal for simulated events (not including radia-
tive photons) is compared with the corresponding distri-
bution obtained with data after background subtraction.
The background contributions are taken from the simula-
tion. The fraction of D0 candidates in the selected mass
range (between 1.83 and 1.89 GeV/c2) is (97.64±0.25)%
in MC and (97.13± 0.29)% in data events. The ratio of
efficiencies is equal to:
ǫ(D0 → K−π+)data
ǫ(D0 → K−π+)MC
= 0.9947± 0.0039. (20)
Since D0 → K−e+νe events have been selected using
a selection requirement on δ(m), we need to confirm that
the distribution of this variable is similar in data and
simulation. This is checked by comparing the distribu-
tions obtained with D0 → K−π+π0 events analyzed as
if they were semileptonic decays. The δ(m) distributions
are compared in Fig. 13. Below 0.16 GeV/c2, there are
0.93552±0.00066 of the D∗+ candidates in the simulation
and 0.93219± 0.00078 for data. The corresponding ratio
of efficiencies (MC/data) is equal to 1.0036± 0.0010.
Using D0 → K−π+π0 events, we also measure the dif-
ference between the fraction of events retained after the
mass-constrained fits. Namely, it is 0.98038± 0.00037 in
the simulation compared to 0.97438 ± 0.00049 in data.
The relative efficiency (MC/data) for this selection is
1.0062 ± 0.0006. Based on these two measured correc-
tions the ratio of efficiencies are:
ǫ(D0 → K−e+νe)MC
ǫ(D0 → K−e+νe)data
= 1.0098± 0.0011. (21)
The quoted uncertainties, in this section, are of statistical
origin and will be included in the statistical uncertainty
on RD. Other differences between the two analyzed chan-
nels are considered in the following section and contribute
to systematic uncertainties.
0.15 0.2 0.25 0.3
0.15 0.2 0.25 0.3
δ(m) (GeV/c2)
FIG. 13: δ(m) distribution for D0 → K−π+π0 events ana-
lyzed as if they were semileptonic decays. Distributions have
been normalized to unity; note that the bin size is not uni-
form. The bottom plot shows the ratio of the two distribu-
tions above.
C. Systematic uncertainties on RD
A summary of the systematic uncertainties on RD are
given in Table VII. They originate from selection criteria
which are different for the two channels. Some of these
uncertainties are the same as those already considered
for the determination of the q2 variation of f+.
1. Correlated systematic uncertainties
Systematic uncertainties on the decay rate coming
from effects that contribute in the measurement of the
q2 dependence of f+(q
2) are evaluated in Section VA
and the full covariance matrix for the measurements of
the number of D0 → K−e+νe signal events and the frac-
tion of the decay spectrum fitted in each of the ten bins
is determined. Among the sources of systematic uncer-
tainties, listed in Table V, those corresponding to:
• the reconstruction algorithm,
• the tuning of the resolution on q2,
• the corrections applied on electron identification,
• the background normalization
are taken as common sources. Corresponding relative
uncertainties on RD are given in Table VII.
Other systematic uncertainties contributing to the
form factor measurement also affect the reference chan-
nel and so their effects on RD cancel. They are related to
the c-hadronization tuning and to the corrections applied
on the kaon identification.
2. Selection requirement on the Fisher discriminant
The stability of the fraction of D0 → K−e+νe events
selected in data and in simulation as a function of the
Fisher discriminant, Fcc, designed to suppress cc̄ back-
ground has been examined. This is done by comparing
the distributions of this variable measured in data and in
simulation as given in Fig. 4 for two selected intervals in
δ(m).
The value corresponding to Fcc > 0 and for events se-
lected in the range δ(m) < 0.16 GeV/c2 is used as the
central result and half the difference between the mea-
surements corresponding to Fcc greater than−0.25 and +
0.25 is taken as systematic uncertainty. This range cor-
responds to a relative change of 40% of the efficiency for
signal events, and gives an uncertainty of ±0.0061 on the
ratio of data and simulated signal candidates.
3. D∗+ counting in D0 → K−π+
D∗+ candidates are selected in the range δ(m) ∈
[0.142, 0.149] GeV/c2. From the simulation it is ex-
pected that the fraction of signal events outside this
interval is equal to 1.4%. Even though the D∗+ sig-
nal is slightly narrower in the simulation, there is not
a large discrepancy in the tails. The fraction of signal
events measured in the sidebands δ(m) ∈ [0.140, 0.142]⊕
[0.149, 0.150] GeV/c2 is 0.4% and 0.5%, respectively, for
simulation and data. An uncertainty of ±0.004, cor-
responding to 30% uncertainty on the total fraction of
events outside the selected δ(m) interval is assumed.
TABLE VII: Summary of systematic uncertainties on the
relative decay rate measurement.
Source Relative variation
Reconstruction algorithm ±0.42%
Resolution on q2 ∼ 0
Electron ID ±0.56%
Background subtraction ±0.63%
Cut on Fisher variable ±0.76%
D∗+ counting (D0 → K−π+) ±0.40%
Total ±1.27%
D. Decay rate measurement
Combining all measured fractions in Eq. (15), the mea-
sured relative decay rate is:
RD = 0.9269± 0.0072± 0.0119. (22)
Using the world average for the branching fraction
BR(D0 → K−π+) = (3.80± 0.07)% [1], gives BR(D0 →
K−e+νe(γ)) = (3.522 ± 0.027 ± 0.045± 0.065)%, where
the last quoted uncertainty corresponds to the accuracy
on BR(D0 → K−π+).
VII. SUMMARY
The decay rate distribution for the channel D0 →
K−e+νe(γ) has been measured in ten bins in Table II.
Several theoretical expectations for the variation of this
form factor with q2 have been considered and values for
the corresponding parameters have been obtained (see
Table III). The q2 variation of the form factor can be
parameterized with a single parameter using different ex-
pressions. The ISGW2 model with expected values for
the parameters is excluded, as is the pole mass parame-
terization with mpole = mD∗
The value of the decay branching fraction has been also
measured independent of a model.
Combining these measurements, the value of the
hadronic form factor is obtained:
f+(0) =
|Vcs|
, (23)
where BR is the measured D0 → K−e+νe branching
fraction, τD0 = (410.1± 1.5)× 10−15 s [1] is the D0 life-
time and I =
∣~pK(q
∣f+(q
2)/f+(0)
dq2. To
account for the variation of the form factor within one
bin, and in particular to extrapolate the result at q2 = 0,
the pole mass and the modified pole ansatze have been
used; the corresponding values obtained for f+(0) differ
by 0.002. Taking the average between these two values
and including their difference in the systematic uncer-
tainty, this gives
f+(0) = 0.727± 0.007± 0.005± 0.007, (24)
where the last quoted uncertainty corresponds to the ac-
curacy on BR(D0 → K−π+), τD0 and |Vcs|. It agrees
with expectations and in particular with LQCD compu-
tations [19]. Using the z expansion of Eq. (6), we find
a0 = (2.98± 0.01± 0.03± 0.03)× 10−2.
The high accuracy of the present measurement will be
a reference test for improved lattice determinations of the
q2 variation of f+.
VIII. ACKNOWLEDGMENTS
The authors wish to thank R. J. Hill, D. Becirevic, C.
Bernard, Ph. Boucaud, S. Descotes-Genon, L. Lellouch,
J.-P. Leroy, A. Le Yaouanc and O. Pène for their help
with the theoretical interpretation of these results.
We are grateful for the extraordinary contributions of
our PEP-II colleagues in achieving the excellent luminos-
ity and machine conditions that have made this work pos-
sible. The success of this project also relies critically on
the expertise and dedication of the computing organiza-
tions that support BABAR. The collaborating institutions
wish to thank SLAC for its support and the kind hospital-
ity extended to them. This work is supported by the US
Department of Energy and National Science Foundation,
the Natural Sciences and Engineering Research Coun-
cil (Canada), Institute of High Energy Physics (China),
the Commissariat à l’Energie Atomique and Institut Na-
tional de Physique Nucléaire et de Physique des Partic-
ules (France), the Bundesministerium für Bildung und
Forschung and Deutsche Forschungsgemeinschaft (Ger-
many), the Istituto Nazionale di Fisica Nucleare (Italy),
the Foundation for Fundamental Research on Matter
(The Netherlands), the Research Council of Norway, the
Ministry of Science and Technology of the Russian Fed-
eration, Ministerio de Educación y Ciencia (Spain), and
the Particle Physics and Astronomy Research Council
(United Kingdom). Individuals have received support
from the Marie-Curie IEF program (European Union)
and the A. P. Sloan Foundation.
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http://arxiv.org/abs/hep-ph/0606023
Introduction
The f+(q2) hadronic form factor
Form factor parameterizations
Taylor expansion
Model-dependent parameterizations
Quantitative expectations
Quark Models
QCD sum rules
Lattice QCD
Analyzed parameterizations
The BABAR Detector and Dataset
Signal reconstruction
Signal selection
Background rejection
q2 measurement
Results on the q2 dependence of the hadronic form factor
Systematic Uncertainties
c-quark hadronization tuning
Reconstruction algorithm
Resolution on q2
Particle identification
Background estimate
Fitting procedure and radiative events
Control of the statistical accuracy in the SVD approach
Summary of systematic errors
Comparison with expectations and with other measurements
Branching fraction measurement
Selection of candidate signal events
Efficiency corrections
Systematic uncertainties on RD
Correlated systematic uncertainties
Selection requirement on the Fisher discriminant
D*+ counting in D0K- +
Decay rate measurement
Summary
Acknowledgments
References
|
0704.0021 | Molecular Synchronization Waves in Arrays of Allosterically Regulated
Enzymes | Molecular Synchronization Waves in Arrays of Allosterically Regulated Enzymes
Vanessa Casagrande,1 Yuichi Togashi,2, ∗ and Alexander S. Mikhailov2, †
1Hahn-Meitner-Institut, Glienicker Straße 100, 14109 Berlin, Germany
2Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
Spatiotemporal pattern formation in a product-activated enzymic reaction at high enzyme con-
centrations is investigated. Stochastic simulations show that catalytic turnover cycles of individual
enzymes can become coherent and that complex wave patterns of molecular synchronization can
develop. The analysis based on the mean-field approximation indicates that the observed patterns
result from the presence of Hopf and wave bifurcations in the considered system.
PACS numbers: 82.40.Ck, 87.18.Pj, 82.39.Fk, 05.45.Xt
Molecular machines, such as molecular motors, ion
pumps and some enzymes, play a fundamental role in
biological cells and can be also used in the emerging soft-
matter nanotechnology [1]. A protein machine is a cyclic
device, where each cycle consists of conformational mo-
tions initiated by binding of an energy-bringing ligand
[2, 3]. In motors, such internal motions generate me-
chanical work [4], while in enzymes they enable or facil-
itate chemical reaction events (see, e.g., [5, 6]). Much
attention has been attracted to studies of biomembranes
with ion pumps and molecular motors, where membrane
instabilities and synchronization effects have been ana-
lyzed [7, 8, 9]. Here, a different class of distributed ac-
tive molecular systems — formed by enzymes — is con-
sidered. The catalytic activity of an allosteric enzyme
protein is activated or inhibited by binding of small reg-
ulatory molecules; the role of such regulatory molecules
can be played by products of the same reaction [10]. Pre-
vious investigations of simple product-regulated enzymic
systems [11, 12] and enzymic networks [13] in small spa-
tial volume with full diffusional mixing have shown that
spontaneous synchronization of molecular turnover cycles
can take place there. External molecular synchroniza-
tion of enzymes of the photosensitive P-450 dependent
monooxygenase system by periodic optical forcing has
been experimentally demonstrated [14].
In this Letter, spatiotemporal pattern formation in en-
zymic arrays is investigated. In such systems, immobile
enzymes are attached to a solid planar support immersed
into a solution through which fresh substrate is supplied
and product molecules are continuously removed. Prod-
uct molecules released by an enzyme diffuse through the
solution and activate catalytic turnover cycles of neigh-
bouring enzymes in the array.
A simple stochastic model [12] of an enzyme as a cyclic
machine (a stochastic phase oscillator), shown in Fig. 1,
is used. Binding of a substrate molecule to an enzyme i
initiates an ordered internal conformational motion, de-
scribed by the conformational phase coordinate φi. The
initial state corresponds to the phase φi = 0. The cat-
alytic conversion event takes place and the product is
released at the state φp inside the cycle. After that, the
substrate
enzyme
regulatory
molecule
feedback
product
FIG. 1: (Color online) A sketch of the model.
conformational motion continues until the equilibrium
state of the enzyme (φi = 1) is finally reached. Initi-
ation of a turnover cycle is a random event, occurring
at a certain probability rate. We assume that substrate
is present in abundance, and its concentration is not af-
fected by the reactions. Conformational motion inside
the cycle is modeled as a stochastic diffusional drift pro-
cess, described by equation
φi = v+ ηi(t), where v is the
mean drift velocity and ηi(t) is an internal white noise
with 〈ηi(t)ηj(t
′)〉 = 2σδijδ(t− t
′) where σ specifies inten-
sity of intramolecular fluctuations.
Allosterically activated enzymes possess a site on their
surface where regulatory molecules can become bound.
Binding of a regulatory molecule leads to conformational
change that enhances catalytic activity of the enzyme. A
regulatory molecule binds to an enzyme with rate con-
stant β and dissociate from it with rate constant κ. Bind-
ing of a regulatory molecule at an enzyme raises its prob-
ability to start a cycle from α0 to α1. We assume that
a regulatory molecule can bind to an enzyme only in its
rest state and this molecule is released when the cycle
is started. The role of regulatory molecules is played
by product molecules of the same reaction. Immobile
enzymes are randomly distributed in space with concen-
tration c. Product diffuses at diffusion constant D and
undergoes decay at rate constant γ. The characteristic
diffusion length of product molecules is ldiff =
In our stochastic 2D simulations, the medium was dis-
cretized into spatial cells (up to 256 × 256), each con-
http://arxiv.org/abs/0704.0021v2
FIG. 2: Stochastic (a,b) and mean-field (c,d) simulations of
2D wave patterns; (a) τp = 0.14, c = 1, and β = 300, (b)
τp = 0.25, c = 10, and β = 10, (c) τp = 0.14, c = 1, and
β = 300, (d) τp = 0.34, c = 100, and β = 1.42. Other
parameters are α0 = 1, α1 = 1000, κ = 10, γ = 10, σ = 0,
D = 100. The linear size of the shown area is L = 40 ldiff in
all panels.
taining a number of enzyme molecules. The cells were
so small that diffusional mixing of product molecules in
a cell within the shortest characteristic time of the reac-
tion could always take place. Each enzyme was described
by the stochastic model given above; diffusion of product
molecules was modeled as a random walk over a discrete
cell lattice. The mean cycle time τ = 1/v was chosen as
the time unit (τ = 1). Systems including up to 655 360
enzymes were used in the simulations.
Figure 2a,b (see also Videos 1 and 2 in ref. [15]) shows
two typical examples of stochastic 2D simulations. Here,
spatial distributions of product molecules are displayed.
Waves of product concentration are propagating through
the medium. In a peak of a wave, many locally present
enzymes are simultaneously releasing product molecules.
Since product release can take place only at a certain
stage inside the cycle, this means that the cycles of en-
zymes are locally synchronized. Not only regular wave
structures, such as rotating spiral waves or target pat-
terns (Fig. 2a), but also complex regimes of wave turbu-
lence (Fig. 2b) have been observed.
To understand and interpret stochastic simulation re-
sults, an analytical study of the system in the mean-field
approximation, which holds in the limit of high enzyme
concentrations, has been performed. In this approxima-
tion, the system is characterised by three continuous vari-
ables n0(r, t), n1(r, t) and m(r, t) which represent local
concentrations of enzymes in the rest state without or
with regulatory molecules attached (n0 and n1) and local
concentration of the product (m). For simplicity, internal
fluctuations in enzymes are neglected (σ = 0). Thus, all
enzymes which have started their cycles at some time t
would release their products at a definite time t+τp (with
τp = φp/v) and finish their cycles, returning to the rest
state, at time t + τ . Therefore, the system is described
by a set of three reaction-diffusion equations with time
delays,
= βmn0 − κn1 − α1n1 (1a)
= −βmn0 + κn1 − α0n0 + α0n0(t− τ)
+α1n1(t− τ) (1b)
= −βmn0 + κn1 + α1n1 − γm+ α0n0(t− τp)
+α1n1(t− τp) +D∇
2m. (1c)
The system always has a uniform stationary state
with certain concentrations n0, n1 and m, which can be
found as solutions of the respective algebraic equations.
This state corresponds to the absence of synchronization.
However, it may become unstable if allosteric activation
is strong enough. To analyze stability, small perturba-
tions δn0, δn1 and δm are added to the stationary state,
equations (1) are linearized and their solutions are sought
as δn0 ∼ δn1 ∼ δm ∼ exp (λqt− iqx) with λq = µq+iωq.
Thus, each spatial mode with wavevector q is character-
ized by its frequency ωq and its rate of growth µq. The
properties µq and ωq are given by the roots of a charac-
teristic equation which is determined by the linearization
matrix of equations (1). The steady state becomes unsta-
ble when at least one spatial mode with some wavenum-
ber q0 starts to grow (µq0 > 0).
As the bifurcation parameter, coefficient β can be cho-
sen. If regulatory molecules cannot bind to enzymes
(β = 0), feedback is absent and instabilities are not pos-
sible. On the other hand, allosteric activation becomes
strong if regulatory molecules can easily bind and, in this
case, emergence of oscillations and wave patterns can be
expected. Our bifurcation analysis reveals that, depend-
ing on the parameters of the system, it can exhibit ei-
ther a Hopf or a wave bifurcation [16]. As a result of
the Hopf bifurcation, uniform oscillations with q = 0 de-
velop. Because of the presence of delays in equations (1),
the characteristic equation is nonpolynomial in terms of
λ and, generally, a number of oscillatory solutions with
different frequencies ω are possible. Physically, such so-
lutions correspond to formation of several synchronous
enzymic groups. This effect has been previously exten-
sively investigated for similar systems in small spatial
volumes with full diffusional mixing [11] and we shall
not further discuss it here. The most robust uniform
oscillations, which we consider, are characterized by the
frequency ω ≈ 2π/τ and correspond to the single-group
synchronization. As the result of a wave bifurcation (also
known as the Hopf bifurcation with a finite wave number
[17]), the first unstable modes are traveling waves with
a certain wavenumber q0. Figure 3 shows the bifurca-
tion diagram in the parameter plane (τp, β). Note the
presence of a codimension-2 bifurcation point where the
boundaries of the Hopf and the wave bifurcations join.
To investigate nonlinear dynamics of the system, nu-
merical simulations of equations (1) have been performed
0 0.1 0.2 0.3 0.4
oscillations
codimension−2
wave−Hopf bifurcation
uniform
ripples
pacemakers/waves
higher frequency/
mixed modes
waves
standing−
traveling
standing waves
FIG. 3: Phase diagram (α0 = 1, α1 = 1000, κ = 10, γ = 10,
c = 100, D = 1000). The Hopf bifurcation (solid line) and the
wave bifurcation (dash-dotted line) boundaries are displayed.
Gray lines show instability of the stationary state with respect
to development of uniform oscillations with two (dashed) and
three (dotted) groups in the well-mixed case. Lines separating
parameter domains with different kinds of patterns are hand-
drawn, based on numerical simulations.
[16]. The explicit Euler integration method has been
used; no-flux boundary conditions were applied. Results
of 1D simulations are summarized in Fig. 3 and examples
of typical observed patterns are shown in Fig. 4. Stand-
ing waves (Fig. 4a) develop when the boundary of the
wave bifurcation (dash-dotted curve) is crossed and uni-
form oscillations are observed above the boundary of the
Hopf bifurcation. Near the codimension-2 point, more
complex behavior was found. This included rippled os-
cillations (Fig. 4b), self-organized pacemakers (Fig. 4c)
and modulated traveling waves (Fig. 4d). The observed
patterns are similar to those previously found in reaction-
diffusion systems with the wave bifurcation [18]. In the
right upper corner of the diagram in Fig. 3, higher fre-
quency oscillations with several synchronous groups take
place.
Two-dimensional simulations of reaction-diffusion
equations (1) with time delay have been performed for
selected parameter values. In 2D simulations, sponta-
neously developing concentric waves (target patterns)
and spiral waves have been observed; target patterns
were however unstable and evolved into pairs of rotat-
ing spiral waves (Fig. 2c and Video 3 [15]). Complex
wave regimes, which can be qualitatively characterized
as turbulence of standing waves, have also been observed
(Fig. 2d and Video 4 [15]).
The mean-field approximation is based on neglect-
ing statistical fluctuations in concentrations of reacting
species [11] and, therefore, it should hold in the high
concentration limit. In Fig. 4, two upper panel rows
display spatiotemporal patterns which are observed in
FIG. 4: Spatiotemporal patterns in a 1D system (in each
panel, the vertical axis is time, running down, and the hor-
izontal axis is the coordinate). The upper two rows are
stochastic simulations (σ = 0) with concentrations c = 1 and
c = 10, the bottom row shows mean-field simulations with
c = 100. (a) τp = 0.3, β = 95/c, (b) τp = 0.14, β = 260/c, (c)
τp = 0.22, β = 600/c, and (d) τp = 0.16, β = 300/c. Other
parameters as in Fig. 3; the system size shown is L = 51
ldiff .
stochastic simulations with parameter values correspond-
ing to the respective mean-field simulations. To compare
mean-field simulations with different enzyme densities,
the following property of equations (1) can be used: in-
troducing relative concentrations ñ0 = n0/c, ñ1 = n1/c
and m̃ = m/c, it can be noticed that they obey the same
equations, but with a rescaled coefficient β̃ = βc. Thus,
essentially the same patterns are observed as long as
the parameter combination βc remains constant. In the
stochastic simulations in Fig. 4, the coefficient β has been
increased to compensate for a decrease in the enzyme
concentration. For larger enzyme concentrations, good
agreement between mean-field predictions and stochas-
tic simulations has been found. In the mean-field equa-
tions (1), intramolecular fluctuations are not taken into
account (σ = 0 and therefore each turnover cycle has
the same fixed duration τ). Stochastic simulations have
been, however, also performed when such fluctuations
were present. Synchronization waves could still be found
even at internal noise levels which corresponded to the
mean relative dispersion ξ of turnover times of about 10%
(with ξ =
/τ ≃ (2στ)
Although the emphasis in this Letter is on the phenom-
ena in two-dimensional enzymic arrays, analogous effects
should be expected for three-dimensional systems repre-
senting aqueous enzymic solutions. The linear stability
analysis, yielding Hopf and wave bifurcation boundaries
(see Fig. 3), is valid also for the 3D geometry. We have
performed preliminary stochastic simulations for thin so-
lution layers with high enzyme concentrations and could
observe synchronization patterns similar to those found
for the enzymic arrays.
A product molecule, released by an enzyme, diffuses
in the solution until it either binds, as a regulatory
molecule, to another enzyme or undergoes a decay. Here,
it should be taken into account that a regulatory molecule
can bind to an allosteric enzyme only at a certain bind-
ing site of characteristic radius R. Using the theory of
diffusion-controlled reactions, the average time ttransit
after which a regulatory product would find a binding
site of one of the enzymes can be roughly estimated
[11] as ttransit = 1/cDR, if enzymes are uniformly dis-
tributed inside the reaction volume with concentration
c. Therefore, binding typically occurs within the dis-
tance Lcorr = (Dttransit)
= (cR)
from the point
where a molecule is released. Obviously, it can only
take place if the product molecule has not undergone
decay until that moment, i.e. if γttransit < 1. This
condition puts a restriction on the enzyme concentration
c, which must be higher than the critical concentration
c∗ = γ/DR. Choosing γ = 103 s−1, D = 10−5 cm2s−1
and R = 10−7 cm, the critical enzyme concentration is
c∗ = 1015 cm−3 = 10−6 M. A similar estimate can be
obtained when enzymes are immobilized on a plane im-
mersed into a reactive solution; in this case the mean dis-
tance between the enzymes on the plane should be less
than lc = (Rldiff)
[22]. Although the required en-
zyme concentrations are relatively large, they are within
the range characteristic for biological cells (glycolytic en-
zymes are present [19] in a cell at even higher concentra-
tion of more than 10−5 M). The characteristic temporal
period of developing patterns is determined by the en-
zyme turnover time τ , which typically varies from mil-
liseconds to seconds. The characteristic length scale of
developing wave patterns is determined by the diffusion
length ldiff , which can vary under these conditions from
a fraction of a micrometer to tens of micrometers.
Our analysis shows that spontaneous molecular syn-
chronization of allosteric product-activated enzymes can
be observed in enzymic arrays. Artificial arrays formed
by immobilized protein machines (molecular motors) are
already used in experiments on active nanoscale trans-
port (see [20]). Many enzymes in biological cells are
membrane-bound, thus forming natural enzymic arrays.
Similar phenomena are possible in dense enzyme solu-
tions. In the study by Petty et al. [21], traveling waves
of NAD(P)H and proton concentrations with the wave-
length of about a micrometer were observed inside neu-
trophil cells. These metabolic waves had the temporal
period of about 300 ms, which is by two orders of magni-
tude shorter than the characteristic period of glycolytic
oscillations in the cells and lies closer to the time scales
of turnover cycles of individual enzymes. An intriguing
question, requiring further detailed analysis, is whether
molecular synchronization waves may have already been
seen in these experiments.
Molecular synchronization waves are principally dif-
ferent from classical concentration waves in reaction-
diffusion systems. Under synchronization conditions,
internal conformational states of individual enzyme
molecules in their turnover cycles become strongly cor-
related. In optics, a similar situation is found when a
transition to coherent laser generation has taken place.
Our theoretical analysis may open a way to the investiga-
tions of a new class of spatio-temporal pattern formation
in chemically active molecular systems.
The authors are grateful to M. Falcke and P. Stange for
valuable discussions. Financial support of Japan Society
for the Promotion of Science through a fellowship for
research abroad (Y. T.) is acknowledged.
∗ Present address: Nanobiology Laboratories, Graduate
School of Frontier Biosciences, Osaka University, 1-3 Ya-
madaoka, Suita, Osaka 565-0871, Japan; Electronic ad-
dress: [email protected]
† Electronic address: [email protected]
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149 (1969).
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[22] Diffusion perpendicular to the plane is considered as di-
lution within a layer of effective thickness ≃ ldiff .
|
0704.0022 | Stochastic Lie group integrators | STOCHASTIC LIE GROUP INTEGRATORS
SIMON J.A. MALHAM∗ AND ANKE WIESE∗
Abstract. We present Lie group integrators for nonlinear stochastic differential equations with
non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given
a Lie group action that generates transport along the manifold, we pull back the stochastic flow on
the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding
Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the
Lie algebra via closed operations and then push back to the Lie group and then to the manifold,
thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas
methods after their deterministic counterparts. We also present stochastic Lie group integration
schemes based on Castell–Gaines methods. These involve using an underlying ordinary differential
integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They
become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying
ordinary differential integrator. Further, we show that some Castell–Gaines methods are uniformly
more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods
by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater
vehicle both perturbed by two independent multiplicative stochastic noise processes.
Key words. stochastic Lie group integrators, stochastic differential equations on manifolds
AMS subject classifications. 60H10, 60H35, 93E20
1. Introduction. We are interested in designing Lie group numerical schemes
for the strong approximation of nonlinear Stratonovich stochastic differential equa-
tions of the form
yt = y0 +
Vi(yτ , τ) dW
τ . (1.1)
HereW 1, . . . ,W d are d independent scalar Wiener processes andW 0t ≡ t. We suppose
that the solution y evolves on a smooth n-dimensional submanifold M of RN with
n ≤ N and Vi : M × R+ → TM, i = 0, 1, . . . , d, are smooth vector fields which in
local coordinates are Vi =
j=1 V
i ∂yj . The flow-map ϕt : M → M of the integral
equation (1.1) is defined as the map taking the initial data y0 to the solution yt at
time t, i.e. yt = ϕt ◦ y0.
Our goal in this paper is to show how the Lie group integration methods developed
by Munthe-Kaas and co-authors can be extended to stochastic differential equations
on smooth manifolds (see Crouch and Grossman [8] and Munthe-Kaas [40]). Suppose
we know that the exact solution of a given system of stochastic differential equations
evolves on a smooth manifold M (see Malliavin [36] or Emery [14]), but we can only
find the solution pathwise numerically. How can we ensure that our approximate
numerical solution also lies in the manifold?
Suppose we are given a finite dimensional Lie group G and Lie group action Λy0
that generates transport across the manifold M from the starting point y0 ∈ M via
elements of G. Then with any given elements ξ in the Lie algebra g corresponding to
the Lie group G, we can associate the infinitesimal action λξ using the Lie group action
Λy0 . The map ξ 7→ λξ is a Lie algebra homomorphism from g to X(M), the Lie algebra
∗Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, UK. ([email protected], [email protected]).
(16/10/2007)
http://arxiv.org/abs/0704.0022v2
2 Malham and Wiese
of vector fields over the manifold M. Further the Lie subalgebra {λξ ∈ X(M) : ξ ∈ g}
is isomorphic to a finite dimensional Lie algebra with the same structure constants
(see Olver [42], p. 56).
Conversely, suppose we know that the Lie algebra generated by the set of govern-
ing vector fields Vi, i = 0, 1, . . . , d, on M is finite dimensional, call this XF (M). Then
we know there exists a finite dimensional Lie group G whose Lie algebra g has the
same structure constants as XF (M) relative to some basis, and there is a Lie group
action Λy0 such that Vi = λξi , i = 0, 1, . . . , d, for some ξi ∈ g (see Olver [42], p. 56 or
Kunita [30], p. 194). The choice of group and action is not unique.
In this paper we assume that there is a finite dimensional Lie group G and action
Λy0 such that our set of governing vector fields Vi, i = 0, 1, . . . , d, are each infinitesimal
Lie group actions generated by some element in g via Λy0 , i.e. Vi = λξi for some ξi ∈ g,
i = 0, 1, . . . , d. They are said to be fundamental vector fields. This means that we can
write down the set of governing vector fields Xξi for a system of stochastic differential
equations on the Lie group G that, via the Lie group action Λy0, generates the flow
governed by the set of vector fields Vi on the manifold. The vector fields Vi on M are
simply the push forward of the vector fields Xξi on G via the Lie group action Λy0 .
Typically the flow on the Lie group also needs to be computed numerically. We thus
want the approximation to remain in the Lie group so that the Lie group action takes
us back to the manifold.
To achieve this, we pull back the set of governing vector fields Xξi on G to the set
of governing vector fields vξi on g, via the exponential map ‘exp’ from g to G. Thus
the stochastic flow generated on g by the vector fields vξi generates the stochastic flow
on G generated by the Xξi . The set of governing vector fields on g are for each σ ∈ g:
vξi ◦ σ ≡
(adσ)
k ◦ ξi , (1.2)
where Bk is the kth Bernoulli number and the adjoint operator adσ is a closed operator
on g, in fact adσ ◦ ζ = [σ, ζ], the Lie bracket on g. Now the essential point is that
ξi ∈ g and so the series on the right or any truncation of it is closed in g. Hence
if we construct an approximation to our stochastic differential equation on g using
the vector fields vξi or an approximation of them achieved by truncating the series
representation, then that approximation must reside in the Lie algebra g. We can then
push the approximation in the Lie algebra forward onto the Lie group and then onto
the manifold. Provided we compute the exponential map and action appropriately, our
approximate solution lies in the manifold (to within machine accuracy). In summary,
for a given ξ ∈ g and any y0 ∈ M we have the following commutative diagram:
∗−−−−→ X(G)
(Λy0 )∗−−−−→ X(M)
exp−−−−→ G
Λy0−−−−→ M
We have implicitly separated the governing set of vector fields Vi, i = 0, 1, . . . , d,
from the driving path process w ≡ (W 1, . . . ,W d). Together they generate the unique
solution process y ∈ M to the stochastic differential equation (1.1). When there
is only one driving Wiener process (d = 1) the Itô map w 7→ y is continuous in
the topology of uniform convergence. When there are two or more driving processes
Stochastic Lie group integrators 3
(d ≥ 2) the Universal Limit Theorem tells us that the Itô map w 7→ y is continuous
in the p-variation topology, in particular for 2 ≤ p < 3 (see Lyons [32], Lyons and
Qian [33] and Malliavin [36]). A Wiener path with d ≥ 2 has finite p-variation for
p > 2. This means that from a pathwise perspective, approximations to y constructed
using successively refined approximations to w are only guaranteed to converge to
the correct solution y, if we include information about the Lévy chordal areas of
the driving path process. Note however that the L2-norm of the 2-variation of a
Wiener process is finite. In the Lie group integration procedure prescribed above we
must solve a stochastic differential system on the Lie algebra g defined by the set
of governing vector fields vξi and the driving path process w ≡ (W 1, . . . ,W d). In
light of the Universal Limit Theorem and with stepsize adaptivity in mind in future
(see Gaines and Lyons [20]), we for instance use in our examples order 1 stochastic
numerical methods—that include the Lévy chordal area—to solve for the flow on the
Lie algebra g.
We have thus explained the idea behind Munthe-Kaas methods and how they can
be generalized to the stochastic setting. The first half of this paper formalizes this
procedure.
In the second half of this paper, we consider autonomous vector fields and con-
struct stochastic Lie group integration schemes using Castell–Gaines methods. This
approach proceeds as follows. We truncate the stochastic exponential Lie series expan-
sion corresponding to the flow ϕt of the solution process y to the stochastic differential
equation (1.1). We then approximate the driving path process w ≡ (W 1, . . . ,W d) by
replacing it by a suitable nearby piecewise smooth path in the appropriate variation
topology. An approximation to the solution yt requires the exponentiation of the
approximate truncated exponential Lie series. This can be achieved by solving the
system of ordinary differential equations driven by the vector field that is the approx-
imate truncated exponential Lie series. If we use ordinary Munthe-Kaas methods as
the underlying ordinary differential integrator the Castell–Gaines method becomes a
stochastic Lie group integrator.
Further, based on the Castell–Gaines approach we then present uniformly accurate
exponential Lie series integrators that are globally more accurate than their stochastic
Taylor counterpart schemes (these are investigated in detail in Lord, Malham and
Wiese [31] for linear stochastic differential equations). They require the assumption
that a sufficiently accurate underlying ordinary differential integrator is used; that
integrator could for example be an ordinary Lie group Munthe-Kaas method. In
the case of two driving Wiener processes we derive the order 1/2, and in the case
of one driving Wiener process the order 1 uniformly accurate exponential Lie series
integrators. As a consequence we confirm the asymptotic efficiency properties for
both these schemes proved by Castell and Gaines [8] (see Newton [41] for more details
on the concept of asymptotic efficiency). We also present in the case of one driving
Wiener process a new order 3/2 uniformly accurate exponential Lie series integrator
(also see Lord, Malham and Wiese [31]).
We present two physical applications that demonstrate the advantage of using
stochastic Munthe-Kaas methods. First we consider a free rigid body which for ex-
ample could model the dynamics of a satellite. We suppose that it is perturbed by
two independent multiplicative stochastic noise processes. The governing vector fields
are non-commutative and the corresponding exact stochastic flow evolves on the unit
sphere. We show that the stochastic Munthe-Kaas method, with an order 1 stochastic
Taylor integrator used to progress along the corresponding Lie algebra, preserves the
4 Malham and Wiese
approximate solution in the unit sphere manifold to within machine error. However
when an order 1 stochastic Taylor integrator is used directly, the solution leaves the
unit sphere. The contrast between these two methods is more emphatically demon-
strated in our second application. Here we consider an autonomous underwater vehicle
that is also perturbed by two independent multiplicative stochastic noise processes.
The exact stochastic flow evolves on the manifold which is the dual of the Euclidean
Lie algebra se(3); two independent Casimirs are conserved by the exact flow. Again
the stochastic Munthe-Kass method preserves the Casimirs to within machine error.
However the order 1 stochastic Taylor integrator is not only unstable for large step-
sizes, but the approximation drifts off the manifold and makes a dramatic excursion
off to infinity in the embedding space R6.
Preserving the approximate flow on the manifold of the exact dynamics may be a
required property for physical or financial systems driven by smooth or rough paths—
for general references see Iserles, Munthe-Kaas, Nørsett and Zanna [25], Hairer, Lubich
and Wanner [22], Elworthy [13], Lyons and Qian [33] and Milstein and Tretyakov [38].
Stochastic Lie group integrators in the form of Magnus integrators for linear stochastic
differential equations were investigated by Burrage and Burrage [5]. They were also
used in the guise of Möbius schemes (see Schiff and Shnider [43]) to solve stochastic
Riccati equations by Lord, Malham and Wiese [31] where they outperformed direct
stochastic Taylor methods. Further applications where they might be applied include:
backward stochastic Riccati equations arising in optimal stochastic linear-quadratic
control (Kohlmann and Tang [28]); jump diffusion processes on matrix Lie groups
for Bayesian inference (Srivastava, Miller and Grenander [44]); fractional Brownian
motions on Lie groups (Baudoin and Coutin [3]) and stochastic dynamics triggered
by DNA damage (Chickarmane, Ray, Sauro and Nadim [10]).
Our paper is outlined as follows. In Section 2 we present the basic geometric setup,
sans stochasticity. In particular we present a generalized right translation vector field
on a Lie group that forms the basis of our subsequent transformation from the Lie
group to the manifold. Using a Lie group action, this vector field pushes forward
to an infinitesimal Lie group action vector field that generates a flow on the smooth
manifold. In Section 3 we specialize to the case of a matrix Lie group and using
the exponential map, derive the pullback of the generalized right translation vector
field on the Lie group to the corresponding vector field on the Lie algebra. To help
give some context to our overall scheme, we provide in Section 4 illustrative examples
of manifolds and natural choices for associated Lie groups and actions that generate
flows on those manifolds. Then in Section 5 we show how a flow on a smooth manifold
corresponding to a stochastic differential equation can be generated by a stochastic
flow on a Lie algebra via a Lie algebra action. We explicitly present stochastic Munthe-
Kaas Lie group integration methods in Section 6. We start the second half of our
paper by reviewing the exponential Lie series for stochastic differential equations in
Section 7. We show in Section 8 how to construct geometric stochastic Castell–Gaines
numerical methods. In particular we also present uniformly accurate exponential Lie
series numerical schemes that not only can be used as geometric stochastic integrators,
but also are always more accurate than stochastic Taylor numerical schemes of the
corresponding order. In Section 9 we present our concrete numerical examples. Finally
in Section 10 we conclude and present some further future applications and directions.
2. Lie group actions. SupposeM is a smooth finite n-dimensional submanifold
of RN with n ≤ N . We use X(M) to denote the Lie algebra of vector fields on the
manifold M, equipped with the Lie–Jacobi bracket [U, V ] ≡ U · ∇V − V · ∇U , for all
Stochastic Lie group integrators 5
U, V ∈ X(M). Let G denote a finite dimensional Lie group.
Definition 2.1 (Lie group action). A left Lie group action of a Lie group G on a
manifold M is a smooth map Λ: G ×M → M satisfying for all y ∈ M and R,S ∈ G:
(1) Λ(id, y) = y; (2) Λ(R,Λ(S, y)) = Λ(RS, y). We denote Λy ◦ S ≡ Λ(S, y).
Hereafter we suppose y0 ∈ M is fixed and focus on the action map Λy0 : G→M.
We assume that the Lie group action Λ is transitive, i.e. transport across the manifold
from any point y0 ∈ M to any other point y ∈ M can always be achieved via a group
element S ∈ G with y = Λy0 ◦ S (Marsden and Ratiu [37], p. 310).
We define the Lie algebra g associated with the Lie group G to be the vector space
of all right invariant vector fields on G. By standard construction this is isomorphic
to the tangent space to G at the identity id ≡ idG (see Olver [42], p. 48 or Marsden
and Ratiu [37], p. 269).
Definition 2.2 (Generalized right translation vector field). Suppose we are
given a smooth map ξ : M→g. With each such map ξ we associate a vector field
Xξ : G → X(G) defined as follows
Xξ ◦ S ≡ ∂τ exp
τ ξ(Λy0 ◦ S)
for S ∈ G, where ‘exp’ is the usual local diffeomorphism exp: g → G from a neigh-
bourhood of the zero element o ∈ g to a neighbourhood of id ∈ G.
Definition 2.3 (Infinitesimal Lie group action). We associate with each vector
field Xξ : G→X(G) a vector field λξ : M→X(M) as the push forward of Xξ from G to
M by Λy0, i.e. λξ ≡
Xξ, so that if S ∈ G and y = Λy0 ◦ S ∈ M, then
λξ ◦ y ≡ ∂τΛy0 ◦ γ(τ)|τ=0 ,
where γ(t) ∈ G, γ(0) = S and ∂τγ(τ) = Xξ ◦ γ(τ) (the flow generated on G by the
vector field Xξ starting at S ∈ G). Naturally, as a vector field λξ is linear, and also
λξ ◦ y ≡ LXξ ◦ Λy0 ◦ S ,
the Lie derivative of Λy0 along Xξ at S ∈ G.
Remarks.
1. The map Λ(S) : M→M defined by y 7→ Λ(S) ◦ y ≡ Λy ◦ S represents a
flow on M. Hence if y = Λ(S) ◦ y0, the push forward of λξ by Λ(S) is given by
λξ ≡ λAdSξ (Marsden and Ratiu [37], p. 317).
2. We define the isotropy subgroup at y0 ∈ M by Gy0 ≡ {S ∈ G : Λy0◦S = y0}; it
is a closed subgroup of G (see Helgason [23], p. 121 or Warner [48], p. 123). We define
the global isotropy subgroup by GM ≡ ∩y0∈MGy0 ≡ {S ∈ G : Λy0 ◦ S = y0, ∀y0 ∈ M};
it is a normal subgroup of G (see Olver [42], p. 38).
3. A Lie group action is said to be is effective/faithful if the map S 7→ Λ(S) from
G to Diff(M), the group of diffeomorphisms on M, is one-to-one. This is equivalent
to the condition that different group elements have different actions, i.e. GM ≡ {idG}.
A Lie group action is said to be free if Gy0 = {idG} for all y0 ∈ M, i.e. Λy0 is a
diffeomorphism from G to M. For more details see Marsden and Ratiu [37], p. 310
and Olver [42], p. 38.
4. The map γ : G/Gy0→M defined by γ : S · Gy0 7→ Λy0 ◦ S is a diffeomorphism,
i.e. M ∼= G/Gy0 for any y0 ∈ M (a manifold M with a Lie group action Λ: G×M→M
defined over it is thus diffeomorphic to a homogeneous manifold ; see Warner [48],
p. 123 or Olver [42], p. 40). Further, the induced action of G/GM on M is effective.
Hence if Λ is not an effective action of G, we can replace it (without loss of generality)
by the induced action of G/GM (see Olver [42], p. 38).
6 Malham and Wiese
5. Our definition for the generalized right translation vector field Xξ on G is
motivated by the standard right translation vector field used to identify g, the vector
space of right invariant vector fields on G, with TidG, the tangent space to G at the
identity. When ξ ∈ g is constant, Xξ ∈ X(G) is right invariant and a Lie bracket
on TidG can be defined via right extension by the corresponding Lie–Jacobi bracket
for the vector fields Xξ on X(G). Unless ξ ∈ g is constant, Xξ is not in general
right invariant. For further details see Varadarajan [47], Olver [42], or Marsden and
Ratiu [37].
6. The infinitesimal generator map ξ 7→ λξ from g to X(M) is a Lie algebra
homomorphism. If we identify g as the vector space of left invariant vector fields on G
this map becomes an anti-homomorphism. The Lie–Jacobi bracket as defined above
gives the right (rather than left) Lie algebra stucture over the group of diffeomorphisms
on M. If in addition we take the Lie–Jacobi bracket to be minus that defined above—
associated with the left Lie algebra structure—then the infinitesimal generator map
becomes a homomorphism again. See for example Marsden and Ratiu [37], p. 324 or
Munthe-Kaas [40].
7. The image of g under the infinitesimal generator map ξ 7→ λξ forms a finite
dimensional Lie algebra of vector fields on M which is isomorphic to the Lie algebra of
the effectively acting quotient group G/GM (see Olver [42], p. 56). Thus the tangent
space to M at any point is g and M inherents a connection from G/GM. Connections
are necessary to define martingales on manifolds, but not for defining semimartingales
(our focus here); see Malliavin [36] and Emery [14].
8. A comprehensive study of the systematic construction of symmetry Lie groups
from given vector fields can be found in Olver [42].
9. We assumed above that the vector fieldsXξ and λξ are autonomous. However
all results in this and subsequent sections up to Section 7 can be straightforwardly
extended to non-autonomous vector fields generated by ξ : M × R→g with (y, t) 7→
ξ(y, t) for all y ∈ M and t ∈ R.
10. For full generality we want to suspend reference to embedding spaces as far as
possible. However in subsequent sections to be concise we will more explicitly reclaim
this context.
3. Pull back to the Lie algebra. For ease of presentation, we will assume in
this section that G is a matrix Lie group. Recall that the exponential map exp: g → G
is a local diffeomorphism from a neighbourhood of o ∈ g to a neighbourhood of id ∈ G.
Let vξ : g→g be the pull back of the vector field Xξ : G→X(G) from G to g via the
exponential mapping exp: g→G, i.e. vξ ◦ σ ≡ exp∗Xξ ◦ σ. If σ ∈ g then
vξ ◦ σ = dexp−1σ ◦ ξ
Λy0 ◦ expσ
. (3.1)
Here dexp−1σ : g→g is the inverse of the right-trivialized tangent map of the exponential
dexpσ : g→g defined as follows. If β(τ) is a curve in g such that β(0) = σ and
β′(0) = η ∈ g then dexp: g× g→g is the local smooth map (Varadarajan [47], p. 108)
dexpσ ◦ η ≡ ∂τ expβ(τ)|τ=0 exp(−σ)
exp(adσ)− id
◦ η .
Note that as a tangent map dexpσ : g→g is linear. The inverse operator dexp
σ is the
operator series (1.2) generated by considering the reciprocal of dexpσ.
Stochastic Lie group integrators 7
To show that (3.1) is true, if exp: g→G with σ 7→ S = expσ, and β(τ) ∈ g with
β(0) = σ and ∂τβ(τ) = vξ ◦ β(τ), then:
exp∗ vξ ◦ S = ∂τ expβ(τ)|τ=0
dexpσ ◦ vξ ◦ σ
exp(σ)
≡ Xξ ◦ S .
Since ‘exp’ is a diffeomorphism in a neighbourhood of o ∈ g, this push forward calcu-
lation establishes the pull back (3.1) for all σ ∈ g in that neighbourhood.
4. Illustrative examples. Suppose the vector field V : M× R→X(M) gener-
ates a flow solution yt ∈ M starting from y0 ∈ M. Then assume there exists a:
1. Lie group G with corresponding Lie algebra g;
2. Lie group action Λy0 : G→M for which a starting point y0 ∈ M is fixed;
3. Vector field λξ : M× R→X(M) such that: V ≡ λξ, i.e. V is a fundamental
vector field corresponding to the action Λy0 .
Let us suppose G is a matrix Lie group (or can be embedded into a matrix Lie
group, for example the Euclidean group SE(3) is naturally embedded into the special
linear group SL(4;R)). We have for all S ∈ G and t ∈ R,
Xξ(S, t) ≡ ξ
Λy0(S), t
S . (4.1)
If V = λξ for some ξ : M→g, some Lie group G and corresponding action Λy0 , then
the flow generated by Xξ on G drives the flow generated by V on M. In each of the
examples below, given the manifold M, we present a natural Lie group and action
associated with the manifold structure, and identify vector fields which generate flows
on the manifold via the Lie group.
Stiefel manifold Vn,k. Suppose M = Vn,k ≡ {y ∈ Rn×k : yTy = I}. Take
G = SO(n), the special orthogonal group, and Λy0(S) ≡ Sy0, the action of left
multiplication. The corresponding Lie algebra g = so(n). Then by direct calculation
λξ(y) = ξ(y, t) y. Hence if the given vector field V (y, t) = ξ(y, t) y, then the push
forward of the flow generated by Xξ(S, t) on G in (4.1) is the flow generated by V on
M. Note that the unit sphere S2 ∼= V3,1, i.e. S2 is just a particular Stiefel manifold.
In Section 9 as an application, we consider rigid body dynamics evolving on S2.
Isospectral manifold Sn. Suppose M = Sn = {y ∈ Rn×n : yT = y}, the set of
n× n real symmetric matrices. Take G = O(n), the orthogonal group and Λy0(S) ≡
T, which is an isospectral action (Munthe-Kaas [40]). The corresponding Lie
algebra is g = so(n). Again, by direct calculation λξ(y) = ξ(y, t) y − y ξ(y, t). Hence
if the given vector field V (y, t) = ξ(y, t) y−y ξ(y, t), then the push forward of the flow
generated by Xξ(S, t) on G in (4.1) is the flow generated by V on M.
Dual of the Euclidean algebra se(3)∗. Suppose M = se(3)∗ ∼= R3, the dual
of the Euclidean algebra se(3) of the Euclidean group SE(3) =
(s, ρ) ∈ SE(3) : s ∈
SO(3), ρ ∈ R3
. Take G = SE(3) so g = se(3) and Λ ≡ Ad∗ : G × g∗→g∗, the
coadjoint action of G on g∗. Then by direct calculation λξ(y) = −ad∗ξ(y). Since
λξ(y) in linear in ξ and −λξ(y) ≡ λ−ξ(y), it follows that if V (y) = ad∗ξ(y), then
the push forward of the flow generated by X−ξ(S, t) = −ξ
Λy0(S), t
S on G is the
flow generated by V on M. For more details see Section 9 where we investigate the
dynamics of an autonomous underwater vehicle evolving on se(3)∗.
8 Malham and Wiese
Grassmannian manifold Gr(k, n). The Grassmannian manifold M = Gr(k, n)
is the space of k-dimensional subspaces of Rn. Take G = GL(n), the general linear
matrix group, where if S ∈ GL(n), we identify
where the block matrices α, β, γ and δ are sizes k × k, k × (n − k), (n− k) × k and
(n − k) × (n − k), respectively (see Schiff and Shnider [43]; Munthe-Kaas [40]). We
choose the action of GL(n) on Gr(k, n) to be the generalized Möbius transformation
Λy0(S) = (αy0 + β)(γy0 + δ)
−1. Hence if
ξ(t) =
a(t) b(t)
c(t) d(t)
then direct calculation reveals that λξ(y) = a(t)y+ b(t)− yc(t)y− yd(t). Hence if the
given vector field V (y) = a(t)y + b(t) − yc(t)y − yd(t), then the push forward of the
flow generated by Xξ(S, t) = ξ(t)S on G is the flow generated by V on Gr(k, n).
5. Stochastic Lie group integration. We show that if a Lie group action
Λ: G ×M→M exists, then for y0 ∈ M fixed, the Lie algebra action Λy0 ◦ exp: g→M
carries a flow on g to a flow on M.
Theorem 5.1. Suppose there exists a Lie group action Λ: G ×M→M. Then if
there exists a process σ ∈ g and a stopping time T∗ such that on [0, T∗), σ satisfies
the Stratonovich stochastic differential equation
vξi ◦ στ dW iτ , (5.1)
then the process y = Λy0 ◦ expσ ∈ M satisfies the Stratonovich stochastic differential
equation on [0, T∗):
yt = y0 +
λξi ◦ yτ dW iτ . (5.2)
Proof. Using Itô’s lemma, if σt ∈ g satisfies (5.1) then Λy0 ◦ expσt satisfies
Λy0 ◦ expσt = Λy0 ◦ exp o+
Lvξi ◦ Λy0 ◦ expστ dW
Now recall that for each i = 0, 1, . . . , d, Xξi is the push forward of vξi from g to G via
the exponential map, and that λξi is the push forward of Xξi from G to M via Λy0
and so the Lie derivative
Lvξi ◦ Λy0 ◦ expσt ≡ λξi ◦ yt .
Then since yt = Λy0 ◦ expσt, we conclude that y ∈ M is a process satisfying the
stochastic differential equation (5.2).
Corollary 5.2. Suppose that for each i = 0, 1, . . . , d there exists ξi : M→g such
that the vector field Vi : M→X(M) and λξi : M→X(M) can be identified, i.e.
Vi ≡ λξi . (5.3)
Stochastic Lie group integrators 9
Then the push forward by ‘Λy0◦exp’ of the flow on the Lie algebra manifold g generated
by the stochastic differential equation (5.1) is the flow on the smooth manifold M
generated by the stochastic differential equation (5.2), whose solution can be expressed
in the form yt = Λy0 ◦ expσt.
Remark. If the action is free then ‘Λy0 ◦ exp’ is a diffeomorphism from a neigh-
bourhood of o ∈ g to a neighbourhood of y0 ∈ M.
6. Stochastic Munthe-Kaas methods. Assuming that the vector fields in our
original stochastic differential equation (1.1) are fundamental and satisfy (5.3), then
stochastic Munthe-Kaas methods are constructed as follows:
1. Subdivide the global interval of integration [0, T ] into subintervals [tn, tn+1].
2. Starting with t0 = 0, repeat the next two steps over successive intervals
[tn, tn+1] until tn+1 = T .
3. Compute an approximate solution σ̂tn,tn+1 to (5.1) across [tn, tn+1] using a
stochastic Taylor, stochastic Runge–Kutta or Castell–Gaines method.
4. Compute the approximate solution ytn+1 ≈ Λytn ◦ exp σ̂tn,tn+1 .
Note that by construction σ̂tn,tn+1 ∈ g because the stochastic differential equa-
tion (5.1) (or any stochastic Taylor or other sensible approximation) evolves the so-
lution locally on the Lie algebra g via the vector fields vξi : g→g. Suitable methods
for approximating the exponential map to ensure it maps g to G appropriately can be
found in Iserles and Zanna [26]. Then by construction ytn+1 ∈ M.
For example, with two Wiener processes and autonomous vector fields vξi ◦ σ, an
order 1 stochastic Taylor Munthe-Kaas method is based on
σ̂tn,tn+1 =
J0vξ0 +J1vξ1 +J2vξ2 +
+J12vξ1vξ2 +J21vξ2vξ1 +
◦o , (6.1)
evaluated at the zero element o ∈ g. Typically ‘dexp−1σ ’ is truncated to only include
the necessary low order terms to maintain the order of the numerical scheme.
Remark. It is natural to invoke Ado’s Theorem (see for example Olver [42] p. 54):
any finite dimensional Lie algebra is isomorphic to a Lie subalgebra of gl(n) (the
general linear algebra) for some n ∈ N. However as Munthe-Kaas [40] points out,
directly using a matrix representation for the given Lie group might not lead to the
optimal computational implementation (other data structures might do so).
7. Exponential Lie series. The stochastic Taylor series is known in different
contexts as the Neumann series, Peano–Baker series or Feynman–Dyson path ordered
exponential. If the vector fields in the stochastic differential equation (1.1) are au-
tonomous (which we assume henceforth), i.e. for all i = 0, 1, . . . , d, Vi = Vi(y) only,
then the stochastic Taylor series for the flow is
Jα1···αm(t)Vα1 · · ·Vαm .
Here Pm is the set of all combinations of multi-indices α = (α1, . . . , αm) of length m
with αi ∈ {0, 1, . . . , d} and
Jα1···αm(t) ≡
· · ·
∫ τm−1
dWα1τm · · · dW
are multiple Stratonovich integrals.
10 Malham and Wiese
The logarithm of ϕt is the exponential Lie series, Magnus expansion (Magnus [34])
or Chen–Strichartz formula (Chen [9], Strichartz [45]). In other words we can express
the flow map in the form ϕt = expψt, where
Ji(t)Vi +
j>i=0
(Jij − Jji)(t)[Vi, Vj ] + · · ·
is the exponential Lie series for our system, and [· , ·] is the Lie–Jacobi bracket on
X(M). See Yamato [49], Kunita [29], Ben Arous [1] and Castell [7] for the derivation
and convergence of the exponential Lie series expansion in the stochastic context;
Strichartz [45] for the full explicit expansion; Sussmann [46] for a related product
expansion and Lyons [32] for extensions to rough paths.
Let us denote the truncated exponential Lie series by
ψ̂t =
Jα cα , (7.1)
where Qm denotes the finite set of multi-indices α for which ‖Jα‖L2 is of order up to
and including tm, where m = 1/2, 1, 3/2, . . .. The terms cα are linear combinations
of finitely many (length α) products of the smooth vector fields Vi, i = 0, 1, . . . , d.
The following asymptotic convergence result can be established along the lines of the
proof for linear stochastic differential equations in Lord, Malham and Wiese [31]; we
provide a proof in Appendix A.
Theorem 7.1. Assume the vector fields Vi have 2m+1 uniformly bounded deriva-
tives, for all i = 0, 1, . . . , d. Then for t ≤ 1, the flow exp ψ̂t ◦ y0 is square-integrable,
where ψ̂t is the truncated Lie series (7.1). Further, if y is the solution of the stochastic
differential equation (1.1), there exists a constant C
m, ‖y0‖2
such that
∥yt − exp ψ̂t ◦ y0
m, ‖y0‖2
tm+1/2 . (7.2)
8. Geometric Castell–Gaines methods. Consider the truncated exponential
Lie series ψ̂tn,tn+1 across the interval [tn, tn+1]. We approximate higher order multiple
Stratonovich integrals across each time-step by their expectations conditioned on the
increments of the Wiener processes on suitable subdivisions (Gaines and Lyons [20]).
An approximation to the solution of the stochastic differential equation (1.1) across
the interval [tn, tn+1] is given by the flow generated by the truncated and conditioned
exponential Lie series ψ̂tn,tn+1 via
ytn+1 ≈ exp
ψ̂tn,tn+1
◦ ytn .
Hence the solution to the stochastic differential equation (1.1) can be approximately
computed by solving the ordinary differential system (see Castell and Gaines [8];
Misawa [39])
u′(τ) = ψ̂tn,tn+1 ◦ u(τ) (8.1)
across the interval τ ∈ [0, 1]. Then if u(0) = ytn we will get u(1) ≈ ytn+1. We
must choose a sufficiently accurate ordinary differential integrator to solve (8.1)—we
implicitly assume this henceforth.
Stochastic Lie group integrators 11
The set of governing vector fields Vi, i = 0, 1, . . . , d, prescribes a map from the
driving path process w ≡ (W 1, . . . ,W d) to the unique solution process y ∈ M to the
stochastic differential equation (1.1). The map w 7→ y is called the Itô map. Recall
that we assume the vector fields are smooth. When there is only one driving Wiener
process (d = 1) the Itô map is continuous in the topology of uniform convergence
(Theorem 1.1.1. in Lyons and Qian [33]). When there are two or more driving pro-
cesses (d ≥ 2) the Universal Limit Theorem (Theorem 6.2.2. in Lyons and Qian [33])
tells us that the Itô map is continuous in the p-variation topology, in particular for
2 ≤ p < 3. A Wiener path with d ≥ 2 has p-variation with p > 2, and the p-variation
metric in this case includes information about the Lévy chordal areas of the path
(Lyons [32]). Hence we must choose suitable piecewise smooth approximations to the
driving path process w. The following result follows from the corresponding result
for ordinary differential equations in Hairer, Lubich and Wanner [22] (p. 112) as well
as directly from Chapter VIII in Malliavin [36] on the Transfer Principle (see also
Emery [15]).
Lemma 8.1. A necessary and sufficient condition for the solution to the stochastic
differential equation (1.1) to evolve on a smooth n-dimensional submanifold M of
RN (n ≤ N) up to a stopping time T∗ is that Vi(y, t) ∈ TyM for all y ∈ M, i =
0, 1, . . . , d.
Hence the stochastic Taylor expansion for the flow ϕt is a diffeomorphism on M.
However a truncated version of the stochastic Taylor expansion for the flow ϕ̂t will not
in general keep you on the manifold, i.e. if y0 ∈ M then ϕ̂t ◦ y0 need not necessarily
lie in M. On the other hand, the exponential Lie series ψt, or any truncation ψ̂t of
it, lies in X(M). By Lemma 8.1 this is a necessary and sufficient condition for the
corresponding flow-map exp ψ̂t to be a diffeomorphism on M. Hence if u(0) = ytn ∈
M, then ytn+1 ≈ u(1) ∈ M. When solving the ordinary differential equation (8.1),
classical geometric integration methods, for example Lie group integrators such as
Runge–Kutta Munthe-Kaas methods, over the interval τ ∈ [0, 1] will numerically
ensure ytn+1 stays in M. Additionally, as the following result reveals, numerical
methods constructed using the Castell–Gaines Lie series approach can also be more
accurate (a proof is provided in Appendix B). We define the strong global error at
time T associated with an approximate solution ŷT as E ≡ ‖yT − ŷT ‖L2 .
Theorem 8.2. In the case of two independent Wiener processes and under the
assumptions of Theorem 7.1, for any initial condition y0 ∈ M and a sufficiently
small fixed stepsize h = tn+1 − tn, the order 1/2 Lie series integrator is globally more
accurate in L2 than the order 1/2 stochastic Taylor integrator. In addition, in the
case of one Wiener process, the order 1 and 3/2 uniformly accurate exponential Lie
series integrators generated by ψ̂
tn,tn+1
= J0V0 + J1V1 +
[V1, [V1, V0]]
(3/2)
tn,tn+1
= J0V0 + J1V1 +
(J01 − J10)[V0, V1] + h
[V1, [V1, V0]]
respectively, are globally more accurate in L2 than their corresponding stochastic Tay-
lor integrators. In other words, if E lsm denotes the global error of the exponential Lie
series integrators of order m above, and Estm is the global error of the stochastic Taylor
integrators of the corresponding order, then E lsm ≤ Estm for m = 1/2, 1, 3/2.
Remarks.
1. The result for ψ̂(3/2) is new. That the order-1/2 Lie series integrator (for two
Wiener processes) and the order 1 integrator generated by ψ̂(1) are uniformly more
accurate confirms the asymptotically efficient properties of these schemes proved by
12 Malham and Wiese
Castell and Gaines [8]. The proof follows along the lines of an analogous result for
linear stochastic systems considered in Lord, Malham and Wiese [31].
2. Consider the order 1/2 exponential Lie series with no vector field commu-
tations. Solving the ordinary differential equation (8.1) using an (ordinary) Euler
Munthe-Kaas method and approximating dexp
σ ≈ id is equivalent to the order 1/2
stochastic Taylor Munthe-Kaas method (for the same Lie group and action).
9. Numerical examples.
9.1. Rigid body. We consider the dynamics of a rigid body such as a satellite
(see Marsden and Ratiu [37]). We will suppose that the rigid body is perturbed by two
independent multiplicative stochastic processes W 1 and W 2 with the corresponding
vector fields Vi(y) ≡ ξi(y) y, for i = 0, 1, 2, with ξi ∈ so(3). If we normalize the initial
data y0 so that |y0| = 1 then the dynamics evolves on M = S2. We naturally suppose
G = SO(3), and Λy0(S) ≡ Sy0 so that λξi(y) = ξi(y) y, and we can pull back the
flow generated by V on M to the flow on G generated by Xξi(S, t) = ξi
Λy0(S)
i = 0, 1, 2. We use the following matrix representation for the ξi(y) ∈ so(3):
ξi(y) =
0 −y3/αi,3 y2/αi,2
y3/αi,3 0 −y1/αi,1
−y2/αi,2 y1/αi,1 0
where the constants αi,j for j = 1, 2, 3 are chosen so that the vector fields Vi and
matrices ξi do not commute for i = 0, 1, 2: α0,1 = 3, α0,2 = 1, α0,3 = 2, α1,1 = 1,
α1,2 = 1/2, α1,3 = 3/2, α2,1 = 1/4, α2,2 = 1, α2,3 = 1/2. The vector fields Vi satisfy
the conditions of Theorem 7.1 since the manifold is compact in this case.
We will numerically solve (1.1) using three different order 1 methods: stochastic
Taylor, stochastic Taylor Munthe-Kaas based on (6.1) and Castell–Gaines (a stan-
dard non-geometric Runge–Kutta method is used to solve the ordinary differential
equation (8.1)). The vector field compositions ViVj needed for the stochastic Taylor
and Castell–Gaines methods are readily computed. For the Munthe-Kaas method we
note that we have vξi ◦ o = ξi(y0) and
vξivξj ◦ o = Â(y0, y0;αi, αj)− 12 [ξi(y0), ξj(y0)] .
Here o ∈ so(3) is the zero element on the Lie algebra, and for all y, z ∈ R3 we define
A(y, z;α, β) ≡
− y3z2
− y1z3
− y2z1
and ˆ : R3→so(3) denotes the vector space isomorphism σ 7→ σ̂ where
0 −σ3 σ2
σ3 0 −σ1
−σ2 σ1 0
Note that ŷ z ≡ y ∧ z (see Marsden and Ratiu [37]). Note also since σ ∈ so(3),
expσ ∈ SO(3) can be conveniently and cheaply computed using Rodrigues’ formula
(see Marsden and Ratiu [37] or Iserles et al. [25]).
In Figure 9.1 we show the distance from the manifold S2 of each the three approx-
imations; we start with initial data y0 = (
2, 0)T. The stochastic Taylor Munthe-
Kaas method can be seen to preserve the solution in the unit sphere to within machine
Stochastic Lie group integrators 13
0 1 2 3 4 5 6 7 8 9 10
Stochastic Taylor
Castell−Gaines
Munthe−Kaas
Fig. 9.1. Rigid body: We show the log-distance of the approximate solution to the unit sphere
as a function of time for each of the methods. Below we show the approximate solutions as a function
of time for the stochastic Taylor (blue) and Munthe-Kaas methods (magenta). The trajectory starts
at the top right and eventually drifts over the left horizon.
14 Malham and Wiese
error. We also see that the stochastic Taylor method clearly drifts off the sphere as
the integration time progresses, as does the non-geometric Castell-Gaines method—
which does however remain markedly closer to the manifold than the stochastic Taylor
scheme.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Stochastic Taylor
Castell−Gaines
Munthe−Kaas
−3.2 −3.1 −3 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2
(stepsize)
Number of sampled paths=100
Stochastic Taylor
Castell−Gaines
Munthe−Kaas
Fig. 9.2. Autonomous underwater vehicle: We show the log-distance of the approximate solution
to the two Casimirs C1 = π ·p (dotted line) and C2 = |p|
2 (solid line) as a function of time for each
of the methods. Below, we also show the global error as a function of stepsize.
Stochastic Lie group integrators 15
9.2. Autonomous underwater vehicle. The dynamics of an ellipsoidal au-
tonomous underwater vehicle is prescribed by the state y = (π, p) ∈ se(3)∗ where
π ∈ so(3)∗ is its angular momentum and p ∈ (R3)∗ its linear momentum (see Holmes,
Jenkins and Leonard [24], Egeland, Dalsmo and Sørdalen [12] and Marsden and
Ratiu [37]). We suppose that the vehicle is perturbed by two independent multi-
plicative stochastic processes. The governing vector fields are for i = 0, 1, 2:
Vi(y) = ad
◦ y .
Here ξi(y) =
ωi(y), ui(y)
∈ se(3) where ωi(y) = I−1i π and ui(y) = M
i p are the
angular and linear velocity, and Ii = diag(αi,1, αi,2, αi,3) andMi = diag(βi,1, βi,2, βi,3)
are the constant moment of inertia and mass matrices, respectively. Explicitly for
ξ ∈ se(3) we have
ad∗ξ ◦ y ≡ (π ∧ ω + p ∧ u, p ∧ ω) .
The system of vector fields Vi, i = 0, 1, 2 represents the Lie–Poisson dynamics on
M = se(3)∗ (Marsden and Ratiu [37]). There are two independent Casimir functions
Ck : se(3)
∗→R, k = 1, 2, namely C1 = π · p and C2 = |p|2; these are conserved by the
flow on se(3)∗. Note that the Hamiltonian, i.e. total kinetic energy 1
(π · ω + p · u),
is also exactly conserved (and helpful for establishing the sufficiency conditions in
Theorem 7.1), but that is not our focus here.
If G = SE(3) ∼= SO(3) × R3, then the coadjoint action of SE(3) on se(3)∗,
: SE(3) × se(3)∗→se(3)∗ is defined for all S = (s, ρ) ∈ SE(3), where s ∈ SO(3)
and ρ ∈ R3, and y ∈ se(3)∗ by: Λy ◦ S = Ad∗S−1 ◦ y ≡
sπ + ρ ∧ (sp), sp
. The
corresponding infinitesimal action λ : se(3)× se(3)∗→se(3)∗ for all ξ ∈ se(3) and y ∈
se(3)∗ is given by (see Marsden and Ratiu [37], p. 477)
λξ ◦ y = −ad∗ξ ◦ y .
Since ad∗ξ(y) = −λξ(y) = λ−ξ(y) the governing set of vector fields on se(3)∗ are
Vi(y) = λ−ξi ◦ y .
We can now pull back this flow on se(3)∗ to a flow on SE(3) via Λy0 . The correspond-
ing flow on SE(3) is generated by the governing set of vector fields for i = 0, 1, 2:
X−ξi ◦ S = −
ωi(y) ∧ s, ωi(y) ∧ ρ+ ui(y)
with y = Λy0(S).
To aid implementation note that SE(3) =
(s, ρ) ∈ SE(3) : s ∈ SO(3), ρ ∈ R3
embeds into SL(4;R) via the map
S = (s, ρ) 7→
where O is the three-vector of zeros. Also se(3) is isomorphic to a Lie subalgebra of
sl(4;R) with elements of the form
σ = (θ, ζ) 7→
16 Malham and Wiese
Hence the governing vector fields on SE(3) are of the form Xξi = −ξi(y)S, where
ξi(y) =
ω̂i(π) ui(p)
The governing vector fields on se(3) are vξi(σ) = −dexpσ ◦ ξi
Λy0(expσ)
. Again the
vector field compositions ViVj needed for the stochastic Taylor and Castell–Gaines
methods can be computed straightforwardly. Direct calculation also reveals that in
block matrix form
vξivξj◦o =
Â(π0, π0;αi, αj) + Â(p0, p0;βi, αj) A(π0, p0;αi, βj)
[ξi(y0), ξj(y0)] .
Here A(y, z;α, β) is defined as for the rigid body example. Note that the exponential
map exp
se(3) : se(3)→SE(3) is defined for all σ = (θ, ζ) ∈ se(3) by
se(3) σ =
so(3) θ̂ f(θ)ζ
where exp
so(3) is the exponential map from so(3) to SO(3) which can be computed
using Rodrigues’ formula and (see Bullo and Murray [4], p. 5)
f(θ) = I3×3 + (1 − cos ‖θ‖)θ̂/‖θ‖2 +
1− (sin ‖θ‖)/‖θ‖
θ̂2/‖θ‖2 .
In Figure 9.2 we show the distance from the manifold se(3)∗ of each the three ap-
proximations; in particular how far the individual trajectories stray from the Casimirs
C1 = π · p and C2 = |p|2. We start with the initial data y0 = (
2, 0, 0,
As before the stochastic Taylor Munthe-Kaas method can be seen to preserve the
Casimirs to within machine error. We also see that the stochastic Taylor method
clearly drifts off the manifold as the integration time progresses and at a particular
time depending on the Wiener path shoots off very rapidly away from the manifold.
Note also that for large stepsizes the stochastic Taylor method is unstable. However
the non-geometric Castell–Gaines and stochastic Munthe-Kaas methods still give reli-
able results in that regime. Lastly, although the the stochastic Munthe-Kaas method
adheres to the manifold to within machine error, the error of the non-geometric
Castell–Gaines method is actually smaller.
10. Conclusions. We have established and implemented stochastic Lie group
integrators based on stochastic Munthe-Kaas methods and also derived geometric
Castell–Gaines methods. We have also revealed several aspects of these integrators
that require further investigation.
1. We could construct a stochastic nonlinear Magnus method by approximating
the solution to the stochastic differential equation (5.1) on the Lie algebra using Picard
iterations (see Casas and Iserles [6]).
2. We would like to develop a practical procedure for implementing ordinary
Munthe-Kaas methods for higher order Castell–Gaines integrators. We need to de-
termine the element ξ : M→g so that in (8.1) we have ψ̂ = λξ.
3. We need to determine the properties of the local and global errors for the
stochastic Munthe-Kaas methods. Also a thorough investigation of the stability prop-
erties of the stochastic Munthe-Kaas and Castell–Gaines methods is required. For
the autonomous underwater vehicle simulations they were both superior to the direct
stochastic Taylor method, especially for larger stepsizes. We also need to compare the
relative efficiency of the methods concerned, in particular to compare an optimally
efficient geometric Castell–Gaines method with the stochastic Munthe-Kaas method.
Stochastic Lie group integrators 17
4. Although we have chiefly confined ourselves to driving paths that are Wiener
processes, we can extend Munthe-Kaas and Castell–Gaines methods to rougher driv-
ing paths (Lyons and Qian [33], Friz [18], Friz and Victoir [19]). Further, what hap-
pens when we consider processes involving jumps? For example Srivastava, Miller and
Grenander [44] consider jump diffusion processes on matrix Lie groups for Bayesian
inference. Or what if we consider fractional Brownian driving paths; Baudoin and
Coutin [3] investigate fractional Brownian motions on Lie groups?
5. Schiff and Shnider [43] have used Lie group methods to derive Möbius schemes
for numerically integrating deterministic Riccati systems beyond finite time removable
singularities and numerical instabilities. They integrate a linear system of equations
on the general linear group GL(n) which corresponds to a Riccati flow on the Grass-
mannian manifold Gr(k, n) via the Möbius action map. Lord, Malham and Wiese [31]
implemented stochastic Möbius schemes and show that they can be more accurate and
cost effective than directly solving stochastic Riccati systems using stochastic Taylor
methods. We would like to investigate further their effectiveness for stochastic Ric-
cati equations arising in Kalman filtering (Kloeden and Platen [27]) and to backward
stochastic Riccati equations arising in optimal stochastic linear-quadratic control (see
for example Kohlmann and Tang [28] and Estrade and Pontier [16]).
6. Other areas of potential application of the methods we have presented in this
paper are for example: term-structure interest rate models evolving on finite dimen-
sional invariant manifolds (see Filipovic and Teichmann [17]); stochastic dynamics
triggered by DNA damage (Chickarmane, Ray, Sauro and Nadim [10]) and stochastic
symplectic integrators for which the gradient of the solution evolves on the symplectic
Lie group (see Milstein and Tretyakov [38]).
Acknowledgments. We thank Alex Dragt, Peter Friz, Anders Hansen, Terry
Lyons, Per-Christian Moan and Hans Munthe–Kaas for stimulating discussions. We
also thank the anonymous referees, whose suggestions and encouragement improved
the original manuscript significantly. SJAM would like to acknowledge the invalu-
able facilities of the Isaac Newton Institute where some of the final touches to this
manuscript were completed.
Appendix A. Proof of Theorem 7.1. We follow the proof for linear stochastic
differential equations in Lord, Malham and Wiese [31] (where further technical details
on estimates for multiple Stratonovich integrals can be found). Suppose ψ̂t ≡ ψ̂t(m)
is the truncated Lie series (7.1). First we show that exp ψ̂t ◦ y0 ∈ L2. We see that
for any number k,
)k ◦ y0 is a sum of |Qm|k terms, each of which is a k-multiple
product of terms Jα cα ◦ y0. It follows that
)k ◦ y0
‖cα ◦ y0‖
αi∈Qm
i=1,...,k
‖Jα1Jα2 · · · Jαk‖L2 . (A.1)
Note that the maximum of the norm of the compositions of vector fields cα◦y0 is taken
over a finite set. Repeated application of the product rule reveals that for i = 1, . . . , k,
each term ‘Jα1Jα2 · · · Jαk ’ in (A.1) is the sum of at most 22mk−1 Stratonovich integrals
Jβ , where for t ≤ 1, ‖Jβ‖L2 ≤ 24mk−1 tk/2. Since the right hand side of equation (A.1)
consists of |Qm|k 22mk−1 Stratonovich integrals Jβ , we conclude that,
)k ◦ y0
‖cα ◦ y0‖ · |Qm| · 26m · t1/2
18 Malham and Wiese
Hence exp ψ̂t ◦ y0 is square-integrable.
Second we prove (7.2). Let ŷt denote the stochastic Taylor series solution, trun-
cated to included terms of order up to and including tm. We have
∥yt − exp ψ̂t ◦ y0
∥yt − ŷt
∥ŷt − exp ψ̂t ◦ y0
We know yt ∈ L2—see Lemma III.2.1 in Gihman and Skorohod [21]. Note that
the assumptions there are fulfilled, since the uniform boundedness of the derivatives
implies uniform Lipschitz continuity of the vector fields by the mean value theorem,
and uniform Lipschitz continuity in turn implies a linear growth condition for the
vector fields since they are autonomous. Note that ŷt is a strong approximation to
yt up to and including terms of order t
m, with the remainder consisting of O(tm+1/2)
terms (see Proposition 5.9.1 in Kloeden and Platen [27]). It follows from the definition
of the exponential Lie series as the logarithm of the stochastic Taylor series, that the
terms of order up to and including tm in exp ψ̂t ◦ y0 correspond with ŷt; the error
consists of O(tm+1/2) terms.
Appendix B. Proof of Theorem 8.2. Our proof follows along the lines of that
for uniformly accurate Magnus integrators for linear constant coefficient systems (see
Lord, Malham & Wiese [31] and Malham and Wiese [35]). Let ϕtn,tn+1 and ϕ̂tn,tn+1
denote the exact and approximate flow-maps constructed on the interval [tn, tn+1] of
length h. We define the local flow remainder as
Rtn,tn+1 ≡ ϕtn,tn+1 − ϕ̂tn,tn+1 ,
and so the local remainder is Rtn,tn+1 ◦ ytn . Let Rls and Rst denote the local flow
remainders corresponding to the exponential Lie series and stochastic Taylor approx-
imations, respectively.
B.1. Order 1/2 integrator: two Wiener processes. For the global order 1/2
integrators we have to leading order Rls = 1
(J12 − J21)[V1, V2] and Rst = J12V1V2 +
J21V2V1. Note that we have included the terms J11V
1 and J22V
2 in the integrators.
A direct calculation reveals that
(Rst ◦ y0)TRst ◦ y0
(Rls ◦ y0)TRls ◦ y0
+ h2mUTBU +O
. (B.1)
Here m = 1/2 (for the order 1/2 integrators), U = (V1V2 ◦ y0, V2V1 ◦ y0)T ∈ R2n, and
B ∈ R2n×2n consists of n× n diagonal blocks of the form bijIn×n where
b = 1
and In×n is the n×n identity matrix. Since b is positive semi-definite, the matrix B =
b⊗In×n is positive semi-definite. Hence the order 1/2 exponential Lie series integrator
is locally more accurate than the corresponding stochastic Taylor integrator.
B.2. Order 1 integrator: one Wiener process. For the global order 1 in-
tegrators we have to leading order Rls = 1
(J01 − J10)[V0, V1] and Rst = J01V0V1 +
J10V1V0 + J111V
h2(V0V
1 + V
1 V0). The terms of order h
2 shown are significant
when we consider the global error in Section B.4 below. The estimate (B.1) also
applies in this case with m = 1 and U = (V0V1 ◦ y0, V1V0 ◦ y0, V 31 ◦ y0)T ∈ R3n; and
B ∈ R3n×3n consists of n× n diagonal blocks of the form bijIn×n where
b = 1
3 3 3
3 3 3
3 3 5
Stochastic Lie group integrators 19
Since b is positive semi-definite, the matrix B = b ⊗ In×n is positive semi-definite.
Hence the order 1 exponential Lie series integrator is locally more accurate than the
corresponding stochastic Taylor integrator.
B.3. Order 3/2 integrator: one Wiener process. The local flow remainders
are Rls = 1
J110−2J101+J011− 12h
[V1, [V1, V0]] and R
st = J011V0V
1 +J101V1V0V1+
J110V
1 V0 + J1111V
1 − 14h
2(V0V
1 + V
1 V0 +
V 41 ). The terms of order h
2 shown are
significant when we consider the global error—but for a different reason this time—see
Section B.4 below. Again, the estimate (B.1) applies in this case with m = 3/2 and
U = (V0V
1 ◦ y0, V1V0V1 ◦ y0, V 21 V0 ◦ y0, V 41 ◦ y0)T ∈ R4n; and B ∈ R4n×4n consists of
n× n diagonal blocks of the form bijIn×n where
b = 1
11 8 5 12
8 8 8 12
5 8 11 12
12 12 12 24
Again, B is positive semi-definite and the order 3/2 exponential Lie series integrator
is locally more accurate than the corresponding stochastic Taylor integrator.
B.4. Global error. Recall that we define the strong global error at time T
associated with an approximate solution ŷT as E ≡ ‖yT − ŷT ‖L2. The exact and
approximate solutions can be constructed by successively applying the exact and
approximate flow maps ϕtn,tn+1 and ϕ̂tn,tn+1 on the successive intervals [tn, tn+1] to
the initial data y0. A straightforward calculation shows for a small fixed stepsize h,
E2 = E (R ◦ y0)TR ◦ y0 , (B.2)
up to higher order terms, where R ≡
n=0 ϕtn+1,tN ◦Rtn,tn+1 ◦ϕt0,tn is the standard
accumulated local error contribution to the global error. The important conclusion is
that when we construct the global error (B.2), the terms of leading order in the local
flow remainders Rls or Rst with zero expectation lose only a half order of convergence
in this accumulation effect. Hence in the local flow remainders shown above, for the
terms of zero expectation, the local superior accuracy for the Lie series integrators
transfers to the corresponding global errors (see Lord, Malham and Wiese [31] for
more details). Terms of non-zero expectation however behave like deterministic er-
ror terms losing a whole order (in the local to global convergence); they contribute
to the global error through their expectations. Hence we include such terms of or-
der h2 in the order 3/2 integrators above and they appear as the terms subtracted
from the remainders shown. For the order 1 integrators we do not need to include
the order h2 terms in the integrator to obtain the correct mean-square convergence.
However to guarantee that the global error for the exponential Lie series integrator
is always smaller than that for the stochastic Taylor scheme, we include this term in
the integrator.
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|
0704.0023 | ALMA as the ideal probe of the solar chromosphere | Astrophysics and Space Science manuscript No.
(will be inserted by the editor)
ALMA as the ideal probe of the solar chromosphere
Maria A. Loukitcheva · Sami K. Solanki · Stephen White
Received: date / Accepted: date
Abstract The very nature of the solar chromosphere,
its structuring and dynamics, remains far from being
properly understood, in spite of intensive research. Here
we point out the potential of chromospheric observa-
tions at millimeter wavelengths to resolve this long-
standing problem. Computations carried out with a so-
phisticated dynamic model of the solar chromosphere
due to Carlsson and Stein demonstrate that millimeter
emission is extremely sensitive to dynamic processes in
the chromosphere and the appropriate wavelengths to
look for dynamic signatures are in the range 0.8-5.0
mm. The model also suggests that high resolution ob-
servations at mm wavelengths, as will be provided by
ALMA, will have the unique property of reacting to
both the hot and the cool gas, and thus will have the
potential of distinguishing between rival models of the
solar atmosphere. Thus, initial results obtained from
the observations of the quiet Sun at 3.5 mm with the
BIMA array (resolution of 12′′) reveal significant oscil-
lations with amplitudes of 50-150 K and frequencies of
1.5-8 mHz with a tendency toward short-period oscil-
lations in internetwork and longer periods in network
regions. However higher spatial resolution, such as that
provided by ALMA, is required for a clean separation
between the features within the solar atmosphere and
M.A. Loukitcheva · S.K. Solanki
Max-Planck-Institut für Sonnensystemforschung, D-37191
Katlenburg-Lindau, Germany
E-mail: [email protected]
M. A. Loukitcheva
Astronomical Institute, St. Petersburg University, 198504 St.
Petersburg, Russia
S. White
Astronomy Department, University of Maryland, College Park,
MD 20742, USA
for an adequate comparison with the output of the com-
prehensive dynamic simulations.
Keywords the Sun · solar chromosphere · millimeter
observations
1 Introduction
The chromosphere remains the least understood layer of
the solar atmosphere, with the very basics of its struc-
ture being hotly debated: is it better described by the
classical picture of a steady temperature rise as a func-
tion of height, with superposed weak oscillations (e.g.
semi empirical models of Vernazza et al. [8], Fontenla
et al. [5]), or does the temperature keep dropping out-
wards, with very hot shocks producing strong localized
heating (radiation hydrodynamic simulations of Carls-
son & Stein [3], [4], and Wedemeyer et al. [9])? The
latter concept is consistent with the IR observations of
carbon monoxide, which require cool gas to be present
at chromospheric heights (see, e.g. Ayres [1]).
Thus, existing models cannot provide a complete de-
scription of the solar chromosphere. Consequently now-
adays two alternative pictures of the chromosphere co-
exist and the role played by chromospheric dynamics in
the structuring of this atmospheric layer is a subject of
intense scientific debate.
One reason for conflicting models is that they are
based either on atomic chromospheric lines and con-
tinua in the UV or on molecular lines in the IR, since
UV observations are practically blind to cool gas in a
dynamic chromosphere, while the IR observations sam-
ple only the cool part of the chromosphere. Improved
and more sensitive diagnostics of the chromospheric
structure and dynamics, that sample both the hot and
http://arxiv.org/abs/0704.0023v1
MDI TRACE 1600
CaII K BIMA 3.5 mm
Fig. 1 Portrait of the solar chromosphere at the center of the Sun’s disk at 4 different wavelengths on May 18, 2004. From top left to
bottom right: MDI longitudinal photospheric magnetogram, UV 1600 A image from TRACE, CaII K line center image from BBSO
and BIMA image at 3.5 mm.
the cool gas and should distinguish between the ri-
val models, are provided by observations at millime-
ter wavelengths with an acceptable spatial resolution
as was proposed by Loukitcheva et al. [6]. In this con-
tribution we review the unique chromospheric obser-
vations at 3.5 mm with the Berkeley-Illinois-Maryland
Array and the analysis of the intensity variations ex-
pected from the model of Carlsson & Stein for mm
wavelengths. We postulate the requirements for mm ob-
servations with the future instruments, with emphasis
on spatial and temporal resolution. Finally we discuss
the prospects for chromospheric studies with ALMA.
2 Results
2.1 Analysis of the BIMA observations at 3.5 mm
The Berkeley-Illinois-Maryland Array (BIMA) operat-
ing at a wavelength of 3.5 mm (frequency of 85 GHz)
has been the only interferometer in the mm range fre-
quently used for solar observations. The BIMA tele-
scopes are now part of the CARMA array which will
also carry out such observations. With the BIMA data
obtained in the years 2003 and 2004 we have constructed
two-dimensional maps of the solar chromosphere with a
resolution of 12′′, which represents the highest spatial
resolution achieved so far at this wavelength for non-
flare solar observations. The BIMA images have led
to new insights in to chromospheric structure and to
the detection of spatially-resolved chromospheric oscil-
lations at mm wavelengths. The details of the restora-
tion procedure and extensive tests of the sensitivity of
the BIMA data to the detection of dynamic signatures
can be found in White et al. [11].
With the currently available resolution the contrast
of the brightness structures is evaluated to be up to
30% of the quiet-sun brightness at 3.5 mm (White et
al. [11]). However, the similarity of brightness struc-
tures, derived from the mm images and seen in other
chromospheric emissions (Fig.1), in spite of the differ-
ence in resolution of the images (1-2′′ resolution of the
UV images), implies that the BIMA resolution is not
enough to resolve the millimeter fine structure and ob-
servations with spatial resolution much higher than 12′′
are required. A detailed analysis of the relations be-
tween the millimeter emission, magnetic field and other
chromospheric diagnostics is in preparation.
In the millimeter brightness we detected intensity
oscillations with typical amplitudes of 50-150 K in the
range of periods from 120 to 700 seconds (frequency
range 1.5-8 mHz). We found a tendency toward short
period oscillations in internetwork and longer periods in
network regions in the quiet Sun, which is in good agree-
ment with the results obtained at other wavelengths. At
3 mm the inner parts of the chromospheric cells exhibit
a behavior typical of the internetwork with the maxi-
mum of the Fourier power in the 3-minute range, how-
ever, most of the oscillations are quasi-periodic, show-
ing up in wave trains of finite duration lasting for typi-
cally 1-3 wave periods (see also Loukitcheva et al. [7]).
2.2 Analysis of the CS model millimeter spectrum
The response of the submillimeter and millimeter ra-
diation to a time-series generated by Carlsson & Stein
(CS) was computed under the assumption of thermal
free-free radiation by Loukitcheva et al. [6]. The results
are depicted in Fig. 2 as the excess intensity as a func-
tion of wavelength and time.
400 720 1040 1360 1680 2000 2320 2640 2960 3280 3600
time(s)
Fig. 2 Evolution of the Carlsson & Stein model millimeter spec-
trum with time. Negative grey scale representing excess intensity
as a function of time and wavelength.
Wave periods of approximately 3 min can be clearly
distinguished in the intensity at all considered wave-
lengths. Though the dominant frequency of the oscilla-
tions changes slightly with wavelength, for all mm wave-
lengths it lies in the range of 3 minutes. The difference
from one period of time to another can be explained
by the presence of merging shocks during certain time
intervals. The differences in the light curves at differ-
ent wavelengths are caused primarily by the difference
in the formation heights of the emitted radiation. In
general the amplitudes of the oscillations compared to
the radiation temperature are large, in this sense mm
wavelength radiation combines the advantages of the
CO lines, which mainly see the cool gas, with those of
atomic lines and UV continua, which mainly sample the
hot gas.
On the whole, the brightness temperatures are ex-
tremely time-dependent at millimeter wavelengths, fol-
lowing changes in the atmospheric parameters. With
increasing wavelength the amplitude of the brightness
oscillations grows significantly, reaches its maximum
value at 2.2 mm (expected to be 15% of the quiet-Sun
brightness temperature), and decreases rapidly towards
longer wavelengths. Thus we can identify the range 0.8-
5.0 mm as the appropriate range of mm wavelengths at
which one can expect the clearest signatures of dynamic
effects. A careful look at the mm brightness spectrum
as a function of time (see Fig. 2) reveals a time delay
between the oscillations at long and short millimeter
wavelengths. Hence, it is possible to study wave modes
traveling in the chromosphere by comparing sub-mm
with mm observations.
3 Discussion
The CS model predicts that spatially and temporally
resolved observations should clearly exhibit the signa-
tures of the strong shock waves. However, a direct com-
parison of the observational data products (RMS val-
ues, histogram skewness, Fourier and wavelet spectra,
etc.), referring to regions with weak magnetic field like
the quiet Sun internetwork, with the corresponding prod-
ucts expected from the simulations of Carlsson & Stein
exhibits large differences. In particular, the RMS of the
brightness temperature is nearly an order of magnitude
larger in the model (800 K at 3 mm) than in the ob-
servations (100 K). Another difference is the absence of
longer periods in the model power spectrum. But these
discrepancies do not rule out the CS models. On the
one hand the model is one dimensional and hence does
not predict a coherence length of the oscillations, while
on the other hand we are not able to resolve individual
oscillating elements due to the limited spatial resolution
of the observations.
Consequently we estimated the influence of the spa-
tial smearing on the model parameters of chromospheric
dynamics and on the observed oscillatory power. Thus
we confirmed that the very limited spatial resolution
currently available hinders a clean separation between
cells and network and typically both network and in-
ternetwork areas contribute to the recorded BIMA ra-
diation. From the analysis of the observational data it
was found that power in all frequency ranges increases
significantly with improving resolution. Consistency be-
tween the power predicted by the CS model and the
observed power is obtained if the coherence length of
oscillating elements is on the order of 1′′.
Our results are consistent with Wedemeyer et al.
[10], who computed the millimeter wave signature re-
sulting from the 3-D simulations of Wedemeyer et al.
[9]. Although the 3-D simulations suffer from the fact
that the radiative transfer of energy is computed en-
tirely in LTE, which becomes a poor assumption at
chromospheric heights, the authors believe that the chro-
mospheric pattern and its temporal evolution is repre-
sentative of the non-magnetic internetwork regions of
the solar chromosphere. The simulations display a com-
plex 3D structure of the chromospheric layers, which is
highly dynamical on temporal scales of 20-25 s and on
spatial scales comparable to solar granulation, which is
in good agreement with the 1′′ size of oscillating ele-
ments that we deduced. According to Wedemeyer et al.
[9] the chromospheric temperature structure is charac-
terized by a pattern of hot shock waves, which originate
from convective motions, and cool gas lying between the
shocks. The intensity distribution at mm wavelengths
follows the pattern of the shocks in the chromosphere
with a sub arcsecond size of the features associated with
the shocks. All this complex and dynamic 3D structure
can be deduced from observations at mm wavelengths
with a sufficiently high spatial resolution of better than
4 Summary
Simultaneous mm-submm observations at different wave-
lengths can be used for the tomography of the solar
atmosphere, as radiation at the different wavelengths
originates from different layers, with the average for-
mation height increasing with wavelength. Such obser-
vations also provide a strong test of present and future
models. However, observations that might be able to
uncover the nature of the chromosphere should meet
the following requirements:
– multiband observations in mm-submm domain (0.8-
5.0 mm) to address shock waves and chromospheric
oscillation modes
– arcsecond spatial resolution to resolve fine structure
– temporal resolution better than a few seconds to
follow its evolution in time
– FOV size of order of 1′
– accurate absolute calibration of the observations (Bas-
tian [2])
These requirements look very similar to the techni-
cal specification of the continuum observations with the
Atacama Large Millimeter Array (ALMA), which rep-
resents an enormous advance over existing instrumenta-
tion operating at mm-submm wavelengths. ALMA will
produce images of the highest resolution available for
the foreseeable future (although the technical problem
of sampling both large and small spatial scales simulta-
neously, required for high–quality imaging of the chro-
mosphere, will remain a challenge) and will be the most
sensitive instrument operating at submm-mm wavelengths.
To summarize, ALMA will be an extraordinarily pow-
erful instrument for studying the solar chromosphere. It
will finally allow the mapping of the three-dimensional
thermal structure of the solar chromosphere which will
be a real breakthrough in solar studies.
Acknowledgements The use of BIMA for scientific research
carried out at the University of Maryland is supported by NSF
grant AST–0028963. Solar research at the University of Maryland
is supported by NSF grant ATM 99-90809 and NASA grants NAG
5-8192, NAG 5-10175, NAG 5-12860 and NAG 5-11872.
References
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Ap. J. 575, 1104-1115 (2002)
2. Bastian, T. S.: ALMA and the Sun. Astronomische
Nachrichten 323, 271-276 (2002)
3. Carlsson, M., & Stein, R.F.: Does a nonmagnetic solar chro-
mosphere exist? Ap. J. 440, L29-L32 (1995)
4. Carlsson, M., & Stein, R.F.: Dynamic Hydrogen Ionization.
Ap. J. 572, 626-635 (2002)
5. Fontenla, J. M.; Avrett, E. H.; Loeser, R.: Energy balance in
the solar transition region. III - Helium emission in hydrostatic,
constant-abundance models with diffusion. Ap. J. 406, 319-345
(1990)
6. Loukitcheva, M., Solanki, S.K., Carlsson, M., Stein, R.F.: Mil-
limeter observations and chromospheric dynamics. A&A 419,
747-756 (2004)
7. Loukitcheva, M., Solanki, S.K., White, S.: The dynamics
of the solar chromosphere: comparison of model predictions
with millimeter-interferometer observations. A&A 456, 713-723
(2006)
8. Vernazza, J. E., Avrett, E. H., Loeser, R.: Structure of the
solar chromosphere. III - Models of the EUV brightness com-
ponents of the quiet-sun. Ap. J. Suppl. 45, 635-725 (1981)
9. Wedemeyer, S., Freytag, B., Steffen, M., Ludwig, H.-G., Hol-
weger, H.: Numerical simulation of the three-dimensional struc-
ture and dynamics of the non-magnetic solar chromosphere.
A&A 414, 1121-1137 (2004)
10. Wedemeyer-Böhm, S., Ludwig, H.-G., Steffen, M., Freytag,
B., Holweger, H.: The shock-patterned solar chromosphere in
the light of ALMA. In: Favata et al. (eds.) Proceedings of
”The 13th Cambridge Workshop on Cool Stars, Stellar Systems
and the Sun” Hamburg, Germany, ESA SP-560, pp. 1035-1038
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Introduction
Results
Discussion
Summary
|
0704.0024 | Formation of quasi-solitons in transverse confined ferromagnetic film
media | Formation of quasi-solitons in transverse confined ferromagnetic
film media
A.A. Serga 1
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
M. Kostylev 2
School of Physics, The University of Western Australia, 35 Stirling
Highway, Crawley WA 6009, Australia
St.Petersburg Electrotechnical University, 197376, St.Petersburg, Russia
B. Hillebrands
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
Abstract The formation of quasi-2D spin-wave waveforms in longitudinally
magnetized stripes of ferrimagnetic film was observed by using time- and space-
resolved Brillouin light scattering technique. In the linear regime it was found
that the confinement decreases the amplitude of dynamic magnetization near the
lateral stripe edges. Thus, the so-called effective dipolar pinning of dynamic mag-
netization takes place at the edges.
In the nonlinear regime a new stable spin wave packet propagating along a
waveguide structure, for which both transversal instability and interaction with
the side walls of the waveguide are important was observed. The experiments and
a numerical simulation of the pulse evolution show that the shape of the formed
waveforms and their behavior are strongly influenced by the confinement.
We report on the observation of a new type of a stable, two-dimensional
nonlinear spin wave packet propagating in a magnetic waveguide structure
and suggest a theoretical description of our experimental findings. Stable
two-dimensional spin wave packets, so-called spin wave bullets, were previ-
ously observed, however solely in long and wide samples of a thin ferrimag-
netic film of yttrium-iron-garnet (YIG) [1, 2, 3], that were practically un-
1Email address: [email protected]
2Email address: [email protected]
bounded in both in-plane directions compared to the lateral size of the spin
wave packets and the wavelength of the carrier spin wave. In a waveguide
structure, where the transverse dimension is comparable to the wavelength,
up to day only quasi one-dimensional nonlinear spin wave objects were ob-
served, which are spin wave envelope solitons. Here a typical system is a
narrow (' 1-2mm) stripe of a YIG ferrite film [4, 5]. Both for solitons and bul-
lets the spreading in dispersion is compensated by the longitudinal nonlinear
compression. Concerning the transverse dimension, solitons have a cosine-
like amplitude distribution due to the lateral confinement in the waveguide,
whereas bullets show a transverse nonlinear instability compensating pulse
widening due to diffraction and leading to transverse confinement.
Here we report on the observation of a new stable spin wave packet prop-
agating along a waveguide structure, for which both transversal instability
and interaction with the side walls of the waveguide are important.
The experiments were carried out using a longitudinally magnetized long
YIG film stripe of 2.5mm width and 7µm thickness. The magnetizing field
was 1831Oe. The spin waves were excited by a microwave magnetic field
created with a microstrip antenna of 25µm width placed across the stripe
and driven by electromagnetic pulses of 20ns duration at a carrier frequency
of 7.125GHz. As is well known the backward volume magnetostatic spin
wave (BVMSW) [6] excited in the given experimental configuration is able to
form both envelope solitons and bullets [4], depending on the geometry. The
spatio-temporal behavior of the traveling BVMSW packets was investigated
by means of space- and time-resolved Brillouin light scattering spectroscopy
The obtained results are demonstrated in Fig. 1 where the spatial distri-
butions of the intensity of the spin wave packets are shown for given moments
of time. The spin wave packets propagate here from left to right and decay
in the course of their propagation along the waveguide because of magnetic
loss. The left set of diagrams corresponds to the linear case. The power of
the driving electromagnetic wave is 20mW. The right set of diagrams corre-
sponding to the nonlinear case was collected for a driving power of 376mW.
Differences between these two cases are clearly observed. First of all the
linear spin wave packet is characterized by a cosine-like lateral profile while
the cross section of the nonlinear pulse is sharply modified relative to the
linear case and has a pronounced bell-like shape. Second, the intensity of the
linear packet decays monotonically with time while the intensity of the non-
linear packet initially increases because of its strong transversal compression
(see the second diagram from the top in Fig. 1).
Both of these nonlinear features provide clear evidence for the develop-
Figure 1: Bullet formation in the transversally confined yttrium-iron-garnet
film.
ment of a transversal instability and bullet formation. It is interesting that
the bell-like cross-section shape survives even at the end of the propaga-
tion distance when the pulse intensity decreases more than ten times and
the nonlinear contribution to the spin wave dynamics should considerably
diminish.
In order to interpret the experimental result we have assumed that the de-
velopment of nonlinear instabilities in a laterally confined medium is strongly
modified by a quantization of the spin wave spectrum. That is why we have
transformed the two-dimensional Nonlinear Schrödinger Equation tradition-
ally used for the analysis of bullet dynamics [4] into a system of coupled
equations for amplitudes of the spin wave width modes. The specific form of
the discrete set of these orthogonal modes is defined by the actual boundary
conditions at the lateral edges of the stripe. We developed a two-dimensional
theory of linear spin-wave dynamics in magnetic stripes. As an important
outcome we found that the Guslienko-Slavins effective boundary condition [8]
for dynamic magnetization at the stripe lateral edges, being initially derived
for spin waves with vanishing longitudinal wavenumbers, is also valid in the
case of propagating width modes with non-vanishing longitudinal wavenum-
bers [9] . The effective boundary condition shows that the magnetization
vector at the lateral stripe edges is highly pinned, that means that the am-
plitude of dynamic magnetization practically vanishes at the edges. For
simplicity it is even possible to consider the stripe width modes to be totally
pinned at the stripe lateral edges. As seen from Fig. 1 this conclusion is in a
good agreement with the experiment.
The analysis of the system of nonlinear equations derived from the Non-
linear Schrödinger Equation shows that the formation of the two-dimensional
waveform can be considered as an enrichment of the spectrum of the width
modes. The partial waveforms carried by the modes have the same carrier
frequencies equal to that of the initial signal and the carrier wave numbers
which satisfy the dispersion relations for the modes. In the linear regime all
the modes are orthogonal to each other and do not interact. In the nonlinear
(high amplitude) regime the width modes become intercoupled by the four-
wave nonlinear interaction, resulting in an intermodal energy transfer and
the mode spectrum enrichment.
As the spin wave input antenna effectively generates only the lowest width
mode, the initial waveform launched in the stripe is determined by it solely.
Therefore to understand the underlaying physics of quasi-bullet formation it
is necessary to consider the nonlinear interaction of higherorder width modes
with it.
Our theoretical analysis shows that the interaction of the lowest width
mode (n = 1) with higherorder modes is different for odd and even higher or-
der modes. While interacting with even modes, the lowest width mode plays
the role of the pumping wave. This parametrically transfers its energy to the
higher width modes. The interaction is purely parametric and therefore a
threshold process. It needs an initial signal to start the process. This signal
usually is a thermally excited mode. Therefore the amplified waveform needs
a large distance of propagation and a group velocity equal to the velocity of
the lowest width mode in order to reach the soliton amplitude level. If there
is a damping of the pumped wave, even modes will never reach an amplitude
comparable with that of the lowest mode. As a result they can contribute to
the nonlinear waveform profile only, if the amplitude of the initial waveform
is far beyond the threshold of soliton formation.
Interaction of modes of the same type of symmetry are described by a
parametric term as well as by an additional pseudo-linear (tri-linear) exci-
tation term, playing the role of an external source of excitation. Such a
pseudo-linear excitation is a threshold-free process. In contrast to paramet-
ric processes it does not need an initial amplitude value to start the the
process. The pseudo-linear excitation is possible only due to the effective
dipolar pinning of the magnetization at the stripe edges. If the edge spins
were unpinned, the interaction of all the width modes would be purely para-
metric.
The purely parametric mechanism of developing a transversal instability
is typical for the process of bullet formation from a plane-wave waveform in
an unconfined medium, which distinguishes it from the process of soliton and
bullet formation in the waveguide structures.
In contrast, the transverse instability of a wave packet in a confined
medium starts as a pseudolinear excitation of higher-order width modes.
This mechanism ensures a rapid growth of the symmetric n = 3 mode up
to the level where the parametric mechanism starts to work. After that the
main mode together with the n = 3 mode are capable to rapidly generate
a large set of yet higher modes through both pseudo-linear and parametric
mechanisms.
Our theory shows that the efficiency of both nonlinear interaction mech-
anisms (parametric and tri-linear) strongly depends on the group velocity
difference of modes and the initial length of the nonlinear pulse. In larger
stripes the group velocities of modes are closer to each other. As a result
the nonlinearly generated higher-order modes longer remain within the pump
pulse. If the pulse is long enough, they reach significant amplitudes and a
bullet-like waveform is formed. In narrower stripes the group velocity differ-
ence is larger, and consequently the nonlinearly generated highorder wave-
forms leave faster the pumping area. As a result, for the same pulse length,
they do not reach significant amplitudes. The nonlinear steepening results
Figure 2: Lateral shapes of the nonlinear SW packets. 1 and 2 – theoretical
results calculated for the ferrite stripes of width of 2.5mm and 1mm , respec-
tively. 3 and 4 – experimental profiles observed in YIG waveguides of width
of 2.5mm and 1mm, respectively. 1 and 3: bullets. 2 and 4: solitons.
in the transformation of the lowest mode into a soliton.
The results of our calculations of the lateral shapes of the nonlinear spin
wave packets in wide (2.5mm) and narrow 1mm ferrite stripes are shown in
Fig. 2. The excellent correspondence with the experimental data provides
good evidence for the validity of the developed theory.
Support by the Deutsche Forschungsgemeinschaft, the Australian Re-
search Council, and Russian Foundation for Basic Research is gratefully ac-
knowledged.
References
[1] O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands, Yu.S. Kivshar,
V. Grimalsky, Yu. Rapoport, A.N. Slavin, 61, 11576 (2000).
[2] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, 92, 117203
(2004).
[3] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, P. Wierzbicki,
V. Vasyuchka, O. Dzyapko, A. Chumak, 94, 167202 (2005).
[4] A.N. Slavin, O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands,
M.P. Kostylev, B.A. Kalinikos, V. Grimalsky, Yu. Rapoport, Chaos 13,
693 (2003).
[5] M. Chen, M.A. Tsankov, J.M. Nash, C.E. Patton, 49, 12773 (1994).
[6] F.R. Morgenthaler, Proceedings of the IEEE 76, 138 (1988).
[7] S.O. Demokritov, B. Hillebrands, A.N. Slavin, Phys. Rep. 348, 441
(2001).
[8] K.Y.Guslienko, S.O.Demokritov, B.Hillebrands, and A.N.Slavin, 66,
132402 (2002).
[9] M.Kostylev, J.-G. Hu, and R.L.Stamps, 90, 012507 (2007).
|
0704.0025 | Spectroscopic Properties of Polarons in Strongly Correlated Systems by
Exact Diagrammatic Monte Carlo Method | arXiv:0704.0025v1 [cond-mat.str-el] 2 Apr 2007
Spectroscopic Properties of Polarons in
Strongly Correlated Systems by Exact
Diagrammatic Monte Carlo Method
A. S. Mishchenko1,2 and N. Nagaosa3
1 CREST, Japan Science and Technology Agency (JST), AIST, 1-1-1, Higashi,
Tsukuba 305-8562, Japan.
2 Russian Research Centre “Kurchatov Institute”, 123182 Moscow, Russia.
3 Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113, Japan.
1 Introduction
Theoretical study of polarons in the strongly correlated system is like an at-
tempt to view contents of a Pandora box embedded into another, even more
sinister and obscure, container of riddles, enigmas and mysteries. This des-
perate situation occurs because solution is not known even for the simplest
polaron problem, i.e. when a perfectly stable quasiparticle (QP) with mo-
mentum as a single quantum number interacts with a well defined bath of
bosonic elementary excitations. To the contrary, the definition of the strongly
correlated system implies that QPs might be highly unstable and the very
notion of QPs, both in electronic and bosonic subsystems, is under question.
Thus, one faces the problem of an interplay between ill defined objects and
it is crucial to solve the problem without approximations. Further difficulty,
pertinent to realistic systems, is an interplay of the momentum and other
quantum numbers characterizing internal states of a QP.
The problem of polaron originally emerged as that of an electron coupled
to phonons (see [1, 2]). In the initial formulation a structureless QP is char-
acterized by the only quantum number, momentum, which changes due to
interaction of the QP with phonons [3, 4]. Later, depending on what can be
called “particle” and “environment”, and how they interact with each other,
the polaron concept was related to extreme diversity of physical phenomena.
There are many other objects which, having nothing to do with phonons, are
isomorphic to simple polaron [5], as, e.g. an exciton-polaron in the intraband
scattering approximation [6, 7, 8, 9]. Another example is the problem of a hole
in the antiferromagnet which is closely related to polaron since hole movement
is accompanied by the spin flips which, in the spin wave approximation, are
equivalent to creation and annihilation of magnons [10, 11].
http://arxiv.org/abs/0704.0025v1
2 A. S. Mishchenko and N. Nagaosa
The concept of polaron was further generalized to include internal degrees
of freedom which, interacting with environment, change their quantum num-
bers. Example of a complex QP is the Jahn-Teller polaron, where electron-
phonon interaction (EPI) changes quantum numbers of degenerate electronic
states [12, 13, 14]. This generalization is important due to it’s relevance to
the colossal magnetoresistance phenomena in the manganese oxides [15, 16].
Another example is the pseudo Jahn-Teller polaron, where EPI is inelastic
and leads to transitions between close in energy electronic levels of a QP
[17, 18, 19]. Further generalization is a system of several QPs which interact
both with each other and environment. For example, effective interaction of
two electrons through exchange by phonons can overcome the Coulomb repul-
sion and form a bound state, bipolaron [20, 21, 22, 23, 24]. On the other hand,
coupling of attracting hole and electron to the lattice vibrations [25, 26, 27]
can create a lot of qualitatively different objects: localized exciton, weakly
bound pair of localized hole and localized electron, etc. [28, 7]. Scattering by
impurities introduces additional complexity to the polaron problem because
interference of impurity potential with lattice distortion, which accompanies
the polaron movement, can contribute either constructively or destructively
to the localization of a QP on impurity [29, 30, 7].
In addition, a bare QP and bosonic bath can not be considered as well
defined in the correlated systems. Angle Resolved Photoemission Spectra
(ARPES), revealing the Lehmann Function (LF) of quasiparticle, demonstrate
broad peaks in many correlated systems: cooper oxide high-temperature su-
perconductors [31, 32, 33], colossal magnetoresistive manganites [34, 35, 36],
quasi-one-dimensional Peierls conductors [37, 38], and Verwey magnetites [39].
Besides, phonons are also broadened in many correlated systems, e.g. in high-
temperature semiconductors [40] and mixed-valent materials [41, 42]. One of
possible reasons for these broadenings is the interaction of the QPs with the
lattice degrees of freedom. However, in many realistic cases other subsystems,
not explicitly included into the polaron Hamiltonian, are responsible for the
decay of QP and phonons, e.g., another electronic bands, phonon anharmonic-
ity, interaction with nuclear spins, etc. Then, if this auxiliary broadening is
known in some approximation, one can formulate an ambitious goal to study
spectral response when “bare” quasiparticle with known damping interacts
with “broadened” bosonic excitations.
No one of traditional numerical methods, to say nothing of analytical ones,
can give approximation free results for measurable quantities of polaron, such
as optical conductivity or angle resolved photoemission spectra, for in macro-
scopic system of arbitrary dimension. Besides, we are not aware of any numer-
ical method which can incorporate in an approximation free way the informa-
tion on the damping of QP and bosonic bath. Below we describe basics of re-
cently developed Diagrammatic Monte Carlo (DMC) method for numerically
exact computation of Green functions and correlation functions in imaginary
time for few polarons in a macroscopic system [43, 44, 45, 46, 47, 48, 49, 50, 51].
Analytic continuation of imaginary time functions to real frequencies is per-
Spectroscopic Properties of Polarons by Exact Monte Carlo 3
formed by a novel approximation free approach of stochastic optimization
(SO) [45, 50, 51], circumventing difficulties of popular Maximal Enthropy
method. Finally we focus on results of application of the DMC-SO machinery
to various problems [52, 53, 54, 55, 56, 57]
The basic models, related to the polaronic objects in correlated systems,
which can be solved by DMC-SO methods, are stated in the next Sect. It
is followed in Sect. 1.2 by description of stumbling blocks encountered by
analytic methods. Sect. 2 concerns the basics of DMC-SO methods. However,
those who are not interested in the details of the methods can briefly look
through the definitions in the introduction for Sect. 2 and turn to Sect. 3
where LF and optical conductivity of Fröhlich polaron are discussed (see also
[58]). Results of studies of the self-trapping phenomenon are presented in Sect.
4 and application of DMC-SO methods to the exciton problem can be found
in Sect. 5. The chapter is completed by Sect. 6 devoted to studies of ARPES
of high temperature superconductors.
1.1 Formulation of a General Model with Interacting Polarons
In general terms, the simplest problem of a complex polaronic object, where
center-of-mass motion does not separate from the rest of degrees of freedom,
is introduced as system of two QPs
εa(k)a
εh(k)hkh
(ak and hk are annihilation operators, and εa(k) and εh(k) are dispersions of
QPs), which interact with each other
Ĥa-h = −N−1
U(p,k,k′)a†
p−khp−k′ap+k′ . (2)
(N is the number of lattice sites) through the instantaneous Coulomb potential
and the scattering by bosons
Ĥpar-bos = i
(b†q,κ − b−q,κ)
γaa,κ(k,q)a
k−qak + γhh,κ(k,q)h
k−qhk + γah,κ(k,q)h
k−qak
+ h.c. (3)
(γ[aa,ah,hh],κ are interaction constants) where quanta of Q different branches
of bosonic excitations are created or annihilated, which are described by
Ĥbos =
ωq,κb
q,κbq,κ . (4)
In general, each QP can be a composite one with internal degree of freedom
represented by T different states
4 A. S. Mishchenko and N. Nagaosa
ĤPJT0 =
ǫi(k)a
i,kai,k, (5)
which quantum numbers can be also changed due to nondiagonal part of
particle-boson interaction
Ĥpar-bos = i
i,j=1
γij,κ(k,q)(b
q,κ − b−q,κ)a
i,k−qaj,k + h.c. (6)
Complicated model (1)-(6) is still too far from the cases encountered in
strongly correlated systems. Due to coupling of QPs (1) and (5) and bosonic
fields (4) to additional degrees of freedom, these excitations are not well de-
fined from the onset. Namely, the dispersion relation of the QP spectrum ǫ(k)
in realistic system is ill-defined. One can speak of a Lehmann Function (LF)
[59, 60, 61] of a QP
Lk(ω) =
δ(ω − Eν(k)) |〈ν|a†k|vac〉|
2 (7)
,which is normalized to unity
dωLk(ω) = 1 and can be interpreted as
a probability that a QP has momentum k and energy ω. (Here {|ν〉} is a
complete set of eigenstates of Hamiltonian Ĥ in a sector of given momentum
k: H |ν(k)〉 = Eν(k) |ν(k)〉.) Only for noninteracting system the LF reduces
to delta function LNONINTk (ω) = δ(ω − ǫ(k)) and, thus, sets up dispersion
relation ω = ǫ(k).
Specific cases of model (1)-(6) describe enormous variety of physical prob-
lems. Hamiltonians (1) and (2), in case of attractive potential U(p,k,k′) > 0,
describe an exciton with static screening [62, 63]. Besides, expressions (1)-(4)
describe bipolaron for repulsive interaction [20, 21, 22, 23, 24] U(p,k,k′) < 0
and exciton-polaron otherwise [25, 26, 27]. The simplest model for exciton-
phonon interaction, when only two (T = 2) lowest states of relative electron-
hole motion are relevant (e.g. in one-dimensional charge-transfer exciton
[64, 65, 66]), is defined by Hamiltonians (4)-(6)). The same relations (4)-(6)
describe the problems of Jahn-Teller [all ǫi in Hamiltonian (5) are the same]
and pseudo Jahn-Teller polaron. The problem of a hole in an antiferromagnet
in spin-wave approximation is expressed in terms of Hamiltonians (4)-(6) with
Q = 1 and T = 1. When hole also interacts with phonons, one has to take into
account one more bosonic branch and set Q = 2 in (4) and (6). Finally, the
simplest nontrivial problem of a polaron, i.e. of a structureless QP interacting
with one phonon branch, is described by noninteracting Hamiltonians of QP
Ĥpar and phonons Ĥph
Ĥ0 =
ǫ(k)a
qbq , (8)
and interaction term
Spectroscopic Properties of Polarons by Exact Monte Carlo 5
Ĥint =
V (k,q)(b†q − b−q)a
k−qak + h.c. . (9)
The simplest polaron problem, in turn, can be subdivided into continuous and
lattice polaron models.
1.2 Limitations of Analytic Methods in Problem of Polarons
Analytic solution for the problem of exciton in a rigid lattice is available
only for small radius Frenkel regime [67] and large radius Wannier regime
[68]. However, even limits of validity for these approximations are not known.
Random phase approximation approaches [62, 63], are capable of obtaining
some qualitative conclusions for intermediate radius regime though its’ quan-
titative results are not reliable due to uncontrolled errors. The situation is
similar with the problem of structureless polaron, where analytic solutions
are known only in the weak and strong coupling regimes. Besides, reliable
results for these regimes are available only for ground state properties.
Although several novel methods, capable of obtaining properties of excited
states, were developed recently, variational coherent-states expansion [69] and
free propagator momentum average summation [70] as a few examples, all of
them, to provide reliable data in a specific regime, need either comparison
with exact sum rules [71, 72] or with exact numerical results.
Application of variational methods to study of excitations is a tricky is-
sue since, strictly speaking, they are valid only for the ground state. As an
example for the importance of sum rules in variational treatment, we refer
to the problem of the optical conductivity of the Fröchlich polaron. Possibil-
ity of existence of Relaxed Excited State (RES), which is a metastable state
where lattice deformation has adjusted to the electronic excitation rendering
stability and narrow linewidth of the spectroscopic response, was briefly men-
tioned by S. I. Pekar in early 50’s. Then, conception of RES was rigorously
formulated by J. T. Devreese with coworkers and has been a subject of ex-
tensive investigations for years [5, 73, 74, 75, 76, 77, 48, 57]. Calculations of
impedance [75] in the framework of technique [78] supported the existence of
a narrow stable peak in the optical conductivity. However, even the authors
of [75] were skeptical about the fact that the width of RES in the strong
coupling regime appeared to be more narrow than the phonon frequency, i.e.
inverse time which is, according to the Heisenberg uncertainty principle, is
required for the lattice readaptation. In consequent paper [77] they realized
the importance of many-phonon processes and studied two-phonon contri-
bution to optical conductivity. Importance of many phonon processes was
confirmed when variational results [75] were compared with exact DMC sim-
ulations [48]. Variational result well reproduced the position of the peak in
exact data though failed in description of the peak width in the strong cou-
pling regime [48]. Finally, when approach [75] was modified and several sum
rules were accurately introduced into variational model [57], both position
6 A. S. Mishchenko and N. Nagaosa
and width of the peak were quantitatively reproduced. Studies [57] (see Sect.
3.1), do not address rather philosophical question whether RES exists or not,
though inevitably prove that, in contrast to the foregoing beliefs, there in no
stable excited state of the Fröhlich polaron in the strong coupling regime.
Note that sometimes excited states can not be handled by analytic methods
even for weak couplings: perturbation theory expression for LF of the Fröhlich
polaron model diverges at the phonon energy ωph [See (34) in Sect. 3.1.] and
more elaborate treatment is necessary.
Difficulties of semianalytic methods enhance in the intermediate coupling
regime where results are sometimes wrong even for ground state properties.
For example, the variatioanl approach [79], which has been considered as an
intermediate coupling theory, appeared to be valid only in the weak coupling
limit [45]. Special interest to the methods, giving reliable information on ex-
cited states, is triggered by the self-trapping phenomenon which occurs just
in the intermediate coupling regime. This phenomenon is a dramatic trans-
formation of QP properties when system parameters are slightly changed
[3, 7, 9, 80]. In the intermediate coupling regime “trapped” QP state with
strong lattice deformation around it and “free” state with weakly perturbed
lattice may hybridize and resonate because of close energies at some critical
value of electron-lattice interaction γc. It is clear that, to study self-trapping,
one has to apply a method giving reliable information on excited states in the
intermediate coupling regime.
2 Diagrammatic Monte Carlo and Stochastic
Optimization Methods
In this section we introduce definitions of exciton-polaron properties which
can be evaluated by DMC and SO methods. An idea of DMC approach for
numerically exact calculation of Green functions (GFs) in imaginary times
is presented in Sect. 2.1, and a short description of SO method, which is
capable of making unbiased analytic continuation from imaginary times to
real frequencies, is given in Sect. 2.2. Using combination of DMC and SO,
one can often circumvent difficulties of analytic and traditional numerical
methods. Therefore, a brief comparative analysis of advantages and drawbacks
of DMC-SO machinery is given in Sect. 2.3.
To obtain information on QPs it is necessary to calculate Matsubara GF
in imaginary time representation and make analytic continuation to the real
frequencies [60]. For the two-particle problem (1)-(4) the relevant quantity is
the two-particle GF [46, 47]
(τ) = 〈vac | ak+p′(τ)hk−p′(τ)h†k−pa
k+p | vac〉 . (10)
(Here hk−p(τ) = e
Ĥτhk−pe
−Ĥτ , τ > 0.) In the case of exciton-polaron, vac-
uum state | vac〉 is the state with filled valence and empty conduction bands.
Spectroscopic Properties of Polarons by Exact Monte Carlo 7
For the bipolaron problem it is a system without particles. In the simpler case
of a QP with two-level internal structure described by (4)-(6) the relevant
quantity is the one-particle matrix GF [52, 47]
Gk,ij(τ) = 〈vac | ai,k(τ)a†j,k | vac〉, i, j = 1, 2. (11)
For a structureless polaron the matrix (11) reduces to one-particle scalar GF
Gk(τ) = 〈vac | ak(τ)a†k | vac〉 . (12)
Information on the response to an external weak perturbation (e.g. optical
absorption) is contained in the current-current correlation function 〈Jβ(τ)Jδ〉
(β/δ are Cartesian indexes).
Lehmann spectral representation of Gk(τ) [60, 61] at zero temperature
Gk(τ) =
dω Lk(ω) e
−ωτ , (13)
with the Lehmann function (LF) Lk(ω) given in (7), reveals information on
the ground and excited states. Here {|ν〉} is a complete set of eigenstates of
Hamiltonian Ĥ in a sector of given momentum k: H |ν(k)〉 = Eν(k) |ν(k)〉.
The LF Lk(ω) has poles (sharp peaks) on the energies of stable (metastable)
states of particle. For example, if there is a stable state at energy E(k), the LF
reads Lk(ω) = Z
(k) δ(ω − E(k)) + . . ., and the state with the lowest energy
Eg.s.(k) in a sector of a given momentum k is highlighted by asymptotic
behavior of GF
Gk(τ ≫ max
ω−1q,κ
) → Z(k) exp[−Eg.s.(k)τ ] , (14)
where Z(k)-factor is the weight of the state. Analyzing the asymptotic behavior
of similar n-phonon GFs [45, 52]
Gk(n, τ ; q1, . . . ,qn) =
〈vac| bqn(τ) · · · bq1(τ) ap(τ)a
· · · b†qn |vac〉 , p = k−
j=1 qj .
one obtains detailed information about lowest state. For example, important
characteristics of the lowest state wave function
Ψg.s.(k) =
q1...qn
θi(k;q1, ...,qn)c
i,k−q1...−qnb
...b†qn | vac〉 (16)
are partial n-phonon contribution
Z(k)(n) ≡
q1...qn
| θi(k;q1, ...,qn) |2 (17)
which is normalized to unity
n=0 Z
(k)(n) ≡ 1, and the average number of
phonons
8 A. S. Mishchenko and N. Nagaosa
〈N〉 ≡ 〈Ψg.s.(k) |
b†qbq | Ψg.s.(k)〉 =
nZ(k)(n) (18)
in polaronic cloud. Another example is the wave function of relative electron-
hole motion of exciton in the lowest state in the sector of given momentum
Ψg.s.(k) =
ξk p(g.s.)a
| vac〉 . (19)
The amplitudes ξk p(g.s.) of this wave function can be obtained [46] from
asymptotic behavior of the following GF (10)
(τ → ∞) =| ξk p(g.s.) |2 e−Eg.s.(k)τ . (20)
Information on the excited states is obtained by the analytic continuation
of imaginary time GF to real frequencies which requires to solve the Fredholm
equation Gk(τ) = F̂ [Lk(ω)] (13)
Lk(ω) = F̂−1ω [Gk(τ)] . (21)
The equation (13) is a rather general relation between imaginary time GF/cor-
relator and spectral properties of the system. For example, the absorption
coefficient of light by excitons I(ω) is obtained as solution of the same equation
I(ω) = F̂−1ω
k=0(τ)
. (22)
Besides, the real part of the optical conductivity σβδ(ω) is expressed [48] in
terms of current-current correlation function 〈Jβ(τ)Jδ〉 by relation
σβδ(ω) = πF̂−1ω [〈Jβ(τ)Jδ〉] /ω . (23)
2.1 Diagrammatic Monte Carlo Method
DMC Method is an algorithm which calculates GF (10)-(12) without any
systematic errors. This algorithm is described below for the simplest case of
structureless polaron [45], and generalizations to more complex cases can be
found in consequent references4. DMC is based on the Feynman expansion of
the Matsubara GF in imaginary time in the interaction representation
4 Generalization of described below technique to the case of exciton (1-2) is given
in [46] and its modification for pseudo-Jahn-Teller polaron (4-6) is developed in
[52, 47]. Method for evaluation of current-current correlation function can be
found in [48] and a case of a polaron interacting with two kinds of bosonic fields
is considered in [49].
Spectroscopic Properties of Polarons by Exact Monte Carlo 9
Gk(τ) =
∣∣∣∣Tτ
ak(τ)a
(0) exp
Ĥint(τ
′)dτ ′
}]∣∣∣∣ vac
; τ > 0 .
Here Tτ is the imaginary time ordering operator, |vac〉 is a vacuum state with-
out particle and phonons, Ĥint is the interaction Hamiltonian in (9). Symbol of
exponent denotes Taylor expansion which results in multiple integration over
internal variables {τ ′1, τ ′2, . . .}. Operators are in the interaction representation
Â(τ) = exp[τ(Ĥpar + Ĥph)]Â exp[−τ(Ĥpar + Ĥph)]. Index “con” means that
expansion contains only connected terms where no one integral over internal
time variables {τ ′1, τ ′2, . . .} can be factorized.
Vick theorem expresses matrix element of time-ordered operators as a
sum of terms, each is a factor of matrix elements of pairs of operators, and
expansion (24) becomes an infinite series of integrals with an ever increasing
number of integration variables
Gk(τ) =
m=0,2,4...
dx′1 · · · dx′m D(ξm)m (τ ; {x′1, . . . , x′m}) . (25)
Here index ξm stands for different Feynman diagrams (FDs) of the same
order m. Term with m = 0 is the GF of the noninteracting QP G
Function D(ξm)m (τ ; {x′1, . . . , x′m}) of any order m can be expressed as a fac-
tor of GFs of noninteracting quasiparticle, GFs of phonons, and interaction
vortexes V (k,q). For the simplest case of Hamiltonian system expressions
for GFs of QP G
(τ2 − τ1) = exp [−ǫ(k)(τ2 − τ1)] (τ2 > τ1) and phonons
q (τ2 − τ1) = exp [−ωq(τ2 − τ1)] (τ2 > τ1) are well known.
An important feature of the DMC method, which is distinct from the
row of other exact numerical approaches, is the explicit possibility to include
renormalized GFs into exact expansion without any change of the algorithm.
For example, if a damping of QP, caused by some interactions not included
in the Hamiltonian, is known, i.e. retarded self-energy of QP Σret(k, ω) is
available, renormalized GF
(τ) =
dωe−ωτ
ImΣret(k, ω)
[ω − ǫ(k)−ReΣret(k, ω)]2 + [ImΣret(k, ω]2
can be introduced instead of bare GF G
(τ). Explicit rules for evaluation of
D(ξm)m do not depend on the order and topology of FD. GFs of noninteracting
(τ2−τ1) (or G̃(0)k (τ2−τ1)) with corresponding times and momenta are
ascribed to horizontal lines and noninteracting GFs of phonon D
q (τ2 − τ1)
(multiplied by the factor of corresponding vortexes V (k′,q)V ∗(k′′,q)) are
attributed to phonon propagator arch (see Fig. 1a). Then, D(ξm)m is the factor
of all GSs. For example, expression for the weight of the second order term
(Fig. 1b) is the following
10 A. S. Mishchenko and N. Nagaosa
D2(τ ; {τ ′2, τ ′1,q}) =
|V (k,q)|2D(0)q (τ ′2 − τ ′1)G
(τ ′1)G
k−q(τ
2 − τ ′1)G
(τ − τ ′2) . (27)
τ’2τ’1
k k-q k
τ0 τ’2τ’4τ’1 τ’3
k k-q-q’ k-q k
Fig. 1. (a) Typical FD contributing into expansion (25). (b) FD of the second
order and (c) forth order.
The DMC process is a numerical procedure which, basing on the Metropo-
lis principle [81, 82], samples different FDs in the parameter space (τ,m, ξm, {x′m})
and collects statistics of external variable τ in a way that the result of this
statistics converges to exact GF Gk(τ). Although sampling of the internal
parameters of one term in (25) and switch between different orders is per-
formed within the the framework of one and the same numerical process,
it is instructive to start with the procedure of evaluation of a specific term
D(ξm)m (τ ; {x′1, . . . , x′m}).
Starting from a set {τ ; {x′1, . . . , x′m}}, an update x
(old)
l → x
(new)
l of an
arbitrary chosen parameter is suggested. This update is accepted or rejected
according to Metropolis principle. After many steps, altering all variables,
statistics of external variable converges to exact dependence of the term on
τ . Suggestion for new value of parameter x
(new)
l = Ŝ
−1(R) is generated by
random number R ∈ [0, 1] with a normalized distribution function W (xl)
in a range x
(min)
l < xl < x
(max)
l . There are only two restrictions for this
otherwise arbitrary function. First, new parameters x
(new)
l must not violate
FD topology, i.e., for example, internal time τ ′1 in Fig. 1c must be in the range
[x(min) = 0, x(max) = τ ′3]. Second, the distribution must be nonzero for the
whole, allowed by FD topology, domain. This ergodicity property is crucial
since it is necessary to sample the whole domain for convergence to exact
answer. At each step, update x
(old)
l → x
(new)
l is accepted with probability
Pacc = M (if M < 1) and always otherwise. The ratio M has the following
D(ξm)m (τ ; {x′1, . . . , x
(new)
l , . . . , x
m})/W (x
(new)
D(ξm)m (τ ; {x′1, . . . , x
(old)
l , . . . , x
m})/W (x
(old)
. (28)
For uniform distribution W = const =
(max)
l − x
(max)
, the probability
of any combination of parameters is proportional to the weight function D.
Spectroscopic Properties of Polarons by Exact Monte Carlo 11
However, for better convergence the distributionW (xnewl ) must be as close as
possible to the actual distribution given by function D(ξm)m ({. . . , x(new)l , . . . , }).
For sampling over FDs of all orders and topologies it is enough to introduce
two complimentary updates. Update A transforms FD D(ξm)m (τ ; {x′1, . . . , x′m})
into higher order FD D(ξm+2)m+2 (τ ; {x′1, . . . , x′m; q′, τ ′3, τ ′4}) with extra phonon
arch, connecting some time points τ ′3 and τ
4 by phonon propagator with mo-
mentum q′ (Fig. 1c). Note that the ratio of weights D(ξm+2)m+2 /D
m is not
dimensionless. Dimensionless Metropolis ratio
D(ξm+2)m+2 (τ ; {x′1, . . . , x′m; q′, τ ′, τ ′′})
D(ξm)m (τ ; {x′1, . . . , x′m})W (q′, τ ′, τ ′′)
. (29)
contains normalized probability function W (q′, τ ′, τ ′′), which is used for gen-
erating of new parameters5. Complementary update B, removing the phonon
propagator, uses ratio M−1 [45].
Note that all updates are local, i.e. do not depend on the structure of the
whole FD. Neither rules nor CPU time, needed for update, depends on the
FD order. DMC method does not imply any explicit truncation of FDs order
due to finite size of computer memory. Ever for strong coupling, where typical
number of phonon propagators Nph, contributing to result, is large, influence
of finite size of memory is not essential. Really, according to Central Limit
Theorem, number of phonon propagators obeys Gauss distribution centered
at N̄ph with half width of the order of
N̄ph [83]. Hence, if a memory for at
least 2N̄ph propagators is reserved, diagram order hardly surpasses this limit.
2.2 Stochastic Optimization Method
The problem of inverting of integral equation (13) is an ill posed problem.
Due to incomplete noisy information about GF Gk(τ), which is known with
statistic errors on a finite number of imaginary times in a finite range [0, τmax],
there is infinite number of approximate solutions which reproduce GF within
some range of deviations and the problem is to chose “the best one”. Another
problem, which is a stumbling block for decades, is the saw tooth noise insta-
bility. It occurs when solution is obtained by a naive method, e.g. by using
least-squares approach for minimizing deviation measure
D[L̃k(ω)] =
∫ τmax
∣∣∣Gk(τ) − G̃k(τ)
∣∣∣G−1k (τ)dτ . (30)
Here G̃k(τ) is obtained from approximate LF L̃k(ω) by applying of integral
operator G̃k(τ) = F
L̃k(ω)
in (13). Saw tooth instability corrupts LF in the
ranges where actual LF is smooth. Fast fluctuations of the solution L̃k(ω) often
5 The factor pA/pB depends on the probability to address add/remove processes.
12 A. S. Mishchenko and N. Nagaosa
have much larger amplitude than the value of actual LF Lk(ω). Standard tools
for saw tooth noise suppression are based on the early 60-es idea of Fillips-
Tikhonov regularization method [84, 85, 86, 87]. A nonlinear functional, which
suppresses large derivatives of approximate solution L̃k(ω), is added to the
linear deviation measure (30). Most popular variant of regularization methods
is the Maximal Entropy Method [61].
However, typical LF of a QP in a boson field consists of δ-functional peaks
and smooth incoherent continuum with a sharp border [45, 54]. Hence, sup-
pression of high derivatives, as a general strategy of the regularization method,
fails. Moreover, any specific implementation of the regularization method uses
predefined mesh in the ω space, which could be absolutely unacceptable for
the case of sharp peaks. If the actual location of a sharp peak is between
predefined discrete points, the rest of spectral density can be distorted be-
yond recognition. Finally, regularization Maximal Entropy approach requires
assumption of Gauss distribution of statistic errors in Gk(τ), which might be
invalid in some cases [61].
Recently, a Stochastic Optimization (SO) method, which circumvents
abovementioned difficulties, was developed [45]. The idea of the SO method
is to generate a large enough number M of statistically independent nonreg-
ularized solutions {L̃(s)
(ω)}, s = 1, ...,M , which deviation measures D(s) are
smaller than some upper limit Du, depending of the statistic noise of the GF
Gk(τ). Then, using linearity of expressions (13), (30), the final solution is
found as the average of particular solutions {L̃(s)
Lk(ω) = M
(ω) . (31)
Particular solution L̃
(ω) is parameterized in terms of sum
(ω) =
χ{Pt}(ω) (32)
of rectangles {Pt} = {ht, wt, ct} with height ht > 0, width wt > 0, and center
ct. Configuration
C = {{Pt}, t = 1, ...,K} , (33)
which satisfies normalization condition
t=1 htwt = 1, defines function
G̃k(τ). The procedure of generating particular solution starts from stochastic
choice of initial configuration Cinits . Then, deviation measure is optimized by
a randomly chosen consequence of updates until deviation is less than Du. In
addition to updates, which do not change number of terms in the sum (32),
there are updates which increase or decrease number K. Hence, since the
number of elements K is not fixed, any spectral function can be reproduced
with desired accuracy.
Spectroscopic Properties of Polarons by Exact Monte Carlo 13
Although each particular solution L̃
(ω) suffers from saw tooth noise at
the area of smooth LF, statistical independence of each solution leads to self
averaging of this noise in the sum (32). Note that suppression of noise happens
without suppression of high derivatives and, hence, sharp peaks and edges
are not smeared out in contrast to regularization approaches. Therefore, saw
tooth noise instability is defeated without corruption of sharp peaks and edges.
Moreover, continuous parameterization (32) does not need predefined mesh in
ω-space. Besides, since the Hilbert space of solution is sampled directly, any
assumption about distribution of statistical errors is not necessary.
SO method was successfully applied to restore LF of Fröhlich polaron
[45], Rashba-Pekar exciton-polaron [54], hole-polaron in t-J-model [53, 49],
and many-particle spin system [88]. Calculation of the optical conductivity of
polaron by SO method can be found in [48]. SO method appeared to be helpful
in cases when GF’s asymptotic limit, giving information about ground state,
can not be reached. For example, sign fluctuations of the terms in expansion
(25) for a hole in the t-J-model lead to poor statistics at large times [53],
though, SO method is capable of recovering energy and Z-factor even from
GF known only at small imaginary times [53].
2.3 Advantages and Drawbacks of DMC-SO Machinery
Among numerical methods, capable of obtaining quantitative results in the
problem of exciton (1) and (2), one can list time-dependent density func-
tional theory [89], Hanke-Sham technique of correcting particle-hole excita-
tion energy [90, 91], and approaches directly solving Bethe-Salpeter equation
[92, 93, 94]. The latter ones provide rather accurate information on the two-
particle GF. However, usage of finite mesh in direct/reciprocal space, which
is avoided in DMC method, leads to its’ failure in Wannier regime [93].
In contrast to DMC method, none of the traditional numeric methods
can give reliable results for measurable properties of excited states of polaron
at arbitrary range of electron-phonon interaction for the macroscopic system
in the thermodynamic limit. Exact diagonalization method [95, 96, 97, 98]
can study excited states though only on rather small finite size systems and
results of this method are not even justified in the variational sense in the
thermodynamic limit [99]. There is a batch of rather effective variational “ex-
act translation” methods [99, 100, 101, 102, 103] where basis is chosen in
the momentum space and, hence, the variational principle is applied in the
thermodynamic limit. Although these methods can reveal few discrete excited
states, its fail for long-range interaction and for dispersive, especially acoustic
phonons due to catastrophic growth of variational basis. A non perturbative
theory, which is able to give information about spectral properties in the ther-
modynamic limit at least for one electron, is Dynamical Mean Field Theory
[104, 105, 106, 107]. However it gives an exact solution only in the case of
infinite dimension which does not correspond to a realistic system and can be
considered only as a guide for extrapolation to finite dimensions [108].
14 A. S. Mishchenko and N. Nagaosa
Recently developed cluster perturbation theory, where exact diagonaliza-
tion of a cluster is further improved by taking into account inter-cluster inter-
action [109, 110, 111, 112, 113], is applicable for study of the excited states,
but limited to one-dimensional lattices or two-dimensional systems with short-
range interaction. Traditional density-matrix renormalization group method
[114, 115, 116, 117, 118] is very effective though mostly limited to one-
dimensional systems and ladders. Finally, recently developed path integral
quantum Monte Carlo algorithm [119, 120, 121, 122] is valid for any dimen-
sion and properly takes into account quasi long-range interactions [123]. Path
integral method is capable of obtaining the density of states [119, 120] and
isotope exponents [121, 124]. However calculations of measurable characteris-
tics of excited states, such as ARPES or optical conductivity, by this method
were never reported.
In conclusion, none of methods, except DMC-SO combination, can obtain
at the moment approximation-free results for measurable physical quantities
for a few QPs interacting with a macroscopic bosonic bath in the thermody-
namic limit. Indeed, there are limitations of the DMC and SO methods. DMC
method does not work in many-fermion systems due to sign problem and SO
method fails at high temperatures, comparable to energies of dominant spec-
tral peaks, because even very small statistical noise of GFs turns Fredholm
equation (13) into essentially “ill defined” problem [84].
3 Spectral Properties of the Fröhlich Polaron
Before development of DMC-SO methods, the information on the excited
states of polaron models, especially the Fröhlich one, was very limited. Knowl-
edge of LF was based on results of infinite-dimensions approximation [125],
exact diagonalization [126, 96, 97, 97], or strong coupling expansion [127]. No
one of the above techniques was capable of obtaining the LF of polaron with-
out approximations, especially for long-range interaction where difficulties of
traditional numerical methods dramatically increase. In a similar way, optical
conductivity (OC) of Fröhlich model was known only in strong coupling ex-
pansion approximation [128], within the framework of the perturbation theory
[129], or was based on the variational Feynman path integral technique [75].
In this sect. we consider exact DMC-SO results on LF [45] and OC [48, 57] of
Fröhlich polaron model.
3.1 Lehmann Function of the Fröhlich Polaron
The perturbation theory expression for the high-energy part (ω > 0) of the
LF for arbitrary interaction potential V (| q |) reads [45] (frequency of the
optical phonon ωph is set to unity)
Lk=0(ω > 0) =
ω − 1
| V (
2(ω − 1)) |2 θ(ω − 1) . (34)
Spectroscopic Properties of Polarons by Exact Monte Carlo 15
Low-energy part of the LF for the short-range interaction V (| q |) =
0 2 4 6
0.000
0.002
0.004
0.006
0 2 4 6
2 4 6
2 4 6
2 4 6
1.0 1.2
L (a)
Fig. 2. Comparison of the numerical results (solid lines) and the perturbation
theory (dashed lines) for the LFs of the Fröhlich model with α = 0.05 (a) and the
short-range interaction model with α = 0.05 and κ = 1 (b). LFs of Fröhlich polaron
for α = 0.5 (c), α = 1 (d) and α = 2 (e). Energy is measured from that of the
ground state of the polaron. The initial fragment of the LF for α = 1 is shown in
the inset (f).
(q2 + κ2)−1/2 , reducing to the Fröhlich one when κ→ 0, is
Lk=0(ω < 0) =
ω + α
. (35)
Comparison of low-energy parts of the LF of the Fröhlich model, obtained
by DMC-SO and taken from (35), shows perfect agreement for α = 0.05: the
accuracy for the polaron energy and Z-factor is about 10−4. On the other hand,
high-energy part of numeric result (Fig. 2) significantly deviates from that of
the analytic expression (35). This is not surprising since for Fröhlich polaron
the perturbation theory expression is diverging as ω → ωph and, therefore
the perturbation theory breaks down. When perturbation theory is obviously
valid, e.g. for the case of finite κ = 1, there is a perfect agreement between
analytic expression and DMC-SO results (Fig. 2b). Note that the high-energy
part of Lk=0(ω) is successfully restored by SO method despite the fact that
the total weight of the feature for α = 0.05 is less than 10−2.
The main deviation of the actual LF from the perturbation theory result
is the extra broad peak in the actual LF at ω ∼ 3.5. To study this feature
Lk=0(ω) was calculated for α = 0.5, α = 1, and α = 2 (Fig. 2c-e). The peak
16 A. S. Mishchenko and N. Nagaosa
0 4 8 12
0 4 8 12
0 4 8 12
=4 =6
Fig. 3. Evolution of spectral density with α in the cross-over region from interme-
diate to strong couplings. The polaron ground state peak is shown only for α = 8.
Note that the spectral analysis still resolves it, despite its very small weight < 10−3.
is seen for higher values of the interaction constant and its weight grows with
α. Near the threshold, ω = 1, LF demonstrates the square-root dependence
ω − 1 (Fig. 2f).
To trace the evolution of the peak at higher values of α the LF was cal-
culated [45] for α = 4, α = 6, and α = 8 (Fig. 3). At α = 4 the peak at
ω ∼ 4 already dominates. Moreover, a distinct high-energy shoulder appears
at α = 4, which transforms into a broad peak at ω ∼ 8.5 in the LF for α = 6.
The LF for α = 8 demonstrates further redistribution of the spectral weight
between different maxima without significant shift of the peak positions.
3.2 Optical Conductivity of the Fröhlich Polaron: Validity of the
Franck-Condon Principle in the Optical Spectroscopy
The FC principle [130, 131] and its validity have been widely discussed in stud-
ies of optical transitions in atoms, molecules [132, 133], and solids [134, 9].
Generally, the FC principle means that if only one of two coupled subsys-
tems, e.g. an electronic subsystem, is affected by an external perturbation,
the second subsystem, e.g., the lattice, is not fast enough to follow the re-
construction of the electronic configuration. It is clear that the justification
for the FC principle is the short characteristic time of the measurement pro-
cess τmp ≪ τic, where τmp is related to the energy gap between the initial
and final states, ∆E, through the uncertainty principle: τmp ≃ h̄/(∆E) and
τic is the time necessary to adjust the lattice when the electronic component
is perturbed. Then, the spectroscopic response considerably depends on the
value of the ratio τmp/τic For example, in mixed valence systems, where the
ionic valence fluctuates between configurations f5 and f6 with characteristic
time τic ≈ 10−13s, spectra of fast and slow experiments are dramatically dif-
ferent [135, 136]. Photoemission experiments with short characteristic times
τmp ≈ 10−16s (FC regime), reveal two lines, corresponding to f5 and f6
Spectroscopic Properties of Polarons by Exact Monte Carlo 17
states. On the other hand slow Mössbauer isomer shift measurements with
τmp ≈ 10−9s show a single broad peak with mean frequency lying between
signals from pure f5 and f6 shells. Finally, according to paradigm of mea-
surement process time, magnetic neutron scattering with τmp ≈ τic revealed
both coherent lines with all subsystems dynamically adjusted and broad inco-
herent remnants of strongly damped excitation of f5 and f6 shells [137, 138].
Actually, the meaning of the times τic and τmp varies with the system and
with the measurement process.
To study the interplay between measurement process time τmp and ad-
justment time τic, the OC of the Fröhlich polaron was studied in [57] from
the weak to the strong coupling regime by three methods. DMC method gives
numerically exact answer which is compared with memory function formalism
(MFF), which is able to take dynamical lattice relaxation into account, and
strong coupling expansion (SCE) which assumes FC approach. It was found
that near critical coupling αc ≈ 8.5 a dramatic change of the OC spectrum
occurs: dominating peak of OC splits into two satellites. In this critical regime
the upper (lower) one quickly decreases (increases) it’s spectral weight as the
value of coupling constant increases. Besides, while OC follows prediction of
MFF at α < αc, its dependence switches to that predicted by SCE for larger
couplings. It was concluded that, for the OC measurement of polaron, the
adjustment time τic ≈ h̄/D is set by typical nonadiabatic energy D. Nona-
diabaticity destroys FC classification at α < αc while FC principle rapidly
regains its validity at large couplings due to fast growth of energy separation
between initial and final states of optical transitions.
Comparison of exact DMC-SO data for OC with existing results of ap-
proximate methods showed [48] that the Feynman path integral technique
[75] of Devreese, De Sitter, and Goovaerts, where OC is calculated starting
from the Feynman variational model [139], is the only successfully describing
evolution of the energy of the main peak in OC with coupling constant α (see
[58]). However, starting from the intermediate coupling regime this approach
fails to reproduce the peak width. Subsequently, the path integral approach
was rewritten in terms of MFF [140]. Then, in [57] the extended MFF for-
malism, which introduces dissipation processes fixed by exact sum rules, was
developed [141].
As shown in Fig. 4a, in the weak coupling regime, the MFF, with or with-
out dissipation, is in very good agreement with DMC data, showing significant
improvement with respect to weak coupling perturbation approach [129] which
provides a good description of OC spectra only for very small values of α. For
1 ≤ α ≤ 8, where standard MFF fails to reproduce peak width (Fig. 4b-d)
and even the peak position (Fig. 4c), the damping, introduced to extended
MFF scheme, becomes crucial. Results of extended MFF are accurate for the
peak energy and quite satisfactory for the peak width (Fig. 4b-e). Note that
the broadening of the peak in DMC data is not a consequence of poor quality
of analytic continuation procedure since DMC-SO methods is capable of re-
vealing such fine features as 2- and 3-phonon thresholds of emission (Fig. 4b).
18 A. S. Mishchenko and N. Nagaosa
0 2 4 6 8
0 2 4 6 8
0 2 4 6 8 10 12
0 5 10 15
0 5 10 15
(a) (b)
=5.25
Fig. 4. Comparison of the optical conductivity calculated by DMC method (circles),
extended MFF (solid line), and DSG [75, 140] (dashed line) for different values of
α. The slanted arrows indicate 2- and 3-phonon thresholds of absorption.
0 5 10 15 20
0 10 20 30
0 10 20 30 40
0 5 10 15
0 5 10 15
(a) (b)
Fig. 5. (a)-(c) Comparison of the optical conductivity calculated within the DMC
method (circles), the extended MFF (solid line), and SCE (dotted line) for different
values of α. (d) The energy of lower- and higher-frequency features (circles and tri-
angles, respectively) compared with the FC transition energy with the SCE (dashed
line) and with the energy of the peak obtained from the extended MFF (solid line).
In the inset, the weights of FC and adiabatically connected transitions are shown as
a function of α (for η = 1.3.)
However, a dramatic change of OC occurs around critical coupling strength
αc ≈ 8.5. The dominating peak of OC splits into two ones, the energy of lover
one corresponding to the predictions of SCR expansion and that of upper
one obeying extended MFF value (Fig. 5a). The shoulder, corresponding to
dynamical extended MFF contribution, rapidly decreases it’s intensity with
increase of α and at large α (Fig. 5b-c) the OC is in good agreement with
strong coupling expansion, assuming FC scheme. Finally, comparing energies
of the peaks, obtained by DMC, extended MFF and FC strong coupling ex-
pansion (Fig. 5d), we conclude that at critical coupling αc ≈ 8.5 the spectral
properties rapidly switch from dynamic, when lattice relaxes at transition, to
FC regime, where nuclei are frozen in initial configuration.
In order get an idea of the FC breakdown authors of [57] consider the fol-
lowing arguments. The approximate adiabatic states are not exact eigenstates
of the system. These states are mixed by nondiagonal matrix elements of the
Spectroscopic Properties of Polarons by Exact Monte Carlo 19
nonadiabatic operator D and exact eigenstates are linear combinations of the
adiabatic wavefunctions. Being interested in the properties of transition from
ground (g) to an excited (ex) state, whose energy correspond to that of the OC
peak, it is necessary to consider mixing of only these states and express exact
wavefunctions as a linear combinations [142, 143] of ground and excited adi-
abatic states. The coefficients of superposition are determined from standard
techniques [142, 143] where nondiagonal matrix elements of the nonadiabatic
operator [142] are expressed in terms of matrix elements of the kinetic energy
operator M , the gap between excited and ground state ∆E = Eex − Eg and
the number nβ of phonons in adiabatic state:
D± =M(∆E)−1
nβ + 1/2± 1/2 +M2(∆E)−2. (36)
The extent to which lattice can follow transition between electronic states,
depends on the degree of mixing between initial and final exact eigenstates
through the nonadiabatic interaction. If initial and ground states are strongly
mixed, the adiabatic classification has no sense and, therefore, the FC pro-
cesses have no place and lattice is adjusted to the change of electronic states
during the transition. In the opposite limit adiabatic approximation is valid
and FC processes dominate. The estimation for the weight of FC component
IFC [57] is equal to unity in the case of zero mixing and zero in the case
of maximal mixing. The weight of adiabatically connected (AC) transition
IAC = 1− IFC is defined accordingly. Non-diagonal matrix element M is pro-
portional to the root square of α with a coefficient η of the order of unity.
In the strong coupling regime, assuming that ∆E ≈ γα2 (γ ≈ 0.1 from MC
data), and nβ ≈ ∆E (nβ ≫ 1), one gets
IFC =
1 + 4(τmp/τic)
, (37)
where τmp = 1/∆E and τic = 1/D. For η of the order of unity one obtains
qualitative description of a rather fast transition from AC- to FC-dominated
transition, when IFC and IAC exchange half of their weights in the range
of α from 7 to 9. The physical reason for such quick change is the faster
growth of energy separation ∆E ∼ α2 compared to that of the matrix ele-
ment M ∼ α1/2. Finally, for large couplings, initial and final states become
adiabatically disconnected. The rapid AC-FC switch has nothing to do with
the self-trapping phenomenon where crossing and hybridization of the ground
and an excited states occurs. This phenomenon is a property of transition
between different states and related to the choice whether lattice can or can
not follow adiabatically the change of electronic state at the transition.
4 Self-Trapping
In this section we consider the self-trapping (ST) phenomenon which, due to
essential importance of many-particle interaction of QP with bosonic bath of
20 A. S. Mishchenko and N. Nagaosa
macroscopic system, was never addressed by exact method before. We start
with a basic definition of the ST phenomenon and introduce the adopted
criterion for it’s existence. Then, generic features of ST are demonstrated
on a simple model of Rashba-Pekar exciton-polaron in Sect. 4.1. It is shown
in Sect. 4.2 that the criterion is not a dogma since even in one dimensional
system, where ST is forbidden by criterion of existence, one can observe all
main features of ST due to peculiar nature of electronic states.
In general terms [7, 80], ST is a dramatic transformation of a QP properties
when system parameters are slightly changed. The physical reason of ST is a
quantum resonance, which happens at some critical interaction constant γc,
between “trapped” (T) state of QP with strong lattice deformation around
it and “free” (F) state. Naturally, ST transition is not abrupt because of
nonadiabatic interaction between T and F states and all properties of the QP
are analytic in γ [144]. At small γ < γc, ground state is an F state which is
weakly coupled to phonons while excited states are T states and have a large
lattice deformation. At critical couplings γ ≈ γc a crossover and hybridization
of these states occurs. Then, for γ > γc the roles of the states exchange. The
lowest state is a T state while the upper one is an F state.
First, and up to now the only quantitative criterion for ST existence was
given in terms of the ground state properties in the adiabatic approximation.
This criterion considers stability of the delocalized state in undistorted lattice
∆ = 0 with respect to the energy gain due to lattice distortion ∆′ 6= 0. ST
phenomenon occurs when completely delocalized state with∆ = 0 is separated
from distorted state with ∆′ 6= 0 by a barrier of adiabatic potential. One of
these states is stable while another one is meta-stable. The criterion of barrier
existence is defined in terms of the stability index
s = d− 2(1 + l) , (38)
where d is the system dimensionality. Index l determines the range of the force
limq→0 ψ(q) ∼ q−l, where ψ(R) is the kernel of interaction U(Rn) = ψ(Rn −
Rn′)ν(Rn′) connecting potential U(Rn) with generalized lattice distortion
ν(Rn′) [7]. The barrier exists for s > 0 and does not exist for s < 0. The
discontinuous change of the polaron state, i.e. ST, occurs in the former case
while does not happen in the latter case. When s = 0, this scaling argument
alone can not conclude the presence or absence of the ST and more detailed
discussion for each model is needed.
4.1 Typical Example of the Self-Trapping: Rasba-Pekar
Exciton-Polaron
Classical example of a system with ST phenomenon is the three dimensional
continuous Rasba-Pekar exciton-polaron in the approximation of intraband
scattering, i.e. when polar electron-phonon interaction (EPI) with dispersion-
less optical phonons ωph = 1 does not change the wave function of internal
Spectroscopic Properties of Polarons by Exact Monte Carlo 21
0 5 10 15 20
0 5 10 15 20
0 5 10 15 20
0 10 20 30 40
Fig. 6. The ground-state energy (a), effective mass (b), and average number of
phonons as function of coupling constant (c). Partial weights of n-phonon states (d)
in the polaron ground state (k = 0) at γ = 18 (circles), γ = 18.35 (squares), and
γ = 19 (diamonds). Dotted line in panel (a) is the result of strong coupling limit
and dashed line is the result of perturbation theory.
electron-hole motion. System is defined as a structureless QP with dispersion
ǫ(k) = k2/2 and short range coupling to phonons [54, 7]. General criterion of
the existence of ST is satisfied for three dimensional system with short range
interaction [54, 7, 50] and, thus, one expects to observe typical features of the
phenomenon.
It is shown [54] that in the vicinity of the critical coupling γc ≈ 18 the
average number of phonons 〈N〉 in (18) and effective mass m∗ quickly in-
crease in the ground state by several orders of magnitude (Fig. 6b-c). Besides,
a quantum resonance between polaronic phonon clouds of F and T state is
demonstrated. Distribution of partial n-phonon contributions Z(k=0)(n) in
(17) has one maximum at n = 0 in the weak coupling regime, which cor-
responds to weak deformation, and one maximum at n ≫ 1 in the strong
coupling regime, which is the consequence of a strong lattice distortion. How-
ever, due to F-T resonance there are two distinct peaks at n = 0 and n ≫ 1
for γ ≈ γc (Fig. 6d).
Near the critical coupling γc the LF of polaron has several stable states
(Fig. 7 a-b) below the threshold of incoherent continuum Egs+ωph. Any state
above the threshold is unstable because emission of a phonon with transition
to the ground state at k = 0 with energy Egs is allowed. On the other hand,
decay is forbidden by conservation laws for states below the threshold. De-
pendence of the energies of ground and excited resonances on the interaction
constant resembles a picture of crossing of several states interacting with each
other (Fig. 7c).
According to the general picture of the ST phenomenon, lowest F state in
the weak coupling regime at k = 0 has small effective mass m∗ ≈ m of the
order of the bare QP mass m. To the contrary, the effective mass of excited
state m∗ ≫ m is large. Hence, below the critical coupling the energy of the
F state, which is lowest at k = 0, has to reach a flat band of T state at
some momentum. Then, F and T state have to hybridize and exchange in
22 A. S. Mishchenko and N. Nagaosa
0 1 2 3
17 18 19
0 2 4
Fig. 7. LF L(k=0)(ω) at critical coupling γ = γc (a) and for γ > γc (b). Energy is
counted from the polaron ground state. (c) Dependence of energy of ground state
(squares) and stable excited states (circles, diamonds, and triangles) on the coupling
constant. Dashed line is the threshold of the incoherent continuum. Dependence
of energy (d) and average number of phonons (e) on the wave vector at γ < γc
(circles and rectangles). Dashed line is the effective mass approximation E(k) =
Egs + k
2/2m∗ for parameters Egs = −3.7946 and m∗ = 2.258, obtained by DMC
estimators for given value of γ. Dotted line is a parabolic dispersion law which is
fitted to last 4 points of energy dispersion curve with parameters E1 = −3.5273 and
m∗1 = 195. Empty square is the energy of first excited stable state at zero momentum
obtained by SO method.
energy. DMC data visualize this picture (Fig. 7 d-e). After F state crosses the
flat band of excited T state, the average number of phonons increases and
dispersion becomes flat.
It is natural to assume that above the critical coupling the situation is
opposite: ground state is the T state with large effective mass while excited
F state has small, nearly bare, effective mass. Indeed, this assumption was
confirmed in the framework of another model which is considered in Sect. 6.1.
Moreover, it was shown that in the strong coupling regime excited resonance
inherits not only bare effective mass around k = 0 but the whole dispersion
law of the bare QP [49].
4.2 Degeneracy Driven Self-Trapping
According to the criterion (38), ST phenomenon in one-dimensional sys-
tem does not occur. Although this statement is probably valid for the
case of single band in relevant energy range, it is not the case for the
generic multi-band cases. This fact has been unnoticed for many years,
Spectroscopic Properties of Polarons by Exact Monte Carlo 23
which prevented the proper explanation of puzzling physics of quasi-one-
dimensional compound Anthracene-PMDA, although it’s optical properties
[65, 145, 146, 147, 66, 148] directly suggested resonance of T and F states.
The reason is that in Anthracene-PMDA, in contrast to conditions at which
criterion (38) is obtained, there are two, nearly degenerate exciton bands.
Then, one can consider quasi-degenerate self-trapping mechanism when ST
phenomenon is driven by nondiagonal interaction of phonons with quaside-
generate exciton levels [52]. Such mechanism was already suggested for expla-
nation of properties of mixed valence systems [143] though it’s relevance was
never proved by an exact approach.
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0 10 20
Fig. 8. Dependence of energy (a) and average number of phonons (b) on the non-
diagonal coupling constant λ12 at λ11 = 0 and λ22 = 0.25. Phonon distributions in
polaron cloud below ST point at λ12 = 1.0125 (c), at ST point at λ12 = 1.0435 (d),
and above ST coupling at λ12 = 1.0625 (e).
The minimal model to demonstrate the mechanism of quasi-degenerate
self-trapping involves one optical phonon branch with frequency ωph = 0.1
and two exciton branches with energies ǫ1,2(q) = ∆1,2 + 2[1 − cos(q)], where
∆1 = 0 and ∆2 = 1. Presence of short range diagonal γ22 and nondiagonal
γ12 interactions (with corresponding dimensionless constants λ22 = γ
22/(2ω)
and λ12 = γ
12/(2ω)) leads to classical self-trapping behavior even in one-
dimensional system [52] (see Fig. 8).
5 Exciton
Despite numerous efforts over the years, there has been no rigorous tech-
nique to solve for exciton properties even for the simplest model (1)-(2) which
treats electron-electron interactions as a static renormalized Coulomb poten-
tial with averaged dynamical screening. The only solvable cases are the Frenkel
small-radius limit [67] and the Wannier large-radius limit [68] which describe
molecular crystals and wide gap insulators with large dielectric constant, re-
spectively. Meanwhile, even the accurate data for the limits of validity of the
24 A. S. Mishchenko and N. Nagaosa
Wannier and Frenkel approximations have not been available. As discussed in
Sects. 1.2 and 2.3, semianalytic approaches has little to add to problem when
quantitative results are needed whereas traditional numerical methods fail to
reproduce them even in the Wannier regime. To the contrary, DMC results
do not contain any approximation.
0.0 0.5 1.0 1.5 2.0
Bandwidth
0.0 20.0 40.0 60.0 80.0
Bandwidth
0 5 10 15 20 25
Coordinate sphere
−0.05
0 200 400 600 800 1000
Electron−hole distance
0 2 4 6 8 10
Coordinate sphere
0 1 2 3 4
Coordinate sphere
Fig. 9. Panel (a): dependence of the exciton binding energy on the bandwidth
Ec = Ev for conduction and valence bands. The dashed line corresponds to the
Wannier model. The solid line is the cubic spline, the derivatives at the right and left
ends being fixed by the Wannier limit and perturbation theory, respectively. Inset
in panel (a): the initial part of the plot. Panel (b): the wave function of internal
motion in real space for the optically forbidden monopolar exciton. Panels (c)-(e):
the wave function of internal motion in real space: (c) Wannier [Ec = Ev = 60]; (d)
intermediate [Ec = Ev = 10]; (e) near-Frenkel [Ec = Ev = 0.4] regimes. The solid
line in the panel (c) is the Wannier model result while solid lines in other panels are
to guide the eyes only.
To study conditions of validity of limiting regimes by DMC method,
electron-hole spectrum of three dimensional system was chosen in the form
of symmetric valence and conduction bands with width Ec and direct gap Eg
Spectroscopic Properties of Polarons by Exact Monte Carlo 25
at zero momentum [46]. For large ratio W = Ec/Eg, when W > 30, exci-
ton binding energy is in good agreement with Wannier approximation results
(Fig. 9a) and probability density of relative electron-hole motion corresponds
(Fig. 9c) to hydrogen-like result. The striking result is the requirement of
rather large valence and conduction bandwidths (W > 20) for applicability of
Wannier approximation. For smaller values ofW the binding energy and wave
function of relative motion (Fig. 9d) deviate from large radius results. In the
similar way, conditions of validity of Frenkel approach are rather restricted
too. Moreover, even strong localization of wave function does not guarantee
good agreement between exact and Frenkel approximation result for binding
energy. At 1 < W < 10 the wave function is already strongly localized though
binding energy considerably differs from Frenkel approximation result. For
example, at W = 0.4 relative motion is well localized (Fig. 9e) whereas the
binding energy of Frenkel approximation is two times larger than exact result
(Inset in Fig. 9a).
A study of conditions necessary for formation of charge transfer exciton in
three dimensional systems is crucial to finalize protracted discussion of numer-
ous models concerning properties of mixed valence semiconductors [149]. A
decade ago unusual properties of SmS and SmB6 were explained by invoking
the excitonic instability mechanism assuming charge-transfer nature of the
optically forbidden exciton [150, 151]. Although this model explained quanti-
tatively the phonon spectra [152, 153], optical properties [154, 155], and mag-
netic neutron scattering data [138], it’s basic assumption has been criticized
as being groundless [156, 157]. To study excitonic wavefunction, dispersions
of valence and conduction bands were chosen as it is typical for mixed valence
materials: almost flat valence band is separated from broad conduction band,
having maximum in the centre and minimum at the border of Brillouin zone
[46]. Results presented in Fig. 9b support assumption of [150, 151] since wave
function of relative motion has almost zero on-site component and maximal
charge density at near neighbors.
6 Polarons in Undoped High Temperature
Superconductors
It is now well established that the physics of high temperature superconduc-
tors is that of hole doping a Mott insulator [158, 159, 160]. Even a single
hole in a Mott insulator, i.e. a hole in an antiferromagnet in case of infinite
Hubbard repulsion U , is substantially influenced by many-body effects [10] be-
cause it’s jump to a neighboring site disturbs antiferromagnetic arrangement
of spins. Hence a thorough understanding of the dynamics of doped holes in
Mott insulators has attracted a great deal of recent interest. The two major
interactions relevant to the electrons in solids are electron-electron interac-
tions (EEI) and electron-phonon interactions (EPI). The importance of the
former at low doping is no doubt essential since the Mott insulator is driven
26 A. S. Mishchenko and N. Nagaosa
by strong Hubbard repulsion, while the latter was considered to be largely
irrelevant to superconductivity based on the observations of a small isotope
effect on the optimal Tc [161] and an absence of a phonon contribution to the
resistivity (for review see [162]).
On the other hand, there are now accumulating evidences that the EPI
plays an important role in the physics of cuprates such as (i) an isotope effect
on superfluid density ρs and Tc away from optimal doping [163], (ii) neutron
and Raman scattering [164, 165, 166] experiments showing strong phonon soft-
ening with both temperature and hole doping, indicating that EPI is strong
[167, 168]. Furthermore, the recent studies of cuprates by the angle resolved
photoemission spectroscopy (ARPES), which spectra are proportional to the
LF (7) [32], resulted in the discovery of the dispersion ”kinks” at around 40-
70meV measured from the Fermi energy, in the correct range of the relevant
oxygen related phonons [169, 170, 171]. These particular phonons - oxygen
buckling and half-breathing modes are known to soften with doping [172, 164]
and with temperature [170, 171, 172, 164, 165, 166] indicating strong cou-
pling. The quick change of the velocity can be predicted by any interaction of
a quasiparticle with a bosonic mode, either with a phonon [170, 171] or with
a collective magnetic resonance mode [173, 174, 175]. However, the recently
discovered “universality” of the kink energy for LSCO over the entire doping
range [176] casts doubts on the validity of the latter scenario as the energy
scale of the magnetic excitation changes strongly with doping.
Besides, measured in undoped high Tc materials ARPES revealed appar-
ent contradiction between momentum dependence of the energy and linewidth
of the QP peak. On the one hand the experimental energy dispersion of the
broad peak in many underdoped compounds [31, 177] obeys the theoretical
predictions [178, 179], whereas the experimental peak width is comparable
with the bandwidth and orders of magnitude larger than that obtained from
theory of Mott insulator [53]. Early attempts to interpret this anomalously
short lifetime of a hole by an interaction with additional nonmagnetic bosonic
excitations, e.g. phonons [180], faced generic question: is it possible that in-
teraction with media leaves the energy dispersion absolutely unrenormalized,
while, induces a decay which inverse life-time is comparable or even larger
than the QP energy dispersion? A possibility of an extrinsic origin of this
width can be ruled out since the doping induces further disorder, while a
sharper peak is observed in the overdoped region.
In order to understand whether phonons can be responsible for peculiar
shape of the ARPES in the undoped cuprates, the LF of an interacting with
phonons hole in Mott insulator was studied by DMC-SO [49]. The case of the
LF of a single hole corresponds to the ARPES in an undoped compound. For
a system with large Hubbard repulsion U , when U is much larger than the
typical bandwidth W of noninteracting QP, the problem reduces to the t-J
model [181, 182, 158, 11]
Spectroscopic Properties of Polarons by Exact Monte Carlo 27
Ĥt-J = −t
〈ij〉s
iscjs + J
(SiSj − ninj/4) . (39)
Here cjσ is projected (to avoid double occupancy) fermion annihilation op-
erator, ni (< 2) is the occupation number, Si is spin 1/2 operator, J is an
exchange integral, and 〈ij〉 denotes nearest-neighbor sites in two dimensional
square lattice. Different theoretical approaches revealed [158, 183, 53] basic
properties of the LF. The LF has a sharp peak in the low energy part of the
spectrum which disperses with a bandwidth WJ/t ∼ 2J and, therefore, the
large QP width in experiment can not be explained. More complicated tt′t′′-J
model takes into account hoppings to the second t′ and third t′′ nearest neigh-
bors and, hence, dispersion of the hole changes [184, 185, 186, 178, 179, 32].
However, for parameters, which are necessary for description of dispersion
in realistic high Tc superconductors [31, 178], peak in the low energy part
remains sharp and well defined for all momenta [187].
After expressing spin operators in terms of Holstein-Primakoff spin wave
operators and diagonalizing the spin part of Hamiltonian (39) by Fourier
and Bogoliubov transformations [188, 10, 189, 190], tt′t′′-J Hamiltonian is
reduced to the boson-holon model, where hole (annihilation operator is hk)
with dispersion ε(k) = 4t′ cos(kx) cos(ky)+2t
′′[cos(2kx)+cos(2ky)] propagates
in the magnon (annihilation operator is αk) bath
Ĥ0t-J =
ε(k)h
αk (40)
with magnon dispersion ωk = 2J
1− γ2
, where γk = (cos kx + cos ky)/2.
The hole is scattered by magnons as described by
Ĥh-mt-J = N
hk−qαk + h.c.
with the scattering vertex Mk,q. Parameters t, t
′ and t′′ are hopping ampli-
tudes to the first, second and third near neighbors, respectively. If hopping
integrals t′ and t′′ are set to zero and bare hole has no dispersion, the problem
(40-41) corresponds to t-J model.
Short range interaction of a hole with dispersionless optical phonons
Ĥe-ph = Ω0
bk of the frequency Ω0 is introduced by Holstein Hamil-
tonian
Ĥe-ph = N−1/2
hk−qbq + h.c.
, (42)
where σ is the momentum and isotope independent coupling constant, M is
the mass of the vibrating lattice ions, and Ω0 is the frequency of dispersionless
phonon. The coefficient in front of square brackets is the standard Holstein in-
teraction constant γ = σ/
(2MΩ0). In the following we characterize strength
of EPI in terms of dimensionless coupling constant λ = γ2/4tΩ0. Note, if in-
teraction with magnetic subsystem (41) is neglected and hole dispersion ε(k)
28 A. S. Mishchenko and N. Nagaosa
is chosen in the form ε(k) = 2t[cos(kx) + cos(ky)], the problem (40), (42) cor-
responds to standard Holstein model where hole with near neighbor hopping
amplitude t interacts with dispersionless phonons.
We consider the evolution of ARPES of a single hole in t-J-Holstein model
(40)-(42) from the weak to the strong coupling regime and dispersion of the LF
in the strong coupling regime in Sect. 6.1. It occurs that properties of the LF in
the strong coupling regime of the EPI explain the puzzle of broad lineshape
in ARPES in underdoped high Tc superconductors. Therefore, in order to
suggest a crucial test for the mechanism of phonon-induced broadening, we
present calculations of the effect of the isotope substitution on the ARPES in
Sect. 6.2.
6.1 Spectral Function of a Hole Interacting with Phonons in the
t-J Model: Self-Trapping and Momentum Dependence
Previously, the LF of t-J-Holstein model was studied by exact diagonalization
method on small clusters [191] and in the non-crossing approximation (NCA)6
for both phonons and magnons [192, 193]. However, the small system size in
exact diagonalization method implies a discrete spectrum and, therefore, the
problem of lineshape could not be addressed. The latter method omits the
FDs with mutual crossing of phonon propagators and, hence, is an invalid
approximation for phonons in strong and intermediate couplings of EPI. This
statement was demonstrated by DMC, which can sum all FDs for Holstein
model both exactly and in the NCA [49]. Exact results and those of NCA are
in good agreement for small values λ ≤ 0.4 and drastically different for λ > 1.
For example, for Ω0/t = 0.1 exact result shows a sharp crossover to strong
coupling regime for λ > λcH ≈ 1.2 whereas NCA result does not undergo such
crossover even at λ = 100. On the other hand, NCA is valid for interaction of a
hole with magnons since spin S=1/2 can not flip more than once and number
of magnons in the polaronic cloud can not be large. Note that the t-J-Holstein
model is reduced to problem of polaron which interacts with several bosonic
fields (3)-(4).
DMC expansion in [49] takes into account mutual crossing of phonon prop-
agators and, in the framework of partial NCA, neglects mutual crossing of
magnon propagators, to avoid sign problem. NCA for magnons is justified for
J/t ≤ 0.4 by good agreement of results of NCA and exact diagonalization on
small clusters [188, 10, 194, 195, 190]. Recently results of exact diagonalization
were compared in the limit of small EPI for t-J-Holstein model, boson-holon
model (40-42) without NCA, and boson-holon model with NCA [196]. Al-
though agreement is not so good as for pure t-J model, it was concluded that
NCA for magnons is still good enough to suggest that one can use NCA for a
qualitative description of the t-J-Holstein model.
6 NCA is equivalent to self-consistent Born approximation (SCBA)
Spectroscopic Properties of Polarons by Exact Monte Carlo 29
0.0 0.2 0.4 0.6
0.0 0.2 0.4
0.0 0.2 0.4
-2.50 -2.25 -2.00
-2 0 2 4
Fig. 10. (a) The LF of a hole in the ground state k = (π/2, π, 2) at J/t = 0.3 and
λ = 0. Low energy part of the LF of a hole in the ground state k = (π/2, π, 2) at
J/t = 0.3: (b) λ = 0; (c) λ = 0.3; (d) λ = 0.4; (e) λ = 0.46. Dependence on coupling
strength λ at J/t = 0.3: (f) energies of lowest LF resonances; (g) Z-factor of lowest
peak; (h) average number of phonons 〈N〉.
Figures 10a-e show low energy part of LF in the ground state at k =
(π/2, π/2) in the weak, intermediate, and strong coupling regimes of inter-
action with phonons. Dependence on the coupling constant of energies of
resonances (Fig. 10f), Zk=(π/2,π/2)-factor of lowest peak (Fig. 10g), and aver-
age number of phonons in the polaronic cloud 〈N〉 (Fig. 10h) demonstrates a
picture which is typical for ST (see [80, 54] and Sect. 4). Two states cross and
hybridize in the vicinity of critical coupling constant λct-J ≈ 0.38, Zk=(π/2,π/2)-
factor of lowest resonance sharply drops and average number of phonons in
polaronic cloud quickly rises. According to the general understanding of the
ST phenomenon, above the critical couplings λ > λct-J one expects that the
lowest state is dispersionless while the upper one has small effective mass.
This assumption is supported by the momentum dependence of the LF in
the strong coupling regime (Fig. 11a-e). Dispersion of upper broad shake-off
Franck-Condon peak nearly perfectly obeys relation
εk = εmin+WJ/t/5{[coskx+cos ky]2+[cos(kx+ky)+cos(kx−ky)]2/4}, (43)
which describes dispersion of the pure t-J model in the broad range of ex-
change constant 0.1 < J/t < 0.9 [194] (Fig. 11f). Note that this property of
the shake-off peak is general for the whole strong coupling regime (Fig. 11f).
Momentum dependence of the shake-off peak, reproducing that of the free
particle, is the direct consequence of the adiabatic regime. Actually, phonon
frequency Ω0 is much smaller than the coherent bandwidth 2J of the t-J
30 A. S. Mishchenko and N. Nagaosa
-2 0 2
-2.5 -2.0
-2.75
-2.50
-2.25
-2.00
-1.75
-10 -5 0 5 10
k=( /2, /2)
/t /t
k=( /4, /4)
k=(0, /4)
k=(0, )
k=( /2, /2)
Fig. 11. The LF of a hole at J/t = 0.3 and λ = 0.46: (a) full energy range
for k = (π/2, π/2); (b–e) low energy part for different momenta. Slanted arrows
show broad peaks which can be interpreted in ARPES spectra as coherent (C) and
incoherent (I) part. Vertical arrows in panels (b)–(e) indicate position of “invisible”
lowest resonance. (f) Dispersion of resonances energies at J/t = 0.3: broad resonance
(filled circles) and lowest polaron pole (filled squares) at λ = 0.46; broad resonance
(open circles) and lowest polaron pole (open squares) at λ = 0.4. The solid curves
are dispersions (43) of a hole in pure t-J model at J/t = 0.3 (WJ/t=0.3 = 0.6):
εmin = −2.396 (εmin = −2.52) for dotted (solid) line. Panel (g) shows ground state
potential Q2/2 (solid line), excited state potential without relaxation D + Q2/2
(dashed line), and relaxed excited state potential D + (Q − λ)2/2 − λ2/2 (dotted
line).
model, giving the adiabatic ratio Ω/2J = 1/6 ≪ 1. Besides, as experience
with the OC of the Fröhlich polaron (Sect. 3.2) shows, there is one more
important parameter in the strong coupling limit. Namely, the ratio between
measurement process time τmp = h̄/∆E where ∆E is the energy separation of
shake-off hump from the ground state pole, and that of characteristic lattice
time τ ≈ 1/Ω0 is much less than unity. Hence, fast photoemission probe sees
the ions frozen in one of possible configurations [197]. The LF in the FC limit
is a sum of transitions between a lower Elow(Q) and an upper Eup(Q) sheets
of adiabatic potential, weighted by the adiabatic wave function of the lower
sheet | ψlow(Q) |2 [198]. If EPI is absent both in initial Elow(Q) = Q2/2
and final Eup(Q) = D + Q2/2 states, the LF is peaked at the energy D.
Then, if there is EPI ∆Eup(Q) = −λQ only in the final state, i.e. when hole
is removed from the Mott insulator, the upper sheet of adiabatic potential
Eup(Q) = D− λ2/2 + (Q− λ)2/2 has the same energy D at Q = 0. Since the
probability function | ψlow(Q) |2 has maximum at Q = 0, the peak of the LF
broadens but it’s energy does not shift [198] (Fig. 11g).
Spectroscopic Properties of Polarons by Exact Monte Carlo 31
Behavior of the LF is the same as observed in the ARPES of undoped
cuprates. The LF consists of a broad peak and a high energy incoherent con-
tinuum (see Fig. 11a). Besides, dispersion of the broad peak “c” in Figs. 11
reproduces that of sharp peak in pure t-J model (Fig. 11b-f). The lowest dis-
persionless peak, corresponding to small radius polaron, has very small weight
and, hence, can not be seen in experiment. On the other hand, according to ex-
periment, momentum dependence of spectral weight Z(k)
of broad resonance
exactly reproduces dispersion of Z(k)-factor of pure t-J model. The reason
for such perfect mapping is that in adiabatic case Ω0/2J ≪ 1 all weight of
the sharp resonance in t-J model without EPI is transformed at strong EPI
into the broad peak. This picture implies that the chemical potential in the
heavily underdoped cuprates is not connected with the broad resonance but
pinned to the real quasiparticle pole with small Z-factor. This conclusion was
recently confirmed experimentally [177].
Comparing the critical EPI for a hole in the t-J-Holstein model (40-42)
λct-J ≈ 0.38 and that for Holstein model λcH ≈ 1.2 with the same value of
hopping t, we conclude that spin-hole interaction accelerates transition into
the strong coupling regime. The reason for enhancement of the role of EPI is
found in [196]. Comparison of the EPI driven renormalization of the effective
mass in t-J-Holstein and Holstein model shows that large effective mass in the
t-J model is responsible for this effect. The enhancement of the role of EPI
by EEI takes place at least for a single hole at the bottom of the t-J band.
Had the comparison been made with half-filled model, the result would have
been smaller enhancement or no enhancement at all [199]. On the other hand,
coupling constant of half-breathing phonon is increased by correlations [200].
Finally, we conclude that effect of enhancement of the effective EPI by EEI is
not unambiguous and depends on details of interaction and filling. However,
this effect is present for small filling in the t-J-Holstein model.
6.2 Isotope Effect on ARPES in Underdoped High-Temperature
Superconductors
The magnetic resonance mode and the phonon modes are the two major
candidates to explain the “kink” structure of the electron energy dispersion
around 40-70 meV below the Fermi energy, and the isotope effect (IE) on
ARPES should be the smoking-gun experiment to distinguish between these
two. Gweon et al. [201] performed the ARPES experiment on O18-replaced
Bi2212 at optimal doping and found an appreciable IE, which however can
not be explained within the conventional weak-coupling Migdal-Eliashberg
theory. Namely the change of the spectral function due to O18-replacement
has been observed at higher energy region beyond the phonon energy (∼
60meV). This is in sharp contrast to the weak coupling theory prediction, i.e.,
the IE should occur only near the phonon energy. Hence the IE in optimal
Bi2212 remains still a puzzle. On the other hand, the ARPES in undoped
materials, as described in Sect. 6.1, has recently been understood in terms of
32 A. S. Mishchenko and N. Nagaosa
the small polaron formation [49, 202, 198]. Therefore, it is essential to compare
experiment in undoped systems with presented in this Sect. DMC-SO data,
where theory can offer quantitative results.
In addition to high-Tc problem, strong EPI mechanism of ARPES spec-
tra broadening was considered as one of alternative scenarios for diatomic
molecules [203], colossal magnetoresistive manganites [34], quasi-one-dimensi-
onal Peierls conductors [37, 38], and Verwey magnetites [39]. Therefore, exact
analysis of the IE on ARPES at strong EPI is of general interest for conclusive
experiments in a broad variety of compound classes.
Dimensionless coupling constant λ = γ2/4tΩ in (42) is an invariant quan-
tity for the simplest case of IE. Indeed, assuming natural relation Ω ∼ 1/
between phonon frequency and mass, we find that λ does not depend on
the isotope factor κiso = Ω/Ω0 =
M0/M , which is defined as the ratio of
phonon frequency in isotope substituted (Ω) and normal (Ω0) systems. We
chose adopted parameters of the tt′t′′-J model which reproduce the experi-
mental dispersion of ARPES [178]: J/t = 0.4, t′/t = −0.34, and t′′/t = 0.23 .
The frequency of the relevant phonon [32] is set to Ω0/t = 0.2 and the isotope
factor κiso =
16/18 corresponds to substitution of O18 isotope for O16.
To sweep aside any doubts of possible instabilities of analytic continuation,
we calculate the LF for normal compound (κnor = 1), isotope substituted
(κiso =
16/18) and “anti-isotope” substituted (κant =
18/16) compounds.
Monotonic dependence of LF on κ ensures stability of analytic continuation
and gives possibility to evaluate the error-bars of a quantity A using quantities
Aiso −Anor, Anor −Aant, and (Aiso −Aant)/2.
Since LF is sensitive to strengths of EPI only for low frequencies [55], we
concentrate on the low energy part of the spectrum. Figure 12 shows IE on the
hole LF for different couplings in nodal and antinodal points, respectively. The
general trend is a shift of all spectral features to larger energies with increase
of the isotope mass (κ < 1). One can also note that the shift of broad FCP is
much larger than that of narrow real-QP peak. Moreover, for large couplings
λ the shift of QP energy approaches zero and only decrease of QP spectral
weight Z is observed for larger isotope mass. On the other hand, the shift
of FCP is not suppressed for larger couplings. Except for the LF in nodal
point at λ = 0.62 (Fig. 12a, b), where LF still has significant weight of QP
δ-functional peak, there is one more notable feature of the IE. With increase
of the isotope mass the height of FCP increases. Taking into account the
conservation law for LF
−∞ Lk(ω) = 1 and insensitivity of high energy part
of LF to EPI strength [55], the narrowing of the FCP for larger isotope mass
can be concluded. To understand the trends of the IE in the strong coupling
regime we analyze the exactly solvable independent oscillators model (IOM)
[60]. The LF in IOM is the Poisson distribution
L(ω) = exp[−ξ0/κ]
[ξ0/κ]
Gκ,l(ω) , (44)
Spectroscopic Properties of Polarons by Exact Monte Carlo 33
Fig. 12. Low energy part of hole LFs: normal compound (solid line), isotope sub-
stituted compound (dotted line) and “antiisotope” substituted compound (dashed
line). LFs at different couplings in the nodal (a, c, e) and antinodal (g, i, l) points.
Insets (b, d, f, h, k) show low energy peak of real QP.
where ξ0 = γ
0 = 4tλ/Ω0 is dimensionless coupling constant for normal
system and Gκ,l(ω) = δ[ω+4tλ−Ω0κl] is the δ-function. The properties of the
Poisson distribution quantitatively explain many features of the IE on LF7.
The energy ωQP = −4tλ of the zero-phonon line l = 0 in (44) depends
only on isotope independent quantities which explains very weak isotope de-
pendence of QP peak energy in insets of Fig. 12. Besides, change of the zero-
phonon line weight Z(0) obeys relation Z
iso /Z
nor = exp [−ξ0(1− κ)/κ] in
IOM. These IOM estimates agree with DMC data within 15% in the nodal
point and within 25% in the antinodal one. IE on FCP in the strong cou-
pling regime follows from the properties of zero M0 =
−∞ L(ω)dω = 1, first
−∞ ωL(ω)dω = 0, and second M2 =
2L(ω)dω = κξ0Ω
0 mo-
ments of shifted Poisson distribution (44). Moments M0 and M2 establish
relation D = hFCPiso /hFCPnor = 1/
κ ≈ 1.03 between heights of FCP in normal
and substituted compounds. DMC data in the antinodal point perfectly agree
with the above estimate for all couplings. This is consistent with the idea that
the anti-nodal region remains in the strong coupling regime even though the
nodal region is in the crossover region. In the nodal point DMC data well
agree with IOM estimate for λ = 0.75 (D ≈ 1.025) whereas at λ = 0.69 and
7 Cautions should be made about approximate form of EPI (42). Strictly speaking,
actual momentum dependence of the interaction constant σ [204, 205] can slightly
change the obtained differences between nodal and antinodal points though the
general trends have to be left intact because ST is caused solely by the short
range part of EPI [80].
34 A. S. Mishchenko and N. Nagaosa
0.65 0.70 0.75
0.65 0.70 0.75
0.65 0.70 0.75
0.5 0.6 0.7
k=( /2, /2)
Fig. 13. (a) Energies of ground state and broad peaks for normal (triangles),
isotope substituted (circles) and “antiisotope” substituted (diamonds) compounds.
Comparison of IOM estimates (lines) with DMC data in the nodal (squares) and
antinodal (diamonds) points: (b) shift of the FCP top, (c) FCP leading edge at 1/2
of height, and (d) FCP leading edge at 1/3 of height.
λ = 0.62 influence of the ST point leads to anomalous values of D: D ≈ 1.07
and D ≈ 0.98, respectively. Shift of the low energy edge at half maximum
∆1/2 must be proportional to change of the root square of second moment
∆√M2 =
ξ0Ω0[1 −
κ]. As we found in numeric simulations of (44) with
Gaussian functions8 Gκ,l(ω), relation ∆1/2 ≈ ∆√M2/2 is accurate to 10% for
0.62 < λ < 0.75. Also, simulations show that the shift of the edge at one
third of maximum ∆1/3 obeys relation ∆1/3 ≈ ∆√M2 . DMC data with IOM
estimates are in good agreement for strong EPI λ = 0.75 (Fig. 13). How-
ever, shift of the FCP top ∆p and ∆1/2 are considerably enhanced in the
self-trapping (ST) transition region. The physical reason for enhancement of
IE in this region is a general property regardless of the QP dispersion, range
of EPI, etc. The influence of nonadiabatic matrix element, mixing excited
and ground states, on the energies of resonances essentially depends on the
phonon frequency. While in the adiabatic approximation ST transition is sud-
den and nonanalytic in λ [80], nonadiabatic matrix elements turn it to smooth
crossover [144]. Thus, as illustrated in Fig. 13a, the smaller the frequency the
sharper the kink in the dependence of excited state energy on the interaction
constant
In the undoped case the present results can be directly compared with
the experiments. It is found that the IE on the ARPES lineshape of a sin-
gle hole is anomalously enhanced in the intermediate coupling regime while
can be described by the simple independent oscillators model in the strong
coupling regime. The shift of FCP top and change of the FCP height are rele-
vant quantities to pursue experimentally in the intermediate coupling regime
since IE on these characteristics is enhanced near the self trapping point. In
8 Results are almost independent on the parameter η of the Gaussian distribution
Gκ,l(ω) = 1/(η
2π) exp(−[ω + 4tλ−Ω0κl]/(2η2)) in the range [0.12, 0.2].
Spectroscopic Properties of Polarons by Exact Monte Carlo 35
contrast, shift of the leading edge of the broad peak is the relevant quantity
in the strong coupling regime since this value increases with coupling as
These conclusions, depending on the fact whether self trapping phenomenon
is encountered in specific case, can be applied fully or partially to another
compounds with strong EPI [34, 37, 38, 39].
6.3 Conclusions and Perspectives
In this article, we have focused mainly on the polaron problem in strongly
correlated systems. This offers an approach from the limit of low carrier con-
centration doped into the (Mott) insulator, which is complementary to the
conventional Eliashberg-Migdal approach for the EPI in metals. In the latter
case, we have the Fermi energy εF as a relevant energy scale, which is usually
much larger than the phonon frequency Ω0. In this case, the adiabatic Migdal
approximation is valid and the vertex corrections, which correspond to the
multi-phonon cloud and are essential to the self-trapping phenomenon, are
suppressed by the ratio Ω0/εF . Therefore an important issue is the crossover
from the strong coupling polaronic picture to the weak coupling Eliashberg-
Migdal picture. This occurs as one increases the carrier doping into the insu-
lator. As is observed by ARPES experiments in high temperature supercon-
ductors, the polaronic states continue to survive even at finite doping [177].
This suggests a novel polaronic metallic state in underdoped cuprates, which
is common also in CMR manganites [36] and is most probably universal in
transition metal oxides. In the optimal and overdoped region, the Eliashberg-
Migdal picture becomes appropriate [170, 171], but still a nontrivial feature of
the EPI is its strong momentum dependence leading to the dichotomy between
the nodal and anti-nodal regions. It is an interesting observation that the high-
est superconducting transition temperature is attained at the crossover region
between the two pictures above, which suggests that both the itinerancy and
strong coupling to the phonons are essential to the quantum coherence. It
should be noted that this crossover occurs in a nontrivial way also in the mo-
mentum space, i.e., the nodal and anti-nodal regions behave quite differently
as discussed in Sect. 6.2. However, the relevance of the EPI to the high Tc
superconductivity is still left for future investigations.
We hope that this article convinces the readers the vital role of ARPES
experiments and numerically exact solutions to the EPI problem, the com-
bination of them offers a powerful tool for the momentum-energy resolved
analysis of these rather complicated strongly correlated electronic systems.
This will pave a new path to the deeper understanding of the many-body
electronic systems.
We thank Y. Toyozawa, Z. X. Shen, T. Cuk, T. Devereaux, J. Zaanen, S.
Ishihara, A. Sakamoto, N. V. Prokofev, B. V. Svistunov, E. A. Burovski, J. T.
Devreese, G. de Filippis, V. Cataudella, P. E. Kornilovitch, O. Gunnarsson,
N. M. Plakida, and K. A. Kikoin, for collaborations and discussions.
36 A. S. Mishchenko and N. Nagaosa
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|
0704.0026 | Placeholder Substructures II: Meta-Fractals, Made of Box-Kites, Fill
Infinite-Dimensional Skies | 7 Placeholder Substructures II: Meta-Fractals,
Made of Box-Kites, Fill Infinite-Dimensional
Skies
Robert P. C. de Marrais ∗
Thothic Technology Partners, P.O.Box 3083, Plymouth MA 02361
March 2, 2022
Abstract
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-
dimensional hypercomplex numbers (N a power of 2, and at least 4) can
represent singularities and, as N → ∞, fractals – and thereby, scale-free net-
works. Any integer > 8 and not a power of 2 generates a meta-fractal or
Sky when it is interpreted as the strut constant (S) of an ensemble of octahe-
dral vertex figures called Box-Kites (the fundamental ZD building blocks).
Remarkably simple bit-manipulation rules or recipes provide tools for trans-
forming one fractal genus into others within the context of Wolfram’s Class
4 complexity.
1 Introduction By Way of Reprise: From Box-Kites
to ETs
The creation of 2N-dimensional analogues of Complex Numbers (and it was not a
trivial insight of 19th Century algebra that legitimate analogs always have dimen-
sion a power of 2) is handled by a now well-known algorithm called the Cayley-
Dickson Process (CDP). Its name suggests a compressed account of its history:
for Arthur Cayley – simultaneously with, but independently of, John Graves –
∗Email address: [email protected]
http://arxiv.org/abs/0704.0026v3
jumped on Hamilton’s initial generalization of the 2-D Imaginaries to the 4-D
Quaternions within weeks of its announcement, producing – by the method later
streamlined into Leonard Dickson’s close-to-modern “cookie-cutter” procedure –
the 8-D Octonions. The hope, voiced by no less than Gauss, had been that an
infinity of new forms of Number were lurking out there, with wondrous proper-
ties just awaiting discovery, whose magical utility would more than compensate
for the loss of things long taken for granted as their seekers ascended into higher
dimensions. But such fantasies were quashed quite abruptly by Adolph Hurwitz’s
proof, just a few years before the 20th Century loomed, that it only took four
dimension-doublings past the Real Number Line to find trouble: the 16-D Sede-
nions had zero-divisors, which meant division algebra itself broke down, which
meant researchers were so at a loss to find anything good to say about such Num-
bers that nobody bothered to even give their 32-D immediate successors a name,
much less investigate them seriously.
But it is with these 32-D “Pathions” (for short for “pathological,” which we’ll
call them from now on) that our own account will pick up in this second part
of our study of “placeholder substructures” (i.e., “zero divisors”) For, due to a
phenomenon we dubbed carrybit overflow in the first installment, strange yet pre-
dictable things are found to be afoot in the ZD equivalent of a “Cayley Table.” As
we’ll see shortly, this is a listing, in a square array, of the ZD “emanations” (or
lack of same) of all ZD “elements” with each other – all, that is, sharing mem-
bership in an ensemble defined not by a shared “identity element,” but a common
strut constant.
What we’ll see is that the lacks are of the essence: for each doubling of N, the
Emanation Table (ET) for the 2N+1-ions of same strut-constant will contain that
of its predecessor, leading to an infinite “boxes-within-boxes” deployment whose
empty cells define, as N grows ever larger, an unmistakable fractal limit. The full
algorithmic analysis of such Matrioshka-doll-like “meta-fractal” aspects – by the
simple rules of what we’ll call “recipe theory” (after the R, C, and P values related
to the Row label, Column label, and their cell-specific Products in such Tables) –
must await our third and last installment. But the colored-quilt-like graphics can
be viewed by any interested readers at their leisure, in the Powerpoint slide-show
online at Wolfram Science from our mid-June presentation at NKS 2006.[1] (The
slide-show’s title is almost identical to that of this monograph, as this latter is
meant to be the “theorem/proof” exposition of that iconic, hence largely intuitive
and empirically driven narration.)
What we’ll need to undertake this voyage is a quick reprise of the results
from Part I [2]. As the hardest part (as a hundred years of denial would imply)
is finding the right way to think about the phenomenology of zero-division, not
understanding its basic workings once they’re hit upon, such a summary can be
much more brief and easy to follow than the proofs required to produce and justify
it. We need but grasp 3 rather simple things. First, we must internalize the path
and vertex structure of an Octahedron – for, properly annotated and storyboarded,
this will provide us with the Box-Kite representation that completely catalogs ZDs
in the 16-D arena where they first emerged (and, as we’ll see in our Roundabout
Theorem herein, underwrites all higher-dimensional ZD emergences as well).
Second, instead of the cumbersome apparatus of CDP that one finds in algebra
texts and the occasional software treatment, we offer two easy algebraic one-liners
which (inspired by Dr. Seuss’s “Thing 1” and “Thing 2”), we simply call “Rule
1” and “Rule 2” – which operate, in almost Pythagorean earnest, on triplets of
integers (indices of associative triplets among our Hypercomplex Units, as we’ll
learn), and which, by so doing, accomplish everything the usual CDP tactics do,
but without the all-too-frequent obfuscation. (There is also a very useful, albeit
quite trivial, “Rule 0,” which merely states that any integer-triple serving to index
an associative triplet for one power of N will continue to do so for all higher pow-
ers. What makes this useful is its allowing us to recursively take triplet “givens”
for lower-level 2N-ions than those of current interest and toss them into the central
circle of the third thing we must grasp.)
We’ll need, that is, to be able to draw the simplest finite projective group’s
7-line, 7-node representation, the so-called PSL(2,7) triangle. The Rules, plus
the Triangle, applied to Box-Kite edge-tracings and nodal indices, are all we’ll
need. Indeed, the Box-Kite itself can be readily derived from the Triangle, by
suppressing the central node, and then recognizing four correspondences. First,
see the Triangle’s 3 triple-noded sides – two vertices plus midpoint – as the sources
of the Box-Kite’s trio of “filled-in” triangles dubbed Trefoil Sails. Second, link
the 1 triple-noded circle (which is a projective line, after all), wrapped around the
suppressed center and threading the midpoints, as the 4th such triangle, the quite
special “Zigzag Sail.” Third, envision the 3 lines from midpoints to angles as
underwriting the ZD-challenged part of the diagram (because ZDs housed at the
midpoint node cannot mutually zero-divide any housed at the opposite, vertex,
node), the struts (whence strut constants). Fourth and last, imagine the other four
triangles of the Box-Kite (meeting, as with the first four, each to each, at corners
only, like same-colored checkerboard squares) as the vents where the wind blows.
They keep the kite afloat, letting the four prettily colored jib-shaped Sails show
off, while the trio of wooden or plastic dowels that form the struts thanklessly
provide the structural stability that makes the kite able to fly in the first place.
As Euclid knew well, 3 points determine a Triangle as well as a Circle – which
is how we can glibly switch gears between representations based on these projec-
tive lines. But the easy convertibility of lines to circles is what projective means
here – and is, as well, at the very heart of linking the above geometrical images
to Imaginary Numbers. From Argand’s diagram to Riemann’s Sphere, this has
been the essence of Complex geometry. On the latter image only, place a sphere
on a flat tabletop, call the point of contact S (for “South”), and then direct rays
from its polar opposite point N. Rays through the equator intersect the table in
a circle whose radius we ascribe an absolute value of 1, with center S = 0. This
circle is just the trace of the usual ei·2π·θ exponential-orbit equation, with the i in
the exponent, of course, being the standard Imaginary. Any diameter through this
circle, extended indefinitely in either direction, is clearly a “projective pencil” of
a circular motion in the plane containing both it and N, and centered on the latter.
What each “line,” then, in the PSL(2,7) triangle represents is a coherent sys-
tem interrelating 3 distinct imaginaries, one per nodal point: that is, a “Quater-
nion copy” sans the Reals (which latter, like our N,S polar axis in the above, must
stand “outside” the Number Space itself, since 3-D visualization is all used up by
the nodes’ dimensional requirements). Hence, the 7 lines are the 7 interconnected
Quaternion copies which constitute the 8-D Octonions. And what makes this espe-
cially rich for our purposes is the built-in recursiveness of this Octonion-labeling
scheme for higher-dimensional isomorphs, embedded in the sorts of ensembles
we’ll be needing ETs to investigate more thoroughly.
To see how this relates to actual integers, take the prototype of the 7 lines
in the Triangle, and consider the Quaternions strictly from the vantage of CDP’s
Rule 1. The first task in studying any system of 2N-ions is generating its units, so
start with N = 0. Treat this singleton as the index of the Real axis: i0, that is, is
identically 1. Add a unit whose index = 20 = 1 and we have the complex plane.
Now, add in a unit whose index is the next available power of 2 – with N = 1, this
is 2 itself. Call this unit and its index G for Generator, and declare this inductive
rule: the index of the product of any two units is always the XOR of the indices of
the units being multiplied; but, for any unit with index u < G, the product of said
unit, written on the left (right), with the Generator written on its right (left), has
index equal to their indices’ simple sum, and sign equal (opposite) to the product
of the signs of their units’: i1 · i2 =+i3, but i2 · i1 =−i3. But this is just a standard
way of summarizing Quaternion multiplication.
Now, set N = 2, making G = 4. Applying the same logic, but slightly general-
ized, we get three more triplets of indices. Dispensing with the tedious overhead
of explicitly writing the indices as subscripts to explicit copies of the letter i, these
are written in cyclical positive order (CPO) as follows: (1,4,5);(2,4,6);(3,4,7).
(CPO is not mysterious: it just means read the triplet listing in left-right order,
and so long as we multiply any unit with any such index by the unit whose index
is to the right of it, the third term will result with signing as specified above: e.g.,
i4 · i5 = +i1; i4 · i3 = −i7.) We now have 4 of the Octonions’ 7 triplets, forming
labels on the nodes of 4 of PSL(2,7)’s lines. Call the central circle spanning the
medians the Rule 0 line (the Quaternions’ “starter kit” we just fed into our Rule 1
induction machine). Putting G = 4 in the center, the 3 lines through it are our Rule
1 triplets. If we further array the Quaternion index-set (1,2,3) in clockwise order
around the 4, starting from the left slope’s midpoint at 10 o’clock, these lines are
all oriented pointing into the angles. Now, with “Rule 2,” let’s construct the lines
along the Triangle’s sides.
Here’s all that Rule 2 says: given an associative index-triplet (henceforth, trip)
like the Quaternions’ (1,2,3), fix any one among them, then take its two CPO
successors and add G to them. Swap the order of the resulting two new units,
and you have a new trip. Hence, fixing 1, 2, and 3 in turn, in that order, Rule 2
gives us these 3 triplets: (1,7,6);(2,5,7);(3,6,5). If you’ve drawn PSL(2,7) with
the Octonion labels per the instructions in the last paragraph, you’ve already seen
these 3 trips are the answers ... and now you know how and why they’re oriented,
too. (Clockwise, in parallel with the Rule 0 circle).
We’ve now laid out all the ingredients we need to do a basic run-through of
Box-Kite properties. We’ll merely state and describe them, rather than prove them
(but we’ll give the Roman numerals of the theorem numbers from last installment,
for those who want to follow them). The first feature in need of elucidating,
which should have those who’ve been reading attentively scratching their heads
just about now, is this: the relations between the indices at the nodes of PSL(2,7)
qua Octonion labeling scheme are clear enough; but how can these same labeled
nodes serve to underwrite the 16-D Sedenion framework that Box-Kites reside in?
The answer has two parts.
First part: since all Imaginaries have negative Reals as squares, Imaginaries
whose products are zero must have different indices – meaning that the simple
case (which we call “primitive” ZDs) will always involve products of pairs of
differently-indexed units, whose respective planes share no points other than 0
[IV]. Second part: given any such ZD dyad, neither index can ever equal G [II];
and, one must have index > G, while the other has index < G [I, III]. The Oc-
tonion labeling scheme maps to the four Sails of a legitimate Sedenion Box-Kite
[V], because it only provides the low-index labels at each of the 6 Octahedral
vertices.
The 4 in the center of our example, meanwhile, is no longer the G for this
setup, since that role in now played by 8 (the next power of 2 in the CDP induc-
tion). In the context of the Box-Kite scheme, it is now represented by a different
letter: S, for strut constant – the only Octonion index not on a Box-Kite vertex.
Which is why, from one vantage, there are 7 distinct (but isomorphic) Box-
Kites in Sedenion space: because we’ve 7 choices of which Octonion to suppress!
6 vertices times 7 gives us the 42 Assessors of our first ZD paper [3], a term
we’ll use interchangeably with dyad throughout. We can, in fact, tug on the net-
work of interconnected lines “wok-cooking” style, stirring things into and out of
the hot oil in the center of the Box-Kite. (S as ”Stir-fry constant”?) To find the
“Octonion copy” labeling low indices on Box-Kite vertices where the 5, say, is
suppressed, trace the line containing it and the 4, and “rotate”: the 1 now goes
from the left slope’s midpoint to the bottom right angle, to be replaced by the 4
while the 5 heads for the middle, with CPO order (and hence, orientation of the
line) remaining unchanged. Of the other 2 trips the 5 belongs to, only one will
preserve midpoint-to-angle orientation along the 6 o’clock-to-midnight vertical:
(2,5,7), as one can check in an instance. (The two possibilities must orient oppo-
sitely when placed along the same line, since one is Rule 1, the other Rule 2.)
From this point, everything is forced. This is obviously a procedure that is
trivial to automate, for any “Octonion copy,” regardless of the ambient dimen-
sionality the Box-Kite it underwrites might float in. This simple insight will be
the basis, in fact, of our proof method, both in this paper and its sequel. Another
simple insight will tell us how to find the high-index term for any vertex’s dyad.
Two indices per vertex leaves 4 that are suppressed: 0 (for the Reals), G and S,
and the XOR (and also simple sum) of the latter two, which we’ll shorthand X.
These four clearly form a Quaternion copy – one, in fact, which has no involve-
ment whatsoever in its containing Box-Kite’s zero-divisions. Putting the index of
the one among these which is itself an L-unit center stage gives us the full array of
L-index sets (trips composed of those indices of a Sail’s 3 vertices <G) associated
with the 4 Sails. Putting in G or X, then, must give us the full array of U-index
sets (“U” as in “upper”).
Since each node belongs to 3 lines in PSL(2,7), the strut constant belongs to
3 trips, each containing one term from the Rule 0 Zigzag Sail’s L-index set, and
one from the Vent which resides opposite it on the Box-Kite’s octahedral frame.
In the Sedenions, three simple rules govern interactions of the Vent and Zigzag
dyads sharing a strut. Writing the U- and L- index terms in upper and lower case
respectively, we can symbolize their dyads as (V, v) and (Z, z) respectively. The
“Three Viziers” (derived as side-effects of [VII], with one for each non-0 member
of our ZD-free index set) read as follows:
VZ1: v · z =V ·Z = S
VZ2: Z · v =V · z = G
VZ3: V · v = z ·Z = X.
The First Vizier motivates the term strut constant: for the same pattern obtains
for it, regardless of the strut being investigated. The Second Vizier shows us
that G connects strut opposites, always by Rule 1 logic. But clearly, the Third
Vizier gives us the simplest way to answer any questions concerning the relations
between indices within a dyad: the L- and U- indices of any dyad belong to the
same trip as X, with CPO ordering determined by whether or not the dyad belongs
to the Zigzag proper or the Vent opposite it.
Beyond the Sedenions, VZ2 is universally true, but VZ1 and VZ3 are only so
up to sign: e.g., the VZ1 L-trip for an arbitrary strut can read (z,c,S) in certain
higher-dimensional contexts. This is ultimately a side-effect of the same “carrybit
overflow” that creates the phenomenon of most interest to us here, the “missing
box-kites” in all 2N-ions, N at least 5, for S > 8 and not a power of 2. Correlated
with such ZD-free structures are “Type II” box-kites with S < 8 (or, more gen-
erally, ¡ G/2), indistinguishable from the standard “Type I” variety but for strut
orientations (with exactly 2 of a “Type II”’s 3 struts always being reversed: see
Appendix B). Their “twist products” (operating similarly on parallel sides of each
of the 3 orthogonal squares or “catamarans” of a box-kite’s orthogonal wire-frame,
as opposed to the 4 triangular “sails” which are our sole focus in this monograph)
let them act as middlemen between the normal and ZD-free structures. Our ar-
guments here will make no use of such “twist product” subtleties (on which, see
Theorem 6 in Part I and the caveat that follows it, and the more developed re-
marks and diagrams in [4]). Indeed, their phenomenology falls “under the radar”
of our Sail-based analysis: strut-opposite Assessors, after all, do not mutually
zero-divide.
Given our limited purposes here, therefore, our toolkit, once the Viziers are
dropped in it, is complete for all our later proofs. (We must simply remember
that invocations of VZ1 and VZ3 implicitly concern sign-free relations between
Vent and Zigzag terms – that is, indices of XOR products only.) What’s left to do
still: get our hands messy with the plumbing, and then clean up with a last grand
construct. Let’s start with the plumbing, and add some notation. Label the Zigzag
dyads with the letters A, B, C; label their strut-opposite terms in the Vent F, E, D
respectively. Specify the diagonal lines containing all and only ZDs in any such
dyad K as (K, /) and (K, \) – for c · (iK + ik) and c · (iK − ik) respectively, c an
arbitrary real scalar. The twelve edges of the octahedral grid are so many pipes,
through which course the two-way streets of edge-currents: for the 3 edges of the
Zigzag (and the 3 defining the opposite Vent), currents joining arbitrary vertices
M and N are called negative, since they have this form:
(M,/) ·(N, \) = (M, \) · (N,/) = 0
Tracing the perimeter of the Zigzag with one’s finger, performing ZD products
in natural sequence – (A, /)·(B, \), followed by the latter times (C, /), then
this times (A, \) and so forth – one should quickly see how the Zigzag’s name
was suggested. Suppressing all letters, one is left with just this cyclically repeating
sequence: /\/\/\.
Currents along all 6 edges joining Zigzag and Vent dyads, on the contrary, con-
nect similarly sloping diagonals, hence are called positive, yielding the shorthand
sequence ///\\\ for Trefoil sail traversals:
(Z,/) ·(V, /)= (Z, \) ·(V,\)= 0
Consider the chain of ZD multiplications one can make along the Zigzag, be-
tween A and B, then B and C, then C and A, for S = 4. The first term of this
6-cycle of zero products, once fully expanded, is writable thus:
(A, /) ·(B, \) = (i1 + i13) · (i2− i14) = (i3 − i15 + i15 − i3) =
(C, /)−(C, /)= (C, \)−(C, \)= 0
We can readily see here where the notion of emanation arises: traversing the
edge between any two vertices in a Sail yields a balance-pan pairing of oppositely
signed instances of the terms at the Sail’s third vertex ... the 0 being, then, an
instance of “balanced bookkeeping” (whence the term “Assessor,” our synonym
for “dyad”). This suggests the spontaneous emanation of particle/anti-particle
pairings from the quantum vacuum, rather than true “emptiness.”
Finally, a side-effect of such “Sail dynamics” is this astonishing phenomenon:
each Sail is an interlacing of 4 associative triplets. For the Zigzag, these are the L-
index (a,b,c), plus the 3 U-index trips obtained by replacing all but one of these
lowercase letters with their uppercase partners: ergo, (a,B,C); (A,b,C); (A,B,c).
Ultimately this tells us that ZDs are extreme preservers of order, since they main-
tain associativity in rigorous lock-step patterns, for all 2N-ions, no matter how
close to ∞ their N might become. Put another way, the century-long aversion re-
action experienced by virtually all mathematicians faced with zero-divisors was
profoundly misguided.
2 Emanation Tables: Conventions for Construction
Theorem 7 guaranteed the simple structure of ETs: because any Assessor’s up-
percase index iU is strictly determined by G and S, once we are given these two
values, the table need only track interactions among the lowercase indices iL. This
will only lead to ambiguities in the very place these are meaningful: in the recur-
sive articulation of a boxes-within-boxes tabulation of meta-fractal or Sky behav-
iors. In such cases, the overlaying will be as rich in significance as the multiplicity
of sheets of a Riemann surface in complex analysis.
An ET does for ZD interactivity what a Cayley Table does for abstract groups:
it makes things visible we otherwise could not see – and in a similar way. Each
Assessor’s L-index is entered (in a manner we’ll soon specify) as a row (R) or
column (C) value, with XOR products (P values) among them being placed in the
“spreadsheet cell” (r,c) uniquely fixed by R and C. We’ve noted such values only
get entered if P is the L-index of a legitimate emanation: that is, the Assessor it
represents mutually zero-divides (forms DMZs with, for “divisors making zero”)
both the Assessors represented by the R and C labels of its cell. (As already
suggested, the natural use of the letters R, C, P here inspired calling the study of
NKS-like “simple rules” for cooking fractals from their bit-strings recipe theory.)
Four conventions are used in building ETs: first, their labeling scheme obeys
the same nested-parentheses ordering we’ve already used in designating Assessors
A through F, with D, E, F the strut opposites of A, B, C in reverse of the order just
written. The L-indices, then, are entered as labels running across the top and down
the left. The label of the lowest L-index is placed flush left (abutting the ceiling),
with the corresponding label of its strut opposite being entered flush right (atop
the floor). As there will always be G− 2 (hence, an even number of) indices to
enter, repeating this procedure after each pair has been copied to horizontal and
vertical labels will completely exhaust them all.
Second convention: As the point of an ET is to display all legitimate DMZs,
any cell whose R and C do not mutually zero-divide is left blank – even if, in
fact, there is a well-defined XOR value. Hence, if R and C reference the same
Assessor, the XOR of their L-indices will be 0; if they reference strut opposites,
the XOR will be S. But in both cases, the cell (hence, the P value) is left blank.
All “normal” ETs, then, will have both long diagonals populated by blank cells,
while all other cells are filled.
Third convention: the two ZD diagonals associated with any Assessor are not
distinguished in the ET, although various protocols are possible that would make
doing so easy. The reasons are parsimony and redundancy: rather than create
longer, or twice as many, entries, we assume both entries for the same Box-Kite
edge will contain the positive-sloping diagonal when the lower L-index appears as
the row label, else the negative-sloping diagonal when the higher L-index appears
first instead. Such niceties won’t concern us much here: the key thing is that, in
fact, all 24 filled cells of a Box-Kite’s ET entries can be mapped one-to-one to its
ZD diagonals. Recall, per Theorem 3, that both ZD diagonals of an Assessor form
DMZs with the same Assessor, according to the same edge-sign logic. This leads
us to the . . .
Fourth convention: Although they are superfluous for many purposes, edge
signs provide critical information for others, and so are indicated in all ETs pro-
vided here. Each of a Box-Kite’s 12 edges conducts two currents – one per ZD
diagonal – and does so according to one or the other orientational option. ZD di-
agonals are conventionally inscribed so that the horizontal axis of their Assessor
plane is the L-indexed unit, while the vertical is the U-indexed unit. But even if
this convention were reversed, the diagonal leading from lower left quadrant to up-
per right would still correspond to the state of synchrony implied by ±k(iL + iU):
for some Assessor U, we write (U,/). Conversely, the orthogonal diagonal in-
dicative of anti-synchrony is written (U,\). If DMZs formed by the Assessors
bounding an edge are both of same kind, then we call the edge blue or notate it
[+]; if Assessors U and V only form DMZs from oppositely oriented ZD diag-
onals – (U,/) · (V,\) = 0 ⇔ (U,\) · (V,/) = 0 – then we call the edge red or
notate it [-]. However, for ET purposes, since the red edges are the most infor-
mative (all-red-edged Zigzags providing the stable basis of Box-Kite structure,
while all-red-edged DEF Vents play a key role in twist-product interpreting – a
deep topic touched upon in Part I, which won’t concern us further here), we leave
them unmarked. The six blue edges bounding the hexagonal view of the Box-Kite,
however, are preceded by an extra mark (best interpreted as a dash, rather than a
minus sign). This has the pragmatic advantage that when zoomed, a large ET will
have its entries with an extra mark become unreadable in many software systems
(e.g., one sees only asterisks) – and so we want the unmarked entries to be those
likely to be of most interest.
Since, given X (or, alternatively, G or N, and S), we can reconstruct a Box-
Kite from just its Zigzag’s L-index trip, gleaning this information from an ET is
worth explaining. If a given row contains the indices of any such Zigzag L-trips,
they will appear as the row label itself, plus two unmarked cell entries, with the
column label of the one appearing as the content of the other. (If either cell in such
a complementary set be marked with a dash, then we are dealing with a DEF Vent
index.) Each Zigzag L-trip will also appear 3 times in an ET, once in each row
whose label is one of its indices, its 2 non-label indices appearing in un-dashed
cell entries each time.
Here is a readily interpreted emanation table. Having 6 = 23 − 2 rows and
columns, G = 8, so N = 4, making this a Sedenion ET (encoding, thereby, a
single Box-Kite). And, since 2⊻ 3 = 4⊻ 5 = 6⊻ 7 = 1, the Strut Constant S = 1
as well. A scan of the first row shows 6 and 5 unmarked, under headings 4 and
7 respectively; however, these two labels appear as cell values which are marked,
making these edges that connect Assessors in the D, E, F Vent. In the fourth row
of entries, though, column labels 5 and 3 contain cell values 3 and 5 respectively,
both unmarked. With their row label 6, then, these form the Zigzag L-index set
(3,6,5), which hence must map to Assessors (A,B,C). Using the mirror-opposite
logic of the labeling scheme to determine strut opposites, it is clear that the six row
and column headings (2,4,6,7,5,3) correspond, in that order, to the Assessors
(F,D,B,E,C,A). (The unmarked contents 6 and 5 in the first row, having labels
(2,4) and (2,7), thereby map to edges FD and FE, connecting DEF Vent Assessors
as claimed.) Finally, the long diagonals are all empty: those cells in the diagonal
beginning at the upper left all have identical row and column labels; those in the
mirror-opposite slots, meanwhile, have labels which are strut-opposites. By our
second convention, all these cells are left blank.
2 4 6 7 5 3
2 6 −4 5 −7
4 6 −2 3 −7
6 −4 −2 3 5
7 5 3 −2 −4
5 −7 3 −2 6
3 −7 5 −4 6
Before beginning an in-depth study of emanation tables by type, there is one
general result that applies to them all – and whose proof will give us the chance
to put the Three Viziers to good use. While seemingly quite concrete, we will use
it in roundabout ways to simplify some otherwise quite complicated arguments,
beginning with next section’s Theorem 9. This Roundabout Theorem is our
Theorem 8. The number of filled cells in any emanation table is a multiple of 24.
Proof. Since 24 is the number of filled cells in a Sedenion Box-Kite, this is equiv-
alent to claiming that CDP zero-divisors come in clusters no smaller than Box-
Kites. We have already seen, in Theorem 5, that the existence of a DMZ implies
the 3-Assessor system of a Sail, which further (as Theorem 7 spelled out) entails a
system of 4 interlocking trips: the Sail’s L-trip, plus 3 trips comprising each L-trip
index plus the U-indices of its Assessor’s 2 “sailing partners.” Since we have an
ET, we have a fixed S and fixed G. Hence, if we suppose our DMZ corresponds
to a Zigzag edge-current, we immediately can derive its L-trip by Theorem 5, and
all 3 Zigzag strut-opposites’ L-indices by VZ 1, and all 6 U-indices by VZ 3. We
then can test whether the Trefoil Sails’ edge-currents are all DMZs as follows. As
we wrote in Theorem 7, (u,v,w) maps to the Zigzag L-trip in CPO, but not neces-
sarily in (a,b,c), order: hence, (uopp,wopp,v) is an L-trip, and can be mapped to
any of the Trefoils. In other words, given the Zigzag’s 3-fold rotational symmetry,
proving the truth of the following arithmetical result proves the DMZ status of all
Trefoil edges. Yet we can avail ourselves of all 3 Zigzag U-trips in proving it.
(wopp −Wopp)
(uopp +Uopp)
−V − v
+v +V
The left bottom result is a given of the trip we started with. The result to its
right is a three-step deduction from one of the Zigzag U-trips: use (uopp,w,vopp);
Rule 2 gives (uopp,vopp+G,w+G); the Second Vizier tells us this is (uopp,V,Wopp);
but the negative inner sign on the upper dyad reverses the sign this trip implies,
yielding +V for the answer.
The top results are derived similarly: find which of the 4 Zigzag trips un-
derwrites the Vizier-derived “harmonic” which contains the pair of terms being
multiplied, and flip signs as necessary. Hence, the top left uses (u,wopp,vopp),
then applies Rule 2 and the Second Vizier to get (−V ), while the top right uses
the Zigzag L-trip itself: (u,v,w)→ (w+G,v,u+G) → (Wopp,v,Uopp) – which,
multiplied by (−1), yields (−v). �
Remark. The implication that, regardless of how large N grows, ZDs only increase
in their interconnectedness, rather than see their basic structures atrophy, flies in
the face of a century’s intuition based on the Hurwitz Proof. That there are no
standalone edge-currents, nor even standalone Sails, bespeaks an astonishing (and
hitherto quite unsuspected) stability in the realm of ZDs.
Corollary. An easy calculation makes it clear that the maximum number of filled
cells in any ET for any 2N-ions is just the square of a row or column’s length in
cells, minus twice the same number (to remove all the blanks in long diagonals):
that is, (2N−1 −2)(2N−1 −2)−2 · 2N−1 +4 = (22N−2 −6 · 2N−1 +8) = (2N−1 −
4)(2N−1 − 2) = 4 · (2N−2 − 1)(2N−2 − 2). By Roundabout, we now know this
number is divisible by 24, hence indicates an integer number of Box-Kites. But
two dozen into this number is just (2N−2 − 1)(2N−2 − 2)/6 – the trip count for
the 2N−2-ions! (See Section 2 of Part I.) We have, then, the very important Trip-
Count Two-Step: The maximum number of Box-Kites that can fill a 2N-ion ET =
TripN−2. We will see just how important this corollary is next section.
3 ETs for N > 4 and S ≤ 7
One of the immediate corollaries of our CDP Rules for creating new triplets from
old ones is something we might call the Zero-Padding Lemma: if two k-bit-long
bitstring representations of two integers R and C being XORed are stuffed with
the same number n of 0s between bits j and j+1, 0 ≤ j ≤ k, their XOR will, but
for the extra n bits of 0s in the same positions, be unchanged – and so will the
sign of the product P of CDP-derived imaginary units with these three bit-strings
representing their respective indices.
Examples. (1,2,3)→ (2,4,6)→ (4,8,12) [Add 1, then 2, 0s to the right of
each bitstring]
(1,2,3)→ (1,4,5)→ (1,8,9) [Add 1, then 2, 0s just before the rightmost bit
in each bitstring]
(3,4,7)→ (3,8,11)→ (3,16,19) [Add 1, then 2, 0s just after the leftmost bit
in each bitstring]
Proof. Rule 1 will create a new unit of index G+L from any unit of index
L < G, regardless of what power of 2 G might be. Rule 2, meanwhile, uses
any power of 2 which exceeds all indices of the trip it would operate on, then
adds this G to two of the members of the trip, creating a new trip with reversed
orientation – one of an infinite series of such, differing only in the power of 2
(hence, position of the leftmost bit) used to construct them. The lemma, then, is
an obvious restatement of the fundamental implications of the CDP Rules.
But creation of U-indices associated with L-indices in Assessor dyads is the
direct result of creating new triplets with G+S as their middle term. Hence, if
we call the current generator g and that of the next higher 2N-ions G (= 2 · g),
then if Assessors with L-indices u and v form DMZs in the Sedenions for a given
strut constant S, their U-indices will increment by g in the Pathions, and zero
division will remain unaffected. By induction, the emanation table contents of the
Sedenion (R,C,P) entries will remain unchanged for all N, for all fixed S≤ 7. This
leads us to
Theorem 9. All non-long-diagonal cell entries in all ETs for all N, for all fixed
S ≤ 7, will be filled.
Proof. Keeping the same notation, the 2N-ions will have g more Assessors than
their predecessors, with indices ranging from g itself to 2g−1 (=G−1). Consider
first some arbitrary Zigzag Assessor with L-index z < g, whose U-index is G+
z ·S. (If it were a Vent Assessor, or a Zigzag on a reversed-orientation strut in a
“Type II” box-kite, the second part of the expression would be reversed: S · z, per
the First Vizier. This effects triplet orientation, but not absolute value of the index,
however, and it is only the latter which matters at the moment.) Now consider the
Assessor whose L-index is the lowest of those new to the 2N-ions, g. We know
it is a Vent Assessor, in all Box-Kites with S < g, of which there are 7 per each
such S in the Pathions, 35 in the 64-D 26-ions, and so on: for it belongs to the trip
(S,g,g+S) (Rule 1), so that its U-index appears on its immediate left in the triplet
(G+g+S,g,G+S) (Rule 2 and last parentheses). Its U-index, then, is G+(g⊻
S), or (recall Rule 1) just G+ g+S. We claim these Assessors form DMZs; or,
writing out the arithmetic, that the following term-by-term multiplication is true:
+g+(G+g+S)
+z+(G+ z·S)
−(G+g+ z ·S)− (z+g)
+(z+g)+(G+g+ z ·S)
Because one Assessor is assumed a Zigzag, while the other is proven a Vent,
the inner signs will be the same. (Simple sign reversals, akin to those involving our
frequently invoked binary variable sg, will let us generalize our proof to include
the Vent-times-Vent case later.) Let’s examine the terms one at a time, starting
with the bottom line. Its left term is an obvious application of Rule 1, as z < g,
the latter being the Generator of the prior CDP level which also contained z as an
L-index. The term on bottom right we derive as follows: we know that z and its
U-index partner in the 2N−1-ions belong to the triplet mediated by g+S: (z,g+
z ·S,g+S). Supplementing this CPO expression by adding G to the right-hand
terms (Rule 2), we get the triplet containing both multiplicands of the bottom-
right quantity: (z,G+g+S,G+g+ z ·S). The multiplicands appear in this trip
in their order of application in forming the product; therefore, their resultant is a
plus-signed copy of the trip’s third term, as shown above.
Moving to the left-hand term of the top line, what trip do the multiplicands
belong to? Within the prior generation, Rule 1 tells us that z’s strut opposite, z ·S,
multiplies g on the left to yield g+ z ·S. Application of Rule 2 to the terms 6= g
reverses order and gives us this: (G+g+z ·S,g,G+z ·S). But what we’ve written
above is the product of multiplying the third and second terms of the trip together,
in CPO-reversed order; hence, the negative sign is correct. Finally, we get the
negative of (z+g) by similar tactics: the term is the U-index of z’s strut-opposite
Assessor in the prior CDP generation, hence belongs to the trip with this CPO
expression: (g+S,z+g,z ·S). Rule 2 gives us (G+g+S,G+z ·S,z+g). Hence,
the product written above is properly signed.
Now, what effect does our initial assumption that z is the L-index of a Zigzag
Assessor have on the argument? The lower-left term is obviously unaffected. But
the upper-left term, perhaps less obviously, also is unchanged: while it seems to
depend on z ·S, in fact this is only used to define the L-index of z’s strut opposite,
which multiplies g on the left to precisely the same effect as z itself, both being less
than it. The two terms on the right, just as clearly, do have their signs changed, for
in both, the order relations of L- and U- indices vis à vis G+S or X are necessarily
invoked. But both signs on the right can be re-reversed to obtain the desired result
if we change the inner sign of the topmost expression – which is to say, we have an
effect analogous to that achieved in earlier arguments by use of the binary variable
sg, as claimed.
Since one CDP level’s G is the g of the next level up, the above demonstration
clearly obtains, by the obvious induction, for all 2N-ions including and beyond the
Pathions. But what if one or both L-indices in a candidate DMZ pairing exceed g?
Rather than answer directly, we use the Roundabout Theorem of last section.
Given a DMZ involving Assessors with L-indices u < g and g, we are assured a
full Box-Kite exists with a Trefoil L-trip (u,g,g+ u). The remaining Assessors,
being their strut opposites, then have L-indices uopp,g+ S, and g+ u · S. As u
varies from 1 to 7, skipping S < 8, zero-padding assures us that all DMZs from
prior CDP generations exist for higher N, for all L-indices u,v < 8. Only those
Box-Kites created by zero-padding from prior-generation Box-Kites (of which
there can be but 1 inherited per fixed S among the 7 found in the Pathions, for
instance) will have all L-indices < g. For all others, the model shown with those
having g as an L-index must obtain. Hence, only one strut will have L-indices
< g, the rest being comprised of some w with L-index ≥ 8, the others deriving
their L-indices from the XOR of w with the strut just mentioned, or with S.
But what will guarantee that any edge-currents will exist between arbitrary
Assessors with L-indices u < g and g+ k,0 < k < g, since there is not even one
DMZ to be found among Assessors with L-indices ≤ g in the candidate Box-
Kite they would share? We can now narrow the focus of our original question
considerably, by making use of the curious computational fact we called the Trip-
Count Two-Step.
In Part I’s preliminary arguments concerning CDP, we showed that the number
of associative triplets in a given generation of 2N-ions, or TripN , can be derived
from a simple combinatoric formula. Call the count of complete Box-Kites in
an ET BKN,S. For S < 8, BKN,S = TripN−2, provided all L-indices g+ k,0 <
k < g, form DMZs in the candidate Box-Kites implied. To begin an induction,
let us consider a new construction along familiar lines, which will provide us an
easy way to comprehend the Pathion trip-systems of all S < 8. Beginning with
N = 5, we designate TripN−2 trips for each S < 8 as type Rule 0, in the manner
the singleton 22-ion trip (1,2,3) was used in our introduction’s ”wok-cooking”
discussion (which Part I, Section 5, used as the basis of its “slipcover proofs”).
But now, instead of putting the Octonions’ G = 4 in the center of the PSL(2,7)
triangle, we put the Sedenions’ 8.
For consistency of examples, we continue to assume S= 1, so we’ll begin with
(3,6,5), the Zigzag L-trip for S = 1 in the Sedenions, and also, by zero-padding,
an L-trip Zigzag for 1 of the 7 Box-Kites with S = 1 among the Pathions. Ex-
tending rays from the (3,6,5) midpoints through the center creates Rule 1 trips
which end in 11,14,13: (a,b,c) get sent to (F,E,D) respectively. The Rule 2
trips along the sides, in order of Zigzag L-index inclusion, then correspond to
Trefoil U-trips, all oriented clockwise. They read symbolically (literally) as fol-
lows: EaD (14,3,13);DbF (13,6,11);FcE (11,5,14). We claim each of these 7
lines, when its nodes are attached to their strut opposites, map 1-to-1 to an S = 1
Pathion Box-Kite. We have this as a given for the Rule 0 trip; we need to ex-
plain this for the Rule 1 trips (which Roundabout already tells us are Box-Kites);
and, we need to prove it for the Rule 2 trips that make the sides. (And, once
we do prove it, and frame the suitable induction for all higher N, the task which
originally motivated us will be done: for these U-trips house the Assessors with
L-indices > g, whose candidate Box-Kites don’t include g.)
The Rule 1 trips, in all instances within this example, correspond to Asses-
sor L-indices (a,d,e). With g = 8 at d, the Third Vizier tells us c = 8+ S =
Sedenion X. (a,b,c) thereby reads, within the Sedenions, as (a,A,X). But in the
Pathions, all 3 terms are less than G, hence can comprise an L-index trip for a Sail
– and specifically, a Zigzag (else the order of A and X would be reversed). Simi-
larly, the old Sedenion ( f ,F) are the new Pathion ( f ,e), with the new trip ( f ,c,e)
being the Third Vizier’s way of saying ( f ,X ,F) from the Sedenions’ vantage.
For the Rule 2 trips, we prove one relation in one of them a DMZ, which
Roundabout tells us implies the whole Box-Kite, while symmetry allows us to
assume the same of the other two. Consider, then, the aDE Trefoil U-trip, in-
stantiated by (3,13,14) in our example; specifically, compute the product of the
Assessors containing a and D = c+ g as L-indices. Their U-indices within the
Pathions must be (G+ a⊻S) = (G+ f ), and (G+ g+ c⊻S) = (G+ g+ d) re-
spectively. We write their dyads when multiplying with opposite inner signs, as
we assume their DMZ is an edge in a Zigzag. We claim the truth of this arithmetic:
+(c+g)− (G+g+d)
+a + (G+ f )
+(G+g+ e)− (b+g)
+(b+g)− (G+g+ e)
Bottom left: (a,b,c)→ (a,c+g,b+g) (Rule 2, with N = 4.)
Bottom right: (a,d,e)→ (a,g+ e,g+d)→ (a,G+g+d,G+g+ e) (Rule 2
twice, N = 4, then N = 5.) Upper dyad’s inner sign reverses that of product.
Top left: ( f ,c,e)→ (e+g,c+g, f )→ (G+ f ,c+g,G+e+g) (Rule 2 twice,
N = 4, then N = 5.)
Top right: ( f ,d,b) → (b+ g,d + g, f ) → (b+ g,G+ f ,G+ g+ d) (Rule 2
twice, N = 4, then N −5.) Upper dyad’s inner sign reverses that of product.
A similar brief exercise with either DMZ formed with the emanated Assessor
will show it, too, has a negative inner sign with respect to a positive in its DMZ
partner. Two negative edge-signs in one Sail means Zigzag (means three negative
edge-signs, in fact). Our proof up through the Pathions is complete; we need only
indicate the existence of a constructive mechanism for pursuing this same strategy
as N grows arbitrarily large.
Consider now the same PSL(2,7) triangle, but in its center put a 16 (= g=G/2
for the 64-D Chingons, after the 64 Hexagrams of the I Ching, to give them a
name). Then, put all 7 of the Pathions’ S = 1 Zigzag L-trips into the Rule 0 circle.
One gets 3 · 7 = 21 Rule 2 Zigzag L-trips, and the 10 integers < g found in them
and the 7 Rule 0 Zigzag L-trips implies there are 10 Rule 1 Trefoil L-trips, each
associated with a distinct Box-Kite. But that would make for 7+ 21+ 10 = 38
Zigzag L-trips, when we know there can only be 35. The extra 3 indicate there’s
some double-duty occurring: specifically, 3 of the Rule 1 Trefoil L-trips in fact
designate not the standard (a,d,e), but ( f ,d,b), with d = g = 16 in each instance.
When (5,14,11) is fed into our “trip machine” as Rule 0 circle, both (11,16,27)
and (14,16,30) map to ( f ,d,b) trips tied to Rule 0 Zigzag L-indices (10,27,17)
and (15,30,17), whose (a,d,e) trips appear as rays on triangles for (3,10,9) and
(3,13,14) respectively. (11,16,27) also shows as an ( f ,d,b) with Rule 0 trip
(6,11,13). (Readers are encouraged to use the code in the appendix to [4], to
generate ETs for low S and N. Trip-machining details for our S = 1 example are
in Appendix A.) For N = 7, use the 35 just-derived S = 1 L-trips as Rule 0 circles
with a central 32, and so on. �
4 The Number Hub Theorem (S = 2N−2) for 2N-ions
Given the lengths required to prove the fullness of ETs for S < 8, it might be
surprising to realize that the infinite number of cases for S = 2N−2 for all 2N-ions
are so simple to handle that they almost prove themselves. Yet the proof of this
Number Hub Theorem, while technically trivial, has far-reaching implications.
Theorem 10. For all 2N-ions with ZDs (N > 3), and S = g = G/2, all non-long-
diagonal entries in the emanation table are filled; more, each such filled cell in
the ET’s upper left quadrant is unmarked (indeed, indicates an edge-current in a
Zigzag); further, the row, column, and cell entries are isomorphic to those found
in an unsigned, CDP-generated, multiplication table for the 2N−2-ions; finally, the
TripN−2 Zigzag L-index sets which underwrite its Box-Kites are precisely all and
only those trips contained in said 2N−2-ions, the ET effectively serving as their
high-level atlas.
Proof. As the largest L-index of any Assessor is 2g− 1, and each S in the ETs
in question is precisely g, then the row (column) labels will ascend from 1 to
g− 1 in simple increments from top to bottom (left to right) in the upper left
quadrant, making its square of filled cells isomorphic to unsigned entries in the
corresponding 2N−2-ion multiplication table. Also, all these filled cells of the ET
will only contain XORs of indices < g. Hence, all and only L-index trips will have
the edges of their (necessarily Zigzag) Sails residing in said quadrant. All non-
long-diagonal cells in the ET are meanwhile filled, since all candidate Assessors
have form M = (m,G+ g+m), and for any CPO triplet (a,b,c) whose row and
column labels plus cell entry are contained in the upper left quadrant, it is easy to
show that the following arithmetic is true:
+b− (G+g+b)
+a + (G+g+a)
+(G+g+ c)− c
+c− (G+g+ c)
Therefore, the TripN−2 Box-Kites, the Zigzag L-index set of each of which is
one of the TripN−2 trips contained in the 2
N−2-ions, all have this simple form:
(a,b,c,d,e, f ) = (a,b,c,g+ c,g+b,g+a) �
Remarks. As will become ever more evident, powers of 2 – which is to say,
singleton 1-bits in indefinitely long binary bitstrings – play a role in ZD number
theory most readily analogized to that of primes in traditional studies. And while
integer triples (from Pythagoras to Fermat) play a central role in prime-factor-
based traditional studies, all XOR triplets at two CDP generations’ remove from
the power of 2 in question are collected by its ET in this new approach. All other
integers sufficiently large (meaning > 8) are meanwhile associated with fractal
signatures, to each of which is linked a unique infinite-dimensional space spanned
by ZD diagonals. But can such a vantage truly be called Number Theory at all?
We say indeed it can: that it is, in fact, the “new kind of number theory” that must
accompany Stephen Wolfram’s New Kind of Science. In his massive 2002 book,
he tells us that, common wisdom to the contrary, complex behavior can be derived
from the simplest arithmetical behavior. The obstacle to seeing this resides in the
common wisdom itself [5, p. 116]:
· · · traditional mathematics makes a fundamental idealization: it as-
sumes that numbers are elementary objects whose only relevant at-
tribute is their size. But in a computer, numbers are not elementary
objects. Instead, they must be represented explicitly, typically by giv-
ing a sequence of digits.
But that ultimately implies strings of 0’s and 1’s, where the matter of impor-
tance becomes which places in the string are held, and which are vacant: the orig-
inal meaning of our decimal notation’s sense of itself as placeholder arithmetic.
The study of zero divisors – placeholder substructures – then becomes the natural
way to investigate the composite characteristics of Numbers qua bitstrings. When
we discover, in what follows, that composite integers (meaning those requiring
multiple bits to be represented) are inherently linked, when seen as strut-constant
bit-strings, with infinite-dimensional meta-fractals, the continuation of the quote
on the following page should ring true:
In traditional mathematics, the details of how operations performed
on numbers affect sequences of digits are usually considered quite
irrelevant. But · · · precisely by looking at such details, we will be
able to see more clearly how complexity develops in systems based
on numbers.
5 The Sand Mandala Flip-Book (8 < S < 16, N = 5)
In the first concrete exploration of ZD phenomenology beyond the Sedenions [6,
pp. 13-19], a startling set of patterns were discovered in the ETs for values of S
beyond the “Bott limit”: that is, for 8 < S < 16 (the upper bound being the G of
the 32-D Pathions), the filled cells sufficed to define not 7, but only 3, Box-Kites
for N = 5; more, the primary geometric figures in each such ET transformed into
each other with each integer increment of S, in a manner exactly reminiscent of the
flip-books which anticipated cartoon animation. While these seemed perplexing
in mid-2002 when they were found, their logic is in fact profoundly simple.
First, each such ET’s S is just the X of one already seen in the Sedenions. We
continue our convention of using g to indicate the G of the prior CDP genera-
tion, employ s for said generation’s S, and reference all prior Assessor indices by
suffixing their letters with asterisks. Then, since S = g+ s, the trip (s,g,g+ s)
mandates, by the First Vizier (whose signed version we invoke due to the direct
derivation from the Sedenions), that g must belong to the Zigzag Sail if it’s to be
an Assessor L-index at all.
Note that this is not a truly legitimate argument, as we’ll see shortly, albeit the
results are correct, as shown by other means in [6]: this is because “Type II” box-
kites first emerge in this current context – but are not among the 3 x 7 “flip-book”
denizens of immediate interest. We will assume, for simplicity of presentation,
that the First Vizier does obtain here: proving that it does, however, requires a
background argument concerning “Type II” box-kites: their S values must be less
than g, hence none of our flip-book candidates can qualify. (But they are just as
numerous as the flip-book box-kites, there being 3 for each of the seven values of
S < g. For their listing, and theoretical framing of “Type II” phenomenology, see
Appendix B.)
We will content ourselves here with giving this as an empirical result, and
assume, therefore, the validity of the signed version of the First Vizier in the
case at hand. Based on this assumption, we can further claim that the Sedenion
Vent L-indices, f∗,e∗,d∗, must also be associated with Zigzag Assessors. By an
argument exactly akin to that of last section, we then have 3 candidate Box-Kites
to consider: since the 3 Vent L-indices are all less than g, they must be mapped to
the 3 Assessors A, g = 8 must adhere to B (and s = 1 to E), while the L-indices of
the C Assessors associated with f∗,e∗,d∗ must be A*, B*, C* respectively. The
proof is easy: taking the new A, C Assessors = ( f∗,G+g+a∗) and (g,G+ s) in
that order as readily generalizable representatives, we do the arithmetic.
+g− (G+ s)
+ f ∗ + (G+g+a∗)
+(G+a∗) − ( f ∗+g)
+( f ∗+g)− (G+a∗)
The bottom left is just Rule 1. For the bottom right, start with the First Vizier:
( f∗,a∗,s)→ ( f∗,G+ s,G+a∗)→ ( f∗) · (−(G+ s)) =−(G+a∗). The top left
is derived thus: (a∗, g, g+a∗)→ (g, G+a∗, G+g+a∗)→ (G+g+a∗) · g =
+(G+a∗). Finally, (a∗,s, f∗)→ (g+a∗,g+ f∗,s)→ (G+g+a∗,G+s,g+ f∗),
but the negative inner sign of the top dyad reverses sign as shown.
The 3 Box-Kites thus derived are the only among the 7 candidates to be viable:
for the Zigzag L-index of the S = 1 Sedenion Box-Kite does not underwrite a Sail;
hence, by what lawyers would call a “fruit of the poisoned tree” argument, neither
do the 3 U-trips associated with the same failed Zigzag. Using A* and B*, then
invoking the Roundabout Theorem, we see this readily:
+b∗+(G+g+ e∗)
+a∗ + (G+g+ f∗)
−(G+g+d∗)− c∗
+c∗ − (G+g+d∗)
NOT ZERO (only c*’s cancel)
With the appending of two successive bits to the left, the bottom-left and top-
right products are identical to those obtaining without the (G+g) being included.
Similarly, the top-left product uses Rule 2 twice, to similar effect, but with (G+g)
included in the outcome: since ( f∗,d∗,b∗) is CPO, we then get −(G+g+d∗).
For the top-right result, meanwhile, the two high bits induce a double reversal,
then are killed by XOR, leaving the product the same as if they hadn’t been there:
( f∗,c∗,e∗)→ (g+e∗,c∗,g+ f∗)→ (G+g+ f∗,c∗,G+g+e∗), hence −c∗. We
have an argument reminiscent of Theorem 2: depending on the inner sign of the
upper dyad, one pair of products cancels or the other, but not both.
We see, then, that the construction given without explanation at the end of
Part I is correct. The arguments given there concerning the vital relationship of
a Box-Kite’s non-ZD structures to semiotic modeling suggest that this “offing”
(to use the appropriately binary slang linked to Mafia hitmen) of a Zigzag’s 4
triplets should have a similarly significant role to play in such modeling. This has
bearing not just on semiotic, but physical models, since the key dynamic fact im-
plicit in the Zigzag L- and U- trips (or just Z-trips henceforth) is their similarity of
orientation: since (a,b,c);(a,B,C);(A,b,C);(A,B,c) are all CPO as written, we
are effectively allowed to do pairwise swaps of upper- and lower- case lettering
among them without inducing anything a physicist might deem observable (e.g.,
a 180◦ reversal or “spin quantum”). This condition of trip sync breaks down as
soon as we attempt to allow similar swapping between Z-trips and their Trefoil
compatriots: in particular, those 2 which don’t share an Assessor with the Zigzag.
The toy model of [7] would use these features to designate the basis of a “Cre-
ation Pressure” that leads to the output of the string theorist’s E8 ×E8 symmetry.
This symmetry, as discussed there, breaks in the standard models when one of the
primordial E8’s decays into an E6 – which has 72 roots to parallel the 72 filled
cells of our Sand Mandalas. For present purposes, the key aspect of this corre-
spondence is that, in ZD theory at least, the explosion of a singleton Box-Kite
into a Sand Mandalic trinity throws the off-switch on the source of the dynamics:
the Z-trips which underwrite trip sync no longer even underwrite Box-Kites. The
whole scenario suggests nothing so much as those boxes which, when opened by
pushing an external lever, emit an arm which pulls up on the same lever, forcing
the box to close and the arm to return to its hiding place inside it.
Let’s turn now to the ET graphics of the flip-book sequence, so suggestive of
cellular automata. For each of the 7 ET’s in question, all labels < g are monotoni-
cally increasing, since S, and hence their strut opposites, exceed them all. But the
only filled (but for long-diagonal crossings) rows and columns will be those with
labels equal to S−g = s and its strut-opposite g, for these L-indices reside at E
and B respectively in all 3 Box-Kites in the ensemble, hence either dyad contain-
ing one of them makes DMZs within each of the trio’s (a,d,e) and ( f ,d,b) Sails,
filling all 12 (= 24−2, minus 2 for diagonals) fillable cells in each row or column
tagged with these Assessors’ label. Thus, as s is incremented, two parallel sets of
perpendicular lines of ET cells start off defining a square missing its corners, then
these parallels move in unit increments toward each other, until they form a 2-ply
crossbar once s = 7 (S = 15). 24 cells each have row label R or column label C
= s; 24 reside in lines with label = g; and 24 more have their contents P = s or
g: these last have an orderliness that is less obvious, but by the last ET in the flip-
book, they have arrayed themselves to form the edges of a diamond, orthogonal to
the long diagonals and meeting up with the crossbar at its four corners, with s = 7
values filling the upward-pointing edges, and g = 8’s those sloping down.
The graphics for the flip-book first appeared in [6, p. 15]; they were recy-
cled on p. 13 of [8]; larger, easily-read versions of these ETs were then included
(along with numerous other Chingon-based flip-books and other graphics we’ll
discuss later) as Slides 25-31 of the Powerpoint presentation comprising [1], de-
livered at Wolfram Science’s June 15-18, 2006, NKS conference in Washington,
D.C. All three of these resources are available online, and the reader is especially
encouraged to explore the last, whose 78 slides can be thought of as the visual
accompaniment to this monograph. (Henceforth, references to numbered Slides
will be to those contained and indexed in it.)
6 64-D Spectrography: 3 Ingredients for “Recipe
Theory”
In a manner clearly related to Bott periodicity, strut constants fall into types de-
marcated by multiples of 8. But unlike the familiar modulo 8 categorization of
types demonstrated, perhaps most familiarly, in the Clifford algebras of various
dimensions, the situation with zero-divisors concerns not typology (which keeps
producing new patterns at all dimensions), but granularity. As we shall see, em-
anation tables for S > 8 (and not a power of 2), aside from diagonally aligned
cells in otherwise empty stretches, display checkerboard layouts of parallel and
perpendicular near-solid lines (NSLs), whose cells all have emanations save for
a pair of long-diagonal crossings, and whose visual rhythms are strictly governed
by S and 8 or the latter’s higher multiples.
The rule we found in the 32-D Pathions for the Sand Mandalas indicates that
the basic pattern (and BK5, S for 8 < S < 16) is “essentially the same” for all of
them. We put the qualifying phrase in quotes, as it is an open question at this
point what features, residing at what depth, are indeed “the same,” and which are
different. For the moment, we will invoke the term spectrographic equivalence as
a sort of promissory note, hoping to stuff ever more elements into its grab-bag of
properties, beginning with two. First is something at once intuitively obvious but
not readily proven. (We will include a corollary to a later theorem when we have
done so). Since the first 8 possible strut-constant values all display maximally-
filled ETs, and since anomalies displayed by higher values are strictly side-effects
of bits to the left of the 8-bit (which are, of course, its multiples), it is natural to
assume that any recursive induction upon simpler forms will echo this “octave”
structure: that each time S passes a new multiple of 8, it participates in a new type.
(As with the Sand Mandalas, we will see this means that BKN, S for the new 7- or
8-element spectral band of new forms will differ from that found in its predecessor
band.) This will lead, in the most clear-cut cases – S = 15, or a multiple of 8 not
a power of 2, say – to grids composed of 8 x 8 boxes some or all of whose borders
are NSLs.
How we determine which cases are clear-cut, meanwhile, and why and how
we might want or need to privilege them, leads to our second property to include
up-front in our grab-bag. In a manner reminiscent of the various tricks – like
minors and cofactors – used in classical matrix theory to prove two matrices are
equivalent, we can transform members of a spectral band into each other by cer-
tain formal methods of hand-waving. With the Sand Mandalas, for instance, we
could replace concrete indices in the row and column labels with abstract desig-
nations referencing the (a,b,c) values of each of their 3 Box-Kites, listed in one
of a number of predetermined orders: by least-first CPO ordering of such (a,b,c)
triplets, in a sequence determined by the Zigzag L-trip of the Sedenion Box-Kite
we can derive them from, for instance (which is equivalent to the 3 sand-mandalic
Box-Kites’ d values, as we’ve seen).
Since which cells are filled is strictly determined by S and G, such desig-
nations eliminate all individuality among the ETs in question. Hence, if certain
display features of one of them seem convenient, we can convert its “tone row”
of indices populating its row and column labels into an abstract layout, governed
by which index is associated with which Assessor, in the manner sketched last
paragraph. We could then use this layout as the template for re-writes of all other
ETs in the same spectral band, knowing that results obtained using the specific
instantiation of the band could thereby be converted into exactly analogous ones
for the other band-members.
We will, in fact, implicitly adopt this tactic by using S = 1 as an exemplary
“for instance” in numerous arguments, while employing the highest-valued S
found among the Sand Mandalas, 15, to simplify the visualizing (and calculat-
ing) of recursive pattern creation for fixed-S, growing N sequences. (S = 15 is
chosen because it has all its low bits filled, hence all XORs are derived by simple
subtraction, leaving carrybit overflow to show itself only in what matters most to
us: the turning off of 4 candidate Box-Kites in the Pathions, and – as we will show
two sections hence – 16 in the Chingons, and 4N−4 in all higher 2N-ions.) Where
we termed, for reasons already explained, the fixed-N, growing S sequences flip-
books, we designate these new displays (for reasons we’ll justify shortly) balloon-
rides.
While there is but one abstract type for the Sedenions, with one Box-Kite for
each of the 7 possible S values, a second spectral band emerges in the Pathions
to include the Sand Mandalas, and two more are added for the 64-D Chingons.
By induction from the universally shared first band for all N > 3, where there
are TripN−2 Box-Kites in each ET, for each S ≤ 8, the first new spectrographic
addition includes the upper multiple of 8 that bounds it, since it is not a power
of 2: 16 < S ≤ 24. The second new range, though, is bounded by G, hence does
not include it, as it is tautologically a power of 2 (which powers, as we saw two
sections ago, comply with a type all their own, with the same Box-Kite-count
formula as for the lowest spectral band): 24 < S < 32.
Each of these two new bands displays a distinctive feature which underwrites
one of the three key ingredients for the recipe theory we are ultimately aiming for.
We call these, for S ascending, (s,g)-modularity and hide/fill involution respec-
tively. The third key ingredient, meanwhile, resides in the band that first emerges
in the Pathions – and whose echo in the Chingons has recapitulative features suffi-
ciently rich as to merit the name of recursivity. We will be devoting Part III’s first
post-introductory section to a thorough treatment of the simplest instance of this
third ingredient, showing how to ascend into the meta-fractal we call the Whor-
fian Sky (named for the great theorist of linguistics, Benjamin Lee Whorf, whose
last-ever lecture on “Language, mind and reality” described the layering of mean-
ing in language in a manner strongly suggesting something akin to it). Among
many visionary passages in his descriptions of a future cross-disciplinary science,
the following seems most apt to serve as the lead-in quote for the third and final
sweep of our argument [9]:
Patterns form wholes, akin to the Gestalten of psychology, which are
embraced in larger wholes in continual progression. Thus the cos-
mic picture has a serial or hierarchical character, that of a progression
of planes or levels. Lacking recognition of such serial order, differ-
ent sciences chop segments, as it were, out of the world, segments
which perhaps cut across the direction of the natural levels, or stop
short when, upon reaching a major change of level, the phenomena
become of quite different type, or pass out of the ken of the older ob-
servational methods. But · · · the facts of the linguistic domain compel
recognition of serial planes, each explicitly given by an order of pat-
terning observed. It is as if, looking at a wall covered with fine tracery
of lacelike design, we found that this tracery served as the ground for
a bolder pattern, yet still delicate, of tiny flowers, and that upon be-
coming aware of this floral expanse we saw that multitudes of gaps
in it made another pattern like scrollwork, and that groups of scrolls
made letters, the letters if followed in a proper sequence made words,
the words were aligned in columns which listed and classified enti-
ties, and so on in continual cross-patterning until we found this wall
to be – a great book of wisdom! [10, p. 248]
Appendix A: Genealogy of S = 1 Box-Kites
N = 4: Unique Quaternion L-index set (1,2,3) fed as Rule 0 circle into PSL(2,7)
with central g = 4, yielding 7 Octonions trips, each with a different S. For S = 1,
have (3,6,5), which becomes singleton Rule 0 for next level.
N = 5: (3,6,5) fed as Rule 0 circle into PSL(2,7) with central g= 8 yields 3 Rule 2
L-trips as triangle’s sides, which (upon affixing their strut opposites as L-indices)
generate (along with zero-padded (3,6,5) ) 4 Box-Kites with X = G+1 = 17.
Triangle’s medians become (a,d,e) Trefoil L-index sets of 3 Rule 1 S = 1 Box-
Kites, making 7 in all. These Zigzag L-index sets become Rule 0 trips for the next
level, and are:
Rule 0: (3,6,5)
Rule 1: (3,10,9); (6,15,9); (5,12,9)
Rule 2: (3,13,14); (6,11,13); (5,14,11)
N = 6: The 7 N = 5 Zigzag L-index sets just listed are fed as Rule 0 circles into
PSL(2,7) triangles with central g = 16, and are Zigzag L-index sets in their own
right for Box-Kites with X = G+1 = 33.
10 Rule 1 medians, 3 redundant (as they generate ( f ,d,b)’s where (a,d,e)’s
are also given: (14,16,30)* and (11,16,27)** in (5,14,11)’s triangle, the latter
also in (6,11,13)’s). They are associated with these 7 Zigzag L-index sets:
(3,18,17); (5,20,17); (6,23,17); (9,24,17);
(10,27,17)∗; (12,29,17); (15,30,17)∗∗
Rule 2 sides: 3 per each Rule 0 trip, as follows:
(3,6,5)→ (3,21,22); (6,19,21); (5,22,19)
(3,10,9)→ (3,25,26); (10,19,25); (9,26,19)
(6,15,9)→ (6,25,31); (15,22,25); (9,31,22)
(5,12,9)→ (5,25,28); (12,21,25); (9,28,21)
(3,13,14)→ (3,30,29); (13,19,30); (14,29,19)
(6,11,13)→ (6,29,27); (11,22,29); (13,27,22)
(5,14,11)→ (5,27,30); (14,21,27); (11,30,21)
N = 7: Feed the just-listed 35 Zigzag L-index sets to PSL(2,7)’s with g = 32, as
Rule 0 circles, thereby generating the 155 S = 1 Zigzags found in the 27-ions, or
Routions – named for the site of the Internet Bubble’s once-famed “Massachusetts
Miracle,” Route 128 – and so on.
Appendix B: A Brief Intro to “Type II” Box-Kites
The recursive generation of Zigzag L-sets just presented calls for some close at-
tention when the box-kites involved are Type II, since they then have the diagonals
of their PSL(2,7) triangles oriented differently: instead of all 3 leading from mid-
points of the Rule 2 sides to the corners, only 1 of these will preserve orientation
for a Type II (with the other two having “reversed VZ1” rules in evidence). We
first give a construction for producing all the Type II box-kites in the Pathions, and
then indicate the manner in which their workings are intimately connected with
the phenomenology of twist products broached in Part I’s Theorem 6.
The construction was presented with different framing in [8], where we de-
ployed a “stereo Fano” representation using side-by-side triangles, the left being a
proper PSL(2,7). Within the Pathions, there are 7 distinct box-kites for each S ex-
cept for the “flip-book” trios, one for each S> 8. And for S= 8 exactly, we saw in
our discussion of the Number Hub Theorem that we can build all 7 by placing 8 in
the center of the standard Fano (what we’ll call PSL(2,7) henceforth), then taking
the Zigzag L-trip for each Sedenion S and placing its units at the sides’ midpoints,
in the usual CPO order (in left, right, and bottom sides respectively). Each of
these 7 lines then generates a new box-kite in the Pathions for the Sedenion S in
question.
If we re-inscribe the starter-kit L-trip, but change G to the Pathion’s 16, ap-
plying VZ2 gives us new U-index terms, but the L-index terms for all 6 Assessors
remain the same as for the Sedenion box-kite: we call this “Rule 0” instance the
zero-padded box-kite (or just ZP) for the S value in question.
If we take the 3 “Rule 1” triplets along the struts, and place them not at the A,
B, C positions of our new Pathion box-kites, but instead at A, D, E (with 8 always
winding up at D), we generate 3 more standard (Type I) box-kites. For S = 1, the
Sedenion Zigzag L-trip is just (3,6,5), and each of its units becomes the low-index
‘A’ for a new Pathion box-kite, with L-indices written in “nested parentheses”
order (that is, A, B, C, D, E, F) as follows: (3,10,9,8,11,2); (6,15,9,8,14,7);
(5,12,9,8,13,4).
But if we take the 3 “Rule 2” triplets along the edges, mapping the Zigzag unit
at the center of each to the low-index ‘A’ of a new box-kite, the 8 doesn’t show
at any Assessor, and two of the three struts will have orientations reversed. These
“Type II” box-kites, again for S = 1, written per the same convention just used for
their 8-bearing siblings, read like this: (3,13,14,15,12,2); (6,11,13,12,10,7);
(5,14,11,10,15,4). Since the A and F low-index terms are the same as in the
same-S Sedenion box-kite, the strut they make obviously has the standard orienta-
tion. (But note that there is nothing essential about the (A,F) strut here: the placing
of the lowest-indexed unit of the Zigzag L-trip at A is a convenient convention,
and its employment in the Pathions suffices to induce this effect; however, it no
longer suffices in higher dimensions, where S can exceed 8 yet still be less than
That their being Type II is an immediate side-effect of “Rule 2” in this method
of deriving them should be obvious. What is less obvious is their special relation-
ship with twist products. Here, we review some of the basics: in the Sedenions,
whenever two Assessors bound an edge, we can swap a pair of corresponding
terms (either L- or U- indices) and then switch the sign joining the L- and U- in-
dices in the resultant pairing, and get an Assessor in another box-kite as a result.
Such “twist products,” then, reverse the edge-sign of a given line of ZDs as we
move between containing box-kites. Moreover, such twists are naturally investi-
gated in the context of the squares, not the triangles, of the octahedral vertex figure
we write Assessors on: the three orthogonal Catamarans, then, instead of the four
touching-only-at-the-vertices Sails. That’s because opposite sides of a Catamaran
twist to Assessors in the same box-kite, so that each Catamaran lets one twist to
two different box-kites – with the terminal Catamaran, in each case, being further
twistable into the box-kite you didn’t twist to in the first instance. As shown in the
“Twisted Sister” and “Royal Hunt” diagrams of [4], these triple transforms can be
represented in their own Fano planes, with the indices placed on their loci now
corresponding to the strut constants of a septet of box-kites.
Each Catamaran comprises the pathways connecting 4 Assessors – meaning
it doesn’t connect up with either term of the third strut in its box-kite. It is not
hard to see that the strut constant of the box-kite one twists to is equal to the strut-
opposite of the term which completes the L-trip of the edge being twisted in the
first place. Hence, any L-index term on a Sedenion box-kite corresponds to the
strut constant of another such box-kite one can twist to. This suggests expanding
the meaning of “twist product” to embrace pairings which share a strut rather than
an edge. For, if we allow this, we can then treat the third strut orthogonal to the
square hosting twists as the “mast” of the Catamaran, giving us an expanded sense
of this latter term which allows us a major simplification: instead of thinking of
the Sedenions’ ZDs as distributed among 7 distinct box-kites, we can see them all
included in one “embroidered” box-kite diagram, which we call a brocade. Each
of the 12 box-kite edges allows twists to a pair of different Assessors – let’s say
(A, b) and (B, a), in the box-kite with S = copp = d. More, the (S,X) pair – which
we can think of as in the box-kite’s center – can be “twisted” with all 6 Assessors
in the original box-kite to yield 12 more. We therefore have 6 + 24 + 12 = the
total set of “42 Assessors” in the Sedenions, all representable, on any one of the 7
component box-kites, as a unitary “brocade.”
It would be nice to be able to generalize the “brocade” notion so as to reduce
the number of basic structures in higher-order contexts: in the Pathions, for in-
stance, there are 77 box-kites, all but 21 of which are “Type I,” with 21 of those
coming in sand-mandala triples, 7 forming the S = 8 “Atlas,” plus 7 ZP’s and 3 ·7
“strongboxes” (so called, because these low-S box-kites contain “pieces of 8”)
completing the collection. But if we also count in the 4 · 7 = 28 “missing” box-
kites for high S, we can collapse our head count from 105 box-kite-like structures
to 15 brocades. Miming the Sedenion situation, the 7 ZP’s form the simplest; the
7 Sand-Mandala trios intermingle with the Atlas septet and the 21 strongboxes to
make 7 more brocades; and the 21 Type II box-kites twist into each other (to fill
out one Catamaran in each) and into the “hidden box-kites” linked with high S
(filling out two more Catamarans per Type II instance), yielding up the final set
of 7 brocades. (We note that the Type II situation is not as mysterious as it might
appear, once we recall the “slipcover proof” logic of Part I, Section 5: with 2 of
3 strut triplets being reversed, “tugging” on a Type II’s Fano will tend to send a
reversed arrow onto an edge 4 times out of 6 – meaning that, in all such cases, the
corollary to Theorem 7, and hence the theorem itself, will fail, thereby explaining
the “why” of “missing” box-kites!)
We gain the generalized “brocade” simplification at a very small price: relax-
ing the notion of “twist product” to embrace source and target L- and U- index
pairs which aren’t necessarily zero-divisors within the context of the G at hand.
But this is an investment which pays dividends, since it allows us to use Type II
structures as “middlemen” to facilitate studying the “hidden box-kite” substruc-
tures of the meta-fractal “white space” in high-S ET’s. Given the semiotic and
semantic importance of “ZD-free” structures (recall that our transcription of Pe-
titot’s analysis of Greimas’ “Semiotic Square” into zero-divisor theory is based
on ZD-free strut opposites), we can expect a richness of results based on Catama-
ran study that should at least equal that we are conducting based on Sails. (For
a “coming attraction,” interested readers should see the online Powerpoint slide-
show linked with our NKS 2007 presentation [11], which will play a role with
respect to our forthcoming and similarly named monograph, “Voyage by Catama-
ran,” akin to that our NKS 2006 slide-show did for the theorem/proof exposition
you are currently reading.)
References
[1] Robert P. C. de Marrais, “Placeholder Substructures: The Road from NKS
to Small-World, Scale-Free Networks Is Paved with Zero-Divisors,” http://
wolframscience.com/conference/2006/ presentations/materials/demarrais.ppt
(Note: the author’s surname is listed under “M,” not “D.”)
[2] Robert P. C. de Marrais, “Placeholder Substructures I: The Road From NKS
to Scale-Free Networks is Paved with Zero Divisors,” Complex Systems, 17
(2007), 125-142; arXiv:math.RA/0703745.
[3] Robert P. C. de Marrais, “The 42 Assessors and the Box-Kites They Fly,”
arXiv:math.GM/0011260.
[4] Robert P. C. de Marrais, “Presto! Digitization,” arXiv:math.RA/0603281
[5] Stephen Wolfram, A New Kind of Science, (Wolfram Media, Champaign IL,
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[6] Robert P. C. de Marrais, “Flying Higher Than A Box-Kite,”
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http://arxiv.org/abs/math/0703745
http://arxiv.org/abs/math/0011260
http://arxiv.org/abs/math/0603281
http://www.wolframscience.com/nksonline
http://arxiv.org/abs/math/0207003
[7] Robert P. C. de Marrais, “The Marriage of Nothing and All: Zero-Divisor
Box-Kites in a ‘TOE’ Sky”, in Proceedings of the 26th International Col-
loquium on Group Theoretical Methods in Physics, The Graduate Center
of the City University of New York, June 26-30, 2006, forthcoming from
Springer–Verlag.
[8] Robert P. C. de Marrais, “The ‘Something From Nothing’ Insertion Point”,
http://www.wolframscience.com/conference/2004/presentations/materials/
rdemarrais.pdf
[9] Robert P. C. de Marrais, “Placeholder Substructures III: A Bit-String-Driven
‘Recipe Theory’ for Infinite-Dimensional Zero-Divisor Spaces,”
arXiv:0704.0112 [math.RA])
[10] Benjamin Lee Whorf, Language, Thought, and Reality, edited by John B.
Carroll (M.I.T. Press, Cambridge MA, 1956).
[11] Robert P. C. de Marrais, “Voyage by Catamaran: Long-Distance Seman-
tic Navigation, from Myth Logic to Semantic Web, Can Be Effected by
Infinite-Dimensional Zero-Divisor Ensembles,” wolframscience.com/
conference/2007/presentations/materials/demarrais.ppt (Note: the author’s
surname is listed this time American style, under “D,” not “M.”)
http://www.wolframscience.com/conference/2004/presentations/materials/
http://arxiv.org/abs/0704.0112
Introduction By Way of Reprise: From Box-Kites to ETs
Emanation Tables: Conventions for Construction
ETs for N > 4 and S 7
The Number Hub Theorem (S = 2N - 2) for 2N-ions
The Sand Mandala Flip-Book (8 < S < 16, N = 5)
64-D Spectrography: 3 Ingredients for ``Recipe Theory''
|
0704.0027 | Filling-Factor-Dependent Magnetophonon Resonance in Graphene | Filling-Factor-Dependent Magnetophonon Resonance in Graphene
M. O. Goerbig,1 J.-N. Fuchs,1 K. Kechedzhi,2 and Vladimir I. Fal’ko2
Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR 8502, F-91405 Orsay, France and
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: October 23, 2018)
We describe a peculiar fine structure acquired by the in-plane optical phonon at the Γ-point
in graphene when it is brought into resonance with one of the inter-Landau-level transitions in
this material. The effect is most pronounced when this lattice mode (associated with the G-band
in graphene Raman spectrum) is in resonance with inter-Landau-level transitions 0 ⇒ +, 1 and
−, 1 ⇒ 0, at a magnetic field B0 ≃ 30T. It can be used to measure the strength of the electron-
phonon coupling directly, and its filling-factor dependence can be used experimentally to detect
circularly polarized lattice vibrations.
PACS numbers: 78.30.Na, 73.43.-f, 81.05.Uw
In metals and semiconductors the spectra of phonons
are renormalized by their interaction with electrons.
Some of the best known examples include the Kohn
anomaly [1] in the phonon dispersion, which originates
from the excitation/de-excitation of electrons across the
Fermi level upon the propagation of a phonon through
the bulk of a metal and a shift in the longitudinal opti-
cal phonon frequency in heavily doped polar semiconduc-
tors [2]. However, despite the transparency of theoretical
models the observation of such effects is often obscured
by the difficulty to change the electron density in a mate-
rial, whereas in semiconductor structures containing two-
dimensional (2D) electrons the density of which can be
varied, the influence of the latter on the phonon modes is
weak due to a negligibly small volume fraction occupied
by the electron gas. In this context, a unique opportunity
arises in graphene-based field-effect transistors [3], where
the density of carriers in an atomically thin film (mono-
layer [4, 5, 6] or a bilayer [7]) can be continuously varied
from 1013cm−2 p-type to 1013cm−2 n-type. Several Ra-
man experiments have already been reported [8, 9] where
the variation of carrier density in graphene changes the
optical phonon frequency, in agreement with theoretical
expectations [10, 11, 12].
When graphene is exposed to a quantizing magnetic
field, its electronic spectrum quenches into discrete Lan-
dau levels (LLs) [13]. Then, the optical phonon energy in
graphene may coincide with the energy of one of the inter-
LL transitions, a condition known as magnetophonon
resonance [14, 15]. Recently, Ando has suggested [16]
that in undoped graphene the magnetophonon resonance
enhances the effect of the electron-phonon coupling on
a spectrum of the in-plane optical phonons - the E2g
modes attributed to the G-band in the Raman spectra
in Refs. [8, 9, 17, 18, 19]. In this paper, we investigate
a rich structure of the anti-crossing experienced by such
lattice modes when a magnetic field makes their energy
equal to the energy of one of the valley-antisymmetric
interband magnetoexcitons [20]. Most saliently, the dif-
ference between circular polarization of various inter-LL
transitions [21, 22] makes the magnetophonon resonance
distinguishable for lattice vibrations of different circular
polarization, which makes the number of split lines in
the fine structure acquired by a phonon and the value of
splitting dependent on the electronic filling factor, ν.
The in-plane optical phonons in graphene [relative dis-
placement u = (ux, uy) of sublattices A and B] have
the energy ω ≈ 0.2eV at the Γ-point (in the center of
the Brillouin zone). These phonons and their coupling to
electrons can be described using the Hamiltonian [10, 11],
Hph =
ωb†µ,qbµ,q + g
2Mω(σxuy − σyux), (1)
u(r) =
2NucMω
bµ,q + b
eµ,qe
−iq·r,
where b
µ,q are annihilation (creation) operators of a
phonon with polarisation eµ,q, M is the mass of a car-
bon atom, and Nuc is the number of unit cells. Here
and below, we use units ~ ≡ 1. Also, we shall uti-
lize a double degeneracy of the E2g mode at the Γ-point
(at q = 0) and describe the in-plane optical phonon in
terms of a degenerate pair of circularly polarized modes,
u = (ux+iuy)/
2 and u� = u
. The constant g in Eq.
(1) characterizes the electron-phonon coupling [23]. This
coupling has the form of the only invariant linear in u per-
mitted by the symmetry group of the honeycomb crystal.
It is constructed using Pauli matrices σ = (σx, σy) acting
in the space of sublattice components of the Bloch func-
tions, [φK+A, φK+B] and [φK−B, φK−A] which describe
electron states in the valleys K± (two opposite corners of
the hexagonal Brillouin zone) and obey the Hamiltonian,
in terms of the electron charge −e < 0 [24],
Hel = ξvσ · p, p =− i∇+ eA, ∂xAy − ∂yAx = B.
Here, ξ = ± distinguishes between K±, and momentum
p is calculated with respect to the center of the corre-
sponding valley. This Hamiltonian represents the dom-
inant term of the next-neighbor tight-binding model of
graphene [25, 26, 27], and the electron-phonon coupling
http://arxiv.org/abs/0704.0027v4
: B sublattice
: A sublattice
(a) (b)
−,(n+1)
+,(n+1)
FIG. 1: (a) Optical phonons are lattice vibrations with an out-
off-phase oscillation of the two sublattices. (b) Interband electron-
hole excitations coupling to phonon modes with different circular
polarization.
in Eq. (1) takes into account the change in the A − B
hopping elements due to the sublattice displacement [28].
In a perpendicular magnetic field, Hel determines [13]
a spectrum of 4-fold (spin and valley) degenerate LLs,
εα=±n = α
2nvλ−1B in the valence band (ε
n>0), con-
duction band (ε+n>0), and at zero energy (ε0 = 0, ex-
actly at the Dirac point in the electron spectrum), in
terms of the magnetic length λB = 1/
eB. Such a
spectrum has been confirmed by recent quantum Hall
effect measurements [4, 5, 6]. In each of the two val-
leys, the LL basis is given by two-component states
1 + δn,0φn,m, iξα(1−δn,0)φn−1,m], where φn,m are
the LL wave functions described by the quantum num-
bers n and m, the latter being related to the guiding
center degree of freedom. Here, we neglect the Zeeman
effect, and simply take into account the two-fold spin de-
generacy.
Excitations of electrons between LLs can be described
in terms of magnetoexcitons (see Fig. 1). Those relevant
for the magnetophonon resonance are
(n, ξ) =
1 + δn,0
+,n,m;ξc−,(n+1),m;ξ,
�(n, ξ) =
1 + δn,0
+,(n+1),m;ξ
c−,n,m;ξ, (2)
where the index A = ,� characterizes the angular mo-
mentum of the excitation and the operators c
α,n,m;ξ
annihilate (create) an electron in the state α, n,m in
the valley Kξ. The normalization factors N n =
[(1 + δn,0)NB(ν̄−,(n+1) − ν̄+,n)]1/2 and N�n = [(1 +
δn,0)NB(ν̄−,n − ν̄+,(n+1))]1/2 are used to ensure the
bosonic commutation relations of the exciton operators,
[ψA(n, ξ), ψ
′, ξ′)] = δA,A′δξ,ξ′δn,n′ , where NB is the
total number of states per LL in a sample, including
the two-fold spin-degeneracy. These commutation rela-
tions are obtained within the mean-field approximation
with 〈c†α,n,m;ξcα′,n′,m′;ξ′〉 = δξ,ξ′δα,α′δn,n′δm,m′(δα,− +
δα,+ν̄α,n), where 0 ≤ ν̄α,n ≤ 1 is the partial filling fac-
tor of the n-th LL. Similarly to magneto-optical selec-
tion rules in graphene [20, 21, 22], α, n ⇒ α′, n ± 1, -
polarized phonons are coupled to electronic transitions
with −, (n + 1) ⇒ +, n, and �-polarized phonons to
−, n ⇒ +, (n + 1) magneto-excitons, at the same en-
ergy Ωn ≡
2(v/λB)(
n+ 1) (Fig. 1), which
follows directly from the composition of the LL in
graphene and the form of the electron-phonon coupling
in Eq. (1). In contrast to photons that couple to
the valley-symmetric mode ψA,s(n) = [ψA(n,K+) +
ψA(n,K−)]/
2, electron-phonon interaction in Eq.(1)
couples phonons to the valley-antisymmetric magnetoex-
citon ψA,as(n) = [ψA(n,K+)− ψA(n,K−)]/
In terms of magnetoexcitons we can, now, rewrite the
electron-phonon Hamiltonian in a bosonized form, as
τ=s,as
A,τ (n)ψA,τ (n) +
AbA (3)
gA(n)
ψA;as(n) + bAψ
A;as(n)
g (n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1),
where gA are the effective coupling constants, with γ =
3a2/2πλ2B and a = 1.4Å (distance between neighbor-
ing carbon atoms). In the Hamiltonian (3), we have omit-
ted electronic excitations with a higher angular momen-
tum which do not couple to the in-plane optical phonon
modes (e.g., n ⇒ n′, with n′ 6= n ± 1). The dressed
phonon operator corresponding to the Hamiltonian (3)
is obtained by solving Dyson’s equation. The pole of the
propagator gives the antisymmetric coupled mode fre-
quencies ω̃A,
ω̃2A − ω2 = 4ω
n=nF+1
ω̃2A − Ω2n
∆nF g
A(nF )
ω̃2A −∆2nF
, (4)
where nF stands for the number of the highest fully occu-
pied LL in the spectrum, and ∆n =
2(v/λB)(
n+ 1−√
n). In Eq. (4), the sum (extended up to the high-
energy cut-off N ∼ (λB/a)2 above which the electronic
dispersion is no longer linear) takes into account inter-
band magnetoexcitons, and the last term gives a small
correction due to an intraband magnetoexciton. In the
small-field limit and large doping (nF ≫ 1), solution of
Eq. (4) reproduces the zero-field result [10, 11] if one
replaces the sum by an integral,
n=0 →
dn, ap-
proximates
n+ 1 ≈ 2
n and ∆nF ≈ 0, and,
then, linearizes Eq. (4) by replacing ω̃A by ω in the de-
nominator,
ω̃ ≃ ω̃0 + λ
ω + 2
2nFv/λB
ω − 2
2nFv/λB
ω̃0 ≃ ω + 2
ω2 − Ω2n
where λ = (2/
3π)(g/t)2 ≃ 3.3× 10−3 is the same as in
Refs. [10, 16] (t = 2v/3a ∼ 3eV is the A−B hopping am-
plitude), and ω̃0 is the renormalized phonon frequency in
an undoped graphene sheet at B = 0. The only variation
arises at high fields, ω̃0 &
2v/λB, where for nF = 0 the
linearized Eq. (4) yields
ω̃ ≃ ω̃0 −
g2(0)
(ω̃0λB/
2v)2 − 1
The strongest effect of the phonon coupling to elec-
tron modes occurs when the frequency of the former
coincides with the frequency Ωn of one of the magne-
toexcitons ψA,as(n). In such a case, the sum on the
right-hand-side of the eigenvalue equation (4) is domi-
nated by the resonance term and may be approximated
by 2ωg2A(n)/ (ω̃A − Ωn). This results in a fine structure
of mixed phonon-magnetoexciton modes, ψA,as(n) cos θ+
bA sin θ with frequency ω̃
A and ψA,as(n) sin θ − bA cos θ
with frequency ω̃−A [where cot 2θ = (Ωn−ω̃0)/2gA], which
are determined for each polarisation (A = ,� ) sepa-
rately,
(n) = 1
(Ωn + ω̃0)∓
(Ωn − ω̃0)2 + g2A(n). (5)
A generic form of the phonon-magnetoexciton anti-
crossing and formation of coupled modes, ω±
(n) in un-
doped graphene (i.e., ν = 0) is illustrated in Fig. 2(a).
Such an anticrossing and mode mixing is simlar to that
described by Ando [16]. It can manifest itself in Raman
spectroscopy: in a fine structure acquired by the G-line
(earlier attributed [8, 9, 17, 18, 19] to the in-plane op-
tical phonon at the Γ-point, E2g mode) at the magneto-
phonon resonance conditions. The effect is the strongest
for the resonance Ωn=0 ≈ ω̃0 between the phonon and
magnetoexciton based upon −, 1 ⇒ 0 and 0 ⇒ +, 1 tran-
sitions. When approaching the resonance (by sweeping a
magnetic field), the phonon line becomes accompanied by
a weak satellite moving towards it and increasing its in-
tensity. Exactly at the magnetophonon resonance, where
both the upper mode [ω̃+A(n)] and the lower mode [ω̃
A(n)]
consist of an equal-weight superposition of the phonon
and the resonant exciton, with cos θ = sin θ = 1/
the G-band in graphene would appear as two lines. For
Ωn=0 =
2v/λB ≈ 36
B[T] meV (see [16, 24]) and
ω̃0 ≃ 200 meV, this resonance occurs in an experimen-
tally accessible field range, B0 ≃ 30 T. For the filling
factor ν = 0, the central LL (n = 0) is always half-filled.
Then, coupling and, therefore, splitting of the �- and -
polarized modes coincide, g� = g , thus, giving rise to a
pair of peaks at the energies ω̃± = ω̃0 ± g� sketched
in part I in Fig. 2(b). For the magnetic field value
B0 ≃ 30 T and g ≃ 0.28eV [12], we estimate this splitting
as 2gA ∼ 16meV (∼ 130cm−1), which largely exceeds the
G-band width observed in Refs. [8, 9, 17, 18, 19].
Doping of graphene changes the strength of the cou-
pling constants g� and g , as shown in Fig. 2(c). This
5 10 15 20 25 30 35
Magnetic Field [T]
3010 20 40
ν = 0
0 < |ν| < 2
|ν| = 2
2g 2g
mode splitting
−6 −4 −2 0
4n−2 4n+2 4n+6
B=Bn>02γ
FIG. 2: (a) Coupled phonon and magneto-excitons as a function
of the magnetic field. Energies are in units of the bare phonon
energy ω. Dashed lines indicate the uncoupled valley-symmetric
modes, with gA = 0. (b) Mode splitting as a function of the filling
factor, as may be seen in Raman spectroscopy, with the resonance
condition Ωn=0 ≈ ω̃0, for ν = 0 in (I), 0 < |ν| < 2 (in II), and
ν = ±2 (in III). The absolute intensity of the modes is in arbitrary
units, but the height and the width reflect the expected relative
intensities. (c) Mode splitting for n = 0, as a function of the filling
factor ν. (d) Same as in (c) for n ≥ 1.
is because a higher (lower) occupancy of the n = 0 LL
reduces (enhances) the oscillator strength of the polar-
ized transition due to the availability of filled and empty
states in the involved LLs, whereas the same change in
the electron density has the opposite effect on g�. As
a result, for an arbitrary filling factor −2 < ν < 2,
we predict that, in the vicinity of magnetophonon reso-
nance, the phonon mode (and, therefore, G-band in Ra-
man spectrum) should split into four lines [part II in Fig.
2(b)], with ω̃±� = ω̃±g� for �-polarized and ω̃± = ω̃±g
for -polarized phonons. In the quantum Hall state at
filling factor ν = 2, the transition −, 1 ⇒ 0 becomes suc-
cessively blocked and no longer affects the frequency of a
-polarized phonon, whereas the transition 0 ⇒ +, 1 ac-
quires the maximum strength, thus, increasing the cou-
pling parameter g�. This leads to the magnetophonon
resonance fine structure consisting of three peaks, with
an even larger splitting between side lines, as sketched in
part III in Fig. 2(b). Interestingly, this may enable one
to directly observe lattice modes with a definite circu-
lar polarization. A further increase of the electron filling
factor reduces the side-line splitting which should com-
pletely disappear at ν = 6, after the transition 0 ⇒ +, 1
becomes blocked by a complete filling of the +, 1 LL [Fig.
2(c)]. The same arguments hold for p-doped graphene,
though in this case the roles of �- and -polarized modes
are interchanged.
Magnetophonon resonances with other possible inter-
LL transitions n ⇒ n+ 1 occur at much lower magnetic
fields, Bn = B0/(
n+ 1)2. For example, a resonant
phonon coupling with the magnetoexciton ψA;as(1) is ex-
pected to occur at B1 ≈ 5T. Its description remains qual-
itatively similar, though for n > 0 the mode splitting is
less pronounced because of the B-field dependence of the
coupling constants in Eq. (3). One finds that g� = g
for |ν| < 2(2n − 1). At ν = 2(2n − 1), filling of the n-
th LL starts changing, which reduces splitting of the -
polarized mode and gives rise to the four-peak structure.
At ν = 2(2n+1), where the +, n LL becomes completely
filled, splitting of the -polarized phonon vanishes, thus,
resulting in the three-peak fine structure [part III in Fig.
2(b)] that would persist up to ν = 2(2n + 3). This is
because the splitting of the �-polarized modes remains
constant up to the filling factor ν = 2(2n + 1), above
which population of the +, (n+ 1) LL starts to suppress
the value of g�, until the latter vanishes at ν = 2(2n+3)
[see Fig. 2(d)].
In conclusion, we have predicted a filling-factor depen-
dence of the fine structure acquired by the in-plane (E2g)
optical phonon in graphene when the latter is in reso-
nance with one of the inter-LL transitions in this ma-
terial. The effect is expected to be most pronounced
when the phonon is resonantly coupled to the 0 ⇒ +, 1
and −, 1 ⇒ 0 transitions, which requires a magnetic field
B0 ≃ 30T. The predicted mode splitting may be used
to measure directly the strength of the electron-phonon
coupling, and also to distinguish between circularly (left-
and right- hand) polarized lattice modes.
We thank D. Abergel, A. Ferrari, P. Lederer, and A.
Pinczuk for useful discussions. This work was suported
by Agence Nationale de la Recherche Grant ANR-06-
NANO-019-03 and EPSRC-Lancaster Portfolio Partner-
ship EP/C511743. We thank the MPI-PKS workshop
‘Dynamics and Relaxation in Complex Quantum and
Classical Systems and Nanostructures’ and the Kavli
Institute for Theoretical Physics, UCSB (NSF PHY99-
07949) for hospitality.
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Erratum
In the previous version (v3) of this Letter, we have
underestimated the numerical value of the mode split-
ting of the magnetophonon resonance [see paragraph af-
ter Eq. (5)] by a factor of 2 (the text above takes into
account the corrected parameters). This is a result of
two mistakes. First, there is a factor of
2, which finds
its origin in an erroneous normalization of the circular
polarized phonons. They should indeed be defined as
u = (ux + iuy)/
2 and u� = (ux − iuy)/
2 [and not
as u = ux + iuy and u� = ux − iuy as incorrectly
assumed on page 1, second column], such that the asso-
ciated phonon operators bA obey the usual commutation
relations [bA, b
] = δA,A′ , with A = ,�. This yields
a factor of
2 in the definition of the effective coupling
constants [Eq. (3)], which read in the corrected form
g (n) = g
(1 + δn,0)γ
ν̄−,(n+1) − ν̄+,n ,
g�(n) = g
(1 + δn,0)γ
ν̄−,n − ν̄+,(n+1) .
As a consequence, the zero-field dimensionless coupling
constant λ [defined in the first column page 3 of our
Letter] is multiplied by a factor of 2 and becomes λ =
3π)(g/t)2.
Second, we also underestimated the numerical value of
the electron-phonon coupling constant g by a factor of√
2. Indeed, g defined in our work [see Eq. (1)] is related
to 〈g2Γ〉F ≃ 0.0405 eV2 computed by Piscanec et al. [2]
as g =
2〈g2Γ〉F ≃ 0.28 eV and not as g =
〈g2Γ〉F ≃ 0.2
eV as incorrectly assumed in our Letter. In addition,
there is a substantial uncertainty in the precise value of
the constant g. In a tight-binding model, the latter may
be related to the derivative of the hopping amplitude t
as a function of the carbon-carbon distance a as g =
(−dt/da)× 3/(2
Mω) [1]. Harrison’s phenomenological
law t ∝ 1/a2 then implies that g ≃ 0.26 eV. Experiments
in graphene [3] and [4] in zero magnetic field give for the
dimensionless coupling constant λ the values 4.4 × 10−3
and 5.3×10−3 respectively. This determines g in between
0.3 eV and 0.36 eV, where we take into account that the
value of t lies between 2.7 and 3 eV. In the end, we have
to take g in the range between 0.26 and 0.36 eV [instead
of g ≃ 0.2 eV] and therefore the dimensionless coupling
constant becomes λ ≃ (2.8 to 5.3)× 10−3 [instead of λ ≃
10−3].
As a result of the two factors of
2, the numerical
estimate for the mode splitting 2gA at ν = 0 and B ≃ 30
T [at the discussed resonance −, 1 ⇒ 0 and 0 ⇒ +, 1, see
second column of page 3] becomes 2gA ∼ 15 meV (∼ 120
cm−1), for g ≃ 0.26 eV and 2gA ∼ 20 meV (∼ 160 cm−1)
for g ≃ 0.36 eV [instead of 2gA ∼ 8 meV]. The effect
is therefore twice larger than initially predicted. The
conclusions of our work remain unaltered.
We would like to thank C. Faugeras and M. Potemski
for having drawn our attention on the underestimated
value of the mode splitting. See also their recent preprint
where they measure the magnetophonon resonance [5].
[1] T. Ando, J. Phys. Soc. Jpn 75, 124701 (2006); ibid 76,
024712 (2007).
[2] S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J.
Robertson, Phys. Rev. Lett. 93, 185503 (2004).
[3] S. Pisana, M. Lazzeri, C. Casiraghi, K. S. Novoselov, A.
K. Geim, A. C. Ferrari, and F. Mauri, Nature Materials
6, 198 (2007).
[4] J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Phys. Rev.
Lett. 98, 166802 (2007).
[5] C. Faugeras, M. Amado, P. Kossacki, M. Orlita, M.
Sprinkle, C. Berger, W.A. de Heer and M. Potemski,
arXiv:0907.5498.
http://arxiv.org/abs/0907.5498
|
0704.0028 | Pfaffians, hafnians and products of real linear functionals | Pfa�ans, hafnians and produ
ts of real linear
fun
tionals
Péter E. Frenkel
Alfréd Rényi Institute of Mathemati
s
Hungarian A
ademy of S
ien
es
P.O.B. 127, 1364 Budapest, Hungary
frenkelp�renyi.hu
Abstra
t
We prove pfa�an and hafnian versions of Lieb's inequalities on deter-
minants and permanents of positive semi-de�nite matri
es. We use the
hafnian inequality to improve the lower bound of Révész and Sarantopou-
los on the norm of a produ
t of linear fun
tionals on a real Eu
lidean
spa
e (this subje
t is sometimes
alled the `real linear polarization
on-
stant' problem).
Mathemati
s Subje
t Classi�
ation: 46C05, 15A15
Keywords: polarization
onstant, real Eu
lidean spa
e, hafnian, pfaf-
�an, positive semi-de�nite matrix
-1. Introdu
tion
The
ontents of this paper are as follows. In Se
tion 0, we sket
h one part of
the histori
ba
kground:
lassi
al inequalities on determinants and permanents
of positive semi-de�nite matri
es. In Se
tion 1, we prove pfa�an and hafnian
versions of these inequalities, and we formulate Conje
ture 1.5, another hafnian
inequality. In Se
tion 2, we apply the hafnian inequality of Theorem 1.4 to
our main goal: improving the lower bound of Révész and Sarantopoulos on the
norm of a produ
t of linear fun
tionals on a real Eu
lidean spa
e (this subje
t
is sometimes
alled the `real linear polarization
onstant' problem, its history is
sket
hed at the end of the paper). This is a
hieved in Theorem 2.3. We point
out that Conje
ture 1.5 would be su�
ient to
ompletely settle the real linear
polarization
onstant problem.
Partially supported by OTKA grants T 046365, K 61116 and NK 72523.
http://arxiv.org/abs/0704.0028v2
0. Old inequalities on determinants and perma-
nents
Re
all that the determinant and the permanent of an n× n matrix A = (ai,j)
are de�ned by
detA =
(−1)π
ai,π(i), per A =
ai,π(i),
where Sn is the symmetri
group on n elements. Throughout this se
tion, we
assume that A is a positive semi-de�nite Hermitian n × n matrix (we write
A ≥ 0). For su
h A, Hadamard proved that
detA ≤
ai,i,
with equality if and only if A has a zero row or is a diagonal matrix. Fis
her
generalized this to
detA ≤ detA′ · detA′′
B∗ A′′
≥ 0, (1)
with equality if and only if detA′ · detA′′ · B = 0.
Con
erning the permanent of a positive semi-de�nite matrix, Mar
us [Mar1,
Mar2℄ proved that
per A ≥
ai,i, (2)
with equality if and only if A has a zero row or is a diagonal matrix. Lieb [L℄
generalized this to
per A ≥ per A′ · per A′′ (3)
for A as in (1), with equality if and only if A has a zero row or B = 0. Moreover,
he proved that in the polynomial P (λ) of degree n′ (=size of A′) de�ned by
P (λ) = per
λA′ B
B∗ A′′
all
oe�
ients ct are real and non-negative. This is indeed a stronger theorem
sin
e it implies
per A = P (1) =
ct ≥ cn′ = per A′ · per A′′.
�okovi¢ [D, Mi℄ gave a simple proof of Lieb's inequalities, and showed also that if
A′ and A′′ are positive de�nite then cn′−t = 0 if and only if all subpermanents of
B of order t vanish. Lieb [L℄ also states an analogous (and analogously provable)
theorem for determinants: for A as in (1), let
D(λ) = det
λA′ B
B∗ A′′
If detA′ · detA′′ = 0, then D(λ) = 0. If A′ and A′′ are positive de�nite, then
(−1)tdn′−t is positive for t ≤ rk B and is zero for t > rk B.
Remark. In all of Lieb's inequalities mentioned above, the
ondition that
the matrix A is positive semi-de�nite
an be repla
ed by the weaker
ondition
that the diagonal blo
ks A′ and A′′ are positive semi-de�nite. The proof goes
through virtually un
hanged. Alternatively, this stronger form of the inequali-
ties
an be easily dedu
ed from the seemingly weaker form above.
1 New inequalities on pfa�ans and hafnians
For an n × n matrix A = (ai,j) and subsets S, T of N := {1, . . . , n}, we write
AS,T := (ai,j)i∈S,j∈T . If |T | = 2t is even, we write
(−1)T := (−1)t+
1.1 Pfa�ans
As far as the appli
ations in Se
tion 2 are
on
erned, this subse
tion may be
skipped.
Re
all that the pfa�an of a 2n × 2n antisymmetri
matrix C = (ci,j) is
de�ned by
pf C =
π∈S2n
(−1)πcπ(1),π(2) · · · cπ(2n−1),π(2n).
We have (pf C)
= detC.
For antisymmetri
A and symmetri
B, both of size n× n, we
onsider the
polynomial
(−1)⌊n/2⌋pf
−λA B
⌊n/2⌋
Theorem 1.1 Let A and B be real n×n matri
es with A antisymmetri
and B
symmetri
. If B is positive semi-de�nite, then pt ≥ 0 for all t. If B is positive
de�nite, then pt > 0 for t ≤ (rk A)/2 and pt = 0 for t > (rk A)/2.
Proof. If B = (bi,j) is positive semi-de�nite, then there exist ve
tors x1, . . . , xn
in a real Eu
lidean spa
e V su
h that (xi, xj) = bi,j. Re
all that in the exterior
tensor algebra
V a positive de�nite inner produ
t (and the
orresponding
Eu
lidean norm) is de�ned by
:= det((vi, wj)).
We have
|S|=2t
|T |=2t
(−1)S(−1)Tpf AS,S · pf AT,T · detBN\S,N\T =
|S|=2t
|T |=2t
(−1)Spf AS,S ·
xi, (−1)Tpf AT,T ·
j 6∈T
|S|=2t
(−1)Spf AS,S ·
Assume that B is positive de�nite. Then the ve
tors xi are linearly independent.
It follows that the tensors
i6∈S xi are also linearly independent as S runs over
the subsets of N . Thus pt = 0 if and only if pf AS,S = 0 for all |S| = 2t, i.e., if
and only if 2t > rk A. �
Theorem 1.2 Let A and B be real n × n matri
es with A antisymmetri
and
B symmetri
. Let λ ≥ 0. If B is positive semi-de�nite, then
(−1)⌊n/2⌋pf
−λA B
≥ detB.
If B is positive de�nite, then equality o
urs if and only if λA = 0.
Proof. The left hand side is
p0 + p1λ+ · · ·+ p⌊n/2⌋λ⌊n/2⌋.
The right hand side is p0. �
I am grateful to the anonymous referee of this paper for the idea of the
following alternative proof of Theorems 1.1 and 1.2. We may assume B > 0,
sin
e every positive semi-de�nite matrix is a limit of positive de�nite ones.
The matrix B−1/2AB−1/2 being real and antisymmetri
, there exists a unitary
matrix U su
h that D := U−1B−1/2AB−1/2U is diagonal with purely imaginary
eigenvalues a1
−1, . . . , an
−1. The real multiset {a1, . . . , an} is invariant
under a ↔ −a. We have
= det
−λA B
= det
BUDU−1
BUDU−1
= det
−λD 1
0 U−1
= det
−1 ai
= detB2 ·
(1 + a2iλ).
Extra
ting square roots, and
hoosing the sign in a
ordan
e with p0 = +detB,
we get
t = (−1)⌊n/2⌋pf
−λA B
= detB ·
(1 + a2iλ),
when
e both theorems immediately follow, sin
e detB > 0.
1.2 Hafnians
Re
all that the hafnian of a 2n× 2n symmetri
matrix C = (ci,j) is de�ned by
haf C =
π∈S2n
cπ(1),π(2) · · · cπ(2n−1),π(2n).
For symmetri
A and B, both of size n× n, we
onsider the polynomial
⌊n/2⌋
Theorem 1.3 Let A and B be symmetri
real n× n matri
es. If B is positive
semi-de�nite, then ht ≥ 0 for all t. If B is positive de�nite, then ht = 0 if and
only if all 2t× 2t subhafnians of A vanish.
Proof. If B = (bi,j) is positive semi-de�nite, then there exist ve
tors x1, . . . , xn
in a real Eu
lidean spa
e V su
h that (xi, xj) = bi,j . Re
all [Mar1, Mar2, MN,
Mi℄ that in the symmetri
tensor algebra SV a positive de�nite inner produ
t
(and the
orresponding Eu
lidean norm) is de�ned by
:= per ((vi, wj)).
We have
|S|=2t
|T |=2t
haf AS,S · haf AT,T · per BN\S,N\T =
|S|=2t
haf AS,S ·
Assume that B is positive de�nite. Then the ve
tors xi are linearly independent.
It follows that the tensors
i6∈S xi are also linearly independent as S runs over
the subsets of N . Thus ht = 0 if and only if haf AS,S = 0 for all |S| = 2t. �
Theorem 1.4 Let A and B be symmetri
real n× n matri
es. Let λ ≥ 0. If B
is positive semi-de�nite, then
≥ per B.
If B is positive de�nite, then equality o
urs if and only if A is a diagonal matrix
or λ = 0.
Proof. The left hand side is
h0 + h1λ+ · · ·+ h⌊n/2⌋λ⌊n/2⌋.
The right hand side is h0. �
Setting A = B and λ = 1, and
ombining with Mar
us's inequality (2), we
arrive at
ase p = 1 of
Conje
ture 1.5 If A = (ai,j) is a positive semi-de�nite symmetri
real n× n
matrix, then the hafnian of the 2pn× 2pn matrix
onsisting of 2p× 2p blo
ks A
is at least (2p− 1)!!n
i,i, with equality if and only if A has a zero row or is
a diagonal matrix.
2 Produ
ts of real linear fun
tionals
In this se
tion, we apply Theorem 1.4 to produ
ts of jointly normal random
variables and then to produ
ts of real linear fun
tionals, whi
h was the main
motivation for this work. The ideas in this se
tion are analogous to those that
Arias-de-Reyna [A℄ used in the
omplex
ase.
Let ξ1, . . . , ξd denote independent random variables with standard Gaussian
distribution, i.e., with joint density fun
tion (2π)−d/2 exp(−|ξ|2/2), where |ξ|2 =
ξ2k.We write Ef(ξ) for the expe
tation of a fun
tion f = f(ξ) = f(ξ1, . . . , ξd).
Re
all that
k = (2p− 1)!! = (2p− 1)(2p− 3) · · · 3 · 1
for k = 1, . . . , d (easy indu
tive proof via integration by parts), and thus
(2pk − 1)!!.
, we write (·, ·) for the standard Eu
lidean inner produ
t. We re
all
the well-known [B2, G, S, Z℄
Wi
k formula. Let x1, . . . , xn be ve
tors in R
with Gram matrix A =
((xi, xj)). Then
(xi, ξ) = haf A. (4)
(For odd n, we de�ne haf A = 0.)
Proof. Both sides are multilinear in the xi, so we may assume that ea
h xi is
an element of the standard orthonormal basis e1, . . . , ed. If there is an ek that
o
urs an odd number of times among the xi, then both sides are zero. If ea
h
ek o
urs 2pk times, then the left hand side is E
k=1 ξ
k , and the right hand
side is
k=1(2pk − 1)!!, whi
h are equal. �
The following theorems are easy
orollaries of Theorem 1.4 together with
the Wi
k formula (4) and Mar
us's theorem (2).
Theorem 2.1 If X1, . . . , Xn are jointly normal random variables with zero
expe
tation, then
X21 · · ·X2n
≥ EX21 · · ·EX2n.
Equality holds if and only if they are independent or at least one of them is
almost surely zero.
Proof. The variables
an be written as Xi = (xi, ξ) with ξ of standard normal
distribution and the xi
onstant ve
tors with a positive semi-de�nite Gram
matrix A = (ai,j) = ((xi, xj)). Then
X2i = E
(xi, ξ)
= haf
≥ per A ≥
ai,i =
E(xi, ξ)
EX2i ,
with equality if and only if A is a diagonal matrix or has a zero row, i.e., the xi
are pairwise orthogonal or at least one of them is zero. �
The generalization of Theorem 2.1 to an arbitrary even exponent 2p is equiv-
alent to Conje
ture 1.5.
Theorem 2.2 For any x1, . . . , xn ∈ Rd, |xi| = 1, the average of
(xi, ξ)
the unit sphere {ξ ∈ Rd : |ξ| = 1} is at least
Γ(d/2)
2nΓ(d/2 + n)
(d− 2)!!
(d+ 2n− 2)!!
d(d+ 2)(d+ 4) . . . (d+ 2n− 2)
with equality if and only if the ve
tors xi are pairwise orthogonal.
Proof. The average on the unit sphere is the
onstant in the theorem times
the expe
tation w.r.t. the standard Gaussian measure (see e.g. [B1℄). By The-
orem 2.1, the latter expe
tation is minimal if and only if the xi are pairwise
orthogonal, in whi
h
ase it is 1. �
Theorem 2.3 For real linear fun
tionals fi on a real Eu
lidean spa
e,
||f1 · · · fn|| ≥
||f1|| · · · ||fn||
n(n+ 2)(n+ 4) · · · (3n− 2)
Here || · || means supremum of the absolute value on the unit sphere. In the
in�nite-dimensional
ase, fun
tionals with in�nite norm may be allowed. Then
the
onvention 0 · ∞ = 0 should be used on the right hand side.
Proof. We may assume that the spa
e is R
with d ≤ n, and the fun
tionals are
given by fi(ξ) = (xi, ξ) with ||fi|| = |xi| = 1. Then ||f1 · · · fn||2 is at least the
average of
f2i (ξ) =
(xi, ξ)
on the unit sphere, whi
h by Theorem 2.2 and
d ≤ n is at least 1/(n(n+ 2)(n+ 4) · · · (3n− 2)). �
It is an unsolved problem, raised by Benítez, Sarantopoulos and Tonge [BST℄
(1998), whether Theorem 2.3 is true with nn under the square root sign in the
denominator on the right hand side. This is
alled the `real linear polarization
onstant' problem. In the
omplex
ase, the a�rmative answer was proved by
Arias-de-Reyna [A℄ in 1998, based on the
omplex analog of the Wi
k formula
[A, B2, G℄ and on Lieb's inequality (3).
Keith Ball [Ball℄ gave another proof
of the a�rmative answer in the
omplex
ase by solving the
omplex plank
problem.
In the real
ase, the a�rmative answer for n ≤ 5 was proved by Pappas and
Révész [PR℄ in 2004. For general n, the best estimate known before the present
paper was that of Révész and Sarantopoulos [RS℄ (2004), based on results of
[MST℄, with (2n)n/4 under the square root sign. See [Mat1, Mat2, MM, R℄ for
a
ounts on this and related questions. Note that
n(n+ 2)(n+ 4) · · · (3n− 2) =
= exp (logn+ log(n+ 2) + log(n+ 4) + · · ·+ log(3n− 2)) <
< exp
log u · du
= exp
[u(log u− 1)]3nn /2
= exp((3n log 3n− 3n− n logn+ n)/2) =
= exp
n(2 logn+ 3 log 3− 2)
and 3
3/e < 3 · 1.8/2.7 = 2, so Theorem 2.3 is an improvement. Note also that
the statement with nn under the square root sign would follow from Conje
-
ture 1.5.
A
knowledgements
I am grateful to Péter Major, Máté Matol
si and Szilárd Révész for helpful
dis
ussions, and to the anonymous referee for useful
omments.
Referen
es
[A℄ J. Arias-de-Reyna, Gaussian variables, polynomials and permanents, Lin.
Alg. Appl. 285 (1998), 107�114.
The referee of the present paper
alled my attention to the fa
t that Arias-de-Reyna used
only the spe
ial
ase of (3) where the matrix A
is of rank 1. This is mu
h simpler than (3) in
general, it
an be proved essentially by the argument Mar
us used in [Mar1, Mar2℄ to prove
the even more spe
ial
ase n
= 1, whi
h still implies inequality (2).
[Ball℄ K. M. Ball, The
omplex plank problem, Bull. London. Math. So
. 33
(2001), 433�442.
[B1℄ A. Barvinok, Estimating L∞ norms by L2k norms for fun
tions on orbits,
Found. Comput. Math. 2 (2002), 393�412.
[B2℄ A. Barvinok, Integration and optimization of multivariate polynomials by
restri
tion onto a random subspa
e, arXiv preprint: math.OC/0502298
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produ
ts of polynomials, Math. Pro
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[D℄ D. �. �okovi¢, Simple proof of a theorem on permanents, Glasgow Math. J.
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[G℄ L. Gurvits, Classi
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omplexity and quantum entanglement, J. Comput.
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onje
tures on permanents, J. Math. Me
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[Mar1℄ M. Mar
us, The permanent analogue of the Hadamard determinant the-
orem, Bull. Amer. Math. So
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[Mar2℄ M. Mar
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[MN℄ M. Mar
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tion as an inner produ
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[Mat1℄ M. Matol
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onstant prob-
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, Permanents, En
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New inequalities on pfaffians and hafnians
Pfaffians
Hafnians
Products of real linear functionals
|
0704.0029 | Understanding the Flavor Symmetry Breaking and Nucleon Flavor-Spin
Structure within Chiral Quark Model | Understanding the Flavor Symmetry Breaking and Nucleon
Flavor-Spin Structure within Chiral Quark Model
Zhan Shu, Xiao-Lin Chen, and Wei-Zhen Deng∗
Department of Physics, Peking University, Beijing 100871, China
Abstract
In χQM, a quark can emit Goldstone bosons. The flavor symmetry breaking in the Goldstone
boson emission process is used to intepret the nucleon flavor-spin structure. In this paper, we
study the inner structure of constituent quarks implied in χQM caused by the Goldstone boson
emission process in nucleon. From a simplified model Hamiltonian derived from χQM, the intrinsic
wave functions of constituent quarks are determined. Then the obtained transition probabilities
of the emission of Goldstone boson from a quark can give a reasonable interpretation to the flavor
symmetry breaking in nucleon flavor-spin structure.
PACS numbers: 12.39.-x, 12.39.Fe, 14.20.Dh
∗Electronic address: [email protected]
http://arxiv.org/abs/0704.0029v2
mailto:[email protected]
I. INTRODUCTION
The measurements of the polarized structure functions of the nucleon in deep inelastic
scattering(DIS) experiments[1, 2, 3, 4] show the complication in proton spin structure. Only
a portion of the proton spin is carried by valence quarks. Moreover, several experiments[5, 6,
7] clearly indicate the ū-d̄ asymmetry as well as the existence of the strange quark content s̄ in
the proton sea. Also the distribution of strange quark in the proton sea is polarized negative.
The DIS results deviate significantly from the näıve quark model (NQM) expectation.
NQM gives many fairly good descriptions of hadron properties. Why does NQM work?
It is a puzzle that the quarks inside a hadron could be treated as non-relativistic particles
in NQM. The chiral quark model (χQM) tries to bridge between QCD and NQM. It was
originated by Weinberg[8] and formulated by Manohar and Georgi[9]. Between the QCD
confinement scale (ΛQCD ≃200MeV) and a chiral symmetry breaking scale (ΛχSB ≃1GeV),
the strong interaction is described by an effective Lagrangian of quarks q, gluons g and
Numbu-Goldstone bosons Π. An important feature of the χQM is that, betweetn ΛQCD and
ΛχSB, the internal gluon effects in a hadron can be small compared to the internal Goldstone
bosons Π and quarks q, so the effective degrees of freedom in this region can be q and Π.
It is interesting that χQM can also be used to explain why NQM does not work in the
above DIS experiments. By the emission of Goldstone boson, χQM allows the fluctuation
of a quark q into a recoiling quark plus a Goldstone boson q → q′Π . The q′Π system then
further splits to generate quark sea through
• the helicity-flipping process
q↑ −→ Π+ q′↓ −→ (qq̄′) + q′↓ (1)
• and the helicity-non-flipping process
q↑ −→ Π+ q′↑ −→ (qq̄′) + q′↑ (2)
where the subscript indicates the helicity of quark. In both the process, q′Π is in a relative P-
wave state. In the helicity-flipping process (1), the orbital angular momentum along helicity
direction must be 〈lz〉 = +1. In the helicity-non-flipping process (2), 〈lz〉 = 0. The process
cause a modification of the spin content of the nucleon because a quark changes its helicity
in (1). Also it causes a modification of the flavor content because the generated quark sea
from Π is flavor dependent[10, 11].
χQM was first used to explain the nucleon sea flavor asymmetry and the smallness of
the quark spin fraction by Eichten, Hinchliffe and Quigg[10]. The flavor asymmetry of sea
quark distribution arises from the mass differences in different quark flavors and in different
Goldstone bosons. Only the lightest Goldstone Boson π was considered since its contribution
dominates. From a perturbation calculation, the probability for an up quark to emit a π+
was estimated to be a = 0.083. This would induce a flavor asymmetry in parton distributions
of nucleon and other hadrons.
However, the estimated transition probability is not enough to full account the flavor
asymmetry in DIS experiments. Contribution from other Π’s and even η′ was considered by
Cheng and Li[11]. Explicit SUf (3) breaking in the transition probabilities was later intro-
duced in refs. 12, 13 and further used by several authors[14, 15, 16, 17, 18, 19]. Nevertheless,
in all these calculations, the transition probabilities were put into model by hand. To fit the
experimental data, the probability of an up quark emitting π+ needs to be set to a >∼ 0.1,
which is about 20% larger than the perturbation calculation. Although the probability of π
emission can be enlarged by using a higher momentum cut off Λ > ΛχQM in the perturbation
calculation [20], however, the chiral quark model is no longer valid at arbitrary high energies
Λ ≫ ΛχQM.
We should not be surprised by this discrepancy since the χQM works in a region right
above the QCD confinement scale ΛQCD. There one may expect the confinement effect is
important and the perturbative calculation of QCD may contain large error. However, there
is another essential difference between the above model calculations and the perturbation
calcultion. In the perturbation calculation, the emitted Goldstone bosons are virtual par-
ticles. In the above model calculations which are closely related to NQM, however, the
Goldstone bosons are close to mass shell under the non-relativistic approximation.
Since χQM can be a bridge between NQM and QCD, it is interesting to explore χQM
from NQM side where we use the wave function method. This will give the above model cal-
culations a concrete foundation in NQM and help us further understand the flavor symmetry
breaking mechanism.
In this paper, we will use wave function method to investigate the flavor symmetry break-
ing in χQM. In a conventional quark model[21], a hadron consists of confined constituent
quarks and its wave function is constructed in the configuration space of the constituent
quarks. To incorporate the transition process of emitting Goldstone boson of χQM into the
quark model, the constituent quarks will have intrinsic wave functions within the configu-
ration q + q′Π.
In Sec. II, we first present the composite wave function of constituent quarks including
components of q′Π. The wave functions and the transition probabilities of q → q′Π are
determined from a simplified χQM Hamiltionian. In Sec. III and Sec. IV, the obtained
transition probabilities are used to calculate nucleon flavor-spin structure and baryon octet
magnetic moments respectively. The numerical results and a brief summary are presented
in Sec. V.
II. THE WAVE FUNCTION OF A CONSTITUENT QUARK
In χQM, the effective Lagrangian below the chiral symmetry breaking scale ΛχQM involves
quarks, gluons, and Goldstone bosons. The first few terms in this Lagrangian are[9]:
LχQM = ψ̄(iDµ + Vµ)γµψ + igAψ̄Aµγµγ5ψ
−mψ̄ψ +
f 2πtr∂
µΣ†∂µΣ+ ... (3)
where Dµ = ∂µ + igGµ is the gauge-covariant derivative of QCD, Gµ the gluon field and
g the strong coupling constant. The dimensionless axial-vector coupling gA = 0.7524 is
determined from the axial charge of the nucleon. m represents the constituent quark masses
due to chiral symmetry breaking. The pseudoscalar decay constant is fπ ≈ 93MeV. The Σ
field, vector currents Vµ and axial-vector currents Aµ are given in terms of the Goldstone
boson fields Φ
π0 + 1√
η π+ K+
π− − 1√
π0 + 1√
K− K̄0 − 2√
, (4)
Σ = exp(i
), (5)
(ξ†∂µξ ± ξ∂µξ†), (6)
ξ = exp(i
). (7)
An expansion of the currents in powers of Φ/fπ yields the effective interaction between Π
and q[10]
LI = −
ψ̄∂µΦγ
µγ5ψ. (8)
This allows the fluctuation of a quark into a recoil quark plus a Goldstone boson q → q′Π.
In quark model, a hadron is built with constituent quarks. In accordance with χQM,
we should treat a constituent quark as a composite particle including such components q′Π.
Here we denote the wave function of a composite constituent quark as |q〉〉. At rest,
|q〉〉 = zq|q〉+
q′Π|q
′Π〉. (9)
In our paper, the state normalization relation is always taken as
〈p|p′〉 = δ3(p− p′). (10)
The above wave function is of essential importance in our work. The square of the mod-
ulus of the coefficient of each q′Π configuration is just the probability for the corresponding
Π emission process
Pq→q′Π = |xqq′Π|
2, (11)
|zq|2 = (1−
Pq→q′Π)
is the probability of no Π emission.
To determine the wave function (9), we first construct a simplified Hamiltonian in the
degrees of freedom q and Π,
H = H0 +HB +HI . (12)
H0 represents the kinetic energies of q and Π. It reads
ψ̄(iα · ∇+m)ψ +
Tr[Φ̇2 + (∇Φ)2] +
m2Π(Φ
, (13)
where mΠ is the physical mass of Π which is nonzero and nondegenerate.
HI = −
d3xLI , (14)
is the χQM interaction. HB is an accessary interaction which is needed to bind the q
together. In our simplified Hamiltonian, we will not disscuss the explicit formalism of HB.
Instead, we will put some physical restriction conditions on it later in this section, which is
sufficient to our calculation.
FromH0, we can expand free fields ψ and Π in terms of creation and annihilation operators
ψq(x) =
(2π)3/2
q(p, s)e−ip·x + bq†
ps(t)v
q(p, s)eip·x
, (15)
ΦΠ(x) =
(2π)3/2
e−ip·x + cΠ†
eip·x
p0=EΠ
, (16)
where
p2 +m2q
is the quark energy of flavor q,
p2 +m2Π
is the energy of Goldstone boson Π. aq†
ps and b
pr are the creation operators of quark q and
anti-quark q̄
pr, a
p′s} = {b
pr, b
p′s} = δ
(3)(p− p′)δrs. (17)
is the creation operator of Π
] = δ(3)(p− p′). (18)
Next, we will replace the field ψ and Φ in the Hamiltonian (12) with the free field of (15)
and (16). Then we can express the Hamiltonian in creation and annihilation operators, for
example
d3p Eq
ps + b
ps] +
d3p EΠ
. (19)
In all the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the emitted Π is assumed
bound to the quark source. To represent that q′Π are bound, we use the well known SHO
function as their spatial wave function
|qΠ〉 =
d3p|p|e−
2λ2 [Y1(θ, φ) c
]1/2 |0〉, (20)
|qΠ ↑〉 = 1√
d3p|p|e−
2λ2 Y11(θ, φ) c
p↓ |0〉
d3p|p|e−
2λ2 Y10(θ, φ) c
p↑ |0〉, (21)
where λ is the “characteristic radius” parameter in Gaussian function. 1/
N is the nor-
malization factor,
dp p4 e
πλ5. (22)
However, we need a binding interaction HB in the Hamiltonian. Yet we do not know how
to write out the explicit form of HB. However, HB should provide enough binding energy.
That is, for the q′Π system, we must have
〈qΠ|H0 +HB|qΠ〉 ≤ mq +mΠ. (23)
That is
EB = 〈qΠ|HB|qΠ〉 ≤ mq +mΠ − 〈qΠ|H0|qΠ〉 = mq − Eq +mΠ − EΠ. (24)
As a rough estimation, we will take the mininum value of EB
EB = −max
{Eq −mq + EΠ −mΠ} = −(Eu −mu + Eπ −mπ). (25)
Then the wave function of a composite constituent quark is determined from Schrödinger
equation
H|q〉〉 =Mq|q〉〉. (26)
After taking the above simplification, we need only solve a matrix eigen-value problem
=Mq
, (27)
where
aδ3(0) = 〈q|H|q〉,
Bq′Πδ
3(0) = 〈q|H|q′Π〉,
Cq′Π;q′′Π′δ
3(0) = 〈q′Π|H|q′′Π′〉,
q′Π = x
For example, let us consider the process u emitting Π. There are four possible |q′Π〉 states
generated by the fluctuations of a u quark: |uπ0〉, |uη〉, |dπ+〉 and |sK+〉. Thus
|u〉〉 = zu|u〉+ xuuπ0 |uπ0〉+ xuuη|uη〉+ xudπ+ |dπ+〉+ xusK+|sK+〉. (28)
Taking these wave functions as basis, we can calculate the matrix of the Hamiltonian in
(27).
a = mu. (29)
C is diagonalized. Its diagonal matrix elements are calculated from H0
Cuπ0;uπ0 =
dp p4 e
p2 +m2u +
) + EB, (30)
Cuη;uη =
dp p4 e
p2 +m2u +
p+m2η) + EB, (31)
Cdπ+;dπ+ =
dp p4 e
p2 +m2d +
) + EB, (32)
CsK+;sK+ =
dp p4 e
p2 +m2s +
) + EB. (33)
B is calculated from HI
Buπ0 = −
dp p4e
, (34)
Buη = −
dp p4e
, (35)
Bdπ+ = −
dp p4e
, (36)
BsK+ = −
dp p4e
. (37)
By diagonalizing this Hamiltonian matrix, we will obtain a new mass of the constituent u
quark Mu and its composite wave function. The constituent masses and wave functions of
d and s quarks can be obtained similarly. We have
|d〉〉 = zd|d〉+ xddπ0 |dπ0〉+ xddη|dη〉+ xduπ−|uπ−〉+ xdsK0|sK0〉, (38)
|s〉〉 = zs|s〉+ xssη|sη〉+ xsdK̄0 |dK̄
0〉+ xsuK−|uK−〉. (39)
From isospin symmetry, mu = md, we have
zd = zu; xddπ0 = −xuuπ0 ; xduπ− = xudπ+ ; ... (40)
However, since mu 6= ms, one should notice that
zs 6= zu; xsdK̄0 6= x
sK0; x
uK− 6= xusK+. (41)
After the diagonalization, the Goldstone bosons Π are separated from quarks q approx-
imately. With only degrees of freedom q one can rebuild the quark model and so Mu, Md,
Ms should be regarded as the constituent quark masses in quark model.
III. FLAVOR AND SPIN STRUCTURE OF PROTON
Having known the wave functions of constituent quark q and the transition amplitudes
of q emitting each Goldstone bosons Π, we are able to calculate the quark distribution in
a constituent quark following refs. 11, 12, 13. In χQM, Π will further split into a quark-
antiquark pair. By substituting the quark contents of Π into wave functions (28), (38) and
(39), we can rewrite the wave functions of constituent quark q as
|u〉〉 = zu|u〉+
xuuη√
|u(uū)〉+
xuuη√
|u(dd̄)〉
2xuuη√
|u(ss̄)〉+ xudπ+ |d(ud̄)〉+ xusK+|s(us̄)〉, (42)
|d〉〉 = zu|d〉+
xuuη√
|d(uū)〉+
xuuη√
|d(dd̄)〉
2xuuη√
|d(ss̄)〉+ xudπ+ |u(dū)〉+ xusK+|s(ds̄)〉, (43)
|s〉〉 = zs|s〉+
xssη√
|s(uū)〉+
xssη√
|s(dd̄)〉 −
2xssη√
|s(ss̄)〉
|d(sd̄)〉+ xsuK−|u(sū)〉. (44)
Then the antiquark and quark flavor contents of the proton (uud) are
ū = 2
xuuη√
xuuη√
+ |xudπ+ |2, u = ū+ 2, (45)
xuuη√
xuuη√
+ 2|xudπ+ |2, d = d̄+ 1, (46)
s̄ = 2|xuuη|2 + 3|xusK+|2, s = s̄. (47)
Some important quantities depending on the above quark distribution are: the Gottfried
sum rule IG =
(ū − d̄) whose deviation indicates the ū-d̄ asymmetry in proton sea;
ū/d̄ measured through the ratio of muon pair production cross sections; and the fractions
of quark flavors in proton fq =
Σ(q+q̄)
, f3 = fu − fd and f8 = fu + fd − 2fs.
We can further calculate the spin structure of proton. Here one should consider the effects
of configuration mixing generated by spin-spin forces[22]. We take the baryon wave functions
from the quark model calculation[23, 24, 25]. The proton wave function for example, is
expressed as
= 0.90|P 28SS〉 − 0.34|P 28S ′S〉 − 0.27|P 28SM〉 (48)
where the baryon SU(6)⊗O(3) wave functions are denoted as |B2S+1N Lσ〉, N is SU(3) mul-
tiplicity. S, L are the total spin and total orbital angular momentum while σ = S,M,A
denotes the permutation symmetry of SU(6). The spin polarization functions will be re-
markably affected by configuration mixing. Following refs. 15, 17, we define the number
operator by
N̂ = nu↑u↑ + nu↓u↓ + nd↑d↑ + nd↓d↓ + ns↑s↑ + ns↓s↓,
where nq↑, nq↓ are the number of q↑, q↓ quarks. The spin structure of the “mixed” proton is
given by
= (0.902 + 0.342)
+ 0.272
. (49)
The spin structure after considering Π-emission is obtained by replacing for every quark in
eq. (49) by
q↑,↓ −→ (1− ΣPi)q↑,↓ + Pflipping(q↑,↓) + Pnon−flipping(q↑,↓), (50)
where Pflipping(q↑,↓) and Pnon−flipping(q↑,↓)| are the probabilities of quark helicity flipping and
non-flipping for q↑,↓ respectively. For example, in the case of u↑ quark we have,
Pflipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↓ + |xudπ+ |2d↓ + |xusK+|2s↓
Pnon−flipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↑ + |xudπ+ |2d↑ + |xusK+|2s↑
Finally the spin polarization functions defined as ∆q = q↑ − q↓ are
∆u = (0.902 + 0.342)
114|xu
|2 + 48|xuuη|2 + 36|xusK+|2
+ 0.272
66|xu
|2 + 24|xuuη|2 + 18|xusK+|2
, (51)
∆d = (0.902 + 0.342)
|2 + 12|xuuη|2 + 9|xusK+|2
+ 0.272
42|xu
|2 + 12|xuuη|2 + 9|xusK+|2
, (52)
∆s = −
. (53)
There are several measured quantities which can be expressed in terms of the above spin
polarization functions. The quantities usually calculated are ∆3 = ∆u−∆d and ∆8 = ∆u+
∆d−2∆s, obtained from the neutron β-decay and the weak decays of hyperons respectively.
Another important quantity is the flavor singlet component of the total quark spin content
defined as 2∆Σ = ∆u + ∆d + ∆s . We also calculate some weak axial-vector form factors
which are also related to the spin polarization functions, (GA/GV )Λ→p =
(2∆u−∆d−∆s),
(GA/GV )Σ−→n = ∆d−∆s, and (GA/GV )Ξ−→Λ = 13(∆u+∆d− 2∆s).
IV. BARYON OCTET MAGNETIC MOMENTS
Considering the relative angular momentum between quark and Goldstone boson Π, the
magnetic moment operator of a qΠ system is
µ̂qΠ =
p2q +m
p2Π +m
p2q +m
p2Π +m
p2Π +m
p2q +m
p2q +m
p2Π +m
l̂ (54)
where eq and eΠ are the electric charges carried by q and Π respectively, ŝ the quark spin
operator and l̂ the relative angular momentum bewteen q and Π. The first term in Eq(54)
is the intrinsic magnetic moment of quark and the other two terms are the contribution of
the orbital angular momentum. Here we have to consider the relativistic effect since the
relative momentum of q or Π are comparable to their masses in the qΠ system
pq,Π ∼ Λ ∼ mq,Π.
With the SHO wave functions of (20), the magnetic moment of qΠ system (54) can be
readily calculated. Then we can recalculate the magnetic moments of constituent quarks
taking into account of the relativistic effect. For example, the magnetic moments of the u
quark is
µu = |zu|2〈u↑|µ̂|u↑〉+ Pu→uπ0〈uπ0|µ̂|uπ0〉+ Pu→uη〈uη|µ̂|uη〉
+ Pu→dπ+〈dπ+|µ̂|dπ+〉+ Pu→sK+〈sK+|µ̂|sK+〉, (55)
where
〈u↑|µ̂|u↑〉 =
, (56)
and the contribution from qΠ systems are
〈uπ0|µ̂|uπ0〉 = − eu
p2 +m2π
p2 +m2u +
p2 +m2π
p2 +m2u
λ2 , (57)
〈uη|µ̂|uη〉 = − eu
p2 +m2η
p2 +m2u +
p2 +m2η
p2 +m2u
λ2 , (58)
〈dπ+|µ̂|dπ+〉 = −
p2 +m2π
p2 +m2d +
p2 +m2π
p2 +m2d
p2 +m2d
p2 +m2d +
p2 +m2π
p2 +m2π
λ2 , (59)
〈sK+|µ̂|sK+〉 = −
p2 +m2K
p2 +m2s +
p2 +m2K
p2 +m2s
p2 +m2s
p2 +m2s +
p2 +m2K
p2 +m2K
λ2 . (60)
The magnetic moments of d and s quarks can be calculated similarly.
One can easily obtain the octet baryon magnetic moments by replacing the valence quarks
inside the baryons with the corresponding constituent quarks. Again we take proton as an
example,
µp = (0.90
2 + 0.342)
+ 0.272
. (61)
If we replace the µq by (55), µp can be further expressed as the baryon magnetic moment in
conventional quark model plus the contribution from the Goldstone boson emission process
[26]. The magnetic moments for other octet baryons can be calculated similarly.
V. NUMERICAL RESULTS AND CONCLUSIONS
In the numerical calculation, most of the parameters can be taken from the experimental
data or the chiral quark model. We collect these fixed input parameters of our calculation
in Table I. Here we have used the the physical masses of Goldstone bosons[27].
TABLE I: The fixed input parameters from chiral quark model and experimental data.
gA fπ(MeV) mπ(MeV) mK(MeV) mη(MeV)
0.7524 93 135 494 548
For the quark masses, since our work focuses on the inner context of the constituent quarks
in quark model, naturally we will refer to the quark masses from quark model, instead of
the chiral quark model values. Here we will use the quark mass values from the widely
accepted Isgur’s quark model[21] as shown in Table II. However, one should be cautious
that, in our model, it is the quark with the Goldstone boson mixing which corresponds to
the constituent quark in quark model. That is, mass values Mq after the diagonalization
process should be set to the quark masses in Isgur’s model. Our strategy is to adjust the
quark masses mq in the model Hamiltonian to fit the Mq values.
Finally we are left only with one free parameter λ which describes the confinement of the
emitted Goldstone boson in our model. An overall fit to the experimental data of nucleon
flavor-spin structure and octet baryon magnetic moments shows that the best value should
be λ=152MeV. With this value of λ and a minimun binding energe EB = −218MeV, the
“bare” values of quark masses mq without Goldstone boson mixing are shown also in Table
Transition probabilities of the light and strange quarks to various q′Π systems are given
in Table III and IV respectively. The probability of a u quark emitting a π+ P (u →
TABLE II: The quark masses with vs. without Goldstone boson mixing.
λ EB(MeV) mu,d(MeV) ms(MeV) Mu,d(MeV) Ms(MeV)
152 −218 288 474 220 419
d + π+)=0.145 is significantly larger than the perturbation calculation a=0.083. One may
notice that the λ parameter value 152MeV in our wave function, which is below ΛQCD, is
rather small than another energy scale ΛχQM in chiral quark model. Surely this will weaken
the interaction between q and q′Π. However, one should also notice that the binding energy
EB = −218MeV will make the energy of a qΠ system much close to the single quark energy.
This will enhance the mixing of q′Π components in a constituent quark.
Also, we notice that the asymmetry between the probabilities of u(d) → s + K and
s → u(d) + K̄. Whether this asymmetry leads to any observable consequence in hadron
structure needs further investigation.
TABLE III: Transition probabilities of a u quark to various q′Π systems and the mass of constituent
u quark.
u → u+ π0 u → u+ η u → d+ π+ u → s+K+ no GB-emission Mu
0.072 0.003 0.145 0.010 0.770 220MeV
TABLE IV: Transition probabilities of a s quark to various q′Π systems and the mass of constituent
s quark.
s → s+ η s → u+K− s → d+ K̄0 no GB-emission Ms
0.012 0.071 0.071 0.846 419MeV
Next, we will compare our calculate results with the experimental data. Since our em-
phasis is on the substructure of a constituent quark in NQM, here we also quote the results
from NQM. In Table V, the calculated flavor and spin structures of the proton are shown.
It should be mentioned that the quark spin polarization functions can be further corrected
by the gluon anomaly[13, 15, 17, 28, 29, 30] through
∆q(Q2) = ∆q − αs(Q
∆g(Q2), (62)
and the flavor singlet component of the total helicity is modified accordingly as
∆Σ(Q2) = ∆Σ− 3αs(Q
∆g(Q2), (63)
where ∆q(Q2) and ∆Σ(Q2) are the experimentally measured quantities, ∆q and ∆Σ corre-
spond to the calculated quantities without gluon correction. Using the experimental data
Σ(Q2 = 5GeV2) = 0.19 ± 0.02[2], αs(Q2 = 5GeV2) = 0.285 ± 0.013[27], and our result
∆Σ=0.346, the gluon polarization ∆g(Q2) is estimated to be 2.293. Both the results with
and without gluon polarization corrections are presented in Table V. The inclusion of gluon
polarization leads to a better agreement with experimental data for the spin structure.
The calculated magnetic moments of octet baryons are given in Table VI. Although the
deviation is somewhat around 30% in the case of Ξ−, our overall fit to octet baryon magnetic
moments is in good agreement with experiments. Also it should be mentioned that even in
the case of Ξ− the fit can perhaps be improved if corrections due to pion loops are taken
into account[32, 33].
In the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the Goldstone boson sector in
χQM is usually extended to include the η′ meson with U(3) symmetry. According to Cheng
and Li[11], in the large Nc limit of QCD, there are nine Goldstone bosons including the usual
octet and the singlet η′. Thus an constituent quark can also transit to a quark-η′ system. We
have also made an U(3) calculation. With the inclusion of η′, we find that the probabilities
for η′-emission from light and strange quarks P (u → u + η′)=P (d → d + η′)=0.0021 and
P (s → s + η′)=0.0018 which are negligibly small as compared to those of octet Goldstone
boson emissions. We therefore conclude that the contribution of η′ is not important, due to
the obvious axial U(1) symmetry breaking in meson mass spectra mη′ > mK,η.
To summarize, the χQM builds a bridge between the QCD and low-energy quark model.
This allows us to understand the mechanism of flavor symmetry breaking and nucleon flavor-
spin structure in NQM through the consideration of the sea quark and Goldstone bosons
in the substructure of constituent quarks. Using the simple SHO wave function, we have
modeled the wave functions of the composite constituent quarks and thus estimated the
transition probabilities for Goldstone boson emissions. These transition probabilities indeed
reflect the flavor SU(3) symmetry breaking in χQM from the differences in quark masses
ms > mu,d and differences in Goldstone bosons masses mK,η > mπ and they are roughly
in agreement with the parametrizations of other model calculations [11, 12, 13, 14, 15, 16,
TABLE V: The calculated values for the quark flavor distribution functions and spin polarization
functions in proton, as compared with experimental data and NQM results.
Data NQM Our Model
With ∆g Without ∆g
∆u 0.85 ± 0.05[2] 1.33 0.864 0.968
∆d −0.41± 0.05[2] −0.33 −0.377 −0.274
∆s −0.07± 0.05[2] 0 −0.107 −0.003
∆3 = (GA/GV )n→p 1.270 ± 0.003[27] 1.67 1.242 1.242
(GA/GV )Λ→p 0.718 ± 0.015[27] 1 0.737 0.737
(GA/GV )Σ→n −0.340 ± 0.017[27] −0.33 −0.270 −0.270
(GA/GV )Ξ→Λ 0.25 ± 0.05[27] 0.33 0.234 0.234
∆8 0.58 ± 0.025[2] 1 0.701 0.701
∆Σ 0.19 ± 0.02[2] 0.5 0.190 0.346
ū − 0.264
d̄ − 0.392
s̄ − 0.036
ū− d̄ −0.118 ± 0.015[6] 0 −0.128
ū/d̄ 0.67 ± 0.06[6] 1 0.674
IG 0.254 ± 0.005[6] 0.33 0.248
fu − 0.577
fd − 0.407
fs 0.10 ± 0.06[31] 0 0.017
f3 − 0.170
f8 − 0.950
f3/f8 0.21 ± 0.05[14] 0.33 0.179
17, 18, 19]. The fit to both the flavor-spin structure of nucleon and octet baryon magnetic
moments are in good agreement with experiments.
TABLE VI: The caculated octet baryon magnetic moments in nuclear magneton, as compared with
experiments and the results of NQM.
Octet baryons Data[27] NQM[34] Our model
p 2.79± 0.00 2.72 2.73
n −1.91 ± 0.00 -1.81 −1.91
Σ− −1.16 ± 0.025 -1.01 −1.23
Σ+ 2.46± 0.01 2.61 2.67
Ξ0 −1.25± 0.0014 −1.41 −1.36
Ξ− −0.65 ± 0.002 −0.50 −0.44
Λ −0.61 ± 0.004 −0.59 −0.56
ΣΛ 1.61± 0.08 1.51 1.63
Acknowledgments
Zhan Shu would like to thank Fan-Yong Zou and Yan-Rui Liu for useful discussions. This
work was supported by the National Natural Science Foundation of China under Grants
10675008.
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INTRODUCTION
The Wave Function of a Constituent quark
FLAVOR AND SPIN STRUCTURE OF PROTON
BARYON OCTET MAGNETIC MOMENTS
NUMERICAL RESULTS AND CONCLUSIONS
Acknowledgments
References
|
0704.0030 | Tuning correlation effects with electron-phonon interactions | Tuning correlation effects with electron-phonon
interactions
J.P.Hague and N.d’Ambrumenil
Department of Physics, University of Warwick, CV4 7AL, U.K.
We investigate the effect of tuning the phonon energy on the correlation ef-
fects in models of electron-phonon interactions using DMFT. In the regime
where itinerant electrons, instantaneous electron-phonon driven correlations
and static distortions compete on similar energy scales, we find several in-
teresting results including (1) A crossover from band to Mott behavior in the
spectral function, leading to hybrid band/Mott features in the spectral func-
tion for phonon frequencies slightly larger than the band width. (2) Since the
optical conductivity depends sensitively on the form of the spectral function,
we show that such a regime should be observable through the low frequency
form of the optical conductivity. (3) The resistivity has a double kondo peak
arrangement [Published as J. Low. Temp. Phys. 140 pp77-89 (2005)].
PACS numbers: 71.10.Fd, 71.27.+a, 71.38.-k
1. Introduction
Mounting experimental evidence from high-Tc cuprates 1, nickelates 2,
manganites 3,4 and other interesting materials suggests that large electron-
phonon interactions may play a more important role in the physics of strongly
correlated electron systems than previously thought. Migdal-Eliashberg and
BCS theories have proved extremely successful in describing the effects of
phonons in many materials. However, if the coupling between electrons and
the underlying lattice is large, and/or the phonons can not be treated within
an adiabatic approximation, conventional approaches fail.
The Holstein model contains most of the fundamental physics of the
electron-phonon problem 5. Tight-binding electrons are coupled to the lat-
tice through a local interaction with Einstein modes. For large phonon
frequencies, electrons interact with a strongly correlated Hubbard-like at-
traction, while for small phonon frequencies the lattice gives rise to a static
http://arxiv.org/abs/0704.0030v1
potential which is essentially uncorrelated. Between these two extreme lim-
its of correlated and uncorrelated behavior, levels of correlation are tuned
by the size of the phonon frequency and novel physics is expected. In par-
ticular, it is normally the strength of interaction which is said to tune the
correlation in e.g. the Hubbard model, whereas in the Holstein model, it can
be seen that both interaction strength and phonon frequency may compete
with each other to play this role.
The dynamical mean-field theory (DMFT) approach has proved suc-
cessful in treating the Holstein and other models 6,7,8. DMFT treats the
self-energy as a momentum-independent quantity and is accurate as long as
the variation across the Brillouin zone is small. For many aspects of the
electron-phonon problem in 3D, correlations are short ranged and DMFT
can be successfully applied. The weak coupling phase diagram was studied
by Ciuchi et al. where competing charge-order (CO) and superconducting
states were found 7. Freericks et al. developed a quantum Monte-Carlo
(QMC) algorithm 8,9 and examined the applicability of several perturba-
tion theory based techniques to the electron-phonon problem 10,11,12. The
prediction of measurable quantities away from certain well-defined limits is
severely restricted owing to difficulties inherent in the analytic continuation.
Dynamic properties such as spectral functions can be computed in the case
of static phonons 13, and close to the static limit 14. Alternatively, the limit
of high phonon frequency (attractive Hubbard model) has been studied with
a QMC algorithm 15.
In the current study we are concerned with the behavior of dynamical
properties that could be measured directly with experiment. We use the iter-
ated perturbation theory approximation, which has been demonstrated to be
accurate for the Hubbard model, and use maximum entropy to analytically
continue the results. We compare the resulting single-particle spectral func-
tions over a wide range of electron-phonon coupling strengths and phonon
frequencies. The results obtained using iterated perturbation theory (IPT)
are promising and capture generic weak and strong coupling behavior for
all phonon frequencies. At intermediate phonon frequencies, we find that
electron-phonon interactions produce a spectral function which is simulta-
neously characteristic of both uncorrelated band (static) and strongly cor-
related Mott/Hubbard regimes. We also find that the competition between
band-like and correlated states causes unusual structures in the optical con-
ductivity and resistivity. Provided a material with high enough phonon
frequency can be identified, it is possible that such a state could be observed
experimentally.
This paper is organized as follows. First, we introduce the Holstein
model, the dynamical mean-field theory and analytic continuation techniques
(1a) (2a)
(2b) (2c)
Fig. 1. Second order contributions to the self-energy. Straight lines repre-
sent electron Green’s functions of the host and wavy lines phonon Green’s
functions.
(section 2.). In section 3., we use IPT to determine the spectral functions of
the Holstein model. We compare IPT with exactly known results in the static
limit. This, in conjunction with the conclusions of Ref. 11 leads us to argue
that IPT is a reasonable approximation for the calculation of dynamical
properties in the intermediate phonon frequency regime. We compute the
density of states, optical conductivity and resistivity, and give a heuristic
explanation for their behavior.
2. Formalism
The Holstein Hamiltonian is written as,
H = −t
<ij>σ
iσcjσ +
(gxi − µ)niσ +
Mω20x
The first term in this Hamiltonian represents a tight binding model with
hopping parameter t. The second term couples the local ion displacement,
xi to the local electron density. The final term can be identified as the
non-interacting phonon Hamiltonian. c
i (ci) create (annihilate) electrons at
site i, pi is the ion momentum, M the ion mass, µ the chemical potential
and g the electron-phonon coupling. The phonons are dispersionless with
frequency ω0.
The perturbation theory of this model may be written down in terms
of electrons interacting via phonons with the effective interaction,
U(iωs) = −
M(ω2s + ω
Here, ωs = 2πsT are the Matsubara frequencies for bosons and s is an
integer. Taking the limit ω0 → ∞, g → ∞, while keeping the ratio g/ω0
finite, leads to an attractive Hubbard model with a non-retarded on-site
interaction U = −g2/Mω20 . Iterated-perturbation theory (IPT) is known to
be a reasonable approximation to the half-filled Hubbard model 16,17. Taking
the opposite limit (ω0 → 0, M → ∞, keeping Mω20 ≡ κ finite) the phonon
kinetic energy term vanishes, and the phonons depend on a static variable
xi. As such, the model may be considered as uncorrelated.
We solve the Holstein model using dynamical mean-field theory (DMFT).
DMFT freezes spatial fluctuations, leading to a theory which is completely
momentum independent, while fully including dynamical effects of excita-
tions. In spite of this simplification, DMFT predicts non-trivial (correlated)
physics and may be used as an approximation to 3D models 18. As discussed
in Ref. 6, DMFT involves the solution of a set of coupled equations which
are solved self-consistently. The Green’s function for the single site problem,
G(iωn) can be written in terms of the self-energy Σ(iωn) as,
G−1(iωn) = G−10 (iωn)− Σ(iωn), (3)
where Σ is a functional of G0, the Green’s function for the host of a single
impurity model. Here ωn = 2πT (n+ 1/2) are the usual Matsubara frequen-
cies. The assumptions of DMFT are equivalent to taking the self-energy of
the original lattice problem to be local, hence G is also given by,
G(iωn) =
dǫD(ǫk)
iωn + µ− Σ(iωn)− ǫk
where D(ǫ) the density of states (DOS) of the non-interacting problem (in
our case g = 0). We work with a Gaussian DOS which corresponds to a
hypercubic lattice 18, D(ǫ) = exp(−ǫ2/2t2)/t
2π. Equations (3) and (4)
are solved according to the following self-consistent procedure: Compute
the Green’s function from equation (4) and the host Green’s function of the
effective impurity problem, G0, from equation (3); then calculate a new self-
energy from the host or full Green’s functions. In the following we will take
the hopping parameter t = 0.5, which sets the energy scale.
Once the algorithm has converged, and after analytically continuing to
the real axis, response functions can be computed. We use the MAXENT
method for the determination of spectral functions from Matsubara axis
data. MAXENT treats the analytic continuation as an inverse problem 19.
The Green’s function, G(z), is given by the integral transform,
G(z) =
z − x
dx (5)
where ρ(x) is the spectral function (ρ(ω) = Im[G(ω + iη)]/π). The problem
of finding ρ is therefore one of inverting the integral transform. Since the
data for Gn are incomplete and noisy for any finite set of Matsubara fre-
quencies, the inversion of the kernel of the discretised problem is ill-defined.
The MAXENT method selects the distribution ρ(x) which assumes the least
structure consistent with the calculated or measured data. These methods
have been extensively reviewed in the context of the inversion of the kernel
in Refs. 19,20. The applicability to the current problem has been thoroughly
tested, and is found to be accurate.
Within the DMFT formalism, many response functions follow directly
from the one-electron spectral function and the electron self-energy (essen-
tially because of the neglect of all connected higher point functions apart
from G0). Here we will be interested in the conductivity 6:
Re[σ(ω)] =
dǫD(ǫ)
dν ρ(ǫ, ν)ρ(ǫ, ν + ω)[f(ν)− f(ν + ω)] (6)
where f(x) is the Fermi-Dirac distribution. Taking the limit, ω → 0, leads
to the DC conductivity. (The conductivity is in units of e2V/ha2, where a
is the lattice spacing, V the volume of the unit cell and e and h are the
electron charge and Planck’s constant respectively.)
3. Results
In this section, we examine the validity of an approximation to the self-
energy constructed from only first and second order terms with respect to the
spectral functions calculated at very high and very low phonon frequencies.
Finally, we calculate the optical conductivity and resistivity.
Spectral functions are shown in figures 2 and 3. The perturbation theory
is carried out in the host Green’s function (i.e. both electrons and phonons
are bare). All diagrams in fig. 1 are considered,
Σ1a(iωn) = −UT
G0(iωn − iωs)D0(iωs) (7)
Σ2a(iωn) = −2U2T 2
D20(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s) (8)
Σ2b(iωn) = U
D0(iωm−s)D0(iωn−m)G0(iωm)G0(iωs)G0(iωn−m+s)
Σ2c(iωn) = U
D0(iωn−m)D0(iωm−s)G20(iωm)G0(iωs) (10)
This also gives the correct weak coupling limit for the electronic Green’s
function.
We consider first the calculation of spectral functions close to the static
and Hubbard limits. In the instantaneous limit the perturbation theories
for the Holstein and Hubbard models are equivalent. It is well known that
second order perturbation theory in the host Green’s function provides a
good approximation to the Hubbard model 21. In the static limit, the exact
solution can easily be calculated 13. Figure 2 shows spectral functions from
the exact solution, computed for a hypercubic lattice and spectral functions,
computed using 2nd order perturbation theory at a temperature of T = 0.08.
The phonon frequency ω0 = T/20 was chosen so that the effects of the
phonon kinetic energy are negligible compared to thermal fluctuations. This
allows a direct comparison to be made between the exact and approximate
results. The comparison shows that the widths and positions of the major
features are closely related.
The results in Figure 2 for the static limit (ω0 → 0), together with the
fact that second order IPT is known to give reasonable results in the instan-
taneous limit (ω0 → ∞), suggest that the calculation of spectral functions
should also be reliable at intermediate frequencies. We note that Freericks et
al. also find a reasonable agreement between the IPT and QMC self-energies
at half-filling, and that this should lead to a good agreement in the Matsub-
ara axis Green’s function. We have therefore solved the IPT equations for
the spectral functions at intermediate frequencies. We show the results in
Fig 3.
The results of the IPT calculations in the regime of intermediate cou-
pling (Fig 3) are consistent with known results for the limiting cases. For
frequencies ω ≫ ω0, the system has the qualitative behavior of the static
limit: The original unperturbed density of states splits into two sub-bands
centered around ±U/2. For small frequencies (ω ≪ U,ω0) and interaction
strength, U , less than some critical value, the system behaves as an in-
teracting electron (Hubbard) model, since the retarded interaction between
particles U(iωs) (see equation 2) is effectively constant for ωs ≪ ω0. There is
then a narrow quasiparticle band at the Fermi energy with density of states
at the Fermi energy pinned at its non-interacting value 6. We also note that
the results for small coupling and small frequencies are in good agreement
with those calculated using ME theory in the metallic phase 22.
The recent renormalization group (NRG) calculations of Meyer et al.
23,24 also report the spectral function in the intermediate regime. The NRG
is in principle an exact method for solving the impurity problem onto which
the DMFT equations map. Our results are largely consistent with theirs
adding further support to the use of IPT in the intermediate regime. When
comparing with the results of Meyer et al 23, one should note that the Hamil-
tonian (1) is exactly the one considered in Ref. 23 but with the quantity
2Mω0 =
Uω0/2 denoted by g in Ref. 23 and with energies measured
in terms of the full bandwidth (instead of the half-bandwidth used here).
In this paper, we work with the Gaussian density of states for the non-
interacting electron DOS, whereas reference 23 uses the semi-elliptic DOS.
In general, we would expect the critical values for the opening of a gap to
be larger for the Gaussian case than for the semi-elliptic case. The critical
coupling for the parameters in Figure 3(c) lies just above U = 2.0, corre-
sponding in the units used in Ref. 23 to g = 1, compared with the critical
value found for the semi-elliptic DOS of g = 0.69 (note that because of the
different energy scales ω0 = 2 in our results corresponds to ω0 = 1 in the
units of 23). However, the shapes of the spectral functions are similar in
both cases, with a five-peaked structure below and four peaked structure
above the transition. The peaks are narrower in the IPT results than in the
NRG results and there is less weight in the high energy peaks. This may
reflect the different DOS, or inaccuracies in the NRG method at frequencies
far from the Fermi energy resulting from the logarithmic discretisation, but
more likely the limitations of the IPT method.
Using the method outlined in section 2. it is possible to calculate the
real-axis self-energy. The temperature evolution of the imaginary part of the
self-energy may be seen in figure 4 for U = 2 and ω0 = 2 The self-energy
at low temperatures and small frequencies shows a quadratic (Fermi-liquid
like) behavior consistent with the narrow quasiparticle peak seen in the spec-
tral function (Fig 3) and develops to a broad central peak at higher tem-
peratures. There are also peaks corresponding to the Hubbard sub-bands.
With increasing temperature these phonon-induced peaks move together and
merge into the single central maximum associated with incoherent on-site
scattering. This peak is naturally characterized within the framework of
the self-consistent impurity model formulation of the DMFT equations 6 in
terms of a Kondo resonance. In this formulation, the dynamical mean field
G0(ω) is written in terms of a hybridization ∆(ω) between the site orbital and
a bath of conduction electrons and is therefore equivalent to an Anderson
impurity model with the added complication that ∆ is frequency-dependent
and needs to be computed self-consistently. However, many of the properties
in the metallic state are similar to those of the Anderson impurity model.
In particular the central peak in the spectral function can be viewed as the
Kondo resonance of the impurity model.
As all connected point-functions with order higher than G0 are neglected
within DMFT, the computation of q = 0 response functions is straightfor-
ward. As an example we show in fig 5 the (real part of the) optical con-
ductivity for various temperatures with U = 2 and ω0 = 2. The structure
seen in the curves reflects the structure of the density of states. There is a
strong response at low frequencies as particle-hole pairs are excited within
the ‘Kondo-like’ quasiparticle resonance at the Fermi energy. The second
peak at frequencies ω ∼ 1 arises from excitations between the quasiparti-
cle resonance and the large satellite (Hubbard band), while the third peak
around ω ∼ 5.0 involves excitations between the satellites. The first dip
at ω = 0.5 is the signature of the small Mott bands close to the Kondo
resonance and is the feature most likely to be observable experimentally.
Also calculated is the resistivity as a function of temperature (figure 6).
The curves reflect the structure of the self-energy shown in Figure 4: At low
temperatures the resistivity rises quadratically as expected for interacting
electrons. The temperature scale is given by the quasiparticle bandwidth
(‘Kondo temperature’). Above this temperature the resistivity drops as the
on-site (Kondo) scattering amplitude for electrons reduces. There is a slight
second peak at higher temperatures. The structures in ρ can be traced back
to the behavior seen in the self-energy. This second peak is the result of an
increase in scattering off the phonons: these soften slowly with increasing
temperature and, around the second peak in the resistivity curve, outweigh
the reduction in Kondo-like scattering as the temperature increases. This
effect clearly involves a partial cancellation between two effects and hence
may be sensitive to the accuracy of the analytic continuation, which at higher
temperatures starts from reduced information (since the majority of Mat-
subara points simply show asymptotic behaviour).
4. Summary
We have discussed the result of changing the ratio of electron and
phonon energies as a method for tuning the amount of correlation in a model
of electron-phonon interactions. We use approximate schemes to solve for
the spectral functions of the Holstein model. On the basis that second-order
iterated perturbation theory predicts the correct qualitative behavior at a
range of couplings in the static limit as well as describing correctly the limit
of infinite phonon frequency, we have computed the spectral function at in-
termediate frequencies and couplings. We have used an adaptation of the
standard maximum entropy scheme to obtain the spectral function, the self-
energy and the conductivity of the model by analytic continuation. These
quantities had not previously been studied.
The results for the intermediate frequency regime are consistent with
what might be expected on the basis of the limiting cases (high and low
frequencies). At energy scales smaller than ω0, the system shows behav-
ior similar to that of the Hubbard model found in the instantaneous limit
ω0 → ∞: there is a narrow central ‘Kondo-resonance’ or quasiparticle band.
At large energies the model behaves as it does in the static regime with a
well-defined band splitting. At intermediate frequencies the picture is com-
plicated by the interplay of the loss of coherence in the quasiparticle band
and the effective renormalization of the phonon frequency as a function of
coupling and temperature. We suggest that if systems with anomalously
large phonon frequencies and couplings exist, then the optical conductivity
should bear the hallmark of the correlation tuned regime.
5. Acknowledgements
The authors would like to thank F.Essler and F.Gebhard for useful
discussions.
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-6 -4 -2 0 2 4 6
(a) Static
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) IPT
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(c) Conserving
U=0.33
U=1.17
U=2.00
U=4.50
Fig. 2. The spectral function in the static limit of the half-filled Holstein
model computed at temperature T = 0.08 (a) using the exact solution and
(b) using 2nd order IPT at a low frequency, ω0 = 0.004. The IPT solution
at this small non-zero frequency is quite close to the exact solution in the
static limit. In particular, the band splitting and the positions of the maxima
agree. To contrast, panel (c) shows the results of the approximation using
the full Green’s function (Diagram 2c from figure 1 is not included to avoid
overcounting)
-6 -4 -2 0 2 4 6
(a) ω0=0.056
U=0.33
U=1.17
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(b) ω0=0.500
U=0.33
U=2.00
U=4.50
-6 -4 -2 0 2 4 6
(c) ω0=2.000
U=0.33
U=2.00
U=4.50
Fig. 3. Spectral functions of the half-filled Holstein model for various
electron-phonon couplings U , approximated using 2nd order perturbation
theory at T = 0.02 and ω0 = 0.056 (top), ω0 = 0.5 (center) and ω0 = 2
(bottom). In the low frequency limit (ω0 = 0.125), the spectral functions
are similar to those in the static limit shown in Fig. 2, with only a small
effect from the non-zero phonon frequency. As the temperature is lower than
the phonon frequency, the central quasiparticle peak is clearly resolved for
U ≤ 2. For the intermediate frequencies (central panel) the peak around
ω = 0 is again clear and has a width ∼ ω0 at low coupling. In the gapped
phase at large couplings two band-splittings are visible. For ω ≫ ω0 the
band splits just as in the static limit, while for ω ≪ U there is a peak at
a renormalized phonon frequency (which is less than the bare phonon fre-
quency). In the ungapped phases for ω0 = 0.5 and 2, the low energy behavior
is similar to that found in the Hubbard model with a narrow quasiparticle
band forming near the Fermi energy with the value at the Fermi energy
pinned to its value in the non-interacting case.
-6 -4 -2 0 2 4 6
T=0.08
T=0.16
T=0.32
Fig. 4. Imaginary part of the self-energy of the half-filled Holstein model
when U = 2 and ω0 = 2 computed using IPT and analytically continued
using MAXENT. At low temperatures the low frequency behavior is Fermi-
liquid like (quadratic dependence on ω) down to quite low frequencies (at
very low frequencies and low temperatures there are some inaccuracies as-
sociated with the truncation in Matsubara frequencies). There are peaks at
the frequencies associated with the phonon energy and with U. As the tem-
perature increases the minimum at the Fermi energy (ω = 0) increases as
incoherent on-site scattering in the corresponding local impurity increases
(see text). At temperatures above the characteristic (Kondo-like) energy
scale the central peak subsides and disappears.
0 1 2 3 4 5 6
T=0.08
T=0.16
T=0.32
Fig. 5. The real part of the optical conductivity for a system with U = 2.0
and ω0 = 2.0 for a range of temperatures. The structure of the spectrum
reflects that in the density of states (see fig 3. At low frequencies, electrons
may be excited within the quasiparticle resonance. The second peak at
ω ∼ 2.0 represents excitations from the Kondo resonance to the large satellite
(Hubbard band), and the peak at ω ∼ 5.0 represents excitations between the
satellites.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
U=2.00
U=2.13
U=2.28
Fig. 6. The resistivity as a function of temperature for the Holstein model
for ω0 = 2 for varying electron-phonon coupling strengths. The resistivity is
in units of e2V/ha2 with V the unit cell volume and a the lattice cell spacing.
The behavior reflects what is seen in the self-energy. At low temperatures
the behavior is similar to that in a Kondo lattice. The resistivity rises
sharply with temperature for temperatures smaller than the quasiparticle
bandwidth. The resistivity then drops for temperatures larger than this
lattice coherence temperature. A simple logarithmic decay with temperature
is not visible because, in addition to the Kondo-like scattering processes,
the electrons are scattered from thermally excited phonons whose spectral
weight broadens and shifts towards lower frequencies as the temperature
rises. This leads to a second peak. In contrast, the second peak is not
visible for the Hubbard model, and indicates the presence of two energy
scales in the Holstein model.
Introduction
Formalism
Results
Summary
Acknowledgements
|
0704.0031 | Crystal channeling of LHC forward protons with preserved distribution in
phase space | Microsoft Word - Diffraction-Arxiv
Crystal channeling of LHC forward protons
with preserved distribution in phase space
V.M. Biryukov♦
Institute for High Energy Physics, Protvino, 142281, Russia
Abstract
We show that crystal can trap a broad (x, x', y, y', E) distribution of particles and
channel it preserved with a high precision. This sampled-and-hold distribution can be
steered by a bent crystal for analysis downstream. In simulations for the 7 TeV Large
Hadron Collider, a crystal adapted to the accelerator lattice traps 90% of diffractively
scattered protons emerging from the interaction point with a divergence 100 times the
critical angle. We set the criterion for crystal adaptation improving efficiency ~100-fold.
Proton angles are preserved in crystal transmission with accuracy down to 0.1 µrad. This
makes feasible a crystal application for measuring very forward protons at the LHC.
1. Introduction
A particle entering the crystal lattice parallel to a major crystallographic direction can be
captured and channeled by the lattice along a crystal axis or plane [1]. For instance, a positive
particle can be channeled between adjacent atomic planes. In a bent crystal, the channeled
particles can follow the bend [2]. This led to elegant technique of beam steering by bent
channeling crystals [3] now experimentally explored over six decades in energy from low MeV
[4] to 1 TeV [5]. The technique is used on permanent basis in IHEP Protvino where crystal
systems extract protons from 70 GeV main ring with efficiency of 85% at intensity up to 4×1012
protons using Si crystals just 2 mm along the beam [6]. Bent crystals channel in good agreement
with predictions up to the highest energies [6-9].
Crystal applications at the 7-TeV Large Hadron Collider are considered for beam collimation
and extraction [10] and in situ calibration of CMS and ATLAS calorimeters [11]. In another
proposal, crystal could capture the particles emerging from the interaction point (IP) with small
angles and channel them out of the beam [12]. This could help to improve on measurement of
small angle elastic and “quasi-elastic” scattering in CMS and ATLAS where lower momentum
transfers might become available for pp elastics scattering and lower proton momentum losses
for diffractive physics [12]. Groups in both CMS (with TOTEM) and ATLAS would like to add
very forward proton detectors, 420 m downstream on both sides, a project FP420 [13]. By
detecting protons that have lost less than 1% of their longitudinal momentum, a rich QCD,
electroweak, Higgs and BSM program becomes accessible, with the potential to make
measurements which are unique at LHC, and difficult even at a future linear collider [13].
The measurement of the displacement x and angle x’ (in the horizontal plane) of the outgoing
protons relative to the beam allows the momentum loss ξ=∆p/p and transverse momentum of the
scattered protons to be reconstructed. Protons emerging from diffractive scattering at LHC have
very small emission angles (10-150 µrad) and fractional momentum loss (ξ = 10-8 – 0.1). Hence
they are very close to the beam and can only be detected in the Roman Pots downstream if their
displacement at the detector location is large enough to escape the beam halo [13].
2. Crystal efficiency
As practice shows, crystal can go into a very limited space and get particles from there [6].
Most efficient crystal applications are based on so called “multipass” mode where particles can
encounter a crystal many times in the ring [6,14]. There are also successful experimental
demonstrations of highly efficient channeling in a single pass, with efficiency up to 60% at
CERN SPS [15]. Throughout this paper we consider only a single-pass channeling.
We show with simulations that at the LHC a crystal can efficiently channel forward protons.
For channeling simulations we apply a Monte Carlo code CATCH [16] successfully used for
prediction of experiments at CERN SPS [9], IHEP U-70 [6], Tevatron [7], RHIC [8] and KEK
[17] and crystal applications at the LHC [10,11].
Crystal capture is very selective in angle. The critical angle θC within which the capture is
possible is as small as about ±5 µrad / E1/2(TeV) at a high energy E in Silicon (110) planes.
Proton divergence of 150 µrad is almost 100 times θC at 7 TeV. Therefore it is not possible to
capture all these protons by a plain crystal. However, we can suggest an efficient solution
benefiting from the fact that all diffractively scattered protons originate from a small region at
the IP.
For standard LHC optics with beta function value at the IP β*=0.55 m, the beam size at the IP
is σbeam≈16 µm rms. The spread in the transverse position of the vertex point where outgoing
protons originate from is determined by the rms spread of the beam and equals σbeam/√2 [13]. As
protons emerge from diffractive scattering at LHC with emission angles up to ≈150 µrad and
interaction width σbeam/√2, the emittance of the beam to be trapped by a crystal is ≤ 2π µrad-mm
only. This corresponds to the acceptance of a Si crystal of ~1 mm transverse size. The match of
the diffractive-protons emittance to the crystal acceptance means that the particles could be
♦ http://mail.ihep.ru/~biryukov/
trapped and channeled efficiently. To realize this, one has to match the crystal design and
location to the application. For the LHC scenario with high luminosity, we find the most efficient
design to be a crystal with a point-to-parallel focusing entry face. Focusing crystal proposed by
A.I Smirnov in 1985 has a face shaped so that the tangents to the crystal planes cross at a focus
line at some distance LF from the crystal [3]. This kind of crystal was successfully tested at IHEP
where it efficiently trapped a beam of ±2 mrad divergence (or ~100 times θC) [18]. The crystal
traps protons emerging from the focus line uniformly from all the angular range if the entry face
has a proper shape [18].
We find that for most efficient channeling in the LHC the focus distance LF of this crystal
should be equal to the effective distance Leff between the crystal location and the IP. In a drift
space, Leff is geometrical distance. In accelerator lattice, Leff =(β*βC)1/2sin∆ψ, where βC is β value
at the crystal and ∆ψ is the phase advance from the IP to the crystal. A plain crystal with a flat
entry face has LF = ∞.
In simulations with the low β* optics settings, a focusing crystal shows best efficiency if
installed at a location with effective distance Leff ≥ 15 m from the IP. It can be a Si (110) or (111)
crystal of ≈(0.15 mrad)×Leff ≥ 2.5 mm transverse size in order to capture efficiently all diffractive
protons. Simulation predicts that a focusing Si(110) crystal with LF=Leff traps 90% of 7 TeV
protons emerging from the IP in the angular range of 150 µrad width into channeling mode. The
efficiency figure is almost independent of the crystal location provided Leff ≥ 15 m.
The reason for high efficiency at high Leff is that, at a distance Leff from the IP, any point at
the crystal entry face sees the beam source of σ size (at the IP) at an angle of σ/Leff. Channeling
efficiency reduces by a factor of about (1-(σ/LeffθC)2)1/2 [3]. The reduction in efficiency by a
factor of ≈1-σ2/2L2effθC2 becomes negligible for L2eff >> σ2/2θC2 =β*ε/4θC2 where ε is beam
emittance. With β*=0.55 m and Si(110) crystal, channeling efficiency saturates for L2eff >> (4
One more idea for efficient channeling of forward protons in the LHC is that a crystal can be
installed with planes parallel to either x’ or y’ plane. For application, it is not critical whether
crystal bends protons in horizontal or vertical plane to produce an offset at the detector. But the
distance Leff in accelerator lattice from the IP to the crystal can be very different in x and y
planes. Then channeling efficiency is very different, whether crystal traps protons in x’ or y’
plane. We suggest in this case to install crystal for channeling in the plane with larger Leff.
For instance, on the location 200 m downstream of the IP5 (CMS) and some 20 m ahead of
the Roman Pot station at 220 m where crystal could be installed, Leff ≈6 m in x’ and ≈20 m in y’
plane. According to the analysis above, channeling efficiency in y’ plane should be great while in
x’ plane moderate. Indeed, our simulations for this location show channeling efficiency of ≈87%
in y’ and only ≈60% in x’ plane, for β*=0.55 m and optimal crystals of Si(110). A plain Si crystal
has channeling efficiency in x’ plane of just 3.5% or 17 times lower than a Si focusing crystal
adapted to the LHC lattice.
For the run-in phase of the LHC with β*=2 m we find that channeling efficiency of ≥ 85% can
be achieved if crystal is located at Leff ≥ 30 m downstream of the IP. The nominal, high
luminosity optics of the LHC is not optimized for forward proton detection. Therefore a
possibility to use a channeling crystal can be very helpful as it offers opportunities for diffractive
physics studies otherwise inaccessible in the nominal LHC settings.
The LHC options with a high β* (1540 and 90 m) are devised for the studies of diffractive
physics. With β*=1540 m, the emittance of diffractively scattered protons increases to ≈50π
µrad-mm. This corresponds to the acceptance of a Si crystal of ≥ 30 mm transverse size. Such a
crystal is not out of question, however the problem is where to fit it in the LHC. In terms of Leff,
good channeling efficiency requires a location with L2eff >> (130 m)2 in this optics.
We simulated channeling on the location 200 m from the IP5. In β*=1540 m optics, a 10-cm
Si(110) crystal trapped and bent protons 0.5 mrad in x’ plane with efficiency of 41%. A Ge(110)
crystal shows there 48% efficiency, i.e. comparable to Si. All figures assumed a perfect match
LF=Leff in crystal. A plain Si crystal gives efficiency of <<1%, or 300 times lower than a Si
focusing crystal adapted to the LHC lattice on this location.
In β*=90 m option on the same location, the choice of plane is important because Leff ≈10 m in
x’ and ≈170 m in y’ plane. Preferred location should have L2eff >> (60 m)2 so we expect very
different efficiencies in x’ and y’ planes. Our simulations give crystal efficiency of 72% in y’ and
only 7% in x’ plane for β*=90 m. Here, crystal application is feasible only with bending in
vertical plane.
Low efficiency may exclude a crystal use for double-pomeron-exchange events (pp→pXp)
with double-arm reconstruction, because the probability to have channeling in both arms in
coincidence becomes small, e.g. (41%)2≈17%. For reconstruction of single-diffraction events
(pp→pX) more detailed studies are required before the benefits (or their absence) from a crystal
use with high β* options can be understood.
In this paper we suggest the use of a single crystal for proton extraction from halo and
delivery to the detector. The use of a 2-stage crystal system [12], first crystal for extracting a
proton and second one for bending it a big angle, would reduce the overall efficiency by a factor
of ~0.6 (ideally) or less. The 2nd crystal traps only part of the protons channeled in the 1st one.
Finally, we notice that one can filter diffraction events with a crystal. Instead of trapping all
forward protons, crystal acceptance can be made smaller and sample e.g. only the most forward
protons emerging from the IP with the angles of a few µrad.
3. Precise transmission in a single (x, x’) plane
Whereas protons are physically delivered from the IP to detector with good efficiency, the
essential question is whether the information on phase space (x, x', y, y', E) distribution of
particles is lost or corrupted while the particles are captured and transmitted in crystal. The
success of experiments on measuring forward high momentum protons at the LHC depends on
the angular precision of proton track reconstruction. A plain crystal would destroy the phase
space information first by selecting particles from just a single direction and then disturbing the
exit angle of particle by coherent and incoherent scattering in crystal. Plain crystal acceptance is
±θC and crystal accuracy in angle transmission is again ±θC. That means, a plain crystal traps and
delivers about zero bit information on angle distribution. In this paper we design a crystal with
the acceptance of ~100 θC and angle transmission accuracy of ~0.1 θC, although it sounds against
the nature of crystal channeling.
Suppose particles are coming with a distribution over (x, x', y, y', E). Ideally, we would like
the crystal to trap all coming particles and preserve their distribution over (x, x', y, y', E), and
then shift an angle of θ each particle towards a physical setup where this distribution can be
analyzed in detail.
One should solve two problems. One problem was to trap and bend a beam with a divergence
much greater than the critical angle. A focusing crystal adapted to the LHC optics solves this
problem. In simulations, a focusing crystal traps with 90% efficiency all protons emerging from
the IP with the angular distribution ~100 times θC. Notice that the trapped particles fully preserve
also their distribution over the angle in the plane orthogonal to the plane of channeling. In such a
crystal, particles are trapped uniformly from a very broad distribution over x’ and y’.
A bent crystal would transform (x, x', y, y') at the entrance into (x, x'+θ, y, y') at the exit. To
do so, each trapped particle has to be channeled over the same distance in crystal. Therefore, the
shape of the crystal exit face must match the entry face. Then in a bent crystal each channeled
particle receives the same bending angle.
Although the crystal described above can solve the idea of sampling a broad distribution of
particles and delivering it to a required destination, second problem is how to preserve the
sampled distribution (x, x', y, y') “frozen” on transmission through the crystal lattice as precise as
possible. The coordinates (x, y) of particles are obviously preserved in crystal, so one should take
care of the accuracy in transmission of angles x’, y’ only.
The protons channeled between atomic planes in crystal are disturbed by (1) oscillations in
the channeling plane with an amplitude up to θC and by (2) scattering on a rarefied electronic gas
(mostly valence electrons) in both planes, x’ and y’. Notice that nuclear scattering will not
disturb the sample of transmitted channeled particles as this process is strongly suppressed for
channeled positive particles. Simply saying, any particle nuclear scattered would be dechanneled
and thus not present in the sample of bent particles.
That gives us the first idea that partially solves the problem of transmission accuracy. The
idea is that the information on crystal-captured particles is very well preserved in one plane, e.g.
(x, x'), while the particles are trapped and bent in another plane, e.g. (y, y’). Notice that particle
distribution in the plane orthogonal to channeling is favored twice. Firstly, they are easily
trapped with a broad angular distribution; secondly they are transmitted with a very little
scattering. Information in this plane will be best preserved. The opportunity to have perfect data
on just one plane is interesting for applications. The reconstruction of the Higgs boson mass in
reaction pp→p+H+p requires (x, x’) data in horizontal plane only [13].
Figure 1 The difference in proton angles, x’ and y’, before and after a Si(110) crystal.
Oscillations in the channeling plane on the atomic coherent potential are a greater problem.
Fig. 1 shows a distribution of the difference in proton angle in x’ and y’ planes before and after a
channeling in crystal, (x’OUT – x’IN) and (y’OUT – y’IN – 0.1 mrad), as obtained in simulations for a
Si(110) bent crystal channeling in y’ plane. The accuracy in x’ transmission in crystal is very
good indeed, ~0.1 µrad rms. The width of (y’OUT – y’IN – 0.1 mrad) distribution is much greater
due to oscillations in the potential of Si (110) planes.
4. Precise transmission in both planes
To solve the problem of accuracy in the other plane, i.e. the plane of channeling, one solution
is to use a channel with a lower critical angle, for instance Si(100) instead of (110) or (111). A
more universal solution is to use a strongly bent crystal. The critical angle θC is gradually
reduced to zero when the crystal curvature approaches a critical value. The strong focusing of a
strongly bent crystal suppresses channeling oscillations to any low level needed in the
application.
Fig. 2 shows the difference in proton angle in x’ and y’ planes before and after channeling in a
crystal, (x’OUT – x’IN– 0.1 mrad) and (y’OUT – y’IN), as obtained in simulations for a 2 mm Si(100)
crystal bent 0.1 mrad. The protons were channeled in x’ plane. The rms value of angle smearing
found in simulations is 0.2 µrad both for x’ and y’.
Figure 2 The same as in Fig. 1 but for a strongly bent Si (100) crystal.
This accuracy should be compared to the angular resolution of the detectors downstream of
the crystal. Measuring proton coordinates with ~10 µm resolution [13] over a base of ~8 m as
allowed by a drift space would give an angular accuracy of ~1.4×10µm/8m=1.8 µrad. Addition
(quadratic) of crystal transmission accuracies doesn’t change this resolution. That would be
perfect for crystal. With a much better resolution on the detector side, down to 0.5 µrad, the
overall resolution becomes ~0.55 µrad, i.e. just slightly disturbed. Crystal transmission in both
planes, x’ and y’, is still almost perfect.
Because of scattering on electronic gas, the protons loose energy in crystal. In simulations, the
energy loss and its fluctuations in crystal are ∆E/E ≈10-7–10-6, i.e. much smaller than even the
nominal energy spread in the LHC beam, 1.1×10-4. Energy losses in bent crystals were studied in
experiments at CERN SPS with protons of 450 GeV and Pb ions of 33 TeV where CATCH
predictions were also validated [9]. The diffractively scattered protons would have energy spread
on the order of 100 GeV, or ∆E/E ≈1.5%, at the crystal entrance. In simulations with β*=0.55-2
m optics, channeling efficiency was completely independent of energy even for ∆E/E≈10%. In
high β* options, crystal efficiency was uniform within ≈0.7% for ∆E/E ≈1.5%. One can say that a
phase space distribution (x, y, x', y', E) can be perfectly preserved in crystal and no information is
lost on transmission in crystal.
Figure 3 An example of beam space (x’, y’) at the entrance to the crystal (a) and at the exit (b).
Fig. 3 shows an example of a (x’, y’) plot at the entrance to the crystal (a) and at the exit of it
(b) where we tried to show how accurately a crystal can transmit a signature in angular space
(semicircle chosen as a probe). The resolution of the image transmitted by a crystal is ~0.2 µrad
in both planes. In terms of the critical channeling angle θC, the obtained resolution is an order of
magnitude finer than θC while the size of the trapped and channeled area can be some orders of
magnitude greater than θC.
In the applications there is no point to have a crystal transmission too perfect. It should match
the other sources of inaccuracy like a multiple scattering in the detectors and vacuum chambers,
etc. By tuning crystal parameters, in principle, one could very much improve in precision of the
beam image downstream of the crystal but loose in brightness of the image, i.e. in statistics rate,
as the efficiency of crystal transmission could be affected.
Finally, we suggest another idea for the channeling plane (a “microscope idea”) that improves
not only the crystal accuracy but even the detector resolution in that plane. Crystal can magnify
beam image in one plane, e.g. transform the entrance values (x, x', y, y') into exit values (x, Nx'
+θ, y, y'). The magnification factor N can be as big as 2 or 10 or even 100, and serve the purpose
to increase strongly the overall angular resolution in x’. In the above examples, the overall
resolution was ~1 µrad defined by detector resolution. With magnification optics, the overall
inaccuracy in x’ would be effectively reduced by factor N, bringing it below 0.1 µrad rms.
Magnification is realized by making the shape of the crystal exit face different from the entry
face. With a magnification factor of 10, e.g., the entry angular opening of 50 µrad would
correspond to the exit opening of 500 µrad.
5. Conclusion
We have shown in simulations that crystal lattice can trap with 90% efficiency a beam with a
(x', y') distribution much broader than a critical angle θC. To achieve that, one has to match the
crystal focus length to the effective length between the particle source and the crystal in the
accelerator lattice. Crystal adaptation to accelerator lattice improved channeling efficiency up to
300-fold. Crystal can transmit the trapped particles in channeled states with the phase space (x,
x', y, y', E) distribution preserved with accuracy an order of magnitude finer than θC. Several
solutions were proposed and supported by simulations for achieving a fine resolution in crystal
transmission.
This may give a beam instrument for collision products in colliders. Usually, accelerator beam
instruments prepare particles for collision: by cooling them, bending, focusing, etc. Detectors
sort out the results of collision. We change this a bit by introducing crystal optics between the
collision point and detectors.
A crystal adapted to the LHC lattice can trap with 90% efficiency all protons emerging from
the IP with divergence of 150 µrad or ~100θC. The trapped protons can be channeled to detectors
with precision down to 0.1 µrad rms. This makes feasible a crystal application for the
measurement of diffractive scattering in CMS and ATLAS at the LHC. While we showed the
physical capabilities of crystal channeling, its actual application in the LHC environment has to
take into account many technical considerations to fit into existing infrastructure of accelerator
and detectors.
Crystal channeling of LHC forward protons can improve proton acceptance in momentum
loss ξ and four-momentum transfer t both in TOTEM and FP420 and allow to reach the smallest
possible value of the scattering angle [9]. Now the sensitive detector area starts at ~12-15σ from
the LHC beam [10]. Crystal can be placed at ~6σ from the LHC beam as it is very small, ~cm Si,
and does not provoke beam instability. Such a crystal can trap and deliver a very useful
information on most forward high momentum “quasi-elastic” and elastic protons at LHC,
unavailable otherwise.
There are practical benefits as well. Crystal would relax tough requirements on β* needed for
TOTEM. Crystal may allow TOTEM to run at the early start of the LHC, possibly running in
parallel to other experiments. Thanks to crystal, FP420 detectors could possibly reside out of the
cold region. The detectors don’t need to be edgeless. Crystal works best with low β*, where
FP420 is interested most. If detectors can be more distanced from the beam, background
conditions may improve. For injection, the active areas of the detectors must be kept away from
the beams and then moved back; instead, one can move a crystal. Crystal can be introduced to
experiment on a later stage in an attempt to expand the horizons of the physics program.
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|
0704.0032 | Probing non-standard neutrino interactions with supernova neutrinos | IFIC/07-03
Probing non-standard neutrino interactions with supernova neutrinos
A. Esteban-Pretel, R. Tomàs and J. W. F. Valle1
1AHEP Group, Institut de F́ısica Corpuscular - C.S.I.C/Universitat de València
Edifici Instituts d’Investigació, Apt. 22085, E-46071 València, Spain
(Dated: November 4, 2018)
We analyze the possibility of probing non-standard neutrino interactions (NSI, for short) through
the detection of neutrinos produced in a future galactic supernova (SN). We consider the effect of
NSI on the neutrino propagation through the SN envelope within a three-neutrino framework, paying
special attention to the inclusion of NSI-induced resonant conversions, which may take place in the
most deleptonised inner layers. We study the possibility of detecting NSI effects in a Megaton water
Cherenkov detector, either through modulation effects in the ν̄e spectrum due to (i) the passage
of shock waves through the SN envelope, (ii) the time dependence of the electron fraction and (iii)
the Earth matter effects; or, finally, through the possible detectability of the neutronization νe
burst. We find that the ν̄e spectrum can exhibit dramatic features due to the internal NSI-induced
resonant conversion. This occurs for non-universal NSI strengths of a few %, and for very small
flavor-changing NSI above a few×10−5.
PACS numbers: 13.15.+g, 14.60.Lm, 14.60.Pq, 14.60.St, 97.60.Bw
I. INTRODUCTION
The very first data of the KamLAND collaboration [1]
have been enough to isolate neutrino oscillations as the
correct mechanism explaining the solar neutrino prob-
lem [2, 3], indicating also that large mixing angle (LMA)
was the right solution. The 766.3 ton-yr KamLAND data
sample further strengthens the validity of the LMA os-
cillation interpretation of the data [4].
Current data imply that neutrino have mass. For an
updated review of the current status of neutrino oscil-
lations see [5]. Theories of neutrino mass [6, 7] typ-
ically require that neutrinos have non-standard prop-
erties such as neutrino electromagnetic transition mo-
ments [8, 9, 10] or non-standard four-Fermi interactions
(NSI, for short) [11, 12, 13]. The expected magnitude of
the NSI effects is rather model-dependent.
Seesaw-type models lead to a non-trivial structure of
the lepton mixing matrix characterizing the charged and
neutral current weak interactions [6]. The NSI which
are induced by the charged and neutral current gauge
interactions may be sizeable [14, 15, 16, 17, 18]. Alter-
natively, non-standard neutrino interactions may arise in
models where neutrinos masses are radiatively “calcula-
ble” [19, 20]. Finally, in some supersymmetric unified
models, the strength of non-standard neutrino interac-
tions may arise from renormalization and/or threshold
effects [21].
We stress that non-standard interactions strengths are
highly model-dependent. In some models NSI strengths
are too small to be relevant for neutrino propagation,
because they are either suppressed by some large mass
scale or restricted by limits on neutrino masses, or both.
However, this need not be the case, and there are many
theoretically attractive scenarios where moderately large
NSI strengths are possible and consistent with the small-
ness of neutrino masses. In fact one can show that
NSI may exist even in the limit of massless neutri-
nos [14, 15, 16, 17, 18]. Such may also occur in the
context of fully unified models like SO(10) [22].
We argue that, in addition to the precision determi-
nation of the oscillation parameters, it is necessary to
test for sub-leading non-oscillation effects that could arise
from non-standard neutrino interactions. These are nat-
ural outcome of many neutrino mass models and can be of
two types: flavor-changing (FC) and non-universal (NU).
These are constrained by existing experiments (see be-
low) and, with neutrino experiments now entering a pre-
cision phase [23], an improved determination of neutrino
parameters and their theoretical impact constitute an im-
portant goal in astroparticle and high energy physics [5].
Here we concentrate on the impact of non-standard
http://arxiv.org/abs/0704.0032v1
neutrino interactions on supernova physics. We show
how complementary information on the NSI parame-
ters could be inferred from the detection of core-collapse
supernova neutrinos. The motivation for the study is
twofold. First, if a future SN event takes place in our
Galaxy the number of neutrino events expected in the
current or planned neutrino detectors would be enor-
mous, O(104 − 105) [24]. Moreover, the extreme con-
ditions under which neutrinos have to travel since they
are created in the SN core, in strongly deleptonised re-
gions at nuclear densities, until they reach the Earth,
lead to strong matter effects. In particular the effect of
small values of the NSI parameters can be dramatically
enhanced, possibly leading to observable consequences.
This paper is planned as follows. In Sec. II we summa-
rize the current observational bounds on the parameters
describing the NSI, including previous works on NSI in
SNe. In Sec. III we describe the neutrino propagation
formalism as well as the SN profiles which will be used.
In Sec. IV we analyze the effect of NSI on the ν propaga-
tion in the inner regions near the neutrinosphere and in
the outer regions of the SN envelope. In Sec. V we discuss
the possibility of using various observables to probe the
presence of NSI in the neutrino signal of a future galactic
SN. Finally in Sec. VI we present our conclusions.
II. PRELIMINARIES
A large class of non-standard interactions may be
parametrized with the effective low-energy four-fermion
operator:
LNSI = −εfPαβ 2
2GF (ν̄αγµLνβ)(f̄γ
µPf) , (1)
where P = L, R and f is a first generation fermion:
e, u, d. The coefficients ε
αβ denote the strength of the
NSI between the neutrinos of flavors α and β and the
P−handed component of the fermion f .
Current constraints on ε
αβ come from a variety of dif-
ferent sources, which we now briefly list.
A. Laboratory
Neutrino scattering experiments [25, 26, 27, 28,
29] provide the following bounds, |εfPµµ | . 10−3 −
10−2, |εfPee | . 10−1 − 1, |εfPµτ | . 0.05, |εfPeτ | . 0.5 at
90 % C.L [30, 31, 32]. On the other hand the analysis
of the e+e− → νν̄γ cross section measured at LEP II
leads to a bound on |εePττ | . 0.5 [33]. Future prospects
to improve the current limits imply the measurement of
sin2 ϑW leptonically in the scattering off electrons in the
target, as well as in neutrino deep inelastic scattering in
a future neutrino factory. The main improvement would
be in the case of |εfPee | and |εfPeτ |, where values as small
as 10−3 and 0.02, respectively, could be reached [31].
The search for flavor violating processes involving
charged leptons is expected to restrict corresponding neu-
trino interactions, to the extent that the SU(2) gauge
symmetry is assumed. However, this can at most give
indicative order-of-magnitude restrictions, since we know
SU(2) is not a good symmetry of nature. Using radiative
corrections it has been argued that, for example, µ − e
conversion on nuclei like in the case of µ−T i also con-
strains |εqPµe | . 7.7× 10−4 [31].
Non-standard interactions can also affect neutrino
propagation through matter, probed in current neutrino
oscillation experiments. The bounds so obtained apply to
the vector coupling constant of the NSI, ε
αβ = ε
since only this appears in neutrino propagation in mat-
ter [91].
B. Solar and reactor
The role of neutrino NSIs as subleading effects on the
solar neutrino oscillations and KamLAND has been re-
cently considered in Ref. [34, 35, 36] with the following
bounds at 90 % CL for ε ≡ − sinϑ23εdVeτ with the al-
lowed range −0.93 . ε . 0.30, while for the diagonal
term ε′ ≡ sin2 ϑ23εdVττ − εdVee , the only forbidden region is
[0.20, 0.78] [36]. Only in the ideal case of infinitely pre-
cise solar neutrino oscillation parameters determination,
the allowed range would “close from the left” for negative
NSI parameter values, at −0.6 for ε and −0.7 for ε′.
C. Atmospheric and accelerator neutrinos
Non-standard interactions involving muon neutrinos
can be constrained by atmospheric neutrino experiments
as well as accelerator neutrino oscillation searches at
K2K and MINOS. In Ref. [37] Super-Kamiokande and
MACRO observations of atmospheric neutrinos were con-
sidered in the framework of two neutrinos. The limits ob-
tained were −0.05 . εdVµτ < 0.04 and |εdVττ − εdVµµ | . 0.17
at 99 % CL. The same data set together with K2K were
recently considered in Refs. [38, 39] to study the nonstan-
dard neutrino interactions in a three generation scheme
under the assumption εeµ = εµµ = εµτ = 0. The al-
lowed region of εττ obtained for values of εeτ smaller
than O(10−1) becomes Σf=u,d,eεfVαβNf/Ne . 0.2 [39] ,
where Nf stands for the fermion number density.
D. Cosmology
If non-standard interactions with electrons were large
they might also lead to important cosmological and as-
trophysical implications. For instance, neutrinos could
be kept in thermal contact with electrons and positrons
longer than in the standard case, hence they would share
a larger fraction of the entropy release from e± annihi-
lations. This would affect the predicted features of the
cosmic background of neutrinos. As recently pointed out
in Ref [40] required couplings are, though, larger than
the current laboratory bounds.
E. NSI in Supernovae
According to the currently accepted supernova (SN)
paradigm, neutrinos are expected to play a crucial role
in SN dynamics. As a result, SN physics provides
a laboratory to probe neutrino properties. Moreover,
many future large neutrino detectors are currently be-
ing discussed [41]. The enormous number of events,
O(104 − 105) that would be “seen” in these detectors in-
dicates that a future SN in our Galaxy would provide a
very sensitive probe of non-standard neutrino interaction
effects.
The presence of NSI can lead to important conse-
quences for the SN neutrino physics both in the highly
dense core as well as in the envelope where neutrinos
basically freely stream.
The role of non-forward neutrino scattering processes
on heavy nuclei and free nucleons giving rise to flavor
change within the SN core has been recently analyzed in
Ref. [42, 43]. The main effect found was a reduction in
the core electron fraction Ye during core collapse. A lower
Ye would lead to a lower homologous core mass, a lower
shock energy, and a greater nuclear photon-disintegration
burden for the shock wave. By allowing a maximum
∆Ye = −0.02 it has been claimed that εeα . 10−3, where
α = µ, τ [43].
On the other hand it has been noted since long ago
that the existence of NSI plays an important role in the
propagation of SN neutrinos through the envelope lead-
ing to the possibility of a new resonant conversion. In
contrast to the well known MSW effect [44, 45] it would
take place even for massless neutrinos [13]. Two basic
ingredients are necessary: universal and flavor changing
NSI. In the original scheme neutrinos were mixed in the
leptonic charged current and universality was violated
thanks to the effect of mixing with heavy gauge singlet
leptons [6, 14]. Such resonance would induce strong neu-
trino flavor conversion both for neutrinos and antineutri-
nos simultaneously, possibly affecting the neutrino sig-
nal of the SN1987A as well as the possibility of having
r−process nucleosynthesis. This was first quantitatively
considered within a two-flavor νe−ντ scheme, and bounds
on the relevant NSI parameters were obtained using both
arguments [46].
One of the main features of the such “internal” or
“massless” resonant conversion mechanism is that it re-
quires the violation of universality, its position being
determined only by the matter chemical composition,
namely the value of the electron fraction Ye, and not by
the density. In view of the experimental upper bounds
on the NSI parameters such new resonance can only take
place in the inner layers of the supernova, near the neu-
trinosphere, where Ye takes its minimum values. In this
region the values of Ye are small enough to allow for
resonance conversions to take place in agreement with
existing bounds on the strengths of non-universal NSI
parameters.
The SN physics implications of another type of NSI
present in supersymmetric R-parity violating models
have also been studied in Ref. [47], again for a system
of two neutrinos. For definiteness NSI on d−quarks were
considered, in two cases: (i) massless neutrinos without
mixing in the presence of flavor-changing (FC) and non-
universal (NU) NSIs, and (ii) neutrinos with eV masses
and FC NSI. Different arguments have been used in
order to constrain the parameters describing the NSI,
namely, the SN1987A signal, the possibility to get suc-
cessful r−process nucleosynthesis, and the possible en-
hancement of the energy deposition behind the shock
wave to reactivate it.
On the other hand several subsequent articles [48, 49,
50] considered the effects of NSI on the neutrino propa-
gation in a three–neutrino mixing scenario for the case
Ye > 0.4, typical for the outer SN envelope. Together
with the assumption that εdVαβ . 10
−2 this prevents the
appearance of internal resonances in contrast to previous
references.
Motivated by supersymmetric theories without R par-
ity, in Ref. [48] the authors considered the effects of
small-strength NSI with d−quarks. Following the for-
malism developed in Refs. [51, 52] they studied the cor-
rections that such NSI would have on the expressions
for the survival probabilities in the standard resonances
MSW-H and MSW-L. A similar analysis was performed
in Ref. [49] assuming Z-induced NSI interactions orig-
inated by additional heavy neutrinos. A phenomeno-
logical generalization of these results was carried out in
Ref. [50]. The authors found an analytical compact ex-
pression for the survival probabilities in which the main
effects of the NSI can be embedded through shifts of the
mixing angles ϑ12 and ϑ13. In contrast to similar expres-
sions found previously these directly apply to all mixing
angles, and in the case with Earth matter effects. The
main phenomenological consequence was the identifica-
tion of a degeneracy between ϑ13 and εeα, similar to the
analogous “confusion” between ϑ13 and the correspond-
ing NSI parameter noted to exist in the context of long-
baseline neutrino oscillations [53, 54].
We have now re-considered the general three–neutrino
mixing scenario with NSI. In contrast to previous
work [48, 49, 50], we have not restricted ourselves to
large values of Ye, discussing also small values present
in the inner layers. This way our generalized descrip-
tion includes both the possibility of neutrinos having the
“massless” NSI-induced resonant conversions in the in-
ner layers of the SN envelope [13, 46, 47], as well as the
“outer” oscillation-induced conversions [48, 49, 50] [92].
III. NEUTRINO EVOLUTION
In this section we describe the main ingredients of our
analysis. Our emphasis will be on the use of astrophys-
ically realistic SN matter and Ye profiles, characterizing
its density and the matter composition. Their details,
in particular their time dependence, are crucial in deter-
mining the way the non-standard neutrino interactions
affect the propagation of neutrinos in the SN medium.
A. Evolution Equation
As discussed in Sec. II in an unpolarized medium the
neutrino propagation in matter will be affected by the
vector coupling constant of the NSI, ε
αβ = ε
αβ [93].
The way the neutral current NSI modifies the neu-
trino evolution will be parametrized phenomenologically
through the effective low-energy four-fermion operator
described in Eq. (1). We also assume ε
αβ ∈ ℜ, neglect-
ing possible CP violation in the new interactions.
Under these assumptions the Hamiltonian describing
the SN neutrino evolution in the presence of NSI can be
cast in the following form [94]
να = (Hkin +Hint)αβ νβ , (2)
where Hkin stands for the kinetic term
Hkin = U
U † , (3)
with M2 = diag(m21,m
3), and U the three-neutrino
lepton mixing matrix [6] in the PDG convention [55] and
with no CP phases.
The second term of the Hamiltonian accounts for the
interaction of neutrinos with matter and can be split into
two pieces,
Hint = H
int +H
int . (4)
The first term, Hstd
describes the standard interaction
with matter and can be written asHstd
= diag (VCC , 0, 0)
up to one loop corrections due to different masses of the
muon and tau leptons [56]. The standard matter poten-
tial for neutrinos is given by
VCC =
2GFNe = V0ρYe , (5)
where V0 ≈ 7.6×10−14 eV, the density is given in g/cm3,
and Ye stands for the relative number of electrons with
respect to baryons. For antineutrinos the potential is
identical but with the sign changed.
The term in the Hamiltonian describing the non-
standard neutrino interactions with a fermion f can be
expressed as,
(Hnsiint )αβ =
f=e,u,d
)αβ , (6)
with (V
)αβ ≡
2GFNfε
αβ. For definiteness and mo-
tivated by actual models, for example, those with broken
R parity supersymmetry we take for f the down-type
quark. However, an analogous treatment would apply
to the case of NSI on up-type quarks, the existence of
NSI with electrons brings no drastic qualitative differ-
ences with respect to the pure oscillation case (see be-
low). Therefore the NSI potential can be expressed as
follows,
(V dnsi)αβ = ε
αβV0ρ(2− Ye) . (7)
From now on we will not explicitely write the superindex
d. In order to further simplify the problem we will rede-
fine the diagonal NSI parameters so that εµµ = 0, as one
can easily see that subtracting a matrix proportional to
the identity leaves the physics involved in the neutrino
oscillation unaffected.
B. Supernova matter profiles
Neutrino propagation depends on the supernova mat-
ter and chemical profile through the effective potential.
This profile exhibits an important time dependence dur-
ing the explosion. Fig. 1 shows the density ρ(t, r) and the
electron fraction Ye(t, r) profiles for the SN progenitor as
well as at different times post-bounce.
Progenitor density profiles can be roughly
parametrized by a power-law function
ρ(r) = ρ0
, (8)
where ρ0 ∼ 104 g/cm3, R0 ∼ 109 cm, and n ∼ 3. The
electron fraction profile varies depending on the matter
composition of the different layers. For instance, typical
values of Ye between 0.42 and 0.45 in the inner regions
are found in stellar evolution simulations [57]. In the in-
termediate regions, where the MSW H and L-resonances
take place Ye ≈ 0.5. This value can further increase in
the most outer layers of the SN envelope due to the pres-
ence of hydrogen.
After the SN core bounce the matter profile is affected
in several ways. First note that a front shock wave starts
to propagate outwards and eventually ejects the SN enve-
lope. The evolution of the shock wave will strongly mod-
ify the density profile and therefore the neutrino propa-
gation [58, 59]. Following Ref. [60] we shall assume that
the structure of the shock wave is more complicated and
an additional “reverse wave” appears due to the collision
of the neutrino-driven wind and the slowly moving mate-
rial behind the forward shock, as seen in the upper panel
of Fig. 1 [95].
On the other hand, the electron fraction is also affected
by the time evolution as the SN explosion proceeds. Once
the collapse starts the core density grows so that the neu-
trinos become eventually effectively trapped within the
so called “neutrinosphere”. At this point the trapped
electron fraction has decreased until values of the order of
0.33 [61]. When the inner core reaches the nuclear density
it can not contract any further and bounces. As a con-
sequence a shock wave forms in the inner core and starts
propagating outwards. When the newly formed super-
nova shock reaches densities low enough for the initially
trapped neutrinos to begin streaming faster than the
shock propagates [62], a breakout pulse of νe is launched.
In the shock-heated matter, which is still rich of elec-
trons and completely disintegrated into free neutrons and
protons, a large number of νe are rapidly produced by
electron captures on protons. They follow the shock on
its way out until they are released in a very luminous
flash, the breakout burst, at about the moment when
the shock penetrates the neutrinosphere and the neutri-
nos can escape essentially unhindered. As a consequence,
the lepton number in the layer around the neutrinosphere
decreases strongly and the matter neutronizes [63]. The
value of Ye steadily decreases in these layers until val-
ues of the order of O(10−2). Outside the neutrinosphere
there is a steep rise until Ye ≈ 0.5. This is a robust
feature of the neutrino-driven baryonic wind. Neutrino
heating drives the wind mass loss and causes Ye to rise
within a few 10 km from low to high values, between 0.45
and 0.55 [64], see bottom panel of Fig. 1. Inspired in the
numerical results of Ref. [60] we have parametrized the
behavior of the electron fraction near the neutrinosphere
phenomenologically as,
Ye = a+ b arctan[(r − r0)/rs] , (9)
where a ≈ 0.23− 0.26 and b ≈ 0.16− 0.20. The param-
eters r0 and rs describe where the rise takes place and
how steep it is, respectively. As can be seen in Fig. 1
both decrease with time.
FIG. 1: Density (upper panel) and electron fraction (bottom
panel) profiles for the SN progenitor and at different instants
after the core bounce, from Ref. [60]. The regions where the
H (yellow) and the L (cyan) resonance take place are also
indicated, as well as the NSI-induced I (gray) resonance for
the parameters εee = 0, εττ . 0.07 and |εµτ | . 0.05
IV. THE TWO REGIMES
In order to study the neutrino propagation through
the SN envelope we will split the problem into two differ-
ent regions: the inner envelope, defined by the condition
VCC ≫ ∆m2atm/(2E) with ∆m2atm ≡ m23 − m22, and the
outer one, where ∆m2atm/(2E) & VCC . From the upper
panel of Fig. 1 one can see how the boundary roughly
varies between r ≈ 108 cm and 109 cm, depending on the
time considered. This way one can fully characterize all
resonances that can take place in the propagation of su-
pernova neutrinos, both the outer resonant conversions
related to neutrino masses and indicated as the upper
bands in Fig. 1, and the inner resonances that follow
from the presence of non-standard neutrino interactions,
indicated by the band at the bottom of the same figure.
Here we pay special attention to the use of realistic mat-
ter and chemical supernova profiles and three-neutrino
flavors thus generalising previous studies.
A. Neutrino Evolution in the Inner Regions
Let us first write the Hamiltonian in the inner layers,
where Hint ≫ Hkin. In this case the Hamiltonian can be
written as
H ≈ Hint = V0ρ(2− Ye)
+ εee εeµ εeτ
εeµ 0 εµτ
εeτ εµτ εττ
When the value of the εαβ is of the same order as the
electron fraction Ye internal resonances can arise [13].
Taking into account the current constraints on the ε’s
discussed in Sec. II one sees that small values of Ye are
required [46, 47]. As a result, these can only take place
in the most deleptonised inner layers, close to the neu-
trinosphere, where the kinetic terms of the Hamiltonian
are negligible.
Given the large number of free parameters εαβ in-
volved we consider one particular case where |εeµ| and
|εeτ | are small enough to neglect a possible initial mixing
between νe and νµ or ντ . Barring fine tuning, this basi-
cally amounts to |εeµ|, |εeτ | ≪ 10−2. According to the
discussion of Sec. II εeµ automatically satisfies the condi-
tion, whereas one expects that the window |εeτ | & 10−2
will eventually be probed in future experiments.
Since the initial fluxes of νµ and ντ are expected to be
basically identical, it is convenient to redefine the weak
basis by performing a rotation in the µ− τ sector:
= U(ϑ′23)
1 0 0
0 c23′ s23′
0 −s23′ c23′
where c23′ and s23′ stand for cos(ϑ
23) and sin(ϑ
23), re-
spectively. The angle ϑ′23 can be written as
tan(2ϑ′23) ≈
. (12)
The Hamiltonian becomes in the new basis
H ′αβ = U
†(ϑ′23)HαβU(ϑ
23) (13)
= V0ρ(2− Ye)
+ εee ε
ε′eµ ε
ε′eτ 0 ε
,(14)
where
ε′eµ = εeµc23′ − εeτs23′ (15)
ε′eτ = εeµs23′ + εeτ c23′ (16)
ε′µµ = (εττ −
ε2ττ + 4ε
µτ )/2 (17)
ε′ττ = (εττ +
ε2ττ + 4ε
µτ )/2 . (18)
With our initial assumptions on εeα one notices that
the new basis ν′α basically diagonalizes the Hamiltonian,
and therefore coincides roughly with the matter eigen-
state basis. A novel resonance can arise if the condition
H ′ee = H
ττ is satisfied, we call this I-resonance, I stand-
ing for “internal” [96]. The corresponding resonance con-
dition can be written as
Y Ie =
1 + εI
, (19)
where εI is defined as ε′ττ − εee. In Fig. 2 we represent
the range of εee and ε
ττ leading to the I-resonance for
an electron fraction profile between different Y mine ’s and
Y maxe = 0.5. It is important to notice that the value
of Y mine depends on time. Right before the collapse the
minimum value of the electron fraction is around 0.4.
Hence the window of NSI parameters that would lead
to a resonance would be relatively narrow, as indicated
by the shaded (yellow) band in Fig. 2. As time goes on
Y mine decreases to values of the order of a few %, and
as a result the region of parameters giving rise to the I-
resonance significantly widens. For example, in the range
|εee| ≤ 10−3 possibly accessible to future experiments one
sees that the I-resonance can take place for values of ε′ττ
of the order of O(10−2). This indicates that the potential
sensitivity on NSI parameters that can be achieved in su-
pernova studies is better than that of the current limits.
FIG. 2: Contours of Y Ie as function of εee and ε
ττ accord-
ing to Eq. (19) for different values of Ye. The region in yel-
low represents the region of parameters that gives rise to I-
resonance before the collapse. The arrows indicate how this
region widens with time.
As seen in Fig. 1 in order to fulfill the I-resonance con-
dition for such small values of the NSI parameters the
values of Ye must indeed lie, as already stated, in the
inner layers.
Several comments are in order: First, in contrast to
the standard H and L-resonances, related to the kinetic
term, the density itself does not explicitly enter into the
resonance condition, provided that the density is high
enough to neglect the kinetic terms. Analogously the en-
ergy plays no role in the resonance condition, which is
determined only by the electron fraction Ye. Moreover,
in contrast to the standard resonances, the I-resonance
occurs for both neutrinos and antineutrinos simultane-
ously [13]. Finally, as indicated in Fig. 3 the νe’s (ν̄e) are
not created as the heaviest (lightest) state but as the in-
termediate state, therefore the flavor composition of the
neutrinos arriving at the H-resonance is exactly the op-
posite of the case without NSI. As we show in Sec. V, this
fact can lead to important observational consequences.
In order to calculate the hopping probability between
matter eigenstates at the I-resonance we use the Landau-
ν m2 ν
FIG. 3: Level-crossing schemes, first panel is for the case of
normal hierarchy (oscillations only), the second includes the
NSI effect. The two lower panels correspond to the inverse hi-
erarchy, oscillations only and oscillations + NSI, respectively.
Zener approximation for two flavors
P ILZ ≈ e−
γI , (20)
where γI stands for the adiabaticity parameter, which
can be generally written as
Em2 − Em1
, (21)
where ϑ̇m ≡ dϑm/dr. If one applies this for-
mula to the e − τ ′ box of Eq. (14) assuming that
tan 2ϑmI = 2H
eτ/(H
ττ − Hee) and Em2 − Em1 =
(H ′ττ −Hee)2 + 4H ′eτ
one gets
4H ′2eτ
(Ḣ ′ττ − Ḣee)
16V0ρε
(1 + εI)3Ẏe
≈ 4× 109rs,5ρ11ε′2eτf(εI) , (22)
where the parametrization of the Ye profile has been de-
fined as in Eq. (9) with b = 0.16. The density ρ11 rep-
resents the density in units of 1011 g/cm3, rs,5 stands
for rs in units of 10
5 cm, and f(εI) is a function whose
value is of the order O(1) in the range of parameters we
are interested in. Taking all these factors into account
it follows that the internal resonance will be adiabatic
provided that ε′eτ & 10
−5, well below the current limits,
in full numerical agreement with, e. g., Ref. [47].
In Fig. 4 we show the resonance condition as well as
the adiabaticity in terms of εττ and εeτ assuming the
other εαβ = 0. In order to illustrate the dependence on
time we consider profiles inspired in the numerical profiles
of Fig. 1 at t = 2 s (upper panel) and 15.7 s (bottom
panel). For definiteness we take Y mine as the electron
fraction at which the density has value of 5× 1011g/cm3.
For comparison with Fig. 2 we have assumed Y mine =
10−2 in the case of 15.7 s. We observe how the border
of adiabaticity depends on εττ through the value of the
density at rI which in turn depends on time.
Before moving to the discussion of the outer resonances
a comment is in order, namely, how does the formalism
change for other non-standard interaction models. First
note that the whole treatment presented above also ap-
plies to the case of NSI on up-type quarks, except that
the position of the internal resonance shifts with respect
to the down-quark case. Indeed, in this case the NSI
potential
(V unsi)αβ = ε
αβV0ρ(1 + Ye) , (23)
0.001 0.01 0.1 1
0.001 0.01 0.1 1
FIG. 4: Contours of constant jump probability at the I-
resonance in terms of εττ and εeτ for two profiles correspond-
ing to Fig. 1 at 2 s with a = 0.235 and b = 0.175 (upper panel)
and 15.7 s with a = 0.26 and b = 0.195 (bottom panel). For
simplicity the other ε’s have been set to zero.
would induce a similar internal resonance for the condi-
tion Ye = ε
I/(1− εI).
In contrast, for the case of NSI with electrons, the
NSI potential is proportional to the electron fraction, and
therefore no internal resonance would appear.
B. Neutrino Evolution in the Outer Regions
In the outer layers of the SN envelope neutrinos can un-
dergo important flavor transitions at those points where
the matter induced potential equals the kinetic terms.
In absence of NSI this condition can be expressed as
VCC ≈ ∆m2/(2E). Neutrino oscillation experiments in-
dicate two mass scales, ∆m2atm and ∆m
⊙ ≡ m22−m21 [5],
hence two different resonance layers arise, the so-called
H-resonance and the L-resonance, respectively.
The presence of NSI with values of |εαβ| . 10−2 modi-
fies the properties of the H and L transitions [48, 49, 50].
In particular one finds that the effects of the NSI can be
described as in the standard case by embedding the ε’s
into effective mixing angles [50]. An analogous “confu-
sion” between sinϑ13 and the corresponding NSI param-
eter εeτ has been pointed out in the context of long-
baseline neutrino oscillations in Refs. [53, 54].
In this section we perform a more general and com-
plementary study for slightly higher values of the NSI
parameters: |εαβ | & few 10−2, still allowed by current
limits, and for which the I-resonance could occur.
The phenomenological assumption of hierarchical
squared mass differences, |∆m2atm| ≫ ∆m2⊙, allows, for
not too large ε’s, a factorization of the 3ν dynamics
into two 2ν subsystems roughly decoupled for the H
and L transitions [65]. To isolate the dynamics of the
H transition, one usually rotates the neutrino flavor ba-
sis by U †(ϑ23), and extracts the submatrix with indices
(1,3) [48, 50]. Whereas this method works perfectly for
small values of εαβ it can be dangerous for values above
10−2. In order to analyze how much our case deviates
from the simplest approximation we have performed a
rotation with the angle ϑ′′23 ≡ ϑ23 − α instead of just
ϑ23. By requiring that the new rotation diagonalizes the
submatrix (2,3) at the H-resonance layer one obtains the
following expression for the correction angle α
tan(2α) =
∆⊙s212s13 + V
ττ s223 − 2V NSIµτ c223
(∆atm +
∆⊙c212(−3 + c213)
+V NSIττ c223 + 2V
µτ s223
, (24)
where ∆atm ≡ ∆m2atm/(2E) and ∆⊙ ≡ ∆m2⊙/(2E). In
our notation sij and s2ij represent sinϑij and sin(2ϑij),
respectively. The parameters cij and c2ij are analogously
defined. In the absence of NSI α is just a small correction
to ϑ23 [97],
tan(2α) ≈ ∆⊙s212s13/∆atmc213 . O(10−3) . (25)
In order to calculate α we need to know the H-
resonance point. To calculate it one can proceed as in
the case without NSI, namely, make the ϑ′′23 rotation and
analyze the submatrix (1, 3). The new Hamiltonian H ′′αβ
has now the form
H ′′ee = V0ρ[Ye + εee(2− Ye)] + ∆atms213
+∆⊙(c
12 + s
13) ,
H ′′ττ = V0ρ(2− Ye)ε′′ττ +∆atmc213c2α
c213c
α + (sαc12 + cαs12s13)
H ′′eτ = V0ρ(2− Ye)ε′′eτ +
∆atms213cα
∆⊙(−c13sαs212 + c212cαs213) . (26)
We have defined ε′′ττ = εττc
23−α + εµτs223−α, and ε
εeτ c23−α+εeµs23−α, where s23−α ≡ sin(ϑ23−α), c23−α ≡
cos(ϑ23 − α), and s223−α ≡ sin(2ϑ23 − 2α), c223−α ≡
cos(2ϑ23 − 2α). The resonance condition for the H tran-
sition, H ′′ee = H
ττ can be then written as
H [Y He + (εee − ε′′ττ )(2− Y He )] = ∆atm(c213c2α − s213)
+∆⊙[c
13 − c2αs213)− s2αs212 + 12s2αs212s13] .(27)
It can be easily checked how in the limit of εαβ → 0 one
recovers the standard resonance condition,
HY He ≈ ∆atmc213 . (28)
In the region where the H-resonance occurs Y He ≈ 0.5.
Taking into account Eqs. (24) and (27) one can already
estimate how the value of α changes with the NSI param-
eters. In Fig. 5 we show the dependence of α on the εττ
after fixing the value of the other NSI parameters. One
can see how for εττ & 10
−2 the approximation of neglect-
ing α significantly worsens. Assuming ϑ23 = π/4 and a
fixed value of εµτ one can easily see that εττ basically
affects the numerator in Eq. (24). Therefore one expects
a rise of α as the value of εττ increases, as seen in Fig. 5.
The dependence of α on εµτ is correlated to the rela-
tive sign of the mass hierarchy and εµτ . For instance,
for normal mass hierarchy and positive values of εµτ the
dependence is inverse, namely, higher values of εµτ lead
to a suppression of α. Apart from this general behav-
ior, α also depends on the diagonal term εee as seen in
Fig. 5. This effect occurs by shifting the resonance point
through the resonance condition in Eq. (27).
One can now calculate the jump probability be-
tween matter eigenstates in analogy to the I-resonance
by means of the Landau-Zener approximation, see
Eqs. (20), (21), and 22,
PHLZ ≈ e−
γH , (29)
where γH represents the adiabaticity parameter at the
FIG. 5: Angle α as function of εττ for different values of εee
and εµτ , in the case of neutrinos of energy 10 MeV, with
normal mass hierarchy, and s213 = 10
−5. The other NSI pa-
rameters take the following values: εeµ = 0 and εeτ = 10
H-resonance, which can be written as
4H ′′2eτ
(Ḣ ′′ττ − Ḣ ′′ee)
, (30)
where the expressions for H ′′αβ are given in Eqs (26).
Let us first consider the case |εαβ| . 10−2. In this case
α ≈ 0 and one can rewrite the adiabaticity parameter as
∆atm sin
cos(2ϑ
)|d ln V/dr|rH
, (31)
where
= ϑ13 + ε
eτ (2− Ye)/Ye (32)
in agreement with Ref. [50]. For slightly larger ε’s there
can be significant differences. In Fig. 6 we show PHLZ in
the εeτ -εττ plane for antineutrinos with energy 10 MeV
in the case of inverse mass hierarchy, using Eq. (29) with
(upper panel) and without (bottom panel) the α cor-
rection. The values of ϑ13 and εeτ have been chosen so
that the jump probability lies in the transition regime be-
tween adiabatic and strongly non adiabatic. In the limit
of small εττ , α becomes negligible and therefore both re-
sults coincide. From Eq. (31) one sees how as the value of
εeτ increases γH gets larger and therefore the transition
becomes more and more adiabatic. For negative values
of εeτ there can be a cancellation between εeτ and ϑ13,
and as a result the transition becomes non-adiabatic.
An additional consequence of Eq. (32) is that a degen-
eracy between εeτ and ϑ13 arises. This is seen in Fig. 7,
which gives the contours of PH
in terms of εeτ and ϑ13
for εττ = 10
−4. One sees clearly that the same Landau-
Zener hopping probability is obtained for different com-
binations of εeτ and ϑ13. This leads to an intrinsic “con-
fusion” between the mixing angle and the corresponding
NSI parameter, which can not be disentangled only in
the context of SN neutrinos, as noted in Ref. [50].
We now turn to the case of |εττ | ≥ 10−2. As |εττ |
increases the role of α becomes relevant. Whereas in the
bottom panel PHLZ remains basically independent of εττ ,
one can see how in the upper panel PHLZ becomes strongly
sensitive to εττ for |εττ | ≥ 10−2.
One sees that for positive values of εττ it tends to adi-
abaticity whereas for negative values to non-adiabaticity.
This follows from the dependence of H ′′eτ on α, essen-
tially through the term −∆⊙c13sαs212, see Eq. (26). For
|εττ | ≥ 10−2 one sees that sinα starts being important,
and as a result this term eventually becomes of the same
order as the others in H ′′eτ . At this point the sign of
εττ , and so the sign of sinα, is crucial since it may con-
tribute to the enhancement or reduction of H ′′eτ . This
directly translates into a trend towards adiabaticity or
non-adiabaticity, seen in Fig. 6. Thus, for the range of
εττ relevant for the NSI-induced internal resonance the
adiabaticity of the outer H resonance can be affected in
a non-trivial way.
Turning to the case of the L transition a similar expres-
sion can be obtained by rotating the original Hamiltonian
by U(ϑ13)
†U(ϑ23)
† [48, 50]. However, in contrast to the
case of the H-resonance, where the mixing angle ϑ13 is
still unknown, in the case of the L transition the angle
ϑ12 has been shown by solar and reactor neutrino exper-
iments to be large [5]. As a result, for the mass scale ∆⊙
this transition will always be adiabatic irrespective of the
values of εαβ, and will affect only neutrinos.
FIG. 6: Landau-Zener jump probability isocontours at the H-
resonance in terms of εeτ and εττ for 10 MeV antineutrinos
in the case of inverted mass hierarchy. Upper panel: α given
by Eq. (24). Bottom panel: α set to zero. The remaining
parameters take the following values: sin2 ϑ13 = 10
, εeτ =
10−3, εee = εeµ = 0. See text.
V. OBSERVABLES AND SENSITIVITY
As mentioned in the introduction one of the major mo-
tivations to study NSI using the neutrinos emitted in a
SN is the enhancement of the NSI effects on the neutrino
propagation through the SN envelope due to the specific
extreme matter conditions that characterize it. In this
section we analyze how these effects translate into ob-
servable effects in the case of a future galactic SN.
Schematically, the neutrino emission by a SN can be di-
vided into four stages: Infall phase, neutronization burst,
accretion, and Kelvin-Helmholtz cooling phase. During
the infall phase and neutronization burst only νe’s are
emitted, while the bulk of neutrino emission is released
in all flavors in the last two phases. Whereas the neutrino
emission characteristics of the two initial stages are basi-
FIG. 7: Landau-Zener jump probability isocontours at the
H-resonance in terms of εeτ and ϑ13 for εττ = 10
−4. An-
tineutrinos with energy 10 MeV and inverted mass hierarchy
has been assumed.
cally independent of the features of the progenitor, such
as the core mass or equation of state (EoS), the details
of the neutrino spectra and luminosity during the ac-
cretion and cooling phases may significantly change for
different progenitor models. As a result, a straightfor-
ward extraction of oscillation parameters from the bulk
of the SN neutrino signal seems hopeless. Only features
in the detected neutrino spectra which are independent
of unknown SN parameters should be used in such an
analysis [66].
The question then arises as to how can one obtain in-
formation about the NSI parameters. Taking into ac-
count that the main effect of NSI is to generate new in-
ternal neutrino flavor transitions, one possibility is to in-
voke theoretical arguments that involve different aspects
of the SN internal dynamics.
In Ref. [47] it was argued that such an internal flavor
conversion during the first second after the core bounce
might play a positive role in the so-called SN shock re-
heating problem. It is observed in numerical simula-
tions [67, 68, 69, 70] that as the shock wave propa-
gates it loses energy until it gets stalled at a few hun-
dred km. It is currently believed that after neutrinos
escape the SN core they can to some extent deposit en-
ergy right behind and help the shock wave continue out-
wards. On the other hand it is also believed that due to
the composition in matter of the protoneutronstar (PNS)
the mean energies of the different neutrino spectra obey
〈Eνe〉 < 〈Eν̄e〉 < 〈Eνµ,ντ 〉. This means that a reso-
nant conversion between νe(ν̄e) and νµ,τ (ν̄µ,τ ) between
the neutrinosphere and the position of the stalled shock
wave would make the νe(ν̄e) spectra harder, and there-
fore the energy deposition would be larger, giving rise to
a shock wave regeneration effect.
Another argument used in the literature was the pos-
sibility that the r−process nucleosynthesis, responsible
for synthesizing about half of the heavy elements with
mass number A > 70 in nature, could occur in the region
above the neutrinosphere in SNe [71, 72]. A necessary
condition is Ye < 0.5 in the nucleosynthesis region. The
value of the electron fraction depends on the neutrino
absorption rates, which are determined in turn by the
νe(ν̄e) luminosities and energy distribution. These can
be altered by flavor conversion in the inner layers due to
the presence of NSI. Therefore by requiring the electron
fraction be below 0.5 one can get information about the
values of the NSI parameters.
While it is commonly accepted that neutrinos will play
a crucial role in both the shock wave re-heating as well
as the r−process nucleosynthesis, there are still other as-
trophysical factors that can affect both. While the issue
remains under debate we prefer to stick to arguments
directly related to physical observables in a large water
Cherenkov detector. There are several possibilities.
(A) the modulations in the ν̄e spectra due to the pas-
sage of shock waves through the supernova [58, 59,
(B) the modulation in the ν̄e spectra due to the time
dependence of the electron fraction, induced by the
I-resonance
(C) the modulations in the ν̄e spectra due to the Earth
matter [73, 74, 75, 76]
(D) detectability of the neutronization νe burst [77, 78]
Three of these observables, 1, 3 and 4 have already been
considered in the literature in the context of neutrino
oscillations. Here we discuss the potential of the above
promising observables in providing information about the
Scheme Hierarchy sin2 ϑ13 NSI Psurv P̄surv
A normal & 10−4 No 0 cos2 ϑ12
B inverted & 10−4 No sin2 ϑ12 0
C any . 10−6 No sin2 ϑ12 cos
AI normal & 10−4 Yes sin2 ϑ12 sin
BI inverted & 10−4 Yes cos2 ϑ12 cos
CIa normal . 10−6 Yes 0 sin2 ϑ12
CIb inverted . 10−6 Yes cos2 ϑ12 0
TABLE I: Definition of the neutrino schemes considered in
terms of the hierarchy, the value of ϑ13, and the presence
of NSI, as described in the text. The values of the survival
probabilities for νe (Psurv) and ν̄e (P̄surv) for each case are
also indicated.
NSI parameters. It is important to pay attention to the
possible ocurrence of the internal I-resonance and to its
effect in the external H and L-resonances. The first can
induce a genuinely new observable effect, item 2 above.
Here we concentrate on neutral current-type non-
standard interactions, hence there will be not effect in the
main reaction in water Cherenkov and scintillator detec-
tors, namely the inverse beta decay, ν̄e+p → e++n [98].
For definiteness we take NSI with d (down) quarks, in
which case the NSI effects will be confined to the neu-
trino evolution inside the SN and the Earth, through the
vector component of the interaction.
From all possible combinations of NSI parameters we
will concentrate on those for which the internal I tran-
sition does take place, namely |εI | & 10−2, see Fig. 2.
Concerning the FC NSI parameters we will consider |ε′eτ |
between few × 10−5 and 10−2, range in which the I-
resonance is adiabatic, see Fig. 4. In the following discus-
sion we will focus on the extreme cases defined in Table I.
One of the motivations for considering these cases is the
fact that the resonances involved become either adiabatic
or strongly non adiabatic, and hence the survival prob-
abilities in the absence of Earth effects or shock wave
passage, become energy independent. This assumption
simplifies the task of relating the observables with the
neutrino schemes.
A. Shock wave propagation
During approximately the first two seconds after the
core bounce, the neutrino survival probabilities are con-
stant in time and in energy for all cases mentioned in Ta-
ble I. Only the Earth effects could introduce an energy
dependence. However, at t ≈ 2 s the H-resonance layer is
reached by the outgoing shock wave, see Fig. 1. The way
the shock wave passage affects the neutrino propagation
strongly depends on the neutrino mixing scenario. In the
absence of NSI cases A and C will not show any evidence
of shock wave propagation in the observed ν̄e spectrum,
either because there is no resonance in the antineutrino
channel as in scenario A, or because the H-resonance is
always strongly non-adiabatic as in scenario C. How-
ever, in scenario B, the sudden change in density breaks
the adiabaticity of the resonance, leading to a time and
energy dependence of the electron antineutrino survival
probability P̄surv(E, t). In the upper panel of Fig. 8 we
show P̄surv(E, t) in the particular case that two shock
waves are present, one forward and a reverse one [60].
The presence of the shocks results in the appearance of
bumps in survival probability at those energies for which
the resonance region is passed by the shock waves. All
these structures move in time towards higher energies, as
the shock waves reach regions with lower density, leading
to observable consequences in the ν̄e spectrum.
We now turn to the case where NSI are present, which
opens the possibility of internal resonances. When such
I-resonance is adiabatic the situation will be similar to
the case without NSI. For normal mass hierarchy, AI and
CIa, ν̄e will not feel the H-resonance and therefore the
adiabaticity-breaking effect will not basically alter their
propagation. In contrast, for inverted mass hierarchy
and large ϑ13, case BI, the H-resonance occurs in the
antineutrino channel and therefore ν̄e will feel the shock
wave passage. However, in contrast to case B now ν̄e will
reach the H-resonance in a different matter eigenstate:
ν̄m1 instead of ν̄
3 , see Fig. 3. That means that before
the shock wave reaches the H-resonance the ν̄e survival
probability will be P̄surv ≈ cos2 ϑ12 ≈ 0.7. Once the
adiabaticity of the H-resonance is broken by the shock
wave then ν̄e will partly leave as ν̄
3 and therefore the
survival probability will decrease. As a consequence one
expects a pattern in time and energy for the survival
FIG. 8: Survival probability P̄surv(E, t) for ν̄e as function
of energy at different times averaged in energies with the en-
ergy resolution of Super-Kamiokande; for the profile shown in
Fig. 1. Upper panel: case B is assumed for sin2 ϑ13 = 10
Bottom panel: case BI , with εττ = 0.07, εeτ = 10
−4 and the
rest of NSI parameters put to zero.
probability in the case BI to be roughly opposite than in
the case B, see bottom panel of Fig. 8. The position of
the peaks and dips en each panel do not exactly coincide
as the value of εττ roughly shifts the position of the H-
resonance.
In the left panels of Fig. 9 we represent in light-shaded
(yellow) the range of εeτ and εττ for which this opposite
shock wave imprint would be observable. In the upper
panels we have assumed a minimum value of the electron
fraction of 0.06, based on the numerical profiles at t =
2 s of Fig. 1. In the bottom panels Y mine is set to 0.01,
inspired in the profiles at t = 15.7 s. It can be seen how as
time goes on the range of εττ ’s for which the I-resonance
takes place widens towards to smaller and smaller values.
This is a direct consequence of the steady deleptonization
of the inner layers.
For smaller ϑ13, case CIb, the situation is different.
Except for relatively large εeτ values theH-resonance will
be strongly non-adiabatic, as in case C. Therefore the
passage of the shock waves will not significantly change
the ν̄e survival probability and will not lead to any ob-
servable effect. In the right panels of Fig. 9 we show
the same as in the left panels but for sin2 ϑ13 = 10
Whereas for large values of ϑ13, left panels, the H-
resonance is always adiabatic and one has only to ensure
the adiabaticity of the I-resonance, for smaller values of
ϑ13 the adiabaticity of the H-resonance strongly depends
on the values of εeτ and εττ , as discussed in Sec. IVB.
This can be seen as a significant reduction of the yel-
low area. Only large values of either εeτ or εττ would
still allow for a clear identification of the opposite shock
wave effects. In dark-shaded (cyan) we show the region
of parameters for which PH lies in the transition region
between adiabatic and strongly non-adiabatic, and there-
fore could still lead to some effect.
A useful observable to detect effects of the shock prop-
agation is the average of the measured positron energies,
〈Ee〉, produced in inverse beta decays. In Fig. 10, we
show 〈Ee〉 together with the one sigma errors expected for
a Megaton water Cherenkov detector and a SN at 10 kpc
distance, with a time binning of 0.5 s, for different neu-
trino schemes: caseB and caseBI with different values of
εττ . For the neutrino fluxes we assumed the parametriza-
tion given by Refs. [79, 80] with 〈E0(ν̄e)〉 = 15 MeV and
〈E0(ν̄µ,τ )〉 = 18 MeV and the following ratio of the total
neutrino fluxes Φ0(ν̄e)/Φ0(ν̄µ,τ ) = 0.8 [99].
One can see how the features of the average positron
energy are a direct consequence of the shape of the sur-
vival probability, where dips have to be translated into
bumps and vice-versa.
Thus, it is important to stress that whereas in case
B one expects the presence of one or two dips (depend-
ing on the structure of the shock wave, see Ref [60]), or
nothing in the other cases, one or two bumps are ex-
pected in case BI, as seen in the upper left panel of
Fig. 10. As discussed in Ref. [60] the details of the
dips/bump will depend on the exact shape of the neu-
trino fluxes, but as long as general reasonable assump-
tions like 〈Eν̄e〉 . 〈Eν̄µ,τ 〉 are considered the dips/bumps
should be observed.
B. Time variation of Ye
We have just seen how the distorsion of the density
profile due to the shock wave passage through the outer
FIG. 9: Range of εττ and εeτ for which the effect of the shock
wave will be observed. In the upper panels a minimum value
of Y mine = 0.06 based on the numerical profiles at t = 2 s
has been assumed, see Fig. 1. In the lower panels we have
considered a case with Y mine = 0.01 inspired in the profile
at t = 15.7 s. The value of sin2 ϑ13 has been assumed to
be 10−2 and 10−7 in the left and right panels, respectively.
We have also superimposed isocontours of constant hopping
probability 0.1 (blue) and 0.9 (red) in the I (solid lines) and
H (dashed lines) resonances for inverted mass hierarchy and
E = 10 MeV and antineutrinos. The area in yellow represents
the parameter space where both resonances will be adiabatic.
In the cyan area the I-resonance is assumed to be adiabatic
whereas H lies in the transition region.
SN envelope can induce a time-dependent modulation in
the ν̄e spectrum in cases B and BI. However the time
dependence of the electron fraction Ye can also reveal the
presence of NSI leaving a clear imprint in the observed
ν̄e spectrum, as we now explain.
As discussed in Sec. IVA the region of NSI parame-
ters leading to I-resonance is basically determined by the
minimum and maximum values of the electron fraction,
Y mine and Y
e . The crucial point is that as the delep-
tonization of the proto-neutron star goes on, the value of
Y mine steadily decreases with time. As a result, the range
of NSI strengths for which the I-resonance takes place
FIG. 10: The average energy of ν̄p → ne+ events binned in
time for case B (dashed blue) and BI (solid red). In each
panel different values of εττ have been assumed. The error
bars represent 1 σ errors in any bin. εeτ = 10
increases with time, as can be seen in Fig. 2.
Let us first discuss the observational consequences of
the time dependence of the electron fraction in case BI.
If εττ (ε
I in general) is large enough the I-resonance will
take place right after the core bounce. In this case, as
seen in the upper left panel of Fig. 10 the two bumps we
have just discussed in Sec. VA would be clearly observed.
However for smaller NSI parameter values it could hap-
pen that the I-resonance occurs only after several sec-
onds. In particular for the specific Ye profile considered
we show how this delay could be of roughly 2, 4 or 9 sec
for values of εττ of 0.025, 0.02 or 0.015, respectively, see
last three panels Fig. 10. As can be inferred from the
figure this delay effect can lead to misidentification of
the pure NSI effect. So, for instance, in the upper right
panel, one sees how the two bumps might also be inter-
preted as two dips, given the astrophysical uncertainties.
This subtle degeneracy can only be solved by extra in-
formation on, for example, the time dependence of the
spectra or the velocity of the shock wave. Given the su-
pernova model, however, the time structure of the signal
could eventually not only point out the presence of NSI
but even potentially indicate a range of NSI parameters.
Let us now turn to the normal mass hierarchy sce-
nario (cases AI and CIa). In analogy to the BI case,
if εI is relatively large the onset of the I-resonance will
take place early on. As can be inferred from Fig. 3 that
implies that ν̄e will escape the SN as ν̄2. For smaller
values, though, it may happen that the I-resonance be-
comes effective only after a few seconds. This means
that during the first seconds of the neutrino signal ν̄e
would leave the star as ν̄1 (cases A and C). Then,
after some point, the electron fraction would be low
enough to switch on the I-resonance, and consequently
ν̄e would enter the Earth as ν̄2. This would result in
a transition in the electron antineutrino survival prob-
ability from P̄surv ≈ cos2 ϑ12 = 0.7 to sin2 ϑ12 = 0.3.
Given the expected hierarchy in the average neutrino en-
ergies 〈Eν̄e〉 . 〈Eν̄µ,τ 〉, it follows that the change in Ye
would lead to a hardening of the observed positron spec-
trum. The effect is quantified in Fig. 11 for different
values of εττ . The figure shows the average energy of
the ν̄p → ne+ events for the case of a Megaton water
Cherenkov detector exactly as in Fig. 10, but for scenar-
ios AI and CIa. One can see how for εττ = 0.07 the I-
resonance condition is always fulfilled and therefore there
is no time dependence. However for smaller values one
can see a rise at a certain moment which depends on the
magnitude of εττ . A similar effect would occur in case
C. Earth matter effects
Before the shock wave reaches the H-resonance layer
the dependence of the neutrino survival probability in the
cases we are considering, on the neutrino energy E is very
weak. However, if neutrinos cross the Earth before reach-
ing the detector, the conversion probabilities may become
energy-dependent, inducing modulations in the neutrino
energy spectrum. These modulations may be observed
in the form of local peaks and valleys in the spectrum of
the event rate σFDν̄e plotted as a function of 1/E. These
modulations arise in the antineutrino channel only when
ν̄e leave the SN as ν̄1 or ν̄2. In the absence of NSI this
happens in cases A and C, where ν̄e leave the star as ν̄1.
FIG. 11: The average energy of ν̄p → ne+ events binned in
time for case AI and CIa and different values of εττ . The
error bars represent 1 σ errors in any bin. εeτ = 10
In the presence of NSI ν̄e will arrive at the Earth as ν̄1
in cases BI, and as ν̄2 in case AI and CIa. Therefore
its observation would exclude cases B and CIb. This
distortion in the spectra could be measured by compar-
ing the neutrino signal at two or more different detectors
such that the neutrinos travel different distances through
the Earth before reaching them [73, 74]. However these
Earth matter effects can be also identified in a single de-
tector [75, 76].
By analyzing the power spectrum of the detected neu-
trino events one can identify the presence of peaks located
at the frequencies characterizing the modulation. These
do not dependend on the primary neutrino spectra, and
can be determined to a good accuracy from the knowl-
edge of the solar oscillation parameters, the Earth matter
density, and the position of the SN in the sky [76]. The
latter can be determined with sufficient precision even if
the SN is optically obscured using the pointing capability
of water Cherenkov neutrino detectors [81].
This method turns out to be powerful in detecting the
modulations in the spectra due to Earth matter effects,
and thus in ruling out cases B and CIb. However, the po-
sition of the peaks does not depend on how ν̄e enters the
Earth, as ν̄1 or ν̄2. Hence it is not useful to discriminate
case AI and CIa from the cases A, C, and BI.
The time dependence of Ye, however, can transform
case B into BI, and C with inverse hierarchy into CIb,
leading respectively to an appearance and disappearance
of these Earth matter effects. In case BI the presence of
the shock wave modulation can spoil a clear identification
of the Earth matter effects. Nevertheless, the disappear-
ance of the Earth matter effects in the transition from
case C to CIb allows us to pin down case CIb.
D. Neutronization burst
The prompt neutronization burst takes place during
the first ∼ 25 ms after the core bounce with a typical
full width half maximum of 5–7ms and a peak luminos-
ity of 3.3–3.5×1053 erg s−1. The striking similarity of the
neutrino emission characteristics despite the variability
in the properties of the pre-collapse cores is caused by
a regulation mechanism between electron number frac-
tion and target abundances for electron capture. This
effectively establishes similar electron fractions in the in-
ner core during collapse, leading to a convergence of the
structure of the central part of the collapsing cores, with
only small differences in the evolution of different pro-
genitors until shock breakout [77, 78].
Taking into account that the SN will be likely to be
obscured by dust and a good estimation of the distance
will not be possible, the time structure of the detected
neutrino signal should be used as signature for the neu-
tronization burst. In Ref. [78] it was shown that such a
time structure can be in principle cleanly seen in the
case of a Megaton water Cherenkov detector. It was
also shown how the time evolution of the signal depends
strongly on the neutrino mixing scheme. In the absence
of NSI the νe peak could be observed provided that the
νe survival probability Pνeνe is not zero. As can be seen
in Table I this happens for cases B and C. However
for case A (normal mass hierarchy and “large” ϑ13), νe
leaves the SN as ν3. This leads to a survival probability
Pνeνe ≈ sin2 ϑ13 . 10−1, and therefore the peak remains
hidden.
Let us now consider the situation where NSI are
prensent. For normal mass hierarchy νe, which is born
as νm2 passes through three different resonances, I, H
and L. Whereas I and L will be adiabatic, the fate of
H will depend on the value of ϑ13. For “large” values,
case AI, the H-resonance will also be adiabatic. This
implies that νe’s will leave as ν2, the survival probability
will be Pνeνe ≈ sin2 ϑ12 ≈ 0.3, and therefore the peak
will be seen, as in cases B and C. If ϑ13 happens to
be very small, case CIa, then H will be strongly non-
adiabatic and therefore νe will leave the star as ν3. As a
consequence the neutronization peak will not be seen.
For inverse mass hierarchy, νe is born as ν
1 and tra-
verses adiabatically I and L. This implies that they will
leave the star as ν1 and therefore the peak will also be
observed. However now the survival probability will be
larger, Pνeνe ≈ cos2 ϑ12 ≈ 0.7. Thus for a given known
normalization, i.e. the distance to the SN, one expects a
larger number of events during the neutronization peak
in this case. In Fig. 12 we show the expected number
of events per time bin in a water Cherenkov detector in
the case of a SN exploding at 10 kpc, for two different
neutrino schemes, C and BI, and for different SN pro-
genitor masses. One can see how the difference due to
the larger survival probability is bigger than the typi-
cal error bars, associated to the lack of knowledge of the
progenitor mass.
Two comments are in order. The neutronization νe
burst takes place during the first milliseconds, before
strong deleptonization takes place. As a result, in con-
trast to other observables we have considered in this pa-
per, here the I-resonance will only occur for εI & 10−1.
On the other hand in the presence of additional NSI with
electrons this would significantly affect the ν − e cross
sections, and consequently the results presented here.
VI. SUMMARY
We have analyzed the possibility of observing clear sig-
natures of non-standard neutrino interactions from the
detection of neutrinos produced in a future galactic su-
pernova.
In Secs. III and IV we have re-considered effect of ν−d
non-standard interactions on the neutrino propagation
through the SN envelope within a three-neutrino frame-
work. In contrast to previous works we have analyzed
the neutrino evolution in both the more deleptonized in-
FIG. 12: Number of events from the elastic scattering on elec-
trons, per time bin in a Megaton water Cherenkov detector for
a SN at 10 kpc for cases C (dashed lines) and BI (solid lines).
Different progenitor masses have been assumed: 13 M⊙ (n13)
in red, 15 M⊙ (s15s7b2) in black, and 25 M⊙ (s25a28) in blue.
1-sigma errors are also shown for the 15 M⊙ case.
ner layers and the outer regions of the SN envelope. We
have also taken into account the time dependence of the
SN density and electron fraction profiles.
First we have found that the small values of the elec-
tron fraction typical of the former allows for internal NSI-
induced resonant conversions, in addition to the standard
MSW-H and MSW-L resonances of the outer envelope.
These new flavor conversions take place for a relatively
large range of NSI parameters, namely |εαα| between
10−2 − 10−1, and |εeτ | & few × 10−5, currently allowed
by experiment. For this range of strengths, in particu-
lar εττ , non-standard interactions can significantly affect
the adiabaticity of the H-resonance. On the other hand
the NSI-induced resonant conversions may also lead to
the modulation of the ν̄e spectra as a result of the time
dependence of the electron fraction.
In Sec. V we have studied the possibility of detecting
NSI effects in a Megaton water Cherenkov detector us-
ing the modulation effects in the ν̄e spectrum due to (i)
the passage of shock waves through the SN envelope, (ii)
the time dependence of the electron fraction and (iii) the
Earth matter effects; and, finally, through the possible
detectability of the neutronization νe burst. Note that
observable (ii) turns out to be complementary to the ob-
servation of the shock wave passage, (i), and offers the
possibility to probe NSI effects also for normal hierarchy
neutrino spectra.
In Table II we summarize the results obtained for dif-
ferent neutrino schemes. We have found that observable
(i) can clearly indicate the existence of NSI in the case
of inverse mass hierarchy and large ϑ13 (case BI). On
the other hand, observable (ii) allows for an identification
of NSI effects in the other cases, normal mass hierarchy
(cases AI and CIa) and inverse mass hierarchy and small
ϑ13 (case CIb). Therefore a positive signal of either ob-
servable (i) or (ii) would establish the existence of NSI. In
the latter case this would, however, leave a degeneracy
among cases AI, CIa, and CIb. Such degeneracy can
be broken with the help of observables (iii) and the ob-
servation of the neutronization νe burst. The detection
of Earth matter effects during the whole supernova neu-
trino signal would rule out case CIb since, as discussed in
Sec. VC, a disappearance of Earth matter effects would
take place due to a transition from C to CIb. Finally,
the (non) observation of the neutronization burst can be
used to distinguish between cases AI and CIa.
Similarly, other degeneracies in Table II may be lifted
by suitably combining different observables. For exam-
ple, a negative of observable (ii) could mean either neg-
ligible NSI strengths or (NU) NSI parameter values so
large that the internal resonance is always present. In
this case one could use the observation of the neutron-
ization burst in order to establish the presence of NSI for
the case of inverse mass hierarchy. In addition the ob-
servation of the shock wave imprint in the ν̄e spectrum
would provide additional information on ϑ13.
In conclusion, by suitably combining all observables
one may establish not only the presence of NSI, but also
the mass hierarchy and probe the magnitude of ϑ13.
Acknowledgments
The authors wish to thank H-Th. Janka, O. Miranda,
S. Pastor, Th. Schwetz, and M. Tórtola for fruitful discus-
sions. Work supported by the Spanish grant FPA2005-
Scheme Hierarchy sin2 ϑ13 NSI shock Ye Earth νe burst
A normal & 10−4 No No No Yes No
B inverted & 10−4 No Yes No No Yes
C any . 10−6 No No No Yes Yes
AI normal & 10−4 Yes No Yes Yes Yes
BI inverted & 10−4 Yes Yes⋆ No Yes Yes⋆
CIa normal . 10−6 Yes No Yes Yes No
CIb inverted . 10−6 Yes No Yes No Yes⋆
TABLE II: Expectations for the observables discussed in the
text: modulation of the ν̄e spectrum due to the shock wave
passage, the time variation of Ye, the Earth effect, and the
observation of the νe burst within various neutrino schemes.
Asterisks indicate that the effect differs from that expected
in the absence of NSI. See text.
01269 and European Network of Theoretical Astroparti-
cle Physics ILIAS/N6 under contract number RII3-CT-
2004-506222. A. E. has been supported by a FPU grant
from the Spanish Government. R. T. has been supported
by the Juan de la Cierva program from the Spanish Gov-
ernment and by an ERG from the European Commission.
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|
0704.0033 | Convergence of the discrete dipole approximation. I. Theoretical
analysis | Convergence of the Discrete Dipole Approximation
Convergence of the discrete dipole approximation.
I. Theoretical analysis.
Maxim A. Yurkin
Faculty of Science, Section Computational Science, of the University of Amsterdam,
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya 3, Novosibirsk 630090 Russia
[email protected]
Valeri P. Maltsev
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya 3, Novosibirsk 630090 Russia
Novosibirsk State University, Pirogova Str. 2, 630090, Novosibirsk, Russia
Alfons G. Hoekstra
Faculty of Science, Section Computational Science, of the University of Amsterdam,
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
[email protected]
Abstract
We performed a rigorous theoretical convergence analysis of the discrete dipole
approximation (DDA). We prove that errors in any measured quantity are bounded by a sum
of a linear and quadratic term in the size of a dipole d, when the latter is in the range of DDA
applicability. Moreover, the linear term is significantly smaller for cubically than for non-
cubically shaped scatterers. Therefore, for small d errors for cubically shaped particles are
much smaller than for non-cubically shaped. The relative importance of the linear term
decreases with increasing size, hence convergence of DDA for large enough scatterers is
quadratic in the common range of d. Extensive numerical simulations were carried out for a
wide range of d. Finally we discuss a number of new developments in DDA and their
consequences for convergence.
Keywords: discrete dipole approximation, non-spherical particle light scattering, convergence
analysis, accuracy
OCIS code: 290.5850, 260.2110, 000.4430
mailto:[email protected]
mailto:[email protected]
1. Introduction
The discrete dipole approximation (DDA) is a well-known method to solve the light
scattering problem for arbitrary shaped particles. Since its introduction by Purcell and
Pennypacker1 it has been improved constantly. The formulation of DDA summarized by
Draine and Flatau2 more than 10 years ago is still most widely used for many applications,3
partly due to the publicly available high-quality and user-friendly code DDSCAT.4 Although
modern improvements of DDA (as discussed in detail in Section 2.F) exist, they are still in the
research stage because they are not widely used in real applications.
DDA directly discretizes the volume of the scatterer and hence is applicable to arbitrary
shaped particles. However, the drawback of this discretization is the extreme computational
complexity of DDA, although it is significantly decreased by advanced numerical
techniques.2,5 That is why the usual application strategy for DDA is “single computation”,
where a discretization is chosen based on available computational resources and some
empirical estimates of the expected errors.3,4 These error estimates are based on a limited
number of benchmark calculations3 and hence are external to the light scattering problem
under investigation. Such error estimates have evident drawbacks, however no better
alternative is available. Some results of analytical analysis of errors in computational
electromagnetics are known, e.g. 6,7, however they typically consider the surface integral
equations. To the best of our knowledge, such analysis has not been done for volume integral
equations (such as DDA).
Usually errors in DDA are studied as a function of the size parameter of the scatterer x
(at a constant or few different total numbers of dipoles N), e.g. 2,8. Only a small number of
papers directly present errors versus discretization parameter (e.g. d – the size of a single
dipole).9-17 The range of d typically studied in those papers is limited to a 5 times difference
between minimum and maximum values, with the exception of two papers11,12 where it is 15
times. Those plots of errors versus discretization parameter are always used to illustrate the
performance of a new DDA formulation and compare it with others. No conclusions about the
convergence properties of DDA, as a function of d, have been made based on these plots. To
our knowledge, no theoretical analysis of DDA convergence has been performed, but only a
few limited empirical studies have appeared in the literature.
In this paper we perform a theoretical analysis of DDA convergence when refining the
discretization (Section 2). We derive rigorous theoretical bounds on the error in any measured
quantity for any scatterer. In Section 3 we present extensive numerical results of DDA
computations for 5 different scatterers using many different discretizations. These results are
discussed in Section 4 to support conclusions of the theoretical analysis. We formulate the
conclusions of the paper in Section 5. In a follow-up paper18 (which from now on we refer to
as Paper 2) the theoretical convergence results are used for an extrapolation technique to
increase the accuracy of DDA computations.
2. Theoretical analysis
In this section we analyze theoretically the errors of DDA computations. We formulate the
volume integral equation for the internal electric field and its operator counterpart in
Section 2.A and its discretization in Section 2.B. Section 2.C contains integral and discretized
formulae for measured quantities that are the final goal of any light scattering simulation. We
derive the main results in Section 2.D, where we consider errors of the traditional DDA
formulation2 without shape errors, which are considered separately in Section 2.E. Finally in
Section 2.F we discuss some recent DDA improvements from the viewpoint of our
convergence theory.
A.Integral Equation
Throughout this paper we assume the )iexp( tω− time dependence of all fields. The scatterer
is assumed dielectric but not magnetic (magnetic permittivity 1=μ ), and the electric
permittivity is assumed isotropic (non-isotropic permittivity will significantly complicate the
derivations but will not principally change the main conclusion of Section 2 – Eqs. (70) and
(87)).
The general form of the integral equation governing the electric field inside the
dielectric scatterer is the following:19,20
)()(),(),()()(),(d)()( 00
rErrLrMrErrrGrErE χχ VVr
∂−+′′′′+= ∫ , (1)
where Einc(r), E(r) are the incident and total electric field at location r; πεχ 4)1)(()( −= rr
is the susceptibility of the medium at point r (ε(r) – relative permittivity). V is the volume of
the particle (more general – the volume, which contains all points where the susceptibility is
not zero), V0 is a smaller volume such that , VV ⊂0 00 \ VV ∂∈r . ),( rrG ′ is the free space
dyadic Green’s function, defined as
−=∇∇+=′
)()(ˆˆ),(
kRgRgk IIIrrG , (2)
where I is the identity dyadic, ck ω= – free space wave vector, rrR ′−= , R=R , and
is a dyadic defined as (μ, ν are Cartesian components of the vector or
tensor), and g(R) is the scalar Green’s function
RR ˆˆ νμμν RRRR =ˆˆ
iexp(
)( = . (3)
M is the following integral associated with the finiteness of the exclusion volume V0
( )∫ ′−′′′′=
)()(),()()(),(d),( s30
rV rErrrGrErrrGrM χχ , (4)
where ),(s rrG ′ is the static limit ( ) of 0→k ),( rrG ′ :
−−=∇∇=′ 23
11ˆˆ),(
IrrG . (5)
L is the so-called self-term dyadic:
rV rL , (6)
where is an external (as viewed from r) normal to the surface ∂Vn′ˆ 0 at point r'.
Eq. (1) can be rewritten in operator form as follows
inc~~~ EEA =⋅ , (7)
where ( 311 )~ CE →=∈ VLH – functions from V to C3 that have finite L1-norm, 2inc~ H∈E –
subspace of H1 containing all functions that satisfy Maxwell equations in free space. A
is a
linear operator . Although the Sobolev norm is physically more sound (based on
the finiteness of energy of the electric field),
21: HH →
6,21 we use the L1-norm. A detailed discussion of
all assumptions made for the electric field is performed in Section 2.D.
B.Discretization
To solve Eq. (1) numerically a discretization is done in the following way.20 Let ,
for . N denotes the number of subvolumes (dipoles). Assuming and
choosing , Eq.
/0=ji VV I ji ≠ iV∈r
iVV =0 (1) can be rewritten as
)()(),(),()()(),(d)()( 3inc rErrLrMrErrrGrErE χχ ii
∂−+′′′′+= ∑ ∫
. (8)
The set of Eq. (8) (for all i) is exact. Further one fixed point ri inside each Vi (its center) is
chosen and is set. irr =
The usual approximation20 is considering E and χ constant inside each subvolume:
iiiii V∈==== rrrErErE for)()(,)()( χχχ . (9)
Eq. (8) can then be rewritten as
( ) iiii
jjjijii V ELMEGEE χχ −++= ∑
inc , (10)
where , )(incinc ii rEE = ),( iii V rLL ∂= ,
( )∫ ′−′′=
iii r ),(),(d
s3 rrGrrGM , (11)
∫ ′′=
ij rV
1 3 rrGG . (12)
A further approximation, which is used in almost all formulations of DDA, is
),()0( jiij rrGG = . (13)
This assumption is made implicitly by all formulations that start by replacing the scatterer
with a set of point dipoles, as was done originally by Purcell and Pennypacker.1 For a cubical
(as well as spherical) cell Vi with ri located at the center of the cell, iL can be calculated
analytically yielding22
=i . (14)
Eq. (10) together with Eqs. (13) and (14) and completely neglecting iM is equivalent to
the original DDA by Purcell and Pennypacker (PP).1 The diagonal terms in Eq. (10) are then
equivalent to the well-known Clausius-Mossotti (CM) polarizability for point dipoles.
Modifications introduced by other DDA prescriptions are discussed in Section 2.F.
In matrix notation Eq. (10) reads
ddd ,incEEA = , (15)
where Ed, Einc,d are elements of (vectors of size N where each element is a 3D complex
vector) and
( )N3C
dA is a N×N matrix where each element is a 3×3 tensor. d is the size of one
dipole. In operator notation Eq. (8) (for irr = ) is as follows
( ) diii ,incinc )(~)(~~ ErErEA == , (16)
We define the discretization error function as
( ) ( )iddidi ,0)(~~ EArEAh −= , (17)
where E0,d is the exact field at the centers of the dipoles – )(
i rEE = , in contrast to E
d that
is only an approximation obtained from solution of Eq. (15) (here we neglect the numerical
error that appears from the solution of Eq. (15) itself, which is acceptable if this error is
controlled to be much less than other errors). Comparing Eqs. (15) and (17) one can
immediately obtain the error in internal fields due to discretization δEd:
( ) ddddd hAEEE 1,0δ −−=−= . (18)
C.Measured quantities
After having determined the internal electric fields, scattered fields and cross sections can be
calculated. Scattered fields are obtained by taking the limit ∞→r of the integral in Eq. (1)
(see e.g. 23)
)iexp(
)(sca nFrE
= , (19)
where rrn = is the unit vector in the scattering direction, and F is the scattering amplitude:
∑∫ ′′⋅′−′−−=
krnnk )()()iexp(d)ˆˆ(i)( 33 rErnrInF χ . (20)
All other differential scattering properties, such as the amplitude and Mueller scattering
matrices, and asymmetry parameter >< θcos can be easily derived from F(n), calculated for
two incident polarizations.24 We consider an incident polarized plane wave:
)iexp()( 0inc rkerE ⋅= , (21)
where , a is direction of incidence, and ak k= 10 =e is assumed. The scattering and
extinction cross sections (Csca, Cext) are derived from the scattering amplitude:23
∫ Ω= nFkC , (22)
( )∗⋅= 02ext )(Re
, (23)
where * denote complex conjugation. The expression for absorption cross section (Cabs)
directly uses the internal fields:23
( )∑ ∫ ′′′=
abs )()(Imd4 rErχπ , (24)
Since only values of the internal field in the centers of dipoles are known, Eqs. (20) and
(24) are approximated by (PP)
∑ ⋅−χ−−=
iii kVnnk )iexp()ˆˆ(i)(
3 nrEInF , (25)
iiiVkC
abs )Im(4 Eχπ . (26)
Corrections to Eq. (26) are discussed in Section 2.F.
Both Eqs. (20) (for each component) and (24) can be generalized as ( )E~~φ (a functional
that is not necessarily linear), which is approximated as:
( ) ( ) ddd φφφ δ~~ += EE , (27)
where ( )dd Eφ corresponds to Eqs. (25) or (26) respectively, and the error δφ d consists of two
parts:
( ) ( )[ ] ( ) ( )[ ]ddddddd EEEE φφφφφ −+−= ,0,0~~δ . (28)
The first one comes from discretization (similar to Eq. (17)), and the second from errors in the
internal fields.
D.Error analysis
In this section we perform error analysis for the PP formulation of DDA. Improvements of
DDA are further discussed in Section 2.F.
We assume cubical subvolumes with size d. We also assume that the shape of the
particle is exactly described by these cubical subvolumes (we call this cubically shaped
scatterer). Moreover, χ is a smooth function inside V (exact assumptions on χ are formulated
below). An extension of the theory to shapes that do not satisfy these conditions is presented
in Section 2.E. If there are several regions with different values of χ (smooth inside each
region), the analysis is still valid but interfaces inside V should be considered the same way as
the outer boundary of V. We further fix the geometry of the scattering problem and incident
field. Therefore we will be interested only in variation of discretization (which is
characterized by the single parameter – d); for reasons that will become clear in the sequel, we
assume that (this bound is not limiting since otherwise DDA is generally
inapplicable
We switch to dimensionless parameters by assuming 1=k , which is equivalent to
measuring all the distances in units of k1 . The unit of electric field can be chosen arbitrary
but constant. In all further derivations we will use two sets of constants: γi and ci. γ1-γ13 are
basic constants that do not depend on the discretization d, but do depend directly upon all
other problem parameters – size parameter eqkRx = (Req – volume-equivalent radius), m,
shape, and incident field – or some of them. On the contrary, c1-c94 are auxiliary values that
either are numerical constants or can be derived in terms of constants γi. Although the
dependencies of ci on γi are not explicitly derived in this paper, one can easily obtain them
following the derivations of this section. That is the main motivation for using such vast
amount of constants instead of an “order of magnitude” formalism. However, such explicit
derivation has limited application because, as we will see further, constants in the final result
depend upon almost all basic constants. Qualitative analysis of these dependencies will be
performed in the end of this section. It should be noted that the main theoretical results
concerning DDA convergence (boundedness of errors by a quadratic function, cf. Eq (70))
can be formulated and applied without consideration of any constants (which is simpler).
However our full derivation enables us to make additional conclusions related to the behavior
of specific error terms.
The total number of dipoles used to discretize the scatterer is
−= dN γ . (29)
We assume that the internal field E
is at least four times differentiable and all these
derivatives are bounded inside V
65432 )(,)(,)(,)(,)( γγγγγ τρνμρνμνμμ ≤∂∂∂∂≤∂∂∂≤∂∂≤∂≤ rErErErErE
for V∈r and τρνμ ,,,∀ .
This assumption is acceptable since there are no interfaces inside V, therefore E
should be a
smooth function. . denotes the Euclidian (L2) norm, which is used for all 3D objects: vectors
and tensors. We use the L1-norm,
. , for N-dimensional vectors and matrices as well as for
functions and operators. Eq. (30) immediately implies that . We require that χ
satisfies Eq.
~ 1 VL∈E
(30) with constants γ7-γ11. Further we will state an estimate for the norm of
)(RG and its derivatives. One can easily obtain from Eq. (2) that for 1>R )(RG satisfies
Eq. (30) (with constants c1-c5), while for 2≤R
6 )(,)(,)(,)(
−−−− ≤∂∂∂≤∂∂≤∂≤ RcRcRcRc RGRGRGRG ρνμνμμ ,
−≤∂∂∂∂ RcRGτρνμ for τρνμ ,,,∀ .
Next we state two auxiliary facts that will be used later. Let Vc be a cube with size d and
with its center at the origin and f(r) a four times differentiable function inside Vc. Then
)(max)()(d
fdcffr
d cVV
∈,∫ , (32)
( ) )(max)(
)()(d
d cVV
∂∂∂∂+∇≤−
=∫ . (33)
Eqs. (32) and (33) are the corollary of expanding f into Taylor series. Odd orders of the Taylor
expansion vanish because of cubical symmetry.
Our first goal is to estimate
dh . Starting from Eq. (17) we write as dih
),()(),(d )0(33 ii
i Vdr
rMPGrPrrGh +
−′′′= ∑ ∫
, (34)
where we have introduced the polarization vector for conciseness
)()()( rErrP χ= , )( ii rPP = . (35)
It is evident that also satisfies Eq. )(rP (30) (with constants c13-c17). We start by estimating
),( iiV rM . Substituting a Taylor expansion of )(rP
( ) ( )( )∑∑ ∂∂+∂+=
ρρ τρ ),,(
)()()( RrP0P0PRP RRR , (36)
where , into Eq μμ Rr ≤≤
~0 (4) gives
( ) ( )(∫ ∑∫ ∂∂+−=
ii RRRRV
τρτρ τρ ),,(
)()(d),( 3s3 RrPRGPRGRGrM ) . (37)
The norms of these two terms can be estimated as
( ) 2183s3 )(d3
)()(d dcRRgR
i ≤=− ∫∫ PIPRGRG , (38)
( ) 21923153 )(d3)),,(~()(d dcRRcRRR
≤≤∂∂ ∫∫ ∑ RGRrPRG
τρτρ τρ . (39)
Eq. (38) follows directly from the definitions in Eqs. (2), (5). To derive Eq. (39) we used
Eq. (31) and the fact that 23RRR ≤∑
τρ . Finally, Eqs. (37)-(39) lead to
20),( dcV ii ≤rM . (40)
To estimate the sum in Eq. (34) we consider separately three cases: 1) dipole j lies in a
complete shell of dipole i (we define the shell below); 2) j lies in a distant shell of dipole i –
1>−= ijijR rr ; 3) all j that fall between the first two cases (see Fig. 1). We define the first
shell (S1(i)) of a cubical dipole as a set of dipoles that touch it (including touching in one point
only). The second shell (S2(i)) is a set of dipoles that touch the outer surface of the first shell,
and so on. The l-th shell (Sl(i)) is then a set of all dipoles that lie on the boundary of the cube
with size and center coinciding with the center of the original dipole. We call a shell
complete if all its elements lie inside the volume of the scatterer V. A shell is called a distant
dl )12( +
Kmax K(i)
(2) (3)
vacuum scatterer
Fig. 1. Partition of the scatterer’s volume into three regions relative to dipole i.
shell if all its elements satisfy , i.e. if its order 1>ijR [ ]dKl 1max => . Let K(i) be the order of
the first incomplete shell, which is an indicator of how close dipole i is to the surface. We
demand to separate cases (1) and (2) described above. All j that fall in the third
case satisfy (the exact value of this constant – slightly larger than
max)( KiK ≤
2<ijR 3 – depends on
d). The number of dipoles in a shell Sl (which can be incomplete) – ns(l) – can be estimated as
33 )12()12()( lclllns ≤−−+≤ . (41)
The sum of the error over all dipoles that lie in complete shells is then
∑ ∑ ∫
)0(33 )(),(d
l iSj
dr PGrPrrG , (42)
Since each shell in Eq. (42) is complete it can be divided into pairs of dipoles that are
symmetric over the center of the shell (j and –j). For convenience we set . The inner
sum in Eq.
0r =i
(42) can then be rewritten as
( ) ( )∑ ∫
+−′−+′′′
)0(33 )()()(d
dr PPGrPrPrG , (43)
Further we introduce the auxiliary function
( ) )()()(
)( 0PrPrPru −′−+′=′ , (44)
which satisfies the following inequalities (follows from Eq. (30) for P(r) and Taylor series)
22 )(,)(,)( crcrc ≤∂∂≤∂≤ rururu νμμ for νμ,∀ . (45)
Then Eq. (43) is equivalent to
V l jl j
drdr PGrGuGrurG ∑ ∫∑ ∫
)0(33
)0(33 )(d)()(d , (46)
where . To estimate the first term we apply Eq. )( jj ruu = (32) to the whole function under
the integral. Using Eqs. (31) and (45) one may obtain
( ) 325)()(max −
≤′′∂∂ ij
, (47)
and hence
)0(33 )()(d
V ll j
ldcRdcdr uGrurG , (48)
where we have used Eq. (41) and for ldRij ≥ )(iSj l∈ .
It is straightforward to show that
∑ ∫∑ ∫
′′=′′
3 )(d
iSj ViSj V l jl j
rgrr IrG , (49)
)0( )(
RgIG . (50)
The derivation is based upon Eq. (2) and the equivalence I
in all sums and integrals
that satisfy cubical symmetry. Then second part of Eq. (46) is transformed to
)0(33 )()(d)(d −
+≤−′′≤
−′′ ∑ ∫∑ ∫ ldcldcRgdrgrcdr
V l jl j
PGrG , (51)
where we apply Eq. (33) to derive the second inequality and use the identity
and the following inequalities
)()(2 rgrg −=∇
( ) 532131 )(, −− ≤∂∂∂∂≤ RcRgRcRg τρνμ for τρνμ ,,,∀ . (52)
Substituting Eqs. (48) and (51) into Eq. (42) one can obtain
( ) 23433
)0(33 )(ln)(),(d diKccdr
l iSj
−′′′∑ ∑ ∫
PGrPrrG , (53)
using the fact that . 1)( ≤diK
We now consider the second part of the sum in Eq. (34) (where ). We first apply
1>ijR
(32), then use Eq. (30) for P(r) and )(rG , and finally invoke Eq. (29):
)0(33 )(),(d dcdNcdcdr
ijij j
i ≤≤≤⎟
−′′′ ∑∑ ∫
PGrPrrG . (54)
To analyze the third part of the sum in Eq. (34) we again sum over shells, however since
they are incomplete we cannot use symmetry considerations. We apply Eq. (33) to the whole
function under the integral and proceed analogous to the derivation of Eq. (51). Using the
identity
)()(2 rGrG −=∇ , (55)
(since we have assumed ) we obtain 1=k
( ) 4372 )()( −= ≤∇ ijRcijRrrPrG , (56)
( ) 738)()(max −
≤′′∂∂∂∂ ij
, (57)
which leads to
)0(33 )(),(d −−
−′′′∑ ∫ lcdlcdr
PGrPrrG , (58)
and then analogous to Eq. (53):
)()()(),(d 442
)( )(
)0(33
iKcidKcdr
iKl iSj
−′′′∑ ∑ ∫ PGrPrrG . (59)
Collecting Eqs. (40), (53), (54), (59) we finally obtain
( ) 24443442141 )(ln)()( diKcciKcidKcdi +++≤ −−h . (60)
Then
( ) ( )442141
max4443
)(ln −−
+++≤= ∑∑ KcdKcKnNdKcc
d hh , (61)
where n(K) is the number of dipoles whose order of the first incomplete shell is equal to K. It
is clear that
NdnKn 12)1()( γ≤≤ , (62)
where γ12 is surface to volume ratio of the scatterer. Finally we obtain
( )[ ]dcddccNd 46245431 ln +−≤h . (63)
The last term in Eq. (63) is mostly determined by dipoles that lie on the surface (or few
dipoles deep) because it comes from the K-4 term in Eq. (61) (which rapidly decreases when
moving from surface). We define surface errors as those associated with the linear term in
Eq. (63). Our numerical simulation (see Section 0) show that this term is small compared to
other terms for “typical” values of d, however it is always significant for small enough values
of d.
From Eq. (18) we directly obtain
δ ddd hAE
≤ . (64)
We assume that a bounded solution of Eq. (7) uniquely exists for any , moreover we
assume that if
inc~ H∈E
inc =E then 131
γ≤E . These assumptions are equivalent to the fact that
1~ −A exists and is finite (the operator 1
~ −A is bounded). Because dA is a discretization of A
one would expect that
( ) 131
lim γ== −
. (65)
Although Eq. (65) seems intuitively correct, its rigorous prove, even if feasible, lies outside
the scope of this paper. For an intuitive understanding one may consult the paper by Rahola,25
where he studied the spectrum of the discretized operator (for scattering by a sphere) and
showed that it does converge to the spectrum of the integral operator with decreasing d. It
should however be noted, that convergence of the spectrum only implies the convergence of
the spectral (L2) norm of the operator and not necessarily the convergence of the L1-norm.
Therefore Eq. (65) should be considered as an assumption. It implies that there exists a d0
such that
for 0dd < ( ) 47
A , (66)
where c47 is an arbitrary constant larger then γ13 (although d0 depends on its choice). For
example 1347 2γ=c should lead to a rather large d0 (a rigorous estimate of d0 does not seem
feasible). Therefore
δ dE satisfies the same constrain as
dh (Eq. (63)) but with constants
c48-c50.
Next we estimate the errors in the measured quantities and start with the discretization
error (first part in Eq. (28)). Examining Eqs. (20) and (24) one can see that Eq. (32) may be
directly applied leading to
( ) ( ) 252551,0~~ dcdc
dd ≤≤− ∑EE φφ . (67)
The second part in Eq. (28) is estimated as
( ) ( ) ( ) dcddccdcdc d
55541
,0 lnδδ +−≤≤≤− ∑ EEEE φφ , (68)
where we used Eq. (29). The estimation of the error for Cabs additionally uses the fact
i c EE δδ 57
c Eδmax
57 .
By combining Eqs. (67) and (68) we obtain the final result of this section:
( ) dcddccd 5625558 lnδ +−≤φ . (69)
It is important to remember that the derivation was performed for constant x, m, shape, and
incident field. There are 13 basic constants (γ1-γ13). γ1 (Eq. (29)) characterizes the total
volume of the scatterer, hence it depends only on x. γ7-γ11 (Eq. (30) for χ(r)) can be easily
obtained given the function χ(r), moreover it is completely trivial in the common case of
homogenous scatterers. γ12 (surface to volume ratio, Eq. (62)) depends on the shape of the
scatterer and is inversely proportional to x. It is not feasible (except for certain simple shapes)
to obtain the values of constants γ2-γ6 (Eq. (30)), since it requires an exact solution for the
internal fields. These constants definitely depend upon all the parameters of the scattering
problem. Moreover, these dependencies can be rapidly varying, especially near the resonance
regions. The same is true for γ13 (L1-norm of the inverse of the integral operator, Eq. (65)).
Finally, there is the important constant d0 that also depends on all the parameters, however
one may expect it to be large enough (e.g. ) for most of the problems – then its
variation can be neglected.
20 ≥d
Before proceeding we introduce the discretization parameter kdmy = . We employ the
commonly used formula as proposed by Draine,8 however the exact dependence on m is not
important because all the conclusions are still valid for constant m. Replacing d by y does not
significantly change the dependence of the constants in Eq. (69) since they all already depend
on m through the basic constants γ2-γ11, γ13. This leads to
( ) ycyyccy 6126059 lnδ +−≤φ . (70)
It is not feasible to make any rigorous conclusions about the variation of the constants in
Eq. (70) with varying parameters because all these constants depend on γ2-γ6, γ13 that in turn
depend in a complex way upon the parameters of the scattering problem. However we can
make one conclusion about the general trend of this dependency.
Following the derivation of the Eq. (70) one can observe that c61 is proportional to γ12,
while c59 and c60 do not directly depend on it (at least part of the contributions to them are
independent of γ12). Therefore the general trend will be a decrease of the ratio 5961 cc with
increasing x (when all other parameters are fixed). This is a mathematical justification of the
intuitively evident fact that surface errors are less significant for larger particles.
In the analysis of the results of the numerical simulations (Section 0) we will neglect the
variation of the logarithm. Eq. (70) then states that error is bounded by a quadratic function of
y (for ). However, keep in mind that our derivation does not lead to an optimal error
estimation, i.e. it overestimates the error and can be improved. For example, the constants γ
0dd ≤
γ6 are usually largest inside a small volume fraction of the scatterer (near the surface or some
internal resonance regions), while in the rest of the scatterer the internal electric field and its
derivatives are bounded by significantly smaller constants. However the order of the error is
estimated correctly, as we will see in the numerical simulations.
It is important to note that Eq. (70) does not imply that δφ y (which is a signed value)
actually depends on y as a quadratic function, but we will see later that it is the case for small
enough y (Section 0, see detailed discussion in Paper 2). Moreover, the coefficients of linear
and quadratic terms for δφ y may have different signs, which may lead to zero error for non-
zero y (however, this y, if it exists, is unfortunately different for each measured quantity).
E.Shape errors
In this section we extend the error analysis as presented in Section 2.D to shapes that cannot
be described exactly by a set of cubical subvolumes. We perform the discretization the same
way as in Section 2.B but some of the Vi are not cubical (for Vi ∂∈ , which denotes that
dipole i lies on the boundary of the volume V). We set ri to be still in the center of the cube
(circumscribing Vi) not to break the regularity of the lattice. The standard PP prescription uses
equal volumes ( ) in Eqs. 3dVi = (10), (14), (25), and (26), i.e. the discretization changes the
shape of the particle a little bit. We will estimate the errors introduced by these boundary
dipoles. These errors should then be added to those obtained in Section 2.D. We start by
estimating
dh . First we consider for dih Vi ∂∉
−′′′=
PGrPrrGh )0(33 )(),(d , (71)
which is just a reduction of Eq. (34). For Vi ∂∈ is the same plus the error in the self-term dih
iiiiii
i VVdr
EΙrLrMPGrPrrGh χ
−′′′= ∑ ∫
),(),()(),(d )0(33 . (72)
Let us define
iij dr
PGrPrrGh )0(33sh )(),(d −′′′= ∫ , (73)
iiiiiiii VV EΙrLrMh χ
⎛ −∂−=
),(),(sh . (74)
We estimate each of the terms in Eq. (73) separately (since there is actually no significant
cancellation, and the error is of the same order of magnitude as the values themselves) using
Eq. (30) for P(r) and )(rG and Eq. (31). This leads to
h (75)
To estimate we assume that the surface of the scatterer is a plane on the scale of the size
of the dipole. A finite radius of curvature only changes the constants in the following
expressions. We will prove that
sh cii ≤h , (76)
therefore we do not need to consider the third term in Eq. (74) (coming from the unity tensor)
at all, since it is bounded by a constant.
( ) ( )∫∫ −′′′+′′−′′=
iiii rrV )()(),(d)(),(),(d),(
s3s3 rPrPrrGrPrrGrrGrM . (77)
−′ ic rrThe function in the first integral is always bounded by
. If the same is true
for the second integral and hence
ii V∈r
dcV ii 66),( ≤rM . (78)
If ii V∉r we introduce an auxilia r ′′ry point that is symmetric to ri over the particle surface
and apply the identity
( ) ( ))()()()()()( ii rPrPrPrPrPrP −′′+′′−′=−′ (79)
to the second integral in Eq. (77). Using Taylor expansion of P near and the fact that r ′′
irrrr −′≤′′−′ for iV∈′r one can show that
∫ ′′+≤ii cdcV ),( s36867r
ir ),(d rrGM , (80)
where the remaining integral can be proven to be equal to ),( iiV rL ∂− . The last prove left
(see Eqs. (74) and (80)) is to demonstrate that ),( iiV rL ∂ is bounded by a constant. The only
potential problem may come from the subsurface of iV∂ th the particle surface
(because it may be close to r
at is part of
ce is i). This subsurfa med planar. We will calculate the
integral in Eq.
(6) over the infinite plane rρrr +=−′ i uch that 0 s =⋅rρ . Then ρρn ±=′
( ) 223
2d),plane.inf(
mm == ∫irL
+∂V r
r , (81)
which is bounded. The rest of the integral (over the part of the cube surface) is bounded by a
constant, which is a manifestation of a more general fact that (by its definition) ), ir ( iVL ∂
does not depend on the size but only on the shape of the volume. Finally we have
69),( cV ii ≤∂ rL ,
which together with Eqs. (74), (78), and (80) prove Eq. (76).
Using Eqs. (75) and (76) we obtain
)ln()( 70
cllnc
+≤+≤ ∑ ∑∑∑ ∑ −hhh 7271
dccNd
, (83)
where we have changed the order of the summation in the double sum and split the
summation over cubical shells for
⎝∂∈∂∈∂∈
maxKl ≤ and . Then we have grouped everything
otal esti
maxKl >
into one sum over boundary dipoles. Eqs. (41) and (62) were used in the last inequality.
Combining Eqs. (63) and (83) one can obtain the t mate of the
dh for any scatterer:
( ) ( )[ ]ddccddccNd lnln 7273245431 −+−≤h . (84)
Using Eq. (66) we immediately obtain the sam
δ de estimate for E .
The derivation of the errors in the measured quantities is slightly modified compared to
(68) e changed to Section 2.D, by the presence of the shape errors. Eqs. (67) and ar
( ) ( )~~ dcdcdcdc
,0 +≤+≤− ∑∑
EE φφ , (85)
( ) )( ( ) ( )ddccddccdddd lnln 2,0 −+−≤−Eφ . dcd δ 7655541
53≤ EEφ (86)
The second term in Eq. (85) comes from surface dipoles for which errors are the same order
as the values themselves. Finally the generalization of Eq. (70) is
( ) ( )yyccyyccy lnlnδ 797826059 −+−≤φ . (87)
The shape errors “reinforce” the surface errors (the linear term of discretization error), and
although they both generally decrease with increasing size parameter x one may expect the
linear term in Eq. (87) to be significant up to higher values of y than in Eq. (70).
All the derivations in this section can in principle be extended to interfaces inside the
particle, i.e. when a surface, which cannot be described exactly as a surface of a set of cubes,
separates two regions where χ(r) varies smoothly. Two parts of the cubical dipole on the
interface should be considered separately the same way as it was done above. This will
however not change the main conclusion of this section – Eq.(87) – but only the constants.
F.Different DDA formulations
In this section we discuss how different DDA formulations modify the error estimates derived
in Sections 2.D and 2.E.
Most of the improvements of PP proposed in the literature are concerned with the self-
term – . They are the Radiative Reaction correction (RR),),( iiV rM
8 the Digitized Green’s
Function (DGF),23 the formulation by Lakhtakia (LAK),26,27 the a1-term method,28,29 the
Lattice Dispersion Relation (LDR),30 the formulation by Peltoniemi (PEL),31 and the
Corrected LDR (CLDR).32 All of them provide an expression for that is of order d),( iiV rM
(except for RR that is of order d3). For instance LDR is equivalent to
( )[ ] iii ddSmbmbbV PrM 3223221 i)32(),( +++= , (88)
(remember that we assumed ) where b1=k 1, b2, b3 are numerical constants and S is a
constant that depends only on the propagation and polarization vectors of the incident field.
However, none of these formulations can exactly evaluate the integral in Eq. (39), because the
variation of the electric field is not known beforehand (PEL solves this problem, but only for
a spherical dipole). Therefore they (hopefully) decrease the constant in Eq. (40), thus
decreasing the overall error in the measured quantities. However, these formulations are not
expected to change the order of the error from d2 to some higher order.
We do not analyze the improvements by Rahmani, Chaumet, and Bryant (RCB)33,34 and
Surface Corrected LDR (SCLDR),17 as they are limited to certain particle shapes.
There exist two improvements of the interaction term in PP: Filtered Coupled Dipoles
(FCD)12 and Integration of Green’s Tensor (IT).35 A rigorous analysis of FCD errors is
beyond the scope of this paper, but it seems that FCD is not designed to reduce the linear term
in Eq. (63) that comes from the incomplete (non-symmetric) shells. This is because FCD
employs sampling theory to improve the accuracy of the overall discretization for regular
cubical grids. FCD does not improve the accuracy of a single ijG calculation (approximation
of an integral over one subvolume).
IT, which numerically evaluates the integral in Eq. (12), has a more pronounced effect
on the error estimate. Consider dipole j from l-th shell (incomplete) of dipole i, then
.),(d),(maxd
)(),(d)(),(d
≤′′′+′∂′′≤
−′′′=−′′′
dlcrrcrrc
jijij
rrGrrG
PrPrrGPGrPrrG
(89)
Here we have used Eq. (36) and Taylor expansion of Green’s tensor up to the first order.
Eq. (89) states that the second term in Eq. (58) is completely eliminated and so is the linear
term in Eqs. (69) and (70) (surface errors). Therefore convergence of DDA with IT for
cubically shaped scatterers is expected to be purely quadratic (neglecting the logarithm).
However, for non cubically shaped scatterers the linear term reappears, due to the shape
errors. Both IT and FCD also modify the self-term, however the effect is basically the same as
for the other formulations.
Several papers aimed to reduce shape errors.10,11,36 The first one – Generalized Semi-
Analytical (GSA) method10 – modifies the whole DDA scheme, while the other two propose
averaging of the susceptibility over the boundary dipoles. We will analyze here Weighted
Discretization (WD) by Piller11 as probably the most advanced method to reduce shape errors
available today.
WD modifies the susceptibility and self-term of the boundary subvolume. We slightly
modify the definition of the boundary subvolume used in Sections 2.B and 2.E to
automatically take into account interfaces inside the scatterer. We define Vi to be always
cubical, but with a possible interface inside. The particle surface, crossing the subvolume Vi,
is assumed planar and divides the subvolume into two parts: the principal volume
(containing the center) and the secondary volume with susceptibilities , and
electric fields , respectively. The electric fields are considered constant inside
each part and related to each other via the boundary condition tensor
iV ii χχ ≡
ii EE ≡
iT :
iii ETE =
s . (90)
In WD the susceptibility of the boundary subvolume is replaced by an effective one, defined
( ) 3ssppe dVV iiiiii TI χχχ += , (91)
which gives the correct total polarization of the cubical dipole. The effective self-term is
directly evaluated starting from Eq. (4), considering χ and E constant inside each part,
( ) ( ) iii
rrV ETrrGrrGrrGrrGrM
′−′′+′−′′= ∫∫ ss3ps3
),(),(d),(),(d),( χχ . (92)
Piller evaluated the integrals in Eq. (92) numerically.11
To take a smooth variation of the electric field and susceptibility into account we define
( r is defined in Section )(s r ′′= χχ i ′′ 2.E) and iT is calculated at the surface between ri and
. and r ′′ ii PP ≡
iiiiii ETEP
ssss χχ == . Then
c rrPrP
−≤−′′
min)( 83
s , (93)
where we have assumed that Eq. (30) for χ(r) and E(r) is also valid in . siV
We start estimating errors of WD with (cf. Eq. shijh (73))
( ) ( )∫∫ −′′′+−′′′=
s)0(3p)0(3sh )(),(d)(),(d
jijiij rr PGrPrrGPGrPrrGh , (94)
Using Taylor expansions of near r)(rP ′ i and r ′′ in and correspondingly and Eq. piV
iV (93)
one may obtain that the main contribution comes from the derivative of Green’s tensor,
leading to (cf. Eq. (75))
h (95)
iih is the following (cf. Eq. (74))
( ) ( )
( ) ( ) ( )
.),(),(
),(d)(),(d)(),(d
),(),(),(d),(),(d
),(),(
pss3s3p3
ess3ps3
iiiiiii
iiiiiii
PrLErL
PPrrGPrPrrGPrPrrG
ErLPrrGrrGPrrGrrG
PrLrMh
−′′+−′′′+−′′′=
∂−′−′′+′−′′−
(96)
The first two integrals can be easily shown to be dс85≤ (cf. Eq. (77)) and third one is
transformed to L the same way as in Eq. (80), thus
iiiiiiiiiiii VVVdc ErLPrLPrLh
esspp
sh ),(),(),( χ∂−∂+∂+≤ , (97)
where the second term comes from the fact that averaged PL is not the same as L times
averaged P. This error depends on the geometry of the interface inside Vi and generally is of
order unity. For example, if the plane interface is described as ε+= izz , taking limit 0→ε
gives the error ( )zii sp2 PP −π (using Eq. (81)). Therefore WD does not principally improve
the error estimate of given by Eq. shiih (76), although it may significantly decrease the
constant. On the other hand, since ),( p iiV rL ∂ and ),(
iiV rL ∂ can be (analytically) evaluated
for a cube intersected by a plane, WD can be further improved to reduce the error in to
linear in d, which is a subject of future research.
Proceeding analogous to the derivation of Eq. (83) one can obtain
Ndcdccllnc
898887
)( ≤⎟⎟
++≤ ∑ ∑
h . (98)
It can be shown that for the scattering amplitude (Eq. (25)) the error estimate given by
Eq. (85) can be improved, since WD correctly evaluates the zeroth order of value for the
boundary dipoles, leading to
( ) ( ) 291490551,0~~ dcdcdc
dd ≤+≤− ∑∑
EE φφ . (99)
In his original paper11 Piller did not specify the expression that should be used for Cabs. Direct
application of the susceptibility provided by WD into Eq. (26) does not reduce the order of
error when compared with the exact Eq. (24) (except when ), since they are not linear
functions of the electric field. However, if we consider separately and (which is
equivalent to replacing
0s =iχ
( )( )∗⋅ iiiiV EEeIm χ by
2ss2pp )Im()Im( iiiiiii VV ETE χχ + ) the same
estimate as in Eq. (99) can be derived for Cabs.
Using Eqs. (98), (99), and the first part of Eq. (86) one can derive the final error
estimate for WD:
( ) ycyyccy 9429392 lnδ +−≤φ , (100)
where the constant before the linear term, as compared to Eq. (87), does not contain a
logarithm and is expected to be significantly smaller, because several factors contributing to it
are eliminated in WD. Although WD has a potential for improving, it does not seem feasible
to completely eliminate the linear term in the shape error. The accuracy of evaluation of the
interaction term over the boundary dipole (cf. Eq. (94)) can be improved by integration of
Green’s tensor over and separately but that would ruin the block-Toeplitz structure of
the interaction matrix and hinder the FFT-based algorithm for the solution of linear
0 30 60 90 120 150 180
Scattering angle θ, deg
cube kD=8
discretized sphere kD=10
sphere kD=3 (x10)
sphere kD=10
sphere kD=30
Fig. 2. S11(θ ) for all 5 test cases in logarithmic scale. The result for the kD = 3 sphere is multiplied by
10 for convenience.
equations.5 Since there is no comparable alternative to FFT nowadays, this method seems
inapplicable.
Minor modifications of the expression for Cabs are possible. Draine8 proposed a
modification of Eq. (26) that was widely used afterwards and which was further modified by
Chaumet et al.35 However, for many cases these expressions are equivalent and, even when
they are not, the difference is of order d3, which is neglected in our error analysis.
3. Numerical simulations
A.Discrete Dipole Approximation
The basics of the DDA method were summarized by Draine and Flatau.2 In this paper we use
the LDR prescription for dipole polarizability,30 which is most widely used nowadays, e.g. in
the publicly available code DDSCAT 6.1.4 We also employ dipole size correction8 for non-
cubically shaped scatterers to ensure that the cubical approximation of the scatterer has the
correct volume; this is believed to diminish shape errors, especially for small scatterers.2 We
use a standard discretization scheme as described in Section 2.E, without any improvements
for boundary dipoles. It is important to note that all the conclusions are valid for any DDA
implementation, but with a few changes for specific improvements as discussed in
Section 2.F.
Our code – Amsterdam DDA (ADDA) – is capable of running on a cluster of computers
(parallelizing a single DDA computation), which allows us to use practically unlimited
number of dipoles, since we are not limited by the memory of a single computer.37,38 We used
a relative error of residual as a stopping criterion for the iterative solution of the DDA
linear system. Tests suggest that the relative error of the measured quantities due to the
iterative solver is then (data not shown) and hence can be neglected (total relative
errors in our simulations are – see Section
810−<
710−<
56 1010 −− ÷> 0). More details about our code can
be found in Paper 2. All DDA simulations were carried out on the Dutch national compute
cluster LISA.39
Table 1. Exact values of Qext for the 5 test cases.
Particle Qext
kD = 8 cube 4.490
discretized kD = 10 sphere 3.916
kD = 3 sphere 0.753
kD = 10 sphere 3.928
kD = 30 sphere 1.985
0.1 1
slope = 0.77
y = kd·m
maximum
θ = 0°
θ = 45°
θ = 90°
θ = 135°
θ = 180°
slope = 0.95
Fig. 3. Relative errors of S11 at different angles θ and maximum over all θ versus y for (a) the kD = 8
cube, (b) the cubical discretization of kD = 10 sphere. A log-log scale is used. A linear fit of maximum
over θ errors is shown. (m = 1.5).
B.Results
We study five test cases: one cube with 8=kD , three spheres with , and a
particle obtained by a cubical discretization of the
30,10,3=kD
10=kD sphere using 16 dipoles per D
(total 2176 dipoles, x equal to that of a sphere; see detailed description in Paper 2). By D we
denote the diameter of a sphere or the edge size of a cube. All scatterers are homogenous with
. Although DDA errors significantly depend on m (see e.g. 5.1=m 14), we limit ourselves to
one single value and study effects of size and shape of the scatterer.
The maximum number of dipoles per D (nD) was 256. The values of nD that we used are
of the form (p is an integer), except for the discretized sphere, where all np2}7,6,5,4{ ⋅ D are
0.1 1
0.1 1
0.01 0.1
y = kd·m
slope 2.29 (c)
slope 0.91
maximum
θ = 0°
θ = 45°
θ = 90°
θ = 135°
θ = 180°
slope 1.05
Fig. 4. Same as Fig. 3 but for (a) kD = 3, (b) kD = 10, and (c) kD = 30 spheres.
multiples of 16 (this is required to exactly describe the shape of the particle composed from a
number of cubes). The minimum values for nD were 8 for the 3=kD sphere, 16 for the cube,
the sphere, and the discretized sphere, and 40 for the 10=kD 30=kD sphere.
All the computations use a direction of incidence parallel to one of the principal axes of
the cubical dipoles. The scattering plane is parallel to one of the face of the cubical dipoles. In
this paper we show results only for the extinction efficiency Qext (for incident light polarized
parallel to one of the principal axes of the cubical dipoles) and phase function S11(θ ) as the
0.01 0.1 1
slope = 0.89
cube kD=8
discretized sphere kD=10
sphere kD=3
sphere kD=10
sphere kD=30
y=kd·m
Fig. 5. Relative errors of Qext versus y for all 5 test cases. A log-log scale is used. A linear fit through 5
finest discretizations of kD = 3 sphere is shown.
most commonly used in applications. However, the theory applies to any measured quantity.
For instance, we have also confirmed it for other Mueller matrix elements (data not shown).
Exact results of S11(θ ) for all 5 test cases are shown in Fig. 2. For spheres this is the
result of Mie theory (the relative accuracy of the code we used24 is at least ) and for the
cube and discretized sphere an extrapolation over the 5 finest discretizations (the
extrapolation technique is presented in Paper 2, together with all details of obtaining these
results, including their estimated errors). We use such ‘exact’ results because analytical theory
is unavailable for these shapes and because errors of the best discretization are larger than that
of the extrapolation. Their use as references for computing real errors (difference between the
computed and the exact value) of single DDA calculations is justified because all these real
errors are significantly larger than the errors of the references themselves (see Paper 2; in
general, real errors obtained this way have an uncertainty of reference error). Exact values of
610−<
ext for all test cases are presented in Table 1.
In the following we show the results of DDA convergence. Fig. 3 and Fig. 4 present
relative errors (absolute values) of S11 at different angles θ and maximum error over all θ
versus y in log-log scale. In many cases the maximum errors are reached at exact
backscattering direction, then these two sets of points overlap. Deep minima that happen at
intermediate values of y for some values of θ (and also sometimes for Qext – Fig. 5) are due to
the fact that the differences between simulated and reference values change sign near these
values of y (see Paper 2 for detailed description of this behavior). The solid lines are linear fits
to all or some points of maximum error. The slopes of these lines are depicted in the figures.
Fig. 5 shows relative errors of Qext for all 5 studied cases in log-log scale. A linear fit through
the 5 finest discretizations of the 3=kD sphere is shown. More results of these numerical
simulations are presented in Paper 2.
4. Discussion
Convergence of DDA for cubically shaped particles (Fig. 3) shows the following trends. All
curves have linear and quadratic parts (the non-monotonic behavior of errors for some θ are
also a manifestation of the fact that signed difference can be approximated by a sum of linear
and quadratic terms that have different signs). The transition between these two regimes
occurs at different y (which indicates the relative importance of linear and quadratic
coefficients). While for maximum errors that are close to those of the backscattering direction
the linear term is significant for larger y, it is much smaller and not significant in the whole
range of y studied for side scattering ( °= 90θ ). R
for l
esults of DDA convergence for spheres (Fig.
4) show a different behavior for different sizes. Errors for the small ( 3=kD ) sphere converge
purely linear (except for small deviation of errors of )90(11 °S values of y). Similar
results are obtained for the 10=kD sphere, but with significant oscillations superimposed
upon the general trend. Convergence for the large ( 30
=kD ) sphere is quadratic or even faster
in the range of y studied, also with significant oscillations.
Comparing Fig. 3 and Fig. 4 (especially Fig. 3(b) and Fig. 4(b) showing results for
almost the same particles) one can deduce the following differences in DDA convergence for
cubically and non-cubically shaped scatterers. The linear term for cubically shaped scatterers
is significantly smaller, resulting in smaller total errors, especially for small y. All these
conclusions, together with the size dependence of the significance of the linear term in the
total errors, are in perfect agreement with the theoretical predictions made in Sections 2.D and
2.E. Errors for non-cubically shaped particles exhibit quasi-random oscillations that are not
present for cubically shaped particles. This can be explained by the sharp variations of shape
errors with changing y (discussed in details in Paper 2). Oscillations for the sphere
Fig. 4(a)) are very small (but still clearly present), which is due to the small size of the
particle and hence featurelessness of its light scattering pattern – the surface structure is not
that important and one may expect rather small shape errors. Results for Qext (Fig. 5) fully
support the conclusions. Errors of Qext for the large sphere at small values of y are
unexpectedly smaller than for smaller spheres. This feature requires further study before
making any firm conclusions, however there is definitely no similar tendency for S11(θ ) (cf.
Fig. 4).
We have also studied a porous cube that was obtained by dividing a cube into
27 smaller cubes and then removing randomly 9 of them. All the conclusions are the same as
those reported for the cube, but with slightly higher overall errors (data not shown).
In this paper we have used a traditional DDA formulation2 for numerical simulations.
However, as we showed in Section 2.F several modern improvements of DDA (namely IT
and WD) should significantly change its convergence behavior. IT should completely
eliminate the linear term for cubically shaped scatterers, which should improve the accuracy
especially for small y. WD should significantly decrease shape and hence total errors for non-
cubically shaped particles, moreover it should significantly decrease the amplitude of quasi-
random error oscillations because it takes into account the location of the interface inside the
boundary dipoles. Numerical testing of DDA convergence using IT and WD is a subject of a
future study.
5. Conclusion
To the best of our knowledge, we conducted for the first time a rigorous theoretical
convergence analysis of DDA. In the range of DDA applicability ( 2<kd ) errors are bounded
by a sum of a linear and quadratic term in the discretization parameter y; the linear term is
significantly smaller for cubically than for non-cubically shaped scatterers. Therefore for
small y errors for cubically shaped particles are much smaller than for non-cubically shaped.
The relative importance of the linear term decreases with increasing size, hence convergence
of DDA for large enough scatterers is quadratic in the common range of y. All these
conclusions were verified by extensive numerical simulations.
Moreover, these simulations showed that errors are not only bounded by a quadratic
function (as predicted in Section 2), but actually can be (with good accuracy) described by a
quadratic function of y. This fact provides a basis for the extrapolation technique presented in
Paper 2.
Our theory predicts that modern DDA improvements (namely IT and WD) should
significantly change the convergence of DDA, however numerical testing of these predictions
is left for future research.
Acknowledgements
We thank Gorden Videen and Michiel Min for valuable comments on earlier version of this
manuscript and anonymous reviewer for helpful suggestions. Our research is supported by the
NATO Science for Peace program through grant SfP 977976.
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|
0704.0034 | Origin of adaptive mutants: a quantum measurement? | Microsoft Word - NatureCorrespondJasmitedit.rtf
Sent to Nature in 1990
Abstract
This is a supplement to the paper arXiv:q-bio/0701050, containing the text of correspondence
sent to Nature in 1990.
Origin of adaptive mutants: a quantum measurement?
Sir, - Several recent works described non-random induction of adaptive mutations by
environmental stimuli 1-3. The most obvious explanation of this striking phenomenon would be
that activation of gene expression leads to the enhancement of its mutation rate4. However, this
does not work with the lacZ mutations described by Cairns and co-workers as the true inducer of
the lac-operon is not lactose as such, but allolactose, a by-product of the β-galactosidase
reaction5. So, in lacZ mutants the operon is not induced by lactose6. Besides, induction of
respective genes would not explain the high fraction, among the revertants, of suppressor
mutations in tRNA genes1,7
Other explanations suggest some special mechanisms for the "acceleration of adaptive
evolution", like selection of "useful" protein coupled to specific reverse transcription1. However,
any mechanism of this type also should have emerged in evolution. I propose that, to explain the
adaptive mutation phenomenon, there is no need for any new ad hoc mechanism. The only thing
that is necessary is to return to the old discussion of the role of quantum concepts in our
understanding of life. This alone will allow the explanation of this manifestly Lamarckian
phenomenon by Darwinian selection, occurring not in a population of organisms as usual, but in a
"population" of virtual, in the direct quantum theory sense, states of each distinct cell. Thus, this
hypothesis may be called "selection of virtual mutations". Detailed substantiation of this concept
will be presented in a special publication; below I briefly show how this explanation might work.
It has been shown by the Cairns group that the mutations ensuring cell growth begin to
accumulate not immediately after plating, but only after conditions are created under which such
mutations become "useful", as if the mutations are induced by these conditions1. I suggest that, to
explain this phenomenon, we should change our ideas about what a cell is, and consider not
actual but virtual mutations. An important distinction of virtual mutations is that they do not
accumulate with time in stationary cell, whereas the number of actual mutants would grow
linearly from the moment of plating, and this would yield drastically different results in
experiments like those shown in Fig. 3 of Ref. 1. Virtual mutations produce "delocalization" of
the cell among different states, similarly to the delocalization of electron in physical space.
However, for a virtual mutation to become an actual one, certain conditions are necessary,
namely the possibility to grow, leading the system away irreversibly from the initial state. Such
conditions arise when, for example, lactose is added to a plate with lacZ bacteria. Briefly, this is
the essence of the proposed explanation.
What is a virtual mutation? The main cause of usual spontaneous mutations is the well-
known base tautomerization8 (having the in vitro frequency of about 10-4). Thus could we reduce
‘virtual mutation’ to such tautomerization? I believe that this view is not consistent with
experiments, as it implies that the same rare tautomeric form should work both in transcription
and in replication. If these processes are considered independent, we logically arrive to the leaky
mutant, which was refuted by Cairns and coworkers1. Thus we need to postulate a correlation
between the recognition of the tautomeric forms in transcription and in replication, making us to
define "virtual mutation" as a certain state of the cell as a whole. Analogous reasoning is
applicable to the "adaptive transpositions" discussed by Cairns. In other words, we consider the
whole cell as a quantum system, with non-negligible nonlocality inherent in such systems. Most
of all it resembles the systems of "generalized rigidity"9, such as superfluid or superconducing
states of matter, whose behavior is linked to quantum correlations; and I believe, similar
correlations take place in the cell too.
I would like to emphasize that the proposed approach does not require detalization of
molecular processes in the cell. Its main focus is the behavior of the cell as a whole. Similarly, to
explain gyroscopic precession there is no need to consider interactions between elements inside
the gyroscope; it’s enough to know some motion invariants, defined by space-time symmetries.
Starting from this general view, one may express the above ideas using the operator
formalism, and considering experiments conducted by Cairns as measurement of the cells’
capability to propagate under given conditions. I suggest that the trait "ability to reproduce on
lactose" (as an example) can be represented by an operator which one may designate "Lac".
Importantly, this new operator will act on the state Ψ of the whole cell because the ability to
reproduce is a property of the cell as a whole, and not of any part of it. Generally, "Lac " will
decompose this Ψ into a superposition of some eigenfunctions. The components of this
superposition are those functions that do not change upon the action of this operator, but are only
multiplied by a constant. It reflects the essence of operator formalism in quantum theory, which
chooses states compatible with given experimental conditions. There are three such
eigenfunctions (I intentionally simplify the situation): ψ1 corresponds to cell death, ψ2 to the
stationary state, and ψ3 to the self-reproduction (that is the virtual mutation, in our case). Each
function will enter the decomposition of Ψ with a coefficient ci related to the probability of this
or that outcome, i.e.:
Ψ = c1 ψ1 + c2 ψ2 + c3 ψ3, where Σ| ci |2 = 1
By plating the cells on lactose agar we, in fact, begin to measure their ability to grow
under these particular conditions. The rate of accumulation of lac revertants, i.e. the probability to
obtain a cell in the mutant state, will correspond to |c3|2, being a small, but finite quantity,
appearing, for example, due to base tautomerization. Here, the role of cell growth is dual: on the
one hand, it is a factor of irreversibility amplifying the "quantum fluctuation", and on the other
hand, it is a selection criterion, as each kind of virtual mutants capable of growth under these
conditions can lead to colony formation. Another situation, i.e. glucose/valine agar, will be
represented by another operator (Valr), which will decompose the same Ψ function according to
another basis, and Valr mutants will be obtained with certain rate. In fact, this is the essence of
adaptive mutation phenomenon, where a particular condition induces emergence of respective
mutants.
Thus, the proposed change of our view on the cell suggests that, in accord with quantum
concepts, we are not dealing with the probability for a cell to mutate by itself, independent of
experimental conditions. Rather, we are dealing with the probability to observe the cell in the
mutant state by plating it on lactose. We are certainly simplifying situation, as spontaneous
mutations that accumulate during cell growth before plating, make our ensemble ‘mixed’.
However, this complication does not change the essence of the explanation, according to which
adaptive mutations emerge by measurement of ‘pure’ state. This resembles the passage of a
polarized photon through a polarizer turned under some angle to the photon polarization. It will
be incorrect to say that the polarization of the photon could turn by the necessary angle by
chance, prior to interaction. It is the specific experimental situation that makes us to decompose
the state vector according to the respective basis states, and to evaluate the fraction of the
component that will pass through polarizer. On the other hand, one may speak about "adaptation"
of photon polarization by selection of "fit" eigenstate, and consider this case as the model for our
phenomenon.
How are all these ideas applicable to the living bacterial cell? Discussion of the possible
role of quantum concepts in biology has a rather long history, initiated by Niels Bohr (‘the
complementarity principle’). Briefly, one might reduce the essence of this discussion to the
principal impossibility to predict precisely the fate of an individual cell. For example, any attempt
to determine, whether it is able to reproduce under certain conditions, will lead to irreversible
change of the state of the cell, even to its death. This is reminiscent of the two-slit diffraction
experiment, where an attempt to determine through which of the two slits the electron actually
passes will lead to disappearance of the interference. The two trajectories of the electron can be
made physically discernable only by the cost of changing the experimental situation. Similarly,
the notorious phenomenon of the "wholeness" of the living organism can be formally expressed
according to the Feynman rules of calculating probabilities: different indiscernible (in the given
experimental conditions) variants should be included in the pure state (i.e. their amplitudes, and
not probabilities, should be summed, leading to interference and other quantum effects). Thus, as
long as a whole cell exists and is alive, we are obligated to treat its different indiscernible states
in this way. Such consideration of operational limitations allows us to explain the adaptive
mutation phenomenon (and hopefully other adaptations too) as the consequence of unavoidable
quantum scatter in measurement of the cell's capability to propagate under given conditions.
In spite of its apparent formal character, this hypothesis allows us to make some
predictions of applied (in particular, medical) interest. It predicts that in processes involving
somatic mutations (e.g. oncogenesis, or specific antibody generation) the mutations may be
induced by conditions allowing the cell that happened to be in the state of virtual mutation to
proliferate irreversibly. I believe, this possibility can be tested experimentally.
References
1. Cairns,J., Overbaugh,J., Miller,S. Nature 335, 142-145 (1988)
2. Shapiro,J.A. Molec. Gen. Genet. 194, 79-90 (1984)
3. Hall,B.J. Genetics 120, 887-897 (1988)
4. Devis,B.D. Proc.Natl.Acad.Sci.USA 86, 5005-5009 (1989)
5. Lewin,B., Genes, p.236 ( J.Wiley & Sons,1985)
6. Burstein,C., Cohn,M., Kepes,A., Monod,J. Bioch.Bioph.Acta 95, 634 (1965)
7. Savic,D.J.& Kanazir,D.T. Molec. Gen. Genet. 137, 143-150 (1975)
8. Topal,M., Fresco,J. Nature 263, 285-289 (1976)
9. Anderson,P.W.,Stein,D.L. in Self-Organizing Systems, ed. by F.E.Yates, pp.451-452
(Plenum Press, 1987)
Comments:
This text was written in 1990. The author translated it to English with the kind help of Dr.
Eugene Koonin (current affiliation: National Center for Biotechnology Information, National
Library of Medicine, National Institutes of Health, Bethesda MD, USA.)
The English version of the text was sent to Nature in 1990 and rejected. At the same time it was
also sent to the following correspondents :
1. JOHN CAIRNS
Department of Cancer Biology, Harvard School of Public Health, Boston, Massachusetts 02115.
2. BARRY HALL
Department of Molecular and Cell Biology, University of Connecticut, Storrs 06269.
3. BERNARD DAVIS
Bacterial Physiology Unit, Harvard Medical School, Boston, MA 02115.
4. KOICHIRO MATSUNO
Department of BioEngineering, Nagaoka University of Technology, Japan.
5. KONSTANTIN CHUMAKOV
Center for Biologics Evaluation and Research, Food and Drug Administration, Rockville,
Maryland 20852, USA.
6. MIKHAIL V. IVANOV
Institute of Microbiology, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 7, k. 2,
Moscow, 117811 Russia.
, as well as to all participants of the discussion ‘Origin of mutants disputed’ (Nature 336, 525 -
526 (08 December 1988)) :
1. D. CHARLESWORTH, B. CHARLESWORTH & J. J. BULL
Department of Ecology and Evolution, University of Chicago, 915 East 57th Street, Chicago,
Illinois 60637, USA
Department of Zoology, University of Texas, Austin, Texas 78712, USA
2. ALAN GRAFEN
Animal Behaviour Research Group, Zoology Department, Oxford University, Oxford OX1 3PS,
3. R. HOLLIDAY & R. F. ROSENBERGER
CSIRO Laboratory for Molecular Biology, North Ryde, Sydney, Australia
Genetics Division, National Institute for Medical Research, Mill Hill, London NW7 1AA, UK
4. LEIGH M. VAN VALEN
Department of Ecology and Evolution, University of Chicago, 915 East 57Street, Chicago,
Illinois 60637, USA
5. ANTOINE DANCHIN
Institut Pasteur, 28 Rue Dr. Roux, 75724 Paris, Cedex 15, France
6. IRWIN TESSMAN
Departments of Biiological Sciences, Purdue University, West Lafayette,
Indiana 47907, USA
|
0704.0035 | Convergence of the discrete dipole approximation. II. An extrapolation
technique to increase the accuracy | Microsoft Word - DDAextrapolation_preprint.doc
Convergence of the discrete dipole approximation.
II. An extrapolation technique to increase the accuracy.
Maxim A. Yurkin
Faculty of Science, Section Computational Science, of the University of Amsterdam,
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya 3, Novosibirsk 630090 Russia
[email protected]
Valeri P. Maltsev
Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya 3, Novosibirsk 630090 Russia
Novosibirsk State University, Pirogova Str. 2, 630090, Novosibirsk, Russia
Alfons G. Hoekstra
Faculty of Science, Section Computational Science, of the University of Amsterdam,
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands
[email protected]
Abstract
We propose an extrapolation technique that allows accuracy improvement of the
discrete dipole approximation computations. The performance of this technique was
studied empirically based on extensive simulations for 5 test cases using many different
discretizations. The quality of the extrapolation improves with refining discretization
reaching extraordinary performance especially for cubically shaped particles. A two
order of magnitude decrease of error was demonstrated. We also propose estimates of
the extrapolation error, which were proven to be reliable. Finally we propose a simple
method to directly separate shape and discretization errors and illustrated this for one
test case.
Keywords: discrete dipole approximation, non-spherical particle light scattering, accuracy,
extrapolation
OCIS code: 290.5850, 260.2110, 000.4430
mailto:[email protected]
mailto:[email protected]
1. Introduction
The discrete dipole approximation (DDA) is a well-known method to solve the light
scattering problem for arbitrary shaped particles. Since its introduction by Purcell and
Pennypacker1 it has been improved constantly. The formulation of DDA summarized by
Draine and Flatau2 more than 10 years ago is still most widely used for different applications,3
partly due to the publicly available high-quality and user-friendly code DDSCAT.4
DDA directly discretizes the volume of the scatterer and hence is applicable to arbitrary
shaped particles. However, the drawback of this discretization is the extreme computational
complexity of DDA of O(N 2), where N is the number of dipoles. This complexity is
decreased to O(NlogN) by advanced numerical techniques.2,5 Still the usual application
strategy for DDA is “single computation”, where a discretization is chosen based on available
computational resources and some empirical estimates of the expected errors.3,4 These error
estimates are based on a limited number of benchmark calculations3 and hence are external to
the light scattering problem under investigation. Such error estimates have evident drawbacks,
however no better alternative is available.
Usually errors in DDA are studied as a function of the size parameter of the scatterer x
(at a constant or few different values of N), e.g. 2,6. Only several papers directly present errors
versus discretization parameter (e.g. d – the size of a single dipole).7-15 The range of d
typically studied in those papers is limited to a 5 times difference between minimum and
maximum values (with the exception of two papers9,10 where it is 15 times). Only two
papers7,15 use extrapolation (to zero d) to get an exact result of some measured quantity,
however they use the simplest linear extrapolation without any theoretical foundation nor
discussion of its capabilities.
It is acknowledged for a long time that DDA errors are due to two different factors:
shape (it is not always possible to describe the particle shape exactly by a collection of
cubical cells) and discretization (finite size of each cell).6 However, the question which of
them is more important in different cases is still open. A discussion on this issue spanned
through several papers16-20 that have not reached any definite conclusions yet. The uncertainty
is due to the indirect methods used that have inherent interpretation problems.
In accompanying paper,21 that from now on we will refer to as Paper 1, we performed a
theoretical analysis of DDA convergence when refining the discretization. It provides the
basis for this paper, where an extrapolation technique is introduced (Section 2) to improve the
accuracy of DDA computations. We thoroughly discuss all free parameters that influence
extrapolation performance and provide a step-by-step prescription, which can be used with
any existing DDA code without any modifications. It is important to note that although
Paper 1 provides a firm theoretical background, it is not necessary to go through all
theoretical details to understand and apply the extrapolation technique that we introduce here.
In Section 3 we present extensive numerical results of DDA computations for 5 different
scatterers using many different discretizations. These results are discussed in Section 4 to
evaluate the performance of the extrapolation technique. We also propose a new method to
directly separate shape and discretization errors of DDA (described and illustrated in
Section 3.B). The results and possible applications are discussed in Section 4. We formulate
the conclusions of the paper in Section 5.
2. Extrapolation
In this section we describe a straightforward technique to significantly increase the accuracy
of a DDA simulation with a relatively small increase of computation time. This technique
does not require any modification of a DDA program but only postprocessing of computed
data. Therefore it can be easily implemented in any existing DDA code.
In Paper 1 we have proven that the error of any measured quantity is bounded by a
quadratic function of the discretization parameter mkdy = (k – free space wave vector, m –
refractive index of the scatterer):
( ) ( )yybayybay lnlnδ 11222 φφφφφ −+−≤ , (1)
where φ y is some measured quantity (e.g. extinction efficiency Qext, Mueller matrix elements
at some scattering angle Sij(θ ), etc.) and δφ y its error (difference between a result of the
numerical simulation and an exact value). , are constants (independent on y), which
are described in detail in Paper 1.
Here we proceed and assume that for sufficiently small y, δφ y can in fact be
approximated by a quadratic function of y (taking the logarithmic term as a constant). The
applicability of this assumption will be tested empirically in Section 3.B. Introduction of
higher-order terms is possible but not necessary (contrary to the quadratic term), and we avoid
it in order to keep our technique as simple and robust as possible. We can now write:
yy yayaa ζφ +++= 2210 , (2)
where a0, a1, a2 are constants that are chosen such that ζ y – the error of the approximation – is
minimized. a0 is then an estimate for the exact value of the measured quantity φ 0. A
procedure to determine a0 is basically fitting of a quadratic function over several points
, which are obtained by a standard DDA simulation. In the ideal case of one
can use any three values of y to obtain the exact value of φ
},{ yy φ 0=yζ
0. However, in practice different
fits will always give different results. We limit ourselves to the usual least-square polynomial
fit of the data. There are three question one should answer before conducting such a fit:
1) how many and which values of y to use?
2) how to weight the influence of different calculated values used in the fitting, i.e. what is
the behavior of expected errors ζ y? (Note that in the polynomial fit we minimize χ2, the
summation of the squared difference between computed values and the fitting function
weighted by the inverse of the expected error ζ y.)
3) how to estimate the difference between a0 and φ 0, i.e. the error of the final result?
It is important to note that, although there are some theoretical hints, answers to these
questions are mainly empirical and should be tested. Our approach is based on the test cases
presented in Section 3.B. These may not be representative for all scattering problems, but they
do show the potential power of our approach. We do not attempt to choose the most suitable
fit options, but merely demonstrate the applicability of the technique.
We start by analyzing the second question, i.e. what is the expected deviation from the
quadratic model, i.e. what is the functional dependence of ζ y on y, to be used as weighting
function in the polynomial fitting procedure? For cubically shaped particles, defined in
Paper 1 as particles whose shape can be exactly discretized using cubical subvolumes, one
expects a smooth variation of the function φ y, and the error can be attributed as a model error,
i.e. coming mainly from neglecting higher order terms in the convergence analysis of Paper 1.
In that case the error ζ y is expected to be a cubical function of y. We have tried cubical,
quadratic and linear error functions when fitting results for cubically shaped particles and
found that, although the differences are small, cubical errors generally lead to the best fits
(data not shown).
Shape errors, which are present for non-cubically shaped particles, are expected to be
very sensitive to y, because they depend upon the position of the particle surface inside the
boundary dipole that changes considerably by a small variation of y (for details see Paper 1).
Therefore shape errors can be viewed as random noise superimposed upon a smooth variation
of φ y. The asymptotic behavior of shape errors is linear in y (see Paper 1). Indeed, in certain
cases we found that using linear errors ζ y results in significantly better fits than when using
cubical errors. However in other cases linear errors performed significantly worse. In our
experience, using a cubical error function is in general always more reliable, even in the
presence of shape errors, because it decreases the influence of points with high values of y,
where the error is larger and less predictable. Since we want the procedure to be as robust as
possible and not to use more complex error functions than strictly needed (e.g. polynomial),
we take a cubical dependence of the error ζ y, both for cubically- and non-cubically shaped
scatterers.
The choice of values of y for computation can be described by the interval [ymin,ymax],
the number of points and their spacing. ymin is usually determined by available computer
hardware (time or memory bounds), that is the best discretization that can be computed for a
given resource. The goal of the extrapolation procedure is to increase the accuracy beyond
this “single DDA boundary”. We will show in Section 3.B that the overall performance of this
technique strongly depends upon ymin.
The choice of ymax is governed by two notions: a larger interval of data points generally
leads to better extrapolation but errors for high values of y are more random and their
significance is anyway much smaller (since we use a cubical error function). We have found
that for cubically shaped scatterers a good choice is minmax 2yy = , while for non-cubically
shaped scatterers increasing the interval to minmax 4yy = does improve the fits. Probably that
is due to the fact that the quality of fit for non-cubically shaped scatterers is determined by
quasi-random shape errors and increasing the range leads to larger statistical significance of
the result. We will also demand that ymax is less than 1, since otherwise DDA is definitely far
from its asymptotic behavior.
Spacing of the sample points depends partly on the problem, especially for cubically
shaped scatterers (in that case an arbitrary number of dipoles cannot be used). We space
computational points approximately uniform on a logarithmic scale, acknowledging the fact
that a relative difference in y is more significant than an absolute. The total number of points
should be large enough for statistical significance. However, a large number of points
increases computational time. We have used 5 points for cubically shaped particles (ratio of
y1 values is 8:7:6:5:4) and 9 points for non-cubically shaped particles (ratio of y1 values is
16:14:12:10:8:7:6:5:4) or less if minmax 4yy < .
The estimation of the error of the final result is difficult since this error is due to model
imperfection and not to some kind of random noise. The standard least-square fitting
technique22 provides a standard error (SE) for the parameter a0, which we use as a starting
point. Numerical simulations (Section 3.B) show that for spheres (the only non-cubical shape
we studied) real errors are less than 2×SE in most cases. That is what one would expect if ζ y
is considered completely random (which is similar to the expected behavior of the shape
errors). For cubical shapes, on the contrary, we have to estimate the error as 10×SE to reliably
describe the real errors. It is important to note that an error estimate based on the SE is the
simplest one can use. Its drawback is that we have to use a large multiplier (based on the real
errors obtained in some of our simulations), which may lead to significant overestimation of
real errors in certain cases.
We can now formulate the step-by-step extrapolation technique. We use abbreviations
(c) and (nc) for cubically and non-cubically shaped scatterers respectively.
1) Select ymin based on your computational resources.
2) Take ymax to be 2 (c) or 4 (nc) times ymin but not larger than 1.
3) Choose 5 (c) or 9 (nc) points over the interval [ymin,ymax] approximately uniformly
spaced on a logarithmic scale.
4) Perform DDA computations for each y.
5) Fit the quadratic function (Eq. (2)) over the points using y},{ yy φ 3 as errors of data
points; a0 is then the estimate of φ 0.
Multiply SE of a0 by 10 (c) or 2 (nc) to obtain an estimate of the extrapolation error.
Results of using this procedure are presented in Section 3, together with computational costs.
The extrapolation procedure is similar to a Romberg integration method,22 which is
adaptive. The error estimate, obtained by extrapolation, is an internal accuracy indicator of
DDA computations that is just as important as the increase in the accuracy itself. Our error
estimate opens the way to adaptive DDA, i.e. a code that will reach a required accuracy, using
minimum computational resources.
3. Numerical simulations
A.Discrete Dipole Approximation
The basics of the DDA method were summarized by Draine and Flatau.2 In this paper we use
the LDR prescription for dipole polarizability,23 which is most widely used nowadays, e.g. in
the publicly available code DDSCAT 6.1.4 We also employ dipole size correction6 for non-
cubically shaped scatterers to ensure that the cubical approximation of the scatterer has the
correct volume; this is believed to diminish shape errors, especially for small scatterers.2 We
use a standard discretization scheme without any improvements for boundary dipoles.
The main numerical challenge of DDA is to solve a large system of 3N linear equations.
This is done iteratively using some Krylov-subspace method,22 while the matrix-vector
products are computed using an FFT-based algorithm.5 Our code – Amsterdam DDA
(ADDA) – is capable of running on a cluster of computers (parallelizing a single DDA
computation), which allows us to use practically an unlimited number of dipoles, since we are
not limited by the memory of a single computer.24,25 We used a relative error of residual
as a stopping criterion. Tests suggest that the relative error of the measured quantities
due to the iterative solver is then (data not shown) and hence can be neglected (total
relative errors in our simulations are – see Section
810−<
710−<
56 1010 −− ÷> 3.B). All DDA simulations
were carried out on the Dutch national compute cluster LISA.26
The execution time of one iteration depends solely on N, it consists of an arithmetic part
which scales linearly with N and an FFT part which scales as NlnN. The number of iterations
only slightly depends on the discretization parameter y for fixed geometry of the scatterer.
Rahola proved this theoretically for any Krylov-subspace method,27 and our own experience
agrees with this conclusion. Therefore the total computational time scales linearly with N
( ) or slightly faster (considering logarithm and imperfect optimization), which is
consistent with our timing results (data not shown).
3−∝ y
We can now estimate the computational overhead of the extrapolation technique
compared to a single DDA computation for ymin (time – t(ymin)). Considering the spacing of
points we used (described in Section 2) the execution time needed for 5 points computation is
and for the 9 points computation – )(5.2 min5 ytt < )(7.2 min9 ytt < . Memory requirements are
the same as for a single computation. For comparison one should note that an 8 times increase
in computational time and memory requirements (for single DDA computation with
Fig. 1. Cubical discretization of a sphere using 16 dipoles per diameter (total 2176 dipoles).
2minyy = ) gives only a 2 to 4 times increase in accuracy (depending in which error regime –
linear or quadratic – ymin is located).
B.Results
We study five test cases: one cube with 8=kD , three spheres with , and a
particle obtained by a cubical discretization of the
30,10,3=kD
10=kD sphere using 16 dipoles per D
(total 2176 dipoles, see Fig. 1, x equal to that of a sphere). By D we denote the diameter of a
sphere or the edge size of a cube. All scatterers are homogenous with . Although
DDA errors significantly depend on m (see e.g.
5.1=m
12), we limit ourselves to one single value and
study the effects of size and shape of the scatterer.
The maximum number of dipoles per D (nD) was 256. The values of nD that we used are
of the form (p is an integer), except for the discretized sphere, where all np2}7,6,5,4{ ⋅ D are
multiples of 16 (this is required to exactly describe the shape of the particle composed from a
number of cubes – see Fig. 1). The minimum values for nD were 8 for the sphere, 16
for the cube, the sphere, and the discretized sphere, and 40 for the sphere.
10=kD 30=kD
Typical computation time for the finest discretization (for the cube with ,
resulting in ) currently is 2.5 hours on a cluster of 64 P4-3.4 GHz processors. We
expect that it can be improved by an order of magnitude by using modern FFT routines (e.g.
fastest Fourier transform in the West – FFTW
047.0=y
7107.1 ⋅=N
28) and a faster iterative solvers (bi-conjugate
gradient stabilized or quasi-minimal residual that were shown to be clearly superior to
CGNR29,30 that we still use). We are currently improving our code along these lines.
All computations use a direction of incidence parallel to one of the principal axes of the
cubical dipoles. The scattering plane is parallel to one of the faces of the cubical dipoles. In
this paper we show results only for the extinction efficiency Qext (for incident light polarized
parallel to one of the principal axes of the cubical dipoles) and phase function S11(θ ) as the
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.
most commonly used in applications. However, the extrapolation technique is equally
applicable to any measured quantity. For instance, we have also applied it for other Mueller
matrix elements (data not shown).
Reference (exact) results of S11(θ ) and Qext for spheres are obtained by Mie theory (the
relative accuracy of the code we use31 is at least ). Unfortunately, no analytical theory
is available for the cube and the discretized sphere, which could provide us with exact results.
Instead, we use extrapolation over the 5 finest discretizations as reference results for these
shapes.
610−<
To justify this choice we discuss, as an example, simulation results of Qext for the cube.
Instead of showing values of Qext itself, we show in Fig. 2a ( )10ext −aQ , with a0 obtained
through fitting the 5 finest discretizations. The extrapolation through these 5 best points
( , ) is also shown. The deviation of the fit from the five best points
(that overlap on
047.0min =y 094.0max =y
Fig. 2(a)) is very small indeed. This is also characterized by a small estimate
of the extrapolation error (see 6108.1 −× Table 1). In Paper 1 we proved that DDA converges
to the exact solution, therefore the result of the best extrapolation should be close to the exact
result. The relative difference between the best discretization and the best extrapolation is
only , therefore it does not make a big difference which one to use as a reference
when evaluating, for instance, the error of the extrapolation through the 5 worst
5100.9 −×
cube kD=8
fit (5 best points)
discretized sphere kD=10
fit (5 best points)
(b) sphere kD=3
fit (9 best points)
sphere kD=10
fit (9 best points)
sphere kD=30
fit (9 best points)
y=kd·m
Fig. 2. Signed relative errors of Qext versus y and their fits by quadratic functions for (a) kD = 8 cube
and discretized kD = 10 sphere, (b) 3 spheres. 5 and 9 best points are used for fits in (a) and (b)
respectively.
Table 1. Extrapolation errors of Qext. Estimate of the extrapolation errors is 10×SE for first two
particles and 2×SE for spheres.
Extrapolation ymin ymax Points Error for ymin Estimate Real
kD = 8 cube
0.047 0.094 5 9.0×10-5 1.8×10-6 ––––
0.094 0.19 5 1.6×10-4 6.6×10-6 4.6×10-6
0.19 0.38 5 2.2×10-4 5.3×10-5 4.0×10-5
0.38 0.75 5 1.1×10-4 3.7×10-4 3.2×10-4
Discretized kD = 10 sphere
0.058 0.12 5 1.0×10-4 2.4×10-5 ––––
0.12 0.23 5 2.0×10-4 9.0×10-6 7.9×10-6
0.23 0.93 4 4.3×10-4 1.2×10-3 5.9×10-4
kD = 3 sphere
0.018 0.070 9 2.2×10-4 1.0×10-5 4.1×10-6
0.035 0.14 9 4.0×10-4 5.9×10-5 4.8×10-5
0.070 0.28 9 6.8×10-4 8.7×10-5 5.7×10-6
0.14 0.54 9 9.0×10-4 3.7×10-4 7.0×10-4
0.28 0.54 5 2.4×10-4 4.3×10-3 1.8×10-3
kD = 10 sphere
0.059 0.23 9 2.7×10-4 2.0×10-4 2.7×10-5
0.12 0.47 9 5.5×10-4 5.5×10-4 3.7×10-4
0.23 0.93 9 1.5×10-3 3.1×10-3 2.1×10-3
kD = 30 sphere
0.18 0.70 9 3.8×10-4 1.3×10-3 1.4×10-3
0.18 0.35 5 3.8×10-4 3.3×10-3 6.9×10-4
Table 2. Comparison of shape and discretization errors of Qext for kD = 10 sphere discretized with
y = 0.93. All errors are relative to the best extrapolation result for the discretized sphere.
Shape Discretization Total
Error 3.1×10-3 8.3×10-3 5.2×10-3
discretizations ( 38.0min =y , 75.0max =y ). Hence all conclusions with respect to the
reliability of the error estimates (as discussed in Section 4) do not depend on the choice of
reference if ymin is large enough. We also apply this reasoning to smaller ymin and assume that
using the reference value obtained by extrapolation of the finest discretizations is a good
enough estimate of the exact value.
The same justification is valid for the discretized sphere (see Table 1 for Qext results).
Comparison of errors of different extrapolations results of S11(θ ) (shown in Fig. 3 and Fig. 4)
is even more convincing. Reference results themselves (both of Qext and S11(θ )) can be found
in Paper 1.
Next we show the results obtained by the extrapolation technique. The dependence of
the signed relative errors of Qext on y for all 5 test cases are shown in Fig. 2. Fig. 2(a) depicts
results for the cube and the discretized sphere. The 5 best points for each scatterer are fitted
by a quadratic function, using the method described in Section 2. Fig. 2(b) depicts
extrapolation results for spheres, using the 9 best points for each of them (cf. Section 2). Since
the exact Mie solution is available, intersection of a fit with a vertical axis is a measure of the
accuracy of extrapolation result. Table 1 summarizes the parameters (ymin, ymax, number of
points) of all the extrapolations, which were carried out, and their performance for Qext.
0 30 60 90 120 150 180
Scattering angle θ, deg
y = 0.75
y = 0.38
extrapolation
estimate
y = 0.19
y = 0.094
extrapolation
estimate
y = 0.094
y = 0.047
extrapolation
(estimate)
Fig. 3. Errors of S11(θ ) in logarithmic scale for extrapolation using 5 values of y in the intervals (a)
[0.047,0.094], (b) [0.094,0.19], and (c) [0.38,0.75] for kD = 8 cube. Estimate of the extrapolation error
is 10×SE.
Next we present some of the extrapolations results for S11(θ ). Results for the cube are
shown in Fig. 3. Each subfigure shows real (compared to the best extrapolation – reference)
and estimated errors together with the errors of the finest and crudest discretizations used.
Only the estimate of the error is shown for the best extrapolation – Fig. 3(a). Fig. 3(b) and (c)
show extrapolation results using 5 points in the intervals [0.094,0.19] and [0.38,0.75]
respectively. The performance of the extrapolation for the discretized sphere is shown in Fig.
4: (a) – best extrapolation, (b) and (c) – results for extrapolation using 5 and 4 points in the
intervals [0.12,0.23] and [0.23,0.93] respectively. The broad spacing of points for
0 30 60 90 120 150 180
Scattering angle θ, deg
y = 0.93
y = 0.23
extrapolation
estimate
y = 0.23
y = 0.12
extrapolation
estimate
y = 0.12
y = 0.058
extrapolation
(estimate)
Fig. 4. Errors of S11(θ ) in logarithmic scale for extrapolation using 5 values of y in the intervals (a)
[0.058,0.12], (b) [0.12,0.23] ((c): 4 values of y in the interval [0.23,0.93]) for the discretized kD = 10
sphere. Estimate of the extrapolation error is 10×SE.
extrapolation depicted in Fig. 4(c) is, as was noted above, due to the complex shape of the
discretized sphere that limits possible values of y to be 0.93 divided by an integer (total time
for computing these 4 points is ). It is important to note once more that we use
10×SE as an estimate of extrapolation error for the cube and discretized sphere and 2×SE for
spheres (cf. Section
)(6.1 minyt<
Extrapolation results for the 3=kD sphere are summarized in Fig. 5: (a) shows the best
extrapolation (using 9 points in the interval [0.018,0.070]), and (b) shows the worst, but still
satisfactory result, i.e. one that shows definite improvement of accuracy over most of the θ
0 30 60 90 120 150 180
Scattering angle θ, deg
y = 0.55
y = 0.14
extrapolation
estimate
y = 0.070
y = 0.018
extrapolation
estimate
Fig. 5. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the intervals (a)
[0.018,0.070], (b) [0.14,0.55] for kD = 3 sphere. Estimate of the extrapolation error is 2×SE.
range. The extrapolation using 5 points from the interval [0.28,0.54] is no longer satisfactory
(data not shown). Errors of the two best extrapolations for the 10=kD sphere (using 9 points
from the intervals [0.059,0.23] and [0.12,0.47]) are shown in Fig. 6(a) and (b) respectively. A
third extrapolation for sphere is not satisfactory (data not shown). Both
extrapolations for the sphere show similar controversial results, only one of them (9
points from the interval [0.18,0.70]) that is overall slightly better is shown in
10=kD
30=kD
Fig. 7. The
estimate of the extrapolation error is overall slightly higher than the real errors of the
extrapolation (data not shown).
Results of S11(θ ) for all extrapolations (see Table 1) support the following trend: the
quality of the extrapolation (defined as decrease of error compared to a single DDA
computation for ymin) rapidly degrades with increasing ymin. The ratio of estimated to real
errors increase with increasing ymin (that can be considered as a degradation of the estimate
quality).
Computation of exact results for both the 10=kD sphere and its cubical discretization
( ) allows us for the first time to directly separate and compare shape and
discretization error of single DDA computations. The shape error is the difference between
some measured quantity for a discretized sphere (calculated to a high accuracy) and that for
the exact sphere. The discretization error is difference between calculation using a limited
number of dipoles (2176) and exact (very accurate) solution for the cubical discretization of
the sphere (first curve in
93.0=y
Fig. 4(c)). The total error is just the sum of the two. These three
0 30 60 90 120 150 180
Scattering angle θ, deg
y = 0.47
y = 0.12
extrapolation
estimate
y = 0.23
y = 0.059
extrapolation
estimate
Fig. 6. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the intervals (a)
[0.059,0.23], (b) [0.12,0.47] for kD = 10 sphere. Estimate of the extrapolation error is 2×SE.
types of errors for S11(θ ) are shown in Fig. 8, all relative to the exact value for discretized
sphere. Errors of Qext are shown in Table 2.
4. Discussion
In their review Draine and Flatau2 gave the condition 1<y for applicability of DDA. Usually
(10 dipoles per wavelength in the medium) is used in applications.6.0=y 3 Smaller y are
used only in studies of DDA errors2,12,13 or of light scattering by particles much smaller than a
wavelength (then d is determined by a shape of a scatterer, and y, being proportional to
scatterer size, can be arbitrarily small).32 However, if one wishes to achieve better (than usual)
accuracy of a DDA simulation, smaller y must be used.
The best extrapolation for the cube (Fig. 3(a)) shows a large improvement compared to
the best single DDA calculation (it should be noted, however, that this result is based on the
empiric error estimate). Maximum errors are decreased more than 2 orders of magnitude. This
would be impossible to reach by a single DDA calculation as it will require over 6 orders of
magnitude increase in execution time and memory, since there is only linear convergence for
such small y. Even for the extrapolation can be called satisfactory because the
maximum error is decreased almost two times when considering the estimate of the error (the
real errors are even less). It is important to note that an estimate of the error is important by
38.0min =y
0 30 60 90 120 150 180
Scattering angle θ, deg
y = 0.18
extrapolation
Fig. 7. Errors of S11(θ ) in logarithmic scale for extrapolation using 9 values of y in the interval
[0.18,0.70] for kD = 30 sphere.
0 30 60 90 120 150 180
Scattering angle θ, deg
discretization
shape
total
Fig. 8. Comparison of discretization and shape errors of S11(θ ) for kD = 10 sphere discretized using 16
dipoles per D ( y = 0.93).
itself (even when it is not less than the error of a single DDA computation) because it does not
require an exact solution (that is usually unavailable in real applications). In general, the
extrapolation decreases large errors better than those that are already small, i.e. it may
significantly decrease maximum errors but prove less satisfactory for certain measured
quantity (e.g. S11 for certain θ). This conclusion holds true for all the extrapolations we
performed (Fig. 3 – Fig. 7 and those not shown).
Extrapolation results for the discretized sphere (Fig. 4) are similar to those for the cube.
Extrapolations for and 0.12 are very good (more than an order of magnitude
decrease of maximum errors), while for
058.0min =y
23.0min =y it is on the edge of being satisfactory.
The latter is strongly influenced by the fact that only 4 points in a broad interval are used
(hence it does not fully comply with the procedure specified in Section 2).
The best extrapolation for the 3=kD sphere (Fig. 5(a)) shows results comparable to
cubically shaped scatterers, however it uses an extremely small 018.0min =y . Already for
(14.0min =y Fig. 5(b)) it only decreased the maximum errors by a factor of two. A similar
boundary value of ymin for satisfactory extrapolation is observed for sphere (10=kD Fig.
6(b)), while the best extrapolation (Fig. 6(a)) does show good results (4 times decrease of
maximum error), although significantly worse than the analogous results for cubically shaped
scatterers. Unfortunately we are currently not able to reach sufficiently small y for the
sphere and the best extrapolation (30=kD Fig. 7) uses rather large , resulting in
almost negligible improvement of accuracy.
18.0min =y
We have also studied a porous cube that was obtained by dividing a cube into
27 smaller cubes and then removing randomly 9 of them. All the conclusions are the same as
those reported for the cube, but with slightly higher overall errors (data not shown).
Extrapolation of Qext (Table 1) shows similar results as discussed above, however the
improvement of accuracy is generally less than for maximum errors in S11(θ ) (which is in
agreement with what we stated above, since errors in Qext are already small). Moreover, one
should take into account that errors of a single DDA calculation for some ymin are
unexpectedly small (e.g. the last extrapolations for the cube and the 3=kD sphere), but these
are just “lucky hits” near the points where the function crosses the horizontal axis
(cf.
)(δ ext yQ
Fig. 2).
Summarizing all results we can conclude that shape errors significantly degrade the
extrapolation performance, because of its abrupt behavior, and therefore the extrapolation
technique is much more suited for cubically shaped particles. One may expect satisfactory
extrapolation for non-cubically shaped particles only when 15.0min <y , while for cubically
shaped particles the condition is 4.0min <y . It is important to note though that extrapolation
can be used for any ymin. The estimate of the error coming from the fitting procedure (SE) can
then be used to decide whether this extrapolation was satisfactory or not. The quality of the
extrapolation significantly increases with decreasing ymin, hence extrapolation is of biggest
value for obtaining (very accurate) benchmark results. The size of the particle for which the
extrapolation technique provides significant improvement is mainly determined by available
computational resources that are required to reach small enough ymin. However, further testing
is required to evaluate the quality of extrapolation for scatterers large compared to the
wavelength.
It is important to note that the linear extrapolation that was applied in two papers7,15 may
lead to completely erroneous results (e.g. if points on the right branch of the parabolas for the
cube and sphere in 3=kD Fig. 2 are used). Quadratic extrapolation, as proposed in this
paper, is much more reliable.
Throughout all the extrapolations we have used error estimates as specified in Section 2:
10×SE and 2×SE for cubically and non-cubically shaped scatterers respectively. All the
results show that these estimates are reliable, i.e. in most cases real errors are less than the
estimates. There are only two exceptions, both for the 3=kD sphere: the fourth extrapolation
of Qext (Table 1) – real error 1.8 times larger than estimate – and second of S11 – real error
1.5-2 times larger than estimate for broad range of θ (data not shown). The existence of such
exceptions is acceptable since the estimates have a statistical nature of a confidence interval.
However, these estimates, though reliable, are definitely not optimal, i.e. they often
significantly overestimate the real errors (e.g. Fig. 5(a)). It also seems to be sensitive to the
spacing of y values used for extrapolation – cf. Fig. 4(c), where unusually broad spacing was
used. Generally this overestimation increases with increasing ymin. We can conclude that the
error estimate should be improved, and this is subject of future research. However, the current
estimate is already suitable for practical applications since they mainly require reliability of
the error estimate, which is demonstrated empirically in this paper.
It is important to note that we limited ourselves to a single value of m. While bounds of
ymin to obtain satisfactory extrapolation definitely dependent on m, other conclusions, such as
the reliability of the error estimate, are expected to hold true for a broad range of m. This can
be easily tested for specific values of m of interest using the methodology put forward in this
paper.
Finally we discuss the results presented in Fig. 8. One cannot conclude that shape errors
dominate over discretization errors (or the other way around): for some θ shape errors are
much larger than discretization, for others – vice versa. However, maximum errors occurring
in backscattering directions are definitely due to shape errors (ratio of maximum shape to
maximum discretization errors is about 4). Errors in Qext (Table 2) are, on the contrary, mostly
due to discretization (although they are almost two orders of magnitude smaller than
maximum errors of S11). One may expect shape errors to become even more important for
smaller values of y, since the linear component of discretization errors is significantly smaller
than that of shape errors (hence for large values of y shape errors scale linearly and
discretization – almost quadratically). Our single result principally shows different angle
dependence of shape and discretization errors of S11: shape errors have a clear tendency to
significantly increase towards backscattering, while the general trend of discretization errors
is uniform over the whole θ range.
We have presented a simple method to directly separate shape and discretization errors
and only one result for illustration. All previous comparisons of shape and discretization
errors had significant inherent interpretation problems that caused a lot of discussions about
their conclusions.16-20 Our method is free of such problems and therefore can be used for
rigorous study of shape errors in DDA. For instance, it can help to directly evaluate the
performance of different techniques to reduce such errors, e.g. weighted discretization (WD).9
Discretization errors are then the limit one can achieve by drastically reducing shape errors.
We have used a traditional DDA formulation2 to show that the extrapolation technique
can be used with current DDA codes (e.g. DDSCAT4) without any modifications. However,
as we showed in Paper 1 several modern improvements of DDA (namely integration of
Green’s tensor (IT)33 and WD) should significantly change the convergence behavior of DDA
computations and hence influence the performance of the extrapolation technique. IT should
completely eliminate the linear term for cubically shaped scatterers. This will improve the
accuracy especially for small y, and probably also improve the quality of the extrapolation for
such scatterers. WD should significantly decrease shape and hence total errors for non-
cubically shaped particles, moreover it should significantly decrease the amplitude of quasi-
random error oscillations because it takes into account the location of the interface inside the
boundary dipoles. Therefore WD should improve the quality of the extrapolation for non-
cubically shaped scatterers. Testing of extrapolation performance of DDA using IT and WD is
a subject of a future study.
5. Conclusion
Based on the theoretical convergence analysis as presented in Paper 1, we proposed an
extrapolation technique together with a step-by-step prescription, which allows accuracy
improvement of DDA computations. The performance of this technique was studied
empirically and we showed that it significantly suppresses maximum errors of S11(θ ) when
and 0.15 for cubically and non-cubically shaped scatterers respectively (for
). The quality of the extrapolation improves with decreasing y
4.0min <y
5.1=m min reaching
extraordinary performance especially for cubically shaped particles – more than two order of
magnitude decrease of error when 05.0min ≈y for wavelength-sized scatterers with 5.1=m
(total computational time for extrapolation is less than 2.7 times that for a single DDA
computation).
The proposed estimates of the extrapolation error were proven to be reliable, although
they can be improved to decrease overestimation of the errors in some cases. This error
estimate is completely internal, and hence can be used to create adaptive DDA – a code that
will automatically refine discretization to reach a required accuracy.
We also proposed a simple method to directly separate shape and discretization errors.
Maximum errors of S11(θ ) for the 10=kD sphere with 5.1=m , discretized using 16 dipoles
per diameter ( ) are mostly due to shape errors, however the same is not true for all
measured quantities. This method can be employed to rigorously study fundamental
properties of these two types of errors and to directly evaluate the performance of different
techniques aimed at reducing shape errors.
93.0=y
Our theory predicts that modern DDA improvements (namely IT and WD) should
significantly change the performance of the extrapolation technique, however numerical
testing of these predictions is left for future research.
Acknowledgements
We thank Gorden Videen and Michiel Min for valuable comments on earlier version of this
manuscript and Denis Shamonin for help with 3D graphics. Our research is supported by the
NATO Science for Peace program through grant SfP 977976.
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|
0704.0036 | A remark on the number of steady states in a multiple futile cycle | A remark on the number of steady states in a multiple futile cycle
Liming Wang and Eduardo D. Sontag
Department of Mathematics
Rutgers University, New Brunswick, NJ, USA
Abstract
The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling.
This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without
direct positive feedbacks. In this paper, we study the number of positive steady states of a general
multisite phosphorylation-dephosphorylation cycle, and how the number of positive steady states varies
by changing the biological parameters. We show analytically that (1) for some parameter ranges, there
are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n − 1
steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there
are at most three steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady
state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard
Michaelis-Menten quasi-steady state conditions, there is at most one steady state.
Keywords: futile cycles, bistability, signaling pathways, biomolecular networks, steady states
1 Introduction
A promising approach to handling the complexity of cell signaling pathways is to decompose pathways into
small motifs, and analyze the individual motifs. One particular motif that has attracted much attention in
recent years is the cycle formed by two or more inter-convertible forms of one protein. The protein, denoted
here by S0, is ultimately converted into a product, denoted here by Sn, through a cascade of “activation”
reactions triggered or facilitated by an enzyme E; conversely, Sn is transformed back (or “deactivated”)
into the original S0, helped on by the action of a second enzyme F . See Figure 1.
S S0 2S1
SSS nn−1n−2
Figure 1: A futile cycle of size n.
Such structures, often called “futile cycles” (also called substrate cycles, enzymatic cycles, or enzymatic
inter-conversions, see [1]), serve as basic blocks in cellular signaling pathways and have pivotal impact on
the signaling dynamics. Futile cycles underlie signaling processes such as GTPase cycles [2], bacterial
two-component systems and phosphorelays [3, 4] actin treadmilling [5]), and glucose mobilization [6], as
well as metabolic control [7] and cell division and apoptosis [8] and cell-cycle checkpoint control [9]. One
very important instance is that of Mitogen-Activated Protein Kinase (“MAPK”) cascades, which regulate
http://arxiv.org/abs/0704.0036v2
primary cellular activities such as proliferation, differentiation, and apoptosis [10–13] in eukaryotes from
yeast to humans.
MAPK cascades usually consist of three tiers of similar structures with multiple feedbacks [14–16].
Each individual level of the MAPK cascades is a futile cycle as depicted in Figure 1 with n = 2. Markevich
et al.’s paper [17] was the first to demonstrate the possibility of multistationarity at a single cascade
level, and motivated the need for analytical studies of the number of steady states. Conradi et al. studied
the existence of multistationarity in their paper [19], employing algorithms based on Feinberg’s chemical
reaction network theory (CRNT). (For more details on CRNT, see [31,32].) The CRNT algorithm confirms
multistationarity in a single level of MAPK cascades, and provides a set of kinetic constants which can
give rise to multistationarity. However, the CRNT algorithm only tests for the existence of multiple steady
states, and does not provide information regarding the precise number of steady states.
In [18], Gunawardena proposed a novel approach to the study of steady states of futile cycles. His
approach, which was focused in the question of determining the proportion of maximally phosphorylated
substrate, was developed under the simplifying quasi-steady state assumption that substrate is in excess.
Nonetheless, our study of multistationarity uses in a key manner the basic formalism in [18], even for the
case when substrate is not in excess.
In Section 2, we state our basic assumptions regarding the model. The basic formalism and background
for the approach is provided in Section 3. The main focus of this paper is on Section 4, where we derive
various bounds on the number of steady states of futile cycles of size n. The first result is a the lower
bound for the number of steady states. Currently available results on lower bounds, as in [29], can only
handle the case when quasi-steady state assumptions are valid; we substantially extend these results to the
fully general case by means of a perturbation argument which allows one to get around these restricted
assumptions. Another novel feature of our results in this paper is the derivation of an upper bound
of 2n − 1, valid for all kinetic constants. Models in molecular cell biology are characterized by a high
degree of uncertainly in parameters, hence such results valid over the entire parameter space are of special
significance. However, when more information of the parameters are available, sharper upper bounds can
obtained, see Theorems 4 and 5. We finally conclude our paper in Section 5 with a conjecture of an n+ 1
upper bound.
We remark that the results given here complement our work dealing with the dynamical behavior
of futile cycles. For the case n = 2, [25] showed that the model exhibits generic convergence to steady
states but no more complicated behavior, at least within restricted parameter ranges, while [27] showed a
persistence property (no species tends to be eliminated) for any possible parameter values. These papers
did not address the question of estimating the number of steady states. (An exception is the case n = 1,
for which uniqueness of steady states can be proved in several ways, and for which global convergence to
these unique equilibria holds [27].)
2 Model assumptions
Before presenting mathematical details, let us first discuss the basic biochemical assumptions that go into
the model. In general, phosphorylation and dephosphorylation can follow either distributive or processive
mechanism. In the processive mechanism, the kinase (phosphatase) facilitates two or more phosphorylations
(dephosphorylations) before the final product is released, whereas in the distributive mechanism, the kinase
(phosphatase) facilitates at most one phosphorylation (dephosphorylation) in each molecular encounter.
In the case of n = 2, a futile cycle that follows the processive mechanism can be represented by reactions
as follows:
S0 + E ←→ ES0 ←→ ES1 −→ S2 + E
S2 + F ←→ FS2 ←→ FS1 −→ S0 + F ;
and the distributive mechanism can be represented by reactions:
S0 + E ←→ ES0 −→ S1 + E ←→ ES1 −→ S2 + E
S2 + F ←→ FS2 −→ S1 + F ←→ FS1 −→ S0 + F.
Biological experiments have demonstrated that both dual phosphorylation and dephosphorylation in MAPK
are distributive, see [14–16]. In their paper [19], Conradi et al. showed mathematically that if either phos-
phorylation or dephosphorylation follows a processive mechanism, the steady state will be unique, which,
it is argued in [19], contradicts experimental observations. So, to get more interesting results, we assume
that both phosphorylations and dephosphorylations in the futile cycles follow the distributive mechanism.
Our structure of futile cycles in Figure 1 also implicitly assumes a sequential instead of a random
mechanism. By a sequential mechanism, we mean that the kinase phosphorylates the substrates in a
specific order, and the phosphatase works in the reversed order. This assumption dramatically reduces the
number of different phospho-forms and simplifies our analysis. In a special case when the kinetic constants
of each phosphorylation are the same and the kinetic constants of each dephosphorylation are the same,
the random mechanism can be easily included in the sequential case. Biologically, there are systems, for
instance the auto-phosphorylation of FGF-receptor-1, that have been experimentally shown to follow a
sequential mechanism [33].
To model the reactions, we assume mass action kinetics, which is standard in mathematical modeling
of molecular events in biology.
3 Mathematical formalism
In this section, we set up a mathematical framework for studying the steady states of futile cycles. Let us
first write down all the elementary chemical reactions in Figure 1:
S0 + E
koff0
kcat0
→ S1 + E
Sn−1 + E
konn−1
koffn−1
kcatn−1
→ Sn + E
S1 + F
loff0
lcat0
→ S0 + F
Sn + F
lonn−1
loffn−1
lcatn−1
→ Sn−1 + F
where kon0 , etc., are kinetic parameters for binding and unbinding, ES0 denotes the complex consisting of
the enzyme E and the substrate S0, and so forth. These reactions can be modeled by 3n + 3 differential-
algebraic equations according to mass action kinetics:
= −kon0s0e+ koff0c0 + lcat0d1
= −konisie+ koffi
ci + kcati−1ci−1 − loni−1sif + loffi−1
di + lcatidi+1, i = 1, . . . , n− 1
= konjsje− (koffj
+ kcatj )cj , j = 0, . . . , n− 1 (1)
= lonk−1skf − (loffk−1
+ lcatk−1)dk, k = 1, . . . , n,
together with the algebraic “conservation equations”:
Etot = e+
Ftot = f +
di, (2)
Stot =
The variables s0, . . . , sn, c0, . . . , cn−1, d1, . . . , dn, e, f stand for the concentrations of
S0, . . . , Sn, ES0, . . . , ESn−1, FS1, . . . , FSn, E, F
respectively. For each positive vector
κ =(kon0 , . . . , konn−1 , koff0
, . . . , koffn−1
, kcat0 , . . . , kcatn−1 ,
lon0 , . . . , lonn−1 , loff0
, . . . , loffn−1
, lcat0 , . . . , lcatn−1) ∈ R
(of “kinetic constants”) and each positive triple C = (Etot, Ftot, Stot), we have a different system Σ(κ, C).
Let us write the coordinates of a vector x ∈ R3n+3+ as:
x = (s0, . . . , sn, c0, . . . , cn−1, d1, . . . , dn, e, f),
and define a mapping
Φ : R3n+3+ × R
+ × R
+ −→ R
with components Φ1, . . . ,Φ3n+3 where the first 3n components are
Φ1(x, κ, C) = −kon0s0e+ koff0c0 + lcat0d1,
and so forth, listing the right hand sides of the equations (1), Φ3n+1 is
ci − Etot,
and similarly for Φ3n+2 and Φ3n+3, we use the remaining equations in (2).
For each κ, C, let us define a set
Z(κ, C) = {x |Φ(x, κ, C) = 0}.
Observe that, by definition, given x ∈ R3n+3+ , x is a positive steady state of Σ(κ, C) if and only if x ∈ Z(κ, C).
So, the mathematical statement of the central problem in this paper is to count the number of elements
in Z(κ, C). Our analysis will be greatly simplified by a preprocessing. Let us introduce a function
Ψ : R3n+3+ × R
+ × R
+ −→ R
with components Ψ1, . . . ,Ψ3n+3 defined as
Ψ1 = Φ1 +Φn+1
Ψi = Φi +Φn+i +Φ2n+i−1 +Ψi−1, i = 2, . . . , n
Ψj = Φj, j = n+ 1, . . . , 3n + 3.
It is easy to see that
Z(κ, C) = {x |Ψ(x, κ, C) = 0},
but now the first 3n equations are:
Ψi = lcati−1di − kcati−1ci−1 = 0, i = 1, . . . , n,
Ψn+1+j = konjsje− (koffj
+ kcatj )cj = 0, j = 0, . . . , n− 1
Ψ2n+k = lonk−1skf − (loffk−1
+ lcatk−1)dk = 0, k = 1, . . . , n,
and can be easily solved as:
si+1 = λi(e/f)si, (3)
di+1 =
fsi+1
, (5)
where
kcatiLMi
KMi lcati
, KMi =
kcati + koffi
, LMi =
lcati + loffi
, i = 0, . . . , n− 1. (6)
We may now express
0 si,
0 ci and
1 di in terms of s0, κ, e and f :
si = s0
1 + λ0
+ λ0λ1
+ · · ·+ λ0 · · ·λn−1
:= s0ϕ
ci = es0
+ · · ·+
λ0 · · ·λn−2
KMn−1
:= es0ϕ
, (7)
di = fs0
+ · · · +
λ0 · · ·λn−1
LMn−1
:= fs0ϕ
Although the equation Ψ = 0 represents 3n+3 equations with 3n+3 unknowns, next we will show that
it can be reduced to two equations with two unknowns, which have the same number of positive solutions
as Ψ = 0. Let us first define a set
S(κ, C) = {(u, v) ∈ R+ × R+ |G
1 (u, v) = 0, G
2 (u, v) = 0},
where G
1 , G
2 : R
+ −→ R are given by
1 (u, v) = v (uϕ
1(u)− ϕ
2(u)Etot/Ftot)− Etot/Ftot + u,
2 (u, v) = ϕ
0(u)ϕ
2 (u)v
2 + (ϕκ0 (u)− Stotϕ
2 (u) + Ftotuϕ
1 (u) + Ftotϕ
2 (u)) v − Stot.
The precise statement is as follows:
Lemma 1 There exists a mapping δ : R3n+3 −→ R2 such that, for each κ, C, the map δ restricted to
Z(κ, C) is a bijection between the sets Z(κ, C) and S(κ, C).
Proof. Let us define the mapping δ : R3n+3 −→ R2 as δ(x) = (e/f, s0), where
x = (s0, . . . , sn, c0, . . . , cn−1, d1, . . . , dn, e, f).
If we can show that δ induces a bijection between Z(κ, C) and S(κ, C), we are done.
First, we claim that δ(Z(κ, C)) ⊆ S(κ, C). Pick any x ∈ Z(κ, C), we have that x satisfies (3)-(5).
Moreover, Φ3n+2(x, κ, C) = 0 yields
Etot = e+ es0ϕ
and thus
1 + s0ϕ
1(e/f)
. (8)
Using Φ3n+1(x, κ, C) = 0 and Φ3n+2(x, κ, C) = 0, we get:
e(1 + s0ϕ
1(e/f))
f(1 + s0ϕ
2(e/f))
, (9)
which is G
1 (e/f, s0) = 0 after multiplying by 1 + s0ϕ
2(e/f) and rearranging terms.
To check that G
2 (e/f, s0) = 0, we start with Φ3n+3(x, κ, C) = 0, i.e.
Stot =
Using (7) and (8), this expression becomes
Stot = s0ϕ
Etots0ϕ
1(e/f)
1 + s0ϕ
1 (e/f)
Ftots0ϕ
2(e/f)
1 + s0ϕ
2(e/f)
= s0ϕ
eFtots0ϕ
1(e/f)
f(1 + s0ϕ
2(e/f))
Ftots0ϕ
2(e/f)
1 + s0ϕ
2(e/f)
where the last equality comes from (9).
After multiplying by 1 + s0ϕ
2 (e/f), and simplifying, we get
ϕκ0 (
)ϕκ2 (
)s20 +
)− Stotϕ
Ftotϕ
) + Ftotϕ
s0 − Stot = 0,
that is, G
2 (e/f, s0) = 0. since both G
1 (e/f, s0) and G
2 (e/f, s0) are zero, δ(x) ∈ S(κ, C).
Next, we will show that S(κ, C) ⊆ δ(Z(κ, C)). For any y = (u, v) ∈ S(κ, C), let the coordinates of x be
defined as:
s0 = v
si+1 = λiusi
1 + s0ϕ
1 (u)
di+1 =
fsi+1
for i = 0, . . . , n − 1. It is easy to see that the vector x = (s0, . . . , sn, c0, . . . , cn−1, d1, . . . , dn, e, f) satisfies
Φ1(x, κ, C) = 0, . . . ,Φ3n+1(x, κ, C) = 0. If Φ3n+2(x, κ, C) and Φ3n+3(x, κ, C) are also zero, then x is an
element of Z(κ, C) with δ(x) = y. Given the condition that G
i (u, v) = 0 (i = 1, 2) and u = e/f, v = s0,
we have G
1 (e/f, s0) = 0, and therefore (9) holds. Since
1 + s0ϕ
1(e/f)
in our construction, we have
Ftot = f(1 + s0ϕ
2(e/f)) = f +
To check Φ3n+3(x, κ, C) = 0, we use
2 (e/f, s0)
1 + s0ϕ
2(e/f)
2 (e/f, s0) = 0 and 1 + s0ϕ
2(e/f) > 0. Applying (7)-(9), we have
di = s0ϕ
0(e/f) +
eFtots0ϕ
1(e/f)
f(1 + s0ϕ
2 (e/f))
Ftots0ϕ
2(e/f)
1 + s0ϕ
2(e/f)
= Stot.
It remains for us to show that the map δ is one to one on Z(κ, C). Suppose that δ(x1) = δ(x2) = (u, v),
where
xi = (si0, . . . , s
0, . . . , c
n−1, d
1, . . . , d
i, f i), i = 1, 2.
By the definition of δ, we know that s10 = s
0 and e
1/f1 = e2/f2. Therefore, s1i = s
i for i = 0, . . . , n.
Equation (8) gives
1 + vϕκ1 (u)
= e2.
Thus, f1 = f2, and c1i = c
i , d
i+1 = d
i+1 for i = 0, . . . , n − 1 because of (3)-(5). Therefore, x
1 = x2, and δ
is one to one.
The above lemma ensures that the two sets Z(κ, C) and S(κ, C) have the same number of elements. From
now on, we will focus on S(κ, C), the set of positive solutions of equations G
1 (u, v) = 0, G
2 (u, v) = 0,
1 (u, v) = v (uϕ
1(u)− ϕ
2(u)Etot/Ftot)− Etot/Ftot + u = 0, (10)
2 (u, v) = ϕ
0(u)ϕ
2 (u)v
2 + (ϕκ0(u)− Stotϕ
2 (u) + Ftotuϕ
1 (u) + Ftotϕ
2(u)) v − Stot = 0. (11)
4 Number of positive steady states
4.1 Lower bound on the number of positive steady states
One approach to solving (10)-(11) is to view G
2 (u, v) as a quadratic polynomial in v. Since G
2 (u, 0) < 0,
equation (11) has a unique positive root, namely
−Hκ,C(u) +
Hκ,C(u)2 + 4Stotϕ
0(u)ϕ
2 (u)
2ϕκ0 (u)ϕ
2 (u)
, (12)
where
Hκ,C(u) = ϕκ0(u)− Stotϕ
2(u) + Ftotuϕ
1(u) + Ftotϕ
2(u). (13)
Substituting this expression for v into (10), and multiplying by ϕκ0 (u), we get
F κ,C(u) :=
−H̃κ,C(u) +
H̃κ,C(u)2 + 4Stotϕ
0 (u)ϕ
2 (u)
2ϕκ2 (u)
uϕκ1(u)−
ϕκ2(u)
ϕκ0(u)+uϕ
0 (u) = 0.
So, any (u, v) ∈ S(κ, C) should satisfy (12) and (14). On the other hand, any positive solution u of (14)
(notice that ϕκ0(u) > 0) and v given by (12) (always positive) provide a positive a solution of (10)-(11),
that is, (u, v) is an element in S(κ, C). Therefore, the number of positive solutions of (10)-(11) is the same
as the number of positive solutions of (12) and (14). But v is uniquely determined by u in (12), which
further simplifies the problem to one equation (14) with one unknown u. Based on this observation, we
have:
Theorem 1 For each positive numbers Stot, γ, there exist ε0 > 0 and κ ∈ R
+ such that the following
property holds. Pick any Etot, Ftot such that
Ftot = Etot/γ < ε0Stot/γ, (15)
then the system Σ(κ, C) with C = (Etot, Ftot, Stot) has at least n + 1 (n) positive steady states when n is
even (odd).
Proof. For each κ, γ, Stot, let us define two functions R+ × R+ −→ R as follows:
κ,γ,Stot(ε, u) = H
κ,(εStot,εStot/γ,Stot)(u) (16)
= ϕκ0(u)− Stotϕ
2(u) + ε
uϕκ1(u) + ε
ϕκ2(u),
κ,γ,Stot(ε, u) = F
κ,(εStot,εStot/γ,Stot)(u) (17)
κ,γ,Stot(ε, u) +
κ,γ,Stot(ε, u)2 + 4Stotϕ
0 (u)ϕ
2 (u)
2ϕκ2 (u)
(uϕκ1(u)− γϕ
2 (u))
− γϕκ0(u) + uϕ
0(u).
By Lemma 1 and the argument before this theorem, it is enough to show that there exist ε0 > 0 and κ ∈
+ such that for all ε ∈ (0, ε0), the equation F̃
κ,γ,Stot(ε, u) = 0 has at least n+1 (n) positive solutions
when n is even (odd). (Then, given Stot, γ, Etot, and Ftot satisfying (15), we let ε = Etot/Stot < ε0,
and apply the result.)
A straightforward computation shows that when ε = 0,
κ,γ,Stot(0, u) = Stot (uϕ
1(u)− γϕ
2(u))− γϕ
0 (u) + uϕ
= λ0 · · ·λn−1u
n+1 + λ0 · · ·λn−2
KMn−1
(1− γβn−1)− γλn−1
+ · · ·+ λ0 · · ·λi−2
KMi−1
(1− γβi−1)− γλi−1
ui + · · · (18)
(1− γβ0)− γλ0
u− γ,
where the λi’s and KMi ’s are defined as in (6), and βi = kcati/lcati . The polynomial F̃
κ,γ,Stot(0, u)
is of degree n + 1, so there are at most n + 1 positive roots. Notice that u = 0 is not a root because
κ,γ,Stot(0, u) = −γ < 0, which also implies that when n is odd, there can not be n + 1 positive roots.
Now fix any Stot and γ. We will construct a vector κ such that F̃
κ,γ,Stot(0, u) has n+ 1 distinct positive
roots when n is even.
Let us pick any n+ 1 positive real numbers u1 < · · · < un+1, such that their product is γ, and assume
(u− u1) · · · (u− un+1) = u
n+1 + anu
n + · · · + a1u+ a0, (19)
where a0 = −γ < 0 keeping in mind that ai’s are given. Our goal is to find a vector κ ∈ R
+ such that
(18) and (19) are the same. For each i = 0, . . . , n − 1, we pick λi = 1. Comparing the coefficients of u
in (18) and (19), we have:
(1 + a0βi) = ai+1 − a0 − 1. (20)
Let us pick KMi > 0 such that
(ai+1 − a0 − 1)− 1 < 0, then take
(ai+1 − a0 − 1)− 1
in order to satisfy (20). From the given
λ0, . . . , λn−1,KM0 , . . . ,KMn−1 , β0, . . . , βn−1,
we will find a vector
κ =(kon0 , . . . , konn−1 , koff0
, . . . , koffn−1
, kcat0 , . . . , kcatn−1 ,
lon0 , . . . , lonn−1 , loff0
, . . . , loffn−1
, lcat0 , . . . , lcatn−1) ∈ R
such that βi = kcati/lcati , i = 0, . . . , n− 1, and (6) holds. This vector κ will guarantee that F̃
κ,γ,Stot(0, u)
has n + 1 positive distinct roots. When n is odd, a similar construction will give a vector κ such that
κ,γ,Stot(0, u) has n positive roots and one negative root.
One construction of κ (given λi,KMi , βi, i = 0, . . . , n − 1) is as follows. For each i = 0, . . . , n − 1, we
start by defining:
LMi =
λiKMi
consistently with the definitions in (6). Then, we take
koni = 1, loni = 1,
koffi
= αiKMi , kcati = (1− αi)KMi , lcati =
1− αi
KMi , loffi
= LMi − lcati ,
where αi ∈ (0, 1) is chosen such that
loffi
= LMi −
1− αi
KMi > 0.
This κ satisfies βi = kcati/lcati , i = 0, . . . , n− 1, and (6).
In order to apply the Implicit Function Theorem, we now view the functions defined by formulas in
(16) and (17) as defined also for ε ≤ 0, i.e. as functions R×R+ −→ R. It is easy to see that F̃
κ,γ,Stot(ε, u)
is C1 on R × R+ because the polynomial under the square root sign in F̃
κ,γ,Stot(ε, u) is never zero. On
the other hand, since F̃
κ,γ,Stot(0, u) is a polynomial in u with distinct roots, ∂F̃
κ,γ,Stot
(0, ui) 6= 0. By the
Implicit Function Theorem, for each i = 1, . . . , n+ 1, there exist open intervals Ei containing 0, and open
intervals Ui containing ui, and a differentiable function
αi : Ei → Ui
such that αi(0) = ui, F̃
κ,γ,Stot(ε, αi(ε)) = 0 for all ε ∈ Ei, and the images αi(Ei)’s are non-overlapping. If
we take
(0, ε0) :=
(0,+∞),
then for any ε ∈ (0, ε0), we have {αi(ε)} as n+ 1 distinct positive roots of F̃
κ,γ,Stot(ε, u). The case when
n is odd can be proved similarly.
The above theorem shows that when Etot/Ftot is sufficiently small, it is always possible for the futile
cycle to have n + 1 (n) steady states when n is even (odd), by choosing appropriate kinetic constants κ.
We should notice that for arbitrary κ, the derivative of F̃ at each positive root may become zero, which
breaks down the perturbation argument. Here is an example to show that more conditions are needed:
n = 2, λ0 = 1, λ1 = 3, γ = 6, β0 = β1 = 1/12, K0 = 1/8, K1 = 1/2, Stot = 5,
we have that
κ,γ,Stot(0, u) = 3u3 − 12u2 + 15u− 6 = 3(u− 1)2(u− 2)
has a double root at u = 1. In this case, even for ε = 0.01, there is only one positive root of F̃
κ,γ,Stot(ε, u),
see Figure 2.
1 2 3
Figure 2: The plot of the function F̃
κ,γ,Stot(0.01, u) on [0, 3]. There is a unique positive real solution
around u = 2.14, the double root u = 1 of F̃
κ,γ,Stot(0, u) bifurcates to two complex roots with non-zero
imaginary parts.
However, the following lemma provides a sufficient condition for ∂F
κ,γ,Stot
(0, ū) 6= 0, for any positive
ū such that F̃
κ,γ,Stot(0, ū) = 0.
Lemma 2 For each positive numbers Stot, γ, and vector κ ∈ R
+ , if
1− γβj
holds for all j = 1, · · · , n − 1, then ∂F̃
κ,γ,Stot
(0, ū) 6= 0.
See Appendix for the proof.
Theorem 2 For each positive numbers Stot, γ, and vector κ ∈ R
+ satisfying condition (21), there exists
ε1 > 0 such that for any Ftot, Etot satisfying Ftot = Etot/γ < ε1Stot/γ, the number of positive steady
states of system Σ(κ, C) is greater or equal to the number of (positive) roots of F̃
κ,γ,Stot(0, u).
Proof. Suppose that F̃
κ,γ,Stot(0, u) has m roots: ū1, . . . , ūm. Applying Lemma 2, we have
κ,γ,Stot
(0, ūk) 6= 0, k = 1, . . . ,m.
By the perturbation arguments as in Theorem 1, we have that there exists ε1 > 0 such that F̃
κ,γ,Stot(ε, u)
has at least m roots for all 0 < ε < ε1.
The above result depends heavily on a perturbation argument, which only works when Etot/Ftot is
sufficiently small. In the next section, we will give an upper bound of the number of steady states with no
restrictions on Etot/Ftot, and independent of κ and C.
4.2 Upper bound on the number of steady states
Theorem 3 For each κ, C, the system Σ(κ, C) has at most 2n− 1 positive steady states.
Proof. An alternative approach to solving (10)-(11) is to first eliminate v from (10) instead of from (11),
Etot/Ftot − u
uϕκ1(u)− (Etot/Ftot)ϕ
, (22)
when uϕκ1(u) − (Etot/Ftot)ϕ
2 (u) 6= 0. Then, we substitute (22) into (11), and multiply by (uϕ
1(u) −
(Etot/Ftot)ϕ
2 (u))
2 to get:
P κ,C(u) := ϕκ0ϕ
+ (ϕκ0 − Stotϕ
2 + Ftotuϕ
1 + Ftotϕ
uϕκ1 −
− Stot
uϕκ1 −
= 0. (23)
Therefore, if uϕκ1(u) − (Etot/Ftot)ϕ
2 (u) 6= 0, the number of positive solutions of (10)-(11) is no greater
than the number of positive roots of P κ,C(u).
In the special case when uϕκ1(u) − (Etot/Ftot)ϕ
2(u) = 0, by (10), we must have u = Etot/Ftot, and
thus ϕκ1 (Etot/Ftot) = ϕ
2 (Etot/Ftot). Substituting into (11), we get a unique v defined as in (12) with
u = Etot/Ftot. But notice that in this case u = Etot/Ftot is also a root of P
κ,C(u), so also in this case
the number of positive solutions to (10)-(11) is no greater than the number of positive roots of P κ,C(u).
It is easy to see that P κ,C(u) is divisible by u. Consider the polynomial Qκ,C(u) := P κ,C(u)/u of
degree 2n + 1. We will first show that Qκ,C(u) has no more than 2n positive roots, then we will prove by
contradiction that 2n distinct positive roots can not be achieved.
It is easy to see that the coefficient of u2n+1 is
(λ0 · · · λn−1)
LMn−1
and the constant term is
FtotKM0
So the polynomial Qκ,C(u) has at least one negative root, and thus has no more than 2n positive roots.
Suppose that S(κ, C) has cardinality 2n, then Qκ,C(u) must have 2n distinct positive roots, and each of
them has multiplicity one. Let us denote the roots as u1, . . . , u2n in ascending order. We claim that none
of them equals Etot/Ftot. If so, we would have ϕ
1(Etot/Ftot) = ϕ
2 (Etot/Ftot), and Etot/Ftot would
be a double root of Qκ,C(u), contradiction.
Since Qκ,C(0) > 0, Qκ,C(u) is positive on intervals
I0 = (0, u1), I1 = (u2, u3), . . . , In−1 = (u2n−2, u2n−1), In = (u2n,∞),
and negative on intervals
J1 = (u1, u2), . . . , Jn = (u2n−1, u2n).
As remarked earlier, ϕκ1 (Etot/Ftot) 6= ϕ
2 (Etot/Ftot), the polynomial Q
κ,C(u) evaluated at Etot/Ftot
is negative, and therefore, Etot/Ftot belongs to one of the J intervals, say Js = (u2s−1, u2s), for some
s ∈ {1, . . . , n} .
On the other hand, the denominator of v in (22), denoted as B(u), is a polynomial of degree n and
divisible by u. If B(u) has no positive root, then it does not change sign on the positive axis of u. But v
changes sign when u passes Etot/Ftot, thus v2s−1 and v2s have opposite signs, and one of (u2s−1, v2s−1)
and (u2s, v2s) is not a solution to (10)-(11), which contradicts the fact that both are in S(κ, C).
Otherwise, there exists a positive root ū of B(u) such that there is no other positive root of B(u)
between ū and Etot/Ftot. Plugging ū into Q
κ,C(u), we see that Qκ,C(ū) is always positive, therefore, ū
belongs to one of the I intervals, say It = (u2t, u2t+1) for some t ∈ {0, . . . , n}. There are two cases:
1. Etot/Ftot < ū. We have
u2s−1 < Etot/Ftot < u2t < ū.
Notice that v changes sign when u passes Etot/Ftot, so the corresponding v2s−1 and v2t have different
signs, and either (u2s−1, v2s−1) /∈ S(κ, C) or (u2t, v2t) /∈ S(κ, C), contradiction.
2. Etot/Ftot > ū. We have
ū < u2t+1 < Etot/Ftot < u2s.
Since v changes sign when u passes Etot/Ftot, so the corresponding v2t+1 and v2s have different
signs, and either (u2t+1, v2t+1) /∈ S(κ, C) or (u2s, v2s) /∈ S(κ, C), contradiction.
Therefore, Σ(κ, C) has at most 2n− 1 steady states.
4.3 Fine-tuned upper bounds
In the previous section, we have seen that any (u, v) ∈ S(κ, C), u 6= Etot/Ftot must satisfy (22)-(23), but
not all solutions of (22)-(23) are elements in S(κ, C). Suppose that (u, v) is a solution of (22)-(23), it is
in S(κ, C) if and only if u, v > 0. In some special cases, for example, when the enzyme is in excess, or the
substrate is in excess, we could count the number of solutions of (22)-(23) which are not in S(κ, C) to get
a better upper bound.
The following is a standard result on continuity of roots; see for instance Lemma A.4.1 in [30]:
Lemma 3 Let g(z) = zn + a1z
n−1 + · · ·+ an be a polynomial of degree n and complex coefficients having
distinct roots
λ1, . . . , λq,
with multiplicities
n1 + · · ·+ nq = n,
respectively. Given any small enough δ > 0 there exists a ε > 0 so that, if
h(z) = zn + b1z
n−1 + · · ·+ bn, |ai − bi| < ε for i = 1, . . . , n,
then h has precisely ni roots in Bδ(λi) for each i = 1, . . . , q.
Theorem 4 For each γ > 0 and κ ∈ R6n−6+ such that ϕ
1 (γ) 6= ϕ
2 (γ), and each Stot > 0, there exists
ε2 > 0 such that for all positive numbers Etot, Ftot satisfying Ftot = Etot/γ < ε2Stot/γ, the system
Σ(κ, C) has at most n+ 1 positive steady states.
Proof. Let us define a function R+ × C −→ C as follows:
κ,γ,Stot(ε, u) = Q
κ,(εStot,εStot/γ,Stot)(u),
and a set B
κ,γ,Stot(ε) consisting of the roots of Q̃
κ,γ,Stot(ε, u) which are not positive or the corresponding
v’s determined by u’s as in (22) are not positive, Since Q̃
κ,γ,Stot(ε, u) is a polynomial of degree 2n + 1,
if we can show that there exists ε2 > 0 such that for any ε ∈ (0, ε2), Q̃
κ,γ,Stot(ε, u) has at least n roots
counting multiplicities that are in B
κ,γ,Stot(ε), then we are done.
In order to apply Lemma 3, we regard the function Q̃
κ,γ,Stot as defined on R× C. At ε = 0:
κ,γ,Stot(0, u) = [ϕκ0ϕ
2(γ − u)
2 + (ϕκ0 − Stotϕ
2 )(uϕ
1 − γϕ
2)(γ − u)− Stot(uϕ
1 − γϕ
= [ϕκ0ϕ
2(γ − u)
2 + ϕκ0(uϕ
1 − γϕ
2)(γ − u)− Stotϕ
1 − γϕ
2)(γ − u)− Stot(uϕ
1 − γϕ
= [ϕκ0 (γ − u)u(ϕ
1 − ϕ
2) + Stotu(uϕ
1 − γϕ
2 )(ϕ
2 − ϕ
1)]/u
= (ϕκ2 − ϕ
1)(uϕ
0 + Stot(uϕ
1 − γϕ
2 )− γϕ
= (ϕκ2 − ϕ
κ,γ,Stot(0, u)
Let us denote the distinct roots of Q̃
κ,γ,Stot(0, u)/u as
u1, . . . , uq,
with multiplicities
n1 + · · ·+ nq = 2n+ 1,
and the roots of ϕκ1 − ϕ
u1, . . . , up, p ≤ q,
with multiplicities
m1 + · · ·+mp = n, ni ≥ mi, for i = 1, . . . , p.
For each i = 1, . . . , p, if ui is real and positive, then there are two cases (ui 6= γ as ϕ
1(γ) 6= ϕ
2(γ)):
1. ui > γ. We have
1(ui)− γϕ
2 (ui) > γ(ϕ
1 (ui)− ϕ
2(ui)) = 0.
2. ui < γ. We have
1(ui)− γϕ
2 (ui) < γ(ϕ
1 (ui)− ϕ
2(ui)) = 0.
In both cases, uiϕ
1(ui)− γϕ
2 (ui) and γ − ui have opposite signs, i.e.
1 (ui)− γϕ
2(ui))(γ − ui) < 0.
Let us pick δ > 0 small enough such that the following conditions hold:
1. For all i = 1, . . . , p, if ui is not real, then Bδ(ui) has no intersection with the real axis.
2. For all i = 1, . . . , p, if ui is real and positive, the following inequality holds for any real u ∈ Bδ(ui):
(uϕκ1(u)− γϕ
2 (u))(γ − u) < 0. (24)
3. For all i = 1, . . . , p, if ui is real and negative, then Bδ(ui) has no intersection with the imaginary
axis.
4. Bδ(uj)
Bδ(uk) = ∅ for all j 6= k = 1, . . . , q.
By Lemma 3, there exists ε3 > 0 such that for all ε ∈ (0, ε3), the polynomial Q̃
κ,γ,Stot(ε, u)/u has exactly
nj roots in each Bδ(uj), j = 1, . . . , q, denoted by u
j (ε), k = 1, . . . , nj .
We pick one such ε, and we claim that none of the roots in Bδ(ui), i = 1, . . . , p with the v defined as
in (22) will be an element in S. If so, we are done, since there are
1 ni ≥
1 mi = n such roots, of
κ,γ,Stot(ε, u) which are in B
κ,γ,Stot(ε).
For each i = 1, . . . , p, there are two cases:
1. ui is not real. Then condition 1 guarantees that u
i (ε) is not real for each k = 1, . . . , ni, and thus is
κ,γ,Stot(ε).
2. ui is real and positive. Pick any root u
i (ε) ∈ Bδ(ui), k = 1, . . . , ni, the corresponding v
i (ε) equals
γ − uki (ε)
uki (ε)ϕ
i (ε)) − γϕ
i (ε))
) < 0
followed from (24). So (uki (ε), v
i (ε)) /∈ S(κ, C), and u
i (ε) ∈ B
κ,γ,Stot(ε).
3. ui is real and negative. By condition 1 and 3, u
i (ε) is not positive for all k = 1, . . . , ni.
The next theorem considers the case when enzyme is in excess:
Theorem 5 For each γ > 0, κ ∈ R6n−6+ such that ϕ
1 (γ) 6= ϕ
2(γ), and each Etot > 0, there exists ε3 > 0
such that for all positive numbers Ftot, Stot satisfying Ftot = Etot/γ > Stot/(ε3γ), the system Σ(κ, C) has
at most one positive steady state.
Proof. For each γ > 0, κ ∈ R6n−6+ such that ϕ
1 (γ) 6= ϕ
2 (γ), and each Etot > 0, we define a function
R+ × C −→ C as follows:
κ,γ,Etot(ε, u) = Q
κ,(Etot,Etot/γ,εEtot)(u).
Let us define the set C
κ,γ,Etot(ε) as the set of roots of Q̄
κ,γ,Etot(ε, u) which are not positive or the
corresponding v’s determined by u’s as in (22) are not positive. If we can show that there exists ε3 > 0
such that for any ε ∈ (0, ε3) there is at most one positive root of Q̄
κ,γ,Etot(ε, u) that is not in C
κ,γ,Etot(ε),
we are done.
In order to apply Lemma 3, we now view the function Q̄
κ,γ,Etot as defined on R× C. At ε = 0:
κ,γ,Etot(0, u) = (γ − u)
(γ − u)ϕκ0ϕ
ϕκ0 +
uϕκ1 +
(uϕκ1 − γϕ
:= (γ − u)R
κ,γ,Etot(u).
Let us denote the distinct roots of Q̄
κ,γ,Etot(0, u)/u as
u1(= γ), u2, . . . , uq,
with multiplicities
n1 + · · ·+ nq = 2n+ 1,
and u2, . . . , uq are the roots of R
κ,γ,Etot(u) other than γ.
Since ϕκ1(γ) 6= ϕ
2(γ), R
κ,γ,Etot(u) is not divisible by u− γ, and thus n1 = 1.
For each i = 2, . . . , q, we have
(γ − ui)ϕ
0(ui)ϕ
2 (ui) = −
ϕκ0(ui) +
1(ui) +
ϕκ2(ui)
1 (ui)− γϕ
2(ui)) .
If ui > 0, then ϕ
0(ui)ϕ
2 (ui) and ϕ
0 (ui) +
1 (ui) +
ϕκ2 (ui) are both positive. Since uiϕ
1(ui)−
γϕκ2(ui) and γ − ui are non zero, uiϕ
1 (ui)− γϕ
2(ui) and γ − ui must have opposite signs, that is
1 (ui)− γϕ
2(ui))(γ − ui) < 0.
Let us pick δ > 0 small enough such that the following conditions hold for all i = 2, . . . , q:
1. If ui is not real, then Bδ(ui) has no intersection with the real axis.
2. If ui is real and positive, then for any real u ∈ Bδ(ui), the following inequality holds:
(uϕκ1(u)− γϕ
2 (u))(γ − u) < 0. (25)
3. If ui is real and negative, then Bδ(ui) has no intersection with the imaginary axis.
4. Bδ(uj)
Bδ(uk) = ∅ for all i 6= k = 2, . . . , q.
By Lemma 3, there exists ε3 > 0 such that for all ε ∈ (0, ε3), the polynomial Q̄
κ,γ,Etot(ε, u) has exactly
nj roots in each Bδ(uj), j = 1, . . . , q, denoted by u
j (ε), k = 1, . . . , nj .
We pick one such ε, and if we can show that all of the roots in Bδ(ui), i = 2, . . . , q are in C
κ,γ,Etot(ε),
then we are done, since the only roots that may not be in C
κ,γ,Etot(ε) are the roots in Bδ(u1), and there
is one root in Bδ(u1).
For each i = 2, . . . , p, there are three cases:
1. ui is not real. Then condition 1 guarantees that u
i (ε) is not real for all k = 1, . . . , ni.
2. ui is real and positive. Pick any root u
i (ε), k = 1, . . . , ni, the corresponding v
i (ε) equals
γ − uki (ε)
uki (ε)ϕ
i (ε))− γϕ
i (ε))
So, uki (ε) is in C
κ,γ,Etot(ε).
3. ui is real and negative. By conditions 1 and 3, u
i (ε) is not positive for all k = 1, . . . , ni.
5 Conclusions and discussions
Here we have set up a mathematical model for multisite phosphorylation-dephosphorylation cycles of size
n, and studied the number of positive steady states based on this model. We reformulated the question
of number of positive steady states to question of the number of positive roots of certain polynomials,
through which we also applied perturbation techniques. Our theoretical results depend on the assumption
of mass action kinetics and distributive sequential mechanism, which are customary in the study of multisite
phosphorylation and dephosphorylation.
An upper bound of 2n−1 steady states is obtained for arbitrary parameter combinations. Biologically,
when the substrate concentration greatly exceeds that of the enzyme, there are at most n + 1 (n) steady
states if n is even (odd). And this upper bound can be achieved under proper kinetic conditions, see
Theorem 1 for the construction. On the other extreme, when the enzyme is in excess, there is a unique
steady state.
As a special case of n = 2, which can be applied to a single level of MAPK cascades. Our results
guarantees that there are no more than three steady states, consistent with numerical simulations in [17].
We notice that there is an apparent gap between the upper bound 2n−1 and the upper bound of n+1
(n) if n is even (odd) when the substrate is in excess. If we think the ratio Etot/Ftot as a parameter ε,
then when ε≪ 1, there are at most n+1 (n) steady states when n is even (odd), which coincides with the
largest possible lower bound. When ε ≫ 1, there is a unique steady state. If the number of steady states
changes “continuously” with respect to ε, then we do not expect the number of steady states to exceed
n + 1 (n) if n is even (odd). So a natural conjecture would be that the number of steady states never
exceed n+ 1 under any conditions.
6 Acknowledgment
We thank Jeremy Gunawardena for very helpful discussions.
7 Appendix
proof of Lemma 2: Recall that (dropping the u’s in ϕκi , i = 0, 1, 2)
κ,γ,Stot(0, u) = uϕκ0 + Stot(uϕ
1 − γϕ
2)− γϕ
κ,γ,Stot
(0, u) = ϕκ0 + Stot(uϕ
1 − γϕ
′ − (γ − u)(ϕκ0 )
Since F̃
κ,γ,Stot(0, ū) = 0,
Stot(ūϕ
1 − γϕ
2 ) = (γ − ū)ϕ
that is,
γ − ū =
Stot(ūϕ
1 − γϕ
Therefore,
κ,γ,Stot
(0, ū) = ϕκ0 + Stot(uϕ
1 − γϕ
Stot(ūϕ
1 − γϕ
(ϕκ0)
= ϕκ0 +
ϕκ0(uϕ
1 − γϕ
′ − (ūϕκ1 − γϕ
= ϕκ0 +
((1 + λ0ū+ λ0λ1ū
2 + · · · + λ0 · · ·λn−1ū
(1− γβ0) + 2
(1− γβ1)ū+ · · ·+ n
λ0 · · · λn−2
KMn−1
(1− γβn−1)ū
λ0 + 2λ0λ1ū+ · · · + nλ0 · · ·λn−1ū
(1− γβ0)ū+
(1− γβ1)ū
2 + · · ·+
λ0 · · ·λn−2
KMn−1
(1− γβn−1)ū
= ϕκ0 +
λ0 · · ·λi−1ū
(j + 1− i)
λ0 · · ·λj−1
(1− γβj)ū
λ0 · · ·λi−1ū
λ0 · · ·λj−1ū
+ Stot
λ0 · · ·λi−1ū
(j + 1− i)
λ0 · · ·λj−1
(1− γβj)ū
λ0 · · ·λi−1ū
λ0 · · ·λn−1ū
λ0 · · ·λj−1ū
1 + Stot(j + 1− i)
1 − γβj
where the product λ0 · · ·λ−1 is defined to be 1 for the convenience of notation.
Because of (21),
(j + 1− i)
1− γβj
so we have ∂F̃
κ,γ,Stot
(0, ū) > 0.
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http://arxiv.org/abs/math/0701575
Introduction
Model assumptions
Mathematical formalism
Number of positive steady states
Lower bound on the number of positive steady states
Upper bound on the number of steady states
Fine-tuned upper bounds
Conclusions and discussions
Acknowledgment
Appendix
|
0704.0037 | The discrete dipole approximation for simulation of light scattering by
particles much larger than the wavelength | The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength
The discrete dipole approximation for simulation of light scattering
by particles much larger than the wavelength
M.A. Yurkina,b∗, V.P. Maltsevb,c, and A.G. Hoekstraa∗
a Section Computational Science, Faculty of Science, University of Amsterdam, Kruislaan
403, 1098 SJ, Amsterdam, The Netherlands
b Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya Str. 3, 630090, Novosibirsk, Russia
c Novosibirsk State University, Pirogova Str. 2, 630090, Novosibirsk, Russia
Abstract
In this manuscript we investigate the capabilities of the Discrete Dipole Approximation
(DDA) to simulate scattering from particles that are much larger than the wavelength of the
incident light, and describe an optimized publicly available DDA computer program that
processes the large number of dipoles required for such simulations. Numerical simulations of
light scattering by spheres with size parameters x up to 160 and 40 for refractive index
and 2 respectively are presented and compared with exact results of the Mie theory.
Errors of both integral and angle-resolved scattering quantities generally increase with m and
show no systematic dependence on x. Computational times increase steeply with both x and
m, reaching values of more than 2 weeks on a cluster of 64 processors. The main distinctive
feature of the computer program is the ability to parallelize a single DDA simulation over a
cluster of computers, which allows it to simulate light scattering by very large particles, like
the ones that are considered in this manuscript. Current limitations and possible ways for
improvement are discussed.
05.1=m
Keywords: discrete dipole approximation, light scattering simulation, computer program
∗ Corresponding authors. Tel.: +31-20-525-7462; fax: +31-20-525-7490.
e-mail addresses: [email protected], [email protected]
1 Introduction
The discrete dipole approximation (DDA) is a general method to calculate scattering and
absorption of electromagnetic waves by particles of arbitrary geometry and composition. The
DDA was first proposed by Purcell and Pennypacker [1] and was reviewed by Draine and
Flatau in 1994 [2]. A recent review [3] describes the current state of the DDA and its
historical development. It also explains the equivalence of the DDA and methods based on the
volume integral equation formulation. The reader is referred to this review for an in-depth
discussion of the DDA.
There are a number of computer programs based on the DDA, some of which were
recently compared by Penttila et al. [4]. The most popular among them is DDSCAT [5],
which has been widely used by many researchers for more than 10 years. In this paper we
present a new program, Amsterdam DDA (ADDA), which recently has been put in the public
domain.1 Its main distinctive feature is the ability to parallelize a single DDA simulation over
a cluster of computers, which allows simulation of light scattering by very large particles.
This is demonstrated for a number of test cases in this manuscript. Validation of ADDA by
simulating light scattering by wavelength-sized particles and comparing it with other DDA
programs was reported elsewhere [4].
Section 2 describes in detail the ADDA computer code, showing its advantages
compared to other codes. A number of numerical tests are shown in Section 3, demonstrating
that DDA is actually capable processing large particles, and showing the current capabilities
of ADDA. Results of these simulations are discussed in Section 4; the errors are compared
with previous results for much smaller particles. Section 5 concludes the manuscript and
discusses possible future work.
2 ADDA computer code
ADDA has been developed over a period of more than 10 years at the University of
Amsterdam [6-8]. Its main feature (distinctive from other DDA codes) has always been the
capability of running on a cluster of computers, parallelizing a single DDA computation, in
contrast with e.g. DDSCAT [5] that allows farming several instantiations of a DDA
simulation to different processors. This allows using a practically unlimited number of
dipoles, since ADDA is not limited by the memory of a single computer [8,9]. Recently the
overall performance of the code has been improved significantly, together with some
optimizations specifically for single-processor mode. ADDA's source code and
documentation is freely available.
Most of ADDA is written in ANSI C, which ensures wide portability on the source-code
level. The code is fully operational under Linux and, in sequential mode, on Windows based
systems. The parallelization over multiple processors is based on a geometric decomposition
of the particle and the single-program-multiple-data paradigm of parallel computing. The
code is written for distributed memory systems using the message passing interface (MPI).2
Note that ADDA should in principle also run on shared memory computers, but so far this
was not explicitly tested. The fast Fourier transform (FFT) used for the matrix-vector products
in the iterative solver is performed either using routines by Temperton [10] or the more
advanced package “Fastest Fourier transform in the West” (FFTW) [11]. The latter is
generally considerably faster but requires a separate package installation.
ADDA has four options implemented for dipole polarizabilities: Clausius-Mossotti [1],
radiative reaction correction [12], lattice dispersion relation (LDR) [13], and corrected LDR
[14]. It includes four iterative methods: conjugate gradient applied to normalized equation
with minimization of residual norm (CGNR) [15], Bi-CG stabilized (Bi-CGSTAB) [15], Bi-
1 http://www.science.uva.nl/research/scs/Software/adda/
2 http://www.mpi-forum.org
http://www.science.uva.nl/research/scs/Software/adda/
http://www.mpi-forum.org/
CG [16], and quasi minimal residual (QMR) [16]. The last two iterative methods employ the
complex-symmetric property of the DDA interaction matrix to halve the calculation time [16].
The default stopping criterion of the iterative method in ADDA is the relative norm of the
residual ε, which must be . 510−<
The usual formulation of DDA can be written as [2,3]:
jijii EPGP =− ∑
−α , (1)
where iα is the tensor of dipole polarizability, is incident electric field,
iE ijG is the free-
space Green’s tensor (complex symmetric), and Pi is the unknown dipole polarization. If the
polarizability tensor is diagonal for all dipoles then there always exists a iβ such that
iii αββ = , i.e. ii αβ = . Moreover, iβ is then complex symmetric, and so is the matrix with
elements
A , (2)
where I is an identity tensor. A is the interaction matrix that is used in ADDA, i.e. the
following system of linear equations is solved:
jjijii
jij ExGxxA βββ =−= ∑∑
, (3)
where iii Px
1−= β is a new unknown vector. Eq. (3) is equivalent to the use of Jacobi-
preconditioning [15] together with keeping the interaction matrix complex-symmetric (for any
distribution of refractive index inside the scatterer and for any of the supported polarization
prescriptions). We have not studied, however, whether this Jacobi-preconditioning improves
the convergence of the iterative solver. Flatau showed [17] that in some test cases it helps,
while in others there is no improvement. It is important to note also that DDA is not limited to
diagonal or symmetric polarizabilities. Any other tensor may be used, but then the interaction
matrix is not complex-symmetric; hence, QMR and Bi-CG are less efficient.
ADDA can perform orientation averaging of the scattering quantities over three Euler
angles (α, β, γ) of the particle orientation. Averaging over the angle α is done with a single
computation of internal fields by computing scattering in different scattering planes, which is
comparably fast. Averaging over the other two Euler angles is done by independent DDA
simulations. The averaging itself is performed using a Romberg integration [18], which may
be used adaptively (i.e. automatically simulating the required number of different orientations
to reach a prescribed accuracy) but limits the possible number of values for each orientation
angle to be , where n is an integer. Moreover, symmetries of the scatterer may be used
to decrease the intervals of Euler angles, over which to average, and hence accelerate the
calculation. This feature of ADDA was tested in a recent benchmark study [4].
12 +n
Other features of ADDA include computation of scattering by a tightly focused
Gaussian beams [6], a checkpoint system to allow for long runs on queuing systems that
enforce upper limits on wall clock time for execution as is usually the case on massively
parallel supercomputers, calculation of radiation forces on each of the dipoles [19], use of
rotational symmetry of the scatterer to halve the simulation time, and an extended command
line interface. Some other features, such as applicability to anisotropic scatterers and a large
set of predefined shapes, are planned to be implemented in the near future.
There are several factors that allow ADDA's performance to compare favorably with
other codes, which was shown in a benchmark study by Penttila et al. [4]. First of all, the
FFTW 3 package that is used automatically adapts itself to optimally perform on any
particular hardware. Moreover, ADDA does not perform complete 3D FFT transforms in one
run, but decomposes them into a set of 1D transforms with data transposition in between. This
allows employing the fact that input data for the forward transform contains many zeros, and
0 20 40 60 80 100 120 140 160
ε ∈(10−5,10−3)
Size parameter x
ε =10−5
70 GB
Fig. 1. Current capabilities of the ADDA for spheres with different x and m. The striped region
corresponds to full convergence and densely hatched region to incomplete convergence. The dashed
lines show two levels of memory requirements for the simulation, according to the “rule of thumb”
(see main text for explanation).
only part of the output data of the backward transform is used [8]. Second, we have
implemented four different Krylov-space-based iterative solvers, allowing us to choose the
most suitable one for a particular application. As is known from the literature [17,20,21] and
demonstrated in Section 3, there is not a best iterative solver for DDA. Depending on all
details of the scattering problem, any of the methods may outperform the others. Third,
dynamic memory allocation and optimized data structures allow all computations, except the
FFT, to be performed only for the real (non-void) dipoles and not for the whole computational
box. This also decreases ADDA's memory consumption. Moreover, symmetry of the
interaction matrix is used to decrease memory required for its Fourier transform. Finally, all
float variables in ADDA are represented in double precision. This accelerates convergence in
cases when machine precision becomes important. Moreover, basic operations with double-
precision numbers can be faster than with single-precision ones on modern processors. This
acceleration comes at a cost of increased memory consumption, which is, however, still lower
than for other computer codes [4].
More information on ADDA can be found in an extensive manual included in the
distribution package.
3 Numerical simulations
3.1 Simulation parameters
In our tests we used ADDA v.0.75, compiled with the Intel C compiler v.9.0 with maximum
possible optimizations (default options in ADDA’s makefile). All the tests were run on the
Dutch compute cluster LISA,3 using 32 nodes (each dual Intel Xeon 3.4 GHz processor with
4 GB RAM). LDR was used as the most common polarization formulation. We have tried
three different iterative solvers: QMR, Bi-CG, and Bi-CGSTAB. For all of them a default
stopping criterion was used. 510−=ε
3 http://www.sara.nl/userinfo/lisa/description/
http://www.sara.nl/userinfo/lisa/description/
Table 1. Parameters of the numerical simulations.
m x λ/md
Number of
dipolesa Iterative method
Number of
iterations
20 9.6 2.6×105 Bi-CGSTAB 6
30 9.6 8.8×105 Bi-CGSTAB 7
40 9.6 2.1×106 Bi-CGSTAB 9
60 9.6 7.1×106 Bi-CGSTAB 14
80 9.6 1.7×107 Bi-CGSTAB 20
100 9.6 3.3×107 Bi-CGSTAB 27
130 10.3 9.0×107 Bi-CGSTAB 40
1.05
160 9.6 1.3×108 Bi-CGSTAB 65
20 10.5 5.1×105 QMR 86
30 11.2 2.1×106 QMR 223
40 10.5 4.1×106 QMR 598
60 9.8 1.1×107 QMR 2120
80 10.5 3.3×107 Bi-CGSTAB 21748
100 10.1 5.7×107 Bi-CGSTAB 6169
130 10.3 1.3×108 Bi-CGSTAB 29200
20 10.8 8.8×105 QMR 1344
30 10.8 3.0×106 QMR 16930
40 10.8 7.1×106 QMR 8164
60 9.6 1.7×107 Bi-CG 127588
20 11.0 1.4×106 QMR 8496 1.6
30 10.5 4.1×106 Bi-CG 69748
20 11.2 2.1×106 QMR 28171 1.8
30 10.2 5.5×106 Bi-CG 118383
2 20 10.1 2.1×106 QMR 58546
a This is the total number of dipoles in the rectangular computational grid, which is the main factor determining
the computation time of one iteration. For spheres the number of dipoles occupied by the scatterer itself is almost
two times smaller.
0 5000 10000 15000 20000 25000
Number of iterations
Fig. 2. Convergence of the QMR iterative solver for the sphere with x = 20 and m = 1.8. The residual
as a function of the iteration number is shown. The system of linear equations contains 3×106
unknowns.
Spheres were used as test objects. Their size parameter x was varied from 20 to 160 and
their refractive index m was varied from 1.05 to 2. We limited ourselves to the case of real m.
The current capabilities of ADDA are shown as a region of the (x,m)-plane in Fig. 1. The
striped region corresponds to full convergence, the densely hatched region corresponds to
those cases where ADDA could not fully converge to the required residual norm, but only to
20 40 60 80 100 120 140 160
106 1 week
1 day
Size parameter x
m =
1.05
1 min
1 hour
Fig. 3. Total simulation wall clock time (on 64 processors) for spheres with different x and m. Time is
shown in logarithmic scale. Horizontal dotted lines corresponding to a minute, an hour, a day, and a
week are shown for convenience.
20 40 60 80 100 120 140 160
Size parameter x
m =
1.05
Fig. 4. Relative errors of the extinction efficiency in logarithmic scale for spheres with different x and
)10,10( 35 −−∈ε . Although this incomplete convergence probably affects the final accuracy of
the scattering quantities only slightly, we remove such results from further consideration
because a separate study is required to quantify this effect (see Section 4). For fully converged
results, the errors of scattering quantities due to the numerical convergence are much smaller
than the total errors (data not shown).
A complete set of (x,m) pairs, for which ADDA converged, is shown in Table 1. It also
shows the number of dipoles per wavelength in the medium ( md/λ where d is the size of the
dipole). We tried to keep it equal to 10 according to the “rule of thumb” as formulated by
Draine and Flatau [2]; however, it was slightly different because we varied the size of the
dipole grid to optimize the parallel efficiency of ADDA.4 The total number of dipoles in a
rectangular computational grid, shown in Table 1, was varied from 2.6×105 to 1.3×108, it can
4 The best parallel performance is obtained when grid size divides the number of processors. However, ADDA
works with any grid size.
20 40 60 80 100 120 140 160
Size parameter x
m =
1.05
Fig. 5. Same as Fig. 4 but now for the asymmetry parameter.
20 40 60 80 100 120 140 160
Size parameter x
m =
1.05
Fig. 6. Maximum relative errors of S11(θ ) in logarithmic scale for spheres with different x and m.
be approximately determined as . Both memory requirements and computation time
of one iteration are proportional to this number. Two dashed lines are shown in
3)18.3( xm
Fig. 1 to
indicate the memory requirements for different x and m. They correspond to typical memory
of a modern desktop computer (2 GB) and the maximum total memory used in our
simulations (70 GB), respectively.
For each sphere we computed the extinction efficiency, the asymmetry parameter, and
all Mueller matrix elements in one scattering plane, which is a symmetry plane of the cubical
discretization of the sphere. Exact results for the same spheres were obtained using the Mie
theory [22]. Spherical symmetry was used by ADDA to get all results from calculations for
only one polarization state of the incident field. Therefore computation time is a factor of two
smaller than for non-symmetric scatterers with the same x and m. We employed a volume
correction to ensure equal volumes of sphere and its dipole representation [2]. Note, however,
that for the very large spheres this correction is extremely small.
3.2 Results
Table 1 shows the iterative solver that provided the best performance for each particular case
and the number of iterations to achieve convergence. Fig. 2 illustrates one specific example of
20 40 60 80 100 120 140 160
Size parameter x
m =
1.05
Fig. 7. Same as Fig. 6 but now for RMS relative errors.
0 30 60 90 120 150 180
170 175 180
Scattering angle θ, deg
Fig. 8. DDA results (dotted line) of S11(θ ) in logarithmic scale for a sphere with x = 160 and m = 1.05,
compared with the results of the Mie theory (solid line).
convergence of the DDA iterative solver. This is QMR applied to the system of 3⋅106 linear
equations obtained for the sphere with 20=x and 8.1=m . The total simulation wall clock
time t for all particles is shown in Fig. 3. Fig. 4 and Fig. 5 show the relative errors of the
extinction efficiency Qext and the asymmetry parameter >< θcos respectively. Maximum -
and root-mean-squared (RMS) relative errors of S11 over the whole range of scattering angle
are shown in Fig. 6 and Fig. 7 respectively. Errors of other non-trivial Mueller matrix
elements behave in a similar way (data not shown).
DDA results of S11(θ) for a sphere with 160=x and 05.1=m are compared with the
Mie theory in Fig. 8. The inset shows a magnification of the backscattering region. This is, to
the best of our knowledge, the largest particle ever simulated with DDA. Fig. 9 and Fig. 10
show the same comparisons but for 60=x , 4.1=m and 20=x , 2=m respectively.
0 30 60 90 120 150 180
Scattering angle θ, deg
Fig. 9. Same as Fig. 8 but now for x = 60 and m = 1.4.
0 30 60 90 120 150 180
Scattering angle θ, deg
Fig. 10. Same as Fig. 8 but now for x = 20 and m = 2.
4 Discussion
The convergence of the QMR iterative solver shown in Fig. 2, featuring plateaus and steep
descents, is in agreement both with its behavior in general [16] and with particular examples
of its application to DDA [20,23]. A distinctive feature of this graph compared to the
literature data is that the convergence slows down with iteration number, i.e. the logarithm of
the residual norm decreases slower than linearly. This is probably due to the large size of the
scatterer and loss of numerical precision (see discussion below).
The total computation times t increase steeply both with x and m (Fig. 3). The time is
displayed in a logarithmic scale covering a range from 4 seconds to more than 2 weeks. For
, the increase of t with x is mostly due to the increasing number of dipoles to model
the scatterer, since the number of iterations increase at a slower pace (
05.1=m
Table 1). For larger m
these two effects are comparable, combining into a very unfavorable scaling, which can be
approximately described by a power law , where )()( mxmCt α≈ 6>α for . It should
be noted that both the number of iterations and t do not always increase monotonically with x.
For example for , and
2.1≥m
80=x 2.1=m 30=x , 4.1=m the execution times are unusually high.
This may be caused by a large condition number of DDA interaction matrices for these two
particular particles. Moreover, when the convergence is slow it may suffer from machine
precision, the latter determining the limit of x and m, for which ADDA will converge at all.
Therefore, current size limitations of the DDA for are due to the practically
unbearable computation times, and not due to memory requirements.
2.1≥m
5 Simulations for larger
m are far from the memory limit shown in Fig. 1. Moreover, simply using more processors
does not solve the problem. Improving numerical performance is required, e.g. dedicated
preconditioning of the iterative solver [15]. On the other hand, extension to larger sizes for
is feasible if more computer resources are available. This facilitates, for example,
simulating scattering of visible light by almost all biological cells in suspension.
2.1<m
The increase of the number of iterations with m is a well-known fact [12,17,21,24];
however, there is still no theoretical foundation to describe it in details. Rahola [24] provided
theoretical predictions of the dependence of the number of iterations on m, valid for scatterers
smaller than the wavelength. However, these conclusions are not applicable to the scatterers
studied in this manuscript. The general reason for the slowing down of the convergence with
increasing m is increased interaction between dipoles and, hence, an increased condition
number of the interaction matrix. Absorption, if present, should decrease the overall
interaction between dipoles in a large scatterer. Therefore, it is expected that convergence for
complex refractive indices should be better than for the purely real ones that we consider here.
The same was suggested by Budko and Samokhin [25] based on the analysis of the spectrum
of the integral scattering operator. However, this proposition is still to be verified by
numerical tests.
Another parameter that may greatly affect the computation time is the convergence
threshold ε. In this paper it is set to a de-facto default value of 10-5 [2], which ensures
negligibly small numerical errors compared to the model errors. However, in many cases
numerical errors are small enough already for , i.e. the difference of the scattering
quantities between simulations with and is significantly smaller than the
difference between the latter and the exact values (data not shown).
310−=ε
310−=ε 510−=ε
Fig. 2 shows that QMR
for a particular case converges to and three and six times faster
respectively than to . Results for other simulated particles and iterative solvers show
similar trends and even larger acceleration with increasing ε in some cases (data not shown).
Therefore, if one can determine an optimum ε for a particular case, it can decrease the
computation time significantly. However, we do not pursue this issue further in this
manuscript.
310−=ε 3102 −×=ε
510−=ε
Fig. 4 shows the deterioration of the accuracy of Qext with increasing m, while there is
no clear dependence on x (the only exception is a single result for ). Results for 2=m
>< θcos (Fig. 5) behave in a similar way. These results are in good agreement with results of
other researchers for smaller size parameters [2,13,26], both in terms of the errors themselves
and their dependence on m. To express errors on the angular dependencies of S11 we use two
integral parameters: the maximum - and RMS relative errors (Fig. 6 and Fig. 7 respectively).
Although these parameters are not completely objective, as they are significantly influenced
by the values of S11 in deep minima, which are completely irrelevant to most real
experiments, they do provide a consistent measure of the DDA accuracy. To relate these
integral parameters to some other criteria, e.g. visual agreement, three examples are presented
in Fig. 8 – Fig. 10. Errors of S11(θ ) show the same tendencies as the integral scattering
quantities, except that errors for are relatively large (larger than those for in
the range ) and generally decrease with x. This is due to the relative nature of the
measured errors and the huge dynamical range of S
05.1=m 2.1=m
11(θ ) for small refractive indices (see Fig.
8). Results for smaller size parameters found in the literature [2,26] show a similar increase of
5 The boundary value of m is not well-defined, as it depends on particular hardware and restrictions on
computation time; 1.2 is just a convenient value to guide the reader.
errors with m: however, the errors themselves are considerably smaller. For instance,
maximum relative errors of S11(θ ) for 10<x and m up to i4.15.2 + are smaller than 0.4. This
is due to the general differences between functions S11(θ ) for particles comparable to and
much larger than the wavelength. The latter has deeper minima and a larger overall dynamic
range. It is important to note that refractive indices as small as 1.05 are rarely used in DDA
simulations [26], therefore it is hard to make any definite conclusions concerning the behavior
of errors in this case.
In what follows, the traditional “rule of thumb” [2] is discussed, which states that for
10/ =mdλ errors of cross sections and asymmetry parameter are expected to be a few
percents, and maximum errors in the angular dependence of S11 on the order of 20-30 %.
Results for both Qext and >< θcos do satisfy the “rule of thumb,” however this rule does not
describe the decrease of errors by two orders of magnitude with decreasing m. The latter can
be used to cut down the number of dipoles and hence computation time in cases when only
integral scattering quantities need to be calculated for small m. Relative errors of S11(θ ) are
much larger than that predicted by the “rule of thumb,” which is due to the fact that the latter
was derived based on test simulations for x smaller than 10 [2]. See, however, the discussion
below on possible changes for complex refractive index and non-spherical shapes. To
conclude, the “rule of thumb” has very limited application for the range of x and m here. More
elaborate empirical functions are required to estimate the number of dipoles needed to reach a
prescribed accuracy. They will also allow a more realistic estimate of DDA computational
complexity, i.e. the computation time needed to reach a certain accuracy of some scattering
quantities for particular x and m. This topic is left for the future study.
The test results shown in this paper are limited to real refractive indices and spherically
shaped scatterers. In the following we try to generalize our conclusions to complex refractive
index and non-spherical shapes. However, we want to stress that this generalization is
speculative, and more numerical tests are clearly needed to verify them. It is expected that
accuracies of integral scattering quantities should not change significantly for more general
cases. Their accuracy should deteriorate both with increasing real and imaginary parts of the
refractive index. The situation for angle-resolved scattering quantities is expected to be
different. Large relative errors observed in this paper are due to deep minima that are a
consequence of both spherical symmetry and purely real refractive index. It is expected that
visual agreement between the DDA results and the exact solution (as shown in Fig. 8 – Fig.
10) should not change significantly for more general cases, however it will result in smaller
relative errors, especially for larger x and smaller m.
5 Conclusion
In this paper we present the ADDA, a computer program to simulate light scattering by
arbitrarily shaped particles. ADDA can parallelize a single DDA simulation, which allows it
not to be limited by the memory of a single computer. Moreover, ADDA is heavily optimized,
which allows it to compare favorably with other programs based on DDA when running on a
single processor. We showed its capabilities for simulating light scattering by spheres with x
up to 160 and m up to 2. The maximum reachable x on a cluster of 64 modern processors
decrease rapidly with increasing m: it is 160 for 05.1=m and only 20-40 (depending on the
convergence threshold) for . This is mostly due to the slow convergence of the iterative
solver leading to practically unbearable computation times. It is expected that larger particle
sizes can be reached if m has a significant imaginary part.
Errors of both integral and angle-resolved scattering quantities show no systematic
dependence on x, but generally increase with m. Errors of Qext and >< θcos range from less
than 0.01 % to 6 %. Maximum - and RMS relative errors of S11(θ ) are in the ranges 0.2–18
and 0.04–1 respectively. Error predictions of the traditional “rule of thumb” have very limited
application in this range of x and m: it describes the upper limit of errors of Qext and >< θcos ,
however it does not account for the decrease of the errors with m.
Currently, the ADDA is capable of simulating light scattering by almost all biological
cells in suspension; however, its performance for other cases can be improved. These
improvements, left for future work, may include improving the convergence of the iterative
solver by preconditioning. It also is desirable to conduct a detailed study of the dependence of
the accuracy of the final results on the size of the dipole and convergence thresholds of the
iterative solver for different scatterers. Such a study should result in a reduction of the
computation time and provide a realistic estimate of DDA complexity over a wide range of x
and m.
Acknowledgements
We thank Gorden Videen for critically reading the manuscript and anonymous reviewer for
valuable comments. Our research is supported by Siberian Branch of the Russian Academy of
Sciences through the grant 2006-03.
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1 Introduction
2 ADDA computer code
3 Numerical simulations
3.1 Simulation parameters
3.2 Results
4 Discussion
5 Conclusion
Acknowledgements
References
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|
0704.0038 | The discrete dipole approximation: an overview and recent developments | The discrete dipole approximation: an overview and recent
developments
M.A. Yurkina,b,∗ and A.G. Hoekstraa
a Section Computational Science, Faculty of Science, University of Amsterdam, Kruislaan
403, 1098 SJ, Amsterdam, The Netherlands
b Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of
Sciences, Institutskaya Str. 3, 630090, Novosibirsk, Russia
Abstract
We present a review of the discrete dipole approximation (DDA), which is a general method
to simulate light scattering by arbitrarily shaped particles. We put the method in historical
context and discuss recent developments, taking the viewpoint of a general framework based
on the integral equations for the electric field. We review both the theory of the DDA and its
numerical aspects, the latter being of critical importance for any practical application of the
method. Finally, the position of the DDA among other methods of light scattering simulation
is shown and possible future developments are discussed.
Keywords: discrete dipole approximation, review, light scattering simulation
∗ Corresponding author: Tel.: +31-20-525-7562; fax: +31-20-525-7490.
E-mail address: [email protected]
Contents
1 Introduction ......................................................................................................................... 2
2 General framework.............................................................................................................. 3
3 Various DDA models .......................................................................................................... 7
3.1 Theoretical base of the DDA........................................................................................ 7
3.2 Accuracy of DDA simulations ................................................................................... 13
3.3 The DDA for clusters of spheres................................................................................ 16
3.4 Modifications and extensions of the DDA................................................................. 18
4 Numerical considerations.................................................................................................. 19
4.1 Direct vs. iterative methods........................................................................................ 19
4.2 Scattering order formulation ...................................................................................... 22
4.3 Block-Toeplitz ........................................................................................................... 23
4.4 FFT............................................................................................................................. 24
4.5 Fast multipole method................................................................................................ 24
4.6 Orientation averaging and repeated calculations ....................................................... 25
5 Comparison of the DDA to other methods ....................................................................... 27
6 Concluding remarks .......................................................................................................... 28
Acknowledgements .................................................................................................................. 28
Appendix. Description of used acronyms and symbols ........................................................... 28
References ................................................................................................................................ 31
1 Introduction
The discrete dipole approximation (DDA) is a general method to compute scattering and
absorption of electromagnetic waves by particles of arbitrary geometry and composition.
Initially the DDA was proposed by Purcell and Pennypacker (PP) [1], who replaced the
scatterer by a set of point dipoles. These dipoles interact with each other and the incident
field, giving rise to a system of linear equations, which is solved to obtain dipole
polarizations. All the measured scattering quantities can be obtained from these polarizations.
The DDA was further developed by Draine and coworkers [2-5], who popularized the method
by developing a publicly available computer code DDSCAT [6]. Later it was shown that the
DDA also can be derived from the integral equation for the electric field, which is discretized
by dividing the scatterer into small cubical subvolumes. This derivation was apparently first
performed by Goedecke and O'Brien [7] and further developed by others (see, for instance,
[8-11]). It is important to note that the final equations, produced by both lines of derivation of
the DDA are essentially the same. The only difference is that derivations based on the integral
equations give more mathematical insight into the approximation, thus pointing at ways to
improve the method, while the model based on point dipoles is physically clearer.
The DDA is called the coupled dipole method or approximation by some researchers
[12,13]. There are also other methods, such as the volume integral equation formulation [14]
and the digitized Green’s function (DGF) [7], which were developed completely
independently from PP. However, later they were shown to be equivalent to DDA [8,15]. In
this review we will use the term DDA to refer to all such methods, since we describe them in
terms of one general framework. However, it is difficult to separate unambiguously the DDA
from other similar methods, based on the volume integral equations for the electromagnetic
fields, such as a broad range of method of moments (MoM) with different bases and testing
functions [16-19]. In our opinion, one fundamental aspect of the DDA is that the solution for
the “physically meaningful” internal fields or their direct derivatives, e.g. polarization, plays
an integral role in the process. In other words, any DDA formulation can be interpreted as
replacing a scatterer by a set of interacting dipoles; this is further discussed in Section 2. An
example of method that is not considered DDA is the MoM with higher-order hierarchical
Legendre basis functions [17].
The DDA is a popular method in the light-scattering community and it has been
reviewed by several authors. An extensive review by Draine and Flatau [4] covers almost all
DDA developments up to 1994. A more recent review by Draine [5] mainly concerns
applications and numerical considerations. DDA theory was discussed together with other
methods for light scattering simulations in reviews by Wriedt [20], Chiappetta and Torresani
[21], and Kahnert [15] and in books by Mishchenko et al. [22] and Tsang et al. [23]. Jones
[24] placed the DDA in context of different methods with respect to particle characterization.
However, many important DDA developments since 1994 are not mentioned in any of these
manuscripts. Those that are mentioned are usually considered as side-steps, and are not placed
into a general framework. Moreover, to the best of our knowledge numerical aspects of the
DDA have never been reviewed extensively – each paper discusses only a few particular
aspects. In this review we try to fill this gap.
A general framework is developed in Section 2 to ease the further discussion of different
DDA models. This framework is based on the integral equation because it allows a uniform
description of all the DDA development. However, connection to a physically clearer model
of point dipoles is discussed throughout the section. The sources of errors in the DDA
formulation are also discussed there.
In Section 3 the physical principles of the DDA are reviewed and results of different
models are compared. In Subsection 3.1 different improvements of polarizabilities and
interaction terms are reviewed from a theoretical point of view. Different expressions for Cabs
also are discussed. Comparison of simulation results using different formulations is given in
Subsection 3.2. Subsection 3.3 covers the special case of a cluster of spheres that allows
particular improvements and simplifications. In Section 3.4 different significant modifications
are reviewed, which do not fall completely into the general framework described in Section 2.
Enhancements of the DDA for some special purposes also are discussed.
Different numerical aspects of the DDA are reviewed in Section 4. These are concerned
primarily with solving very large systems of linear equations (Subsection 4.1). Subsection 4.2
describes the simplest iterative procedure to solve DDA linear system, which has a clear
physical meaning. The special structure of the DDA interaction matrix for a rectangular grid
and its application to decrease computational costs are described in subsections 4.3 and 4.4
respectively. General methods to accelerate calculations, which do not require a rectangular
grid, are discussed in Subsection 4.5. Subsection 4.6 covers special techniques to increase the
efficiency of repeated calculations (e.g. in orientation averaging).
A numerical comparison of the DDA with other methods is reviewed in Section 5; its
strong and weak points are discussed. Section 6 concludes the review and discusses future
development of the DDA.
2 General framework
The )iexp( tω− time dependence of all fields is assumed throughout this review. The scatterer
is assumed dielectric but not magnetic (magnetic permittivity 1=μ ). The electric permittivity
is assumed isotropic to simplify the derivations; however, extension to arbitrary dielectric
tensors is straightforward.1
The general form of the integral equation governing the electric field inside the
dielectric scatterer is the following [8,15]:
)()(),(),()()(),(d)()( 00
rErrLrMrErrrGrErE χχ VVr
∂−+′′′′+= ∫ , (1)
1 In most formulae scalar values can be replaced directly by tensors, but there are exceptions. Extensions of
DDA to optically anisotropic scatterers are discussed in Section 3.4.
where Einc(r) and E(r) are the incident and total electric field at location r;
πεχ 4)1)(()( −= rr is the susceptibility of the medium at point r (ε(r) – relative
permittivity). V is the volume of the particle, i.e., the volume that contains all points where the
susceptibility is not zero. V0 is a smaller volume such that , VV ⊂0 00 \ VV ∂∈r . ),( rrG ′ is
the free space dyadic Green’s function, defined as
−=∇∇+=′ 222
i1ˆˆ)iexp()iexp(ˆˆ),(
k IIIrrG , (2)
where I is the identity dyadic, ck ω= is the free space wave vector, rrR ′−= , R=R ,
and is a dyadic defined as (μ and ν are Cartesian components of the vector
or tensor). M is the following integral associated with the finiteness of the exclusion volume
RR ˆˆ νμμν RRRR =ˆˆ
( )∫ ′−′′′′=
)()(),()()(),(d),( s30
rV rErrrGrErrrGrM χχ , (3)
where ),(s rrG ′ is the static limit ( ) of 0→k ),( rrG ′ :
−−=∇∇=′
11ˆˆ),(
IrrG . (4)
L is the so-called self-term dyadic:
rV rL , (5)
where is an external normal to the surface ∂Vn′ˆ 0 at point r'. L is always a real, symmetric
dyadic with trace equal to 4π [25]. It is important to note that L does not depend on the size
of the volume V0, but only on its shape (and location of the point r inside it). On the contrary,
M does depend on the size of the volume, moreover it approaches zero when the size of the
volume decreases [8] (if both χ(r) and E(r) are continuous inside V0).
When deriving Eq. (1) the singularity of the Green’s function has been treated explicitly,
therefore it is preferable to the commonly used formulation [8,15]:
∫ ′′′′+=
r )()(),(d)()( 3inc rErrrGrErE χ . (6)
Moreover, Yanghjian noted [25] that there exist several methods for treating the singularity in
Eq. (6) leading to different results. He also proved that the derivation of Eq. (6) is false in the
vicinity of the singularity of ),( rrG ′ . Hence it can be considered correct only if the
singularity is then treated in a way similar to that of Lakhtakia [8], resulting in the correct
Eq. (1).
Discretization of Eq. (1) is done in the following way [15]. Let , U
= /0=ji VV I
for ji ≠ ; N denotes number of subvolumes.2 Although the formulation is applicable to any
set of subvolumes Vi, in most applications standard (equal) cells are used. Then the shape of
the scatterer cannot always be described exactly by such standard cells. Hence, the
discretization may be only approximately correct. Assuming iV∈r and choosing iVV =0 ,
Eq. (1) can be rewritten as
)()(),(),()()(),(d)()( 3inc rErrLrMrErrrGrErE χχ ii
∂−+′′′′+= ∑ ∫
. (7)
2 In the framework of the DDA we usually call a subvolume a dipole.
The set of Eq. (7) (for all i) is exact. Further, one fixed point ri inside each Vi (its center) is
chosen and is set. In many cases the following assumptions can be made: irr =
)()()()(),(d3 jjijj
rErGrErrrG χχ =′′′′∫ , (8)
)()(),( iiiiiV rErMrM χ= , (9)
which state that integrals in Eq. (7) linearly depend upon the values of χ and E at point ri.
Eq. (7) can then be rewritten as
( ) iiii
jjjijii V ELMEGEE χχ −++= ∑
inc , (10)
where , , )( ii rEE = )(
incinc
ii rEE = )( ii rχχ = , ),( iii V rLL ∂= .
The usual approximation [15] is to consider E and χ constant inside each subvolume:
iii V∈== rrErE for)(,)( χχ , (11)
which automatically implies Eqs. (8), (9) and
( )∫ ′−′′=
iii r ),(),(d
s3)0( rrGrrGM , (12)
∫ ′′=
ij rV
1 3)0( rrGG . (13)
Superscript (0) denotes approximate values of the dyadics. A further approximation, which is
used in almost all formulations of the DDA, including e.g. [8], is
),()0( jiij rrGG = . (14)
This assumption is made implicitly by all formulations that start by replacing the scatterer
with a set of point dipoles. It is important to note that Eq. (10) and derivations resulting from
it require weaker assumptions (Eqs. (8), (9)) than imposed by Eq. (11) and, moreover,
Eq. (14). It is possible to formulate the DDA based on Eq. (10), e.g. the Peltoniemi
formulation [26] that is described in Section 3.1. We postulate Eq. (10) as a distinctive feature
of the DDA, i.e. a method is called the DDA if and only if its main equation is equivalent to
Eq. (10) with any Vi, χi, iM , iL , and ijG .
Kahnert [15] distinguished the DDA from the MoM by the fact that the MoM solves
directly Eq. (10) for unknown Ei, while the DDA seeks not the total, but the exciting electric
fields
( )( ) selfexc iiiiiii EEEMLIE −=−+= χ , (15)
( ) iiiii ELME χ−=self , (16)
where is the field induced by the subvolume on itself. Eq. selfiE (10) is then equivalent to
jjijii
excexcinc EαGEE , (17)
where iα is the polarizability tensor defined as
( )( ) 1−−+= iiiiii V χχ MLIα . (18)
However, an alternative formulation of the DDA exists [4] seeking a solution for unknown
polarizations Pi:
iiiiii V EEαP χ==
exc , (19)
jijiii PGPαE
1inc . (20)
It is important to note that Pi, defined by Eq. (19), is only an approximation to the polarization
of the subvolume Vi. This approximation is exact only under the assumption of Eq. (11), while
the formulation itself does not require it. The formulation, using Eq. (20), can be thought as
an intermediary between the DDA and the MoM as classified by Kahnert [15], therefore
revealing complete equivalence of these two formulations. The special structure of the matrix
ijG makes Eq. (20) preferable over Eqs. (10), (17) to find a numerical solution. This is
discussed in Section 4.
Lakhtakia [8] classified strong and weak forms of the DDA as those accounting for or
neglecting iM respectively. The weak form approaches the strong form when the size of the
cell decreases, because iM approaches zero. For a cubical cell Vi and with ri located at the
center of the cell, iL can be calculated analytically yielding [25]
=i . (21)
Using Eq. (18), this results in the well-known Clausius-Mossotti (CM) polarizability (used
originally by Purcell and Pennypacker [1]) for the weak form of the DDA:
ii d ε
α IIα , (22)
where )( ii rεε = , and d is the size of the cubical cell.
After the internal electric fields are determined, the scattered fields and cross sections
can be calculated. The scattered fields are obtained by taking the limit ∞→r of the integral
in Eq. (1) (see e.g. [7]):
)iexp(
)(sca nFrE
= , (23)
where rrn = is the unit vector in the scattering direction, and F is the scattering amplitude:
∑∫ ′′⋅′−′−−=
krnnk )()()iexp(d)ˆˆ(i)( 33 rErnrInF χ . (24)
All other differential scattering properties, such as amplitude and Mueller scattering matrices,
and asymmetry parameter >< θcos can be derived from F(n), calculated for two incident
polarizations [27]. Radiation forces also can be calculated [28-30]. Consider an incident
polarized plane wave3
)iexp()( 0inc rkerE ⋅= , (25)
where , a is the incident direction, and ak k= 10 =e . The scattering cross section Csca is [27]
2sca )(d
C . (26)
Absorption and extinction cross sections (Cabs, Cext) are derived [7,14] directly from the
internal fields:
( )∑ ∫ ′′′=
abs )()(Imd4 rErχπ , (27)
[ ]( ) ( )∗∗ ⋅=′⋅′′′= ∑∫ 02inc3ext )(Re4)()()(Imd4 eaFrErEr krkC i Vi
χπ , (28)
where * denotes a complex conjugate. Conservation of energy necessitates that
absextsca CCC −= . (29)
However, as was noted by Draine [2], use of Eq. (29) for evaluation of Csca can lead to larger
errors than Eq. (26), especially when . scaabs CC >>
The easiest way to express Eqs. (24) and (27) in terms of the internal fields in the
subvolumes centers is to assume Eq. (11), yielding
∑ ∫ ⋅′−′−−=
krnnk )iexp(d)ˆˆ(i)( 33)0( nrEInF χ , (30)
3 DDA can be used for any incident wave, e.g. Gaussian beams [31]; however, we do not discuss this here.
∑∑ ∗==
iii kVkC )Im(4)Im(4
abs EPE πχπ . (31)
Further approximation of Eq. (30), leaving only the lowest order expansion of the exponent
around ri, leads to
∑ ⋅−−−=
ii knnk )iexp()ˆˆ(i)(
3)0( nrPInF , (32)
which together with Eq. (28), leads to
( )∑ ∗⋅=
inc)0(
ext Im4 EPπ . (33)
Eqs. (32) and (33) are identical to those used by Purcell and Pennypacker [1] and then by
Draine [2], while expressions for Cabs (compared to Eq. (31)) are slightly different. These
differences are discussed in Subsection 3.1. Unfortunately, many researchers do not specify
explicitly how the scattering quantities are obtained from the computed internal fields or
polarizations. Those who do usually use Draine’s prescription (Eqs. (26), (32), (33), and
(35)).
Errors of the formulation can be classified as associated with the finite cell size d
(discretization errors), and with approximating the particle shape with a set of standard cells,
e.g. cubical (shape errors). Discretization errors result from considering E constant inside
each cell and the approximate evaluation of iM and ijG . Shape errors also can be considered
as resulting from the assumption of constant χ and E inside bordering cells, which is false
since the edge of the particle crosses these cells. On the other hand, shape errors can be
viewed as a difference of the results for the exact particle shape and for that comprised of the
set of standard cells. Both errors approach zero when ∞→N , while the geometry of the
scatterer and parameters of the incident field are fixed. However, the same does not apply if
while N is fixed, i.e. the DDA is not exact in the long-wavelength limit. Moreover,
both errors are sensitive to the size of the scatterer in the resonance region (see discussion in
Subsection
3.2). The behavior of these errors was studied by Yurkin et al. [32].
3 Various DDA models
3.1 Theoretical base of the DDA
Since the original manuscript by Purcell and Pennypacker [1], many attempts have been made
to improve the DDA. The first stage (1988-1993) of these improvements was reviewed by
Draine and Flatau [4]. It has been noted [2] that Eq. (22) does not satisfy energy conservation,
and results obtained using this formulation do not satisfy the optical theorem. Based on the
well-known [33] “radiative reaction” (RR) electric field, a correction to the polarizability for a
finite dipole was added [2]:
i)32(1 α
= . (34)
Draine [2] also proposed the following expression for the absorption cross section:
[ ]∑ ⋅−⋅=
iiii kkC
*3*exc)0(
abs )32()Im(4 PPEPπ , (35)
derived from Eq. (29) applied to a single point dipole. The PP formulation uses Eq. (35)
without the second part. It can be verified that Eq. (35) results in zero absorption for any
scatterer if the polarizability is of the following form:
IAα 31 i)32( kii −=
− , Hii AA = , (36)
where H denotes the conjugate transpose of a tensor. For real refractive index m, RR and all
other expressions specified below result in α satisfying Eq. (36), which makes Eq. (35)
clearly favorable over e.g. the PP formulation. It must be noted however that the original PP
formulation, where CM polarizability was used, also results in zero absorption for real m.
The correction in Eq. (34) is ( )3)(O kd . Several other corrections of ( )2)(O kd have been
proposed. The first one was proposed by Goedecke and O’Brien [7] and independently in two
other manuscripts [34,35]. They started from Eqs. (10)-(12) and used the following
simplifying fact for a cubical cell (also valid for spherical cells), resulting from symmetry:
)(d IRRf
RRf , (37)
where the origin is in the center of the cube. Eq. (37) is valid for any f(R) that has a singularity
of less than third order for , i.e. the integrals on both sides are defined. They obtained 0→R
32)0( )iexp(d
Rki IM . (38)
By expanding exp(ikR) in Taylor series one can obtain
( )⎟⎟
++= ∫ 423
2)0( Oi
ki IM . (39)
The remaining integral was evaluated by approximating the cube by a volume-equivalent
sphere, resulting in
( )( )432DGF1)0( )(O))i(32()( kdkdkdbi ++= IM , (40)
611992.1)34( 31DGF1 ≈= πb . (41)
An exact evaluation, obtained without expanding the exponent, of Eq. (38) for the equivolume
sphere with radius 31)43( πda = was performed by Livensay and Chen [36] and
implemented into the DGF formulation of the DDA by Hage and Greenberg [14,35] and later
Lakhtakia [37]:
[ ]1)iexp()i1()38()0( −−= kakai IM π . (42)
In terms of the first two orders of expansion, this yields an identical result as Eq. (40). Finally
the polarizability is obtained as
( )( )32DGF13CM
)i()32()(1 kdkdbd +−
α . (43)
We denote the method based on Eq. (42) as LAK. Differences between LAK and DGF should
be noticeable only for large values of kd.
Dungey and Bohren [38], using results by Doyle [39], proposed the following treatment
of the polarizability. First, each cubic cell is replaced by the inscribed sphere that is called a
dipolar subunit with a higher relative electric permittivity εs as determined by the Maxwell-
Garnett effective medium theory [27]:
f , (44)
where 6π=f is the volume filling factor. Other effective medium theories also may be used
[40]. Next, the dipole moment of the equivalent sphere is determined using the Mie theory,
and the polarizability is defined as [39]
i= , (45)
where α1 is the electric dipole coefficient from the Mie theory (see e.g. [41]):
)()()()(
)()()()(
sssssss
sssssss
1 xmxxxmm
xmxxxmm
α , (46)
where ψ, ξ are Riccati-Bessel functions; 2s kdx = and ss ε=m are the size parameter and
the relative refractive index of the equivalent sphere. We denote this formulation for the
polarizability as the a1-term method (note that this terminology was introduced later [42]). It
has the particular property that 1constCMM ≠→αα when , contrary to all other
polarization prescription, for which this ratio approaches 1. It should be noted that the Mie
theory is based on the assumption that the external electric field is a plane wave. In most
applications of the DDA this is true for the incident electric field, but not for the field created
by other subvolumes. Therefore the a
1-term method is expected to be correct only for very
small cell size. Hence it is not clear whether this method has advantages even compared to
CM. On the other hand, this method may be more justified for clusters of small spheres,
where each sphere can be considered as a dipole (see Subsection 3.3).
Draine and Goodman [3] pointed out that considering electric fields constant for
evaluating integrals over a cell introduces errors of order ( )( )2O kd . This represents a problem
for many polarizability corrections, based on integral equations. Draine and Goodman
approached this problem from a different angle. They determined the optimal polarizability in
the sense that an infinite lattice of point dipoles with such polarizability would lead to the
same propagation of plane waves4 as in a medium with a given refractive index. This
polarizability was called LDR (Lattice Dispersion Relation) and is, as expected, CM plus
high-order corrections. These corrections in turn depend on the direction of propagation a and
the polarization of the incident field e0:
( ) ( )[ ]322LDR32LDR2LDR13CM
)i()32()(1 kdkdSmbmbbd +++−
α , (47)
8915316.1LDR1 ≈b , , , 1648469.0
2 −≈b 7700004.1
3 ≈b (48)
( )∑=
20eaS . (49)
We use a reverse sign convention in the denominator of Eq. (47) and the LDR coefficients as
compared to the original paper [3].
Recently it has been shown [43] that the LDR derivation is not completely accurate,
since the resulting dipole moment does not satisfy the transversality condition, for which a
correction was proposed. This corrected LDR (CLDR) differs principally in the fact that the
polarizability tensor can not be made isotropic but only diagonal [43], though not dependent
on the incident polarization:
( ) ( )[ ]3222LDR32LDR2LDR13CM
)i()32()(1 kdkdambmbbd +++−
α . (50)
Another flaw of LDR is that it is evidently not correct for dipoles near the particle surface.
However, it is not clear how to evaluate the effect of these mistreated surface dipoles on the
overall results, e.g. on the scattering cross section.
Further improvement of the DDA was initiated by Peltoniemi [26] (PEL) who showed
that the term M(Vi) in Eq. (7) can be evaluated exactly up to the third order of kd by
expanding the term )()( rEr ′′χ under the integral in a Taylor series over the point irr =′ ,
yielding
( ) ( ,)(O3i3)iexp(d
)iexp(
EkdERRRRkRRk
ERkRRk
REMVM
ντρτρνμ
+∂∂−+−
∂−++=
(51)
where χ, E and their derivatives are all considered at the point ri. Eq. (51) is correct up to the
third order of kd since the third term in the Taylor series vanishes because of symmetry. For
spherical Vi of radius a, the integrals can be evaluated exactly [26] in a way similar to
4 with certain direction of propagation and polarization state.
obtaining Eq. (42), but only terms of less than fourth order of kd are significant, which results
( EkaakakaVi χχχχπ 42232 )(O)(10
)( +⎥
⎛ ⋅∇∇−∇−⎟
⎛ += EEEM ). (52)
If χ is constant inside the cell then the Maxwell equations state that
EE 222 km−=∇ , 0=⋅∇ E . (53)
Hence Eq. (9) is valid up to the third order of ka and
( )[ ]322 )(i)32()()101(1)34( kakami ++= IM π . (54)
Piller and Martin [44] proposed using sampling theory to evaluate the integrals in
Eq. (1). The electric field and the susceptibility is sampled:
∑ −′=′′
iiih )()()()()(
r rErrrrEr χχ , (55)
where hr(r) is the impulse response function of an antialiasing filter defined as
)cos()sin(
qrqrqr
=r , (56)
where dq π2= . Eq. (1) is then transformed to Eq. (10) with the so-called filtered Green’s
function, defined as
∫ −′′′=
r3 )(),(d
ij hrV
rrrrGG . (57)
Eq. (57) can be viewed as a generalization of Eq. (13). The latter is obtained if a pulse
function is considered instead of hr. The integral in Eq. (57) is evaluated analytically [44],
taking V0 to be infinitesimally small. The filtered Green’s function does not have a singularity
when , therefore ji rr = iiii V GM = . It was shown that the Fourier spectrum of E(r) lies on a
sphere with radius m(r)k, if m is constant in the vicinity of r. Therefore at least two sampling
points per wavelength in the scatterer are required. The susceptibility is also filtered, either by
a mean value filter or a more complicated one, e.g. a Hanning window. This approach is
called FCD (filtered coupled dipoles), and a computer code library for evaluation of filtered
Green’s function is available [45].
Chaumet et al. [11] proposed direct integration of the Green’s tensor (IT) in Eqs. (12),
(13). A Weyl expansion of the Green’s tensor is performed, transforming it to a form allowing
efficient numerical computation of the self-term ( LM − ). They also proposed a correction to
the second term in Draine’s expression for Cabs (Eq. (35)). Extension of their results to a non-
isotropic self-term is
( ) ( )[ ]∑ −⋅+⋅=
iiiiiii VkC /)(ImIm4
***exc)0(
abs PLMPEPπ
The corrected second term is based on radiation energy of a finite dipole [11]: , in
contrast to a point dipole used in the derivation of Eq.
)Im( self ∗⋅ ii PE
(35). One can see that Eqs. (58) and
(31) are equivalent. Moreover, both of them are equivalent to Eq. (35) if and only if
IAM iii Vk
3i)32(+= , Hii AA = . (59)
This condition is similar, but not equivalent, to Eq. (36) and is always satisfied for RR, DGF,
and LAK. Other polarizability prescriptions satisfy Eq. (59) for real m, then both Eqs. (58)
and (35) result in zero absorption.
Rahmani, Chaumet, and Bryant [46] proposed a new method (RCB) to determine
polarizability based on the known solution of the electrostatic problem for the same scatterer.
In the static limit the electric field at any point is linearly related to the incident field
)()()( 01 rErCrE −= . (60)
Substituting Eq. (60) into Eq. (20) with the static Green’s tensor, one can obtain the
polarizability, which would give an exact solution in the static limit, as
1RCB −= iiii V Λχα , (61)
ii CCrrGCΛ
1),( −
∑+= χ , (62)
where )( ii rCC = . This static polarizability then replaces the CM polarizability, and the RR
(Eq. (34)) is applied to it [46] to obtain the final polarizability for DDA simulations. It was
later shown that RCB polarizabilities differ significantly from CM only for dipoles closer than
2d to the interface [47].
In their next manuscript [48] Rahmani et al. stated that the previous derivation is correct
only if the tensor C is constant inside the particle (e.g. for ellipsoids), since otherwise the
polarizability tensor obtained from Eq. (61) is generally not symmetric, which is physically
impossible in the static case. This shows that a particle with a non-constant C is not
equivalent to any set of physical point dipoles even in the static regime. However, it is
equivalent to a set of non-physical dipoles with an asymmetric polarizability. Therefore, the
polarization defined by Eq. (61) formally can be used, by itself or with RR, even when C is
not constant.
Collinge and Draine [47] empirically combined the RCB prescription with CLDR to get
the surface-corrected LDR (SCLDR):
( )( ) 13RCBRCBSCLDR −−= BαIαα d , (63)
where B is the correction matrix (analogous to Eq. (50)):
( )[ ]3222LDR32LDR2LDR1 )i()32()( kdkdambmbbB +++= μμνμν δ . (64)
All methods based on the paper by Rahmani et al. [46] are initially limited to very specific
shapes of the scatterer (ellipsoids, infinite slabs and cylinders). Expansion of its applicability
to other shapes is debatable [48] and would anyway require a preliminary solution of the
electrostatic problem for the same shape, which is generally not trivial.
All DDA formulations are schematically depicted in Fig. 1, which also shows
interrelations between them. Some formulations can be compared unambiguously in terms of
theoretical soundness: one is an improvement of the other, i.e. it employs fewer
approximations. Such formulations are depicted in the same column on Fig. 1, while others
cannot be compared directly with each other; they give rise to different columns. Comparison
between formulations from different columns can and has been made almost exclusively
empirically by comparing the accuracy of the simulation results (see Subsection 3.2).
All the above techniques are aimed at reducing discretization errors; only a few aim at
reducing shape errors. Some of them employ adaptive discretization (different dipole sizes) to
better describe the shape of the scatterer (see Subsection 3.4). Another approach is to average
susceptibility in boundary subvolumes. The simplest averaging using the Lorentz-Lorenz
mixing rule was proposed by Evans and Stephens [49] for the case of the boundary between
the scatterer and its surrounding medium
3434 e
, (65)
where is the effective susceptibility, and f is the volume fraction of the subvolume actually
occupied by scatterer.
A more advanced averaging, called the weighted discretization (WD), was proposed by
Piller [13]. It modifies the susceptibility and self-term of the boundary subvolume.5 The
particle surface, crossing the subvolume Vi, is assumed linear and divides the subvolume into
two parts: the principal that contains the center and a secondary with susceptibilities piV
5 any subvolume that has non-zero intersection with both the scatterer and the outer medium. All such
subvolumes are accounted for.
Integral Eq. (1)
discretization
(no assumptions)
Eq. (7)
General
formulation of
DDA – Eq. (20)
Eqs. (8), (9)
DGF, LAK
Eqs. (11)
Eq. (14)
(weak form)
CLDR
a1 term
SCLDR FCD
sampling with
antialiasing
filter
removing
antialiasing
filter
improving polarizability starting
from dipole formulation
complies
Eq. (14)
simplifies to
Fig. 1. Scheme of interrelation between the different DDA models discussed in Section 3.1. Arrows
down correspond to assumptions employed. Vertical position of the method qualitatively corresponds
to its accuracy (higher = better), however methods in different columns cannot be compared directly.
iχ , and electric fields , respectively. Electric fields are considered constant
inside each part and related to each other via a boundary condition tensor
iχ ii EE ≡
iT :
iii ETE =
s . (66)
Then the total polarization of the subvolume can be evaluated as follows:
iiiiiiiii
i VVVr
EEErErP essspp3 )()(d χχχχ =+=′′′= ∫ , (67)
( ) iiiiiii VVV TI ssppe χχχ += . (68)
The susceptibility of the boundary subvolume is replaced by an effective one.
The effective self-term is evaluated directly starting from Eq. (3), considering χ and E
constant inside each part:
( ) ( ) ii
rr TrrGrrGrrGrrGM ss3ps3ee
),(),(d),(),(d χχχ ∫∫ ′−′′+′−′′= . (69)
Piller [13] evaluated the integrals in Eq. (69) numerically. The final equations are the same as
Eq. (20), where polarizabilities are obtained from Eq. (18) using effective susceptibilities and
self-terms for boundary subvolumes. Hence, WD does not modify the general numerical
scheme.
Currently, there are no rigorous theoretical reasons for preferring one formulation over
others. However, theoretical analyses of DDA convergence when refining discretization
recently conducted by Yurkin et al. [32], showed that IT and WD significantly improve the
convergence of shape and discretization errors, respectively. Experimental verification of
these theoretical conclusions is still to be performed.
Table 1. Accuracy of different DDA formulations for a sphere.a
Value Method x a/d y m Error, % Ref.
Cext a1-term 1÷2 2÷4
c 0.65
0.85
1.33+0.05i
1.7+0.1i
CSec, S11 LAK 9
0.44
0.42
0.51
1.05
1.33+0.01i
2.5+1.4i
0.05, 37
0.5, 35
4, 15
Csca, Cabs DGF ≤3.2c 16 ≤1 4+3i 5, 10÷30 [3]
CSec LDR ≤8c 16 ≤0.5, ≤0.1 m-1≤1 1, 2
Csca
LDR ≤7c 16 ≤1
≤0.5, ≤1
2+i 1.5
3, 4
CSec ≤16c
25 ≤1 1.6+0.0008i
2.5+0.02i
LDR [51]
[4]eCSec
LDR any any ≤1 |m|≤3 5
20÷30
Csca LDR ≤10c 16 kd≤0.63 0.69
0.41
0.29
[148]
S11 LDR ≤10c 24 kd≤0.42 0.69
0.41
Cext, RMS11S
3.2 Accuracy of DDA simulations
Over the years many results on the accuracy of DDA simulations have been published. It is,
however, generally hard to systematically compare the relevant manuscripts because they all
use different independent parameters, such as the size parameter x, refractive index m, or
discretization, as a function of which the error is measured. We will describe discretization by
the parameter kdmy = or . The former is used wherever possible; however,
in some cases a description of results is more straightforward in terms of y
kdmy )Re(Re =
Re. Accuracy results
LDR 20÷160
20÷130
20÷60
20÷30
20÷30
32÷256c
40÷256c
48÷128c
56÷80c
64÷88c
0.61÷0.65d
0.56÷0.64d
0.58÷0.65d
0.57÷0.60d
0.56÷0.62d
0.62d
1.05
0.04, 38
0.4, 23
1, 59
4.4, 56
5.7, 105
2.0, 86
[113]
Ψ FCD π, 2π 2.8, 5.6c 1.7 1.5 1 [44]
WD-FCD 0.5÷3.2c
1.5÷3.8c
0.9÷1.5c
yRe=0.63 |m|<7b
|m|<2.5b
|m|<4b
[10]Ψ
IT ≤5.2c CSec
≤2.1c
≤1.1c
8 ≤1 1.5+0.3i
3.5+1.4i
7.1+0.7i
Cabs
CSec RCB-RR ≤8.2c
≤7.5c
≤5.9c
≤3.4c
≤1.3c
16 ≤1 1.8+0.4i
1.9+i
2.5+i
2.5+4i
7.4+9.4i
CSec SCLDR
SCLDR
≤7.2c
≤1.5c
≤1.5c
12 ≤0.8 1.33+0.1i
5+4i
5+4i
a All errors are relative. CSec denotes the maximum error over all cross sections, S11 and correspond to
maximum and root mean square error over the range of scattering angles, Ψ is the normalized mean error of the
far-field electric fields [44]. In some cases two errors are shown in one cell separated by a coma. They
correspond to two values of one of the parameters in the same row.
b approximate description of the range.
c this value is determined by other values in the same row.
d this value is slightly different for different size parameters.
e this corresponds to the “rule of thumb” for spheres.
for scattering by a sphere are summarized in Table 1. All manuscripts on this subject can be
divided into two classes: those that fix x and vary N (or equivalently, the number of dipoles
per sphere radius a/d) with y, and those that fix a/d and vary the size parameter with y. The
former is easier to interpret; the latter is easier to simulate. To facilitate comparison between
different methods we provide both x and a/d, however one of them is dependent on the other.
Some additional information on these results follows below.
Draine and Goodman [3] compared RR, DGF, and LDR for cross sections of a sphere
with . DGF is generally more accurate than RR. For 16/ =da 1|1| ≤−m LDR gives superior
or comparable results to DGF, for i2+=m LDR and DGF are comparable, and for
DGF is preferable over LDR. In the review of LDR DDA, Draine and Flatau [4]
summarized that for cross sections can be evaluated to accuracies of a few percent
provided . In that case differential cross sections have satisfactory accuracy: relative
errors up to 20-30%, but only where the absolute value of the differential cross sections is
small. For spheres, such results are obtained even for
i34 +=m
2|| ≤m
3|| ≤m . Comparison of CLDR to LDR
[43] only results in minor differences. Generally CLDR results in slightly better accuracy for
Csca, but worse for Cabs.
Piller and Martin [44] compared FCD to LAK by studying the dependence of the mean
relative error of the far-field electric fields (Ψ) on y for spheres with π=x , 2π and . It
was shown that FCD (with a Hanning window filter for the electric permittivity ε) is roughly
3 times more accurate than LAK in the range
5.1=m
5.27.0 ≤≤ y and gives similar accuracies for
(for larger spheres). Comparison of WD to traditional methods [13] was performed
for spheres with
4.0≤y
π=x , 2π and 32.1=m , i7.01.2 + . LAK was used to determine
polarizabilities. For in the range 32.1=m 3.14.0 ≤≤ y overall accuracy was only slightly
improved, but error peaks for certain values of y were smoothed out. For i7.01.2 +=m
accuracy was improved 4-5 times over the whole range 3.1≤y . Piller also showed [10] that a
combination of WD and FCD gives even better results. Generally FCD decreases the negative
effects of Re(ε) on accuracy and WD those of Im(ε).
Rahmani et al. [48] showed that RCB was clearly superior to CM in calculating cross
sections for fixed and m from 16/ =da i4.08.1 + to i4.94.7 + in the range . Two
corrections (LDR and RR) over the static case were compared, and they gave similar overall
results. Improvement of overall accuracy compared with CM was 2-5 times in all cases
studied. For a thin slab, it was shown [46,48] that the internal fields calculated using RCB
differ from those by CM mostly near the interfaces, where RCB yields much smaller errors,
almost the same as far from interfaces.
Collinge and Draine [47] compared LDR, RCB, and SCLDR in calculations of cross
sections of spheres with . It was shown that for 12/ =da i01.033.1 +=m , LDR and SCLDR
are superior in the range , while for 8.0≤y i45 +=m , SCLDR and RCB are superior.
Convergence of cross sections for spheres and ellipsoids for increasing N with fixed x and
different m (from to i01.033.1 + i45+ ) also was studied. SCLDR showed the most stable
results for all cases, being the most or close to the most accurate one; however, for ellipsoids
with large Im(m) RCB gave significantly more accurate results for Csca, especially for larger y.
Performance of the DDA for more complex shapes also was studied by different
authors. Flatau et al. [50] compared DDA simulations for a bisphere with an exact solution
from a multipole expansion. For i01.033.1 +=m , 16/ =da , and 8.0≤y , LDR was several
times more accurate than DGF and resulted in errors of less than 0.5% for both Csca and Cabs.
Xu and Gustafson [51] made a similar but much more extended study of LDR. For
, i008.06.1 +=m 25/ =da , and , errors in C4.0≤y ext, Cabs, and θcos are within 10%. For
, errors in the angular dependence of S81.0=y 11 are up to 20% while S12 and S21 were
completely wrong. For , errors in cross sections exceed 10% for . i02.05.2 +=m 3.0≥y
Errors in the angular dependencies of the Mueller matrix elements are within 10-20% for
and increase rapidly with increasing y. For a fixed 3.0=y 3=x and , errors
i004.06.1 +=m
ext, Cabs, and >< θcos decrease from 10% to 1% while y decrease from 1 to 0.2. For
, the angular dependence of S33.0=y 11 is in good agreement with the rigorous solution,
while S12 and S21 differ significantly for certain orientations of the bisphere.
Hage and Greenberg [14] compared LAK to experimental results obtained from
microwave experiments on porous cubes. Using i005.0362.1 +=m , 64.0=y and ,
they obtained a difference of less than 40% with the experimental results of angular scattering
patterns, except for deep minima. Light scattering of cubes, tiles, and cylinders with similar
parameters also was studied and comparable differences between experiment and theory were
obtained. Theoretical errors were estimated to be less than 10%, except for deep minima.
5504=N
Iskander et al. [34] conducted a limited test of LAK for small elongated spheroids,
comparing the results to those obtained using an iterative extended boundary condition
method. Using , calculations were performed for aspect ratios up to 20 with maximum
size parameter of the long axis being 10 and 0.5 for
i01.033.1 +=m and
respectively. Errors in scattering cross section were 21% and 11%, respectively. Ku [52]
compared LAK with CM and the a
i28.076.1 +
1-term for different shapes, but his conclusions are based
on a large parameter y (up to 2), and are therefore suspicious and not further discussed here.
Andersen et al. [53] studied the performance of the DDA for Rayleigh-sized clusters of
a few spheres (most DDA formulations are then equivalent to CM). Several constituent
materials were tested, all with high refractive indices in the studied region. It was shown that
the DDA failed to converge using the fixed computational resources for very high (up to 13.0)
and very low (down to 0.12) Re(m); up to 30 dipoles were used per diameter of a single
sphere.
It can be concluded that particles with more complex shapes than spheres are more
difficult to model with the DDA, leading to larger errors for the same m and y. This effect can
be explained in general by the increase of surface to volume ratio and hence larger fraction of
boundary subvolumes [32]. Another possible reason is complex regions, e.g. contact between
two particles in a cluster, where rapid variation of the electric field deteriorates the overall
accuracy. There is, however, a notable exception from this general tendency. Shapes, which
can be modeled exactly by a set of cubical dipoles, e.g. a cube, can be simulated using the
DDA much more accurately than spheres, especially for small y [32].
Draine and Flatau [4] have introduced a “rule of thumb” for discretization: use 10
dipoles per wavelength in the medium (i.e. either y or yRe equal to 0.63, depending on the
interpretation). Though it is widely used, the accuracy of the results, when using such
discretization, is hard to deduce a priori. Draine and Flatau themselves derived an estimate of
the error based on a set of test simulations. This estimate is described above and mentioned in
Table 1; it is usually cited as a “few percent accuracy in cross sections.” However, it may
significantly over- or under-estimate the error, especially for large size parameters. Moreover,
it does not completely account for the dependence on m, even in the stated range of its
application ( ), since DDA accuracy deteriorates rapidly with increasing m (see 2|| ≤m Table
1). Still, the rule of thumb is good first guess for many applications.
Most studies of DDA accuracy are limited to integral scattering quantities and, at most,
the angular dependence of S11. In only a few manuscripts are other scattering quantities
studied. For instance, Singham [54] simulated the angular dependence of Mueller matrix
element S34 for spheres and less compact particles, using CM polarizability. It was shown that
an accurate simulation of this element requires smaller values of y than for S11. For 55.1=x
and a calculation of S33.1=m 11 was accurate already for 8.0=y , while was
required for S
2.0≤y
34. It was also reported that for less compact objects like discs and rods, the
required y was larger, 0.4 and 0.55 respectively, because of the smaller interaction between
the dipoles. However, Hoekstra and Sloot argued [55] that this effect is mostly caused by the
pronounced S34 sensitivity to surface roughness, which is significant for smaller size if y is
fixed. They showed that for and 7.10=x 05.1=m , very high accuracy is achieved with
because of the larger number of dipoles used. 66.0=y
Internal fields are an intermediate result in the DDA. They cannot be directly compared
to the experimental results; however, all measured scattering quantities are derived from
them. Therefore, a study of their accuracy can reveal greater understanding of the nature of
DDA errors. Hoekstra et al. [56] performed such a study for LAK polarizability. Three
spheres were examined with , 9, 5 and 9=x 05.1=m , i01.033.1 + , respectively.
Values of y were 0.44, 0.42, and 0.51 respectively. The most significant errors in the
amplitude of the internal field were localized at the boundary of the spheres with maximum
relative errors of 3.4%, 19%, and 120% respectively. Errors in S
i4.15.2 +
12, S33, S34 were significant
only for the third sphere. It was shown that for a given yRe these errors rapidly increase with m
but only slightly depend upon x in the range from 1 to 10. Moreover, the DDA is capable of
reproducing resonances of Mie theory, although their positions are slightly shifted (less than
1% in m).
Druger and Bronk [57] studied the accuracy of the internal fields for single and coated
spheres. They used 5.1=x , , and CM polarizability. Errors in the internal fields were
localized at the interfaces, with average errors larger than 30% for a single sphere with
and , and less than 7% for a single and concentric sphere with and
. The core of the concentric sphere has
8.1≤m
8.1=m 17.0=y 3.1=m
08.0=y 1.1=m and its diameter is half the total
diameter. The angular dependence of the absolute values of S1 and S2 had significant errors in
the side- and backscattering. It can be concluded that shape errors contribute mostly to the
internal fields near the boundary, and increase with m.
All the literature discussing DDA accuracy shows errors as a function of input
parameters and discretization, which is the most straightforward way. The only exception so
far is the rule of thumb, which is too general and approximate to be applied in many particular
cases. A more useful way to present errors is to fix the desired accuracy for certain input
parameters and find the discretization that results in such accuracy. Such an analysis can be
applied directly to practical calculations and can be used to derive rigorous estimates of DDA
computational requirements [58].
In a number of manuscripts the origin of errors in the DDA was examined to try to
separate and compare shape and discretization errors [49,59-62]; however, no definite
conclusions were reached. The uncertainty was due to the indirect methods used that have
inherent interpretation problems. Recently, Yurkin et al. [63] proposed a direct method to
separate shape and discretization errors, which can be used to study their fundamental
properties. This method also can be applied to study the performance of different formulations
aimed at decreasing shape errors, e.g. WD. For example, it has been shown that the maximum
errors of S11(θ) for a sphere with and 5=x 5.1=m , discretized using 16 dipoles per diameter
( ), are mostly due to shape errors. However the same is not true for all measured
quantities. In another manuscript [32] it was suggested that the discretization error should
decrease more rapidly with decreasing y than shape errors. However, it is still hard to deduce
a priori the importance of shape errors for a certain scatterer and y; hence, further systematic
quantitative study is required.
93.0=y
3.3 The DDA for clusters of spheres
There are two main peculiarities when the DDA is applied to clusters of spheres. First, such
particles are generally less compact, yielding smaller interactions between dipoles. This leads
to a smaller condition number of the DDA interaction matrix and hence faster convergence of
the iterative solver (see Section 4.1). Second, when the constituent spheres are small
compared to the wavelength, each sphere can be modeled as one spherical subvolume,
yielding some theoretical simplifications.
A general theory exists [64] based on the Mie theory (generalized multiparticle Mie
solution (GMM) [65]) that allows for highly accurate simulations of clusters of spheres.
However, when many small spheres are used one wants to minimize the number of unknowns
in the linear system. Direct reduction of the GMM to the lowest order (using only the first
order expansion coefficients) leads to DDA + CM [64]. Improving accuracy in the GMM is
done by accounting for higher multipole moments, while the DDA introduces higher order
corrections to the coefficients of the linear system. It is not clear how the accuracy of these
two methods compare with each other; however, the former should lead to a formulation
similar to a coupled multipole method (Subsection 3.4) with a larger number of unknowns.
DDA-based methods (starting usually with the integral equations introduced in Section 2)
should be successful in making the formulation more accurate without increasing the number
of unknowns, which is the goal for large clusters of small spheres. Moreover, the DDA may
employ fast algorithms for solving the linear system. In this setting, the fast multipole method
(FMM) (see Subsection 4.5) seems most promising.
It should be noted, however, that a cluster having a small size parameter (i.e. in the
electrostatic approximation) does not imply that all expansion coefficients, except the first
one, are negligible. This is because the size of the constituent particles is also very small and
the fields inside them are far from constant, especially when the spheres are located close to
each other and have large refractive indices [66]. Therefore, the DDA does have some
principal difficulties of calculating scattering by clusters of spheres. Mackowski [67], for
instance, found that for some systems composed of spheres much smaller than the
wavelength, up to 10 expansion terms were necessary to achieve convergence. In studies of
osculating spheres, Ngo et al. [68] proved that the GMM could be chaotic and were able to
calculate Lyapunov exponents, and that the slow convergence for the touching spheres was
the result of the system lying in an attractor region. A recent paper by Markel et al. [69]
presented computationally efficient modifications of the GMM in the static limit and
demonstrated the insufficiency of the DDA to compute scattering properties of fractal
aggregates accurately. However, Kim et al. [70] showed that the DDA is satisfactory in
calculating the static polarizability of dielectric nanoclusters, especially of clusters with a
large number of constituents.
The development of DDA-based methods for calculating light scattering by clusters of
small spheres was started by Jones [71,72], who developed a method similar to CM. Iskander
et al. [34] used a method equivalent to LAK to calculate scattering of chained aerosol
clusters. This subject was further investigated by Kosaza [73,74]. Lou and Charalampopoulos
[75] (LC) further improved the calculations of the interaction term and scattering quantities.
Starting from an integral equation for the internal field equivalent to Eq. (1), they assumed
Eq. (11). After that the integrals in Eqs. (12) and (13) over spherical subvolumes can be
evaluated analytically. The result for the interaction term is the following:
),()()0( jiij ka rrGG η= , (70)
where a correction function η is defined as
)O()101(1
cossin
3)( 42
x +−=
=η . (71)
Eq. (30) also is evaluated analytically, yielding
∑ ⋅−−−=
ii knnkak )iexp()ˆˆ)((i)(
3)0( nrPInF η , (72)
( )∑ ∗⋅=
iikakC
inc)0(
ext Im)(4 EPηπ . (73)
The following expression for Cabs is stated without derivation:
iikakC )Im()(4
abs EPηπ . (74)
Markel et al. [76] applied the DDA to fractal clusters of spheres, and studied their
optical properties. However, they have not fixed the polarizability of a single dipole but rather
treated it as a variable, calculating the dependence of a cluster’s optical characteristics upon it.
Pustovit et al. [77] argued that the DDA is inaccurate for touching spheres. They developed a
hybrid of the DDA and the GMM, which considers only pair interactions between spheres (as
the DDA) but, when calculating them, accounts for higher multipole terms. This formulation
can be considered as the one providing a more accurate evaluation of the interaction term
(Eq. (13)), and hence similar to LC.
LC was compared to DGF and LAK in a Csca computation of a cluster of 10 particles for
and . Differences between DGF and LAK are less than 1% (as
expected), while the difference between LC and LAK increases quadratically with ka,
reaching 10% for
i7.07.1 +=m 5.005.0 ≤≤ ka
5.0=ka . However, as no exact (e.g. GMM) solution is presented, the
accuracy of each individual method is not clear.
Okamoto [42] tested the a1-term method for clusters of up to 3 touching spheres. No
effective medium is needed in this case, making the method sounder. It was shown that the a1-
term is clearly superior to LDR in cross-sections calculations, when each sphere is treated as a
single dipole. Errors of the a1-term are less than 10% for 2.1≤y when . For
three collinear touching spheres the errors are 30% and 40% for and 2.8 when
and respectively. However, errors do not seem to diminish
significantly for small y (results are presented only down to
i01.033.1 +=m
9.1≤y
i01.033.1 +=m i2 +
2.0=y ). Therefore, the a1-term
seems suitable for obtaining quick crude estimations of cross sections.
In the sequel of this subsection we mention several applications of the DDA to
scattering from clusters of spheres. It was applied to describe the scattering by astrophysical
dust aggregates [78,79] using the a1-term method. Hull et al. [80] applied CM DDA to Diesel
soot particles. LC was applied [81] to the computation of light scattering by randomly
branched chain aggregates. Lumme and Rahola [40] studied scattering properties of clusters
of large spheres (each modeled by a set of dipoles) with the a1-term method considering
astrophysical applications. Hage and Greenberg [35] studied scattering by porous particles,
which were modeled as clusters of cubical cells making their method equivalent to standard
LAK. Recently the DDA with LDR was used [82] to model scattering by porous dust grains
and compare them to approximate theories, e.g. effective medium theories. It also was used to
study light scattering by fractal aggregates [83], especially its dependence on the internal
structure [84].
3.4 Modifications and extensions of the DDA
Bourrely et al. [85] proposed to use small d to minimize surface roughness, but larger dipoles
inside the particle. Starting with small dipoles with CM polarizability, one dipole is combined
with 6 adjacent ones (if they all have the same polarizability) producing a dipole, located at
the same point but with a 7 times larger polarizability. This operation is repeated while
possible. Interaction terms are considered in their simplest form (Eq. (14)). This method
allows the decrease of the shape errors with only a minor increase in the number of dipoles.
The authors showed that this method is more than two times more accurate than CM for some
test cases.
Rouleau and Martin [86] proposed a generalized semi-analytical method. A dynamic
grid is used to evaluate the integral in Eq. (1). First, a static grid is built inside the particle.
Then each point on the static grid is used as an origin of a spherical coordinate system, and
the particle is approximated by an ensemble of volume elements in these spherical
coordinates. As usual, the polarization inside each subvolume is assumed constant, but
Eq. (13) can be evaluated analytically in spherical coordinates. Polarization inside a
subvolume is obtained by interpolation of its values at the points of the static grid. In addition,
adaptive gridding is employed, where smaller subvolumes are used at the boundary of the
particle.
Mulholland et al. [87] proposed a coupled electric and magnetic dipole method
(CEMD), where a magnetic dipole is considered at each subvolume together with an electric
dipole. Polarizabilities are derived from the a1 and b1 terms of the Mie theory. CEMD
requires two times more variables in the linear system, since the electric and magnetic fields
are interconnected. Lemaire [88] went further and developed the coupled multipole method,
considering also the electric quadrupole. Addition of the electric quadrupole can be
considered as a more accurate evaluation of the interaction term in Eq. (13), as compared to
Eq. (14). It results in even better accuracy than CEMD, but at the expense of additional
computation time. The major disadvantage of all these four methods is that the matrix of the
system of linear equations does not seem to have any special form, suitable for faster
algorithms (see Section 4). Therefore computational costs are much larger compared to
regular methods, thus limiting their practical use. In what follows, several DDA extensions
are mentioned without further discussion.
The theoretical basis for application of the DDA to optically anisotropic particles was
summarized by Lakhtakia [89]. Loiko and Molochko [90] applied the DDA to study light
scattering by liquid-crystal spherical droplets. Smith and Stokes [91] used the DDA to
calculate the Faraday effect for nanoparticles. Researchers in the electrical engineering
community applied MoM (in a variation that is equivalent to the DDA) to anisotropic
scatterers [92,93].
Rectangular parallelepipeds can be used as subvolumes in the DDA [11,23,43]. This
allows an accurate description of light scattering by particles with large aspect ratios, using
fewer dipoles and is also compatible with FFT techniques (Subsection 4.4).
Khlebtsov [94] proposed a simplification of the DDA, based on the assumption that all
polarizations are parallel to the incident electric field. The number of variables is thus reduced
three times, however at a cost of accuracy. Moreover, depolarization is completely ignored.
Markel [95] analytically solved the DDA equations for scattering by an infinite one-
dimensional periodic dipole array. This approach is similar to the one used in obtaining the
LDR formulation for dipole polarizability [3].
Chaumet et al. [96] generalized the DDA to periodic structures, and further to defects in
a periodic grating on a surface [97]. The idea of using the complex Green’s tensor in the
standard DDA formulation was summarized by Martin [98].
Yang et al. [99] used the DDA to calculate surface electromagnetic fields and determine
Raman intensities for small metal particles of arbitrary shape.
Lemaire and Bassrei [100] showed that the shape of an object can be reconstructed from
the measured angle dependence of scattered intensities. This procedure can be thought of as
an inversion of the dependence between dipole polarizabilities and scattering. This
dependence is taken from the DDA. A similar idea is used in recent manuscripts on optical
tomography [101-103].
Zubko et al. [104] modified the Green’s tensor used in the DDA to study the
backscattering of debris particles. They showed that the far-field part of the Green’s tensor is
responsible for both the backscattering brightness surge and the negative polarization branch.
4 Numerical considerations
In this section the numerical aspects of the DDA are discussed. One should keep in mind,
however, that final simulation times depend not only on the chosen numerical methods but
also on the particular implementation. Recently, Penttila et al. [105] have compared four
different computer programs for the DDA. These are based on almost identical numerical
methods: the Krylov-subspace iterative method (Section 4.1) combined with a FFT
acceleration of the matrix-vector product (Section 4.4). However, simulation times may differ
by several factors. Optimizations of computer codes are not further discussed in this review.
4.1 Direct vs. iterative methods
There are two general types of methods to solve linear systems of equations , where x
is an unknown vector and A and y are known matrix and vector, respectively: direct and
yAx =
iterative [106]. Direct methods give results in a fixed number of steps, while the number of
iterations required in iterative methods is generally not known a priori. The most usual
example of a direct method is LU decomposition, which allows quick solving for multiple y
once the decomposition is performed. Iterative methods are usually faster, less memory
consuming and numerically more stable. However, iterative methods cannot be considered
superior over direct, since they strongly depend on the problem to solve [107].
For a general n×n matrix (in DDA Nn 3= ) computation time of LU decomposition is
O(n3) and storage requirements O(n2), while computation time for one iteration is O(n2) [107].
Iterative methods for a general matrix converge in O(n) iterations, although some of them
may not converge at all. However, in many cases satisfactory accuracy can be obtained after a
much smaller number of iterations. In these cases, iterative methods can provide significant
increases in speed, especially for large n. Most iterative methods access the matrix A only
through matrix-vector multiplication (sometimes also with the transposed matrix), which
allows the construction of special routines for calculation of these products. Such routines
may decrease memory requirements, since it is no longer necessary to store the entire matrix,
especially for matrices of special form (see Subsection 4.3). A special structure of the matrix
may also allow acceleration of the matrix-vector product from O(n2) to O(nlnn) (see
subsections 4.4, 4.5). However, the same applies to direct methods (see Subsection 4.3).
Throughout DDA history, mostly iterative methods were employed (however see
Subsection 4.6). At first, they were used to accelerate computations [1], but they also allowed
larger numbers of dipoles to be simulated [6,108], since storage of the entire matrix is
prohibitive for direct methods. The most widely used iterative methods in the DDA are
Krylov-space methods, such as [107] conjugate gradient (CG), CG applied to the Normalized
equation with minimization of Residual norm (CGNR), Bi-CG, Bi-CG stabilized (Bi-
CGSTAB), CG squared (CGS), generalized minimal residual (GMRES), quasi-minimal
residual (QMR), transpose free QMR (TFQMR), and generalized product-type methods based
on Bi-CG (GPBi-CG) [109].
An important part of the iterative solver is preconditioning, which effectively decreases
the condition number of the matrix A and therefore speeds up convergence. However, this
requires additional computational time during both initialization and each iteration.
Preconditioning of the initial system can be summarized as [107]
yMxMAMM 12
21 )( =
− , (75)
where M1 and M2 are left and right preconditioners, respectively. Preconditioners should
either allow fast inversion or be integrated into the iteration process. The simplest
preconditioner of the first type is the Jacobi (point), which is just the diagonal part of matrix
A. An example of the second type of preconditioner is the Neumann polynomial
preconditioner of order l:
)( AIM . (76)
QMR and Bi-CG can be made to employ the complex symmetric (CS) property of the
DDA interaction matrix to halve the number of matrix-vector multiplications [110] (and thus
computational time). Lumme and Rahola [40] were the first to apply QMR(CS) to the DDA
and compared it with CGNR. They used m from i1.06.1 + to i43+ , and x from 1.3 to 13.5,
corresponding to N from 136 to 20336. For all cases studied QMR(CS) was 2-4 times faster
than CGNR.
Rahola [9] further studied QMR(CS) and compared it to CGNR, Bi-CG(CS), Bi-
CGSTAB, CGS, GMRES (full and with different memory length). For a “typical small
problem” (parameters were not specified, unfortunately) the convergence of different methods
was tested and QMR(CS) along with Bi-CG(CS) showed the best results. Although full
GMRES was able to converge in fewer iterations, GMRES with as much as 40 memory
lengths was slower than QMR(CS).
Flatau [111] reviewed the use of iterative algorithms in the DDA and tested many of
them, together with several preconditioners. He calculated scattering of a homogenous sphere
with and m from 1.33 up to 1.0=x i0001.05+ , 1=x and m from 1.33 up to and
. Left (L) and right (R) Jacobi-, and first-order Neumann polynomial
preconditioners were tested. Unfortunately the number of dipoles N was not specified, which
hampers comparison with other studies. For small particles CG(L) was superior for all
refractive indices studied. CG and CG(R) showed similar results, while CGNR(L) and Bi-
CGSTAB(L) were about 4 times slower. For
i33.1 +
i0001.03+
1=x Bi-CGSTAB(L) was superior while Bi-
CGSTAB,(R) and CGS,(L),(R) were slightly worse. TFQMR (both with and without Jacobi
preconditioner) was 3-4 times slower. The first-order Neumann preconditioner showed
unsatisfactory results. It was concluded that Bi-CGSTAB(L) is the most satisfactory choice
for the DDA, and that method is the default one used in the DDSCAT program [6].
Recently Fan et al. [112] have compared GMRES, QMR(CS), Bi-CGSTAB, GPBi-CG,
and Bi-CG(CS). They tested them on wavelength-sized scatterers (x up to 10) with m up to
, and concluded that GMRES with memory depth 30 was the fastest, although it
required four times more memory than the other methods. However, only the times of the
matrix-vector product was compared, while other parts of the iteration may also take
significant time, especially for GMRES(30). Choosing from less memory-consuming
methods, QMR(CS) and Bi-CG(CS) showed a better convergence rate than Bi-CGSTAB and
GPBi-CG, especially when
i2.05.4 +
2>m . Moreover, the authors pointed out some flaws in the
comparison by Flatau [111], making his conclusions insufficient.
Yurkin et al. [113] employed QMR(CS), Bi-CG(CS), and Bi-CGSTAB to simulate light
scattering by spheres with x up to 160 and 40 for 05.1=m and 2, respectively. It was shown
that convergence of the iterative methods becomes very slow with increasing x and m (up to
105 iterations are required), and none of them is clearly preferable to the others. Moreover,
there seems to be no systematic dependency of the choice of the best iterative solver on x and
m; however, the difference in computational time was less than a factor of two, except for the
largest x and m studied.
Rahola [114] showed that the spectrum of the integral scattering operator for any
homogenous scatterer is a line in the complex plane going from 1 to m2, except for a small
amount of points, which corresponds to refractive indices that cause resonances for the
specific shape. The spectrum of A is similar, since this matrix is obtained in the DDA by
discretization of the integral operator (see also [9]). Assuming that the spectrum of A exactly
lies on the specified line, it was shown that an estimate for the optimal reduction factor6 γ can
be given as
Eq. (77) is an approximation valid for small particle sizes, where no, or only few, resonances
are present. However, in all cases the spectrum of A resembles the spectrum of the linear
operator, which is defined by shape, size and refractive index of the scatterer. Therefore, the
spectrum, and thus convergence, should not depend significantly on the discretization. This
fact was confirmed empirically in other manuscripts [9,63].
Budko and Samokhin [115] generalized Rahola’s results to arbitrary inhomogeneous
and anisotropic scatterers. They described a region in the complex plane that contains the
whole spectrum of the integral scattering operator. This region depends only on the values of
m inside the scatterer and does not depend on x. They showed that for purely real m or for m
with very small imaginary part this region may come close to the origin, therefore the
spectrum may contain very small eigenvalues for particles larger than the wavelength. This
6 Norm of the residual is decreased by this factor every iteration.
may explain the extremely slow convergence of the iterative solver for real m and large x,
which was recently obtained in numerical simulations [113]. Based on the analysis of the
spectrum of the integral scattering operator for particles much smaller than the wavelength,
Budko et al. [116] proposed an efficient iteration method for this particular case.
It can be concluded that there are several modern iterative methods (QMR(CS), Bi-
CG(CS), and Bi-CGSTAB) that have proved to be efficient when applied to the DDA.
However, none of them can be claimed superior to the others, and one should test them for
particular light-scattering problems. Moreover, except for the simplest cases, preconditioning
of the DDA interaction matrix is almost not studied, while there is a need for it for large x and
m, since then all methods converge extremely slowly or even diverge. It seems to us that the
next major numerical advance in the DDA will be achieved by developing a powerful
preconditioner for the DDA matrix.
A large number of dipoles requires large computational power and, hence, parallel
computers are commonly used, e.g. [108,113]. Parallel efficiency is not discussed here, but
for iterative solvers, it is generally close to 1 [117]. However, this is not true for all
preconditioners [107], and hence heavy preconditioners requiring large computational time in
combination with a parallel DDA implementation should be employed with caution.
4.2 Scattering order formulation
The Rayleigh-Debye-Gans (RDG) approximation [27] consists in considering E(r) equal to
Einc(r). F(n) is then obtained directly from Eq. (24). Generalization of the RDG approach is
obtained by iteratively solving the integral equation (1), which can be rewritten as
)()()( inc rΛErErE += , (78)
where Λ is a linear integral operator describing the scatterer. The iterative scheme is readily
obtained by inserting the current (l-th) iteration of the electric field E(l)(r) into the right side of
Eq. (78) and calculating the next iteration in the left side:
)()()( )(inc)1( rΛErErE ll +=+ . (79)
The starting value is taken the same as in RDG, , and the general formula for
the solution is the following:
)()( inc)0( rErE =
inc )()(
l rEΛrE , (80)
which is a direct implementation of the well-known Neumann series:
lΛΛI , (81)
where I is the unitary operator. A necessary and sufficient condition for Neumann-series
convergence is
1<Λ . (82)
Physical sense of this iterative method lies in successive calculations of interaction
between different parts of the scatterer. The zeroth approximation (or RDG) accounts for no
interaction; the first approximation considers the influence of scattering of each dipole on the
others once, and so on. Eq. (82) states that the interaction inside the scatterer should be small,
but not as small as required for the applicability of RDG ( 1<<Λ ). In scattering problems,
especially in quantum physics, Eq. (80) is called the Born expansion.
Although theoretically clear, the Born expansion is not directly applicable [118], since
each successive iteration requires analytical evaluation of multidimensional integrals with
rising complexity, which quickly becomes unfeasible even for the simplest scatterers. The
latest result is probably that of Acquista [118], who evaluated the Born expansion for a
homogenous sphere up to second order. Therefore, realistic application of the Born expansion
does require discretization of the integral operator, which is naturally done in the DDA.
A scattering order formulation (SOF) of the DDA was developed independently by
Chiappetta [119] and Singham and Bohren [12,120] by applying the Neumann series to
Eq. (17). Λ is then a matrix defined as jijij αGΛ = , where each element is a dyadic, which
can be expressed as a 3×3 matrix. An explicit check of Eq. (82) for a certain scatterer is not
feasible numerically, however de Hoop [121] derived a sufficient condition for scalar waves:
1)(max)(2 20 <r
χπ kR , (83)
where R0 is the radius of the smallest sphere circumscribing the scatterer. Although not
directly applicable to light scattering, Eq. (83) can be used as an estimate.
The range of size parameter and refractive index where SOF converges is limited [120].
Moreover, even when SOF converges, more advanced iterative methods converge faster (see
Subsection 4.1). However, SOF has clear physical sense and can be used to study the
importance of multiple scattering.
4.3 Block-Toeplitz
A square matrix A is called Toeplitz if jiij aA −= , i.e. matrix elements on any line parallel to
the main diagonal are the same [106]. In a block-Toeplitz (BT) matrix (of order K) elements
ai are not numbers, but square matrices themselves:
. (84)
A 2-level BT matrix has BT matrices as components ai. Proceeding recursively a multilevel
BT (MBT) matrix for any number of levels is defined.
Let us consider a rectangular lattice nx×ny×nz, numbered in the following way
zzyyzxxzy ininninnni +−+−= )1()1( , (85)
where indicates the position of the element along the axes. Let us also define
the vector index . Then one can verify that the interaction matrix in Eq.
},...,1{ μμ ni ∈
),,( zyx iii=i (20),
defined by Eq. (13), satisfies the following:
jiGGG −′== jiij . (86)
This equation alone can be used to greatly reduce the storage requirements of iterative
methods by use of indirect addressing. Further improvement is to note that Eq. (86) defines a
symmetric 3-level BT matrix (orders of subsequent levels – nx, ny, nz) whose smallest blocks
are 3×3 matrices (dyadics) ijG .
A rectangular lattice is not much of a restriction, since any scatterer can be embedded in
an appropriate rectangular grid. However, additional “empty” dipoles should be introduced to
build up the grid up to the full parallelepiped. Moreover, position and size of the dipoles
cannot be chosen arbitrarily to better describe the shape of the scatterer. This is especially
problematic for highly porous particles or clusters of particles, where the monomer has a size
comparable to a single dipole. For all other cases these restrictions are minor compared to the
large increase in computational speed, imposed by the BT-structure of the interaction matrix.
A matrix-vector multiplication can be transformed to a convolution, which is computed using
a fast Fourier transform (FFT) technique in O(nln(n)) operations (see Subsection 4.4). Note
however, that alternative techniques exist that do not require a regular grid (see Subsection
4.5).
The BT-structure also permits acceleration of direct methods. Flatau et al. [122] used an
algorithm for inversion of symmetric BT-matrices. It has complexity )(O 3 xnn and storage
requirements )(O 2 xnn , since only 2 block columns of the inverse matrix need to be stored.
In this case the x-axis is oriented along the longest particle dimension. Recently Flatau [123]
studied the special case of 1D DDA where all dipoles are located on a straight line and
equally spaced, in which systems of equations for different components can be separated. The
interaction matrix for each component is symmetric Toeplitz, and a modern fast algorithm can
be applied for its inversion. This method requires preliminarily solving linear equations for
two right sides (e.g., by some iterative technique); then multiplication of the inverse matrix by
any vector (i.e., a solution of the linear system for any right part) requires only O(nln(n))
operations. However, Flatau pointed out a strict limitation for all methods for fast calculation
of the inverse of the interaction matrix: they are applicable only when polarizabilities of all
dipoles are the same, since otherwise the first term on the right side of Eq. (20) ruins the BT
structure on the diagonal of the interaction matrix. Therefore, they are currently limited to
homogenous rectangular scatterers. Fortunately, it is not a problem for matrix-vector
multiplication, since the diagonal term can be evaluated independently and added to the final
result.
4.4 FFT
Goodman et al. [124] showed that multiplication of the interaction matrix for a rectangular
lattice (see Subsection 4.3) by a vector can be transformed into a discrete convolution
jii PGPGPGy ′′=′== ∑∑∑
)2,2,2(
)1,1,1(
)1,1,1(1
zyxzyx nnnnnnN
jij , (87)
where iG′ is defined by Eq. (86) (and 0=′0G ) for μμ ni ≤ and
⎧ ≤≤∀
otherwise,0
1:, μμμ njj
P . (88)
Both G ′ and are then regarded as periodic in each dimension μ with period 2nP′ μ. A
discrete convolution can be transformed with a FFT to an element-wise product of two
vectors, which is easily computed. It requires evaluation of a direct and inverse FFT for each
matrix-vector product. Each of them is a 3D FFT of order 2nx×2ny×2nz. This operation is done
for each of the 3 Cartesian components of P′ and preliminary calculations is performed for 6
independent tensor components of G ′ .
A slightly different method can be devised based on the paper by Barrowes et al. [125],
who developed an algorithm for multiplication of any MBT by a vector. The multiplication is
brought down to a 1D convolution that is evaluated by two 1D FFTs of order
)12)(12)(12( −−− zyx nnn . Flatau [123] proposed an algorithm of matrix-vector
multiplication for BT interaction matrix (e.g. 1D DDA), which requires twice as many FFTs
as the standard algorithm, but of order n instead of 2n. Although Flatau stated that an
extension of this algorithm to the general 3D case is straightforward, it is at least not trivial
and probably its complexity will scale the same as standard methods.
4.5 Fast multipole method
The fast multipole method (FMM) was developed by Greengard and Rokhlin [126] for
efficient evaluation of the potential and force fields in N-body simulations where all pairwise
interactions of N particles are computed. The FMM is based on truncated potential expansions
[127]. It is also called a hierarchical tree method because particles are grouped together in a
hierarchical way, and the interaction between single particles and this hierarchy of particle
groups is calculated [128]. However, some researchers distinguish between single- and
multilevel FMM [129,130]; only the latter is truly hierarchical. The FMM naturally fits the
DDA, since the matrix-vector multiplication is actually computing the total field on each
single dipole due to all other dipoles, as was noted by Hoekstra and Sloot [128]. The
computational complexity of the FMM (see below) is similar to FFT-based methods (see
Subsection 4.4), but it does not require any regularity of the grid, thus making it applicable to
any scatterer. The drawback is that the FMM is conceptually more complex, making it much
harder to code. Nonetheless, the FMM was implemented in the DDA by Rahola [9,127].
Error analysis is critical for the FMM, since the acceleration is obtained by using
approximations, in contrast to exact FFT-based methods. Approximation parameters are
chosen to keep an error, calculated according to some estimate, in certain bounds. The more
exact the error estimate is, the less computations are required; thus, the faster the whole
algorithm. Therefore, algorithm complexity is directly connected to error analysis [131].
Koc and Chew [129] described the application of multilevel FMM to the DDA. They
used semi-empirical formulae to determine the number of terms in multipole series, and
obtained O(N) complexity. However, rigorous, close to exact, error analysis is still lacking for
the FMM applied to the DDA. It will allow obtaining a real algorithm complexity with
guaranteed accuracy. Such an analysis has been conducted for 2D acoustic scattering [130],
and for light scattering formulated in terms of surface integrals [131]. In both cases the FMM
was proven to have an asymptotic complexity O(Nln2(N)). Application of the FMM to
surface-integrals formulation of light scattering was reviewed by Dembart and Yip [132].
Another problem of implementing the FMM is that it is completely dependent upon the
exact form of the interaction potential ijG . All manuscripts mentioned above deal with
interaction between point dipoles, i.e. Eq. (14). If a more complex expression for ijG is used
(e.g. IT), most of the FMM should be developed anew. This makes integration of the FMM
and the DDA a formidable problem.
The FMM is a promising method to calculate light scattering by particles that cannot be
mapped effectively on a rectangular grid; however, there is still space for improving its theory
to make it more robust and guarantee certain accuracy.
The FMM is not the only hierarchical tree method available. For instance, a very
intuitively simple method was proposed by Barnes and Hut [133,134]. Multipole expansions
over the center of mass in gravitational computations are used, contrary to geometrical center
in the FMM. It automatically eliminates the second term in the multipole expansion, and
allows fast evaluation of monopole terms. Though this method is much simpler and clearer
than the FMM, it has very little control over the errors that can be studied almost exclusively
empirically. It can be applied to the DDA without significant increase in the total
computational errors.7
An alternative approach was proposed by Ding and Tsang [135]. They studied scattering
from trees and used a sparse matrix iterative approach. The interaction matrix is divided into a
strong part, which accounts for interaction between nearby dipoles, and a complement weak
part: . The strong part is sparse and therefore allows quick solution of the linear
system. The weak part is a small correction that is accounted for iteratively:
ws AAA +=
yxA =)0(s , . )(w)1(s ll xAyxA −=+ (89)
The authors demonstrate potential of this approach for some test cases.
4.6 Orientation averaging and repeated calculations
In many physical applications, one is interested in optical properties of an ensemble of
randomly oriented particles. When the concentration of particles is small, multiple scattering
is negligible and the optical properties are obtained by averaging single-particle scattering
over different particle orientations. More general problems, where particles are not identical
or multiple scattering is significant, are not considered here.
Orientation averaging of any scattering property can be described as the integral over
the Euler’s orientation angles (including a probability distribution function if necessary),
7 Hoekstra AG, unpublished results
which is brought down to a sum by appropriate quadrature. The problem therefore consists in
calculation of some scattering property for a set of different orientations of the same particle.
The easiest way is to calculate it by solving sequentially and independently each problem
from the set. However, the large size of this set calls for some means of reducing the
calculations. This is especially relevant when the particle is asymmetric; hence, its optical
properties are sensitive to particle orientation. Let us further assume for clarity that we are
interested in the scattering matrix at a certain scattering angle. All the discussion for other
scattering properties is analogous or even simpler.
Singham et al. [136] noted that the set of problems described above is physically
equivalent to a fixed orientation of the particle and different incident and scattering directions.
The latter are determined by transformation of the laboratory reference frame to the reference
frame associated with the particle. The amplitude scattering matrix, and hence the Mueller
matrix, also is transformed along with the reference frame (see e.g. [137] for transformation
formulae). There are two immediate advantages of using a fixed particle orientation. First, A
is kept constant (see though discussion below), and therefore the construction of A is done
only once. Second, the amplitude matrix for any scattering angle is quickly obtained after the
linear system is solved (for two incident polarizations). Hence, integration over one Euler
angle is relatively fast.
The constancy of A can be exploited to further reduce the time of orientation-averaging.
If or its LU decomposition is obtained [75,136], a single solution for any right part y can
be obtained in n
2 operations – the same or less time than required for one iteration using
general iterative methods (see Subsection 4.1). Moreover, Singham et al. [136] and McClain
and Ghoul [138] independently proposed an analytical way of averaging the scattering matrix
at any scattering angle, which requires O(n2) operations once is known. Khlebtsov [139]
extended this technique to averaging of extinction and absorption cross sections.
However, by employing special properties of the matrix A in the DDA allows
computing matrix-vector products in O(nln(n)) operations (see subsections 4.4, 4.5). Although
some acceleration of direct methods also can be performed (see Subsection 4.3), they are still
O(n2) or slower. For large n, iterative methods (assuming that they converge in much less than
n iterations) are clearly preferable, even if many quadrature points are used. Moreover, large n
is unattainable by direct methods because of storage requirements. Another improvement
could be using a heavy preconditioner, which has large initialization cost and greatly
increases convergence rate. Initialization cost is then justified because it is computed only
once. Possible candidates are incomplete factorization preconditioners [107].
Above it was stated that A is constant for a fixed-orientation particle. However, modern
DDA formulations (e.g. LDR) take into account the direction of light incidence. Hence A
depends upon this direction, but only weakly through ( )( )2O kd corrections. This complicates
the techniques described above, however probably they still may be used together with some
special methods to correct for small changes in A on every step. Such methods have not been
developed as yet.
Another possibility to perform orientation averaging is to first compute the T-matrix of
the particle, which then allows analytical averaging [140]. The T-matrix formalism is based
on the multipole expansion which is truncated at some order N0. Although N0 is hard to
deduce a priori, usually it is several times x [141,142]. The number of rows in the T-matrix
equals . The simplest way to evaluate the T-matrix based on the DDA is to solve
for every incident spherical wave (i.e. for each row of the T-matrix) independently [141].
Then the above discussion about optimizing this repeated calculation is relevant. Using
iterative techniques with N
)2(2 00 +NN
iter number of iterations, computation time is
( ) ( )[ ]20iter20 O)ln(O NNNNN + , where the first term in the sum is the time for solving the linear
system, and the second one is the actual computation of the values in the row of the T-matrix.
A new method to obtain the T-matrix from the DDA interaction matrix was proposed by
Mackowski [141]. This requires two summations with computational time ( ))ln(O 20 NNN and
( )NN 40O . Mackowski showed that for 5=x his method is an order of magnitude faster than
the straightforward one.
Recently Muinonen and Zubko [143] have proposed a way to optimize ensemble
averaging of DDA results over different sizes and refractive indices. It is based on calculating
a “good guess” for the initial vector in the iterative solver using results of the calculations
with similar parameters. Similar ideas can be used to optimize simulation of a set of slightly
different shapes or orientational averaging.
Use of repeated calculations to increase the accuracy of DDA simulations was proposed
recently by Yurkin et al. [63]. Several independent simulations with different discretization
parameter were performed and results were extrapolated to the infinite discretization giving
better accuracy than those of a single DDA simulation.
5 Comparison of the DDA to other methods
Hovenier et al. [144] compared the DDA, the extended boundary condition method (EBCM),
and the separation of variables method (SVM) for calculations of scattering by spheroids,
finite cylinders and bispheres. Parameters of the problems were as follows: ,
equivolume size parameter ,
i01.05.1 +=m
5=x 6.0=y . The angular dependencies of scattering matrix
elements were calculated. The EBCM and SVM seemed to achieve an exact solution, and the
DDA showed little errors, except for backscattering angles, where they were up to 10-20%.
Wriedt and Comberg [145] compared the DDA, EBCM, and finite difference time
domain (FDTD) method for a cube with 33.1=m , 1.5 and 9.2=x , 4.9, 9.7. For and
4.9. The DDA and EBCM achieved good accuracy in calculation of scattering intensity angle
dependence; the DDA was 2-5 times faster, but consumed 8-16 times more memory (y was in
the range 0.3-0.5). The FDTD had similar computational requirements as the DDA but was
less accurate. For the DDA was the only one to achieve little errors within the given
computational resources.
9.2=x
7.9=x
Comberg and Wriedt [146] compared the DDA, GMM (see Subsection 3.3) and the
generalized multipole technique (GMT) for clusters of a few spheres. A single sphere had x in
the range 4–20 and 33.1=m , 1.5. All the methods managed to achieve good accuracy, but the
GMM was one order of magnitude (and for large x even several orders) faster than the other
two. The DDA and GMT also were used to compute scattering by a cluster of two oblate
spheroids with and . The DDA was less accurate and consumed 4 times more
memory, but was 6 times faster than the GMT.
5=x 33.1=m
Wriedt et al. [147] compared the DDA, FDTD, GMT, and discrete sources method
(DSM) for the calculation of light scattering by a red blood cell (RBC) with and
. Accuracy of all methods was similar. The DDA and GMT showed similar
calculation times; they were 7 times faster than the FDTD and 12 times slower than the DSM.
It should be noted that the latter employed the axisymmetric property of RBC.
06.1=m
Recently Yurkin et al. [58] systematically compared the DDA and the FDTD for
spheres with m from 1.02 to 2 and x from 10 to 100, depending on m. It was shown that
numerical performance of the DDA is much more sensitive to the refractive index than that of
the FDTD. Therefore, the DDA is preferable for small m, the FDTD for larger m. Cleary, the
crossover point is not well defined and will depend on the details of the problem at hand as
well as on the particular implementations of both methods.
The main advantage of the DDA is that it is one of the most general methods, having a
very broad range of applicability, limited only by available computational power. The reverse
of this advantage is that it has almost no means to use the symmetry of the scatterer. Thus the
DDA is not able to compete with the EBCM for homogenous axisymmetric scatterers. For
homogenous non-axisymmetric scatterers the DDA is competitive with the EBCM for single-
particle orientation, but the latter allows much faster orientation averaging. The EBCM has
little applicability to inhomogeneous scatterers, where the DDA can be applied without any
changes. Comparison between the FDTD and the DDA suggests that the DDA is more
suitable for small m. It also should be noted that the FDTD is even more general, being easily
applicable to non-harmonic incident electric fields. Moreover, simulation of one pulse
incident wave with the FDTD gives the solution for a complete spectrum of incident harmonic
plane waves, but with a limitation on accuracy.
6 Concluding remarks
The DDA has been reviewed using a general framework based on the integral equation for the
electric field. Although mainstream DDA algorithms as used in several production computer
programs, has not changed significantly since 1994, many different improvements have been
proposed since that time. Some of them do improve the accuracy or numerical performance of
the DDA; however, they still wait for a wide acceptance. It seems that a critical mass of new
improvements is building up, hopefully resulting in a next breakthrough in the field of the
DDA.
In our opinion, future major improvements in the DDA computer implementations will
be connected with one of the following:
1) Decreasing shape errors by implementing WD or similar techniques.
2) Improving polarizability and interaction terms by techniques that are still to be
developed similar to IT and PEL.
3) Studying different preconditioners for the DDA interaction matrix, either trying some of
the known types or developing one considering the special structure of the matrix.
Item (1) should improve the overall accuracy of the DDA, especially for cases where shape
errors are dominant, item (2) should expand the DDA applicability region to higher refractive
indices, and item (3) should boost overall performance, especially for large size parameters
and/or refractive indices.
Acknowledgements
We thank Dan Mackowski for clarifying discussion on the simulations of scattering by
clusters of spheres and Gorden Videen for critically reading the manuscript and for valuable
discussions. Our research is supported by Siberian Branch of the Russian Academy of
Sciences through the grant 2006-03.
Appendix. Description of used acronyms and symbols
See Tables A1 and A2.
Table A1. Acronyms in alphabetical order.
Acronym Description Sectiona
(L) left Jacobi preconditioner 4.1
I right Jacobi preconditioner 4.1
a1-term (M) dipole term in the Mie theory 3.1
Bi-CGSTAB Bi-CG stabilized 4.1
BT block-Toeplitz 4.3
CEMD coupled electric and magnetic dipole 3.4
CG conjugate gradient 4.1
CGNR CG applied to normalized equation with minimization of residual norm 4.1
CGS CG squared 4.1
CS complex symmetric 4.1
CLDR corrected LDR 3.1
CM Clausius-Mossotti 2
DDA discrete dipole approximation 1
DGF digitized Green’s function 1
DSM discrete sources method 5
EBCM extended boundary condition method 5
FCD filtered coupled dipoles 3.1
FDTD finite difference time domain 5
FFT fast Fourier transform 4.4
FMM fast multipole method 4.5
GMM generalized multiparticle Mie solution 3.3
GMRES generalized minimal residual 4.1
GMT generalized multipole technique 5
GPBi-CG generalized product-type methods based on Bi-CG 4.1
IT integration of Green’s tensor 3.1
LAK Lakhtakia 3.1
LC Lou and Charalampopoulos 3.3
LDR lattice dispersion relation 3.1
MBT multilevel BT 4.3
MoM method of moments 3.1
PEL Peltoniemi 3.1
PP Purcell and Pennypacker 1
QMR quasi-minimal residual 4.1
RBC red blood cell 5
RCB Rahmani, Chaumet, and Bryant 3.1
RDG Rayleigh-Debye-Gans approximation 4.2
RR radiative reaction correction 3.1
SCLDR surface-corrected LDR 3.1
SOF scattering order formulation 4.2
SVM separation of variables method 5
TFQMR transpose free QMR 4.1
WD weighted discretization 3.1
a where it is explained or first appears (if no explanation is given).
Table A2. Symbols used, Latin and Greek letters in alphabetical order.a
Symbols Description Section
(0) superscript: approximate value (usually under constant field assumption) 2
(n) superscript: after n-th iteration 4.2
>< θcos asymmetry parameter 2
* superscript: complex conjugate 2
A a matrix 3.1
a kk 2
a radius of (equivalent) sphere 3.1
B correction matrix in SCLDR 3.1, Eq. (64)
b1 – b3 numerical coefficients in polarization prescriptions 3.1
C tensor of electrostatic solution 3.1, Eq. (60)
Csca, Cabs, Cext scattering, absorption, extinction cross section 2
c speed of light in vacuum 2
d size of a cubical cell 2
E, Einc, Eexc, Eself, Esca (total) electric field, incident, exciting, self-induced, scattered 2
e0 polarization vector of the incident wave, 10 =e 2
e superscript: effective 3.1
F scattering amplitude 2
f a function;
volume filling factor
G free space dyadic Green’s function (tensor) 2
2sG G in static limit
ijG interaction term
H superscript: conjugate transpose 3.1
hr impulse response function of a filter 3.1
2, 4.1I , I identity dyadic (tensor), operator (matrix)
i, j subscript: vector indices 4.3
i imaginary unity 2
i, j subscript: number of the dipole 2
K order of a BT matrix 4.3
k free space wave vector 2
L self-term dyadic 2
M integral associated with finiteness of V0;
preconditioner
M dyadic associated with M 2
m refractive index (relative) 3.1
N total number of dipoles 2
n rr 2
n size of a matrix 4.1
n′ˆ external normal to the surface 2
nx, ny, nz sizes of the rectangular lattice 4.3
P polarization 2
p superscript: principal 3.1
q dπ2 3.1
R rr ′− 2
R0 radius of the smallest sphere circumscribing the scatterer 4.2
2r, r′ radius-vectors
S LDR coefficient dependent on incident polarization 3.1, Eq. (49)
Si amplitude matrix element 3.2
Sij Mueller matrix element 3.2
s superscript: secondary;
strong;
subscript: equivalent spherical dipole
T boundary condition tensor 3.1, Eq. (66)
t time 2
V volume of the scatterer 2
V0 exclusion volume 2
w superscript: weak 4.5
x unknown vector 4.1
Table A2 (continued)
Symbols Description Section
x size parameter of scatterer 3.2
x, y, z Cartesian coordinates 4.3
y a known vector (right side of a linear system) 4.1
y kdm || 3.2
yRe kdm)Re( 3.2
2α, α polarizability, tensor
4.1γ optimal reduction factor
3.1δ Kronecker symbol
2ε electric permittivity (relative)
3.3η correction function
Λ intermediate tensor in RCB method 3.1, Eq. (62)
4.2Λ linear integral operator, its matrix
μ, ν, ρ, τ, … sub-, superscript: Cartesian components of vectors (tensors) 2
ξ, ψ Riccati-Bessel functions 3.1
2χ electric susceptibility
3.2Ψ mean relative error of far-field electric field
2Ω solid angle
2ω circular frequency of the harmonic electric field
a common sub- and superscripts are given on their own. For all vectors – the same symbol but in italic (instead of
bold) denotes Euclidian norm of the vector (except unitary vectors).
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1 Introduction
2 General framework
3 Various DDA models
3.1 Theoretical base of the DDA
3.2 Accuracy of DDA simulations
3.3 The DDA for clusters of spheres
3.4 Modifications and extensions of the DDA
4 Numerical considerations
4.1 Direct vs. iterative methods
4.2 Scattering order formulation
4.3 Block-Toeplitz
4.4 FFT
4.5 Fast multipole method
4.6 Orientation averaging and repeated calculations
5 Comparison of the DDA to other methods
6 Concluding remarks
Acknowledgements
Appendix. Description of used acronyms and symbols
References
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0704.0039 | Scalar radius of the pion and zeros in the form factor | Scalar radius of the pion and zeros in the form factor
José A. Oller and Luis Roca
Departamento de F́ısica. Universidad de Murcia.
E-30071, Murcia. Spain.
[email protected] , [email protected]
Abstract
The quadratic pion scalar radius, 〈r2〉πs , plays an important role for present precise deter-
minations of ππ scattering. Recently, Ynduráin, using an Omnès representation of the null
isospin(I) non-strange pion scalar form factor, obtains 〈r2〉πs = 0.75± 0.07 fm2. This value is
larger than the one calculated by solving the corresponding Muskhelishvili-Omnès equations,
〈r2〉πs = 0.61± 0.04 fm2. A large discrepancy between both values, given the precision, then
results. We reanalyze Ynduráin’s method and show that by imposing continuity of the re-
sulting pion scalar form factor under tiny changes in the input ππ phase shifts, a zero in the
form factor for some S-wave I=0 T−matrices is then required. Once this is accounted for,
the resulting value is 〈r2〉πs = 0.65± 0.05 fm2. The main source of error in our determination
is present experimental uncertainties in low energy S-wave I=0 ππ phase shifts. Another
important contribution to our error is the not yet settled asymptotic behaviour of the phase
of the scalar form factor from QCD.
http://arxiv.org/abs/0704.0039v2
1 Introduction
The scalar form factor of the pion, Γπ(t), corresponds to the matrix element
Γπ(t) =
d4x e−i(q
′−q)x〈π(q′)|
muū(x)u(x) +mdd̄(x)d(x)
|π(q)〉 , t = (q′ − q)2 . (1.1)
Performing a Taylor expansion around t = 0,
Γπ(t) = Γπ(0)
t〈r2〉πs +O(t2)
, (1.2)
where 〈r2〉πs is the quadratic scalar radius of the pion.
The quantity 〈r2〉πs contributes around 10% [1] to the values of the S-wave ππ scattering lengths
a00 and a
0 as determined in ref.[1], by employing Roy equations and χPT to two loops. If one
takes into account that this reference gives a precision of 2.2% in its calculation of the scattering
lengths, a 10% of contribution from 〈r2〉πs is a large one. Related to that, 〈r2〉πs is also important
in SU(2)×SU(2) χPT since it gives the low energy constant ℓ̄4 that controls the departure of Fπ
from its value in the chiral limit [2, 3] at leading order correction.
Based on one loop χPT , Gasser and Leutwyler [2] obtained 〈r2〉πs = 0.55 ± 0.15 fm2. This
calculation was improved later on by the same authors together with Donoghue [4], who solved
the corresponding Muskhelishvili-Omnès equations with the coupled channels of ππ and KK̄. The
update of this calculation, performed in ref.[1], gives 〈r2〉πs = 0.61±0.04 fm2, where the new results
on S-wave I=0 ππ phase shifts from the Roy equation analysis of ref.[5] are included. Moussallam
[6] employs the same approach and obtains values in agreement with the previous result.
One should notice that solutions of the Muskhelishvili-Omnès equations for the scalar form
factor rely on non-measured T−matrix elements or on assumptions about which are the channels
that matter. Given the importance of 〈r2〉πs , and the possible systematic errors in the analyses
based on Muskhelishvili-Omnès equations, other independent approaches are most welcome. In
this respect we quote the works [7, 8, 9], and Ynduráin’s ones [10, 11, 12]. These latter works have
challenged the previous value for 〈r2〉πs , shifting it to the larger 〈r2〉πs = 0.75 ± 0.07 fm2. From
ref.[1] the equations,
δa00 = +0.027∆r2 , δa
0 = −0.004∆r2 , (1.3)
give the change of the scattering lengths under a variation of 〈r2〉πs defined by 〈r2〉πs = 0.61(1 +
∆r2) fm
2. For the difference between the central values of 〈r2〉πs given above from refs.[1, 10], one
has ∆r2 = +0.23. This corresponds to δa
0 = +0.006 and δa
0 = −0.001, while the errors quoted
are a00 = 0.220 ± 0.005 and a20 = −0.0444 ± 0.0010. We then adduce about shifting the central
values for the predicted scattering lengths at the level of one sigma.
The value taken for 〈r2〉πs is also important for determining the O(p4) χPT coupling ℓ̄4. The
value of ref.[1] is ℓ̄4 = 4.4±0.2 while that of ref.[10] is ℓ̄4 = 5.4±0.5. Both values are incompatible
within errors.
The papers [10, 11, 12] have been questioned in refs.[13, 14]. The value of the Kπ quadratic
scalar radius, 〈r2〉Kπs , obtained by Ynduráin in ref.[10], 〈r2〉Kπs = 0.31± 0.06 fm2, is not accurate,
because he relies on old experiments and on a bad parameterization of low energy S-wave I=1/2Kπ
phase shifts by assuming dominance of the κ resonance as a standard Breit-Wigner pole [15]. Fur-
thermore, 〈r2〉Kπs was recently fixed by high statistics experiments in an interval in agreement with
the sharp prediction of [15], based on dispersion relations (three-channel Muskhelishvili-Omnès
equations from the T−matrix of ref.[16]) and two-loop χPT [17]. From the recent experiments
[18, 19], one has for the charged kaons [18] 〈r2〉K±πs = 0.235 ± 0.014 ± 0.007 fm2, and for the
neutral ones [19] 〈r2〉KLπs = 0.165 ± 0.016 fm2. The prediction of [15], in an isospin limit, is
〈r2〉Kπs = 0.192± 0.012 fm2, lying just in the middle of the experimental determinations. Another
issue is Ynduráin’s more sound determination of the pionic scalar radius, whose (in)correctness is
not settled yet.
In this paper we concentrate on the approach of Ynduráin [10, 11, 12] to evaluate the quadratic
scalar radius of the pion based on an Omnés representation of the I=0 non-strange pion scalar form
factor. Our main conclusion will be that this approach [10] and the solution of the Muskhelishvili-
Omnès equations [4], with ππ and KK̄ as coupled channels, agree between each other if one
properly takes into account, for some T−matrices, the presence of a zero in the pion scalar form
factor at energies slightly below the KK̄ threshold. Precisely these T−matrices are those used in
[10] and favoured in [11]. Once this is considered we conclude that 〈r2〉πs = 0.63± 0.05 fm2.
The contents of the paper are organized as follows. In section 2 we discuss the Omnès rep-
resentation of Γπ(t) and derive the expression to calculate 〈r2〉πs . This calculation is performed
in section 3, where we consider different parameterizations for experimental data and asymptotic
phases for the scalar form factor. Conclusions are given in the last section.
2 Scalar form factor
The pion scalar form factor Γπ(t), eq.(1.1), is an analytic function of t with a right hand cut,
due to unitarity, for t ≥ 4m2π. Performing a dispersion relation of its logarithm, with the possible
zeroes of Γπ(t) removed, the Omnès representation results,
Γπ(t) = P (t) exp
s(s− t)
. (2.1)
Here, P (t) is a polynomial made up from the zeroes of Γπ(t), with P (0) = Γπ(0). In the previous
equation, φ(s) is the phase of Γπ(t)/P (t), taken to be continuous and such that φ(4m
π) = 0. In
ref.[10] the scalar form factor is assumed to be free of zeroes and hence P (t) is just the constant
Γπ(0) (the exponential factor is 1 for t = 0). Thus,
Γπ(t) = Γπ(0) exp
s(s− t)
. (2.2)
From where it follows that,
〈r2〉πs =
ds . (2.3)
One of the features of the pion scalar form factor of refs.[4, 6, 8], as discussed in ref.[13], is the
presence of a strong dip at energies around the KK̄ threshold. This feature is also shared by the
strong S-wave I=0 ππ amplitude, tππ. This is so because tππ is in very good approximation purely
elastic below the KK̄ threshold and hence, neglecting inelasticity altogether in the discussion that
follows, it is proportional to sin δπe
iδπ , with δπ the S-wave I=0 ππ phase shift. It is an experimental
fact that δπ is very close to π around the KK̄ threshold, as shown in fig.1. Therefore, if δπ = π
happens before the opening of this channel the strong amplitude has a zero at that energy. On
the other hand, if δπ = π occurs after the KK̄ threshold, because inelasticity is then substantial,
see eq.(2.4) below, there is not a zero but a pronounced dip in |tππ|. This dip can be arbitrarily
close to zero if before the KK̄ threshold δπ approaches π more and more, without reaching it.
400 600 800 1000 1200 1400 1600
(MeV)
Eq. (3.13), [20]
PY [24]
CGL [1]
Sol. A of [27]
Sol. B of [27]
Sol. C of [27]
Sol. D of [27]
Sol. E of [27]
Kaminski et al. [21]
BNL-E865 Coll. [25]
NA48/2 Coll. [26]
300 350 400 450
BL-E865 Coll. [25]
NA48/2 Coll. [26]
PY [24]
CGL [1]
Figure 1: S-wave I = 0 ππ phase shift, δπ(s). Experimental data are from refs.[21, 25, 26, 27].
Because of Watson final state theorem the phase φ(s) in eq.(2.1) is given by δπ(s) below the
KK̄ threshold, neglecting inelasticity due to 4π or 6π states as indicated by experiments [20]. The
situation above the KK̄ threshold is more involved. Let us recall that
tππ = (η e
2iδπ − 1)/2i , (2.4)
with 0 ≤ η ≤ 1 and the inelasticity is given by 1− η2, with η the elasticity coefficient. We denote
by ϕ(s) the phase of tππ, required to be continuous (below 4m
K it is given by δπ(s)). By continuity,
close enough to the KK̄ threshold and above it, η → 1 and then we are in the same situation as in
the elastic case. As a result, because of the Watson final state theorem and continuity, the phase
φ(s) must still be given by ϕ(s). For δπ(sK) < π, sK = 4m
K , ϕ(s) does not follow the increasing
trend with energy of δπ(s) but drops as a result of eq.(2.4), see fig.2 for δπ(sK) < π. This is easily
seen by writing explicitely the real and imaginary parts of tππ in eq.(2.4),
tππ =
η sin 2δπ +
(1− η cos 2δπ) . (2.5)
400 600 800 1000 1200 1400 1600 1800 2000
(MeV)
1000 1100 1200 1300 1400 1500 1600
(MeV)
( δπ(sK)< 180
( δπ(sK)> 180
Figure 2: Left panel: Strong phase ϕ(s), eigenvalue phase δ(+)(s) and asymptotic phase φas(s). Right
panel: Integrand of 〈r2〉πs in eq.(3.12) for parameterization I (dashed line) and II (solid line). For more
details see the text. Notice that the uncertainty due to φas(s) is much reduced in the integrand.
The imaginary part is always positive (η < 1 above the KK̄ threshold and 1.1 GeV [20]) while the
real part is negative for δπ < π, but in an interval of just a few MeV the real part turns positive
as soon as δπ > π, fig.1. As a result, ϕ(s) passes quickly from values below but close to π to
the interval [0, π/2]. This rapid motion of φ(s) gives rise to a pronounced minimum of |Γπ(t)| at
this energy, as indicated in ref.[13] and shown in fig.3. The drop in φ(s) becomes more and more
dramatic as δπ(sK) → π− (with the superscript +(−) indicating that the limit is approached from
values above(below), respectively); and in this limit, φ(sk) = ϕ(sK) is discontinuous at sK . This is
easily understood from eq.(2.5). Let us call s1 the point at which δπ(s1) = π with s1 > sK . Close
and above s1, ϕ(s) ∈ [0, π/2], for the reasons explained above, and ϕ(s) has decreased very rapidly
from almost π at the KK̄ threshold to values below π/2 just after s1. Then, in the limit s1 → s+K
one has φ(s−K) = ϕ(s
K) = π on the left, while on the right φ(s
K) = ϕ(s
K) < π/2. As a result ϕ(s)
is discontinuous at s = sK . We stress that this discontinuity of ϕ(s) at sK when δπ(sK) → π−
applies rigorously to φ(sK) as well since η(sK) = 1. This discontinuity at s = sK implies also that
the integrand in the Omnès representation for Γπ(t) develops a logarithmic singularity as,
φ(s−K)− φ(s
, (2.6)
with δ → 0+. When exponentiating this result one has a zero for Γπ(sK) as (δ/sK)ν , ν = (φ(s−K)−
φ(s+K))/π > 0 and δ → 0+. This zero is a necessary consequence when evolving continuously from
δπ(sK) < π to δπ(sK) > π.
#1 This in turn implies rigorously that in the Omnès representation of
Γπ(t), eq.(2.1), P (t) must be a polynomial of first degree for those cases with δπ(sK) ≥ π,#2
P (t) = Γπ(0)
s1 − t
, (2.7)
with s1 the position of the zero. Notice that the degree of the polynomial P (t) is discrete and thus
by continuity it cannot change unless a singularity develops. This is the case when δπ(sK) = π,
changing the degree from 0 to 1. Hence, if δπ(sK) ≥ π for a given tππ, instead of eqs.(2.2) and
(2.3) one must then consider,
Γπ(t) = Γπ(0)
s1 − t
s(s− t)
, (2.8)
〈r2〉πs = −
ds . (2.9)
For those tππ for which δπ(sK) > π then ϕ(s) follows δπ(s) just after the KK̄ threshold and there
is no drop, as emphasized in ref.[11], see fig.2.
Summarizing, we have shown that Γπ(t) has a zero at s1 when δπ(sK) ≥ π as a consequence of
the assumption that φ(s) follows ϕ(s) above the KK̄ threshold, along the lines of ref.[11], and by
imposing continuity in Γπ(t) under small changes in δπ(sK) ≃ π. As a result eqs.(2.8) and (2.9)
should be used in the latter case, instead of eqs.(2.2) and (2.3), valid for δπ(sK) < π. This solution
was overlooked in refs.[10, 11, 12]. We show in appendix A why the previous discussion on the
zero of Γπ(t) for δπ(sK) ≥ π at s1 cannot be applied to all pion scalar form factors, in particular
to the strange one.
If eq.(2.2) were used for those tππ with δπ(sK) ≥ π then a strong maximum of |Γπ(t)| would
be obtained around the KK̄ threshold, instead of the aforementioned zero or the minimum of
refs.[4, 6], as shown in fig.3 by the dashed-dotted line. That is also shown in fig.10 of ref.[22]
or fig.2 of [13]. This is the situation for the Γπ(t) of refs.[10, 11], and it is the reason why 〈r2〉πs
obtained there is much larger than that of refs.[4, 1, 6]. That is, Ynduráin uses eqs.(2.2), (2.3) for
δπ(sK) ≥ π, instead of eqs.(2.8), (2.9) (solid line in fig.3). The unique and important role played
by δπ(sK) (for elastic tππ below the KK̄ threshold) is perfectly recognised in ref.[11]. However, in
this reference the astonishing conclusion that Γπ(t) has two radically different behaviours under
tiny variations of tππ was sustained. These variations are enough to pass from δπ(sK) < π to
δπ(sK) ≥ π [10], while the T− or S−matrix are fully continuous. Because of this instability of the
solution of refs.[10, 11] under tiny changes of δπ(s), we consider ours, that produces continuous
Γπ(t), to be certainly preferred. We also stress that our solutions, either for δπ(sK) ≥ π and
δπ(sK) < π, are the ones that agree with those obtained by solving the Muskhelishvili-Omnès
equations [4, 1, 6] and Unitary χPT [8].
#1It can be shown from eq.(2.5) that φ(s−
) − φ(s+
) = π. Here we are assuming η = 1 for s ≤ sK , which is a
very good approximation as indicated by experiment [20, 21].
#2We are focusing in the physically relevant region of experimental allowed values for δπ(sK), which can be larger
or smaller than π but close to.
0 200 400 600 800 1000 1200
(MeV)
δπ(sK)<π
δπ(sK)>π, P(t)=Γπ(0)(s1- t)/s1
δπ(sK)>π, P(t)=Γπ(0)
ref. [8] (δπ(sK)>π)
PSfrag replacements
Figure 3: |Γπ(t)/Γπ(0)| from eq.(2.2) with δπ(sK) < π, dashed-line, and δπ(sK) > π, dashed-dotted line.
The solid line corresponds to use eq.(2.8) for the latter case. For this figure we have used parameterization
II (defined in section 3) with α1 = 2.28 (dashed line) and 2.20 (dashed-dotted and solid lines). The
dashed-double-dotted line is the scalar form factor of ref.[8] that has δπ(sK) > π.
Let us now show how to fix s1 in terms of the knowledge of δπ(s) with δπ(sK) ≥ π. For this
purpose let us perform a dispersion relation of Γπ(t) with two subtractions,
Γπ(t) = Γπ(0) +
〈r2〉πs t+
ImΓπ(s)
s2(s− t)
ds , (2.10)
From asymptotic QCD [23] one expects that the scalar form factor vanishes at infinity [10, 12],
then the dispersion integral in eq.(2.10) should converge rather fast. Eq.(2.10) is useful because
it tells us that the only point around 1 GeV where there can be a zero in Γπ(t) is at the energy
s1 for which the imaginary part of Γπ(t) vanishes. Otherwise, the integral in the right hand side
of eq.(2.10) picks up an imaginary part and there is no way to cancel it as Γπ(0), 〈r2〉πs and t are
all real. Since |ImΓπ(t)| = |Γπ(t) sin δπ(t)| for t ≤ sK , it certainly vanishes at the point s1 where
δπ(s1) = π. As there is only one zero at such energies, this determines s1 exactly in terms of the
given parameterization for δπ(s).
One could argue against the argument just given to determine s1 that this energy could be
complex. However, this would imply two zeroes at s1 and s
1, and then the degree of P (t) would
be two instead of one. Notice that the degree of the polynomial P (t) is discrete and thus, by
softness in the continuous parameters of the T−matrix, its value should stay at 1 for some open
domain in the parameters with δπ(sK) > π until a discontinuity develops. Physically, the presence
of two zeroes would in turn require that φ(s) → 3π so as to guarantee that Γπ(t) still vanishes as
−1/t, as required by asymptotic QCD [23, 10]. This value for the asymptotic phase seems to be
rather unrealistic as ϕ(s) only reaches 2π at already quite high energy values, as shown in fig.2.
3 Results
Our main result from the previous section is the sum rule to determine 〈r2〉πs ,
〈r2〉πs = −
θ(δπ(sK)− π) +
ds , (3.11)
where θ(x) = 0 for x < 0 and 1 for x ≥ 0. We split 〈r2〉πs in two parts:
〈r2〉πs = QH +QA ,
QH = −
θ(δπ(sK)− π) +
, (3.12)
with sH = 2.25 GeV
2. Reasons for fixing sH to this value are given below.
The main issue in the application of eq.(3.11) is to determine φ(s) in the integrand. Below the
KK̄ threshold and neglecting inelasticity, one has that φ(s) = δπ(s), 4m
π ≤ s ≤ 4m2K . This follows
because of the Watson final state theorem, continuity and the equality φ(4m2π) = δπ(4m
π) = 0.
For practical applications we shall consider the S-wave I=0 ππ phase shifts given by the
K−matrix parameterization of ref.[20] (from its energy dependent analysis of data from 0.6 GeV
up to 1.9 GeV) and the parameterizations of ref.[1] (CGL) and ref.[24] (PY). The resulting δπ(s)
for all these parameterizations are shown in fig.1. We use CGL from ππ threshold up to 0.8 GeV,
because this is the upper limit of its analysis, while PY is used up to 0.9 GeV, because at this
energy it matches well inside the experimental errors with the data of [20]. The K−matrix of
ref.[20] is used for energies above 0.8 GeV, when using CGL below this energy (parameterization
I), and above 0.9 GeV, when using PY for lower energies (parameterization II). We take the pa-
rameterizations CGL and PY as their difference below 0.8 GeV accounts well for the experimental
uncertainties in δπ, see fig.1, and they satisfy constraints from χPT (the former) and dispersion re-
lations (both). The reason why we skip to use the parameterization of ref.[20] for lower energies is
because one should be there as precise as possible since this region gives the largest contribution to
〈r2〉πs , as it is evident from the right panel of fig.2. It happens that the K−matrix of [20], that fits
data above 0.6 GeV, is not compatible with data from Ke4 decays [25, 26]. We show in the insert
of fig.1 the comparison of the parameterizations CGL and PY with the Ke4 data of [25, 26]. We
also show in the same figure the experimental points on δπ from refs.[20, 21, 27]. Both refs.[20, 21]
are compatible within errors, with some disagreement above 1.5 GeV. This disagreement does not
affect our numerical results since above 1.5 GeV we do not rely on data.
The K−matrix of ref.[20] is given by,
Kij(s) = αiαj/(x1 − s) + βiβj/(x2 − s) + γij , (3.13)
where
1 = 0.11± 0.15 x
2 = 1.19± 0.01
α1 = 2.28± 0.08 α2 = 2.02± 0.11
β1 = −1.00± 0.03 β2 = 0.47± 0.05
γ11 = 2.86± 0.15 γ12 = 1.85± 0.18 γ22 = 1.00± 0.53 ,
(3.14)
with units given in appropriate powers of GeV. In order to calculate the contribution from the
phase shifts of this K−matrix we generate Monte-Carlo gaussian samples, taking into account the
errors shown in eq.(3.14), and evaluate QH according to eq.(3.12). The central value of δπ(sK)
for the K−matrix of ref.[20] is 3.05, slightly below π. When generating Monte-Carlo gaussian
samples according to eq.(3.14), there are cases with δπ(sK) ≥ π, around 30% of the samples. Note
that for these cases one also has the contribution −6/s1 in eq.(3.11).
The application of Watson final state theorem for s > 4m2K is not straightforward since inelastic
channels are relevant. The first important one is the KK̄ channel associated in turn with the
appearance of the narrow f0(980) resonance, just on top of its threshold. This implies a sudden
drop of the elasticity parameter η, but it again rapidly raises (the f0(980) resonance is narrow
with a width around 30 MeV) and in the region 1.12 . s . 1.52 GeV2 is compatible within
errors with η = 1 [20, 21]. For η ≃ 1, the Watson final state theorem would imply again that
φ(s) = ϕ(s), but, as emphasized by [13], this equality only holds, in principle, modulo π. The
reason advocated in ref.[13] is the presence of the region sK < s < 1.1
2 GeV2 where inelasticity
can be large, and then continuity arguments alone cannot be applied to guarantee the equality
φ(s) ≃ ϕ(s) for s & 1.12 GeV2. This argument has been proved in ref.[11] to be quite irrelevant
in the present case. In order to show this a diagonalization of the ππ and KK̄ S−matrix is done.
These channels are the relevant ones when η is clearly different from 1, between 1 and 1.1 GeV.
Above that energy one also has the opening of the ηη channel and the increasing role of multipion
states.
We reproduce here the arguments of ref.[11], but deliver expressions directly in terms of the
phase shifts and elasticity parameter, instead of K−matrix parameters as done in ref.[11]. For
two channel scattering, because of unitarity, the T−matrix can be written as:
(ηe2iδπ − 1) 1
1− η2ei(δπ+δK)
1− η2ei(δπ+δK) 1
(ηe2iδK − 1)
, (3.15)
with δK the elastic S-wave I=0KK̄ phase shift. In terms of the T -matrix the S-wave I=0 S−matrix
is given by,
S = I + 2iT , (3.16)
satisfying SS† = S†S = I. The T -matrix can also be written as
T = Q1/2
K−1 − iQ
Q1/2 , (3.17)
where the K−matrix is real and symmetric along the real axis for s ≥ 4m2π and Q = diag(qπ, qK),
with qπ(qK) the center of mass momentum of pions(kaons). This allows one to diagonalize K with
a real orthogonal matrix C, and hence both the T− and S−matrices are also diagonalized with
the same matrix. Writing,
cos θ sin θ
− sin θ cos θ
, (3.18)
one has
cos θ =
[(1− η2)/2]1/2
1− η2 cos2∆− η| sin∆|
1− η2 cos2∆
]1/2 ,
sin θ = −
sin∆√
1 + (1− η2) cot2∆
1− η2 cos2∆− η| sin∆|
1− η2 cos2∆
]1/2 , (3.19)
with ∆ = δK − δπ. On the other hand, the eigenvalues of the S−matrix are given by,
e2iδ(+) = S11
1 + e2i∆
1 + (1− η2) cot2∆
(3.20)
e2iδ(−) = S22
1 + e−2i∆
1 + (1− η2) cot2∆
. (3.21)
The eigenvalue phase δ(+) satisfies δ(+)(sK) = δπ(sK). The expressions above for exp 2iδ(+) and
exp 2iδ(−) interchange between each other when tan∆ crosses zero and simultaneously the sign in
the right hand side of eq.(3.19) for sin θ changes. This diagonalization allows to disentangle two
elastic scattering channels. The scalar form factors attached to every of these channels, Γ′1 and
Γ′2, will satisfy the Watson final state theorem in the whole energy range and then one has,
= CTQ1/2Γ = CTQ1/2
Γπ = q
λ cos θ |Γ′1|eiδ(+) ± sin θ |Γ′2|eiδ(−)
ΓK = q
± cos θ |Γ′2|eiδ(−) − λ sin θ |Γ′1|eiδ(+)
. (3.22)
The ± in front of |Γ′2| is due to the fact that Γ′2 = 0 at sK , as follows from its definition in the
equation above. Since Watson final state theorem only fixes the phase of Γ′2 up to modulo π, and
the phase is not defined in the zero, we cannot fix the sign in front at this stage. Next, Γ′1 has a
zero at s1 when δπ(sK) ≥ π. For this case, −|Γ′1| must appear in the previous equation, so as to
guarantee continuity of its ascribed phase, and this is why λ = (−1)θ(δπ(sK)−π).
Now, when η → 1 then sin θ → 0 as
(1− η)/2 and φ(s) is then the eigenvalue phase δ(+).
This eigenvalue phase can be calculated given the T−matrix. For those T−matrices employed
here, and those of refs.[10, 11, 4, 13], δ(+)(s) follows rather closely ϕ(s) in the whole energy range.
This is shown in fig.2 and already discussed in detail in ref.[11]. In this way, one guarantees
that φ(s) and ϕ(s) do not differ between each other in an integer multiple of π when η ≃ 1,
1.12 . s . 1.52 GeV2.
For the calculation of QH in eq.(3.12) we shall equate φ(s) = ϕ(s) for 4m
K < s < 1.5
2 GeV2.
Denoting,
= I1 + I2 + I3 ,
∫ 1.12
ds , (3.23)
QH ≃ IH −
θ(δπ(sK)− π) . (3.24)
Now, eq.(3.22) can also be used to estimate the error of approximating φ(s) by ϕ(s) in the range
4m2K < s < 1.5
2 GeV2 to calculate I2 and I3 as done in eq.(3.23). We could have also used δ(+)(s)
in eq.(3.23). However, notice that when η . 1 then ϕ(s) ≃ δ(+)(s) and when inelasticity could be
substantial the difference between δ(+)(s) and ϕ(s) is well taken into account in the error analysis
that follows. Remarkably, consistency of our approach also requires φ(s) to be closer to ϕ(s) than
to δ(+)(s). The reason is that ϕ(s) for δπ(sK) ≥ π is in very good approximation the ϕ(s) for
δπ(sK) < π plus π, this is clear from fig.2. This difference is precisely the required one in order
to have the same value for 〈r2〉πs either for δπ(sK) < π or δπ(sK) ≥ π from eq.(3.11). However, the
difference for δ(+)(s) between δπ(sK) < π and δπ(sK) ≥ π is smaller than π. Indeed, we note that
φ(s) follows closer ϕ(s) than δ(+)(s) for the explicit form factors of refs.[8, 4].
Let us consider first the range 1.12 < s < 1.52 GeV2 where from experiment [20] η ≃ 1 within
errors. With ǫ = ± tan θ|Γ′2/Γ′1| and ρ = δ(−) − δ(+), eq.(3.22) allows us to write,
Γπ = λ cos θ |Γ′1|eiδ(+)(1 + ǫ cos ρ)
1 + i
ǫ sin ρ
1 + ǫ cos ρ
. (3.25)
When η → 1 then ǫ → 0, according to the expansion,#3
tan θ =
(1− η)/2
1− 1 + 3 cos 2∆
8 sin2∆
(1− η)
(1− η)5/2
. (3.26)
Rewriting,
1 + i
ǫ sin ρ
1 + ǫ cos ρ
= exp
ǫ sin ρ
1 + ǫ cos ρ
+O(ǫ2) , (3.27)
which from eqs.(3.25) and (3.27) implies a shift in δ(+) because of inelasticity effects,
δ(+) → δ(+) +
ǫ sin ρ
1 + ǫ cos ρ
. (3.28)
#3The the ratio |Γ′2/Γ′1|, present in ǫ, is not expected to be large since the f0(1300) couples mostly to 4π and
similarly to ππ and KK̄, and the f0(1500) does mostly to ππ [28].
Using η = 0.8 in the range 1.12 . s . 1.52 GeV, η ≃ 1 from the energy dependent analysis of
ref.[20] given by the K−matrix of eq.(3.13), one ends with ǫ ≃ 0.3. Taking into account that δ(+)
is larger than & 3π/2 for δπ(sK) ≥ π (in this case δ(+) ≃ δπ), and around 3π/4 for δπ(sK) < π,
see fig.2, one ends with relative corrections to δ(+) around 6% for the former case and 13% for the
latter. Although the K−matrix of ref.[20], eq.(3.13), is given up to 1.9 GeV, one should be aware
that to take only the two channels ππ and KK̄ in the whole energy range is an oversimplification,
particularly above 1.2 GeV. Because of this we finally double the previous estimate. Hence I3 is
calculated with a relative error of 12% for δπ(sK) ≥ π and 25% for δπ(sK) < π.
In the narrow region between sK < s < 1.1
2 GeV2, η can be rather different from 1, due to
the f0(980) that couples very strongly to the just open KK̄ channel. However, from the direct
measurements of ππ → KK̄ [29], where 1 − η2 is directly measured,#4 one has a better way to
determine η than from ππ scattering [20, 21]. It results from the former experiments, as shown
also by explicit calculations [30, 31, 32], that η is not so small as indicated in ππ experiments [20],
and one has η ≃ 0.6 − 0.7 for its minimum value. Employing η = 0.6 in eq.(3.28) then ǫ ≃ 0.5.
Taking δ(+) around π/2 when δπ(sK) < π this implies a relative error of 30%. For δπ(sK) ≥ π one
has instead δ(+) & π, and a 15% of estimated error. Regarding the ratio of the moduli of form
factors entering in ǫ we expect it to be . 1 (see appendix A). Therefore, our error in the evaluation
of I2 is estimated to be 30% and 15% for the cases δπ(sK) < π and δπ(sK) ≥ π, respectively.
As a result of the discussion following eq.(3.24), we consider that the error estimates done
for I2 and I3 in the case δπ(sK) < π are too conservative and that the relative errors given for
δπ(sK) > π are more realistic. Nonetheless, since the absolute errors that one obtains for I2 and
I3 are the same in both cases (because I2 and I3 for δπ(sK) < π are around a factor 2 smaller than
those for δπ(sK) ≥ π) we keep the errors as given above. To the previous errors for I2 and I3 due
to inelasticity, we also add in quadrature the noise in the calculation of QH due to the error in
tππ from the uncertainties in the parameters of the K−matrix eqs.(3.13), (3.14), and those in the
parameterizations CGL and PY.
We finally employ for s > 2.25 GeV2 the knowledge of the asymptotic phase of the pion scalar
form factor in order to evaluate QA in eq.(3.12). The function φ(s) is determined so as to match
with the asymptotic behaviour of Γπ(t) as −1/t from QCD. The Omnès representation of the
scalar form factor, eqs.(2.2) and (2.8), tends to t−q/π and t−q/π+1 for t → ∞, respectively. Here, q
is the asymptotic value of the phase φ(s) when s → ∞. Hence, for δπ(sK) < π the function φ(s)
is then required to tend to π while for δπ(sK) ≥ π the asymptotic value should be 2π. The way
φ(s) is predicted to approach the limiting value is somewhat ambiguous [11, 12],
φas(s) ≃ π
log(s/Λ2)
. (3.29)
In this equation, 2dm = 24/(33 − 2nf ) ≃ 1, Λ2 is the QCD scale parameter and n = 1, 2 for
δπ(4m
K) < π, ≥ π, respectively. The case n = 2 was not discussed in refs.[10, 11, 12, 13, 14] for
the form factor given in eq.(1.1). There is as well a controversy between [14] and [12] regarding
the ± sign in eq.(3.29). If leading twist contributions dominate [11, 12] then the limiting value is
reached from above and one has the plus sign, while if twist three contributions are the dominant
ones [14] the minus sign has to be considered [12]. In the left panel of fig.2 we show with the wide
#4Neglecting multipion states.
φ(s) I I II II
δπ(sK) ≥ π < π ≥ π < π
I1 0.435± 0.013 0.435± 0.013 0.483± 0.013 0.483± 0.013
I2 0.063± 0.010 0.020± 0.006 0.063± 0.010 0.020± 0.006
I3 0.143± 0.017 0.053± 0.013 0.143± 0.017 0.053± 0.013
QH 0.403± 0.024 0.508± 0.019 0.452± 0.024 0.554± 0.019
QA 0.21± 0.03 0.10± 0.03 0.21± 0.03 0.10± 0.03
〈r2〉πs 0.61± 0.04 0.61± 0.04 0.66± 0.04 0.66± 0.04
Table 1: Different contributions to 〈r2〉πs as defined in eqs.(3.12) and (3.23). All the units are fm2.
In the value for 〈r2〉πs the errors due to I1, I2, I3 and QA are added in quadrature.
bands the values of φ(s)as for s > 2.25 GeV
2 from eq.(3.29), considering both signs, for n = 1
(δπ(sK) < π) and 2 (δπ(sK) ≥ π). We see in the figure that above 1.4−1.5 GeV (1.96−2.25 GeV2)
both ϕ(s) and φ(s)as phases match and this is why we take sH = 2.25 GeV
2 in eq.(3.11), similarly
as done in refs.[10, 11]. In this way, we also avoid to enter into hadronic details in a region where
η < 1 with the onset of the f0(1500) resonance. The present uncertainty whether the + or − sign
holds in eq.(3.29) is taken as a source of error in evaluating QA. The other source of uncertainty
comes from the value taken for Λ2, 0.1 < Λ2 < 0.35 GeV2, as suggested in ref.[10]. From fig.2 it is
clear that our error estimate for φas(s) is very conservative and should account for uncertainties
due to the onset of inelasticity for energies above 1.4 − 1.5 GeV and to the appearance of the
f0(1500) resonance. In the right panel of fig.2 we show the integrand for 〈r2〉πs , eq.(3.12), for
parameterization I (dashed line) and II (solid line). Notice as the large uncertainty in φas(s) is
much reduced in the integrand as it happens for the higher energy domain.
In table 1 we show the values of I1, I2, I3, QH , QA and 〈r2〉πs for the parameterizations I and
II and for the two cases δπ(sK) ≥ π and δπ(sK) < π. This table shows the disappearance of the
disagreement between the cases δπ(sK) ≥ π and δπ(sK) < π from the ππ and KK̄ T−matrix
of eq.(3.13), once the zero of Γπ(t) at s1 < sK is taken into account for the former case. This
disagreement was the reason for the controversy between Ynduráin and ref.[13] regarding the value
of 〈r2〉πs . The fact that the parameterization II gives rise to a larger value of 〈r2〉πs than I is because
PY follows the upper δπ data below 0.9 GeV, while CGL follows lower ones, as shown in fig.1.
The different errors in table 1 are added in quadrature. The final value for 〈r2〉πs is the mean
between those of parameterizations I and II and the error is taken such that it spans the interval
of values in table 1 at the level of two sigmas. One ends with:
〈r2〉πs = 0.63± 0.05 fm
2 . (3.30)
The largest sources of error in 〈r2〉πs are the uncertainties in the experimental δπ and in the
asymptotic phase φas. This is due to the fact that the former are enhanced because of its weight
in the integrand, see fig.2, and the latter due to its large size.
Our number above and that of refs.[1, 4], 〈r2〉πs = 0.61± 0.04 fm2, are then compatible. On the
other hand, we have also evaluated 〈r2〉πs directly from the scalar form factor obtained with the
dynamical approach of ref.[8] from Unitary χPT and we obtain 〈r2〉πs = 0.64±0.06 fm2, in perfect
agreement with eq.(3.30). Notice that the scalar form factor of ref.[8] has δπ(sK) > π and we have
checked that it has a zero at s1, as it should. This is shown in fig.3 by the dashed-double-dotted line.
The value 〈r2〉πs = 0.75±0.07 fm2 from refs.[10, 11] is much larger than ours because the possibility
of a zero at s1 was not taking into account there and other solution was considered. This solution,
however, has an unstable behaviour under the transition δπ(sK) = π− 0+ to δπ(sK) = π+0+ and
it cannot be connected continuously with the one for δπ(sK) < π. Our solution for Γπ(t) from
Ynduráin’s method does not have this unstable behaviour and it is continuous under changes in
the values of the parameters of the K−matrix, eqs.(3.13) and (3.14). This is why, from our results,
it follows too that the interesting discussion of ref.[11], regarding whether δπ(sK) < π or ≥ π,
is not any longer conclusive to explain the disagreement between the values of refs.[10, 11] and
ref.[1] for 〈r2〉πs .
We can also work out from our determination of 〈r2〉πs , eq.(3.30), values for the O(p4) SU(2)
χPT low energy constant ℓ̄4. We take the two loop expression in χPT for 〈r2〉πs [1],
〈r2〉πs =
8π2f 2π
ℓ̄4 −
+ ξ∆r
, (3.31)
where fπ = 92.4 MeV is the pion decay constant, ξ = (Mπ/4πfπ)
2 and Mπ is the pion mass. First,
at the one loop level calculation ∆r = 0 and then one obtains,
ℓ̄4 = 4.7± 0.3 . (3.32)
We now move to the determination of ℓ̄4 based on the full two loop relation between 〈r2〉πs and ℓ̄4.
The expression for ∆r can be found in Appendix C of ref.[1]. ∆r is given in terms of one O(p6)
χPT counterterm, r̃S2 , and four O(p4) ones. Taking the values of all these parameters, but for ℓ̄4,
from ref.[1], and solving for ℓ̄4, one arrives to
ℓ̄4 = 4.5± 0.3 . (3.33)
This number is in good agreement with ℓ̄4 = 4.4± 0.2 [1].
Ref.[12] also points out that one loop χPT fits to the S-, P- and D-wave scattering lengths and
effective ranges give rise to much larger values for ℓ̄2 and ℓ̄4 than those of ref.[1]. For more details
we refer to [12].
4 Conclusions
In this paper we have addressed the issue of the discrepancies between the values of the quadratic
pion scalar radius of Leutwyler et al. [4, 13], 〈r2〉πs = 0.61 ± 0.04 fm2, and Ynduráin’s papers
[10, 11, 12], 〈r2〉 = 0.75±0.07 fm2. One of the reasons of interest for having a precise determination
of 〈r2〉πs is its contribution of a 10% to a00 and a20, calculated with a precision of 2% in ref.[1]. The
value taken for 〈r2〉πs is also important for determining the O(p4) χPT coupling ℓ̄4.
From our study it follows that Ynduráin’s method to calculate 〈r2〉πs [10, 11], based on an
Omnès representation of the pion scalar form factor, and that derived by solving the two(three)
coupled channel Muskhelishvili-Omnès equations [4, 1, 6], are compatible. It is shown that the
reason for the aforementioned discrepancy is the presence of a zero in Γπ(t) for those S-wave I=0
T−matrices with δπ(sK) ≥ π and elastic below the KK̄ threshold, with sK = 4m2K . This zero
was overlooked in refs.[10, 11], though, if one imposes continuity in the solution obtained under
tiny changes of the ππ phase shifts employed, it is necessarily required by the approach followed
there. Once this zero is taken into account the same value for 〈r2〉πs is obtained irrespectively
of whether δπ(sK) ≥ π or δπ(sK) < π. Our final result is 〈r2〉πs = 0.63 ± 0.05 fm2. The error
estimated takes into account experimental uncertainty in the values of δπ(s), inelasticity effects
and present ignorance in the way the phase of the form factor approaches its asymptotic value π,
as predicted from QCD. Employing our value for 〈r2〉πs we calculate ℓ̄4 = 4.5 ± 0.3. The values
〈r2〉πs = 0.61± 0.04 fm2 and ℓ̄4 = 4.5± 0.3 of ref.[1] are then in good agreement with ours.
Acknowledgements
We thank Miguel Albaladejo for providing us numerical results from some unpublished T−matrices
and Carlos Schat for his collaboration in a parallel research. We also thank F.J. Ynduráin for
long discussions and B. Anathanarayan, I. Caprini, G. Colangelo, J. Gasser and H. Leutwyler for
a critical reading of a previous version of the manuscript. This work was supported in part by the
MEC (Spain) and FEDER (EC) Grants FPA2004-03470 and Fis2006-03438, the Fundación Séneca
(Murcia) grant Ref. 02975/PI/05, the European Commission (EC) RTN Network EURIDICE
under Contract No. HPRN-CT2002-00311 and the HadronPhysics I3 Project (EC) Contract No
RII3-CT-2004-506078.
Appendices
A Coupled channel dynamics
We take ππ and KK̄ coupled channels and denote by F1 and F2 their respective I=0 scalar form
factors. Unitarity requires,
ImFi =
Fjρjθ(t− s′j)t∗ji , (A.1)
where ||tij|| is the I=0 S-wave T−matrix, s′i is the threshold energy square of channel i and
ρi = qi/8π
s, with qi its center of mass three momentum.
A general solution to the previous equations is given by,
F = T G , F =
, G =
, (A.2)
where the functions Gi(t) do not have right hand cut. This equation is interesting as tells us that
if pion dynamics dominate, |G1| >> |G2|, then F1 ≃ G1t11 and the form factor phase φ(s) follows
ϕ(s). As a result, like t11, it has a zero at s1 below the KK̄ threshold for δπ(sK) ≥ π, as shown
in section 3. On the other hand, if kaon dynamics dominates, |G2| >> |G1|, then F1 ≃ G2t12 and
φ(s) follows the phase of t12, that above the KK̄ threshold is clearly above π. This is why for
the pion strange scalar form factor there is no zero at s1 . sK for δπ(sK) ≥ π, indeed there is a
maximum like that shown in fig.3 by the dashed-dotted line.
As in section 3 we now proceed to the diagnolization above the KK̄ threshold of the renormal-
ized T−matrix T ′,
T ′ = ρ1/2Tρ1/2 , ρ =
, T̃ = CTT ′C =
t̃11 0
0 t̃22
t̃11 = sin δ(+)e
iδ(+) , t̃22 = sin δ(−)e
iδ(−) . (A.3)
The corresponding diagonal form factors F ′1 and F
2, collected in the vector F
′, are
F ′ = CTρ1/2F = T̃CTρ−1/2G =
cos θ ρ
1 G1 − sin θ ρ
t̃11{
sin θ ρ
1 G1 + cos θ ρ
. (A.4)
The previous expressions allow to obtaining F1 directly in terms of the eigenphases and with clean
separation between pion, G1, and kaon dynamics, G2. From eq.(3.22) it follows that,
cos2 θ ρ−1G1 − cos θ sin θ ρ−1/22 ρ
sin2 θ ρ−11 G1 + cos θ sin θ ρ
t̃22 .
(A.5)
For δπ(sK) ≥ π typical values, somewhat above the KK̄ threshold, are e2iδ(+) ≃ +i, e2iδ(−) ≃ −i
and sin θ > 0. For dominance of G1 one has F1/G1 ≃ ρ−11 (i + cos 2θ)/2 while for dominance
of G2 the result is F1/G2 ≃ − sin θ cos θ ρ−1/22 ρ
1 < 0. The factors G1,2 do not introduce
any change in φ(s) with respect to its value before the opening of the KK̄ threshold since they
are smooth functions in s.#5 In both cases the phase φ(s) is larger than π and F1 follows the
upper trend of phases shown in fig.2 (note that in this case t̃11 is in the first quadrant though
δπ > π). Now, doing the same exercise for δπ(sK) < π, one has the typical values e
2iδ(+) ≃ −i,
e2iδ(−) ≃ +i and sin θ < 0. For pion dominance then F1/G1 ≃ ρ−11 (i− cos 2θ)/2 and for the kaon
one F1/G2 ≃ + sin θ cos θρ−1/22 ρ
1 < 0. Thus, in the former case the phase is & π/2, and follows
the lower trend of phases of fig.2, while in the latter is & π and follows again the upper trend (this
is the case of the strange scalar form factor).
The demonstration in section 3 that φ(sK) is discontinuous in the limit δπ(sK) → π− by taking
s1 → s+K , cannot be applied in the case of kaon dominance (e.g. pion strange scalar form factor).
From eq.(A.5) it follows that,
F1(t) ≃ − cos θ sin θρ−1/22 ρ
t̃11 − t̃22
. (A.6)
The point is that t̃22 for t ≥ s1 (s1 → s+K) is of size comparable with that of t̃11 (both tend to zero)
and the phase does not follow δ(+). This is not the case for pion dominance because for s1 → s+K
then sin2 θ → 0, F1(t) ≃ cos2 θ ρ−11 G1t̃11, eq.(A.5), and φ(s) follows δ(+).
From eq.(A.4) we can also write |Γ′2/Γ′2| ≃ |t̃11 tan θ/t̃22| for the case of pion dominance. Since
typically |t̃11/t̃22| ≃ 1, as shown above for energies somewhat above the KK̄ threshold, then
|Γ′2/Γ′1| ≃ | tan θ| < 1. This is why we consider that equating it to 1 in section 3 is a conservative
estimate.
#5Due to the Adler zeroes this is not necessarily case close to the ππ threshold.
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http://arxiv.org/abs/hep-ex/0610071
Introduction
Scalar form factor
Results
Conclusions
Coupled channel dynamics
|
0704.0040 | Multilinear function series in conditionally free probability with
amalgamation | arXiv:0704.0040v3 [math.OA] 5 Sep 2008
MULTILINEAR FUNCTION SERIES IN CONDITIONALLY
FREE PROBABILITY WITH AMALGAMATION
MIHAI POPA
Abstract. As in the cases of freeness and monotonic independence, the
notion of conditional freeness is meaningful when complex-valued states are
replaced by positive conditional expectations. In this framework, the paper
presents several positivity results, a version of the central limit theorem and
an analogue of the conditionally free R-transform constructed by means of
multilinear function series.
1. Introduction
The paper addresses a topic related to conditionally free (or, shortly, using the
term from [2], c-free) probability. This notion was developed in the ’90’s (see [1],
[2]) as an extension of freeness within the framework of ∗-algebras endowed with
not one, but two states. Namely, given a family of unital algebras {A}i∈I, each Ai
endowed with two expectations ϕi, ψi : Ai −→ C, their c-free product is the triple
(A, ϕ, ψ), where:
(i) A = ∗i∈IAi is the free product of the algebras Ai.
(ii) ψ = ∗i∈Iψi and ϕ = ∗(ψi)i∈Iϕi are expectations given by the relations
(a) ψ(a1 · · · an) = 0
(b) ϕ(a1 · · · an) = ϕε(1)(a1) · · ·ϕε(n)(an)
for all aj ∈ Aε(j), j = 1, . . . , n such that ψε(j)(aj) = 0 and ε(1) 6= · · · 6= ε(n).
A key result is that if the Ai are ∗-algebras and ϕi, ψi are positive functionals,
then ϕ and ψ are also positive.
In [6], the positivity of the free product maps ϕ, ψ is proved for the case
when ϕ1, ϕ2 are positive conditional expectations in a common C∗-subalgebra,
but ψ1, ψ2 remain positive C-valued maps. A more general situation is indeed
discussed (see Theorem 3, Section 6, from [6]), but the question if ϕ, ψ re positive
for ϕ1,2, ψ1,2 arbitrary positive conditional expectations is left unanswered.
A first answer was given in [8], where we showed that for A a ∗-algebra, the
analogous construction with both ϕ and ψ valued in a C∗-subalgebra B of A still
retains the positivity. The present paper further develops this result (see Theorem
2.3) and also demonstrates the use of multilinear function series in c-free setting.
2000 Mathematics Subject Classification. Primary 45L53; Secondary 46L08.
Key words and phrases. conditional freeness, conditional expectation, R-transform, multi-
linear function series.
http://arxiv.org/abs/0704.0040v3
2 MIHAI POPA
In [2] is constructed a c-free version of Voiculescu’s R-transform, which we will
call the cR-transform, with the property that cRX+Y =
cRX +
cRY if X and Y
are c-free elements from the algebra A relative to ϕ and ψ (i.e. the relations (a)
and (b) from the definition of the c-free product hold true for the subalgebras
generated by X and Y .)
The apparatus of multilinear function series is used in recent work of K. Dykema
([3] and [4]) to construct suitable analogues for the R and S-transforms in the
framework of freeness with amalgamation. We will show that this construction
is also appropriate for the cR-transform mentioned above. The techniques used
differ from the ones of [3], the Fock space type construction being substituted by
combinatorial techniques similar to [2] and [7]. Particularly, Theorems 3.3 and 3.6
contain new (shorter) proves of the results 6.1–6.13 from [3].
The paper is structured in four sections. In Section 2 are stated the basic
definitions and are proved the main positivity results. Section 3 describes the
construction and the basic property of the multilinear function series cR-transform
and Section 4 treats the central limit theorem and a related positivity property.
2. Definitions and positivity results
Definition 2.1. Let {Ai}i∈I be a family of algebras, all containing the subalgebra
B. Suppose D is a subalgebra of B and Ψi : Ai −→ D and Φi : Ai −→ B are con-
ditional expectations, i ∈ I. We say that the triple (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi)
is the conditionally free product with amalgamation over (B,D), or shortly, the
c-free product, of the triples (Ai,Φi,Ψi)i∈I if
(1) A is the free product with amalgamation over B of the family (Ai)i∈I
(2) Ψ = ∗i∈IΨi and Φ = ∗(Ψi)i∈IΦi are determined by the relations
Ψ(a1a2 . . . an) = 0
Φ(a1a2 . . . an) = Φ(a1)Φ(a2) . . .Φ(an),
for all ai ∈ Aε(i), ε(i) ∈ I such that ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψε(i)(ai) = 0.
When D = C, this definition reduces to the one given in [6]. When both B and
D are equal to C, this definition was given in [2].
When discussing positivity, we need a ∗-structure on our algebras. We will
demand that B and D be C∗-algebras, while Ai and A are only required to be
∗-algebras.
The following results are slightly modified versions of Lemma 6.4 and Theorem
6.5 from [8].
Lemma 2.2. Let B be a C∗-algebra and A1, A2 be two ∗-algebras containing B
as a ∗-subalgebra, endowed with positive conditional expectations Φj : Aj −→ B,
j = 1, 2. If a1, . . . , an ∈ A1, an+1, . . . , an+m ∈ A2 and A = (Ai,j)i,j ∈ Mn+m(B)
is the matrix with the entries
Ai,j =
i aj) if i, j ≤ n
i )Φ2(aj) if i ≤ n, j > n
i )Φ1(aj) if i > n, j ≤ n
i aj) if i, j > n
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 3
then A is positive.
Proof. The vector space E = B ⊕ ker(Φ1) ⊕ ker(Φ2) has a B-bimodule structure
given by the algebraic operations on A1 and A2. Consider the B-sesquilinear
pairing
〈·, ·〉 : E× E −→ B
determined by the relations:
〈b1, b2〉 = b∗1b2, for b1, b2 ∈ B
〈uj, vj〉 = Φj(u∗jvj), for uj , vj ∈ ker(Φj), j = 1, 2
〈u1, u2〉 = 〈u2, u1〉 = 0 for u1 ∈ ker(Φ1), and u2 ∈ ker(Φ2).
〈b, uj〉 = 〈uj , b〉 = 0 for all b ∈ B, uj ∈ Aj
With this notation, we have that Ai,j = 〈ai, aj〉, hence it suffices to show that
〈a, a〉 ≥ 0 for all a ∈ E.
Indeed, for an element a = b + u1 + u2 with b ∈ B, uj ∈ ker(Φj), j = 1, 2, we
have:
〈a, a〉 = 〈b + u1 + u2, b+ u1 + u2〉
= 〈b, b〉+ 〈u1, u1〉+ 〈u2, u2〉
= b∗b+Φ1(u
1u1) + Φ2(u
Theorem 2.3. Let B be a C∗-algebra and D a C∗-subalgebra of B. Suppose that
A1, A2 are ∗-algebras containing B, each endowed with two positive conditional
expectations Φj : Aj −→ B, and Ψj : Aj −→ D, j = 1, 2 and consider the c-free
product (A,Φ,Ψ) = ∗i=1,2(Ai,Φi,Ψi).
Then the maps Φ and Ψ are positive.
Proof. The positivity of Ψ is by now a classical result in the theory of free proba-
bility with amalgamation over a C∗-algebra (for example, see [9], Theorem 3.5.6).
For the positivity of Φ we have to show that Φ(a∗a) ≥ 0 for any a ∈ A.
Any element of A can be written as
s1,k . . . sn(k),k,
where sj,k ∈ Aε(j,k) ε(1, k) 6= ε(2, k) 6= · · · 6= ε(n(k), k).
Writing
s(j,k) = s(j,k) −Ψ(s(j,k)) +Ψ(s(j,k))
and expanding the product, we can consider a of the form
a = d+
a1,k . . . an(k),k
with d ∈ D ⊂ B and aj,k ∈ Aε(j,k) such that Ψε(j,k)(aj,k) = 0 and ε(1, k) 6=
ε(2, k) 6= · · · 6= ε(n(k), k).
4 MIHAI POPA
Therefore
Φ(a∗a) = Φ
d+ d∗
a1,k . . . an(k),k
a1,k . . . an(k),k
a1,k . . . an(k),k
]∗[ N∑
a1,k . . . an(k),k
Since Φ is a conditional expectation and d ∈ D ⊂ B, the above equality becomes
Φ(a∗a) = d∗d+
d∗Φ(a1,k . . . an(k),k) +
Φ(a∗n(k),k . . . a
1,k)d
k,l=1
a∗n(k),k . . . a
1,ka1,l . . . an(l),l
Using the definition of the conditionally free product with amalgamation over B
and that Ψε(j,k)(aj,k) = 0 for all j, k, one further has
Φ(a∗a) = d∗d+
Φ(d∗a1,k)Φ(a2,k) . . .Φ(an(k),k)
Φ(an(k),k)
. . .Φ(a∗2,k)Φ(a
1,kd)
k,l=1
Φ(an(k),k)
∗ . . .Φ(a∗2,k)
Φ(a∗1,ka1,l)Φ(a2,l) . . .Φ(an(l),l)
that is
Φ(a∗a) = d∗d+
Φ(d∗a1,k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,kd)
k,l=1
Φ(a2,k) . . .Φ(an(k),k)
Φ(a∗1,ka1,l)
Φ(a2,l) . . .Φ(an(l),l)
From Lemma 2.2, the matrix S =
Φ(a∗1,ia1,j)
i,j=1
is positive in MN+1(B),
therefore
S = T ∗T, for some T ∈MN+1(B).
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 5
Denote now a1,N+1 = d and vk = Φ(a2,k) . . .Φ(an(k),k).The identity for Φ(a
becomes:
Φ(a∗a) = (v1, . . . , vN , 1)
∗T ∗T (v1, . . . , vN , 1)
≥ 0, as claimed.
Theorem 2.4. Assume that I =
j∈J Ij is a partition of I. Then:
∗j∈J (∗i∈Ij (Ai,Φi,Ψi)) = ∗i∈I(Ai,Φi,Ψi)
Proof. The proof is identical to the proofs of similar results in [6] and [2]. Consider
ai ∈ Aε(i), 1 ≤ i ≤ m such that ε(1) 6= ε(2 6= · · · 6= ε(m) and Ψε(i)(ai) = 0. Let
1 = i0 < i1 < · · · < ik = m and Jl = {ε(i), il−1 ≤ i < il}.
Since
(∗j∈JlΨj) ((ail−1 · · · ail)) = 0,
it suffices to show that
Φ(a1 · · · am) =
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail)] .
Φ(a1 · · · am) = Φε(1)(a1) · · ·Φε(m)(am)
while, since Ψε(i)(ai) = 0,
(∗(Ψj),j∈JlΦj)(ail−1 · · · ail) = Φil−1(ail−1) · · ·Φil(ail)
and the conclusion follows. �
Definition 2.5. Let A be an algebra (respectively a ∗-algebra),B a subalgebra (∗-
subalgebra) of A and D a subalgebra (∗-subalgebra) of B. Suppose A is endowed
with the conditional expectations Ψ : A −→ D and Φ : A −→ D.
(i) The subalgebras (∗-subalgebras) (Ai)i∈I of A are said to be c-free with
respect to (Φ,Ψ) if:
(a) (Ai)i∈I are free with respect to Ψ.
(b) if ai ∈ Aε(i), 1 ≤ i ≤ m, are such that ε(1) 6= · · · 6= ε(m) and
Ψ(ai) = 0, then Φ(a1 · · ·am) = Φ(a1) · · ·Φ(am).
(ii) The elements (Xi)i∈I of A are said to be c-free with respect to (Φ,Ψ) if
the subalgebras (∗-subalgebras) generated by B and Xi are c-free with
respect to (Φ,Ψ).
We will denote by B〈ξ〉 the non-commutative algebra of polynomials in the
symbol ξ and with coefficients from B (the coefficients do not commute with
the symbol ξ). If I is a family of indices, B〈{ξi}i∈I〉 will denote the algebra of
polynomials in the non-commuting variables {ξ}i∈I and with coefficients from B.
We will identify B〈{ξi}i∈I〉 with the free product with amalgamation over B of
the family {B〈ξi〉}i∈I .
6 MIHAI POPA
If A is a ∗-algebra and B is with the C∗-algebra, B〈ξ〉 will also be considered
with a ∗-algebra structure, by taking ξ∗ = ξ. If X is a selfadjoint element from A,
we define the conditional expectations
ΦX ,ΨX : B〈ξ〉 −→ B
given by ΦX(f(ξ)) = Φ(f(X)) and ΨX(f(ξ)) = Ψ(f(X)) , for any f(ξ) ∈ B〈ξ〉.
Corollary 2.6. Suppose that A is a ∗-algebra and X and Y are c-free selfadjoint
elements of A such that the maps ΦX ,ΨX and ΦY ,ΨY are positive. Then the
maps ΦX+Y and ΨX+Y are also positive.
Proof. The positivity of ΨX+Y is an immediate consequence of the fact that X
and Y are free with amalgamation over B with respect to Ψ. It remains to prove
the positivity of ΦX+Y .
Since the maps ΦX : B〈ξ1〉 −→ B and ΦY : B〈ξ2〉 −→ B are positive, from
Theorem 2.3 so is
Φx ∗(ΨX ,ΨY ) ΦY : B〈ξ1〉 ∗B B〈ξ2〉 = B〈ξ1, ξ2〉 −→ B
Remark also that
iZ : B〈ξ〉 ∋ f(ξ) 7→ f(X + Y ) ∈ B〈ξ1〉 ∗B B〈ξ2〉
is a positive B-functional.
The conclusion follows from the fact that the c-freeness of X and Y is equivalent
ΦX+Y = (ΦX ∗(ΨX ,ΨY ) ΦY ) ◦ iX+Y .
3. Multilinear function series and the cR-transform
Let A be a ∗-algebra containing the C∗-algebra B, endowed with a conditional
expectation Ψ : A −→ B. If X is a selfadjoint element of A, then by the moment
of order n of X we will understand the map
X : B× · · · ×B︸ ︷︷ ︸
n−1 times
X (b1, . . . , bn−1) = Ψ(Xb1X . . .Xbn−1X)
If B = C, then the moment-generating series of X
mX(z) =
Ψ(Xn)zn
encodes all the information about the moments of X . For B 6= C, the straightfor-
ward generalization
mX(z) =
Ψ(Xn)zn
generally fails to keep track of all the possible moments of X . A solution to
this inconvenience was proposed in [3], namely the moment-generating multilin-
ear function series of X . Before defining this notion, we will briefly recall the
construction and several results on multilinear function series.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 7
Let B be an algebra over a field K. We set B̃ equal to B if B is unital and
to the unitization of B otherwise. For n ≥ 1, we denote by Ln(B) the set of all
K-multilinear maps
ωn : B× · · · ×B︸ ︷︷ ︸
n times
A formal multilinear function series over B is a sequence ω = (ω0, ω1, . . . ),
where ω0 ∈ B̃ and ωn ∈ Ln(B) for n ≥ 1. According to [3], the set of all
multilinear function series over B will de denoted by Mul[[B]].
For α, β ∈Mul[[B]], the formal sum α+ β and the formal product αβ are the
elements from Mul[[B]] defined by:
(α+ β)n(b1, . . . , bn) = αn(b1, . . . , bn) + βn(b1, . . . , bn)
(αβ)n(b1, . . . , bn) =
αk(b1, . . . , bk)βn−k(bk+1, . . . , bn)
for any b1, . . . , bn ∈ B.
If β0 = 0, then the formal composition α ◦ β ∈Mul[[B]] is defined by
(α ◦ β)0 = α0
and, for n ≥ 1, by
(α ◦ β)n(b1, . . . , bn) =
βp1(b1, . . . , bp1), . . . , βpk(bqk+1, . . . , bqk+pk)
where the second summation is done over all k-tuples p1, . . . , pk ≥ 1 such that
p1 + · · ·+ pk = n and qj = p1 + · · ·+ pj−1.
One can work with elements ofMul[[B]] as if they were formal power series. The
relevant properties are described in [3], Proposition 2.3 and Proposition 2.6. As in
[3], we use 1, respectively I, to denote the identity elements of Mul[[B]] relative
to multiplication, respectively composition. In other words, 1 = (1, 0, 0, . . . ) and
I = (0, idB, 0, 0, . . . ). We will also use the fact that an element α ∈Mul[[B]] has
an inverse with respect to formal composition, denoted α〈−1〉, if and only if α has
the form (0, α1, α2, . . . ) with α1 an invertible element of L1(B).
Definition 3.1. With the above notation, the moment-generating multilinear func-
tion series MX of X is the element of Mul[[B]] such that:
MX,0 = Ψ(X)
MX,n(b1, . . . , bn) = Ψ(Xb1X · · ·XbnX).
Given an element α ∈ Mul[[B]], the multilinear function series Rα is defined
by the following equation (see [3], Def 6.1):
(1 + αI)
◦ (I + IαI)〈−1〉. (3.1)
A key property of R is that for any X,Y ∈ A free over B, we have
RMX+Y = RMX +RMY . (3.2)
8 MIHAI POPA
These relations were proved earlier in the particular case B = C. One can also
describe Rα by combinatorial means, via the recurrence relation
αn(b1, . . . , bn) =
[b1αp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)αp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)αn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
Following an idea from [2], the above equation can be graphically illustrated by
the picture:
In the case of scalar c-free probability, an analogue of the Voiculescu’s R-
transform is developed in [2]. In order to avoid confusions, we will denote it
by cR.
The cR-transform has the property that it linearizes the c-free convolution of
pairs of compactly supported measures. In particular, if X and Y are c-free
elements from some algebra A, then
cRX+Y =
cRX +
cRY .
If the ∗-algebraA is endowed with the C-valued states ϕ, ψ andX is a selfadjoint
element of A, then (see [2]), the coefficients {cRm}m ≥ 0 of cRX are defined by
the recurrence:
ϕ(Xn) =
l(1),...,l(k)≥0
l(1)+···+l(k)=n−k
cRk · ψ(X l(1)) · · ·ψ(X l(k−1))ϕ(X l(k))
equation that can be graphically illustrated by the picture, were the dark boxes
stand for the application of ϕ and the light ones for the application of ψ:
The above considerations lead to the following definition:
Definition 3.2. Let β, γ ∈Mul[[B]]. The multilinear function series cRβ,γ is the
element of Mul[[B]] defined by the recurrence relation
βn(b1, . . . , bn) =
cRβ,γ,k
[b1γp(1)(b3, . . . , bi1−2)bi1−1], . . .
. . . , [bi(k−1)γp(k)(bi(k−1)+1, . . . , bi(k)−2)bi(k)−1]
bi(k)βn−ik(bik+1 , . . . , bn)
where the second summation is done over all 1 = i(0) < i(1) < · · · < i(k) ≤ n and
p(k) = i(k)− i(k − 1)− 2.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 9
The following analytical description of cRβ,γ also shows that it is unique and
well-defined:
Theorem 3.3. For any β, γ ∈Mul[[B]],
Rβ,γ =
β(1 + Iβ)−1
◦ (I + IγI)〈−1〉 (3.3)
Before proving the theorem, remark that the right-hand side of (3.3) is well-
defined and unique, since 1 + Iγ is invertible with respect to the formal multipli-
cation, I+ IβI is invertible with respect to formal composition and its inverse has
0 as first component (see [3]). We will need the following auxiliary result:
Lemma 3.4. Let β be an element of Mul[[B]] and I the identity element with
respect to formal composition, I = (0, idB, 0, 0 . . . ).
(i) the multilinear function series Iβ is given by:
(Iβ)0 = 0
(Iβ)n(b1, . . . , bn) = b1βn−1(b2, . . . , bn)
(ii) the multilinear function series IβI is given by
(IβI)0 = 0
(IβI)1(b1) = 0
(IβI)n(b1, . . . , bn) = b1βn−2(b2, . . . , bn−1)bn
Proof. Since I = (0, idB, 0, . . . ), one has:
(Iβ)0 = I0β0 = 0.
If n ≥ 1,
(Iβ)n(b1, . . . , bn) =
Ik(b1, . . . , bk)βn−k(bk+1, . . . , bn)
= I1(b1)βn−1(bk+1, . . . , bn)
= b1βn−1(bk+1, . . . , bn).
For IβI, the same computations give:
(IβI)0 = (Iβ)0I0 = 0
(IβI)1 = (Iβ)0I1(b1) + (Iβ)1(b1)I0
If n ≥ 2, one has:
(IβI)n(b1, . . . , bn) =
(Iβ)k(b1, . . . , bk)In−k(bk+1, . . . , bn)
= (Iβ)n−1(b1, . . . , bk)I1(b1)
= b1βn−2(b2, . . . , bn−1)bn
10 MIHAI POPA
Proof of the Theorem 3.3: Set σ = I + IβI. Then
(cRβ,γ ◦ σ)n (b1, . . . , bn) =
p1,...,pk≥1
p1+···+pk=n
Rβ,γ,k
σp1 (b1, . . . , bp1), . . . ,
σpk(bqk+1, . . . , bqk+pk)
where qi = p1 + · · ·+ pi−1.
From Lemma (3.4)(ii), we have that
σn(b1, . . . , bn) = (I + IβI)n(b1, . . . , bn)
therefore Definition 3.2 is equivalent to
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) bk+1βn−k−2(bk+2, . . . , bn)
Considering now Lemma 3.4(i), the above relation becomes
βn(b1, . . . , bn) =
(cRβ,γ ◦ (I + IβI)k(b1, . . . , bk)) (I + Iβ)n−k(bk+1, . . . , bn)
therefore
β = [cRβ,γ ◦ (I + IγI)] (1 + Iβ)
which is equivalent to (3.3). �
Remark 3.5. Up to a shift in the coefficients, equation (3.3) is similar to the result
in the case B = C from [2], Theorem 5.1.
Let X be a selfadjoint element of A. If A is endowed with two B-valued condi-
tional expectations Φ,Ψ, the element X will have two moment-generating multi-
linear function series, one with respect to Ψ, that we will denote by MX , and one
with respect to Φ, denoted MX . For brevity, we will use the notation
cRX for the
multilinear function series cRMX ,MX .
Theorem 3.6. Let X and Y be two elements of A that are c-free with respect to
the pair of conditional expectations (Φ,Ψ). Then
cRX+Y =
cRX +
Proof. Let A be an algebra containing B as a subalgebra and endowed with the
conditional expectations Φ,Ψ : A −→ B. Consider the set A0 = A \ B (set
difference). For n ≥ 1 define the maps
cr : A0 × · · · × A0︸ ︷︷ ︸
n times
given by the recurrence formula:
Φ(a1 · · · an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
rk(a1[Ψ(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ(al(k−1)+1 · · · al(k)1)], al(k)[Φ(al(k)+1 · · · an)])
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 11
Note that crn is well defined, and that, for any b1, . . . , bn ∈ B,
crn+1(X, b1X, . . . , bnX) =
cRX,n(b1, . . . , bn). (3.4)
As in Section 2, consider B〈ξi〉, the noncommutative algebras of polynomi-
als in the symbols ξi, i = 1, 2 and with coefficients from B and the conditional
expectations
ΦX ,ΨX : B〈ξ1〉 −→ B
given by
ΦX(f(ξ1)) = Φ(f(X))
ΨX(f(ξ1)) = Ψ(f(X))
and their analogues ΦY ,ΨY for B〈ξ2〉.
OnB〈ξ1, ξ2〉, identified toB〈ξ1〉∗BB〈ξ2〉, consider the conditional expectations
Ψ0,Φ0, ϕ given by:
Ψ0 = ΨX ∗ΨY
Φ0(f(ξ1, ξ2)) = Φ(f(X,Y ))
ϕ(a1a2 . . . an) =
l(1)<···<l(k)
1<l(1),l(k)≤n
ρk(a1[Ψ0(a2 · · · al(1)−1)], . . . ,
. . . , al(k−1)[Ψ0(al(k−1)+1 · · ·al(k)1 )], al(k)[ϕ(al(k)+1 · · · an)])
where a1, . . . , an are elements of the set B〈ξ1, ξ2〉0 = B〈ξ1〉 ∪B〈ξ2〉 \B, and the
ρn : B〈ξ1, ξ2〉0 × . . .B〈ξ1, ξ2〉0︸ ︷︷ ︸
n times
are given by:
ρn(a1, . . . , an) =
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ1〉
cr(a1, . . . , an) if all a1, . . . , an ∈ B〈ξ2〉
0 otherwise
We will show that ϕ = Φ0, in particular ϕ is also well-defined. Consider the
element a ∈ B〈ξ1, ξ2〉 of the form a = a1 · · · an with aj ∈ B〈ξε(j)〉, such that
ε(1) 6= ε(2) 6= · · · 6= ε(n) and Ψ0(aj) = 0. The computation of ϕ(a1 · · · an) is done
via the recurrence relation above. Because of the definition of ρ and the fact that
Ψ0 = ΨX ∗ΨY , only the term with k = 1 contribute at the sum, i.e.
ϕ(a1 · · · an) = ϕ(a1ϕ(a2 · · · an))
= ϕε(1)(a1ϕ(a2 · · · an)
= ϕε(1)(a1)ϕ(a2 · · · an)
and the identity between ϕ and Φ0 follows by induction over n.
Since ϕ = Φ0, the maps ρn and
crn are satisfying the same recurrence relation,
hence
ρn(a1, . . . , an) =
cr(a1, . . . , an).
12 MIHAI POPA
In particular
RX+Y,n(b1, . . . , bn) =
rn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X + Y )b1(X + Y ) . . . (X + Y )bn(X + Y ))
= ρn+1((X)b1(X) . . . (X)bn(X)) + ρn+1((Y )b1(Y ) . . . (Y )bn(Y ))
= cRX,n(b1, . . . , bn) +
cRY,n(b1, . . . , bn).
4. Central limit theorem
Consider the ordered set 〈n〉 = {1, 2, . . . , n} and π a partition of 〈n〉 with blocks
B1, . . . , Bm:
〈n〉 = B1 ⊔B2 ⊔ · · · ⊔Bm.
The blocks Bp and Bq of π are said to be crossing if there exist i < j < k < l
in 〈n〉 such that i, k ∈ Bp and j, l ∈ Bq.
The partition π is said to be non-crossing if all pairs of distinct blocks of π
are not crossing. We will denote by NC2(n) the set of all non-crossing partitions
of 〈n〉 whose blocks contain exactly 2 elements and by NC≤s(n) the set of all
non-crossing partitions of 〈n〉 whose blocks contain at most s elements.
Let now γ be a non-crossing partition of 〈n〉 and B and C be two blocks of
π. We say that B is interior to C if there exist two indices i < j in 〈n〉 such
that i, j ∈ C and B ⊂ {i + 1, . . . , j − 1}. The block B is said to be outer if it is
not interior to any other block of γ. In a non-crossing partition of 〈n〉, the block
containing 1 is always outer.
Consider now an element X of A. Let π be a partition from NC2(n+ 1) (n =
odd) and B1 = (1, k) be the block of π containing 1. We define, by recurrence,
the following expressions:
Vπ(X, b1, . . . , bn) = Ψ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Vπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Wπ(X, b1, . . . , bn) = Φ(Xb1Vπ|{2,...,j−1}(X, b2, . . . , bk−2)bk−1X)bk
Wπ|{k+1,...,n+1}(X, bk+1, . . . , bn)
Theorem 4.1. (Central Limit Theorem) Let (Xn)n≥1 be a sequence of c-free
elements of A such that:
(1) all Xn have the same moment-generating multilinear function series, M
with respect to Φ and M with respect to Ψ.
(2) Ψ(Xn) = Φ(Xn) = 0.
X1 + · · ·+XN√
Then:
(i) lim
RSN = (0,M1(·), 0, . . . )
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 13
(ii) lim
RSN = (0,M1(·), 0, . . . )
(iii) there exist two conditional expectations ν : B〈ξ〉 −→ B, depending only
on M1(·), and µ : B〈ξ〉 −→ B, depending only on M1(·) and M1(·), such
ΨSN = ν
ΦSN = µ
in the weak sense; in particular,
ν(ξb1ξ . . . bnξ) =
π∈NC2(n)
Vπ(X1, b1, . . . , bn)
µ(ξb1ξ . . . bnξ) =
π∈NC2(n)
Wπ(X1, b1, . . . , bn).
Proof. Let X be an element of A with the same moment generating series as Xj,
j ≥ 1. As shown in [3],
RSN =
R Xk√
= NR X√
Also, from Theorem 2.4 and Theorem 3.6, it follows that
cRSN =
cR Xk√
= N cR X√
Since R and cR are multilinear and M0 = M0 = 0, we have that
cRSN ,n = lim
cRX,n
0 if n 6= 1
M1(·) if n = 1
and the similar relations for RSN ,n, hence (i) and (ii) are proved.
For (iii) it suffices to check the relations for ν(ξb1ξ . . . bnξ) and µ(ξb1ξ . . . bnξ),
which are a trivial corollary of (i), (ii), and the recurrence formulas that define R
and cR. �
Remark 4.2. For B = C, the theorem is a weaker version of Theorem 4.3 from [2].
If Ψ is C-valued, then the result is similar to Corollary 5.1 from [6]. Also, under
the assumptions that for some a, b ∈ B we have that:
NΨ(X1 · · ·XN ) = a
NΨ(X1 · · ·XN ) = b
the same techniques lead to a Poisson-type limit Theorem, similar to Corollary 2,
Section 5 of [6].
14 MIHAI POPA
In the following remaining pages we will describe the positivity of the limit
functionals µ and ν in terms of Φ and Ψ. The central result is Corollary 4.4.
For simplicity, suppose that B is a unital ∗-algebra (otherwise, we can replace
B by its unitisation). Consider the symbol ξ, the ∗-algebra B〈ξ〉 of polynomials
in ξ with coefficients from B, as defined before, and consider also the linear space
BξB generated by the set {b1ξb2; b1, b2 ∈ B} with the B-bimodule structure
given by
a1b1ξb2a2 = (a1b1)ξ(b2a2)
for all a1, a2, b1, b2 ∈ B.
Lemma 4.3. For any positive B-sesquilinear pairing 〈·, ·〉 on BξB there exists a
positive conditional expectation ϕ : B〈ξ〉 −→ B such that for any b1, b2 ∈ B one
has that
ϕ(ξb∗1b2ξ) = 〈b1ξ, b2ξ〉
Proof. Without loss of generality, we can suppose that B is unital (otherwise we
can replace B by its unitization).
Consider the Full Fock bimodule over BξB
F〈ξ〉 = B⊕
BξB⊗B · · · ⊗B BξB︸ ︷︷ ︸
n times
with the pairing given by
〈a, b〉 = a∗b
〈a1ξ ⊗ · · · ⊗ anξ, b1ξ ⊗ · · · ⊗ bmξ〉 = δm,n〈anξ, 〈. . . , 〈a1ξ, b1ξ〉b2ξ〉, . . . bnξ〉.
(a, aj , b, bj ∈ B, j = 1, . . . , n)
Note that the B-linear operators A1, A2 : F〈ξ〉 −→ F〈ξ〉 described by the
relations
A1b = ξb
A1(a1ξ ⊗ · · · ⊗ anξb) = ξ ⊗ a1ξ ⊗ · · · ⊗ anξb
A2b = 0
A2(a1ξ ⊗ · · · ⊗ anξb) = 〈ξ, a1ξ〉a2ξ ⊗ · · · ⊗ anξb
are self-adjoint to each other, in the sense that
〈A1ζ̃1, ζ̃2〉 = 〈ζ̃1, A2ζ̃2〉
for any ζ̃1, ζ̃2 ∈ F〈ξ〉, therefore S = A1 +A2 is selfadjoint.
Moreover, for any a, b ∈ B,
〈1, Sa∗bS1〉 = 〈aS1, bS1〉
= 〈a(A1 +A2)1, b(A1 +A2)1〉
= 〈aξ, bξ〉
and the conclusion follows by setting ϕ(p(ξ)) = 〈1, p(S)1〉 for all p ∈ B〈ξ〉. �
Corollary 4.4. The maps µ and ν from Theorem 4.1 are positive if and only if
for any b ∈ B one has that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0.
MULTILINEAR FUNCTION SERIES, C-FREE PROBABILITY WITH AMALGAMATION 15
Proof. One implication is trivial, since, if ν and µ are positive, then
Ψ(Xb∗bX) = ν(Xb∗bX) = ν((bX)∗bX) ≥ 0
Φ(Xb∗bX) = µ(Xb∗bX) = µ((bX)∗bX) ≥ 0.
Suppose now that Φ(Xb∗bX) ≥ 0 and Ψ(Xb∗bX) ≥ 0 for all b ∈ B. We will
use the same argument as in [9] and [8].
Consider the set of selfadjoint symbols {ξi}i≥1. On each B-bimodule BξiB we
have the positive B-sesquilinear pairings 〈·, ·〉Φ and 〈·, ·〉Ψ determined by
〈aξi, bξi〉Φ = Φ(Xa∗bX)
〈aξi, bξi〉Ψ = Ψ(Xa∗bX).
As shown in Lemma 4.3, the above B-sesquilinear pairings determine positive
conditional expectations ϕ1, ψi : Ai −→ B, where Ai = B〈ξi〉 be the ∗-algebras of
polynomials in ξ with coefficients from B, i ≥ 1.
For τ : B〈ξ〉 −→ B a conditional expectation, and λ ≥ 0, note with Dλτ the
dilation with λ of τ , i.e.
Dλτ(ξb1ξ · · · bnξ) = λn+1τ(ξb1ξ · · · bnξ)
Remark that if τ is positive, then Dλτ is also positive.
With the notations above, consider, as in Definition 2.1, the conditionally free
product (A,Φ,Ψ) = ∗i∈I(Ai,Φi,Ψi). The elements {ξi}i≥1 are conditionally free
in A, so Theorem 4.1 implies that:
µ = lim
Φ ξ1+···+ξN√
= D 1√
Φξ1+···+ξN
ν = lim
Ψ ξ1+···+ξN√
= D 1√
Ψξ1+···+ξN
= D 1√
(∗Ni=1Ψξi
We have that ∗Ni=1Ψξi ≥ 0 since it is the free product of states (see, for example
[9]), hence the positivity of ν.
Also, Theorem 2.4 and Corollary 2.6 imply that Φξ1+···+ξN ≥ 0, therefore µ ≥ 0.
Acknowledgment. This research was partially supported by the Grant 2-CEx06-
11-34 of the Romanian Government. I am thankful to Marek Bożejko for present-
ing me the basics of c-freeness and bringing to my attention the references [2] and
[6]. I thank also Hari Bercovici for his constant support and his many advices
during the work on this paper.
16 MIHAI POPA
References
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Probability and Related Topics, VI (1991), 219-236, World Scientific, Singapore.
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tionally free Random Variables., in: Pac. J. Math. 175 (1996), 357-388.
3. Dykema, K.: Multilinear function series and transforms in Free Probability theory, in: Ad-
vances in Mathematics 208 (2007) 351407.
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231 (2006) 90110
5. Lance, E. C.: Hilbert C*-modules. A toolkit for operator algebraists, London Mathematical
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matics 153 (1) (2002)
7. Nica, A. and Speicher, R.: Lectures on the Combinatorics of the Free Probability, London
mathematical Society Lecture Note Series 335, Cambridge University Press 2006
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Mihai Popa: Department of Mathematics, Indiana University at Bloomington,
Rawles Hall, 931 E 3rd St, Bloomington, IN 47405
E-mail address: [email protected]
|
0704.0041 | Quantum Group of Isometries in Classical and Noncommutative Geometry | arXiv:0704.0041v4 [math.QA] 26 Oct 2007
Quantum Group of Isometries in Classical and
Noncommutative Geometry
Debashish Goswami 1
Stat-Math Unit, Kolkata Centre,
Indian Statistical Institute
203, B. T. Road, Kolkata 700 108, India
Abstract
We formulate a quantum generalization of the notion of the group of
Riemannian isometries for a compact Riemannian manifold, by intro-
ducing a natural notion of smooth and isometric action by a compact
quantum group on a classical or noncommutative manifold described
by spectral triples, and then proving the existence of a universal object
(called the quantum isometry group) in the category of compact quan-
tum groups acting smoothly and isometrically on a given (possibly
noncommutative) manifold satisfying certain regularity assumptions.
In fact, we identify the quantum isometry group with the universal ob-
ject in a bigger category, namely the category of ‘quantum families of
smooth isometries’, defined along the line of Woronowicz and Soltan.
We also construct a spectral triple on the Hilbert space of forms on a
noncommutative manifold which is equivariant with respect to a nat-
ural unitary representation of the quantum isometry group. We give
explicit description of quantum isometry groups of commutative and
noncommutative tori, and in this context, obtain the quantum double
torus defined in [11] as the universal quantum group of holomorphic
isometries of the noncommutative torus.
1 Introduction
Since the formulation of quantum automorphism groups by Wang ([15],
[16]), following suggestions of Alain Connes, many interesting examples of
such quantum groups, particularly the quantum permutation groups of finite
sets and finite graphs, have been extensively studied by a number of mathe-
maticians (see, e.g. [1], [2], [17] and references therein), who have also found
applications to and interaction with areas like free probability and subfactor
theory. The underlying basic principle of defining a quantum automorphism
group corresponding to some given mathematical structure (for example, a
1The author gratefully acknowldges support obtained from the Indian National
Academy of Sciences through the grants for a project on ‘Noncommutative Geometry
and Quantum Groups’, and also wishes to thank The Abdus Salam ICTP (Trieste), where
a major part of the work was done during a visit as Junior Assciate.
http://arxiv.org/abs/0704.0041v4
finite set, a graph, a C∗ or von Neumann algebra) consists of two steps :
first, to identify (if possible) the group of automorphisms of the structure as
a universal object in a suitable category, and then, try to look for the univer-
sal object in a similar but bigger category by replacing groups by quantum
groups of appropriate type. However, most of the work done so far concern
some kind of quantum automorphism groups of a ‘finite’ structure, for ex-
ample, of finite sets or finite dimensional matrix algebras. It is thus quite
natural to try to extend these ideas to the ‘infinite’ or ‘continuous’ mathe-
matical structures, for example classical and noncommutative manifolds. In
the present article, we have made an attempt to formulate and study the
quantum analogues of the groups of Riemannian isometries, which play a
very important role in the classical differential geometry. The group of Rie-
mannian isometries of a compact Riemannian manifold M can be viewed as
the universal object in the category of all compact metrizable groups acting
on M , with smooth and isometric action. Therefore, to define the quantum
isometry group, it is reasonable to consider a category of compact quantum
groups which act on the manifold (or more generally, on a noncommutative
manifold given by spectral triple) in a ‘nice’ way, preserving the Riemannian
structure in some suitable sense, to be precisely formulated. In this article,
we have given a definition of such ‘smooth and isometric’ action by a com-
pact quantum group on a (possibly noncommutative) manifold, extending
the notion of smooth and isometric action by a group on a classical mani-
fold. Indeed, the meaning of isometric action is nothing but that the action
should commute with the ‘Laplacian’ coming from the spectral triple, and
we should mention that this idea was already present in [2], though only in
the context of a finite metric space or a finite graph. The universal object
in the category of such quantum groups, if it exists, should be thought of
as the quantum analogue of the group of isometries, and we have been able
to prove its existence under some regularity assumptions, all of which can
be verified for a general compact connected Riemannian manifold as well
as the standard examples of noncommutative manifolds. Motivated by the
ideas of Woronowicz and Soltan, we actually consider a bigger category. The
isometry group of a classical manifold, viewed as a compact metrizable space
(forgetiing the group structure), can be seen to be the universal object of a
category whose object-class consists of subsets (not necessarily subgroups)
of the set of smooth isometries of the manifold. Then it can be proved
that this universal compact set has a canonical group structure. A natural
quantum analogue of this has been formulated by us, called the category of
‘quantum families of smooth isometries’. The underlying C∗-algebra of the
quantum isometry group has been identified with its universal object and
moreover, it is shown to be equipped with a canonical coproduct making it
into a compact quantum group.
We believe that a detailed study of quantum isometry groups will not
only give many new and interesting examples of compact quantum groups,
it will also contribute to the understanding of quantum group covariant
spectral triples. In fact, we have made some progress in this direction already
by constructing a spectral triple (which is often closely related to the original
spectral triple) on the Hilbert space of forms which is equivriant with respect
to a canonical unitary representation of the quantum isometry group.
In a companion article [3] with J. Bhowmick, we provide explicit compu-
tations of quantum isometry groups of a few classical and noncommutative
manifolds. However, we briefly quote some of main results of [3] in the
present article. One interesting observation is that the quantum isometry
group of the noncommutative two-torus Aθ (with the canonical spectral
triple) is (as a C∗ algebra) a direct sum of two commutative and two non-
commutative tori, and contains as a quantum subgroup (which is univer-
sal for certain class of isometric actions called holomorphic isometries) the
‘quantum double-torus’ discovered and studied by Hajac and Masuda ([11]).
2 Definition of the quantum isometry group
2.1 Isometry groups of classical manifolds
We begin with a well-known characterization of the isometry group of a (clas-
sical) compact Riemannian manifold. Let (M,g) be a compact Riemannian
manifold and let Ω1 = Ω1(M) be the space of smooth one-forms, which has
a right Hilbert-C∞(M)-module structure given by the C∞(M)-valued inner
product << ·, · >> defined by
<< ω, η >> (m) =< ω(m), η(m) > |m,
where < ·, · > |m is the Riemannian metric on the cotangent space T
mM at
the pointm ∈M . The Riemannian volume form allows us to make Ω1 a pre-
Hilbert space, and we denote its completion by H1. Let H0 = L
2(M,dvol)
and consider the de-Rham differential d as an unbounded linear map from
H0 toH1, with the natural domain C
∞(M) ⊂ H0, and also denote its closure
by d. Let L := −d∗d. The following identity can be verified by direct and
easy computation using the local coordinates :
(∂L)(φ,ψ) ≡ L(φ̄ψ)−L(φ̄)ψ−φ̄L(ψ) = 2 << dφ, dψ >> for φ,ψ ∈ C∞(M) (∗).
Proposition 2.1 A smooth map γ : M → M is a Riemannian isometry if
and only if γ commutes with L in the sense that L(f ◦ γ) = (L(f)) ◦ γ for
all f ∈ C∞(M).
Proof :
If γ commutes with L then from the identity (*) we get for m ∈ M and
φ,ψ ∈ C∞(M) :
< dφ|γ(m), dψ|γ(m) > |γ(m)
= << dφ, dψ >> (γ(m))
(∂L(φ,ψ) ◦ γ)(m)
∂L(φ ◦ γ, ψ ◦ γ)(m)
= << d(φ ◦ γ), d(ψ ◦ γ) >> (m)
= < d(φ ◦ γ)|m, d(ψ ◦ γ)|m > |m
= < (dγ|m)
∗(dφ|γ(m)), (dγ|m)
∗(dψ|γ(m)) > |m,
which proves that (dγ|m)
∗ : T ∗
M → T ∗mM is an isometry. Thus, γ is a
Riemannian isometry.
Conversely, if γ is an isometry, both the maps induced by γ on H0 and
H1, i.e. U
γ : H0 → H0 given by U
γ (f) = f ◦ γ and U
γ : H
1 → H1 given
by U1γ (fdφ) = (f ◦ γ)d(φ ◦ γ) are unitaries. Moreover, d ◦ U
γ = U
γ ◦ d on
C∞(M) ⊂ H0. From this, it follows that L = −d
∗d commutes with U0γ . ✷
Now let us consider a compact metrizable (i.e. second countable) space
Y with a continuous map θ : M × Y → M . We abbreviate θ(m, y) as ym
and denote by ξy the map M ∋ m 7→ ym. Let α : C(M) → C(M)⊗C(Y ) ∼=
C(M × Y ) be the map given by α(f)(m, y) := f(ym) for y ∈ Y , m ∈ M
and f ∈ C(M). For a state φ on C(Y ), denote by αφ the map (id⊗ φ) ◦ α :
C(M) → C(M). We shall also denote by C the subspace of C(M) ⊗ C(Y )
generated by elements of the form α(f)(1⊗ψ), f ∈ C(M), ψ ∈ C(Y ). Since
C(M) and C(Y ) are commutative algebras, it is easy to see that C is a
∗-subalgebra of C(M)⊗ C(Y ). Then we have the following
Theorem 2.2 (i) C is norm-dense in C(M)⊗C(Y ) if and only if for every
y ∈ Y , ξy is one-to-one.
(ii) The map ξy is C
∞ for every y ∈ Y if and only if αφ(C
∞(M)) ⊆ C∞(M)
for all φ.
(iii) Under the hypothesis of (ii), each ξy is also an isometry if and only if
αφ commutes with (L − λ)
−1 for all state φ and all λ in the resolvent of L
(equivalently, αφ commutes with the Laplacian L on C
∞(M)).
Proof :
(i) First, assume that ξy is one-to-one for all y. By Stone-Weirstrass Theo-
rem, it is enough to show that C separates points. Take (m1, y1) 6= (m2, y2)
in M × Y . If y1 6= y2, we can choose ψ ∈ C(Y ) which separates y1 and y2,
hence (1 ⊗ ψ) ∈ C separates (m1, y1) and (m2, y2). So, we can consider the
case when y1 = y2 = y (say), but m1 6= m2. By injectivity of ξy, we have
ym1 6= ym2, so there exists f ∈ C(M) such that f(ym1) 6= f(ym2), i.e.
α(f)(m1, y) 6= α(f)(m2, y). This proves the density of C.
For the converse, we argue as in the proof of Proposition 3.3 of [14].
Assume that C is dense in C(M)⊗ C(Y ), and let y ∈ Y , m1,m2 ∈ M such
that ym1 = ym2. That is, α(f)(1 ⊗ ψ)(m1, y) = α(f)(1 ⊗ ψ)(m2, y) for all
f ∈ C(M), ψ ∈ C(Y ). By the density of C we get χ(m1, y) = χ(m2, y) for
all χ ∈ C(M × Y ), so (m1, y) = (m2, y), i.e. m1 = m2.
(ii) The ‘if part’ of (ii) follows by considering the states corresponding to
point evaluation, i.e. C(Y ) ∋ ψ 7→ ψ(y), y ∈ Y . For the converse, we note
that an arbitrary state φ corresponds to a regular Borel measure µ on Y
so that φ(h) =
hdµ, and thus, αφ(f)(m) =
f(ym)dµ(y) for f ∈ C(M).
From this, by interchanging differentiation and integation (which is allowed
by the Dominated Convergence Theorem, since µ is a finite measure) we can
prove that αφ(f) is C
∞ whenever f is so.
The assertion (iii) follows from Proposition 2.1 in a straghtforward way.
Let us recall a few well-known facts about the Laplacian L, viewed as a
negative self-adjoint operator on the Hilbert space L2(M,dvol). It is known
(see [12] and references therein) that L has compact resolvents and all its
eigenvectors belong to C∞(M). Moreover, it follows from the Sobolev Em-
bedding Theorem that
Dom(Ln) = C∞(M).
Let {eij , j = 1, ..., di; i = 1, 2, ...} be the set of (normalised) eigenvectors of
L, where eij ∈ C
∞(M) is an eigenvector corresponding to the eigenvalue λi,
|λ1| < |λ2| < .... We have the following:
Lemma 2.3 The complex linear span of {eij} is norm-dense in C(M).
Proof :
This is a consequence of the asymptotic estimates of eigenvalues λi, as
well as the uniform bound of the eigenfunctions eij . For example, it is
known ([9],Theorem 1.2) that there exist constants C,C ′ such that ‖eij‖∞ ≤
C|λi|
4 , di ≤ C
′|λi|
2 , where n is the dimension of the manifoldM . Now,
for f ∈ C∞(M) ⊆
k≥1Dom(L
k), we write f as an a-priori L2-convergent
series
ij fijeij (fij ∈ C), and observe that
|fij |
2|λi|
2k < ∞ for every
k ≥ 1. Choose and fix sufficiently large k such that
i≥0 |λi|
n−1−2k < ∞,
which is possible due to the well-known Weyl asymptotics of eigenvalues of
L. Now, by the Cauchy-Schwarz inequality and the estimate for di, we have
|fij|‖eij‖∞ ≤ C(C
|fij |
2|λi|
n−1−2k
Thus, the series
ij fijeij converges to f in sup-norm, so Sp{eij , j = 1, 2, ..., di ; i =
1, 2, ...} is dense in sup-norm in C∞(M), hence in C(M) as well. ✷
Let us denote Sp{eij , j = 1, ..., di; i ≥ 1} by A
0 from now on. We shall
now show that C∞(M) can be replaced by the smaller subspace A∞0 in
Theorem 2.2. We need a lemma for this, which will be useful later on too.
Lemma 2.4 Let H1,H2 be Hilbert spaces and for i = 1, 2, let Li be (possibly
unbounded) self-adjoint operator on Hi with compact resolvents, and let Vi
be the linear span of eigenvectors of Li. Moreover, assume that there is an
eigenvalue of Li for which the eigenspace is one-dimensional, say spanned by
a unit vector ξi. Let Ψ be a linear map from V1 to V2 such that L2Ψ = ΨL1
and Ψ(ξ1) = ξ2. Then we have
〈ξ2,Ψ(x)〉 = 〈ξ1, x〉 ∀x ∈ V1. (1)
Proof:
By hypothesis on Ψ, it is clear that there is a common eigenvalue, say λ0, of
L1 and L2, with the eigenvectors ξ1 and ξ2 respectively. Let us write the set
of eigenvalues of Li as a disjoint union {λ0}
Λi (i = 1, 2), and let the corre-
sponding orthogonal decomposition of Vi be given by Vi = Cξi
Vλi ≡
Cξi ⊕ V
i, say, where V
i denotes the eigenspace of Li corresponding to the
eigenvalue λ. By assumption, Ψ maps Vλ1 to V
2 whenever λ is an eigenvalue
of L2, i.e. V
2 6= {0}, and otherwise it maps V
1 into {0}. Thus, Ψ(V
1) ⊆ V
Now, (1) is obviously satisfied for x = ξ1, so it is enough to prove (1) for all
x ∈ V ′1. But we have 〈ξ, x〉 = 0 for x ∈ V
1, and since Ψ(x) ∈ V
2 = V2
it follows that 〈ξ2,Ψ(x)〉 = 0 = 〈ξ1, x〉. ✷
Lemma 2.5 Let Y and α be as in Theorem 2.2. Then the following are
equivalent.
(a) For every y ∈ Y , ξy is smooth isometric.
(b) For every state φ on C(Y ), we have αφ(A
0 ) ⊆ A
0 , and αφL = Lαφ on
A∞0 .
Proof:
We prove only the nontrivial implication (b) ⇒ (a). Assume (b) that αφ
leaves A∞0 invariant and commutes with L on it, for every state φ. To
prove that α is a smooth isometric action, it is enough (see the proof of
Theorem 2.2) to prove that αy(A
∞) ⊆ A∞ for all y ∈ Y , where αy(f) :=
(id⊗evy)(f) = f ◦ξy, evy being the evaluation at the point y. LetM1, ...,Mk
be the connected components of the compact manifoldM . Thus, the Hilbert
space L2(M,dvol) admits an orthogonal decomposition ⊕ki=1L
2(Mi,dvol),
and the Laplacian L is of the form ⊕iLi where Li denotes the Laplacian on
Mi. Since each Mi is connected, we have Ker(Li) = Cχi, where χi is the
constant function on Mi equal to 1. Now, we note that for fixed y and i,
the image of Mi under the continuous function ξy must be mapped into a
component, sayMj. Thus, by applying Lemma 2.4 with H1 = L
2(Mi),H2 =
L2(Mj), Ψ = ξy and the L
2-continuity of the map f 7→ αy(f) = f ◦ ξy, we
αy(f)(x)dvol(x) =
f(x)dvol(x)
for all f in the linear span of eigenvectors of Li, hence (by density) for all f
in L2(Mi). It follows that
αy(f)dvol =
fdvol for all f ∈ L2(M), in
particular for all f ∈ C(M). Since αy is a ∗-homomorphism on C(M), we
〈αy(f), αy(g)〉 =
αy(fg)dvol =
fgdvol = 〈f, g〉,
for all f, g ∈ C(M). Thus, αy extends to an isometry on L
2(M), to be
denoted by the same notation, which by our assumption commutes with the
self-adjoint operator L on the core A∞0 , and hence αy commutes with L
n for
all n. In particular it leaves invariant the domains of each Ln, which implies
∞) ⊆ A∞. ✷
In view of the fact that the set of isometries of M , denoted by ISO(M),
is a compact second countable (i.e. compact metrizable) group, we see that
ISO(M) is the maximal compact second countable group acting on M such
that the action is smooth and isometric. In other words, if we consider a
catogory whose objects are compact metrizable groups acting smoothly and
isometrically on M , and morphisms are the group homomorphisms com-
muting with the actions on M , then ISO(M) (with its canonical action on
M) is the initial object of this cateogory. However, one can take a more
general viewpoint and consider the category of compact metrizable spaces
Y equipped with a continuous map θ : M × Y → M satisfying (i)-(iii) of
Theorem 2.2, or equivalently, the pair of commutative unital C∗-algebras
B = C(Y ) and a unital C∗-homomorphism α : C(M) → C(M) → B satisfy-
ing the conditions (i)-(iii). The set of isometries ISO(M) (as a topological
space) can be identified with the universal object of this category, and then
one can prove that it has a group structure.
It is quite natural to formulate a quantum analogue of the above, by con-
sidering, in the spirit of Woronowicz and Soltan (see [19] and [13]), ‘quantum
families of isometries’, which can be defined to be a pair (B, α) where B is
a (not necessarily commutative) C∗-algebra and α : C(M) → C(M)⊗ B is
unital C∗-homomorhism satisfying (i)-(iii) of Theorem 2.2, i.e. the linear
span of α(C(M))(1⊗B) (which is not necessarily a ∗-subalgebra any more,
B being possibly noncommutative) is norm-dense in C(M)⊗ B and for ev-
ery state φ on B, the map αφ keeps C
∞(M) invariant and commutes with
the Laplacian L. The morphisms of this category are obvious. We shall
prove that this category has a universal object, and this universal object
can be equipped with a canonical quantum group structure. This will define
the quantum isometry group of a manifold. However, we shall go beyond
classical manifolds and define quantum isometry group QISO(A∞,H,D)
for a spectral triple (A∞,H,D), with A∞ being unital, and satisfying cer-
tain assumptions. To this end, we need to carefully formulate the notion
of Laplacian in noncommutative geometry, which is the goal of the next
subsection.
2.2 Laplacian in noncommutative geometry
Given a spectral triple (A∞,H,D), we recall from [10] and [6] the con-
struction of the space of one-forms. We have a derivation from A∞ to the
A∞-A∞ bimodule B(H) given by a 7→ [D, a]. This induces a bimodule
morphism π from Ω1(A∞) (the bimodule of universal one-forms on A∞) to
B(H), such that π(δ(a)) = [D, a], where δ : A∞ → Ω1(A∞) denotes the
universal derivation map. We set Ω1D ≡ Ω
∞) := Ω1(A∞)/Ker(π) ∼=
π(Ω1(A∞)) ⊆ B(H). Assume that the spectral triple is of compact type
and has a finite dimension in the sense of Connes ([6]), i.e. there is some
p > 0 such that the operator |D|−p (interpreted as the inverse of the re-
striction of |D|p on the closure of its range, which has a finite co-dimension
since D has compact resolvents) has finite nonzero Dixmier trace, denoted
by Trω (where ω is some suitable Banach limit, see, e.g. [6], [10]). Con-
sider the canonical ‘volume form’ τ coming from the Dixmier trace, i.e.
τ : B(H) → C defined by τ(A) := 1
Trω(|D|−p)
Trω(A|D|
−p). Let us at this
point assume that the spectral triple is QC∞, i.e. A∞ and {[D, a], a ∈ A∞}
are contained in the domains of all powers of the derivation [|D|, ·]. Under
this assumption, τ is a positive faithful trace on the C∗-subalgebra gener-
ated by A∞ and {[D, a] a ∈ A∞}, and the GNS Hilbert space L2(A∞, τ) is
denoted by H0D. Similarly, we equip Ω
D with a semi-inner product given by
< η, η′ >:= τ(η∗η′), and denote the Hilbert space obtained from it by H1D.
The map dD : H
D → H
D given by dD(·) = [D, ·] is an unbounded densely
defined linear map. Let us assume the following:
Assumption(i) (a) dD is closable (the closure is denoted again by dD);
(b) A∞ ⊆ Dom(L), where L := −d∗DdD and A
∞ is viewed as a dense sub-
space of H0D;
At this point, let us show that this assumption is valid under a very
natural condition on the spectral triple.
Lemma 2.6 Suppose that for every element a ∈ A∞, the map R ∋ t 7→
αt(X) := exp(itD)Xexp(−itD) is differentiable at t = 0 in the norm-
topology of B(H), where X = a or [D, a]. Then the assumption (i) is sat-
isfied. Moreover, in this case, L maps A∞ into the weak closure of A∞ in
B(H0D).
Proof :
We first observe that τ(αt(A)) = τ(A) for all t and for all A ∈ B(H), since
exp(itD) commutes with |D|−p. If moreover, A belongs to the domain of
norm-differentiability (at t = 0) of αt, i.e.
αt(A)−A
→ i[D,A] in operator-
norm, then it follows from the property of the Dixmier trace that τ([D,A]) =
limt→0
τ(αt(A))−τ(A)
= 0. Now, since by assumption we have the norm-
differentiability at t = 0 of αt(A) for A belonging to the ∗-subalgebra (say
B) generated by A∞ and [D,A∞], it follows that τ([D,A]) = 0 ∀A ∈ B. Let
us now fix a, b, c ∈ A∞ and observe that
< a dD(b), dD(c) >
= τ((a dD(b))
∗dD(c) >
= −τ([D, [D, b∗]a∗c]) + τ([D, [D, b∗]a∗]c)
= τ([D, [D, b∗]a∗]c),
using the fact that τ([D, [D, b∗]a∗c]) = 0. This implies
| < a dD(b), dD(c) > | ≤ ‖[D, [D, b
∗]a∗]‖τ(c∗c)
2 = ‖[D, [D, b∗]a∗]‖‖c‖2,
where ‖c‖2 = τ(c
2 denotes the L2-norm of c ∈ H0D. This proves that
a dD(b) belongs to the domain of d
D for all a, b ∈ A
∞, so in particular
d∗D is dense, i.e. dD is closable. Moreover, taking a = 1, we see that
∞) ⊆ Dom(d∗D), or in other words, A
∞ ⊆ Dom(d∗DdD). This proves
(i)(a) and (i)(b). The last sentence in the statement of the lemma can be
proven along the line of Theorem 2.9, page 129, [10]. ✷
We need few more assumptions on the operator L to define the quantum
isometry group.
Assumption (ii): L has compact resolvents,
Assumption(iii): L(A∞) ⊆ A∞;
Assumption(iv): Each eigenvector of L (which has a discrete spectrum,
hence a complete set of eigenvectors) belongs to A∞;
Assumption(v)(‘connectedness assumption’): the kernel of L is one-dimensional,
spanned by the identity 1 of A∞, viewed as a unit vector in H0D.
We call L the noncommutative Laplacian and Tt the noncommutative heat
semigroup. We summarize some simple observations in form of the following
Lemma 2.7 (a) If the assumptions (i)-(v) are valid, then for x ∈ A∞, we
have L(x∗) = (L(x))∗.
(b) If Tt := exp(tL) maps H
D into A
∞ for all t > 0, the the assumption
(iv) is satisfied.
Proof :
It follows by simple calculation using the facts that τ is a trace and dD(x
−(dD(x))
∗ that
τ(L(x∗)∗y)
= −τ(dD(x)dD(y)) = −τ(dD(y)dD(x)) = τ((dD(y
∗))∗dD(x))
= < y∗,L(x) >= τ(yL(x)) = τ(L(x)y),
for all y ∈ A∞. By density of A∞ in H0D (a) follows. To prove (b), we note
that if x ∈ H0D is an eigenvector of L, say L(x) = λx (λ ∈ C), then we have
Tt(x) = e
λtx, hence x = e−λtTt(x) ∈ A
Since by assumption, L has a countable set of eigenvalues each with finite
multiplicity, let us denote them by λ0 = 0, λ1, λ2, ... with V0 = C 1, V1, V2, ...
be corresponding eigenspaces (finite dimensional), and for each i, let {eij , j =
1, ..., di} be an orthonormal basis of Vi. By Assumption (iv), Vi ⊆ A
∞ for
each i, Vi is closed under ∗, and moreover, {e
ij , j = 1, ..., di} is also an or-
thonormal basis for Vi, since τ(x
∗y) = τ(yx∗) for x, y ∈ A∞. We also make
the following
Assumption (vi) The complex linear span of {eij , i = 0, 1, ...; j = 1, ..., di},
say A∞0 , is norm-dense in A
Definition 2.8 We say that a spectral triple satisfying the assumptions (i)-
(vi) admissible.
Remark 2.9 We have just seen that classical spectral triple (A∞ = C∞(M),H,D),
where M is compact connected spin manifold, H is the L2 space of square
integrable spinors and D is the Dirac operator, is indeed admissible in our
sense. Later on we shall discuss how we can weaken the connectedness
assumption as well, thus accommodating a general classical (commutative)
spectral triple in our set-up. Moreover, the standard examples of noncom-
mutative spectral triples, e.g. those on Aθ, quantum Heisenberg manifold
etc., do belong to the admissible class.
Lemma 2.10 Let us assume that the spectral triple (A∞,H,D) is admis-
sible. Let Ψ : A∞0 → A
0 be a (norm-) bounded linear map, such that
Ψ(1) = 1, and Ψ ◦L = L◦Ψ on the subspace A∞0 spanned (algebraically) by
Vi, i = 1, 2, .... Then τ(Ψ(x)) = τ(x) for all x ∈ A
Proof :
By Lemma 2.4 with H1 = H2 = H
D, ξ1 = ξ2 = 1, we have τ(Ψ(x)) = τ(x)
for all x ∈ A∞0 . By the norm-continuity of Ψ and τ it extends to the whole
of A∞. ✷
2.3 Definition and existence of the quantum isometry group
We begin by recalling the definition of compact quantum groups and their
actions from [18]. A compact quantum group is given by a pair (S,∆), where
S is a unital separable C∗ algebra equipped with a unital C∗-homomorphism
∆ : S → S ⊗ S (where ⊗ denotes the injective tensor product) satisfying
(ai) (∆⊗ id) ◦∆ = (id⊗∆) ◦∆ (co-associativity), and
(aii) the linear span of ∆(S)(S ⊗ 1) and ∆(S)(1 ⊗ S) are norm-dense in
S ⊗ S.
It is well-known (see [18]) that there is a canonical dense ∗-subalgebra S0
of S, consisting of the matrix coefficients of the finite dimensional unitary
(co)-representations of S, and maps ǫ : S0 → C (co-unit) and κ : S0 → S0
(antipode) defined on S0 which make S0 a Hopf ∗-algebra.
We say that the compact quantum group (S,∆) acts on a unital C∗
algebra B, if there is a unital C∗-homomorphism α : B → B ⊗ S satisfying
the following :
(bi) (α⊗ id) ◦ α = (id⊗∆) ◦ α, and
(bii) the linear span of α(B)(1 ⊗ S) is norm-dense in B ⊗ S.
Let us now recall the concept of universal quantum groups as in [17],
[15] and references therein. We shall use most of the terminologies of [15],
e.g. Woronowicz C∗ -subalgebra, Woronowicz C∗-ideal etc, however with
the exception that we shall call the Woronowicz C∗ algebras just compact
quantum groups, and not use the term compact quantum groups for the
dual objects as done in [15]. For Q ∈ GLn(C), let Au(Q) denote the uni-
versal compact quantum group generated by uij, i, j = 1, ..., n satisfying the
relations
uu∗ = In = u
∗u, u′QuQ−1 = In = QuQ
−1u′,
where u = ((uij)), u
′ = ((uji)) and u = ((u
ij)). The coproduct, say ∆̃, is
given by,
∆̃(uij) =
uik ⊗ ukj.
We refer the reader to [17] for a detailed discussion on the structure and clas-
sification of such quantum groups. Let us denote by Ui the quantum group
Adi(I), where di is dimension of the subspace Vi. We fix a representation
βi : Vi → Vi⊗Ui of Ui on the Hilbert space Vi, given by βi(eij) =
k eik⊗u
for j = 1, ..., di, where u
(i) ≡ u
are the generators of Ui as discussed before.
Thus, both u(i) and ¯u(i) are unitaries. It follows from [15] that the represen-
tations βi canonically induce a representation β = ∗iβi of the free product
U := ∗iUi (which is a compact quantum group, see [15] for the details) on
the Hilbert space H0D, such that the restriction of β on Vi coincides with βi
for all i.
In view of the characterization of smooth isometric action on a classical
manifold, we make the following definitions.
Definition 2.11 A quantum family of smooth isometries of a noncommu-
tative manifold A∞ (or, more precisely on the corresponding spectral triple)
is a pair (S, α) where S is a separable unital C∗-algebra, α : A → A ⊗ S
(where A denotes the C∗ algebra obtained by completing A∞ in the norm of
B(H0D)) is a unital C
∗-homomorphism, satisfying the following:
(a) Sp
α(A)(1⊗ S)
= A⊗ S,
(b) αφ := (id ⊗ φ) ◦ α maps A
0 into itself and commutes with L on A
for every state φ on S.
In case the C∗-algebra S has a coproduct ∆ such that (S,∆) is a compact
quantum group and α is an action of (S,∆) on A, we say that (S,∆) acts
smoothly and isometrically on the noncommutative manifold.
Fix a spectral triple (A∞,H,D). Consider the category Q with the
object-class consisting of all quantum families of isometries (S, α) of the
given noncommutative manifold, and the set of morphismsMor((S, α), (S ′, α′))
being the set of unital C∗-homomorphisms φ : S → S ′ satisfying (id⊗φ)◦α =
α′. We also consider another category Q′ whose objects are triplets (S,∆, α)
where (S,∆) is a compact quantum group acting smoothly and isometrically
on the given noncommutative manifold, with α being the corresponding ac-
tion. The morphisms are the homomorphisms of compact quantum groups
which are also morphisms of the underlying quantum families. The forget-
ful functor F : Q′ → Q is clearly faithful, and we can view F (Q′) as a
subcategory of Q.
Let us assume from now on that the spectral triple (A∞,H,D) is admis-
sible. Our aim is to prove the existence of a universal object in Q. We shall
also prove that the (unique upto isomorphism) universal object belongs to
F (Q′), and its pre-image in Q′ is a universal object in the category Q′. To
this end, we need some preparatory results.
Lemma 2.12 Consider an admissible spectral triple (A∞,H,D) and let
(S, α) be a quantum family of smooth isometries of the spectral triple. More-
over, assume that the action α is faithful in the sense that there is no proper
C∗-subalgebra S1 of S such that α(A
∞) ⊆ A∞ ⊗ S1. Then α̃ : A
∞ ⊗ S →
A∞ ⊗ S defined by α̃(a⊗ b) : α(a)(1 ⊗ b) extends to an S-linear unitary on
the Hilbert S-module H0D ⊗S, denoted again by α̃. Moreover, we can find a
C∗-isomorphism φ : U/I → S between S and a quotient of U by a C∗-ideal
I of U , such that α = (id ⊗ φ) ◦ (id ⊗ ΠI) ◦ β on A
∞ ⊆ H0D, where ΠI
denotes the quotient map from U to U/I.
If, furthermore, there is a compact quantum group structure on S given
by a coproduct ∆ such that (S,∆, α) is an object in Q′, the map α : A∞ →
A∞⊗S extends to a unitary representation (denoted again by α) of the com-
pact quantum group (S,∆) on H0D. In this case, the ideal I is a Woronowicz
C∗-ideal and the C∗-isomorphism φ : U/I → S is a morphism of compact
quantum groups.
Proof :
Let ω be any state on S. Since the action α : A∞ → A∞ ⊗ S is smooth
and isometric, we conclude by Lemma 2.10 that τ(αω(x)) = τ(x)ω(1) for all
x ∈ A. Since ω is arbitrary, we have (τ ⊗ id)α(x) = τ(x)1S for all x ∈ A.
So, < α(x), α(y) >S=< x, y > 1S , where < ·, · >S denotes the S-valued
inner product of the Hilbert module H0D ⊗S. This proves that α̃ defined by
α̃(x⊗ b) := α(x)(1⊗ b) (x ∈ A∞, b ∈ S) extends to an S-linear isometry on
the Hilbert S-module H0D⊗S. Moreover, since α(A
∞)(1⊗S) is norm-dense
in Ā⊗S, it is clear that the S-linear span of the range of α(A∞) is dense in
the Hilbert module H0D ⊗ S, or in other words, the isometry α̃ has a dense
range, so it is a unitary.
Since αω leaves each Vi invariant, it is clear that α maps Vi into Vi ⊗ S
for each i. Let v
(j, k = 1, ..., di) be the elements of S such that α(eij) =
k eik ⊗ v
. Note that vi := ((v
)) is a unitary in Mdi(C)⊗S. Moreover,
the ∗-subalgebra generated by all {v
, i, j, k ≥ 1} must be dense in S by
the assumption of faithfulness.
We have already remarked that {e∗ij} is also an orthonormal basis of
Vi, and since α, being a C
∗-action on A, is ∗-preserving, we have α(e∗ij) =
(α(eij))
, and therefore ((v
)) is also unitary. By univer-
sality of Ui, there is a C
∗-homomorphism from Ui to S sending u
and by definition of the free product, this induces a C∗-homomorphism, say
Π, from U onto S, so that U/I ∼= S, where I := Ker(Π).
In case S has a coproduct ∆ making it into a compact quantum group
and α is a quantum group action, it is easy to see that the subalgebra of
S generated by v
is a Hopf algebra, with ∆(v
. From
this, it follows that Π is Hopf-algebra morphism, hence I is a Woronowicz
C∗-ideal. ✷
Before we state and prove the main theorem, let us note the following
elementary fact about C∗-algebras.
Lemma 2.13 Let C be a C∗ algebra and F be a nonempty collection of
C∗-ideals (closed two-sided ideals) of C. Then for any x ∈ C, we have
‖x+ I‖ = ‖x+ I0‖,
where I0 denotes the intersection of all I in F and ‖x + I‖ = inf{‖x −
y‖ : y ∈ I} denotes the norm in C/I.
Proof :
It is clear that supI∈F ‖x + I‖ defines a norm on C/I0, which is in fact a
C∗-norm since each of the quotient norms ‖ · +I‖ is so. Thus the lemma
follows from the uniqueness of C∗ norm on the C∗ algebra C/I0. ✷
Theorem 2.14 For any admissible spectral triple (A∞,H,D), the category
Q of quantum families of smooth isometries has a universal (initial) object,
say (G, α0). Moreover, G has a coproduct ∆0 such that (G,∆0) is a com-
pact quantum group and (G,∆0, α0) is a universal object in the category Q
of compact quantum groups acting smoothly and isometrically on the given
spectral triple. The action α0 is faithful.
Proof :
Recall the C∗-algebra U considered before, and the map β from H0D to
H0D⊗U . By our definition of β, it is clear that β(A
0 ) ⊆ A
0 ⊗algU . However,
β is only a linear map (unitary) but not necessarily a ∗-homomorphism. We
shall construct the universal object as a suitable quotient of U . Let F
be the collection of all those C∗-ideals I of U such that the composition
ΓI := (id⊗ΠI) ◦ β : A
0 → A
0 ⊗alg (U/I) extends to a C
∗-homomorphsim
from Ā to Ā ⊗ (U/I), where ΠI denotes the quotient map from U onto
U/I. This collection is nonempty, since the trivial one-dimensional C∗-
algebra C gives an object in Q and by Lemma 2.12 we do get a member of
F . Now, let I0 be the intersection of all ideals in F . We claim that I0 is
again a member of F . Since any C∗-homomorphism is contractive, we have
‖ΓI(a)‖ ≡ ‖β(a) + Ā ⊗ I‖ ≤ ‖a‖ for all a ∈ A
0 and I ∈ F . By Lemma
2.13, we see that ‖ΓI0(a)‖ ≤ ‖a‖ for a ∈ A
0 , so ΓI0 extends to a norm-
contractive map on Ā by the density of A∞0 in Ā. Moreover, for a, b ∈ Ā
and for I ∈ F , we have ΓI(ab) = ΓI(a)ΓI(b). Since ΠI = ΠI ◦ ΠI0 , we can
rewrite the homomorphic property of ΓI as
ΓI0(ab)− ΓI0(a)ΓI0(b) ∈ Ā ⊗ (I/I0).
Since this holds for every I ∈ F , we conclude that ΓI0(ab)−ΓI0(a)ΓI0(b) ∈
I∈F Ā⊗(I/I0) = (0), i.e. ΓI0 is a homomorphism. In a similar way, we can
show that it is a ∗-homomorphism. Since each βi is a unitary representation
of the compact quantum group Ui on the finite dimensional space Vi, it
follows that βi(Vi)(1 ⊗ Ui) is total in Vi ⊗ Ui. In particular, for any vi ∈ Vi
(i arbitrary), the element vi ⊗ 1Ui = vi ⊗ 1U belongs to the linear span of
βi(Vi)(1⊗Ui) ⊂ β(Vi)(1⊗U). Thus, A
0 ⊗1U is contained in the linear span of
β(A∞0 )(1⊗U) and henceA
0 ⊗1 U
is linearly spanned by ΓI0(A
0 )(1⊗U/I0).
By the norm-denisty of A∞0 in A and the contractivity of the quotient map,
it follows that A ⊗ U/I0 is the closed linear span of ΓI0(A
0 )(1 ⊗ U/I0).
This completes the proof that (U/I0,ΓI0) is indeed an object of Q.
We now show that G := U/I0 is a universal object in Q. To see this, con-
sider any object (S, α) of Q. Without loss of generality we can assume the
action to be faithful, since otherwise we can replace S by the C∗-subalgebra
generated by the elements {v
} appearing in the proof of Lemma 2.12. But
by Lemma 2.12 we can further assume that S is isomorphic with U/I for
some I ∈ F . Since I0 ⊆ I, we have a C
∗-homomorphism from U/I0 onto
U/I, sending x+I0 to x+I, which is clearly a morphism in the category Q.
This is indeed the unique such morphism, since it is uniquely determined on
the dense subalgebra generated by {u
+ I0, i, j, k ≥ 1} of G.
To construct the coproduct on G = U/I0, we first consider α
(2) = (ΓI0 ⊗
id) ◦ΓI0 : A → A⊗G ⊗G. It is easy to verify that (G ⊗G, α
(2)) is an object
in the category Q, so by the universality of (G,ΓI0), we have a unique unital
C∗-homomorphism ∆0 : G → G ⊗ G satisfying
(id⊗∆0) ◦ ΓI0(x) = α
(2)(x) ∀x ∈ A.
Taking x = eij, we get
eil ⊗ (πI0 ⊗ πI0)
eil ⊗∆0(πI0(u
Comparing coefficients of eil, and recalling that ∆̃(u
(where ∆̃ denotes the coproduct on U), we have
(πI0 ⊗ πI0) ◦ ∆̃ = ∆0 ◦ πI0 (2)
on the linear span of {u
, i, j, k ≥ 1}, and hence on the whole of U . This
implies that ∆0 maps I0 = Ker(πI0) into Ker(πI0⊗πI0) = (I0⊗1+1⊗I0) ⊂
U ⊗ U . In other words, I0 is a Hopf C
∗-ideal, and hence G = U/I0 has the
canonical compact quantum group structure as a quantum subgroup of U . It
is clear from the relation (2) that ∆0 coincides with the canonical coproduct
of the quantum subgroup U/I0 inherited from that of U . It is also easy to
see that the object (G,∆0,ΓI0) is universal in the category Q
′, using the fact
that (by Lemma 2.12) any compact quantum group (G,Φ) acting smoothly
and isometrically on the given spectral triple is isomorphic with a quantum
subgroup U/I, for some Hopf C∗-ideal I of U .
Finally, the faithfulness of α0 follows from the universality by standard
arguments which we briefly sketch. If G1 ⊂ G is a ∗-subalgebra of G such
that α0(A) ⊆ A ⊗ G1, it is easy to see that (G1,∆0, α0) is also a universal
object, and by definition of universality of G it follows that there is a unique
morphism, say j, from G to G1. But the map j ◦ i is a morphism from G to
itself, where i : G1 → G is the inclusion. Again by universality, we have that
j ◦ i = idG , so in particular, i is onto, i.e. G1 = G. ✷
Definition 2.15 We shall call the universal object (G,∆0) obtained in the
theorem above the quantum isometry group of (A∞,H,D) and denote it
by QISO(A∞,H,D), or just QISO(A∞) (or sometimes QISO(Ā)) if the
spectral triple is understood from the context.
Remark 2.16 Assume that an admissible spectral triple (A∞,H,D) also
satisfies the condition (i) of Lemma 2.5, i.e.
Dom(Ln) = A∞. Let α :
A → A⊗S be a smooth isometric action on A∞ by a compact quantum group
S. We recall from the proof of Lemma 2.12 that the map α̃ from A⊗ S to
itself extends to an S-linear unitary on the Hilbert S-module H0D ⊗ S, i.e.
α̃ can be viewed as a unitary in B(H0D) ⊗ S. Clearly, for any state φ on
S, we have αφ = (id ⊗ φ)(α̃) ∈ B(H
D). Now, by the definition of a smooth
isometric action, the bounded operator αφ commutes with the self-adjoint
operator L on A∞0 , which is a core for L. So, αφ must commute with L
for all n, and in particular keeps A∞ =
nDom(L
n) invariant.
Remark 2.17 Let us now briefly indicate how one can weaken the hy-
pothesis of connectedness. Such an extension of our results is desirable
to accommodate the classical spaces, including the finite sets and graphs,
in our framework. One possibile approach could be to consider the cate-
gory of compact quantum group actions α which are not only ‘smmoth’ and
‘isometric’ in our sense, but also satisfy the τ -invariance condition, i.e.
(τ ⊗ id)(α(a)) = τ(a)1. It is easy to see that the connectedness assumption
has been used by us only to prove that the τ -invariance is automatic for
smooth isometric actions. Thus, if we work in the smaller cateogory of such
τ -invraiant actions only, the proof of Theorem 2.14 does go through and thus
we can prove the existence of a universal object, to be defined as the quantum
isometry group. It is easy to see that for the algebra of functions on a finite
set, with the spectral triple given by D = 0, this quantum isometry group
coincides with thw quantum permutation group defined by Wang.
Remark 2.18 It is easy to see how to extend our formulation and results
to spectral triples which are not necessarily of type II, i.e. when the trace
τ is replaced by some non-tracial positive functional. Indeed, our construc-
tion will go through in such a situation more or less verbatim, by replacing
the universal quantum groups Adi(I) by Adi(Qi) for some suitable choice of
matrices Qi coming from the modularity property of τ .
2.4 Construction of quantum group-equivariant spectral triples
In this subsection, we shall briefly discuss the relevance of quantum isometry
group to the problem of constructing quantum group equivariant spectral
triples, which is important to understand the role of quantum groups in the
framework of noncommutative geometry. There has been a lot of activity
in this direction recently, see, for example, the articles by Chakraborty and
Pal ([5]), Connes ([7]), Landi et al ([8]) and the references therein. In the
classical situation, there exists a natural unitary representation of the isom-
etry group G = ISO(M) of a manifold M on the Hilbert space of forms, so
that the operator d+d∗ (where d is the de-Rham differential operator) com-
mutes with the representation. Indeed, d+d∗ is also a Dirac operator for the
spectral triple given by the natural representation of C∞(M) on the Hilbert
space of forms, so we have a canonical construction of G-equivariant spectral
triple. Our aim in this subsection is to generalize this to the noncommuta-
tive framework, by proving that dD + d
D is equivariant with respect to a
canonical unitary representation on the Hilbert space of ‘noncommutative
forms’ (see, for example, [10] for a detailed discussion of such forms).
Consider an admissible spectral triple (A∞,H,D) and moreover, make
the assumption of Lemma 2.6, i.e. assume that t 7→ eitDxe−itD is norm-
differentiable at t = 0 for all x in the ∗-algebra B generated by A∞ and
[D,A∞].
Lemma 2.19 In the notation of Lemma 2.6, we have the following (where
b, c ∈ A∞):
d∗D(dD(b)c) = −
(bL(c)− L(b)c− L(bc)) . (3)
Proof:
Denote by χ(b, c) the right hand side of euqation (3) and fix any a ∈ A∞.
Using the facts the the functional τ is a faithful trace on the ∗-algebra B,
L = −d∗DdD and that [D,X] = 0 for any X in B, we have,
τ(a∗χ(b, c))
{τ(a∗bL(c)) + τ(ca∗L(b)) + τ(a∗L(bc))}
{τ([D, a∗b][D, c]) − τ([D, ca∗][D, b])− τ([D, a∗][D, bc])}
{τ(a∗[D, b][D, c]) − τ([D, c]a∗[D, b])− τ(c[D, a∗][D, b])− τ([D, a∗][D, b]c)}
= −τ([D, a∗][D, b]c)
= τ([D, a]∗[D, b]c)
= 〈dD(a), dD(b)c〉
= τ(a∗(d∗D(dD(b)c))).
From this, we get the following by a simple computation:
〈adD(b), a
′dD(b
′)〉 = −
τ(b∗Ψ(a∗a, b′)), (4)
for a, b, a′, b′ ∈ A∞, and where Ψ(x, y) := L(xy)−L(x)y+xL(y). Now, let us
denote the quantum isometry group of the given spectral triple (A∞,H,D)
by (G,∆, α). Let A0 denote the ∗-algebra generated by A
0 , G0 denote ∗-
algebra of G generated by matrix elements of irreducible representations.
Clearly, α : A0 → A0 ⊗alg G0 is a Hopf-algebraic action of G0 on A0. Define
Ψ̃ : (A0 ⊗alg G0)× (A0 ⊗alg G0) → A0 ⊗alg G0 by
Ψ̃((x⊗ q), (x′ ⊗ q′)) := Ψ(x, x′)⊗ (qq′).
It follows from the relation (L ⊗ id) ◦ α = α ◦ L on A0 that
Ψ̃(α(x), α(y)) = α(Ψ(x, y)). (5)
We now define a linear map α(1) from the linear span of {adD(b) : a, b ∈ A0}
to H1D ⊗ G by setting
α(1)(adD(b)) :=
i dD(b
j )⊗ a
where for any x ∈ A0 we write α(x) =
i ∈ A0⊗algG0 (summation
over finitely many terms). We shall sometimes use the Sweedler convention
of writing the above simply as α(x) = x(1) ⊗ x(2). It then follows from the
identities (4) and (5), and also the fact that (τ ⊗ id)(α(a)) = τ(a)1 for all
a ∈ A0 that
〈adD(b), a
′dD(b
(τ ⊗ id)(α(b∗)Ψ̃(α(a∗a′), α(b′)))
(τ ⊗ id)(α(b∗)α(Ψ(a∗a′, b′)))
(τ ⊗ id)(α(b∗Ψ(a∗a′, b′)))
τ(b∗Ψ(a∗a′, b′))1G
= 〈adD(b), a
′dD(b
′)〉1G .
This proves that α(1) is indeed well-defined and extends to a G-linear isom-
etry on H1D ⊗ G, to be denoted by U
(1), which sends (adD(b)) ⊗ q to
α(1)(adD(b))(1 ⊗ q), a, b ∈ A0, q ∈ G. Moreover, since the linear span
of α(A∞0 )(1 ⊗ G) is dense in H
D ⊗ G, it is easily seen that the range of the
isometry U (1) is the whole of H1D ⊗G, i.e. U
(1) is a unitary. In fact, from its
definition it can also be shwon that U (1) is a unitary representation of the
compact quantum group G on H1D.
In a similar way, we can construct unitary representation U (n) of G on
the Hilbert space of n-forms for any n ≥ 1, by defining
U (n)((a0dD(a1)dD(a2)...dD(an))⊗q) = a
0 dD(a
1 )...dD(a
n )⊗(a
1 ...a
n q), ai ∈ A
(using Sweedler convention) and verifying that it extends to a unitary.
We also denote by U (0) the unitary representation α̃ on H0D discussed be-
fore. Finally, we have a unitary representation U =
n≥0 U
(n) of G on
H̃ :=
D, and also extend dD as a closed densely defined operator on H̃
in the obvious way, by defining dD(a0dD(a1)...dD(an)) = dD(a0)...dD(an).
It is now straightforward to see the following:
Theorem 2.20 The operator D′ := dD+d
D is equivariant in the sense that
U(D′ ⊗ 1) = (D′ ⊗ 1)U .
We point out that there is a natural representation π of A on H̃ given
by π(a)(a0dD(a1)...dD(an)) = aa0dD(a1)...dD(an), and (π(A
∞), H̃,D′) is
indeed a spectral triple, which is G-equivariant.
Although the relation between spectral properties of D and D′ is not
clear in general, in many cases of interest (e.g. when there is an underlying
type (1, 1) spectral data in the sense of [10]) these two Dirac operators are
closely related. As an illustration, consider the canonical spectral on the
noncommutative 2-torus Aθ, which is discussed in some details in the next
section. In this case, the Dirac operator D acts on L2(Aθ, τ) ⊗ C
2, and it
can easily be shown (see [10]) that the Hilbert space of forms is isomorphic
with L2(Aθ, τ)⊗C
4 ∼= L2(Aθ)⊗C
2; thus D′ is essentially same as D in this
case.
3 Examples and computations
We give some simple yet interesting explicit examples of quantum isometry
groups here. However, we give only some computational details for the first
example, and for the rest, the reader is referred to a companion article ([3]).
Example 1 : commutative tori
Consider M = T, the one-torus, with the usual Riemannian structure. The
∗-algebra A∞ = C∞(M) is generated by one unitary U , which is the multi-
plication operator by z in L2(T). The Laplacian is given by L(Un) = −n2Un.
If a compact quantum group (S,∆S) acts on A
∞ smoothly, let An, n ∈ Z be
elements of S such that α0(U) =
n ⊗An (here α0 : A
∞ → A∞ ⊗alg S
is the S-action on A∞). Note that this infinite sum converges at least
in the topology of the Hilbert space L2(T) ⊗ L2(S), where L2(S) denotes
the GNS space for the Haar state of S. It is clear that the condition
(L ⊗ id) ◦ α0 = α0 ◦ L forces to have An = 0 for all but n = ±1. The
conditions α0(U)α0(U)
∗ = α0(U)
∗α0(U) = 1 ⊗ 1 further imply the follow-
A∗1A1 +A
−1A−1 = 1 = A1A
1 +A−1A
A∗1A−1 = A
−1A1 = A1A
−1 = A−1A
1 = 0.
It follows that A±1 are partial isometries with orthogonal domains and
ranges. Say, A1 has domain P and range Q. Hence the domain and
range of A−1 are respectively 1 − P and 1 − Q. Consider the unitary
V = A + B, so that V P = A, V (1 − P ) = B. Now, from the fact that
(L⊗ id)(α0(U
2)) = α0(L(U
2)) it is easy to see that the coefficient of 1⊗1 in
the expression of α0(U)
2 must be 0, i.e. AB+BA = 0. From this, it follows
that V and P commute and therefore P = Q. By straightforward calculation
using the facts that V is unitary, P is a projection and V and P commute, we
can verify that α0 given by α0(U) = U ⊗V P +U
−1⊗V (1−P ) extends to a
∗-homomorphsim from A∞ to A∞⊗C∗(V, P ) satisfying (L⊗id)◦α0 = α0◦L.
It follows that the C∗ algebra QISO(T) is commutative and generated by a
unitary V and a projection P , or equivalently by two partial isometries A,
B such that A∗A = AA∗, B∗B = BB∗, AB = BA = 0. So, as a C∗ algebra
it is isomorphic with C(T) ⊕ C(T) ∼= C(T × Z2). The coproduct (say ∆0)
can easily be calculated from the requirement of co-associativity, and the
Hopf algebra structure of QISO(T) can be seen to coincide with that of
the semi-direct product of T by Z2, where the generator of Z2 acts on T by
sending z 7→ z̄.
We summarize this in form of the following.
Theorem 3.1 The universal quantum group of isometries QISO(T) of the
one-torus T is isomorphic (as a quantum group) with C(T >⊳Z2) = C(ISO(T)).
We can easily extend this result to higher dimensional commutative tori,
and can prove that the quantum isometry group coincides with the classical
isometry group. This is some kind of rigidity result, and it will be interest-
ing to investigate the nature of quantum isometry groups of more general
classical manifolds.
Example 2 : Noncommutative torus; holomorphic isomrtries
Next we consider the simplest and well-known example of noncommutative
manifold, namely the noncommutative two-torus Aθ, where θ is a fixed
irrational number (see [6]). It is the universal C∗ algebra generated by
two unitaries U and V satisfying the commutation relation UV = λV U ,
where λ = e2πiθ. There is a canonical faithful trace τ on Aθ given by
τ(UmV n) = δmn. We consider the canonical spectral triple (A
∞,H,D),
where A∞ is the unital ∗-algebra spanned by U, V , H = L2(τ)⊕ L2(τ) and
D is given by
0 d1 + id2
d1 − id2 0
where d1 and d2 are closed unbounded linear maps on L
2(τ) given by
mV n) = mUmV n, d2(U
mV n) = nUmV n. It is easy to compute the
space of one-forms Ω1D (see [4], [10], [6]) and the Laplacian L = −d
∗d is
given by L(UmV n) = −(m2 + n2)UmV n. For simplicity of computation,
instead of the full quantum isometry group we at first concentrate on an
interesting quantum subgroup G = QISOhol(A∞,H,D), which is the uni-
versal quantum group which leaves invariant the subalgebra ofA∞ consisting
of polynomials in U , V and 1, i.e. span of UmV n with m,n ≥ 0. The proof
of existence and uniqueness of such a universal quantum group is more or
less identical to the proof of existence and uniqueness of QISO. We call G
the quantum group of “holomorphic” isometries, and observe in the theorem
stated below without proof (see [3]) that this quantum group is nothing but
the quantum double torus studied in [11].
Theorem 3.2 Consider the following co-product ∆B on the C
∗ algebra B =
C(T2)⊕A2θ, given on the generators A0, B0, C0,D0 as follows ( where A0,D0
correspond to C(T2) and B0, C0 correspond to A2θ)
∆B(A0) = A0 ⊗A0 + C0 ⊗B0, ∆B(B0) = B0 ⊗A0 +D0 ⊗B0,
∆B(C0) = A0 ⊗ C0 +C0 ⊗D0, ∆B(D0) = B0 ⊗ C0 +D0 ⊗D0.
Then (B,∆0) is a compact quantum group and it has an action α0 on Aθ
given by
α0(U) = U ⊗A0 + V ⊗B0, α0(V ) = U ⊗C0 + V ⊗D0.
Moreover, (B,∆B) is isomorphic (as quantum group) with G = QISO
hol(A∞,H,D).
We refer to [3] for a proof of the above result, and to [11] for the computation
of the Haar stat and representation theory of the compact quantum group
Example 3 : Noncommutative Torus; full quantum isometry group
By similar but somewhat tedious calculations (see [3]) one can also describe
explicitly the full quantum isometry group QISO(A∞,H,D). It is as a
C∗ algebra has eight direct summands, four of which are isomorphic with
the commutative algebra C(T2), and the other four are irrational rotation
algebras.
Theorem 3.3 QISO(Aθ) = ⊕
∗(Uk1, Uk2) (as a C
∗ algebra), where for
odd k, Uk1, Uk2 are the two commuting unitary generators of C(T
2), and for
even k, Uk1Uk2 = exp(4πiθ)Uk2Uk1, i.e. they generate A2θ. The (co)-action
on the generators U, V (say) of Aθ are given by the following :
α0(U) = U⊗(U11+U31)+V⊗(U52+U62)+U
−1⊗(U21+U41)+V
−1⊗(U72+U82),
α0(V ) = U⊗(U51+U71)+V⊗(U12+U22)+U
−1⊗(U61+U81)+V
−1⊗(U32+U42).
From the co-associativity condition, the co-product of QISO(Aθ) can easily
be calculated. For the detailed description of the coproduct, counit, an-
tipode and study of the representation theory of QISO(Aθ), the reader is
referred to [3]. It is interesting to mention here that the quantum isometry
group of Aθ is a Rieffel type deformation of the isometry group (which is
same as the quantum isometry group) of the commutative two-torus. The
commutative two-torus is a subgroup of its isometry group, but when the
isometry group is deformed into QISO(Aθ), the subgroup relation is not
respected, and the deformation of the commutative torus, which is A2θ, sits
in QISO(Aθ) just as a C
∗ subalgebra (in fact a direct summand) but not
as a quantum subgroup any more. This perhaps provides some explanation
of the non-existence of any Hopf algebra structure on the noncommutative
torus.
Acknowledgement : The author would like to thank P. Hajac for draw-
ing his attention to the article [11], and S.L. Woronowicz for many valuable
comments and suggestions which led to substantial improvement of the pa-
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0704.0042 | General System theory, Like-Quantum Semantics and Fuzzy Sets | Microsoft Word - Like-QuantumSemantics.doc
General System theory, Like-Quantum Semantics and Fuzzy Sets
Ignazio Licata
Isem, Institute for Scientific Methodology, Pa, Italy
[email protected]
Abstract: It is outlined the possibility to extend the quantum formalism in relation to the
requirements of the general systems theory. It can be done by using a quantum semantics
arising from the deep logical structure of quantum theory. It is so possible taking into account the
logical openness relationship between observer and system. We are going to show how
considering the truth-values of quantum propositions within the context of the fuzzy sets is here
more useful for systemics . In conclusion we propose an example of formal quantum coherence.
Key-words: Quantum Theory; Fuzzy Sets; System Theory; Syntax and Semantics of Scientific
Theories; Logical Openness.
Published in Systemics of Emergence. Research and Development, Minati G., Pessa E., Abram M.,
Springer, 2006, pages 723-734.
1.The role of syntactics and semantics in general system theory
The omologic element breaks specializations up, forces taking into account different things at the same time, stirs up the
interdependent game of the separated sub-totalities, hints at a broader totality whose laws are not the ones of its
components. In other words, the omologic method is an anti-separatist and reconstructive one, which thing makes it
unpleasant to specialists.
F. Rossi-Landi 1985
The systemic-cybernetic approach ( Wiener, 1961; von Bertalannfy,1968; Klir, 1991) requires a
careful evaluation of epistemology as the critical praxis internal to the building up of the scientific
discourse. That is why the usual referring to a “connective tissue” shared in common by different
subjects could be misleading. As a matter of fact every scientific theory is the outcome of a
complex conceptual construction aimed to the problem peculiar features, so what we are
interested in is not a framework shaping an abstract super-scheme made by the “filtering” of the
particular sciences, but a research focusing on the global and foundational characteristics of
scientific activity in a trans-disciplinary perspective. According to such view, we can understand
the General System Theory (GST) by the analogy to metalogic. It deals with the possibilities and
boundaries of various formal systems to a more higher degree than any specific structure.
A scientific theory presupposes a certain set of relations between observer and system, so
GST has the purpose to investigate the possibility of describing the multeity of system-observer
relationships. The GST main goal is delineating a formal epistemology to study the scientific
knowledge formation, a science able to speak about science. Succeeding to outline such
panorama will make possible analysing those inter-disciplinary processes which are more and
more important in studying complex systems and they will be guaranteed the “transportability”
conditions of a modellistic set from a field to another one. For instance, during a theory developing,
syntax gets more and more structured by putting univocal constraints on semantics according to
the operative requirements of the problem. Sometimes it can be useful generalising a syntactic tool
in a new semantic domain so as to formulate new problems. Such work, a typically trans-
disciplinary one, can only be done by the tools of a GST able to discuss new relations between
syntactics (formal model) and semantics ( model usage). It is here useful to consider again the
omologic perspective, which not only identifies analogies and isomorphisms in pre-defined
structures, but aims to find out a structural and dynamical relation among theories to an higher
level of analysis, so providing new use possibilities (Rossi-Landi, 1985). Which thing is particularly
useful in studying complex systems, where the very essence of the problem itself makes a
dynamic use of models necessary to describe the emergent features of the system (Minati &
Brahms, 2002; Collen, 2002).
We want here to briefly discuss such GST acceptation, and then showing the possibility of
modifying the semantics of Quantum Mechanics (QM) so to get a conceptual tool fit for the
systemic requirements.
2. Observer as emergence surveyor and semantic ambiguity solver
What we look at is not Nature in itself, but Nature unveiling to our questioning methods.
W. Heisenberg, 1958
A very important and interesting question in system theory can be stated as follows: given a set
of measurement systems M and of theories T related to a system S, is it always possible to order
them, such that Ti-1 �Ti, where the partial order symbol � is used to denote the relationship
“physically weaker than” ? We shall point out that, in this case, the ith theory of the chain contains
more information than the preceding ones. This consequently leads to a second key question: can
an unique final theory Tf describe exhaustively each and every aspect of system S ? From the
informational and metrical side, this is equivalent to state that all of the information contained in a
system S can be extracted, by means of adequate measurement processes.
The fundamental proposition for reductionism is, in fact, the idea that such a theory chain will
be sufficient to give a coherent and complete description for a system S. Reductionism, in the light
of our definitions, coincides therefore with the highest degree of semantic space “compression”;
each object D ∈ Ti in S has a definition in a theory Ti belonging to the theory chain, and the latter
is - on its turn - related to the fundamental explanatory level of the “final” theory Tf. This implies that
each aspect in a system S is unambiguously determined by the syntax described in Tf. Each
system S can be described at a fundamental level, but also with many phenomenological
descriptions, each of these descriptions can be considered an approximation of the “final” theory.
Anyway, most of the “interesting” systems we deal with cannot be included in this chained-
theory syntax compatibility program: we have to consider this important aspect for a correct
epistemic definition of systems “complexity”. Let us illustrate this point with a simple reasoning,
based upon the concepts of logical openness and intrinsic emergence (Minati, Pessa, Penna,
1998; Licata, 2003b).
Each measurement operation can be theoretically coded on a Turing machine. If a coherent
and complete fundamental description Tf exists, then there will also exist a finite set - or, at most,
countably infinite - of measurement operations M which can extract each and every single
information that describes the system S. We shall call such a measurement set Turing-observer.
We can easily imagine Turing-observer as a robot that executes a series of measurements on a
system. The robot is guided by a program built upon rules belonging to the theory T. It can be
proved, though, that this is only possible for logically closed systems, or at most for systems with a
very low degree of logical openness. When dealing with highly logically open systems, no recursive
formal criterion exists that can be as selective as requested (i.e., automatically choose which
information is relevant to describe and characterize the system, and which one is not), simply
because it is not possible to isolate the system from the environment. This implies that the Turing-
observer hypothesis does not hold for fundamental reasons, strongly related to Zermelo-Fraenkel's
choice axiom and to classical Godel's decision problems. In other words, our robot executes the
measurements always following the same syntactics, whereas the scenario showing intrinsic
emergence is semantically modified. So it is impossible thinking to codify any possible
measurement in a logically open system!
The observer therefore plays a key rule, unavoidable as a semantic ambiguity solver: only the
observer can and will single out intrinsic-observational emergence properties ( Bass &
Emmeche,1997; Cariani, 1991), and subsequently plan adequate measurement processes to
describe what – as a matter of fact- have turned in new systems. System complexity is
structurally bound to logical openness and is, at the same time, both an expression of highly
organized system behaviours (long-range correlations, hierarchical structure, and so on) and an
observer's request for new explanatory models.
So, a GST has to allow - in the very same theoretical context – to deal with the observer as an
emergence surveyor in a logical open system. In particular, it is clear that the observer itself is a
logical open system.
Moreover, it has to be pointed out that the co-existence of many description levels – compatible but
not each other deductible – leads to intrinsic uncertainty situations, linked to the different
frameworks by which a system property can be defined.
3. Like-quantum semantics
I’m not happy with all the analyses that go with just the classical theory, because nature isn’t classical, damm it,
and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful
problem, because it doesn’t look so easy. Thank you.
R. P. Feyman, 1981
When we modify and/or amplify a theory so as to being able to speak about different systems
from the ones they were fitted for, it could be better to look at the theory deep structural features so
as to get an abstract perspective able to fulfil the omologic approach requirements, aiming to point
out a non-banal conceptual convergence.
As everybody knows, the logic of classical physics is a dichotomic language (tertium non
datur), relatively orthocomplemented and able to fulfil the weak distributivity relations by the logical
connectives AND/OR. Such features are the core of the Boolean commutative elements of this
logic because disjunctions and conjunctions are symmetrical and associative operations. We shall
here dwell on the systemic consequences of these properties. A system S can get or not to get a
given property P. Once we fix the P truth-value it is possible to keep on our research over a new
preposition P subordinated to the previous one’s truth-value. Going ahead, we add a new piece of
information to our knowledge about the system. So the relative orthocomplementation axiom
grants that we keep on following a successions of steps, each one making our uncertainty about
the system to diminish or, in case of a finite amount of steps, to let us defining the state of the
system by determining all its properties. Each system’s property can be described by a countable
infinity of atomic propositions. So, such axiom plays the role of a describable axiom for classical
systems.
The unconstrained use of such kind of axiom tends to hide the conceptual problems spreading
up from the fact that every description implies a context, as we have seen in the case of Turing-
observer analysis, and it seems to imply that systemic properties are independent of the observer,
it surely is a non-valid statement when we deal with open logical systems. In particular, the
Boolean features point out that it is always possible carrying out exhaustively a synchronic
description of the properties of a systems. In other words, every question about the system is not
depending on the order we ask it and it is liable to a fixed answer we will indicate as 0- false / 1-
true. It can be suddenly noticed that the emergent features otherwise get a diachronic nature and
can easily make such characteristics not taken for granted. By using Venn diagrams it is possible
providing a representation of the complete descriptiveness of a system ruled by classical logics. If
the system’s state is represented by a point and a property of its by a set of points, then it is always
possible a complete “blanketing” of the universal set I, which means the always universally true
proposition. (see fig. 1).
The quantum logics shows deep differences which could be extremely useful for our goals
(Birkhoff & von Neumann, 1936; Piron, 1964). At the beginning it was born to clarify some QM’s
counter-intuitive sides , later it has developed as an autonomous field greatly independent from the
matters which gave birth to it. We will abridge here the formal references to an essential survey,
focusing on some points of general interest in systemics.
The quantum language is a non-Boolean orthomodular structure, which is to say it is relatively
orthocomplemented but non-commutative, for the crack down of the distributivity axiom. Such thing
comes naturally from the Heisenberg Indetermination Principle and binds the truth- value of an
assertion to the context and the order by which it has been investigated (Griffiths, 1995). A well-
known example is the one of a particle’s spin measurement along a given direction. In this case we
deal with semantically well defined possibilities and yet intrinsically uncertain. Let put xΨ
the spin measurement along the direction x. For the indetermination principle the value yΨ will be
totally uncertain, yet the proposition yΨ =0 ∨ yΨ =1 is necessarily true. In general, if P is a
proposition , (-P ) its negation and Q the property which does not commute with P, then we will get
a situation that can be represented by a “patchy” blanketing of the set I (see fig.2).
Such configuration finds its essential meaning just in its relation with the observer. So we can
state that when a situation can be described by a quantum logics, a system is never completely
defined a priori. The measurement process by which the observer’s action takes place is a choice
fixing some system’s characteristics and letting other ones undefined. It happens just for the nature
itself of the observer-system inter-relationship. Each observation act gives birth to new descriptive
possibilities. The proposition Q – in the above example – describes properties that cannot be
defined by any implicational chain of propositions P. Since the intrinsic emergence cannot be
regarded as a system property independent of the observer action- as in naïve classical
emergentism - , Q can be formally considered the expression of an emergent property. Now we are
strongly tempted to define as emergent the undefined proposition of quantum-like anti-
commutative language. In particular, it can be showed that a non-Boolean and irreducible
orthomodular language arises infinite propositions. It means that for each couple of propositions P1
and P2 such that non of them imply the other , there exists infinite propositions Q which imply P1
∨ P2 without necessarily implying the two of them separately: tertium datur. In a sense, the
disjunction of the two propositions gets more information than their mere set-sum, that is the
entirely opposite of what happens in the Boolean case. It is now easy to comprehend the deep
relation binding the anti-commutativity, indetermination principles and system’s holistic global
structure. A system describable by a Boolean structure can be completely “solved” by analysing
the sub-systems defined by a fit decomposition process( Heylighen, 1990; Abram, 2002). On the
contrary, in the anti-commutative case studying any sub-system modifies the entire system in an
irreversible and structural way and produces uncertainty correlated to the gained information,
which think makes absolutely natural extending the indetermination principles to a big deal of
spheres of strong interest for systemics (Volkenshtein , 1988).
A particularly key-matter is how to conceptually managing the infinite cardinality of emergent
propositions in a lik-quantum semantics. As everybody knows traditional QM refers to the
frequentistic probability worked out within the Copenhagen Interpretation (CIQM). It is essentially a
sub specie probabilitatis Boolean logics extension. The values between [ ]1,0 - i.e. between the
completely and always true proposition I and the always false one O – are meant as expectation
values, or the probabilities associated to any measurable property. Without dwelling on the
complex – and as for many questions still open – debate on QM interpretation, we can here ask if
the probabilistic acception of truth-values is the fittest for system theory. As it usually happens
when we deal with trans-disciplinary feels, it will bring us to add a new, and of remarkable interest
for the “ordinary” QM too, step to our search.
4. A Fuzzy Interpretation of Quantum Languages
A slight variation in the founding axioms of a theory can give way to huge changings on the frontier.
S. Gudder, 1988
The study of the structural and logical facets of quantum semanics does not provide any
necessary indications about the most suitable algebraic space to implement its own ideas. One of
the thing which made a big merit of such researches has been to put under discussion the key role
of Hilbert space. In our approach we have kept the QM “internal” problems and its extension to
systemic questions well separated. Anyway, the last ones suggest an interpretative possibility
bounded to fuzzy logic, which thing can considerably affect the traditional QM too. The fuzzy set
theory is , in its essence, a formal tool created to deal with information characterized with
vagueness and indeterminacy. The by-now classical paper of Lotfi Zadeh (Zadeh, 1965) brings to
a conclusion an old tradition of logics, which counts Charles S. Peirce, Jan C. Smuts, Bertrand
Russell, Max Black and Ian Lukasiewicz among its forerunners. At the core of the fuzzy theory lies
the idea that an element can belong to a set to a variable degree of membership; the same goes
for a proposition and its variable relation to the true and false logical constants. We underline here
two aspects of particular interest for our aims. The fuzziness’ definition concerns single elements
and properties, but not a statistical ensemble, so it has to be considered a completely different
concept from the probability one, it should –by now- be widely clarified (Mamdani, 1977; Kosko,
1990). A further essential – even maybe less evident – point is that fuzzy theory calls up a non-
algorithmic “oracle”, an observator (i.e. a logical open system and a semantic ambiguity solver) to
make a choice as for the membership degree. In fact, the most part of the theory in its structure is
free-model; no equation and no numerical value create constraints to the quantitative evaluation,
being the last one the model builder’s task. There consequently exists a deep bound between
systemics and fuzziness successfully expressed by the Zadeh’s incompatibility principle (Zadeh,
1972) which satisfies our requirement for a generalized indeterminacy principle. It states that by
increasing the system complexity (i.e. its logical openness degree), it will decrease our ability to
make exact statements and proved predictions about its behaviour. There already exists many
examples of crossing between fuzzy theory and QM (Dalla Chiara, Giuntini, 1995; Cattaneo, Dalla
Chiara, Giuntini 1993). We want here to delineate the utility of fuzzy polyvalence for systemic
interpretation of quantum semantics.
Let us consider a complex system, such as a social group, a mind and a biological organism.
Each of these cases show typical emergent features owed both to the interaction among its
components and the inter-relations with the environment. An act of the observer will fix some
properties and will let some others undetermined according to a non-Boolean logic. The recording
of such properties will depend on the succession of the measurement acts and their very nature.
The kind of complexity into play, on the other hand, prevents us by stating what the system state is
so as to associate to the measurement of a property an expectation probabilistic value. In fact, just
the above-mentioned examples are related to macroscopic systems for which the probabilistic
interpretation of QM is patently not valid. Moreover, the traditional application of the probability
concept implies the notion of “possible cases”, and so it also implies a pre-defined knowledge of
systems’ properties. However, the non-commutative logical structure here outlined does not
provide any cogent indication on probability usage.
Therefore, it would be proper to look at a fuzzy approach so to describe the measurement acts.
We can state that given a generic system endowed with high logical openness and an indefinite set
of properties able of describing it, each of them will belong to the system in a variable degree.
Such viewpoint expressing the famous theorem of fuzzy “subsetness” – also known as “the whole
into the part” principle – could seem to be too strong , indeed it is nothing else than the most
natural expression of the actual scientific praxis facing intrinsic emergent systems. At the
beginning, we have at our disposal indefinite information progressively structuring thanks to the
feedback between models and measurements. It can be shown that any logically open model of
degree n – where n is an integer – will let a wide range of properties and propositions
indeterminate (the Qs in fig. 2).The above-mentioned model is a “static” approximation of a
process showing aspects of variable closeness and openness. The latter ones varies in time,
intensity, different levels and context. It is remarkable pointing out how such systems are “flexible”
and context-sensitive, change the rules and make use of “contradictions” . This point has to be
stressed to understand the link between fuzzy logic and quantum languages. By increasing the
logical openness and the unsharp properties of a system, it will be less and less fit to be described
by a Boolean logic. It brings as a consequence that for a complex system the intersection between
a set (properties, propositions) and its complement is not equal to the empty set, but it includes
they both in a fuzzy sense. So we get a polyvalent semantic situation which is well fitted for being
described by a quantum language. As for our systemic goal it is the probabilistic interpretation to
be useless, so we are going to build a fuzzy acception of the semantics of the formalism. In our
case, given a system S and a property Q,, let Ψ be a function which associates Q to S, the
expression ( ) [ ]1,0∈Ψ QS has not to be meant as a probability value, but as a degree of
membership. Such union between the non-commutative sides of quantum languages and fuzzy
polyvalence appears to be the most suitable and fecund for systemics.
Let us consider the traditional expression of quantum coherence (the property expressing the
QM global and non-local characteristics, i.e. superposition principle, uncertainty, interference of
probabilities), 2211 Ψ+Ψ=Ψ aa . In the fuzzy interpretation, it means that the properties 1Ψ e 2Ψ
belong to Ψ with degrees of membership 1a e 2a respectively. In other words, for complex
systems the Schrödinger’s cat can be simultaneously both alive and dead ! Indeed the recent
experiments with SQUIDs and the other ones investigating the so-called macroscopic quantum
states suggest a form of macro-realism quite close to our fuzzy acception (Leggett, 1980; Chiatti,
Cini, Serva, 1995). It can provide in nuce an hint which could show up to be interesting for the QM
old-questioned interpretative problems.
In general, let x be a position coordinate of a quantum object and Ψ its wave function,
( ) dVx 2Ψ is usually meant as the probability of finding the particle in a region dV of space. On the
contrary, in the fuzzy interpretation we will be compelled to look at the Ψ square modulus as the
degree of membership of the particle to the region dV of space. How unusual it may seem, such
idea has not to be regarded thoughtlessly at. As a matter of fact, in Quantum Field Theory and in
other more advanced quantum scenarios, a particle is not only a localized object in the space, but
rather an event emerging from the non-local networks elementary quantum transition (Licata,
2003a). Thus, the measurement is a “defuzzification” process which, according to the stated,
reduces the system ambiguity by limiting the semantic space and by defining a fixed information
quantity.
If we agree with such interpretation we will easily and immediately realize that we will able to
observate quantum coherence behaviours in non-quantum and quite far from the range of Plank’s
h constant situations. We reconsider here a situation owed to Yuri Orlov (Orlov, 1997).
Let us consider a Riemann’s sphere (Dirac, 1947) – see fig. 3 - and let assume that each point
on the sphere represents a single interpretation of a given situation, i.e. the assigning of a
coherent set of truth-values to a given proposition. Alternatively, we can consider the choosing of
a vector v from the centre O to a point on the sphere as a logical definition of a world. If we
choose a different direction, associated to a different vector w , we can now set the problem about
the meaning of the amplitude between the logical descriptions of the two worlds. It is known that
such amplitude is expressed by ( )ϑcos121 + , where ϑ is the angle between the two
interpretations. The amplitude corresponds to a superposition of worlds, so producing the typical
interference patterns which in vectorial terms are related to v
w . In this case, the traditional use of
probability is not necessary because our knowledge of one of the two world with probability equal
to p =1 (certainity), say nothing us about the other one probability. An interpretation is not a
quantum object in the proper sense, and yet we are forced to formally introduce a wave-function
and interference terms whose role is very obscure a one. The fuzzy approach, instead, clarifies
the quantum semantics of this situation by interpreting interference as a measurement where the
properties of the world wv wv Ψ+Ψ are owed to the global and indissoluble (non-local)
contribution of the v and w overlapping.
In conclusion, the generalized using of quantum semantics associated to new interpretative
possibilities gives to systemics a very powerful tool to describe the observator-environment relation
and to convey the several, partial attempts - till now undertaken - of applying the quantum
formalism to the study of complex systems into a comprehensive conceptual root.
ACKNOWLEDGEMENTS
A special thank to Prof. G. Minati for his kindness and his supporting during this paper drafting. I
owe a lot to the useful discussing on structural Quantum Mechanics and logics with my good
friends Prof. Renato Nobili (who let me use the figs. 1 and 2 from his book “Dai Quark alla Mente”,
to be published) and Prof. Eliano Pessa. Dedicated to M.V.
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|
0704.0043 | Nonequilibrium entropy limiters in lattice Boltzmann methods | Nonequilibrium entropy limiters
in lattice Boltzmann methods 1
R. A. Brownlee, A. N. Gorban ∗, J. Levesley
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Abstract
We construct a system of nonequilibrium entropy limiters for the lattice Boltz-
mann methods (LBM). These limiters erase spurious oscillations without blurring
of shocks, and do not affect smooth solutions. In general, they do the same work
for LBM as flux limiters do for finite differences, finite volumes and finite elements
methods, but for LBM the main idea behind the construction of nonequilibrium en-
tropy limiter schemes is to transform a field of a scalar quantity — nonequilibrium
entropy. There are two families of limiters: (i) based on restriction of nonequilibrium
entropy (entropy “trimming”) and (ii) based on filtering of nonequilibrium entropy
(entropy filtering). The physical properties of LBM provide some additional bene-
fits: the control of entropy production and accurate estimate of introduced artificial
dissipation are possible. The constructed limiters are tested on classical numeri-
cal examples: 1D athermal shock tubes with an initial density ratio 1:2 and the
2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse
100 × 100 grid. All limiter constructions are applicable for both entropic and non-
entropic quasiequilibria.
Key words: lattice Boltzmann method, numerical regularisation, entropy
PACS: 47.11.Qr, 47.20.-k, 47.11.-j, 51.10.+y
1 Introduction
In 1959, S.K. Godunov [17] demonstrated that a (linear) scheme for a PDE
could not, at the same time, be monotone and second order accurate. Hence,
∗ Corresponding author.
Email addresses: [email protected] (R. A. Brownlee),
[email protected] (A. N. Gorban), [email protected]
(J. Levesley).
1 This work is supported by EPSRC grant number GR/S95572/01.
Preprint submitted to Physica A 24 October 2018
http://arxiv.org/abs/0704.0043v1
we should choose between spurious oscillation in high order non-monotone
schemes and additional dissipation in first order schemes. Flux limiter schemes
are invented to combine high resolution schemes in areas with smooth fields
and first order schemes in areas with sharp gradients.
The idea of flux limiters can be illustrated by computation of the flux F0,1
of the conserved quantity u between a cell marked by 0 and one of two its
neighbour cells marked by ±1:
F0,1 = (1− φ(r))f low0,1 + φ(r)f
0,1 ,
where f low0, 1 , f
0, 1 are low and high resolution scheme fluxes, respectively, r =
(u0 − u−1)/(u1 − u0), and φ(r) ≥ 0 is a flux limiter function. For r close to 1,
the flux limiter function φ(r) should be also close to 1.
Many flux limiter schemes have been invented during the last two decades [43].
No particular limiter works well for all problems, and a choice is usually made
on a trial and error basis.
Below are several examples of flux limiter functions:
φmm(r) = max [0,min (r, 1)] (minmod, [36]);
φos(r) = max [0,min (r, β)] , (1 ≤ β ≤ 2) (Osher, [10]);
φmc(r) = max [0,min (2r, 0.5(1 + r), 2)] (monotonised central [42]);
φsb(r) = max [0,min (2r, 1) ,min (r, 2)] (superbee, [36]);
φsw(r) = max [0,min (βr, 1) , (r, β)] , (1 ≤ β ≤ 2) (Sweby, [40]).
The lattice Boltzmann method has been proposed as a discretization of Boltz-
mann’s kinetic equation and is now in wide use in fluid dynamics and beyond
(for an introduction and review see [38]). Instead of fields of moments M , the
lattice Boltzmann method operates with fields of discrete distributions f . This
allows us to construct very simple limiters that do not depend on slopes or
gradients.
All the limiters we construct are based on the representation of distributions
f in the form:
f = f ∗ + ‖f − f ∗‖
f − f ∗
‖f − f ∗‖
where f ∗ is the correspondent quasiequilibrium (conditional equilibrium) for
given moments M , f − f ∗ is the nonequilibrium “part” of the distribution,
which is represented in the form “norm×direction” and ‖f − f ∗‖ is the norm
of that nonequilibrium component (usually this is the entropic norm). Lim-
iters change the norm of the nonequilibrium component f − f ∗, but do not
touch its direction or the equilibrium. In particular, limiters do not change the
macroscopic variables, because moments for f and f ∗ coincide. All limiters we
use are transformations of the form
f 7→ f ∗ + φ× (f − f ∗) (1)
with φ > 0. If f − f ∗ is too big, then the limiter should decrease its norm.
The outline of the paper is as follows. In Sec. 2 we introduce the notions and
notations from lattice Boltzmann theory we need, in Sec. 3 we elaborate the
idea of entropic limiters in more detail and construct several nonequilibrium
entropy limiters for LBM, in Sec. 4 some numerical experiments are described:
(1) 1D athermal shock tube examples;
(2) steady state vortex centre locations and observation of first Hopf bifur-
cation in 2D lid-driven cavity flow.
Concluding remarks are given in Sec. 5.
2 Background
The essence of lattice Boltzmann methods was formulated by S. Succi in the
following maxim: “Nonlinearity is local, non-locality is linear” 2 . We should
even strengthen this statement. Non-locality (a) is linear; (b) is exactly and
explicitly solvable for all time steps; (c) space discretization is an exact oper-
ation.
The lattice Boltzmann method is a discrete velocity method. The finite set
of velocity vectors {vi} (i = 1, ...m) is selected, and a fluid is described by
associating, with each velocity vi, a single-particle distribution function fi =
fi(x, t) which is evolved by advection and interaction (collision) on a fixed
computational lattice. The values fi are named populations. If we look at all
lattice Boltzmann models, one finds that there are two steps: free flight for
time δt and a local collision operation.
The free flight transformation for continuous space is
fi(x, t+ δt) = fi(x− viδt, t).
After the free flight step the collision step follows:
fi(x) 7→ Fi({fj(x)}), (2)
2 S. Succi, “Lattice Boltzmann at all-scales: from turbulence to DNA transloca-
tion”, Mathematical Modelling Centre Distinguished Lecture, University of Leices-
ter, Leicester UK, 15th November 2006.
or in the vector form
f(x) 7→ F (f(x)).
Here, the collision operator F is the set of functions Fi({fj}) (i = 1, ...m).
Each function Fi depends on all fj (j = 1, ...m): new values of the populations
fi at a point x are known functions of all previous population values at the
same point.
The lattice Boltzmann chain “free flight → collision → free flight → collision
· · · ” can be exactly restricted onto any space lattice which is invariant with
respect to space shifts of the vectors viδt (i = 1, ...m). Indeed, free flight trans-
forms the population values at sites of the lattice into the population values
at sites of the same lattice. The collision operator (2) acts pointwise at each
lattice site separately. Much effort has been applied to answer the questions:
“how does the lattice Boltzmann chain approximate the transport equation for
the moments M?”, and “how does one construct the lattice Boltzmann model
for a given macroscopic transport phenomenon?” (a review is presented in
book [38]).
In our paper we propose a universal construction of limiters for all possible
collision operators, and the detailed construction of Fi({fj}) is not important
for this purpose. The only part of this construction we use is the local equilibria
(sometimes these states are named conditional equilibria, quasiequilibria, or
even simpler, equilibria).
The lattice Boltzmann models should describe the macroscopic dynamic, i.e.,
the dynamic of macroscopic variables. The macroscopic variables Mℓ(x) are
some linear functions of the population values at the same point: Mℓ(x) =
imℓifi(x), or in the vector form, M(x) = m(f(x)). The macroscopic vari-
ables are invariants of collisions:
mℓifi =
mℓiFi({fj}) (or m(f) = m(F (f))).
The standard example of the macroscopic variables are hydrodynamic fields
(density–velocity–energy density): {n, nu, E}(x) := ∑i{1, vi, v2i /2}fi(x). But
this is not an obligatory choice. If we would like to solve, by LBM methods,
the Grad equations [22] or some extended thermodynamic equations [25], we
should extend the list of moments (but, at the same time, we should be ready
to introduce more discrete velocities for a proper description of these extended
moment systems). On the other hand, the athermal lattice Boltzmann models
with a shortened list of macroscopic variables {n, nu} are very popular.
The quasiequilibrium is the positive fixed point of the collision operator for
the given macroscopic variablesM . We assume that this point exists, is unique
and depends smoothly on M . For the quasiequilibrium population vector for
given M we use the notation f ∗M , or simply f
∗, if the correspondent value of
M is obvious. We use Π∗ to denote the equilibration projection operation of
a distribution f into the corresponding quasiequilibrium state:
Π∗(f) = f ∗m(f).
For some of the collision models an entropic description of equilibrium is pos-
sible: an entropy density function S(f) is defined and the quasiequilibrium
point f ∗M is the entropy maximiser for given M [26,39].
As a basic example we shall consider the lattice Bhatnagar–Gross–Krook
(LBGK) model with overrelaxation (see, e.g., [3,12,23,28,38]). The LBGK col-
lision operator is
F (f) := Π∗(f) + (2β − 1)(Π∗(f)− f), (3)
where β ∈ [0, 1]. For β = 0, LBGK collisions do not change f , for β = 1/2
these collisions act as equilibration (this corresponds to the Ehrenfests’ coarse
graining [15] further developed in [14,19,20]), for β = 1, LBGK collisions act
as a point reflection with the center at the quasiequilibrium Π∗(f).
It is shown [8] that under some stability conditions and after an initial period
of relaxation, the simplest LBGK collision with overrelaxation [23,38] provides
second order accurate approximation for the macroscopic transport equation
with viscosity proportional to δt(1− β)/β.
The entropic LBGK (ELBM) method [5,20,26,39] differs in the definition
of (3): for β = 1 it should conserve the entropy, and in general has the following
form:
F (f) := (1− β)f + βf̃ , (4)
where f̃ = (1 − α)f + αΠ∗(f). The number α = α(f) is chosen so that the
constant entropy condition is satisfied: S(f) = S(f̃). For LBGK (3), α = 2. Of
course, for ELBM the entropic definition of quasiequilibrium should be valid.
In the low-viscosity regime, LBGK suffers from numerical instabilities which
readily manifest themselves as local blow-ups and spurious oscillations.
The LBM experiences the same spurious oscillation problems near sharp gra-
dients as high order schemes do. The physical properties of the LBM schemes
allows one to construct new types of limiters: the nonequilibrium entropy lim-
iters. In general, they do the same work for LBM as flux limiters do for finite
differences, finite volumes and finite elements methods, but for LBM the main
idea behind the construction of nonequilibrium entropy limiter schemes is to
limit a scalar quantity — nonequilibrium entropy (and not the vectors or ten-
sors of spatial derivatives, as it is for flux limiters). These limiters introduce
some additional dissipation, but all this dissipation could easily be evaluated
through analysis of nonequilibrium entropy production.
Two examples of such limiters have been recently proposed: the positivity
rule [6,31,41] and the Ehrenfests’ regularisation [7]. The positivity rule just
provides positivity of distributions: if a collision step produces negative popu-
lations, then the positivity rule returns them to the boundary of positivity. In
the Ehrenfests’ regularisation, one selects the k sites with highest nonequilib-
rium entropy (the difference between entropy of the state f and entropy of the
corresponding quasiequilibrium state f ∗ at a given space point) that exceed a
given threshold and equilibrates the state in these sites.
The positivity rule and Ehrenfests’ regularisation provide rare, intense and
localised corrections. It is easy and also computationally cheap to organise
more gentle transformation with smooth shift of highly nonequilibrium states
to quasiequilibrium. The following regularisation transformation distributes
its action smoothly: we can just choose in (1) φ = φ(∆S(f)) with sufficiently
smooth function φ(∆S(f)). Here f is the state at some site, f ∗ is the corre-
sponding quasiequilibrium state, S is entropy, and ∆S(f) := S(f ∗)− S(f).
The next step in the development of the nonequilibrium entropy limiters is in
the usage of local entropy filters. The filter of choice here is the median filter: it
does not erase sharp fronts, and is much more robust than convolution filters.
An important problem is: “how does one create nonequilibrium entropy lim-
iters for LBM with non-entropic quasiequilibria?”. We propose a solution
of this problem based on the nonequilibrium Kullback entropy. For entropic
quasiequilibrium the Kullback entropy approach gives the same entropic lim-
iters. In thermodynamics, Kullback entropy belongs to the family of Massieu–
Planck–Kramers functions (canonical or grandcanonical potentials).
3 Nonequilibrium entropy limiters for LBM
3.1 Positivity rule
There is a simple recipe for positivity preservation [6,31,41]: to substitute
nonpositive I
0 (f)(x) by the closest nonnegative state that belongs to the
straight line
λf(x) + (1− λ)Π∗(f(x))| λ ∈ R
defined by the two points, f(x) and corresponding quasiequilibrium. This op-
eration is to be applied pointwise, at points of the lattice where positivity
is violated. The coefficient λ depends on x too. Let us call this recipe the
positivity rule (Fig. 1). This recipe preserves positivity of populations and
probabilities, but can affect accuracy of approximation. The same rule is nec-
F(f )
Positivity fixation
Positivity domain
Fig. 1. Positivity rule in action. The motions stops at the positivity boundary.
essary for ELBM (4) when the positive “mirror state” f̃ with the same entropy
as f does not exists on the straight line (5).
3.2 Ehrenfests’ regularisation
To discuss methods with additional dissipation, the entropic approach is very
convenient. Let entropy S(f) be defined for each population vector f = (fi)
(below we use the same letter S for local in space entropy, and hope that
context will make this notation always clear). We assume that the global
entropy is a sum of local entropies for all sites. The local nonequilibrium
entropy is
∆S(f) := S(f ∗)− S(f), (6)
where f ∗ is the corresponding local quasiequilibrium at the same point.
The Ehrenfests’ regularisation [6,7] provides “entropy trimming”: we moni-
tor local deviation of f from the corresponding quasiequilibrium, and when
∆S(f)(x) exceeds a pre-specified threshold value δ, perform local Ehrenfests’
steps to the corresponding quasiequilibrium: f 7→ f ∗ at those points.
So that the Ehrenfests’ steps are not allowed to degrade the accuracy of LBGK
it is pertinent to select the k sites with highest ∆S > δ. The a posteriori
estimates of added dissipation could easily be performed by analysis of entropy
production in Ehrenfests’ steps. Numerical experiments show (see, e.g., [6,7])
that even a small number of such steps drastically improve stability.
To avoid the change of accuracy order “on average”, the number of sites with
this step should be ≤ O(Nh/L) where N is the total number of sites, h is
the step of the space discretization and L is the macroscopic characteristic
length. But this rough estimate of accuracy in average might be destroyed
by concentration of Ehrenfests’ steps in the most nonequilibrium areas, for
example, in the boundary layer. In that case, instead of the total number of
sites N in O(Nh/L) we should take the number of sites in a specific region.
The effects of concentration could be easily analysed a posteriori.
3.3 Smooth limiters of nonequilibrium entropy
The positivity rule and Ehrenfests’ regularisation provide rare, intense and
localised corrections. Of course, it is easy and also computationally cheap to
organise more gentle transformation with a smooth shift of highly nonequilib-
rium states to quasiequilibrium. The following regularisation transformation
distributes its action smoothly:
f 7→ f ∗ + φ(∆S(f))(f − f ∗). (7)
The choice of function φ is highly ambiguous, for example, φ = 1/(1+α∆Sk)
for some α > 0 and k > 0. There are two significantly different choices: (i)
ensemble-independent φ (i.e., the value of φ depends on local value of ∆S
only) and (ii) ensemble-dependent φ, for example
φ(∆S) =
1 + (∆S/(αE(∆S)))k−1/2
1 + (∆S/(αE(∆S)))k
, (8)
where E(∆S) is the average value of ∆S in the computational area, k ≥ 1,
and α & 1. For small ∆S, φ(∆S) ≈ 1 and for ∆S ≫ αE(∆S), φ(∆S) tends
αE(∆S)/∆S. It is easy to select an ensemble-dependent φ with control
of total additional dissipation.
3.4 Monitoring of total dissipation
For given β, the entropy production in one LBGK step in quadratic approxi-
mation for ∆S is:
δLBGKS ≈ [1− (2β − 1)2]
∆S(x),
where x is the grid point, ∆S(x) is nonequilibrium entropy (6) at point x,
δLBGKS is the total entropy production in a single LBGK step. It would be
desirable if the total entropy production for the limiter δlimS was small relative
to δLBGKS:
δlimS < δ0δLBGKS. (9)
A simple ensemble-dependent limiter (perhaps, the simplest one) for a given
δ0 operates as follows. Let us collect the histogram of the ∆S(x) distribution,
and estimate the distribution density, p(∆S). We have to estimate a value
∆S0 that satisfies the following equation:
p(∆S)(∆S −∆S0) d∆S = δ0[1− (2β − 1)2]
p(∆S)∆S d∆S. (10)
In order not to affect distributions with small expectation of ∆S, we choose
a threshold ∆St = max{∆S0, δ}, where δ is some predefined value (as in
the Ehrenfests’ regularisation). For states at sites with ∆S ≥ ∆St we pro-
vide homothety with quasiequilibrium center f ∗ and coefficient
∆St/∆S (in
quadratic approximation for nonequilibrium entropy):
f(x) 7→ f ∗(x) +
(f(x)− f ∗(x)). (11)
3.5 Median entropy filter
The limiters described above provide pointwise correction of nonequilibrium
entropy at the “most nonequilibrium” points. Due to the pointwise nature,
the technique does not introduce any nonisotropic effects, and provides some
other benefits. But if we involve the local structure, we can correct local non-
monotone irregularities without touching regular fragments. For example, we
can discuss monotone increase or decrease of nonequilibrium entropy as regular
fragments and concentrate our efforts on reduction of “speckle noise” or “salt
and pepper noise”. This approach allows us to use the accessible resource of
entropy change (9) more thriftily.
Among all possible filters, we suggest the median filter. The median is a more
robust average than the mean (or the weighted mean) and so a single very
unrepresentative value in a neighborhood will not affect the median value
significantly. Hence, we suppose that the median entropy filter will work better
than entropy convolution filters.
The median filter considers each site in turn and looks at its nearby neighbours.
It replaces the nonequilibrium entropy value ∆S at the point with the median
of those values ∆Smed, then updates f by the transformation (11) with the
homothety coefficient
∆Smed/∆S. The median, ∆Smed, is calculated by first
sorting all the values from the surrounding neighbourhood into numerical order
and then replacing that being considered with the middle value. For example,
if a point has 3 nearest neighbors including itself, then after sorting we have
3 values ∆S: ∆S1 ≤ ∆S2 ≤ ∆S3. The median value is ∆Smed = ∆S2. For 9
nearest neighbors (including itself) we have after sorting ∆Smed = ∆S5. For
27 nearest neighbors ∆Smed = ∆S14.
We accept only dissipative corrections (those resulting in a decrease of ∆S,
∆Smed < ∆S) because of the second law of thermodynamics. The analogue
of (10) is also useful for acceptance of the most significant corrections.
Median filtering is a common step in image processing [34] for the smoothing
of signals and the suppression of impulse noise with preservation of edges.
3.6 Entropic steps for non-entropic quasiequilibria
Beyond the quadratic approximation for nonequilibrium entropy all the logic of
the above mentioned constructions remain the same. There exists only one sig-
nificant change: instead of a simple homothety (11) with coefficient
∆St/∆S
the transformation (7) should be applied, where the multiplier φ is a solution
of the nonlinear equation
S(f ∗ + φ(f − f ∗)) = S(f ∗)−∆St.
This is essentially the same equation that appears in the definition of ELBM
steps (4).
More differences emerge for LBM with non-entropic quasiequilibria. The main
idea here is to reason that non-entropic quasiequilibria appear only because of
technical reasons, and approximate continuous physical entropic quasiequilib-
ria. This is not an approximation of a density function, but an approximation
of measure, i.e., from the cubature formula:
f(v) ≈
fiδ(v − vi)
ϕ(v)f(v) dv ≈
ϕ(vi)fi.
The discrete populations fi are connected to continuous (and sufficiently
smooth) densities f(v) by cubature weights fi ≈ wif(vi). These weights for
quasiequilibria are found by moment and flux matching conditions [37]. It
is impossible to approximate the BGS entropy
f ln fdv just by discretiza-
tion (to change integration by summation, and continuous distribution f by
discrete fi), because cubature weights appear as additional variables. Never-
theless, the approximate discretization of the Kullback entropy SK [30] does
not change its form:
SK(f) = −
f(v) ln
f ∗(v)
dv ≈ −
fi ln
, (12)
because fi/f
i approximates the ratio of functions f(v)/f
∗(v) and
i fi . . .
gives the integral
f(v) . . .dv approximation. Here, in (12), the state f ∗ is the
quasiequilibrium with the same values of the macroscopic variables as f . More-
over, for given values of the macroscopic variables, SK(f) achieves its maxi-
mum at the point f = f ∗ (both for continuous and for discrete distributions).
The corresponding maximal value is zero. Below, SK is the discrete Kullback
entropy. If the approximate discrete quasiequilibrium f ∗ is non-entropic, we
can use −SK(f) instead of ∆S(f).
For entropic quasiequilibria with perfect entropy the discrete Kullback entropy
gives the same ∆S: −SK(f) = ∆S(f). Let the discrete entropy have the
standard form for an ideal (perfect) mixture [27].
S(f) = −
fi ln
After the classical work of Zeldovich [44], this function is recognised as a
useful instrument for the analysis of kinetic equations (especially in chemical
kinetics [21]). If we define f ∗ as the conditional entropy maximum for given
k mjkfk, then
ln f ∗k =
µjmjk,
where µj(M) are the Lagrange multipliers (or “potentials”). For this entropy
and conditional equilibrium we find
∆S = S(f ∗)− S(f) =
fi ln
, (13)
if f and f ∗ have the same moments, m(f) = m(f ∗). The right hand side
of (13) is −SK(f).
In thermodynamics, the Kullback entropy belongs to the family of Massieu–
Planck–Kramers functions (canonical or grandcanonical potentials). There is
another sense of this quantity: SK is the relative entropy of f with respect to
f ∗ [18,35].
In quadratic approximation,
−SK(f) =
fi ln
(fi − f ∗i )2
3.7 ELBM collisions as a smooth limiter
On the base of numerical tests, the authors of [41] claim that the positivity
rule provides the same results (in the sense of stability and absence/presence
of spurious oscillations) as the ELBM models, but ELBM provides better
accuracy.
For the formal definition of ELBM (4) our tests do not support claims that
ELBM erases spurious oscillations (see below). Similar observation for Burgers
equation was previously published in [4]. We understand this situation in the
following way. The entropic method consists at least of three components:
(1) entropic quasiequilibrium, defined by entropy maximisation;
(2) entropy balanced collisions (4) that have to provide proper entropy bal-
ance;
(3) a method for the solution of the transcendental equation S(f) = S(f̃) to
find α = α(f) in (4).
It appears that the first two items do not affect spurious oscillations at all,
if we solve the equation for α(f) with high accuracy. Additional viscosity
is, potentially, added by explicit analytic formulas for α(f). In order not to
decrease entropy, errors in these formulas always increase dissipation. This
can be interpreted as a hidden transformation of the form (7), where the
coefficients in φ depend also on f ∗.
3.8 Monotonic and double monotonic limiters
Two monotonicity properties are important in the theory of nonequilibrium
entropy limiters:
(1) a limiter should move the distribution to equilibrium: in all cases of (1)
0 ≤ φ ≤ 1. This is the dissipativity condition which means that limiters
never produce negative entropy.
(2) a limiter should not change the order of states on the line: if for two
distributions with the same moments, f and f ′, ∆S(f) > ∆S(f ′) before
the limiter transformation, then the same inequality should hold after the
limiter transformation too. For example, for the limiter (7) it means that
∆S(f ∗ + xφ(∆S(f ∗ + x(f − f ∗))(f − f ∗)) is a monotonically increasing
function of x > 0.
In quadratic approximation,
∆S(f ∗ + x(f − f ∗)) = x2∆S(f),
∆S(f ∗ + xφ(∆S(f ∗ + x(f − f ∗))(f − f ∗)) = x2φ2(x2∆S(f)),
and the second monotonicity condition transforms into the following require-
ment: yφ(y2s) is a monotonically increasing (not decreasing) function of y > 0
for any s > 0.
If a limiter satisfies both monotonicity conditions, we call it “double mono-
tonic”. For example, Ehrenfests’ regularisation satisfies the first monotonicity
condition, but obviously violates the second one. The limiter (8) violates the
first condition for small ∆S, but is dissipative and satisfies the second one in
quadratic approximation for large ∆S. The limiter with φ = 1/(1+α∆Sk) al-
ways satisfies the first monotonicity condition, violates the second if k > 1/2,
and is double monotonic (in quadratic approximation for the second condi-
tion), if 0 < k ≤ 1/2. The threshold limiters (11) are also double monotonic.
Of course, it is not forbidden to use any type of limiters under the local and
global control of dissipation, but double monotonic limiters provide some nat-
ural properties automatically, without additional care.
4 Numerical experiment
To conclude this paper we report some numerical experiments conducted to
demonstrate the performance of some of the proposed nonequilibrium entropy
limiters for LBM from Sec. 3.
4.1 Velocities and quasiequilibria
We will perform simulations using both entropic and non-entropic quasiequi-
libria, but we always work with an athermal LBM model. Whenever we use
non-entropic quasiequilibria we employ Kullback entropy (13).
In 1D, we use a lattice with spacing and time step δt = 1 and a discrete
velocity set {v1, v2, v3} := {0,−1, 1} so that the model consists of static, left-
and right-moving populations only. The subscript i denotes population (not
lattice site number) and f1, f2 and f3 denote the static, left- and right-moving
populations, respectively. The entropy is S = −H , with
H = f1 log(f1/4) + f2 log(f2) + f3 log(f3),
(see, e.g., [27]) and, for this entropy, the local entropic quasiequilibrium state
f ∗ is available explicitly:
f ∗1 =
1 + 3u2
f ∗2 =
(3u− 1) + 2
1 + 3u2
f ∗3 = −
(3u+ 1)− 2
1 + 3u2
where
fi, u :=
vifi. (15)
The standard non-entropic polynomial quasiequilibria [38] are:
f ∗1 =
f ∗2 =
(1− 3u+ 3u2),
f ∗3 =
(1 + 3u+ 3u2).
In 2D, we employ a uniform 9-speed square lattice with discrete velocities
{vi | i = 0, 1, . . . 8}: v0 = 0, vi = (cos((i − 1)π/2), sin((i − 1)π/2)) for i =
1, 2, 3, 4, vi =
2(cos((i − 5)π
), sin((i − 5)π
)) for i = 5, 6, 7, 8. The
numbering f0, f1, . . . , f8 are for the static, east, north, west, south, north-
east, northwest, southwest and southeast-moving populations, respectively.
As usual, the entropic quasiequilibrium state, f ∗, can be uniquely determined
by maximising an entropy functional
S(f) = −
fi log
subject to the constraints of conservation of mass and momentum [2]:
f ∗i = ρWi
1 + 3u2j
2uj +
1 + 3u2j
1− uj
. (17)
Here, the lattice weights, Wi, are given lattice-specific constants: W0 = 4/9,
W1,2,3,4 = 1/9 and W5,6,7,8 = 1/36. Analogously to (15), the macroscopic vari-
ables ρ and u = (u1, u2) are the zeroth and first moments of the distribution
f , respectively. The standard non-entropic polynomial quasiequilibria [38] are:
f ∗i = ρWi
1 + 3viu+
9(viu)
. (18)
4.2 LBGK and ELBM
The governing equations for LBGK are
fi(x+ vi, t+ 1) = f
i (x, t) + (2β − 1)(f ∗i (x, t)−fi(x, t)), (19)
where β = 1/(2ν + 1).
For ELBM (4) the governing equations are:
fi(x+ vi, t+ 1) = (1− β)f ∗i (x, t) + βf̃i(x, t), (20)
with β as above and f̃ = (1−α)f+αf ∗. The parameter, α, is chosen to satisfy
a constant entropy condition. This involves finding the nontrivial root of the
equation
S((1− α)f + αf ∗) = S(f). (21)
To solve (21) numerically we employ a robust routine based on bisection. The
root is solved to an accuracy of 10−15 and we always ensure that the returned
value of α does not lead to a numerical entropy decrease. We stipulate that
if, at some site, no nontrivial root of (21) exists we will employ the positivity
rule instead (Fig. 1).
4.3 Shock tube
The 1D shock tube for a compressible athermal fluid is a standard benchmark
test for hydrodynamic codes. Our computational domain will be the interval
[0, 1] and we discretize this interval with 801 uniformly spaced lattice sites.
We choose the initial density ratio as 1:2 so that for x ≤ 400 we set ρ = 1.0
else we set ρ = 0.5. We will fix the kinematic viscosity of the fluid at ν = 10−9.
4.3.1 Comparison of LBGK and ELBM
In Fig. 2 we compare the shock tube density profile obtained with LBGK
(using entropic quasiequilibria (14)) and ELBM. On the same panel we also
display both the total entropy S(t) :=
x S(x, t) and total nonequilibrium
entropy ∆S(t) :=
x∆S(x, t) time histories. As expected, by construction,
we observe that total entropy is (effectively) constant for ELBM. On the other
hand, LBGK behaves non-entropically for this problem. In both cases we ob-
serve that nonequilibrium entropy grows with time.
As we can see, the choice between the two collision formulas LBGK (19)
or ELBM (20) does not affect spurious oscillation, and reported regularisa-
tion [29] is, perhaps, the result of approximate analytical solution of the equa-
tion (21). Inaccuracy in the solution of (21) can be interpreted as a hidden
nonequilibrium entropy limiter. But it should be mentioned that the entropic
method consists not only of the collision formula, but, what is important, in-
cludes a special choice of quasiequilibrium that could improve stability (see,
e.g., [13]). Indeed, when we compare ELBM with LBGK using either entopic or
standard polynomial quasiequilibria, there appears to be some gain in employ-
ing entropic quasiequilibria (Fig. 3). We observe that the post-shock region
for the LBGK simulations is more oscillatory when polynomial quasiequilibria
are used. In Fig. 3 we have also included a panel with the simulation result-
ing from a much higher viscosity (ν = 3.3333 × 10−2). Here, we observe no
appreciable differences in the results of LBGK and ELBM.
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
Fig. 2. Density and profile of the 1:2 athermal shock tube simulation with ν = 10−9
after 400 time steps using (a) LBGK (19); (b) ELBM (20). In this example, no
negative population are produced by any of the methods so the positivity rule is
redundant. For ELBM in this example, (21) always has a nontrivial root. Total
entropy and nonequilibrium entropy time histories are shown in panels (c), (d) and
(e), (f) for LBGK and ELBM, respectively.
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
0 0.5 1
Fig. 3. Density and velocity profile of the 1:2 isothermal shock tube simula-
tion after 400 time steps using (a) LBGK (19) with polynomial quasiequilib-
ria (16) [ν = 3.3333 × 10−2]; (b) LBGK (19) with entropic quasiequilibria (14)
[ν = 3.3333 × 10−2]; (c) ELBM (20) [ν = 3.3333 × 10−2]; (d) LBGK (19) with
polynomial quasiequilibria (16) [ν = 10−9]; (e) LBGK (19) with entropic quasiequi-
libria (14) [ν = 10−9]; (f) ELBM (20) [ν = 10−9].
4.3.2 Nonequilibrium entropy limiters.
Now, we would like to demonstrate just a representative sample of the many
possibilities of limiters suggested in Sec. 3. In each case the limiter is im-
plemented by a post-processing routine immediately following the collision
step (either LBGK (19) or ELBM (20)). Here, we will only consider LBGK
collisions and entropic quasiequilibria (14).
The post-processing step adjusts f by the update formula:
f 7→ f ∗ + φ(∆S)(f − f ∗),
where ∆S is defined by (6) and φ is a limiter function.
For the Ehrenfests’ regularisation one would choose
φ(∆S)(x) =
1, ∆S(x) ≤ δ,
0, otherwise,
where δ is a pre-specified threshold value. Furthermore, it is pertinent to select
just k sites with highest ∆S > δ. This limiter has been previously applied to
the shock tube problem in [6,7,8] and we will not reproduce those results here.
Instead, our first example will be the following smooth limiter:
φ(∆S) =
1 + α∆Sk
. (22)
For this limiter, we will fix k = 1/2 (so that the limiter is double monotonic in
quadratic approximation to entropy) and compare the density profiles for α =
δ/(E(∆S)k), δ = 0.1, 0.01, 0.001. We have also ensured an ensemble-dependent
limiter because of the dependence of α on the average E(∆S). As with Fig. 2,
we accompany each panel with the total entropy and nonequilibrium entropy
histories. Note the different scales for nonequilibrium entropy. Note also that
entropy (necessarily) now grows due to the additional dissipation.
Our next example (Fig. 5) considers the threshold filter (10). In this example
we choose the estimates ∆S0 = 5E(∆S), 10E(∆S), 20E(∆S) and fix the tol-
erance δ = 0 so that the influence of the threshold alone can be studied. Only
entropic adjustments are accepted in the limiter: ∆St ≤ ∆S. As the threshold
increases, nonequilibrium entropy grows faster and spurious begin to appear.
Finally, we test the median filter (Fig. 6). We choose a minimal filter so that
only the nearest neighbours are considered. As with the threshold filter, we
introduce a tolerance δ and we try the values δ = 10−3, 10−4, 10−5. Only
entropic adjustments are accepted in the limiter: ∆Smed ≤ ∆S.
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0.025
0 0.5 1
0 100 200 300 400
0 100 200 300 400
Fig. 4. Density and profile of the 1:2 athermal shock tube simulation with ν = 10−9
after 400 time steps using LBGK (19) and the smooth limiter (22) with k = 1/2,
α = δ/(E(∆S)k) and (a) δ = 0.1; (b) δ = 0.01 and (c) δ = 0.001. Total entropy and
nonequilibrium entropy time histories for each parameter set {k, α(δ)} are displayed
in the adjacent panels.
We have seen that each of the examples we have considered (Fig. 4, Fig. 5
and Fig. 6) is capable of subduing spurious post-shock oscillations compared
with LBGK (or ELBM) on this problem (cf. Fig. 2). Of course, by limiting
nonequilibrium entropy the result is necessarily an increase in entropy.
From our experiences our recommendation is that the median filter is the
superior choice amongst all the limiters suggested in Sec. 3. The action of the
median filter is found to be both extremely gentle and, at the same time, very
effective.
4.4 Lid-driven cavity
Our second numerical example is the classical 2D lid-driven cavity flow. A
square cavity of side length L is filled with fluid with kinematic viscosity ν
(initially at rest) and driven by the cavity lid moving at a constant velocity
(u0, 0) (from left to right in our geometry).
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
Fig. 5. Density and profile of the 1:2 athermal shock tube simulation with ν = 10−9
after 400 time steps using LBGK (19) and the threshold limiter (10) with (a)
∆St = 5E(∆S); (b) ∆St = 10E(∆S) and (c) ∆St = 20E(∆S). Total entropy and
nonequilibrium entropy time histories for each threshold ∆St are displayed in the
adjacent panels.
We will simulate the flow on a 100 × 100 grid using LBGK regularised with
the median filter limiter. Unless otherwise stated, we use entropic quasiequilib-
ria (17). The implementation of the filter is as follows: the filter is not applied
to boundary nodes; for nodes which immediately neighbour the boundary the
stencil consists of the 3 nearest neighbours (including itself) closest to the
boundary; for all other nodes the minimal stencil of 9 nearest neighbours is
used.
We have purposefully selected such a coarse grid simulation because it is read-
ily found that, on this problem, unregularised LGBK fails (blows-up) for all
but the most modest Reynolds numbers Re := Lu0/ν.
4.4.1 Steady-state vortex centres
For modest Reynolds number the system settles to a steady state in which the
dominant features are a primary central rotating vortex, with several counter-
rotating secondary vortices located in the bottom-left, bottom-right (and pos-
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
0 0.5 1
0 100 200 300 400
0 100 200 300 400
Fig. 6. Density and profile of the 1:2 athermal shock tube simulation with ν = 10−9
after 400 time steps using LBGK (19) and the minimal median limiter with (a)
δ = 10−5; (b) δ = 10−4 and (c) δ = 10−3. Total entropy and nonequilibrium
entropy time histories for each tolerance δ are displayed in the adjacent panels.
sibly top-left) corners.
Steady state has been extensively investigated in the literature. The study
of Hou et al [24] simulates the flow over a range of Reynolds numbers using
unregularised LBGK on a 256×256 grid. Primary and secondary vortex centre
data is provided. We compare this same statistic for the present median filtered
coarse grid simulation. We will employ the same convergence criteria used
in [24]. Namely, we deem that steady state has been reached by ensuring
that the difference between the maximum value of the stream function for
successive 10, 000 time steps is less that 10−5. The stream function, which is
not a primary variable in the LBM simulation, is obtained from the velocity
data by integration using Simpson’s rule. Vortex centres are characterised as
local extrema of the stream function.
We compare our results with the LBGK simulations in [24] and [41]. To align
ourselves with these studies we specify the following boundary condition: lid
profile is constant; remaining cavity walls are subject to the “bounce-back”
condition [38]. In our simulations, the initial uniform fluid density profile is
ρ = 2.7 and the velocity of the lid is u0 = 1/10 (in lattice units).
Collected in Table 1, for Re = 2000, 5000 and 7500, are the coordinates of
the primary and secondary vortex centres using (a) unregularised LBGK; (b)
LBGK with median filter limiter (δ = 10−3); (c) LBGK with median filter lim-
iter (δ = 10−4), all with non-entropic polynomial quasiequilibria (18). Lines
(d), (e) and (f) are the same but with entropic quasiequilibria (17). The re-
maining lines of Table 1 are as follows: (g) literature data [24] (unregularised
LBGK on a 256×256 grid); (h) literature data [41] (positivity rule); (i) litera-
ture data [41] (ELBM). With the exception of (g), all simulation are conducted
on a 100 × 100 grid. The top-left vortex does not appear at Re = 2000 and
no data was provided for it in [41] at Re = 5000. The unregularised LBGK
Re = 7500 simulation blows-up in finite time and the simulation becomes
meaningless. The y-coordinate of the two lower-vortices at Re = 5000 in (i)
appear anomalously small and were not reproduced by our experiments with
the positivity rule (not shown).
We have conducted two runs of the experiment with the median filter param-
eter δ = 10−3 and δ = 10−4. Despite the increased number of realisations the
vortex centre locations remain effectively unchanged and we detect no signif-
icant variation between the two runs. This demonstrates the gentle nature of
the median filter. At Reynolds Re = 2000 the median filter has no effect at all
on the vortex centres compared with LBGK.
We find no significant differences between the experiments with entropic and
non-entropic polynomial quasiequilibria in this test.
The coordinates of the primary vortex centre for unregularised LBGK at Re =
5000 are already quite inaccurate as LBGK begins to lose stability. Stability
is lost entirely at some critical Reynolds number 5000 < Re ≤ 7500 and the
simulation blows-up.
Furthermore, we have agreement (within grid resolution) with the data given
in [24]. Also compiled in Table 1 is the data from the limiter experiments
conducted in [41] (although not explicitly discussed in the language of limiters
by the authors of that work). In [41] the authors give vortex centre data for
the positivity rule (Fig. 1) and for ELBM (which we interpret as containing a
hidden limiter). In [41] the positivity rule is called FIX-UP.
As Reynolds number increases the flow in the cavity is no longer steady and a
more complicated flow pattern emerges. On the way to a fully developed tur-
bulent flow, the lid-driven cavity flow is known to undergo a series of period
doubling Hopf bifurcations. On our coarse grid, we observe that the coordi-
nates of the primary vortex centre (maximum of the stream function) is a very
robust feature of the flow, with little change between coordinates (no change
in y-coordinates) computed at Re = 5000 and Re = 7500 with the median fil-
ter. On one hand, because of this observation it becomes inconclusive whether
Table 1
Primary and secondary vortex centre coordinates for the lid-driven cavity flow at
Re = 2000, 5000, 7500.
Primary Lower-left Lower-right Top-left
Re x y x y x y x y
2000 (a) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (b) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (c) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (d) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (e) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (f) 0.5253 0.5455 0.0909 0.1010 0.8384 0.1010 Not applicable
2000 (g) 0.5255 0.5490 0.0902 0.1059 0.8471 0.0980 Not applicable
2000 (h) 0.5200 0.5450 0.0900 0.1000 0.8300 0.0950 Not applicable
2000 (i) 0.5200 0.5500 0.0890 0.1000 0.8300 0.1000 Not applicable
5000 (a) 0.5152 0.6061 0.0808 0.1313 0.7980 0.0707 0.0505 0.8990
5000 (b) 0.5152 0.5354 0.0808 0.1313 0.8081 0.0808 0.0606 0.8990
5000 (c) 0.5152 0.5354 0.0808 0.1313 0.8081 0.0808 0.0707 0.8889
5000 (d) 0.5152 0.5960 0.0808 0.1313 0.8081 0.0808 0.0505 0.8990
5000 (e) 0.5152 0.5354 0.0808 0.1313 0.8081 0.0808 0.0606 0.8990
5000 (f) 0.5152 0.5354 0.0808 0.1313 0.8081 0.0808 0.0707 0.8889
5000 (g) 0.5176 0.5373 0.0784 0.1373 0.8078 0.0745 0.0667 0.9059
5000 (h) 0.5150 0.5680 0.0950 0.0100 0.8450 0.0100 Not available
5000 (i) 0.5150 0.5400 0.0780 0.1350 0.8050 0.0750 Not available
7500 (a) — — — — — — — —
7500 (b) 0.5051 0.5354 0.0707 0.1515 0.7879 0.0707 0.0606 0.8990
7500 (c) 0.5051 0.5354 0.0707 0.1515 0.7879 0.0707 0.0707 0.8889
7500 (d) — — — — — — — —
7500 (e) 0.5051 0.5354 0.0707 0.1515 0.7879 0.0707 0.0606 0.8990
7500 (f) 0.5051 0.5354 0.0707 0.1515 0.7879 0.0707 0.0707 0.8889
7500 (g) 0.5176 0.5333 0.0706 0.1529 0.7922 0.0667 0.0706 0.9098
the median limiter is adding too much additional dissipation. On the other
hand, a more studious choice of control criteria may indicate that the first
bifurcation has already occurred by Re = 7500.
4.4.2 First Hopf bifurcation
A survey of available literature reveals that the precise value of Re at which
the first Hopf bifurcation occurs is somewhat contentious, with most current
studies (all of which are for incompressible flow) ranging from around Re =
7400–8500 [9,32,33]. Here, we do not intend to give a precise value because
it is a well observed grid effect that the critical Reynolds number increases
(shifts to the right) with refinement (see, e.g., Fig. 3 in [33]). Rather, we
will be content to localise the first bifurcation and, in doing so, demonstrate
that limiters are capable of regularising without effecting fundamental flow
features.
To localise the first bifurcation we take the following algorithmic approach.
Entropic quasiequilibria are in use. The initial uniform fluid density profile
is ρ = 1.0 and the velocity of the lid is u0 = 1/10 (in lattice units). We
record the unsteady velocity data at a single control point with coordinates
(L/16, 13L/16) and run the simulation for 5000 non-dimensionless time units
(5000L/u0 time steps). Let us denote the final 1% of this signal by (usig, vsig).
We then compute the energy Eu (ℓ2-norm normalised by non-dimensional
signal duration) of the deviation of usig from its mean:
Eu :=
u0|usig|
(usig − usig)
, (23)
where |usig| and usig denote the length and mean of usig, respectively. We
choose this robust statistic instead of attempting to measure signal amplitude
because of numerical noise in the LBM simulation. The source of noise in LBM
is attributed to the existence of an inherently unavoidable neutral stability
direction in the numerical scheme (see, e.g., [8]).
We opt not to employ the “bounce-back” boundary condition used in the pre-
vious steady state study. Instead we will use the diffusive Maxwell boundary
condition (see, e.g., [11]), which was first applied to LBM in [1]. The essence
of the condition is that populations reaching a boundary are reflected, propor-
tional to equilibrium, such that mass-balance (in the bulk) and detail-balance
are achieved. The boundary condition coincides with “bounce-back” in each
corner of the cavity.
To illustrate, immediately following the advection of populations consider the
situation of a wall, aligned with the lattice, moving with velocity uwall and
with outward pointing normal to the wall in the negative y-direction (this is
the situation on the lid of the cavity with uwall = u0). The implementation
of the diffusive Maxwell boundary condition at a boundary site (x, y) on this
wall consists of the update
fi(x, y, t+ 1) = γf
i (uwall), i = 4, 7, 8,
f2(x, y, t) + f5(x, y, t) + f6(x, y, t)
f ∗4 (uwall) + f
7 (uwall) + f
8 (uwall)
Observe that, because density is a linear factor of the quasiequilibria (17),
the density of the wall is inconsequential in the boundary condition and can
therefore be taken as unity for convenience. As is usual, only those populations
pointing in to the fluid at a boundary site are updated. Boundary sites do not
undergo the collisional step that the bulk of the sites are subjected to.
We prefer the diffusive boundary condition over the often preferred “bounce-
back” boundary condition with constant lid profile. This is because we have
experienced difficulty in separating the aforementioned numerical noise from
the genuine signal at a single control point using “bounce-back”. We remark
that the diffusive boundary condition does not prevent unregularised LBGK
from failing at some critical Reynolds number Re > 5000.
Now, we conduct an experiment and record (23) over a range of Reynolds
numbers. In each case the median filter limiter is employed with parameter
δ = 10−3. Since the transition between steady and periodic flow in the lid-
driven cavity is known to belong to the class of standard Hopf bifurcations
we are assured that E2u ∝ Re [16]. Fitting a line of best fit to the resulting
data localises the first bifurcation in the lid-driven cavity flow to Re = 7135
(Fig. 7). This value is within the tolerance of Re = 7402±4% given in [33] for
a 100×100 grid. We also provide a (time averaged) phase space trajectory and
Fourier spectrum for Re = 7375 at the monitoring point (Fig. 8 and Fig. 9)
which clearly indicate that the first bifurcation has been observed.
5 Conclusions
Entropy and thermodynamics are important for stability of the lattice Boltz-
mann methods. It is now clear: after almost 10 years of work since the pub-
lication of [26] proved this statement (the main reviews are [5,28,39]). The
question is now: “how does one utilise, optimally, entropy and thermody-
namic structures in lattice Boltzmann methods?”. In our paper we attempt to
propose a solution (temporary, at least). Our approach is applicable to both
entropic as well as for non-entropic polynomial quasiequilibria.
5750 6000 6250 6500 6750 7000 7250 7500 7750 8000
0.005
0.015
0.025
0.035
0.045
(7135,0)
Fig. 7. Plot of energy squared, E2u (23), as a function of Reynolds number, Re, using
LBGK regularised with the median filter limiter with δ = 10−3 on a 100× 100 grid.
Straight lines are lines of best fit. The intersection of the sloping line with the x-axis
occurs close to Re = 7135.
We have constructed a system of nonequilibrium entropy limiters for the lattice
Boltzmann methods (LBM):
• the positivity rule that provides positivity of distribution;
• the pointwise entropy limiters based on selection and correction of most
nonequilibrium values;
• filters of nonequilibrium entropy, and the median filter as a filter of choice.
All these limiters exploit physical properties of LBM and allow control of total
additional entropy production. In general, they do the same work for LBM as
flux limiters do for finite differences, finite volumes and finite elements meth-
ods, and come into operation when sharp gradients are present. For smoothly
changing waves, the limiters do not operate and the spatial derivatives can be
represented by higher order approximations without introducing non-physical
oscillations. But there are some differences too: for LBM the main idea behind
the construction of nonequilibrium entropy limiter schemes is to limit a scalar
quantity — the nonequilibrium entropy — or to delete the “salt and pepper”
noise from the field of this quantity. We do not touch the vectors or tensors
of spatial derivatives, as it is for flux limiters.
Standard test examples demonstrate that the developed limiters erase spurious
oscillations without blurring of shocks, and do not affect smooth solutions. The
limiters we have tested do not produce a noticeable additional dissipation and
Fig. 8. Velocity components as a function of time for the signal (usig, vsig) at the
monitoring point (L/16, 13L/16) using LBGK regularised with the median filter
limiter with δ = 10−3 on a 100 × 100 grid (Re = 7375). Dots represent simulation
results and the solid line is a 100 step time average of the signal.
allow us to reproduce the first Hopf bifurcation for 2D lid-driven cavity on a
coarse 100× 100 grid. At the same time the simplest median filter deletes the
spurious post-shock oscillations for low viscosity.
Perhaps, it is impossible to find one best nonequilibrium entropy limiter for
all problems. It is a special task to construct the optimal limiters for a specific
classes of problems.
Acknowledgments
Discussion of the preliminary version of this work with S. Succi and par-
ticipants of the lattice Boltzmann workshop held on 15th November 2006
in Leicester (UK) was very important. Author A. N. Gorban is grateful to
S. K. Godunov for the course of numerical methods given many years ago at
Novosibirsk University. This work is supported by Engineering and Physical
Sciences Research Council (EPSRC) grant number GR/S95572/01.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency
Fig. 9. Relative amplitude spectrum for the signal usig at the monitoring point
(L/16, 13L/16) using LBGK regularised with the median filter limiter with δ = 10−3
on a 100 × 100 grid (Re = 7375). We measure a dominant frequency of ω = 0.525.
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Introduction
Background
Nonequilibrium entropy limiters for LBM
Positivity rule
Ehrenfests' regularisation
Smooth limiters of nonequilibrium entropy
Monitoring of total dissipation
Median entropy filter
Entropic steps for non-entropic quasiequilibria
ELBM collisions as a smooth limiter
Monotonic and double monotonic limiters
Numerical experiment
Velocities and quasiequilibria
LBGK and ELBM
Shock tube
Lid-driven cavity
Conclusions
References
|
0704.0044 | Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in
magnetized weakly collisional plasmas | THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 182:310 (2009) [e-print arXiv:0704.0044]
Preprint typeset using LATEX style emulateapj v. 08/22/09
ASTROPHYSICAL GYROKINETICS: KINETIC AND FLUID TURBULENT CASCADES IN MAGNETIZED WEAKLY
COLLISIONAL PLASMAS
A. A. SCHEKOCHIHIN,1,2 S. C. COWLEY,2,3 W. DORLAND,4 G. W. HAMMETT,5 G. G. HOWES,6 E. QUATAERT,7 AND T. TATSUNO4
Submitted April 1, 2007; accepted February 21, 2009; published May 6, 2009
ABSTRACT
This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It
is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium
and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many
others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in
a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the
mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron
frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at
this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic
cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the
kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are
four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path
and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem
is systematically reduced to a more physically transparent and computationally tractable system of equations,
which are derived in a rigorous way. In the “inertial range” above the ion gyroscale, the kinetic cascade
separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-field-
strength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at
both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field
lines associated with the Alfvénic component (in the collisional limit, these compressive fluctuations become
the slow and entropy modes of the conventional MHD). In the “dissipation range” below ion gyroscale, there
are again two cascades: the kinetic-Alfvén-wave (KAW) cascade governed by two fluid-like Electron Reduced
Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space
and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at
the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly
damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are
derived for all of these cascades. The relationship between the theoretical models proposed in this paper and
astrophysical applications and observations is discussed in detail.
Subject headings: magnetic fields—methods: analytical—MHD—plasmas—turbulence
1. INTRODUCTION
As observations of velocity, density and magnetic fields in
astrophysical plasmas probe ever smaller scales, turbulence—
i.e., broadband disordered fluctuations usually characterized
by power-law energy spectra—emerges as a fundamental and
ubiquitous feature.
One of the earliest examples of observed turbulence in
space was the detection of a Kolmogorov k−5/3 spectrum
of magnetic fluctuations in the solar wind over a fre-
quency range of about three decades (first reported by
Matthaeus & Goldstein 1982; Bavassano et al. 1982 and con-
firmed to a high degree of accuracy by a multitude of subse-
Electronic address: [email protected]
1 Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
Oxford OX1 3NP, UK.
2 Plasma Physics, Blackett Laboratory, Imperial College, Lon-
don SW7 2AZ, UK.
3 Euratom/UKAEA Fusion Association, Culham Science Centre, Abing-
ton OX14 3DB, UK.
4 Department of Physics, University of Maryland, College Park,
MD 20742-3511.
5 Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451.
6 Department of Physics and Astronomy, University of Iowa, Iowa City,
IA 52242-1479.
7 Department of Astronomy, University of California, Berkeley, CA
94720-3411.
quent observations, e.g., Marsch & Tu 1990a; Horbury et al.
1996; Leamon et al. 1998; Bale et al. 2005; see Fig. 1). An-
other famous example in which the Kolmogorov power law
appears to hold is the electron density spectrum in the inter-
stellar medium (ISM)—in this case it emerges from observa-
tions by various methods in several scale intervals and, when
these are pieced together, the power law famously extends
over as many as 12 decades of scales (Armstrong et al. 1981,
1995; Lazio et al. 2004), a record that has earned it the name
of “the Great Power Law in the Sky.” Numerous other mea-
surements in space and astrophysical plasmas, from the mag-
netosphere to galaxy clusters, result in Kolmogorov (or con-
sistent with Kolmogorov) spectra but also show steeper power
laws at very small (microphysical) scales (these observations
are discussed in more detail in § 8).
Power-law spectra spanning broad bands of scales are
symptomatic of the fundamental role of turbulence as a mech-
anism of transferring energy from the outer scale(s) (hence-
forth denoted L), where the energy is injected to the inner
scale(s), where it is dissipated. As these scales tend to be
widely separated in astrophysical systems, one way for the
system to bridge this scale gap is to fill it with fluctuations; the
power-law spectra then arise due to scale invariance at the in-
termediate scales. Besides being one of the more easily mea-
http://arxiv.org/abs/0704.0044v4
http://arxiv.org/abs/0704.0044
mailto:[email protected]
2 SCHEKOCHIHIN ET AL.
surable characteristics of the multi-scale nature of turbulence,
power-law (and, particularly, Kolmogorov) spectra evoke a
number of fundamental physical ideas that lie at the heart of
the turbulence theory: universality of small-scale physics, en-
ergy cascade, locality of interactions, etc. In this paper, we
shall revisit and generalize these ideas for the problem of ki-
netic plasma turbulence,8 so it is perhaps useful to remind the
reader how they emerge in a standard argument that leads to
the k−5/3 spectrum (Kolmogorov 1941; Obukhov 1941).
1.1. Kolmogorov Turbulence
Suppose the average energy per unit time per unit volume
that the system dissipates is ε. This energy has to be trans-
ferred from some (large) outer scale L at which it is injected
to some (small) inner scale(s) at which the dissipation occurs
(see § 1.5). It is assumed that in the range of scales interme-
diate between the outer and the inner (the inertial range), the
statistical properties of the turbulence are universal (indepen-
dent of the macrophysics of injection or of the microphysics
of dissipation), spatially homogeneous and isotropic and the
energy transfer is local in scale space. The flux of kinetic en-
ergy through any inertial-range scale λ is independent of λ:
∼ ε = const, (1)
where the (constant) density of the medium is absorbed into
ε, uλ is the typical velocity fluctuation associated with the
scale λ, and τλ is the cascade time.
9 Since interactions are
assumed local, τλ must be expressed in terms of quantities
associated with scale λ. It is then dimensionally inevitable
that τλ ∼ λ/uλ (the nonlinear interaction time, or turnover
time), so we get
uλ ∼ (ελ)1/3. (2)
This corresponds to a k−5/3 spectrum of kinetic energy.
1.2. MHD Turbulence and Critical Balance
That astronomical data appear to point to a ubiquitous na-
ture of what, in its origin, is a dimensional result for the turbu-
lence in a neutral fluid, might appear surprising. Indeed, the
astrophysical plasmas in question are highly conducting and
support magnetic fields whose energy is at least comparable
to the kinetic energy of the motions. Let us consider a situa-
tion where the plasma is threaded by a uniform dynamically
strong magnetic field B0 (the mean, or guide, field; see § 1.3
for a brief discussion of the validity of this assumption). In
the presence of such a field, there is no dimensionally unique
way of determining the cascade time τλ because besides the
nonlinear interaction time λ/uλ, there is a second character-
istic time associated with the fluctuation of size λ, namely the
Alfvén time l‖λ/vA, where vA is the Alfvén speed and l‖λ is
the typical scale of the fluctuation along the magnetic field.
The first theories of magnetohydrodynamic (MHD) turbu-
lence (Iroshnikov 1963; Kraichnan 1965; Dobrowolny et al.
1980) calculated τλ by assuming an isotropic cascade (l‖λ ∼
λ) of weakly interacting Alfvén-wave packets (τλ ≫ l‖λ/vA)
8 An outline of a Kolmogorov-style approach to kinetic turbulence was
given in a recent paper by Schekochihin et al. (2008b). It can be read as a
conceptual introduction to the present paper, which is much more detailed
and covers a much broader set of topics.
9 This is the version of Kolmogorov’s theory due to Obukhov 1941.
FIG. 1.— Spectra of electric and magnetic fluctuations in the solar wind
at 1 AU (see Table 1 for the solar-wind parameters corresponding to this
plot). This figure is adapted with permission from Fig. 3 of Bale et al. (2005)
(copyright 2005 by the American Physical Society). We have added the
reference slopes for Alfvén-wave and kinetic-Alfvén-wave turbulence in
bold dashed (red) lines and labeled “KRMHD,” “GK ions,” and “ERMHD”
the wavenumber intervals where these analytical descriptions are valid (see
§ 3, § 5 and § 7).
and obtained a k−3/2 spectrum. The failure of the ob-
served spectra to conform to this law (see references above)
and especially the observational (see references at the end
of this subsection) and experimental (Robinson & Rusbridge
1971; Zweben et al. 1979) evidence of anisotropy of MHD
fluctuations led to the isotropy assumption being discarded
(Montgomery & Turner 1981).
The modern form of MHD turbulence theory is commonly
associated with the names of Goldreich & Sridhar (1995,
1997, henceforth, GS). It can be summarized as follows. As-
sume that (a) all electromagnetic perturbations are strongly
anisotropic, so that their characteristic scales along the mean
field are much larger than those across it, l‖λ ≫ λ, or, in terms
of wavenumbers, k‖ ≪ k⊥; (b) the interactions between the
Alfvén-wave packets are strong and the turbulence at suffi-
ciently small scales always arranges itself in such a way that
the Alfvén timescale and the perpendicular nonlinear interac-
tion timescale are comparable to each other, i.e.,
ω ∼ k‖vA ∼ k⊥u⊥, (3)
where ω is the typical frequency of the fluctuations and u⊥
is the velocity fluctuation perpendicular to the mean field.
Taken scale by scale, this assumption, known as the critical
balance, removes the dimensional ambiguity of the MHD tur-
bulence theory. Thus, the cascade time is τλ ∼ l‖λ/vA ∼λ/uλ,
whence
uλ∼ (εl‖λ/vA)1/2 ∼ (ελ)1/3 , (4)
l‖λ∼ l
2/3, (5)
where l0 = v
A/ε. The scaling relation (4) is equivalent to a
⊥ spectrum of kinetic energy, while Eq. (5) quantifies the
anisotropy by establishing the relationship between the per-
pendicular and parallel scales. Note that Eq. (4) implies that
in terms of the parallel wavenumbers, the kinetic-energy spec-
trum is ∼ k−2‖ .
The above considerations apply to Alfvénic fluctuations,
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 3
i.e., perpendicular velocities and magnetic-field perturbations
from the mean given (at each scale) by δB⊥ ∼ u⊥
4πρ0,
where ρ0 is the mean mass density of the plasma (see
Fig. 1 and discussion in § 8.1.1). Other low-frequency
MHD modes—slow waves and the entropy mode—turn
out to be passively advected by the Alfvénic component
of the turbulence (this follows from the anisotropy; see
Lithwick & Goldreich 2001, and §§ 2.4-2.6, § 5.5, and § 6.3
for further discussion of the compressive fluctuations).
As we have mentioned above, the anisotropy was, in
fact, incorporated into MHD turbulence theory already by
Montgomery & Turner (1981). However, these authors’ view
differed from the GS theory in that they thought of MHD
turbulence as essentially two dimensional, described by a
Kolmogorov-like cascade (Fyfe et al. 1977), with an admix-
ture of Alfvén waves having some spectrum in k‖ unrelated
to the perpendicular structure of the turbulence (note that
Higdon 1984, while adopting a similar view, anticipated the
scaling relation (5), but did not seem to consider it to be any-
thing more than the confirmation of an essentially 2D nature
of the turbulence). In what we are referring to here as GS
turbulence, the 2D and Alfvénic fluctuations are not separate
components of the turbulence. The turbulence is three dimen-
sional, with correlations parallel and perpendicular to the (lo-
cal) mean field related at each scale by the critical balance
assumption.
Indeed, intuitively, we cannot have k‖vA ≪ k⊥u⊥: the tur-
bulence cannot be any more 2D than allowed by the critical
balance because fluctuations in any two planes perpendicu-
lar to the mean field can only remain correlated if an Alfvén
wave can propagate between them in less than their perpen-
dicular decorrelation time. In the opposite limit, weakly inter-
acting Alfén waves with fixed k‖ and ω = k‖vA ≫ k⊥u⊥ can be
shown to give rise to an energy cascade towards smaller per-
pendicular scales where the turbulence becomes strong and
Eq. (3) is satisfied (Goldreich & Sridhar 1997; Galtier et al.
2000; Yousef et al. 2009). Thus, there is a natural tendency
towards critical balance in a system containing nonlinearly in-
teracting Alfvén waves. We will see in what follows that crit-
ical balance may, in fact, be taken as a general physical prin-
ciple relating parallel scales (associated with linear propaga-
tion) and perpendicular scales (associated with nonlinear in-
teraction) in anisotropic plasma turbulence (see § 7.5, § 7.9.4,
§ 7.10.3).
We emphasize that, the anisotropy of astrophysical plasma
turbulence is an observed phenomenon. It is seen most clearly
in the spacecraft measurements of the turbulent fluctuations
in the solar wind (Belcher & Davis 1971; Matthaeus et al.
1990; Bieber et al. 1996; Dasso et al. 2005; Bigazzi et al.
2006; Sorriso-Valvo et al. 2006; Horbury et al. 2005, 2008;
Osman & Horbury 2007; Hamilton et al. 2008) and in the
magnetosheath Sahraoui et al. (2006); Alexandrova et al.
(2008b). In a recent key development, solar-wind data anal-
ysis by Horbury et al. (2008) approaches quantitative cor-
roboration of the critical balance conjecture by confirm-
ing the scaling of the spectrum with the parallel wavenum-
ber ∼ k−2
that follows from the first scaling relation in
Eq. (4). Anisotropy is also observed indirectly in the
ISM (Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al.
2002; Dennett-Thorpe & de Bruyn 2003), including recently
in molecular clouds (Heyer et al. 2008), and, with unambigu-
ous consistency, in numerical simulations of MHD turbulence
(Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac
2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al.
2003).10
1.3. MHD Turbulence with and without a Mean Field
In the discussion above, treating MHD turbulence as tur-
bulence of Alfvénic fluctuations depended on assuming the
presence of a mean (guide) field B0 that is strong compared to
the magnetic fluctuations, δB/B0 ∼ u/vA ≪ 1. We will also
need this assumption in the formal developments to follow
(see § 2.1, § 3.1). Is it legitimate to expect that such a spatially
regular field will be generically present? Kraichnan (1965) ar-
gued that in a generic situation in which all magnetic fields are
produced by the turbulence itself via the dynamo effect, one
could assume that the strongest field will be at the outer scale
and that this field will play the role of an (approximately) uni-
form guide field for the Alfvén waves in the inertial range.
Formally, this amounts to assuming that in the inertial range,
≪ 1, k‖L ≪ 1. (6)
It is, however, by no means obvious that this should be true.
When a strong mean field is imposed by some external mech-
anism, the turbulent motions cannot bend it significantly, so
only small perturbations are possible and δB ≪ B0. In con-
trast, without a strong imposed field, the energy density of the
magnetic fluctuations is at most comparable to the kinetic-
energy density of the plasma motions, which are then suffi-
ciently energetic to randomly tangle the field, so δB ≫ B0.
In the weak-mean-field case, the dynamically strong
stochastic magnetic field is a result of saturation of the
small-scale, or fluctuation, dynamo—amplification of mag-
netic field due to random stretching by the turbulent mo-
tions (see review by Schekochihin & Cowley 2007). The
definitive theory of this saturated state remains to be dis-
covered. Both physical arguments and numerical evidence
(Schekochihin et al. 2004; Yousef et al. 2007) suggest that
the magnetic field in this case is organized in folded flux
sheets (or ribbons). The length of these folds is compara-
ble to the outer scale, while the scale of the field-direction
reversals transverse to the fold is determined by the dissipa-
tion physics: in MHD with isotropic viscosity and resistiv-
ity, it is the resistive scale.11 Although Alfvén waves prop-
10 The numerical evidence is much less clear on the scaling of the
spectrum. The fact that the spectrum is closer to k
than to k
in numerical simulations (Maron & Goldreich 2001; Müller et al. 2003;
Mason et al. 2007; Perez & Boldyrev 2008, 2009; Beresnyak & Lazarian
2008b) prompted Boldyrev (2006) to propose a scaling argument that allows
an anisotropic Alfvénic turbulence with a k
spectrum. His argument is
based on the conjecture that the fluctuating velocity and magnetic fields tend
to partially align at small scales, an idea that has had considerable numeri-
cal support (Maron & Goldreich 2001; Beresnyak & Lazarian 2006, 2008b;
Mason et al. 2006; Matthaeus et al. 2008a). The alignment weakens nonlin-
ear interactions and alters the scalings. Another modification of the GS the-
ory leading to an anisotropic k
spectrum was proposed by Gogoberidze
(2007), who assumed that MHD turbulence with a strong mean field is dom-
inated by non-local interactions with the outer scale. However, in both argu-
ments, the basic assumption that the turbulence is strong is retained. This is
the main assumption that we make in this paper: the critical balance conjec-
ture (3) is used below not as a scaling prescription but in a weaker sense of
an ordering assumption, i.e., we simply take the wave propagation terms in
the equations to be comparable to the nonlinear terms. It is not hard to show
that the results derived in what follows remain valid whether or not the align-
ment is present. We note that observationally, only in the solar wind does one
measure the spectra with sufficient accuracy to state that they are consistent
with k
but not with k
(see § 8.1.1).
11 In weakly collisional astrophysical plasmas, such a description is not
4 SCHEKOCHIHIN ET AL.
agating along the folds may exist (Schekochihin et al. 2004;
Schekochihin & Cowley 2007), the presence of the small-
scale direction reversals means that there is no scale-by-scale
equipartition between the velocity and magnetic fields: while
the magnetic energy is small-scale dominated due to the di-
rection reversals,12 the kinetic energy should be contained pri-
marily at the outer scale, with some scaling law in the inertial
range.
Thus, at the current level of understanding we have to as-
sume that there are two asymptotic regimes of MHD turbu-
lence: anisotropic Alfvénic turbulence with δB ≪ B0 and
isotropic MHD turbulence with small-scale field reversals and
δB ≫ B0. In this paper, we shall only discuss the first regime.
The origin of the mean field may be external (as, e.g., in the
solar wind, where it is the field of the Sun) or due to some
form of mean-field dynamo (rather than small-scale dynamo),
as usually expected for galaxies (see, e.g., Shukurov 2007).
Note finally that the condition δB ≪ B0 need not be satis-
fied at the outer scale and in fact is not satisfied in most space
or astrophysical plasmas, where more commonly δB ∼ B0 at
the outer scale. This, however, is sufficient for the Kraich-
nan hypothesis to hold and for an Alfénic cascade to be set
up, so at small scales (in the inertial range and beyond), the
assumptions (6) are satisfied.
1.4. Kinetic Turbulence
The GS theory of MHD turbulence (§ 1.2) allows us to
make sense of the magnetized turbulence observed in cosmic
plasmas exhibiting the same statistical scaling as turbulence
in a neutral fluid (although the underlying dynamics are very
different in these two cases!). However, there is an aspect of
the observed astrophysical turbulence that undermines the ap-
plicability of any type of fluid description: in most cases, the
inertial range where the Kolmogorov scaling holds extends to
scales far below the mean free path deep into the collisionless
regime. For example, in the case of the solar wind, the mean
free path is close to 1 AU, so all scales are collisionless—
an extreme case, which also happens to be the best studied,
thanks to the possibility of in situ measurements (see § 8).
The proper way of treating such plasmas is using kinetic
theory, not fluid equations. The basis for the application of the
MHD fluid description to them has been the following well
known result from the linear theory of plasma waves: while
the fast, slow and entropy modes are damped at the mean-
free-path scale both by collisional viscosity (Braginskii 1965,
see § 6.1.2) and by collisionless wave–particle interactions
(Barnes 1966, see § 6.2.2), the Alfvén waves are only damped
at the ion gyroscale. It has, therefore, been assumed that the
MHD description, inasmuch as it concerns the Alfvén-wave
cascade, can be extended to the ion gyroscale, with the un-
derstanding that this cascade is decoupled from the damped
cascades of the rest of the MHD modes. This approach and
its application to the turbulence in the ISM are best explained
by Lithwick & Goldreich (2001).
While the fluid description may be sufficient to under-
stand the Alfvénic fluctuations in the inertial range, it is cer-
applicable: the field reversal scale is most probably determined by more
complicated and as yet poorly understood kinetic plasma effects; below this
scale, an Alfvénic turbulence of the kind discussed in this paper may exist
(Schekochihin & Cowley 2006).
12 See Haugen et al. (2004) for an alternative view. Note also that the
numerical evidence cited above pertains to forced simulations. In decaying
MHD turbulence simulations, the magnetic energy does indeed appear to be
at the outer scale (Biskamp & Müller 2000), so one might expect an Alfvénic
cascade deep in the inertial range.
tainly inadequate for everything else: the compressive fluctu-
ations in the inertial range and turbulence in the dissipation
range (below the ion gyroscale), where power-law spectra are
also detected (e.g., Denskat et al. 1983; Leamon et al. 1998;
Czaykowska et al. 2001; Smith et al. 2006; Sahraoui et al.
2006; Alexandrova et al. 2008a,b, see also Fig. 1). The fun-
damental challenge that a comprehensive theory of astrophys-
ical plasma turbulence must meet is to give the full account of
how the turbulent fluctuation energy injected at the outer scale
is cascaded to small scales and deposited into particle heat.
We shall see (§ 3.4 and § 3.5) that the familiar concept of an
energy cascade can be generalized in the kinetic framework
as the kinetic cascade of a single quantity that we call the
generalized energy (see also Schekochihin et al. 2008b, and
references therein). The small scales developed in the pro-
cess are small scales both in the position and velocity space.
The fundamental reason for this is the low collisionality of
the plasma: since heating cannot ultimately be accomplished
without collisions, large gradients in phase space are neces-
sary for the collisions to be effective.
The idea of a generalized energy cascade in phase space
as the engine of kinetic plasma turbulence is the central con-
cept of this paper. In order to understand the physics of the
kinetic cascade in various scale ranges, we derive in what fol-
lows a hierarchy of simplified, yet rigorous, reduced kinetic,
fluid and hybrid descriptions. While the full kinetic theory
of turbulence is very difficult to handle either analytically or
numerically, the models we derive are much more tractable.
For all, the regimes of applicability (scale/parameter ranges,
underlying assumptions) are clearly stated. In each of these
regimes, the kinetic cascade splits into several channels of en-
ergy transfer, some of them familiar (e.g., the Alfvénic cas-
cade, § 5.3 and § 5.4), others conceptually new (e.g., the ki-
netic cascade of collisionless compressive fluctuations, § 6.2,
or the entropy cascade, §§ 7.9-7.12).
So as to introduce this theoretical framework in a way that
is both analytically systematic and physically intelligible, let
us first consider the characteristic scales that are relevant to
the problem of astrophysical turbulence (§ 1.5). The models
we derive are previewed in § 1.6, at the end of which the plan
of further developments is given.
1.5. Scales in the Problem
1.5.1. Outer Scale
It is a generic feature of turbulent systems that energy is
injected via some large-scale mechanism: “large scale” here
means some scale (or a range of scales) comparable to the
size of the system, depending on its global properties, and
much larger than the microphysical scales at which energy
is dissipated and converted into heat (§ 1.5.2). Examples of
large-scale stirring of turbulent fluctuations include the solar
activity in the corona (launching Alfvén waves to produce
turbulence in the solar wind); supernova explosions in the
ISM (e.g., Norman & Ferrara 1996; Ferrière 2001); the mag-
netorotational instability in accretion disks (Balbus & Hawley
1998); merger events, galaxy wakes and active galactic
nuclei in galaxy clusters (e.g., Subramanian et al. 2006;
Enßlin & Vogt 2006; Chandran 2005a). Since in this paper
we are concerned with the local properties of astrophysical
plasmas, let us simply assume that energy injection occurs at
some characteristic outer scale L. All further considerations
will apply to scales that are much smaller than L and we will
assume that the particular character of the energy injection
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 5
FIG. 2.— Partition of the wavenumber space by characteristic scales. The wavenumbers are normalized by l0 ∼ v
A/ε, where ε is the total power input (see
§ 1.2). Dotted line shows the path an Alfvén-wave cascade starting at the outer scale L ∼ l0 takes through the wavenumber space. We also show the regions of
validity of the three tertiary approximations. They all require k‖ ≪ k⊥ (anisotropic fluctuations) and k‖ρi ≪ 1 (i.e., k‖vthi ≪ Ωi, low-frequency limit). Reduced
MHD (RMHD, § 2) is valid when k⊥ρi ≪ k‖λmfpi ≪ (me/mi)
1/2 (strongly magnetized collisional limit, adiabatic electrons). The regions of validity of Kinetic
Reduced MHD (KRMHD, § 5) and Electron Reduced MHD (ERMHD, § 7) lie within that of the isothermal electron/gyrokinetic ion approximation (Fig. 4) with
the additional requirement that k⊥ρi ≪ min(1,k‖λmfpi) (strongly magnetized ions) for KRMHD or k⊥ρi ≫ 1 (unmagnetized ions) for ERMHD. The collisional
limit of KRMHD (§ 6.1 and Appendix D), (me/mi)1/2 ≪ k‖λmfpi ≪ 1, is similar to RMHD, except electrons are isothermal. The dotted line is the scaling of k‖
vs k⊥ from critical balance in both the Alfvén-wave [§ 1.2, Eq. (5)] and kinetic-Alfvén-wave [§ 7.5, Eq. (241)] regimes.
does not matter at these small scales.
In most astrophysical situations, one cannot assume that
equilibrium quantities such as density, temperature, mean ve-
locity and mean magnetic field are uniform at the outer scale.
However, at scales much smaller than L, the gradients of the
small-scale fluctuating fields are much larger than the outer-
scale gradients (although the fluctuation amplitudes are much
smaller; for the mean magnetic field, this assumption is dis-
cussed in some detail in § 1.3), so we may neglect the equi-
librium gradients and consider the turbulence to be homoge-
neous. Specifically, this is a good assumption if k‖L ≫ 1
[Eq. (6)], i.e., not only the perpendicular scales but also the
much larger parallel ones are still shorter than the outer scale.
Note that we cannot generally assume that the outer-scale en-
ergy injection is anisotropic, so the anisotropy is also the prop-
erty of small scales only.
1.5.2. Microscales
There are four microphysical scales that mark the transi-
tions between distinct physical regimes:
Electron diffusion scale. — At k‖λmfpi(mi/me)
1/2 ≫ 1, the
electron response is isothermal (§ 4.4, Appendix A.4). At
k‖λmfpi(mi/me)
1/2 ≪ 1, it is adiabatic (§ 4.8.4, Appendix A.3).
Mean free path. — At k‖λmfpi ≫ 1, the plasma is collisionless.
In this regime, wave–particle interactions can damp compres-
sive fluctuations via Barnes damping (§ 6.2.2), so kinetic de-
scription becomes essential. At k‖λmfpi ≪ 1, the plasma is
collisional and fluid-like (§ 6.1, Appendices A and D).
Ion gyroscale. — At k⊥ρi ≪ 1, ions (as well as the electrons)
are magnetized and the magnetic field is frozen into the ion
flow (the E × B velocity field). At k⊥ρi ∼ 1, ions can ex-
change energy with electromagnetic fluctuations via wave–
particle interactions (and ion heating eventually occurs via a
kinetic ion-entropy cascade, see §§ 7.9-7.10). At k⊥ρi ≫ 1,
the ions are unmagnetized and have a Boltzmann response
(§ 7.2). Note that the ion inertial scale di = ρi/
βi is compara-
ble to the ion gyroscale unless the plasma beta βi = 8πniTi/B
is very different from unity. In the theories developed below,
di does not play a special role except in the limit of Ti ≪ Te,
which is not common in astrophysical plasmas (see further
discussion in § 7.1 and Appendix E).
Electron gyroscale. — At k⊥ρe ≪ 1, electrons are magnetized
and the magnetic field is frozen into the electron flow (§ 4,
§ 7, Appendix C). At k⊥ρe ∼ 1, the electrons absorb the
energy of the electromagnetic fluctuations via wave–particle
interactions (leading to electron heating via a kinetic electron-
entropy cascade, see § 7.12).
Typical values of these scales and of several other key pa-
rameters are given in Table 1. In Fig. 2, we show how the
wavenumber space, (k⊥,k‖), is divided by these scales into
several domains, where the physics is different. Further parti-
tioning of the wavenumber space results from comparing k⊥ρi
and k‖λmfpi (k⊥ρi ≪ k‖λmfpi is the limit of strong magnetiza-
6 SCHEKOCHIHIN ET AL.
TABLE 1
REPRESENTATIVE PARAMETERS FOR ASTROPHYSICAL PLASMAS.
Parameter
Solar
1 AU(a)
ionized
ISM(b)
Accretion
flow near
Sgr A∗(c)
Galaxy
clusters
(cores)(d)
ne = ni, cm−3 30 0.5 106 6× 10−2
Te, K ∼ Ti(e) 8000 1011 3× 107
Ti, K 5× 105 8000 ∼ 1012(f) ?(e)
B, G 10−4 10−6 30 7× 10−6
βi 5 14 4 130
vthi, km/s 90 10 10
5 700
vA, km/s 40 3 7× 10
U , km/s(f) ∼ 10 ∼ 10 ∼ 104 ∼ 102
L, km(f) ∼ 105 ∼ 1015 ∼ 108 ∼ 1017
(mi/me)1/2λmfpi, km 10
10 2× 108 4× 1010 4× 1016
λmfpi, km
(g) 3× 108 6× 106 109 1015
ρi, km 90 1000 0.4 104
ρe, km 2 30 0.003 200
a Values for slow wind (mean flow speed Vsw = 350 km/s in this case)
measured by Cluster spacecraft and taken from Bale et al. (2005), ex-
cept the value of Te, which they do not report, but which is expected
to be of the same order as Ti (Newbury et al. 1998). Note that the
data interval studied by Bale et al. (2005) is slightly atypical, with βi
higher than usual in the solar wind (the full range of βi variation in
the solar wind is roughly between 0.1 and 10; see Howes et al. 2008a
for another, perhaps more typical, fiducial set of slow-wind parameters
and Appendix A of the review by Bruno & Carbone 2005 for slow- and
fast-wind parameters measured by Helios 2). However, we use their pa-
rameter values as our representative example because the spectra they
report show with particular clarity both the electric and magnetic fluc-
tuations in both the inertial and dissipation ranges (see Fig. 1). See
further discussion in § 8.1 and § 8.2.
b Typical values (see, e.g., Norman & Ferrara 1996; Ferrière 2001). See
discussion in § 8.4.
c Values based on observational constraints for the radio-emitting
plasma around the Galactic Center (Sgr A∗) as interpreted by
Loeb & Waxman (2007) (see also Quataert 2003). See discussion in
§ 8.5.
d Values for the core region of the Hydra A cluster taken from
Enßlin & Vogt (2006); see Schekochihin & Cowley 2006 for a consis-
tent set of numbers for the hot plasmas outside the cores. See discussion
in § 8.6.
e We assume Ti ∼ Te for these estimates.
f Rough order-of-magnitude estimate.
g Defined λmfpi = vthi/νii, where νii is given by Eq. (52).
tion, see Appendix A.2) and, most importantly, from com-
paring parallel and perpendicular wavenumbers. As we ex-
plained above, observational and numerical evidence tells us
that Alfvénic turbulence is anisotropic, k‖ ≪ k⊥. In Fig. 2,
we sketch the path the turbulent cascade is expected to take
in the wavenumber space (we use the scalings of k‖ with k⊥
that follow from the GS argument for the Alfvén waves and an
analogous argument for the kinetic Alfvén waves, reviewed in
§ 1.2 and § 7.5, respectively).
1.6. Kinetic and Fluid Models
What is the correct analytical description of the turbulent
plasma fluctuations along the (presumed) path of the cascade?
As we promised above, it is going to be possible to simplify
the full kinetic theory substantially. These simplifications can
be obtained in the form of a hierarchy of approximations and
as these emerge, specific physical mechanisms that control the
turbulent cascade in various physical regimes become more
transparent.
Gyrokinetics (§ 3). — The starting point for these develop-
ments and the primary approximation in the hierarchy is gy-
rokinetics, a low-frequency kinetic theory resulting from av-
eraging over the cyclotron motion of the particles. Gyroki-
netics is appropriate for the study of subsonic plasma turbu-
lence in virtually all astrophysically relevant parameter ranges
(Howes et al. 2006). For fluctuations at frequencies lower
than the ion cyclotron frequency, ω ≪ Ωi, gyrokinetics can
be systematically derived by making use of the following two
assumptions, which also underpin the GS theory (§ 1.2): (a)
anisotropy of the turbulence, so ǫ∼ k‖/k⊥ is used as the small
parameter, and (b) strong interactions, i.e., the fluctuation am-
plitudes are assumed to be such that wave propagation and
nonlinear interaction occur on comparable timescales: from
Eq. (3), u⊥/vA ∼ ǫ. The first of these assumptions implies that
fluctuations at Alfvénic frequencies satisfy ω ∼ k‖vA ≪ Ωi
even when their perpendicular scale is such k⊥ρi ∼ 1. This
makes gyrokinetics an ideal tool both for analytical theory and
for numerical studies of astrophysical plasma turbulence; the
numerical approaches are also made attractive by the long ex-
perience of gyrokinetic simulations accumulated in the fusion
research and by the existence of publicly available gyroki-
netic codes (Kotschenreuther et al. 1995; Jenko et al. 2000;
Candy & Waltz 2003; Chen & Parker 2003). A concise re-
view of gyrokinetics is provided in § 3 (see Howes et al. 2006
for a detailed derivation). The reader is urged to pay partic-
ular attention to § 3.4 and § 3.5, where the concept of the ki-
netic cascade of generalized energy is introduced and the par-
ticle heating in gyrokinetics is discussed (Appendix F intro-
duces additional conservation laws that arise in 2D and some-
times also in 3D). This establishes the conceptual framework
in which most of the subsequent physical arguments are pre-
sented. The region of validity of gyrokinetics is illustrated
in Fig. 3: it covers virtually the entire path of the turbulent
cascade, except the largest (outer) scales, where one cannot
assume anisotropy. Note that the two-fluid theory, which is
the starting point for the MHD theory (see Appendix A), is
not a good description at collisionless scales. It is important
to mention, however, that the formulation of gyrokinetics that
we adopt, while appropriate for treating fluctuations at col-
lisionless scales, does nevertheless require a certain (weak)
degree of collisionality (see discussion in § 3.1.3 and an ex-
tended treatment of collisions in gyrokinetics in Appendix B).
Isothermal Electron Fluid (§ 4). — While gyrokinetics con-
stitutes a significant simplification, it is still a fully kinetic
description. Further progress towards simpler models is
achieved by showing that, for parallel scales smaller than the
electron diffusion scale, k‖λmfpi ≫ (me/mi)1/2, and perpen-
dicular scales larger than the electron gyroscale, k⊥ρe ≪ 1,
the electrons are a magnetized isothermal fluid while ions
must be treated (gyro)kinetically. This is the secondary
approximation in our hierarchy, derived in § 4 via an asymp-
totic expansion in (me/mi)
1/2 (see also Appendix C.1). The
plasma is described by the ion gyrokinetic equation and two
fluid-like equations that contain electron dynamics—these
are summarized in § 4.9. The region of validity of this
approximation is illustrated in Fig. 4: it does not capture the
dissipative effects around the electron diffusion scale or the
electron heating, but it remains uniformly valid as the cascade
passes from collisional to collisionless scales and also as it
crosses the ion gyroscale.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 7
In order to elucidate the nature of the turbulence above and
below the ion gyroscale, we derive two tertiary approxima-
tions, one of which is valid for k⊥ρi ≪ 1 (§ 5 and § 6) and the
other for k⊥ρi ≫ 1 (§ 7; see also Appendix C, which gives
a non-rigorous, non-gyrokinetic, but perhaps more intuitive,
derivation of the results of § 4 and § 7.2).
Kinetic Reduced MHD (§ 5 and § 6). — On scales above the ion
gyroscale, known as the “inertial range” we demonstrate that
the decoupling of the Alfvén-wave cascade and its indiffer-
ence to both collisional and collisionless damping are explicit
and analytically provable properties. We show rigorously that
the Alfvén-wave cascade is governed by a closed set of two
fluid-like equations for the stream and flux functions—the Re-
duced Magnetohydrodynamics (RMHD)—independently of
the collisionality (§ 5.3 and § 5.4; the derivation of RMHD
from MHD and its properties are presented in § 2). The cas-
cade proceeds via interaction of oppositely propagating wave
packets and is decoupled from the density and magnetic-field-
strength fluctuations (the “compressive” modes; in the colli-
sional limit, these are the entropy and slow modes; see § 6.1
and Appendix D). The latter are passively mixed by the
Alfvén waves, but, unlike in the fluid (collisional) limit, this
passive cascade is governed by a (simplified) kinetic equa-
tion for the ions (§ 5.5). Together with RMHD, it forms
a hybrid fluid-kinetic description of magnetized turbulence
in a weakly collisional plasma, which we call Kinetic Re-
duced MHD (KRMHD). The KRMHD equations are sum-
marized in § 5.7. Their collisional and collisionless limits
are explored in § 6.1 and § 6.2, respectively. Whereas the
Alfvén waves are undamped in this approximation, the com-
pressive fluctuations are subject to damping both in the col-
lisional (Braginskii 1965 viscous damping, § 6.1.2) and colli-
sionless (Barnes 1966 damping, § 6.2.2) limits. In the colli-
sionless limit, the compressive component of the turbulence is
a simple example of an essentially kinetic turbulence, includ-
ing such features as conservation of generalized energy de-
spite collisionless damping and (parallel) phase mixing, pos-
sibly leading to ion heating (§§ 6.2.3-6.2.5). How strongly
the compressive fluctuations are damped depends on the par-
allel scale of these fluctuations. Since the ion kinetic equation
turns out to be linear along the moving field lines associated
with the Alfvén waves, the compressive fluctuations do not, in
the absence of finite-gyroradius effects, develop small parallel
scales and their cascade may be only weakly damped above
the ion gyroscale—this is discussed in § 6.3.
Electron Reduced MHD (§ 7). — At the ion gyroscale, the
Alfvénic and the compressive cascades are no longer decou-
pled and their energy is partially damped via collisionless
wave–particle interactions (§ 7.1). This part of the energy
is channeled into ion heat. The rest of it is converted into
a cascade of kinetic Alfvén waves (KAW). This cascade ex-
tends through what is known as the “dissipation range” to
the electron gyroscale, where its turn comes to be damped via
wave–particle interaction and transferred into electron heat.
The KAW turbulence is again anisotropic with k‖ ≪ k⊥. It
is governed by a pair of fluid-like equations, also derived
from gyrokinetics. We call them Electron Reduced MHD
(ERMHD). In the high-beta limit, they coincide with the re-
duced (anisotropic) form of the previously known Electron
MHD (Kingsep et al. 1990). The ERMHD equations are de-
rived in § 7.2 (see also Appendix C.2) and the KAW cas-
cade is considered in §§ 7.3-7.5. The fate of the inertial-
range energy collisionlessly damped at the ion gyroscale is
investigated in §§ 7.9-7.11; an analogous consideration for
the KAW energy damped at the electron gyroscale is pre-
sented in § 7.12. In these sections, we introduce the no-
tion of the entropy cascade—a nonlinear phase-mixing pro-
cess whereby the collisionless damping occurring at the ion
and electron gyroscales is made irreversible and particles are
heated. This part of the cascade is purely kinetic and its salient
feature is the particle distribution functions developing small
scales in the gyrokinetic phase space. Note that besides deriv-
ing rigorous sets of equations for the dissipation-range turbu-
lence, § 7 also presents a number of Kolmogorov-style scaling
predictions—both for the KAW cascade (§ 7.5) and for the en-
tropy cascade (§ 7.9.2, § 7.10.2, § 7.10.4, § 7.12).
Hall Reduced MHD (Appendix E). — The reduced (anisotropic)
form of the popular Hall MHD system can be derived as a
special limit of gyrokinetics (k⊥ρi ≪ 1, Ti ≪ Te, βi ≪ 1).
The resulting Hall Reduced MHD (HRMHD) equations
are a convenient model for some purposes because they
simultaneously capture the cold-ion, low-beta limits of both
the KRMHD and ERMHD systems. However, they are
usually not strictly applicable in space and astrophysical
plasmas of interest, where ions are rarely cold and βi is not
particularly low. The HRMHD equations are derived in § E.1,
the kinetic cascade of generalized energy in the Hall limit is
discussed in § E.2, and the circumstances under which the ion
inertial and ion sound scales become important in theories of
plasma turbulence are summarized in § E.4. Theories of the
dissipation-range turbulence based on Hall MHD are briefly
discussed in § 8.2.6.
The regions of validity of the tertiary approximations—
KRMHD and ERMHD—are illustrated in Fig. 2. In this fig-
ure, we also show the region of validity of the RMHD sys-
tem derived from the standard compressible MHD equations
by assuming anisotropy of the turbulence and strong inter-
actions. This derivation is the fluid analog of the derivation
of gyrokinetics. We present it in § 2, before embarking on
the gyrokinetics-based path outlined above, in order to make
a connection with the conventional MHD treatment and to
demonstrate with particular simplicity how the assumption of
anisotropy leads to a reduced fluid system in which the decou-
pling of the cascades of the Alfvén waves and of the compres-
sive modes is manifest (Appendix A extends this derivation
to Braginskii 1965 two-fluid equations in the limit of strong
magnetization; it also works out rigorously the transition from
the fluid limit to the KRMHD equations).
The main formal developments of this paper are contained
in §§ 3-7. The outline given above is meant to help the reader
navigate these sections. In § 8, we discuss at some length
how our results apply to various astrophysical plasmas with
weak collisionality: the solar wind and the magnetosheath,
the ISM, accretion disks, and galaxy clusters (§ 8.1 and § 8.2
can also be read as an overall summary of the paper in light
of the evidence available from space-plasma measurements).
Finally, in § 9, we provide a brief epilogue and make a few
remarks about future directions of inquiry.
2. REDUCED MHD AND THE DECOUPLING OF TURBULENT
CASCADES
8 SCHEKOCHIHIN ET AL.
Consider the equations of compressible MHD
= −ρ∇·u, (7)
B ·∇B
, (8)
= 0, s =
, γ =
, (9)
= B ·∇u − B∇·u, (10)
where ρ is the mass density, u velocity, p pressure, B magnetic
field, s the entropy density, and d/dt = ∂/∂t + u ·∇ (the con-
ditions under which these equations are valid are discussed in
Appendix A). Consider a uniform static equilibrium with a
straight mean field in the z direction, so
ρ = ρ0 + δρ, p = p0 + δp, B = B0ẑ + δB, (11)
where ρ0, p0, and B0 are constants. In what follows, the sub-
scripts ‖ and ⊥ will be used to denote the projections of fields,
variables and gradients on the mean-field direction ẑ and onto
the plane (x,y) perpendicular to this direction, respectively.
2.1. RMHD Ordering
As we explained in the Introduction, observational and nu-
merical evidence makes it safe to assume that the turbulence
in such a system will be anisotropic with k‖ ≪ k⊥ (at scales
smaller than the outer scale, k‖L ≫ 1; see § 1.3 and § 1.5.1).
Let us, therefore, introduce a small parameter ǫ ∼ k‖/k⊥ and
carry out a systematic expansion of Eqs. (7-10) in ǫ. In this
expansion, the fluctuations are treated as small, but not arbi-
trarily so: in order to estimate their size, we shall adopt the
critical-balance conjecture (3), which is now treated not as a
detailed scaling prescription but as an ordering assumption.
This allows us to introduce the following ordering:
∼ δB⊥
∼ ǫ, (12)
where vA = B0/
4πρ0 is the Alfvén speed. Note that this
means that we order the Mach number
, (13)
where cs = (γp0/ρ0)
1/2 is the speed of sound and
is the plasma beta, which is ordered to be order unity in the ǫ
expansion (subsidiary limits of high and low β can be taken
after the ǫ expansion is done; see § 2.4).
In Eq. (12), we made two auxiliary ordering assump-
tions: that the velocity and magnetic-field fluctuations
have the character of Alfvén and slow waves (δB⊥/B0 ∼
u⊥/vA, δB‖/B0 ∼ u‖/vA) and that the relative amplitudes
of the Alfvén-wave-polarized fluctuations (δB⊥/B0, u⊥/vA),
slow-wave-polarized fluctuations (δB‖/B0, u‖/vA) and den-
sity/pressure/entropy fluctuations (δρ/ρ0, δp/p0) are all the
same order. Strictly speaking, whether this is the case depends
on the energy sources that drive the turbulence: as we shall
see, if no slow waves (or entropy fluctuations) are launched,
none will be present. However, in astrophysical contexts, the
outer-scale energy input may be assumed random and, there-
fore, comparable power is injected into all types of fluctua-
tions.
We further assume that the characteristic frequency of the
fluctuations is ω∼ k‖vA [Eq. (3)], meaning that the fast waves,
for which ω ≃ k⊥(v2A + c2s )1/2, are ordered out. This restric-
tion must be justified empirically. Observations of the solar-
wind turbulence confirm that it is primarily Alfvénic (see,
e.g., Bale et al. 2005) and that its compressive component is
substantially pressure-balanced (Roberts 1990; Burlaga et al.
1990; Marsch & Tu 1993; Bavassano et al. 2004, see Eq. (22)
below). A weak-turbulence calculation of compressible MHD
turbulence in low-beta plasmas (Chandran 2005b) suggests
that only a small amount of energy is transferred from the fast
waves to Alfvén waves with large k‖. A similar conclusion
emerges from numerical simulations (Cho & Lazarian 2002,
2003). As the fast waves are also expected to be subject to
strong collisionless damping and/or to strong dissipation after
they steepen into shocks, we eliminate them from our con-
sideration of the problem and concentrate on low-frequency
turbulence.
2.2. Alfvén Waves
We start by observing that the Alfvén-wave-polarized
fluctuations are two-dimensionally solenoidal: since, from
Eq. (7),
∇·u = − d
= O(ǫ2) (15)
and ∇·δB = 0 exactly, separating the O(ǫ) part of these diver-
gences gives ∇⊥ ·u⊥ = 0 and ∇⊥ · δB⊥ = 0. To lowest order
in the ǫ expansion, we may, therefore, express u⊥ and δB⊥ in
terms of scalar stream (flux) functions:
u⊥ = ẑ×∇⊥Φ,
= ẑ×∇⊥Ψ. (16)
Evolution equations for Φ and Ψ are obtained by substituting
the expressions (16) into the perpendicular parts of the induc-
tion equation (10) and the momentum equation (8)—of the
latter the curl is taken to annihilate the pressure term. Keep-
ing only the terms of the lowest order, O(ǫ2), we get
+{Φ,Ψ}= vA
, (17)
∇2⊥Φ+
Φ,∇2⊥Φ
∇2⊥Ψ+
Ψ,∇2⊥Ψ
, (18)
where {Φ,Ψ} = ẑ · (∇⊥Φ×∇⊥Ψ) and we have taken into
account that, to lowest order,
+ u⊥ ·∇⊥ =
+{Φ, · · ·} , (19)
b̂ ·∇= ∂
·∇⊥ =
{Ψ, · · ·} . (20)
Here b̂ = B/B0 is the unit vector along the perturbed field line.
Equations (17-18) are known as the Reduced Magne-
tohydrodynamics (RMHD). The first derivations of these
equations (in the context of fusion plasmas) are due to
Kadomtsev & Pogutse (1974) and to Strauss (1976). These
were followed by many systematic derivations and gener-
alizations employing various versions and refinements of
the basic expansion, taking into account the non-Alfvénic
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 9
modes (which we will do in § 2.4), and including the ef-
fects of spatial gradients of equilibrium fields (e.g., Strauss
1977; Montgomery 1982; Hazeltine 1983; Zank & Matthaeus
1992; Kinney & McWilliams 1997; Bhattacharjee et al. 1998;
Kruger et al. 1998). A comparative review of these expansion
schemes and their (often close) relationship to ours is outside
the scope of this paper. One important point we wish to em-
phasize is that we do not assume the plasma beta [defined in
Eq. (14)] to be either large or small.
Equations (17) and (18) form a closed set, meaning that the
Alfvén-wave cascade decouples from the slow waves and den-
sity fluctuations. It is to the turbulence described by Eqs. (17-
18) that the GS theory outlined in § 1.2 applies.13 In § 5.3, we
will show that Eqs. (17) and (18) correctly describe inertial-
range Alfvénic fluctuations even in a collisionless plasma,
where the full MHD description [Eqs. (7-10)] is not valid.
2.3. Elsasser Fields
The MHD equations (7-10) in the incompressible limit
(ρ = const) acquire a symmetric form if written in terms of
the Elsasser fields z± = u± δB/
4πρ (Elsasser 1950). Let
us demonstrate how this symmetry manifests itself in the re-
duced equations derived above.
We introduce Elsasser potentials ζ± = Φ±Ψ, so that z±⊥ =
ẑ×∇⊥ζ±. For these potentials, Eqs. (17-18) become
∇2⊥ζ±∓ vA
∇2⊥ζ± = −
ζ+,∇2⊥ζ−
ζ−,∇2⊥ζ+
∓∇2⊥ {ζ+, ζ−}
. (21)
These equations show that the RMHD has a simple set of ex-
act solutions: if ζ− = 0 or ζ+ = 0, the nonlinear term vanishes
and the other, non-zero, Elsasser potential is simply a fluc-
tuation of arbitrary shape and magnitude propagating along
the mean field at the Alfvén speed vA: ζ
± = f±(x,y,z∓ vAt).
These solutions are finite-amplitude Alfvén-wave packets of
arbitrary shape. Only counterpropagating such solutions can
interact and thereby give rise to the Alfvén-wave cascade
(Kraichnan 1965). Note that these interactions are conserva-
tive in the sense that the “+” and “−” waves scatter off each
other without exchanging energy.
Note that the individual conservation of the “+” and “−”
waves’ energies means that the energy fluxes associated
with these waves need not be equal, so instead of a sin-
gle Kolmogorov flux ε assumed in the scaling arguments
13 The Alfvén-wave turbulence in the RMHD system has been stud-
ied by many authors. Some of the relevant numerical investigations are
due to Kinney & McWilliams (1998), Dmitruk et al. (2003), Oughton et al.
(2004), Rappazzo et al. (2007, 2008), Perez & Boldyrev (2008, 2009). An-
alytical theory has mostly been confined to the weak-turbulence paradigm
(Ng & Bhattacharjee 1996, 1997; Bhattacharjee & Ng 2001; Galtier et al.
2002; Lithwick & Goldreich 2003; Galtier & Chandran 2006; Nazarenko
2008). We note that adopting the critical balance [Eq. (3)] as an ordering
assumption for the expansion in k‖/k⊥ does not preclude one from subse-
quently attempting a weak-turbulence approach: the latter should simply be
treated as a subsidiary expansion. Indeed, implementing the anisotropy as-
sumption on the level of MHD equations rather than simultaneously with
the weak-turbulence closure (Galtier et al. 2000) significantly reduces the
amount of algebra. One should, however, bear in mind that the weak-
turbulence approximation always breaks down at some sufficiently small
scale—namely, when k⊥ ∼ (vA/U)
L, where L is the outer scale of the
turbulence, U velocity at the outer scale, and k‖ the parallel wavenum-
ber of the Alfvén waves (see Goldreich & Sridhar 1997 or the review by
Schekochihin & Cowley 2007). Below this scale, interactions cannot be as-
sumed weak.
reviewed in § 1.2, we could have ε+ 6= ε−. The GS the-
ory can be generalized to this case of imbalanced Alfvénic
cascades (Lithwick et al. 2007; Beresnyak & Lazarian 2008a;
Chandran 2008), but here we will focus on the balanced tur-
bulence, ε+ ∼ ε−. If one considers the turbulence forced
in a physical way (i.e., without forcing the magnetic field,
which would break the flux conservation), the resulting cas-
cade would always be balanced. In the real world, imbal-
anced Alfvénic fluxes are measured in the fast solar wind,
where the influence of initial conditions in the solar atmo-
sphere is more pronounced, while the slow-wind turbulence
is approximately balanced (Marsch & Tu 1990a; see also re-
views by Tu & Marsch 1995; Bruno & Carbone 2005 and ref-
erences therein).
2.4. Slow Waves and the Entropy Mode
In order to derive evolution equations for the remaining
MHD modes, let us first revisit the perpendicular part of the
momentum equation and use Eq. (12) to order terms in it. In
the lowest order, O(ǫ), we get the pressure balance
B0δB‖
= 0 ⇒ δp
. (22)
Using Eq. (22) and the entropy equation (9), we get
, (23)
where s0 = p0/ρ
0 . Now, substituting Eq. (15) for ∇·u in the
parallel component of the induction equation (10), we get
− b̂ ·∇u‖ = 0. (24)
Combining Eqs. (23) and (24), we obtain
1 + c2s/v
b̂ ·∇u‖, (25)
1 + v2A/c2s
b̂ ·∇u‖. (26)
Finally, we take the parallel component of the momentum
equation (8) and notice that, due to the pressure balance (22)
and to the smallness of the parallel gradients, the pressure
term is O(ǫ3), while the inertial and tension terms are O(ǫ2).
Therefore,
= v2Ab̂ ·∇
. (27)
Equations (26-27) describe the slow-wave-polarized fluctu-
ations, while Eq. (23) describes the zero-frequency entropy
mode, which is decoupled from the slow waves.14 The non-
linearity in Eqs. (26-27) enters via the derivatives defined in
14 For other expansion schemes leading to reduced sets of equations for
these “compressive” fluctuations see references in § 2.2. Note that the na-
ture of the density fluctuations described above is distinct from the so called
“pseudosound” density fluctuations that arise in the “nearly incompress-
ible” MHD theories (Montgomery et al. 1987; Matthaeus & Brown 1988;
Matthaeus et al. 1991; Zank & Matthaeus 1993). The “pseudosound” is es-
sentially the density response caused by the nonlinear pressure fluctuations
calculated from the incompressibility constraint. The resulting density fluc-
tuations are second order in Mach number and, therefore, order ǫ2 in our
expansion [see Eq. (13)]. The passive density fluctuations derived in this sec-
tion are order ǫ and, therefore, supersede the “pseudosound” (see review by
Tu & Marsch 1995 for a discussion of the relevant solar-wind evidence).
10 SCHEKOCHIHIN ET AL.
Eqs. (19-20) and is due solely to interactions with Alfvén
waves. Thus, both the slow-wave and the entropy-mode cas-
cades occur via passive scattering/mixing by Alfvén waves, in
the course of which there is no energy exchange between the
cascades.
Note that in the high-beta limit, cs ≫ vA [see Eq. (14)], the
entropy mode is dominated by density fluctuations [Eq. (23),
cs ≫ vA], which also decouple from the slow-wave cascade
[Eq. (25), cs ≫ vA]. and are passively mixed by the Alfvén-
wave turbulence:
= 0. (28)
The high-beta limit is equivalent to the incompressible ap-
proximation for the slow waves.
In § 5.5, we will derive a kinetic description for the inertial-
range compressive fluctuations (density and magnetic-field
strength), which is more generally valid in weakly collisional
plasmas and which reduces to Eqs. (26-27) in the collisional
limit (see Appendix D). While these fluctuations will in gen-
eral satisfy a kinetic equation, they will remain passive with
respect to the Alfvén waves.
2.5. Elsasser Fields for the Slow Waves
The original Elsasser (1950) symmetry was derived for in-
compressible MHD equations. However, for the “compres-
sive” slow-wave fluctuations, we may introduce generalized
Elsasser fields:
= u‖±
. (29)
Straightforwardly, the evolution equation for these fields is
∓ vA√
1 + v2A/c2s
1∓ 1√
1 + v2A/c2s
ζ+,z±
1± 1√
1 + v2A/c2s
ζ−,z±
. (30)
In the high-beta limit (vA ≪ cs), the generalized Elsasser
fields (29) become the parallel components of the conven-
tional incompressible Elsasser fields. We see that only in this
limit do the slow waves interact exclusively with the counter-
propagating Alfvén waves, and so only in this limit does set-
ting ζ− = 0 or ζ+ = 0 gives rise to finite-amplitude slow-wave-
packet solutions z±
= f±(x,y,z∓ vAt) analogous to the finite-
amplitude Alfvén-wave packets discussed in § 2.3.15 For gen-
eral β, the phase speed of the slow waves is smaller than that
of the Alfvén waves and, therefore, Alfvén waves can “catch
up” and interact with the slow waves that travel in the same
direction. All of these interactions are of scattering type and
involve no exchange of energy.
2.6. Scalings for Passive Fluctuations
15 Obviously, setting both ζ± = 0 does always enable these finite-
amplitude slow-wave solutions. More non-trivially, such finite-amplitude so-
lutions exist in the Lagrangian frame associated with the Alfvén waves—this
is discussed in detail in § 6.3.
The scaling of the passively mixed scalar fields introduced
above is slaved to the scaling of the Alfvénic fluctuations.
Consider for example the entropy mode [Eq. (23)]. As
in Kolmogorov–Obukhov theory (see § 1.1), one assumes a
local-in-scale-space cascade of scalar variance and a constant
flux εs of this variance. Then, analogously to Eq. (1),
v2thi
∼ εs. (31)
Since the cascade time is τ−1λ ∼ u⊥ ·∇⊥ ∼ vA/l‖λ ∼ ε/u2⊥λ,
)1/2 u⊥λ
, (32)
so the scalar fluctuations have the same scaling as the turbu-
lence that mixes them (Obukhov 1949; Corrsin 1951). In GS
turbulence, the scalar-variance spectrum should, therefore, be
⊥ (Lithwick & Goldreich 2001). The same argument ap-
plies to all passive fields.
It is the (presumably) passive electron-density spectrum
that provides the main evidence of the k−5/3 scaling in the in-
terstellar turbulence (Armstrong et al. 1981, 1995; Lazio et al.
2004, see further discussion in § 8.4.1). The explanation of
this spectrum in terms of passive mixing of the entropy mode,
originally proposed by Higdon (1984), was developed on the
basis of the GS theory by Lithwick & Goldreich (2001). The
turbulent cascade of the compressive fluctuations and the rel-
evant solar-wind data is discussed further in § 6.3. In partic-
ular, it will emerge that the anisotropy of these fluctuations
remains a non-trivial issue: is there an analog of the scaling
relation (5)? The scaling argument outlined above does not
invoke any assumptions about the relationship between the
parallel and perpendicular scales of the compressive fluctu-
ations (other than the assumption that they are anisotropic).
Lithwick & Goldreich (2001) argue that the parallel scales of
the Alfvénic fluctuations will imprint themselves on the pas-
sively advected compressive ones, so Eq. (5) holds for the
latter as well. In § 6.3, we examine this conclusion in view
of the solar-wind evidence and of the fact that the equations
for the compressive modes become linear in the Lagrangian
frame associated with the Alfvénic turbulence.
2.7. Five RMHD Cascades
Thus, the anisotropy and critical balance (3) taken as
ordering assumptions lead to a neat decomposition of the
MHD turbulent cascade into a decoupled Alfvén-wave cas-
cade and cascades of slow waves and entropy fluctuations pas-
sively scattered/mixed by the Alfvén waves. More precisely,
Eqs. (23), (21) and (30) imply that, for arbitrary β, there are
five conserved quantities:16
W±AW =
d3rρ0|∇⊥ζ±|2 (Alfven waves), (33)
W±sw =
d3rρ0|z±‖ |
2 (slow waves), (34)
(entropy fluctuations).(35)
16 Note that magnetic helicity of the perturbed field is not an invariant of
RMHD, except in two dimensions (see Appendix F.4). In 2D, there is also
conservation of the mean square flux,
d3r |Ψ|2 (see Appendix F.2).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 11
W +AW and W
AW are always cascaded by interaction with each
other, Ws is passively mixed by W
AW and W
AW, W
sw are pas-
sively scattered by W∓AW and, unless β ≫ 1, also by W
This is an example of splitting of the overall energy cascade
into several channels (recovered as a particular case of the
more general kinetic cascade in Appendix D.2)—a concept
that will repeatedly arise in the kinetic treatment to follow.
The decoupling of the slow- and Alfvén-wave cascades in
MHD turbulence was studied in some detail and confirmed
in direct numerical simulations by Maron & Goldreich (2001,
for β ≫ 1) and by Cho & Lazarian (2002, 2003, for a range
of values of β). The derivation given in § 2.2 and § 2.4 (cf.
Lithwick & Goldreich 2001) provides a straightforward theo-
retical basis for these results, assuming anisotropy of the tur-
bulence (which was also confirmed in these numerical stud-
ies).
It turns out that the decoupling of the Alfvén-wave cascade
that we demonstrated above for the anisotropic MHD turbu-
lence is a uniformly valid property of plasma turbulence at
both collisional and collisionless scales and that this cascade
is correctly described by the RMHD equations (17-18) all the
way down to the ion gyroscale, while the fluctuations of den-
sity and magnetic-field strength do not satisfy simple fluid
evolution equations anymore and require solving the kinetic
equation. In order to prove this, we adopt a kinetic descrip-
tion and apply to it the same ordering (§ 2.1) as we used to
reduce the MHD equations. The kinetic theory that emerges
as a result is called gyrokinetics.
3. GYROKINETICS
The gyrokinetic formalism was first worked out
for linear waves by Rutherford & Frieman (1968)
and by Taylor & Hastie (1968) (see also Catto 1978;
Antonsen & Lane 1980; Catto et al. 1981) and subsequently
extended to the nonlinear regime by Frieman & Chen (1982).
Rigorous derivations of the gyrokinetic equation based on
the Hamiltonian formalism were developed by Dubin et al.
(1983, electrostatic) and Hahm et al. (1988, electromagnetic).
This approach is reviewed in Brizard & Hahm (2007). A
more pedestrian, but perhaps also more transparent exposition
of the gyrokinetics in a straight mean field can be found in
Howes et al. (2006), who also provide a detailed explanation
of the gyrokinetic ordering in the context of astrophysical
plasma turbulence and a treatment of the linear waves and
damping rates. Here we review only the main points so as
to allow the reader to understand the present paper without
referring elsewhere.
In general, a plasma is completely described by the distribu-
tion function fs(t,r,v)—the probability density for a particle
of species s (= i,e) to be found at the spatial position r mov-
ing with velocity v. This function obeys the kinetic Vlasov–
Landau (or Boltzmann) equation
+ v ·∇ fs +
· ∂ fs
, (36)
where qs and ms are the particle’s charge and mass, c is the
speed of light, and the right-hand side is the collision term
(quadratic in f ). The electric and magnetic fields are
E = −∇ϕ−
, B = ∇×A. (37)
The first equality is Faraday’s law uncurled, the second
the magnetic-field solenoidality condition; we shall use the
Coulomb gauge, ∇·A = 0. The fields satisfy the Poisson and
the Ampère–Maxwell equations with the charge and current
densities determined by fs(t,r,v):
∇·E = 4π
qsns = 4π
d3v fs, (38)
∇×B − 1
d3vv fs. (39)
3.1. Gyrokinetic Ordering and Dimensionless Parameters
As in § 2 we set up a static equilibrium with a uniform mean
field, B0 = B0ẑ, E0 = 0, assume that the perturbations will be
anisotropic with k‖ ≪ k⊥ (at scales smaller than the outer
scale, k‖L ≫ 1; see § 1.3 and § 1.5.1), and construct an expan-
sion of the kinetic theory around this equilibrium with respect
to the small parameter ǫ ∼ k‖/k⊥. We adopt the ordering ex-
pressed by Eqs. (3) and (12), i.e., we assume the perturbations
to be strongly interacting Alfvén waves plus electron density
and magnetic-field-strength fluctuations.
Besides ǫ, several other dimensionless parameters are
present, all of which are formally considered to be of order
unity in the gyrokinetic expansion: the electron–ion mass ra-
tio me/mi, the charge ratio
Z = qi/|qe| = qi/e (40)
(for hydrogen, this is 1, which applies to most astrophysical
plasmas of interest to us), the temperature ratio17
τ = Ti/Te, (41)
and the plasma (ion) beta
v2thi
8πniTi
, (42)
where vthi = (2Ti/mi)
1/2 is the ion thermal speed and the total
β was defined in Eq. (14) based on the total pressure p = niTi +
neTe. We shall occasionally also use the electron beta
8πneTe
βi. (43)
The total beta is β = βi +βe.
3.1.1. Wavenumbers and Frequencies
As we want our theory to be uniformly valid at all (perpen-
dicular) scales above, at or below the ion gyroscale, we order
k⊥ρi ∼ 1, (44)
where ρi = vthi/Ωi is the ion gyroradius, Ωi = qiB0/cmi the ion
cyclotron frequency. Note that
ρi. (45)
17 It can be shown that equilibrium temperatures change on the timescale
∼ (ǫ2ω)−1 (Howes et al. 2006). On the other hand, from standard theory
of collisional transport (e.g., Helander & Sigmar 2002), the ion and elec-
tron temperatures equalize on the timescale ∼ ν−1ie ∼ (mi/me)
1/2ν−1ii [see
Eq. (51)]. Therefore, τ can depart from unity by an amount of order
ǫ2(ω/νii)(mi/me)1/2. In our ordering scheme [Eq. (49)], this is O(ǫ2) and,
therefore, we should simply set τ = 1 + O(ǫ2). However, we shall carry the
parameter τ because other ordering schemes are possible that permit arbitrary
values of τ . These are appropriate to plasmas with very weak collisions. For
example, in the solar wind, τ appears to be order unity but not exactly 1
(Newbury et al. 1998), while in accretion flows near the black hole, some
models predict τ ≫ 1 (see § 8.5).
12 SCHEKOCHIHIN ET AL.
FIG. 3.— Regions of validity in the wavenumber space of two primary approximations—the two-fluid (Appendix A.1) and gyrokinetic (§ 3). The gyrokinetic
theory holds when k‖ ≪ k⊥ and ω ≪ Ωi [when k‖ ≪ k⊥ < ρ
i , the second requirement is automatically satisfied for Alfvén, slow and entropy modes; see
Eq. (46)]. The two-fluid equations hold when k‖λmfpi ≪ 1 (collisional limit) and k⊥ρi ≪ 1 (magnetized plasma). Note that the gyrokinetic theory holds for all
but the very largest (outer) scales, where anisotropy cannot be assumed.
Assuming Alfvénic frequencies implies
∼ k⊥ρi√
ǫ. (46)
Thus, gyrokinetics is a low-frequency limit that averages over
the timescales associated with the particle gyration. Because
we have assumed that the fluctuations are anisotropic and have
(by order of magnitude) Alfvénic frequencies, we see from
Eq. (46) that their frequency remains far below Ωi at all scales,
including the ion and even electron gyroscale—the gyroki-
netics remains valid at all of these scales and the cyclotron-
frequency effects are negligible (cf. Quataert & Gruzinov
1999).
3.1.2. Fluctuations
Equation (3) allows us to order the fluctuations of the scalar
potential: on the one hand, we have from Eq. (3) u⊥ ∼ ǫvA; on
the other hand, the plasma mass flow velocity is (to the lowest
order) the E×B drift velocity of the ions, u⊥ ∼ cE⊥/B0 ∼
ck⊥ϕ/B0, so
ǫ. (47)
All other fluctuations (magnetic, density, parallel velocity) are
ordered according to Eq. (12).
Note that the ordering of the flow velocity dictated by
Eq. (3) means that we are considering the limit of small Mach
numbers:
M ∼ u
. (48)
This means that the gyrokinetic description in the form used
below does not extend to large sonic flows that can be
present in many astrophysical systems. It is, in principle,
possible to extend the gyrokinetics to systems with sonic
flows (e.g., in the toroidal geometry; see Artun & Tang 1994;
Sugama & Horton 1997). However, we do not follow this
route because such flows belong to the same class of non-
universal outer-scale features as background density and tem-
perature gradients, system-specific geometry etc.—these can
all be ignored at small scales, where the turbulence should be
approximately homogeneous and subsonic (as long as k‖L ≫
1, see discussion in § 1.5.1).
3.1.3. Collisions
Finally, we want our theory to be valid both in the colli-
sional and the collisionless regimes, so we do not assume
ω to be either smaller or larger than the (ion) collision fre-
quency νii:
k‖λmfpi√
∼ 1, (49)
where λmfpi = vthi/νii is the ion mean free path (this order-
ing can actually be inferred from equating the gyrokinetic en-
tropy production terms to the collisional entropy production;
see extended discussion in Howes et al. 2006). Note that the
ordering (49) holds on the understanding that we have ordered
k⊥ρi ∼ 1 [Eq. (44)] because the fluctuation frequency can de-
pend on k⊥ρi in the dissipation range (see § 7.3).
Other collision rates are related to νii via a set of standard
formulae (see, e.g., Helander & Sigmar 2002), which will be
useful in what follows:
νei = Zνee =
τ 3/2
νii, (50)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 13
νie =
τ 3/2
νii, (51)
νii =
2πZ4e4ni lnΛ
, (52)
where lnΛ is the Coulomb logarithm and the numerical factor
in the definition of νie has been inserted for future notational
convenience (see Appendix A). We always define
λmfpi =
, λmfpe =
λmfpi. (53)
The ordering of the collision frequency expressed by
Eq. (49) means that collisions, while not dominant as in
the fluid description (Appendix A), are still retained in
the version of the gyrokinetic theory adopted by us. Their
presence is required in order for us to be able to assume
that the equilibrium distribution is Maxwellian [Eq. (54)
below] and for the heating and entropy production to be
treated correctly (§ 3.4 and § 3.5). However, our ordering of
collisions and of the fluctuation amplitudes (§ 3.1.2) imposes
certain limitations: thus, we cannot treat the class of nonlinear
phenomena involving particle trapping by parallel-varying
fluctuations, non-Maxwellian tails of particle distributions,
plasma instabilities arising from the equilibrium pressure
anisotropies (mirror, firehose) and their possible nonlinear
evolution to large amplitudes (see discussion in § 8.3).
The region of validity of the gyrokinetic approximation in
the wavenumber space is illustrated in Fig. 3—it embraces
all of the scales that are expected to be traversed by the
anisotropic energy cascade (except the scales close to the
outer scale).
As we explained above, me/mi, βi, k⊥ρi and k‖λmfpi (or
ω/νii) are assigned order unity in the gyrokinetic expansion.
Subsidiary expansions in small me/mi (§ 4) and in small or
large values of the other three parameters (§§ 5-7) can be car-
ried out at a later stage as long as their values are not so large
or small as to interfere with the primary expansion in ǫ. These
expansions will yield simpler models of turbulence with more
restricted domains of validity than gyrokinetics.
3.2. Gyrokinetic Equation
Given the gyrokinetic ordering introduced above, the ex-
pansion of the distribution function up to first order in ǫ can
be written as
fs(t,r,v) = F0s(v) −
qsϕ(t,r)
F0s(v) + hs(t,Rs,v⊥,v‖). (54)
To zeroth order, it is a Maxwellian:18
F0s(v) =
(πv2ths)
v2ths
, vths =
, (55)
with uniform density n0s and temperature T0s and no mean
flow. As will be explained in more detail in § 3.5, F0s has a
slow time dependence via the equilibrium temperature, T0s =
T0s(ǫ
2t). This reflects the slow heating of the plasma as the tur-
bulent energy is dissipated. However, T0s can be treated as a
constant with respect to the time dependence of the first-order
18 The use of isotropic equilibrium is a significant idealization—this is
discussed in more detail in § 8.3.
distribution function (the timescale of the turbulent fluctua-
tions). The first-order part of the distribution function is com-
posed of the Boltzmann response [second term in Eq. (54), or-
dered in Eq. (47)] and the gyrocenter distribution function hs.
The spatial dependence of the latter is expressed not by the
particle position r but by the position Rs of the particle gy-
rocenter (or guiding center)—the center of the ring orbit that
the particle follows in a strong guide field:
Rs = r +
v⊥× ẑ
. (56)
Thus, some of the velocity dependence of the distribution
function is subsumed in the Rs dependence of hs. Explicitly,
hs depends only on two velocity-space variables: it is cus-
tomary in the gyrokinetic literature for these to be chosen as
the particle energy εs = msv
2/2 and its first adiabatic invari-
ant µs = msv
⊥/2B0 (both conserved quantities to two lowest
orders in the gyrokinetic expansion). However, in a straight
uniform guide field B0ẑ, the pair (v⊥,v‖) is a simpler choice,
which will mostly be used in what follows (we shall some-
times find an alternative pair, v and ξ = v‖/v, useful, especially
where collisions are concerned). It must be constantly kept in
mind that derivatives of hs with respect to the velocity-space
variables are taken at constant Rs, not at constant r.
The function hs satisfies the gyrokinetic equation:
{〈χ〉Rs ,hs} =
qsF0s
∂〈χ〉Rs
where
χ(t,r,v) = ϕ−
v⊥ ·A⊥
, (58)
the Poisson brackets are defined in the usual way:
{〈χ〉Rs ,hs} = ẑ ·
∂〈χ〉Rs
× ∂hs
, (59)
and the ring average notation is introduced:
〈χ(t,r,v)〉Rs =
t,Rs −
v⊥× ẑ
, (60)
where ϑ is the angle in the velocity space taken in the plane
perpendicular to the guide field B0ẑ. Note that, while χ is
a function of r, its ring average is a function of Rs. Note
also that the ring averages depend on the species index, as
does the gyrocenter variable Rs. Equation (57) is derived by
transforming the first-order kinetic equation to the gyrocenter
variable (56) and ring averaging the result (see Howes et al.
2006, or the references given at the beginning of § 3). The
ring-averaged collision integral (∂hs/∂t)c is discussed in Ap-
pendix B.
3.3. Field Equations
To Eq. (57), we must append the equations that determine
the electromagnetic field, namely, the potentials ϕ(t,r) and
A(t,r) that enter the expression for χ [Eq. (58)]. In the
non-relativistic limit (vthi ≪ c), these are the plasma quasi-
neutrality constraint [which follows from the Poisson equa-
tion (38) to lowest order in vthi/c]:
qsδns =
n0s +
d3v〈hs〉r
14 SCHEKOCHIHIN ET AL.
and the parallel and perpendicular parts of Ampère’s law
[Eq. (39) to lowest order in ǫ and in vthi/c]:
∇2⊥A‖ = −
j‖ = −
d3vv‖〈hs〉r, (62)
∇2⊥δB‖ = −
∇⊥× j⊥
d3v〈v⊥hs〉r
, (63)
where we have used δB‖ = ẑ · (∇⊥×A⊥) and dropped the dis-
placement current. Since field variables ϕ, A‖ and δB‖ are
functions of the spatial variable r, not of the gyrocenter vari-
able Rs, we had to determine the contribution from the gy-
rocenter distribution function hs to the charge distribution at
fixed r by performing a gyroaveraging operation dual to the
ring average defined in Eq. (60):
〈hs(t,Rs,v⊥,v‖)〉r =
t,r +
v⊥× ẑ
,v⊥,v‖
In other words, the velocity-space integrals in Eqs. (61-63)
are performed over hs at constant r, rather than constant Rs.
If we Fourier transform hs in Rs, the gyroaveraging operation
takes a simple mathematical form:
〈hs〉r =
〈eik·Rs〉rhsk(t,v⊥,v‖)
eik·r
ik · v⊥× ẑ
hsk(t,v⊥,v‖)
eik·rJ0(as)hsk(t,v⊥,v‖), (65)
where as = k⊥v⊥/Ωs and J0 is a Bessel function that arose
from the angle integral in the velocity space. In Eq. (63), an
analogous calculation taking into account the angular depen-
dence of v⊥ leads to
δB‖ = −
eik·r
d3vmsv
J1(as)
hsk(t,v⊥,v‖).
Note that Eq. (63) [and, therefore, Eq. (66)] is the gyroki-
netic equivalent of the perpendicular pressure balance that ap-
peared in § 2 [Eq. (22)]:
B0δB‖
= ∇⊥ ·
d3v〈ẑ× v⊥hs〉r
= ∇⊥ ·
t,r +
v⊥× ẑ
,v⊥,v‖
= −∇⊥∇⊥ :
d3vms〈v⊥v⊥ hs〉r = −∇⊥∇⊥ : δP⊥,(67)
where we have integrated by parts with respect to the gyroan-
gle ϑ and used ∂v⊥/∂ϑ = ẑ× v⊥, ∂2v⊥/∂ϑ2 = −v⊥ (cf. the
Appendix of Roach et al. 2005).
Once the fields are determined, they have to be substi-
tuted into χ [Eq. (58)] and the result ring averaged [Eq. (60)].
Again, we emphasize that ϕ, A‖ and δB‖ are functions of r,
while 〈χ〉Rs is a function of Rs. The transformation is ac-
complished via a calculation analogous to the one that led to
Eqs. (65) and (66):
〈χ〉Rs =
eik·Rs〈χ〉Rs ,k, (68)
〈χ〉Rs ,k = J0(as)
v‖A‖k
v2ths
J1(as)
. (69)
The last equation establishes a correspondence between the
Fourier transforms of the fields with respect to r and the
Fourier transform of 〈χ〉Rs with respect to Rs.
3.4. Generalized Energy and the Kinetic Cascade
As promised in § 1.4, the central unifying concept of this
paper is now introduced.
If we multiply the gyrokinetic equation (57) by T0shs/F0s
and integrate over the velocities and gyrocenters, we find that
the nonlinear term conserves the variance of hs and
d3Rs qs
∂〈χ〉Rs
T0shs
. (70)
Let us now sum this equation over all species. The first term
on the right-hand side is
∂〈χ〉Rs
d3v〈hs〉r −
d3v〈vhs〉r
d3rE · j, (71)
where we have used Eq. (61) and Ampère’s law [Eqs. (62-
63)] to express the integrals of hs. The second term on the
right-hand side is the total work done on plasma per unit time.
Using Faraday’s law [Eq. (37)] and Ampère’s law [Eq. (39)],
it can be written as
d3rE · j = − d
|δB|2
+ Pext, (72)
where Pext ≡ −
d3rE · jext is the total power injected into the
system by the external energy sources (outer-scale stirring; in
terms of the Kolmogorov energy flux ε used in the scaling
arguments in § 1.2, Pext = Vmin0iε, where V is the system vol-
ume). Combining Eqs. (70-72), we find (Howes et al. 2006)
T0s〈h2s 〉r
|δB|2
= Pext +
T0shs
. (73)
W is a positive definite quantity—this becomes explicit if we
use Eq. (61) to express it in terms of the total perturbed distri-
bution function δ fs = −qsϕF0s/T0s + hs [see Eq. (54)]:
T0sδ f
|δB|2
. (74)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 15
We will refer to W as the generalized energy. We use this
term to emphasize the role of W as the cascaded quantity in
gyrokinetic turbulence (see below). This quantity is, in fact,
the gyrokinetic version of a collisionless kinetic invariant var-
iously referred to as the generalized grand canonical poten-
tial (see Hallatschek 2004, who points out the fundamental
role of this quantity in plasma turbulence simulations) or free
energy (e.g., Fowler 1968; Scott 2007). The non-magnetic
part of W is related to the perturbed entropy of the sys-
tem (Krommes & Hu 1994; Sugama et al. 1996; Howes et al.
2006; Schekochihin et al. 2008b, see discussion in § 3.5).19
Equation (73) is a conservation law of the generalized en-
ergy: Pext is the source and the second term on the right-hand
side, which is negative definite, represents collisional dissi-
pation. This suggests that we might think of kinetic plasma
turbulence in terms of the generalized energy W injected by
the outer-scale stirring and dissipated by collisions. In or-
der for the dissipation to be important, the collisional term in
Eq. (73) has to become comparable to Pext. This can happen
in two ways:
1. At collisional scales (k‖λmfpi ∼ 1) due to deviations of
the perturbed distribution function from a local per-
turbed Maxwellian (see § 6.1 and Appendix D);
2. At collisionless scales (k‖λmfpi ≫ 1) due the develop-
ment of small scales in the velocity space—large gra-
dients in v‖ (see § 6.2.4) or v⊥ (which is accompanied
by the development of small perpendicular scales in the
position space; see § 7.9.1).
Thus, the dissipation is only important at particular (small)
scales, which are generally well separated from the outer
scale. The generalized energy is transferred from the outer
scale to the dissipation scales via a nonlinear cascade. We
shall call it the kinetic cascade. It is analogous to the energy
cascade in fluid or MHD turbulence, but a conceptually new
feature is present: the small scales at which dissipation hap-
pens are small scales both in the velocity and position space.
Whereas the large gradients in v‖ are produced by the lin-
ear parallel phase mixing, whose role in the kinetic dissipa-
tion processes has been appreciated for some time (Landau
1946; Hammett et al. 1991; Krommes & Hu 1994; Krommes
1999; Watanabe & Sugama 2004, see § 6.2.4), the emergence
of large gradients in v⊥ is due to an essentially nonlinear
phase mixing mechanism (§ 7.9.1). At spatial scales smaller
than the ion gyroradius, this nonlinear perpendicular phase
mixing turns out to be a faster and, therefore, presumably the
dominant way of generating small-scale structure in the veloc-
ity space. It was anticipated in the development of gyrofluid
moment hierarchies by Dorland & Hammett (1993). Here we
treat it for the first time as a phase-space turbulent cascade:
this is done in § 7.9 and § 7.10 (see also Schekochihin et al.
2008b).
In the sections that follow, we shall derive particular forms
of W for various limiting cases of the gyrokinetic theory
(§ 4.7, § 5.6, § 6.2.5, § 7.8, Appendices D.2 and E.2). We
shall see that the kinetic cascade of W is, indeed, a direct
generalization of the more familiar fluid cascades (such as
19 Note also that a quadratic form involving both the perturbed distribution
function and the electromagnetic field appears, in a more general form than
Eq. (74), in the formulation of the energy principle for the Kinetic MHD
approximation (Kruskal & Oberman 1958; Kulsrud 1962, 1964). Regarding
the relationship between Kinetic MHD and gyrokinetics, see footnote 23.
the RMHD cascades discussed in § 2) and that W contains
the energy invariants of the fluid models in the appropriate
limits. In these limits, the cascade of the generalized en-
ergy will split into several decoupled cascades, as it did in
the case of RMHD (§ 2.7). Whenever one of the physically
important scales (§ 1.5.2) is crossed and a change of physical
regime occurs, these cascades are mixed back together into
the overall kinetic cascade of W , which can then be split in
a different way as it emerges on the “opposite side” of the
transition region in the scale space. The conversion of the
Alfvénic cascade into the KAW cascade and the entropy cas-
cade at k⊥ρi ∼ 1 is the most interesting example of such a
transition, discussed in § 7.
The generalized energy appears to be the only quadratic
invariant of gyrokinetics in three dimensions; in two dimen-
sions, many other invariants appear (see Appendix F).
3.5. Heating and Entropy
In a stationary state, all of the the turbulent power injected
by the external stirring is dissipated and thus transferred into
heat. Mathematically, this is expressed as a slow increase in
the temperature of the Maxwellian equilibrium. In gyrokinet-
ics, the heating timescale is ordered as ∼ (ǫ2ω)−1.
Even though the dissipation of turbulent fluctuations may
be occurring “collisionlessly” at scales such that k‖λmfpi ≫ 1
(e.g., via wave–particle interaction at the ion gyroscale; § 7.1),
the resulting heating must ultimately be effected with the help
of collisions. This is because heating is an irreversible process
and it is a small amount of collisions that make “collisionless”
damping irreversible. In other words, slow heating of the
Maxwellian equilibrium is equivalent to entropy production
and Boltzmann’s H-theorem rigorously requires collisions to
make this possible. Indeed, the total entropy of species s is
Ss = −
d3v fs ln fs
F0s lnF0s +
δ f 2s
+ O(ǫ3), (75)
where we took
d3rδ fs = 0. It is then not hard to show that
T0shs
where the overlines mean averaging over times longer than
the characteristic time of the turbulent fluctuations ∼ ω−1
but shorter than the typical heating time ∼ (ǫ2ω)−1 (see
Howes et al. 2006; Schekochihin et al. 2008b for a detailed
derivation of this and related results on heating in gyroki-
netics; see also earlier discussions of the entropy production
in gyrokinetics by Krommes & Hu 1994; Krommes 1999;
Sugama et al. 1996). We have omitted the term describing the
interspecies collisional temperature equalization. Note that
both sides of Eq. (76) are order ǫ2ω.
If we now time average Eq. (73) in a similar fashion, the
left-hand side vanishes because it is a time derivative of a
quantity fluctuating on the timescale ∼ ω−1 and we confirm
that the right-hand side of Eq. (76) is simply equal to the av-
erage power Pext injected by external stirring. The import of
Eq. (76) is that it tells us that heating can only be effected
by collisions, while Eq. (73) implies that the injected power
gets to the collisional scales in velocity and position space by
means of a kinetic cascade of generalized energy.
16 SCHEKOCHIHIN ET AL.
The first term in the expression for the generalized energy
(74) is −
s T0sδSs, where δSs is the perturbed entropy [see
Eq. (75)]. The second term in Eq. (74) is magnetic energy.
Collisionless damping of electromagnetic fluctuations can be
thought of as a redistribution of the generalized energy, trans-
ferring the electromagnetic energy into entropy fluctuations,
while the total W is conserved (a simple example of how that
happens for collisionless compressive fluctuations in the iner-
tial range is worked out in § 6.2.3).
The contribution to the perturbed entropy from the gy-
rocenter distribution is the integral of −h2s/2F0s, whose
evolution equation (70) can be viewed as the gyrokinetic
version of the H-theorem. The first term on the right-hand
side of this equation represents the wave–particle interaction
(collisionless damping). Under time average, it is related to
the work done on plasma [Eq. (71)] and hence to the average
externally injected power Pext via time-averaged Eq. (72).
In a stationary state, this is balanced by the second term in the
right-hand side of Eq. (70), which is the collisional-heating,
or entropy-production, term that also appears in Eq. (76).
Thus, the generalized energy channeled by collisionless
damping into entropy fluctuations is eventually converted
into heat by collisions. The sub-gyroscale entropy cascade,
which brings the perturbed distribution function hs to col-
lisional scales, will be discussed further in § 7.9 and § 7.10
(see also Schekochihin et al. 2008b).
This concludes a short primer on gyrokinetics necessary
(and sufficient) for adequate understanding of what is to fol-
low. Formally, all further analytical derivations in this paper
are simply subsidiary expansions of the gyrokinetics in the pa-
rameters we listed in § 3.1: in § 4, we expand in (me/mi)
in § 5 in k⊥ρi (followed by further subsidiary expansions in
large and small k‖λmfpi in § 6), and in § 7 in 1/k⊥ρi.
4. ISOTHERMAL ELECTRON FLUID
In this section, we carry out an expansion of the electron gy-
rokinetic equation in powers of (me/mi)
1/2 ≃ 0.02 (for hydro-
gen plasma). In virtually all cases of interest, this expansion
can be done while still considering
βi, k⊥ρi, and k‖λmfpi to
be order unity.21 Note that the assumption k⊥ρi ∼ 1 together
with Eq. (45) mean that
k⊥ρe ∼ k⊥ρi(me/mi)1/2 ≪ 1, (77)
i.e., the expansion in (me/mi)
1/2 means also that we are
considering scales larger than the electron gyroradius. The
idea of such an expansion of the electron kinetic equation
has been utilized many times in plasma physics literature.
The mass-ratio expansion of the gyrokinetic equation in a
form very similar to what is presented below is found in
Snyder & Hammett (2001).
20 Note that Eq. (72) is valid not only in the integral form but also indi-
vidually for each wavenumber: indeed, using the Fourier-transformed Fara-
day and Ampère’s laws, we have Ek · j
k + E
k · jk = Ek · j
ext,k + E
k · jext,k −
(1/4π)∂|δBk|2/∂t. In a stationary state, time averaging eliminates the time
derivative of the magnetic-fluctuation energy, so Ek · j∗k + E
k · jk = 0 at all k
except those corresponding to the outer scale, where the external energy in-
jection occurs. This means that below the outer scale, the work done on one
species balances the work done on the other. The wave–particle interaction
term in the gyrokinetic equation is responsible for this energy exchange.
21 One notable exception is the LAPD device at UCLA, where β ∼ 10−4 −
10−3 (due mostly to the electron pressure because the ions are cold, τ ∼
0.1, so βi ∼ βe/10; see, e.g., Morales et al. 1999; Carter et al. 2006). This
interferes with the mass-ratio expansion.
The primary import of this section will be technical: we
shall dispense with the electron gyrokinetic equation and thus
prepare the necessary ground for further approximations. The
main results are summarized in § 4.9. A reader who is only
interested in following qualitatively the major steps in the
derivation may skip to this summary.
4.1. Ordering the Terms in the Kinetic Equation
In view of Eq. (77), ae ≪ 1, so we can expand the Bessel
functions arising from averaging over the electron ring mo-
tion:
J0(ae) = 1 −
a2e + · · · ,
J1(ae)
a2e + · · ·
. (78)
Keeping only the lowest-order terms of the above expansions
in Eq. (69) for 〈χ〉Re , then substituting this 〈χ〉Re and qe =
−e in the electron gyrokinetic equation, we get the following
kinetic equation for the electrons, accurate up to and including
the first order in (me/mi)
1/2 (or in k⊥ρe):
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
. (79)
Note that ϕ, A‖, δB‖ in Eq. (79) are taken at r = Re. We
have indicated the lowest order to which each of the terms
enters if compared with v‖∂he/∂z. In order to obtain these
estimates, we have assumed that the physical ordering intro-
duced in § 3.1 holds with respect to the subsidiary expansion
in (me/mi)
1/2 as well as for the primary gyrokinetic expansion
in ǫ, so we can use Eqs. (3) and (12) to order terms with re-
spect to (me/mi)
1/2. We have also made use of Eqs. (45), (47),
and of the following three relations:
∼ vthe
, (80)
(v‖/c)A‖
∼ vtheδB⊥
, (81)
v2the
βi. (82)
The collision term is estimated to be zeroth order because [see
Eqs. (49), (50)]
k‖λmfpi
. (83)
The consequences of other possible orderings of the collision
terms are discussed in § 4.8. We remind the reader that all
dimensionless parameters except k‖/k⊥ ∼ ǫ and (me/mi)1/2
are held to be order unity.
We now let he = h
e + h
e + . . . and carry out the expansion
to two lowest orders in (me/mi)
4.2. Zeroth Order
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 17
To zeroth order, the electron kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(0)e
, (84)
where we have assembled the terms in the left-hand side to
take the form of the derivative of the distribution function
along the perturbed magnetic field:
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
. (85)
We now multiply Eq. (84) by h(0)e /F0e and integrate over v and
r (since we are only retaining lowest-order terms, the distinc-
tion between r and Re does not matter here). Since ∇·B = 0,
the left-hand side vanishes (assuming that all perturbations are
either periodic or vanish at the boundaries) and we get
h(0)e
∂h(0)e
‖e = 0.
The right-hand side of this equation is zero because the
electron flow velocity is zero in the zeroth order, u(0)
(1/n0e)
d3vv‖h
e = 0. This is a consequence of the paral-
lel Ampére’s law [Eq. (62)], which can be written as follows
u‖e =
4πen0e
∇2⊥A‖ + u‖i, (87)
where
u‖i =
eik·r
d3vv‖J0(ai)hik. (88)
The three terms in Eq. (87) can be estimated as follows
∼ ǫvthe
ǫ, (89)
∼ ǫ, (90)
c∇2⊥A‖
4πen0evA
∼ k⊥ρi
ǫ, (91)
where we have used the fundamental ordering (12) of the slow
waves (u‖i ∼ ǫvA) and Alfvén waves (δB⊥ ∼ ǫB0). Thus, the
two terms in the right-hand side of Eq. (87) are one order of
(me/mi)
1/2 smaller than u(0)
‖e , which means that to zeroth order,
the parallel Ampère’s law is u(0)
‖e = 0.
The collision operator in Eq. (86) contains electron–
electron and electron–ion collisions. To lowest order in
(me/mi)
1/2, the electron–ion collision operator is simply the
pitch-angle scattering operator [see Eq. (B20) in Appendix B
and recall that u‖i is first order]. Therefore, we may then
rewrite Eq. (86) as follows
h(0)e
Cee[h
νeiD (v)
1 − ξ2
∂h(0)e
= 0. (92)
Both terms in this expression are negative definite and must,
therefore, vanish individually. This implies that h(0)e must be
a perturbed Maxwellian distribution with zero mean veloc-
ity (this follows from the proof of Boltzmann’s H theorem;
see, e.g., Longmire 1963), i.e., the full electron distribution
function to zeroth order in the mass-ratio expansion is [see
Eq. (54)]:
fe = F0e +
+ h(0)e =
2πTe/me
, (93)
where ne = n0e + δne, Te = T0e + δTe. Expanding around the
unperturbed Maxwellian F0e, we get
h(0)e =
v2the
F0e, (94)
where the fields are taken at r = Re. Now substitute this so-
lution back into Eq. (84). The collision term vanishes and the
remaining equation must be satisfied at all values of v. This
gives
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (95)
b̂ ·∇δTe
= 0. (96)
The collision term is neglected in Eq. (95) because, for h(0)e
given by Eq. (94), it vanishes to zeroth order.
4.3. Flux Conservation
Equation (95) implies that the magnetic flux is conserved
and magnetic-field lines cannot be broken to lowest order in
the mass-ratio expansion. Indeed, we may follow Cowley
(1985) and argue that the left-hand side of Eq. (95) is minus
the projection of the electric field on the total magnetic field
[see Eq. (37)], so we have
E · b̂ = −b̂ ·∇
; (97)
hence the total electric field is
Î − b̂b̂
and Faraday’s law becomes
= −c∇×E = ∇× (ueff ×B) , (99)
ueff =
E +∇T0e
×B, (100)
i.e., the magnetic field lines are frozen into the velocity field
ueff. In Appendix C.1, we show that this effective velocity is
the part of the electron flow velocity ue perpendicular to the
total magnetic field B [see Eq. (C6)].
The flux conservation is broken in the higher orders of the
mass-ratio expansion. In the first order, Ohmic resistivity for-
mally enters in Eq. (95) (unless collisions are even weaker
than assumed so far; if they are downgraded one order as is
done in § 4.8.3, resistivity enters in the second order). In the
second order, the electron inertia and the finiteness of the elec-
tron gyroradius also lead to unfreezing of the flux. This can be
seen formally by keeping second-order terms in Eq. (79), mul-
tiplying it by v‖ and integrating over velocities. The relative
importance of these flux unfreezing mechanisms is evaluated
in § 7.7.
18 SCHEKOCHIHIN ET AL.
4.4. Isothermal Electrons
Equation (96) mandates that the perturbed electron temper-
ature must remain constant along the perturbed field lines.
Strictly speaking, this does not preclude δTe varying across
the field lines. However, we shall now assume δTe = const (has
no spatial variation), which is justified, e.g., if the field lines
are stochastic. Assuming that no spatially uniform perturba-
tions exist, we may set δTe = 0. Equation (94) then reduces
h(0)e =
F0e(v), (101)
or, using Eq. (54),
δ fe =
F0e(v). (102)
Hence follows the equation of state for isothermal electrons:
δpe = T0eδne. (103)
4.5. First Order
We now integrate Eq. (79) over the velocity space and retain
the lowest (first) order terms only. Using Eq. (101), we get
A‖,u‖e
= 0, (104)
where the parallel electron velocity is first order:
u‖e = u
d3vv‖h
e . (105)
The velocity-space integral of the collision term does not enter
because it is subdominant by at least one factor of (me/mi)
indeed, as shown in Appendix B.1, the velocity integration
leads to an extra factor of k2⊥ρ
e , so that
∼ νeik2⊥ρ2e
i νii
, (106)
where we have used Eqs. (45) and (50). The collision term
is subdominant because of the ordering of the ion collision
frequency given by Eq. (49).
4.6. Field Equations
Using Eq. (101) and qi = Ze, n0e = Zn0i, T0e = T0i/τ ,
we derive from the quasi-neutrality equation (61) [see also
Eq. (65)]
eik·r
d3vJ0(ai)hik, (107)
and, from the perpendicular part of Ampère’s law [Eq. (66),
using also Eq. (107)],
eik·r
J0(ai) +
v2thi
J1(ai)
. (108)
The parallel electron velocity, u‖e, is determined from the par-
allel part of Ampère’s law, Eq. (87).
The ion distribution function hi that enters these equations
has to be determined by solving the ion gyrokinetic equation:
Eq. (57) with s = i.
4.7. Generalized Energy
The generalized energy (§ 3.4) for the case of isothermal
electrons is calculated by substituting Eq. (102) into Eq. (74):
T0iδ f
n0eT0e
|δB|2
, (109)
where δ fi = hi −
Zeϕ/T0i
F0i [see Eq. (54)].
4.8. Validity of the Mass-Ratio Expansion
Let us examine the range of spatial scales in which the
equations derived above are valid. In carrying out the ex-
pansion in (me/mi)
1/2, we ordered k⊥ρi ∼ 1 [Eq. (77)] and
k‖λmfpi ∼ 1 [Eq. (83)]. Formally, this means that the perpen-
dicular and parallel wavelengths of the perturbations must not
be so small or so large as to interfere with the mass ratio ex-
pansion. We now discuss the four conditions that this require-
ment leads to and whether any of them can be violated without
destroying the validity of the equations derived above.
4.8.1. k⊥ρi ≪ (mi/me)
This is equivalent to demanding that k⊥ρe ≪ 1, a condition
that was, indeed, essential for the expansion to hold [Eq. (78)].
This is not a serious limitation because electrons can be con-
sidered well magnetized at virtually all scales of interest for
astrophysical applications. However, we do forfeit the de-
tailed information about some important electron physics at
k⊥ρe ∼ 1: for example such effects as wave damping at the
electron gyroscale and the electron heating (although the total
amount of the electron heating can be deduced by subtracting
the ion heating from the total energy input). The breaking of
the flux conservation (resistivity) is also an effect that requires
incorporation of the finite electron gyroscale physics.
4.8.2. k⊥ρi ≫ (me/mi)
If this condition is broken, the small-k⊥ρi expansion, car-
ried out in § 5, must, formally speaking, precede the mass-
ratio expansion. However, it turns out that the small-
k⊥ρi expansion commutes with the mass-ratio expansion
(Schekochihin et al. 2007, see also footnote 23), so we
may use the equations derived in §§ 4.2-4.6 when k⊥ρi .
(me/mi)
4.8.3. k‖λmfpi ≪ (mi/me)
Let us consider what happens if this condition is broken
and k‖λmfpi & (mi/me)
1/2. In this case, the collisions be-
come even weaker and the expansion procedure must be mod-
ified. Namely, the collision term picks up one extra order of
(me/mi)
1/2, so it is first order in Eq. (79). To zeroth order,
the electron kinetic equation no longer contains collisions: in-
stead of Eq. (84), we have
v‖b̂ ·∇h(0)e = v‖
. (110)
We may seek the solution of this equation in the form h(0)e =
H(t,Re)F0e + h
e,hom, where H(t,Re) is an unknown function to
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 19
FIG. 4.— Region of validity in the wavenumber space of the secondary approximation—isothermal electrons and gyrokinetic ions (§ 4). It is the region
of validity of the gyrokinetic approximation (Fig. 3) further circumscribed by two conditions: k‖λmfpi ≫ (me/mi)
1/2 (isothermal electrons) and k⊥ρe ≪ 1
(magnetized electrons). The region of validity of the strongly magnetized two-fluid theory (Appendix A.2) is also shown. It is the same as for the full two-fluid
theory plus the additional constraint k⊥ρi ≪ k‖λmfpi. The region of validity of MHD (or one-fluid theory) is the subset of this with k‖λmfpi ≪ (me/mi)
(adiabatic electrons).
be determined and h(0)e,hom is the homogeneous solution satis-
fying
b̂ ·∇h(0)e,hom = 0, (111)
i.e., h(0)e,hom must be constant along the perturbed magnetic
field. This is a generalization of Eq. (96). Again assuming
stochastic field lines, we conclude that h(0)e,hom is independent
of space. If we rule out spatially uniform perturbations, we
may set h(0)e,hom = 0. The unknown function H(t,Re) is readily
expressed in terms of δne and ϕ:
d3vh(0)e ⇒ H =
, (112)
so h(0)e is again given by Eq. (101), so the equations derived
in §§ 4.2-4.6 are unaltered. Thus, the mass-ratio expansion
remains valid at k‖λmfpi & (mi/me)
4.8.4. k‖λmfpi ≫ (me/mi)
If the parallel wavelength of the fluctuations is so long that
this is violated, k‖λmfpi . (me/mi)
1/2, the collision term in
Eq. (79) is minus first order. This is the lowest-order term in
the equation. Setting it to zero obliges h(0)e to be a perturbed
Maxwellian again given by Eq. (94). Instead of Eq. (84), the
zeroth-order kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(1)e
. (113)
Now the collision term in this order contains h(1)e , which
can be determined from Eq. (113) by inverting the colli-
sion operator. This sets up a perturbation theory that in due
course leads to the Reduced MHD version of the general
MHD equations—this is what was considered in § 2. Equa-
tion (96) no longer needs to hold, so the electrons are not
isothermal. In this true one-fluid limit, both electrons and
ions are adiabatic with equal temperatures [see Eq. (115) be-
low]. The collisional transport terms in this limit (parallel
and perpendicular resistivity, viscosity, heat fluxes, etc.) were
calculated [starting not from gyrokinetics but from the gen-
eral Vlasov–Landau equation (36)] in exhaustive detail by
Braginskii (1965). His results and the way RMHD emerges
from them are reviewed in Appendix A.
In physical terms, the electrons can no longer be isothermal
if the parallel electron diffusion time becomes longer than the
characteristic time of the fluctuations (the Alfvén time):
vtheλmfpik2‖
⇔ k‖λmfpi .
. (114)
Furthermore, under a similar condition, electron and ion tem-
peratures must equalize: this happens if the ion–electron col-
lision time is shorter than the Alfvén time,
⇔ k‖λmfpi .
(115)
(see Lithwick & Goldreich 2001 for a discussion of these con-
ditions in application to the ISM).
4.9. Summary
The original gyrokinetic description introduced in § 3 was
a system of two kinetic equations [Eq. (57)] that evolved the
electron and ion distribution functions he, hi and three field
20 SCHEKOCHIHIN ET AL.
equations [Eqs. (61-63)] that related ϕ, A‖ and δB‖ to he and
hi. In this section, we have taken advantage of the smallness
of the electron mass to treat the electrons as an isothermal
magnetized fluid, while ions remained fully gyrokinetic.
In mathematical terms, we solved the electron kinetic equa-
tion and replaced the gyrokinetics with a simpler closed sys-
tem of equations that evolve 6 unknown functions: ϕ, A‖, δB‖,
δne, u‖e and hi. These satisfy two fluid-like evolution equa-
tions (95) and (104), three integral relations (107), (108), and
(87) which involve hi, and the kinetic equation (57) for hi.
The system is simpler because the full electron distribution
function has been replaced by two scalar fields δne and u‖e.
We now summarize this new system of equations: denoting
ai = k⊥v⊥/Ωi, we have
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (116)
+ b̂ ·∇u‖e = −
,(117)
eik·r
d3vJ0(ai)hik, (118)
u‖e =
4πen0e
∇2⊥A‖ +
eik·r
d3vv‖J0(ai)hik, (119)
eik·r
J0(ai) +
v2thi
J1(ai)
, (120)
and Eq. (57) for s = i and ion–ion collisions only:
{〈χ〉Ri ,hi} =
∂〈χ〉Ri
F0i + 〈Cii[hi]〉Ri ,
(121)
where 〈Cii[. . .]〉Ri is the gyrokinetic ion–ion collision oper-
ator (see Appendix B) and the ion–electron collisions have
been neglected to lowest order in (me/mi)
1/2 [see Eq. (51)].
Note that Eqs. (116-117) have been written in a compact form,
where
+ uE ·∇ =
{ϕ, · · ·} (122)
is the convective derivative with respect to the E×B drift ve-
locity, uE = −c∇⊥ϕ× ẑ/B0, and
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
(123)
is the gradient along the total magnetic field (mean field plus
perturbation).
The generalized energy conserved by Eqs. (116-121) is
given by Eq. (109).
It is worth observing that the left-hand side of Eq. (116) is
simply minus the component of the electric field along the to-
tal magnetic field [see Eq. (37)]. This was used in § 4.3 to
prove that the magnetic flux described by Eq. (116) is exactly
conserved (see § 7.7 for a discussion of scales at which this
conservation is broken). Equation (116) is the projection of
the generalized Ohm’s law onto the total magnetic field—the
right-hand side of this equation is the so-called thermoelec-
tric term. This is discussed in more detail in Appendix C.1,
where we also show that Eq. (117) is the parallel part of Fara-
day’s law and give a qualitative non-gyrokinetic derivation of
Eqs. (116-117).
We will refer to Eqs. (116-121) as the equations of isother-
mal electron fluid. They are valid in a broad range of scales:
the only constraints are that k‖ ≪ k⊥ (gyrokinetic order-
ing, § 3.1), k⊥ρe ≪ 1 (electrons are magnetized, § 4.8.1) and
k‖λmfpi ≫ (me/mi)1/2 (electrons are isothermal, § 4.8.4). The
region of validity of Eqs. (116-121) in the wavenumber space
is illustrated in Fig. 4. A particular advantage of this hybrid
fluid-kinetic system is that it is uniformly valid across the
transition from magnetized to unmagnetized ions (i.e., from
k⊥ρi ≪ 1 to k⊥ρi ≫ 1).
5. TURBULENCE IN THE INERTIAL RANGE: KINETIC RMHD
Our goal in this section is to derive a reduced set of equa-
tions that describe the magnetized plasma in the limit of small
k⊥ρi. Before we proceed with an expansion in k⊥ρi, we need
to make a formal technical step, the usefulness of which will
become clear shortly. A reader with no patience for this or
any of the subsequent technical developments may skip to the
summary at the end of this section (§ 5.7).
5.1. A Technical Step
Let us formally split the ion gyrocenter distribution function
into two parts:
v⊥ ·A⊥
F0i + g
eik·Ri
J0(ai)
v2thi
J1(ai)
F0i + g. (124)
Then g satisfies the following equation, obtained by substitut-
ing Eq. (124) and the expression for ∂A‖/∂t that follows from
Eq. (116) into the ion gyrokinetic equation (121):
{〈χ〉Ri ,g}− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+ b̂ ·∇
v⊥ ·A⊥
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
. (125)
In the above equation, we have used compact notation in
writing out the nonlinear terms: e.g.,
A‖,ϕ− 〈ϕ〉Ri
A‖(r),ϕ(r)
〈A‖〉Ri ,〈ϕ〉Ri
, where the first Poisson
bracket involves derivatives with respect to r and the second
with respect to Ri.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 21
The field equations (118-120) rewritten in terms of g are
−Γ1(αi)
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, (126)
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vv‖J0(ai)gk
︸ ︷︷ ︸
= u‖ki, (127)
︸ ︷︷ ︸
Γ2(αi) +
︸ ︷︷ ︸
1 −Γ1(αi)
]Zeϕk
︸ ︷︷ ︸
v2thi
J1(ai)
︸ ︷︷ ︸
, (128)
where ai = k⊥v⊥/Ωi, αi = k
i /2 and we have defined
Γ0(αi) =
d3v [J0(ai)]
2 F0i
= I0(αi)e
−αi = 1 −αi + · · · , (129)
Γ1(αi) =
v2thi
J0(ai)
J1(ai)
F0i = −Γ′0(αi)
= [I0(αi) − I1(αi)] e−αi = 1 −
αi + · · · , (130)
Γ2(αi) =
v2thi
J1(ai)
F0i = 2Γ1(αi). (131)
Underneath each term in Eqs. (125-128), we have indicated
the lowest order in k⊥ρi to which this term enters.
5.2. Subsidiary Ordering in k⊥ρi
In order to carry out a subsidiary expansion in small k⊥ρi,
we must order all terms in Eqs. (95-104) and (125-128) with
respect to k⊥ρi. Let us again assume, like we did when ex-
panding the electron equation (§ 4), that the ordering intro-
duced for the gyrokinetics in § 3.1 holds also for the sub-
sidiary expansion in k⊥ρi. First note that, in view of Eq. (47),
we must regard Zeϕ/T0i to be minus first order:
. (132)
Also, as δB⊥/B0 ∼ ǫ [Eq. (12)],
(v‖/c)A‖
∼ vthiδB⊥
βi, (133)
so ϕ and (v‖/c)A‖ are same order.
Since u‖ = u‖i (electrons do not contribute to the mass flow),
assuming that slow waves and Alfvén waves have comparable
energies implies u‖i ∼ u⊥. As u‖i is determined by the second
equality in Eq. (127), we can order g [using Eq. (12)]:
, (134)
so g is zeroth order in k⊥ρi. Similarly, δne/n0e ∼ δB‖/B0 ∼ ǫ
are zeroth order in k⊥ρi—this follows directly from Eq. (12).
Together with Eq. (3), the above considerations allow us to
order all terms in our equations. The ordering of the collision
term involving ϕ is explained in Appendix B.2.
5.3. Alfvén Waves: Kinetic Derivation of RMHD
We shall now show that the RMHD equations (17-18) hold
in this approximation. There is a simple correspondence be-
tween the stream and flux functions defined in Eq. (16) and
the electromagnetic potentials ϕ and A‖:
ϕ, Ψ = −
4πmin0i
. (135)
The first of these definitions says that the perpendicular flow
velocity u⊥ is the E×B drift velocity; the second definition
is the standard MHD relation between the magnetic flux func-
tion and the parallel component of the vector potential.
5.3.1. Derivation of Eq. (17)
Deriving Eq. (17) is straightforward: in Eq. (95), we retain
only the lowest—minus first—order terms (those that contain
ϕ and A‖). The result is
= 0. (136)
Using Eq. (135) and the definition of the Alfvén speed, vA =
4πmin0i, we get Eq. (17). By the argument of § 4.3,
Eq. (136) expresses the fact that that magnetic-field lines are
frozen into the E×B velocity field, which is the mean flow
velocity associated with the Alfvén waves (see § 5.4).
5.3.2. Derivation of Eq. (18)
As we are about to see, in order to derive Eq. (18), we have
to separate the first-order part of the k⊥ρi expansion. The
easiest way to achieve this, is to integrate Eq. (125) over the
velocity space (keeping r constant) and expand the resulting
equation in small k⊥ρi. Using Eqs. (126) and (127) to express
the velocity-space integrals of g, we get
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
−Γ1(αi)
︸ ︷︷ ︸
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vJ0(ai){〈χ〉Ri ,g}k
︸ ︷︷ ︸
d3vJ0(ai)
v⊥ ·A⊥
︸ ︷︷ ︸
22 SCHEKOCHIHIN ET AL.
. (137)
Underneath each term, the lowest order in k⊥ρi to which it
enters is shown. We see that terms containing ϕ are all first
order, so it is up to this order that we shall retain terms. The
collision term integrated over the velocity space picks up two
extra orders of k⊥ρi (see Appendix B.1), so it is second order
and can, therefore, be dropped. As a consequence of quasi-
neutrality, the zeroth-order part of the above equation exactly
coincides with Eq. (104), i.e, δni/n0i = δne/n0e satisfy the
same equation. Indeed, neglecting second-order terms (but
not first-order ones!), the nonlinear term in Eq. (137) (the last
term on the left-hand side) is
d3vv‖g
v2thi
, (138)
and, using Eqs. (126-128) to express velocity-space integrals
of g in the above expression, we find that the zeroth-order part
of the nonlinearity is the same as the nonlinearity in Eq. (104),
while the first-order part is
ρ2i ∇2⊥
4πen0e
∇2⊥A‖
, (139)
where we have used the expansion (129) of Γ0(αi) and con-
verted it back into x space.
Thus, if we subtract Eq. (104) from Eq. (137), the remain-
der is first order and reads
ρ2i ∇2⊥
ρ2i ∇2⊥
4πen0e
∇2⊥A‖ −
4πen0e
∇2⊥A‖
= 0. (140)
Multiplying Eq. (140) by 2T0i/Zeρ
i and using Eq. (135), we
get the second RMHD equation (18).
We have established that the Alfvén-wave component of the
turbulence is decoupled and fully described by the RMHD
equations (17) and (18). This result is the same as that in
§ 2.2 but now we have proven that collisions do not affect the
Alfvén waves and that a fluid-like description only requires
k⊥ρi ≪ 1 to be valid.
5.4. Why Alfvén Waves Ignore Collisions
Let us write explicitly the distribution function of the ion
gyrocenters [Eq. (124)] to two lowest orders in k⊥ρi:
〈ϕ〉Ri F0i +
v2thi
F0i + g + · · · , (141)
where, up to corrections of order k2⊥ρ
i , the ring-averaged
scalar potential is 〈ϕ〉Ri = ϕ(Ri), the scalar potential taken at
the position of the ion gyrocenter. Note that in Eq. (141), the
first term is minus first order in k⊥ρi [see Eq. (132)], the sec-
ond and third terms are zeroth order [Eq. (134)], and all terms
of first and higher orders are omitted. In order to compute the
full ion distribution function given by Eq. (54), we have to
convert hi to the r space. Keeping terms up to zeroth order,
we get
〈ϕ〉Ri ≃
ϕ(Ri) =
ϕ(r) +
v⊥× ẑ
·∇ϕ(r) + · · ·
ϕ(r) +
2v⊥ ·uE
v2thi
+ . . . , (142)
where uE = −c∇ϕ(r)× ẑ/B0, the E×B drift velocity. Sub-
stituting Eq. (142) into Eq. (141) and then Eq. (141) into
Eq. (54), we find
fi = F0i +
2v⊥ ·uE
v2thi
F0i +
v2thi
F0i + g + · · · . (143)
The first two terms can be combined into a Maxwellian
with mean perpendicular flow velocity u⊥ = uE . These are
the terms responsible for the Alfvén waves. The remaining
terms, which we shall denote δ f̃i, are the perturbation of the
Maxwellian in the moving frame of the Alfvén waves—they
describe the passive (compressive) component of the turbu-
lence (see § 5.5). Thus, the ion distribution function is
(πv2thi)
(v⊥ − uE)2 + v2‖
+ δ f̃i. (144)
This sheds some light on the indifference of Alfvén waves
to collisions: Alfvénic perturbations do not change the
Maxwellian character of the ion distribution. Unlike in a neu-
tral fluid or gas, where viscosity arises when particles trans-
port the local mean momentum a distance ∼ λmfpi, the parti-
cles in a magnetized plasma instantaneously take on the lo-
cal E×B velocity (they take a cyclotron period to adjust, so,
roughly speaking, ρi plays the role of the mean free path).
Thus, there is no memory of the mean perpendicular motion
and, therefore, no perpendicular momentum transport.
Some readers may find it illuminating to notice that
Eq. (140) can be interpreted as stating simply ∇·j = 0: the first
two terms represent the divergence of the polarization current,
which is perpendicular to the magnetic field;22 the last two
terms are b̂ ·∇ j‖. No contribution to the current arises from
the collisional term in Eq. (137) as ion–ion collisions cause
no particle transport to lowest order in k⊥ρi.
5.5. Compressive Fluctuations
The equations that describe the density (δne) and magnetic-
field-strength (δB‖) fluctuations follow immediately from
Eqs. (125-128) if only zeroth-order terms are kept. In these
equations, terms that involve ϕ and A‖ also contain factors
∼ k2⊥ρ2i and are, therefore, first-order [with the exception of
the nonlinearity on the left-hand side of Eq. (125)]. The fact
that 〈Cii[〈ϕ〉Ri F0i]〉Ri in Eq. (125) is first order is proved in Ap-
pendix B.2. Dropping these terms along with all other contri-
butions of order higher than zeroth and making use of Eq. (69)
22 The polarization-drift velocity is formally higher order than uE in the
gyrokinetic expansion. However, since uE does not produce any current,
the lowest-order contribution to the perpendicular current comes from the
polarization drift. The higher-order contributions to the gyrocenter distri-
bution function did not need to be calculated explicitly because the informa-
tion about the polarization charge is effectively carried by the quasi-neutrality
condition (61). We do not belabor this point because, in our approach, the no-
tion of polarization charge is only ever brought in for interpretative purposes,
but is not needed to carry out calculations. For further qualitative discussion
of the role of the polarization charge and polarization drift in gyrokinetics,
we refer the reader to Krommes 2006 and references therein.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 23
to write out 〈χ〉Ri , we find that Eq. (125) takes the form
+ v‖ b̂ ·∇
v2thi
v2thi
, (145)
where we have used definitions (122-123) of the convective
time derivative d/dt and the total gradient along the magnetic
field b̂ · ∇ to write our equation in a compact form. Note
that, in view of the correspondence between Φ, Ψ and ϕ, A‖
[Eq. (135)], these nonlinear derivatives are the same as those
defined in Eqs. (19-20). The collision term in the right-hand
side of the above equation is the zeroth-order limit of the gy-
rokinetic ion–ion collision operator: a useful model form of it
is given in Appendix B.3 [Eq. (B18)].
To zeroth order, Eqs. (126-128) are
d3vg, (146)
d3vv‖g, (147)
v2thi
g. (148)
Note that u‖ is not an independent quantity—it can be com-
puted from the ion distribution but is not needed for the deter-
mination of the latter.
Equations (145-148) evolve the ion distribution function
g, the “slow-wave quantities” u‖, δB‖, and the density fluc-
tuations δne. The nonlinearities in Eq. (145), contained in
d/dt and b̂ ·∇, involve the Alfvén-wave quantities Φ and Ψ
(or, equivalently, ϕ and A‖) determined separately and inde-
pendently by the RMHD equations (17-18). The situation
is qualitatively similar to that in MHD (§ 2.4), except now
a kinetic description is necessary—Eqs. (145-148) replace
Eqs. (25-27)—and the nonlinear scattering/mixing of the slow
waves and the entropy mode by the Alfvén waves takes the
form of passive advection of the distribution function g. The
density and magnetic-field-strength fluctuations are velocity-
space moments of g.
Another way to understand the passive nature of the com-
pressive component of the turbulence discussed above is to
think of it as the perturbation of a local Maxwellian equilib-
rium associated with the Alfvén waves. Indeed, in § 5.4, we
split the full ion distribution function [Eq. (144)] into such a
local Maxwellian and its perturbation
δ f̃i = g +
v2thi
F0i. (149)
It is this perturbation that contains all the information about
the compressive component; the second term in the above ex-
pression enforces to lowest order the conservation of the first
adiabatic invariant µi = miv
⊥/2B. In terms of the function
(149), Eqs. (145-148) take a somewhat more compact form
(cf. Schekochihin et al. 2007):
δ f̃i −
v2thi
+ v‖b̂ ·∇
δ f̃i +
δ f̃i
, (150)
FIG. 5.— Channels of the kinetic cascade of generalized energy (§ 3.4)
from large to small scales: see § 2.7 and Appendix D.2 (inertial range,
collisional regime), § 5.6 and § 6.2.5 (inertial range, collisionless regime),
§ 7.8 and § 7.12 (dissipation range). Note that some ion heating probably
also results from the collisional and collisionless damping of the compressive
fluctuations in the inertial range (see § 6.1.2 and § 6.2.4).
d3vδ f̃i, (151)
v2thi
δ f̃i. (152)
5.6. Generalized Energy: Three KRMHD Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≪ 1 is cal-
culated by substituting into Eq. (109) the perturbed ion dis-
tribution function δ fi = 2v⊥ · uEF0i/v2thi + δ f̃i [see Eqs. (143)
and (149)]. After performing velocity integration, we get
min0iu
n0iT0i
δ f̃ 2i
=WAW +Wcompr. (153)
We see that the kinetic energy of the Alfvénic fluctuations
has emerged from the ion-entropy part of the generalized en-
ergy. The first two terms in Eq. (153) are the total (kinetic
plus magnetic) energy of the Alfvén waves, denoted WAW. As
we learned from § 5.3, it cascades independently of the rest of
the generalized energy, Wcompr, which contains the compres-
sive component of the turbulence (§ 5.5) and is the invariant
conserved by Eqs. (150-152).
In terms of the potentials used in our discussion of RMHD
in § 2, we have
WAW =
min0i
|∇⊥Φ|2 + |∇⊥Ψ|2
min0i
|∇⊥ζ+|2 + |∇⊥ζ−|2
=W +AW +W
AW (154)
whereW +AW and W
AW are the energies of the “+” and “−” waves
[Eq. (33)], which, as we know from § 2.3, cascade by scatter-
ing off each other but without exchanging energy.
Thus, the kinetic cascade in the limit k⊥ρi ≪ 1 is split, in-
dependently of the collisionality, into three cascades: of W +AW,
24 SCHEKOCHIHIN ET AL.
W −AW and Wcompr. The compressive cascade is, in fact, split
into three independent cascades—the splitting is different in
the collisional limit (Appendix D.2) and in the collisionless
one (§ 6.2.5). Figure 5 schematically summarizes both the
splitting of the kinetic cascade that we have worked out so far
and the upcoming developments.
5.7. Summary
In § 4, gyrokinetics was reduced to a hybrid fluid-kinetic
system by means of an expansion in the electron mass, which
was valid for k⊥ρe ≪ 1. In this section, we have further re-
stricted the scale range by taking k⊥ρi ≪ 1 and as a result have
been able to achieve a further reduction in the complexity of
the kinetic theory describing the turbulent cascades. The re-
duced theory derived here evolves 5 unknown functions: Φ,
Ψ, δB‖, δne and g. The stream and flux functions, Φ and Ψ
are related to the fluid quantities (perpendicular velocity and
magnetic field perturbations) via Eq. (16) and to the electro-
magnetic potentials ϕ, A‖ via Eq. (135). They satisfy a closed
system of equations, Eqs. (17-18), which describe the decou-
pled cascade of Alfvén waves. These are the same equations
that arise from the MHD approximations, but we have now
proven that their validity does not depend on the assumption
of high collisionality (the fluid limit) and extends to scales
well below the mean free path, but above the ion gyroscale.
The physical reasons for this are explained in § 5.4. The den-
sity and magnetic-field-strength fluctuations (the “compres-
sive” fluctuations, or the slow waves and the entropy mode in
the MHD limit) now require a kinetic description in terms of
the ion distribution function g [or δ f̃i, Eq. (149)], evolved by
the kinetic equation (145) [or Eq. (150)]. The kinetic equation
contains δne and δB‖, which are, in turn calculated in terms
of the velocity-space integrals of g via Eqs. (146) and (148)
[or Eqs. (151) and (152)]. The nonlinear evolution (turbulent
cascade) of g, δB‖ and δne is due solely to passive advection
of g by the Alfvén-wave turbulence.
Let us summarize the new set of equations:
= vAb̂ ·∇Φ, (155)
∇2⊥Φ= vAb̂ ·∇∇2⊥Ψ, (156)
+ v‖ b̂ ·∇
v2thi
v2thi
, (157)
v2thi
(158)
v2thi
g, (159)
where
+{Φ, · · ·} , b̂ ·∇ = ∂
{Ψ, · · ·} . (160)
An explicit form of the collision term in the right-hand side of
Eq. (157) is provided in Appendix B.3 [Eq. (B18)].
The generalized energy conserved by Eqs. (155-159) is
given by Eq. (153). The kinetic cascade is split, the Alfvénic
cascade proceeding independently of the compressive one
(see Fig. 5).
The decoupling of the Alfvénic cascade is manifested by
Eqs. (155-156) forming a closed subset. As already noted in
§ 4.9, Eq. (155) is the component of Ohm’s law along the total
magnetic field, B ·E = 0. Equation (156) can be interpreted as
the evolution equation for the vorticity of the perpendicular
plasma flow velocity, which is the E×B drift velocity.
We shall refer to the system of equations (155-159) as Ki-
netic Reduced Magnetohydrodynamics (KRMHD).23 It is a
hybrid fluid-kinetic description of low-frequency turbulence
in strongly magnetized weakly collisional plasma that is uni-
formly valid at all scales satisfying k⊥ρi ≪ min(1,k‖λmfpi)
(ions are strongly magnetized)24 and k‖λmfpi ≫ (me/mi)1/2
(electrons are isothermal), as illustrated in Fig. 2. Therefore,
it smoothly connects the collisional and collisionless regimes
and is the appropriate theory for the study of the turbulent cas-
cades in the inertial range. The KRMHD equations generalize
rather straightforwardly to plasmas that are so collisionless
that one cannot assume a Maxwellian equilibrium distribu-
tion function (Chen et al. 2009)—a situation that is relevant
in some of the solar-wind measurements (see further discus-
sion in § 8.3).
KRMHD describe what happens to the turbulent cascade at
or below the ion gyroscale—we shall move on to these scales
in § 7, but first we would like to discuss the turbulent cascades
of density and magnetic-field-strength fluctuations and their
damping by collisional and collisionless mechanisms.
6. COMPRESSIVE FLUCTUATIONS IN THE INERTIAL RANGE
Here we first derive the nonlinear equations that govern
the evolution of the compressive (density and magnetic-field-
strength) fluctuations in the collisional (k‖λmfpi ≪ 1, § 6.1 and
Appendix D) and collisionless (k‖λmfpi ≫ 1, § 6.2) limits, dis-
cuss the linear damping that these fluctuations undergo in the
two limits and work out the form the generalized energy takes
for compressive fluctuations (which is particularly interesting
in the collisionless limit, §§ 6.2.3-6.2.5). As in previous sec-
tions, an impatient reader may skip to § 6.3 where the results
of the previous two subsections are summarized and the im-
plications for the structure of the turbulent cascades of the
density and field-strength fluctuations are discussed.
6.1. Collisional Regime
6.1.1. Equations
In the collisional regime, k‖λmfpi ≪ 1, the fluid limit is re-
covered by expanding Eqs. (155-159) in small k‖λmfpi. The
calculation that is necessary to achieve this is done in Ap-
pendix D (see also Appendix A.4). The result is a closed set
23 The term is introduced by analogy with a popular fluid-kinetic system
known as Kinetic MHD, or KMHD (see Kulsrud 1964, 1983). KMHD is de-
rived for magnetized plasmas (ρi ≪ λmfpi) under the assumption that kρs ≪ 1
and ω ≪ Ωs but without assuming either strong anisotropy (k‖ ≪ k⊥) or
small fluctuations (|δB| ≪ B0). The KRMHD equations (155-159) can be
recovered from KMHD by applying to it the GK-RMHD ordering [Eq. (12)
and § 3.1] and an expansion in (me/mi)1/2 (Schekochihin et al. 2007). This
means that the k⊥ρi expansion (§ 5), which for KMHD is the primary ex-
pansion, commutes with the gyrokinetic expansion (§ 3) and the (me/mi)1/2
expansion (§ 4), both of which preceded it in this paper.
24 The condition k⊥ρi ≪ k‖λmfpi must be satisfied because in our esti-
mates of the collision terms (Appendix B.2) we took k⊥ρi ≪ 1 while assum-
ing that k‖λmfpi ∼ 1.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 25
of three fluid equations that evolve δB‖, δne and u‖:
= b̂ ·∇u‖ +
, (161)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (162)
+κ‖ib̂ ·∇
b̂ ·∇δTi
, (163)
where
ν‖ib̂ ·∇u‖
, (164)
and ν‖i and κ‖i are the coefficients of parallel ion viscosity
and thermal diffusivity, respectively. The viscous and ther-
mal diffusion are anisotropic because plasma is magnetized,
λmfpi ≫ ρi (Braginskii 1965). The method of calculation of
ν‖i and κ‖i is explained in Appendix D.3. Here we shall ig-
nore numerical prefactors of order unity and give order-of-
magnitude values for these coefficients:
ν‖i ∼ κ‖i ∼
v2thi
∼ vthiλmfpi. (165)
If we set ν‖i = κ‖i = 0, Eqs. (161-164) are the same as the
RMHD equations of § 2 with the sound speed defined as
cs = vA
. (166)
This is the natural definition of cs for the case of adiabatic
ions, whose specific heat ratio is γi = 5/3, and isothermal elec-
trons, whose specific heat ratio is γe = 1 [because δpe = T0eδne;
see Eq. (103)]. Note that Eq. (164) is equivalent to the
pressure balance [Eq. (22) of § 2] with p = niTi + neTe and
δpe = T0eδne.
As in § 2, the fluctuations described by Eqs. (161-164) sep-
arate into the zero-frequency entropy mode and the left- and
right-propagating slow waves with
ω = ±
1 + v2A/c2s
(167)
[see Eq. (30)]. All three are cascaded independently of each
other via nonlinear interaction with the Alfvén waves. In Ap-
pendix D.2, we show that the generalized energy Wcompr for
this system, given in § 5.6, splits into the three familiar invari-
ants W +sw, W
sw, and Ws, defined by Eqs. (34-35) (see Fig. 5).
6.1.2. Dissipation
The diffusion terms add dissipation to the equations. Be-
cause diffusion occurs along the field lines of the total mag-
netic field (mean field plus perturbation), the diffusive terms
are nonlinear and the dissipation process also involves interac-
tion with the Alfvén waves. We can estimate the characteristic
parallel scale at which the diffusion terms become important
by balancing the nonlinear cascade time and the typical diffu-
sion time:
k‖vA ∼ vthiλmfpik2‖ ⇔ k‖λmfpi ∼ 1/
βi, (168)
where we have used Eq. (165).
Technically speaking, the cutoff given by Eq. (168) always
lies in the range of k‖ that is outside the region of validity
of the small-k‖λmfpi expansion adopted in the derivation of
Eqs. (161-163). In fact, in the low-beta limit, the collisional
cutoff falls manifestly in the collisionless scale range, i.e.,
the collisional (fluid) approximation breaks down before the
slow-wave and entropy cascades are damped and one must use
the collisionless (kinetic) limit to calculate the damping (see
§ 6.2.2). The situation is different in the high-beta limit: in
this case, the expansion in small k‖λmfpi can be reformulated
as an expansion in small 1/
βi and the cutoff falls within the
range of validity of the fluid approximation. Equations (161-
163) in this limit are
= b̂ ·∇u‖, (169)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (170)
1 + Z/τ
5/3 + Z/τ
κ‖ib̂ ·∇
b̂ ·∇δne
. (171)
As in § 2 [Eq. (28)], the density fluctuations [Eq. (171)] have
decoupled from the slow waves [Eqs. (169-170)]. The former
are damped by thermal diffusion, the latter by viscosity. The
corresponding linear dispersion relations are
ω = −i
1 + Z/τ
5/3 + Z/τ
‖, (172)
ω =±k‖vA
ν‖ik‖
. (173)
Equation (172) describes strong diffusive damping of the den-
sity fluctuations. The slow-wave dispersion relation (173) has
two distinct regimes:
1. When k‖ < 2vA/ν‖i, it describes viscously damped slow
waves. In particular, in the limit k‖λmfpi ≪ 1/
βi, we
ω ≃±k‖vA − i
. (174)
2. For k‖ > 2vA/ν‖i, both solutions become purely imag-
inary, so the slow waves are converted into aperiodic
decaying fluctuations. The stronger-damped (diffusive)
branch has ω ≃ −iν‖ik2‖, the weaker-damped one has
ω ≃ −i v
∼ − i
λmfpi
∼ − i√
λmfpi
. (175)
This damping effect is called viscous relaxation. It is
valid until k‖λmfpi ∼ 1, where it is replaced by the col-
lisionless damping discussed in § 6.2.2 [see Eq. (190)].
The viscous and thermal-diffusive dissipation mechanisms
described above lead, in the limits where they are efficient, to
ion heating via the standard fluid (collisional) route, involving
the development of small parallel scales in the position space,
but not in velocity space (see § 3.4 and § 3.5).
6.2. Collisionless Regime
6.2.1. Equations
In the collisionless regime, k‖λmfpi ≫ 1, the collision inte-
gral in the right-hand side of the kinetic equation (157) can be
26 SCHEKOCHIHIN ET AL.
neglected. The v⊥ dependence can then be integrated out of
Eq. (157). Indeed, let us introduce the following two auxiliary
functions:
Gn(v‖) = −
dv⊥ v⊥
v2thi
g, (176)
GB(v‖) = −
dv⊥ v⊥
v2thi
g. (177)
In terms of these functions,
dv‖Gn,
dv‖GB (178)
and Eq. (157) reduces to the following two coupled one-
dimensional kinetic equations
+ v‖b̂ ·∇Gn = −
v‖FM(v‖)
×b̂ ·∇
, (179)
+ v‖b̂ ·∇GB =
v‖FM(v‖)
×b̂ ·∇
, (180)
where FM(v‖) = (1/
πvthi)exp(−v2‖/v
thi) is a one-dimensional
Maxwellian. This system can be diagonalized, so it splits into
two decoupled equations
+v‖b̂ ·∇G± =
v‖FM(v‖)
b̂ ·∇
dv′‖ G
±(v′‖), (181)
where
± = −
(182)
and we have introduced a new pair of functions
G+ = GB +
Gn, G
− = Gn +
GB, (183)
where
σ = 1 +
. (184)
Equation (181) describes two decoupled kinetic cascades,
which we will discuss in greater detail in §§ 6.2.3-6.2.5.
6.2.2. Collisionless Damping
Fluctuations described by Eq. (181) are subject to collision-
less damping. Indeed, let us linearize Eq. (181), Fourier trans-
form in time and space, divide through by −i(ω − k‖v‖), and
integrate over v‖. This gives the following dispersion relation
(the “−” branch is for G−, the “+” branch for G+)
ζiZ (ζi) = Λ
± − 1, (185)
FIG. 6.— Schematic log-log plot (artist’s impression) of the ratio of the
damping rate of magnetic-field-strength fluctuations to the Alfvén frequency
k‖vA in the high-beta limit [see Eqs. (173) and (190)]. In Barnes et al. (2009),
this plot is reproduced via a direct numerical solution of the linearized ion
gyrokinetic equation with collisions.
where ζi = ω/|k‖|vthi = ω/|k‖|vA
βi and we have used the
plasma dispersion function (Fried & Conte 1961)
Z (ζi) =
x − ζi
(186)
(the integration is along the Landau contour). This function is
not to be confused with the ion charge parameter Z = qi/e.
Formally, Eq. (185) has an infinite number of solutions.
When βi ∼ 1, they are all strongly damped with damping rates
Im(ω) ∼ |k‖|vthi ∼ |k‖|vA, so the damping time is compara-
ble to the characteristic timescale on which the Alfvén waves
cause these fluctuations to cascade to smaller scales.
It is interesting to consider the high- and low-beta limits.
High-Beta Limit. — When βi ≫ 1, we have in Eq. (185)
− − 1≃−2
, G− ≃ Gn, (187)
+ − 1≃
, G+ ≃ GB +
Gn. (188)
The “−” branch corresponds to the density fluctuations. The
solution of Eq. (185) has Im(ζi) ∼ 1, so these fluctuations are
strongly damped:
ω ∼ −i|k‖|vA
βi. (189)
The damping rate is much greater than the Alfvénic rate k‖vA
of the nonlinear cascade. In contrast, for the “+” branch, the
damping rate is small: it can be obtained by expanding Z(ζi) =
π + O (ζi), which gives25
ω = −i
|k‖|vthi√
|k‖|vA√
. (190)
Since Gn is strongly damped, Eq. (188) implies G
+ ≃ GB, i.e.,
the fluctuations that are damped at the rate (190) are predom-
inantly of the magnetic-field strength. The damping rate is a
25 This is the gyrokinetic limit (k‖/k⊥ ≪ 1) of the more general damping
effect known in astrophysics as the Barnes (1966) damping and in plasma
physics as transit-time damping. We remind the reader that our approach was
to carry out the gyrokinetic expansion (in small k‖/k⊥) first, and then take
the high-beta limit as a subsidiary expansion. A more standard approach in
the linear theory of plasma waves is to take the limit of high βi while treating
k‖/k⊥ as an arbitrary quantity. A detailed calculation of the damping rates
done in this way can be found in Foote & Kulsrud (1979).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 27
constant (independent of k‖) small fraction ∼ 1/
βi of the
Alfvénic cascade rate.
In Fig. 6, we give a schematic plot of the damping rate of the
magnetic-field-strength fluctuations (slow waves) connecting
the fluid and kinetic limits for βi ≫ 1.
Low-Beta Limit. — When βi ≪ 1, we have
− − 1≃−
, G− ≃ Gn +
GB, (191)
+ − 1≃ 2
, G+ ≃ GB. (192)
For the “−” branch, we again have Im(ζi) ∼ 1, so
ω ∼ −i|k‖|vA
βi, (193)
which now is much smaller than the Alfvénic cascade rate
k‖vA. For the “+” branch (predominantly the field-strength
fluctuations), we seek a solution with ζ = −iζ̃i and ζ̃i ≫ 1.
Then Eq. (185) becomes ζiZ(ζi) ≃ 2
π ζ̃i exp(ζ̃i) = 2/βi. Up
to logarithmically small corrections, this gives ζ̃i ≃
| lnβi|,
whence
ω ∼ −i|k‖|vA
βi| lnβi|. (194)
While this damping rate is slightly greater than that of the “−”
branch, it is still much smaller than the Alfvénic cascade rate.
6.2.3. Collisionless Invariants
Equation (181) obeys a conservation law, which is very easy
to derive. Multiplying Eq. (181) by G±/FM and integrating
over space and velocities and performing integration by parts
in the right-hand side, we get
(G±)2
b̂ ·∇
dv‖v‖G
±. (195)
On the other hand, integrating Eq. (181) over v‖ gives
± = −b̂ ·∇
dv‖v‖G
±. (196)
Using this to express the right-hand side of Eq. (195) as a full
time derivative, we find
dW±compr
= 0, (197)
where the two invariants are
W±compr =
n0iT0i
(G±)2
(198)
It is useful (and always possible) to split
G± = FM
± + G̃±, (199)
where
dv‖G̃
± = 0 by construction. Then
W±compr =
n0iT0i
(G̃±)2
. (200)
Written in this form, the two invariants W±compr are mani-
festly positive definite quantities because Λ+ > 1 and Λ− < 0.
The invariants regulate the two decoupled kinetic cascades of
compressive fluctuations in the collisionless regime. The col-
lisionless damping derived in § 6.2.2 leads to exponential de-
cay of the density and field-strength fluctuations, or, equiva-
lently, of
±, while conserving W±compr. This means that
the damping is merely a redistribution of the conserved quan-
tity W±compr: the first term in Eq. (200) grows to compensate
for the decay of the second.
6.2.4. Linear Parallel Phase Mixing
In dynamical terms, how does the kinetic system Eq. (181)
arrange for the integral of the distribution function G±(v‖) to
decay while allowing its norm to grow? This is a very well
known phenomenon of (linear) phase mixing (Landau 1946;
Hammett et al. 1991; Krommes & Hu 1994; Krommes 1999;
Watanabe & Sugama 2004). To put it in simple terms, the
solution of the linearized Eq. (181) consists of the inhomoge-
neous part, which contains the collisionless damping and the
homogeneous part (solution of the left-hand side = 0) given by
G± ∝ e−ik‖v‖t , the so-called ballistic response (this is also the
nonlinear solution if t and k‖ are interpreted as Lagrangian
variables in the frame of the Alfvén waves; see § 6.3). As
time goes on, this part of the solution becomes increasingly
oscillatory in v‖, so its velocity integral tends to zero, while
its amplitude does not decay. It is such ballistic contributions
that make up the G̃± term in Eq. (200).
As the velocity gradient of G̃± increases with time,
∂G̃±/∂v‖ ∼ k‖tG±, at some point it can become sufficiently
large to activate the collision integral [the right-hand side of
Eq. (157)], which has so far been neglected. This way the col-
lisionless damping of compressive fluctuations can be turned
into ion heating—a simple example of a more general prin-
ciple of how electromagnetic fluctuation energy is transferred
into heat via the entropy part of the generalized energy (§ 3.5).
Indeed, we will prove in § 6.2.5 that the invariants W±compr are
constituent parts of the overall generalized energy functional
for the compressive fluctuations, so their cascade to small
scales in phase space is part of the overall kinetic cascade in-
troduced in § 3.4.
It is not entirely clear how efficient is the parallel-phase-
mixing route to ion heating and, therefore, whether the colli-
sionlessly damped energy of compressive fluctuations ends up
in the ion heat or rather reaches the ion gyroscale and couples
back to the Alfvénic component of the turbulence (§ 7.1). The
answer to this question will depend on whether compressive
fluctuations can develop large k‖—a non-trivial issue further
discussed in § 6.3.
6.2.5. Generalized Energy: Three Collisionless Cascades
We will now show how the generalized energy for com-
pressive fluctuations in the collisionless regime incorporates
the two invariants derived in § 6.2.3.
Rewriting the compressive part of the KRMHD generalized
energy [Eq. (153)] in terms of the function g [see Eq. (149)],
we get
Wcompr =
n0iT0i
. (201)
28 SCHEKOCHIHIN ET AL.
Using Eqs. (178) and (183), we can express δne and δB‖ in
terms of
± as follows
, (202)
, (203)
where σ was defined in Eq. (184) and
. (204)
In order to express g in terms of G±, we have to reconstruct
the v⊥ dependence of g, which we integrated out at the begin-
ning of § 6.2.1.
Let us represent the distribution function as follows
πv2thi
e−xĝ(x,v‖), ĝ(x,v‖) =
Ll(x)Gl(v‖), (205)
where x = v2⊥/v
thi and we have expanded ĝ in Laguerre poly-
nomials Ll(x) = (e
x/l!)(dl/dxl)xle−x. Since Laguerre polyno-
mials are orthogonal, the first term in Eq. (201) splits into a
sum of “energies” associated with the expansion coefficients:
. (206)
The expansion coefficients are determined via the Laguerre
transform:
Gl(v‖) =
dxe−xLl(x)ĝ(x,v‖). (207)
As L0 = 1 and L1 = 1 − x, it is easy to see that δne and δB‖ can
be expressed as linear combinations of
dv‖G0 and
dv‖G1
[see Eqs. (176-178)]. Using Eqs. (176), (177), and (183), we
can show that
G0 = −
+G+ +
σ − 1 −
, (208)
σΛ+G+ −
, (209)
where G± satisfy Eq. (181). As follows from Eq. (157) (ne-
glecting the collision integral), all higher-order expansion co-
efficients satisfy a simple homogeneous equation:
+ v‖b̂ ·∇Gl = 0, l > 1. (210)
Thus, the distribution function can be explicitly written in
terms of G±:
G0(v‖) +
v2thi
G1(v‖)
πv2thi
thi + g̃, (211)
where G0 and G1 are given by Eqs. (208-209) and g̃ com-
prises the rest of the Laguerre expansion (all Gl with l > 1),
i.e., it is the homogeneous solution of Eq. (157) that does not
contribute to either density or magnetic-field strength:
+ v‖b̂ ·∇g̃ = 0,
d3v g̃ = 0,
v2thi
g̃ = 0. (212)
Now substituting Eqs. (208) and (209) into Eq. (206) and
then substituting the result and Eqs. (202-203) into Eq. (201),
we find after some straightforward manipulations
Wcompr =
T0ig̃
(Λ+)2W +compr
(Λ−)2W −compr, (213)
where κ is defined by Eq. (204) and W±compr are the two inde-
pendent invariants that we derived in § 6.2.3. Thus, the gener-
alized energy for compressive fluctuations splits into three in-
dependently cascading parts: W±compr associated with the den-
sity and magnetic-field-strength fluctuations and a purely ki-
netic part given by the first term in Eq. (213) (see Fig. 5).
The dynamical evolution of this purely kinetic component is
described by Eq. (212)—it is a passively mixed, undamped
ballistic-type mode.
All three cascade channels lead to small perpendicular spa-
tial scales via passive mixing by the Alfvénic turbulence and
also to small scales in v‖ via the parallel phase mixing pro-
cess discussed in § 6.2.4 (note that g̃ is subject to this process
as well).
6.3. Parallel and Perpendicular Cascades
Let us return to the kinetic equation (157) and transform
it to the Lagrangian frame associated with the velocity field
u⊥ = ẑ×∇⊥Φ of the Alfvén waves: (t,r) → (t,r0), where
r(t,r0) = r0 +
dt ′u⊥(t
′,r(t ′,r0)). (214)
In this frame, the convective derivative d/dt defined in
Eq. (160) turns into ∂/∂t, while the parallel spatial gradient
b̂ ·∇ can be calculated by employing the Cauchy solution for
the perturbed magnetic field δB⊥ = ẑ×∇⊥Ψ:
b̂(t,r) = ẑ +
δB⊥(t,r)
= b̂(0,r0) ·∇0r, (215)
where r is given by Eq. (214) and ∇0 = ∂/∂r0. Then
b̂ ·∇ = b̂(0,r0) ·
·∇ = b̂(0,r0) ·∇0 =
, (216)
where s0 is the arc length along the perturbed magnetic field
taken at t = 0 [if δB⊥(0,r0) = 0, s0 = z0]. Thus, in the La-
grangian frame associated with the Alfvénic component of
the turbulence, Eq. (157) is linear. This means that, if the
effect of finite ion gyroradius is neglected, the KRMHD sys-
tem does not give rise to a cascade of density and magnetic-
field-strength fluctuations to smaller scales along the moving
(perturbed) field lines, i.e., b̂ · ∇δne and b̂ · ∇δB‖ do not in-
crease. In contrast, there is a perpendicular cascade (cascade
in k⊥): the perpendicular wandering of field lines due to the
Alfvénic turbulence causes passive mixing of δne and δB‖ in
the direction transverse to the magnetic field (see § 2.6 for a
quick recapitulation of the standard scaling argument on the
passive cascade that leads to a k
⊥ in the perpendicular di-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 29
FIG. 7.— Lagrangian mixing of passive fields: fluctuations develop small
scales across, but not along the exact field lines.
rection). Figure 7 illustrates this situation.26
We emphasize that this lack of nonlinear refinement of the
scale of δne and δB‖ along the moving field lines is a particu-
lar property of the compressive component of the turbulence,
not shared by the Alfvén waves. Indeed, unlike Eq. (157), the
RMHD equations (155-156), do not reduce to a linear form
under the Lagrangian transformation (214), so the Alfvén
waves should develop small scales both across and along the
perturbed magnetic field.
Whether the density and magnetic-field-strength fluctua-
tions develop small scales along the magnetic field has direct
physical and observational consequences. Damping of these
fluctuations, both in the collisional and collisionless regimes,
discussed in § 6.1.2 and § 6.2.2, respectively, depends pre-
cisely on their scale along the perturbed field: indeed, the
linear results derived there are exact in the Lagrangian frame
(214). To summarize these results, the damping rate of δne
and δB‖ at βi ∼ 1 is
γ∼ vthiλmfpik2‖0, k‖0λmfpi ≪ 1, (217)
γ∼ vthik‖0, k‖0λmfpi ≫ 1, (218)
where k‖0 ∼ b̂ ·∇ is the wavenumber along the perturbed field
(i.e., if there is no parallel cascade, the wavenumber of the
large-scale stirring).
Whether this damping cuts off the cascades of δne and δB‖
depends on the relative magnitudes of the damping rate γ for
a given k⊥ and the characteristic rate at which the Alfvén
waves cause δne and δB‖ to cascade to higher k⊥. This rate
is ωA ∼ k‖AvA, where k‖A is the parallel wave number of the
Alfvén waves that have the same k⊥. Since the Alfvén waves
do have a parallel cascade, assuming scale-by-scale critical
balance (3) leads to [Eq. (5)]
k‖A ∼ k
0 . (219)
If, in contrast to the Alfvén waves, δne and δB‖ have no par-
allel cascade, k‖0 does not grow with k⊥, so, for large enough
k⊥, k‖0 ≪ k‖A and γ≪ωA. This means that, despite the damp-
ing, the density and field-strength fluctuations should have
perpendicular cascades extending to the ion gyroscale.
The validity of the argument at the beginning of this sec-
tion that ruled out the parallel cascade of δne and δB‖ is not
quite as obvious as it might appear. Lithwick & Goldreich
(2001) argued that the dissipation of δne and δB‖ at the ion
gyroscale would cause these fluctuations to become uncorre-
lated at the same parallel scales as the Alfvénic fluctuations by
which they are mixed, i.e., k‖0 ∼ k‖A. The damping rate then
becomes comparable to the cascade rate, cutting off the cas-
cades of density and field-strength fluctuations at k‖λmfpi ∼ 1.
The corresponding perpendicular cutoff wavenumber is [see
26 Note that effectively, there is also a cascade in k‖ if the latter is mea-
sured along the unperturbed field—more precisely, a cascade in kz. This is
due to the perpendicular deformation of the perturbed magnetic field by the
Alfvén-wave turbulence: since ∇⊥ grows while b̂ ·∇ remains the same, we
have from Eq. (123) ∂/∂z ≃ −(δB⊥/B0) ·∇⊥.
Eq. (219)]
k⊥ ∼ l1/20 λ
mfpi . (220)
Asymptotically speaking, in a weakly collisional plasma,
this cutoff is far above the ion gyroscale, k⊥ρi ≪ 1. How-
ever, the relatively small value of λmfpi in the warm ISM,
which was the main focus of Lithwick & Goldreich 2001,
meant that the numerical value of the perpendicular cutoff
scale given by Eq. (220) was, in fact, quite close both to
the ion gyroscale (see Table 1) and to the observational es-
timates for the inner scale of the electron-density fluctuations
in the ISM (Spangler & Gwinn 1990; Armstrong et al. 1995).
Thus, it was not possible to tell whether Eq. (220), rather than
k⊥ ∼ ρ−1i , represented the correct prediction.
The situation is rather different in the nearly collision-
less case of the solar wind, where the cutoff given by
Eq. (220) would mean that very little density or field-
strength fluctuations should be detected above the ion gy-
roscale. Observations do not support such a conclu-
sion: the density fluctuations appear to follow a k−5/3 law
at all scales larger than a few times ρi (Lovelace et al.
1970; Woo & Armstrong 1979; Celnikier et al. 1983, 1987;
Coles & Harmon 1989; Marsch & Tu 1990b; Coles et al.
1991), consistently with the expected behavior of an un-
damped passive scalar field (see § 2.6). An extended range
of k−5/3 scaling above the ion gyroscale is also observed for
the fluctuations of the magnetic-field strength (Marsch & Tu
1990b; Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Alexandrova et al. 2008a).
These observational facts suggest that the cutoff formula
(220) does not apply. This does not, however, conclusively
vitiate the Lithwick & Goldreich (2001) theory. Heuristically,
their argument is plausible, although it is, perhaps, useful
to note that in order for the effect of the perpendicular dis-
sipation terms, not present in the KRMHD equations (157-
159), to be felt, the density and field-strength fluctuations
should reach the ion gyroscale in the first place. Quanti-
tatively, the failure of the compressive fluctuations in the
solar wind to be damped could still be consistent with the
Lithwick & Goldreich (2001) theory because of the relative
weakness of the collisionless damping, especially at low beta
(§ 6.2.2)—the explanation they themselves favor. The way to
check observationally whether this explanation suffices would
be to make a comparative study of the compressive fluctua-
tions for solar-wind data with different values of βi. If the
strength of the damping is the decisive factor, one should al-
ways see cascades of both δne and δB‖ at low βi, no cascades
at βi ∼ 1, and a cascade of δB‖ but not δne at high βi (in
this limit, the damping of the density fluctuations is strong,
of the field-strength weak; see § 6.2.2). If, on the other hand,
the parallel cascade of the compressive fluctuations is intrin-
sically inefficient, very little βi dependence is expected and a
perpendicular cascade should be seen in all cases.
Obviously, an even more direct observational (or numer-
ical) test would be the detection or non-detection of near-
perfect alignment of the density and field-strength structures
with the moving field lines (not with the mean magnetic
field—see footnote 26), but it is not clear how to measure
this reliably. It is interesting, in this context, that in near-
the-Sun measurements, the density fluctuations are reported
to have the form of highly anisotropic filaments aligned with
the magnetic field (Armstrong et al. 1990; Grall et al. 1997;
Woo & Habbal 1997). Another intriguing piece of observa-
30 SCHEKOCHIHIN ET AL.
tional evidence is the discovery that the local structure of the
magnetic-field-strength and density fluctuations at 1 AU is, in
a certain sense, correlated with the solar cycle (Kiyani et al.
2007; Hnat et al. 2007; Wicks et al. 2009)—this suggests a
dependence on initial conditions that is absent in the Alfvénic
fluctuations and that presumably should also disappear in the
compressive fluctuations if the latter are fully mixed both in
the perpendicular and parallel directions.
7. TURBULENCE IN THE DISSIPATION RANGE: ELECTRON RMHD
AND THE ENTROPY CASCADE
7.1. Transition at the Ion Gyroscale
The validity of the theory discussed in § 5 and § 6 breaks
down when k⊥ρi ∼ 1. As the ion gyroscale is approached,
the Alfvén waves are no longer decoupled from the rest of
the plasma dynamics. All modes now contain perturbations
of density and magnetic-field strength and can be collision-
lessly damped. Because of the low-frequency nature of the
Alfvén-wave cascade, ω ≪ Ωi even at k⊥ρi ∼ 1 [Eq. (46)],
so the ion cyclotron resonance (ω − k‖v‖ = ±Ωi) is not im-
portant, while the Landau one (ω = k‖v‖) is. The linear the-
ory of this collisionless damping in the gyrokinetic approx-
imation is worked out in detail in Howes et al. (2006) (see
also Gary & Borovsky 2008). Figure 8 shows the solutions of
their dispersion relation that illustrate how the Alfvén wave
becomes a dispersive kinetic Alfvén wave (KAW) (see § 7.3)
and collisionless damping becomes important as the ion gy-
roscale is reached.
We stress that this transition occurs at the ion gyroscale, not
at the ion inertial scale di = ρi/
βi (except in the limit of cold
ions, τ = T0i/T0e ≪ 1; see Appendix E). This statement is true
even when βi is not order unity, as illustrated in Fig. 8: for the
three cases plotted there, k⊥di = 1 corresponds to k⊥ρi = 0.1,
1 and 10 for βi = 0.01, 1 and 100, respectively, but there is
no trace of the ion inertial scale in the solutions of the linear
dispersion relation. Nonlinearly, in the limit βi ≪ 1, we may
consider the scales k⊥di ∼ 1 and expand the gyrokinetics in
k⊥ρi = k⊥di
βi ≪ 1 in a way similar to how it was done in § 5
and obtain precisely the same results: Alfvénic fluctuations
described by the RMHD equations and compressive fluctua-
tions passively advected by them and satisfying the reduced
kinetic equation derived in § 5.5. Thus, even though di ≫ ρi
at low beta, there is no change in the nature of the turbulent
cascade until k⊥ρi ∼ 1 is reached.
The nonlinear theory of what happens at k⊥ρi ∼ 1 is very
poorly understood. It is, however, possible to make progress
by examining what kind of fluctuations emerge on the other
side of the transition, at k⊥ρi ≫ 1. As we will demonstrate
below, it turns out that another turbulent cascade—this time
of KAW—is possible in this so-called dissipation range. It
can transfer the energy of KAW-like fluctuations down to the
electron gyroscale, where electron Landau damping becomes
important (see Howes et al. 2006). Some observational evi-
dence of KAW is, indeed, available in the solar wind and the
magnetosphere (Bale et al. 2005; Grison et al. 2005, see fur-
ther discussion in § 8.2.4). Below we derive the equations that
describe KAW-like fluctuations in the scale range k⊥ρi ≫ 1,
k⊥ρe ≪ 1 (§ 7.2) and work out a Kolmogorov-style scaling
theory for this cascade (§ 7.5).
Because of the presence of the collisionless damping at the
ion gyroscale, only a certain fraction of the turbulent power
arriving there from the inertial range is converted into the
KAW cascade, while the rest is Landau-damped. The damp-
ing leads to the heating of the ions, but the process of deposit-
ing the collisionlessly damped fluctuation energy into the ion
heat is non-trivial because, as we explained in § 3.5, collisions
do need to play a role in order for true heating to occur. As
we explained in § 3.5 and will see specifically for the dissi-
pation range in § 7.8, the electromagnetic-fluctuation energy
does not disappear as a result of the Landau damping but is
converted into ion entropy fluctuations, while the generalized
energy is conserved. Collisions are then accessed and ion
heating achieved via a purely kinetic phenomenon: the ion
entropy cascade in phase space (nonlinear phase mixing), for
which a theory is developed in § 7.9 and § 7.10. A similar pro-
cess of conversion of the KAW energy into electron entropy
fluctuations and then electron heat is treated in § 7.12.
Figure 5 illustrates the routes energy takes from the ion gy-
roscale towards heating. Crucially, it is at k⊥ρi ∼ 1 that it
is decided how much energy would eventually go into the
ions and how much into electrons.27 How this distribution
of energy depends on plasma parameters (βi and T0i/T0e)
is an open theoretical question28 of considerable astrophys-
ical interest: e.g., the efficiency of ion heating is a key un-
known in the theory of advection-dominated accretion flows
(Quataert & Gruzinov 1999, see discussion in § 8.5) and of
the solar corona (e.g., Cranmer & van Ballegooijen 2003); we
will also see in § 7.11 that it may determine the form of the
observed dissipation-range spectra in space plasmas.
A short summary of this section is given in § 7.14.
7.2. Equations of Electron Reduced MHD
The derivation is straightforward: when ai ∼ k⊥ρi ≫ 1, all
Bessel functions in Eqs. (118-120) are small, so the integrals
of the ion distribution function vanish and Eqs. (118-120) be-
, (221)
u‖e =
4πen0e
∇2⊥A‖ = −
ρi∇2⊥Ψ√
, u‖i = 0, (222)
, (223)
where we used the definitions (135) of the stream and flux
functions Φ and Ψ.
These equations are a reflection of the fact that, for k⊥ρi ≫
1, the ion response is effectively purely Boltzmann, with the
gyrokinetic part hi contributing nothing to the fields or flows
[see Eq. (54) with hi omitted; hi does, however, play an impor-
tant role in the energy balance and ion heating, as explained
in §§ 7.8-7.10 below]. The Boltzmann response for ion den-
sity is expressed by Eq. (221). Equation (222) states that the
parallel ion flow velocity can be neglected. Finally, Eq. (223)
expresses the pressure balance for Boltzmann (and, therefore,
isothermal) electrons [Eq. (103)] and ions: if we write
B0δB‖
= −δpi − δpe = −T0iδni − T0eδne, (224)
27 Some of the energy of compressive fluctuations may go into ion heat via
collisional (§ 6.1.2) or collisionless (§ 6.2.2) damping of these fluctuations
in the inertial range. Whether this is a significant ion heating mechanism
depends on the efficiency of the parallel cascade (see § 6.2.4 and § 6.3).
28 How much energy is converted into ion entropy fluctuations in the pro-
cess of a nonlinear turbulent cascade is not necessarily directly related to the
strength of the linear collisionless damping.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 31
FIG. 8.— Numerical solutions of the linear gyrokinetic dispersion relation (for a detailed treatment of the linear theory, see Howes et al. 2006) showing the
transition from the Alfvén wave to KAW between the inertial range (k⊥ρi ≪ 1) and the dissipation range (k⊥ρi & 1). We show three cases: low beta (βi = 0.01),
βi = 1, and high beta (βi = 100). In all three cases, τ = 1 and Z = 1. Bold solid lines show the real frequency ω, bold dashed lines the damping rate γ, both
normalized by k‖vA (in gyrokinetics, ω/k‖vA and γ/k‖vA are functions of k⊥ only). Dotted lines show the asymptotic KAW solution (230). Horizontal solid line
shows the Alfvén wave ω = k‖vA. Vertical solid lines show k⊥ρi = 1 and k⊥ρe = 1. Note that the damping can be considered strong if the characteristic decay
time is comparable or shorter than the wave period, i.e., γ/ω & 1/2π. Thus, in these plots, the damping at k⊥ρi ∼ 1 is relatively weak for βi = 1, relatively
strong for low beta and very strong for high beta.
it follows that
, (225)
which, combined with Eq. (221), gives Eq. (223). We remind
the reader that the perpendicular Ampère’s law, from which
Eq. (223) was derived [Eq. (66) via Eq. (120)] is, in gyrokinet-
ics, indeed equivalent to the statement of perpendicular pres-
sure balance (see § 3.3).
Substituting Eqs. (221-223) into Eqs. (116-117), we obtain
the following closed system of equations
b̂ ·∇Φ, (226)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
. (227)
Note that, using Eq. (223), Eqs. (226) and (227) can be recast
as two coupled evolution equations for the perpendicular and
parallel components of the perturbed magnetic field, respec-
tively [Eqs. (C10) in Appendix C.2].
We shall refer to Eqs. (226-227) as Electron Reduced MHD
(ERMHD). They are related to the Electron Magnetohydrody-
namics (EMHD)—a fluid-like approximation that evolves the
magnetic field only and arises if one assumes that the mag-
netic field is frozen into the electron flow velocity ue, while
the ions are immobile, ui = 0 (Kingsep et al. 1990):
4πen0e
∇× [(∇×B)×B] . (228)
As explained in Appendix C.2, the result of applying the
RMHD/gyrokinetic ordering (§ 2.1 and § 3.1) to Eq. (228),
where B = B0ẑ + δB and
ẑ×∇⊥Ψ+ ẑ
, (229)
coincides with our Eqs. (226-227) in the effectively incom-
pressible limits of βi ≫ 1 or βe = βiZ/τ ≫ 1. When betas are
arbitrary, density fluctuations cannot be neglected compared
to the magnetic-field-strength fluctuations [Eq. (225)] and
give rise to perpendicular ion flows with ∇·ui 6= 0. Thus, our
ERMHD system constitutes the appropriate generalization of
EMHD for low-frequency anisotropic fluctuations without the
assumption of incompressibility.
A (more tenuous) relationship also exists between our
ERMHD system and the so-called Hall MHD, which, like
EMHD, is based on the magnetic field being frozen into
the electron flow, but includes the ion motion via the stan-
dard MHD momentum equation [Eq. (8)]. Strictly speak-
ing, Hall MHD can only be used in the limit of cold ions,
τ = T0i/T0e ≪ 1 (see, e.g., Ito et al. 2004; Hirose et al. 2004,
and Appendix E), in which case it can be shown to reduce
to Eqs. (226-227) in the appropriate small-scale limit (Ap-
pendix E). Although τ ≪ 1 is not a natural assumption for
most space and astrophysical plasmas, Hall MHD has, due to
its simplicity, been a popular theoretical paradigm in the stud-
ies of space and astrophysical plasma turbulence (see § 8.2.6).
We have therefore devoted Appendix E to showing how this
approximation fits into the theoretical framework proposed
here: namely, we derive the anisotropic low-frequency ver-
sion of the Hall MHD approximation from gyrokinetics under
the assumption τ ≪ 1 and discuss the role of the ion inertial
and ion sound scales, which acquire physical significance in
this limit. However, outside this Appendix, we assume τ ∼ 1
everywhere and shall not use Hall MHD.
The validity of the ERMHD equations as a model for
plasma dynamics in the dissipation range is further discussed
in § 7.6.
7.3. Kinetic Alfvén Waves
The linear modes supported by ERMHD are kinetic Alfvén
waves (KAW) with frequencies
ωk = ±
1 + Z/τ
2 +βi
1 + Z/τ
) k⊥ρik‖vA. (230)
This dispersion relation is illustrated in Fig. 8: note that the
transition from Alfvén waves to dispersive KAW always oc-
curs at k⊥ρi ∼ 1, even when βi ≪ 1 or βi ≫ 1. In the latter
case, there is a sharp frequency jump at the transition (accom-
panied by very strong ion Landau damping).
The eigenfunctions corresponding to the two waves with
32 SCHEKOCHIHIN ET AL.
FIG. 9.— Polarization of the kinetic Alfvén wave, see Eqs. (232) and (233).
frequencies (230) are
2 +βi
∓ k⊥Ψk. (231)
Using Eqs. (229) and (223), the perturbed magnetic-field vec-
tor can be expressed as follows
= −iẑ× k⊥
1 + Z/τ
2 +βi
1 + Z/τ
(232)
so, for a single “+” or “−” wave (corresponding to Θ−k = 0 or
k = 0, respectively), δBk rotates in the plane perpendicular
to the wave vector k⊥ clockwise with respect to the latter,
while the wave propagates parallel or antiparallel to the guide
field (Fig. 9).
The waves are elliptically right-hand polarized. Indeed, us-
ing Eq. (223), the perpendicular electric field is:
E⊥k = −ik⊥ϕ+
−ik⊥ + ẑ×k⊥
ϕ (233)
(cf. Gary 1986; Hollweg 1999). The second term is small in
the gyrokinetic expansion, so this is a very elongated ellipse
(Fig. 9).
7.4. Finite-Amplitude Kinetic Alfvén Waves
As we are about to argue for a critically balanced KAW
turbulence in a fashion analogous to the GS theory for the
Alfvén waves (§ 1.2), it is a natural question to ask how simi-
lar the nonlinear properties of a putative KAW cascade will be
to an Alfvén-wave cascade. As in the case of Alfvén waves,
there are two counterpropagating linear modes [Eqs. (230)
and (231)], and it turns out that certain superpositions of these
modes (KAW packets) are also exact nonlinear solutions of
Eqs. (226-227). Let us show that this is the case.
We might look for the nonlinear solutions of Eqs. (226-227)
by requiring that the nonlinear terms vanish. Since b̂ · ∇ =
∂/∂z + (1/vA){Ψ, · · ·}, this gives
{Ψ,Φ} = 0 ⇒ Ψ = c1Φ, (234)
{Ψ,ρ2i ∇2⊥Ψ} = 0 ⇒ ρ2i ∇2⊥Ψ = c2Ψ, (235)
where c1 and c2 are constants. Whether such solutions are
possible is determined by substituting Eqs. (234) and (235)
into Eqs. (226) and (227) and demanding that the two result-
ing linear equations be consistent with each other (both equa-
tions now just evolve Ψ). This is achieved if29
c21 = −
2 +βi
, (236)
so real solutions exist if c2 < 0. In particular, wave pack-
ets consisting of KAW given by one of the linear eigen-
modes (231) with an arbitrary shape in z but confined to a
single shell |k⊥| = k⊥ = const, satisfy Eqs. (234-236) with
c2 = −k2⊥ρ
i . This outcome is, in fact, only mildly non-trivial:
in gyrokinetics, the Poisson bracket nonlinearity [Eq. (59)]
vanishes for any monochromatic (in k⊥) mode because the
Poisson bracket of two modes with wavenumbers k⊥ and k′⊥
is ∝ ẑ · (k⊥ × k′⊥). Therefore, any monochromatic solution
of the linearized equations is also an exact nonlinear solution.
As we have shown above, a superposition of monochromatic
KAW that have a fixed k⊥, or, somewhat more generally, sat-
isfy Eq. (235) with a fixed c2, is still an exact solution.
Note that a similar procedure applied to the RMHD equa-
tions (17-18) returns the Elsasser solutions: perturbations of
arbitrary shape that satisfy Φ = ±Ψ. The physical difference
between these finite-amplitude Alfven-wave packets and the
finite-amplitude KAW packets discussed above is that non-
linear interactions can occur not just between counterpropa-
gating KAW but also between copropagating ones—a natural
conclusion because KAW are dispersive (their group velocity
along the guide field is ∝ vAk⊥ρi), so copropagating waves
with different k⊥ can “catch up” with each other and inter-
act.30
7.5. Scalings for KAW Turbulence
A scaling theory for the turbulence described by Eqs. (221-
227) can be constructed along the same lines as the GS theory
for the Alfvén-wave turbulence (§ 1.2). Namely, we shall as-
sume that the turbulence below the ion gyroscale consists of
KAW-like fluctuations with k‖ ≪ k⊥ (Quataert & Gruzinov
1999) and that the interactions between them are critically
balanced (Cho & Lazarian 2004), i.e., that the propagation
time and nonlinear interaction time are comparable at every
scale. We stress that none of these assumptions are, strictly
speaking, inevitable31 (and, in fact, neither were they in-
evitable in the case of Alfvén waves). Since we have de-
rived Eqs. (226-227) from gyrokinetics, the anisotropy of
the fluctuations described by these equations is hard-wired,
but it is not guaranteed that the actual physical cascade be-
low the ion gyroscale is indeed anisotropic, although anal-
ysis of solar-wind measurements does seem to indicate that
29 Formally speaking, c1 and c2 can depend on t and z. If this is allowed,
we still recover Eq. (236), but in addition to it, we get the evolution equation
c1∂c1/∂t = vA(1 + Z/τ )∂c1/∂z. This allows c1 = const, but there are, of
course, other solutions. We shall not consider them here.
30 The calculation above is analogous to the calculation by
Mahajan & Krishan (2005) for incompressible Hall MHD (i.e., essen-
tially, the high-βe limit of the equations discussed in Appendix E), but
the result is more general in the sense that it holds at arbitrary ion and
electron betas. The Mahajan–Krishan solution in the EMHD limit amounts
to noticing that Eq. (228) becomes linear for force-free (Beltrami) magnetic
perturbations, ∇× δB = λδB. Substituting Eq. (229) into this equation
and using Eq. (223), we see that the force-free equation is equivalent
to Eqs. (234-236) if c2 = −λ2 and the incompressible limit (βi ≫ 1 or
βe = βiZ/τ ≫ 1) is taken.
31 In fact, the EMHD turbulence was thought to be weak by several au-
thors, who predicted a k−2 spectrum of magnetic energy assuming isotropy
(Goldreich & Reisenegger 1992) or k
for the anisotropic case (Voitenko
1998; Galtier & Bhattacharjee 2003; Galtier 2006).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 33
at least a significant fraction of it is (see Leamon et al.
1998; Hamilton et al. 2008). Numerical simulations based
on Eq. (228) (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005)
have revealed that the spectrum of magnetic fluctuations
scales as k
⊥ , the outcome consistent with the assumptions
stated above. Let us outline the argument that leads to this
scaling.
First assume that the fluctuations are KAW-like and that Θ+
and Θ− [Eq. (231)] have similar scaling. This implies
1 +βi
Φλ (237)
(for the purposes of scaling arguments and order-of-
magnitude estimates, we set Z/τ = 1, but keep the βi de-
pendence so low- and high-beta limits could be recovered if
necessary). The fact that fixed-k⊥ KAW packets, which sat-
isfy Eq. (237) with λ = 1/k⊥, are exact nonlinear solutions
of the ERMHD equations (§ 7.4) lends some credence to this
assumption.
Assuming scale-space locality of interactions implies a
constant-flux KAW cascade: analogously to Eq. (1),
(Ψλ/λ)
τKAWλ
∼ (1 +βi)(Φλ/ρi)
τKAWλ
∼ εKAW = const, (238)
where τKAWλ is the cascade time and εKAW is the KAW energy
flux proportional to the fraction of the total flux ε (or the total
turbulent power Pext; see § 3.4) that was converted into the
KAW cascade at the ion gyroscale.
Using Eqs. (226-227) and Eq. (237), it is not hard to see
that the characteristic nonlinear decorrelation time is λ2/Φλ.
If the turbulence is strong, then this time is comparable to
the inverse KAW frequency [Eq. (230)] scale by scale and we
may assume the cascade time is comparable to either:
τKAWλ ∼
1 +βi
. (239)
In other words, this says that ∂/∂z ∼ (δB⊥/B0) ·∇⊥ and so
δB⊥λ/B0 ∼ λ/l‖λ (note that the last relation confirms that
our scaling arguments do not violate the gyrokinetic ordering;
see § 2.1 and § 3.1). Equation (239) is the critical-balance as-
sumption for KAW. As in the case of the Alfvén waves (§ 1.2),
we might argue physically that the critical balance is set up be-
cause the parallel correlation length l‖λ is determined by the
condition that a wave can propagate the distance l‖λ in one
nonlinear decorrelation time corresponding to the perpendic-
ular correlation length λ.
Combining Eqs. (238) and (239), we get the desired scaling
relations for the KAW turbulence:
(εKAW
)1/3 vA
(1 +βi)1/3
2/3, (240)
(1 +βi)1/6
, (241)
where l0 = v
A/ε, as in § 1.2. The first of these scaling relations
is equivalent to a k
⊥ spectrum of magnetic energy, the sec-
ond quantifies the anisotropy (which is stronger than for the
GS turbulence). Both scalings were confirmed in the numer-
ical simulations of Cho & Lazarian (2004)—it is their detec-
tion of the scaling (241) that makes a particularly strong case
that KAW turbulence is not weak and that the critical balance
hypothesis applies.
For KAW-like fluctuations, the density [Eq. (221)] and
magnetic field [Eqs. (223) and (231)] have the same spec-
trum as the scalar potential, i.e., k
⊥ , while the electric field
E ∼ k⊥ϕ has a k−1/3⊥ spectrum. The solar-wind fluctuation
spectra reported by Bale et al. (2005) indeed are consistent
with a transition to KAW turbulence around the ion gyroscale:
k−5/3 magnetic and electric-field power spectra at kρi ≪ 1 are
replaced, for kρi & 1, with what appears to be consistent with
a k−7/3 scaling for the magnetic-field spectrum and a k−1/3 for
the electric one (see Fig. 1). A similar result is recovered in
fully gyrokinetic simulations with βi = 1, τ = 1 (Howes et al.
2008b). However, not all solar-wind observations are quite as
straightforwardly supportive of the notion of the KAW cas-
cade and much steeper magnetic-fluctuation spectra have also
been reported (e.g., Denskat et al. 1983; Leamon et al. 1998;
Smith et al. 2006). Possible reasons for this will emerge in
§ 7.6 and § 7.11 and the solar-wind data are further discussed
in § 8.2.4 and § 8.2.5.
7.6. Validity of the Electron RMHD and the Effect of Electron
Landau Damping
The ERMHD equations derived in § 7 are valid provided
k⊥ρi ≫ 1 and also provided it is sufficient to use the leading
order in the mass-ratio expansion (isothermal electrons; see
§ 4). In particular, this means that the electron Landau damp-
ing is neglected. Asymptotically speaking, this is a rigorous
limit, but one must be cautious in applying it to real plas-
mas. Since the width of the scale range where k⊥ρi ≫ 1 and
k⊥ρe ≪ 1 is only ∼ (mi/me)1/2 ≃ 43, for some values of the
plasma parameters (T0i/T0e and βi) there may not be a very
broad interval of scales where the electron Landau damping
is truly negligible. Consider, for example, the low-beta limit,
βi ≪ 1. In this limit, the KAW frequency is ω ∼ k⊥ρik‖vA
[Eq. (230)]. The electron Landau damping becomes impor-
tant when ω ∼ k‖vthe, or k⊥ρe ∼
βi ≪ 1, so the ERMHD
approximation breaks down and, consequently, the KAW cas-
cade, if any, should be interrupted well before the electron
gyroscale is reached. Figure 8 shows the solution of the
full gyrokinetic dispersion relation (Howes et al. 2006) for
small, unity and large βi. One can judge for which scales and
how well (or how badly) the ERMHD approximation holds
from the precision with which the exact frequency follows the
asymptotic solution Eq. (230) and from the relative strength
of the damping compared to the real frequency of the waves.
Non-negligible electron Landau damping may affect turbu-
lence spectra because one can no longer assume a constant
flux of KAW energy as we did in § 7.5. To evaluate the conse-
quences of this effect, Howes et al. (2008a) constructed a sim-
ple model of spectral energy transfer and concluded that Lan-
dau damping leads to steepening of the KAW spectra—one
of several possible reasons for steep dissipation-range spectra
observed in space plasmas (see also § 7.11).
7.7. Unfreezing of Flux
As ERMHD is a limit of the isothermal-electron-fluid sys-
tem (§ 4), the magnetic-field lines remain unbroken (see
§ 4.3). Within the orderings employed above (small mass ra-
tio, νii ∼ ω, βi ∼ 1, τ ∼ 1), the flux unfreezes only in the
vicinity of the electron gyroscale. It is interesting to evaluate
somewhat more precisely the scale at which this happens as a
function of plasma parameters.
34 SCHEKOCHIHIN ET AL.
Physically, there are three kinds of mechanisms by which
the flux conservation is broken: electron inertia, the effects of
finite electron gyroradius, and Ohmic resistivity. Let us take
the v‖ moment of the electron gyrokinetic equation [Eq. (57),
s = e, integration at constant r] and use Eq. (222) to evaluate
the inertial term in the resulting parallel electron momentum
equation:
d2e∇2⊥A‖, (242)
where de = ρe/
βe is the electron inertial scale and βe =
Zβi/τ . Comparing this with the ∂A‖/∂t term in the right-
hand side of the electron momentum equation, we see that the
electron inertia becomes important when k⊥ρe ∼
βe. The
finite-gyroradius effects enter when k⊥ρe ∼ 1. Thus, at low
βe, the electron inertia becomes important above the electron
gyroscale, whereas at high βe, the finite-gyroradius effects en-
ter first. Finally, the Ohmic resistivity comes from the colli-
sion term (see Appendix B.4):
d3vv‖
νeiu‖e ∼ νeik2⊥d2e A‖. (243)
Thus, resistivity starts to act when k⊥de ∼ (ω/νei)1/2. Using
the KAW frequency [Eq. (230)] to estimate ω and assuming
that τ is not small, we get
k⊥ρe ∼ k‖λmfpi
1 +βi
. (244)
Thus, the resistive scale can only be larger the electron gy-
roscale if the plasma is collisional (k‖λmfpi ≪ 1) and/or elec-
trons are much colder than ions (τ ≫ 1) and/or βi ≪ 1. Note
if only the last of these conditions is satisfied, the electron
inertia still becomes important at larger scales than resistivity.
7.8. Generalized Energy: KAW and Entropy Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≫ 1 is cal-
culated by substituting Eqs. (221) and (223) into Eq. (109):
T0i〈h2i 〉r
n0iT0i
=Whi +WKAW. (245)
Here the first term, Whi , is the total variance of hi, which is
proportional to minus the entropy of the ion gyrocenter distri-
bution (see § 3.5) and whose cascade to collisional scales will
be discussed in § 7.9 and § 7.10. The remaining two terms are
the independently cascaded KAW energy:
WKAW =
min0i
|∇⊥Ψ|2
min0i
|Θ+|2 + |Θ−|2
. (246)
Although we can write WKAW as the sum of the energies of
the “+” and “−” linear KAW eigenmodes [Eq. (231)], which
are also exact nonlinear solutions (§ 7.4), the two do not cas-
cade independently and can exchange energy. Note that the
ERMHD equations also conserve
d3rΨΦ, which is readily
interpreted as the helicity of the perturbed magnetic field (see
Appendix F.3). However, it does not affect the KAW cascade
discussed in § 7.5 because it can be argued to have a tendency
to cascade inversely (Appendix F.6).
Comparing the way the generalized energy is split above
and below the ion gyroscale (see § 5.6 for the k⊥ρi ≪ 1 limit),
we interpret what happens at the k⊥ρi ∼ 1 transition as a redis-
tribution of the power that arrived from large scales between
a cascade of KAW and a cascade of the (minus) gyrocenter
entropy in the phase space (see Fig. 5). The latter cascade
is the way in which the energy diverted from the electromag-
netic fluctuations by the collisionless damping (wave–particle
interaction) can be transferred to the collisional scales and de-
posited into heat (§ 7.1). The concept of entropy cascade as
the key agent in the heating of the plasma was introduced in
§ 3.5, where we promised a more detailed discussion later on.
We now proceed to this discussion.
7.9. Entropy Cascade
The ion-gyrocenter distribution function hi satisfies the ion
gyrokinetic equation (121), where ion–electron collisions are
neglected under the mass-ratio expansion. At k⊥ρi ≫ 1, the
dominant contribution to 〈χ〉Ri comes from the electromag-
netic fluctuations associated with KAW turbulence. Since
the KAW cascade is decoupled from the entropy cascade, hi
is a passive tracer of the ring-averaged KAW turbulence in
phase space. Expanding the Bessel functions in the expres-
sion for 〈χ〉Ri ,k [ai ≫ 1 in Eq. (69) with s = i] and making
use of Eqs. (222-223) and of the KAW scaling Ψ ∼ Φ/k⊥ρi
[Eq. (231)], it is not hard to show that
〈χ〉Ri ,k ≃
〈ϕ〉Ri ,k =
J0(ai)Φk
, (247)
where
J0(ai) ≃
, ai = k⊥ρi
, (248)
so hi satisfies [Eq. (121)]
+{〈Φ〉Ri ,hi} =
βiρivA
∂〈Φ〉Ri
F0i + 〈Cii[hi]〉Ri
(249)
with the conservation law [Eq. (70), s = i]
βi ρivA
∂〈Φ〉Ri
hi 〈Cii[hi]〉Ri
. (250)
7.9.1. Nonlinear Perpendicular Phase Mixing
The wave–particle interaction term (the first term on the
right hand sides of these two equations) will shortly be seen
to be subdominant at k⊥ρi ≫ 1. It represents the source of
the invariant Whi due to the collisionless damping at the ion
gyroscale of some fraction of the energy arriving from the in-
ertial range. In a stationary turbulent state, we should have
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 35
FIG. 10.— Nonlinear perpendicular phase-mixing mechanism: the
gyrocenter distribution function at Ri of particles with velocities v⊥ and v
is mixed by turbulent fluctuations of the potential Φ (E×B flows) averaged
over particle orbits separated by a distance greater than the correlation length
of Φ.
dWhi/dt = 0 and this source should be balanced on average by
the (negative definite) collisional dissipation term ( = heating;
see § 3.5). This balance can only be achieved if hi develops
small scales in the velocity space and carries the generalized
energy, or, in this case, entropy, to scales in the phase space at
which collisions are important. A quick way to see this is by
recalling that the collision operator has two velocity deriva-
tives and can only balance the terms on the left-hand side of
Eq. (249) if
∼ ω ⇒ δv
, (251)
where ω is the characteristic frequency of the fluctuations
of hi. If νii ≪ ω, δv/vthi ≪ 1. This is certainly true for
k⊥ρi ∼ 1: taking ω ∼ k‖vA and using k‖λmfpi ≫ 1 (which
is the appropriate limit at and below the ion gyroscale for
most of the plasmas of interest; cf. footnote 24), we have
νii/ω ∼
βi/k‖λmfpi ≪ 1.
The condition (251) means that the collision rate can be ar-
bitrarily small—this will always be compensated by the suf-
ficiently fine velocity-space structure of the distribution func-
tion to produce a finite amount of entropy production (heat-
ing) independent of νii in the limit νii → +0. The situa-
tion bears some resemblance to the emergence of small spa-
tial scales in neutral-fluid turbulence with arbitrarily small
but non-zero viscosity (Kolmogorov 1941). The analogy
is not perfect, however, because the ion gyrokinetic equa-
tion (249) does not contain a nonlinear interaction term that
would explicitly cause a cascade in the velocity space. In-
stead, the (ring-averaged) KAW turbulence mixes hi in the
gyrocenter space via the nonlinear term in Eq. (249), so hi
will have small-scale structure in Ri on characteristic scales
much smaller than ρi. Let us assume that the dominant non-
linear effect is a local interaction of the small-scale fluctua-
tions of hi with the similarly small-scale component of 〈Φ〉Ri .
Since ring averaging is involved and k⊥ρi is large, the val-
ues of 〈Φ〉Ri corresponding to two velocities v and v′ will
come from spatially decorrelated electromagnetic fluctuations
if k⊥v⊥/Ωi and k⊥v
⊥/Ωi [the argument of the Bessel function
in Eq. (247)] differ by order unity, i.e., for
|v⊥ − v′⊥|
(252)
(see Fig. 10). This relation gives a correspondence between
the decorrelation scales of hi in the position and velocity
space. Combining Eqs. (252) and (251), we see that there is
a collisional cutoff scale determined by k⊥ρi ∼ (ω/νii)1/2 ≫
1.32 The cutoff scale is much smaller than the ion gyroscale.
In the range between these scales, collisional dissipation is
small. The ion entropy fluctuations are transferred across this
scale range by means of a cascade, for which we will con-
struct a scaling theory in § 7.9.2 (and, for the case without the
background KAW turbulence, in § 7.10).
It is important to emphasize that no matter how small the
collisional cutoff scale is, all of the generalized energy chan-
neled into the entropy cascade at the ion gyroscale eventually
reaches it and is converted into heat. Note that the rate at
which this happens is in general amplitude-dependent because
the process is nonlinear, although we will argue in § 7.9.4 (see
also § 7.10.3) that the nonlinear cascade time and the parallel
linear propagation (particle streaming) time are related by a
critical-balance-like condition (we will also argue there that
the linear parallel phase mixing, which can generate small
scales in v‖, is a less efficient process than the nonlinear per-
pendicular one discussed above).
It is interesting to note the connection between the entropy
cascade and certain aspects of the gyrofluid closure formal-
ism developed by Dorland & Hammett (1993). In their the-
ory, the emergence of small scales in v⊥ manifested itself as
the growth of high-order v⊥ moments of the gyrocenter distri-
bution function. They correctly identified this effect as a con-
sequence of the nonlinear perpendicular phase mixing of the
gyrocenter distribution function caused by a perpendicular-
velocity-space spread in the ring-averaged E ×B velocities
(given by 〈uE〉Ri = ẑ×∇〈Φ〉Ri in our notation) arising at and
below the ion gyroscale.
7.9.2. Scalings
Since entropy is a conserved quantity, we will follow the
well trodden Kolmogorov path, assume locality of interac-
tions in scale space and constant entropy flux, and conclude,
analogously to Eq. (1),
v8thi
h2iλ ∼ εh = const, (253)
where εh is the entropy flux proportional to the fraction of the
total turbulent power ε (or Pext; see § 3.4) that was diverted
into the entropy cascade at the ion gyroscale, and is the cas-
cade time that we now need to find.
By the critical-balance assumption, the decorrelation time
of the electromagnetic fluctuations in KAW turbulence is
comparable at each scale to the KAW period at that scale and
to the nonlinear interaction time [Eq. (239)]:
τKAWλ ∼
(1 +βi)1/3
. (254)
The characteristic time associated with the nonlinear term in
Eq. (249) is longer than τKAWλ by a factor of (ρi/λ)
1/2 due to
the ring averaging, which reduces the strength of the nonlinear
interaction. This weakness of the nonlinearity makes it pos-
sible to develop a systematic analytical theory of the entropy
32 Another source of small-scale spatial smoothing comes from the per-
pendicular gyrocenter-diffusion terms ∼ −νii(v/vthi)2k2⊥ρ
i hik that arise in
the ring-averaged collision operators, e.g., the second term in the model
operator (B13). These terms again enforce a cutoff wavenumber such that
k⊥ρi ∼ (ω/νii)
1/2 ≫ 1.
36 SCHEKOCHIHIN ET AL.
cascade (Schekochihin & Cowley 2009). It is also possible
to estimate the cascade time via a more qualitative argument
analogous to that first devised by Kraichnan (1965) for the
weak turbulence of Alfvén waves: during each KAW correla-
tion time τKAWλ, the nonlinearity changes the amplitude of hi
by only a small amount:
∆hiλ ∼ (λ/ρi)1/2hiλ ≪ hiλ; (255)
these changes accumulate with time as a random walk,
so after time t, the cumulative change in amplitude is
∆hiλ(t/τKAWλ)
1/2; finally, the cascade time t = is the time
after which the cumulative change in amplitude is compara-
ble to the amplitude itself, which gives, using Eq. (254),
τKAWλ ∼
(1 +βi)1/3
. (256)
Substituting this into Eq. (253), we get
hiλ ∼
v3thi
)1/2(
(1 +βi)1/6√
(257)
which corresponds to a k
⊥ spectrum of entropy.
In the argument presented above, we assumed that the scal-
ing of hi was determined by the nonlinear mixing of hi by
the ring-averaged KAW fluctuations rather than by the wave–
particle interaction term on the right-hand side of Eq. (249).
We can now confirm the validity of this assumption. The
change in amplitude of hi in one KAW correlation time τKAWλ
due to the wave–particle interaction term is
∆hiλ∼
v3thi
βiρivA
∼ n0i
v3thi
(εKAW
)1/3 1√
βi (1 +βi)1/3
7/6, (258)
where we have used Eq. (240). Comparing this with Eq. (255)
and using Eq. (257), we see that ∆hiλ in Eq. (258) is a factor
of (λ/ρi)
1/2 smaller than ∆hiλ due to the nonlinear mixing.
7.9.3. Phase-Space Cutoff
To work out the cutoff scales both in the position and veloc-
ity space, we use Eqs. (251) and (252): in Eq. (251), ω ∼ 1/,
where is the characteristic decorrelation time of hi given by
Eq. (256); using Eq. (252), we find the cutoffs:
∼ (νiiτρi )3/5 = Do−3/5, (259)
where τρi is the cascade time [Eq. (256)] taken at λ = ρi.
By a recently established convention, the dimensionless num-
ber Do = 1/νiiτρi is called the Dorland number. It plays
the role of Reynolds number for kinetic turbulence, mea-
suring the scale separation between the ion gyroscale and
the collisional dissipation scale (Schekochihin et al. 2008b;
Tatsuno et al. 2009a,b).
7.9.4. Parallel Phase Mixing
Another assumption, which was made implicitly, was that
the parallel phase mixing due to the second term on the left-
hand side of Eq. (249) could be ignored. This requires jus-
tification, especially because it is with this “ballistic” term
that one traditionally associates the emergence of small-scale
structure in the velocity space (e.g., Krommes & Hu 1994;
Krommes 1999; Watanabe & Sugama 2004). The effect of
the parallel phase mixing is to produce small scales in veloc-
ity space δv‖ ∼ 1/k‖t. Let us assume that the KAW turbu-
lence imparts its parallel decorrelation scale to hi and use the
scaling relation (241) to estimate k‖ ∼ l−1‖λ. Then, after one
cascade time [Eq. (256)], hi is decorrelated on the parallel
velocity scales
βi(1 +βi)
∼ 1. (260)
We conclude that the nonlinear perpendicular phase mixing
[Eq. (259)] is more efficient than the linear parallel one. Note
that up to a βi-dependent factor Eq. (260) is equivalent to a
critical-balance-like assumption for hi in the sense that the
propagation time is comparable to the cascade time, or k‖v‖ ∼
−1 [see Eq. (249)].
7.10. Entropy Cascade in the Absence of KAW Turbulence
It is not currently known how one might determine ana-
lytically what fraction of the turbulent power arriving from
the inertial range to the ion gyroscale is channeled into the
KAW cascade and what fraction is dissipated via the kinetic
ion-entropy cascade introduced in § 7.9 (perhaps it can only
be determined by direct numerical simulations). It is cer-
tainly a fact that in many solar-wind measurements, the rel-
atively shallow magnetic-energy spectra associated with the
KAW cascade (§ 7.5) fail to appear and much steeper spectra
are detected (close to k−4; see Leamon et al. 1998; Smith et al.
2006). In view of this evidence, it is interesting to ask what
would be the nature of electromagnetic fluctuations below the
ion gyroscale if the KAW cascade failed to be launched, i.e.,
if all (or most) of the turbulent power were directed into the
entropy cascade (i.e., if W ≃Whi in § 7.8).
7.10.1. Equations
It is again possible to derive a closed set of equations for all
fluctuating quantities.
Let us assume (and verify a posteriori; § 7.10.4) that the
characteristic frequency of such fluctuations is much lower
than the KAW frequency [Eq. (230)] so that the first term in
Eq. (116) is small and the equation reduces to the balance of
the other two terms. This gives
, (261)
meaning that the electrons are purely Boltzmann [he = 0 to
lowest order; see Eq. (101)]. Then, from Eq. (118),
ρivthi
eik·r
d3vJ0(ai)hik (262)
Using Eq. (262), we find from Eq. (120) that the field-
strength fluctuations are
eik·r
v2thi
J1(ai)
hik, (263)
which is smaller than Zeϕ/T0i by a factor of βi/k⊥ρi.
Therefore, we can neglect δB‖/B0 compared to δne/n0e in
Eq. (117). Using Eq. (261), we get what is physically the
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 37
electron continuity equation:
+ b̂ ·∇
4πen0e
∇2⊥A‖ + u‖i
= 0, (264)
u‖i =
eik·r
d3vv‖J0(ai)hik. (265)
Note that in terms of the stream and flux functions, Eq. (264)
takes the form
ρ2i ∇2⊥Ψ =
, (266)
where we have approximated b̂ · ∇ ≃ ∂/∂z, which will, in-
deed, be shown to be correct in § 7.10.4.
Together with the ion gyrokinetic equation, which deter-
mines hi, Eqs. (261-264) form a closed set. They describe
low-frequency fluctuations of the density and electromagnetic
field due solely to the presence of fluctuations of hi below the
ion gyroscale.
It follows from Eq. (263) that δB‖/B0 contributes subdom-
inantly to 〈χ〉Ri [Eq. (69) with s = i and ai ≫ 1]. It will be
verified a posteriori (§ 7.10.4) that the same is true for A‖.
Therefore, Eqs. (247) and (249) continue to hold, as in the
case with KAW. This means that Eqs. (249) and (262) form
a closed subset. Thus the kinetic ion-entropy cascade is self-
regulating in the sense that hi is no longer passive (as it was
in the presence of KAW turbulence; § 7.9) but is mixed by
the ring-averaged “electrostatic” fluctuations of the scalar po-
tential, which themselves are produced by hi according to
Eq. (262).
The magnetic fluctuations are passive and determined by
the electrostatic and entropy fluctuations via Eqs. (263)
and (264).
7.10.2. Scalings
From Eq. (262), we can establish a correspondence between
Φλ and hiλ (the electrostatic fluctuations and the fluctuations
of the ion-gyrocenter distribution function):
Φλ ∼ ρivthi
hiλλ, (267)
where the factor of (λ/ρi)
1/2 comes from the Bessel function
[Eq. (248)] and the factor of (δv⊥/vthi)
1/2 results from the
v⊥ integration of the oscillatory factor in the Bessel function
times hi, which decorrelates on small scales in the velocity
space and, therefore, its integral accumulates in a random-
walk-like fashion. The velocity-space scales are related to the
spatial scales via Eq. (252), which was arrived at by an ar-
gument not specific to KAW-like fluctuations and, therefore,
continues to hold.
Using Eq. (267), we find that the wave–particle interaction
term in the right-hand side of Eq. (249) is subdominant: com-
paring it with ∂hi/∂t shows that it is smaller by a factor of
(λ/ρi)
3/2 ≪ 1. Therefore, it is the nonlinear term in Eq. (249)
that controls the scalings of hiλ and Φλ.
We now assume again the scale-space locality and con-
stancy of the entropy flux, so Eq. (253) holds. The cascade
(decorrelation) time is equal to the characteristic time associ-
ated with the nonlinear term in Eq. (249): ∼ (ρi/λ)1/2λ2/Φλ.
Substituting this into Eq. (253) and using Eq. (267), we ar-
rive at the desired scaling relations for the entropy cascade
(Schekochihin et al. 2008b):
v3thi
)1/3 1√
1/6, (268)
)1/3 vthi√
7/6, (269)
)1/3 √
1/3, (270)
where l0 = v
A/ε, as in § 1.2. Note that since the existence
of this cascade depends on it not being overwhelmed by the
KAW fluctuations, we should have εKAW ≪ ε and εh = ε −
εKAW ≈ ε.
The scaling for the ion-gyrocenter distribution function,
Eq. (268), implies a k
⊥ spectrum—the same as for the KAW
turbulence [Eq. (257)]. The scaling for the the cascade time,
Eq. (270), is also similar to that for the KAW turbulence
[Eq. (256)]. Therefore the velocity- and gyrocenter-space cut-
offs are still given by Eq. (259), where τρi is now given by
Eq. (270) taken at λ = ρi.
A new feature is the scaling of the scalar potential, given by
Eq. (269), which corresponds to a k
−10/3
⊥ spectrum (unlike the
KAW spectrum, § 7.5). This is a measurable prediction for the
electrostatic fluctuations: the implied electric-field spectrum
⊥ . From Eq. (261), we also conclude that the density
fluctuations should have the same spectrum as the scalar po-
tential, k
−10/3
⊥ —another measurable prediction.
The scalings derived above for the spectra of the ion
distribution function and of the scalar potential have been
confirmed in the numerical simulations by Tatsuno et al.
(2009a,b), who studied decaying electrostatic gyrokinetic tur-
bulence in two spatial dimensions. They also found velocity-
space scalings in accord with Eq. (252) (using a spectral
representation of the correlation functions in the v⊥ space
based on the Hankel transform of the distribution function;
see Plunk et al. 2009).
7.10.3. Parallel Cascade and Parallel Phase Mixing
We have again ignored the ballistic term (the second on
the left-hand side) in Eq. (249). We will estimate the effi-
ciency of the parallel spatial cascade of the ion entropy and of
the associated parallel phase mixing by making a conjecture
analogous to the critical balance: assuming that any two per-
pendicular planes only remain correlated provided particles
can stream between them in one nonlinear decorrelation time
(cf. § 1.2 and § 7.9.4), we conclude that the parallel particle-
streaming frequency k‖v‖ should be comparable at each scale
to the inverse nonlinear time −1, so
k‖vthi ∼ 1. (271)
As we explained in § 7.9.4, the parallel scales in the velocity
space generated via the ballistic term are related to the parallel
wavenumbers by δv‖ ∼ 1/k‖t. From Eq. (271), we find that
after one cascade time , the typical parallel velocity scale is
δv‖/vthi ∼ 1, so the parallel phase mixing is again much less
efficient than the perpendicular one.
Note that Eq. (271) combined with Eq. (270) means that the
anisotropy is again characterized by the scaling relation k‖ ∼
⊥ , similarly to the case of KAW turbulence [see Eq. (241)
and § 7.9.4].
38 SCHEKOCHIHIN ET AL.
7.10.4. Scalings for the Magnetic Fluctuations
The scaling law for the fluctuations of the magnetic-field
strength follows immediately from Eqs. (263) and (269):
ρivthi
−11/6
13/6, (272)
whence the spectrum of these fluctuations is k
−16/3
The scaling of A‖ (the perpendicular magnetic fluctuations)
depends on the relation between k‖ and k⊥. Indeed, the ratio
between the first and the third terms on the left-hand side of
Eq. (264) [or, equivalently, between the first and second terms
on the right-hand side of Eq. (266)] is ∼
k‖vthi
. For a crit-
ically balanced cascade, this makes the two terms comparable
[Eq. (271)]. Using the first term to work out the scaling for the
perpendicular magnetic fluctuations, we get, using Eq. (269),
ρivthi
−11/6
13/6, (273)
which is the same scaling as for δB‖/B0 [Eq. (272)].
Using Eq. (273) together with Eqs. (269) and (270), it is
now straightforward to confirm the three assumptions made
in § 7.10.1 that we promised to verify a posteriori:
1. In Eq. (116), ∂A‖/∂t ≪ cb̂ ·∇ϕ, so Eq. (261) holds (the
electrons remain Boltzmann). This means that no KAW
can be excited by the cascade.
2. δB⊥/B0 ≪ k‖/k⊥, so b̂ ·∇ ≃ ∂/∂z in Eq. (264). This
means that field lines are not significantly perturbed.
3. In the expression for 〈χ〉Ri [Eq. (69)], v‖A‖/c ≪ ϕ, so
Eq. (249) holds. This means that the electrostatic fluc-
tuations dominate the cascade.
7.11. Cascades Superposed?
The spectra of magnetic fluctuations obtained in § 7.10.4
are very steep—steeper, in fact, than those normally observed
in the dissipation range of the solar wind (§ 8.2.5). One might
speculate that the observed spectra may be due to a superposi-
tion of the two cascades realizable below the ion gyroscale: a
high-frequency cascade of KAW (§ 7.5) and a low-frequency
cascade of electrostatic fluctuations due to the ion entropy
fluctuations (§ 7.10). Such a superposition could happen if
the power going into the KAW cascade is relatively small,
εKAW ≪ ε. One then expects an electrostatic cascade to be
set up just below the ion gyroscale with the KAW cascade
superseding it deeper into the dissipation range. Comparing
Eqs. (240) and (269), we can estimate the position of the spec-
tral break:
k⊥ρi ∼
ε/εKAW
. (274)
Since ρi/ρe ∼ (τmi/me)1/2/Z is not a very large number, the
dissipation range is not very wide. It is then conceivable that
the observed spectra are not true power laws but simply non-
asymptotic superpositions of the electrostatic and KAW spec-
tra with the observed range of “effective” spectral exponents
due to varying values of the spectral break (274) between the
two cascades.33
33 Several alternative theories that aim to explain the dissipation-range
spectra exist: see § 8.2.6.
The value of εKAW/ε specific to any particular set of param-
eters (βi, τ , etc.) is set by what happens at k⊥ρi ∼ 1 (§ 7.1;
see § 8.2.2, § 8.2.5, and § 8.5 for further discussion).
7.12. Below the Electron Gyroscale: The Last Cascade
Finally, let us consider what happens when k⊥ρe ≫ 1. At
these scales, we have to return to the full gyrokinetic sys-
tem of equations. The quasi-neutrality [Eq. (61)], parallel
[Eq. (62)] and perpendicular [Eq. (66)] Ampère’s law become
eik·r
d3vJ0(ae)hek, (275)
4πen0e
∇2⊥A‖ =
eik·r
d3vv‖J0(ae)hek, (276)
eik·r
v2the
J1(ae)
hek, (277)
where βe = βiZ/τ . We have discarded the velocity integrals
of hi both because the gyroaveraging makes them subdom-
inant in powers of (me/mi)
1/2 and because the fluctuations
of hi are damped by collisions [assuming the collisional cut-
off given by Eq. (259) lies above the electron gyroscale]. To
Eqs. (275-277), we must append the gyrokinetic equation for
he [Eq. (57) with s = e], thus closing the system.
The type of turbulence described by these equations is very
similar to that discussed in § 7.10. It is easy to show from
Eqs. (275-277) that
. (278)
Hence the magnetic fluctuations are subdominant in the ex-
pression for 〈χ〉Re [Eq. (69) with s = e and ae ≫ 1], so
〈χ〉Re ≃ 〈ϕ〉Re . The electron gyrokinetic equation then is
{〈ϕ〉Re ,he} =
, (279)
where the wave–particle interaction term in the right-hand
side has been dropped because it can be shown to be small
via the same argument as in § 7.10.2.
Together with Eq. (275), Eq. (279) describes the kinetic cas-
cade of electron entropy from the electron gyroscale down to
the scale at which electron collisions can dissipate it into heat.
This cascade the result of collisionless damping of KAW at
k⊥ρe ∼ 1, whereby the power in the KAW cascade is con-
verted into the electron-entropy fluctuations: indeed, in the
limit k⊥ρe ≫ 1, the generalized energy is simply
= Whe (280)
(see Fig. 5).
The same scaling arguments as in § 7.10.2 apply and scaling
relations analogous to Eqs. (268-270), and (272) duly follow:
v3the
(εKAW
1/6, (281)
(εKAW
vthe l
7/6, (282)
)1/3(
, (283)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 39
(εKAW
−11/6
13/6, (284)
where l0 = v
A/ε, as in § 1.2. The formula for the collisional
cutoffs in the wavenumber and velocity space is analogous to
Eq. (259):
∼ (νeiτρe )3/5, (285)
where τρe is the cascade time (283) taken at λ = ρe.
7.13. Validity of Gyrokinetics in the Dissipation Range
As the kinetic cascade takes the (generalized) energy to ever
smaller scales, the frequency ω of the fluctuations increases.
In applying the gyrokinetic theory, one must be mindful of
the need for this frequency to stay smaller than Ωi. Using
the scaling formulae for the characteristic times of the fluc-
tuations derived above [Eqs. (254), (270) and (283)], we can
determine the conditions for ω ≪ Ωi. Thus, for the gyroki-
netic theory to be valid everywhere in the inertial range, we
must have
k⊥ρi ≪ β3/4i
(286)
at all scales down to k⊥ρi ∼ 1, i.e., ρi/l0 ≪ β3/2i , not a very
stringent condition.
Below the ion gyroscale, the KAW cascade (§ 7.5) remains
in the gyrokinetic regime as long as
k⊥ρi ≪
i (1 +βi)
(287)
(we are assuming Ti/Te ∼ 1 everywhere). The condition for
this still to be true at the electron gyroscale is
i (1 +βi)
. (288)
The ion entropy fluctuations passively mixed by the KAW tur-
bulence (§ 7.9) satisfy Eq. (287) at all scales down to the ion
collisional cutoff [Eq. (259)] if
λmfpi
i (1 +βi)
. (289)
Note that the condition for the ion collisional cutoff to lie
above the electron gyroscale is
λmfpi
βi(1 +βi)1/3
)5/6(
(290)
In the absence of KAW turbulence, the pure ion-entropy cas-
cade (§ 7.10) remains gyrokinetic for
k⊥ρi ≪ β3/2i
. (291)
This is valid at all scales down to the ion collisional cutoff
provided λmfpi/l0 ≪ β3i (l0/ρi), an extremely weak condition,
which is always satisfied. This is because the ion-entropy
fluctuations in this case have much lower frequencies than in
the KAW regime. The ion collisional cutoff lies above the
electron gyroscale if, similarly to Eq. (290),
λmfpi
)5/6(
. (292)
If the condition (290) is satisfied, all fluctuations of the
ion distribution function are damped out above the electron
gyroscale. This means that below this scale, we only need
the electron gyrokinetic equation to be valid, i.e., ω ≪ Ωe.
The electron-entropy cascade (§ 7.12), whose characteristic
timescale is given by Eq. (283), satisfies this condition for
k⊥ρe ≪
β3/2e
. (293)
This is valid at all scales down to the electron
collisional cutoff [Eq. (285)] provided λmfpe/l0 ≪
(ε/εKAW)
2β3e (mi/me)
3(l0/ρe), which is always satisfied.
Within the formal expansion we have adopted (k⊥ρi ∼ 1
and k‖λmfpi ∼
βi), it is not hard to see that λmfpi/l0 ∼ ǫ2
and ρi/l0 ∼ ǫ3. Since all other parameters (me/mi, βi, βe
etc.) are order unity with respect to ǫ, all of the above con-
ditions for the validity of the gyrokinetics are asymptotically
correct by construction. However, in application to real as-
trophysical plasmas, one should always check whether this
construction holds. For example, substituting the relevant pa-
rameters for the solar wind shows that the gyrokinetic ap-
proximation is, in fact, likely to start breaking down some-
where between the ion and electron gyroscales (Howes et al.
2008a).34 This releases a variety of high-frequency wave
modes, which may be participating in the turbulent cascade
around and below the electron gyroscale (see, e.g., the recent
detailed observations of these scales in the magnetosheath by
Mangeney et al. 2006; Lacombe et al. 2006 or the early mea-
surements of high-frequency fluctuations in the solar wind by
Denskat et al. 1983; Coroniti et al. 1982).
7.14. Summary
In this section, we have analyzed the turbulence in the dissi-
pation range, which turned out to have many more essentially
kinetic features than the inertial range.
At the ion gyroscale, k⊥ρi ∼ 1, the kinetic cascade rear-
ranged itself into two distinct components: part of the (gener-
alized) energy arriving from the inertial range was collision-
lessly damped, giving rise to a purely kinetic cascade of ion-
entropy fluctuations, the rest was converted into a cascade of
Kinetic Alfvén Waves (KAW) (Fig. 5; see § 7.1 and § 7.8).
The KAW cascade is described by two fluid-like equa-
tions for two scalar functions, the magnetic flux function
Ψ = −A‖/
4πmin0i and the scalar potential, expressed, for
continuity with the results of § 5, in terms of the function
Φ = (c/B0)ϕ. The equations are (see § 7.2)
b̂ ·∇Φ, (294)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
, (295)
where b̂ · ∇ = ∂/∂z + (1/vA){Ψ, · · ·}. The density and
34 See this paper also for a set of numerical tests of the validity of gy-
rokinetics in the dissipation range, a linear theory of the conversion of KAW
into ion-cyclotron-damped Bernstein waves, and a discussion of the potential
(un)importance of ion cyclotron damping for the dissipation of turbulence.
40 SCHEKOCHIHIN ET AL.
magnetic-field-strength fluctuations are directly related to the
scalar potential:
. (296)
We call Eqs. (294-296) the Electron Reduced Magnetohydro-
dynamics (ERMHD).
The ion-entropy cascade is described by the ion gyrokinetic
equation:
+{〈Φ〉Ri ,hi} = 〈Cii[hi]〉Ri . (297)
The ion distribution function is mixed by the ring-averaged
scalar potential and undergoes a cascade both in the velocity
and gyrocenter space—this phase-space cascade is essential
for the conversion of the turbulent energy into the ion heat,
which can ultimately only be done by collisions (see § 7.9).
If the KAW cascade is strong (its power εKAW is an order-
unity fraction of the total injected turbulent power ε), it de-
termines Φ in Eq. (297), so the ion-entropy cascade is passive
with respect to the KAW turbulence. Equations (294-295) and
(297) form a closed system that determines the three func-
tions Φ, Ψ, hi, of which the latter is slaved to the first two.
One can also compute δne and δB‖, which are proportional
to Φ [Eq. (296)]. The generalized energy conserved by these
equations is given by Eq. (245).
If the KAW cascade is weak (εKAW ≪ ε), the ion-entropy
cascade dominates the turbulence in the dissipation range and
drives low-frequency mostly electrostatic fluctuations, with a
subdominant magnetic component. These are given by the
following relations (see § 7.10)
ρivthi
2(1 + τ/Z)
eik·r
d3vJ0(ai)hik, (298)
ρivthi
, (299)
eik·r ×
1 + Z/τ
J0(ai)
hik, (300)
eik·r
v2thi
J1(ai)
hik, (301)
where ai = k⊥v⊥/Ωi, Equations (297) and (298) form a closed
system for Φ and hi. The rest of the fields, namely δne, Ψ and
δB‖, are slaved to hi via Eqs. (299-301).
The fluid and kinetic models summarized above are valid
between the ion and electron gyroscales. Below the electron
gyroscale, the collisionless damping of the KAW cascade con-
verts it into a cascade of electron entropy, similar in nature to
the ion-entropy cascade (§ 7.12).
The KAW cascade and the low-frequency turbulence asso-
ciated with the ion-entropy cascade have distinct scaling be-
haviors. For the KAW cascade, the spectra of the electric,
density and magnetic fluctuations are (§ 7.5)
EE (k⊥) ∝ k−1/3⊥ , En(k⊥) ∝ k
⊥ , EB(k⊥) ∝ k
⊥ . (302)
For the ion- and electron-entropy cascades (§ 7.9 and § 7.12),
EE (k⊥) ∝ k−4/3⊥ , En(k⊥) ∝ k
−10/3
⊥ , EB(k⊥) ∝ k
−16/3
(303)
We argued in § 7.11 that the observed spectra in the dissipa-
tion range of the solar wind could be the result of a superpo-
sition of these two cascades, although a number of alternative
theories exist (§ 8.2.6).
8. DISCUSSION OF ASTROPHYSICAL APPLICATIONS
We have so far only occasionally referred to some relevant
observational evidence for space and astrophysical plasmas.
We now discuss in more detail how the theoretical framework
laid out above applies to real plasma turbulence in space.
Although we will discuss the interstellar medium, accre-
tion disks and galaxy clusters towards the end of this sec-
tion, the most rewarding source of observational information
about plasma turbulence in astrophysical conditions is the so-
lar wind and the magnetosheath because only there direct in
situ measurements of all the interesting quantities are possi-
ble. Measurements of the fluctuating magnetic and velocity
fields in the solar wind have been available since the 1960s
(Coleman 1968) and a vast literature now exists on their spec-
tra, anisotropy, Alfvénic character and many other aspects (a
short recent review is Horbury et al. 2005; two long ones are
Tu & Marsch 1995; Bruno & Carbone 2005). It is not our
aim here to provide a comprehensive survey of what is known
about plasma turbulence in the solar wind. Instead, we shall
limit our discussion to a few points that we consider impor-
tant in light of the theoretical framework proposed in this pa-
per.35 As we do this, we shall provide copious references to
the main body of the paper, so this section can be read as a
data-oriented guide to it, aimed both at a thorough reader who
has arrived here after going through the preceding sections
and an impatient one who has skipped to this one hoping to
find out whether there is anything of “practical” use in the
theoretical developments above.
8.1. Inertial-Range Turbulence in the Solar Wind
In the inertial range, i.e., for k⊥ρi ≪ 1, the solar-wind turbu-
lence should be described by the reduced hybrid fluid-kinetic
theory derived in § 5 (KRMHD). Its applicability hinges on
three key assumptions: (i) the turbulence is Alfvénic, i.e., con-
sists of small (δB/B0 ≪ 1) low-frequency (ω ∼ k‖vA ≪ Ωi)
perturbations of an ambient mean magnetic field and corre-
sponding velocity fluctuations; (ii) it is strongly anisotropic,
k⊥ ≫ k‖; (iii) the equilibrium distribution can be approxi-
mated or, at least, reasonably modeled by a Maxwellian with-
out loss of essential physics (this will be discussed in § 8.3).
If these assumptions are satisfied, KRMHD (summarized in
§ 5.7) is a rigorous set of dynamical equations for the inertial
range, a set of Kolmogorov-style scaling predictions for the
Alfvénic component of the turbulence can be produced (the
GS theory, reviewed in § 1.2), while to the compressive fluc-
tuations, the considerations of § 6 apply. So let us examine
the observational evidence.
8.1.1. Alfvénic Nature of the Turbulence
The presence of Alfvén waves in the solar wind was re-
ported already the early works of Unti & Neugebauer (1968)
and Belcher & Davis (1971). Alfvén waves are detected al-
ready at very low frequencies (large scales)—and, at these
35 An extended quantitative discussion of the applicability of the gyroki-
netic theory to the turbulence in the slow solar wind was given by Howes et al.
(2008a).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 41
low frequencies, have a k−1 spectrum.36 This spectrum cor-
responds to a uniform distribution of scales/frequencies of
waves launched by the coronal activity of the Sun. Nonlin-
ear interaction of these waves gives rise to an Alfvénic tur-
bulent cascade of the type that was discussed above. The ef-
fective outer scale of this cascade can be detected as a spec-
tral break where the k−1 scaling steepens to the Kolmogorov
slope k−5/3 (see Bavassano et al. 1982; Marsch & Tu 1990a;
Horbury et al. 1996 for fast-wind results on the spectral break;
for a discussion of the effective outer scale in the slow wind
at 1 AU, see Howes et al. 2008a). The particular scale at
which this happens increases with the distance from the Sun
(Bavassano et al. 1982), reflecting the more developed state
of the turbulence at later stages of evolution. At 1 AU, the
outer scale is roughly in the range of 105 − 106 km; the k−5/3
range extends down to scales/frequencies that correspond to a
few times the ion gyroradius (102 − 103 km; see Table 1).
The range between the outer scale (the spectral break) and
the ion gyroscale is the inertial range. In this range, δB/B0 de-
creases with scale because of the steep negative spectral slope.
Therefore, the assumption of small fluctuations, δB/B0 ≪ 1,
while not necessarily true at the outer scale, is increasingly
better satisfied further into the inertial range (cf. § 1.3).
Are these fluctuations Alfvénic? In a plasma such as the
solar wind, they ought to be because, as showed in § 5.3, for
k⊥ρi ≪ 1, these fluctuations are rigorously described by the
RMHD equations. The magnetic flux is frozen into the ion
motions, so displacing a parcel of plasma should produce a
matching (Alfvénic) perturbation of the magnetic field line
and vice versa: in an Alfvén wave, u⊥ = ±δB⊥/
4πmin0i.
The strongest confirmation that this is indeed true for the
inertial-range fluctuations in the solar wind was achieved by
Bale et al. (2005), who compared the spectra of electric and
magnetic fluctuations and found that they both scale as k−5/3
and follow each other with remarkable precision (see Fig. 1).
The electric field is a very good measure of the perpendicular
velocity field because, for k⊥ρi ≪ 1, the plasma velocity is
the E×B drift velocity, u⊥ = cE× ẑ/B0 (see § 5.4).
This picture of agreement between basic theory and ob-
servations is upset in a disturbing fashion by an extraordi-
nary recent result by Chapman & Hnat (2007); Podesta et al.
(2006) and J. E. Borovsky (2008, private communication),
who claim different spectral indices for velocity and mag-
netic fluctuations—k−3/2 and k−5/3, respectively. This result
is puzzling because if it is asymptotically correct in the iner-
tial range, it implies either u⊥ ≫ δB⊥ or u⊥ ≪ δB⊥ and it is
not clear how perpendicular velocity fluctuations in a near-
ideal plasma could fail to produce Alfvénic displacements
and, therefore, perpendicular magnetic field fluctuations with
matching energies. Plausible explanations may be either that
the velocity field in these measurements is polluted by a non-
Alfvénic component parallel to the magnetic field (although
data analysis by Chapman & Hnat 2007 does not support this)
or that the flattening of the velocity spectrum is due to some
form of a finite-gyroradius effect or even an energy injection
into the velocity fluctuations at scales approaching the ion
gyroscale (e.g., from the pressure-anisotropy-driven instabili-
36 Inferred from the frequency spectrum f −1 via the Taylor (1938) hypoth-
esis, f ∼ k ·Vsw , where Vsw is the mean velocity at which the wind blows
past the spacecraft. The Taylor hypothesis is a good assumption for the so-
lar wind because Vsw (∼ 800 km/s in the fast wind, ∼ 300 km/s in the slow
wind) is highly supersonic, super-Alfvénic and far exceeds the fluctuating
velocities.
ties, § 8.3).
8.1.2. Energy Spectrum
How solid is the statement that the observed spectrum
has a k−5/3 scaling? In individual measurements of the
magnetic-energy spectra, very high accuracy is claimed
for this scaling: the measured spectral exponent is be-
tween 1.6 and 1.7; agreement with Kolmogorov value 1.67
is often reported to be within a few percent (see, e.g.,
Horbury et al. 1996; Leamon et al. 1998; Bale et al. 2005;
Narita et al. 2006; Alexandrova et al. 2008a; Horbury et al.
2008)). There is a somewhat wider scatter of spectral in-
dices if one considers large sets of measurement intervals
(Smith et al. 2006), but overall, the observational evidence
does not appear to be consistent with a k
⊥ spectrum consis-
tently found in the MHD simulations with a strong mean field
(Maron & Goldreich 2001; Müller et al. 2003; Mason et al.
2007; Perez & Boldyrev 2008, 2009; Beresnyak & Lazarian
2008b) and defended on theoretical grounds in the recent
modifications of the GS theory by Boldyrev (2006) and by
Gogoberidze (2007) (see footnote 10). This discrepancy be-
tween observations and simulations remains an unresolved
theoretical issue. It is probably best addressed by numeri-
cal modeling of the RMHD equations (§ 2.2) and by a de-
tailed comparison of the structure of the Alfvénic fluctuations
in such simulations and in the solar wind.
8.1.3. Anisotropy
Building up evidence for anisotropy of turbulent fluctua-
tions has progressed from merely detecting their elongation
along the magnetic field (Belcher & Davis 1971)—to fitting
data to an ad hoc model mixing a 2D perpendicular and a
1D parallel (“slab”) turbulent components in some propor-
tion37 (Matthaeus et al. 1990; Bieber et al. 1996; Dasso et al.
2005; Hamilton et al. 2008)—to formal systematic unbiased
analyses showing the persistent presence of anisotropy at all
scales (Bigazzi et al. 2006; Sorriso-Valvo et al. 2006)— to di-
rect measurements of three-dimensional correlation functions
(Osman & Horbury 2007)—and finally to computing spectral
exponents at fixed angles between k and B0 (Horbury et al.
2008). The latter authors appear to have achieved the first
direct quantitative confirmation of the GS theory by demon-
strating that the magnetic-energy spectrum scales as k
wavenumbers perpendicular to the mean field and as k−2
wavenumbers parallel to it [consistent with the first scaling
relation in Eq. (4)]. This is the closest that observations have
got to confirming the GS relation k‖ ∼ k
⊥ [see Eq. (5)] in a
real astrophysical turbulent plasma.
8.1.4. Compressive Fluctuations
According to the theory developed in § 5, the density and
magnetic-field-strength fluctuations are passive, energetically
decoupled from and mixed by the Alfvénic cascade (§ 5.5;
these are slow and entropy modes in the collisional MHD
limit—see § 2.4 and § 6.1). These fluctuations are expected to
be pressure-balanced, as expressed by Eq. (22) or, more gen-
erally in gyrokinetics, by Eq. (67). There is, indeed, strong
37 These techniques originate from the view of MHD turbulence as a su-
perposition of a 2D turbulence and an admixture of Alfvén waves (Fyfe et al.
1977; Montgomery & Turner 1981). As we discussed in § 1.2, we consider
the Goldreich & Sridhar (1995, 1997) view of a critically balanced Alfvénic
cascade to be better physically justified.
42 SCHEKOCHIHIN ET AL.
evidence that magnetic and thermal pressures in the solar
wind are anticorrelated, although there are some indications
of the presence of compressive, fast-wave-like fluctuations as
well (Roberts 1990; Burlaga et al. 1990; Marsch & Tu 1993;
Bavassano et al. 2004).
Measurements of density and field-strength fluctua-
tions done by a variety of different methods both at
1 AU (Celnikier et al. 1983, 1987; Marsch & Tu 1990b;
Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Kellogg & Horbury 2005; Alexandrova et al. 2008a) and
near the Sun (Lovelace et al. 1970; Woo & Armstrong 1979;
Coles & Harmon 1989; Coles et al. 1991) show fluctuation
levels of order 10% and spectra that appear to have a k−5/3
scaling above scales of order 102 − 103 km, which approxi-
mately corresponds to the ion gyroscale. The Kolmogorov
value of the spectral exponent is, as in the case of Alfvénic
fluctuations, measured quite accurately in individual cases
(1.67 ± 0.03 in Celnikier et al. 1987). Interestingly, the
higher-order structure function exponents measured for the
magnetic-field strength show that it is a more intermittent
quantity than the velocity or the vector magnetic field (i.e.,
than the Alfvénic fluctuations) and that the scaling expo-
nents are quantitatively very close to the values found for
passive scalars in neutral fluids (Bershadskii & Sreenivasan
2004; Bruno et al. 2007). One might argue that this lends
some support to the theoretical expectation of passive
magnetic-field-strength fluctuations.
Considering that in the collisionless regime these fluctua-
tions are supposed to be subject to strong kinetic damping
(§ 6.2.2), the presence of well-developed Kolmogorov-like
and apparently undamped turbulent spectra is more surprising
than has perhaps been publicly acknowledged. An extended
discussion of this issue was given in § 6.3. Without the in-
clusion of the dissipation effects associated with the finite ion
gyroscale, the passive cascade of the density and field strength
is purely perpendicular to the (exact) local magnetic field and
does not lead to any scale refinement along the field. This im-
plies highly anisotropic field-aligned structures, whose length
is determined by the initial conditions (i.e., conditions in the
corona). The kinetic damping is inefficient for such fluctua-
tions. While this would seem to explain the presence of fully
fledged power-law spectra, it is not entirely obvious that the
parallel cascade is really absent once dissipation is taken into
account (Lithwick & Goldreich 2001), so the issue is not yet
settled. This said, we note that there is plenty of evidence of
a high degree of anisotropy and field alignment of the den-
sity microstructure in the inner solar wind and outer corona
(e.g., Armstrong et al. 1990; Grall et al. 1997; Woo & Habbal
1997). There is also evidence that the local structure of the
compressive fluctuations at 1 AU is correlated with the coro-
nal activity, implying some form of memory of initial condi-
tions (Kiyani et al. 2007; Hnat et al. 2007; Wicks et al. 2009).
We note, finally, that whether compressive fluctuations in
the inertial range can develop short parallel scales should also
tell us how much ion heating can result from their damping
(see § 6.2.4).
8.2. Dissipation-Range Turbulence in the Solar Wind and the
Magnetosheath
At scales approaching the ion gyroscale, k⊥ρi ∼ 1, effects
associated with the finite extent of ion gyroorbits start to
matter. Observationally, this transition manifests itself as a
clear break in the spectrum of magnetic fluctuations, with the
inertial-range k−5/3 scaling replaced by a steeper slope (see
Fig. 1). While the electrons at these scales can be treated as
an isothermal fluid (as long as we are considering fluctuations
above the electron gyroscale, k⊥ρe ≪ 1; see § 4), the fully
gyrokinetic description (§ 3) has to be adopted for the ions.
It is, indeed, to understand plasma dynamics at and around
k⊥ρi ∼ 1 that gyrokinetics was first designed in fusion plasma
theory (Frieman & Chen 1982; Brizard & Hahm 2007). In or-
der for gyrokinetics and further dissipation-range approxima-
tions that follow from it (§ 7) to be a credible approach in
the solar wind and other space plasmas, it has to be estab-
lished that fluctuations at and below the ion gyroscale are still
strongly anisotropic, k‖ ≪ k⊥. If that is the case, then their
frequencies (ω∼ k‖vAk⊥ρi, see § 7.3) will still be smaller than
the cyclotron frequency in at least a part of the “dissipation
range”38—the range of scales k⊥ρi & 1 (see § 7.13).
Note that additional information about the dissipation-
range turbulence can be extracted from the measurements in
the magnetosheath—while scales above the ion gyroscale are
probably non-universal there, the dissipation range appears to
display universal behavior, mostly similar to the solar wind
(see, e.g., Alexandrova 2008). This complements the obser-
vational picture emerging from the solar-wind data and al-
lows us to learn more as fluctuation amplitudes in the mag-
netosheath are larger and much smaller scales can be probed
than in the solar wind (Mangeney et al. 2006; Lacombe et al.
2006; Alexandrova et al. 2008b).
8.2.1. Anisotropy
We know with a fair degree of certainty that the fluctu-
ations that cascade down to the ion gyroscale from the in-
ertial range are strongly anisotropic (§ 8.1.3). While it ap-
pears likely that the anisotropy persists at k⊥ρi ∼ 1, it is ex-
tremely important to have a clear verdict on this assumption
from solar wind measurements. While Leamon et al. (1998)
and, more recently, Hamilton et al. (2008) did present some
evidence that magnetic fluctuations in the solar wind have a
degree of anisotropy below the ion gyroscale, no definitive
study similar to Horbury et al. (2008) or Bigazzi et al. (2006);
Sorriso-Valvo et al. (2006) exists as yet. In the magne-
tosheath, where the dissipation-range scales are easier to mea-
sure than in the solar wind, recent analysis by Sahraoui et al.
(2006); Alexandrova et al. (2008b) does show evidence of
strong anisotropy.
Besides confirming the presence of the anisotropy, it would
be interesting to study its scaling characteristics: e.g., check
the scaling prediction k‖ ∼ k
⊥ [Eq. (241); see also § 7.9.4
and § 7.10.3] in a similar fashion as the GS relation k‖ ∼ k
[Eq. (5)] was corroborated by Horbury et al. (2008).
In this paper, we have proceeded on the assumption that
the anisotropy, and, therefore, low frequencies (ω ≪ Ωi) do
characterize fluctuations in the dissipation range—or, at least,
that the low-frequency anisotropic fluctuations are a signifi-
cant energy cascade channel and can be considered decoupled
from any possible high-frequency dynamics.
8.2.2. Transition at the Ion Gyroscale: Collisionless Damping and
Heating
38 This term, customary in the space-physics literature, is somewhat of
a misnomer because, as we have seen in § 7, rich dissipationless turbulent
dynamics are present in this range alongside what is normally thought of as
dissipation.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 43
If the fluctuations at the ion gyroscale have k‖ ≪ k⊥ and
ω ≪ Ωi (§ 8.2.1), they are not subject to the cyclotron res-
onance (ω − k‖v‖ = ±Ωi), but are subject to the Landau one
(ω = k‖v‖). Alfvénic fluctuations at the ion gyroscale are
no longer decoupled from the compressive fluctuations and
can be Landau-damped (§ 7.1). It seems plausible that it
is the inflow of energy from the Alfvénic cascade that ac-
counts for a pronounced local flattening of the spectrum of
density fluctuations in the solar wind observed just above
the ion gyroscale (Woo & Armstrong 1979; Celnikier et al.
1983, 1987; Coles & Harmon 1989; Marsch & Tu 1990b;
Coles et al. 1991; Kellogg & Horbury 2005).39
In energetic terms, Landau damping amounts to a redis-
tribution of generalized energy from electromagnetic fluctu-
ations to entropy fluctuations (§ 3.4, § 7.8). This gives rise
to the entropy cascade, ultimately transferring the Landau-
damped energy into ion heat (§ 3.5, § 7.9 and § 7.10). How-
ever, only part of the inertial-range cascade is so damped be-
cause an alternative, electron, cascade channel exists: the ki-
netic Alfvén waves (§§ 7.2-7.8). The energy transferred into
the KAW-like fluctuations can cascade to the electron gy-
roscale, where it is Landau damped on electrons, converting
first into the electron entropy cascade and then electron heat
(§ 7.12).
Thus, the transition at the ion gyroscale ultimately de-
cides in what proportion the turbulent energy arriving from
the inertial range is distributed between the ion and electron
heat. How the fraction of power going into either depends on
parameters—βi, Ti/Te, amplitudes, . . . —is a key unanswered
question both in space and astrophysical (see, e.g., § 8.5) plas-
mas. Gyrokinetics appears to be an ideal tool for addressing
this question both analytically and numerically (Howes et al.
2008b). Within the framework outlined in this paper, the min-
imal model appropriate for studying the transition at the ion
gyroscale is the system of equations for isothermal electrons
and gyrokinetic ions derived in § 4 (it is summarized in § 4.9).
8.2.3. Ion Gyroscale vs. Ion Inertial Scale
It is often assumed in the space physics literature that it is
at the ion inertial scale, di = ρi/
βi, rather than at the ion gy-
roscale ρi that the spectral break between the inertial and dis-
sipation range occurs. The distinction between di and ρi be-
comes noticeable when βi is significantly different from unity,
a relatively rare occurrence in the solar wind. While some at-
tempts to determine at which of these two scales a spectral
break between the inertial and dissipation ranges occurs have
produced claims that di is a more likely candidate (Smith et al.
2001), more comprehensive studies of the available data sets
conclude basically that it is hard to tell (Leamon et al. 2000;
Markovskii et al. 2008).
In the gyrokinetic approach advocated in this paper, the ion
inertial scale does not play a special role (see § 7.1). The only
parameter regime in which di does appear as a special scale
is Ti ≪ Te (“cold ions”), when the Hall MHD approximation
can be derived in a systematic way (see Appendix E). This,
however, is not the right limit for the solar wind or most other
astrophysical plasmas of interest because ions are rarely cold.
Hall MHD is discussed further in § 8.2.6 and Appendix E.
8.2.4. KAW Turbulence
39 Celnikier et al. (1987) proposed that the flattening might be a k−1 spec-
trum analogous to Batchelor’s spectrum of passive scalar variance in the
viscous-convective range. We think this analogy cannot apply because den-
sity is not passive at or below the ion gyroscale.
If gyrokinetics is valid at scales k⊥ρi & 1 (i.e., if k‖ ≪ k⊥,
ω ≪ Ωi and it is acceptable to at least model the equilibrium
distribution as a Maxwellian; see § 8.3), the electromagnetic
fluctuations below the ion gyroscale will be described by the
fluid approximation that we derived in § 7.2 and referred to
ERMHD. The wave solutions of this system of equations are
the kinetic Alfvén waves (§§ 7.3-7.4) and it is possible to ar-
gue for a GS-style critically balanced cascade of KAW-like
electromagnetic fluctuations (§ 7.5) between the ion and elec-
tron gyroscales (Landau damped on electrons at k⊥ρe ∼ 1; the
expression for the KAW damping rate in the gyrokinetic limit
is given in Howes et al. 2006; see also Fig. 8).
Individual KAW have, indeed, been detected in space plas-
mas (e.g., Grison et al. 2005). What about KAW turbulence?
How does one tell whether any particular spectral slope one is
measuring corresponds to the KAW cascade or fits some alter-
native scheme for the dissipation-range turbulence (§ 8.2.6)?
It appears to be a sensible program to look for specific rela-
tionships between different fields predicted by theory (§ 7.2)
and for the corresponding spectral slopes and scaling relations
for the anisotropy (§ 7.5). This means that simultaneous mea-
surements of magnetic, electric, density and magnetic-field-
strength fluctuations are needed.
For the solar wind, the spectra of electric and magnetic
fluctuations below the ion gyroscale reported by Bale et al.
(2005) are consistent with the k−1/3 and k−7/3 scalings pre-
dicted for an anisotropic critically balanced KAW cascade
(§ 7.5; see Fig. 1 for theoretical scaling fits superimposed
on a plot taken from Bale et al. 2005; note, however, that
Bale et al. 2005 themselves interpreted their data in a some-
what different way and that their resolution was in any case
not sufficient to be sure of the scalings). They were also able
to check that their fluctuations satisfied the KAW dispersion
relation—for critically balanced fluctuations, this is, indeed,
plausible. Magnetic-fluctuation spectra recently reported by
Alexandrova et al. (2008a) are only slightly steeper than the
theoretical k−7/3 KAW spectrum. These authors also find a
significant amount of magnetic-field-strength fluctuations in
the dissipation range, with a spectrum that follows the same
scaling—this is again consistent with the theoretical picture
of KAW turbulence [see Eq. (223)]. Measurements reported
by Czaykowska et al. (2001); Alexandrova et al. (2008b) for
the magnetosheath appear to present a similar picture.
The density spectra measured by Celnikier et al. (1983,
1987) steepen below the ion gyroscale following the flattened
segment around k⊥ρi ∼ 1 (discussed in § 8.2.2). For a KAW
cascade, the density spectrum should be k−7/3 (§ 7.5); with-
out KAW, k−10/3 (§ 7.10.2). The slope observed in the papers
cited above appears to be somewhat shallower even than k−2
(cf. a similar result by Spangler & Gwinn 1990 for the ISM;
see § 8.4.1), but, given imperfect resolution, neither seriously
in contradiction with the prediction based on the KAW cas-
cade, nor sufficient to corroborate it. Unfortunately, we have
not found published simultaneous measurements of density-
and magnetic- or electric-fluctuation spectra.
8.2.5. Variability of the Spectral Slope
While many measurements consistent with the KAW pic-
ture can be found, there are also many in which the spectra
are much steeper (Denskat et al. 1983; Leamon et al. 1998).
Analysis of a large set of measurements of the magnetic-
fluctuation spectra in the dissipation range of the solar wind
reveals a wide spread in the spectral indices: roughly between
44 SCHEKOCHIHIN ET AL.
−1 and −4 (Smith et al. 2006). There is evidence of a weak
positive correlation between steeper dissipation-range spectra
and higher ion temperatures (Leamon et al. 1998) or higher
cascade rates calculated from the inertial range (Smith et al.
2006). This suggests that a larger amount of ion heating may
correspond to a fully or partially suppressed KAW cascade,
which is in line with our view of the ion heating and the KAW
cascade as the two competing channels of the overall kinetic
cascade (§ 7.8). With a weakened KAW cascade, all or part of
the dissipation range would be dominated by the ion entropy
cascade—a purely kinetic phenomenon manifested by pre-
dominantly electrostatic fluctuations and very steep magnetic-
energy spectra (§ 7.10). This might account both for the steep-
ness of the observed spectra and for the spread in their indices
(§ 7.11), although many other theories exist (see § 8.2.6).
While we may thus have a plausible argument, this is not
yet a satisfactory quantitative theory that would allow us to
predict when the KAW cascade is present and when it is not or
what dissipation-range spectrum should be expected for given
values of the solar-wind parameters (βi, Ti/Te, etc.). Resolu-
tion of this issue again appears to hinge on the question of how
much turbulent power is diverted into the ion entropy cascade
(equivalently, into ion heat) at the ion gyroscale (see § 8.2.2).
8.2.6. Alternative Theories of the Dissipation Range
A number of alternative theories and models have been put
forward to explain the observed spectral slopes (and their vari-
ability) in the dissipation range. It is not our aim to review or
critique them all in detail, but perhaps it is useful to provide a
few brief comments about some of them in light of the theo-
retical framework constructed in this paper.
This entire theoretical framework hinges on adopting gy-
rokinetics as a valid description or, at least, a sensible model
that does not miss any significant channels of energy cascade
and dissipation. While we obviously believe this to be the
right approach, it is worth spelling out what effects are left
out “by construction.”
Parallel Alfvén-wave cascade and ion cyclotron damping. — The
use of gyrokinetics assumes that fluctuations stay anisotropic
at all scales, k‖ ≪ k⊥, and, therefore, ω ≪ Ωi, so the
cyclotron resonances are ordered out. However, if one
insists on routing the Alfvén-wave energy into a paral-
lel cascade, e.g., by forcibly setting k⊥ = 0, it is pos-
sible to construct a weak turbulence theory in which it
is dissipated by the ion cyclotron damping (Yoon & Fang
2008). Numerical simulations of 3D MHD turbulence do
not support the possibility of a parallel Alfvén-wave cascade
(Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac
2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al.
2003). Solar-wind evidence that the perpendicular cascade
dominates is quite strong for the inertial range (§ 8.1.3) and
less so for the dissipation range (§ 8.2.1). While, as stated
in § 8.2.1, one cannot yet definitely claim that observations
tell us that ω ≪ Ωi at k⊥ρi ∼ 1, it has been argued that
observations do not appear to be consistent with cyclotron
damping being the main mechanism for the dissipation of
the inertial-range Alfvénic turbulence at the ion gyroscale
(Leamon et al. 1998, 2000; Smith et al. 2001). Ion-cyclotron
resonance could conceivably be reached somewhere in the
dissipation range (see § 7.13). At this point gyrokinetics will
formally break down, although, as argued by Howes et al.
(2008a, see their § 3.6), this does not necessarily mean that
ion cyclotron damping will become the dominant dissipation
channel for the turbulence.
Parallel whistler cascade. — A parallel magnetosonic/whistler
cascade eventually damped by the electron cyclotron
resonance (Stawicki et al. 2001) is also excluded in the
construction of gyrokinetics. The whistler cascade has
been given some consideration in the Hall MHD approxi-
mation (further discussed at the end of this section). Both
weak-turbulence theory (Galtier 2006) and 3D numerical
simulations (Cho & Lazarian 2004) concluded that, like
in MHD, the turbulent cascade is highly anisotropic, with
perpendicular energy transfer dominating over the parallel
one.40 The same conclusion appears to have been reached
in recent 2D kinetic PIC simulations by Gary et al. (2008);
Saito et al. (2008). Thus, the turbulence again seems to be
driven into the gyrokinetically accessible regime.
While theory and numerical simulations appear to make
arguing in favor of a parallel cascade and cyclotron heat-
ing difficult, there exists some observational evidence in sup-
port of them, especially for the near-Sun solar wind (e.g.,
Harmon & Coles 2005). Thus, the presence or relative im-
portance of the cyclotron heating in the solar wind and, more
generally, the mechanism(s) responsible for the observed per-
pendicular ion heating (Marsch et al. 1983) remain a largely
open problem. Besides the theories mentioned above, many
other ideas have been proposed, some of which attempted
to reconcile the dominance of the low-frequency perpendic-
ular cascade with the possibility of cyclotron heating (e.g.,
Chandran 2005b; Markovskii et al. 2006; see Hollweg 2008
for a concise recent review of the problem).
Mirror cascade. — Sahraoui et al. (2006) analyzed a set of
Cluster multi-spacecraft measurements in the magnetosheath
and reported a broad power-law (∼ k−8/3) spectrum of mirror
structures at and below the ion gyroscale. They claim that
these are not KAW-like fluctuations because their frequency
is zero in the plasma frame. Although these structures are
highly anisotropic with k‖ ≪ k⊥, they cannot be described by
the gyrokinetic theory in its present form because δB‖/B0 is
very large (∼ 40%, occasionally reaching unity) and because
the particle trapping by fluctuations, which is likely to be
important in the nonlinear physics of the mirror instabil-
ity (Kivelson & Southwood 1996; Pokhotelov et al. 2008;
Rincon et al. 2009), is ordered out in gyrokinetics. Thus, if a
“mirror cascade” exists, it is not captured in our description.
More generally, the effect of the pressure-anisotropy-driven
instabilities on the turbulence in the dissipation range is a
wide open area, requiring further analytical effort (see § 8.3).
If k‖ ≪ k⊥, ω ≪ Ωi, and δB/B0 ≪ 1 are accepted for the
dissipation range and plasma instabilities at the ion gyroscale
(§ 8.3) are ignored, the formal gyrokinetic theory and its
asymptotic consequences derived above should hold. There
are two essential features of the linear physics at and below
the ion gyroscale that must play some role: the collisionless
(Landau) damping and the dispersive nature of the wave so-
lutions (see Fig. 8 and § 7.3; cf., e.g., Leamon et al. 1999;
Stawicki et al. 2001). Both of these features have been em-
ployed to explain the spectral break at the ion gyroscale and
the spectral slopes below it.
40 It is possible to produce a parallel cascade artificially by running 1D
simulations (Matthaeus et al. 2008b).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 45
Landau damping and instrumental effects. — In most of our dis-
cussion, (§ 7, §§ 8.2.4-8.2.5), we effectively assumed that the
Landau damping is only important at k⊥ρi ∼ 1 and k⊥ρe ∼ 1,
but not in between, so we could talk about asymptotic scal-
ings and dissipationless cascades. However, as was noted
in § 7.6, a properly asymptotic scaling behavior in the dis-
sipation range is probably impossible in nature because the
scale separation between the ion and electron gyroscales is
only about (mi/me)
1/2 ≃ 43. In particular, there is not always
a wide scale interval where the kinetic damping is negligi-
bly small (especially at low βi; see Fig. 8; cf. Leamon et al.
1999). Howes et al. (2008a) proposed a model of how the
presence of damping combined with instrumental effects (a
resolution floor) could lead to measured spectra that look like
power laws steeper than k−7/3, with the effective spectral ex-
ponent depending on plasma parameters (we refer the reader
to that paper for a discussion of how this compares with pre-
vious models of a similar kind, e.g., Li et al. 2001). A key
physical assumption of theirs and similar models is that the
amount of power drained from the Alfvén-wave and KAW
cascades into the ion heat is set by the strength of the linear
damping. Whether this is justified is not yet clear.
Hall and Electron MHD. — If Landau damping is deemed
unimportant in some part of the dissipation range (which
can be true in some regimes; see Fig. 8 and Howes et al.
2006, 2008a,b) and the wave dispersion is considered to
be the salient feature, it might appear that a fluid, rather
than kinetic, description should be sufficient. Hall MHD
(Mahajan & Yoshida 1998) or its kdi ≫ 1 limit the Electron
MHD (Kingsep et al. 1990) have been embraced by many au-
thors as such a description, suitable both for analytical argu-
ments (Goldreich & Reisenegger 1992; Krishan & Mahajan
2004; Gogoberidze 2005; Galtier & Bhattacharjee 2003;
Galtier 2006; Alexandrova et al. 2008a) and numerical sim-
ulations (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005;
Galtier & Buchlin 2007; Matthaeus et al. 2008b).
To what extent does this constitute an approach alterna-
tive to (and better than?) gyrokinetics (as suggested, e.g., by
Matthaeus et al. 2008b)? For fluctuations with k‖ ≪ k⊥, Hall
MHD is merely a particular limit of gyrokinetics: βi ≪ 1 and
Ti/Te ≪ 1 (cold-ion limit; see Appendix E). If k‖ is not small
compared to k⊥, then the gyrokinetics is not valid, while Hall
MHD continues to describe the cold-ion limit correctly (e.g.,
Ito et al. 2004; Hirose et al. 2004), capturing in particular the
whistler branch of the dispersion relation. However, as we
have already mentioned above, the dominance of the perpen-
dicular energy transfer (k‖ ≪ k⊥) is supported both by weak-
turbulence theory for Hall MHD (Galtier 2006) and by 3D
numerical simulations of the Electron MHD (Cho & Lazarian
2004).
Thus, the gyrokinetic theory and its rigorous limits, such
as ERMHD (§ 7.2), supersede Hall MHD for anisotropic tur-
bulence. Since ions are generally not cold in the solar wind
(or any other plasma discussed here), Hall MHD is not for-
mally a relevant approximation. It also entirely misses the
kinetic damping and the associated entropy cascade channel
leading to particle heating (§ 7.1, § 7.9 and § 7.10). However,
Hall MHD does capture the Alfvén waves becoming disper-
sive and numerical simulations of it do show a spectral break,
although, technically speaking, at the wrong scale (di instead
of ρi; see § 7.1). Although Hall MHD cannot be rigorously
used as quantitative theory of the spectral break and the asso-
ciated change in the nature of the turbulent cascade, the Hall
MHD equations in the limit kdi ≫ 1 are mathematically sim-
ilar to our ERMHD equations (see § 7.2 and Appendix E) to
within constant coefficients probably not essential for quali-
tative models of turbulence. Therefore, results of numerical
simulations of Hall and Electron MHD cited above are di-
rectly useful for understanding the KAW cascade—and, in-
deed, in the limit kdi ≫ 1, kde ≪ 1, they are mostly consistent
with the scaling arguments of § 7.5.
Alfvén vortices. — Finally we mention an argument pertaining
to the dissipation-range spectra that is not based on energy
cascades at all. Based on the evidence of Alfvén vortices in
the magnetosheath, Alexandrova (2008) speculated that steep
power-law spectra observed in the dissipation range at least
in some cases could reflect the geometry of the ion-gyroscale
structures rather than a local energy cascade. If Alfvén vor-
tices are a common feature, this possibility cannot be ex-
cluded. However, the resulting geometrical spectra are quite
steep (k−4 and steeper), so they can become important only
if the KAW cascade is weak or suppressed—somewhat simi-
larly to the steep spectra associated with the entropy cascade
(§ 7.11).
8.3. Is Equilibrium Distribution Isotropic and Maxwellian?
In rigorous theoretical terms, the weakest point of this pa-
per is the use of a Maxwellian equilibrium. Formally, this is
only justified when the collisions are weak but not too weak:
we ordered the collision frequency as similar to the fluctu-
ation frequency [Eq. (49)]. This degree of collisionality is
sufficient to prove that a Maxwellian equilibrium distribution
F0s(v) does indeed emerge in the lowest order of the gyroki-
netic expansion (Howes et al. 2006). This argument works
well for plasmas such as the ISM (§ 8.4), where collisions are
weak (λmfpi ≫ ρi) but non-negligible (λmfpi ≪ L). In space
plasmas, the mean free path is of the order of 1 AU—the dis-
tance between the Sun and the Earth (see Table 1). Strictly
speaking, in so highly collisionless a plasma, the equilib-
rium distribution does not have to be either Maxwellian or
isotropic.
The conservation of the first adiabatic invariant, µ = v2⊥/2B,
suggests that temperature anisotropy with respect to the
magnetic-field direction (T0⊥ 6= T0‖) may exist. When the
relative anisotropy is larger than (roughly) 1/βi, it triggers
several very fast growing plasma instabilities: most promi-
nently the firehose (T0⊥ < T0‖) and mirror (T0⊥ > T0‖) modes
(e.g., Gary et al. 1976). Their growth rates peak around the
ion gyroscale, thus giving rise to additional energy injection
at k⊥ρi ∼ 1.
No definitive analytical theory of how these fluctuations sat-
urate, cascade and affect the equilibrium distribution has been
proposed. It appears to be a reasonable expectation that the
fluctuations resulting from temperature anisotropy will satu-
rate by limiting this anisotropy. This idea has some support
in solar-wind observations: while the degree of anisotropy
of the core particle distribution functions varies consider-
ably between data sets, the observed anisotropies do seem
to populate the part of the parameter plane (T0⊥/T0‖,βi) cir-
cumscribed in a rather precise way by the marginal stabil-
ity boundaries for the mirror and firehose (Gary et al. 2001;
Kasper et al. 2002; Marsch et al. 2004; Hellinger et al. 2006;
Matteini et al. 2007).41
41 Note that Kellogg et al. (2006) measure the electric-field fluctuations
46 SCHEKOCHIHIN ET AL.
If we want to study turbulence in data sets that do not lie
too close to these stability boundaries, assuming an isotropic
Maxwellian equilibrium distribution [Eq. (54)] is probably
an acceptable simplification, although not an entirely rigor-
ous one. Further theoretical work is clearly possible on this
subject: thus, it is not a problem to formulate gyrokinetics
with an arbitrary equilibrium distribution (Frieman & Chen
1982) and starting from that, once can generalize the results
of this paper (for the KRMHD system, § 5, this has been done
by Chen et al. 2009). Treating the instabilities themselves
might prove more difficult, requiring the gyrokinetic order-
ing to be modified and the expansion carried to higher orders
to incorporate features that are not captured by gyrokinetics,
e.g., short parallel scales (Rosin et al. 2009), particle trap-
ping (Pokhotelov et al. 2008; Rincon et al. 2009), or nonlin-
ear finite-gyroradius effects (Califano et al. 2008). Note that
the theory of the dissipation-range turbulence will probably
need to be modified to account for the additional energy in-
jection from the instabilities and for the (yet unclear) way in
which this energy makes its way to dissipation and into heat.
Besides the anisotropies, the particle distribution functions
in the solar wind (especially the electron one) exhibit non-
Maxwellian suprathermal tails (see Maksimovic et al. 2005;
Marsch 2006, and references therein). These contain small
(∼ 5% of the total density) populations of energetic particles.
Both the origin of these particles and their effect on turbulence
have to be modeled kinetically. Again, it is possible to formu-
late gyrokinetics for general equilibrium distributions of this
kind and examine the interaction between them and the turbu-
lent fluctuations, but we leave such a theory outside the scope
of this paper.
Thus, much remains to be done to incorporate realistic equi-
librium distribution functions into the gyrokinetic description
of the solar wind plasma. In the meanwhile, we believe that
the gyrokinetic theory based on a Maxwellian equilibrium dis-
tribution as presented in this paper, while idealized and imper-
fect, is nevertheless a step forward in the analytical treatment
of the space-plasma turbulence compared to the fluid descrip-
tions that have prevailed thus far.
8.4. Interstellar Medium
While the solar wind is unmatched by other astrophysical
plasmas in the level of detail with which turbulence in it can
be measured, the interstellar medium (ISM) also offers an ob-
server a number of ways of diagnosing plasma turbulence,
which, in the case of the ISM, is thought to be primarily ex-
cited by supernova explosions (Norman & Ferrara 1996). The
accuracy and resolution of this analysis are due to improve
rapidly thanks to many new observatories, e.g., LOFAR,42
Planck (Enßlin et al. 2006), and, in more distant future, the
SKA (Lazio et al. 2004).
The ISM is a spatially inhomogeneous environment consist-
ing of several phases that have different temperatures, densi-
ties and degrees of ionization (Ferrière 2001).43 We will use
the Warm ISM phase (see Table 1) as our fiducial interstel-
lar plasma and discuss briefly what is known about the two
main observationally accessible quantities—the electron den-
sity and magnetic fields—and how this information fits into
in the ion-cyclotron frequency range, estimate the resulting velocity-space
diffusion and argue that it is sufficient to isotropize the ion distribution
42 http://www.lofar.org
43 And, therefore, different degrees of importance of the neutral particles
and the associated ambipolar damping effects—these will not be discussed
here; see Lithwick & Goldreich 2001.
the theoretical framework proposed here.
8.4.1. Electron Density Fluctuations
The electron-density fluctuations inferred from the inter-
stellar scintillation measurements appear to have a spectrum
with an exponent ≃ −1.7, consistent with the Kolmogorov
scaling (Armstrong et al. 1981, 1995; Lazio et al. 2004; see,
however, dissenting evidence by Smirnova et al. 2006, who
claim a spectral exponent closer to −1.5). This holds over
about 5 decades of scales: λ ∈ (105,1010) km. Other observa-
tional evidence at larger and smaller scales supports the case
for this presumed inertial range to be extended over as many
as 12 decades: λ ∈ (102,1015) km, a fine example of scale
separation that prompted an impressed astrophysicist to dub
the density scaling “The Great Power Law in the Sky.” The
upper cutoff here is consistent with the estimates of the su-
pernova scale of order 100 pc—presumably the outer scale of
the turbulence (Norman & Ferrara 1996) and also roughly the
scale height of the galactic disk (obviously the upper bound
on the validity of any homogeneous model of the ISM tur-
bulence). The lower cutoff is an estimate for the inner scale
below which the logarithmic slope of the density spectrum
steepens to about −2 (Spangler & Gwinn 1990).
Higdon (1984) was the first to realize that the electron-
density fluctuations in the ISM could be attributed to a cas-
cade of a passive tracer mixed by the ambient turbulence (the
MHD entropy mode; see § 2.6). This idea was brought to ma-
turity by Lithwick & Goldreich (2001), who studied the pas-
sive cascades of the slow and entropy modes in the frame-
work of the GS theory (see also Maron & Goldreich 2001).
If the turbulence is assumed anisotropic, as in the GS theory,
the passive nature of the density fluctuations with respect to
the decoupled Alfvén-wave cascade becomes a rigorous re-
sult both in MHD (§ 2.4) and, as we showed above, in the
more general gyrokinetic description appropriate for weakly
collisional plasmas (§ 5.5). Anisotropy of the electron-density
fluctuations in the ISM is, indeed, observationally supported
(Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al. 2002;
Dennett-Thorpe & de Bruyn 2003; Heyer et al. 2008, see also
Lazio et al. 2004 for a concise discussion), although detailed
scale-by-scale measurements are not currently possible.
If the underlying Alfvén-wave turbulence in the ISM has
⊥ spectrum, as predicted by GS, so should the elec-
tron density (see § 2.6). As we discussed in § 6.3, the phys-
ical nature of the inner scale for the density fluctuations de-
pends on whether they have a cascade in k‖ and are effi-
ciently damped when k‖λmfpi ∼ 1 or fail to develop small
parallel scales and can, therefore, reach k⊥ρi ∼ 1. The ob-
servationally estimated inner scale is consistent with the ion
gyroscale, ρi ∼ 103 km (see Table 1; note that the ion iner-
tial scale di = ρi/
βi is similar to ρi at the moderate values
of βi characteristic of the ISM—see further discussion of the
(ir)relevance of di in § 7.1, § 8.2.3 and Appendix E). How-
ever, since the mean free path in the ISM is not huge (Ta-
ble 1), it is not possible to distinguish this from the perpen-
dicular cutoff k−1⊥ ∼ λ
mfpiL
−1/2 ∼ 500 km implied by the par-
allel cutoff at k‖λmfpi ∼ 1 [see Eq. (220)], as advocated by
Lithwick & Goldreich (2001). Note that the relatively short
mean free path means that much of the scale range spanned by
the Great Power Law in the Sky is, in fact, well described by
the MHD approximation either with adiabatic (§ 2) or isother-
mal (§ 6.1 and Appendix D) electrons.
Below the ion gyroscale, the −2 spectral exponent reported
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 47
by Spangler & Gwinn (1990) is measured sufficiently impre-
cisely to be consistent with the −7/3 expected for the density
fluctuations in the KAW cascade (§ 7.5). However, given the
high degree of uncertainty about what happens in this “dis-
sipation range” even in the much better resolved case of the
solar wind (§ 8.2), it would probably be wise to reserve judg-
ment until better data are available.
8.4.2. Magnetic Fluctuations
The second main observable type of turbulent fluctuations
in the ISM are the magnetic fluctuations, accessible indirectly
via the measurements of the Faraday rotation of the polar-
ization angle of the pulsar light travelling through the ISM.
The structure function of the rotation measure (RM) should
have the Kolmogorov slope of 2/3 if the magnetic fluctua-
tions are due to Alfvénic turbulence described by the GS the-
ory. There is a considerable uncertainty in interpreting the
available data, primarily due to insufficient spatial resolution
(rarely better than a few parsec). Structure function slopes
consistent with 2/3 have been reported (Minter & Spangler
1996), but, depending on where one looks, shallower struc-
ture functions that seem to steepen at scales of a few parsec
are also observed (Haverkorn et al. 2004).
A recent study by Haverkorn et al. (2005) detected an in-
teresting trend: the RM structure functions computed for re-
gions that lie in the galactic spiral arms are nearly perfectly
flat down to the resolution limit, while in the interarm regions,
they have detectable slopes (although these are mostly shal-
lower that 2/3). Observations of magnetic fields in external
galaxies also reveal a marked difference in the magnetic-field
structure between arms and interarms: the spatially regular
(mean) fields are stronger in the interarms, while in the arms,
the stochastic fields dominate (Beck 2007). This qualitative
difference between the magnetic-field structure in the arms
and interarms has been attributed to smaller effective outer
scale in the arms (∼ 1 pc, compared to ∼ 102 pc in the in-
terarms; see Haverkorn et al. 2008) or to the turbulence in the
arms and interarms belonging to the two distinct asymptotic
regimes described in § 1.3: closer to the anisotropic Alfvénic
turbulence with a strong mean field in the interarms and to the
isotropic saturated state of small-scale dynamo in the arms
(Schekochihin et al. 2007).
8.5. Accretion Disks
Accretion of plasma onto a central black hole or neutron
star is responsible for many of the most energetic phenomena
observed in astrophysics (see, e.g., Narayan & Quataert 2005
for a review). It is now believed that a linear instability of dif-
ferentially rotating plasmas—the magnetorotational instabil-
ity (MRI)—amplifies magnetic fields and gives rise to MHD
turbulence in astrophysical disks (Balbus & Hawley 1998).
Magnetic stresses due to this turbulence transport angular mo-
mentum, allowing plasma to accrete. The MRI converts the
gravitational potential energy of the inflowing plasma into
turbulence at the outer scale that is comparable to the scale
height of the disk. This energy is then cascaded to small
scales and dissipated into heat—powering the radiation that
we see from accretion flows. Fluid MHD simulations show
that the MRI-generated turbulence in disks is subsonic and
has β ∼ 10 − 100. Thus, on scales much smaller than the scale
height of the disk, homogeneous turbulence in the parameter
regimes considered in this paper is a valid idealization and
the kinetic models developed above should represent a step
forward compared to the purely fluid approach.
Turbulence is not yet directly observable in disks, so mod-
els of turbulence are mostly used to produce testable predic-
tions of observable properties of disks such as their X-ray and
radio emission. One of the best observed cases is the (pre-
sumed) accretion flow onto the black hole coincident with the
radio source Sgr A∗ in the center of our Galaxy (see review
by Quataert 2003).
Depending on the rate of heating and cooling in the inflow-
ing plasma (which in turn depend on accretion rate and other
properties of the system under consideration), there are differ-
ent models that describe the physical properties of accretion
flows onto a central object. In one class of models, a geometri-
cally thin optically thick accretion disk (Shakura & Sunyaev
1973), the inflowing plasma is cold and dense and well de-
scribed as an MHD fluid. When applied to Sgr A∗, these
models produce a prediction for its total luminosity that is
several orders of magnitude larger than observed. Another
class of models, which appears to be more consistent with the
observed properties of Sgr A∗, is called radiatively inefficient
accretion flows (RIAFs; see Rees et al. 1982; Narayan & Yi
1995 and review by Quataert 2003 of the applications and ob-
servational constraints in Sgr A∗). In these models, the in-
flowing plasma near the black hole is believed to adopt a two-
temperature configuration, with the ions (Ti ∼ 1011 − 1012 K)
hotter than the electrons (Te ∼ 109 − 1011 K).44 The electron
and ion thermodynamics decouple because the densities are
so low that the temperature equalization time ∼ ν−1ie is longer
than the time for the plasma to flow into the black hole. Thus,
like the solar wind, RIAFs are macroscopically collisionless
plasmas (see Table 1 for plasma parameters in the Galactic
center; note that these parameters are so extreme that the gy-
rokinetic description, while probably better than the fluid one,
cannot be expected to be rigorously valid; at the very least, it
needs to be reformulated in a relativistic form). At the high
temperatures appropriate to RIAFs, electrons radiate energy
much more efficiently than the ions (by virtue of their much
smaller mass) and are, therefore, expected to contribute dom-
inantly to the observed emission, while the thermal energy of
the ions is swallowed by the black hole. Since the plasma
is collisionless, the electron heating by turbulence largely de-
termines the thermodynamics of the electrons and thus the
observable properties of RIAFs. The question of which frac-
tion of the turbulent energy goes into ion and which into elec-
tron heating is, therefore, crucial for understanding accretion
flows—and the answer to this question depends on the de-
tailed properties of the small-scale kinetic turbulence (e.g.,
Quataert & Gruzinov 1999; Sharma et al. 2007), as well as on
the linear properties of the collisionless MRI (Quataert et al.
2002; Sharma et al. 2003).
Since all of the turbulent power coming down the cascade
must be dissipated into either ion or electron heat, it is re-
ally the amount of generalized energy diverted at the ion gy-
roscale into the ion entropy cascade (§§ 7.8-7.9) that decides
how much energy is left to heat the electrons via the KAW
cascade (§§ 7.2-7.5, § 7.12). Again, as in the case of the solar
wind (§ 8.2.2 and § 8.2.5), the transition around the ion gy-
roscale from the Alfvénic turbulence at k⊥ρi ≪ 1 to the KAW
turbulence at k⊥ρi ≫ 1 emerges as a key unsolved problem.
8.6. Galaxy Clusters
44 It is partly with this application in mind that we carried the general
temperature ratio in our calculations; see footnote 17.
48 SCHEKOCHIHIN ET AL.
Galaxy clusters are the largest plasma objects in the Uni-
verse. Like the other examples discussed above, the intraclus-
ter plasma is in the weakly collisional regime (see Table 1).
Fluctuations of electron density, temperature and of magnetic
fields are measured in clusters by X-ray and radio observa-
tories, but the resolution is only just enough to claim that a
fairly broad scale range of fluctuations exists (Schuecker et al.
2004; Vogt & Enßlin 2005). No power-law scalings have yet
been established beyond reasonable doubt.
What fundamentally hampers quantitative modeling of tur-
bulence and related effects in clusters is that we do not have
a definite theory of the basic properties of the intracluster
medium: its (effective) viscosity, magnetic diffusivity or ther-
mal conductivity. In a weakly collisional and strongly mag-
netized plasma, all of these depend on the structure of the
magnetic field (Braginskii 1965), which is shaped by the tur-
bulence. If (or at scales where) a reasonable a priori assump-
tion can be made about the field structure, further analytical
progress is possible: thus, the theoretical models presented in
this paper assume that the magnetic field is a sum of a slowly
varying in space “mean field” and small low-frequency per-
turbations (δB ≪ B0).
In fact, since clusters do not have mean fields of any mag-
nitude that could be considered dynamically significant, but
do have stochastic fields, the outer-scale MHD turbulence in
clusters falls into the weak-mean-field category (see § 1.3).
The magnetic field should be highly filamentary, organized
in long folded direction-reversing structures. It is not cur-
rently known what determines the reversal scale.45 Obser-
vations, while tentatively confirming the existence of very
long filaments (Clarke & Enßlin 2006), suggest that the re-
versal scale is much larger than the ion gyroscale: thus, the
magnetic-energy spectrum for the Hydra A cluster core re-
ported by Vogt & Enßlin (2005) peaks at around 1 kpc, com-
pared to ρi ∼ 105 km. Below this scale, an Alfvén-wave cas-
cade should exist (as is, indeed, suggested by Vogt & Enßlin’s
spectrum being roughly consistent with k−5/3 at scales below
the peak). As these scales are collisionless (λmfpi ∼ 100 pc in
the cores and ∼ 10 kpc in the bulk of the clusters), it is to this
turbulence that the theory developed in this paper should be
applicable.
Another complication exists, similar to that discussed in
§ 8.3: pressure anisotropies could give rise to fast plasma
instabilities whose growth rate peaks just above the ion gy-
roscale. As was pointed out by Schekochihin et al. (2005),
these are, in fact, an inevitable consequence of any large-scale
fluid motions that change the strength of the magnetic field.
Although a number of interesting and plausible arguments
can be made about the way the instabilities might determine
the magnetic-field structure (Schekochihin & Cowley 2006;
Schekochihin et al. 2008a; Rosin et al. 2009; Rincon et al.
2009), it is not currently understood how the small-scale
fluctuations resulting from these instabilities coexist with the
Alfvénic cascade.
The uncertainties that result from this imperfect under-
standing of the nature of the intracluster medium are exempli-
fied by the problem of its thermal conductivity. The magnetic-
field reversal scale in clusters is certainly not larger than the
electron diffusion scale, (mi/me)
1/2λmfpi, which varies from a
45 See Schekochihin & Cowley (2006) for a detailed presentation of our
views on the interplay between turbulence, magnetic field and plasma ef-
fects in cluster; for further discussions and disagreements, see Enßlin & Vogt
(2006); Subramanian et al. (2006); Brunetti & Lazarian (2007).
few kpc in the cores to a few hundred kpc in the bulk. There-
fore, one would expect that the approximation of isothermal
electron fluid (§ 4) should certainly apply at all scales below
the reversal scale, where δB ≪ B0 presumably holds. Even
this, however, is not absolutely clear. One could imagine
the electrons being effectively adiabatic if (or in the regions
where) the plasma instabilities give rise to large fluctuations
of the magnetic field (δB/B0 ∼ 1) at the ion gyroscale re-
ducing the mean free path to λmfpi ∼ ρi (Schekochihin et al.
2008a; Rosin et al. 2009; Rincon et al. 2009). Such fluctu-
ations cannot be described by the gyrokinetics in its cur-
rent form. The current state of the observational evidence
does not allow one to exclude either of these possibilities.
Both isothermal (Fabian et al. 2006; Sanders & Fabian 2006)
and non-isothermal (Markevitch & Vikhlinin 2007) coherent
structures that appear to be shocks are observed. Disordered
fluctuations of temperature can also be detected, which allows
one to infer an upper limit for the scale at which the isothermal
approximation can start being valid: thus, Markevitch et al.
(2003) find temperature variations at all scales down to ∼
100 kpc, which is the statistical limit that defines the spa-
tial resolution of their temperature map. In none of these or
similar measurements is the magnetic field data available that
would make possible a pointwise comparison of the magnetic
and thermal structure.
Because of this lack of information about the state of the
magnetized plasma in clusters, theories of the intracluster
medium are not sufficiently constrained by observations, so
no one theory is in a position to prevail. This uncertain state
of affairs might be improved by analyzing the observationally
much better resolved case of the solar wind, which should be
quite similar to the intracluster medium at very small scales
(except for somewhat lower values of βi in the solar wind).
9. CONCLUSION
In this paper, we have considered magnetized plasma tur-
bulence in the astrophysically prevalent regime of weak col-
lisionality. We have shown how the energy injected at the
outer scale cascades in phase space, eventually to increase the
entropy of the system and heat the particles. In the process,
we have explained how one combines plasma physics tools—
in particular, the gyrokinetic theory—with the ideas of a tur-
bulent cascade of energy to arrive at a hierarchy of tractable
models of turbulence in various physically distinct scale in-
tervals. These models represent the branching pathways of a
generalized energy cascade in phase space (the “kinetic cas-
cade”; see Fig. 5) and make explicit the “fluid” and “kinetic”
aspects of plasma turbulence.
A detailed outline of these developments was given in the
Introduction. Intermediate technical summaries were pro-
vided in § 4.9, § 5.7, and § 7.14. An astrophysical summary
and discussion of the observational evidence was given in § 8,
with a particular emphasis on space plasmas (§§ 8.1-8.3). Our
view of how the transformation of the large-scale turbulent
energy into heat occurs was encapsulated in the concept of
a kinetic cascade of generalized energy. It was previewed in
§ 1.4 and developed quantitatively in §§ 3.4-3.5, § 4.7, § 5.6,
§§ 6.2.3-6.2.5, §§ 7.8-7.12, Appendices D.2 and E.2.
Following a series of analytical contributions that set
up a theoretical framework for astrophysical gyrokinetics
(Howes et al. 2006, 2008a; Schekochihin et al. 2007, 2008b,
and this paper), an extensive program of fluid, hybrid fluid-
kinetic, and fully gyrokinetic46 numerical simulations of mag-
netized plasma turbulence is now underway (for the first re-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 49
sults of this program, see Howes et al. 2008b; Tatsuno et al.
2009a,b). Careful comparisons of the fully gyrokinetic
simulations with simulations based on the more readily
computable models derived in this paper (RMHD—§ 2,
isothermal electron fluid—§ 4, KRMHD—§ 5, ERMHD—
§ 7, HRMHD—Appendix E) as well as with the numerical
studies based on various Landau fluid (Snyder et al. 1997;
Goswami et al. 2005; Ramos 2005; Sharma et al. 2006, 2007;
Passot & Sulem 2007) and gyrofluid (Hammett et al. 1991;
Dorland & Hammett 1993; Snyder & Hammett 2001; Scott
2007) closures appear to be the way forward in developing a
comprehensive numerical model of the kinetic turbulent cas-
cade from the outer scale to the electron gyroscale. Of the
many astrophysical plasmas to which these results apply, the
solar wind and, perhaps, the magnetosheath, due to the high
quality of turbulence measurements possible in them, appear
to be the most suitable test beds for direct and detailed quan-
titative comparisons of the theory and simulation results with
observational evidence. The objective of all this work remains
a quantitative characterization of the scaling-range properties
(spectra, anisotropy, nature of fluctuations and their interac-
tions), the ion and electron heating, and the transport proper-
ties of the magnetized plasma turbulence.
We thank O. Alexandrova, S. Bale, J. Borovsky, T. Carter,
S. Chapman, C. Chen, E. Churazov, T. Enßlin, A. Fabian,
A. Finoguenov, A. Fletcher, M. Haverkorn, B. Hnat, T. Hor-
bury, K. Issautier, C. Lacombe, M. Markevitch, K. Osman,
T. Passot, F. Sahraoui, A. Shukurov, and A. Vikhlinin for
helpful discussions of experimental and observational data;
I. Abel, M. Barnes, D. Ernst, J. Hastie, P. Ricci, C. Roach,
and B. Rogers for discussions of collisions in gyrokinetics;
and G. Plunk for discussions of the theory of gyrokinetic tur-
bulence in two spatial dimensions. The authors’ travel was
supported by the US DOE Center for Multiscale Plasma Dy-
namics and by the Leverhulme Trust (UK) International Aca-
demic Network for Magnetized Plasma Turbulence. A.A.S.
was supported in part by a PPARC/STFC Advanced Fellow-
ship and by the STFC Grant ST/F002505/1. He also thanks
the UCLA Plasma Group for its hospitality on several occa-
sions. S.C.C. and W.D. thank the Kavli Institute for The-
oretical Physics and the Aspen Center for Physics for their
hospitality. G.W.H. was supported by the US DOE contract
DE-AC02-76CH03073. G.G.H. and T.T. were supported by
the US DOE Center for Multiscale Plasma Dynamics. E.Q.
and G.G.H. were supported in part by the David and Lucille
Packard Foundation.
46 Using the publicly available GS2 code (developed originally for fusion
applications; see http://gs2.sourceforge.net) and the purpose-built AstroGK
code (see http://www.physics.uiowa.edu/~ ghowes/astrogk/).
APPENDIX
A. BRAGINSKII’S TWO-FLUID EQUATIONS AND REDUCED MHD
Here we explain how the standard one-fluid MHD equations used in § 2 and the collisional limit of the KRMHD system
(§ 6.1, derived in Appendix D) both emerge as limiting cases of the two-fluid theory. For the case of anisotropic fluctuations,
k‖/k⊥ ≪ 1, all of this can, of course, be derived from gyrokinetics, but it is useful to provide a connection to the more well
known fluid description of collisional plasmas.
A.1. Two-Fluid Equations
The rigorous derivation of the fluid equations for a collisional plasma was done in the classic paper of Braginskii (1965). His
equations, valid for ω/νii ≪ 1, k‖λmfpi ≪ 1, k⊥ρi ≪ 1 (see Fig. 3), evolve the densities ns, mean velocities us and temperatures
Ts of each plasma species (s = i,e):
+ us ·∇
ns = −ns∇·us, (A1)
+ us ·∇
us = −∇ps −∇· Π̂s + qsns
us ×B
+ Fs, (A2)
+ us ·∇
Ts = −ps∇·us −∇·Γs − Π̂s : ∇us + Qs, (A3)
where ps = nsTs and the expressions for the viscous stress tensor Π̂s, the friction force Fs, the heat flux Γs and the interspecies heat
exchange Qs are given in Braginskii (1965). Equations (A1-A3) are complemented with the quasi-neutrality condition, ne = Zni,
and the Faraday and Ampère laws, which are (in the non-relativistic limit)
= −c∇×E, j = ene(ui − ue) =
∇×B. (A4)
Because of quasi-neutrality, we only need one of the continuity equations, say the ion one. We can also use the electron momen-
tum equation [Eq. (A2), s = e] to express E, which we then substitute into the ion momentum equation and the Faraday law. The
resulting system is
= −ρ∇·u, (A5)
−∇· Π̂+ B ·∇B
+ ue ·∇
ue, (A6)
u×B − j×B
c∇· Π̂e
+ ue ·∇
, (A7)
http://gs2.sourceforge.net
http://www.physics.uiowa.edu/~
50 SCHEKOCHIHIN ET AL.
where ρ = mini, u = ui, p = pi + pe, Π̂ = Π̂i + Π̂e, ue = u − j/ene, ne = Zni, d/dt = ∂/∂t + u ·∇. The ion and electron temperatures
continue to satisfy Eq. (A3).
A.2. Strongly Magnetized Limit
In this form, the two-fluid theory starts resembling the standard one-fluid MHD, which was our starting point in § 2: Eqs. (A5-
A7) already look similar to the continuity, momentum and induction equations. The additional terms that appear in these equations
and the temperature equations (A3) are brought under control by considering how they depend on a number of dimensionless
parameters: ω/νii, k‖λmfpi, k⊥ρi, (me/mi)
1/2. While all these are small in Braginskii’s calculation, no assumption is made as to
how they compare to each other. We now specify that
k‖λmfpi√
, k⊥ρi ≪ k‖λmfpi ∼
≪ 1 (A8)
(see Fig. 4). Note that the first of these relations is equivalent to assuming that the fluctuation frequencies are Alfvénic—the same
assumption as in gyrokinetics [Eq. (49)]. The second relation in Eq. (A8) will be referred to by us as the strongly magnetized
limit. Under the assumptions (A8), the two-fluid equations reduce to the following closed set:47
= −ρ∇·u, (A10)
b̂b̂ : ∇u − 1
b̂b̂ρν‖i
b̂b̂ : ∇u − 1
B ·∇B
, (A11)
= B ·∇u − B∇·u, (A12)
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
− νie (Ti − Te) +
miν‖i
b̂b̂ : ∇u − 1
, (A13)
Te∇·u +
b̂ρκ‖eb̂ ·∇Te
νie (Te − Ti) , (A14)
where ν‖i = 0.90vthiλmfpi is the parallel ion viscosity, κ‖i = 2.45vthiλmfpi parallel ion thermal diffusivity, κ‖e = 1.40vtheλmfpe ∼
Z2/τ 5/2
(mi/me)
1/2κ‖i parallel electron thermal diffusivity [here λmfpi = vthi/νii with νii defined in Eq. (52)], and νie ion–electron
collision rate [defined in Eq. (51)]. Note that the last term in Eq. (A13) represents the viscous heating of the ions.
A.3. One-Fluid Equations (MHD)
If we now restrict ourselves to the low-frequency regime where ion–electron collisions dominate over all other terms in the
ion-temperature equation (A13),
k‖λmfpi√
≪ 1 (A15)
[see Eqs. (A8) and (51)], we have, to lowest order in this new subsidiary expansion, Ti = Te = T . We can now write p = (ni +ne)T =
(1 + Z)ρT/mi and, adding Eqs. (A13) and (A14), find the equation for pressure:
p∇·u = ∇·
b̂neκ‖eb̂ ·∇T
miν‖i
b̂b̂ : ∇u − 1
, (A16)
where we have neglected the ion thermal diffusivity compared to the electron one, but kept the ion heating term to maintain
energy conservation. Equation (A16) together with Eqs. (A10-A12) constitutes the conventional one-fluid MHD system. With
the dissipative terms [which are small because of Eq. (A15)] neglected, this was the starting point for our fluid derivation of
RMHD in § 2.
Note that the electrons in this regime are adiabatic because the electron thermal diffusion is small
∼ k‖λmfpi
≪ 1, (A17)
47 The structure of the momentum equation (A11) is best understood by realizing that ρν‖i
b̂b̂ : ∇u −∇·u/3
= p⊥ − p‖ , the difference between the perpen-
dicular and parallel (ion) pressures. Since the total pressure is p = (2/3)p⊥ + (1/3)p‖ , Eq. (A11) can be written
p⊥ − p‖
B ·∇B
. (A9)
This is the general form of the momentum equation that is also valid for collisionless plasmas, when k⊥ρi ≪ 1 but k‖λmfpi is order unity or even large.
Equation (A9) together with the continuity equation (A11), the induction equation (A12) and a kinetic equation for the particle distribution function (from the
solution of which p⊥ and p‖ are determined) form the system known as Kinetic MHD (KMHD, see Kulsrud 1964, 1983). The collisional limit, k‖λmfpi ≪ 1, of
KMHD is again Eqs. (A10-A14).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 51
provided Eq. (A15) holds and βi is order unity. If we take βi ≫ 1 instead, we can still satisfy Eq. (A15), so Ti = Te follows from
the ion temperature equation (A13) and the one-fluid equations emerge as an expansion in high βi. However, these equations now
describe two physical regimes: the adiabatic long-wavelength regime that satisfies Eq. (A17) and the shorter-wavelength regime
in which (me/mi)
βi ≪ k‖λmfpi ≪ (me/mi)1/2
βi, so the fluid is isothermal, T = T0 = const, p = [(1+Z)T0/mi]ρ = c2sρ [Eq. (9)
holds with γ = 1].
A.4. Two-Fluid Equations with Isothermal Electrons
Let us now consider the regime in which the coupling between the ion and electron temperatures is small and the electron
diffusion is large [the limit opposite to Eqs. (A15) and (A17)]:
k‖λmfpi√
∼ k‖λmfpi
≫ 1, (A18)
Then the electrons are isothermal, Te = T0e = const (with the usual assumption of stochastic field lines, so b̂ · ∇Te = 0 implies
∇Te = 0, as in § 4.4), while the ion temperature satisfies
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
miν‖i
b̂b̂ : ∇u − 1
. (A19)
Equation (A19) together with Eqs. (A10-A12) and p = ρ(Ti + ZT0e)/mi are a closed system that describes an MHD-like fluid
of adiabatic ions and isothermal electrons. Applying the ordering of § 2.1 to these equations and carrying out an expansion in
k‖/k⊥ ≪ 1 entirely analogously to the way it was done in § 2, we arrive at the RMHD equations (17-18) for the Alfvén waves
and the following system for the compressive fluctuations (slow and entropy modes):
+ b̂ ·∇u‖ = 0, (A20)
− v2Ab̂ ·∇
= ν‖i b̂ ·∇
b̂ ·∇u‖ +
, (A21)
= κ‖ib̂ ·∇
b̂ ·∇
, (A22)
and the pressure balance
b̂ ·∇u‖ +
. (A23)
Recall that these equations, being the consequence of Braginskii’s two-fluid equations (§ A.1), are an expansion in k‖λmfpi ≪ 1
correct up to first order in this small parameter. Since the dissipative terms are small, we can replace (d/dt)δρ/ρ0 in the viscous
terms of Eqs. (A21) and (A23) by its value computed from Eqs. (A20), (A22) and (A23) in neglect of dissipation: (d/dt)δρ/ρ0 =
−b̂ · ∇u‖/(1 + c2s/v2A) [cf. Eq. (25)], where the speed of sound cs is defined by Eq. (166). Substituting this into Eqs. (A21)
and (A23), we recover the collisional limit of KRMHD derived in Appendix D, see Eqs. (D18-D20) and (D22).
B. COLLISIONS IN GYROKINETICS
The general collision operator that appears in Eq. (36) is (Landau 1936)
= 2π lnΛ
q2s q
fs′ (v
∂ fs(v)
fs(v)
∂ fs′ (v′)
, (B1)
where w = v − v′ and lnΛ is the Coulomb logarithm. We now take into account the expansion of the distribution function (54),
use the fact that the collision operator vanishes when it acts on a Maxwellian, and retain only first-order terms in the gyrokinetic
expansion. This gives us the general form of the collision term in Eq. (57): it is the ring-averaged linearized form of the Landau
collision operator (B1), (∂hs/∂t)c = 〈Cs[h]〉Rs , where
Cs[h] = 2π lnΛ
q2s q
F0s′(v
hs(v) − F0s(v)
hs′(v
. (B2)
Note that the velocity derivatives are taken at constant r, i.e., the gyrocenter distribution functions that appear in the integrand
should be understood as hs(v)≡ hs(t,r+v⊥× ẑ/Ωs,v⊥,v‖). The explicit form of the gyrokinetic collision operator can be derived
in k space as follows:
eik·Rhk
eik·rCs
e−ik·ρhk
eik·Rs
eik·ρs(v)Cs
e−ik·ρhk
, (B3)
52 SCHEKOCHIHIN ET AL.
where ρs(v) = −v⊥× ẑ/Ωs and Rs = r−ρs(v). Angle brackets with no subscript refer to averages over the gyroangle ϑ of quantities
that do not depend on spatial coordinates. Note that inside the operator Cs[. . .], h occurs both with index s and velocity v and
with index s′ and velocity v′ (over which summation/integration is done). In the latter case, ρ = ρs′(v′) = −v′⊥× ẑ/Ωs′ in the
exponential factor inside the operator.
Most of the properties of the collision operator that are used in the main body of this paper to order the collision terms
can be established in general, already on the basis of Eq. (B3) (§§ B.1-B.2). If the explicit form of the collision operator is
required, we could, in principle, perform the ring average on the linearized operator C [Eq. (B2)] and derive an explicit form of
(∂hs/∂t)c. In practice, in gyrokinetics, as in the rest of plasma physics, the full collision operator is only used when it is absolutely
unavoidable. In most problems of interest, further simplifications are possible: the same-species collisions are often modeled by
simpler operators that share the full collision operator’s conservation properties (§ B.3), while the interspecies collision operators
are expanded in the electron–ion mass ratio (§ B.4).
B.1. Velocity-Space Integral of the Gyrokinetic Collision Operator
Many of our calculations involve integrating the gyrokinetic equation (57) over the velocity space while keeping r constant.
Here we estimate the size of the integral of the collision term when k⊥ρs ≪ 1. Using Eq. (B3),
d3veik·r−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r2π
dv⊥ v⊥
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
d3vJ0(as)e
ik·ρs(v)Cs
e−ik·ρhk
eik·r
1 − ik · v⊥× ẑ
k · v⊥× ẑ
+ . . .
e−ik·ρhk
. (B4)
Since the (linearized) collision operator Cs conserves particle number, the first term in the expansion vanishes. The operator
Cs = Css +Css′ is a sum of the same-species collision operator [the s′ = s part of the sum in Eq. (B2)] and the interspecies collision
operator (the s′ 6= s part). The former conserves total momentum of the particles of species s, so it gives no contribution to the
second term in the expansion in Eq. (B4). Therefore,
d3v〈〈Css[hs]〉Rs〉r ∼ νssk
s δns. (B5)
The interspecies collisions do contribute to the second term in Eq. (B4) due to momentum exchange with the species s′. This
contribution is readily inferred from the standard formula for the linearized friction force (see, e.g., Helander & Sigmar 2002):
d3vvCss′
e−ik·ρhk
S (v)e
−ik·ρs(v)hsk + ms′νs
S (v)e
−ik·ρs′ (v)hs′k
, (B6)
S (v) =
2πn0s′q
s′ lnΛ
(vths
vths′
vths′
erf ′
vths′
, (B7)
where erf(x) = (2/
dy exp(−y2) is the error function. From this, via a calculation of ring averages analogous to Eq. (B17),
we get
−ik ·
v⊥× ẑ
e−ik·ρhk
S (v)
ik ·ρs(v)e−ik·ρs(v)
hsk +
S (v)
ik ·ρs′(v)e−ik·ρs′ (v)
S (v)asJ1(as)hsk +
S (v)as′J1(as′)hs′k
∼ νss′k2⊥ρ2sδns + νs′sk2⊥ρ2s′δns′ . (B8)
For the ion–electron collisions (s = i, s′ = e), using Eqs. (45) and (51), we find that both terms are ∼ (me/mi)1/2νiik2⊥ρ2i δni.
Thus, besides an extra factor of k2⊥ρ
i , the ion–electron collisions are also subdominant by one order in the mass-ratio expansion
compared to the ion–ion collisions. The same estimate holds for the interspecies contributions to the third and fourth terms in
Eq. (B4). In a similar fashion, the integral of the electron–ion collision operator (s = e, s′ = i), is ∼ νeik2⊥ρ2eδne, which is the same
order as the integral of the electron–electron collisions.
The conclusion of this section is that, both for ion and for electron collisions, the velocity-space integral (at constant r) of the
gyrokinetic collision operator is higher order than the collision operator itself by two orders of k⊥ρs. This is the property that we
relied on in neglecting collision terms in Eqs. (104) and (137).
B.2. Ordering of Collision Terms in Eqs. (125) and (137)
In § 5, we claimed that the contribution to the ion–ion collision term due to the (Ze〈ϕ〉Ri/T0i)F0i part of the ion distribution
function [Eq. (124)] was one order of k⊥ρi smaller than the contributions from the rest of hi. This was used to order collision
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 53
terms in Eqs. (125) and (137). Indeed, from Eq. (B3),
Ze〈ϕ〉Ri
eik·Ri
eik·ρiCii
e−ik·ρi J0(ai)F0i
]〉 Zeϕk
eik·Ri
eik·ρiCii
1 − ik ·ρi −
(k ·ρi)2 −
+ · · ·
∼ νiik2⊥ρ2i
F0i. (B9)
This estimate holds because, as it is easy to ascertain using Eq. (B2), the operator Cii annihilates the first two terms in the
expansion and only acts non-trivially on an expression that is second order in k⊥ρi. With the aid of Eq. (47), the desired ordering
of the term (B9) in Eq. (125) follows. When Eq. (B9) is integrated over velocity space, the result picks up two extra orders in
k⊥ρi [a general effect of integrating the gyroaveraged collision operator over the velocity space; see Eq. (B4)]:
Ze〈ϕ〉Ri
∼ νiik4⊥ρ4i
, (B10)
so the resulting term in Eq. (137) is third order, as stated in § 5.3.
B.3. Model Pitch-Angle-Scattering Operator for Same-Species Collisions
A popular model operator for same-species collisions that conserves particle number, momentum, and energy is constructed
by taking the test-particle pitch-angle-scattering operator and correcting it with an additional term that ensures momentum con-
servation (Rosenbluth et al. 1972; see also Helander & Sigmar 2002):
CM[hs] = ν
D (v)
1 − ξ2
) ∂hs
1 − ξ2
2v ·U[hs]
v2ths
, U[hs] =
d3vvνssD (v)hs
d3v (v/vths)2νssD (v)F0s(v)
, (B11)
νssD (v) = νss
(vths
v2ths
erf ′
, νss =
2πn0sq
s lnΛ
, (B12)
where the velocity derivatives are at constant r. The gyrokinetic version of this operator is (cf. Catto & Tsang 1977;
Dimits & Cohen 1994)
〈CM[hs]〉Rs =
eik·RsνssD (v)
1 − ξ2
) ∂hsk
v2(1 + ξ2)
4v2ths
s hsk + 2
v⊥J1(as)U⊥[hsk] + v‖J0(as)U‖[hsk]
v2ths
, (B13)
U⊥[hsk] =
d3vv⊥J1(as)νssD (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
, U‖[hsk] =
d3vv‖J0(as)ν
D (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
where as = k⊥v⊥/Ωs. The velocity derivatives are now at constant Rs. The spatial diffusion term appearing in the ring-averaged
collision operator is physically due to the fact that a change in a particle’s velocity resulting from a collision can lead to a change
in the spatial position of its gyrocenter.
In order to derive Eq. (B13), we use Eq. (B3). Since, ρs(v) =
1 − ξ2 sinϑ+ ŷv
1 − ξ2 cosϑ
/Ωs, it is not hard to see
e−ik·ρs(v)hsk = e
−ik·ρs(v)
1 − ξ2
ik⊥ ·
v⊥× ẑ
e−ik·ρs(v)hsk = e
−ik·ρ(v)
ik⊥ ·v⊥
hsk. (B14)
Therefore,
eik·ρs(v)
1 − ξ2
e−ik·ρs(v)hsk
1 − ξ2
) ∂hsk
k2⊥hsk,
eik·ρs(v)
e−ik·ρs(v)hsk
1 − ξ2
k2⊥hsk.
(B15)
Combining these formulae, we obtain the first two terms in Eq. (B13). Now let us work out the U term:
eik·ρs(v)v ·
d3v′ v′νssD (v
′)e−ik·ρs(v
′)hsk
v′⊥,v
veik·ρs(v)
dv′⊥ v
dv′‖ν
v′e−ik·ρs(v
v′⊥,v
(B16)
Since
ve±ik·ρs(v)
= ẑv‖
e±ik·ρs(v)
v⊥e±ik·ρs(v)
, where
e±ik·ρs(v)
= J0(as) and
±ik·ρs(v)
= ẑ×
v⊥× ẑ
∓ik⊥ ·
v⊥× ẑ
= ±iΩsẑ×
∓ik⊥ ·
v⊥× ẑ
= ±i ẑ×k⊥
v⊥J1(as), (B17)
we obtain the third term in Eq. (B13).
54 SCHEKOCHIHIN ET AL.
It is useful to give the lowest-order form of the operator (B13) in the limit k⊥ρs ≪ 1:
〈CM[hs]〉Rs = ν
D (v)
1 − ξ2
) ∂hs
d3v′v′
νssD (v
′)hs(v
d3v′v′2νssD (v
′)F0s(v′)
+ O(k2⊥ρ
s ). (B18)
This is the operator that can be used in the right-hand side of Eq. (145) (as, e.g., is done in the calculation of collisional transport
terms in Appendix D.3).
In practical numerical computations of gyrokinetic turbulence, the pitch-angle scattering operator is not sufficient because the
distribution function develops small scales not only in ξ but also in v (M. Barnes, W. Dorland and T. Tatsuno 2006, unpublished).
This is, indeed, expected because the phase-space entropy cascade produces small scales in v⊥, rather than just in ξ (see § 7.9.1).
In order to provide a cut off in v, an energy-diffusion operator must be added to the pitch-angle-scattering operator derived above.
A numerically tractable model gyrokinetic energy-diffusion operator was proposed by Abel et al. (2008); Barnes et al. (2009).48
B.4. Electron–Ion Collision Operator
This operator can be expanded in me/mi and to the lowest order is (see, e.g., Helander & Sigmar 2002)
Cei[h] = ν
D (v)
1 − ξ2
) ∂he
1 − ξ2
2v ·ui
v2the
, νeiD (v) = νei
(vthe
. (B19)
The corrections to this form are O(me/mi). This is second order in the expansion of § 4 and, therefore, we need not keep these
corrections. The operator (B19) is mathematically similar to the model operator for the same-species collisions [Eq. (B13)]. The
gyrokinetic version of this operator is derived in the way analogous to the calculation in Appendix B.3. The result is
〈Cei[h]〉Re =
eik·ReνeiD (v)
1 − ξ2
) ∂hek
v2(1 + ξ2)
4v2the
v2the
J1(ae)
2v′2⊥
v2thi
hik +
2v‖J0(ae)u‖ki
v2the
. (B20)
At scales not too close to the electron gyroscale, namely, such that k⊥ρe ∼ (me/mi)1/2, the second and third terms are manifestly
second order in (me/mi)
1/2, so have to be neglected along with other O(me/mi) contributions to the electron–ion collisions.
49 The
remaining two terms are first order in the mass-ratio expansion: the first term vanishes for he = h
e [Eq. (101)], so its contribution
is first order; in the fourth term, we can use Eq. (87) to express u‖i in terms of quantities that are also first order. Keeping only
the first-order terms, the gyrokinetic electron–ion collision operator is
〈Cei[h]〉Re = ν
D (v)
1 − ξ2
) ∂h(1)e
2v‖u‖i
v2the
. (B21)
Note that the ion drag term is essential to represent the ion–electron friction correctly and, therefore, to capture the Ohmic
resistivity (which, however, is rarely more important for unfreezing flux than the electron inertia and the finiteness of the electron
gyroradius; see § 7.7).
C. A HEURISTIC DERIVATION OF THE ELECTRON EQUATIONS
Here we show how the equations (116-117) of § 4 and the ERMHD equations (226-227) of § 7 can be derived heuristically
from electron fluid dynamics and a number of physical assumptions, without the use of gyrokinetics (§ C.1). This derivation is
not rigorous. Its role is to provide an intuitive route to the isothermal electron fluid and ERMHD approximations.
C.1. Derivation of Eqs. (116-117)
We start with the following three equations:
= −c∇×E, ∂ne
+∇· (neue) = 0, E +
ue ×B
. (C1)
These are Faraday’s law, the electron continuity equation, and the generalized Ohm’s law, which is the electron momentum
equation with all electron inertia terms neglected (i.e., effectively, the lowest order in the expansion in the electron mass me). The
electron pressure is assumed to be scalar by fiat (this can be justified in certain limits: for example in the collisional limit, as in
Appendix A, or for the isothermal electron fluid approximation derived in § 4). The electron-pressure term in the right-hand side
of Ohm’s law is sometimes called the thermoelectric term. We now assume the same static uniform equilibrium, E0 = 0, B0 = B0ẑ,
that we have used throughout this paper and apply to Eqs. (C1) the fundamental ordering discussed in § 3.1.
48 The collision operator now used the GS2 and AstroGK codes (see footnote 46) is their energy-diffusion operator plus the pitch-angle-scattering opera-
tor (B13).
49 The third term in Eq. (B20) is, in fact, never important: at the electron scales, k⊥ρe ∼ 1, it is negligible because of the Bessel function in the velocity
integral (Abel et al. 2008).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 55
First consider the projection of Ohm’s law onto the total magnetic field B, use the definition of E [Eq. (37)], and keep the
leading-order terms in the ǫ expansion:
E · b̂ = − 1
b̂ ·∇pe ⇒
+ b̂ ·∇ϕ = b̂ ·∇ δpe
. (C2)
This turns into Eq. (116) if we also assume isothermal electrons, δpe = T0eδne [see Eq. (103)].
With the aid of Ohm’s law, Faraday’s law turns into
= ∇× (ue ×B) = −ue ·∇B + B ·∇ue − B∇·ue. (C3)
Keeping the leading-order terms, we find, for the components of Eq. (C3) perpendicular and parallel to the mean field,
+ u⊥e ·∇⊥
= b̂ ·∇u⊥e,
+ u⊥e ·∇⊥
= b̂ ·∇u‖e. (C4)
In the last equation, we have used the electron continuity equation to write
∇·ue = −
+ u⊥e ·∇⊥
. (C5)
From Ohm’s law, we have, to lowest order,
u⊥e = −ẑ×
E⊥ +∇⊥
= ẑ×∇⊥
. (C6)
Using this expression in the second of the equations (C4) gives
− b̂ ·∇u‖e =
, (C7)
where d/dt is defined in the usual way [Eq. (122)]. Assuming isothermal electrons (δpe = T0eδne) annihilates the second term on
the right-hand side and turns the above equation into Eq. (117). As for the first of the equations (C4), the use of Eq. (C6) and
substitution of δB⊥ = −ẑ×∇⊥A‖ turns it into the previously derived Eq. (C2), whence follows Eq. (116).
Thus, we have shown that Eqs. (116-117) can be derived as a direct consequence of Faraday’s law, electron fluid dynamics
(electron continuity equation and the electron force balance, a. k. a. the generalized Ohm’s law), and the assumption of isothermal
electrons—all taken to the leading order in the gyrokinetic ordering given in § 3.1 (i.e., assuming strongly interacting anisotropic
fluctuations with k‖ ≪ k⊥).
We have just proved that Eqs. (116) and (117) are simply the perpendicular and parallel part, respectively, of Eq. (C3). The
latter equation means that the magnetic-field lines are frozen into the electron flow velocity ue, i.e., the flux is conserved, the
result formally proven in § 4.3 [see Eq. (99)].
C.2. Electron MHD and the Derivation of Eqs. (226-227)
One route to Eqs. (226-227), already explained in § 7.2, is to start with Eqs. (C2) and (C7) and assume Boltzmann electrons
and ions and the total pressure balance. Another approach, more standard in the literature on the Hall and Electron MHD, is to
start with Eq. (C3), which states that the magnetic field is frozen into the electron flow. The electron velocity can be written in
terms of the ion velocity and the current density, and the latter then related to the magnetic field via Ampère’s law:
ue = ui −
= ui −
4πene
∇×B. (C8)
To the leading order in ǫ, the perpendicular and parallel parts of Eq. (C3) are Eqs. (C4), respectively, where the perpendicular and
parallel electron velocities are [from Eq. (C8)]
u⊥e = u⊥i +
4πen0e
ẑ×∇⊥δB‖, u‖e = u‖i +
4πen0e
∇2⊥A‖. (C9)
The relative size of the two terms in each of these expressions is controlled by the size of k⊥di, where di = ρi/
βi is the ion
inertial scale. When k⊥di ≫ 1, we may set ui = 0. Note, however, that the ion motion is not totally neglected: indeed, in the
second of the equations (C4), the δne/ne terms comes, via Eq. (C5), from the divergence of the ion velocity [from Eq. (C8),
∇·ui = ∇·ue]. To complete the derivation, we relate δne to δB‖ via the assumption of total pressure balance, as explained in
§ 7.2, giving us Eq. (225). Substituting this equation and Eqs. (C9) into Eqs. (C4), we obtain
= v2Adi b̂ ·∇
1 + 2/βi(1 + Z/τ )
b̂ ·∇∇2⊥Ψ, (C10)
where Ψ = −A‖/
4πmin0i. Equations (C10) evolve the perturbed magnetic field. These equations become the ERMHD equations
(226-227) if δB‖/B0 is expressed in terms of the scalar potential via Eq. (223).
56 SCHEKOCHIHIN ET AL.
Note that there are two special limits in which the assumption of immobile ions suffices to derive Eqs. (C10) from Eq. (C3)
without the need for the pressure balance: βi ≫ 1 (incompressible ions) or τ = T0i/T0e ≪ 1 (cold ions) but βe = βiZ/τ ≫ 1. In both
cases, Eq. (225) shows that δne/n0e ≪ δB‖/B0, so the density perturbation can be ignored and the coefficient of the right-hand
side of the second of the equations (C10) is equal to 1. The limit of cold ions is discussed further in Appendix E.
D. FLUID LIMIT OF THE KINETIC RMHD
Taking the fluid (collisional) limit of the KRMHD system (summarized in § 5.7) means carrying out another subsidiary
expansion—this time in k‖λmfpi ≪ 1. The expansion only affects the equations for the density and magnetic-field-strength
fluctuations (§ 5.5) because the Alfvén waves are indifferent to collisional effects.
The calculation presented below follows a standard perturbation algorithm used in the kinetic theory of gases and in plasma
physics to derive fluid equations with collisional transport coefficients (Chapman & Cowling 1970). For magnetized plasma,
this calculation was carried out in full generality by Braginskii (1965), whose starting point was the full plasma kinetic theory
[Eqs. (36-39)]. While what we do below is, strictly speaking, merely a particular case of his calculation (see Appendix A), it has
the advantage of relative simplicity and also serves to show how the fluid limit is recovered from the gyrokinetic formalism—a
demonstration that we believe to be of value.
It will be convenient to use the KRMHD system written in terms of the function δ f̃i = g + (v2⊥/v
thi)(δB‖/B0)F0i, which is the
perturbation of the local Maxwellian in the frame of the Alfvén waves [Eqs. (150-152)]. We want to expand Eq. (150) in powers
of k‖λmfpi, so we let δ f̃i = δ f̃
i + δ f̃
i + . . ., δB‖ = δB
+ δB(1)
+ . . ., etc.
D.1. Zeroth Order: Ideal Fluid Equations
Since [see Eq. (49)]
k‖λmfpi√
k‖vthi
∼ k‖λmfpi, (D1)
to zeroth order Eq. (150) becomes
δ f̃ (0)i
= 0. The zero mode of the collision operator is a Maxwellian. Therefore, we
may write the full ion distribution function up to zeroth order in k‖λmfpi as follows [see Eq. (144)]
2πTi/mi
mi[(v⊥ − uE)2 + (v‖ − u‖)2]
, (D2)
where ni = n0i + δni and Ti = T0i + δTi include both the unperturbed quantities and their perturbations. The E×B drift velocity uE
comes from the Alfvén waves (see § 5.4) and does not concern us here. Since the perturbations δni, u‖ and δTi are small in the
original gyrokinetic expansion, Eq. (D2) is equivalent to
δ f̃ (0)i =
δn(0)e
v2thi
δT (0)i
v2thi
F0i, (D3)
where we have used quasi-neutrality to replace δni/n0i = δne/n0e. This automatically satisfies Eq. (151), while Eq. (152) gives us
an expression for the ion-temperature perturbation:
δT (0)i
δn(0)e
δB(0)
. (D4)
Note that this is consistent with the interpretation of the perpendicular Ampère’s law [Eq. (63), which is the progenitor of
Eq. (152)] as the pressure balance [see Eq. (67)]: indeed, recalling that the electron pressure perturbation is δpe = T0eδne
[Eq. (103)], we have
= −δpe − δpi = −δneT0e − δniT0i − n0iδTi, (D5)
whence follows Eq. (D4) by way of quasi-neutrality (Zni = ne) and the definitions of Z, τ , βi [Eqs. (40-42)].
Since the collision operator conserves particle number, momentum and energy, we can obtain evolution equations for δn(0)e /n0e,
and δB(0)
/B0 by multiplying Eq. (150) by 1, v‖, v
2/v2thi, respectively, and integrating over the velocity space. The three
moments that emerge this way are
d3vδ f̃ (0)i =
δn(0)e
d3vv‖δ f̃
i = u
v2thi
δ f̃ (0)i =
δn(0)e
δT (0)i
. (D6)
The three evolution equations for these moments are
δn(0)e
δB(0)
+ b̂ ·∇u(0)
= 0, (D7)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 57
du(0)
− v2A b̂ ·∇
δB(0)
= 0, (D8)
δn(0)e
δT (0)i
δB(0)
b̂ ·∇u(0)
= 0. (D9)
These allow us to recover the fluid equations we derived in § 2.4: Eq. (D8) is the parallel component of the MHD momentum
equation (27); combining Eqs. (D7), (D9) and (D4), we obtain the continuity equation and the parallel component of the induction
equation—these are the same as Eqs. (25) and (26):
δn(0)e
1 + c2s/v
b̂ ·∇u(0)
δB(0)
1 + v2A/c2s
b̂ ·∇u(0)
, (D10)
where the sound speed cs is defined by Eq. (166). From Eqs. (D7) and (D9), we also find the analog of the entropy equation (23):
δT (0)i
δn(0)e
δs(0)
δs(0)
δT (0)i
δn(0)e
δn(0)e
δB(0)
. (D11)
This implies that the temperature changes due to compressional heating only.
D.2. Generalized Energy: Five RMHD Cascades Recovered
We now calculate the generalized energy by substituting δ f̃i from Eq. (D3) into Eq. (153) and using Eqs. (D4) and (D11):
min0iu
min0iu
n0iT0i
1 + Z/τ
5/3 + Z/τ
=W +AW +W
AW +W
sw +W
n0iT0i
1 + Z/τ
5/3 + Z/τ
Ws. (D12)
The first two terms are the Alfvén-wave energy [Eq. (154)]. The following two terms are the slow-wave energy, which splits into
the independently cascaded energies of “+” and “−” waves (see § 2.5):
WSW = W
sw +W
min0i
|z+‖|
2 + |z−‖|
. (D13)
The last term is the total variance of the entropy mode. Thus, we have recovered the five cascades of the RMHD system (§ 2.7;
Fig. 5 maps out the fate of these cascades at kinetic scales).
D.3. First Order: Collisional Transport
Now let us compute the collisional transport terms for the equations derived above. In order to do this, we have to determine
the first-order perturbed distribution function δ f̃ (1)i , which satisfies [see Eq. (150)]
δ f̃ (1)i
δ f̃ (0)i −
v2thi
δB(0)
+ v‖ b̂ ·∇
δ f̃ (0)i +
δn(0)e
. (D14)
We now use Eq. (D3) to substitute for δ f̃ (0)i and Eqs. (D10-D11) and (D8) to compute the time derivatives. Equation (D14)
becomes
δ f̃ (1)i
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D15)
where ξ = v‖/v. Note that the right-hand side gives zero when multiplied by 1, v‖ or v
2 and integrated over the velocity space, as
it must do because the collision operator in the left-hand side conserves particle number, momentum and energy.
Solving Eq. (D15) requires inverting the collision operator. While this can be done for the general Landau collision operator
(see Braginskii 1965), for our purposes, it is sufficient to use the model operator given in Appendix B.3, Eq. (B18). This simplifies
calculations at the expense of an order-one inaccuracy in the numerical values of the transport coefficients. As the exact value of
these coefficients will never be crucial for us, this is an acceptable loss of precision. Inverting the collision operator in Eq. (D15)
then gives
δ f̃ (1)i =
ν iiD(v)
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D16)
58 SCHEKOCHIHIN ET AL.
where ν iiD(v) is a collision frequency defined in Eq. (B12) and we have chosen the constants of integration in such a way that the
three conservation laws are respected:
d3vδ f̃ (1)i = 0,
d3vv‖δ f̃
i = 0,
d3vv2δ f̃ (1)i = 0. These relations mean that δn
e = 0,
= 0, δT (1)i = 0 and that, in view of Eq. (152), we have
δB(1)
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖, (D17)
where ν‖i is defined below [Eq. (D21)]. Equations (D16-D17) are now used to calculate the first-order corrections to the moment
equations (D7-D9). They become
+ b̂ ·∇u‖ = 0, (D18)
− v2Ab̂ ·∇
2/3 + c2s/v
1 + c2s/v
ν‖i b̂ ·∇
b̂ ·∇u‖
, (D19)
= κ‖ib̂ ·∇
b̂ ·∇δTi
, (D20)
where we have introduced the coefficients of parallel viscosity and parallel thermal diffusivity:
ν‖i =
ν iiD(v)v
F0i(v), κ‖i =
ν iiD(v)v
v2thi
F0i(v). (D21)
All perturbed quantities are now accurate up to first order in k‖λmfpi. Note that in Eq. (D19), we used Eq. (D17) to express
δB(0)
= δB‖ − δB
. We do the same in Eq. (D4) and obtain
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖
. (D22)
This equation completes the system (D18-D20), which allows us to determine δne, u‖, δTi and δB‖. In § 6.1, we use the equations
derived above, but absorb the prefactor (2/3 + c2s/v
A)/(1 + c
A) into the definition of ν‖i. The same system of equations can also
be derived from Braginskii’s two-fluid theory (Appendix A.4), from which we can borrow the quantitatively correct values of the
viscosity and ion thermal diffusivity: ν‖i = 0.90v
thi/νii, κ‖i = 2.45v
thi/νii, where νii is defined in Eq. (52).
E. HALL REDUCED MHD
The popular Hall MHD approximation consists in assuming that the magnetic field is frozen into the electron flow velocity
[Eq. (C3)]. The latter is calculated from the ion flow velocity and the current determined by Ampère’s law [Eq. (C8)]:
4πen0e
, (E1)
where the ion flow velocity ui satisfies the conventional MHD momentum equation (8). The Hall MHD is an appealing theoretical
model that appears to capture both the MHD behavior at long wavelengths (when ue ≃ ui) and some of the kinetic effects that
become important at small scales due to decoupling between the electron and ion flows (the appearance of dispersive waves)
without bringing in the full complexity of the kinetic theory. However, unlike the kinetic theory, it completely ignores the
collisionless damping effects and suggests that the key small-scale physical change is associated with the ion inertial scale
di = ρi/
βi (or, when βe ≪ 1, the ion sound scale ρs = ρi
Z/2τ ; see § E.3), rather than the ion gyroscale ρi. Is this an acceptable
model for plasma turbulence? Figure 8 illustrates the fact that at τ ∼ 1, the ion inertial scale does not play a special role linearly,
the MHD Alfvén wave becomes dispersive at the ion gyroscale, not at di, and that the collisionless damping cannot in general
be neglected. A detailed comparison of the Hall MHD linear dispersion relation with full hot plasma dispersion relation leads to
the conclusion that Hall MHD is only a valid approximation in the limit of cold ions, namely, τ = T0i/T0e ≪ 1 (Ito et al. 2004;
Hirose et al. 2004). In this Appendix, we show that a reduced (low-frequency, anisotropic) version of Hall MHD can, indeed, be
derived from gyrokinetics in the limit τ ≪ 1.50 This demonstrates that the Hall MHD model fits into the theoretical framework
proposed in this paper as a special limit. However, the parameter regime that gives rise to this special limit is not common in
space and astrophysical plasmas of interest.
E.1. Gyrokinetic Derivation of Hall Reduced MHD
Let us start with the equations of isothermal electron fluid, Eqs. (116-121), i.e., work within the assumptions that allowed us
to carry out the mass-ratio expansion (§ 4.8). In Eq. (120) (perpendicular Ampère’s law, or gyrokinetic pressure balance), taking
50 Note that, strictly speaking, our ordering of the collision frequency does not allow us to take this limit (see footnote 17), but this is a minor betrayal of rigor,
which does not, in fact, invalidate the results.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 59
the limit τ ≪ 1 gives
eik·r
d3vJ0(ai)hik
, (E2)
where we have used Eq. (118) to express the hi integral and the expression for the electron beta βe = βiZ/τ . Note that the above
equation is simply the statement of a balance between the magnetic and electron thermal pressure (the ions are relatively cold,
so they have fallen out of the pressure balance). Using Eq. (E2) to express δne in terms of δB‖ in Eqs. (116) and (117) and also
substituting for u‖e from Eq. (119) [or, equivalently, Eq. (87)], we get
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E3)
where we have used our usual definitions of the stream and flux functions [Eq. (135)] and of the full derivatives [Eq. (160)].
These equations determine the evolution of the magnetic field, but we still need the ion gyrokinetic equation (121) to calculate
the ion motion (Φ = cϕ/B0 and u‖i) via Eqs. (118) and (88). There are two limits in which the ion kinetics can be reduced to
simple fluid models.
E.1.1. High-Ion-Beta Limit, βi ≫ 1
In this limit, k⊥ρi = k⊥di
βi ≫ 1 as long as k⊥di is not small. Then the ion motion can be neglected because it is averaged out
by the Bessel functions in Eqs. (118) and (88)—in the same way as in § 7.2. So we get Φ = (τ/Z)vAdiδB‖/B0 [using Eq. (E2);
this is the τ ≪ 1 limit of Eq. (223)] and u‖i = 0. Noting that βe = βiZ/τ ≫ 1 in this limit, we find that Eqs. (E3) reduce to
= v2Adi b̂ ·∇
= −di b̂ ·∇∇2⊥Ψ, (E4)
which is the τ ≪ 1 limit of our ERMHD equations (226-227) [or, equivalently, Eqs. (C10)].
E.1.2. Low-Ion-Beta Limit, βi ∼ τ ≪ 1 (the Hall Limit)
This limit is similar to the RMHD limit worked out in § 5: we take, for now, k⊥di ∼ 1 and βe ∼ 1 (in which subsidiary
expansions can be carried out later), and expand the ion gyrokinetics in k⊥ρi = k⊥di
βi ≪ 1. Note that ordering βe ∼ 1 means
that we have ordered βi ∼ τ ≪ 1. We now proceed analogously to the way we did in § 5: express the ion distribution in terms of
the g function defined by Eq. (124) and, using the relation (E2) between δB‖/B0 and δne/n0e, write Eqs. (125-127) as follows:
︸ ︷︷ ︸
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+b̂ ·∇
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
F0i +
v⊥ ·A⊥
︸ ︷︷ ︸
,(E5)
Γ1(αi) +
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, u‖ki
d3vv‖J0(ai)gk
︸ ︷︷ ︸
. (E6)
All terms in these equations can be ordered with respect to the small parameter
βi (an expansion subsidiary to the gyrokinetic
expansion in ǫ and the Hall expansion in τ ≪ 1). The lowest order to which they enter is indicated underneath each term. The
ordering we use is the same as in § 5.2, but now we count the powers of
βi and order formally k⊥di ∼ 1 and βe ∼ 1. It is easy
to check that this ordering can be summarized as follows
and that the ion and electron terms in Eqs. (E3) are comparable under this ordering, so their competition is retained (in fact, this
could be used as the underlying assumption behind the ordering). The fluctuation frequency continues to be ordered as the Alfvén
frequency, ω ∼ k‖vA. The collision terms are ordered via ω/νii ∼ k‖λmfpi/
βi and k‖λmfpi ∼ 1, although the latter assumption is
not essential for what follows, because collisions turn out to be negligible and it is fine to take k‖λmfpi ≫ 1 from the outset and
neglect them completely.
In Eqs. (E6), we use Eqs. (129) and (130) to write 1 −Γ0(αi) ≃ αi = k2⊥ρ2i /2 and Γ1(αi) ≃ 1. These equations imply that if we
expand g = g(−1) + g(0) + . . ., we must have
d3vg(−1) = 0, so the contribution to the right-hand side of the first of the equations
60 SCHEKOCHIHIN ET AL.
(E6) (the quasi-neutrality equation) comes from g(0), while the parallel ion flow is determined by g(−1). Retaining only the lowest
(minus first) order terms in Eq. (E5), we find the equation for g(−1), the v‖ moment of which gives an equation for u‖i:
∂g(−1)
{ϕ,g(−1)} = 2
v‖b̂ ·∇
F0i ⇒
= v2Ab̂ ·∇
. (E8)
Now integrating Eq. (E5) over the velocity space (at constant r), using the first of the equations (E6) to express the integral of
g(0), and retaining only the lowest (zeroth) order terms, we find
ρ2i ∇2⊥
+ b̂ ·∇u‖i = 0 ⇒
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ, (E9)
where we have used the second of the equations (E3) to express the time derivative of δB‖/B0.
Together with Eqs. (E3), Eqs. (E8) and (E9) form a closed system, which it is natural to call Hall Reduced MHD (HRMHD)
because these equations can be straightforwardly derived by applying the RMHD ordering (§ 2.1) to the MHD equations (8-10)
with the induction equation (10) replaced by Eq. (E1). Indeed, Eqs. (E8) and (E9) exactly coincide with Eqs. (27) and (18), which
are the parallel and perpendicular components of the MHD momentum equation (8) under the RMHD ordering; Eqs. (E3) should
be compared Eqs. (17) and (26) while noticing that, in the limit τ ≪ 1, the sound speed is cs = vA
βe/2 [see Eq. (166)]. The
incompressible case (Mahajan & Yoshida 1998) is recovered in the subsidiary limit βe ≫ 1 (i.e., 1 ≫ βi ≫ τ ).
E.2. Generalized Energy for Hall RMHD and the Passive Entropy Mode
To work out the generalized energy (§ 3.4) for the HRMHD regime, we start with the generalized energy for the isothermal
electron fluid [Eq. (109)] and use Eq. (E2) to express the density perturbation:
T0iδ f
, (E10)
where δB⊥ = ẑ×∇⊥Ψ. The perturbed ion distribution function can be written in the same form as it was done in § 5.4 [Eq. (143)]:
to lowest order in the
βi expansion (§ E.1.2),
δ f (−1)i =
2v⊥ ·u⊥
v2thi
F0i + g(−1) =
2v⊥ ·u⊥
v2thi
F0i +
2v‖u‖i
v2thi
F0i + g̃, (E11)
where u⊥ = ẑ×∇⊥Φ. The last equality above is achieved by noticing that, since g(−1) satisfies Eq. (E8), we may split it into a
perturbed Maxwellian with parallel velocity u‖i and the remainder: g
(−1) = 2v‖u‖iF0i/v
thi + g̃. Then g̃ is the homogeneous solution
of the leading-order kinetic equation [see Eq. (E8)]:
+{Φ, g̃} = 0,
d3v g̃ = 0. (E12)
Substituting Eq. (E11) into Eq. (E10) and keeping only the leading-order terms in the
βi expansion, we get
min0iu
min0iu
T0ig̃
. (E13)
The first four terms are the energy of the Alfvénic and slow-wave-polarized fluctuations [cf. Eq. (D12)]. Unlike in RMHD, these
are not decoupled in HRMHD, unless a further subsidiary long-wavelength limit is taken (see § E.4). It is easy to verify that the
sum of these four terms is indeed conserved by Eqs. (E3), (E8) and (E9). The last term in Eq. (E13) is an individually conserved
kinetic quantity. Its conservation reflects the fact that g̃ is decoupled from the wave dynamics and passively advected by the
Alfvénic velocities via Eq. (E12).51
The passive kinetic mode g̃ can be thought of as a kinetic version of the MHD entropy mode and, indeed, reduces to it if the
collision operator in Eq. (E5) is upgraded to the leading order by orderingω/νii ∼ 1 (i.e., by considering long parallel wavelengths,
k‖λmfpi ∼
βi). In such a collisional limit, g̃ has to be a perturbed Maxwellian with no density or velocity perturbation [because
d3vg̃ = 0, while the velocity perturbation is explicitly separated from g̃ in Eq. (E11)]. Therefore,
v2thi
F0i ⇒
T0ig̃
n0iT0i
δT 2i
T 20i
. (E14)
This is to be compared with the βi ∼ τ ≪ 1 limit of Eqs. (D11) and (D12). As we have established, in the
βi expansion,
δTi = δT
i , δni = δn
i , δB‖ = δB
, so to lowest order δs/s0 = δTi/T0i and Eq. (E14) describes the entropy mode in the Hall limit.
51 A similar splitting of the generalized energy cascade into a fluid-like cascade plus a passive cascade of a zero-density part of the distribution function occurs
in the Hasegawa–Mima regime, which is the electrostatic version of the Hall limit (Plunk et al. 2009).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 61
E.3. Hall RMHD Dispersion Relation
Linearizing the Hall RMHD equations (E3), (E8) and (E9) (derived in § E.1.2 assuming the ordering βi ∼ τ ≪ 1), we obtain
the following dispersion relation:52
ω2 − k2‖v
1 + 2/βe
= ω2k2‖v
1 + 2/βe
. (E15)
When the coupling term on the right-hand side is negligible, k⊥di/
1 + 2/βe ≪ 1, we recover the MHD Alfvén wave, ω2 = k2‖v
and the MHD slow wave, ω2 = k2
v2A/(1 + v
s ) [Eq. (167)], where cs = vA
βe/2 in the limit τ ≪ 1 [Eq. (166)]. In the opposite
limit, we get the kinetic Alfvén wave, ω2 = k2
i /(1 + 2/βe) [same as Eq. (230) with τ ≪ 1].
The solution of the dispersion relation (E15) is
1 + 2/βe
. (E16)
The corresponding eigenfunctions then satisfy53
Ψ = −
Φ+ vAdi
, u‖i = −
, Φ = −
Ψ. (E17)
Equation (E16) takes a particularly simple form in the subsidiary limits of high and low electron beta βe = βiZ/τ :
βe ≫ 1 : ω2 = k2‖v
, βe ≪ 1 : ω2 = k2‖v
1 + k2⊥ρ
and ω2 =
1 + k2⊥ρ
, (E18)
where ρs = di
βe/2 = ρi
Z/2τ = cs/Ωi is called the ion sound scale. The Alfvén wave and the slow wave (known as the ion
acoustic wave in the limit of τ ≪ 1, βe ≪ 1) become dispersive at the ion inertial scale (k⊥di ∼ 1) when βe ≫ 1 and at the ion
sound scale (k⊥ρs ∼ 1) when βe ≪ 1.
E.4. Summary of Hall RMHD and the Role of the Ion Inertial and Ion Sound Scales
We have shown that in the limit of cold ions and low ion beta (βi ∼ τ ≪ 1, “the Hall limit”), gyrokinetic turbulence can be
described by five scalar functions: the stream and flux functions Φ and Ψ for the Alfvénic fluctuations, the parallel velocity and
magnetic-field perturbations u‖i and δB‖ for the slow-wave-polarized fluctuations, and g̃, the zero-density, zero-velocity part of
the ion distribution function, which is the kinetic version of the MHD entropy mode. The first four of these functions satisfy a
closed set of four fluid-like equations, derived in § E.1 and collected here:
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E19)
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ,
= v2Ab̂ ·∇
. (E20)
We call these equations the Hall Reduced Magnetohydrodynamics (HRMHD). To fully account for the generalized energy cas-
cade, one must append to the four HRMHD equations the fifth, kinetic equation (E12) for g̃, which is energetically decoupled
from HRMHD and slaved to the Alfvénic velocity fluctuations (§ E.2).
The equations given above are valid above the ion gyroscale, k⊥ρi ≪ 1. They contain a special scale, di/
1 + 2/βe, which
is the ion inertial scale di for βe ≫ 1 and the ion sound scale ρs = cs/Ωi for βe ≪ 1. As becomes clear from the linear theory
(§ E.3), the Alfvén and slow waves become dispersive at this scale. Nonlinearly, this scale marks the transition from the regime
in which the Alfvénic and slow-wave-polarized fluctuations are decoupled to the regime in which they are mixed. Namely, when
k⊥di/
1 + 2/βe ≪ 1, HRMHD turns into RMHD: Eqs. (E19) become Eqs. (17) and (26), while Eqs. (E20) remain unchanged
and identical to Eqs. (18) and (27); in the opposite limit, k⊥di/
1 + 2/βe ≫ 1, the ion motion decouples from the magnetic-field
evolution and Eqs. (E19) turn into the ERMHD equations (226-227).
Since we are considering the case βi ≪ 1, both di and ρs are much larger than the ion gyroscale ρi. In the opposite limit of
βi ≫ 1 (§ E.1.1), while di is the only scale that appears explicitly in Eqs. (E4), we have di ≪ ρi and the equations themselves
represent the dynamics at scales much smaller than the ion gyroscale, so the transition between the RMHD and ERMHD regimes
occurs at k⊥ρi ∼ 1. The same is true for βi ∼ 1, when di ∼ ρi. The ion sound scale ρs ≫ ρi does not play a special role when
52 The full gyrokinetic dispersion relation in a similar limit was worked out in Howes et al. (2006), Appendix D.2.1.
53 Note that wave packets with |k⊥| = k⊥ and satisfying Eq. (E17) with k‖vA/ω as a function of k⊥ given by Eq. (E16) are exact nonlinear solutions of
the HRMHD equations (E3) and (E8-E9). This can be shown via a calculation analogous to that in § 7.3 (for the incompressible Hall MHD, this was done by
Mahajan & Krishan 2005).
62 SCHEKOCHIHIN ET AL.
βi is not small: it is not hard to see that for k⊥ρs ∼ 1, the ion motion terms in Eqs. (E19) dominate and we simply recover the
inertial-range KRMHD model (§ 5) by expanding in k⊥ρi = k⊥ρs
2τ/Z ≪ 1.
Various theories of the dissipation-range turbulence based on Hall and Electron MHD are further discussed in § 8.2.6.
F. TWO-DIMENSIONAL INVARIANTS IN GYROKINETICS
Since gyrokinetics is in a sense a “quasi-two-dimensional” approximation, it is natural to inquire if this gives rise to additional
conservation properties (besides the conservation of the generalized energy discussed in § 3.4) and how they are broken by the
presence of parallel propagation terms. It is important to emphasize that, except in a few special cases, these invariants are only
invariants in 2D, so gyrokinetic turbulence in 2D and 3D has fundamentally different properties, despite its seemingly “quasi-2D”
nature. It is, therefore, generally not correct to think of the gyrokinetic turbulence (or its special case the MHD turbulence) as
essentially a 2D turbulence with an admixture of parallel-propagating waves (Fyfe et al. 1977; Montgomery & Turner 1981).
In this Appendix, we work out the 2D invariants. Without attempting to present a complete analysis of the 2D conservation
properties of gyrokinetics, we limit our discussion to showing how some more familiar fluid invariants (most notably, magnetic
helicity) emerge from the general 2D invariants in the appropriate asymptotic limits.
F.1. General 2D Invariants
In deriving the generalized energy invariant, we used the fact that
d3Rs hs{〈χ〉Rs ,hs} = 0, so Eq. (57) after multiplication
by T0shs/F0s and integration over space contains no contribution from the Poisson-bracket nonlinearity. Since we also have∫
d3Rs 〈χ〉Rs{〈χ〉Rs ,hs} = 0, multiplying Eq. (57) by qs〈χ〉Rs and integrating over space has a similar outcome. Subtracting the
latter integrated equation from the former and rearranging terms gives
qs〈χ〉Rs
= qsv‖
d3Rs 〈χ〉Rs
qs〈χ〉Rs
. (F1)
We see that in a purely 2D situation, when ∂/∂z = 0, we have an infinite family of invariants Is = Is(v⊥,v‖) whose conservation (for
each species and for every value of v⊥ and v‖!) is broken only by collisions. In 3D, the parallel particle streaming (propagation)
term in the gyrokinetic equation generally breaks these invariants, although special cases may arise in which the first term on the
right-hand side of Eq. (F1) vanishes and a genuine 3D invariant appears.
F.2. “A2
-Stuff”
Let apply the mass-ratio expansion (§ 4.1) to Eq. (F1) for electrons. Using the solution (101) for the electron distribution
function, we find
T0eF0e
v2the
d3rA‖
v2the
+ · · ·
= −ev‖
v2the
F0e −
∂h(1)e
d3rA‖
, (F2)
where we have kept terms to two leading orders in the expansion. To lowest order, the above equation reduces to
d3rA‖
. (F3)
This equation can also be obtained directly from Eq. (116) (multiply by A‖ and integrate). In 2D, it expresses a well known
conservation law of the “A2‖-stuff.” As this 2D invariant exists already on the level of the mass-ratio expansion of the electron
kinetics, with no assumptions about the ions, it is inherited both by the RMHD equations in the limit of k⊥ρi ≪ 1 (§ 5.3) and
by the ERMHD equations in the limit of k⊥ρi ≫ 1 (§ 7.2). In the former limit, δne/n0e on the right-hand side of Eq. (F3) is
negligible (under the ordering explained in § 5.2); in the latter limit, it is expressed in terms of ϕ via Eq. (221). The conservation
of “A2‖-stuff” is a uniquely 2D feature, broken by the parallel propagation term in 3D.
F.3. Magnetic Helicity in the Electron Fluid
If we now divide Eq. (F2) through by ev‖/c and integrate over velocities, we get, after some integrations by parts, another
relation that becomes a conservation law in 2D and that can also easily be derived directly from the equations of the isothermal
electron fluid (116-117):
d3rA‖
. (F4)
In the ERMHD limit k⊥ρi ≫ 1 (§ 7.2), we use Eqs. (221-223) to simplify the above equation and find that the integral on the
right-hand side vanishes and we get a genuine 3D conservation law:
d3rA‖δB‖ = 0. (F5)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 63
This can also be derived directly from the ERMHD equations (226-227) [using Eq. (223)]. The conserved quantity is readily seen
to be the helicity of the perturbed magnetic field:
d3rA · δB =
∇⊥×A‖ẑ
+ A‖δB‖
A‖ẑ · (∇⊥×A⊥) + A‖δB‖
d3rA‖δB‖. (F6)
F.4. Magnetic Helicity in the RMHD Limit
Unlike in the case of ERMHD, the helicity of the perturbed magnetic field in RMHD is conserved only in 2D. This is because
the induction equation for the perturbed field has an inhomogeneous term associated with the mean field [Eq. (10) with B =
B0ẑ + δB] (this issue has been extensively discussed in the literature; see Matthaeus & Goldstein 1982; Stribling et al. 1994;
Berger 1997; Montgomery & Bates 1999; Brandenburg & Matthaeus 2004). Directly from the induction equation or from its
RMHD descendants Eqs. (17) and (26), we obtain [note the definitions (135)]
d3rA‖δB‖ =
1 + v2A/c2s
, (F7)
so helicity is conserved only if ∂/∂z = 0.
For completeness, let us now show that this 2D conservation law is a particular case of Eq. (F1) for ions. Let us consider the
inertial range (k⊥ρi ≪ 1). We substitute Eq. (124) into Eq. (F1) for ions and expand to two leading orders in k⊥ρi using the
ordering explained in § 5.2:
v‖〈A‖〉Ri
Z2e2v2
d3rA‖g + · · ·
Z2e2v2
d3rA‖
v2thi
+ Zev‖
d3rA‖
. (F8)
The lowest-order terms in the above equations (all proportional to v2‖F0i) simply reproduce the 2D conservation of “A
‖-stuff,”
given by Eq. (F3). We now subtract Eq. (F3) multiplied by (Zev‖/c)
2F0i/T0i from Eq. (F8). This leaves us with
d3rA‖g = c
+ v‖F0i
v2thi
d3rA‖
. (F9)
This equation is a general 2D conservation law of the KRMHD equations (see § 5.7) and can also be derived directly from them.
If we integrate it over velocities and use Eqs. (146) and (147), we simply recover Eq. (F4). However, since Eq. (F9) holds for
every value of v‖ and v⊥, it carries much more information than Eq. (F4).
To make connection to MHD, let us consider the fluid (collisional) limit of KRMHD worked out in Appendix D. The distribu-
tion function to lowest order in the k‖λmfpi ≪ 1 expansion is g = −(v2⊥/v2thi)δB‖/B0 +δ f̃
i , where δ f̃
i is the perturbed Maxwellian
given by Eq. (D3). We can substitute this expression into Eq. (F9). Since in this expansion the collision integral is applied to δ f̃ (1)i
and is the same order as the rest of the terms (see § D.3), conservation laws are best derived by taking 1, v‖, and v
2/v2thi moments
of Eq. (F9) so as to make the collision term vanish. In particular, multiplying Eq. (F9) by 1 + (2τ/3Z)v2/v2thi, integrating over
velocities and using Eqs. (D4) and (D6), we obtain the evolution equation for
d3rA‖δB‖, which coincides with Eq. (F7). Note
that, either proceeding in an analogous way, one can derive similar equations for
d3rA‖δne and
d3rA‖u‖—these are also 2D
invariants of the RMHD system, broken in 3D by the presence of the propagation terms. The same result can be derived directly
from the evolution equations (D8) and (D10).
F.5. Electrostatic Invariant
Interestingly, the existence of the general 2D invariants introduced in § F.1 alongside the generalized energy invariant given by
Eq. (73) means that one can construct a 2D invariant of gyrokinetics that does not involve any velocity-space quantities. In order
to do that, one must integrate Eq. (F1) over velocities, sum over species, and subtract Eq. (73) from the resulting equation (thus
removing the h2s integrals). The result is not particularly edifying in the general case, but it takes a simple form if one considers
electrostatic perturbations (δB = 0). In this case, χ = ϕ, and the manipulations described above lead to the following equation
d3v Is −W
q2s n0s
1−Γ0(αs)
|ϕk|2 =
d3rE‖ j‖ −
d3Rs 〈ϕ〉Rs
, (F10)
where E‖ = −∂ϕ/∂z, αs = k2⊥ρ
s/2 and Γ0 is defined by Eq. (129). In 2D, E‖ = 0 and the above equation expresses a conservation
law broken only by collisions. The complete derivation and analysis of 2D conservation properties of gyrokinetics in the electro-
static limit, including the invariant (F10), the electrostatic version of Eq. (F1), and their consequences for scalings and cascades,
was given by Plunk et al. (2009). Here we briefly consider a few relevant limits.
For k⊥ρi ≪ 1, we have Γ0(α) = 1 −αs + . . ., so the invariant given by Eq. (F10) is simply the kinetic energy of the E×B flows:
s(msn0s/2)
d3r |∇⊥Φ|2, where Φ = cϕ/B0. In the limit k⊥ρi ≫ 1, k⊥ρe ≪ 1, we have Y = −n0i
d3rZ2e2ϕ2/2T0i. In
64 SCHEKOCHIHIN ET AL.
the limit k⊥ρe ≫ 1, we have Y = −(1 + Z/τ )n0e
d3re2ϕ2/2T0e. Whereas we are not interested in electrostatic fluctuations in the
inertial range, electrostatic turbulence in the dissipation range was discussed in § 7.10 and § 7.12. The electrostatic 2D invariant
in the limits k⊥ρi ≫ 1, k⊥ρe ≪ 1 and k⊥ρe ≫ 1 can also be derived directly from the equations given there [in the former limit,
use Eq. (264) to express u‖i in terms of j‖ in order to get Eq. (F10)].
Note that, taken separately and integrated over velocities, Eq. (F1) for ions (when k⊥ρi ≫ 1, k⊥ρe ≪ 1) and for electrons
(when k⊥ρe ≫ 1), reduces to lowest order to the statement of 3D conservation of
d3Ri T0ih2i /2F0i [Whi in Eq. (245)] and∫
d3Re T0eh2e/2F0e [Eq. (280)], respectively.
F.6. Implications for Turbulent Cascades and Scalings
Since invariants other than the generalized energy or its constituent parts are present in 2D and, in some limits, also in 3D, one
might ask how their presence affects the turbulent cascades and scalings. As an example, let us consider the magnetic helicity in
KAW turbulence, which is a 3D invariant of the ERMHD equations (§ F.3).
A Kolmogorov-style analysis of a local KAW cascade based on a constant flux of helicity gives (proceeding as in § 7.5):
τKAWλ
1 +βi
τKAWλ
1 +βi
∼ εH = const ⇒ Φλ ∼
(1 +βi)1/6
1/3, (F11)
where εH is the helicity flux (omitting constant dimensional factors, the helicity is now defined as
d3rΨΦ and assumed to be
non-zero). This corresponds to a k
⊥ spectrum of magnetic energy.
In order to decide whether we expect the scalings to be determined by the constant-helicity flux or by the constant-energy
flux (as assumed in § 7.5), we adapt a standard argument originally due to Fjørtoft (1953). If the helicity flux of the KAW
turbulence originating at the ion gyroscale (via partial conversion from the inertial-range turbulence; see § 7) is εH , its energy
flux is εKAW ∼ εH [set λ = ρi in Eq. (F11) and compare with Eq. (238)]. If the cascade between the ion and electron gyroscales
is controlled by maintaining a constant flux of helicity, then the helicity flux arriving to the electron gyroscale is still εH , while
the associated energy flux is εHρi/ρe ≫ εKAW, i.e., more energy arrives to ρe than there was at ρi! This is clearly impossible in
a stationary state. The way to resolve this contradiction is to conclude that the helicity cascade is, in fact, inverse (i.e., directed
towards larger scales), while the energy cascade is direct (to smaller scales). A similar argument based on the constancy of the
energy flux εKAW then leads to the conclusion that the helicity flux arriving to the electron gyroscale is εKAWρe/ρi ≪ εH ∼ εKAW,
i.e., the helicity indeed does not cascade to smaller scales. It does not, in fact, cascade to large scales either because the ERMHD
equations are not valid above the ion gyroscale and the helicity of the perturbed magnetic field in the inertial range is not a 3D
invariant (§ F.4). The situation would be different if an energy source existed either at the electron gyroscale or somewhere in
between ρe and ρi. In such a case, one would expect an inverse helicity cascade and the consequent shallower scaling [Eq. (F11)]
between the energy-injection scale and the ion gyroscale.
Other invariants introduced above can in a similar fashion be argued to give rise to inverse cascades in the hypothetical 2D
situations where they are valid and provided there is energy injection at small scales (for the electrostatic case, see Plunk et al.
2009 and numerical simulations by Tatsuno et al. 2009b). The view of turbulence advanced in this paper does not generally
allow for this to happen. First, the fundamentally 3D nature of the turbulence is imposed via the critical balance conjecture and
supported by the argument that “two dimensionality” can only be maintained across parallel distances that do not exceed the
distance a parallel-propagating wave (or parallel-streaming particles) travels over one nonlinear decorrelation time (see § 1.2,
§ 7.5 and § 7.10.3). Secondly, the lack of small-scale energy injection was assumed at the outset. This can, however, be violated
in real astrophysical plasmas by various small-scale plasma instabilities (e.g., triggered by pressure anisotropies; see discussion
in § 8.3). Treatment of such effects falls outside the scope of this paper and remains a matter for future work.
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|
0704.0045 | Evolution of solitary waves and undular bores in shallow-water flows
over a gradual slope with bottom friction | Evolution of solitary waves and undular bores in
shallow-water flows over a gradual slope with bottom
friction
G.A. El1, R.H.J. Grimshaw2
Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU, UK
1 e-mail: [email protected] 2 e-mail: [email protected]
A.M. Kamchatnov
Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow Region, 142190 Russia
e-mail: [email protected]
Abstract
This paper considers the propagation of shallow-water solitary and nonlinear peri-
odic waves over a gradual slope with bottom friction in the framework of a variable-
coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using
a recent development of this theory for perturbed integrable equations. This general
approach enables us not only to improve known results on the adiabatic evolution of
isolated solitary waves and periodic wave trains in the presence of variable topography
and bottom friction, modeled by the Chezy law, but also importantly, to study the
effects of these factors on the propagation of undular bores, which are essentially un-
steady in the system under consideration. In particular, it is shown that the combined
action of variable topography and bottom friction generally imposes certain global
restrictions on the undular bore propagation so that the evolution of the leading soli-
tary wave can be substantially different from that of an isolated solitary wave with
the same initial amplitude. This non-local effect is due to nonlinear wave interactions
within the undular bore and can lead to an additional solitary wave amplitude growth,
which cannot be predicted in the framework of the traditional adiabatic approach to
the propagation of solitary waves in slowly varying media.
1 Introduction
There have been many studies of the propagation of water waves over a slope, sometimes
also subject to the effects of bottom friction. Many of these works have considered linear
waves, or have been numerical simulations in the framework of various nonlinear long-wave
model equations. Our interest here is in the propagation of weakly nonlinear long water
http://arxiv.org/abs/0704.0045v1
waves over a slope, simultaneously subject to bottom friction, a combination apparently
first considered by Miles (1983a,b) albeit for the special case of a single solitary wave, or a
periodic wavetrain. An appropriate model equation for this scenario is the variable-coefficient
perturbed Korteweg-de Vries (KdV) equation (see Grimshaw 1981, Johnson 1973a,b),
At + cAx +
AAx +
Axxx = −CD
|A|A. (1)
Here A(x, t) is the free surface elevation above the undisturbed depth h(x) and c(x) =
gh(x) is the linear long wave phase speed. The bottom friction term on the right-hand
side is represented by the Chezy law, modelling a turbulent boundary layer. Here CD is a
non-dimensional drag coefficient, often assumed to have a value around 0.01 (Miles 1983a,b).
Other forms of friction could be used (see, for instance Grimshaw et al 2003) but the Chezy
law seems to be the most appropriate for water waves in a shallow depth. In (1) the first
two terms on the left-hand side are the dominant terms, and by themselves describe the
propagation of a linear long wave with speed c. The remaining terms on the left-hand
side represent, respectively, the effect of varying depth, weakly nonlinear effects and weak
linear dispersion. The equation is derived using the usual KdV balance in which the linear
dispersion, represented by ∂2/∂x2, is balanced by nonlinearity, represented by A. Here we
have added to this balance weak inhomogeneity so that cx/c scales as h
2∂3/∂x3, and weak
friction so that CD scales with h∂/∂x. Within this basic balance of terms, we can cast (1)
into the asymptotically equivalent form
AAX +
AXXX = −CD
|A|A, (2)
where τ =
c(x′)
, X = τ − t. (3)
Here we have h = h(x(τ)), explicitly dependent on the variable τ which describes evolution
along the path of the wave.
The governing equation (2) can be cast into several equivalent forms. That most com-
monly used is the variable-coefficient KdV equation, obtained here by putting
B = (gh)1/4A (4)
so that Bτ +
2g1/4h5/4
BBX +
BXXX = −CD
|B|B . (5)
This form shows that, in the absence of friction term, i.e. when CD ≡ 0, equation (2)
has two integrals of motion with the densities proportional to h1/4A and h1/2A2. These
are often referred to as laws for the conservation of “mass” and “momentum”. However,
these densities do not necessarily correspond to the corresponding physical entities. Indeed,
to leading order, the “momentum” density is proportional to the wave action flux, while
the “mass” density differs slightly from the actual mass density. This latter issue has been
explored by Miles (1979), where it was shown that the difference is smaller than the error
incurred in the derivation of equation (4), and is due to reflected waves.
Our main concern in this paper is with the behaviour of an undular bore over a slope
in the presence of bottom friction, using the perturbed KdV equation (2), where we were
originally motivated by the possibility that the behaviour of a tsunami approaching the
shore might be modeled in this way. The undular bore solution to the unperturbed KdV
equation can be constructed using the well-known Gurevich-Pitaevskii (GP) (1974) approach
(see also Fornberg and Whitham 1978). In this approach, the undular bore is represented
as a modulated nonlinear periodic wave train. The main feature of this unsteady undular
bore is the presence of a solitary wave (which is the limiting wave form of the periodic
cnoidal wave) at its leading edge. The original initial-value problem for the KdV equation is
then replaced by a certain boundary-value problem for the associated modulation Whitham
equations. We note, however, that so far, the simplest, “(x/t)”-similarity solutions of the
modulation equations have been used for the modelling of undular bores in various contexts
(see Grimshaw and Smyth 1986, Smyth 1987 or Apel 2003 for instance). These solutions,
while effectively describing many features of undular bores, are degenerate and fail to cap-
ture, even qualitatively, some important effects associated with non-self-similar modulation
dynamics. In particular, in the classical GP solution for the resolution of an initial jump
in the unperturbed KdV equation, the amplitude of the lead solitary wave in the undular
bore is constant (twice the value of the initial jump). On the other hand, the modulation
solution for the undular bore evolving from a general monotonically decreasing initial profile
shows that the lead solitary wave amplitude in fact grows with time (Gurevich, Krylov and
Mazur 1989; Gurevich, Krylov and El 1992; Kamchatnov 2000). As we shall see, the very
possibility of such variations in the modulated solutions of the unperturbed KdV equation
has a very important fluid dynamics implication: in a general setting, the undular bore lead
solitary wave cannot be treated as an individual KdV solitary wave but rather represents a
part of the global nonlinear wave structure. In other words, while at every particular moment
of time the lead solitary wave has the spatial profile of the familiar KdV soliton, generally,
the temporal dependence of its amplitude cannot be obtained in the framework of single
solitary wave perturbation theory.
In the unperturbed KdV equation, the growth of the lead solitary wave amplitude is
caused by the spatial inhomogeneity of the initial data. Here, however, the presence of a
perturbation due to topography and/or friction serves as an alternative and/or additional
cause for variation of the lead solitary wave amplitude. Thus, in the present case, the
variation in the amplitude will have two components (which generally, of course, cannot be
separated because of the nonlinear nature of the problem); one is local, described by the
adiabatic perturbation theory for a single solitary wave, and the other one is nonlocal, which
in principle requires the study of the full modulation solution. Depending on the relative
values of the small parameters associated with the slope, friction and spatial non-uniformity
of the initial modulations, we can take into account only one of these components, or a
combination of them.
The structure of the paper is as follows. First, in Section 2, we reformulate the basic model
(1) as a constant-coefficient KdV equation perturbed by terms representing topography
and friction. Then we derive in Section 3 the associated perturbed Whitham modulation
equations using methods recently developed by Kamchatnov (2004). Next, in Section 4, this
Whitham system is integrated in the solitary-wave limit. Our purpose here is primarily to
obtain the equation of a multiple characteristic, which defines the leading edge of a shoaling
undular bore in the case when the modulations due to the combined action of the slope and
bottom friction are small compared to the existing spatial modulations due to non-uniformity
of the initial data. As a by-product of this integration, we reproduce and extend the known
results on the adiabatic variation of a single solitary wave (Miles 1983a,b). Then, in Section
5, we carry out an analogous study of a cnoidal wave, propagating over a gradual slope and
subject to friction, a case studied previously by Miles (1983 b) but under the restriction of
zero mean flow, which is removed here. Finally, in Section 6 we study effects of a gradual
slope and bottom friction on the front of an undular bore which represents a modulated
cnoidal wave transforming into a system of weakly interacting solitons near its leading edge.
2 Problem formulation
For the purpose of the present paper it is convenient to recast (2) into the standard KdV
equation form with constant coefficients, modified by certain perturbation terms. Thus we
introduce the new variables
A, T =
hdτ =
6g3/2
h(x)dx. (6)
so that UT + 6UUX + UXXX = R = F (T )U −G(T )|U |U, (7)
where F (T ) = −9hT
, G(T ) = 4CD
. (8)
In this form, the governing equation (7) has the structure of the integrable KdV equation
on the left-hand side, while the separate effects of the varying depth and the bottom friction
are represented by the two terms on the right-hand side. This structure enables us to use
the general theory developed in Kamchatnov (2004) for perturbed integrable systems.
For much of the subsequent discussion, it is useful to assume that h(x) = constant,
CD = 0 for x < 0 in the original equation (1), which corresponds to F (T ) = G(T ) = 0 for
T < 0 in (7). We shall also assume that A = 0 for x > 0 at t = 0, which corresponds to
U = 0 for X > 0 on X = τ(T ) (see (6)). Then we shall propose two types of initial-value
problem for (1), and correspondingly for (7).
(a) Let a solitary wave of a given amplitude a0 initially propagating over a flat bottom
without friction (i.e a soliton described by an unperturbed KdV equation), enter the variable
topography and bottom friction region at t = 0, x = 0 (Fig. 1 a).
(b) Let an undular bore of a given intensity propagate over a flat bottom without friction
(the corresponding solution of the unperturbed KdV equation will be discussed in Section
5). Let the lead solitary wave of this undular bore have the same amplitude a0 and enter
the variable topography and bottom friction region at t = 0, x = 0 (Fig. 1b).
In particular, we shall be interested in the comparison of the slow evolution of these
two, initially identical, solitary waves in the two different problems described above. The
expected essential difference in the evolution is due to the fact that the lead solitary wave
in the undular bore is generally not independent of the remaining part of the bore and can
exhibit features that cannot be captured by a local perturbation analysis. The well-known
example of such a behaviour, when a solitary wave is constrained by the condition of being
a part of a global nonlinear wave structure, is provided by the undular bore solution of the
KdV-Burgers (KdV-B) equation
ut + 6uux + uxxx = µuxx, µ ≪ 1 . (9)
( )h x
a)
( )h x
Figure 1: Isolated solitary wave (a) and undular bore (b) entering the variable topogra-
phy/bottom friction region.
Indeed, the undular bore solution of the KdV-B equation (9) is known to have a solitary
wave at its leading edge (see Johnson 1970; Gurevich & Pitaevskii 1987; Avilov, Krichever
& Novikov 1987) and this solitary wave: (a) is asymptotically close to a soliton solution
of the unperturbed KdV equation; and (b) has the amplitude, say a0, that is constant in
time. At the same time, it is clear that if one takes an isolated KdV soliton of the same
amplitude a0 as initial data for the KdV-Burgers equation it would damp with time due
to dissipation. The physical explanation of such a drastic difference in the behaviour of an
isolated soliton and a lead solitary wave in the undular bore for the same weakly dissipative
KdV-B equation is that the action of weak dissipation on an expanding undular bore is
twofold: on the one hand, the dissipation tends to decrease the amplitude of the wave
locally but, on the other hand, it “squeezes” the undular bore so that the interaction (i.e.
momentum exchange) between separate solitons within the bore becomes stronger than in
the absence of dissipation and this acts as the amplitude increasing factor. The additional
momentum is extracted from the upstream flow with a greater depth (see Benjamin and
Lighthill 1954). As a result, in the case of the KdV-B equation, an equilibrium non-zero
value for the lead solitary wave amplitude in the undular bore is established. Of course,
for other types of dissipation, a stationary value of the lead soliton amplitude would not
necessarily exist, but in general, due to the expected increase of the soliton interactions near
the leading edge, the amplitude of the lead soliton of the undular bore would decay slower
than that of an isolated soliton. Indeed, the presence here of variable topography as well
can result in an additional “nonlocal” amplitude growth.
While the problem (a) can be solved using traditional perturbation analysis for a single
solitary wave, which leads to an ordinary differential equation along the solitary wave path
(see Miles 1983a,b), the undular bore evolution problem (b) requires a more general approach
which can be developed on the basis of Whitham’s modulation theory leading to a system of
three nonlinear hyperbolic partial differential equations of the first order. Since the Whitham
method, being equivalent to a nonlinear multiple scale perturbation procedure, contains the
adiabatic theory of slow evolution of a single solitary wave as a particular (albeit singular)
limit, it is instructive for the purposes of this paper to treat both problems (a) and (b) using
the general Whitham theory.
3 Modulation equations
The original Whithammethod (Whitham 1965, 1974) was developed for conservative constant-
coefficient nonlinear dispersive equations and is based on the averaging of appropriate con-
servation laws of the original system over the period of a single-phase periodic travelling
wave solution. The resulting system of quasi-linear equations describes the slow evolution
of the modulations (i.e. of the mean value, the wavenumber, the amplitude etc.) of the pe-
riodic travelling wave. Here, that approach is extended to the perturbed KdV equation (6)
following the general approach of Kamchatnov (2004), which extends earlier results for cer-
tain specific cases (see Gurevich and Pitaevskii (1987, 1991), Avilov, Krichever and Novikov
(1987) and Myint and Grimshaw (1995) for instance).
We suppose that the evolution of the nonlinear wave is adiabatically slow, that is, the
wave can be locally represented as a solution of the corresponding unperturbed KdV equation
(i.e. (7) with zero on the right-hand side) with its parameters slowly varying with space and
time. The one-phase periodic solution of the KdV equation can be written in the form
U(X, T ) = λ3 − λ1 − λ2 − 2(λ3 − λ2)sn2(
λ3 − λ1 θ,m) (10)
where sn(y,m) is the Jacobi elliptic sine function, λ1 ≤ λ2 ≤ λ3 are parameters and the
phase variable θ and the modulus m are given by
θ = X − V T, V = −2(λ1 + λ2 + λ3) , (11)
λ3 − λ2
λ3 − λ1
, (12)
and L =
−P (µ)
2K(m)√
λ3 − λ1
, (13)
where K(m) is the complete elliptic integral of the first kind, L is the “wavelength” along the
X-axis (which is actually a retarded time rather than a true spatial co-ordinate). Here we
have used the representation of the basic ordinary differential equation for the KdV travelling
wave solution (10) in the form (see Kamchatnov (2000) for a general motivation behind this
representation)
−P (µ), (14)
where
µ = 1
(U + s1), s1 = λ1 + λ2 + λ3 (15)
P (µ) =
(µ− λi) = µ3 − s1µ2 + s2µ− s3, (16)
that is the solution (10) is parameterized by the zeroes λ1, λ2, λ3 of the polynomial P (µ).
In a modulated wave, the parameters λ1, λ2, λ3 are allowed to be slow functions of X and
T , and their evolution is governed by the Whitham equations. For the unperturbed KdV
equation, the evolution of the modulation parameters is due to a spatial non-uniformity
of the initial distributions for λj, j = 1, 2, 3 and the typical spatio-temporal scale of the
modulation variations is determined by the scale of the initial data.
In the case of the perturbed KdV equation (7), the evolution of the parameters λ1, λ2, λ3
is caused not only by their initial spatial non-uniformity, but also by the action of the
weak perturbation, so that, generally, at least two independent spatio-temporal scales for
the modulations can be involved. However, at this point we shall not introduce any scale
separation within the modulation theory and derive general perturbed Whitham equations
assuming that the typical values of F (T ) and G(T ) are O(∂λj/∂T, ∂λj/∂X) within the
modulation theory.
It is instructive to first introduce the Whitham equations for the perturbed KdV equation
(7) using the traditional approach of averaging the (perturbed) conservation laws. To this
end, we introduce the averaging over the period (13) of the cnoidal wave (10) by
〈F〉 =
Fdθ =
−P (µ)
. (17)
In particular,
〈U〉 = 2〈µ〉 − s1 = 2(λ3 − λ1)
+ λ1 − λ2 − λ3, (18)
〈U2〉 = 8[−s1
(λ3 − λ1)
s1λ1 +
(λ21 − λ2λ3)] + s21 , (19)
where E(m) is the complete elliptic integral of the second kind. Now, one represents the
KdV equation (7) in the form of the perturbed conservation laws
= Rj , j = 1, 2, 3 , Rj ≪ 1 , (20)
where Pj and Qj are the standard expressions for the conserved densities (Kruskal integrals)
and “fluxes” of the unperturbed KdV equation. Just as in the Whitham (1965) theory for
unperturbed dispersive systems, the number of conservation laws required is equal to the
number of free parameters in the travelling wave solution, which is three in the present case.
Next, one applies the averaging (17) to the system (20) to obtain (see Dubrovin and Novikov
1989)
∂〈Pj〉
∂〈Qj〉
= 〈Rj〉 , j = 1, 2, 3 . (21)
The system (21) describes slow evolution of the parameters λj in the cnoidal wave solution
(10).
Along with these derived perturbed conservative form of the Whitham equations, we
introduce the wave conservation law which is a general condition for the existence of slowly
modulated single-phase travelling wave solutions (10) (see for instance Whitham 1974) and
must be consistent with the modulation system (21). This conservation law has the form
= 0 , (22)
where k =
, ω = kV (23)
are the “wavenumber” and the “frequency” respectively (we have put quotation marks here
because the actual wavenumber and frequency related to the physical variables x, t are
different quantities from those in (23), but are related through the transformations (3, 6) ).
The wave conservation law (22) can be introduced instead of any of three inhomogeneous
averaged conservation laws comprising the Whitham system (21).
It is known that the Whitham system for the homogeneous constant-coefficient KdV
equation can be represented in diagonal (Riemann) form (Whitham 1965, 1974) by an ap-
propriate choice of the three parameters characterising the periodic travelling wave solution.
In fact, in our solution (7) the parameters λj have already been chosen so that they coincide
with the Riemann invariants of the unperturbed KdV modulation system. Introducing them
explicitly into the perturbed system (21) we obtain (see Kamchatnov 2004)
∂L/∂λi
〈(2λi − s1 − U)R〉
j 6=i(λi − λj)
, i = 1, 2, 3, (24)
where R is the perturbation term on the right-hand side of the KdV equation (7) and
vi = −2
∂L/∂λi
, i = 1, 2, 3, (25)
are the Whitham characteristic velocities corresponding to the unperturbed KdV equation.
It should be noted that the straightforward realisation of the above lucid general algo-
rithm for obtaining perturbed modulation system in diagonal form is quite a laborious task.
In fact, to derive system (24), the so-called finite-gap integration method incorporating the
integrable structure of the unperturbed KdV equation has been used. The modulation sys-
tem (24) in a more particular form corresponding to specific choices of the perturbation term
was obtained by Myint and Grimshaw (1995) using a multiple-scale perturbation expansion.
In that latter setting, the wave conservation law (22) is an inherent part of the construction,
while in the averaging approach used here, it can be obtained as a consequence of the system
(24).
To obtain an explicit representation of the Whitham equations for the present case of
equation (7), we must substitute the perturbation R from the right-hand side of (7) and
perform the integration (17) with U given by (10). From now on, we are going to consider
only the flows where U ≥ 0 so that the perturbation term assumes the form
R(U) = G(T )U − F (T )U2 . (26)
Substituting (26) into (24) we obtain, after some detailed calculations (see Appendix),
the perturbed Whitham system in the form
= ρi = Ci[F (T )Ai −G(T )Bi], i = 1, 2, 3 (27)
where C1 =
, C2 =
E − (1−m)K
, C3 =
; (28)
(5λ1 − λ2 − λ3)E +
(λ2 − λ1)K,
(5λ2 − λ1 − λ3)E − (λ2 − λ1)
λ3 − λ1
(5λ3 − λ1 − λ2)E −
(λ2 − λ1)
(−27λ21 − 7λ22 − 7λ23 + 2λ1λ2 + 2λ1λ3 + 22λ2λ3)E
(λ2 − λ1)(3λ1 + λ2 + λ3)K,
(−7λ21 − 27λ22 − 7λ23 + 2λ1λ2 + 22λ1λ3 + 2λ2λ3)E
λ2 − λ1
λ3 − λ1
(7λ21 + 15λ
2 + 11λ
3 − 6λ1λ2 − 18λ1λ3 + 6λ2λ3)K,
(−7λ21 − 7λ22 − 27λ23 + 22λ1λ2 + 2λ1λ3 + 2λ2λ3)E
(7λ21 + 11λ
2 + 15λ
3 − 18λ1λ2 − 6λ1λ3 + 6λ2λ3)K;
and the characteristic velocities are:
v1 = −2
4(λ3 − λ1)(1−m)K
v2 = −2
4(λ3 − λ2)(1−m)K
E − (1−m)K
v3 = −2
4(λ3 − λ2)K
The equations (27) – (31) provide a general setting for studying the nonlinear modulated
wave evolution over variable topography with bottom friction. In the absence of the pertur-
bation terms (i.e. when F (T ) ≡ 0, G(T ) ≡ 0), the system (27), (31) indeed coincides with
the original Whitham equations (Whitham 1965) for the integrable KdV dynamics. In that
case the variables λ1, λ2, λ3 become Riemann invariants, so in this general (perturbed) case
we shall call them Riemann variables.
It is important to study the structure of the perturbed Whitham equations (27) – (31) in
two limiting cases when the underlying cnoidal wave degenerates into (i) a small-amplitude
sinusoidal wave (linear limit), when λ2 = λ3 (m = 0), and (ii) into a solitary wave when
λ2 = λ1 (m = 1). Since in both these limits the oscillations do not contribute to the mean
flow (they are infinitely small in the linear limit and the distance between them becomes
infinitely long in the solitary wave limit) one should expect that in both cases one of the
Whitham equations will transform into the dispersionless limit of the original perturbed
KdV equation (7) i.e.
UT + 6UUX = F (T )U −G(T )U2, (32)
Indeed, using formulae (27) – (31) we obtain for m = 0:
λ2 = λ3 ,
− 6λ1
= λ1F + λ
+ (6λ1 − 12λ3)
= λ1F + λ
Similarly, for m = 1, one has
λ2 = λ1 ,
− (4λ1 + 2λ3)
(4λ1 − λ3)F +
(7λ23 − 24λ1λ3 + 32λ21)G,
− 6λ3
= λ3F + λ
We see that, in both cases, one of the Riemann variables (taken with inverted sign) coincides
with the solution of the dispersionless equation (32) (recall that in the derivation of the
Whitham equations we assumed U ≥ 0 everywhere), namely U = 〈U〉 = −λ1 when λ2 = λ3
(m = 0) and U = 〈U〉 = −λ3 when λ2 = λ1 (m = 1).
To conclude this section, we present expressions for the physical wave parameters such as
the surface elevation wave amplitude a, mean elevation 〈A〉 speed and wavenumber in terms
of the modulation solution λj(X, T ). Using (6) and (10) we obtain for the wave amplitude
(peak to trough) and the mean elevation
(λ3 − λ2) , 〈A〉 =
〈U〉 , (35)
where the dependence of 〈U〉 on λj(X, T ), j = 1, 2, 3 is given by (18) and X = X(x, t),
T = T (x, t) by (3, 6). In order to obtain the physical wavenumber κ and the frequency Ω
we first note that the phase function θ(X, T ) defined in (11) is replaced by a more general
expression defined so that k = θX and kV = −θT are the “wavenumber” and “frequency” in
the X −T coordinate system. Then we define the physical phase function Θ(x, t) = θ(X, T )
so that we get
κ = Θx , Ω = −Θt . (36)
It now follows that
(1− hV
) , Ω = k , and
1− hV/6g
. (37)
Note that the physical frequency is the “wavenumber” in the X − T coordinate system,
and that the physical phase speed is Ω/κ. Since the validity of the KdV model (1) requires
inter alia that the wave be right-going it follows from this expression that the modulation
solution remains valid only when hV < 6g. Of course, the validity of (1) also requires that
the amplitude remains small, and this would normally also ensure that V remains small.
4 Modulation solution in the solitary wave limit
In this section, we shall integrate the perturbed modulation system (27) along the multiple
characteristic corresponding to the merging of two Riemann variables λ2 and λ1. As we shall
see later, this characteristic specifies the motion of the leading edge of the shoaling undular
bore in the case when the perturbations due to variable topography and bottom friction can
be considered as small compared with the existing spatial modulations within the bore. At
the same time, as the case λ2 = λ1 ( i.e. m = 1) corresponds to the solitary wave limit in the
travelling wave solution (10), our results here are expected to be consistent with the results
from the traditional perturbation approach to the adiabatic variation of a solitary wave due
to topography and bottom friction (see Miles 1983a,b).
In the limit m → 1 the periodic solution (10) of the KdV equation goes over to its solitary
wave solution
U(X, T ) = U0sech
λ3 − λ1(X − VsT )]− λ3, (38)
where
U0 = 2(λ3 − λ1) , Vs = −(4λ1 + 2λ3) (39)
are the solitary wave amplitude and “velocity” respectively. The solution (38) depends on two
parameters λ1 and λ3 whose adiabatic slow evolution is governed by the reduced modulation
system (34). It is important that the second equation in this system is decoupled from
the first one. Hence, evolution of the pedestal −λ3 on which the solitary wave rides, can
be found from the solution of this dispersionless equation by the method of characteristics.
When λ3(X, T ) is known, evolution of the parameter λ1 can be found from the solution of
the first equation (34). As a result, we arrive at a complete description of adiabatic slow
evolution of the solitary wave parameters taking account of its interaction with the (given)
pedestal.
However, it is important to note here that while this description of the adiabatic evolution
of a solitary wave is complete as far as the solitary wave itself is concerned, it fails to describe
the evolution of a trailing shelf, which is needed to conserve total “mass” (see, for instance,
Johnson 1973b, Grimshaw 1979 or Grimshaw 2006). This trailing shelf has a very small
amplitude, but a very large length scale, and hence can carry the same order of “mass” as
the solitary wave. But note that the “momentum” of the trailing shelf is much smaller than
that of the solitary wave, whose adiabatic deformation is in fact governed to leading order by
conservation of “momentum”, or more precisely, by conservation of wave action flux (strictly
speaking, conservation only in the absence of friction).
The situation simplifies if the solitary wave propagates into a region of still water so
that there is no pedestal ahead of the wave, that is λ3 = 0 in X > τ(T ). But then, since
λ3 = 0 is an exact solution of the degenerate Whitham system (34) for this solitary wave
configuration, we can put λ3 = 0 both in the solitary wave solution,
U(X, T ) = −2λ1sech2[
−λ1 (X − VsT )], Vs = −4λ1, (40)
and in equation (34) for the parameter λ1 to obtain,
− 4λ1
Fλ1 +
Gλ21 , (41)
As we see, the solitary wave moves with the instant velocity
= −4λ1, (42)
and the parameter λ1 changes with T along the solitary wave trajectory according to the
ordinary differential equation
F (T )λ1 +
G(T )λ21. (43)
It can be shown that equation (43) is consistent with the equation for the solitary wave half-
width γ =
−λ1 obtained by the traditional perturbation approach (see Grimshaw (1979)
for instance).
Next, we re-write equation (43) in terms the original independent x-variable. For that,
we find from (6), that
dT = (h1/2/6g3/2)dx (44)
and F = −27
)3/2 dh
, G = 4CD
. (45)
Then substituting these expressions into (43) yields the equation
= −31
λ21 (46)
which can be easily integrated to give
−C0 −
, (47)
where C0 is an integration constant and x = 0 is a reference point where h = h0. According
to (40), U0 = −2λ1 is the amplitude of the soliton expressed in terms of variable U(X, T ).
Returning to the original surface displacement A(x, t) by means of (6) and denoting C0 =
4/(3ga0h0), we find the dependence of the surface elevation soliton amplitude a = (2h
2/3g)U0
on x in the form
a = a0
CDa0h0
, (48)
where a0 is the solitary wave amplitude at x = 0. We note that for CD = 0 this reduces to
the classical Boussinesq (1872) result a ∼ h−1, while for h = h0 it reduces to the well-known
algebraic decay law a ∼ 1/(1 + constant x) due to Chezy friction. Miles (1983a,b) obtained
this expression for a linear depth variation, although we note that there is a factor of 2
difference from (48) (in Miles (1983a,b) the factor 16CD/15 is 8CD/15). The trajectory of
the soliton can be now found from (42) and (47):
− t =
dx′h−5/2(x′)
CDa0h0
h3(x)
. (49)
This expression determines implicitly the dependence of x on t along the solitary wave path
and provides the desired equation for the multiple characteristic of the modulation system
for the case m = 1.
It is instructive to derive an explicit expression for the solitary wave speed by computing
the derivative dx/dt from (49), or more simply, directly from (37),
1− a/2h
. (50)
The formula (50) yields the restriction for the relative amplitude γ = a/h < 2 which is
clearly beyond the applicability of the KdV approximation (wave breaking occurs already
at γ = 0.7 (see Whitham 1974)). In the frictionless case (CD = 0) equation (48) gives
a/h = a0h0/h
2, and so the expression (50) for the speed must fail as h → 0. It is interesting
to note that this failure of the KdV model as h → 0 due to appearance of infinite (and
further negative!) solitary wave speeds is not apparent from the expression (48) for the
solitary wave amplitude, and the implication is that the model cannot be continued as
h → 0. Curiously this restriction of the KdV model seems never to have been noticed before
in spite of numerous works on this subject. Note that taking account of bottom friction
leads to a more complicated formula for the solitary wave speed as a function of h but the
qualitative result remains the same.
It is straightforward to show from (46) or (48) that
= −hx
CDa0h0
CDa0h0
. (51)
It follows immediately that for a wave advancing into increasing depth (hx > 0), the ampli-
tude decreases due to a combination of increasing depth and bottom friction. However, for
a wave advancing into decreasing depth, there is a tendency to increase the amplitude due
to the depth decrease, but to decrease the amplitude due to bottom friction. Hence whether
or not the amplitude increases is determined by which of these effects is larger, and this in
turn is determined by the slope, the depth, and the consolidated drag parameter CDa0/h0.
To illustrate, let us consider the bottom topography in the form
h(x) = h1−α0 (h0 − δx)α , α > 0 , (52)
which satisfies the condition h(0) = h0; the parameter δ characterizes the slope of the bot-
tom. In this case the formula (48) becomes
a = a0
δ(3α− 1)h0
)(3α−1)/α
if α 6= 1/3. One can see now that if α < 1/3, then the bottom friction term is relatively
unimportant due to the smallness of CD. Of course, for this case we again recover the
Boussinesq result, now slightly modified,
a ≈ a0
δ(1− 3α)h20
, 0 < α <
, h ≪ h0. (54)
Of course, this result is impractical in the KdV context as the KdV approximation used here
requires the ratio a/h to remain small.
If α > 1/3 now obtain asymptotic formula
15(3α− 1)δ
, h ≪ h0 , (55)
which is independent of the initial amplitude a0. This expression is consistent with the small-
amplitude KdV approximation as long as (3α− 1)δ/CD is order unity. Simple inspection of
(55) shows that the solitary wave amplitude
• increases as h → 0 if 1
< α < 1
• is constant as h → 0 if α = 1
• decreases as h → 0 if α > 1
Thus for 1/3 < α < 1/2, as for the case α < 1/3, the amplitude will increase as the depth
decreases, in spite of the presence of (sufficiently small) friction. However, for α > 1/3, even
although there is usually some initial growth in the amplitude, eventually even small bottom
friction will take effect and the amplitude decreases to zero. We note that if α = 1/3 then
the integral
h−3dx in (48) diverges logarithmically as h → 0, which just slightly modifies
the result (55) for h ≪ h0 and implies growth of the amplitude ∝ ln h/h as h → 0.
Of particular interest is the case α = 1. In that case formula (53) becomes
a = a0
. (56)
and a ≈ 15
h , h ≪ h0 (57)
These expressions (56, 57) were obtained by Miles (1983a,b) using wave energy conservation
(as above, note, however, that in Miles (1983a,b) the numerical coefficient is 15/4 rather
than 15/8). Thus, these results obtained from the Whitham theory are indeed consistent, at
the leading order, with the traditional perturbation approach for a slowly-varying solitary
wave.
5 Adiabatic deformation of a cnoidal wave
Next we consider a modulated cnoidal wave (10) in the special case when the modulation does
not depend on X . While this case is, strictly speaking, impractical as it assumes there is an
infinitely long wavetrain, it can nevertheless provide some useful insights into the qualitative
effects of gradual slope and friction on undular bores which are locally represented as cnoidal
waves. In the absence of friction, the slow dependence of the cnoidal wave parameters on T
was obtained by Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979) (see also Grimshaw
2006), assuming that the surface displacement had a zero mean (i.e. 〈U〉 = 0), while,
the effects of friction were taken into account by Miles (1983b) using the same zero-mean
displacement assumption. However, this assumption is inconsistent with our aim to study
undular bores where the value of 〈U〉 is essentially nonzero. Hence, we need to develop a
more general theory enabling us to take into account variations in all the parameters in the
cnoidal wave. Such a general setting is provided by the modulation system (27).
Thus we consider the case when the Riemann variables in (27) do not depend on the
variable X so that the general Whitham equations become ordinary differential equations
in T , which can be conveniently reformulated in terms of the original spatial x-coordinate
using the relationship (44):
, i = 1, 2, 3, (58)
where all variables are defined above in section 3 (see 28, 29, 30). This system can be readily
solved numerically. But it is instructive, however, to first indicate some general properties
of the solution.
First, the solution to the system (58) must have the property of conservation of “wave-
length” L (or “wavenumber” k=2π/L)
2K(m)√
λ3 − λ1
= constant (59)
Indeed, the wave conservation law (22) in absence of X-dependence assumes the form
= 0 , (60)
which yields (59). Thus the system of three equations (58) can be reduced to two equations.
Next, applying Whitham averaging directly to (7) yields
P̃ , M = 〈U〉 , P̃ = 〈|U |U〉 . (61)
P − 4CD
Q̃ , P = 〈U2〉 , Q̃ = 〈|U |3〉 . (62)
The equation set (59), (61), (62) comprise a closed modulation system for three independent
modulation parameters, say M , P̃ and m. While this system is not as convenient for further
analysis as the system (27) in Riemann variables, it does not have a restriction U > 0 inherent
in (27), and allows for some straightforward inferences regarding the possible existence of
modulation solutions with zero mean elevation, that is with M = 0. Indeed, one can see
that the solution with the zero mean is actually not generally permissible when CD 6= 0, a
situation overlooked in Miles (1983b). Indeed, M = 0 immediately then implies that P̃ = 0
by (61). But then due to (59) we have all three modulation parameters fixed which is clearly
inconsistent with the remaining equation (62) (except for the trivial case M = 0, P = 0,
Q̃ = 0). However, in the absence of friction, when CD = 0, equation (61) uncouples and
permits a nontrivial solution with a zero mean. In general, when CD = 0 equations (61),
(62) can be easily integrated to give
d = Mh9/4 = constant; σ = Ph9/2 = constant. (63)
Then, using (18, 19, 59) one readily gets the formula for the variation of the modulus m,
and hence of all the other wave parameters, as a function of h
K2[2(2−m)EK − 3E2 − (1−m)K2] =
(σ − d2)L4
. (64)
200
400
600
800
0.
2
0.
4
0.
6
0.
8
C = 0
C = 0.01
Figure 2: Dependence of the modulus m on the physical space coordinate x in the cases
without and with bottom friction in the X-independent modulation solution.
Formula (64) generalises to the case M 6= 0 (i.e. d 6= 0) the expressions of Ostrovsky &
Pelinovsky (1970, 1975), Miles (1979) and Grimshaw (2006) (note that in Grimshaw (2006)
the zero mean restriction in actually not necessary). We note here that, again with CD = 0,
equation (5) implies conservation of 〈B〉 and 〈B2〉 (the averaged wave action flux), which,
together with (59), also yield (64).
The physical frequency Ω and wavenumber κ in the modulated periodic wave under study
are given by the formula (37), and we recall here that k = 2π/L is constant (see (59)). As
discussed before at the end of Section 3 we must require that the phase speed stays positive as
the wave evolves, and here that requires that the physical wavenumber κ > 0. Since a/h (and
hence hV/6g) is supposed to be small within the range of applicability of the KdV equation
(2) the expression (37) implies the behaviour κ ≃ Ω/
gh which of course agrees with the
well known result for linear waves on a sloping beach (see Johnson 1997 for instance). This
effect will be slightly attenuated for the nonlinear cnoidal wave, since V h/6g > 0, but the
overall effect will be a “squeezing” of the cnoidal wave, a result important for our further
study of undular bores. Next we study numerically the combined effect of slope and friction
on a cnoidal wave.
As we have shown, in the presence of Chezy friction M 6= 0, and we have also assumed
that U > 0, which is necessary when we come to study undular bores. Now we use the
stationary modulation system (58) in Riemann variables, which was derived using this as-
sumption. We solve the coupled ordinary differential equation system (58) for the case of a
linear slope
h(x) = h0 − δx (65)
with h0 = 10, δ = 0.01, and with the initial conditions
λ1 = −0.441, λ2 = 0.147, λ3 = 0.294 at x = 0, (66)
which corresponds to a nearly harmonic wave with m = 0.2, a/h0 = 0.2, 〈A〉/h0 ≈ 0.3
at x = 0 (see (35)). Also we note that for the chosen parameters we have V = 0, so at
x = 0 we have κ = Ω/
gh0 as in linear theory. It is instructive to compare solutions with
(CD = 0.01) and without (CD = 0) friction. In Fig. 2 the dependence of the modulus m
100 200 300 400 500
C = 0
C = 0.01
100 200 300 400 500
h <A>
1/4 C = 0
C = 0.01
Figure 3: Left: Dependence of the mean value 〈A〉 in theX-independent modulation solution
on the physical space coordinate x without (dashed line) and with (solid line) bottom friction;
Right: Same but multiplied by the Green’s law factor, h1/4
100
200
300
400
500
1.
4
1.
6
1.
8
2.
2
2.
4
C = 0
C = 0.01
Figure 4: Dependence of the surface elevation amplitude a on the space coordinate x. Dashed
line corresponds to the frictionless case and solid line to the case with bottom friction.
on x is shown for both cases. We see that for the frictionless case m → 1 with decrease
of depth, i.e. the wave crests assume the shape of solitary waves when one approaches the
shoreline. When CD 6= 0 the modulus also grows with decrease of depth but never reaches
unity. The dependence on x of the mean surface elevation 〈A〉 for the cases without and
with friction is shown in Fig. 3. We have checked that the “wavelength” L (59) is constant
for both solutions. Also, one can see from Fig. 3 (right) that the value h1/4〈A〉 ∝ d is indeed
conserved in the frictionless case but is not constant if friction is present (the same holds
true for the value h1/2〈A2〉 ∝ σ but we do not present the graph here). Finally, in Fig. 4 the
dependence of the physical elevation wave amplitude a on the spatial coordinate x is shown.
One can see that the amplitude adiabatically grows with distance in the frictionless case
due to the effect of the slope (without friction) but, not unexpectedly, gradually decreases
in the case when bottom friction is present, where the decrease for these parameter settings
is comparable in magnitude to the effect of the slope. In both cases the main qualitative
changes occur in the wave shape and the wavelength.
Overall, we can infer from these results that the main local effect of a slope and bottom
friction on a cnoidal wave, along with the adiabatic amplitude variations, is twofold: a wave
with a m < 1 at x = 0 tends to transform into a sequence of solitary waves as x decreases,
and at the same time the distance between subsequent wave crests tends to decrease. This
is in sharp contrast with the behaviour of modulated cnoidal waves in problems described
by the unperturbed KdV equation, where growth of the modulus m is accompanied by an
increase of the distance between the wave crests. Generally, in the study of behaviour of
unsteady undular bores in the presence of a slope and bottom friction we will have to deal
with the combination of these two opposite tendencies.
6 Undular bore propagation over variable topography
with bottom friction
6.1 Gurevich-Pitaevskii problem for flat-bottom zero-friction case
We now turn to the problem (b) outlined in Section 2. We study the evolution of an undular
bore developing from an initial surface elevation jump ∆ > 0, located at some point x0 < 0.
As discussed below, the undular bore will expand with time so that at some t = t0 its lead
solitary wave enters the gradual slope region, which begins at x = 0 (see Fig. 1b). We assume
that for x < 0 one has h = h0 = constant and CD ≡ 0. We shall first present a formulation
of the Gurevich-Pitaevskii problem for the perturbation-free KdV equation and reproduce
the well-known similarity modulation solution describing the evolution of the undular bore
until the moment it enters the slope. We emphasize that, although this formulation and,
especially, this similarity solution are known very well and have been used by many authors,
some of the inferences important for the present application to fluid dynamics have not been
widely appreciated, as far as we can discern. Pertinent to our main objective in this paper,
we undertake a detailed study of the characteristics of the Whitham modulation system in
the vicinity of the leading edge of the undular bore solution, and show that the boundary con-
ditions of Gurevich-Pitaevskii type permit only two possible characteristics configurations,
implying two qualitatively different types of the leading solitary wave behaviour. Next, we
shall show how this Gurevich-Pitaevskii formulation of the problem applies to the perturbed
modulation system in the form (27) and finally we will study the effects of the perturbation
on the modulations in the vicinity of the leading edge of the undular bore.
In the case of a flat, frictionless bottom the original equation (1) becomes the constant-
coefficient KdV equation which can be cast into the standard form
ηζ + 6ηηξ + ηξξξ = 0 (67)
by introducing the new variables
A , ξ =
(x+ x0 −
gh0t) , ζ =
t , (68)
where x0 < 0 is an arbitrary constant. In the Gurevich-Pitaevskii (GP) approach, one
considers a large-scale initial disturbance η(ξ, 0) = f(ξ), in the form of a decreasing profile,
f ′(ξ) < 0 (e.g. a smooth step: f(ξ) → 0 as ξ → +∞; f(ξ) → η0 > 0 as ξ → −∞), whose
initial evolution until some critical (breaking) time ζb can be described by the dispersionless
limit of the KdV equation, i.e. by the Hopf equation,
ζ < ζb : η ≈ r(ξ, ζ), rζ + 6rrξ = 0 , r(ξ, 0) = f(ξ) . (69)
The evolution (69) leads to wave-breaking of the r(ξ)-profile at some ζ = ζb, with the
consequence that the dispersive term in the KdV equation then comes into play, and an
undular bore forms, which can be locally represented as a single-phase travelling wave. This
travelling wave is modulated in such a way that it acquires the form of a solitary wave at the
leading edge ξ = ξ+(ζ) and gradually degenerates, via the nonlinear cnoidal-wave regime, to
a linear wave packet at the trailing edge ξ = ξ−(ζ). It is important that this undular bore
is essentially unsteady, i.e. the region ξ−(ζ) < ξ < ξ+(ζ) expands with time ζ .
The single-phase travelling wave solution of the KdV equation (67) has the form (cf.
(10))
η(ξ, ζ) = r3 − r1 − r2 − 2(r3 − r2)sn2(
r3 − r1θ,m) (70)
θ = ξ + 2(r1 + r2 + r3)ζ , m =
r3 − r2
r3 − r1
. (71)
The parameters r1 ≤ r2 ≤ r3 ≤ 0 in the undular bore are slowly varying functions of ξ, ζ ,
whose evolution is governed by the Whitham equations
+ vj(r1, r2, r3)
= 0 , j = 1, 2, 3. (72)
The characteristic velocities in (72) are given by (31). We stress that, although analytical
expressions (70) and (10) (as well as (72) and the homogeneous version of (27)) are identical,
they are written for completely different sets of variables, both dependent and independent.
The Riemann invariants rj(ξ, ζ) are subject to special matching conditions at the free
boundaries, ξ = ξ±(ζ) defined by the conditions m = 0 (trailing edge) and m = 1 (leading
edge), formulated in Gurevich and Pitaevskii (1974) (see also Kamchatnov (2000) or El
(2005) for a detailed description).
At the trailing (harmonic) edge, where the wave amplitude a = 2(r3 − r2) vanishes and
m = 0, one has
ξ = ξ−(ζ) : r2 = r3 , −r1 = r . (73)
At the leading (soliton) edge, where m = 1 one has
ξ = ξ+(ζ) : r2 = r1 , −r3 = r . (74)
In both (73) and (74), r(ξ, ζ) is the solution of the Hopf equation (69).
The curves ξ = ξ±(ζ) are defined for the solution of the GP problem (72), (73), (74) by
the ordinary differential equations
= v−(ξ−, ζ) ,
= v+(ξ+, ζ) , (75)
where v± are calculated as the values of double characteristic velocities of the modulation
system at the undular bore edges,
v− = v2(r1, r3, r3)|ξ=ξ−(ζ) = v3(r1, r3, r3)|ξ=ξ−(ζ), (76)
v+ = v2(r1, r1, r3)|ξ=ξ+(ζ) = v1(r1, r1, r3)|ξ=ξ+(ζ) (77)
These equations (75) essentially represent kinematic boundary conditions for the undular
bore (see El 2005). Indeed, the double characteristic velocity v2(r1, r3, r3) = v3(r1, r3, r3) can
be shown to coincide with the linear group velocity of the small-amplitude KdV wavepacket
while the double characteristic velocity v2(r1, r1, r3) = v1(r1, r1, r3) is the soliton speed.
One might infer from this GP formulation of the problem that, since the leading edge of
the undular bore specified by (75), (77) is a characteristic of the modulation system, then
the value of the double Riemann invariant r+ ≡ r2 = r1 is constant. Then, on considering an
undular bore propagating into still water, where r = 0, one would obtain from the matching
condition (74) at the leading edge that r3|ξ=ξ+ = 0 and thus, the amplitude of the lead solitary
wave a+ = 2(r3−r1)|ξ=ξ+ = −r+ would always be constant as well. However, this contradicts
the general physical reasoning that the amplitude of the lead solitary wave should be allowed
to change in the case of general initial data. The apparent contradiction is resolved by noting
that the leading edge specified by (75), (77) can be an envelope of the characteristic family,
i.e. a caustic, rather than necessarily a regular characteristic, and hence there is no necessity
for the double Riemann invariant r+ to be constant along the curve ξ = ξ+(ζ) in general case.
On the other hand, since the leading edge is defined by the condition m = 1, the wave form
at the leading edge will coincide with the spatial profile of the standard KdV soliton. Thus
we arrive at the conclusion that, in general, the amplitude of the leading KdV solitary wave
will vary, even in the absence of the perturbation terms. Of course, in the unperturbed KdV
equation, such varying solitary waves cannot not exist on their own, and require the presence
of the rest of the undular bore. We also stress that these variations of the leading solitary
wave in the undular bore, as described here, have a completely different physical nature to
the variations of the parameters of an individual solitary wave due to small perturbations as
described in Section 4. They are caused by nonlinear wave interactions within the undular
bore rather than by a local adiabatic response of the solitary wave to a perturbation induced
by topography and friction. Importantly for our study, however, it will transpire that the
action of these same perturbation terms on the undular bore can lead to both a local and a
nonlocal response of the leading solitary wave.
6.2 Undular bore developing from an initial jump
Next we consider the simplest solution of the modulation system, which describes an undular
bore developing from an initial discontinuity placed at the point x = −x0. In (η; ξ, ζ) -
variables we have the initial conditions
η(ξ, 0) = ∆ for ξ < 0 ; η(ξ, 0) = 0 for ξ > 0 , (78)
where ∆ > 0 is a constant. Then, on using (69), the initial conditions (78) are readily
translated into the free-boundary matching conditions (73), (74) for the Riemann invariants.
Because of the absence of a length scale in this problem, the corresponding solution of the
modulation system must depend on the self-similar variable τ = ξ/ζ alone, which reduces
the modulation system to the ordinary differential equations
(vi − τ)
= 0 , i = 1, 2, 3. (79)
-4
0
-3
0
-2
0
-1
0
10
20
-0.8
-0.6
-0.4
-0.2
-30 -20 -10 10
η(ξ, ζ = 5)
Figure 5: Left: Riemann invariants behaviour in the similarity modulation solution for the
flat-bottom zero-friction case ; Right: corresponding undular bore profile η(ξ).
The boundary conditions for (79) follow from the matching conditions (73), (74) using the
initial condition (78):
τ = τ− : r2 = r3 , r1 = −∆
τ = τ+ : r2 = r1 , r3 = 0 .
where τ± are self-similar coordinates (speeds) of the leading and trailing edges, ξ± = τ±ζ .
Taking into account the inequality r1 ≤ r2 ≤ r3 one obtains the well-known modulation
solution of Gurevich and Pitaevskii (1974) (see also Fornberg and Whitham 1978) in the
r1 = −∆ , r3 = 0 , r2 = −m∆ , (81)
= v2(−∆,−m∆, 0) = 2∆[(1 +m)−
2m(1−m)K(m)
E(m)− (1−m)K(m)
] . (82)
This modulation solution (81), (82) (see Fig. 5a) represents the replacement, due to averag-
ing over the oscillations, of the unphysical formal three-valued solution of the dispersionless
KdV equation (i.e. of the Hopf equation) which would have taken place in the absence of
the dispersive regularisation by the undular bore. We see that (82) describes an expansion
fan in the characteristic (ξ, ζ)-plane and thus is a global solution. Substituting (81), (82)
into the travelling wave solution (70) one obtains the asymptotic wave form of the undular
bore (see Fig. 5b), which then can be readily represented in terms of the original physical
variables using the relationships (68).
The equations of the trailing and leading edges of the undular bore are determined from
(82) by putting m = 0 and m = 1 respectively
= τ− = v2(−∆, 0, 0) = −6∆ ,
= τ+ = v2(−∆,−∆, 0) = 4∆ . (83)
The leading solitary wave amplitude is η0 = 2(r3−r1) = 2∆, which is exactly twice the height
of the initial jump. This corresponds to the amplitude of the surface elevation a = 3h0∆ (see
(68)). Note that, to get the leading solitary wave of the same initial amplitude a0 as for the
separate solitary wave considered in Section 4, one should use the jump value ∆0 = a0/3h0,
which of course is just 2∆̃, where ∆̃ = 3h0∆/2 is the initial discontinuity in the surface
elevation.
6.3 Structure of the undular bore front
We are especially interested in the behaviour of the modulation solution (81), (82) in the
vicinity of the leading edge ξ = ξ+(ζ). This behaviour is essentially determined by the
manner in which the pair of characteristics corresponding to the velocities v2 and v1 merge
into a multiple eigenvalue v+ of the modulation system at ξ = ξ+(ζ).
First, one can readily infer from the modulation solution (81), (82) that the phase velocity
c = −2(r1 + r2 + r3) = 2∆(1 +m) > v2(−∆,−m, 0) for m < 1 and c = v2 for m = 1. Thus,
any individual wave crest generated at the trailing edge of the undular bore moves towards
the leading edge, i.e. for any crest m → 1 as ζ → ∞. Thus, for any particular wave crest,
except for the very first one, the solitary wave ‘status’ is achieved only asymptotically as
ζ → ∞.
Without loss of generality we assume in this section that ∆ = 1 in (81), (82). First, as
we have already mentioned, the characteristic family Γ2 : dξ/dζ = v2 is an expansion fan in
the ξ, ζ - plane,
Γ2 : ξ = C2ζ , (84)
parameterised by a constant C2, −6 ≤ C2 ≤ 4 . Next, in (82) we make an asymptotic
expansion of v2(−1,−m, 0) for small (1−m) ≪ 1, to get
2(1−m) ln(16/(1−m)) ≃ τ+ − ξ/ζ (85)
or, with logarithmic accuracy,
(τ+ − ξ/ζ) ≪ 1 : 1−m ≃
τ+ − ξ/ζ
2 ln[1/(τ+ − ξ/ζ)]
. (86)
Next, expanding v1(−1,−m, 0) for (1 − m) ≪ 1 and using (86) we get the asymptotic
equation for the characteristics family Γ1,
= v1 = τ
+ + (τ+ − ξ/ζ) +O(1−m) , (87)
which is readily integrated to leading order to give
Γ1 : ξ ≃ τ+ζ −
, (88)
where C1 ≥ 0 is an arbitrary constant ‘labeling’ the characteristics; C1 = 0 corresponds to
the leading edge of the undular bore. This asymptotic formula (88) is valid as long as ζ ≫ 1.
The behaviour of the characteristics belonging to the families Γ1 and Γ2 near the leading
edge is shown in Fig. 6a.
Next, expanding the equation for the third characteristic family, Γ3: dξ/dζ = v3(−1,−m, 0)
for (1−m) ≪ 1, we get on using (86)
τ+ − ξ/ζ
ln(1/(τ+ − ξ/ζ))
+O(τ+ − ξ/ζ) . (89)
Figure 6: Characteristics behaviour for the similarity modulation solution near the leading
edge ξ+(ζ): (a) families Γ1: dξ/dζ = v1 and Γ2 : ξ = C2ζ , (b) family Γ3: dξ/dζ = v3.
Integrating (89) we obtain to first order
Γ3 : ξ ≃ C3 − g(ζ) , (90)
where g(ζ) =
τ+ζ − C3
ln |τ+ζ − C3| − ln ζ
dζ , g(C3/τ
+) = 0 , (91)
C3 being an arbitrary constant. The asymptotic formula (90) is valid as long as g(ζ)/C3 ≪
1. Since the characteristics Γ3 intersect the leading edge ξ = τ
+ζ we must indicate their
behaviour outside the undular bore. It follows from the matching condition (74) and the
limiting structure (34) of the characteristic velocities of the Whitham system, that the
characteristics from the family Γ3 match with the Hopf equation characteristics dξ/dζ = 6r
carrying the value of the Riemann invariant r = 0 corresponding to still water upstream the
undular bore. Therefore, the sought external characteristics are simply vertical lines ξ = C3.
The qualitative behaviour of the characteristics from the family Γ3 is shown in Fig. 6b.
It is clear from the asymptotic behaviour of the characteristics that the edge characteristic
ξ = τ+ζ corresponding to the motion of the leading solitary wave intersects only with
characteristics of the family Γ3 carrying the Riemann invariant value r3 = 0 into the undular
bore domain. Since, according to the matching condition (80), r3 ≡ 0 everywhere along the
edge characteristic one can infer that the leading solitary wave motion is completely specified
by its amplitude at ζ = 0. Indeed, in this case, the leading edge represents a genuine multiple
characteristic of the modulation system, along which the Riemann invariant r+ = r2 = r1 is
a constant. Given the constant value of r1 = −1 for the solution (82), one infers that the
amplitude of the lead soliton of the self-similar undular bore, η0 = 2(r3 − r+) = 2 is also a
constant value. Thus, in the undular bore evolving from an initial jump, the leading solitary
wave represents an independent soliton of the KdV equation. Of course, this fact follows
directly from the modulation solution (82) but now we have established its meaning in the
context of the characteristics, which will play an important role below.
Next we discuss the structure of the undular bore front in the case when the initial
profile η(ξ, 0) is not a simple jump discontinuity, and instead has the form of a monotonically
decreasing function, for instance, (−ξ)1/2 when ξ ≤ 0 and η(ξ, 0) = 0 for ξ > 0. In that case,
the modulation solution for the undular bore no longer possesses x/t-similarity as in the
Figure 7: a) Leading edge ξ+(ζ) of non-self-similar undular bore as an envelope of pairwise
merging characteristics from the families dξ/dζ = v1 and dξ/dζ = v2; b) behaviour of the
Riemann invariants in non-self-similar modulation solution with r3 ≡ 0.
jump resolution case and, as a result, the speed (and therefore, the amplitude) of the lead
solitary wave is not constant. For instance, for the afore-mentioned square-root initial profile
the amplitude of the lead solitary wave grows as ζ2 (see Gurevich, Krylov and Mazur 1989,
or Kamchatnov 2000). Clearly, such an amplitude variation is impossible if the leading edge
ξ+(ζ) was a regular characteristic carrying a constant value of the Riemann invariant r+. As
discussed above, however, the GP matching conditions (73) -(77) admit another possibility;
the leading edge curve is the envelope of the characteristic families Γ1: dξ/dζ = v1 and Γ2:
dξ/dζ = v2 merging when m = 1. This configuration is shown in Fig. 7a. In this case, the
behaviour of the modulus m in the vicinity of the leading edge is given by the asymptotic
formula found in Gurevich & Pitaevskii (1974):
(1−m)2
(r+)2
(ξ+ − ξ) (92)
where the function r+(ζ) 6= constant is assumed to be known. Another specific feature of
this (general) configuration is that dr1,2/dξ → ±∞ as ξ → ξ+ (see Fig. 7b - also found in
Gurevich & Pitaevskii 1974, see also Kamchatnov 2000), which is in drastic contrast with
similarity solution (see Fig. 6a). This particular difference was discussed in relation with
undular bores in the KdV-Burgers equation in Gurevich and Pitaevskii (1987).
In summary, we see from (92) that the structure of the modulation solution in the vicin-
ity of the leading edge of an undular bore defined as a characteristic envelope is qualitatively
different compared to that for the similarity case (see (85)). The more general (but qual-
itatively similar to (92)) asymptotic formula which takes into account small perturbations
due to a variable topography and bottom friction will be derived later. At the moment,
it is important for us that in this configuration, when the leading edge is a characteristic
envelope rather than just a characteristic, the value r+, and thus, the leading solitary wave
amplitude are allowed to vary.
The analysis of the corresponding modulation solution in Gurevich, Krylov and Mazur
(1989) showed that, while in the case of an initial jump the wave crests generated at the
trailing edge reach the leading edge (and therefore, transform into solitary waves) only
asymptotically as t → ∞, for the more general case of decreasing initial data each wave
crest generated at the trailing edge reaches the leading edge in finite time and replaces
(overtakes) the existing leading solitary wave. This process is manifested as a continuous
amplitude growth of the (apparent) leading solitary wave. As in classical soliton theory,
an alternative explanation of the leading solitary wave amplitude growth can be made in
terms of the momentum exchange between the “instantaneous” leading solitary wave and
solitary waves of greater amplitude coming from the left. Indeed, as the rigorous analysis of
Lax, Levermore and Venakides showed (see Lax, Levermore and Venakides (1994) and the
references therein), the whole modulated structure of the undular bore can be asymptotically
described in terms of the interactions of a large number of KdV solitons initially ‘packed’
into a non-oscillating large-scale initial profile.
This latter interpretation is especially instructive for our purposes. Our point is that
the specific cause of the enhanced soliton interactions resulting in amplitude growth at the
leading edge is not essential; it can be large-scale spatial variations of the initial profile as
just described, but it could also equally well be an effect of small perturbations in the KdV
equation itself. Indeed, in the weakly perturbed KdV equation, the local wave structure
of the undular bore must be described to leading order by the periodic solution (70) of
the unperturbed KdV equation, so if one assumes the GP boundary conditions analogous to
(73) – (77) for the perturbed modulation system (27), one invariably will have to deal with
one of the two possible types of the characteristics behaviour (shown in Figs. 7a and 8a)
in the vicinity of the leading edge of the undular bore, because this qualitative behaviour
is determined only by the structure of the GP boundary conditions and by the associated
asymptotic structure of the characteristic velocities of the Whitham system for (1−m) ≪ 1,
which are the same for both unperturbed and perturbed modulation systems. Next, we will
show that, by using the knowledge of this qualitative behaviour of the characteristics, one
is able to construct the asymptotic modulation solution for the undular bore front in the
presence of variable topography and bottom friction even if the full solution of the perturbed
modulation system is not available.
6.4 Gurevich-Pitaevskii problem for perturbed modulation sys-
We investigate now how the GP matching problem applies to the perturbed modulation
system (27). As in the original GP problem, we postulate the natural physical requirement
that the mean value 〈U〉 is continuous across the undular bore edges, which represent free
boundaries and are defined by the conditions m = 0 (trailing edge X = X−(T )) and m = 1
(leading edge X = X+(T )). Also, we consider propagation of the undular bore into still
water, hence 〈U〉|X=X+(T ) = 0. Now, using the explicit expression (18) for 〈U〉 in terms of
complete elliptic integrals and calculating its limits as m → 0 and m → 1 one has
X = X−(T ) : λ2 = λ3 , 〈U〉 = −λ1 = u ,
X = X+(T ) : λ2 = λ1 , 〈U〉 = −λ3 = 0 ,
where u(X, T ) is solution of the dispersionless perturbed KdV equation (7), i.e.
uT + 6uuX = F (T )u−G(T )u2, (94)
with the boundary conditions
∆0 if τ < τ0; u
= 0 if τ > τ0 , (95)
where τ0 = −x0/
gh0. The boundary conditions (95) correspond to a discontinuous initial
surface elevation A(x, t) at x = −x0, obtained by using transformations (3) and (6) where
one sets t = 0. As earlier, ∆0 = a0/(3h0) is the value of the discontinuity in A, chosen in
such a way that the amplitude of the lead solitary wave in the undular bore was exactly a0
in the flat-bottom zero-friction region (see Section 6.2).
This free-boundary matching problem is then complemented by the kinematic conditions
explicitly defining the boundaries X = X±(T ). These are formulated using the multiple
characteristic directions of the perturbed modulation system (27) in the limits as m → 0
and m → 1 (cf. (75) - (77)),
= V −(X−, T ) ,
= V +(X+, T ) , (96)
where V − = v2(u, λ
−, λ−) = v3(u, λ
−, λ−), (97)
V + = v2(λ
+, λ+, 0) = v1(λ
+, λ+, 0) , (98)
and λ− = λ2(X
−, T ) = λ3(X
−, T ) , λ+ = λ2(X
+, T ) = λ1(X
+, T ). (99)
Thus, for the perturbed KdV equation the leading and trailing edges of the undular bore are
defined mathematically in the same way as for the unperturbed one, albeit for a different
set of variables.
6.5 Deformation of the undular bore front due to variable topog-
raphy and bottom friction
Finally we study the effects of gradual slope and bottom friction on the leading front of the
self-similar expanding undular bore described in Sections 6.2, 6.3. The result will essentially
depend on the relative values of the small parameters appearing in the problem. We note
that in general there are three distinct relevant small parameters,
≪ 1 , δ = max(hx) ≪ 1, CD ≪ 1 (100)
The first small parameter is determined by the ratio of the equilibrium depth in the flat
bottom region, to the distance from the beginning of the slope region to the location of the
initial jump discontinuity in the surface displacement. This measures the typical relative
spatial variations of the modulation parameters in the undular bore when it reaches the
beginning of the slope. The second and third parameters are contained in the KdV equation
(1) itself and measure the values of the slope and bottom friction respectively. In terms of
the transformed variables appearing in (7), |F (T )| ∼ δ, |G(T )| ∼ CD (see (8)). Generally
we assume δ ∼ CD (the possible orderings δ ≪ CD or CD ≪ δ can be then considered as
particular cases).
To obtain a quantitative description of the vicinity of the leading edge of the undular
bore we perform an expansion of the Whitham modulation system (27) for (1 − m) ≪ 1.
We first introduce the substitutions
λi(X, T ) = λ
+(T ) + li(X̃, T ) , vi = V
+ + v′i , ρi = ρ
+ + ρ′i, i = 1, 2. (101)
where X̃ = X+ −X , V + = −4λ+ , ρ+ =
F (T )λ+ +
G(T )(λ+)2. (102)
Since λ2 ≥ λ1, v2 ≥ v1 one always has l2 ≥ l1, v′2 ≥ v′1. Assuming X̃/X+ ≪ 1 ⇔ 1−m ≪ 1
and using that λ3 = 0 to leading order in the vicinity of the leading edge (see the matching
condition (93)), we have from asymptotic expansions of (28) – (31) as (1−m) ≪ 1
v′1 = M1(l2 − l1) ≡ −2
ln(16/(1−m))
1 + 1
(1−m) ln(16/(1−m))
(l2 − l1),
v′2 = M2(l2 − l1) ≡ −2
1− ln(16/(1−m))
(1−m) ln(16/(1−m))
(l2 − l1),
(103)
ρ′1 = N1(l2 − l1) ≡
1 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
− 3λ+
(l2 − l1)
ρ′2 = N2(l2 − l1) ≡
5 + ln
l2 − l1
−16λ+
2λ+ ln
l2 − l1
−16λ+
+ 13λ+
(l2 − l1).
(104)
Naturally, v′i and ρ
i vanish when l2 = l1. Now, substituting (101), (102) into the modulation
system (27) we obtain
− (V + + v′i)
= ρ+ + ρ′i, i = 1, 2. (105)
On using the kinematic condition (96) at the leading edge, this reduces to
− v′i
= ρ+ + ρ′i, i = 1, 2. (106)
There are two qualitatively different cases to consider:
(i) limX̃→0 |dli/dX̃| < ∞, i = 1, 2 (Fig. 8a)
(ii) limX̃→0 |dli/dX̃| = ∞, i = 1, 2 (Fig. 8b)
The case (i) implies that to leading order (106) reduces to
= ρ+ , (107)
which, together with the kinematic condition dX+/dT = −4λ+, defines the leading edge
curve X+(T ). One can observe that this system coincides with (43), (42) defining the
Figure 8: Riemann variables behaviour in the vicinity of the leading edge of the undular
bore propagating over gradual slope with bottom friction (a) Adiabatic variations of the
similarity GP regime, δ ≪ ǫ, CD ≪ ǫ; (b) General case, δ ∼ CD ∼ ǫ.
motion of a separate solitary wave over a gradual slope with bottom friction. Its integral
expressed in terms of original physical x, t-variables is given by (49). Therefore, in the case
(i) the lead solitary wave in the undular bore to leading order is not restrained by interactions
with the remaining part of the bore and behaves as a separate solitary wave. Physically this
case corresponds to adiabatic deformation of the similarity modulation solution (81), (82)
and implies the following small parameter ordering : δ ≪ ǫ, CD ≪ ǫ.
Next, we study the structure of this weakly perturbed similarity modulation solution in
the vicinity of the leading edge. The next leading order of the system (106) yields
− v′i
= ρ′i, i = 1, 2, (108)
that is
= −N1
= −N2
. (109)
Subtraction of one equation (109) from another with account of the relationship l2 − l1 ∼=
−λ+(1−m) leads consistently to leading order to the differential equation for 1−m
∂(1 −m)
F (T )
16G(T )
, (110)
This equation should be solved with the initial condition
1−m = 0 at X̃ = 0 . (111)
Elementary integration gives with the accuracy O(1−m) (cf. (85))
(1−m) ln 16
F (T )− 16
λ+G(T )
X+ −X
. (112)
This formula determines the dependence of the modulus m on T and X (as long as 1−m ≪
Now, we make use of the solution λ+ of equation (107) given by (47) with C0 =
4/(3ga0h0) (see (48)). Under supposition that the integral
h−3dx diverges as h → 0,
so that the turbulent bottom friction plays an essential role in the undular bore front be-
haviour (see Section 4 for a similar approximation for an isolated solitary wave), we obtain
for h ≪ h0
(1−m) ln
2 + 3h2
(X+ −X). (113)
At last, if the bottom topography is approximated by the dependence (52), we get with the
same accuracy
(1−m) ln 16
(3α− 1)δ
(X+ −X) , (114)
where α > 1/3. The second term in square brackets tends to zero as h → 0. However, the
region where it can be neglected may be very narrow because of smallness of the parameter
δ. We recall that in this formula X+ is given by (49) and X is defined by (3) in terms of
the original physical independent variables x and t.
Summarising, if the conditions δ, CD ≪ ǫ are satisfied, the lead solitary wave of the
undular bore behaves as an individual (noninteracting) solitary wave adiabatically varying
under small perturbation due to variable topography and bottom friction. The modulation
solution in the vicinity of the leading edge also varies adiabatically, however, its qualitative
structure considered in Section 6.4 (see Figs 5,6) remains unchanged.
In a sharp contrast with the described case of adiabatic deformation of an undular bore
front is case (ii) when the second term in the left-hand side of (106) contributes to the
leading order, i.e. to the motion of the leading edge itself. Namely, we have
= ρ+ + v′i
, i = 1, 2. (115)
Now dλ+/dT 6= ρ+ which means that the amplitude of the lead solitary wave a = −2λ+ varies
essentially differently compared to the case of an isolated solitary wave. Indeed, the term
ρ+ in the right-hand side of (115) is responsible for local adiabatic variations of the solitary
wave while the term v′i∂li/∂X̃ describes nonlocal parts of the variations associated with
the wave interactions within the undular bore. Using asymptotic formulae (103) implying
v′2 ≥ 0, v′1 ≤ 0, and the condition limX̃→0 |dl1,2/dX̃| = ∞ along with l2 ≥ l1, it is not
difficult to show that this nonlocal term is always nonnegative , i.e. the lead solitary wave
in the undular bore propagating over a gradual slope with bottom friction always moves
faster (and, therefore, has greater amplitude) than an isolated solitary wave of the same
initial amplitude in the beginning of the slope. Indeed, as we have shown in Section 5, the
presence of a slope and bottom friction always result in “squeezing” the cnoidal wave, hence
increasing momentum exchange between solitary waves in the vicinity of the leading edge
of the undular bore and acceleration of the lead solitary wave itself. The situation here is
qualitatively analogous to that described in Section 6.4 where the general global modulation
solution for the unperturbed KdV equation was discussed. Similarly to that case, the leading
edge now represents a characteristic envelope – a caustic (otherwise we are back in the case
(i) implying dλ+/dT = ρ+) (see Fig. 6a).
Unlike the case of adiabatic variations of the leading edge, determination of the function
λ+(T ) requires now knowledge of the full solution of the perturbed modulation system (27)
with the matching conditions (93). While the analytic methods to construct such a solution
for inhomogeneous quasilinear systems are not available presently, it is instructive to assume
that dλ+/dT − ρ+ is a known function of T and to study the structure of the solution in
close vicinity of the leading edge. With an account of the explicit form (103) of the velocity
corrections, equations (115) assume the form
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
, (116)
= −dλ
+/dT − ρ+
2(l2 − l1)
ln[16/(1−m)]
(1−m)
. (117)
Taking the difference of (116) and (117) we transform it to the form
∂(1 −m)
dλ+/dT − ρ+
(λ+)2
(1−m) ln[16/(1−m)]
. (118)
This equation can be readily integrated with the initial condition (111) to give
(1−m)2
2(dλ+/dT − ρ+)
(λ+)2
(X+ −X). (119)
This solution coincides with the asymptotic formula (92) for the behaviour of the modulus
in the vicinity of the leading edge of the undular bore in general unperturbed GP problem
[16] but instead of the derivative dλ+/dT in (92) we have the difference dλ+/dT −ρ+ (which
is always positive as we have established).
7 Conclusions
We have studied the effects of a gradual slope and turbulent (Chezy) bottom friction on the
propagation of solitary waves, nonlinear periodic waves and undular bores in shallow-water
flows in the framework of the variable-coefficient perturbed KdV equation. The analysis has
been performed in the most general setting provided by the associated Whitham equations
describing slow modulations of a periodic travelling wave due to the slope, bottom friction
and spatial nonuniformity of initial data. This modulation theory, developed in general form
for perturbed integrable equations in Kamchatnov (2004) was applied here to the perturbed
KdV equation and allowed us to take into account slow variations of all three parameters
in the cnoidal wave solution. The particular time-independent solutions of the perturbed
modulation equations were shown to be consistent with the adiabatically varying solutions
for a single solitary wave and for a periodic wave propagating over a slope without bottom
friction obtained in Ostrovsky & Pelinovsky (1970, 1975) and Miles (1979, 1983a). It was
shown, however, that the assumption of zero mean elevation used in these papers for the
description of slow variations of a cnoidal wave, ceases to be valid in the case when the
turbulent bottom friction is present. In this case, a more general solution was obtained
numerically improving the results of Miles (1983b).
Further, the derived full time-dependent modulation system was used for the descrip-
tion of the effects of variable topography and bottom friction on the propagation of undular
bores, in particular on the variations of the undular bore front representing a system of
weakly interacting solitary waves. By the analysis of the characteristics of the Whitham
system in the vicinity of the leading edge of the undular bore, two possible configurations
have been identified depending on whether the leading edge of the undular bore represents a
regular characteristic of the modulation system or its singular characteristic, i.e. a caustic.
The first case was shown to correspond to adiabatically slow deformations of the classi-
cal Gurevich-Pitaevskii modulation solution and is realised when the perturbations due to
variable topography and bottom friction are small compared with the existing spatial non-
uniformity of modulations in the undular bore (which is supposed to be formed outside the
region of variable topography/bottom friction). In the case when modulations due to the
external perturbations are comparable in magnitude with the existing modulations in the
undular bore, the leading edge becomes a caustic, and this situation was shown to corre-
spond to enhanced solitary wave interactions within the undular bore front. These enhanced
interactions have been shown to lead to a “nonlocal” leading solitary wave amplitude growth,
which cannot be predicted in the frame of the traditional local adiabatic approach to prop-
agation of an isolated solitary wave in a variable environment. As we mentioned in the
Introduction, one of our original motivations for this study was the possibility to model a
shoreward propagating tsunami as an undular bore. In this context, we would suggest that
the second scenario described above is the more relevant, which has the implication that
the growth, and eventual breaking of the leading waves in a tsunami wavetrain, cannot be
modeled as a local effect for that particular wave, but is determined instead by the whole
structure of the wavetrain.
Acknowledgements
This work was started during the visit of A.M.K. at the Department of Mathematical Sci-
ences, Loughborough University, UK. A.M.K. is grateful to EPSRC for financial support.
Appendix A: Derivation of the perturbed modulation system
We express the integrand function in the right-hand side of (24) in terms of the µ-variable
(15):
(2λi − s1 − U)R = 8Gµ3 − [8Gλi + 4(F + 2s1G)]µ2
+ [4(F + 2s1G)λi + 2s1(s1G+ F )]µ− 2s1(s1G+ F )λi.
(120)
Then we obtain with the use of (13), (14), and (16) the following expressions:
〈µ〉 =
µdθ =
−P (µ)
〈µ2〉 = 1
µ2dθ =
〈µ3〉 =
µ3dθ = −
+ s1〈µ2〉 − s2〈µ〉+ s3,
(121)
where I is a known integral
(λ3 − µ)(µ− λ2)(µ− λ1) dµ
(λ3 − λ1)5/2[(1−m+m2)E(m)− (1−m)(1−m/2)K(m)],
(122)
K(m) and E(m) being the complete elliptic integrals of the first and second kind, respec-
tively. The derivatives of I with respect to λi are also known table integrals (Gradshtein &
Ryzhik 1980):
(λ3 − µ)(µ− λ2)
µ− λ1
λ3 − λ1[(λ2 + λ3 − 2λ1)E − 2(λ2 − λ1)K],
(λ3 − µ)(µ− λ1)
µ− λ2
λ3 − λ1[(λ3 − λ1)K + (λ1 + λ3 − 2λ2)E],
(µλ2)(µ− λ1)
λ3 − µ
λ3 − λ1[(2λ3 − λ1 − λ2)E − (λ2 − λ1)K].
(123)
We can easily express the si-derivatives in terms of λi derivatives by differentiation of the
formulae (see (16))
s1 = λ1 + λ2 + λ3, s2 = λ1λ2 + λ1λ3 + λ2λ3, s3 = λ1λ2λ3 (124)
and solving the linear system for differentials. Simple calculation gives
(−1)3−k
j 6=i(λi − λj)
. (125)
Then, combining (123) and (125), we obtain the derivatives ∂I/∂si and hence the expressions
(λ3 − λ1)
(s21 − 3s2)
(λ2 − λ1)(λ2 + λ3 − 2λ1)
+ s1λ1 + λ
1 − λ2λ3
(λ3 − λ1)
(126)
To complete the calculation of the right-hand side of (24), we need also expressions
∂L/∂λ1
= 2(λ2 − λ1)
∂L/∂λ2
2(λ3 − λ2)(1−m)K
E − (1−m)K
∂L/∂λ3
2(λ3 − λ2)K
(127)
Collecting all contributions into perturbations terms, we obtain the Whitham equations in
the form
= ρi = Ci[F (T )Ai −G(T )Bi], (128)
where Cj , Aj , Bj and vj , j = 1, 2, 3 are specified by formulae (28) - (30).
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Introduction
Problem formulation
Modulation equations
Modulation solution in the solitary wave limit
Adiabatic deformation of a cnoidal wave
Undular bore propagation over variable topography with bottom friction
Gurevich-Pitaevskii problem for flat-bottom zero-friction case
Undular bore developing from an initial jump
Structure of the undular bore front
Gurevich-Pitaevskii problem for perturbed modulation system
Deformation of the undular bore front due to variable topography and bottom friction
Conclusions
|
0704.0046 | A limit relation for entropy and channel capacity per unit cost | A limit relation for entropy
and channel capacity per unit cost
Imre Csiszár1,4, Fumio Hiai2,5 and Dénes Petz3,4
4 Alfréd Rényi Institute of Mathematics,
H-1364 Budapest, POB 127, Hungary
5 Graduate School of Information Sciences, Tohoku University
Aoba-ku, Sendai 980-8579, Japan
Abstract: In a quantum mechanical model, Diósi, Feldmann and Kosloff
arrived at a conjecture stating that the limit of the entropy of certain mixtures
is the relative entropy as system size goes to infinity. The conjecture is
proven in this paper for density matrices. The first proof is analytic and uses
the quantum law of large numbers. The second one clarifies the relation to
channel capacity per unit cost for classical-quantum channels. Both proofs
lead to generalizations of the conjecture.
Key words: Shannon entropy, von Neumann entropy, relative entropy, ca-
pacity per unit cost, Holevo bound.
1E-mail: [email protected]. Partially supported by the Hungarian Research Grant OTKA T068258.
2E-mail: [email protected]. Partially supported by Grant-in-Aid for Scientific Research
(B)17340043.
3E-mail: [email protected]. Partially supported by the Hungarian Research Grant OTKAT068258.
http://arxiv.org/abs/0704.0046v1
1 Introduction
It was conjectured by Diósi, Feldmann and Kosloff in [4], based on thermodynamical
considerations, that the von Neumann entropy of a quantum state equal to a mixture
Rn :=
σ ⊗ ρ⊗(n−1) + ρ⊗ σ ⊗ ρ⊗(n−2) + · · · + ρ⊗(n−1) ⊗ σ
exceeds the entropy of a component asymptotically by the Umegaki relative entropy
S(σ‖ρ), that is,
S(Rn) − (n− 1)S(ρ) − S(σ) → S(σ‖ρ) (1)
as n → ∞. Here ρ and σ are density matrices acting on a finite dimensional Hilbert
space. Recall that S(σ) = −Tr σ log σ and
S(σ‖ρ) =
Tr σ(log σ − log ρ) if supp σ ≤ supp ρ
+∞ otherwise.
Concerning the background of quantum entropy quantities, we refer to [10, 12].
Apparently no exact proof of (1) has been published even for the classical case, al-
though for that case a heuristic proof is offered in [4].
In the paper first an analytic proof of (1) is given for the case supp σ ≤ supp ρ, using
an inequality between the Umegaki and the Belavkin-Staszewski relative entropies, and
the weak law of large numbers in the quantum case. In the second part of the paper, it
is clarified that the problem is related to the theory of classical-quantum channels. The
essential observation is the fact that S(Rn) − (n− 1)S(ρ) − S(σ) in the conjecture is a
Holevo quantity (classical-quantum mutual information) for a certain channel for which
the relative entropy emerges as the capacity per unit cost.
The two different proofs lead to two different generalizations of the conjecture.
2 An analytic proof of the conjecture
In this section we assume that supp σ ≤ supp ρ for the support projections of σ and ρ.
One can simply compute:
S(Rn‖ρ
⊗n) = Tr(Rn logRn − Rn log ρ
= −S(Rn) − (n− 1)Tr ρ log ρ− Trσ log ρ.
Hence the identity
S(Rn‖ρ
⊗n) = −S(Rn) + (n− 1)S(ρ) + S(σ‖ρ) + S(σ)
holds. It follows that the conjecture (1) is equivalent to the statement
S(Rn‖ρ
⊗n) → 0 as n → ∞
when supp σ ≤ supp ρ.
Recall the Belavkin-Staszewski relative entropy
SBS(ω‖ρ) = Tr(ω log(ω
1/2ρ−1ω1/2)) = −Tr(ρ η(ρ−1/2ωρ−1/2))
if suppω ≤ supp ρ, where η(t) := −t log t, see [1, 10]. It was proved by Hiai and Petz
S(ω‖ρ) ≤ SBS(ω‖ρ), (2)
see [6], or Proposition 7.11 in [10].
Theorem 1. If supp σ ≤ supp ρ, then S(Rn)− (n−1)S(ρ)−S(σ) → S(σ‖ρ) as n → ∞.
Proof: We want to use the quantum law of large numbers, see Proposition 1.17 in
[10]. Assume that ρ and σ are d × d density matrices and we may suppose that ρ is
invertible. Due to the GNS-construction with respect to the limit ϕ∞ of the product
states ϕn(A) = Tr ρ
⊗nA on the n-fold tensor product Md(C)
⊗n, n ∈ N, all finite tensor
products Md(C)
⊗n are embedded into a von Neumann algebra M acting on a Hilbert
space H. If γ denotes the right shift and X := ρ−1/2σρ−1/2, then Rn is written as
Rn = (ρ
1/2)⊗n
γi(X)
(ρ1/2)⊗n.
By inequality (2), we get
0 ≤ S(Rn‖ρ
⊗n) ≤ SBS(Rn‖ρ
= −Tr
ρ⊗n η
(ρ−1/2)⊗nRn(ρ
−1/2)⊗n
γi(X)
, (3)
where Ω is the cyclic vector in the GNS-construction.
The law of large numbers gives
γi(X) → I
in the strong operator topology in B(H), since ϕ(X) = Tr ρρ−1/2σρ−1/2 = 1.
Since the continuous functional calculus preserves the strong convergence (simply due
to approximation by polynomials on a compact set), we obtain
γi(X)
→ η(I) = 0 strongly.
This shows that the upper bound (3) converges to 0 and the proof is complete.
By the same proof one can obtain that for
Rm,n :=
σ⊗m ⊗ ρ⊗(n−1) + ρ⊗ σ⊗m ⊗ ρ⊗(n−2) + · · · + ρ⊗(n−1) ⊗ σ⊗m
the limit relation
S(Rm,n) − (n− 1)S(ρ) −mS(σ) → mS(σ‖ρ) (4)
holds as n → ∞ when m is fixed.
In the next theorem we treat the probabilistic case in a matrix language. The proof
includes the case when supp σ ≤ supp ρ is not true. Those readers who are not familiar
with the quantum setting of the previous theorem are suggested to follow the arguments
below.
Theorem 2. Assume that ρ and σ are commuting density matrices. Then S(Rn)− (n−
1)S(ρ) − S(σ) → S(σ‖ρ) as n → ∞.
Proof: We may assume that ρ = Diag(µ1, . . . , µℓ, 0, . . . , 0) and σ = Diag(λ1, . . . , λd)
are d×d diagonal matrices, µ1, . . . , µℓ > 0 and ℓ < d. (We may consider ρ, σ in a matrix
algebra of bigger size if ρ is invertible.) If supp σ ≤ supp ρ, then λℓ+1 = · · · = λd = 0;
this will be called the regular case. When supp σ ≤ supp ρ is not true, we may assume
that λd > 0 and we refer to the singular case.
The eigenvalues of Rn correspond to elements (i1, . . . , in) of {1, . . . , d}
(λi1µi2 · · ·µin + µi1λi2µi3 · · ·µin + · · · + µi1 · · ·µin−1λin). (5)
We divide the eigenvalues in three different groups as follows:
(a) A corresponds to (i1, . . . , in) ∈ {1, . . . , d}
n with 1 ≤ i1, . . . , in ≤ ℓ,
(b) B corresponds to (i1, . . . , in) ∈ {1, . . . , d}
n which contains exactly one d,
(c) C is the rest of the eigenvalues.
If the eigenvalue (5) is in group A, then it is
(λi1/µi1) + · · · + (λin/µin)
µi1µi2 · · ·µin .
First we compute
η(κ) =
i1,...,in
(λi1/µi1) + · · · + (λin/µin)
µi1 · · ·µin
Below the summations are over 1 ≤ i1, . . . , in ≤ ℓ:
i1,...,in
(λi1/µi1) + · · · + (λin/µin)
µi1 · · ·µin
i1,...,in
(λi1/µi1) + · · · + (λin/µin)
µi1 · · ·µin
log(µi1 · · ·µin) + Qn
i1,...,in
λi1µi2 · · ·µin log µik +
i1,...,in
λi1µi2 · · ·µin logµik
+ · · · +
i1,...,in
λi1µi2 · · ·µin log µik
(n− 1)
µik logµik +
λik logµik
= (n− 1)S(ρ) −
λi logµi + Qn,
where
Qn :=
i1,...,in
(µi1 · · ·µin)η
(λi1/µi1) + · · · + (λin/µin)
Consider a probability space
(Ω,P) :=
{1, . . . , ℓ}N, (µ1, . . . , µℓ)
where (µ1, . . . , µℓ)
N is the product of the measure on {1, . . . , ℓ} with the distribution
(µ1, . . . , µℓ). For each n ∈ N let Xn be a random variable on Ω depending on the
nth {1, . . . , ℓ} so that the value of Xn at i ∈ {1, . . . , ℓ} is λi/µi. Then X1, X2, . . . are
identically distributed independent random variables and Qn is the expectation value of
X1 + · · · + Xn
The strong law of large numbers says that
X1 + · · · + Xn
→ E(X1) =
λi almost surely.
Since η((X1 + · · · + Xn)/n) is uniformly bounded, the Lebesgue bounded convergence
theorem implies that
Qn → η
as n → ∞.
In the regular case
i=1 λi = 1, Qn → 0 and all non-zero eigenvalues are in group A.
Hence we have
S(Rn) − (n− 1)S(ρ) − S(σ) = −
λi logµi +
λi log λi + Qn = S(σ‖ρ) + Qn
and the statement is clear.
Next we consider the singular case, when we have
η(κ) = (n− 1)S(ρ) + O(1),
and we turn to eigenvalues in B. If the eigenvalue corresponding to (i1, . . . , in) ∈
{1, . . . , d}n is in group B and i1 = d, then the eigenvalue is
λdµi2 . . . µin .
It follows that
i2,...,in
(λdµi2 · · ·µin
(λdµi2 · · ·µin
i2,...,in
(µi2 · · ·µin) log(µi2 · · ·µin) −
(n− 1)S(ρ) −
When i2 = d, . . . , in = d, we get the same quantity, so this should be multiplied with n:
η(κ) = λd(n− 1)S(ρ) − λd log
We make a lower estimate to the entropy of Rn in such a way that we compute
κ η(κ)
when κ runs over A and B. It is clear now that
S(Rn) − (n− 1)S(ρ) − S(σ) ≥
η(κ) +
η(κ) − (n− 1)S(ρ) − S(σ)
≥ λd(n− 1)S(ρ) + λd log n + O(1) → +∞
as n → ∞.
3 Interpretation as capacity
A classical-quantum channel with classical input alphabet X transfers the input x ∈ X
into the output W (x) ≡ ρx which is a density matrix acting on a Hilbert space K. We
restrict ourselves to the case when X is finite and K is finite dimensional.
If a classical random variable X is chosen to be the input, with probability distribution
P = {p(x) : x ∈ X}, then the corresponding output is the quantum state ρX :=
x∈X p(x)ρx. When a measurement is performed on the output quantum system, it
gives rise to an output random variable Y which is jointly distributed with the input X .
If a partition of unity {Fy : y ∈ X} in B(K) describes the measurement, then
Prob(Y = y |X = x) = Tr ρxFy (x, y ∈ X ). (6)
According to the Holevo bound, we have
I(X ∧ Y ) := H(Y ) −H(Y |X) ≤ I(X,W ) := S(ρX) −
p(x)S(ρx), (7)
which is actually a simple consequence of the monotonicity of the relative entropy un-
der state transformation [7], see also [11]. I(X,W ) is the so-called Holevo quantity or
classical-quantum mutual information, and it satisfies the identity
p(x)S(ρx‖ρ) = I(X,W ) + S(ρX‖ρ), (8)
where ρ is an arbitrary density.
The channel is used to transfer sequences from the classical alphabet; x = (x1, x2, . . . , xn) ∈
X n is transferred into the quantum state W⊗n(x) = ρx := ρx1⊗ρx2⊗. . .⊗ρxn . A code for
the channel W⊗n is defined by a subset An ⊂ X
n, which is called a codeword set. The de-
coder is a measurement {Fy : y ∈ X
n}. The probability of error is Prob(X 6= Y ), where
X is the input random variable uniformly distributed on An and the output random
variable is determined by (6), where x and y are replaced by x and y.
The essential observation is the fact that S(Rn)−(n−1)S(ρ)−S(σ) in the conjecture
is a Holevo quantity in case of a channel with input sequences (x1, x2, . . . , xn) ∈ {0, 1}
and outputs ρx1 ⊗ ρx2 ⊗ . . . ⊗ ρxn, where ρ0 = σ, ρ1 = ρ and the codewords are all
sequences containing exactly one 0. More generally, we shall consider Holevo quantities
I(A, ρ0, ρ1) := S
S(ρx).
defined for any set A ⊂ {0, 1}n of binary sequences of length n.
The concept related to the conjecture we study is the channel capacity per unit cost
which is defined next for simplicity only in the case where X = {0, 1}, the cost of a
character 0 ∈ X is 1, while the cost of 1 ∈ X is 0.
For a memoryless channel with a binary input alphabet X = {0, 1} and an ε > 0, a
number R > 0 is called an ε-achievable rate per unit cost if for every δ > 0 and for any
sufficiently large T , there exists a code of length n > T with at least eT (R−δ) codewords
such that each of the codewords contains at most T 0’s and the error probability is at
most ε. The largest R which is an ε-achievable per unit cost for every ε > 0 is the
channel capacity per unit cost.
Lemma 1. For an arbitrary A ⊂ {0, 1}n,
I(A, ρ0, ρ1) ≤ c(A)S(ρ0‖ρ1)
holds, where
c(A) :=
|{i : xi = 0}|.
Proof: Let c(x) := |{i : xi = 0}| for x ∈ A. Since I(A, ρ0, ρ1) is a particular Holevo
quantity I(X,W⊗n), we can use the identity (8) to get an upper bound
S(ρx‖ρ
1 ) =
c(x)S(ρ0‖ρ1) = c(A)S(ρ0‖ρ1)
for I(A, ρ0, ρ1).
Lemma 2. If A ⊂ {0, 1}n is a code of the channel W⊗n, whose probability of error (for
some decoding scheme) does not exceed a given 0 < ε < 1, then
(1 − ε) log |A| − log 2 ≤ I(A, ρ0, ρ1).
Proof: The right-hand side is a bound for the classical mutual information I(X∧Y ) =
H(Y ) − H(Y |X), where Y is the channel output, see (7). Since the error probability
Prob(X 6= Y ) is smaller than ε, application of the Fano inequality (see [3]) gives
H(X|Y ) ≤ ε log |A| + log 2.
Therefore
I(X ∧ Y ) = H(X) −H(X|Y ) ≥ (1 − ε) log |A| − log 2,
and the proof is complete.
The above two lemmas shows that the relative entropy S(ρ0‖ρ1) is an upper bound
for the channel capacity per unit cost of the channel W (0) = ρ0 and W (1) = ρ1 with
a binary input alphabet. In fact, assume that R > 0 is an ε-achievable rate. For every
δ > 0 and T > 0 there is a code A ⊂ {0, 1}n for which we get by Lemmas 1 and 2
TS(ρ0‖ρ1) ≥ c(A)S(ρ0‖ρ1) ≥ I(A, ρ0, ρ1)
≥ (1 − ε) log |A| − log 2
≥ (1 − ε)T (R− δ) − log 2.
Since T is arbitrarily large and ε, δ are arbitrarily small, R ≤ S(ρ0‖ρ1) follows. That
S(ρ0‖ρ1) equals the channel capacity per unit cost will be verified below.
Theorem 3. Let the classical-quantum channel W : X = {0, 1} → B(K) be defined as
W (0) = ρ0 ≡ σ and W (1) = ρ1 ≡ ρ. Assume that An ⊂ {0, 1}
n is chosen such that
(a) each element x = (x1, x2, . . . , xn) ∈ An contains at most ℓ copies of 0,
(b) log |An|/ logn → c as n → ∞,
c(An) :=
|{i : xi = 0}| → c as n → ∞
for some real number c > 0 and for some natural number ℓ. If the random variable Xn
has a uniform distribution on An, then
S(ρXn) −
S(ρx)
= cS(σ‖ρ).
The proof of the theorem is divided into lemmas. We need the direct part of the
so-called quantum Stein lemma obtained in [6], see also [2, 5, 9, 12].
Lemma 3. Let ρ0 and ρ1 be density matrices. For every η > 0 and 0 < R < S(ρ0‖ρ1),
if N is sufficiently large, then there is a projection E ∈ B(K⊗N) such that
αN [E] := Tr ρ
0 (I − E) < η
and for βN [E] := Tr ρ
1 E the estimate
log βN [E] < −R
holds.
Note that αN is called the error of the first kind, while βN is the error of the second
kind.
Lemma 4. Assume that ε > 0, 0 < R < S(ρ0‖ρ1), ℓ is a positive integer and the
sequences x in An ⊂ {0, 1}
n contain at most ℓ copies of 0. Let the codewords be the
N-fold repetitions xN = (x,x, . . . ,x) of the sequences x ∈ An. If N is the integer part
and n is large enough, then there is a decoding scheme such that the error probability is
smaller than ε.
Proof: We follow the probabilistic construction in [13]. Let the codewords be the N -
fold repetitions xN = (x,x, . . . ,x) of the sequences x ∈ An. The corresponding output
density matrices act on the Hilbert space K⊗Nn ≡ (K⊗n)⊗N . We decompose this Hilbert
space into an N -fold product in a different way. For each 1 ≤ i ≤ n, let Ki be the
tensor product of the factors i, i + n, i + 2n, . . . , i + (N − 1)n. So K is identified with
K1 ⊗K2 ⊗ . . .⊗Kn.
For each 1 ≤ i ≤ n we perform a hypothesis testing on the Hilbert space Ki. The
0-hypothesis is that the ith component of the actually chosen x ∈ An is 0. Based on
the channel outputs at time instances i, i + n, . . . , i + (N − 1)n, the 0-hypothesis is
tested against the alternative hypothesis that the ith component of x is 1. According
to the quantum Stein lemma (Lemma 3), given any η > 0 and 0 < R < S(σ‖ρ), for N
sufficiently large, there exists a test Ei such that the probability of error of the first kind
is smaller than η, while the probability of error of the second kind is smaller than e−NR.
The projections Ei and I − Ei form a partition of unity in the Hilbert space Ki, and
the n-fold tensor product of these commuting projection will give a partition of unity in
K⊗Nn. Let y ∈ {0, 1}n and set Fy := ⊗
i=1Fyi , where Fyi = Ei if yi = 0 and Fyi = I −Ei
if yi = 1. Therefore, the result of decoding can be an arbitrary 0–1 sequence in {0, 1}
The decoding scheme gives y ∈ {0, 1}n in such a way that yi = 0 if the tests accepted
the 0-hypothesis for i and yi = 1 if the alternative was accepted. The error probability
should be estimated:
Prob(Y 6= X|X = x) =
y:y 6=x
Tr ρ⊗N
y:y 6=x
Tr ρ⊗Nxi Fyi
y:yi 6=xi
Tr ρ⊗Nxj Fyj ≤
Tr ρ⊗Nxi (I − Fxi).
If xi = 0, then
Tr ρ⊗Nxi (I − Fxi) = Tr ρ
0 (I −Ei) ≤ η,
because it is an error of the first kind. When xi = 1,
Tr ρ⊗Nxi (I − Fxi) = Tr ρ
1 Ei ≤ e
from the error of the second kind. It follows that ℓη + ne−NR is a bound for the error
probability. The first term will be small if η is small. The second term will be small
if N is large enough. If both terms are majorized by ε/2, then the statement of the
lemma holds. We can choose n so large that N defined by the statement should be large
enough.
Proof of Theorem 3: Since Lemma 1 gives an upper bound, that is,
lim sup
S(ρXn) −
S(ρx)
≤ cS(σ‖ρ),
it remains to prove that
lim inf
S(ρXn) −
S(ρx)
≥ cS(σ‖ρ).
Lemma 4 is about the N -times repeated input XN and describes a decoding scheme
with error probability at most ε. According to Lemma 2 we have
(1 − ε) log |An| − 1 ≤ S(ρXN ) −
S(ρxN ).
From the subadditivity of the entropy we have
S(ρXN ) ≤ NS(ρX)
S(ρxN ) = NS(ρx)
holds due to the additivity for product. It follows that
(1 − ε)
log |An|
≤ S(ρX) −
S(ρx).
From the choice of N in Lemma 4 we have
log |An|
log n
logn + log 2 − log ε
log |An|
and the lower bound is arbitrarily close to cR. Since R < S(ρ0‖ρ1) was arbitrary, the
proof is complete.
References
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1019–1030.
Introduction
An analytic proof of the conjecture
Interpretation as capacity
|
0704.0047 | Intelligent location of simultaneously active acoustic emission sources:
Part I | Intelligent location of simultaneously active acoustic
emission sources:
Part I
Tadej Kosel and Igor Grabec
Faculty of Mechanical Engineering, University of Ljubljana,
Aškerčeva 6, POB 394, SI-1001 Ljubljana, Slovenia
e-mail: [email protected]; [email protected]
Abstract— The intelligent acoustic emission locator is described
in Part I, while Part II discusses blind source separation,
time delay estimation and location of two simultaneously active
continuous acoustic emission sources.
The location of acoustic emission on complicated aircraft frame
structures is a difficult problem of non-destructive testing. This
article describes an intelligent acoustic emission source locator.
The intelligent locator comprises a sensor antenna and a general
regression neural network, which solves the location problem
based on learning from examples. Locator performance was
tested on different test specimens. Tests have shown that the
accuracy of location depends on sound velocity and attenuation
in the specimen, the dimensions of the tested area, and the
properties of stored data. The location accuracy achieved by
the intelligent locator is comparable to that obtained by the
conventional triangulation method, while the applicability of the
intelligent locator is more general since analysis of sonic ray paths
is avoided. This is a promising method for non-destructive testing
of aircraft frame structures by the acoustic emission method.
INTRODUCTION
Acoustic emission (AE) concerns non-destructive testing
methods and is used to locate and characterize developing
cracks and defects in material. In non-destructive testing of
aviation frame structures, acoustic emission is a well accepted
method [8]. The location problem is usually solved by various
triangulation techniques based on the analysis of ultrasonic ray
trajectories [10], [1], [3]. Solving and programming the related
equation is rather cumbersome and cannot be simply per-
formed if the structure of the tested specimen is geometrically
complicated. Acoustic emission testing of aircraft structures
is a challenging and difficult problem. The structures involve
bolts, fasteners and plates, all of which move relative to one
another due to differential structural loading during flight. The
complex geometry of the airframe results in multiple mode
conversions of AE source signals, compounding the difficulty
of relating the source event to the detected signal.
In order to avoid difficulties with equation solving and
programming of the triangulation procedure, several empirical
approaches based on learning from examples have already
been proposed [5]. We developed an intelligent locator capable
of learning from examples which we therefore called an
intelligent locator. The purpose of developing the intelligent
Manuscript generated: January 31, 2007
locator is to replace information obtained from the analysis
of sonic ray trajectories by information obtained directly from
simulated AE events on the specimen under test. In this way,
the calibration procedure, which has to be performed anyway,
could be generalized to the training of the intelligent locator.
The development of such an intelligent locator has been
described elsewhere [4]. In the locator developed a general
regression neural network (GRNN) is employed [9], which
acquires data about the detected AE signals and parameters
of their sources during learning. The GRNN uses these data
in testing when estimating the unknown source position from
detected AE signals. For this purpose, associative GRNN
operation is utilized. The basis of such operation is statistical
estimation determined by the conditional average [6]. Conse-
quently, the accuracy of the intelligent locator also depends on
the learning procedure, and must be examined before testing.
This article describes the results obtained by testing the
intelligent locator on experimental continuous AE sources. The
purpose of this study was to test and examine the advantages
of the intelligent locator compared to a conventional locator.
as described in Part I. In Part II an experiment will be
explained in which an intelligent locator was used to locate
two simultaneously active continuous AE sources generated
by leakage air flow. Location of more than one source at the
same time on the test specimen is a new approach in acoustic
emission testing, and is a very promising method for aircraft
and airspace structural testing.
When preparing the experiments, we focused on locating
evolving defects in stressed materials and constructions, and
leakage of vessels. We therefore performed location exper-
iments on four different specimens with three different AE
sources. The specimens comprised bands, plates, rings, and
vessels, while the AE sources were simulated by rupture of a
pencil lead (pen test), material deformation during tensile test,
and leakage air flow through a small hole in a sample. The
positions of AE sources used in testing were well specified.
Actual positions were compared with estimated ones, and
the discrepancy was used to describe the inaccuracy of the
locator. In this article, only the experiment with leakage air
flow through a small hole in a sample is explained. In Part I,
location of one continuous AE source is explained. This Part
is intended for better understanding of Part II and comparison
of results. In Part II, a new approach to the location of two
http://arxiv.org/abs/0704.0047v1
simultaneously active continuous AE sources is explained.
Below, the article first explains the theoretical background
for application of the conditional average to the location
problem, then describes auxiliary AE signal processing, and
finally demonstrates performance of the experimental intelli-
gent locator.
THEORETICAL BACKGROUND
In this section we describe a non-parametric approach to
empirical modeling of AE phenomena and solving the location
problem. This modeling stems from a description of physical
laws in terms of probability distributions. Since it has been
explained in detail elsewhere, we present here just its basic
concepts [6], [5].
The object of empirical modeling is the relationship between
variables which are simultaneously measured by a set of
sensors. In our example the variables are source coordinates
and AE signal characteristics. Let them be represented by a
vector of M components: x = (ξ1, . . . , ξM ). In the empirical
description of an AE phenomenon we repeat the observation N
times to create a database of prototype vectors {x1, . . . ,xN}.
Instead of formulating a relation between the components of x
we instead treat this vector as a random variable and express
the joint probability density function f by the estimator
f(x) =
δ(x− xn) . (1)
Here δ denotes Dirac’s delta function. For the purposes of
modelling, we must also estimate the probability density in
the space between the prototype points. This is achieved by
expressing the singular delta function in Eqs. 1 by a smooth
function, such as for example the Gaussian
wn(x− xn, σ) = exp
−‖x− xn‖
, n = 1, . . . , N .
in which σ denotes the smoothing parameter.
The data vectors determine an empirical model of the
probability density function. Their acquisition corresponds to
the learning phase of the empirical modeling. Let us further
assume that observation of AE phenomenon provides only
partial information that is given by a truncated vector
g = (ξ1, . . . , ξS ; ∅) , (3)
in which ∅ denotes missing components. The problem is
to estimate the complementary vector of missing or hidden
components:
h = (∅; ξS+1, . . . , ξM ); (4)
such that the complete data vector is determined by concate-
nation
x = g ⊕ h = (ξ1, . . . , ξS , ξS+1, . . . , ξM ) . (5)
A statistically optimal solution to this problem is determined
by the conditional average estimator, which is expressed by a
superposition of terms [6]
Bn(g)hn, where (6)
Bn(g) =
w(g − gn, σ)
w(g − gk, σ)
. (7)
The basis functions Bn(g) represent a measure of similarity
between the truncated vector g given by a particular ob-
servation and truncated vectors from the database gn. The
higher the value of Bn(g) the higher the contribution of hn
to the sum 7 estimating ĥ. Hence, estimation of the hidden
vector ĥ resembles associative recall, which is characteristic
of intelligence. The conditional average represents a general
non-parametric regression [6].
During the learning phase of operation an intelligent locator
of AE sources accepts AE signals and source coordinates
and stores prototype data vectors, while during application
it accepts only AE signals and estimates the corresponding
source position. Each of these phases can be performed in a
separate unit which can be interpreted as a layer of a sensory-
neural network.
In order to ensure acceptable properties of the locator,
the smoothing parameter σ must be properly chosen[2]. The
purpose of δ function smoothing is to estimate the probability
density function between the prototype data points. A unique
method for optimal specification of the smoothing parameter
is as yet unknown. In this case, it is numerically simpler to
specify σ by the half distance to the closest neighbor point:
σn = 0.5 min
‖gi − gn‖ , for all i 6= n . (8)
Signal pre-processing
The intelligent locator comprised a sensor antenna, signal
pre-processing unit and source locating unit, as shown in
Fig. 1. The first unit calculates the time delay ∆t from
AE signals y1(t) and y2(t), while the second unit estimates
the source position ẑ from the time delay ∆t. AE signals
y1(t) and y2(t) are detected by sensors and filtered using a
Butterworth bandpass filter. Without the bandpass filter, time
delays cannot be easily mapped to source positions on the
sample band, and therefore the applicability of this method
depends on the proper choice of bandpass filter function H(f).
We found on dispersive specimens that information in the
continuous AE signal about source position is located in a
narrow frequency band. A wave packet with approximately
constant wave velocity along the specimen must be extracted
by this filter. The filter function H(f) is determined during
training procedure of the locator.
PSfrag replacements
y1(t)
y2(t)
y1(t)
y2(t)
Ry1y2 ∆tCross-
correlator
detector
Locator ẑSensor
Sensor
Bandpass
filter
Test specimen
#2 H(f)
Signal pre-processing unit Source location
Fig. 1. AE signal processing by the intelligent locator
Two conventional methods for time delay estimation be-
tween two signals are known: threshold function and cross-
correlation function. Estimation of time delay by the threshold
function is simple, but only applicable in the case of discrete
AE. More general, but also more demanding, is time delay
estimation from the cross-correlation function of AE signals
[11]. The cross-correlation function:
Ry1y2(τ) =
y1(t) y2(t+ τ) , (9)
generally exhibits a peak when parameter τ corresponds to
the time delay ∆t between signals y1(t) and y2(t). The time
delay is thus determined from the position of the peak of the
cross-correlation function. One advantage of the application
of the cross-correlation function is that it does not depend
on the discrete or continuous character of AE signals. This
method for time delay estimation is only applicable when one
AE source is active at the time of detection. In the event of
two or more simultaneously active continuous AE sources, a
different approach should be used which will be discussed in
the Part II.
A filter function is calculated during calibration of the
intelligent locator as follows. During calibration, a set of
prototype sources is generated on the test specimen by a pen
test at a prepared coordinate net[8]. This net in most cases
has linear sections, where the prototype sources are positioned
on a straight line. In this case, we know that time delays
between signals are also linearly dependent. If we have a test
specimen with a complicated geometrical structure, then a pre-
calibration process has to be performed in which we have to
choose a geometrically simple part of the specimen and carry
out a pre-calibration procedure on this part such that time
delays between signals are linearly dependent.
For calibration we used AE signals generated by a pen
test. We obtained 12 pairs of AE signals from two sensors
concatenated with known coordinates of sources. The posi-
tions of simulated sources were uniformly distributed along a
straight line on a specimen. In such cases, time delay ∆t is
linearly related to source position z. This is of advantage for
optimal determination of bandpass filter because the reference
is a straight line. Calculation of time delays on the same set
of prototype AE signals was repeated 70 times. The bandpass
filter of ∆f = 10 kHz was shifted by 1 kHz at each repetition
from 5 to 75 kHz. Time delays were calculated at each
repetition and the distribution obtained was compared with a
straight line, as shown in Fig. 2. The frequency bandwidth was
considered optimal when the root mean square error (RMSE)
was minimal, as shown in Fig. 3(a). The optimal frequency
band for this specimen was 35-45 kHz and the velocity of
elastic waves was 1.7 km s−1. The filter was further used for
pre-processing samples of prototype as well as test sources. As
shown in Fig. 3(b), the pairs (z,∆t), estimated from filtered
signals, fit a straight line, except one outlier, which results
from experimental error.
EXPERIMENT
The intelligent AE source locator is shown schematically
in Fig. 4. It includes an automatic data-acquisition system
−1 0 1
PSfrag replacements
l [m]
5–15 kHz
−1 0 1
PSfrag replacements
l [m]
∆t [ms]
5–15 kHz
15–25 kHz
−1 0 1
PSfrag replacements
l [m]
∆t [ms]
5–15 kHz
15–25 kHz
25–35 kHz
−1 0 1
PSfrag replacements
l [m]
∆t [ms]
5–15 kHz
15–25 kHz
25–35 kHz
35–45 kHz
−1 0 1
PSfrag replacements
l [m]
∆t [ms]
5–15 kHz
15–25 kHz
25–35 kHz
35–45 kHz
45–55 kHz
−1 0 1
PSfrag replacements
l [m]
∆t [ms]
5–15 kHz
15–25 kHz
25–35 kHz
35–45 kHz
45–55 kHz
55–65 kHz
Fig. 2. Distribution of time delays and their linear approximation along
the band specimen. By this procedure an optimal bandpass filter can be
determined.
controlled by computer and a network of AE sensors.
The AE sensors are piezoelectric transducers (pinducers).
The diameter of the transducer active area is 1.3 mm, And so it
can be considered a point-like sensor. The signals from sensors
are fed to a digital oscilloscope where they are digitized and
transferred to a PC. Operation of the intelligent locator is
determined by software in the PC that controls data acquisition
and estimates the position of unknown AE sources.
The locator operates in two different modes:
1) In learning or calibration mode, a set of N pen tests is
performed in which complete information about the AE
phenomenon is acquired. The operator must prepare an
orientation net the shape of which depends on the shape
of the test specimen. The recommended shape is an
equidistant net, since such position of prototype sources
yield a minimum error of the locator. ¿From source
coordinates and time delays between pre-processed AE
signals, the prototype vectors are created and stored in
the memory of the neural network as a data base.
2) In application mode, only time delays between AE
signals are provided. There are then associated in the
neural network with the estimated source coordinates.
In the case of discrete AE, the time delay can visually be
estimated from a marked jump in the burst of the AE signal, or
can be instrumentally determined using a threshold function.
Hence, in the case of continuous AE, time delays cannot
be simply estimated, although a cross-correlation function
has already been used for this purpose. In our approach, we
therefore applied a cross-correlation function. The purpose of
this experiment was to determine the accuracy of location of
continuous AE sources on a one-dimensional specimen.
Two experiments on aluminum band specimen are explained
in this article. We tested the locator on an aluminum band
specimen of dimensions 4000 × 40 × 5mm3. Reflection of
AE signals at the ends of the band specimen was reduced
by sharpening the ends. For testing we selected a test area
15−25 35−45 55−65 75−85
PSfrag replacements
∆f [kHz] - Frequency band
E ∆fopt
−1000 −500 0 500 1000
PSfrag replacements
z [mm] - Actual location
-outlier
Fig. 3. Time delays for prototype and test sources by using the bandpass
filter of frequency 35-45 kHz. a) Deviation of prototype source position from
a straight line for different filter frequency bandwidth. b) Time delays of
prototype and test sources; Legend: + prototype source, ◦ test source
in the middle of the band specimen where 23 holes were
prepared. The distance between holes was 100 mm and the
diameter of holes was 2 mm. Two AE sensors were mounted
100 mm away from the terminal holes. For the purpose of
locator training, we generated 12 prototype sources separated
by 200 mm, while all 23 holes were applied for locator testing.
In this experiment, we calibrate the locator by pen test and
examine it by continuous AE generated by air flow. The air
flow was produced by expansion of compressed air through
nozzle of 1 mm diameter. The nozzle was mounted 1 mm
above the band specimen surface.
Two experiments were performed. In the first experiment,
only one continuous AE source was active on the band
specimen, while in the second experiment two continuous AE
sources were active simultaneously on the band specimen.
Successive simultaneous location of two sources is explained
in Part II.
Signals were processed as shown in Fig. 1. The first step
in processing was calculation of cross-correlation function
of AE signals. The corresponding signal was sent through a
bandpass Butterworth filter of bandpass from 35 to 45 kHz.
Determination of this filter is explained earlier in this article.
RESULTS
The results of locator testing are shown in Fig. 5(a). The
absolute location error for each test source is shown in
Fig. 5(b). Location error in the experiment ranges from 1.3 mm
to 60 mm with average value εa = 20mm (ignoring the
outlier). If we describe the error with respect to the distance
between sensors (2.4 m), the relative value is less than 1%.
Increasing the number of prototype sources can reduce the
error. Despite the complexity of continuous AE signals, the
location problem was solved satisfactorily with respect to The
accuracy required in non-destructive testing. Results also show
that a standard calibration procedure with discrete AE signals
generated by pen test can be used for locator training.
PSfrag replacements
Sensors
Operator
Analog
Signals
#2 Digital
oscilloscope
Parameter set
Computer
Calibration by simulated AE sources
Fig. 4. Experimental setup of intelligent locator
−1000 −500 0 500 1000
−1000
PSfrag replacements
x [mm] - Actual location
-outlier
−1000 −500 0 500 1000
PSfrag replacements
x [mm] - Actual location
-outlier
Fig. 5. Result of continuous AE source location on the band. a) Estimated
versus actual location of test sources; Legend: + prototype source, ◦ test source.
b) Absolute location error; εa - average error.
DISCUSSION AND CONCLUSION
Estimation of source coordinates by the conditional average
is subject to systematic error caused by smoothing of the
delta function [5]. This error can be reduced by increasing
the number of prototype sources. Since it is not always
possible to increase the number of prototype sources due to
the complexity of experiments, a compromise must be found
by trial and error.
Experimental error is acceptable, so we decided to make
additional tests, as will be discussed in Part II.
This study shows that a conventional AE locator operating
on the triangulation method can be successfully replaced by an
intelligent locator that learns from examples. The results show
that the intelligent locator can locate sources with acceptable
accuracy in cases of: (1) discrete AE on band and plate, (2)
continuous AE on band, (3) discrete AE on plate with hole
(ring), (4) discrete AE generated by specimen rupture during
the tensile test, and (5) discrete AE on pressure vessel. Is has
been also shown that the locator can perform zonal locating[7].
Comparing mean errors of all experiments and the distances
between prototype sources, we find that the average error
is always less than 30% of the distance between prototype
sources, while the maximal error is always less than 50% of
the distance between prototype sources. The accuracy of the
locator can be controlled by the number of prototype sources
excited during training. The experimental error of the locator is
a consequence of wave dispersion on a specimen that operates
as a waveguide, reflections from boundaries, and attenuation.
We found for dispersive waves that an optimal wave packet
must be found which has approximately constant velocity
along the test specimen. Estimation of time delay between AE
signals by the cross-correlation function is only applicable for
one active AE source. If there are several simultaneously active
AE sources, then blind source separation should be used, as
will be shown in Part II.
REFERENCES
[1] Chan, Y. T. Ho, K. C. 1994 , A simple and efficient estimator for hy-
perbolic location, IEEE Transactions on Signal Processing 42(8), 1905–
1915.
[2] Cherkassky, V. Mulier, F. 1998 , Leraning from Data: Concepts, Theory,
and Methods, John Wiley & Sons inc., New York.
[3] Friedlander, B. 1987 , A passive localization algorithm and its accuracy
analysis, IEEE Journal of Oceanic Engineering OE-12(1), 234–245.
[4] Grabec, I. Antolovič, B. 1994 , Intelligent locator of AE sources, in
T. Kishi, Y. Mori M. Enoki, eds, The 12th International Acoustic Emission
Symposium, Vol. 7 of Progress in Acoustic Emission, The Japanese
Society for Non-Destructive Inspection, Tokyo, Japan, pp. 565–570.
[5] Grabec, I. Sachse, W. 1991 , ‘Automatic modeling of physical phenomena:
Application to ultrasonic data’, J. Appl. Phys. 69(9), 6233–6244.
[6] Grabec, I. Sachse, W. 1997 , Synergetics of Measurement, Prediction and
Control, Springer-Verlag, Berlin.
[7] Kosel, T. Grabec, I. 1998 , Intelligent locator of discrete and continuous
acoustic emission sources, in J. Grum, ed., Application of Contemporary
Non-destructive Testing in Engineering, The 5th International Conference
of Slovenian Society for Nondestructive Testing, Slovenian Society for
Nondestructive Testing, Ljubljana, Slovenia, pp. 39–54.
[8] McIntire, P. Miller, R. K., eds 1987 , Acoustic Emission Testing, Vol. 5
of Nondestructive Testing Handbook, 2 edn, American Society for Non-
destructive Testing, Philadelphia, USA.
[9] Specht, D. F. 1991 , A general regression neural network, IEEE Trans.
on Neural Networks 2(6), 568–576.
[10] Tobias, A. 1976 , Acoustic emission source location in two dimensions
by an array of three sensors, Non-Destructive Testing 9(2), 9–12.
[11] Ziola, S. M. Gorman, M. R. 1991 , Source location in thin plates using
cross-correlation, J. Acoust. Soc. Am. 90(5), 2551–2556.
References
|
0704.0048 | Inference on white dwarf binary systems using the first round Mock LISA
Data Challenges data sets | compiled:
25 October
Inference on white dwarf binary systems using the
first round Mock LISA Data Challenges data sets
Alexander Stroeer1,2, John Veitch1, Christian Röver3,
Ed Bloomer4, James Clark4, Nelson Christensen5,
Martin Hendry4, Chris Messenger4, Renate Meyer3,
Matthew Pitkin4, Jennifer Toher4, Richard Umstätter3,
Alberto Vecchio1,2 and Graham Woan4
1 School of Physics & Astronomy, University of Birmingham, Birmingham, UK
2 Department of Physics & Astronomy, Northwestern University, Evanston, IL, USA
3 Department of Statistics, The University of Auckland, Auckland, New Zealand
4 Department of Physics & Astronomy, University of Glasgow, Glasgow, UK
5 Physics & Astronomy, Carleton College, Northfield, MN, USA
Abstract. We report on the analysis of selected single source data sets from the
first round of the Mock LISA Data Challenges (MLDC) for white dwarf binaries. We
implemented an end-to-end pipeline consisting of a grid-based coherent pre-processing
unit for signal detection, and an automatic Markov Chain Monte Carlo post-processing
unit for signal evaluation. We demonstrate that signal detection with our coherent
approach is secure and accurate, and is increased in accuracy and supplemented
with additional information on the signal parameters by our Markov Chain Monte
Carlo approach. We also demonstrate that the Markov Chain Monte Carlo routine is
additionally able to determine accurately the noise level in the frequency window of
interest.
PACS numbers: 04.80.Nn, 02.70.Uu.
Submitted to: Classical and Quantum Gravity
1. Introduction
The data obtained from LISA [1] will contain a large number of white dwarf binary
systems across the whole observational window [2]. At frequencies below ∼ 3 mHz the
sources are so abundant that they produce a stochastic foreground whose intensity
dominates the instrumental noise [3]. The closer (and louder) sources will still be
sufficiently bright to be individually resolvable. Above ∼ 3 mHz the sources become
sufficiently sparse in parameter space (and in particular in the frequency domain) that
the detectable sources become individually resolvable. The identification of white dwarfs
in the LISA data set represents one of the most interesting analysis problems posed by
the mission: the total number of signals in the data set is unknown, the effective noise
http://arxiv.org/abs/0704.0048v2
WD MLDC1 2
level affecting the measurements is not easily estimated from the data streams, and
there is a large number of overlapping sources to the limit of confusion.
Bayesian inference provides a clear framework to tackle such a problem [4, 5, 6].
Some of us have carried out exploratory studies and “proof of concept” analyses
on simplified problems that have demonstrated that Bayesian techniques do indeed
show good potential for LISA applications [11, 10, 12]. Similarly other authors have
successfully implemented techniques using Bayesian inference [18, 17, 16]. In this paper
we present the first results of an end-to-end analysis pipeline developed in the context of
the Mock LISA Data Challenges that has evolved from our earlier work. This pipeline
is applied to the simplest single-source challenge data sets 1.1.1a and 1.1.1b and all the
results presented here are obtained after the release of the key files. In a companion
paper [19], we present results that we have obtained for the analysis of the data sets
containing gravitational radiation from a massive-black-hole binary inspiral. Our group
submitted an entry for the MLDC analysing the blind data set 1.1.1c [13, 14]: however
that result suffered from the fact that the pipeline was not complete, the analysis code
was inefficient and we encountered hardware problems with the Beowulf cluster used to
perform the analysis.
The results that we present here are obtained with a two-stage end-to-end analysis
pipeline: (i) we first process the data set with a grid-based coherent algorithm to identify
candidate signals; (ii) we then follow up the candidate signals with a Markov Chain
Monte Carlo code to obtain probability density function on the model parameters. Our
method differs from other MCMC methods that have been proposed and applied to
the MLDC data in the context of white dwarf binaries [18, 17, 16]: the MCMC is not
used to search, but only in the final stage of the analysis to produce posterior density
functions of the model parameters. The noise spectral level is included as one of the
unknown parameters and is estimated together with the parameters of the gravitational
wave source(s).
2. Analysis method
In this section we describe the two stage approach that we have adopted for the analysis.
The signal produced by a white dwarf binary system is modelled as monochromatic in
the source reference frame, following the conventions adopted in the first MLDC [7, 8, 9].
It is described by 7 parameters: ecliptic latitude ϑe and longitude ϕe, inclination ι and
polarisation angle Ψ, frequency at a reference time f0 and corresponding overall phase
Φ0 and amplitude A.
The data distributed for the MLDC are the three TDI v1.5 Michelson observables
X , Y and Z ‡. From those we construct the two orthogonal TDI outputs
A = (2X − Y − Z)/3 (1)
E = (Z − Y )/
3 (2)
‡ In our MCMC analysis we use the data set produced using the LISA Simulator.
WD MLDC1 3
by diagonalizing the noise covariance matrix following the procedure presented in [23].
The noise affecting the channels A and E is uncorrelated and described by the one-sided
noise spectral density Sn(f). We model the LISA response function in the low frequency
limit in order to improve the computational efficiency of our analysis.
2.1. First stage: Grid based search
The first stage of the pipeline consists of a fast search of the data for the best
matched filter based on the well-known F -statistic algorithm, first developed for triaxial
pulsar signals in the context of ground-based observations [20]. This exploits the Fast
Fourier Transform to perform matching in the frequency domain to templates which are
generated at an array of fixed points in the parameter space.
The data from an individual detector in the frequency domain d̃(f) is supposed to
contain a signal plus Gaussian noise, d̃(f) = h̃(f) + ñ(f). We define the logarithmic
likelihood as a measure of match, as given by logL ≈ (d̃|h̃)− 1
(h̃|h̃) with (·|·) denoting
the scalar product as defined in [20]. A single signal in the F -statistic algorithm is
re-parameterised as a linear function of four orthogonal variables, and the frequency
f0. The detection statistic is based on four parameters AF , BF , CF and DF , found by
integrating over the response functions a(t) and b(t) to the two polarisation states of
the gravitational wave signal [20],
∫ Tobs
a(t)2dt (3)
b(t)2dt (4)
a(t)b(t)dt, (5)
DF = AFBF − C2F (6)
Tobs denotes the total observed time for the data set being analysed. The optimal
detection statistic 2F , which is pre-maximised over the nuisance parameters h0, ι, φ0
and ψ is
2F = 8
Sn(f)Tobs
BF |Fa|2 + AF |Fb|2 − 2CF ×R(FaFb)
. (7)
Fa and Fb are the demodulated Fourier transforms of the data,
∫ Tobs
d(t)a(t)e−iΦ(t)dt; Fb =
∫ Tobs
d(t)b(t)e−iΦ(t)dt, (8)
Φ(t) is the phase of the gravitational wave signal, as is described in [22].
As the LISA array moves in space, the frequency f0 is affected by Doppler
modulations. This modulation changes with differing position of the source in the sky,
implying the need to recalculate the modulations and thus a(t) and b(t) for each sky
position that is tested - a significant factor in the performance of this approach. The
differing modulation structure however also allows us to estimate the location of the
WD MLDC1 4
source in the sky by maximising the 2F value. The resolution possible on the sky with
this method is not as good as from a full Bayesian posterior probability calculation as
performed in the parameter estimation stage, as shown in an example for Challenge
1.1.1a in figure 1. Nevertheless, since this statistic can be computed fairly quickly it
serves as a useful way of finding initial values to feed into the MCMC routine, as adopted
within the pipeline. The resolution achievable on the sky increases with frequency, which
implies that the mismatch between filter and signal falls off more rapidly at higher
frequencies, requiring a greater number of templates to cover the sky. Therefore for
challenge 1.1.1b at f ≈ 3mHz a sky grid of size 5,752 points was used, in comparison
with 765 points for challenge 1.1.1a at f ≈ 1mHz.
The F -statistic search was implemented using the LIGO “Lalapps” suite of software
[24], in which the pulsar search code was modified by Reinhard Prix and John Whelan
to use the LISA response function for the TDI variables X , Y , and Z [21]. These input
data streams were given in the form of Short Fourier Transforms, each of length one day,
created from the MLDC1 challenge data. For each challenge the full specified range of
frequencies was searched for the signal as it would be in a blind search. The code was
run on a single CPU and executed in a few hours, with the run-time increasing at higher
frequency due to the higher resolution of sky and frequency grid that had to be used.
The candidate chosen to pass to the MCMC stage was simply that which triggered the
highest value of 2F .
2.2. Second stage: Markov Chain Monte Carlo follow-up
According to Bayes’ theorem, the posterior probability, p(m̃|d̃) of a model m̃ given the
data d̃ depends on the prior distribution p(m̃), containing the information known before
the analysis, the likelihood L(d̃|m̃) of the model and a normalisation factor p(d̃)
p(m̃|d̃) = L(d̃|m̃)p(m̃)
p(d̃)
The posterior probability density function shows the joint probability density of given
values of parameters of the model m̃, conditional on the data d̃.
We implemented Bayes’ theorem using data in the form of TDI variables A and E
and modelled our template according to the Long Wavelength Approximation directly
in the Fourier domain [25] to gain computational speed. The logarithmic likelihood
L(d̃|m̃) in this stage explicitly included its dependence on the one-sided noise spectral
density Sn(f)
logL(d̃|m̃) = const. −
log Sn(f) − (d̃− h̃|d̃− h̃), (10)
shown here for either A or E, with the combined likelihood as sum of the individual
likelihoods. We restricted our analysis to a sufficiently narrow frequency window in
order to be able to approximate the noise spectral density as constant, Sn(f) = S0. This
window was set as the interval in frequency that contains at least 98% of the power of our
WD MLDC1 5
Ecliptic Longitude
2F as a function of sky position, at a frequency 0.001063 Hz
1 2 3 4 5 6
Figure 1. The variation of 2F values for the search for unknown signal 1.1.1a, as a
function of sky position, parameterised by ecliptic latitude β and longitude λ. The
distribution is multi-modal and non-Gaussian, and has a poor resolution in comparison
with that can be achieved with the MCMC and a Bayesian likelihood, but by finding the
maximum it serves well as a starting point for the more refined parameter estimation
below.
model m̃, with the interval’s upper and lower limits given by f±(2/year)(5+2πf0AU/c)
[25]. S0 is therefore an additional parameter to be inferred within the model m̃ in Eq. 10.
We implemented an automatic Random Walk Metropolis sampler (Stroeer &
Vecchio 2007, in. prep.) to sample from the posterior probability density function in
form of a Markov chain. Metropolis sampling eliminates the need to explicitly calculate
the normalisation constant in Bayes’ theorem, and the evolving Markov chain gives
easy access to joint as well as marginalised posterior density distribution. The sampler
was started from the parameter set which triggered the highest value of 2F in our grid
based coherent run of the analysis (see former section). The automated function of the
Metropolis sampling is achieved by controlling the sampling step-size with adaptive
acceptance probability techniques [26]. The sampler therefore does not depend on
assumptions about the signal in the data set in order to perform successfully and reliably;
it develops a suitable algorithm and approach by itself based on the properties of the
likelihood as found on the fly, in the initial steps of the sampler. The length of our
Markov chain was pre-set to 106, with the initial 104 chain states discarded as the
“burn-in” phase of our sampler. The runtime for one data analysis run is 5 hours on a
single 2 GHz CPU on the Tsunami cluster of the University of Birmingham.
WD MLDC1 6
Figure 2. The marginalised posterior probability density functions of the eight
unknown parameters – the seven parameters that describe the signal and the noise
spectral density S0 – for the the challenge data set 1.1.1a. The vertical black solid
line denotes the true value of the parameter (for the polarisation angle the true value
modulo π/2), and the grey dashed line the initial value for the MCMC analysis as
determined by the template of the first-stage that produces the maximum value of the
F -statistic. In the case of the noise spectral density the first stage of the analysis does
not provide an estimate; the true value of this parameter is taken to be the value of
the instrumental noise spectrum used to generate the data set and provided in [9].
WD MLDC1 7
Figure 3. The marginalised posterior probability density functions of the eight
unknown parameters for the the challenge data set 1.1.1b. Labels are as in Figure
3. Results
We found that the most promising candidate signal from the F -statistic search already
matched the true embedded signal to high accuracy, particularly in frequency and
sky location. Our MCMC sampler, as a post-processing unit, thus only needed 1000
iterations to burn in and to establish a reliable sampling from the posterior. The
marginalised posteriors are shown in Figs. 2 and 3. We found, as seen in latter
figures, that the MCMC sampler further refined the initial guesses from the F -statistic,
as measured by the absolute difference between the true value of a given parameter
and the median of the marginalised posterior recovered for that parameter. Table 1
WD MLDC1 8
Table 1. Details about the results from Challenge 1.1.1a and Challenge 1.1.1b. S0,
the constant one-sided noise spectral density within our narrow frequency window, is
compared to the true one sided noise spectral density at the true frequency of the
signal, Ψ is given modulo π/2. Int90 denotes the minimum interval to include 90% of
MCMC states for given parameter, ∆mode denotes the absolute difference between the
true value of a signal parameter and the mode of its recovered posterior; ∆median and
∆mean denote the equivalent absolute difference for median and mean of the posterior
respectively; σ denotes the sampled standard deviation of the posterior as derived
from the median. We further quote the signal-to-noise ratio (SNR) for a template
using the true values of the source and the recovered values of the data analysis run,
as derived from the median of the individual posterior distributions, and the correlation
C between these two templates.
Int90 ∆mode ∆median ∆mean σ
Challenge 1.1.1a
10−41Hz−1
(3.53257, 4.72639) -0.42084 -0.440278 -0.452456 0.36704
ϑe/ rad (0.958409, 1.03165) -0.0147383 -0.0149381 -0.0148725 0.0222861
ϕe/ rad (5.05376, 5.13528) -0.00550139 -0.00569547 -0.00579889 0.0247886
Ψ/ rad (1.32475, 0.500553) 0.1768 0.1823 0.1902 0.1908
ι/ rad (0.097761, 1.0008) -0.0459747 0.190001 0.23459 0.295211
A/10−22 (1.61976, 2.67967) 0.664371 0.358844 0.298978 0.368524
f0/ mHz (1.06273, 1.06273) -1.19664e-06 -1.22207e-06 -1.22259e-06 1.04422e-06
Φ0/ rad (3.10668, 5.808) -0.164989 0.00998525 0.229659 0.829146
SNR true = 51.024497 recovered = 50.648600
C true vs. recovered = 0.99689
Challenge 1.1.1b
10−41Hz−1
(0.876833, 1.38959) -0.0679571 -0.0906557 -0.0996144 0.16017
ϑe/ rad (-0.121611, 0.0116916) -0.0343353 -0.151185 -0.150328 0.0406552
ϕe/ rad (4.60969, 4.63537) 0.00265723 0.00305564 0.00302203 0.00779893
Ψ/ rad (0.246328, 0.362409) 0.0301541 0.0311747 0.0311268 0.0353938
ι/ rad (1.22036, 1.33338) -0.0430412 -0.040458 -0.0394818 0.0348383
A/10−22 (0.45001, 0.542454) -0.016442 -0.0151921 -0.0149907 0.0281154
f0/ mHz (3.00036, 3.00036) 3.1221e-07 2.49289e-07 2.42807e-07 8.18111e-07
Φ0/ rad (5.83869, 6.19411) 0.137219 0.119301 0.119921 0.502384
SNR true = 36.587444 recovered = 37.368806
C true vs. recovered = 0.97897
shows details of the statistics of recovered posterior distributions. We highlight that
the majority of the true values of the parameters are within one standard deviation
of the median of the posterior, with a small percentage within two sampled standard
deviations. In addition, every true value of a parameter of the signal is within the
minimum interval of the posterior to cover 90% of all MCMC state values. Recovered
signal-to-noise ratios are measured as SNR = (s|h)/
(h|h), and the match C =
(htrue|hmed)/
(htrue|htrue) (hmed|hmed) between a template constructed from the true
values and a template from the median values of the individual posterior distributions,
yielding a correlation that is always higher than 0.97. Noise levels are determined
accurately and within 1 to 1.5 sampled standard deviations. Nevertheless we note that
WD MLDC1 9
our run on Challenge 1.1.1a shows a lower match and higher differences between true
value and recovered value of parameters as compared to the run on Challenge 1.1.1b. It
also exhibits tailing posterior distributions in inclination and amplitude, although the
SNR of Challenge 1.1.1a is twice the value of Challenge 1.1.1b.
4. Conclusions
We have presented a new approach to LISA data analysis in the form of an end-to-end
pipeline. We first detected and identified candidate signals in the LISA data stream with
a grid-based coherent algorithm, and then post-processed the most promising candidate
signals with an automatic Markov Chain Monte Carlo code to obtain probability
densities for the model’s parameters. We demonstrated successful identification and
post-processing of the signals from the double white dwarf single source MLDC data
sets 1.1.1a and 1.1.1b. Furthermore, the automatic Markov Chain Monte Carlo code
successfully identified the noise level within a small frequency window of interest in
these data sets. We note that a parallel approach to the data analysis of binary inspiral
signals is being developed by Röver et al, with a Markov Chain Monte Carlo method
that can successfully post-process a candidate signal generated from the true parameters
of the signal. Signal detection in a pre-processing stage is currently being tested within
parallel tempered MCMC methods and/or time-frequency analyses [19].
We identify two prominent and promising features of our pipeline: its ability to
determine good initial conditions for the MCMC and its ability to run the MCMC
automatically. As we have demonstrated in this paper, the width of the marginalised
posterior density for the frequency parameter is extremely narrow. It is therefore vital
that the initial estimate of the frequency is within this region, as the almost flat structure
of the posterior PDF outside this region gives little to no information on the location of
the peak. The chances of finding the mode through a random sampling are decreased
further still with a larger prior range for the parameter. Adding an F -statistic search as
the first stage in the pipeline solves this problem, since the frequency and position in the
sky are recovered very accurately, to within the limits of the posterior probability region
of interest, before the MCMC performs post-processing and parameter estimation. The
automatic feature of the MCMC ensures a successful post-processing for the other
astrophysical parameters that may have been located outside the posterior region of
interest by the F -statistic approach, as in the case for the amplitude of Challenge
1.1.1a. Convergence is aided by the ability of our code to increase or decrease sampling
step-sizes according to its experience of the sampling quality of the posterior during the
burn-in phase.
We are working on an extension of the pipeline as shown in this document to
successfully tackle multi-source data sets, required for the second round of the MLDC.
Current work includes the exploration of our grid-based coherent search on such data
streams in order to automatically identify the most promising individual candidate
signals, and the implementation of an automatic Reversible Jump Markov Chain Monte
WD MLDC1 10
Carlo routine (e.g. as already demonstrated in [10]) to find the trans-dimensional
probability density functions of the parameters of an unknown total number of signals.
We highlight that the noise level determination presented here already serves as a key
ingredient to round 2, where the simulation of a galactic white dwarf binary population
introduces additional confusion noise levels from unresolvable sources.
Acknowledgements
Nelson Christensen’s work was supported by the National Science Foundation grant
PHY-0553422 and the Fulbright Scholar Program. Alberto Vecchio’s work was partially
supported by the Packard Foundation and the National Science Foundation. The
University of Auckland group was supported by the Royal Society of New Zealand
Marsden Fund Grant UOA-204.
References
[1] Bender B L et al 1998 LISA Pre-Phase A Report; Second Edition, MPQ 233
[2] Nelemans G, Yungelson L R and Portegies Zwart S F 2001 Astron. and Astrophys. 375 890
[3] Farmer A J and Phinney E S 2003 Mon. Not. R. Astron. Soc 346 1197
[4] Jaynes E T Probability theory: The logic of science 2003 Cambridge University Press
[5] Gregory P C Bayesian logical data analysis for the physical sciences 2005 Cambridge University
Press
[6] Gelman A, Carlin J B, Stern H, and Rubin D B Bayesian data analysis 1997 Chapman & Hall
CRC Boca Raton
[7] Arnaud K A et al 2006 AIP Conf. Proc. 873 619 Preprint gr-qc/0609105
[8] Arnaud K A et al 2006 AIP Conf. Proc. 873 625 Preprint gr-qc/0609106
[9] Mock LISA Data Challenge Task Force, “Document for Challenge 1,”
svn.sourceforge.net/viewvc/lisatools/Docs/challenge1.pdf.
[10] Stroeer A, Gair J and Vecchio A 2006 Automatic Bayesian inference for LISA data analysis
strategies Preprint gr-qc/0609010
[11] Umstätter R, Christensen N, Hendry M, Meyer R, Simha V, Veitch J, Vigeland S and Woan G
2005 Phys Rev D 72 022001
[12] Wickham E D L, Stroeer A and Vecchio A 2006 Class Quantum Grav 23 819
[13] Bloomer E et al Report on MLDC1 available at http://astrogravs.nasa.gov/docs/mldc/round1/entries.html
[14] Arnaud K A et al 2007 Preprint gr-qc/0701139
[15] Arnaud K A et al 2007 Preprint gr-qc/0701170
[16] Crowder, J., and Cornish, N. J. 2007 Phys. Rev. D 75 043008
[17] Crowder J, Cornish N J and Reddinger J L 2006 Phys. Rev. D 73 063011
[18] Cornish N J and Crowder J 2005 Phys. Rev. D 72 043005
[19] Röver C et al in this volume
[20] Jaranowski P, Królak A and Schutz B F 1998 Phys. Rev. D 58 063001
[21] Prix R and Whelan J 2006 Technical note
[22] Brady P R, Creighton T, Cutler C and Schutz B F 1997 Phys. Rev. D 57 2101
[23] Prince T A, Tinto M, Larson S L and Armstrong J W 2002 Phys. Rev. D 66 122002
[24] LAL Home Page: http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html
[25] Cornish N J, Larson S L 2003 Phys. rev. D 67 103001
[26] Atchade Y F, Rosenthal J S 2005 Bernoulli 11 815-828
http://arxiv.org/abs/gr-qc/0609105
http://arxiv.org/abs/gr-qc/0609106
http://arxiv.org/abs/gr-qc/0609010
http://astrogravs.nasa.gov/docs/mldc/round1/entries.html
http://arxiv.org/abs/gr-qc/0701139
http://arxiv.org/abs/gr-qc/0701170
http://www.lsc-group.phys.uwm.edu/daswg/projects/lal.html
Introduction
Analysis method
First stage: Grid based search
Second stage: Markov Chain Monte Carlo follow-up
Results
Conclusions
|
0704.0049 | An algorithm for the classification of smooth Fano polytopes | An algorithm for the classification of smooth Fano
polytopes
Mikkel Øbro
October 24, 2018
Abstract
We present an algorithm that produces the classification list of
smooth Fano d-polytopes for any given d ≥ 1. The input of the algo-
rithm is a single number, namely the positive integer d. The algorithm
has been used to classify smooth Fano d-polytopes for d ≤ 7. There
are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256
isomorphism classes of smooth Fano 7-polytopes.
1 Introduction
Isomorphism classes of smooth toric Fano varieties of dimension d correspond
to isomorphism classes of socalled smooth Fano d-polytopes, which are fully
dimensional convex lattice polytopes in Rd, such that the origin is in the
interior of the polytopes and the vertices of every facet is a basis of the
integral lattice Zd ⊂ Rd. Smooth Fano d-polytopes have been intensively
studied for the last decades. They have been completely classified up to
isomorphism for d ≤ 4 ([1], [18], [3], [15]). Under additional assumptions
there are classification results valid in every dimension.
To our knowledge smooth Fano d-polytopes have been classified in the fol-
lowing cases:
• When the number of vertices is d+ 1, d+ 2 or d+ 3 ([9],[2]).
• When the number of vertices is 3d, which turns out to be the upper
bound on the number of vertices ([6]).
• When the number of vertices is 3d− 1 ([19]).
• When the polytopes are centrally symmetric ([17]).
• When the polytopes are pseudo-symmetric, i.e. there is a facet F ,
such that −F is also a facet ([8]).
• When there are many pairs of centrally symmetric vertices ([5]).
http://arxiv.org/abs/0704.0049v1
2 2 SMOOTH FANO POLYTOPES
• When the corresponding toric d-folds are equipped with an extremal
contraction, which contracts a toric divisor to a point ([4]) or a curve
([16]).
Recently a complete classification of smooth Fano 5-polytopes has been an-
nounced ([12]). The approach is to recover smooth Fano d-polytopes from
their image under the projection along a vertex. This image is a reflexive
(d− 1)-polytope (see [3]), which is a fully-dimensional lattice polytope con-
taining the origin in the interior, such that the dual polytope is also a lattice
polytope. Reflexive polytopes have been classified up to dimension 4 using
the computer program PALP ([10],[11]). Using this classification and PALP
the authors of [12] succeed in classifying smooth Fano 5-polytopes.
In this paper we present an algorithm that classifies smooth Fano d-polytopes
for any given d ≥ 1. We call this algorithm SFP (for Smooth Fano Poly-
topes). The input is the positive integer d, nothing else is needed. The
algorithm has been implemented in C++, and used to classify smooth Fano
d-polytopes for d ≤ 7. For d = 6 and d = 7 our results are new:
Theorem 1.1. There are 7622 isomorphism classes of smooth Fano 6-
polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
The classification lists of smooth Fano d-polytopes, d ≤ 7, are available on
the authors homepage: http://home.imf.au.dk/oebro
A key idea in the algorithm is the notion of a special facet of a smooth Fano
d-polytope (defined in section 3.1): A facet F of a smooth Fano d-polytope
is called special, if the sum of the vertices of the polytope is a non-negative
linear combination of vertices of F . This allows us to identify a finite subset
Wd of the lattice Z
d, such that any smooth Fano d-polytope is isomorphic to
one whose vertices are contained in Wd (theorem 3.6). Thus the problem of
classifying smooth Fano d-polytopes is reduced to the problem of considering
certain subsets of Wd.
We then define a total order on finite subsets of Zd and use this to define a
total order on the set of smooth Fano d-polytopes, which respects isomor-
phism (section 4). The SFP-algorithm (described in section 5) goes through
certain finite subsets of Wd in increasing order, and outputs smooth Fano
d-polytopes in increasing order, such that any smooth Fano d-polytope is
isomorphic to exactly one in the output list.
As a consequence of the total order on smooth Fano d-polytopes, the algo-
rithm needs not consult the previous output to check for isomorphism to
decide whether or not to output a constructed polytope.
2 Smooth Fano polytopes
We fix a notation and prove some simple facts about smooth Fano polytopes.
The convex hull of a set K ∈ Rd is denoted by convK. A polytope is
the convex hull of finitely many points. The dimension of a polytope P is
the dimension of the affine hull, affP , of the polytope P . A k-polytope is
a polytope of dimension k. A face of a polytope is the intersection of a
supporting hyperplane with the polytope. Faces of polytopes are polytopes.
Faces of dimension 0 are called vertices, while faces of codimension 1 and 2
are called facets and ridges, respectively. The set of vertices of a polytope
P is denoted by V(P ).
Definition 2.1. A convex lattice polytope P in Rd is called a smooth Fano
d-polytope, if the origin is contained in the interior of P and the vertices of
every facet of P is a Z-basis of the lattice Zd ⊂ Rd.
We consider two smooth Fano d-polytopes P1, P2 to be isomorphic, if there
exists a bijective linear map ϕ : Rd → Rd, such that ϕ(Zd) = Zd and
ϕ(P1) = P2.
Whenever F is a (d−1)-simplex in Rd, such that 0 /∈ affF , we let uF ∈ (R
be the unique element determined by 〈uF , F 〉 = {1}. For every w ∈ V(F )
we define uw
∈ (Rd)∗ to be the element where 〈uw
, w〉 = 1 and 〈uw
, w′〉 = 0
for every w′ ∈ V(F ), w′ 6= w. Then {uw
|w ∈ V(F )} is the basis of (Rd)∗
dual to the basis V(F ) of Rd.
When F is a facet of a smooth Fano polytope and v ∈ V(P ), we certainly
have 〈uF , v〉 ∈ Z and
〈uF , v〉 = 1 ⇐⇒ v ∈ V(F ) and 〈uF , v〉 ≤ 0 ⇐⇒ v /∈ V(F ).
The lemma below concerns the relation between the elements uF and uF ′ ,
when F and F ′ are adjacent facets.
Lemma 2.2. Let F be a facet of a smooth Fano polytope P and v ∈ V(F ).
Let F ′ be the unique facet which intersects F in a ridge R of P , v /∈ V(R).
Let v′ = V(F ′) \ V(R).
1. 〈uv
, v′〉 = −1.
2. 〈uF , v
′〉 = 〈uF ′ , v〉.
3. 〈uF ′ , x〉 = 〈uF , x〉+ 〈u
, x〉(〈uF , v
′〉 − 1) for any x ∈ Rd.
4. In particular,
• 〈uv
, x〉 < 0 iff 〈uF ′ , x〉 > 〈uF , x〉.
• 〈uv
, x〉 > 0 iff 〈uF ′ , x〉 < 〈uF , x〉.
• 〈uv
, x〉 = 0 iff 〈uF ′ , x〉 = 〈uF , x〉.
for any x ∈ Rd.
4 2 SMOOTH FANO POLYTOPES
5. Suppose x 6= v′ is a vertex of P where 〈uv
, x〉 < 0. Then 〈uF , v
〈uF , x〉.
Proof. The sets V(F ) and V(F ′) are both bases of the lattice Zd and the
first statement follows.
We have v + v′ ∈ span(F ∩ F ′), and then the second statement follows.
Use the previous statements to calculate 〈uF ′ , x〉.
〈uF ′ , x〉 = 〈uF ′ ,
w∈V(F )
〈uwF , x〉w〉
w∈V(F )\{v}
〈uwF , x〉+ 〈u
F , x〉〈uF ′ , v〉
= 〈uF , x〉+ 〈u
F , x〉
〈uF ′ , v〉 − 1
= 〈uF , x〉+ 〈u
F , x〉
〈uF , v
′〉 − 1
As 〈uF , v
′〉 − 1 < 0 the three equivalences follow directly.
Suppose there is a vertex x ∈ V(P ), such that 〈uv
, x〉 < 0 and 〈uF , v
〈uF , x〉. Then
〈uF ′ , x〉 = 〈uF , x〉+ 〈u
F , x〉(〈uF , v
′〉 − 1) ≥ 〈uF , x〉 − (〈uF , v
′〉 − 1) ≥ 1.
Hence x is on the facet F ′. But this cannot be the case as V(F ′) = {v′} ∪
V(F ) \ {v}. Thus no such x exists.
And we’re done.
In the next lemma we show a lower bound on the numbers 〈uw
, v〉, w ∈ V(F ),
for any facet F and any vertex v of a smooth Fano d-polytope.
Lemma 2.3. Let F be a facet and v a vertex of a smooth Fano polytope P .
〈uwF , v〉 ≥
0 〈uF , v〉 = 1
−1 〈uF , v〉 = 0
〈uF , v〉 〈uF , v〉 < 0
for every w ∈ V(F ).
Proof. When 〈uF , v〉 = 1 the statement is obvious.
Suppose 〈uF , v〉 = 0 and 〈u
, v〉 < 0 for some w ∈ V(F ). Let F ′ be the
unique facet intersecting F in the ridge conv{V(F ) \ {w}}. By lemma 2.2
〈uF ′ , v〉 > 0. As 〈uF ′ , v〉 ∈ Z we must have 〈uF ′ , v〉 = 1. This implies
〈uF , v〉 = −1.
Suppose 〈uF , v〉 < 0 and 〈u
, v〉 < 〈uF , v〉 ≤ −1 for some w ∈ V(F ). Let
F ′ 6= F be the facet containing the ridge conv{V(F ) \ {w}}, and let w′ be
the unique vertex in V(F ′) \ V(F ). Then by lemma 2.2
〈uF ′ , v〉 = 〈uF , v〉 + 〈u
F , v〉(〈uF , w
′〉 − 1) ≥ 〈uF , v〉 − 〈u
F , v〉.
If 〈uF , v〉 − 〈u
, v〉 > 0, then v is on the facet F ′. But this is not the case
as 〈uw
, v〉 < −1. We conclude that 〈uw
, v〉 ≥ 〈uF , v〉.
When F is a facet and v a vertex of a smooth Fano d-polytope P , such that
〈uF , v〉 = 0, we can say something about the face lattice of P .
Lemma 2.4 ([7] section 2.3 remark 5(2), [13] lemma 5.5). Let F be a facet
and v be vertex of a smooth Fano polytope P . Suppose 〈uF , v〉 = 0.
Then conv{{v} ∪ V(F ) \ {w}} is a facet of P for every w ∈ V(F ) with
, v〉 = −1.
Proof. Follows from the proof of lemma 2.3.
3 Special embeddings of smooth Fano polytopes
In this section we find a concrete finite subset Wd of Z
d with the nice prop-
erty that any smooth Fano d-polytope is isomorphic to one whose vertices
are contained in Wd. The problem of classifying smooth Fano d-polytopes
is then reduced to considering subsets of Wd.
3.1 Special facets
The following definition is a key concept.
Definition 3.1. A facet F of a smooth Fano d-polytope P is called special,
if the sum of the vertices of P is a non-negative linear combination of V(F ),
that is
v∈V(P )
w∈V(F )
aww , aw ≥ 0.
Clearly, any smooth Fano d-polytope has at least one special facet.
Let F be a special facet of a smooth Fano d-polytope P . Then
0 ≤ 〈uF ,
v∈V(P )
v〉 = d+
v∈V(P ),〈uF ,v〉<0
〈uF , v〉,
which implies −d ≤ 〈uF , v〉 ≤ 1 for any vertex v of P . By using the lower
bound on the numbers 〈uw
, v〉, w ∈ V(F ) (see lemma 2.3), we can find an
explicite finite subset of the lattice Zd, such that every v ∈ V(P ) is contained
in this subset. In the following lemma we generalize this observation to
subsets of V(P ) containing V(F ).
Lemma 3.2. Let P be a smooth Fano polytope. Let F be a special facet of
P and let V be a subset of V(P ) containing V(F ), whose sum is ν.
〈uF , ν〉 ≥ 0
6 3 SPECIAL EMBEDDINGS OF SMOOTH FANO POLYTOPES
〈uwF , ν〉 ≤ 〈uF , ν〉+ 1
for every w ∈ V(F ).
Proof. For convenience we set U = V(P ) \ V and µ =
v∈U v. Since F is a
special facet we know that
0 ≤ 〈uF ,
v∈V(P )
v〉 = 〈uF , ν〉+ 〈uF , µ〉.
The set V(F ) is contained in V so 〈uF , v〉 ≤ 0 for every v in U , hence
〈uF , ν〉 ≥ 0.
Suppose that for some w ∈ V(F ) we have 〈uw
, ν〉 > 〈uF , ν〉+ 1. By lemma
2.3 we know that
〈uwF , v〉 ≥
−1 〈uF , v〉 = 0
〈uF , v〉 〈uF , v〉 < 0
for every vertex v ∈ V(P ) \ V(F ). There is at most one vertex v of P ,
〈uF , v〉 = 0, with negative coefficient 〈u
, v〉 (lemma 2.4). So
〈uwF , µ〉 ≥ 〈uF , µ〉 − 1.
Now, consider 〈uw
v∈V(P ) v〉.
〈uwF ,
v∈V(P )
v〉 = 〈uwF , ν〉+ 〈u
F , µ〉 > 〈uF , ν〉+ 〈uF , µ〉 = 〈uF ,
v∈V(P )
But this implies that 〈ux
v∈V(P ) v〉 is negative for some x ∈ V(F ). A
contradiction.
Corollary 3.3. Let F be a special facet and v any vertex of a smooth Fano
d-polytope. Then −d ≤ 〈uF , v〉 ≤ 1 and
〈uF , v〉
≤ 〈uwF , v〉 ≤
1 , 〈uF , v〉 = 1
d− 1 , 〈uF , v〉 = 0
d+ 〈uF , v〉 , 〈uF , v〉 < 0
for every w ∈ V(F ).
Proof. For 〈uF , v〉 = 1 the statement is obvious. When 〈uF , v〉 = 0 the
coefficients of v with respect to the basis V(F ) is bounded below by −1
(lemma 2.3), so no coefficient exceeds d− 1.
So the case 〈uF , v〉 < 0 remains. The lower bound is by lemma 2.3. Use
lemma 3.2 on the subset V = V(F ) ∪ {v} to prove the upper bound.
3.2 Special embeddings 7
3.2 Special embeddings
Let (e1, . . . , ed) be a fixed basis of the lattice Z
d ⊂ Rd.
Definition 3.4. Let P be a smooth Fano d-polytope. Any smooth Fano
d-polytope Q, with conv{e1, . . . , ed} as a special facet, is called a special
embedding of P , if P and Q are isomorphic.
Obviously, for any smooth Fano polytope P , there exists at least one special
embedding of P . As any polytope has finitely many facets, there exists only
finitely many special embeddings of P .
Now we define a subset of Zd which will play an important part in what
follows.
Definition 3.5. By Wd we denote the maximal set (with respect to inclu-
sion) of lattice points in Zd such that
1. The origin is not contained in Wd.
2. The points in Wd are primitive lattice points.
3. If a1e1 + . . .+ aded ∈ Wd, then −d ≤ a ≤ 1 for a = a1 + . . .+ ad and
≤ ai ≤
1 , a = 1
d− 1 , a = 0
d+ a , a < 0
for every i = 1, . . . , d.
The next theorem is one of the key results in this paper. It allows us to
classify smooth Fano d-polytopes by considering subsets of the explicitely
given set Wd.
Theorem 3.6. Let P be an arbitrary smooth Fano d-polytope, and Q any
special embedding of P . Then V(Q) is contained in the set Wd.
Proof. Follows directly from corollary 3.3 and the definition of Wd.
4 Total ordering of smooth Fano polytopes
In this section we define a total order on the set of smooth Fano d-polytopes
for any fixed d ≥ 1.
Throughout the section (e1, . . . , ed) is a fixed basis of the lattice Z
8 4 TOTAL ORDERING OF SMOOTH FANO POLYTOPES
4.1 The order of a lattice point
We begin by defining a total order � on Zd.
Definition 4.1. Let x = x1e1 + . . . + xded, y = y1e1 + . . . + yded be two
lattice points in Zd. We define x � y if and only if
(−x1 − . . .− xd, x1, . . . , xd) ≤lex (−y1 − . . .− yd, y1, . . . , yd),
where ≤lex is the lexicographical ordering on the product of d + 1 copies of
the ordered set (Z,≤).
The ordering � is a total order on Zd.
Example. (0, 1) ≺ (−1, 1) ≺ (1,−1) ≺ (−1, 0).
Let V be any nonempty finite subset of lattice points in Zd. We define max V
to the maximal element in V with respect to the ordering�. Similarly, minV
is defined to be the minimal element in V .
A important property of the ordering is shown in the following lemma.
Lemma 4.2. Let P be a smooth Fano d-polytope, such that conv{e1, . . . , ed}
is a facet of P . For every 1 ≤ i ≤ d, let vi 6= ei denote the vertex of P , such
that conv{e1, . . . , ei−1, vi, ei+1, . . . , ed} is a facet of P .
Then vi = min{v ∈ V(P ) | 〈u
, v〉 < 0}.
Proof. By lemma 2.2.(1) the vertex vi is in the set {v ∈ V(P ) | 〈u
, v〉 < 0},
and by lemma 2.2.(5) and the definition of the ordering �, vi is the minimal
element in this set.
In fact, we have chosen the ordering � to obtain the property of lemma 4.2,
and any other total order on Zd having this property can be used in what
follows.
4.2 The order of a smooth Fano d-polytope
We can now define an ordering on finite subsets of Zd. The ordering is
defined recursively.
Definition 4.3. Let X and Y be finite subsets of Zd. We define X � Y if
and only if X = ∅ or
Y 6= ∅ ∧ (minX ≺ minY ∨ (minX = minY ∧X\{minX} � Y \{min Y })).
Example. ∅ ≺ {(0, 1)} ≺ {(0, 1), (−1, 1)} ≺ {(0, 1), (1,−1)} ≺ {(−1, 1)}.
When W is a nonempty finite set of subsets of Zd, we define maxW to be the
maximal element in W with respect to the ordering of subsets �. Similarly,
minW is the minimal element in W .
Now, we are ready to define the order of a smooth Fano d-polytope.
4.3 Permutation of basisvectors and presubsets 9
Definition 4.4. Let P be a smooth Fano d-polytope. The order of P ,
ord(P ), is defined as
ord(P ) := min{V(Q) | Q a special embedding of P}.
The set is non-empty and finite, so ord(P ) is well-defined.
Let P1 and P2 be two smooth Fano d-polytopes. We say that P1 ≤ P2 if
and only if ord(P1) � ord(P2). This is indeed a total order on the set of
isomorphism classes of smooth Fano d-polytopes.
4.3 Permutation of basisvectors and presubsets
The group Sd of permutations of d elements acts on Z
d is the obvious way
by permuting the basisvectors:
σ.(a1e1 + . . .+ aded) := a1eσ(1) + . . .+ adeσ(d) , σ ∈ Sd.
Similarly, Sd acts on subsets of Z
σ.X := {σ.x | x ∈ X}.
In this notation we clearly have for any special embedding P of a smooth
Fano d-polytope
ord(P ) � min{σ.V(P ) | σ ∈ Sd}.
Let V and W be finite subsets of Zd. We say that V is a presubset of W , if
V ⊆ W and v ≺ w whenever v ∈ V and w ∈ W \ V .
Example. {(0, 1), (−1, 1)} is a presubset of {(0, 1), (−1, 1), (1,−1)}, while
{(0, 1), (1,−1)} is not.
Lemma 4.5. Let P be a smooth Fano polytope. Then every presubset V of
ord(P ) is the minimal element in {σ.V | σ ∈ Sd}.
Proof. Let ord(P ) = {v1, . . . , vn}, v1 ≺ . . . ≺ vn. Suppose there exists a
permutation σ and a k, 1 ≤ k ≤ n, such that
σ.{v1, . . . , vk} = {w1, . . . , wk} ≺ {v1, . . . , vk},
where w1 ≺ . . . ≺ wk. Then there is a number j, 1 ≤ j ≤ k, such that
wi = vi for every 1 ≤ i < j and wj ≺ vj.
Let σ act on {v1, . . . , vn}.
σ.{v1, . . . , vn} = {x1, . . . , xn} , x1 ≺ . . . ≺ xn.
Then xi � vi for every 1 ≤ i < j and xj ≺ vj. So σ.ord(P ) ≺ ord(P ), but
this contradicts the definition of ord(P ).
10 5 THE SFP-ALGORITHM
5 The SFP-algorithm
In this section we describe an algorithm that produces the classification list
of smooth Fano d-polytopes for any given d ≥ 1. The algorithm works by
going through certain finite subsets of Wd in increasing order (with respect
to the ordering defined in the previous section). It will output a subset V
iff convV is a smooth Fano d-polytope P and ord(P ) = V .
Throughout the whole section (e1, . . . , ed) is a fixed basis of Z
d and I denotes
the (d− 1)-simplex conv{e1, . . . , ed}.
5.1 The SFP-algorithm
The SFP-algorithm consists of three functions,
SFP, AddPoint and CheckSubset.
The finite subsets of Wd are constructed by the function AddPoint, which
takes a subset V , {e1, . . . , ed} ⊆ V ⊆ Wd, together with a finite set F ,
I ∈ F , of (d − 1)-simplices in Rd as input. It then goes through every v in
the set
{v ∈ Wd | max V ≺ v}
in increasing order, and recursively calls itself with input V ∪ {v} and some
set F ′ of (d − 1)-simplices of Rd, F ⊆ F ′. In this way subsets of Wd are
considered in increasing order.
Whenever AddPoint is called, it checks if the input set V is the vertex set of
a special embedding of a smooth Fano d-polytope P such that ord(P ) = V ,
in which case the polytope P = convV is outputted.
For any given integer d ≥ 1 the function SFP calls the function AddPoint
with input {e1, . . . , ed} and {I}. In this way a call SFP(d) will make the
algorithm go through every finite subset of Wd containing {e1, . . . , ed}, and
smooth Fano d-polytopes are outputted in strictly increasing order.
It is vital for the effectiveness of the SFP-algorithm, that there is some
efficient way to check if a subset V ⊆ Wd is a presubset of ord(P ) for some
smooth Fano d-polytope P . The function AddPoint should perform this
check before the recursive call AddPoint(V,F ′).
If P is any smooth Fano d-polytope, then any presubset V of ord(P ) is the
minimal element in the set {σ.V |σ ∈ Sd} (by lemma 4.5). In other words, if
there exists a permutation σ such that σ.V ≺ V , then the algorithm should
not make the recursive call AddPoint(V ).
But this is not the only test we wish to perform on a subset V before the
recursive call. The function CheckSubset performs another test: It takes
a subset V , {e1, . . . , ed} ⊆ V ⊆ Wd as input together with a finite set of
(d−1)-simplices F , I ∈ F , and returns a set F ′ of (d−1)-simplices containing
F , if there exists a special embedding P of a smooth Fano d-polytope, such
5.2 An example of the reasoning in CheckSubset 11
1. V is a presubset of V(P )
2. F is a subset of the facets of P
This is proved in theorem 5.1. If no such special embedding exists, then
CheckSubset returns false in many cases, but not always! Only when
CheckSubset(V,F) returns a set F ′ of simplices, we allow the recursive
call AddPoint(V,F ′).
Given input V ⊆ Wd and a set F of (d − 1)-simplices of R
d, the function
CheckSubset works in the following way: Suppose V is a presubset of V(P )
for some special embedding P of a smooth Fano d-polytope and F is a subset
of the facets of P . Deduce as much as possible of the face lattice of P and
look for contradictions to the lemmas stated in section 2. The more facets
we know of P , the more restrictions we can put on the vertex set V(P ), and
then on V . If a contradiction arises, return false. Otherwise, return the
deduced set of facets of P .
The following example illustrates how the function CheckSubset works.
5.2 An example of the reasoning in CheckSubset
Let d = 5 and V = {v1, . . . , v8}, where
v1 = e1 , v2 = e2 , v3 = e3 , v4 = e4 , v5 = e5
v6 = −e1 − e2 + e4 + e5 , v7 = e2 − e3 − e4 , v8 = −e4 − e5.
Suppose P is a special embedding of a smooth Fano 5-polytope, such that
V is a presubset of V(P ). Certainly, the simplex I is a facet of P .
Notice, that V does not violate lemma 3.2.
v1 + . . . + v8 = e2 + e5.
If V did contradict lemma 3.2, then the polytope P could not exist, and
CheckSubset(V, {I}) should return false.
For simplicity we denote any k-simplex conv{vi1 , . . . , vik} by {i1, . . . , ik}.
Since 〈uI , v6〉 = 0, the simplices F1 = {2, 3, 4, 5, 6} and F2 = {1, 3, 4, 5, 6}
are facets of P (lemma 2.4).
There are exactly two facets of P containing the ridge {1, 2, 4, 5}. One of
them is I. Suppose the other one is {1, 2, 4, 5, 9}, where v9 is some lattice
point not in V , v9 ∈ V(P ). Then 〈uI , v9〉 > 〈uI , v7〉 by lemma 2.2.(5)
and then v9 ≺ v7 by the definition of the ordering of lattice points Z
But then V is not a presubset of V(P ). This is the nice property of the
ordering of Zd, and the reason why we chose it as we did. We conclude that
F3 = {1, 2, 4, 5, 7} is a facet of P , and by similar reasoning F4 = {1, 2, 3, 5, 8}
and F5 = {1, 2, 3, 4, 8} are facets of P .
12 5 THE SFP-ALGORITHM
Now, for each of the facets Fi and every point vj ∈ V , we check if 〈uFi , vj〉 =
0. If this is the case, then by lemma 2.4 conv({vj} ∪ V(Fi) \ {w}) is a facet
of P for every w ∈ V(Fi) where 〈u
, vj〉 < 0. In this way we get that
{2, 4, 5, 6, 7} , {1, 4, 5, 6, 7} , {1, 2, 3, 7, 8} , {1, 3, 5, 7, 8}
are facets of P .
We continue in this way, until we cannot deduce any new facet of P . Every
time we find a new facet F we check that v is beneath F (that is 〈uF , v〉 ≤ 1)
and that lemma 2.3 holds for any v ∈ V . If not, then CheckSubset(V, {I})
should return false.
If no contradiction arises, CheckSubset(V, {I}) returns the set of deduced
facets.
5.3 The SFP-algorithm in pseudo-code
Input: A positive integer d.
Output: A list of special embeddings of smooth Fano d-polytopes, such that
1. Any smooth Fano d-polytope is isomorphic to one and only one poly-
tope in the output list.
2. If P is a smooth Fano d-polytope in the output list, then V(P ) =
ord(P ).
3. If P1 and P2 are two non-isomorphic smooth Fano d-polytopes in the
output list and P1 preceeds P2 in the output list, then ord(P1) ≺
ord(P2).
SFP ( an integer d ≥ 1 )
1. Construct the set V = {e1, . . . , ed} and the simplex I = convV .
2. Call the function AddPoint(V, {I}).
3. End program.
AddPoint ( a subset V where {e1, . . . , ed} ⊆ V ⊆ Wd , a set of (d − 1)-
simplices F in Rd where I ∈ F )
1. If P = conv(V(V )) is a smooth Fano d-polytope and V(V ) = ord(P ),
then output P .
2. Go through every v ∈ Wd, maxV(V ) ≺ v, in increasing order with
respect to the ordering ≺:
(a) If CheckSubset(V ∪ {v},F) returns false, then goto (d). Oth-
erwise let F ′ be the returned set of (d− 1)-simplices.
5.4 Justification of the SFP-algorithm 13
(b) If V ∪ {v} 6= min{σ.(V ∪ {v}) | σ ∈ Sd}, then goto (d).
(c) Call the function AddPoint(V ∪ {v},F ′).
(d) Let v be the next element in Wd and go back to (a).
3. Return
CheckSubset ( a subset V where {e1, . . . , ed} ⊆ V ⊆ Wd , a set of (d− 1)-
simplices F in Rd where I ∈ F )
1. Let ν =
v∈V v.
2. If 〈uI , ν〉 < 0, then return false.
3. If 〈u
, ν〉 > 1 + 〈uI , ν〉 for some i, then return false.
4. Let F ′ = F .
5. For every i ∈ {1, . . . , d}: If the set {v ∈ V |〈u
, v〉 < 0} is equal to
{max V }, then add the simplex conv({max V } ∪ V(I) \ {ei}) to F
6. If there exists F ∈ F ′ such that V(F ) is not a Z-basis of Zd, then
return false.
7. If there exists F ∈ F ′ and v ∈ V such that 〈uF , v〉 > 1, then return
false.
8. If there exists F ∈ F ′, v ∈ V and w ∈ V(F ), such that
〈uwF , v〉 <
0 〈uF , v〉 = 1
−1 〈uF , v〉 = 0
〈uF , v〉 〈uF , v〉 < 0
then return false.
9. If there exists F ∈ F ′, v ∈ V and w ∈ V(F ), such that 〈uF , v〉 = 0 and
, v〉 = −1, then consider the simplex F ′ = conv({v}∪V(F ) \ {w}).
If F ′ /∈ F ′, then add F ′ to F ′ and go back to step 6.
10. Return F ′.
5.4 Justification of the SFP-algorithm
The following theorems justify the SFP-algorithm.
Theorem 5.1. Let P be a special embedding of a smooth Fano d-polytope
and V a presubset of V(P ), such that {e1, . . . , ed} ⊆ V . Let F be a set of
facets of P .
Then CheckSubset(V,F) returns a subset F ′ of the facets of P and F ⊆ F ′.
14 6 CLASSIFICATION RESULTS AND WHERE TO GET THEM
Proof. By lemma 3.2 the subset V will pass the tests in step 2 and 3 in
CheckSubset.
The function CheckSubset constructs a set F ′ of (d−1)-simplices contain-
ing the input set F . We now wish to prove that every simplex F in F ′ is a
facet of P : By the assumptions the subset F ⊆ F ′ consists of facets of P .
Consider the addition of a simplex Fi, 1 ≤ i ≤ d, in step 5:
Fi = conv({max V } ∪ V(I) \ {ei}).
As maxV is the only element in the set {v ∈ V |〈uei
, v〉 < 0} and V is a
presubset of V(P ), Fi is a facet of P by lemma 4.2.
Consider the addition of simplices in step 9: If F is a facet of P , then by
lemma 2.4 the simplex conv({v} ∪ V(F ) \ {w}) is a facet of P .
By induction we conclude, that every simplex in F ′ is a facet of P . Then
any simplex F ∈ F ′ will pass the tests in steps 6–8 (use lemma 2.3 to see
that the last test is passed).
This proves the theorem.
Theorem 5.2. The SFP-algorithm produces the promised output.
Proof. Let P be a smooth Fano d-polytope. Clearly, P is isomorphic to at
most one polytope in the output list.
Let Q be a special embedding of P such that V(Q) = ord(P ). We need to
show that Q is in the output list. Let V(Q) = {e1, . . . , ed, q1, . . . , qk}, where
q1 ≺ . . . ≺ qk, and let Vi = {e1, . . . , ed, q1, . . . , qi} for every 1 ≤ i ≤ k.
Certainly the function AddPoint has been called with input {e1, . . . , ed}
and {I}.
By theorem 5.1 the function call CheckSubset(V1 , {I}) returns a set F1 of
(d − 1)-simplices which are facets of Q, I ⊂ F1. By lemma 4.5 the set V1
passes the test in 2b in AddPoint. Then AddPoint is called recursively
with input V1 and F1.
The call CheckSubset(V1,F1) returns a subset F2 of facets of Q, and the
set V2 passes the test in 2b in AddPoint. So the call AddPoint(V2,F2) is
made.
Proceed in this way to see that the call AddPoint(Vk ,Fk) is made, and then
the polytope Q = convVk is outputted in step 1 in AddPoint.
6 Classification results and where to get them
A modified version of the SFP-algorithm has been implemented in C++,
and used to classify smooth Fano d-polytopes for d ≤ 7. On an average
home computer our program needs less than one day (january 2007) to con-
struct the classification list of smooth Fano 7-polytopes. These lists can be
downloaded from the authors homepage: http://home.imf.au.dk/oebro
REFERENCES 15
An advantage of the SFP-algorithm is that it requires almost no memory:
When the algorithm has found a smooth Fano d-polytope P , it needs not
consult the output list to decide whether to output the polytope P or not.
The construction guarentees that V(P ) = min{σ.V(P ) | σ ∈ Sd} and it
remains to check if V(P ) = ord(P ). Thus there is no need of storing the
output list.
The table below shows the number of isomorphism classes of smooth Fano
d-polytopes with n vertices.
n d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7
4 2 1
5 1 4 1
6 1 7 9 1
7 4 28 15 1
8 2 47 91 26 1
9 27 268 257 40
10 10 312 1318 643
11 1 137 2807 5347
12 1 35 2204 19516
13 5 771 26312
14 2 186 14758
15 39 4362
16 11 1013
17 1 214
18 1 43
Total 1 5 18 124 866 7622 72256
References
[1] V. V. Batyrev, Toroidal Fano 3-folds, Math. USSR-Izv. 19 (1982), 13–
[2] V. V. Batyrev, On the classification of smooth projective toric varieties,
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[3] V. V. Batyrev, On the classification of toric Fano 4-folds, J. Math. Sci.
(New York) 94 (1999), 1021–1050.
[4] L. Bonavero, Toric varieties whose blow-up at a point is Fano. Tohoku
Math. J. 54 (2002), 593–597.
16 REFERENCES
[5] C. Casagrande, Centrally symmetric generators in toric Fano varieties,
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Fourier 56 (2006), 121–130.
[7] O. Debarre, Toric Fano varieties in Higher dimensional varieties and
rational points, lectures of the summer school and conference, Budapest
2001, Bolyai Society Mathematical Studies 12, Springer, 2001.
[8] G. Ewald, On the classification of toric Fano varieties, Discrete Com-
put. Geom. 3 (1988), 49–54.
[9] P. Kleinschmidt, A classification of toric varieties with few generators,
Aequationes Math 35 (1988), no.2-3, 254–266.
[10] M. Kreuzer & H. Skarke, Classification of reflexive polyhedra in three
dimensions, Adv. Theor. Math. Phys. 2 (1998), 853–871.
[11] M. Kreuzer & H. Skarke, Complete classification of reflexive polyhedra
in four dimensions, Adv. Theor. Math. Phys. 4 (2000), 1209–1230.
[12] M. Kreuzer & B. Nill, Classification of toric Fano 5-folds, Preprint,
math.AG/0702890.
[13] B. Nill, Gorenstein toric Fano varieties, Manuscr. Math. 116 (2005),
183–210.
[14] B. Nill. Classification of pseudo-symmetric simplicial reflexive poly-
topes, Preprint, math.AG/0511294, 2005.
[15] H. Sato, Toward the classification of higher-dimensional Toric Fano
varieties,. Tohoku Math. J. 52 (2000), 383–413.
[16] H. Sato, Toric Fano varieties with divisorial contractions to curves.
Math. Nachr. 261/262 (2003), 163–170.
[17] V.E. Voskresenskij & A. Klyachko, Toric Fano varieties and systems of
roots. Math. USSR-Izv. 24 (1985), 221–244.
[18] K. Watanabe & M. Watanabe, The classification of Fano 3-folds with
torus embeddings, Tokyo Math. J. 5 (1982), 37–48.
[19] M. Øbro, Classification of terminal simplicial reflexive d-polytopes with
3d− 1 vertices, Preprint, math.CO/0703416.
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http://arxiv.org/abs/math/0703416
REFERENCES 17
Department of Mathematics
University of Århus
8000 Århus C
Denmark
E-mail address : [email protected]
Introduction
Smooth Fano polytopes
Special embeddings of smooth Fano polytopes
Special facets
Special embeddings
Total ordering of smooth Fano polytopes
The order of a lattice point
The order of a smooth Fano d-polytope
Permutation of basisvectors and presubsets
The SFP-algorithm
The SFP-algorithm
An example of the reasoning in CheckSubset
The SFP-algorithm in pseudo-code
Justification of the SFP-algorithm
Classification results and where to get them
|
0704.0050 | Intelligent location of simultaneously active acoustic emission sources:
Part II | Intelligent location of two simultaneously active
acoustic emission sources:
Part II
Tadej Kosel and Igor Grabec
Faculty of Mechanical Engineering, University of Ljubljana,
Aškerčeva 6, POB 394, SI-1001 Ljubljana, Slovenia
e-mail: [email protected]; [email protected]
Abstract— Part I describes an intelligent acoustic emission
locator, while Part II discusses blind source separation, time
delay estimation and location of two continuous acoustic emission
sources.
Acoustic emission (AE) analysis is used for characterization
and location of developing defects in materials. AE sources
often generate a mixture of various statistically independent
signals. A difficult problem of AE analysis is separation and
characterization of signal components when the signals from
various sources and the mode of mixing are unknown. Recently,
blind source separation (BSS) by independent component analysis
(ICA) has been used to solve these problems. The purpose of
this paper is to demonstrate the applicability of ICA to locate
two independent simultaneously active acoustic emission sources
on an aluminum band specimen. The method is promising for
non-destructive testing of aircraft frame structures by acoustic
emission analysis.
INTRODUCTION
A common goal of many non-destructive testing methods is
to detect defects in materials. Acoustic emission analysis (AE)
is a passive testing method used to locate and characterize
defects which emit sound[10].
There are many ways to deduce the location of an AE
source from electrical signals detected by a chain of sensors.
The corresponding problems may be classified by the type of
acoustic source mechanism as the location of a continuous
emission source, such as that generated by a leak, or as the
location of discrete emission, such as an AE burst caused by a
growing crack. This paper describes a method for processing
continuous AE signals to determine the time delay (T-D)
between signals and thus to provide information for location
of AE sources. It should be pointed out that application of
AE source characteristics, such as count, count rate, ampli-
tude distribution, and conventional time delay measurement,
becomes meaningless when dealing with continuous acoustic
sources.
The basic information for AE source location consists of
T-D between stress waves detected at different positions on
a specimen. In the case of only one active AE source, T-
D of continuous acoustic waves can be estimated using the
cross-correlation function (CCF) of sensor signals described
in Part I of this article[10], [7]. In the case of two (or
Manuscript generated: January 31, 2007
more) simultaneously active AE sources, this method is not
applicable, since analysis of the CCF leads only to the T-D
of the most powerful AE signal. Detection of simultaneously
active independent AE source signals therefore requires a more
sophisticated approach.
The purpose of our study was to find a suitable method for
processing a mixture of two simultaneously active continuous
AE signals to determine the T-D and, related to this, the
coordinates of both AE sources. We found that the Blind
Source Separation (BSS) method solves this problem satisfac-
torily. BSS is a general signal processing method involving
the recovery of the contributions of different sources from
a finite set of observations recorded by sensors, independent
of the propagation medium and without any prior knowledge
of the sources. BSS has already been successfully applied
in medicine, telecommunications, image processing etc[8].
However, it is also a promising method for AE analysis of
aircraft structures, because AE signals are often hidden in a
mixture of signals from various sources. BSS could extract
the specific signature of each AE source, which can further
be used for location and characterization purposes, or to
isolate AE sources from background noise. We conducted
experiments with BSS on an aluminum beam on which two
continuous AE sources were generated simultaneously by air
flow.
METHODS
In this section we explain two different methods for time
delay estimation of AE sources. The first method is based on
analysis of the CCF and is convenient for T-D estimation of
one active continuous AE source as is described in Part I[10],
[7], [12]. The CCF exhibits a peak when the delay parameter
compensates the T-D between the sensor signals [10]. The T-D
is thus determined by the position of the highest peak of the
CCF. The second method is based on BSS algorithm and is
convenient for T-D estimation of two (or more) simultaneously
active continuous AE sources[9]. Location of two simulta-
neously active AE sources was performed by an intelligent
locator based on a general regression neural network[5] as is
described in Part I.
Multichannel Blind Source Separation has recently received
increased attention due to the importance of its potential
http://arxiv.org/abs/0704.0050v1
applications[3]. It occurs in many fields of engineering and
applied sciences, including processing of signals from antenna
array, speech and geophysical data processing, noise reduction,
biological system analysis, etc. It consists of recovering signals
emitted by unknown sources and mixed by an unknown
medium (material where waves propagate), using only several
observations of the mixtures. The only assumptions made
are the linearity of the mixing system and the statistical
independence of original signals.
BSS methods may be classified in several ways. One
possible classification that can be made depends on whether
the mixtures are instantaneous or convolutive [4]. Convo-
lutive mixtures correspond to a mixing system with time
dependent memory. They represent a more general case than
instantaneous mixtures, and they have in particular acoustic
applications. Recently, the principle of independent component
analysis (ICA) was applied in BSS, and it was found to
be a simple and powerful tool[6]. This study deals with the
separation of two convolutively mixed independent continuous
AE signals by ICA and the intelligent locator was used to
locate two independent continuous AE sources based on T-D
The mixing and filtering processes of unknown input signals
sj(t) may have different mathematical or physical back-
grounds, depending on specific applications. In this paper,
we focus mainly on the simplest cases with n signals xi(t)
linearly mixed in n unknown statistically independent, zero
mean source signals sj(t). The composition is expressed in
matrix notation as x = A ∗ s [8], where ‘*’ denotes a
convolution, x = [x1(t), . . . , xn(t)]
T is the vector of sensor
signals, s = [s1(t), . . . , sn(t)]
T is the vector of source signals
and A is an unknown full rank n × n mixing matrix whose
elements are finite inpulse response (FIR) filters. We assume
that only vector x is available. The goal of ICA is to find a
matrix W , by which vector x can be transformed into source
signals u = W ∗ x.
Matrix W is simply the inverse of A. However, when noise
corrupts the signals, matrix W must be found by an optimal
statistical treatment of the inverse problem. The optimal ma-
trix W can be estimated by a feed-forward neural network
operating in the frequency domain. A learning algorithm with
Amari’s natural gradient can be written as[1]: ũ = W̃ · x̃,
W̃ (τ + 1) = W̃ (τ) + α∆W̃ (τ) + η∆W̃ (τ − 1), ∆W̃ =
[I − ỹ · ũH] W̃ , ỹ = tanh(ℜ[ũ]) + ı tanh(ℑ[ũ]), where α is
the learning rate, η is the constant of learning, I is the identity
matrix and the tilde ‘˜ ’represents a frequency domain.
The ICA algorithm runs off-line and proceeds as follows
[11] (Fig. 1):
1) Pre-process the time-domain input signals, x(t): sub-
stract the mean from each signal.
2) Initialize the frequency domain unmixing filters, W̃ .
3) Take a block of input data and convert it into the
frequency domain using the Fast Fourier Transform
(FFT).
4) Filter the frequency domain input block, x̃, through W̃
to get the estimated source signals, ũ.
5) Pass ũ through the frequency domain nonlinearity, ỹ.
6) Use W̃ , ũ and ỹ along with the natural gradient
extension [2] to compute the change in the unmixing
PSfrag replacements
pre-process initialize
unmixing
filters
filter
update rule
Fig. 1. Block diagram of ICA algorithm
filter, ∆W̃ .
7) Take the next block of input data, covert it into the
frequency domain, and proceed from step 4. Repeat this
process until the unmixing filters have converged upon
a solution, passing several times through the data.
8) Normalize W̃ and convert it back into the time domain,
using the Inverse Fast Fourier Transform (IFFT).
9) Convolve the time domain unmixing filters, W , with x
to get the estimated sources.
EXPERIMENTS
We performed experiments with two independent continu-
ous AE sources on an aluminum band of dimensions 4000×
40× 5mm3. Reflections at the end of the band were reduced
by wrapping the ends in putty. The testing area was on the
longitudinal axis in the middle of the band, where 23 holes of
diameter 2 mm and mutual separation 100 mm were prepared
as shown in Fig. 2.
PSfrag replacements
φ 2 mm bandl air flow
Fig. 2. AE generation by air flowing through the hole
Two AE sensors were mounted 100 mm away from the
terminal holes, that is 2.4 m from each other. The origin of
the coordinate system was in the middle of the band and the
testing area extended from −1.1m to +1.1m. AE signals
were excited by two independent air jets flowing through
the holes. The source position was arbitrarily selected at
+100mm and +800mm. Air jets were formed by two nozzles
of diameter 1 mm using pressure 7 bar. The experimental set-
up consisted of the test specimen (aluminum band), two AE
sensors (pinducers), two AE sources (air jets), two amplifiers,
a digital oscilloscope (A/D converter) and a computer (BSS
module, locator, plotter) as shown in Fig. 3. Three experiments
were performed : (1) T-D estimation using a CCF of two AE
signals that were not simultaneously active; (2) T-D estimation
using a CCF of two AE signals which were simultaneously
active and (3) T-D estimation of AE signals using ICA.
Location of sources, based on T-D, by the intelligent locator
was performed in all three cases.
PSfrag replacements
sensor
AE source
locator
plotter
Fig. 3. Experimental set-up
In the first experiment only one air jet was activated for
a particular measurement. In the second experiment both air
jets were activated. Sensor signals were linear convolutive
mixtures of two independent continuous AE sources as shown
in Fig. 4. The auto-correlation R11, R22 and cross-correlation
functions R12, R21 were calculated from sensor signals. Only
one T-D of two signals can be estimated from the highest
peak in both CCF, regardless of the number of independent
AE sources on the test specimen as shown in Fig. 5. This
means that a CCF can not be used for automatic T-D estima-
tion of multiple AE signals on the test specimen. The CCF
exhibits various peaks which belong to various independent
AE sources, but it is ussually impossible to relate these peaks
to corresponding coordinates of AE sources.
0 0.1 0.2 0.3 0.4
PSfrag replacements
t [ms]
(a) Sensory signal #1
0 0.1 0.2 0.3 0.4
PSfrag replacements
t [ms]
(b) Sensory signal #2
Fig. 4. Mixtures of two independent continuous AE sources aquired by two
sensors
In the third experiment the ICA algorithm was used to
solve this problem satisfactorily. The ICA algorithm results
in demixing FIR filters which extract the independent source
signals from sensory signals. By inverting the demixing filters
W we obtain mixing filters A. In the case of two independent
0 5000 10000 15000
PSfrag replacements
PSfrag replacements
Rx1x1
PSfrag replacements
Rx1x1
Rx1x2
0 5000 10000 15000
PSfrag replacements
Rx1x1
Rx1x2
Rx2x1
Fig. 5. Auto- and cross-correlation functions of sensory signals; down-arrow
marks the highest peak
AE sources and two sensors, the components of A are four
FIR mixing filters, as shown in Fig. 6. There are two direct
a11, a22 and two cross mixing filters a12, a21. The first index
of the filter represents the number of the sensor, while the
second index represents the number of the source. The position
of the highest peak of the cross FIR filters determines the T-
D between two signals from two sensors. If we substract the
coordinate of the highest peak of a direct mixing FIR filter
a11 from the coordinate of the highest peak of cross filter a21
we obtain the T-D of first independent AE source, since each
of the highest peaks in the FIR filters belongs to different
independent AE signals.
RESULTS
The results of T-D estimation of two continuous independent
AE sources are shown in Fig. 7. Three experiments were done.
In the first experiment, the T-D was estimated by a CCF of two
AE sources which were not active simultaneously as marked
by ‘◦’. Locations of these two sources estimated by the
intelligent locator were +181 mm and +784 mm. The second
experiment was performed with both AE sources active simul-
taneously. T-D were also estimated by a CCF. The highest peak
0 5000 10000 15000
PSfrag replacements
PSfrag replacements
PSfrag replacements
0 5000 10000 15000
0.5PSfrag replacements
Fig. 6. Mixing filters obtained by ICA of sensory signals; down-arrow marks
the highest peak
position corresponds to the source location marked by ‘− −’
and was +784 mm. The third experiment was performed using
ICA for T-D estimation and location by intelligent locator. The
result is marked by ‘�’. Estimated positions of this two sources
were +179 mm and +784 mm respectively. If we compare the
coordinates of both independent AE sources estimated by
the first experiment and by the third experiment, we find a
good correspondence. If we compare estimated AE source
coordinates with actual coordinates, which were +100 mm and
+800 mm respectively, we observe a slight disagreement due to
experimental error. Experimental error is about 3% regarding
the distance between sensors. Absolute error in this case is
79 mm and 16 mm respectively. The results also depend on
the number and distribution of prototype sources marked by
‘•’, which are essential for operation of the intelligent locator.
If the number of prototype sources is increased, location error
is reduced. In our case the prototype sources were distributed
along the beam from −1.1m to +1.1m separated by 0.1m, so
that systematic error of the locator was set to several procents.
PSfrag replacements
actual position l [m]
correlation function
Fig. 7. Results of location of two continuous independent AE sources.
Symbols: ‘�’ – AE sources obtained by ICA; ‘◦’ – estimated AE sources
obtained by cross-correlation function in two steps, when just one of two
AE sources was active at time of measurement; ‘− −’ – estimated AE
sources obtained by cross-correlation function when two AE sources were
active simultaneously; ‘•’ – prototype AE sources required for location using
intelligent locator; ‘−’ – distribution of actual sources.
DISCUSSION AND CONCLUSION
CCF is applicable to T-D estimation only in the case of one
active AE source. The goal of our research is to develop a new
method to estimate T-D between AE signals in the case of mul-
tiple simultaneously active continuous AE sources. We have
shown that, for this purpose, ICA is an applicable option. ICA
finds a linear coordinate system (the unmixing filters) such that
the resulting signals are statistically independent. This is an
advantage of ICA over CCF. It represents a new approach to
processing of AE data and further expands the applicability of
AE analysis in the field of non-destructive testing. In machines
or in an industrial environment, multiple sources are usually
active Simultaneously, often representing environmental dis-
turbances. The corresponding complex signals are not directly
applicable to characterization of particular sources. However,
separation of contributions by ICA analysis in fact represents a
kind of filtering, increasing the applicability of filtered signals
to characterization of sources in complex environments. Future
research will be focused on location of multiple AE sources
on two-dimensional and three-dimensional specimens.
REFERENCES
[1] Amari, S.-I. 1998 , Natural gradient works efficiently in learning, Neural
Computation 10, 251–276.
[2] Amari, S.-I., Cichocki, A. Yang, H. H. 1996 , A new learning algorithm
for blind signal separation, in D. Touretzky, M. Mozer M. Hasselmo, eds,
‘Advances in Neural Information Processing Systems’, Vol. 8, MIT Press,
Cambridge MA, pp. 752–763.
[3] Burel, G. 1992 , Blind separation of sources: A nonlinear algorithm,
Neural Networks 5, 937–947.
[4] Deville, Y. Charkani, N. 1997 , Analysis of the stability of time-
domain source separation algorithms for convolutively mixed signals, in
International Comference on Acoustics, Speech, and Signal Processing,
pp. 1835–1838.
[5] Grabec, I. Sachse, W. 1997 , Synergetics of Measurement, Prediction and
Control, Springer-Verlag, Berlin.
[6] Hyvarinen, A. Oja, E. 2000 , Independent component analysis: algorithms
and applications, Neural Networks 13, 411–430.
[7] Kosel, T., Grabec, I. Mužič, P. 2000 , Location of continuous acoustic
emission sources generated by air flow, Ultrasonics 38(1–8), 824–826.
[8] Lee, T.-W. 1998 , Independent Component Analysis, Theory and Appli-
cations, Kluwer Academic Publishers, Boston etc.
[9] Lee, T.-W., Bell, A. J. Lambert, R. 1997 , Blind separation of convolved
and delayed sources, Advances in Neural Information Processing Systems
9, 758–764.
[10] McIntire, P. Miller, R. K., eds 1987 , Acoustic Emission Testing,
Vol. 5 of Nondestructive Testing Handbook, 2 edn, American Society
for Nondestructive Testing, Philadelphia, USA.
[11] Westner, A. G. 1996 , Object-based audio capture: Separating
acoustically-mixed sources, MSc Thesis, Massachusetts Institute of Tech-
nology.
[12] Ziola, S. M. Gorman, M. R. 1991 , Source location in thin plates using
cross-correlation, J. Acoust. Soc. Am. 90(5), 2551–2556.
References
|
0704.0051 | Visualizing Teleportation | Visualizing Teleportation
Scott M. Cohen∗
Department of Physics, Duquesne University, Pittsburgh, Pennsylvania 15282 and
Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
A novel way of picturing the processing of quantum information is described, allowing a direct vi-
sualization of teleportation of quantum states and providing a simple and intuitive understanding
of this fascinating phenomenon. The discussion is aimed at providing physicists a method of ex-
plaining teleportation to non-scientists. The basic ideas of quantum physics are first explained in
lay terms, after which these ideas are used with a graphical description, out of which teleportation
arises naturally.
I. INTRODUCTION
One of the most exciting and fastest-growing fields of physics today is quantum information. Especially since the
discovery by Shor [1, 2] that there exist calculations for which a quantum computer is apparently far more efficient than
a classical computer, interest in understanding quantum information has increased at an impressive rate. One widely
publicized discovery that has emerged from work in this field is teleportation [3]. While not precisely equivalent to the
process enjoying widespread fame amongst fans of Star Trek (“Beam me up, Scotty”), the phenomenon referred to here
is nonetheless fascinating, and perhaps even astonishing. The reason for the widespread publicity of this rigorously
proven (and experimentally tested [4, 5, 6, 7, 8], though not yet unambiguously demonstrated) scientific prediction
is almost certainly in large part due to the fact that it shares the same name as the just-mentioned, intriguing idea
from science-fiction.
The usual way of describing teleportation is through mathematical equations, and this mathematics is relatively
straightforward, as has been amply demonstrated elsewhere [3, 9]. Hence, an understanding of this phenomenon is
accessible to physicists, other scientists, and those possessing a reasonably strong level of mathematical skill. There
does, on the other hand, seem to be a good deal of misunderstanding of teleportation amongst non-scientists, with the
notion floating around that the amazing phenomenon shown regularly in episodes of Star Trek – that is, of material
objects being teleported from one place to another – has actually turned out to be possible in real life. Nothing could
be further from the truth, of course, so we are left wondering how to rectify this unfortunate state of affairs. The
question I address here is the following: can the true (scientific) phenomenon of teleportation be understood by others,
those without much skill in mathematics? The usual explanations will certainly fail in this regard, even if carefully
presented by a competent physicist, because mathematics has a well-known tendency to scare people away, and in any
case, the mathematics of teleportation is not all that simple. The paper is addressed to physicists possessing a solid
understanding of quantum physics (including graduate students), with the aim to provide a method by which such a
physicist can explain teleportation to someone who is not mathematically inclined. Thus, the objective is ultimately,
though indirectly, to educate the general public about teleportation, and by extension, quantum mechanics itself. The
approach involves only the most basic ideas about quantum physics, and while it does not entirely avoid mathematical
expressions, it uses only the simplest mathematics (one only needs to accept that certain objects are either 0 or 1) and
relies almost entirely on “pictures”, allowing the layperson to visualize – and thus, understand – what is happening.
In the following sections, I will describe my method of directly visualizing teleportation. These sections are written
as if addressed to the layperson. The next section explains the probabilistic nature of quantum physics by considering
“quantum coins”, which are examples of two-level systems. This section describes how one should think about
measurements, what is meant by probabilities for classical systems, and then how these ideas can be used to describe
quantum systems. Then, in Section III, I present my graphical approach to understanding the dynamics of quantum
information processing, which is then used in Section III B to explain in pictures how teleportation of quantum states
is possible. One of the crucial observations will be that a shared entangled state on, say, systems a and b, provides
the parties with multiple “images” of the state of an additional system A. The ability to manipulate these images
– independently by each party, and differently from one image to the next – is what allows teleportation to be
accomplished. More generally, these ideas provide important insights into why entanglement is a valuable resource, as
I have described in detail elsewhere, and they have been useful in understanding other aspects of quantum information
∗Electronic address: [email protected]
http://arxiv.org/abs/0704.0051v2
mailto:[email protected]
processing [10, 11].
II. PROBABILITIES
Perhaps the most fundamental aspect of quantum theory is that it can only make predictions in terms of probabil-
ities. In general even if one has a complete description of the state of a quantum system, one will not know ahead of
time what the outcome of a given measurement will be. This is in direct contradiction with our everyday experience,
which we refer to as “classical”. For example, a flipped classical coin which lands heads (“heads” is then a complete
description of the state of this coin), is known with certainty to be heads, and also with certainty to not be tails.
That is, if we know the state of a classical coin (in this case “heads”), we can predict with certainty the answer to
any reasonable question we choose to ask (or “measure”) about that coin (for example, “Is it tails?”). We therefore
need to understand what is meant by the “state” of a quantum system and how this state relates to probabilities and
outcomes of measurements. The following definition of a measurement will be adequate for our purposes.
Definition: A measurement is a procedure that provides answers to a collection of yes-no questions, which is
both mutually exclusive (when the answer to one of the questions is “yes”, the answer to all the others is “no”) and
complete (all possibilities are included; that is, one of the questions will always be answered in the affirmative). The
single question that receives the “yes” answer is referred to as the outcome of the measurement.
For example, since a classical coin is either heads or tails, and these two possibilities are mutually exclusive, a
measurement on a classical coin is a procedure that answers the two questions “Is it heads?” and “Is it tails?” Since
the coin will always be one or the other, there will always be a “yes” answer to one of these questions, and then the
other question is always answered “no”. Hence these two questions do indeed constitute a measurement according to
the above definition. If “Is it heads?” is answered affirmatively, then “heads” is the outcome of the measurement.
It turns out that these two questions also constitute a measurement on quantum coins. However, in contrast to
the classical case in which this is the only possible measurement, there is a vast array of possible measurements on
quantum coins. This will become clearer from the discussion in the following sections, where we introduce a compact
way of describing these things, a way commonly used in quantum mechanics.
A. Classical coins and classical probabilities
Consider again a flipped classical coin. The coin lands either heads or tails. It will be useful to use a somewhat
abbreviated notation: |H〉 for heads and |T 〉 for tails. The statement that “if it is heads, it is not tails” (that is, has
zero probability of being tails) will be represented as
〈T |H〉 = 0.
The left-facing bracket |H〉 represents the known initial state (“It is heads.”) and the right-facing bracket 〈T | represents
the question (“Is it tails?”). The number (0) appearing on the right-hand side of the equal sign then gives the
probability that with this initial state, the answer to this question will be yes. For the above example, we have that
the probability is 0, which is as expected since when the coin is H it will never be T . Note that it is useful to use the
left- and right-facing brackets, so that we can easily read off what is the initial state and what is the question being
asked about it. Simply writing TH = 0 in the above equation would lead to confusion when we discuss two coins (see
below), which might have an initial state where one is tails, the other heads, represented by |TH〉.
Perhaps an even more trivial statement “if it is heads, then it is heads” (with certainty, or with probability one),
will similarly be represented as
〈H |H〉 = 1.
Again, the right-facing bracket contains the question 〈H |, or “Is it heads?”, and the fact that the expression is equal
to 1 indicates that the answer to this question will always be “yes” when the initial state is |H〉. These statements
are trivial because if we know the state of a classical coin, we can predict with certainty whether it will be heads or
tails when we look at it.
Although the remaining equations will look a bit more involved, the only mathematics the reader need understand
is contained in the above two equations, along with two others that are almost exactly the same. The discussion in the
remainder of this paper will follow from the four simple statements, 〈H |H〉 = 1, 〈T |T 〉 = 1, 〈T |H〉 = 0, 〈H |T 〉 = 0.
Next let us consider two coins. In this case, a complete list of mutually exclusive possibilities is HH, HT, TH, TT .
We can make statements in exactly the same way we did above, for example “if they are HH , then they are not HT ”,
which in our notation is written
〈H1T2|H1H2〉 = 〈H1|H1〉 × 〈T2|H2〉 = (1)× (0) = 0,
where the subscripts (1, 2) have been inserted for clarity to indicate which coin is which. Note that in this equation,
we have equated the expression 〈H1T2|H1H2〉 with the product of two expressions, 〈H1|H1〉 and 〈T2|H2〉. This is
because any question about the two coins jointly is the same as two questions, one about each of the coins separately.
It is obviously also true that “if they are HH , then they are HH”, so
〈H1H2|H1H2〉 = 〈H1|H1〉 × 〈H2|H2〉 = (1)× (1) = 1.
For three coins, there are eight possibilities (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT ) and the same
notation will readily account for this case, as well. We will not need to consider more than three coins here, though
it is in principle straightforward to do so.
B. Quantum coins and quantum probabilities
Quantum coins behave very differently as compared to their classical counterparts, and quantum probabilities must
be understood in very different ways. We still have heads and tails, |H〉 and |T 〉, as possible states of a quantum coin.
We refer to these two states as being “orthogonal” to each other, by which we simply mean that they are mutually
exclusive: if the quantum coin is H , it is definitely (with certainty) not T , and vice-versa. We note that the four
equations appearing in the previous section are equally true for both quantum and classical coins. However, there
now exist some very strange possibilities. If I were to suggest that a classical coin can be both H and T at one and
the same time, you would be completely justified in thinking I’d gone slightly crazy. I am going to tell you, though,
that at least in a certain (though very real) sense, this is the case for quantum coins (though you may still wonder a
bit about my sanity). The point is that, in the quantum case, it makes complete sense to ask questions such as: “If
the coin is H , is it half H and half T ?”; or we can turn this around and ask “If the coin is half H and half T , is it
H?” Neither of these questions makes any sense whatsoever when referred to a classical coin. On the other hand, for
a quantum coin these are not only legitimate questions, but they are in fact very important ones (we do not consider
the negligible possibility of a classical coin landing on its edge, and in any case this bears no relationship to what we
mean by a quantum coin being half H and half T ).
To represent these questions, we can write the state (Q) of a quantum coin that is half H and half T as
|Q〉 =
|T 〉.
Then the answer to the question, “If the coin is half H and half T , is it H?” is answered by the equation,
〈H |Q〉 = 〈H |
〈H |H〉+
〈H |T 〉
(1) +
(0) =
which should be interpreted as meaning “yes, with probability 1/2”, implying also “no, with probability 1−1/2 = 1/2”
[In quantum mechanics, it is actually the square of the object on the left-hand side of the foregoing equation that
represents the probability, rather than that object itself, which is known as the “probability amplitude”; however,
although the difference between probabilities and probability amplitudes is crucial to the understanding of quantum
mechanics, I have chosen in the present discussion to overlook this distinction for the benefit of the layperson to whom
these ideas are aimed, as they would only serve to complicate matters, causing unnecessary confusion amongst the
intended audience]. The left-facing bracket |Q〉 represents the known initial state, and the right-facing bracket 〈H |
represents the question (“Is it heads?”). The number 1/2 appearing on the right-hand side of the last line then gives
the probability that with this initial state, the answer to this question will be yes. The point to understand here is
that even though we have a complete description (Q) of the state of the quantum coin, we do not generally know in
advance whether the coin will be H or T when we look at it. We can only predict in terms of probabilities: if we
perform this experiment many times, half the time the answer will be yes and the other half of the time it will be no.
Furthermore, there are many more questions we can ask in the quantum, as compared to the classical, case. We are
no longer restricted to asking “is the coin H?” or “is it T ?”, but we can ask other questions, such as the reverse of
the question we just answered,
〈Q|H〉 =
〈H |+
〈H |H〉+
〈T |H〉
(1) +
(0) =
We see that the question “If the coin is H , is it half H and half T ?” has the same answer as the previous question:
“yes, with probability 1/2; and no, with probability 1/2.”
We note that in the remainder of the paper, instead of phrasing questions as “is the coin half H and half T ?”, we
instead ask whether it is “equal parts” H and T . While there is no real difference between these two questions, this
rephrasing allows us to simplify the notation by dispensing with the factors of 1/2 that have appeared in the above
discussion. In doing so, the equations will not yield the same numbers as probabilities for the various questions, but
this will not hamper the presentation since the numerical values of the probabilities are not crucial to the ideas we
wish to convey: we just need to remember that certain objects are equal to 1 and others are equal to 0.
III. TELEPORTATION
What exactly do we mean by teleportation in the context of quantum information? It is not a material object that
is being teleported, but rather the state of a quantum system. We will assume that the system is a quantum coin,
with a complete set of mutually exclusive (orthogonal) states being “heads” and “tails”, which we may denote as
|H〉 and |T 〉. Suppose Alice and Bob are physicists in locations widely separated from each other. They each have a
quantum coin – labeled a and b, respectively – and these two coins are in the state
|B0〉ab = |HaHb〉+ |TaTb〉,
where the subscripts used here refer to system a (b) in Alice’s (Bob’s) possession. This state of two quantum coins has
a very strange property, which is known as entanglement, and the state itself is an example of a maximally entangled
state. Entanglement is a rather strange sort of correlation between quantum systems, which manifests itself in the
state B0 by the fact that neither system a nor b can be considered to have a definite “state of its own” independent
of the other system: whatever is the state of coin a, coin b will have the same state, but one cannot say anything
about the state of either coin independent of the other one. It is this property of entanglement that is credited with
enabling Alice and Bob to accomplish teleportation.
Alice is given another coin (system A), prepared in a state
|SA〉 = cH |HA〉+ cT |TA〉
with arbitrary coefficients cH and cT that are completely unknown to her and to Bob. If cH = 1/2 and cT = 1/2,
we have the case discussed in the previous section, where the coin is equally likely to be found to be H or T . For
other values of these coefficients, the two possibilities will in general not be equally likely. Alice’s task is to perform
operations on the systems in her possession (a and A) in such a way that Bob will end up with his system (b) in
precisely the state |Sb〉, which is the same state as |SA〉, but now on the distant system b. It turns out that this task
can be accomplished if Alice communicates information to Bob (perhaps via a telephone) about what she ended up
doing to her systems, after which Bob performs a rather simple quantum operation, dependent on the information
obtained from Alice, on system b.
An important point to understand in what follows is that nothing either of them does in this process provides
even the slightest information about the coefficients cH and cT , so the state (S) that has been teleported remains
completely unknown to the parties. This aspect of teleportation becomes even more amazing if one considers the
amount of information that is conveyed: the information contained in a quantum state is far greater than the amount
actually transmitted from Alice to Bob via the telephone (as we will see below, the amount transmitted via the
telephone is two classical bits, enough to convey which one of four possibilities has been chosen). True, the classical
information one can encode in a two-level quantum system cannot exceed one bit (one bit is the amount of information
needed to choose between two possibilities, such as |H〉 and |T 〉). But if Alice wanted to tell Bob how to create the
state in his own lab by communicating with him over a phone line, this would require an infinite amount of classical
information; that is, enough information to completely describe the arbitrary numbers, cH and cT (it is infinite because
one of these numbers might well be an irrational number such as π, having a decimal expansion that is unending,
never repeating itself). Of course, Alice and Bob are both completely ignorant of what these numbers are, so even if
it were possible to transmit an infinite amount of information, they don’t even know what information they would
need to send! Nonetheless, when they share entanglement, it is possible for the two of them, by working together, to
create the unknown state on Bob’s coin b with the communication of only two classical bits.
A. Visualizing quantum information processing
Let us now introduce the pictorial method which will be used to visualize teleportation. The simple diagrams we
will use to depict states of multiple quantum coins, held by two different parties, are familiar to many researchers
working in quantum information. We will now illustrate how these diagrams are used to represent quantum states,
and then how they can be used to follow what happens to these coins when measurements are performed by one of
the parties. Then, we will be ready to use them for visualizing teleportation.
1. States of quantum coins
To depict the state of a single quantum coin labeled A (standing for Alice; she will also have the other coin labeled
a, while Bob’s single coin is labeled b), we may use a simple box diagram,
|SA〉 = cH |HA〉 + cT |TA〉 =
The coefficients cH and cT appearing in the boxes indicate “how much” is in that part of the state SA of coin A. The
next example illustrates the case where there are two coins (A and b) held by two different parties. Then, the state
of these two coins might be
|SAHb〉 =
|Hb〉 |Tb〉
with SA as given above. The empty squares on the right-hand side of this diagram represent the fact that system b is
“not T ” (has zero probability of being tails); the cH in the upper-left corner represents the probability the coins are
both heads; and the cT in the lower-left, the probability Bob’s coin is heads and Alice’s is tails.
If there are three parties involved, a three-dimensional cube could be used to represent this situation. However, it
will serve our present purposes to represent both of Alice’s systems along the vertical dimension of the diagram. We
might have coins A and b as in the previous example, and coin a being heads, the overall state of these three coins
represented as
|SAHaHb〉 =
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
If instead the a,b systems are both T , this picture is
|SATaTb〉 =
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
Now consider what happens if we add the previous two equations together. Then our two coins a,b are “equal parts in
HH and in TT ”, which is what we previously referred to as the “maximally entangled state” |B0〉ab = |HaHb〉+|TaTb〉.
The corresponding diagram looks like
|SA〉(|HaHb〉 + |TaTb〉) =
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
|Hb〉 |Tb〉
Notice how there are now two images of the state |SA〉. This observation turns out to be rather useful in understanding
entanglement [10, 11], but we will not need to discuss such issues here. Let us now look at how to represent
measurements by use of these diagrams.
2. Measurements on quantum coins
Suppose Alice and Bob share three quantum coins in the state represented in the last equation of the previous
section, and Alice wants to know something about her coins. If she measures coin a and discovers it is H , then we
〈Ha| ×
|Hb〉 |Tb〉
|Hb〉 |Tb〉
Recall that when the right-facing bracket 〈Ha| is attached to the left-facing one |Ha〉 on the left of this equation, we get
〈Ha|Ha〉 = 1, which “preserves” the upper row, whereas 〈Ha|Ta〉 = 0, indicating that the bottom row is annihilated
(multiplied by 0), which is why it no longer appears on the far right of this equation. The interpretation is as follows:
when the question “Is coin a heads?” is answered in the affirmative the other coins are left in the state |SAHb〉. We
see how this measurement acts on both of the images simultaneously, rather than on the two independently. The
upper-left image has been preserved intact, but the other image was annihilated, disappearing altogether.
On the other hand, if the outcome of Alice’s measurement had been that coin a was T , this would be represented
〈Ta| ×
|Hb〉 |Tb〉
|Hb〉 |Tb〉
In this case, the upper-left image has disappeared and the one in the lower-right has been preserved intact. In each
of these cases, the state of coin A is unchanged, but that of coin b is left in a state that corresponds directly to the
outcome of Alice’s measurement on a. If she discovers that coin a was H (or T ), then coin b ends up H (or T ).
Alternatively, she could do a measurement that includes the question “Is coin a equal parts H and T ?” If the answer
to this question is yes, then
(〈Ha|+ 〈Ta|) ×
|Hb〉 |Tb〉
|Hb〉 |Tb〉
SA SA = |SA〉 (|Hb〉 + |Tb〉) ,
which is just a sum of the previous two equations (notice how after each of the three measurement outcomes we have
just considered, the two images have been collapsed into a single row). Once again we see that the state of coin b
ends up corresponding to the outcome of Alice’s measurement on coin a. This illustrates some of the strangeness
that resides in entangled states of quantum systems: no matter what measurement Alice makes on coin a and no
matter what outcome she obtains from that measurement, the resulting state of coin b will correspond directly to
that outcome.
The way the images of SA appear in the diagram is crucial. The fact that the two start out in different rows
and in different columns will be important in what is to come. If entanglement between systems a,b was absent, for
example if they were in the (unentangled) state (|Ha〉+ |Ta〉) |Hb〉, then this would be represented by (recall that
|SA〉 = cH |HA〉+ cT |TA〉)
|SA〉 (|Ha〉 + |Ta〉) |Hb〉 =
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
|Hb〉 |Tb〉
Under these circumstances, Bob’s view of the lower image of SA is “obstructed” by the presence of the upper image;
the two images effectively appear as one to him. As will become clear in the following section, the presence of
entanglement between the a,b coins will be necessary for them to accomplish teleportation. We will see that it is
Bob’s (and Alice’s) ability to “see” the two images separately, and the consequent ability for each of them to act
differently on one of the images as compared to the other, that is crucial to their success.
In the next section, we turn to the task of teleporting the state SA onto Bob’s coin b. To begin this process,
Alice will perform a measurement that asks “joint” questions; that is, questions about both coins in her possession
simultaneously. As an example, she could ask if they are both H . That is,
〈HAHa| ×
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
|Hb〉 |Tb〉
The cH appearing in the box on the right corresponds to the probability that the answer to this question will be
“yes”. More important for our purposes is to recognize that when this is the outcome of the measurement, coin b
ends up H , once again a consequence of the initial entanglement between coins a and b.
Now let us see how teleportation is possible.
B. Visualizing teleportation
Teleportation is accomplished with the aid of the extra systems a, b in the entangled state |B0〉ab. System A starts
in state |SA〉, discussed above, and this is the state they will teleport. Alice will ask a set of joint questions, which
together constitute a measurement, about the state of the two coins in her possession, a and A. The first question
she asks is whether these two coins are equal parts HH and TT . When the answer is yes, we have
(〈HAHa| + 〈TATa|) ×
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
|Hb〉 |Tb〉
cH cT
Notice how the middle two rows are annihilated by this outcome (because these rows correspond to a situation where
the two coins are different – one H and one T – whereas we are asking if they are the same), and the remaining
rows are collapsed into a single row. Now, if we look carefully (or perhaps, not even so carefully) at the final
diagram in this picture, we will arrive at a rather startling conclusion. We see that the state of Bob’s system b
is now |Sb〉 = cH |Hb〉 + cT |Tb〉. That is, the unknown state |SA〉, originally on system A, is now on Bob’s system
b. Furthermore, the question asked by Alice had nothing whatsoever to do with the coefficients cH and cT , which
determine what the original state of coin A was. Hence, the parties remain completely ignorant of the state S, yet
that state has been successfully teleported!
We are not quite finished, however, since we would like for Alice and Bob to be able to teleport no matter which
joint question ends up being the outcome of Alice’s measurement. Because of the probabilistic nature of the quantum
world, she cannot choose the outcome of her measurement. Instead, Alice effectively asks all of the questions in her
chosen measurement and then must wait for Nature to decide which question she (Nature, that is) will choose as the
outcome. The nice thing about Nature is that she will tell Alice which question was chosen.
There must be four questions in a complete set of questions making up a joint measurement on coins A, a. Let me
illustrate with one other question how Alice and Bob can succeed with teleportation, and then the reader is asked to
believe that they can also succeed with either of the remaining two questions (these can be treated in a very similar
way to the one shown here [12]). The second question is: Are coins A, a equal parts TH and HT ? The corresponding
diagram is
(〈TAHa| + 〈HATa|) ×
|Hb〉 |Tb〉
|TAHa〉
|HAHa〉
|TATa〉
|HATa〉
|Hb〉 |Tb〉
cT cH
Here, the first and last rows are annihilated by this outcome, and the middle two are collapsed into a single row.
Looking at the final diagram, we see that coin b is left in the state cT |Hb〉 + cH |Tb〉, which has the coefficients cH
and cT exchanged in comparison to the state S that we want it to be in. However, Alice knows that this is the
question to which Nature answered yes, and she can call Bob on the telephone and inform him of this fact. Once
he knows this is the question that was answered affirmatively, all he needs to do is “flip his quantum coin”. Recall
that this is a quantum coin, which he cannot simply pick up and turn over. Instead what we mean by this is that he
exchanges H for T and vice-versa. In turns out that this is a legitimate action that can be performed on a quantum
coin, and it will leave coin b in the desired state: cT |Hb〉 + cH |Tb〉 → cT |Tb〉 + cH |Hb〉 = |Sb〉. Notice also that they
again remain completely ignorant of the coefficients cH and cT – nothing that has happened has provided them with
any such information, nor have they needed it. It turns out that no matter which of the four outcomes of Alice’s
measurement she obtains, once she informs Bob of that outcome, he will be able to perform a legitimate action on
his quantum coin that will leave his coin in the state |Sb〉. All Bob needs to know, in order to choose which action
to perform, is the outcome of Alice’s measurement: Alice needs only to send him two bits of information, enough to
choose between one of the four possible outcomes. Furthermore, none of the four outcomes provides either party with
any information about the coefficients, cH and cT , so they both remain completely ignorant of the original state they
have just successfully teleported.
The diagrams provide a great deal of insight into what is going on. The crucial observation is the presence of two
images of the state S, resulting from the entanglement between coins a, b. Alice does a measurement that, while not
acting independently on these two images, does act differently on them, as we alluded to above. This measurement
picks out different parts of S from the different images in a way such that all of S is preserved and none of it is
repeated. For example, in the previous example, the H part is preserved from the lower-right image, and the T part
from the upper-left one (and vice-versa in the example before that). This suggests (and it is indeed the case) that for
coins that have more than two sides (a six-sided quantum die, for example), the parties can teleport the state of such
an n-sided coin by sharing a maximally-entangled state that is “large enough” to provide them with n images of the
unknown state S. Then Alice can design her measurement such that for each outcome: (1) a different part of S is
extracted from each image; and (2) the whole state is preserved across the n images. Afterward, Bob can recover the
state simply by rearranging the various parts, which he will be able to do once Alice informs him of the outcome of
her measurement. Alice’s measurement does not provide them with any information about the original state they are
attempting to teleport, nor does Bob’s rearrangement require that they know anything about it. In all cases, they
remain ignorant of the state they are teleporting. The reader is encouraged to draw a diagram (perhaps for coins
with n = 3 sides each; see the following paragraph) and follow through the argument to be sure it is clear how this
is done. The diagram will have n × n = n2 horizontal rows (representing Alice’s two n-sided coins A, a, each row
corresponding to one of the combinations of sides of these two coins: HH, HT, HU, HV, · · ·, where H,T, U, V, · · ·
label the various sides), and n vertical columns (representing Bob’s coin b).
A complete measurement for the n = 3 case will include n2 = 9 outcomes, but an essentially complete understanding
can be gained if the reader considers only the three outcomes corresponding to (1) 〈HAHa| + 〈TATa| + 〈UAUa|; (2)
〈HATa|+ 〈TAUa|+ 〈UAHa|; and (3) 〈HAUa|+ 〈TAHa|+ 〈UATa|, where U is the third side of these coins (the other six
outcomes involve additional complications that I have not explained here, but these outcomes are not crucial for the
general kind of understanding we are aiming for here). In this case the appropriate generalization of B0 is the state
|HaHb〉+ |TaTb〉+ |UaUb〉, and the three terms in this expression yield the three images needed for teleportation.
IV. TELEPORTING CLASSICAL COINS
In this section, I consider teleportation of classical coins, which turns out to be possible using a method that bears
a striking resemblance to the method used for quantum coins. Imagine that Chloe prepares a classical coin (labeled
A) as either H or T , and gives it to Alice, who is not allowed to look at the coin. Chloe also prepares classical coins
a and b such that they are either HH (both H) or TT (both T ). She then gives coin a to Alice and coin b to Bob,
but again does not allow these parties to look at their coins. Alice now asks Chloe the following two questions: Are
coins A and a the same? Or are they different? This pair of “yes-no” questions represents a measurement, as defined
earlier, on this pair of coins. If Chloe informs her they are the same, then Alice knows that coin b, which is guaranteed
to be the same as a, is also the same as A; if, on the other hand, Chloe says coins A and a are different, then coin b
is also different from A. Alice now calls Bob on the telephone and tells him to “flip” or “don’t flip”. In the first case
(A same as a) she tells him not to flip, while when A and a are different, she tells him to flip. After he follows her
instruction, Bob’s coin b will with certainty match coin A. The state of coin A has been teleported onto coin b.
It is instructive to look at why the quantum case is astonishing while the classical one is rather mundane. There
are three important differences between classical and quantum teleportation. The first difference has to do with the
information that Alice would need to transmit to Bob in order to inform him of the state of coin A, if she happened
to know that state. For a classical coin, there are only two possibilities, H or T , so she would need to transmit only
one bit to Bob. This is the same amount of information that is actually transmitted when she tells him “flip” or
“don’t flip” – again, two possibilities. In contrast, as was discussed at the beginning of Section III for the case of
quantum coins, it would require an infinite amount of information for Alice to inform Bob of the state of coin A,
whereas she only actually transmits two bits of information when informing him which of her four questions was the
outcome of her measurement. We see that the two cases, classical vs. quantum, are dramatically different in terms of
the amounts of information involved.
The second difference between these two cases is a bit more subtle. In the classical case, if Alice were to cheat
and actually look at coin A, she would automatically know what state that coin is in and be able to tell Bob what
to do with his coin – turn it H or turn it T ; another one-bit message encompassing these two possibilities. This
absolutely will not work for a quantum coin, which Alice cannot simply “look” at to discover its state. The reason
is the following: To begin with, in contrast to a classical coin, when Alice looks at her quantum coin, she invariably
disturbs it in the process. That is, no matter what state the coin was in before she looked at it, the state after she
looks at it is with certainty given by the outcome of her measurement. For example, even if the state is “equal parts
H and T ” before she asks if it is H or T , if the answer is H (T ), then the coin is now H (T ). Or if it is H to begin
with and she does a measurement that answers yes to the question “Is it equal parts H and T ?”, then the state of
the coin will now be equal parts H and T . Hence, when she looks at it, she will get one of two answers (those being
the two possible outcomes of her measurement) as to the state of the coin, but if she looks at it wrong, that answer
will not tell her what the state was beforehand, but only what it is now. Furthermore, since she has now disturbed
the state, there is no way to go back and try again, since the coin is now in a completely different state than the one
she is trying to discover. The moral of this story is twofold: With quantum coins (1) don’t bother trying to cheat;
and (2) there’s no point in asking for a “do-over”.
The third difference between the quantum and classical cases is even more subtle and is related to entanglement,
for which there is no counterpart with classical coins. For classical teleportation, Chloe must tell Alice whether or not
coins a,A are the same or different. When these coins are classical, and since Chloe is the one that prepared them,
she is certainly able to do so. However, in the quantum case coins a, b are entangled, which means that neither one
has a definite state of its own. Since coin a does not have a definite state, the question whether coins a,A are the
same (have the same state) has no answer ! Even if we assume that Chloe prepared coins a, b in their entangled state,
there is nothing whatsoever that she (or anyone else) can say about the state of coin a, except that it is entangled
with b and has no definite state of its own. It is worth noting that in both the quantum and classical cases, coins a, b
are correlated with each other in ways that at first glance appear to be very similar – when one is measured and found
to be H (T ), the other one will also be H (T ). Nonetheless, the correlations present in the entangled state B0 of
quantum coins a, b have no analog in the case of classical coins. One reason is precisely what we have just discussed:
that the quantum coins can be correlated in this way even though neither of the individual coins has a definite state
of its own (a classical coin always has a definite state of its own).
V. CONCLUSION
I have described a novel way to visualize the processing of quantum information, and used this picture to give a simple
way to “see” how teleportation is possible. The picture turns out to be useful beyond just providing an understanding
of previously known phenomena (teleportation), however. Indeed, it has given us a deeper understanding of the
process of deterministically implementing non-local unitaries by local operations and classical communication (when
shared entanglement is available as a resource), allowing us to construct new protocols [10] that go far beyond what
was previously known to be possible [13]. We have also used this picture to study the question of what entanglement
resources are required to locally implement other non-local operations, such as measurement protocols for the purpose
of distinguishing sets of quantum states that are indistinguishable without the extra entangled resource [11].
Acknowledgments
This work has been supported in part by the National Science Foundation through Grant No. PHY-0456951. I am
very grateful for numerous discussions with Bob Griffiths and others in his research group.
[1] P. W. Shor, in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, edited by S. Goldwasser
(IEEE Computer Society, Los Alamitos, CA, 1994).
[2] E. Gerjuoy, Am. J. Phys. 73, 521 (2005).
[3] C. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).
[4] M. Barrett et al., Nature 429, 737 (2004).
[5] M. Riebe et al., Nature 429, 734 (2004).
[6] D. Bouwmeester et al., Nature 390, 575 (1997).
[7] A. Furusawa et al., Science 282, 706 (1998).
[8] J. F. Sherson et al., Nature 443, 557 (2006).
[9] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge,
UK, 2000).
[10] L. Yu, R. B. Griffiths, and S.M. Cohen, to be published.
[11] S. M. Cohen, Phys. Rev. A 77, 012304: 1 (2008).
[12] The other two questions involve minus signs and are represented as (3) 〈HAHa| − 〈TATa| and (4) 〈HATa| − 〈TAHa|,
respectively. Multiplying one or the other of the states H and T by a minus sign is another valid quantum operation,
which Bob can perform on his coin. By doing so, he effectively turns each of these cases into one of the two already shown
explicitly in the paper. When question (3) is answered affirmatively, multiplication of Tb by the minus sign is all that is
needed to complete the teleportation; and for question (4), the minus sign operation need only be followed by Bob flipping
his quantum coin.
[13] B. Reznik, Y. Aharonov, and B. Groisman, Phys. Rev. A 65, 032312: 1 (2002).
Introduction
Probabilities
Classical coins and classical probabilities
Quantum coins and quantum probabilities
Teleportation
Visualizing quantum information processing
States of quantum coins
Measurements on quantum coins
Visualizing teleportation
Teleporting classical coins
Conclusion
Acknowledgments
References
|
0704.0052 | Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries | QUANTUM FIELD THEORY ON CURVED
BACKGROUNDS. II.
SPACETIME SYMMETRIES
ARTHUR JAFFE AND GORDON RITTER
Abstract. We study space-time symmetries in scalar quantum field
theory (including interacting theories) on static space-times. We first
consider Euclidean quantum field theory on a static Riemannian mani-
fold, and show that the isometry group is generated by one-parameter
subgroups which have either self-adjoint or unitary quantizations. We
analytically continue the self-adjoint semigroups to one-parameter uni-
tary groups, and thus construct a unitary representation of the isometry
group of the associated Lorentzian manifold. The method is illustrated
for the example of hyperbolic space, whose Lorentzian continuation is
Anti-de Sitter space.
1. Introduction
The extension of quantum field theory to curved space-times has led to
the discovery of many qualitatively new phenomena which do not occur
in the simpler theory on Minkowski space, such as Hawking radiation; for
background and historical references, see [2, 6, 18].
The reconstruction of quantum field theory on a Lorentz-signature space-
time from the corresponding Euclidean quantum field theory makes use of
Osterwalder-Schrader (OS) positivity [15, 16] and analytic continuation. On
a curved background, there may be no proper definition of time-translation
and no Hamiltonian; thus, the mathematical framework of Euclidean quan-
tum field theory may break down. However, on static space-times there is a
Hamiltonian and it makes sense to define Euclidean QFT. This approach was
recently taken by the authors [11], in which the fundamental properties of
Osterwalder-Schrader quantization and some of the fundamental estimates
of constructive quantum field theory1 were generalized to static space-times.
The previous work [11], however, did not address the analytic continuation
which leads from a Euclidean theory to a real-time theory. In the present
article, we initiate a treatment of the analytic continuation by constructing
unitary operators which form a representation of the isometry group of the
Lorentz-signature space-time associated to a static Riemannian space-time.
Our approach is similar in spirit to that of Fröhlich [4] and of Klein and
Date: February 22, 2007.
1For background on constructive field theory in flat space-times, see [8, 9].
http://arxiv.org/abs/0704.0052v1
2 ARTHUR JAFFE AND GORDON RITTER
Landau [13], who showed how to go from the Euclidean group to the Poincaré
group without using the field operators on flat space-time.
This work also has applications to representation theory, as it provides
a natural (functorial) quantization procedure which constructs nontrivial
unitary representations of those Lie groups which arise as isometry groups
of static, Lorentz-signature space-times. These groups are typically non-
compact. For example, when applied to AdSd+1, our procedure gives a
unitary representation of the identity component of SO(d, 2). Moreover,
our procedure makes use of the Cartan decomposition, a standard tool in
representation theory.
2. Classical Space-Time
2.1. Structure of Static Space-Times.
Definition 2.1. A quantizable static space-time is a complete, con-
nected orientable Riemannian manifold (M,gab) with a globally-defined (smooth)
Killing field ξ which is orthogonal to a codimension-one hypersurface Σ ⊂M ,
such that the orbits of ξ are complete and each orbit intersects Σ exactly
once.
Throughout this paper, we assume that M is a quantizable static space-
time. Definition 2.1 implies that there is a global time function t defined
up to a constant by the requirement that ξ = ∂/∂t. Thus M is foliated by
time-slices Mt, and
M = Ω− ∪ Σ ∪Ω+
where the unions are disjoint, Σ =M0, and Ω± are open sets corresponding
to t > 0 and t < 0 respectively. We infer existence of an isometry θ which
reverses the sign of t,
θ : Ω± → Ω∓ such that θ
2 = 1, θ|Σ = id.
Fix a self-adjoint extension of the Laplacian, and let C = (−∆ +m2)−1
be the resolvent of the Laplacian (also called the free covariance), where
m2 > 0. Then C is a bounded self-adjoint operator on L2(M). For each s ∈
R, the Sobolev space Hs(M) is a real Hilbert space, defined as completion
of C∞c (M) in the norm
(2.1) ‖f‖2s = 〈f,C
−sf〉.
The inclusion Hs →֒ Hs+k for k > 0 is Hilbert-Schmidt. Define S :=⋂
s<0Hs(M) and S
s>0Hs(M). Then
S ⊂ H−1(M) ⊂ S
form a Gelfand triple, and S is a nuclear space.
Recall that S ′ has a natural σ-algebra of measurable sets (see for instance
[7, 8, 17]). There is a unique Gaussian probability measure µ with mean
zero and covariance C defined on the cylinder sets in S ′ (see [7]).
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES3
More generally, one may consider a non-Gaussian, countably-additive
measure µ on S ′ and the space
E := L2(S ′, µ).
We are interested in the case that the monomials of the form A(Φ) =
Φ(f1) . . .Φ(fn) for fi ∈ S are all elements of E , and for which their span
is dense in E . This is of course true if µ is the Gaussian measure with
covariance C.
For an open set Ω ⊂ M , let EΩ denote the closure in E of the set of
monomials A(Φ) =
iΦ(fi) where supp(fi) ⊂ Ω for all i. Of particular
importance for Euclidean quantum field theory is the positive-time subspace
E+ := EΩ+ .
2.2. The Operator Induced by an Isometry. Isometries of the under-
lying space-time manifold act on a Hilbert space of classical fields arising in
the study of a classical field theory. For f ∈ C∞(M) and ψ : M → M an
isometry, define
fψ ≡ (ψ−1)∗f = f ◦ ψ−1.
Since det(dψ) = 1, the operation f → fψ extends to a bounded operator
on H±1(M) or on L
2(M). A treatment of isometries for static space-times
appears in [11].
Definition 2.2. Let ψ be an isometry, and A(Φ) = Φ(f1) . . .Φ(fn) ∈ E a
monomial. Define the induced operator
(2.2) Γ(ψ)A ≡ Φ(f1
ψ) . . .Φ(fn
and extend Γ(ψ) by linearity to the domain of polynomials in the fields,
which is dense in E .
3. Osterwalder-Schrader Quantization
3.1. Quantization of Vectors (The Hilbert Space H of Quantum
Theory). In this section we define the quantization map E+ → H , where
H is the Hilbert space of quantum theory. The existence of the quantization
map relies on a condition known as Osterwalder-Schrader (or reflection)
positivity. A probability measure µ on S ′ is said to be reflection positive if
(3.1)
Γ(θ)F F dµ ≥ 0
for all F in the positive-time subspace E+ ⊂ E . Let Θ = Γ(θ) be the
reflection on E induced by θ. Define the sesquilinear form (A,B) on E+×E+
as (A,B) = 〈ΘA,B〉E , so (3.1) states that (F,F ) ≥ 0.
Assumption 1 (O-S Positivity). Any measure dµ that we consider is re-
flection positive with respect to the time-reflection Θ.
4 ARTHUR JAFFE AND GORDON RITTER
Definition 3.1 (OS-Quantization). Given a reflection-positive measure dµ,
the Hilbert space H of quantum theory is the completion of E+/N with
respect to the inner product given by the sesquilinear form (A,B). Denote
the quantization map Π for vectors E+ → H by Π(A) = Â, and write
(3.2) 〈Â, B̂〉H = (A,B) = 〈ΘA,B〉E for A,B ∈ E+ .
3.2. Quantization of Operators. The basic quantization theorem gives a
sufficient condition to map a (possibly unbounded) linear operator T on E
to its quantization, a linear operator T̂ on H . Consider a densely-defined
operator T on E , the unitary time-reflection Θ, and the adjoint T+ = ΘT ∗Θ.
A preliminary version of the following was also given in [10].
Definition 3.2 (Quantization Condition I). The operator T satisfies QC-I
i. The operator T has a domain D(T ) dense in E .
ii. There is a subdomain D0 ⊂ E+ ∩D(T )∩D(T
+), for which D̂0 ⊂ H
is dense.
iii. The transformations T and T+ both map D0 into E+.
Theorem 3.3 (Quantization I). If T satisfies QC-I, then
i. The operators T ↾D0 and T
+↾D0 have quantizations T̂ and T̂+ with
domain D̂0.
ii. The operators T̂ ∗ =
T̂ ↾D̂0
and T̂+ agree on D̂0.
iii. The operator T̂ ↾D0 has a closure, namely T̂
Proof. We wish to define the quantization T̂ with the putative domain D̂0
(3.3) T̂ Â = T̂A .
For any vector A ∈ D0 and for any B ∈ (D0 ∩ N ), it is the case that
 = Â+B. The transformation T̂ is defined by (3.3) iff T̂A = ̂T (A+B) =
T̂A+ T̂B. Hence one needs to verify that T : D0 ∩ N → N , which we now
The assumption D0 ⊂ D(T
+), along with the fact that Θ is unitary,
ensures that ΘD0 ⊂ D(T
∗). Therefore for any F ∈ D0,
(3.4)
〈ΘF, TB〉E = 〈T
∗ΘF,B〉E = 〈Θ(ΘT
∗ΘF ) , B〉E = 〈ΘT
+F,B〉E = 〈T̂
+F, B̂〉H .
In the last step we use the fact assumed in part (iii) of QC-I that T+ :
D0 → E+, yielding the inner product of two vectors in H . We infer from
the Schwarz inequality in H that
|〈ΘF, TB〉E | ≤ ‖T̂+F‖H ‖B̂‖H = 0 .
As 〈ΘF, TB〉E = 〈F̂ , T̂B〉H , this means that T̂B ⊥ D̂0. As D̂0 is dense in
H by QC-I.ii, we infer T̂B = 0. In other words, TB ∈ N as required to
define T̂ .
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES5
In order show that D̂0 ⊂ D(T̂
∗), perform a similar calculation to (3.4)
with arbitrary A ∈ D0 replacing B, namely
(3.5)
〈F̂ , T̂ Â〉H = 〈ΘF, TA〉E = 〈Θ(ΘT
∗ΘF ) , A〉E = 〈ΘT
+F,A〉E = 〈T̂+F, Â〉H .
The right side is continuous in  ∈ H , and therefore F̂ ∈ D(T ∗). Further-
more T ∗F̂ = T̂+F . This identity shows that if F ∈ N , then T̂+F = 0.
Hence T+↾D0 has a quantization T̂
+, and we may write (3.5) as
(3.6) T ∗F̂ = T̂+F̂ , for all F ∈ D0 .
In particular T̂ ∗ is densely defined so T̂ has a closure. This completes the
proof. �
Definition 3.4 (Quantization Condition II). The operator T satisfies QC-II
i. Both the operator T and its adjoint T ∗ have dense domains D(T ),D(T ∗) ⊂
ii. There is a domain D0 ⊂ E+ in the common domain of T , T
+, T+T ,
and TT+.
iii. Each operator T , T+, T+T , and TT+ maps D0 into E+.
Theorem 3.5 (Quantization II). If T satisfies QC-II, then
i. The operators T ↾D0 and T
+↾D0 have quantizations T̂ and T̂+ with
domain D̂0.
ii. If A,B ∈ D0, one has 〈B̂, T̂ Â〉H = 〈T̂+B̂, Â〉H .
Remarks.
i. In Theorem 3.5 we drop the assumption that the domain D̂0 is dense,
obtaining quantizations T̂ and T̂+ whose domains are not necessarily
dense. In order to compensate for this, we assume more properties
concerning the domain and the range of T+ on E .
ii. As D̂0 need not be dense in H , the adjoint of T̂ need not be defined.
Nevertheless, one calls the operator T̂ symmetric in case one has
(3.7) 〈B̂, T̂ Â〉H = 〈T̂ B̂, Â〉H , for all A,B ∈ D0 .
iii. If Ŝ ⊃ T̂ is a densely-defined extension of T̂ , then Ŝ∗ = T̂+ on the
domain D̂0.
Proof. We define the quantization T̂ with the putative domain D̂0. As in
the proof of Theorem 3.3, this quantization T̂ is well-defined iff it is the case
that T : D0 ∩ N → N . For any F ∈ D0 ∩ N , by definition ‖F̂‖H = 0.
〈TF, TF 〉H = (TF, TF ) = 〈ΘTF, TF 〉E = 〈F, T
∗ΘTF 〉
where one uses the fact that D0 ⊂ D(T
+T ). Thus
〈TF, TF 〉H =
ΘF, T+TF
= 〈F, T+TF 〉H .
6 ARTHUR JAFFE AND GORDON RITTER
Here we use the fact that T+T maps D0 to E+. Thus one can use the
Schwarz inequality on H to obtain
〈TF, TF 〉H ≤ ‖F̂‖H ‖T̂
= 0 .
Hence T : D0 ∩ N → N , and T has a quantization T̂ with domain D̂0.
In order verify that T+↾D0 has a quantization, one needs to show that
T+ : D0 ∩N ⊂ N . Repeat the argument above with T
+ replacing T . The
assumption TT+ : D0 → E+ yields for F ∈ D0 ∩ N ,
〈T+F, T+F 〉H = 〈T
∗ΘF, T+F 〉E = 〈ΘF, TT
+F 〉E = 〈F̂ , T̂ T
+F 〉H .
Use the Schwarz inequality in H to obtain the desired result that
〈T+F, T+F 〉H ≤ ‖F̂‖H ‖T̂ T
= 0 .
Hence T+ has a quantization T̂+ with domain D̂0, and for B ∈ D0 one has
T̂+B = T̂+B̂. In order to establish (ii), assume that A,B ∈ D0. Then
〈B̂, T̂ Â〉H = 〈ΘB,TA〉E = 〈Θ(ΘT
∗ΘB) , A〉E = 〈ΘT
+B,A〉E
= 〈T̂+B, Â〉H = 〈T̂+B̂, Â〉H .(3.8)
This completes the proof.
4. Structure and Representation of the Lie Algebra of
Killing Fields
For the remainder of this paper we assume the following, which is clearly
true in the Gaussian case as the Laplacian commutes with the isometry
group G. (A further explanation was given in [11].)
Assumption 2. The isometry groups G that we consider leave the measure
dµ invariant, in the sense that Γ, defined above, is a unitary representation
of G on E .
4.1. The Representation of g on E .
Lemma 4.1. Let Gi be an analytic group with Lie algebra gi (i = 1, 2),
and let λ : g1 → g2 be a homomorphism. There cannot exist more than one
analytic homomorphism π : G1 → G2 for which dπ = λ. If G1 is simply
connected then there is always one such π.
Let D = d/dt denote the canonical unit vector field on R. Let G be a
real Lie group with algebra g, and let X ∈ g. The map tD → tX(t ∈ R)
is a homomorphism of Lie(R) → g, so by the Lemma there is a unique
analytic homomorphism ξX : R → G such that dξX(D) = X. Conversely,
if η is an analytic homomorphism of R → G, and if we let X = dη(D),
it is obvious that η = ξX . Thus X 7→ ξX is a bijection of g onto the set
of analytic homomorphisms R → G. The exponential map is defined by
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES7
exp(X) := ξX(1). For complex Lie groups, the same argument applies,
replacing R with C throughout.
Since g is connected, so is exp(g). Hence exp(g) ⊆ G0, where G0 denotes
the connected component of the identity in G. It need not be the case for
a general Lie group that exp(g) = G0, but for a large class of examples
(the so-called exponential groups) this does hold. For any Lie group, exp(g)
contains an open neighborhood of the identity, so the subgroup generated
by exp(g) always coincides with G0.
We will apply the above results with G = Iso(M), the isometry group of
M , and g = Lie(G) the algebra of global Killing fields. Thus we have a bijec-
tive correspondence between Killing fields and 1-parameter groups of isome-
tries. This correspondence has a geometric realization: the 1-parameter
group of isometries
φs = ξX(s) = exp(sX)
corresponding to X ∈ g is the flow generated by X.
Consider the two different 1-parameter groups of unitary operators:
(1) the unitary group φ∗s on L
2(M), and
(2) the unitary group Γ(φs) on E .
Stone’s theorem applies to both of these unitary groups to yield densely-
defined self-adjoint operators on the respective Hilbert spaces.
In the first case, the relevant self-adjoint operator is simply an extension
of −iX, viewed as a differential operator on C∞c (M). This is because for
f ∈ C∞c (M) and p ∈M , we have:
Xpf = (LXf)(p) =
f(φs(p))|s=0.
Thus −iX is a densely-defined symmetric operator on L2(M), and Stone’s
theorem implies that −iX has self-adjoint extensions.
In the second case, the unitary group Γ(φs) on E also has a self-adjoint
generator Γ(X), which can be calculated explicitly. By definition,
e−isΓ(X)
Φ(fi)
Φ(fi ◦ φ−s).
Now replace s→ −s and calculate d/ds|s=0 applied to both sides of the last
equation to see that
Φ(fi)
Φ(f1) . . .Φ(−iXfj)Φ(fj+1) . . .Φ(fn) .
One may check that Γ is a Lie algebra representation of g, i.e. Γ([X,Y ]) =
[Γ(X),Γ(Y )].
4.2. The Cartan Decomposition of g. For each ξ ∈ g, there exists some
dense domain in E on which Γ(ξ) is self-adjoint, as discussed previously.
8 ARTHUR JAFFE AND GORDON RITTER
However, the quantizations Γ̂(ξ) acting on H may be hermitian, anti-
hermitian, or neither depending on whether there holds a relation of the
(4.1) Γ(ξ)Θ = ±ΘΓ(ξ),
with one of the two possible signs, or whether no such relation holds.
Even if (4.1) holds, to complete the construction of a unitary representa-
tion one must prove that there exists a dense domain in H on which Γ̂(ξ) is
self-adjoint or skew-adjoint. This nontrivial problem will be dealt with in a
later section using Theorems 3.3 and 3.5 and the theory of symmetric local
semigroups [12, 4]. Presently we determine which elements within g satisfy
relations of the form (4.1).
Let ϑ := θ∗ as an operator on C∞(M), and consider a Killing field X ∈ g
also as an operator on C∞(M). Define T : g → g by
(4.2) T (X) := ϑXϑ.
From (4.2) it is not obvious that the range of T is contained in g. To prove
this, we recall some geometric constructions.
Let M,N be manifolds, let ψ : M → N be a diffeomorphism, and X ∈
Vect(M). Then
(4.3) ψ−1∗Xψ∗ = X(· ◦ ψ) ◦ ψ−1.
defines an operator on C∞(N). One may check that this operator is a
derivation, thus (4.3) defines a vector field on N . The vector field (4.3) is
usually denoted
ψ∗X = dψ(Xψ−1(p))
and referred to as the push-forward of X.
We now wish to show that g = g+⊕ g−, where g± are the ±1-eigenspaces
of T . This is proven by introducing an inner product (X,Y )g on g with
respect to which T is self-adjoint.
Theorem 4.2. Consider g as a Hilbert space with inner product (X,Y )g.
The operator T : g → g is self-adjoint with T 2 = I; hence
(4.4) g = g+ ⊕ g−
as an orthogonal direct sum of Hilbert spaces, where g± are the ±1-eigenspaces
of T . Further, ∂t ∈ g− hence dim(g−) ≥ 1. Elements of g− have hermitian
quantizations, while elements of g+ have anti-hermitian quantizations.
Proof. Write (4.2) as
(4.5) T (X) = θ−1∗Xθ∗ = θ∗X .
Thus T is the operator of push-forward by θ. The push-forward of a Killing
field by an isometry is another Killing field, hence the range of T is contained
2It is not the case that g− consists only of ∂t. In particular, dim(g−) = 2 for M = H 2.
It can occur that dim g+ = 0.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES9
in g. Also, T must have a trivial kernel since T 2 = I, and this implies that
T is surjective. It follows from (4.5) that T is a Hermitian operator on
g. Hence T is diagonalizable and has real eigenvalues which are square
roots of 1. This establishes the decomposition (4.4). That elements of
g− have hermitian quantizations, while elements of g+ have anti-hermitian
quantizations follows from Theorem 3.3. �
A Cartan involution is a Lie algebra homomorphism g → g which squares
to the identity. It follows from (4.2) that T is a Lie algebra homomorphism;
thus, Theorem 4.2 implies that T is a Cartan involution of g. This implies
that the eigenspaces (g+, g−) form a Cartan pair, meaning that
(4.6) [g+, g+] ⊂ g+, [g+, g−] ⊂ g−, and [g−, g−] ⊂ g+ .
Clearly g+ is a subalgebra while g− is not, and any subalgebra contained in
g− is abelian.
5. Reflection-Invariant and Reflected Isometries
Let G = Iso(M) denote the isometry group of M , as above. Then G
has a Z2 subgroup containing {1, θ}. This subgroup acts on G by conjuga-
tion, which is just the action ψ → ψθ := θψθ. Conjugation is an (inner)
automorphism of the group, so
(ψφ)θ = ψθφθ, (ψθ)−1 = (ψ−1)θ.
Definition 5.1. We say that ψ ∈ G is reflection-invariant if
ψθ = ψ,
and that ψ is reflected if
ψθ = ψ−1.
Let GRI denote the subgroup of G consisting of reflection-invariant elements,
and let GR denote the subset of reflected elements.
Note that GRI is the stabilizer of the Z2 action, hence a subgroup. An
alternate proof of this proceeds usingGRI = exp(g+). Although GR is closed
under the taking of inverses and does contain the identity, the product of two
reflected isometries is no longer reflected unless they commute. Generally,
the product of an element of GR with an element ofGRI is neither an element
of GR nor of GRI . The only isometry that is both reflection-invariant and
reflected is θ itself. Thus we have:
GR ∩GRI = {1, θ} ⊂ GR ∪GRI ( G.
Theorem 5.2. Let G0 denote the connected component of the identity in
G. Then G0 is generated by GR ∪ GRI . (This is a form of the Cartan
decomposition for G.)
10 ARTHUR JAFFE AND GORDON RITTER
Proof. Since g = g+ ⊕ g− as a direct sum of vector spaces (though not of
Lie algebras), we have
exp(g)
exp(g+) ∪ exp(g−)
Choose bases {ξ±,i}i=1,...,n± for g± respectively. Then we have:
{exp(sξ+,i) : 1 ≤ i ≤ n+, s ∈ R}∪{exp(sξ−,j) : 1 ≤ j ≤ n−, s ∈ R}
Furthermore, exp(sξ−,i) is reflected, while exp(sξ+,i) is reflection-invariant,
completing the proof. �
Corollary 5.3. The Lie algebra of the subgroup GRI is g+.
To summarize, the isometry group of a static space-time can always be
generated by a collection of n (= dim g) one-parameter subgroups, each of
which consists either of reflected isometries, or reflection-invariant isome-
tries.
6. Construction of Unitary Representations
6.1. Self-adjointness of Semigroups. In this section, we recall several
known results on self-adjointness of semigroups. Roughly speaking, these
results imply that if a one-parameter family Sα of unbounded symmetric
operators satisfies a semigroup condition of the form SαSβ = Sα+β, then
under suitable conditions one may conclude essential self-adjointness.
A theorem of this type appeared in a 1970 paper of Nussbaum [14], who
assumed that the semigroup operators have a common dense domain. The
result was rediscovered independently by Fröhlich, who applied it to quan-
tum field theory in several important papers [5, 3]. For our intended appli-
cation to quantum field theory, it turns out to be very convenient to drop
the assumption that ∃ a such that the Sα all have a common dense domain
for |α| < a, in favor of the weaker assumption that
α>0D(Sα) is dense.
A generalization of Nussbaum’s theorem which allows the domains of the
semigroup operators to vary with the parameter, and which only requires the
union of the domains to be dense, was later formulated and two independent
proofs were given: one by Fröhlich [4], and another by Klein and Landau [12].
The latter also used this theorem in their construction of representations of
the Euclidean group and the corresponding analytic continuation to the
Lorentz group [13].
In order to keep the present article self-contained, we first define symmet-
ric local semigroups and then recall the refined self-adjointness theorem of
Fröhlich, and Klein and Landau.
Definition 6.1. Let H be a Hilbert space, let T > 0 and for each α ∈ [0, T ],
let Sα be a symmetric linear operator on the domain Dα ⊂ H , such that:
(i) Dα ⊃ Dβ if α ≤ β and D :=
0<α≤T Dα is dense in H ,
(ii) α→ Sα is weakly continuous,
(iii) S0 = I, Sβ(Dα) ⊂ Dα−β for 0 ≤ β ≤ α ≤ T , and
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES11
(iv) SαSβ = Sα+β on Dα+β for α, β, α + β ∈ [0, T ].
In this situation, we say that (Sα,Dα, T ) is a symmetric local semigroup.
It is important that Dα is not required to be dense in H for each α; the
only density requirement is (i).
Theorem 6.2 ([12, 4]). For each symmetric local semigroup (Sα,Dα, T ),
there exists a unique self-adjoint operator A such that3
Dα ⊂ D(e
−αA) and Sα = e
−αA|Dα for all α ∈ [0, T ].
Also, A ≥ −c if and only if ‖Sαf‖ ≤ e
cα‖f‖ for all f ∈ Dα and 0 < α < T .
6.2. Reflection-Invariant Isometries.
Lemma 6.3. Let ψ be a reflection-invariant isometry and assume ∃ p ∈ Ω+
such that ψ(p) ∈ Ω+. Then ψ preserves the positive-time subspace, i.e.
ψ(Ω+) ⊆ Ω+.
Proof. We first prove that ψ(Σ) ⊆ Σ. Suppose not; then ∃ p ∈ Σ with ψ(p) 6∈
Σ. Assume ψ(p) ∈ Ω+ (without loss of generality: we could repeat the same
argument with ψ(p) ∈ Ω−). Then Ω+ contains (θψθ)(p) = θψ(p) ∈ Ω−, a
contradiction since Ω−∩Ω+ = ∅. We used the fact that θ|Σ = id so θ(p) = p.
Hence ψ restricts to an isometry of Σ. It follows that the restriction of ψ to
M ′ =M \Σ is also an isometry. However, M ′ = Ω− ⊔Ω+, where ⊔ denotes
the disjoint union. Therefore ψ(Ω+) is wholly contained in either Ω+ or Ω−,
since ψ is a homeomorphism and so ψ(Ω+) is connected. The possibility
that ψ(Ω+) ⊆ Ω− is ruled out by our assumption, so ψ(Ω+) ⊆ Ω+. �
Lemma 6.3 has the immediate consequence that if ξ ∈ g+ then the one-
parameter group associated to ξ is positive-time-invariant. This result plays
a key role in the proof of Theorem 6.4.
6.3. Construction of Unitary Representations. The rest of this section
is devoted to proving that the theory of symmetric local semigroups can be
applied to the quantized operators on H corresponding to each of a set of
1-parameter subgroups of G = Iso(M). The proof relies upon Lemma 6.3,
and Theorems 3.3, 3.5 and 6.2.
Theorem 6.4. Let (M,gab) be a quantizable static space-time. Let ξ be a
Killing field which lies in g+ or g−, with associated one-parameter group of
isometries {φα}α∈R. Then there exists a densely-defined self-adjoint opera-
tor Aξ on H such that
Γ̂(φα) =
e−αAξ , if ξ ∈ g−
eiαAξ if ξ ∈ g+.
3The authors of [4, 12] also showed that
bD :=
0<α≤S
0<β<α
Sβ(Dα)
, where 0 < S ≤ T,
is a core for A, i.e. (A, bD) is essentially self-adjoint.
12 ARTHUR JAFFE AND GORDON RITTER
Proof. First suppose that ξ ∈ g−, which implies that the isometries φα are
reflected, and so Γ(φα)
+ = Γ(φα). Define
Ωξ,α := φ
α (Ω+).
For all α in some neighborhood of zero, Ωξ,α is a nonempty open subset of
Ω+, and moreover, as α → 0
+, Ωξ,α increases to fill Ω+ with Ωξ,0 = Ω+.
These statements follow immediately from the fact that, for each p ∈ Ω+,
φα(p) is continuous with respect to α, and φ0 is the identity map.
Since φα(Ωξ,α) ⊆ Ω+, we infer that Γ(φα)EΩξ,α ⊆ E+. By Theorem 3.5,
Γ(φα) has a quantization which is a symmetric operator on the domain
Dξ,α := Π(EΩξ,α).
Note that Dξ,α is not necessarily dense in H .
4 We now show that Theorem
6.2 can be applied.
Fix some positive constant a with Ωξ,a nonempty. Note that
0<α≤a
Ωξ,α = Ω+ ⇒
0<α≤a
EΩξ,α = E+.
It follows that
Dξ :=
0<α≤a
is dense in H . This establishes condition (i) of Definition 6.1, and the
other conditions are routine verifications. Theorem 6.2 implies existence of
a densely-defined self-adjoint operator Aξ on H , such that
Γ̂(φα) = exp(−αAξ) for all α ∈ [0, a] .
This proves the theorem in case ξ ∈ g−.
Now suppose that ξ ∈ g+, implying that the isometries φα are reflection-
invariant, and
Γ(φα)
+ = Γ(φα)
−1 = Γ(φ−α) on E .
Lemma 6.3 implies that Γ(φα)E+ ⊆ E+. By Theorem 3.3, Γ(φα) has a
quantization Γ̂(φα) which is defined and satisfies
Γ̂(φα)
∗ = Γ̂(φα)
on the domain Π(E+), which is dense in H by definition. In this case we
do not need Theorem 6.2; for each α, Γ̂(φα) extends by continuity to a one-
parameter unitary group defined on all of H (not only for a dense subspace).
By Stone’s theorem,
Γ̂(φα) = exp(iαAξ)
for Aξ self-adjoint and for all α ∈ R. The proof is complete. �
4Density of Dξ,α would be implied by a Reeh-Schlieder theorem, which we do not prove
except in the free case. Theorem 6.2 removes the need for a Reeh-Schlieder theorem in
this argument.
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES13
7. Analytic Continuation
Each Riemannian static space-time (M,gab) has a Lorentzian continuation
Mlor, which we construct as follows. In adapted coordinates, the metric gab
on M takes the form
(7.1) ds2 = F (x)dt2 + Gµν(x)dx
µdxν .
The analytic continuation t → −it of (7.1) is standard and gives a metric
of Lorentz signature, ds2lor = −F dt
2 + G dx2, by which we define the
Lorentzian space-time Mlor. Einstein’s equation Ricg = k g is preserved
by the analytic continuation, but we do not use this fact anywhere in the
present paper.
Let {ξ
i : 1 ≤ i ≤ n±} be bases of g±, respectively. Let A
i = Aξ(±)
the densely-defined self-adjoint operators on H , constructed by Theorem
6.4. Let
(7.2) U
i (α) = exp(iαA
i ) , for 1 ≤ i ≤ n±
be the associated one-parameter unitary groups on H .
We claim that the group generated by the n = n+ + n− one-parameter
unitary groups (7.2) is isomorphic to the identity component of
Glor := Iso(Mlor),
the group of Lorentzian isometries. Since locally, the group structure is
determined by its Lie algebra, it suffices to check that the generators satisfy
the defining relations of glor := Lie(Glor).
Since quantization of operators preserves multiplication, we have
(7.3) X,Y,Z ∈ g, [X,Y ] = Z ⇒ [Γ̂(X), Γ̂(Y )] = Γ̂(Z).
In what follows, we will use the notation ĝ± for {Γ̂(X) : X ∈ g±}.
Quantization converts the elements of g− from skew operators into Her-
mitian operators; i.e. elements of ĝ− are Hermitian on H and hence, ele-
ments of i ĝ− are skew-symmetric on H . Thus ĝ+ ⊕ i ĝ− is a Lie algebra
represented by skew-symmetric operators on H .
Theorem 7.1. We have an isomorphism of Lie algebras:
(7.4) glor ∼= ĝ+ ⊕ i ĝ− .
Proof. LetMC be the manifold obtained by allowing the t coordinate to take
values in C. Define ψ :MC →MC by t 7→ −it. Then glor is generated by
i }1≤i≤n+ ∪ {ηj}1≤j≤n− , where ηj := iψ
It is possible to define a set of real structure constants fijk such that
(7.5) [ξ
fijkξ
14 ARTHUR JAFFE AND GORDON RITTER
Applying ψ∗ to both sides of (7.5), the commutation relations of glor are
seen to be
(7.6) [ηi, ηj ] = −fijkξ
together with the same relations for g+ as before. Now (7.3) implies that
(7.6) are the precisely the commutation relations of ĝ+ ⊕ i ĝ−, completing
the proof of (7.4). �
Corollary 7.2. Let (M,gab) be a quantizable static space-time. The unitary
groups (7.2) determine a unitary representation of G0lor on H .
7.1. Conclusions. We have obtained the following conclusions. There is a
unitary representation of the group G0lor on the physical Hilbert space H
of quantum field theory on the static space-time M . This representation
maps the time-translation subgroup into the unitary group exp(itH), where
the energy H ≥ 0 is a positive, densely-defined self-adjoint operator corre-
sponding to the Hamiltonian of the theory. The Hilbert space H contains a
ground state Ψ0 = 1̂ which is such that HΨ0 = 0 and Ψ0 is invariant under
the action of all spacetime symmetries. We obtain these results via analytic
continuation from the Euclidean path integral, under mild assumptions on
the measure which should include all physically interesting examples. This is
done without introducing the field operators; nonetheless, Theorems 3.3 and
3.5 do suffice to construct field operators. In the special case M = Rd with
G = SO(4), we obtain a unitary representation of the proper orthochronous
Lorentz group, G0lor = L
+ = SO
0(3, 1).
8. Hyperbolic Space and Anti-de Sitter Space
Consider Euclidean quantum field theory on M = H d. The metric is
ds2 = r−2
dx2i ,
where we define r = xd for convenience. The Laplacian is
(8.1) ∆
d = (2− d)r
+ r2∆
The d−1 coordinate vector fields {∂/∂xi : i 6= d} are all static Killing fields,
and any one of the coordinates xi (i 6= d) is a satisfactory representation of
time in this space-time. It is convenient to define t = x1 as before, and to
identify t with time.
The time-zero slice is M0 = H
d−1. From
H d = {v ∈ Rd,1 | 〈v, v〉 = −1, v0 > 0}
it follows that Isom(H d) = O+(d, 1) and the orientation-preserving isometry
group is SO+(d, 1).
QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES15
Figure 1. Flow lines of the Killing field ζ = (t2 − r2)∂t +
2tr ∂r on H
For constant curvature spaces, one may solve Killing’s equation LKg = 0
explicitly. Let us illustrate the solutions and their quantizations for d = 2.
The three Killing fields
(8.2) ξ = ∂t, η = t∂t + r∂r, ζ = (t
2 − r2)∂t + 2tr ∂r
are a convenient basis for g. Any d-dimensional manifold satisfies dim g ≤
d(d + 1)/2, manifolds saturating the bound are said to be maximally sym-
metric, and H d is maximally symmetric.
Now, ∂tf(−t) = −f
′(−t) so ∂tΘ = −Θ∂t, hence ∂t ∈ g−. Similar calcula-
tions show [Θ, η] = 0 and Θζ = −ζΘ. Thus η spans g+, while ∂t, ζ span g−.
The commutation relations5 for g are:
[η, ζ] = ζ, [η, ∂t] = −∂t, [ζ, ∂t] = −2η.
These calculations verify that (g+, g−) forms a Cartan pair, as defined in
(4.6).
The flows associated to (8.2) are easily visualized: ξ is a right-translation,
and η flow-lines are radially outward from the Euclidean origin. The flows
of ζ are Euclidean circles, indicated by the darker lines in Figure 1. Hence
the flows of η are defined on all of E+, while the flows of ζ are analogous to
space-time rotations in R2, and hence, must be defined on a wedge of the
Wα = {(t, r) : t, r > 0, tan
−1(r/t) < α}.
The simple geometric idea of Section 6.2 is nicely confirmed in this case: the
flows of η (the generator of g+) preserve the t = 0 plane, and are separately
isometries of Ω+ and Ω−.
Corollary 7.2 implies that the procedure outlined above defines a uni-
tary representation of the identity component of Iso(AdS2) on the physical
Hilbert space H for quantum field theory on this background, including
theories with interactions that preserve the symmetry. Since Iso(AdSd+1) =
5Note that quite generally [g−, g−] ⊆ g+ so it’s automatic that [ζ, ∂t] is proportional
to η.
16 ARTHUR JAFFE AND GORDON RITTER
SO(d, 2), we have a unitary representation of SO0(1, 2). The latter is a non-
compact, semisimple real Lie group, and thus it has no finite-dimensional
unitary representations, but a host of interesting infinite-dimensional ones.
Appendix A. Euclidean Reeh-Schlieder Theorem
We prove the Euclidean Reeh-Schlieder property for free theories on curved
backgrounds. It is reasonable to expect this property to extend to interact-
ing theories on curved backgrounds, but it would have to be established for
each such model since it depends explicitly on the two-point function.
The Reeh-Schlieder theorem guarantees the existence of a dense quanti-
zation domain based on any open subset of Ω+. For this reason, one could
use the Reeh-Schlieder (RS) theorem with Nussbaum’s theorem [14] to con-
struct a second proof of Theorem 6.4 under the additional assumption that
M is real-analytic.
Fortunately, our proof of Theorem 6.4 is completely independent of the
Reeh-Schlieder property. This has two advantages: we do not have to assume
M is a real-analytic manifold and, more importantly, our proof of Theorem
6.4 generalizes immediately and transparently to interacting theories as long
as the Hilbert space H is not modified by the interaction.
We state and prove this using the one-particle space; however, the result
clearly extends to the quantum-field Hilbert space.
Theorem A.1. Let M be a quantizable static space-time endowed with a
real-analytic structure, and assume that gab is real-analytic. Let O ⊂ Ω+
and D = C∞(O) ⊂ L2(Ω+). Then D̂
⊥ = {0}.
Proof. Let f ∈ L2(Ω+) with f̂ ⊥ D . For x ∈ Ω+, define
η(x) := 〈f̂ , δ̂x〉H = 〈Θf,Cδx〉L2 .
Real-analyticity of η(x) follows from the real-analyticity of (the integral
kernel of) C, which in turn follows from the elliptic regularity theorem in the
real-analytic category (see for instance [1, Sec. II.1.3]). Now by assumption,
for any g ∈ C∞c (O), we have
0 = 〈ĝ, f̂〉H = 〈Θf,Cg〉L2(M).
Let g → δx for x ∈ O. Then 0 = 〈Θf,Cδx〉L2 ≡ η(x). Since η|O = 0, by
real-analyticity we infer the vanishing of η on Ω+, completing the proof. �
Acknowledgements. We are grateful to Hanno Gottschalk and Alexander
Strohmaier for helpful discussions, and G.R. is grateful to the Universität
Bonn for their hospitality during February 2007.
References
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QUANTUM FIELD THEORY ON CURVED BACKGROUNDS. II.SPACETIME SYMMETRIES17
[2] N. D. Birrell and P. C. W. Davies. Quantum fields in curved space, vol-
ume 7 of Cambridge Monographs on Mathematical Physics. Cambridge
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[3] W. Driessler and J. Fröhlich. The reconstruction of local observable
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[4] J. Fröhlich. Unbounded, symmetric semigroups on a separable Hilbert
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[5] Jürg Fröhlich. The pure phases, the irreducible quantum fields, and
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field theories. Ann. Physics, 97(1):1–54, 1976.
[6] Stephen A. Fulling. Aspects of quantum field theory in curved space-
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[8] James Glimm and Arthur Jaffe. Quantum physics. Springer-Verlag,
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[10] Arthur Jaffe. Introduction to Quantum Field Theory. 2005. Lecture
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[11] Arthur Jaffe and Gordon Ritter. Quantum field theory on curved
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[12] Abel Klein and Lawrence J. Landau. Construction of a unique self-
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Green’s functions. Comm. Math. Phys., 31:83–112, 1973.
[16] Konrad Osterwalder and Robert Schrader. Axioms for Euclidean
Green’s functions. II. Comm. Math. Phys., 42:281–305, 1975. With
an appendix by Stephen Summers.
http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf
18 ARTHUR JAFFE AND GORDON RITTER
[17] Barry Simon. The P (φ)2 Euclidean (quantum) field theory. Princeton
University Press, Princeton, N.J., 1974. Princeton Series in Physics.
[18] Robert M. Wald. Quantum field theory in curved space-time. In Grav-
itation et quantifications (Les Houches, 1992), pages 63–167. North-
Holland, Amsterdam, 1995.
E-mail address: arthur [email protected]
Harvard University, 17 Oxford St., Cambridge, MA 02138
E-mail address: [email protected]
Harvard University, 17 Oxford St., Cambridge, MA 02138
1. Introduction
2. Classical Space-Time
2.1. Structure of Static Space-Times
2.2. The Operator Induced by an Isometry
3. Osterwalder-Schrader Quantization
3.1. Quantization of Vectors (The Hilbert Space H of Quantum Theory)
3.2. Quantization of Operators
4. Structure and Representation of the Lie Algebra of Killing Fields
4.1. The Representation of g on E
4.2. The Cartan Decomposition of g
5. Reflection-Invariant and Reflected Isometries
6. Construction of Unitary Representations
6.1. Self-adjointness of Semigroups
6.2. Reflection-Invariant Isometries
6.3. Construction of Unitary Representations
7. Analytic Continuation
7.1. Conclusions
8. Hyperbolic Space and Anti-de Sitter Space
Appendix A. Euclidean Reeh-Schlieder Theorem
Acknowledgements
References
|
0704.0053 | A Global Approach to the Theory of Special Finsler Manifolds | A GLOBAL APPROACH TO THE
THEORY OF
SPECIAL FINSLER MANIFOLDS
Nabil L. Youssef†, S. H. Abed† and A. Soleiman‡
†Department of Mathematics, Faculty of Science,
Cairo University, Giza, Egypt.
[email protected], [email protected]
‡Department of Mathematics, Faculty of Science,
Benha University, Benha, Egypt.
[email protected]
Dedicated to the memory of Prof. Dr. A. TAMIM
Abstract. The aim of the present paper is to provide a global presentation of the
theory of special Finsler manifolds. We introduce and investigate globally (or in-
trinsically, free from local coordinates) many of the most important and most com-
monly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg,
general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible,
P ∗-Finsler, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, Sv-recurrent of
the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric, h-isotropic,
of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature.
The global definitions of these special Finsler manifolds are introduced. Various
relationships between the different types of the considered special Finsler manifolds
are found. Many local results, known in the literature, are proved globally and
several new results are obtained. As a by-product, interesting identities and properties
concerning the torsion tensor fields and the curvature tensor fields are deduced.
Although our investigation is entirely global, we provide; for comparison rea-
sons, an appendix presenting a local counterpart of our global approach and the local
definitions of the special Finsler spaces considered. 1
Keywords and phrases. Berwald, Landesberg, P -reducible, C-reducible, Semi-C-
reducible, Quasi-C-reducible, P ∗-Finsler, Ch-recurrent, Cv-recurrent, Sv-recurrent,
C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric, h-isotropic, Of scalar curva-
ture, Of constant curvature, Of p-scalar curvature, Of s-ps-curvature.
2000 AMS Subject Classification. 53C60, 53B40.
ArXiv Number: 0704.0053
http://arxiv.org/abs/0704.0053v3
Introduction
In Finsler geometry all geometric objects depend not only on positional coordi-
nates, as in Riemannian geometry, but also on directional arguments. In Riemannian
geometry there is a canonical linear connection on the manifold M , while in Finsler
geometry there is a corresponding canonical linear connection, due to E. Cartan,
which is not a connection on M but is a connection on π−1(TM), the pullback of
the tangent bundle TM by π : TM −→ M (the pullback approach). Moreover, in
Riemannian geometry there is one curvature tensor and one torsion tensor associated
with a given linear connection on the manifold M , whereas in Finsler geometry there
are three curvature tensors and five torsion tensors associated with a given linear
connection on π−1(TM).
Most of the special spaces in Finsler geometry are derived from the fact that
the π-tensor fields (torsions and curvatures) associated with the Cartan connection
satisfy special forms. Consequently, special spaces of Finsler geometry are more
numerous than those of Riemannian geometry. Special Finsler spaces are investigated
locally (using local coordinates) by many authors: M. Matsumoto [16], [18], [15], [14]
and others [6], [19], [8], [7]. On the other hand, the global (or intrinsic, free from
local coordinates) investigation of such spaces is very rare in the literature. Some
considerable contributions in this direction are due to A. Tamim [24], [25].
In the present paper, we provide a global presentation of the theory of special
Finsler manifolds. We introduce and investigate globally many of the most important
and most commonly used special Finsler manifolds : locally Minkowskian, Berwald,
Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-
C-reducible, P ∗-Finsler, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, Sv-
recurrent of the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric,
h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-
curvature.
The paper consists of two parts, preceded by a preliminary section (§1), which
provides a brief account of the basic concepts of the pullback approach to Finsler
geometry necessary to this work. For more detail, the reader is referred to [1], [3], [5]
and [24].
In the first part (§2), we introduce the global definitions of the aforementioned
special Finsler manifolds in such a way that, when localized, they yield the usual
local definitions current in the literature (see the appendix). The definitions are
arranged according to the type of the defining property of the special Finsler manifold
concerned.
In the second part (§3), various relationships between the different types of the
considered special Finsler manifolds are found. Many local results, known in the
literature, are proved globally and several new results are obtained. As a by-product
of some of the obtained results, interesting identities and properties concerning the
torsion tensor fields and the curvature tensor fields are deduced, which in turn play
a key role in obtaining other results.
Among the obtained results are: a characterization of Riemannian manifolds, a
characterization of Sv-recurrent manifolds, a characterization of P -symmetric
manifolds, a characterization of Berwald manifolds (in certain cases), the equivalence
of Landsberg and general Landsberg manifolds under certain conditions, a classifica-
tion of h-isotropic Ch-recurrent manifolds and a presentation of different conditions
under which an R3-like Finsler manifold becomes a Finsler manifold of s-ps curvature.
The above results are just a non-exhaustive sample of the global results obtained in
this paper.
It should finally be noted that some important results of [8], [9], [11], [13], [19],
[20],...,etc. (obtained in local coordinates) are immediately derived from the obtained
global results (when localized).
Although our investigation is entirely global, we conclude the paper with an ap-
pendix presenting a local counterpart of our global approach and the local definitions
of the special Finsler spaces considered. This is done to facilitate comparison and to
make the paper more self-contained.
1. Notation and Preliminaries
In this section, we give a brief account of the basic concepts of the pullback
formalism of Finsler geometry necessary for this work. For more details refer to [1], [3],
[5] and [24]. We make the general assumption that all geometric objects we consider
are of class C∞. The following notations will be used throughout this paper:
M : a real differentiable manifold of finite dimension n and of class C∞,
F(M): the R-algebra of differentiable functions on M ,
X(M): the F(M)-module of vector fields on M ,
πM : TM −→M : the tangent bundle of M ,
π : TM −→M : the subbundle of nonzero vectors tangent to M ,
V (TM): the vertical subbundle of the bundle TTM ,
P : π−1(TM) −→ TM : the pullback of the tangent bundle TM by π,
P ∗ : π−1(T ∗M) −→ TM : the pullback of the cotangent bundle T ∗M by π,
X(π(M)): the F(TM)-module of differentiable sections of π−1(TM).
Elements of X(π(M)) will be called π-vector fields and will be denoted by barred
letters X. Tensor fields on π−1(TM) will be called π-tensor fields. The fundamental
π-vector field is the π-vector field η defined by η(u) = (u, u) for all u ∈ TM . The
lift to π−1(TM) of a vector field X on M is the π-vector field X defined by X(u) =
(u,X(π(u))). The lift to π−1(TM) of a 1-form ω on M is the π-form ω defined by
ω(u) = (u, ω(π(u))).
The tangent bundle T (TM) is related to the pullback bundle π−1(TM) by the
short exact sequence
0 −→ π−1(TM)
−→ T (TM)
−→ π−1(TM) −→ 0,
where the bundle morphisms ρ and γ are defined respectively by ρ = (πT M , dπ) and
γ(u, v) = ju(v), where ju is the natural isomorphism ju : TπM (v)M −→ Tu(TπM (v)M).
Let ∇ be a linear connection (or simply a connection) in the pullback bundle
π−1(TM). We associate to ∇ the map
K : TTM −→ π−1(TM) : X 7−→ ∇Xη,
called the connection (or the deflection) map of ∇. A tangent vector X ∈ Tu(TM)
is said to be horizontal if K(X) = 0 . The vector space Hu(TM) = {X ∈ Tu(TM) :
K(X) = 0} of the horizontal vectors at u ∈ TM is called the horizontal space to M
at u . The connection ∇ is said to be regular if
Tu(TM) = Vu(TM)⊕Hu(TM) ∀u ∈ TM.
If M is endowed with a regular connection, then the vector bundle maps
γ : π−1(TM) −→ V (TM),
ρ|H(T M) : H(TM) −→ π
−1(TM),
K|V (T M) : V (TM) −→ π
−1(TM)
are vector bundle isomorphisms. Let us denote β = (ρ|H(T M))
−1, then
ρoβ = idπ−1(T M), βoρ =
idH(T M) on H(TM)
0 on V(TM)
(1.1)
For a regular connection∇ we define two covariant derivatives
∇ and
∇ as follows:
For every vector (1)π-form A, we have
∇ A)(øX, øY ) := (∇βøXA)(øY ) , (
∇ A)(øX, øY ) := (∇γøXA)(øY ).
The classical torsion tensor T of the connection ∇ is defined by
T(X, Y ) = ∇XρY −∇Y ρX − ρ[X, Y ] ∀X, Y ∈ X(TM).
The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted respectively by Q
and T , are defined by
Q(X, Y ) = T(βXβY ), T (X, Y ) = T(γX, βY ) ∀X, Y ∈ X(π(M)).
The classical curvature tensor K of the connection ∇ is defined by
K(X, Y )ρZ = −∇X∇Y ρZ +∇Y∇XρZ +∇[X,Y ]ρZ ∀X, Y, Z ∈ X(TM).
The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted respec-
tively by R, P and S, are defined by
R(X, Y )øZ = K(βXβY )øZ, P (X, Y )øZ = K(βX, γY )øZ, S(X, Y )øZ = K(γX, γY )øZ.
We also have the (v)h-, (v)hv- and (v)v-torsion tensors, denoted respectively by R̂,
P̂ and Ŝ, defined by
R̂(X, Y ) = R(X, Y )øη, P̂ (X, Y ) = P (X, Y )øη, Ŝ(X, Y ) = S(X, Y )øη.
Theorem 1.1. [25] Let (M,L) be a Finsler manifold. There exists a unique regular
connection ∇ in π−1(TM) such that
(a) ∇ is metric : ∇g = 0,
(b) The horizontal torsion of ∇ vanishes : Q = 0,
(c) The mixed torsion T of ∇ satisfies g(T (X, Y ), Z) = g(T (X,Z), Y ).
Such a connection is called the Cartan connection associated to the Finsler man-
ifold (M,L).
One can show that the torsion T of the Cartan connection has the property that
T (X, η) = 0 for all X ∈ X(π(M)) and associated to T we have:
Definition 1.2. [25] Let ∇ be the Cartan connection associated to (M,L). The
torsion tensor field T of the connection ∇ induces a π-tensor field of type (0, 3),
called the Cartan tensor and denoted again T , defined by :
T (X, Y , Z) = g(T (X, Y ), Z), for all X, Y , Z ∈ X(TM).
It also induces a π-form C, called the contracted torsion, defined by :
C(X) := Tr{Y 7−→ T (X, Y )}, for all X ∈ X(TM).
Definition 1.3. [25] With respect to the Cartan connection ∇ associated to (M,L),
we have
– The horizontal and vertical Ricci tensors Rich and Ricv are defined respectively by:
Rich(X, Y ) := Tr{Z 7−→ R(X,Z)Y }, for all X, Y ∈ X(TM),
Ricv(X, Y ) := Tr{Z 7−→ S(X,Z)Y }, for all X, Y ∈ X(TM).
– The horizontal and vertical Ricci maps Rich0 and Ric
0 are defined respectively by:
g(Rich0(X), Y ) := Ric
h(X, Y ), for all X, Y ∈ X(TM),
g(Ricv0(X), Y ) := Ric
v(X, Y ), for all X, Y ∈ X(TM).
– The horizontal and vertical scalar curvatures Sch , Scv are defined respectively by:
Sch := Tr(Rich0), Sc
v := Tr(Ricv0),
where R and S are respectively the horizontal and vertical curvature tensors of ∇.
Proposition 1.4. [12] Let (M,L) be a Finsler manifold. The vector field G deter-
mined by iGΩ = −dE is a spray, called the canonical spray associated to the energy
E, where E := 1
L2 and Ω := ddJE.
One can show, in this case, that G = βoη, and G is thus horizontal with respect
to the Cartan connection ∇.
Theorem 1.5. [26] Let (M,L) be a Finsler manifold. There exists a unique regular
connection D in π−1(TM) such that
(a) D is torsion free,
(b) The canonical spray G = βoη is horizontal with respect to D,
(c) The (v)hv-torsion tensor P̂ of D vanishes.
Such a connection is called the Berwald connection associated to the Finsler
manifold (M,L).
2. Special Finsler spaces
In this section, we introduce the global definitions of the most important and
commonly used special Finsler spaces in such a way that, when localized, they yield
the usual local definitions existing in the literature (see the Appendix). Here we
simply set the definitions, postponing investigation of the mutual relationships be-
tween these special Finsler spaces to the next section. The definitions are arranged
according to the type of defining property of the special Finsler space concerned.
Throughout the paper, g, ĝ, ∇ and D denote respectively the Finsler metric in
π−1(TM), the induced metric in π−1(T ∗M), the Cartan connection and the Berwald
connection associated to a given Finsler manifold (M,L). Also, T denotes the torsion
tensor of the Cartan connection (or the Cartan tensor) and R, P and S denote
respectively the horizontal curvature, the mixed curvature and the vertical curvature
of the Cartan connection.
Definition 2.1. A Finsler manifold (M,L) is :
(a) Riemannian if the metric tensor g(x, y) is independent of y or, equivalently, if
T (X, Y ) = 0, for all X, Y ∈ X(π(M)).
(b) locally Minkowskian if the metric tensor g(x, y) is independent of x or, equiva-
lently, if
∇βX T = 0 and R = 0.
Definition 2.2. A Finsler manifold (M,L) is said to be :
(a) Berwald [24] if the torsion tensor T is horizontally parallel. That is,
∇βX T = 0.
(b) Ch-recurrent if the torsion tensor T satisfies the condition
∇βX T = λo(X) T,
where λo is a π-form of order one.
(c) P ∗-Finsler manifold if the π-tensor field ∇βηT is expressed in the form
∇βη T = λ(x, y) T,
where λ(x, y) =
bg(∇βη C,C)
g(∇βηøC,øC)
and C2 := ĝ(C,C) = C(C) 6= 0; C
being the π-vector field defined by g(C,X) = C(X).
Definition 2.3. A Finsler manifold (M,L) is said to be:
(a) Cv-recurrent if the torsion tensor T satisfies the condition
(∇γXT )(Y , Z) = λo(X)T (Y , Z).
(b) C0-recurrent if the torsion tensor T satisfies the condition
(DγXT )(Y , Z) = λo(X)T (Y , Z).
Definition 2.4. [25] A Finsler manifold (M,L) is said to be :
(a) semi-C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) =
{~(X, Y )C(Z) + ~(Y , Z)C(X) + ~(Z,X)C(Y )}+
C(X)C(Y )C(Z),
where µ and τ are scalar functions satisfying µ + τ = 1, ~ = g − ℓ ⊗ ℓ and ℓ(X) :=
L−1g(X, η).
(b) C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) =
{~(X, Y )C(Z) + ~(Y , Z)C(X) + ~(Z,X)C(Y )}.
(c) C2-like if dimM ≥ 2 and the Cartan tensor T has the form
T (X, Y , Z) =
C(X)C(Y )C(Z).
Definition 2.5. A Finsler manifold (M,L), where dimM ≥ 3, is said to be quasi-C-
reducible if the Cartan tensor T is written as :
T (X, Y , Z) = A(X, Y )C(Z) + A(Y , Z)C(X) + A(Z,X)C(Y ),
where A is a symmetric indicatory (2) π-form (A(X, η) = 0 for all X).
Definition 2.6. [25] A Finsler manifold (M,L) is said to be :
(a) S3-like if dim(M) ≥ 4 and the vertical curvature tensor S(X, Y , Z,W )
:= g(S(X, Y )Z,W ) has the form :
S(X, Y , Z,W ) =
(n− 1)(n− 2)
{~(X,Z)~(Y ,W )− ~(X,W )~(Y , Z)}.
(b) S4-like if dim(M) ≥ 5 and the vertical curvature tensor S(X, Y , Z,W ) has the
form :
S(X, Y , Z,W ) =~(X,Z)F(Y ,W )− ~(Y , Z)F(X,W )+
+ ~(Y ,W )F(X,Z)− ~(X,W )F(Y , Z),
(2.1)
where F is the (2)π-form defined by F =
{Ricv −
Scv ~
2(n− 2)
Definition 2.7. A Finsler manifold (M,L) is said to be :
(a) Sv-recurrent if the v-curvature tensor S satisfies the condition
(∇γXS)(Y , Z,W ) = λ(X)S(Y , Z)W,
where λ is a π-form of order one.
(b) Sv-recurrent of the second order if the v-curvature tensor S satisfies the condition
∇ S)(øY, øX,Z,W,U) = Θ(X, Y )S(Z,W )U,
where Θ is a π-form of order two.
Definition 2.8. [24] A Finsler manifold (M,L) is said to be :
(a) a Landsberg manifold if
P̂ (X, Y ) = P (X, Y )η = 0 ∀X, Y ∈ X(π(M)), or equivalently ∇βη T = 0.
(b) a general Landsberg manifold if
Tr{Y −→ P̂ (X, Y )} = 0 ∀X,∈ X(π(M)), or equivalently ∇βη C = 0.
Definition 2.9. A Finsler manifold (M,L) is said to be P -symmetric if the mixed
curvature tensor P satisfies
P (X, Y )Z = P (Y ,X)Z, ∀ øX, øY, øZ ∈ X(π(M)).
Definition 2.10. A Finsler manifold (M,L), where dimM ≥ 3, is said to be P2-like
if the mixed curvature tensor P has the form :
P (X, Y , Z, øW ) = α(Z)T (X, Y , øW )− α(W ) T (X, øY, Z),
where α is a (1) π-form (positively homogeneous of degree 0).
Definition 2.11. [25] A Finsler manifold (M,L), where dimM ≥ 3, is said to be
P -reducible if the π-tensor field P (X, Y , Z) := g(P (X, Y )η, Z) can be expressed in
the form :
P (X, Y , Z) = δ(X)~(Y , Z) + δ(Y )~(Z,X) + δ(Z)~(X, Y ),
where δ is a (1) π-form satisfying δ(øη) = 0.
Definition 2.12. [2] A Finsler manifold (M,L), where dimM ≥ 3, is said to be
h-isotropic if there exists a scalar ko such that the horizontal curvature tensor R has
the form
R(X, Y )Z = ko{g(Y , Z)X − g(X,Z)Y }.
Definition 2.13. [2] A Finsler manifold (M,L), where dimM ≥ 3, is said to be :
(a) of scalar curvature if there exists a scalar function k : TM −→ R such that
the horizontal curvature tensor R(X, Y , Z,W ) := g(R(X, Y )Z,W ) satisfies the
relation
R(η,X, η, Y ) = kL2~(X, Y ).
(b) of constant curvature if the function k in (a) is constant.
Definition 2.14. A Finsler manifold (M,L) is said to be R3-like if dimM ≥ 4 and
the horizontal curvature tensor R(X, Y , Z,W ) is expressed in the form
R(X, Y , Z,W ) =g(X,Z)F (Y ,W )− g(Y , Z)F (X,W )+
+ g(Y ,W )F (X,Z)− g(X,W )F (Y , Z),
(2.2)
where F is the (2)π-form defined by F = 1
{Rich − Sc
2(n−1)
3. Relationships between different types of special
Finsler spaces
This section is devoted to global investigation of some mutual relationships
between the special Finsler spaces introduced in the preceding section. Some conse-
quences are also drawn from these relationships.
We start with some immediate consequences from the definitions:
(a) A Locally Minkowskian manifold is a Berwald manifold.
(b) A Berwald manifold is a Landsberg manifold.
(c) A Landsberg manifold is a general Landsberg manifold.
(d) A Berwald manifold is Ch-recurrent (resp. P ∗-Finsler).
(e) A P ∗-manifold is a Landsberg manifold.
(f) A C-reducible (resp. C2-like) manifold is semi-C-reducible.
(g) A semi-C-reducible manifold is quasi-C-reducible.
(h) A Finsler manifold of constant curvature is of scalar curvature.
The following two lemmas are useful for subsequent use.
Lemma 3.1. [25] For every øX, øY ∈ X(π(M)), we have:
(a) P (øη, øX)øY = 0, (b) P (øX, øη)øY = 0, (c) P (øX, øY )øη = (∇βøηT )(øX, øY ).
Lemma 3.2. If φ is the vector π-form defined by
φ(øX) := øX − L−1ℓ(øX)øη, or φ := I − L−1ℓ⊗ øη, (3.1)
where ℓ is the π-form given by ℓ(X) = L−1g(X, η), then we have:
(a) ~(øX, øY ) = g(φ(øX), øY ), (b) φ(øη) = 0, (c) φ o φ = φ,
(d) Tr(φ) = n− 1, (e) ∇βøX φ = 0, (f) ∇βøX ~ = 0.
As we have seen, a Landsberg manifold is general Landsberg. The converse is
not true. Nevertheless, we have
Proposition 3.3. A C-reducible general Landsberg manifold (M,L) is a Landsberg
manifold.
Proof. Since (M,L) is a C-reducible manifold, then, by Definition 2.4, Lemma 3.2,
the symmetry of ~ and the non-degeneracy of g, we get
T (øX, øY ) =
{~(øX, øY )øC + C(øX)φ(øY ) + C(øY )φ(øX)},
where øC is the π-vector field defined by g(øC, øX) := C(øX). Taking the h-covariant
derivative ∇βøZ of both sides of the above equation, we obtain
(∇βøZ T )(øX, øY ) =
{(∇βøZ ~)(øX, øY )øC + ~(øX, øY )∇βøZ øC + C(øX)(∇βøZ φ)(øY ) +
+(∇βøZ C)(øX)φ(øY ) + C(øY )(∇βøZ φ)(øX) + (∇βøZ C)(øY )φ(øX)},
from which, by setting øZ = øη and taking into account the fact that ∇βøZ ~ = 0 and
that ∇βøZ φ = 0 ( Lemma 3.2), we get
(∇βøη T )(øX, øY ) =
{~(øX, øY )∇βøη øC+(∇βøη C)(øX)φ(øY )+(∇βøη C)(øY )φ(øX)}.
Now, under the given assumption that the (M,L) is a general Landsberg manifold,
then ∇βøη C = 0 (Definition 2.8) and hence ∇βøη øC = 0. Hence ∇βøη T = 0 and the
result follows. �
Also, a Berwald manifold is Landsberg. The converse is by no means true,
although we have no counter-examples. Finding a Landsberg manifold which is not
Berwald is still an open problem. Nevertheless, we have
Proposition 3.4. [25] A C-reducible Landsberg manifold (M,L) is a Berwald
manifold.
Combining the above two Propositions, we obtain the more powerful result :
Proposition 3.5. A C-reducible general Landsberg manifold (M,L) is a Berwald
manifold.
Summing up, we get:
Theorem 3.6. Let (M,L) be a C-reducible Finsler manifold. The following assertion
are equivalent :
(a) (M,L) is a Berwald manifold.
(b) (M,L) is a Landsberg manifold.
(c) (M,L) is a general Landsberg manifold.
We retrieve here a result of Matsumuoto [15], namely
Corollary 3.7. If the h-curvature tensor R and hv-curvature tensor P of a C-
reducible manifold vanish, then the manifold is Locally Minkowskian.
Remark 3.8. [15] It may be conjectured that a Finsler manifold will be Minkowskian
if the h-curvature tensor R and hv-curvature tensor P vanish. As above seen the
conjecture is verified already under somewhat strong condition “C-reducibility”.
Theorem 3.9. Let (M,L) be a Finsler manifold. Then we have :
(a) A C-reducible manifold is P -reducible.
(b) A P -reducible general Landsberg manifold is Landsberg.
Proof.
(a) Since (M,L) is C-reducible, then by Definition 2.4, we have
T (øX, øY, øZ) =
SøX,øY,øZ{~(øX, øY )C(øZ)}.
Applying the h-covariant derivative ∇βøW on both sides of the above equation, taking
into account the fact that (∇βøW T )(øX, øY, øZ) = g((∇βøW T )(øX, øY ), øZ) and that
∇βøW ~ = 0, we obtain
g((∇βøWT )(øX, øY ), øZ) =
SøX,øY,øZ{~(øX, øY )(∇βøW C)(øZ)}.
From which, by setting øW = øη and noting that P (øX, øY )øη = (∇βøη T )(øX, øY ),
the result follows.
(b) Since (M,L) is a P -reducible manifold, then by Definition 2.11, taking into
account the fact that g is nondegenerate, we obtain
P (øX, øY )øη = δ(øX)φ(øY ) + δ(øY )φ(øX) + ø~(øX, øY ) øζ, (3.2)
where øζ is the π-vector field defined by g(øζ, øX) := δ(øX).
Since δ(øη) = 0, then Tr{øY 7−→ δ(øY )φ(øX) + ~(øX, øY ) øζ} = 2δ(øX). Taking
the trace of both sides of (3.2), using the fact that P (øX, øY )øη = (∇βøη T )(øX, øY )
(Lemma 3.1) and that Tr{øY 7−→ (∇βøη T )(øX, øY )} = (∇βøη C)(øX), we get
δ(øX) =
n + 1
(∇βøη C)(øX). (3.3)
Now, from Equations (3.2) and (3.3), we have
g(P (øX, øY )øη, øZ) =
n + 1
SøX,øY,øZ{~(øX, øY )(∇βøη C)(øZ)}. (3.4)
According to the given assumption that the manifold is general Landsberg, then
∇βøη C = 0. Therefore, from (3.4), we get P (øX, øY )øη = 0 and hence the manifold
is Landsberg. �
Proposition 3.10.
(a) A Ch-recurrent manifold is a P ∗-Finsler manifold.
(b) A general Landsberg P ∗-Finsler manifold is a Landsberg manifold.
Proof. The proof is straightforward and we omit it. �
Proposition 3.11. A C2-like Finsler manifold is a Berwald manifold if, and only if,
the π-tensor field C is horizontally parallel.
Proof. Let (M,L) be C2-like. Then, T (øX, øY, øZ) =
C(øC)
C(øX)C(øY )C(øZ),
from which T (øX, øY ) = 1
C(øC)
C(øX)C(øY )øC. Taking the h-covariant derivative of
both sides, we get
(∇βøZT )(øX, øY ) =
−∇βøZC(øC)
C(øX)C(øY )øC +
C(øC)
(∇βøZC)(øX)C(øY )øC +
C(øC)
(∇βøZC)(øY )C(øX)øC +
C(øC)
C(øX)C(øY )∇βøZøC.
In view of this relation, ∇βøZ T = 0 if, and only if, ∇βøZ C = 0. Hence the result. �
Corollary 3.12. A C2-like general Landsberg manifold is a Landsberg manifold.
In view of the above Theorems, we have:
Corollary 3.13. The two notions of being Landsberg and general Landsberg coincide
in the case of C-reducibility, P -reducibility, C2-likeness or P
∗-Finsler.
As we know, a C-reducible Landsberg manifold is a Berwald manifold (Proposi-
tion 3.4 ). Moreover, A C2-like Finsler manifold is a Berwald manifold if, and only
if, the π-tensor field C is horizontally parallel (Proposition 3.11). We shall try to
generalize these results to the case of semi-C-reduciblity.
Theorem 3.14. A semi-C-reducible Finsler manifold is a Berwald manifold if, and
only if, the characteristic scalar µ and the π-tensor field C are horizontally parallel.
Proof. Firstly, if (M,L) is semi-C-reducible, then
T (øX, øY, øZ) =
SøX,øY,øZ{~(øX, øY )C(øZ)}+
C(øC)
C(øX)C(øY )C(øZ).
Taking the h-covariant derivative of both sides, noting that ∇βøX~ = 0, we get
(∇βøWT )(øX, øY, øZ) =
n + 1
SøX,øY,øZ{~(øX, øY ){µ(∇βøWC)(øZ) + (∇βøWµ)C(øZ)}}+
SøX,øY,øZ{(∇βøWC)(øX)C(øY )C(øZ)} −
∇βøW µ
τ ∇βøWC(øC)
}C(øX)C(øY )C(øZ).
Now, if the characteristic scalar µ and the π-tensor field C are horizontally par-
allel, then ∇βøWT = 0 and (M,L) is a Berwald manifold.
Conversely, if (M,L) is a Berwald manifold, then∇βøXT = 0 and hence ∇βøXC =
0, ∇βøXøC = 0. These, together with the above equation, give
∇βøWµ{
SøX,øY,øZ{~(øX, øY )C(øZ)} −
C(øX)C(øY )C(øZ)} = 0,
which implies immediately that ∇βøWµ = 0. �
The following lemmas are useful for subsequent use
Lemma 3.15. For all X, Y ∈ X(π(M)), we have :
(a) [γX, γY ] = γ(∇γXY −∇γYX)
(b) [γX, βY ] = −γ(P (Y ,X)η +∇βYX) + β(∇γXY − T (X, Y ))
(c) [βX, βY ] = γ(R(X, Y )η) + β(∇βXY −∇βYX)
Lemma 3.16. For all øX, øY, øZ, øW ∈ X(π(M)) and W ∈ X(TM), we have :
(a) g((∇WT )(øX, øY ), øZ) = g((∇WT )(øX, øZ), øY ),
(b) g(S(øX, øY )øZ, øW ) = −g(S(øX, øY )øW, øZ).
Proof.
(a) From the definition of the covariant derivative, we get
g((∇WT )(øX, øY ), øZ) = g(∇WT (øX, øY ), øZ)− g(T (∇WøX, øY ), øZ)−
−g(T (øX,∇WøY ), øZ).
(3.5)
Now, we have
g(∇WT (øX, øY ), øZ) = W · g(T (øX, øY ), øZ)− g(T (øX, øY ),∇WøZ)
= W · g(T (øX, øY ), øZ)− g(T (øX,∇WøZ), øY ),
Similarly,
g(T (øX,∇WøY ), øZ) = W · g(T (øX, øZ), øY )− g(∇WT (øX, øZ), øY ).
Substituting these two equations into (3.5), noting the property that g(T (∇WøX, øY ), øZ)
= g(T (∇WøX, øZ), øY ) (cf. §1), the result follows.
(b) follows directly from the general formula (which can be easily proved)
g(K(X, Y )øZ, øW ) + g(K(X, Y )øW, øZ) = 0
by setting X = γøX and Y = γøY , where K is the classical curvature tensor of the
Cartan connection as a linear connection in the pull-back bundle (cf. §1). �
Proposition 3.17. Let (M,L) be a Ch-recurrent Finsler manifold (∇βøXT = λ0(øX)T ).
Then, we have:
(a) If Ko := λo(øη) = 0, then the hv-curvature tensor P is expressed in the form:
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)
and the (v)hv-torsion P̂ vanishes.
(b) If Ko 6= 0, then the v(hv)-torsion tensor P̂ is recurrent:
(∇βøZP̂ )(øX, øY ) = (λo(øZ) +
∇βøZKo
)P̂ (øX, øY ).
Proof.
(a) The hv-curvature tensor P can be written in the form [25]:
P (øX, øY, øZ, øW ) = g((∇βøZT )(øX, øY ), øW )− g((∇βøWT )(øX, øY ), øZ)+
+g(T (øX, øZ), P̂(øW, øY ))− g(T (øX, øW ), P̂(øZ, øY )).
Then, by using P̂ (øX, øY ) = (∇βøηT )(øX, øY ) (Lemma 3.1) and the C
h-recurrence
condition, we get
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)−
−λo(øη){g(T (øX, øW ), T (øY, øZ))− g(T (øX, øZ), T (øY, øW ))}
= λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)− λo(øη)S(øX, øY, øZ, øW ).
Now, if λo(øη) = 0, then (a) follows from the above relation.
(b) If Ko := λo(øη) 6= 0, then by Lemma 3.1 and the recurrence condition, we have
P̂ (øX, øY ) = KoT (øX, øY ),
from which
(∇βøZP̂ )(øX, øY ) = {∇βøZKo +Koλo(øZ)}T (øX, øY ).
Then, (b) follows from the above two equations. �
Theorem 3.18. Assume that (M,L) is Ch-recurrent. Then, the v-curvature tensor S
is recurrent with respect to the h-covariant differentiation : ∇βøXS = θ(øX)S, where
θ is a π-form of order one.
Proof. One can easily show that : For all X, Y, Z ∈ X(TM),
SX,Y,Z{K(X, Y )ρZ +∇XT(Y, Z) +T(X, [Y, Z])} = 0.
Setting X = γøX, Y = γøY and Z = βøZ in the above equation, we get
S(øX, øY )øZ = ∇γøY T (øX, øZ)−∇γøXT (øY, øZ)−∇βøZT(γøX, γøZ)−
−T(γøX, [γøY, βøZ]) +T(γøY, [γøX, βøZ]) +T([γøX, γøY ], βøZ).
Using Lemma 3.15 and the fact that T(γøX, γøZ) = 0, the above equation reduces
S(øX, øY )øZ = (∇γøY T )(øX, øZ)− (∇γøXT )(øY, øZ)+
+T (øX, T (øY, øZ))− T (øY, T (øX, øZ)).
(3.6)
From which, since g(T (øX, øY ), øZ) = g(T (øX, øZ), øY ), we have
g(S(øX, øY )øZ, øW ) = g((∇γøY T )(øX, øZ), øW )− g((∇γøXT )(øY, øZ), øW )+
+g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)).
Similarly,
g(S(øX, øY )øW, øZ) = g((∇γøY T )(øX, øW ), øZ)− g((∇γøXT )(øY, øW ), øZ)+
+g(T (øX, øZ), T (øY, øW ))− g(T (øY, øZ), T (øX, øW )).
The above two equations, together with Lemma 3.16, yield
g((∇γøXT )(øY, øZ), øW ) = g((∇γøY T )(øX, øZ), øW ). (3.7)
By (3.6) and (3.7), we obtain
S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)). (3.8)
Now, using the given assumption that the manifold is Ch-recurrent, Equation
(3.8) implies that
(∇βøXS)(øY, øZ, øV, øW ) = ∇βøXS(øY, øZ, øV, øW )−
−S(∇βøXøY, øZ, øV, øW )− S(øY,∇βøXøZ, øV, øW )−
−S(øY, øZ,∇βøXøV, øW )− S(øY, øZ, øV,∇βøXøW ).
= +∇βøXg(T (øY, øW ), T (øZ, øV ))−∇βøXg(T (øZ, øW ), T (øY, øV ))−
−g(T (∇βøXøY, øW ), T (øZ, øV )) + g(T (øZ, øW ), T (∇βøXøY, øV ))−
−g(T (øY, øW ), T (∇βøXøZ, øV )) + g(T (∇βøXøZ, øW ), T (øY, øV ))−
−g(T (øY, øW ), T (øZ,∇βøXøV )) + g(T (øZ, øW ), T (øY,∇βøXøV ))−
−g(T (øY,∇βøXøW ), T (øZ, øV )) + g(T (øZ,∇βøXøW ), T (øY, øV )).
= g((∇βøXT )(øY, øW ), T (øZ, øV )) + g(T (øY, øW ), (∇βøXT )(øZ, øV ))−
−g((∇βøXT )(øZ, øW ), T (øY, øV ))− g(T (øZ, øW ), (∇βøXT )(øY, øV )).
= 2λo(øX)S(øY, øZ, øV, øW ) =: θ(øX)S(øY, øZ, øV, øW ).
Hence, the result follows. �
Corollary 3.19. In the course of the proof of Theorem 3.18, we have shown that
(Equations (3.7) and (3.8)) :
(a) (∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ),
(b) S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)).
Corollary 3.20. Let (M,L) be a C2-like Finsler manifold. Then the the v-curvature
tensor S vanishes.
Proof. Substituting T (øX, øY ) = 1
C(øC)
C(øX)C(øY )øC in Corollary 3.19(b), we
get the result. �
Corollary 3.21. Let (M,L) be a C-reducible manifold. Then,
(a) the v-curvature tensor S has the form
S(øX, øY, øZ, øW ) =
(n + 1)2
{C2~(øX, øW )~(øY, øZ)− C2~(øY, øW )~(øX, øZ) +
+~(øX, øW )C(øY )C(øZ) + ~(øY, øZ)C(øX)C(øW )−
−~(øY, øW )C(øX)C(øZ)− ~(øX, øZ)C(øY )C(øW )}.
(b) the vertical Ricc tensor Ricv has the form
Ricv(øX, øY ) =
(3− n)
(n + 1)2
C(øX)C(øY )−
(n− 1)
(n+ 1)2
C2~(øX, øY ).
(c) the vertical scalar curvature Scv has the form
Scv =
(2− n)
(n+ 1)
Theorem 3.22. A Finsler manifold (M,L) is P -Symmetric if, and only if, the
v-curvature tensor S satisfies the equation ∇βøηS = 0.
Proof. One can show that: For all X, Y, Z ∈ X(TM),
SX,Y,Z{∇ZK(X, Y )−K(X, Y )∇Z −K([X, Y ], Z)} = 0. (3.9)
Setting X = γøX, Y = γøY and Z = βøZ in the above equation, we get
∇βøZS(øX, øY )øW +∇γøY P (øZ, øX)øW −∇γøXP (øZ, øY )øW−
−S(øX, øY )∇βøZøW + P (øZ, øY )∇γøXøW − P (øZ, øX)∇γøY øW−
−K([γøX, γøY ], βøZ)øW −K([γøY, βøZ], γøX)øW −K([βøZ, γøX ], γøY )øW = 0.
By using Lemma 3.15, the above relation reduces to
(∇βøZS)(øX, øY, øW ) + (∇γøY P )(øZ, øX, øW )− (∇γøXP )(øZ, øY, øW )+
+S(P (øZ, øY )øη, øX)øW − S(P (øZ, øX)øη, øY )øW+
+P (T (øY, øZ), øX)øW − P (T (øX, øZ), øY )øW = 0.
(3.10)
Setting øZ = øη in the above equation, taking into account Lemma 3.1 and the fact
that T (øX, øη) = 0 and that (∇γøXP )(øη, øY, øZ) = −P (øX, øY )øZ, we get
P (øX, øY )øZ = P (øY, øX)øZ − (∇βøηS)(øX, øY, øZ). (3.11)
The result follows immediately from (3.11). �
According to (3.11) and Lemma 3.1, we have :
Corollary 3.23. Let P̂ (øX, øY ) := P (øX, øY )øη and T̂ (øX, øY ) := (∇βøηT )(øX, øY ).
Then the π-tensor fields P̂ and T̂ are symmetric.
Theorem 3.18 and Theorem 3.22 give rise the following result.
Theorem 3.24. Assume that a Finsler manifold (M,L) is Ch-recurrent and P -
symmetric. If θ(øη) 6= 0, then the v-curvature tensor S vanishes identically.
Now, we shall prove the following lemma which provides some important and
useful properties of the torsion tensor T and the v-curvature S :
Lemma 3.25. For every øX, øY, øZ and øW ∈ X(π(M)), we have
(a) T (øX, øY ) = T (øY, øX),
(b) T (øη, øX) = 0,
(c) SøX,øY,øZS(øX, øY )øZ = 0,
(d) g(S(øX, øY )øZ, øW ) = g(S(øZ, øW )øX, øY ),
(e) S(øη, øX)øY = 0 = S(øX, øη)øY ,
(f) (∇γøXS)(øη, øY )øZ = −S(øX, øY )øZ, (∇γøXS)(øη, øX)øη = 0 .
(g) S(øX, øY )øZ = −1
{(DγXT )(Y , Z)− (DγY T )(X,Z)}.
Consequently, S vanishes if and only if (DγXT )(Y , Z) = (DγY T )(X,Z).
Proof.
(a) From Corollary 3.19(a), we have
(∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ).
Setting øZ = øη and using the fact that T (øX, øη) = 0 and that K oγ = idX(π(M)),
the result follows.
(b) Follows from (a) together with the relation T (øX, øη) = 0.
(c) Setting X = γøX, Y = γøY and Z = γøZ in (3.9) and using Lemma 3.15, we
SøX,øY,øZ(∇γøXS)(øY, øZ, øW ) = 0.
Again, setting øW = øη in the above equation and using the fact that S(øX, øY )øη =
0 and that K oγ = idX(π(M)), the result follows.
(d) Follows from Corollary 3.19(b), noting that T is symmetric.
(e) and (f) are clear.
(g) From the relation DγXøY = ∇γXøY − T (øX, øY ) [27], we get
(DγXT )(øY, øZ) = (∇γXT )(øY, øZ)−T (øX, T (øY, øZ))+T (T (øX, øY ), øZ)+T (øY, T (øX, øZ)),
(DγY T )(øX, øZ) = (∇γY T )(øX, øZ)−T (øY, T (øX, øZ))+T (T (øY, øX), øZ)+T (øX, T (øY, øZ)).
The result follows from the above two equations, using Corollary 3.19 and the sym-
metry of T . �
As a direct consequence of the above lemma, we have the
Corollary 3.26. A P2-like Finsler manifold is P -symmetric.
Proposition 3.27. Assume that (M,L) is Cv-recurrent. Then, the v-curvature ten-
sor S is v-recurrent : ∇γøXS = Ψ(øX)S, Ψ being a (1)π-form. Consequently, S
vanishes identically.
Proof. Taking the v-covariant derivative of both sides of the relation in Corollary
3.19(b) and, then, using the assumption that ∇γXT = λ0(X)T , we get
(∇γøXS)(øY, øZ, øV, øW ) = 2λo(øX)S(øY, øZ, øV, øW ) =: ψ(øX)S(øY, øZ, øV, øW ),
which shows that S is v-recurrent.
Now, setting øV = øη in the last equation, using the properties of S and noting
that K oγ = idX(π(M)), we conclude that S = 0. �
The following result gives a characterization of Riemannian manifolds in terms
of Cv-recurrence and C0-recurrence.
Theorem 3.28.
(a) A Cv-recurrent Finsler manifold is Riemannian,
(b) A C0-recurrent Finsler manifold is Riemannian.
Proof. (a) Since (M,L) is Cv-recurrent, then (∇γXT )(Y , Z) = λo(X)T (Y , Z), from
which, by setting øX = øη and noting that ∇γøηT = −T , we get
T (Y , Z) = −λo(η)T (Y , Z). (3.12)
But since (∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ) (Corollary 3.19), then λo(øX)T (øY, øZ) =
λo(øY )T (øX, øZ). Hence,
λo(η)T (Y , Z) = 0. (3.13)
Then, the result follows from (3.12) and (3.13).
(b) can be proved similarly. �
Theorem 3.29. For a Finsler manifold (M,L), the following assertions are
equivalent :
(a) (M,L) is Sv-recurrent.
(b) The v-curvature tensor S vanishes identically.
(c) (M,L) is Sv-recurrent of the second order.
Proof.
(a) =⇒ (b) : If (M,L) is Sv-recurrent, then by Definition 2.7(a) we have
(∇γøWS)(øX, øY, øZ) = λ(øW )S(øY, øX)øZ,
from which, by setting øZ = øη, taking into account the fact that S(øX, øY )øη = 0
and that Koγ = idπ−1(TM), the result follows.
(b) =⇒ (a) : Trivial.
(b) =⇒ (c) : Trivial.
(c) =⇒ (b) : If the given manifold (M,L) is Sv-recurrent of the second order, then
by Definition 2.7(b) we get
Θ(øX, øY )S(øZ, øV )øW = (
∇ S)(øY, øX, øZ, øV, øW )
= ∇γøY (∇γøXS)(øZ, øV, øW )− (∇γ∇γøY øXS)(øZ, øV, øW )−
−(∇γøXS)(∇γøY øZ, øV, øW )− (∇γøXS)(øZ,∇γøY øV, øW )−
−(∇γøXS)(øZ, øV,∇γøY øW ).
By substituting øZ = øη = øW in the above equation and using Lemma 3.25 and the
fact that S(øX, øY )øη = 0, we get
S(øX, øY )øZ = −S(øZ, øY )øX and S(øX, øY )øZ = −S(øX, øZ)øY.
From this, together with the identity SøX,øY,øZS(øX, øY )øZ = 0, the v-curvature
tensor S vanishes identically. �
In view of the above theorem we have :
Corollary 3.30.
(a) An Sv-recurrent (resp. a second order Sv-recurrent) manifold (M,L) is S3-like,
provided that dimM ≥ 4.
(b) An Sv-recurrent (resp. a second order Sv-recurrent) manifold (M,L) is S4-like,
provided that dimM ≥ 5.
Theorem 3.31. If (M,L) is a P2-like Finsler manifold, then the v-curvature tensor
S vanishes or the hv-curvature tensor P vanishes. In the later case, the h-covariant
derivative of S vanishes.
Proof. As (M,L) is P2-like, then P (X, Y , η, øW ) = α(η)T (X, Y , øW ) =: αoT (X, Y , øW )
and hence
P̂ (øX, øY ) = αoT (X, Y ). (3.14)
Now, setting øW = øη into (3.10), we get
(∇γøY P̂ )(øZ, øX)− (∇γøX P̂ )(øZ, øY )− P (øZ, øX)øY + P (øZ, øY )øX−
−P̂ (T (øX, øZ), øY ) + P̂ (T (øY, øZ), øX) = 0.
Hence,
g((∇γøY P̂ )(øZ, øX), øW )− g((∇γøXP̂ )(øZ, øY ), øW )− P (øZ, øX, øY, øW )+
+P (øZ, øY, øX, øW )− g(P̂ (T (øX, øZ), øY ), øW ) + g(P̂ (T (øY, øZ), øX), øW ) = 0.
From which, together with (3.14) and Definition 2.10, taking into account the relation
(∇γøY P̂ )(øZ, øX) = (∇γøY αo)T (øZ, øX) + αo(∇γøY T )(øZ, øX), we obtain
g((∇γøY αo)T (øZ, øX) + αo(∇γøY T )(øZ, øX), øW )− g((∇γøXαo)T (øZ, øY )+
+αo(∇γøXT )(øZ, øY ), øW ) + α(X)T (Z, Y , øW )− α(W ) T (Z, øY,X)− α(Y )T (Z,X, øW )
+α(W ) T (X, øY, Z)− g(αoT (T (øX, øZ), øY ), øW ) + g(αoT (T (øY, øZ), øX), øW ) = 0.
Therefore, using Corollary 3.19,
(∇γøY α)(øη)T (øX, øZ, øW )− (∇γøXα)(øη)T (øY, øZ, øW ) = αoS(øX, øY, øW, øZ).
It is to be observed that the left-hand side of the above equation is symmetric in
the arguments øZ and øW while the right-hand side is skew-symmetric in the same
arguments. Hence we have
αoS(øX, øY, øW, øZ) = 0, (3.15)
ε(øY )T (øX, øZ, øW )− ε(øX)T (øY, øZ, øW ) = 0, (3.16)
where ε is the π-form defined by ε(øY ) := (∇γøY α)(øη).
Now, If ε 6= 0, it follows from (3.16) that there exists a scalar function Υ such that
T (øX, øY, øZ) = Υ ε(øX)ε(øY )ε(øZ). Consequently, T (øX, øY ) = Υ ε(øX)ε(øY )øε,
where g(øε, øX) := ε(øX). From which
S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ))
= Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε)−Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) = 0.
On the other hand, if the v-curvature tensor S 6= 0, then it follows from (3.15)
that ε = 0 and α(øη) = 0. Hence, α = 0 and the hv-curvature tensor P vanishes. In
this case, it follows from the identity (3.10) that ∇βøXS = 0. �
Proposition 3.32. A P2-like Finsler manifold (M,L) is a P
∗-Finsler manifold.
Proof. As (M,L) is P2-like, then from (3.14), we have P̂ (X, Y ) = αoT (X, Y ). Using
Lemma 3.1, we get (∇βøηT )(øX, øY ) = α0T (øX, øY ), from which, by taking the trace,
∇βøηC = α0T , where α0 =
bg(∇βη C,C)
. Hence the result. �
The next definition will be useful in the sequel.
Definition 3.33. A π-tensor field Θ is positively homogenous of degree r in the
directional argument y (symbolically, h(r)) if it satisfies the condition
∇γη Θ = rΘ, or Dγη Θ = rΘ.
Lemma 3.34. Let (M,L) be a Finsler manifold, then we have
(a) The Finsler metric g (the angular metric tensor ~) is homogenous of degree 0,
(b) The v-curvature tensor S is homogenous of degree −2,
(c) The hv-curvature tensor P is homogenous of degree −1,
(d) The h-curvature tensor R is homogenous of degree 0,
(e) The (h)hv-torsion tensor T is homogenous of degree −1,
(f) The (v)hv-torsion tensor P̂ is homogenous of degree 0,
(g) The (v)h-torsion tensor R̂ is homogenous of degree 1.
Lemma 3.35. For every vector (1)π-form A, we have
∇ A)(øX, øY, øZ)− (
∇ A)(øY, øX, øZ) = A(R(øX, øY )øZ)− R(øX, øY )A(øZ)+
γ bR(øX,øY )
A)(øZ).
Deicke theorem [4] can be formulated globally as follows:
Lemma 3.36. Let (M,L) be a Finsler manifold. The following assertions are
equivalent:
(a) (M,L) is Riemannian,
(b) The (h)hv-torsion tensor T vanishes,
(c) The π-form C vanishes.
Theorem 3.37. Let (M,L) be Finsler manifold which is h-isotropic (of scalar k0)
and Ch-recurrent (of recurrence vector λ0). Then, (M,L) is necessarily one of the
following:
(a) A Riemannian manifold of constant curvature,
(b) A Finsler manifold of dimension 2,
(c) A Finsler manifold of dimensions n ≥ 3 with vanishing scalar k0 and
(∇βøXλo)(øY ) = (∇βøY λo)(øX).
Proof. For a Ch-recurrent manifold, one can easily show that
∇ T )(øX, øY, øZ, øW )− (
∇ T )(øY, øX, øZ, øW ) =
= {(∇βøXλo)(øY )− (∇βøY λo)(øX)}T (øZ, øW ) =: Ψ(øX, øY )T (øZ, øW ).
From which, taking into account Lemma 3.35, we obtain
Ψ(øX, øY )T (øZ, øW ) = T (R(øX, øY )øZ, øW ) + T (øZ,R(øX, øY )øW )−
−R(øX, øY )T (øZ, øW ) + (∇
γ bR(øX,øY )
T )(øZ, øW ).
Now, as (M,L) is h-isotropic of scalar k0, then the h-curvature tensor R has the form
R(øX, øY )øZ = k0{g(øX, øZ)øY − g(øY, øZ)øX} ; (n ≥ 3).
From the above two equations, we get
Ψ(øX, øY )T (øZ, øW ) = k0g(øX, øZ)T (øY, øW )− k0g(øY, øZ)T (øX, øW ) + k0g(øX, øW )T (øZ, øY )−
−k0g(øY, øW )T (øZ, øX)− k0g(øX, T (øZ, øW ))øY + k0g(øY, T (øZ, øW ))øX
+k0g(øX, øη)(∇γøY T )(øZ, øW )− k0g(øY, øη)(∇γøXT )(øZ, øW ).
(3.17)
Setting øY = øη, noting that T is h(−1) and g(øη, øη) = L2, we get
Ψ(øX, øη)T (øZ, øW ) = −k0g(øη, øZ)T (øX, øW )− k0g(øη, øW )T (øZ, øX)− k0T (øX, øZ, øW )øη −
−k0g(øX, øη)T (øZ, øW )− k0L
2(∇γøXT )(øZ, øW ).
From which, we have
g(øY, øη)Ψ(øX, øη)T (øZ, øW ) = −k0g(øY, øη)g(øη, øZ)T (øX, øW )− k0g(øY, øη)g(øη, øW )T (øZ, øX)−
−k0g(øY, øη)T (øX, øZ, øW )øη− k0g(øY, øη)g(øX, øη)T (øZ, øW )−
2g(øY, øη)(∇γøXT )(øZ, øW ),
(3.18)
whereas
g(øX, øη)Ψ(øY, øη)T (øZ, øW ) = −k0g(øX, øη)g(øη, øZ)T (øY, øW )− k0g(øX, øη)g(øη, øW )T (øZ, øY )−
−k0g(øX, øη)T (øY, øZ, øW )øη− k0g(øX, øη)g(øY, øη)T (øZ, øW )−
2g(øX, øη)(∇γøY T )(øZ, øW ).
(3.19)
Now, from (3.17), (3.18) and (3.19), we obtain
T (øZ, øW ){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} =
= UøX,øY k0L
2{~(øX, øZ)T (øY, øW ) + ~(øX, øW )T (øY, øZ)− φ(øY ) T (øX, øZ, øW )}.
Taking the trace of both sides of the above equation, we get
C(øZ){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} =
= 2k0L
2{~(øX, øZ)C(øY )− ~(øY, øZ)C(øX)}.
(3.20)
Setting øZ = øC, taking into account the fact that ~(øX, øC) = C(øX), the above
equation reduces to
C(øC){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} = 0.
Now, if C(øC) = g(øC, øC) = 0, then øC = 0 and so C = 0. Consequently, by
Lemma 3.36, (M,L) is a Riemannian manifold of constant curvature.
On the other hand, if (M,L) is not Riemannian, then we have
L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη) = 0.
From which, together with (3.20), we get
k0{~(øX, øZ)C(øY )− ~(øY, øZ)C(øX)} = 0. (3.21)
If k0 6= 0, then, by (3.21), ~(øX, øZ)C(øY ) = ~(øY, øZ)C(øX). Setting øY = øC,
we get ~(øX, øZ) = 1
C(øX)C(øZ), which implies that dimM = 2.
If k0 = 0, then R = 0 and (3.17) yields Ψ(øX, øY ) = 0, which means that
(∇βøXλo)(øY ) = (∇βøY λo)(øX). �
Now, we focus our attention to the interesting case (c) of the above theorem. In
this case, the h-curvature tensor R = 0 and hence the (v)h-torsion tensor R̂ = 0.
Therefore, the equation (deduced from (3.9))
(∇γøXR)(øY, øZ, øW ) + (∇βøY P )(øZ, øX, øW )− (∇βøZP )(øY, øX, øW )−
−P (øZ, P (øY, øX)øη)øW +R(T (øX, øY ), øZ)øW − S(R(øY, øZ)øη, øX)øW+
+P (øY, P (øZ, øX)øη)øW − R(T (øX, øZ), øY )øW = 0.
reduces to
(∇βøY P )(øZ, øX, øW )− (∇βøZP )(øY, øX, øW )−
−P (øZ, P̂ (øY, øX))øW + P (øY, P̂ (øZ, øX))øW = 0.
Setting øW = øη, we get
(∇βøY P̂ )(øZ, øX)− (∇βøZP̂ )(øY, øX)− P̂ (øZ, P̂ (øY, øX)) + P̂ (øY, P̂ (øZ, øX)) = 0.
(3.22)
Since (M,L) is Ch-recurrent, then, by Proposition 3.17, the (v)hv-torsion tensor
P̂ satisfies the relations (∇βøZP̂ )(øX, øY ) = (Koλo(øZ) + ∇βøZKo)T (øX, øY ) and
P̂ (øX, øY ) = λo(øη)T (øX, øY ) = KoT (øX, øY ). From these, together with (3.22),
we get
(Koλo(øY ) +∇βøYKo)T (øZ, øX)− (Koλo(øZ) +∇βøZKo)T (øX, øY )−
−K2oT (øZ, T (øX, øY )) +K
oT (øY, T (øX, øZ)) = 0.
Hence, by Corollary 3.19,
K2oS(øY, øZ, øX, øW ) = UøY,øZ{(Koλo(øY ) +∇βøYKo)T (øX, øZ, øW )}.
As S(øY, øZ, øX, øW ) is skew-symmetric in the arguments øX and øW while the
right-hand side is symmetric in the same arguments, we obtain
K2oS(øY, øZ, øX, øW ) = 0, (3.23)
UøY,øZ{(Koλo(øY ) +∇βøYKo)T (øZ, øX, øW )} = 0. (3.24)
It follows from (3.23) and () that
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ).
On the other hand, if Ko 6= 0, then the v-curvature tensor S vanishes from (3.23).
Next, it is seen from (3.24) that, if V(øY ) := Koλo(øY ) +∇βøYKo 6= 0, then there
exists a scalar function Υ =
T (øX,øZ,øW )T (øX,øY,øZ)T (øY,øZ,øW )
(T (øX,øY,øW ))2(V(øZ))3
such that
T (øX, øY, øW ) = ΥV(øX)V(øY )V(øW ).
Summing up, we have
Theorem 3.38. Let (M,L) be a Finsler manifold of dimensions n ≥ 3. If (M,L) is
h-isotropic and Ch-recurrent, then
(a) the recurrence vector λo satisfies : (∇βøXλo)(øY ) = (∇βøY λo)(øX),
(b) the h-curvature tensor R = 0 and the (v)h-torsion tensor R̂ = 0,
(c) the hv-curvature tensor P has the property that
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ),
(d) the (v)hv-torsion tensor P̂ (øX, øY ) = KoT (øX, øY ).
Moreover, if Ko 6= 0, then
(e) the v-curvature tensor S vanishes,
(f) the (h)hv-torsion tensor T satisfies : T (øX, øY, øW ) = ΥV(øX)V(øY )V(øW ).
By Definition 2.10 and Theorem 3.38, we immediately have :
Corollary 3.39. A Finsler manifold (M,L) of dimension n ≥ 3 which is h-isotropic
and Ch-recurrent is necessarily P2-like.
Now, we define an operator P which aids us to investigate the R3-like manifolds.
Definition 3.40.
(a) If ω is a π-tensor field of type (1,p), then P · ω is a π-tensor field of the same
type defined by :
(P · ω)(øX1, ..., øXp) := φ(ω(φ(øX1), ..., φ(øXp))),
where φ is the vector π-form defined by (3.1).
(b) If ω is a π-tensor field of type (0,p), then P · ω is a π-tensor field of the same
type defined by :
(P · ω)(øX1, ..., øXp) := ω(φ(øX1), ..., φ(øXp)).
Remark 3.41. Since φ(φ(øX)) = φ(øX) for every øX ∈ X(π(M)) (Lemma 3.2),
then the operator P is a projector (i.e. P · (P · ω) = P · ω).
Definition 3.42. A π-tensor field ω is said to be indicatory if it satisfies the
condition : P · ω = ω.
The following result gives a characterization of the indicatory property for certain
types of π-tensor fields :
Lemma 3.43.
(a) A vector (2)π-form ω is indicatory if, and only if, ω(øX, øη) = 0 = ω(øη, øX)
and g(ω(øX, øY ), øη) = 0.
(b) A scaler (2) π-form ω is indicatory if, and only if, ω(øX, øη) = 0 = ω(øη, øX).
Proof.
(a) Let ω be a vector (2)π-form. By Definition 3.40(a) and taking into account (3.1),
we get
(P · ω)(øX, øY ) = φ(ω(φ(øX), φ(øY )))
= φ{ω(øX − L−1ℓ(øX)øη, øY − L−1ℓ(øY )øη)}
= φ{ω(øX, øY )− L−1ℓ(øY )ω(øX, øη)−
−L−1ℓ(øX)ω(øη, øY ) + L−2ℓ(øX)ℓ(øY )ω(øη, øη)}
= ω(øX, øY )− L−2g(ω(øX, øY ), øη)øη − φ{L−1ℓ(øY )ω(øX, øη)+
+L−1ℓ(øX)ω(øη, øY )− L−2ℓ(øX)ℓ(øY )ω(øη, øη)}
(3.25)
Now, if ω(øX, øη) = 0 = ω(øη, øX) and g(ω(øX, øY ), øη) = 0, then (3.25) implies
that (P · ω)(øX, øY ) = ω(øX, øY ) and hence ω is indicatory.
On the other hand, if ω is indicatory, then ω(øX, øY ) = φ(ω(φ(øX), φ(øY ))).
From which, setting øX = øη (resp. øY = øη) and taking into account the fact that
φ(øη) = 0 (Lemma 3.2), we get ω(øη, øY ) = 0 (resp. ω(øX, øη) = 0). From this, to-
gether with (P·ω)(øX, øY ) = ω(øX, øY ), Equation (3.25) implies that L−2g(ω(øX, øY ), øη)øη =
0. Consequently, g(ω(øX, øY ), øη) = 0.
(b) The proof is similar to that of (a) and we omit it. �
Proposition 3.44. For a Finsler manifold (M,L), the following tensors are
indicatory :
(a) The π-tensor field φ,
(b) The mixed torsion tensor T ,
(c) The v-curvature tensor S,
(d) The angular metric tensor ~,
(e) The π-tensor field P · ω for every π-tensor field ω.
Now, we define the following π-tensor fields:
F : F (X, Y ) := 1
{Rich(X, Y )−
Schg(X,Y )
2(n−1)
Fo : g(Fo(øX), øY ) := F (øX, øY ),
F a : F a(øX) := F (øη, øX),
F b : F b(øX) := F (øX, øη),
m : m(øX, øY ) := (P · F )(øX, øY ),
mo : g(mo(øX), øY ) := m(øX, øY ),
a : a(øX) := L−1(P · F a)(øX),
øa : g(øa, øY ) := a(øX),
b : b(øX) := L−1(P · F b)(øX),
øb : g(øb, øX) := b(øX),
c : c := L−2F (øη, øη),
R̂ : R̂(øX, øY ) := R(øX, øY )øη,
H : H(øX) := R(øη, øX)øη = R̂(øη, øX).
(3.26)
Remark 3.45. One can show that m, mo, a and b are indicatory and H(øη) = 0.
Proposition 3.46. If (M,L) is an R3-like Finsler manifold, then the π-tensor field
F can be written in the form
F (øX, øY ) = m(øX, øY ) + ℓ(øX)a(øY ) + ℓ(øY )b(øX) + c ℓ(øX)ℓ(øY ). (3.27)
Proof. The proof follows from Definitions 2.14 and 3.40(b), taking into account
Equations (3.1) and (3.26). In more details :
(P · F )(øX, øY ) = F (φ(øX), φ(øY ))
= F (øX − L−1ℓ(øX)øη, øY − L−1ℓ(øY )øη)
= F (øX, øY )− L−1ℓ(øY )F (øX, øη)−
−L−1ℓ(øX)F (øη, øY ) + L−2ℓ(øX)ℓ(øY )F (øη, øη)
= F (øX, øY )− L−1ℓ(øY ){(P · F b)(øX) + L−1ℓ(øX)F (øη, øη)}−
−L−1ℓ(øX){(P · F a)(øY ) + L−1ℓ(øY )F (øη øη)}+ L−2ℓ(øX)ℓ(øY )F (øη, øη)
= F (øX, øY )− ℓ(øX)a(øY )− ℓ(øY )b(øX)− c ℓ(øX)ℓ(øY ). �
Remark 3.47. One can show that the π-tensor fields a and b satisfy the following
relations
F a(øX) = L{a(øX) + c ℓ(øX)},
F b(øX) = L{b(øX) + c ℓ(øX)}.
(3.28)
Proposition 3.48. In an R3-like Finsler manifold (M,L), we have :
(a) R(øX, øY )øZ = g(øX, øZ)Fo(øY )+F (øX, øZ)øY−g(øY, øZ)Fo(øX)−F (øY, øZ)øX.
(b) R̂(øX, øY ) = g(øX, øη)Fo(øY )+F (øX, øη)øY −g(øY, øη)Fo(øX)−F (øY, øη)øX.
(c) H(øY ) = L2Fo(øY ) + c L
2øY − g(øY, øη)Fo(øη)− F (øY, øη)øη.
(d) Fo(øX) = mo(øX) + øa ℓ(øX) + L
−1b(øX)øη + c L−1ℓ(øX)øη.
Consequently,
(e) R̂(øX, øY ) = L{ℓ(øX)(mo(øY ) + c φ(øY )) + b(øX)φ(øY )}−
− L{ℓ(øY )(mo(øX) + c φ(øX)) + b(øY )φ(øX)}.
(f) H(øY ) = L2{mo(øY ) + c φ(øY )}.
Proof.
(a) Since (M,L) is an R3-like manifold, then by Definition 2.14, we have
R(X, Y , Z,W ) =g(X,Z)F (Y ,W )− g(Y , Z)F (X,W )+
+ g(Y ,W )F (X,Z)− g(X,W )F (Y , Z).
From which, using the fact that g(Fo(øX), øY ) = F (øX, øY ) and that the Finsler
metric g is non-degenerate, the result follows.
(b) Follows from (a) by setting øZ = øη.
(c) Follows from (b) by setting øX = øη.
(d) By (3.27) and (3.26), we get
g(Fo(øX), øY ) = g(mo(øX), øY )+g(øa, øY ) ℓ(øX)+L
−1b(øX)g(øη, øY )+c L−1ℓ(øX)g(øη, øY ).
Hence, the result follows, from the non-degeneracy of g.
(e) Follows by substituting Fo(øX) (from (d)) and F
b(øX) (from (3.28)) into (b).
(f) Follows from (e) by setting øX = øη, taking into account Remark 3.45 and the
fact that ℓ(øη) = L. �
Remark 3.49. In view of (3.26) and Lemma 3.2, Definition 2.13(a) can be reformu-
lated as follows:
A Finsler manifold (M,L) is of scaler curvature if the π-tensor field H satisfies the
relation H(øX) = L2κφ(øX), where κ is a scalar function on TM.
Definition 3.50. A Finsler manifold (M,L) is said to be of perpendicular scalar
(or of p-scalar) curvature if the h-curvature tensor R satisfies the condition
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øX, øW )~(øY, øZ)}, (3.29)
where Ro is a function called the perpendicular scalar curvature.
Definition 3.51. A Finsler manifold (M,L) is said to be of s-ps curvature if (M,L)
is both of scalar curvature and of p-scalar curvature.
Proposition 3.52. If mo(øX) = t φ(øX), then an R3-like Finsler manifold is a
Finsler manifold of s-ps curvature.
Proof. Under the given assumption and taking into account Proposition 3.48(f), we
H(øX) = L2κφ(øX), with κ = t + c.
Thus, the considered manifold is of scalar curvature.
Now, we prove that the given manifold is of p-scalar curvature. Applying the
projection P on the h-curvature tensor R of an R3-like manifold, we get
(P · R)(øX, øY, øZ, øW ) = R(φ(øX), φ(øY ), φ(øZ), φ(øW ))
= g(φ(øX), φ(øZ))(P · F )(øY, øW ) + g(φ(øY ), φ(øW ))(P · F )(øX, øZ)−
−g(φ(øY ), φ(øZ))(P · F )(øX, øW )− g(φ(øX), φ(øW ))(P · F )(øY, øZ)
= g(φ(øX), φ(øZ))m(øY, øW ) + g(φ(øY ), φ(øW ))m(øX, øZ)−
−g(φ(øY ), φ(øZ))m(øX, øW )− g(φ(øX), φ(øW ))m(øY, øZ).
(3.30)
Since
g(φ(øX), φ(øY )) = g(φ(øX), øY − L−1ℓ(øY )øη) = g(φ(øX), øY )− L−1ℓ(øY )g(φ(øX), øη)
= ~(øX, øY )− L−1ℓ(øY )~(øX, øη) = ~(øX, øY ),
then, by using again the given assumption (mo = t φ =⇒ m = t~), Equation (3.30)
reduces to
(P · R)(øX, øY, øZ, øW ) = ~(øX, øZ)m(øY, øW ) + ~(øY, øW )m(øX, øZ)−
−~(øY, øZ)m(øX, øW )− ~(øX, øW )m(øY, øZ)
= 2t{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}.
Therefore, by taking Ro = 2t, we have
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}.
Consequently, the given manifold is of p-scalar curvature. �
Theorem 3.53. If an R3-like Finsler manifold (M,L) is of p-scalar curvature, then
it is of s-ps curvature.
Proof. Since the considered manifold is R3-like, then, by the same procedure as in
the proof of Proposition 3.52, we have
(P · R)(øX, øY, øZ, øW ) = ~(øX, øZ)m(øY, øW ) + ~(øY, øW )m(øX, øZ)−
−~(øY, øZ)m(øX, øW )− ~(øX, øW )m(øY, øZ).
(3.31)
On the other hand, since the considered manifold is of p-scalar curvature, then the
h-curvature tensor satisfies
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}. (3.32)
Now, from Equations (3.31) and (3.32), we obtain
UøX,øY {Ro~(øX, øZ)~(øY, øW )− ~(øX, øZ)m(øY, øW )− ~(øY, øW )m(øX, øZ)} = 0.
Using (3.26) and the non-degeneracy of the metric tensor g, the above equation
reduces to
UøX,øY {Ro~(øX, øZ)φ(øY )− ~(øX, øZ)mo(øY )−m(øX, øZ)φ(øY )} = 0. (3.33)
Since the π-tensor fields φ,m and mo are indicatory, then
Tr{øY 7−→ ~(øX, øY )φ(øZ)} = g(øX, φ(øZ)) = ~(øX, øZ),
Tr{øY 7−→ ~(øX, øY )mo(øZ)} = m(øX, øZ),
Tr{øY 7−→ m(øX, øY )φ(øZ)} = m(øX, øZ).
Consequently, if we take the trace of both sides of Equation (3.33), making use of
Lemma 3.43, we get
(n− 2)Ro~(øX, øZ)− (n− 3)m(øX, øZ)− (n− 1)t ~(øX, øZ) = 0,
where t := 1
Tr(mo). From which, using (3.26) and Lemma 3.2, we get
(n− 2)Roφ− (n− 3)mo − (n− 1)t φ = 0. (3.34)
Again, taking the trace of the above equation, we obtain
(n− 1)(n− 2)(Ro − 2t) = 0.
Substituting the above relation into (3.34), we get mo = t φ. Hence, by Proposition
3.52, the result follows. �
Theorem 3.54. If an R3-like Finsler manifold (M,L) is of scalar curvature, then it
is of s-ps curvature.
Proof. Since the given manifold is R3-like, then the π-tensor H is given by (cf.
Proposition 3.48):
H(øX) = L2{mo(øX) + c φ(øX)}. (3.35)
And since the considered manifold is of scalar curvature, then
H(øX) = L2κφ(øX). (3.36)
From Equations (3.35) and (3.36), we deduce thatmo(øX) = (κ−c)φ(øX) =: tφ(øX).
Hence, by Proposition 3.52, the result follows. �
Now, let us define the π-tensor field
Ψ(øX, øY, øZ, øW ) = R(øX, øY, øZ, øW )− 1
UøX,øY {g(øX, øZ)Ric
h(øY, øW )+
+g(øY, øW )Rich(øX, øZ)− rg(øX, øZ)g(øY, øW )},
(3.37)
where r = 1
Sch. From Definition 2.14 and (3.37), we immediately obtain
Theorem 3.55. An R3-like Finsler manifold is characterized by
Ψ(øX, øY, øZ, øW ) = 0.
The tensor field Ψ in the above theorem being of the same form as the Weyl
conformal tensor in Riemannian geometry, we draw the following
Theorem 3.56. An R3-like Riemannian manifold is conformally flat.
Remark 3.57. It should be noted that some important results of [8], [9], [11], [13],
[19], [20],...,etc. (obtained in local coordinates) are retrieved from the above mentioned
global results (when localized).
Appendix. Local formulae
For the sake of completeness, we present in this appendix a brief and concise
survey of the local expressions of some important geometric objects and the local
definitions of the special Finsler manifolds treated in the paper.
Let (U, (xi)) be a system of local coordinates on M and (π−1(U), (xi, yi)) the
associated system of local coordinates on TM . We use the following notations :
(∂i) := (
): the natural basis of TxM, x ∈M ,
(∂̇i) := (
): the natural basis of Vu(TM), u ∈ TM ,
(∂i, ∂̇i): the natural basis of Tu(TM),
(ø∂i): the natural basis of the fiber over u in π
−1(TM) (ø∂i is the lift of ∂i at u).
To a Finsler manifold (M,L), we associate the geometric objects :
gij :=
∂̇i∂̇jL
2 = ∂̇i∂̇jE: the Finsler metric tensor,
Cijk :=
∂̇k gij : the Cartan tensor,
~ij := gij − ℓiℓj (ℓi := ∂L/∂y
i): the angular metric tensor,
Gh: the components of the canonical spray,
Ghi := ∂̇iG
Ghij := ∂̇jG
i = ∂̇j ∂̇iG
(δi) := (∂i −G
i ∂̇h): the basis of Hu(TM) adapted to G
(δi, ∂̇i): the basis of Tu(TM) = Hu(TM)⊕ Vu(TM) adapted to G
We have :
γ(ø∂i) = ∂̇i,
ρ(∂i) = ø∂i, ρ(∂̇i) = 0, ρ(δi) = ø∂i,
β(ø∂i) = δi,
J(∂i) = ∂̇i, J(∂̇i) = 0, J(δi) = ∂̇i,
h := βoρ = dxi ⊗ ∂i −G
j ⊗ ∂̇i v := γoK = dy
i ⊗ ∂̇i +G
j ⊗ ∂̇i.
We define :
γhij :=
ghℓ(∂i gℓj + ∂j giℓ − ∂ℓ gij),
Chij :=
ghℓ(∂̇i gℓj + ∂̇j giℓ − ∂̇ℓ gij) =
ghℓ ∂̇i gjℓ = g
hℓCijℓ,
Γhij :=
ghℓ(δi gℓj + δj giℓ − δℓ gij) .
Then, we have :
• The canonical spray G: Gh = 1
γhij y
• The Barthel connection Γ: Ghi = ∂̇iG
h = Γhijy
j = Ghijy
• The Cartan connection CΓ: ( Γhij, G
i , C
The associated h-covariant (resp. v-covariant) derivative is denoted by p (resp. |),
where Ki
j|k := δkK
mk −K
jk and K
j |k := ∂̇kK
mk −K
• The Berwald connection BΓ: ( Ghij, G
i , 0).
The associated h-covariant (resp. v-covariant) derivative is denoted by
p(resp.
where Ki
:= δkK
mk −K
jk and K
:= ∂̇kK
We also have Ghij = Γ
ij + C
ij |k y
k = Γhij + C
ij |o, where C
ij |o = C
ij |k y
For the Cartan connection, we have :
(v)h-torsion : Rijk = δkG
j − δjG
k = Ujk{δkG
(v)hv-torsion : P ijk = G
jk − Γ
jk = C
jk|my
m = C i
jk|0,
(h)hv-torsion : C ijk = 1/2{g
ri∂̇rgjk},
h-curvature : Rihjk = Ujk{δkΓ
hj + Γ
mk} − C
hv-curvature : P ihjk = ∂̇kΓ
hj − C
+ C ihmP
v-curvature : Sihjk = C
mj − C
mk = Ujk{C
For the Berwald connection, we have :
(v)h-torsion : R∗ijk = δkG
j − δjG
k = Ujk{δkG
h-curvature : R∗ihjk = Ujk{δkG
hj +G
hv-curvature : P ∗ihjk = ∂̇kG
hj =: G
In the following, we give the local definitions of the special Finsler spaces treated
in the paper. For each special Finsler space (M,L), we set its name, its defining
property and a selected reference in which the local definition is located:
• Rimaniann manifold [22] : gij(x, y) ≡ gij(x) ⇐⇒ Cijk = 0 ⇐⇒ Ci := C
ik = 0
(Deicke’s theorem [4]).
• Minkowaskian manifold [22]: gij(x, y) ≡ gij(y) ⇐⇒ C
= 0 and Rhijk = 0.
• Berwald manifold [22]: Γhij(x, y) ≡ Γ
ij(x) (i.e. ∂̇kΓ
ij = 0) ⇐⇒ C
• Ch-recurrent manifold [13]: Chij|k = µkChij ,
where µj is a covariant vector field.
• P ∗-Finsler manifold [7]: Ch
ij|0 = λ(x, y)C
where λ(x, y) = PiC
; Pi := P
ik = C
ik|0 = Ci|0 and C
2 = CiC
i 6= 0.
• Cv-recurrent manifold [13]: C ijk|l = λl C
jk or Cijk|l = λl Cijk.
• C0-recurrent manifold [13]: C ijk
= λl C
jk or Cijk
= λl Cijk.
• Semi-C-reducible manifold (dimM ≥ 3) [18]:
Cijk =
(n+ 1)
(~ijCk + ~jkCi + ~kiCj) +
CiCjCk, C
2 6= 0,
where µ and τ are scalar functions satisfying µ+ τ = 1.
• C-reducible manifold (dimM ≥ 3) [15]: Cijk =
(~ijCk + ~jkCi + ~kiCj).
• C2-like manifold (dimM ≥ 2) [17]: Cijk =
CiCjCk, C
2 6= 0.
• quasi-C-reducible manifold (dimM ≥ 3) [23]: Cijk = AijCk + AjkCi + AkiCj ,
where Aij(x, y) is a symmetric tensor field satisfying Aijy
i = 0.
• S3-like manifold (dimM ≥ 4) [6]: Slijk =
(n−1)(n−2)
{~ik~lj − ~ij~lk},
where S is the vertical scalar curvature.
• S4-like manifold (dimM ≥ 5) [6]: Slijk = ~ljFik − ~lkFij + ~ikFlj − ~ijFlk,
where Fij :=
{Sij −
2(n−2)
S~ij}; Sij being the vertical Ricci tensor.
• Sv-recurrent manifold [20], [11]: Shijk|m = λmShijk,
where λj(x, y) is a covariant vector field.
• Second order Sv-recurrent manifold [20], [11]: Shijk|m|n = ΘmnShijk,
where Θij(x, y) is a covariant tensor field.
• Landsberg manifold [7]: P hkji y
k = 0 ⇐⇒ (∂̇iΓ
k = 0 ⇐⇒ Ch
yk = 0.
• General Landsberg manifold [10]: P rijry
i = 0 ⇐⇒ Cj|o = 0.
• P -symmetric manifold [19]: Phijk = Phikj.
• P2-like manifold (dimM ≥ 3) [14]: Phijk = αhCijk − αiChjk,
where αk(x, y) is a covariant vector field.
• P -reducible manifold (dimM ≥ 3) [19]: Pijk =
(~ij Pk + ~jk Pi + ~ki Pj),
where Pijk = ghiP
• h-isotropic manifold (dimM ≥ 3) [13]: Rhijk = ko{ghjgik − ghkgij},
for some scalar ko, where Rhijk = gilR
• Manifold of scalar curvature [21]: Rijkl y
iyk = kL2~jl,
for some function k : TM −→ R .
• Manifold of constant curvature [21]: the function k in the above definition is
constant.
• Manifold of perpendicular scalar (or of p-scalar ) curvature [8], [9]:
P ·Rhijk := ~
k Rlmnr = Ro{~ik~hj − ~ij~hk},
where Ro is a function called a perpendicular scalar curvature.
• Manifold of s-ps curvature [8], [9]: (M,L) is both of scalar curvature and of
p-scalar curvature.
• R3-like manifold (dimM ≥ 4) [8]: Rhijk = ghjFik − ghkFij + gikFhj − gijFhk,
where Fij :=
{Rij −
r gij}; Rij := R
ijh, r :=
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|
0704.0054 | The Hardy-Lorentz Spaces $H^{p,q}(R^n)$ | The Hardy-Lorentz Spaces Hp,q(Rn)
Wael Abu-Shammala and Alberto Torchinsky
Abstract
In this paper we consider the Hardy-Lorentz spaces Hp,q(Rn), with
0 < p ≤ 1, 0 < q ≤ ∞. We discuss the atomic decomposition of
the elements in these spaces, their interpolation properties, and the
behavior of singular integrals and other operators acting on them.
The real variable theory of the Hardy spaces represents a fruitful setting for
the study of maximal functions and singular integral operators. In fact, it
is because of the failure of these operators to preserve L1 that the Hardy
space H1 assumes its prominent role in harmonic analysis. Now, for many
of these operators, the role of L1 can just as well be played by H1,∞, or
Weak H1. However, although these operators are amenable to H1 − L1 and
H1,∞ − L1,∞ estimates, interpolation between H1 and H1,∞ has not been
available. Similar considerations apply to Hp and Weak Hp for 0 < p < 1.
The purpose of this paper is to provide an interpolation result for the
Hardy-Lorentz spaces Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞, including the case of
Weak Hp as and end point for real interpolation. The atomic decomposition
is the key ingredient in dealing with interpolation since in this context neither
truncations are available, nor reiteration applies.
The paper is organized as follows. The Lorentz spaces, including criteria
that assure membership in Lp,q, 0 < p < ∞, 0 < q ≤ ∞, are discussed in
Section 1. In Section 2 we show that distributions in Hp,q have an atomic
decomposition in terms ofHp atoms with coefficients in an appropriate mixed
norm space. An interesting application of this decomposition is to Hp,q−Lp,∞
estimates for Calderón-Zygmund singular integral operators, p < q ≤ ∞.
Also, by manipulating the different levels of the atomic decomposition, we
show that, for 0 < q1 < q < q2 ≤ ∞, H
p,q is an intermediate space between
Hp,q1 and Hp,q2. This result applies to Calderón-Zygmund singular integral
operators, including those with variable kernels, Marcinkiewicz integrals, and
other operators.
http://arxiv.org/abs/0704.0054v1
1 The Lorentz spaces
The Lorentz space Lp,q(Rn) = Lp,q, 0 < p <∞, 0 < q ≤ ∞, consists of those
measurable functions f with finite quasinorm ‖f‖p,q given by
‖f‖p,q =
[t1/pf ∗(t)]q
, 0 < q <∞ ,
‖f‖p,∞ = sup
[t1/pf ∗(t)] , q = ∞ .
The Lorentz quasinorm may also be given in terms of the distribution func-
tion m(f, λ) = |{x ∈ Rn : |f(x)| > λ}|, loosely speaking, the inverse of the
non-increasing rearrangement f ∗ of f . Indeed, we have
‖f‖p,q =
λq−1m(f, λ)q/p dλ
2km(f, 2k)1/p
when 0 < q <∞, and
‖f‖p,∞ = sup
2km(f, 2k)1/p , q = ∞ .
Note that, in particular, Lp,p = Lp, and Lp,∞ is weak Lp.
The following two results are useful in verifying that a function is in Lp,q.
Lemma 1.1. Let 0 < p <∞, and 0 < q ≤ ∞. Assume that the non-negative
sequence {µk} satisfies {2
kµk} ∈ ℓ
q. Further suppose that the non-negative
function ϕ verifies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0 + ηk0, where ψk0 is essentially
bounded and satisfies ‖ψk0‖∞ ≤ c 2
k0, and
2k0εpm(ηk0 , 2
k0) ≤ c
[2kεµk]
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
Proof. It clearly suffices to verify that ‖{2k |{ϕ > γ 2k}|1/p}‖ℓq <∞, where γ
is an arbitrary positive constant. Now, given k0, let ψk0 and ηk0 be as above,
and put γ = c+ 1, where c is the constant in the above inequalities; for this
choice of γ, {ϕ > γ 2k0} ⊆ {ηk0 > 2
When q = ∞, we have
2k0εm(ηk0 , 2
k0)1/p ≤ c
[2−k(1−ε) 2k µk]
≤ c 2−k0(1−ε) sup
[ 2k µk] .
Thus, 2k0 m(ηk0 , 2
k0)1/p ≤ supk≥k0[ 2
k µk] , and, consequently,
2k0 m(ϕ, γ 2k0)1/p ≤ c ‖{2kµk}‖ℓ∞ , all k0.
When 0 < q < ∞, let 1 − ε = 2δ and rewrite the right-hand side above
[2k(1−δ)µk]
When p < q, by Hölder’s inequality with exponent r = q/p and its conjugate
r′, this expression is dominated by
2k δpr
)1/r′(
2k(1−δ)µk
]rp )1/r
≤ c 2−k0 δp
2k(1−δ)µk
]q )p/q
and, when 0 < q ≤ p, r < 1, and we get a similar bound by simply observing
that it does not exceed
2−k0δp
[2k(1−δ)µk]
≤ 2−k0δp
2k(1−δ)µk
]q )p/q
Whence, continuing with the estimate, we have
2k0εpm(ηk0 , 2
k0) ≤ c 2−k0δp
2k(1−δ)µk
]q )p/q
which yields, since 1− ε = 2 δ,
2k0 m(ϕ, γ 2k0)1/p ≤ c 2k0 δ
2k(1−δ)µk
]q )1/q
Thus, raising to the q and summing, we get
2k0 m(ϕ, γ 2k0)1/p
2k0 δ q
2k(1−δ)µk
which, upon changing the order of summation in the right-hand side of the
above inequality, is bounded by
2k(1−δ)µk
k0=−∞
2k0 δ q
The reader will have no difficulty in verifying that, for Lemma 1.1 to hold,
it suffices that ψx0 satisfies
m(ψx0 , 2
k0)1/p ≤ c µk0 , all k0 .
This holds, for instance, when ‖ψx0‖
r ≤ c 2
, 0 < r < ∞. In fact, the
assumptions of Lemma 1.1 correspond to the limiting case of this inequality
as r → ∞.
Another useful condition is given by our next result, the proof is left to
the reader.
Lemma 1.2. Let 0 < p < ∞, and let the non-negative sequence {µk} be
such that {2kµk} ∈ ℓ
q, 0 < q ≤ ∞. Further, suppose that the non-negative
function ϕ satisfies the following property: there exists 0 < ε < 1 such that,
given an arbitrary integer k0, we have ϕ ≤ ψk0+ηk0, where ψk0 and ηk0 satisfy
2k0pm(ψk0 , 2
k0)ε ≤ c
2kµεk
, 0 < ε < min(1, q/p) ,
2k0ε|{ηk0 > 2
k0}| ≤ c
2kεµk
Then, ϕ ∈ Lp,q, and ‖ϕ‖p,q ≤ c ‖{2
kµk}‖ℓq.
We will also require some basic concepts from the theory of real interpo-
lation. Let A0, A1, be a compatible couple of quasinormed Banach spaces,
i.e., both A0 and A1 are continuously embedded in a larger topological vector
space. The Peetre K functional of f ∈ A0 + A1 at t > 0 is defined by
K(t, f ;A0, A1) = inf
f=f0+f1
‖f0‖0 + t ‖f1‖1 ,
where f = f0 + f1, f0 ∈ A0 and f1 ∈ A1.
In the particular case of the Lq spaces, the K functional can be computed
by Holmstedt’s formula, see [12]. Specifically, for 0 < q0 < q1 ≤ ∞, let α be
given by 1/α = 1/q0 − 1/q1. Then,
K(t, f ;Lq0, Lq1) ∼
f ∗(s)q0ds
)1/q0
f ∗(s)q1ds
)1/q1
The intermediate space (A0, A1)η, q, 0 < η < 1, 0 < q < ∞, consists of
those f ’s in A0 + A1 with
‖f‖(A0,A1)η, q =
t−ηK(t, f ;A0, A1)
]q dt
‖f‖(A0,A1)η,∞ = sup
t−ηK(t, f ;A0, A1)
<∞ , q = ∞ .
Finally, for the Lq and Lp,q spaces, we have the following result. Let
0 < q1 < q < q2 ≤ ∞, and suppose that 1/q = (1 − η)/q1 + η/q2. Then,
Lq = (Lq1 , Lq2)η,q, and, L
1,q = (L1,q1, L1,q2)η,q, see [4].
2 The Hardy-Lorentz spaces Hp,q
In this paper we adopt the atomic characterization of the Hardy spaces Hp,
0 < p ≤ 1. Recall that a compactly supported function a with [n(1/p− 1)]
vanishing moments is an Hp atom with defining interval I (of course, I is
a cube in Rn), if supp(a) ⊆ I, and |I|1/p |a(x)| ≤ 1. The Hardy space
Hp(Rn) = Hp consists of those distributions f that can be written as f =
λjaj , where the aj ’s are H
p atoms,
p < ∞, and the convergence is
in the sense of distributions as well as in Hp. Furthermore,
‖f‖Hp ∼ inf
where the infimum is taken over all possible atomic decompositions of f .
This last expression has traditionally been called the atomic Hp norm of f .
C. Fefferman, Rivière and Sagher identified the intermediate spaces be-
tween the Hardy space Hp0, 0 < p0 < 1, and L
∞, as
(Hp0, L∞)η,q = H
p,q, 1/p = (1− η)/p0 , 0 < q ≤ ∞ ,
where Hp,q consists of those distributions f whose radial maximal function
Mf(x) = supt>0 |(f ∗ ϕt)(x)| belongs to L
p,q. Here ϕ is a compactly sup-
ported, smooth function with nonvanishing integral, see [10]. R. Fefferman
and Soria studied in detail the space H1,∞, which they called Weak H1, see
[11].
Just as in the case of Hp, Hp,q can be characterized in a number of
different ways, including in terms of non-tangential maximal functions and
Lusin functions. In what follows we will calculate the quasinorm of f in Hp,q
by the means of the expression
2km(Mf, 2k)1/p
, 0 < p ≤ 1, 0 < q ≤ ∞ ,
where Mf is an appropriate maximal function of f .
Passing to the atomic decomposition of Hp,q, the proof is divided in two
parts. First, we construct an essentially optimal atomic decomposition; Par-
ilov has obtained independently this result for H1,q when 1 ≤ q, see [14].
Also, R. Fefferman and Soria gave the atomic decomposition of Weak H1,
see [11], and Alvarez the atomic decomposition of Weak Hp, 0 < p < 1, see
Theorem 2.1. Let f ∈ Hp,q, 0 < p ≤ 1, 0 < q ≤ ∞. Then f has an atomic
decomposition f =
j,k λj,kaj,k, where the aj,k’s are H
p atoms with defining
intervals Ij,k that have bounded overlap uniformly for each k, the sequence
{λj,k} satisfies
j |λj,k|
< ∞, and the convergence is in the
sense of distributions. Furthermore,
j |λj,k|
∼ ‖f‖Hp,q .
Proof. The idea of constructing an atomic decomposition using Calderón’s
reproducing formula is well understood, so we will only sketch it here, for
further details, see [5] and [18]. Let Nf(x) = sup{|(f ∗ ψt)(y)| : |x− y| < t}
denote the non-tangential maximal function of f with respect to a suitable
smooth function ψ with nonvanishing integral. One considers the open sets
Ok = {Nf > 2
k}, all integers k, and builds the atoms with defining interval
associated to the intervals, actually cubes, of the Whitney decomposition
of Ok, and hence satisfying all the required properties. More precisely, one
constructs a sequence of bounded functions fk with norm not exceeding c 2
for each k, and such that f −
|k|≤n fk → 0 as n→ ∞ in the sense of distri-
butions. These functions have the further property that fk(x) =
j αj,k(x) ,
where |αj,k(x)| ≤ c 2
k, c is a constant, each αj,k has vanishing moments up
to order [n(1/p − 1)] and is supported in Ij,k - roughly one of the Whitney
cubes -, where the Ij,k’s have bounded overlaps for each k, uniformly in k. It
only remains now to scale αj,k,
αj,k(x) = λj,k aj,k(x) ,
and balance the contribution of each term to the sum. Let λj,k = 2
k|Ij,k|
Then, aj,k(x) is essentially an H
p atom with defining interval Ij,k, and one
j |λj,k|
∼ 2k |Ok|
1/p. Thus,
|λj,k|
)1/p∥
2k |Ok|
∼ ‖f‖Hp,q , 0 < q ≤ ∞ . �
As an application of this atomic decomposition, the reader should have
no difficulty in showing directly the C. Fefferman, Rivière, Sagher character-
ization of Hp,q, see [10].
Another interesting application of this decomposition is to Hp,q − Lp,∞
estimates for Calderón-Zygmund singular integral operators T , p < q ≤ ∞.
This approach combines the concept of p-quasi local operator of Weisz, see
[17], with the idea of variable dilations of R. Fefferman and Soria, see [11].
Intuitively, since Hörmander’s condition implies that T maps H1 into L1, say,
for T to be defined in H1,s, 1 < s ≤ ∞, some strengthening of this condition
is required. This is accomplished by the variable dilations. Moreover, since
we will include p < 1 in our discussion, as p gets smaller, more regularity of
the kernel of T will be required. This justifies the following definition.
Given 0 < p ≤ 1, let N = [n(1/p − 1)], and, associated to the kernel
k(x, y) of a Calderón-Zygmund singular integral operator T , consider the
modulus of continuity ωp given by
ωp(δ) = sup
Rn\(2/δ)I
| k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)| dy
where 0 < δ ≤ 1, and the sup is taken over the collection of arbitrary intervals
I of Rn centered at yI . Here, for a multi-index α = (α1, . . . , αn),
kα(x, yI) =
Dαk(x, y)
ωp(δ) controls the behavior of T on atoms. More precisely, if a is an H
p atom
with defining interval I, and 0 < δ < 1, observe that
T (a)(x) =
[k(x, y)−
|α|≤N
(y − yI)
αkα(x, yI)] a(y) dy ,
and, consequently,
Rn\(2/δ)I
|T (a)(x)|p dx ≤ ωp(δ) .
We are now ready to prove the Hp,q − Lp,∞ estimate for a Calderón-
Zygmund singular integral operator T with kernel k(x, y).
Theorem 2.2. Let 0 < p ≤ 1, and p < q ≤ ∞. Assume that a Calderón-
Zygmund singular integral operator T is of weak-type (r, r) for some 1 < r <
∞, and that the modulus of continuity ωp of the kernel k satisfies a Dini
condition of order q/(q − p), namely,
Ap,q =
ωp(δ)
q/(q−p)dδ
](q−p)/q
Then T maps Hp,q continuously into Lp,∞, and ‖Tf‖p,∞ ≤ cA
p,q ‖f‖Hp,q .
Proof. We need to show that
2k0pm(Tf, 2k0) ≤ c ‖f‖
Hp,q , all k0 .
Let f =
j λj,kaj,k , be the atomic decomposition of f given in Theorem
2.1, and set f1 =
j λj,kaj,k, and f2 = f − f1. Further, let µk =
j |λj,k|
, and recall that ‖{µk}‖ℓq ∼ ‖f‖Hp,q .
Since ‖f1‖
r ≤ c 2
k0(r−p)‖f‖
Hp,∞, we have
2pk0m(Tf1, 2
k0) ≤ c ‖f‖
Hp,∞ .
Next, put I∗j,k = 2
1/n(3/2)p(k−k0)/nIj,k, and let
I∗j,k .
Since |I∗j,k| = 2(3/2)
p(k−k0)|Ij,k| ∼ 2
−k0p(3/4)p(k−k0)|λj,k|
p, we get
|Ω| ≤
|I∗j,k| ≤ c 2
(3/4)p(k−k0)
|λj,k|
≤ c 2−k0p
≤ c 2−k0p‖f‖
Hp,∞ .
Also, since 0 < p ≤ 1, it readily follows that
|T (f2)(x)|
|λj,k|
p|T (aj,k)(x)|
and, by Tonelli and the estimate for T (a), we have
|T (f2)(x)|
p dx ≤
|λj,k|
Rn\I∗
|T (aj,k)(x)|
)p(k−k0)/n
)pk/n)q/(q−p))(q−p)/q
ωp(δ)
q/(q−p)dδ
](q−p)/q
Hp,q .
This bound gives at once
2pk0 |{x /∈ Ω : |T (f2)(x)| > 2
k0}| ≤ cAp,q ‖f‖
Hp,q ,
which implies that
2pk0m(Tf2, 2
k0−1) ≤ 2pk0
|Ω|+ |{x /∈ Ω : |T (f2)(x)| > 2
k0−1}|
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q .
Finally,
2k0pm(Tf, 2k0) ≤ 2k0pm(Tf1, 2
k0−1) + 2k0pm(Tf2, 2
k0−1)
≤ c ‖f‖
Hp,∞ + cAp,q ‖f‖
Hp,q ,
and, since ‖f‖Hp,∞ ≤ c ‖f‖Hp,q for all q, we have finished. �
We pass now to the converse of Theorem 2.1. It is apparent that a
condition that relates the coefficients λj with the corresponding atoms aj
involved in an atomic decomposition of the form
j λjaj(x) is relevant here.
More precisely, if Ij denotes the supporting interval of aj , let
Ik = {j : 2
k ≤ |λj|/|Ij|
1/p < 2k+1} ,
and, for λ = {λj}, put
‖λ‖[p,q] =
]q/p)1/q
We then have,
Theorem 2.3. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is in
the sense of distributions. Further, assume that the family {Ij} consisting of
the supports of the aj’s has bounded overlap at each level Ik uniformly in k,
and ‖λ‖[p,q] <∞. Then, f ∈ H
p,q, and ‖f‖Hp,q ≤ c ‖λ‖[p,q].
Proof. Let Mf(x) = supt>0 |(f ∗ ψt)(x)| denote the radial maximal function
of f with respect to a suitable smooth function ψ with support contained in
{|x| ≤ 1} and nonvanishing integral. We will verify that Mf satisfies the
conditions of Lemma 1.1 and is thus in Lp,q.
Fix an integer k0 and let
g(x) =
λjaj(x) .
Since ‖Mg‖∞ ≤ ‖g‖∞ it suffices to estimate |g(x)|. Let C be the bounded
overlap constant for the family of the supports of the aj’s. Then, for j ∈ Ik,
|λj| |aj(x)| =
|Ij|1/p
|λj | |Ij|
1/p |aj(x)| ≤ 2
kχIj (x) ,
and, consequently,
|g(x)| ≤
χIj(x) ≤ C 2
Next, let
h(x) =
λjaj(x) .
Since aj has N = [n(1/p − 1)] vanishing moments, it is not hard to see
that, if Ij is the defining interval of aj and Ij is centered at xj , and γ =
(n+N+1)/n > 1/p, then, with c independent of j, ϕj(x) =Maj(x) satisfies
ϕj(x) ≤ c
|Ij |
γ−1/p
(|Ij|+ |x− xj |n)γ
Thus, if 1/γ < εp < 1,
Mh(x)εp ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
which, upon integration, yields
Mh(x)εp dx ≤ c
j∈Ik,k≥k0
(|λj| |Ij|
γ−1/p)εp
(|Ij |+ |x− xj |n)γεp
The integrals in the right-hand side above are of order |Ij|
1−γεp and, conse-
quently, by Chebychev’s inequality,
2k0εp|{Mh > 2k0}| ≤ c
j∈Ik,k≥k0
εp |Ij|
1−ε ≤ c
|Ij| .
Thus, Lemma 1.1 applies with ϕ = Mf , ψk0 = Mg, ηk0 = Mh, and µk =
, and we get
2km(Mf, 2k)1/p
)1/p}∥
which, since
|Ij| ∼
, j ∈ Ik ,
is bounded by c ‖λ‖[p,q], 0 < q ≤ ∞. �
The next result is of interest because it applies to arbitrary decomposi-
tions in Hp,q. The proof relies on Lemma 1.2, and is left to the reader.
Theorem 2.4. Let 0 < p ≤ 1, 0 < q ≤ ∞, and let f be a distribution given
by f =
j λj aj(x) , where the aj’s are H
p atoms, and the convergence is
in the sense of distributions. Further, assume that ‖λ‖[η,q] < ∞ for some
0 < η < min(p, q). Then, f ∈ Hp,q, and ‖f‖Hp,q ≤ c ‖λ‖[η,q].
2.1 Interpolation between Hardy-Lorentz spaces
We are now ready to identify the intermediate spaces of a couple of Hardy-
Lorentz spaces with the same first index p ≤ 1.
Theorem 2.5. Let 0 < p ≤ 1. Given 0 < q1 < q < q2 ≤ ∞, define 0 < η < 1
by the relation 1/q = (1− η)/q1 + η/q2. Then, with equivalent quasinorms,
Hp,q = (Hp,q1, Hp,q2)η,q .
Proof. Since the non-tangential maximal function Nf of a distribution f in
Hp,q1 is in Lp,q1, and that of f in Hp,q2 is in Lp,q2, we have
K(t, Nf ;Lp,q1, Lp,q2) ≤ cK(t, f ;Hp,q1, Hp,q2) .
Thus,
‖Nf‖p,q ∼ ‖Nf‖(Lp,q1 ,Lp,q2)η,q ≤ c ‖f‖(Hp,q1 ,Hp,q2 )η,q ,
and (Hp,q1, Hp,q2)η,q →֒ H
To show the other embedding, with the notation in the proof of Theorem
2.1, write f =
j λj,kaj,k , and recall that for every integer k, the level
set Ik = {j : |λj,k|/|Ij,k|
1/p ∼ 2k} contains exclusively the sequence {λj,k}.
Let µ
|λj,k|
p. By construction,
k ∼ ‖f‖
Hp,q . Now, rearrange
{µk} into {µ
l }, and, for each l ≥ 1, let kl be such that µkl = µ
l . For
l0 ≥ 1, let Kl0 = {k1, . . . , kl0}, and put f1,l0 =
k∈Kl0
j λj,kaj,k and f2,l0 =
f − f1,l0 . Then, by Theorem 2.2, f1,l0 ∈ H
p,q1, f2,l0 ∈ H
p,q2, and, with the
usual interpretation for q2 = ∞,
‖f1,l0‖Hp,q1 ≤ c
)1/q1
, ‖f2,l0‖Hp,q2 ≤ c
)1/q2
So, for t > 0 and every positive integer l0, we have
K(t, f ;Hp,q1, Hp,q2) ≤ c
)1/q1
)1/q2
Now, by Homstedt’s formula, there is a choice of l0 such that the right-hand
side above ∼ K(t, {µk}; ℓ
q1, ℓq2), and, consequently,
K(t, f ;Hp,q1, Hp,q2) ≤ cK(t, {µk}; ℓ
q1, ℓq2) .
Thus,
‖f‖(Hp,q1 ,Hp,q2 )η,q ≤ c ‖{µk}‖(ℓq1 ,ℓq2)η,q
≤ c ‖{µk}‖ℓq ≤ c ‖f‖Hp,q ,
and Hp,q →֒ (Hp,q1, Hp,q2)η,q. �
The reader will have no difficulty in verifying that Theorem 2.5 gives that
if T is a continuous, sublinear map fromH1 into L1, and fromH1,∞ into L1,∞,
then ‖Tf‖1,q ≤ c ‖f‖H1,q for 1 < q < ∞. This observation has numerous
applications. For instance, consider the Calderón-Zygmund singular integral
operators with variable kernel defined by
TΩ(f)(x) = p.v.
Ω(x, x−y)
|x− y|n
f(y) dy .
Under appropriate growth and smoothness assumptions on Ω, TΩ maps H
continuously into L1, see [6], and H1,∞ continuously into L1,∞, see [8]. Thus,
if Ω satisfies the assumptions of both of these results, TΩ maps H
1,q contin-
uously into L1,q for 1 < q < ∞. A similar result follows by invoking the
characterization of H1,q given by C. Fefferman, Rivière and Sagher. How-
ever, in this case the Hp−Lp estimate requires additional smoothness of Ω, as
shown, for instance, in [6]. Similar considerations apply to the Marcinkiewicz
integral, see [9], and [7].
Finally, when p < 1, our results cover, for instance, the δ-CZ operators
satisfying T ∗(1) = 0 discussed by Alvarez and Milman, see [3]. These oper-
ators, as well as a more general related class introduced in [15], preserve Hp
and Hp,∞ for n/(n + δ) < p ≤ 1, and, consequently, by Theorem 2.5, they
also preserve Hp,q for p in that same range, and q > p.
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DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY,
BLOOMINGTON, IN 47405
E-mail: [email protected], [email protected]
The Lorentz spaces
The Hardy-Lorentz spaces Hp,q
Interpolation between Hardy-Lorentz spaces
|
0704.0055 | Potassium intercalation in graphite: A van der Waals density-functional
study | Potassium intercalation in graphite: A van der Waals density-functional study
Eleni Ziambaras,1 Jesper Kleis,1 Elsebeth Schröder,1 and Per Hyldgaard1, 2, ∗
Department of Applied Physics, Chalmers University of Technology, SE–412 96 Göteborg, Sweden
Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE–412 96 Göteborg, Sweden
(Dated: April 1, 2007)
Potassium intercalation in graphite is investigated by first-principles theory. The bonding in
the potassium-graphite compound is reasonably well accounted for by traditional semilocal density
functional theory (DFT) calculations. However, to investigate the intercalate formation energy
from pure potassium atoms and graphite requires use of a description of the graphite interlayer
binding and thus a consistent account of the nonlocal dispersive interactions. This is included
seamlessly with ordinary DFT by a van der Waals density functional (vdW-DF) approach [Phys.
Rev. Lett. 92, 246401 (2004)]. The use of the vdW-DF is found to stabilize the graphite crystal,
with crystal parameters in fair agreement with experiments. For graphite and potassium-intercalated
graphite structural parameters such as binding separation, layer binding energy, formation energy,
and bulk modulus are reported. Also the adsorption and sub-surface potassium absorption energies
are reported. The vdW-DF description, compared with the traditional semilocal approach, is found
to weakly soften the elastic response.
I. INTRODUCTION
Graphite with its layered structure is easily interca-
lated by alkali metals (AM) already at room tempera-
ture. The intercalated compound has two-dimensional
layers of AM between graphite layers,1,2,3,4,5 giving rise
to interesting properties, such as superconductivity.6,7
The formation of an AM-graphite intercalate proceeds
with adsorption of AM atoms on graphite and absorption
of AM atoms below the top graphite layer, after which
further exposure to AM atoms leads the AM intercalate
compound.
Recent experiments8,9 on the structure and elec-
tronic properties of AM/graphite systems use samples
of graphite that are prepared by heating SiC crystals
to temperatures around ∼ 1400◦ C.10 This heat-induced
graphitization is of great value for spectroscopic studies
of graphitic systems, since the resulting graphite overlay-
ers are of excellent quality.11 The nature of the bonding
between the SiC surfaces and graphite has been explored
experimentally with photoemission spectroscopy12 and
theoretically13 with a van der Waals density functional
(vdW-DF) theory approach that accounts for the van der
Waals (vdW) forces.14,15,16,17
Here we investigate with density functional theory
(DFT) the effects on the graphite structure and the
energetics and the elastic response when potassium
is intercalated. The final intercalate compound is
C8K. The AM intercalate system is interesting in it-
self and has been the focus of numerous experimen-
tal investigations.18,19,20,21,22 Graphitic systems are also
ideal test materials in ongoing theory development that
aims at improving the description of the nonlocal inter-
layer bonds in sparse systems.14,23,24 Standard DFT ap-
proaches are based on local (local density approximation,
LDA) and semilocal approximations (generalized gradi-
ent approximation, GGA)25,26,27,28 for the electron ex-
change and correlation. Such regular DFT tools do not
treat correctly the weak vdW binding, e.g., the cohe-
sion between (adjacent) graphite layers. The failure of
traditional DFT for graphite makes it impossible to ob-
tain a meaningful comparison of the energetics in on-
surface AM adsorption and subsurface AM absorption.
Conversely, investigations of graphitic systems like C8K
permit us to test the accuracy of our vdW-DF develop-
ment work.
We explore the nature of the bonding of graphite,
the process leading to intercalation via adsorption and
absorption of potassium, and the nature of potassium-
intercalated graphite C8K using a recently developed
vdW-DF density functional.16 This choice of functional is
essential for a comparison of graphite and C8K properties
because of the inability of traditional GGA-based DFT
to describe graphite. We calculate the structure and elas-
tic response (bulk modulus B0) of pristine graphite and
potassium intercalated graphite and we present results
for the formation energies of the C8K system.
The intercalation of potassium in graphite is preceded
by the adsorption of potassium on top of a graphite
surface and potassium absorption underneath the top
graphite layer of the surface. In this work we study
how potassium bonds to graphite in these two parts of
the process towards intercalation. Our vdW-DF inves-
tigations of the binding of potassium in or on graphite
supplements corresponding vdW-DF studies of the bind-
ing of polycyclic aromatic hydrocarbon dimers, of the
polyethylene crystal, of benzene dimers, and of poly-
cyclic aromatic hydrocarbon and phenol molecules on
graphite.29,30,31,32,33,34
The outline of the paper is as follows. Section II con-
tains a short description of the materials of interest here:
graphite, C8K, and graphite with an adsorbed or ab-
sorbed K atom layer. The vdW-DF scheme is described
in Sec. III. Section IV presents our results, Sec. V the
discussion, and conclusions are drawn in Sec. VI.
http://arxiv.org/abs/0704.0055v1
FIG. 1: (Color online) Simple hexagonal graphite (AA stack-
ing) and natural hexagonal graphite (AB stacking). The two
structures differ by that each second carbon layer in AB-
stacked graphite is shifted, whereas in AA-stacked graphite
all planes are directly above each other. The experimentally
obtained in-plane lattice constant and sheet separation of nat-
ural graphite is (Ref. 40) a = 2.459 Å and dC-C = 3.336 Å,
respectively.
II. MATERIAL STRUCTURE
Graphite is a semimetallic solid with strong intra-plane
bonds and weakly coupled layers. The presence of these
two types of bonding results in a material with different
properties along the various crystallographic directions.35
For example, the thermal and electrical conductivity
along the carbon sheets is two orders of magnitude higher
than that perpendicular to the sheets. This specific prop-
erty allows heat to move directionally, which makes it
possible to control the heat transfer. The relatively weak
vdW forces between the sheets contribute to another in-
dustrially important property: graphite is an ideal lubri-
cant. In addition, the anisotropic properties of graphite
make the material suitable as a substrate in electronic
studies of ultrathin metal films.36,37,38,39
The natural structure of graphite is an AB stacking,
with the graphite layers shifted relative to each other,
as illustrated in Fig. 1. The figure also shows hexagonal
graphite, consisting of AA-stacked graphite layers. The
in-plane lattice constant a and the layer separation dC-C
is also illustrated. In natural graphite the primitive unit
cell is hexagonal, includes four carbon atoms in two lay-
ers, and has unit cell side lengths a and height c = 2dC-C.
The physical properties of graphite have been studied
in a variety of experimental40,41,42 and theoretical43,44
work. Some of the DFT work has been performed in
LDA, which does not provide a physically meaningful
account of binding in layered systems.15,45 At the same
time, using GGA is not an option because it does not
bind the graphite layers. For a good description of the
FIG. 2: (Color online) Crystalline structure of C8K show-
ing the AA-stacking of the carbon layers (small balls) and
the αβγδ-stacking of the potassium layers (large balls) per-
pendicular to the graphene sheets. The potassium layers are
arranged in a p(2× 2) structure, with the K atoms occupying
the sites over the hollows of every fourth carbon hexagon.
graphite structure and nature the vdW interactions must
be included.45
Alkali metals (AM), except Na, easily penetrate the
gallery of the graphite forming alkali metal graphite in-
tercalation compounds. These intercalation compounds
are formed through electron exchange between the inter-
calated layer and the host carbon layers, resulting in a
different nature of the interlayer bonding type than that
of pristine graphite. The intercalate also affects the con-
ductive properties of graphite, which becomes supercon-
ductive in the direction parallel to the planes at critical
temperatures below 1 K.6,7
The structure of AM graphite intercalation compounds
is characterized by its stage n, where n is the number of
graphite sheets located between the AM layers. In this
work we consider only stage-1 intercalated graphite C8K,
in which the layers of graphite and potassium alternate
throughout the crystal. The primitive unit cell of C8K
is orthorhombic and contains sixteen C atoms and two
K atoms. In the C8K crystal the K atoms are ordered
in a p(2× 2) registry with K-K separation 2a, where a is
the in-plane lattice constant of graphite. This separation
of the potassium atoms is about 8% larger than that in
the natural K bcc crystal (based on experimental values).
The carbon sheet stacking in C8K is of AA type, with the
K atoms occupying the sites over the hollows of every
fourth carbon hexagon, each position denoted by α, β,
γ, or δ, and the stacking of the K atoms perpendicular
to the planes being described by the αβγδ-sequence as
illustrated in Fig. 2.
III. COMPUTATIONAL METHODS
The first-principle total-energy and electronic struc-
ture calculations are performed within the framework
of DFT. The semilocal Perdew-Burke-Ernzerhof (PBE)
flavor26 of GGA is chosen for the exchange-correlation
functional for the traditional self-consistent calculations
underlying the vdW-DF calculations. For all GGA cal-
culations we use the open source DFT code Dacapo,46
which employs Vanderbilt ultrasoft pseudopotentials,47
periodic boundary conditions, and a plane-wave basis set.
An energy cut-off of 500 eV is used for the expansion of
the wave functions and the Brillouin zone (BZ) of the
unit cells is sampled according to the Monkhorst-Pack
scheme.48 The self-consistently determined GGA valence
electron density n(r) as well as components of the energy
from these calculations are passed on to the subsequent
vdW-DF calculation of the total energy.
For the adsorption and absorption studies a graphite
surface slab consisting of 4 layers is used, with a surface
unit cell of side lengths twice those in the graphite bulk
unit cell (i.e., side lengths 2a). The surface calculations
are performed with a 4×4×1 k-point sampling of the BZ.
The (pure) graphite bulk GGA calculations are per-
formed with a 8×8×4 k-point sampling of the BZ,
whereas for the C8K bulk structure, in a unit cell at least
double the size in any direction, 4×4×2 k-points are used,
consistent also with the choice of k-point sampling of the
surface slabs.
We choose to describe C8K by using a hexagonal unit
cell with four formula units, lateral side lengths approxi-
mately twice those of graphite and with four graphite and
four K-layers in the direction perpendicular to the layers.
C8K can also be described by the previously mentioned
primitive orthorhombic unit cell containing two formula
units of atoms but we retain the orthorhombic cell for
ease of description and for simple implementation of nu-
merically robust vdW-DF calculations.
In all our studies, except test cases, the Fast Fourier
Transform (FFT) grids are chosen such that the separa-
tion of neighboring points is maximum ∼0.13 Å in any
direction in any calculation.
A. vdW density function calculations
In graphite, the carbon layers bind by vdW interac-
tions only. In the intercalated compound a major part of
the attraction is ionic, but also here the vdW interactions
cannot be ignored. In order to include the vdW interac-
tions systematically in all of our calculations we use the
vdW-DF of Ref. 16. There, the correlation energy func-
tional is divided into a local and a nonlocal part,
Ec ≈ E
c + E
c , (1)
where the local part is approximated in the LDA and the
nonlocal part Enlc is consistently constructed to vanish
for a homogeneous system. The nonlocal correlation Enlc
is calculated from the GGA-based n(r) and its gradients
by using information about the many-body response of
the weakly inhomogeneous electron gas:
Enlc =
dr′n(r)φ(r, r′)n(r′). (2)
The nonlocal kernel φ(r, r′) can be tabulated in terms
of the separation |r − r′| between the two fragments at
positions r and r′ through the parameters D = (q0 +
q′0)|r − r
′|/2 and δ = (q0 − q
0)/(q0 + q
0). Here q0 is a
local parameter that depends on the electron density and
its gradient at position r. The analytic expression for the
kernel φ in terms of D and δ can be found in Ref. 16.
For periodic systems, such as bulk graphite, C8K,
and the graphite surface (with adsorbed or absorbed K-
atoms), the nonlocal correlation per unit cell is simply
evaluated from the interaction of the points in the unit
cell V0 with points everywhere in space (V ) in the three
(for bulk graphite and C8K) or two (for the graphite
surface) dimensions of periodicity. Thus, the V -integral
in Eq. (2) in principle requires a representation of the
electron density infinitely repeated in space. In prac-
tice, the nonlocal correlation rapidly converges31 and it
suffices with repetitions of the unit cell a few times in
each spatial direction. For graphite bulk the V -integral
is converged when we use a V that extends 9 (7) times
the original unit cell in directions parallel (perpendicular)
to the sheets. For the potassium investigation a signif-
icantly larger original unit cell is adopted (see Fig. 2);
here a fully converged V corresponds to a cell extending
five (three) times the original cell in the direction parallel
(perpendicular) to the sheets for C8K bulk. To describe
the nonlocal correlations (2) for the graphite surface a
sufficient V extends five times the original unit cell along
the carbon sheets.
For the exchange energy Ex we follow the choice of
Ref. 16 of using revPBE27 exchange. Among the func-
tionals that we have easy access to, the revPBE has
proved to be the best candidate for minimizing the ten-
dency of artificial exchange binding in graphite.15
Using the scheme described above to evaluate Enlc , the
total energy finally reads:
EvdW−DF = EGGA − EGGAc + E
c + E
c , (3)
where EGGA is the GGA total energy with the revPBE
choice for the exchange description and EGGAc (E
the GGA (LDA) correlation energy. As our GGA calcu-
lations in this specific application of vdW-DF are carried
out in PBE, not revPBE, we further need to explicitly
replace the PBE exchange in EGGA by that of revPBE
for the same electron charge density distribution.
B. Convergence of the local and nonlocal energy
variation
DFT calculations provide physically meaningful results
for energy differences between total energies (3). To un-
derstand materials and processes we must compare total
energy differences between a system with all constituents
at relatively close distance and a system of two or more
fragments at “infinite” separation (the reference system).
Since the total energy (3) consists both of a long-range
term and shorter-ranged GGA and LDA terms it is nat-
ural to choose different ways to represent the separated
fragments for these different long- or short-range energy
terms.
For the shorter-range energy parts (LDA and GGA
terms) the reference system is a full system with vacuum
between the fragments. For LDA and GGA calculations
it normally suffices to make sure that the charge den-
sity tails of the fragments do not overlap, but here we
find that the surface dipoles cause a slower convergence
with layer separation. We use a system with the layer
separation between the potassium layer and the nearest
graphite layer(s) dC-K = 12 Å (8 Å) as reference for the
adsorption (absorption) study.
The evaluation of the nonlocal correlationsEnlc requires
additional care. This is due to technical reasons per-
taining to numerically stability in basing the Enlc eval-
uation on the FFT grid used to converge the underly-
ing traditional-DFT calculations. The evaluation of the
nonlocal correlation energy, Eq. (2), involves a weighted
double integral of a kernel with a significant short-range
variation16. The shape of the kernel makes the Enlc eval-
uation sensitive to the particulars of FFT-type griding,49
for example, to the relative position of FFT grid points
relative to the nuclei position (for a finite grid-point spac-
ing).
However, robust evaluation of binding- or cohesive-
energy contributions by nonlocal correlations can gen-
erally be secured by a further splitting of energy differ-
ences into steps that minimize the above-mentioned grid
sensitivity. The problem of FFT sensitivity of the Enlc
evaluation is accentuated because the binding in the Enlc
channel arises as a smaller energy difference between siz-
ableEnlc contributions of the system and of the fragments.
Conversely, convergence in vdW-DF calculations of bind-
ing and cohesive energies can be obtained even at a mod-
erate FFT grid accuracy (0.13 Å used here) by devising a
calculational scheme that always maintains identical po-
sition of the nuclei relative to grid points in the combined
systems as well as in the fragment reference system.
Thus we obtain a numerically robust evaluation of the
Enlc energy differences by choosing steps for which we
can explicitly control the FFT griding. For adsorption
and absorption cases we calculate the reference systems
as a sum of Enlc -contributions for each fragment and we
make sure to always position the fragment at the exact
same position in the system as in the interacting system.
For bulk systems we choose steps in which we exclusively
adjust the inter-plane or in-plane lattice constant. Here
the reference system is then simply defined as a system
with either double (or in some cases quadruple) lattice
constant and with a corresponding doubling of the FFT
griding along the relevant unit-cell vector.
The cost of full convergence is that, in practice, we of-
ten do three or more GGA calculations and subsequent
Enlc calculations for each point on the absorption, absorp-
tion, or formation-energy curve. In addition to the cal-
culations for the full system we have to do one for each of
the isolated fragments at identical position in the adsorp-
tion/absorption cases and one or more for fragments in
the doubled unit-cell and doubled griding reference. We
have explicitly tested that using a FFT grid spacing of
< 0.13 Å (but not larger) for such reference calculations
is sufficient to ensure full convergence in the reported
Enlc (and E
vdW−DF total) energy variation for graphitic
systems.
C. Material formation and sorption energies
The cohesive energy of graphite (G) is the energy gain,
per carbon atom, of creating graphite at in-plane lattice
constant a and layer separation dC-C from isolated (spin-
polarized) carbon atoms.
EG,coh(a, dC-C) = EG,tot(a, dC-C)− EC-atom,tot (4)
where EG,tot and EC-atom,tot are total energies per carbon
atom. The graphite structure is stable at the minimum
of the cohesive energy, at lattice constants a = aG and
2dC-C = cG.
The adsorption (absorption) energy for a p(2 × 2) K-
layer over (under) the top layer of a graphite surface is
the difference in total energy [from Eq. (3)] for the system
at hand minus the total energy of the initial system, i.e.,
a clean graphite surface and isolated gas-phase potassium
atoms. However, due to the above mentioned technical
issues in using the vdW-DF we calculate the adsorption
and absorption energy as a sum of (artificial) stages lead-
ing to the desired system: First the initially isolated,
spin-polarized potassium atoms are gathered into a free
floating potassium layer with the structure correspond-
ing to a full cover of potassium atoms. By this the total
system gains the energy ∆EK-layer(aG), with
∆EK-layer(a) = EK,tot(a)− EK-atom,tot . (5)
In adsorption the potassium layer is then simply placed
on top of the four-layer (2 × 2) graphite surface (with
the K atoms above graphite hollows) at distance dC-K.
The system thereby gains a further energy contribution
∆EK-G(dC-K). This leads to an adsorption energy per
K-atom
Eads(dC-K) = ∆EK-layer(aG) + ∆EK-G(dC-K) . (6)
In absorption the top graphite layer is peeled off the
(2 × 2) graphite surface and moved to a distance far
from the remains of the graphite surface. This process
costs the system an (“exfoliation”) energy −∆EC-G =
−[Etot,C-G(dC-C = cG/2)− Etot,C-G(dC-C → ∞)]. At the
far distance the isolated graphite layer is moved into AA
stacking with the surface, at no extra energy cost. Then,
the potassium layer is placed midway between the far-
away graphite layer and the remains of the graphite sur-
face. Finally the two layers are gradually moved towards
the surface. At distance 2dC-K between the two topmost
graphite layers (sandwiching the K-layer) the system has
further gained an energy ∆EC-K-G(dC-K). The absorp-
tion energy per K-atom is thus
Eabs(dC-K) = −∆EC-G+∆EK-layer(aG)+∆EC-K-G(dC-K) .
Similarly, the C8K intercalate compound is formed
from graphite by first moving the graphite layers far
apart accordion-like (and there shift the graphite stack-
ing from ABA . . . to AAA . . . at no energy cost), then
changing the in-plane lattice constant of the isolated
graphene layers from aG to a, then intercalating K-layers
(in stacking αβγδ) between the graphite layers, and fi-
nally moving all the K- and graphite layers back like an
accordion, with in-plane lattice constant a (which has the
value aC8K at equilibrium).
In practice, a unit cell of four periodically repeated
graphite layers is used in order to accommodate the
potassium αβγδ-stacking. The energy gain of creating
a (2× 2) graphene sheet from 8 isolated carbon atoms is
defined similarly to that of the K-layer:
∆EC-layer(a) = EC-layer,tot(a)− 8EC-atom,tot . (8)
The formation energy for the C8K intercalate com-
pound per K atom or formula unit, Eform, is thus found
from the energy cost of moving four graphite layers
apart by expanding the (2 × 2) unit cell to large height,
−∆EG-acc, the cost of changing the in-plane lattice con-
stant from aG to a in each of the four isolated graphene
layers, 4(∆EC-layer(a)−∆EC-layer(aG)), the gain of creat-
ing four K-layers from isolated K-atoms, 4∆EK-layer(a),
plus the gain of bringing four K-layers and four graphite
layers together in the C8K structure, ∆EC8K-acc(a, dC-K),
yielding
Eform(a, dC-K)
−∆EG-acc + 4∆EC-layer(a)− 4∆EC-layer(aG)
+ 4∆EK-layer(a) + ∆EC8K-acc(a, dC-K)
. (9)
The relevant energies to use for comparing the three
different mechanisms of including potassium (adsorp-
tion, absorption and intercalation) are thus Eads(dC-K),
Eabs(dC-K) and Eform(a, dC-K) at their respective mini-
mum values.
IV. RESULTS
Experimental observations indicate that the intercala-
tion of potassium into graphite starts with the absorption
of evaporated potassium into an initially clean graphite
surface.50 This subsurface absorption is preceded by ini-
tial, sparse potassium adsorption onto the surface, and
proceeds with further absorption into deeper graphite
voids. The general view is that the K atoms enter
graphite at the graphite step edges.20 The amount and
position of intercalated K atoms is controlled by the tem-
perature and time of evaporation.
Below, we first describe the initial clean graphite sys-
tem, and the energy gain in (artificially) creating free-
floating K-layers from isolated K-atoms. Then we present
and discuss our results on potassium adsorption and sub-
surface absorption, followed by a characterization of bulk
For the adsorption (absorption) system we calculate
the adsorption (absorption) energy curve, including the
equilibrium structure. As a demonstration of the need
for a relatively fine FFT griding in the vdW-DF cal-
culations we also calculate and compare the absorption
curve for a more sparse FFT grid. For the bulk sys-
tems (graphite and C8K) we determine the lattice pa-
rameters and the bulk modulus. We also calculate the
formation energy of C8K and the energy needed to peel
off one graphite layer from the graphite surface and com-
pare with experiment.51
A. Graphite bulk structure
The present calculations on pure graphite are for the
natural, AB-stacked graphite (lower panel of Fig. 1). The
cohesive energy is calculated at a total of 232 structure
values (a, dC-C) and the equilibrium structure and bulk
modulus B0 are then evaluated using the method de-
scribed in Ref. 52.
Figure 3 shows a contour plot of the graphite cohesive
energy variation EG,coh as a function of the layer sep-
aration dC-C and the in-plane lattice constant a, calcu-
lated within the vdW-DF scheme. The contour spacing
is 5meV per carbon atom, shown relative to the energy
minimum located at (a, dC-C) = (aG, cG/2) =(2.476 Å,
3.59 Å). These values are summarized in Table I together
with the results obtained from a semilocal PBE cal-
culation. As expected, and discussed in Ref. 14, the
semilocal PBE calculation yields unrealistic results for
the layer separation. The table also presents the cor-
responding experimental values. Our calculated lattice
values obtained using vdW-DFT are in good agreement
with experiment,40 and close to those found from the
older vdW-DF of Refs. 14 and 15, (in which we for Enlc as-
sume translational invariance of n(r) along the graphite
planes,) at (2.47 Å, 3.76 Å).
Consistent with experimental reports18 and our previ-
ous calculations14,15,45 we find graphite to be rather soft,
indicated by the bulk modulus B0 value. Since in-plane
compression is very hard in graphite most of the softness
suggested by (the isotropic) B0 comes from compression
perpendicular to the graphite layers, and the value of
B0 is expected to be almost identical to the C33 elastic
3 3.5 4 4.5 5 5.5 6
dC−C [Å]
FIG. 3: Graphite cohesive energy EG,coh (AB-stacked), based
on vdW-DF, as a function of the carbon layer separation dC-C
and the in-plane lattice constant a. The energy contours are
spaced by 5meV per carbon atom.
TABLE I: Optimized structure parameters and elastic
properties for natural hexagonal graphite (AB-stacking)
and the potassium-intercalated graphite structure C8K in
AαAβAγAδAα . . . stacking. The table shows the calcu-
lated optimal values of the in-plane lattice constant a, the
(graphite-)layer-layer separation dC-C, and the bulk modulus
B0. In C8K the value if dC-C is twice the graphite-potassium
distance dC-K.
Graphite C8K
PBE vdW-DF Exp. PBE vdW-DF Exp.
a (Å) 2.473 2.476 2.459a 2.494 2.494 2.480b
dC-C (Å) ≫ 4 3.59 3.336
a 5.39 5.53 5.35c
B0 (GPa) 27 37
de 37 26 47de
aRef. 40. bRef. 53. cRef. 4. dRef. 18.
eValue presented is for C33; for laterally rigid materials, like
graphite and C8K, C33 is a good approximation of B0.
coefficient.14,18
We find the energy cost of peeling off a graphite layer
from the graphite surface (the exfoliation energy) to be
∆EC-G = −435 meV per (2 × 2) unit cell, i.e., −55
meV per surface carbon atom (Table II). A recent
experiment51 measured the desorption energy of poly-
cyclic aromatic hydrocarbons (basically flakes of graphite
sheets) off a graphite surface. From this experiment
the energy cost of peeling off a graphite layer from the
graphite surface was deduced to −52± 5meV/atom.
Our value −55 meV/C-atom is also consistent with
a separate vdW-DF determination29 of the binding
(−47meV per in-plane atom) between two (otherwise)
isolated graphene sheets.
For the energies of the absorbate system and of the
C8K intercalate a few other graphite-related energy con-
tributions are needed. The energy of collecting C atoms
to form a graphene sheet at lattice constant a from iso-
lated (spinpolarized) atoms is given by ∆EC-layer(a); we
find that changing the lattice constant a from aG to
the equilibrium value aC8K of C8K causes this energy
to change a mere 30 meV per (2 × 2) sheet. The contri-
bution ∆EG-acc is the energy of moving bulk graphite
layers (in this case four periodically repeated layers)
far away from each other, by expanding the unit cell
along the direction perpendicular to the layers. Thus,
∆EG-acc = 32∆EG,coh(aG, cG/2) − 4∆EC-layer(aG) tak-
ing the number of atoms and layers per unit cell into
account. We find the value ∆EG-acc = −1600 meV per
(2×2) four-layer unit cell. This corresponds to −50 meV
per C atom, again consistent with our result for the ex-
foliation energy, ∆EC-G/8 = −55 meV.
B. Creating a layer of K-atoms
The (artificial) step of creating a layer of potassium
atoms from isolated atoms releases a significant energy
∆EK-layer. This energy contains the energy variation
with in-plane lattice constant and the energy cost of
changing from a spin-polarized to a spin-balanced elec-
tron configuration for the isolated atom.54
The creation of the K-layer provides an energy gain
which is about half an eV per potassium atom, depending
on the final lattice constant. With the graphite lattice
constant aG the energy change, including the spin-change
cost, is ∆EK-layer(aG) = −476meV per K atom in vdW-
DF (−624meV when calculated within PBE), whereas
∆EK-layer(aC8K) = −473meV in vdW-DF.
C. Graphite-on-surface adsorption of potassium
The potassium atoms are adsorbed on a usual
ABA . . .-stacked graphite surface. We consider here full
(one monolayer) coverage, which is one potassium atom
per (2 × 2) graphite surface unit cell. This orders the
potassium atoms in a honeycomb structure with lattice
constant 2aG, and a nearest-neighbor distance within the
K-layer of aG.
The unit cell used in the standard DFT calculations
for adsorption and absorption has a height of 40 Å and
includes a vacuum region sufficiently big that no interac-
tions (within GGA) can occur between the top graphene
sheet and the slab bottom in the periodically repeated
image of the slab. The vacuum region is also large in
order to guarantee that the separation from any atom to
the dipole layer55 always remains larger than 4 Å.
In the top panel of Fig. 4 we show the adsorption en-
ergy per potassium atom. The adsorption energy at equi-
librium is −937 meV per K atom at distance dC-K = 3.02
Å from the graphite surface.
For comparison we also show the adsorption curve cal-
culated in a PBE-only traditional DFT calculation. Since
vdW−DF
2 3 4 5 6 7
dC−K [Å]
vdW−DF
43.532.5
dC−K [Å]
sparse
FIG. 4: Potassium adsorption and absorption energy at the
graphite surface as a function of the separation dC-K of the
K-atom layer and the nearest graphite layer(s) (at in-plane
lattice constant corresponding to that of the surface, aG).
Top panel: Adsorption curve based on vdW-DF calculations
(solid line with black circles) and PBE GGA calculations
(dashed line). The horizontal lines to the left show the en-
ergy gain in creating the isolated K layer from isolated atoms,
∆EK-layer(aG), the asymptote of Eads(dC-K) in this plot.
Bottom panel: Absorption curve based on vdW-DF calcu-
lations. The asymptote is here the sum ∆EK-layer(aG) −
∆EC-G. Inset: Binding energy of the K-layer and the
top graphite layer (“C-layer”) on top of the graphite slab,
∆EC-K-G. The dashed curve shows our results when in E
ignoring every second FFT grid point (in each direction) of
the charge density from the underlying GGA calculations, the
solid curve with black circles shows the result of using every
available FFT grid point.
the interaction between the K-layer and the graphite sur-
face has a short-range component to it, even GGA calcu-
lations, such as the PBE curve, show significant binding
(−900 meV/K-atom at dC-K = 2.96 Å). This is in con-
trast to the pure vdW binding between the layers in clean
graphite.14,15 Note that the asymptote of the PBE curve
is different from that of the vdW-DF curve, this is due
to the different energy gains (∆EK-layer) in collecting a
potassium layer from isolated atoms when calculated in
PBE or in vdW-DF.
For K-adsorption the vdW-DF and PBE curves agree
reasonably well, and the use of vdW-DF for this spe-
cific calculation is not urgently necessary. However, in
order to compare the adsorption results consistently to
absorption, intercalation and clean graphite, it is neces-
sary to include the long-range interactions through vdW-
DF. As shown for the graphite bulk results above, PBE
yields quantitatively and qualitatively wrong results for
the layer separation.
D. Graphite-subsurface absorption of potassium
The first subsurface adsorption of K takes place in the
void under the top-most graphite layer. The surface ab-
sorption of the first K-layer causes a lateral shift of the
top graphite sheet, resulting in a A/K/ABAB . . . stack-
ing of the graphite. We have studied the bonding nature
of this absorption process by considering a full p(2× 2)-
intercalated potassium layer in the subsurface of a four
layer thick graphite slab.
Following the receipt of Section III for the absorption
energy (7) the energies ∆EC-K-G are approximated by
those from a four-layer intercalated graphite slab with
the stacking A/K/ABA, and the values are shown in the
inset of Fig. 4. The absorption energy Eabs is given by
the curve in the bottom panel of Fig. 4, and its minimum
is −952meV per K atom at dC-K = 2.90 Å.
To investigate what grid spacing is sufficiently dense
to obtain converged total-energy values in vdW-DF we
do additional calculations in the binding distance region
with a more sparse grid. Specifically, the inset of Fig. 4
compares the vdW-DF calculated at full griding with one
that uses only every other FFT grid point in each direc-
tion, implying a grid spacing for Enlc (but not for the lo-
cal terms) which is maximum 0.26 Å. We note that using
the full grid yields smaller absolute values of the absorp-
tion energy. We also notice that the effect is more pro-
nounced for small separations than for larger distances.
Thus given resources, the dense FFT grid calculations
are preferred, but even the less dense FFT grid calcu-
lations yield reasonably well-converged results. In all
calculations (except tests of our graphitic systems) we
use a spacing with maximum 0.13 Å between grid points.
This is a grid spacing for which we have explicitly tested
convergence of the vdW-DF for graphitic systems given
the computational strategy described and discussed in
Sec. III.
E. Potassium-intercalated graphite
When potassium atoms penetrate the gallery of the
graphite, they form planes that are ordered in a p(2× 2)
fashion along the planes. The K intercalation causes a
shift of every second carbon layer resulting into an AA
stacking of the graphite sheets. The K atoms then simply
occupy the sites over the hollows of every fourth carbon
hexagon. The order of the K atoms perpendicular to the
planes is described by the αβγδ stacking, illustrated in
Fig. 2.
For the potassium intercalated compound C8K we cal-
culate in standard DFT using PBE the total energy at
132 different combinations of the structural parameters a
TABLE II: Comparison of the graphite exfoliation energy per
surface atom, EC-G/8, graphite layer binding energy per car-
bon atom, ∆EC-acc/32, the energy gain per K atom of col-
lecting K- and graphite-layers at equilibrium to form C8K,
∆EC8K-acc/4, and the equilibrium formation energy of C8K,
Eform.
∆EC-G/8 ∆EC-acc/32 ∆EC8K-acc/4 Eform
[meV/atom] [meV/atom] [meV/C8K] [meV/C8K]
vdW-DF −55 −50 −818 −861
PBE − − −511 −
Exp. −52± 5a −1236b
aRef. 51. bRef. 1.
5 5.2 5.4 5.6 5.8 6
dC−C [Å]
FIG. 5: Formation energy of C8K, Eform, as a function of
the carbon-to-carbon layer separation dC-C and half the in-
plane lattice constant, a. The energy contours are spaced by
20meV per formula unit.
and dC-C. The charge densities and energy terms of these
calculations are then used as input to vdW-DF. The equi-
librium structure and elastic properties (B0) both for the
vdW-DF results and for the PBE results are then evalu-
ated with the same method as in the graphite case.52
Figure 5 shows a contour plot of the C8K formation
energy, calculated in vdW-DF, as a function of the C-C
layer separation (dC-C) and the in-plane periodicity (a)
of the graphite-layer structure. The contour spacing is
20 meV per formula unit and are shown relative to the
energy minimum at (a, dC-C) = (2.494 Å, 5.53 Å).
V. DISCUSSION
Table I presents an overview of our structural results
obtained with the vdW-DF for graphite and C8K. The
table also contrasts the results with the corresponding
values calculated with PBE where available. The vdW-
DF value dC-C = 5.53 Å for the C8K C − C layer sep-
aration is 3% larger than the experimentally observed
value whereas the PBE value corresponds to less than a
1% expansion. Our vdW-DF result for the C8K bulk
modulus (26 GPa) is also softer than the PBE result
(37 GPa) and further away from the experimental esti-
mates (47 GPa) based on measurements of the C33 elastic
response.18 A small overestimation of atomic separation
is consistent with the vdW-DF behavior that has been
documented in a wide range of both finite and extended
systems.14,15,16,17,29,30,33,34 This overestimation results,
at least in part, from our choice of parametrization of
the exchange behavior — an aspect that lies beyond
the present vdW-DF implementation which focuses on
improving the account of the nonlocal correlations, per
se. It is likely that systematic investigations of the ex-
change effects can further refine the accuracy of vdW-DF
implementations.56 In any case, vdW-DF theory calcu-
lations represent, in contrast to PBE, the only approach
to obtain a full ab initio characterization of the AM in-
tercalation process.
The C8K system is more compact than graphite and
this explains why PBE alone can here provide a good de-
scription of the materials structure and at least some ma-
terials properties, whereas it fails completely for graphite.
The distance between the graphene sheets upon interca-
lation of potassium atoms is stretched compared to that
of pure graphite, but the (K-)layer to (graphite-)layer
separation, dC-K = dC-C/2 = 2.77 Å, is significantly less
than the layer-layer separation in pure graphite. This in-
dicates that C8K is likely held together, at least in part,
by shorter-ranged interactions.
Table II documents that the vdW binding neverthe-
less plays an important role in the binding and forma-
tion of C8K. The table summarizes and contrasts our
vdW-DF and PBE results for graphite exfoliation and
layer binding energies as well as C8K interlayer binding
and formation energies. The vdW-DF result for the C8K
formation energy is smaller than experimental measure-
ments by 31% but it nevertheless represent a physically
motivated ab initio calculation. In contrast, the C8K
formation energy is simply unavailable in PBE because
PBE, as indicated, fails to describe the layer binding in
graphite. Moreover, for the vdW-DF/PBE comparisons
that we can make — for example, of the C8K layer inter-
action ∆EC8K-acc — the vdW-DF is found to significantly
strengthen the bonding compared with PBE.
It is also interesting to note that the combination of
shorter-ranged and vdW bonding components in C8K
yields a layer binding energy that is close to that of
the graphite case. In spite of the difference in nature
of interactions, we find almost identical binding energies
per layer for the case of the exfoliation and accordion in
graphite and for the accordion in C8K. This observation
testifies to a perhaps surprising strength of the so-called
soft-matter vdW interactions.
In a wider perspective our vdW-DF permits a first
comparison of the range of AM-graphite systems from
adsorption over absorption to full intercalation and thus
insight on the intercalation progress. Assuming a dense
2×2 configuration, we find that the energy for potassium
adsorption and absorption is nearly degenerate with an
indication that absorption is slightly preferred, consis-
tent with experimental behavior. We also find that the
potassium absorption may eventually proceeds towards
full intercalation thanks to a significant release of forma-
tion energy.
VI. CONCLUSIONS
The potassium intercalation process in graphite has
been investigated by means of the vdW-DF density func-
tional method. This method includes the dispersive in-
teractions needed for a consistent investigation of the
intercalation process. For clean graphite the vdW-DF
predicts — contrary to standard semilocal DFT imple-
mentations — a stabilized bulk system with equilibrium
crystal parameters in close agreement with experiments.
Two limits of the absorption process have been inves-
tigated by the vdW-DF, namely single layer subsurface
absorption and the fully potassium intercalated stage-1
crystal C8K. Here the vdW-DF is shown to enhance the
(semi-)local type of bonding described by traditional ap-
proaches. The significant impact on the materials behav-
ior indicates that the vdW-DF is needed not only for a
consistent description of sparse matter systems that are
solely stabilized by dispersion forces, but also for their
intercalates.
We thank D.C. Langreth and B.I. Lundqvist for stim-
ulating discussions. Partial support from the Swedish
Research Council (VR), the Swedish National Graduate
School in Materials Science (NFSM), and the Swedish
Foundation for Strategic Research (SSF) through the
consortium ATOMICS is gratefully acknowledged, as
well as allocation of computer time at UNICC/C3SE
(Chalmers) and SNIC (Swedish National Infrastructure
for Computing).
∗ Electronic address: [email protected]
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|
0704.0056 | Phase diagram of Gaussian-core nematics | Phase diagram of Gaussian-core nematics
Santi Prestipino1, ∗ and Franz Saija2, †
Università degli Studi di Messina, Dipartimento di Fisica, Contrada Papardo, 98166 Messina, Italy
Istituto per i Processi Chimico-Fisici del CNR, Sezione di Messina, Via La Farina 237, 98123 Messina, Italy
(Dated: November 4, 2018)
We study a simple model of a nematic liquid crystal made of parallel ellipsoidal particles interact-
ing via a repulsive Gaussian law. After identifying the relevant solid phases of the system through
a careful zero-temperature scrutiny of as many as eleven candidate crystal structures, we determine
the melting temperature for various pressure values, also with the help of exact free energy calcu-
lations. Among the prominent features of this model are pressure-driven reentrant melting and the
stabilization of a columnar phase for intermediate temperatures.
PACS numbers: 05.20.Jj, 61.20.Ja, 61.30.-v, 64.70.Md
Keywords: Liquid crystals; columnar phase; solid-liquid and solid-solid transitions; isothermal-isobaric Monte
Carlo; exact free-energy calculations
INTRODUCTION
Since five decades now, numerical simulation has im-
posed as an invaluable tool for the determination of equi-
librium statistical properties of many-particle systems.
Despite a long history, however, a precise numerical eval-
uation of the Helmholtz free energy of a simple model
fluid in its solid phase has resisted all attacks for many
years until, in a remarkable 1984 paper [1], Frenkel and
Ladd showed how a reference Einstein solid can be used
right to this purpose. Since then, it has become possi-
ble to trace a numerically accurate and complete equi-
librium phase diagram for simple-fluid systems by Monte
Carlo simulation methods. The only real limitation of
the Frenkel-Ladd method is the necessity of a preliminary
identification of all relevant solid structures. Depending
on the complexity of the model potential, some struc-
ture could be skipped, neither it does necessarily show
up spontaneously in the simulation due to the effective
fragmentation of the system phase space into inescapable
ergodic basins.
In a series of papers [2, 3, 4], we have employed
the Frenkel-Ladd technique in combination with the
standard thermodynamic-integration method in order to
trace the phase diagram of some reference simple-fluid
models. In particular, we have provided the first accu-
rate determination of the phase diagram for the so-called
Gaussian-core model, which is meant to describe dilute
solutions of polymer coils [5, 6]. The thermodynamics
of this model is ruled by the competition between the
fluid and two different, body-centered-cubic (BCC) and
face-centered-cubic (FCC), crystal structures; its pecu-
liar features are reentrant melting by isothermal com-
pression and, in a narrow range of temperatures, BCC
reentrance in the solid sector.
Following earlier simulational work by Frenkel and col-
laborators on hard ellipsoids and spherocylinders [7, 8, 9,
10], as well as by other authors on hard dumbbells [11],
we aim here to provide another demonstration of the use
of simulation for the description of thermodynamic prop-
erties of elongated particles. Such molecules can exist in a
number of partially-ordered mesophases with long-range
orientational order, possibly in combination with one- or
two-dimensional translational order (as in smectic and
in columnar liquid crystals, respectively) [12, 13]. Liquid
crystals do also usually give rise to numerous solid phases
which, as a rule, can hardly be anticipated from just a
glance at the interaction potential between the molecules.
Very recently, an interesting liquid-crystal model was
introduced by de Miguel and Martin del Rio [14] whose
phase diagram shows a stable smectic phase as well as
pressure-driven reentrance of the nematic phase. The
model consists of equally-oriented hard ellipsoids that are
further equipped with an attractive spherical well (there
is no isotropic phase in this model since the particles
are artificially constrained to stay parallel to each other,
hence the fluid phase is a nematic liquid crystal). Ini-
tially, we thought of this model as an ideal candidate for
a complete reconstruction of the phase diagram. Unfor-
tunately, the model potential turns out to be not simple
enough to allow for a straightforward identification of
the structure of its solid phase(s) and, in this respect,
the original paper is in fact reticent. We have made
an attempt to resolve the solid structure in terms of
stretched cubic lattices but a direct inspection of many
equilibrated solid configurations reveals more complicate,
yet periodically repeated patterns. Probably, this results
from a difficult matching between the optimization re-
quirements of the different pair-potential components,
i.e., a cylindrically-symmetric hard-core repulsion and a
spherically-symmetric step-like attraction.
To retain nematic reentrance and, possibly, also the
smectic phase, we have considered a more tractable test
case, that is a uniaxial deformation of the repulsive Gaus-
sian potential, which we expect to provide a model ne-
matic fluid whose phase diagram can fully be worked out,
also in its solid region. It can plausibly be argued on sym-
metry grounds and also expected from the smoothness of
http://arxiv.org/abs/0704.0056v1
the potential that all solid phases of the model will now
be found within the class of uniaxially-stretched cubic
crystals.
The rest of the paper is organized as follows: In Sec-
tion II, we present our liquid-crystal model together with
a catalogue of crystal structures that are possibly rele-
vant to it. Next, in Section III, we outline the numer-
ical methods by which the phase diagram of the model
is being drawn. Results are exposed in Section IV while
further comments and conclusions are deferred to Section
MODEL
We consider a nematic fluid of N parallel ellipsoids
of revolution whose geometric boundaries are smeared
out by a pair interaction u that smoothly depends on
the ratio between the center-to-center distance r and the
“contact distance” σ, which is the distance of closest ap-
proach in case of sharp boundaries. σ is a function of the
angle θ that the ray r joining the two molecular centers
forms with the direction ẑ of the axis of revolution. Its
closed-form expression is easily found to be:
σ(θ) =
L2 sin2 θ +D2 cos2 θ
, (2.1)
D and L being the transversal (with respect to ẑ) and the
longitudinal diameter respectively (we hereafter consider
only the prolate case L > D). For uniaxial particles, the
functional dependence of σ is actually on cos θ, as ex-
emplified by Eq. (2.1). We also note that hard ellipsoids
do correspond to an interaction strenght being +∞ for
r < σ(θ) and zero otherwise.
For the efficiency of numerical calculation, sufficiently
short-range interaction in all directions is highly desir-
able and, among smooth interactions, a good choice is a
Gaussian-decaying two-body repulsion,
u(r, θ) = ǫ exp
σ(θ)2
, (2.2)
ǫ > 0 being an arbitrary energy scale. Eq. (2.2) defines
the Gaussian-core nematic (GCN) fluid. It is evident
that, upon increasing the aspect ratio L/D, larger and
larger system sizes are needed in order to pull down any
rounding-off error that is implicit e.g. in the numerical
calculation of the total energy.
Another crucial quantity to determine in a simulation
is the pressure. For a V -volume system of N parallel
ellipsoids in contact with a heat bath at temperature
T , the equilibrium pressure P can be calculated from
a virial theorem that generalizes the one valid for a sim-
ple fluid. Let the total potential energy of the system
be of the general form U =
i<j u(|Ri − Rj |, cos θij),
where Ri is the center-of-mass position of particle i and
cos θij = (Ri−Rj)·ẑ. Upon switching to scaled V −1/3Ri
coordinates, one readily gets:
P = kBTρ−
1(Rij , cos θij)
, (2.3)
where u′1 is the u derivative with respect to its first
argument, ρ = N/V is the (number) density, and kB
is Boltzmann’s constant. Clearly, 〈. . .〉 is a canonical-
ensemble average. Upon introducing the T - and ρ-
dependent, two-body distribution function g2(R1,R2) =
g(|R1−R2|, cos θ12), the system pressure can also be ex-
pressed as
P = kBTρ−
dr r3g(r, τ)u′1(r, τ) . (2.4)
In particular, for a system of hard ellipsoids the pressure
reads:
P = kBTρ+
dτ σ(τ)3g(σ(τ)+, τ) . (2.5)
Anyway, a practical implementation of Eq. (2.4) or (2.5)
in a simulation requires a precise evaluation of the two-
argument function g which, ordinarily, is a difficult task
to accomplish with negligible statistical errors. A much
better solution is to switch to the isothermal-isobaric en-
semble, by simulating the system under constant-T and
constant-P conditions, see Section III.
As was mentioned in the Introduction, one main in-
convenience of liquid-crystal simulations is the correct
identification of the solid phase(s) of the system, since
a plethora of such phases are conceivable and there is
no unfailing criterion for choosing those that are really
relevant to the specific model under investigation. The
actual importance of a given crystal phase can only be
judged a posteriori, after proving its mechanical stability
in a long simulation run and, ultimately, on the basis of
the calculation of its Gibbs free energy, but nothing can
nevertheless ensure that no important phase was skipped.
Besides these vague indications, we adopted a more strin-
gent test in order to select the phases for which it is worth
performing the numerically-expensive calculation of the
free energy. With specific reference to the model (2.2),
we did a comprehensive T = 0 study of the chemical po-
tential µ as a function of the pressure for many stretched
cubic and hexagonal phases, in such a way as to iden-
tify the stable ground states and leave out from further
consideration all solids with a very large µ at zero tem-
perature. In fact, it is unlikely that such phases can ever
play a role for the thermodynamics at non-zero temper-
atures.
For the interaction potential describing the GCN
model, we surmise that all of its stable crystal phases
are to be sought among the structures obtained from
the common cubic and hexagonal lattices by a suit-
able stretching along a high-symmetry crystal axis, with
optimal stretching ratios α that are probably close to
L/D. Take e.g. the case of BCC. We can stretch
it along [001], [110], or [111], this way defining new
BCC001(α), BCC110(α), and BCC111(α) lattices (the
number within parentheses is the stretching ratio; for in-
stance, BCC001(2) is a BCC crystal whose unit cell has
been expanded by a factor of 2 along ẑ). The same can
be done with the simple-cubic (SC) and FCC structures.
We further consider hexagonal-close-packed (HCP) and
simple-hexagonal (SH) lattices that are stretched along
[111], this way arriving at a total of eleven potentially
relevant crystal phases.
METHOD
For fixed T and P values, the most stable of several
thermodynamic phases is the one with lowest chemical
potential µ (Gibbs free energy per particle). At T =
0, only crystal phases are involved in this competition
and, once a list of relevant phases has been compiled,
searching for the optimal one at a given P becomes a
simple computational exercise. An exact property of the
Gaussian-core model (which is the L/D = 1 limit of the
GCN model) is that, on increasing pressure, the BCC
crystal takes over the FCC crystal at P ∗ ≡ PD3/ǫ ≃
0.055 [3]. Hence, in the GCN model with L/D > 1 a
leading role is naturally expected for the stretched FCC
and BCC crystals.
For an assigned crystal structure, we calculate the T =
0 chemical potential µ(P ) of the GCN model for a given
pressure P by adjusting the stretching ratio α(P ) and the
density ρ(P ) until the minimum of (U+PV )/N is found.
Once the profile of µ as a function of P is known for each
structure, it is straightforward to draw the T = 0 phase
diagram for the given L/D.
The known thermodynamic behavior at zero tempera-
ture provides the general framework for the further simu-
lational study at non-zero temperatures. In fact, it is safe
to say that the same crystals that are stable at T = 0
also give the underlying lattice structure for the stable
solid phases at T > 0. As we shall see in more detail in
the next Section, the only complication is the existence
of three degenerate T = 0 structures for not too small
pressures, which obliged us to consider each of them as a
potentially relevant low-temperature GCN phase.
We perform a Monte Carlo (MC) simulation of the
GCN model with L/D = 3 in the isothermal-isobaric
ensemble, using the standard Metropolis algorithm with
periodic boundary conditions and the nearest-image con-
vention. For the solid phase, four different types of
lattices are considered, namely FCC001(3), BCC110(3),
BCC111(3), and BCC001(3) (see Section IV). The num-
ber of particles in a given direction is chosen so as to guar-
antee a negligible contribution to the interaction energy
from pairs of particles separated by half a simulation-
box length in that direction. More precisely, our samples
consist of 10× 20× 8 = 1600 particles in the FCC001(3)
phase, of 8 × 24 × 6 = 1152 particles in the fluid and in
the solid BCC110(3) phase, of 10× 12× 18 = 2160 parti-
cles in the BCC111(3) phase, and of 12× 12× 10 = 1440
particles in the BCC001(3) phase. Considering the large
system sizes employed, we made no attempt to extrapo-
late our finite-size results to infinity.
At given T and P , equilibration of the sample typically
took a few thousand MC sweeps, a sweep consisting of
one average attempt per particle to change its center-of-
mass position plus one average attempt to change the
volume by a isotropic rescaling of particle coordinates.
The maximum random displacement of a particle and
the maximum volume change in a trial MC move are ad-
justed once a sweep during the run so as to keep the
acceptance ratio of moves close to 50% and 40%, respec-
tively. While the above setup is sufficient when simu-
lating a (nematic) fluid system, it could have harmful
consequences on the sampling of a solid state to operate
with a fixed box shape since this would not allow the
system to release the residual stress. That is why, after a
first rough optimization with a fixed box shape, the equi-
librium MC trajectory of a solid state is generated with a
modified (so called constant-stress) Metropolis algorithm
which makes it possible to adjust the length of the vari-
ous sides of the box independently from each other (see
e.g. [8]). Ordinarily, however, the simulation box will de-
viate only very little from its original shape. When the
opposite occurs, this indicates a mechanic instability of
the solid in favor of the fluid, hence it gives a clue as to
where melting is located. We note that MC simulations
with a varying box shape are not well suited for the fluid
phase since in this case one side of the box usually be-
comes much larger or smaller than the other two, a fact
that seriously prejudicates the reliability of the simula-
tion results.
In order to locate the melting point for a given pres-
sure, we generate separate sequences of simulation runs,
starting from the cold solid on one side and from the hot
fluid on the other side. The last configuration produced
in a given run is taken to be the first of the next run at a
slightly different temperature. The starting configuration
of a “solid” chain of runs was always a perfect crystal with
α = 3 and a density equal to its T = 0 value. Usually,
this series of runs is carried on until a sudden change is
observed in the difference between the energies/volumes
of solid and fluid, so as to prevent us from averaging over
heterogeneous thermodynamic states. Thermodynamic
averages are computed over trajectories 104 sweeps long.
Much longer trajectories are constructed for estimating
the chemical potential of the fluid (see below).
Estimating statistical errors is a critical issue whenever
different candidate solid structures so closely compete for
thermodynamic stability. To this aim, we divide the MC
trajectory into ten blocks and estimate the length of the
error bars to be twice as large as the standard deviation of
the block averages. Typically, the relative errors affecting
the energy and the volume of the fluid are found to be
very small, a few hundredths percent at the most (for a
solid, they are even smaller).
A more direct clue about the nature of the phase(s)
expressed by the system for intermediate temperatures
can be got from a careful monitoring across the state
space of a “smectic” order parameter (OP) and of two
different, transversal and longitudinal (with respect to ẑ)
distribution functions (DFs). The smectic OP is defined
τ(λ) =
. (3.1)
This quantity is able to notice the existence of a layered
structure along ẑ in the system, be it solid-like or smectic-
like. In particular, the λ at which τ takes its largest value
gives the nominal distance λmax between the layers. A
large value of τ at λmax signals a strong layering along z
with period λmax. In order to discriminate between solid
and smectic (fluid) layers, we can rely on the in-plane
DF g⊥(r⊥), with r⊥ = r − (r · ẑ)ẑ, which informs on
how much rapid is the decay of crystal-like spatial corre-
lations in directions perpendicular to ẑ. The persistence
of crystal order along ẑ is measured through another DF,
g‖(z), which gives similar indications as τ(λ). A liquid-
like profile of g⊥ along with a sharply peaked τ or g‖ will
be faithful indication of a smectic phase. Conversely,
a sharply peaked g⊥ along with a structureless g‖ will
be the imprints of a columnar phase. Both g⊥(r⊥) and
g‖(z) are normalized in such a way as to approach 1 at
large distances in case of fully disordered center-of-mass
distributions in the respective directions. Slight devia-
tions from this asymptotic value may occur as a result
of the variation of box sidelengths during a simulation
run. The two DFs were constructed with a spatial reso-
lution of ∆r⊥ = D/20 and ∆z = L/20 respectively, and
updated every 10 MC sweeps.
We compute the difference in chemical potential be-
tween any two equilibrium states of the system – say,
1 and 2 – within the same phase (or even in different
phases, provided they are separated by a second-order
boundary) by the standard thermodynamic-integration
method as adapted to the isothermal-isobaric ensemble,
i.e., via the combined use of the formulas:
µ(T, P2)− µ(T, P1) =
dP v(T, P ) (3.2)
µ(T2, P )
µ(T1, P )
u(T, P ) + Pv(T, P )
(3.3)
To prove really useful, however, the above equations re-
quire an independent estimate of µ for at least one ref-
erence state in each phase. For the fluid, a reference
state can be any state characterized by a very small den-
sity (a nearly ideal gas), since then the excess chemical
potential can be estimated accurately through Widom’s
particle-insertion method [15]. The use of this technique
for small but finite densities avoids the otherwise neces-
sary extrapolation to the ideal gas limit as a reference
state for thermodynamic integration.
In order to calculate the excess Helmholtz free energy
of a solid, we resort to the method proposed by Frenkel
and Ladd [1], based on a different kind of thermody-
namic integration (see Ref. [4] for a full description of this
method and of its implementation on a computer). We
note that the ellipsoidal symmetry of the GCN particles
is not a complication at all, since the particle axes are
frozen and the only degrees of freedom been left are the
centers of mass. The solid excess Helmholtz free energy is
calculated through a series of NV T simulation runs, i.e.,
for fixed density and temperature. As far as the density
is concerned, its value is chosen in a way such that com-
plies with the pressure of the low-temperature reference
state, that is the one from which the NPT sequence of
runs is started. We wish to emphasize that, thanks to the
large sample sizes employed, the density histogram in a
NPT run always turned out to be sharply peaked, indi-
cating very limited density fluctuations (hence, negligible
ensemble dependence of statistical averages).
RESULTS
Zero-temperature calculations
For various L/D values in the interval between 1.1
and 3, we have calculated the T = 0 chemical poten-
tial µ(P ) for our eleven candidate ground states, with
P ranging from 0 to 0.20. We report in Table 1 the
results relative to L/D = 3 for two values of P , 0.05
and 0.20. An emergent aspect of this Table is the exis-
tence of a rich degeneracy that is only partly a result of
the effective identity of crystal structures up to a dila-
tion. Take e.g. the five structures with the minimum µ
(and with the same density). While the BCC001 lattice
with α = 3 is obtained from the FCC001 lattice with
α = 3/
2 = 2.12 . . . by a simple
2 dilation, there is
no homothety transforming BCC001(3) into BCC110(3)
or into BCC111(3) (in turn equivalent to SC111(1.5)):
Points in these three lattices have different local envi-
ronments, as can be checked by counting the nth-order
neighbors for n up to 4, yet the three stretched BCC crys-
tals of minimum µ share the same U/N . Also the pairs
FCC110(3), FCC111(3) and SC001(3), SC110(3) consist
of topologically-different degenerate structures. This fact
is an emergent phenomenon whose deep reason remains
unclear to us; it should deal with the dependence of u on
the ratio r/σ(θ), since the same symmetry holds with a
polynomial, rather than Gaussian, dependence.
For the case of L/D = 3, we show in Fig. 1 the over-
all P dependence at T = 0 of the chemical potential µ
for the various solids. The solid with the minimum µ is
either of the type FCC001 (with α = 3) or, say, of the
type BCC001 (with α = 3), a fact that holds true, but
with α = L/D, for all 1 < L/D < 3. Other solids are
definitely ruled out, and the same will probably hold for
T > 0. On increasing L/D, the transition from a FCC-
type to a BCC-type phase occurs at a lower and lower
pressure, whose reduced value is slightly less than 0.02
for L/D = 3.
Monte Carlo simulation
In order to investigate the thermodynamic behavior of
the GCN model at non-zero temperatures, we have car-
ried out a number of MC simulation runs for a GCN
system with L/D = 3, which is the system with the
strongest liquid-crystalline features that we can still man-
age numerically.
We have effected scans of the phase diagram for six
different pressure values, P ∗ = 0.01, 0.02, 0.03, 0.05, 0.12,
and 0.20. With all probability, FCC001(3) is the sta-
ble system phase only in a very small pocket of T -P
plane nearby the origin. However, we decided not to
embark on a free-energy study of the relative stability of
fluid, FCC001(3), and BCC-type phases at such low pres-
sures since this would require a numerical accuracy that
is beyond our capabilities. To a first approximation, the
boundary line between FCC001(3) and, say, BCC111(3)
can be assumed to run at constant pressure. For relating
data obtained at different pressures, we have carried out
two further sequences of MC runs along the isothermal
paths for T ∗ = 0.002 (solids) and T ∗ = 0.015 (fluid).
The Frenkel-Ladd computation of the excess
Helmholtz free energy per particle fex confirms that
the BCC001(3), BCC110(3), and BCC111(3) solids
are nearly degenerate at low temperature. We take
T ∗ = 0.002, P ∗ = 0.05 as a reference state for the
calculation of solid free energies. With the density
fixed at ρ = 0.08562D−3, in every case corresponding
to P ∗ = 0.05, we find βfex = 144.461(2), 144.470(2),
and 144.453(3), for the three above solids respectively,
implying a weak preference for the BCC111(3) phase.
Then, using thermodynamic integration along the
T ∗ = 0.002 isotherm (see Eq. (3.2)), we have studied
the relative stability of the three solids as a function
of pressure, up to P ∗ = 0.20. The results, depicted
in Fig. 2, suggest that BCC111(3) is the stable phase
throughout the low-temperature region, the other solids
being very good solutions anyway with near-optimal
chemical potentials.
We then follow the thermal disordering of the BCC-
type solids for fixed pressure (with three cases consid-
ered, P ∗ = 0.05, 0.12, and 0.20) through sequences of
isothermal-isobaric runs, all starting from T ∗ = 0.002,
with steps of 0.001. Any such sequence is stopped when
the values of potential energy and specific volume have
collapsed onto those of the fluid, thus informing that the
ultimate bounds of solid stability are reached (usually, a
solid can hardly be overheated). The stability thresholds
detected this way are fairly consistent with the indica-
tion coming from the DF profiles which, upon increas-
ing temperature, will eventually show a fluid-like appear-
ance. Thermodynamic integration (see Eq. (3.3)) is used
to propagate the calculated µ for T ∗ = 0.002 to higher
temperatures.
As far as the (nematic) fluid is concerned, we have
first generated a sequence of NPT simulation runs for
P ∗ = 0.05, starting from T ∗ = 0.015. At this initial
point, the excess chemical potential µex was estimated
by Widom’s insertion method, obtaining µex = 0.986(5).
It is worth noting that, in a long simulation run of as
many as 5×104 MC sweeps at equilibrium, the chemical-
potential value relaxed very soon, with small fluctuations
around the average and no significant drift observed. Our
analysis of the fluid phase is completed by further sim-
ulation runs along the isobaric paths for P ∗ = 0.12 and
0.20, for which we did not have the need to compute the
chemical potential again since this could be deduced from
the volume data along the T ∗ = 0.015 isotherm.
Chemical-potential results along the three isobars on
which we focussed are reported in Figs. 3 to 5. As is clear,
with increasing temperature the fluid eventually takes
over the solids. Among the solids, the BCC111(3) phase
is the preferred one for any temperature and pressure, al-
though the chemical potential of the other solid phases is
only slightly larger. On increasing pressure, the melting
temperature goes down, like in the Gaussian-core model.
The necessity of a matching with the zero-temperature
melting point for P = 0 will then imply reentrant melt-
ing in the GCN model too. The maximum error on the
melting temperature Tm, which we estimate to be about
0.003 (hence not that small), entirely depends on the lim-
ited precision of the fluid µex, which then constitutes a
major source of error on Tm.
The only conclusion we can draw from the above
chemical-potential study is that BCC111(3) is the most
stable solid phase of the system (provided the pressure is
not too low). However, a closer look at the DF profiles
obtained from the simulation of BCC111(3) raises some
doubts about the absolute stability of this phase at in-
termediate temperatures, whatever the pressure, calling
for a different interpretation of the hitherto considered
as BCC111(3) MC data. Take, for instance, the case of
P = 0.05. Upon increasing temperature, while g⊥ keeps
strongly peaked all the way to melting, the solid-like os-
cillations of g‖ undergo progressive damping until they
are washed out completely, suggesting a second-order
(or very weak first-order at the most) transformation of
BCC111(3) into a columnar phase before melting. This
is illustrated in Figs. 6 and 7, where the DFs are plotted
for a number of temperatures. A similar indication is got
from the behavior of the smectic OP, see Fig. 8, whose
highest maximum eventually deflates at practically the
same temperature, T ∗ ≈ 0.005, at which the oscillations
of g‖ disappear. Note that no appearance of a columnar
phase is seen during the simulation of either BCC110(3)
or BCC001(3), nor in the simulation of FCC001(3) for
P ∗ = 0.01. A slice of the columnar phase is depicted in
Fig. 9 (right panels). In this phase, columns of stacked
particles are arranged side by side, tightly packed to-
gether so as to project a triangular solid on the x-y plane.
Neighboring columns are not commensurate with each
other, as implied by a completely featureless g‖.
The probable reason for the instability of the smectic
phase in the GCNmodel is the absence of an ad hocmech-
anism for lateral attraction between the molecules, which
is present instead in the model of Ref. [14]. By the way,
hard ellipsoids do not show a smectic phase either [7], at
variance with (long) hard spherocylinders where particle
geometry alone proves sufficient to stabilize a periodic
modulation of the number density along ẑ [10].
Given the compelling evidence of a columnar phase
in the GCN model, one may now ask whether the con-
clusions drawn from the chemical-potential data are all
flawed. In particular, the µ curves that are tagged as
BCC111(3) in Figs. 3 to 5 would be meaningless beyond
a certain temperature Tc < Tm. In fact they are not, i.e.,
they retain full validity up to melting since the (nearly)
continuous character of the transition from BCC111(3)
to columnar allows one to safely continuate thermody-
namic integration across the boundary, with the proviso
that what previously treated as the BCC111(3) chemi-
cal potential beyond Tc is to be assigned instead to the
columnar phase.
As pressure goes up, the transition from BCC111(3) to
columnar takes place at lower and lower temperatures.
In order to exclude that the columnar phase too, like-
wise the fluid, will show reentrant behavior at low pres-
sure, we have simulated the disordering of a BCC111(3)
solid also for P ∗ = 0.02 and 0.03 (in fact, no reentrance
of the columnar phase is observed). Further points on
the melting line for P = 0.01, 0.02, and 0.03 are fixed
through the behavior of g⊥ as a function of temperature.
All in all, the overall GCN phase diagram appears as
sketched in Fig. 10. This is similar to the phase portrait
of the Gaussian-core model, see Fig. 1 of Ref. [4], with
the obvious exception of the columnar phase. There is a
small discrepancy between the melting points as located
through free-energy calculations (full dots in Fig. 10) and
those assessed from the evolution of g⊥ (open dots). In
our opinion, this would mostly be attributed to the sta-
tistical error associated with the µex of the fluid in its
reference state. Notwithstanding their limited precision,
however, free-energy calculations are all but useless in
identifying the structure of the solid phase. In conclu-
sion, although some aspects of the equilibrium behavior
of the GCN model remain still uncertain, especially with
regard to the exact location of the solid-solid transition
at low pressure, we are confident that the main features
of the GCN phase diagram are correctly accounted for
by Fig. 10.
Summing up, there are at least two conceivable and
mutually exclusive paths for the thermal disordering of
a liquid-crystal solid (aside from a direct transformation
of it into a nematic phase). One is through the forma-
tion of a smectic phase, which eventually transforms into
a nematic fluid. A second possibility is a more gradual
release of crystalline order by the appearance of a colum-
nar phase as intermediate stage between the solid and the
nematic phase. Our study showed that it is this second
scenario that occurs in the GCN model, with no evidence
whatsoever of a smectic phase.
CONCLUSIONS
We have introduced a liquid-crystal model of softly-
repulsive parallel ellipsoids, named the Gaussian-core ne-
matic (GCN) model, aiming at a complete characteriza-
tion of its phase behavior, including the solid sector. This
requires a preliminary identification of all relevant solid
structures, which is generally a far-from-trivial task to be
accomplished for model liquid crystals [16]. Through a
careful scrutiny of as many as eleven uniaxially-deformed
cubic and hexagonal phases, we obtained a thorough de-
scription of the T = 0 equilibrium phase portrait of the
GCN model, identifying its ground state at any given
pressure. In doing so, we discovered a rich and absolutely
unexpected structural degeneracy, which is only lifted by
going to T > 0. At low temperature, and for not too
low pressures, our free-energy calculations indicate that a
GCN system with an aspect ratio of 3 is found in just one
solid phase, i.e., a stretched BCC solid with the molecules
oriented along [111]. Only near zero pressure, the stable
phase becomes a stretched FCC solid. With increasing
temperature, the BCC-type solid first undergoes a weak
transition into a columnar phase, which still retains par-
tial crystalline order, before melting completely into the
nematic fluid.
It is worth emphasizing that our interest in the GCN
model is purely theoretical, hard-core ellipsoids provid-
ing a more physically realistic model liquid crystal. One
could even argue that a Gaussian repulsion is highly irre-
alistic for a liquid crystal. In real atomic systems, super-
position of particle cores is strongly obstructed, whence
the consideration of hard-core or steep inverse-power re-
pulsion in the more popular models. However, unless the
system density is very high, higher than considered in
our study, repulsive Gaussian particles would effectively
be blind to an inner hard core, which thus may or may
not exist, as evidenced e.g. in the snapshots of Fig. 9
where particles appear well spaced out.
The GCN model is a “deformation” of Stillinger’s
Gaussian-core model, well known for exhibiting a
reentrant-melting transition. Various instances of reen-
trant behavior are also known for nematics [17] and in-
deed one of the original motivations for the present work
was searching for a new kind of reentrance, i.e., re-
appearance of a more disordered phase with increasing
pressure. With this study, we provide yet another exam-
ple of reentrant behavior in a model nematic: While this
is nothing but the analog of fluid-phase reentrance in the
Gaussian-core model, the absolute novelty of our findings
is in the nature of the intermediate phase, this being sur-
prisingly columnar in a range of pressures rather than
genuinely solid.
∗ Electronic address: [email protected]
† Electronic address: [email protected]
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[5] F. H. Stillinger, J. Chem. Phys., 65, 3968 (1976).
[6] A. Lang, C. N. Likos, M. Watzlawek, and H. Löwen, J.
Phys.: Condens. Matter, 12, 5087 (2000).
[7] D. Frenkel, B. M. Mulder, and J. P. McTague, Phys. Rev.
Lett. 52, 287 (1984).
[8] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel,
Phys. Rev. A 36, 2929 (1987).
[9] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237
(1990); ibidem, 43, 4334 (1991).
[10] P. Bolhuis and D. Frenkel, J. Chem. Phys. 106, 666
(1997).
[11] C. Vega, E. P. A. Paras, and P. A. Monson, J. Chem.
Phys. 96, 9060 (1992); ibidem, 97, 8543 (1992).
[12] P. Pasini and C. Zannoni eds., Advances in the Computer
Simulations of Liquid Crystals (NATO-ASI Series, 1998).
[13] S. Singh, Phys. Rep. 324, 107 (2000).
[14] E. de Miguel and E. Martin del Rio, Phys. Rev. Lett. 95,
217802 (2005).
[15] B. Widom, J. Chem. Phys. 39, 2808 (1963).
[16] After completion of this paper, we became aware of
the discovery, reported in P. Pfleiderer and T. Schilling,
cond-mat/0612151, of a new stable crystal phase in
freely-standing hard ellipsoids. This further demonstrates
that the solid structure of liquid crystals is generally dif-
ficult to anticipate, even when the model system is the
simplest as possible.
[17] The first example of such behavior was discovered by
P. E. Cladis, Phys. Rev. Lett. 35, 48 (1975); see also
Ref. [14] and references therein.
mailto:[email protected]
mailto:[email protected]
TABLE I: GCN model for L/D = 3: T = 0 chemical poten-
tial µ(P ) for eleven different solids and two values of P ∗, 0.05
and 0.20. Nx, Ny , Nz are the number of lattice points along
the three spatial directions, ρ = NxNyNz/V is the density,
and α is the stretching ratio (for the SH111 lattice, α is the
so-called c/a ratio). Nx, Ny , Nz have been chosen so large
that the rounding-off error on the total potential energy per
particle, U/N , due to the finite lattice size is negligible. The
numerical precision on ρ and α is of one unit on the last deci-
mal digit. Looking at the Table, the most stable structures at
both pressures are five degenerate crystals, actually belonging
to three distinct types which are exemplified by BCC001(3)
(equivalent to FCC001(2.12) up to a dilation), BCC110(3),
and BCC111(3) (equivalent to SC111(1.5)) – within brackets
is the value of α.
crystal Nx, Ny , Nz ρ(0.05) α(0.05) µ(0.05) ρ(0.20) α(0.20) µ(0.20)
FCC001 10,20,10 0.086 2.12 0.855724 0.157 2.12 2.093695
BCC001 14,14,10 0.086 3.00 0.855724 0.157 3.00 2.093695
SC001 20,20,8 0.086 3.00 0.881586 0.158 3.00 2.105241
FCC110 16,12,12 0.086 3.00 0.856391 0.157 3.00 2.094368
BCC110 10,28,8 0.086 3.00 0.855724 0.157 3.00 2.093695
SC110 14,18,10 0.086 3.00 0.881586 0.158 3.00 2.105241
FCC111 16,18,9 0.086 3.00 0.856391 0.157 3.00 2.094368
BCC111 12,12,18 0.086 3.00 0.855724 0.157 3.00 2.093695
SC111 12,12,18 0.086 1.50 0.855724 0.157 1.50 2.093695
HCP111 18,20,10 0.086 3.00 0.856429 0.157 3.02 2.094474
SH111 18,20,9 0.086 2.75 0.870014 0.158 2.69 2.099565
FIG. 1: T = 0 equilibrium behavior of the GCN model with
L/D = 3. Left: T = 0 chemical potential µ(P ∗) of var-
ious crystals relative to BCC110(3), which thus serves as
the zero or reference level. The reduced pressure P ∗ is in-
cremented by steps of 0.01. Note that, for all P , the five
crystals FCC001(2.12), BCC001(3), BCC110(3), BCC111(3),
and SC111(1.5) are degenerate (∆µ = 0). Other data points
are for FCC001 (continuous line; α = 3 for P ∗ = 0.01, be-
ing α = 2.12 otherwise), FCC110(3) and FCC111(3) (dotted
line), HCP111 (open dots), SH111 (open squares), SC001(3)
and SC110(3) (dashed line). Right: Resulting equation of
state in the pressure range from 0 to 0.30. FCC001(3) (open
triangle) is stable at very low pressure, up to slightly less
than 0.02, while FCC001(2.12), BCC001(3), etc. (open dots)
prevail for higher pressures.
FIG. 2: GCN model with L/D = 3, chemical-potential results
for T ∗ = 0.002. In the picture, we plot the reduced chemical
potential of the three T = 0 degenerate structures that exist
for not too low pressure, taking BCC111(3) for reference. The
latter phase gives the most stable solid for any P in the range
from 0.05 to 0.20 (and, most likely, even further). The µ
curves are obtained by thermodynamic integration of volume
MC data, using as initial conditions those specified by the
Frenkel-Ladd calculations that were carried out at P ∗ = 0.05.
Though the reported µ values for the BCC-type solids are very
close to each other and also affected by some numerical noise,
the higher stability of BCC111(3) cannot be truly called into
question – a regular pattern is clearly seen behind each curve.
FIG. 3: GCN model with L/D = 3, chemical-potential
results for P ∗ = 0.05: Chemical potential of the fluid
phase (dotted line) as compared with those of the competing
solid phases for that pressure (BCC001(3), long-dashed line;
BCC110(3), dashed line; BCC111(3), continuous line). While
the BCC111(3) solid is stable at low temperature, the fluid
phase overcomes it in stability for higher temperatures. This
is more clearly seen in the inset, where chemical-potential
differences are reported, taking the fluid µ for reference. The
melting temperature for P ∗ = 0.05, which is where the con-
tinuous line crosses zero, is estimated to be T ∗ ≃ 0.0073.
FIG. 4: GCN model with L/D = 3, chemical-potential results
for P ∗ = 0.12. Same notation as in Fig. 3, except for the
absence of data for BCC001(3), which were not computed.
Despite this, a look at the results in Figs. 2 and 3 give us
confidence that the chemical potential of BCC001(3) will be
closer to that of BCC110(3) than is for P ∗ = 0.05.
FIG. 5: GCN model with L/D = 3, chemical-potential results
for P ∗ = 0.20. Same notation as in Figs. 3 and 4.
FIG. 6: GCN model with L/D = 3, distribution functions of
BCC111(3) for P ∗ = 0.05. Left: T ∗ = 0.002. Right: T ∗ =
0.003. The strenght of crystalline order along ẑ, as measured
by the amplitude of g‖ oscillations, reduces with increasing
temperature, until complete disorder is left above T ∗ ≃ 0.005
(see next Fig. 7). Considering that the crystallinity within
the x-y plane persists well beyond T ∗ = 0.005 (the spatial
modulation of g⊥ remains solid-like beyond this temperature
and up to melting), we conclude that the GCN system is found
in a columnar phase for 0.005 < T < Tm.
FIG. 7: GCN model with L/D = 3, distribution functions
of BCC111(3) for P ∗ = 0.05. Left: T ∗ = 0.004. Right:
T ∗ = 0.005.
FIG. 8: GCN model with L/D = 3, smectic order parame-
ter τ (λ) of BCC111(3) for P ∗ = 0.05. The behavior of τ (λ)
faithfully reproduces that seen for g‖(z) (cf. Figs. 6 and 7):
The deflating of the highest τ (λ) maximum with increasing
temperature closely follows the thermal damping of g‖(z) os-
cillations.
FIG. 9: GCN model with L/D = 3, some snapshots of the
particle configuration taken at low temperature (T ∗ = 0.002,
BCC111(3) solid phase) and at intermediate temperature
(T ∗ = 0.006, columnar phase). The reduced pressure is
P ∗ = 0.05 in both cases. Above: side view, i.e., projection
of particle coordinates onto the x-z plane. Below: top view,
i.e., projection of particle coordinates onto the x-y plane. For
clarity, in spite of their mutual interaction being soft, the par-
ticles are given sharp ellipsoidal boundaries, corresponding to
a unitary short axis (D) and a long axis of L = 3D. While the
crystalline order along z is lost already at T ∗ = 0.005 (hence,
it is there in the top-left panel while it is absent in the top-
right panel), the triangular order within the x-y plane is main-
tained up to the melting temperature (here, Tm ≃ 0.0073).
FIG. 10: GCN model with L/D = 3, sketch of the phase
diagram on the T -P plane. The full dots mark the location
of the melting transition as extracted from our free-energy
calculations. Open symbols refer instead to the transition
thresholds as given by a visual inspection of the DF profiles.
Though the latter melting-point estimates are more easily ob-
tained than the former, the free-energy study was essential to
identify the correct solid structure of the GCN model at not
too low pressure. To help the eye, tentative phase bound-
aries are drawn as continuous (i.e., first-order) and dashed
(nearly second-order) lines through the transition points. In
the low-pressure region, the solid-solid boundary is highly hy-
pothetical since we have no data there.
|
0704.0057 | High-spin to low-spin and orbital polarization transitions in
multiorbital Mott systems | High-spin to low-spin and orbital polarization transitions in multiorbital Mott systems
Philipp Werner and Andrew J. Millis
Columbia University, 538 West, 120th Street, New York, NY 10027, USA
(Dated: June 30, 2007)
We study the interplay of crystal field splitting and Hund coupling in a two-orbital model which
captures the essential physics of systems with two electrons or holes in the eg shell. We use single site
dynamical mean field theory with a recently developed impurity solver which is able to access strong
couplings and low temperatures. The fillings of the orbitals and the location of phase boundaries
are computed as a function of Coulomb repulsion, exchange coupling and crystal field splitting. We
find that the Hund coupling can drive the system into a novel Mott insulating phase with vanishing
orbital susceptibility. Away from half-filling, the crystal field splitting can induce an orbital selective
Mott state.
PACS numbers: 71.10.Fd, 71.10.Fd, 71.28.+d, 71.30.+h
The Mott metal-insulator transition plays a fundamen-
tal role in electronic condensed matter physics [1]. Much
attention has focused on the one-orbital case, in part be-
cause of its presumed relevance to high temperature su-
perconductivity [2] and in part because appropriate the-
oretical tools for the multiorbital case have until recently
not been available. In most Mott systems, however, more
than one orbital is relevant [3] and the redistribution of
electrons among different orbitals leads to new phenom-
ena such as orbital ordering or “orbital selective” Mott
transitions. Recent studies of nickelates [4], titanates [5],
cobaltates [6], manganates [7], vanadates [8, 9, 10] and
ruthenates [3, 11, 12, 13] have focused interest on the in-
terplay between the Mott metal-insulator transition and
orbital degeneracy. A fundamental question in this field,
relevant in particular to the issue of lattice distortions
in strongly correlated materials, is the response of multi-
orbital systems to a perturbation which breaks the orbital
degeneracy. In this paper, we show that a two orbital
model with Hund coupling and crystal field splitting ex-
hibits two fundamentally different Mott phases, one char-
acterized by a vanishing orbital susceptibility, and one
adiabatically connected to the band insulating state. We
characterize these phases in terms of the atomic ground
states.
Multiorbital models are more difficult to study both
because of the larger number of degrees of freedom,
and because the physically important exchange and
pair-hopping terms are not easy to treat by standard
Hubbard-Stratonovich methods [14]. Weak coupling ap-
proaches [12] have been used to show that exchange and
pair hopping interactions act to suppress the response to
a crystal field splitting, and some authors have studied
the model without exchange and pair hopping terms [9],
but a reliable extension of these results to physically rel-
evant Slater-Kanamori interactions and the strong cou-
pling regime has been lacking.
Dynamical mean field theory (DMFT) provides a non-
perturbative and computationally tractable framework to
study correlation effects and has allowed insights into
the Mott metal-insulator transition [15]. In its sin-
gle site version, DMFT ignores the momentum depen-
dence of the self-energy and reduces the original lat-
tice problem to the self-consistent solution of a quan-
tum impurity model given by the Hamiltonian HQI =
Hloc + Hhyb + Hbath. For multi-orbital models Hloc =∑
m ǫmc
j,k,l,m U
jklmc
l cm, where m = (i, σ)
denotes both orbital and spin indices, and U jklm some
general four-fermion interaction. Hhyb and Hbath are
the impurity-bath mixing and bath Hamiltonians, respec-
tively. While the DMFT approximation simplifies the
problem enormously (replacing a 3 + 1 dimensional field
theory by a quantum impurity model plus a self consis-
tency condition), the extra complications associated with
exchange couplings in multiorbital systems have until re-
cently prohibited extensive numerical work. Interesting
progress has been made using a finite temperature exact
diagonalization technique [6, 13], but this approach re-
quires a truncation of Hbath to a small number of levels.
In Refs. [16, 17] we have introduced a continuous-time
impurity solver which can handle the general interactions
in Hloc. The method, which is free from systematic er-
rors, is based on a diagrammatic expansion of the parti-
tion function in the impurity-bath hybridization Hhyb.
Here, we employ this solver to study the physically
relevant case in which the number of electrons matches
the number of orbitals. The local Hamiltonian is
Hloc = −
α=1,2
µnα,σ +
∆(n1,σ − n2,σ)
α=1,2
Unα,↑nα,↓ +
U ′n1,σn2,−σ
(U ′ − J)n1,σn2,σ
− J(ψ†
2,↑ψ2,↓ψ1,↑ + ψ
2,↓ψ1,↑ψ1,↓ + h.c.), (1)
with µ the chemical potential, ∆ the crystal field split-
ting, U the intra-orbital and U ′ the inter-orbital Coulomb
interaction, and J the coefficient of the Hund coupling.
We adopt the conventional choice of parameters, U ′ =
http://arxiv.org/abs/0704.0057v2
0 1 2 3 4 5 6 7 8 9
∆/t=0.2
∆/t=0.6
∆/t=1
FIG. 1: Filling of orbital 1 as a function of U for ∆/t =
0.2, 0.6, 1 and several values of J/U . The different curves
for given ∆ correspond (from bottom to top) to J/U = 0,
(0.01), (0.02), 0.05, 0.1, 0.15, 0.25, respectively. Open (full)
symbols correspond to metallic (insulating) solutions. The
metal-insulator transition is characterized by a jump in filling
and a coexistence region where both insulating and metallic
solutions exist. Our data show the region of stability of the
metallic phase.
U − 2J , which follows from symmetry considerations for
d-orbitals in free space and is also assumed to hold in
solids. With this choice the Hamiltonian (1) is rotation-
ally invariant in orbital space and the condition for half-
filling becomes µ = µ1/2 ≡ 32U −
J . In the DMFT self-
consistency loop we use a semi-circular density of states
of bandwith 4t (Bethe lattice). The temperature, unless
otherwise noted, is T/t = 0.02 and we suppress magnetic
order by averaging over spin up and down in each orbital.
No sign problem is encountered in the simulations.
The main result is shown in Fig. 1, which for several
values of ∆ and J/U plots the filling per spin, n1, of or-
bital 1 as a function of interaction strength. The half
filling, non-magnetic condition implies n2 = 1 − n1. In
the T → 0 limit, three phases are found: a metallic phase
(which may have any value of n1 between 0 and 0.5), an
orbitally polarized insulator favored by large ∆ and small
J , and a Mott insulator (with n1 = 0.5 = n2) favored
by large U , small ∆ and large J . If U is increased from
zero to a small value, the orbital splitting either increases
(small J/U) or decreases (large J/U), consistent with the
findings of Ref. [12]. As interaction strength is further in-
creased, one of several things may happen: at very small
J/U , n1 continues to decrease, and the system eventually
undergoes a transition to an orbitally polarized insulator
(for large ∆ essentially a band insulator). For somewhat
larger J/U , the occupancy n1, after an initial decrease,
goes through a minimum and begins to increase. At even
stronger interactions, one then observes a transition ei-
ther to an orbitally polarized insulator (where n1 may
take a range of values) or into a special type of insulator
0 0.2 0.4 0.6 0.8 1
J/U=0
0.05 0.1
metal
insulator
0 0.5 1 1.5 2 2.5 3 3.5
J/U=0 J/U=0.25
Mott insulator
(spin triplet for J/U=0.25)
metal
orbitally polarized
insulator
FIG. 2: Phase diagram in the plane of crystal field splitting ∆
and intraorbital Coulomb repulsion U for indicated values of
J/U . For J = 0 the phase boundary is a monotonic function
of ∆, whereas for J/U > 0 it peaks near ∆ =
2J (indicated
by the dotted lines). The insulating state in the region ∆ .√
2J is characterized by a vanishing orbital susceptibility.
with n1 = 0.5.
Figure 2 shows the metal-insulator phase diagram in
the space of crystal field splitting and Coulomb repul-
sion for several values of J/U . In the absence of a crystal
field splitting (∆ = 0), we observe a metal-insulator tran-
sition at a strongly J-dependent critical U . This finding
is consistent with data presented in Ref. [18]. As ∆ is
increased, the critical U changes. For J = 0 and fixed
U , n1 decreases until the band is emptied and a metal-
insulator transition occurs. The monotonic decrease of
the critical U with ∆ at J = 0 is a special case. For
J > 0, the first effect of a small ∆ is to stabilize the
metallic phase. Then, at larger ∆, a reentrant insulating
phase occurs. We shall show below that this behavior
arises from the unusual nature of the insulating state at
J > 0 and small ∆, which is characterized at T = 0 by a
strictly vanishing orbital susceptibility. If ∆ is increased
at large U , this state makes a transition to an orbitally
polarized insulator at ∆ ≈
2J . We therefore plot in
Fig. 2 the curves ∆ =
2J as dotted lines, and sug-
gest that they correspond to the T = 0 phase boundary
0.05
0.15
0.25
0.35
0.45
0 0.2 0.4 0.6 0.8 1 1.2
J/U=0.1
0 0.05 0.1 0.15 0.2
J/U=0
J/U=0.002
J/U=0.005
J/U=0.010
J/U=0.020
FIG. 3: Filling of orbital 1 as a function of ∆ for fixed U
and indicated values of J/U . Top panel: U/t = 6. Open
(full) symbols correspond to metallic (insulating) solutions.
Bottom panel: U/t = 9. Here, all solutions are insulating.
For crystal field splittings smaller than ∆c =
2J (indicated
by a vertical line) the orbital susceptibility in the T → 0 limit
is completely suppressed. Solid lines are for βt = 50, dotted
lines show results for βt = 12.5, 25 and 100, respectively.
between two distinct insulating states.
Figure 3 plots the filling of orbital 1 as a function of
crystal field splitting for fixed U/t and several values of
J/U . The leftmost curve in the upper panel shows the
density variation for J/U = 0.02. At ∆ = 0, the model
is metallic. The rapid variation of n1 with ∆ reflects the
large, but finite orbital susceptibility of the metal, which
for this small value of J is strongly enhanced by U . At
∆/t ≈ 0.325 >
2J , an apparently first order transition
occurs to the orbitally polarized insulating state, which
then evolves smoothly (as ∆ is increased) to the band
insulator (n1 → 0). The two larger J values reveal a
different behavior. For ∆ < ∆c ≈
2J the insulating
state is characterized by an orbital occupancy which is
independent of crystal field splitting. Then, an appar-
ently discontinuous transition occurs to a metallic state
with a large orbital susceptibility, which at even larger ∆
exhibits a first order transition to the orbitally polarized
insulating state. The lower panel of Fig. 3 shows the be-
2 4 6 8 10 12 14 16
basis state
∆/t=0.3
∆/t=0.5
∆/t=0.7
FIG. 4: Weight of the different eigenstates of Hloc for U/t =
6, J/U = 0.05 and ∆/t = 0.3, 0.5 and 0.7. The smallest
crystal field splitting corresponds to an insulating state with
suppressed orbital susceptibility, the intermediate value to a
metallic state and the largest splitting to a “band insulator”
(see Fig. 3).
havior for larger U , where the model is always insulating.
Our data for J = 0 exhibit a rapid variation of n1 with
∆. The slope is set by the inverse of the Kugel-Khomskii
superexchange ∼ t2/U ; thermal effects are unimportant
at βt = 50. For J > 0 and small ∆, the model is insu-
lating, with a vanishing orbital susceptibility, then (near
2J) makes a transition to the orbitally polarized
insulating phase with a differential susceptibility ∂n1/∂∆
determined by Kugel-Khomskii physics. Note that the
transition between the two insulators is sharp only at
T = 0; for T > 0 a rapid (but smooth) crossover occurs.
To gain insight into these phenomena, we look at the
contribution to the partition function from the differ-
ent eigenstates of the local Hamiltonian. Hloc has 16
eigenstates, which we number essentially as in Table II
of Ref. [17]. For the following discussion it is important to
note that |6〉, |7〉 and |8〉 are the three spin triplet states
(with energy U − 3J − 2µ), while |10〉 and |11〉 are lin-
ear combinations of the states |↑↓, 0〉 and |0,↑↓〉 with two
electrons in one orbital and none in the other. The latter
two states are coupled by the pair hopping and affected
by the crystal field splitting. Here, we choose them to
be eigenstates of Hloc corresponding to the eigenenergies
J2 + 4∆2 − 2µ: (1 + α2±)−1/2(| ↑↓, 0〉+ α±|0,↑↓〉),
α± = ±(
J2 + 4∆2 ∓ 2∆)/J . In particular, we choose
|10〉 to be the eigenstate with lower energy.
Figure 4 shows the weights of these states for the three
phases found at U/t = 6, J/U = 0.05 (see Fig. 3).
In the small-∆ phase, the triplet states are occupied,
with small excursions into states with occupancy 1 or
3. We therefore call this phase the triplet Mott insu-
lator. The triplet states of course have one electron in
each orbital and gain no energy from orbital polarization
(the remarkable fact is that this feature is preserved af-
0.55
0.65
0.75
3.5 4 4.5 5 5.5 6
orbital 1
orbital 2
FIG. 5: Filling n1(µ), n2(µ) for U/t = 4, J/U = 0.25 and
∆/t = 0.4. Full (open) symbols correspond to insulating
(metallic) solutions. At half-filling (µ/t = 3.5), the system
is in a triplet Mott insulating state, for 3.9 . µ/t . 4.6 in
an orbital selective Mott state, and for µ/t & 4.6 metallic in
both bands.
ter coupling to the lattice). In the metallic phase, a large
number of states is visited, while in the orbitally polar-
ized insulator, the dominant local state (whose weight
increases continuously with ∆) is a singlet (|10〉). The
triplet states are almost completely suppressed in the or-
bitally polarized phase. The large-U insulator-insulator
transition exhibits the same features, but without the
intermediate metallic phase, and is therefore also a tran-
sition between high and low spin states. Comparison of
the eigenenergies of the spin triplet states and |10〉 show
that these levels cross at ∆ =
2J . Thus, the transition
from triplet Mott insulator to orbitally polarized insula-
tor occurs at ∆c =
2J , consistent with our numerical
data. We also note that the wave-function of state |10〉
depends on the ratio J/∆, leading in the large-∆ limit
to n1(∆) ≈ (J/4∆)2. In the low spin phase, the or-
bital susceptibility has therefore two contributions: one
originating from Kugel-Khomskii physics and one of or-
der J2/∆3 from Hloc. The latter explains the roundings
seen in the right most curve of Fig. 3.
We briefly address the issue of the orbital selective
Mott transition, which provides a mechanism for local
moment formation in correlated materials, and has been
the subject of much recent debate [11]. Previous studies
focused on two-orbital models with different band-widths
and integer number of electrons. We find that in the
presence of a crystal field splitting, shifting the chemical
potential can drive the system into an orbital selective
Mott state, even if the band-widths are equal. Figure 5
shows the filling per spin in both orbitals as a function
of µ, for U/t = 4, J/U = 0.25 and ∆/t = 0.4. Doping
occurs first in one of the bands, leaving the other in a
Mott state with a magnetic moment. Further change of
the chemical potential drives the second band metallic.
In conclusion, we have shown that multiorbital impu-
rity models with realistic couplings can be efficiently sim-
ulated with the method of Ref. [17]. We have presented
numerical evidence, based on single site DMFT calcula-
tions, for the existence of two distinct Mott insulating
phases in a half-filled two-orbital model with Hund cou-
pling and crystal field splitting. At strong interactions
and J <
2∆, the system, in the T → 0 limit, is in
a phase characterized by a vanishing orbital susceptibil-
ity, and a spin 1 moment on each site. For J >
an orbitally polarized insulator is found. The exchange
terms promote insulating behavior at ∆ = 0 but can
stabilize a metallic phase at values of ∆ for which the
non-interacting model is a band-insulator.
It is interesting to compare our results to recent work
on the bilayer Hubbard model [19, 20]. The model
which these authors study is equivalent to our model with
U = U ′ = J , and ∆ replaced by the interlayer hopping.
In the low energy sector of this model, only four states
(essentially our three triplets and the pair hopping state
|10〉) are relevant, and what these authors describe as the
Mott insulator to band insulator crossover corresponds
to our transition (apparently sharp at T = 0) between
triplet Mott insulator and orbitally polarized insulator.
The existence of two distinct insulating phases raises
many interesting questions including the theory of an
insulator with strictly vanishing orbital susceptibility
(which should exhibit an orbital gauge symmetry) and
the nature and properties of the different metal-insulator
transitions. The physics near the “triple point” remains
to be studied. Our results away from half-filling suggest
that lightly doped La2NiO4 is in an orbitally selective
Mott phase.
The calculations have been performed on the Hreidar
beowulf cluster at ETH Zürich, using the ALPS-library
[21]. We thank M. Troyer for the generous allocation of
computer time, A. Georges and A. Poteryaev for stimu-
lating discussions and NSF-DMR-040135 for support.
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0704.0058 | Intelligent Life in Cosmology | arXiv:0704.0058v1 [physics.pop-ph] 31 Mar 2007
Intelligent Life in Cosmology
Frank J. Tipler
Department of Mathematics and Department of Physics
Tulane University
New Orleans, Louisiana 70118 USA
Abstract
I shall present three arguments for the proposition that intelligent life is very rare in the universe. First,
I shall summarize the consensus opinion of the founders of the Modern Synthesis (Simpson, Dobzhanski,
and Mayr) that the evolution of intelligent life is exceedingly improbable. Second, I shall develop the Fermi
Paradox: if they existed, theyd be here. Third, I shall show that if intelligent life were too common, it
would use up all available resources and die out. But I shall show that the quantum mechanical principle of
unitarity (actually a form of teleology!) requires intelligent life to survive to the end of time. Finally, I shall
argue that, if the universe is indeed accelerating, then survival to the end of time requires that intelligent
life, though rare, to have evolved several times in the visible universe. I shall argue that the acceleration is
a consequence of the excess of matter over antimatter in the universe. I shall suggest experiments to test
these claims.
Keywords: extraterrestrial prokaryotes, extraterrestrial intelligent life, cosmological acceleration, uni-
tarity, teleology, future of universe, closed universe, black hole evaporation, baryogenesis
1. Introduction
Martin Rees is fond of arguing, absence of evidence is not evidence of absence. How could anyone
disagree? But on the question of the existence of extraterrestrial intelligent life, we have an undeniable fact:
they aren’t here. That is, extraterrestrial intelligent beings are not obviously present on our planet, or in
our solar system. I think even Martin will agree with this! But I claim this fact allows us to conclude that
extraterrestrial intelligence (ETI) is absence from our Galaxy and from the Local Group of galaxies. In other
words, if they existed, theyd be here!
This argument has often been called the Fermi Paradox. I think it is analogous to Olbers’ Paradox in
cosmology, which uses an equally obvious fact, known to all of us — the fact that the sky is dark at night
— to conclude that the universe must have evolved to its present state. The universe cannot have been
the same as it appears now for all eternity. I shall outline in Section 2 the reasons that the absence of ETI
on Earth allows us to conclude that they don’t exist in our galactic neighborhood. I have developed this
argument is much more detail elsewhere, addressing all counter-arguments that have been proposed. So I
shall only outline my argument in Section 2. I shall also only outline the evolutionary argument against ETI
here. Mayr, Dobzhanski, Simpson and Ayala have defended this position at length over the past 40 years,
and I’m sure this argument is quite familiar to the readers of this journal. What I want to develop in this
paper is a new argument against the existence of ETI.
I shall call it the Limited Resources Argument. It is related to the Fermi Paradox in that it assumes
that an intelligent life form will inevitability expand off its planet of origin and once this expansion begins, it
will never stop. But if intelligent life were common in the cosmos, the expansion of technological civilization
would use up resources so fast that intelligent life would die out. If intelligent life is rare, the speed of light
barrier will prevent life from using up the resources too fast.
The immediate reaction to this argument is, so what if intelligent life uses up the resources too fast and
dies out? Do we have any reason for believing that intelligent has some guarantee for survival that other
species do not? Most species that have evolved are now extinct, and have left no descendants. Why should
Homo sapiens be any different? There is no evidence from evolutionary biology that intelligence should
survive indefinitely.
But there is evidence from physics for the importance of intelligence life in cosmology. Not of course in
the current phase of universal history, but instead near the end of the universe.
http://arXiv.org/abs/0704.0058v1
II. Why Intelligent Life Must Be Rare
A. The Improbable Evolution Argument
The argument against ETI that most readers of this journal will be familiar with goes back to Alfred
Russell Wallace, and has more recently been defended by such major evolutionists as George Gaylord Simp-
son, Theodosius Dobzhanski, and Ernst Mayr. These scientists point out that according to the Modern
Synthesis, evolution has no knowledge of goals. Instead, natural selection acts on random mutations, muta-
tions which never appear with the intent of achieving a goal in the distant future. There are an enormous
number of evolutionary pathways, and so few of these lead to intelligent life, that it is unlikely intelligent
life will appear more than once in the visible universe, which is the part of the universe within 13.7 billion
light years. The universe is observed to be 13.7 billion years old, and so we cannot see out a distance greater
than 13.7 billion light years, the distance light could have traveled in that time. (Actually, we can see out
a bit further than 13.7 billion light years because of the expansion of the universe, but let me ignore this
minor technicality.) Even if we were to assume that all the matter and energy in the visible universe were in
the form of Earthlike planets, there would be only (!) about 1028 Earthlike planets in the visible universe.
This number assumes that “earthlike” means only that the mass of the planet is greater than or equal to
the mass of the Earth. No assumption is made about the planet’s star, atmosphere, or orbital radius.
The well-known evolutionist Francisco Ayala has recently made this argument quantitative. He estimates
that the probability of an intelligent species evolving on an Earthlike planet upon which one-cell organisms
have appeared is less than 10 to the minus one million power! This number is so tiny that the evolution
of intelligent life is exceedingly unlikely to have occurred even once. Ayala’s number is not contradicted
by the fact that intelligent life exists on Earth. It is just exceedingly improbable that it exists anywhere
in the universe (at least if the universe is finite in spatial size, as I shall argue in Section IV that it is).
Ayala’s number depends on the assumption that gene changes upon which natural selection operates are
essentially random. Evolution has no foresight. Mayr has emphasized that intelligence on earth is limited to
the chordate lineage, so, he argues that if the chordates never appeared on Earth, neither would intelligence.
But chordates first evolved more than half a billion years ago. These animals did not know that they had to
evolve so that Homo sapiens would eventually appear. Natural selection can only operate during an animal’s
lifetime. It cannot select a genome with the intent of using the genome a billion yeas later.
There is an important caveat to this; a caveat first pointed out by Charles Darwin himself in the last
pages of his book The Variation of Animals and Plants under Domestication. Darwin noted that at the
ultimate level of physics, the universe is deterministic. This means that at the ultimate level, there are no
random events. In particular, the evolution of Homo sapiens was inevitable, determined by the initial state
of the universe and the universes initial conditions. “Random” variation does not mean uncaused. It just
means unpredictable for human beings. Therefore, at this ultimate physical level, Darwin claims that his
own theory is only an approximation. Darwin noted that the advance of science might enable us to obtain
enough information to predict these “random” variations. I shall argue below that this time has now come.
B. If They Existed, They’d Be Here
The argument against the existence of extraterrestrial intelligent life that I have developed in most
detail is sometimes called the Fermi Paradox: if they existed, they’d be here. The force of this argument is
not usually appreciated, because most people — and even most scientists (! — tacitly assume that any alien
civilization, no matter when they evolved or how long they have had advanced technology, will nevertheless
have essentially the technology of the late 20th century. The reason for this tacit assumption is the usual
human weakness: we have an unfortunate habit of trying to impose our current human perspectives on the
physical universe.
But let consider the consequences of only slightly more advanced computer technology than we now
have. According to most computer experts, within a century or so we should have computer programs which
have human level intelligence, computers which can run such programs and also make copies of themselves
and the programs. Imagine such a machine combined with our rocket technology into a space probe. Such a
space probe can reach the nearest star in 40,000 years. Once there in the nearest star system, the probe could
make several copies of itself, using the asteroid material which we now know is present in almost all star
systems, sending these daughter probes to further star systems, where the process would be repeated. Even
with our rocket technology, every star system in the entire Galaxy would have a probe within 100 million
years. With a more advanced rocket technology, a rocket technology which is even today been experimented
with, it should be possible to send a probe between the stars at 1/10 light speed. With such a speed, probes
would cover the entire galaxy within a few million years. And all for the cost of a single probe!
Almost any motivation we can imagine would lead an intelligent species with the technology to launch
that single probe. Suppose for example, ET wants to contact other intelligent life forms. Then rather than
send out radio signals, they should send out that single probe. With radio, one has to send out the signals
to many stars, over many thousands of years. (We would expect evolution to intelligence to require billions
of years, as it did on Earth.) But once the probe is launched, coverage of the entire galaxy is automatic.
Once in a target star system, the intelligent probe can contact any intelligent life forms that happen to have
evolved on any planet in the system. Or if no intelligent life is found, the probe can study the entire system
and transmit the results back to Earth. This on the spot investigation is obviously impossible if radio signals
are sent out instead of a space probe.
One might think an intelligent species would be reluctant to use probes because of the worry that
these machines would eventually escape from the control of the original transmitting species. But the same
objection can be made to sending out radio signals. It is impossible to predict what use a recipient species
would make of the information in the signal. Many scientists here on Earth have opposed the transmission
of signals, fearing that hostile aliens may use the signals to home in on our planet. The fear of losing control
of the probes — which, since these machines are rational beings, should be regarded as our mind children
— apply with equal force to our biological descendants. “No species now existing will transmit its unaltered
likeness to a distant futurity” was how Darwin put in the closing pages of Origin of Species. We do not know
whether they will be good or bad by our standards. We do know that in the far future they won’t be Homo
sapiens.
But in the long run, our descendants, whatever they look like, whether they are silicon machines or
the more familiar DNA devices, must leave the Earth if they are to survive. Within 6 billion years, the
Sun’s atmosphere will expand out and engulf the Earth, which will spiral into the Sun and be vaporized. A
similar fate is in store for any and all intelligent species that evolve on a water planet. Making the reasonable
Darwinian assumption that survival will be a central motivation of all intelligent species, all intelligent species
will eventually develop space travel, leave their planet, and colonize their own star system. The universe
is 13.7 billion years old, and most stars and their planets are billions of years older than our own. Thus,
whatever the probability intelligent life evolves on an earthlike planet on which one-cell organisms appear,
most intelligent species would be billions of years older than we are. They should have left their mother
planet billions of years ago. Once they leave their planet, nothing can stop their expansion into interstellar
space. If they existed, they would be here.
C. The Limited Resources Argument
Once an intelligent species begins its expansion into interstellar space, there is only the speed of light
barrier to stop the expansion. Furthermore, as Dyson has emphasized, intelligent life will eventually develop
the ability to convert any form of matter into living matter and life support devices. Given time, intelligent
life can take apart no only asteroids, but also entire Jupiter-sized planets and even stars. Thus a galaxy
which has been invaded (infected?) by a space travelling intelligent life form will start to disappear. This,
by the way, is yet another argument for human uniqueness in the visible universe. We have never observed
galaxies in the process of controlled disintegration. Intelligent life, in the long term, ought to appear as a
horde of locusts, devouring all matter in its domain. A galactic wide government cannot be set up to stop
such behavior because of the speed of light barrier, but even if it could be set up, it would have no choice
but to allow such behavior. Survival requires the conversion of matter into energy. Setting an ultimate limit
to how much matter can be so converted would merely doom life to extinction.
However, the speed of light barrier, which prevents a galactic scale government from being set up to
prevent life from devouring all matter, itself imposes a limitation on how fast life can use up resources. The
disc of our galaxy is some 100,000 light years across; we not use up the material resources of our galaxy in less
than 100,000 years. The Virgo cluster is some 60 million light years away. We cannot use up the resources
of the Virgo cluster in less than 60 million years. If the universe were closed and decelerating, a single
intelligent life form could not devour the entire universe until after the universe had begun to recollapse.
Actually the universe is currently accelerating. If this acceleration were to continue forever at its present
rate, our descendants could devour only the region currently within at most 10 billion light years. This limit
is imposed by the speed of light barrier modified by the universal acceleration.
But the more intelligent life there is in the universe, the more planets upon which intelligent life inde-
pendently evolves, the more rapidly resources will be used up. When all the material resources are used up,
intelligent life will die. The more common intelligent life is in the universe, the more rapidly it will become
extinct.
Conversely, if intelligent life is quite rare — a single intelligent species, if the universe were closed and
always decelerating — intelligent life would be forced by the laws of physics to use resources at just the right
rate to survive to the very end of time. And even more intelligent species could so survive if the universe
were to have a period of acceleration in its expansion phase, as the universe is indeed observed to have.
But why should the universe adjust the number of intelligent species so that the descendants of the
species would survive to the end of time? As Darwin pointed out in the closing pages of Origin of Species,
almost all species that have ever existed on Earth have died out, leaving no descendants. Why should an
intelligent life form have a survival probability utterly different from almost all other species? I claim that
intelligent life will survive until the end of time because the laws of physics require it. Or to put it another
ways, because such survival is one of the goals of the universe.
III. Unitarity is Teleology
Teleology has been completely rejected by evolutionary biologists. This rejection is unfortunate, because,
teleology is alive and well in physics, under the name of unitarity. Unitarity is an absolutely central postulate
of quantum mechanics, and it has many consequences. One of these consequences is the CPT theorem, which
implies that the g-factors of particles and antiparticles must be exactly equal. This equality (for electrons
and positrons) has been verified experimentally to 13 decimal places, the most precise experimental number
we have. Which is why very few physicists are willing to give up the postulate of unitarity! Furthermore,
unitarity is closely related to the law of conservation of energy, and a violation of unitarity has been shown
to result usually in the gigantic creation of energy out of nothing. One model (due to Leonard Susskin) of
unitarity violation had the implication that whenever a microwave oven was turned on, so much energy was
created that the Earth was blown apart. So physicists are very reluctant to abandon unitarity.
Unitarity is most often applied to what physicists call the S-matrix, which is the quantum mechanical
linear operator that transforms any state in the ultimate past to a unique state in the ultimate future. But
unitarity more generally applies to the time evolution operator, a linear operator that carries the quantum
state of the universe at any initial time uniquely into the quantum state of the universe at any chosen future
time. Uniquely is a key word. It means that unitarity is the quantum mechanical version of determinism.
Contrary to what is generally thought, determinism is alive and well in quantum mechanics. Determinism,
however, applies to wave functions (quantum states) rather than to individual particles. Alternatively, we
can say that determinism applies to coherent collections of worlds rather than to individuals. There is a sense,
which I won’t have room to discuss here, in which quantum mechanics is more deterministic than classical
mechanics, and that Schrödinger derived his famous equation by requiring that classical mechanics in it
most general expression (Hamilton-Jacobi theory) be deterministic. (See Tipler (2005) for the mathematical
details.)
But the usual past-to-future determinism is not the fundamental meaning of unitarity. What unitarity
really means is that the inverse of the time evolution operator exists, and is easily computed from the time
evolution operator itself by forming the time evolution operator’s hermitian conjugate. Any operator whose
inverse is obtained in this manner is said to be a unitary operator. But in the present context, the important
point is that the inverse of the time evolution operator exists. The inverse of any operator is an operator
that undoes the effect of the original operator. In the case of the time evolution operator, which generates
past-to-future evolution, the inverse operator generates future-to-past evolution. In other words, it carries
future quantum states uniquely into past quantum states. Therefore, unitarity tells us that any complete
statement of usual past-to-future causation is mathematically equivalent to some complete statement of
future-to-past causation. In more traditional language, a complete list of all efficient causes is equivalent to
some complete list of final causes. Teleology is reborn!
Nevertheless, the Second Law of Thermodynamics says that the complexity of the universe at the
microlevel is increasing with time. This means that it will usually be the case that past-to-future causation
will be the simpler explanation of the two causal languages. But this will not always be the case. We should
always remember that for physical reality the two causation languages are mathematically equivalent. It
might occasionally be the case that we humans can understand where the evolution of the universe is taking
us only by using future-to-past causation. That is, we can understand what is happening now only by
considering the ultimate goal of the universe.
To reject this possibility is a terrible mistake. Humans naturally think in terms of past-to-future
causation because our memories are designed (by the laws of physics) to work in this time direction. But the
universe is not similarly restricted. It is a mistake to impose human limitations on the physical universe. It
was a terrible mistake to require that solar system mechanics look simple in a geocentric frame of reference.
Let me now use this future-to-past causation to show that biological evolution cannot be completely
random. I shall now argue that the laws of physics require intelligent life to evolve somewhere, and survive
to the very end of time.
IV. Why Intelligent Life MUST Exist in the Far Future
The necessity of intelligent life in the far future is an automatic consequence of the laws of physics,
specifically quantum mechanics, general relativity, the Standard Model of particle physics, and most impor-
tantly, the Second Law of Thermodynamics. I shall show that the mutual consistency of these laws requires
three things. First, the universe must be closed (the universe’s spatial topology must be a three-sphere).
Second, life must survive to the very end of time. Third, the knowledge possessed by life must increase to
infinity as the end of time is approached. I do not assume life survives to the end of time. Life’s survival
follows from the laws of physics. If the laws of physics be for us, who can be against us?
But before I prove that the laws of physics require life to survive, let me first show that it is possible
for life to survive. To survive for infinite experiential time, life requires an unlimited supply of energy. That
is, the supply of available energy must diverge to infinity as the end of time is approached. Nevertheless,
conservation of energy requires the total energy of the universe to be constant. In fact, Roger Penrose has
shown that the total energy of any closed universe is ZERO! The total energy is zero now, was zero in the
past, and will be zero at all times in the future. One might wonder how this is possible. After all, we are
now receiving energy from the Sun, we are using food energy as we read this, and we can extract energy
from coal, oil, and uranium. Energy, in other words, seems to be non-zero.
However, the forms of energy just listed are not all the forms of energy in the universe. There is also
gravitational energy, which is negative. So if we were to add all the positive forms of energy — radiant
energy, the stored energy in coal, oil, and uranium, and most importantly, the mass-energy of matter —
to the negative gravitational energy, the sum is zero. This means that if we can make the gravitational
energy even more negative, the positive energy, that is, the energy available for life, necessarily increases,
even though the total energy in the universe stays zero. The key property of energy that must always be
kept in mind is that it transforms from one form to another. Once we realize that gravitational energy can
transformed into available energy, we understand where life can obtain the unlimited source of available
energy it needs for survival: life must make the total gravitational energy approach minus infinity.
Life can do this only if the universe is closed, and collapses to zero size as the end of time is approached.
Conversely, if the universe is closed and collapses to zero size, then the total gravitational energy goes to
minus infinity, since the gravitational energy of a system is inversely proportional the size of the system.
I have shown in my book (Tipler, 1994) that life can in fact extract unlimited available energy from the
collapse of the universe.
Now let me outline the proof of my three claims above. I can give here only a bare outline. For complete
details, the reader is referred to my book (Tipler, 1994) and to papers ((Tipler et al, 2000), and (Tipler
2001)) on arXiv, the physics preprint database (available on the Internet at http://arxiv.org/). Black holes
exist, but Hawking proved that were black holes to evaporate completely — as they necessarily would if the
universe were to expand forever — the black holes would violate unitarity, the fundamental law of quantum
mechanics which I described in the previous section. Hence the universe must eventually stop expanding,
collapse, and end in a final singularity. If this final singularity were to be accompanied by event horizons, then
the Bekenstein Bound (another law of quantum mechanics, basically the Heisenberg Uncertainty Principle
http://arxiv.org/
expressed in the language of information theory) would have the following effect. It would force that all
the microstate information in the universe to go to zero as the universe approaches the final singularity.
But the microstate information going to zero would imply that the entropy of the universe would have to
go to zero, and this would contradict the Second Law of Thermodynamics, which says that the entropy of
the universe can never decrease. But if event horizons do not exist, then the Bekenstein Bound allows the
information in the microstates to diverge to infinity as the final singularity is approached. Conversely, ONLY
if event horizons do not exist can quantum mechanics (the Bekenstein Bound) be consistent with the Second
Law of Thermodynamics. Therefore, event horizons cannot exist, and by Seifert’s Theorem (see (Tipler,
1994), p. 435) the non-existence of event horizons requires the universe to be spatially closed. In Penrose’s
c-boundary construction (Tipler, 1994), (Hawking and Ellis, 1973), a singularity without event horizons is a
single point. I call such a final singularity the OMEGA POINT. At a Windsor Castle conference, Martin Rees
objected that many physicists (in particular, himself) do not accept Hawking’s proof that unitarity would
be violated were a black hole to evaporate to completion. But most of the physicists who reject Hawking’s
argument nevertheless accept that there is nevertheless a Black Hole Information Problem: i.e., that we
must explain how the information that falls into a black hole gets out. Many solutions to the Information
Problem have been proposed but all of these solutions (except the one I shall advance) have one feature in
common. They all involve proposed new laws of physics. My proposal — that there are no event horizons
at all, hence no black hole event horizons, so ALL information at all events are accessible to all observers
in the far future — does NOT involve new physical laws. Only classical general relativity is used. I use
Hawkings unitarity argument only to infer the non-existence of event horizons. If we resolve the Black Hole
Information Problem by simply assuming the non-existence of event horizons, then I don’t need to use either
the Bekenstein Bound or the Second Law of Thermodynamics to infer the existence of the Omega Point,
or spatial closure. Resolving the Information Problem using known physics automatically yields no event
horizons and spatial closure for the universe.
If the universe were to evolve into an Omega Point type final singularity without life being present to
guide its evolution, then the non-existence of event horizons would mean that the universe would be evolving
into an infinitely improbable state. Such an evolution would contradict the Second Law of Thermodynamics,
which requires the universe to evolve from less probable to more probable states. On the other hand, if life
is present guiding the evolution of the universe into the final singularity, then the absence of event horizons
is actually the MOST probable state, because the absence of event horizons is exactly what life requires
in order to survive (details in my book (Tipler 1994)). In other words, the validity of the Second Law
of Thermodynamics REQUIRES life to be present all the way into the final singularity, and further, the
Second Law requires life to guide the universe in such a way as to eliminate the event horizons. Life is the
only process consistent with known physical law capable of eliminating event horizons without the universe
evolving into an infinitely improbable state. Exactly how life eliminates the event horizons is described in
my book (Tipler, 1994). Roughly speaking, life nudges the universe so as to allow light to circumnavigate
the universe first in one direction, and then another. This is done repeatedly, an infinite number of times.
There are thus an INFINITE number of circumnavigations of light before the Omega Point is reached. If
we were to regard a single circumnavigation as a single tick of the light clock there would be an infinite
amount of such time between now and the Omega Point. An even more physical time would be the number
of experiences which life has between now and the Omega Point. This “experiential time” — the time
experienced by life in the far future — is the most appropriate physical time to use near the Omega Point.
It is far more appropriate than the human based proper time we now use in our clocks.
V. Life in the Future of an Accelerating Universe
As anyone who has read the science columns of the newspapers over the past decade knows, the universe
is now accelerating. The most recent WMAP observations of the Cosmic Microwave Background Radiation
provide the strongest evidence for acceleration, but there are several independent lines of evidence that lead
to the conclusion that the universe is accelerating. The evidence is also strong that the mechanism for the
acceleration is due to a positive cosmological constant. If this acceleration were to continue forever, then as
Barrow and I showed in our book (Barrow and Tipler, 1986), intelligent life will eventually die out, and the
entire theory, which I described in section III, would be false. If intelligent life is to continue until the very
end of time — as it must if the laws of physics are to hold at all times — then the universe must eventually
stop accelerating, slow down until the expansion stops, and then recollapse to a final singularity. In this
section, I shall outline a mechanism which can cancel the acceleration. My proposal assumes the validity of
the Standard Model of particle physics, a theory which is so far supported by all experiments conducted to
date, and which provides only one mechanism for a universal acceleration.
The latest WMAP observations of the Cosmic Microwave Background Radiation (CMBR) have provided
the following facts. First, the universe is 13.7 billion years old. Second, in the present epoch, the density
parameters of the curvature, the ordinary matter, the dark matter, and the dark energy are respectively
Ωk << 0.01, Ωm = 0.04, ΩDM = 0.23, and ΩΛ = 0.73. Notice that the subscript on the dark energy is Λ. I
use this subscript to emphasize that the WMAP data indicate the dark energy looks observationally like the
effect of a positive cosmological constant, traditionally written Λ. Any correct cosmological theory must be
consistent with these observations.
The Standard Model, minimally coupled to gravity, necessarily has a positive cosmological constant. I
predicted in my book (Tipler, 1994) that this cosmological constant would cause the universe to undergo an
acceleration. I argued that this acceleration would occur in the collapsing phase of universal history. I did
not realize that an acceleration could also occur in the expanding phase. Though I should have, since the
Standard Model requires such an acceleration.
The Standard Model requires a positive cosmological constant to cancel the effect of the Higgs vacuum.
Recall that according to the Standard Model, the universe is permeated with a non-zero value of the Higgs
field, and it is this non-zero value that breaks the electroweak symmetry and gives mass to all the particles.
But this symmetry breaking is accomplished via the Higgs potential, which for constant Higgs field, acts
exactly a very strong negative cosmological constant. Initially, at the Big Bang singularity, the Higgs field,
and hence the Higgs potential, was zero. But zero is not the lowest value of the potential, so as the universe
expanded, the Higgs potential dropped to its lowest value, corresponding to a negative cosmological constant.
Now in special relativity, this negative constant can be re-normalized out of existence. Not so in general
relativity. Any constant in the matter Lagrangian multiples the invariant volume element, and is equivalent
to putting in a cosmological constant in the Lagrangian (Weinberg, 1988).
The value of the negative cosmological constant corresponding to the Higgs potential can be set by
experiment, and it is enormous: −1.0× 1026 gm/cm3, as compared to the energy density of the dark matter
and dark energy, only 10−29 gm/cm3. The only way to make the Standard Model consistent with general
relativity is to add a positive cosmological constant of the same magnitude to the Lagrangian. We would
expect the value of the added positive cosmological constant to precisely cancel the value of the Higgs
potential, when the Higgs is in its true ground state (the absolute lowest energy density of the potential).
But the Higgs field cannot presently be in its true ground state, for a very simple reason: there is more
matter than antimatter in the universe. The Standard Model has a mechanism of generating this observed
excess of matter over antimatter, but most cosmologists believe that this cannot be the main mechanism
to generate matter, because they think, incorrectly, that it will generate too many photons to baryons. I
have shown that this large number of photons to baryons is a consequence of imposing the wrong boundary
conditions in the very early universe. If the only boundary conditions consistent with the Bekenstein Bound
(a.k.a. quantum field theory) are imposed, the photon to baryon ratio turns out fine. The Standard Model
generation of matter works by electroweak vacuum tunneling. And if this tunneling yields an excess of matter
over antimatter, the Higgs field cannot be in its true vacuum. Thus the excess of matter over antimatter in
the universe ultimately causes the observed acceleration of the universe!
Conversely, if the excess of matter over antimatter were to disappear — if matter were converted into
energy via electroweak tunneling — and if this disappearance were to occur rapidly enough, then the Higgs
potential would fall toward its true ground state, the positive cosmological constant would be progressively
cancelled, and the universe would cease to accelerate. If he universe were a spatially a three-sphere — and
I have argued in the previous section that it is — then once the acceleration stops, the universe will expand
to a maximum size, and then recollapse into the final singularity.
Provided, of course, than a mechanism can be found to convert matter into energy via electrweak
quantum tunneling. The mechanism would have to be the inverse of the process that created the matter
excess in the early universe. But a large amount of matter was created in the early universe because the
gauge field energy density was enormous. The gauge field energy density is tiny today: 10−31 gm/cm3, and
getting smaller as the universe expands. If the acceleration is to stop, another mechanism must annihilate
the matter.
I claim that our future descendants will annihilate the matter. Once again, they will annihilate the
matter in order to survive. Survival requires energy. If baryon number is conserved, then only a small
fraction of the energy content of matter can be extracted. If hydrogen is converted into helium, as in the
Sun, only 0.7% of the mass of the hydrogen is converted into energy. But if our descendants use the inverse of
baryogenesis (the technical term for the process that generated matter in the early universe), ALL the energy
in matter can be extracted. I predict that in the future, a way will be found to use inverse baryogenesis,
our descendants will use this process as their main energy source, and as a consequence of using up there
matter resources, they will save both themselves, and the entire universe. Because if the acceleration can be
cancelled and universal recollapse induced, then the gravitational collapse energy can provide an unlimited
energy source, as I showed above.
But in an accelerating universe, life can only travel to the cosmological event horizon, which is about 10
billion light years away at the present time, given the observed value of the dark energy. (Actually, I should
call it the “pseudo event horizon”, since it would be a true event horizon only if life never stops the expansion,
and the Omega Point never develops. The Omega Point, recall, means that there are no event horizons.) But
quantum non-locality means that the quantum tunneling responsible for baryogenesis generates a uniform
density of baryons on large scales. (And since it is the creation of baryons that generate perturbations in the
CMBR, the perturbation spectrum must be scale invariant.) This means that the baryons have essentially
the same density on large scales everywhere in the universe. This means that the acceleration must be
universal. This means that if the universe is to recollapse, the baryons must be annihilated everywhere, even
at distances greater than 10 billion light years, where our descendants cannot travel, even were rockets based
on baryon annihilation to be constructed. Such rockets could approach light speed. I have shown (Tipler,
1994) that such rockets can travel cosmological distances, using the expansion of the universe itself to slow
down the rocket. Our descendants can reach the pseudo event horizon but no farther.
Thus the laws of physics require there to exist other intelligent species in the universe. Because of the
Limited Resources Argument, the different intelligent life forms must be rare, roughly one species per Hubble
volume. The nearest other intelligent life form must be roughly 10 billion light years away. But were we to
look for them, we would not see them, because at 10 billion light years, we would see their galaxy as it was
10 billion years ago, probably long before their planetary system formed.
VI. Conclusion and Proposed Experiments
But sufficiently advanced radio telescopes MIGHT be able to detect their future presence. In other
words, I shall now argue that there is a role for SETI! If we cannot detect alien civilizations, we might be
able to detect the one-cell organisms out of which they will eventually evolve. Provided that these organisms
already existed 10 billion years ago.
There is some evidence that the one-cell organism that were our own ancestors were around billions of
years before the Earth formed 4.6 billion years ago. William Schopf (1999, p. 77) has discovered structures
in the 3,465 ± 5 million-year-old Apex chert of Australia that closely resemble modern cyanobacteria. Schopf
identified these structures as fossil cyanobacteria, an identification that has been recently challenged. But I
shall assume that his identification is correct, so I can consider the consequences.
Now cyanobacteria are actually very sophisticated biochemical machines. If the fossil found by Schopf are
indeed cyanobacteria, then all the machinery of prokaryotes, including photosynthetic ability, must have been
present on Earth almost as soon as the Earth became capable of sustaining life, about 3.8 billion years ago.
Schopf himself remarks (1999, p. 98) that it seems extraordinary to suppose that this much sophistication
could have evolved in the geologically short period between the solidification of the Earth and the date of
the Apex fossils. I agree with Schopf. If indeed the Apex structures are fossils of cyanobacteria, then these
organisms cannot have evolved on Earth. They must have evolved their observed level of sophistication on
some other planet whose star long ago left the main sequence, and in the process, scattered the cyanobacteria
throughout interstellar space.
At the Windsor Castle conference, Paul Davies emphasized the consensus opinion that cyanobacteria
could survive a trip from one of Solar System’s planets, but because of the amount of radiation that they
would receive, they could not survive an interstellar journey. But the evidence Paul cited was theoretical,
rather than experimental. Cyanobacteria are capable of surviving nuclear explosions, and they have been
known to live inside nuclear reactors (Schopf, 1999, pp. 232-234). Given the ability of cyanobacteria to
survive radiation, their biochemical complexity, and the evidence that they appeared almost instantaneously
on Earth, I think that the preponderance of evidence says that cyanobacteria evolved billions of years before
the Earth formed, on a star that has long since disappeared.
This hypothesis has consequences. First, our interplanetary space probes should find cyanobacteria
wherever in the Solar System there is, or has been, liquid water. But if cyanobacteria have indeed been
dispersed throughout interstellar space billions of years before the Earth formed, we would expect to find
cyanobacteria, with the same DNA codons and cellular machinery, wherever there is liquid water in the
entire Galaxy. This hypothesis can be rigorously tested only with interstellar space probes. Incidentally,
notice that I’ve given in passing yet another reason why interstellar probes will eventually be sent out by
any intelligent species: to check how related life is in the Galaxy.
But if photosynthetic organisms have existed for billions of years before the Earth formed — for the
order of 10 billion years — and if our evolution is typical, we would expect intelligent life near the pseudo
event horizon to have evolved from organisms, some of which have photosynthetic ability, which existed on
liquid water planets 10 billion years ago. We would also expect there to have been time for the photosynthetic
organisms to convert some of these ancient planets’ atmospheres into oxygen atmospheres. This is what we
should search for in distant galaxies: the spectral lines of free oxygen. It has long been known that the
oxygen in Earth’s atmosphere can be seen at a distance of 10 light years by a one meter orbiting telescope.
A million-kilometer telescope would be able to see free oxygen lines in planetary atmospheres near the pseudo
event horizon. From the arguments above, some such atmospheres must exist.
A million-kilometer telescope is not going to be built in the immediate future. In the short run, I would
propose testing the hypothesis that the excess of matter over antimatter is responsible for the universal
acceleration, and that a special boundary condition on the fields of the Standard Model generate the excess
of matter over antimatter. This can be done rather easily, using a modification of the original equipment that
discovered the CMBR. I have shown in (Tipler, 2001, 2005) that if Standard Model physics is responsible
for both the dark matter and the dark energy, then the CMBR should not couple to right-handed electrons,
and this can be seen by sending the CMBR through filters consisting of poor conductors. Through such a
filter, the CMBR would be more penetrating than thermal radiation of the same temperature. I have shown
elsewhere that the same effect is visible in the Sunyaev-Zel-dovich effect (Tipler, 2005), and it is responsible
for the great penetrating power of ultrahigh energy cosmic rays (Tipler, 2001, 2005).
Two of the arguments against the existence of ETI have been around for a long time. The evolutionary
argument goes back to Alfred Wallace, with Darwin the co-discoverer of the principle of natural selection.
The Fermi Paradox goes back to Enrico Fermi. I’ve added a third, the “Limited Resources Argument” which
connects the rarity of intelligent life in the universe to the unlimited survival of intelligence in the far future.
But to appreciate the power of this argument, we must learn to give up anthropocentric ways of thinking.
We must abandon the (usually tacit) idea that our technology exhausts what is possible using the known
laws of physics. We must abandon the idea that the universe acts according to human thought patterns,
that causality works from past to future. We must abandon the idea that the universe evolves us as the
highest level of intelligence, and that all other intelligent species will be as limited in space as we are. Finally,
we must abandon the idea that there is a limit to what intelligence can accomplish, and that intelligence
will never play a role on the cosmological scale. Once we give up these human ways of thinking, we can
appreciate the true relation between intelligent life and the cosmos.
References
Barrow, J.D., Tipler, F. J. 1986 The Anthropic Cosmological Principle, Oxford University Press.
Hawking, S.W., Ellis, G.F.R. 1973 The Large-Scale Structure of Space-Time, Cambridge University Press.
Schopf, W. 1999 Cradle of Life: the Discovery of Earths Earliest Fossils, Princeton University Press.
Tipler, F. J. 1994 The Physics of Immortality, Doubleday.
Tipler, F. J., Graber, J., McGinley, M., Nichols-Barrer, J., Staecker 2000 gr-qc/0003082.
Tipler, F.J. 2001 astro-ph/0111520.
http://arXiv.org/abs/gr-qc/0003082
http://arXiv.org/abs/astro-ph/0111520
Tipler, F. J. 2005, Reports Prog. Phys. 68, pp. 897–964.
Weinberg, S. 1989, Rev. Mod. Phys., 61, pp. 1–22.
|
0704.0059 | The Mass and Radius of the Unseen M-Dwarf Companion in the Single-Lined
Eclipsing Binary HAT-TR-205-013 | THE MASS AND RADIUS OF THE UNSEEN M-DWARF
COMPANION IN THE SINGLE-LINED ECLIPSING BINARY
HAT-TR-205-013
Thomas G. Beatty1, José M. Fernández2,3, David W. Latham2, Gáspár Á.
Bakos2,4, Géza Kovács5, Robert W. Noyes2, Robert P. Stefanik2, Guillermo
Torres2, Mark E. Everett6, Carl W. Hergenrother7,2
[email protected]
ABSTRACT
We derive masses and radii for both components in the single-lined eclipsing
binary HAT-TR-205-013, which consists of a F7V primary and a late M-dwarf
secondary. The system’s period is short, P = 2.230736 ± 0.000010 days, with
an orbit indistinguishable from circular, e = 0.012 ± 0.021. We demonstrate
generally that the surface gravity of the secondary star in a single-lined binary
undergoing total eclipses can be derived from characteristics of the light curve
and spectroscopic orbit. This constrains the secondary to a unique line in the
mass-radius diagram with M/R2 = constant. For HAT-TR-205-013, we assume
the orbit has been tidally circularized, and that the primary’s rotation has been
synchronized and aligned with the orbital axis. Our observed line broadening,
Vrot sin irot = 28.9± 1.0 km s−1, gives a primary radius of RA = 1.28 ± 0.04 R⊙.
Our light curve analysis leads to the radius of the secondary, RB = 0.167± 0.006
R⊙, and the semimajor axis of the orbit, a = 7.54± 0.30 R⊙ = 0.0351± 0.0014
AU. Our single-lined spectroscopic orbit and the semimajor axis then yield the
individual masses, MB = 0.124 ± 0.010 M⊙ and MA = 1.04 ± 0.13 M⊙. Our
result for HAT-TR-205-013 B lies above the theoretical mass-radius models from
1Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138
2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
3Department of Astronomy, Pontificia Universidad Católica, Casilla 306, Santiago 22, Chile
4Hubble Fellow
5Konkoly Observatory, Budapest, P.O. Box 67, H-1125, Hungary
6Planetary Science Institute, 1700 East Fort Lowell Road, Suite 106, Tucson, AZ 85719
7Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85719
http://arxiv.org/abs/0704.0059v2
– 2 –
the Lyon group, consistent with results from double-lined eclipsing binaries. The
method we describe offers the opportunity to study the very low end of the stellar
mass-radius relation.
Subject headings: binaries: eclipsing — binaries: spectroscopic — stars: funda-
mental parameters — stars: low-mass — stars: rotation
1. INTRODUCTION
Solving for the masses and radii of stars has traditionally been accomplished through the
analysis of double-lined eclipsing binaries, where the light of both components is detected.
Masses and radii determined this way are fundamental and can be very accurate, because
they rely only on Newton’s laws and geometry for the analysis of the spectroscopic orbit and
light curve, and not on models of stellar structure and evolution. In particular, analysis of the
eclipse light curve yields the orbital inclination, and when combined with the double-lined
spectroscopic orbit, this yields individual masses for both stars.
There are dozens of double-lined eclipsing binaries with very accurate mass and ra-
dius determinations (e.g. see Andersen 1991, for a review), but only 10 M dwarfs (in
5 systems) with accuracies better than 3 percent. In order of increasing mass, the 5
systems are: CM Draconis (Lacy 1977; Metcalfe et al. 1996), CU Cancri (Ribas 2003),
NSVS01031772 (López-Morales et al. 2006), YY Geminorum (Torres & Ribas 2002), and
GU Boötis (López-Morales & Ribas 2005). Figure 1 shows the observational results for these
10 M dwarfs on the mass-radius diagram, along with the predicted theoretical models from
Baraffe et al. (1998). All of the observed radii are larger than the theoretical predictions,
typically by 5 to 10 percent. Another striking feature of Figure 1 is the lack of accurate
mass-radius determinations in the lower half of the M-dwarf mass range, from CM Dra B
(0.214 M⊙), all the way down to the substellar limit near 0.075 M⊙. Recently, however, the
growing number of short-period single-lined eclipsing binaries with F- and G-star primaries
and M-dwarf secondaries identified by wide-angle photometric surveys for transiting planet
promises to provide a way to fill in this gap in the mass-radius diagram (Bouchy et al. 2005;
Pont et al. 2005a,b, 2006).
One approach to using a single-lined eclipsing binary to solve for the mass and radius of
the unseen companion is to use stellar models together with spectroscopic and photometric
observations of the primary to estimate its mass and radius. Then the radius ratio from the
light curve yields the radius of the secondary, and the mass function from the spectroscopic
orbit with the orbital inclination from the light curve can be combined to yield the mass of
– 3 –
the secondary. However, this approach is not fundamental, because it relies on stellar models
to characterize the primary. Tests of the mass-radius relation with the low-mass secondaries
in such systems are no better than the validity of the models for the primaries. In particular,
the theoretical isochrones depend on metallicity, and significant errors can result if the wrong
metallicity is adopted.
An alternative, and more fundamental, approach relies on the expectation that short-
period binaries whose orbits have been circularized by tidal mechanisms must also have the
axial rotation of both stars synchronized to the orbital period, and both rotational axes
aligned with the normal to the orbit, so that the two inclinations are equal, irot = iorb (e.g.
see Hut 1981; Zahn 1989). In this case, a measurement of the spectral line broadening due to
rotation, Vrot sin irot, combined with a value for iorb from an analysis of the light curve, yields
the actual equatorial rotational velocity, Vrot. If the rotation is synchronized with the orbit,
then Prot = Porb, and the radius of the primary can be solved. The key to this approach
is the ability to derive accurate values for the rotational broadening of the spectral lines,
because the primary radius can be no more accurate than the value derived for Vrot. The
radius of the primary then sets the scale for the rest of the system, yielding the radius of the
secondary and the semi-major axis of the orbit in actual length units, as well as the orbital
inclination. Then Newton’s revised version of Kepler’s Third Law can be used to derive the
sum of the masses, and the mass function from the single-lined spectroscopic orbit allows the
individual masses to be determined. This approach depends on the predictions from stellar
models only in minor ways: limb darkening coefficients are needed for the detailed analysis
of the eclipse light curve, and the rotational broadening that is derived from the observed
spectra can depend weakly on the metallicity that is adopted.
The advantage of using single-lined eclipsing binaries is that it dramatically increases
the number of low-mass stars whose masses and radii can be determined. Indeed, over the
past 30 years only seven double-lined eclipsing systems composed of low-mass stars have been
identified (the aforementioned CM Dra, CU Cnc, NSVS01031772, YY Gem, and GU Boo,
plus OGLE-BW3-V38 (Maceroni & Montalbán 2004), and TrES-Her0-07621 (Creevey et al.
2005)). Meanwhile, more than 75 single-lined eclipsing binaries with M-dwarf secondaries
have been discovered by wide-angle photometric transiting planet surveys such as Vulcan,
TrES, and HAT in just the last five years (Latham 2007).
Increasing the number of low-mass stars with fundamental determinations of masses and
radii is worthwhile because of the insights these systems can yield into stellar structure. Low-
mass stars near the hydrogen-burning limit are cool enough that their interior temperatures
are on the order of the electron Fermi temperature (Chabrier & Baraffe 1997), causing parts
of the stellar interior to be in the state of a partially degenerate electron gas, which means
– 4 –
a classical Maxwell-Boltzmann description of the interior does not apply. Furthermore, the
electron number density is such that the mean inter-ionic distance is itself on the order of the
Thomas-Fermi screening length, meaning that the electron gas is polarized by the external-
ionic field (Chabrier & Baraffe 1997). Add to this the further complexity that magnetic
fields may play a role in the inner workings of low mass stars (Mullan & MacDonald 2001),
and it becomes clear that any attempt to describe the interior of low mass stars must take
into account the physics of both magnetic fields and partially degenerate polarized plasmas.
Not only is it difficult to model the interiors of low-mass stars, but the usual grey
model atmospheres are no longer applicable. The relatively low temperatures of the stellar
atmospheres allow for the recombination of molecular hydrogen and other molecules, such
as TiO. Therefore, accurate non-grey model atmospheres that take into account the effects
of molecules must be derived and matched to the interior models (Baraffe et al. 2002).
Thus stars near the bottom of the main sequence pose a challenge to dense matter physi-
cists and stellar astronomers. One of the few methods of testing models for low-mass stars
is by confronting theoretical predictions of the mass-radius relation with observations. Up
until now there has been little data with which to constrain the possible theories. Measuring
the masses and radii of M-dwarfs in single-lined eclipsing binaries therefore provides a new
opportunity to test the mass-radius relation near the bottom of the main sequence.
In this paper, we determine masses and radii for the components of HAT-TR-205-
013, an eclipsing single-lined binary identified by the HATnet (Hungarian-made Automated
Telescope Network, see Bakos et al. (2004)). The unseen M-dwarf secondary has a mass and
radius of 0.124 M⊙ and 0.167 R⊙, with 1-σ errors of 9 and 4 percent, respectively. This
result places the M dwarf about 10 percent above the radius predicted by the Baraffe et al.
(1998) models. In this first paper we describe our analysis techniques in detail. In future
papers we will present the results for additional M dwarf secondaries.
2. OBSERVATIONS AND DATA REDUCTION
2.1. HAT Photometry
The HATnet project,1 initiated in 2003 by G. Á. B, is a wide-field survey that aims for
the discovery of transiting planets around bright stars. It currently comprises 6 small wide-
field automated telescopes, each of which monitors 8◦ × 8◦ of sky, typically containing 5000
1www.hatnet.hu
– 5 –
stars bright enough to permit detection of planetary transits via the typical 1% photometric
dips they induce on their parent stars. The instruments are deployed in a two-station,
longitude-distributed network, with four telescopes at the Fred L. Whipple Observatory in
Arizona, and two telescopes at the Submillimeter Array at Hawaii. For a more detailed
description of HAT’s instrumentation, observations, and data flow, see Bakos et al. (2002,
2004).
The HAT-TR-205-013 system lies in HATnet survey field G205, centered at α = 22h55m,
and δ = +37◦30′. 3357 observations were made of this field by the HATnet telescopes between
5 October 2003 and 30 January 2004. Exposure times were 5 min at a cadence of 5.5 min.
Light curves were derived by aperture photometry for the 6400 stars in G205 bright enough
to yield photometric precision of better than 2% (reaching 0.3% in some cases). In deriving
the light curves, we made use of the Trend Filtering Algorithm (Kovács, Bakos, & Noyes
2005) to correct for spurious trends in the data. We then searched all the light curves for
characteristic transit signals, using the Box-fitting Least Squares (Kovács, Zucker, & Mazeh
2002) algorithm, which searches for box-shaped dips in the parameter space of frequency,
transit duration, and phase of ingress. Candidate transit signals with the highest detection
significance were then examined individually, to isolate those with the best combination of
stellar type (preferably main-sequence stars of spectral type mid-F or later), and depth,
shape, and duration of transit. One of these was HAT-TR-205-013, for which we identified
a prominent periodicity with period 2.2307 days and transit depth of 0.02 mag. Figure 2
shows the phase-folded and flux-normalized data for HAT-TR-205-013.
Following our determination of HAT-TR-205-013 as a potential planetary system, we
identified the star in the 2MASS catalog as 2MASS 23080834+3338039, which yielded J
and K magnitudes of J = 9.691 and K = 9.408. We also found HAT-TR-205-013 in the
Tycho catalog, as TYC 2755-36-1, with magnitudes BT = 11.355 and VT = 10.729. We
then scheduled HAT-TR-205-013 for follow-up spectroscopic observations to determine if
the photometric dip was caused by a stellar companion.
2.2. Follow-up Spectroscopy
Our usual strategy for following up transiting-planet candidates identified by wide-field
photometric surveys is to start with an initial spectroscopic reconnaissance, to see if there
is evidence for a stellar companion that is responsible for the observed light curve. We have
used the CfA Digital Speedometers (Latham 1992) on the 1.5-m Wyeth Reflector at the
Oak Ridge Observatory in the Town of Harvard, Massachusetts and on the 1.5-m Tillinghast
Reflector at the Fred L. Whipple Observatory on Mount Hopkins, Arizona to obtain single-
– 6 –
order echelle spectra in a wavelength window of 45 Å centered at 5187 Å with a resolution of
8.5 km s−1 and a typical signal-to-noise ratio per resolution element of 15 to 20. For slowly
rotating solar-type stars these spectra deliver radial velocities accurate to about 0.5 km s−1,
which is sufficient to detect orbital motion due to companions with masses down to about 5
or 10 MJ for orbital periods of a few days.
Spectroscopic follow-up observations of transiting-planet candidates are easier to sched-
ule than photometric observations, because the radial velocity varies throughout the entire
orbital phase, while the transit light curve has a duty cycle of only a few percent, e.g. 3
hours out of 3 days. Spectroscopy has the additional advantage over photometry that we
can use our spectra to classify the parent star, deriving effective temperature, rotational
velocity, and surface gravity by correlating the observed spectra with a library of synthetic
spectra. This information is often useful in rejecting host stars that are too hot, or are
rotating too rapidly, or are evolved (in which case we assume the giant must be the bright
star in a blended triple). Thus at CfA we usually start with a spectroscopic reconnaissance
before attempting follow-up photometry or very precise radial-velocity observations (e.g. see
O’Donovan et al. 2007).
Starting in 1999 with transiting-planet candidates provided by the Vulcan team (Borucki et al.
2001), we quickly learned that the vast majority of the planet candidates were actually eclips-
ing binaries masquerading as transiting planets (Latham 2003). One of the most common
imposters was an F- or G-star primary eclipsed by a late M-dwarf secondary. Such systems
produce light curves similar to planets transiting solar-type dwarfs, because M dwarfs near
the bottom of the main sequence have roughly the same size as giant planets. Nevertheless,
they are easy to distinguish from planets, because their masses are two orders of magnitude
larger, and the reflex orbital motion they induce in their parent stars has an amplitude of
at least several km s−1.
We determined the rotational and radial velocities of HAT-TR-205-013 by cross corre-
lation of the observed spectra against templates drawn from a library of synthetic spectra
calculated by Jon Morse for a grid of Kurucz (1992) stellar atmospheres. The library grid
has a spacing of 250 K in effective temperature, Teff ; 0.5 in log surface gravity, log g; and
0.5 in log metallicity relative to the sun, [Fe/H]. For the final radial-velocity determinations
we adopted the template with the Teff , log g, and Vrot values that gave the highest value for
the peak of the correlation coefficient, averaged over all the observed spectra, assuming solar
metallicity. For the correlation analysis we used xcsao (Kurtz & Mink 1998) running inside
the IRAF2 environment.
2IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Ob-
– 7 –
Our very first spectrum of HAT-TR-205-013 revealed that the spectral lines of the
primary were broadened by about 30 km s−1 of rotation. In our experience this is a strong
suggestion that the companion is a star with enough mass to synchronize the rotation of
the primary with the orbital period. The second exposure showed that the velocity varied
by several km s−1. Additional spectra soon revealed a spectroscopic orbit with a period
that matched the photometric period and an orbital semi-amplitude, K, of about 20 km s−1
implying that the secondary was a small M dwarf. Altogether we accumulated 23 spectra
yielding the radial velocities reported in Table 1, which were calculated using a template
with Teff = 6250 K, log g = 4.0, [Fe/H]=0.0, and Vrot = 30 kms
−1. The parameters of our
orbital solution for HAT-TR-205-013 are reported in Table 2 and the corresponding velocity
curve and observed velocities are plotted in Figure 3.
Because the value of the rotational velocity is critical for our determination of the radius
of the primary star, we evaluated both the internal precision of our determinations and
possible systematic errors resulting from uncertainties in the surface gravity and metallicity
of the primary star. Our final results for the mass and radius of the primary star imply a
surface gravity log g = 4.24, halfway between the nearest templates in our library of synthetic
spectra. Therefore we evaluated the rotational velocity at both log g = 4.0 and 4.5, and also
at two metallicities, [Fe/H] = 0.0 and −0.5. At each of the four combinations of log g and
[Fe/H] we ran correlations over a wide range of temperatures, 5500 to 7250 K, and rotational
velocities, 10 to 50 km s−1. At each gravity and metallicity we interpolated to find the
temperature and rotation that gave the highest value of the correlation coefficient averaged
over all the observed spectra weighted by the photon statistics. For these experiments we
did not use two of our observed spectra that had much lower exposure levels than the other
21 observations. The results are reported in Table 3. They show that the accuracy of our
rotational-velocity determination at a given log g and [Fe/H] is not seriously limited by the
scatter from the individual exposures, despite the relatively low signal-to-noise ratio of our
observed spectra. The uncertainty in the mean rotational velocity at each log g and [Fe/H]
due to internal scatter is less than 0.3 km s−1.
The results reported in Table 3 also show that the systematic errors due to uncertainties
in the gravity and metallicity are not serious. For example, the dependence of rotational
velocity on metallicity is rather weak, only 0.2 km s−1 for a change of 0.5 in [Fe/H]. On the
other hand, the dependence on gravity is rather strong, about 1.0 km s−1 for a change of 0.5
in log g. Fortunately, the uncertainty in the actual gravity of the primary star is quite small,
less than 0.2 in log g, so the rotational velocity interpolated between the two gravities should
servatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under
contract with the National Science Foundation.
– 8 –
be good to perhaps 0.2 km s−1. Unfortunately we do not yet have a direct measure of the
metallicity of HAT-TR-205-013, but the fact that the templates with [Fe/H] = 0.0 give a
better match to the observed spectra than the templates with [Fe/H] =−0.5 suggests that the
metallicity is probably within 0.5 dex of solar. In the absence of an accurate metallicity we
adopt a value of [Fe/H] = −0.2, which is typical of the solar neighborhood (Nordström et al.
2004). The temperature corresponding to this metallicity that we derive from our spectra is
Teff = 6295 K, but with a systematic error that could exceed 200 K because of the degeneracy
between temperature and metallicity in the analysis of our spectra. The alternative approach
of deriving a temperature from photometric indices is hampered by the lack of accurate
photometry and the possibility that there may be significant reddening. The distance that
we derive from the Stefan-Boltzmann law using R = 1.27 ± 0.04 R⊙, Teff = 6295 ± 200 K,
VT = 10.729 ± 0.048 (which corresponds to V = 10.67 ± 0.05), and bolometric correction
= −0.011 (Flower 1996) is 232 ± 18 pc. From our spectroscopic temperature and log g, we
estimate the spectral type of the primary to be F7V (Cox 2000).
Thus our experiments with different gravities and metallicities suggest that the system-
atic errors due to uncertainties in the template parameters may be less than 0.4 km s−1.
When combined in quadrature with the internal precision estimate of 0.3 km s−1, this sug-
gests that the total error in our final interpolated rotational velocity could be as small as 0.5
km s−1. Previous experience however, would indicate that the actual uncertainty is higher
than this internal estimate. Therefore, to be conservative we adopt an uncertainty of 1.0
km s−1 and use Vrot sin irot = 28.9 ± 1.0 km s−1 for our determination of the radius of the
primary and subsequent analysis.
One could also determine the value of Vrot sin irot by spectroscopically measuring the
amplitude of the Rossiter-McLaughlin Effect during eclipse (Gaudi & Winn 2007). In the
case of HAT-TR-205-013 the amplitude of this effect would be about 0.5 km s−1, too small to
be measured with the CfA Digital Speedometers, but potentially observable with an accuracy
of a few percent by instruments such as HIRES on Keck 1.
Our approach relies on the assumption that the stellar rotation has been synchronized to
the orbital period. One way to test this assumption might be to use very precise photometry
to derive a rotational period for the star. We did look for sinusoidal variations in the HAT
photometry but did not find anything significant near the orbital period. We did detect a
marginally significant variation with half the orbital period and amplitude of about 1 mmag,
which is consistent with the the expected ellipsoidal deformation of the primary star. A
second, more global test would be to use stellar models to estimate the mass and radius of
the primary, but a prerequisite for such an analysis would be accurate determinations of the
temperature and metallicity, presumably from high-quality spectra, and such results are not
– 9 –
yet available.
2.3. Follow-up KeplerCam Photometry
To provide a high-quality light curve for the analysis of the primary eclipse of HAT-
TR-205-013 we used KeplerCam on the 1.2-m telescope at the Fred L. Whipple Observatory
on Mount Hopkins, Arizona. KeplerCam utilizes a monolithic 4K×4K Fairchild 486 CCD
that provides a 23′ field and a pixel size of 0.34′′. We used the predicted eclipse times from
our spectroscopic orbit to schedule observations on the night of 22-23 October 2005. We
successfully observed a full eclipse, alternating between the Sloan g and i bands.
The seeing was approximately 2′′ FWHM throughout the night, but we deliberately
defocused the telescope to get images with 3′′ FWHM in order to keep the peak counts
in the images well below saturation. Over the course of the observations the focus of the
telescope, which depends weakly on temperature, was adjusted three times to keep the image
size near 3′′ FWHM. Because we used automatic guiding, the centroid of the images moved
less than 3 pixels over the duration of the observations. For readout we binned the pixels
2 × 2, which gave a total readout time including overhead of 12 seconds. Exposure times
were 30 seconds for the g band and 10 seconds for the i band. The images were obtained in
sequences of 3 g-band exposures followed by 6 i-band exposures. All told, we were able to
collect a total of 297 images in the g band and 588 in the i band. Some thin cirrus clouds
were present at the beginning of the night, with no noticeable degradation of the light curves.
A quarter Moon rose during the observations after the end of egress, and this contributed
to a slight increase in photometric scatter after egress.
All the images were reduced by applying an overscan correction and then subtracting
the two-dimensional residual bias pattern. After correcting for shutter effects, we flattened
each image using a normalized set of combined twilight images. To produce the light curve,
we used the first image from each filter in an observing sequence as our astrometric reference
for identifying the same stars in subsequent images. We then determined the relative shift
between images to relocate each star in the following images. We measured the flux of each
star in a 6.7′′ circular aperture around the position derived from the astrometric fit using
daophot/phot within IRAF (Tody 1986, 1993). We estimated the sky in an annulus around
each star with inner and outer radii of 9.4′′ and 13.4′′ using the sigma-rejection mode.
We iteratively selected comparison stars by removing any that showed unusual noise or
variability in their differential light curves, obtaining 21 comparison stars for the g band and
37 for the i band. Based on a weighted mean flux from the comparison stars, we calculated
– 10 –
an extinction correction, and then applied this to each comparison star. The typical rms
residual for both bands was 1.5 mmag, a value that increased slightly toward the end of the
night due to the increased sky brightness from the rising Moon. The main contributor of noise
to the photometry from individual images was atmospheric scintillation, which accounted
for over 65% of our calculated rms (Young 1967). The individual photometric measurements
in the g and i bands are reported in Tables 4 and 5, respectively.
3. LIGHT-CURVE ANALYSIS
Our KeplerCam light curves for HAT-TR-205-013 give good coverage of the eclipse
centered at HJD 2453666.747±0.001, with more than two hours of coverage both before the
start of ingress and after the end of egress. The eclipse itself lasts about 3 hours between
first and fourth contact and is about 2 percent deep, in agreement with the discovery light
curve from HAT. Our KeplerCam light curves clearly show the effects of limb darkening on
eclipse shape as a function of wavelength: the i-band light curve is slightly shallower and
possesses a flatter bottom than the g-band curve. The portions of the light curves before
and after eclipse showed a slight drift, which we removed with a linear fit. The resulting
light curves are plotted in Figure 4.
From our fits to the KeplerCam light curves we derive transit centers of 2453666.7465±
0.0005 and 2453666.7473 ± 0.0005 in the g and i bands, respectively. When we com-
bine these transit times with the results from the HAT photometry, we get the ephemeris
2453666.74748± 0.00018 +N × 2.230736± 0.000010.
Knowing the duration and depth of the eclipse, together with the orbital period, we
were able to derive rough values for the ratio of semimajor axis and radius of the secondary
to the radius of the primary star using the relations
δ (2)
where a is the orbital semimajor axis, RA and RB are the primary and secondary stars’ radii,
P is the orbital period, ∆ttr is the transit length, and δ is the transit depth in relative flux.
For our values of P ∼ 2.23 days, ∆ttr ∼ 2.5 hours and δ ∼ 0.02, we obtained a/RA ∼ 5.93
and RB/RA ∼ 0.133 from this first approximation.
We then built a two dimensional grid of light-curve fits in a/RA and RB/RA, centered
on the rough values obtained above. The impact parameter b was introduced as a third
– 11 –
dimension in the grid, varying from b = 0 (central transit) to b = 1 + RB/RA (grazing
transit). The grid steps were 0.002 in a/RA, 0.00001 in RB/RA, and 0.001 in b. We generated
synthetic light-curves for each combination of parameters using the routines provided by
Mandel & Agol (2002), together with the quadratic limb-darkening coefficients derived by
Claret (2004), using the temperature and surface gravity for the primary we had derived
spectroscopically. We adopted solar values for the metallicity and surface turbulence of the
primary star. The exact coefficients we used were u1 = 0.4238 and u2 = 0.3250 in the g,
and u1 = 0.1814 and u2 = 0.3723 in the i band. Uncertainties in our values of the limb-
darkening coefficients, largely a result of the uncertainty in the spectroscopic temperature
measurement, had a negligible effect on our final results.
To identify the synthetic light curve that gave the best fit to the actual data, we looked
for the fit with the smallest value of χ2, inspecting the χ2 contours to ensure that we had
found the global minimum. We also inspected plots of the fits to the data as a qualitative
check. We applied this procedure to both the g and i light curves. The results are summarized
in Table 6.
To calculate the uncertainties for our values of a/RA, RB/RA, and b, we performed
additional fits wherein we fixed one of the parameters at a value differing slightly from the
best-fit value, and fit for the other two parameters. We changed this fixed parameter until
the eventual fit achieved a χ2 that was 1-σ away from our best fit parameters’ χ2. We
then used the difference between this 1-σ value and our best fit value as the uncertainty
for that particular parameter. We examined the correlation between the three different
parameters by making a final fit to the data using the Levenberg-Marquardt method – a
combination of the Inverse-Hessian and Steepest Descent methods (Press et al. 1992). This
allowed us to compute the covariance matrix for the three fit parameters, and thus the
correlation coefficients for the parameters. For all three, the correlation with any of the
other two was always smaller than 0.5 in both bands. While this is not negligible, we found
that the contribution to the total estimated error was minor. Of course, the Levenberg-
Marquardt method can also be used to calculate the uncertainties in a/RA, RB/RA, and b.
However, the uncertainties that we derived from our grid analysis were substantially larger.
To be conservative, we adopted the grid uncertainties instead of the Levenberg-Marquardt
uncertainties. Figure 5 shows contour plots of χ2 for our fits. Table 6 reports the final fitting
results and errors, and Table 7 lists the correlation coefficients. The KeplerCam data and
best-fit light curves are shown in Figure 4. Simultaneous fitting of the light curves in both
bands produced fits with inferior χ2 statistics, a result of the slight difference in RB/RA in
the two bands. We therefore use a simple average of the light curve results as our adopted
values for the parameters in Table 6.
– 12 –
3.1. MASSES AND RADII FOR HAT-TR-205-013
Using the values that we have measured from the eclipse light curves, as well as the
observed spectroscopic orbit parameters, we may restrict the location of HAT-TR-205-013
A and B on the mass-radius diagram to unique curves described completely by these observ-
ables. To emphasize this, we have written the observable values in brackets in the following
derivation.
To begin, we use Newton’s revised version of Kepler’s Third Law and the spectroscopic
mass function from the orbital solution:
[P ]2 =
G (MA +MB)
a3 (3)
sin iorb
(MA +MB)
1− [e]2 (4)
where [KA] is the semi-amplitude of the spectroscopic orbit for the primary in km s
−1, [e]
the eccentricity of the orbit, and G is the Gravitational Constant. With these two equations,
the two unknowns (MA and MB) may be found:
[P ]2
[P ] [KA]
1− [e]2
2πa sin iorb
sin iorb
1− [e]2 (6)
Both the orbital inclination and the semimajor axis can be expressed in terms of the
observables [a/RA], [RB/RA] and [b], and the unknown stellar radii RA and RB:
By taking our value for the impact parameter [b], which measures the projected separa-
tion of the primary and secondary stars at the mid-point of the eclipse, the orbital inclination
can be expressed in terms of the observables [a/RA] and [b]:
[b] = [a/RA] cos iorb (7)
sin iorb =
[a/RA]2
Similarly, the semimajor axis may be written in terms of the observables [a/RA], [RB/RA]
and [b], and the unknown stellar radii RA and RB:
a = [a/RA] RA =
[a/RA]
[RB/RA]
RB (9)
– 13 –
Substituting these values into Eq.(5) and (6) gives expressions for the mass of each
component as functions of only the observables and the respective stellar radii:
G[P ]2
[a/RA]
[P ] [KA]
1− [e]2
2π(1− [b]2/[a/RA]2)1/2 [a/RA] RA
R3A (10)
G[P ]
[a/RA]
[RB/RA]
1− [e]2
(1− [b]2/[a/RA]2)
R2B (11)
Therefore, solely from what we are able to measure using the eclipse light curves and
spectroscopy, we may confine HAT-TR-205-013 B to a single curve on the mass-radius dia-
gram that goes as MB/R
B = constant. This constant is directly proportional to the surface
gravity of the object, which has been noted previously by Southworth et al. (2004). In our
case, the quality of our photometry allows us to measure this constant extremely accurately;
the uncertainty region around this surface gravity line is on the order of the line width itself
when it is plotted.
Through the assumed synchronization of the primary’s rotation to the orbital period we
may locate HAT-TR-205-013 B on this gravity determination curve, but it is important to
note that the line itself is defined without having made any assumptions about the system.
Indeed, it is possible to calculate a similar gravity curve for any eclipsing system, or for a
system containing a transiting planet. All that is required is a good-quality light curve and
spectroscopic orbit. This has previously been done for the case of a transiting planet by
Winn et al. (2006).
To place HAT-TR-205-013 on the mass-radius diagram more specifically, we note that
that the measured eccentricity of our spectroscopic orbit for HAT-TR-205-013 is indistin-
guishable from circular, and therefore, for the reasons described in the Introduction, we
assume that the spin axes of both stars have been aligned with the orbital normal and that
the rotation of both stars has been synchronized to the orbital period. This allows us to use
the observed rotational line broadening of the primary to solve for the radius of the primary
in linear units, which in turn allows us to convert the orbital size and secondary radius into
linear units from the values of [a/RA] and [RB/RA] derived from the light curves.
Using the assumption of synchronization, and that iorb = irot, we see by inspection that
[Vrot sin irot]
sin iorb
[Vrot sin irot]
sin iorb
[RB/RA] (13)
where [Vrot sin irot] is the projected rotational broadening of the primary derived from its
observed spectra. We may now substitute in Eq.(8) for sin iorb to get both radii in terms of
– 14 –
our observables:
2π (1− [b]2/[a/RA]2)
[Vrot sin irot] (14)
2π (1− [b]2/[a/RA]2)1/2
[RB/RA] [Vrot sin irot] (15)
By combining these two statements with Eq.(9) and (10), we arrive at expressions for
the masses of each component in terms of just the observable quantities:
[a/RA]
(1− [b]2/[a/RA]2)3/2
1− [e]2
[a/RA][Vrot sin irot]
[Vrot sin irot]
3 (16)
[a/RA]
(1− [b]2/[a/RA]2)3/2
1− [e]2 [Vrot sin irot]2 (17)
The results for the masses and radii for both components of HAT-TR-205-013 are pre-
sented in Table 8. The errors were estimated using Monte-Carlo simulations and were com-
pared with the results of formal error propagation, including the correlation coefficients
derived from the light-curve fits. Both approaches delivered similar results. The mass and
radius obtained for the primary star are essentially the same for both the g and i light curves,
but the mass and radius for the secondary differ by 0.8 and 3 percent, respectively. This
radius difference between the two light curves is close to 1-σ, and may be due to uncertainties
in the limb-darkening coefficients. Our adopted values are based on the average values of
the light curve parameters.
4. DISCUSSION
In Figure 6 we plot our mass and radius for the M-dwarf secondary HAT-TR-205-
013 B on a mass-radius diagram, together with isochrones for ages of 0.5 and 5 Gyr from
Baraffe et al. (1998). We also plot the results for 11 M dwarf secondaries from the sample
of OGLE planetary candidates analyzed by Bouchy et al. (2005); Pont et al. (2005a,b, 2006)
and listed in Table 9. For the systems OGLE-TR-34 (Bouchy et al. 2005), OGLE-TR-120
(Pont et al. 2005b), and the low mass systems OGLE-TR-122 (Pont et al. 2005a) and OGLE-
TR-123 (Pont et al. 2006) the authors had to use stellar models to to estimate the masses
and radii of the primaries without the assumption of synchronization, as synchronization
implied masses and radii that were inconsistent with the spectroscopic observations. For the
other seven systems, they were able to assume synchronization and to derive the radius of the
primary from the observed rotational line broadening. In general the agreement between the
– 15 –
OGLE results and the Baraffe et al. (1998) isochrones looks promising, but the observational
uncertainties are still too large to allow a critical test of the theoretical models. The OGLE
systems are all much fainter than HAT-TR-205-013, which presents significant challenges
for both the spectroscopic and photometric follow-up observations. Spectroscopy with the
resolution and signal-to-noise ratio suitable for determining accurate values for rotational
broadening requires time on large telescopes, and photometry for high-quality light curves
also requires large telescopes to achieve the needed photon statistics. Eclipsing binaries
identified by wide-angle surveys are much brighter and therefore less challenging on both
counts.
Our value for the radius of the M-dwarf secondary in HAT-TR-205-013 lies 11 percent,
about 3-σ, above the theoretical isochrones. This divergence is further reinforced by Eq.(11),
which, as has been previously noted, restricts the position of HAT-TR-205-013 B to lie on a
single line that is determined by the surface gravity of the object. This gravity curve does
not rely on any prior assumptions about the HAT-TR-205-013 system, nor does it depend
upon our measured value of Vrot sin irot, which is the biggest contributor of uncertainty to our
final results. We use the assumption of synchronization and the spectroscopically measured
Vrot sin irot to place HAT-TR-205-013 B at a specific location along the curve, but it is
important to note that in the region that we find HAT-TR-205-013 B, the curve of allowable
locations runs nearly parallel to the theoretical models. This is illustrated in Figure 6 by
the red line that passes through our point for HAT-TR-205-013 B. Thus the conclusion that
the theoretical models predict a radius for HAT-TR-205-013 B that is too small by about 10
percent is on much firmer ground than the error bar in the observed radius might suggest.
Indeed, it would require a 6-σ difference in Vrot sin irot to place HAT-TR-205-013 B onto the
Baraffe models.
Our result for HAT-TR-205-013 B supports the suggestion from the results for double-
lined eclipsing binaries plotted in Figure 1 that the models predict radii for M dwarfs that
are too small by up to 10 percent. This discrepancy has been noted before, for example
by Torres & Ribas (2002) in the case of YY Gem. Torres et al. (2006) raised the issue of
whether short-period eclipsing binaries are representative of isolated field stars and wide
binaries where tidal forces are negligible. They suggested that the rapid rotation of the
stars in these systems caused by tidal synchronization might give rise to enhanced magnetic
activity, thus decreasing the efficiency of energy transport in the convective envelopes and
leading to inflated stellar radii. For low mass stars, this effect is examined in more detail by
Lopez-Morales (2007).
In the case of HAT-TR-205-013, we see no evidence in the photometry of star spots on
the primary star, which would be tell-tale indicators of enhanced stellar magnetic activity.
– 16 –
Though HAT-TR-205-013 A is rapidly rotating, the lack of magnetic ativity is not suprising,
given its spectral type (F7). The star’s outer convective layer is relatively shallow, and
it is not unusual for rapidly rotating stars of this type to lack strong magnetic activity
(Torres et al. 2006).
In some instances it may be possible to independently determine the rotational period of
the primary through high-quality light curves used to definitively identify photometric varia-
tion outside of eclipse. This would serve as a check to the assumption of tidal synchronization
in the system.
In future papers we will present the results for several additional single-lined eclipsing
binaries with circularized orbits.
We thank Joe Zajac, Perry Berlind, and Mike Calkins for obtaining some of the spectro-
scopic observations; Bob Davis for maintaining the database for the CfA Digital Speedome-
ters; and John Geary, Andy Szentgyorgyi, Emilio Falco, Ted Groner, and Wayne Peters for
their contribution to making KeplerCam such an effective instrument for obtaining high-
quality light curves. TGB thanks the Harvard University Origins of Life Initiative for sup-
port. GK thanks the support of OTKA K-60750. The HATnet project is supported by
NASA Grant NNG04GN74G. This research was supported in part by the Kepler Mission
under NASA Cooperative Agreement NCC2-1390.
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Table 1. Individual Radial Velocities
HJD Vrad σ(Vrad)
(days) (km s−1) (km s−1)
2453034.45642 −2.02 1.38
2453035.47574 −10.11 1.61
2453035.58018 −18.52 1.01
2453036.48778 −12.93 1.53
2453037.46565 −0.47 1.43
2453037.61215 −6.83 0.91
2453038.46413 −25.72 1.48
2453038.57874 −19.76 1.15
2453040.47360 −28.58 1.24
2453042.58686 −27.79 0.91
2453043.58338 +4.84 1.23
2453044.58422 −19.73 1.14
2453045.57911 −6.42 0.73
2453046.46373 −2.11 0.80
2453046.60000 −10.47 0.77
2453047.50881 −20.45 1.49
2453047.58731 −14.83 0.95
2453543.94910 −4.01 1.10
2453658.69572 −20.53 1.16
2453659.75967 +3.14 2.28
2453659.78398 +2.85 1.09
2453660.70213 −25.98 1.57
2453664.70202 −19.39 1.21
– 20 –
Table 2. Spectroscopic Orbital Parameters
Parameter Value
P (days) 2.23072± 0.00005
γ (km s−1) −9.83± 0.30
K (km s−1) 18.33± 0.47
e 0.012± 0.021
ω (◦) 143± 90
Epoch (HJD) 2, 453, 198.61± 0.56
Nobs 23
O − C rms (km s−1) 1.06
f(M) (M⊙3) 0.00142± 0.00023
aA sin i (Gm) 0.562± 0.030
– 21 –
Table 3. Rotational Velocity Results
log g,[Fe/H] Teff < Vrot > σ(< Vrot >) Correlation
(K) (km s−1) (km s−1) Coefficient
4.0,0.0 6340 29.4 0.25 0.826
4.5,0.0 6540 28.4 0.24 0.823
4.0,−0.5 5960 29.2 0.21 0.821
4.5,−0.5 6150 28.2 0.24 0.816
Adopted:
4.24,−0.2 6295 28.9 1.0
– 22 –
Table 4. g Band Photometry
HJD Flux
2453666.575985 1.00054
2453666.576472 0.99856
2453666.576946 1.00108
2453666.579226 1.00084
2453666.579712 1.00008
2453666.580198 0.99985
2453666.582501 0.99998
2453666.582976 0.99882
2453666.583474 1.00159
Note. — Table 4 is pre-
sented in its entirety in the
electronic edition of the As-
trophysical Journal. A por-
tion is shown here for guid-
ance regarding its form and
content.
column (1): Heliocentric
Julian Date,
column (2): Normalized in-
strumental flux.
– 23 –
Table 5. i Band Photometry
HJD Flux
2453666.574226 0.99951
2453666.574469 0.99803
2453666.574724 1.00096
2453666.574967 0.99846
2453666.575233 0.99628
2453666.575488 1.00014
2453666.577432 1.00062
2453666.577698 0.99886
2453666.577965 0.99941
2453666.578208 0.99613
2453666.578474 0.99957
2453666.578728 0.99962
2453666.580719 0.99837
2453666.580974 0.99976
2453666.581228 0.99761
2453666.581494 1.00025
2453666.581772 1.00113
2453666.582027 0.99823
Note. — Table 5 is pre-
sented in its entirety in the
electronic edition of the As-
trophysical Journal. A por-
tion is shown here for guid-
ance regarding its form and
content.
column (1): Heliocentric
Julian Date,
– 24 –
column (2): Normalized in-
strumental flux.
– 25 –
Table 6. Light-Curve Fit Results
Parameter g Band i Band Adopted
a/RA 5.93± 0.15 5.91± 0.16 5.92± 0.11
RB/RA 0.1330± 0.0010 0.1288± 0.0007 0.1309± 0.0006
b 0.36± 0.06 0.37± 0.07 0.365± 0.046
– 26 –
Table 7. Correlation Coefficients
Coefficient g Band i Band
(a/RA,RB/RA) 0.28 0.27
(a/RA,b) −0.21 −0.42
(RB/RA,b) 0.04 0.01
– 27 –
Table 8. Physical Parameters for HAT-TR-205-013
Parameter g Band i Band Adopted
MA (M⊙) 1.04± 0.14 1.03± 0.14 1.04± 0.13
RA (R⊙) 1.28± 0.04 1.28± 0.04 1.28± 0.04
MB (M⊙) 0.124± 0.011 0.123± 0.011 0.124± 0.010
RB (R⊙) 0.169± 0.006 0.164± 0.006 0.167± 0.006
a (AU) 0.0351± 0.0015 0.0351± 0.0015 0.0351± 0.0014
– 28 –
Table 9. Masses and Radii for Low-Mass Stars
Name M (M⊙) R (R⊙) Type Ref.
OGLE-TR-123 B 0.085± 0.011 0.133± 0.009 SB1 EB 1
OGLE-TR-122 B 0.092± 0.009 0.120± 0.018 SB1 EB 2,3
OGLE-TR-106 B 0.116± 0.021 0.181± 0.013 SB1 EB 3
HAT-TR-205-013 B 0.123± 0.011 0.167± 0.007 SB1 EB 13
OGLE-TR-125 B 0.209± 0.033 0.211± 0.027 SB1 EB 3
CM Dra B 0.2136± 0.0010 0.2347± 0.0019 SB2 EB 4,5
CM Dra A 0.2307± 0.0010 0.2516± 0.0020 SB2 EB 4,5
OGLE-TR-78 B 0.243± 0.015 0.240± 0.013 SB1 EB 3
OGLE-TR-5 B 0.271± 0.035 0.263± 0.012 SB1 EB 6
OGLE-TR-7 B 0.281± 0.029 0.282± 0.013 SB1 EB 6
OGLE-TR-6 B 0.359± 0.025 0.393± 0.018 SB1 EB 6
OGLE-TR-18 B 0.387± 0.049 0.390± 0.040 SB1 EB 6
CU Cnc B 0.3890± 0.0014 0.3908± 0.0094 SB2 EB 7
OGLE-BW3-V38 B 0.41± 0.09 0.44± 0.06 SB2 EB 8
CU Cnc A 0.4333± 0.0017 0.4317± 0.0052 SB2 EB 7
OGLE-BW3-V38 A 0.44± 0.07 0.51± 0.04 SB2 EB 8
OGLE-TR-120 B 0.47± 0.04 0.42± 0.02 SB1 EB 3
TrES-Her0-07621 B 0.489± 0.003 0.452± 0.050 SB2 EB 9
TrES-Her0-07621 A 0.493± 0.003 0.453± 0.060 SB2 EB 9
NSVS01031772 B 0.4982± 0.0025 0.5088± 0.0030 SB2 EB 10
OGLE-TR-34 B 0.509± 0.038 0.435± 0.033 SB1 EB 6
NSVS01031772 A 0.5428± 0.0027 0.5260± 0.0028 SB2 EB 10
YY Gem A & B 0.5992± 0.0047 0.6191± 0.0057 SB2 EB 11
GU Boo B 0.599± 0.006 0.620± 0.020 SB2 EB 12
GU Boo A 0.610± 0.007 0.623± 0.016 SB2 EB 12
References. — 1. Pont et al. (2006); 2. Pont et al. (2005a); 3. Pont et al.
(2005b); 4. Lacy (1977); 5. Metcalfe et al. (1996); 6. Bouchy et al. (2005);
– 29 –
7. Ribas (2003); 8. Maceroni & Montalbán (2004); 9. Creevey et al.
(2005); 10. López-Morales et al. (2006); 11. Torres & Ribas (2002); 12.
López-Morales & Ribas (2005); 13. This paper
– 30 –
Fig. 1.— The mass-radius diagram for 10 stars in 5 double-lined eclipsing binaries each
composed of two M dwarfs, and with errors better than 3 percent.
– 31 –
9.98
10.02
10.04
10.06
10.08
10.1
-0.4 -0.2 0 0.2 0.4
Phase
HAT-5 TFA data
HAT-8 TFA data
10.01
10.02
10.03
10.04
10.05
10.06
-0.1 -0.05 0 0.05 0.1
Fig. 2.— The phase-folded HATnet light curve for HAT-TR-205-013.
– 32 –
Fig. 3.— The velocity curve for our orbital solution for HAT-TR-205-013, together with the
individual observed velocities. The lower panel shows the O-C velocity residuals from the
orbital solution.
– 33 –
Fig. 4.— KeplerCam light curves for HAT-TR-205-013 in the SDSS g and i bands. Contin-
uous lines show the best fit synthetic light curves for each.
– 34 –
Fig. 5.— Contours of χ2 for the results from fits to the light curves in the g-band (left
panels) and i-band (right panels). For each band the three panels show the projections onto
the three possible planes involving b, a/RA, and RB/RA. The 1-σ, 2-σ, and 3-σ contours are
plotted.
– 35 –
Fig. 6.— The mass-radius diagram for M dwarfs in single-lined eclipsing binaries. The M
dwarfs from Pont et al. (2005a,b, 2006) are plotted as open circles. The red line passing
through the point for HAT-TR-205-013 B shows the constraint imposed on its location by
Eq.(11) and our observed quantities, without making any explicit assumptions (such as
synchronization) about the system. Assuming synchronization, the hash marks on the line
show the effect that differences of ± 1, 2, & 3 km s−1 in Vrot have on our final results.
INTRODUCTION
OBSERVATIONS AND DATA REDUCTION
HAT Photometry
Follow-up Spectroscopy
Follow-up KeplerCam Photometry
LIGHT-CURVE ANALYSIS
MASSES AND RADII FOR HAT-TR-205-013
DISCUSSION
|
0704.0060 | Coulomb excitation of unstable nuclei at intermediate energies | Coulomb excitation of unstable nuclei at intermediate energies
C.A. Bertulani1∗, G. Cardella2, M. De Napoli2,3, G. Raciti2,3, and E. Rapisarda2,3
1 Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA
2 Istituto Nazionale di Fisica Nucleare, Sezione di Catania,
via Santa Sofia 64, I-95123, Catania, Italy
3 Dipartimento di Fisica e Astronomia, Universitá Catania,
via Santa Sofia 64, I-95123, Catania, Italy
Abstract
We investigate the Coulomb excitation of low-lying states of unstable nuclei in intermediate
energy collisions (Elab ∼ 10−500 MeV/nucleon). It is shown that the cross sections for the E1 and
E2 transitions are larger at lower energies, much less than 10 MeV/nucleon. Retardation effects
and Coulomb distortion are found to be both relevant for energies as low as 10 MeV/nucleon and
as high as 500 MeV/nucleon. Implications for studies at radioactive beam facilities are discussed.
PACS numbers: 25.60.-t, 25.70.-z, 25.70.De
Keywords: Coulomb excitation, cross sections, unstable nuclei.
∗ [email protected].
http://arxiv.org/abs/0704.0060v2
Unstable nuclei are often studied with reactions induced by secondary radioactive beams.
Examples of these reactions are elastic scattering, fragmentation and Coulomb excitation
by heavy targets. Coulomb excitation is specially useful since the interaction mechanism is
very well known [1]. It is the result of electromagnetic interactions of a projectile (ZP ,AP )
with a target (ZT ,AT ). One of the participating nuclei is excited as it passes through the
electromagnetic field of the other. Here we will only consider the excitation of the pro-
jectile as is of interest in studies carried out in heavy ion facilities around the world, e.g.
LNS/Catania, NSCL/MSU, GSI, GANIL, RIKEN, etc. In Coulomb excitation a virtual pho-
ton with energy E is absorbed by the projectile. Because in pure Coulomb excitation the
participating nuclei stay outside the range of the nuclear strong force, the excitation cross
section can be expressed in terms of the same multipole matrix elements that character-
ize excited-state gamma-ray decay, or the reduced transition probabilities, B(πλ; Ji → Jf).
Hence, Coulomb excitation amplitudes are strongly coupled with valuable nuclear struc-
ture information. Therefore, this mechanism has been used for many years to study the
electromagnetic properties of low-lying nuclear states [1].
Coulomb excitation cross sections are large if the adiabacity parameter satisfies the con-
dition
ξ = ωfi
< 1 , (1)
where a0 is half the distance of closest approach in a head-on collision for a projectile
velocity v, and Ex = ~ωfi is the excitation energy. This adiabatic cut-off limits the possible
excitation energies below 1-2 MeV in sub-barrier collisions. A possible way to overcome this
limitation, and to excite high-lying states, is to use higher projectile energies. In this case,
the closest approach distance, at which the nuclei still interact only electromagnetically, is
of order of the sum of the nuclear radii, R = RP + RT , where P refers to the projectile
and T to the target. For very high energies one has also to take into account the Lorentz
contraction of the interaction time by means of the Lorentz factor γ = (1− v2/c2)−1/2, with
c being the speed of light. For such collisions the adiabacity condition, Eq. (1), becomes
ξ(R) =
< 1 . (2)
From this relation one obtains that for bombarding energies around and above 100
MeV/nucleon, states with energy up to 10-20 MeV can be readily excited [3].
An appropriate description of Coulomb excitation at intermediate energies (Elab =
10 − 500 MeV/nucleon) has been described in ref. [2]. In this energy region neither the
non-relativistic Coulomb excitation formalism described in ref. [1], nor the relativistic one
formulated in refs. [3, 4] are appropriate. This is discussed in details in ref. [2] where it
is shown that the correct values of the Coulomb excitation cross sections differ by up to
30-40% when compared to the non-relativistic and relativistic treatments used to calculate
experimental observables (cross sections, gamma-ray angular distributions, etc.).
We follow the formalism of ref. [2] to calculate cross sections for Coulomb excitation from
energies varying from 10 to 500 MeV/nucleon. These are the energies where most radioactive
beam facilities are or will be operating around the world. The calculated cross sections will
be of useful guide for future experiments. We also compare the accurate calculations with
those obtained by using simple analytical formulas and test the regime of their validity.
The cross sections for the transition Ji → Jf in the projectile are calculated using the
equation [2]
dσi→f
4π2Z2T e
B(πλ, Ji → Jf)
(2λ+ 1)3
| S(πλ, µ) |2 , (3)
where π = E or M stands for the electric or magnetic multipolarity, and
B(πλ, Ji −→ Jf) =
2Ji + 1
|〈Jf ‖M(πλ)‖Ji〉|
are the reduced transition probabilities. In these equations, ǫ = 1/ sin(Θ/2), with Θ being
the deflection angle, a0 = ZPZT e
2/m0v
2 and a = a0/γ. The complex functions S(πλ, µ) are
integrals along Coulomb trajectories corrected for retardation. Their calculation and how
they relate to the non-relativistic and relativistic theories are described in details in ref. [2].
Here we will introduce another comparison tool for the total cross section, which is obtained
by integration of eq. 3 over scattering angles. The code COULINT [2] was used to calculate
the orbital integrals S(πλ, µ) and the cross sections of eq. 3 (for more details, see ref. [2]).
Using the theory described in ref. [4], it is easy to show that approximate values of the
cross sections for E1, E2, and M1 transitions can be obtained by means of the relations
(app)
B (E1)
ξK0K1 −
K21 −K
(app)
E3xB (E2)
K21 + ξ
K0K1 −
K21 −K
(app)
B (M1)
ξK0K1 −
K21 −K
, (5)
where Kn are the modified Bessel functions of the second order, as a function of ξ given by
eq. 2, with R corrected for recoil by the modification R → R + πa/2 [3].
Here we will only consider the excitation of the lowest lying states in light and medium
heavy nuclei. For nuclear masses A < 20, the TUNL nuclear data evaluation web site
was of great help [5]. The electromagnetic transition rates at the TUNL database are
given in Weisskopf units and are transformed to the appropriate B(πλ, Ii → If)-values
by means of the standard Weisskopf relations BW (E1; Ji → Jgs) = 0.06446A
2/3 e2fm2,
BW (E2; Ji → Jgs) = 0.05940A
4/3 e2fm4, and BW (M1; Ji → Jgs) = 1.79 (e~/2mnc)
. For
comparison, a few medium mass nuclei, as well as a few stable nuclei, were included in the
calculation. Other data were taken from refs. [6, 7, 8, 9].
Some cases of nuclei far from the stability line are very interesting and deserve further
study, possibly using the method of Coulomb excitation. For example, it is well known that
nuclei with open shells tend to have B(E2) values greater than 10 W.u., whereas nuclei with
shell closure of neutrons or protons tend to have distinctly smaller B(E2) values. Typical
examples of the latter category are the doubly magic nuclei, 16O and 48Ca, which B(E2)
values are 3.17 and 1.58 W.u., respectively. According to an empirical formula adjusted
to a global fit of the known transition rates, the values of first excited 2+ level, E2+ , and
B(E2; 0+ → 2+) are related by [10] (E2+ in keV)
B(E2; 0+ → 2+) = 26
A2/3E2+
e2fm4. (6)
The value of B(E2) for 16C based on this formula is at least one order of magnitude
larger than what is observed experimentally in a Coulomb dissociation experiment [9]. The
anomalously strong hindrance of the 16C transition is not well explained theoretically. This
is just an example of the power of Coulomb excitation as a tool to access the new physics
inherent of poorly known rare nuclear species.
Another example is the strong E1 transition in 11Be. 11Be is an archetype of a halo nu-
cleus and exhibits the fastest known dipole transition between bound states in nuclei. The
B(E1) transition strength between the ground and the only bound excited state (at 0.32
MeV) was determined from lifetime measurements by Millener et al. to be 0.116 e2fm2 [11].
However, Coulomb excitation experiments have obtained a much smaller value of the B(E1)
which is still a matter of investigation [12, 13, 14]. It is thus seems clear that predictions
based on traditional nuclear structure and reaction theory often yields results in disagree-
ment with experimental data. In spite of that, when proper corrections are accounted for
(e.g. channel-coupling, nuclear excitation, relativistic corrections), Coulomb excitation of ra-
dioactive beams is a powerful complementary tool to investigate electromagnetic properties
of nuclei far from the stability line.
In Table 1 we compare our calculations with several experimentally obtained cross sections
for Coulomb excitation of unstable nuclei. The units of energy are MeV, the laboratory
energy is in MeV/nucleon, the B-values are in units of e2 fm2λ, and the cross sections are
in millibarns. The last two columns give the calculated cross sections obtained by using
eqs. 3 and 5, respectively. Since the cross sections of eq. 5 are functions of the minimum
impact parameter, the values reported in the Table have been calculated according to the
experimental angular ranges reported in the seventh column. Except for the 11Be case, for
which the discrepancy between theory and experiment is known (see discussion above), the
calculated cross sections are close to the experimental values. Nonetheless, the calculated
cross sections tend to be smaller than the experimental ones for 17Ne, 32Mg, 38S, 40S, 42S,
44Ar, and 46Ar projectiles. This is worrisome because the B(πλ) values were extracted
from the experimentally obtained cross sections, using equations similar to eq. 5. These
experimental B-values would have to be larger by 10− 30% according to our calculations.
It is important to stress the fact that many experimental data on unstable nuclei collected
up to now have been analyzed by means of theoretical tools (DWBA and coupled-channels
codes) which do not include relativistic dynamics (the inclusion of relativistic kinematics
is straightforward). This problem was first addressed in ref. [15], where it was shown
that the analysis of experimental data at intermediate energies without a proper treatment
of relativistic dynamics leads to wrong values of electromagnetic transition probabilities.
We should stress that a full theoretical treatment of relativistic dynamics of strong and
electromagnetic interactions in many-body systems is very difficult and still does not exist
[15].
Data Projectile Target Elab πλ B(πλ) θrange Ex σexp σth σapp
1 [16] 11Be Pb 43. E1 0.115 < 5◦ 0.32 (1
) 191± 26 328. 323.
2 [16] 11Be Pb 59.4 E1 0.094 < 3.8◦ 0.32 (1
) 304± 43 213. 211.
3 [18] 11Be Au 57.6 E1 0.079 < 3.8◦ 0.32 (1
) 244± 31 170. 168.
4 [17] 11Be Pb 64. E1 0.099 < 3.8◦ 0.32 (1
) 302± 31 217. 215.
5 [19, 20] 17Ne Au 60. M1 0.163 < 4.5◦ 1.29 (1
) 12± 4 12.6 13.0
6 [6] 32Mg Pb 49.2 E2 454 < 4◦ 0.885 (0+ → 2+) 91.7± 14.4 137. 128.
7 [19] 38S Au 39.2 E2 235 < 4.1◦ 1.29 (0+ → 2+) 59± 7 48. 45.0
8 [19] 40S Au 39.5 E2 334 < 4.1◦ 0.91 (0+ → 2+) 94± 9 75.5 70.4
9 [19] 42S Au 40.6 E2 397 < 4.1◦ 0.89 (0+ → 2+) 128± 19 101. 94.3
10 [19] 44Ar Au 33.5 E2 345 < 4.1◦ 1.14 (0+ → 2+) 81± 9 62.3 58.3
11 [19] 46Ar Au 35.2 E2 196 < 4.1◦ 1.55 (0+ → 2+) 53± 10 40.9 38.2
12 [8] 46Ar Au 76.4 E2 212 < 2.9◦ 1.55 (0+ → 2+) 68± 8 50.0 47.4
Table 1. Cross sections for Coulomb excitation of unstable nuclei. The units of energy
are MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ,
and the cross sections are in millibarns. The data for different experiments (numbered 1 to
12) were collected from the references listed in column 1. The last two columns give the
calculated cross sections obtained by using eqs. 3 and 5, respectively.
In figure 1 we show a comparison between the experimental data and our calculations.
We notice that the cross sections calculated with help of eq. 5 are not much different than
those calculated with eq. 3. They are systematically lower, up to 10%, than the exact
calculation following eq. 3. As we discuss below, this is not always the case, specially for
the excitation of high-lying states. In fact, this is a good check of eq. 3, which is done in a
very different way than the analytical calculations of eq. 5. But as we will see below, this
agreement is not always the case, specially when one includes small impact parameters for
which the sensitivity to the relativistic corrections is higher (see ref. [2]). The dashed curve
in figure 1 is a guide to the eye. It helps to see that the experimental cross sections are on
average larger than the calculated ones, either with eq. 3 (open circles), or with eq. 5 (open
triangles).
2 4 6 8 10 12
Data set
FIG. 1: Comparison between experimental Coulomb excitation cross sections (solid stars with
error bars) and theoretical ones, calculated either with eq. 3 (open circles), or with eq. 5 (open
triangles).
Ex [MeV] J
i → J
f πλ B(πλ) [e
2 fm2λ] 10 20 30 50 100 200 500
11Be 0.32 1
E1 0.115 1128 653 473 315 187 115 69.6
11B 2.21 3
M1 2.40×10−2 0.301 0.799 1.15 1.63 2.33 3.08 4.17
11C 2.00 3
M1 1.52×10−2 0.196 0.551 0.793 1.12 1.57 2.07 2.76
12B 0.953 1+ → 2+ M1 4.62×10−3 0.227 0.395 0.490 0.607 0.762 0.917 1.13
12C 4.44 0+ → 2+ E2 37.9 34.6 38.6 31.3 21.6 12.1 6.93 3.81
13C 3.09 1
E1 1.39×10−2 8.37 11.3 11.0 9.61 7.28 5.39 3.89
13N 2.37 1
E1 3.56×10−2 38.2 43.6 39.6 32.5 23.2 16.4 11.4
15C 0.74 1
E2 2.90 8.79 4.04 2.65 1.59 0.839 0.475 0.267
16C 1.77 0+ → 2+ E2 2.12 8.81 4.41 2.92 1.76 0.920 0.517 0.285
16N 0.12 0− → 2− E2 10.2 31.0 14.1 9.21 5.53 2.91 1.64 0.926
17N 1.37 1
M1 5.15×10−3 0.153 0.304 0.397 0.516 0.680 0.848 1.09
17O 0.87 5
E2 2.07 6.30 2.88 1.87 1.12 0.588 0.332 0.184
17F 0.5 5
E2 21.6 68.3 29.7 19.3 11.6 6.08 3.44 1.92
18O 1.98 0+ → 2+ E2 44.8 109 60.7 40.9 24.8 11.6 7.27 3.99
18F 0.94 1+ → 3+ E2 37.9 115 52.5 34.1 20.4 10.7 6.01 3.34
18Ne 1.89 0+ → 2+ E2 248 615 342 229 138 72.0 40.1 22.1
19O 0.1 5
M1 2.34×10−4 0.0495 0.0615 0.0673 0.0737 0.0799 0.779 0.799
19F 0.11 1
E1 5.51×10−4 8.07 4.36 3.06 1.97 1.10 0.592 0.337
19Ne 0.24 1
E2 119 361 157 102 61.6 32.5 18.5 10.5
20O 1.67 0+ → 2+ E2 28.0 72 37.4 24.9 15.1 7.86 4.41 2.43
20F 0.656 2+ → 3+ M1 3.56×10−3 0.237 0.385 0.465 0.560 0.683 0.803 0.959
20Ne 1.63 0+ → 2+ E2 319 834 433 287 173 89.8 50.3 27.6
30Ne 0.791 0+ → 2+ E2 460 1167 550 361 218 115 65.0 35.2
32Mg 0.885 0+ → 2+ E2 454 1151 541 355 214 112 63.0 36.7
42S 0.89 0+ → 2+ E2 397 945 445 292 175 91.9 52 29.7
46Ar 1.55 0+ → 2+ E2 190 399 209 140 84.4 44.1 24.7 13.6
54Ni 1.40 0+ → 2+ E2 626 1319 677 447 268 139 78.1 43.1
Table 2 - Cross sections (in mb) for Coulomb excitation of projectiles incident on Pb
targets at bombarding energies ranging from 10 to 500 MeV/nucleon. The energy units are
MeV, the laboratory energy is in MeV/nucleon, the B(πλ)-values are in units of e2fm2λ.
The cross sections for Coulomb excitation of numerous projectiles incident on Pb targets
at bombarding energies ranging from 10 to 500 MeV/nucleon are shown in Table 2. These
cross sections were calculated assuming that the detectors collect events from all possible
Coulomb scattering events. In a real experimental situation, the angular distribution is
restricted to angular windows, reducing the available cross sections. Only the lowest lying
transitions have been considered, i.e. from the ground to the first excited states. One
observes that some cross sections are very large, specially for 11Be, 18Ne, 30Ne and 54Ni.
For these and other similar cases, the measurements are easy to perform, with a large
number of events/second even with modest intensities. Cases such as 16C are well within
the experimental possibilities in most radioactive beam facilities.
Table 2 also shows that, except for M1 excitations, the Coulomb excitation cross sections
decrease steadily as the energy increases from 10 to 500 MeV/nucleon. Based on these
numbers alone, one could conclude that Coulomb excitation of low-lying states (in contrast
to the case of high-lying states, e.g. giant resonances [4]) are better suited for studies at low
energies. However, reactions at lower energies while are less influenced by contamination due
to nuclear breakup [12, 14] can give rise to large high-order effects [21]. The interpretation
of data could be distorted as in the case of Coulomb dissociation of 8B at low energy [24],
which was completely misinterpreted in terms of first-order calculations. In some situations,
when higher-order effects are relevant, the effect of the nuclear breakup cannot be neglected
either [22, 23]. Thus, the choice of the incident energy would depend on the experimental
conditions. Identification of gamma-rays from de-excitation using Doppler shift techniques
are often more advantageous at higher energies. Moreover, except for few cases (e.g. 11C), the
magnetic dipole transitions are much smaller than those for E1 and E2 transitions. Even
for M1 transitions the measurements are under the possibility of most new experimental
facilities.
The comparison of the exact calculations, using eq. 3 (solid lines), and the approximations
5 (dashed lines) are shown in figs. 2(a-d), for 11Be, 11B, 54Ni and 16O, respectively. The 16O
case (as well as for 12C in Table 2) was included for comparison, with a high-lying excited
state. We see from figs 2a and 2b that the approximations in eq. 5 work quite well for
the M1 multipolarity and reasonably well (within 20% at 10 MeV/nucleon and 5% at 50
MeV/nucleon) for the E1 cases. But they fail badly at low and intermediate energies for
(c) (d)
FIG. 2: Coulomb excitation cross section of the first excited state in 11Be, 11B and 54Ni and of the
13.05 MeV sate in 16O projectiles incident on Pb targets as a function of the laboratory energy.
the E2 ( fig. 2c). The reason is that the E2 Coulomb field (“tidal field”) is very sensitive
to the details of the collision dynamics at low energies. These conclusions can be deceiving
since even for the E1 and M1 cases the approximations in eq. 5 may strongly differ from
the exact calculations if the excitation energy is large (see discussion in ref. [2]). This is
shown in figure 2d, where we plot the Coulomb excitation cross section of the Ex = 13.09
MeV state in 16O. In this case, the cross sections based on eq. 5 is a factor of 10 smaller
than the exact calculation at 10 MeV/nucleon. At 100 MeV/nucleon this difference drops
to 10%, which still needs to be considered with care.
In summary, in this article we have used the formalism of ref. [2] to predict the cross
sections for Coulomb excitation of several light projectiles with electromagnetic transitions
found in the literature, listed in the TUNL database [5], and for a few other selected cases.
These estimates will be useful for planing Coulomb excitation experiments at present and
future heavy ion facilities. It is evident that the inclusion of relativistic effects combined
with Coulomb distortion are of the utmost relevance. The cross section inferred by using
non-relativistic or pure relativistic treatments can be wrong by up to 30% even at 100
MeV/nucleon, as shown here and in ref. [2]. Finally, the use of Coulomb excitation to
produce nuclei in high-lying states is an important tool to study particle emission processes.
For example, the excitation of 18Ne and its subsequent decay by two-proton emission is a
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References
|
0704.0061 | Intersection Bodies and Generalized Cosine Transforms | arXiv:0704.0061v2 [math.FA] 3 May 2007
INTERSECTION BODIES AND GENERALIZED
COSINE TRANSFORMS
BORIS RUBIN
Abstract. Intersection bodies represent a remarkable class of
geometric objects associated with sections of star bodies and in-
voking Radon transforms, generalized cosine transforms, and the
relevant Fourier analysis. The main focus of this article is interrela-
tion between generalized cosine transforms of different kinds in the
context of their application to investigation of a certain family of
intersection bodies, which we call λ-intersection bodies. The latter
include k-intersection bodies (in the sense of A. Koldobsky) and
unit balls of finite-dimensional subspaces of Lp-spaces. In particu-
lar, we show that restrictions onto lower dimensional subspaces of
the spherical Radon transforms and the generalized cosine trans-
forms preserve their integral-geometric structure. We apply this
result to the study of sections of λ-intersection bodies. New char-
acterizations of this class of bodies are obtained and examples are
given. We also review some known facts and give them new proofs.
Contents
1. Introduction.
2. Preliminaries.
3. Analytic families of the generalized cosine transforms.
4. Positive definite homogeneous distributions.
5. λ-intersection bodies.
6. Examples of λ-intersection bodies.
7. (q, ℓ)-balls.
8. The generalized cosine transforms and comparison of volumes.
9. Appendix.
2000 Mathematics Subject Classification. Primary 44A12; Secondary 52A38.
Key words and phrases. Spherical Radon transforms, cosine transforms, inter-
section bodies.
The research was supported in part by the NSF grant DMS-0556157 and the
Louisiana EPSCoR program, sponsored by NSF and the Board of Regents Support
Fund.
http://arxiv.org/abs/0704.0061v2
2 BORIS RUBIN
1. Introduction
This is an updated and extended version of our previous preprint
[R5].
Intersection bodies interact with Radon transforms and encompass
diverse classes of geometric objects associated to sections of star bodies.
The concept of intersection body was introduced in the remarkable
paper by Lutwak [Lu] and led to a breakthrough in the solution of the
long-standing Busemann-Petty problem; see [G], [K4], [Lu], [Z2] for
references and historical notes.
We remind some known facts that will be needed in the following.
An origin-symmetric (o.s.) star body in Rn, n ≥ 2, is a compact set
K with non-empty interior such that tK ⊂ K ∀t ∈ [0, 1], K = −K,
and the radial function ρK(θ) = sup{λ ≥ 0 : λθ ∈ K} is continuous
on the unit sphere Sn−1. In the following, Kn denotes the set of all
o.s. star bodies in Rn, Gn,i is the Grassmann manifold of i-dimensional
linear subspaces of Rn, and voli(·) denotes the i-dimensional volume
function. The Minkowski functional of a body K ∈ Kn is defined by
||x||K = min{a ≥ 0 : x ∈ aK}, so that ||θ||K = ρ−1K (θ), θ ∈ Sn−1.
Definition 1.1. [Lu] A body K ∈ Kn is an intersection body of a body
L ∈ Kn if ρK(θ) = voln−1(L ∩ θ⊥) for every θ ∈ Sn−1, where θ⊥ is the
central hyperplane orthogonal to θ.
By taking into account that voln−1(L ∩ θ⊥) in Definition 1.1 is a
constant multiple of the Minkowski-Funk transform
(Mf)(θ) =
Sn−1∩θ⊥
f(u) dθu, f(u) = ρ
L (u),
Goodey, Lutwak and Weil [GLW] generalized Definition 1.1 as follows.
Definition 1.2. A body K ∈ Kn is an intersection body if ρK = Mµ
for some even non-negative finite Borel measure µ on Sn−1.
A sequence of bodies Kj ∈ Kn is said to be convergent to K ∈ Kn
in the radial metric if lim
||ρKj − ρK ||C(Sn−1) = 0.
Proposition 1.3. The class of intersection bodies is the closure of the
class of intersection bodies of star bodies in the radial metric.
Proposition 1.4. If K is an intersection body in Rn, n > 2, then for
every i = 2, 3, . . . , n − 1 and every η ∈ Gn,i, K ∩ η is an intersection
body in η.
Regarding these two important propositions see [FGW], [GW] and
a nice historical survey in [G].
INTERSECTION BODIES 3
Different generalizations of the concept of intersection body associ-
ated to lower dimensional sections were suggested in the literature; see,
e.g., [K4], [RZ], [Z1]. The following one, which plays an important role
in the study of the lower dimensional Busemann-Petty problem, is due
to Zhang [Z1].
Definition 1.5. We say, that a body K ∈ Kn belongs to Zhang’s class
Zni if there is a non-negative finite Borel measure m on the Grassmann
manifold Gn,i such that ρ
K = R
im, where R
i is the dual spherical
Radon transform; see (2.2), (2.5).
Another generalization was suggested by Koldobsky [K2] and de-
scribed in detail in [K4]. This class of bodies will be our main concern.
Definition 1.6. [K4, p. 71] A body K ∈ Kn is a k-intersection body
of a body L ∈ Kn (we write K = IBk(L)) if
(1.1) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
We denote by IBk,n the set of all bodies K ∈ Kn satisfying (1.1) for
some L ∈ Kn.
When k = 1, this definition coincides with Definition 1.1 up to a
constant multiple. An analog of Definition 1.2 was given in the Fourier
analytic terms as follows.
Definition 1.7. [K4, Definition 4.7] A body K ∈ Kn is a k-intersection
body if there is a non-negative finite Borel measure µ on Sn−1, so that
for every Schwartz function φ,
||x||−kK φ(x) dx =
tk−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
The set of all k-intersection bodies in Rn will be denoted by Ink .
Keeping in mind Proposition 1.3 for k = 1, one can alternatively
define the class Ink as a closure of IBk,n in the radial metric; cf. [Mi1,
p. 532]. However, to apply results from [K4] to such class, equivalence
of this definition to Definition 1.7 must be proved. We will do this in
the more general situation in Section 5.2.
From Definitions 1.6 and 1.7 it is not clear, for which bodies L ∈ Kn
the relevant k-intersection body K = IBk(L) does exist. It is also not
obvious which bodies actually constitute the class Ink . The following
important characterization is due to Koldobsky.
4 BORIS RUBIN
Theorem 1.8. [K4, Theorem 4.8] A body K ∈ Kn is a k-intersection
body if and only if || · ||−kK represents a positive definite tempered dis-
tribution on Rn, that is, the Fourier transform (|| · ||−kK )∧ is a positive
tempered distribution on Rn.
The concept of k-intersection body is related to another important
development. For K ∈ Kn, the quasi-normed space (Rn, || · ||K) is said
to be isometrically embedded in Lp, p > 0, if there is a linear operator
T : Rn → Lp([0, 1]) so that ||x||K = ||Tx||Lp([0,1]).
Theorem 1.9. [K4, Theorem 6.10] The space (Rn, || · ||K) embeds iso-
metrically in Lp, p > 0, p 6= 2, 4, . . . , if and only if Γ(−p/2)(|| · ||pK)∧
is a positive distribution on Rn \ {0}.
Following Theorems 1.9 and 1.8, one can formally say that K ∈ Ink if
and only if (Rn, || · ||K) embeds isometrically in L−k. This observation,
combined with Definition 1.7, was used by A. Koldobsky to define the
concept of “isometric embedding in Lp” for negative p.
Definition 1.10. [K4, Definition 6.14] Let 0 < p < n, K ∈ Kn. The
space (Rn, || · ||K) is said to be isometrically embedded in L−p if there
is a non-negative finite Borel measure µ on Sn−1, so that for every
Schwartz function φ,
||x||−pK φ(x) dx =
tp−1φ̂(tθ) dt
dµ(θ),
where φ̂ denotes the Fourier transform of φ.
Origin-symmetric bodiesK in this definition can be regarded as “unit
balls of n-dimensional subspaces of L−p”. Comparing Definitions 1.10
and 1.7, one might call these bodies “p-intersection bodies”. Since the
meaning of the space L−p itself is not specified in Definition 1.10 and
since our paper is mostly focused on geometric properties of bodies
(rather than embeddings in Lp), in the following we prefer to adopt
another name “λ-intersection body”, where λ is a real number, that
will be specified in due course. We denote the set of all λ-intersection
bodies in Rn by Inλ .
Contents of the paper. We will focus on intimate connection
between intersection bodies, spherical Radon transforms, and general-
ized cosine transforms; see definitions in Section 2.2. This approach
is motivated by the fact that the volume of a central cross section of
a star body is expressed through the spherical Radon transform, and
the latter is a member of the analytic family of the generalized cosine
transforms. These transforms were introduced by Semyanistyi [Se] and
INTERSECTION BODIES 5
arise (up to naming and normalization) in different contexts of analysis
and geometry; see, e.g., [K4], [R1]-[RZ], [Sa2], [Sa3], [Str1], [Str2].
Sections 2-4 provide analytic background for geometric considera-
tions in Sections 5-7. In Section 2 we establish our notation and define
the generalized cosine transforms on the sphere and the relevant dual
transforms on Grassmann manifolds. In Section 3 we present basic
properties of these transforms, establish new relations between spheri-
cal Radon transforms and the generalized cosine transforms, and prove
“restriction theorems”, which are akin to trace theorems in Sobolev
spaces. Section 4 deals with positive definite homogeneous distribu-
tions, that can be characterized in terms of the generalized cosine
transforms. This section serves as a preparation for the forthcoming
definition of the concept of λ-intersection body. We investigate which
λ’s are appropriate and why. In Section 5 we switch to geometry and
define the class Inλ of λ-intersection bodies. The case 0 < λ < n cor-
responds to the “unit balls of L−p-spaces” in the spirit of Definition
1.10. The reader will find in this section new proofs of some known
facts. We introduce the notion of λ-intersection body of a star body
in Rn, which extends Definition 1.6 to all λ < n, λ 6= 0. The class
of all such bodies will be denoted by IBnλ . We will prove that for
all λ < n, λ 6= 0,−2,−4, . . . , the class Inλ is the closure of IBnλ in
the radial metric. The case λ = 1 gives Proposition 1.3. It will be
proved that all m-dimensional central sections of λ-intersection bod-
ies are λ-intersection bodies in the corresponding m-planes provided
λ < m, λ 6= 0.
The natural question arises: How to construct λ-intersection bodies?
In Section 6 we give a series of examples; some of them are known and
some are new. They can be obtained by utilizing auxiliary statements
from Section 3. In particular, the famous embedding of Zhang’s class
Znn−k into Ink , which was first established in [K3] and studied in [Mi1],
[Mi2], will be generalized to the case, when k is replaced by any λ ∈
(0, n). Section 7 is devoted to the so called (q, ℓ)-balls, defined by
Bnq,ℓ = {x = (x′, x′′) : |x′|q + |x′′|q ≤ 1; x′ ∈ Rn−ℓ, x′′ ∈ Rℓ}, q > 0.
We show that if 0 < q ≤ 2, then Bnq,ℓ ∈ Inλ for all λ ∈ (0, n). If
q > 2 and n − 3 ≤ λ < n, we still have Bnq,ℓ ∈ Inλ . If q > 2 and
0 < λ < λ0 = max(n − ℓ, ℓ) − 2, then Bnq,ℓ 6∈ Inλ . The case, when
q > 2, ℓ > 1, and λ0 ≤ λ < n− 3 represents an open problem.
In Section 8 we remind the generalized Busemann-Petty problem
(GBP) for i-dimensional central sections of o.s. convex bodies in Rn.
This challenging problem is still open for i = 2 and i = 3 (n ≥ 5).
It actually inspires the whole investigation. Using properties of the
6 BORIS RUBIN
generalized cosine transforms, we give a short direct proof of the fact
that an affirmative answer to GBP implies that every smooth o.s. con-
vex body in Rn with positive curvature is an (n− i)-intersection body.
This fact was discovered by A. Koldobsky. The original proof in [K3]
is based on the embedding Inn−i ⊂ Zni and Zhang’s result [Z1, Theorem
6]. The latter heavily relies on the Hahn-Banach separation theorem.
Our proof is more constructive and almost self-contained. We conclude
the paper by Appendix, which is added for convenience of the reader.
The list of references at the end of the paper is far from being com-
plete. Further references can be found in cited books and papers.
Acknowledgement. I am grateful to Professor Alexander Koldob-
sky, who shared with me his knowledge of the subject. Special thanks
go to Professors Erwin Lutwak, Deane Yang, and Gaoyong Zhang for
useful discussions.
2. Preliminaries
2.1. Notation. In the following, N = {1, 2, . . . } is the set of all nat-
ural numbers, Sn−1 is the unit sphere in Rn with the area σn−1 =
2πn/2/Γ(n/2); Ce(S
n−1) is the space of even continuous functions on
Sn−1; SO(n) is the special orthogonal group of Rn; for θ ∈ Sn−1 and
γ ∈ SO(n), dθ and dγ denote the relevant invariant probability mea-
sures; D(Sn−1) is the space of C∞-functions on Sn−1 equipped with
the standard topology, and D′(Sn−1) stands for the corresponding dual
space of distributions. The subspaces of even test functions (distribu-
tions) are denoted by De(Sn−1) ( D′e(Sn−1)); Gn,i denotes the Grass-
mann manifold of i-dimensional subspaces ξ of Rn with the SO(n)-
invariant probability measure dξ; D(Gn,i) is the space of infinitely dif-
ferentiable functions on Gn,i.
We write M(Sn−1) and M(Gn,i) for the spaces of finite Borel mea-
sures on Sn−1 and Gn,i; M+(Sn−1) and M+(Gn,i) are the relevant
spaces of non-negative measures; Me+(Sn−1) denotes the space of even
measures µ ∈ M+(Sn−1). Given a function ϕ on Gn,i, we denote
ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−i. Similarly, given a measure µ ∈ M(Gn,n−i),
the corresponding “orthogonal measure” µ⊥ in M(Gn,i) is defined by
(µ⊥, ϕ) = (µ, ϕ⊥), ϕ ∈ C(Gn,i).
Let {Yj,k} be an orthonormal basis of spherical harmonics on Sn−1.
Here j = 0, 1, 2, . . . , and k = 1, 2, . . . , dn(j), where dn(j) is the di-
mension of the subspace of spherical harmonics of degree j. Each
function ω ∈ D(Sn−1) admits a decomposition ω =
j,k ωj,kYj,k with
the Fourier-Laplace coefficients ωj,k =
ω(θ)Yj,k(θ)dθ, which decay
rapidly as j → ∞. Each distribution f ∈ D′(Sn−1) can be defined by
INTERSECTION BODIES 7
(f, ω) =
j,k fj,kωj,k where fj,k = (f, Yj,k) grow not faster than j
m for
some integer m. We will need the Poisson integral, which is defined for
f ∈ L1(Sn−1) by
(2.1) (Πtf)(θ) = (1− t2)
f(u)|θ − tu|−ndu, 0 < t < 1,
and has the Fourier-Laplace decomposition Πtf =
j,k t
jfj,kYj,k [SW].
For f ∈ D′(Sn−1), this decomposition serves as a definition of Πtf . For
harmonic analysis on the unit sphere, the reader is referred to [Le],
[Mü], [Ne], [SW], and a survey article [Sa3].
2.2. Basic integral transforms. For integrable functions f on Sn−1
and ϕ onGn,i, 1 ≤ i ≤ n−1, the spherical Radon transform (Rif)(ξ), ξ ∈
Gn,i, and its dual (R
iϕ)(θ), θ ∈ Sn−1, are defined by
(2.2) (Rif)(ξ) =
θ∈Sn−1∩ξ
f(θ) dξθ, (R
iϕ)(θ) =
ϕ(ξ) dθξ,
where dξθ and dθξ denote the probability measures on the manifolds
Sn−1 ∩ ξ and {ξ ∈ Gn,i : ξ ∋ θ}, respectively. The precise meaning of
the second integral is
(2.3) (R∗iϕ)(θ) =
SO(n−1)
ϕ(rθγp0) dγ, θ ∈ Sn−1,
where p0 is an arbitrarily fixed coordinate i-plane containing the north
pole en and rθ ∈ SO(n) is a rotation satisfying rθen = θ.
Operators Ri and R
i extend to finite Borel measures in a canonical
way, using the duality
(2.4)
(Rif)(ξ)ϕ(ξ)dξ =
f(θ)(R∗iϕ)(θ)dθ.
Specifically, for µ ∈ M(Sn−1) and m ∈ M(Gn,i), we define Riµ ∈
M(Gn,i) and R∗im ∈ M(Sn−1) by
(2.5) (Riµ, ϕ)=
(R∗iϕ)(θ)dµ(θ), (R
im, f)=
(Rif)(ξ)dm(ξ),
where ϕ ∈ C(Gn,i), f ∈ C(Sn−1).
The generalized cosine transforms are defined by
(2.6) (Rαi f)(ξ) = γn,i(α)
|Prξ⊥θ|α+i−n f(θ) dθ,
(2.7) (
αϕ)(θ) = γn,i(α)
|Prξ⊥θ|α+i−n ϕ(ξ) dξ,
8 BORIS RUBIN
γn,i(α) =
σn−1 Γ((n− α− i)/2)
2π(n−1)/2 Γ(α/2)
, Re α > 0, α+i−n 6= 0, 2, 4, . . . .
Here Prξ⊥θ stands for the orthogonal projection of θ onto ξ
⊥, the or-
thogonal complement of ξ ∈ Gn,i. If f and ϕ are smooth enough, then
integrals (2.2) can be regarded (up to a constant multiple) as members
of the relevant analytic families (2.6) and (2.7); cf. Lemma 3.1. The
particular case i = n − 1 in (2.2) corresponds to the Minkowski-Funk
transform
(2.8) (Mf)(u) =
{θ : θ·u=0}
f(θ) duθ = (Rn−1f)(u
⊥), u ∈ Sn−1,
which integrates a function f over great circles of codimension 1. This
transform is a member of the analytic family
(2.9) (Mαf)(u) = (Rαn−1f)(u
⊥) = γn(α)
f(θ)|θ · u|α−1 dθ,
(2.10) γn(α)=
σn−1 Γ
(1−α)/2
2π(n−1)/2Γ(α/2)
, Re α>0, α 6=1, 3, 5, . . . .
The values α = 1, 3, 5, . . . are poles of the Gamma function Γ((1−α)/2).
In some occasions we include these values into consideration and set
(2.11) (M̃αf)(u) =
f(θ)|θ · u|α−1 dθ.
Historical notes. Regarding spherical Radon transforms (2.2) and
the Minkowski-Funk transform (2.8), see [GGG], [He], [R2], [R3]. The
first detailed investigation of the analytic family {Mα} is due to Se-
myanistyi [Se], who showed that these operators naturally arise in the
Fourier analysis of homogeneous functions. The case α = 2 in (2.11)
was known before, thanks to W. Blaschke, A.D. Alexandrov, and P.
Lévy. Integrals (2.9) (sometimes with different normalization) arise
in diverse areas of analysis and geometry; see [K4], [R1] - [R3], [Sa3],
[Str1], and references therein. In convex geometry and Banach space
theory, operators (2.11) with α − 1 replaced by p are known as the p-
cosine transforms. More general analytic families (2.6) and (2.7) were
introduced in [R2].
INTERSECTION BODIES 9
3. Analytic Families of the Generalized Cosine
Transforms
3.1. Basic properties. Below we review basic properties of integrals
(2.6), (2.7), (2.9); see [R2], [R3] for more details. For integrable func-
tions f and ϕ and Reα > 0, integrals (2.6), (2.7) and (2.9) are ab-
solutely convergent. When f and ϕ are infinitely differentiable, these
integrals extend meromorphically to all α ∈ C.
Lemma 3.1. If f and ϕ are continuous functions, then
Rαi f = R
i f = ciRif, ci =
2π(i−1)/2
;(3.1)
0ϕ = ciR
iϕ,(3.2)
Mαf = M0f = cn−1Mf, cn−1 =
2π(n−2)/2
.(3.3)
Hence, the Radon transform, its dual, and the Minkowski-Funk trans-
form can be regarded (up to a constant multiple) as members of the
corresponding analytic families {Rαi }, {
α}, {Mα}.
Proof. Formulas (3.2) and (3.3) follow from (3.1). To prove (3.1), we
write (2.6) in bi-spherical coordinates θ = u sin ψ + vcosψ, where
u ∈ Sn−1 ∩ ξ ∼ Si−1, v ∈ Sn−1 ∩ ξ⊥ ∼ Sn−i−1, 0 ≤ ψ ≤ π/2.
dθ = c sini−1 ψ cosn−i−1ψ dψdudv, c = σi−1σn−i−1/σn−1.
This gives
(Rαi f)(ξ) = c γn,i(α)
∫ π/2
sini−1 ψ cosα−1ψ dψ
Sn−1∩ξ⊥
Sn−1∩ξ
f(u sin ψ+vcosψ) du
ci(α)
Γ(α/2)
tα/2−1F (t) dt,
where
ci(α) =
c γn,i(α) Γ(α/2)
σi−1σn−i−1
Γ((n− α− i)/2)
2π(n−1)/2
2π(i−1)/2
as α → 0, and
F (t) = (1− t2)i/2−1
Sn−1∩ξ⊥
Sn−1∩ξ
1− t2+vt) du.
10 BORIS RUBIN
Since
Γ(α/2)
tα/2−1F (t) dt = F (0) =
Sn−1∩ξ
f(u)du = (Rif)(ξ),
we are done. �
Analytic continuation of integrals (2.9) can be realized in spherical
harmonics as Mαf=
mj,αfj,kYj,k, where
(3.4) mj,α=
(−1)j/2
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
if j is even,
0 if j is odd;
see [R1], [R3]. If f ∈D′(Sn−1), then Mαf is a distribution defined by
(Mαf, ω)=(f,Mαω)=
mj,α fj,k ωj,k, ω∈D(Sn−1); α 6=1, 3, 5, . . . .
Lemma 3.2. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If α + β = 2 − n and
f ∈ De(Sn−1) (or f ∈ D′e(Sn−1)), then
(3.5) MαMβf = f.
If α, 2−n−α 6= 1, 3, 5, . . ., then Mα is an automorphism of the spaces
De(Sn−1) and D′e(Sn−1).
Proof. The equality (3.5) is equivalent to mj,αmj,β = 1, α+β = 2−n.
The latter follows from (3.4). The second statement is a consequence of
the standard theory of spherical harmonics [Ne], because the Fourier-
Laplace multiplier mj,α has a power behavior as j → ∞. �
Corollary 3.3. The Minkowski-Funk transform on the spaces De(Sn−1)
and D′e(Sn−1) can be inverted by the formula
(3.6) (M)−1 = cn−1M
2−n, cn−1 =
2π(n−2)/2
Note that there is a wide variety of diverse inversion formulas for
the Minkowski-Funk transform (see [GGG], [He], [R3] and references
therein), but all of them are, in fact, different realizations of (3.6),
depending on classes of functions.
3.2. Auxiliary statements. We establish some connections between
operator families defined above.
INTERSECTION BODIES 11
Lemma 3.4. Let α, β ∈ C; α, β 6= 1, 3, 5, . . . . If Reα > Reβ, then
Mα =MβAα,β, where Aα,β is a spherical convolution operator with the
Fourier-Laplace multiplier
(3.7) aα,β(j) =
Γ(j/2 + (1− α)/2)
Γ(j/2 + (n− 1 + α)/2)
Γ(j/2 + (n− 1 + β)/2)
Γ(j/2 + (1− β)/2)
so that aα,β(j) ∼ (j/2)β−α as j → ∞. If α and β are real numbers
satisfying α > β > 1− n, α + β < 2, then Aα,β is an integral operator
such that Aα,βf ≥ 0 for every non-negative f ∈ L1(Sn−1).
Proof. The first statement follows from (3.4). To prove the second one,
we consider integral operators
+ f)(x) =
Γ(µ/2)
(1− t2)µ/2−1(Πtf)(x) tn−νdt,(3.8)
− f)(x) =
Γ(µ/2)
(t2 − 1)µ/2−1(Π1/tf)(x) t1−νdt,(3.9)
expressed through the Poisson integral (2.1). The Fourier-Laplace mul-
tipliers of Q
+ and Q
− are
(3.10) q̂
+ (j)=
Γ((j+n−ν+1)/2)
Γ((j+n−ν+1+µ)/2)
− (j)=
Γ((j+ν−µ)/2)
Γ((j+ν)/2)
They can be easily computed by taking into account that Πt ∼ tj in
the Fourier-Laplace terms. If f ∈ L1(Sn−1) and 0 < µ < ν < n, then
integrals (3.8) and (3.9) are absolutely convergent and obey Q
± f ≥ 0
when f ≥ 0. Comparing (3.10) and (3.7), we obtain a factorization
Aα,β = Q
α−β,1−β
α−β,1−β
− (set µ = α − β, ν = 1 − β), which implies
the second statement of the lemma. �
It is convenient to introduce a special notation for the spherical
Radon transform and the generalized cosine transform with orthogonal
argument. Assuming ξ ∈ Gn,i, we denote
(3.11) (Rn−i,⊥f)(ξ) = (Rn−if)(ξ
⊥), (Rαn−i,⊥f)(ξ) = (R
n−if)(ξ
Lemma 3.5. Let f ∈ L1(Sn−1), Re α > 0; α 6= 1, 3, 5, . . . . Then
(3.12) (RiM
αf)(ξ) = c (Rα+i−1n−i,⊥ f)(ξ), ξ ∈ Gn,i, c =
2π(i−1)/2
or (replace i by n− i)
(3.13) (Rn−i,⊥M
αf)(ξ) =
2π(n−i−1)/2
σn−i−1
(Rα+n−i−1i f)(ξ).
If f ∈ De(Sn−1), then (3.12) and (3.13) extend to Reα ≤ 0 by analytic
continuation.
12 BORIS RUBIN
Proof. For Reα > 0,
αf)(ξ) = γn(α)
Sn−1∩ξ
f(θ)|θ · u|α−1 dθ.
Since |θ · u| = |Prξθ||vθ · u| for some vθ ∈ Sn−1 ∩ ξ, by changing the
order of integration, we obtain
αf)(ξ) = γn(α)
f(θ)|Prξθ|α−1 dθ
Sn−1∩ξ
|vθ · u|α−1dξu.
The inner integral is independent on vθ and can be easily evaluated:
Sn−1∩ξ
|vθ · u|α−1dξu =
|t|α−1(1− t2)(i−3)/2 dt
2π(i−1)/2 Γ(α/2)
σi−1 Γ((i+ α− 1)/2)
This implies (3.12). �
The following statement is dual to Lemma 3.5.
Lemma 3.6. Let µ ∈ M(Gn,i), α 6= 1, 3, 5, . . . . Then
(3.14) MαR∗iµ = c
Rα+i−1n−i µ
⊥, c = 2π(i−1)/2/σi−1,
in the D′(Sn−1)-sense. If Reα > 0 and µ is absolutely continuous with
density ϕ ∈ L1(Gn,i), then
(3.15) MαR∗iϕ = c
Rα+i−1n−i ϕ
almost everywhere on Sn−1. If ϕ ∈ D(Gn,i), then (3.15) extends to all
complex α 6= 1, 3, 5, . . . by analytic continuation.
Proof. Let ω ∈ De(Sn−1) (it suffices to consider only even test func-
tions). By (2.4) and (3.12),
(MαR∗iµ, ω) = (µ,RiM
αω) = c (µ,Rα+i−1n−i,⊥ ω) = c (µ
⊥, Rα+i−1n−i ω).
This gives the result. �
The next statement contains explicit representations of the right in-
verse of the dual Radon transform R∗i (note that R
i is non-injective on
D(Gn,i) when 1 < i < n− 1).
Lemma 3.7. Every function f ∈De(Sn−1) is represented as f=R∗iAf ,
where A : De(Sn−1) → D(Gn,i),
(3.16) Af = c1R
i f = c2Rn−i,⊥M
2−nf,
π(1−i)/2σn−2
σn−i−1
Γ((n− i)/2)
Γ((n− 1)/2)
, c2 =
2πn/2−1
INTERSECTION BODIES 13
Proof. The coincidence of expressions in (3.16) follows from (3.13). To
prove the first equality, we invoke spherical convolutions defined by
analytic continuation of the integral
(3.17) (Qαf)(θ)=
σn−1Γ((n−1−α)/2)
2π(n−1)/2Γ(α/2)
(1−|u·θ|2)(α−n+1)/2f(u)du,
Reα > 0, α − n 6= 0, 2, 4, . . . , so that Q0f = f [R2]. By Theorem
1.1 from [R2], R∗iR
i f = c
α+i−1f , and therefore (set α = 1 − i),
i f = c
1 f , as desired. �
The next statement provides an intriguing factorization of the Minkowski-
Funk transform in terms of Radon transforms associated to mutually
orthogonal subspaces. This factorization can be useful in different oc-
currences.
Theorem 3.8. For f ∈ L1(Sn−1) and 0 < i < n,
(3.18) Mf = R∗iRn−i,⊥f.
Proof. By (2.3),
(R∗iRn−i,⊥f)(θ) =
SO(n−1)
(Rn−i,⊥f)(rθγR
i) dγ
SO(n−1)
(Rn−if)(rθγR
n−i) dγ
SO(n−1)
Sn−1∩rθγR
f(v) dv
Sn−1∩Rn−i
SO(n−1)
f(rθγw) dγ.
The inner integral is independent on w ∈ Sn−1 ∩ Rn−i and equals
(Mf)(θ). This gives (3.18). �
3.3. Restriction theorems. Theorems of such type deal with traces
of functions on lower dimensional subspaces and are well known, for in-
stance, in the theory of function spaces. To the best of our knowledge,
traces of functions represented by Radon transforms or, more generally,
by the generalized cosine transforms , were not studied systematically
and deserve particular attention, because they provide analytic back-
ground to a series of results related to sections of star bodies; cf. [R3,
Sec. 3.5], [FGW]. Given a subspace η ∈ Gn,m and k < m, we denote
by Gk(η) the manifold of all k-dimensional subspaces of η.
14 BORIS RUBIN
Theorem 3.9. Let f ∈ Ce(Sn−1), 1 ≤ k < m < n, λ 6= 0,−2,−4, . . . .
If Reλ < k, then for every η ∈ Gn,m and every ξ ∈ Gk(η),
(3.19) (Rk−λn−kf)(ξ
⊥) = (Rk−λm−kT
η f)(ξ
⊥ ∩ η),
where
(3.20) (T λη f)(u) = c̃
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ = π(m−n)/2 σn−m/2.
In particular (let λ→ k),
(3.21) (Rn−kf)(ξ
⊥)=c (Rm−kT
η f)(ξ
⊥ ∩ η), c= π
(n−m)/2 σm−k−1
σn−k−1
Proof. By (2.6),
(Rk−λn−kf)(ξ
⊥)=γn,n−k(k − λ)
|Prξθ|−λ f(θ) dθ.
We represent θ in bi-spherical coordinates as
(3.22) θ = ucosψ + v sinψ,
where
u ∈ Sn−1 ∩ η ∼ Sm−1, v ∈ Sn−1 ∩ η⊥ ∼ Sn−m−1, 0 ≤ ψ ≤ π/2,
dθ = c′′ sinn−m−1 ψ cosm−1ψ dψdudv, c′′ = σm−1σn−m−1/σn−1.
If ξ ⊂ η, then |Prξθ| = |Prξ[Prηθ]| = |Prξu| cosψ, and therefore,
(Rk−λn−kf)(ξ
⊥) = γm,m−k(k − λ)
Sn−1∩η
|Prξu|−λ(T λη f)(u) du,
where
(T λη f)(u) =
c′′ γn,n−k(k − λ)
γm,m−k(k − λ)
∫ π/2
sinn−m−1 ψ cosm−λ−1ψ dψ
Sn−1∩η⊥
f(ucosψ+v sin ψ) dv
π(m−n)/2 σn−m
Sn−1∩(η⊥⊕Ru)
f(w)|u · w|m−λ−1 dw.
Formula (3.21) follows from (3.19) by (3.1). �
INTERSECTION BODIES 15
Theorem 3.10. Let f ∈De(Sn−1), η∈Gn,m, 1<m<n. Suppose that
f =M1−λg, where Reλ < m, λ 6= 0,−2,−4, . . . . Then the restriction
of f onto η is represented as f =M1−λ
Sn−1∩η
T λη g, where T
η has the form
(3.20) andM1−λ
Sn−1∩η
denotes the same operator M1−λ, but on the sphere
Sn−1 ∩ η.
Proof. For Reλ < 1, the statement is a particular case of Theorem
3.9 (set k = 1). For other values of λ, the result follows by analytic
continuation. �
Remark 3.11. The restriction λ 6= 0,−2,−4, . . . in Theorems 3.9 and
3.10 is caused by the Gamma function Γ(λ/2) in the numerator of the
corresponding normalizing factor. It is evident from the proof, that
both theorems remain true also for λ = −2ℓ, ℓ ∈ N, if we remove the
normalizing factor. Then M1−λ in Theorem 3.10 will be substituted
for M̃1+2ℓ; see (2.11).
We will need the following generalization of Theorem 3.10.
Theorem 3.12. Let f ∈Ce(Sn−1), µ ∈ Me+(Sn−1), and let η∈Gn,m,
1<m<n. Suppose that f =M1−λµ, if λ < m, λ 6= −2ℓ, ℓ ∈ N, and
f=M̃1+2ℓµ, if λ = −2ℓ.
(i) There is a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of
f onto Sn−1 ∩ η is represented as f =M1−λ
Sn−1∩η
(ii) If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then (i) holds with dν(θ) =
(T λη g)(θ)dθ, where T
η g has the form (3.20).
(iii) If λ = −2ℓ, ℓ ∈ N, then (i) and (ii) hold with M1−λ
Sn−1∩η
substituted
for M̃1+2ℓ
Sn−1∩η
Proof. STEP 1. Let first λ < m, λ 6= 0,−2,−4, . . . . We invoke the
Poisson integral (2.1) so that
Πtf = ΠtM
1−λµ =M1−λgt, gt = Πtµ ∈ De(Sn−1), t ∈ (0, 1).
Since f is continuous, then Πtf converges to f as t → 0 uniformly
on Sn−1, and therefore, uniformly on Sn−1 ∩ η. Hence, for any test
function ω ∈ D(Sn−1 ∩ η), owing to Theorem 3.10, we have
(f, ω) = lim
(Πtf, ω) = lim
(M1−λgt, ω)
= lim
(M1−λ
Sn−1∩η
T λη gt, ω) = lim
(T λη gt,M
Sn−1∩η
= lim
(νt,M
Sn−1∩η
ω), νt = T
η gt.(3.23)
Thus, lim
(νt,M
Sn−1∩η
ω) exists for every ω∈D(Sn−1∩η). If ω is even,
i.e., ω ∈ De(Sn−1 ∩ η), then, by Lemma 3.2, we can replace ω by
16 BORIS RUBIN
M1−m+λ
Sn−1∩η
ω and conclude that the limit lim
(νt, ω) is well-defined for every
ω ∈ De(Sn−1∩η). Since νt = T λη Πtµ is an even function and the generic
test function ω ∈ D(Sn−1 ∩ η) can be represented as ω+ + ω−, where
ω± are even and odd, respectively, it follows that the limit lim
(νt, ω) =
(νt, ω+) is well-defined for every ω ∈ D(Sn−1∩ η) (not only for even
ω, as stated above). Since D′(Sn−1∩ η) is weakly complete, there is an
even distribution ν in D′(Sn−1 ∩ η) so that
(ν, ω) = lim
(νt, ω), ω ∈ D(Sn−1 ∩ η).
Furthermore, since (νt, ω) = (T
η Πtµ, ω) is non-negative for every non-
negative ω ∈ D(Sn−1 ∩ η) and every t ∈ (0, 1), then ν is a positive
distribution and, by Theorem 9.1, ν is a measure in Me+(Sn−1 ∩ η).
Thus, by (3.23), (f, ω) = lim
(νt,M
Sn−1∩η
ω) = (ν,M1−λ
Sn−1∩η
ω), which
means that f =M1−λ
Sn−1∩η
ν, as desired.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then νt = T λη Πtg tends to T λη g uni-
formly on Sn−1∩η as t→ 0. Hence, by (3.23), (f, ω) = (T λη g,M1−λSn−1∩ηω),
which means f =M1−λ
Sn−1∩η
T λη g.
STEP 2. Consider the case λ = −2ℓ, ℓ ∈ N, when f = M̃1+2ℓµ,
µ ∈ Me+(Sn−1), and the operator T λη = T−2ℓη has the form
(T−2ℓη h)(u) = c̃
Sn−1∩(η⊥⊕Ru)
|u · w|m+2ℓ−1 h(w) dw,
cf. (3.20). For any functions h ∈ C(Sn−1) and ω ∈ C(Sn−1 ∩ η),
(3.24) (T−2ℓη h, ω) = (h,
T−2ℓη ω),
where
T−2ℓη ω)(θ)=
Γ(m/2)
2Γ(n/2)
( Prηθ
|Prηθ|
|Prηθ|2ℓ ∈ C(Sn−1).
INTERSECTION BODIES 17
Indeed, using bi-spherical coordinates (see (3.22)), we have
(T−2ℓη h, ω) = c̃
Sn−1∩η
ω(u)du
Sn−1∩(η⊥⊕Ru)
h(w)|u · w|m+2ℓ−1 dw
c̃ σn−m−1
Sn−1∩η
ω(u)du
sinn−m−1 ψ cosm+2ℓ−1ψ dψ
Sn−1∩η⊥
h(ucosψ+v sin ψ) dv
c̃ σn−m−1
c′′ σn−m
h(θ)ω
( Prηθ
|Prηθ|
|Prηθ|2ℓ dθ = (h,
T−2ℓη ω).
Let h = Πtµ and observe that the limit lim
(T−2ℓη Πtµ, ω) exists, be-
cause, by (3.24), (T−2ℓη Πtµ, ω) = (Πtµ,
T−2ℓη ω) → (µ,
T−2ℓη ω). Note
that (T−2ℓη Πtµ, ω) ≥ 0 for any non-negative ω ∈ C(Sn−1∩η). Applying
the standard completeness argument (as in Step 1), we conclude, that
there is a measure ν ∈ M+(Sn−1 ∩ η) such that
(T−2ℓη Πtµ, ω) = (ν, ω) ∀ω ∈ C(Sn−1 ∩ η).
Using this equality, for f = M̃1+2ℓµ we obtain
(f, ω) = lim
(Πtf, ω) = lim
(ΠtM̃
1+2ℓµ, ω) = lim
(M̃1+2ℓΠtµ, ω)
(use Theorem 3.10 and Remark 3.11)
= lim
(M̃1+2ℓ
Sn−1∩η
T−2ℓη Πtµ, ω) = lim
(T−2ℓη Πtµ, M̃
Sn−1∩η
= (ν, M̃1+2ℓ
Sn−1∩η
This gives the result.
If dµ(θ) = g(θ)dθ, g ∈ Ce(Sn−1), then, by Theorem 3.10 and Remark
3.11, for θ ∈ Sn−1 ∩ η we have
(Πtf)(θ) = (ΠtM̃
1+2ℓg)(θ) = (M̃1+2ℓΠtg)(θ) = (M̃
Sn−1∩η
T−2ℓη Πtg)(θ).
Owing to continuity of the operators M̃1+2ℓ
Sn−1∩η
, T−2ℓη , and Πt in the
relevant spaces of continuous functions, by passing to the limit as t→ 0,
we obtain f(θ) = (M̃1+2ℓ
Sn−1∩η
T−2ℓη g)(θ), θ ∈ Sn−1 ∩ η, as desired. �
18 BORIS RUBIN
4. Positive Definite Homogeneous Distributions
We remind some known facts; see, e.g., [GS], [Le]. Let S(Rn) be the
Schwartz space of rapidly decreasing C∞-functions on Rn and S ′(Rn)
its dual. The Fourier transform of F ∈ S ′(Rn) is defined by
〈F̂ , φ̂〉 = (2π)n〈F, φ〉, φ̂(y) =
φ(x) eix·y dx, φ ∈ S(Rn).
A distribution F ∈ S ′(Rn) is homogeneous of degree λ ∈ C if for any
φ ∈ S(Rn) and any a > 0, 〈F, φ(x/a)〉 = aλ+n 〈F, φ〉. Homogeneous dis-
tributions on Rn are intimately connected with distributions on Sn−1.
Let first f ∈ L1(Sn−1), (Eλf)(x) = |x|λf(x/|x|), x ∈ Rn \ {0}. The
operator Eλ generates a meromorphic S ′-distribution
〈Eλf, φ〉= a.c.
rλ+n−1u(r)dr, u(r) =
f(θ)φ(rθ)dθ,
where “a.c.” denotes analytic continuation in the λ-variable. The dis-
tribution Eλf is regular if Reλ > −n and admits simple poles at
λ = −n,−n − 1, . . .. The above definition extends to all distributions
f ∈ D′(Sn−1) by the formula
〈Eλf, φ〉 = a.c.
rλ+n−1u(r)dr, u(r) = (f, φ(rθ)), 1
and the map Eλ : D′(Sn−1) → S ′(Rn) is weakly continuous. If f
is orthogonal to all spherical harmonics of degree j, then the deriv-
ative u(j)(r) equals zero at r = 0 and the pole at λ = −n − j is
removable. In particular, if f is an even distribution, i.e., (f, ϕ) =
(f, ϕ−), ϕ−(θ) = ϕ(−θ) ∀ϕ ∈ D(Sn−1), then the only possible poles
of Eλf are −n,−n− 2,−n− 4, . . . .
The Fourier transform of homogeneous distributions was extensively
studied by many authors; see [Sa3] and references therein. We restrict
our consideration to even distributions, when the operator family {Mα}
defined by (2.9) naturally arises thanks to the formula
(4.1) [E1−n−αf ]
∧ = 21−απn/2Eα−1M
This formula amounts to Semyanistyi [Se]. If f ∈ De(Sn−1), then (4.1)
holds pointwise for 0 < Reα < 1 (see, e.g., Lemma 3.3 in [R1] ) and
extends in the S ′-sense to all α ∈ C satisfying
(4.2) α /∈ {1, 3, 5, . . .} ∪ {1− n,−n− 1,−n− 3, . . .}.
1Here and on, different notations 〈·, ·〉 and (·, ·) are used for distributions on Rn
and Sn−1, respectively.
INTERSECTION BODIES 19
Since De(Sn−1) is dense in D′e(Sn−1) and the maps E1−n−α and Eα−1
are weakly continuous from D′e(Sn−1) to S ′(Rn), then (4.1) extends to
all f ∈ D′e(Sn−1).
Regarding the cases excluded in (4.2), we note that if α = 1+ 2ℓ for
some ℓ = 0, 1, . . ., then (4.1) is meaningful if and only if f is orthogonal
to all spherical harmonics of degree 2ℓ. If α = 1 − n − 2ℓ for some
ℓ = 0, 1, . . ., then, according to the spherical harmonic decomposition
j,k fj,kYj,k, j even, formula (4.1) is substituted for the following:
[E2ℓf ]
∧(ξ) = (2π)n
fj,k(−∆)ℓ−j/2Yj,k(i∂) δ(ξ)(4.3)
+2n+2ℓπn/2E−n−2ℓM
1−n−2ℓ
fj,kYj,k
where −∆ is the Laplace operator, ∂ = (∂/∂ξ1, . . . , ∂/∂ξn), and δ(ξ)
is the delta function. It is worth noting that for α = 1, 3, 5, . . ., the
distribution [E1−n−αf ]
∧ can also be understood in the regularized sense
without any orthogonality assumptions. However, such regularization
does not preserve homogeneity; see [Sa1], [Sa3].
Our main concern is positivity and positive definiteness of even ho-
mogeneous distributions. The reader is referred to [GV] for the general
theory. A distribution F ∈ S ′(Rn) is positive if 〈F, φ〉 ≥ 0 for all non-
negative φ ∈ S(Rn). A similar definition holds for distributions on the
sphere and on Rn \ {0}. A distribution F ∈ S ′(Rn) is positive definite
if F̂ is positive. For our purposes, it is important to know, which even
homogeneous distributions are positive definite. Let us rewrite (4.1)
and (4.2) with 1− n− α replaced by −λ. We have
(4.4) [E−λf ]
∧ = 2n−λπn/2Eλ−nM
1+λ−nf,
(4.5) λ /∈ Λ0, Λ0 = {n, n + 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 4.1. Let λ ∈ R \ Λ0, f ∈ D′e(Sn−1).
(i) If λ < 0 and E−λf is a positive definite distribution, then f = 0.
(ii) For all λ ∈ R \ Λ0, the following statements are equivalent:
(a) [E−λf ]
∧ is a positive distribution on Rn \{0} (for λ > 0, this can
be replaced by “E−λf is a positive definite distribution on R
(b) M1+λ−nf ∈ Me+(Sn−1);
(c) f =M1−λµ for some measure µ ∈ Me+(Sn−1).
Furthermore, for any real λ 6= 0,−2,−4, . . ., and any i = 1, 2, . . . , n−1,
(c) is equivalent to
(d) Rif = R
n−i,⊥µ for some measure µ ∈ Me+(Sn−1).
20 BORIS RUBIN
Proof. (i) Choose φ(x) = exp(−|x|m) pt,θ(x/|x|), where m ∈ 2N and
pt,θ(·) is the Poisson kernel
(4.6) pt,θ(u) =
1− t2
(1− 2tu · θ + t2)n/2
, 0 < t < 1; u, θ ∈ Sn−1.
Then 〈Eλ−nM1+λ−nf, φ〉 = cλ(ΠtM1+λ−nf)(θ), where
cλ = a.c.
rλ−1 exp(−rm) dr = m−1Γ(λ/m)
and (ΠtM
1+λ−nf)(θ) is the Poisson integral of M1+λ−nf . If E−λf is a
positive definite distribution, then, by (4.4), Eλ−nM
1+λ−nf is a positive
distribution. On the other hand, if λ < 0 and m > −λ, then cλ < 0.
Hence 〈Eλ−nM1+λ−nf, φ〉 can be non-negative for every non-negative
φ ∈ S(Rn) only if (ΠtM1+λ−nf)(θ) = 0 for every 0 < t < 1 and
θ ∈ Sn−1. The latter implies M1+λ−nf = 0, which is equivalent to
f = 0 because M1+λ−n is injective; see Lemma 3.2.
(ii) Let [E−λf ]
∧ be a positive distribution on Rn \{0}. It means that
for every φ ∈ S(Rn) such that φ ≥ 0 and 0 /∈ suppφ, 〈[E−λf ]∧, φ〉 ≥ 0
or, by (4.4), 〈Eλ−nM1+λ−nf, φ〉 ≥ 0. Choose φ(x) = ψ(|x|)ω(x/|x|),
where ω ∈ D(Sn−1), ω ≥ 0, and ψ is a smooth non-negative function
such that
rα+n−2ψ(r)dr = 1 and 0 /∈ suppψ. Then
〈Eλ−nM1+λ−nf, φ〉 = (M1+λ−nf, ω) ≥ 0,
and therefore, M1+λ−nf ∈ Me+(Sn−1); see Theorem 9.1.
Conversely, let µ = M1+λ−nf ∈ Me+(Sn−1) and let φ ∈ S(Rn);
φ ≥ 0. In the case λ < 0 we additionally assume 0 /∈ suppφ. By (4.4),
〈[E−λf ]∧, φ〉 = 2n−λπn/2 〈Eλ−nµ, φ〉
= 2n−λπn/2
rλ−1dr
φ(rθ)dµ(θ) ≥ 0.
This proves equivalence of (a) and (b). Equivalence of (b) and (c)
follows from Lemma 3.2.
Let us prove the equivalence of (c) and (d). If Rif = R
n−i,⊥µ,
µ ∈ Me+(Sn−1), then, by (3.15),
(f, R∗iϕ) = (Rif, ϕ) = (R
n−i,⊥µ, ϕ) = (µ,
Ri−λn−iϕ
= c−1(µ,M1−λR∗iϕ), ϕ ∈ D(Gn,i).
Since any function ω ∈ De(Sn−1) can be expressed as ω = R∗iϕ for
some ϕ ∈ D(Gn,i) (see Lemma 3.7), this gives (f, ω) = c−1(µ,M1−λ, ω)
which is (c). Conversely, let f = M1−λµ, µ ∈ Me+(Sn−1), that is,
(f, ω) = (µ,M1−λ, ω) for every ω ∈ De(Sn−1). Choose ω = R∗iϕ, ϕ ∈
INTERSECTION BODIES 21
D(Gn,i). Then, as above, (f, R∗iϕ) = (µ,M1−λR∗iϕ) = c (µ,
Ri−λn−iϕ
which gives (d). �
5. λ-intersection bodies
5.1. Definitions and comments. We remind that Kn is the set of
all origin-symmetric star bodies K in Rn, n ≥ 2; ρK and || · ||K are the
radial function and the Minkowski functional of K. The following defi-
nitions and statements are motivated by Theorem 4.1 and the previous
consideration. Let λ be a real number,
(5.1) sλ =
1 if λ > 0, λ 6= n, n+ 2, n+ 4, . . . ,
Γ(λ/2) if λ < 0, λ 6= −2,−4, . . . .
The values λ = 0, n, n + 2, n + 4, . . . will not be considered in the
following, but values λ = −2,−4, . . . will be included. They become
meaningful if we change normalization. For λ 6= 0, n, n + 2, n + 4 . . . ,
let Inλ be the set of bodies K ∈ Kn, for which there is a measure
µ ∈ Me+(Sn−1) such that sλρK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρK = M̃
1−λµ ≡ M̃1+2ℓµ, otherwise. The equality sλρK = M1−λµ
means that for any ϕ ∈ D(Sn−1),
ρkK(θ)ϕ(θ) dθ =
(M1−λϕ)(θ) dµ(θ),
where for λ ≥ 1, (M1−λϕ)(θ) is understood in the sense of analytic
continuation. We remind the notation
Λ0 = {n, n+ 2, n+ 4 . . .} ∪ {0,−2,−4, . . .}.
Theorem 5.1. For λ ∈ R\Λ0, the following statements are equivalent:
(a) K ∈ Inλ ;
(b) The Fourier transform [sλ || · ||−λK ]∧ is a positive distribution on
n\{0} (for λ > 0, this can be replaced by “|| · ||−λK is a positive definite
distribution on Rn”);
(c) sλM
1+λ−nρλK ∈ Me+(Sn−1);
The theorem is an immediate consequence of Theorem 4.1 if the lat-
ter is applied to f = sλρ
K . Another useful characterization is provided
by Theorem 4.1 (d).
Theorem 5.2. Let λ ∈ R \ Λ0. If K ∈ Inλ , then for every i ∈
{1, 2, . . . , n−1} there is a measure µ ∈ Me+(Sn−1) such that sλRiρλK =
Ri−λn−i,⊥µ. Conversely, if
sλRiρ
K = R
n−i,⊥µ
for some i ∈ {1, 2, . . . , n−1} and some µ ∈ Me+(Sn−1), then K ∈ Inλ .
22 BORIS RUBIN
Although Inλ was called “the set of bodies”, the definition of this set
is purely analytic and extra work is needed to understand what bodies
(if any) actually constitute the class Inλ .
The following comments will be helpful.
1. The case λ > n is not so interesting, because by Theorem 5.1(c),
Inλ is either empty (if Γ((n − λ)/2) < 0) or coincides with the whole
class Kn (if Γ((n− λ)/2) > 0).
2. The case λ ∈ (0, n) agrees with the concept of isometric embed-
ding of the space (Rn, || · ||K) into L−p, p = λ; see Introduction. In the
framework of this concept, all bodies K ∈ Inλ can be regarded as “unit
balls of n-dimensional subspaces of L−λ”.
3. If K ∈ Inλ , where λ < 0 (one can replace λ by = −p, p > 0), then
||u||pK =
|θ · u|p dµ(θ)
for some µ ∈ Me+(Sn−1). This is the well known Lévy representation,
characterizing isometric embedding of the space (Rn, || · ||K) into Lp;
see Lemma 6.4 in [K4]. Statement (b) in Theorem 5.1 agrees with
Theorem 1.9. Keeping this terminology, we can state the following
Proposition 5.3. Let p > −n, p 6= 0. Then (Rn, || · ||K) embeds
isometrically in Lp if and only if K ∈ In−p.
4. If λ = k ∈ {1, 2, . . . , n−1}, then Inλ = Ink coincides with the class
of k-intersection bodies; see Definition 1.7 and Theorem 1.8. Theorems
5.1 and 5.2 provide new characterizations of this class.
These comments inspire the following
Definition 5.4. Let λ < n, λ 6= 0. A body K ∈ Kn is said to be a
λ-intersection body if K ∈ Inλ , or, in other words, if there is a measure
µ ∈ Me+(Sn−1) such that sλρλK = M1−λµ if λ 6= −2ℓ, ℓ ∈ N, and
ρ−2ℓK = M̃
1+2ℓµ, otherwise.
The result of Theorem 5.2 for λ = i = k can serve as an alternative
definition of k-intersection bodies in terms of Radon transforms. This
definition agrees with Definition 1.6 and mimics Definition 1.2.
Definition 5.5. Let k ∈ {1, 2, . . . , n − 1}. A body K ∈ Kn is a k-
intersection body if there is a non-negative measure µ on Sn−1 such
(5.2) (Rkρ
K)(ξ) = (Rn−kµ)(ξ
⊥), ξ ∈ Gn,k.
INTERSECTION BODIES 23
Equality (5.2) is understood in the weak sense according (2.5). Namely,
for ϕ ∈ C(Gn,k) and ϕ⊥(η) = ϕ(η⊥), η ∈ Gn,n−k, (5.2) means
(5.3)
K)(ξ)ϕ(ξ) dξ =
(R∗n−kϕ
⊥)(θ) dµ(θ).
5.2. λ-intersection bodies of star bodies and closure in the ra-
dial metric. As we mentioned in Introduction, the class of intersection
bodies, which coincides with Inλ when λ = 1, is the closure in the ra-
dial metric of the class of intersection bodies of star bodies. Below
we extend this result to all λ < n, λ 6= 0, in the framework of the
unique approach. We remind (see Definition 1.6) that K ∈ Kn is a
k-intersection body of a body L ∈ Kn and write K = IBk(L) if
(5.4) volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) ∀ξ ∈ Gn,k.
Let IBk,n be the set of all bodies K ∈ Kn satisfying (5.4) for some
L ∈ Kn.
How can we extend the purely geometric property (5.4) to non-
integer values of k? To this end, we first express (5.4) in terms of
the generalized cosine transforms (2.9).
Lemma 5.6. If K = IBk(L) is infinitely smooth, then
(5.5) ρn−kL =cM
1−n+kρkK , ρ
−1M1−kρn−kL ,
c = πk−n/2(n− k)/k.
Proof. We make use of (3.13), where we set i = k, α = 1 − n + k and
f = ρkK . By (3.1), this gives
(5.6) Rkρ
K = c̃Rn−k,⊥M
1−n+kρkK , c̃ =
πk−n/2 σn−k−1
On the other hand, if K = IBk(L) is infinitely smooth, then, according
to (5.4) and the equality
(5.7) volk(K ∩ ξ) =
K)(ξ),
we have
(5.8) Rkρ
k σn−k−1
(n− k) σk−1
Rn−k,⊥ρ
Comparing (5.6) and (5.8), owing to injectivity of the Radon transform,
we obtain the first equality in (5.5). The second equality follows from
the first one by (3.5). �
24 BORIS RUBIN
Equalities (5.5) are extendable to non-integer values of k. We denote
cλ,n = π
λ−n/2(n−λ)/λ,
and let sλ be defined by (5.1).
Definition 5.7. Let λ < n, λ 6= 0; K,L ∈ Kn. We say that K is a
λ-intersection body of L and write K = IBλ(L) if sλρλK=c−1λ,nM1−λρ
in the case λ 6= −2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓρ
L , otherwise. The
set of all λ-intersection bodies of star bodies will be denoted by IBλ,n.
We also denote
(5.9) IB∞λ,n={K ∈ IBλ,n : ρK ∈ De(Sn−1)}.
By (3.5), equality sλρ
K = c
1−λρn−λL is equivalent to ρ
sλ cλ,nM
1−n+λρλK . Both equalities are generally understood in the sense
of distributions, for instance,
K , ϕ) = c
λ,n(ρ
1−λϕ), ϕ ∈ D(Sn−1).
If K (or L) is smooth, then sλρ
K(θ)=c
λ,n(M
1−λρn−λL )(θ) pointwise for
every θ∈Sn−1.
Theorem 5.8. Let λ < n, λ 6= 0. If λ 6= −2ℓ, ℓ ∈ N, then the class
Inλ of λ-intersection bodies is the closure of the classes IBλ,n and IB∞λ,n
of λ-intersection bodies of star bodies in the radial metric:
(5.10) Inλ = cl IBλ,n = cl IB∞λ,n.
If λ = −2ℓ, ℓ ∈ N, then Inλ ⊂ cl IBλ,n = cl IB∞λ,n.
Proof. STEP 1. We first prove that Inλ ⊂ cl IB∞λ,n. Let K ∈ Inλ , i.e.,
(a) sλρ
1−λµ, µ ∈ Me+(Sn−1), if λ 6= −2ℓ, ℓ ∈ N, and
(b) ρ−2ℓK = M̃
1+2ℓµ, otherwise.
Our aim is to define a sequence Kj ∈ IB∞λ,n such that ρKj → ρK in the
C-norm. Consider the Poisson integral Πtρ
K (see (2.1)), that converges
to ρλK in the C-norm when t→ 1. In the case (a), for any test function
ω ∈ D(Sn−1) we have
K , ω) = (ρ
K ,Πtω) = s
λ (µ,M
1−λΠtω) = s
1−λΠtµ, ω).
Similarly, in the case (b), we have a pointwise equality (Πtρ
K )(θ) =
(M̃1+2ℓΠtµ)(θ), θ ∈ Sn−1. Choose Kj so that ρλKj = Πtjρ
K , where tj is
a sequence in (0, 1) approaching 1. Clearly, Kj converges to K in the
radial metric. Moreover, Kj ∈ IB∞λ,n, because ρλKj = s
1−λρn−λLj
and ρ−2ℓKj = M̃
1+2ℓρn+2ℓLj , where the bodies Lj are defined by ρ
cλ,nΠtjµ in the case (a), and ρ
= Πtjµ in the case (b), respectively.
INTERSECTION BODIES 25
Conversely, let K ∈ cl IB∞λ,n, λ 6= −2,−4, . . . . It means that there
is a sequence of Kj ∈ IB∞λ,n such that lim
||ρK − ρKj ||C(Sn−1) = 0 and
= c−1λ,nM
1−λρn−λLj , ρLj ∈ De+(S
n−1). If j → ∞, then for every
ω ∈ D(Sn−1),
(5.11) sλ(ρ
,M1−n+λω) → sλ(ρλK ,M1−n+λω)=sλ(M1−n+λρλK , ω).
The right-hand side of (5.11) is non-negative, because by (3.5), for
every j and every ω ∈ De+(Sn−1),
,M1−n+λω) = c−1λ,n(M
1−λρn−λLj ,M
1−n+λω) = c−1λ,n(ρ
, ω) ≥ 0.
By Theorem 9.1, it follows that sλM
1−n+λρλK is a non-negative mea-
sure. We denote it by µ. By (3.5), for any ω ∈ D(Sn−1),
K , ω) = sλ(M
1−n+λρλK ,M
1−λω) = (µ,M1−λω) = (M1−λµ, ω),
i.e., K∈Inλ . This gives IB∞λ,n⊂Inλ and, by above, Inλ =cl IB∞λ,n.
STEP 2. It remains to prove that cl IB∞λ,n = cl IBλ,n. Since IB∞λ,n ⊂
IBλ,n, then cl IB∞λ,n ⊂ cl IBλ,n. To prove the opposite inclusion, let
K ∈ cl IBλ,n and consider the case λ 6= −2,−4, . . . . We have to show
that there is a sequence of smooth bodies Kj, which converges to K in
the radial metric and such that sλρ
= c−1λ,nM
1−λρn−λLj for some bodies
Lj ∈ Kn. Since K ∈ cl IBλ,n, there is a sequence K̃j ∈ Kn such that
||ρK̃j −ρK ||C(Sn−1) = 0 and sλρ
= c−1λ,nM
1−λρn−λ
for some bodies
L̃j ∈ Kn. We define smooth bodies Kj and Lj by
ρλKj = Π1−1/jρ
, ρn−λLj = Π1−1/jρ
where Π1−1/j stands for the Poisson integral with parameter 1 − 1/j.
Since operators Π1−1/j andM
1−λ commute, then sλρ
=c−1λ,nM
1−λρn−λLj ,
and therefore, Kj ∈ IB∞λ,n. On the other hand, by the properties of the
Poisson integral [SW],
|ρλKj − ρ
K | ≤ |Π1−1/jρλK̃j − Π1−1/jρ
K |+ |Π1−1/jρλK − ρλK | → 0
as j → ∞. It means, that K ∈ cl IB∞λ,n or cl IBλ,n ⊂ cl IB∞λ,n. Hence,
by above, cl IBλ,n = cl IB∞λ,n. For λ = −2,−4, . . . , the argument is
similar. �
Remark 5.9. If λ = −2,−4, . . . , we cannot prove the coincidence of
Inλ and cl IB∞λ,n, because the proof of the embedding cl IB∞λ,n ⊂ Inλ
relies heavily on the fact that M1−λ is an isomorphism of De(Sn−1). If
λ = −2,−4, . . . , this is not so, and the operator M̃1−λ has a nontrivial
kernel, which consists of spherical harmonics of degree > 2ℓ; see [R1]
for details.
26 BORIS RUBIN
It is interesting to translate Theorem 5.8 for λ = −p, p > 0, into
the language of isometric embeddings. Ignoring a non-important pos-
itive constant factor and using polar coordinates, one can replace the
equalities sλρ
K = c
1−λρn−λL and ρ
K = M̃
1+2ℓρn+2ℓL in Definition
5.7 by
(5.12) ||u||pK =
|x · u|p dx, u ∈ Sn−1.
Corollary 5.10.
(i) A unit ball of every n-dimensional subspace of Lp, can be approxi-
mated in the radial metric by bodies K, defined by (5.12), where L ∈ Kn
has a C∞ boundary.
(ii) If, moreover, p 6= 2, 4, . . . , then the set of unit balls of all n-
dimensional subspaces of Lp, can be identified with the closure in the
radial metric of the set of bodies K satisfying (5.12) for some smooth
body L ∈ Kn (one can also consider arbitrary bodies L ∈ Kn).
5.3. Central sections of λ-intersection bodies. It is known, that
a cross-section K ∩ η of a body K ∈ Ink by any m-dimensional central
plane η is a k-intersection body in η provided 1 ≤ k < m < n. This
fact was established in [Mi1, Proposition 3.17] by using Theorem 1.8
and a certain approximation procedure. Below we present more general
results, including sections of k-intersection bodies of star bodies and
the case of non-integer k = λ. These results are consequences of the
restriction theorems from Section 3.3.
Theorem 5.11. Let 1 ≤ k < m < n, η ∈ Gn,m. If K = IBk(L) in Rn,
then K ∩ η = IBk(L̃) in η, where the body L̃ is defined by
(5.13) ρm−k
(u) = ck,m,n
Sn−1∩(η⊥⊕Ru)
ρn−kL (w)|u · w|
m−k−1 dw,
u ∈ Sn−1 ∩ η, ck,m,n =
(m− k) σn−m
2(n− k)
Proof. By (5.7) and (3.21) (with f = ρn−kL ),
volk(K ∩ ξ) = voln−k(L ∩ ξ⊥) =
σn−k−1
(Rn−kρ
L )(ξ
c σn−k−1
(Rm−kT
L )(ξ
⊥ ∩ η)(5.14)
σm−k−1
(Rm−kρ
)(ξ⊥ ∩ η) = volm−k(L̃ ∩ ξ⊥),
as desired. �
INTERSECTION BODIES 27
Theorem 5.11 has the following generalization.
Theorem 5.12. Let 1 < m < n, η ∈ Gn,m and suppose that λ < m,
λ 6= 0. If K = IBλ(L) in Rn, then K ∩ η = IBλ(L̃) in η, where the
body L̃ is defined by
(5.15) ρm−λ
(u) = c̃
Sn−1∩(η⊥⊕Ru)
ρn−λL (w)|u · w|
m−λ−1 dw,
u ∈ Sn−1 ∩ η, c̃ =
(m− λ) σn−m
2(n− λ)
if λ 6= −2ℓ, ℓ ∈ N,
π(m−n)/2 σn−m/2 otherwise.
Moreover, if K ∈ Inλ in Rn, then K ∩ η ∈ Imλ in η.
Proof. Let λ 6= −2ℓ, ℓ ∈ N, and let θ ∈ Sn−1 ∩ η. By Definition 5.7,
K = c
1−λρn−λL , and Theorem 3.12 (with f = sλρ
K and g =
c−1λ,nρ
L ) yields
K(θ) = (M
Sn−1∩η
T λη [c
L ])(θ) = c
λ,m(M
Sn−1∩η
)(θ),
where ρm−λ
= c T λη ρ
L , c = π
(n−m)/2(m − λ)/(n − λ). By Definition
5.7 and (3.20), we are done. If λ = −2ℓ, ℓ ∈ N, then, as above,
ρ−2ℓK (θ) = (M̃
Sn−1∩η
T−2ℓη ρ
L )(θ) = (M
Sn−1∩η
ρm+2ℓ
where ρm+2ℓ
= T−2ℓη ρ
L . This gives (5.15).
Furthermore, if K ∈ Inλ , λ 6= −2ℓ, ℓ ∈ N, then, by Definition 5.4,
K = M
1−λµ, µ ∈ Me+(Sn−1). Hence, by Theorem 3.12, there is
a measure ν ∈ Me+(Sn−1 ∩ η) such that the restriction of sλρλK onto
Sn−1∩η is represented as sλρλK =M1−λSn−1∩ην. It means that K∩η ∈ I
in η. In the case λ = −2ℓ, ℓ ∈ N, the argument is similar. �
6. Examples of λ-intersection bodies
The definition of the classes Inλ and IBλ,n and all known characteri-
zations are purely analytic. Unlike the case λ = 1, when an intersection
body of a star body is explicitly defined by a simple geometric proce-
dure, it is not clear how can we construct λ−intersection bodies in the
general case. Below we give some examples, when the radial function
of a λ−intersection body can be explicitly determined. These examples
utilize the generalized cosine transforms.
Example 6.1. Let λ < 1, λ 6= 0. This case is the simplest. Indeed,
given a non-negative measure µ on Sn−1, the relevant λ−intersection
28 BORIS RUBIN
body can be directly constructed by the formula ρλK = M
1−λµ, if λ 6=
−2ℓ, ℓ ∈ N, and ρ−2ℓK = M̃1+2ℓµ, otherwise. In other words (cf. (2.11)),
(6.1) ρλK(u) =
|θ · u|−λ dµ(θ).
This fact (with λ replaced by −p) is a reformulation of Theorem 6.17
from [K4], which was stated in the language of isometric embeddings
and relies on the P. Lévy characterization; see also Lemma 6.4 and
Theorem 4.11 in [K4].
Example 6.2. If n − 3 ≤ λ < n, λ > 0, then Inλ includes all origin-
symmetric convex bodies in Rn.
This fact is due to Koldobsky [K4, Corollary 4.9]. It can be proved
using a slight modification of the argument from [R3, Sec. 7] as follows.
By Theorem 5.1 (c), it suffices to check that for any o.s. convex body
K,M1+λ−nρλK ∈ Me+(Sn−1). For λ ≥ n−1, this is obvious. To handle
the case n− 3 ≤ λ < n− 1, suppose first that K is infinitely smooth.
Using polar coordinates, for Reα > 0, we can write
(6.2) (Mαρα+n−1K )(u) = (α + n− 1) γn(α)
|x · u|α−1 dx.
Then M1+λ−nρλK can be realized as analytic continuation (a.c.) at α =
1 + λ− n of the right-hand side of (6.2). The latter can be written as
I(α) = 2(α+ n− 1)γn(α)
tα−1AK,u(t) dt,
AK,u(t) = voln−1(K∩{tu+u⊥}). Taking analytic continuation (see [GS,
Chapter 1]), for −2 < α < 0 (which is equivalent to n−3 ≤ λ < n−1)
we get
a.c.I(α) = c1
tα−1[AK,u(t)−AK,u(0)] dt.
Similarly, a.c.I(α) at α = −2 (which corresponds to λ = n − 3) is
K,u(0). Following [GS], one can easily check that constants c1 and
c2 are negative. Since K is convex, both analytic continuations are
positive, and thereforeM1+λ−nρλK > 0. If K is an arbitrary o.s. convex
body, we approximate it in the radial metric by smooth o.s. convex
bodies Kj. Then for any test function ω ∈ D+(Sn−1), by the previous
step we have
(M1+λ−nρλK , ω) = (ρ
1+λ−nω) = lim
(ρλKj ,M
1+λ−nω)
= lim
(M1+λ−nρλKj , ω) ≥ 0.
INTERSECTION BODIES 29
Hence, by Theorem 9.1, M1+λ−nρλK is a non-negative measure and the
proof is complete.
Example 6.3. If ρλK =
Ri−λn−iν for some ν ∈ M+(Gn,n−i) and λ ≤ i < n,
then K ∈ Inλ .
Indeed, for any test function ω ∈ D(Sn−1), by (3.12) (with α = 1−λ)
we have
(ρλK , ω) = (
Ri−λn−iν, ω) = (ν, R
n−iω) = (ν
⊥, Ri−λn−i,⊥ω)
= c−1(ν⊥, RiM
1−λω) = c−1(R∗i ν
⊥,M1−λω), c =
2π(i−1)/2
It means that for 0 < λ ≤ i < n and ν ∈ M+(Gn,n−i),
(6.3) ρλK =
Ri−λn−iν ⇐⇒ {ρλK =M1−λµ, µ = c−1R∗i ν⊥}.
By Definition 5.4, this gives the result. The particular case λ = i
implies the embedding into Ini of the Zhang’s class Znn−i; see Definition
1.5. This embedding was proved in [K3] and [Mi1] in a different way;
see also [Mi2], where it is proved that Znn−i is a proper subset of Ini if
2 ≤ i ≤ n− 2.
Example 6.4. If 0 < (i− 1)/2 < λ ≤ i < n and ρλK = M i−λµ for some
µ ∈ M+(Sn−1), then K ∈ Inλ .
Indeed, by Lemma 3.4 (with α = i− λ, β = 1− λ), ρλK =M i−λµ =
M1−λAi−λ,1−λ, where Ai−λ,1−λ is an integral operator which preserves
positivity provided i− λ > 1− λ > 1− n, (i− λ) + (1− λ) < 2. This
is just our case.
Example 6.5. One can construct bodies K ∈ Inλ from bodies L ∈ Inδ
by the formula ρK = ρ
L provided 0 < δ < λ < n.
Indeed, by Definition 5.4, there is a measure µ ∈ M+(Sn−1) so that
ρδL = M
1−δµ. Then, by Lemma 3.4 (with α = 1 − δ, β = 1 − λ),
ρλK = ρ
L = M
1−δµ = M1−λA1−δ,1−λµ, and we are done. This example
generalizes the corresponding result from [Mi1, p. 533, Statement (c)],
which was obtained in a different way for the case, when λ and δ are
integers.
Example 6.6. Let
(6.4) Bnq = {x ∈ Rn : ||x||q =
|xk|q
≤ 1}.
If 0 < q ≤ 2, then Bnq ∈ Inλ for all λ ∈ (0, n). If 2 < q <∞, λ ∈ (0, n),
then Bnq ∈ Inλ if and only if λ ≥ n− 3.
30 BORIS RUBIN
Both statements are due to Koldobsky. The first one follows from
the fact that for 0 < q ≤ 2 the Fourier transform of ||x||−λq is a positive
S ′-distribution (see Lemmas 3.6 and 2.27 in [K4]). The second state-
ment is a reformulation of Theorem 4.13 from [K4]. The “if” part is a
consequence of Example 6.2.
7. (q, ℓ)-balls
In this section we consider one more example, which resembles Ex-
ample 6.6, but does not fall into its scope and requires a separate
consideration. Let
x = (x′, x′′) ∈ Rn, x′ ∈ Rn−ℓ =
Rej , x
′′ ∈ Rℓ =
j=n−ℓ+1
Rej ,
e1, . . . , εn being coordinate unit vectors. Consider the (q, ℓ)-ball
(7.1) Bnq,ℓ = {x : ||x||q,ℓ = (|x′|q + |x′′|q)1/q ≤ 1}, q > 0.
We wonder, for which triples (q, ℓ, n), Bnq,ℓ is a λ-intersection body. To
study this problem, we need some preparation. Consider the Fourier
integral
(7.2) γq,ℓ(η) =
e−|y|
eiy·η dy, η ∈ Rℓ, q > 0.
The function γq,ℓ(η) is uniformly continuous on R
ℓ and vanishes at
infinity.
Lemma 7.1. If 0 < q ≤ 2, then γq,ℓ(η) > 0 for all η ∈ Rℓ.
Proof. (Cf. [K4, p. 44, for ℓ = 1]). For η = 0, the statement is obvious.
It is known (see, e.g., [SW]), that
(7.3) [e−t|· |
]∧(η) = πℓ/2t−ℓ/2e−|η|
2/4t, t > 0.
This gives the result for q = 2. Let 0 < q < 2. By Bernstein’s theorem
[F, Chapter 18, Sec. 4], there is a non-negative finite measure µq on
[0,∞) so that e−zq/2 =
e−tz dµq(t), z ∈ [0,∞). Replace z by |y|2 to
(7.4) e−|y|
e−t|y|
dµq(t).
Then (7.3) yields
γq,ℓ(η) =
eiy·ηdy
e−t|y|
dµq(t) =
dµq(t)
eiy·ηe−t|y|
= πℓ/2
t−ℓ/2e−|η|
2/4t dµq(t) > 0.
INTERSECTION BODIES 31
The Fubini theorem is applicable here, because, by (7.4),
|eiy·η|dy
e−t|y|
dµq =
e−|y|
dy <∞.
Our next concern is the behavior of γq,ℓ(η) when |η| → ∞. If q
is even, then e−|·|
is a Schwartz function and therefore, γq,ℓ is infin-
itely smooth and rapidly decreasing. In the general case, we have the
following.
Lemma 7.2. For any q > 0,
(7.5) lim
|η|→∞
|η|ℓ+qγq,ℓ(η) = 2ℓ+qπℓ/2−1Γ(1+ q/2)Γ((ℓ+ q)/2) sin(πq/2).
Proof. For ℓ = 1, this statement can be found in [PS, Chapter 3, Prob-
lem 154] and in [K4, p. 45]. In the general case, the proof is more
sophisticated and relies on the properties of Bessel functions. By the
well-known formula for the Fourier transform of a radial function (see,
e.g., [SW]), we write γq,ℓ(η) = I(|η|), where
I(s) = (2π)ℓ/2s1−ℓ/2
rℓ/2Jℓ/2−1(rs) dr
= (2π)ℓ/2s−ℓ
[(rs)ℓ/2Jℓ/2(rs)] dr.
Integration by parts yields
I(s) = q(2π)ℓ/2s−ℓ/2
rℓ/2+q−1Jℓ/2(rs) dr.
Changing variable z = sqrq, we obtain
sℓ+qI(s) = (2π)ℓ/2A(s1/q), A(δ) =
e−zδzℓ/2qJℓ/2(z
1/q) dz.
We actually have to compute the limit A0 = lim
A(δ). To this end, we
invoke Hankel functions H
ν (z), so that Jν(z) = ReH
ν (z) if z is real
[Er]. Let hν(z) = z
ν (z). This is a single-valued analytic function
in the z-plane with cut (−∞, 0]. Using the properties of the Bessel
functions [Er], we get
(7.6) lim
hν(z) = 2
νΓ(ν)/πi,
(7.7) hν(z) ∼
2/π zν−1/2eiz−
(ν+ 1
), z → ∞.
Then we write A(δ) as A(δ) = Re
e−zδhℓ/2(z
1/q) dz and change the
line of integration from [0,∞) to ℓθ = {z : z = reiθ, r > 0} for
32 BORIS RUBIN
small θ < πq/2. By Cauchy’s theorem, owing to (7.6) and (7.7), we
obtain A(δ) = Re
e−zδhℓ/2(z
1/q) dz. Since for z = reiθ, hℓ/2(z
1/q) =
O(1) when r = |z| → 0 and hℓ/2(z1/q) = O(r(ℓ−1)/2qe−r
1/q sin(θ/q)) as
r → ∞, by the Lebesgue theorem on dominated convergence, we get
A0 = Re
hℓ/2(z
1/q) dz. To evaluate the last integral, we again use
analyticity and replace ℓθ by ℓπq/2 = {z : z = reiπq/2, r > 0} to get
A0 = Re
eiπq/2
hℓ/2(r
1/qeiπ/2) dr
To finalize calculations, we invoke McDonald’s function Kν(z) so that
hν(z) = z
νH(1)ν (z) = −
(ze−iπ/2)νKν(ze
−iπ/2).
This gives
sin(πq/2)
sℓ/2+q−1Kℓ/2(s) ds.
The last integral can be explicitly evaluated by the formula 2.16.2 (2)
from [PBM], and we obtain the result. �
Now we can proceed to studying (q, ℓ)-balls Bnq,ℓ; see (7.1). There is
an intimate connection between geometric properties of the balls Bnq,ℓ
and the Fourier transform of the power function || · ||pq,ℓ. The case q = 2
is well-known and associated with Riesz potentials; see, e.g., [St]. The
relevant case of ℓnq -balls, which agrees with ℓ = 1 was considered in
Example 6.6.
Lemma 7.3. Let q > 0, ξ = (ξ′, ξ′′) ∈ Rn, γq,ℓ(ξ′′) and γq,n−ℓ(ξ′) be
the functions of the form (7.2). We define
(7.8) hp,q,ℓ(ξ) =
Γ(−p/q)
tn+p−1 γq,n−ℓ(ξ
′t) γq,ℓ(ξ
′′t) dt.
(i) Let ξ′ 6= 0 and ξ′′ 6= 0. If q is even, then the integral (7.8) is abso-
lutely convergent for all p > −n. Otherwise, it is absolutely convergent
when −n < p < 2q. In these cases, hp,q,ℓ (ξ) is a locally integrable
function away from the coordinate subspaces Rℓ and Rn−ℓ.
(ii) If −n < p < 0, then hp,q,ℓ (ξ) ∈ L1loc(Rn)∩S ′(Rn) and (||·||
∧(ξ) =
hp,q,ℓ(ξ) in the sense of S ′-distributions. Specifically, for ϕ ∈ S(Rn),
(7.9) 〈hp,q,ℓ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉.
INTERSECTION BODIES 33
Proof. (i) For any 0 < ε < a <∞,
ε<|ξ′|<a
ε<|ξ′′|<a
|hp,q,ℓ (ξ; , ξ′′)| dξ′′
|Γ(−p/q)|
tn+p−1 dt
ε<|ξ′|<a
|γq,n−ℓ (ξ′t)| dξ′
ε<|ξ′′|<a
|γq,ℓ (ξ′′t)| dξ′′
|Γ(−p/q)|
tp−1 dt
tε<|z′|<ta
|γq,n−ℓ (z′)| dz′
tε<|z′′|<ta
|γq,ℓ (z′′)| dz′′
|Γ(−p/q)|
(...) =
|Γ(−p/q)|
(I1 + I2).
The first integral is dominated by
tn+p−1 dt, c = σn−ℓ−1σℓ−1max
|γq,n−ℓ (z′)| max
|γq,ℓ (z′′)|
and is finite for p > −n. The second integral can be estimated by
making use of Lemma 7.2. Specifically, if q is not an even integer, then
I2 ≤ cε
tp−1 dt
|z′|>tε
|z′|n−ℓ+q
|z′′|>tε
|z′′|ℓ+q
tp−2q−1 dt.
If q is even, then γq,ℓ and γq,n−ℓ are rapidly decreasing and I2 ≤
tp−2m−1 dt for any m > 0. This gives what we need.
(ii) If −n < p < 0, the same argument is applicable with ε = 0. In
this case, I2 does not exceed ||γq,n−ℓ||1||γq,ℓ||1
tp−1 dt. The latter is
finite when p < 0, because, by Lemma 7.2, γq,n−ℓ and γq,ℓ are integrable
functions on respective spaces. When ξ → ∞, one can readily check
that hp,q,ℓ (ξ) = O(|ξ|m) for some m > 0, and therefore, hp,q,ℓ ∈ S ′(Rn).
To compute the Fourier transform (|| · ||pq,ℓ)∧(ξ), we replace ||x||
q,ℓ by
the formula
||x||pq,ℓ =
Γ(−p/q)
tp−1 e−|x
′/t|q−|x′′/t|q dt, p < 0,
34 BORIS RUBIN
and note that the Fourier transform of the function x→ e−|x′/t|q−|x′′/t|q
is just γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t). Then
〈|| · ||pq,ℓ)
∧ , ϕ̂〉 = (2π)n〈|| · ||pq,ℓ , ϕ〉
(2π)nq
Γ(−p/q)
tp−1 dt
′/t|q−|x′′/t|qϕ(x) dx
Γ(−p/q)
tn+p−1 dt
γq,n−ℓ (ξ
′t) γq,ℓ (ξ
′′t) ϕ̂(ξ) dξ.
Interchange of the order of integration in this argument can be easily
justified using absolute convergence of integrals under consideration.
Theorem 7.4. If 0 < q ≤ 2, 0 < ℓ < n, then Bnq,ℓ is a λ-intersection
body for any 0 < λ < n.
Proof. Owing to Lemma 7.1, the function (7.8) (with p replaced by −λ)
is positive, and therefore, by Lemma 7.3, || · ||−λq,ℓ represents a positive
definite distribution. Now the result follows by Theorem 5.1. �
Consider the case q > 2. In this case Bnq,ℓ is convex, and, owing to
Example 6.2, Bnq,ℓ ∈ Inλ for all n− 3 ≤ λ < n. What about λ < n− 3?
This case is especially intriguing.
Proposition 7.5. If q > 2 and 0 < λ < max(n− ℓ, ℓ)− 2, then || · ||−λq,ℓ
is not a positive definite distribution and therefore, Bnq,ℓ 6∈ Inλ .
Proof. Let 0 < λ < n− ℓ− 2 and suppose the contrary, that Bnq,ℓ ∈ Inλ .
Consider the section of Bnq,ℓ by the (n − ℓ + 1)-dimensional plane η =
Ren ⊕ Rn−ℓ. By Theorem 5.12, Bnq,ℓ ∩ η ∈ In−ℓ+1λ in η, and therefore
||xnen + x′′||λq,ℓ = (|xn|q + |x′′|q)−λ/q
is a positive definite distribution in η. By the second derivative text (see
[K4, Theorem 4.19]) this is impossible if 0 < λ < n− ℓ− 2. A similar
contradiction can be obtained if we assume 0 < λ < ℓ− 2 and consider
the section of Bnq,ℓ by the (ℓ+ 1)-dimensional plane Re1 ⊕ Rℓ. �
Proposition 7.5 can be proved without using the second derivative
text and Theorem 5.12 on sections of λ-intersection bodies; see [R4].
The bounds for λ appear to be the same.
Open problem. Let q > 2, ℓ > 1. Is Bnq,ℓ a λ-intersection body if
max(n− ℓ, ℓ)− 2 < λ < n− 3?
This problem does not occur in the case ℓ = 1 as in Example 6.6.
INTERSECTION BODIES 35
8. The generalized cosine transforms and comparison of
volumes
For 1 < i < n, let voli(·) denote the i-dimensional volume function.
Suppose that i is fixed, and let A and B be o.s. convex bodies in Rn
satisfying
(8.1) voli(A ∩ ξ) ≤ voli(B ∩ ξ) ∀ξ ∈ Gn,i.
Does it follow that
(8.2) voln(A) ≤ voln(B) ?
This question is known as the Generalized Busemann-Petty Problem
(GBP); see [G], [RZ], [Z1].
Theorem 8.1. If GBP (8.1)-(8.2) has an affirmative answer, then
every smooth origin-symmetric convex body with positive curvature in
n is an (n− i)-intersection body.
Proof. Suppose that B is an o.s. convex body in Rn so that the radial
function ρB is infinitely smooth, the boundary of B has a positive curva-
ture and B /∈ Inn−i. By Definition 5.4, there is a function ϕ ∈ De(Sn−1),
which is negative on some open origin-symmetric set Ω ⊂ Sn−1 and
such that ρn−iB = M
1+i−nϕ. We choose a function h ∈ De(Sn−1) so
that h 6≡ 0, h(θ) ≥ 0 if θ ∈ Ω and h(θ) ≡ 0 otherwise. Define an o.s.
smooth body A by ρiA = ρ
B − εM1−ih, ε > 0. If ε is small enough,
then A is convex. Since by (3.12), RiM
1−ih = cR0n−i,⊥h ≥ 0, then
A ≤ RiρiB, which gives (8.1). On the other hand, by (3.5),
(ρn−iB , ρ
B − ρiA) = ε(M1+i−nϕ,M1−ih) = ε(ϕ, h) < 0,
or (ρn−iB , ρ
B) < (ρ
B , ρ
A). By Hölder’s inequality, this implies voln(B) <
voln(A), which contradicts (8.2). �
Remark 8.2. As we noted in Introduction, Theorem 8.1 is not new, and
its proof given in [K3] relies on a sequence of deep facts from functional
analysis. The proof presented above is much more elementary and
constructive. For instance, it allows us to keep invariance properties of
the bodies under control. This advantage was essentially used in our
paper [R4].
Theorem 8.1 and Proposition 7.5 imply the following
Corollary 8.3. Let 1 ≤ ℓ ≤ n/2; i > ℓ+2, B = Bn4,ℓ (see (7.1)). Then
there is a smooth o.s. convex body A in Rn so that (8.1) holds but (8.2)
fails.
36 BORIS RUBIN
Setting ℓ = 1 in this statement, we obtain the well-known Bourgain-
Zhang theorem, which states that GBP has a negative answer when
3 < i < n; see [BZ], [K4], [RZ] on this subject. For i = 2 and i = 3
(n ≥ 5) the GBP is still open. An affirmative answer in these cases
was obtained in [R4] for bodies having a certain additional symmetry.
9. Appendix
Every positive distribution F ∈ S ′(Rn) is given by a tempered non-
negative measure µ, i.e., 〈F, φ〉 =
φ(x)dµ(x); see, e.g., [GV, p.147]).
For convenience of the reader, we present a similar fact for the sphere.
Theorem 9.1. A distribution f ∈ D′(Sn−1) is positive if and only if
there is a measure µ ∈ M+(Sn−1) such that
(f, ϕ) =
ϕ(θ)dµ(θ) ∀ϕ ∈ D(Sn−1).
Proof. This statement is known, however, we could not find precise ref-
erence and decided to give a proof for convenience of the reader. The
“if” part is obvious. To prove the “ only if” part, we write a test func-
tion ϕ ∈ D(Sn−1) as a sum ϕ = ϕ1+iϕ2, where ϕ1 = Reϕ, ϕ2 = Imϕ.
Since −||ϕ||C(Sn−1) ≤ ϕj ≤ ||ϕ||C(Sn−1), j = 1, 2, and f is positive, then
−(f, 1) ||ϕ||C(Sn−1) ≤ (f, ϕj) ≤ (f, 1) ||ϕ||C(Sn−1),
and therefore, |(f, ϕ)| ≤ |(f, ϕ1)|+ |(f, ϕ2)| ≤ 2(f, 1) ||ϕ||C(Sn−1). Since
D(Sn−1) is dense in C(Sn−1), then f extends as a linear continuous
functional f̃ on C(Sn−1) and, by the Riesz theorem, there is a mea-
sure µ on Sn−1 such that (f̃ , ω) =
ω(θ)dµ(θ) for every ω ∈
C(Sn−1). In particular, (f, ϕ) = (f̃ , ϕ) =
ϕ(θ)dµ(θ) for every
ϕ ∈ D(Sn−1). By taking into account that every non-negative function
ω ∈ C(Sn−1) can be uniformly approximated by non-negative functions
ϕk ∈ D(Sn−1) (for instance, by Poisson integrals of ω), we get
ω(θ)dµ(θ) = lim
ϕk(θ)dµ(θ) = lim
(f, ϕk) ≥ 0.
The latter means that µ is non-negative. �
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Department of Mathematics, Louisiana State University, Baton Rouge,
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E-mail address : [email protected]
|
0704.0062 | On-line Viterbi Algorithm and Its Relationship to Random Walks | arXiv:0704.0062v1 [cs.DS] 31 Mar 2007
On-line Viterbi Algorithm and Its Relationship to Random Walks
Rastislav Šrámek1, Broňa Brejová2, and Tomáš Vinař2
1 Department of Computer Science, Comenius University,
842 48 Bratislava, Slovakia, e-mail: [email protected]
2 Department of Biological Statistics and Computational Biology, Cornell University,
Ithaca, NY 14853, USA, e-mail: {bb248,tv35}@cornell.edu
Abstract. In this paper, we introduce the on-line Viterbi algorithm for decoding hidden Markov
models (HMMs) in much smaller than linear space. Our analysis on two-state HMMs suggests that the
expected maximum memory used to decode sequence of length n with m-state HMM can be as low
as Θ(m log n), without a significant slow-down compared to the classical Viterbi algorithm. Classical
Viterbi algorithm requires O(mn) space, which is impractical for analysis of long DNA sequences (such
as complete human genome chromosomes) and for continuous data streams. We also experimentally
demonstrate the performance of the on-line Viterbi algorithm on a simple HMM for gene finding on
both simulated and real DNA sequences.
Keywords: hidden Markov models, on-line algorithms, Viterbi algorithm, gene finding
1 Introduction
Hidden Markov models (HMMs) are generative probabilistic models that have been succesfuly
used for annotation of sequence data, such as DNA and protein sequences, natural langauge texts,
and sequences of observations or measurements. Their numerous applications include gene finding
[1], protein secondary structure prediction [2], and speech recognition [3]. The linear-time Viterbi
algorithm [4] is the most commonly used algorithm for these tasks. Unfortunately, the space required
by the Viterbi algorithm grows linearly with the length of the sequence (with a high constant factor),
which makes it unsuitable for analysis of continuous or very long sequences. For example, DNA
sequence of a single chromosome can be hundreds of megabases long. In this paper, we address this
problem by proposing an on-line Viterbi algorithm that on average requires much less memory and
that can annotate continuous streams of data on-line without reading the complete input sequence
first.
An HMM, composed of states and transitions, is a probabilistic model that generates sequences
over a given alphabet. In each step of this generative process, the current state generates one symbol
of the sequence according to the emission probabilities associated with that state. Then, an outgoing
transition is randomly chosen according to the transition probability table, and this transition is
followed to the new state. This process is repeated until the whole sequence is generated.
The states in the HMM represent distinct features of the observed sequences (such as protein
coding and non-coding sequences in a genome), and the emission probabilities in each state represent
statistical properties of these features. The HMM thus defines a joint probability Pr(X,S) over all
possible sequences X and all state paths S through the HMM that could generate these sequences.
To annotate a given sequence X, we want to recover the state path S that maximizes this joint
probability. For example, in an HMM with one state for protein-coding sequences, and one state
for non-coding sequences, the most probable state path marks each symbol of the input sequence
X as either protein coding or non-coding.
http://arxiv.org/abs/0704.0062v1
To compute the most probable state path, we use the Viterbi dynamic programming algorithm
[4]. For every prefix X1 . . . Xi of the given sequence X and for every state j, we compute the most
probable state path generating this prefix ending in state j. We store the probability of this path
in table P (i, j) and its second last state in table B(i, j). These values can be computed from left to
right, using the recurrence P (i, j) = maxk{P (i− 1, k) · tk(j) · ej(Xi)}, where tk(j) is the transition
probability from state k to state j, and ej(Xi) is the emission probability of the i-th symbol
of X in state j. Back pointer B(i, j) is the value of k that maximizes P (i, j). After computing
these values, we can recover the most probable state path S = s1, . . . , sn by setting the last state
as sn = argmaxk{P (n, k)}, and then following the back pointers B(i, j) from right to left (i.e.,
si = B(i + 1, si+1)). For an HMM with m states and a sequence X of length n, the running time
of the Viterbi algorithm is Θ(nm2), and the space is Θ(nm).
This algorithm is well suited for sequences and models of moderate size. However, to annotate
all 250 million symbols of the human chromosome 1 with a gene finding HMM consisting of hundred
states, we would require 25 GB of memory just to store the back pointers B(i, j). This is clearly
impractical on most computational platforms.
Several solutions are used in practice to overcome this problem. For example, most practical
gene finding programs process only sequences of limited size. The long input sequence is split into
several shorter sequences which are processed separately. Afterwards, the results are merged and
conflicts are resolved heuristically. This approach leads to suboptimal solutions, especially if the
genes we are looking for cross the boundaries of the split.
Grice et al. [5] proposed a practical checkpointing algorithm that trades running time for space.
We divide the input sequence into K blocks of L symbols, and during the forward pass, we only
keep the first column of each block. To obtain the most probable state path, we recompute the last
block of L columns, and use back pointers to recover the last L states of the most probable path, as
well as the last state of the previous block. The information about this last state can now be used to
recompute the most probable state path within the previous block in the same way, and the process
is repeated for all blocks. Since every value of P (i, j) will be computed twice, this means two-fold
slow-down compared to the Viterbi algorithm, but if we set K = L =
n, this algorithm only
requires Θ(
nm) memory. Checkpointing can be further generalized to trade L-fold slow-down for
memory of Θ( L
nm) [6, 7].
In this paper, we propose and analyze an on-line Viterbi algorithm that does not use fixed
amount of memory for a given sequence. Instead, the amount of memory varies depending on
the properties of the HMM and the input sequence. In the worst case, our algorithm still requires
Θ(nm) memory; however, in practice the requirements are much lower. We prove, by demonstrating
analogy to random walks and using results from the theory of extreme values, that in simple cases
the expected space for a sequence of length n is as low as Θ(m log n). We also experimentally
demonstrate that the memory requirements are low for more complex HMMs.
2 On-line Viterbi algorithm
In our algorithm, we represent the back pointer matrix B in the Viterbi algorithm by a tree structure
(see [4]), with node (i, j) for each sequence position i and each state j. Parent of node (i, j) is the
node (i − 1, B(i, j)). In this data structure, the most probable state path is a path from the leaf
node (n, j) with the highest probability P (n, j) to the root of the tree (see Figure 1).
This tree is built as the Viterbi algorithm progresses from left to right. After processing sequence
position i, all edges that do not lie on one of the paths ending in a level i node can be removed;
sequence positions
Fig. 1. Example of the back pointer tree structure. Dashed lines mark the edges that cannot be part of the
most probable state path. The square node marks the coalescence point of the remaining paths.
these edges will not be used in the most probable path [8]. The remaining m paths represent all
possible initial segments of the most probable state path. These paths are not necessarily edge
disjoint; in fact, often all the paths share the same prefix up to some node called coalescence point
(see Figure 1).
Left of the coalescence point, there is only a single candidate for the initial segment of the most
probable state path. Therefore we can output this segment and remove all edges and nodes of the
tree up to the coalescence point. Forney [4] describes an algorithm that after processing D symbols
of the input sequence checks whether a coalescence point has been reached; in such case, the initial
segment of the most probable state path is outputted. If the coalescence point was not reached,
one potential initial segment is chosen heuristicaly. Several studies [9, 10] suggest how to choose D
to limit the expected error caused by such heuristic steps in the context of convolution codes.
Here we show how to detect the existence of a coalescence point dynamically without introducing
significant overhead to the whole computation. We maintain a compressed version of the back
pointer tree, where we omit all internal nodes that have less than two children. Any path consisting
of such nodes will be contracted to a single edge. This compressed tree has m leaves and at most
m− 1 internal nodes. Each node stores the number of its children and a pointer to its parent node.
We also keep a linked list of all the nodes of the compressed tree ordered by the sequence position.
Finally, we also keep the list of pointers to all the leaves.
When processing the k-th sequence position in the Viterbi algorithm, we update the compressed
tree as follows. First, we create a new leaf for each node at position i, link it to its parent (one of
the former leaves), and insert it into the linked list. Once these new leaves are created, we remove
all the former leaves that have no children, and recursively all of their ancestors that would not
have any children.
Finally, we need to compress the new tree: we examine all the nodes in the linked list in order
of decreasing sequence position. If the node has zero or one child and is not a current leaf, we
simply delete it. For each leaf or node that has at least two children, we follow the parent links
until we find its first ancestor (if any) that has at least two children and link the current node
directly to that ancestor. A node (ℓ, j) that does not have an ancestor with at least two children
is the coalescence point; it will become a new root. We can output the most probable state path
for all sequence positions up to ℓ, and remove all results of computation for these positions from
memory.
The running time of this update is O(m) per sequence position, and the representation of the
compressed tree takes O(m) space. Thus the asymptotic running time of the Viterbi algorithm is
not increased by the maintanance of the compressed tree. Moreover, we have implemented both
the standard Viterbi algorithm and our new on-line extension, and the time measurements suggest
that the overhead required for the compressed tree updates is less than 5%.
The worst-case space required by this algorithm is still O(nm). However, this is rarely the case
for realistic data; required space changes dynamically depending on the input. In the next section,
we show that for simple HMMs the expected maximum space required for processing sequence of
length n is Θ(m log n). This is much better than checkpointing, which requires space of Θ(m
with a significant increase in running time. We conjecture that this trend extends to more complex
cases. We also present experimental results on a gene finding HMM and real DNA sequences showing
that the on-line Viterbi algorithm leads to significant savings in memory.
Another advantage of our algorithm is that it can construct initial segments of the most probable
state path before the whole input sequence is read. This feature makes it ideal for on-line processing
of signal streams (such as sensor readings).
3 Memory requirements of the on-line Viterbi algorithm
In this section, we analyze the memory requirements of the on-line Viterbi algorithm. The memory
used by the algorithm is variable throughout the execution of the algorithm, but of special interest
are asymptotic bounds on the expected maximum amount of memory used by the algorithm while
decoding a sequence of length n.
We use analogy to random walks and results in extreme value theory to argue that for a symmet-
ric two-state HMMs, the expected maximum memory is Θ(m log n). We also conduct experiments
on an HMM for gene finding, and both real and simulated DNA sequences.
3.1 Symmetric two-state HMMs
Consider a two-state HMM over a binary alphabet as shown in Figure 2a. For simplicity, we assume
t < 1/2 and e < 1/2. The back pointers between the sequence positions i and i+1 can form one of
the configurations i–iii shown in Figure 2b. Denote pA = log P (i, A) and pB = logP (i, B), where
P (i, j) is the table of probabilities from the Viterbi algorithm. The recurrence used in the Viterbi
algorithm implies that the configuration i occurs when log t−log(1−t) ≤ pA−pB ≤ log(1−t)−log t,
configuration ii occurs when pA−pB ≥ log(1−t)−log t, and configuration iii occurs when pA−pB ≤
log t− log(1− t). Configuration iv never happens for t < 1/2.
Note that for a two-state HMM, a coalescence point occurs whenever one of the configurations
ii or iii occur. Thus the memory used by the HMM is proportional to the length of continuous
sequence of configurations i. We will call such a sequence of configurations a run.
First, we analyze the length distribution of runs under the assumption that the input sequence
X is a sequence of uniform i.i.d. binary random variables. In such case, we represent the run by a
symmetric random walk corresponding to a random variable X = pA−pB
log(1−e)−log e
− (log t− log(1− t)).
Whenever this variable is within the interval (0,K), where K =
log(1−t)−log(t)
log(1−e)−log(e)
, the configuration
i occurs, and the quantity pA−pB is updated by log(1−e)−log e, if the symbol at the corresponding
sequence position is 0, or log e− log(1− e), if this symbol is 1. These shifts correspond to updating
the value of X by +1 or −1.
When X reaches 0, we have a coalescence point in configuration iii, and the pA−pB is initialized
to log t− log(1 − t) ± (log e − log 1 − e), which either means initialization of X to +1, or another
0: 1−e
1−t 1−t
1: 1−e
configuration i:
configuration ii:
configuration iii: configuration iv:
(a) (b)
Fig. 2. (a) Symmetric two-state HMM with two parameters: e for emission probabilities and t for transitions
probabilities. (b) Possible back-pointer configurations for the two-state HMM.
coalescence point, depending on the symbol at the corresponding sequence position. The other case,
when X reaches K and we have a coalescence point in configuration ii, is symmetric.
We can now apply the classical results from the theory of random walks (see [11, ch.14.3,14.5])
to analyze the expected length of runs.
Lemma 1. Assuming that the input sequence is uniformly i.i.d., the expected length of a run of a
symmetrical two-state HMM is K − 1.
Therefore the larger is K, the more memory is required to decode the HMM. The worst case is
achieved as e approaches 1/2. In such case, the two states are indistinguishable and being in state
A is equivalent to being in state B. Using the theory of random walks, we can also characterize the
distribution of length of runs.
Lemma 2. Let Rℓ be the event that the length of a run of a symmetrical two-state HMM is either
2ℓ + 1 or 2ℓ + 2. Then, assuming that the input sequence is uniformly i.i.d., for some constants
b, c > 0:
b · cos2ℓ π
≤ Pr(Rℓ) ≤ c · cos2ℓ
Proof. For a symmetric random walk on interval (0,K) with absorbing barriers and with starting
point z, the probability of event Wz,n that this random walk ends in point 0 after n steps is zero,
if n− z is odd, and the following quantity, if n− z is even [11, ch.14.5]:
Pr(Wz,n) =
0<v<K/2
cosn−1
Using symmetry, note that the probability of the same random walk ending after n steps at barrier
K is the same as probability of WK−z,n. Thus, if K is odd, we can state:
Pr(Rℓ) = Pr(W1,2ℓ+1) + Pr(WK−1,2ℓ+1)
0<v<K/2
cos2ℓ
+ (−1)v+1 sin πv
0<v<K/2, v odd
cos2ℓ
There are at most K/4 terms in the sum and they can all be bounded from above by cos2ℓ πv
Thus, we can give both upper and lower bounds on Pr(Rℓ) using only the first term of the sum as
follows:
cos2ℓ
≤ Pr(Rℓ) ≤ cos2ℓ
Similarly, if K is even, we can state:
Pr(Rℓ) = Pr(W1,2ℓ+1) + Pr(WK−1,2ℓ+2)
0<v<K/2
cos2ℓ
1 + (−1)v+1 cos
and thus we have a similar bound:
1 + cos
cos2ℓ
≤ Pr(Rℓ) ≤ 2 cos2ℓ
The previous lemma characterizes the length distribution of a single run. However, to analyze
memory requirements for a sequence of length n, we need to consider maximum over several runs
whose total length is n. Similar problem was studied for the runs of heads in a sequence of n coin
tosses [12, 13]. For coin tosses, the length distribution of runs is geometric, while in our case the
runs are only bounded by geometricaly decaying functions. Still, we can prove that the expected
length of the longest run grows logarithmically with the length of the sequence, as is the case for
the coin tosses.
Lemma 3. Let X1,X2, . . . be a sequence of i.i.d. random variables drawn from a geometrically
decaying distribution over positive integers, i.e. there exist constants a, b, c, a ∈ (0, 1), 0 < b ≤ c,
such that for all integers k ≥ 1, bak ≤ Pr(Xi > k) ≤ cak.
Let Nn be the largest index such that
i=1...Nn
Xi ≤ n, and let Yn be max{X1,X2, . . . ,XNn , n−
i=1Xi}. Then
E[Yn] = log1/a n+ o(log n) (7)
Proof. Let Zn = maxi=1...nXn be the maximum of the first n runs. Clearly, Pr(Zn ≤ k) = Pr(Xi ≤
k)n, and therefore (1− cak)n ≤ Pr(Zn ≤ k) ≤ (1− bak)n for all integers k ≥ log1/a(c).
Lower bound: Let tn = log1/a n−
lnn. If Yn ≤ tn, we need at least n/tn runs to reach the sum n,
i.e. Nn ≥ n/tn − 1 (discounting the last incomplete run). Therefore
Pr(Yn ≤ tn) ≤ Pr(Z n
−1 ≤ tn) ≤ (1− batn)
= (1− batn)a
−tnatn ( n
Since limn→∞ a
tn(n/tn−1) = ∞ and limx→0(1− bx)1/x = e−b, we get limn→∞Pr(Yn ≤ tn) = 0.
Note that E[Yn] ≥ tn(1− Pr(Yn ≤ tn)), and thus we get the desired bound.
Upper bound: Clearly, Yn ≤ Zn and so E[Yn] ≤ E[Zn]. Let Z ′n be the maximum of n i.i.d. geometric
random variables X ′1, . . . ,X
n such that Pr(X
i ≤ k) = 1− ak.
We will compare E[Zn] to the expected value of variable Z
n. Without loss of generality, c ≥ 1.
For any real x ≥ log1/a(c) + 1 we have:
Pr(Zn ≤ x) ≥ (1− ca⌊x⌋)n
1− a⌊x⌋−log1/a(c)
1− a⌊x−log1/a(c)−1⌋
= Pr(Z ′n ≤ x− log1/a(c)− 1)
= Pr(Z ′n + log1/a(c) + 1 ≤ x)
This inequality holds even for x < log1/a(c) + 1, since the right-hand side is zero in such case.
Therefore, E[Zn] ≤ E[Z ′n+log1/a(c)+1] = E[Z ′n]+O(1). Expected value of Z ′n is log1/a(n)+o(log n)
[14], which proves our claim. ⊓⊔
Using results of Lemma 3 together with the characterization of run length distributions by
Lemma 2, we can conclude that for symmetric two-state HMMs, the expected maximum memory
required to process a uniform i.i.d. input sequence of length n is (1/ ln(1/ cos(π/K)))·ln n+o(log n).
3 Using the Taylor expansion of the constant term as K grows to infinity, 1/ ln(1/ cos(π/K))) =
2K2/π2 +O(1), we obtain that the maximum memory grows approximately as (2K2/π2) lnn.
The asymptotic bound Θ(log n) can be easily extended to the sequences that are generated by
the symmetric HMM, instead of uniform i.i.d. The underlying process can be described as a random
walk with approximately 2K states on two (0,K) lines, each line corresponding to sequence symbols
generated by one of the two states. The distribution of run lengths still decays geometrically as
required by Lemma 3; the base of the exponent is the largest eigenvalue of the transition matrix
with absorbing states omitted (see e.g. [15, Claim 2]).
The situation is more complicated in the case of non-symmetric two-state HMMs. Here, our
random walks proceed in steps that are arbitrary real numbers, different in each direction. We are
not aware of any results that would help us to directly analyze distributions of runs in these models,
however we conjecture that the size of the longest run is still Θ(log n). Perhaps, to obtain bounds
on the length distribution of runs, one can approximate the behaviour of such non-discrete random
walks by a different model (for example, [16, ch.7]).
3.2 Multi-state HMMs
Our analysis technique cannot be easily extended to HMMs with many states. In two-state HMMs,
each new coalescence event clears the memory, and thus the execution of the algorithm can be
divided into more or less independent runs. A coalescent event in a multi-state HMM results in a
non-trivial tree left in memory, sometimes with a substantial depth. Thus, the sizes of consecutive
runs are no longer independent (see Figure 3a).
3 We omitted the first run, which has a different starting point and thus does not follow the distribution outlined in
Lemma 2. However, the expected length of this run does not depend on n and thus contributes only a lower-order
term. We also omitted the runs of length one that start outside the interval (0,K); these runs again contribute
only to lower order terms of the lower bound.
15.2M 15.3M 15.4M 15.5M
Section of chromosome 1
0 5M 10M 15M 20M
Sequence length
Human genome (35)
HMM generated (100)
Random i.i.d. (35)
Fig. 3. Memory requirements of a gene finding HMM. a) Actual length of table used on a segment of human
chromosome 1. b) Average maximum table length needed for prefixes of 20 MB sequences.
To evaluate the memory requirements of our algorithm for multi-state HMMs, we have im-
plemented the algorithm and performed several experiments on both simulated and biological se-
quences. First, we generalized the symmetric HMMs from the previous section to multiple states.
The symmetric HMM with m states emits symbols over m-letter alphabet, where each state
emits one symbol with higher probability than the other symbols. The transition probabilities
are equiprobable, except for self-transitions. We have tested the algorithm for m ≤ 6 and sequences
generated both by a uniform i.i.d. process, and by the HMM itself. Observed data are consistent
with the logarithmic growth of average maximum memory needed to decode a sequence of length
n (data not shown).
We have also evaluated the algorithm using a simplified HMM for gene finding with 265 states.
The emission probabilities of the states are defined using at most 4-th order Markov chains, and
the structure of the HMM reflects known properties of genes (similar to the structure shown in
[17]). The HMM was trained on RefSeq annotations of human chromosomes 1 and 22.
In gene finding, we segment the input DNA sequence into exons (protein-coding sequence in-
tervals), introns (non-coding sequence separating exons within a gene), and intergenic regions (se-
quence separating genes). Common measure of accuracy is exon sensitivity (how many of real exons
we have succesfuly and exactly predicted). The implementation used here has exon sensitivity 37%
on testing set of genes by Guigo et al. [18]. A realistic gene finder, such as ExonHunter [19], trained
on the same data set achieves sensitivity of 53%. This difference is due to additional features that
are not implemented in our test, namely GC content levels, non-geometric length distributions, and
sophisticated signal models.
We have tested the algorithm on 20 MB long sequences: regions from the human genome,
simulated sequences generated by the HMM, and i.i.d. sequences. Regions of the human genome
were chosen from hg18 assembly so that they do not contain sequencing gaps. The distribution for
the i.i.d. sequences mirrors the distribution of bases in the human chromosome 1.
The results are shown in Figure 3b. The average maximum length of the table over several
samples appears to grow faster than logarithmically with the length of the sequence, though it
seems to be bounded by a polylogarithmic function. It is not clear whether the faster growth is an
artifact that would disapear with longer sequences or higher number of samples.
The HMM for gene finding has a special structure, with three copies of the state for introns
that have the same emission probabilities and the same self-transition probability. In two-state
symmetric HMMs, similar emission probabilities of the two states lead to increase in the length of
individual runs. Intron states of a gene finder are an extreme example of this phenomenon.
Nonetheless, on average a table of length roughly 100,000 is sufficient to to process sequences
of length 20 MB, which is a 200-fold improvement compared to the trivial Viterbi algorithm. In
addition, the length of the table did not exceed 222,000 on any of the 20MB human segments. As
we can see in Figure 3a, most of the time the program keeps only relatively short table; the average
length on the human segments is 11,000. The low average length can be of a significant advantage
if multiple processes share the same memory.
4 Conclusion
In this paper, we introduced the on-line Viterbi algorithm. Our algorithm is based on efficient detec-
tion of coalescence points in trees representing the state-paths under consideration of the dynamic
programming algorithm. The algorithm requires variable space that depends on the HMM and
on the local properties of the analyzed sequence. For two-state symmetric HMMs, we have shown
that the expected maximum memory used for analysis of sequence of length n is approximately
only (2K2/π2) ln n. Our experiments on both simulated and real data suggest that the asymptotic
bound Θ(m lnn) also extend to multi-state HMMs, and in fact, for most of the time throughout
the execution of the algorithm, much less memory is used.
Further advantage of our algorithm is that it can be used for on-line processing of streamed
sequences; all previous algorithms that are guaranteed to produce the optimal state path require
the whole sequence to be read before the output can be started.
There are still many open problems. We have only been able to analyze the algorithm for two-
state HMMs, though trends predicted by our analysis seem to generalize even to more complex cases.
Can our analysis be extended to multi-state HMMs? Apparently, design of the HMM affects the
memory needed for the decoding algorithm; for example, presence of states with similar emission
probabilities tends to increase memory requirements. Is it possible to characterize HMMs that
require large amounts of memory to decode? Can we characterize the states that are likely to serve
as coalescence points?
Acknowledgments: Authors would like to thank Richard Durrett for useful discussions. Recently, we
have found out that parallel work on this problem is also performed by another research group [20].
Focus of their work is on implementation of an algorithm similar to our on-line Viterbi algorithm
in their gene finder, and possible applications to parallelization, while we focus on the expected
space analysis.
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|
0704.0063 | Experimental efforts in search of 76Ge Neutrinoless Double Beta Decay | Experimental efforts in search for
- 1 -
Experimental efforts in search of 76Ge Neutrinoless Double Beta Decay
Somnath Choudhury †
Department of Physics & Meteorology
Indian Institute of Technology, Kharagpur – 721302, India
Abstract
Neutrinoless double beta decay is one of the most sensitive approaches in non-accelerator particle physics to take
us into a regime of physics beyond the standard model. This article is a brief review of the experiments in search
of neutrinoless double beta decay from 76Ge. Following a brief introduction of the process of double beta decay from
76Ge, the results of the very first experiments IGEX and Heidelberg-Moscow which give indications of the existence of
possible neutrinoless double beta decay mode has been reviewed. Then ongoing efforts to substantiate the early findings are
presented and the Majorana experiment as a future experimental approach which will allow a very detailed study of the 0ν
decay mode is discussed.
Keywords: neutrinoless, Majorana particle, pulse shape discrimination.
1. Introduction
Neutrinoless double beta decay is one of the most sensitive approaches with great perspectives to test particle
physics beyond the Standard Model. There is immense scope to use 0νββ decay for constraining neutrino masses,
left–right–symmetric models, interactions involving R-parity breaking in the supersymmetric model and
leptoquark scenarios, as well as effective lepton number violating couplings. Experimental limits on 0νββ decay
are not only complementary to accelerator experiments but at least in some cases competitive or superior to the
best existing direct search limits. The steadily improving experimental limits on the half-life of 0νββ can be
translated into more stringent limits on the parameters of these new physics scenarios.
In the process of beta decay an unstable nucleus decays by converting a neutron in the nucleus to a proton and
emitting an electron and an anti-neutrino. In order for beta decay to be possible the final nucleus must have a
larger binding energy than the original nucleus. For some nuclei, such as Germanium-76 the nuclei with atomic
number one higher have a smaller binding energy, preventing beta decay from occurring. In the case of
Germanium-76 the nuclei with atomic number two higher, Selenium-76 has a larger binding energy, so the
"double beta decay" process is allowed. In double beta decay two neutrons in the nuclei are converted to protons,
and two electrons and two anti-neutrinos are emitted. It is the rarest known kind of radioactive decay; it was
observed for only ten isotopes. For some nuclei, the process occurs as conversion of two protons to neutrons, with
emission of two neutrinos and absorption of two orbital electrons (double electron capture). If mass difference
between the parent and daughter atoms is more than 1022 keV (two electron masses), another branch of the
process becomes possible, with capture of one orbital electron and emission of one positron. And, at last, when
the mass difference is more then 2044 keV (four electron masses), the third branch of the decay arises, with
emission of two positrons (β+β+ decay).
The processes described above are also known as two neutrino double beta decay, as two neutrinos (or anti-
neutrinos) are emitted. If the neutrino is a Majorana particle, meaning that the anti-neutrino and the neutrino are
actually the same particle then it is possible for neutrinoless double beta decay to occur. In 0νββ decay the emitted
neutrino is immediately absorbed (as its anti-particle) by another nucleon of the nucleus, so the total kinetic
energy of the two electrons would be exactly the difference in binding energy between the initial and final state
nuclei.
† Now at Indiana University – Bloomington, USA
Experiments have been carried out and proposed to search for 0νββ decay mode, as its discovery would indicate
that neutrinos are indeed Majorana particles and allow a calculation of neutrino mass. While the two-neutrino
mode (1.1) is allowed by the Standard Model of particle physics, the neutrinoless mode (0νββ) (1.2) requires
violation of lepton number (∆L=2). This mode is possible only, if the neutrino is a Majorana particle, i.e. the
neutrino is its own antiparticle. Double beta decay, the rarest known nuclear decay process, can occur in different
modes:
_
2νββ -decay : A(Z,N) → A(Z+2, N-2)+2e ⎯ +2 ν (1.1)
0 ν ββ -decay : A(Z,N) → A(Z+2, N-2) + 2e⎯ (1.2)
0 ν(2) χ ββ -decay : A(Z,N) → A(Z+2, N-2)+2e ⎯ + (2) χ (1.3)
2. Double Beta Decay: A Rare Process
The process arises in certain cases of even-A nuclei, where A is the mass number and is the sum of the number of
protons and neutrons (A = Z + N). For even-A nuclei, the strong pairing force between like nucleons (neutrons
like to be paired with other neutrons in a given nucleus, with the same true for protons), the binding energy of
even-even nuclei (even number of protons and even number neutrons) is larger than that of odd-odd nuclei (odd
numbers of protons and neutrons). This fact results in two separate parabolas on a plot of binding energy, one
parabola for even-even nuclei and one for odd-odd. Consequently, one occasionally finds a situation where two
even-even nuclei for a given mass number A are stable against ordinary beta decay. However, the heavier nucleus
is not fully stable and can decay to the lighter nucleus via normal double beta decay, a second-order process
whereby the nuclear charge changes by two units. The ground state of the even-even nuclei is 0+ (positive parity)
and the nuclear transition is 0+ +→0 .
One particular type of experimental approach that hopes to determine if the neutrino is a massive Majorana
particle is the search for neutrinoless double beta decay. This type of experiment is perhaps the only feasible
method for determining if the neutrino is a Majorana or Dirac particle. While neutrinoless double beta decay has
not yet been experimentally discovered, searches have been conducted for many years, with many continuing
today. In fact, the next generation of double beta decay experiments is currently being designed and developed
and involves a tremendous increase in the amount of source material to be studied (on the order of a half-ton or
more). In neutrinoless double beta decay an antineutrino emitted at the first vertex is absorbed at the second as
- 2 -
seen in the figure below or that a virtual neutrino emitted by a neutron is absorbed by the second neutron
participating in the double beta decay.
The two neutrino mode is allowed in standard model. The neutrinoless mode can occur only if neutrinos have
masses of the Majorana type. The decay rate is proportional to the squared mass. In other words, the half life is
inversely proportional to the squared mass. Experimentally one can distinguish the two modes. In the two
neutrino mode the electrons take away only a fraction of the energy Q released in the decay. The sum energy
spectrum is continuous, extending from 0 to Q. In the neutrinoless mode the total energy Q is carried away by the
electrons, and the sum energy spectrum is a peak centered at Q, with a width given by the instrumental resolution.
allowed in the Standard Model of physics is given by
(2.1)
the exchange of the Majorana neutrino in the absence of right-handed
(2.2)
tively. The nuclear
y in the
The decay rate for 2ν ββ decay which is
The decay rate for the process involving
currents can be expressed as follows:
2/1 ),()00(
MZEGT −=→
[ ] 2
2/1 ),()00( ><−=→
νννν mM
MZEGT F
The M and MGT F are the nuclear matrix elements of Gamow-Teller and Fermi transitions respec
atrix elements of the 0+→0+ Gamow-Teller and Fermi transition for the two neutrino mode in weak theorm
second order perturbation is given by
- 3 -
(2
∑ Δ+−= n in
∑∑ ++++++ iknnjf 0110 ττ
- 4 -
nd is given by and the Gamow-Teller transition operator
respectively.
complete orthonormal set of intermediate excited states have been introduce denot
eutrino double beta decay mode has been expressed in terms of single beta transitions through the introduction of
termediate excited states via which the transition from the initial 0+ to the final 0+ state occurs.
or the neutrinoless mode, the nuclear matrix elements resulting from Fermi and Gamow-Teller transitions are
(2.6)
(2.7)
here, R ro=1.2 fm.
he par e lepton phase space, and gV and gA are the weak vector and
xial-vector coupling constants respectively. The <mν> is the effective electron neutrino ma . If the light
j« few MeV) exchange is the dominant mechanism at both
ton number
mj) and the mixing
arameters (
est possible construction cost. The next sections review experiments
3. International Germa iment (IGEX)
is a unique ground to investigate the nature and properties of the neutrino. The
tern. To
(2.4)
where Δ denotes the average energy a
∑ Δ+−= n in
∑∑ ++++++ ikknnjjf 0110 τστσ
and the Fermi transition operator is given by.
A d ed by . Thus the two
given by
(2.5)
The function H depends on the distance between the nucleons and approximately has the form
w = ro A
⅓, A being the mass number and
T t G2ν and G0ν results from integrating over th
neutrino (m for the 0νββ-decay process and th
the neutrino currents are left-handed, then the 0νββ-decay amplitude is proportional to the lep
iolating parameters. This effective mass is related to the light neutrino mass eigenvalues (v
p Uej) and is give by the relation
(2.8)
The effective light neutrino mass <mν> may be suppressed by a destructive interference between the different
contributions in the sum of equation (2.8) if CP is conserved. In this case the mixing matrix satisfies the condition
Uej= Uej*.ζj, where ζj = ±i is the CP parity of the Majorana neutrino νj. The absolute value has thus been inserted
for convenience, since the quantity inside it is squared in equation (2.8) and is complex if CP is violated.
The ideal 0νββ-decay experiment has the following dream features: the lowest possible background, the best
possible energy resolution, the greatest possible mass of the parent isotope, detection efficiency near 100% for
valid events, a unique signature and the low
in such an effort from the isotope 76Ge.
nium EXper
The nuclear Double Beta Decay
neutrinoless decay mode, if it exists, would provide an unambiguous evidence of the Majorana nature of the
neutrino, its non-zero mass, and the non-conservation of lepton number. After implication from solar and
atmospheric neutrino oscillation results that neutrinos have non-zero mass, the process of neutrinoless Double
Beta Decay has become the most relevant place to test the neutrino mass scale and its hierarchy pat
achieve high sensitivity limits of the effective Majorana electron neutrino mass derived from the neutrinoless half-
life lower bound required for such new objectives, it will require a large number of double beta emitter nuclei, a
ejj Umm
++++ ∑= ik
jjkfF ErHM 0),(0
0 ττν
++++ ∑= ikjjkfGT ErHM 0.),(00 ττν kj σσ
2)( fi EEΔ = −
∑ ∑ ++
jj ττσ and
∫ +−+=
}2/)({
fi EEE
sin2 qrqR
- 5 -
of this type of search was the IGEX. The International Germanium
Xperiment (IGEX) was a search for the neutrinoless double beta decay of 76Ge employing large amounts of
In the first phase of the experiment three detectors of 0.7
ents (the most
sites. It provided a rejection of ~ 60 % of the events in the region of
IGEX spectrum with and without the PSD background rejection.
The IGEX detectors had the initial objective of the detection of the double beta decay of 76Ge. At the end of 1999
certain modifications were made to adapt the detectors to the detection at low energy where the signal of WIMPs
(Weak Interacting Massive Particles) is relevant. The shielding, shared by three IGEX detectors (2 kg germanium
detectors isotopically enriched to 86% in 76Ge) and the COSME detector, included from inside to outside 40 cm
of lead, a PVC box (silicone sealed and flushed with nitrogen), 2 mm of cadmium, plastic scintillators working in
anticoincidence with the Ge detectors and 20 cm of polyethylene. The shielding was modified on July 2001 as it
included only one 2 kg germanium detector inside a more efficient neutron shielding. These techniques of passive
very low background and a sharp energy resolution in the Q-value region, and effective methods to disentangle
signal from noise. A typical example
HPGe detectors, isotopically enriched to 86% in 76Ge.
kg active volume each were operated: one in the Homestake gold mine (4000 m.w.e.), other in the Baksan
Neutrino Observatory (660 m.w.e.) and the other in the Canfranc underground laboratory (Laboratory 2 at 1380
m.w.e.). A conservative lower bound on the neutrinoless half-life of about 1024 years was derived.
The International Germanium EXperiment (IGEX) took data at the Canfranc Underground Laboratory in Spain at
a depth of 2450 m.w.e. in a search of neutrinoless double beta decay. Three Germanium detectors (RG1, RG2
and RG3), of ~2 kg each, enriched to 86% in 76Ge were used. Efforts were made to reduce part of the radioactive
background by discriminating it from the expected signal by comparison of the shape of the pulses (PSD) of both
types of events. The method was applied to the data recorded by two Ge detectors of the IGEX, which has
produced one of the two best current sensitivity limits for the Majorana neutrino mass parameter. In the second
phase, three large detectors (2 kg each) were fabricated (with improvements derived from the analysis of data of
Phase 1). They are installed in the Canfranc underground laboratory (Laboratory 3 at 2450 m.w.e.) inside a low
background shielding consisting of 40 cm of lead, a PVC box (silicone sealed and flushed with nitrogen), 2mm of
cadmium, 20 cm of polyethylene and an active veto (plastic scintillators). A pulse shape discrimination (PSD)
technique capable to distinguish single site events (ββ decay events for example) from multisite ev
dominant background events) is implemented. New limits on the neutrinoless half-life and the neutrino mass
parameter were thus obtained from here.
In large intrinsic Ge detectors, the charge carriers take 300 - 500 ns to reach their respective electrodes. These
drift times are long enough for the current pulses to be recorded at a sufficient sampling rate. The current pulse
contributions from electrons and holes are displacement currents, and therefore dependent on their instantaneous
velocities and locations. Accordingly, events occurring at a single site (ββ-decay events for example) have
associated current pulse characteristics which reflect the position in the crystal where the event occurred. More
importantly, these single-site events (SSE) frequently have pulse shapes that differ significantly from those due to
the background events that produce electron-hole pairs at several sites by multi-Compton-scattering process, for
example (the so-called Multi-Site Events (MSE)). Consequently, pulse-shape analysis was used to distinguish
between these two types of energy depositions since DBD events belong to the SSE class of events and will
deposit energy at a single site in the detector while most of the background events belong to the MSE class of
events and will deposit energy at several
interest, accepting the criterion that those events having more than two lobes cannot be due to DBD event.
- 6 -
and active shielding, along with the extreme radiopurity of the detectors and their components, allowed a low
energy background as well as a low enough threshold which are unique in this type of detectors. So, very stringent
contour limits for cross sections and masses of dark matter particles interacting with Ge nuclei through spin-
independent interactions were derived from here. The need to understand and reject backgrounds in Ge-diode
detector double-beta decay experiments thus gave rise to the development of the pulse shape analysis technique in
such detectors to distinguish DBD single-site energy deposits from the multiple-site deposits. Henceforth the
analysis was extended by DBD people to segmented Ge detectors to study the effectiveness of combining
segmentation with pulse shape analysis to identify the multiplicity of the energy deposits.
The IGEX calculations for a lower bound to the half-life for the neutrinoless mode where there were fewer than
3.1 candidate events (90% Confidence Level) under a peak having FWHM = 4 keV and centered at 2038.56 keV
corresponded to:
he requirements for a next generation experiment can easily be deduced by reference to (3.1)
where N is the number of parent nuclei, t is the counting time, and c is the upper limit on the number of 0νββ-
decay counts consistent with the observed background. To improve the sensitivity of ‹mν› by a factor of 100, the
quantity Nt/c must be increased by a factor of 104. The quantity N can feasibly be increased by a factor of ~102
over present experiments, so that t/c must also be improved by that amount. Since practical counting times can
only be increased by a factor of 2 to 4, the background should be reduced by a factor of 25 to 50 below present
levels. These are approximately the target parameters of the next generation neutrinoless double-beta decay
experiments.
Histogram of the IGEX data in the energy region of interest for the 0ν -ββ decay. The limits on the half-life and
neutrino mass parameter are also shown.
The Effective ν Mass: The section of KKDK on effective neutrino mass (“Critical View to the IGEX neutrinoless
ouble-beta decay experiment...” published in Phys. Rev. D, Volume 65 (2002) 092007, by H. V. Klapdor-
Kleingrothaus, A. Dietz, and I. V. Krivosheina) begins with: “Starting from their incorrectly determined half-life
limit the authors claim a range of effective neutrino mass of (0.33-1.35) eV.” In response the IGEX collaboration,
came out stating that KKDK selected only the 52.51 mole·years of the IGEX data that had been subjected to PSD
and obtained T½0ν > 7.1×1024 y using the maximum number of counts, 3.1, from the entire 117 mole·years of data
which was erroneous and unjustified. In another case, KKDK also decided to arbitrarily use the entire IGEX data
set prior to PSD selection from which they obtained 0ν a bound of T½0ν > 1.1 × 1025 y for which there was no
scientific justification for selecting only PSD corrected data on one hand and totally ignoring the PSD corrected
data on the other hand. In the conclusion of KKDK it states: “the IGEX paper - apart from the too high half-life
limits presented, as a consequence of an arithmetic error - is rather incomplete in its presentation”. In response to
this paper the IGEX collaboration published the article “The IGEX experiment revisited: a response to the critique
of Klapdor-Kleingrothaus, Dietz, and Krivosheina” where they stated that there was absolutely no arithmetic error
yryrGe 25
76 1057.1
1087.4
>T 0 (ν2/1 1.3
).2(ln0
2/1 =
- 7 -
and that the analysis of the published IGEX data presented in KKDK stands illegitimate. To obtain a much shorter
bound on the half-life, they arbitrarily analyzed two ~ halves of the data separately. Instead of having 4.88×1025 y
in the numerator (ln2 N.t) they used 2.2×1025 y. Yet they used the 90% CL upper limit on the number of counts
under the peak, obtained by IGEX from all of the data. In another analysis, they ignored the fact that 52.51
mole·years were corrected with PSD and treated the complete uncorrected data set. Naturally, the lower limits on
T1/2oν (76Ge) obtained by these completely unjustified procedures are shorter than that obtained from properly
analyzing the complete data set. This paper henceforth states “the lower limit quoted by IGEX, T1/20ν ≥ 1.57 × 1025
years, is correct and that there was no arithmetical error as claimed in the Critical View article.”
4. The HEIDELBERG - MOSCOW Experiment
The Heidelberg-Moscow experiment at the Gran Sasso underground laboratory is now claimed to be the most
sensitive neutrinoless double beta decay experiment worldwide. It has contributed in an extraordinary way to the
research in neutrino physics and particularly to beyond standard model physics, and limits for the latter are
competing with those from the largest high-energy accelerators. The emphasis on the first indication for
y is found in the Heidelberg-Moscow experiment ving first evidence of the lepton neutrinoless double beta deca gi
number violation and a Majorana nature of the neutrinos. The neutrinoless double beta decay could answer
questions to the absolute scale of the neutrino mass and the fundamental character of the neutrino whether it is a
Dirac or a Majorana particle.
Entrance of the highway tunnel under Gran Sasso mountain.
With the support of the LNGS the experimental building of the experiment was built between Halls A and B in
Gran Sasso, into which the first enriched 76 76Ge detector (the first high-purity enriched Ge detector worldwide)
was installed in July 1990 . First preparation work had been done since 1989 in a provisional tent in Hall C. The
ll amount of five enriched 76Ge detectors of in total 11 kg was finally installed in 1995 and were operated since
n method
lead (detectors ## 1,2,3,5). Each setup is coated with stainless steal casing. Non-
dioactive pure nitrogen was blown through casings to reduce radon emanation contribution. To reduce neutron
background the casing with detectors ##1,2,3,5 was coated with borated polyethylene and two anticoincidence
plates of plastic scintillator were located over the casing in order to reduce muon component. The setup was
located in Gran Sasso underground laboratory, Italy at a depth of 3500 metres of water equivalent of the lab
1996 with a newly developed pulse shape discriminatio
High purity germanium crystals, enriched by Germanium-76 isotope up to 86% are used as the main detecting
elements. Five coaxial detectors with the total weight of 11.5 kg (125 moles in the active volume of detectors) are
used. Each detector is located in a separate cryostat made of electrolytic copper with low content of radioactive
impurities. The quantity of other designed materials (iron, bronze, light material insulators) is minimized in order
to reduce the feasible radioactive impurities contribution to the total background of the detectors. The detectors
were located in two separate shielded boxes. One of them, 270 mm thick is made of electrolytic copper (detector
#4), the other consists of two layers of lead – inner -100mm of high purity LCD2-grade lead and outer – 200 mm
of low background Boliden
reduces influence of cosmic rays on background conditions of the experiment. The electronics and the system of
collecting data allow to record each event – the number (or numbers) of acted detector, amplitude and pulse shape,
and anticoincidence veto. The Heidelberg-Moscow experiment, with five enriched 86%-88% high-purity p-type
Germanium detectors, of in total 10.96 kg of active volume, used the largest source strength of all double beta
experiments at present, and reached a record low level of background. The detectors were the first high-purity Ge
detectors ever produced. The degree of enrichment was checked by investigation of tiny pieces of Ge after crystal
production using the Heidelberg MP-Tandem accelerator as a mass spectrometer.
The detectors, except detector # 4, were operated in a common Pb shielding of 30 cm, which consisted of an inner
shielding of 10 cm radiopure LC2-grade Pb followed by 20 cm of Boliden lead. The whole setup was placed in an
air-tight steel box and flushed with radiopure nitrogen in order to suppress the 222Rn contamination of the air. The
shielding was improved in the course of the measurement. The steel box operated since 1994 centered inside a 10-
cm boron-loaded polyethylene shielding to decrease the neutron flux from outside. An active anticoincidence
shielding was placed on top of the setup since 1995 to reduce the effect of muons. Detector # 4 was installed in a
separate setup, which had an inner shielding of 27.5 cm electrolytical Cu, 20 cm lead, and boron-loaded
below the steel box, but no muon shielding. The setup was kept air-tight closed since
stallation of detector #5 in February’95. Since then no radioactive contaminations of the inner of the
he sensitivity for the 0ν ββ half-life is given by (4.1)
ent are: energy resolution, background and
e strength ever operated in a double beta decay
xperiment. The background reached to the experiment, was 0.113 ± 0.007 events/kg y keV (in the period 1995-
was the lowest lim
polyethylene shielding
experimental setup by air and dust from the tunnel could occur.
- 8 -
With denoting the degree of enrichment, ε the efficiency of the detector for detection of a double beta event, M
the detector (source) mass, ∆E the energy resolution, B the background and t the measuring time, the sensitivity of
our 11 kg of enriched 76Ge experiment corresponds to that of an at least 1.2 ton natural Ge experiment. After
enrichment - the other most important parameters of a ββ experim
source strength. The high energy resolution of the Ge detectors of 0.2% or better, assures no background for a
0νββ line from the two-neutrino double beta decay in this experiment (5.5 × 10-9 events expected in the energy
range 2035-2039.1keV), in contrast to most other present experimental approaches, where limited energy
resolution is a severe drawback.
The efficiency of Ge detectors for detection of 0ν ββ decay events is close to 100%. The source strength in the
Heidelberg-Moscow experiment of 11kg was the largest sourc
~0 〉〈×ν ε mand
. 02/1 Δ ννTBE
2003) in the 0ν ββ decay region (around Qββ). This it ever obtained in such type of experiment.
- 9 -
he statistics collected in this experiment during 13 years of stable running is the largest ever collected in a
presented a paper concerning “Measurement of the Bi spectrum in the energy
gion around the Q-value of Ge neutrinoless double-beta decay”. In this work they presented the measurements
f the 214Bi spectrum from a 226Ra source with a high purity germanium detector. Their attention was mostly
focused on the energy region around the Q-value of 76Ge neutrinoless double-beta decay (2039.006 keV). The
results of the measurement strongly relates to the first indication for neutrinoless double beta decay of 76Ge. An
analysis of the data collected during ten years of measurements by the Heidelberg-Moscow experiment, at Gran-
Sasso Underground Laboratory, yields a first indication for the neutrinoless double beta decay of 76Ge. An
important point of this analysis is the interpretation of the background, in the region around the Q-value of the
double beta decay (2039.006 keV), as containing several weak photopeaks. It was suggested and has been shown
that four of these peaks are produced by a contamination from the isotope 214Bi, whose lines are present
throughout the Heidelberg-Moscow background spectrum.
In this work they performed a measurement of a 226Ra source with a high-purity germanium detector. The aim of
this work was to study the spectral shape of the lines in the energy region from 2000 to 2100keV and, most
important, to show the difference in this spectral shape when changing the position of the source with respect to
the detector, and to verify the effect of TCS (True Coincidence Summing) for the weak 214Bi lines seen in the
Heidelberg-Moscow experiment. The activity of the 226Ra source is 95.2kBq. The isotope 226Ra appears in the
238U natural decay chain and from its decays also 214Bi is produced. The γ-spectrum of 214Bi is clearly visible in
the 226Ra measured spectrum. 214Bi is a naturally occurring isotope: it is produced in the 238U natural decay chain
through the β- decay of 214Pb and the alpha decay of 218At. With a subsequent β- reaction, 214Bi decays then into
214Po (the branching ratio with respect to the α decay into 210Tl is 99.979%). The decay, however, does not lead
d state of 214Po, but to its excited states. From the decays of those excited states to the ground
tate the well known γ-spectrum of 214Bi is obtained, which contains more than hundred lines.
the table given below, one can see in the energy region around the Q-value of the 0νββ decay (2000-2100keV),
ur γ-lines and one E0 transition with energy 2016.7keV are expected. The E0 transition can produce a
double beta decay experiment. The experiment took data during ~ 80% of its installation time. The Q value for
neutrinoless double beta decay was recently determined with high precision.
The background of the experiment: (1) primordial activities of the natural decay chains from 238 232U, Th, and 40K;
(2) anthropogenic radio nuclides, like 137 134 125 207Cs, Cs, Sb, Bi; (3) cosmogenic isotopes, produced by activation
due to cosmic rays during production and transport; (4) the bremsstrahlungs spectrum of 210Bi (daughter of 210Pb);
(5) elastic and inelastic neutron scattering; and (6) direct muon-induced events.
H.V. Klapdor-Kleingrothaus, O. Chkvorez, I.V. Krivosheina and C. Tomei at Max-Planck-Institut fur Kernphysik
in the Heidelberg-Moscow group 214
directly to the groun
conversion electron or an electron-positron pair but it could not contribute directly to the γ-spectrum in the
considered energy region if the source is located outside the detector active volume.
0.0502010.71
Intensity(%)Energy (keV)
0.0782052.94
0.05020889.7
0.0202021.8
0.00582016.7
0.0502010.71
Intensity(%)Energy (keV)
0.0202021.8
0.00582016.7
0.0782052.94
0.05020889.7
The intensity of each line is defined as the number of emitted photons, with the corresponding energy, per 100
decays of the parent nuclide. The considerations for the measurement were the efficiency of the detector (which
depends on the size of the detector and on the distance source-detector) and the effect called True Coincidence
Summing (TCS). The lifetimes of the atomic excited levels are much shorter than the resolving time of the
detector. If two gamma-rays are emitted in cascade, there is a certain probability that they will be detected
- 10 -
lled in
. The measurement of Bi
pectrum, with a high purity germanium detector, in the energy region around the Q-value of 76Ge neutrinoless
ta decay (2039.006keV) was done with the 226Ra source used for the measurements positioned, in a first
step the source was positioned on the top of the detector, directly in contact with the copper cap (close geometry)
and in a second step the source was moved 15cm away from the detector cap (far geometry). The results of the
measurements show that, if the source is close to the detector, the intensities of the weak Bi lines in the energy
region 2000- 2100keV are not in the same ratio as reported by Table of Isotopes. The results of the analysis of the
data collected by the Heidelberg-Moscow experiment with all the five detectors, yielding a first indication for the
neutrinoless double beta decay of 76Ge, shows that four 214Bi lines are present in the energy region from 2000 to
2080keV (many other strong lines from the same isotope are present in the spectrum), due to the presence of
bismuth in the experimental setup, especially in the copper in the vicinity of the Ge crystals.
together. If this happens, then a pulse will be recorded which represents the sum of the energies of the two
individual photons, instead of two separated pulses with different energies. The TCS effect can result both in
lower peak-intensity for full-energy peaks and in bigger peak-intensity for those transitions whose energy can be
given by the sum of two lower-energy gamma-rays. In this case, the lines at 2010.7 keV and 2016.7 keV can be
given by the coincidence of the 609.312 keV photon (strongest line, intensity = 46.1%) with the 1401.50keV
photon (intensity = 1.27%) or with the 1407.98keV photon (intensity = 2.15%). The degree of TCS depends on
the probability that two gamma-rays emitted simultaneously will be detected simultaneously which is a function
of the detector geometry and of the solid angle subtended at the detector by the source and for this the intensities
of the two lines mentioned above (2010.71keV and 2016.7keV) are expected to depend on the position of the
source with respect to the detector.
The 226Ra γ-ray spectra were measured using a γ-ray spectroscopy system based on an HPGe detector insta
the operation room of the HEIDELBERG-MOSCOW experiment in Gran Sasso Underground Laboratory, Italy.
The coaxial germanium detector had an external diameter of 5.2cm and 4.9cm height. The distance between the
top of the detector and the copper cap was kept at 3.5cm. The relative detection efficiency of the detector was
23% and the energy resolution being 3.6keV for the energy range 2000-2100keV 214
double-be
The above figure shows the sum spectrum of the 76Ge detectors 1,2,3,4 and 5 over the period August
1990 to May 2003 as recorded by the Heidelberg-Moscow experiment.
- 11 -
There is no null hypothesis analysis demonstrating that the data require a peak. Furthermore, no simulation has
to demonstrate that the analysis correctly finds true peaks or that it would find no peaks if none
existed. Monte Carlo simulations of spectra containing different numbers of peaks are needed to confirm the
significance of any found peaks.
2. There are three unidentified peaks in the region of analysis that have greater significance than the 2039-keV
peak. There is no discussion of the origin of these peaks.
3. There is no discussion of how sensitive the conclusions are to different mathematical models. There is a
previous Heidelberg-Moscow publication that gives a lower limit of 1.9 × 1025 y (90% confidence level). This is
in conflict with the “best value” of a newer KDHK paper of 1.5 × 1025 y. This indicates a dependence of the
results on the analysis model and the background evaluation.
In this paper they state that a number of other cross checks of the result should also be performed. For example,
there is no discussion of how a variation of the size of the chosen analysis window affects the significance of the
hypothetical peak. There is no relative peak strength analysis of all the 214Bi peaks. Quantitative evaluations
should be made on the four 214Bi peaks in the region of interest. There is no statement of the net count rate of the
peaks other than the 2039-keV peak. There being no presentation of the entire spectrum, is difficult to compare
relative strengths of peaks. There is no discussion of the relative peak strengths before and after the single-site-
event cut.
On the other hand the Heidelberg-Moscow group claims that the signal found at Qββ is consisting of single site
events and is not a γ line. The signal does not occur in the Ge experiments not enriched in the double beta emitter
76Ge, while neighbouring background lines appear consistently in these experiments. On this basis they translated
the observed numbers of events into half-lives for neutrinoless double beta decay.
The Heidelberg-Moscow experiment continued regularly from 1990 till 2003. The analysis of the full data taken
with the Heidelberg-Moscow experiment in the period 2 August 1990 until 20 May 2003 is presented. The
completed Heidelberg-Moscow 76Ge Experiment -71.7 kg y after 13 years of operation presents their mass
calculation limit status as mν (eV) = 0.24 - 0.58 ( 99.997% C.L.) with the best value of 0.4 eV (95% C.L.).
hile an unambiguous interpretation of all of the neutrino oscillation experiments is not yet possible, it is
bundantly clear that neutrinos exhibit properties not included in the standard model, namely mass and flavor
arch which will employ 500 kg of Ge,
a. The Majorana experiment is proposed for a US deep underground laboratory,
eriments. Furthermore, new segmented Ge detector
yogenic performance and background reduction and
Moscow and IGEX experiments both utilized Germanium enriched to 86% in Ge and operated deep
In a paper by Klapdor-Kleingrothaus, Dietz, Harney, and Krivosheina (hereafter referred to as KDHK) evidence is
claimed for zero-neutrino double-beta decay in 76 Ge. The high quality data, upon which this claim is based, was
compiled by the 2 careful efforts of the Heidelberg-Moscow collaboration, and is well documented. However, the
analysis in KDHK makes an extraordinary claim, and therefore requires very solid substantiation according to
another paper “Comment on Evidence for Neutrinoless Double Beta Decay” C.E.Aalseth et al. They state that a
large number of issues were not addressed in KDHK some of which are:
been presented
5. The proposed MAJORANA experiment
mixing. Accordingly, sensitive searches for neutrinoless double-beta decay (0νββ-decay) are more important than
ever. Experiments with large quantities of Ge, isotopically enriched in 76Ge, have thus far proven to be the most
sensitive, specifically the Heidelberg-Moscow and IGEX experiments with lower limits in half-life sensitivities
1.9×1025 y and 1.6×1025 y respectively. A new generation of experiments will be required to make significant
improvements in sensitivity one of which is the proposed Majorana Experiment.
The Majorana Experiment is a next-generation Ge double-beta decay se
isotopically enriched to 86% in Ge, in the form of ~200 detectors in a close-packed array for high granularity.
Each crystal will be electronically segmented, with each region fitted with pulse-shape analysis electronics. A
half-life sensitivity is predicted of 4.2 × 1027 years or < mν> ~ 0.02 - 0.07 eV, depending on the nuclear matrix
elements used to interpret the dat
and requires very little R&D as it stands on the technical shoulders of the IGEX experiment and other previous
successful double-beta decay and low-background exp
technology has recently become commercially available, while Pacific Northwest National Laboratory
(PNNL)/University of South Carolina (USC) researchers have developed new pulse-shape discrimination
techniques.
Several configurations have been evaluated with respect to cr
rejection. It will concentrate on a conventional modular design using ultra-low background cryostat technology
developed by IGEX. It will also utilize new pulse-shape discrimination hardware and software techniques
developed by the Majorana collaboration and detector segmentation to reduce background. The Heidelberg-
underground. The projection for the Majorana is that the background will be reduced by a factor of 65 over the
early IGEX results prior to pulse shape analysis (from 0.2 to ~0.003 keV-1
- 12 -
germanium by limiting the time above ground after crystal growth, careful material selection
marily comprised of multiple
o or
ore of the independent segments. When coincidences are found, the output from all detector segments is
nly of the full-energy peak lying above a featureless
kg-1 y-1). This will occur mainly by the
decay of the internal background due to cosmogenic neutron spallation reactions that produce 56 58 60Co, Co, Co,
65Zn and 68Ge in the
and electroforming copper cryostats. One component of the background reduction will arise from the
segmentation and granularity of the detector array.
Most of the Compton continuum consists of single Compton scatterings followed by escape of the scattered
gamma ray, whereas full-energy events at typical gamma-ray energies are pri
scattering sequences followed by a photoelectric absorption. The peak-to-Compton ratio can therefore be
enhanced by requiring a recorded event to correspond to more than one interaction within the detector before its
acceptance. In germanium detectors, this selection is usually accomplished by subdividing the detector into
several segments (or providing several adjacent independent detectors) and seeking coincident pulses from tw
summed and recorded. The resulting spectrum is made up o
continuum that is greatly suppressed and has no abrupt Compton edges. New Ge experiments must not simply be
a volume expansion of IGEX or Heidelberg–Moscow. They must have superior background rejection and better
electronic stability. The summing of 200 individual energy spectra can result in serious loss of energy resolution
for the overall experiment which can be avoided by segmenting n-type intrinsic Ge detectors, advanced PSD
techniques and electronic stability in measurement.
The above figure depicts a standard Ge detector segmentation scheme. This is the configuration of the SEGA
detector undergoing tests by the Majorana collaboration. A configuration with six-azimuthal-segment by two-
axial-segment geometry is shown in the above figure.
Efforts are thus on with the Majorana experiment for the search of neutrinoless double beta decay that would give
a new shape to the standard model of physics. Majorana cannot not simply be a volume expansion of IGEX, but
must have superior background rejection. As it was conclusively shown that the limiting background in at least
some previous experiments has been cosmogenic activation of the germanium itself, it is necessary to mitigate
those background sources. Cosmogenic activity fortunately has certain factors which discriminate it from the
signal of interest. For example, while 0νββ -decay would deposit 2 MeV between two electrons in a small,
perhaps 1 mm3 volume, internal 60Co decay deposits about 318 keV (endpoint) in beta energy near the decaying
atom, while simultaneous 1173 keV and 1332 keV gammas can deposit energy elsewhere in the crystal, most
probably both in more than one location, for a total energy capable of reaching the 2039 keV region-of-interest. A
similar situation exists for internal 68Ge decay. Thus deposition-location multiplicity distinguishes double-beta
decay from the important long lived cosmogenics in germanium. Isotopes such as 56 57 58Co, Co, Co and 68Ge are
produced at a rate of roughly 1 atom per day per kilogram on the earth’s surface. Only 60Co and 68Ge have both
the energy and half-life to be of concern. To pursue the multiplicity parameter, firstly, the detector current pulse
shape carries with it the record of energy deposition along the electric field lines in the crystal; that is, the radial
- 13 -
imension of cylindrical detectors. This information may be exploited through pulse-shape discrimination.
econdly, the electrical contacts of the detector may be divided to produce independent regions of charge
collection. The ability of new techniques to be easily calibrated for individual detectors makes them practical for
large detector arrays. Calibration for single-site event pulses was trivially accomplished by collecting pulses from
thorium ore; the 2614.47-keV gamma ray from 208Tl produces a largely single-site double-escape peak at 1592.47
keV. The PSD discriminator was then calibrated to the properties of the double-escape peak A slightly improved
double-escape peak was be made from the 26Al gamma ray of 2938.22-keV. The double-escape appears at
1916.22 keV, only about 120 keV away from the expected region of interest for 0νββ-decay. The obvious and
direct use of pulse- shape discrimination and segmentation is the rejection of cosmogenic pulses in the germanium
itself. However, the approach should be also effective on gamma rays from the shielding and structural materials.
The background effects of neutrons of both high energy (cosmic muon generated) and low energy (fission and
(α,n) from rock) could be protected by the segmentation and granularity of the detectors. These neutrons could
also produce other unwanted activities like the formation of 3H and 14C in nitrogen from high and low energy
neutrons, respectively. Fortunately, Majorana detectors will not be surrounded by nitrogen at high density.
The GERDA (GERmanium Detector Assembly), which is another next generation 76Ge double beta decay
experiment at the Gran Sasso Underground Laboratory, has projected a sensitivity in the half-life of the 0νββ-
decay mode which is less than the proposed Majorana experiment. In conclusion, the Majorana project has been
designed in a compact, modular way such that it can be built and operated with high confidence in the approach
and the technology. The initial years of construction will allow alternate cooling methods to be employed if they
have an advantage and should they be shown to overcome long-term concerns due to surface contamination,
muon-induced ions, and diffusion. The Majorana Collaboration has made an extensive analysis of the predicted
backgrounds and their impact on the final sensitivity of the experiment. The Majorana experiment represents a
great increase in Ge mass over IGEX with new segmented Ge detectors and the newest electronic systems for
pulse-shape discrimination. Their conclusion is that with 500 kg of Ge, enriched to 86% in the isotope 76Ge, the
Majorana array operating over 10 years including construction time, can reach a lower limit on T1/20ν of 4×1027
years. This corresponds to an upper bound of < m > of 0.038 ± 0.007eV. One advantage of 76ν Ge is that it may
well be a candidate for a future more reliable microscopic calculation of the 0ν ββ- decay nuclear matrix element.
6 Conclusion
eutrinoless double beta decay is thus one of the most sensitive approaches with great perspectives to test pN article
hysics beyond the Standard Model. The possibilities to use 0νββ decay for constraining neutrino masses, left–
ght symmetric models, SUSY and leptoquark scenarios, as well as effective lepton number violating couplings,
have been reviewed. It is a very sensitive probe to the lepton number violating terms in the Lagrangian such as the
Majorana mass of the light neutrinos, right-handed weak couplings involving heavy Majorana neutrinos, as well
as Higgs and other interactions involving violation of chirality conservation.
- 14 -
In search for neutrinoless double beta decay 76Ge as the source material has multiple advantages. It has high
resolution (< 4 keV at Qββ) with no background from 2ν mode. A huge leap in sensitivity is possible applying
ultra-low background techniques and 0ν- ββ signal discrimination. There can be a phased approach in the
experiment with the increment of target mass. The source and detector are the same material thereby reducing
background and maintaining the 4π geometry and the only way to scrutinize 0ν – DBD claim on short time scale:
since it tests T1/2 and not mν. The consequences of Neutrinoless Double Beta Decay are- [1] Total Lepton number
violation: The most important consequence of the observation of neutrinoless double beta decay is that lepton
number is not conserved. This is fundamental for particle physics. [2] Majorana nature of neutrino: Another
fundamental consequence is that the neutrino is a Majorana particle. Both of these conclusions are independent of
any discussion of nuclear matrix elements. [3] Effective neutrino mass: The matrix element enters when we derive
a value for the effective neutrino mass - making the most natural assumption that the 0νββ decay amplitude is
dominated by exchange of a massive Majorana neutrino.
Acknowledgements
I would like to thank the IGEX collaboration, the Heidelberg-Moscow collaboration and the Majorana
n for having used information from th imental works to write up this bri review.
eferences
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Lett. A(2002), hep-ph/0202018 C. E.
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[9] H.V. Klapdor-Kleingrothaus et al., (Heidelberg-Moscow Collaboration.), Eur. Phys. J. A 12(2001)147.
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[12] “Heidelberg - Moscow Experiment. First Evidence for Lepton Number Violation and the Majorana Character of
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[16.] “The IGEX experiment revisited: a response to the critique of Klapdor-Kleingrothaus, Dietz, and Krivosheina”,
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[23] “Latest Results fro
|
0704.0064 | Nilpotent symmetry invariance in the superfield formulation: the
(non-)Abelian 1-form gauge theories | arXiv:0704.0064v5 [hep-th] 24 Oct 2008
arXiv:0704.0064 [hep-th]
CAS-PHYS-BHU/Preprint
NILPOTENT SYMMETRY INVARIANCE IN THE SUPERFIELD
FORMULATION: THE (NON-)ABELIAN 1-FORM GAUGE THEORIES
R. P. MALIK
Centre of Advanced Studies, Physics Department,
Banaras Hindu University, Varanasi- 221 005, (U. P.), India
E-mails: [email protected] ; [email protected]
Abstract: We capture the off-shell as well as the on-shell nilpotent Becchi-Rouet-Stora-
Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian densities of the
four (3 + 1)-dimensional (4D) (non-)Abelian 1-form gauge theories within the framework
of the superfield formalism. In particular, we provide the geometrical interpretations for (i)
the above nilpotent symmetry invariance, and (ii) the above Lagrangian densities, in the
language of the specific quantities defined in the domain of the above superfield formalism.
Some of the subtle points, connected with the 4D (non-)Abelian 1-form gauge theories, are
clarified within the framework of the above superfield formalism where the 4D ordinary
gauge theories are considered on the (4, 2)-dimensional supermanifold parametrized by the
four spacetime coordinates xµ (with µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ
and θ̄. One of the key results of our present investigation is a great deal of simplification
in the geometrical understanding of the nilpotent (anti-)BRST symmetry invariance.
PACS numbers: 11.15.-q, 12.20.-m, 03.70.+k
Keywords: Superfield formalism; (non-)Abelian 1-form gauge theories; (anti-)BRST sym-
metries; symmetry invariance; horizontality condition; geometrical interpretations
http://arxiv.org/abs/0704.0064v5
1 Introduction
The geometrical superfield approach [1-8] to Becchi-Rouet-Stora-Tyutin (BRST) formalism
is one of the most attractive and intuitive approaches which enables us to gain some physical
insights into the beautiful (but abstract mathematical) structures that are associated with
the nilpotent (anti-)BRST symmetry transformations and their corresponding generators.
The latter quantities play a very decisive role in (i) the covariant canonical quantization of
the gauge theories, (ii) the proof of the unitarity of the “quantum” gauge theories at any
arbitrary order of perturbative computations for a given physical process (that is allowed
by the theory), (iii) the definition of the physical states of the “quantum” gauge theories
in the quantum Hilbert space, and (iv) the cohomological description of the physical states
of the quantum Hilbert space w.r.t. the conserved and nilpotent BRST charge.
To be specific, in the superfield formulation [1-8] of the 4D 1-form gauge theories, one
defines the super curvature 2-form F̃ (2) = d̃Ã(1)+ i Ã(1)∧ Ã(1) in terms of the super exterior
derivative d̃ = dxµ∂µ + dθ∂θ + dθ̄∂θ̄ (with d̃
2 = 0) and the super 1-form connection Ã(1) on
a (4, 2)-dimensional supermanifold parametrized by the usual spacetime variables xµ (with
µ = 0, 1, 2, 3) and a pair of anticommuting (i.e. θ2 = θ̄2 = 0, θθ̄ + θ̄θ = 0) Grassmannian
variables θ and θ̄. The above super 2-form is subsequently equated, due to the so-called
horizontality condition [1-8], to the ordinary curvature 2-form F (2) = dA(1) + iA(1) ∧ A(1)
defined on the ordinary 4D flat Minkowski spacetime manifold in terms of the ordinary
exterior derivative d = dxµ∂µ (with d
2 = 0) and the 1-form connection A(1) = dxµAµ. The
above super exterior derivative d̃ and super 1-form connection Ã(1) are the generalization of
the 4D ordinary exterior derivative d and 1-form connection A(1) to the (4, 2)-dimensional
supermanifold because d̃ → d, Ã(1) → A(1) in the limit (θ, θ̄) → 0.
The above horizontality condition (HC) has been referred to as the soul-flatness con-
dition in [9] which amounts to setting equal to zero all the Grassmannian components of
the (anti)symmetric second-rank super tensor that constitutes the super curvature 2-form
F̃ (2) on the (4, 2)-dimensional supermanifold. The key consequences, that emerge from
the HC, are (i) the derivation of the nilpotent (anti-)BRST symmetry transformations for
the gauge and (anti-)ghost fields of a given 4D 1-form gauge theory, (ii) the geometrical
interpretation of the (anti-)BRST symmetry transformations for the 4D local fields as the
translation of the corresponding superfields along the Grassmannian directions of the su-
permanifold, (iii) the geometrical interpretation of the nilpotency property as a pair of
successive translations of the superfield along a particular Grassmannian direction of the
supermanifold, and (iv) the geometrical interpretation of the anticommutativity property
of the (anti-)BRST symmetry transformations for a 4D local field as the sum of (a) the
translation of the corresponding superfield first along the θ-direction followed by the trans-
lation along the θ̄-direction, and (b) the translation of the same superfield first along the
θ̄-direction followed by the translation along the θ-direction.
It will be noted that the above HC (i.e. F̃ (2) = F (2)) is valid for the non-Abelian (i.e.
A(1)(n)∧A(1)(n) 6= 0) 1-form gauge theory as well as the Abelian (i.e. A(1)∧A(1) = 0) 1-form
gauge theory. As expected, for both types of theories, the HC leads to the derivation of the
nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of
the respective theories. We lay emphasis on the fact that the HC does not shed any light
on the derivation of the nilpotent (anti-)BRST symmetry transformations associated with
the matter fields of the interacting 4D (non-)Abelian 1-form gauge theories.
In a recent set of papers [10-17], the above HC condition has been generalized, in a
consistent manner, so as to compute the nilpotent (anti-)BRST symmetry transformations
associated with the matter fields of a given 4D interacting 1-form gauge theory (along with
the well-known nilpotent transformations for the gauge and (anti-)ghost fields) without
spoiling the cute geometrical interpretations of the (anti-)BRST symmetry transformations
(and their corresponding generators) that emerge from the HC alone. The latter approach
has been christened as the augmented superfield approach to BRST formalism where the
restrictions imposed on the (4, 2)-dimensional superfields are (i) the HC plus the invariance
of the (super) matter Noether conserved currents [10-14], (ii) the HC plus the equality of
any (super) conserved quantities [15], (iii) the HC plus a restriction that owes its origin
to the gauge invariance and the (super) covariant derivatives on the matter (super)fields
[16,17], and (iv) an alternative to the HC where the gauge invariance and the property
of a pair of (super) covariant derivatives on the (super) matter fields (and their intimate
connection with the (super) curvatures) play a crucial role [18-20].
In all the above approaches [1-20], however, the invariance of the Lagrangian densities
of the 4D (non-)Abelian 1-form gauge theories, under the nilpotent (anti-)BRST symmetry
transformations, has not yet been discussed at all. Some attempts in this direction have
been made in our earlier works where the specific topological features [21,22] of the 2D
free (non-)Abelian 1-form gauge theories have been captured in the superfield formulation
[23-25]. In particular, the invariance of the Lagrangian density under the nilpotent and
anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations has been ex-
pressed in terms of the superfields and the Grassmannian derivatives on them. These are,
however, a bit more involved in nature because of the existence of a new set of nilpotent
(anti-)co-BRST symmetries in the theory. The geometrical interpretations for the La-
grangian densities and the symmetric energy-momentum tensor (for the above topological
theory) have also been provided within the framework of the superfield formulation.
The purpose of our present paper is to capture the (anti-)BRST symmetry invariance of
the Lagrangian density of the 4D (non-)Abelian 1-form gauge theories within the framework
of the superfield approach to BRST formalism and to demonstrate that the above symme-
try invariance could be understood in a very simple manner in terms of the translational
generators along the Grassmannian directions of the (4, 2)-dimensional supermanifold on
which the above 4D ordinary gauge theories are considered. In addition, the reason behind
the existence (or non-existence) of any specific nilpotent symmetry transformation could
also be explained within the framework of the above superfield approach. We demonstrate
the uniqueness of the existence of the nilpotent (anti-)BRST symmetry transformations
for the Lagrangian density of a U(1) Abelian 1-form gauge theory. We go a step further
and show the existence of the nilpotent BRST symmetry transformations for the specific
Lagrangian densities (cf. (4.1) and (4.4) below) of the 4D non-Abelian 1-form gauge theory
and clarify the non-existence of the anti-BRST symmetry transformations for these spe-
cific Lagrangian densities within the framework of the superfield formulation (cf. section
5 below). Finally, we provide the geometrical basis for the existence of the off-shell nilpo-
tent and anticommuting (anti-)BRST symmetry transformations (and their corresponding
generators) for the specifically defined Lagrangian densities (cf. (4.7) and/or (4.8) below)
of the 4D non-Abelian 1-form gauge theory in the Feynman gauge.
The motivating factors that have propelled us to pursue our present investigation are
as follows. First and foremost, to the best of our knowledge, the property of the symmetry
invariance of a given Lagrangian density has not yet been captured in the language of the
superfield approach to BRST formalism. Second, the above (anti-)BRST invariance of the
theory has never been shown, in as simplified fashion, as we demonstrate in our present
endeavour. The geometrical interpretations for (i) the existence of the above nilpotent
(anti-)BRST symmetry invariance, and (ii) the on-shell conditions of the on-shell nilpotent
(anti-)BRST symmetries, turn out to be quite transparent in our present work. Third, we
establish the uniqueness of the existence of the (anti-)BRST symmetry invariance in their
various forms. The non-existence of the specific symmetry transformation is also explained
within the framework of the superfield approach to BRST formalism. Finally, our present
investigation is the first modest step in the direction to gain some insights into the existence
of the nilpotent symmetry transformations and their invariance for the higher form (e.g.
2-form, 3-form, etc.) gauge theories within the framework of the superfield formulation.
The contents of our present paper are organized as follows. In section 2, we recapitulate
some of the key points connected with the nilpotent (anti-)BRST symmetry transformations
for the free 4D Abelian 1-form gauge theory (having no interaction with matter fields) in
the Lagrangian formulation. The above symmetry transformations as well as the symmetry
invariance of the Lagrangian densities are captured in the geometrical superfield approach
to BRST formalism in section 3 where the HC on the gauge superfield plays a crucial role.
Section 4 deals with the bare essentials of the nilpotent (anti-)BRST symmetry transfor-
mations for the 4D non-Abelian 1-form gauge theory in the Lagrangian formulation. The
subject matter of section 5 concerns itself with the superfield formulation of the symmetry
invariance of the appropriate Lagrangian densities of the above 4D non-Abelian 1-form
gauge theory. Finally, in section 6, we summarize our key results, make some concluding
remarks and point out a few future directions for further investigations.
2 (Anti-)BRST symmetries in Abelian theory: Lagrangian formulation
Let us begin with the following (anti-)BRST invariant Lagrangian density of the 4D Abelian
1-form gauge theory∗ in the Feynman gauge [26,27,9]
B = −
F µνFµν + B (∂µA
B2 − i ∂µC̄ ∂
µC, (2.1)
where Fµν = ∂µAν − ∂νAµ is the antisymmetric (Fµν = −Fνµ) curvature tensor that con-
stitutes the Abelian 2-form F (2) = dA(1) ≡ 1
(dxµ ∧ dxν)Fµν , B is the Nakanishi-Lautrup
auxiliary multiplier field and (C̄)C are the anticommuting (i.e. C2 = C̄2 = 0, CC̄+C̄C = 0)
(anti-)ghost fields of the theory. The above Lagrangian density respects the off-shell nilpo-
tent (s2(a)b = 0) (anti-)BRST symmetry transformations s(a)b (with sbsab + sabsb = 0)
sbAµ = ∂µC, sbC = 0, sbC̄ = iB, sbB = 0, sbFµν = 0,
sabAµ = ∂µC̄, sabC̄ = 0, sabC = −iB, sabB = 0, sabFµν = 0.
(2.2)
It is clear that, under the nilpotent (anti-)BRST symmetry transformations s(a)b, the cur-
vature tensor Fµν is found to be invariant. In other words, the 2-form F
(2), owing its
origin to the cohomological operator d = dxµ∂µ, is an (anti-)BRST invariant object for the
Abelian U(1) 1-form gauge theory and is, therefore, a physically meaningful (i.e. gauge-
invariant) quantity. These observations will play an important role in our discussion on the
horizontality condition that would be exploited in the context of our superfield approach
to (anti-)BRST invariance of the Lagrangian densities in sections 3 and 5 (see below).
A noteworthy point, at this stage, is the observation that the gauge-fixing and Faddeev-
Popov ghost terms can be written, modulo a total derivative, in the following fashion
−i C̄ {(∂µA
B}], sab
+i C {(∂µA
sb sab
(2.3)
The above equation establishes, in a very simple manner, the (anti-)BRST invariance of
the 4D Lagrangian density (2.1). The simplicity ensues due to (i) the nilpotency s2(a)b = 0
of the (anti-)BRST symmetry transformations, (ii) the anticommutativity property (i.e.
sbsab + sabsb = 0) of s(a)b, and (iii) the invariance of the Fµν term under s(a)b.
As a side remark, it is interesting to note that the following on-shell (i.e. ✷C = ✷C̄ = 0)
nilpotent (s̃2(a)b = 0) (anti-)BRST symmetry transformations (with s̃bs̃ab + s̃abs̃b = 0)
s̃bAµ = ∂µC, s̃bC = 0, s̃bC̄ = −i(∂µA
µ), s̃bFµν = 0,
s̃abAµ = ∂µC̄, s̃abC̄ = 0, s̃abC = +i(∂µA
µ), s̃abFµν = 0,
(2.4)
∗We adopt here the notations and conventions such that the flat Minkowski metric in 4D is ηµν = diag
(+1,−1,−1,−1) so that AµB
µ = ηµνA
µBν = A0B0 − AiBi for two non-null 4-vectors Aµ and Bµ. The
Greek indices µ, ν...... = 0, 1, 2, 3 and Latin indices i, j, k.... = 1, 2, 3 stand for the 4D spacetime and 3D
space directions on the 4D Minkowski spacetime manifold, respectively, and the symbol ✷ = (∂0)
2 − (∂i)
†We follow here the notations and conventions adopted in [27]. In its full blaze of glory, the nilpotent
(anti-)BRST transformations δ(A)B are a product of an anticommuting spacetime independent parameter
η and s(a)b (i.e. δ(A)B = ηs(a)b) where the nilpotency property is encoded in the operators s(a)b.
are the symmetry transformations for the following Lagrangian density
b = −
F µνFµν −
µ)2 − i ∂µC̄ ∂
µC. (2.5)
The above transformations (2.4) and the Lagrangian density (2.5) have been derived from
(2.2) and (2.1) by the substitution B = −(∂µA
µ). An interesting point, connected with the
on-shell nilpotent symmetry transformations, is to express the analogue of (2.3) as ‡
C̄ (∂µA
µ) + i Aµ∂
µC̄], s̃ab
C (∂µA
µ)− i Aµ∂
s̃b s̃ab
(2.6)
It should be noted that, in the above precise computation, one has to take into account
the on-shell (✷C = ✷C̄ = 0) conditions so that, for all practical purposes s̃(a)b(∂µA
µ) = 0.
The above nilpotent (anti-)BRST symmetry transformations (i.e. sr, s̃r with r = b, ab)
are connected with the conserved and nilpotent generators (i.e. Qr, Q̃r with r = b, ab).
This statement can be succinctly expressed, in the mathematical form, as
sr Ω = −i [ Ω, Qr ](±), s̃r Ω̃ = −i [ Ω̃, Q̃r ](±), r = b, ab, (2.7)
where the subscripts (with the signatures (±)) on the square bracket stand for the bracket
to be an (anti)commutator, for the generic fields Ω = Aµ, C, C̄, B and Ω̃ = Aµ, C, C̄ (of
the Lagrangian densities (2.1) and (2.5)), being (fermionic)bosonic in nature. The above
charges Qr, Q̃r are found to be anticommuting (i.e. QbQab+QabQb = 0, Q̃bQ̃ab+Q̃abQ̃b = 0)
and off-shell as well as on-shell nilpotent (Q2(a)b = 0, Q̃
(a)b = 0) in nature, respectively.
3 (Anti-)BRST invariance in Abelian theory: superfield formalism
In this section, we exploit the geometrical superfield approach to BRST formalism, endowed
with the theoretical arsenal of the horizontality condition, to express the (anti-)BRST
symmetry transformations and the Lagrangian densities (cf. (2.1) and (2.5)) in terms of
the superfields defined on the (4, 2)-dimensional supermanifold. The latter is parametrized
by the spacetime coordinates xµ (with µ = 0, 1, 2, 3) and a pair of Grassmannian variables θ
and θ̄. As a consequence, the generalization of the 4D ordinary exterior derivative d = dxµ∂µ
and the 1-form connection A(1) = dxµAµ(x) on the (4, 2)-dimensional supermanifold, are
d → d̃ = dxµ ∂µ + dθ ∂θ + dθ̄ ∂θ̄, d̃
2 = 0,
A(1) → Ã(1) = dxµ Bµ(x, θ, θ̄) + dθ F̄(x, θ, θ̄) + dθ̄ F(x, θ, θ̄),
(3.1)
where the mapping from the 4D local fields to the superfields are: Aµ(x) → Bµ(x, θ, θ̄),
C(x) → F(x, θ, θ̄) and C̄(x) → F̄(x, θ, θ̄). The super-expansion of the superfields, in terms
‡We lay emphasis on the fact that (2.6) cannot be derived directly from (2.3) by the simple substitution
B = −(∂µA
µ). One has to be judicious to deduce the precise expression for (2.6). The logical reasons
behind the derivation of (2.6) are encoded in the superfield formulation (cf. (3.9) below).
of the basic fields as well as the secondary fields, are (see, e.g., [4-7, 10-12]):
Bµ(x, θ, θ̄) = Aµ(x) + θ R̄µ(x) + θ̄ Rµ(x) + i θ θ̄ Sµ(x),
F(x, θ, θ̄) = C(x) + i θ B̄1(x) + i θ̄ B1(x) + i θ θ̄ s(x),
F̄(x, θ, θ̄) = C̄(x) + i θ B̄2(x) + i θ̄ B2(x) + i θ θ̄ s̄(x).
(3.2)
It can be readily seen that, in the limiting case of (θ, θ̄) → 0, we get back our 4D basic
fields (Aµ, C, C̄). Furthermore, on the r.h.s. of the above super expansion, the bosonic (i.e.
Aµ, Sµ, B1, B̄1, B2, B̄2) and the fermionic (Rµ, R̄µ, C, C̄, s, s̄) fields do match.
At this juncture, we have to recall our observations after equation (2.2). The nilpotent
(anti-)BRST symmetry transformations basically owe their origin to the cohomological
operator d. This is capitalized in the horizontality condition where we impose the restriction
d̃Ã(1) = dA(1) on the super 1-form connection Ã(1) that contains the superfields defined on
the (4, 2)-dimensional supermanifold. The latter condition yields the following relationships
(see, e.g., for details, in our earlier works [21-25]):
B1 = B̄2 = s = s̄ = 0, B̄1 +B2 = 0, (3.3)
where we are free to choose the secondary fields (B2, B̄1) (i.e. B2 = B ⇒ B̄1 = −B) in
terms of the Nakanishi-Lautrup auxiliary field B of the BRST invariant Lagrangian density
(2.1). The other relations, that emerge from the above HC (i.e. d̃Ã(1) = dA(1)), are
Rµ = ∂µC, R̄µ = ∂µC̄, Sµ = ∂µB, ∂µBν − ∂νBµ = ∂µAν − ∂νAµ. (3.4)
At this stage, the super-curvature tensor F̃µν = ∂µBν − ∂νBµ is not equal to the ordinary
curvature tensor Fµν = ∂µAν−∂νAµ as the former contains Grassmannian dependent terms.
The substitution of the above values (cf. (3.3),(3.4)) of the secondary fields, in terms
of the basic and auxiliary fields of the Lagrangian density (2.1), leads to
B(h)µ (x, θ, θ̄) = Aµ + θ ∂µC̄ + θ̄ ∂µC + i θ θ̄ ∂µB,
F (h)(x, θ, θ̄) = C − i θ B, F̄ (h)(x, θ, θ̄) = C̄ + i θ̄ B,
(3.5)
where the superscript (h) has been used to denote that the above expansions have been
obtained after the application of the HC. It can be seen that, due to (3.5), we get
ν − ∂νB
µ = ∂µAν − ∂νAµ, (3.6)
where there is no Grassmannian θ and θ̄ dependence on the l.h.s.
In the language of the geometry on the (4, 2)-dimensional supermanifold, the expansions
(3.5) imply that the (anti-)BRST symmetry transformations s(a)b (and their corresponding
generators Q(a)b) for the 4D local fields (cf. (2.7)) are connected with the translational
generators (∂/∂θ, ∂/∂θ̄) because the translation of the corresponding (4, 2)-dimensional
superfields, along the Grassmannian directions of the supermanifold, produces it. Thus,
the Grassmannian independence of the super curvature tensor F̃ (h)µν = ∂µB
ν −∂νB
µ implies
that the 4D curvature tensor Fµν is an (anti-)BRST (i.e. gauge) invariant physical quantity.
In terms of the superfields, equations (2.3) can be expressed as
Limθ→0
−i F̄ (h) { (∂µB(h)µ +
Limθ̄→0
+ iF (h) { (∂µB(h)µ +
Bµ(h)B(h)µ +
F (h) F̄ (h)
(3.7)
These equations are unique because there is no other way to express the above equations in
terms of the derivatives w.r.t. Grassmannian variables θ and θ̄. Thus, besides (2.3), there
is no other possibility to express the gauge-fixing and the Faddeev-Popov ghost terms in
the language of the off-shell nilpotent (anti-)BRST symmetry transformations (2.2). The
superfield approach to BRST formulation, therefore, establishes the uniqueness of (2.3).
To express (2.6) in terms of the superfields, one has to substitute B = −(∂µA
µ) in (3.5).
Thus, the expansion (3.5), in terms of the transformations (2.4), becomes§
µ(o)(x, θ, θ̄) = Aµ + θ ∂µC̄ + θ̄ ∂µC − i θ θ̄ ∂µ(∂
ρAρ),
≡ Aµ + θ (s̃abAµ) + θ̄ (s̃bAµ) + θ θ̄(s̃bs̃abAµ),
(o) (x, θ, θ̄) = C + i θ (∂µA
µ) ≡ C + θ (s̃abC),
(o) (x, θ, θ̄) = C̄ − i θ̄ (∂µA
µ) ≡ C̄ + θ̄ (s̃bC̄).
(3.8)
We note that (3.5) and (3.8) are the super expansions (after the application of the HC)
which lead to the derivation of the off-shell nilpotent (anti-)BRST symmetry transforma-
tions s(a)b as well as the on-shell nilpotent (anti-)BRST symmetry transformations s̃(a)b,
respectively, for the basic fields Aµ, C and C̄ of the theory.
The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian density (2.5) can
also be expressed in terms of the superfields (3.8). In other words, (vis-à-vis (3.7)), we have
the following equations that are the analogue of (2.6), namely;
Limθ→0
(o) (∂
µAµ) + i B
µ(o) ∂
(o) )
Limθ̄→0
(o) (∂
µAµ)− i B
µ(o) ∂
(o) )
(o) B
µ(o) +
(o) F̄
(3.9)
We know that, for all practical computational purposes, it is essential to take into account
s̃(a)b(∂µA
µ) = 0 because of the on-shell conditions ✷C = ✷C̄ = 0. The logical reason behind
such a restriction (i.e. s̃(a)b(∂µA
µ) = 0) in (2.6) is encoded in the superfield approach to
BRST formalism as can be seen from a close look at (3.9).
The Lagrangian density (2.1) can be expressed, in terms of the (4, 2)-dimensional
superfields, in the following distinct and different forms
B = −
F̃ (h)µν F̃
µν(h) + Limθ→0
−i F̄ (h)(∂µB(h)µ +
, (3.10)
§The on-shell nilpotent (anti-)BRST symmetry transformations s̃(a)b can also be obtained by invoking
the (anti-)chiral superfields on the appropriately chosen supermanifolds (see, e.g. [23] for details).
B = −
F̃ (h)µν F̃
µν(h) + Limθ̄→0
+i F (h)(∂µB(h)µ +
, (3.11)
B = −
F̃ (h)µν F̃
µν(h) +
Bµ(h)B(h)µ +
F (h)F̄ (h)
. (3.12)
It would be noted that the kinetic energy term −(1/4)F̃ (h)µν F̃
µν(h) is independent of the
variables θ and θ̄ because F̃ (h)µν = Fµν . In exactly similar fashion, the Lagrangian density of
(2.5) can be expressed, with the help of the super expansion (3.8), as
b = −
µν(o)F̃
µν(h)
(o) + Limθ→0
(o) (∂
µAµ) + i B
µ(o) ∂
(o) )
, (3.13)
b = −
µν(o)F̃
µν(h)
(o) + Limθ̄→0
(o) (∂
µAµ)− i B
µ(0) ∂
(o) )
, (3.14)
b = −
µν(o)F̃
µν(h)
(o) +
(o) B
µ(o) +
(o) F̄
. (3.15)
The form of the Lagrangian densities (e.g. from (3.10) to (3.15)) simplify the proof for the
(anti-)BRST invariance of the Lagrangian densities in (2.1) and (2.5).
In the above forms (e.g. from (3.10) to (3.12)) of the Lagrangian density, the BRST
invariance sbLB = 0 and the anti-BRST invariance sabLB = 0 become very transparent
and simple because the following equalities and mappings exist, namely;
B = 0 ⇒ Limθ→0
B = 0, sb ⇔ Limθ→0
, s2b = 0 ⇔
= 0, (3.16)
B = 0 ⇒ Limθ̄→0
B = 0, sab ⇔ Limθ̄→0
, s2ab = 0 ⇔
= 0. (3.17)
Similarly, the most beautiful relation (3.12), leads to the (anti-)BRST invariance together.
Here one has to use the anticommutativity property sbsab + sabsb = 0 in the language of
the translational generators (i.e. (∂/∂θ̄), (∂/∂θ)) along the Grassmannian directions of the
supermanifold, for its proof. This statement can be mathematically expressed as
s(a)bL
B = 0 ⇒
B = 0, sbsab + sabsb = 0 ⇔
= 0. (3.18)
In exactly similar fashion, the on-shell nilpotent (anti-)BRST symmetry invariance (i.e.
s̃(a)bL
b = 0) of the Lagrangian density (2.5) can also be captured in the language of the
superfields if we exploit the expressions (3.13) to (3.15) for the Lagrangian density. In the
latter case, the on-shell nilpotent (anti-)BRST invariance turns out to be like (3.16), (3.17)
and (3.18) with the replacements: s(a)b → s̃(a)b, L
B → L
b , L̃
(1,2,3)
B → L̃
(1,2,3)
Mathematically, the (anti-)BRST invariance of the Lagrangian density (2.1) is captured
in the equations (3.16) to (3.18). In the language of geometry on the (4, 2)-dimensional
supermanifold, the (anti-)BRST invariance corresponds to the Grassmannian independence
of the supersymmetric versions of the Lagrangian density (2.1). In other words, the trans-
lation of the super Lagrangian densities (i.e. (3.10) to (3.12)), along the (θ)θ̄ directions of
the supermanifold, is zero. This observation captures the (anti-)BRST invariance of (2.1).
4 (Anti-)BRST symmetries in non-Abelian theory: Lagrangian approach
We begin with the following BRST-invariant Lagrangian density, in the Feynman gauge,
for the four (3 + 1)-dimensional non-Abelian 1-form gauge theory¶ (see, e.g. [26,27,9])
B = −
F µν · Fµν +B · (∂µA
B · B − i∂µC̄ ·D
µC, (4.1)
where the curvature tensor (Fµν) is defined through the 2-form F
(2)(n) = dA(1)(n)+iA(1)(n)∧
A(1)(n). Here the non-Abelian 1-form gauge connection is A(1)(n) = dxµ(Aµ · T ) and the
exterior derivative is d = dxµ∂µ. The Nakanishi-Lautrup auxiliary field B = B · T is
required for the linearization of the gauge-fixing term and the (anti-)ghost fields (C̄)C are
essential for the proof of the unitarity in the theory. The latter fields are fermionic (i.e.
(Ca)2 = 0, (C̄a)2 = 0, CaCb + CbCa = 0, CaC̄b + C̄bCa = 0, etc.) in nature.
The above Lagrangian density respects the following off-shell nilpotent ((s
2 = 0)
BRST symmetry transformations s
b , namely;
b Aµ = DµC, s
b C = −
(C × C), s
b C̄ = iB,
b B = 0, s
b Fµν = i(Fµν × C).
(4.2)
It will be noted that (i) the curvature tensor Fµν · T transforms here under the BRST
symmetry transformation. However, it can be checked explicitly that the kinetic energy
term −(1/4)Fµν · F
µν remains invariant under the BRST symmetry transformations, (ii)
the nilpotent anti-BRST symmetry transformations corresponding to the above BRST
symmetry transformations (4.2) cannot be defined for the Lagrangian density (4.1), and
(iii) the on-shell nilpotent version of the above BRST symmetry transformations is also
possible if we substitute, in the above symmetry transformations, B = −(∂µA
µ). The
ensuing on-shell (i.e. ∂µD
µC = 0) nilpotent BRST symmetry transformations s̃
b are
b Aµ = DµC, s̃
b C = −
(C × C),
b C̄ = −i(∂µA
µ), s̃
b Fµν = i(Fµν × C).
(4.3)
The above on-shell nilpotent transformations leave the following Lagrangian density
b = −
F µν · Fµν −
µ) · (∂ρA
ρ)− i∂µC̄ ·D
µC, (4.4)
¶For the non-Abelian 1-form gauge theory, the notations used in the Lie algebraic space are: A · B =
AaBa, (A ×B)a = fabcAbBc, DµC
a = ∂µC
a + ifabcAbµC
c ≡ ∂µC
a + i(Aµ × C)
a, Fµν = ∂µAν − ∂νAµ +
iAµ×Aν , Aµ = Aµ ·T, [T
a, T b] = fabcT c where the Latin indices a, b, c = 1, 2, 3....N are in the SU(N) Lie
algebraic space. The structure constant fabc can be chosen to be totally antisymmetric for any arbitrary
semi simple Lie algebra that includes SU(N), too (see, e.g., [27]).
quasi-invariant because it transforms to a total derivative.
The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.1) and
(4.4) can be written, modulo a total derivative, as a BRST-exact quantity in terms of
the off-shell and on-shell nilpotent BRST symmetry transformations (4.2) and (4.3). This
statement can be mathematically expressed as follows
−i C̄ · {(∂µA
= B · (∂µA
B · B − i ∂µC̄ ·D
µC, (4.5)
C̄ · (∂µA
µ) + i Aµ · ∂
µ) · (∂ρA
ρ)− i ∂µC̄ ·D
µC. (4.6)
It will be noted that one has to take into account s̃
b (∂µA
µ) = ∂µD
µC = 0 in the above
proof of the exactness of the expression in (4.6).
The Lagrangian densities that respect the off-shell nilpotent (i.e. (s
(a)b)
2 = 0) and
anticommuting (s
ab + s
b = 0) (anti-)BRST symmetry transformations are
(1)(n)
b = −
F µν · Fµν +B · (∂µA
(B · B + B̄ · B̄)− i∂µC̄ ·D
µC, (4.7)
(2)(n)
b = −
F µν · Fµν − B̄ · (∂µA
(B · B + B̄ · B̄)− iDµC̄ · ∂
µC. (4.8)
Here auxiliary fields B and B̄ satisfy the Curci-Ferrari condition B+B̄ = −(C×C̄) [28,29].
It is also evident, from this relation, that B ·(∂µA
µ)−i∂µC̄ ·D
µC = −B̄ ·(∂µA
µ)−iDµC̄ ·∂
Furthermore, it should be re-emphasized that the Lagrangian densities (4.1) and (4.4) do
not respect the anti-BRST symmetry transformations of any kind. The BRST and anti-
BRST symmetry transformations, for the above Lagrangian densities, are
b Aµ = DµC, s
b C = −
(C × C), s
b C̄ = iB,
b B = 0, s
b Fµν = i(Fµν × C), s
b B̄ = i(B̄ × C),
(4.9)
ab Aµ = DµC̄, s
ab C̄ = −
(C̄ × C̄), s
ab C = iB̄,
ab B̄ = 0, s
ab Fµν = i(Fµν × C̄), s
ab B = i(B × C̄).
(4.10)
The above off-shell nilpotent (anti-)BRST symmetry transformations leave the Lagrangian
densities (4.7) as well as (4.8) quasi-invariant as they transform to some total derivatives.
The gauge-fixing and Faddeev-Popov ghost terms of the Lagrangian densities (4.7) and
(4.8) can be written, in a symmetrical fashion with respect to s
b and s
ab , as
Aµ ·A
µ + C · C̄
= B · (∂µA
(B ·B + B̄ · B̄)− i∂µC̄ ·D
≡ −B̄ · (∂µA
(B · B + B̄ · B̄)− iDµC̄ · ∂
(4.11)
This demonstrates the key fact that the above gauge-fixing and Faddeev-Popov ghost terms
are (anti-)BRST invariant together because of the nilpotency and anticommutativity of the
(anti-)BRST symmetry transformations s
(a)b that are present in the theory.
5 (Anti-)BRST invariance in non-Abelian theory: superfield approach
To capture (i) the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry transfor-
mations, and (ii) the invariance of the Lagrangian densities, in the language of the superfield
approach to BRST formalism, we have to consider the 4D 1-form non-Abelian gauge theory
on a (4, 2)-dimensional supermanifold. As a consequence, we have the following mappings:
d → d̃ = dxµ ∂µ + dθ ∂θ + dθ̄ ∂θ̄, d̃
2 = 0,
A(1)(n) → Ã(1)(n) = dxµ(Bµ · T )(x, θ, θ̄) + dθ(F̄ · T )(x, θ, θ̄) + dθ̄(F · T )(x, θ, θ̄),
(5.1)
where the (4, 2)-dimensional superfields (Bµ ·T,F ·T, F̄ ·T ) are the generalizations of the 4D
basic local fields (Aµ ·T, C ·T, C̄ ·T ) of the Lagrangian density (4.1), (4.7) and (4.8). These
superfields can be expanded along the Grassmannian directions of the supermanifold, in
terms of the basic 4D fields, auxiliary fields and secondary fields as [4,16,19]
(Bµ · T )(x, θ, θ̄) = (Aµ · T )(x) + θ (R̄µ · T )(x) + θ̄ (Rµ · T )(x) + i θ θ̄ (Sµ · T )(x),
(F · T )(x, θ, θ̄) = (C · T )(x) + i θ (B̄1 · T )(x) + i θ̄ (B1 · T )(x) + i θ θ̄ (s · T )(x),
(F̄ · T )(x, θ, θ̄) = (C̄ · T )(x) + i θ (B̄2 · T )(x) + i θ̄ (B2 · T )(x) + i θ θ̄ (s̄ · T )(x).
(5.2)
To determine the exact expressions for the secondary fields, in terms of the basic and
auxiliary fields of the theory, we have to exploit the HC. The horizontality condition, for
the non-Abelian gauge theory is the requirement of the equality of the Maurer-Cartan
equation on the (super) manifolds. In other words, the covariant reduction of the super
2-form curvature F̃ (2)(n) to the ordinary 2-form curvature (i.e. d̃Ã(1)(n)+ iÃ(1)(n) ∧ Ã(1)(n) =
dA(1)(n)+ iA(1)(n)∧A(1)(n)) leads to the determination of the secondary fields in terms of the
basic and auxiliary fields of the theory. The ensuing expansions, in terms of the basic and
auxiliary fields, lead to (i) the derivation of the (anti-)BRST symmetry transformations
for the basic fields of the theory, and (ii) the geometrical interpretations of the nilpotent
(anti-)BRST symmetry transformations (and their corresponding nilpotent generators) for
the basic fields of the theory as the translations of the corresponding superfields along the
Grassmannian directions of the (4, 2)-dimensional supermanifold (see, e.g., [16,19]).
With the identifications B2 = B and B̄1 = B̄, the following relationships emerge after
the application of the horizontality condition ‖ (see, e.g., [16]):
Rµ = DµC, R̄µ = DµC̄, B + B̄ = −(C × C̄), s = i(B̄ × C),
Sµ = DµB +DµC × C̄ ≡ −DµB̄ −DµC̄ × C,
s̄ = −i(B × C̄), B1 = −
(C × C), B̄2 = −
(C̄ × C̄).
(5.3)
‖In the rest of our present text, we shall be using the short-hand notations for all the fields e.g.:
Aµ · T = Aµ, C · T = C, B · T = B, etc., for the sake of brevity.
The substitution of the above expressions, which are obtained after the application of the
horizontality condition, leads to the following expansions
B(h)µ (x, θ, θ̄) = Aµ + θ DµC̄ + θ̄ DµC + i θ θ̄ (DµB +DµC × C̄),
F (h)(x, θ, θ̄) = C + i θ B̄ −
θ̄ (C × C)− θ θ̄ (B̄ × C),
F̄ (h)(x, θ, θ̄) = C̄ −
θ (C̄ × C̄) + i θ̄ B + θ θ̄ (B × C̄).
(5.4)
The above expansions (see, e.g., our earlier works [16,19]) can be expressed in terms of the
off-shell nilpotent (anti-)BRST symmetry transformations (4.9) and (4.10).
With the above expansion at our disposal, the gauge-fixing and Faddeev-Popov terms
of the Lagrangian density (4.1) can be written, modulo a total ordinary derivative, as
Limθ→0
−iF̄ (h) · ∂µB(h)µ −
F̄ (h) · B
= B · (∂µA
B · B − i ∂µC̄ ·D
µC. (5.5)
Furthermore, it can be seen that, due to the validity and consequences of the horizontality
condition, the super curvature tensor F̃µν has the following form [16,4]
F̃ (h)µν = Fµν + iθ(Fµν × C̄) + iθ̄(Fµν × C)− θ θ̄ (Fµν × B + Fµν × C × C̄). (5.6)
It is clear from the above relationship that the kinetic energy term of the present 4D
non-Abelian 1-form gauge theory remains invariant, namely;
F̃ (h)µν · F̃
µν(h) = −
Fµν · F
µν . (5.7)
The Grassmannian independence of the l.h.s. of (5.7) has deep meaning as far as physics is
concerned. It implies immediately that the kinetic energy term of the non-Abelian gauge
theory is an (anti-)BRST (i.e. gauge) invariant physical quantity.
At this juncture, it is worthwhile to point out that one can also capture the equation
(4.6) in the superfield approach to BRST formalism where the on-shell nilpotent version
of the BRST symmetry transformations (i.e. s̃
b ) plays an important role. For this pur-
pose, we have to express the superfield expansion (5.4) for the on-shell nilpotent BRST
symmetry transformation where one has to exploit the replacement B = −(∂µA
µ). With
this substitution, the equation (5.4) for the superfield expansion becomes
µ(o)(x, θ, θ̄) = Aµ + θ DµC̄ + θ̄ DµC + i θ θ̄ [−Dµ(∂
ρAρ) +DµC × C̄],
(o) (x, θ, θ̄) = C + i θ B̄ −
θ̄ (C × C)− θ θ̄ (B̄ × C),
(o) (x, θ, θ̄) = C̄ −
θ (C̄ × C̄)− i θ̄ (∂µA
µ)− θ θ̄ [(∂µA
µ)× C̄)].
(5.8)
Now, the equation (4.6) can be expressed in terms of the above superfields, as:
Limθ→0
(o) · (∂
µAµ) + i B
µ(o) · ∂
µ) · (∂ρA
ρ)− i ∂µC̄ ·D
(5.9)
Furthermore, it will be noted that the analogue of (5.6), for the on-shell nilpotent BRST
symmetry transformation (i.e. F̃
µν(o)), can be obtained by the replacement B = −(∂µA
Once again, the equality (5.7) would remain intact even if we take into account the on-shell
nilpotent BRST symmetry transformations. Thus, we note that the kinetic energy term
(i.e. (−(1/4)F µν · Fµν = −(1/4)F̃
µν(h)
(o) · F̃
µν(o)) of the non-Abelian gauge theory remains
independent of the Grassmannian variables θ and θ̄ after the application of the HC. This
statement is true for the off-shell as well as the on-shell nilpotent (anti-)BRST symmetry
transformations. Physically, it implies that the kinetic energy term for the gauge field of
the non-Abelian theory is an (anti-)BRST (i.e. gauge) invariant quantity.
The above key observation helps in expressing the Lagrangian density (4.1) and (4.4)
in terms of the superfields (obtained after the application of HC), as
B = −
F̃ (h)µν · F̃
µν(h) + Limθ→0
−iF̄ (h) · ∂µB(h)µ −
F̄ (h) · B
b = −
µν(o) · F̃
µν(h)
(o) + Limθ→0
(o) · (∂
µAµ) + i B
µ(o) · ∂
(5.10)
This result, in turn, simplifies the BRST invariance of the above Lagrangian density (4.1)
and (4.4) (describing the 4D 1-form non-Abelian gauge theory) as follows
Limθ→0
B = 0 ⇒ s
B = 0, Limθ→0
b = 0 ⇒ s̃
b = 0. (5.11)
This is a great simplification because the total super Lagrangian densities (5.10) remain
independent of the Grassmannian variable θ̄. This key result is encoded in the mapping
b , s̃
b ) ⇔ Limθ→0(∂/∂θ̄) and the nilpotency (s
2 = 0, (s̃
2 = 0, (∂/∂θ̄)2 = 0.
It can be readily checked that the analogues of (5.5) and (5.9) cannot be expressed as
the derivative w.r.t. the Grassmannian variable θ. To check this, one has to exploit the
super expansions (5.4) and (5.8) obtained after the application of the HC (in the context
of the derivation of the off-shell as well as the on-shell nilpotent BRST symmetry transfor-
mations s
b and s̃
b ). It can be clearly seen that the operation of the derivative w.r.t. the
Grassmannian variable θ, on any combination of the superfields from the expansions (5.4)
and (5.8), does not lead to the derivation of the r.h.s. of (5.5) and (5.9). In the language
of the superfield approach to BRST formalism, this is the reason behind the non-existence
of the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4).
The form of the gauge-fixing and Faddeev-Popov terms (4.11), expressed in terms of
the BRST and anti-BRST symmetry transformations together, can be represented in the
language of the superfields (obtained after the application of HC), as
B(h)µ · B
µ(h) + F (h) · F̄ (h)
= B · (∂µA
(B · B + B̄ · B̄)− i∂µC̄ ·D
(5.12)
As a consequence of the above expression, the Lagrangian densities (4.7) (as well as (4.8))
can be presented, in terms of the superfields, as
(1,2)(n)
b = −
F̃ µν(h) · F̃ (h)µν +
B(h)µ · B
µ(h) + F (h) · F̄ (h)
. (5.13)
The BRST and anti-BRST invariance of the above super Lagrangian density (and that of
the ordinary 4D Lagrangian densities (4.7) and (4.8)) is encoded in the following simple
equations that are expressed in terms of the translational generators along the Grassman-
nian directions of the (4, 2)-dimensional supermanifold, namely;
Limθ→0
(1,2)(n)
b = 0 ⇒ s
(1)(n)
b = 0, Limθ̄→0
(1,2)(n)
b = 0 ⇒ s
(2)(n)
b = 0.
(5.14)
This is a tremendous simplification of the (anti-)BRST invariance of the Lagrangian den-
sities (4.7) and (4.8) in the language of the superfield approach to BRST formalism. In
other words, if one is able to show the Grassmannian independence of the super Lagrangian
densities of the theory, the (anti-)BRST invariance of the 4D theory follows automatically.
In the language of the geometry on the supermanifold, the (anti-)BRST invariance of
a 4D Lagrangian density is equivalent to the statement that the translation of the super
version of the above Lagrangian density, along the Grassmannian directions of the (4, 2)-
dimensional supermanifold, is zero. Thus, the super Lagrangian density of an (anti-)BRST
invariant 4D theory is a Lorentz scalar, constructed with the help of (4, 2)-dimensional
superfields (obtained after the application of HC), such that, when the partial derivatives
w.r.t. the Grassmannian variables (θ and θ̄) operate on it, the result is zero.
The nilpotency and anticommutativity properties (that are associated with the con-
served (anti-)BRST charges and (anti-)BRST symmetry transformations) are found to be
captured very naturally (cf. (3.16)-(3.18)) when we consider the superfield formulation of
the (anti-)BRST invariance of the Lagrangian density of a given 1-form gauge theory. We
mention, in passing, that one could also derive the analogue of the equations (3.16), (3.17)
and (3.18) for the 4D non-Abelian 1-form gauge theory in a straightforward manner.
6 Conclusions
In our present investigation, we have concentrated mainly on the (anti-)BRST invariance
of the Lagrangian densities of the free 4D (non-)Abelian 1-form gauge theories (having no
interaction with matter fields) within the framework of the superfield approach to BRST
formalism. We have been able to provide the geometrical basis for the existence of the
(anti-)BRST invariance in the above 4D theories. To be more specific, we have been able
to show that the Grassmannian independence of the (4, 2)-dimensional super Lagrangian
density, expressed in terms of the appropriate superfields, is a clear-cut proof that there is
an (anti-)BRST invariance (cf. (3.16), (3.17), (3.18), (5.11), (5.14)) in the 4D theory.
If the super Lagrangian density could be expressed as a sum of (i) a Grassmannian
independent term, and (ii) a derivative w.r.t. the Grassmannian variable, then, the cor-
responding 4D Lagrangian density will automatically respect BRST and/or anti-BRST
invariance. In the latter piece of the above super Lagrangian density, the derivative could
be either w.r.t. θ or w.r.t. θ̄ or w.r.t. both of them put together. More specifically,
(i) if the derivative is w.r.t. θ̄, the nilpotent symmetry would correspond to the BRST,
(ii) if the derivative is w.r.t. θ, the nilpotent symmetry would be that of the anti-BRST
type, and (iii) if both the derivatives are present together, both the nilpotent (anti-)BRST
symmetries would be present together (and they would turn out to be anticommuting).
For the 4D (non-)Abelian 1-form gauge theories, that are considered on the (4, 2)-
dimensional supermanifold, it is the HC on the 1-form super connection Ã(1) that plays a
very important role in the derivation of the (anti-)BRST symmetry transformations. The
cohomological origin for the above HC lies in the (super) exterior derivatives (d̃)d. This
point has been made quite clear in our discussions after the off-shell as well as the on-shell
nilpotent (anti-)BRST symmetry transformations (2.2), (2.4), (4.2), (4.3), (4.9) and (4.10).
In fact, it is the full kinetic energy term of the above theories (owing its origin to the
cohomological operator d = dxµ∂µ) that remains invariant under the above on-shell as well
the off-shell nilpotent (anti-)BRST symmetry transformations.
The HC produces specifically the nilpotent (anti-)BRST symmetry transformations for
the gauge and (anti-)ghost fields because of the fact that the super 1-form connection
Ã(1)/Ã(1)(n) (cf. (3.1) and (5.1)) is constructed with a super vector multiplet (Bµ,F , F̄)
which is the generalization of the gauge field Aµ and the (anti-)ghost fields (C̄)C (of the
ordinary 4D (non-)Abelian 1-form gauge theories) to the (4, 2)-dimensional supermanifold.
As a consequence, only the nilpotent and anticommuting (anti-)BRST symmetry transfor-
mations for the 4D local fields Aµ, C and C̄ are obtained when the full potential of the HC
is exploited within the framework of the above superfield formulation.
It is worthwhile to point out that geometrically the super Lagrangian densities, ex-
pressed in terms of the (4, 2)-dimensional superfields, are equivalent to the sum of the
kinetic energy term and the translations of some composite superfields (obtained after the
application of the HC) along the Grassmannian directions (i.e. θ and/or θ̄) of the (4, 2)-
dimensional supermanifold. This observation is distinctly different from our earlier works
on the superfield approach to 2D (non-)Abelian 1-form gauge theories [24,25,23] which are
found to correspond to the topological field theories. In fact, for the latter theories, the
total super Lagrangian density turns out to be a total derivative w.r.t. the Grassmannian
variables (θ and/or θ̄). That is to say, even the kinetic energy term of the latter theories,
is able to be expressed as the total derivative w.r.t. the variables θ and/or θ̄.
In our present endeavour, within the framework of the superfield approach to BRST
formalism, we have been able to provide (i) the logical reason behind the non-existence of
the anti-BRST symmetry transformations for the Lagrangian densities (4.1) and (4.4) for
the 4D non-Abelian 1-form gauge theory, (ii) the explicit explanation for the uniqueness
of the equations (2.3) and (2.6) for the 4D Abelian 1-form gauge theory, (iii) the convinc-
ing proof for the on-shell nilpotent (anti-)BRST invariance of the gauge-fixing term (i.e.
s̃(a)b(∂µA
µ) = 0, s̃
(a)b(∂µA
µ) = 0) for the (non-)Abelian 1-form gauge theories, and (iv) the
compelling arguments for the non-existence of the exact analogue(s) of (2.3) and (2.6) for
the non-Abelian 1-form gauge theory. To the best of our knowledge, the logical explana-
tions for the above subtle points (connected with the 1-form gauge theories) are completely
new. Thus, the results of our present work are simple, beautiful and original.
It is worthwhile to mention that our superfield construction and its ensuing geometrical
interpretations are not specific to the Feynman gauge (which has been taken into account
in our present endeavor). To corroborate this assertion, we take the simple case of the 4D
Abelian 1-form gauge theory and write the Lagrangian density (2.1) in the arbitrary gauge
(a,ξ)
B = −
F µνFµν + B (∂µA
B2 − i ∂µC̄ ∂
µC, (6.1)
where ξ is the gauge parameter. It is elementary to check that, in the limit ξ → 1, we get
back our Lagrangian density (2.1) for the Abelian theory in the Feynman gauge.
The analogue of the equation (2.3) (for the gauge-fixing and Faddeev-Popov ghost terms
in the case of the arbitrary gauge) can be expressed as
−i C̄ {(∂µA
B}], sab
+i C {(∂µA
sb sab
(6.2)
The above expression can be easily generalized to the analogues of the equations (3.10)—
(3.12) in terms of the superfields by taking the help of (3.8). Thus, the geometrical inter-
pretations remain intact even in the case of the arbitrary gauge.
In a similar fashion, for the 4D non-Abelian 1-form gauge theory, the equations (4.5),
(4.6) and (4.11) can be generalized to the case of arbitrary gauge and, subsequently, can
be expressed in terms of superfields as the analogues of (5.5), (5.9) and (5.12). Finally,
we can obtain the analogues of (5.7), (5.10) and (5.13) which will lead to the derivation of
the analogues of (5.11) and (5.14). Thus, we note that geometrical interpretations, in the
arbitrary gauge, remain the same for the 4D (non-)Abelian 1-form gauge theory within the
framework of our superfield approach to BRST formalism.
Our present work can be generalized to the case of the interacting 4D (non-)Abelian
1-form gauge theories where there exists an explicit coupling between the gauge field and
the matter fields. In fact, our earlier works [14-18] might turn out to be quite handy in
attempting the above problems. It seems to us that it is the combination of the HC and
the restrictions, owing their origin to the (super) covariant derivative on the matter (super)
fields and their intimate connection with the (super) curvatures, that would play a decisive
role in proving the existence of the (anti-)BRST invariance for the above gauge theories.
It is gratifying to state that we have accomplished the above goals in our very recent
endeavours [30-32]. In fact, we have been able to provide the geometrical basis for the
existence of the (anti-)BRST invariance, in the context of the interacting (non-)Abelian
1-form gauge theories with Dirac as well as complex scalar fields, within the framework
of the augmented superfield approach to BRST formalism. As it turns out, here too, the
super Lagrangian density is found to be independent of the Grassmannian variables.
In our earlier works [33-35], we have been able to show the existence of the nilpotent
(anti-)BRST and (anti-)co-BRST symmetry transformations for the 4D free Abelian 2-form
gauge theory. We have also established the quasi-topological nature of it in [35]. In a recent
work [36], the nilpotent (anti-)BRST symmetry transformations have been captured in the
framework of the superfield formulation. It would be a very nice endeavour to study the
(anti-)BRST and (anti-)co-BRST invariance of the 4D Abelian 2-form gauge theory within
the framework of superfield formulation. At present, this issue and connected problems in
the context of the 4D free Abelian 2-form gauge theory are under intensive investigation
and our results would be reported in our forthcoming future publications [37].
Acknowledgement: Financial support from the Department of Science and Technology
(DST), Government of India, under the SERC project sanction grant No: - SR/S2/HEP-
23/2006, is gratefully acknowledged.
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|
0704.0065 | Littlewood-Richardson polynomials | Littlewood–Richardson polynomials
A. I. Molev
School of Mathematics and Statistics
University of Sydney, NSW 2006, Australia
[email protected]
Abstract
We introduce a family of rings of symmetric functions depending on an in-
finite sequence of parameters. A distinguished basis of such a ring is comprised
by analogues of the Schur functions. The corresponding structure coefficients
are polynomials in the parameters which we call the Littlewood–Richardson
polynomials. We give a combinatorial rule for their calculation by modifying
an earlier result of B. Sagan and the author. The new rule provides a formula
for these polynomials which is manifestly positive in the sense of W. Graham.
We apply this formula for the calculation of the product of equivariant Schu-
bert classes on Grassmannians which implies a stability property of the struc-
ture coefficients. The first manifestly positive formula for such an expansion
was given by A. Knutson and T. Tao by using combinatorics of puzzles while
the stability property was not apparent from that formula. We also use the
Littlewood–Richardson polynomials to describe the multiplication rule in the
algebra of the Casimir elements for the general linear Lie algebra in the basis
of the quantum immanants constructed by A. Okounkov and G. Olshanski.
http://arxiv.org/abs/0704.0065v3
1 Introduction
Let a = (ai), i ∈ Z be a sequence of variables. Consider the ring of polynomials
Z[a] in the variables ai with integer coefficients. Introduce another infinite set of
variables x = (x1, x2, . . . ) and for each nonnegative integer n denote by Λn the ring
of symmetric polynomials in x1, . . . , xn with coefficients in Z[a]. The ring Λn is
filtered by the usual degrees of polynomials in x1, . . . , xn with the ai considered to
have the zero degree. The evaluation map
ϕn : Λn → Λn−1, P (x1, . . . , xn) 7→ P (x1, . . . , xn−1, an) (1.1)
is a homomorphism of filtered rings so that we can define the inverse limit ring Λ by
Λ = lim
Λn, n → ∞, (1.2)
where the limit is taken with respect to the homomorphisms (1.1) in the category of
filtered rings. When a is specialized to the sequence of zeros, this reduces to the usual
definition of the ring of symmetric functions; see e.g. Macdonald [14]. In that case,
a distinguished basis of Λ is comprised by the Schur functions sλ(x) parameterized
by all partitions λ. The respective analogues of the sλ(x) in the general case are the
double Schur functions sλ(x||a) which form a basis of Λ over Z[a]. We introduce the
Littlewood–Richardson polynomials cνλµ(a) as the structure coefficients of the ring Λ
in the basis of double Schur functions,
sλ(x||a) sµ(x||a) =
cνλµ(a) sν(x||a). (1.3)
In the specialization a = (0) the polynomials cνλµ(a) become the classical Littlewood–
Richardson coefficients cνλµ; see [12]. These are remarkable nonnegative integers which
occupy a prominent place in combinatorics, representation theory and geometry; see
e.g. Fulton [5], Macdonald [14] and Sagan [21].
The main result of this paper is a combinatorial rule for the calculation of the
Littlewood–Richardson polynomials which provides a manifestly positive formula in
the sense that cνλµ(a) is written as a polynomial in the differences ai − aj , i < j, with
positive integer coefficients.
We consider two applications of the rule. The results of Knutson and Tao [9]
imply that under an appropriate specialization, the polynomials cνλµ(a) describe the
multiplication rule for the equivariant Schubert classes on Grassmannians; see also
Fulton [6] for a more direct argument. Let n and N be nonnegative integers with
n 6 N and let Gr(n,N) denote the Grassmannian of the n-dimensional vector sub-
spaces of CN . The torus T = (C∗)N acts naturally on Gr(n,N). The equivariant
cohomology ring H∗T (Gr(n,N)) is a module over the polynomial ring Z[t1, . . . , tN ]
which can be identified with H∗T ({pt}), the equivariant cohomology ring of a point.
This module has a basis of the equivariant Schubert classes σλ parameterized by all
diagrams λ contained in the n×m rectangle, m = N − n; see e.g. [5, 6]. Then
σλ σµ =
d νλµ σν , (1.4)
where d νλµ = c
λµ(a) with the sequence a specialized by
a−m+1 = −t1, a−m+2 = −t2, . . . , an = −tN , (1.5)
while the remaining parameters ai are set to zero (the ti should be replaced with
yi in the notation of [9]). The coefficients d
λµ are given explicitly as polynomials
in the ti − tj , i > j, with positive integer coefficients. This positivity property was
established by Graham [8] in the general context of the equivariant Schubert calculus.
The first manifestly positive formula for the coefficients in the expansion (1.4) was
obtained by Knutson and Tao [9] by using combinatorics of puzzles. An earlier rule
of Molev and Sagan [17] also calculates d νλµ but lacks the explicit positivity property.
Our new rule implies a stability property of the coefficients d νλµ (see Corollary 3.1
below). Even though this property was not pointed out in [9], it can be derived
directly from the puzzle rule; see also Fulton [6] for its geometrical interpretation and
an extension to the equivariant Schubert calculus on the flag variety.
As another application, we obtain a rule for the positive integer expansion of the
product of two (virtual) quantum immanants (or the corresponding higher Capelli
operators) of Okounkov and Olshanski [18, 19]; cf. [17]. The quantum immanants
Sλ|n are elements of the center Z(gln) of the universal enveloping algebra U(gln)
parameterized by partitions λ with at most n parts; see [18]. The elements Sλ|n form
a basis of Z(gln) so that we can define the coefficients f
λµ by the expansion
Sλ|n Sµ|n =
f νλµ Sν|n.
Then f νλµ = c
λµ(a) for the specialization ai = −i for i ∈ Z. As n → ∞ this yields
a multiplication rule for the virtual quantum immanants Sλ; see Section 3.2 for the
definitions.
We define the double Schur function sλ(x||a) as the sequence of the double Schur
polynomials
sλ(x1, . . . , xn ||a), n = 1, 2, . . . , (1.6)
which are compatible with respect to the homomorphisms (1.1),
ϕn : sλ(x1, . . . , xn ||a) 7→ sλ(x1, . . . , xn−1 ||a). (1.7)
The polynomials (1.6) are closely related to the “factorial” or “double” Schur poly-
nomials sλ(x|u) with x = (x1, . . . , xn). The latter were introduced by Goulden and
Greene [7] and Macdonald [13] as a generalization of the factorial Schur polynomials
of Biedenharn and Louck [1, 2], and they are also a special case of the double Schu-
bert polynomials of Lascoux and Schützenberger; see Lascoux [11]. We follow Chen,
Li and Louck [4] and Fulton [6] and use the name “double Schur polynomials” for
the related polynomials sλ(x||a) as well.
In a more detail, consider a partition λ which is a sequence λ = (λ1, . . . , λn)
of integers λi such that λ1 > · · · > λn > 0. We will identify λ with its diagram
represented graphically as the array of left justified rows of unit boxes with λ1 boxes
in the top row, λ2 boxes in the second row, etc. The total number of boxes in λ will
be denoted by |λ|. The transposed diagram λ′ = (λ′1, . . . , λ
p) is obtained from λ by
applying the symmetry with respect to the main diagonal, so that λ′j is the number
of boxes in the j-th column of λ.
Let u = (u1, u2, . . . ) be a sequence of variables. The polynomials sλ(x|u) can be
defined by
sλ(x|u) =
(xT (α) − uT (α)+c(α)), (1.8)
where T runs over all semistandard (column-strict) tableaux of shape λ with entries
in {1, . . . , n}, T (α) is the entry of T in the box α ∈ λ and c(α) = j − i is the content
of the box α = (i, j) in row i and column j.
By a reverse λ-tableau T we will mean the tableau obtained by filling in the boxes
of λ with the numbers 1, 2, . . . , n in such a way that the entries weakly decrease along
the rows and strictly decrease down the columns. If α = (i, j) is a box of λ we let
T (α) = T (i, j) denote the entry of T in the box α. We define the double Schur
polynomials sλ(x||a) by
sλ(x||a) =
(xT (α) − aT (α)−c(α)), (1.9)
summed over the reverse λ-tableaux T . Then we have
sλ(x||a) = sλ(x|u) (1.10)
for the sequences a and u related by an−i+1 = ui with i = 1, 2, . . . . In particular, the
polynomial sλ(x||a) only depends on the variables ai with i 6 n, i ∈ Z. The relation
(1.10) is verified easily by replacing xi with xn−i+1 in (1.8) for all i = 1, . . . , n and using
the fact that sλ(x|u) is a symmetric polynomial in x. The property (1.7) of the double
Schur polynomials is immediate from their definition. In the specialization of the
sequence a with ai = −i, i ∈ Z, formula (1.9) defines the shifted Schur polynomials
of Okounkov and Olshanski [18, 19] in the variables yi = xi+ i. The use of the reverse
tableaux was significant in their study of the vanishing and stability properties of
these polynomials and associated central elements of the universal enveloping algebra
for the Lie algebra gln; see also Section 3.2 below.
Note that the stability property (1.7) extends to the double Schubert polynomials
(and to the equivariant Schubert calculus on the flag manifold). This follows easily
from the Cauchy formula for the Schubert polynomials (e.g., put x1 = y1 in [15,
Formula in 2.5.5]). In a more general context, this was also pointed out in [3].
The double Schur polynomials sλ(x||a) parameterized by the diagrams λ with at
most n rows form a basis of the ring Λn. Due to the stability property (1.7), the
Littlewood–Richardson polynomials cνλµ(a) can be defined by the expansion (1.3),
where x is understood as the set of variables x = (x1, . . . , xn) for any positive integer
n such that the diagrams λ, µ and ν have at most n rows. This allows us to work
with a finite set of variables for the determination of the polynomials cνλµ(a). For
the proof of the main theorem (Theorem 2.1) we follow the general approach of [17],
using the techniques of “barred” tableaux and modify the corresponding arguments
in order to obtain manifestly positive polynomials. This is achieved by imposing a
boundness condition on the barred tableaux.
It was observed by Goulden and Greene [7] and Macdonald [13] that sλ(x|u),
regarded as a formal power series in the infinite sets of variables x and u, admits
a “supertableaux” representation. We show that this representation has its “finite”
counterpart where x is a finite set of variables. We derive the corresponding formula
by choosing a certain specialization of the 9th Variation in [13]. This representa-
tion leads to a “supertableau” expression for the Littlewood–Richardson polynomials
cνλµ(a), although that expression is neither manifestly positive, nor stable.
After the first version of this paper was completed we have learned of an indepen-
dent work of V. Kreiman [10], where a positive equivariant Littlewood–Richardson
rule was given. That rule is equivalent to our Theorem 2.1 although the proof in [10]
is different. Moreover, Kreiman’s paper also provides a weight-preserving bijection
between the Knutson–Tao puzzles and the barred tableaux used in Theorem 2.1.
This work was inspired by Bill Fulton’s lectures [6]. I am grateful to Bill for
stimulating discussions.
2 Multiplication rule
Let R denote a sequence of diagrams
µ = ρ(0) → ρ(1) → · · · → ρ(l−1) → ρ(l) = ν, (2.1)
where ρ → σ means that σ is obtained from ρ by adding one box. Let ri denote
the row number of the box added to the diagram ρ(i−1). The sequence r1r2 . . . rl is
called the Yamanouchi symbol of R. Introduce the ordering on the set of boxes of
a diagram λ by reading them by columns from left to right and from bottom to top
in each column. We call this the column order . We shall write α ≺ β if α (strictly)
precedes β with respect to the column order. Given a sequence R, construct the
set T (λ,R) of barred reverse λ-tableaux T with entries from {1, 2, . . . } such that T
contains boxes α1, . . . , αl with
α1 ≺ · · · ≺ αl and T (αi) = ri, 1 6 i 6 l.
We will distinguish the entries in α1, . . . , αl by barring each of them. So, an element
of T (λ,R) is a pair consisting of a reverse λ-tableau and a chosen sequence of barred
entries compatible with R. We shall keep the notation T for such a pair. For example,
let R be the sequence
(3, 1) → (3, 2) → (3, 2, 1) → (3, 3, 1) → (4, 3, 1)
so that the Yamanouchi symbol is 2 3 2 1. Then for λ = (5, 5, 3) the following barred
λ-tableau belongs to T (λ,R):
For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, set ρ(α) = ρ
(i). The barred entries
r1, . . . , rl divide the tableau into regions marked by the elements of the sequence R,
as illustrated:
ρ(0) ρ(1)
r1 r2
· · ·
Finally, a reverse λ-tableau T will be called ν-bounded if
T (1, j) 6 ν ′j for all j = 1, . . . , λ1.
Note that ν-bounded λ-tableaux exist only if λ ⊆ ν.
We are now in a position to state a rule for the calculation of the Littlewood-
Richardson polynomials cνλµ(a) defined by (1.3).
Theorem 2.1. The polynomial cνλµ(a) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
, (2.2)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). Moreover, for each factor occurring in the formula (2.2) we have
ρ(α)T (α) > c(α).
Before proving the theorem, let us point out some properties of the Littlewood-
Richardson polynomials which are immediate from the rule and consider some exam-
ples. The polynomial cνλµ(a) is zero unless both diagrams λ and µ are contained in
ν and |λ| + |µ| > |ν|. In this case cνλµ(a) is a homogeneous polynomial in the ai of
degree |λ| + |µ| − |ν|. If |λ| + |µ| − |ν| = 0 then the theorem reproduces a version
of the classical Littlewood-Richardson rule; see Corollary 2.9 below. Note also that
by the definition, the polynomials have the symmetry cνλµ(a) = c
µλ(a) which is not
apparent from the rule.
Example 2.2. For the product of the double Schur functions s(2)(x||a) and s(2,1)(x||a)
we have
s(2)(x||a) s(2,1)(x||a) = s(4,1)(x||a) + s(3,2)(x||a) + s(3,1,1)(x||a) + s(2,2,1)(x||a)
a−1 − a2 + a−2 − a0
s(3,1)(x||a) +
a−1 − a2
s(2,2)(x||a)
a−1 − a0
s(2,1,1)(x||a) +
a−1 − a2
a−1 − a0
s(2,1)(x||a).
For instance, the coefficient of s(3,1)(x||a) is calculated by the following barred (2)-
tableaux
1 1 1 1 2 1
compatible with the sequence (2, 1) → (3, 1). They contribute respectively a−1 − a1,
a−2 − a0, a1 − a2 which sums up to the coefficient a−1 − a2 + a−2 − a0. Alternatively,
using the symmetry cνλµ(a) = c
µλ(a) we can calculate the coefficient of s(3,1)(x||a) by
considering the barred (2, 1)-tableaux
compatible with the sequences (2) → (3) → (3, 1) and (2) → (2, 1) → (3, 1), respec-
tively. Their contributions to the coefficient are a−2 − a0 and a−1 − a2.
Example 2.3. For the calculation of c
(5,2,2)
(4,2,1)(2,2)
(a) take λ = (4, 2, 1), µ = (2, 2) and
ν = (5, 2, 2). We have ten sequences R of the form (2.1) but the set T (λ,R) contains
ν-bounded tableaux only for three of them. For the sequence R1 with the Yamanouchi
symbol 1 3 3 1 1, the set T (λ,R1) contains two bounded barred tableaux
3 1 1
3 1 1
whose contributions to the Littlewood–Richardson polynomial are (a0 − a3)(a0 − a2)
and (a0 − a3)(a−2 − a1), respectively. For the sequence R2 with the Yamanouchi
symbol 1 3 1 3 1, the set T (λ,R2) contains the bounded tableaux
3 1 1
3 1 1
with the respective contributions (a0 − a3)(a−4 − a−2) and (a0 − a3)(a−3 − a−1). For
the sequence R3 with the Yamanouchi symbol 3 1 3 1 1, the set T (λ,R3) contains the
only bounded tableau
3 1 1
with the contribution (a−1 − a3)(a0 − a3). Hence,
(5,2,2)
(4,2,1)(2,2)
(a) = (a0 − a3) (a−4 + a−3 + a0 − a1 − a2 − a3).
Taking λ = (2, 2), µ = (4, 2, 1) and ν = (5, 2, 2) we get two sequences with the
Yamanouchi symbols 1 3 and 3 1. The corresponding sets T (λ,R) consist of five
and four bounded barred tableaux, respectively, thus leading to a slightly longer
calculation.
Proof of Theorem 2.1. We present the proof as a sequence of lemmas. Due to the
stability property (1.7), we may (and will) work with a finite set of variables x =
(x1, . . . , xn). Accordingly, possible entries of the tableaux are now elements of the set
{1, . . . , n}. Introduce another sequence of variables b = (bi), i ∈ Z, and define the
Littlewood–Richardson type coefficients cνλµ(a, b) by the expansion
sλ(x||b) sµ(x||a) =
cνλµ(a, b) sν(x||a). (2.3)
Lemma 2.4. The coefficient cνλµ(a, b) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a, b) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− bT (α)−c(α)
, (2.4)
summed over all sequences R of the form (2.1) and all reverse λ-tableaux T ∈ T (λ,R).
Proof. This is essentially a reformulation of the main result of [17] (Theorem 3.1).
Note that the summation in (2.4) is taken over all barred tableaux T ∈ T (λ,R) (not
just over the ν-bounded ones as in (2.2)). Rather than repeating the whole argument
of [17], we only sketch the main steps of the proof and indicate the necessary changes
to be made. We refer the reader to [17] for the details.
We assume that all diagrams here have at most n rows. If ρ = (ρ1, . . . , ρn) is a
such diagram, we set
aρ = (a1−ρ1 , . . . , an−ρn) and |aρ| = a1−ρ1 + · · ·+ an−ρn.
Under the correspondence (1.10) we have aρ = uρ = (uρ1+n, . . . , uρn+1), the latter
notation was used in [17].
The starting point is the Vanishing Theorem of [18] whose proof was also repro-
duced in [17]. By that theorem,
sλ(aρ ||a) = 0 unless λ ⊆ ρ,
sλ(aλ ||a) =
(i,j)∈λ
ai−λi − aλ′j−j+1
The first claim of the lemma follows from the Vanishing Theorem which also implies
λµ(a, b) = sλ(aµ ||b).
This proves (2.4) for the case ν = µ. Now we suppose that |ν| − |µ| > 1 and proceed
by induction on |ν| − |µ|. The induction step is based on the recurrence relation
cνλµ(a, b) =
|aν | − |aµ|
cνλµ+(a, b)−
λµ (a, b)
(2.5)
which was proved in [17, Proposition 3.4]; see also [9]. Suppose that the diagram ν
is obtained from µ by adding one box in row r. Then
cνλµ(a, b) =
sλ(aν ||b)− sλ(aµ ||b)
(aν)r − (aµ)r
. (2.6)
Now use the definition (1.9) of the double Schur polynomials. Since the n-tuples aν
and aµ only differ at the r-th component, the ratio on the right hand side of (2.6)
can be expanded by taking into account the entries r of the reverse λ-tableaux T .
We need the following formula, where we are thinking of y = (aν)r, z = (aµ)r and
mi = bT (α)−c(α) as α runs over the boxes of T with T (α) = r in column order:
i=1(y −mi)−
i=1(z −mi)
y − z
(z −m1) . . . (z −mj−1)(y −mj+1) . . . (y −mk).
The right hand side of (2.6) can now be interpreted as the right hand side of (2.4),
where R is the only sequence µ → ν and the sum is taken over the reverse λ-tableaux
T with one barred entry r, as illustrated:
µ r ν
Here ρ(α) = µ for all boxes α preceding the box occupied by the barred r, and
ρ(α) = ν for all boxes α which follow that box in column order. Note that the
variables y and z are now swapped on the right hand side of the above expansion,
as compared to [17] (this does not change the polynomial due to the symmetry in y
and z). Consequently, the column order used in [17] is the opposite to the order on
the boxes of λ we use here.
We can represent the above calculation of cνλµ(a) by the “diagrammatic” relation
|aν | − |aµ|
µ r ν = ν − µ
Consider now the next case where |ν| − |µ| = 2 and apply the recurrence relation
(2.5). We have three subcases: the diagram ν is obtained from µ by adding two boxes
in different rows and columns; by adding two boxes in the same row; or by adding
two boxes in the same column. The first two subcases are dealt with in a way similar
to the case |ν|− |µ| = 1. An additional care is needed for the third subcase where we
suppose that ν is obtained from µ by adding the boxes in rows r and r + 1. Denote
by ρ the diagram obtained from µ by adding the box in row r. Then (2.5) gives
cνλµ(a, b) =
cνλρ(a, b)− c
λµ(a, b)
|aν | − |aµ|
Set s = r+1. Exactly as in the case |ν|−|µ| = 1, we have the following diagrammatic
relations:
|aρ| − |aµ|
sρ ν = ρ s ν − µ s ν
|aν | − |aρ|
sρ ν = µ r ν − µ r ρ
Hence, the desired formula for cνλµ(a, b) will follow if we prove the relation
µ r ν = µ s ν
We construct a weight-preserving bijection between the barred reverse λ-tableaux
which are represented by the left and right hand sides of this diagrammatic relation.
Here the weight is the product on the right hand side of (2.4) corresponding to a
barred tableau. Let such a tableau with a barred entry r in the box (i, j) be given.
Suppose first that the box (i − 1, j) belongs to the diagram and it is occupied by
s = r+1. Then the image of the tableau under the map is the same tableau but the
entry T (i, j) = r is now unbarred while T (i− 1, j) = r + 1 is barred. Since
(aν)r+1 = (aµ)r and T (i− 1, j)− c(i− 1, j) = T (i, j)− c(i, j),
the weights of the tableaux are preserved under the map.
Suppose now that the entry in the box (i − 1, j) is greater than r + 1, or this
box is outside the diagram. Consider all entries r in the row i to the left of the box
(i, j) and suppose that they occupy the boxes (i, j −m), (i, j −m+1), . . . , (i, j− 1).
Then the image of the tableau under the map is the tableau obtained by replacing
the entries in each of the boxes (i, j −m), . . . , (i, j) with s = r + 1 and barring the
entry in the box (i, j −m). The weights of the tableaux are again preserved.
The inverse map is described in a similar way. This gives the desired weight-
preserving bijection. The general argument uses similar calculations with the barred
diagrams and a similar bijection described in [17].
Remark 2.5. (i) A cohomological interpretation of the coefficients cνλµ(a, b) and their
puzzle computation can be found in [9].
(ii) The definition (2.3) of the coefficients cνλµ(a, b) can be extended to the case
where λ is a skew diagram. Lemma 2.4 and its proof remain valid; see [17].
(iii) In contrast with the Littlewood–Richardson polynomials cνλµ(a), the coeffi-
cients cνλµ(a, b) do not have the stability property as they depend on n.
Lemma 2.4 implies that the Littlewood–Richardson polynomials can be calcu-
lated by (2.4) with b = a, that is, cνλµ(a) = c
λµ(a, a). Our strategy now is to show
that (unlike the formula of Theorem 3.1 in [17]), the formula (2.4) (with b = a) is
“nonnegative” in the sense that all nonzero products which occur in the formula are
polynomials in the ai − aj with i < j. Then we demonstrate that the ν-boundness
condition serves to eliminate the unwanted zero terms.
Lemma 2.6. Let R be a sequence of the form (2.1) and let T ∈ T (λ,R). Suppose
that ∏
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
6= 0. (2.7)
Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. Suppose on the contrary that there exists a box α = (i, j) with an unbarred
T (i, j) and the condition ρ(i, j)T (i,j) < j − i; the equality ρ(i, j)T (i,j) = j − i is
excluded since this would violate (2.7). Choose such a box with the minimum possible
value of j. If all the entries T (i, 1), . . . , T (i, j − 1) of T are barred then ρ(i, j) is
obtained from µ by adding boxes in rows T (i, 1) > · · · > T (i, j − 1) and, possibly,
by adding other boxes. Since T (i, j − 1) > T (i, j), we have ρ(i, j)T (i,j) > j − 1, a
contradiction. So, at least one of the entries T (i, 1), . . . , T (i, j−1) must be unbarred.
Take such an unbarred entry T (i, k) which is the closest to T (i, j), that is, all entries
T (i, k+1), . . . , T (i, j − 1) are barred. Then ρ(i, j) is obtained from ρ(i, k) by adding
boxes in rows T (i, k + 1) > · · · > T (i, j − 1) and, possibly, by adding other boxes.
Hence,
ρ(i, j)T (i,j) > ρ(i, k)T (i,k) + j − k − 1
which implies ρ(i, k)T (i,k) < k − i + 1. However, if ρ(i, k)T (i,k) = k − i then the
factor in (2.7) corresponding to α = (i, k) is zero, which is impossible. Therefore
ρ(i, k)T (i,k) < k − i which contradicts the choice of j.
Lemma 2.7. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R). If
(2.7) holds then T is ν-bounded.
Proof. By Lemma 2.6, for all unbarred entries T (1, k) of the first row of the tableau
T we have ρ(1, k)T (1,k) > k. This implies νT (1,k) > k. If the entry T (1, j) is barred
then ρ(1, k)T (1,k) > k for the nearest unbarred entry T (1, k) on its left (if it exists).
Then ν is obtained from ρ(1, k) by adding boxes in rows T (1, k + 1) > · · · > T (1, j)
and, possibly, by adding other boxes. This implies νT (1,j) > j. Thus, this inequality
holds for all j = 1, . . . , λ1. This is equivalent to the ν-boundness of T .
Lemma 2.8. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R) is
ν-bounded. Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. We argue by contradiction. Taking into account Lemma 2.6, we find that for
some α = (i, j) with unbarred T (α) we have ρ(i, j)T (i,j) = j − i. Set t = T (i, j)
and consider all barred entries of T (assuming for now they exist) which are equal
to t and occur to the right of the column j. Since T is a reverse tableau, these
entries t̄ can only occur in rows 1, 2, . . . , i. Let (r, k) be the box with the maximum
column number k containing t̄. Then the total number of such entries t̄ does not
exceed k− j. This implies that the number of boxes νt in row t of ν does not exceed
ρ(i, j)t + k − j = k − i. Hence, ν
k 6 t − 1. On the other hand, by the ν-boundness
of T we have t = T (r, k) 6 T (1, k) 6 ν ′k, a contradiction.
If none of the boxes to the right of the column j contains t̄ then νt = ρ(i, j)t = j−i.
However, by the assumption, νt > νT (1,j) > j, a contradiction.
This completes the proof of the theorem.
By the column word of a tableau T we will mean the sequence of all entries of T
written in the column order.
Corollary 2.9. Suppose that |ν| = |λ| + |µ|. The Littlewood–Richardson coefficient
cνλµ equals the number of ν-bounded reverse λ-tableaux T whose column word coincides
with the Yamanouchi symbol of a certain sequence R of the form (2.1).
This can be shown to be equivalent to a well-known version of the Littlewood–
Richardson rule. Corollary 2.9 also holds with the ν-boundness condition dropped;
see Lemma 2.7. By the corollary, cνλµ counts the cardinality of the intersection of two
finite sets: the set of column words of ν-bounded reverse λ-tableaux and the set of
Yamanouchi symbols of the sequences of the form (2.1).
Remark 2.10. Due to (1.10), the multiplication rule for the polynomials sλ(x|u) is
obtained from Theorem 2.1 by replacing ai with un−i+1 for each i. The corresponding
coefficients are polynomials in the ui−uj, i > j, with positive integer coefficients.
Corollary 2.11. Suppose that the polynomials cνλµ(a) are defined by the expansion
(1.3) with x = (x1, . . . , xn). Then c
λµ(a) is independent of n as soon as n > ν
Moreover, if n < ν ′1 then c
λµ(a) = 0.
Proof. This follows from the boundness condition on the reverse tableaux.
3 Applications
3.1 Equivariant Schubert calculus on the Grassmannian
As in the Introduction, consider the equivariant cohomology ring H∗T (Gr(n,N)) as
a module over Z[t1, . . . , tN ]. Let x1, . . . , xn denote the Chern roots of the dual S
of the tautological subbundle S of the trivial bundle CNGr(n,N) so that for the total
equivariant Chern class of S we have
cT (S) =
(1− xi).
Then, due to [6, Lecture 8, Proposition 1.1] (see also [16]), the equivariant Schubert
classes σλ can be expressed by
σλ = sλ(x|u), u = (−tN , . . . ,−t1, 0, . . . ).
Hence, Theorem 2.1 yields a multiplication rule for the equivariant Schubert classes.
The corresponding stability property is implied by Corollary 2.11.
Corollary 3.1. We have
σλ σµ =
d νλµ σν ,
where
d νλµ =
T (α) unbarred
tm+T (α)−c(α) − tm+T (α)−ρ(α)
T (α)
, (3.1)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the d νλµ are polynomials in the ti− tj, i > j, with positive
integer coefficients. Moreover, the coefficients d νλµ, regarded as polynomials in the
variables ai defined in (1.5), are independent of n and m, as soon as the inequalities
n > λ′1 + µ
1 and m > λ1 + µ1 hold.
Example 3.2. For any n > 3 and m > 4 we have
σ(2) σ(2,1) = σ(4,1) + σ(3,2) + σ(3,1,1) + σ(2,2,1)
+ (tm+2 − tm−1 + tm − tm−2) σ(3,1) + (tm+2 − tm−1) σ(2,2)
+ (tm − tm−1) σ(2,1,1) + (tm+2 − tm−1) (tm − tm−1) σ(2,1).
This follows from Example 2.2.
The first manifestly positive rule for the expansion of σλ σµ was given by Knutson
and Tao [9] by using combinatorics of puzzles. Although the stability property was not
pointed out in [9], it can be deduced directly from the puzzle rule or by applying the
weight-preserving bijection between the puzzles and the barred tableaux constructed
by Kreiman [10].
3.2 Quantum immanants and higher Capelli operators
Let gln denote the general linear Lie algebra over C. Consider the center Z(gln) of
the universal enveloping algebra U(gln). The algebra U(gln) is equipped with the
natural filtration. For all n we identify gln−1 as a subalgebra of gln in a usual way
and denote by gl∞ the corresponding inductive limit
gl∞ =
Due to Olshanski [20], there exist filtration-preserving homomorphisms
on : Z(gln) → Z(gln−1), n > 1, (3.2)
which allow one to define the algebra Z of the virtual Casimir elements for the Lie
algebra gl∞ as the inverse limit
Z = lim
Z(gln), n → ∞,
in the category of filtered algebras.
The quantum immanants Sλ|n are elements of the center Z(gln) of the universal
enveloping algebra U(gln) parameterized by the diagrams λ with at most n rows;
see [18]. The elements Sλ|n form a basis of Z(gln) and they are consistent with the
Olshanski homomorphisms (3.2) so that
on : Sλ|n 7→ Sλ|n−1, (3.3)
where we assume Sλ|n = 0 if the number of rows of λ exceeds n. For any diagram λ,
the corresponding virtual quantum immanant Sλ is then defined as the sequence
Sλ = ( Sλ|n |n > 0).
The elements Sλ parameterized by all diagrams λ form a basis of the algebra Z so
that we can define the coefficients f νλµ by the expansion
Sλ Sµ =
f νλµ Sν .
Note that the same coefficients f νλµ determine the multiplication rule for the higher
Capelli operators ∆λ, which are defined as the sequences of the images of the quantum
immanants Sλ|n, where each image is taken under a natural representation of gln by
differential operators; see [18, 19].
Corollary 3.3. The coefficient f νλµ is zero unless µ ⊆ ν. If µ ⊆ ν then
f νλµ =
T (α) unbarred
ρ(α)T (α) − c(α)
, (3.4)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the f νλµ are nonnegative integers.
Proof. Due to the stability property (3.3) of the quantum immanants, it suffices to
calculate the corresponding coefficients for the expansion of the products Sλ|n Sµ|n.
The images of the quantum immanants Sλ|n under the Harish-Chandra isomorphism
can be identified with the double Schur polynomials sλ(x||a) where the sequence a is
specialized to ai = −i; see [18]. Therefore, the coefficients in question coincide with
the corresponding specializations of the Littlewood–Richardson polynomials cνλµ(a).
Example 3.4. Using Example 2.2 we get
S(2) S(2,1) = S(4,1) + S(3,2) + S(3,1,1) + S(2,2,1) + 5 S(3,1) + 3 S(2,2) + S(2,1,1) + 3 S(2,1).
In the course of the proof of Corollary 3.3 we also calculated the coefficients
for the expansion of the products Sλ|n Sµ|n for any n. Some other formulas for these
coefficients were obtained in [17]. In particular, it was shown that the f νλµ are integers,
although their positivity property was not established there.
Note also that the algebra of virtual Casimir elements Z is isomorphic to the
algebra of shifted symmetric functions Λ∗; see [19]. The latter can be regarded as the
specialization of Λ (or rather, its extension over C) at ai = −i for all i ∈ Z.
4 Supertableau formulas for sλ(x||a) and c
λµ(a)
Here we obtain one more rule for the calculation of the Littlewood–Richardson poly-
nomials cνλµ(a). It relies on a supertableau representation of the double Schur poly-
nomials sλ(x||a) which is implied by the results of [13]. This representation provides
a “finite” version of the supertableau formulas of [7] and [13]; cf. [4].
Fix a positive integer n. For r > 1 set u(r) = (u1, . . . , ur) and use the 9th Variation
in [13] with the indeterminates hrs specialized by
hrs = hr(u
(n−r−s+1)) if r + s 6 n,
and 0 otherwise, where hr denotes the r-th complete symmetric polynomial. Let us
write ŝλ/µ(u) for the corresponding Schur functions. Then (8.2) and (9.1) in [13] give
ŝλ/µ(u) =
α∈λ/µ
uT (α),
summed over semistandard tableaux T of shape λ/µ, such that the entries of the i-th
row do not exceed n− λi + i. Furthermore, using (6.18)
1 and (9.6 ′) in [13] we get
sλ(x|u) =
sµ(x) ŝλ′/µ′(−u). (4.5)
Equivalently, this can be interpreted as a combinatorial expression for the polynomials
sλ(x|u) in terms of “supertableaux”. Identify the indices of u with the symbols
1′, 2′, . . . . A supertableau T is obtained by filling in the diagram of λ with the indices
1, . . . , n, 1′, 2′, . . . in such a way that in each row (resp. column) each primed index is
to the right (resp. below) of each unprimed index; unprimed indices weakly increase
along the rows and strictly increase down the columns; primed indices strictly increase
along the rows and weakly increase down the columns; primed indices in column j
do not exceed n− λ′j + j. Relation (4.5) implies the following.
Proposition 4.1. We have
sλ(x|u) =
T (α) unprimed
xT (α)
T (α) primed
(−uT (α)), (4.6)
summed over all λ-supertableaux T .
Using (1.10), we get an analogous representation for the double Schur polynomials
sλ(x||a). A reverse supertableau T is obtained by filling in the diagram of λ with
1This formula in [13] should be corrected by replacing a(λj+n−j) with a(λi+n−i).
the indices 1, . . . , n, n′, (n − 1)′, . . . (including non-positive primed indices) in such
a way that in each row (resp. column) each primed index is to the right (resp.
below) of each unprimed index; unprimed indices weakly decrease along the rows and
strictly decrease down the columns; primed indices strictly decrease along the rows
and weakly decrease down the columns; primed indices in column j are not less than
λ′j − j + 1. The following supertableau representation of the polynomials sλ(x||a)
follows from Proposition 4.1.
Corollary 4.2. We have
sλ(x||a) =
T (α) unprimed
xT (α)
T (α) primed
(−aT (α)), (4.7)
summed over all reverse λ-supertableaux T .
Example 4.3. Let n = 2 and λ = (2, 1). By the definition (1.9),
s(2,1)(x||a) = (x2 − a2)(x1 − a0)(x1 − a2) + (x2 − a2)(x2 − a1)(x1 − a2).
On the other hand, the reverse (2, 1)-supertableaux are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
1 0 ′
1 1 ′
1 2 ′
2 ′ 0 ′
2 ′ 1 ′
which yield
s(2,1)(x||a) = x
1x2 + x1x
2 − x1x2a2 − x
2a2 − x
1a2 − x1x2a0 − x1x2a1 − x1x2a2
+ x2a0a2 + x2a1a2 + x2a
2 + x1a0a2 + x1a1a2 + x1a
2 − a0a
2 − a1a
Formula (4.5) implies a supertableau representation of the coefficients cνλµ(a, b)
and hence, of the Littlewood–Richardson polynomials cνλµ(a). The representation
for the latter is neither manifestly positive, nor stable; it provides an expression for
cνλµ(a) as an alternating sum of monomials in the ai. Given a sequence R of the form
(2.1), construct the set S(λ,R) of barred reverse λ-supertableaux by analogy with
T (λ,R). A tableau T ∈ S(λ,R) must contain boxes α1, . . . , αl occupied by unprimed
indices r1, r2, . . . , rl listed in the column order which is restricted to the subtableau of
T formed by the unprimed indices. As before, we distinguish the entries in α1, . . . , αl
by barring each of them. For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, which is
occupied by an unprimed index, set ρ(α) = ρ(i).
Corollary 4.4. The coefficients cνλµ(a, b) defined in (2.3) can be given by
cνλµ(a, b) =
T (α) unprimed, unbarred
aT (α)−ρ(α)
T (α)
T (α) primed
(−bT (α)), (4.8)
summed over sequences R of the form (2.1) and reverse supertableaux T ∈ S(λ,R).
Proof. Applying formula (4.5) we can reduce the calculation of cνλµ(a, b) to the par-
ticular case of the sequence b = (0). Now (4.8) follows from Lemma 2.4.
Example 4.5. In order to calculate the Littlewood–Richardson polynomial c
(2,1)
(2,1) (2)
we may take n = 2; see Corollary 2.11. The barred reverse supertableaux compatible
with the sequence (2) → (2, 1) are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
so that
(2,1)
(2,1) (2)
(a) = a2−1 + a−1a1 + a−1a2 − a−1a2 − a1a2 − a
− a−1a0 − a−1a1 − a−1a2 + a0a2 + a1a2 + a
= a2−1 − a−1a0 − a−1a2 + a0a2,
which agrees with Example 2.2.
References
[1] L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined
in terms of tableaux, Advances in Appl. Math. 10 (1989), 396–438.
[2] L. C. Biedenharn and J. D. Louck, Inhomogeneous basis set of symmetric poly-
nomials defined by tableaux, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), 1441–1445.
[3] A. S. Buch and R. Rimányi, Specializations of Grothendieck polynomials, C. R.
Acad. Sci. Paris, Ser. I 339 (2004), 1–4.
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Alg. Comb. 15 (2002), 7-26.
[5] W. Fulton, Young tableaux. With applications to representation theory and ge-
ometry. London Mathematical Society Student Texts, 35. Cambridge University
Press, Cambridge, 1997.
[6] W. Fulton, Equivariant cohomology in algebraic geometry, Eilenberg lectures,
Columbia University, Spring 2007. Available at
http://www.math.lsa.umich.edu/∼dandersn/eilenberg
[7] I. Goulden and C. Greene, A new tableau representation for supersymmetric
Schur functions, J. Algebra. 170 (1994), 687–703.
[8] W. Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109
(2001), 599–614.
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ans, Duke Math. J. 119 (2003), 221–260.
[10] V. Kreiman, Equivariant Littlewood-Richardson tableaux, preprint
arXiv:0706.3738.
[11] A. Lascoux, Interpolation, Lectures at Tianjin University, June 1996. Available
at http://www-igm.univ-mlv.fr/∼al/pub−engl.html
[12] D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos.
Trans. Roy. Soc. London Ser. A 233 (1934), 49–141.
[13] I. G. Macdonald, Schur functions: theme and variations, in “Actes 28-e
Séminaire Lotharingien”, pp. 5–39. Publ. I.R.M.A. Strasbourg, 1992, 498/S–27.
[14] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University
Press, Oxford, 1995.
[15] L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci,
SMF/AMS Texts and Monographs, Vol. 6, 1998.
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the Grassmannian, preprint math.CO/0506335.
[17] A. I. Molev and B. E. Sagan, A Littlewood-Richardson rule for factorial Schur
functions, Trans. Amer. Math. Soc, 351 (1999), 4429–4443.
[18] A. Okounkov, Quantum immanants and higher Capelli identities, Transform.
Groups 1 (1996), 99–126.
http://www.math.lsa.umich.edu/~dandersn/eilenberg
http://arxiv.org/abs/0706.3738
http://www-igm.univ-mlv.fr/~al/pub$_-$engl.html
http://arxiv.org/abs/math/0506335
[19] A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math.
J. 9 (1998), 239–300.
[20] G. I. Olshanski, Representations of infinite-dimensional classical groups, limits
of enveloping algebras, and Yangians, in “Topics in Representation Theory”,
Advances in Soviet Math. 2, Amer. Math. Soc., Providence RI, 1991, pp. 1–66.
[21] B. E. Sagan, The symmetric group. Representations, combinatorial algorithms,
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Verlag, New York, 2001.
Introduction
Multiplication rule
Applications
Equivariant Schubert calculus on the Grassmannian
Quantum immanants and higher Capelli operators
Supertableau formulas for s(x || a) and c(a)
|
0704.0066 | Lagrangian quantum field theory in momentum picture. IV. Commutation
relations for free fields | Lagrangian quantum field theory
in momentum picture
IV. Commutation relations for free fields
Bozhidar Z. Iliev
∗ † ‡
Short title: QFT in momentum picture: IV
Produced: → November 4, 2018
http://www.arXiv.org e-Print archive No. : arXiv:0704.0066[hep-th]
R© TM
Subject Classes:
Quantum field theory
2000 MSC numbers:
81Q99, 81S05, 81T99
2003 PACS numbers:
03.70.+k, 11.10.Ef, 11.10.-z,
11.90.+t, 12.90.+b
Key-Words:
Commutation relations, Anticommutation relations, Free quantum fields
Paracommutation relations, Parafermi and parabose commutation relations
Heisenberg relations (equations), Euler-Lagrange equations, Equations of motion
Lagrangians for free fields, Momentum operator, Angular momentum operator
Spin and orbital angular momentum operators, Normal ordering
∗Laboratory of Mathematical Modeling in Physics, Institute for Nuclear Research and Nuclear
Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussée 72, 1784 Sofia, Bulgaria
†E-mail address: [email protected]
‡URL: http://theo.inrne.bas.bg/∼bozho/
http://arxiv.org/abs/0704.0066v1
http://www.arXiv.org
http://arxiv.org/abs/0704.0066
http://theo.inrne.bas.bg/~bozho/
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations i
Contents
1 Introduction 1
2 The momentum picture 3
3 Lagrangians, Euler-Lagrange equations and dynamical variables 5
4 On the uniqueness of the dynamical variables 10
5 Heisenberg relations 14
6 Types of possible commutation relations 20
6.1 Restrictions related to the momentum operator . . . . . . . . . . . . . . . . . 21
6.2 Restrictions related to the charge operator . . . . . . . . . . . . . . . . . . . . 25
6.3 Restrictions related to the angular momentum operator(s) . . . . . . . . . . . 27
7 Inferences 31
8 State vectors, vacuum and mean values 37
9 Commutation relations for several coexisting different free fields 43
9.1 Commutation relations connected with the momentum operator. Problems
and their possible solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
9.2 Commutation relations connected with the charge and angular momentum
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.3 Commutation relations between the dynamical variables . . . . . . . . . . . . 50
9.4 Commutation relations under the uniqueness conditions . . . . . . . . . . . . 52
10 Conclusion 54
References 55
This article ends at page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Abstract
Possible (algebraic) commutation relations in the Lagrangian quantum theory of free
(scalar, spinor and vector) fields are considered from mathematical view-point. As sources of
these relations are employed the Heisenberg equations/relations for the dynamical variables
and a specific condition for uniqueness of the operators of the dynamical variables (with
respect to some class of Lagrangians). The paracommutation relations or some their gen-
eralizations are pointed as the most general ones that entail the validity of all Heisenberg
equations. The simultaneous fulfillment of the Heisenberg equations and the uniqueness re-
quirement turn to be impossible. This problem is solved via a redefinition of the dynamical
variables, similar to the normal ordering procedure and containing it as a special case. That
implies corresponding changes in the admissible commutation relations. The introduction of
the concept of the vacuum makes narrow the class of the possible commutation relations;
in particular, the mentioned redefinition of the dynamical variables is reduced to normal
ordering. As a last restriction on that class is imposed the requirement for existing of an
effective procedure for calculating vacuum mean values. The standard bilinear commutation
relations are pointed as the only known ones that satisfy all of the mentioned conditions and
do not contradict to the existing data.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 1
1. Introduction
The main subject of this paper is an analysis of possible (algebraic) commutation relations
in the Lagrangian quantum theory1 of free fields. These relations are considered only from
mathematical view-point and physical consequence of them, like the statistics of many-par-
ticle systems, are not investigated.
The canonical quantization method finds its origin in the classical Hamiltonian mechan-
ics [9, 10] and naturally leads to the canonical (anti)commutation relations [3, 11,12]. These
relations can be obtained from different assumptions (see, e.g., [1,13–15]) and are one of the
basic corner stones of the present-day quantum field theory.
Theoretically there are possible also non-canonical commutation relations. The best
known example of them being the so-called paracommutation relations [16–18]. But, however,
it seems no one of the presently known particles/fields obeys them.
In the present work is shown how different classes of commutation relations, understood
in a broad sense as algebraic connections between creation and/or annihilation operators,
arise from the Lagrangian formalism, when applied to three types of Lagrangians describing
free scalar, spinor and vector fields. Their origin is twofold. One one hand, a requirement for
uniqueness of the dynamical variables (that can be calculated from Lagrangians leading to
identical Euler-Lagrange equation) entails a number of specific commutation relations. On
another hand, any one of the so-called Heisenberg relations/equations [3, 11], implies cor-
responding commutation relations; for example, the paracommutation relations arise from
the Heisenberg equations regarding the momentum operator, when ‘charge symmetric’ La-
grangian is employed.2 The combination of the both methods leads to strong, generally
incompatible, restrictions on the admissible types of commutation relations.
The introduction of the concept of vacuum, combined with the mentioned uniqueness of
the operators of the dynamical variables, changes the situation and requires a redefinition
of these operators in a way similar to the one known as the normal ordering [1, 3, 11, 12],
which is its special case. Some natural assumptions reduce the former to the letter one; in
particular, in that way are excluded the paracommutation relations. However, this does not
reduce the possible commutation relations to the canonical ones. Further, the requirement
to be available an effective procedure for calculating vacuum mean (expectation) values, to
which reduce all predictable results in the theory, puts new restriction, whose only realistic
solution at the time being seems to be the standard canonical (anti)commutation relations.
The layout of the work is as follows.
Sect. 2 gives an idea of the momentum picture of motion and discusses the relations
between the creation and annihilation operators in it and in Heisenberg picture. In Sect. 3
are reviewed some basic results from [13–15], part of which can be found also in papers like [1,
3, 11, 12]. In particular, the explicit expression of the dynamical variables via the creation
1 In this paper we considered only the Lagrangian (canonical) quantum field theory in which the quantum
fields are represented as operators, called field operators, acting on some Hilbert space, which in general
is unknown if interacting fields are studied. These operators are supposed to satisfy some equations of
motion, from them are constructed conserved quantities satisfying conservation laws, etc. From the view-point
of present-day quantum field theory, this approach is only a preliminary stage for more or less rigorous
formulation of the theory in which the fields are represented via operator-valued distributions, a fact required
even for description of free fields. Moreover, in non-perturbative directions, like constructive and conformal
field theories, the main objects are the vacuum mean (expectation) values of the fields and from these are
reconstructed the Hilbert space of states and the acting on it fields. Regardless of these facts, the Lagrangian
(canonical) quantum field theory is an inherent component of the most of the ways of presentation of quantum
field theory adopted explicitly or implicitly in books like [1–8]. Besides, the Lagrangian approach is a source
of many ideas for other directions of research, like the axiomatic quantum field theory [3,7,8].
2 Ordinary [3,11], the commutation relations are postulated and the validity of the Heisenberg relations is
then verified. We follow the opposite method by postulating the Heisenberg equations and, then, looking for
commutation relations that are compatible with them.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 2
and annihilation operators are presented (without assuming some commutation relations or
normal ordering) and it is pointed to the existence of a family of such variables for a given
system of Euler-Lagrange equations for free fields. The last fact is analyzed in Sect. 4,
where a number of its consequences, having a sense of commutation relations, are drawn.
The Heisenberg relations and the commutation relations between the dynamical variables
are reviewed and analyzed in Sect. 5. It is pointed that the letter should be consequences
from the former ones. Arguments are presented that the Heisenberg equation concerning
the angular momentum operator should be split into two independent ones, representing its
‘orbital’ and ‘spin’ parts, respectively.
Sect. 6 contains a method for assigning commutation relations to the Heisenberg equa-
tions. It is shown that the Heisenberg equation involving the ‘orbital’ part of the angular
momentum gives rise to a differential, not algebraic, commutation relation and the one con-
cerning the ‘spin’ part of the angular momentum implies a complicated integro-differential
connections between the creation and annihilation operators. Special attention is paid to
the paracommutation relations, whose particular kind are the ordinary ones, which ensure
the validity of the Heisenberg equations concerning the momentum operator. Partially is
analyzed the problem for compatibility of the different types of commutation relations de-
rived. It is proved that some generalization of the paracommutation relations ensures the
fulfillment of all of the Heisenberg relations.
Sect. 7 is devoted to consequences from the commutation relations derived in Sect. 6
under the conditions for uniqueness of the dynamical variables presented in Sect. 4. Gen-
erally, these requirements are incompatible with the commutation relations. To overcome
the problem, it is proposed a redefinition of the dynamical variables via a method similar to
(and generalizing) the normal ordering. This, of course, entails changes in the commutation
relations, the new versions of which happen to be compatible with the uniqueness conditions
and ensure the validity of the Heisenberg relations.
The concept of the vacuum is introduced in Sect. 8. It reduces (practically) the redefini-
tion of the operators of the dynamical variables to the one obtained via the normal ordering
procedure in the ordinary quantum field theory, but, without additional suppositions, does
not reduce the commutation relations to the standard bilinear ones. As a last step in specify-
ing the commutation relations as much as possible, we introduce the requirement the theory
to supply an effective way for calculating vacuum mean values of (anti-normally ordered)
products of creation and annihilation operators to which are reduced all predictable results,
in particular the mean values of the dynamical variables. The standard bilinear commutation
relation seems to be the only ones know at present that survive that last condition, however
their uniqueness in this respect is not investigated.
Sect. 9 deals with the same problems as described above but for systems containing at
least two different quantum fields. The main obstacle is the establishment of commutation
relations between creation/annihilation operators concerning different fields. Argument is
presented that they should contain commutators or anticommutators of these operators.
The major of corresponding commutation relations are explicitly written and the results
obtained turn to be similar to the ones just described, only in ‘multifield’ version.
Section 10 closes the paper by summarizing its main results.
The books [1–3] will be used as standard reference works on quantum field theory. Of
course, this is more or less a random selection between the great number of (text)books and
papers on the theme to which the reader is referred for more details or other points of view.
For this end, e.g., [4, 12,19] or the literature cited in [1–4,12,19] may be helpful.
Throughout this paper ~ denotes the Planck’s constant (divided by 2π), c is the velocity of
light in vacuum, and i stands for the imaginary unit. The superscripts † and ⊤ mean respec-
tively Hermitian conjugation and transposition (of operators or matrices), the superscript ∗
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 3
denotes complex conjugation, and the symbol ◦ denotes compositions of mappings/operators.
By δfg, or δ
f or δ
fg (:= 1 for f = g, := 0 for f = g) is denoted the Kronecker δ-symbol,
depending on arguments f and g, and δn(y), y ∈ Rn, stands for the n-dimensional Dirac
δ-function; δ(y) := δ1(y) for y ∈ R.
The Minkowski spacetime is denoted by M . The Greek indices run from 0 to dimM −1 =
3. All Greek indices will be raised and lowered by means of the standard 4-dimensional
Lorentz metric tensor ηµν and its inverse ηµν with signature (+ − −−). The Latin indices
a, b, . . . run from 1 to dimM − 1 = 3 and, usually, label the spacial components of some
object. The Einstein’s summation convention over indices repeated on different levels is
assumed over the whole range of their values.
At last, we ought to give an explanation why this work appears under the general title
“Lagrangian quantum field theory in momentum picture” when in it all considerations are
done, in fact, in Heisenberg picture with possible, but not necessary, usage of the creation and
annihilation operators in momentum picture. First of all, we essentially employ the obtained
in [13–15] expressions for the dynamical variables in momentum picture for three types of
Lagrangians. The corresponding operators in Heisenberg picture, which in fact is used in this
paper, can be obtained via a direct calculation, as it is partially done in, e.g., [1] for one of
the mentioned types of Lagrangians. The important point here is that in Heisenberg picture
it suffice to be used only the standard Lagrangian formalism, while in momentum picture one
has to suppose the commutativity between the components of the momentum operator and
the validity of the Heisenberg relations for it (see below equations (2.6) and (2.7)). Since for
the analysis of the commutation relations we intend to do the fulfillment of these relations is
not necessary (they are subsidiary restrictions on the Lagrangian formalism), the Heisenberg
picture of motion is the natural one that has to be used. For this reason, the expression for the
dynamical variables obtained in [13–15] will be used simply as their Heisenberg counterparts,
but expressed via the creation and annihilation operators in momentum picture. The only
real advantage one gets in this way is the more natural structure of the orbital angular
momentum operator. As the commutation relations considered below are algebraic ones, it
is inessential in what picture of motion they are written or investigated.
2. The momentum picture
Since the momentum picture of motion will be used only partially in this work, below is
presented only its definition and the connection between the creation/annihilation operators
in it and in Heisenberg picture. Details concerning the momentum picture can be found
in [20,21] and in the corresponding sections devoted to it in [13–15].
Let us consider a system of quantum fields, represented in Heisenberg picture of motion
by field operators ϕ̃i(x) : F → F , i = 1, . . . , n ∈ N, acting on the system’s Hilbert space
F of states and depending on a point x in Minkowski spacetime M . Here and henceforth,
all quantities in Heisenberg picture will be marked by a tilde (wave) “˜” over their kernel
symbols. Let P̃µ denotes the system’s (canonical) momentum vectorial operator, defined via
the energy-momentum tensorial operator T̃ µν of the system, viz.
P̃µ :=
x0=const
T̃0µ(x) d3x. (2.1)
Since this operator is Hermitian, P̃†µ = P̃µ, the operator
U(x, x0) = exp
(xµ − xµ0 ) P̃µ
, (2.2)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 4
where x0 ∈ M is arbitrarily fixed and x ∈ M ,3 is unitary, i.e. U†(x0, x) := (U(x, x0))† =
U−1(x, x0) := (U(x, x0))−1 and, via the formulae
X̃ 7→ X (x) = U(x, x0)( X̃ ) (2.3)
Ã(x) 7→ A(x) = U(x, x0) ◦ ( Ã(x)) ◦ U−1(x, x0), (2.4)
realizes the transition to the momentum picture. Here X̃ is a state vector in system’s Hilbert
space of states F and Ã(x) : F → F is (observable or not) operator-valued function of x ∈ M
which, in particular, can be polynomial or convergent power series in the field operators
ϕ̃i(x); respectively X (x) and A(x) are the corresponding quantities in momentum picture.
In particular, the field operators transform as
ϕ̃i(x) 7→ ϕi(x) = U(x, x0) ◦ ϕ̃i(x) ◦ U−1(x, x0). (2.5)
Notice, in (2.2) the multiplier (xµ − xµ0 ) is regarded as a real parameter (in which P̃µ is
linear). Generally, X (x) and A(x) depend also on the point x0 and, to be quite correct, one
should write X (x, x0) and A(x, x0) for X (x) and A(x), respectively. However, in the most
situations in the present work, this dependence is not essential or, in fact, is not presented
at all. For that reason, we shall not indicate it explicitly.
The momentum picture is most suitable in quantum field theories in which the compo-
nents P̃µ of the momentum operator commute between themselves and satisfy the Heisenberg
relations/equations with the field operators, i.e. when P̃µ and ϕ̃i(x) satisfy the relations:
[ P̃µ, P̃ν ] = 0 (2.6)
[ ϕ̃i(x), P̃µ] = i~∂µ ϕ̃i(x). (2.7)
Here [A,B]± := A ◦ B ± B ◦ A, ◦ being the composition of mappings sign, is the commuta-
tor/anticommutator of operators (or matrices) A and B.
However, the fulfillment of the relations (2.6) and (2.7) will not be supposed in this paper
until Sect. 6 (see also Sect. 5).
Let a±s (k) and a
s (k) be the creation/annihilation operators of some free particular field
(see Sect. 3 below for a detailed explanation of the notation). We have the connections
ã±s (k) = e
xµkµ U−1(x, x0) ◦ a±s (k) ◦ U(x, x0)
ㆱs (k) = e
xµkµ U−1(x, x0) ◦ a†±s (k) ◦ U(x, x0)
m2c2 + k2 (2.8)
whose explicit form is
ã±s (k) = e
kµa±s (k)
ㆱs (k) = e
kµa†±s (k)
m2c2 + k2. (2.9)
Further it will be assumed ã±s (k) and ã
s (k) to be defined in Heisenberg picture, indepen-
dently of a±s (k) and a
s (k), by means of the standard Lagrangian formalism. What concerns
the operators a±s (k) and a
s (k), we shall regard them as defined via (2.9); this makes them
independent from the momentum picture of motion. The fact that the so-defined operators
a±s (k) and a
s (k) coincide with the creation/annihilation operators in momentum picture
(under the conditions (2.6) and (2.7)) will be inessential in the almost whole text.
3 The notation x0, for a fixed point in M , should not be confused with the zeroth covariant coordinate
µ of x which, following the convention xν := ηνµx
µ, is denoted by the same symbol x0. From the context,
it will always be clear whether x0 refers to a point in M or to the zeroth covariant coordinate of a point
x ∈ M .
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 5
3. Lagrangians, Euler-Lagrange equations
and dynamical variables
In [13–15] we have investigated the Lagrangian quantum field theory of respectively scalar,
spin 1
and vector free fields. The main Lagrangians from which it was derived are respectively
(see loc. cit. or, e.g. [1, 3, 11,12]):
L̃′sc = L̃′sc( ϕ̃, ϕ̃†) =−
1 + τ( ϕ̃)
m2c4 ϕ̃(x) ◦ ϕ̃†(x) + 1
1 + τ( ϕ̃)
c2~2(∂µ ϕ̃(x)) ◦ (∂µ ϕ̃†(x))
(3.1a)
L̃′sp = L̃′sp( ψ̃,
ψ) =− 1
i~c{ ˜̆ψ
(x)C−1γµ ◦ (∂µ ψ̃(x))
− (∂µ ˜̆ψ
(x))C−1γµ ◦ ψ̃(x)}+mc2 ˜̆ψ
(x)C−1 ◦ ψ̃(x)
(3.1b)
L̃′v = L̃′v( Ũ , Ũ†) =
1 + τ( Ũ)
Ũ†µ ◦ Ũ
1 + τ( Ũ)
−(∂µ Ũ
ν) ◦ (∂
µ Ũν) + (∂µ Ũ
) ◦ (∂ν Ũ
(3.1c)
Here it is used the following notation: ϕ̃(x) is a scalar field, a tilde (wave) over a symbol
means that it is in Heisenberg picture, the dagger † denotes Hermitian conjugation, ψ̃ :=
( ψ̃0, ψ̃1, ψ̃2, ψ̃3) is a 4-spinor field,
ψ := C ψ̃
:= C( ψ̃†γ0) is its charge conjugate with γµ
being the Dirac gamma matrices and the matrix C satisfies the equations C−1γµC = −γµ
and C⊤ = −C, Uµ is a vector field, m is the field’s mass (parameter) and the function
τ(A) :=
1 for A† = A (Hermitian operator)
0 for A† 6= A (non-Hermitian operator)
, (3.2)
with A : F → F being an operator on the systems Hilbert space F of states, takes care of
is the field charged (non-Hermitian) or neutral (Hermitian, uncharged). Since a spinor field
is a charged one, we have τ( ψ̃) = 0; sometimes below the number 0 = τ( ψ̃) will be written
explicitly for unification of the notation.
We have explored also the consequences from the ‘charge conjugate’ Lagrangians
L̃′′sc = L̃′′sc( ϕ̃, ϕ̃†) := L̃′sc( ϕ̃†, ϕ̃) (3.3a)
L̃′′sp = L̃′′sp( ψ̃,
ψ) := L̃′sp(
ψ, ψ̃) (3.3b)
L̃′′v = L̃′′v( Ũ , Ũ†) := L̃′v( Ũ†, Ũ), (3.3c)
as well as from the ‘charge symmetric’ Lagrangians
L̃′′′sc = L̃′′′sc( ϕ̃, ϕ̃†) :=
L̃′sc + L̃′′sc
L̃′sc( ϕ̃, ϕ̃†) + L̃′sc( ϕ̃†, ϕ̃)
(3.4a)
L̃′′′sp = L̃′′′sp( ψ̃,
ψ) :=
L̃′sp + L̃′′sp
L̃′sp( ψ̃,
ψ) + L̃′sp(
ψ, ψ̃)
(3.4b)
L̃′′′v = L̃′′′v ( Ũ , Ũ†) :=
L̃′v + L̃′′v
L̃′v( Ũ , Ũ†) + L̃′v( Ũ†, Ũ)
. (3.4c)
It is essential to be noted, for a massless, m = 0, vector field to the Lagrangian formalism
are added as subsidiary conditions the Lorenz conditions
∂µ Ũµ = 0 ∂
µ Ũ†µ = 0 (3.5)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 6
on the solutions of the corresponding Euler-Lagrange equations. Besides, if the opposite is
not stated explicitly, no other restrictions, like the (anti)commutation relations, are supposed
to be imposed on the above Lagrangians. And a technical remark, for convenience, the fields
ϕ̃, ψ̃ and Ũ and their charge conjugate ϕ̃†, ˜̆ψ and Ũ†, respectively, are considered as
independent field variables.
Let L̃′ denotes any one of the Lagrangians (3.1) and L̃′′ (resp. L̃′′′) the corresponding
to it Lagrangian given via (3.3) (resp. (3.4)). Physically the difference between L̃′ and L̃′′ is
that the particles for L̃′ are antiparticles for L̃′′ and vice versa. Both of the Lagrangians L̃′
and L̃′′ are not charge symmetric, i.e. the arising from them theories are not invariant under
the change particle↔antiparticle (or, in mathematical terms, under some of the changes
ϕ̃ ↔ ϕ̃†, ψ̃ ↔ ˜̆ψ, Ũ ↔ Ũ†) unless some additional hypotheses are made. Contrary to
this, the Lagrangian L̃′′′ is charge symmetric and, consequently, the formalism on its base
is invariant under the change particle↔antiparticle.4
The Euler-Lagrange equations for the Lagrangians L̃′, L̃′′ and L̃′′′ happen to coin-
cide [13–15]:5
∂ L̃′
( ∂ L̃′
∂(∂µχ)
≡ ∂ L̃
( ∂ L̃′′
∂(∂µχ)
≡ ∂ L̃
( ∂ L̃′′′
∂(∂µχ)
= 0, (3.6)
where χ = ϕ̃, ϕ̃†, ψ̃,
ψ, Ũ , Ũ† for respectively scalar, spinor and vector field.
Since the creation and annihilation operators are defined only on the base of Euler-La-
grange equations [1, 3, 11–15], we can assert that these operators are identical for the La-
grangians L̃′, L̃′′ and L̃′′′. We shall denote these operators by a±s (k) and a
s (k) with the
convention that a+s (k) (resp. a
s (k)) creates a particle (resp. antiparticle) with 4-momen-
tum (
m2c2 + k2,k), polarization s (see below) and charge (−q) (resp. (+q))6 and a†−s (k)
(resp. a−s (k)) annihilates/destroys such a particle (resp. antiparticle). Here and henceforth
k ∈ R3 is interpreted as (anti)particle’s 3-momentum and the values of the polarization index
s depend on the field considered: s = 1 for a scalar field, s = 1 or s = 1, 2 for respectively
massless (m = 0) or massive (m 6= 0) spinor field, and s = 1, 2, 3 for a vector field.7 Since
massless vector field’s modes with s = 3 may enter only in the spin and orbital angular mo-
menta operators [15], we, for convenience, shall assume that the polarization indices s, t, . . .
take the values from 1 to 2j+1− δ0m(1− δ0j), where j = 0, 12 , 1 is the spin for scalar, spinor
and vector field, respectively, and δ0m := 1 for m = 0 and δ0m := 0 for m 6= 0;8 if the value
s = 3 is important when j = 1 and m = 0, it will be commented/considered separately. Of
course, the creation and annihilation operators are different for different fields; one should
write, e.g., ja
(k) for a±s (k), but we shall not use such a complicated notation and will
assume the dependence on j to be an implicit one.
4 Besides, under the same assumptions, the Lagrangian L̃′′′ does not admit quantization via anticommu-
tators (commutators) for integer (half-integer) spin field, while L̃′ and L̃′′ do not make difference between
integer and half-integer spin fields.
5 Rigorously speaking, the Euler-Lagrange equations for the Lagrangian (3.4b) are identities like 0 = 0 —
see [22]. However, bellow we shall handle this exceptional case as pointed in [14].
6 For a neutral field, we put q = 0.
7 For convenience, in [14], we have set s = 0 if m = 0 and s = 1, 2 if m 6= 0 for a spinor field. For a massless
vector field, one may set s = 1, 2, thus eliminating the ‘unphysical’ value s = 3 for m = 0 — see [1, 11, 15].
In [13], for a scalar field, the notation ϕ±
(k) and ϕ
(k) is used for a±
(k) and a
(k), respectively.
8 In this way the case (j, s,m) = (1, 3, 0) is excluded from further considerations; if (j,m) = (1, 0) and
q = 0, the case considered further in this work corresponds to an electromagnetic field in Coulomb gauge,
as the modes with s = 3 are excluded [15]. However, if the case (j, s,m) = (1, 3, 0) is important for some
reasons, the reader can easily obtain the corresponding results by applying the ones from [15].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 7
The following settings will be frequently used throughout this chapter:
0 for scalar field
for spinor field
1 for vector field
1 for q = 0 (neutral (Hermitian) field)
0 for q 6= 0 (charged (non-Hermitian) field)
ε := (−1)2j =
+1 for integer j (bose fields)
−1 for half-integer j (fermi fields)
(3.7)
[A,B]± := [A,B]±1 := A ◦B ±B ◦A, (3.8)
where A and B are operators on the system’s Hilbert space F of states.
The dynamical variables corresponding to L̃′, L̃′′ and L̃′′′ are, however, completely dif-
ferent, unless some additional conditions are imposed on the Lagrangian formalism [13–15].
In particular, the momentum operators P̃ωµ , charge operators Q̃ω, spin operators S̃ωµν and
orbital operators L̃ωµν , where ω = ′, ′′, ′′′, for these Lagrangians are [13–15]:
P̃ ′µ =
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a†+s (k) ◦ a−s (k) + εa†−s (k) ◦ a+s (k)}
(3.9a)
P̃ ′′µ =
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a+s (k) ◦ a†−s (k) + εa−s (k) ◦ a†+s (k)}
(3.9b)
P̃ ′′′µ =
2(1 + τ)
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{[a†+s (k), a−s (k)]ε + [a+s (k), a†−s (k)]ε}
(3.9c)
Q̃′ = +q
2j+1−δ0m(1−δ0j)
d3k{a†+s (k) ◦ a−s (k)− εa†−s (k) ◦ a+s (k)} (3.10a)
Q̃′′ = −q
2j+1−δ0m(1−δ0j)
d3k{a+s (k) ◦ a†−s (k)− εa−s (k) ◦ a†+s (k)} (3.10b)
Q̃′′′ = 1
2j+1−δ0m(1−δ0j )
d3k{[a†+s (k), a−s (k)]ε − [a+s (k), a†−s (k)]ε} (3.10c)
S̃ ′µν =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)
+ σss
µν (k)a
s (k) ◦ a+s′(k)
(3.11a)
S̃ ′′µν = ε
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s′(k) ◦ a
s (k)
+ σss
µν (k)a
(k) ◦ a†+s (k)
(3.11b)
S̃ ′′′µν =
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)[a
s (k), a
(k)]ε
+ σss
µν (k)[a
s (k), a
s′(k)]ε
(3.11c)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 8
L̃′µν =x0µ P̃ ′ν − x0 ν P̃ ′µ
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)
+ lss
µν (k)a
s (k) ◦ a+s′(k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(3.12a)
L̃′′µν =x0µ P̃ ′′ν − x0 ν P̃ ′′µ
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s′(k) ◦ a
s (k)
+ lss
µν (k)a
(k) ◦ a†+s (k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
m2c2+k2
(3.12b)
L̃′′′µν =x0µ P̃ ′′′ν − x0 ν P̃ ′′′µ
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)[a
s (k), a
(k)]ε
+ lss
µν (k)[a
s (k), a
(k)]ε
4(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k) + a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(3.12c)
Here we have used the following notation: (−1)n+1/2 := (−1)ni for all n ∈ N and i := +
←−−−−−→
◦B(k) := −
∂A(k)
◦B(k) +
A(k) ◦ kµ
∂B(k)
←−−−→
◦B(k)
(3.13)
for operators A(k) and B(k) having C1 dependence on k,9 and σ
ss′,±
µν (k) and l
ss′,±
µν (k) are
9 More generally, if ω : {F → F} → {F → F} is a mapping on the operator space over the system’s Hilbert
space, we put A
ω ◦ B := −ω(A) ◦ B + A ◦ ω(B) for any A,B : F → F . Usually [2, 12], this notation is used
for ω = ∂µ.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 9
some functions of k such that10
µν (k) = −σss
νµ (k) l
ss′,±
µν (k) = −lss
νµ (k)
µν (k) = l
ss′,±
νµ (k) = 0 for j = 0 (scalar field)
µν (k) = −σss
µν (k) =: σ
µν (k) = −σs
µν (k) = −σss
νµ (k) for j = 1 (vector field)
µν (k) = −lss
µν (k) =: l
µν (k) = −ls
µν (k) = −lss
νµ (k) for j = 1 (vector field).
(3.14)
A technical remark must be make at this point. The equations (3.9)–(3.12) were de-
rived in [13–15] under some additional conditions, represented by equations (2.6) and (2.7),
which are considered bellow in Sect. 5 and ensure the effectiveness of the momentum pic-
ture of motion [21] used in [13–15]. However, as it is partially proved, e.g., in [1], when the
quantities (3.9)–(3.12) are expressed via the Heisenberg creation and annihilation operators
(see (2.9)), they remain valid, up to a phase factor, and without making the mentioned
assumptions, i.e. these assumptions are needless when one works entirely in Heisenberg pic-
ture. For this reason, we shall consider (3.9)–(3.12) as pure consequence of the Lagrangian
formalism.
We should emphasize, in (3.11) and (3.12) with S̃ωµν and L̃ωµν , ω = ′, ′′, ′′′, are denoted
the spin and orbital, respectively, operators for L̃ω, which are the spacetime-independent
parts of the spin and orbital, respectively, angular momentum operators [14, 23]; if the last
operators are denoted by S̃ωµν and L̃
µν , the total angular momentum operator of a system
with Lagrangian L̃ω is [23]
M̃ωµν = L̃
µν + S̃
µν = L̃ωµν + S̃ωµν , ω = ′, ′′, ′′′ (3.15)
and S̃ωµν = S̃ωµν (and hence L̃
µν = L̃ωµν) iff S̃
µν is a conserved operator or, equivalently, iff
the system’s canonical energy-momentum tensor is symmetric.11
Going ahead (see Sect. 6), we would like to note that the expressions (3.9c) and, conse-
quently, the Lagrangian L̃′′′ are the base from which the paracommutation relations were
first derived [16].
And a last remark. Above we have expressed the dynamical variables in Heisenberg picture
via the creation and annihilation operators in momentum picture. If one works entirely in
Heisenberg picture, the operators (2.9), representing the creation and annihilation operators
in Heisenberg picture, should be used. Besides, by virtue of the equations
(a±s (k))
† = a†∓s (k) (a
s (k))
† = a∓s (k) (3.16)
ã±s (k)
= ã†∓s (k)
ㆱs (k)
= ã∓s (k), (3.17)
some of the relations concerning a
s (k), e.g. the Euler-Lagrange and Heisenberg equations,
are consequences of the similar ones regarding a±s (k). In view of (2.9), we shall consider (3.9)–
(3.12) as obtained form the corresponding expressions in Heisenberg picture by making the
replacements ã±s (k) 7→ a±s (k) and ã
s (k) 7→ a†±s (k). So, (3.9)–(3.12) will have, up to a
phase factor, a sense of dynamical variables in Heisenberg picture expressed via the cre-
ation/annihilation operators in momentum picture.
10 For the explicit form of these functions, see [13–15]; see also equation (6.57) below.
11 In [14,23] the spin and orbital operators are labeled with an additional left superscript ◦, which, for brevity,
is omitted in the present work as in it only these operators, not S̃
µν and L̃
µν , will be considered. Notice,
the operators S̃
µν and L̃
µν are, generally, time-dependent while the orbital and spin ones are conserved, as
a result of which the total angular momentum is a conserved operator too [14,23].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 10
4. On the uniqueness of the dynamical variables
Let D = Pµ, Q, Sµν , Lµν denotes some dynamical variable, viz. the momentum, charge,
spin, or orbital operator, of a system with Lagrangian L. Since the Euler-Lagrange equations
for the Lagrangians L′, L′′ and L′′′ coincide (see (3.6)), we can assert that any field satisfying
these equations admits at least three classes of conserved operators, viz. D′, D′′ and D′′′ =
D′+D′′
.Moreover, it can be proved that the Euler-Lagrange equations for the Lagrangian
Lα,β := αL′ + β L′′ α+ β 6= 0 (4.1)
do not depend on α, β ∈ C and coincide with (3.6). Therefore there exists a two parameter
family of conserved dynamical variables for these equations given via
Dα,β := αD′ + βD′′ α+ β 6= 0. (4.2)
Evidently L′′′ = L 1
and D′′′ = D 1
. Since the Euler-Lagrange equations (3.6) are linear
and homogeneous (in the cases considered), we can, without a lost of generality, restrict the
parameters α, β ∈ C to such that
α+ β = 1, (4.3)
which can be achieved by an appropriate renormalization (by a factor (α+β)−1/2) of the field
operators. Thus any field satisfying the Euler-Lagrange equations (3.6) admits the family
Dα,β, α + β = 1, of conserved operators. Obviously, this conclusion is valid if in (4.1) we
replace the particular Lagrangians L′ and L′′ (see (3.1) and (3.3)) with any two Lagrangians
(of one and the same field variables) which lead to identical Euler-Lagrange equations. How-
ever, the essential point in our case is that L′ and L′′ do not differ only by a full divergence,
as a result of which the operators Dα,β are different for different pairs (α, β), α+ β = 1.12
Since one expects a physical system to possess uniquely defined dynamical characteristics,
e.g. energy and total angular momentum, and the Euler-Lagrange equations are considered
(in the framework of Lagrangian formalism) as the ones governing the spacetime evolution of
the system considered, the problem arises when the dynamical operators Dα,β, α+β = 1, are
independent of the particular choice of α and β, i.e. of the initial Lagrangian one starts off.
Simple calculation show that the operators (4.2), under the condition (4.3), are independent
of the particular values of the parameters α and β if and only if
D′ = D′′. (4.4)
Some consequences of the condition(s) (4.4) will be considered below, as well as possible ways
for satisfying these restrictions on the Lagrangian formalism.
Combining (3.9)–(3.12) with (4.4), for respectively D = Pµ, Q, Sµν , Lµν , we see that a
free scalar, spinor or vector field has a uniquely defined dynamical variables if and only if
the following equations are fulfilled:
2j+1−δ0m(1−δ0j )
d3k kµ
m2c2+k2
a†+s (k) ◦ a−s (k)− εa−s (k) ◦ a†+s (k)
− a+s (k) ◦ a†−s (k) + εa†−s (k) ◦ a+s (k)
= 0 (4.5)
12 Note, no commutativity or some commutation relations between the field operators and their charge (or
Hermitian) conjugate are presupposed, i.e., at the moment, we work in a theory without such relations and
normal ordering.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 11
2j+1−δ0m(1−δ0j )
a†+s (k) ◦ a−s (k)− εa−s (k) ◦ a†+s (k)
+ a+s (k) ◦ a†−s (k)− εa†−s (k) ◦ a+s (k)
= 0 (4.6)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)− εσ
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εσss′,+µν (k)a+s′(k) ◦ a
s (k) + σ
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.7)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k)− εl
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εlss′,+µν (k)a+s′(k) ◦ a
s (k) + l
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦a−s (k)+εa−s (k)
←−−−−−→
←−−−−−→
◦a†+s (k)
−a+s (k)
←−−−−−→
←−−−−−→
◦a†−s (k)−εa†−s (k)
←−−−−−→
←−−−−−→
◦a+s (k)
m2c2+k2
(4.8)
In (4.6) is retained the constant factor q as in the neutral case it is equal to zero and,
consequently, the equation (4.6) reduces to identity.
Since the Euler-Lagrange equations do not impose some restrictions on the creation and
annihilation operators, the equations (4.5)–(4.8) can be regarded as subsidiary conditions
on the Lagrangian formalism and can serve as equations for (partial) determination of the
creation and annihilation operators. The system of integral equations (4.5)–(4.8) is quite
complicated and we are not going to investigate it in the general case. Below we shall
restrict ourselves to analysis of only those solutions of (4.5)–(4.8), if any, for which the
integrands in (4.5)–(4.8) vanish. This means that we shall replace the system of integral
equations (4.5)–(4.8) with respect to creation and annihilation operators with the following
system of algebraic equations (do not sum over s and s′ in (4.12) and (4.13)!):
a†+s (k) ◦ a−s (k) − εa−s (k) ◦ a†+s (k) − a+s (k) ◦ a†−s (k) + εa†−s (k) ◦ a+s (k) = 0 (4.9)
a†+s (k) ◦ a−s (k) − εa−s (k) ◦ a†+s (k) + a+s (k) ◦ a†−s (k) − εa†−s (k) ◦ a+s (k) = 0 if q 6= 0
(4.10)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k) + εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
−a+s (k)
←−−−−−→
←−−−−−→
◦a†−s (k)−εa†−s (k)
←−−−−−→
←−−−−−→
◦a+s (k)
m2c2+k2
(4.11)
µν (k)a
s (k) ◦ a−s′(k)− εσ
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εσss′,+µν (k)a+s′(k) ◦ a
s (k) + σ
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.12)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 12
µν (k)a
s (k) ◦ a−s′(k)− εl
ss′,−
µν (k)a
(k) ◦ a†+s (k)
− εlss′,+µν (k)a+s′(k) ◦ a
s (k) + l
ss′,+
µν (k)a
s (k) ◦ a+s′(k)
= 0 (4.13)
Here: s = 1, . . . , 2j + 1 − δ0m(1 − δ0j) in (4.9)–(4.11) and s, s′ = 1, . . . , 2j + 1 − δ0m(1 −
δ1j) in (4.12) and (4.13). (Notice, by virtue of (3.14), the equations (4.12) and (4.13) are
identically valid for j = 0, i.e. for scalar fields.) Since all polarization indices enter in (4.5)
and (4.6) on equal footing, we do not sum over s in (4.9)–(4.11). But in (4.12) and (4.13)
we have retain the summation sign as the modes with definite polarization cannot be singled
out in the general case. One may obtain weaker versions of (4.9)–(4.13) by summing in them
over the polarization indices, but we shall not consider these conditions below regardless of
the fact that they also ensure uniqueness of the dynamical variables.
At first, consider the equations (4.9)–(4.11). Since for a neutral field, q = 0, we have
s (k) = a
s (k), which physically means coincidence of field’s particles and antiparticles,
the equations (4.9)–(4.11) hold identically in this case.
Let consider now the case q 6= 0, i.e. the investigated field to be charged one. Using the
standard notation (cf. (3.8))
[A,B]η := A ◦B + ηB ◦A, (4.14)
for operators A and B and η ∈ C, we rewrite (4.9) and (4.10) as
[a†+s (k), a
s (k)]−ε − [a+s (k), a†−s (k)]−ε = 0 (4.9′)
[a†+s (k), a
s (k)]−ε + [a
s (k), a
s (k)]−ε = 0 if q 6= 0, (4.10′)
which are equivalent to
[a†±s (k), a
s (k)]−ε = 0 if q 6= 0. (4.15)
Differentiating (4.15) and inserting the result into (4.11), one can verify that (4.11) is
tantamount to
a†+s (k),
◦ a−s (k)
a+s (k),
◦ a†−s (k)
m2c2+k2
= 0 if q 6= 0, (4.16)
Consider now (4.12) and (4.13). By means of the shorthand (4.14), they read
µν (k)[a
s (k), a
(k)]−ε + σ
ss′,+
µν (k)[a
s (k), a
(k)]−ε
= 0 (4.17)
µν (k)[a
s (k), a
(k)]−ε + l
ss′,+
µν (k)[a
s (k), a
(k)]−ε
= 0. (4.18)
For a scalar field, j = 0, these conditions hold identically, due to (3.14). But for j 6= 0 they
impose new restrictions on the formalism. In particular, for vector fields, j = 1 and ε = +1
they are satisfied iff (see (3.14))
[a†+s (k), a
(k)]−ε − [a†−s (k), a+s′(k)]−ε − [a
(k), a−s (k)]−ε + [a
(k), a+s (k)]−ε = 0. (4.19)
One can satisfy (4.17) and (4.18) if the following generalization of (4.15) holds
[a†±s (k), a
(k)]−ε = 0. (4.20)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 13
For spin j = 1
(and hence ε = −1 – see (3.7)), the conditions (4.12) and (4.13) cannot
be simplified much, but, if one requires the vanishment of the operator coefficients after
ss′,±
µν (k) and l
ss′,±
µν (k), one gets
a†±s (k) ◦ a∓s′(k) = 0 j =
ε = −1. (4.21)
Excluding some special cases, e.g. neutral scalar field (q = 0 and j = 0), the equa-
tions (4.15) and (4.21) are unacceptable from many viewpoints. The main of them is that they
are incompatible with the ordinary (anti)commutation relations (see, e.g., e.g. [1, 11, 12, 18]
or Sect. 6, in particular, equations (6.13) bellow); for example, (4.21) means that the acts of
creation and annihilation of (anti)particles with identical characteristics should be mutually
independent, which contradicts to the existing theory and experimental data.
Now we shall try another way for achieving uniqueness of the dynamical variables for
free fields. Since in (4.9)–(4.13) naturally appear (anti)commutators between creation and
annihilation operators and these (anti)commutators vanish under the standard normal or-
dering [1,11,12,18], one may suppose that the normally ordered expressions of the dynamical
variables may coincide. Let us analyze this method.
Recall [1, 3, 11, 12], the normal ordering operator N (for free field theory) is a linear
operator on the operator space of the system considered such that to a product (composition)
c1 ◦ · · · ◦ cn of n ∈ N creation and/or annihilation operators c1, . . . cn it assigns the operator
(−1)f cα1 ◦ · · · cαn . Here (α1, . . . , αn) is a permutation of (1, . . . , n), all creation operators
stand to the left of all annihilation ones, the relative order between the creation/annihilation
operators is preserved, and f is equal to the number of transpositions among the fermion
operators (j = 1
) needed to be achieved the just-described order (“normal order”) of the
operators c1 ◦ · · · ◦ cn in cα1 ◦ · · · cαn .13 In particular this means that
a+s (k) ◦ a
t (p)
= a+s (k) ◦ a
t (p) N
a†+s (k) ◦ a−t (p)
= a†+s (k) ◦ a−t (p)
a−s (k) ◦ a
t (p)
t (p) ◦ a−s (k) N
a†−s (k) ◦ a+t (p)
= εa+t (p) ◦ a†−s (k) (4.22)
and, consequently, we have
[a†±s (k), a
t (p)]−ε
= 0 N
[a±s (k), a
t (p)]−ε
= 0, (4.23)
due to ε := (−1)2j = ±1 (see (3.7)). (In fact, below only the equalities (4.22) and (4.23),
not the general definition of a normal product, will be applied.)
Applying the normal ordering operator to (4.9′), (4.10′), (4.17) and (4.18), we, in view
of (4.23), get the identity 0 = 0, which means that the conditions (4.9), (4.10), (4.12)
and (4.13) are identically satisfied after normal ordering. This is confirmed by the application
of N to (3.9) and (3.10), which results respectively in (see (4.22))
N ( P̃ ′µ) = N ( P̃ ′′µ)
1 + τ
2j+1−δ0m(1−δ0j )
d3kkµ|
m2c2+k2
{a†+s (k) ◦ a−s (k) + a+s (k) ◦ a†−s (k)} (4.24)
13 We have slightly modified the definition given in [1,3,11,12] because no (anti)commutation relations are
presented in our exposition till the moment. In this paper we do not concern the problem for elimination of
the ‘unphysical’ operators a±
(k) and a
(k) from the spin and orbital momentum operators when j = 1; for
details, see [15], where it is proved that, for an electromagnetic field, j = 1 and q = 0, one way to achieve this
is by adding to the number f above the number of transpositions between a±s (k), s = 1, 2, and a
(k) needed
for getting normal order.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 14
N ( Q̃′) = N ( Q̃′′) = 1
1 + τ
2j+1−δ0m(1−δ0j )
d3k{a†+s (k) ◦ a−s (k)− a+s (k) ◦ a†−s (k)}.
(4.25)
Therefore the normal ordering ensures the uniqueness of the momentum and charge operators,
if we redefine them respectively as
P̃µ := N ( P̃ ′µ) Q̃ := N ( Q̃′). (4.26)
Putting ωµν := kµ
− kν ∂∂kµ and using (4.22), one can verify that
a+s (k)
←−−−→
ωµν ◦ a†−s (k)
= a+s (k)
←−−−→
ωµν ◦ a†−s (k)
a†+s (k)
←−−−→
ωµν ◦ a−s (k)
= a†+s (k)
←−−−→
ωµν ◦ a−s (k)
a−s (k)
←−−−→
ωµν ◦ a†+s (k)
= −εa†+s (k)
←−−−→
ωµν ◦ a−s (k)
a†−s (k)
←−−−→
ωµν ◦ a+s (k)
= −εa+s (k)
←−−−→
ωµν ◦ a†−s (k).
(4.27)
As a consequence of these equalities, the action of N on the l.h.s. of (4.11) vanishes. Com-
bining this result with the mentioned fact that the normal ordering converts (4.12) and (4.13)
into identities, we see that the normal ordering procedure ensures also uniqueness of the spin
and orbital operators if we redefine them respectively as:
S̃µν := N ( S̃ ′µν) := N ( S̃ ′′µν) =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k) + εσ
ss′,+
µν (k)a
s′(k) ◦ a
s (k)
(4.28)
L̃µν := N ( L̃′µν) := N ( L̃′′µν) = x0µ P̃ν − x0 ν P̃µ +
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k)a
s (k) ◦ a−s′(k) + εl
ss′,+
µν (k)a
(k) ◦ a†−s (k)
2(1 + τ)
2j+1−δ0m(1−δ0j )
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
+ a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
m2c2+k2
(4.29)
where (3.14) was applied.
5. Heisenberg relations
The conserved operators, like momentum and charge operators, are often identified with the
generators of the corresponding transformations under which the action operator is invari-
ant [1, 3, 11, 12]. This leads to a number of commutation relations between the components
of these operators and between them and the field operators. The relations of the letter
set are known/referred as the Heisenberg relations or equations. Both kinds of commuta-
tion relations are from pure geometric origin and, consequently, are completely external to
the Lagrangian formalism; one of the reasons being that the mentioned identification is, in
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 15
general, unacceptable and may be carried out only on some subset of the system’s Hilbert
space of states [23, 24]. Therefore their validity in a pure Lagrangian theory is questionable
and should be verified [11]. However, the considered relations are weaker conditions than
the identification of the corresponding operators and there are strong evidences that these
relations should be valid in a realistic quantum field theory [1,11]; e.g., the commutativity be-
tween the momentum and charge operators (see below (5.18)) expresses the experimental fact
that the 4-momentum and charge of any system are simultaneously measurable quantities.
It is known [1,11], in a pure Lagrangian approach, the field equations, which are usually
identified with the Euler-Lagrange, 14 are the only restrictions on the field operators. Besides,
these equations do not determine uniquely the field operators and the letter can be expressed
through the creation and annihilation operators. Since the last operators are left completely
arbitrary by a pure Lagrangian formalism, one is free to impose on them any system of
compatible restrictions. The best known examples of this kind are the famous canonical
(anti)commutation relations and their generalization, the so-called paracommutation rela-
tions [16,18]. In general, the problem for compatibility of such subsidiary to the Lagrangian
formalism system of restrictions with, for instance, the Heisenberg relations is open and
requires particular investigation [11]. For example, even the canonical (anti)commutation
relations for electromagnetic field in Coulomb gauge are incompatible with the Heisenberg
equation involving the (total) angular momentum operator unless the gauge symmetry of this
field is taken into account [11, § 84]. However, the (para)commutation relations are, by con-
struction, compatible with the Heisenberg relations regarding momentum operator (see [16]
or below Subsect. 6.1). The ordinary approach is to be imposed a system of equations on
the creation and annihilation operators and, then, to be checked its compatibility with, e.g.,
the Heisenberg relations. In the next sections we shall investigate the opposite situation:
assuming the validity of (some of) the Heisenberg equations, the possible restrictions on
the creation and annihilation operators will be explored. For this purpose, below we briefly
review the Heisenberg relations and other ones related to them.
Consider a system of quantum fields ϕ̃i(x), i = 1, . . . , N ∈ N, where ϕ̃i(x) denote the
components of all fields (and their Hermitian conjugates), and P̃µ, Q̃ and M̃µν be its
momentum, charge and (total) angular momentum operators, respectively. The Heisenberg
relations/equations for these operators are [1, 3, 11,12]
[ ϕ̃i(x), P̃µ] = i~
∂ ϕ̃i(x)
(5.1)
[ ϕ̃i(x), Q̃] = e( ϕ̃i)q ϕ̃i(x) (5.2)
[ ϕ̃i(x), M̃µν ] = i~{xµ∂ν ϕ̃i(x)− xν∂µ ϕ̃i(x)}+ i~
ϕ̃i′(x). (5.3)
Here: q = const is the fields’ charge, e( ϕ̃i) = 0 if ϕ̃
i = ϕ̃i, e( ϕ̃i) = ±1 if ϕ̃
i 6= ϕ̃i with
e( ϕ̃i)+e( ϕ̃
i ) = 0, and the constants I
iµν = −Ii
iνµ characterize the transformation properties
of the field operators under 4-rotations. (If ε( ϕ̃i) 6= 0, it is a convention whether to put
ε( ϕ̃i) = +1 or ε( ϕ̃i) = −1 for a fixed i.)
We would like to make some comments on (5.3). Since its r.h.s. is a sum of two operators,
the first (second) characterizing the pure orbital (spin) angular momentum properties of
the system considered, the idea arises to split (5.3) into two independent equations, one
involving the orbital angular momentum operator and another concerning the spin angular
momentum operator. This is supported by the observation that, it seems, no process is known
for transforming orbital angular momentum into spin one and v.v. (without destroying the
14 Recall, there are Lagrangians whose classical Euler-Lagrange equations are identities. However, their
correct and rigorous treatment [22] reveals that they entail field equations which are mathematically correct
and physically sensible.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 16
system). So one may suppose the existence of operators M̃orµν and M̃
µν such that
[ ϕ̃i(x), M̃orµν ] = i~{xµ∂ν ϕ̃i(x)− xν∂µ ϕ̃i(x)} (5.4)
[ ϕ̃i(x), M̃spµν ] = i~
iµν ϕ̃i′(x) (5.5)
M̃µν = M̃orµν + M̃spµν . (5.6)
However, as particular calculations demonstrate [5,14,15], neither the spin (resp. orbital)
nor the spin (resp. orbital) angular momentum operator is a suitable candidate for M̃spµν
(resp. M̃orµν). If we assume the validity of (5.1), then equations (5.4) and (5.5) can be
satisfied if we choose
M̃orµν(x) = L̃extµν := xµ P̃ν − xν P̃µ (5.7)
M̃spµν(x) = M̃(0)µν (x) := M̃µν − L̃extµν = S̃µν + L̃µν − {xµ P̃ν − xν P̃µ} (5.8)
with M̃µν satisfying (5.3). These operators are not conserved ones. Such a representation
is in agreement with the equations (3.12), according to which the operator (5.7) enters addi-
tively in the expressions for the orbital operator.15 The physical sense of the operator (5.7)
is that it represents the orbital angular momentum of the system due to its movement as a
whole. Respectively, the operator (5.8) describes the system’s angular momentum as a result
of its internal movement and/or structure.
Since the spin (orbital) angular momentum is associated with the structure (movement)
of a system, in the operator (5.8) are mixed the spin and orbital angular momenta. These
quantities can be separated completely via the following representations of the operators
Morµν and M
µν in momentum picture (when (5.1) holds)
Morµν = xµ Pν − xµPµ + Lintµν (5.9)
Mspµν = Mµν − (xµ Pν − xµ Pµ)− Lintµν , (5.10)
where Lintµν describes the ‘internal’ orbital angular momentum of the system considered and
depends on the Lagrangian we have started off. Generally said, Lintµν is the part of the
orbital angular momentum operator containing derivatives of the creation and annihilation
operators. In particular, for the Lagrangians L′, L′′ and L′′′ (see Sect. 3), the explicit forms
of the operators (5.9) and (5.10) respectively are:
M′ orµν =xµP ′ν − xν P ′µ
2(1 + τ)
2j+1−δ0m(1−δ0j)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(5.11a)
M′′ orµν =xµP ′′ν − xν P ′′µ
2(1 + τ)
2j+1−δ0m(1−δ0j)
a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k)
m2c2+k2
(5.11b)
15 This is evident in the momentum picture of motion, in which xµ stands for x0µ in (3.12) — see [13–15].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 17
M′′′ orµν =xµP ′′′ν − xν P ′′′µ
4(1 + τ)
2j+1−δ0m(1−δ0j)
a†+s (k)
←−−−−−→
←−−−−−→
◦ a−s (k)
− εa−s (k)
←−−−−−→
←−−−−−→
◦ a†+s (k) + a+s (k)
←−−−−−→
←−−−−−→
◦ a†−s (k)
− εa†−s (k)
←−−−−−→
←−−−−−→
◦ a+s (k)
m2c2+k2
(5.11c)
M′ spµν =
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,−
µν (k))a
s (k) ◦ a−s′(k)
+ (σss
µν (k) + l
ss′,+
µν (k))a
s (k) ◦ a+s′(k)
(5.12a)
M′′ spµν = ε
(−1)j−1/2j~
1 + τ
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,+
µν (k))a
(k) ◦ a†−s (k)
+ (σss
µν (k) + σ
ss′,−
µν (k))a
(k) ◦ a†+s (k)
(5.12b)
M′′′ spµν =
(−1)j−1/2j~
2(1 + τ)
2j+1−δ0m(1−δ1j )
s,s′=1
µν (k) + l
ss′,−
µν (k))[a
s (k), a
s′(k)]ε
+ (σss
µν (k) + l
ss′,+
µν (k))[a
s (k), a
(k)]ε
(5.12c)
Obviously (see Sect. 2), the equations (5.12) have the same form in Heisenberg picture in
terms of the operators (2.9) (only tildes over M and a must be added), but the equa-
tions (5.11) change substantially due to the existence of derivatives of the creation and
annihilation operators in them [13–15]:
M̃′ orµν =
2(1 + τ)
2j+1−δ0m(1−δ0j )
ã†+s (k)
←−−−−−→
←−−−−−→
◦ ã−s (k)
− εã†−s (k)
←−−−−−→
←−−−−−→
◦ ã+s (k)
m2c2+k2
(5.13a)
M̃′′ orµν =
2(1 + τ)
2j+1−δ0m(1−δ0j )
ã+s (k)
←−−−−−→
←−−−−−→
◦ ã†−s (k)
− εã−s (k)
←−−−−−→
←−−−−−→
◦ ã†+s (k)
m2c2+k2
(5.13b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 18
M̃′′′ orµν =
4(1 + τ)
2j+1−δ0m(1−δ0j )
ã†+s (k)
←−−−−−→
←−−−−−→
◦ ã−s (k)
− εã−s (k)
←−−−−−→
←−−−−−→
◦ ã†+s (k) + ã+s (k)
←−−−−−→
←−−−−−→
◦ ã†−s (k)
− εã†−s (k)
←−−−−−→
←−−−−−→
◦ ã+s (k)
m2c2+k2
(5.13c)
From (5.13) and (5.12) is clear that the operators M̃orµν and M̃
µν so defined are conserved
(contrary to (5.7) and (5.8)) and do not depend on the validity of the Heisenberg rela-
tions (5.1) (contrary to expressions (5.11) in momentum picture).
The problem for whether the operators (5.12) and (5.13) satisfy the equations (5.4)
and (5.5), respectively, will be considered in Sect. 6.
There is an essential difference between (5.4) and (5.5): the equation (5.5) depends on
the particular properties of the operators ϕ̃i(x) under 4-rotations via the coefficients I
(see (5.25) below), while (5.4) does not depend on them. This is explicitly reflected in (5.11)
and (5.12): the former set of equations is valid independently of the geometrical nature of the
fields considered, while the latter one depends on it via the ‘spin’ (‘polarization’) functions
ss′,±
µν (k) and l
ss′,±
µν (k). Similar remark concerns (5.3), on one hand, and (5.1) and (5.2), on
another hand: the particular form of (5.3) essentially depends on the geometric properties
of ϕ̃i(x) under 4-rotations, the other equations being independent of them.
It should also be noted, the relation (5.3) does not hold for a canonically quantized
electromagnetic field in Coulomb gauge unless some additional terms it its r.h.s., reflecting
the gauge symmetry of the field, are taken into account [11, § 84].
As it was said above, the relations (5.1)–(5.3) are from pure geometrical origin. However,
the last discussion, concerning (5.4)–(5.8), reveals that the terms in braces in (5.3) should be
connected with the momentum operator in the (pure) Lagrangian approach. More precisely,
on the background of equations (3.11a)–(3.12c), the Heisenberg relation (5.3) should be
replaced with
[ ϕ̃i(x), M̃µν ] = xµ[ ϕ̃i(x), P̃ν ] − xν [ ϕ̃i(x), P̃µ] + i~
iµν ϕ̃i′(x), (5.14)
which is equivalent to (5.3) if (5.1) is true. An advantage of the last equation is that it is valid
in any picture of motion (in the same form) while (5.3) holds only in Heisenberg picture.16
Obviously, (5.14) is equivalent to (5.5) with M̃spµν defined by (5.8).
The other kind of geometric relations mentioned at the beginning of this section are
connected with the basic relations defining the Lie algebra of the Poincaré group [7, pp. 143–
147], [8, sect. 7.1]. They require the fulfillment of the following equations between the com-
ponents P̃µ of the momentum and M̃µν of the angular momentum operators [3, 5, 7, 8]:
[ P̃µ, P̃ν ] = 0 (5.15)
[M̃µν , P̃λ] = −i~(ηλµ P̃ν − ηλν P̃µ). (5.16)
[M̃κλ, M̃µν ] = −i~
ηκµ M̃λν − ηλµ M̃κν − ηκν M̃λµ + ηλν M̃κµ
. (5.17)
We would like to pay attention to the minus sign in the multiplier (−i~) in (5.16) and (5.17)
with respect to the above references, where i~ stands instead of −i~ in these equations. When
16 In other pictures of motion, generally, additional terms in the r.h.s. of (5.3) will appear, i.e. the functional
form of the r.h.s. of (5.3) is not invariant under changes of the picture of motion, contrary to (5.14).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 19
(a representation of) the Lie algebra of the Poincaré group is considered, this difference in the
sign is insignificant as it can be absorbed into the definition of M̃µν . However, the change
of the sign of the angular momentum operator, M̃µν 7→ −M̃µν , will result in the change
i~ 7→ −i~ in the r.h.s. of (5.3). This means that equations (5.15), (5.16) and (5.3), when
considered together, require a suitable choice of the signs of the multiplier i~ in their right
hand sides as these signs change simultaneously when M̃µν is replaced with −M̃µν . Since
equations (5.3), (5.16) and (5.17) hold, when M̃µν is defined according to the Noether’s
theorem and the ordinary (anti)commutation relations are valid [13–15], we accept these
equations in the way they are written above.
To the relations (5.15)–(5.17) should be added the equations [3, p. 78]
[ Q̃, P̃µ] = 0 (5.18)
[ Q̃, M̃µν ] = 0, (5.19)
which complete the algebra of observables and express, respectively, the translational and
rotational invariance of the charge operator Q̃; physically they mean that the charge and
momentum or the charge and angular momentum are simultaneously measurable quantities.
Since the spin properties of a system are generally independent of its charge or momentum,
one may also expect the validity of the relations17
[ S̃µν , P̃µ] = 0 (5.20)
[ S̃µν , Q̃] = 0. (5.21)
But, as the spin describes, in a sense, some of the rotational properties of the system,
equality like [ S̃µν , L̃κλ] = 0 is not likely to hold. Indeed, the considerations in [13–15] reveal
that (5.20) and (5.21), but not the last equation, are true in the framework of the Lagrangian
formalism with added to it standard (anti)commutation relations. Notice, if (5.20) and (5.21)
hold, then, respectively, (5.16) and (5.19) are equivalent to
[ L̃µν , P̃λ] = −i~(ηλµ P̃ν − ηλν P̃µ). (5.22)
[ Q̃, L̃µν ] = 0. (5.23)
It is intuitively clear, not all of the commutation relations (5.1)–(5.3) and (5.15)–(5.21)
are independent: if D̃ denotes some of the operators P̃µ, Q̃, M̃µν , S̃µν or L̃µν and the
commutators [ ϕ̃i(x), D̃] , i = 1, . . . , N , are known, then, in principle, one can calculate the
commutators [Γ( ϕ̃1(x), . . . , ϕ̃N (x)), D̃] , where Γ( ϕ̃1(x), . . . , ϕ̃N (x)) is, for example, any
function/functional bilinear in ϕ̃1(x), . . . , ϕ̃N (x); to prove this fact, one should apply the
identity [A,B ◦ C] = [A,B] ◦ C + B ◦ [A,C] a suitable number of times. In particular, if
D̃1 and D̃2 denote any two (distinct) operators of the dynamical variables, and [ ϕ̃i(x), D̃1]
is known, then the commutator [ D̃1, D̃2] can be calculated explicitly. For this reason, we
can expect that:
(i) Equation (5.1) implies (5.15), (5.16), (5.18), (5.20) and (5.22).
(ii) Equation (5.2) implies (5.18), (5.19), (5.21), and (5.23).
(iii) Equation (5.3) implies (5.16), (5.17), and (5.19).
Besides, (5.3) may, possibly, entail equations like (5.17) with S or L forM , with an exception
of M̃µν in the l.h.s., i.e.
[ S̃κλ, M̃µν ] = −i~
ηκµ S̃λν − ηλµ S̃κν − ηκν S̃λµ + ηλν S̃κµ
[ L̃κλ, M̃µν ] = −i~
ηκµ L̃λν − ηλµ L̃κν − ηκν L̃λµ + ηλν L̃κµ
} (5.24)
17 Recall, S̃µν (resp. L̃µν) is the conserved spin (resp. orbital) operator, not the generally non-conserved
spin (resp. orbital) angular momentum operator [23].
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 20
The validity of assertions (i)–(iii) above for free scalar, spinor and vector fields, when respec-
tively
ϕ̃i(x) 7→ ϕ̃(x), ϕ̃†(x) Ii
iµν 7→ Iµν = 0 e( ϕ̃) = −e( ϕ̃†) = +1 (5.25a)
ϕ̃i(x) 7→ ψ̃(x), ˜̆ψ(x) Ii
iµν 7→ Iψµν = Iψ̆µν = −
σµν e( ψ̃) = −e( ˜̆ψ) = +1 (5.25b)
ϕ̃i(x) 7→ Ũµ(x), Ũ†µ(x) Ii
iµν 7→ Iσρµν = I†σρµν = δσµηνρ − δσν ηµρ e( Ũµ) = −e( Ũ†µ) = +1,
(5.25c)
where σµν := i
[γµ, γν ] with γµ being the Dirac γ-matrices [1, 25], is proved in [13–15],
respectively. Besides, in loc. cit. is proved that equations (5.24) hold for scalar and vector
fields, but not for a spinor field.18
Thus, we see that the Heisenberg relations (5.1)–(5.3) are stronger than the commutation
relations (5.15)–(5.23), when imposed on the Lagrangian formalism as subsidiary restrictions.
6. Types of possible commutation relations
In a broad sense, by a commutation relation we shall understand any algebraic relation
between the creation and annihilation operators imposed as subsidiary restriction on the
Lagrangian formalism. In a narrow sense, the commutation relations are the equations (6.13),
with ε = −1, written below and satisfied by the bose creation and annihilation operators. As
anticommutation relations are known the equations (6.13), with ε = +1, written below and
satisfied by the fermi creation and annihilation operators. The last two types of relations
are often referred as the bilinear commutation relations [18]. Theoretically are possible also
trilinear commutation relations, an example being the paracommutation relations [16, 18]
represented below by equations (6.18) (or (6.20)).
Generally said, the commutation relations should be postulated. Alternatively, they
could be derived from (equivalent to them) different assumptions added to the Lagrangian
formalism. The purpose of this section is to be explored possible classes of commutation
relations, which follow from some natural restrictions on the Lagrangian formalism that are
consequences from the considerations in the previous sections. Special attention will be paid
on some consequences of the charge symmetric Lagrangians as the free fields possess such a
symmetry [1, 3, 11,12].
As pointed in Sect 3, the Euler-Lagrange equations for the Lagrangians L̃′, L̃′′ and L̃′′′
coincide and, in quantum field theory, the role of these equations is to be singled out the
independent degrees of freedom of the fields in the form of creation and annihilation operators
a±s (k) and a
s (k) (which are identical for L̃′, L̃′′ and L̃′′′). Further specialization of these
operators is provided by the commutation relations (in broad sense) which play a role of field
equations in this situation (with respect to the mentioned operators).
Before proceeding on, we would like to simplify our notation. As a spin variable, s say, is
always coupled with a 3-momentum one, k say, we shall use the letters l, m and n to denote
pairs like l = (s,k), m = (t,p) and n = (r, q). Equipped with this convention, we shall write,
e.g., a±l for a
s (k) and a
l for a
s (k). We set δlm := δstδ
3(k−p) and a summation sign like
l should be understood as
d3k, where the range of the polarization variable s will
be clear from the context (see, e.g., (3.9)–(3.12)).
18 The problem for the validity of assertions (i)–(iii) or equations (5.24) in the general case of arbitrary
fields (Lagrangians) is not a subject of the present work.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 21
6.1. Restrictions related to the momentum operator
First of all, let us examine the consequences of the Heisenberg relation (5.1) involving the
momentum operator. Since in terms of creation and annihilation operators it reads [1,13–15]
[a±s (k), Pµ] = ∓kµa±s (k) [a†±s (k), Pµ] = ∓kµa†±s (k) k0 =
m2c2 + k2, (6.1)
the field equations in terms of creation and annihilation operators for the Lagrangians (3.1),
(3.3) and (3.4) respectively are (see [13–15] or (6.1) and (3.9)):
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), a
t (q) ◦ a−t (q) + εa
t (q) ◦ a+t (q)
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.2a)
2j+1−δ0m(1−δ0j )
m2c2+q2
a†±s (k), a
t (q) ◦ a−t (q) + εa
t (q) ◦ a+t (q)
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0
(6.2b)
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), a
t (q) ◦ a
t (q) + εa
t (q) ◦ a
t (q)
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.3a)
2j+1−δ0m(1−δ0j )
m2c2+q2
a†±s (k), a
t (q) ◦ a
t (q) + εa
t (q) ◦ a
t (q)
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0
(6.3b)
2j+1−δ0m(1−δ0j )
m2c2+q2
a±s (k), [a
t (q), a
t (q)]ε + [a
t (q), a
t (q)]ε
± (1 + τ)a±s (k)δstδ3(k − q)
d3q = 0
(6.4a)
2j+1−δ0m(1−δ0j)
m2c2+q2
a†±s (k), [a
t (q), a
t (q)]ε + [a
t (q), a
t (q)]ε
± (1 + τ)a†±s (k)δstδ3(k − q)
d3q = 0,
(6.4b)
where j and ε are given via (3.7), the generalized commutation function [·, ·]ε is defined
by (4.14), and the polarization indices take the values
s, t = 1, . . . , 2j + 1− δ0m(1− δ0j) =
1 for j = 0 or for j = 1
and m = 0
1, 2 for j = 1
and m 6= 0 or for j = 1 and m = 0
1, 2, 3 for j = 1 and m 6= 0
(6.5)
The “b” versions of the equations (6.2)–(6.4) are consequences of the “a” versions and the
equalities
(a±l )
† = a
l ) = a
l (6.6)
[A,B]η
= η[A†, B†]η for [A,B]η = η[B,A]η η = ±1. (6.7)
Applying (6.2)–(6.4) and the identity
[A,B ◦ C] = [A,B]η ◦ C − ηB ◦ [A,C]η for η = ±1 (6.8)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 22
for the choice η = −1, one can prove by a direct calculation that
[ P̃µ, P̃ν ] = 0 [ Q̃, P̃µ] = 0 [ S̃µν , P̃λ] = 0
[ L̃µν , P̃λ] = −i~{ηλµ P̃ν − ηλν P̃µ} [M̃µν , P̃λ] = −i~{ηλµ P̃ν − ηλν P̃µ},
(6.9)
where the operators P̃µ, Q̃, S̃µν , L̃µν , and M̃µν denote the momentum, charge, spin,
orbital and total angular momentum operators, respectively, of the system considered and
are calculated from one and the same initial Lagrangian. This result confirms the supposition,
made in Sect. 5, that the assertion (i) before (5.24) holds for the fields investigated here.
Below we shall study only those solutions of (6.2)–(6.4) for which the integrands in them
vanish, i.e. we shall replace the systems of integral equations (6.2)–(6.4) with the following
systems of algebraic equations (see the above convention on the indices l and m and do not
sum over indices repeated on one and the same level):
a±l , a
m ◦ a−m + εa†−m ◦ a+m
± (1 + τ)δlma±l = 0 (6.10a)
l , a
m ◦ a−m + εa†−m ◦ a+m
± (1 + τ)δlma†±l = 0 (6.10b)
a±l , a
m ◦ a†−m + εa−m ◦ a†+m
± (1 + τ)δlma±l = 0 (6.11a)
l , a
m ◦ a†−m + εa−m ◦ a†+m
± (1 + τ)δlma†±l = 0 (6.11b)
a±l , [a
m , a
m]ε + [a
± 2(1 + τ)δlma±l = 0 (6.12a)
, [a†+m , a
m]ε + [a
± 2(1 + τ)δlma†±l = 0. (6.12b)
It seems, these are the most general and sensible trilinear commutation relations one may
impose on the creation and annihilation operators.
First of all, we should mentioned that the standard bilinear commutation relations, viz. [1,
3, 11–15]
[a±l , a
m]−ε = 0 [a
l , a
m ]−ε = 0
[a∓l , a
m]−ε = (±1)2j+1τδlm idF [a
l , a
m ]−ε = (±1)2j+1τδlm idF
[a±l , a
m ]−ε = 0 [a
l , a
m]−ε = 0
, a†±m ]−ε = (±1)2j+1δlm idF [a
, a±m]−ε = (±1)2j+1δlm idF , (6.13)
provide a solution of any one of the equations (6.10)–(6.12) in a sense that, due to (3.7)
and (6.8), with η = −ε any set of operators satisfying (6.13) converts (6.10)–(6.12) into
identities.
Besides, this conclusion remains valid also if the normal ordering is taken into account,
i.e. if, in this particular case, the changes a
m ◦ a+m 7→ εa+m ◦ a
m and a
m ◦ a
m 7→ εa†+m ◦ a−m
are made in (6.10)–(6.12).
Now we shall demonstrate how the trilinear relations (6.12) lead to the paracommuta-
tion relations. Equations (6.12) can be ‘split’ into different kinds of trilinear commutation
relations into infinitely many ways. For example, the system of equations
a±l , [a
± (1 + τ)δlma±l = 0 (6.14a)
a±l , [a
m , a
± (1 + τ)δlma±l = 0 (6.14b)
, [a+m, a
± (1 + τ)δlma†±l = 0 (6.14c)
l , [a
m , a
± (1 + τ)δlma†±l = 0 (6.14d)
provides an evident solution of (6.12). However, it is a simple algebra to be seen that
these relations are incompatible with the standard (anti)commutation relations (6.13) and,
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 23
in this sense, are not suitable as subsidiary restrictions on the Lagrangian formalism. For
our purpose, the equations
a+l , [a
+ 2δlma
l = 0 (6.15a)
a+l , [a
m , a
+ 2τδlma
l = 0 (6.15b)
a−l , [a
− 2τδlma−l = 0 (6.15c)
a−l , [a
m , a
− 2δlma−l = 0 (6.15d)
and their Hermitian conjugate provide a solution of (6.12), which is compatible with (6.13),
i.e. if (6.13) hold, the equations (6.15) are converted into identities.
The idea of the paraquantization is in the following generalization of (6.15)
a+l , [a
+ 2δlna
m = 0 (6.16a)
a+l , [a
m , a
+ 2τδlna
m = 0 (6.16b)
a−l , [a
− 2τδlma−n = 0 (6.16c)
a−l , [a
m , a
− 2δlma−n = 0 (6.16d)
which reduces to (6.15) for n = m and is a generalization of (6.13) in a sense that any set
of operators satisfying (6.13) converts (6.16) into identities, the opposite being generally not
valid.19
Suppose that the field considered consists of a single sort of particles, e.g. electrons or
photons, created by b
and annihilated by bl := a
. Then the equation Hermitian
conjugated to (6.15a) reads
[bl, [b
m, bm]ε] = 2δlmbm. (6.17)
This is the main relation from which the paper [16] starts. The basic paracommutation
relations are [16–18,26]:
[bl, [b
m, bn]ε] = 2δlmbn (6.18a)
[bl, [bm, bn]ε] = 0. (6.18b)
The first of them is a generalization (stronger version) of (6.17) by replacing the second index
m with an arbitrary one, say n, and the second one is added (by ”hands”) in the theory as
an additional assumption. Obviously, (6.18) are a solution of (6.15) and therefore of (6.12)
in the considered case of a field consisting of only one sort of particles.
The equations (6.15) contain also the relativistic version of the paracommutation rela-
tions, when the existence of antiparticles must be respected [18, sec. 18.1]. Indeed, noticing
that the field’s particles (resp. antiparticles) are created by b
:= a+
(resp. c
) and
annihilated by bl := a
(resp. cl := a
), from (6.15) and the Hermitian conjugate to them
equations, we get
[bl, [b
m, bm]ε] = 2δlmbm [cl, [c
m, cm]ε] = 2δlmcm (6.19a)
, [c†m, cm]ε] = −2τδlmb†m [c
, [b†m, bm]ε] = −2τδlmc†m. (6.19b)
Generalizing these equations in a way similar to the transition from (6.17) to (6.18), we
obtain the relativistic paracommutation relations as (cf. (6.16))
[bl, [b
m, bn]ε] = 2δlmbn [bl, [bm, bn]ε] = 0 (6.20a)
[cl, [c
m, cn]ε] = 2δlmcn [cl, [cm, cn]ε] = 0 (6.20b)
l , [c
m, cn]ε] = −2τδlnb†m [c
l , [b
m, bn]ε] = −2τδlnc†m. (6.20c)
19 Other generalizations of (6.15) are also possible, but they do not agree with (6.13). Moreover, it is easy
to be proved, any other (non-trivial) arrangement of the indices in (6.16) is incompatible with (6.13).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 24
The equations (6.20a) (resp. (6.20b)) represent the paracommutation relations for the field’s
particles (resp. antiparticles) as independent objects, while (6.20c) describe a pure relativistic
effect of some “interaction” (or its absents) between field’s particles and antiparticles and
fixes the paracommutation relations involving the bl’s and cl’s, as pointed in [18, p. 207]
(where bl is denoted by al and cl by bl). The relations (6.17) and (6.20) for ε = +1 (resp.
ε = −1) are referred as the parabose (resp. parafermi) commutation relations [18]. This
terminology is a natural one also with respect to the commutation relations (6.16), which
will be referred as the paracommutation relations too.
As first noted in [16], the equations (6.13) provide a solution of (6.20) (or (6.18) in the
nonrelativistic case) but the latter equations admit also an infinite number of other solutions.
Besides, by taking Hermitian conjugations of (some of) the equations (6.18) or (6.20) and
applying generalized Jacobi identities, like
α[[A,B]ξ , C]η + ξη[[A,C]−α/ξ , B]−α/η − α2[[B,C]ξη/α, A]1/α = 0 αξη 6= 0
β[A, [B,C]α, ]−βγ + γ[B, [C,A]β , ]−γα + α[C, [A,B]γ , ]−αβ = 0 α, β, γ = ±1
[[A,B]η, C]− + [[B,C]η, A]− + [[C,A]η , B]− = 0 η = ±1
[[A,B]ξ, [C,D]η ]− = [[A,B]ξ , C]−,D]η + η[[A,B]ξ ,D]−, C]1/η η 6= 0,
(6.21)
one can obtain a number of other (para)commutation relations for which the reader is referred
to [16,18,26].
Of course, the paracommutation relations (6.16), in particular (6.18) and (6.20) as their
stronger versions, do not give the general solution of the trilinear relations (6.12). For
instance, one may replace (6.12) with the equations
a+l , [a
m , a
n ]ε + [a
+ 2(1 + τ)δlna
m = 0 (6.22a)
a−l , [a
m , a
n ]ε + [a
− 2(1 + τ)δlma−n = 0. (6.22b)
and their Hermitian conjugate, which in terms of the operators bl and cl introduced above
[bl, [b
m, bn]ε + [c
m, cm]ε] = 2(1 + τ)δlmbn (6.23a)
[cl, [b
m, bn]ε + [c
m, cm]ε] = 2(1 + τ)δlmcn, (6.23b)
and supplement these relations with equations like (6.18b). Obviously, equations (6.16) con-
vert (6.22) into identities and, consequently, the (standard) paracommutation relations (6.20)
provide a solution of (6.23). On the base of (6.23) or other similar equations that can be
obtained by generalizing the ones in (6.10)–(6.12), further research on particular classes of
trilinear commutation relations can be done, but, however, this is not a subject of the present
work.
Let us now pay attention to the fact that equations (6.10), (6.11) and (6.12) are generally
different (regardless of existence of some connections between their solutions). The cause for
this being that the momentum operators for the Lagrangians L′, L′′ and L′′′ are generally
different unless some additional restrictions are added to the Lagrangian formalism (see
Sect. 4). A necessary and sufficient condition for (6.10)–(6.12) to be identical is
[a±l , [a
m , a
m]−ε − [a+m, a†−m ]−ε] = 0, (6.24)
which certainly is valid if the condition (4.9′), viz.
[a†+m , a
m]−ε − [a+m, a†−m ]−ε = 0, (6.25)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 25
ensuring the uniqueness of the momentum operator are, holds. If one adopts the standard
bilinear commutation relations (6.13), then (6.25), and hence (6.24), is identically valid, but
in the framework of, e.g., the paracommutation relations (6.16) (or (6.20) in other form) the
equations (6.25) should be postulated to ensure uniqueness of the momentum operator and
therefore of the field equations.
On the base of (6.10) or (6.11) one may invent other types of commutation relations,
which will not be investigated in this paper because we shall be interested mainly in the
case when (6.10), (6.11) and (6.12) are identical (see (6.24)) or, more generally, when the
dynamical variables are unique in the sense pointed in Sect. 4.
6.2. Restrictions related to the charge operator
The consequences of the Heisenberg relations (5.2), involving the charge operator for a charged
field, q 6= 0 (and hence τ = 0 – see (3.7)), will be examined in this subsection. In terms of
creation and annihilation operators it is equivalent to [1, 13–15]
[a±s (k), Q] = qa±s (k) [a†±s (k), Q] = −qa†±s (k), (6.26)
the values of the polarization indices being specified by (6.5). Substituting here (3.10), we
see that, for a charged field, the field equations for the Lagrangians L′, L′′ and L′′′ (see
Sect. 3) respectively are:
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), a
t (p) ◦ a−t (p)− εa
t (p) ◦ a+t (p)] − a±s (k)δstδ3(k − p)} = 0
(6.27a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), a
t (p) ◦ a−t (p)− εa
t (p) ◦ a+t (p)] + a†±s (k)δstδ3(k − p)} = 0
(6.27b)
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), a+t (p) ◦ a
t (p)− εa−t (p) ◦ a
t (p)] + a
s (k)δstδ
3(k − p)} = 0
(6.28a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), a+t (p) ◦ a
t (p)− εa−t (p) ◦ a
t (p)] − a†±s (k)δstδ3(k − p)} = 0
(6.28b)
2j+1−δ0m(1−δ0j )
d3p{[a±s (k), [a
t (p), a
t (p)]ε − [a+t (p), a
t (p)ε] − 2a±s (k)δstδ3(k − p)} = 0
(6.29a)
2j+1−δ0m(1−δ0j )
d3p{[a†±s (k), [a
t (p), a
t (p)]ε − [a+t (p), a
t (p)ε] + 2a
s (k)δstδ
3(k − p)}=0.
(6.29b)
Using (6.27)–(6.29) and (6.8), with η = ε = −1, or simply (6.26), one can easily verify
the validity of the equations
[ P̃µ, Q̃] = 0 [ L̃µν , Q̃] = 0
[ S̃µν , Q̃] = 0 [M̃µν , Q̃] = 0,
(6.30)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 26
where the operators P̃µ, Q̃, S̃µν , L̃µν and M̃µν are calculated from one and the same
initial Lagrangian according to (3.9)–(3.12). This result confirms the validity of assertion (ii)
before (5.24) for the fields considered.
Following the above considerations, concerning the momentum operator, we shall now
replace the systems of integral equations (6.27)–(6.29) with respectively the following stronger
systems of algebraic equations (by equating to zero the integrands in (6.27)–(6.29)):
a±l , a
m ◦ a−m − εa†−m ◦ a+m
− δlma±l = 0 (6.31a)
, a†+m ◦ a−m − εa†−m ◦ a+m
+ δlma
= 0 (6.31b)
a±l , a
m ◦ a†−m − εa−m ◦ a†+m
+ δlma
l = 0 (6.32a)
, a+m ◦ a†−m − εa−m ◦ a†+m
− δlma†±l = 0 (6.32b)
a±l , [a
m , a
m]ε − [a+m, a†−m ]ε
− 2δlma±l = 0 (6.33a)
, [a†+m , a
m]ε − [a+m, a†−m ]ε
+ 2δlma
= 0. (6.33b)
These trilinear commutation relations are similar to (6.10)–(6.12) and, consequently, can be
treated in analogous way.
By invoking (6.8), it is a simple algebra to be proved that the standard bilinear commu-
tation relations (6.13) convert (6.31)–(6.33) into identities. Thus (6.13) are stronger version
of (6.31)–(6.33) and, in this sense, any type of commutation relations, which provide a
solution of (6.31)–(6.33) and is compatible with (6.13), is a suitable candidate for general-
izing (6.13). To illustrate that idea, we shall proceed with (6.33) in a way similar to the
‘derivation’ of the paracommutation relations from (6.12).
Obviously, the equations (cf. (6.14) with τ = 0, as now q 6= 0)
, [a+m, a
m ]ε] + δlma
m = 0 (6.34a)
, [a†+m , a
m]ε] − δlma±m = 0 (6.34b)
and their Hermitian conjugate provide a solution of (6.33), but, as a direct calculations shows,
they do not agree with the standard (anti)commutation relations (6.13). A solution of (6.33)
compatible with (6.13) is given by the equations (6.15), with τ = 0 as the field considered is
charged one — see (3.7). Therefore equations (6.16), with τ = 0, also provide a compatible
with (6.13) solution of (6.33), from where immediately follows that the paracommutation
relations (6.20), with τ = 0, convert (6.33) into identities. To conclude, we can say that the
paracommutation relations (6.20), in particular their special case (6.13), ensure the simul-
taneous validity of the Heisenberg relations (5.1) and (5.2) for free scalar, spinor and vector
fields.
Similarly to (6.22), one may generalize (6.33) to
a+l , [a
m , a
n ]ε − [a+m, a†−n ]ε
− 2δlna+m = 0 (6.35a)
a−l , [a
m , a
n ]ε − [a+m, a†−n ]ε
− 2δlma−n = 0. (6.35b)
which equations agree with (6.13), (6.15), (6.16) and (6.20), but generally do not agree
with (6.22), with τ = 0, unless the equations (6.16), with τ = 0, hold.
More generally, we can assert that (6.33) and (6.12), with τ = 0, hold simultaneously if
and only if (6.15), with τ = 0, is fulfilled. From here, again, it follows that the paracommu-
tation relations ensure the simultaneous validity of (5.1) and (5.2).
Let us say now some words on the uniqueness problem for the Heisenberg equations
involving the charge operator. The systems of equations (6.31)–(6.33) are identical iff
a±l , [a
m , a
m]−ε + [a
m ]−ε
= 0, (6.36)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 27
which, in particular, is satisfied if the condition
[a†+m , a
m]−ε + [a
m ]−ε = 0, (6.37)
ensuring the uniqueness of the charge operator (see (4.10′)), is valid. Evidently, equa-
tions (6.36) and (6.24) are compatible iff
a+l , [a
m , a
a−l , [a
m , a
= 0 (6.38)
which is a weaker form of (4.15) ensuring simultaneous uniqueness of the momentum and
charge operator.
6.3. Restrictions related to the angular momentum operator(s)
It is now turn to be investigated the restrictions on the creation and annihilation operators
that follow from the Heisenberg relations (5.3) concerning the angular momentum operator.
They can be obtained by inserting the equations (3.11) and (3.12) into (5.3). As pointed
in Sect. 5, the resulting equalities, however, depend not only on the particular Lagrangian
employed, but also on the geometric nature of the field considered; the last dependence
being explicitly given via (5.25) and the polarization functions σss
µν (k) and l
ss′m±
µν (k) (see
also (3.14)).
Consider the terms containing derivatives in (5.3),
L̃orµν := i~
ϕ̃i(x). (6.39)
If ϕ̃
(k) denotes the Fourier image of ϕ̃i(x), i.e.
ϕ̃i(x) = Λ
d4ke−
kµxµ ϕ̃
(k), (6.40)
with Λ being a normalization constant, then the Fourier image of (6.39) is
(k). (6.41)
Comparing this expression with equations (3.12), we see that the terms containing derivatives
in (3.12) should be responsible for the term (6.39) in (5.3).20 For this reason, we shall suppose
that the momentum operator M̃µν admits a representation
M̃µν = M̃orµν + M̃spµν (6.42)
such that the operators M̃orµν and M̃
µν satisfy the relations (5.4) and (5.5), respectively.
Thus we shall replace (5.3) with the stronger system of equations (5.4)–(5.5). Besides,
we shall admit that the explicit form of the operatorsM̃orµν and M̃
µν are given via (5.13)
and (5.12) for the fields investigated in the present work.
Let us consider at first the ‘orbital’ Heisenberg relations (5.4), which is independent
of the particular geometrical nature of the fields studied. Substituting (5.13) and (6.40)
into (5.4), using that ϕ̃
(±k), with k2 = m2c2, is a linear combination of ã±s (k) with classical,
not operator-valued, functions of k as coefficients [1, 13–15] and introducing for brevity the
operator
ωµν(k) := kµ
, (6.43)
20 The terms proportional to the momentum operator in (3.12) disappear if the creation and annihilation
operators (2.9) in Heisenberg picture are employed (see also [13–15]).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 28
we arrive to the following integro-differential systems of equations:
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), ã
t (p) ◦ ã−t (q)
− εã†−t (p) ◦ ã+t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.44a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ㆠ±s (k), ã
t (p) ◦ ã−t (q)
− εã†−t (p) ◦ ã+t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.44b)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), ã+t (p) ◦ ã
t (q)
− εã−t (p) ◦ ã
t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.45a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ㆠ±s (k), ã+t (p) ◦ ã
t (q)
− εã−t (p) ◦ ã
t (q)] )
m2c2+p2
= 2(1 + τ)ωµν(k)(ã
s (k)) (6.45b)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ã±s (k), [ã
t (p), ã
t (q)]ε
+ [ã+t (p), ã
t (q)]ε] )
m2c2+p2
= 4(1 + τ)ωµν(k)(ã
s (k)) (6.46a)
2j+1−δ0m(1−δ0j )
(−ωµν(p) + ωµν(q))([ㆠ±s (k), [ã
t (p), ã
t (q)]ε
+ [ã+t (p), ã
t (q)]ε] )
m2c2+p2
= 4(1 + τ)ωµν(k)(ã
s (k)), (6.46b)
where k0 =
m2c2 + k2 is set after the differentiations are performed (see (6.43)). Follow-
ing the procedure of the previous considerations, we replace the integro-differential equa-
tions (6.44)–(6.46) with the following differential ones:
(−ω◦µν(m) + ω◦µν(n))([ã±l , ã
m ◦ ã−n − εã†−m ◦ ã+n ] )
= 2(1 + τ)δlmω
µν(l)(ã
l ) (6.47a)
(−ω◦µν(m)+ω◦µν(n))([ã
l , ã
m ◦ ã−n − εã†−m ◦ ã+n ] )
= 2(1+ τ)δlmω
µν(l)(ã
l ) (6.47b)
(−ω◦µν(m) + ω◦µν(n))([ã±l , ã
m ◦ ã†−n − εã−m ◦ ã†+n ] )
= 2(1 + τ)δlmω
µν(l)(ã
) (6.48a)
(−ω◦µν(m)+ω◦µν(n))([ã
l , ã
m ◦ ã†−n − εã−m ◦ ã†+n ] )
= 2(1+ τ)δlmω
µν(l)(ã
l ) (6.48b)
(−ω◦µν(m) + ω◦µν(n))([ã±l , [ã
m , ã
n ]ε + [ã
m, ã
n ]ε] )
= 4(1 + τ)δlmω
µν(l)(ã
l ) (6.49a)
(−ω◦µν(m) + ω◦µν(n))([ã
, [ã†+m , ã
n ]ε + [ã
m, ã
n ]ε] )
= 4(1 + τ)δlmω
µν(l)(ã
(6.49b)
where we have set (cf. (6.43))
ω◦µν(l) := ωµν(k) = kµ
if l = (s,k) (6.50)
and k0 =
m2c2 + k2 is set after the differentiations are performed.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 29
Remark. Instead of (6.47)–(6.49) one can write similar equations in which the operator
−ω◦µν(m) or +ω◦µν(n) is deleted and the factor +12 or −
, respectively, is added on their right
hand sides. These manipulations correspond to an integration by parts of some of the terms
in (6.44)–(6.46).
The main difference of the obtained trilinear relations with respect to the previous ones
considered above is that they are partial differential equations of first order.
The relations (6.49) agree with the equations (6.16) in a sense that if (6.16) hold,
then (6.49) become identically valid. Indeed, since
(−ω◦µν(m) + ω◦µν(n))(ã±mδln)
= −2δlmω◦µν(m)(ã±m)
(−ω◦µν(m) + ω◦µν(n))(ã±n δlm)
= +2δlmω
µν(m)(ã
(6.51)
due to (6.50), (6.43) and the equality
dδ(x)
f(x) = −δ(x)df(x)
for a C1 function f , the
application of the operator (−ω◦µν(m) + ω◦µν(n)) to (6.16) and subsequent setting n = m
entails (6.49). In particular, this means that the paracommutation relations (6.20) and,
moreover, the standard (anti)commutation relations (6.13) convert (6.49) into identities.
Therefore the ‘orbital’ Heisenberg relations (5.4) hold for scalar, spinor and vector fields
satisfying the bilinear or para commutation relations.
It should be noted, the paracommutation relations are not the only trilinear commutation
relations that are solutions of (6.49). As an example, we shall present the trilinear relations
a+l , [a
a+l , [a
m , a
= −(1 + τ)δlna+m (6.52a)
a−l , [a
a−l , [a
m , a
= +(1 + τ)δlma
n , (6.52b)
which reduce to (6.14) for n = m, do not agree with (6.13), but convert (6.49) into identities
(see (6.51)). Other example is provided by the equations (6.22), which are compatible with
the paracommutation relations and, as a result of (6.51), convert (6.49) into identities. Prima
facie one may suppose that any solution of (6.12) provides a solution of (6.49), but this is
not the general case. A counterexample is provided by the commutation relations
a±l , [a
m , a
n ]ε + [a
± 2(1 + τ)δlna±m = 0, (6.53)
which reduce to (6.12) for n = m, satisfy (6.49) with ã+l for ã
l , and do not satisfy (6.49)
with ã−l for ã
l (see (6.51) and cf. (6.22)).
From (5.13) follows that the operator M̃orµν is independent of the Lagrangian L′, L′′ or
L′′′ one starts off if and only if (see (4.11))
(−ω◦µν(m) + ω◦µν(n))
[ã†+m , ã
n ]−ε − [ã+m, ã†−n ]−ε
= 0. (6.54)
This condition ensures the coincidence of the systems of equations (6.47), (6.48) and (6.49)
too. However, the following necessary and sufficient condition for the coincidence of these
systems is expressed by the weaker equations
(−ω◦µν(m) + ω◦µν(n))
ã±l , [ã
m , ã
n ]−ε − [ã+m, ã†−n ]−ε
= 0. (6.55)
It is now turn to be considered the ‘spin’ Heisenberg relations (5.5).
Recall, the field operators ϕi for the fields considered here admit a representation [13–15]
ϕi = Λ
i (p)a
t (p) + v
i (p)a
t (p)
, (6.56)
where Λ is a normalization constant and v
i (p) are classical, not operator-valued, complex
or real functions which are linearly independent. The particular definition of v
i (p) depends
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 30
on the geometrical nature of ϕi and can be found in [13–15] (see also [1]), where the reader
can find also a number of relations satisfied by v
i (p). Here we shall mention only that
i (p) = 1 for a scalar field and v
i (p) = v
i (p) =: v
i(p) = (v
i(p))
∗ for a vector field.
The explicit form of the polarization functions σ
ss′,±
µν (k) and l
ss′,±
µν (k) (see Sect. 3, in
particular (3.14)) through v
i (k) are [13–15]:
µν (k) =
(−1)j
j + δj0
i (k))
∗Iii′µνv
µν (k) =
(−1)j
2j + δj0
i (k))
←−−−−−→
←−−−−−→
i (k),
(6.57)
with an exception that σ
ss′,±
0a (k) = σ
ss′,±
a0 (k) = 0, a = 1, 2, 3, for a spinor field, j =
, [14].
Evidently, the equations (3.14) follow from the mentioned facts (see also (5.25)).
Substituting (6.56) and (5.12) into (5.5), we obtain the following systems of integral
equations (corresponding respectively to the Lagrangians L′, L′′ and L′′′):
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))[a
t (p), a
s (k) ◦ a−s′(k)]
+ (σss
µν (k) + l
ss′,+
µν (k))[a
t (p), a
s (k) ◦ a+s′(k)]
d3pIi
(p)a±t (p) (6.58)
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,+
µν (k))[a
t (p), a
(k) ◦ a†−s (k)]
+ (σss
µν (k) + l
ss′,−
µν (k))[a
t (p), a
(k) ◦ a†+s (k)]
d3pIi
(p)a±t (p) (6.59)
(−1)j+1j
2(1 + τ)
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))
a±t (p), [a
s (k), a
(k)]ε
+ (σss
µν (k) + l
ss′,+
µν (k))
a±t (p), [a
s (k), a
(k)]ε
d3pIi
(p)a±t (p).
(6.60)
For the difference of all previously considered systems of integral equations, like (6.2)–
(6.4), (6.27)–(6.29) and (6.44)–(6.46), the systems (6.58)–(6.60) cannot be replaced by ones
consisting of algebraic (or differential) equations. The cause for this state of affairs is that
in (6.58)–(6.60) enter polarization modes with arbitrary s and s′ and, generally, one cannot
‘diagonalize’ the integrand(s) with respect to s and s′; moreover, for a vector field, the modes
with s = s′ are not presented at all (see (3.14)). That is why no commutation relations can
be extracted from (6.58)–(6.60) unless further assumptions are made. Without going into
details, below we shall sketch the proof of the assertion that the commutation relations (6.16)
convert (6.60) into identities for massive spinor and vector fields.21 In particular, this entails
that the paracommutation and the bilinear commutation relations provide solutions of (6.60).
Let (6.16) holds. Combining it with (6.60), we see that the latter splits into the equations
21 The equations (6.58)–(6.60) are identities for scalar fields as for them Iµν = 0 and v
i (k) = 1, which
reflects the absents of spin for these fields.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 31
(−1)jj
1 + τ
i (p)
τ(σst,−µν (p) + l
µν (p)) + ε(σ
µν (p) + l
µν (p))
a+s (p),
(p)a+s (p) (6.61a)
(−1)j+1j
1 + τ
i (p)
(σts,−µν (p) + l
µν (p)) + ετ(σ
µν (p) + l
µν (p))
a−s (p),
i′ (p)a
s (p). (6.61b)
Inserting here (6.57), we see that one needs the explicit definition of v
i (k) and formulae for
sums like ρii′(k) :=
i (k)(v
(k))∗, which are specific for any particular field and can
be found in [13–15]. In this way, applying (5.25), (3.7) and the mentioned results from [13–15],
one can check the validity of (6.61) for massive fields in a way similar to the proof of (5.3)
in [13–15] for scalar, spinor and vector fields, respectively.
We shall end the present subsection with the remark that the equations (4.17) and (4.18),
which together with (4.15) ensure the uniqueness of the spin and orbital operators, are
sufficient conditions for the coincidence of the equations (6.58), (6.59) and (6.60).
7. Inferences
To begin with, let us summarize the major conclusions from Sect. 6. Each of the Heisenberg
equations (5.1)–(5.3), the equations (5.3) being split into (5.4) and (5.5), induces in a natural
way some relations that the creation and annihilation operators should satisfy. These rela-
tions can be chosen as algebraic trilinear ones in a case of (5.1) and (5.2) (see (6.10)–(6.12)
and (6.31)–(6.33), respectively). But for (5.4) and (5.5) they need not to be algebraic and
are differential ones in the case of (5.4) (see (6.47)–(6.49)) and integral equations in the case
of (5.5) (see (6.58)–(6.60)). It was pointed that the cited relations depend on the initial
Lagrangian from which the theory is derived, unless some explicitly written conditions hold
(see (6.24), (6.37) and (6.55)); in particular, these conditions are true if the equations (4.9)–
(4.13), ensuring the uniqueness of the corresponding dynamical operators, are valid. Since
the ‘charge symmetric’ Lagrangians (3.4) seem to be the ones that best describe free fields,
the arising from them (commutation) relations (6.12), (6.33), (6.49) and (6.60) were stud-
ied in more details. It was proved that the trilinear commutation relations (6.16) convert
them into identities, as a result of which the same property possess the paracommutation
relations (6.20) and, in particular, the bilinear commutation relations (6.13). Examples of tri-
linear commutation relations, which are neither ordinary nor para ones, were presented; some
of them, like (6.14), (6.34) and (6.52), do not agree with (6.13) and other ones, like (6.16),
(6.22) and (6.35), generalize (6.20) and hence are compatible with (6.13). At last, it was
demonstrated that the commutators between the dynamical variables (see (5.15)–(5.23)) are
uniquely defined if a Heisenberg relation for one of the operators entering in it is postulated.
The chief aim of the present section is to be explored the problem whether all of the
reasonable conditions, mentioned in the previous sections and that can be imposed on the
creation and annihilation operators, can hold or not hold simultaneously. This problem is
suggested by the strong evidences that the relations (5.1)–(5.3) and (5.15)–(5.23), with a
possible exception of (5.3) (more precisely, of (5.5)) in the massless case, should be valid in
a realistic quantum field theory [1, 3, 7, 8, 11, 12]. Besides, to the arguments in loc. cit., we
shall add the requirement for uniqueness of the dynamical variables (see Sect. 4).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 32
As it was shown in Sect. 6, the relations (5.1), (5.2), (5.4) and (5.5) are compatible if
one starts from a charge symmetric Lagrangian (see (3.4)), which best describes a free field
theory; in particular, the commutation relations (6.16) (and hence (6.20) and (6.13)) ensure
their simultaneous validity.22 For that reason, we shall investigate below only commutation
relations for which (5.1), (5.2), (5.4) and (5.5) hold. It will be assumed that they should be
such that the equations (6.10)–(6.12), (6.31)–(6.33), (6.47)–(6.49) and (6.58)–(6.60), respec-
tively, hold.
Consider now the problem for the uniqueness of the dynamical variables and its consis-
tency with the commutation relations just mentioned for a charged field. It will be assumed
that this uniqueness is ensured via the equations (4.9)–(4.11).
The equation (4.15), viz.
[a†±m , a
m]−ε = 0, (7.1)
is a necessary and sufficient conditions for the uniqueness of the momentum and charge
operators (see Sect. 4 and the notation introduced at the beginning of Sect. 6). Before
commenting on this relation, we would like to derive some consequences of it. Applying
consequently (6.8) for η = −ε, (7.1) and the identity
[A,B ◦ C]+ = [A,B]η ◦ C − ηB ◦ [A,C]−η η = ±1 (7.2)
for η = +ε,−ε, we, in view of (7.1), obtain
[a+m, [a
m ]ε] = [a
m , [a
m]−ε]+ = (1− ε)[a†−m , a+m]ε ◦ a+m
[a−m, [a
m , a
m]ε] = ε[a
m , [a
m]−ε]+ = ε(1 − ε)[a†+m , a−m]ε ◦ a−m.
(7.3)
Forming the sum and difference of (6.12a), for τ = 0, and (6.33a), we see that the system
of equations they form is equivalent to
[a+l , [a
m , a
m]ε] = 0 [a
l , [a
m ]ε] = 0 (7.4a)
, [a+m, a
m ]ε] + 2δlma
= 0 [a−
, [a†+m , a
m]ε] − 2δlma−l = 0. (7.4b)
Combining (7.4b), for l = m, with (7.3), we get
(1− ε)[a†−m , a+m]ε ◦ a+m + 2a+m = 0 ε(1− ε)[a†+m , a−m]ε ◦ a−m − 2a−m = 0. (7.5)
Obviously, these equations reduce to
a±m = 0 (7.6)
for bose fields as for them ε = +1 (see (3.7)). Since the operators (7.6) describe a completely
unobservable field, or, more precisely, an absence of a field at all, the obtained result means
that the theory considered cannot describe any really existing physical field with spin j = 0, 1.
Such a conclusion should be regarded as a contradiction in the theory. For fermi fields, j = 1
and ε = −1, the equations (7.5) have solutions different from (7.6) iff a±m are degenerate
operators, i.e. with no inverse ones, in which case (7.4a) is a consequence of (7.5) and (7.1)
(see (6.8) and (7.3) too).
The source of the above contradiction is in the equation (7.1), which does not agree with
the bilinear commutation relations (6.13) and contradicts to the existing correlation between
creation and annihilation of particles with identical characteristics (m = (t,p) in our case)
as (7.1) can be interpreted physically as mutual independence of the acts of creation and
annihilation of such particles [1, § 10.1].
At this point, there are two ways for ‘repairing’ of the theory. On one hand, one can
forget about the uniqueness of the dynamical variables (in a sense of Sect. 4), after which
22 The special case(s) when (5.5) may not hold for a massless field will not be considered below.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 33
the formalism can be developed by choosing, e.g., the charge symmetric Lagrangians (3.4)
and following the usual Lagrangian formalism; in fact, this is the way the parafield theory is
build [16,18]. On another hand, one may try to change something at the ground of the theory
in such a way that the uniqueness of the dynamical variables to be ensured automatically.
We shall follow the second method. As a guiding idea, we shall have in mind that the
bilinear commutation relations (6.13) and the related to them normal ordering procedure
provide a base for the present-day quantum field theory, which describes sufficiently well
the discovered elementary particles/fields. On this background, an extensive exploration of
commutation relations which are incompatible with (6.13) is justified only if there appear
some evidences for fields/particles that can be described via them. In that connection it
should be recalled [17, 18], it seems that all known particles/fields are described via (6.13)
and no one of them is a para particle/field.
Using the notation introduced at the beginning of Sect. 4, we shall look for a linear
mapping (operator) E on the operator space over the system’s Hilbert space F of states such
E(D′) = E(D′′). (7.7)
As it was shown in Sect. 4, an example of an operator E is provided by the normal ordering
operator N . Therefore an operator satisfying (7.7) always exists. To any such operator E
there corresponds a set of dynamical variables defined via
D = E(D′). (7.8)
Let us examine the properties of the mapping E that it should possess due to the re-
quirement (7.7).
First of all, as the operators of the dynamical variables should be Hermitian, we shall
require
= E(B†) (7.9)
for any operator B, which entails
D† = D, (7.10)
due to (3.9)–(3.12) and (7.8).
As in Sect. 4, we shall replace the so-arising integral equations with corresponding alge-
braic ones. Thus the equations (4.5)–(4.20) remain valid if the operator E is applied to their
left hand sides.
Consider the general case of a charged field, q 6= 0. So, the analogue of (4.15) reads
[a†±m , a
= 0, (7.11)
which equation ensures the uniqueness of the momentum and charge operators. Respectively,
the condition (4.11) transforms into
(−ω◦µν(m) + ω◦µν(n))
E([a†+m , a−n ]−ε)− E([a+m, a†−n ]−ε)
= 0, (7.12)
which, by means of (7.11) can be rewritten as (cf. (4.16))
ω◦µν(n)
E([a†+m , a−n ]−ε)− E([a+m, a†−n ]−ε)
= 0. (7.13)
At the end, equations (4.17) and (4.18) now should be written as
µν (k) E
[a†+s (k), a
(k)]−ε
+ σss
µν (k) E
[a†−s (k), a
(k)]−ε
= 0 (7.14)
µν (k) E
[a†+s (k), a
(k)]−ε
+ lss
µν (k) E
[a†−s (k), a
(k)]−ε
= 0. (7.15)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 34
These equations can be satisfied if we generalize (7.11) to (cf. (4.20))
[a†±s (k), a
(k)]−ε
= 0 (7.16)
for any s and s′. At last, the following stronger version of (7.16)
[a†±m , a
n ]−ε
= 0, (7.17)
for any m = (t,p) and n = (r, q), ensures the validity of (7.14) and (7.15) and thus of the
uniqueness of all dynamical variables.
It is time now to call attention to the possible commutation relations. The replacement
D′, D′′, D′′′ 7→ D := E(D′) = E(D′′) = E(D′′′) results in corresponding changes in the
whole of the material of Sect. 6. In particular, the systems of commutation relations (6.10)–
(6.12), (6.31)–(6.33), (6.47)–(6.49) and (6.58)–(6.60) should be replaced respectively with:23
a±l , E(a
m ◦ a−m) + ε E(a†−m ◦ a+m)
± (1 + τ)δlma±l = 0 (7.18)
a±l , E(a
m ◦ a−m)− ε E(a†−m ◦ a+m)
− δlma±l = 0 (7.19)
(−ω◦µν(m) + ω◦µν(n))([ã±l , E(ã
m ◦ ã−n )− ε E(ã†−m ◦ ã+n )] )
= 2(1 + τ)δlmω
µν(l)(ã
(7.20)
(−1)j+1j
1 + τ
s,s′,t
i (p)
µν (k) + l
ss′,−
µν (k))[a
t (p), E(a†+s (k) ◦ a−s′(k))]
+ (σss
µν (k) + l
ss′,+
µν (k))[a
t (p), E(a†−s (k) ◦ a+s′(k))]
d3pIi
i′ (p)a
t (p).
(7.21)
Due to the uniqueness conditions (7.11)–(7.14), one can rewrite the terms E(a†±m ◦ a∓m)
in (7.18)–(7.21) in a number of equivalent ways; e.g. (see (7.11))
E(a†±m ◦ a∓m) = ε E(a∓m ◦ a†±m ) =
E([a†±m , a∓m]ε). (7.22)
Consider the general case of a charged field, q 6= 0 (and hence τ = 0). The system of
equations (7.18)–(7.19) is then equivalent to
, E(a†±m ◦ a∓m)
= 0 (7.23a)
, E(a†−m ◦ a+m)
+ εδlma
= 0 (7.23b)
a−l , E(a
m ◦ a−m)
− δlma−l = 0. (7.23c)
These (commutation) relations ensure the simultaneous fulfillment of the Heisenberg rela-
tions (5.1) and (5.2) involving the momentum and charge operators, respectively. To ensure
also the validity of (7.20), with τ = 0, and, consequently, of (5.4), we generalize (7.23) to
a±l , E(a
m ◦ a∓n )
= 0 (7.24a)
, E(a†−m ◦ a+n )
+ εδlma
n = 0 (7.24b)
, E(a†+m ◦ a−n )
− δlma−n = 0, (7.24c)
for any l = (s,k), m = (t,p) and n = (t, q) (see also (6.51)). In the way pointed in
Sect. 6, one can verify that (7.24) for any l = (s,k), m = (t,p) and n = (r,p) entails (7.21)
and hence (5.5). At last, to ensure the validity of all of the mentioned conditions and a
23 To save some space, we do not write the Hermitian conjugate of the below-written equations.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 35
suitable transition to a case of Hermitian field, for which q = 0 and τ = 1 (see (3.7)), we
generalize (7.24) to
a+l , E(a
m ◦ a−n )
+ τδlna
m = 0 (7.25a)
a−l , E(a
m ◦ a+n )
− ετδlna−m = 0 (7.25b)
, E(a†−m ◦ a+n )
+ εδlma
n = 0, (7.25c)
, E(a†+m ◦ a−n )
− δlma−n = 0 (7.25d)
where l, m and n are arbitrary. As a result of (7.17), which we assume to hold, and τa
τa±l (see (3.7)), the equations (7.25a) and (7.25c) (resp. (7.25b) and (7.25d)) become identical
when τ = 1 (and hence a
l = a
l ); for τ = 0 the system (7.25) reduces to (7.24). Recalling
that ε = (−1)2j (see (3.7)), we can rewrite (7.25) in a more compact form as
a±l , E(a
m ◦ a∓n )
+ (±1)2j+1τδlna±m = 0 (7.26a)
a±l , E(a
m ◦ a±n )
− (∓1)2j+1τδlma±n = 0. (7.26b)
Since the last equation is equivalent to (see (7.17)) and use that ε = (−1)2j)
, E(a±m ◦ a†∓n )
+ (±1)2j+1δlna±m = 0, (7.26b′)
it is evident that the equations (7.26a) and (7.26b) coincide for a neutral field.
Let us draw the main moral from the above considerations: the equations (7.17) are
sufficient conditions for the uniqueness of the dynamical variables, while (7.26) are such
conditions for the validity of the Heisenberg relations (5.1)–(5.5), in which the dynamical
variables are redefined according to (7.8). So, any set of operators a±
and E , which are
simultaneous solutions of (7.17) and (7.26), ensure uniqueness of the dynamical variables
and at the same time the validity of the Heisenberg relations.
Consider the uniqueness problem for the solutions of the system of equations consisting
of (7.17)and (7.26). Writing (7.17) as
E(a†±m ◦ a∓n ) = ε E(a∓n ◦ a†±m ) =
E([a†±m , a∓n ]ε), (7.27)
which reduces to (7.22) for n = m, and using ε = (−1)2j (see (3.7)), one can verify that (7.26)
is equivalent to
a+l , E([a
n ]ε)
+ 2δlna
m = 0 (7.28a)
a+l , E([a
m , a
n ]ε)
+ 2τδlna
m = 0 (7.28b)
a−l , E([a
n ]ε)
− 2τδlma−n = 0 (7.28c)
a−l , E([a
m , a
n ]ε)
− 2δlma−n = 0. (7.28d)
The similarity between this system of equations and (6.16) is more than evident: (7.28) can
be obtained from (6.16) by replacing [·, ·]ε with E([·, ·]ε).
As it was said earlier, the bilinear commutation relations (6.13) and the identification of
E with the normal ordering operator N ,
E = N , (7.29)
convert (7.27)–(7.28) into identities; by invoking (6.8), for η = −ε, the reader can check
this via a direct calculation (see also (4.23)). However, this is not the only possible solution
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 36
of (7.27)–(7.28). For example, if, in the particular case, one defines an ‘anti-normal’ ordering
operator A as a linear mapping such that
A(a+m ◦ a†−n ) := εa†−n ◦ a+m A(a†+m ◦ a−n ) := εa−n ◦ a†+m
A(a−m ◦ a†+n ) := a−m ◦ a†+n A(a†−m ◦ a+n ) := a†−m ◦ a+n ,
(7.30)
then the bilinear commutation relations (6.13) and the setting E = A provide a solution
of (7.27)–(7.28); to prove this, apply (6.8) for η = −ε. Evidently, a linear combination of
N and A, together with (6.13), also provides a solution of (7.27)–(7.28).24 Other solution
of the same system of equations is given by E = id and operators a±
satisfying (6.16), in
particular the paracommutation relations (6.20), and a
m ◦ a,∓n = εa,∓n ◦ a†±m . The problem
for the general solution of (7.27)–(7.28) with respect to E and a±l is open at present.
Let us introduce the particle and antiparticle number operators respectively by (see (7.27),
(7.9) and (3.16))
Nl :=
= E(a+
◦ a†−
) = (Nl)† =: N †l
†Nl :=
l , a
= E(a†+l ◦ a
l ) = (
†Nl)† =: †Nl†.
(7.31)
As a result of the commutation relations (7.28), with n = m, they satisfy the equations25
[Nl, a+m]− = δlma+l (7.32a)
[ †Nl, a+m]− = τδlma+l (7.32b)
[Nl, a†+m ]− = τδlma
l (7.32c)
[ †Nl, a†+m ]− = δlma
l . (7.32d)
Combining (3.9)–(3.12) and (5.11)–(5.13) with (7.8), (7.27) and (7.31), we get the following
expressions for the operators of the (redefined) dynamical variables:
P̃µ =
1 + τ
m2c2+k2
(Nl + †Nl) l = (s,k) (7.33)
Q̃ = q
(−Nl + †Nl) (7.34)
S̃µν =
(−1)j−1/2j~
1 + τ
{εσmn,+µν Nnm + σmn,−µν †Nmn)}
m=(s,k)
n=(s′,k)
(7.35)
L̃µν = x0µ P̃ν − x0 ν P̃µ +
(−1)j−1/2j~
1 + τ
{εlmn,+µν Nnm + lmn,−µν †Nmn)}
m=(s,k)
n=(s′,k)
2(1 + τ)
−ω◦µν(l) + ω◦µν(m)
(Nl + †Nl)
m=l=(s,k)
(7.36)
M̃spµν =
(−1)j−1/2j~
1 + τ
{ε(σmn,+µν + lmn,+µν )Nnm + (σmn,−µν + lmn,−µν ) †Nmn)}
m=(s,k)
n=(s′,k)
(7.37)
M̃orµν =
2(1 + τ)
−ω◦µν(l) + ω◦µν(m)
(Nl + †Nl)
m=l=(s,k)
. (7.38)
24 If we admit a±
to satisfy the ‘anomalous” bilinear commutation relations (8.27) (see below), i.e. (6.13)
with ε for −ε and (±1)2j for (±1)2j+1, then E = N , A also provides a solution of (7.27)–(7.28). However,
as it was demonstrated in [13–15], the anomalous commutation relations are rejected if one works with the
charge symmetric Lagrangians (3.4).
25 The equations (7.32a) and (7.32b) correspond to (7.28a) and (7.28b), respectively, and (7.32c) and (7.32d)
correspond to the Hermitian conjugate to (7.28c) and (7.28d), respectively.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 37
Here ω◦µν(l) is defined via (6.50), we have set
σmn,±µν := σ
ss′,±
µν (k) l
µν := l
ss′,±
µν (k) for m = (s,k) and n = (s
′,k), (7.39)
and (see (7.27))
Nlm :=
[a+l , a
= E(a+l ◦ a
m ) = (Nml)† =: N
†Nlm :=
l , a
= E(a†+l ◦ a
m) = (
†Nml)† =: †Nml†
(7.40)
are respectively the particle and antiparticle transition operators (cf. [26, sec. 1] in a case of
parafields). Obviously, we have
Nl = Nll †Nl = †Nll. (7.41)
The choice (7.29), evidently, reduces (7.33)–(7.36) to (4.24), (4.25), (4.28) and (4.29), respec-
tively.
In terms of the operators (7.38), the commutation relations (7.28) can equivalently be
rewritten as (see also (7.9))
[Nlm, a+n ]− = δmna+l (7.42a)
[ †Nlm, a+n ]− = τδmna+l (7.42b)
[Nlm, a†+n ]− = τδmna
(7.42c)
[ †Nlm, a†+n ]− = δmna
l . (7.42d)
If m = l, these relations reduce to (7.32), due to (7.39).
We shall end this section with the remark that the conditions for the uniqueness of
the dynamical variables and the validity of the Heisenberg relations are quite general and
are not enough for fixing some commutation relations regardless of a number of additional
assumptions made to reduce these conditions to the system of equations (7.27)–(7.28).
8. State vectors, vacuum and mean values
Until now we have looked on the commutation relations only from pure mathematical view-
point. In this way, making a number of assumptions, we arrived to the system (7.27)–(7.28) of
commutation relations. Further specialization of this system is, however, almost impossible
without making contact with physics. For the purpose, we have to recall [1, 3, 11, 12] that
the physically measurable quantities are the mean (expectation) values of the dynamical
variables (in some state) and the transition amplitudes between different states. To make
some conclusions from these basic assumption of the quantum theory, we must rigorously
said how the states are described as vectors in system’s Hilbert space F of states, on which
all operators considered act.
For the purpose, we shall need the notion of the vacuum or, more precisely, the assumption
of the existence of unique vacuum state (vector) (known also as the no-particle condition).
Before defining rigorously this state, which will be denoted by X0, we shall heuristically
analyze the properties it should possess.
First of all, the vacuum state vector X0 should represent a state of the field without any
particles. From here two conclusions may be drawn: (i) as a field is thought as a collection
of particles and a ‘missing’ particle should have vanishing dynamical variables, those of the
vacuum should vanish too (or, more generally, to be finite constants, which can be set equal
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 38
to zero by rescaling some theory’s parameters) and (ii) since the operators a−l and a
l are
interpreted as ones that annihilate a particle characterize by l = (s,k) and charge −q or +q,
respectively, and one cannot destroy an ‘absent’ particle, these operators should transform
the vacuum into the zero vector, which may be interpreted as a complete absents of the field.
Thus, we can expect that
D(X0) = 0 (8.1a)
a−l (X0) = 0 a
l (X0) = 0. (8.1b)
Further, as the operators a+l and a
l are interpreted as ones creating a particle charac-
terize by l = (s,k) and charge −q or +q, respectively, state vectors like a+l (X0) and a
l (X0)
should correspond to 1-particle states. Of course, a necessary condition for this is
X0 6= 0, (8.2)
due to which the vacuum can be normalize to unit,
〈 X0| X0〉 = 1, (8.3)
where 〈·|·〉 : F × F → C is the Hermitian scalar (inner) product of F . More generally, if
, . . .) is a monomial only in i ∈ N creation operators, the vector
ψl1l2... := M(a+l1 , a
, . . .)(X0) (8.4)
may be expected to describe an i-particle state (with i1 particles and i2 antiparticles, i1+i2 =
i, where i1 and i2 are the number of operators a
l and a
l , respectively, in M(a
, . . .)).
Moreover, as a free field is intuitively thought as a collection of particles and antiparticles,
it is natural to suppose that the vectors (8.4) form a basis in the Hilbert space F . But the
validity of this assumption depends on the accepted commutation relations; for its proof,
when the paracommutation relations are adopted, see the proof of [18, p. 26, theorem I-1].
Accepting the last assumption and recalling that the transition amplitude between two
states is represented via the scalar product of the corresponding to them state vectors, it
is clear that for the calculation of such an amplitude is needed an effective procedure for
calculation of scalar products of the form
〈ψl1l2...|ϕm1m2...〉 := 〈 X0|(M(a+l1 , a
, . . .))† ◦ M′(a+m1 , a
, . . .)X0〉, (8.5)
with M and M′ being monomials only in the creation operators. Similarly, for computation
of the mean value of some dynamical operator D in a certain state, one should be equipped
with a method for calculation of scalar products like
〈ψl1l2...| Dϕm1m2...〉 := 〈X0|(M(a+l1 , a
, . . .))† ◦ D ◦ M′(a+m1 , a
, . . .)X0〉. (8.6)
Supposing, for the moment, the vacuum to be defined via (8.1), let us analyze (8.1)–(8.6).
Besides, the validity of (7.27)–(7.28) will be assumed.
From the expressions (7.8) and (3.9)–(3.12) for the dynamical variables, it is clear that
the condition (8.1a) can be satisfied if
E(a†±m ◦ a∓n )(X0) = 0, (8.7)
which, in view of (7.27), is equivalent to any one of the equations
E(a±m ◦ a†∓n )(X0) = 0 (8.8a)
E([a±m, a†∓n ]ε)(X0) = 0. (8.8b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 39
Equation (8.7) is quite natural as it expresses the vanishment of all modes of the vacuum
corresponding to different polarizations, 4-momentum and charge. It will be accepted here-
after.
By means of (8.8) and the commutation relations (7.28) in the form (7.42), in particu-
lar (7.32), one can explicitly calculate the action of any one of the operators (7.33)–(7.38)
on the vectors (8.4): for the purpose one should simply to commute the operators Nlm (or
Nl = Nll) with the creation operators in (8.4) according to (7.42) (resp. (7.32)) until they
act on the vacuum and, hence, giving zero, as a result of (8.8) and (7.42) (resp. (7.32)). In
particular, we have the equations (k0 =
m2c2 + k2):
a+l (X0)
= kµa
l (X0) P̃µ
l (X0)
= kµa
l (X0) l = (s,k) (8.9)
= −qa+
(X0) Q̃
= +qa
(X0) (8.10)
l=(s,k)
(−1)j−1/2j~
1 + τ
{εσlm,+µν + τσml,−µν }
m=(t,k)
a+m|m=(t,k)(X0)
l=(s,k)
(−1)j−1/2j~
1 + τ
{ετσlm,+µν + σml,−µν }
m=(t,k)
a†+m |m=(t,k)(X0)
(8.11)
l=(s,k)
= (x0 µkν − x0 νkµ)(a+l )(X0)− i~
ω◦µν(l)(a
(−1)j−1/2j~
1 + τ
{εllm,+µν + τ lml,−µν }
m=(t,k)
a+m|m=(t,k)(X0)
l=(s,k)
= (x0 µkν − x0 νkµ)(a†+l )(X0)− i~
ω◦µν(l)(a
(−1)j−1/2j~
1 + τ
{ετ llm,+µν + lml,−µν }
m=(t,k)
a†+m |m=(t,k)(X0)
(8.12)
M̃spµν
l=(s,k)
(−1)j−1/2j~
1 + τ
{ε(σlm,+µν + llm,+µν )
+ τ(σml,−µν + l
µν )}
m=(t,k)
a+m|m=(t,k)(X0)
M̃spµν
l=(s,k)
(−1)j−1/2j~
1 + τ
{ετ(σlm,+µν + llm,+µν )
+ (σml,−µν + l
µν )}
m=(t,k)
a†+m |m=(t,k)(X0)
(8.13)
M̃orµν
ã+l (X0)
= −i~
ω◦µν(l)(ã
(X0) M̃orµν
l (X0)
= −i~
ω◦µν(l)(ã
(X0). (8.14)
These equations and similar, but more complicated, ones with an arbitrary monomial in the
creation operators for a+
are the base for the particle interpretation of the quantum
theory of free fields. For instance, in view of (8.9) and (8.10), the state vectors a+l (X0) and
l (X0) are interpreted as ones representing particles with 4-momentum (
m2c2 + k2,k)
and charges −q and +q, respectively; similar multiparticle interpretation can be given to the
general vectors (8.4) too.
The equations (8.9)–(8.12) completely agree with similar ones obtained in [13–15] on the
base of the bilinear commutation relations (6.13).
By means of (8.7), the expression (8.6) can be represented as a linear combination of
terms like (8.5). Indeed, as D is a linear combinations of terms like E(a†±m ◦a∓n ), by means of
the relations (7.28) we can commute each of these terms with the creation (resp. annihilation)
operators in the monomial M′(a+m1 , a
m2 , . . .) (resp. (M(a+l1 , a
, . . .))† = M′′(a†−
, . . .))
and thus moving them to the right (resp. left) until they act on the vacuum X0, giving
the zero vector — see (8.7). In this way the matrix elements of the dynamical variables,
in particular their mean values, can be expressed as linear combinations of scalar products
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 40
of the form (8.5). Therefore the supposition (8.7) reduces the computation of mean values
of dynamical variables to the one of the vacuum mean value of a product (composition)
of creation and annihilation operators in which the former operators stand to the right of
the latter ones. (Such a product of creation and annihilation operators can be called their
‘antinormal’ product; cf. the properties (7.30) of the antinormal ordering operator A.)
The calculation of such mean values, like (8.5) for states ψ,ϕ 6= X0, however, cannot
be done (on the base of (7.27)–(7.28), (8.7) and (8.1a)) unless additional assumption are
made. For the purpose one needs some kind of commutation relations by means of which
the creation (resp. annihilation) operators on the r.h.s. of (8.5) to be moved to the left (resp.
right) until they act on the left (resp. right) vacuum vector X0; as a result of this operation,
the expressions between the two vacuum vectors in (8.5) should transform into a linear
combination of constant terms and such with no contribution in (8.5). (Examples of the last
type of terms are E(a†±m ◦ a∓) and normally ordered products of creation and annihilation
operators.) An alternative procedure may consists in defining axiomatically the values of all
or some of the mean values (8.5) or, more stronger, the explicit action of all or some of the
operators, entering in the r.h.s. of (8.5), on the vacuum.26 It is clear, both proposed schemes
should be consistent with the relations (7.27)–(7.28), (8.1b) and (8.7)–(8.8).
Let us summarize the problem before us: the operator E in (7.27)–(7.28) has to be fixed
and a method for computation of scalar products like (8.5) should be given provided the
vacuum vector X0 satisfies (8.1b), (8.2), (8.3) and (8.7). Two possible ways for exploration
of this problem were indicated above.
Consider the operator E . Supposing E(a†±m ◦ a∓n ) to be a function only of a
m and a
we, in view of (8.1b), can write E(a†±m ◦ a∓n ) = f±(a
m ◦ a∓n ) ◦ b with b = a−n (upper sign) or
b = a
m (lower sign) and some functions f
±. Applying (7.27), we obtain (do not sum over l)
E(a†+m ◦ a−l ) = f
+(a†+m , a
l ) ◦ a
l E(a
m ◦ a
l ) = f
−(a+m, a
l ) ◦ a
◦ a†+m ) = εf+(a†+m , a−l ) ◦ a
E(a†−
◦ a+m) = εf−(a+m, a
) ◦ a†−
Since E is a linear operator, the expression E(a†±m ◦a∓n ) turns to be a linear and homogeneous
function of a
m and a
n , which immediately implies f
±(A,B) = λ±A for operators A and
B and some constants λ± ∈ C. For future convenience, we assume λ± = 1, which can be
achieved via a suitable renormalization of the creation and annihilation operators.27 Thus,
the last equations reduce to
E(a†+m ◦ a−l ) = a
m ◦ a−l E(a
m ◦ a
) = a+m ◦ a
(8.15a)
E(a−l ◦ a
m ) = εa
m ◦ a−l E(a
l ◦ a
m) = εa
m ◦ a
l . (8.15b)
Evidently, these equations convert (7.27), (8.7) and (8.8) into identities. Comparing (8.15)
and (4.22), we see that the identification
E = N (8.16)
of the operator E with the normal ordering operator N is quite natural. However, for our
purposes, this identification is not necessary as only the equations (8.15), not the general
definition of N , will be employed.
26 Such an approach resembles the axiomatic description of the scattering matrix [1,7,8].
27 Since λ+ = 0 or/and λ− = 0 implies D = 0, due to (7.8), these values are excluded for evident reasons.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 41
As a result of (8.15), the commutation relations (7.28) now read:
[a+l , a
m ◦ a†−n ] + δlna+m = 0 (8.17a)
, a†+m ◦ a−n ] + τδlna+m = 0 (8.17b)
, a+m ◦ a†−n ] − τδlma−n = 0 (8.17c)
[a−l , a
m ◦ a−n ] − δlma−n = 0. (8.17d)
(In a sense, these relations are ‘one half’ of the (para)commutation relations (6.16): the
latter are a sum of the former and the ones obtained from (8.17) via the changes a+m ◦a
n ◦ a+m and a
m ◦ a−n 7→ εa−n ◦ a
m ; the last relations correspond to (7.28) with E = A, A
being the antinormal ordering operator — see (7.30). Said differently, up to the replacement
a±i 7→
for all l, the relations (8.17) are identical with (6.16) for ε = 0; as noted in [26, the
remarks following theorem 2 in sec. 1], this is a quite exceptional case from the view-point
of parastatistics theory.) By means of (6.8) for η = −ε, one can verify that equations (8.17)
agree with the bilinear commutation relations (6.13), i.e. (6.13) convert (8.17) into identities.
The equations (8.15) imply the following explicit forms of the number operators (7.31)
and the transition operators (7.40):
Nl = a+l ◦ a
†Nl = a†+l ◦ a
l (8.18)
Nlm = a+l ◦ a
†Nlm = a†+l ◦ a
m. (8.19)
As a result of them, the equations (7.33)–(7.36) are simply a different form of writing of (4.24),
(4.25), (4.28) and (4.29), respectively.
Let us return to the problem of calculation of vacuum mean values of antinormal ordered
products like (8.5). In view of (8.1b) and (8.3), the simplest of them are
〈 X0|λ idF (X0)〉 = λ 〈X0|M±(X0)〉 = 0 (8.20)
where λ ∈ C and M+ (resp. M−) is any monomial of degree not less than 1 only in the
creation (resp. annihilation) operators; e.g. M± = a±l , a
l , a
◦a±l2 , a
◦a†±l2 . These equations,
with λ = 1, are another form of what is called the stability of the vacuum: if Xi denotes an
i-particle state, i ∈ N∪{0}, then, by virtue of (8.20) and the particle interpretation of (8.4),
we have
〈 Xi| X0〉 = δi0, (8.21)
i.e. the only non-forbidden transition into (from) the vacuum is from (into) the vacuum.
More generally, if Xi′,0 and X0,j′′ denote respectively i′-particle and j′′-antiparticle states,
with X0,0 := X0, then
〈Xi′,0| X0,j′′〉 = δi′0δ0j′′ , (8.22)
i.e. transitions between two states consisting entirely of particles and antiparticles, respec-
tively, are forbidden unless both states coincide with the vacuum. Since we are dealing with
free fields, one can expect that the amplitude of a transitions from an (i′-particle + j′-an-
tiparticle) state Xi′,j′ into an (i′′-particle + j′′-antiparticle) state Xi′′,j′′ is
〈 Xi′,j′| Xi′′,j′′〉 = δi′i′′δj′j′′ , (8.23)
but, however, the proof of this hypothesis requires new assumptions (vide infra).
Let us try to employ (8.17) for calculation of expressions like (8.5). Acting with (8.17)
and their Hermitian conjugate on the vacuum, in view of (8.1b), we get
a+m ◦ (−a†−n ◦ a+l + δln idF)(X0) = 0 a
n ◦ (a−m ◦ a
− δlm idF )(X0) = 0
a†+m ◦ (−a−n ◦ a+l + τδln idF )(X0) = 0 a
n ◦ (a†−m ◦ a
l − τδlm idF )(X0) = 0.
(8.24)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 42
These equalities, as well as (8.17), cannot help directly to compute vacuum mean values
of antinormally ordered products of creation and annihilation operators. But the equa-
tions (8.24) suggest the restrictions28
◦ a+m(X0) = δlm X0 a−l ◦ a
m (X0) = δlm X0
a−l ◦ a
m(X0) = τδlm X0 a
l ◦ a
m (X0) = τδlm X0
(8.25)
to be added to the definition of the vacuum. These conditions convert (8.24) into identities
and, in this sense agree with (8.17) and, consequently, with the bilinear commutation rela-
tions (6.13). Recall [16, 18], the relations (8.25) are similar to ones accepted in the parafield
theory and coincide with that for parastatistics of order p = 1; however, here we do not sup-
pose the validity of the paracommutation relations (6.20) (or (6.16)). Equipped with (8.25),
one is able to calculate the r.h.s. of (8.5) for any monomial M (resp. M′) and monomials
M′ (resp. M) of degree 1, degM′ = 1 (resp. degM = 1).29 Indeed, (8.25), (8.1b) and (8.3)
entail:
〈 X0|a†−l ◦ a
m(X0)〉 = 〈X0|a−l ◦ a
m (X0)〉 = δlm
〈 X0|a−l ◦ a
m(X0)〉 = 〈X0|a
◦ a†+m (X0)〉 = τδlm
〈 X0|(M(a+l1 , a
, · · · ))† ◦ a+m(X0)〉 = 〈 X0|(M(a+l1 , a
, · · · ))† ◦ a†+m (X0)〉 = 0 degM≥ 2
〈 X0|a−l ◦ M(a
, a†+m2 , · · · )(X0)〉 = 〈X0|a
l ◦ M(a
, a†+m2 , · · · )(X0)〉 = 0 degM≥ 2.
(8.26)
Hereof the equation (8.23) for i′ + j′ = 1 (resp. i′′ + j′′ = 1) and arbitrary i′′ and j′′ (resp. i′
and j′) follows.
However, it is not difficult to be realized, the calculation of (8.5) in cases more general
than (8.20) and (8.26) is not possible on the base of the assumptions made until now.30 At
this point, one is free so set in an arbitrary way the r.h.s. of (8.5) in the mentioned general
case or to add to (8.17) (and, possibly, (8.25)) other (commutation) relations by means of
which the r.h.s. of (8.5) to be calculated explicitly; other approaches, e.g. some mixture
of the just pointed ones, for finding the explicit form of (8.5) are evidently also possible.
Since expressions like (8.5) are directly connected with observable experimental results, the
only criterion for solving the problem for calculating the r.h.s. of (8.5) in the general case
can be the agreement with the existing experimental data. As it is known [1, 3, 11, 12], at
present (almost?) all of them are satisfactory described within the framework of the bilinear
commutation relations (6.13). This means that, from physical point of view, the theory
should be considered as realistic one if the r.h.s. of (8.5) is the same as if (6.13) are valid
or is reducible to it for some particular realization of an accepted method of calculation,
e.g. if one accepts some commutation relations, like the paracommutation ones, which are
a generalization of (6.13) and reduce to them as a special case (see, e.g., (6.20)). It should
be noted, the conditions (8.1b)–(8.3) and (8.25) are enough for calculating (8.5) if (6.16),
or its versions (6.17) or (6.20), are accepted (cf. [16]). The causes for that difference are
replacements like [a+m, a
n ] 7→ 2a+m◦a
n , when one passes from (6.16) to (8.17); the existence
of terms like a
n ◦ a+ma+l in (6.16) are responsible for the possibility to calculate (8.5).
28 Since the operators a±
and a
are, generally, degenerate (with no inverse ones), we cannot say that (8.24)
implies (8.25).
29 For degM′ = 0 (resp. degM′ = 0) — see (8.20).
30 It should be noted, the conditions (8.1b)–(8.3) and (8.25) are enough for calculating (8.5) if the rela-
tions (6.16), or their version (6.20), are accepted (cf. [16]). The cause for that difference is in replacements
like [a+m, a
n ] 7→ 2a
m ◦ a
n , when one passes from (6.16) to (8.17); the existence of terms like a
n ◦ a
m ◦ a
in (6.16) is responsible for the possibility to calculate (8.5), in case (6.16) hold.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 43
If evidences appear for events for which (8.5) takes other values, one should look, e.g.,
for other commutation relations leading to desired mean values. As an example of the last
type can be pointed the following anomalous bilinear commutation relations (cf. (6.13))
, a±m]ε = 0 [a
, a†±m ]ε = 0
[a∓l , a
m]ε = (±1)2jτδlm idF [a
l , a
m ]ε = (±1)2jτδlm idF
[a±l , a
m ]ε = 0 [a
l , a
m]ε = 0
[a∓l , a
m ]ε = (±1)2jδlm idF [a
l , a
m]ε = (±1)2jδlm idF , (8.27)
which should be imposed after expressions like E(a†±m ◦ a∓n ) are explicitly calculated. These
relations convert (8.17) and (8.25) into identities and by their means the r.h.s. of (8.5) can be
calculated explicitly, but, as it is well known [1,3,11,12,27] they lead to deep contradictions
in the theory, due to which should be rejected.31
At present, it seems, the bilinear commutation relations (6.13) are the only known com-
mutation relations which satisfy all of the mentioned conditions and simultaneously provide
an evident procedure for effective calculation of all expressions of the form (8.5). (Besides,
for them and for the paracommutation relations the vectors (8.4) form a base, the Fock
base, for the system’s Hilbert space of states [18].) In this connection, we want to mention
that the paracommutation relations (6.16) (or their conventional version (6.20)), if imposed
as additional restrictions to the theory together with (8.17), reduce in this particular case
to (6.13) as the conditions (8.25) show that we are dealing with a parafield of order p = 1,
i.e. with an ordinary field [17,18].32
Ending this section, let us return to the definition of the vacuum X0. It, generally,
depends on the adopted commutation relations. For instance, in a case of the bilinear com-
mutation relations (6.13) it consists of the equations (8.1a)–(8.3), while in a case of the
paracommutation relations (6.16) (or other ones generalizing (6.13)) it includes (8.1a)–(8.3)
and (8.25).
9. Commutation relations for
several coexisting different free fields
Until now we have considered commutation relations for a single free field, which can be
scalar, or spinor or vector one. The present section is devoted to similar treatment of a
system consisting of several, not less than two, different free fields. In our context, the
fields may differ by their masses and/or charges and/or spins; e.g., the system may consist
of charged scalar field, neutral scalar field, massless spinor field, massive spinor field and
massless neural vector field. It is a priori evident, the commutation relations regarding only
one field of the system should be as discussed in the previous sections. The problem is to be
derived/postulated commutation relations concerning different fields. It will be shown, the
developed Lagrangian formalism provides a natural base for such an investigation and makes
superfluous some of the assumptions made, for example, in [17, p. B 1159, left column] or
in [18, sec. 12.1], where systems of different parafields are explored.
To begin with, let us introduce suitable notation. With the indices α, β, γ = 1, 2, . . . , N
will be distinguished the different fields of the system, with N ∈ N, N ≥ 2, being their
number, and the corresponding to them quantities. Let qα and jα be respectively the charge
31 As it was demonstrated in [13–15], a quantization like (8.27) contradicts to (is rejected by) the charge
symmetric Lagrangians (3.4).
32 Notice, as a result of (8.17), the relations (6.16) correspond to (7.28) for E = A, with A being the
antinormal ordering operator (see (7.30)).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 44
and spin of the α-th field. Similarly to (3.7), we define
jα :=
0 for scalar α-th field
for spinor α-th field
1 for vector α-th field
τα :=
1 for qα = 0 (neutral (Hermitian) field)
0 for qα 6= 0 (charged (non-Hermitian) field)
εα := (−1)2jα =
+1 for integer jα (bose fields)
−1 for half-integer jα (fermi fields)
(9.1)
Suppose Lα is the Lagrangian of the α-field. For definiteness, we assume Lα for all
α to be given by one and the same set of equations, viz. (3.1), or (3.3) or (3.4). To save
some space, below the case (3.4), corresponding to charge symmetric Lagrangians, will be
considered in more details; the reader can explore other cases as exercises.
Since the Lagrangian of our system of free fields is
Lα, (9.2)
the dynamical variables are
Dα (9.3)
and the corresponding system of Euler-Lagrange equations consists of the independent equa-
tions for each of the fields of the system (see (3.6) with Lα for L). This allows an introduction
of independent creation and annihilation operators for each field. The ones for the α-th field
will be denoted by a±α,sα(k) and a
α,sα(k); notice, the values of the polarization variables
generally depend on the field considered and, therefore, they also are labeled with index
α for the α-th field. For brevity, we shall use the collective indices lα, mα and nα, with
lα := (α, sα,k) etc., in terms of which the last operators are a±
and a
, respectively. The
particular expressions for the dynamical operators Dα are given via (3.9)–(3.12) in which
the following changes should be made:
τ 7→ τα j 7→ jα ε 7→ εα s 7→ sα s′ 7→ s′α
µν (k) 7→ σs
αs′α,±
µν (k) l
ss′,±
µν (k) 7→ ls
αs′α,±
µν (k).
(9.4)
The content of sections 4 and 5 remains valid mutatis mutandis, viz. provided the just
pointed changes (9.4) are made and the (integral) dynamical variables are understood in
conformity with (9.3).
9.1. Commutation relations connected with the momentum operator.
Problems and their possible solutions
In sections 6–8, however, substantial changes occur; for instance, when one passes from (6.12)
or (6.15) to (6.16). We shall consider them briefly in a case when one starts from the charge
symmetric Lagrangians (3.4).
The basic relations (6.12), which arise from the Heisenberg relation (5.1) concerning the
momentum operator, now read (here and below, do not sum over α, and/or β and/or γ if
the opposite is not indicated explicitly!)
a±lα , [a
]εβ + [a
± (1 + τ)δlαmβa±lα = 0 (9.5a)
lα , [a
]εβ + [a
± (1 + τ)δlαmβa
lα = 0. (9.5b)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 45
It is trivial to be seen, the following generalizations of respectively (6.14) and (6.15)
a±lα , [a
± (1 + τβ)δlαmβa±lα = 0 (9.6a)
± (1 + τβ)δlαmβa±lα = 0 (9.6b)
lα , [a
± (1 + τβ)δlαmβa
lα = 0 (9.6c)
± (1 + τβ)δlαmβa
= 0 (9.6d)
a+lα , [a
+ 2δlαmβa
lα = 0 (9.7a)
+ 2τβδlαmβa
= 0 (9.7b)
a−lα , [a
− 2τβδlαmβa−lα = 0 (9.7c)
− 2δlαmβa−lα = 0 (9.7d)
provide a solution of (9.5) in a sense that they convert it into identity. As it was said in
Sect. 6, the equations (9.6) (resp. (9.7)) for a single field, i.e. for β = α, agree (resp. disagree)
with the bilinear commutation relations (6.13).
The only problem arises when one tries to generalize, e.g., the relations (9.7) in a way
similar to the transition from (6.15) to (6.16). Its essence is in the generalization of expres-
sions like [a
]εβ and τ
βδlαmβa
. When passing from (6.15) to (6.16), the indices l and
m are changed so that the obtained equations to be consistent with (6.13); of course, the
numbers ε and τ are preserved because this change does not concern the field regarded. But
the situation with (9.7) is different in two directions:
(i) If we change the pair (mβ,mβ) in [a
]εβ with (m
β, nγ), then with what the num-
ber εβ should be replace? With εβ , or εγ or with something else? Similarly, if the mentioned
changed is performed, with what the multiplier τβ in τβδlαmβa
lα should be replaced? The
problem is that the numbers εβ and τβ are related to terms like a
and a±
◦ a†∓
in the momentum operator, as a whole and we cannot say whether the index β in εβ and τβ
originates from the first of second index mβ in these expressions.
(ii) When writing (mβ , nγ) for (mβ ,mβ) (see (i) above), then shall we replace δlαmβa
with δlαmβa
nγ , or δlαnγa
, or δmβnγa
? For a single field, γ = β = α, this problem is
solved by requiring an agreement of the resulting generalization (of (6.16) in the particular
case) with the bilinear commutation relations (6.13). So, how shall (6.13) be generalized for
several, not less than two, different fields? Obviously, here we meet an obstacle similar to
the one described in (i) above, with the only change that −εβ should stand for εβ .
Let blα and clα denote some creation or annihilation operator of the α-field. Consider the
problem for generalizing the (anti)commutator [blα , clα ]±εα . This means that we are looking
for a replacement
[blα , clα ]±εα 7→ f±(blα , cmβ ;α, β), (9.8)
where the functions f± are such that
f±(blα , cmβ ;α, β)
= [blα , clα ]±εα . (9.9)
Unfortunately, the condition (9.9) is the only restriction on f± that the theory of free fields
can provide. Thus the functions f±, subjected to equation (9.9), become new free parameters
of the quantum theory of different free fields and it is a matter of convention how to choose/fix
them.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 46
It is generally accepted [18, appendix F], the functions f± to have forms ‘maximum’
similar to the (anti)commutators they generalize. More precisely, the functions
f±(blα , cmβ ;α, β) = [blα , cmβ ]±εαβ (9.10)
where εαβ ∈ C are such that
εαα = εα, (9.11)
are usually considered as the only candidates for f±. Notice, in (9.10), εαβ are functions in
α and β, not in lα and/or mβ. Besides, if we assume εαβ to be function only in εα and εβ ,
then the general form of εαβ is
εαβ = uαβεα + (1− uαβ)εβ + vαβ(1− εαεβ) uαβ , vαβ ∈ C, (9.12)
due to (9.1) and (9.11). (In view of (6.13), the value εαβ = +1 (resp. εαβ = −1) corresponds
to quantization via commutators (resp. anticommutators) of the corresponding fields.)
Call attention now on the numbers τα which originate and are associated with each term
[blα , cmα ]±εα . With every change (9.8) one can associate a replacement
τα 7→ g(blα , cmβ ;α, β), (9.13)
where the function g is such that
g(blα , cmβ ;α, β)
= τα. (9.14)
Of course, the last condition does not define g uniquely and, consequently, the function
g, satisfying (9.14), enters in the theory as a new free parameter. Suppose, as a working
hypothesis similar to (9.10)–(9.11), that g is of the form
g(blα , cmβ ;α, β) = τ
αβ , (9.15)
where ταβ are complex numbers that may depend only on α and β and are such that
ταα = τα. (9.16)
Besides, if we suppose ταβ to be functions only in τα and τβ, then
ταβ = xαβτα + yαβτβ + (1− xαβ − yαβ)τατβ xαβ , yαβ ∈ C, (9.17)
as a result of (9.1) and (9.16).
Let us summarize the above discussion. If we suppose a preservation of the algebraic
structure of the bilinear commutation relations (6.13) for a system of different free fields,
then the replacements
[blα , clα ]±εα 7→ [blα , cmβ ]±εαβ εαα = εα (9.18a)
τα 7→ ταα ταα = τα (9.18b)
should be made; accordingly, the relations (6.13) transform into:
]−εαβ = 0 [a
]−εαβ = 0
[a∓lα , a
]−εαβ = τ
αβδlαmβ idF ×
lα , a
]−εαβ = τ
αβδlαmβ idF ×
[a±lα , a
]−εαβ = 0 [a
lα , a
]−εαβ = 0
]−εαβ = δlαmβ idF ×
]−εαβ = δlαmβ idF ×
, (9.19)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 47
where 1 (resp. −εαβ) in
corresponds to the choice of the upper (resp. lower) signs. If
we suppose additionally εαβ (resp. ταβ) to be a function only in εα and εβ (resp. in τα and
τβ), then these numbers are defined up to two sets of complex parameters:
εαβ = uαβεα + (1− uαβ)εβ + vαβ(1− εαεβ) uαβ, vαβ ∈ C (9.20a)
ταβ = xαβτα + yαβτβ + (1− xαβ − yαβ)τατβ xαβ, yαβ ∈ C. (9.20b)
A reasonable further specialization of εαβ and ταβ may be the assumption their ranges
to coincide with those of εα and τα, respectively. As a result of (9.1), this supposition is
equivalent to
vαβ = −uαβ,−uαβ + 1, uαβ − 1, uαβ uαβ ∈ C (9.21a)
(xαβ , yαβ) = (0, 0), (0, 1), (1, 0), (1, 1). (9.21b)
Other admissible restriction on (9.20) may be the requirement εαβ and ταβ to be symmetric,
εαβ(εα, εβ) = εβα(εα, εβ) = εαβ(εβ , εα) (9.22a)
ταβ(τα, τβ) = τβα(τα, τβ) = ταβ(τβ, τα), (9.22b)
which means that the α-th and β-th fields are treated on equal footing and there is no a
priori way to number some of them as the ‘first’ or ‘second’ one.33 In view of (9.20), the
conditions (9.22) are equivalent to
uαβ =
vαβ ∈ C (9.23a)
yαβ = xαβ. (9.23b)
If both of the restrictions (9.21) and (9.23) are imposed on (9.20), then the arbitrariness of
the parameters in (9.20) is reduced to:
(uαβ , uαβ) =
(9.24a)
(xαβ , yαβ) = (0, 0), (1, 1) (9.24b)
and, for any fixed pair (α, β), we are left with the following candidates for respectively εαβ
and ταβ:
(+1 + εα + εβ − εαεβ) (9.25a)
(−1 + εα + εβ + εαεβ) (9.25b)
0 := τ
α + τβ (9.25c)
1 := τ
α + τβ − τατβ. (9.25d)
When free fields are considered, as in our case, no further arguments from mathematical
or physical nature can help for choosing a particular combination (εαβ , ταβ) from the four
possible ones according to (9.25) for a fixed pair (α, β). To end the above considerations of
εαβ and ταβ, we have to say that the choice
(εαβ , ταβ) = (ε
+ , τ
0 ) =
(+1 + εα + εβ − εαεβ), τα + τβ
(9.26)
33 However, nothing can prevent us to make other choices, compatible with (9.18), in the theory of free
fields; for instance, one may set εαβ = εαεβεβα and ταβ = 1
(τα + τβ)τβα.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 48
is known as the normal case [18, appendix F]; in it the relative behavior of bose (resp.
fermi) fields is as in the case of a single field, i.e. they are quantized via commutators (resp.
anticommutators) as (εαβ , ταβ) = (+1, 0) (resp. (εαβ , ταβ) = (−1, 0)), and the one of bose and
fermi field is as in the case of a single fermi field, viz. the quantization is via commutators as
(εαβ , ταβ) = (+1, 0). All combinations between ε
± and τ
0,1 different from (9.26) are referred
as anomalous cases. Above we supposed the pair (α, β) to be fixed. If α and β are arbitrary,
the only essential change this implies is in (9.25), where the choice of the subscripts +, −, 0
and 1 may depend on α and β. In this general situation, the normal case is defined as the one
when (9.26) holds for all α and β. All other combinations are referred as anomalous cases;
such are, for instance, the ones when some fermi and bose operators satisfy anticommutation
relations, e.g. (9.19) with εαβ = −1 for εα + εβ = 0, or some fermi fields are subjected to
commutation relations, like (9.19) with εαβ = +1 for εα = εβ = −1. For some details on this
topic, see, for instance, [18, appendix F], [7, chapter 20] and [27, sect 4-4]. Fields/operators
for which εαβ = +1 (resp. εαβ = −1), with β 6= α, are referred as relative parabose (resp.
parafermi) in the parafield theory [17,18]. One can transfer this terminology in the general
case and call the fields/operators for which εαβ = +1 (resp. εαβ = −1), with β 6= α, relative
bose (resp. fermi) fields/operators.
Further the relations (9.19) will be referred as the multifield bilinear commutation rela-
tions and it will be assumed that they represent the generalization of the bilinear commuta-
tion relations (6.13) when we are dealing with several, not less than two, different quantum
fields. The particular values of εαβ and ταβ in them are insignificant in the following; if one
likes, one can fix them as in the normal case (9.26). Moreover, even the definition (9.19)
of ταβ is completely inessential at all, as ταβ always appears in combinations like ταβδlαmβ
(see (9.19) or similar relations, like (9.27), below), which are non-vanishing if β = α, but
then ταα = τα; so one can freely write τα for ταβ in all such cases.
Equipped with (9.19) and (9.18), we can generalize (9.7) in different ways. For example,
the straightforward generalization of (6.16) is:
, [a+
nγ ]εβγ
+ 2δlαnγa
= 0 (9.27a)
a+lα , [a
, a−nγ ]εβγ
+ 2ταγδlαnγa
= 0 (9.27b)
, [a+
nγ ]εβγ
− 2ταβδlαmβa−nγ = 0 (9.27c)
a−lα , [a
, a−nγ ]εβγ
− 2δlαmβa−nγ = 0. (9.27d)
However, generally, the relations (9.19) do not convert (9.27) into identities. The reason is
that an equality/identity like (cf. (6.8))
[blα , cmβ ◦ dnγ ] = [blα , cmβ ]−εαβ ◦ dnγ + λαβγcmβ ◦ [blα , dnγ ]−εαγ , (9.28)
where blα , cmβ and dnγ are some creation/annihilation operators and λ
αβγ ∈ C, can be valid
only for
λαβγ = εαβ εαγ = 1/εαβ (εαβ 6= 0), (9.29)
which, in particular, is fulfilled if γ = β and εαβ = ±1. So, the agreement between (9.19)
and (9.27) depends on the concrete choice of the numbers εαβ . There exist cases when even
the normal case (9.26) cannot ensure (9.19) to convert (9.27) into identities; e.g. when the
α-th field and β-th fields are fermion ones and the γ-th field is a boson one. Moreover, it can
be proved that (9.19) and (9.27) are compatible in the general case if unacceptable equalities
like a±
◦ a±m = 0 hold.
One may call (9.27) the multifield paracommutation relations as from them a correspond-
ing generalization of (6.18) and/or (6.20) can be derived. For completeness, we shall record
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 49
the multifield version of (6.20):
[blα , [b
, bnγ ]εβγ ] = 2δlαmβbnγ [blα , [bmβ , bnγ ]εβγ ] = 0 (9.30a)
[clα , [c
, cnγ ]εβγ ] = 2δlαmβcnγ [clα , [cmβ , cnγ ]εβγ ] = 0 (9.30b)
lα , [c
, cnγ ]εβγ ] = −2ταγδlαnγb
lα , [b
, bnγ ]εβγ ] = −2ταγδlαnγc
. (9.30c)
For details regarding these multifield paracommutation relations, the reader is referred to [17,
18], where the case τα = τβ = ταβ = 0 is considered.
We leave to the reader as exercise to write down the multifield versions of the commuta-
tion relations (6.22) or (6.23), which provide examples of generalizations of (9.7) and hence
of (9.19) and (9.27).
9.2. Commutation relations connected with the charge and
angular momentum operators
In a case of several, not less than two, different fields, the basic trilinear commutation rela-
tions (6.33), which ensure the validity of the Heisenberg relation (5.2) concerning the charge
operator, read:
a±lα , [a
]εβ − [a+mβ , a
− 2δlαmβa±lα = 0 (9.31a)
lα , [a
]εβ − [a+mβ , a
+ 2δlαmβa
lα = 0. (9.31b)
Of course, these relations hold only for those fields which have non-vanishing charges, i.e.
in (9.31) is supposed (see (9.1))
τα = 0 τβ = 0 (⇐⇒ qαqβ 6= 0). (9.32)
The problem for generalizing (9.31) for these fields is similar to the one for (9.7) in the
case of non-vanishing charges, τβ = 0. Without repeating the discussion of Subsect. 9.1,
we shall adopt the rule (9.18) for generalizing (anti)commutation relations between cre-
ation/annihilation operators of a single field. By its means one can obtain different general-
izations of (9.31). For instance, the commutation relations.
, a−nγ ]εβγ − [a+mβ , a
nγ ]εβγ
− 2δlαnγa+mβ = 0 (9.33a)
a−lα , [a
, a−nγ ]εβγ − [a+mβ , a
nγ ]εβγ
− 2δlαmβa−nγ = 0 (9.33b)
and their Hermitian conjugate contain (9.31) and (6.35) as evident special cases and agree
with (9.19) if γ = β and εαβεβγ = +1. Besides, the multifield paracommutation rela-
tions (9.27) for charged fields, τα = τβ = τγ = 0, convert (9.33) into identities and, in this
sense, (9.33) agree with (contain as special case) (9.27) for charged fields. As an example
of commutation relations that do not agree with (9.27) for charged fields and, consequently,
with (9.33), we shall point the following ones:
a±lα , [a
nγ ]εβγ
+ δlαnγa
= 0 (9.34a)
, a−nγ ]εβγ−
− δlαnγa±mβ = 0, (9.34b)
which are a multifield generalization of (6.34).
The consideration of commutation relations originating from the ‘orbital’ Heisenberg
equation (5.4) is analogous to the one of the same relations regarding the charge operator.
The multifield version of (6.49) is:
(−ω◦µν(mβ) + ω◦µν(nγ))([ã±lα , [ã
, ã−nγ ]εβγ
+ [ã+
nγ ]εβγ ] )
nγ=mβ
= 4(1 + ταβ)δlαmβω
α)(ã±
) (9.35a)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 50
(−ω◦µν(mβ) + ω◦µν(nγ))([ã
lα , [ã
, ã−nγ ]εβγ
+ [ã+
nγ ]εβγ ] )
nγ=mβ
= 4(1 + ταβ)δlαmβω
α)(ã
lα ) (9.35b)
where
ω◦µν(l
α) := ωµν(k) = kµ
if lα = (α, sα,k). (9.36)
Applying (6.51), with mβ for m and nγ for n, one can check that the multifield paracom-
mutation relations (9.27) convert (9.35) into identities and hence provide a solution of (9.35)
and ensure the validity of (5.4), when system of different free fields is considered. An example
of a solution of (9.35) which does not agree with (9.27) is provided by the following multifield
generalization of (6.52):
a+lα , [a
nγ ]εβγ
a+lα , [a
, a−nγ ]εβγ
= −(1 + ταγ)δlαnγa+mβ (9.37a)
a−lα , [a
nγ ]εβγ
a−lα , [a
, a−nγ ]εβγ
= +(1 + ταβ)δlαmβa
nγ , (9.37b)
which provides a solution of (9.5). Notice, the evident multifield version of (6.53) agrees
with (9.5), but disagrees with (9.35) when the lower signs are used.
At last, the multifield exploration of the ‘spin’ Heisenberg relations (5.5) is a mutatis
mutandis (see (9.35)) version of the corresponding considerations in the second part of Sub-
sect. 6.3. The main result here is that the multifield bilinear commutation relations (9.19),
as well as their para counterparts (9.27), ensure the validity of (5.5).
9.3. Commutation relations between the dynamical variables
The aim of this subsection is to be discussed/proved the commutation relations (5.15)–(5.24)
for a system of at least two different quantum fields from the view-point of the commutation
relations considered in subsections 9.1 and 9.2.
To begin with, we rewrite the Heisenberg relations (5.1), (5.2) and (5.4) in terms of
creation and annihilation operators for a multifield system [1,11]:
, Pµ] = ∓kµa±lα [a
, Pµ] = ∓kµa†±lα (9.38)
[a±lα , Q] = qa
lα [a
lα , Q] = −qa
lα (9.39)
,Morµν ] = i~ω◦µν(lα)
,Morµν ] = i~ω◦µν(lα)
, (9.40)
where lα = (α, sα,k), ω◦(lα) is defined by (9.36) and k0 =
m2c2 + k2 is set in (9.38)
and (9.40) (after the differentiations are performed in the last case). The corresponding
version of (5.5) is more complicated and depends on the particular field considered (do not
sum over sα!):
[a±α,sα(k),Mspµν ] = i~gα
αtα,+
µν (k)a
α,tα(k) +
αtα,−
µν (k)a
α,tα(k)
α,sα(k),Mspµν ] = i~hα
αtα,−
µν (k)a
α,tα(k) +
αtα,+
µν (k)a
α,tα(k)
(9.41)
where fsα = −1, 0,+1 (depending on the particular field), gα := −hα := 1jα+δjα0 (−1)
jα+1 and
sαtα,+
µν (k) and
sαtα,−
µν (k) are some functions which strongly depend on the particular field
considered, with ±σ
sαtα,±
µν (k) being related to the spin (polarization) functions σ
sαtα,±
µν (k)
(see (3.14) and (3.11)).34 As a result of (5.6), (9.40) and (9.41), one can easily write the
Heisenberg relations (5.3) in a form similar to (9.38)–(9.41).
34 If φ̃αi (k) are the Fourier images of the α-th field and
i (k) =
i (k)ã
α,sα(k) + v
i (k)ã
α,sα(k)
, (9.42)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 51
The commutation relations involving the momentum operator are:
[Pµ, Pν ] = 0 [Q, Pµ] = 0
[Sµν , Pλ] = [Mspµν , Pλ] = 0
[Lµν , Pλ] = [Morµν , Pλ] = [Mµν , Pλ] = −i~{ηλµ Pν − ηλν Pµ}.
(9.45)
We claim that these equations are consequences from (9.38) and the explicit expressions (3.9)–
(3.12) and (5.11)–(5.13) for the operators of the dynamical variables of the free fields con-
sidered in the present work. In fact, since (9.38) implies
[b±lα ◦ c
, Pµ] = 0 lα = (α, sα,k), mβ = (β, sβ ,k) (9.46a)
[b±lα
←−−−→
ω◦µν (l
α) ◦ c∓
, Pµ] = ±2(kµηνλ − kνηµλ)b±lα ◦ c
, (9.46b)
where b±lα , c
lα = a
lα , a
lα and
←−−−→
ω◦µν (l
α) is defined via (9.36) and (3.13), the verification of (9.45)
reduces to almost trivial algebraic calculations. Further, we assert that any system of commu-
tation relations considered in Subsect. 9.1 entails (9.45): as these relations always imply (9.5)
(or similar multifield versions of (6.10) and (6.11) in the case of the Lagrangians (3.1) or (3.3),
respectively) and, on its turn, (9.5) implies (5.1), the required result follows from the last
assertion and the remark that (5.1) and (9.38) are equivalent. As an additional verification
of the validity of (9.45), the reader can prove them by invoking the identity (6.8) and any
system of commutation relations mentioned in Subsect. 9.1, in particular (9.19) and (9.27).
The commutation relations concerning the charge operator read:
[Pµ, Q] = 0 [Q, Q] = 0
[Lµν , Q] = [Sµν , Q] = 0
[Morµν , Q] = [Mspµν , Q] = [Mµν , Q] = 0.
(9.47)
These equations are trivial corollaries from (3.9)–(3.12) and (5.11)–(5.13) and the observation
that (9.39) implies
lα ◦ a
, Q] = [a±lα ◦ a
, Q] = 0 if qα = qβ, (9.48)
due to (6.8) for η = −1. Since any one of the systems of commutation relations mentioned in
Subsect. 9.2 entails (9.31) (or systems of similar multifield versions of (6.31) and (6.32), if the
Lagrangians (3.1) or (3.3) are employed), which is equivalent to (9.39), the equations (9.47)
hold if some of these systems is valid. Alternatively, one can prove via a direct calculation
that the commutation relations arising from the charge operator entail the validity of (9.47);
where v
i (k) are linearly independent functions normalize via the condition
i (k)
i (k) = δ
, (9.43)
with fs
= 1 for jα = 0, 1
and fs
= 0,−1 for (jα, sα) = (1, 3) or (jα, sα) = (1, 1), (1, 2), respectively, then
µν (k) :=
i (k)
µν (k) :=
i (k)
(9.44)
with Ii
iµν given via (5.25). Besides, σ
µν (k) =
µν (k) with an exception that σ
µν (k) = 0 for
α = 1
and (µ, ν) = (a, 0), (0, a) with a = 1, 2, 3.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 52
for the purpose the identity (6.8) and the explicit expressions for the dynamical variables via
the creation and annihilation operators should be applied.
At last, consider the commutation relations involving the different angular momentum
operators:
[Pλ, Sµν ] = [Pλ,Mspµν ] = 0
[Pλ, Lµν ] = [Pλ,Morµν ] = [Pλ,Mµν ] = +i~{ηλµ Pν − ηλν Pµ}
[Q, Lµν ] = [Q, Sµν ] = [Q,Morµν ] = [Q,Mspµν ] = [Q,Mµν ] = 0
[Sκλ,Mµν ] = −i~
ηκµ Sλν − ηλµ Sκν − ηκν Sλµ + ηλν Sκµ
[Lκλ,Mµν ] = −i~
ηκµ Lλν − ηλµ Lκν − ηκν Lλµ + ηλν Lκµ
[Mκλ,Mµν ] = −i~
ηκµMλν − ηλµMκν − ηκνMλµ + ηλνMκµ
(9.49)
(The other commutators, that can be form from the different angular momentum operators,
are complicated and cannot be expressed in a ‘closed’ form.) The proof of these relations is
based on equations like (see (9.40) and (6.8))
[blα ◦ cmβ ,Morµν ] = i~ω◦µν(lα)
blα ◦ cmβ
lα = (α, sα,k), mβ = (β, sβ ,k), (9.50)
with blα , clα = a
lα , a
lα , a
lα , a
lα , and similar, but more complicated, ones involving the other
angular momentum operators. It, generally, depends on the particular field considered and
will be omitted.
As it was said in Subsect. 6.3, the Heisenberg relations concerning the angular momentum
operator(s) do not give rise to some (algebraic) commutation relations for the creation and
annihilation operators. For this reason, the only problem is which of the commutation
relations discussed in subsections 9.1 and 9.2 imply the validity of the equations (9.49) (or
part of them). The general answer of this problem is not known but, however, a direct
calculation by means of (9.7), if it holds, and (6.8) shows the validity of (9.49). Since (9.19)
and (9.27) imply (9.7), this means that the multifield bilinear and para commutation relations
are sufficient for the fulfillment of (9.49).
To conclude, let us draw the major moral of the above material: the multifield bilinear
commutation relations (9.19) and the multifield paracommutation relations (9.27) ensure
the validity of all ‘standard’ commutation relations (9.45), (9.47) and (9.49) between the
operators of the dynamical variables characterizing free scalar, spinor and vector fields.
9.4. Commutation relations under the uniqueness conditions
As it was said at the end of the introduction to this section, the replacements (9.4) ensure the
validity of the material of Sect. 4 in the multifield case. Correspondingly, the considerations
in Sect. 7 remain valid in this case provided the changes
l 7→ lα m 7→ mβ n 7→ nγ
τδlm 7→ ταβδlαmβ = ταδlαmβ
[bm, bm]ε 7→ [bmβ , bmβ ]εβ [bm, bn]ε 7→ [bmβ , bnγ ]εβγ ,
(9.51)
with bm (or bmβ ) being any creation/annihilation operator, and, in some cases, (9.4) are
made.35 Without going into details, we shall write the final results.
The multifield version of (7.27)–(7.28) is:
E(a†±
◦ a∓nγ ) = εβγ E(a∓nγ ◦ a
E([a†±
, a∓nγ ]εβγ) (9.52)
35 As a result of (7.11), (7.16) and (7.17), in expressions like (7.18)–(7.26) the number ε should be replace
by εαβ, where α and β are the corresponding field indices of the creation/annihilation operators on which the
operator E acts, i.e. ε E(bm ◦ bn) 7→ ε
βγ E(bmβ ◦ bnγ ).
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 53
a+lα , E([a
nγ ]εβγ)
+ 2δlαnγa
= 0 (9.53a)
, E([a†+
, a−nγ ]εβγ )
+ 2ταγδlαnγa
= 0 (9.53b)
a−lα , E([a
nγ ]εβγ)
− 2ταβδlαmβa−nγ = 0 (9.53c)
, E([a†+
, a−nγ ]εβγ )
− 2δlαmβa−nγ = 0 (9.53d)
γ =β. (9.53e)
As one can expect, the relations (9.53a)–(9.53d) can be obtained from the multifield paracom-
mutation relations (9.27) via the replacement [·, ·]ε 7→ E([·, ·]εβγ ). It should be paid special
attention on the equation (9.53e). It is due to the fact that in the expressions for the dynami-
cal variables do not enter ‘cross-field-products’, like a
for β 6= α, and it corresponds to
the condition (ii) in [17, p. B 1159]. The equality (9.53e) is quite important as it selects only
that part of the ‘ E-transformed’ multifield paracommutation relations (9.27) which is com-
patible with the bilinear commutation relations (9.19) (see (9.28) and (9.29)). Besides, (9.53e)
makes (9.53a)–(9.53d) independent of the particular definition of εαβ (see (9.11)).
The equations (9.52) are the only restrictions on the operator E ; examples of this operator
are provided by the normal (resp. antinormal) ordering operator N (resp. A), which has the
properties (cf. (4.22) (resp. (7.30))
◦ a†−nγ
:= a+
◦ a†−nγ N
◦ a−nγ
◦ a−nγ
◦ a†+nγ
:= εβγa
nγ ◦ a−mβ N
◦ a+nγ
:= εβγa+nγ ◦ a
(9.54)
◦ a†−nγ
:= εβγa
nγ ◦ a+mβ A
◦ a−nγ
:= εβγa−nγ ◦ a
◦ a†+nγ
:= a−
◦ a†+nγ A
◦ a+nγ
◦ a+nγ .
(9.55)
The material of Sect. 8 has also a multifield variant that can be obtained via the re-
placements (9.51) and (9.4). Here is a brief summary of the main results found in that
The operator E should possess the properties (9.54) and, in this sense, can be identified
with the normal ordering operator,
E = N . (9.56)
As a result of this fact and εββ = εβ (see (9.11)), the commutation relations (9.53) take the
final form:
a+lα , a
◦ a†−
+ δlαnβa
= 0 (9.57a)
a+lα , a
+ ταβδlαnβa
= 0 (9.57b)
a−lα , a
◦ a†−
− ταβδlαmβa−nβ = 0 (9.57c)
− δlαmβa−nβ = 0 (9.57d)
which is the multifield version of (8.17) and corresponds, up to the replacement a±lα 7→
2a±lα ,
to (9.27) with εβγ = 0.
The vacuum state vector X0 is supposed to be uniquely defined by the following equations
(cf. (8.1b)–(8.3)):
a−lα X0 = 0 a
lα X0 = 0 (9.58a)
X0 6= 0 (9.58b)
〈 X0| X0〉 = 1 (9.58c)
lα ◦ a
(X0) = δlαmβ X0 a−lα ◦ a
(X0) = δlαmβ X0
(X0) = ταβδlαmβ X0 a−lα ◦ a
(X0) = ταβδlαmβ X0.
(9.58d)
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 54
The Hilbert space F of state vectors is a direct sum of the Hilbert spaces Fα of the
different fields and it is supposed to be spanned by the vectors
... = M(a
, . . . )(X0) (9.59)
with M(a+
, . . . ) being arbitrary monomial only in the creation operators.
Since (9.58a), (9.56) and (9.54) imply the multifield version of (8.7), the computation of
the mean values of (8.6), with l1 7→ lα11 etc., of the dynamical variables is reduced to the one
of scalar products like (cf. (8.5))
〈ψlα1
...|φmβ1
〉 = 〈 X0|
, . . . )
)† ◦ M′(a+
, . . . )(X0)〉 (9.60)
of basic vectors of the form (9.59). By means of the basic properties (9.58) of the vacuum,
one is able to calculate the simplest forms of the vacuum mean values (9.60), viz. the mul-
tifield versions (see (9.51)) of (8.20) and (8.26). But more general such expression cannot
be calculated by means of (9.57)–(9.58). Prima facie one can suppose that the multifield
commutation relations (9.19), which ensure the vectors (9.59) to form a base of the system’s
Hilbert space of states, can help for the calculation of (9.60) in more complicated cases. In
fact, this is the case which works perfectly well and covers the available experimental data.
In this connection, we must mention that the applicability of (9.19) for calculation of (9.60)
is ensured by the compatibility/agreement between (9.19) and (9.57): by means of (6.8) for
η = −εαβ, one can check that (9.19) converts (9.57) into identities.36
The commutation relations (9.57) admit as a solution also the multifield version of the
anomalous bilinear commutation relations (8.27) but it, as we said earlier, leads to contradic-
tions and must be rejected. The existence of solutions of (9.57) different from it and (9.19)
seems not to be investigated. If there appear date which do not fit into the description by
means of (9.19), one should look for other, if any, solutions of (9.57) or compatible with (9.57)
effective procedures for calculating vacuum mean values like (9.60).
10. Conclusion
In this paper we have investigated two sources of (algebraic) commutation relations in the
Lagrangian quantum theory of free scalar, spinor and vector fields: the uniqueness of the
dynamical variables (momentum, charge and angular momentum) and the Heisenberg rela-
tions/equations for them. If one ignores the former origin, which is the ordinary case, the
paracommutation relations or some their generalizations seems to be the most suitable can-
didates for the most general commutation relations that ensure the validity of all Heisenberg
equations. The simultaneous consideration of the both sources mentioned reveals, however,
their incompatibility in the general case. The outlet of this situation is in the redefinition
of the operators of the dynamical variables, similar to the normal ordering procedure and
containing it as a special case. That operation ensures the uniqueness of the new (redefined)
dynamical variables and changes the possible types of commutation relations. Again, the
commutation relations, connected with the Heisenberg relations concerning the (redefined)
momentum operator, entail the validity of all Heisenberg equations.
36 Recall, equations (9.19) and (9.27), or (9.53a)–(9.53d), for γ 6= β are generally incompatible. For instance,
excluding some special cases, like systems consisting of only fermi (bose) fields or one fermi (bose) field and
arbitrary number of bose (fermi) fields, the only operators satisfying (9.19) and (9.27) for γ 6= β and having
normal spin-statistics connection are such that bmβ ◦ bnγ = 0, with γ 6= β and bmβ and cnγ being any
creation/annihilation operators, which, in particular, means that no states with two particles from different
fields can exist.
Bozhidar Z. Iliev: QFT in momentum picture: IV. Commutation relations 55
Further constraints on the possible commutation relations follow from the definition/in-
troduction of the concept of the vacuum (vacuum state vector). They practically reduce the
redefined dynamical variables to the ones obtained via normal ordering procedure, which
results in the explicit form (8.17) of the admissible commutation relations. In a sense,
they happen to be ‘one half’ of the paracommutation ones. As a last argument in the way
for finding the ‘unique true’ commutation relations, we require the existence of procedure
for calculation of vacuum mean values of anti-normally ordered products of creation and
annihilation operators, to which the mean values of the dynamical variables and the transition
amplitudes between different states are reduced. We have pointed that the standard bilinear
commutation relations are, at present, the only known ones that satisfy all of the conditions
imposed and do not contradict to the existing experimental data.
The consideration of a system of at least two different quantum free fields meets a new
problem: the general relations between creation/annihilation operators belonging to differ-
ent fields turn to be undefined. The cause for this is that the commutation relations for any
fixed field are well defined only on the corresponding to it Hilbert subspace of the system’s
Hilbert space of states and their extension on the whole space, as well as the inclusion in
them of creation/annihilation operators of other fields, is a matter of convention (when free
fields are concerned); formally this is reflected in the structure of the dynamical variables
which are sums of those of the individual fields included in the system under consideration.
We have, however, presented argument by means of which the a priori existing arbitrari-
ness in the commutation relations involving different field operators can be reduced to the
‘standard’ one: these relations should contain either commutators or anticommutators of
the creation/annihilation operators belonging to different fields. A free field theory cannot
make difference between these two possibilities. Accepting these possibilities, the admissible
commutation relations (9.57) for system of several different fields are considered. They turn
to be corresponding multifield versions of the ones regarding a single field. Similarly to the
single field case, the standard multifield bilinear commutation relations seem to be the only
known ones that satisfy all of the imposed restrictions and are in agreement with the existing
data.
Acknowledgments
This research was partially supported by the National Science Fund of Bulgaria under Grant
No. F 1515/2005.
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Introduction
The momentum picture
Lagrangians, Euler-Lagrange equations and dynamical variables
On the uniqueness of the dynamical variables
Heisenberg relations
Types of possible commutation relations
Restrictions related to the momentum operator
Restrictions related to the charge operator
Restrictions related to the angular momentum operator(s)
Inferences
State vectors, vacuum and mean values
Commutation relations for several coexisting different free fields
Commutation relations connected with the momentum operator. Problems and their possible solutions
Commutation relations connected with the charge and angular momentum operators
Commutation relations between the dynamical variables
Commutation relations under the uniqueness conditions
Conclusion
References
This article ends at page
|
0704.0067 | Order of Epitaxial Self-Assembled Quantum Dots: Linear Analysis | Order of Epitaxial Self-Assembled Quantum Dots: Linear Analysis
Lawrence H. Friedman
November 4, 2018
Dept. of Engineering Science and Mechanics, Pennsylvania State University, 212 Earth and Engineering Science
Building, University Park, Pennsylvania 16802
[email protected]
keywords: quantum dots, strained films, epitaxial growth, semiconductors
Abstract
Epitaxial self-assembled quantum dots (SAQDs) are of interest for nanostructured optoelectronic and electronic
devices such as lasers, photodetectors and nanoscale logic. Spatial order and size order of SAQDs are important
to the development of usable devices. It is likely that these two types of order are strongly linked; thus, a study of
spatial order will also have strong implications for size order. Here a study of spatial order is undertaken using a linear
analysis of a commonly used model of SAQD formation based on surface diffusion. Analytic formulas for film-height
correlation functions are found that characterize quantum dot spatial order and corresponding correlation lengths that
quantify order. Initial atomic-scale random fluctuations result in relatively small correlation lengths (about two dots)
when the effect of a wetting potential is negligible; however, the correlation lengths diverge when SAQDs are allowed
to form at a near-critical film height. The present work reinforces previous findings about anisotropy and SAQD order
and presents as explicit and transparent mechanism for ordering with corresponding analytic equations. In addition,
SAQD formation is by its nature a stochastic process, and various mathematical aspects regarding statistical analysis
of SAQD formation and order are presented.
1 Introduction
Epitaxial self-assembled quantum dots (SAQDs) represent an important step in the advancement of semiconductor
fabrication at the nanoscale that will allow breakthroughs in optoelectronics and electronics. [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12] Most frequent optoelectronic applications are high efficiency lasers with exotic wavelengths or photode-
tectors. [1, 3, 4, 5, 6, 7, 8, 10, 11, 12] SAQDs are the result of a transition from 2D growth to 3D growth in strained
epitaxial films such as SixGe1−x/Si and InxGa1−xAs/GaAs. This process is known as Stranski-Krastanow growth or
Volmer-Webber growth. [13, 1, 14, 15].
In applications, order is a key factor. There are two types of order, spatial and size. Spatial order refers to
the regularity of SAQD dot placement, and it is necessary for nano-circuitry applications. Size order refers to the
uniformity of SAQD size which determines the voltage and/or energy level quantization of SAQDs. It is reasonable
to expect that these type of order are linked, and it is important to understand the factors that determine SAQD order.
Further understanding should help in the design and simulation of both spontaneous “bottom up” self-assembly and
directed or guided self-assembly to enhance SAQD order. [16, 17, 18, 19, 20, 21, 22, 23]Here, an elaboration of and
further application of a linear analysis of SAQD order [24] is presented. The work reported here forms the basis of a
non-linear theory and modeling of SAQD order that will be reported in future work.
In [24] it was reported that one could calculate a correlation function using a linearized model of SAQD formation.
This correlation function included two correlation lengths that could be used to describe SAQD order. It was also found
that one effect of a hypothesized wetting potential was to enhance SAQD order when growth occurs near the critical
film height for 3D growth. Here, these results are expanded to create a more rigorous linearized theory of SAQD order
that will inform non-linear theories. In particular, the model is generalized to any model that combines local energy
effects such as surface energy density and non-local elastic destabilization, and the procedure for predicting order
based on any linear theory with peak wavelengths is presented. The hypothesized effect of elastic anisotropy in [24]
is verified with calculations using linear anisotropic elasticity theory. [25, 26] Details such as statistical fluctuation
and convergence are also addressed along with a discussion of the possible forms of linear anisotropic terms in SAQD
growth kinetics, and the effect of an atomic-scale cutoff in the continuum theory is addressed. Finally, the order
enhancing effect of growing near the critical threshold is explored in more detail using calculations appropriate to
Ge/Si SAQDs.
In the literature, two modes of SAQD formation are generally discussed, the thermal nucleation mode and the nu-
cleationless mode. [27, 28, 29] In the thermal nucleation mode, a 2D film surface is metastable, and the formation of
individual quantum dots is thermally activated. [27]. This growth mode leads to the formation of individual quantum
dots as uncorrelated or loosely correlated discrete events at essentially random locations. In the nucleationless mode,
the 2D film surface transitions from stable (or metastable) to unstable. In this mode, dots form everywhere at once
appearing at first as a cross-hatched ripple-like disturbance on the 2D film surface and then maturing into recogniz-
able individual dots.[27, 30, 28, 31, 32] These two modes are probably connected via an encompassing conceptual
and mathematical model1, and perhaps some of what is observed experimentally is in fact a hybrid mechanism. In
agreement with intuition, it appears that the nucleationless mode leads to a more ordered dot pattern than the thermal
nucleation mode that is dominated by randomness. 2 Thus, the presented analysis applies to the nucleationless mode.
There are various implementations of nucleationless growth models [28, 37, 38, 39, 40, 18, 34], although, there is
also a great deal of commonality among these models. In particular, they all include a non-local elastic effect and local
surface energies and/or local wetting energies. Here, a linear analysis of quantum dot order resulting from this class of
model is presented. Particular note is taken of the effects of stochastic initial conditions crystal anisotropy in general,
elastic anisotropy in particular, and the effect of varying film height as a control parameter as first introduced in [33].
A simple model similar to [28, 37, 38, 40, 18] is presented to produce numerical examples and explore the effects
of the average film height. Concurrently, a more abstract and general model is presented and analyzed that includes
non-local elastic strain effects, and a local combined surface and wetting energy. The linear model with stochastic
initial conditions and deterministic film height evolution will pave the way for more sophisticated analysis involving a
non-linear model of stochastic film height evolution.
As previously stated, one of the goals in the present work is to further explore the role of the wetting poten-
tial during growth near the stability threshold in film height. A wetting potential has been included in the analysis
and simulations in [38, 33, 37, 28]. Although somewhat controversial, the wetting potential plays an important phe-
nomenological role. It ensures that growth takes place in the Stranski-Krastanow mode: that a 3D unstable growth
occurs only after a critical layer thickness is achieved, and that a residual wetting layer persists. The physical origins
and consequences of the wetting potential are discussed in [41, 28]. The analysis presented here is usable in models
that neglect the wetting potential by simply setting it to zero. Another possibility is simply that the wetting potential
is simply an approximation to the stabilizing effect of intermixing. [42] That said, if the wetting potential is real, the
present analysis shows that it is beneficial to SAQD order to grow near the critical layer thickness.
The presented analytic formulas and linear analysis are intended to complement existing numerical models of
SAQD order. [43, 37, 44, 45] and to form a basis for future non-linear analytic analysis of SAQD order. The current
findings agree with previous work on the beneficial effects of elastic anisotropy to enhance in-plane order.
The linear analysis, of course, represents a simplification of the film evolution, and it applies only to the initial
stages of SAQD formation when the nominally flat film surface becomes unstable and transitions to three-dimensional
growth. However, the small surface fluctuation stage of SAQD growth determines the initial seeds of order or disorder
in an SAQD array; thus, the small fluctuation stage should have an important influence on the final outcome. At later
stages when surface fluctuations are large, there is a natural tendency of SAQDS to either order or ripen [33, 37, 46,
39, 47] Ordering systems tend to evolve slowly due to critical slowing down [39], while ripening tends to diminish
order further. [37] Thus, it is possible that the linear model could, in fact, yield good predictions of SAQD order.
The simplification and linearizion facilitates the development of analytic solutions that are most transparent, easily
portable to multiple material systems and have no effective limit on system size. Finally, it is virtually impossible to
have a thorough understanding of the full non-linear model without first having a thorough understanding of the linear
behavior.
The remainder of the paper is organized as follows. Section 2 presents the physical assumptions and mathematical
1It is likely that there a transition from stable, to metastable and finally to unstable. The analysis presented in [33] would appear to support such
a view where the film height acts as the control parameter driving the transition. There is also some controversy regarding whether all dot growth is
nucleationless or not. [34, 32]
2Compare various figures in [29, 35, 14, 31, 36].
approximations used to model film growth. Section 3 discusses the stochastic initial conditions and the resulting
correlation functions and correlation lengths. Section 4, presents a procedure for estimating SAQD order with an
application to Ge dots on a Si substrate. Section 5 presents conclusions, while Appendices A-F present additional
calculational details.
2 Modeling
The formation of SAQDs is modeled as a deterministic surface diffusion process with stochastic initial conditions. The
resulting equations and ultimately the sought after correlation functions are different depending on whether the film
surface is treated as one-dimensional isotropic, two-dimensional isotropic or two-dimensional anisotropic. The 1D
and 2D isotropic cases are discussed first, and then the essential differences of the 2D anisotropic model are presented.
The stochastic initial conditions need to be expressed in terms of the correlation functions that are also use to analyze
order; consequently, the discussion of the initial conditions is deferred to Sec. 3.2.
It should be noted that the results presented here are fairly general. There has been a good deal of recent work
refining the modeling of nucleationless growth processes to incorporate various phenomenological aspects of SAQD
growth. For example, the inclusion of orientation-dependent surface energy [38], strain-dependent surface energy [34]
and explicit modeling of atomic species segregation and film-substrate inter diffusion. [48] Two models are presented
here. One is a simple concrete example. It is the simplest model one can use including elastic effects surface energy
and wetting energy. The second model is more abstract and describes the general case of a local potential energy
that depends on both the film height and film height gradient. One effect that is not examined here is that of mixed
4-fold and two-fold symmetry. Such a mixing can occur due to diffusional anisotropy or surface energy anisotropy.
(Sec. 2.2.1.2 and Appendix D). However, a similar analysis procedure should work for these cases as well. The general
procedure for possible application to other models is discussed in Sec. 3.5.
The following discussion will use abstract vector notation, e.g. k instead of ki, etc. Also, because it is sometimes
computational expedient to perform one-dimensional modeling [24, 39, 17, 42], the case of a one dimensional surface
with two dimensional volume is discussed along with the case of an isotropic 2D surface. To facilitate this combined
discussion, the dimensionality of the surface will be denoted as d. In Secs. 3.3 and 3.4, d = 1, 2 will be substituted as
appropriate. Finally, much of the calculation involves reciprocal space. The convention used for the Fourier transforms
f(x) =
ddk eik·xfk, and fk = (2π)
ddx e−ik·xf(x)
following the example of [28].
2.1 1D and 2D Isotropic model
This discussion pertains to both the 1D model and the 2D isotropic model. The formation of SAQDs is modeled as a
surface diffusion process where the film height is a function of the lateral position and time. The system is treated as
deterministic with stochastic initial conditions. First, the general non-linear governing equations are presented. Then,
the linearized form is presented. Finally, the key behavior is reviewed.
The mathematical model uses film height, H(x, t) as the dependent variable and the horizontal position x and
time t as the independent variables. The film height evolves over time due to surface diffusion driven by a diffusion
potential µ(x, t) and a flux of new material Q. The surface velocity is thus
vn = nz∂tH = −∇S · D∇Sµ(x, t) +Q (1)
where nz is the vertical component of the surface normal nz = [1 + (∇H)2]−1/2, ∇S is the surface gradient, D is the
diffusivity, and ∇S · is the surface divergence.
2.1.1 Energetics
The diffusion potential µ(x, t) must produce Stranski-Krastanow growth. Thus, it must contain an elastic term that
destabilizes film growth, a surface energy term that stabilizes planar growth and a wetting energy that ensures a wetting
layer. The diffusion potential can be derived from a total free energy.
F = Felast + Fsurf. + Fwet
volume
dV ω +
surface
dAsurf. γ +
dAW (H)
where ω is the elastic energy density, γ is the surface energy density, W (H) is the wetting energy density. The
last integral corresponds to Fwet, and whether the integral should be taken over the film surface or the substrate is
ambiguous. The “simple” model (Sec. 2.1.1.1) assumes that the integral is over the substrate, while the “general”
model (Sec. 2.1.1.2) can accommodate both cases.
2.1.1.1 simple form The simplest possible model results if the integral corresponding to Fwet is taken over the
lateral positions x rather than over the actual free-surface. In concrete terms, one can use dV = d2xdz and dAsurf. =
1 + (∇H(x))2
to obtain the expression,
volume
d2xdz ω[H](x, z) +
x-plane
1 + (∇H(x))2
γ +W (H(x))
, (2)
where the “ω[H]” indicates that the elastic energy density is a non-local functional of the film height,H. The diffusion
potential µ can be found, similar to [15], by differentiating F with respect to the surface motion (Appendix A.1),
µ(x) = ΩδF/δH(x). Doing so for Eq. (2) (Appendix A.2),
µ(x) = Ω [ω(x)− γκ(x) +W ′ (H(x))] . (3)
where Ω is the atomic volume, ω(x) is the elastic energy density at the film surface (implicitly ω[H] (x,H(x))),
κ = ∇ ·
∇H(x)
1 + (∇H(x))2
]−1/2}
is the total surface curvature, and W ′(H) = ∂H(x)W (H(x)) is the
derivative of W (H(x)) evaluated at x.
2.1.1.2 general form It should be noted that Eq. (3) is not the same diffusion potential used in [38]. The wetting
potential used there can be derived by taking W (H) as an energy density of the free surface, not a density in the
x-plane. Expressions like Eq. (3) and Eq. (1) in [38] are part of a larger class of surface evolution models with more
or less the same linear behavior.
The surface and wetting energy can be combined and incorporated into a more general form, with a total free
energy Fsw and a free energy density Fsw(H,∇H) that depends on the film heightH(x) and the film height slope or
orientation ∇H(x). The total free energy is thus
F = Felast. + Fsw (4)
volume
d2xdz ω[H](x, z) +
x−plane
d2xFsw (H(x),∇H(x)) .
Fsw may not necessarily be the sum of separate surface energy and wetting energy contributions. It need only be a
local function ofH and ∇H. The corresponding diffusion potential is
µ(x) = Ω
ω(x) + F (10)sw (x)−∇ · F
sw (x)
, (5)
where F (mn)sw indicates the mth derivative with respect to H and the nth derivative with respect to ∇H. F
sw (x) =
∂H(x)Fsw (H(x),∇H(x)) and each vector component of F
sw (x) is
F(01)sw (x)
= ∂[∇H(x)]
Fsw (H(x),∇H(x)).
One can obtain the results of the simple model (Eqs. (2) and (3)) by setting
Fsw =
1 + (∇H(x))2
γ +W (H(x)) . (6)
A diffusion potential like Eq. (1) in [38] can be obtained by setting
Fsw =
1 + (∇H(x))2
[γ (∇H(x)) +W (H(x))] .
This is different from Eq. (6) in two ways. First, the surface energy density depends on the surface orientation. Second,
the Jacobian, J =
1 + (∇H(x))2
multiplies both the surface energy density and the wetting potential. Despite
these differences, the common form of the diffusion potential (Eq. (5)) among different models suggests that they
might all lead to similar linearized forms and behavior.
2.1.1.3 Linearization The diffusion potential is now linearized about the average film height H̄. In general, one
can control the amount of deposited material, and thus the average film height H̄. It is therefore useful to decompose
H(x) into the spatially averaged mean value and fluctuations about the average. Similar to [28],
H = H̄+ h(x, t). (7)
In the present calculation, H̄ is specified as constant in time. This assumption corresponds physically to a fast deposi-
tion and then an anneal. It is not too difficult to generalize to a time dependent H̄, but that is beyond the scope of this
manuscript. In [38, 49], deposition and evaporation is explicitly modeled.
All terms in µ(x, t) are now kept to only linear order in h(x, t). The elastic energy density ω is a non-local
functional of h(x, t) [40]; however, the equations generating ω(x) are translationally invariant. Thus, it is convenient
to use reciprocal space for the linearization. The curvature is trivially linearized as κ(x) → ∇2h(x) in real space or
κk → −k2hk in reciprocal space. The linearized elastic strain energy ω can be found in reciprocal space as in [15]
to be ωk = −2M(1 + ν)�2mhk, where M = E/(1 − ν) is the biaxial modulus, E is the Young modulus, ν is the
Poisson ratio, and �m is the film-substrate mismatch strain. This formula neglects possible differences in elastic moduli
between the film and substrate as in [28], but a similar method of analysis should apply to that case as well. Linearizing
Eqs. (3) and (5) in reciprocal space, µk is proportional to hk with a proportionality coefficient that depends on k and
µlin,k = f(k, H̄)hk (8)
where f(k, H̄) for three different isotropic cases, corresponding to Eqs. (3) and (5), and an abstracted general form, is
given by
f(k, H̄) =
−2M(1 + ν)�2mk + γk2 +W ′′(H̄)
; case a (Eq. (3))
−2M(1 + ν)�2mk + F 02k2 + F 20
; case b (Eq. (5))
−ak + bk2 + c ; case c (general)
. (9)
Due to isotropy, f(k, H̄) is independent of the direction of k, and only the wave number, k = ‖k‖, appears in the
right hand side. F (20)sw is the second derivative of Fsw with respect to H, and F
sw the second derivative of Fsw with
respect to ∇H. F (20)sw and F
sw depend on H̄ only; thus they are constants in the present analysis. See Appendix B.2
for more precise definitions and the derivation of f(k, H̄). Using Eq. (6), produces F (02)sw = γ and F
sw = W ′′(H̄)
which is identical to the simple case of Eq. (9), a. Case c, labeled as “general” where a, b, and c depend implicitly on
H̄ shows that f(k, H̄) for cases a and b have the same relatively simple form. It also emphasizes the dynamic effects
as opposed to the physical causes. There is a destabilizing term, −ak, a short wavelength cutoff term, bk2, and a term
that stabilizes the entire spectrum, c.
Despite the label “general,” there are of course limitations to the application of Eqs. (8) and (9). For example,
there has been recent work on the effects of strain-dependent surface energies. [34] The second form can not represent
such an effect because the derivation assumes that the surface energy only depends on local quantities, (H and ∇H)
whereas the strain effect is non-local. However, it is reasonable to conjecture that a more detailed analysis of the
effects of a strain dependent surface energy term would produce a coefficient function f(k, H̄) not very different from
the case c “general” form of Eq. (9). Thus, the following analysis may very well apply to this more exotic model, but
more study is needed to be certain.
2.1.2 Dynamics
As discussed in Sec. 2.1.1, the dynamics are derived assuming no flux of new material (Q = 0) and keeping only
terms to linear order in the height fluctuation, h(x, t). Under these assumptions, Eq. (1) can be decomposed into a
Table 1: Characteristic wave-numbers, characteristic times and associated dimensionless variables for the three cases
addressed in Eq. (9).
kc tc α β
case a 2M(1+ν)�
16DΩM4(1+ν)4�8m
γW ′′(H̄)
4M2(1+ν)2�4m
case b 2M(1+ν)�
(F (02)sw )
16DΩM4(1+ν)4�8m
F (02)sw F
4M2(1+ν)2�4m
case c a/b b3/(DΩa4) k/kc cb/a2
trivial equation for H̄ and an equation for the film height fluctuation by inserting Eq. (7).
dH̄/dt = 0 (10)
∂th(x) = −∇ · D∇µlin(x) (11)
where µlin(x) is the inverse Fourier transform of Eqs. (8) and (9), and it depends implicitly on the average film height
H̄. Note that the time dependence is implicitly while the coordinate dependence is explicit. The explicit coordinate
dependence serves to distinguish Assuming that the diffusivity D is constant, the Fourier transform of Eq. (11) gives
the linearized differential equation for the evolution of each Fourier component.
∂thk = −Dk2µk = −Dk2f(k, H̄)hk. (12)
Solving Eq. (12),
hk(t) = hk(0)e
σkt; (13)
σk = −Dk2f(k, H̄). (14)
The surface evolves in reciprocal space as an initial condition, hk(0) multiplied by an envelope function, eσkt. For
most values of H̄, this envelope function has a peak. As time passes, this peak narrows and can be approximated by a
gaussian. To analyze this behavior, appropriate dimensionless variables are defined. Then, the stability of the film is
discussed. Finally, σk is expanded about its peak to aid analytic calculations.
The time dependent behavior of the film height fluctuations is facilitated by using a characteristic wave number,
characteristic time and related dimensionless variables. For the “general” case c of Eq. (9), the characteristic wavenum-
ber is kc = a/b, and the characteristic time is tc = 1/(DΩbk4c ) = b3/(DΩa4). These characteristic dimensions can be
used to define a dimensionless wave vector, α = k/kc and a dimensionless wetting parameter β = c/(bk2c ) = cb/a
One can also define a dimensionless time, τ = t/tc. To obtain the corresponding characteristic scales for cases a and
b, one merely has to plug in the appropriate substitutes for a, b and c and follow the pattern. For example, for case
a, make the substitution a → Ω2M(2 + ν)�2m, etc. Table 1 summarizes these values for all three cases. For all three
cases, f(k, H̄) and the growth constant σk reduce to the following forms:
f(k, H̄) = f(kcα, H̄) = Ωbk2c
−α+ α2 + β
σk = σkcα = t
α− α2 − β
, (16)
where α = ‖α‖ = k/kc is the dimensionless wave number. These forms are plotted in Figs. 1a and 2. Fig. 1a shows
f(k, H̄)/Ωbk2c vs. α for an isotropic or one dimensional surface. Figs. 2 shows tcσk vs. α for a 2D anisotropic
surface (Sec. 2.2). However, the curves marked 0◦ are identical to the dispersion relation for a 1D or 2D isotropic
surface (compare Eqs. (9) and (23)).
2.1.3 Peaks
The peak growth rate and the corresponding wavenumber k can be found from Eq. (16). σk has a peak at k0 = kcα0
where
9− 32β
. (17)
Expanding σk about this peak to second order in k − k0,
σk ≈ σ0 −
σ2(k − k0)2
! ! 0.25!
increasing θk
Figure 1: Dimensionless diffusion potential prefactors vs. dimensionless wave number. (a) The one dimensional or
isotropic case with β = 0.3. (b) The elastically isotropic case with anisotropy �A = 0.1 (see Eq. (22)).
increasing θk
22.5◦
increasing θk
22.5◦
Figure 2: Dimensionless growth constant vs. dimensionless wave number. Curves are plotted for the elastically
anisotropic case, but the curves marked 0◦ are the same as for the isotropic cases. In (a), β = 0. In (b) β = 0.2.
Figure 3: Exponential Envelope eσkt as function of α for β = 0.208 and t/tc = 100. (a) 2D isotropic surface. (b) 2D
anisotropic surface with � = 0.1236.
The two constants are
t−1c α
0 (α0 − 2β) , (18)
σ2 = t
c (3α0 − 4β) . (19)
Inserting this approximation for σk into Eq. (13),
hk(t) = hk(0)e
σ0te−
2σ2t(k−k0)
. (20)
The individual initial surface fluctuation components grow with a gaussian shaped envelope. An example of this
envelope is plotted in Fig. 3(a). Notice that in two dimensions, the envelope forms a ring as the peak is about the
wave-number k0 but not about any particular point in the k-plane.
2.1.4 Stability and wetting potential
Stranski-Krastanow growth is marked by a transition from stable two-dimensional growth to unstable three-dimensional
growth once a critical height Hc is reached. [1] Eqs. (17), (18) and (20) are useful for analyzing the transition from
stable to unstable growth. In order for this transition to occur, there must be some stabilizing term in the diffusion
potential. In the present model, this means that there must be some surface energy-like term that varies strongly with
film height. This condition equates to stating that W ′′(H̄) or F 20sw or c (Eq. (9)) must be rather large if H̄ < Hc.
However, as H̄ increases, these terms are reduced. Finally, when H̄ > Hc, this term is no longer capable of stabilizing
the film against fluctuations of all possible wavelengths.
The critical value Hc can be found using the analysis from [33]. By inspection of Eqs. (8), (9) and (12), modes
with f > 0 increase the total free energy F as they grow; thus, they are stable and decay with time. Modes with
f < 0 decrease the total free energy F as they grow; thus, they are unstable and grow with time. This growth and
decay rule is easily verified by inspection of Eq. (14). Thus, stable growth occurs when f(k, H̄) > 0 for all values of
k, and unstable growth occurs when f(k, H̄) < 0 for some values of k. Thus, the transition from stable to unstable
growth occurs when the minimum value of f(k, H̄) just becomes negative. Using the same dimensional analysis as in
the previous section and following the discussion of [33], one finds that the minimum value, fmin = Ωbk2c (β − 1/4),
occurs at kmin/kc = αmin = 1/2. fmin first becomes negative, and the transition to unstable growth occurs when
the dimensionless wetting parameter (Table 1) drops to a critical value, β = 1/4 . β > 1/4 stable 2D growth, and
β < 1/4 unstable 3D growth. It is reasonable to suppose that W (H̄), W ′′(H̄), and thus β are positive monotonically
decreasing functions of H̄ so that the interface becomes less important for large values of H̄. For example, in [50] it is
assumed that W (H) = B/H, where B is constant. When β → 0, corresponding to large H̄, the case discussed in [28]
is obtained. A similar analysis can be done for cases b and c once one specifies how the terms F (20)sw and F
sw or a, b
and c depend on H̄.
Using a guessed form for a wetting potential, one can find the critical film heightHc by setting β = 1/4 . Applying
this condition to case a in Eq. (3)
W ′′(Hc) = γk2c/4.
Using the wetting potential of [50] as an example, W (H) = B/H,
Hc = 3
8B/(γk2c ) =
8Bγ/(2M(1 + ν)�2m)2. (21)
Conversely, one can fit a wetting potential to an observed or reasonable critical layer thickness from the same condition.
Using the example wetting potential from [50],
(2M(1 + ν)�2m)
as stated in [50].3
2.2 2D Anisotropic case
Crystal anisotropy leads to a dispersion relation σk that is both quantitatively and qualitatively different from the
isotropic case. Here the effect of elastic anisotropy is discussed in most detail. Other sources of anisotropy are
the surface and wetting energies. For example, in [38] the surface energy density is orientation dependent which
introduces a possible anisotropy in the dispersion relation. Possible sources of anisotropy are an anisotropic elastic
stiffness tensor, an orientation dependent surface energy or wetting potential or anisotropic diffusion. As discussed
below, the form of anisotropy to linear order in the height fluctuation, h, is somewhat restricted. Results are presented
for 4-fold symmetric surfaces, that is surfaces that have invariant dynamic evolution laws when rotated by 90◦. Possible
complications arising from 2-fold symmetric anisotropic terms (with 180◦ rotational symmetry) are also discussed. As
for the isotropic case, first the energetics are discussed, then the dynamics, and finally the expansion about the peaks
in the dispersion relation, σk.
2.2.1 Energetics
The discussion of energetics will first treat the effects of elastic anisotropy and then anisotropy resulting from surface
or wetting like terms.
3This result from [50] corresponds to the choice Fsw(H,∇H) =
1 + (∇H)2γ+W (H). However, the numerical model in [50] appears to
use Fsw(H,∇H) =
1 + (∇H)2 [γ(∇H) +W (H)]. This difference should lead to a slightly different critical film height in their numerical
model from the one that they predicted (Eq. (21)).
Figure 4: Plot of Eθk/(M�
m) for various materials. Symbols indicate values calculated using Appendix C. Solid lines
are the interpolation (Eq. (22)) using the values from Table 2.
Table 2: Elastic constants [51] and calculated values (see Appendix C) for various materials of interest at T = 300K.
c11 c12 c44 M
1011 ergcm3 10
11 erg
cm3 10
11 erg
cm3 10
11 erg
Ge 12.60 4.40 6.77 13.93 2.16 1.906 0.1176
Si 16.60 6.40 7.96 18.07 2.22 1.997 0.1005
InAs 8.34 4.54 3.95 7.94 2.70 2.09 0.226
GaAs 11.90 5.34 5.96 12.45 2.15 1.87 0.1302
2.2.1.1 Elastic anisotropy One would like to obtain a simple symbolic expression for the elastic energy density at
the free surface, ωk, to first order in hk for the elastically anisotropic case. Similar discussions can be found in [25, 26].
For the isotropic case, ωk = −2M(1 + ν)�2mhk. For the anisotropic case,
ωk = −Eθkkhk
where the prefactor Eθk is the decrease in elastic energy at the surface per unit wave number (k → 1) and unit amplitude
(hk → 1) . It is not constant, but instead depends on the θk, the angle that k makes with the x−direction. The case of a
cube-symmetry elastic stiffness tensor such as for Si is considered where one must specify three elastic constants c11,
c12 and c44. [51]. Growth on a (100) surface will produce an elastic energy prefactor Eθk that is four-fold symmetric
(symmetric upon rotations by 90◦). A procedure similar to [25, 26] based on a first order perturbation analysis is
followed (Appendix C). A relatively simple interpolation formula [24] is hypothesized and then verified numerically.
The interpolation procedure, suggested in [24] uses the lowest possible order expansion in sin(θk) and cos(θk)
that has the appropriate four-fold symmetry and then interpolates between θk = 0◦ and θk = 45◦. Thus,
Eθk = E0◦
1− �A sin2 (2θk)
where �A = (E0◦ − E45◦)/E0◦ is an anisotropy factor. This lowest order form turns out to be a very good fit to
numerical calculations (Fig. 4). Table 2 gives values of E0◦ and �A for some systems of interest. In the elastically
isotropic case, E0◦ = E45◦ = 2M(1 + ν) so that �A = 0.
There are two important differences from the elastically isotropic case. The first is obvious, that Eθk depends on
angular orientation, θk. The second is that the peak value of ωk is not the same as that for the elastically isotropic case
because in general, E0◦ 6= 2M(1 + ν). In [24], where the purpose was simply to investigate the mechanism by which
elastic anisotropy effects order, this second difference was neglected.
2.2.1.2 Surface and Wetting Energy Anisotropy The surface energy and wetting potential can be additional
sources of anisotropy if they depend on the surface orientation so that γ → γ(∇H) or W (H) → W (H,∇H) (for
example, [52, 38]). Then, to first order in h ,
µsurf.,k = Ω
γk2 + k · γ̃′′ · k
where γ̃′′ is the (2×2) matrix or Hessian matrix that results from taking the second derivatives of γ(∇H) with respect
to the two components of ∇H (Appendix B.1). Similarly
µwet,k = Ω
W (20) + k · W̃(02) · k
where W (20) and W̃(02) are the second derivatives of W (H,∇H) with respect to H and ∇H (Appendix B.1). For
both µsurf.,k and µwet,k, the first term is isotropic, and the second term contains any possible anisotropy.
The rank of the γ̃′′ and W̃(02) matrices greatly restricts the possible forms of the additional anisotropy. These
(2 × 2) matrices must be either two-fold symmetric or perfectly isotropic. Thus, if the surface energy and wetting
potential are four-fold symmetric as Eθk is, then γ̃
′′ → γ′′, a scalar, and W̃(02) → W (02), a scalar, and neither one
contributes any additional anisotropy. They do, however, help to stabilize or further destabilize the 2D surface as they
add terms proportional to k2. The effect of these additional terms is indistinguishable from the effect of varying the
value of the surface energy density, γ. [52, 31]
It should be noted that the (100) surface of a diamond or zinc-blend structures allows for anisotropy that is only 2-
fold symmetric (rotations by 180◦). Thus, they could “break” the four-fold symmetry that occurs when one considers
the elastic anisotropy alone. However, this “broken” symmetry is somewhat dubious because even the diamond and
zinc-blend structures have a screw symmetry (rotations by 90◦ and translation in the [100] direction by half a lattice
vector). Thus, if for example, W (H,∇H) is anisotropic with two-fold symmetry to linear order, there must be a
fast oscillation with changes in the film height H. In Appendix D, a similar term related to anisotropic diffusion is
discussed. There does not appear to be any evidence for this two-fold symmetry in the case of (100) surfaces of IV/IV
systems such as Ge/Si, but in III-V/III-V systems the four-fold symmetry of the (100) surface may indeed be “broken”
in this way corresponding to either a surface energy anisotropy or a diffusional anisotropy. [53, 54]. Further analysis
of such terms in any more detail would greatly complicate the present discussion, so it is left for future work. Most
of the modeling literature avoids this complication by not including the symmetry-breaking of the zinc-blend surface,
for example [25, 26, 38].
One can perform a similar analysis of the combined surface and wetting potential, Fsw(H,∇H) (case b). To linear
order the resulting anisotropic diffusion potential is (Appendix B.2)
µsw,k = Ω
F (20)sw + k · F̃
sw · k
Again, F̃(02)sw is a rank 2 tensor, and all of the same symmetry considerations apply here as well.
Because the two-fold symmetry anisotropic terms are excluded from the current discussion, and isotropic terms
simply “renormalize” the effective of surface energy, there will be no further consideration of anisotropy resulting
from the surface energy or wetting potential in this discussion. Further calculations will proceed assuming that the
surface energy density, γ, nor the wetting potential,W (H), depend on ∇H or similarly that Fsw(H,∇H) has a purely
isotropic dependence on ∇H. This assumption can be made without affecting any of the qualitative results.
2.2.1.3 total diffusion potential Having dispensed with the discussion of the various sources of anisotropy, the
total diffusion potential is stated for the case of 4-fold symmetric elastic anisotropy and a completely isotropic surface
energy and wetting potential. µk = f(k, H̄) with
f(k, H̄) =
1− �A sin2(2θk)
k + γk2 +W ′′(H̄)
; case a (Eq. (3))
1− �A sin2(2θk)
k + F (02)sw k2 + F
; case b (Eq. (5))
1− �A sin2(2θk)
+ bk2 + c ; case c (general)
. (23)
Table 3: Characteristic wave-numbers, characteristic times and associated dimensionless variables for the three cases
addressed in Eq. (9)
kc tc α β
case a E0◦/γ γ3/(DΩE40◦) k/kc γW ′′(H̄)/E20◦
case b E0◦/F
/(DΩE40◦) k/kc F
sw /E20◦
case c a/b b3/(DΩa4) k/kc cb/a2
2.2.2 Dynamics
The dynamics is governed by surface diffusion, just as for the fully isotropic case. It is assumed that the diffusivity
is isotropic as was done for the surface energy and the wetting energies; thus, all anisotropy in the film evolution
dynamics comes from elastic effects alone. The possibility and effects of an anisotropic diffusion potential is discussed
in Appendix D (also see [54]). The time dependence of the surface perturbations simply follows Eqs. (13) and (14),
but with Eq. (23) used for f(k, H̄). As for the isotropic case, appropriate characteristic wave numbers (kc) and
time scales (tc) can be found for each of the three cases along with the associated dimensionless wave vector α and
dimensionless wetting parameter β. These are listed in Table 3. The dispersion relation, σk can be expressed in terms
of these dimensionless variables (α and β), giving
σk = σkcα = t
1− �A sin2(2θk)
− α2 − β
. (24)
The stability behavior is essentially the same as for the isotropic case with a transition occurring at β = 1/4 corre-
sponding to H̄ = Hc.
2.2.3 Expansion about peaks
σk has 4 peaks at (k, θk) = (k0, π[n− 1]/2) with k0 = kcα0 (Eq. (17)) and n = 1 . . . 4. In vector form, there are four
peaks at
kn = k0 (cos(π(n− 1)/2)i + sin(π(n− 1)/2)j) .
Similar to the isotropic case, σk can be expanded about individual peaks so that in the vicinity of peak n, σk ≈ σn
σn = σ0 −
σ‖(k − k0)2 −
0(θk − nπ/2)
where σ0 is given by Eq. (18), σ‖ = σ2 given by Eq. (19), and
σ⊥ = 8�Aα0t
In terms of the vector components parallel and perpendicular to kn, k‖ and k⊥ respectively,
σn = σ0 −
σ‖(k‖ − k0)2 −
k‖ = cos[π(n − 1)/2]kx + sin[π(n − 1)/2]ky , and k⊥ = − sin[π(n − 1)/2]kx + cos[π(n − 1)/2]ky . The time
evolution of hkin the vicinity of one of the kn is
hk(t) ≈ hk(0)et(σ0−
2σ2(k‖−k0)
2− 12σ⊥k
3 Correlation Functions
Correlation functions and associated constants such as correlation lengths can be very useful for characterizing order.
In particular, the autocorrelation function (Eq. (25)) and its Fourier transform (Eq. (26)) also known as the spectrum
function can give a very good characterization of dot order (Figs. 6a and c and 5b, e and h). The autocorrelation
function is denoted CA(∆x) where ∆x is the difference vector between two points in the x−plane. The spectrum
function is a function of k, and it is denoted CAk . The goal here is to be able to predict these two functions and to
h(nm)
h(nm)
h(nm)
describe them quantitatively in a manner that can be used to characterize SAQD order with just a few numbers. The au-
tocorrelation function is the result of a spatial average over one experiment or one simulation (numerical experiment).
It is regular and repeatable because it is closely tied to the correlation function and spectrum function that results from
an ensemble average (Eqs. X and X). These are denoted as C(∆x) and the spectrum Ck respectively. Note that the
ensemble averaged functions do not have a superscript “A.” These ensemble average correlation functions are useful
in the analysis of stochastic ordinary and partial differential equations. [55, 56]. From a strictly technical viewpoint,
the spatial average and the ensemble average are not exactly the same; however, they are closely enough connected
that it is reasonable to use one as a substitute for the other (Sec. 3.1 and Appendix 3).
In the following, the analysis of SAQD order via autocorrelation and correlation functions is discussed (Sec. 3.1).
Then, the stochastic initial conditions are discussed (Sec. 3.2). Then, the prediction of the Fourier transforms of the
correlation functions is discussed (Sec. 3.3). The real-space correlation functions are presented (Sec. 3.4). Finally,
there are some notes regarding generalizing the analysis method to any dispersion relation that has peaks (Sec. 3.5),
for example, peaks related to broken four-fold symmetry or growth on a miscut substrate.
3.1 Correlation Functions and SAQD order
Auto-correlation functions are well-suited for investigating SAQD order. The autocorrelation function is defined as
CA(∆x) =
d2x′ h(∆x + x′)h(x′)∗. (25)
Its Fourier transform sometimes called the spectrum [56], spectrum function or power spectrum is
CAk =
(2π)d
d2∆x e−ik·∆xC(∆x) =
(2π)d
, (26)
where A is the projected area of the film in the x − y−plane. A periodic array of SAQDs leads to a periodic auto-
correlation function. A nearly periodic array leads to a range-limited periodic auto-correlation function. The ensemble-
mean of these autocorrelation functions can be calculated, and it is a good predictor of a SAQD order.
3.1.1 Periodic array
Consider a perfectly periodic height fluctuation corresponding to a perfect lattice of SAQDs,
h(x) =
exp [ikn · (x− xO)] (27)
plus higher order harmonic, where the dots have a height proportional to h0, N is the degree of symmetry, probably,
4-fold or 6-fold, xO is a random origin offset.
kn = k0
2π(n− 1)
i + sin
2π(n− 1)
In a linear analysis, the higher order harmonics do not come into play, so they are neglected here. In reciprocal space,
e−ikn·xOδd(k− kn)
plus higher order harmonic. The autocorrelation function is found by plugging Eq. (27) into Eq. (25) and simplifying,
CA(∆x) =
)2 N∑
exp [ikn ·∆x] (28)
plus higher order harmonic. In finding Eq. (28), the relation∫
d2x′ ei(km−kn)·x
= Aδkmkn = (2π)
dδd(km − kn) (29)
has been used. δkk′ is the Kronecker Delta, and δd(k− k′) is the Dirac Delta. Eq. (29) will be helpful whenever it is
necessary to take an areal average or sum over Dirac Delta functions. In reciprocal space,
CAk =
(2π)2
m,n=1
δ2(k− km)δ2(k− kn)
δ2(k− ki) (30)
plus higher order harmonics, where δd(k − kn) = (A/(2π)d)δkkn .4. Thus, the order of the SAQD lattice manifests
itself as periodic functions in real-space (Eq. (28)) and sharp peaks in reciprocal space (Eq. (30)).
3.1.2 Nearly Periodic array
A nearly periodic arrays shows deviation from perfect order. This deviation is shows itself by broadening of the peaks
of the spectrum function, CAk , and by range limited periodicity of the real-space autocorrelation function, C
A(∆x).
These two measure of disorder are naturally related.
The disorder in lateral dot size ∆size and spacing, ∆spacing are related to each other and to the broadening of the
peaks in CAk (Fig. 6.a and c). Prior to ripening, the size and spatial order should be related, as the volume of a dot
should be proportional to the amount of nearby material. If the SAQDs have nearly uniform size and spacing (peak-
to-peak distance) L0, the reciprocal space autocorrelation function will be tightly clustered around the wavenumber
characterizing the dot spacing k0 = 2π/L0. There are a number of such peaks depending on the system symmetry
(Fig. 6.a and c), but consider just one. Since the order is not perfect, the peak will have a finite width. Consequently,
there will be a scatter in the dot size. Since L0 = 2π/k0, the scatter in dot spacing (∆spacing) is related to the scatter in
Fourier components (∆k). Taking the derivative of the spacing-wavenumber relation and rearranging,
∆spacing
It is reasonable to expect that the fractional disorder in size (∆size /Lsize) is given by a similar (if not exactly the same)
number.
Another way to view spatial order (periodicity) is not by dot-dot distances, but the distance over which the dot array
can be considered periodic. This limited periodicity is evident in the film height autocorrelation function (Eq. (25) and
Figs. 5.b, e and h). Consider two distant dots. Their position will be completely uncorrelated, so it will be completely
random as to whether one position corresponds to a peak or a valley. Thus, for a large differences in position the
autocorrelation function tends to zero.
CA(∆xlarge) = 0
Similarly, the mean-square fluctuation of the film height can be large so that
CA(∆x = 0)� 0.
The distance over which the autocorrelation function, CA(∆x) decays to 0 is the correlation length, Lcor. Thus, Lcor
is a reasonable measure of spatial order.
The two measures of order ∆spacing and Lcor are intrinsically linked. The well known rule of Fourier transforms
states that the product of the real-space and reciprocal space widths must be greater than or equal to unity. Thus,
∆kLcor ≥ 1, or ∆spacing ≥ 2πL20/Lcor. Similarly, one can expect that ∆size ∼ L2size/Lcor. Thus, assuming that dot size
is governed by the amount of nearby material, small dispersions in dot size are only possible if there is long correlation
length.
3.1.3 Ensemble Correlation Functions / ergodicity
SAQDs are seeded by random fluctuations. Consequently, each experiment or simulation must be treated as just
one possible realization, and the autocorrelation function will be different for each realization. Thus, for analytic
4Eq. (29) has been used to help with summation
predictions, one must rely on ensemble averages. In [24], it was assumed that the ensemble average correlation
function was a good description of a SAQD order, an assumption that was born out by numerical calculations. Now,
this relation is put on a more solid ground. In particular, it is found that the ensemble correlation functions provide
good estimates of the auto correlation function and spectrum function produced by any particular realization. First,
it is shown that the mean value of the film-height fluctuation is zero. Then the method to calculate the ensemble-
averaged autocorrelation function and spectrum function is presented. Additional mathematical details are presented
in Appendix 3.
3.1.3.1 Mean fluctuation It is fairly straightforward to show that the ensemble mean film-height fluctuation is
zero. The governing dynamics (Eq. (12)) is invariant upon the substitution h(x, t) → −h(x, t). Thus, assuming that
one does not bias the initial conditions the mean fluctuations must be zero for all time,
〈h(x, t)〉 = 〈−h(x, t)〉 = 0, and 〈hk(t)〉 = 0.
This is a common situation, and it is most appropriate to characterize the film height fluctuations using the two-point
correlation function (or simply “the correlation function”). [55]
3.1.3.2 Correlation Function The autocorrelation function can be estimated by its ensemble average. Further-
more, this ensemble average is equivalent to the correlation function that can be easily calculated analytically. These
relations are first discussed for the real-space correlation functions and then their Fourier transforms. First, the statis-
tical properties of the autocorrelation function are discussed. Then the statistical properties of the spectrum function.
Finally, the method to The main results are reported here, and details of derivations are reported in Appendix E.
First it is noted that the autocorrelation function averaged over all realizations is equal to the ensemble correlation
function. 〈
CA(∆x)
= C(∆x), where C(∆x) = 〈h(∆x)h(0)〉 , (31)
where 〈. . . 〉 indicate an ensemble average. Eq. (31) assumes that the model of film-growth is translationally invariant.5
This relationship is fortunate, in that it allows one to predict the “typical” autocorrelation function using analytic tools
that apply only to ensemble averages.
Second, it is noted that as the area that is used to calculate the autocorrelation function becomes large, the autocor-
relation function tends towards it mean value,
CA(∆x) ≈ C(∆x) +O[A−1/2], (32)
where O[A−1/2] indicates statistical fluctuations about the mean value that become smaller and smaller as the area in
an experiment or the simulation area in a numerical experiment becomes larger. These fluctuations or noise die out as
A1/2. For example, the autocorrelation functions in Figs. 5.e and h are very close to the ensemble average autocorre-
lation functions Figs. 5.f and h, but have random fluctuations that are most visible far from the origin. This property,
that averaging over a parameter such as position is equivalent to averaging over all realizations, is known as ergodicity.
Individual realizations are tightly distributed about a “typical” behavior. This tight distribution lends credibility to the
notion that one can have representative experiments or simulations. Unfortunately, the “demonstration” of Eq. (32) in
Appendix E is not as general as one might like. Rigorously, it applies when the Fourier components of film height (hk)
are independent and normally distributed; however, it is reasonable to conjecture that a relationship like Eq. (32) holds
whenever the statistical distribution of film heights is suitably bounded as the boundedness of CAk plays an important
role in the derivations.
In reciprocal space, one finds that the ensemble-mean spectrum function is〈
= Ck, (33)
where Ck is defined as the prefactor appearing in the reciprocal-space two-point correlation function.
Ckk = 〈hkh∗k〉 = Ckδ
d(k− k′) = Ck
(2π)d
δkk′ , (34)
5A quick survey of literature will find that, virtually all published continuum models of SAQD formation are translationally invariant.
where Eq. (29) has been used. This form for the two-point correlation function in reciprocal space occurs if and only
if the system is translationally invariant. Eq. (33) is valuable because one can solve for Ck analytically in the linear
case or using various analytic approximations in the non-linear case. Unlike the autocorrelation function, the spectrum
function fluctuates greatly about its mean. In fact, the fluctuations are about 100% (Appendix E.2). These large
fluctuations result in the commonly observed speckle pattern for the spectrum function CAk (Figs. 6.a and c). Contrast
this pattern with ensemble-mean spectrum function Ck shown in Figs. 6.b and d. These speckles can be removed by a
smoothing operation, and a relation similar to Eq. (32) results (Appendix E.2.2). Finally, it should be noted that just
as CAk is the Fourier transform of C
A(∆x), Ck is the Fourier transform of C(∆x) (Appendix E.1).
3.2 Stochastic Initial Conditions
To model or simulate the formation of SAQDs, it is absolutely essential to include some sort of stochastic effect. An
initially flat film h(x, 0) = 0 is in unstable equilibrium. Thus, to seed the formation of quantum dots, it is necessary
to perturb the flat surface. The simplest method to do this is to use stochastic initial conditions with deterministic
evolution. One can tenuously suppose that white noise initial conditions do not “bias” the ultimate evolution of the
film. [57] Thus, the initial conditions are taken from an ensemble with zero mean,
〈h(x, 0)〉 = 0. (35)
and a spatial correlation function,
C(x,x′, 0) = 〈h(x, 0)h(x′, 0)∗〉 = ∆2δd (x− x′) , (36)
where the brackets 〈. . . 〉 indicate an ensemble average, ∆ is the noise amplitude, and δd(x) is the d−dimensional
Dirac Delta function. White noise conditions have an infinite amplitude which is not physical. Thus, a minimum
modification can be made to “cut off” the infinite fluctuations.
C(x,x′, 0) =
(2πb20)
(x− x′)2
In the limit b0 → 0, this correlation function reverts to the white noise correlation functions.
In reciprocal space,
Ckk′(0) = 〈hk(0)h∗k′(0)〉
= (2π)−2d
ddx′ e(−ik·x+ik
′·x′)C(x,x′, 0)
(2π)d
δd(k− k′)
Letting b0 → 0, the white noise reciprocal space correlation function is obtained. Thus, the initial spectrum function
Ck(0) =
(2π)d
The atomic-scale has a small and short-lived influence on the final film morphology (Appendix F), but the cutoff
procedure is useful for choosing a reasonable value of ∆2. It seems reasonable to choose ∆2 so that the initial r.m.s.
fluctuation
C(0, 0) = 〈h(0, 0)h(0, 0)∗〉1/2is one monolayer (1 ML). Also, choosing b0 = 1 ML as the atomic scale
cutoff is
∆2 = (2π)d/2(1 ML)2+d, (38)
where the natural unit 1 ML is, of course, material dependent.
Using stochastic initial conditions, one can integrate individual initial conditions to obtain representative samples
and then average over many realizations, the Monte Carlo approach, or one can calculate analytically, the statistical
measures of the ensemble. The ensemble statistical measures are strongly related to the statistical measures of order
for an individual realization, so the second approach is opted for here. Thus, the predicted SAQD order is ultimately
stated in terms of ensemble correlation functions.
3.3 Reciprocal Space Correlation Functions
The reciprocal space correlation function, Ckk′ , and spectrum function, Ck, are calculated for the 1D and 2D isotropic
case and then for the 2D anisotropic case. Generally Ck includes the length scales introduced in Sec. 2.1.3 as well as
the atomic scale cutoff b0.
Ckk′ = 〈hk(t)h∗k′(t)〉 = e
(σk+σk′ )t 〈hk(0)hk′(0)∗〉
(2π)2
e(σk+σk′ )t−
δ2(k− k′). (39)
Without much error, b0 can be neglected in the exponential (Appendix F). Using Eq. (34), the spectrum function is
then identified as
(2π)d
e2σkt. (40)
Ck is now calculated for each model: 1D isotropic, 2D isotropic and 2D anisotropic.
3.3.1 one-dimensional
The one dimensional surface is the simplest, so it is treated first. The spectrum function is simply
e2σ0t−
2 (2σ2t)(k−k0)
Ck has a peak at k = ±k0i. One can easily read off the correlation length as
Lcor =
2σ2t = k
2(3α0 − 4β)(t/tc). (41)
so that
e2σ0t−
cor(k−k0)
This approximation is valid when k0Lcor � 1. In terms of kx,
e2σ0t
cor(kx−k0)
cor(kx+k0)
3.3.2 2D isotropic
The 2D isotropic case is very similar;
(2π)2
e2σ0t−
cor(k−k0)
, (42)
where Lcor is the same as in Eq. (41). It has maximum that forms a ring in the k−plane as graphed in Fig. 6.b.
3.3.3 anisotropic
The anisotropic spectrum function is
(2π)2
e2σ0t
‖(k‖−k0)
2− 12L
⊥ , (43)
where
2σ‖t = k
(6α0 − 8β)(t/tc), (44)
2σ⊥t = k
16�α0(t/tc), (45)
k‖ = cos[π(n− 1)/2]kx + sin[π(n− 1)/2]ky , and k⊥ = − sin[π(n− 1)/2]kx + cos[π(n− 1)/2]ky and it is graphed
in Fig. 6.d. This approximation is valid when k0L‖ � 1 and k0L⊥ � 1.
Figure 6: CAk and Ck for Ge/Si as discussed in Sec. 4. (a,b) 2D isotropic surface. Eq. (42) is used for Ck. (c,d) 2D
anisotropic surface. Eq. (43) is used for Ck.
Figure 7: 1D isotropic surface in real space for Ge/Si as discussed in Sec. 4. (a) Example of h(x) plotted over a length
of 8Lcor. (b) corresponding reals space correlation functions plotted for range ±4Lcor. Filled plot is an example of
CA(∆x). Solid line isC(∆x) (Eq. (46)).
Loosely speaking, one can argue that the isotropic case is similar to letting �A → 0 in Eq. (45) so that the
perpendicular correlation length is always 0 regardless of time. A more conservative approach would be to argue that
L⊥ ≈ 2π/k0 for the isotropic model via inspection of Figs. 6(a) and (b). Even still, the more conservative result
guarantees that the perpendicular correlation length will always be the same as the dot spacing; thus, it will always
limit SAQD order to the first nearest neighbor at best.
3.4 Real Space Correlation Functions
The real space correlation functionsC(∆x) are now calculated for the 1D and 2D isotropic cases and the 2D elastically
anisotropic case.
3.4.1 one-dimensional
In one dimension,
C(∆x) =
dkx e
ikx∆xCk
e2σ0t−
2/L2cor2 cos (k0x) . (46)
Thus, C(∆x) has a damped periodicity indicating that it is imperfectly periodic (Fig. 7).
3.4.2 2D isotropic
In two dimensions with elastic isotropy,
C(∆x) =
d2k eik·∆xCk
(2π)2
e2σ0t
dk kei(k∆x cos(θk−θ∆x)e−
cor(k−k0)
Performing the angular integration first,
C(∆x) =
e2σ0t
dk kJ0(k∆x)e
− 12L
cor(k−k0)
where J0 is the zeroth Bessel function. In general, this integral is best performed numerically; however, it can be
solved in two important cases: ∆x→ 0 and Lcor →∞ (corresponding to long times). In the first case,
C(∆x) =
e2σ0t
dk ke−
cor(k−k0)
Under the same conditions that Eq. (42) is valid (k0Lcor � 1), the lower limit of the integral can be approximated as
−∞ so that
C(∆x = 0) =
∆2k0√
2πLcor
e2σ0t. (47)
This function gives the mean square surface height fluctuation. In the second case where Lcor →∞, e−
cor(k−k0)
(2π)1/2L−1cor δ(k − k0), so that
C(∆x) =
∆2k0√
2πLcor
e2σ0tJ0 (k0∆x) . (48)
This correlation function is the most ordered case for a 2D isotropic surface. It is graphed in Fig. 5c.
3.4.3 anisotropic
To find the real-space correlation function for the elastically anisotropic case, it is best to find the contribution from
each peak and then sum so that
C(∆x) =
(2π)2
e2σ0t
Cn(∆x) (49)
where
Cn(∆x) =
d2k eik·xe−
‖(k‖−k0)
2− 12L
∆x can be decomposed into the directions parallel and perpendicular to kn, so that ∆x‖ = cos(π(n − 1)/2)∆x +
sin(π(n− 1)/2)∆y and ∆x⊥ = − sin(π(n− 1)/2)∆x+ cos(π(n− 1)/2)∆y. Thus,
Cn(∆x) =
dk‖ e
ik‖∆x‖− 12L
‖(k‖−k0)
dk⊥ e
ik⊥∆x⊥− 12L
2 (x2‖/L2‖+x2⊥/L2⊥)eik0x‖ .
Plugging into Eq. (49),
C(∆x) =
πL‖L⊥
e2σ0t
2 (x2/L2‖+y2/L2⊥) cos(k0x) + e
− 12 (x2/L2⊥+y2/L2‖) cos(k0y)
. (50)
3.5 Generalizability
The dynamics and analysis used here were for a specific model, but the general procedure for analyzing the order
resulting from a linearized model should hold for any model with well-separated peaks in the dispersion relation, σk.
The procedure to follow is:
1. Generate the dispersion relation, σk as some function of k.
2. Find the peaks in the dispersion relation, kn, (n = 1 . . . N )
3. Expand about the peaks to generate the peak values, σn, and local Hessian matrix,(
∂ki∂kj
The spectrum function is then approximately
Ck(t) ≈
(2π)2
e2σnt exp
t (k− kn) · H̃n · (k− kn)
. (51)
4. Find the Eigenvalues of the local Hessian matrix, (Hn)I and (Hn)II . They should be negative, if there is a
peak at kn
5. Use the eigenvalues to determine the correlation lengths, (Ln)I =
2 |(Hn)I | t and (Ln)II =
2 |(Hn)II | t.
The real-space correlation function is
C(∆x, t) ≈
(Hn)I (Hn)II
e2σnt exp
x · H̃−1n · x
eikn·x. (52)
The “goodness” of these approximate forms requires that (Ln)
I and (Ln)
II be much less than the spacing
between peaks in the correlation function so that the gaussians do not overlap greatly. A reasonable test for
this no-overlap condition is ‖kn‖ (Ln)I � 1 and ‖kn‖ (Ln)II � 1, assuming that the peaks are not large in
number or very closely spaced.
4 Order Predictions
The real-space correlation function formulas (Eqs. (46), (47), and (50)) and correlation length formulas (Eqs. (41), (44)
and (45)) can now be used to estimate the order of SAQDs. Ge on Si is chosen for this example because this system has
received the most attention from theoretical work [58, 38, 31, 18, 39, 41, 27, 25, 26, and others], and it is the simplest
since it involves the diffusion of a single species. The procedure described below tries to predict the amount of order
when an initial atomic-scale fluctuation becomes “large”. “Large” is taken to be greater than atomic-scale. Beyond
this point, one would expect non-linear terms to become important. An example is presented for Ge on Si at 600K to
compare and contrast the 2D anisotropic results with the 1D isotropic and 2D isotropic results. The predictions are
also compared with a linear numerical calculation on a discrete reciprocal-space grid to test the approximations made
and to illustrate the relation between the surface profile (h(x)), the example autocorrelation functions (CA(x) and
CAk ) and the ensemble correlation functions (C(∆x) and Ck). Figs. 6, 7 and 5 show these results. Finally, the relation
between average film height and order is investigated.
4.1 Ge at 600K
The formulations for the three discussed cases are implemented for Ge/Si at 600K. The correlation lengths are esti-
mated for the end of the linear regime where fluctuations become large (greater than atomic scale). First, appropriate
physical constants are used to give the corresponding correlation length and correlation functions vs. time. These in-
clude an initial average film height H̄ and a white noise amplitude ∆ (Eq. (38)). These initial conditions approximate
a film at the beginning of an anneal that immediately follows a rapid deposition. The time tlarge is found by solving for
the time where the mean-square fluctuations are atomic scale,
h(x, t)2
= C(∆x = 0) = 1 ML2. At this point, the
correlation lengths are calculated.
Physical constants for the 2D anisotropic calculation are taken as follows. The elastic constants for Ge at 600 K
are c11 = 1.199 × 1012, c12 = 4.01 × 1011(from cS = 3.991), c44 = 6.73. [51] Using aGe = 0.5658nm and
aSi = 0.5431nm, it is found that �m = 0.0418. Using the procedure from (Appendix C), M = 1.332× 1012dyn/cm2.
E0◦ = 4.96 × 109erg/cm3, and E45◦ = 4.35 × 109erg/cm3 , giving �A = 0.1236. The atomic volume is Ω =
2.27 × 10−23 cm3. The estimated surface energy density is γ = 1927 erg/cm2. The wetting potential is estimated
by picking a plausible critical surface height, Hc ≈ 4 ML = 1.132 nm and setting W (H) = E20◦H3c/(8γH) =
2.315 × 10−6/H erg/cm2. The resulting characteristic wave number is kc = 0.257 nm−1. The initial film height is
taken to be H̄ = Hc + 0.25 ML = 1.203 nm and then allowed to evolve naturally. Thus, β = 0.208, α0 = 0.5658,
k0 = 0.1456 nm−1, σ0 = 0.1192/tc, σ‖ = 0.864/(k2c tc), σ⊥ = 0.559/(k
c tc), L‖ = 0.744k
0 (t/tc)
1/2, and
L⊥ = 0.599k
0 (t/tc)
1/2. The unspecified diffusivity has been absorbed into the characteristic time tc. From Eq. (38),
∆2 = 0.0403 nm4, and Eq. (50) gives
C(0) =
1.223× 10−3tc/t
e0.02385t/tc nm2.
The initial infinitely rough surface undergoes a smoothing described by the tc/t factor. Then the surface roughens due
to the exponential. The initial divergent roughness is an artifact of the non-physical white noise with the atomic scale
cutoff b0 neglected (Appendix F). The time for the fluctuations to become “large” again are found by setting
C(0) = h2large (53)
where hlarge = 1 ML = 0.283 nm. The solutions are t1 = 0.01527tc or t2 = 430tc. The first solution is discarded
since it is due to the non-physical white noise. At tlarge = t2, L‖ = 105.8 nm, and L⊥ = 85.2 nm. Taking L⊥ as
more limiting, the correlation spans about n = k0L⊥/π = 3.95 islands across. The corresponding reciprocal space
(Eq. (43)) and real-space correlation function (Eq. (50)) are shown in Figs. 6.d and 5.f respectively.
A corresponding numerical experiment is performed. A periodic surface of size l = 96(2π/k0) is used. Random
initial conditions consistent with Eq. (38) are used for k−space points on a square grid bounded by kx, ky = ±2k0.
The relation between discrete and continuous Fourier components is used, (hk)discrete = [(2π)d/A]hk. Eqs. (13)
and (14) are used without any additional approximation to find hk at time t = tlarge. The resulting CAk , a portion of
the height profile h(x) and CA(∆x) are plotted in Figs. 6(c), 5(d) and 5(f) respectively.
Similar calculations can be performed for the one-dimensional and two-dimensional elastically isotropic cases.
Isotropic values used previously [58, 24] are about E = 1.361 × 1012 dyn/cm2 and ν = 0.198 giving M = E/(1 −
ν) = 1.697 × 1012 dyn/cm2 and E = 2M(1 + ν) = 7.10 × 109 erg/cm3. Using the same critical surface height,
Hc = 4 ML, W (H) = 4.74×10−6/H erg/cm2. The resulting characteristic wave number is kc = 0.368 nm−1. If the
film is grown to H̄ = Hc+0.25 ML = 1.203 nm and then allowed to evolve naturally, β = 0.208; thus, α0 = 0.5658,
k0 = 0.208 nm−1, σ0 = 0.1192/tc, σ2 = 0.864/(k2c tc), and Lcor = 0.744k
0 (t/tc)
1/2. In one dimension, Eq. (46)
is used to find the mean square height fluctuation. Using Eq. (38) with d = 1, ∆2 = 0.0568 nm3, and
C(0, t) = 0.01271(t/tc)
−1/2e0.0238t/tc .
Setting C(0, t) = (1 ML)2 = 0.0801 nm2, t1 = 0.0252tc, and t2 = 186.9tc. At t2, Lcor = 48.8 nm, and n =
k0Lcor/π = 3.24, so about 3 dots in a row should be well correlated. The corresponding numerical calculation of
size l = 96(2π/k0) is performed. A portion of h(x), CA(∆x) and C(∆x) are shown in Fig. 7. In two dimensions,
Eq. (47) is used to find
h(x, t)2
C(0, t) = 9.40× 10−4(t/tc)−1/2e0.0238t/tc .
Setting C(0, t) = 0.0801 nm2, t1 = 1.376 × 10−4tc, and t2 = 306tc. At t2, Lcor = 62.4 nm, and n = k0Lcor/π =
4.14, and correlation is expected to extend about 4 dots. However, it should be noted that this correlation is not
lattice-like. Corresponding numerical results and ensemble correlation functions are shown in Figs. 6 and 5.a-c.
4.2 General case of β
In [24] it was suggested that allowing the film to evolve with β close to the stability threshold could enhance the SAQD
correlation. It is interesting to note what happens for different values of β. Similar analytic and numerical calculations
are performed for the large film-height limit, β = 0, for the 2D anisotropic Ge/Si surface. For β = 0, tlarge = 40.3tc,
L⊥ = 30.0 nm, and n = k0L⊥/π = 1.84, so one to two dots in a row are expected to be well correlated. h(x)
and real-space correlation functions are shown in Figs. 5g-i. The range of order is significantly less than for the case
β = 0.208 (Sec. 4.1). For Si/Ge at 600K, the 2D anisotropic predictions for tlarge and L⊥ are shown in Fig. 8. In
general, the closer β is to the critical value 0.25, the longer the correlation length. One can manipulate equation (53)
to find that tlarge/tc varies approximately but not exactly as (β − 1/4)−1 × ln[h2large
�A/(∆2k2c )]. Consequently,
L⊥ ∼ (β − 1/4)−1/2. Furthermore, the appearance of hlarge and ∆2 inside the logarithm shows that the final order
estimates are not overly sensitive to the guesses for ∆2 and h2large. The divergence of L⊥ with β − 1/4 is initially
encouraging, but it is clear that for the parameters used for Ge/Si, subatomic control of the film height is needed to
yield significantly enhanced long range correlations. Also as one approaches this threshold, one can probably expect
thermal activation to nucleate subcritical SAQDs whose effect on supercritically formed SAQDs is uncertain. There
should be some interesting phenomena at the theH → Hc.
l arge c
l a r g e
k 0L �
� 0 �
l a r g e � �
Figure 8: tlarge and L⊥ vs. β for Si/Ge using the 2D anisotropic model as described in Sec. 4. Units are normalize to
characteristic time tc and predicted number of correlated dots (n = k0L⊥/π).
5 Discussion/Conclusions
The order of epitaxial self-assembled quantum dots during initial stages of growth has been studied using a common
model of surface diffusion with stochastic initial conditions. It has been shown that correlation functions of small
surface height fluctuations can be predicted analytically using corresponding ensemble average correlation functions.
These correlation functions are characterized by correlation lengths that can be predicted by analytic formulas given
certain reasonable assumptions about the diffusion potential and the height and lateral scale of initial atomic scale
random fluctuations. Thus, the linear model of film surface height evolution via surface diffusion has enabled analytic
predictions of epitaxial SAQD order that are valid for small film height fluctuations. To what extent the initial degree
of order persists into later stages of growth remains to be studied, but the order of initial stages should certainly have
a strong influence on final outcomes. Furthermore, the linear analysis should provide insight into the less tractable
non-linear behavior. These predictions of SAQD order have been used to investigate the role of crystal anisotropy and
initial film height.
Crystal anisotropy has been shown to play an important role in enhancing SAQD order as observed in previous
numerical simulations continuum and atomistic numerical simulations. [43, 37, 44, 45] If a four-fold symmetry is
assumed for the governing dynamics, the effect of crystal anisotropy to linear order is felt through elastic anisotropy
alone. It is shown that elastic anisotropy is required to produce a lattice-like structure of SAQDs. The enhanced spatial
order should in turn lead to enhanced size order, a consequence that must be confirmed with non-linear studies, but
appears to be true based on the present available literature.
The role of initial film height has been shown to greatly influence order. Growth near the critical film height for
dot formation can enhance order. This order enhancement comes from increasing the duration of the linear small-
fluctuation stage of growth. In fact, the predicted correlation lengths diverge when the initial film height approaches
the critical film height from above. Achieving large correlation lengths in this manor is of course practically limited
by ability to control film heights to subatomic accuracy. Additionally, one should be careful when interpreting the
continuum model in such a context, as the effect of atomic discreteness might be greater at the transition film height.
Finally, it is likely that additional randomizing effects of thermal activation will effectively cut off this divergence
when the critical film height is approached from below during deposition.
Finally, the presented method may be useful as a first step in the analysis of methods to enhance SAQD order. It is
reasonable to suppose that under some circumstances initial growth stages will be very important while for others they
will not. For example, prior work on vertical stacking appears to confirm the presented ordering mechanism. [44].
Vertical stacking not only achieves vertical correlation of dots, but each layer is more ordered horizontally than the
one below. Additionally, a “growth window” was found, whereby to achieve enhanced order, the evolution of each
layer be terminated before ripening begins. The reported simulation [44] supports the following scenario for SAQD
order development. Order is enhanced during the small fluctuation stage as described here. Once the fluctuations are
sufficiently large, the seeded dots evolve towards their equilibrium shapes. Finally, the dots begin to ripen and order
diminishes. Order is transfered via strain to the next layer so that the next layer gets a head start on its initial ordering.
Thus, the multiple layers of dots effectively draws out the linear growth stage. It may be possible to modify the present
model to predict the correlation length of each SAQD layer.
A Diffusion Potential
The diffusion potential is calculated in terms of the film height H that is a function of the in plane coordinates x =
xi + yj. The elastic and surface energy portions of the diffusion potential can be found in [15]
µelast(x) = Ωω(x), and µsurf = −Ωγκ(x),
where Ω is the atomic volume, ω(x) is the elastic energy density at the film surface, γ is the surface energy density,
and κ is the total surface curvature. However, other calculations need to be included:
1. µwet for the two wetting potential cases, Eq. (3) and (5),
2. and µsurf and µwet when the surface energy density γ and wetting energy density W also depend on surface
orientation.
Before these case are addressed, a general form for the diffusion potential is justified.
A.1 General Form µ = ΩδF/δH(x)
The diffusion potential, µ(x), is the change in free energy, F , when a particle is added at a position, x. Note that µ(x)
and F are relative energies. They can be used to compare the binding energy of one site on the surface in comparison
with another site, but should not be interpreted as an absolute binding energy or total formation energy of the surface.
If a particle has a volume Ω, then the diffusion potential at x is related to the variation of free energy with volume,
δF = Ω−1
ddxµ(x)δV (x), (54)
where δV (x) is the volume variation at x. Calculating δV (x), V =
ddxH(x).Therefore, δV (x) = δH(x).
Substituting into δF (Eq. (54)), δF = Ω−1
ddxµ(x)δH(x) or µ(x) = ΩδF/δH(x).
A.2 Simple Model
Starting from Eq. (2), µ(x) is found by taking the variational derivative,
µelast.(x) = Ω
δH(x)
volume
ddxdz ω[H](x, z) = Ωω (x)
where the “[H]” indicates that the elastic energy, ω, is a nonlocal functional of the film height H, and ω(x) =
ω[H] (x,H(x)), the elastic energy density evaluated above lateral position x at the free surface (z = H(x)). See [15]
for details of the derivation. The surface energy diffusion potential is
µsurf.(x,t) = Ω
δH(x)
1 + (∇H(x))2
= −Ω∇ ·
1 + (∇H(x))2
γ = −Ωγκ(x).
The wetting energy diffusion potential is
µwet(x) = Ω
δH(x)
ddxW (H(x))
= ΩW ′(H(x))
Putting these three terms together, one obtains Eq. (3)
A.3 General Model
Consider the general form for the combined surface energy and wetting potential,
Fsw =
ddxFsw(H(x),∇H(x))
as in Eq. (4) so that the free energy is an integral over the x−plane of an energy density that depends on H(x) and
∇H(x) locally. The corresponding diffusion potential is
µ(x) = Ω
δH(x)
F (10)sw (H(x),∇H(x))−∇ · F
sw (H(x),∇H(x))
B Linearized Diffusion Potential and Anisotropy
The linearized diffusion potential µlin, k is found by finding µ(x) to first order in height fluctuations (h), to get µlin(x)
and then taking the Fourier transform to get µlin,k. The linearization of the simple isotropic diffusion potential corre-
sponding to Eqs. (2) and (3) was discussed in Sec. 2.1.1.1. Here, the more general diffusion potential corresponding
to Eqs (4) and (5) is linearized and then applied to the anisotropic simple model and the anisotropic general model.
Only the surface and wetting parts of the diffusion potential are discussed in this appendix. See ref. [15], Sec. 2.2.1.1
and Appendix C for discussion of µelast..
B.1 Linearizing the simple model
Consider a wetting potential and diffusion potential that both depend on the film height gradient ∇H, γ → γ(∇H)
and W (H) → W (H,∇H). Starting from Eq. (6) and expanding to second order in the film height fluctuation using
H(x) = H̄+ h(x) (Eq. (7)),
1 + (∇H)2
]−1/2
γ(∇H) =
(∇h)2 + . . .
γ + γ′ ·∇h+ γ̃′′ : ∇h∇h+ . . .
= γ + γ′ ·∇h−
γ (∇h)2 + γ̃′′ : ∇h∇h+O[h3]
where γ is γ(0), and the primes indicate the derivatives with respect to the surface height gradient.
γ′ = ∂∇Hγ(∇H)|∇H=0 , and γ̃
′′ = ∂∇H∂∇Hγ(∇H)|∇H=0 .
Taking the derivative with respect to ∇h results in a tensor of rank equal to the order of the derivative because ∇h is
a vector (tank 1 tensor). Taking the variational derivative, µsurf.(x) = ΩδFsurf./δh(x),
µsurf., lin(x) = Ω
γ∇2h(x)− γ̃′′ : ∇∇h(x)
The term with γ′ vanishes because it is the divergence of a constant (∇ · γ′). Taking the inverse Fourier transform,
µsurf., lin,k = Ω
−γk2 + k · γ̃′′ · k
hk. (55)
The first term is isotropic. The second term is parameterized by a rank 2 symmetric tensor.
Going through the same process, one finds essentially the same result for an orientation dependent wetting energy.
The step details are so close to the details for linearizing the more general form, Fsw(H,∇H), they are deferred to
(Appendix B.2). One finds that
µwet,lin,k = Ω
W (20) + k · W̃(02) · k
. (56)
where W (mn) = ∂mH∂
∇HW (H,∇H)|H=H̄,∇H=0 is the m
th and nth derivative of the wetting energy density with
respect toH and ∇H evaluated for a perfectly flat film of height H̄. W (mn) is a tensor of rank n.
B.1.1 isotropic case
In the isotropic case, γ̃′′ → γ′′Ĩ, where Ĩ is the identity operator, and γ′′ is a scalar. Similarly, W̃(02) →W (02)Ĩ. One
thus gets for the combined surface and wetting parts of the diffusion potential,
µsw,lin,k = Ω
−γ + γ′′ +W (02)
k2 +W (20)
Thus, in the isotropic case, the linear order effect of introducing a surface orientation to either the surface energy or
the wetting potential is simply to change the apparent surface energy density by γ → γ − γ′′ −W (02).
B.1.2 anisotropic case
The surface and wetting parts of the diffusion potential (Eqs. (55) and (56)) can admit only a limited anisotropy.
They both contain rank 2 symmetric tensors, γ̃′′ and W̃(02) in the x−plane. For a two-dimensional surface, this
means that they can either have two-fold-symmetric (rotations by 180◦) anisotropy or none at all. Thus, for the case
considered in Sec. 2.2.1.2, four-fold-symmetric anisotropy , the surface and wetting parts of the diffusion potential
must be completely isotropic. As discussed in Sec. 2.2.1.2, the (100) surface of zinc-blend structures, such as the
mentioned Ge, Si, InAs and GaAs present a rather complicated situation. For simplicity, it is assumed here that the
surface and wetting energies are at least four-fold symmetric. Consequently, they are completely isotropic.
Finally, it should be noted that if Fsw depends on higher order derivatives, then the discussion is greatly compli-
cated and a larger class of anisotropic terms is admissible. For example, whenFsw → Fsw(H,∇H,∇∇H,∇∇∇H, . . . )
is expanded aboutH(x) = H̄ to quadratic order in h, it would contains tensors of rank 6 and maybe even higher.
B.2 Linearizing the general model
The elastic part of the linearized diffusion potential was discussed in Sec. 2.2.1.1 and Appendix C . Eq. (56) can be
found by using all of the following steps with the substitution Fsw →W . The surface-wetting part of the diffusion po-
tential µ(x) is found by expanding Fsw to second order in the film-height fluctuation, h, and then taking the variational
derivative. Expanding Fsw about h = 0 and ∇h = 0,
Fsw(H̄+ h,∇h) = F (00)sw + F
sw h+ F
sw ·∇h+ hF
sw ·∇h . . .
· · ·+
F (20)sw h
F̃(02)sw : ∇h∇h+O[h
Note that in this expansion, all the F (mn)sw terms are constant with respect to h and depend implicitly on the average
film height, H̄. The first index indicates the mth derivative with respect to h. The second index indicates the nth
derivative with respect to ∇h. The derivatives are evaluated for a perfectly flat surface of height H̄. Thus,
F (mn)sw = ∂
∇HFsw (H,∇H)|H=H̄,∇H=0 .
Since ∇h is a vector in the x−plane, F (mn)sw is a tensor of rank n. Taking the variational derivative of Fsw =∫
ddxFsw(H,∇H) and keeping terms to order h1,
δh(x)
= F (10)sw −∇ · F
sw + F
sw h−∇ ·
F̃(02)sw ·∇h
Note that the F (00)sw term vanishes because it is constant, and the F
sw term vanishes upon simplification. Additionally,
the F (10)sw can be neglected if one enforces the condition that the film-height fluctuations do not add or subtract material
from the surface, namely that
ddx δh(x, t) = 0. Alternatively, one can discard it in anticipation of taking the gradient
of the diffusion potential, since it is a constant. The term ∇ · F(01)sw = 0 for the same reasons, or because F
sw is a
constant. Multiplying through by the atomic volume,
µlin(x) = Ω
F (20)sw h− F̃
sw : ∇∇h
. (57)
B.2.1 isotropic case
In the isotropic case, F̃(02)sw must be proportional to the identity so that F̃
sw = F
sw Ĩ; thus,
µsw,lin(x) = Ω
F (20)sw h(x)− F
2h(x)
Taking the inverse Fourier transform of this equation,
µsw,lin,k = Ω
F (20)sw + F
This gives case b in Eq. (9).
B.2.2 anisotropic case
If the surface is anisotropic, then F̃(02)sw in Eq. (57) is a rank 2 symmetric tensor in the x−plane. Thus, it can have two
distinct eigenvalues, and automatically has 2-fold rotational symmetry (rotations by 180◦). If any other symmetry is
assumed such as 4-fold symmetry (rotations by 90◦), then F̃(02)sw must be fully isotropic. Taking the inverse Fourier
transform,
µsw,lin,k = Ω
F (20)sw + k · F̃
sw · k
In Eq. (23), case b, it is assumed that there is four-fold symmetry, resulting in a surface-wetting part of the diffusion
potential that is completely isotropic.
C Elastic Anisotropy
In principal, the anisotropic elastic energy ωk is found in the same fashion as the isotropic elastic energy. [15] The
flat film, initially in a state of biaxial stress, is perturbed by a small periodic surface fluctuation of amplitude h0. An
appropriate elastic field is added to satisfy the perturbed traction-free boundary condition at the free surface. Finally,
the elastic energy is evaluated at the free surface to first order in h0. The coefficient h0 is the sought after ωk. The
equations themselves are cumbersome and best solved using a numeric implementation, so an abstract procedure for
calculating ωk is outlined here. ωk is found for k = 1 but arbitrary θk.
Let the surface have a height variation
h(x) = h0e
To first order in h0, the surface normal is
n(x) = −ikh0eikxi + k.
The elastic energy needs to be calculated to first order in h0. To find the elastic energy, it is necessary to find the
perturbing elastic field to first order in h0.
The initial unperturbed stress state is
σ̃m =
σm 0 00 σm 0
0 0 0
where σm =
c11 + c12 − 2c212/c11
�m. Note that this stress state is isotropic in the x−y-plane and thus independent
of rotations about the vertical axis. Under this stress state, a flat surface is traction-free. With the height perturbation,
the traction is
tj = (n · σ̃m)j = −ikh0M�mδj1e
ikx. (58)
Next to find the perturbing elastic fields. These are not isotropic in the x − y−plane, and it is necessary to take
into account the angle. First, the 3× 3× 3× 3 elastic stiffness tensor cijkl is constructed for the cube orientation from
the compact 9× 9 matrix cij . The tensor representation aids in rotation. The stiffness tensor is then passively rotated
in the x− y−plane by an angel θk,
cijkl(θk) =
m,n,p,q=1
R(θk)imR(θk)jnR(θk)kpR(θk)lqcmnpq
where
R(θk) =
cos(θk) sin(θk) 0− sin(θk) cos(θk) 0
0 0 1
This passive rotation of cijkl is equivalent to actively rotating the wave vector k = ki by θk.
The appropriate form for the perturbing displacement field is found. Assume a displacement of the form
ui(x, y, z) = Uie
k(ix+κz),
where κ can have a complex value. The elastic equilibrium equations are
i,k,l=1
cijkl(θk)
ul = 0; j = 1 . . . 3.
Cjl(θk, κ)Ul
k2ek(ix+κz) = 0 (59)
where
Cjl(θk, κ) =
i,k=1
cijkl(θk)(iδi1 + δi3κ)(iδk1 + δk3κ).
Factoring out k2ek(ix+κz), the part in parenthesis must be identically zero.
To obtain a non-trivial solution, the determinant of Cjl(θk, κ) to zero. Six complex values of κ are found. The
values of κ with Re[κ] < 0 are discarded since the corresponding displacements blow up as z → −∞. Each of
the remaining values κ = κp with p = 1 . . . 3 is substituted back into Cjl(θk, κ), and Eq. (59) is solved to find the
corresponding eigenvectors, Upl . The total displacement is thus
ul(x, y, z) = i�mh0
k(ix+κpz),
where it is assumed that the perturbing elastic displacement field is proportional to h0and σm, and the factor of i is
put in for convenience. The coefficients Ap can be found from the traction-free boundary condition at the free surface.
The traction formula is
i,k,l=1
nicijkl(θk)
ul(x, y, z) = ik�mh0
i,k,l,p=1
nicijkl(θk)ApU
l (iδk1 + κ
pδk3)e
k(ix+κpz) (60)
The traction is already proportional to h0. Thus, all terms in the sum must be kept to zeroth order in h0 so that
h(x) = 0, and n(x) = k.
Thus, plugging z = 0 to Eq. (60),
tj = ik�mh0
(ic3j1l(θk) + κ
pc3j3l(θk))ApU
ikx. (61)
Since the total traction (Eqs. (58) and (61)) must be zero, the coefficients Ap are found from
KjpAp = Rj ,
where
Kjp =
(ic3j1l(θk) + κ
pc3j3l(θk))U
Rj = Mδj1
for j = 1 . . . 3. It is worth noting that only for the symmetry directions, θk = 0◦ and θk = 45◦ is the strain purely
plane-strain as it is for the elastically isotropic case.
The elastic energy at the film surface is found to order O(h0). If the stress and strain are expanded to first order in
h0, σ̃ = σ̃0 + σ̃1, and �̃ = �̃0 + �̃1, then
�̃ : c̃ : �̃ =
σ̃0 : �0 + σ̃0 : �̃1 +O(h
Thus,
U = U0 +M�m ((�1)11 + (�1)22)
(�1)11 =
= −�mkh0
(�1)22 = ∂u2/∂y = 0. Thus,
U = U0 − Eθkkh0e
where
Eθk = M�
where Apand U
1 are implicitly functions of θk. This procedure has been used to find the values of E0◦ and E45◦ for
Table. 2 and Sec. 4.
D Diffusional Anisotropy
In general, the surface diffusivity can depend on the film height H(x) and the surface orientation ∇H(x) so that the
surface current is
JS(x) = D̃(H(x),∇H(x)) ·∇sµ(x)
where ∇s is the surface gradient, and D̃ is a rank 2 tensor in the two-dimensional space tangent to the film surface at
x. Linearizing the surface current about a flat surface,
JS(x) = D̃(H̄) ·∇µlin(x)
where the diffusivity must be evaluated for h = 0 and ∇h = 0, since µlin(x) is already proportional to h(x). The lin-
earized diffusivity is a symmetric rank 2 tensor in the x−plane. Thus, it is similar to F̃sw discussed in Appendix B.2.2.
It is automatically either two-fold symmetry (rotations by 180◦) or it is completely isotropic. In Eq. (23), four-fold
symmetry of the surface is assumed. Thus, the diffusivity must be completely isotropic; D̃ → D, a scalar. Sec-
tion 2.2.1.2 and Appendix B.2.2 contain discussions of the symmetry properties of the various rank 2 tensors that
appear in the linear evolution equations. A limited case of diffusional anisotropy has been modeled via kinetic Monte
Carlo technique. [54]
E Correlation Functions
E.1 Mean Values
Equations (31) and (33) are central to the presented analysis. Here, they are derived. The two-point correlation func-
tions for a stochastic system are introduced. Then, the average of the autocorrelation function is taken and expressed
in terms of the two-point correlation functions. Finally, this average is simplified using the translational invariance of
the system (governing equations and ensemble of initial conditions).
The two-point real-space space correlation function is
C(x,x′) = 〈h(x)h(x′)∗〉 ,
and the reciprocal space correlation function is
Ckk′ = 〈hkh∗k′〉 .
These are related by the double Fourier transform,
Ckk′ = 1(2π)2d
ddxddx′ e−ik·x+ik
′·x′C(x,x′); (62)
C(x,x′) =
ddkddk′ eik·x−ik
′·x′Ckk′ . (63)
These ensemble correlation functions can be used to give the ensemble-mean autocorrelation function and spec-
trum function. In real space,
CA(∆x)
d2x′ 〈h(∆x + x′)h(x′)〉
d2x′ C(∆x + x′,x′). (64)
(2π)d
〈hkh∗k〉 =
(2π)d
Ckk. (65)
Fortunately, the translational invariance of the system simplifies these relations. Inspecting the governing equations
and invoking the translational invariance of the stochastic initial conditions, the resulting ensemble and its statistical
measures must also be translationally invariant. Thus under the translation by x′,
C(∆x + x′,x′) = C(∆x,0) = C(∆x), (66)
so that the independent variable is reduced to just the difference vector ∆x = x − x′. This relation can be used to
simplify both the real and reciprocal space relations.
The real space relation simplifies as follows.Inserting Eq. (66) into Eq. (64),
CA(∆x)
d2x′ C(∆x,0) = C(∆x). (67)
The reciprocal space relation (Eq. (62)) simplifies to
Ckk′ = Ckδ
2(k− k′) = Ck
(2π)d
δkk′ , (68)
where
(2π)d
d2∆x e−ik·∆xC(∆x).
One can see immediately from Eq. (67) that Ck is the Fourier transform of
CA(∆x)
= C(∆x), or one can plug
Eq. (68) into Eq. (65), to get
= Ck.
E.2 Variance and Convergence
The ergodic hypothesis is that an average with respect to a parameter such as position or time tends towards an
ensemble average. In this case,
CAk ≈
= Ck, (69)
and CA(∆x) ≈
CA(∆x)
= C(∆x).
when the surface area is very large. The ensemble average is a good substitute if the variance about the average
vanishes as the substrate area A becomes large. It is found that in reciprocal space,
Var(CAk ) =
= C2k. (70)
Thus, the ergodic hypothesis does not hold for CAk . In practice, C
k is a speckled version of Ck (Fig. 6) However, if
one smooths CAk by averaging over a small patch in reciprocal space of size ksmooth = 1/∆s, so that
CAk (∆s) =
)d/2 ∫
ddk′ e−
′−k)2CAk′ , (71)
then Var
CAk (∆s)
diminishes as 1/A. For sufficiently large ∆s,〈
CAk (∆s)
≈ Ck, (72)
CAk (∆s)
πd/2∆ds
C2k. (73)
Thus, the ergodic hypothesis (Eq. (69)) only holds for a smoothed version of CAk .
In real space,
CA(∆x)
CA(∆x)
CA(∆x)
(2π)d
e2ik·∆xC2k + C
, (74)
where the integral is bounded (finite) provided that either t > 0 or the atomic scale cutoff b0 > 0. Thus, the ergodic
hypothesis holds for the real space autocorrelation function.
E.2.1 Eq. (70)
First,
CAk C
is calculated.
CAk C
(2π)d
〈hkh∗khk′h
k′〉 .
Assume that he distribution of hk is gaussian. Also, assume that h(x) is real so that hkh−k = |hk|
2. Then,〈
= Ck1Ck2δ
d(k1 − k4)δd(k2 − k3) . . .
. . . +Ck1Ck2δ
d(k1 + k3)δ
d(k2 + k4) . . .
. . . +Ck1Ck3δ
d(k1 − k2)δd(k3 − k4).
Thus,
CAk C
(2π)d
δd(k− k′)
. . .
. . . +C2k
δd(k + k′)
+ CkCk′
δd(0)
. (75)
= C2k
δkk′ + δk(−k′)
+ CkCk′ , (76)
where Eq. (29) has been used liberally. Setting k = k′, results in Eq. (70).
E.2.2 Eq. (73)
Now consider CAk smoothed over a length ∆s (Eq. (71)). The mean value is
CAk (∆s)
)d/2 ∫
ddk′ e−
′−k)2 〈CAk′〉 .
)d/2 ∫
ddk′ e−
′−k)2Ck′ .
For sufficiently small ksmooth, (sufficiently large ∆s), Eq. (72) results.
The variance of CAk (∆s) is now calculated. First, it is necessary to calculate
CAk (∆s)
CAk (∆s)
ddk′ e−
′−k)2 . . .
. . . ×
ddk′′ e−
′′−k)2 〈CAk′CAk′′〉 .
Using Eq. (75) and Eq. (29) as needed,
CAk (∆s)
ddk′ddk′′ e−
′−k)2e−
′′−k)2
(2π)d
. . .
· · · ×
δd(k′ − k′′)
+ C2k
δd(k′ + k′′)
+ CkCk′
δd(0)
′−k)2C2k′ + e
− 12 ∆
s[(k′−k)2+(k′+k)2]C2k′
. . .
· · ·+
)d/2 ∫
ddk′ e−
′−k)2Ck′
The first integral is bounded (finite) because Ck is bounded. Let its finite value be denoted I . The second integral is
simply
CAk (∆s)
.Thus,
Var(CAk (∆s)) =
∆2ds I
a finite value that decreases as A−1 as required for the ergodic hypothesis to hold. For sufficiently small ksmooth (large
∆s), I ≈ (π/∆2s)d/2C2k, and Eq. (73) results. It should also be noted that the large ∆s required for this approximation
also creates a more stringent requirement that A be large.
E.2.3 Eq. (74)
Now, consider the real space auto-correlation function. First,
CA(∆x)CA(∆x)
is needed.
CA(∆x)CA(∆x)
ddkddk′ ei(k+k
′)·∆x 〈CAk CAk′〉
Proceeding in a fashion similar to the previous section (making use of Eqs. (75) and (29) as needed) ,
CA(∆x)CA(∆x)
(2π)2d
ddkddk′ ei(k+k
′)·∆x
δd(k− k′)
. . .
· · ·+ C2k
δd(k + k′)
+ CkCk′
δd(0)
(2π)d
e2ik·∆xC2k + C
. . .
· · ·+
ddk eik·∆xCk
ddk′ eik
′·∆xCk
(2π)d
e2ik·∆xC2k + C
CA(∆x)
Thus, Eq. (74) results. For the variance to be vanishing, the integral in Eq. (74) must be bounded (finite). If time,
t > 0, the exponential in Eq. (77) guarantees that the integral is bounded. For time t = 0, the integral is only bounded
if the atomic scale cutoff b0 > 0.
F Atomic Scale Cutoff
Starting from Eq. (39),
(2π)d
e2σkt−
. (77)
The effect of the small scale cutoff is both small and short-lived, as it only works to suppress fluctuations with large
wavenumbers. The most important fluctuations have wavenumbers between 0 and 2kc. Thus, the typical size of the
cutoff term is about b20k
c . If a typical dot size or spacing size 10 nm, and a typical atomic scale is 10
−1 nm, a typical
value for this term is about 10−3 − 10−2. To calculate the effect of the cutoff, it can absorbed into the time-dependent
part with the substitution
so that its effect lasts only as long as a perturbation with atomic scale curvature (κ = b0). Thus, Eq. (40) is a good
approximation.
Acknowledgement
Thanks to L. Fang and C. Kumar for useful comments during the writing of this article.
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Introduction
Modeling
1D and 2D Isotropic model
Energetics
simple form
general form
Linearization
Dynamics
Peaks
Stability and wetting potential
2D Anisotropic case
Energetics
Elastic anisotropy
Surface and Wetting Energy Anisotropy
total diffusion potential
Dynamics
Expansion about peaks
Correlation Functions
Correlation Functions and SAQD order
Periodic array
Nearly Periodic array
Ensemble Correlation Functions / ergodicity
Mean fluctuation
Correlation Function
Stochastic Initial Conditions
Reciprocal Space Correlation Functions
one-dimensional
2D isotropic
anisotropic
Real Space Correlation Functions
one-dimensional
2D isotropic
anisotropic
Generalizability
Order Predictions
Ge at 600K
General case of
Discussion/Conclusions
Diffusion Potential
General Form =F/H(x)
Simple Model
General Model
Linearized Diffusion Potential and Anisotropy
Linearizing the simple model
isotropic case
anisotropic case
Linearizing the general model
isotropic case
anisotropic case
Elastic Anisotropy
Diffusional Anisotropy
Correlation Functions
Mean Values
Variance and Convergence
Eq. (??)
Eq. (??)
Eq. (??)
Atomic Scale Cutoff
|
0704.0068 | A Note About the {Ki(z)} Functions | A NOTE ABOUT THE {Ki(z)}
FUNCTIONS
Branko J. Malešević
In the article [10], A. Petojević verified useful properties of the Ki(z) functions
which generalize Kurepa’s [1] left factorial function. In this note, we present
simplified proofs of two of these results and we answer the open question stated
in [10]. Finally, we discuss the differential transcendency of the Ki(z) functions.
A. Petojević [7, p. 3.] considered the family of functions:
vMm(s; a, z) =
(−1)k−1
z +m+ 1− k
L[s; 2F1(a, k − z,m+ 2; 1− t)],(1)
for ℜ(z) > v−m−2, where v∈N is a positive integer; m∈{−1, 0, 1, 2, . . .} is an integer;
s, a, z are complex variables; L[s;F (t)] is Laplace transform and 2F1(a, b, c;x) is
the hypergeometric function (|x| < 1). D- .Kurepa has considered in the articles
[1, p. 151.] and [2, p. 297.] a complex function defined by the integral:
K(z) =
tz − 1
dt,(2)
for ℜ(z)>0. Especially, forKurepa’s functionK(z), it is true thatK(z)=1M0(1; 1, z),
for ℜ(z)>0, according to [10]. For various of values of parameters v,m, s, a, z from (1),
different special functions, as presented in [10], are obtained. A. Petojević has con-
sidered in the article [10, p. 1640.] the following sequence of functions:
Ki(z) =
1M0(1; 1, z + i− 1)− 1M0(1; 1, i− 1)
1M−1(1; 1, i)
for i∈N and ℜ(z)>−i. On the basis of the definition in (3), the following represen-
tation via Kurepa’s function is true:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
for i∈N and ℜ(z)>−i+1. Note that K(0)=0 [2, p. 297.] and therefore K1(z)=K(z)
for ℜ(z)>0. Analytical and differential–algebraic properties of Kurepa’s function
K(z) are considered in articles [1− 12] and in many other articles. On the basis
of well-known statements for Kurepa’s function K(z), using representation (4), in
many cases we can get simple proofs for analogous statements for Ki(z) functions.
For example, it is a well-known fact that it is possible to analytically continue Ku-
repa’s function to a meromorphic function with simple poles at integer points z = −1
and z = −m, (m ≥ 3) [2, p. 303.], [3, p. 474.]. Residues of Kurepa’s function at
these poles have the following form [2]:
Research partially supported by the MNTRS, Serbia, Grant No. 144020.
http://arxiv.org/abs/0704.0068v2
2 Branko J. Malešević
z = −1
K(z) = −1 and res
z = −m
K(z) =
(−1)k−1
, (m≥3).(5)
For Kurepa’s function K(z) the infinite point is an essential singularity [3]. Hence,
on the basis of (4), each function Ki(z) is meromorphic with simple poles at integer
points z=−i and z=−(i+m), (m≥2). On the basis of (4) we have:
z = −(i+m)
Ki(z) =
(i−1)!
· res
z = −(i+m)
K(z + i− 1) = 1
(i−1)!
· res
z = −(m+1)
K(z),(6)
where m = 0 or m≥2. Hence:
z = −i
Ki(z)=−
(i−1)!
and res
z = −(i+m)
Ki(z)=
(i−1)!
(−1)k−1
, (m≥2).(7)
For each Ki(z) function the infinite point is an essential singularity. Therefore, we get
Theorem 3.3. from [10]. Next, it is a well-known fact that for Kurepa’s function
the following asymptotic relation K(x) ∼ Γ(x) is true for real x such that x → ∞
and where Γ(x) is the gamma function [2, p. 299.]. Hence, for fixed i ∈N and real
x>−i+1, on the basis of (4), we get:
Ki(x)
Γ(x+ i− 1)
(i−1)!
· K(i+ x− 1)−K(i− 1)
Γ(x+ i− 1)
(i−1)!
Ki(x)
Γ(x+ i)
(i−1)!
· K(i+ x− 1)−K(i− 1)
(x+ i− 1)Γ(x+ i− 1)
0.(9)
Therefore, we get Theorem 3.6. from [10]. Next we give a solution to the open
problem stated in Question 3.7. in [10]. Namely, the following formula in the article
[8, p. 35.] is given:
K(z) =
Ei(1) + iπ
(−1)zΓ(1 + z)Γ(−z,−1)
,(10)
for values z ∈ C\{−1,−2,−3,−4, . . .} and i =
−1. In the previous formula Ei(z)
and Γ(z, a) are exponential integral and incomplete gamma function respectively [8].
Then, for fixed i∈N and values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}, on the basis of
(4) and (10), we get:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
Ei(1) + iπ
e(i−1)!
(−1)z+i−1Γ(1 + z + i− 1)Γ(−z − i+ 1,−1)
e(i−1)!
−Ei(1) + iπ
e(i−1)!
− (−1)
i−1Γ(i)Γ(−i+ 1,−1)
e(i−1)!
= (−1)ie−1
Γ(1− i,−1)− (−1)z Γ(1− i− z,−1)Γ(i+ z)
(i−1)!
Therefore, the affirmative answer for Question 3.7. from [10] is true for complex
values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}.
A note about the {Ki(z)}∞i=1 functions 3
Finally, at the end of this note let us emphasize one differential–algebraic fact for
the sequence of functions Ki(z). On the basis of the formula (17) from the article
[10], we can conclude that each Ki(z) function satisfies the following recurrence re-
lation (i−1)!Ki(z + 1)− (i−1)!Ki(z) = Γ(z + i). The previous relation can be used
to verify the differential transcendency of these functions as discussed in [11, 12].
Therefore, we can conclude that each Ki(z) function is a differential transcendental
function, i.e. it satisfies no algebraic differential equation over the field of complex ra-
tional functions.
REFERENCES
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297 − 307.
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[7] A. Petojević: The function vMm(s; a, z) and some well-known sequences, Journal of
Integer Sequences, Article 02.1.6, Vol. 5 (2002).
[8] B. Malešević: Some considerations in connection with Kurepa’s function, Univerzitet u
Beogradu, Publikacije Elektrotehničkog Fakulteta, Serija Matematika, 14 (2003), 26−36,
available at http://pefmath.etf.bg.ac.yu/ .
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Pure and Applied Mathematics, Vol. 5, Issue 4, Article 84, (2004), available at
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[10] A. Petojević: The {Ki(z)}
i=1 functions, Rocky Mountain Journal of Mathematics,
Vol. 36, No. 5, (2006), 1637-1650.
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in Bulletin of the Belgian Mathematical Society − Simon Stevin 2007, available at
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[12] Ž. Mijajlović, B. Malešević: Analytical and differential – algebraic properties of
Gamma function, to appear in International Journal of Applied Mathematics &
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J.Rassias (ed.), Functional Equations, Integral Equations, Differen-
tial Equations & Applications, http://www.ceser.res.in/ijamas/cont/fida.html
Special Issues dedicated to the Tri-Centennial Birthday Anniversary of L. Euler, 2007.,
available at http://arxiv.org/abs/math.GM/0605430 .
University of Belgrade, (Received : 04/01/2007 )
Faculty of Electrical Engineering, (Accepted : 05/25/2007 )
P.O.Box 35-54, 11 120 Belgrade, Serbia
[email protected], [email protected]
http://elib.mi.sanu.ac.yu/pages/browse_journals.php
http://gauss.elfak.ni.ac.yu
http://pefmath.etf.bg.ac.yu/
http://jipam.vu.edu.au/
http://arxiv.org/abs/math.GM/0412354
http://www.ceser.res.in/ijamas/cont/fida.html
http://arxiv.org/abs/math.GM/0605430
|
0704.0069 | Dynamical Objects for Cohomologically Expanding Maps | Dynamical Objects for Cohomologically
Expanding Maps.
John W. Robertson
November 4, 2018
John W. Robertson1
Wichita State University
Wichita, Kansas 67260-0033
Phone: 316-978-3979
Fax: 316-978-3748
[email protected]
Abstract
The goal of this paper is to construct invariant dynamical objects
for a (not necessarily invertible) smooth self map of a compact mani-
fold. We prove a result that takes advantage of differences in rates of
expansion in the terms of a sheaf cohomological long exact sequence
to create unique lifts of finite dimensional invariant subspaces of one
term of the sequence to invariant subspaces of the preceding term.
This allows us to take invariant cohomological classes and under the
right circumstances construct unique currents of a given type, includ-
ing unique measures of a given type, that represent those classes and
are invariant under pullback. A dynamically interesting self map may
have a plethora of invariant measures, so the uniquess of the con-
structed currents is important. It means that if local growth is not
too big compared to the growth rate of the cohomological class then
the expanding cohomological class gives sufficient “marching orders”
to the system to prohibit the formation of any other such invariant
1Research partially supported by a Wichita State University ARCS Grant.
current of the same type (say from some local dynamical subsystem).
Because we use subsheaves of the sheaf of currents we give conditions
under which a subsheaf will have the same cohomology as the sheaf
containing it. Using a smoothing argument this allows us to show
that the sheaf cohomology of the currents under consideration can be
canonically identified with the deRham cohomology groups. Our main
theorem can be applied in both the smooth and holomorphic setting.
MSC: 37C05, 32H50, 18F20, 55N30
1 Introduction
Our purpose is to construct invariant dynamical objects for a self map f : X →
X of a compact topological space. We make use of sheaf cohomology and
differences in rates of expansion in different terms of a long exact sequence
to construct invariant sections of a sheaf. We will show that there are in-
variant degree 1 currents (or eigencurrents) corresponding to each expanding
eigenvector of H1(X,R). We also show that successive preimages of suffi-
ciently regular degree one currents converge to one of these eigencurrents.
We show that if most of the expansion f : X → X is ”along” an invariant
cohomological class v ∈ Hk(X,R) then there is an invariant current c in that
cohomology class and other sufficiently regular currents in the same class
converge to c under successive pullback.
The group cohomology of Z acting on a space of functions on X via pull-
back has been studied in the context of dynamical systems [Kat03]. This
work seems related to ours, but to be pursued in an essentially different di-
rection. Our map f is not assumed to be invertible, so there is not necessarily
a Z action, only an N action. Also, we use sheaves rather than functions and
make substantial use of cohomological tools. Most importantly, we are par-
ticularly interested in the construction of invariant currents, especially when
the current is some sense unique.
Our results are actually motivated by results in higher dimensional holo-
morphic dynamics showing the existence of a unique closed positive (1, 1) cur-
rent under a variety of circumstances (just about any recent paper on higher
dimensional holomorphic dynamics either proves such results or makes essen-
tial use of such results, see e.g. [FS92], [HOV94], [HOV95], [BS91a], [BS91b],
[BS92], [BLS93], [BS98a], [BS98b], [BS99], [Can01], [McM02], [FS94a], [FS94b],
[FS95b], [FS01], [FS95a], [JW00], [FJ03], [Ued94], [Ued98], [Ued97], and
[DS05]).
While invariant measures have been a focal point in dynamics, it seems
that invariant currents also have an imporant role to play. We will show under
mild conditions that if some degree one cohomological class of a smooth self
map f of a compact manifold is invariant and expanded there is necessarily
a invariant degree one current of a certain type representing that class. We
obtain analogous results for higher degree currents given bounds on the local
growth rates of f . The uniqueness of these classes is significant. It seems
clear that one could modify a map locally near a fixed point to obtain other
invariant currents of the same type without affecting the topology. Thus
our results also say that any local modification that created an invariant
current of the given type must violate the local growth conditions. In other
words, as long as things do not grow too fast compared to the growth rate of
the cohomology class, the expansion of the cohomology class gives sufficient
“marching orders” to points that no other invariant cohomological class of the
given type can be created by purely local dynamical behavior. Our results
give explicit conditions under which uniqueness is guaranteed. For degree
one currents, no restriction on local growth rates is necessary for our results.
2 Cohomomorphisms
We will make use of sheaves in this paper. There are two standard def-
initions of sheaves on a topological space X, one as a topological space
([Bre97],[GR84]), and one as a functor on the category TopX satisfying var-
ious axioms ([Har77],[Wei97]). Since we will often want to make use of a
topology on sections of a sheaf A that differs from the topology these inherit
using the topological definition of a sheaf, we will instead use the functor
definition of a sheaf.
Our sheaves will always be sheaves of K modules over some fixed field K.
We will require that K have an absolute value for which K is complete.
Given a continuous map f : X → Y and sheaves A and B on X and
Y respectively, an f -cohomomorphism is a generalized notion of a pullback
from B to A through f . Different types of geometric objects pull back
differently, and this allows us to handle all cases at once.
We take the following facts from from [Bre97] page 14–15.
Definition 1. If A and B are sheaves onX and Y then an “f-cohomomorphism”
k : B → A is a collection of homomorphisms kU : B(U) → A (f−1(U)), for
U open in Y , compatible with restrictions.
Note that if A is a sheaf on X and f : X → Y is continuous then there
is a canonical cohomomorphism f∗A ; A where f∗A is the direct image of
A , i.e. given an open U ⊂ Y , f∗A (U) = A (f−1(U)).
Remark. Given a continuous map f : X → Y of topological spaces X and
Y and sheaves A and B on X and Y respectively, all f -cohomomorphisms
f : B ; A are given by a composition of the form
j→ f∗A
f∗→ A
where j : B → f∗A is a sheaf homomorphism, and each such composition is
seen to given an f -cohomomorphism.
The usual notion of “a morphism of sheaves on X” is the same as an idX
cohomomorphism of sheaves on X.
2.1 Cohomomorphisms and Γ.
The functor Γ returns the global sections of that sheaf. Given a morphism
φ : A → A ′ of sheaves on X, Γφ is just the homomorphism A (X)→ A ′(X).
Given sheaves A and B on X and Y and given f : X → Y continuous
then for a sheaf cohomomorphism F : B → A one defines ΓF to be the
homomorphism B(Y ) → A (X). This extends Γ to be a functor on the
category of topological spaces with an associated sheaf where morphisms are
given by cohomomorphisms.
3 Invariant Global Sections
Fix a continuous self map f : X → X of a topological space X. We will
be interested in f self cohomomorphisms of sheaves A on X. As we will
typically have several sheaves of interest on X, each with a corresponding
f self cohomomorphism, we let fA : A ; A be the default notation for an
f -cohomomorphism of A .
Assume that X is a manifold and that
p→ B q→ C
is a short exact sequence of sheaves on X. Let f : X → X be a continuous
self map of X and assume further that we are given f self cohomomorphisms
of each of these sheaves and that
// B q
commutes. We will say that a commutative diagram as in (1) is an f self-
cohomomorphism of the sequence A → B → C .
Applying the functor Γ to this diagram, the rows can be extended in the
usual long exact sequence. The resulting diagram is commutative ([Bre97]
page 62).
0 // A (X)
C (X)
H1(X,A )
· · ·
0 // A (X)
// B(X)
// C (X)
// H1(X,A )
// · · ·
One can think of B as providing local potentials for members of C and
of A as being those potentials which give rise to the zero member of C .
It will be assumed that the reader is familiar with interpreting H1(X,A )
as classifying equivalence classes of bundles with transition functions in A .
We will frequently refer to members of H1(X,A ) as bundles. Sections of
such bundles will be assumed to be given locally by local sections of B,
so that every member c of Γ(C ) is given locally by potentials in B, and
these potentials, taken together, are a section of the corresponding bundle
δ(c) ∈ H1(X,A ).
Convention 1. We will frequently refer to a member v of H1(X,A ) as a
bundle, to a member c ∈ Γ(C ) as a divisor and if δ(c) = v we will call c a
divisor of the bundle v. We think this substantially adds to the readability of
the paper.
Definition 2. The support of a divisor c ∈ Γ(C ) is defined to be the com-
plement of the union of all open sets U such that c
Lemma 3. If an open set U lies outside the support of some c ∈ Γ(C ) then
f−1(U) lies outside the support of fC (c)
Proof. We note that by the definition of an f -cohomomorphism fC : C → C ,
since the cohomomorphism fC on C (U) is a homomorphism from C (U) to
C (f−1(U)) and the induced action of fC on Γ(C ) restricted to U must agree
with its action C (U) → C (f−1(U)), then if an open set U is outside the
support of c then f−1(U) is outside the support of of fC (c).
The following conditions for a given v ∈ H1(X,A ) will be of interest:
Definition 4. We will refer to a bundle v ∈ H1(X,A ) for which (H1p)(v) =
0 as being closed.
Note that this notion depends upon the exact sequence A → B → C ,
and not just on v. If B is γ acyclic then every member of H1(X,A ) is closed.
Definition 5. We will call a bundle v ∈ H1(X,A ) base point free if for every
x ∈ X there is some divisor c ∈ Γ(C ) associated to v whose support does
not contain x.
Lemma 6. If B is soft, X is a regular topological space, and a ∈ H1(X,A )
is a closed bundle then a is base point free.
Proof. From the long exact sequence there is some c′ ∈ Γ(C ) with δ(c′) = a
and given any point x ∈ X, from the fact that B
� C the germ c′x of c
at x is the image under qx of some germ b
x of Γ(B) at x. Choose an open
neighborhood U of x on which there is some b′ ∈ B(U) with b′x = b′′x. The
topological assumption on X implies that there is a neighborhood V b U
of x. The fact that B is soft implies there is some b ∈ Γ(B) such that
. Then c = c′ − b ∈ Γ(C ) has δ(c) = a and x 6∈ Supp(c).
Definition 7. We will refer to a bundle a ∈ H1(X,A ) such that fA (a) = λ·a
for some λ ∈ C as a λ eigenbundle.
We also find it useful to introduce a relevant notion of expansiveness
of a map f : X → X relative to a base point free closed eigenbundle v ∈
H1(X,A ).
Definition 8. Given a base point free closed eigenbundle v ∈ H1(X,A )
then we say that f is cohomologically expansive at x for v if for any open
neighborhood U of x and any divisor c ∈ Γ(C ) of v, the set U intersects the
support of fkC (c) for all sufficiently large k.
Remark. It is a corollary of the definition that the set of points at which
f is cohomologically expansive for v is closed and forward invariant. If
Supp fkC (c) = f
−k(Supp(c)) for each c ∈ Γ(C ) then the set of cohomolog-
ically expansive points is totally invariant.
The notion of being cohomologically expansive at x for v means roughly
that under iteration by f small neighborhoods U of x always grow to cover
enough of X that the pullback of the bundle v to the set fk(U) is a nontrivial
bundle on fk(U) whenever k is large.
We show that if B is soft and X is a compact metric space then some
minimal expansion takes place at points where f is cohomologically expansive
for a closed eigenbundle a ∈ H1(X,A ).
We use B�(x) to denote the ball of radius � about x.
Lemma 9. Let X be a compact metric space. If B is soft and v is a closed
eigenbundle then there exists δ > 0 such that for every � > 0 there exists
some K > 0 such that if f is cohomologically expanding at x then for every
k > K, diam fk(B�(x)) > δ.
Proof. The bundle v is base point free by Lemma 6. Using compactness we
can conclude that there is a finite open cover U1, . . . , U` of X such that for
each j, Uj is disjoint from Supp cj for some cj ∈ Γ(C ) with δ(cj) = v. We
will prove the lemma by contradiction. Let δ be the Lebesgue number of the
cover U1, . . . , U`. If the lemma is false there is some � > 0 and some increasing
sequence kn and points xn at which f is cohomologically expansive such that
diam fkn(B�(xn)) ≤ δ for each n. By going to a subsequence if necessary we
can assume xn converges to a point x∞. Letting U = B 1
�(x∞) we see that
U ⊂ B�(xn) for all large n and thus there is some one cj of c1, . . . , c` such that
fkn(U) is disjoint from Supp cj for infinitely many values of n. Consequently
U is disjoint from Supp fknC (cj) for infinitely many n, contrary to x∞ being
a point at which f is cohomologically expansive for v.
We included Lemma 9 to show that our notion of cohomological expansion
is genuinely expansive. However, depending on the nature of A , being coho-
mologically expansive can imply that neighborhoods grow a great deal under
iteration indeed. In Lemma 10 we show that given any closed set K such that
the pullback of a fixed point free closed eigenbundle a ∈ H1(X,A ) to K is a
trivial bundle then any neighborhood U of a point at which f is cohomolog-
ically expanding for a is so expanded under iteration that fk(U) 6⊂ intK for
all sufficiently large k. The collection of such sets K typically contains very
large sets about every point so no matter where fk(x) is the conclusion that
fk(U) does not lie in any intK implies some points of fk(U) must lie far away
from fk(x). The point is roughly that large iterates of any neighborhood of
x can not be homotopically contracted to a point in X.
Lemma 10. If B is soft, then for any closed set K ⊂ X such that the image
of H1(X,A ) → H1(K,A
) is zero, given any divisor c ∈ Γ(C ), there is
another divisor c′ ∈ Γ(C ) associated to the same bundle and c′ is supported
outside the interior of K. Consequently, if f is cohomologically expansive
at x ∈ X for some base point free closed eigenbundle a ∈ H1(X,A ) then
necessarily for any neighborhood U of x, fk(U) 6⊂ intK for all large k, where
intK is the interior of K.
Proof. We use the commutative diagram
H0(X,B)
Γq //
H0(X,C )
H1(X,A )
H0(K,B
Γq // H0(K,C
δ // 0
which we have written using H0 instead of Γ so it is clear what the ambi-
ent space is in each case. From exactness there exists some β ∈ H0(K,B
such that δ(β) = c
. Then since B is soft the mapH0(X,B)→ H0(K,B
is surjective so there is some b ∈ Γ(B) = H0(X,B) such that b
= β. Then
c′ = c − (Γq)(b) has δ(c′) = δ(c) and c′
= 0 so Supp(c′) is disjoint from
the interior of K.
It is easy to see that if f is cohomologically expansive at x ∈ X for some
fixed point free closed eigenbundle a ∈ H1(X,A ) then necessarily for any
neighborhood U of x, fk(U)∩Supp c 6= ∅ for all large k for any c ∈ Γ(C ) such
that δ(c) = a. Hence fk(U) can not lie in the interior of K for any large
Convention 2. We let K be either R or C, although our central theorems
only require K to be a complete field with an absolute value.
The following Theorem takes advantage of the fact that in an exact se-
quence the eigenvalues of members of nonadjacent members of the sequence
do not have to agree to give conditions under which one can uniquely “lift”
fixed members of one term of the exact sequence to a fixed member of the pre-
ceding term. Interpreted as a statement in the context of sheaf cohomology
we will be able to use this Theorem to make dynamical conclusions.
The theorem shows that each closed eigenbundle of the induced map
fA : H
1(X,A ) → H1(X,A ) with sufficiently large eigenvalue has a unique
associated invariant divisor c ∈ Γ(C ).
Definition 11. Given any finite dimensional K vector space V along with
a linear map g : V → V and any positive real number r, we let the r chron-
ically expanding subspace of V be the span of the subspaces associated2 to
eigenvalues of absolute value greater than r. We refer to the 1 chronically
expanding subspace simply as the chronically expanding subspace.
Theorem 12 (Unique Invariant Subspace Theorem). We will assume the
following:
• f : X → X is a continuous self map of a topological space X.
• We are given an f self cohomomorphism of a short exact sequence of
sheaves on X,
p→ B q→ C
• Γ(B) is a Banach space over K, and there exists some α, d ∈ R>0 such
that ‖ΓfBk(B)‖ ≤ d · αk‖B‖ for k ∈ N, B ∈ Γ(B),
• Γ(C ) is a topological vector space over K.
• If a sequence Ci ∈ Γ(C ) of divisors converges to another divisor C∞
then the support of C∞ is contained in the closure of the union of the
supports of Ci.
• The maps ΓfC and Γq are continuous.
• We are given a finite dimensional H1(fA ) invariant subspace W of the
α chronically expanding subspace of H1(X,A ). We also require W to
be comprised only of closed bundles.
2Meaning for each eigenvector λ we include not just the λ eigenspace, but also every
v ∈ V such that (g − λ · idV )n(v) = 0 for some positive integer n.
Then given any K linear map s : W → Γ(C ) such that δs = idW there is a
K linear map τ : W → Γ(B) satisfying
κ := lim
(ΓfC )
ksgk = s+ (Γq)τ (3)
where g : W → W is the inverse of H1fA
. Under iterated pullback the
rescaled pullbacks of any divisor C ∈ Γ(C ) of a bundle w ∈ W converge
toward the invariant plane of divisors κ(W ) ⊂ Γ(C ). The map κ : W →
Γ(C ) is the unique map making the diagram
wwo o
Γ(C )
// H1(X,A )
wwo o
// Γ(C )
// H1(X,A )
commute. Finally, for any basepoint free eigenbundle v ∈ W the support of
the corresponding invariant divisor κ(v) ∈ Γ(C ) is contained in the set of
points on which f is cohomologically expansive for v.
Proof. We note that δ
(ΓfC )sg−s
= 0 and so there is a map σ : W → Γ(B)
such that (Γq)σ = (ΓfC )sg − s.
Define Φ: Hom(W,Γ(B)) → Hom(W,Γ(B)) by Φ(σ) = (ΓfB)σg−1. We
will show that the sequence of maps Φk is exponentially contracting on
Hom(W,Γ(B)). Fix a norm ‖ · ‖ on W . The assumption that W lies in
the α chronically expanding subspace of H1(X,A ) implies that there exists
some β > α and some c > 0 such that ‖g−k(w)‖ ≤ cβ−k‖w‖ for k ∈ N,
w ∈ W . This with the assumption on the rate of expansion of ΓfB easily
implies that
‖Φk(φ)(w)‖ = ‖(ΓfB)k(φ(g−k(w)))‖ ≤ cd
‖φ‖ · ‖w‖
Thus Φk is an operator of norm no more than cd
, where α < β.
Letting τk = σ + Φ(σ) + Φ
2(σ) + · · · + Φk(σ) then limk→∞ τk converges
to some map τ . It is easily confirmed that (Γq)τk = (ΓfC )
ksg−k − s. Equa-
tion (3) then follows by continuity of Γq.
The conclusions about the map κ are easy consequences of its definition.
For the final conclusion note that if we just let W be the span of v then
we have already shown that if C is the unique invariant member of Γ(C )
associated to v then for any divisor c′ ∈ Γ(C ) satisfying δ(c′) = v letting
λ be the eigenvalue of v we can write c′ = κ(v) + (Γq)(b) and equation 3
becomes (ΓfC )
kc′/λk = κ(v) + (Γq)(ΓfB)
kbλk where the final term goes to
zero as k →∞ (by our assumptions on growth rates of g−1 and ΓfB). Hence
(ΓfC )
k(c′)/λk converges to c = κ(v). If U is any open subset of X and if the
support of c′ is disjoint from fn(U) for arbitrarily large values of n, then the
support of (ΓfC )
n(c′) must be disjoint from U for arbitrarily large values of
n. Since, rescaled, these converge to c then U must lie outside the support
of c.
Remark. While we have not formally required X to be compact, the re-
quirement that Γ(B) be a Banach space makes this the main case in which
Theorem 12 is apt to have interesting applications.
Theorem 12 shows that among all members of Γ(C ) representing a coho-
mology class in W there is a unique invariant linear subspace which can be
identified with W and all other such members of Γ(C ) are contracted to this
invariant copy of W in Γ(C ) under (rescaled) pullback.
Corollary 13. Assume that the hypothesis of Theorem 12 are satisfied, and
that g : W → W is dominated by a single simple real eigenvalue r > 0 with
eigenvector v. Let C ≡ κ(v) be the unique invariant divisor of v. Then given
a divisor C′ ∈ Γ(C ) of any w ∈ W the successive rescaled pullbacks fkC (C
′)/rk
converge to a multiple (possibly zero) of C.
Proof. This is a direct consequence of equation (3).
The assumption that g : W → W is dominated by a single simple real
eigenvalue is meant to handle the most typical situation, and is not an es-
sential restriction.
Remark. Given that for a fixed f : X → X the category of SC sheaves A
on X endowed with an f self cohomomorphism F is an abelian category
with enough injectives, then the functor Fixed Γ which gives the fixed global
sections of A under F will be left exact and its right derived functors should
be of dynamical interest. In the case where A is a sheaf of functions and
f is invertible this is just group cohomology with the group Z acting on
Γ(A ) and has been an object of study for some time (see, e.g. [Kat03]). We
anticipate studying the case of more general sheaves A and the right derived
functors of the composition Fixed Γ in a future paper, including the case of
endomorphisms.
3.1 Regularity and Positivity
Typically our regularity results for the members invariant plane κ(W ) will
be most easily described in terms of B rather than C . We therefore make
the following definition.
Definition 14. Given a subsheaf B′ ⊂ B we will say a divisor C ∈ Γ(C ) has
local B′ potentials if C ∈ Γ(q(B′)). This is equivalent to requiring that about
each point x ∈ X there is an open neighborhood U and some B′ ∈ B′(U)
such that q(B′) = C
The proof of Theorem 12 implicitly provides a method to prove regularity
results for members of the invariant plane κ(W ). We make this explicit as a
corollary (of the proof).
Corollary 15. Assume we are given f : X → X and a short exact sequence
of sheaves A
p→ B q→ C satisfying the hypothesis of Theorem 12. Assume
that B′ is a subsheaf of B and that ΓfB(B
′) ⊂ B′. Let C ′ be the image of
B′ under q : B → C . Let A ′ ⊂ A be the kernel of q : B → C ′. Assume
that the canonical map H1(X,A ′) → H1(X,A ) is injective. Assume that
there are basis members w1, . . . , wk of W with divisors each of which has local
potentials in B′. Let r be the the inverse of the absolute value of the largest
eigenvalue of g−1 (so for all j ≥ 0, g−j is an operator of norm no more
than cr−j for some c > 0) Finally assume that for any sequence of numbers
aj, j = 0, 1, 2, . . . such that |aj| is no more than a constant times r−j as
j →∞ then for B ∈ Γ(B′) the exponentially decaying sequence
a0 B + a1 (ΓfB)(B) + a2 (ΓfB)
2(B) + · · · (4)
converges in the Banach space structure on Γ(B) to a member of Γ(B′).
Then the map κ : W → Γ(C ) lands in Γ(C ′).
Proof. Since W lies in the α chronically expanding subspace of W then neces-
sarily α/r < 1. Thus the terms of equation (4) have exponentially decreasing
norms and the series is exponentially decaying.
By the assumption of a divisor in Γ(C ′) for each member wj of a basis
then the map s : W → Γ(C ) in Theorem 12 can be assumed to land in Γ(C ′).
Then (ΓfC )sg
−1− s lands in Γ(C ′) and satisfies δ((ΓfC )sg−1− s) = 0. Since
H1(X,A ′) → H1(X,A ) injects it easily follows that for each wj one can
choose σ(wj) to be a member Bj of Γ(B)
′. Using the basis w1, . . . , wk to write
g−1 as a matrix A, and letting aij,` be the ij entry of A
` (so for each ij, aij,` is
bounded by a constant times r−`) we see that τ`(wj) = Bj + (ΓfB)(a1j,1B1 +
· · ·+ akj,1Bk) + (ΓfB)2(a1j,2B2 + · · ·+ akj,2Bk) + · · ·+ (ΓfB)`(a1j,`B1 + · · ·+
akj,`Bk). Gathering all the B1 terms, B2 terms, etc... from the right hand
side we see that τ = limk→∞ τk is a member of Γ(B
′) and thus that κ lands
in Γ(C ′) by equation (3).
The following trivial observation will suffice for our needed positivity
conclusions.
Observation. Assume we have an f self cohomomorphism of a short exact
sequence of sheaves A
p→ B q→ C satisfying the hypothesis of Theorem 12,
and also a subsheaf C ′ ⊂ C such that
1. C ′ is closed under multiplication by R>0. Note that C ′ is not necessarily
a sheaf of K modules, or even of groups.
2. fC (C
′) ⊂ C ′
3. Γ(C ′) is closed in Γ(C ).
Then for any closed eigenbundle v ∈ H1(X,A ) with eigenvalue in K0 and at
least one divisor C′ ∈ Γ(C ′) the unique invariant divisor C ∈ Γ(C ) of v also
lies in Γ(C ′).
Proof. The proof is trivial since C = limk→∞(ΓfC )
k(C′)/λk where λ ∈ R>0 is
the eigenvalue of v.
4 Subsheaf Cohomology
In applications of Theorem 12 it is common that there is a well understood
exact sequence of sheaves
d0→ S1
d1→ S2
d2→ · · · (5)
and that B is a subsheaf of Sk for some k, A is the kernel of dk
: B →
Sk+1 and C is the image of B in Sk+1. Moreover, in these cases the self co-
homomorphism f on A → B → C is induced by an f self cohomomorphism
of the sequence (5). In order to apply Theorem 12 to these cases we need to
understand the R module H1(X,A ) and its induced self map.
There does not seem to be a computationally useful way to extract an
injective resolution of A using subsheaves of S0
d0→ S1
d1→ · · · even if this
last sequence is acyclic. Consider for example the case where for each n,
Sn is the sheaf of currents of degree n and B ⊂ Sk is a subsheaf of mildly
regular currents. It is not clear one could make the regularization method of
[dR84] work to compare H1(X,A ) to deRham cohomology groups because
his chain homotopy operator A does not restrict well to B since dA does not
preserve regularity. We use a standard sheaf cohomological trick, which we
include here as a proposition which we will need and which we expect to be
commonly used in conjuction with Theorem 12 because of the requirement
that Γ(B) be a Banach space.
Theorem 16 (Subsheaf Cohomology). Assume we are given an exact se-
quence of sheaves S0
d0→ S1
d1→ S2
d2→ · · · and that B is a subsheaf of Sk for
some k ≥ 1. Let A = ker dk
, and B′ be the preimage of B under dk−1.
Further assume that for each j ≥ 1 we have Hj(X,B′) = 0, Hj(X,B) = 0
and for any m satisfying 0 ≤ m ≤ k − 1 we have Hj(X,Sm) = 0 for j ≥ 1.
Then for each n ≥ 1 there is a canonical isomorphism
Hn(X,A ) ∼= Hn+k(X, ker d0).
Proof. While this result is essential for us, its proof is a standard cohomo-
logical trick. First one notes that ker dk−1
= ker dk−1 by the definition of
B′. One has the short exact sequences of sheaves:
ker dk−1 → B′ → (dk(B′) = A )
ker dj → Sj → ker dj+1, j = 0, . . . , k − 2.
Considering the long exact sequences for these shows that the induced maps
Hn(X,A )→ Hn+1(X, ker dk−1) andHn+j(X, ker dk−j)→ Hn+j−1(X, ker dk−j−1)
are isomorphisms for j = 1, . . . , k−1. Composing each of these canonical iso-
morphisms gives a canonical isomorphism fromHn(X,A )→ Hn+k(X, ker d0).
Remark. We take it as clear from the functorality of the δ map in the long
exact sequence that given an f -self cohomomorphism of S0
d0→ S1
d1→ S2
· · · which maps B to itself that the induced map of H1(X,A ) is identified
with the induced map of Hk+1(X, ker d0) via the above isomorphism.
We will need one more tool be able to make effective use of Theorem 16
for calculating sheaf cohomology of subsheaves of sheaves of currents.
Definition 17. By an interval flow h on a bounded open interval I ⊂ R we
will mean the flow obtained by integrating a vector field of the form σ(t) ∂
where σ is positive exactly on I and zero elsewhere. We use h(x, t) to denote
the location of x ∈ R after following the flow for time t.
Definition 18. By an n-box in Rn we will mean an open subset which is
a product of n bounded open intervals I1, . . . , In. By an n-box in an n
dimensional manifold we will mean an n-box which is compactly supported
in some coordinate patch. By an n-subbox of an n box U = I1 × · · · × In we
will mean an n box of the form I ′1 × · · · × I ′n where I ′k is a subinterval of Ik
for each k ∈ 1, . . . , n.
Definition 19. By an n-box flow we will mean the Rn action h on Rn
which is the product of n interval flows h1(t1), . . . , hn(tn) on Rn. That is
h(x, t) = (h1(x1, t1), . . . , hn(xn, tn)) where x = (x1, . . . , xn), t = (t1, . . . , tn)
and h1, . . . , hn are interval flows on I1, . . . , In respectively. We refer to the
n-box I1 × · · · × In as the open support of the n-box flow. We will often ht
to denote the diffeomorphism h(·, t) : Rn → Rn.
Definition 20. Let h be an n-box flow on an n-box B. Let ρ be a compactly
supported smooth volume form on Rn. With this data we define an operator
Sh,ρ on smooth k forms on any n box U containing B by
Sh,ρ(φ) =
h∗t (φ)ρ(t) (6)
We say Sh,ρ defines a box smear on U , or smears U . We will omit the
subscript from Sh,ρ when the meaning is clear from context. It is clear S(φ)
is compactly supported in U if φ is.
It is clear from the definition of S that if ψ is an n− k form on U then∫
SH,ρ(φ) ∧ ψ =
φ ∧ S−H,ρ(ψ)
where−H is the family Ht with the parameter negated. From this motivation
we define a smear of a current.
Definition 21. Given h, ρ defining a smear on an n box U we define the
smear Sh,ρ on currents on U via
< Sh,ρ(C), φ >≡< C,S−h,ρ(φ) > .
Lemma 22. Given h, ρ defining a smear S on an n box U then d
S(dC) for currents C on any open subset of U containing the open support
of the smear. Also, restricted to the open support of the smear, S(C) is a
smooth form on V .
Proof. We remark that it is clear that d
= S(dφ) for forms φ, and
consequently for currents φ via the definition.
Because on the open support of the smear, a smear is just convolution
with a smooth function, then we see that if V is an open subset of the open
support of smear S on U then for any current C on U , S(C)
is a smooth
form on V .
Proposition 23. Let B be a sheaf of degree k currents. Assume that B
contains the sheaf of smooth k forms on X, and that B(U) is closed under
smears on any n-box U ⊂ X. Let B′ be the preimage under d of B in the
sheaf of degree k − 1 currents. Then B′ is soft, and therefore, Γ-acyclic.
Proof. To show that B′ is soft it is sufficient to show that B′ is locally soft
([Bre97] page 69). Given an n-box U in X we therefore only need to show
that if K is a closed subset of X in U and if W is an open neighborhood
of K then given any member B′0 of B
′(W ) there is an open neighborhood
W0 ⊂ W of K and a member B′ ∈ B′(U) such that B′
= B′0
Choose any pair of open sets V1, V2 such that K b V1 b V2 b W . Then
V2 \ V1 is compact and can therefore be covered by finitely many (open) n-
subboxes Y1 . . . , YN of U . Moreover these subboxes can all be chosen to be
disjoint from K and to lie inside W . Letting S1, . . . ,Sn be smears on U with
open support Y1, . . . , YN respectively then let B = S1(S2(· · · (SN(B′0)) · · · )).
Then on each Yj, B is given by a smooth k form. Also, B
= B′0
. Finally,
we choose a smooth function ψ : U → [0, 1] which is one on a neighborhood
of V1 and zero on a neighborhood of U \ V2. Then the current B′ ≡ ψB
extends (by zero) to a current on all of U . Then for each Yj, B
smooth function times a smooth form. Thus d(B′
) is a smooth form and
Figure 1: A current comprised of parallel submanifolds smeared and cropped.
lies in B(Yj). The boxes Yj cover V2 \ V1. Outside V2, B′ is identically zero.
We know that dB ∈ B(W ) by Lemma 22. We also know that ψ ≡ 1 on
an open neighborhood W1 of V1. Thus d(B
) = d(B
) ∈ B(W1). We
thus conclude that B′ ∈ B′(U) since its restriction to each Yj, to W1 and to
U \ V2 is a section of B′. Letting W0 = V2 \ (Y1 ∪ Y2 ∪ · · · ∪ YN) then W0 is
an open neighborhood of K, then W0 ⊂ W1 so B′
= B′0 since
W0 is disjoint from the open support of each of the smears S1, . . . ,SN . This
completes the proof that B′ is soft.
The following gives a broad generalization of the equalivalence of the co-
homology of currents with the deRham cohomology groups. To the author’s
knowledge, this result is new.
Corollary 24. Let B be a sheaf of degree k currents. Assume that B con-
tains the sheaf of smooth k forms on X, and that B(U) is closed under
smears on any n-box U ⊂ X. Letting A be the subsheaf of d closed members
of B, then
Hm(X,A ) = Hm+k(X,K),
where K is R or C depending on whether or not we allow complex valued
currents and forms.
Proof. This is an immediate consequence of Proposition 23 and Theorem 16.
5 Invariant Currents
Notation 1. If G is some sheaf of functions on a smooth orientable manifold
X we will use F k(G ) to denote the sheaf of k forms on X with coefficients
in G . We will let F kc (G ) be the subsheaf of closed (in the sense of currents)
members of F k(G ).
It will be convenient to use either degree or dimension of a current de-
pending on the context (just as dimension and codimension are useful for
discussing manifolds), so we will not stick to just one of these terms. We will
let C k denote the sheaf of degree k currents with the index written above
as is typical for cohomology since d increases the degree. We will similarly
write Ck for the sheaf of dimension k currents with the index written below
since d decreases dimension as is common for homology. We use the following
convention to realize a form α as a current so that if α is C1 then dα is the
same whether computed as a current or a form.
Definition 25. Given an k form α with L1 coefficients on an n manifold X
we realize α as a degree k current via
β 7→ (−1)(
α ∧ β
Definition 26. Given a (possibly complex) nonzero deRham cohomology
class c ∈ HkdeRham(X) with f
∗(c) = α · c for some scalar α ∈ C we will refer
to a current C in the same cohomology class as α as an eigencurrent for f if
f ∗(C) = αC.
Currents naturally pushforward, rather than pullback. Because we are
considering maps which are not necessarily invertible we need to address
how this pullback is performed. If f has critical points it is impossible to
define a continuous pullback operation f ∗ on all currents in a way that agrees
with expected cases. For instance, consider f(x) = x2 and let Ca be the
dimension one current on R with Ca(h(x)dx) = h(a), i.e. Ca is a unit mass
vector. Then the pullback f ∗(Ca) should be the sum of weighted unit masses
at the two preimages of this vector (just like the pullback of a point mass
is a sum of point masses each weighted by multiplicity), that is, f ∗(Ca) =
C√a − C−√a
. However, these pullbacks do not converge to a current as
a → 0 so f ∗(C0) is not defined. Since we want f ∗ to be continuous, we are
forced to work with currents that have some extremely mild regularity. We
address this in the next section.
5.1 Nimble Forms and Lenient Currents
Finding a good set of currents to use to study smooth finite self maps (not
necessarily invertible) of compact manifolds turns out to be rather delicate.
Our solution is to first expand our class of forms to include pushforwards (in
the sense of currents) of forms through an appropriate class of smooth maps.
Then we restrict our attention to currents which act on this extended class
of forms.
This solution has the very nice property that it can potentially be adapted
directly to study the dynamics of other various other categories of smooth
maps (by simply changing which forms are considered nimble, according
to the class of maps used). It will convenient to first define the natural
pushforward operator on forms:
Definition 27. Given a compact orientable manifold X we let SX be the
category of smooth maps f : X → X of nonzero degree and having the
property that the critical set has measure zero. We use critical set here to
mean the points at which Df is not invertible.
It follows from our definition that the image of any set of positive measure
under some f ∈ SX has positive measure.
Definition 28. Given a compact orientable manifold X we define N k to be
those currents ϕ which are a finite sum of currents of the form p∗(σ) where
p : X → X is a map in SX and σ is a form of degree k. The pushforward
p∗(σ) is computed in the sense of currents.
We will later show that nimble forms are also, in fact, bona fide forms.
Definition 29. We topologize N k by saying ϕj → ϕ in N k if for sufficiently
large j there are maps f1, . . . , fk and k forms σ1j, . . . , σkj as well as forms
σ1, . . . , σk such that
i fi∗(σij) = ϕj and
i fi∗(σi) = ϕ (where pushfor-
wards are taken in the sense of currents) and for each i ∈ 1, . . . , k, the forms
σij converge to σi in the strong sense (i.e. all derivatives converge uniformly).
Lemma 30. Given a compact orientable manifold Y , N k(Y ) is a topological
vector space.
Proof. This follows easily from our definition of the topology.
We now define the corresponding space of currents.
Definition 31. We define the dimension k lenient currents Lk(Y ) to be
the topological dual of N k(Y ). Every member of Lk(Y ) is a dimension k
current, but with the added structure of its action on all nimble k forms. We
give Lk the weak topology, i.e. Ci → C in Lk iff < Ci, ϕ >→< C, ϕ > for
every ϕ ∈ N k. We write L k for the lenient currents of degree k.
We define operations of wedge products with smooth forms as is usual for
currents. It is clear that the lenient dimension k currents give a sheaf on X.
The following properties of nimble forms are also immediately clear.
Lemma 32. Let f : X → X be a member of SX . The pushforward (as
a current) of a nimble k form by f is again a nimble form. Moreover
f∗ : N
k(X)→ N k(X) is continuous (in the topology of nimble forms). Also
the exterior derivative of a nimble form (as a current) is a nimble form and
d : N k(X)→ N k+1(X) is continuous.
The basic necessary facts about pulling back lenient currents are then
immediate. We state them here:
Lemma 33. Given f : X → X a member of SX the induced map f ∗ on the
sheaf of lenient degree k currents is an f cohomomorphism of sheaves. Both
f ∗ : L k(X) → L k(X) and d : L k(X) → L k+1(X) are continuous. Lastly,
f ∗d = df ∗ : L k(Y )→ L k+1(X).
Proposition 34. Assume that f : X → X is a member of SX . Let R be the
regular set of f . By Sard’s theorem R has full measure. Since the critical set
is compact then R is an open subset of X. Since the preimage of a measure
zero set has measure zero for SX maps then f−1(R) is also a full measure
open set in X. There is a well defined operation f? which maps k forms on
f−1(R) to k forms on R. Given a k form β on X, f?(β) is defined on any
open subset V ⊂ R such that each component U1, . . . , Um of f−1(V ) maps
diffeomorphically onto V by the formula
f?(β)
deg f
(β) · σi (7)
where σi ∈ {±1} is the oriented degree of f
: Ui → V . The pushforward
f? satisfies:
• f?d = df? (keeping in mind that f? returns a current on R)
• f?(1) = 1
• f?(f ∗(β) ∧ α) = β ∧ f?(α)
• (f?)n = (fn)?
• The formula ∫
f ∗(β) ∧ α =
β ∧ f?(α) (8)
holds for any k form β with L∞loc coefficients on Y and any smooth n−k
form α on X. This justifies using f? to pullback currents. (Part of the
conclusion is that both sides are integrable.)
Proof. Each statement is a consequence of formula (7) except the integrabil-
ity conclusion for equation (8). Local charts can be given which are bounded
subsets of Rn and for which Df remains uniformly bounded (over each of the
charts) and thus f ∗(β) will be a form with L∞loc coefficients in these charts.
Thus the left hand side of (8) is the integral of a bounded function over a
finite union of bounded charts and is therefore absolutely integrable. Since
∗(β) ∧ α) = β ∧ f?(α) it is sufficient to show that if γ is an n form with
L∞loc coefficients then ∫
f−1(R)
f?(γ). (9)
Typicaly f?(γ) is unbounded so we need to show that the right hand side of
(9) is integrable. About any point x ∈ R we can find an open V such that
each of the preimages U1, . . . , Uk of V is mapped diffeomorphically onto V .
Since X is orientable and n dimensional there is a well defined notion of the
absolute value of an n form. Then∫
|f?(γ)| ≤
deg f
((f
= ∑
|γ| =
f−1(V )
NowR is covered by countably many such sets V and listing them as V0, V1, V2, . . . ,
we can let V ′0 = V0, V
1 = V1 \V0, V ′2 = V2 \ (V0∪V1), . . . . Then R is the union
of the countable collection of disjoint measurable sets V ′j and∫
|f?(γ)| =
|f?(γ)| ≤
f−1(Vj)
|γ| =
f−1(R)
Since
f−1(R)
|γ| is finite then f?(γ) is an L1 form. Using precisely the same
argument but with the absolute values removed and the inequalities replaced
with equalities then shows
f?(γ) =
f−1(R)
Since R and f−1(R) are open and full measure then f? is an operator
which takes in forms on X and returns forms defined almost everywhere on
We now show that nimble forms are bona fide forms.
Lemma 35. If g : X → X is a map in SX and σ is a smooth k form on X
then the current g∗(σ) is the current of integration against the form g?(σ).
Proof. If ϕ is a smooth n − k form then by definition < g∗(σ), ϕ >=<
σ, g∗(ϕ) >= (−1)(
σ ∧ g∗(ϕ) = (−1)(
g?(σ) ∧ ϕ =< g?(σ), ϕ >
by formula (8) of Proposition 34
As described in [Fed69], an inner product on a vector space V can be
viewed as an isomorphism ` : V → V ∗ satisfying certain properties. The
inverse of ` gives the induced inner product on V ∗. The fact that < v,w >≤
‖v‖ · ‖w‖ with equality iff v and w are scalar multiples implies that the inner
product norm on V ∗ is the same as the operator norm of V ∗ acting on V .
The induced map
V ∗ gives an inner product on
We call this the canonical inner product on
V induced by the inner prod-
uct on V . Hence, given a Riemannian metric on X, there are canonical
smoothly varying inner products on
TxX and
T ∗xX for each x ∈ X.
At any point x ∈ X we define ‖
Dxf‖ to be the operator norm of the
linear function
Dxf :
TxX →
Tf(x)X. We define ‖
Df‖ to be
the L∞loc norm of the map x 7→ ‖
Dxf‖. Also, given a k form ϕ we define
the comass ‖ϕ‖L∞loc of ϕ to be the L
loc norm of the function x 7→ ‖
ϕx‖. It
is clear that the k forms with the comass norm is a Banach space. We now
show that the k forms with L∞loc coefficients are naturally lenient currents.
We start by defining the action on nimble forms.
Definition 36. Given an n− k form C with L∞loc coefficients we define
< C, p∗(σ) >= (−1)(
n−k+1
C ∧ p?(σ)
Lemma 37. The space F n−k(L∞loc) of n−k forms with L
loc coefficients under
the comass norm includes continuously into Lk(X) where the action of C ∈
F n−k(L∞loc) on some ϕ =
i fi∗(σi) ∈ N
k(X), with each fi ∈ SX and each
σi ∈ F k(C∞) is given by
< C, ϕ >≡
f ∗i (C) ∧ σi.
Proof. The assumption that X is compact means that any two Rieman-
nian metrics on X are comparable. Choose one so the notion of the comass
norm makes sense. The result is then a straightforward consequence of equa-
tion (8), Lemma 35, and our definitions.
Remark. It follows that a current with local F k(L∞loc) potentials is also a
lenient current.
Remark. Given a member C of F k(L∞loc) then f
∗(C) is the same whether
done as a lenient current or as a form. This, along with the fact that df ∗ =
f ∗d justifies the ad hoc pullback of closed positive (1, 1) currents used so
successfully in holomorphic dynamics. Similarly dC gives the same result
whether calculated as a lenient current or a form if C ∈ F k(C1).
5.2 Hölder Lemmas
We will want to apply Corollary 15 to show that each eigencurrent we con-
struct has local d potentials (or ddc potentials in the holomorphic case) which
are forms with Hölder continuous coefficients. In order to do this we will need
a few facts which we include here in order to avoid having to include regu-
larization results as afterthoughts to our main theorems.
Observation. Let Hα be the functions with coefficients that are Hölder of
exponent at least equal to some fixed α > 0. Since diffeomorphisms preserve
Hölder exponents and averages of Hölder functions are Hölder then we take
it as clear that Corollary 24 applies to show that H1(X,A ′) = H1(X,A )
where A ′ is the closed members of F k(Hα)) and A is the closed degree k
currents.
Lemma 38. Let X be a compact manifold (real or complex) with a Rieman-
nian metric and of real dimension n. Let f : X → X be a smooth map. Then
local coordinate charts Ui can be chosen on X (each representing a convex
open subset of Rn) so that there is a positive constant 1 < M so that for any
k form ϕ, there exist constants c, C > 0 such that writing each fk∗(ϕ) in any
of the charts Ui as
fk∗(ϕ) =
akidx
then each function aki satisfies
|aki| ≤ c · ‖fk∗(ϕ)‖comass (10)
and for each j ∈ 1, . . . , n,
∂aki
≤ C ·Mk.
Proof. Equation (10) is a basic fact.
The rest is a straightforward consequence of realizing a self map of a
manifold as being made up of a bunch of maps between different coordinate
patches in Rn. That is, one chooses an open cover of patches Ui of X. Each
patch is realized in Rn as a round ball. Thinking of each patch as lying in Rn
then we can find explicit maps from between open subsets of Rn of the form
pij : Ui ∩ f−1(Uj)→ Uj. By shrinking each open ball Ui a small amount the
resulting patches still cover X but the derivatives of the maps pij are all now
bounded (since we are working on relatively compact subsets of the previous
maps pij).
Then given any x we can keep track of which patch fk(x) is in at each time
and can then realize the map fk(x) as a composition pi1i2 ◦ pi2i3 ◦ · · · ◦ pik−1ik .
Since each partial derivative of each pij is uniformly bounded then any partial
derivative of the composition grows at most exponentially with k and we are
done.
The following observation will also be useful:
Lemma 39. If there are positive constants c, C,m,M with m < 1 < M
such that a sequence of smooth functions hk on an open convex set U ⊂ Rn
satisfies
‖hk‖sup < c ·mk
and www∂hk
< C ·Mk
for all k ∈ 0, 1, 2, . . . then h1+h2+h3+. . . converges to a bounded continuous
function which is Hölder of any exponent α <
log(m)
log(m/M)
Proof. The proof is elementary.
5.3 Eigencurrents for Cohomologically Expanding Smooth
We will call a section V of
TX a k-vector field. We define ‖V ‖L∞loc to be
the L∞loc norm of the function x 7→ ‖Vx‖. Whether Theorem 12 applies to a
map will depend the size of ‖
Df‖. Replacing f with an iterate does not
affect the needed estimate so we make the following definition.
Definition 40. We define Υk to be the limit supremum as j → ∞ of
D(f j)‖
j . It follows that Υ1 ≥ eλ for any Lyapunov exponent λ and
that Υk ≤ Υk1 ([Fed69] page 33).
We let B be the sheaf F k−1(L∞loc). The norm ‖ · ‖∞ clearly makes Γ(B)
into a Banach space. Given a member B ∈ Γ(B), since the operator norm on
TxX is equal to the norm already defined on
T ∗xX for each x ∈ X
then ‖B‖∞ is equal to supremum of the L∞loc norm of the function x 7→ B(Vx)
as V varies over all L∞loc k-vector fields of norm no more than one.
Theorem 41. Given f : X → X an a map in SX for the compact orientable
manifold X, assume that c ∈ HkdeRham(X) is a cohomology class (using ei-
ther real or complex deRham cohomology) which is an eigenvector for f ∗ with
eigenvalue β. Assume also that |β| > Υk−1. Then there exists a unique eigen-
current C with local F k−1(L∞loc) potentials representing the class c. Moreover
C has local F k−1(H) potentials.
Also, given any neighborhood U ⊂ X of any point in the support of C,
then for every lenient current C′ with local F k−1(L∞loc) potentials and which
represents the cohomology class c then fk(U) ∩ Supp C′ 6= ∅ for all large k.
Assume that the linear map f ∗ : HkdeRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r. Given C′ any current which has local
F k−1(L∞loc) potentials and which represents a cohomology class in the Υ
chronically expanding subspace of HkdeRham(X), then the successive rescaled
pullbacks fk∗(C′)/rk of C′ converge to a multiple of C in the sense of lenient
currents (and thus also in the sense of currents).
Proof. We let B = F k−1(L∞loc), A and C be the kernel and image respec-
tively of B
d→ L k. By Theorem 24, H1(X,A ) can be canonically identified
with Hk(X,K). Since B is Γ-acyclic then every member of H1(X,A ) is a
closed bundle with respect to the short exact sequence A → B → C .
From Lemma 33 there is an induced f cohomorphism of the short ex-
act sequence A
ι→ B d→ C . Also Γ(C ) is a space of lenient currents by
Lemma 33 and thus has a natural structure as a topological vector space. If
a sequence Bi ∈ Γ(B) converges to B ∈ Γ(B) then < dBi, ϕ >=
Bi∧dϕ =∫
B ∧ dϕ =< dB, ϕ > so the map d : Γ(B)→ Γ(C ) is continuous.
The cohomomorphism ΓfB is pullback f
∗ of differential forms. Fixing any
real α satisfying Υk−1 < α < |β| it is clear from the definition of Υk−1 that one
can choose a real d > 0 such that ‖
D(f `)‖ ≤ d ·α` for all ` ∈ N. The `th
pullback f `∗(B) of B ∈ Γ(B) satisfies ‖f `∗(B)‖∞ = supV ‖B(
D(f `)(V ))‖∞
where the supremum is taken over all k-covector fields V with ‖V ‖∞ ≤ 1.
However
D(f `)(V ) is a k-covector field of norm no more than ‖
D(f `)‖,
so ‖f `∗(B)‖∞ ≤ ‖B‖∞ ·
D(f `)‖∞ ≤ d · α`‖B‖∞.
Given any W in the Υk−1 chronically expanding subspace of H
k(X,K),
we can alter our choice of α > Υk−1 so that W also lies in the α chronically
expanding subspace of Hk(X,K).
We can therefore apply Theorem 12 to conclude that there is a (unique)
map κ : W → Γ(C ) such that f ∗κ = κf ∗, where the first f ∗ is pullback of
currents and the second is pullback on Hk(X,K).
In fact κ(W ) lies in the space of currents with locally Hölder potentials
(meaning F k−1(H) potentials) by applying Corollary 15 in conjunction with
Observation 5.2, Lemma 38 and Lemma 39. The second half of the Theorem
is a consequence of equation (3).
Remark. Theorem 41 gives regular degree one eigencurrents for every eigen-
value of f ∗ : H1(X,K) → H1(X,K) of norm greater one without requiring
any constraints on the local behavior of f . The degree one eigencurrents
seem to be, in some sense, more robust than currents of lower dimension,
including invariant measures. Moreover since codimension one closed sub-
manifolds are closed currents with local F 0(L∞loc) potentials then successive
rescaled preimages of such manifolds in the right cohomological class will
converge to the eigencurrent.
Remark. The fact that eigencurrents constructed via Theorem 41 have local
potentials which are forms does not imply their support has positive Lebesgue
measure as the classical example of a monotonic nonconstant function which
is constant on a set of full measure shows.
Remark. The assumption that f ∗ : H1deRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r is not essential, but just meant to handle
the simplest case. In fact the proof actually shows that if W lies in the
Υk−1 chronically expanding subspace of H
k(X,K) then every current in the
invariant plane κ(W ) ⊂ Γ(C ) of currents has local F k−1(H) potentials and
any current with cohomological class in W with local F k−1(L∞loc) potentials
is attracted to κ(W ) under successive rescaled pullback.
Since measures are of particular interest in dynamics, we note thatH1(X,F n−1(L∞loc)) =
Hn(X,K) = K by Corollary 24 so there is a unique f ∗ eigenvalue and it is
precisely the topological degree of f . We thus obtain:
Corollary 42. Given that Υn−1 < deg f then there is a unique dimension
zero eigencurrent C with F k−1(L∞loc) potentials (and in fact it has F
k−1(H)
potentials) and the successive rescaled preimages of any C′ with F k−1(L∞loc)
potentials converge to C. If additionally there is no point x ∈ X about which
f is locally an orientation reversing diffeomorphism then C (and every other
member of κ(W )) is a positive distribution and is therefore a Radon measure.
Proof. Since f ∗ pulls back dimension zero currents (i.e. distributions) which
are positive to distributions which are positive then by Corollary 3.1 the
distribution C is positive. It is therefore a Radon measure (see e.g. [HL99]
page 270).
Remark. In the case where f is orientation reversing on some parts of X (but
not on all of X) some special remarks apply. If it happens that successive
rescaled images of some point converge to a dimension zero eigencurrent then
since preimages of points are counted with multiplicity then when pulled back
through a portion of X on which f reverses orientation the sign of a point
is flipped. Thus in this case the eigencurrent may not describe so much
the distribution of preimages as the relative density of preimages counted
negatively as compared to those counted positively. The number of actual
preimages of a point may grow exponentially faster than the degree of the
map in such cases so that dividing by the degree does not yield a measure in
the limit unless some such “cancellation” takes place in the limit. One would
expect that the corresponding eigencurrents have local potentials which are
not of bounded variation in such a case.
5.4 Eigencurrents for Smooth Covering Maps
We will call a covering map which is locally a diffeomorphism a smooth
covering map. We now consider the special case of smooth self covering
maps f : X → X of a compact smooth orientable manifold X. We show
that in this case we have a substantially broader collection of currents whose
successive pullbacks converge to an eigencurrent, albeit we need different
estimates for Theorem 12 to apply. We will pull back currents by pushing
forward forms with f?. Since the regular set of f is all of X then f? is a well
defined operator from smooth forms to smooth forms.
Definition 43. For a map satisfying the hypothesis of Proposition 34 we
define the operation f ∗ from currents on X to currents on Y by
< f ∗(C), α >≡< C, f?(α) > .
Clearly f ∗ preserves the dimension of a current.
Let Mk−1 be the sheaf for which Mk−1(U) is the Banach space of bounded
linear operations on the topological vector space comprised of F k−1(C∞)(U)
with the ‖ · ‖∞ norm. Equivalently, Mk−1 is the sheaf of dimension k − 1
currents of finite mass.
Choose a Riemannian metric on X. If f : X → X is a smooth cover
then for each x ∈ X and each ` ∈ N, Dx(f `) : TxX → Tf`(x)X is invertible.
We let νk(x, `) be the operator norm of the inverse of
TxX →∧k
Tf`(x)X. We define νk(`) = supx∈X νk(x, `)
1/`. We define νk = lim sup`→∞ νk(`).
The iterated pushforward operation f `? : F
k−1(C∞)(X) → F k−1(C∞)(X)
satisfies ‖f `?(ϕ)‖∞ ≤ νk(`) · ‖ϕ‖∞ as is straightforward to verify. If f is in-
vertible then νk is a bound on the growth of the k
th wedge product of the
derivative under f−1. For non-invertible f , νk represents a bound on the
growth of the kth wedge product of the derivative under any sequence of
successive branches of f−1.
Theorem 44. Given f : X → X a smooth self covering map and that
c ∈ HkdeRham(X) is a cohomology class (using either real or complex deR-
ham cohomology) which is an eigenvector for f ∗ with eigenvalue β. Assume
also that |β| > νk−1. Then there exists a unique eigencurrent C with local
Mk−1 potentials representing the class c. Moreover C has local F
k−1(C0)
potentials. Consequently C is a current of order one.
Also, given any neighborhood U ⊂ X of any point in the support of C, then
for every lenient current C′ with local Mk−1 potentials and which represents
the cohomology class c then fk(U) ∩ Supp C′ 6= ∅ for all large k.
Assume that the linear map f ∗ : HkdeRham(X)→ H
deRham(X) is dominated
by a single simple real eigenvalue r. Given C′ any current which has local
Mk−1 potentials and which represents a cohomology class in the ν
k−1 chroni-
cally expanding subspace of HkdeRham(X), then the successive rescaled pullbacks
fk∗(C′)/rk of C′ converge a multiple of C.
Proof. We let A and C be the kernel and image respectively of d : Mk−1 →
C k. Since df? = f?d then pullback of currents gives an f cohomomorphism
of the short exact sequence of sheaves A →Mk−1 → C .
Since ΓMk−1 is the continuous linear operators on a normed vector space
then it is a Banach space. From the observations previous to the statement
of Theorem 44 one concludes that for any α > νk−1 there is a constant d > 0
such that ‖f `∗(B)‖ ≤ d · αk‖B‖ for all ` ∈ N.
Since Γ(C ) is a space of currents it is naturally a topological vector space
over K.
The map f ∗ : Γ(C ) → Γ(C ) is continuous since if Ci → C in Γ(C ) then
< f ∗(Ci), ϕ >=< Ci, f?(ϕ) >→< C, f?(ϕ) >=< f ∗(C), ϕ >.
If Pi → P in ΓMk−1 (using the Banach space structure) then ‖Pi−P‖ →
0 by assumption then ‖P(dϕ) − Pi(dϕ)‖ ≤ ‖P − Pi‖ · ‖dϕ‖ → 0. Hence
< dPi, ϕ >= Pi(dϕ)→ P(dϕ) =< dP, ϕ > and so we conclude that the map
d : ΓMk−1 → Γ(C ) is continuous.
Given any W in the νk−1 chronically expanding subspace of H
k(X,K),
we can alter our choice of α > νk−1 so that W also lies in the α chronically
expanding subspace of Hk(X,K).
We can therefore apply Theorem 12 to conclude that there is a (unique)
map κ : W → Γ(C ) such that f ∗κ = κf ∗, where the first f ∗ is pullback of
currents and the second is pullback on Hk(X,K).
In fact κ(W ) in the currents with locally continuous potentials by apply-
ing applying Corollary 15 in conjunction with Observation 5.2, Lemma 38
and Lemma 39. The second half of the Theorem is a consequence of equa-
tion (3).
Proposition 45. Let Y be an oriented codimension k submanifold of X. If
the cohomological class of Y (as a current) lies in the νk−1 chronically expand-
ing subspace of Hk(X,K) then the successive rescaled preimages of Y con-
verge to the invariant plane of currents κ(W ). If f ∗ : Hk(X,K)→ Hk(X,K)
is dominated by a single real eigenvalue r > νk−1 then the successive rescaled
preimages of Y converge to a multiple (possibly zero) of the r eigencurrent.
In particular, if νn−1 < deg f then the successive rescaled preimages of any
point converge to the unique invariant measure with Mn−1 potentials.
Proof. This follows immediately from Theorem 44 if we show that Y has
local potentials in Mk−1. This is equivalent to showing that locally Y = dP
where < P,ϕ >≤ a · ‖ϕ‖∞ for some a > 0. Let B be a ball in Rn and Y0
a k-plane in Rn. Then there is a k + 1 half plane P such that, as currents
in U , ∂P = Y0. Moreover it is clear that < P,ϕ >≤ a‖ϕ‖∞ for some real
a > 0. (There are also local potentials for Y which are given by forms with
L1loc coefficients. These can be constructed by choosing a projection π from
U \ Y0 to a codimension one cylinder C with axis Y0, and choosing a volume
form σ on C. The local potential is the pullback π∗(σ).)
Remark. As with Theorem 41, Theorem 44 gives regular degree one eigencur-
rents for every eigenvalue of f ∗ : H1(X,K)→ H1(X,K) of norm greater one
without requiring any constraints on the local behavior of f . In holomorphic
dynamics much progress has been made in constructing degree one eigencur-
rents and then constructing dynamically important invariant measures via
a generalized wedge product (see the references cited at the beginning of
Section 6).
Remark. The proof of Proposition 45 could clearly be modified to apply to
many singular manifolds as well.
6 Holomorphic Endomorphisms
We now restrict our interest to holomorphic dynamics. Thus all manifolds
are assumed to be complex manifolds and all maps are assumed to be holo-
morphic unless stated otherwise.
Holomorphic endomorphisms of the Riemann sphere have been studied
in great detail. For endomorphisms much of the theory is still in its be-
ginnings. Much attention has been paid to holomorphic automorphisms of
C2 [FM89], [FS92], [HOV94], [HOV95], [BS91a], [BS91b], [BS92], [BLS93],
[BS98a], [BS98b], [BS99] or K3 surfaces [Can01], [McM02], the major de-
velopments for endomorphisms have been on Pn, [FS94a], [FS94b], [FS95b],
[FS01], [FS95a], [JW00], [FJ03], [Ued94], [Ued98], [Ued97]. Recent signifi-
cant developments have been made for endomorphisms of Kahler manifolds
in [DS05]. The paper [DS05] shows existence of eigencurrents (or Green’s
currents) for endomorphisms of Kahler manifolds under a simple condition
on the comparative rates of growth of volume in two different dimensions.
They also show that a specific weighted sum of an arbitrary closed positive
smooth current will converge to the Green’s current, and that the Green’s
current has a Hölder continuous potential. In this setting our theorem shows
that arbitrary (rescaled) preimages of a broader class of currents will con-
verge to the Green’s current. A wide variety of results have been proven
in these various circumstances either showing the existence of invariant cur-
rents, showing convergence of currents to invariant currents, or studying the
properties of these invariant currents. We include here results that follow
from the method of this paper, which we are sure substantially overlap with
existing results. Presumably our cohomologicaly lifting theorem could be
used in conjuction with Theorem 12 to show existence of higher degree (k, k)
currents given certain bounds on local growth rates.
6.1 ddc Cohomology
Let Z be a complex manifold and let f : Z → Z be a holomorphic self map
of Z. Let H be the sheaf of pluriharmonic functions, let L∞loc be the sheaf
of locally bounded functions, and let C be the sheaf of currents with local
potentials in L∞loc, i.e. currents locally of the form dd
cb, for b a locally bounded
function. The members of C are closed (1, 1) currents on Z.
Using the usual pullback on functions, and the induced pullback on cur-
rents with function potentials (i.e. pullback the current by pulling back its
local potentials), then we get a self cohomomorphism of the exact sequence
of sheaves
H → L∞loc
ddc→ C . (11)
We note that H1(Z,H ) is a finite dimensional R vector space as can be
seen from the long exact sequence for the short exact sequence R→ O →H
where the first map is inclusion and the second takes the imaginary part. The
terms H1(Z,O) → H1(Z,H ) → H2(Z,R) give the finite dimensionality
since O is a coherent analytic sheaf (see e.g. [Tay02] page 302).
Then from Theorem 12 we obtain:
Corollary 46. Given v any closed eigenbundle of H1(Z,H ) for f ∗ with
eigenvalue r > 1, there is a unique closed (1, 1) current C such that limk→∞ f
k∗(C′)/rk
converges to C for any divisor C′ of v.
Remark. We note that the terms “closed eigenbundle” and “divisor” in Corol-
lary 46 are understood using the long exact sequence for (11).
We can apply Corollary 15 to show that
Corollary 47. Any such invariant current C so obtained has Hölder contin-
uous local potentials.
Proof. The result follows from Lemma 5.2, Lemma 38, the fact that the
ddc closed Hölder continuous functions are the same as the ddc closed L∞loc
functions and from Corollary 15.
Also from Observation 3.1,
Corollary 48. If v has a plurisubharmonic section the current C is positive.
7 Result via Invariant Sections
We stated early on that our construction of invariant members of H0(C ) for
a self cohomomorphism of a short exact sequence A → B → C of sheaves
could be done in terms of finding invariant sections of bundles. We illustrate
this here in a specific case where we can take advantage of geometry to make
further conclusions. Finding an invariant section of a bundle is equivalent to
finding an invariant trivialization of the bundle, and we will make our initial
statement in terms of a trivialization.
Let Z be a compact complex manifold. Let f : Z → Z be a holomorphic
endomorphism. Let p ∈ H1(Z,H ) be an eigenvector for f ∗ with real eigen-
value λ of norm greater than one. If f ∗ were to have complex eigenvalues of
interest, an analogous construction can be made to the one that follows.
We note that there is a canonical bundle map f̃ : f ∗(p) → p which gives
the map f on the base space. It is easy to show that there is a map σ : p→ λp
which is the identity on the base space and takes the form r 7→ λr+ b on the
fibers, where b is a constant. What is more, the map τλ is easily seen to be
unique up to the addition of a constant. Then define the map f̌ : p → p to
be the composition of
τλ→ λp = f ∗(p) f̃→ p.
Then f̌ is the map f on the base space and takes the form r 7→ λr + b on
the fibers.
Since every pluriharmonic bundle is trivial as a smooth bundle, then we
can choose a smooth trivialization t : p→ R, i.e. t(a+ r) = σ(a) + r for any
a ∈ p, r ∈ R, where a+ r is computed in the fiber containing a.
Theorem 49. There is a unique continuous trivialization g : p → R such
that:
g(a+ r) = g(a) + r for a ∈ p and r ∈ R,
g(f̌(a)) = λ · g(a) for a ∈ p,
moreover
g = lim
λ−k ◦ t ◦ f̌ ◦k
and the limit converges uniformly. Finally, the zero set of g is the image of a
section g : Z → p and is exactly the set of points whose forward image under
f̌ remains bounded.
Proof. Define a function T : p→ R by
T (a) ≡ t
f̌(a)
− λ · t(a).
Note that T descends to a well defined continuous function T : Z → R since
for an arbitrary r ∈ R one has T (a + r) = t
f̌(a + r)
− λ · t(a + r) =
t(f̌(a) + λr)− λ · (t(a) + r) = T (a).
One notes that since the function T is necessarily bounded if Z is compact
then defining
g(a) ≡ t(a) + λ−1 · T (a) + λ−2T (f̌(a)) + λ−3T (f̌ ◦2(a)) + · · ·
gives a continuous function g : p→ R satisfying the above two properties.
Assume g1 and g2 are two such functions. Then ∆ ≡ g1− g2 : p→ R is a
function satisfying ∆(a+ r) = ∆(a) for a ∈ p and r ∈ R so ∆ descends to a
continuous function ∆: p→ R satisfying ∆(f̌(a)) = λ ·∆(a). However since
λ > 1 one concludes that this is only possible if ∆ ≡ 0 since M is compact
so ∆(M) has compact image in R.
It is easy to check using the definition of T that λ−k ◦ t ◦ f̌ ◦k(a) is exactly
a partial sum of the first k terms of the above series and this gives the
convergence result. The conclusion about the section g is trivial.
The above construction can be carried through almost without modifi-
cation for any subspace of H1(Z,H ) on which f ∗ is expanding. This gives
an alternate way of understanding the convergence of preimages of sections.
The point is that if s is any section of p, i.e. the potential of a current C,
then 1
f ∗(C) is a current with potential which is the setwise preimage of s
under f̌ (this is easy to confirm from the construction of f). The Green’s
trivialization g shows that f̌ is uniformly repelling away from the image of
the invariant section g. Thus as long as s is bounded in p, (not even neces-
sarily continuous), then the successive preimages of s will converge uniformly
to the section g. Since uniform convergence of potentials implies convergence
of currents then the rescaled pullbacks of a current C converge to the cur-
rent with potential g. We already have this as a theorem, so we have not
restated it as such here. This is just an alternative approach. Note that in
the case where Z = P2 [FJ03] has given far more precise control of when the
successive rescaled preimages of a current will converge to the eigencurrent.
7.1 Sections version with an Invariant Ample Bundle
It is also interesting to consider the special case where there is an invariant
ample bundle with eigenvalue λ ≥ 2 an integer. Without loss of generality
we assume ` is very ample. The morphism of sheaves log | · | : O∗ → H
induces a map from holomorphic line bundles to pluriharmonic bundles. We
let p = log |`| be the corresponding pluriharmonic bundle.
It is easy to see that there is a holomorphic map ` → `λ which is of the
form σλ : z 7→ azλ, a ∈ C∗ on each fiber and is the identity on the base
space. There is also a canonical holomorphic map f̃ : f ∗(`) → ` which is a
line bundle map and is f on the base space.
One then defines the holomorphic map f̆ : `→ ` which is the composition
σk→ `k = f ∗(`) f̃→ `.
This map is of the form z 7→ azk on each fiber and is equal to the map
f : Z → Z on the base space. Let `∗ denote ` with its zero section removed,
so that log | · | : ` → p is a well defined continuous map. Since the preimage
of the zero section of ` under f̆ is the zero section then f̆ is a holomorphic
self map of `∗. It is easy to confirm that f̆ : ` → ` can be rescaled so that
the diagram
log |·|
log |·|
commutes.
Our Greens trivialization g : p→ R can be pulled back to give a Green’s
function G : `∗ → R on the punctured bundle `∗. It satisfies G(f̃(w)) =
λ · G(w) and G(βw) = G(w) + log |β| for w ∈ ` and β ∈ C∗. Since g is
a trivialization of an R bundle over a compact space, g is proper. Since
log | · | : `∗ → p is proper then G is proper. Thus, in this setting one can
construct a Greens function that is exactly analogous to the Green’s function
constructed on Cn+1 for a holomorphic endomorphism of Pn. Potentially one
could take advantage of the special geometry of very ample bundles to get
information about the dynamics in this situation.
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Introduction
Cohomomorphisms
Cohomomorphisms and .
Invariant Global Sections
Regularity and Positivity
Subsheaf Cohomology
Invariant Currents
Nimble Forms and Lenient Currents
Hölder Lemmas
Eigencurrents for Cohomologically Expanding Smooth Maps
Eigencurrents for Smooth Covering Maps
Holomorphic Endomorphisms
ddc Cohomology
Result via Invariant Sections
Sections version with an Invariant Ample Bundle
Bibliography
|
0704.0070 | Coincidence of the oscillations in the dipole transition and in the
persistent current of narrow quantum rings with two electrons | Coincidence of the oscillations in the dipole transition and in the persistent current of
narrow quantum rings with two electrons
Y. Z. He and C. G. Bao∗
State Key Laboratory of Optoelectronic Materials and Technologies,
and Department of Physics, Sun Yat-Sen University, Guangzhou, 510275, P.R. China
The fractional Aharonov-Bohm oscillation (FABO) of narrow quantum rings with two electrons
has been studied and has been explained in an analytical way, the evolution of the period and
amplitudes against the magnetic field can be exactly described. Furthermore, the dipole transition
of the ground state was found to have essentially two frequencies, their difference appears as an
oscillation matching the oscillation of the persistent current exactly. A number of equalities relating
the observables and dynamical parameters have been found.
PACS numbers: 73.23.Ra, 78.66.-w
* The corresponding author
Quantum rings containing only a few electrons can
be now fabricated in laboratories1,2. When a magnetic
field B is applied, interesting physical phenomena, e.g.,
Aharonov-Bohm oscillation (ABO) and fractional ABO
(FABO)of the ground state (GS) energy Eo and persis-
tent current Jo, have been observed
2−4,13. In the the-
oretical aspect, a number of calculations based on exact
diagonalization5−8, local-spin-density approximation9,10,
and the diffusion Monte Carlo method11 have been
performed. These calculations can in general repro-
duce the experimental data. For examples, in the
calculation of 4-electron ring6,11, the period of oscilla-
tion Φ0/4 found in experiments was recovered (Φ0 =
hc/eisthefluxquantum).
In addition to the oscillations in Eo and Jo, the oscil-
lation in the optical properties is noticeable.16,17. In this
paper a new kind of oscillation found in the dipole tran-
sition of two-electron (2-e) narrow rings is reported. The
emitted (absorbed) photon of the dipole transition of the
GS was found to have essentially two energies, their dif-
ference is exactly equal to hJo, where h is the Planck’s
constant. In other words the difference of the two photon
energies appears as an oscillation which matches exactly
the oscillation of Jo. This finding is approved by both
numerical calculation and analytical analysis as follows.
The narrow 2-e ring is first considered as one-
dimensional, then the effect of the width of the ring is
further evaluated afterward. The Hamiltonian reads
H = T + V12 +HZeeman (1)
G(−i ∂
+Φ)2, G = ~
2m∗R2
where m∗ the effective mass, θj the azimuthal an-
gle of the j − th electron, Φ = πR2B/Φ0, where B
is a magnetic field perpendicular to the plane of the
ring, V12 the e-e Coulomb interaction, HZeeman =
−SZµΦ the well known Zeeman energy where SZ is
the Z-component of the total spin S, and µ = g
πR2Φ0
where g∗ is the effective g-factor and µB is the Bohr
magneton. The interaction is adjusted as 7 V12 =
e2/(2ε
d2 +R2 sin2((θ1 − θ2)/2))
−1, where ε is the di-
electric constant and the parameter d is introduced to
account for the effect of finite thickness of the ring.
We first perform a numerical calculation so that all
related quantities can be evaluated quantitatively. m∗ =
0.063me, ε = 12.4 (for InGaAs), d = 0.05R , and the
units meV , nm , Tesla and Φ0 are used. Accordingly,
G = 604.8/R2,and µ = 33.53/R2.
A set of basis functions φk1k2 = e
i(k1θ1+k2θ2)/2π is in-
troduced to diagonalize the Hamiltonian, where k1 and
k2 must be integers to assure the periodicity, the sum of
k1 and k2 is just the total orbital angular momentum L.
φk1k2 must be further (anti-)symmetrized when S = 0(1).
When about three thousand basis functions are adopted,
accurate solutions (at least six effective digits) can be ob-
tained. The low-lying spectrum is plotted in Fig.1, where
the oscillation of the GS energy and the transition of the
GS angular momentum Lo can be clearly seen.
Let θC = (θ2 + θ1)/2 , and ϕ = θ2 − θ1. Then
H = Hcoll +Hint (2)
whereHcoll =
G(−i ∂
+2Φ)2+HZeeman andHint =
2G(−i ∂
)2+V12, they are for the collective and internal
motions, respectively. Our numerical results lead to the
following points.
(i) Separability: The separability of one-dimensional
ring is well known5. However, for the convenience of the
following description, it is briefly summarized as follows.
Each eigenenergy E can be exactly divided as a sum of
three terms
E = 1
G(L+ 2Φ)2 + Eint − SZµΦ (3)
where the first term is the kinetic energy of collective
motion, Eint is the internal energy .
Since the basis functions can be rewritten as
http://arxiv.org/abs/0704.0070v1
φk1k2 = e
iLθCei
(k2−k1)ϕ/2π (4)
the spatial part of each eigenstate Ψ is strictly separa-
ble as Ψ = 1√
eiL θCψint where the first part describes
the collective motion, while ψint is a normalized internal
state depending only on ϕ. In particular, both Eint and
ψint do not depend on B (or Φ).
0 2 4 6 8
E (meV)
FIG. 1: Low-lying levels of a 2-e ring against Φ/Φ0 in the
FABO region. When Φ/ is positive, Lo is negative, the num-
bers by the curves are −Lo.
(ii) Classification of ψint: When L is even (odd), (k2−
k1)/2 is an integer (half-integer), thus the period of ϕ as
shown in (4) is 2π (4π). Therefore, the periodicity of the
internal states have two choices. In fact, the difference in
the periodicity is closely related to the dependence of the
domains of the new variables θC and ϕ, this point has
been discussed in detail in ref.[14,15]. Let Q = (−1)L,
then the four cases (Q,S) = (1,0), (-1,0), (-1,1) and (1,1)
are associated with four types of states labeled by a, b, c,
and d , respectively. The internal states of Type a
are denoted as ψa, ψa∗ , · · · and the associated internal
energies as Ea < Ea∗ , · · · and so on. Examples of
ψint and Eint are plotted in Fig.2 and listed in Table 1,
respectively.
Table 1, The lowest and second lowest internal
energies (in meV ) of Type a to d, R = 30nm.
Type a b c d
Eint 2.626 4.247 2.630 4.272
E∗int 6.342 8.912 6.435 9.158
Due to the e-e repulsion, a dumbbell shape (DB), i.e.,
ϕ = 180◦, is advantageous in energy because the two
electrons are farther away from each other meanwhile.
However, a rotation of this geometry by π is equivalent
to an interchange of particles, these operations will cre-
ate the factors (−1)L and (−1)S, respectively, from the
wave function. Therefore, the equivalence leads to a con-
straint, accordingly the DB is allowed only for the states
Type a Type b
0 90 180 270 360
Type c
0 90 180 270 360
Type d
FIG. 2: Four types of ψint against ϕ , R = 40nm. The lowest
three of each type are shown, the higher state has more nodes.
with L + S even, i.e., only for Type a and c. Other-
wise, the states would have an inherent node at the DB
and therefore be higher in energy as shown in Table 1,
where Ea << Eb, Ec << Ed, and Ea ≈ Ec. In Fig.2
the patterns of Type a are one-to-one similar to Type c
, they all have a peak at the DB. On the contrary, all
those of Type (b) and(d) have the inherent node at the
DB. It is noticeable that Type b and c are not continuous
at ϕ = 0 and 2π due to their periods are not equal to 2π.
It was found that the internal states of all the GSs are
either ψa or ψc without exceptions because the favorable
DB is allowed in them. When the dynamical parame-
ters vary in reasonable ranges, the qualitative features of
Fig.2 remain the same.
According to (3), an appropriate Lo would be chosen to
minimize the GS energy. When Φ increases, Lo will un-
dergo even-odd transitions repeatedly and become more
negative as shown in Fig.1. Correspondingly, the total
spin So undergoes singlet-triplet transitions, and ψa and
ψc appear in the GS alternatively. However, due to the
Zeeman effect, when Φ is larger than a critical value Φcrit
, only So = 1 states will be dominant, and accordingly
only ψc will appear in the GS. The region Φ < (>) Φcrit
is called the FABO (ABO) region.
(iii) Persistent Current : Let J1 be the current of the
particle e1. The expression of J1 is well known.
5 However,
since it does not depend on the azimuthal angle, it equals
to its average over θ1. Thus the total current J = J1+J2
J = 1
dθ1dθ2 [Ψ
∗(−i ∂
+2Φ)Ψ+c.c.] (5)
where g = ~/(m∗R2). Using the arguments θC and ϕ
and making use of the separability, the integration over
θC and ϕ can be performed. Thus we have
J = g(L+ 2Φ)/2π (6)
This equation demonstrates explicitly the mechanism
of the oscillation of the persistent current, it is caused
by the step-by-step transition of L during the increase
of Φ. Examples of J are shown in Fig.3, where each
stronger oscillation (associated with a L odd and S = 1
GS) is followed by a weaker oscillation (associated with
a L even and S = 0 GS).
0 2 4 6 8
0 2 4 6 8
0 2 4 6 8
FIG. 3: The oscillation of the persistent current and the two
photon energies of the ground states against Φ/Φ0 in the
FABO region. The unit of current is 10−5C/R, where C
is the velocity of light. In the lowest panel, the black square
(white circle) denotes ~ω+ (~ω−), namely, the energy associ-
ated with Lo to Lo + 1 (Lo − 1) transition.
(iv) Relations among the internal states : Define
Om = e
im(θ1−θC)+ eim(θ2−θC) = 2 cos(mϕ/2). By an-
alyzing the numerical data, we found
Ñ(O1ψa) = ψb + ξa and Ñ(O1ψc) = ψd + ξc (7)
where Ñ is the operator of normalization, both ξa and
ξc are very small functions and depend on the dynamical
parameters very weakly. E.g., when R varies from 30 to
90, the weights of ξa and ξc vary from 0.0004 to 0.0002.
They are so small that in fact can be neglected. Since
O1 contains a node at the DB, it must cause a change of
type from a to b, or from c to d. Thus it is not surprising
that (7) holds. Since O1 is the operator of the dipole
transition (see below), eq.(7) provides an additional rule
of selection as discussed later.
(v) Dipole transition: The probability of dipole tran-
sition reads P
(o),± =
(ω±/c)
3R2 |A
(o),±|
2, where ω±
is the frequency of the photon,
= 〈Ψ(f)±|e
±iθ1 + e±iθ2 |Ψ(o)〉
= δL(f), L(o) ±1〈ψ
int |O1|ψ
int〉 (8)
where (f) and (o) denote the final and initial states,
respectively, the signs ± are associated with L(f) =
L(o) ± 1.
Let the initial state be the GS with Lo, then ψ
must be ψa or ψc depending on Lo is even or odd .
Let α denotes the type of the initial state. Due to (7)
int |O1|ψ
int〉 = δ(f),α < O1ψα|O1ψα >
1/2, where
δ(f),α implies that the final state must be ψb (ψd) if α = a
(c), otherwise the amplitude is zero. Thus, due to the ad-
ditional rule of selection eq.(7), the dipole strength of the
GS is completely concentrated in two final states having
L(f) = L(o) ± 1 and both having the same internal state
specified by eq.(7). Accordingly, only the photons with
the two energies
~ω± = E(f)± − E(o)
= G[ 1
(1± 2(Lo + 2Φ)) + ∆α/G] (9)
can be emitted (absorbed), where ∆α = Eb − Ea or
Ed − Ec depending on α = a or (c). The oscillation
of ~ω± is plotted in the lowest panel of Fig.3. It turns
out that ∆α/G depends on R very weakly, thus ~ω± is
nearly proportional to R−2. Accordingly, a smaller ring
will have a larger probability of transition with a higher
energy.
(vi) FABO region: The oscillation in this region is com-
plicated as shown in Fig.1 and 3. It is noted that the GS
energy (3), persistent current (6), and the photon ener-
gies (9) all contain the factor Lo + 2Φ, thus their FABO
are completely in phase and have the same mechanism
caused by the transition of Lo against Φ. In Fig.1 the
abscissa Φ can be divided into segments, in each the
GS has a specific Lo and the GS energy is given by a
piece of a parabolic curve. The segment is called an
even (odd) segment if Lo is even (odd). At the border of
two neighboring segments the two GS energies are equal.
From the equality and based on (3), the right and left
boundaries of the segment with Lo can be obtained as
Φright(Lo) = (1−(µ/2G)
2)−1[ 1−µ(Ec−Ea)/G
2Lo+(−1)
Lo(2(Ec−Ea)+µ(Lo−1/2))/G]/4 (10)
Φleft(Lo) = (1 − (µ/2G)
2)−1[−1 − µ(Ec − Ea)/G
2Lo − (−1)
Lo(2(Ec − Ea) + µ(Lo + 1/2))/G]/4 (11)
where Lo ≤ 0 and Φright(Lo) = Φleft(Lo − 1), µ arises
from the HZeeman. The length of the segment reads
dLo = Φright(Lo) − Φleft(Lo) = (1 − (µ/2G)
2)−1[1 +
(−1)Lo(2(Ec − Ea) + µLo)/G]/2 (12)
which is related to the period of the FABO. When Φ
increases, the magnitude of Lo would increase. Since µLo
is negative, it is clear from eq.(12) that the length of even
(odd) segments would become shorter (longer) when Φ
increases.
The location of a segment with a given Lo can be
known from the inequality Φleft(Lo) ≤ Φ ≤ Φright(Lo).
Once the relation between Lo and the segments of Φ is
clear, every details of the FABO can be analytically and
exactly explained via the eq.(3), (6), and (9). In particu-
lar, the extrema in each segment can be known by giving
Φ = Φright or Φleft. For an example, the maximal cur-
rent is g(Lo+2Φright)/2π. Incidentally, the minimum of
the GS energy in a segment is Emin = Ec−µ
2/8G+µLo/2
(if So = 1), or just equal to Ea (if S0 = 0).
It is noted that Ec − Ea (cf. Table 1) and µ/G (it
is 0.0554 in our case) are both small. When Φ is small
the magnitude of |Lo| would be also small . In this case
eq.(12) leads to dLo ≈ 1/2, i.e., the period is a half of
the one of the normal ABO. In fact, (12) provides an
quantitative description of the variation of the period of
the FABO.
(vii) ABO region: When Φ becomes sufficiently large,
Lo will become very negative, the even segments will
disappear due to their lengths dLo ≤ 0 . We can de-
fine a critical odd integer Lcrit so that dLcrit−1 ≤ 0
while dLcrit+1 > 0, thereby the critical flux separating
the FABO and ABO region can be defined as
Φcrit = Φleft(Lcrit) (13)
Once Φ > Φcrit, Lo remains odd and the system keeps
polarized. Let IX be the largest even integer smaller
than −(G + 2(Ec − Ea))/µ. It turns out from eq.(12)
that Lcrit = IX + 1. With our parameters, Lcrit = −19
and accordingly Φcrit = 9.003 ( refer to Fig.1). Both
Lcrit and Φcrit depend on R very weakly, but sensitively
on the effective mass m∗.
In the ABO region (Φ > Φcrit), eqs.(10) to (12) do not
hold. Instead we have Φright = −(Lo − 1)/2, Φleft =
−(Lo + 1)/2, and dLo = 1. Thus the normal ABO re-
covers. Evaluated from (6), the magnitude of current is
from −g/2π to g/2π (for a comparison, it is from −g/4π
to g/4π for 1-e rings). From (9) the photon energies ~ω+
is from ∆c −G/2 to ∆c + 3G/2, at the same time ~ω−
is from ∆c + 3G/2 to ∆c −G/2.
(viii) Relations between the photon energies and other
physical quantities : Due to (7), the emitted (absorbed)
dipole photon has only two frequencies , therefore it is
meaningful to define ∆~ω = ~(ω+ − ω−). Directly from
(9) and (6), we have
∆~ω = hJo (14)
where h is the Planck’s constant and Jo is the persis-
tent current of the GS. To compare with 1-e rings, the
latter has ∆~ω = 2hJoĖq.(14) demonstrates that the os-
cillation of ∆~ω and the oscillation of Jo are matched
with each other exactly, they keep strictly proportional
to each other during the variation of Φ.
The maxima of ∆~ω measured in the ABO and FABO
regions, respectively, read
(∆~ω)
max = 2G (15)
(∆~ω)
max = 2G(Lo + 2Φright) (16)
Obviously, (15) provides a way to determine G, m∗can
be thereby obtained. (16) can be rewritten as
Ec−Ea = (G−µLo)/2−(2G−µ)/(4G)(∆~ω)
max (17)
This equation can be used to determine Ec −Ea. Fur-
thermore, we define
Γ~ω = ~(ω+ + ω−) = G+ 2∆α (18)
Once G has been known, (18) can be used to determine
Eb−Ea andEd−Ec. Since the spectrum can be generated
from the internal energies via (3), the evolutions of the
spectrum and the persistent current against Φ can be
understood simply by measuring the photon energies.
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
B (Tesla)
50 to 120nm
FIG. 4: Evolution of hJo(solid line) and ∆~ω (dotted line)
against B for a 2-e ring with ra = 50 and rb = 120nm
(ix) Effect of the width: We now consider a two-
dimensional model in which the two electrons are strictly
confined in an annular region by a potential U(r), which
is zero if ra < r < rb or is infinite otherwise. Under
this model we have performed numerical calculation to
obtain ∆~ω and hJo, where Jo is now the total angular
current inside the ring (from ra to rb). The result is
shown in Fig.4 where ra = 50 and rb = 120 are assumed,
and the two quantities are slightly different from each
other. However, when the width becomes smaller, say
rb − ra < 30, the two curves overlap. Thus (14) works
not only for one-dimensional but also for two-dimensional
narrow rings. Let us define r = ~/
m∗(∆~ω)ABmax.
For one-dimensional rings and from (15), we have r= R,
where R is the radius of the ring. For two-dimensional
rings, it was found from our numerical calculation that
r ≈ (rb + ra)/2 if rb − ra < 30. E.g., when rb = 100
and ra = 70, r =85.03. When rb = 100 and ra = 90, r
=95.00. Thus (15) works also well for two-dimensional
narrow rings if the R in G is replaced by the average
radius.
It is noted that the band-structure and related optical
properties of 2-e rings have already been studied in de-
tail by Wendler and coauthors18. They classify the eigen-
states according to their radial motion, relative angular
motion, and collective rotation. In our paper the relative
angular motion is further classified into four types ac-
cording to the inherent nodal structures and periodicity
of their wave functions, i.e., according to whether the DB
shape is allowed and whether the wave function is contin-
uous at ϕ = 2π. The DB-accessibility turns out to be im-
portant because it affects the eigenenergies decisively. In
fact, the classification of states based on inherent nodal
structures was found to be crucial in atomic physics,19
this would be also true in two-dimensional systems. Fur-
thermore, the rule of selection for the dipole transition
has been proposed in ref.[18]. In our paper, an addi-
tional rule (namely,eq.(7)) is further proposed based on
the possible transition of internal structures. This rule
would affect the dipole spectrum seriously because the
emission (absorption) is thereby concentrated into two
frequencies. The difference of these two frequencies turns
out to be proportional to the persistent current. There-
fore the measurement of this difference can be used to
determine the magnitude of the current.
In summary, we have studied the FABO both analyt-
ically and numerically. The analytical formalism pro-
vides not only a base for qualitative understanding, but
also provides a number of formulae for quantitative de-
scription. The domain of Φ is divided into segments,
each corresponds to a Lo. This division describes ex-
actly how Lo would transit against |Phi, which causes
directly the FABO. Thereby the variation of the period
and amplitude of the oscillation of the GS energy, persis-
tent current, and the frequencies of dipole transition in
the FABO region can be described exactly. A number of
equalities to relate the physical quantities and dynamical
parameters have been found. In particular, a new oscilla-
tion, namely, the oscillation of ∆~ω was found to match
exactly the oscillation of Jo. Since the photon energies
can be more accurately measured, other observables and
parameters can be thereby determined via the equali-
ties. Since the separability of the Hamiltonian and the
existence of inherent nodes are common, the above de-
scription can be more or less generalized to N−electron
rings, this deserves to be further studied.
Acknowledgment, This work is supported by the
NSFC of China under the grants 10574163 and 90306016.
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|
0704.0071 | Pairwise comparisons of typological profiles (of languages) | Microsoft Word - Pairwise comparisons of typological profiles6.doc
Pairwise comparisons of typological profiles
Søren Wichmann
Max Planck Institute for Evolutionary Anthropology & Leiden University
Eric W. Holman
University of California, Los Angeles
0. Introduction
Being rare or ‘exotic’ is a relative phenomenon. From a Samoan point of view Burushaski is an
extremely exotic language, but from the point of view of Telugu much less so. In this brief note we
want to look a how different and how similar languages turn out to be in pairwise comparisons and
the role that genealogical relatedness plays in this regard. We are interested in knowing whether
there is a cut-off point Shigh in the amount of similarities such that we can be sure that language pairs
that have more than Shigh similarities are all generally thought to be related and also whether there
is a cut-of point Slow at the other end of the scale such that all languages having less similarities than
Slow are thought to be unrelated. In other words, if a language is ‘normal’ relative to some other
language (as Burushaski is to Telugu), does this imply that the two languages are related according
to commonly accepted classifications? Or, if two languages are mutually very exotic (as Burushaski
and Samoan), does this imply that they are not thought to be related in commonly accepted
classifications?
The data we use, as well as the genealogical classification, are from the World Atlas of
Language Structures (Haspelmath et al., ed., henceforth WALS). The conclusions must of course be
seen in relation to this particular dataset. Thus, when we observe a certain amount of typological
similarity between two languages, this is strictly and only similarity in terms of the kinds of features
investigated in WALS. The dataset includes 134 nonredundant features, each of which has from two
to nine discrete values. All of these are quite generic typological features. Our conclusions are also
limited to the amount of data available. We have required that for any language pair in our sample
there should be 45 or more features attested for both members of the pair (a motivation for this
precise number follows shortly). This has limited our sample to 320 languages and 29,810 pairs of
languages compared. Among these pairs, there are 1,099 which are related according to the
We would like to thank Bernard Comrie, Cecil Brown, and Dietrich Stauffer for comments on this manuscript.
classification used in WALS. Henceforth we substitute ‘related’ for the more cumbersome ‘related
according to the WALS classification’. We follow this classification because it seeks to meet a
consensus view.
1. Results
Figure 1 presents the overall results of the investigation. As can be seen, the more similar languages
get, the greater the probability is that they are related. The figures on which the curve is based are
presented in Table 1. Percent similarity was defined as the percentage of attested features for which
both languages have the same value. We have binned language pairs in 5% intervals from 10% to
90% similarity. For the plot in Figure 1 the mean percent similarity in each interval was used. Table
1 gives some additional information: it also shows how many language pairs belong in each
interval. This is important for the interpretation of the results, as we shall see shortly.
Before giving our interpretation let us explain why we have chosen the criterion that
language pairs should have 45 or more features attested for both languages. It turns out that for a
criterion of 30 or more features the curve is rather similar but not quite as steep, showing less
dependence between the amount of similarity and the probability of finding related pairs. This
indicates that the fewer features one operates with, the more prominent is random sampling
variability in percent similarity. When operating with a criterion of 60 or more attested features the
curve becomes uneven, indicating that the higher criterion passes too few pairs for stable results.
This becomes even more pronounced when the criterion is 75 or more features. Obviously, with a
more extended database the number of features taken to be criterial could be raised, but 45 is a
number that suits the data available in WALS.
Figure 1. The probability of finding related languages
Probability of finding related languages
0 10 20 30 40 50 60 70 80 90 100
% similarity
Table 1. Data. (%SIM = % typological similarity between members of pairs; %REL = % of
language pairs that are related; PAIRS = number of language pairs in range)
%SIM %REL PAIRS
10.0-14.9 0 11
15.0-19.9 0 91
20.0-24.9 0 443
25.0-29.9 0.26 1566
30.0-34.9 0.33 3904
35.0-39.9 0.4 6019
40.0-44.9 1.2 6772
45.0-49.9 3.26 4873
50.0-54.9 6.68 3520
55.0-59.9 15.41 1551
60.0-64.9 23.72 666
65.0-69.9 38.24 238
70.0-74.9 54.26 94
75.0-79.9 61.54 39
80.0-84.9 85 20
85.0-89.9 100 2
90.0-94.9 100 1
It may be of interest to mention the language pairs that fall in the lower and upper
ranges of the percentage of shared values. Collectors of linguistic trivia may find it interesting that
the members of the most divergent language pair in the world (in our dataset), i.e. Tümpisa
Shoshone and Wari’, are found in the same general area, namely the Americas, that someone who is
tired of Romance linguistics should turn to Nivkh and someone fed up with Swedish should visit
the Koasatis when looking for something as radically different as it gets. Lists of the 20 most
divergent language pairs and the 20 most similar ones are provided in tables 2 and 3.
Table 2. The 20 most divergent language pairs in the sample
Language A Language B Number of
features
compared
% Similarity
Tümpisa
Shoshone
Wari' 48 10.4
Archi Tukang Besi 46 13
Maybrat Limbu 45 13.3
Italian Nivkh 51 13.7
Burushaski Samoan 49 14.3
Tzutujil Burmese 49 14.3
Ju|'hoan Yup'ik (Central) 56 14.3
Maybrat Tamil 55 14.5
Nubian
(Dongolese)
Acehnese 48 14.6
Swedish Koasati 47 14.9
Klamath Wari' 47 14.9
Kongo Ladakhi 46 15.2
Bashkir Maori 46 15.2
Berber (Middle
Atlas)
Waorani 45 15.6
Lango Archi 45 15.6
Archi Thai 45 15.6
Thai Retuarã 45 15.6
Ijo (Kolokuma) Kutenai 50 16
Kongo Evenki 56 16.1
Arabic (Egyptian) Tümpisa
Shoshone
48 16.7
Table 3. The 20 most similar language pairs in the sample
Language A Language B Relatedness Number of
features
compared
% Similarity
Lango Luo same genus 46 80.4
Luvale Zulu same genus 97 80.4
Khmer Vietnamese same family,
different genera
89 80.9
Vietnamese Thai different families 110 80.9
Khalkha Tuvan same family,
different genera
48 81.3
Lithuanian Russian same family,
different genera
64 81.3
Greek (Modern) Bulgarian same family,
different genera
64 81.3
Khmer Thai different families 91 81.3
Polish Russian same genus 71 81.7
Russian Serbian-Croatian same genus 45 82.2
Swahili Zulu same genus 107 82.2
Dagur Turkish same family,
different genera
46 82.6
Telugu Kannada same family,
different genera
47 83
Kongo Nkore-Kiga same genus 48 83.3
Dutch German same genus 56 83.9
Italian Spanish same genus 63 84.1
Drehu Iaai same genus 46 84.8
English Swedish same genus 60 85
French Italian same genus 64 85.9
Hindi Panjabi same genus 49 91.8
While Table 2 does not point in any specific direction and remains a curiosity, Table 3 provides
fragments of information which fits into the larger picture that emerges from our study. We note
that two pairs of unrelated languages, Vietnamese-Thai and Khmer-Thai, turn up in this list, which
otherwise consists of genealogically unrelated language pairs. Furthermore, the rest of the pairs
represent a mixture of languages related to different degrees (see Dryer 1992, 2005 for a definition
of ‘genera’).
Returning to Figure 1 and the associated data in Table 1 let us proceed to overall
interpretations. We set out asking whether there is some degree of similarity in typological profiles
beyond which it is certain that languages are related. The answer is positive, but nevertheless
discouraging. Members of language pairs in the sample that are 81.5% or more similar are all
related. But only 12 pairs of languages are that similar, in spite of the fact that there are 1099 pairs
of related languages in the sample! On the other hand, if there are less than 25% shared feature
values all language pairs will be unrelated, and this goes for 545 pairs in the sample. If one allows
for a very small margin of error (around 1%), it can predicted that less than 40% shared feature
values implies unrelatedness. That goes for 12,034 language pairs in the sample—close to half of
the total of 29,810. Thus, lack of similarity is a good predictor of unrelatedness, but presence of
similarity is a bad predictor of relatedness.
2. Are there ways of improving the results?
We next consider the question of whether the prediction of relatedness could be improved
somehow. In other studies (Holman et al. 2006a,b, Brown et al. 2006) we have made exact
quantitative explorations of the relationship between typological similarity and geographical
distance among languages. Not surprisingly, the greater the geographical proximity is between
languages, the more similar they tend to be (this goes for both related and unrelated languages). If
one takes into account the areal factor, this might move the cut-off point to allow more accurate
predictions of relatedness. Testing this strategy was unsuccessful. We were not able to obtain
markedly different results by adjusting the measure of similarity relative to geographical distance:
the correlation between adjusted and unadjusted measures was 0.96. The reason for this is probably
that the distance measure, as given in the WALS database, identifies the location of a given
language (roughly) with its center of extension. This means that some neighbouring languages, such
as German and Dutch, are treated as having a certain distance between them when in reality they
don’t have any. The more widespread the languages compared are, the bigger this problem gets.
Since it is impossible to provide adequate measure of geographical distances for 29,810 language
pairs, and not just take recourse to a mechanical measure of distance from one WALS dot to
another, it is not viable to improve on the cut-off point in such a way.
Also, the 134 features differ appreciably in the distribution of rarity and commonness
among their values. It is possible to imagine that taking into account the relative rarity of feature
values might improve the predictions. We again failed to obtain markedly different results by
adjusting the measure of similarity relative to differences among features: the correlation between
adjusted and unadjusted measures was 0.98. The probable reason is that differences among features
tend to average out in a sample of at least 45 attested features.
Another strategy to try to improve the power of prediction concerning relatedness
would be to weight different features or values of features according to their stability. We have
explored ways of measuring stability and have come out with a ranked order of stability for WALS
features (Wichmann et al. 2006). Conceivably, if the features shared among languages were
weighted for their stability the cut-off point could be pushed a bit. We expect, however, that the
results would be similar to the results for taking into account rarity, since stable and unstable
features would also average out.
A final strategy to improve the results would be to take into account the areality of
features. The linguistic typological literature abounds with statements concerning the susceptibility
to diffusion of certain features as opposed to others. In practice, however, it turns out to be virtually
impossible to define areas and measure areality in a consistent way. A major contribution of WALS
has been to show that most typological features are ‘areal’ to various extents. Browsing the maps
will make it clear to anyone that almost any feature can spread and that whatever features diffuse
are the features that happen to exist in an area. Thus, ‘areality’ is not amenable to quantification in
any straightforward way.
3. Deviant language pairs
The results reported on in Figure 1 and Table 1 show that there are a few pairs of languages which
are related even though showing less than 40% similarities, which is the point where pairs tend
overwhelmingly not to be related. It serves the record to provide a list of the pairs of related
languages that are deviant in the sense that they show less similarities than related languages
normally do. This list is provided in Table 4.
Table 4. Related languages that have unusually different typological profiles (less than 40%
similarities)
Language A Language B Language family Number of
features
compared
% Similarity
Luvale Ijo (Kolokuma) Niger-Congo 52 28.8
Zulu Ijo (Kolokuma) Niger-Congo 52 28.8
Maidu
(Northeast)
Tsimshian
(Coast)
Penutian 48 29.2
Ngiti Koyra Chiini Nilo-Saharan 47 29.8
Yoruba Ijo (Kolokuma) Niger-Congo 51 31.4
Mundari Semelai Austro-Asiatic 66 31.8
Swahili Ijo (Kolokuma) Niger-Congo 50 32
Maung Yidiny Australian 81 32.1
Mundari Khmer Austro-Asiatic 78 32.1
Koyraboro Senni Murle Nilo-Saharan 65 32.3
Koromfe Ijo (Kolokuma) Niger-Congo 49 32.7
Beja Margi Afro-Asiatic 45 33.3
Sango Ijo (Kolokuma) Niger-Congo 51 33.3
Nandi Koyraboro Senni Nilo-Saharan 47 34
Nandi Koyra Chiini Nilo-Saharan 52 34.6
Marathi Spanish Indo-European 52 34.6
Margi Amharic Afro-Asiatic 49 34.7
Mundari Vietnamese Austro-Asiatic 88 35.2
Garo Cantonese Sino-Tibetan 51 35.3
Berber (Middle
Atlas)
Kera Afro-Asiatic 65 35.4
Irish Marathi Indo-European 45 35.6
Paamese Acehnese Austronesian 45 35.6
Limbu Mandarin Sino-Tibetan 45 35.6
Mandarin Bawm Sino-Tibetan 76 36.8
Ijo (Kolokuma) Diola-Fogny Niger-Congo 46 37
Ngiti Nubian
(Dongolese)
Nilo-Saharan 54 37
Miwok (Southern
Sierra)
Tsimshian
(Coast)
Penutian 62 37.1
Mundari Khmu' Austro-Asiatic 70 37.1
Bagirmi Nubian
(Dongolese)
Nilo-Saharan 64 37.5
Beja Hausa Afro-Asiatic 82 37.8
Koromfe Kisi Niger-Congo 45 37.8
Yidiny Tiwi Australian 90 37.8
Limbu Meithei Sino-Tibetan 45 37.8
Kera Amharic Afro-Asiatic 50 38
Zulu Yoruba Niger-Congo 104 38.5
Beja Kera Afro-Asiatic 57 38.6
Ngiyambaa Maranungku Australian 74 39.2
Malagasy Acehnese Austronesian 56 39.3
Ngiti Nandi Nilo-Saharan 48 39.6
Lugbara Lango Nilo-Saharan 53 39.6
Fur Ngiti Nilo-Saharan 58 39.7
Experts in the different families involved will surely have good explanations for these deviant
cases. In some cases a pair may in reality not belong to the same family, as in the case of large and
not altogether uncontroversial families such as Australian and Nilo-Saharan. In other cases, such as
the two pairs featuring Marathi, a wide separation both temporally and geographically and
interaction with widely different types of languages may conspire to make a related pair stand out as
unusually different. In any case, measuring the amount of typological similarity provides a clue that
‘something is going on’—either the classification is potentially wrong or heavy language contact is
involved. So the method of comparing typological profiles is potentially useful for someone
wishing to probe into the behavior of different languages within a proposed family.
4. Conclusions
The results reported on in this note were, in part, unsurprising and, in part, unexpected. Figure 1
showed a close correlation between relatedness and typological similarity. This is what we had
expected. But we also expected to find some minimal amount of typological similarity among
language pairs which would suffice to predict that two languages are related. It turned out to be the
case, however, that the amount of similarity required to make this prediction is so high (81.5%) that
only few language pairs qualify. In practice, this means that typological features such as those of
WALS are not useful for identifying relatedness among languages when it comes to comparisons of
single pairs (when groups of languages are compared the situation may be different, but this issue is
beyond the scope of this paper). At the other end of the scale we found that typological dissimilarity
is a good predictor of unrelatedness: with only a small margin of error one can predict that
languages which have 60% or more differences are not related according to the WALS
classification. Our finding that a certain amount of typological differences can be used to predict
that languages are not commonly believed to be related means that typological differences are a
yardstick for gauging the limits of the traditional comparative method.
While it was was not surprising to find a correlation between relatedness and the
amount of typological differences among language pairs, this finding may nevertheless steer us in
new directions. Presumably there is a correlation between the amount of shared basic vocabulary
and relatedness as well. If so, the amount of shared basic vocabulary and the amount of typological
similarity should also be correlated, and it may even be possible to start considering whether there
is such a thing as a ‘typological clock’ such that the time of separation of languages of a given
family may be inferred from the amount of typological differences within the family. The fact that
unrelated languages may be as similar typologically as related ones indicates that for a ‘typological
clock’ to work reasonably well, several pairwise comparison should be made. How, in practice, this
kind of methodology could be developed would be an item for future research.
References
Brown, Cecil H., Eric W. Holman, Christian Schulze, Dietrich Stauffer, and Søren Wichmann.
2006. Are similarities among languages of the Americas due to diffusion or
inheritance? An exploration of the WALS evidence. Paper presented at the conference
“Genes and Languages”, University of California Santa Barbara, September 8-10,
2006.
Dryer, Matthew S. 1992. The Greenbergian word order correlations. Language 68:81-138.
Dryer, Matthew S. 2005. “Genealogical language list,” in The World Atlas of Language Structures,
edited by Martin Haspelmath, Matthew S. Dryer, David Gil, and Bernard Comrie, pp.
584-643. Oxford: Oxford University Press.
Haspelmath, Martin, Matthew S. Dryer, David Gil, and Bernard Comrie (eds.). 2005. The World
Atlas of Language Structures. Oxford: Oxford University Press.
Holman, Eric W., Dietrich Stauffer, Christian Schulze, and Søren Wichmann. 2006. On the relation
between structural diversity and geographical distance among languages: observations
and computer simulations. (Revised version under review for Linguistic Typology).
Holman, Eric W., Søren Wichmann, and Cecil H. Brown. 2006. Linguistic and cultural diffusion in
a comparative perspective. Submitted.
Wichmann, Søren, Eric W. Holman, & Hans-Jörg Bibiko. 2006. How computer simulations may
help linguists: recent progress and prospects for more. Paper presented at the
conference “Language and physics”, Warsaw, September 11-15, 2006.
|
0704.0072 | The decomposition method and Maple procedure for finding first integrals
of nonlinear PDEs of any order with any number of independent variables | The decomposition method and Maple procedure
for finding first integrals of nonlinear PDEs
of any order with any number of independent variables
Yu. N. Kosovtsov
Lviv Radio Engineering Research Institute, Ukraine
email: [email protected]
Abstract
In present paper we propose seemingly new method for finding solu-
tions of some types of nonlinear PDEs in closed form. The method is
based on decomposition of nonlinear operators on sequence of operators
of lower orders. It is shown that decomposition process can be done by
iterative procedure(s), each step of which is reduced to solution of some
auxiliary PDEs system(s) for one dependent variable. Moreover, we find
on this way the explicit expression of the first-order PDE(s) for first inte-
gral of decomposable initial PDE. Remarkably that this first-order PDE
is linear if initial PDE is linear in its highest derivatives.
The developed method is implemented in Maple procedure, which can
really solve many of different order PDEs with different number of in-
dependent variables. Examples of PDEs with calculated their general
solutions demonstrate a potential of the method for automatic solving of
nonlinear PDEs.
1 Introduction
Nonlinear partial differential equations (PDEs) play very important role in many
fields of mathematics, physics, chemistry, and biology, and numerous applica-
tions. If for nonlinear ordinary differential equations (ODEs) one can observe
incontestable progress in their automatic solving, the situation for nonlinear
PDEs seems as nearly hopeless one.
Despite the fact that various methods for solving nonlinear PDEs have been
developed in 19-20 centuries as the suitable groups of transformations, such as
point or contact transformations, differential substitutions, and Backlund trans-
formations etc., the most powerful method for explicit integration of second-
order nonlinear PDEs in two independent variables remains the method of Dar-
boux [1]-[4]. The original Darboux method (as already Darboux stated in [1])
is extendable in principle to equations of all orders in an arbitrary number of
independent variables, even to systems of equations; however, in [1]-[2] and sub-
sequent papers by many authors, the detailed calculations were performed only
http://arxiv.org/abs/0704.0072v1
for a single second-order equation with one dependent and two independent
variables.
The Darboux method was refined in recent years into more precise and effi-
cient (although not completely algorithmic) form [5]-[8] and references therein.
Nevertheless this approaches suffer from high complexity and necessitate to use
some tricks.
There were some partially successful attempts to extend modern variants of
the Darboux method based on Laplace cascade method on higher-order PDEs
and PDEs in the space of more than two independent variables [10]-[13] but
they suffer from high complexity too.
There is an original approach to the problem, based on the special type
of local change of variables which leads to the order reduction of initial PDE,
proposed in [14], which is suitable for high dimensions problems but of very
special class though.
In present paper we propose seemingly new method for finding solutions
of some types of nonlinear PDEs in closed form. The method is based on
decomposition of nonlinear operators on sequence of operators of lower orders. It
is shown that decomposition process can be done by iterative procedure(s), each
step of which is reduced to solution of some auxiliary PDEs system(s) for one
dependent variable. Moreover, we find on this way the explicit expression of the
first-order PDE(s) for first integral of decomposable initial PDE. Remarkably
that this first-order PDE is linear if initial PDE is linear in its highest derivatives.
The developed method is implemented in Maple procedure, which can really
solve many of different order PDEs with different number of independent vari-
ables. Examples of PDEs with calculated their general solutions demonstrate a
potential of the method for automatic solving of nonlinear PDEs.
2 Bases of the method
2.1 Decomposable PDEs
The simplest second-order non-linear PDE for w = w(t, x)
can be easily transformed to the following decomposed form
) = 0 , (2)
from which we can without difficulty obtain the general solution to PDE (1) in
two steps. First step gives us
d ln(G(x))
, (3)
where G(x) is an arbitrary function. And then, solving the equation (3) on the
second step, we obtain
w(t, x) = F (t)G(x) , (4)
where F (t) is one more arbitrary function.
The main observations on analyzing the grounds of solvability of the PDE
(1) by the above method are that
1. The PDE (1) is ”decomposable”, i.e., it can be represented as a composi-
tion of successive differential operators of type (5) (not necessarily linear). It is
clear that such type of decomposition can be done for some PDEs of any order
and with any number of independent variables in the following manner
D1(w) = u1 ,
D2(u1) = u2 ,
. . . . . . , (5)
Dn(un−1) = 0 ,
where ~x = (x1, . . . , xm), w = w(~x), ui = ui(~x) and
Di(u) = Vi(~x, u,
, . . . ,
Assuming that Vi are arbitrary functions, and eliminating ui by successive sub-
stitutions in system (5), we get a family of PDEs for w of nth order
Dn(Dn−1(. . . D1(w) . . . )) = 0 . (6)
which are ”decomposable” and in principle their solutions general or particular
can be obtained by integration of split system (5). The PDE (6) is nonlinear if
at least one of the operators Di is nonlinear. Not all PDEs admit such repre-
sentation. And in positive cases such representation is not unique in general.
Note that as a matter of fact Di need not be the first-order differential
operators. So the composition procedure for nth order PDE, when n > 2 can
be as follows
(w) = u ,
(u) = 0 , (7)
where n1, n2 are integers and n1 + n2 = n, w = w(~x), u = u(~x), and (k ≤ j)
i (u) = Vi(~x, u,
, . . . ,
. . . ∂x
m |k1+···+km=k≤j
, . . . ,
The late representation allows us to carry out the PDE‘s decomposition or
order reduction gradually bit by bit.
We have to stress here that in general representations (5) and (7) may have
different meaning. For example, some PDEs do not admit representation (5)
but permit the form (7) with both solvable DEs.
2. Each step of the solving process for decomposed PDE is faced with the
necessity to solve differential equation Di(ui−1) = ui (or D
i (ui−1) = ui), so all
such DEs must be solvable. Note that only first step Dn(un−1) = 0 is free from
arbitrary functions.
So one of the PDEs solving strategies may be as follows. First of all we
try to decompose given PDE. In order to do so we have to solve corresponding
auxiliary nonlinear PDE system for unknown functions Vi, it is sufficient to
find a particular solution here. And, if it is successful, then, deciding between
the variants, try to solve each arising DE from the chain (5). Main obstacle
here, beginning at the second step is just mentioned necessity to solve DEs with
arbitrary functions. There are sufficiently narrow circle of solvable (in sense of
the general solutions) DEs with an arbitrary function as a parameter.
Another (classification) approach can be based on the usage of only solvable
DEs. That is, we can form a composition of successive solvable differential
operators and as a result obtain a families of solvable PDEs. Such a way leads to
extensive nontrivial families for different types of nonlinear PDEs which general
solutions can be expressed in closed form. But on this way we encounter a
difficulty to circumscribe such families integrally and are forced to consider
particular subfamilies. Nevertheless it yields extensive field of PDEs for methods
testing [15].
2.2 Decomposition algorithm for decomposable PDEs
For nth order PDE, when n > 2 there are some slightly different approaches
which are dictated by goals of the problem. If the goal is to decompose given
nonlinear operator then we have to use the scheme (7) with n1 = 1, n2 = n− 1.
And conversely we have to use the scheme (7) with n1 = n − 1, n2 = 1 if the
goal is to solve given PDE. The last procedure in some features resembles the
well-known technics of reducing ODEs order, e.g., by first integral method. Of
course, it is possible to use intermediate cases.
All above cases can be treated by the same way as we consider below but
each of them leads to auxiliary PDEs systems of different order, viz n2+1, with
corresponding calculation complexity.
In sequel we will consider for shortness only the case with n1 = n−1, n2 = 1,
as more practical for PDEs solving.
Let us consider the decomposition of type (7) with Dn−1
(w) as a solution
of the following equation with respect of u = u(~x)
J(u, ~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n−1
, . . . ,
∂n−1w
) = 0 (8)
D2(u) = V (~x, u,
, . . . ,
) . (9)
If substitute u = Dn−1
(w) into (9) we obtain decomposable n-th order PDE
V (~x, U0, Ux1 , . . . , Uxm) = 0 , (10)
where (we use below the following notation w = W0 and
...∂x
= Wk1,...,km)
(w) = U0 , (11)
∂Wk1,...,km
,...,k∗
= Uxi (i = 1, . . . ,m) , (12)
where k∗j = kj + 1 if j = i and k
j = kj otherwise, and it is supposed that
differentiations in sum are carried out on all indexed W ‘s which are involved in
Here we can introduce U0 and Ux1 , . . . , Uxm as new independent variables
if express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n using
linear system (12).
Assuming that given PDE of order n
F (~x, w,
, . . . ,
. . . ∂x
m |k1+···+km=k≤n
, . . . ,
) = 0 (13)
is decomposable, we receive, that after substitution of the new variables, left-
hand side of given PDE must turn into (10) with some V .
Left-hand side of given PDE expressed in new variables is the first-order
differential expression with respect to
J(U0, ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and must not depend on all indexed W ‘s, that is derivatives of F expressed in
new variables with respect to all indexed W ‘s are equal to zero. Sequence of
such derivatives of F equated to zero form a second-order PDE system for J .
So a solution (particular as well) the PDE system gives possible expression of
differential operator Dn−1
(w) through (8) and differential operator D2(u) by
substituting the solution of J into left-hand side of given PDE expressed in new
variables.
Of course, there are problems where a operator decomposition is required
only. But in most cases obtained decomposition is intended for finding solutions
for given PDE. If in obtained decomposition the corresponding PDE D2(u) = 0
is solvable, then substitution of obtained u into J expressed in original variables
gives us a first integral (see its definition in the next subsection) of given PDE.
It is easy to see that for decomposable PDEs the first integral is a differential
equation, so we can try to solve it or to find a first integral for this new DE (or
decompose it) by the scheme described above until we come to the first-order
Remarkably that in the approach under consideration the finding of first
integrals can be done more directly and effectively.
2.3 Differential equation for first integral of decomposable
The first integral I of the PDE is an expression, involving one arbitrary func-
tion, which is equivalent in some sense to the given PDE. The first integral
vanishes on the set of solutions of given PDE. And (in accordance with [4]) all
differential consequences of the equation I = 0 coincide with respective differen-
tial consequences of given PDE (e.g., elimination of the arbitrary function leads
to the given PDE).
Our goal here is to find PDE for first integral of a decomposable PDE. To
do so we first of all have to take into account that u(~x) is the solution of the
corresponding PDE
V (~x, u,
, . . . ,
) = 0 ,
so u(~x) depends only on ~x but in no way on indexed W ‘s. Secondly, the depen-
dent variable in this case, namely
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
of given PDE (13) expressed in new variables do not to depend on Ux1 , . . . , Uxm
and is a first integral of given PDE.
If now consider u(~x) as an unknown function, we can denote the first integral
I(~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1) =
J(u(~x), ~x,W0,W1,0,...,0, . . . ,Wk1,...,km |k1+···+km=k≤n−1, . . . ,W0,0,...,n−1)
and instead of (12) in the form
∂Wk1,...,km
,...,k∗
= −Uxi
(i = 1, . . . ,m)
we arrive to the following system
∂Wk1,...,km
,...,k∗
= 0 (i = 1, . . . ,m) . (14)
If express m variables from the set {Wk1,...,km} with k1 + · · ·+ km = n (at
least one of which is actual for given PDE - note that there are some variants
here as a rule, so we can obtain some consistent PDEs on this stage) using linear
system (14) and substitute them into given PDE (13) we receive a first-order
(even linear if PDE (13) is linear in its highest derivatives) PDE with respect to
first integral I. And it remains only to solve this PDE(s) to find a first integral
of given PDE.
Note, given PDE is decomposable iff exists a solution of such first-order
PDE(s).
3 Examples
To facilitate necessary calculations in the process of finding first integrals I
have implemented above described method in prototype of Maple procedure
reduce PDE order (see Appendix). The input data of the procedure are given
PDE of any order and dependent variable of the PDE with any number of
independent variables. The procedure tries to find first integral(s) of the input
linear or nonlinear PDE.
The Maple built-in procedure pdsolve is used inside my procedure to solve
the first-order PDE for first integral. As different Maple versions have different
PDE solving abilities so the output depends on Maple version. In the following
examples I refer to Maple 11.
The procedure reduce PDE order is able to find first integrals for many
known and unknown linear and nonlinear PDEs. Here we give examples of
PDEs for which it is possible to find finally their general solutions. More exam-
ples one can find in collection of solvable nonlinear PDEs [15].
3.1 Second-order PDE with two independent variables
For PDE (w = w(t, x))
−kw− bc
= 0 (15)
with a 6= 0 and 4ak − b2 6= 0 the procedure reduce PDE order outputs the
following first integral
I = F1
4ak − b2 − 2 arctan
c+ 2a∂w
4ak − b2
4ak − b2
with arbitrary function F1.
The ODE I = 0 can be solved and one obtains (after some hand simplifica-
tions and edition) the following general solution to (15)
w(t, x) =
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
G(t)} exp
exp(t
b2 − 4ak)F (x)(b +
b2 − 4ak)−
b2 − 4ak + b
1 + exp(t
b2 − 4ak)F (x)
where F (x) and G(t) are arbitrary functions.
3.2 Second-order PDE with four independent variables
For PDE
∂x1∂x4
∂x2∂x4
∂x3∂x4
+ C0 +B1
C1(A1
+B1w +B0)+
C2(A1
+B1w +B0)
2 = 0 , (16)
where w = w(x1, x2, x3, x4) and Ai, Bi, Ci are constants, the procedure re-
duce PDE order outputs the following first integral
I = F1
x1, x2, x3, x4 +
2 arctan
2C2(A1
+B1w +B0) + C1
4C0C2 − C21
4C0C2 − C21
with arbitrary function F1.
The PDE I = 0 can be solved and one obtains the following general solution
to (16)
w(x1, x2, x3, x4) =
2A1C2
exp(−B1x1
)(2B0C2 + C1 + tan[
4C0C2 − C21
+G(ξ, (A2ξ +A1x2 −A2x1), (A3ξ +A1x3 −A3x1))]
4C0C2 − C21 )dξ
+ exp(−B1x1
)F [(A1x2 − A2x1), (A1x3 −A3x1), x4] ,
where F (t1, t2, t3) and G(t1, t2, t3) are arbitrary functions, c is arbitrary con-
stant.
3.3 Third order PDE with two independent variables
For PDE (w = w(t, x))
∂t∂x2
− 2w ∂
− w∂w
− aw3 = 0 (17)
the procedure reduce PDE order outputs the following first integrals
I1 = F1
− axw2), 1
ax2w2 + 2w(
− x ∂
) + 2x
I2 = F1
− atw2 −
with arbitrary function F1.
We can form some PDEs from I1 and to solve them we can repeat the process
of order reduction with the procedure reduce PDE order. The ODE I2 = 0 can
be solved directly and one obtains in any way the following general solution to
w(t, x) = F (t) exp
− xH(t) + x
G(x)dx −
xG(x)dx
where F (t), H(t) and G(x) are arbitrary functions.
3.4 Fourth order PDE with two independent variables
For PDE (w = w(t, x))
∂t2∂x2
− 2w2
∂t2∂x
∂t∂x2
− 2∂w
= 0 (18)
the procedure reduce PDE order outputs the following first integrals
I1 = F1(t,
∂t2∂x
− 2w∂w
− x ∂
∂t2∂x
w − 2x∂w
I2 = F1(x,
∂t∂x2
w + 2
− t ∂
∂t∂x2
w − 2t∂w
with arbitrary function F1.
The wealth of first integrals here allows us to operate with them in many
different ways. Apart from aforesaid subsequent order reduction we can, for
example, from
∂t∂x2
− 2w∂w
w + 2
= F (x)
∂t∂x2
] = G(x) ,
where F (x) and G(x) are arbitrary functions, algebraically eliminate mixed
derivative and obtain the following ODE
+ [tF (x)−G(x)]w2 = 0 ,
which gives the general solution to (18)
w(t, x) = H(t) exp
xF (x) dx − tx
F (x) dx+
G(x) dx −
xG(x) dx + xK(t)
where F (x), H(t), G(x) and K(t) are arbitrary functions.
4 Conclusion
The method have considered above is efficient enough for solving decomposable
PDEs of relatively high order with many independent variables. The main
limitation here is concerned with abilities to solve corresponding auxiliary first-
order PDEs for first integrals.
An adaptability of the method to PDEs which are not decomposable but
which general solutions can be expressed in closed form remains unsolved yet.
But it can be shown on examples that there are some ways to extend the method
for some types of such PDEs. These approaches deserve further thorough study
in another publication.
References
[1] G. Darboux, Sur les equations aux derivees partieles du second ordre. Ann.
Sci. Ecole Norm. Sup. 1870, v. 7, pp. 163-173.
[2] G. Darboux, Lecons sur la theorie generale des surfaces. v.II. Paris: Her-
mann, 1915.
[3] E. Goursat, Lecons sur l’integration des equations aux derivees partieles
du second ordre a deux variables independantes. V.I,II. Paris: Hermann,
1896, 1898.
[4] A.R. Forsyth, Theory of differential equations. Part 4. Partial differential
equations, vol. 6, Dover Press, New York, 1959.
[5] M. Juras, Generalized Laplace invariants and classical integration meth-
ods for second-order scalar hyperbolic partial differential equations in the
plane, Differential Geometry and Applications: Proc., Conf. Brno (Czech
Republic), 28 Aug.-1 Sept. 1995, Brno: Masaric Univ., 1966, pp. 275-284.
[6] M. Juras, Geometric aspects of second-order scalar hyperbolic partial dif-
ferential equations in the plane, Ph.D. thesis, 1997, Utah State University,
[7] V.V. Sokolov, A.V. Zhiber, On the Darboux integrable hyperbolic equa-
tions. Phys. Lett. A, v. 208, pp. 303-308, 1995.
[8] A.V. Zhiber, V.V. Sokolov, Exact integrable Liouville type hyperbolic equa-
tions [in Russian], Uspekhi Mat. Nauk, Vol. 56, No. 1, pp. 64-104, 2001.
[9] S.P.Tsarev, On Darboux integrable nonlinear partial differential equations,
Proc. Steklov Institute of Mathematics, v. 225, p. 372-381, 1999.
[10] J. Le Roux. Extensions de la methode de Laplace aux equations lin-
eaires aux derivees partielles dordre superieur au second. Bull. Soc. Math.
de France, v. 27, p. 237262, 1899. A digitized copy is obtainable from
http://www.numdam.org/
[11] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine
con un numero qualunque di variabili. Atti Acc. Naz. dei Lincei. Mem.
Classe fis., mat., nat. (ser. 5) v. 4, 1901, p. 121178. Also Opere v. III, p.
489566.
[12] U. Dini, Sopra una classe di equazioni a derivate parziali di secondordine.
Atti Acc. Naz. dei Lincei. Mem. Classe fis., mat., nat. (ser. 5) v. 4, 1902,
p. 431467. Also Opere v. III, p. 613660.
[13] S.P. Tsarev, On factorization and solution of multidimensional linear par-
tial differential equations. http://arxiv.org/abs/cs.SC/0609075, 2006.
[14] V.M. Boyko, W.I. Fushchych, Lowering of order and general solutions of
some classes of partial differential equations, Reports on Math. Phys., V.
41, No. 3, pp. 311-318, 1998.
[15] Yu.N. Kosovtsov, The general solutions of some nonlinear second
and third order PDEs with constant and nonconstant parameters.
http://arxiv.org/abs/math-ph/0609003 , 2006.
http://www.numdam.org/
http://arxiv.org/abs/cs.SC/0609075
http://arxiv.org/abs/math-ph/0609003
5 Appendix.
Maple procedure reduce PDE order
reduce PDE order:=proc(pde,unk)
local B,W,N,NN,ARG,acargs,i,M,pde0,DN,IND,IND2,IND3,IND4,ARGS,SUB,SUB0,
Z0,Bargs,EQS,XXX,WW,BB,PP,pdeI,IV,s,AN;
option ‘Copyright (c) 2006-2007 by Yuri N. Kosovtsov. All rights reserved.‘;
N:=PDETools[difforder](op(1,[selectremove(has,indets(pde,function),unk)]));
NN:=op(1,[selectremove(has,op(1,[selectremove(has,indets(pde,function),unk)]),diff)]);
ARG:=[op(unk)];
acargs:={};
for i from 1 to nops(ARG) do
if PDETools[difforder](NN,op(i,ARG))=0 then else acargs:=acargs union {op(i,ARG)}
fi; od;
acargs:=convert(acargs,list);
M:=op(0,unk)(op(acargs));
if type(pde,equation)=true then
pde0:=lhs(subs(unk=M,pde))-rhs(subs(unk=M,pde)) else pde0:=subs(unk=M,pde)
DN:=[seq(seq(i,i=1..nops(acargs)),j=1..N)];
IND:=seq(op(combinat[choose](DN,i)),i=1..N);
IND2:=seq(op(combinat[choose](DN,i)),i=1..N-2);
IND3:=op(combinat[choose](DN,N-1));
IND4:=op(combinat[choose](DN,N));
ARGS:=op(unk),M,seq(convert(D[op(op(i,[IND2]))](op(0,unk))
(op(acargs)),diff),i=1..nops([IND2]));
SUB:={M=W[0],seq(convert(D[op(op(i,[IND]))](op(0,unk))
(op(acargs)),diff)=W[op(op(i,[IND]))],i=1..nops([IND]))};
SUB0:={W[0]=op(0,unk)(op(ARG)),
seq(W[op(op(i,[IND]))]=subs(M=op(0,unk)(op(ARG)),
convert(D[op(op(i,[IND]))](op(0,unk))(op(acargs)),diff)),i=1..nops([IND]))};
Z0:=B(ARGS,seq(convert(D[op(op(i,[IND3]))](op(0,unk))(op(acargs)),diff),
i=1..nops([IND3])));
Bargs:=op(indets(subs(SUB,Z0),name));
EQS:=convert(subs(SUB,{seq(diff(Z0,op(i,acargs))=0,i=1..nops(acargs))}),diff);
XXX:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
WW:=select(type,indets(subs(SUB,pde0)), ’name’) intersect
{seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
BB:=select(has,combinat[choose](XXX, nops(acargs)),WW);
PP:={};
pdeI:={seq({subs(subs(solve(EQS,op(i,BB)),subs(SUB,pde0)))},i=1..nops(BB))};
IV:={seq(W[op(op(i,[IND4]))],i=1..nops([IND4]))};
for s from 1 to nops(pdeI) do
AN:=pdsolve(op(s,pdeI),{B},ivars=IV);
for i from 1 to nops(AN) do
if op(0,lhs(op(i,AN)))=B then
PP:=PP union {rhs(op(i,AN))}
catch:
end try;
PP:=subs(SUB0,PP);
RETURN(PP);
end proc:
Calling Sequence: reduce PDE order(PDE, f(~x));
PDE - partial differential equation;
f(~x) - indeterminate function with its arguments.
Introduction
Bases of the method
Decomposable PDEs
Decomposition algorithm for decomposable PDEs
Differential equation for first integral of decomposable PDEs
Examples
Second-order PDE with two independent variables
Second-order PDE with four independent variables
Third order PDE with two independent variables
Fourth order PDE with two independent variables
Conclusion
Appendix. Maple procedure reduce_PDE_order
|
0704.0074 | Injective Morita contexts (revisited) | Injective Morita Contexts (Revisited)
Dedicated to Prof. Robert Wisbauer
J. Y. Abuhlail ∗ S. K. Nauman
Department of Mathematics & Statistics Department of Mathematics
King Fahd University of Petroleum King AbdulAziz University
& Minerals, Box # 5046 P.O.Box 80203
31261 Dhahran (KSA) 21589 Jeddah (KSA)
[email protected] [email protected]
Abstract
This paper is an exposition of the so-called injective Morita contexts (in which
the connecting bimodule morphisms are injective) and Morita α-contexts (in which
the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-
Huisgen). Motivated by situations in which only one trace ideal is in action, or
the compatibility between the bimodule morphisms is not needed, we introduce the
notions of Morita semi-contexts and Morita data, and investigate them. Injective
Morita data will be used (with the help of static and adstatic modules) to establish
equivalences between some intersecting subcategories related to subcategories of cat-
egories of modules that are localized or colocalized by trace ideals of a Morita datum.
We end up with applications of Morita α-contexts to ∗-modules and injective right
wide Morita contexts.
1 Introduction
Morita contexts, in general, and (semi-)strict Morita contexts (with surjective con-
necting bilinear morphisms), in particular, were extensively studied and developed expo-
nentially during the last few decades (e.g. [AGH-Z1997]). However, we sincerely feel that
there is a gap in the literature on injective Morita contexts (i.e. those with injective con-
necting bilinear morphisms). Apart from the results in [Nau1994-a], [Nau1994-b] (where
the second author initially explored this notion) and from an application to Grothendieck
groups in the recent paper ([Nau2004]), it seems that injective Morita contexts were not
studied systematically at all.
∗Corresponding Author
http://arxiv.org/abs/0704.0074v2
We noticed that in several results of ([Nau1993], [Nau1994-a] and [Nau1994-b]) that are
related to Morita contexts, only one trace ideal is used. Observing this fact, we introduce
the notions of Morita semi-contexts and Morita data and investigate them. Several results
are proved then for injective Morita semi contexts and/or injective Morita data.
Consider a Morita datum M = (T, S, P,Q,<,>T , <,>S), with not necessarily compat-
ible bimodule morphisms <,>T : P ⊗S Q→ T and <,>S: Q⊗T P → S. We say that M is
injective, iff <,>T and <,>S are injective, and to be a Morita α-datum, iff the associated
dual pairings Pl := (Q, TP ), Pr := (Q,PS), Ql := (P, SQ) and Qr := (P,QT ) satisfy
the α-condition (which is closely related to the notion of local projectivity in the sense of
Zimmermann-Huisgen [Z-H1976]). The α-condition was introduced in [AG-TL2001] and
further investigated by the first author in [Abu2005].
While (semi-)strict unital Morita contexts induce equivalences between the whole mod-
ule categories of the rings under consideration, we show in this paper how injective Morita
(semi-)contexts and injective Morita data play an important role in establishing equiva-
lences between suitable intersecting subcategories of module categories (e.g. intersections
of subcategories that are localized/colocalized by trace ideals of a Morita datum with sub-
categories of static/adstatic modules, etc.). Our main applications in addition to equiv-
alences related to the Kato-Ohtake-Müller localization-colocalization theory (developed in
[Kat1978], [KO1979] and [Mül1974]), will be to ∗-modules (introduced by Menini and Or-
satti [MO1989]) and to right wide Morita contexts (introduced by F. Castaño Iglesias and
J. Gómez-Torrecillas [C-IG-T1995]).
Most of our results will be stated for left modules, while deriving the “dual” versions for
right modules is left to the interested reader. Moreover, for Morita contexts, some results
are stated/proved for only one of the Morita semi-contexts, as the ones corresponding to
the second semi-context can be obtained analogously. For the convenience of the reader, we
tried to make the paper self-contained, so that it can serve as a reference on injective Morita
(semi-)contexts and their applications. In this respect, and for the sake of completeness, we
have included some previous results of the authors that are (in most cases) either provided
with new shorter proofs, or are obtained under weaker conditions.
This paper is organized as follows: After this brief introduction, we give in Section 2
some preliminaries including the basic properties of dual α-pairings, which play a central
role in rest of the work. The notions of Morita semi-contexts and Morita data are intro-
duced in Section 3, where we clarify their relations with the dual pairings and the so-called
elementary rngs. Injective Morita (semi-)contexts appear in Section 4, where we study
their interplay with dual α-pairings and provide some examples and a counter-example.
In Section 5 we include some observations regarding static and adstatic modules and use
them to obtain equivalences among suitable intersecting subcategories of modules related
to a Morita (semi-)context. In the last section, more applications are presented, mainly to
subcategories of modules that are localized or colocalized by a trace ideal of an injective
Morita (semi-)context, to ∗-modules and to injective right wide Morita contexts.
2 Preliminaries
Throughout, R denotes a commutative ring with 1R 6= 0R and A,A
′, B, B′ are unital
R-algebras. We have reserved the term “ring” for an associative ring with a multiplicative
unity, and we will use the term “rng” for a general associative ring (not necessarily with
unity). All modules over rings are assumed to be unitary, and ring morphisms are assumed
to respect multiplicative unities. If T and S are categories, then we write T ≤ S (T ≤ S)
to mean that T is a (full) subcategory of S, and T ≈ S to indicate that T and S are
equivalent.
Rngs and their modules
2.1. By an A-rng (T, µT ), we mean an (A,A)-bimodule T with an (A,A)-bilinear mor-
phism µT : T ⊗A T → T, such that µT ◦ (µT ⊗A idT ) = µT ◦ (idT ⊗A µT ). We call an A-rng
(T, µT ) an A-ring, iff there exists in addition an (A,A)-bilinear morphism ηT : A → T,
called the unity map, such that µT ◦ (ηT ⊗A idT ) = ϑ
T and µT ◦ (idT ⊗A ηT ) = ϑ
T (where
A⊗A T
≃ T and T ⊗A A
≃ T are the canonical isomorphisms). So, an A-ring is a unital
A-rng; and an A-rng is (roughly speaking) an A-ring not necessarily with unity.
2.2. A morphism of rngs (ψ : δ) : (T : A) → (T ′ : A′) consists of a morphism of R-algebras
δ : A→ A′ and an (A,A)-bilinear morphism ψ : T → T ′, such that µT ′◦χ
(A,A′)
(T ′,T ′)
◦(ψ⊗Aψ) =
ψ ◦µT (where χ
(A,A′)
(T ′,T ′)
: T ′ ⊗A T
′ → T ′ ⊗A′ T
′ is the canonical map induced by δ). By RNG
we denote the category of associative rngs with morphisms being rng morphisms, and
by URNG < RNG the (non-full) subcategory of unital rings with morphisms being the
morphisms in RNG which respect multiplicative unities.
2.3. Let (T, µT ) be an A-rng. By a left T -module we mean a left A-module N with a left
A-linear morphism φNT : T ⊗AN → N, such that φ
T ◦ (µT ⊗A idN) = φ
T ◦ (idT ⊗A φ
T ). For
left T -modulesM,N, we call a left A-linear morphism f :M → N a T -linear morphism,
iff f(tm) = tf(m) for all t ∈ T. The category of left T -modules and left T -linear morphisms
is denoted by TM. The category MT of right T -modules is defined analogously. Let (T : A)
and (T ′ : A′) be rngs. We call an (A,A′)-bimodule N a (T, T ′)-bimodule, iff (N, φNT ) is
a left T -module and (N, φNT ′) is a right T
′-module, such that φNT ′ ◦ (φ
T ⊗A′ idT ′) = φ
(idT ⊗Aφ
T ′). For (T, T
′)-bimodulesM,N, we call an (A,A′)-bilinear morphism f :M → N
(T, T ′)-bilinear, provided f is left T -linear and right T ′-linear. The category of (T, T ′)-
bimodules is denoted by TMT ′ . In particular, for any A-rng T, a left (right) T -module
M has a canonical structure of a unitary right (left) S-module, where S := End(TM)
(S := End(MT )); and moreover, with this structure M becomes a (T, S)-bimodule (an
(S, T )-bimodule).
Remark 2.4. Similarly, one can define rngs over arbitrary (not-necessarily unital) ground
rngs and rng morphisms between them. Moreover, one can define (bi)modules over such
rngs and (bi)linear morphisms between them.
Notation. Let T be an A-rng. We write TU (UT ) to denote that U is a left (right) T -
module. For a left (right) T -module TU, we consider the set
∗U := HomT−(U, T ) (U
Hom−T (U, T )) of all left (right) T -linear morphisms from U to T with the canonical right
(left) T -module structure.
Generators and cogenerators
Definition 2.5. Let T be an A-rng. For a left T -module TU consider the following sub-
classes of TM :
Gen(TU) := {TV | ∃ a set Λ and an exact sequence U
(Λ) → V → 0};
Cogen(TU) := {TW | ∃ a set Λ and an exact sequence 0 → W → U
Pres(TU) := {TV | ∃ sets Λ1,Λ2 and an exact sequence U
(Λ2) → U (Λ1) → V → 0};
Copres(TU) := {TW | ∃ sets Λ1,Λ2 and an exact sequence 0 →W → U
Λ1 → UΛ2};
A left T -module in Gen(TU) (respectively Cogen(TU), Pres(TU), Copres(TU)) is said to be
U-generated (respectively U-cogenerated, U-presented, U-copresented). Moreover,
we say that TU is a generator (respectively cogenerator, presentor, copresentor), iff
Gen(TU) = TM (respectively Cogen(TU) = TM, Pres(TU) = TM, Copres(TU) = TM).
Dual α-pairings
In what follows we recall the definition and properties of dual α-pairings introduced
in [AG-TL2001, Definition 2.3.] and studied further in [Abu2005].
2.6. Let T be an A-rng. A dual left T -pairing Pl = (V, TW ) consists of a left T -module
W and a right T -module V with a right T -linear morphism κPl : V →
∗W (equivalently
a left T -linear morphism χPl : W → V
∗). For dual left pairings Pl = (V, TW ), P
l = (V
′), a morphism of dual left pairings (ξ, θ) : (V ′,W ′) → (V,W ) consists of a triple
(ξ, θ : ς) : (V, TW ) → (V
′, T ′W
where ξ : V → V ′ and θ : W ′ → W are T -linear and ς : T → T ′ is a morphism of rngs,
such that considering the induced maps <,>T : V ×W → T and <,>T ′: V
′ ×W ′ → T ′ we
< ξ(v), w′ >T ′= ς(< v, θ(w
′) >T ) for all v ∈ V and w
′ ∈ W ′. (1)
The dual left pairings with the morphisms defined above build a category, which we denote
by Pl. With Pl(T ) ≤ Pl we denote the full subcategory of dual T -pairings. The category
Pr of dual right pairings and its full subcategory Pr(T ) ≤ Pr of dual right T -pairings are
defined analogously.
Remark 2.7. The reader should be warned that (in general) for a non-commutative rng T
and a dual left T -pairing Pl = (V, TW ), the following map induced by the right T -linear
morphism κPl : V →
<,>T : V ×W → T, < v, w >T := κPl(v)(w)
is not necessarily T -balanced, and so does not induce (in general) a map V ⊗T W → T. In
fact, for all v ∈ V, w ∈ W and t ∈ T we have
< vt, w > = κPl(vt)(w) = [κPl(v)t](w) = [κPl(v)(w)]t = < v,w >T t;
< v, tw > = κPl(v)(tw) = t[κPl(v)(w)] = t < v, w >T .
2.8. Let T be an A-rng, N,W be left T -modules and identify NW with the set of all
mappings fromW toN. Considering N with the discrete topology andNW with the product
topology, the induced relative topology on HomT−(W,N) →֒ N
W is a linear topology (called
the finite topology), for which the basis of neighborhoods of 0 is given by the set of
annihilator submodules:
Bf (0) := {F
⊥(HomT−(W,N)) | F = {w1, ..., wk} ⊂W is a finite subset},
where
F⊥(HomT−(W,N)) := {f ∈ HomT−(W,N)) | f(W ) = 0}.
2.9. Let T be an A-rng, Pl = (V, TW ) a dual left T -pairing and consider for every right
T -module UT the following canonical map
U : U ⊗T W → Hom−T (V, U),
ui ⊗T wi 7→ [v 7→
ui < v,wi >T ]. (2)
We say that Pl = (V, TW ) ∈ Pl(T ) satisfies the left α-condition (or is a dual left α-
pairing), iff α
U is injective for every right T -module UT . By P
l (T ) ≤ Pl(T ) we denote the
full subcategory of dual left T -pairings satisfying the left α-condition. The full subcategory
of dual right α-pairings Pαr (T ) ≤ Pr(T ) is defined analogously.
Definition 2.10. Let T be an A-rng, Pl = (V, TW ) be a dual left T -pairing and consider
κPl : V →
∗W and α
V : V ⊗T W → End(VT ).
We say Pl ∈ Pl(T ) is
dense, iff κPl(V ) ⊆
∗W is dense (w.r.t. the finite topology on ∗W →֒ TW );
injective (resp. semi-strict, strict), iff α
V is injective (resp. surjective, bijective);
non-degenerate, iff V
→֒ ∗W and W
→֒ V ∗ canonically.
2.11. Let T be an A-rng. We call a T -module W locally projective (in the sense of B.
Zimmermann-Huisgen [Z-H1976]), iff for every diagram of T -modules
0 // F
g′◦ι
ι //W
// N // 0
with exact rows and finitely generated T -submodule F ⊆W : for every T -linear morphism
g : W → N, there exists a T -linear morphism g′ : W → L, such that g ◦ ι = π ◦ g′ ◦ ι.
For proofs of the following basic properties of locally projective modules and dual
α-pairings see [Abu2005] and [Z-H1976]:
Proposition 2.12. Let T be an A-ring and Pl = (V, TW ) ∈ Pl(T ).
1. The left T -module TW is locally projective if and only if (
∗W,W ) is an α-pairing.
2. The left T -module TW is locally projective, iff for any finite subset {w1, ..., wk} ⊆ W,
there exists {(fi, w̃i)}
i=1 ⊂
∗W ×W such that wj =
fi(wj)w̃i for all j = 1, ..., k.
3. If TW is locally projective, then TW is flat and T -cogenerated.
4. If Pl ∈ P
l (T ), then TW is locally projective.
5. If TW is locally projective and κP (V ) ⊆
∗W is dense, then Pl ∈ P
l (T ).
6. Assume TT is an injective cogenerator. Then Pl ∈ P
l (T ) if and only if TW is locally
projective and κPl(V ) ⊆
∗W is dense.
7. If T is a QF ring, then Pl ∈ P
l (T ) if and only if TW is projective and W
→֒ V ∗.
The following result completes the nice observation [BW2003, 42.13.] about locally
projective modules:
Proposition 2.13. Let T be a ring, TW a left T -module, S := End(TW )
op and consider
the canonical (S, S)-bilinear morphism
[, ]W :
∗W ⊗T W → End(TW ), f ⊗T w 7→ [w̃ 7→ f(w̃)w].
1. TW is finitely generated projective if and only if [, ]W is surjective.
2. TW is locally projective if and only if Im([, ]W ) ⊆ End(TW ) is dense.
Proof. 1. This follows by [Fai1981, 12.8.].
2. Assume TW is locally projective and consider for every left T -module N the canonical
mapping
[, ]WN :
∗ W ⊗T N → HomT (W,N), f ⊗T n 7→ [w̃ 7→ f(w̃)n].
It follows then by [BW2003, 42.13.], that Im([, ]WN ) ⊆ HomT (W,N) is dense. In
particular, setting N = W we conclude that Im([, ]W ) ⊆ End(TW ) is dense. On
the other hand, assume Im([, ]W ) ⊆ End(TW ) is dense. Then for every finite subset
{w1, ..., wk} ⊆ W, there exists
g̃i ⊗T w̃i ∈
∗W ⊗T W with
wj = idW (wj) = [, ]W (
g̃i ⊗T w̃i)(wj) =
g̃i(wj)w̃i for j = 1, ..., k.
It follows then by Proposition 2.12 “2” that TW is locally projective.�
3 Morita (Semi)contexts
We noticed, in the proofs of some results on equivalences between subcategories of
module categories associated to a given Morita context, that no use is made of the com-
patibility between the connecting bimodule morphisms (or even that only one trace ideal
is used and so only one of the two bilinear morphisms is really in action). Some results of
this type appeared, for example, in [Nau1993], [Nau1994-a] and [Nau1994-b]. Moreover,
in our considerations some Morita contexts will be formed for arbitrary associative rngs
(i.e. not necessarily unital rings). These considerations motivate us to make the following
general definitions:
3.1. By a Morita semi-context we mean a tuple
mT = ((T : A), (S : B), P, Q,<,>T , I), (3)
where T is an A-rng, S is a B-rng, P is a (T, S)-bimodule, Q is an (S, T )-bimodule,
<,>T : P ⊗S Q → T is a (T, T )-bilinear morphism and I := Im(<,>T ) ⊳ T (called the
trace ideal associated to mT ).We drop the ground rings A,B and the trace ideal I ⊳ T,
if they are not explicitly in action. If mT (3) is a Morita semi-context and T, S are unital
rings, then we call mT a unital Morita semi-context.
3.2. Let mT = ((T : A), (S : B), P, Q,<,>T ), mT ′ = ((T
′ : A′), (S ′ : B′), P ′, Q′, <,>T ′) be
Morita semi-contexts. By a morphism of Morita semi-contexts from mT to mT ′ we
mean a four fold set of morphisms
((β : δ), (γ : σ), φ, ψ) : ((T : A), (S : B), P, Q) → ((T ′ : A′), (S ′ : B′), P ′, Q′),
where (β : δ) : (T : A) → (T ′ : A′) and (γ : σ) : (S : B) → (S ′ : B′) are rng morphisms,
φ : P → P ′ is (T, S)-bilinear and ψ : Q→ Q′ is (S, T )-bilinear, such that
β(< p, q >T ) =< φ(p), ψ(q) >T ′ for all p ∈ P, q ∈ Q .
Notice that we consider P ′ as a (T, S)-bimodule and Q′ as an (S, T )-bimodule with actions
induced by the morphism of rngs (β : δ) and (γ : σ). By MSC we denote the category
of Morita semi-contexts with morphisms defined as above, and by UMSC < MSC the
(non-full) subcategory of unital Morita semi-contexts.
Morita semi-contexts are closely related to dual pairings in the sense of [Abu2005]:
3.3. Let (T, S, P,Q,<,>T ) ∈ MSC and consider the canonical isomorphisms of Abelian
groups
Hom(S,T )(Q,
≃ Hom(T,T )(P ⊗S Q, T )
≃ Hom(T,S)(P,Q
This means that we have two dual T -pairings Pl := (Q, TP ) ∈ Pl(T ) and Qr := (P,QT ) ∈
Pr(T ), induced by the canonical T -linear morphisms
κPl := ξ
−1(<,>T ) : Q→
∗P and κQr := ζ(<,>T ) : P → Q
On the other hand, let (S, T,Q, P,<,>S) ∈ MSC and consider the canonical isomorphisms
of Abelian groups
Hom(S,T )(Q,P
≃ Hom(S,S)(Q⊗T P, S)
≃ Hom(T,S)(P,
Then we have two dual S-pairings Pr := (Q,PS) ∈ Pr(S) and Ql := (P, SQ) ∈ Pl(S),
induced by the canonical morphisms
κPr := ξ
′−1(<,>S) : Q→ P
∗ and κQr := ζ
′(<,>S) : P →
3.4. By a Morita datum we mean a tuple
M = ((T : A), (S : B), P, Q,<,>T , <,>S, I, J), (4)
where the following are Morita semi-contexts.
MT := ((T : A), (S : B), P, Q,<,>T , I) and MS := ((S : B), (T : A), Q, P,<,>S, J) (5)
If, moreover, the bilinear morphisms <,>T : P ⊗S Q → T and < −, >S: Q ⊗T P → S are
compatible, in the sense that
< q, p >S q
′ = q < p, q′ >T and p < q, p
′ >S =< p, q >T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q, (6)
then we call M a Morita context. If T, S in a Morita datum (context) M are unital,
then we call M a unital Morita datum (context).
3.5. LetM = ((T : A), (S : B), P, Q,<,>T , <,>S) andM
′ = ((T ′ : A′), (S ′ : B′), P ′, Q′, <
,>T ′, <,>S′) be Morita contexts. Extending [Ami1971, Page 275], we mean by a mor-
phism of Morita contexts from M to M′ a four fold set of maps
((β : δ), (γ : σ), φ, ψ) : ((T : A), (S : B), P, Q) → ((T ′ : A′), (S ′ : B′), P ′, Q′),
where (β : δ) : (T : A) → (T ′ : A′), (γ : σ) : (S : B) → (S ′ : B′) are rng morphisms,
φ : P → P ′ is (T, S)-bilinear and ψ : Q→ Q′ is (S, T )-bilinear, such that
β(< p, q >T ) =< φ(p), ψ(q) >T ′ and γ(< q, p >S) =< ψ(q), φ(p) >S′ ∀ p ∈ P, q ∈ Q.
By MC we denote the category of Morita contexts with morphisms defined as above, and
by UMC <MC the (non-full) subcategory of unital Morita contexts.
Example 3.6. If R is commutative, then any Morita semi-context (R,R, P,Q,<,>R) yields
a Morita context (R,R, P,Q,<,>R, [, ]R), where [, ]R := Q⊗R P ≃ P ⊗R Q
−→ R.�
3.7. We call a Morita semi-context mT = (T, S, P,Q,<,>T ) semi-derived (derived),
iff S := End(TP )
op (and Q = ∗P ). We call a Morita datum, or a Morita context, M =
(T, S, P,Q,<,>T , <,>S) semi-derived (derived), iff S = End(TP )
op, or T = End(PS)
(S = End(TP )
op and Q = ∗P, or T = End(PS) and Q = P
Remark 3.8. Following [Cae1998, 1.2.] (however, dropping the condition that the bilinear
map <,>T : P ⊗SQ→ T is surjective), Morita semi-contexts (T, S, P,Q,<>T ) in our sense
were called dual pairs in [Ver2006]. However, we think the terminology we are using is more
informative and avoids confusion with other notions of dual pairings in the literature (e.g.
the ones studied by the first author in [Abu2005]). The reason for this specific terminology
(i.e. Morita semi-contexts) is that every Morita context contains two Morita semi-contexts
as clear from the definition; and that any Morita semi-context can be extended to a (not
necessarily unital) Morita context in a natural way as explained below.
Elementary rngs
In what follows we demonstrate how to build new Morita (semi-)contexts from a given
Morita semi-context. These constructions are inspired by the notion of elementary rngs in
[Cae1998, 1.2.] (and [Ver2006, Remark 3.8.]):
Lemma 3.9. Let mT := ((T : A), (S : B), P, Q,<,>T ) ∈ MSC.
1. The (T, T )-bimodule T := P ⊗S Q has a structure of a T -rng (A-rng) with multipli-
cation
(p⊗S q) ·T (p
′ ⊗S q
′) :=< p, q >T p
′ ⊗S q
′ ∀ p, p′ ∈ P, q, q′ ∈ Q,
such that <,>T : T → T is a morphism of A-rngs, P is a (T, S)-bimodule and Q is
an (S,T)-bimodule, where
(p⊗S q)⇀ p̃ :=< p, q >T p̃ and q̃ ↼ (p⊗S q) := q̃ < p, q >T .
Moreover, we have morphisms of T -rngs (A-rngs)
ψ : T → End(PS), p⊗S q 7→ [p̃ 7→< p, q >T p̃];
φ : T → End(SQ)
op, p⊗S q 7→ [q̃ 7→ q̃ < p, q >T ],
((T : A), (S : B), P, Q, idT) ∈ MSC and we have a morphism of Morita semi-contexts
(<,>T , idS, , idP , idQ) : (T, S, P,Q, idT) → (T, S, P,Q,<,>T ).
2. The (S, S)-bimodule S := Q⊗T P has a structure of an S-rng (B-rng) with multipli-
cation
(q⊗T p) ·S (q
′ ⊗T p
′) := q < p, q′ >T ⊗T p
′ = q⊗T < p, q
′ >T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q,
such that <,>S: S → S is a morphism of B-rngs, P is a (T,S)-bimodule and Q is
an (S, T )-bimodule, where
p̃ ↼ (q ⊗T p) :=< p̃, q >T p and (q ⊗T p)⇀ q̃ := q < p, q̃ >T .
Moreover, we have morphisms of S-rngs (B-rngs)
Ψ : S → End(TP )
op, q ⊗T p 7→ [p̃ 7→< p̃, q >T p],
Φ : S → End(QT ), q ⊗T p 7→ [q̃ 7→ q < p, q̃ >T ],
and M := ((T : A), (S : B), P, Q,<,>T , idS) is a Morita context.
Remarks 3.10. 1. Given ((S : B), (T : A), Q, P,<,>S) ∈ MSC, the (S, S)-bimodule
S := Q⊗T P becomes an S-rng with multiplication
(q ⊗T p) ·S (q
′ ⊗T p
′) :=< q, p >S q
′ ⊗T p
′ ∀ p, p′ ∈ P, q, q′ ∈ Q;
and the (T, T )-bimodule T := P ⊗S Q becomes a T -rng with multiplication
(p⊗S q) ·T (p
′⊗S q
′) := p < q, p′ >S ⊗S q
′ = p⊗S < q, p
′ >S q
′ ∀ p, p′ ∈ P, q, q′ ∈ Q.
Analogous results to those in Lemma 3.9 can be obtained for the S-rng S and the
T -rng T.
2. Given a Morita semi-context (T, S, P,Q,<,>T ) several equivalent conditions for the
T -rng T := P ⊗S Q to be unital and the modules TP, QT to be firm can be found in
[Ver2006, Theorem 3.3.]. Analogous results can be formulated for the S-rng Q⊗T P
and the S-modules PS, SQ corresponding to any (S, T,Q, P,<,>S) ∈ MSC.
Proposition 3.11. 1. Let mT = (T, S, P,Q,<,>T ) ∈ UMSC and assume the A-rng
T := P ⊗S Q to be unital. If <,>T : T → T respects unities (and mT is injective),
then <,>T is surjective (T
≃ T as A-rings).
2. Let mS = (S, T,Q, P,<,>S) ∈ UMSC and assume the B-rng S := Q ⊗S P to
be unital. If <,>S: S → S respects unities (and mS is injective), then <,>S is
surjective (S
≃ S as B-rings).
3. Let M = (T, S, P,Q,<,>T , <,>S) ∈ UMC and assume the rngs T := P ⊗S Q, T,
S := Q⊗S P to be unital. If <,>T : P ⊗S Q → T and <,>S: S → S respect unities,
then T
≃ T as A-ring, S
≃ S as B-rings and we have equivalences of categories
TM ≈ SM (and MT ≈ MS).
Proof. Assume T is unital with 1T =
pi ⊗S qi. If <,>T respects unities, then we
< pi, qi >T= 1T , and so for any t ∈ T we get t = t1T =
t < pi, qi >T=∑n
< tpi, qi >T∈ Im(<,>T ). One can prove “2” analogously. As for “3”, it is well
known that a unital Morita context with surjective connecting bimodule morphisms is
strict (e.g. [Fai1981, 12.7.]), hence T
≃ T, S
≃ S. The equivalences of categories
TM ≃ TM ≈ SM ≃ SM (and MT ≃ MT ≈ MS ≃ MS) follow then by classical Morita
Theory (e.g. [Fai1981, Chapter 12]).�
Definition 3.12. Let T be an A-rng, VT a right T -module and consider for every left
T -module TL the annihilator
ann⊗L(VT ) := {l ∈ L | V ⊗T l = 0}.
Following [AF1974, Exercises 19], we say VT is L-faithful, iff ann
L(VT ) = 0; and to be
completely faithful, iff VT is L-faithful for every left T -module SL. Similarly, we can
define completely faithful left T -modules.
Under suitable conditions, the following result characterizes the Morita data, which
are Morita contexts:
Proposition 3.13. Let M = (T, S, P,Q,<,>T , <,>S) be a Morita datum.
1. If M ∈ MC, then S
≃ S and T
≃ T as rngs.
2. Assume TP is Q-faithful and QT is P -faithful. Then M ∈ MC if and only if S
and T
≃ T as rngs.
Proof. 1. Obvious.
2. Assume S
≃ S and T
≃ T as rngs. If p ∈ P and q, q′ ∈ Q are arbitrary, then we have
for any p̃ ∈ P :
< q, p >S q
′ ⊗T p̃ = (q ⊗T p) ·S (q
′ ⊗T p̃) = (q ⊗T p) ·S (q
′ ⊗T p̃) = q < p, q
′ >T ⊗T p̃,
hence < q, p >S q
′ − q < p, q′ >T∈ annQ(P ) = 0 (since TP is Q-faithful), i.e.
< q, p >S q
′ = q < p, q′ >T for all p ∈ P and q, q
′ ∈ Q. Assuming QT is P -faithful,
one can prove analogously that < p, q >T p
′ = p < q, p′ >S for all p, p
′ ∈ P and
q ∈ Q. Consequently, M is a Morita context.�
4 Injective Morita (Semi-)Contexts
Definition 4.1. We call a Morita semi-context mT = (T, S, P,Q,<,>T , I) :
injective (resp. semi-strict, strict), iff <,>T : P ⊗S Q→ T is injective (resp. surjec-
tive, bijective);
non-degenerate, iff Q →֒ ∗P and P →֒ Q∗ canonically;
Morita α-semi-context, iff Pl := (Q, TP ) ∈ P
l (T ) and Qr := (P,QT ) ∈ P
r (T ).
Notation. By MSCα ≤ MSC (UMSCα ≤ UMSC) we denote the full subcategory of
(unital) Morita semi-contexts satisfying the α-condition. Moreover, we denote by IMSC ≤
MSC (IUMSC ≤ UMSC) the full subcategory of injective (unital) Morita semi-contexts.
Definition 4.2. We say a Morita datum (context) M = (T, S, P,Q,<,>T , <,>S, I, J) :
is injective (resp. semi-strict, strict), iff <,>T : P⊗SQ→ T and <,>S: Q⊗T P → S
are injective (resp. surjective, bijective);
is non-degenerate, iff Q →֒ ∗P, P →֒ Q∗, Q →֒ P ∗ and P →֒ ∗Q canonically;
satisfies the left α-condition, iffPl := (Q, TP ) ∈ P
l (T ) andQl := (P, SQ) ∈ P
l (S);
satisfies the right α-condition, iff Qr := (P,QT ) ∈ P
r (T ) and Pr := (Q,PS) ∈
Pαr (S);
satisfies the α-condition, or M is a Morita α-datum (Morita α-context), iff M
satisfies both the left and the right α-conditions.
Notation. By MCαl < MC (UMC
l < UMC) we denote the full subcategory of Morita
contexts satisfying the left α-condition, and by MCαr < MC (UMC
r < UMC) the full
subcategory of (unital) Morita contexts satisfying the right α-condition. Moreover, we set
α := MCαl ∩MC
r and UMC
α := UMCαl ∩ UMC
Lemma 4.3. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ MC. Consider the Morita semi-
context MS := (S, T,Q, P,<,>S), the dual pairings Pl := (Q, TP ) ∈ Pl(T ), Qr :=
(P,QT ) ∈ Pr(T ) and the canonical morphisms of rings
ρP : S → End(TP )
op and λQ : S → End(QT ).
1. If Qr is injective (semi-strict), then MS is injective (ρP : S → End(TP )
op is a
surjective morphism of B-rngs).
2. Assume PS is faithful and let Qr be semi-strict. Then S ≃ End(TP )
op (an isomor-
phism of unital B-rings) and MS is strict.
3. If Pl is injective (semi-strict), then MS is injective (λQ : S → End(QT ) is a surjec-
tive morphism of B-rngs).
4. Assume SQ is faithful and let Pl is semi-strict. Then S ≃ End(QT ) (an isomorphism
of unital B-rings) and MS is strict.
Proof. We prove only “1” and “2”, as “3” and “4” can be proved analogously.
Consider the following butterfly diagram with canonical morphisms
Q⊗T Q
∗P ⊗T P
Q⊗T P
idQ⊗TκQr
llYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
⊗T idP
22eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
uujjj
Hom−T (
∗P,Q)
(κPl ,Q)
Hom−T (Q
∗, P )
(κQr ,P )
ttjjjj
**TTT
End(QT ) End(TP )
qi ⊗T pi ∈ Q⊗T P be arbitrary. For every p̃ ∈ P we have
[(κQr , P ) ◦ α
qi ⊗T pi)](p̃) =
< p̃, qi >T pi
p̃ < qi, pi >S
= ρP (
< qi, pi >S)(p̃)
= (ρP◦ <,>S)(
qi ⊗T pi)(p̃),
i.e. α
P := (κQr , P ) ◦ α
P = ρP◦ <,>S; and
[, ]lP ◦ (κPl ⊗T idP ))(
qi ⊗T pi)](p̃) =
κPl(qi)(p̃)pi
< p̃, qi >T pi
p̃ < qi, pi >S
= ρP (
< qi, pi >S)(p̃)
= [(ρP◦ <,>S)(
qi ⊗T pi)](p̃),
i.e. [, ]lP ◦ (κPl ⊗T idP ) = ρP◦ <,>S . On the other hand, for every q̃ ∈ Q we have
((κPl , Q) ◦ α
qi ⊗T pi)(q̃) =
qi < pi, q̃ >T
< qi, pi >S)q̃
= λQ(
< qi, pi >S)(q̃)
= (λQ◦ <,>S)(
qi ⊗T pi),
i.e. α
Q := (κPl , Q) ◦ α
Q = λQ◦ <,>S and
([, ]rQ ◦ (idQ ⊗T κQr))(
qi ⊗T pi)](q̃) =
qiκQr(pi)(q̃)
qi < pi, q̃ >T
< qi, pi >S q̃
= λQ(
< qi, pi >S)(q̃)
= [(λQ◦ <,>S)(
qi ⊗T pi)](q̃),
i.e. [, ]rQ ◦ (idQ ⊗T κQr) = λQ◦ <,>S . Hence Diagram (7) is commutative.
(1) Follows directly from the assumptions and the equality α
P = ρP◦ <,>S .
(2) Let PS be faithful, so that the canonical left S-linear map ρP : S → End(TP )
is injective. Assume now that Qr is semi-strict. Then ρP is surjective by “1” , whence
bijective. Since rings of endomorphisms are unital, we conclude that S ≃ End(TP )
op is a
unital B-ring as well (with unity ρ−1P (idP )). Moreover, the surjectivity of α
P = ρP◦ <,>S
implies that <,>S is surjective (since ρP is injective), say 1S =
< q̃j, p̃j >S for some
{(q̃j , p̃j)}J ⊆ Q× P. For any
qi ⊗T pi ∈ Ker(<,>S), we have then
qi ⊗T pi = (
qi ⊗T pi) · 1S =
(qi ⊗T pi) · (
< q̃j, p̃j >S)
qi ⊗T pi < q̃j, p̃j >S =
qi⊗T < pi, q̃j >T p̃j
qi < pi, q̃j >T ⊗T p̃j =
< qi, pi >S q̃j ⊗T p̃j
< qi, pi >S)q̃j ⊗T p̃j = 0,
i.e. <,>S is injective, whence an isomorphism.�
The following result shows that Morita α-contexts are injective:
Corollary 4.4. MCαl ∪MC
r ≤ IMC.
Example 4.5. Let mT = (T, S, P,Q,<,>T ) be a non-degenerate Morita semi-context. If
T is a QF ring and the T -modules TP, QT are projective, then by Proposition 2.12 “7”
Pl := (Q, TP ) ∈ P
l (T ) and Qr := (P,QT ) ∈ P
r (T ) (i.e. mT is a Morita α-semi-
context, whence injective). On the other hand, let M = (T, S, P,Q,<,>T , <,>S) be a
non-degenerate Morita datum. If T, S are QF rings and the modules TP, QT , PS, SQ are
projective, then M is an Morita α-datum (whence injective).�
Every semi-strict unital Morita context is injective (whence strict, e.g. [Fai1981, 12.7.]).
The following example, which is a modification of [Lam1999, Example 18.30]), shows that
the converse is not necessarily true:
Example 4.6. Let T = M2(Z2) be the ring of 2× 2 matrices with entries in Z2. Notice that
∈ T is an idempotent, and that eTe ≃ Z2 as rings. Set
P := Te = {
| a′, c′ ∈ Z2} and Q := eT = {
| a, b ∈ Z2}.
Then P = Te is a (T, eTe)-bimodule and Q = eT is an (eTe, T )-bimodule. Moreover, we
have a Morita context
Me = (T, eTe, T e, , eT, <,>T , < . >eTe),
where the connecting bilinear maps are
<,>T : Te⊗eTe eT → T,
a′a a′b
c′a c′b
<,>eTe : eT ⊗T Te → eTe
aa′ + bc′ 0
Straightforward computations show that<,>T is injective but not surjective (as
Im(<,>T )) and that <,>eTe is in fact an isomorphism. This means that Me is an injective
Morita context that is not semi-strict (whence not strict).�
Definition 4.7. Let T be a rng and I ⊳ T an ideal. For every left T -module TV consider
the canonical T -linear map
ζI,V : V → HomT (I, V ), v 7→ [t 7→ tv].
We say T I is strongly V -faithful, iff annV (I) := Ker(ζI,V ) := 0. Moreover, we say I is
strongly faithful, if T I is V -faithful for every left T -module TV. Strong faithfulness of I
w.r.t. right T -modules can be defined analogously.
Remark 4.8. Let T be a rng, I ⊳ T an ideal and TU a left ideal. It’s clear that ann
U(IT ) ⊆
annU(I) := Ker(ζI,U). Hence, if T I is strongly U-faithful, then IT is U-faithful (which
justifies our terminology). In particular, if T I is strongly faithful, then IT is completely
faithful.
Morita α-contexts are injective by Corollary 4.4. The following result gives a
partial converse:
Lemma 4.9. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ MC and assume the Morita semi-
context MS := (S, T,Q, P,<,>S, J) is injective.
1. If SJ is strongly faithful, then Qr := (P,QT ) ∈ P
r (T ).
2. If JS is strongly faithful, then Pl := (Q, TP ) ∈ P
l (T ).
Proof. We prove only “1”, since “2” can be proved similarly. Assume MS is injective and
consider for every left T -module U the following diagram
Q⊗T U
ζJ,Q⊗T U ((QQ
HomT−(P, U)
ψQ,Uuukkk
HomS−(J,Q⊗T U)
where for all f ∈ HomT−(P, U) and
< qj , pj >S∈ J we define
ψQ,U(f)(
< qj , pj >S) :=
qj ⊗T f(pj).
Then we have for every
q̃i ⊗T ũi ∈ Q⊗T U and s =
< qj, pj >S∈ J :
(ψQ,U ◦ α
q̃i ⊗T ũi)(s) =
qj ⊗T [α
q̃i ⊗T ũi)](pj)
qj ⊗T
< pj , q̃i >T ũi]
qj⊗T < pj, q̃i >T ũi
qj < pj , q̃i >T ⊗T ũi
< qj , pj >S q̃i ⊗T ũi
= ζJ,Q⊗TU(
q̃i ⊗T ũi)(s),
i.e. diagram (8) is commutative. If SJ is strongly faithful, then Ker(ζJ,Q⊗TU) = annQ⊗TU(J) =
0, hence ζJ,Q⊗TU is injective and it follows then that α
U is injective.�
Proposition 4.10. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ IMC. If T I, IT , SJ and JS
are strongly faithful, then M ∈ MCα.
5 Equivalences of Categories
In this section we give some applications of injective Morita (semi-)contexts and in-
jective Morita data to equivalences between suitable subcategories of modules arising in
the Kato-Müller-Ohtake localization-colocalization theory (as developed in (e.g. [Kat1978],
[KO1979], [Mül1974]). All rings, hence all Morita (semi-)contexts and data, in this section
are unital.
Static and Adstatic Modules
5.1. ([C-IG-TW2003]) Let A and B be two complete cocomplete Abelian categories, R :
A → B an additive covariant functor with left adjoint L : B → A and let
ω : LR → 1A and η : 1B → RL
be the induced natural transformations (called the counit and the unit of the adjunction,
respectively). Related to the adjoint pair (L,R) are two full subcategories of A and B :
Stat(R) := {X ∈ A | LR(X)
≃ X} and Adstat(R) := {Y ∈ B | Y
≃ RL(Y )},
whose members are called R-static objects and R-adstatic objects, respectively. It is
evident (from definition) that we have equivalence of categories Stat(R) ≈ Adstat(R).
A typical situation, in which static and adstatic objects arise naturally is the
following:
5.2. Let T, S be rings, TUS a (T, S)-bimodule and consider the covariant functors
HlU := HomT (U,−) : TM → SM and T
U := U ⊗S − : SM → TM.
It is well-known that (TlU ,H
U) is an adjoint pair of covariant functors via the natural
isomorphisms
HomT (U ⊗S M,N) ≃ HomS(M,HomT (U,N)) for all M ∈ SM and N ∈ TM
and the natural transformations
ωlU : U ⊗S HomT (U,−) → 1TM and η
U : 1SM → HomT (U, U ⊗S −)
yield for every TK and SL the canonical morphisms
ωlU,K : U ⊗S HomT (U,K) → K and η
U,L : L→ HomT (U, U ⊗S L). (9)
We call the HlU-static modules U-static w.r.t. S and set
Statl(TUS) := Stat(H
U) = {TK | U ⊗S HomT−(U,K)
≃ K};
and the HlU-adstatic modules U-adstatic w.r.t. S and set
Adstatl(TUS) := Adstat(H
U) = {SL | L
≃ HomT−(U, U ⊗S L)}.
By [Nau1990a] and [Nau1990b], there are equivalences of categories
Statl(TUS) ≈ Adstat
l(TUS). (10)
On the other hand, one can define the full subcategories Statr(TUS) ≈ Adstat
r(TUS) :
Statr(TUS) := {KS | Hom−S(U,K)⊗T U ≃ K};
Adstatr(TUS) := {LT | L ≃ Hom−S(U, L⊗T U)}.
In particular, setting
Stat(TU) := Stat
l(TUEnd(TU)op); Adstat(TU) := Adstat
l(TUEnd(TU)op);
Stat(US) := Stat
r(End(SU)US); Adstat(US) := Adstat
r(End(SU)US),
there are equivalences of categories:
Stat(TU) ≃ Adstat(TU) and Stat(US) ≃ Adstat(US). (11)
Remark 5.3. The theory of static and adstatic modules was developed in a series of papers
by the second author (see the references). They were also considered by several other
authors (e.g. [Alp1990], [CF2004]). For other terminologies used by different authors, the
interested reader may refer to a comprehensive treatment of the subject by R. Wisbauer
in [Wis2000].
Intersecting subcategories
Several intersecting subcategories related to Morita contexts were introduced in
the literature (e.g. [Nau1993], [Nau1994-b]). In what follows we introduce more and we
show that many of these coincide, if one starts with an injective Morita semi-context.
Moreover, other results on equivalences between some intersecting subcategories related
to an injective Morita context will be reframed for arbitrary (not necessarily compatible)
injective Morita data.
Definition 5.4. 1. For a right T -module X, a T -submodule X ′ ⊆ X is called K-pure
for some left T -module TK, iff the following sequence of Abelian groups is exact
0 → X ′ ⊗T K → X ⊗T K → X/X
′ ⊗T K → 0;
2. For a left T -module Y, a T -submodule Y ′ ⊆ Y is called L-copure for some left
T -module TL, iff the following sequence of Abelian groups is exact
0 → HomT (Y/Y
′, L) → HomT (Y, L) → HomT (Y
′, L) → 0.
Definition 5.5. (Compare [KO1979, Theorems 1.3., 2.3.]) Let T be a ring, I ⊳ T an
ideal, U a left T -module and consider the canonical T -linear morphisms
ζI,U : U → HomT (I, U) and ξI,U : I ⊗T U → U.
1. We say TU is I-divisible, iff ξI,U is surjective (equivalently, iff IU = U).
2. We say TU is I-localized, iff U
≃ HomT (I, U) canonically (equivalently iff T I is
strongly U -faithful and T I ⊆ T is U -copure).
3. We say a left T -module U is I-colocalized, iff I⊗TU
≃ U canonically (equivalently,
iff TU is I-divisible and IT ⊆ T is U -pure).
Notation. For a ring T, an ideal I ⊳ T, and with morphisms being the canonical ones,
we set
ID := {TU | IU = U}; IF := {TU | U →֒ HomT−(I, U)};
IL := {TU | U ≃ HomT (I, U}; IC := {TU | I ⊗T U ≃ U};
DI := {UT | UI = U}; FI := {UT | U →֒ Hom−T (I, U)};
LI := {UT | U ≃ HomT (I, U}; CI := {UT | U ⊗T I ≃ U}; .
The following result is due to T. Kato, K. Ohtake and B. Müller (e.g. [Mül1974],
[Kat1978], [KO1979]):
Proposition 5.6. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then there are equiv-
alences of categories
IC ≈ JC, CI ≈ CJ , IL ≈ JL and LI ≈ LJ .
5.7. Let mT = (T, S, P,Q,<,>T , I) ∈ UMSC and consider the dual pairings Pl := (Q,
TP ) ∈ Pl(T ) and Qr := (P,QT ) ∈ Pr(T ). For every left (right) T -module U consider the
canonical S-linear morphism induced by <,>T :
U : Q⊗T U → HomT−(P, U) (α
U : U ⊗T P → Hom−T (Q,U)).
We define
Dl(mT ) := {TU | Q⊗T U
≃ HomT−(P, U)};
Dr(mT ) := {UT | U ⊗T P
≃ Hom−T (Q,U)}.
Moreover, set
Ul(mT ) := Stat
l(TPS) ∩Adstat
l(SQT ); Ur(mT ) := Stat
r(SQT ) ∩Adstat
r(TPS);
Vl(mT ) := Stat
l(TPS) ∩ Dl(mT ); Vr(mT ) := Stat
r(SQT ) ∩ Dr(mT );
Vl(mT ) := IC ∩ Dl(mT ); Vr(mT ) := CI ∩ Dr(mT );
V̂l(mT ) := Vl(mT )∩ IL; V̂r(mT ) := Vr(mT ) ∩ LI ;
Wl(mT ) := Adstat
l(SQT ) ∩ Dl(mT ); Wr(mT ) := Adstat
r(TPS) ∩ Dr(mT );
Wl(mT ) := IL ∩ Dl(mT ); Wr(mT ) := LI ∩ Dr(mT );
Ŵl(mT ) := Wl(mT )∩ IC; Ŵr(mT ) := Wr(mT ) ∩ CI ;
Xl(mT ) := Vl(mT ) ∩Wl(mT ); Xr(mT ) := Vr(mT ) ∩Wr(mT );
Xl(mT ) := Vl(mT ) ∩Wl(mT ); Xr(mT ) := Vr(mT ) ∩Wr(mT ).
X ∗l (mT ) := {S(Q⊗T U) | V ∈ Xl(mT )}; X
r (mT ) := {(U ⊗T P )S | V ∈ Xr(mT )};
l (mT ) := {S(Q⊗T U) | V ∈ Xl(mT )}; X
r(mT ) := {(U ⊗T P )S | V ∈ Xr(mT )}.
Given mS = (S, T,Q, P,<,>S, J) ∈ UMSC one can define analogously, the corresponding
intersecting subcategories of SM and MS.
As an immediate consequence of Proposition 5.6 we get
Corollary 5.8. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ IUMC and consider the asso-
ciated Morita semi-contexts MT and MS (5).
1. If IC ≤ Dl(MT ) and JC ≤ Dl(MS), then Vl(MT ) ≈ Vl(MS). Similarly, if CI ≤
Dr(MT ) and CJ ≤ Dr(MS), then Vr(MT ) ≈ Vr(MS).
2. If IL ≤ Dl(MT ) and JL ≤ Dl(MS), then Wl(MT ) ≈ Wl(MS). Similarly, if LI ≤
Dr(MT ) and LJ ≤ Dr(MS), then Wr(MT ) ≈ Wr(MS).
Starting with a Morita context, the following result was obtained in [Nau1993,
Theorem 3.2.]. We restate the result for an arbitrary (not necessarily compatible) Morita
datum and sketch its proof:
Lemma 5.9. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be a unital Morita datum and con-
sider the associated Morita semi-contexts MT and MS in (5). Then there are equivalences
of categories
Xl(MT )
HomT−(P,−)
HomS−(Q,−)
Xl(MS) and Xr(MT )
Hom−T (Q,−)
Hom−S(P,−)
Xr(MS).
Proof. Let TV ∈ Xl(MT ). By the equivalence Stat
l(TPS)
HomT (P,−)
≈ Adstatl(TPS) in 5.2 we
have HomT−(P, V ) ∈ Adstat
l(TPS). Moreover, V ∈ Dl(M), hence HomT−(P, V ) ≃ Q⊗T V
canonically and it follows then from the equivalence Adstatl(SQT )
≈ Statl(SQT ) that
HomT−(P, V ) ∈ Stat
l(SQT ). Moreover, we have the following natural isomorphisms
P ⊗S HomT−(P, V ) ≃ V ≃ HomS−(Q,Q⊗T V ) ≃ HomS−(Q,HomT−(P, V )), (13)
i.e. HomT−(P, V ) ∈ Dl(MS). Consequently, HomT−(P, V ) ∈ Xl(MS). Moreover, (13)
yields a natural isomorphism V ≃ HomS−(Q,HomT−(P, V )). Analogously, one can show for
everyW ∈ Xl(MS) that HomS−(Q,W ) ∈ Xl(MT ) and thatW ≃ HomT−(P,HomS−(Q,W ))
naturally. Consequently, Xl(MT ) ≈ Xl(MS). The equivalences Xr(MT ) ≈ Xr(MS) can
be proved analogously.�
Proposition 5.10. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be a unital injective Morita
datum and consider the associated Morita semi-contexts MT and MS in (5).
1. There are equivalences of categories
Statl(T IT ) ≈ Adstat
l(T IT ); Stat
l(SJS) ≈ Adstat
l(SJS);
Statr(T IT ) ≈ Adstat
r(T IT ); Stat
r(SJS) ≈ Adstat
r(SJS).
2. If Statl(T IT ) ≤ X
l (MS) and Stat
l(SJS) ≤ X
l (MT ), then there are equivalences of
categories
Statl(T IT ) ≈ Stat
l(SJS) and Adstat
l(T IT ) ≈ Adstat
l(SJS).
3. If Statr(T IT ) ≤ X
r (MS) and Stat
r(SJS) ≤ X
r (MT ), then there are equivalences of
categories
Statr(T IT ) ≈ Stat
r(SJS) and Adstat
r(T IT ) ≈ Adstat
r(SJS).
Proof. To prove “1”, notice that since M is an injective Morita datum, P ⊗S Q
and Q⊗T P
≃ J as bimodules and so the four equivalences of categories result from 5.2.
To prove “2”, one can use an argument similar to that in [Nau1994-b, Theorem 3.9.] to
show that the inclusion Statl(T IT ) = Stat
l(T (P ⊗S Q)T ) ≤ X
l (MS) implies Stat
l(T IT ) =
Statl(T (P ⊗S Q)T ) = Xl(MT ) and that the inclusion Stat
l(SJS) = Stat
l(S(Q ⊗T P )S) ≤
X ∗l (MT ) implies Stat
l(SJS) = Stat
l(S(Q ⊗T P )S) = Xl(MS). The result follows then by
Lemma 5.9. The proof of “3” is analogous to that of “2”.�
For injective Morita semi-contexts, several subcategories in (12) are shown in the
following result to be equal:
Theorem 5.11. Let mT = (T, S, P,Q,<,>T , I) ∈ IUMS. Then
1. Vl(mT ) = Vl(mT ), Wl(mT ) = Wl(mT ), whence
V̂l(mT ) = Ŵl(mT ) = Xl(mT ) = Xl(mT ) = IC∩Dl(mT )∩ IL and X
l (mT ) = X
l (mT ).
2. Vr(mT ) = Vr(mT ), Wr(mT ) = Wr(mT ), whence
V̂r(mT ) = Ŵr(mT ) = Xr(mT ) = Xr(mT ) = CI∩Dr(mT )∩LI and X
r (mT ) = X
r(mT ).
Proof. We prove only “1” as “2” can be proved analogously. Assume the Morita semi-
context mT = (T, S, P,Q,<,>T , I) is injective. By our assumption we have for every
V ∈ Dl(mT ) the commutative diagram
P ⊗S (Q⊗T V )
idP⊗S(α
(P ⊗S Q)⊗T V
<,>T⊗T idV≃
P ⊗S HomT−(P, V )
// V I ⊗T VξI,V
Then it becomes obvious that ωlP,V : P ⊗S HomT (P, V ) → V is an isomorphism if and only
if ξI,V : I ⊗T V → V is an isomorphism. Consequently
V(mT ) = Dl(mT ) ∩ Stat
l(TPS) = Dl(mT ) ∩ IC = V(mT ).
On the other hand, we have for every V ∈ Dl(mT ) the following commutative diagram
HomS−(Q,HomT−(P, V ))
// HomT−(P ⊗S Q, V )
HomS−(Q,Q⊗T V )
// HomT−(I, V )
(<,>T ,V )≃
It follows then that ηlP,L : V → HomS(Q,Q ⊗T P ) is an isomorphism if and only if
ζI,V : V → HomT (I, V ) is an isomorphism. Consequently,
W(mT ) = Dl(mT ) ∩ Adstat
l(TPS) = Dl(mT ) ∩I L = W(mT ).
Moreover, we have
V̂l(mT ) := Vl(mT ) ∩ IL = Vl(mT ) ∩ IL = IC ∩ Dl(mT )∩ IL
= IC ∩Wl(mT ) = IC ∩Wl(mT ) = Ŵl(mT ).
On the other hand, we have
Xl(mT ) = Vl(mT ) ∩Wl(mT ) = Vl(mT ) ∩Wl(mT ) = Xl(mT )
and so the equalities V̂l(mT ) = Ŵl(mT ) = Xl(mT ) = Xl(mT ) and X
l (mT ) = X
l (mT ) are
established.�
In addition to establishing several other equivalences of intersecting subcategories,
the following results reframe the equivalence of categories V̂ ≈ Ŵ in [Nau1994-b, Theorem
4.9.] for an arbitrary (not necessarily compatible) injective Morita datum:
Theorem 5.12. Let M = (T, S, P,Q,<,>T , <,>S, I, J) be an injective Morita datum and
consider the associated Morita semi-contexts MT and MS (5).
1. The following subcategories are mutually equivalent:
V̂l(MT ) = Ŵl(MT ) = Xl(MT ) = Xl(MT ) ≈ Xl(MS) = Xl(MS) = Ŵl(MS) = V̂l(MS).
2. If Vl(MT ) ≤ IL and Wl(MS) ≤ JC, then Vl(MT ) ≈ Wl(MS). If Wl(MT ) ≤ IC
and Vl(MS) ≤ JL, then Wl(MT ) ≈ Vl(MS).
3. The following subcategories are mutually equivalent:
V̂r(MT ) = Ŵr(MT ) = Xr(MT ) = Xr(MT ) ≈ Xr(MS) = Xr(MS) = Ŵr(MS) = V̂r(MS).
4. If Vr(MT ) ≤ LI and Wr(MT ) ≤ CJ , then Vr(MT ) ≈ Wr(MS). If Wr(MT ) ≤ CJ
and Vr(MS) ≤ LI , then Vr(MS) ≈ Wr(MT ).
Proof. By Lemma 5.9, Xl(MT ) ≈ Xl(MS) and so “1” follows by Theorem 5.11. If
Vl(MT ) ≤ IL and Wl(MS) ≤ JC, then we have
Vl(MT ) = Vl(MT ) ∩ IL = V̂l(MT ) ≈ Ŵl(MS) = Wl(MS) ∩ JC = Wl(MS).
On the other hand, if Wl(MT ) ≤ IL and Vl(MS) ≤ JC, then
Wl(MT ) = Wl(MT ) ∩ IC = Ŵl(MT ) ≈ V̂l(MS) = Vl(MS) ∩ JL = Vl(MS).
So we have established “2”. The results in “3” and “4” can be obtained analogously.�
6 More applications
In this final section we give more applications of Morita α-(semi-)contexts and
injective Morita (semi-)contexts. All rings in this section are unital, whence all Morita
(semi-)contexts are unital. Moreover, for any ring T we denote with TE an arbitrary, but
fixed, injective cogenerator in TM.
Notation. Let T be an A-ring. For any left T -module TV, we set
#V := HomT (V, TE). If
moreover, TVS is a (T, S)-bimodule for some B-ring S, then we consider
S V with the left
S-module structure induced by that of VS.
Lemma 6.1. (Compare [Col1990, Lemma 3.2.], [CF2004, Lemmas 2.1.2., 2.1.3.]) Let T be
an A-ring, S a B-ring and TVS a (T, S)-bimodule,
1. A left T -module TK is V -generated if and only if the canonical T -linear morphism
ωlV,K : V ⊗S HomT (V,K) → K (18)
is surjective. Moreover, V ⊗S W ⊆ Pres(TV ) ⊆ Gen(TV ) for every left S-module
2. A left S-module SL is
S V -cogenerated if and only if the canonical S-linear morphism
ηlV,L : L→ HomT (V, V ⊗S L) (19)
is injective. Moreover, HomT (V,M) ⊆ Copres(
S V ) ⊆ Cogen(
S V ) for every left
T -module TM.
Remark 6.2. Let T be an A-ring, S a B-ring and TVS a (T, S)-bimodule. Notice that for
any left S-module SL we have
ann⊗L(VS) := {l ∈ L | V ⊗S l = 0} = Ker(η
V,L),
whence (by Lemma 6.1 “2” ) VS is L-faithful if and only if SL is
S V -cogenerated. It follows
then that VS is completely faithful if and only if
S V is a cogenerator.
Localization and colocalization
In what follows we clarify the relations between static (adstatic) modules and subcate-
gories colocalized (localized) by a trace ideal of a Morita context satisfying the α-condition.
Recall that for any (T, S)-bimodule TPS we have by Lemma 6.1:
Statl(TPS) ⊆ Gen(TP ) and Adstat
l(TPS) ⊆ Cogen(
S P ). (20)
Theorem 6.3. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then we have
IC ⊆ ID ⊆ Gen(TP ). (21)
Assume Pr := (Q,PS) ∈ P
r (S). Then
1. Gen(TP ) = Stat
l(TPS) ⊆ IF.
2. If Gen(TP ) ⊆ IC, then IC = ID = Gen(TP ) = Stat
l(TPS).
3. If Qr := (P,QT ) ∈ P
r (T ), then T I ⊆ TT is pure and IC = ID.
Proof. For every left T -module TK, consider the following diagram with canonical mor-
phisms and let α2 := ζI,K ◦ ω
P,K. It is easy to see that both rectangles and the two right
triangles commutes:
P ⊗S Q⊗T K
idP⊗Sα
<,>T⊗T idK
P ⊗S HomT (P,K)
HomT (P,K)//
HomS(Q,HomT (P,K))
HomT (P ⊗S Q,K)
I ⊗T K
// HomT (I,K)
(<,>T ,K)
It follows directly from the definitions that IC ⊆ ID and Stat
l(TPS) ⊆ Gen(TP ). If TK
is I-divisible, then ξI,K◦ <,>T ⊗T idK = ω
P,K ◦ idP ⊗S α
K is surjective, whence ω
is surjective and we conclude that TK is P -generated by Lemma 6.1 “1”. Consequently,
ID ⊆ Gen(TP ).
Assume now that Pr ∈ P
r (S). Considering the canonical map ρQ : T → End(SQ)
the map ρQ◦ <,>T= α
Q is injective and so the bilinear map <,>T is injective (i.e.
P ⊗S Q
≃ I). Define α1 := (idP ⊗S α
K ) ◦ (<,>T ⊗T idK)
−1, so that the left triangles
commute. Notice that αPr
HomT (P,K)
is injective and the commutativity of the upper right
triangle in Diagram (22) implies that α2 is injective (whence ω
P,K is injective by the
commutativity of the lower right triangle).
1. If K ∈ Statl(TPS), then the commutativity of the lower right triangle (22) and the
injectivity of α2 show that ζI,K is injective; hence, Stat
l(TPS) ⊆ IF. On the other
hand, if TK is P -generated, then ω
P,K is surjective by Lemma 6.1 (1), thence bijective,
i.e. K ∈ Statl(TPS). Consequently, Gen(TP ) = Stat
l(TPS).
2. This follows directly from the inclusions in (21) and “1”.
3. Assume Qr := (P,QT ) ∈ P
r (T ). Since Pr ∈ P
r (S), it follows by analogy to Propo-
sition 2.12 “3” that PS is flat, hence idP ⊗S α
K is injective. The commutativity of
the upper left triangle in Diagram (22) implies then that α1 is injective, thence ξI,K
is injective by commutativity of the lower left triangle (i.e. T I ⊆ TT is K-pure). If
TK is divisible, then K ⊗T I
≃ K (i.e. K ∈ IC).�
Theorem 6.4. Let M = (T, S, P,Q,<,>T , <,>S, I, J) ∈ UMC. Then we have
JL ⊆ JF ⊆ Cogen(
S P ) and Adstat
l(TPS) ⊆ Cogen(
S P ).
Assume Qr := (P,QT ) ∈ P
r (T ). Then
1. JS ⊆ SS is pure and JC ⊆ Cogen(
S P ).
2. If Pr := (Q,PS) ∈ P
r (S), then JL ⊆ Adstat
l(TPS) ⊆ Cogen(
S P ) ⊆ JF.
3. If Pr ∈ P
r (S) and Cogen(
S P ) ⊆ JL, then JL = Cogen(
S P ) = Adstat
l(TPS).
Proof. For every right S-module L consider the commutative diagram with canonical
morphisms and let α3 be so defined, that the left triangles become commutative
J ⊗S L
ξJ,L //
ζJ,L // HomS(J, L)
(<,>S ,L)
HomS(Q⊗T P, L)
≃ can
Q⊗T P ⊗S L
(<,>S)⊗S idL
// HomT (P, P ⊗S L)
HomT (P,HomS(Q,L))
By definition JL ⊆ JF and Adstat
l(TPS) ⊆ Cogen(
S P ). If SL ∈ JF, then ζJ,L is injective
and it follows by commutativity of the right rectangle in Diagram (23) that ηlP,L is injective,
hence SL is
S P -cogenerated by Lemma 6.1 “2”. Consequently, JF ⊆ Cogen(
S P ).
Assume now that Qr ∈ P
r (T ). Then it follows from Lemma 4.3 that <,>S is injective
(hence Q⊗T P
≃ J) and so α4 := (can ◦ (<,>S, L))
−1 ◦ (P, αPrL ) is injective.
1. Since α3 is injective, ξJ,L is also injective for every SL, i.e. JS ⊆ SS is pure. If SL ∈
JC, then it follows from the commutativity of the left rectangle in Diagram (23) that
ηlP,L is injective, hence L ∈ Cogen(
S P ) by Lemma 6.1 (2).
2. Assume that Pr ∈ P
r (S), so that α4 is injective. If SL ∈ JL, then ζJ,L is an
isomorphism, thence ηlP,L is surjective (notice that α4 is injective). Consequently,
JL ⊆ Adstat
l(TPS).
3. This follows directly from the assumptions and “2”.�
∗-Modules
To the end of this section, we fix a unital ring T, a left T -module TP and set S :=
End(TP )
Definition 6.5. ([MO1989]) We call TP a ∗-module, iff Gen(TP ) ≈ Cogen(
S P ).
Remark 6.6. It was shown by J. Trlifaj [Trl1994] that all ∗-modules are finitely generated.
By definition, Statl(TPS) ≤ TM and Adstat
l(TPS) ≤ SM are the largest subcat-
egories between which the adjunction (P ⊗S −,HomT (P,−)) induces an equivalence. On
the other hand, Lemma 6.1 shows that Gen(TP ) ≤ TM and Cogen(
S P ) ≤ SM are the
largest such subcategories (see [Col1990, Section 3] for more details). This suggests the
following observation:
Proposition 6.7. ([Xin1999, Lemma 2.3.]) We have
TP is a ∗ -module ⇔ Stat(TP ) = Gen(TP ) and Adstat(TP ) = Cogen(
S P ).
Definition 6.8. A left T -module TU is said to be
semi-
-quasi-projective (abbr. s-
-quasi-projective), iff for any left T -module
TV ∈ Pres(TU) and any U-presentation
U (Λ) → U (Λ
′) → V → 0
of TV (if any), the following induced sequence is exact:
HomT (U, U
(Λ)) → HomT (U, U
(Λ′)) → HomT (U, V ) → 0;
weakly-
-quasi-projective (abbr. w-
-quasi-projective), iff for any left T -
module TV and any short exact sequence
0 → K → U (Λ
′) → V → 0
with K ∈ Gen(TU) (if any), the following induced sequence is exact:
0 → HomT (U,K) → HomT (U, U
(Λ′)) → HomT (U, V ) → 0;
self-tilting, iff TU is w-
-quasi-projective and Gen(TU) = Pres(TU);∑
-self-static, iff any direct sum U (Λ) is U -static.
(self)-small, iff HomT (U,−) commutes with direct sums (of TU);
Proposition 6.9. Assume M = (T, S, P,Q,<,>T , <,>S) is a unital Morita context.
1. If Pr := (Q,PS) ∈ P
r (S), then:
(a) Gen(TP ) = Stat
l(TPS);
(b) there is an equivalence of categories Gen(TP ) ≈ Cop(
S P );
(c) TP is
-self-static and Statl(TPS) is closed under factor modules.
(d) Gen(TP ) = Pres(TP );
2. If M ∈ UMCαr and Cogen(
S P ) ⊆ JL, then:
(a) Gen(TP ) = Stat
l(TPS) and Cogen(
S P ) = Adstat
l(TPS);
(b) there is an equivalence of categories Cogen(
S P ) ≈ Gen(TP );
(c) TP is a ∗-module;
(d) TP is self-tilting and self-small.
Proof. 1. If Pr ∈ P
r (S), then it follows by Theorem 6.3 that Gen(TP ) = Stat
l(TPS),
which is equivalent to each of “b” and “c” by [Wis2000, 4.4.] and to “d” by [Wis2000,
4.3.].
2. It follows by the assumptions, Theorems 6.3, 6.4 and 5.2 that Gen(TP ) = Stat
l(TPS) ≈
Adstatl(TPS) = Cogen(
S P ), whence Gen(TP ) ≈ Cogen(
S P ) (which is the definition
of ∗-modules). Hence “a” ⇔“b” ⇔“c”. The equivalence “a” ⇔ “d” is evident by
[Wis2000, Corollary 4.7.] and we are done.�
Wide Morita Contexts
Wide Morita contexts were introduced by F. Castaño Iglesias and J. Gómez-Torrecillas
[C-IG-T1995] and [C-IG-T1996] as an extension of classical Morita contexts to Abelian
categories.
Definition 6.10. Let A and B be Abelian categories. A right (left) wide Morita
context between A and B is a datum Wr = (G,A,B, F, η, ρ), where G : A ⇄ B : F are
right (left) exact covariant functors and η : F ◦ G −→ 1A, ρ : G ◦ F −→ 1B (η : 1A −→
F ◦ G, ρ : 1B −→ G ◦ F ) are natural transformations, such that for every pair of objects
(A,B) ∈ A× B the compatibility conditions G(ηA) = ρG(A) and F (ρB) = ηF (B) hold.
Definition 6.11. Let A and B be Abelian categories and W = (G,A,B, F, η, ρ) be a right
(left) wide Morita context. We call W injective (respectively semi-strict, strict), iff η
and ρ are monomorphisms (respectively epimorphisms, isomorphisms)
Remarks 6.12. Let W = (G,A,B, F, η, ρ) be a right (left) wide Morita context.
1. It follows by [CDN2005, Propositions 1.1., 1.4.] that if either η or ρ is an epimorphism
(monomorphism), then W is strict, whence A ≈ B.
2. The resemblance of injective left wide Morita contexts is with the Morita-Takeuchi
contexts for comodules of coalgebras, i.e. the so called pre-equivalence data for cate-
gories of comodules introduced in [Tak1977] (see [C-IG-T1998] for more details).
Injective Right wide Morita contexts
In a recent work [CDN2005, 5.1.], Chifan, et. al. clarified (for module categories) the
relation between classical Morita contexts and right wide Morita contexts. For the conve-
nience of the reader and for later reference, we include in what follows a brief description
of this relation.
6.13. Let T, S be rings, A := TM and B := SM. Associated to each Morita context
M = (T, S, P,Q,<,>T , <,>S) is a wide Morita context as follows: Define G : A ⇄ B : F
by G(−) = Q ⊗T − and F (−) = P ⊗S −. Then there are natural transformations η :
F ◦G −→ 1
and ρ : G ◦ F −→ 1
such that for each TV and WS :
ηV : P ⊗S (Q⊗T V ) → V,
pi ⊗S (qi ⊗T vi) 7→
< pi, qi >T vi,
ρW : Q⊗T (P ⊗S W ) → W,
qi ⊗T (pi ⊗S wi) 7→
< qi, pi >S wi.
Then the datum Wr(M) := (G, TM, SM, F, η, ρ) is a right wide Morita context.
Conversely, let T ′, S ′ be two rings and W ′r = (G
′, T ′M, S′M, F
′, η′, ρ′) be a right wide
Morita context between T ′M and S′M such that the right exact functors G
′ : T ′M ⇄
S′M : F
′ commute with direct sums. By Watts’ Theorems (e.g. [Gol1979]), there exists a
(T, S)-bimodule P ′ (e.g. F ′(S ′)) such that F ′ ≃ P ′⊗S′ −, an (S, T )-bimodule Q
′ such that
G′ ≃ Q′ ⊗T ′ − and there should exist two bilinear forms
<,>T ′: P
′ ⊗S′ Q
′ → T ′ and <,>S′: Q
′ ⊗T ′ P
′ → S ′,
such that the natural transformations η′ : F ′ ◦G′ → 1
, ρ : G′ ◦ F ′ → 1
are given by
η′V ′(p
′ ⊗S′ q
′ ⊗T ′ v
′) =< p′, q′ >T ′ v
′ and ρ′W ′(q
′ ⊗T p
′ ⊗S w
′) =< q′, p′ >S′ w
for all V ′ ∈ T ′M, W
′ ∈ S′M, p
′ ∈ P ′, q′ ∈ Q′, v′ ∈ V ′ and w′ ∈ W ′. It can be shown that
in this way one obtains a Morita context M′ = M′(W ′r) := (T
′, S ′, P ′, Q′, <,>T ′, <,>S′).
Moreover, it turns out that given a wide Morita context Wr, we have Wr ≃ Wr(M(Wr)).
The following result clarifies the relation between injective Morita contexts and
injective right wide Morita contexts.
Theorem 6.14. Let M = (T, S, P,Q,<,>T , <,>S) be a Morita context, A := TM, B :=
SM and consider the induced right wide Morita context Wr(M) := (G,A,B, F, η, ρ).
1. If Wr(M) is an injective right wide Morita context, then M is an injective Morita
context.
2. If M ∈ UMCαr , then Wr(M) is an injective right wide Morita context.
Proof. 1. Let Wr(M) be an injective right wide Morita context. Then in particular,
<,>T= ηT and <,>S= ρS are injective, i.e. M is an injective Morita context.
2. Assume that M satisfies the right α-condition. Suppose there exists some TV and∑
pi ⊗S (qi ⊗T vi) ∈ Ker(ηV ). Then for any q ∈ Q we have
0 = q ⊗T ηV (
(pi ⊗S qi)⊗T vi) =
q⊗T < pi, qi >T vi
q < pi, qi >T ⊗T vi =
< q, pi >S qi ⊗T vi
< q, pi >S (qi ⊗T vi) = α
pi ⊗S (qi ⊗T vi))(q).
Since Pr := (Q,PS) ∈ P
r (S), the morphism α
is injective and so
pi⊗S (qi⊗T
vi) = 0, i.e. ηV is injective. Analogously, suppose
qi ⊗T (pi ⊗S wi) ∈ Ker(ρW ).
Then for any p ∈ P we have
0 = p⊗S ρW (
qi ⊗T (pi ⊗S wi) =
p⊗S < qi, pi >S wi
p < qi, pi >S ⊗Swi =
< p, qi >T pi ⊗S wi
< p, qi >T (pi ⊗S wi) = α
qi ⊗T (pi ⊗S wi))(p).
Since Qr := (P,QT ) ∈ P
r (T ), the morphism α
is injective and so
qi ⊗T
(pi ⊗S wi) = 0, i.e. ρW is injective. Consequently, the induced right wide Morita
context Wr(M) is injective.�
Acknowledgement: The authors thank the referee for his/her careful reading of
the paper and for the fruitful suggestions, comments and corrections, which helped in
improving several parts of the paper. Moreover, they acknowledge the excellent research
facilities as well as the support of their respective institutions, King Fahd University of
Petroleum and Minerals and King AbdulAziz University.
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Introduction
Preliminaries
Morita (Semi)contexts
Injective Morita (Semi-)Contexts
Equivalences of Categories
More applications
|
0704.0075 | Strong decays of charmed baryons | Strong decays of charmed baryons
Chong Chen, Xiao-Lin Chen, Xiang Liu,∗ Wei-Zhen Deng, and Shi-Lin Zhu†
Department of physics, Peking University, Beijing, 100871, China
(Dated: October 29, 2018)
There has been important experimental progress in the sector of heavy baryons in the past several
years. We study the strong decays of the S-wave, P-wave, D-wave and radially excited charmed
baryons using the 3P0 model. After comparing the calculated decay pattern and total width with
the available data, we discuss the possible internal structure and quantum numbers of those charmed
baryons observed recently.
PACS numbers: 13.30.Eg, 12.39.Jh
I. INTRODUCTION
Babar and Belle collaborations observed sev-
eral excited charmed baryons: Λc(2880, 2940)
Ξc(2980, 3077)
+,0 and Ωc(2768)
0 last year [1, 2, 3, 4, 5],
which inspired several investigations of these states
in literature [6, 7, 8, 9]. We collect the experimental
information of these recently observed hadrons in Table
I. Their quantum numbers have not been determined
except Λc(2880)
+. In order to understand their struc-
tures using the present experimental information, we
study the strong decay pattern of the excited charmed
baryons systematically in this work. In the past decades,
there has been some research work on heavy baryons
[8, 11, 12].
The quantum numbers and decay widths of S-wave and
some P-wave charmed baryons are known [13]. We first
systematically analyze their strong decays in the frame-
work of the 3P0 strong decay model. Accordingly one
can extract the parameters and estimate the accuracy of
the 3P0 model when it’s applied in the charmed baryon
system. Then we go one step further and extend the
same formalism to study the decay patterns of these new
charmed baryons Λc(2880, 2940)
+, Ξ(2980, 3077)+,0 un-
der different assignments of their quantum numbers. Af-
ter comparing the theoretical results with the available
experimental data, we can learn their favorable quantum
numbers and assignments in the quark model.
State Mass and Width (MeV) Decay channels in experiments Other information
2881.5 ± 0.3, < 8 [10] Λcπ
2881.9 ± 0.1± 0.5 , 5.8± 1.5± 1.1 [1] D0p
JP favors 5
Λc(2880)
2881.2 ± 0.2+0.4
−0.3, 5.5
−0.5 ± 0.4 [2] Σ
⋆0,++
c (2520)π
Γ(Σ⋆c(2520)π
Γ(Σc(2455)π±)
= 0.225 ± 0.062 ± 0.025 [2]
2939. ± 1.3± 1.0, 17.5± 5.2± 5.9 [1] D0pΛc(2940)
2937.9 ± 1.0+1.8
−0.4, 10± 4± 5 [2] Σc(2455)
0,++π+,−
2967.1 ± 1.9± 1.0, 23.6± 2.8± 1.3 [3] Λ+c K
−π+Ξc(2980)
2978.5 ± 2.1± 2.0, 43.5± 7.5± 7.0 [4] Λ+c K
Ξc(2980)
0 2977.1 ± 8.8± 3.5, 43.5 [4] Λ+c K
3076.4 ± 0.7± 0.3, 6.2± 1.6± 0.5 [3] Λ+c K
−π+Ξc(3077)
3076.7 ± 0.9± 0.5, 6.2± 1.2± 0.8 [4] Λ+c K
Ξc(3077)
0 3082.8 ± 1.8± 1.5, 5.2± 3.1± 1.8 [4] Λ+c K
Ωc(2768)
0 2768.3 ± 3.0 [5] Ω0cγ J
P = 3
TABLE I: A summary of recently observed charmed baryons by Babar and Belle collaborations.
Very recently CDF collaboration reported four par-
ticles [14, 15], which are consistent with Σ±b and Σ
predicted in the quark model [16]. Their masses are
= 5808+2.0−2.3± 1.7 MeV, MΣ−
= 5816+1.0−1.0± 1.7MeV,
= 5829+1.6−1.8±1.7,MΣ∗−
= 5837+2.1−1.9±1.7MeV. The
http://arxiv.org/abs/0704.0075v3
mass splitting between Σb and Σ
b was discussed in Refs.
[17, 18] while the strong decays of Σ
b were studied in
Ref. [19]. As a byproduct, we also calculate the strong
decays of Σ
b and other S-wave bottom baryons in this
work.
This paper is organized as follows. We give a short the-
oretical review of S-wave, P-wave and D-wave charmed
baryons and introduce our notations for them in Section
II. Then we give a brief review of 3P0 model in Section
III. We present the strong decay amplitudes of charmed
baryons in Section IV. Section V is the numerical results.
The last section is our discussion and conclusion. Some
lengthy formulae are collected in the Appendix.
II. THE NOTATIONS AND CONVENTIONS OF
CHARMED BARYON
We first introduce our notations for the excited
charmed baryons. Inside a charmed baryon there are one
charm quark and two light quarks (u, d or s). It belongs
to either the symmetric 6F or antisymmetric 3̄F flavor
representation (see Fig. 1). For the S-wave charmed
baryons, the total color-flavor-spin wave function and
color wave function must be symmetric and antisymmet-
ric respectively. Hence the spin of the two light quarks
is S=1 for 6F or S=0 for 3̄F . The angular momentum
and parity of the S-wave charmed baryons are JP = 1
for 6F and J
P = 1
for 3̄F . The names of S-wave
charmed baryons are listed in Fig. 1, where we use the
star to denote 3
baryons and the prime to denote the
JP = 1
baryons in the 6F representation.
c (ssc)
(∗)++
c (uuc)
c (ddc) Ξ
′(∗)0
c (dsc)
c (udc) Ξ
′(∗)+
c (usc)
Ξ0c(dsc)
Λ+c (udc) Ξ
c (usc)
FIG. 1: The SU(3) flavor multiplets of charmed baryons
In Fig. 2 we introduce our notations and conventions
for the P-wave charmed baryons. lρ is the orbital angular
momentum between the two light quarks while lλ denotes
the orbital angular momentum between the charm quark
and the two light quark system. We use the prime to
label the ΞcJl baryons in the 6F representation and the
tilde to discriminate the baryons with lρ = 1 from that
with lλ = 1.
The notation for D-wave charmed baryons is more
complicated (see Fig. 3). Besides the prime, lρ and lλ
(a) lρ = 0, lλ = 1
(b) lρ = 1, lλ = 0
Jl = 1: Σc1(
) Ξ′c1(
Jl = 0: Σc0(
) Ξ′c0(
Jl = 2: Σc2(
) Ξ′c2(
fS(6): L = 1 ⊗ Sq1q2 = 1
fA(3̄): L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Λc1(
) Ξc1(
Jl = 1: Λ̃c1(
) Ξ̃c1(
Jl = 0: Λ̃c0(
) Ξ̃c0(
Jl = 2: Λ̃c2(
) Ξ̃c2(
fA(3̄): L = 1 ⊗ Sq1q2 = 1
fS(6): L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Σ̃c1(
) Ξ̃′c1(
FIG. 2: The notations for P-wave charmed baryons. fS(6F )
and fA(3̄F ) denote the SU(3) flavor representation. Sq1q2 is
the total spin of the two light quarks. L denotes the total
orbital angular momentum of charmed baryon system.
defined above, we use the hat and check to denote the
charmed baryons with lρ = 2 and lρ = 1 respectively. For
the baryons with lρ = 1 and lλ = 1, we use the super-
script L to denote the different total angular momentum
in Λ̌LcJl , Σ̌
and Ξ̌LcJl .
III. THE
3P0 MODEL
The 3P0 model was first proposed by Micu [20] and
further developed by Yaouanc et al. later [21, 22, 23].
Now this model is widely used to study the strong decays
of hadrons [24, 25, 26, 27, 28, 29, 30, 31].
According to this model, a pair of quarks with JPC =
0++ is created from the vacuum when a hadron decays,
which is shown in Fig. 4 for the baryon decay process
A → B + C. The new qq̄ pair created from the vacuum
together with the qqq within the the initial baryon re-
group into the outgoing meson and baryon via the quark
rearrangement process. In the non-relativistic limit, the
transition operator is written as
T = −3γ
〈1 m; 1 −m|0 0〉
d3k4 d
3(k4 + k5)
k4 − k5
χ451,−m ϕ
4i(k4) d
5j(k5) (1)
where i and j are the color indices of the created quark
and anti-quark. ϕ450 = (uū+ dd̄ + ss̄)/
3 and ω450 = δij
for the flavor and color singlet respectively. χ451,−m is for
(a) lρ = 0, lλ = 2
(b) lρ = 2, lλ = 0
Jl = 2: Σc2(
) Ξ′c2(
Jl = 1: Σc1(
) Ξ′c1(
Jl = 3: Σc3(
) Ξ′c3(
fS(6): L = 2 ⊗ Sq1q2 = 1
fA(3̄): L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Λc2(
) Ξc2(
Jl = 2: Σ̂c2(
) Ξ̂′c2(
Jl = 1: Σ̂c1(
) Ξ̂′c1(
Jl = 3: Σ̂c3(
) Ξ̂′c3(
fS(6): L = 2 ⊗ Sq1q2 = 1
fA(3̄): L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Λ̂c2(
) Ξ̂c2(
(c) lρ = 1, lλ = 1
Jl = 1: Λ̌
) Ξ̌1c1(
Jl = 0: Λ̌
) Ξ̌1c0(
Jl = 2: Λ̌
) Ξ̌1c2(
L = 1 ⊗ Sq1q2 = 1
L = 0 ⊗ Sq1q2 = 1 =⇒ Jl = 1: Λ̌
) Ξ̌0c1(
)fA(3̄)
Jl = 2: Λ̌
) Ξ̌2c2(
Jl = 1: Λ̌
) Ξ̌2c1(
Jl = 3: Λ̌
) Ξ̌2c3(
L = 2 ⊗ Sq1q2 = 1
L = 0 ⊗ Sq1q2 = 0 =⇒ Jl = 0: Σ̌
) Ξ̌′0c0(
L = 1 ⊗ Sq1q2 = 0 =⇒ Jl = 1: Σ̌
) Ξ̌′1c1(
L = 2 ⊗ Sq1q2 = 0 =⇒ Jl = 2: Σ̌
) Ξ̌′2c2(
fS(6)
FIG. 3: The notations for the D-wave charmed baryons.
the spin triplet state. Ym1 (k) ≡ |k|Y m1 (θk, φk) is a solid
harmonic polynomial corresponding to the p-wave quark
pair. γ is a dimensionless constant related to the strength
of the quark pair creation from the vacuum, which was
extracted by fitting to data. The hadron and meson state
are defined as respectively according to the definition of
FIG. 4: The decay process of A → B +C in 3P0 model.
mock state [32]
|A(nA2SA+1LA JAMJA )(PA)〉
MLA ,MSA
〈LAMLASAMSA |JAMJA〉
d3k1d
3(k1 + k2 + k3−PA)
×ψnALAMLA(k1,k2,k3)χ
SAMSA
ϕ123A ω
×| q1(k1)q2(k2)q3(k3)〉,
|B(nB2SB+1LB JBMJB )(PB)〉
MLB ,MSB
〈LBMLBSBMSB |JBMJB 〉
d3kad
3(ka + kb−PB)ψnBLBMLB(ka,kb)
×χabSBMSBϕ
B | qa(ka)q̄b(kb)〉 (3)
and satisfy the normalization condition
〈A(PA)|A(P′A)〉 = 2EAδ3(PA −P
〈B(PB)|B(P′B)〉 = 2EBδ3(PB −P
B) . (4)
The subscripts 1, 2, 3 denote the quarks of parent
hadron A. a and b refer to the quark and antiquark within
the meson B respectively. ki(i = 1, 2, 3) are the momen-
tum of quarks in hadron A. ka and kb are the momentum
of the quark and antiquark in meson B. PA(B) represents
the momentum of state A(B). SA(B) and JA(B) denote
the total spin and the total angular momentum of state
A(B).
The S-matrix is defined as
〈f |S|i〉 = I − i2πδ(Ef − Ei)MMJAMJBMJC . (5)
The helicity amplitude of the process A→ B + C in the
center of mass frame of meson A is
MMJAMJBMJC (A→ BC)
8EAEBEC γ
MLA ,MSA ,
MLB ,MSB ,
MLC ,MSC ,m
〈LAMLASAMSA |JAMJA〉
×〈LBMLBSBMSB |JBMJB 〉〈LCMLCSCMSC |JCMJC 〉
×〈1 m; 1 −m| 0 0〉 〈χ235SCMSC χ
SBMSB
|χ123SAMSAχ
×〈ϕ235C ϕ14B |ϕ123A ϕ450 〉 I
MLA ,m
MLB ,MLC
(p) (6)
where the spatial integral I
MLA ,m
MLB ,MLC
(p) is defined as
MLA ,m
MLB ,MLC
d3k1d
3(k4 + k5)
×δ3(k1 + k2 + k3 −PA)δ3(k1 + k4 −PB)
×δ3(k2 + k3 + k5 −PC)
×ψ∗nBLBMLB(k1,k4)ψ
nCLCMLC
(k2,k3,k5)
×ψnALAMLA(k1,k2,k3) Y
(k4 − k5
. (7)
〈χ235SCMSC χ
SBMSB
|χ123SAMSAχ
1−m〉 and 〈ϕ235C ϕ14B |ϕ123A ϕ450 〉
denote the spin and flavor matrix element respectively.
The decay width of the process A→ B + C is
Γ = π2
2JA + 1
MJA ,MJB ,MJC
∣MMJAMJBMJC
where |p| is the momentum of the daughter baryon in
the parent’s center of mass frame. s = 1/(1 + δBC) is a
statistical factor which is needed if B and C are identical
particles.
IV. THE STRONG DECAYS OF CHARMED
BARYON
According to the 3P0 model, the decay occurs through
the recombination of the five quarks from the initial
charmed baryon and the created quark pair. So there
are three ways of regrouping:
A(q1, q2, c3) + P(q4, q̄5) → B(q2, q4, c3) + C(q1, q̄5), (8)
A(q1, q2, c3) + P(q4, q̄5) → B(q1, q4, c3) + C(q2, q̄5), (9)
A(q1, q2, c3) + P(q4, q̄5) → B(q1, q2, q4) + C(c3, q̄5)(10)
where qi and c3 denote the light quark and charm quark
respectively.
When the excited charmed baryon decays into a
charmed baryon plus a light meson as shown in Eq. (8)
and (9), the total decay amplitude reads
MMJAMJBMJC
= −2γ
8EAEBEC
m1,m3,m4,m
×〈J12M12; s3m3|JAMJA〉〈lρAmρA; lλAmλA|LAMLA〉〈LAMLA ; S1 2m1 2|J12M12〉
×〈s1m1; s2m2|S1 2m1 2〉 〈J14M14; s3m3|JB MJB 〉 〈lρBmρB; lλBmλB|LB MLB〉
×〈LBMLB ; S1 4m1 4|J14M14〉〈s1m1; s4m4|S1 4m1 4〉〈1m; 1 −m|00〉 〈s4m4; s5m5|1 −m〉
×〈LC MLC ; SC MC |JC MJC〉〈s2m2; s5m5|SCMC〉 × 〈φ
1,4,3
1,2,3
A 〉 × I
MLA ,m
MLB ,MLC
(p), (11)
where the pre-factor 2 in front of γ arises from the fact
that the amplitude from the Eq. (8) is the same as that
from Eq. (9).
The overlap integral in the momentum space is
MLA ,m
MLB ,MLC
= δ3(PB −PC)
d3p1d
B(lρB ,mρB, lλB,mλB)
×ψ∗C(LC MLC)Ym1
(p4 − p5
ψA(lρA,mρA, lλA,mλA).
Since all hadrons in the final states are S-wave in this
work, eq. (12) can be further expressed as
MLA ,m
MLB ,MLC
= δ3(PB −PC)Π(lρA,mρA, lλA,mλA,m), (13)
where we have used the harmonic oscillator wave func-
tions for both the meson and baryon. The expressions
of Π(lρA,mρA, lλA,mλA,m) for the decays of S-wave, P-
wave and D-wave charmed baryons are collected in the
Appendix. We also move the lengthy expressions of mo-
mentum space integration of S-wave, P-wave and D-wave
charmed baryons to the Appendix.
V. NUMERICAL RESULTS
The decay widths of charmed baryons from the 3P0
model involve several parameters: the strength of quark
pair creation from vacuum γ, the R value in the harmonic
oscillator wave function of meson and the αρ,λ in the
baryon wave functions. We follow the convention of Ref.
[34] and take γ = 13.4, which is considered as a universal
parameter in the 3P0 model. The R value of π and K
mesons is 2.1 GeV−1 [34] while it’s R = 2.3 GeV−1 for
the D meson [35]. αρ = αλ = 0.5 GeV for the proton and
Λ [31]. For S-wave charmed baryons, the parameters αρ
and αλ in the harmonic oscillator wave functions can be
fixed to reproduce the mass splitting through the contact
term in the potential model [33]. Their values are αρ =
0.6 GeV and αλ = 0.6 GeV. For P-wave and D-wave
charmed baryons, αρ and αλ are expected to lie in the
range 0.5 ∼ 0.7 GeV. In the following, our numerical
results are obtained with the typical values αρ = αλ =
0.6 GeV.
The strong decay widths of the S-wave charmed
baryons Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645)
are listed in Table II. Accordingly the decay widths of
S-wave bottomed baryons are presented in Table III. Be-
cause Ξb, Ξ
b and Ξ
b have not been observed so far, their
masses are taken from the theoretical estimate in Ref.
[36], which are mΞb = 5805.7 MeV, mΞ′
= 5950 MeV
and mΞ∗
= 5966.1 MeV.
The quantum number and internal structure of the
following P-wave charmed baryons Λ+c (2593), Λ
c (2625),
Ξ+,0c (2790) and Ξ
c (2815) are relatively known exper-
imentally [13]. Their strong decay modes and widths
from the 3P0 model are collected in Table IV. The quan-
tum number of Σ++c (2800) is still unknown [13]. Thus
under different P-wave assignments of Σ++c (2800), we
present the strong decay widths of its possible decay
modes in Table V. In the heavy quark limit, the pro-
cess Σ++c (2800) → Λ+c π+ is forbidden if Σ++c (2800) is
assigned as Σc1(
), Σc1(
), Σ̃c1(
) and Σ̃c1(
which is observed in our calculation as can be seen from
Table V.
Λc(2880)
+ and Λc(2940)
+ are observed in the invariant
mass spectrum of D0p [1]. The first radial excitation of
Λc does not decay into D
0p from the 3P0 model. Hence
the possibility of Λc(2880)
+ and Λc(2940)
+ being a radial
excitation is excluded. We calculate their strong decays
assuming they are D-wave charmed baryons. The results
are shown in Table VI and VII.
With positive parity, Ξ(2980)+,0 and Ξ(3077)+,0 can
be either the first radially excited charmed baryons or the
D-wave charmed baryons. With different assumptions of
their quantum numbers we present their strong decay
widths in Table VIII, IX and Fig. 8.
The numerical results depend on the parameters αρ
and αλ in the harmonic oscillator wave functions of
the charmed baryons. We illustrate such a dependence
in Figs. 5, 6 and 7 using several typical decay chan-
nels: Σ++c (2455) → Λ+c π+, Λ+c (2593) → Σ++c (2455)π−
and Λ+c (2880) → Σ∗++c (2520)π−, where Σ++c (2455),
Λ+c (2593) and Λ
c (2880) are S-wave, P-wave and D-wave
baryons respectively.
0.70 0.50
(GeV
(GeV)
FIG. 5: The variation of the decay width of Σ++c (2455) →
Λ+c π
+ with αρ and αλ.
0.70 0.56
FIG. 6: The variation of decay width of Λ+c (2593) →
Σ++c (2455)π
− with αρ and αλ. Here Λ
c (2593) is assigned
as Λc1(
0.640
(GeV)
FIG. 7: The variation of decay width of Λ+c (2880) →
Σ∗++c (2520)π
− with αρ and αλ. Here Λ
c (2880) is assigned
as Λc2(
TABLE II: The strong decay widths of S-wave charmed
baryons Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645).
Here all results are in units of MeV.
JP Channel Width Total width (Exp) [13]
Σ++c (2455)
Λ+c π
+ 1.24 2.23 ± 0.30
Σ+c (2455)
Λ+c π
0 1.40 < 4.6
Σ0c(2455)
Λ+c π
− 1.24 2.2± 0.40
Σ∗++c (2520)
Λ+c π
+ 11.9 14.9 ± 1.9
Σ∗+c (2520)
Λ+c π
0 12.1 < 17
Σ∗0c (2520)
Λ+c π
− 11.9 16.1 ± 2.1
Ξ∗+c (2645)
Ξ+c π
0 0.64
Ξ∗+c (2645)
+ 0.49
< 3.1
Ξ∗0c (2645)
Ξ+c π
− 0.54
Ξ∗0c (2645)
0 0.54
< 5.5
VI. DISCUSSION AND CONCLUSION
At present it is still too difficult to calculate the strong
decay widths of hadrons from the first principles of QCD.
For this purpose, some phenomenological strong decay
models were proposed such as the 3P0 model, flux tube
model, QCD sum rule, lattice QCD etc, among which
only the first two approaches can be applied to the strong
decays of excited hadrons. To a large extent, the predic-
tions from the 3P0 and flux tube models roughly agree
with each other.
TABLE III: The strong decay widths of S-wave bottom
baryons Σb, Σ
b , Ξ
b and Ξ
b . Here all results are in units
of MeV.
JP Channel Width Experimental results [14]
+ 3.5
− 4.7
+ 7.5
− 9.2
Ξbπ 0.10 -
Ξbπ 0.85 -
TABLE IV: The decay widths of P-wave charmed baryons
Λ+c (2593, 2625) and Ξ
c (2790, 2815) with the fixed structure
and quantum number assignments. Here all results are in
units of MeV.
Assignment Channel Γ ΓExp [13]
Σ++c π
− 3.4
Σ+c π
0 6.4Λ
c (2593) Λc1(
+ 3.4
3.6+2.0
Σ++c π
− 1.9× 10−3 < 0.10
Σ+c π
0 2.6× 10−3 < 1.9Λ
c (2625) Λc1(
+ 1.9× 10−3 < 0.10
Ξ′+c π
0 5.0
Ξ+c (2790) Ξc1(
Ξ′0c π
+ 4.9
Ξ′+c π
− 5.2
Ξ0c(2790) Ξc1(
Ξ′0c π
0 5.1
Ξ⋆+c π
0 2.7
Ξ+c (2815) Ξc1(
Ξ⋆0c π
+ 2.6
< 3.5
Ξ⋆+c π
− 2.7
Ξ0c(2815) Ξc1(
Ξ⋆0c π
0 2.8
< 6.5
The 3P0 model possesses inherent uncertainties [21, 29,
34]. In certain cases, the result from the 3P0 model may
be a factor of 2 ∼ 3 off the experimental width. The un-
certainty source of the 3P0 model arises from the strength
of the quark pair creation from the vacuum γ, the approx-
imation of non-relativity, and assuming the simple har-
monic oscillator radial wave functions for the hadrons.
Even with the above uncertainty, the 3P0 model is still
the most systematic, effective and widely used framework
to study the hadron strong decays.
In this work, we have calculated the strong decay
widths of charmed baryons using the 3P0 model. Our
numerical results do not strongly depend on the param-
eters αρ and αλ as shown in Figs. 5, 6 and 7. Thus the
following qualitative features and conclusions remain es-
sentially unchanged with reasonable variations of αρ and
Our results for the S-wave charmed baryons
Σ++,+,0c (2455), Σ
∗++,+,0
c (2520) and Ξ
c (2645) are
roughly consistent with experimental data within the
inherent uncertainty of the 3P0 model. As a byproduct,
we have also calculated the strong decays of Σ±b and Σ
TABLE V: The decay widths of Σ++c (2800) in dif-
ferent P-wave charmed baryons assignments. R =
Σ+,++c π
+,0/Σ⋆+,++c π
+,0. The total width of Σ++c (2800) is
75+22
−17 MeV [13]. Here all results are in units of MeV.
Assignment Λ+c π
+ Σ+,++c π
+,0 Σ⋆+,++c π
) 307 0.0 0.0 -
) 0.0 296 0.4 740
) 0.0 0.7 220 3× 10−3
) 8.1 1.3 0.3 4.3
) 8.1 0.6 0.5 1.2
Σ̃c1(
) 0.0 75 69 1.1
Σ̃c1(
) 0.0 75 69 1.1
observed by CDF Collaboration recently. The numerical
results are consistent with the experimental values too.
The decay width of P-wave baryon Λ+c (2593) is three
times larger than the experimental value. With the
large experimental uncertainty and the inherent the-
oretical uncertainty of the the 3P0 model, such a
deviation is still acceptable. The decay widths of
Λ+c (2625) and Ξ
c (2790, 2815) are compatible with the
experimental upper bound. By comparing our results
with the experimental total width, we tend to exclude
the Σc0(
) assignment for Σ++c (2800). Since the
Σ++c (2800) is observed in Λ
+ channel [37], there are
only two assignments left for Σ++c (2800), i.e. Σc2(
or Σc2(
). More experimental information such as the
ratio
Γ[Σ++c (2800)→Σ
c (2800)→Σ
⋆+,++
c π+,0]
will be helpful in the deter-
mination of the quantum number of Σ++c (2800).
We have also calculated the strong decay widths of
newly observed Λc(2880, 2940)
+, Ξ(2980, 3077)+,0 as-
suming they are candidates of D-wave charmed baryons.
We find that the only possible assignment of Λc(2880)
Λ̌2c3(
) after considering both its total decay width and
the ratio Γ(Σ⋆cπ
±)/Γ(Σcπ
±), which agrees very well with
the indication from Belle experiment that Λc(2880)
+ fa-
vors JP = 5
by the analysis of the angular distribution
Unfortunately the experiment information about the
Λc(2940)
+, Ξ(2980, 3077)+,0 is scarce at present. From
their calculated decay widths, we can only exclude some
assignments which are marked with crosses in Tables
VII, VIII and IX. The decay width ratios of Λc(2940)
Ξ(2980, 3077)+,0 from the 3P0 model will be useful in
the identification of their quantum numbers in the fu-
ture since the inherent uncertainty cancels largely.
We have also discussed the strong decays of
Ξ(2980, 3077)+,0 assuming they are radial excitations.
Unfortunately the numerical results in Fig. 8 depend
quite strongly on the node of the spatial wave function
which is related to the parameters of the harmonic oscil-
lator wave functions as shown in Fig. 8. We are unable
to make strong conclusions here.
Appendix
A. The harmonic oscillator wave functions used in
our calculation
For the S-wave charmed baryon,
ψ(0, 0, 0, 0) = 33/4 (
4 exp
For the P-wave charmed baryon,
ψ(1,m, 0, 0) = −i 33/4
)1/2(
1/α2ρ
Ym1 (pρ)
, (15)
ψ(0, 0, 1,m) = −i 33/4
)1/2( 1
Ym1 (pλ)
. (16)
For the D-wave charmed baryon,
ψ(2,m, 0, 0) = 33/4
)1/2( 1
Ym2 (pρ)
,(17)
ψ(0, 0, 2,m) = 33/4
)1/2( 1
Ym2 (pλ)
,(18)
ψ(1,m, 1,m′) = −33/4
)1/2( 1
Ym1 (pρ)
)1/2( 1
1 (pλ)
× exp
. (19)
Here Yml (p) is the solid harmonic polynomial.
The ground state wave function of meson is
ψ(0, 0) =
R2(p2 − p5)2
. (20)
The the wave function of the first radially excited
charmed baryon ψ(nρ, nλ) reads as
ψ(1, 0)
= 33/4
π2αραλ
ψ(0, 1)
= 33/4
π2αραλ
where nρ and nλ denote the radial quantum number be-
tween the two light quarks and between heavy quark and
the two light quarks respectively. Here pρ =
(p1−p2)
and pλ =
(p1 + p2 − 2p3) for the above expressions.
All the above harmonic oscillator wave functions can be
normalized as
dp1dp2dp3|ψ|2 = 1.
B. The momentum space integration
The momentum space integration
Π(lρA,mρA, lλA,mλA,m) includes:
For the S-wave charmed baryon decay,
Π(0, 0, 0, 0, 0) = β|p| ∆0,0. (21)
For the P-wave charmed baryon decay,
Π(0, 0, 1, 0, 0) =
f2β|p|2 − ζ
∆0,1,
Π(0, 0, 1, 1,−1) = Π(0, 0, 1,−1, 1) = ζ
∆0,1,
Π(1, 0, 0, 0, 0) =
β̟|p|2 + 1
4λ1f1
∆1,0,
Π(1, 1, 0, 0,−1) = Π(1,−1, 0, 0, 1) = β̟|p|2 ∆1,0.
For the D-wave charmed baryon decay,
Π(0, 0, 2, 0, 0) = − f2
β |p|3 + ζ|p|
×∆0,2,
Π(0, 0, 2, 1,−1) = Π(0, 0, 2,−1, 1) =
|p|∆0,2,
Π(2, 0, 0, 0, 0) = −2
(β̟2|p|3 +
̟ |p|
2λ1f1
∆2,0,
Π(2, 1, 0, 0,−1) = Π(2,−1, 0, 0, 1)
2λ1f1
∆2,0,
Π(1, 0, 1, 0, 0) =
β̟|p|3 + 1
β + ζ̟
4λ1f1
)|p|+ f2
2λ1f1
∆1,1,
Π(1, 1, 1,−1, 0) = Π(1,−1, 1, 1, 0) =
4λ1f1
∆1,1,
Π(1, 0, 1, 1,−1) = Π(1, 0, 1,−1, 1) =
̟ ζ|p|
∆1,1,
Π(1, 1, 1, 0,−1) = Π(1,−1, 1, 0, 1)
4λ1f1
ζ)× f2
∆1,1.
For the strong decay of the radial excitation, the mo-
mentum space integrals denoted as Π(nρ, nλ) are:
Π(0, 1) =
4α2λf
k − 3β
2f21α
∆0,0,
Π(1, 0) =
(β̟2k3 −
3βα2ρ
k − λ2̟ζ
∆0,0.
where
R2, λ2 = −
R2, λ4 =
λ5 = −(
12α2λ
f1 = λ3 −
, f2 = λ5 −
2λ2λ4
f3 = λ6 −
, ζ =
4λ1f1
β = (1 +
3λ2f2 − 2
3λ4f1 + 2λ1f2
6λ1f1
∆0,0 = (
× exp
− (f3 −
)|p|2
∆0,1 = (
− (f3 −
)|p|2
∆1,0 = (
− (f3 −
)|p|2
∆0,2 = (
− (f3 −
)|p|2
∆2,0 = (
− (f3 −
)|p|2
∆1,1 = (
− (f3 −
)|p|2
− ( 3
In the above expressions, |p| reads as
|p| =
(m2A − (mB +mC)2)(m2A − (mB −mC)2)
Acknowledgments
C.C. thanks W.J. Fu for the help in the numeri-
cal calculation and Y.R. Liu and B. Zhang for useful
discussions. This project was supported by the Na-
tional Natural Science Foundation of China under Grants
10421503 and 10625521, Chinese Ministry of Educa-
tion and the China Postdoctoral Science foundation (No.
20060400376).
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[12] D. Pirjol and T.M. Yan, Phys. Rev. D 56, 5483 (1997).
[13] W.M. Yao et al., Particle Data Group, J. Phys. G 33, 1
(2006).
[14] CDF Collaboration, I.V.Gorelov, arXiv:
hep-ex/0701056.
[15] I.V. Gorelov, arXiv: hep-ex/0701056.
[16] E. Jenkins, Phys. Rev. D 54, 4515 (1996); ibid.
55, 10 (1997); M. Karlinear and H.J. Lipkin, arXiv:
hep-ph/0307243; M. Karlinear and H.J. Lipkin, Phys.
Lett. B 575, 249 (2003).
[17] J.L. Rosner, Phys. Rev. D 75, 013009 (2007).
[18] M. Karliner and H.J. Lipkin, arXiv: hep-ph/0611306.
[19] C.W. Hwang, arXiv: hep-ph/0611221.
[20] L. Micu, Nucl. Phys. B10, 521 (1969).
[21] A. Le Yaouanc, L. Oliver, O. Pène and J. Raynal, Phys.
Rev. D8, 2223 (1973); D9, 1415 (1974); D11, 1272
(1975); Phys. lett. B71, 57 (1977); B71, 397 (1977).
[22] A. Le Yaouanc, L. Oliver, O. Pène and J. Raynal, Phys.
Lett. B72, 57 (1977).
[23] A. Le Yaouanc, L. Oliver, O. Pène and J. Raynal, Hadron
Transitions in the Quark Model, Gordon and Breach Sci-
ence Publishers, New York, 1987.
[24] H.G. Blundell and S. Godfrey, Phys. Rev. D53, 3700
(1996).
[25] P.R. Page, Nucl. Phys. B446, 189 (1995); S. Capstick
and N. Isgur, Phys. Rev. D34, 2809 (1986).
http://arxiv.org/abs/hep-ex/0608043
http://arxiv.org/abs/hep-ex/0607042
http://arxiv.org/abs/hep-ex/0608055
http://arxiv.org/abs/hep-ph/0612332
http://arxiv.org/abs/hep-ph/0609195
http://arxiv.org/abs/hep-ph/0606166
http://arxiv.org/abs/hep-ph/0606015
http://arxiv.org/abs/hep-ph/0703257
http://arxiv.org/abs/hep-ex/0701056
http://arxiv.org/abs/hep-ex/0701056
http://arxiv.org/abs/hep-ph/0307243
http://arxiv.org/abs/hep-ph/0611306
http://arxiv.org/abs/hep-ph/0611221
TABLE VI: The decay widths of Λ+c (2880) with different D-wave assignments. All results are in units of MeV.
Assignment Σ0,+,++c π
+,0,− Σ⋆0,+,++c π
+,0,− Γ(Σ
Γ(Σcπ±)
D0p Remark
) 7.8 0.9 0.11 0.0 ×
) 0.06 5.34 89 0.0 ×
Λ̂c2(
) 78.3 59.1 0.75 0.0 ×
Λ̂c2(
) 78.3 59.1 0.75 0.0 ×
Λ̌0c1(
) 0.9 2.3 2.6 2.3 ×
Λ̌0c1(
) 0.22 6.0 27 2.3 ×
Λ̌1c0(
) 132 144 1.1 0.0 ×
Λ̌1c1(
) 66.3 18.0 0.27 150 ×
Λ̌1c1(
) 16.5 45.0 2.7 150 ×
Λ̌1c2(
) 82.8 9.0 0.10 0.0 ×
Λ̌1c2(
) 0.0 54.1 − 0.0 ×
Λ̌2c1(
) 25.7 8.1 0.32 64 ×
Λ̌2c1(
) 6.5 20.4 3.1 64 ×
Λ̌2c2(
) 57.9 14.2 0.24 0.0 ×
Λ̌2c2(
) 9.4 47.1 5.0 0.0 ×
Λ̌2c3(
) 10.8 5.5 0.51 12
Λ̌2c3(
) 6.1 7.4 1.2 12 ×
[26] S. Capstick and W. Roberts, Phys. Rev. D49, 4570
(1994).
[27] E.S. Ackleh, T. Barnes and E.S. Swanson, Phys. Rev.
D54, 6811 (1996).
[28] H.Q. Zhou, R.G. Ping and B.S. Zou, Phys. Lett. B611,
123 (2005).
[29] X.H. Guo, H.W. Ke, X.Q. Li, X. Liu and S.M. Zhao,
arXiv: hep-ph/0510146.
[30] J. Lu, W.Z. Deng, X.L. Chen and S.L. Zhu, Phys.
Rev. D 73 054012, (2006); B. Zhang, X. Liu and
S.L. Zhu, DOI: 10.1140/epjc/s10052-007-0221-y, arXiv:
hep-ph/0609013.
[31] S. Capstick and W. Roberts, Phys. Rev. D 47, 1994
(1993).
[32] C. Hayne and N. Isgur, Phys. Rev. D 25, 1944 (1982).
[33] S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
[34] H. G. Blundell, S. Godfrey, Phys. Rev. D 53, 3700
(1996).
[35] F.E. Close and E.S. Swanson, Phys. Rev. D 72, 094004
(2005).
[36] E. Jenkins, Phys. Rev. D 54, 4515 (1996).
[37] Belle Collaboration, R. Mizuk, Phys. Rev. Lett. 94,
122002 (2005).
http://arxiv.org/abs/hep-ph/0510146
http://arxiv.org/abs/hep-ph/0609013
TABLE VII: The decay widths of Λ+c (2940) with different D-wave assignments. Here all results are in units of MeV.
Assignment Σ0,+,++c π
+,0,− Σ⋆0,+,++c π
+,0,− Γ(Σ
Γ(Σcπ±)
D0p Remark
) 11.7 9.1 0.77 0.0 ×
) 0.2 9.1 46 0.0 ×
Λ̂c2(
) 170 150 0.88 0.0 ×
Λ̂c2(
) 170 150 0.88 0.0 ×
Λ̌0c1(
) 2.2 0.5 0.23 11
Λ̌0c1(
) 0.6 1.4 2.3 11
Λ̌1c0(
) 212 259 1.2 0.0 ×
Λ̌1c1(
) 106 32.4 0.31 340 ×
Λ̌1c1(
) 26.5 81.0 3.1 340 ×
Λ̌1c2(
) 142 16.2 0.11 0.0 ×
Λ̌1c2(
) 0.0 97.0 − 0.0 ×
Λ̌2c1(
) 34.5 12.6 0.37 95 ×
Λ̌2c1(
) 8.6 31.7 3.7 95 ×
Λ̌2c2(
) 77.7 27.7 0.36 0.0 ×
Λ̌2c2(
) 19.5 75.6 3.9 0.0 ×
Λ̌2c3(
) 22.2 12.9 0.58 49 ×
Λ̌2c3(
) 12.4 17.5 1.4 49 ×
0.2 0.4 0.6 0.8 1.0 1.2
(GeV)
(3077):
(1/2+) with (n =1, n =0)
0.4 0.6 0.8 1.0
(3077):
(1/2+) with (n =0, n =1)
(GeV)
0.2 0.4 0.6 0.8 1.0
(GeV)
(3077): '
(1/2+) with (n =1, n =0)
(a) (b) (c)
0.4 0.5 0.6 0.7 0.8 0.9 1.0
(GeV)
(3077): '
(1/2+) with (n =0, n =1)
0.2 0.4 0.6 0.8 1.0
(3077):
(3/2+) with (n =1, n =0)
(GeV)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(3077):
(3/2+) with (n =0, n =1)
(GeV)
(d) (e) (f)
FIG. 8: The dependence of the total decay width of Ξc(3077) on the parameter αλ or αρ if Ξc(3077) is a radial excitation. In
these figures, we fix αρ = 0.6 GeV for the case with nλ = 1, and αλ = 0.6 GeV for nρ = 1. The situation of Ξc(2980) as a
radial excitation is very similar.
TABLE VIII: The decay widths of Ξ+c (2980) with different D-wave assignments. Here all results are in units of MeV.
Assignment Ξ0cπ
+ Ξ′0c π
+ Ξ⋆0c π
+ Σ++c k
− Λ+c k̄
0 Remark
) 0.0 1.1 0.11 0.37 0.0 ×
) 0.0 0.12 × 10−2 0.67 0.11 × 10−3 0.0 ×
Ξ′c1(
) 4.4 0.72 0.18 0.25 5.3
Ξ′c1(
) 4.4 0.18 0.46 0.062 5.3
Ξ′c2(
) 0.0 0.16 0.17 0.56 0.0 ×
Ξ′c2(
) 0.0 0.47 × 10−2 1.0 0.71 × 10−4 0.0 ×
Ξ′c3(
) 0.054 0.53 × 10−2 0.14 × 10−2 0.82 × 10−4 0.053 ×
Ξ′c3(
) 0.054 0.30 × 10−2 0.19 × 10−2 0.46 × 10−4 0.053 ×
Ξ̂c2(
) 0.0 9.5 6.1 0.61 0.0
Ξ̂c2(
) 0.0 9.5 6.1 0.61 0.0
Ξ̂′c1(
) 74 6.3 1.0 0.40 78 ×
Ξ̂′c1(
) 74 1.6 2.5 0.10 78 ×
Ξ̂′c2(
) 0.0 14 4.5 0.91 0.0
Ξ̂′c2(
) 0.0 6.3 7.1 0.40 0.0
Ξ̂′c3(
) 48 7.2 2.9 0.46 50 ×
Ξ̂′c3(
) 48 4.1 3.9 0.26 50 ×
) 0.0 0.30 1.4 1.3 0.0 ×
Ξ̌0c1(
) 1.0 0.40 0.46 1.7 0.46 ×
Ξ̌0c1(
) 1.0 0.10 1.2 0.43 0.46 ×
) 0.0 18 4.4 5.5 0.0
) 0.0 4.5 11 1.4 0.0
Ξ̌1c0(
) 0.0 18 18 5.5 0.0
Ξ̌1c1(
) 62 9.1 2.2 2.8 72 ×
Ξ̌1c1(
) 62 2.3 5.5 0.69 72 ×
Ξ̌1c2(
) 0.0 11 1.1 0.34 0.0 ×
Ξ̌1c2(
) 0.0 0.0 6.6 0.0 0.0 ×
) 0.0 5.6 1.8 2.4 0.0
) 0.0 1.7 4.32 0.24 0.0
Ξ̌2c1(
) 19 3.7 1.1 1.6 23
Ξ̌2c1(
) 19 0.93 2.6 0.40 23
Ξ̌2c2(
) 0.0 8.4 1.7 0.36 0.0
Ξ̌2c2(
) 0.0 1.2 6.0 0.16 0.0
Ξ̌2c3(
) 8.1 1.3 60 0.19 8.7 ×
Ξ̌2c3(
) 8.1 0.75 0.81 0.10 8.7
TABLE IX: The decay widths of Ξ+c (3077) with different D-wave assignments. Here all results are in units of MeV.
Assignment Ξ0cπ
+ Ξ′0c π
+ Ξ⋆0c π
+ Σ++c k
− Σ++c k
− Λ+c k̄
0 D+Λ Remark
) 0.0 2.1 0.30 0.73 0.054 0.0 0.0
) 0.0 0.037 1.7 0.42 × 10−2 0.32 0.0 0.0
Ξ′c1(
) 7.0 1.4 0.46 0.49 0.089 4.4 3.2
Ξ′c1(
) 7.0 0.36 1.1 0.12 0.22 4.4 3.2
Ξ′c2(
) 0.0 3.2 0.43 1.1 0.081 0.0 0.0
Ξ′c2(
) 0.0 0.025 × 10−2 2.5 0.28 × 10−2 0.48 0.0 0.0 ×
Ξ′c3(
) 0.19 0.029 0.012 0.32 × 10−2 0.32× 10−3 0.12 0.026 ×
Ξ′c3(
) 0.19 0.016 × 10−3 0.016 0.18 × 10−2 0.44× 10−3 0.12 0.026 ×
Ξ̂c2(
) 0.0 34 29 6.0 2.0 0.0 0.0 ×
Ξ̂c2(
) 0.0 34 29 6.0 2.0 0.0 0.0 ×
Ξ̂′c1(
) 201 23 4.8 4.0 0.33 130 38 ×
Ξ̂′c1(
) 201 5.7 12 1.0 0.83 130 38 ×
Ξ̂′c2(
) 0.0 51 22 8.9 1.5 0.0 0.0 ×
Ξ̂′c2(
) 0.0 23 34 4.0 2.3 0.0 0.0 ×
Ξ̂′c3(
) 129 26 14 4.5 0.94 84 25 ×
Ξ̂′c3(
) 129 15 19 2.6 0.13 84 25 ×
) 0.0 0.69 0.13 0.29 1.2 0.0 0.0
Ξ̌0c1(
) 15 0.92 0.044 0.39 0.38 11 0.64× 10−3 ×
Ξ̌0c1(
) 15 0.23 0.11 0.096 0.96 11 0.64× 10−3 ×
) 0.0 39 12 12 0.21 0.0 0.0 ×
) 0.0 9.9 30 3.0 5.2 0.0 0.0 ×
Ξ̌1c0(
) 0.0 39 47 12 8.3 0.0 0.0 ×
Ξ̌1c1(
) 110 20 5.9 6.1 1.0 69 42 ×
Ξ̌1c1(
) 110 5.0 15 1.5 2.6 69 42 ×
Ξ̌1c2(
) 0.0 25 3.0 7.6 0.52 0.0 0.0 ×
Ξ̌1c2(
) 0.0 0.0 18 0.0 3.1 0.0 0.0 ×
) 0.0 9.2 6.0 3.9 0.75 0.0 0.0
) 0.0 5.8 10 1.1 2.1 0.0 0.0
Ξ̌2c1(
) 22 6.1 2.3 2.6 0.54 14 15 ×
Ξ̌2c1(
) 22 1.5 5.6 0.64 1.3 14 15 ×
Ξ̌2c2(
) 0.0 14 5.2 5.8 0.77 0.0 0.0 ×
Ξ̌2c2(
) 0.0 3.9 14 0.74 3.0 0.0 0.0 ×
Ξ̌2c3(
) 21 4.4 2.5 0.85 0.23 14 4.3 ×
Ξ̌2c3(
) 21 2.5 3.4 0.48 0.31 14 4.3 ×
Introduction
The notations and conventions of charmed baryon
The 3P0 model
The strong decays of charmed baryon
Numerical results
Discussion and conclusion
Appendix
The harmonic oscillator wave functions used in our calculation
The momentum space integration
Acknowledgments
References
|
0704.0076 | CP violation in beauty decays | October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
International Journal of Modern Physics A
c© World Scientific Publishing Company
CP VIOLATION IN BEAUTY DECAYS∗
MICHAEL GRONAU
Physics Department, Technion – Israel Institute of Technology
32000 Haifa, Israel
[email protected]
Precision tests of the Kobayashi-Maskawa model of CP violation are discussed, pointing
out possible signatures for other sources of CP violation and for new flavor-changing
operators. The current status of the most accurate tests is summarized.
Keywords: CP violation; CKM; New Physics.
PACS Nos.: 13.25.Hw, 11.30.Er, 12.15.Ji, 14.40.Nd
1. Introduction
It took thirty-seven years from the discovery of a tiny CP violating effect of order
10−3 inKL → π+π− 1 to a first observation of a breakdown of CP symmetry outside
the strange meson system. A large CP asymmetry of order one between rates of
initial B0 and B̄0 decays to J/ψKS was measured in summer 2001 by the Babar and
Belle Collaborations.2 A sizable however smaller asymmetry had been anticipated
twenty years earlier 3 in the framework of the Kobayashi-Maskawa (KM) model
of CP violation,4 in the absence of crucial information on b quark couplings. The
asymmetry was observed in a time-dependent measurement as suggested,5 thanks
to the long B0 lifetime and the large B0-B̄0 mixing.6 The measured asymmetry,
fixing (in the standard phase convention7) the sine of the phase 2β (≡ 2φ1) ≡
2arg(VtbV
td) of the top-quark dominated B
0-B̄0 mixing amplitude, was found to
be in good agreement with other determinations of Cabibbo-Kobayasi-Maskawa
(CKM) parameters,8,9 including a recent precise measurement of Bs-B̄s mixing.
This showed that the CKM phase γ (≡ φ3) ≡ arg(V ∗ub), which seems to be unable
to account for the observed cosmological baryon asymmetry,11 is the dominant
source of CP violation in flavor-changing processes. With this confirmation the next
pressing question became whether small contributions beyond the CKM framework
occur in CP violating flavor-changing processes, and whether such effects can be
observed in beauty decays.
One way of answering this question is by over-constraining the CKM unitarity
triangle through precise CP conserving measurements related to the lengths of the
∗Based partially on review talks given at recent conferences.
http://arxiv.org/abs/0704.0076v2
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
2 M. Gronau
sides of the triangle. An alternative and more direct way, focusing on the origin
of CP violation in the CKM framework, is to measure β and γ in a variety of B
decay modes. Different values obtained from asymmetries in several processes, or
values different from those imposed by other constraints, could provide clues for
new sources of CP violation and for new flavor-changing interactions. Such phases
and interactions occur in the low energy effective Hamiltonian of extensions of the
Standard Model (SM) including models based on supersymmetry.12
In this presentation we will focus on the latter approach based primarily on
CP asymmetries, using also complementary information on hadronic B decay rates
which are expected to be related to each other in the CKM framework. In the
next section we outline several of the most relevant processes and the theoretical
tools applied for their studies, quoting numerous papers where these ideas have
been originally proposed and where more details can be found.13 Sections 3, 4 and
5 describe a number of methods in some detail, summarizing at the end of each
section the current experimental situation. Section 6 discusses several tests for NP
effects, while Section 7 concludes.
2. Processes, methods and New Physics effects
Whereas testing the KM origin of CP violation in most hadronic B decays requires
separating strong and weak interaction effects, in a few “golden modes” CP asym-
metries are unaffected by strong interactions. For instance, the decay B0 → J/ψKS
is dominated by a single tree-level quark transition b̄ → c̄cs̄, up to a correction
smaller than a fraction of a percent.14,15,16,17 Thus, the asymmetries measured
in this process and in other decays dominated by b̄ → c̄cs̄ have already provided a
rather precise measurement of sin 2β,18,19,20
sin 2β = 0.678± 0.025 . (1)
This value permits two solutions for β at 21.3◦ and at 68.7◦. Time-dependent an-
gular studies of B0 → J/ψK∗0,21 and time-dependent Dalitz analyses of B0 →
Dh0 (D → KSπ+π−, h0 = π0, η, ω)22 measuring cos 2β > 0 have excluded the
second solution at a high confidence level, implying
β = (21.3± 1.0)◦ . (2)
Since B0 → J/ψKS proceeds through a CKM-favored quark transition, contribu-
tions to the decay amplitude from physics at a higher scale are expected to be
very small, potentially identifiable by a tiny direct asymmetry in this process or in
B+ → J/ψK+.23
Another process where the determination of a weak phase is not affected
by strong interactions is B+ → DK+, proceeding through tree-level amplitudes
b̄ → c̄us̄ and b̄ → ūcs̄. The interference of these two amplitudes, from D̄0 and D0
which can always decay to a common hadronic final state, leads to decay rates and
a CP asymmetry which measure very cleanly the relative phase γ between these
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 3
amplitudes.24,25 The trick here lies in recognizing the measurements which yield
this fundamental CP-violating quantity. Physics beyond the SM is expected to have
a negligible effect on this determination of γ which relies on the interference of two
tree amplitudes.
B decays into pairs of charmless mesons, such as B → ππ (or B → ρρ) and
B → Kπ (or B → K∗ρ), involve contributions of both tree and penguin ampli-
tudes which carry different weak and strong phases.14,26,27 Contrary to the case
of B → DK, the determination of β and γ using CP asymmetries in charmless B
decays involves two correlated aspects which must be considered: its dependence
on strong interaction dynamics and its sensitivity to potential New Physics (NP)
effects. This sensitivity follows from the CKM and loop suppression of penguin am-
plitudes, implying that new heavy particles at the TeV mass range, replacing the
W boson and the top-quark in the penguin loop, may have sizable effects.28. In
order to claim evidence for physics beyond the SM from a determination of β and
γ in these processes one must handle first the question of dynamics. There are two
approaches for treating the dynamics of charmless hadronic B decays:
(1) Study systematically strong interaction effects in the framework of QCD.
(2) Identify by symmetry observables which do not depend on QCD dynamics.
The first approach faces the difficulty of having to treat precisely long distance
effects of QCD including final state interactions. Remarkable theoretical progress
has been made recently in proving a leading-order (in 1/mb) factorization formula
for these amplitudes in a heavy quark effective theory approach to perturbative
QCD.29,30,31 However, there remain differences between ways of treating in differ-
ent approaches power counting, the scale of Wilson coefficients, end-point quark dis-
tribution functions of light mesons, and nonperturbative contributions from charm
loops.32 Also, the nonperturbative input parameters in these calculations involve
non-negligible uncertainties. These parameters include heavy-to-light form factors
at small momentum transfer, light-cone distribution amplitudes, and the average
inverse momentum fraction of the spectator quark in the B meson. The resulting
inaccuracies in calculating magnitudes and strong phases of amplitudes prohibit a
precise determination of γ from measured decay rates and CP asymmetries. Also,
the calculated rates and asymmetries cannot provide a clear case for physics be-
yond the SM in cases where the results of a calculation deviate slightly from the
measurements.
In the second approach one applies isospin symmetry to obtain relations among
several decay amplitudes. For instance, using the distinct behavior under isospin of
tree and penguin operators contributing in B → ππ, a judicious choice of observ-
ables permits a determination of γ or α (≡ φ2) = π − β − γ. 33 The same analysis
applies in B decays to pairs of longitudinally polarized ρ mesons. In case that an
observable related to the subdominant penguin amplitude is not measured with
sufficient precision, it may be replaced in the analysis by a CKM-enhanced SU(3)-
related observable, in which a large theoretical uncertainty is translated to a small
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
4 M. Gronau
error in γ. The precision of this method is increased by including contributions of
higher order electroweak penguin amplitudes, which are related by isospin to tree
amplitudes.34,35 With sufficient statistics one should also take into account isospin-
breaking corrections of order (md−mu)/ΛQCD ∼ 0.02,36,37 and an effect caused by
the ρ meson width.38 A similar analysis proposed for extracting γ in B → Kπ 39,40
requires using flavor SU(3) instead of isospin for relating electroweak penguin con-
tributions and tree amplitudes.35,41 While flavor SU(3) is usually assumed to be
broken by corrections of order (ms − md)/ΛQCD ∼ 0.3, in this particular case a
rather precise recipe for SU(3) breaking is provided by QCD factorization, reducing
the theoretical uncertainty in γ to only a few degrees.42
Charmless B decays, which are sensitive to physics beyond the SM 28, provide a
rich laboratory for studying various signatures of NP. A large variety of theories have
been studied in this context, including supersymmetric models, models involving
tree-level flavor-changing Z or Z ′ couplings, models with anomalous three-gauge-
boson couplings and other models involving an enhanced chromomagnetic dipole
operator.43,44 The following effects have been studied and will be discussed in
Section 6 in a model-independent manner:
(1) Within the SM, the three values of γ extracted from B → ππ, B → Kπ and
B+ → DK+ are equal. As we will explain, these three values are expected to be
different in extensions of the SM involving new low energy four-fermion operators
behaving as ∆I = 3/2 in B → ππ and as ∆I = 1 in B → Kπ.
(2) Other signatures of anomalously large ∆I = 1 operators contributing to
B → Kπ are violations of isospin sum rules, holding in the SM for both decay rates
and CP asymmetries in these decays.45,46,47
(3) Time-dependent asymmetries in B0 → π0KS , B0 → φKS and B0 → η′KS
and in other b → s penguin-dominated decays may differ substantially from the
asymmetry sin 2β sin∆mt, predicted approximately in the SM.26,43,48 Significant
deviations are expected in models involving anomalous |∆S| = 1 operators behaving
as ∆I = 0 or ∆I = 1.
(4) An interesting question, which may provide a clue to the underlying New
Physics once deviations from SM predictions are observed, is how to diagnose the
value of ∆I in NP operators contributing to |∆S| = 1 charmless B decays. We will
discuss an answer to this question which has been proposed recently.49
3. Determining γ in B → DK
In this section we will discuss in some length a rather rich and very precise method
for determining γ in processes of the form B → D(∗)K(∗), which uses both charged
and neutral B mesons and a large variety of final states. It is based on a broad idea
that any coherent admixture of a state involving a D̄0 from b̄ → c̄us̄ and a state
with D0 from b̄ → ūcs̄ can decay to a common final state.24,25 The interference
between the two channels, B → D(∗)0K(∗), D0 → fD and B → D̄(∗)0K(∗), D̄0 →
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 5
fD, involves the weak phase difference γ, which may be determined with a high
theoretical precision using a suitable choice of measurements. Effects of D0-D̄0
mixing are negligible.50 While some of these processes are statistically limited,
combining them together is expected to reduce the experimental error in γ. In
addition to (quasi) two-body B decays, the D or D∗ in the final state may be
accompanied by any multi-body final state with quantum numbers of a kaon.25
Each process in this large class of neutral and charged B decays is characterized
by two pairs of parameters, describing complex ratios of amplitudes for D0 and D̄0
for the two steps of the decay chain (we use a convention rB , rf ≥ 0, 0 ≤ δB, δf <
A(B → D(∗)0K(∗))
A(B → D̄(∗)0K(∗))
= rBe
i(δB+γ) ,
A(D0 → fD)
A(D̄0 → fD)
= rfe
iδf . (3)
In three-body decays ofB andD mesons, such asB → DKπ andD → Kππ, the two
pairs of parameters (rB , δB) and (rf , δf ) are actually functions of two corresponding
Dalitz variables describing the kinematics of the above three-body decays. The
sensitivity of determining γ depends on rB and rf because this determination relies
on an interference of D0 and D̄0 amplitudes. For D decay modes with rf ∼ 1 (see
discussion below) the sensitivity increases with the magnitude of rB.
For each of the eight sub-classes of processes, B+,0 → D(∗)K(∗)+,0, one may
study a variety of final states in neutral D decays. The states fD may be divided
into four families, distinguished qualitatively by their parameters (rf , δf ) defined in
Eq. (3):
(1) fD = CP-eigenstate
24,25,51 (K+K−,KSπ
0, etc.); rf = 1, δf = 0, π.
(2) fD = flavorless but non-CP state
52 (K+K∗−,K∗+K−, etc.); rf = O(1).
(3) fD = flavor state
53 (K+π−,K+π−π0, etc.); rf ∼ tan2 θc.
(4) fD = 3-body self-conjugate state
54 (KSπ
+π−); rf , δf vary across the Dalitz
plane.
In the first family, CP-odd states occur in Cabibbo-favored D0 and D̄0 decays,
while CP-even states occur in singly Cabibbo-suppressed decays. The second family
of states occurs in singly Cabibbo-suppressed decays, the third family occurs in
Cabibbo-favored D̄0 decays and in doubly Cabibbo-suppressed D0 decays, while
the last state is formally a Cabibbo-favored mode for both D0 and D̄0.
The parameters rB and δB in B → D(∗)K(∗) depend on whether the B meson
is charged or neutral, and may differ for K vs K∗,55 and for D vs D∗, where a
neutral D∗ can be observed in D∗ → Dπ0 or D∗ → Dγ.56 The ratio rB involves
a CKM factor |VubVcs/VcbVus| ≃ 0.4 in both B+ and B0 decays, and a color-
suppression factor in B+ decays, while in B0 decays both b̄ → c̄us̄ and b̄ → ūcs̄
amplitudes are color-suppressed. A rough estimate of the color-suppression factor in
these decays may be obtained from the color-suppression measured in corresponding
CKM-favored decays, B → Dπ,D∗π,Dρ,D∗ρ, where the suppression is found to
be in the range 0.3 − 0.5.57 Thus, one expects rB(B0) ∼ 0.4, rB(B+) = (0.3 −
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
6 M. Gronau
0.5)rB(B
0) in all the processes B+,0 → D(∗)K(∗)+,0. We note that three-body B+
decays, such as B+ → D0K+π0, are not color-suppressed, making these processes
advantageous by their potentially large value of rB which varies in phase space.
58,59
The above comparison of rB(B
+) and rB(B
0) may be quantified more precisely
by expressing the four ratios rB(B
0)/rB(B
+) in B → D(∗)K(∗) in terms of recip-
rocal ratios of known magnitudes of amplitudes:60
0 → D(∗)K(∗)0)
rB(B+ → D(∗)K(∗)+)
B+ → D̄(∗)0K(∗)+)
B0 → D̄(∗)0K(∗)0)
. (4)
This follows from an approximation,
A(B0 → D(∗)0K(∗)0) ≃ A(B+ → D(∗)0K(∗)+) , (5)
where the B0 and B+ processes are related to each other by replacing a spectator
d quark by a u quark. While formally Eq. (5) is not an isospin prediction, it may
be obtained using an isospin triangle relation,61
A(B0 → D(∗)0K(∗)0) = A(B+ → D(∗)0K(∗)+) +A(B+ → D(∗)+K(∗)0), (6)
and neglecting the second amplitude on the right-hand-side which is “pure
annihilation”.62 This amplitude is expected to be suppressed by a factor of four or
five relative to the other two amplitudes appearing in (6) which are color-suppressed.
Evidence for this kind of suppression is provided by corresponding ratios of CKM-
favored amplitudes,57 |A(B0 → D−s K+)/
2A(D̄0π0)| = 0.23 ± 0.03, |A(B0 →
D∗−s K
2A(D̄∗0π0)| < 0.24.
Applying Eq. (4) to measured branching ratios,57,63 one finds
rB(B+)
B → DK B → DK∗ B → D∗K B → D∗K∗
2.9± 0.4 3.7± 0.3 > 2.2 > 3.0 (7)
This agrees with values of rB(B
0) near 0.4 and rB(B
+) between 0.1 and 0.2. Note
that in spite of the expected larger values of rB in B
0 decays, from the point
of view of statistics alone (without considering the question of flavor tagging and
the efficiency of detecting a KS in B
0 → D(∗)K0), B+ and B0 decays may fare
comparably when studying γ. This follows from (5) because the statistical error on
γ scales roughly as the inverse of the smallest of the two interfering amplitudes.
We will now discuss the actual manner by which γ can be determined using
separately three of the above-mentioned families of final states fD. We will men-
tion advantages and disadvantages in each case. For illustration of the method we
will consider B+ → fDK+. We will also summarize the current status of these
measurements in all eight decay modes B+,0 → D(∗)K(∗)+,0.
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CP violation in beauty decays 7
3.1. fD = CP-eigenstates
One considers four observables consisting of two charge-averaged decay rates for
even and odd CP states, normalized by the decay rate into a D0 flavor state,
RCP± ≡
Γ(DCP±K
−) + Γ(DCP±K
Γ(D0K−)
, (8)
and two CP asymmetries for even and odd CP states,
ACP± ≡
Γ(DCP±K
−)− Γ(DCP±K+)
Γ(DCP±K−) + Γ(DCP±K+)
. (9)
In order to avoid dependence of RCP± on errors in D
0 and DCP branching ratio
measurements one uses a definition of RCP± in terms of ratios of B decay branching
ratios intoDK andDπ final states.59 The four observablesRCP± and ACP± provide
three independent equations for rB, δB and γ,
RCP± = 1 + r
B ± 2rB cos δB cos γ , (10)
ACP± = ±2rB sin δB sin γ/RCP± . (11)
While in principle this is the simplest and most precise method for extracting
γ, up to a discrete ambiguity, in practice this method is sensitive to r2B , because
(RCP+ + RCP−)/2 = 1 + r
B . This becomes very difficult for charged B decays
where one expects rB ∼ 0.1− 0.2, but may be feasible for neutral B decays where
rB ∼ 0.4. An obvious signature for a non-zero value of rB would be observing a
difference between RCP+ and RCP− which is linear in this quantity.
Studies of B+ → DCPK+, B+ → DCPK∗+ and B+ → D∗CPK+ have been car-
ried out recently,64,65,66 each consisting of a few tens of events. A nonzero difference
RCP+ −RCP− at 2.6 standard deviations, measured in B+ → DCPK∗+,64 is prob-
ably a statistical fluctuation. A larger difference is anticipated in B0 → DCPK∗0,
as the value of rB in this process is expected to be three or four times larger than
in B+ → DK∗+. [See Eq. (7).] Higher statistics is required for a measurement of γ
using this method.
3.2. fD = flavor state
Consider a flavor state fD in Cabibbo-favored D̄
0 decays, accessible also to doubly
Cabibbo-suppressed D0 decays, such that one has rf ∼ tan2 θc in Eq. (3). One
studies the ratio of two charge-averaged decay rates, for decays into f̄DK and fDK,
Γ(fDK
−) + Γ(f̄DK
Γ(f̄DK−) + Γ(fDK+)
, (12)
and the CP asymmetry,
Γ(fDK
−)− Γ(f̄DK+)
Γ(fDK−) + Γ(f̄DK+)
. (13)
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8 M. Gronau
These observables are given by
Rf = r
B + r
f + 2rB rf cos(δB − δf ) cos γ , (14)
Af = 2r rf sin(δB − δf ) sin γ/Rf , (15)
where a multiplicative correction 1 +O(rBrf ) ∼ 1.01 has been neglected in (14).
These two observables involve three unknowns, rB , δB − δf and γ. One assumes
rf to be given by the measured ratio of doubly Cabibbo-suppressed and Cabibbo-
favored branching ratios. Thus, one needs at least two flavor states, fD and f
for which two pairs of observables (Rf , Af ) and (Rf ′ , Af ′) provide four equations
for the four unknowns, rB, δB − δf , δB − δf ′ , γ. The strong phase differences δf , δf ′
can actually be measured at a ψ′′ charm factory,67 thereby reducing the number of
unknowns to three.
While the decay rate in the numerator of Rf is rather low, the asymmetry Af
may be large for small values of rB around 0.1, as it involves two amplitudes with
a relative magnitude rf/rB.
So far, only upper bounds have been measured for Rf implying upper limits
on rB in several processes, rB(B
+ → DK+) < 0.2,68,69,70 rB(B+ → D∗K+) <
0.2,68 r(B+ → DK∗+) < 0.4,71 and rB(B0 → DK∗0) < 0.4.63,72 Further con-
straints on rB in the first three processes have been obtained by studying D decays
into CP-eigenstates and into the state KSπ
+π−. Using rB(B
0 → DK∗0)/rB(B+ →
DK∗+) = 3.7 ± 0.3 in (7) and assuming that rB(B+ → DK∗+) is not smaller
than about 0.1, one may conclude that a nonzero measurement of rB(B
0 → DK∗0)
should be measured soon. The signature for B0 → D0K∗0 events would be two
kaons with opposite charges.
3.3. fD = KSπ
The amplitude for B+ → (KSπ+π−)DK+ is a function of the two invariant-mass
variables, m2
≡ (pKS + pπ±)2, and may be written as
A(B+ → (KSπ+π−)DK+) = f(m2+,m2−) + rBei(δB+γ)f(m2−,m2+) . (16)
In B− decay one replaces m+ ↔ m−, γ → −γ. The function f may be written as a
sum of about twenty resonant and nonresonant contributions modeled to describe
the amplitude for flavor-tagged D̄0 → KSπ+π− which is measured separately.73,74
This introduces a model-dependent uncertainty in the analysis. Using the measured
function f as an input and fitting the rates for B± → (KSπ+π−)DK± to the
parameters, rB , δB and γ, one then determines these three parameters.
The advantage of using D → KSπ+π− decays over CP and flavor states is
being Cabibbo-favored and involving regions in phase space with a potentially large
interference between D0 and D̄0 decay amplitudes. The main disadvantage is the
uncertainty introduced by modeling the function f .
Two recent analyses of comparable statistics by Belle and Babar, combining
B± → DK±, B± → D∗K± and B± → DK∗±, obtained values 73 γ = [53+15
−18 ± 3±
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CP violation in beauty decays 9
9(model)]◦ and γ = [92±41±11±12(model)]◦.74 [This second value does not use the
process B+ → D(KSπ+)K∗ , also studied by the same group,75.] The larger errors in
the second analysis are correlated with smaller values of the extracted parameters rB
in comparison with those extracted in the first study. The model-dependent errors
may be reduced by studying at CLEO-c the decays DCP± → KSπ+π−, providing
further information on strong phases in D decays.67
Conclusion: The currently most precise value of γ is γ = [53+15
−18 ± 3± 9(model)]◦,
obtained from B± → D(∗)K(∗)± using D → KSπ+π−. These errors may be reduced
in the future by combining the study of all D decay modes in B+,0 → D(∗)K(∗)+,0.
The decay B0 → DK∗0 seems to carry a high potential because of its expected
large value of rB. Decays B
0 → D(∗)K0 may also turn useful, as they have been
shown to provide information on γ without the need for flavor tagging of the initial
B0.60,76
4. The currently most precise determination of γ: B → ππ, ρρ, ρπ
4.1. B → ππ
The amplitude for B0 → π+π− contains two terms, conventionally denoted “tree”
(T ) and “penguin” (P ) amplitudes, 14,26 involving a weak CP-violating phase γ
and a strong CP-conserving phase δ, respectively:
A(B0 → π+π−) = |T |eiγ + |P |eiδ . (17)
Time-dependent decay rates, for an initial B0 or a B
, are given by
Γ(B0(t)/B
(t) → π+π−) = e−ΓtΓπ+π− [1± C+− cos∆mt∓ S+− sin∆mt] , (18)
where
S+− =
2Im(λππ)
1 + |λππ|2
, C+− =
1− |λππ |2
1 + |λππ |2
, λππ ≡ e−2iβ
0 → π+π−)
A(B0 → π+π−)
. (19)
One has14
S+− = sin 2α+ 2|P/T | cos2α sin(β + α) cos δ +O(|P/T |2) ,
C+− = 2|P/T | sin(β + α) sin δ +O(|P/T |2) . (20)
This tells us two things:
(1) The deviation of S+− from sin 2α and the magnitude of C+− increase with
|P/T |, which can be estimated to be |P/T | ∼ 0.5 by comparing B → ππ rates with
penguin-dominated B → Kπ rates.77
(2) Γπ+π− , S+− and C+− are insufficient for determining |T |, |P |, δ and γ (or α).
Further information on these quantities may be obtained by applying isospin sym-
metry to all B → ππ decays.
In order to carry out an isospin analysis,33 one uses the fact that the three
physical B → ππ decay amplitudes and the three B̄ → ππ decay amplitudes,
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10 M. Gronau
depending each on two isospin amplitudes, obey triangle relations of the form,
A(B0 → π+π−)/
2 +A(B0 → π0π0)−A(B+ → π+π0) = 0 . (21)
Furthermore, the penguin amplitude is pure ∆I = 1/2; hence the ∆I = 3/2 am-
plitude carries a week phase γ, A(B+ → π+π0) = e2iγA(B− → π−π0). Defin-
ing sin 2αeff ≡ S+−/(1 − C2+−)1/2, the difference αeff − α is then determined by
an angle between corresponding sides of the two isospin triangles sharing a com-
mon base, |A(B+ → π+π0)| = |A(B− → π−π0)|. A sign ambiguity in αeff − α is
resolved by two model-independent features which are confirmed experimentally,
|P |/|T | ≤ 1, |δ| ≤ π/2. This implies α < αeff .78
Table I. Branching ratios and CP asymmetries in B → ππ, B → ρρ.
Decay mode Branching ratio (10−6) ACP = −C S
B0 → π+π− 5.16± 0.22 0.38 ± 0.07 −0.61± 0.08
B+ → π+π0 5.7± 0.4 0.04 ± 0.05
B0 → π0π0 1.31± 0.21 0.36
+0.33
−0.31
B0 → ρ+ρ− 23.1
0.11 ± 0.13 −0.06± 0.18
B+ → ρ+ρ0 18.2± 3.0 −0.08± 0.13
B0 → ρ0ρ0 1.07± 0.38
Current CP-averaged branching ratios and CP asymmetries for B → ππ and
B → ρρ decays are given in Table I,20 where ACP ≡ −C for decays to CP eigen-
states. An impressive experimentally progress has been achieved in the past two
years in extracting a precise value for αeff , αeff = (110.6
−3.2)
◦. However, the er-
ror on αeff − α using the isospin triangles is still large. An upper bound, given by
CP-averaged rates and a direct CP asymmetry in B0 → π+π−,79,80
cos 2(αeff − α) ≥
Γ+− + Γ+0 − Γ00
)2 − Γ+−Γ+0
Γ+−Γ+0
1− C2+−
, (22)
leads to 0 < αeff − α < 31◦ at 1σ. Adding in quadrature the error in αeff and the
uncertainty in α−αeff , this implies α = (95± 16)◦ or γ = (64± 16)◦ by . A similar
central value but a smaller error, α = (97± 11)◦, has been reported recently by the
Belle Collaboration.81 The possibility that a penguin amplitude in B0 → π+π−
may lead to a large CP asymmetry S for values of α near 90◦ where sin 2α = 0 was
anticipated fifteen years ago.82
The bound on αeff − α may be improved considerably by measuring a nonzero
direct CP asymmetry in B0 → π0π0. This asymmetry can be shown to be large and
positive (see Eq. (46) in Sec. 5.2), implying a large rate for B̄0 but a small rate for
B0. Namely, the triangle (21) is expected to be squashed, while the B̄ triangle is
roughly equal sided.
An alternative way of treating the penguin amplitude in B0 → π+π− is by
combining within flavor SU(3) the decay rate and asymmetries in this process with
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CP violation in beauty decays 11
rates and asymmetries in B0 → K0π+ or B0 → K+π−.77 The ratio of ∆S = 1 and
∆S = 0 tree amplitudes in these processes, excluding CKM factors, is taken to be
given by fK/fπ assuming factorization, while the ratio of corresponding penguin
amplitudes is allowed to vary by ±0.22 around one. A current update of this rather
conservative analysis obtains 83
γ = (73± 4+10
◦ , (23)
where the first error is experimental, while the second one is due to an uncertainty
in SU(3) breaking. A discussion of SU(3) breaking factors relating B0 → π+π− and
B0 → K+π− is included in Section 5.2.
4.2. B → ρρ
Angular analyses of the pions in ρ decays have shown that B0 → ρ+ρ− is dominated
almost 100% by longitudinal polarization 20. This simplifies the isospin analysis of
CP asymmetries in these decays to becoming similar to B0 → π+π−. The advantage
of B → ρρ over B → ππ is the relative small value of (
ρ0ρ0) in comparison with
ρ+ρ−) and (
ρ+ρ0) (see Table I), indicating a smaller |P/T | in B → ρ+ρ− (|P/T | <
0.3 8) than in B0 → π+π− (|P/T | ∼ 0.5 77). Eq. (22) leads to an upper bound on
αeff − α in B → ρρ, 0 < αeff − α < 17◦ (at 1σ). The asymmetries for longitudinal
ρ’s given in Table I imply αeff = (91.7
−5.2)
◦. Thus, one finds α = (83 ± 10)◦ or
γ = (76± 10)◦ by adding errors in quadrature.
A stronger bound on |P/T | in B0 → ρ+ρ−, leading to a more precise value of γ,
may be obtained by relating this process to B+ → K∗0ρ+ within flavor SU(3). 84
One uses the branching ratio and fraction of longitudinal rate measured for this
process 20, (
K∗0ρ+) = (9.2 ± 1.5) × 10−6, fL(K∗0ρ+) = 0.48 ± 0.08, to normalize
the penguin amplitude in B0 → ρ+ρ−. Including a conservative uncertainty from
SU(3) breaking and smaller amplitudes, one finds a value
γ = (71.4+5.8
−1.7)
◦ , (24)
where the first error is experimental and the second one theoretical.
The current small theoretical error in γ requires including isospin breaking effects
in studies based on isospin symmetry. The effect of electroweak penguin amplitudes
on the isospin analyses of B → ππ and B → ρρ has been calculated and was found
to move γ slightly higher by an amount ∆γEWP = 1.5
◦.34,35 Other corrections,
relevant to methods using π0 and ρ0, includng π0-η-η′ mixing, ρ-ω mixing, and a
small I = 1 ρρ contribution allowed by the ρ-width, are each smaller than one
degree.36,37,38
Conclusion: Taking an average of the two values of γ in (23) and (24) obtained
from B0 → π+π− and B0 → ρ+ρ−, and including the above-mentioned EWP
correction, one finds
γ = (73.5± 5.7)◦ . (25)
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12 M. Gronau
A third method of measuring γ (or α) in time-dependent Dalitz analyses of B0 →
(ρπ)0 involves a much larger error,85 and has a small effect on the overall averaged
value of the weak phase. We note that sin γ is close to one and its relative error
is only 3%, the same as the relative error in sin 2β and slightly smaller than the
relative error in sinβ.
5. Rates, asymmetries, and γ in B → Kπ
5.1. Extracting γ in B → Kπ
The four decays B0 → K+π−, B0 → K0π0, B+ → K0π+, B+ → K+π0 involve
a potential for extracting γ, provided that one is sensitive to interference between
a dominant isoscalar penguin amplitude and a small tree amplitude contributing
to these processes. This idea has led to numerous suggestions for determining γ in
these decays starting with a proposal made in 1994.86,87 An interference between
penguin and tree amplitudes may be identified in two ways:
(1) Two different properly normalized B → Kπ rates.
(2) Nonzero direct CP asymmetries.
Table II. Branching ratios and asymmetries in B → Kπ.
Decay mode Branching ratio (10−6) ACP
B0 → K+π− 19.4± 0.6 −0.097± 0.012
B+ → K+π0 12.8± 0.6 0.047± 0.026
B+ → K0π+ 23.1± 1.0 0.009± 0.025
B0 → K0π0 10.0± 0.6 −0.12± 0.11
Current branching ratios and CP asymmetries are summarized in Table II.20 Three
ratios of rates, calculated using the ratio of B+ and B0 lifetimes, τ+/τ0 = 1.076±
0.008,20 are:
R ≡ Γ(B
0 → K+π−)
Γ(B+ → K0π+)
= 0.90± 0.05 ,
2Γ(B+ → K+π0)
Γ(B+ → K0π+)
= 1.11± 0.07 ,
Γ(B0 → K+π−)
2Γ(B0 → K0π0) = 0.97± 0.07 . (26)
The largest deviation from one, observed in the ratio R at 2σ, is insufficient for
claiming unambiguous evidence for a non-penguin contribution. An upper limit,
R < 0.965 at 90% confidence level, would imply γ ≤ 79◦ using sin2 γ ≤ R,88
which neglects however “color-suppressed” EWP contributions.89 As we will argue
now, these contributions and “color-suppressed” tree amplitudes are actually not
suppressed as naively expected.
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CP violation in beauty decays 13
The nonzero asymmetry measured in B0 → K+π− provides first evidence for
an interference between penguin (P ′) and tree (T ′) amplitudes with a nonzero rel-
ative strong phase. Such an interference occurs also in B+ → K+π0 where no
asymmetry has been observed. An assumption that other contributions to the lat-
ter asymmetry are negligible has raised some questions about the validity of the
CKM framework. In fact, a color-suppressed tree amplitude (C′), also occurring in
B+ → K+π0,86 resolves this “puzzle” if this amplitude is comparable in magnitude
to T ′. Indeed, several studies have shown that this is the case,90,91,92,93,94 also im-
plying that color-suppressed and color-favored EWP amplitudes are of comparable
magnitudes.35 For consistency between the two CP asymmetries in B0 → K+π−
and B+ → K+π0, the strong phase difference between C′ and T ′ must be negative
and cannot be very small.95 This seems to stand in contrast to QCD calculations
using a factorization theorem.29,31,94
The small asymmetry ACP (B
+ → K+π0) implies bounds on the sine of the
strong phase difference δc between T
′ +C′ and P ′. The cosine of this phase affects
Rc − 1 involving the decay rates for B+ → K0π+ and B+ → K0π+. A question
studied recently is whether the two upper bounds on | sin δc| and | cos δc| are con-
sistent with each other or, perhaps, indicate effects of NP. Consistency was shown
by proving a sum rule involving ACP (B
+ → K+π0) and Rc − 1, in which an elec-
troweak penguin (EWP) amplitude plays an important role. We will now present a
proof of the sum rule, which may provide important information on γ.95
The two amplitudes for B+ → K0π+,K+π0 are given in terms of topological
contributions including P ′, T ′ and C′,
A(B+ → K0π+) = (P ′ − 1
P ′cEW ) +A
A(B+ → K+π0) = (P ′ − 1
P ′cEW ) + (T
′ + P ′cEW ) + (C
′ + P ′EW ) +A
′ , (27)
where P ′EW and P
EW are color-favored and color-suppressed EWP contributions.
The small annihilation amplitude A′ and a small u quark contribution to P ′ involv-
ing a CKM factor V ∗ubVus will be neglected (|V ∗ubVus|/|V ∗cbVcs| = 0.02). Evidence for
the smallness of these terms can be found in the small CP asymmetry measured for
B+ → K0π+. Large terms would require rescattering and a sizable strong phase
difference between these terms and P ′.
Flavor SU(3) symmetry relates ∆I = 1, I(Kπ) = 3/2 electroweak penguin and
tree amplitudes through a calculable ratio δEW
35,41,
T ′ + C′ + P ′EW + P
EW = (T
′ + C′)(1 − δEW e−iγ) ,
δEW = −
c9 + c10
c1 + c2
|V ∗tbVts|
|V ∗ubVus|
= 0.60± 0.05 . (28)
The error in δEW is dominated by the current uncertainty in |Vub|/|Vcb| = 0.104±
0.007 57, including also a smaller error from SU(3) breaking estimated using QCD
factorization. Eqs. (27) and (28) imply 96
Rc = 1− 2rc cos δc(cos γ − δEW) + r2c (1− 2δEW cos γ + δ2EW) , (29)
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14 M. Gronau
ACP (B
+ → K+π0) = −2rc sin δc sin γ/Rc , (30)
where rc ≡ |T ′ + C′|/|P ′ − 13P
EW | and δc is the strong phase difference between
T ′ + C′ and P ′ − 1
P ′cEW .
The parameter rc is calculable in terms of measured decay rates, using bro-
ken flavor SU(3) which relates T ′ + C′ and T + C dominating B+ → π+π0 by a
factorization factor fK/fπ (neglecting a tiny EWP term in B
+ → π+π0),87
|T ′ + C′| =
|A(B+ → π+π0)| . (31)
Using branching ratios from Tables I and II, one finds
B+ → π+π0)
B+ → K0π+)
= 0.198± 0.008 . (32)
The error in rc does not include an uncertainty from assuming factorization for
SU(3) breaking in T ′ + C′. While this assumption should hold well for T ′, it may
not be a good approximation for C′ which as we have mentioned is comparable in
magnitude to T ′ and carries a strong phase relative to it. Thus one should allow a
10% theoretical error when using factorization for relating B → Kπ and B → ππ
T + C amplitudes, so that
rc = 0.20± 0.01 (exp)± 0.02 (th) . (33)
Eliminating δc in Eqs. (29) and (30) by retaining terms which are linear in rc,
one finds
Rc − 1
cos γ − δEW
ACP (B
+ → K+π0)
sin γ
= (2rc)
2 +O(r3c ) . (34)
This sum rule implies that at least one of the two terms whose squares occur on
the left-hand-side must be sizable, of the order of 2rc = 0.4. The second term,
|ACP (B+ → K+π0)|/ sin γ, is already smaller than ≃ 0.1, using the current 2σ
bounds on γ and |ACP (B+ → K+π0)|. Thus, the first term must provide a dominant
contribution. For Rc ≃ 1, this implies γ ≃ arccos δEW ≃ (53.1± 3.5)◦. This range is
expanded by including errors in Rc and ACP (B
+ → K+π0). For instance, an upper
bound Rc < 1.1 would imply an inportant upper limit, γ < 70
◦. Currently one only
obtains an upper limit γ ≤ 88◦ at 90% confidence level.95 This bound is consistent
with the value obtained in (25) from B → ππ and B → ρρ, but is not competitive
with the latter precision.
Conclusion: The current constraint obtained from Rc and ACP (B
+ → K+π0) is
γ ≤ 88◦ at 90% confidence level. Further improvement in the measurement of Rc
(which may, in fact, be very close to one) is required in order to achieve a precision
in γ comparable to that obtained in B → ππ, ρρ. (A conclusion concerning the
different CP asymmetries measured in B0 → K+π− and B+ → K+π0 will be given
at the end of the next subsection.)
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CP violation in beauty decays 15
5.2. Symmetry relations for B → Kπ rates and asymmetries
The following two features imply rather precise sum rules in the CKM framework,
both for B → Kπ decay rates and CP asymmetries:
(1) The dominant penguin amplitude is ∆I = 0.
(2) The four decay amplitudes obey a linear isospin relation,39
A(K+π−)−A(K0π+)−
2A(K+π0) +
2A(K0π0) . (35)
An immediate consequence of these features are two isospin sum rules, which
hold up to terms which are quadratic in small ratios of non-penguin to penguin
amplitudes,45,46,47
Γ(K+π−) + Γ(K0π+) = 2Γ(K+π0) + 2Γ(K0π0) , (36)
∆(K+π−) + ∆(K0π+) = 2∆(K+π0) + 2∆(K0π0) , (37)
where
∆(Kπ) ≡ Γ(B̄ → K̄π̄)− Γ(B → Kπ) . (38)
Quadratic corrections to (36) have been calculated in the SM and were found to
be a few percent.97,98,99 This is the level expected in general for isospin-breaking
corrections which must therefore also be considered. The above two features imply
that these ∆I = 1 corrections are suppressed by a small ratio of non-penguin to
penguin amplitudes and are therefore negligible.100 Indeed, this sum rule holds
experimentally within a 5% error.101 One expects the other sum rule (37) to hold
at a similar precision.
The CP rate asymmetry sum rule (37), relating the four CP asymmetries, leads
to a prediction for the asymmetry in B0 → K0π0 in terms of the other three
asymmetries which have been measured with higher precision,
ACP (B
0 → K0π0) = −0.140± 0.043 . (39)
While this value is consistent with experiment (see Table II), higher accuracy in
this asymmetry measurement is required for testing this straightforward prediction.
Relations between CP asymmetries in B → Kπ and B → ππ following from
approximate flavor SU(3) symmetry of QCD 102 are not expected to hold as pre-
cisely as isospin relations, but may still be interesting and useful. An important
question relevant to such relations is how to include SU(3)-breaking effects, which
are expected to be at a level of 20-30%. Here we wish to discuss two SU(3) rela-
tions proposed twelve years ago,103,104 one of which holds experimentally within
expectation, providing some lesson about SU(3) breaking, while the other has a an
interesting implication for future applications of the isospin analysis in B → ππ.
A most convenient proof of SU(3) relations is based on using a diagramatic
approach, in which diagrams with given flavor topologies replace reduced SU(3)
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16 M. Gronau
matrix elements.86 In this language, the amplitudes for B0 decays into pairs of
charged or neutral pions, and pairs of charged or neutral π and K, are given by:
−A(B0 → π+π−) = T + (P + 2P cEW /3) + E + PA ,
2A(B0 → π0π0) = C − (P − PEW − P cEW /3)− E − PA ,
−A(B0 → K+π−) = T ′ + (P ′ + 2P ′cEW /3) ,
2A(B0 → K0π0) = C′ − (P ′ − P ′EW − P ′cEW /3) . (40)
The combination E + PA, representing exchange and penguin annihilation topolo-
gies, is expected to be 1/mb-suppressed relative to T and C,
31,62 as demonstrated
by the small branching ratio measured for B0 → K+K−.20 This term will be
neglected.
Expressing topological amplitudes in terms of CKM factors, SU(3)-invariant
amplitudes and SU(3) invariant strong phases, one may write
T ≡ V ∗ubVud|T + Puc| , P + 2P cEW /3 ≡ V ∗tbVtd|Ptc|eiδ ,
T ′ ≡ V ∗ubVus|T + Puc| , P ′ + 2P ′cEW /3 ≡ V ∗tbVts|Ptc|eiδ , (41)
C ≡ V ∗ubVud|C − Puc| , P − PEW − P cEW /3 ≡ V ∗tbVtd|P̃tc|eiδ̃ ,
C′ ≡ V ∗ubVus|C − Puc| , P ′ − P ′EW − P ′cEW /3 ≡ V ∗tbVts|P̃tc|eiδ̃ .
Unitarity of the CKM matrix, V ∗cbVcd(s) = −V ∗tbVtd(s) − V ∗ubVud(s), has been used
to absorb in T (
′) and C(
′) a penguin term Puc ≡ Pu − Pc multiplying V ∗ubVud(s),
while Ptc ≡ Pt − Pc and P̃tc ≡ P̃t − P̃c contain two distinct combinations of EWP
contributions. Using the identity
Im (V ∗ubVudVtbV
td) = −Im (V ∗ubVusVtbV ∗ts) , (42)
one finds103,104
∆(B0 → K+π−) = −∆(B0 → π+π−) (43)
∆(B0 → K0π0) = −∆(B0 → π0π0) , (44)
where ∆ is the CP rate difference defined in (38).
Quoting products of branching ratios and asymmetries from Tables I and II,
Eq. (43) reads
− 1.88± 0.24 = −1.96± 0.37 . (45)
This SU(3) relation works well and requires no SU(3)-breaking. An SU(3) breaking
factor fK/fπ in T but not in P , or in both T and P , are currently excluded at a
level of 1.0σ, or 1.75σ. More precise CP asymmetry measurements in B0 → K+π−
and B0 → π+π− are required for determining the pattern of SU(3) breaking in tree
and penguin amplitudes.
Using the prediction (39) of the B → Kπ asymmetry sum rule, Eq. (44) predicts
ACP (B
0 → π0π0) = 1.07± 0.38 . (46)
The error is dominated by current errors in CP asymmetries for B+ → K0π+
and B+ → K+π0, and to a less extent by the error in (
π0π0). SU(3) breaking in
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CP violation in beauty decays 17
amplitudes could modify this prediction by a factor fπ/fK if this factor applies to
C, and less likely by (fπ/fK)2. A large positive CP asymmetry, favored in all three
cases, will affect future applications of the isospin analysis in B → ππ. It implies
that while the B̄ isospin triangle is roughly equal-sided, the B triangle is squashed.
A twofold ambiguity in the value of γ disappears in the limit of a flat B triangle.24
Conclusion: The isospin sum rule for B → Kπ decay rates holds well, while the
CP asymmetry sum rule predicts ACP (B
0 → K0π0) = −0.140±0.043. The different
asymmetries in B0 → K+π− and B+ → K+π0 can be explained by an amplitude
C′ comparable to T ′ and involving a relative negative strong phase, and should
not be considered a “puzzle”. An SU(3) relation for B0 → ππ and B0 → Kπ CP
asymmetries works well for charged modes. The corresponding relation for neutral
modes predicts a large positive asymmetry in B0 → π0π0. Improving asymmetry
measurements can provide tests for SU(3) breaking factors.
6. Tests for small New Physics effects
6.1. Values of γ
We have described three ways for extracting a value for γ relying on interference
of distinct pairs of quark amplitudes, (b → cūs, b → uc̄s), (b → cc̄s, b → uūs) and
(b → cc̄d, b → uūd). The three pairs provide a specific pattern for CP violation in
the CKM framework, which is expected to be violated in many extensions of the
SM. The rather precise value of γ (25) extracted from B → ππ, ρρ, ρπ is consistent
with constraints on γ from CP conserving measurements related to the sides of the
unitarity triangle.8,9 The values of γ obtained in B → D(∗)K(∗) and B → Kπ
are consistent with those extracted in B → ππ, ρρ, ρπ, but are not yet sufficiently
precise for testing small NP effects in charmless B decays. Further experimental
improvements are required, in particular in the former two types of processes.
While the value of γ in B → D(∗)K∗) is not expected to be affected by NP,
the other two classes of processes involving penguin loops are susceptible to such
effects. The extraction of γ in B → ππ ρρ assumes that γ is the phase of a ∆I =
3/2 tree amplitude, while an additional ∆I = 3/2 EWP contribution is included
using isospin. The extracted value could be modified by a new ∆I = 3/2 effective
operator originating in physics beyond the SM, but not by a new ∆I = 1/2 operator.
Similarly, the value of γ extracted in B → Kπ is affected by a potential new ∆I = 1
operator, but not by a new ∆I = 0 operator, because the amplitude (28), playing
an essential role in this method, is pure ∆I = 1.
6.2. B → Kπ sum rule
Charmless |∆S| = 1 B and Bs decays are particularly sensitive to NP effects, as
new heavy particles at the TeV mass range may replace the the W boson and top-
quark in the penguin loop dominating these amplitudes.28 The sum rule (36) for
B → Kπ decay rates provides a test for such effects. However, as we have argued
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
18 M. Gronau
from isospin considerations, it is only affected by quadratic ∆I = 1 amplitudes
including NP contributions. Small NP amplitudes, contributing quadratically to
the sum rule, cannot be separated from SM corrections, which are by themselves at
a level of a few percent. This is the level to which the sum rule has already been
tested. We will argue below for evidence showing that potential NP contributions
to |∆S| = 1 charmless decays must be suppressed by roughly an order of magnitude
relative to the dominant b→ s penguin amplitudes.
6.3. Values of S,C in |∆S| = 1 B0 → fCP decays
A class of b → s penguin-dominated B0 decays to CP-eigenstates has recently at-
tracted considerable attention. This includes final statesXKS andXKL, whereX =
φ, π0, η′, ω, f0, ρ
0,K+K−,KSKS , π
0π0, for which measured asymmetries −ηCPS
and C are quoted in Table III. [The asymmetries S and C = −ACP were de-
fined in (18) for B0 → π+π−. Observed modes with KL in the final states obey
ηCP (XKL) = −ηCP (XKS).] In these processes, a value S = −ηCP sin 2β (for states
Table III. Asymmetries S and C in B0 → XKS .
X φ π0 η′ ω f0(980)
−ηCP S 0.39± 0.18 0.33± 0.21 0.61± 0.07 0.48± 0.24 0.42 ± 0.17
C 0.01± 0.13 0.12± 0.11 −0.09± 0.06 −0.21± 0.19 −0.02± 0.13
X ρ0 K+K− KSKS π
−ηCP S 0.20± 0.57 0.58
+0.18
−0.13
0.58± 0.20 −0.72± 0.71
C 0.64± 0.46 0.15± 0.09 −0.14± 0.15 0.23± 0.54
with CP-eigenvalue ηCP ) is expected approximately.
26,43 These predictions involve
hadronic uncertainties at a level of several percent, of order λ2, λ ∼ 0.2. It has been
pointed out some time ago105 that it is difficult to separate these hadronic uncer-
tainties within the SM from NP contributions to decay amplitudes if the latter are
small. In the next subsection we will discuss indirect experimental evidence showing
that NP contributions to S and C must be small. Corrections to S = −ηCP sin 2β
and values for the asymmetries C have been calculated in the SM using methods
based on QCD factorization106,107 and flavor SU(3),90,108,109 and were found to
be between a few percent up to above ten percent within hadronic uncertainties.
Whereas the deviation of S from −ηCP sin 2β is process-dependent, a generic
result has been proven a long time ago for both S and C, to first order in |c/p|,14
∆S ≡ −ηCPS − sin 2β = 2
cos 2β sin γ cos∆ ,
C = 2
sin γ sin∆ . (47)
Here p and c are penguin and color-suppressed tree amplitudes involving a small ra-
tio and relative weak and strong phases γ and ∆, respectively. This implies ∆S > 0
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CP violation in beauty decays 19
for |∆| < π/2, which can be argued for several of the above decays using QCD
arguments106,107 or SU(3) fits.109 (Note that while |p| is measurable in certain
decay rates up to first order corrections, |c| and ∆ involve sizable hadronic uncertain-
ties in QCD calculations.) In contrast to this expectation, the central values mea-
sured for ∆S are negative for all decays. (See Table III.) Consequently, one finds an
averaged value sin 2βeff = 0.53±0.05,20 to be compared with sin 2β = 0.678±0.025.
Two measurements which seem particularly interesting are−ηCPSφKS = 0.39±0.18,
where a positive correction of a few percent to sin 2β is expected in the SM,106,107
and −ηCPSπ0KS = 0.33± 0.21, where a rather large positive correction to sin 2β is
expected shifting this asymmetry to a value just above 0.8.90
While the current averaged value of sin 2βeff is tantalizing, experimental errors
in S and C must be reduced further to make a clear case for physics beyond the
SM. Assuming that the discrepancy between improved measurements and calcu-
lated values of S and C persists beyond theoretical uncertainties, can this pro-
vide a clue to the underlying New Physics? Since many models could give rise to
a discrepancy,28,43,44 one would seek signatures characterizing classes of models
rather than studying the effects in specific models. One way of classifying extensions
of the SM is by the isospin behavior of the new effective operators contributing to
b→ sqq̄ transitions.
6.4. Diagnosis of ∆I for New Physics operators
Four-quark operators in the effective Hamiltonian associated with NP in b → sqq̄
transitions can be either isoscalar or isovector operators. We will now discuss a study
proposed recently in order to isolate ∆I = 0 or ∆I = 1 operators, thus determining
corresponding NP amplitudes and CP violating phases.49 We will show that since S
and C in the above processes combine ∆I = 0 or ∆I = 1 contributions, separating
these contributions requires using also information from other two asymmetries,
which are provided by isospin-reflected decay processes.
Two |∆S| = 1 charmless B (or Bs) decay processes, related by isospin reflection,
RI : u ↔ d, ū ↔ −d̄, can always be expressed in term of common ∆I = 0 and
∆I = 1 amplitudes B and A in the form:
A(B+ → f) = B +A , A(B0 → RIf) = ±(B −A) . (48)
A proof of this relation uses a sign change of Clebsch-Gordan coefficients underm↔
−m.49 The description (48) applies, in particular, to pairs of processes involving
all the B0 decay modes listed in Table III, and B+ decay modes where final states
are obtained by isospin reflection from corresponding B0 decay modes. Decay rates
for pairs of isospin-reflected B decay processes, and for B̄ decays to corresponding
charge conjugate final states are therefore given by (we omit inessential common
kinematic factors),
Γ+ = |B +A|2 , Γ0 = |B − A|2 ,
Γ− = |B̄ + Ā|2 , Γ0̄ = |B̄ − Ā|2 . (49)
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20 M. Gronau
The amplitudes B̄ and Ā are related to B and A by a change in sign of all weak
phases, whereas strong phases are left unchanged.
For each pair of processes one defines four asymmetries: an isospin-dependent
CP-conserving asymmetry,
Γ+ + Γ− − Γ0 − Γ0̄
Γ+ + Γ− + Γ0 + Γ0̄
, (50)
two CP-violating asymmetries for B+ and B0,
A+CP ≡
Γ− − Γ+
Γ− + Γ+
, − C ≡ A0CP ≡
Γ0̄ − Γ0
Γ0̄ + Γ0
, (51)
and the time-dependent asymmetry S in B0 decays,
1 + |λ|2
, λ ≡ ηCP
B̄ − Ā
e−2iβ , (52)
In the Standard Model, the isoscalar amplitude B contains a dominant penguin
contribution, BP , with a CKM factor V
cbVcs. The residual isoscalar amplitude,
∆B ≡ B −BP , (53)
and the amplitude A, consist each of contributions smaller than BP by about an
order of magnitude.29,30,31,32,86 These contributions include terms with a much
smaller CKM factor V ∗ubVus, and a higher order electroweak penguin amplitude with
CKM factor V ∗tbVts. Thus, one expects
|∆B| ≪ |BP | , |A| ≪ |BP | . (54)
Consequently, the asymmetries AI , A
CP and ∆S are expected to be small, of or-
der 2|A|/|B| and 2|∆B|/|BP |. In contrast, potentially large contributions to ∆B
and A from NP, comparable to BP , would most likely lead to large asymmetries
of order one. An unlikely exception is the case when both ∆B/BP and A/BP are
purely imaginary, or almost purely imaginary. This would require very special cir-
cumstances such as fine-tuning in specific models. Excluding cancellations between
NP and SM contributions in both CP-conserving and CP violating asymmetries,
tests for the hierarchy (54) become tests for the smallness of corresponding potential
NP contributions to B and A.
There exists ample experimental information showing that asymmetries A+CP
are small in processes related by isospin reflection to the decay modes in Table III.
Upper limits on the magnitudes of most asymmetries are at a level of ten or fifteen
percent [e.g., A+CP (K
+φ) = 0.034±0.044,A+CP (K+η′) = 0.031±0.026], while others
may be as large as twenty or thirty percent [A+CP (K
+ρ0) = 0.31+0.11
−0.10]. Similar values
have been measured for isospin asymmetries AI [e.g., AI(K
+φ) = −0.037± 0.077,
+η′) = −0.001± 0.033, AI(K+ρ0) = −0.16± 0.10].49 Since these two types
of asymmetries are of order 2|∆B|/|BP | and 2|A|/|BP |, this confirms the hierarchy
(54), which can be assumed to hold also in the presence of NP.
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CP violation in beauty decays 21
We will take by convention the dominant penguin amplitude BP to have a zero
weak phase and a zero strong phase, referring all other strong phases to it. Writing
B = BP +∆B , B̄ = BP +∆B̄ , (55)
and expanding the four asymmetries to leading order in ∆B/BP or A/BP , one has
∆S = cos 2β
Im(Ā−A)
− Im(∆B̄ −∆B)
, (56)
Re(Ā+A)
, (57)
A+CP =
Re(Ā−A)
Re(∆B̄ −∆B)
, (58)
A0CP = −
Re(Ā−A)
Re(∆B̄ −∆B)
. (59)
The four asymmetries provide the following information:
• The ∆I = 0 and ∆I = 1 contributions in CP asymmetries are separated
by taking sums and differences,
A∆I=0CP ≡
(A+CP +A
CP ) =
Re(∆B̄ −∆B)
, (60)
A∆I=1CP ≡
(A+CP −A
CP ) =
Re(Ā−A)
. (61)
• ReA/BP and ReĀ/BP may be separated by using information from A∆I=1CP
and AI .
• ∆S is governed by an imaginary part of a combination of ∆I = 0 and ∆I =
1 terms which cannot be separated in B decays. Such a separation is possible
in Bs decays to pairs of isospin-reflected decays, e.g. Bs → K+K−,KSKS
or Bs → K∗+K∗−,K∗0K̄∗0, where 2β in the definition of ∆S (47) is now
replaced by the small phase of Bs-B̄s mixing.
One may take one step further under the assumption that strong phases as-
sociated with NP amplitudes are small relative to those of the SM and can be
neglected.110 This assumption, which must be confronted by data, is reasonable
because rescattering from a leading b → scc̄ amplitude is likely the main source
of strong phases, while rescattering from a smaller b → sqq̄ NP amplitude is then
a second-order effect. In the convention (55), where the strong phase of BP is set
equal to zero, ∆B and A have the same CP-conserving strong phase δ, and involve
CP-violating phases φB and φA, respectively,
∆B = |∆B|eiδeiφB , A = |A|eiδeiφA . (62)
Since the four asymmetries (56)-(59) are first order in small ratios of amplitudes,
one may take BP in their expression to be given by the square root of Γ+ or Γ0,
thereby neglecting second order terms. These four observables can then be shown to
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22 M. Gronau
determine |A|, φA and |∆B| sinφB .49 The combination |∆B| cosφB adds coherently
to BP and cannot be fixed independently.
The amplitudes ∆B and A consist of process-dependent SM and potential NP
contributions. Assuming that the former are calculable, either using methods based
on QCD-factorization or by fitting within flavor SU(3) these and other B decay
rates and asymmetries, the four asymmetries determine the magnitude and CP
violating phase of a ∆I = 1 NP amplitude and the imaginary part of a ∆I = 0 NP
amplitude. In certain cases, e.g., B → φK or B → η′KS , stringent upper bounds on
SM contributions to ∆B and Amay suffice if some of the four measured asymmetries
are larger than permitted by these bounds. In the pair B+ → K+π0, B0 → K0π0,
the four measured asymmetries [using the predicted value (39)] are AI = 0.087 ±
0.038, A∆I=0CP = −0.047± 0.025, A∆I=1CP = 0.094± 0.025,∆S = −0.35± 0.21. Some
reduction of errors is required for a useful implementation of this method.
Conclusion: There exists ample experimental evidence in pairs of isospin-reflected
b → s penguin-dominated decays that potential NP amplitudes must be small.
Assuming that these amplitudes involve negligible strong phases, and assuming that
small SM non-penguin contributions are calculable or can be strictly bounded, one
may determine the magnitude and CP violating phase of a NP ∆I = 1 amplitude,
and the imaginary part of a NP ∆I = 0 amplitude in each pair of isospin-reflected
decays.
6.5. Null or nearly-null tests
We have not discussed null tests of the CKM framework.111 Evidence for physics
beyond the Standard Model may show-up as (small) nonzero asymmetries in pro-
cesses where they are predicted to be extremely small in the CKM framework. A
well-known example is B+ → π+π0, where the CP asymmetry is expected to be a
small fraction of a percent including EWP amplitudes.34,35 We have only discussed
exclusive hadronic B decays, where QCD calculations involve hadronic uncertain-
ties. A more robust calculation exists for the direct CP asymmetry in inclusive
radiative decays B → Xsγ, found to be smaller than one percent.112 The current
upper limit on this asymmetry is at least an order of magnitude larger.113
Time-dependent asymmetries in radiative decays B0 → KSπ0γ, for a KSπ0
invariant-mass in the K∗ region and for a larger invariant-mass range including
this region, are interesting because they test the photon helicity, predicted to be
dominantly right-handed in B0 decays and left-handed in B̄0 decays.105,114 The
asymmetry, suppressed by ms/mb, is expected to be several percent in the SM,
and can be very large in extensions where spin-flip is allowed in b → sγ. While
dimensional arguments seem to indicate a possible larger asymmetry in the SM,
of order ΛQCD/mb ∼ 10%,115 calculations using perturbative QCD116 and QCD
factorization117 find asymmetries of a few percent. The current averaged values,
for the K∗ region and for a larger invariant-mass range including this region, are
S((KSπ
0)K∗γ) = −0.28 ± 0.26 and S(KSπ0γ) = −0.09 ± 0.24.20,118 These mea-
October 27, 2018 17:34 WSPC/INSTRUCTION FILE CP-review
CP violation in beauty decays 23
surements must be improved in order to become sensitive to the level predicted in
the SM, or to provide evidence for physics beyond the SM.
7. Summary
The Standard Model passed with great success numerous tests in the flavor sector,
including a variety of measurements of CP asymmetries related to the CKM phases
β and γ. Small potential New Physics corrections may occur in ∆S = 0 and |∆S| =
1 penguin amplitudes, affecting the extraction of γ and modifying CP-violating
and isospin-dependent asymmetries in |∆S| = 1 B0 decays and isospin-related B+
decays. Higher precision than achieved so far is required for claiming evidence for
such effects and for sorting out their isospin structure.
Similar studies can be performed with Bs mesons produced at hadron colliders
and at e+e− colliders running at the Υ(5S) resonance. Time-dependence in Bs →
D−s K
+ and Bs → J/ψφ or Bs → J/ψη measures γ and the small phase of the
Bs-B̄s mixing amplitude.
119 Comparing time-dependence and angular analysis in
Bs → J/ψφ with b → s penguin-dominated processes including Bs → φφ,Bs →
K∗+K∗−, Bs → K∗0K̄∗0 provides a methodic search for potential NP effects. Work
on Bs decays has just begun at the Tevatron.
120 One is looking forward to first
results from the LHC.
Acknowledgments
I am grateful to numerous collaborators, in particular to Jonathan Rosner whose
collaboration continued without interruption for many years. This work was sup-
ported in part by the Israel Science Foundation under Grant No. 1052/04 and by
the German-Israeli Foundation under Grant No. I-781-55.14/2003.
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|
0704.0077 | Universal Forces and the Dark Energy Problem | arXiv:0704.0077v1 [physics.gen-ph] 1 Apr 2007
UNIVERSAL FORCES AND THE DARK
ENERGY PROBLEM
AFSAR ABBAS
Centre for Theoretical Physics
JMI, Jamia Nagar, New Delhi - 110025, India
(e-mail : [email protected])
Abstract
The Dark Energy problem is forcing us to re-examine our models and our
understanding of relativity and space-time. Here a novel idea of Fundamen-
tal Forces is introduced. This allows us to perceive the General Theory of
Relativity and Einstein’s Equation from a new pesrpective. In addition to
providing us with an improved understanding of space and time, it will be
shown how it leads to a resolution of the Dark Energy problem.
http://arxiv.org/abs/0704.0077v1
Dark Energy is certainly the most puzzling problem in physics and as-
tronomy today [1]. All kind of proposals to solve the problem are being put
forward, but we are nowhere near a resolution of the issues involved. In this
paper the author suggests incorporation of a new concept in our understand-
ing of Nature and which helps in explaining the Dark Energy problem and
has the potential of providing us with an improved understanding of Nature.
The new concept is called ”Universal Force”. It was first proposed by
philosopher-scientist Hans Reichenbach [2]. Although this novel concept was
actually suggested in what may be termed as the philosophical context, here
the author would like to emphasize as to how the same can be used as a
powerful tool in the physical context as well. Actually we shall see that
we have to improve upon the original idea of Reichenbach [3] in a significant
manner to be able to use it in physics and astronomy. As to what changes are
necessary would be discussed below after the original idea of the Universal
Forces of Reichenbach is introduced.
To the sceptics who may believe that philosophy of science has no rele-
vance to the actual science itself, it may be pointed out that one never loses
by being open to ideas from any quarters whatsoever. Obviously the only
criteria would be relevance and proper applicability of the idea in whatever
science one is talking about. A recent Conference Proceeding [4] would attest
to this fact. In the case of the Dark Energy conundrum, wherein we do not
even know where we stand, this point becomes all the more pertinent.
Within the discipline of philosophy of science the Universal Force concept
of Reichenbach has had mixed reactions. While a few philosophers have been
supportive of the concept [5,6,7], some others have been critical of it [8,9,10].
We shall not delve too deep into the philosophical issues of the same, as
that would take us farther away from our main purpose here and which is
to see how to properly utilize the concept as scientists. Hence we shall use
only those points as would be found to be relevant for our purpose here. It
would suffice to mention here as to what Rudolf Carnap has stated in the
Introduction of Reichenbach’s book [3]. He called the concept as ” ... of great
interest for the methodology of physics but what has so far not received the
attention it deserves”. In this paper we shall try to rectify for this failure
of appreciating the concept of the Universal Force - albeit in a somewhat
altered and improved form.
Reichenbach defines two kind of forces - Differential Forces and Univer-
sal Forces. It may be pointed out that the term ”force” here should not be
taken strictly as defined in physics but in a broad and general framework.
In fact Carnap has suggested that the term ”effect” instead of ”force’ would
better serve the purpose [5] and which allows it be used in different frame-
works. Hence to conform with the accepted practice, though in this paper
we shall continue to use the term ”Universal Force” the reader may do well
to remember that what we really mean is ”Universal Effect”.
One calls a force Differetial if it acts differently on different substances. It
is called Universal if it is quantitatively the same for all the substances [3,5].
If we heat a rod of initial length l0 from initial temperature T0 to tempetature
T then its length is given as
l = l0[1 + β(T − T0)] (1)
where β the coefficient for thermal expansion is different for different
materials. Hence this is a Differential Force. Now the correction factor due
to the influence of gravitation on the length of the rod is
l = l0[1− C
φ] (2)
Here the rod is placed at a distance r from sun whose mass is m and φ is
the angle of the rod with respect to the the line sun to rod. C is a universal
constant ( in CGS unit C= 3.7 x 10−29 ). As this acts in the same manner
for any material of mass m, gravity is a Universal Force as per the above
definition.
Reichenbach also gives a general definition of the Universal Forces [3,p
12] as: (1) affecting all the materials in the same manner and (2) there are
no insulating walls against it. We saw above that gravity is such a force,
Indeed gravity is a Universal Force par excellance. It affects all matter
in the same manner. The equality of the gravitational and inertial masses is
what ensures this physically. If the gravitational and inertial masses were not
found to be equal, then one would not have been able to visualize of the paths
of freely falling mass points as geodesics in the four dimentional space-time.
In that case different geodesics would have resulted from different materials
of mass points [3].
Therefore the universal effect of gravitation on different kinds of measur-
ing instruments is to define a single geometry for all of them. Viewed this
way, one may say that gravity is geometerized. ”It is not theory of gravitation
that becomes geometry, but it is geometry that becomes the experience of the
gravitational field” [3, p 256]. Why does the planet follow the curved path?
Not because it is acted upon by a force but because the curved space-time
manifold leaves it with no other choice!
So as per Einstein’s theory of relativity, one does not speak of a change
produced by the gravitational field in the measuring instruments, but regard
the measuring instruments as free from any deforming forces. Gravity being a
Universal Force, in the Einstein’s Theory of Relativity, it basically disappears
and is replaced by geometry.
In fact Reichenbach [3, p 22] shows how one can give a consistent defi-
nition of a rigid rod - the same rigid rods which are needed in relativity to
measure all lengths. ”Rigid rods are solid bodies which are not affected by
Differential Forces, or concerning which the influence of Differential Forces
has been eliminated by corrections; Universal Forces are disregarded. We do
not neglect Universal Forces. We set them to zero by definition. Without
such a rule a rigid body cannot be defined.” In fact this rule also helps in
defining a closed system as well.
All this was formalized in terms of a theorem by Reichenbach [3, p 33]
THEOREM θ :
Given the geometry G0 to which the measuring instruments conform,
we can imagine a Universal Force F which affects the instruments in such
a way that the actual geometry is an arbitrary geometry G, while the ob-
served deviation from G is due to universal deformation of the measuring
instruments.”
G0 + F = G (3)
Hence only the combination G0+F is testable. As per Reichenbach’s prin-
ciple one prefers the theory wherein we put F=0. If we accept Reichenbach
principle of putting the Universal Force of gravity to zero, then the arbitrari-
ness in the choice of the measuring procedure is avoided and the question of
the geometrical structure of the physical space has a unique answer deter-
mined by physical measurement. It is this principle which Carnap praises
highly [5, p 171], ” Whenever there is a system of physics in which a certain
universal effect is asserted by a law that specifies under what conditions in
what amount the effect occurs, then the theory should be transformed so that
the amount of effect would be reduced to zero. This is what Einstein did
in regard to contraction and expansion of bodies in gravitational field.” The
left hand side of Einstein’s equation (below) gives the relevant non-Euclideon
geometry
Gµν = 8πG〈φ|Tµν|φ〉 (4)
In the case of gravity, and in as much as Einsteins’s Theory of Relativ-
ity has been well tested experimentally, we treat the above concept as well
placed empirically. But from this single success Reichenbach generalizes this
as a fundamental principle for all cases where Universal forces may arise. As
Carnap states [5, p 171], ” Whenever universal effects are found in physics,
Reichenbach maintained that it is always possible to eliminate them by suit-
able transformation of theory; such a transformation should be made because
of the overall simplicity that would result. This is a useful general principle,
deserving more attention than it has received. It applies not only to relativ-
ity theory, but also to situations that may arise in the future in which other
universal effects may be observed. Without the adoption of this rule there
is no way to give unique answer to the question - what is the structure of
space?”.
As such Reichenbach goes ahead and tries to apply this principle of elimi-
nation of Universal Forces to another universal effect that he finds and which
arises from considerations of topology ( as an additional consideration over
and above that of geometry ) of space-time of the universe.
The Theorem θ is limited to talking about the geometry of space-time
only. It does not take account of specific topological issues that may arise.
To take account of topology of the space-time we shall have to extend the
said theorem appropriately.
What would one experience if space had different topological properties.
To make the point home Reichenbach considers a torus-space [3, p 63]. This
is quite detailed and extensive. However for the purpose of simplifying the
and shortening the discussion here we shall talk of a two dimensional being
who lives on the surface of a sphere. His measurements tell him so. But in
spite of this he insists that he lives on a plane. He may actually do so as per
our discussion above if he confines himself to metrical relations only. With
an appropriate Universal Force he can he can justify living on a plane. But
the surface of a sphere is topologically different from that of a plane. On a
sphere if he starts at a point X and goes on a world tour he may come back
to the same point X. But this is impossible on a plane. And hence to account
for coming back to the ”same point” he has to maintain that on the plane
he actually has come back to a different point Y - which though is identical
to X in all other respects. One option for him is to accept that he is actually
living on a sphere. However if he still wants to maintain his position that he
is living on a plane then he has to explain as to how point Y is physically
identical to point X in spite of the fact that X and Y are different and distinct
points of space. Indeed he can do so by visualizing a fictitious force as an
effect of some kind of ”pre-established harmony” [3, p 65] by proposing that
everything that occurs at X also occurs at the point Y. As it would affect all
matter in the same manner this corresponds to a Universal Force/Effect as
per Reichenbach’s definition.
This interdependence of corresponding points which is essential in this
”pre-established” harmony cannot be interpreted as ordinary causality, as
it does not require ordinary time to transmit it and also does not spread
continuously through intervening space. Hence there is no mysterious causal
connection between the points X and point Y. Thus this necessarily entails
proposing a ”causal anomaly” [3, p 65]. In short connecting different topolo-
gies through a fictitious Universal Effect of ”pre-established harmony” neces-
sarly calls for introduction of ”causal anomalies”. Call this new hypothesize
Universal Force as A and the Theorem θ be extended to read
G0 + F + A = G (5)
where on the right had side we have given a different capital G which
reduces to G of the original Theorem θ when A is set equal to zero.
Now as per Reichenbach’s law of preferring that physical reality wherein
all Universal Forces are put to zero, he advocates of putting A to zero. He
pointed out that this has the advantage of retaining physical ”causality ”
in our science, This he takes as a success of his methodology. As per Re-
ichenbach [3, p 65] ” The principle of causality is one of its (physics) sacred
laws, which it will not abandon lightly; pre-established harmony, however is
incompatible with this law”.
However, as the said ’causal anomaly” is of topological origin we cannot
be sure in what manner it will manifest itself physically. In addition will not
the Universal Force/Effect of ”pre-established harmony” compensate for it in
some manner? So what one is saying is that it is possible that Reichenbach
was wrong in putting all Universal Forces to zero. It was OK to put F to
zero which justified the geometrical interpretation of gravity. But in the case
of this new topological Universal Force we really do not know enough and
let us not be governed by any theoretical prejudice and let the Nature decide
as to what is happening. So to say, let us look at modern cosmology to see
if it is throwing up any new Universal Forces which may be identified with
our ”pre-established harmony” here.
To understand this let us look at the Einstein’s Equation given above.
Harvey and Schucking [11] correcting for Einstein’s error in understanding
the role of the cosmological term λ have derived the most general equation
of motion to be
Gµν + λgµν = 8πG〈φ|Tµν|φ〉 (6)
They showed that [11] the Cosmological Constant λ above provides a
new repulsive force proportional to mass m, repelling every particle of mass
m with a force
F = mc2
x (7)
Recent data [1] on λ is what leads to the crisis of Dark Energy.
Quite clearly this repulsive force is a new Universal Force as per our
definition and hence conforms to the ”pre-established harmony” aspect of
the ”causal anomaly”. Thus we see that indeed as per the recent data on
accelerating universe we have stumbled upon this new Universal Force which
is of topological origin. Hence the source of dark energy is due to ”causal
anomaly” arising from the unique topological structure of our universe. This
solves the mystery of the origin of Dark Energy.
So we would like to emphasize that it is the accelerating universe ( and
hence the Dark Energy ) which is forcing us to accept the incorporation of
this ”causal anomaly” of topological origin. Implications of this new concept
in physics have now to be explored.
Note that as per Theorem θ when one puts F to be zero then one obtains
the proper non-Euclidean Geometry of Einstein’s equation. But now we
know that full structure is the sum of this non-Euclidean geometry plus A
, the new Universal Force ( as per the modified theorem above ) and this is
what the accelerating universe is forcing us to accept. This is what we called
capital G above. We feel that the DASI data on Ω0 being close to one and
thus showing that the Universe is flat [1] is consistent with capital G being
equal to G+ A. In principle just as per the original Theorem θ one may
add a Universal Force F to Einstein’s non-Euclidean geometry to obtain a
physically relevant Euclidean geometry, so in the same manner given a non-
Euclidean geometry of Einstein on can add an appropriate Universal Force
A to provide a flat universe. And this is exactly what capital G is telling us.
Thus the observed flatness of the universe may be treated as a success of the
new idea proposed here.
One would like to ask as to in what other manner incorporation of this
new ”causal anomaly” may help us in understanding Nature better? Will
it provide new perspectives as answers to quantum mechanical puzzles of
quantum jumps, non-locality etc. These are open questions to be tackled in
future.
REFERENCES
1. M S Turner, ”Making sense of the new cosmology”, Int J Mod Phys,
A17S1 (2002) 180-196
2. Hans Reichenbach (1891-1953) can properly be called a philosopher-
scientist. As a leading philosopher of science he was founder of the Berlin
Circle and a proponent of logical positivism. Among his teachers were David
Hilbert, Max Planck, Max Born and Albert Einstein. He wrote extensively
on the theory of probability, theory of relativity and quantum mechanics.
His philosophical writings have a definite scientific touch in them, very much
akin to that of Descartes, Leibniz and Huygens.
3. H Reichenbach, ”The philosophy of space and time”, Dover, New York
(1957) (Original German edition in 1928)
4. C Callender and N Huggett, ” Physics meets philosophy at the Planck
scale”, Cambridge University Press, UK (2001)
5. R Carnap, ”An introduction to the philosophy of science”, Basic Books,
New York (1966)
6. E Nagel, ”The structure of science”, Routledge and Kegan Paul, Lon-
don (1961)
7. D Dieks, ”Gravitation as a Universal Force”, Synthese, 73 (1987)
381-397
8. B Ellis, ”Universal and Differential Forces”, Brit J Phil Sc, 14 (1963)
177-194
9. A Gruenbaum, ”Philosophical problems of space and time”, Dordrecht,
Holland; D Reidel (1973) or Alfred A Knopf, New York (1963)
10. R Torretti, ”Relativity and geometry”, Pergamon Press (1983)
11. A Harvey and E Schucking, ”Einstein’s mistake and the cosmological
constant”, Am J Phys, 68 (2000) 723-727
|
0704.0078 | Linear perturbations of matched spacetimes: the gauge problem and
background symmetries | Linear perturbations of matched spacetimes: the
gauge problem and background symmetries
Marc Mars∗, Filipe C. Mena♭ and Raül Vera‡
∗Facultad de Ciencias, Universidad de Salamanca,
Plaza de la Merced s/n, 37008 Salamanca, Spain
♭Departamento de Matemática, Universidade do Minho,
Campus de Gualtar, 4710 Braga, Portugal
♭Mathematical Institute, University of Oxford,
24-29 St Giles’, Oxford OX1 3LB, UK
‡Fisika Teorikoaren Saila, Euskal Herriko
Unibertsitatea, Apt. 644, Bilbao 48080, Basque Country, Spain
June 12, 2018
Abstract
We present a critical review about the study of linear perturbations of matched
spacetimes including gauge problems. We analyse the freedom introduced in the
perturbed matching by the presence of background symmetries and revisit the par-
ticular case of spherically symmetry in n-dimensions. This analysis includes settings
with boundary layers such as brane world models and shell cosmologies.
PACS numbers: 0420, 0240
1 Introduction
An important aspect in any geometric gravitational theory is the analysis of how to match
two spacetimes. This is true in particular for General Relativity and its perturbation
theory. Despite the relevance and maturity of the matching theory one often finds papers
where the matching conditions are not properly used. Most of the difficulties arise from
the fact that the matching conditions are imposed in specific coordinate systems in a
manner which is not completely coordinate independent. More specifically, matching two
spacetimes requires identifying the boundaries pointwise, and sometimes this identification
is done implicitly by fixing spacetime coordinates, without paying enough attention to the
fact that solving the matching involves finding an identification of the boundary and that
this should not be fixed a priori.
In perturbation theory this problem also arises, and it gets complicated by the fact
that the fields to be matched (as the perturbed metric) are gauge dependent. So, in
addition to a priori choices of identifications of the boundary, there is also the problem
http://arxiv.org/abs/0704.0078v1
that particular gauges are often used. It may be argued that the matching theory must
be gauge independent and therefore it can be performed in any gauge. This is true, but
only when due care is taken to ensure that the choice of gauge does not restrict, a priori,
the perturbed identification of the boundaries.
A complete description of the linearized matching conditions has been achieved only
recently by Carter and Battye [5] and independently by Mukohyama [6]. To second order,
the matching conditions have been recently found in [7]. Despite these papers, we believe
that some confusion still lingers in the field, in particular with respect to the existing gauge
invariant formulations. The aim of this paper is to try to clarify these issues. In order
to do that, we will critically discuss some of the approaches proposed in the literature
trying to make clear which are the implicit assumptions made and to what extent are
they justified.
The first papers discussing the perturbed matching theory are, as far as we know,
the classic papers by Gerlach and Sengupta [2, 3]. However, as explained below, their
description of the perturbed matching theory contains imprecisions, and we will therefore
start discussing their approach pointing out the difficulties they encounter. A first attempt
to justify the claims in [2, 3] is due to Mart́ın-Garćıa and Gundlach [4], who propose a
different but nevertheless closely related set of linearized matching conditions. Pointing
out the implicit assumption made by these authors will also help us to try to explain the
subtleties inherent to the perturbed matching theory.
In [6] the linearized matching conditions are described for arbitrary backgrounds,
perturbations and matching hypersurfaces, and then applied to the case of two background
spacetimes with a high degree of symmetry, namely those which admit a maximal group of
isometries acting on codimension two spacelike submanifolds (e.g. spherically symmetric
spacetimes). In order to simplify the matching conditions, Mukohyama derives a set of
matching conditions for so-called doubly gauge invariants. However, a gap arises in his
final conclusions as the presented set of conditions for doubly gauge invariant quantities
for the linearized matching of spacetimes are only shown to be necessary conditions.
Analysing sufficiency touches directly on the issue we are trying to emphasize in this
paper, so we devote one section to clarify this point, where we show how these conditions
are, stricktly speaking, not sufficient. Since the matching conditions in terms of doubly
gauge invariants are widely used in the literature, we consider important to close this
gap. Moreover, the constructions of gauge invariant quantities using spherical harmonic
decompositons leaves out the l = 0 and l = 1 sectors. We will discuss this issue and its
consequences.
The paper is organized as follows. We start by summarising the perturbed matching
conditions in Section 2, where we also describe the gauge freedom involved. Then, the
procedures used in the classic papers [2, 3] together with the justifications and further
developments in [4] are reviewed in Section 3. Section 4 focuses on the consequences of
the existence of symmetries in the background configuration, which will have relevance
in our final discussion. Section 5 has three subsections. The first one is devoted to
present briefly the procedure and results discussed in [6] particularised to the case of
spherically symmetric backgrounds. In the second subsection we analyse the sufficiency
of the doubly gauge invariant matching conditions in [6]. The last subsection is devoted
to the study of the freedom left in the perturbation of the matching hypersurface once
the metric perturbations have been fixed at both sides. We finish with an appendix where
explicit expressions for the discontinuities of the perturbed second fundamental forms in
the spherical case are given. Some of these expressions are used in the main text.
2 Linearized matching
In this section we describe the gauge freedom involved in the linearised spacetime matching
and summarise the perturbed matching conditions.
2.1 Gauge freedom
The purpose of the matching theory is to construct a new spacetime out of two spacetimes
M± with boundary by finding a suitable diffeomorphism between the boundaries which
allows for their pointwise identification. In particular, the matched spacetime cannot
be thought to exist beforehand. Another aspect to bear in mind is that the matching
conditions involve exclusively tensors on the identified boundary Σ and hence any coordi-
nate system in M± is equally valid. This is well-known but it is still source of confusion
sometimes.
In perturbed matching theory, not only the metrics are perturbed but also the match-
ing hypersurfaces may be deformed. Furthermore, as for the metric, the “deviation” of the
matching hypersurface is also a gauge dependent quantity. This can be best understood
by viewing perturbations as ε-derivatives (at ε = 0) of a one-parameter family of space-
times (M+ε , g
ε ) with boundary Σ
ε . It is convenient to embed M
ε within a larger manifold
(without boundary) V +ε to clarify the discussion. A priori, the manifolds (M
ε , gε) are
completely distinct so it makes no direct sense to talk about ε-derivatives. It is necessary
to identify first the different manifolds so that a single point p refers to one point on each
of the manifolds. Obviously, there are infinite ways to identify the manifolds, all of them
equally valid a priori. This freedom leads to the gauge dependence of the perturbed met-
ric (and of any other geometrically defined tensor). The identification above may, or may
not, map the boundaries Σ+ε among themselves. A priori, a point in Σ
0 may be mapped,
for ε 6= 0, to a point on Σ+ε , to a point interior to M+ε or to a point exterior to M+ε (within
the extension V +ε ) which is not part of the manifold. How can we then take derivatives
with respect to ε at those later points? Since only derivatives at ε = 0 are needed, re-
stricting to infinitesimal values of ε entails no loss of generality. Then, if for some small
ε, a point q ∈ Σ+0 is mapped to the exterior of M+ε , it follows from differentiability with
respect to ε that q is mapped, for the reverse value −ε, to a point interior to M+ε . Thus,
perturbations can be defined at the boundary by taking one sided derivatives, i.e. to take
limits ε → 0, with a sign restriction on ε (c.f. [7] for an alternative discussion).
However, an important issue remains: How do we describe the deformation of the
boundary Σ+0 ? As a set of points each boundary Σ
ε maps, with the above identification,
into a hypersurface of the background spacetime, which we call Σ̂+ε . In general, this
hypersurface will not coincide with Σ+0 and may well touch it or cross it. This gives us
an idea of how the boundary is deformed, but only as a subset, not pointwise. In order
to know how the boundary actually moves within the background, we need to prescribe a
priori a pointwise identification of Σ+0 with Σ
ε . This identification is completely different
and independent from the one described above involving spacetime points, and involves
only the points on the boundaries. As before, there are infinitely many ways to identify the
boundaries, and this defines a second gauge freedom, which involves objects intrinsically
defined on the boundary. This gauge freedom will be referred as hypersurface gauge, as
opposed to the usual spacetime gauge described above.
With both identifications chosen, the deformation of the boundary within the back-
ground can already be described: Fix a point q on the background boundary Σ+0 . The
identification of the boundaries defines a point qε on Σ
ε , for each ε. The spacetime iden-
tification takes this point qε and maps it into a point q̂ε of the background M
0 (perhaps
after a sign restriction on ε). Obviously q̂ε belongs to the perturbed hypersurface Σ̂
ε . We
have therefore not only a deformation of the background hypersurface as a set of points,
but also pointwise information. It only remains to take the tangent vector of the curve
q̂ε at ε = 0, i.e. ~Z
+ = dq̂ε
|ε=0 which encodes completely the deformation of the matching
hypersurface as seen from the background spacetime. Two final remarks are in order:
(i) ~Z+ is defined exclusively on Σ+0 , no extension thereof is defined or required and (ii)
~Z+ depends on both the spacetime and hypersurface gauges, since its defining curve is
constructed using both identifications. However, decomposing ~Z+ = Q+~n 0+ +
~T+, where
~n 0+ is the unit normal of Σ
0 (assumed non-null anywhere) and
~T+ is tangent to it, it turns
out that Q+ depends on the spacetime gauge but not on the hypersurface gauge. This
is because changing the hypersurface gauge reorganizes the points within each Σ̂+ε , but
cannot modify any of them as a set of points.
Tensors defined intrinsically on the boundaries Σ+ε are completely unaffected by the
spacetime identification, and are therefore invariant under spacetime gauge transforma-
tions. Recall that the matching conditions involve only objects intrinsic to the match-
ing hypersurfaces. Since the perturbed matching conditions are, formally, just their ε-
derivatives, it follows by construction that the perturbed matching conditions must be
gauge invariant under spacetime gauge transformations. This may seem surprising at first
sight since the matching conditions must involve the perturbed metric, which is obviously
gauge dependent. However, the conditions turn out to be gauge independent because they
also involve the deformation vector ~Z+, which is spacetime gauge dependent. This vector
is therefore of fundamental importance and must be taken into account in any sensible
approach to the problem, as we shall see next.
2.2 Matching conditions
Let (M±0 , g
0 ) be n-dimensional spacetimes with non-null boundaries Σ
0 . Matching them
requires an identification of the boundaries, i.e. a pair of embeddings Φ± : Σ0 −→ M±0
with Φ±(Σ0) = Σ
0 , where Σ0 is an abstract copy of any of the boundaries. Let ξ
(i, j, . . . = 1, . . . , n−1) be a coordinate system on Σ0. Tangent vectors to Σ±0 are obtained
by e±αi =
(α, β, . . . = 0, . . . , n − 1). There are also unique (up to orientation) unit
normal vectors n
α to the boundaries. We choose them so that if n
α points towards
M+ then n
α points outside of M− or viceversa. The first and second fundamental are
simply q(0)ij
± ≡ e±αi e
, K(0)ij
± = −n(0)±αe
i ∇±β e
j . The matching conditions
(in the absence of shells) require the equality of the first and second fundamental forms
on Σ±0 , i.e.
q(0)ij
+ = q(0)ij
−, K(0)ij
+ = K(0)ij
−. (1)
Under a perturbation of the background metric g±pert = g
(0)± + g(1)± and of the matching
hypersurfaces via ~Z± = Q± ~n(0)± + ~T
±, the matching conditions will be perturbatively
satisfied if and only if [6]
q(1)+ij = q
ij , K
ij = K
ij, (2)
q(1)ij
± = L~T±q
± + 2Q±K(0)ij
± + e±αi e
±, (3)
K(1)ij
± = L~T±K
± − ǫDiDjQ± +Q±(−n(0)± µn
αµβνe
±K(0)
g(1)αβ
βK(0)ij
± − n(0)± µS
(1)±µ
j , (4)
where ǫ = n
(0)α, D is the covariant derivative of (Σ, q(0)±) and S(1)
βγ ≡ 12(∇
(1)±α
∇±γ g(1)±αβ −∇±α g(1)
In these equations, Q± and ~T± are a priori unknown quantities and fulfilling the
matching conditions requires showing that two vectors ~Z± exist such that (2) are satisfied.
The spacetime gauge freedom can be exploited to fix either or both vectors ~Z± a priori,
but this should be avoided (or at least carefully analysed) if additional spacetime gauge
choices are made, in order not to restrict a priori the possible matchings. Regarding the
hypersurface gauge, this can be used to fix one of the vectors ~T+ or ~T−, but not both.
As already stressed the linearized matching conditions are by construction spacetime
gauge invariant (in fact each of the tensors q(1)ij
±, K(1)ij
± is). Moreover, the set of
conditions (2) are hypersurface gauge invariant, provided the background is properly
matched, since [6] under such a gauge transformation given by the vector ~ζ on Σ0, q
transforms as q(1)ij + L~ζ q(0)ij , and similarly for K(1)ij .
3 On previous spacetime gauge invariant formalisms
The first attempt to derive a general formalism for the matching conditions in linearized
gravity is, to our knowledge, due to Gerlach and Sengupta [2]. Their approach is based
on the description of the matching hypersurface Σ as a level set of a function f defined
on the spacetime. Assuming the level sets {f = const} to be timelike, a field of spacelike
unit normals is defined as nµ = (g
αβf,αf,β)
−1/2f,µ. The unperturbed matching conditions
correspond to the continuity everywhere (in particular across Σ) of the tensors
qαβ ≡ gαβ − nαnβ, Kαβ ≡ qαµqβν∇µnν , (5)
which are the spacetime versions of the first and second fundamental forms introduced
above. Being f defined everywhere, it makes sense to perturb it in order to describe the
variation of the matching hypersurface. Obviously, by perturbing f one also perturbs nµ.
The perturbed matching conditions proposed in [2] read
β△(qαβ)+ = qµαqνβ△(qαβ)−, qµαqνβ△(Kαβ)+ = qµαqνβ△(Kαβ)−, (6)
where qβ
α is the projector onto Σ, △ stands for perturbation and + and − denote the
quantities as computed from either side of the matching hypersurface Σ. These expres-
sions involve the projections of the perturbations of qαβ and Kαβ onto Σ. The need of
1We will abuse slightly the notation and refer to vectors on Σ0 and their images on spacetime with
the same symbol. The meaning should be clear from the context.
considering only the projected components is justified in [2] since the matching condi-
tions need to be intrinsic to the matching hypersurfaces. However, Gerlach and Sengupta
themselves note that conditions (6) are not gauge2 invariant.
Since the main interest in [2, 3] refers to spherically symmetric backgrounds, this
“ambiguity” is fixed in that case by finding suitable gauge invariant combinations of
the linearized matching conditions, which turn out to give a correct set of necessary
perturbed matching conditions in spherical symmetry. It should be stressed however, that
the authors consider these gauge invariant subset to be sufficient also, with no further
justification.
We know from the discussion in Sect. 2.1 above that (6) cannot be correct as it leads
to a set of gauge dependent conditions. Since, on the other hand the proposal (6) may
look plausible, it is of interest to point out where, and in which sense, it fails to be correct.
The first source of problems comes from assuming that the matched spacetime is given
beforehand. Indeed, qαβ and Kαβ are spacetime tensors and they can only exist (and be
continuous) once the matched spacetime is constructed. But this is precisely the purpose
of the matching conditions, so the conditions become circular. Another aspect of the same
problem is that one can only talk about continuity once the pointwise identification of
the boundaries is chosen. But a level set of a function defines only a set of points and not
the way those points must be identified. A third instance of the same issue is that tensor
components must be expressed in some basis, e.g. a common coordinate system covering
both sides of Σ. But again this cannot be assumed a priori. It needs to be constructed.
Let us however mention that once the pointwise identification of the boundaries is
chosen, the use of spacetime tensors is allowed provided they are finally projected onto
the hypersurface. In that sense, and when properly used, using spacetime indices may
simplify some calculations notably (see Carter and Battye, [5] where this notation is used
to derive the perturbed matching conditions).
Besides this aspect (which already affects the background matching) the perturbed
equations (6) suffer from one extra problem. The perturbations △(qαβ)(p) and△(Kαβ)(p)
at a point p in the background can be defined by taking ε-derivatives at fixed p and ε = 0
of the corresponding tensors (defined by gαβ(ε) and fε). For each value of ε, the matching
conditions impose the continuity of qαβ(ε) andKαβ(ε) everywhere (with the caveat already
mentioned regarding the identification of the boundaries). However, continuity of △(qαβ)
and △(Kαβ) at p would only follow if derivatives of continuous functions with respect
to an external parameter were necessarily continuous (in our case, the derivative with
respect to ε), which is not true in general. A trivial example is given by the function
u(ε, x) = |x + ε|, whith x ∈ R. For each ε this function is continuous. However, the
derivative with respect to ε does not even exist at x = 0, ε = 0. This reflects the fact that
subtracting continuous tensors at a fixed spacetime point p leads to objects that need not
be continuous. This is in fact the main problem of (6) as linearized matching conditions.
An immediate question arises: Why is the gauge invariant subset of matching condi-
tions found in [2, 3] for spherically symmetric backgrounds correct? In order to understand
this, let us rewrite (6) using the formalism of section 2.2. First of all, since △(nαnβ) will
contain, at least, one free n
α , we have
β△(qαβ)± = qµαqνβg(1)
αβ . (7)
2Throughout this section gauge will refer to spacetime gauge. Hypersurface gauges will only appear
briefly towards the end of the section.
Moreover, a simple calculation gives △(∇αnβ) = ∇α(△nβ) − S(1)µαβ n
µ and △(qαβ ) =
−g(1)αµn(0)µ n(0)β + g(0)αµ△(nµ)n
(0)α△(nβ). These, together with standard properties
of the projector, lead to
β△(Kαβ)± =
a(0)ν qµ
α△(nα) + qµαqνβ∇α(△nβ)− qµαqνβS(1)ραβ n
±, (8)
where a
ν ≡ n(0)α∇αn(0)ν . In general, these expressions do not agree with (3) and (4).
However, when the gauges are chosen so that ~Z± = 0, then △f ≡ 0 on Σ because
the matching hypersurface is unperturbed as seen from the background. Consequently
∂α(△f) ∝ n(0)α on Σ, which implies △(nα)
α for some function h. Imposing ~n(ε) to
be unit for all ε fixes h = ǫ
β . Inserting into (8) the matching conditions (6)
become
βg(1)αβ
βg(1)αβ
, (9)
n(0)α n
β Kµν − qµ
n(0)α n
β Kµν − qµ
which agree with (2) (with the exception that (9) refers to spacetime tensors and (2) are
defined on Σ). Since Gerlach and Sengupta derive a subset of gauge invariant matching
conditions out of (6) in the spherically symmetric case and their conditions are correct in
one gauge, it follows that the invariant subset is correct in any gauge. This is the reason
why the results in [2, 3] involving spherically symmetric backgrounds turn out to be fine.
Substantial progress in the linearized matching problem was made by Mart́ın-Garćıa
and Gundlach [4]. These authors pointed out the lack of justification in [2, 3] for the choice
of (6) as matching conditions. It was also argued that for spacetimes with boundary it
only makes sense to define perturbations by using gauges where the perturbed matching
hypersurface is mapped onto the background matching hypersurface. Perturbations in this
gauge, called “surface gauge” (not to be confused with hypersurface gauge) are denoted
by △̄, and its defining property is △̄f = 0. The idea was to write down the matching
conditions in this gauge and then transform into any other gauge if necessary. As noticed
by the authors, the surface gauge is not unique since there are still three degrees of freedom
left, which correspond to the three directions tangent to Σ.
A relevant observation made in [4] was that the continuity of tensorial perturbations
may depend on the index position in the tensors. The authors argue that the tensors
truly intrinsic to the hypersurfaces are qαβ, Kαβ (with indices upstairs) and propose the
following perturbed matching conditions
△̄(qαβ)+ = △̄(qαβ)−, △̄(Kαβ)+ = △̄(Kαβ)−, (10)
which are demonstrated to become exactly (9). This shows the equivalence of both pro-
posals in the surface gauge, as explicitly stated in [4]. This justifies partially the validity of
both approaches in the surface gauge. However, the justification is not complete because
of the issue we discuss next.
Indeed, conditions (10) still carry one implicit assumption that needs to be clarified.
As already stressed the perturbed matching conditions have two inherent and independent
degrees of gauge freedom. The approach by Mart́ın-Garćıa and Gundlach involves only
spacetime objects, and therefore can only notice the spacetime gauge freedom. This leads
to an incorrect statement in [4], as it is not true that the linearized matching conditions
read (10) in any surface gauge. Conditions (10) will only be valid when the spacetime
gauge maps pairs of background points (identified, via the background matching) to pairs
of points on the perturbed boundaries Σ±ε which are also identified through the matching.
Notice that not all surface gauges have this property. In explicit terms, this means that
the vectors ~Z± must (i) only have tangential components (so that we are in surface gauge)
and (ii) have the same components when written in terms of an intrinsic basis of Σ0. In
less precise, but more intuitive terms, condition (ii) states that ~Z+ and ~Z− are the same
vector, i.e. that the gauges in both regions are chosen such that the displacement of
one fixed point of the background hypersurface is identical in both regions (the displaced
point, of course, stays on the unperturbed hypersurface, due to the choice of surface
gauge). Observe finally that if Q± = 0 and ~T+ = ~T−, then the linearized matching
conditions (2) truly reduce to conditions (9), once the latter are projected on Σ. This
shows the correctness of the approaches by Gerlach and Sengupta and Mart́ın-Garćıa and
Gundlach in special gauges.
4 Freedom in matching due to symmetries
We devote this section to the study of the consequences of the existence of background
symmetries on perturbed spacetime matchings.
The existence of symmetries in the background configuration introduces two issues
which are important to take into consideration: the first corresponds to the freedom in-
troduced by the matching procedure, when preserving the symmetries, at the background
level [9], c.f. [10] for an application. The second issue corresponds to the consequences
that the symmetries in the background configuration may have on the perturbation of the
matching.
It must be stressed here that the arbitrariness introduced by the presence of symmetries
in the background configuration is completely independent from both the hypersurface and
spacetime gauge freedoms. However, that arbitrariness is gauge dependent and therefore
a gauge choice can be made to remove it. As we will show, an isometry in the background
implies that there is a direction along which the difference [~T ] ≡ ~T+ − ~T− cannot be
determined by the perturbed matching equations. But, as we have discussed at the end
of section 2, one could eventually choose part of the spacetime gauges (if there is any
freedom left) to fix [~T ]. Note, finally, that a change of hypersurface gauge leaves [~T ]
invariant.
4.1 Isometries
We shall now consider the presence of isometries in the background configuration. So,
let us assume that one of the sides, say (M+0 , g
(0)+), admits an isometry generated by
the Killing vector field ~ξ+ tangent to the boundary Σ+0 . The commutation of the Lie
derivative and the pull-back implies [9]
L~ξ+q
+ = e+αi e
j L~ξ+g
+|Σ0 = 0,
which means that ~ξ+ is a Killing vector of (Σ0, q
+). This implies from expression (3)
that q(1)ij
+ is invariant under the transformation ~T+ → ~T+ + ~ξ+|Σ0.
As for K(1)ij
+, from its expression (4), it is again clear that the previous transforma-
tions of ~T+ will leave K(1)ij
+ invariant provided L~ξ+K(0)ij+ = 0. But this is precisely the
case since ~ξ+ is a Killing vector orthogonal to n
+ , which implies L~ξ+n
+ β|Σ+
= 0, and
hence
L~ξ+K
+ = e+αi e
j L~ξ+(∇n
+ )αβ|Σ0 = e+αi e
j ∇αL~ξ+n
+ β|Σ0 = 0.
Of course, all this discussion also applies to the (−) side.
The combination of the invariance of q(1)ij
± and K(1)ij
± leads to the fact that the
first order perturbed matching conditions are invariant under a change of the vectors
~T± along the direction of any isometry of the background configuration (preserved by
the matching). Then, as expected, when symmetries are present the linearized matching
conditions cannot determine the difference [~T ] completely: they leave undetermined the
relative (between the two sides) deformation of the hypersurface along the direction of
the symmetry. Note that, still, the shape of the perturbed hypersurface is completely
determined, since that is driven by Q±.
The overall picture is as follows: at the background level we have the arbitrariness of
the identification of Σ+0 with Σ
0 [9], which can be seen as a “sliding” between Σ
0 and
Σ−0 . The perturbation adds to this an arbitrary shift of the deformation of the matching
hypersurface at each side along the orbits of the isometry group. As an example, in
the description of stationary and axisymmetric compact bodies discussed in [10, 9], the
background sliding corresponds to an arbitrary constant rotation of the interior with
respect to the exterior. Note that, in that case, this rotation is only relevant because
the exterior is taken to be asymptotically flat. As a result, two identical interiors can,
in principle, give rise to two exteriors that differ by a constant rate rotation [10]. The
shift of the surface deformation would, in principle, lead to an arbitrary constant rotation
along the axial coordinate of the surface deformation of the body. Likewise, two identical
perturbations in the interior of the body may produce two different perturbations in the
exterior, which may differ by a relative constant rate rotation. A choice of spacetime
gauge could be used to relate the deformations inside and outside. However, this may
interfere with other gauge fixings that may have been made.
5 n-dimensional spherically symmetric backgrounds
In this section we shall revisit Mukohyama’s theory for linearized matching in the special
case of spherical symmetry. Similar results [6] hold for backgrounds admitting isometry
groups of dimension (n−1)(n−2)/2 acting on non-null codimension-two orbits of arbitrary
topology (strictly speaking the orbits need to be compact).
5.1 The approach of Mukohyama
Concentrating on one of the two spacetimes to be matched, either + or −, we consider a
spherically symmetric background metric of the form
αdxβ = γabdx
adxb + r2ΩABdθ
AdθB, (11)
where γab (a, b, .. = 0, 1) is a Lorentzian two-dimensional metric (depending only on {xa}),
r > 0 is a function of {xa}, and ΩABdθAdθB is the n − 2 dimensional unit sphere metric
with coordinates {θA} (A,B, . . . = 2, 3, . . . , n− 1).
A general spherically symmetric background hypersurface can be given in parametric
form as
Σ0 := {x0 = Z(0)0(λ), x1 = Z(0)1(λ), θA = ϑA}, (12)
where {ξi} = {λ, ϑA} is a coordinate system in Σ0 adapted to the spherical symmetry.
The tangent vectors to Σ0 read
~eλ =
˙Z(0)0∂x0 +
˙Z(0)1∂x1
, ~eϑA = ∂θA |Σ0 , (13)
where dot is derivative w.r.t. λ. With N2 ≡ −ǫeλaeλbγab|Σ0, so that ǫ = 1 (ǫ = −1)
corresponds to a timelike (spacelike) hypersurface, the unit normal to Σ0 reads
(0) =
− det γ
− ˙Z(0)1dx0 + ˙Z(0)0dx1
, (14)
where the sign choice of N corresponds to the choice of orientation of the normal. The
background induced metric and second fundamental form on Σ0 read
q(0)ijdξ
idξj = −ǫN2dλ2 + r2|Σ0ΩAB|Σ0dϑAdϑB, (15)
K(0)ijdξ
idξj = N2Kdλ2 + r2K̄|Σ0ΩAB|Σ0dϑAdϑB, (16)
where
K ≡ N−2eλaeλb∇an(0)b , K̄ = n
(0)a∂xa ln r.
It follows that the background matching conditions (1) are
N2+ = N
+|Σ0 = r2−|Σ0, K+ = K−, K̄+ = K̄−. (17)
Using (3) and (4) we could now compute the first order perturbations q(1)ij and K
ij in
terms of the above quantities and ~Z (or equivalently Q and ~T ), c.f. Eqs. (45) and (46)
in [6]. Let us recall (see subsection 2.2) that while the individual tensors q(1)ij and K
are not hypersurface gauge invariant, their respective differences from the + and − sides
(i.e. the linearized matching conditions) are. Those tensors depend of the hypersurface
gauge through the tangent vectors ~T+ and ~T−, which under a gauge change transform
simply by adding the gauge vector. It follows that only their difference [~T ] can appear
in the linearized matching conditions. Consequently there are three degrees of freedom
that cannot be fixed by the equations, but can be fixed by choosing the hypersurface
gauge, for instance to set ~T+. Thus, the linearized matching conditions can be looked at
as equations for the difference [~T ] as well as for Q+ and Q−, i.e. for five objects. If these
equations admit solutions, then the linearized matching is possible and it is impossible
otherwise.
Mukohyama emphasizes the convenience to look for doubly gauge invariant quantities
to write down the linearized matching conditions, however the matching conditions are
already gauge invariant (both for the spacetime and hypersurfaces gauges). Looking for
gauge invariant combinations on each side amounts to writing equations where the dif-
ference vector [~T ] simply drops. Indeed, in many cases, knowing the value of such vector
in a specific matching is not interesting. In that sense, using doubly gauge invariant
quantities is useful as it lowers the number of equations to analyse. However, we want to
stress that this is not related to obtaining gauge invariant linearized matching equations.
It is just related to not solving for superfluous information. In fact, a set of equations
where also Q+ and Q− have disappeared would be even more convenient from this point
of view, provided one is not interested in knowing how the hypersurfaces are deformed in
the specific spacetime gauge being used.
Since the use of doubly gauge invariant matching conditions is used extensively, let us
recall its main ingredients in order to discuss if they really are equivalent to the full set
of linearized matching equations and in which sense.
To that aim Mukohyama [6], decomposes the perturbation tensors q(1)ij and K
ij in
terms of scalar Y , vector VA and tensor harmonics TAB on the sphere, as
q(1)ijdξ
idξj =
(σ00Y dλ
2 + σ(Y )T(Y )ABdϑ
AdϑB) +
2(σ(T )0V(T )A + σ(L)0V(L)A)dλdϑ
(σ(T )T(T )AB + σ(LT )T(LT )AB + σ(LL)T(LL)AB)dϑ
AdϑB, (18)
K(1)ijdξ
idξj =
(κ00Y dλ
2 + κ(Y )T(Y )ABdϑ
AdϑB) +
2(κ(T )0V(T )A + κ(L)0V(L)A)dλdϑ
(κ(T )T(T )AB + κ(LT )T(LT )AB + κ(LL)T(LL)AB)dϑ
AdϑB, (19)
where all the scalar coefficients depend only on λ. Each coefficient in the decomposition
has indices l and m which have been dropped for notational simplicity. Notice that
each coefficient σ and κ is defined in the range of l’s appearing in the corresponding
summatory. By construction, each of the σ and κ are spacetime-gauge invariant (but
not hypersurface-gauge invariant). For l ≥ 2 they can even be written down [6] explicity
in terms of spacetime-gauge invariant quantities. In a similar way, the doubly gauge-
invariant quantities presented in [6], are only defined for l ≥ 2 (except k(T )0, which is also
defined for l = 1), and read
l ≥ 2 : f00 ≡ σ00 − 2N∂λ
l ≥ 2 : f ≡ σ(Y ) + ǫN−2χ∂λ
r2|Σ0
k2l σ(LL),
l ≥ 2 : f0 ≡ σ(T )0 − r2|Σ0∂λ
r−2|Σ0σ(LT )
l ≥ 2 : f(T ) ≡ σ(T ),
l ≥ 2 : k00 ≡ κ00 + ǫKσ00 + ǫχ∂λK,
l ≥ 1 : k(T )0 ≡ κ(T )0 − K̄σ(T )0, (20)
l ≥ 2 : k(L)0 ≡ κ(L)0 +
(ǫK − K̄)σ(L)0 +
(ǫK + K̄)
χ− r2|Σ0∂λ(r−2|Σ0σ(LL))
l ≥ 2 : k(LT ) ≡ κ(LT ) − K̄σ(LT ),
l ≥ 2 : k(LL) ≡ κ(LL) − K̄σ(LL),
l ≥ 2 : k(Y ) ≡ κ(Y ) − K̄σ(Y ) + ǫN−2r2|Σ0χ∂λK̄,
l ≥ 2 : k(T ) ≡ κ(T ) − K̄σ(T ),
3The ranges of l’s are not made explicit in [6] in order to include also non-compact homogeneous
spaces, where the index l is continuous. However, to discuss sufficiency of the equations we need to be
precise on the range of validity of each equation.
where k2l = l(l + n− 3) and
l ≥ 2 : χ ≡ σ(L)0 − r2|Σ0∂λ(r−2|Σ0σ(LL)).
The orthogonality properties of the scalar, vector and tensor harmonics imply that
the equalities of the coefficients σ and κ for each l and m is equivalent to the equality of
the perturbation tensors (18) and (19) at both sides of Σ0. Thus, recalling the notation
[f ] ≡ f+|Σ0 − f−|Σ0 , the equations
l ≥ 0 : [σ00] = [σ(Y )] = 0
l ≥ 1 : [σ(L)0] = [σ(T )0] = 0
l ≥ 2 : [σ(T )] = [σ(LT )] = [σ(LL)] = 0
l ≥ 0 : [κ00] = [κ(Y )] = 0
l ≥ 1 : [κ(L)0] = [κ(T )0] = 0
l ≥ 2 : [κ(T )] = [κ(LT )] = [κ(LL)] = 0
are equivalent to (2) and therefore correspond exactly to the linearized matching condi-
tions in this setting. Notice that each of the equalities in (21) and (22) is in fact one
equation for each l and m in the appropriate range. We will however refer to them simply
as equations.
The full linearized matching conditions obviously imply the following equalities in
terms of the doubly-gauge invariant quantities (20),
l ≥ 2 : [f00] = [f ] = [f0] = [f(T )] = 0 (23)
l ≥ 1 : [k(T )0] = 0
l ≥ 2 : [k00] = [k(Y )] = [k(L)0] = [k(LL)] = [k(LT )] = [k(T )] = 0.
Whether these equations can be regarded as the full set of linearized matching conditions
or not requires studying their sufficiency, i.e. whether they imply (21)-(22) or not. This
point was not mentioned in [6] and in fact the answer turns out to be negative, although
in a mild way, as we discuss in the next subsection.
5.2 On the sufficiency of the continuity of the doubly-gauge in-
variants
Let us recall that fulfilling the matching conditions requires finding two ~Z± such that
(21)-(22) are satisfied. The key issue for the matching is therefore to show existence of
deformation vectors ~Z± so that all the equations hold.
A plausibility argument in favour of the sufficiency of (23)-(24) comes from simple
equation counting. Indeed, as already discussed, the linearized matching conditions are
spacetime and hypersurface gauge invariant and therefore can only involve the difference
vector [~T ], i.e. three quantities. Since constructing double gauge invariant quantities on
each side eliminates this vector, the number of equations should be reduced exactly by
three if they are to remain equivalent to the original set. This is precisely what happens
as we go from the original forteen equations in (21)-(22) down to eleven equations in
(23)-(24). This argument however is not conclusive, both because it is not rigorous and
because each equation in those expressions is, in fact, a set of equations depending on l
and m, and the range of l’s changes with the equations. Let us therefore analyse this issue
in detail. In particular we need to discuss what are the consequences of the non-existence
of doubly gauge-invariant variables for l = 0 and l = 1 (except for k(T )0 which exists for
l = 1), something not mentioned in [6].
Let us start by finding explicit expressions for σ’s valid in the whole range of l’s. As
in [6], we decompose g(1) in harmonics as
g(1)αβdx
αdxβ =
(habY dx
adxb + h(Y )T(Y )ABdθ
AdθB)
2(h(T )aV(T )A + h(L)aV(L)A)dx
(h(T )T(T )AB + h(LT )T(LT )AB + h(LL)T(LL)AB)dθ
AdθB, (25)
and ~Z as
zaY dx
(z(T )V(T )A + z(L)V(L)A)dθ
QY n(0) − ǫN−2zλY eλ
(z(T )V(T )A + z(L)V(L)A)dθ
A, (26)
which implies Tαdx
l=0(−ǫN−2zλY eλ) +
l=1(z(T )V(T )A + z(L)V(L)A)dθ
A. Inserting
these expressions into (2) and expanding in spherical harmonics it is straightforward to
l ≥ 0 : [σ00] = 0 ⇔ [hλλ] + 2[Q]N2K + 2N∂λ
N−1[zλ]
l ≥ 1 : [σ(L)0] = 0 ⇔ [zλ] + [h(L)λ] + r2|Σ0∂λ(r−2|Σ0[z(L)]) = 0, (27)
l ≥ 2 : [σ(LL)] = 0 ⇔ [z(L)] + [h(LL)] = 0, (28)
l ≥ 0 : [σ(Y )] = 0 ⇔ [h(Y )] + 2[Q]r2|Σ0K̄ − ǫN−2[zλ]∂λ(r2|Σ0)−
k2l [z(L)] = 0,
l ≥ 1 : [σ(T )0] = 0 ⇔ [h(T )λ] + r2|Σ0∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [σ(LT )] = 0 ⇔ [z(T )] + [h(LT )] = 0, (29)
l ≥ 2 : [σ(T )] = 0 ⇔ [h(T )] = 0,
where [hλλ], [h(L)λ], etc. denote eλ
b[hab], eλ
a[h(L)a], etc. Later on we will also write
down the explicit expressions for (22) but they are not needed in this subsection.
It is obvious by the form of f ’s and κ’s (20) that the set of equations (21)-(22) are
equivalent to (23)-(24) together with
l ≥ 2 : [σ(L)0] = [σ(LT )] = [σ(LL)] = 0 (30)
l = 0, 1 : [σ00] = [σ(Y )] = 0
l = 1 : [σ(L)0] = [σ(T )0] = 0
l = 0, 1 : [κ00] = [κ(Y )] = 0
l = 1 : [κ(L)0] = 0.
Sufficiency of Mukohyama’s doubly gauge invariant matching conditions would follow if
these equations serve exclusively to determine the discontinuity [~T ], i.e. [zλ] for l ≥ 0
and [z(T )], [z(L)] for l ≥ 1. Now, the explicit expressions (27), (29), (28) show that (30)
determine uniquely [zλ], [z(T )] and [z(L)] for l ≥ 2. So, restricted to the sector l ≥ 2
Mukoyama’s doubly gauge invariant matching conditions can be regarded as equivalent
to the full set of matching conditions. Taking all l’s into account, however, the equations
turn out not to be sufficient. To show this, it is enough to display one equation involving
the discontinuity of the background metric perturbations and [Q] (but not [~T ]) which holds
as a consequence of the full set of matching conditions (21)-(22) but not as a consequence
of (23)-(24). Using the fact that each l = 1 expression refers to n − 1 objects (one for
each m), the number of equations in (31)-(32) is 7n− 3, while the number of unkowns in
[~T ] not yet determined by (30), i.e. [zλ] for l = 0, 1 and [z(T )], [z(L)] for l = 1 is 3n − 2,
which is smaller. It is to be expected, therefore, that (31), (32) imply conditions where
these variables do not appear. This can be made explicit, for instance, by combining
[σ00]l=0 = 0 with [σ(Y )]l=0 = 0 which yields
l = 0 : [hλλ] + 2[Q]N
2K + 2N∂λ
∂λ(r2|Σ0)
[h(Y )] + 2[Q]r
2|Σ0K̄
whenever ∂λ(r
2|Σ0) 6= 0. (If ∂λ(r2|Σ0) = 0 it is enough to consider [σ(Y )]l=0 = 0.) This
relation is enough to show that the continuity of the doubly-gauge invariant variables of
Mukohyama is not sufficient to ensure the existence of the perturbed matching. Of course,
this does not invalidate Mukohyama’s approach in any way, which remains interesting
and useful. It only means that, when using this approach to solve linearized matchings,
one still needs to look more carefully into the l = 0 and l = 1 sector to make sure that
the remaining equations (31) and (32) hold.
On the other hand, equations (31), (32) do not completely determine [~T ]. The variable
[z(T )]l=1 only appears in [σ(T )0]l=1 = 0, in the term ∂λ(r
−2|Σ0 [z(T )]). As a result, the
matching conditions do not fix [z(T )]l=1 completely, but up to a constant factor times
r2|Σ0 (for each m). Recalling that V(T )AdϑA for l = 1 correspond to the three Killing
vectors on the sphere, this arbitrary constant (for each m) accounts for the addition to
[~T ] of an arbitrary Killing vector of the sphere. This is in accordance with the discussion
in Section 4. We devote the following subsection to complete the study of the freedom
left in the matching.
5.3 Freedom in the matching
As already emphasized, solving the linearized matching amounts to finding perturbation
vectors ~Z+ and ~Z−. Assume now that a linearized matching between two given back-
grounds and perturbations has been done. It is natural to ask what is the most general
matching between those two spaces, i.e. what is the most general solution for ~Z+ and ~Z−
of the matching conditions. Geometrically, this means finding all the possible deforma-
tions of the matching hypersurface Σ0 which allow the two spaces to be matched.
Since this problem is of interest not only when the full matching conditions are imposed
but also in situations where layers of matter are present (e.g. in brane-world or shell
cosmologies) so that jumps in the second fundamental forms are allowed, we will analyse
this issue in two steps. First, we will study the equations involving the perturbed first
fundamental forms and will determine the freedom they admit. On a second step we
will write down the extra conditions coming from the equality of the second fundamental
forms.
Thus, let us consider two perturbation configurations of the same background matching
and denote their respective sets of difference variables on Σ0 as [f ] and [f ]
′ for any given
variable f . Now, we will define the difference between the two configurations as < f >≡
[f ]′ − [f ] for any variable f . The assumption that the perturbation on each side is fixed
once and for all implies < g(1) >= 0. We are assuming that the linearized matching
conditions are satisfied in each case, and so we can subtract them. Linearity implies that
the differences of the linearised matching equations become equations for the difference
vector < ~Z >. The general solution of these equations clearly determines the freedom in
the deformation of the hypesurface.
The difference of the equations in (21) for the two configurations using < g(1) >= 0
give the following set of equations
l ≥ 0 : < σ00 >= 0 → < Q > N2K +N∂λ
N−1 < zλ >
= 0, (33)
l ≥ 1 : < σ(L)0 >= 0 → < zλ > +r2|Σ0∂λ(r−2|Σ0 < z(L) >) = 0, (34)
l ≥ 2 : < σ(LL) >= 0 → < z(L) >= 0, (35)
l ≥ 0 : < σ(Y ) >= 0 → 2 < Q > r2|Σ0K̄ − ǫN−2 < zλ > ∂λ(r2|Σ0)
k2l < z(L) >= 0, (36)
l ≥ 1 : < σ(T )0 >= 0 → ∂λ(r−2|Σ0 < z(T ) >) = 0, (37)
l ≥ 2 : < σ(LT ) >= 0 → < z(T ) >= 0, (38)
l ≥ 2 : < σ(T ) >= 0 → 0 = 0.
Expressions (35) and (38) readily determine < z(L) >l≥2=< z(T ) >l≥2= 0, which sub-
stituted in (34) give < zλ >l≥2= 0. As a result, (36) for l ≥ 2 lead to < Q >l≥2= 0.
Clearly all the equations for l ≥ 2 are now satisfied. We now concentrate on the l = 1
equations. Equation (37) implies that < z(T ) >l=1= ar
2|Σ0, where a is a constant for each
m. Combining equations (33), (34) and (36) for l = 1 we obtain the following equation
for r−2|Σ0 < z(L) >l=1,
K̄∂2λ(r−2|Σ0 < z(L) >l=1) +
(2K̄ + ǫK)∂λ(ln r|Σ0)− K̄∂λ lnN
−2|Σ0 < z(L) >l=1)
r−2|Σ0 < z(L) >l=1= 0, (39)
while (34) and (33) determine < zλ >l=1 and < Q >l=1 respectively (provided K 6= 0,
which occurs generically). The two equations for l = 0 can be rearranged onto
K̄∂λ(N−1 < zλ >l=0) +N−1 < zλ >l=0 ǫK∂λ ln(r|Σ0) = 0 (40)
plus the equation (33) for l = 0, which determines < Q >l=0.
Summarizing, we have found that the freedom in the deformation of the hypersurface
compatible with the linearized matching conditions involving the first fundamental form
[~Z]′ − [~Z] =
< Q > Y ~n(0) − ǫN−2 < zλ > Y ~eλ
+am~V
(T ) + r
−2|Σ0 < z(L) >l=1,m ~V m(L),
where r−2|Σ0 < z(L) >l=1,m, satisfy (39), < zλ >l=0 satisfy (40) and the rest of the
variables are completely determined as described above. The term in am corresponds
to adding Killing vectors on the sphere, something already discussed in Section 4. The
rest of terms involve combinations (with functions) of the conformal Killing vectors on
the sphere and tangential vectors along λ. Notice that the coefficients of the conformal
Killing (i.e. < z(L) >l=1,m ) determine all the rest of the l = 1 coefficients. In particular
when < z(L) >l=1,m vanishes, then all the l = 1 terms vanish and the freedom becomes
radially symmetric.
We now add to the analysis the difference of the equations in (22). Due to the fact
that all coefficients in < ~Z > vanish for l ≥ 2 we only need to consider the equations for
l = 0, 1, i.e. (32). We refer the reader to Appendix A for the explicit expressions of (32)
in terms of the metric perturbations and ~Z. For the sake of completeness we also include
all the explicit expressions of (22) in Appendix A. The difference of equations (32), see
(44)-(46), whenever < g(1) >= 0 read
l = 0, 1 : < κ00 >= 0 ⇔ (41)
− < QR(γ)dbac > n
(0)dn(0)aeλ
c − ǫ∂2λ < Q > +
2∂λ < Q > −ǫ < Q > K2N2
−ǫKN2∂λ(N−2 < zλ >)− ǫ∂λ(K < zλ >) = 0,
l = 1 : < κ(L)0 >= 0 ⇔ (42)
−ǫ∂λ < Q > +ǫK < zλ > +ǫ < Q > ∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0 < z(L) >) = 0
l = 0, 1 : < κ(Y ) >= 0 ⇔ (43)
N−2∂λ(r
2|Σ0) (∂λ < Q > +K < zλ >) +
< Qn(0)an(0)b∇a∇br2 >
N−2eλ
a < zλn
(0)b∇b∇ar2 > +
l(l + n− 3)
ǫ < Q > −2K̄ < z(L) >
It can be checked that in general these equations overdetermine the previous equations,
i.e. (39) and (40), although there may be particular cases for which they are compatible.
Therefore, generically, they will imply that < z(L) >l=1,m= 0 and < zλ >l=0= 0, and
thence all the rest of the variables vanish, < zλ >l=1,m=< Q >l=1,m= 0, < zλ >l=0=<
Q >l=0= 0, so that the only freedom left is given by
[~Z]′ − [~Z] = am~V m(T ).
Finding in which particular cases equations (39)-(43) are compatible is straightforward
but tedious and will not be carried out explicitly here.
Acknowledgements
FM and MM thank CRUP(Portugal)/MCT(Spain) for grant E-113/04. FM thanks FCT
(Portugal) for grant SFRH/BPD/12137/2003 and CMAT, University of Minho, for sup-
port. MM was supported by the projects FIS2006-05319 of the Spanish Ministerio de
Educación y Tecnoloǵıa and SA010CO of the Junta de Castilla y León. RV was supported
by the Irish IRCSET, Ref. PD/2002/108, and now is funded by the Basque Government
Ref. BFI05.335.
A Appendix
For the sake of completeness we devote this appendix to present the explicit expressions
of (22) in terms of the metric perturbations and ~Z, which read
l ≥ 0 : [κ00] = 0 ⇔ (44)
N2K[hnn]−
n(0)aeλ
c(2∇c[hab]−∇a[hbc])− [QR(γ)dbac]n
(0)dn(0)aeλ
−ǫ∂2λ[Q] +
2∂λ[Q]− ǫ[Q]K2N2 − ǫKN2∂λ(N−2[zλ])− ǫ∂λ(K[zλ]) = 0,
l ≥ 1 : [κ(L)0] = 0 ⇔ (45)
[hnλ]−
n(0)aeλ
b(∂b[h(L)a]− ∂a[h(L)b])− ǫ∂λ[Q]− ǫK[zλ]
+(ǫ[Q] + [h(L)n])∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(L)]) = 0
l ≥ 0 : [κ(Y )] = 0 ⇔ (46)
r2|Σ0K̄[hnn] +
N−2∂λ(r
2|Σ0) (ǫ[hnλ] + ∂λ[Q] +K[zλ]) +
[n(0)a∂ah(Y )]
[Qn(0)an(0)b∇a∇br2]−
N−2eλ
a[zλn
(0)b∇b∇ar2]
l(l + n− 3)
[h(L)n] + ǫ[Q]− 2K̄[z(L)]
l ≥ 1 : [κ(T )0] = 0 ⇔
n(0)aeλ
b(∂b[h(T )a]− ∂a[h(T )b]) + [h(T )n]∂λ ln(r|Σ0) + r2|Σ0K̄∂λ(r−2|Σ0[z(T )]) = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(T )n] +
n(0)a∂a[h(LT )] + K̄[z(T )] = 0,
l ≥ 2 : [κ(LT ] = 0 ⇔ −
[h(L)n] +
n(0)a∂a[h(LL)] + K̄[z(L)]−
[Q] = 0,
l ≥ 2 : [κ(T ] = 0 ⇔
n(0)a∂a[h(T )] = 0.
References
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1509-1519
Introduction
Linearized matching
Gauge freedom
Matching conditions
On previous spacetime gauge invariant formalisms
Freedom in matching due to symmetries
Isometries
n-dimensional spherically symmetric backgrounds
The approach of Mukohyama
On the sufficiency of the continuity of the doubly-gauge invariants
Freedom in the matching
Appendix
|
0704.0079 | Operator algebras associated with unitary commutation relations | 7 Operator Algebras Associated with Unitary
Commutation Relations
Stephen C. Power†
Lancaster University
Department of Mathematics and Statistics
Lancaster, United Kingdom LA1 4YF
E-mail: [email protected]
Baruch Solel‡
Technion
Department of Mathematics
Haifa 32000, Israel
E-mail: [email protected]
November 25, 2021
∗2000 Mathematics Subject Classification. 47L55, 47L30, 47L75, 46L05.
†SCP is supported by EPSRC grant EP/E002625/1
‡BS is supported by the Fund for the Promotion of Research at the Technion and by
EPSRC grant EP/E002625/1
http://arxiv.org/abs/0704.0079v1
Abstract
We define nonselfadjoint operator algebras with generators
Le1 , . . . , Len , Lf1 , . . . , Lfm subject to the unitary commutation rela-
tions of the form
LeiLfj =
ui,j,k,lLflLek
where u = (ui,j,k,l) is an nm × nm unitary matrix. These algebras,
which generalise the analytic Toeplitz algebras of rank 2 graphs with
a single vertex, are classified up to isometric isomorphism in terms of
the matrix u.
1 Introduction
The unilateral shift on complex separable Hilbert space generates two funda-
mental operator algebras, namely the norm closed (unital) algebra and the
weak operator topology closed algebra. The former is naturally isomorphic to
the disc algebra of holomorphic functions on the unit disc, continuous to the
boundary, while the latter is isomorphic to H∞. The freely noncommuting
multivariable generalisations of these algebras arise from the freely noncom-
muting shifts Le1 , . . . , Len given by the left creation operators on the Fock
space Fn =
k=0⊕(Cn)⊗k. Here the generated operator algebras, denoted
An and Ln for the norm and weak topologies, are known as the noncommu-
tative disc algebra and the freesemigroup algebra. They have been studied
extensively with respect to operator algebra structure, representation theory
and the multivariable operator theory of row contractions. See for example
[2], [9].
Higher rank generalisations of these algebras arise when one considers
several families of freely noncommuting generators between which there are
commutation relations. In the present paper we consider a very general form
of such relations, namely
LeiLfj =
ui,j,k,lLflLek
where Le1 , . . . , Len and Lf1 , . . . , Lfm are freely noncommuting and u = (ui,j,k,l)
is an nm×nm unitary matrix. The associated operator algebras are denoted
Au and Lu and we classify them up to various forms of isomorphism in terms
of the unitary matrices u. Such unitary relations arose originally in the con-
text of the general dilation theorem proven in Solel ([12], [13]) for two row
contractions [T1 · · ·Tn] and [S1 · · ·Sm] satisfying the unitary commutation
relations.
For n = m = 1, we have u = [α] with |α| = 1 and Au is the subalgebra of
the rotation C*-algebra for the relations uv = αvu. When u is a permutation
unitary matrix arising from a permutation θ in Snm then the relations are
those associated with a single vertex rank 2 graph in the sense of Kumjian
and Pask, and the algebras in this case have been considered in Kribs and
Power [5] and Power [10]. In particular, in [10] it was shown that there are 9
operator algebras Aθ arising from the 24 permutations in case n = m = 2. In
contrast, we see below in Section 6 that for general 2 by 2 unitaries u there
are uncountably many isomorphism classes of the unitary relation algebras
Au expressed in terms of a nine fold real parametrisation of isomorphism
types.
The algebras Aθ are easily defined; they are determined by the left regular
representation of the semigroup F+θ whose generators are e1, . . . , en, f1, . . . , fm
subject to the relations eifj = flek where θ(i, j) = (k, l). On the other hand
the unitary relation algebras Au are generated by creation operators on a
Z2+-graded Fock space
k,l ⊕(Cn)⊗k⊗ (Cm)⊗l with relations arising from the
identification u : Cn ⊗Cm → Cm ⊗Cn. In particular, Au is a representation
of the non-selfadjoint tensor algebra of a rank 2 correspondence (or a product
system over N2) in the sense of [13]. See also [3]
In the main results, summarised partly in Theorem 5.10, we see that if
Au and Av are isomorphic then the two families of generators have match-
ing cardinalities. Furthermore, if n 6= m then the algebras are isomorphic if
and only if the unitaries u, v in Mnm(C) are unitary equivalent by a unitary
A ⊗ B in Mn(C) ⊗Mm(C). As in [10] we term this product unitary equiv-
alence (with respect to the fixed tensor product decomposition). The case
n = m admits an extra possibility, in view of the possibility of generator
exchanging isomorphisms, namely that u, ṽ are product unitary equivalent,
where ṽi,j,k,l = v̄l,k,j,i.
The theorem is proven as follows. After some preliminaries we identify, in
Section 3, the character space M(Au) and the set of w*-continuous charac-
ters on Lu. These are subsets of the closed unit ball product Bn ×Bm which
are associated with a variety Vu in C
n×Cm determined by u. We then define
the core Ω0u, a closed subset of the realised character space Ωu =M(Au), and
we identify this intrinsically (algebraically) in terms of representations of Au
into T2, the algebra of upper triangular matrices in M2(C). The importance
of the core is that we are able to show that the interior is a minimal automor-
phism invariant subset on which automorphisms act transitively. This allows
us to infer the existence of graded isomorphisms from general isomorphisms.
To construct automorphisms we first review, in Section 4, Voiculescu’s con-
struction of a unitary action of the Lie group U(1, n) on the Cuntz algebraOn
and the operator algebras An and Ln. This provides, in particular, unitary
automorphisms Θα, for α ∈ Bn, which act transitively on the interior ball,
Bn, of the character space of An. For these explicit unitary automorphisms
of the ei-generated copy of An in Au, we establish unitary commutation re-
lations for the tuples Θα(Le1), . . . ,Θα(Len) and Lf1 , . . . , Lfm , when (α, 0) is
a point in the core. This enables us to define natural unitary automorphisms
of Au itself, and in Theorem 4.8 the relative interior of the core is identi-
fied as an automorphism invariant set in the Gelfand space Ωu. In Section
5 we determine the graded and bigraded isomorphisms in terms of product
unitary equivalence. To do this we observe that such isomorphisms induce
an origin preserving biholomorphic map between the cores Ω0u and Ω
v and
that these maps, by a generalised Schwarz’s Lemma, are implemented by a
product unitary. We then prove the main classification theorem.
In Section 6 we analyse in detail the case n = m = 2 and consider the
special case of permutation unitaries.
Finally, in Section 7 we show that the algebra Au is contained in a tensor
algebra T+(X), associated with a correspondence X as in [7]. Moreover, at
least when n 6= m, every automorphism of Au extends to an automorphism
of T+(X). The advantage of the tensor algebra is that its representation
theory is known ([7]) while this is not the case yet for the algebra Au.
2 Preliminaries
Fix two finite dimensional Hilbert spaces E = Cn and F = Cm and a
unitary mn × mn matrix u. The rows and columns of u are indexed by
{1, . . . , n}×{1, . . . , m} (u = (u(i,j),(k,l))) and when we write u as an mn×mn
matrix we assume that {1, . . . , n} × {1, . . . , m} is ordered lexicographically
(so that, for example, the second row is the row indexed by (1, 2)). We also
fix orthonormal bases {ei} and {fj} for E and F respectively and the matrix
u is used to identify E ⊗ F with F ⊗ E through the equation
ei ⊗ fj =
u(i,j),(k,l)fl ⊗ ek. (1)
Equivalently, we write
fl ⊗ ek =
ū(i,j),(k,l)ei ⊗ fj. (2)
For every k, l ∈ N, we write X(k, l) for E⊗k ⊗ F⊗l. Using succesive applica-
tions of (1), we can identify X(k, l) with E⊗k1 ⊗ F⊗l1 ⊗ E⊗k2 ⊗ · · · ⊗ F⊗lr
whenever k =
ki and l =
Let F(n,m, u) be the Fock space given by the Hilbert space direct sum
X(k, l) =
E⊗k ⊗ F⊗l,
and, for e ∈ E and f ∈ F , write Le and Lf for the “shift” operators
Leei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl = e⊗ei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl
Lfei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl = f⊗ei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl
where, in the last equation, we use (1) to identify the resulting vector as a
vector of E⊗k ⊗ F⊗(l+1).
The unital semigroup generated by {I, Le, Lf : e ∈ E, f ∈ F} is
denoted F+u and the algebra it generates denoted C[F
u ]. The norm closure
of C[F+u ] will be written Au and its closure in the weak* operator topology
will be written Lu. In particular, the algebras Lθ and Aθ studied in [10] are
the algebras Lu and Au for u which is a permutation matrix.
The results of Section 2 in [5] hold here too with minor changes. Every
A ∈ Lu is the limit (in the strong operator topology) of its Cesaro sums
Σp(A) =
(1− k
)Φk(A)
where Φk(A) lies in Lu and is “supported” on
l ⊕E⊗l ⊗ F⊗(k−l). In fact,
let Qk be the projection of F(n,m, u) onto
l ⊕E⊗l ⊗ F⊗(k−l), form the
one-parameter unitary group {Ut} defined by Ut :=
k=0 e
iktQk and set
γt = AdUt. Then {γt}t∈R is a w∗-continuous action of R on L(F(n,m, u))
that normalizes both Au and Lu and
Φk(a) =
e−iktγt(a)dt
for all a ∈ L(F(n,m, u)). Then Φk leaves Lu invariant.
We can define the algebra Ru generated by the right shifts Re and Rf
defined by
Reei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl = ei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fjl⊗e
Rfei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fil = ei1⊗ei2⊗· · ·⊗eik⊗fj1⊗fj2⊗· · ·⊗fil⊗f.
The techniques of the proof of Proposition 2.3 of [5] can be applied here to
show that the commutant of Ru is Lu. Also, mapping ei1 ⊗ ei2 ⊗ · · · ⊗ eik ⊗
fj1 ⊗ fj2 ⊗ · · · ⊗ fjl to fjl ⊗ fjl−1 ⊗ · · · ⊗ fj1 ⊗ eik ⊗ eik−1 ⊗ · · · ⊗ ei1 , we get a
unitary operator
W : F(n,m, u) → F(n,m, u∗)
implementing a unitary equivalence of Lu and Ru∗ . In fact, it is easy to
check that ReiW = WLei and RfjW = WLfj for every i, j. To see that the
commutation relation in the range is given by u∗, apply W to (2) to get (in
the range of W ) ek⊗ fl =
i,j ū(i,j),(k,l)fj ⊗ ei =
i,j(u
∗)(k,l),(i,j)fj ⊗ ei which
is equation (1) with u∗ instead of u.
As in [5], we conclude that (Lu)′ = Ru and (Lu)′′ = Lu.
3 The character space and its core
In the following proposition we describe the structure of the character spaces
M(Lu) and M(Au) (equipped with the weak∗ topology). Similar results
were obtained in [5] for algebras defined for higher rank graphs and in [2] for
analytic Toeplitz algebras. (See also [10].)
It will be convenient to write
Vu = {(z, w) ∈ Cn × Cm : ziwj =
u(i,j),(k,l)zkwl } (3)
Ωu = Vu ∩ (Bn × Bm) (4)
where Bn is the open unit ball of C
n. We refer to Vu as the variety associated
with u.
Proposition 3.1 (1) The linear multiplicative functionals on C[F+u ] are in
one-to-one correspondence with points (z, w) in Vu.
(2) M(Au) is homeomorphic to Ωu.
(3) For (z, w) ∈ Ωu, write α(z,w) for the corresponding character of Au.
Then α(z,w) extends to a w
∗-continuous character on Lu if and only if
(z, w) ∈ Bn × Bm.
Proof. Part (1) follows immediately from (1). Fix α ∈ M(Au) and
write zi = α(Lei), 1 ≤ i ≤ n, and wi = α(Lfj ), 1 ≤ j ≤ m. From the
multiplicativity and linearity of α and (1), it follows that (z, w) ∈ Vu. Since
α is contractive and maps
i aiLei to
i aizi, it follows that ‖z‖ ≤ 1 and
similarly ‖w‖ ≤ 1. Thus (z, w) ∈ Ωu.
For the other direction, fix first (z, w) ∈ Ωu with ‖z‖ < 1 and ‖w‖ < 1.
It follows from the definition of Ωu and from (1) that (z, w) defines a linear
and multiplicative map α on the algebra C[F+u ] such that Lei is mapped into
zi and α(Lfj) = wj . Abusing notation slightly, we write α(x) for α(Lx) for
every x ∈ E⊗k ⊗ F⊗l. Also, for i = (i1, . . . , ik) and j = (j1, . . . , jl), we write
eifj for ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl. These elements form an orthonormal
basis for E⊗k ⊗ F⊗l and we now set
α(eifj)eifj ∈ F(X).
If pi ≥ 0 and p1 + . . . + pn = k then there are k!p1!···pn! terms ei1 ⊗ · · · ⊗ eik
with α(ei1 ⊗ · · · ⊗ eik) = z
2 · · · z
k . It follows that
i |α(ei)|2 =
i=(i1,...,ik)
|α(ei1)|2 · · · |α(eik)|2. Thus
‖wα‖2 =
i,j,k,l
|α(eifj)|2 = (1− ‖z‖2)−1(1− ‖w‖2)−1 <∞
Note that, for every x ∈ E⊗k ⊗ F⊗l,
〈x, wα〉 = α(x).
Thus, for e ∈ E, 〈x, L∗ewα〉 = 〈Lex, wα〉 = α(e⊗x) = α(e)α(x) = 〈α(e)wα, x〉
and, similarly 〈x, L∗fwα〉 = 〈α(f)wα, x〉 for f ∈ F . Thus 〈wα, L∗ewα〉 =
α(e)α(wα) = α(e)
|α(eifj)|2 = α(e)‖wα‖2. Similarly, 〈wα, L∗fwα〉 =
α(f)α(wα) = α(f)
|α(eifj)|2 = α(f)‖wα‖2 for f ∈ F . Thus if we write
να = wα/‖wα‖ then
α(x) = 〈Lxνα, να〉
for every x ∈ E⊗k ⊗ F⊗l (for every k, l). This shows that α is contractive
and is w∗-continuous. We can, therefore, extend it to an element of M(Lu),
also denoted α.
The analysis above shows that the image of the map α 7→ (z, w) ∈ Ωu
defined above (onM(Au)) contains Vu∩(Bn×Bm). Since M(Au) is compact
and the map is w∗-continuous, its image contains (and, thus, is equal to) Ωu.
This completes the proof of (2). To complete the proof of (3), we need
to show that, if (z, w) ∈ Ωu and the corresponding character extends to a
w∗-continuous character on Lu, then ‖z‖ < 1 and ‖w‖ < 1.
For this, write L for the w∗-closed subalgebra of Lu generated by {Le :
e ∈ E} ∪ {I}. Let P be the projection of F(E, F, u) onto F(E) = C ⊕
E ⊕ (E ⊗ E) ⊕ · · ·. Then PLP = PLuP and the map T 7→ PTP , is a
w∗-continuous isomorphism of L onto PLuP . The latter algebra is unitarily
equivalent to the algebra Ln studied in [2]. A w∗-continuous character of Lu
gives rise, therefore, to a w∗-continuous character on Ln. It follows from [2,
Theorem 2.3] that z ∈ Bn. Similarly, one shows that w ∈ Bm. �
To state the next result, we first write u(i,j) for the n×m matrix whose
k, l-entry is u(i,j),(k,l). Thus, the (i, j) row of u provides the n rows of u(i,j).
We then compute
u(i,j),(k,l)zkwl =
u(i,j),(k,l)wl)zk =
(u(i,j)w)kzk = 〈u(i,j)w, z̄〉.
Write Ei,j for the n×m matrix whose i, j-entry is 1 and all other entries are
0 (so that 〈Ei,jw, z̄〉 = ziwj) and write C(i,j) for the matrix u(i,j)−Ei,j . Then
the computation above yields the following.
Lemma 3.2 With C(i,j) defined as above, we have
Vu = {(z, w) ∈ Cn × Cm : 〈C(i,j)w, z̄〉 = 0, for all i, j}.
Definition 3.3 The core of Ωu is the subset given by
Ω0u := {(z, w) ∈ Bn × Bm : C(i,j)w = 0, Ct(i,j)z = 0 for all i, j}.
Fix (z, w) ∈ Ω0u. We have u(i,j)w = Ei,jw for all i, j. Thus, for every k,
u(i,j),(k,l)wl = δi,kwj (6)
(where δi,k is 1 if i = k and 0 otherwise) and, for a1, a2, . . . , an, in C we have
k,l u(i,j),(k,l)akwl = aiwj. Hence, if we let w̃
(i) be the vector in Cmn defined
by w̃
(k,l)
= δk,iwl, we get uw̃
(i) = w̃(i). Similarly, for z, we have
u(i,j),(k,l)zk = δj,lzi (7)
and for scalars b1, . . . , bm we have
k,l u(i,j),(k,l)blzk = bjzi. Thus, writing z̃(j)
for the vector defined by (z̃(j))(k,l) = δl,jzk, we have uz̃(j) = z̃(j). The vector
w̃(i) in Cnm = Cn ⊗ Cm is also expressible as δi ⊗ w where {δ1, . . . , δn} is
the standard basis of Cn, and, similarly, z̃(j) = z ⊗ δj . We therefore obtain
Lemma 3.4 which will be useful in Section 6.
We note also the following companion formula. Suppose (z, w) ∈ Ω0u.
Then, as we noted above, uz̃(j) = z̃(j) and, thus, u
∗z̃(j) = z̃(j). Writing this
explicitly, we have, for all i, j, l,
u(k,l),(i,j)z̄k = δj,lz̄i. (8)
Lemma 3.4 Let (z, w) be a vector in the core Ω0u. Then
span{z̃(j), w̃(i) : 1 ≤ i ≤ n, 1 ≤ j ≤ m} ⊆ Ker(u− I).
In particular,
(i) If the core contains a vector (z, w) with z 6= 0, then dim(Ker(u−I)) ≥
(ii) If the core contains a vector (z, w) with w 6= 0 then dim(Ker(u−I)) ≥
(iii) If the core contains a vector (z, w) with z 6= 0 and w 6= 0, then
dim(Ker(u− I)) ≥ m+ n− 1.
We now characterise the core in an algebraic manner in terms of repre-
sentations into the algebra T2 of upper triangular 2×2 matrices. We remark
that nest representations such as these have proven useful in the algebraic
structure theory of nonself-adjoint algebra [?], [11].
Let ρ : C[F+u ] → T2 with
ρ(T ) =
ρ1,1(T ) ρ1,2(T )
0 ρ2,1(T )
Then ρ1,1 and ρ2,2 are characters and ρ1,2 is a linear functional that satisfies
ρ1,2(TS) = ρ1,1(T )ρ1,2(S) + ρ1,2(T )ρ2,2(S) (9)
for T, S ∈ C[F+u ].
We now restrict to the case where ρ1,1 = ρ2,2. By Proposition 3.1(1),
both are associated with a point (z, w) in Vu. It follows from (9) that ρ1,2
is determined by its values on Lei and Lfj . Setting λi = ρ1,2(Lei) and µj =
ρ1,2(Lfj ), we associate with each homomorphism ρ (as discussed above) a
quadruple (z, w, λ, µ) where (z, w) ∈ Vu and, for every i, j,
ziµj + λiwj =
u(i,j),(k,l)(wlλk + µlzk). (10)
(The last equation follows from (1)). Using (5) we can write the last equation
〈u(i,j)w, λ̄〉+ 〈u(i,j)µ, z̄〉 = ziµj + λiwj = 〈Ei,jw, λ̄〉+ 〈Ei,jµ, z̄〉.
That is,
〈C(i,j)w, λ̄〉+ 〈µ, Ct(i,j)z〉 = 0. (11)
The following lemma now follows from the definition of the core.
Lemma 3.5 A point (z, w) ∈ Ωu lies in the core Ω0u if and only if every
(λ, µ) ∈ Cn × Cm defines a homomorphism ρ : C[F+u ] → T2 such that
ρ(Lei) =
zi λi
ρ(Lfj ) =
wj µj
for all i, j.
4 Automorphisms of Ln and Lu
We first derive the unitary automorphisms of Ln and An associated with
U(1, n). These were obtained by Voiculescu [14] in the setting of the Cuntz-
Toeplitz algebra. However the automorphisms restrict to an action of U(1, n)
on the free semigroup algebra. The result is rather fundamental, being a
higher dimensional version of the familiar Möbius automorphism group on
H∞. For the reader’s convenience we provide complete proofs. See also the
discussion in Davidson and Pitts [2], and in [1], [10].
Lemma 4.1 Let α ∈ Bn and write
(i) x0 = (1− ‖α‖2)−1/2,
(ii) η = x0α, and
(iii) X1 = (ICn + ηη
∗)1/2.
(1) ‖η‖2 = |x0|2 − 1,
(2) X1η = x0η, and
(3) X21 = I + ηη
In particular, the matrix X =
satisfies X∗JX = J , where J =
Proof. Part (1) is an easy computation and part (3) follows from the
definition of X1. For (2), note that X
1η = (I + ηη
∗)η = η + ‖η‖2η = x20η
and, for every ζ ∈ η⊥, X1ζ = ζ . Suppose X1η = aη + ζ (ζ ∈ η⊥). Then
x20η = X
1η = a
2η + ζ and it follows that a = x0 (as X1 ≥ 0) and ζ = 0. �
The lemma exhibits specific matrices (X1 is nonnegative) in U(1, n) asso-
ciated with points in the open ball. One can similarly check (see [2] or [10] for
example) that the general form of a matrix Z in U(1, n) is Z =
η2 Z1
where
‖η1‖2 = ‖η2‖2 = |z0|2 − 1,
Z1η1 = z̄0η2, Z
1η2 = z0η1,
Z∗1Z1 = In + η1η
1 , Z1Z
1 = In + η2η
It is these equations that are equivalent to the single matrix equation Z∗JZ =
It is well known that the map θX defined on Bn by
θX(λ) =
X1λ+ η
x0 + 〈λ, η〉
, λ ∈ Bn.
is an automorphism of Bn with inverse θX−1 . See Lemma 4.9 of [2] and Lemma
8.1 of [10] for example. We make use of this in the proof of Voiculescu’s
theorem below.
Let L1, . . . , Ln be the generators of the norm closed algebraAn and for ζ ∈
Cn write Lζ =
ζiLi. Recall that the character space M(An) is naturally
identifiable with the closed ball B̄n, with λ in this ball providing a character
φλ for which φλ(Li) = λi. The proof is a reduced version of that given above
for M(Aθ).
Theorem 4.2 Let α ∈ Bn and let X1, x0, η and X be associated with α as
in Lemma 4.1. Then
(i) there is an automorphism ΘX of Ln such that
Θα(Lζ) = (x0I + Lη)
−1(LX1ζ + 〈ζ, η̄〉I), (12)
(ii) the inverse automorphism Θ−1X is ΘX−1, and X
−1 is the matrix in
U(1, n) associated with −α,
(iii) there is a unitary UX on Fn such that for a ∈ An,
UXaξ0 = Θα(a)(x0I + Lη)
and ΘX(a) = UXaU
Proof. Let Fn be the Fock space for Ln, In = IFn , and let L̃ =
[In L1 · · ·Ln] viewed as an operator from (C⊕ Cn)⊗Fn = Fn ⊕ (Cn ⊗Fn)
to Fn. Then
L̃(J ⊗ I)L̃∗ = In − L̃L̃∗ = In − (L1L∗1 + . . . LnL∗n) = P0
where P0 is the vacuum vector projection from Fn to C. Also, since XJX =
J , we have
L̃(J ⊗ I)L̃∗ = L̃(X ⊗ In)(J ⊗ I)(X ⊗ In)L̃∗ = [Y0 Y1](J ⊗ I)[Y0 Y1]∗
where
[Y0 Y1] = [In L]
x0 ⊗ In η∗ ⊗ In
η ⊗ In X1 ⊗ In
Thus Y0Y
0 − Y1Y ∗1 = P0. Also
Y0 = x0 ⊗ In + L(η ⊗ In) = x0In + Lη,
Y1 = η
∗ ⊗ In + L(X1 ⊗ In) = η∗ ⊗ In + [LX1e1 . . . LX1en]
where, here, e1, . . . , en is the standard basis for C
The operator V = Y −10 Y1 is a row isometry [V1 · · · Vn], from Cn ⊗Fn to
Fn with defect 1. To see this we compute
I − V V ∗ = I − Y −10 Y1Y ∗1 Y ∗−10 = I − Y −10 (−P0 + Y0Y ∗0 )Y ∗−10
= I + Y −10 P0Y
0 − I = ξ
0 = Y
0 ξ0 = (x0In + Lη)
−1ξ0 = x
(x−10 Lη)
and so
‖ξ′0‖ = |x0|−2
|x0|−2j‖η‖2j =
x20 − ‖η‖2
Considering the path t → tα for 0 ≤ t ≤ 1 and the corresponding path of
partial isometries V it follows from the stability of Fredholm index that the
index of V and L coincide and so in fact V is a row isometry. Thus V1, . . . , Vn
are isometries with orthogonal ranges.
We now have a contractive algebra homomorphism An → L(Fn) deter-
mined by the correspondence Lei → Vi, i = 1, . . . , n. In fact it is an algebra
endomorphism Θ : An → An. Indeed, for ξ = (ξ1, . . . , ξn) we have
Θ(Lξ) =
ξiVi =
0 Y1(ei ⊗ In)
ζi(x0In + Lη)
−1(η∗ ⊗ In + [LX1e1 . . . LX1en])[In · · · In]t
= (x0In + Lη)
−1(〈ζ, η〉In + LX1ζ).
Thus far we have followed Voiculescu’s proof [14]. The following argument
shows that Θ is an automorphism and is an alternative to the calculation
suggested in [14]. The calculation shows that
φλ ◦ΘX = φθ
We have
φλ ◦ΘX(Lζ) = φλ((x0In + Lη)−1(〈ζ, η〉In + LX1ζ))
= (x0 + 〈λ, η〉)−1(〈ζ, η〉+ 〈X1ζ, λ〉) = φµ(Lζ)
where
X∗1λ + η
x0 + 〈λ, η〉
X1λ+ η
x0 + 〈λ, η〉
= θX(λ).
Write ΘX for the contractive endomorphism Θ of An as constructed
above. It follows that the composition Φ = ΘX−1 ◦ ΘX is a contractive
endomorphism which, by the remarks preceding the statement of the theo-
rem, induces the identity map on the character space, so that φλ = φλ ◦Φ−1
for all λ ∈ Bn. Such a map must be the identity. Indeed, suppose that we
have the Fourier series representation Φ−1(Le1) = a1Le1 + . . . + anLen + X
where X is a series with terms of total degree greater than one. It follows
t−1φ(t,0,...,0)(Φ
−1(Le1)) = a1
while
t−1φ(t,0,...,0)(Le1) = 1.
Since the induced map is the identity, we have a1 = 1 and ak = 0 for k ≥ 2.
In this way we see that the image of each Li has the form Li+Ti where Ti has
only terms of total degree greater than one. Since Liξ0 is orthogonal to Tiξ0
and Φ−1(Li) is a contraction, we have 1 ≥ ‖Φ−1(Li)ξ0‖2 = ‖Liξ0 + Tiξ0‖2 =
‖Liξ0‖2 + ‖Tiξ0‖2 = 1 + ‖Tiξ0‖2. Thus Tiξ0 = 0 and, consequently, Ti = 0
and so the composition Φ is the identity map.
Finally, we show that Θα is unitarily implemented. Define UX on Anξ0
by UXaξ0 = ΘX(a)ξ
0 = ΘX(a)(x0I + Lη)
−1ξ0 for a ∈ A. Since ΘX is an
automorphism, (UXa)bξ0 = UXabξ0 = ΘX(a)ΘX(b)ξ
0 = ΘX(a)UXbξ0, for
a, b ∈ An, and it follows that UXa = ΘX(a)UX , as linear transformations on
the dense space Anξ0.
Now, V = [V1, . . . , Vn] is a row isometry with defect space spanned by ξ
The map UX maps ξi = Liξ0 to ΘX(Li)ξ
0 = Viξ
0 and, if w = w(e1, . . . , en) is
a word in e1, . . . , en , then
UXξw = UXw(L1, . . . , Ln)ξ0 = ΘX(w(L1, . . . , Ln))ξ
0 = w(V1, . . . , Vn)ξ
Since V is a row isometry and ξ′0 is a unit wandering vector for V , it follows
that {w(V1, . . . , Vn)ξ′0} is an orthonormal set. Thus, UX is an isometry. Since
the range of UX contains UXAnξ0 = ΘX(An)ξ′0 = An(x0I + Lη)−1ξ0 = Anξ0
we see that UX is unitary. �
Remark 4.3 With the same calculations as in the proof above and slightly
more notation, one can show that each invertible matrix Z ∈ U(1, n) defines
an automorphism ΘZ and that Z → ΘZ is an action of U(1, n) on An and,
in particular, ΘZΘX = ΘZX . Moreover, Z → UZ is a unitary representation
of U(1, n) implementing this as the following calculation indicates.
Let W =
be the matrix in U(1, n) associated with β ∈ Bn
as in Lemma 4.1. Then
UXUWaξ0 = UX(Θβ(a)(w0 + Lω)
−1ξ0)
= Θα(Θβ(a)(w0 + Lω)
−1)(x0In + Lη)
= Θα(Θβ(a))Θα((w0 + Lω)
−1)(x0In + Lη)
= ΘXW (a)Θα((w0 + Lω)
−1)(x0In + Lη)
= ΘXW (a)[w0In + (x0In + Lη)
−1(LX1ω + 〈ω, η〉In)]
(x0In + Lη)
= ΘXW (a)[w0x0In + w0Lη + LX1ω + 〈ω, η〉In)]
= ΘXW (a)[(w0x0In + 〈ω, η〉)In + Lω0η+X1ω]
One readily checks that this is the same as UXW (a)ξ0
It is evident from the last theorem and its proof that the unitary auto-
morphisms of An and Ln act transitively on the open subset Bn associated
with the weak star continuous characters. We shall show that a version of
this holds for the unitary relation algebras with respect to the open core of
the character space. As a first step to constructing automorphisms of Au we
obtain unitary commutation relations for the n-tuples [Θ(Le1), . . . ,Θ(Len)]
and [Lf1 , . . . , Lfm ] for certain automorphisms Θ of the copy of An in Au.
Lemma 4.4 Suppose (z, w) ∈ Ω0u∩(Bn×Bm). Write α for z̄ and let Θ := Θα
be as in (12). Then, for every 1 ≤ i ≤ n and 1 ≤ j ≤ m,
Θ(Lei)Lfj =
u(i,j),(k,l)LflΘ(Lek). (13)
Proof. Write Y for ηη∗ and β for (x0 + 1)
−1. Since X21 = I + ηη
X1 = I + βηη
∗ = I + βY and Y = (Yi,j) where Yi,j = ηiη̄j = x
0z̄izj . We now
compute
(X1ei)fj = eifj +
βYt,ietfj = eifj +
t,k,l
βYt,iu(t,j),(k,l)flek
u(i,j),(k,l)flek +
t,k,l
βx20z̄tziu(t,j),(k,l)flek
u(i,j),(k,l)flek + βx
t,k,l
z̄tu(t,j),(k,l)flek.
Using the core equation (8), the last expression is equal to
u(i,j),(k,l)flek + βx
δj,lz̄kflek
u(i,j),(k,l)flek + βx
z̄kfjek
u(i,j),(k,l)flek + βx
(δj,lzi)z̄kflek.
Using the core equation (7), this is equal to
u(i,j),(k,l)flek + βx
u(i,j),(t,l)zt)z̄kflek
u(i,j),(k,l)flek + βx
k,l,t
u(i,j),(k,l)zkz̄tflet
u(i,j),(k,l)flek + β
k,l,t
u(i,j),(k,l)Yt,kflet
u(i,j),(k,l)flek + β
u(i,j),(k,l)flY ek
u(i,j),(k,l)flX1ek.
LX1eiLfj =
u(i,j),(k,l)LflLX1ek . (14)
Next, we compute
i z̄ieifj =
i,k,l u(i,j),(k,l)z̄iflek. Using (8), this is equal
k,l δj,lz̄kflek =
k z̄kfjek. Thus
z̄ieifj =
z̄ifjei (15)
and, hence, Lη commutes with Lfj . It follows that
Lfj (x0I − Lη)−1 = (x0I − Lη)−1Lfj . (16)
We have, using (14) and (16),
(x0I − Lη)−1LX1eiLfj =
u(i,j),(k,l)(x0I − Lη)−1LflLX1ek
u(i,j),(k,l)Lfl(x0I − Lη)
−1LX1ek .
Also, applying (7) and (16), we get
(x0I − Lη)−1〈ei, η〉Lfj = ziLfj (x0I − Lη)−1
δj,lziLfl(x0I − Lη)
u(i,j),(k,l)zkLfl(x0I − Lη)
Subtracting the last two equations, we get (13). �
Corollary 4.5 In the notation of Lemma 4.4, for every i, j,
L∗fjΘ(Lei) =
u(i,l),(k,j)Θ(Lek)L
Proof. It follows from (13) that Θ(Lei)Lfl =
k,t u(i,l),(k,t)LftΘ(Lek) for
every i, l. Thus, for i, j, l,
L∗fjΘ(Lei)LflL
u(i,l),(k,t)L
LftΘ(Lek)L
u(i,l),(k,t)δj,tΘ(Lek)L
u(i,l),(k,j)Θ(Lek)L
Summing over l, we get
L∗fjΘ(Lei)(
u(i,l),(k,j)Θ(Lek)L
l LflL
= I − P where P is the projection onto the subspace C ⊕
E ⊕ (E ⊗ E) ⊕ . . .. Note that P is left invariant under the operators in
the algebra generated by {Lei : 1 ≤ i ≤ n} and, in particular, by Θ(Lei).
Thus L∗fjΘ(Lei)P = L
PΘ(Lei)P = 0 =
k,l u(i,l),(k,j)Θ(Lek)L
P . This
completes the proof of the corollary. �
Proposition 4.6 Suppose (z, w) ∈ Ω0u ∩ (Bn × Bm). Then there is a auto-
morphism Θ̃z of Au that is unitarily implemented and such that, for every
X ∈ Au,
α(0,w)(Θ̃
z (X)) = α(z,w)(X) (17)
where α(z,w) is the character associated with (z, w) by Proposition 3.1.
Proof. Let U be the unitary operator implementing Θ. We can view
F(n,m, u) as the sum
F(n,m, u) =
F⊗k ⊗ F(E)
where F(E) = C⊕E⊕ (E⊗E)⊕· · ·. We now let V be the unitary operator
whose restriction to F⊗k ⊗F(E) is Ik ⊗U (where Ik is the identity operator
on F⊗k). It is easy to check that, for every fj ,
V LfjV
∗ = Lfj .
Now, fix i. We shall show, by induction, that, for every k and every ξ ∈
F⊗k ⊗ F(E),
(Ik ⊗ U)Leiξ = Θ(Lei)(Ik ⊗ U)ξ. (18)
For k = 0 this is just the fact that U implements Θ. Suppose we know this
for k and fix fj ∈ F . Then, for ξ ∈ F⊗k ⊗ F(E) we have,
(Ik+1 ⊗ U)LeiLfjξ =
u(i,j),(k,l)(Ik+1 ⊗ U)LflLekξ
u(i,j),(k,l)Lfl(Ik ⊗ U)Lekξ.
Applying the induction hypothesis, this is equal to
k,l u(i,j),(k,l)LflΘ(Lek)(Ik⊗
U)ξ. Using (13), this is Θ(Lei)Lfj (Ik ⊗ U)ξ = Θ(Lei)(Ik ⊗ U)Lfjξ. Since
F⊗(k+1) ⊗ F(E) is spanned by elements of the form Lfjξ (as above) the
equality follows. From the relations of Lemma 4.4 it follows that the map
Θ̃z : X → V XV ∗ defines a unitary endomorphism of Au. Since Θ is an
automorphism of An it follows that Θ̃z gives the desired automorphism. �
Clearly, in Proposition 4.6, we can interchange z and w to get the follow-
ing, where Θz,w = Θ̃zΘ̃w.
Proposition 4.7 Suppose (z, w) ∈ Ω0u ∩ (Bn ×Bm). Then there is a unitary
automorphism Θz,w of Lu which is a homeomorphism with respect to the
w∗-topologies and which restricts to an automorphism of Au. Moreover, for
every X ∈ Lu,
α(0,0)(Θ
z,w(X)) = α(z,w)(X) (19)
where α(z,w) is the character associated with (z, w) as in Proposition 3.1.
An automorphism Ψ of Au, defines a map on the character space of Au,
namely φ 7→ φ ◦Ψ−1. Thus using Proposition 3.1 we have a homeomorphism
θΨ of Ωu. Also, since Ωu ∩ (Bn × Bm) is the interior of Ωu, θΨ maps Ωu ∩
(Bn × Bm) onto itself.
Similarly, if Ψ is an automorphism of Lu which is a homeomorphism with
respect to the w∗-topologies, then θΨ is a homeomorphism of Ωu∩ (Bn×Bm).
In the following theorem we identify the relative interior of the core as the
orbit of (0, 0) under the group of maps θΨ associated with automorphisms Ψ.
Theorem 4.8 For (z, w) ∈ Bn×Bm the following conditions are equivalent.
(1) (z, w) ∈ Ω0u.
(2) There exists a completely isometric automorphism Ψ of Lu that is a
homeomorphism with respect to the w∗-topologies and restricts to an
automorphism of Au, such that θΨ(0, 0) = (z, w).
(3) There exists an algebraic automorphism Ψ of Au such that θΨ(0, 0) =
(z, w).
Proof. The proof that (1) implies (2) follows from Proposition 4.7. Clearly
(2) implies (3). It is left to show that (3) implies (1).
Given a point (z, w) ∈ Ωu, we saw in Lemma 3.5 that, for every (λ, µ)
satisfying (11) there is a homomorphism ρz,w,λ,µ : C[F
u ] → T2. For (z, w) =
(0, 0) equation (11) holds for every pair (λ, µ). Since ρ0,0,λ,µ vanishes off a
finite dimensional subspace, it is a bounded homomorphism. In fact, for
every (λ, µ), ‖ρ0,0,λ,µ‖ ≤ 1 + ‖λ‖+ ‖µ‖.
Given Ψ and (z, w) as in (3), for every (λ, µ) ∈ Cn × Cm, ρ0,0,λ,µ ◦ Ψ−1
is a homomorphism on C[F+u ] and, thus, it is of the form ρz,w,λ′,µ′ for some
(unique) (λ′, µ′) satisfying (11). Write ψ(λ, µ) = (λ′, µ′) and note that this
defines a continuous map. To prove the continuity, suppose (λn, µn) → (λ, µ)
and write ρn for ρ0,0,λn,µn and ρ for ρ0,0,λ,µ. Then (using the estimate on the
norm of ρ0,0,λ,µ) there is someM such that ‖ρn‖ ≤M for all n and ‖ρ‖ ≤M .
For every Y ∈ C[F+u ], ρn(Y ) → ρ(Y ). Now fix X ∈ Au and ǫ > 0. There is
some Y ∈ C[F+u ] such that ‖X − Y ‖ ≤ ǫ and there is some N such that for
n ≥ N ‖ρn(Y )− ρ(Y )‖ ≤ ǫ. Thus, for such n, ‖ρn(X)− ρ(X)‖ ≤ (2M +1)ǫ.
Setting X = Ψ(Lei), we get λ
n → λ′ and similarly for µ′.
If (z, w) is not in Ω0u, then the set of all (λ, µ) satisfying (11) is a subspace
of Cn ×Cm of dimension strictly smaller than n+m and, as is shown above,
it contains the continuous image (under the injective map ψ) of Cn × Cm.
This is impossible. �
5 Isomorphic algebras
In this section we shall find conditions for algebras Au and Av to be (isomet-
rically) isomorphic. The characterisation also applies to the weak star closed
algebras Lu.
We start by considering a special type of isomorphism. We shall now
assume that the set {n,m} for both algebras is the same. In fact, by inter-
changing E and F , we can assume that the corresponding dimensions are the
same and the algebras are defined on F(n,m, u) and F(n,m, v) respectively.
This assumption will be in place in the discussion below up to the end of
Lemma 5.5.
The algebra Au carries a natural Z2+-grading, with the (k, l) labeled sub-
space being spanned by products of the form Lei1Lei2 . . . LeikLfi1Lfi2 . . . Lfil .
Also, the total length of such operators provides a natural Z+-grading. Note
that an algebra isomorphism Ψ : Au → Av which respects the Z+-grading is
determined by a linear map between the spans of the generators
Le1 , . . . , Len , Lf1 , . . . , Lfm . Here we use the same notation for the generators
of Au and Av. Such an isomorphism will be called graded.
We now consider two types of graded isomorphisms, namely, either bi-
graded, as in the following definition, or, in case n = m, bigraded after
relabeling generators.
Definition 5.1 (i) An isomorphism Ψ : Au → Av is said to be bigraded
isomorphism if there are unitary matrices A (n × n) and B (m ×m)
such that
Ψ(Lei) =
ai,jLej , Ψ(Lfk) =
bk,lLfl.
(ii) If m = n and Ψ is a graded isomorphism such that
Ψ(Lei) =
ai,jLfj , Ψ(Lfk) =
bk,lLel
for n × n unitary matrices A and B then we say that Ψ is a graded
exchange isomorphism.
We write ΨA,B for the bigraded isomorphism (as in (i)) and Ψ̃A,B for the
graded exchange isomorphism.
Abusing notation, we write Ψ(ei) =
j ai,jej instead of Ψ(Lei) =
j ai,jLej
for a bigraded isomorphism (and similarly for the other expressions).
For unitary permutation matrices the following lemma was proved in [10,
Theorem 5.1(iii)].
Lemma 5.2 (i) If ΨA,B is a bigraded isomorphism then
(A⊗ B)v = u(A⊗B) (20)
where A⊗B is the mn×mn matrix whose (i, j), (k, l) entry is ai,kbj,l.
(ii) If m = n and Ψ̃A,B is a graded exchange isomorphism then
(A⊗ B)ṽ = u(A⊗B) (21)
where ṽ(i,j),(k,l) = v̄(l,k),(j,i).
Proof. Assume Ψ = ΨA,B is a bigraded isomorphism. For i, j,
Ψ(ei ⊗ fj) = (
ai,kek)⊗ (
bj,lfl) =
(A⊗ B)(i,j),(k,l)ek ⊗ fl =
k,l,r,t
(A⊗ B)(i,j),(k,l)v(k,l),(r,t)ft ⊗ er =
((A⊗ B)v)(i,j),(r,t)ft ⊗ er.
On the other hand,
Ψ(ei ⊗ fj) = Ψ(
u(i,j),(k,l)fl ⊗ ek) =
k,l,t,r
u(i,j),(k,l)bl,tak,rft ⊗ er =
(u(A⊗B))(i,j),(r,t)ft ⊗ er.
This proves equation (20). A similar argument can be used to verify equation
(21). �
Definition 5.3 If u, v are mn×mn unitary matrices and there exist unitary
matrices A and B satisfying (20), we say that u and v are product unitary
equivalent.
Now suppose that A and B are unitary matrices satisfying (20). The
same computation as in Lemma 5.2 shows that WA,B : E ⊗u F → E ⊗v F
defined by
WA,B(ei ⊗ fj) =
(A⊗ B)(i,j),(k,l)ek ⊗ fl
is a well defined unitary operator. Here the notation E ⊗u F indicates that
this is E ⊗ F as a subspace of F(n,m, u). Similarly, one defines a unitary
operator, also denoted WA,B, from E
⊗k ⊗F⊗l in F(n,m, u) to E⊗k ⊗ F⊗l in
F(n,m, v) by
WA,B(ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl) =
ai1,r1 · · · aik,rkbj1,t1 · · · bjl,tler1 ⊗ · · · ⊗ erk ⊗ ft1 ⊗ · · · ⊗ ftl .
This gives a well defined unitary operator
WA,B : F(n,m, u) → F(n,m, v).
Lemma 5.4 For every i, j, write Aei =
k ai,kek and Bfj =
l bj,lfl.
Then, for g1, g2, . . . , gr in {e1, . . . , en, f1, . . . , fm},
WA,B(g1 ⊗ g2 ⊗ · · · ⊗ gr) = Cg1 ⊗ Cg2 ⊗ · · · ⊗ Cgr (22)
where Cgi = Agi if gi ∈ {e1, . . . , en} and Cgi = Bgi if gi ∈ {f1, . . . , fm}.
Proof. If the gi’s are ordered such that the first ones are from E and the
following vectors are from F , then the result is clear from the definition of
WA,B. Since we can get any other arrangement by starting with one of this
kind and interchanging pairs gl, gl+1 successively (with gl ∈ {e1, . . . , en} and
gl+1 ∈ {f1, . . . , fm}), it is enough to show that that if (22) holds for a given
arrangement of e’s and f ’s and we apply such an interchange, then it still
holds. So, we assume gl = ek, gl+1 = fs and we write g
′ = g1 ⊗ · · · ⊗ gl−1,
g′′ = gl+2 ⊗ · · · ⊗ gr, Cg′ = Cg1 ⊗ · · · ⊗ Cgl−1 and Cg′′ = Cgl+2 ⊗ · · · ⊗ Cgr
and compute
WA,B(g
′ ⊗ fs ⊗ ek ⊗ g′′) = WA,B(
ū(i,j),(k,s)g
′ ⊗ ei ⊗ fj ⊗ g′′).
Using our assumption, this is equal to
ū(i,j),(k,s)Cg
′ ⊗ (
ai,tet)⊗ (
bj,qfq)⊗ Cg′′ =
i,j,t,q
ū(i,j),(k,s)ai,tbj,qCg
′ ⊗ et ⊗ fq ⊗ Cg′′ =
i,j,t,q,d,p
ū(i,j),(k,s)ai,tbj,qv(t,q),(d,p)Cg
′ ⊗ fp ⊗ ed ⊗ Cg′′ =
(u∗)(k,s),(i,j)(A⊗ B)(i,j),(t,q)v(t,q),(d,p)Cg′ ⊗ fp ⊗ ed ⊗ Cg′′ =
(A⊗ B)(k,s),(d,p)Cg′ ⊗ fp ⊗ ed ⊗ Cg′′ =
ak,dbs,pCg
′ ⊗ fp ⊗ ed ⊗ Cg′′ =
Cg′ ⊗ Bfs ⊗ Aek ⊗ Cg′′
completing the proof. �
The following lemma was proved in [10, Section 7] and it shows that the
necessary conditions of Lemma 5.2 are also sufficient conditions on A⊗B for
the existence of a unitarily implemented isomorphism ΨA,B.
Lemma 5.5 For unitary matrices A,B satisfying (20) and X ∈ Au, the
X 7→WA,BXW ∗A,B
is the bigraded isomorphism ΨA,B : Au → Av. Moreover ΨA,B extends to
a unitary isomorphism Lu → Lv, and similar statements holds for graded
exchange isomorphisms (when m = n).
Proof. It will suffice to show the equality
ΨA,B(X)WA,B = WA,BX
for X = Lei and for X = Lfj . Let X = Lfj and apply both sides of the
equation to ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl. Using Lemma 5.4, we get
ΨA,B(Lfj )WA,B(ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl)
bj,rLfr(Aei1 ⊗ · · · ⊗ Aeik ⊗Bfj1 ⊗ · · · ⊗ Bfjl)
= Bfj ⊗Aei1 ⊗ · · · ⊗Aeik ⊗ Bfj1 ⊗ · · · ⊗ Bfjl
=WA,B(fj ⊗ ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl)
=WA,BLfj (ei1 ⊗ · · · ⊗ eik ⊗ fj1 ⊗ · · · ⊗ fjl).
This proves the equality for X = Lfj . The proof for X = Lei is similar. �
At this point we drop our assumption that the set {n,m} is the same for
both algebras and write {n′, m′} for the dimensions associated with Av. We
shall see in Proposition 5.8 (and Remark 5.11(i)) that, if the algebras are
isomorphic, then necessarily {n,m} = {n′, m′}.
Given an isomorphism Ψ : Au → Av we get a homeomorphism θΨ : Ωu →
Ωv (as in the discussion preceeding Theorem 4.8). The arguments used in
the proof of Theorem 4.8 to show that part (3) implies part (1) apply also
to isomorphisms and thus, θΨ(0, 0) ∈ Ω0v.
Proposition 5.6 Let Ψ : Au → Av be an (algebraic) isomorphism. Then
u) = Ω
v and θΨ(Ω
u ∩ (Bn × Bm)) = Ω0v ∩ (Bn × Bm).
Proof. Fix (z, w) in Ω0u and use Theorem 4.8 to get an automorphism Φ
of Au such that θΦ(0, 0) = (z, w). But then θΨ◦Φ(0, 0) = θΨ(z, w) and, as we
noted above, this implies that θΨ(z, w) ∈ Ω0v. It follows that θΨ(Ω0u) ⊆ Ω0v
and, applying this to Ψ−1, the lemma follows. �
Lemma 5.7 The map θΨ is a biholomorphic map.
Proof. The coordinate functions for θΨ are (z, w) 7→ α(z,w)(Ψ−1(ei)) (and
(z, w) 7→ α(z,w)(Ψ−1(fj))) where α(z,w) is the character associated with (z, w)
by Proposition 3.1. For every Y ∈ C[F+v ], α(z,w)(Y ) is a polynomial in (z, w)
(for (z, w) ∈ Ωv) and, therefore, an analytic function. Each X ∈ Av is a norm
limit of elements in C[F+v ] and, thus, α(z,w)(X) is an analytic function being
a uniform limit of analytic functions on compact subsets of Ωv. Hence, for
every (z, w) ∈ Ωv, there is a power series that converges in some, non empty,
circular, neighborhoodC of (z, w) that represents α(z,w)(X) onC∩Ωv. Taking
for X the operators Ψ−1(ei) and Ψ
−1(fj), we see that θ is analytic. The same
arguments apply to θ−1. �
The facts in the following proposition obtained in [10] in the case of
permutation matrices.
Proposition 5.8 Let Ψ : Au → Av be an algebraic isomorphism and let
θΨ : Ωu → Ωv be the associated map between the character spaces. Suppose
θΨ(0, 0) = (0, 0). Then we have the following.
(1) {n,m} = {n′, m′} and we shall assume that n = n′ and m = m′
(interchanging E and F and changing u to u∗ if necessary).
(2) There are unitary matrices U (n×n) and V (m×m) such that θΨ(z, w) =
(Uz, V w) for (z, w) ∈ Ωu. (If n = m it is also possible that θΨ(z, w) =
(V w, Uz).)
(3) If Ψ is an isometric isomorphism, then Ψ is a bigraded isomorphism.
(Or, if m = n, it may be a graded exchange isomorphism).
Proof. The proof of Proposition 6.3 in [10] giving (1) and (2) in the
permutation case is based essentially on Schwarz’s lemma for holomorphic
map from the unit disc. It applies without change to the case of unitary
matrices.
For (3) we may assume m = m′ and n = n′. From (2) we have for each
Φ(Lei) = LUei +X where X is a sum of higher order terms. Since Φ(Lei) is a
contraction and LUei is an isometry it follows, as in the proof of Voiculescu’s
theorem, that X = 0. Similarly, Φ(Lfj ) = LV fj and it follows that Φ is
bigraded. �
Since every graded isomorphism Ψ satisfies θΨ(0, 0) = (0, 0), we conclude
the following.
Corollary 5.9 Every graded isometric isomorphism is bigraded if n 6= m
and otherwise is either bigraded or is a graded exchange isomorphism.
Theorem 5.10 The following statements are equivalent for unitary matrices
u, v in Mn(C)⊗Mm(C).
(i) There is an isometric isomorphism Ψ : Au → Av.
(ii) There is a graded isometric isomorphism from Ψ : Au → Av.
(iii) The matrices u, v are product unitary equivalent or (in case n = m)
the matrices u, ṽ are product unitary equivalent, where ṽ(i,j),(k,l) = v̄(l,k),(j,i).
(iv) There is an isometric w*-continuous isomorphism Γ : Lu → Lv.
Proof. Given Ψ in (i), let (z, w) = θΨ(0, 0). By Proposition 5.6 (z, w) lies in
the interior of Ω0v. By Theorem 4.8 there is a completely isometric automor-
phism Φ ofAv such that θΦ(0, 0) = (z, w) and, therefore, θΦ−1◦Ψ(0, 0) = (0, 0).
By Proposition 5.8, Φ−1 ◦Ψ is a graded isometric isomorphism and (ii) holds.
Lemma 5.2 shows that (ii) implies (iii) and Lemma 5.5 that (iii) implies (i).
Finally, (iii) implies (iv) follows from Lemma 5.5, and (iv) implies (ii) is
entirely similar to (i) implies (ii). �
Remark 5.11 The argument at the beginning of the proof of Theorem 5.10
shows that, whenever Au and Av are isomorphic, we have {n,m} = {n′, m′}.
Theorem 5.12 For n 6= m the isometric automorphisms of Au are of the
form ΨA,BΘz,w where (z, w) ∈ Ω0u and (A⊗B)u = u(A⊗B). In case n = m
the isometric automorphisms include, in addition, those of the form Ψ̃A,BΘz,w
where (A⊗ B)ũ = u(A⊗ B).
6 Special cases
6.1 The case n = m = 2
Even in the low dimensions n = m = 2 there are many isomorphism classes
and special cases. Note that the product unitary equivalence class orbit O(u)
of the 4× 4 unitary matrix u takes the form
O(u) = {(A⊗B)u(A⊗ B)∗ : A,B ∈ SU2(C)},
and so the product unitary equivalence classes are parametrised by the set of
orbits, U4(C)/Ad(SU2(C)×SU2(C)). This set admits a 10-fold parametrisa-
tion, since, as is easily checked, U4(C) and SU2(C)×SU2(C) are real algebraic
varieties of dimension 16 and 6 respectively. It follows that the isometric iso-
morphism types of the algebras Au admit a 10 fold real parametrisation, with
coincidences only for pairs O(u),O(v) with u = ṽ
We now look at some special cases in more detail. Let d = dimKer(u−I).
Case I: d = 0
For every (z, w) ∈ B2 × B2, we have (z, w) ∈ Ωu if and only if the vector
(z1w1, z1w2, z2w1, z2w2)
t lies in Ker(u− I). Thus, in case I, Ωu is as small as
possible and is equal to
Ωmin := (B2 × {0}) ∪ ({0} × B2).
It follows from Lemma 3.4 that, in this case,
Ω0u = {(0, 0)}.
By Proposition 5.8 every isometric automorphism of Au is graded and the
isometric automorphisms of Au are given by pairs (A,B) of unitary matrices
such that A⊗ B either commutes with u or intertwines u and ũ.
Case II: d = 1
When d = 1 it still follows from Lemma 3.4 that
Ω0u = {(0, 0)}
but now it is possible for Ωu to be larger than Ωmin. In fact, if the non zero
vector (a, b, c, d)t spanning Ker(u− I) satisfies ad 6= bc then Ωu = Ωmin but
if ad = bc then the matrix
is of rank one and can be written as
(z1, z2)
t(w1, w2). Thus, (z, w) ∈ Vu and Ωu contains some (z, w) with non
zero z and w.
Since Ω0u = {(0, 0)}, it is still true that isometric isomorphisms and auto-
morphisms of these algebras are graded.
Case III: d = 2
When d = 2 it is possible that Ω0u will contain non zero vectors (z, w) but, as
Lemma 3.4 shows, it does not contain a vector with both z 6= 0 and w 6= 0.
All other possibilities may occur. For example write u1, u2 and u3 for the
three diagonal matrices:
u1 = diag(1,−1,−1, 1), u2 = diag(1,−1, 1,−1)
u3 = diag(1, 1,−1,−1).
Using the definition of the core, we easily see that
Ω0u1 = {(0, 0)}, Ω
= {(0, 0, w1, 0) : |w1| ≤ 1}
Ω0u3 = {(z1, 0, 0, 0) : |z1| ≤ 1}.
Thus, the only isometric automorphisms of Au1 are graded, the isomet-
ric automorphisms of Au2 are formed by composing graded automorphisms
with automorphisms of the type described in Proposition 4.7 (with z = (0, 0)
and w = (w1, 0)). Similarly, for the automorphisms of Au3, we use Proposi-
tion 4.6.
Case IV: d = 3
In this case we are able to obtain an explicit 2-fold parametrization of the
isomorphism types of the algebra Au.
Every 4×4 unitary matrix u with dim(Ker(u− I)) = 3 is determined by
a unit eigenvector x and its (different from 1) eigenvalue. So that ux = λx,
‖x‖ = 1, |λ| = 1 and λ 6= 1. Suppose u and v are product unitary equivalent;
that is
(A⊗ B)u = v(A⊗B)
for unitary matrices A,B, and write x, λ for the unit eigenvector and eigen-
value of u. (Of course, x is determined only up to a multiple by a scalar
of absolute value 1). Then y = (A ⊗ B)x is a unit eigenvector of v with
eigenvalue λ. For unit vectors x, y (in C4) we write x ∼ y if there are unitary
(2 × 2) matrices A,B with y = (A ⊗ B)x. For the statement of the next
lemma recall that the entries of the vectors x and y in C4 are indexed by
{(i, j) : 1 ≤ i, j ≤ 2}.
Lemma 6.1 For a vector x = {x(i,j)} in C4, write c(x) for the 2× 2 matrix
c(x) =
x(1,1) x(1,2)
x(2,1) x(2,2)
Then x ∼ y if and only if there are unitary matrices A,B such that c(x) =
Ac(y)B. (In this case, we shall write c(x) ∼ c(y).)
Proof. Suppose y = (A⊗B)x for some unitary matrices A = (ai,j) and B =
(bi,j). Then c(y)i,j = y(i,j) =
(A ⊗ B)(i,j),(k,l)x(k,l) =
k,l ai,kbj,lc(x)k,l =
(Ac(x)B)i,j. �
Using the polar decomposition c(x) = U |c(x)| and diagonalizing |c(x)| =
V ∗, we find that c(x) ∼
= c(y) where y = (a, 0, 0, d)
and a, d ≥ 0. Then a, d (the eigenvalues of |c(x)|) are uniquely determined
once we choose them such that a ≤ d and, if ‖x‖ = 1, then a2 + d2 = 1
(so that 0 ≤ a ≤ 1/
2 and a determines d). In this way, we associate to
each unitary matrix u as above a pair (a, λ) with 0 ≤ a ≤ 1/
2, λ 6= 1
and |λ| = 1. Using Lemma 6.1 and the discussion preceeding it, we have the
following.
Corollary 6.2 For every 4× 4 unitary matrix u with dim(Ker(u− I)) = 3,
there are numbers λ (with |λ| = 1 and λ 6= 1) and a (0 ≤ a ≤ 1/
2) such
that u and v are product unitary equivalent if and only if they have the same
a, λ.
Proof. Let u and v be unitary matrices with dim(Ker(u− I)) = 3 and let
(a, λ), (b, µ) be the pairs associated to u and v (respectively) as above. Also
write x for the unit eigenvector of u associated to the eigenvalue λ and let y
be the unit eigenvector of v associated to µ.
Suppose u and v are product unitarily equivalent. Then they are unitary
equivalent and, thus, λ = µ. Write (A⊗B)u = v(A⊗B) for unitary matrices
A,B. As we saw above, y can be chosen to be (A⊗ B)x so that x ∼ y and,
by Lemma 6.1, c(x) ∼ c(y). It follows that a = b.
Conversely, assume that a = b and λ = µ. Then c(x) ∼ c(y) and, thus,
x ∼ y so we can write y = (A⊗B)x for some unitary matrices A,B. Writing
v′ = (A⊗B)u(A⊗B)∗, we find that y is the unit eigenvector of v′ associated
to λ. Thus v = v′, completing the proof. �
For every a, λ as in Corollary 6.2 we let u(a, λ) be the following 4 × 4
matrix.
u(a, λ) =
(λ− 1)a2 + 1 0 0 (λ− 1)a(1− a2)1/2
0 1 0 0
0 0 1 0
(λ− 1)a(1− a2)1/2 0 0 λ+ (1− λ)a2
It is a straightforward computation to verify that dim(Ker(u− I)) = 3 and
that λ is an eigenvalue of u(a, λ) with eigenvector (a, 0, 0, (1− a2)1/2)t. Thus
the pair associated to u(a, λ) is a, λ and we have
Corollary 6.3 Every matrix u with dim(Ker(u−I)) = 3 is product unitary
equivalent to a unique matrix of the form u(a, λ) (with 0 ≤ a ≤ 1/
2, |λ| = 1
and λ 6= 1).
Using the definition of the core, we immediately get the following.
Proposition 6.4 If a = 0, |λ| = 1, λ 6= 1, then Ωu(0,λ) is the union
{(z1, z2, w1, 0) : z ∈ B2; |w1| ≤ 1} ∪ {(z1, 0, w1, w2) : w ∈ B2; |z1| ≤ 1},
Ω0u(0,λ) = {(z1, 0, w1, 0) : |z1| ≤ 1; |w1| ≤ 1}.
If a 6= 0 then
Ωu(a,λ) = {(z1, z2, w1, w2) : az1w1 + (1− a2)1/2z2w2 = 0, (z, w) ∈ B2 × B2}
Ω0u(a,λ) = {(0, 0)}.
Proof. The space Ωu(a,λ) consists of points (z, w) for which
(z1w1, z1w2, z2w1, z2w2)
t = u(a, λ)(z1w1, z1w2, z2w1, z2w2)
that is, for which
((λ− 1)a2 + 1)z1w1 + (λ− 1)a(1− a2)1/2z2w2 = z1w1,
(λ− 1)a(1− a2)1/2z1w1 + (λ+ (λ− 1)a2)z2w2 = z2w2.
If a = 0 this implies z2w2 = 0, while if a 6= 0 then (z1w1, 0, 0, z2w2) is a fixed
vector for u(a, λ) and so for some scalar µ (z1w1, z2w2) = µ((1− a2)1/2,−a).
The descriptions of Ωu(a,λ) follows.
From the definition of the core and the fact that here C12 = C21 = 0 and
C11 =
(λ− 1) 0
0 (λ− 1)a(1− a2)1/2
C22 =
(λ− 1)a(1− a2)1/2 0
0 (λ− 1) + (λ− 1)a2
we see that for a = 0 we have w2 = z2 = 0 while for a 6= 0, z1 = z2 = w1 =
w2 = 0. �
Recall that, for a 4 × 4 unitary matrix v we defined the matrix ṽ by
ṽ(i,j),(k,l) = v̄(l,k),(j,i) and showed (Corollary 5.10) that Au and Av are isomet-
rically isomorphic if and only if either u and v or u and ṽ are product unitary
equivalent.
Now, it is easy to check that ũ(a, λ) = u(a, λ̄) and so, using Proposi-
tion 3.3 and previous results, we obtain the following.
Theorem 6.5 Let 0 ≤ a, b ≤ 1/
2, |λ| = |µ| = 1, λ, µ 6= 1. Then
(1) Au(a,λ) and Au(b,µ) are isometrically isomorphic if and only if a = b and
λ equals either µ or µ̄.
(2) When a 6= 0 the isometric automorphisms of Au(0,λ) are all bigraded
(3) If a = 0 then there are isometric isomorphisms that are not graded
Case V: d = 4
This is the case where u = I. We have Ωu = Ω
u = Bn×Bm and the isometric
automorphisms are obtained by composing graded automorphisms and the
automorphisms described by Proposition 4.6, Proposition 4.7.
6.2 Permutation unitary relation algebras
With more structure assumed for a class of unitaries u it may be possible to
derive an appropriately more definitive classification of the algebras Au. We
indicate this now for the class of permutation unitaries. A fuller discussion
is in [10].
Let θ ∈ S4, viewed as a permutation of the product set {1, 2} × {1, 2} =
{11, 12, 21, 22}. Associate with θ the matrix uθ = u(i,j),(k,l) where u(i,j),(k,l) =
1 if (k, l) = θ(i, j) and is zero otherwise. If τ ∈ S4 is product conjugate to θ in
the sense that τ = σθσ−1 with σ in S2×S2, then it follows that uτ and uθ are
product unitarily equivalent. Thus we need only consider product conjugacy
classes. It turns out that these classes are the same as the product unitary
equivalence classes of the matrices uθ.
It can be helpful to view a permutation θ in Snm as a permutation of the
entries of an n ×m rectangular array, since product conjugacy corresponds
to conjugation through row permutations and column permutations. Con-
sidering this for n = m = 2 one can verify firstly that there are at most
9 isomorphism types for the algebras Atheta corresponding to the following
permutations:
θ1 = id, θ2 = (11, 12), θ3 = (11, 22),
θ4a = (11, 22, 12), θ4b = θ
4a = (11, 12, 22), θ5 = ((11, 12), (21, 22)),
θ6 = ((11, 22), (12, 21)), θ7 = (11, 12, 22, 21), θ8 = (11, 12, 21, 22).
The Gelfand spaces of the algebras Aθ (and Lθ) distinguish all of these al-
gebras except for the pairs {θ4a, θ4b} and {θ7, θ8}. However, one can verify
in both cases that neither the pair u, v nor the pair u, ṽ are product unitary
equivalent. Theorem 5.10 now applies to yield the following result from [10].
Theorem 6.6 For n = m = 2 there are 9 isometric isomorphism classes for
the algebras Aθ and for the algebras Lθ.
To a higher rank graph (Λ, d) in the sense of Kumjian and Pask [6] one can
associate nonself-adjoint Toeplitz algebra AΛ,LΛ, as in Kribs and Power [5].
In the single vertex rank 2 case it is easy to see that AΛ is equal to the algebra
Au for some permutation matrix u = θ in Snm. Thus Theorem 5.10 classifies
these algebras in terms of product unitary equivalence restricted to Snm as
stated formally in the next theorem. In the rank 2 case this is a significant
improvement on the results in [10] which, although covering general rank,
were restricted to the case of trivial core for the character space. With θ̃ the
permutation for the permutation matrix ũθ (which corresponds to generator
exchange) we have:
Theorem 6.7 Let Λ1 and Λ2 be single vertex 2-graphs with relations de-
termined by the permutations θ1 and θ2. Then the rank 2 graph algebras
AΛ1,AΛ2 are isometrically isomorphic if and only if the pair θ1, θ2 or the
pair θ1, θ̃2 are product unitary equivalent
It is natural to expect that as in the (2, 2) case product unitary equiva-
lence will correspond to product conjugacy.
7 Au as a subalgebra of a tensor algebra
Let En be the Toeplitz extension of the Cuntz algebra On and write H for
the Fock space associated with E (that is, H = C ⊕ E ⊕ (E ⊗ E) ⊕ · · ·).
Note that En acts naturally on H ( by the “shift” or “creation” operators
Li = Lei, 1 ≤ i ≤ n). In fact, Le1, . . . , Len generate En as a C∗-algebra.
Consider also the space F(F )⊗H = H⊕(F⊗H)⊕((F⊗F )⊗H)⊕· · ·. This
space is isomorphic to F(E, F, u) and we write w : F(F )⊗H → F(E, F, u)
for the isomorphism. It will be convenient to write wk for the restriction of
w to the summand F⊗k⊗H (which is an isomorphism onto its image). Note
that, for a fixed k, {w∗kLeiwk : 1 ≤ i ≤ n } is a set of n isometries with
orthogonal ranges. Thus it defines a representation ρk of En on F⊗k⊗H (with
ρk(Lei) = w
kLeiwk). (Note that we are using Lei for the creation operators
both on H and on F(E, F, u). This should cause no confusion). We also
write ρ∞ for the representation
k ⊕ρk of En on F(F )⊗H (where ρ0 is the
representation of En on H).
Let X be the column space Cm(En). This is a C∗-module over En. As a
vector space it is the direct sum of m copies of En. The right module action
of En on X is given by (ai) · b = (aib) and the En-valued inner product is
〈(ai), (bi)〉 =
i bi. For every 1 ≤ i ≤ n, we write S̃i for the operator in
L(X) defined by
S̃i(aj)
j=1 = (
u(i,j),(k,l)Lekaj)
Note that
u(i,j),(k,l)Lekaj)
l=1, (
j′,k′
u(i,j′),(k′,l)Lek′ bj′)
l=1〉 =
j,j′,k,k′,l
ū(i,j),(k,l)a
Lek′ bj′u(i,j′),(k′,l) =
(uu∗)(i,j′),(i,j)a
jbj′ =
a∗jbj
= 〈(aj), (bj′)〉.
Thus S̃i is an isometry. A similar computation shows that these isometries
have orthogonal ranges and, thus, this family defines a ∗-homomorphism
ϕ : En → L(X), with ϕ(Lei) = S̃i, 1 ≤ i ≤ n, making X a C∗-correspondence
over En (in the sense of [8] and [7]). Once we have a correspondence we can
formX⊗X and, more generally, X⊗k. Recall that to define X⊗X one defines
the sesquilinear form 〈x⊗y, x′⊗y′〉 = 〈y, ϕ(〈x, x′〉)y′〉 on the algebraic tensor
product and then lets X ⊗X be the Hausdorff completion. The right action
of En on X ⊗X is (x⊗ y) · a = x⊗ (y · a) and the left action is given by the
map ϕ2.
ϕ2(a)(x⊗ y) = ϕ(a)x⊗ y.
The definition of X⊗k is similar (and the left action map is denoted ϕk)
For k = 0 we set X⊗0 = En and ϕ0 is defined by left multiplication . Also
write ϕ∞ for
k ⊕ϕk, the left action of En on F(X).
One can then define the Hilbert spaceX⊗k⊗EnH by defining the sesquilin-
ear form 〈x⊗h, y⊗k〉 = 〈h, 〈x, y〉k〉 (x, y ∈ X⊗k) and applying the Hausdorff
completion.
Now define the map
v : X ⊗En H → F ⊗H
by setting
v((ai)⊗ h) =
fi ⊗ aih.
It is straightforward to check that this map is a well defined Hilbert space
isomorphism. By induction, we also define maps vk : X
⊗k⊗En H → F⊗k⊗H
vk+1((aj)⊗ z) =
fj ⊗ vk((ϕk(aj)⊗ IH)z) (23)
for z ∈ X⊗k ⊗En H and v0 is the identity map from En ⊗En H (which is
isomorphic to H) and F⊗0 ⊗ H = H . Assume that vk is a Hilbert space
isomorphism of X⊗k ⊗En H onto F⊗k ⊗ H and compute, for (aj), (bj) ∈ X
and z, z′ ∈ X⊗k ⊗H ,
〈vk+1((aj)⊗z), vk+1((bj)⊗z)〉 =
〈fj⊗vk((ϕk(aj)⊗IH)z), fj′⊗vk((ϕk(bj′)⊗IH)z′)〉 =
〈vk((ϕk(aj)⊗ IH)z), vk((ϕk(bj)⊗ IH)z′)〉 =
〈z, (ϕk(a∗jbj)⊗ IH)z′)〉 =
〈(aj)⊗ z, (bj)⊗ z′〉.
Thus, by induction, each map vk is a Hilbert space isomorphism and, sum-
ming up, we get a Hilbert space isomorphism
v∞ :=
⊕vk : F(X)⊗En H → F(F )⊗H.
Lemma 7.1 v∞ is a Hilbert space isomorphism and intertwines the actions
of En. That is,
v∞ ◦ (ϕ∞(a)⊗ IH) = ρ∞(a) ◦ v∞
for a ∈ En.
Proof. We show that, for every p ≥ 0 and a ∈ En, we have
vp ◦ (ϕp(a)⊗ IH) = ρp(a) ◦ vp. (24)
The proof will proceed by induction on p. For p = 0 this is clear so we now
assume that it holds for p. For 1 ≤ i ≤ n, (aj) ∈ X and z ∈ X⊗p⊗H , we have
vp+1((ϕp+1(Lei)⊗ IH)((aj)⊗ z)) = vp+1(ϕ(Lei)(aj)⊗ z) =
l,k,j u(i,j),(k,l)fl ⊗
vp((ϕp(Lekaj)⊗ IH)z). Using the induction hypothesis, this is equal to
l,k,j
u(i,j),(k,l)fl ⊗ ρp(Lek)ρp(aj)vpz =
l,k,j
u(i,j),(k,l)fl ⊗ w∗pLekwpρp(aj)vpz =
l,k,j
u(i,j),(k,l)fl ⊗ ekρp(aj)vpz = w∗∞
ei ⊗ fj ⊗ ρp(aj)vpz =
ρp+1(Lei)w
fj ⊗ ρp(aj)vpz.
Using the induction hypothesis again, we get ρp+1(Lei)w
j fj⊗vp((ϕp(aj)⊗
IH)z) = ρp+1(Lei)vp+1((aj)⊗z). This proves (24) for p+1 and the generators
of En. Since both ρp+1 and vp+1(ϕp+1(·)⊗IH)v∗p+1 are ∗-homomorphisms, (24)
holds for p + 1 and every a ∈ En, completing the induction step. Thus, (24)
holds for every p and this implies the statement of the lemma. �
Write δl for the vector (aj) in X such that al = I and aj = 0 if l 6= j.
The tensor algebra T+(X) is generated by the operators Tδl (where Tδl is the
creation operator on F(X) associated with δl) and the C∗-algebra ϕ∞(En).
The latter algebra is generated (as a C∗-algebra) by the operators ϕ∞(Li)
where {Li} is the set of generators of En.
We have
Lemma 7.2 For every 1 ≤ i ≤ n and 1 ≤ j ≤ m and k ≥ 0,
(i) w ◦ vk ◦ (ϕ∞(Li)⊗ IH) = Lei ◦ w ◦ vk.
(ii) w ◦ vk+1 ◦ (Tδj ⊗ IH) = Lfj ◦ w ◦ vk.
Proof. Part (i) follows from (24) and part (ii) from (23) (with δj in place
of (aj)). �
Recalling that w ◦ v∞ is a unitary operator mapping F(X) ⊗ H onto
F(E, F, u), we get
Theorem 7.3 (1) The algebra Au is unitarily isomorphic to the (norm
closed) subalgebra of the tensor algebra T+(X) that is generated by
{ϕ∞(Li), Tδj : 1 ≤ i ≤ n, 1 ≤ j ≤ m}.
(2) The (norm closed) subalgebra of B(F(E, F, u)) that is generated by
{Lei, L∗ei, Lfj : 1 ≤ i ≤ n, 1 ≤ j ≤ m } is unitarily isomorphic to the
tensor algebra T+(X) (and contains Au).
(2) The (norm closed) subalgebra of B(F(E, F, u)) that is generated by
{Lei, L∗fj , Lfj : 1 ≤ i ≤ n, 1 ≤ j ≤ m } is unitarily isomorphic to a
tensor algebra T+(Y ) (and contains Au).
Proof. Parts (1) and (2) follow from Lemma 7.2. For part (3), note
that one can interchange the roles of E and F . More precisely, one defines
the C∗-module Y over Em to be Y = Cn(Em) and the left action of Em on
Y by ϕY (Lfl)(bk)
k=1 = (
j,k ū(i,j),(k,l)Lfjbk)
i=1. This makes Y into a C
correspondence over Em and the rest of the proof proceeds along similar lines
as above. �
Suppose m = 1. Then X is the correspondence associated with the
automorphism α of En given by mapping Ti to
j=1 ui,jTj (note that u, in
this case, is an n × n matrix). The tensor algebra T+(X) is the analytic
crossed product En ×α Z+ and Au is unitarily isomorphic to the subalgebra
of this analytic crossed product that can be written An×α Z+. One can also
embed Au in T+(Y ) (as in Corollary 7.3(3)). Here Em is simply the (classical)
Toeplitz algebra T and Y = Cn(T ) with ϕY (Tz)(bk)k = (
k ūi,kTzbk)i (where
Tz is the generator of T ).
Remark 7.4 Since the automorphisms Θz,w and ΨA,B of Au are both uni-
tarily implemented, they can be extended to T+(X). It is easy to check that
they map T+(X) into itself and, thus, are automorphisms of T+(X). Hence,
at least when n 6= m, every automorphism of Au can be extended to an auto-
morphism of the tensor algebra T+(X) that contains it (see Theorem 5.12).
References
[1] K.R. Davidson, Free Semigroup Algebras : a survey. Systems, approxi-
mation, singular integral operators, and related topics (Bordeaux, 2000),
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[2] K.R. Davidson and D.R. Pitts, The algebraic structure of noncommuta-
tive analytic Toeplitz algebras, Math. Ann. 311 (1998), 275-303.
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(1991), 292-304.
[10] S.C. Power, Classifying higher rank analytic Toeplitz algebras, preprint
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tions of commuting CP maps, J. Funct. Anal.235 (2006), 593-618.
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http://arxiv.org/abs/math/0604630
http://arxiv.org/abs/math/0504129
Introduction
Preliminaries
The character space and its core
Automorphisms of Ln and Lu
Isomorphic algebras
Special cases
The case n=m=2
Permutation unitary relation algebras
Au as a subalgebra of a tensor algebra
|
0704.0080 | Shaping the Globular Cluster Mass Function by Stellar-Dynamical
Evaporation | THE ASTROPHYSICAL JOURNAL, 679:1272–1287, 2008 JUNE 1
Preprint typeset using LATEX style emulateapj v. 08/22/09
SHAPING THE GLOBULAR CLUSTER MASS FUNCTION BY STELLAR-DYNAMICAL EVAPORATION
DEAN E. MCLAUGHLIN1,2 AND S. MICHAEL FALL3,4
The Astrophysical Journal, 679:1272–1287, 2008 June 1
ABSTRACT
We show that the globular cluster mass function (GCMF) in the Milky Way depends on cluster half-mass
density, ρh, in the sense that the turnover mass MTO increases with ρh while the width of the GCMF decreases.
We argue that this is the expected signature of the slow erosion of a mass function that initially rose towards low
masses, predominantly through cluster evaporation driven by internal two-body relaxation. We find excellent
agreement between the observed GCMF—including its dependence on internal density rhoh, central concen-
tration c, and Galactocentric distance rgc—and a simple model in which the relaxation-driven mass-loss rates
of clusters are approximated by −dM/dt = µev ∝ ρ
h . In particular, we recover the well-known insensitivity
of MTO to rgc. This feature does not derive from a literal “universality” of the GCMF turnover mass, but rather
from a significant variation of MTO with ρh—the expected outcome of relaxation-driven cluster disruption—
plus significant scatter in ρh as a function of rgc. Our conclusions are the same if the evaporation rates are
assumed to depend instead on the mean volume or surface densities of clusters inside their tidal radii, as
µev ∝ ρ
t or µev ∝ Σ
t —alternative prescriptions that are physically motivated but involve cluster properties
(ρt and Σt) that are not as well defined or as readily observable as ρh. In all cases, the normalization of µev
required to fit the GCMF implies cluster lifetimes that are within the range of standard values (although falling
towards the low end of this range). Our analysis does not depend on any assumptions or information about
velocity anisotropy in the globular cluster system.
Subject headings: galaxies: star clusters—globular clusters: general
1. INTRODUCTION
The mass functions of star cluster systems provide an im-
portant point of reference for attempts to understand the con-
nection between old globular clusters (GCs) and the young
massive clusters that form in local starbursts and galaxy merg-
ers. When expressed as the number per unit logarithmic mass,
dN/d log M, the GC mass function (GCMF) is character-
ized by a peak, or turnover, at a mass MTO ≈ 1–2× 10
that is empirically very similar in most galaxies. By con-
trast, the mass functions of young clusters show no such fea-
ture but instead rise monotonically towards low masses over
the full observed range (106 M⊙ & M & 10
4 M⊙ in the best-
studied cases), in a way that is well described by a power law,
dN/d log M ∝ M1−β with β ≃ 2 (e.g., Zhang & Fall 1999).
At the same time, for high M > MTO, old GCMFs
closely resemble the mass functions of young clusters, and
of molecular clouds in the Milky Way and other galaxies
(Harris & Pudritz 1994; Elmegreen & Efremov 1997); and
it is well known that a number of dynamical processes
cause star clusters to lose mass and can lead to their com-
plete destruction as they orbit for a Hubble time in the
potential wells of their parent galaxies (e.g., Fall & Rees
1977; Caputo & Castellani 1984; Aguilar, Hut, & Ostriker
1988; Chernoff & Weinberg 1990; Gnedin & Ostriker 1997;
Murali & Weinberg 1997). It is therefore natural to ask
whether the peaks in GCMFs can be explained by the deple-
1 Dept. of Physics and Astronomy, University of Leicester, University
Road, Leicester, UK LE1 7RH
2 Permanent address: Astrophysics Group, Lennard-Jones Lab-
oratories, Keele University, Keele, Staffordshire, UK ST5 5BG;
[email protected]
3 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08450
4 Permanent address: Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218; [email protected]
tion over many Gyr of globulars from initial mass distribu-
tions that were similar to those of young clusters below MTO
as well as above.
Our chief purpose in this paper is to establish and inter-
pret an aspect of the Galactic GCMF that appears fundamen-
tal but has gone largely unnoticed to date: dN/d log M has
a strong and systematic dependence on GC half-mass den-
sity, ρh ≡ 3M/8πr
h (rh being the cluster half-mass radius), in
the sense that the turnover mass MTO increases and the width
of the distribution decreases with increasing ρh. As observed
facts, these must be explained by any theory of the GCMF. We
argue here that they are an expected signature of slow dynam-
ical evolution from a mass function that initially increased
towards M < MTO, if the long-term mass loss from surviv-
ing GCs has been dominated by stellar escape due to internal,
two-body relaxation (which we refer to from now on as either
relaxation-driven evaporation or simply evaporation).
Fall & Zhang (2001; hereafter FZ01) explain in detail why
cluster evaporation dominates the long-term evolution of the
low-mass shape of observable GCMFs. Briefly, stellar evolu-
tion removes (through supernovae and winds) the same frac-
tion of mass from all clusters of a given age, and so cannot
change the shape of dN/d log M (unless special initial con-
ditions are invoked; cf. Vesperini & Zepf 2003). Meanwhile,
for GCs like those that have survived for a Hubble time in
the Milky Way, the mass loss from gravitational shocks dur-
ing disk crossings and bulge passages is generally less than
that due to evaporation for M < MTO (FZ01; Prieto & Gnedin
2006).5
As we discuss further in §2 below, the evaporation of tidally
5 It is possible that there existed a past population of GCs with low den-
sities or concentrations, or perhaps on extreme orbits, that were destroyed in
less than a Hubble time by shocks or stellar evolution. Our discussion does
not cover such clusters.
http://arxiv.org/abs/0704.0080v4
2 McLAUGHLIN & FALL
limited clusters proceeds at a rate, µev ≡ −dM/dt, that is ap-
proximately constant in time and primarily determined by
cluster density. FZ01 show that a constant mass-loss rate
leads to a power-law scaling dN/d log M ∝ M1−β with β → 0
(corresponding to a flat distribution of clusters per unit lin-
ear mass) at sufficiently low M < µevt in the evolved mass
function of coeval GCs that began with any nontrivial ini-
tial dN/d log M0.
6 To accommodate this when dN/d log M0
originally increased towards low masses as a power law, a
time-dependent peak must develop in the GCMF at a mass
of order MTO(t) ∼ µevt (FZ01). But then, since µev depends
fundamentally on cluster density, so too must MTO.
A β ≃ 0 power-law scaling below the turnover mass has
been confirmed directly in the GCMFs of the Milky Way
(FZ01) and the giant elliptical M87 (Waters et al. 2006), while
Jordán et al. (2007) show it to be consistent with dN/d log M
data for 89 Virgo Cluster galaxies, and it is apparent in deep
observations of some other GCMFs (e.g., in the Sombrero
galaxy, M104; Spitler et al. 2006). As regards the peak itself,
old GCs are observed (e.g., Jordán et al. 2005) to have rather
similar densities on average—and, therefore, similar typical
µev—in galaxies with widely different total luminosities and
Hubble types. (Inasmuch as cluster densities are set by tides,
this is probably related to the mild variation of mean galaxy
density with total luminosity; see FZ01, and also Jordán et al.
2007.) Thus, an evaporation-dominated evolutionary origin
for a turnover in the GCMF appears to be consistent with
the well-known fact that the mass scale MTO generally dif-
fers very little among galaxies (e.g., Harris 2001; Jordán et al.
2006).
If this picture is basically correct, it implies that, even
though MTO may appear nearly universal when considering
the global mass functions of entire GC systems, in fact the
GCMFs of subsamples of clusters with similar ages but dif-
ferent densities should have different turnovers. In §2, we
show—working for definiteness and relatively easy observ-
ability with the half-mass density, ρh—that this is the case for
globulars in the Milky Way. We fit the observed dN/d log M
for GCs in bins of different ρh with models assuming that
(1) the initial distribution increased as a β = 2 power law at
low masses and (2) the mass-loss rates of individual clusters
can be estimated from their half-mass densities by the rule
µev ∝ ρ
h . In §3 we discuss the validity of this prescription
for µev, which is certainly approximate but captures the main
physical dependence of relaxation-driven mass loss. In par-
ticular, we show that the alternative mass-loss laws µev ∝ ρ
and µev ∝ Σ
t —where ρt and Σt are the mean volume and
surface densities inside cluster tidal radii—lead to models for
the GCMF that are essentially indistinguishable from those
based on µev ∝ ρ
h . The normalization of µev required to fit
the observed GCMF implies cluster lifetimes that are within
a factor of ≈ 2 (perhaps slightly on the low side, if the ini-
tial power-law exponent at low masses was β = 2) of typical
values in theories and simulations of two-body relaxation in
tidally limited GCs.
We also show in §2 that when the observed densities of in-
dividual clusters are used in our models to predict GCMFs
in different bins of Galactocentric radius (rgc), they fit the
6 Throughout this paper, we use “initial” to mean at a relatively early time
in the development of long-lived clusters, after they have dispersed any rem-
nants of their natal gas clouds, survived the bulk of stellar-evolution mass
loss, and come into virial equilibrium in the tidal field of a galaxy.
much weaker variation of dN/d log M and MTO as functions
of rgc, which is well-known in the Milky Way and other large
galaxies (see Harris 2001; Harris, Harris, & McLaughlin
1998; Barmby, Huchra, & Brodie 2001; Vesperini et al. 2003;
Jordán et al. 2007). Similarly, applying our models to the
GCs in two bins of central concentration, with only the mea-
sured ρh of the clusters in each subsample as input, suffices
to account for previously noted differences between the mass
functions of low- and high-concentration Galactic globulars
(Smith & Burkert 2002). The most fundamental feature of the
GCMF therefore appears to be its dependence on cluster den-
sity, which can be understood at least qualitatively (and even
quantitatively, to within a factor of 2) in terms of evaporation-
dominated cluster disruption.
There is a widespread perception that if the GCMF evolved
slowly from a rising power law at low masses, then a weak
or null variation of MTO with rgc can be achieved only in GC
systems with strongly radially anisotropic velocity distribu-
tions, which are not observed (see especially Vesperini et al.
2003). This apparent inconsistency has been cited to bol-
ster some recent attempts to identify a mechanism by which
a “universal” peak at MTO ∼ 10
5 M⊙ might have been im-
printed on the GCMF at the time of cluster formation, or
very shortly afterwards, and little affected by the subsequent
destruction of lower-mass GCs (e.g., Vesperini & Zepf 2003;
Parmentier & Gilmore 2007). However, given the real suc-
cesses of an evaporation-dominated evolutionary scenario for
the origin of MTO, as summarized above and added to below,
it would be premature to reject the idea in favor of requiring a
near-formation origin, solely on the basis of difficulties with
GC kinematics. (And, in any event, formation-oriented mod-
els must now be reconsidered in light of the non-universality
of MTO as a function of cluster density.)
We are not concerned in this paper with velocity anisotropy
in GC systems, because we only predict an evaporation-
evolved dN/d log M as a function of cluster density (and age)
and take the observed distribution of ρh versus rgc in the Milky
Way as a given, to show consistency with the observed be-
havior of MTO as a function of rgc. Most other models (FZ01;
Vesperini et al. 2003; and references therein) predict dynam-
ically evolved GCMFs directly in terms of rgc, and in doing
so are forced also to derive theoretical dependences of cluster
density on rgc. It is only at this stage that GC orbital distri-
butions enter the problem, and then only in conjunction with
several other assumptions and simplifications. As we discuss
further in §3 below, the radially biased GC velocity distribu-
tions that appear in such models could well be consequences
of one or more of these other assumptions, rather than of the
main hypothesis about evaporation-dominated GCMF evolu-
tion.
2. THE GALACTIC GCMF AS A FUNCTION OF
CLUSTER DENSITY
In this section we define and model the dependence of
the Galactic GCMF on cluster density. First, we describe
the dependence that is expected to arise from evaporation-
dominated evolution.
Two-body relaxation in a tidally limited GC leads to a
roughly steady rate of mass loss, µev ≡ −dM/dt ≃ constant
in time. Thus, the total cluster mass decreases approxi-
mately linearly, as M(t) ≃ M0 −µevt. This behavior is exact
in some classic models of GC evolution (Hénon 1961) and
is found to be a good approximation in most other calcula-
tions (e.g., Lee & Ostriker 1987; Chernoff & Weinberg 1990;
SHAPING THE GCMF BY EVAPORATION 3
Vesperini & Heggie 1997; Gnedin, Lee, & Ostriker 1999;
Baumgardt 2001; Giersz 2001; Baumgardt & Makino 2003;
Trenti, Heggie, & Hut 2007). The result comes from a variety
of computational methods (semi-analytical, Fokker-Planck,
Monte Carlo, and N-body simulation) applied to clusters with
different initial conditions (densities and concentrations) on
different kinds of orbits (circular and eccentric; with and with-
out external gravitational shocks) and with different internal
processes and ingredients (with or without stellar mass spec-
tra, binaries, and central black holes). To be sure, deviations
from perfect linearity in M(t) do occur, but these are generally
small—especially away from the endpoints of the evolution,
i.e., for 0.9&M(t)/M0 & 0.1—and neglecting them to assume
an approximately constant dM/dt is entirely appropriate for
our purposes.
When gravitational shocks are subdominant to relaxation-
driven evaporation, as they generally appear to be for
extant GCs, they work to boost the mass-loss rate µev
slightly without altering the basic linearity of M(t) (e.g.,
Vesperini & Heggie 1997; Gnedin, Lee, & Ostriker 1999; see
also Figure 1 of FZ01). A time-dependent mass scale ∆ ≡
µevt is then associated naturally with any system of coeval
clusters having a common mass-loss rate: all those with initial
M0 ≤ ∆ are disrupted by time t, and replaced with the rem-
nants of objects that began with M0 >∆. As we mentioned
in §1, if the initial GCMF increased towards low masses as a
power law, then ∆ is closely related to a peak in the evolved
distribution, which eventually decreases towards low M <∆
as dN/d log M ∝ M1−β with β = 0 (FZ01).
In standard theory (e.g., Spitzer 1987; Binney & Tremaine
1987, Section 8.3), the lifetime of a cluster against evapora-
tion is a multiple of its two-body relaxation time, trlx. For
a total mass M of stars within a radius r, this scales to first
order (ignoring a weak mass dependence in the Coulomb log-
arithm) as trlx(r) ∝ (Mr
3)1/2 ∝ M/ρ1/2, where ρ∝ M/r3. In a
concentrated cluster with an internal density gradient, trlx(r)
of course varies throughout the cluster, and the global re-
laxation timescale is an average of the local values (see the
early discussion by King 1958). This can still be written as
trlx ∝ M/ρ
1/2, with M the total cluster mass and ρ an appro-
priate reference density. We then have for the instantaneous
mass-loss rate, µev ≡ −dM/dt ∝ M/trlx ∝ ρ1/2. Insofar as this
is approximately constant in time, a GCMF evolving from an
initial β > 1 power law at low masses should therefore de-
velop a peak at a mass that depends on cluster density and age
through the parameter ∆∝ ρ1/2t.
It remains to identify the best measure of ρ in this context.
A standard choice in the literature, and the one that we even-
tually make to derive our main results in this paper, is the
half-mass density ρh = 3M/8πr
h. However, in a steady tidal
field, the mean density ρt inside the tidal radius of a cluster is
constant by definition, and thus choosing ρ = ρt instead is the
simplest way to ensure that µev ∝ ρ
1/2 and µev ≃ constant in
time are mutually consistent. In fact, King (1966) found from
direct calculations of the escape rate at each radius within his
standard (lowered Maxwellian) models, that the coefficient
in µev ∝ ρ
t is only a weak function of the internal density
structure (concentration) of the models, and thus only a weak
function of time for a cluster evolving quasistatically through
a series of such models.
The rule µev ∝ ρ
t is routinely used to set the GC mass-
loss rates in models for the dynamical evolution of the GCMF,
although such studies normally express µev immediately in
terms of orbital pericenters, rp, most often by assuming ρt ∝
r−2p as for GCs in galaxies whose total mass distributions fol-
low a singular isothermal sphere (e.g., Vesperini 1997, 1998,
2000, 2001; Vesperini et al. 2003; Baumgardt 1998; FZ01).
This bypasses any explicit examination of the GCMF as a
function of cluster density, which is our main goal in this pa-
per. But it is done in part because tidal radii are the most
poorly constrained of all structural parameters for GCs in the
Milky Way (their theoretical definition is imprecise and their
empirical estimation is highly model-dependent and sensitive
to low-surface brightness data), and they are exceedingly dif-
ficult if not impossible to measure in distant galaxies. We
deal with this here by focusing on the GCMF as a function of
cluster density ρh inside the less ambiguous, empirically bet-
ter determined and more robust half-mass radius, asking how
simple models with µev ∝ ρ
h fare against the data.
Taking µev ∝ ρ
h in place of µev ∝ ρ
t , which we do to
construct evaporation-evolved model GCMFs in §2.2, is most
appropriate if the ratio ρt/ρh is the same for all clusters and
constant in time. This is the case in Hénon’s (1961) model
of GC evolution, and in this limit (adopted by FZ01 in their
models for the Galactic GCMF) our analysis is rigorously jus-
tified. However, real clusters are not homologous (ρt/ρh dif-
fers among clusters) and they do not evolve self-similarly (ρh
may vary in time even if ρt does not). The key assumption in
our models is that µev is approximately independent of time
for any GC, which is well-founded in any case. By using cur-
rent ρh values to estimate µev, we do not suppose that the half-
mass densities are also constant, but we in effect use a single
number for all GCs to represent a range of (ρt/ρh)
1/2. Equiv-
alently, we ignore a dependence on cluster concentration in
the normalization of µev ∝ ρ
h . As we discuss further in §3,
it is reasonable to neglect this complication in a first approx-
imation because (ρt/ρh)
1/2 varies much less among Galactic
globulars than ρt and ρh do separately. We demonstrate this
explicitly by repeating our analysis with ρh replaced by ρt and
recover essentially the same results for the GCMF.
In §3 we also discuss some recent results, which indicate
that the timescale for relaxation-driven evaporation depends
on a slightly less-than-linear power of trlx (Baumgardt 2001;
Baumgardt & Makino 2003). We point out that this implies
that µev may increase as a modest power of the average sur-
face density of a cluster as well as (or, in an important special
case, instead of) the usual volume density. However, we show
in detail that making the appropriate changes throughout the
rest of the present section to reflect this possibility does not
change any of our conclusions.
2.1. Data
Figure 1 shows the distribution of mass against half-mass
density and against Galactocentric radius for 146 Milky Way
GCs in the catalogue of Harris (1996),7 along with the dis-
tribution of ρh versus rgc linking the two mass plots. The
Harris catalogue actually records the absolute V magni-
tudes of the GCs. We obtain masses from these by apply-
ing the population-synthesis model mass-to-light ratios ΥV
computed by McLaughlin & van der Marel (2005) for indi-
vidual clusters based on their metallicities and an assumed
age of 13 Gyr. However, we first multiplied all of the
7 Feb. 2003 version; see http://physwww.mcmaster.ca/∼harris/mwgc.dat .
http://physwww.mcmaster.ca/~harris/mwgc.dat
4 McLAUGHLIN & FALL
FIG. 1.— Left: Mass versus three-dimensional half-mass density, ρh ≡ 3M/8πr
h , and versus Galactocentric radius, rgc, for 146 Milky Way GCs in the
catalogue of Harris (1996). The dashed line in the first panel is M ∝ ρ
h , a locus of approximately constant lifetime against evaporation. Right: Half-mass
density versus rgc for the same clusters.
McLaughlin & van der Marel ΥV values by a factor of 0.8 so
as to obtain a median Υ̂V ≃ 1.5M⊙L
⊙ in the end,
8 consistent
with direct dynamical estimates (see McLaughlin 2000 and
McLaughlin & van der Marel 2005; also Barmby et al. 2007).
By assigning mass-to-light ratios to GCs in this way, we
allow for expected differences between clusters with differ-
ent metallicities. Our application of a corrective factor to
the population-synthesis values, ΥpopV , is motivated empiri-
cally by the fact that their distribution among Galactic GCs
is strongly peaked around a median Υ̂popV ≃ 1.9 M⊙ L
⊙ , while
the observed (dynamical) ΥdynV lie in a fairly narrow range
around Υ̂dynV ≃ 1.5 M⊙ L
⊙ (McLaughlin & van der Marel
2005). However, it is worth noting that the size of this differ-
ence is similar to what is found in some numerical simulations
of two-body relaxation over a Hubble time in clusters with a
spectrum of stellar masses (e.g., Baumgardt & Makino 2003).
In such simulations, ΥdynV falls below Υ
V due to the prefer-
ential escape of low-mass stars with high individual M∗/L∗
(population-synthesis models do not incorporate this or any
other stellar-dynamical effect). Thus, a median Υ̂dynV < Υ̂
may itself be a signature of cluster evaporation. We might
then also expect that more dynamically evolved clusters—that
is, those with shorter relaxation times—could have systemat-
ically lower ratios of ΥdynV /Υ
V . However, this is a relatively
small effect, which is not well quantified theoretically and
is not clearly evident in real data (the numbers published by
McLaughlin & van der Marel 2005 show no significant corre-
lation between Υ
V and trh for Galactic globulars). We
therefore proceed, as stated, with a single ΥdynV /Υ
V = 0.8
assumed for all GCs.
Harris (1996) gives the projected half-light radius Rh for
141 of the clusters with a mass estimated in this way, and for
these we obtain the three-dimensional half-mass radius from
the general rule rh = (4/3)Rh (Spitzer 1987), which assumes
no internal mass segregation. The remaining five objects have
mass estimates but no size measurements. To each of these
clusters, we assign an rh equal to the median value for those
of the other 141 GCs having masses within a factor two of the
one with unknown rh. In all cases, the half-mass density is
8 Throughout this paper, we use bx to denote the median of any quantity x.
ρh ≡ 3M/8πr
The leftmost panel in Figure 1 shows immediately that
the cluster mass distribution has a strong dependence on
half-mass density: the median M̂ increases with ρh while
the scatter in log M—that is, the width of the GCMF—
decreases. The first of these points is related to the fact that
rh correlates poorly with M (e.g., Djorgovski & Meylan 1994;
McLaughlin 2000). The second point, that the dispersion of
dN/d log M decreases with increasing ρh, is behind the find-
ing (Kavelaars & Hanes 1997; Gnedin 1997) that the GCMF
is broader at very large Galactocentric radii. We return to this
in §2.2.
A natural concern, when plotting M against ρh as we have
done here, is that any apparent correlation might only be a
trivial reflection of the definition ρh ∝ M/r
h. This may seem
particularly worrisome because, as we just mentioned, it is
known that size does not correlate especially well with mass
for GCs in the Milky Way (or, indeed, in other galaxies).
However, the lack of a tight M–rh correlation does not imply
that all GCs have the same rh, even within the unavoidable
measurement errors. The root-mean-square (rms) scatter of
log rh about its average value is ±0.3 for Galactic GCs, and
the 68-percentile spread in log rh is slightly greater than 0.5,
or more than a factor of 3 in linear terms (from the data in
Harris 1996; see, e.g., Figure 8 of McLaughlin 2000). This
compares to an rms random measurement error (from formal,
χ2 fitting uncertainties) of δ(log rh) ≈ 0.05, or about 10% rel-
ative error; and an rms systematic measurement error (i.e.,
differences in the rh inferred from fitting different structural
models to a single cluster) of perhaps δ(log rh) . 0.03; see
McLaughlin & van der Marel (2005). Most of the scatter in
plots of observed half-light radius versus mass is therefore
real and contains physical information. The left-hand panel
of Figure 1 displays this information in a form that highlights
clear, nontrivial overall trends requiring physical explanation.
The dashed line in the plot of mass against density traces
the proportionality M ∝ ρ
h , or Mr
h = constant. Insofar as
the half-mass relaxation time scales as trh ∝ (Mr
1/2, and
to the extent that µev ∝ M/trh ∝ ρ
h approximates the av-
erage rate of relaxation-driven mass loss, this line is one of
equal evaporation time. That such a locus nicely bounds
the lower envelope of the observed cluster distribution is
SHAPING THE GCMF BY EVAPORATION 5
TABLE 1
MILKY WAY GC PROPERTIES IN BINS OF DENSITY AND GALACTOCENTRIC RADIUS
Bin N bρh
brgc a Mmin Mmax bM
a MTO
[M⊙ pc−3] [kpc] [M⊙] [M⊙] [M⊙] [M⊙]
ρh bins
0.034 ≤ ρh ≤ 76.5 M⊙ pc
−3 48 8.48 12.9 5.63× 102 8.84× 105 4.12× 104 3.98× 104
78.8 ≤ ρh ≤ 526 M⊙ pc
−3 49 232 5.6 8.37× 103 1.67× 106 1.22× 105 1.58× 105
579 ≤ ρh ≤ 5.65× 10
4 M⊙ pc−3 49 973 3.2 1.93× 104 1.30× 106 2.82× 105 2.88× 105
rgc bins
0.6 ≤ rgc ≤ 3.2 kpc 47 597 1.9 4.47× 103 1.02× 106 1.15× 105 2.14× 105
3.3 ≤ rgc ≤ 9.4 kpc 50 261 5.2 2.02× 103 1.67× 106 1.27× 105 1.66× 105
9.6 ≤ rgc ≤ 123 kpc 49 18.4 18.3 5.63× 102 1.30× 106 7.42× 104 8.71× 104
a The notation bx represents the median of quantity x.
b MTO is the peak mass of the model GCMFs traced by the solid curves in each panel of Figure 2, which
are given by equation (3) of the text with β = 2, Mc = 10
6 M⊙, and individual ∆ given by the observed ρh
of each cluster through equation (4).
itself a strong hint that relaxation-driven cluster disruption
has significantly modified the GCMF at low masses (re-
call that Mr3h = constant defines one side of the GC “sur-
vival triangle” when the M–ρh plot is recast as rh versus M:
Fall & Rees 1977; Okazaki & Tosa 1995; Ostriker & Gnedin
1997; Gnedin & Ostriker 1997). It is also further evidence
that the weak correlation of observed rh with M is due to sig-
nificant and real differences in cluster radii, since if rh were
intrinsically the same for all GCs, then we would see M ∝ ρh
instead.
The middle panel of Figure 1 shows the well-known result
that the typical GC mass depends weakly if at all on Galacto-
centric radius, at least until large rgc & 30–40 kpc, where there
are too few clusters to discern any trend. The right-hand panel
of the figure shows why this is true even though the GCMF
depends significantly on cluster density: although there is a
correlation between half-mass density and Galactocentric po-
sition, the large scatter about it is such that convolving the
observed M versus ρh with the observed ρh versus rgc results
in an almost null dependence of M on rgc.
We now divide the GC sample in Figure 1 roughly into
thirds, in two different ways: first on the basis of half-mass
density, and second by Galactocentric radius. These ρh and
rgc bins are defined in Table 1, which also gives a few sum-
mary statistics for the globulars in each subsample. We count
the clusters in every subsample in about 10 equal-width bins
of log M to obtain histogram representations of dN/d log M,
first as a function of ρh and then as a function of rgc. These
GCMFs are shown by the points in Figure 2, with errorbars
indicating standard Poisson uncertainties. The curves in the
figure trace model GCMFs, which we describe in §2.2. For
the moment, it is important to note that the dashed curve is the
same in every panel, apart from minor differences in normal-
ization, and is proportional to the GCMF for the whole sample
of 146 GCs. (In the middle-left panel of Figure 2, which per-
tains to clusters distributed tightly around the median ρh of
the entire GC system, the dashed curve is coincident with the
solid curve running through the data.)
The left-hand panels of Figure 2 show directly that the
GCMF is peaked for clusters at any density, and that the mass
of the peak increases systematically with ρh (see also the last
column of Table 1, but note that the turnover masses there
refer to the model GCMFs that we develop below). The sta-
tistical significance of this is very high, and qualitatively it
is the behavior expected if MTO owes its existence to cluster
disruption at a rate that increases with ρh, as is the case with
relaxation-driven evaporation.
The right-hand panels of Figure 2 confirm once again that
the GCMF peak mass is a very weak function of Galactocen-
tric position. In fact, the observed distributions in the two rgc
bins inside ≃10 kpc are statistically indistinguishable in their
entirety, and the main difference at larger rgc & 10 kpc is a
slightly higher proportion of low-mass clusters rather than a
large change in MTO. All of this is consistent with the pri-
mary dependence of the GCMF being that on ρh, since Fig-
ure 1 shows that the GC density distribution is not sensitive to
Galactocentric position for rgc . 10–20 kpc but has a substan-
tial low-density tail at larger radii (with a broader associated
GCMF, as seen in the upper-left panel of Figure 2).
2.2. Simple Models
We now assess more quantitatively whether these results
are consistent with evaporation-dominated evolution of the
GCMF from an initial distribution like that observed for
young clusters in the local universe. We model the time-
evolution of the distribution of M versus ρh in Figure 1 but do
not attempt this for the distribution of ρh over rgc—the details
of which likely depend on a complicated interplay between
the tidal field of the Galaxy, the present and past orbital pa-
rameters of clusters, and the structural nonhomology of GCs.
To compare our models to the current GCMF as a function of
rgc, we simply calculate them using the observed ρh of indi-
vidual clusters in different ranges of Galactocentric radius.
We assume that the initial GCMF was independent of clus-
ter density, and that all globulars surviving to the present day
have been losing mass for the past Hubble time at constant
rates. We use the current half-mass density of each cluster
to estimate µev ∝ ρ
h . As we discussed earlier, an approxi-
mately time-independentµev is indicated by most calculations
of two-body relaxation in tidally limited GCs. We give a more
detailed, a posteriori justification in §3 for using ρh, rather
than other plausible measures of cluster density, to estimate
Consider first a group of coeval GCs with an initial mass
function dN/d log M0 and a single, time-independent mass-
loss rate µev. The mass of every cluster decreases linearly as
M(t) = M0 −µevt, and at any later time each has lost the same
amount ∆ ≡ M0 − M(t) = µevt. FZ01 show rigorously that in
6 McLAUGHLIN & FALL
FIG. 2.— GCMF as a function of half-mass density, ρh ≡ 3M/8πr
h (left panels), and as a function of Galactocentric radius, rgc (right panels), for 146 Milky
Way GCs in the catalogue of Harris (1996). The dashed curve in all cases is an evolved Schechter function for the entire GC system (Jordán et al. 2007): equation
(3) with β = 2, Mc = 106 M⊙, and ∆≡ 2.3×105 M⊙ for all clusters (from equation [4] and a median bρh = 246 M⊙ pc
−3), giving a peak at MTO = 1.6×10
5 M⊙ .
Solid curves are the GCMFs predicted by equation (3) with β = 2 and Mc = 106 M⊙ but individual ∆ given by the observed ρh of each cluster (equation [4]) in
the different subsamples.
this case, the evolved and initial GCMFs are related by
d log M
d log M0
(M +∆)
d log (M +∆)
. (1)
This is the basis for the claim that the mass function scales
generically as dN/d log M ∝ M+1 (a β = 0 power law) at low
enough M(t)<∆—that is, for the surviving remnants of clus-
ters with M0 ≈∆—just so long as the initial distribution was
not a delta function.
We follow FZ01 (see also Jordán et al. 2007) in adopting a
Schechter (1976) function for the initial GCMF:
dN/d log M0 ∝ M
0 exp
−M0/Mc
. (2)
With β ≃ 2, this distribution describes the power-law mass
functions of young massive clusters in systems like the Anten-
nae galaxies (e.g., Zhang & Fall 1999). An exponential cut-
off at Mc ∼ 10
6 M⊙ is generally consistent with such data,
even if not always demanded by them; here we require it
mainly to match the curvature observed at high masses in old
GCMFs (e.g., Burkert & Smith 2000; Jordán et al. 2007).
Combining equations (1) and (2) gives the probability den-
sity that a single GC with known evaporation rate and age has
an instantaneous mass M. The time-dependent GCMF of a
system of N GCs with a range of µev (or ages, or both) is
then just the sum of all such individual probability densities:
d log M
[M +∆i]
M +∆i
. (3)
Here the total mass losses ∆i = (µevt)i may differ from clus-
ter to cluster (ti being the age of a single GC) but both β and
Mc are assumed to be constants, independent of ρh in particu-
lar.9 Given each ∆i, the normalizations Ai in equation (3) are
defined so that the integral over d log M of each term in the
summation is unity.
Jordán et al. (2007) have introduced a specialization of
equation (3) in which all clusters have the same ∆. They refer
to this as an evolved Schechter function and describe its prop-
erties in detail (including giving a formula for the turnover
mass MTO as a function of ∆ and Mc) for the case β = 2. Here
we note only that, at very young cluster ages or for slow mass-
loss rates, such that ∆ ≪ Mc and only the low-mass, power-
law part of the initial GCMF is significantly eroded, any one
evolved Schechter function has a peak at MTO ≃ ∆/(β − 1)
(for β > 1). As ∆ increases relative to Mc, the turnover at
first increases proportionately and the width of the distribu-
tion decreases (since the high-mass end at M & MTO is largely
unchanged). For large ∆≫Mc, however, the peak is bounded
above by MTO →Mc and the width approaches a lower limit.
Thus, the dependence of MTO on ∆ is weaker than linear when
Mc is finite in the initial GCMF of equation (2). Any peak in
the full equation (3) for a system of GCs with individual ∆
values is an average of N different turnovers and must be cal-
culated numerically.
In their modeling of the Milky Way GC system, FZ01 ef-
9 Note that Mc appears to take on different values in the GCMFs of other
galaxies, varying systematically with the total luminosity Lgal (Jordán et al.
2007). The reasons for this are unclear, as is the origin of this mass scale in
the first place.
10 The increase of MTO and the decrease of the full width of dN/d log M
for increasing ∆ eventually saturate when the mass loss per GC is so high
that it affects clusters in the exponential part of the initial Schechter-function
GCMF. This is because dN/d log M ∝ M+1 exp(−M/Mc) is a self-similar
solution to equation (1).
SHAPING THE GCMF BY EVAPORATION 7
fectively compute mass functions of the type (3)—based on
the same initial conditions and dynamical evolution—with a
distribution of ∆ values determined by the orbital parame-
ters of clusters in an idealized, spherical and static logarith-
mic Galaxy potential (used both to fix µev in terms of clus-
ter tidal densities and to estimate additional mass loss due
to gravitational shocks). Jordán et al. (2007) fit GCMF data
in the Milky Way and scores of Virgo Cluster galaxies with
their version of equation (3) in which all GCs have the same
∆. They thus estimate the dynamical mass loss from typi-
cal clusters in these systems. Here, we construct models for
the Milky Way GCMF using ∆ values given directly by the
observed half-mass densities of individual GCs.
We adopt β = 2 for the initial low-mass power-law index
in equation (2), which carries over into equation (3) for the
evolved dN/d log M. Jordán et al. (2007) have fitted the full
Galactic GCMF with an evolved Schechter function assuming
β = 2 and a single ∆ ≡ ∆̂ for all surviving globulars. They
find Mc ≃ 10
6 M⊙ and ∆̂ = 2.3× 10
5 M⊙. We use this value
of Mc in equation (3) and we associate ∆̂ with the mass loss
from clusters at the median half-mass density of the entire GC
system, which is ρ̂h = 246 M⊙ pc
−3 from the data in Figure 1.
Since we are assuming that ∆ = µevt ∝ ρ
h t for coeval GCs,
we therefore stipulate
∆ = 1.45× 104 M⊙
ρh/M⊙ pc
−3)1/2 (4)
for globulars with arbitrary ρh. Assuming a typical GC age of
t = 13 Gyr, this corresponds to a mass-loss rate of
µev ≃ 1100 M⊙ Gyr
−1 (ρh/M⊙ pc−3
. (5)
In §3 we discuss the cluster lifetimes implied by this value
of µev. We emphasize here that the scaling of µev and ∆
with ρ
h follows rather generically from our hypothesis of
evaporation-dominated cluster evolution, while the numerical
coefficients in equations (4) and (5) are specific to the assump-
tion of β = 2 for the power-law index at low masses in the
initial GCMF.
The dashed curve shown in every panel of Figure 2 is
the evolved Schechter function fitted to the entire GCMF of
the Milky Way by Jordán et al. (2007). This has a peak at
MTO ≃ 1.6× 10
5 M⊙ (magnitude MV ≃ −7.4 for a typical V -
band mass-to-light ratio of 1.5 in solar units) and gives a very
good description of the observed dN/d log M in the middle
density bin, 79 . ρh . 530 M⊙ pc
−3, and in the two inner ra-
dius bins, rgc ≤ 9.4 kpc. This is expected, since the median
half-mass density in each of these cluster subsamples is very
close to the system-wide median ρ̂h = 246 M⊙ pc
−3 (see Ta-
ble 1). Even in the outermost rgc bin, a Kolmogorov-Smirnov
(KS) test only marginally rejects the dashed-line model (at
the ≃95% level), because this subsample still includes many
GCs at or near the global median ρ̂h (see Figure 1). By con-
trast, the average GCMF is strongly rejected as a model for
the lowest- and highest-density GCs on the left-hand side of
Figure 2: the KS probabilities that these data are drawn from
the dashed distribution are <10−4 in both cases. This is also
expected since, by construction, these bins only contain clus-
ters with densities well away from the median of the full GC
system, for which the total mass lost by evaporation should be
significantly different from the typical ∆̂ = ∆(ρ̂h).
The solid curves in Figure 2, which are different in ev-
ery panel, are the superpositions of many different evolved
Schechter functions, as in equation (3), with distinct ∆ values
given by equation (4) using the observed ρh of each cluster in
the corresponding subsample. These models provide excel-
lent matches to the observed dN/d log M in every ρh and rgc
bin, with χ2 < 1.3 per degree of freedom in all cases. This is
the main result of this paper.
The last column of Table 1 gives the mass MTO at which
each of the solid model GCMFs in Figure 2 peaks. We note
that these turnovers increase roughly as MTO ∼ ρ̂
0.3−0.4
h for
our specific binnings in ρh and rgc, somewhat shallower than
the ρ
h scaling of the cluster mass-loss rate that defines the
models. This is partly because of the averaging over indi-
vidual turnovers implied by the summation of many evolved
Schechter functions in each GC bin, and partly because—as
we discussed just after equation (3)—the turnover mass of any
one evolved Schechter function cannot increase indefinitely in
direct proportion to ∆ ∝ ρ
h , but has a strict upper limit of
MTO ≤ Mc.
Our models are naturally consistent with the fact that the
GCMF is narrower for clusters with higher densities. This
is obvious in the left-hand panels of Figure 2; in the dis-
cussion immediately after equation (3), we described how it
follows from the increase of MTO with ∆ ∝ ρ
h for a sin-
gle evolved Schechter function. In addition, the superposi-
tion of many such functions with separate, density-dependent
turnovers and widths results in wider GCMFs for cluster sub-
samples spanning larger ranges of ρh. This accounts in partic-
ular for the breadth of the mass function at rgc ≥ 9.4 kpc. The
globulars at these radii have 0.034 ≤ ρh ≤ 4.1×10
3 M⊙ pc
corresponding to individual evolved Schechter functions with
turnovers at 2.7× 103 . MTO . 4.0× 10
5 M⊙. The compos-
ite GCMF in the lower-right panel of Figure 2 is therefore
extremely broad and shows a very flat peak, such that an over-
all MTO cannot be established precisely from the data alone.
This explains the findings of Kavelaars & Hanes (1997), who
pointed out that the GCMF of the outermost third of the Milky
Way cluster system has a turnover that is statistically consis-
tent with the full-Galaxy average, but a larger dispersion (see
also Gnedin 1997).
Finally, if the GCMF evolved dynamically from initial con-
ditions similar to those we have adopted, then the data and
models in the left-hand panels of Figure 2 argue against
the notion that external gravitational shocks, rather than in-
ternal two-body relaxation, were primarily responsible for
shaping the present-day GCMF. This is because the mass-
loss rate caused by shocks alone, −dM/dt = µsh ∝ M/ρh,
differs significantly from that caused by evaporation alone,
−dM/dt = µev ∝ ρ
h . The direct dependence of µsh on M en-
sures that shocks become progressively less important com-
pared to evaporation as clusters lose mass (at a given ρh), and
consequently shocks are not likely to have had much effect on
the observed GCMF for M < MTO. Furthermore, the inverse
dependence of µsh on ρh is contrary to the direct dependence
of MTO on ρh shown in Figure 2. The different roles played
by shocks and evaporation in shaping the observed GCMF are
discussed more fully by FZ01. We note here that gravitational
shocks may have been important in destroying very massive
or very low-density clusters early in the history of our Galaxy.
2.3. Other Cluster Properties
If the current shape of the GCMF is fundamentally the re-
sult of long-term cluster disruption according to a mass-loss
8 McLAUGHLIN & FALL
rule like µev ∝ ρ
h , then it should be possible to reproduce
the distribution as a function of any other cluster attribute by
using the observed ρh of individual GCs in equations (3) and
(4) to build model dN/d log M for subsamples of the Galactic
cluster system defined by that attribute—as we did for the rgc
binning of §2.2. Here we explore one example in which dif-
ferences in the GCMFs of two groups of globulars can be seen
in this way to follow from differences in their ρh distributions.
Smith & Burkert (2002) have shown that the mass function
of Galactic globulars with King (1966) model concentrations
c < 0.99 has a less massive peak than that for c ≥ 0.99. [Here
c ≡ log(rt/r0), where rt is the fitted tidal radius and r0 a core
scale.] They further find that a power-law fit to the low-
c GCMF just below its peak returns dN/d log M ∝ M+0.5—
shallower than the M+1 expected generically for a mass-loss
rate that is constant in time—but they confirm that the latter
slope applies for the GCMF at c ≥ 0.99. They discuss various
options to explain these results, including a suggestion that, if
the mass functions of both low- and high-concentration clus-
ters evolved slowly from the same, young-cluster–like initial
distribution, then the mass-loss law for low-c GCs may have
differed from that for high-c clusters. However, they give no
physical explanation for such a difference, and we can show
now that none is required.
The upper panel of Figure 3 plots concentration against
half-mass density for the same 146 GCs from Figure 1; the
filled circles distinguish 24 clusters with c < 0.99. There
is a correlation of sorts between c and ρh, which either de-
rives from or causes the better-known correlation between c
and M (e.g., Djorgovski & Meylan 1994; McLaughlin 2000).
The important point here is that the ρh distribution is off-
set to lower values and has a higher dispersion at c < 0.99.
Following the discussion in §2.2, we therefore expect the
low-concentration GCMF to have a smaller MTO, a flatter
shape around the peak, and a larger full width than the high-
concentration GCMF.
The lower panel of Figure 3 shows the GCMFs for c < 0.99
(filled circles) and c ≥ 0.99 (open circles). The curves are
again given by equation (3) with β = 2, Mc = 10
6 M⊙, and in-
dividual ∆ calculated from the observed cluster ρh through
equation (4). These models peak at MTO ≃ 4.3× 10
for the c < 0.99 subsample but at MTO ≃ 1.8× 10
5 M⊙ for
c ≥ 0.99, entirely as a result of the different ρh involved.
The larger width of dN/d log M and its shallower slope at
any M . 105 M⊙ for the low-concentration GCs are also
clear, in the model curves as well as the data. It is further
evident that there are no low-c Galactic globulars observed
with M & 2× 105 M⊙, above the nominal turnover of the full
GCMF (as Smith & Burkert 2002 noted). But this is not sur-
prising, given that there are so few low-concentration clusters
in total and they are expected to be dominated by low-mass
objects because of their generally low densities. Thus, the
solid curve in Figure 3 predicts perhaps ≃ 3 high-mass clus-
ters with c < 0.99, where none is found.
The apparent variation of the Milky Way GCMF with in-
ternal concentration is therefore consistent with the same
density-based model for evaporation-dominated dynamical
evolution that we compared to dN/d log M as a function of
ρh and rgc in §2.2. To show this, we have made use of the
densities ρh exactly as observed within the two concentration
bins indicated in Figure 3—just as we also took ρh directly
from the data for GCs in different ranges of rgc to construct
models for comparison with the observed dN/d log M in the
FIG. 3.— Top: Concentration parameter as a function of half-mass density
for 146 Galactic GCs. The line of points at c ≡ 2.5 comes from the practice of
assigning this value to core-collapsed clusters in the Harris (1996) catalogue
and its sources. Bottom: GCMF data and models (eqs. [3] and [4]) for 24
clusters with c < 0.99 (filled circles and solid curve) and 122 clusters with
c ≥ 0.99 (open circles and dashed curve).
right-hand panels of Figure 2. Of course, this is not the same
as explaining the distribution of ρh versus rgc or c. Doing so
would certainly be of interest in its own right, but it is beyond
the scope of our work here.
3. DISCUSSION
In this section, we first show that the mass-loss rate in equa-
tion (5) above implies cluster lifetimes that compare favorably
with those expected from relaxation-driven evaporation. Then
we discuss why it is reasonable to approximate µev ∝ ρ
the first place. Finally, we address the issue of possible con-
flict, in some other models for evaporation-dominated GCMF
evolution, between the near-constancy of MTO as a function
of rgc and the observed kinematics of GC systems.
3.1. Cluster Lifetimes
The disruption time of a GC with mass M and a steady
mass-loss rate µev is just tdis = M/µev. It is convenient,
for purposes of comparison with evaporation times in the
literature, to normalize tdis to the relaxation time of a
cluster at its half-mass radius. In general, this is trh =
0.138M1/2r
G1/2m∗ ln (γM/m∗)
, where m∗ is the mean
SHAPING THE GCMF BY EVAPORATION 9
stellar mass. For clusters of stars with a single mass,
m∗ ≃ 0.7M⊙ and γ ≃ 0.4 are appropriate (Spitzer 1987;
Binney & Tremaine 1987, equation [8-72]), in which case
equation (5) for µev from our GCMF modeling implies
µevtrh
0.57M/M⊙
0.57× 105
. (6)
Clusters with realistic stellar mass spectra will have slightly
different values of m∗ and a smaller γ in the calculation of
the relaxation time (Giersz & Heggie 1996), which changes
the numerical value of tdis/trh somewhat but does not alter any
scalings.
We obtained the normalization of µev ∝ ρ
h in §2.2 by
fitting to observed GCMFs constructed by applying a spe-
cific mass-to-light ratio ΥV to every cluster, with models as-
suming a specific form for the initial dN/d log M0. Thus,
the result in equation (6) depends both on the median Υ̂V
and on the power-law index β at low masses in the original
Schechter-function GCMF. The net scaling, for either single-
or multiple-mass clusters, is
tdis/trh ∝ Υ̂
V (β − 1)
−1 . (7)
To see the dependence of this dimensionless lifetime on
Υ̂V , note that we require µev ∝∆∝ΥV to fit the mass losses
of clusters with a given distribution of luminosities (the di-
rect observables), whereas M/trh is proportional to ρ
V (L/r
1/2. Therefore, tdis/trh ∝ (M/trh)/µev ∝Υ
V . The
mass-to-light ratios adopted in this paper, with a median value
Υ̂V ≃ 1.5 M⊙ L
⊙ , are tied directly to dynamical determina-
tions (§2.1).
To understand the dependence on β in equation (7), recall
first that the coefficients in our expressions for ∆ and µev
as functions of ρh (eqs. [4] and [5]) followed from choos-
ing β = 2 for the power-law exponent at low masses in the
initial GCMF (equation [2]). As we mentioned just after
equation (3), the turnover mass of an evolved Schechter func-
tion with any β > 1 is MTO ≃ ∆/(β − 1) in the limit of low
∆ ∝ ρ
h , and MTO → Mc for very high ∆. In this sense,
the strongest observational constraints on the normalizations
of ∆ and µev come from the low-density clusters. All other
things being equal, their GCMF can be reproduced with β 6= 2
if ∆ and µev are multiplied by (β − 1) at fixed ρh. Therefore,
tdis ∝ 1/µev ∝ 1/(β− 1). Observations of young massive clus-
ters (e.g., Zhang & Fall 1999) indicate that β is near 2; but if
it were slightly shallower, then the cluster lifetimes we infer
from the old GCMF would increase accordingly. Even a rela-
tively minor change to β = 1.5 would double tdis/trh from ≈10
to ≈20.
In the model of Hénon (1961) for single-mass clusters
evolving self-similarly (fixed ratio ρt/ρh of mean densities in-
side the tidal and half-mass radii) in a steady tidal field (ρt
constant in time), a cluster loses 4.5% of its remaining mass
every half-mass relaxation time. The time to complete disrup-
tion is therefore tdis/trh = 1/0.045≃ 22. For non-homologous
clusters in a steady tidal field, tdis/trh is a function of central
concentration and can differ from the Hénon value by factors
of about two. From one-dimensional Fokker-Planck calcula-
tions, Gnedin & Ostriker (1997) find tdis/trh ≃ 10–40 for King
(1966) model clusters with c values similar to those found in
real GCs and with gravitational shocks suppressed (see their
Figure 6). Thus, even though the evaporation time in equation
(6) may be slightly shorter than is typically found in theoreti-
cal calculations, it is certainly within the range of such calcu-
lations. Moreover, the assumptions of a steady tidal field and
a single stellar mass in Hénon (1961) and Gnedin & Ostriker
(1997) are important. Part of the difference between the typ-
ical lifetimes in these particular theoretical treatments and
our estimate of tdis/trh from the GCMF is that the former do
not include gravitational shocks, which may have accelerated
somewhat the evolution of real clusters (although we stress
again that shocks do not appear in general to have dominated
the evolution of extant Galactic GCs and are not expected
to affect the basic time-independence of the net mass-loss
rate; see Vesperini & Heggie 1997, Gnedin, Lee, & Ostriker
1999, FZ01, and Prieto & Gnedin 2006). A spectrum of stel-
lar masses in the clusters may also have contributed to an in-
crease in evaporation rate over the single-mass values (e.g.,
Johnstone 1993; Lee & Goodman 1995).
Estimates of evaporation times from other numerical
methods and for models of multimass clusters can be rather
sensitive to the detailed computational techniques and input
assumptions and approximations, and differences at roughly
the factor-of-two level in tdis/trh between different analyses
are not uncommon; see, e.g., Vesperini & Heggie (1997),
Takahashi & Portegies Zwart (1998, 2000), Baumgardt
(2001), Joshi, Nave, & Rasio (2001), Giersz (2001), and
Baumgardt & Makino (2003). Thus, although the lifetimes
in these studies tend to be broadly comparable to those in
Hénon (1961) and Gnedin & Ostriker (1997), noticeably
shorter values do occur in some models. In any case, we
are encouraged by consistency to within factors of two or
three between estimates of tdis or µev by such vastly different
methods—one purely observational, based on the mass
functions of cluster systems; the other purely theoretical,
based on idealized models for the evolution of individual
clusters—particularly since each method involves several
uncertain inputs and parameters.
3.2. Approximating µev ∝ ρ
3.2.1. Half-mass versus Tidal Density
The dimensionless disruption time in equation (6) is inde-
pendent of any cluster property other than the Coulomb log-
arithm because we have used GC half-mass densities to esti-
mate tdis = M/µev ∝ M/ρ
h , while trh also scales as M/ρ
However, as we mentioned above, the Fokker-Planck calcu-
lations of Gnedin & Ostriker (1997) in particular show that
tdis/trh is actually a function of central concentration, c, for
King (1966) model clusters in steady tidal fields. The constant
of proportionality in µev ∝ ρ
h should therefore also depend
on c, a detail that we have neglected to this point. We show
now that this has not biased any of our analysis or affected our
conclusions.
The dotted curve in Figure 4 illustrates the dependence of
tdis/trh on c for single-mass King models, as given by equation
(30) of Gnedin & Ostriker (1997). The solid curve is propor-
tional to (ρh/ρt)
1/2 = (r3t /2r
1/2, which we have calculated
as a function of c for these models and multiplied by a con-
stant to compare directly with tdis/trh. Evidently, there is an
approximate equality tdis/trh ≈ 2.15(ρh/ρt)
1/2, which holds to
within <15% over the range of concentrations shown in Fig-
ure 4 (note that all but 6 Galactic GCs have 0.7 ≤ c ≤ 2.5,
corresponding to central potentials 3 .W0 . 11). Thus, if the
10 McLAUGHLIN & FALL
FIG. 4.— Dependence of tdis/trh (dotted line; from Gnedin & Ostriker
1997) and (ρh/ρt )
1/2 (solid line; after scaling by a factor of 2.15) on cen-
tral concentration for single-mass King-model clusters. Over the range of c
shown, which includes nearly all Galactic globulars, the approximate propor-
tionality tdis/trh ∝ (ρh/ρt )
1/2 holds to within better than 15%. Thus, to this
level of accuracy the evaporation time tdis is roughly the same multiple of
t for clusters with any internal density profile.
evaporation time is written as tdis ∝ trh(ρh/ρt)
1/2 ∝ M/ρ
then the constant of proportionality in the mass-loss rate µev ∝
M/tdis ∝ ρ
t should be nearly independent of c. In fact, King
(1966) originally concluded, from quite basic arguments, that
the evaporation rate of a cluster with a lowered-Maxwellian
velocity distribution would take the form µev ∝ ρ
t with
only a weak dependence on c. An essentially concentration-
independent scaling of µev with ρ
t is also found in N-
body simulations of tidally limited, multimass clusters (e.g.,
Vesperini & Heggie 1997) and so is not an artifact of any as-
sumptions specific to the calculations of either King (1966) or
Gnedin & Ostriker (1997).
This suggests that it might have been more natural to spec-
ify cluster evaporation rates proportional to ρ
t rather than
h when developing our GCMF models in §2. For any clus-
ter in a steady tidal field, with a constant ρt , such a choice
would also have been automatically consistent with an ap-
proximately time-independent µev and the corresponding lin-
ear M(t) dependence that we have adopted throughout this
paper. As we discussed at the beginning of §2, our deci-
sion to work with ρh rather than ρt was motivated by the
fact that the half-mass density is much better defined in prin-
ciple and more accurately observed in practice. Neverthe-
less, re-writing µev ∝ ρ
t as µev ∝ (ρt/ρh)
1/2 × ρ
h makes
it clear that the validity of our models, with a fixed coefficient
in µev ∝ ρ
h , depends on the extent to which variations in
(ρt/ρh)
1/2 can safely be ignored.
Figure 4 shows that the full range of possible values for
(ρh/ρt)
1/2 in King-model clusters with c ≥ 0.7 is only a factor
of ≃ 4 between minimum and maximum. Therefore, using
a single, intermediate value of this density ratio to describe
all GCs (or a single GC evolving in time through a series of
quasi-static King models)—which we have effectively done
by using a GCMF fit to normalize ∆ and µev in equations (4)
and (5)—should never be in error by more than a factor of 2 or
so. This is a relatively small inaccuracy, given that measured
GC densities range over four to five orders of magnitude.
To confirm more directly that our models with µev ∝ ρ
are good approximations to GCMF evolution under a mass-
loss law µev ∝ ρ
t , we have repeated the analysis of §2 in full
but using the GC tidal densities ρt (derived from the values of
rt listed by Harris 1996) in place of ρh throughout. All of our
main results persist.
For example, the two panels of Figure 5, which are analo-
gous to the left- and rightmost panels of Figure 1 above, show
that (1) the GC mass distribution has a clear dependence on
ρt , with a lower envelope that is well matched by a line of
constant evaporation time, M ∝ ρ
t (the dashed line in the
plot); and (2) although the scatter in the distribution of ρt
over Galactocentric radius is smaller than the scatter in ρh ver-
sus rgc, it is still significant. Because the M–rgc distribution
can now be viewed as the convolution of the M–ρt distribu-
tion with the ρt–rgc distribution, the scatter in ρt versus rgc is
again critical in explaining the weak or null dependence of the
GCMF on Galactocentric radius. (The M–rgc distribution is,
of course, unchanged from that shown in the middle panel of
Figure 1.)11
Figure 6 shows the Milky Way GCMF for globulars in three
equally populated bins of tidal density (defined as indicated in
the left-hand panels of the plot) and in the same three bins of
Galactocentric radius that we used in §2.2 above. Our mod-
els for these distributions are based as before on equation (3)
with β = 2, but now the total mass lost from any GC is esti-
mated from its tidal density rather than its half-mass density.
Specifically, we take
∆ = 2.1× 105 M⊙
ρt/M⊙ pc
−3)1/2 . (8)
The numerical coefficient in equation (8) is such that it gives a
∆ identical to that in equation (4) for a GC with ρh/ρt = 210,
which is the median value of this density ratio for the 146 GCs
in the Harris (1996) catalogue.
As in Figure 2, the dashed curve in every panel of Figure
6 is the same, representing a fit to the average dN/d log M of
the entire Galactic GC system. Thus, it is immediately clear
that the peak mass of the GCMF increases significantly and
systematically with increasing ρt , just as it does with increas-
ing ρh. Meanwhile, the solid curves are subsample-specific
model GCMFs, obtained by using the observed tidal density
of each cluster in any ρt or rgc bin to specify individual ∆ val-
ues via equation (8) for each of the evolved Schechter func-
tions in the summation of equation (3). As expected, there is
no appreciable difference, in terms of the fits to any of the ob-
served GCMFs, between these models based on evaporation
rates µev ∝ ρ
t and our original models with µev ∝ ρ
3.2.2. Retarded Evaporation
Another potential concern comes from recent arguments
(see especially Baumgardt 2001; Baumgardt & Makino 2003)
11 As was also the case with our earlier plots involving ρh in Figure 1,
the scatter and structure in both panels of Figure 5 are real, since the rms
scatter of log rt about the best-fit lines to either of log M or log rgc is 0.3–0.35
while the rms errorbars based on formal fitting uncertainties are in the range
δ(log rt ) ≃ 0.05–0.15 for a variety of models (McLaughlin & van der Marel
2005).
SHAPING THE GCMF BY EVAPORATION 11
FIG. 5.— Scatter plots of mass M versus mean density inside the tidal radius (ρt ≡ 3M/4πr3t ) and of ρt versus Galactocentric radius rgc, for 146 Galactic GCs
from the Harris (1996) catalogue. These plots are analogous to the left- and rightmost panels of Figure 1. The dashed line in the left-hand plot traces the relation
M ∝ ρ
t , which defines a locus of constant evaporation time for µev ∝ ρ
FIG. 6.— Observed GCMF (points, with Poisson errorbars) and models (curves) as a function of mean cluster density inside the tidal radius, ρt ≡ 3M/4πr3t (left-
hand panels), and as a function of Galactocentric radius, rgc (right-hand panels). The dashed curve in every panel is an evolved Schechter function representing
the entire GC system: equation (3) with β = 2, Mc = 106 M⊙, and a single ∆, common to all clusters, evaluated from equation (8) using the median bρt of all 146
Galactic GCs. Solid curves are subsample-specific models using equation (3) with β = 2 and Mc = 106 M⊙ but a different ∆ value for every cluster (obtained
from equation [8] using individual observational estimates of ρt ) in any ρt or rgc bin.
12 McLAUGHLIN & FALL
that the total evaporation time of a tidally limited cluster
is not simply a multiple of an internal two-body relaxation
time, trlx ∝ (Mr
3)1/2, but depends on both trlx and the crossing
time tcr ∝ (M/r
3)−1/2 through the combination tdis ∝ t
with x < 1. The mass-loss rate µev ∝ M/tdis then scales as
M3/2−xr−3/2, which for x 6= 1 differs from the rates µev ∝ ρ
and µev ∝ ρ
t that we have so far adopted. However, our
GCMF models are still meaningful, because postulating tdis ∝
txrlxt
cr implies a dependence of µev on a measure of cluster
density that is, once again, well approximated by ρ
h for
Galactic GCs. Before showing this, we briefly discuss the
reasons and the evidence for a possible dependence of tdis on
both trlx and tcr.
If stars are assumed to escape a cluster as soon as they
have attained energies above some critical value as a result
of two-body relaxation, then tdis ∝ trlx is expected (and con-
firmed by N-body simulations; e.g., Baumgardt 2001). How-
ever, more complicated behavior may arise when escape not
only depends on stars satisfying such an energy criterion, but
also requires them to cross a spatial boundary. Then, al-
though the stars are still scattered to near- and above-escape
energies on the timescale trlx, they require some additional
time to actually leave the cluster. This escape timescale is
related fundamentally to tcr (but also depends on details of
the stellar orbits, the external tidal field, and the shape of
the zero-energy surface). The longer this extra time, the
higher is the probability that further encounters with bound
cluster stars may scatter any potential escapers back down
to sub-escape energies. The net result is a slow-down (“re-
tardation”) of the overall evaporation rate (Chandrasekhar
1942; King 1959; Takahashi & Portegies Zwart 1998, 2000;
Fukushige & Heggie 2000; Baumgardt 2001) and a length-
ening of the cluster lifetime tdis, by a factor that can be ex-
pected to increase with the ratio tcr/trlx. If this factor scales as
(tcr/trlx)
1−x for some x < 1, then tdis ∝ trlx (tcr/trlx)
1−x = txrlxt
While such a retardation of evaporation can be expected
to occur at some level in all clusters, there are physical sub-
tleties in the effect that are probably not captured adequately
by a simple re-parametrization of lifetimes as tdis ∝ t
In particular, it is unlikely that this expression can hold for
clusters of all masses with a single value of x < 1. Since
tcr/trlx ∝ M
−1, very massive clusters have tcr ≪ trlx, and stars
scattered to greater than escape energies by relaxation cross
the tidal boundary effectively instantaneously—implying that
the standard tdis ∝ trlx, or x→ 1, applies in the high-mass limit.
Indeed, if this were not the case, and a fixed x < 1 held for all
M, then an unphysical tdis < trlx would obtain at high enough
masses; see Baumgardt (2001) for further discussion. Unfor-
tunately, “very massive” is not well quantified in this context,
and it is not yet clear if a single value of x is accurate for the
entire GC mass regime. So far, it has been checked directly
only for initial cluster masses below the current peak of the
GCMF.
It is also worth noting that the analysis and simulations
aimed at this problem to date have dealt with clusters on cir-
cular or moderately eccentric orbits in galactic potentials that
are static and spherical. This means that any tidal perturba-
tions felt by stars within the clusters are relatively weak and/or
slow compared to their own orbital periods, leading to nearly
adiabatic or at least non-impulsive responses. In more realis-
tic situations, the galactic potential would be time-dependent
and non-spherical and there might be additional tidal pertur-
bations, including disk and bulge shocks. These perturbations
could in some cases accelerate the escape of weakly bound
stars from the clusters and thus counteract the retardation ef-
fect to some degree. Further study is therefore needed to de-
termine the regime of validity of the formula tdis ∝ t
cr and
its possible modification outside this regime.
In the meantime, Baumgardt (2001) and
Baumgardt & Makino (2003; hereafter BM03) have fit-
ted this formula to the lifetimes of a suite of N-body clusters
with initial masses M0 . 7 × 10
4 M⊙ and several different
initial concentrations and orbital eccentricities. BM03 at
first write tdis in terms of the relaxation and crossing times
of clusters at their half-mass radii, so that trlx ∝ (Mr
tcr ∝ (M/r
−1/2, and tdis ∝ M
x−1/2r
h (see their equation
[5]). However, they immediately take a factor of (rt/rh)
out from the normalization of this scaling—in effect to obtain
tdis ∝M
x−1/2r
t with a different constant of proportionality—
and then use a simple definition of the tidal radius (their
equation [1], r3t = GMr
c , which is appropriate for a
circular orbit of radius rp in a logarithmic potential with
circular speed Vc; see Innanen, Harris, & Webbink 1983)
to obtain the total lifetime of a cluster as a function of its
initial mass, perigalactic distance, and Vc (their equation
[7]). A single exponent x ≃ 0.75 and a single normalization
in this function then suffice to predict to within 10% the
lifetimes of the simulated clusters, regardless of their initial
concentrations. By implication, if trlx and tcr were fixed at rh
rather than rt , then tdis would have an additional concentration
dependence, related to the ratio (rt/rh)
3/2—very similar to
what we discussed in §3.2.1 for the case x = 1.
We now re-examine the Milky Way GCMF in terms of
this prescription for retarded evaporation (bearing in mind the
caveats mentioned above). To avoid any explicit dependences
on concentration, we also focus on the tidal radius and write
tdis ∝ M
x−1/2r
t for general x ≤ 1; but we do not substitute a
potential- and orbit-specific formula for rt in terms of rp and
galactic properties such as Vc. Instead, to keep the empha-
sis entirely on cluster densities, we re-write the scaling of the
lifetime in terms of the mean surface density inside the tidal
radius, Σt ≡ M/πr
t , and the corresponding volume density
ρt = 3M/4πr
t . This leads to tdis ∝ MΣ
−3(1−x)
−2(x−3/4)
t , which
then implies
µev ≡ −dM/dt ∝ M/tdis ∝ Σ
3(1−x)
2(x−3/4)
t . (9)
Clearly, the standard µev ∝ ρ
t , which we have already dis-
cussed, is recovered for x = 1; while for x = 0.75, we have the
equally straightforward µev ∝ Σ
BM03 find that, even with the retarded evaporation implied
by x ≃ 0.75, the masses of their simulated clusters still de-
crease approximately linearly with time after stellar-evolution
effects (which are only important for the first few 108 yr) are
separated out; see especially their Figure 6, equation (12), and
related discussion. Thus, if the GCMF initially rose towards
low masses and has been eroded by slow, relaxation-driven
cluster destruction, then in this modified description of evap-
oration we might expect the current mass function to depend
fundamentally on Σt rather than ρh or ρt . But because M(t)
still decreases nearly linearly with t, only now with µev ∝Σ
for each cluster, the shape of the evolved GCMF and its de-
pendence on Σt should resemble our earlier results for ρh and
SHAPING THE GCMF BY EVAPORATION 13
We have confirmed this expectation by repeating all of our
analyses in §2 again, now using µev ∝Σ
t to estimate cluster
mass-loss rates. As before, we calculate Σt from the data in
the Harris (1996) catalogue, although we caution once more
that the tidal radii, and thus the derived Σt , are more uncertain
than rh and ρh.
Figure 7, which should be compared to Figures 1 and 5
above, shows that the average Galactic GC mass increases
systematically with Σt; that the lower envelope of the M–Σt
distribution is described well by M ∝ Σ
t (the dashed line in
the left-hand panel of Figure 7), which is a locus of constant
lifetime against evaporation for µev ∝ Σ
t ; and that the scat-
ter in the distribution of cluster Σt versus Galactocentric ra-
dius (right-hand panel of the figure) is substantial, as required
to account for the almost non-existent correlation between M
and rgc.
The left-hand side of Figure 8 shows the mass functions
of globulars in three bins of Σt , as defined in each panel.
The right-hand side of the figure shows dN/d log M in the
same three intervals of rgc as in Figures 2 and 6 above. As
in those earlier plots, the dashed curve in all panels of Figure
8 is a model GCMF with the same parameters in every case,
representing the mass function of the entire Galactic GC sys-
tem. Once again, compared to the average MTO, the observed
turnover mass is significantly lower for clusters in the lowest
Σt bin and higher for clusters in the highest Σt bin, while the
width of dN/d log M decreases noticeably as Σt increases.
The solid curves in Figure 8 are again different in every
panel. They are the sums of evaporation-evolved Schechter
functions as in equation (3), with the usual β = 2 assumed
but with total mass losses estimated individually for each GC
in any Σt or rgc bin according to ∆ ∝ Σ
t rather than ∆ ∝
h or ∆ ∝ ρ
t . However, it turns out not to be necessary
to change the normalization of ∆ ∝ ρ
h in equation (4) to
achieve good fits to the observed GCMF as a function of either
Σt or rgc. Thus, in Figure 8 we have simply used
∆ = 1.45× 104 M⊙
Σt/M⊙ pc
−2)3/4 . (10)
The fits of these models, based on tdis ∝ t
cr with x ≃
0.75, are indistinguishable from the fits of our original mod-
els based on the standard tdis ∝ trlx, i.e., x = 1. (We have con-
firmed that adopting individual ∆ given by equation [10] also
reproduces the GCMFs of low-and high-concentration GCs
in Figure 3 as well as before.) It was somewhat unexpected
that equation (10) and equation (4) should have the same nu-
merical coefficient, but we note that this follows empirically
from the fact that the measured ρh and Σt of Galactic GCs
are consistent with the simple near-equality, ρh/M⊙ pc
(Σt/M⊙ pc
−2)1.5 in the mean. This is illustrated in Figure 9,
which also shows that there is significant scatter about the re-
lation.12 However, this scatter does not correlate with clus-
ter mass or Galactocentric radius. From a pragmatic point of
view, therefore, ρ
h and Σ
t are near enough to interchange-
able for our purposes, and there is no practical difference be-
12 Although it may be only a coincidence that the constant of proportion-
ality in ρh ∝ Σ
t is so near unity, the basic scaling itself holds because
combining the observed correlation between cluster mass and central con-
centration (Djorgovski & Meylan 1994; McLaughlin 2000) with the intrinsic
dependence of rt/rh on c in King models leads roughly to (rt/rh) ∝ M
tween GCMF models based on one or the other measure of
GC density.
One further check on this is to verify that the mass-loss rate
associated with equation (10) is roughly in keeping with that
implied by the N-body simulations pointing to x = 0.75 in the
first place. Thus, we compare the rate
µev = ∆/(13 Gyr) ≃ 1100 M⊙ Gyr
−1 (Σt/M⊙ pc
−2)3/4 (11)
to a formula implicit in BM03. Starting with their equa-
tion (7) for the lifetime tdis as a function of initial cluster
mass and perigalactic distance and circular speed in a loga-
rithmic halo potential; using their x = 0.75 and their normal-
ization of 1.91× 106 yr, multiplied as in their equation (9)
by (1 + e) to allow for eccentric orbits with apo- and peri-
galactic distances related by e ≡ (ra − rp)/(ra + rp); insert-
ing their equation (1) for rt ; taking the mean mass of clus-
ter stars to be m∗ = 0.55M⊙, as they do; using γ = 0.02 as
they do in the Coulomb logarithm, ln(γM0/m∗); and defining
Σt,0 ≡ M0/πr
t,0 (the subscript 0 denoting initial values), we
obtain
µev(BM03)≃
0.7M0
1 + e
M⊙ Gyr
0.036M0/M⊙
0.036× 105
]3/4 (
M⊙ pc−2
This is appropriate for clusters that just fill their Roche lobes
at perigalacticon, which is where Σt,0 is specified. The factor
of 0.7 in the first equality accounts for mass loss due to stellar
evolution in the BM03 simulations, which, as they discuss,
can be treated as having occurred almost immediately and in
full at the beginning of a cluster’s life.
Our GCMF-based µev is a factor of ≈ 2 faster than the N-
body value for clusters on circular orbits (with e = 0 and in
steady tidal fields) in the simulations; and our µev is still
within a factor of about three of the N-body rate for clusters on
eccentric orbits with e = 0.5 in BM03 (e ≃ 0.5–0.6 is typical
for tracers with an isotropic velocity distribution in a logarith-
mic potential; van den Bosch et al. 1999). This is very similar
to the comparison of lifetimes in §3.1 for our original models
based on µev ∝ ρ
h . Moreover, our new estimate of µev and
that in BM03 are still subject to their own, separate uncertain-
ties and reflect different idealizations and assumptions. For
example, our rate still depends on the exact power-law expo-
nent β at low masses in the initial GCMF, as discussed after
equation (7); while the rate from BM03 still neglects grav-
itational shocks from disk crossings and passages by a dis-
crete galactic bulge, and may additionally be biased low for
M0 > 10
5 M⊙ if x > 0.75 at such masses. All of this—not
to mention again the large uncertainties and possible system-
atics in the estimates of tidal radii needed to calculate Σt—
makes the near agreement between equations (11) and (12)
more striking than any apparent discrepancy.
In summary, although the relation µev ∝ ρ
h ≃ constant
in time is rigorously correct only in rather specific circum-
stances, our GCMF models based on it in §2 are good proxies,
in all respects, for models based on other plausible characteri-
zations of relaxation-driven cluster mass loss. This result will
likely be important for future studies of the mass functions of
extragalactic cluster systems, where it may well be necessary
14 McLAUGHLIN & FALL
FIG. 7.— Scatter plots of mass M versus mean surface density inside the tidal radius (Σt ≡ M/πr2t ) and of Σt versus Galactocentric radius rgc, for 146 Galactic
GCs from the Harris (1996) catalogue. These plots are analogous to the left- and rightmost panels of Figure 1, and the two panels of Figure 5. The dashed line in
the left-hand plot traces the relation M ∝ Σ
t , which defines a locus of constant evaporation time for µev ∝ Σ
FIG. 8.— Observed GCMF (points, with Poisson errorbars) and models (curves) as a function of mean surface density inside the tidal radius, Σt ≡ M/πr2t (left-
hand panels), and as a function of Galactocentric radius, rgc (right-hand panels). The dashed curve in every panel is an evolved Schechter function representing
the entire GC system: equation (3) with β = 2, Mc = 106 M⊙, and a single ∆, common to all clusters, evaluated from equation (10) using the median bΣt of all
146 Galactic GCs. Solid curves are subsample-specific models using equation (3) with β = 2 and Mc = 106 M⊙ but a different ∆ value for every cluster (obtained
from equation [10] using individual observational estimates of Σt ) in any Σt or rgc bin.
SHAPING THE GCMF BY EVAPORATION 15
FIG. 9.— Half-mass density, ρh = 3M/8πr
h , against mean surface den-
sity inside the tidal radius, Σt = M/πr2t , for 146 clusters with data in Harris
(1996). The straight line is ρh = Σ
to adopt procedures based on ρh rather than ρt or Σt because
of the difficulty or impossibility of estimating tidal radii.
3.3. MTO versus rgc, and Velocity Anisotropy in GC Systems
In this paper we have directly modeled dN/d log M as a
function only of GC density and age, and used the observed ρh
(or ρt , or Σt) of clusters in relatively narrow ranges of Galac-
tocentric position to show that such models are consistent with
the current near-constancy of the GCMF as a function of rgc.
Most other models in the literature for evaporation-dominated
GCMF evolution, in either the Milky Way or other galaxies,
instead predict the distribution explicitly as a function of rgc at
any time. They therefore need, in effect, to derive theoretical
density–position relations for clusters in galaxies alongside
their main GCMF calculations. This usually begins with the
adoption of analytical potentials to describe the parent galax-
ies of GCs. Taking these to be spherical and static for a Hub-
ble time allows the use of standard tidal-limitation formulae
to write GC densities ab initio in terms of the (fixed) peri-
centers rp of unique orbits in the adopted potentials. Cluster
relaxation times and mass-loss rates µev then follow as func-
tions of rp as well. Finally, specific initial mass, space, and
velocity (or orbital eccentricity) distributions are chosen for
entire GC systems, so that at all later times it is known what
the dynamically evolved dN/d log M is for globulars with any
single rp; how many clusters with a given rp survive; and what
the distributions of rp and all dependent cluster properties are
at any instantaneous position rgc.
In this approach, if the GCMF began with a power-law rise
towards low masses and its current peak is due entirely to clus-
ter disruption, then a dependence of MTO on rp is expected in
general, because the densities of tidally limited GCs decrease
with increasing rp. Thus, models along these lines that as-
sume the orbit distribution of a GC system to be the same
at all radii in a galaxy (i.e., that the time average of the ra-
tio rgc/rp is independent of position) have typically had diffi-
culty in accounting for the observed weak or non-correlation
between MTO and present rgc in large galaxies. This is partic-
ularly a problem if it is assumed that the initial GCMF was a
pure power law, with the same index at arbitrarily high masses
as low (e.g., Baumgardt 1998; Vesperini 2001). It is poten-
tially less of a concern if dN/d log M started as a Schechter
function with an exponential cut-off at masses M > Mc, as
we have assumed, since then the existence of a strict upper
bound MTO ≤ Mc (§2.2) means that the dependence of an
evaporation-evolved MTO on rp and rgc must saturate for small
enough galactocentric radii (high enough GC densities). Even
so, the “scale-free” models of FZ01, in which Mc ≃ 10
and all GCs in a Milky Way-like galaxy potential have the
same time-averaged rgc/rp, still predict a gradient in MTO ver-
sus rgc that is stronger than observed.
FZ01 showed that, if they left all of their other assumptions
unchanged, then a dependence of GCMF peak mass on rgc
could be effectively erased by an appropriately varying radial
velocity anisotropy in the initial GC system. Thus, in their
“Eddington” models the eccentricity of a typical cluster or-
bit increases with galactocentric distance (the time average of
rgc/rp increases with radius), such that globulars spread over
a larger range of current rgc can have more similar rp and asso-
ciated MTO. However, the initial velocity-anisotropy gradient
required to fit the Milky Way GCMF data specifically is only
marginally consistent with the observed kinematics of the GC
system (e.g., Dinescu, Girard, & van Altena 1999).13 Subse-
quently, Vesperini et al. (2003) constructed broadly similar
models for the GCMF of the Virgo elliptical M87 and con-
cluded that there, too, a variable radial velocity anisotropy is
required to match the observed MTO versus rgc; but the model
anisotropy profile in this case is clearly inconsistent with the
true velocity distribution of the GC system, which is observed
to be isotropic out to large rgc (Romanowsky & Kochanek
2001; Côté et al. 2001).
These results certainly suggest that some element is lacking
in rgc-oriented GCMF models developed as outlined above.
But they do not mean that the fault lies with the main hypoth-
esis, that the difference between the mass functions of young
clusters and old GCs is due to the effects of slow, relaxation-
driven disruption in the latter case. Any conclusions about
velocity anisotropy depend on the totality of steps taken to
connect the densities and positions of clusters; and it is possi-
ble that reasonable changes to one or more of these ancillary
assumptions could make the models compatible with the ob-
served kinematics of GCs in both the Milky Way and M87,
without abandoning a basic physical picture of evaporation-
dominated GCMF evolution that is otherwise quite success-
One issue is that previous models have always specified
evaporation rates a priori as functions of cluster density (or or-
bital pericenter), usually normalizing µev so that tdis/trh ≃ 20–
40 as in standard treatments of two-body relaxation. How-
ever, following our discussion in §3.1 and §3.2, it would seem
worthwhile to investigate these models with µev increased at
fixed ρh or rp to allow tdis/trh ≈ 10 (if β ≃ 2 for the low-mass
power-law part of the initial GCMF).
FZ01 and Vesperini et al. (2003) both consider velocity dis-
tributions parametrized by a galactocentric anisotropy radius,
RA, inside of which a cluster system is essentially isotropic
and beyond which it is increasingly dominated by radial or-
13 The fact that clusters on radial orbits are preferentially disrupted lessens
any inconsistency between the radial anisotropy required in the initial veloc-
ity distribution and observational constraints on the present velocity distribu-
tion.
16 McLAUGHLIN & FALL
bits. In these terms, the difficulty with the published models
is that, to reproduce the observed insensitivity of MTO to rgc
given standard normalizations of µev, they require values of
RA that are smaller than allowed by observations (especially
for M87). Increasing RA to more realistic values while keep-
ing the normalization of µev fixed leads to a stronger gra-
dient in MTO: the orbits of GCs at small rgc . RA remain
closely isotropic and the typical rp and MTO are essentially
unchanged, while at large galactocentric distances the clus-
ter orbits are on average less radial than before, with larger
rp, lower densities, and lower evolved MTO for a given rgc.
This effect is illustrated, for example, in Figure 9 of FZ01.
However, it can be compensated at least in part by increas-
ing µev by a common factor for all GCs, with the new, larger
RA fixed, if the initial mass function is assumed to have been
a Schechter function rather than a pure power law extend-
ing to arbitrarily high masses. A faster evaporation rate
will then lead to a (roughly) proportionate increase in the
evolved GCMF peak mass for GCs with relatively low den-
sities, i.e., those at large rgc and rp; but the increase in MTO
will be smaller, and eventually even negligible, for higher-
density clusters at progressively smaller rgc—again because
MTO grows less than linearly with µev ∝ ρ
h when there is
an upper limit MTO < Mc due to an exponential cut-off in the
initial dN/d log M0. Thus, the qualitative effect of increasing
the normalization of µev in models with radially varying GC
velocity anisotropy is to weaken the amount of radial-orbit
bias required to fit an observed MTO versus rgc.
Another point, emphasized by FZ01, has to do with the
standard starting assumption that GCs orbit in galaxies that
are perfectly static and spherical. In reality, galaxies grow
hierarchically. In this case, even if the values of µev are not
changed, much of the burden for the weakening or erasing of
any initial gradients in MTO versus rgc may be transferred from
velocity anisotropy to the time-dependent evolution of the
galaxies themselves. Violent relaxation, major mergers, and
smaller accretion events all work to move clusters between
different parts of galaxies and between different progenitors,
scrambling and combining any number of pericenter–density–
MTO relations. Any position dependences in the GC ρh dis-
tribution and in MTO itself for the final galaxy are therefore
bound to be weaker, more scattered, and more difficult to re-
late accurately to a cluster velocity distribution than in the
case of a monolithic, non-evolving potential. Allowing for
a non-spherical galaxy potential would have qualitatively the
same effect, because in this case every cluster explores a range
of pericenters and different maximum tidal fields on each of
its orbits.
In this situation, it may be important to ask how evapora-
tion rates can still be approximately constant in time—so that
cluster masses still decrease approximately linearly with t as
our models assume—if the tidal field around any given GC
changes significantly over time. Thus, consider first a sys-
tem of GCs in a single, static galaxy potential. The mass-
evolution curve for each cluster is approximately a straight
line, M(t) ≃ M0 −µevt, with µev depending on some measure
of internal density, which may be ρ
h , ρ
t , or Σ
t . The av-
erage mass-evolution curve for the entire system of clusters
is also approximately linear, 〈M(t)〉 ≃ 〈M0〉 − 〈µev〉t. If now
a merger or other event rearranges the clusters in the galaxy,
then after the event the mass-loss rates of some clusters will
be higher than before and the rates of other clusters will be
lower than before. However, if the mean density of the galaxy
as a whole is roughly the same after the event as before, then
so too will be the average of the GC densities, because of
tidal limitation. The average 〈µev〉 ∝ 〈ρ
h 〉 (say) will differ
even less between the pre- and post-merger systems. Thus, al-
though using instantaneous densities to estimate the past µev
of individual clusters may err on the high side for some clus-
ters and on the low side for others, these errors will average
away to a small or even zero net bias. The approximation
µev ≃ constant in time in our GCMF models will then still be
valid in the mean, and the average 〈M(t)〉 dependence of suf-
ficiently large numbers of clusters will remain roughly linear.
This type of scenario might be expected to pertain at least
to galaxies that evolve on the fundamental plane, since this
entails a connection between the total (baryonic plus dark)
masses and circular speeds of galaxies, of the form Mgal ∝V
or Mgal ∝ V
c . By the virial theorem, the average densities
scale as ρgal ∝ V
2, and thus ρgal ∝ M
gal or ρgal ∝ M
gal .
Insofar as 〈ρh〉 ∝ 〈ρt〉 ∝ ρgal for the GCs, the system-wide av-
erage 〈µev〉 ∝ 〈ρ
h 〉 should therefore not change drastically
even after a major merger between two fundamental-plane
galaxies; at most, the ratio of final to initial 〈µev〉 will be
roughly of order the −1/4 power of the ratio of final to ini-
tial Mgal. Note that this line of reasoning is closely related to
that applied by FZ01 to explain the small observed galaxy-
to-galaxy differences in the average turnover masses of entire
GC systems (although non-zero differences do exist, and can
be accomodated in these sorts of arguments; see Jordán et al.
2006, 2007).
A full exploration of questions such as these, about the wide
range of ingredients in current GC-plus-galaxy models, will
most likely require large N-body simulations set in a realistic,
cold dark matter cosmology. Until these can be carried out,
it is our view that the kinematics of globular cluster systems
cannot be used as decisive side constraints on theories for the
GCMF.
4. CONCLUSIONS
We have shown that the mass function dN/d log M of glob-
ular clusters in the Milky Way depends significantly on clus-
ter half-mass density, ρh, with the peak or turnover mass MTO
increasing and the width of the distribution decreasing as ρh
increases. This behavior is expected if the GCMF initially
rose towards masses below the present turnover scale—as the
mass functions of young cluster systems like that in the An-
tennae galaxies do—and has evolved to its current shape via
the slow depletion of low-mass clusters over Gyr timescales,
primarily through relaxation-driven evaporation. The fact that
MTO increases with cluster density favors evaporation over
external gravitational shocks as the primary mechanism of
low-mass cluster disruption, since the mass-loss rates asso-
ciated with shocks depend inversely on cluster density and
directly on cluster mass. Our results therefore add to previ-
ous arguments supporting an interpretation of the GCMF in
terms of evaporation-dominated evolution, based on the fact
that dN/d log M scales as M1−β with β ≃ 0 in the low-mass
limit (Fall & Zhang 2001).
The observed GCMF as a function of ρh is fitted well
by simple models in which the initial distribution was
a Schechter function, dN/d log M0 ∝ M
0 exp
−M0/Mc
with β = 2 and Mc ≃ 10
6 M⊙ assumed, and in which clusters
have been losing mass for a Hubble time at roughly steady
rates that can be estimated from their current half-mass den-
SHAPING THE GCMF BY EVAPORATION 17
sities as µev ∝ ρ
h . We have shown that, although this pre-
scription is approximate, it captures the main physical depen-
dence of relaxation-driven evaporation. In particular, it leads
to model GCMFs that are entirely consistent with those re-
sulting from alternative characterizations of evaporation rates
in terms of cluster tidal densities ρt or mean surface densities
Σt (§3.2). The normalization of µev at a given ρh (or ρt , or Σt)
required to fit the GCMF implies total cluster lifetimes that
are within range of the lifetimes typically obtained in theoret-
ical studies of two-body relaxation, although our values may
be slightly shorter than the theoretical ones if the low-mass,
power-law part of the initial cluster mass function was as steep
as we have assumed.
Taking clusters in various bins of central concentration c
and Galactocentric radius rgc and using their (individual) ob-
served densities as direct input to our models yields dynam-
ically evolved GCMFs as functions of c and rgc that agree
well with all data. This again indicates that the most fun-
damental physical dependence in the GCMF is that on clus-
ter density. Moreover, our models for dN/d log M versus
rgc obtained in this way are consistent in particular with the
well-known insensitivity of the GCMF peak mass to Galac-
tocentric position. This is seen to follow from a significant
variation of MTO with ρh (or ρt , or Σt)—due in our analysis
to evaporation-dominated cluster disruption—combined with
substantial scatter in the GC densities at any Galactocentric
position.
We have not invoked an anisotropic GC velocity distribu-
tion to explain the observed weak variation of MTO with rgc;
indeed, we have made no predictions or assumptions what-
soever about velocity anisotropy. We have emphasized that,
when velocity anisotropy enters other long-term dynamical-
evolution models for the GCMF, it is only in conjunction with
several additional, interrelated assumptions made as part of
larger efforts to derive theoretical density–rgc relations for
GCs—which we have not attempted to do here. The appar-
ent need in some current models for a strong bias towards
high-eccentricity cluster orbits to explain the near-constancy
of MTO versus rgc might well be avoided by changing one or
more ancillary assumptions in the models, without having to
discard the underlying idea that the peak and low-mass shape
of the GCMF are the result of relaxation-driven cluster dis-
ruption.
It clearly will be of interest to test and refine the main
ideas in this paper through modeling of the GCMFs in other
galaxies. For the time being at least, doing so will re-
quire the estimation of approximate mass-loss rates using
cluster half-mass densities rather than tidal quantities, sim-
ply because GC half-light radii can be measured accurately
in many systems beyond the Local Group, whereas tidal
radii are much more model-dependent and difficult to ob-
serve. Chandar, Fall, & McLaughlin (2007) have recently
shown that the peak mass of the GCMF in the Sombrero
galaxy (M104) increases with ρh in a way that is reasonably
well described by sums of evolved Schechter (1976) functions
as in the models presented in this paper. It should be rela-
tively straightforward to pursue similar studies in other nearby
galaxies.
We thank Michele Trenti, Douglas Heggie, Bill Harris, Ru-
pali Chandar, and Bruce Elmegreen for helpful discussions
and comments. SMF acknowledges support from the Am-
brose Monell Foundation and from NASA grant AR-09539.1-
A, awarded by the Space Telescope Science Institute, which is
operated by AURA, Inc., under NASA contract NAS5-26555.
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http://arxiv.org/abs/astro-ph/0608069
|
0704.0081 | Quantum Deformations of Relativistic Symmetries | Quantum Deformations of Relativistic Symmetries∗
V.N. Tolstoy†
Institute of Nuclear Physics, Moscow State University,
119 992 Moscow, Russia; e-mail: [email protected]
Abstract
We discussed quantum deformations of D = 4 Lorentz and Poincaré algebras. In
the case of Poincaré algebra it is shown that almost all classical r-matrices of S. Za-
krzewski classification correspond to twisted deformations of Abelian and Jordanian
types. A part of twists corresponding to the r-matrices of Zakrzewski classification
are given in explicit form.
1 Introduction
The quantum deformations of relativistic symmetries are described by Hopf-algebraic
deformations of Lorentz and Poincaré algebras. Such quantum deformations are classified
by Lorentz and Poincaré Poisson structures. These Poisson structures given by classical
r-matrices were classified already some time ago by S. Zakrzewski in [1] for the Lorentz
algebra and in [2] for the Poincaré algebra. In the case of the Lorentz algebra a complete
list of classical r-matrices involves the four independent formulas and the corresponding
quantum deformations in different forms were already discussed in literature (see [3, 4, 5,
6, 7]). In the case of Poincaré algebra the total list of the classical r-matrices, which satisfy
the homogeneous classical Yang-Baxter equation, consists of 20 cases which have various
numbers of free parameters. Analysis of these twenty solutions shows that each of them
can be presented as a sum of subordinated r-matrices which almost all are of Abelian
and Jordanian types. A part of twists corresponding to the r-matrices of Zakrzewski
classification are given in explicit form.
2 Preliminaries
Let r be a classical r-matrix of a Lie algebra g, i.e. r ∈
∧ g and r satisfies to the classical
Yang–Baxter equation (CYBE)
[r12, r13 + r23] + [r13, r23] = Ω , (2.1)
∗Invited talk at the XXII Max Born Symposium ”Quantum, Super and Twistors”, September 27-29,
2006 Wroclaw (Poland), in honour of Jerzy Lukierski.
†Supported by the grants RFBR-05-01-01086 and FNRA NT05-241455GIPM.
http://arxiv.org/abs/0704.0081v1
where Ω is g-invariant element, Ω ∈ (
∧ g)g. We consider two types of the classical
r-matrices and corresponding twists.
Let the classical r-matrix r = rA has the form
xi ∧ yi , (2.2)
where all elements xi, yi (i = 1, . . . , n) commute among themselves. Such an r-matrix is
called of Abelian type. The corresponding twist is given as follows
= exp
= exp
xi ∧ yi
. (2.3)
This twisting two-tensor F := Fr
satisfies the cocycle equation
F 12(∆⊗ id)(F ) = F 23(id⊗∆)(F ) , (2.4)
and the ”unital” normalization condition
(ǫ⊗ id)(F ) = (id⊗ ǫ)(F ) = 1 . (2.5)
The twisting element F defines a deformation of the universal enveloping algebra U(g)
considered as a Hopf algebra. The new deformed coproduct and antipode are given as
follows
∆(F )(a) = F∆(a)F−1 , S(F )(a) = uS(a)u−1 (2.6)
for any a ∈ U(g), where ∆(a) is a co-product before twisting, and u =
i S(f
i ) if
i ⊗ f
Let the classical r-matrix r = rJ(ξ) has the form
rJ(ξ) = ξ
xν ∧ yν
, (2.7)
where the elements xν , yν (ν = 0, 1, . . . , n) satisfy the relations
[x0, y0] = y0 , [x0, xi] = (1− ti)xi , [x0, yi] = tiyi ,
[xi, yj] = δijy0 , [xi, xj ] = [yi, yj] = 0 , [y0, xj ] = [y0, yj] = 0 ,
(2.8)
(i, j = 1, . . . , n), (ti ∈ C). Such an r-matrix is called of Jordanian type. The corresponding
twist is given as follows [8, 9]
= exp
xi ⊗ yi e
−2tiσ
exp(2x0 ⊗ σ) , (2.9)
1Here entering the parameter deformation ξ is a matter of convenience.
2It is easy to verify that the two-tensor (2.7) indeed satisfies the homogenous classical Yang-Baxter
equation (2.1) (with Ω = 0), if the elements xν , yν (ν = 0, 1, . . . , n) are subject to the relations (2.8).
where σ := 1
ln(1 + ξy0).
Let r be an arbitrary r-matrix of g. We denote a support of r by Sup(r)4. The
following definition is useful.
Definition 2.1 Let r1 and r2 be two arbitrary classical r-matrices. We say that r2 is
subordinated to r1, r1 ≻ r2, if δr1(Sup(r2)) = 0, i.e.
(x) := [x⊗ 1 + 1⊗ x, r1] = 0 , ∀x ∈ Sup(r2) . (2.10)
If r1 ≻ r2 then r = r1 + r2 is also a classical r-matrix (see [15]). The subordination
enables us to construct a correct sequence of quantizations. For instance, if the r-matrix
of Jordanian type (2.7) is subordinated to the r-matrix of Abelian type (2.2), rA ≻ rJ ,
then the total twist corresponding to the resulting r-matrix r = rA+ rJ is given as follows
Fr = Fr
. (2.11)
The further definition is also useful.
Definition 2.2 A twisting two-tensor Fr(ξ) of a Hopf algebra, satisfying the conditions
(2.4) and (2.5), is called locally r-symmetric if the expansion of Fr(ξ) in powers of the
parameter deformation ξ has the form
Fr(ξ) = 1 + c r +O(ξ
2) . . . (2.12)
where r is a classical r-matrix, and c is a numerical coefficient, c 6= 0.
It is evident that the Abelian twist (2.3) is globally r-symmetric and the twist of Jordanian
type (2.9) does not satisfy the relation (2.12), i.e. it is not locally r-symmetric.
3 Quantum deformations of Lorentz algebra
The results of this section in different forms were already discussed in literature (see
[3, 4, 5, 6, 7]).
The classical canonical basis of the D = 4 Lorentz algebra, o(3, 1), can be described
by anti-Hermitian six generators (h, e±, h
′, e′±) satisfying the following non-vanishing
commutation relations5:
[h, e±] = ±e± , [e+, e−] = 2h , (3.1)
[h, e′±] = ±e
± , [h
′, e±] = ±e
± , [e±, e
∓] = ±2h
′ , (3.2)
[h′, e′±] = ∓e± , [e
−] = −2h , (3.3)
and moreover
x∗ = −x (∀ x ∈ o(3, 1)) . (3.4)
3The corresponding twists for Lie algebras sl(n), so(n) and sp(2n) were firstly constructed in the
papers [10, 11, 12, 13].
4The support Sup(r) is a subalgebra of g generated by the elements {xi, yi} if r =
xi ∧ yi.
5Since the real Lie algebra o(3, 1) is standard realification of the complex Lie sl(2,C) these relations
are easy obtained from the defining relations for sl(2,C), i.e. from (3.1).
A complete list of classical r-matrices which describe all Poisson structures and generate
quantum deformations for o(3, 1) involve the four independent formulas [1]:
r1 = α e+ ∧ h , (3.5)
r2 = α (e+ ∧ h− e
+ ∧ h
′) + 2β e′+ ∧ e+ , (3.6)
r3 = α (e
+ ∧ e− + e+ ∧ e
−) + β (e+ ∧ e− − e
+ ∧ e
−)− 2γ h ∧ h
′ , (3.7)
r4 = α
e′+ ∧ e− + e+ ∧ e
− − 2h ∧ h
± e+ ∧ e
+ . (3.8)
If the universal R-matrices of the quantum deformations corresponding to the classical
r-matrices (3.5)–(3.8) are unitary then these r-matrices are anti-Hermitian, i.e.
r∗j = −rj (j = 1, 2, 3, 4) . (3.9)
Therefore the ∗-operation (3.4) should be lifted to the tensor product o(3, 1) ⊗ o(3, 1).
There are two variants of this lifting: direct and flipped, namely,
(x⊗ y)∗ = x∗ ⊗ y∗ (∗ − direct) , (3.10)
(x⊗ y)∗ = y∗ ⊗ x∗ (∗ − flipped) . (3.11)
We see that if the ”direct” lifting of the ∗-operation (3.4) is used then all parameters in
(3.5)–(3.8) are pure imaginary. In the case of the ”flipped” lifting (3.11) all parameters
in (3.5)–(3.8) are real.
The first two r-matrices (3.5) and (3.6) satisfy the homogeneous CYBE and they are
of Jordanian type. If we assume (3.10), the corresponding quantum deformations were
described detailed in the paper [6] and they are entire defined by the twist of Jordanian
type:
= exp (h⊗ σ) , σ =
ln(1 + αe+) (3.12)
for the r-matrix (3.5), and
= exp
σ ∧ ϕ
exp (h⊗ σ − h′ ⊗ ϕ) , (3.13)
(1 + αe+)
2+ (αe′+)
, ϕ = arctan
1 + αe+
(3.14)
for the r-matrix (3.6). It should be recalled that the twists (3.12) and (3.13) are not
locally r-symmetric. A locally r-symmetric twist for the r-matrix (3.5) was obtained in
[14] and it has the following complicated formula:
= exp
∆(h)−
sinhαe+
⊗ e−αe++ eαe+⊗ h
sinhαe+
α∆(e+)
sinhα∆(e+)
, (3.15)
where ∆ is a primitive coproduct.
The last two r-matrices (3.7) and (3.8) satisfy the non-homogeneous (modified) CYBE
and they can be easily obtained from the solutions of the complex algebra o(4,C) ≃
sl(2,C)⊕ sl(2,C) which describes the complexification of o(3, 1). Indeed, let us introduce
the complex basis of Lorentz algebra (o(3, 1) ≃ sl(2;C) ⊕ sl(2,C)) described by two
commuting sets of complex generators:
(h + ıh′) , E1± =
(e± + ıe
±) , (3.16)
(h− ıh′) , E2± =
(e± − ıe
±) , (3.17)
which satisfy the relations (compare with (3.1))
[Hk, Ek±] = ±Ek± , [Ek+, Ek−] = 2Hk (k = 1, 2) . (3.18)
The ∗-operation describing the real structure acts on the generatorsHk, and Ek± (k = 1, 2)
as follows
H∗1 = −H2 , E
1± = −E2± , H
2 = −H1 , E
2± = −E1± . (3.19)
The classical r-matrix r3, (3.7), and r4, (3.8), in terms of the complex basis (3.16), (3.17)
take the form
r3 = r
1 + r
r′3 := 2(β + ıα)E1+ ∧ E1− + 2(β − ıα)E2+ ∧ E2− ,
r′′3 := 4ıγ H2 ∧H1 ,
(3.20)
r4 = r
4 + r
r′4 := 2ıα(E1+ ∧ E1− −E2+ ∧ E2− − 2H1 ∧H2) ,
r′′4 := 4ıλE1+ ∧ E2+
(3.21)
For the sake of convenience we introduce parameter6λ in r′′4 . It should be noted that
r′3, r
3 and r
4 are themselves classical r-matrices. We see that the r-matrix r
3 is simply
a sum of two standard r-matrices of sl(2;C), satisfying the anti-Hermitian condition
r∗ = −r. Analogously, it is not hard to see that the r-matrix r4 corresponds to a Belavin-
Drinfeld triple [15] for the Lie algebra sl(2;C) ⊕ sl(2,C)). Indeed, applying the Cartan
automorphism E2± → E2∓, H2 → −H2 we see that this is really correct (see also [16]).
We firstly describe quantum deformation corresponding to the classical r-matrix r3
(3.20). Since the r-matrix r′′3 is Abelian and it is subordinated to r
3 therefore the algebra
o(3, 1) is firstly quantized in the direction r′3 and then an Abelian twist corresponding
to the r-matrix r′′3 is applied. We introduce the complex notations z± := β ± ıα. It
should be noted that z− = z
+ if the parameters α and β are real, and z− = −z
the parameters α and β are pure imaginary. From structure of the classical r-matrix r′3
it follows that a quantum deformation Ur′
(o(3, 1)) is a combination of two q-analogs of
U(sl(2;C)) with the parameter qz
and qz
, where qz
:= exp z±. Thus Ur′3(o(3, 1))
(sl(2;C))⊗Uq
(sl(2;C)) and the standard generators q±H1z
, E1± and q
, E2± satisfy
6We can reduce this parameter λ to ± 1
by automorphism of o(4,C).
the following non-vanishing defining relations
qH1z+ E1± = q
E1± q
, [E1+, E1−] =
q2H1z+ − q
qz+ − q
, (3.22)
qH2z− E2± = q
E2± q
, [E2+, E2−] =
q2H2z− − q
qz− − q
. (3.23)
In this case the co-product ∆r′
and antipode Sr′
for the generators q±H1z
, E1± and q
E2± can be given by the formulas:
(q±H1z+ ) = q
⊗ q±H1z+ , ∆r′1
(E1±) = E1± ⊗ q
+ q−H1z+ ⊗ E1± , (3.24)
(q±H2z− ) = q
⊗ q±H2z− , ∆r′1
(E2±) = E2± ⊗ q
+ q−H2z− ⊗ E2± , (3.25)
(q±H1z+ ) = q
, Sr′
(E1±) = −q
E1± , (3.26)
(q±H2z
) = q∓H2z− , Sr′1
(E2±) = −q
E2± . (3.27)
The ∗-involution describing the real structure on the generators (3.8) can be adapted to
the quantum generators as follows
(q±H1z+ )
∗ = q∓H2
, E∗1± = −E2± , (q
)∗ = q∓H1
, E∗2± = −E1± , (3.28)
and there exit two ∗-liftings: direct and flipped, namely,
(a⊗ b)∗ = a∗ ⊗ b∗ (∗ − direct) , (3.29)
(a⊗ b)∗ = b∗ ⊗ a∗ (∗ − flipped) (3.30)
for any a ⊗ b ∈ Ur′
(o(3, 1)) ⊗ Ur′
(o(3, 1)), where the ∗-direct involution corresponds to
the case of the pure imaginary parameters α, β and the ∗-flipped involution corresponds
to the case of the real deformation parameters α, β. It should be stressed that the Hopf
structure on Ur′
(o(3, 1)) satisfy the consistency conditions under the ∗-involution
(a∗) = (∆r′
(a))∗, Sr′
((Sr′
(a∗))∗) = a (∀x ∈ Ur′
(o(3, 1)) . (3.31)
Now we consider deformation of the quantum algebra Ur′
(o(3, 1)) (secondary quan-
tization of U(o(3, 1))) corresponding to the additional r-matrix r′′3 , (3.20). Since the
generators H1 and H2 have the trivial coproduct
(Hk) = Hk ⊗ 1 + 1⊗Hk (k = 1, 2) , (3.32)
therefore the unitary two-tensor
:= qH1∧H2ıγ (F
= F−1
) (3.33)
satisfies the cocycle condition (2.4) and the ”unital” normalization condition (2.5). Thus
the complete deformation corresponding to the r-matrix r3 is the twisted deformation of
(o(3, 1)), i.e. the resulting coproduct ∆r
is given as follows
(x) = Fr′′
(x)F−1
(∀x ∈ Ur′
(o(3, 1)) . (3.34)
and in this case the resulting antipode Sr
does not change, Sr
. Applying the
twisting two-tensor (3.33) to the formulas (3.24) and (3.25) we obtain
(q±H1z+ ) = q
⊗ q±H1z+ , ∆r′1(q
) = q±H2z− ⊗ q
, (3.35)
(E1+) = E1+ ⊗ q
qH2ıγ + q
q−H2ıγ ⊗ E1+ , (3.36)
(E1−) = E1− ⊗ q
q−H2ıγ + q
qH2ıγ ⊗ E1− , (3.37)
∆r3(E2+) = E2+ ⊗ q
q−H1ıγ + q
qH1ıγ ⊗ E2+ , (3.38)
∆r3(E2−) = E2− ⊗ q
qH1ıγ + q
q−H1ıγ ⊗ E2− . (3.39)
Next, we describe quantum deformation corresponding to the classical r-matrix r4
(3.21). Since the r-matrix r′4(α) := r
4 is a particular case of r3(α, β, γ) := r3, namely
r′4(α) = r3(α, β = 0, γ = α), therefore a quantum deformation corresponding to the r-
matrix r′4 is obtained from the previous case by setting β = 0, γ = α, and we have the
following formulas for the coproducts ∆r′
) = q
(k = 1, 2) , (3.40)
(E1+) = E1+ ⊗ q
H1+H2
+ q−H1−H2
⊗ E1+ , (3.41)
(E1−) = E1− ⊗ q
H1−H2
+ q−H1+H2
⊗ E1− , (3.42)
(E2+) = E2+ ⊗ q
−H1−H2
ξ + q
H1+H2
ξ ⊗ E2+ , (3.43)
(E2−) = E2− ⊗ q
H1−H2
ξ + q
−H1+H2
ξ ⊗ E2− , (3.44)
where we set ξ := ıα.
Consider the two-tensor
:= exp
λE1+q
H1+H2
ξ ⊗ E2+q
H1+H2
. (3.45)
Using properties of q-exponentials (see [17]) is not hard to verify that Fr′′
satisfies the co-
cycle equation (2.4). Thus the quantization corresponding to the r-matrix r4 is the twisted
q-deformation Ur′
(o(3, 1)). Explicit formulas of the co-products ∆r
(·) = F
(·)F−1
and antipodes Sr4(·) in the complex and real Cartan-Weyl bases of Ur′4(o(3, 1)) will be
presented in the outgoing paper [7].
4 Quantum deformations of Poincare algebra
The Poincaré algebra P(3, 1) of the 4-dimensional space-time is generated by 10 elements:
the six-dimensinal Lorentz algebra o(3, 1) with the generators Mi, Ni (i = 1, 2, 3):
[Mi, Mj ] = ıǫijk Mk, [Mi, Nj ] = ıǫijk Nk, [Ni, Nj ] = −ıǫijk Mk, (4.1)
and the four-momenta Pj, P0 (j = 1, 2, 3) with the standard commutation relations:
[Mj , Pk] = ıǫjkl Pl , [Mj , P0] = 0 , (4.2)
[Nj, Pk] = −ıδjk P0 , [Nj , P0] = −ıPj . (4.3)
The physical generators of the Lorentz algebra, Mi, Ni (i = 1, 2, 3), are related with the
canonical basis h, h′, e±, e
± as follows
h = ıN3 , e± = ı(N1 ± M2), (4.4)
h′ = −ıM3 , e
± = ı(±N2 −M1). (4.5)
The subalgebra generated by the four-momenta Pj, P0 (j = 1, 2, 3) will be denoted by P
and we also set P± := P0 ± P3.
S. Zakrzewski has shown in [2] that each classical r-matrix, r ∈ P(3, 1) ∧ P(3, 1), has
a decomposition
r = a+ b+ c , (4.6)
where a ∈ P ∧P, b ∈ o(3, 1) ∧P, c ∈ o(3, 1) ∧ o(3, 1) satisfy the following relations
[[c, c]] = 0 , (4.7)
[[b, c]] = 0 , (4.8)
2[[a, c]] + [[b, b]] = tΩ (t ∈ R) , (4.9)
[[a, b]] = 0 . (4.10)
Here [[·, ·]] means the Schouten bracket. Moreover a total list of the classical r-matrices
for the case c 6= 0 and also for the case c = 0, t = 0 was found.7 It was shown that
there are fifteen solutions for the case c = 0, t = 0, and six solutions for the case c 6= 0
where there is only one solution for t 6= 0. Thus Zakrzewski found twenty r-matrices which
satisfy the homogeneous classical Yang-Baxter equation (t = 0 in (4.9)). Analysis of these
twenty solutions shows that each of them can be presented as a sum of subordinated r-
matrices which almost all are of Abelian and Jordanian types. Therefore these r-matrices
correspond to twisted deformations of the Poincaré algebra P(3, 1). We present here
r-matrices only for the case c 6= 0, t = 0:
r1 = γh
′ ∧ h+ α(P+ ∧ P− − P1 ∧ P2) , (4.11)
r2 = γe
+ ∧ e+ + β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + β2h
′ ∧ P+ , (4.12)
r3 = γe
+ ∧ e+ + β(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + αP1 ∧ P+ , (4.13)
r4 = γ(e
+ ∧ e+ + e+ ∧ P1+ e
+ ∧ P2− P1 ∧ P2) + P+ ∧ (α1P1+ α2P2) , (4.14)
r5 = γ1(h ∧ e+ − h
′ ∧ e′+) + γ2e+ ∧ e
+ . (4.15)
The first r-matrix r1 is a sum of two subordinated Abelian r-matrices
r1 := r
1 + r
1 , r
1 ≻ r
r′1 = α(P+ ∧ P− − P1 ∧ P2) , r
1 := γh
′ ∧ h .
(4.16)
Therefore the total twist defining quantization in the direction to this r-matrix is the
ordered product of two the Abelian twits
= Fr′′
= exp
γh′ ∧ h
α(P+ ∧ P− − P1 ∧ P2)
. (4.17)
7Classification of the r-matrices for the case c = 0, t 6= 0 is an open problem up to now.
The second r-matrix r2 is a sum of three subordinated r-matrices where two of them
are of Abelian type and one is of Jordanian type
r2 := r
3 + r
2 + r
2 , r
2 ≻ r
2 ≻ r
r′2 := β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) ,
r′′2 := γe
+ ∧ e+ , r
2 := β2h
′ ∧ P+ .
(4.18)
Corresponding twist is given by the following formulas
= Fr′′′
, (4.19)
where
= exp
β1(e+ ⊗ P1 − e
+ ⊗ P2)
exp(2h⊗ σ+) ,
= exp(γe′+ ∧ e+) , Fr′′′2 = exp(β2h
′ ∧ σ+) .
(4.20)
Here and below we set σ+ :=
ln(1 + β1P+).
The third r-matrix r3 is a sum of two subordinated r-matrices where one is of Abelian
type and another is a more complicated r-matrix which we call mixed Jordanian-Abelian
r3 := r
3 + r
3 , r
3 ≻ r
r′3 := β1(e+ ∧ P1 − e
+ ∧ P2 + h ∧ P+) + αP1 ∧ P+ ,
r′′3 := γe
+ ∧ e+ .
(4.21)
Corresponding twist is given by the following formulas
= Fr′′
, (4.22)
where
= exp
β1(e+ ⊗ P1 − e
+ ⊗ P2)
exp(αP1 ∧ σ+) exp(2h⊗ σ+) ,
= exp(γe′+ ∧ e+) .
(4.23)
The fourth r-matrix r4 is a sum of two subordinated r-matrices of Abelian type
r4 := r
4 + r
4 , r
4 ≻ r
r′4 := P+ ∧ (α1P1 + α+P2) ,
r′′4 := γ(e
+ − P1) ∧ (e+ + P2) .
(4.24)
Corresponding twist is given by the following formulas
= Fr′′
, (4.25)
where
= exp
(P+ ⊗ (α1P1 + α2P2)
= exp
γ(e′+ − P1) ∧ (e+ + P2)
(4.26)
The fifth r-matrix r5 is the r-matrix of the Lorentz algebra, (3.6), and the correspond-
ing twist is given by the formula (3.13).
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[15] A. Belavin and V. Drinfeld, Functional Anal. Appl., 16(3), 159 (1983); translated
from Funktsional. Anal. i Prilozhen, 16, 1 (1982) (Russian).
[16] A.P. Isaev and O.V. Ogievetsky, Phys. Atomic Nuclei, 64(12), 2126 (2001);
math.QA/0010190.
[17] S.M. Khoroshkin and V. Tolstoy, Comm. Math. Phys., 141(3), 599 (1991).
http://arxiv.org/abs/q-al/9602001
http://xxx.lanl.gov/abs/hep-th/0301033
arXiv:hep-th/0604146
http://xxx.lanl.gov/abs/math.QA/0402433
http://xxx.lanl.gov/abs/math.QA/0701079
Introduction
Preliminaries
Quantum deformations of Lorentz algebra
Quantum deformations of Poincare algebra
|
0704.0082 | Matter-Wave Bright Solitons with a Finite Background in Spinor
Bose-Einstein Condensates | arXiv:0704.0082v1 [cond-mat.other] 1 Apr 2007
Typeset with jpsj2.cls <ver.1.2> Full Paper
Matter-Wave Bright Solitons with a Finite Background in Spinor
Bose-Einstein Condensates
Tetsuo Kurosaki∗ and Miki Wadati
Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033
We investigate dynamical properties of bright solitons with a finite background in the
F = 1 spinor Bose-Einstein condensate (BEC), based on an integrable spinor model which
is equivalent to the matrix nonlinear Schrödinger equation with a self-focusing nonlineality.
We apply the inverse scattering method formulated for nonvanishing boundary conditions.
The resulting soliton solutions can be regarded as a generalization of those under vanishing
boundary conditions. One-soliton solutions are derived in an explicit manner. According to
the behaviors at the infinity, they are classified into two kinds, domain-wall (DW) type and
phase-shift (PS) type. The DW-type implies the ferromagnetic state with nonzero total spin
and the PS-type implies the polar state, where the total spin amounts to zero. We also
discuss two-soliton collisions. In particular, the spin-mixing phenomenon is confirmed in a
collision involving the DW-type. The results are consistent with those of the previous studies
for bright solitons under vanishing boundary conditions and dark solitons. As a result, we
establish the robustness and the usefulness of the multiple matter-wave solitons in the spinor
BECs.
KEYWORDS: spinor Bose-Einstein condensate, internal degrees of freedom, matrix nonlinear
Schrödinger equation, integrable model, bright soliton, nonvanishing boundary
condition, multi-component Gross-Pitaevskii equation, spin exchange coupling,
spin switching
∗E-mail address: [email protected]
http://arxiv.org/abs/0704.0082v1
J. Phys. Soc. Jpn. Full Paper
1. Introduction
In 2002, matter-wave bright solitons in quasi-one-dimensional (1D) Bose-Einstein conden-
sates (BECs) were observed experimentally.1, 2) Bright solitons propagate in most cases with
much larger amplitudes than dark solitons,3, 4) and are expected to have the potential for
various applications such as coherent transport and atom interferometry. Soliton propagation
in BEC can be described by the Gross-Pitaevskii (GP) equation. The GP equation, called the
nonlinear Schrödinger (NLS) equation in nonlinear science, is integrable and has soliton solu-
tions in a one-dimensional and uniform system. Recent experimental and theoretical advances
about matter-wave bright solitons are reviewed, for instance, in ref. 5.
The experimental creation of matter-wave solitons has been so far achieved only for single-
component BEC. It is, nevertheless, very interesting to consider soliton propagation in BEC
with internal degrees of freedom, so-called, spinor BEC. When BEC of ultracold alkali atoms
is trapped exclusively by optical means, the hyperfine spin of atoms remains liberated. The
spinor BEC was realized in such a way.6–8) Internal degrees of freedom endow solitons with
a multiplicity. The multiple solitons will show a rich variety of dynamics. Here, we focus on
the boson system in the F = 1 hyperfine spin state, exemplified by 23Na, 39K and 87Rb. The
multi-component GP equation for F = 1 spinor BEC turns to an integrable model at special
points, which is mathematically equivalent to the matrix NLS equation. An integrable model
with a self-focusing nonlineality enables one to perform exact analysis via the inverse scatter-
ing method (ISM) for the matrix NLS equation.9) In particular, bright soliton solutions under
vanishing boundary conditions (VBC) are obtained, whose properties are investigated in refs.
10 and 11. Recently, the ISM for the matrix NLS equation under nonvanishing boundary con-
ditions (NVBC) is formulated.12) Dark solitons in the F = 1 spinor BEC can be investigated
by applying the ISM under NVBC to an integrable model with a self-defocusing nonlineal-
ity.13) Although the ISM under NVBC is dedicated mainly to the self-defocusing case, we note
that this technique is also applicable to an integrable model with a self-focusing nonlineality,
which makes us available to bright soliton solutions with a finite background.
In this paper, the detail of matter-wave bright solitons in the quasi-1D F = 1 spinor BEC
is further investigated, based on an integrable model. We consider matter-wave spinor bright
solitons traveling on a finite background of the condensate. We write down explicitly new
soliton solutions, and verify that the obtained soliton solutions have the similar properties
compared to those without a background. In the usual experimental setups, the condensates
are confined in a finite-size regime, and the matter-wave bright solitons will accompany a
finite background. The study given in this paper is meaningful in such realistic circumstances.
The paper is organized as follows. In § 2, the GP equation for quasi-1D F = 1 spinor BEC
is introduced. In particular, the integrable model is presented. There, the interactions between
two atoms are supposed to be inter-atomic attractive and ferromagnetic, which lead to bright
J. Phys. Soc. Jpn. Full Paper
solitons. In § 3, the inverse scattering method under nonvanishing boundary conditions is
applied to the integrable model. This application leads to bright soliton solutions with a finite
background. Several conserved quantities of the model are also provided. One-soliton solutions
are investigated in § 4. The spin states of one-solitons are classified, assuming that discrete
eigenvalues are purely imaginary. Two-soliton solutions are discussed in § 5. The last section,
§ 6, is devoted to the concluding remarks.
2. GP Equation for F = 1 Spinor Bose-Einstein Condensates
For BEC of ultracold alkali atoms, the mean-field theory works well, because almost all
atoms go into condensation and the condensate is dilute. In this paper, we deal with the
quasi-one-dimensional system. Atoms in the F = 1 hyperfine spin state have three magnetic
substates labeled by the magnetic quantum number mF = 1, 0,−1. The system is charac-
terized by a vectorial field operator with the components corresponding to each substate,
Φ̂ = (Φ̂1, Φ̂0, Φ̂−1)
T , satisfying equal-time commutation relations:
[Ψ̂α(x, t), Ψ̂
(x′, t)] = δαβδ(x− x′), (1)
where the subscripts α, β take on 1, 0,−1. In the framework of the mean-field theory for BEC,
the quantum field is replaced with the order parameter:
Φ(x, t) ≡ 〈Φ̂(x, t)〉 = (Φ1(x, t),Φ0(x, t),Φ−1(x, t))T . (2)
Φ(x, t) is often called the spinor condensate wavefunction, which is normalized to the total
number of atoms NT:
dxΦ†(x, t)Φ(x, t) = NT. (3)
The spinor condensate wavefunction obeys a set of coupled evolution equations, namely, the
multi-component GP equation:
i~∂tΦ1 = −
∂2xΦ1 + (c̄0 + c̄2)
|Φ1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ−1|2Φ1 + c2Φ∗−1Φ20,
i~∂tΦ0 = −
∂2xΦ0 + (c̄0 + c̄2)
|Φ1|2 + |Φ−1|2
+c̄0|Φ0|2Φ0 + 2c̄2Φ∗0Φ1Φ−1,
i~∂tΦ−1 = −
∂2xΦ−1 + (c̄0 + c̄2)
|Φ−1|2 + |Φ0|2
+(c̄0 − c̄2)|Φ1|2Φ−1 + c̄2Φ∗1Φ20, (4)
where c̄0 = (ḡ0 + 2ḡ2)/3 and c̄2 = (ḡ2 − ḡ0)/3 denote effective 1D coupling constants for the
mean-field and the spin-exchange interaction, respectively. Here, the effective 1D coupling
constants ḡf are given by
ḡf =
4~2af
1− Caf/a⊥
, (5)
J. Phys. Soc. Jpn. Full Paper
where af are the s-wave scattering lengths in the total hyperfine spin f channel, a⊥ is the
size of the transverse ground state, m is the atomic mass, and C = −ζ(1/2) ≃ 1.46. Note that
one may change the values of c̄0 and c̄2 by tuning a⊥.
Equation (4) is derived as follows. The interaction between two atoms in the F = 1
hyperfine spin state has a form,15, 16)
V̂ (x1 − x2) = δ(x1 − x2)
c̄0 + c̄2F̂ 1 · F̂ 2
, (6)
where F̂ i is the spin operator. The Gross-Pitaevskii energy functional is thus given by
EGP[Φ] =
α∂xΦα +
α′Φα′Φα +
αβ · fα′β′Φβ′Φβ
, (7)
where repeated subscripts (α, β, α′, β′ = 1, 0,−1) should be summed up and f = (fx, fy, fz)T
with fi(i = x, y, z) being 3× 3 spin-1 matrices. Then, the variational principle: i~∂tΦα(x, t) =
δEGP[Φ]/δΦ
α(x, t), for α = 1, 0,−1, yields eq. (4).
An important fact is that eq. (4) possesses a completely integrable point when c̄0 = c̄2 ≡
−c < 0, equivalently, 2ḡ0 = −ḡ2 > 0.10, 11) This condition is realized when
a⊥ = 3C
2a0 + a2
, (8)
assuming that a0a2(a2−a0) > 0 holds. The situation corresponds to attractive (c̄0 < 0) and fer-
romagnetic (c̄2 < 0) interaction. When we change the wavefunction by Φ = (φ1,
2φ0, φ−1)
and measure time and length in units of t̄ = ~a⊥/c and x̄ = ~
a⊥/2mc, respectively, we can
rewrite eq. (4) with c̄0 = c̄2 ≡ −c < 0 into the dimensionless form,
i∂tφ1 = −∂2xφ1 − 2(|φ1|2 + 2|φ0|2)φ1 − 2φ∗−1φ20,
i∂tφ0 = −∂2xφ0 − 2(|φ−1|2 + |φ0|2 + |φ1|2)φ0 − 2φ∗0φ1φ−1,
i∂tφ−1 = −∂2xφ−1 − 2(|φ−1|2 + 2|φ0|2)φ−1 − 2φ∗1φ20. (9)
Then, eq. (9) is found to be equivalent to a 2× 2 matrix version of the NLS equation with a
self-focusing nonlinearity:
i∂tQ+ ∂
xQ+ 2QQ
†Q = O, (10)
with an identification,
φ1 φ0
φ0 φ−1
. (11)
The matrix NLS equation (10) is integrable in the sense that the initial value problem can
be solved via the inverse scattering method.9, 12) The integrability of the reduced equations
(9) is thus proved automatically. Thus, we have derived the integrable spinor model. Another
integrable point of eq. (4) is c̄0 = c̄2 ≡ c > 0, i.e., the matrix NLS equation with a self-
defocusing nonlineality.13) Special solutions for generic coupling constants c̄0, c̄2 are given in
ref. 17.
J. Phys. Soc. Jpn. Full Paper
3. Bright Solitons with a Finite Background
We consider bright soliton solutions of the integrable model (9) under NVBC, whereas
those under VBC are studied in refs. 10 and 11. We summarize briefly the results of the inverse
scattering method for eq. (9) with NVBC.12)
We define the nonvanishing boundary conditions as
Q(x, t) → Q±, x→ ±∞,
±Q± = Q±Q
± = λ
0I, (12)
where λ0 is a positive real constant and I denotes a 2 × 2 unit matrix. Note that vanishing
boundary conditions are recovered as λ0 → 0. The analysis of the ISM under NVBC yields
the standard form of the multiple soliton solutions of the 2 × 2 matrix NLS equation with a
self-focusing nonlineality (10) as
Q(x, t) = λ0 e
I + 2i(I · · · I
︸ ︷︷ ︸
. (13)
Here, a 2 × 2 complex matrix Πj is called the polarization matrix. S is a 2N × 2N matrix
defined by
Sij =
ζj + λj
δijI +
i(ζi + ζj)
ζi + λi
ζj + λj
1 ≤ i, j ≤ N, (14)
where λj is a complex discrete eigenvalue for the bound state and ζj = (λ
j + λ
1/2 with
Im ζj > 0 for j = 1, . . . , N . It is required for the ISM under NVBC that a two-sheet Riemann
surface is introduced appropriately, due to a double-valued function ζj. The phase of the
carrier wave, φ(x, t), is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ, (15)
and the coordinate function is given by
χj ≡ χj(x, t) = 2iζj(x− 2(λj + k)t). (16)
The above solution is the M(= N/2)-soliton solution. The ISM under NVBC for the self-
focusing case results in pairs of discrete eigenvalues corresponding to each Riemann sheet.
The constraint should be imposed on λj and ζj (j = 1, · · · , N) such that λ2l−1 = λ∗2l and
ζ2l−1 = −ζ∗2l for l = 1, · · · , N/2. At the same time, Πj must satisfy that Π2l−1 = Π
For our reduction to the integrable model for F = 1 spinor BEC, we must make the
potential Q symmetric, noting eq. (11). The symmetry of Q is naturally reflected in Πj.
When we take every Πj to be symmetric in eq. (13), soliton solutions of the integrable model
(9) under NVBC are obtained.
J. Phys. Soc. Jpn. Full Paper
The form (13) is the standard form of soliton solutions in the sense that the boundary
value at x→ ∞ ⇔ eχj → 0 is supposed to be fixed as
Q(x, t) e−iφ(x,t) → λ0I, x→ ∞. (17)
The spinor model, however, allows the SU(2) transformation of the solutions, if they are kept
symmetric. To be concrete, let U be a 2× 2 unitary matrix. When Q is a solution of eq. (10)
with eq. (11), then
Q′ = UQUT , (18)
is also a solution. Assuming that Q is the standard form (13), the limit x→ ∞ of Q′ becomes
Q′ e−iφ(x,t) → λ0 UUT ≡ λ0Q′+, x→ ∞. (19)
Q′+ = UUT is the so-called Cholesky decomposition. The arbitrary boundary conditions Q′+
other than eq. (17) are thus realized via the SU(2) transformation. On the other hand, the
behavior in the limit x → −∞ varies depending on whether detΠ = 0 or not, which will be
discussed later for the one-soliton case.
There is another important concept about an integrable model. Due to the integrability,
the model has the infinite conservation laws, which restrict the dynamics of the system in an
essential way. Several conserved quantities, related to the physical quantities of the system,
are listed below:
Total number : N̄T =
dx n̄(x, t),
n̄(x, t) = tr (Q†Q)− tr (Q†±Q±). (20)
Total spin : FT =
dx f(x, t),
f(x, t) = tr (Q†σQ). (21)
Total momentum : P̄T =
dx p̄(x, t),
p̄(x, t) = −i~[tr (Q†Qx)− tr (Q†±Q±,x)]. (22)
Total energy : ĒT =
dx ē(x, t),
ē(x, t) = c[tr (Q†xQx −Q†QQ†Q)
−tr (Q†±,xQ±,x −Q
±Q±)]. (23)
Here, σ = (σx, σy, σz)
T are the Pauli matrices. To avoid the divergence of integrals, we should
subtract the contribution of the background from the physical quantities, except the total spin,
to which the background does not contribute explicitly. These subtractions are emphasized
by the bars on the conserved densities and quantities. The local spin density f = (fx, fy, fz)
is covariant under the SU(2) transformation (18), whereas the other densities such as n̄(x, t),
J. Phys. Soc. Jpn. Full Paper
p̄(x, t) and ē(x, t) are invariant. The total macroscopic spin is directed to face to an arbitrary
direction by the global spin rotation. The SU(2) symmetry of the system causes the energy
degenerated states of solitons for this spin rotation.
4. One-Soliton Solutions
In this section, one-soliton solutions of the integrable spinor model (9) are investigated in
detail. We can derive the explicit form of one-soliton solutions by setting N = 2 (M = 1) in
the formula (13). The calculation is complicated but straightforward. The result is as follows:
Q = λ0 e
iφ(x,t)
I + 2i
, (24)
where detS is given by
detS = κ21κ
2 − eχ1ν1κ1κ22 trΠ− ex2ν2κ21κ2 trΠ†
+eχ1+χ2κ1κ2
̟ + ν1ν2(|tr Π|2 − 1)
+e2χ1ν21κ
2 detΠ + e
2χ2ν22κ
1 detΠ
− e2χ1+χ2ν1κ2̟ trΠ† detΠ− eχ1+2χ2ν2κ1̟ trΠdetΠ†
+e2χ1+2χ2̟2|detΠ|2, (25)
and T is a 2× 2 matrix such that
T = eχ1κ1κ
2 Π+ e
χ2κ21κ2 Π
− eχ1+χ2κ1κ2
ς1 trΠ · Π† + ς2 trΠ† ·Π+ µ(|trΠ|2 − 1)I
− e2χ1µ1κ22 detΠ · I − e2χ2µ2κ21 detΠ† · I
+e2χ1+χ2κ2 detΠ
ς21 Π
† +̟ trΠ† · I
+eχ1+2χ2κ1 detΠ
ς22 Π+̟ trΠ · I
− e2χ1+2χ2̟ (ς1 + ς2) |detΠ|2 · I. (26)
We explain physical meanings of notations. The phase of the carrier wave is given by
φ(x, t) = kx− (k2 − 2λ20)t+ δ. (27)
Let λj and ζj = (λ
j + λ
1/2 for j = 1, 2 be complex constants satisfying λ1 = λ
2 and
ζ1 = −ζ∗2 . Without loss of generality, we assume that (λ1, ζ1) ((λ2, ζ2)) belongs to the upper
(lower) Riemann sheet, which is characterized such that Im ζ Im λ > 0 (Im ζ Im λ < 0).
χj(x, t) is expressed in terms of them as
χ1 ≡ χ1(x, t) = 2iζ1(x− 2(λ1 + k)t), (28)
χ2 ≡ χ2(x, t) = 2iζ2(x− 2(λ2 + k)t). (29)
Note that χ1 = χ
2 ≡ χ holds. Reχ thus denotes the coordinate of the envelope soliton,
whereas Imχ implies the self-modulation phase. The polarization matrix Π is a 2×2 symmetric
J. Phys. Soc. Jpn. Full Paper
matrix. Here, the normalization in a sense of the square norm is imposed on Π as a matter of
convenience:
, 2|α|2 + |β|2 + |γ|2 = 1. (30)
The other parameters are expedient functions of λ0, λ1 and λ2:
ζj + λj
, µ = iλ0
κ1 + κ2
ζ1 + ζ2
, νj =
iλ0κj
, ̟ = ν1ν2 − µ2, ςj = νj − µ, (31)
for j = 1, 2. We list the meaning of each parameter as follows:
k : wave number of soliton’s carrier wave.
λ0 : amplitude of soliton’s carrier wave.
φ(x, t) : phase of soliton’s carrier wave.
Re χ(x, t) : coordinate of soliton’s envelope.
Im χ(x, t) : self-modulation phase of soliton.
Π : symmetric polarization matrix of soliton.
Equations (24)-(26) are new soliton solutions which had never been written down explicitly
in the literatures. If we take the vanishing limit λ0 → 0, ζ1 and ζ2 converge at λ1 and −λ2,
respectively. Then,
· 2kR
Πe−(χR+ρ/2) + (σyΠ†σy) eχR+ρ/2detΠ
e−(2χR+ρ) + 1 + e2χR+ρ|detΠ|2
eiχI , (32)
with notations
eρ/2 ≡ 1
, (33)
χR ≡ χR(x, t) = kR(x− 2kI t)− ǫ, (34)
χI ≡ χI(x, t) = kIx+ (k2R − k2I )t, (35)
each of which holds the following correspondence respectively:
kR = −2Imλ1, (36)
kI = 2Reλ1 + k, (37)
ǫ = − ln(4|λ1|). (38)
Equations (32)-(35) are the same forms as those in ref. 11, except a phase factor. This con-
sequence is natural but non-trivial, because the formula of solitons under VBC9) is quite
different from that under NVBC (13), in particular, in the form of the matrix S. Actually, the
initial displacement ǫ can be arbitrarily changed, regardless of eq. (38), by the parallel shift
of the position x. We have shown that the soliton solutions (24)-(26) can be regarded as a
general form of bright soliton solutions, including the case of VBC.
J. Phys. Soc. Jpn. Full Paper
4.1 Classification by the boundary conditions
We shall show that there are two kinds of one-soliton solutions depending on the boundary
conditions, detΠ = 0 or detΠ 6= 0. The similar classification about the boundary conditions
also exists for dark solitons.13) The examples of snapshots of one-soliton density profiles are
shown in Fig. 1. The upper row is for detΠ = 0, and the lower row is for detΠ 6= 0. The
shape of envelope solitons looks a locally-oscilating wave rather than, literally, a solitary wave,
because of the self-modulation due to the complex velocity.
For detΠ = 0, the boundary conditions of the standard form (24)-(26) are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
I − 2i ς1 trΠ · Π
† + ς2 trΠ
† · Π+ µ(|trΠ|2 − 1)I
̟ + ν1ν2(|tr Π|2 − 1)
, x→ −∞. (39)
The left and right boundary values differ in not only the global phase but also the population
of each component, in general. That is, those are the SU(2) rotated boundary conditions.
In the upper row of Fig. 1, we see that the envelope soliton of each component forms the
domain-wall (DW) shape, although it does not manifest in the total number density.
On the other hand, for detΠ 6= 0, the boundary conditions are
Q e−iφ → λ0I, x→ ∞,
Q e−iφ → λ0
1− 2i
ς1 + ς2
I, x→ −∞. (40)
In contrast to the case that detΠ = 0, both boundary values are diagonal matrices, and only
the phase-shift (PS) occurs. That is, those are the U(1) rotated boundary conditions.
For the above reasons, we call one-soliton solutions the DW-type for detΠ = 0, and the
PS-type for detΠ 6= 0. Remark the following; the spin density profile of DW-type suggests
that the total spin is nonzero, whereas that of PS-type is dipole-shape, implying that the total
spin amounts to zero. See the right panel of Fig. 1. This observation will be solidified in § 4.3.
4.2 Case of purely imaginary discrete eigenvalues
The ISM performed on Riemann sheets involves a double-valued function of the spectral
parameter, and it usually renders a very complicated representation of N -soliton solutions
even for N = 1, as is seen from eqs. (24)-(26). To simplify an explicit representation, it is
convenient to assume that λj and ζj are purely imaginary.
18) The similar approach is employed
for the analysis of N -soliton solutions of the derivative NLS equation under NVBC.20)
If we take a pair of discrete eigenvalues as
(λ1, ζ1) ≡ (iλ0λ, iλ0ζ), (λ2, ζ2) ≡ (−iλ0λ, iλ0ζ), (41)
where λ and ζ are positive real numbers such that
λ > 1, ζ =
λ2 − 1, (42)
J. Phys. Soc. Jpn. Full Paper
(a) (b) (c)
Fig. 1. Snapshots of one-soliton density profiles. The upper row is plotted for detΠ = 0 at the moment
t = 0, with k = 0, λ0 = 1, λ1 = 1+i, ξ1 = 1.27+0.79i (χ(x, t) = −(1.57−2.54i)x− (8.23−1.94i)t)
and Π =
4/5 2/5
2/5 1/5
. The lower row is plotted for detΠ 6= 0 at the moment t = 0, with the
same parameters except for Π =
2 2/5
2/5 3/(5
. The left panel (a) depicts the local density
for each component, |φ1|2 (solid line), |φ0|2 (chain line) and |φ−1|2 (dotted line). The center panel
(b) depicts the local number density n, where the contribution of the background is included. The
right panel (c) depicts the local spin densities, fx (solid line) and fz (dotted line). fy vanishes
identically due to a choice of a real matrix Π.
we obtain a relatively simple form of one-soliton solutions as
Q = λ0 e
iφ(x,t)
I + 2i
, (43)
detS = 1−
(tr Π(t) + trΠ†(t)) + e2χP
|trΠ(t)|2
(detΠ(t) + detΠ†(t))−
trΠ†(t)detΠ(t) + trΠ(t)detΠ†(t)
e4χP |detΠ(t)|2, (44)
2i T =
2(|tr Π(t)|2 − 1) e2χP + 2e2χP
detΠ(t) +
detІ(t)
tr Π†(t)detΠ(t) +
trΠ(t)detΠ†(t)
10/18
J. Phys. Soc. Jpn. Full Paper
−2(ζ + λ) eχP + 2
e2χP trΠ†(t) + 2
e3χP detΠ†(t)
−2(ζ − λ) eχP − 2
e2χP trΠ(t) + 2
e3χP detΠ(t)
Π†(t), (45)
where the coordinate of the envelope soliton is given by
χP (x, t) = −2λ0ζ(x− 2kt), (46)
and the time dependence of the phase modulation is embedded in the polarization matrix,
namely,
Π(t) ≡ Πe4iλ20ζλt. (47)
When we take the limit λ0 → 0 with λ0λ and λ0ζ kept finite in eqs. (43)-(45), Q converges
to the form of eqs. (32)-(35), accompanying the parameters kR = −2λ0λ, kI = k and ǫ =
− ln(4λ0λ). Here, kR and kI are independent free parameters, apart from the trivial initial
displacement ǫ. In this sense, in spite of the reduction, we can still regard the form of one-
soliton solutions (43)-(45) as a general form of those under VBC.
We can also take another limit. That is, we consider the reduction to the single-component
case. If we set
k = 0, Π =
eiθ 0
, (48)
the (1,1)-component of Q becomes
Q11 · e−i(2λ
t+δ) = λ0 − 2λ0ζ
ζ cos(4λ20ζλt+ θ) + iλ sin(4λ
0ζλt+ θ)
λ cosh(2λ0ζx+ ψ)− cos(4λ20ζλt+ θ)
, (49)
where eψ = ζ/λ. The form (49) was given in ref. 18. We thus verify that our soliton solutions
are also generalization of those for the single-component NLS equation under NVBC.19)
4.3 Spin states
In this subsection, we discuss the spin states of one-soliton solutions with a finite back-
ground, by calculating the total spin. The conservation law guarantees that we obtain the
total spin FT from integrating at arbitrary time. Therefore, we can select the time so that the
calculation becomes easier. We concentrate on the case of purely imaginary discrete eigenval-
ues. As a result, we see that the DW-type is associated with the ferromagnetic state, whereas
the PS-type is associated with the polar state. One-soliton solutions for purely imaginary dis-
crete eigenvalues (43)-(45) include those under VBC apart from an initial displacement, and
therefore the classification about the spin states presented below is wider than that performed
before.10, 11)
4.3.1 Ferromagnetic state
For detΠ = 0 (DW-type), we substitute eqs. (43)-(45) into eq. (21) and calculate the total
spin. The time t′ such that trΠ(t′) + trΠ†(t′) = 0 is suitable for the calculation. The result is
11/18
J. Phys. Soc. Jpn. Full Paper
as follows:
FT = 4λ0τ
2λRe{α∗(β + γ)}
−2ζIm{α∗(β − γ)}
λ(|β|2 − |γ|2)
, τ ≡
|β + γ|2
, (50)
with the modulus
|F T|2 = (4λλ0)2τ. (51)
The total number of the particles transformed into a soliton, N̄T, is calculated by eq. (20),
N̄T = 4λ0ζ, (52)
and the range of the value taken by |F T|2 is expressed in terms of N̄T,
N̄2T ≤ |F T|2 ≤ N̄2T + (4λ0)2. (53)
Remark that, in the vanishing limit λ0 → 0, the modulus of the total spin is always equal to
the total number of the particles, namely, |F T| → NT.
With nonzero total spin, the DW-type of solitons belongs to the ferromagnetic state.
Since inter-atomic ferromagnetic interactions are supposed here, solitons tend to take the
ferromagnetic state or DW-type. In various contexts of physics, the domain-walls are topo-
logical solitons related with the symmetry breaking. Here, resulting from the domain-walls,
the magnetic entity emerges as the spontaneously broken symmetry. It is worthy to notice
that the case |N̄T| < |FT| may happen. The background is spinless, but its internal spin state
appears to be affected on the ground that the ferromagnetic soliton runs over the background.
Thereby, the background contributes to the total spin.
4.3.2 Polar state
If detΠ 6= 0 (PS-type), the solitons show the other magnetic property. The time t′ such
that detΠ(t′) = detΠ†(t′) > 0 is suitable for the analysis. After lengthy calculations, the local
spin density at such time is derived as
= 8λ20 e
−3υΞ−2(χP ′)
β − γ
ζ e2υΞ(χP ′) sinh(χP ′)
(ζ2 − λ2)tr Π(t′) + (ζ2 + λ2)trΠ†(t′)
sinh(2χP ′)
λ2/ζ · ((tr Π†(t′))2 − |tr Π(t′)|2) + 4ζdetΠ(t′) + 2ζ(|tr Π(t′)|2 − 1)
sinh(χP ′)
+h.c., (54)
fy = 32λ
−3υIm{α∗(β − γ)}Ξ−2(χP ′)
2ζ eυ sinh(2χP ′)− (trΠ(t′) + trΠ†(t′)) sinh(χP ′)
, (55)
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J. Phys. Soc. Jpn. Full Paper
where Ξ(χP ′) is an even function of χP ′ , χP ′(x, t) is a parallel-shifted coordinate, and υ is a
constant:
Ξ(χP ′) ≡ 2 cosh(2χP ′)−
2 e−υ
(trΠ(t′) + tr Π†(t′)) cosh(χP ′)
ζ2 + |trΠ(t′)|2
λ2detΠ(t′)
, (56)
χP ′ ≡ χP + υ = −2λ0ζ(x− 2kt′) + υ, (57)
υ ≡ ln
(detΠ(t′))1/2
. (58)
Note that fx and fz share the same functional form. Each component of the above local spin
density is an odd function of χP ′ and, in particular, it has the node at the same point x0 such
that χP ′(x0, t
′) = 0, namely, x0 = 2kt
′ + (2λ0ζ)
−1υ. Consequently, the total spin amounts to
zero:
dxf(χP ′) = (0, 0, 0)
T . (59)
For this reason, the PS-type of solitons, on the average, belongs to the polar state.15)
5. Two-Soliton Collision
We proceed to the discussion of two-soliton collisions in the integrable spinor model (9).
Two-soliton solutions are obtained by setting N = 4 (M = 2) in the formula (13). There
exist two independent discrete eigenvalues and symmetric polarization matrices, respectively,
i.e., λ1 = λ
2 and Π1 = Π
2 for one of solitons, λ3 = λ
4 and Π3 = Π
4 for the other. Each
soliton is separated at t = ±∞. Then, a two-soliton is asymptotically two one-solitons. The
classification of one-soliton solutions based on the values of the determinants of polarization
matrices, discussed in the previous section, is thus valid for two-soliton solutions.
The derivation of the explicit form is more complicated than that in the case for one-soliton
solutions. For the derivative NLS equation under NVBC, explicit two-soliton solutions and
shifts of soliton positions due to collisions between solitons have been analytically obtained,
in the case of purely imaginary eigenvalues, where complexity of calculation is considerably
reduced.20) This strategy, however, does not stand in our NLS equation under NVBC. The
reason is understood from eq. (46). In the spinor model, purely imaginary eigenvalues give
two solitons with the same velocity 2k, and they do not collide with each other. Accordingly,
we can not investigate the properties of collisions for purely imaginary discrete eigenvalues.
No one has studied explicit multi-soliton solutions of the NLS equation under NVBC, even in
the single-component case, because of the computational complexity.
Here, we graphically show the characteristic behaviors of two-soliton collisions in the
spinor model, by use of the exact solutions given by the ISM. Referring to them, we carry
out the qualitative discussions. Although the presented graphs are depicted for the specific
parameters, much the same behaviors are observed for arbitrarily selected parameters.
13/18
J. Phys. Soc. Jpn. Full Paper
Figure 2 illustrates the behavior of a mutual collision between two PS-types, where
detΠ1 6= 0 and detΠ3 6= 0. One can see that, in all three components, Both solitons re-
tain their shapes before and after the collision, which is the common property with solitons
in the single-component case. In this sense, PS-PS soliton collision is essentially equivalent to
two-soliton collision of the single-component NLS equation.
Figure 3 illustrates the behavior of a mutual collision between DW-type and PS-type,
where detΠ1 = 0 and detΠ3 6= 0. The behavior of collisions between DW-type and PS-type is
qualitatively alien from that between two PS-types. One observes that, in PS-type, much of
the population initially inhabiting the hyperfine substate |F = 1,mF = ±1〉 is transferred into
the hyperfine substate |F = 1,mF = 0〉 due to the collision. In contrast, in DW-type, such
spin transfer does not occur, and the domain-wall shape is preserved against the collision. This
phenomenon can be interpreted as follows. DW-type, with nonzero spin, can affect the internal
spin state of PS-type, whereas PS-type, which is expected to have zero spin in total, does not
affect the internal spin state of DW-type. This kind of spin-transfer phenomenon, called the
spin-switching, has been first predicted for the case of VBC.11) Due to the conservation laws,
the total number, the total spin and so on are invariant throughout the collision process.
Population-mixing among internal degrees of freedom is permitted, as far as the conservation
laws are not violated. The spin-switching is one of the dynamical processes which make the
spinor solitons more interesting.
Finally, for a mutual collision between two DW-types, where detΠ1 = detΠ3 = 0, the
shapes of both solitons are expected to be deformed by the collision since each soliton carries
nonzero total spin. In fact, drastic population-mixing is seen in Fig. 4, which shows an example
of this kind of collisions. One finds that domain-walls ”repel” at the collision region, rather
than collide.
(a) (b) (c)
Fig. 2. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two PS-types.
The parameters used here are k = 1, λ0 = 1, λ1 = 1.03i, λ3 = 1.05 + i, Π1 =
2 i/2
i/2 0
0 i/2
i/2 1/
. The velocity of the right (left) moving soliton is 2.00 (−3.41). The collision
takes place at t = 0.
14/18
J. Phys. Soc. Jpn. Full Paper
(a) (b) (c)
Fig. 3. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between DW-
type and PS-type. The parameters used here are the same as those of Fig. 2, except for
2i/3 −1/3
, Π3 =
0 −1/
. The right (left) moving soliton is DW-type
(PS-type).
(a) (b) (c)
|φ1|2 |φ0|2 |φ−1|2
t t t
x x x
Fig. 4. Density plots of |φ1|2 (a), |φ0|2 (b) and |φ−1|2 (c) for a mutual collision between two DW-
types. The parameters used here are the same as those of Fig. 2, except for Π1 =
1/2 i/2
i/2 −1/2
. The values more than 2 are colored white.
6. Concluding Remarks
In this paper, we have investigated dynamical properties of matter-wave bright solitons
with a finite background in the F = 1 spinor Bose-Einstein condensate. To perform our anal-
ysis concretely, we have exploited an integrable spinor model with a self-focusing nonlinearity
and the inverse scattering method under nonvanishing boundary conditions. The situation
15/18
J. Phys. Soc. Jpn. Full Paper
that matter-wave solitons are located on a finite background fits to the experiments.
One-soliton solutions are derived explicitly and studied in detail. From the point of the
mathematical view, they offer general forms of bright soliton solutions of the NLS equation.
We have confirmed that our one-soliton solutions include those obtained in the previous works.
One-soliton solutions are classified into two kinds by the difference of boundary conditions;
DW-type and PS-type. The spin density profiles of one-solitons vary depending on the bound-
ary conditions. In the case of purely imaginary discrete eigenvalues, we have analytically shown
that DW-type is in the ferromagnetic state, whereas PS-type is in the polar state. The exis-
tence of two distinct magnetic properties for one-soliton solutions also gives rise to fascinating
phenomena in the case for two-soliton collisions, for example, the spin-switching. The above
results for bright solitons with a finite background agree with those for bright solitons under
VBC10, 11) and dark solitons.13)
Several problems still remain. It is desirable to extend our analysis to the case of general
discrete eigenvalues. The computations of the conserved quantities other than the total spin
are also required. (One approach is given in Appendix.) In addition, we wish to investigate
analytical properties of general N -soliton solutions under NVBC in the spinor model. Needless
to say, too complicated calculations are inevitable for the above problems. The remaining
problems should be discussed elsewhere.
We conclude that the properties of the multiple matter-wave solitons in the spinor BECs
are interesting and should be useful in various applications. Bright solitons are preferable to
dark solitons for applications, because of the advantage in the propagation distance. We hope
that our analysis contributes to illuminating dynamical properties of solitons in the spinor
BECs, which should be demonstrated experimentally in near future.
Acknowledgment
One of the authors (TK) acknowledges Dr. J. Ieda and Dr. M. Uchiyama for valuable
comments and discussions.
Appendix: Several Conserved Quantities of One-Soliton Solutions
The conserved quantities help us to understand the dynamics of the system. In this ap-
pendix, we calculate the total number, the total spin, the total momentum and the total
energy of the spinor model. We assume that, in addition to purely imaginary discrete eigen-
values, Π is a real symmetric 2× 2 matrix. The condition that Π is a real symmetric matrix
is inherent in the self-defocusing case, i.e., dark solitons.12)
For Π = Π†, one-soliton solutions of purely imaginary discrete eigenvalues (43)-(45) be-
come the following form at t = t′ = (4n− 1)π/8λ20ξλ for n = 0,±1, . . . :
Q = λ0 e
iφ(x,t)
I + 4iζ
Πe−(χP+ρ
′/2) + (σyΠσy) eχP+ρ
′/2detΠ
e−(2χP+ρ
′) + 1 + e2χP+ρ
(detΠ)2
, (A·1)
16/18
J. Phys. Soc. Jpn. Full Paper
where eρ
′/2 ≡ λ/ζ. This form is suitable for calculations, since the imaginary part is separated
from the real one. One can see clearly that the one-soliton solutions under VBC (32) are located
on a finite background in the form (A·1). Note that the domain-wall shape is lost even for
detΠ = 0 there.
Several conserved quantities of the solitons (A·1) are calculated by use of eqs. (20)-(23).
The results for detΠ = 0 are given by
N̄T = 4λ0ζ, (A·2)
FT = N̄T
2α(β + γ)
β2 − γ2
, |F T| = N̄T, (A·3)
P̄T = N̄T~k, (A·4)
ĒT = N̄Tc
(k2 − 2λ20)−
, (A·5)
and those for detΠ 6= 0 are given by
N̄T = 8λ0ζ, (A·6)
FT = (0, 0, 0)
T , (A·7)
P̄T = N̄T~k, (A·8)
ĒT = N̄Tc
(k2 − 2λ20)−
. (A·9)
It is intriguing that, for fixed amplitude and discrete eigenvalue, N̄T, P̄T and ĒT of the
PS-type (detΠ 6= 0) have just twice values as those of the DW-type (detΠ = 0), respectively.
This enables us to interpret that the PS-type of solitons is a bound state of the two DW-types
of solitons. Additionally, for fixed amplitude and total number, the total energy ĒT of the
DW-type is lower than that of the PS-type: ĒDWT − ĒPST = −N̄3Tc/16 < 0, which suggests that
the DW-type is physically preferable. This result is consistent with inter-atomic ferromagnetic
interaction, i.e., c̄2 < 0.
17/18
J. Phys. Soc. Jpn. Full Paper
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18/18
|
0704.0083 | Why there is something rather than nothing (out of everything)? | Why there is something rather than nothing (out of everything)?
A.O.Barvinsky
Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, 119991 Moscow, Russia
The path integral over Euclidean geometries for the recently suggested density matrix of the
Universe is shown to describe a microcanonical ensemble in quantum cosmology. This ensemble
corresponds to a uniform (weight one) distribution in phase space of true physical variables, but
in terms of the observable spacetime geometry it is peaked about complex saddle-points of the
Lorentzian path integral. They are represented by the recently obtained cosmological instantons
limited to a bounded range of the cosmological constant. Inflationary cosmologies generated by these
instantons at late stages of expansion undergo acceleration whose low-energy scale can be attained
within the concept of dynamically evolving extra dimensions. Thus, together with the bounded
range of the early cosmological constant, this cosmological ensemble suggests the mechanism of
constraining the landscape of string vacua and, simultaneously, a possible solution to the dark energy
problem in the form of the quasi-equilibrium decay of the microcanonical state of the Universe.
PACS numbers: 04.60.Gw, 04.62.+v, 98.80.Bp, 98.80.Qc
Euclidean quantum gravity (EQG) is a lame duck in
modern particle physics and cosmology. After its sum-
mit in early and late eighties (in the form of the cosmo-
logical wavefunction proposals [1, 2] and baby universes
boom [3]) the interest in this theory gradually declined,
especially, in cosmological context, where the problem of
quantum initial conditions was superseded by the con-
cept of stochastic inflation [4]. EQG could not stand the
burden of indefiniteness of the Euclidean gravitational
action [5] and the cosmology debate of the tunneling vs
no-boundary proposals [6].
Thus, a recently suggested EQG density matrix of the
Universe [7] is hardly believed to be a viable candidate
for the initial state of the Universe, even though it avoids
the infrared catastrophe of small cosmological constant
Λ, generates an ensemble of universes in the limited
range of Λ, and suggests a strong selection mechanism
for the landscape of string vacua [7, 8]. Here we want to
justify this result by deriving it from first principles of
Lorentzian quantum gravity applied to a microcanonical
ensemble of closed cosmological models.
Thermal properties in quantum cosmology [9] are in-
corporated by a mixed physical state, which is dynam-
ically more preferable than a pure state of the Hartle-
Hawking type. This follows from the path integral for
the EQG statistical sum [7, 8]. It can be cast into the
form of the integral over a minisuperspace of the lapse
function N(τ) and scale factor a(τ) of spatially closed
FRW metric ds2 = N2(τ) dτ2 + a2(τ) d2Ω(3),
e−Γ =
periodic
D[ a,N ] e−ΓE [ a,N ], (1)
e−ΓE [ a,N ] =
periodic
Dφ(x) e−SE [ a,N ;φ(x) ]. (2)
Here ΓE [ a, N ] is the Euclidean effective action of all
inhomogeneous “matter” fields which include also met-
ric perturbations on minisuperspace background Φ(x) =
(φ(x), ψ(x), Aµ(x), hµν (x), ...). SE [a,N ;φ(x)] is the clas-
sical Eucidean action, and the integration runs over pe-
riodic fields on the Euclidean spacetime with a compact-
ified time τ (of S1 × S3 topology).
For free matter fields φ(x) conformally coupled to grav-
ity (which are assumed to be dominating in the sys-
tem) the effective action has the form [7] ΓE [ a,N ] =
dτ NL(a, a′) + F (η), a′ ≡ da/Ndτ . Here NL(a, a′) is
the effective Lagrangian of its local part including the
classical Einstein term (with the cosmological constant
Λ = 3H2) and the contribution of the conformal anomaly
of quantum fields and their vacuum (Casimir) energy,
L(a, a′) = −aa′2−a+H2a3+B
. (3)
F (η) is the free energy of their quasi-equilibrium excita-
tions with the temperature given by the inverse of the
conformal time η =
dτ N/a. This is a typical boson or
fermion sum F (η) = ±
1∓ e−ωη
over field oscil-
lators with energies ω on a unit 3-sphere. We work in
units of mP = (3π/4G)
1/2, and B is the constant deter-
mined by the coefficient of the Gauss-Bonnet term in the
overall conformal anomaly of all fields φ(x).
Semiclassically the integral (1) is dominated by the
saddle points — solutions of the Friedmann equation
−H2 − C
, (4)
modified by the quantum B-term and the radiation term
C/a4 with the constant C satisfying the bootstrap equa-
tion C = B/2 + dF (η)/dη. Such solutions represent
garland-type instantons which exist only in the limited
range 0 < Λmin < Λ < 3m
P/B [7, 8] and eliminate the
infrared catastrophe of Λ = 0. The period of these quasi-
thermal instantons is not a freely specifiable parameter,
but as a function of Λ follows from this bootstrap. There-
fore this is not a canonical ensemble.
Contrary to the construction above, the density ma-
trix that we advocate here is given by the canonical path
integral of Lorentzian quantum gravity. Its kernel in the
http://arxiv.org/abs/0704.0083v2
representation of 3-metrics and matter fields denoted be-
low as q reads
ρ(q+, q−) = e
q(t±)= q±
D[ q, p,N ] e
dt (p q̇−NµHµ)
, (5)
where the integration runs over histories of phase-space
variables (q(t), p(t)) interpolating between q± at t± and
the Lagrange multipliers of the gravitational constraints
Hµ = Hµ(q, p) — lapse and shift functionsN(t) = N
µ(t).
The measure D[ q, p,N ] includes the gauge-fixing factor
containing the delta function δ(χ) =
µ δ(χ
µ) of gauge
conditions χµ and the ghost factor [10, 11] (condensed
index µ includes also continuous spatial labels). It is
important that the integration range of Nµ
−∞ < N < +∞, (6)
is such that it generates in the integrand the delta-
functions of these constraints δ(H) =
µ δ(Hµ). As a
consequence the kernel (5) satisfies the set of quantum
Dirac constraints — Wheeler-DeWitt equations
q, ∂/i∂q
ρ( q, q− ) = 0, (7)
and the density matrix (5) can be regarded as an operator
delta-function of these constraints
ρ̂ ∼ “
δ(Ĥµ)
”. (8)
This notation should not be understood literally because
this multiple delta-function is not well defined, for the
operators Ĥµ do not commute and form a quasi-algebra
with nonvanishing structure functions. Moreover, ex-
act operator realization Ĥµ is not known except the
first two orders of a semiclassical ~-expansion [12]. For-
tunately, we do not need a precise form of these con-
straints, because we will proceed with their path-integral
solutions well adjusted to the semiclassical perturbation
theory. This strategy does not extend beyond typical
field-theoretic considerations when the path integral is
regarded more fundamental than the Schrodinger equa-
tion marred with the problems of divergent equal-time
commutators, operator ordering, etc.
The very essence of our proposal is the interpretation
of (5) and (8) as the density matrix of a microcanonical
ensemble in spatially closed quantum cosmology. A sim-
plest analogy is the density matrix of an unconstrained
system having a conserved Hamiltonian Ĥ in the micro-
canonical state with a fixed energy E, ρ̂ ∼ δ(Ĥ − E).
Major distinction of (8) from this case is that spatially
closed cosmology does not have freely specifiable con-
stants of motion like the energy or other global charges.
Rather it has as constants of motion the Hamiltonian and
momentum constraints Hµ, all having a particular value
— zero. Therefore, the expression (8) can be considered
as a most general and natural candidate for the quantum
state of the closed Universe. Below we confirm this fact
by showing that in the physical sector the correspond-
ing statistical sum is just a uniformly distributed (with
a unit weight) integral over entire phase space of true
physical degrees of freedom. Thus, this is a sum over
Everything. However, in terms of the observable quanti-
ties, like spacetime geometry, this distribution turns out
to be nontrivially peaked around a particular set of uni-
verses. Semiclassically this distribution is given by the
EQG density matrix and the saddle-point instantons of
the above type [7].
From the normalization of the density matrix in the
physical Hilbert space the statistical sum follows as the
path integral
1 = Trphys ρ̂ =
q, ∂/i∂q
ρ(q, q′)
periodic
D[ q, p,N ] e i
dt(p q̇−NµHµ), (9)
where the integration runs over periodic in time histo-
ries of q = q(t). Here µ
q, ∂/i∂q
= µ̂ is the mea-
sure which distinguishes the physical inner product in
the space of solutions of the Wheeler-DeWitt equations
(ψ1|ψ2) = 〈ψ1|µ̂|ψ2〉 from that of the space of square-
integrable functions, 〈ψ1|ψ2〉 =
dq ψ∗1ψ2. This measure
includes the delta-function of unitary gauge conditions
and an operator factor built with the aid of the relevant
ghost determinant [12].
On the other hand, in terms of the physical phase space
variables the Faddeev-Popov path integral equals [10, 11]
D[ q, p,N ] e i
dt (p q̇−NµHµ)
DqphysDpphys e
dt (pphys q̇phys−Hphys(t))
= Trphys
T e−i
dt Ĥphys(t)
, (10)
where T denotes the chronological ordering. Here the
physical Hamiltonian Hphys(t) and its operator realiza-
tion Ĥphys(t) are nonvanishing only in unitary gauges ex-
plicitly depending on time [12], ∂tχ
µ(q, p, t) 6= 0. In static
gauges, ∂tχ
µ = 0, they identically vanish, because in spa-
tially closed cosmology the full Hamiltonian reduces to
the combination of constraints.
The path integral (10) is gauge-independent on-shell
[10, 11] and coincides with that in the static gauge.
Therefore, from Eqs.(9)-(10) with Ĥphys = 0, the sta-
tistical sum of our microcanonical ensemble equals
e−Γ = Trphys Iphys =
dqphys dpphys
= sum over Everything. (11)
This ultimate equipartition, not modulated by any non-
trivial density of states, is a result of general covariance
and closed nature of the Universe lacking any freely speci-
fiable constants of motion. The volume integral of entire
physical phase space, whose structure and topology is
not known, is very nontrivial. However, below we show
that semiclassically it is determined by EQG methods
and supported by instantons of [7] spanning a bounded
range of the cosmological constant.
Integration over momenta in (9) yields a Lagrangian
path integral with a relevant measure and action
e−Γ =
D[ q,N ] eiSL[ q, N ]. (12)
Integration runs over periodic fields (not indicated ex-
plicitly but assumed everywhere below) even despite the
Lorentzian signature of the underlying spacetime. Sim-
ilarly to the procedure of [7, 8] leading to (1)-(2) in
the Euclidean case, we decompose [ q,N ] into a min-
isuperspace [ aL(t), NL(t) ] and the “matter” φL(x) vari-
ables, the subscript L indicating their Lorentzian na-
ture. With a relevant decomposition of the measure
D[ q,N ] = D[ aL, NL ]×DφL(x), the microcanonical sum
takes the form
e−Γ =
D[ aL, NL ] e
iΓL[ aL, NL ], (13)
eiΓL[ aL, NL ] =
DφL(x) e
iSL[ aL, NL;φL(x)], (14)
where ΓL[ aL, NL ] is a Lorentzian effective action. The
stationary point of (13) is a solution of the effective equa-
tion δΓL/δNL(t) = 0. In the gauge NL = 1 it reads as
a Lorentzian version of the Euclidean equation (4) and
the bootstrap equation for the radiation constant C with
the Wick rotated τ = it, a(τ) = aL(t), η = i
dt/aL(t).
However, with these identifications C turns out to be
purely imaginary (in view of the complex nature of the
free energy F (i
dt/aL)). Therefore, no periodic solu-
tions exist in spacetime with a real Lorentzian metric.
On the contrary, such solutions exist in the Euclidean
spacetime. Alternatively, the latter can be obtained with
the time variable unchanged t = τ , aL(t) = a(τ), but
with the Wick rotated lapse function
NL = −iN, iSL[ aL, NL;φL] = −SE[ a,N ;φ ]. (15)
In the gauge N = 1 (NL = −i) these solutions exactly
coincide with the instantons of [7]. The corresponding
saddle points of (13) can be attained by deforming the
integration contour (6) of NL into the complex plane to
pass through the point NL = −i and relabeling the real
Lorentzian t with the Euclidean τ . In terms of the Eu-
clidean N(τ), a(τ) and φ(x) the integrals (13) and (14)
take the form of the path integrals (1) and (2) in EQG,
iΓL[ aL, NL] = −ΓE [ a, N ]. (16)
However, the integration contour for the Euclidean N(τ)
runs from −i∞ to +i∞ through the saddle point N = 1.
This is the source of the conformal rotation in Euclidean
quantum gravity, which is called to resolve the problem of
unboundedness of the gravitational action and effectively
renders the instantons a thermal nature, even though
they originate from the microcanonical ensemble. This
mechanism implements the justification of EQG from
canonical quantization of gravity [14] (see also [15] in
black hole context).
To show this we calculate (1) in the one-loop approx-
imation with the measure inherited from the canonical
path integral (5) D[ a,N ] = DaDN µ[ a,N ] δ[χ ] DetQ.
Here µ[ a,N ] is a local measure determined by the La-
grangian NL(a, a′), (3), in the local part of ΓE [ a,N ],
µ1−loop =
∂2(NL)
∂ȧ ∂ȧ
N a2a′2
D = a a′2(a2 −B +B a′2). (17)
The Faddeev-Popov factor δ[χ ] DetQ contains a gauge
condition χ = χ(a,N) fixing the one-dimensional dif-
feomorphism, τ → τ̄ = τ − f/N , which for infinitesi-
mal f = f(τ) has the form ∆fN ≡ N̄(τ) − N(τ) = ḟ ,
∆fa ≡ ā(τ) − a(τ) = ȧ f/N , and Q = Q(d/dτ) is a
ghost operator determined by the gauge transformation
of χ(a,N), ∆fχ = Q(d/dτ) f(τ).
The conformal mode σ of the perturbations δa = σa0
and δN = σN0 on the saddle-point background (labeled
below by zero, a = a0 + δa, N = N0 + δN) origi-
nates from imposing the background gauge χ(a,N) =
δN − (N0/a0) δa. In this gauge Q = a(d/dτ)a−1, and
the quadratic part of ΓE takes the form [13]
δ2σΓE = −
3πm2P
, (18)
where D is given by (17). As is known from [7] for the
background instantons a20(τ) ≥ a2− > B (a− is the turn-
ing point with the smallest value of a0(τ)), so that D > 0
everywhere except the turning points where D degener-
ates to zero. Therefore δ2σΓE < 0 for real σ, but the
Gaussian integration runs along the imaginary axes and
yields the functional determinant of the positive operator
— the kernel of the quadratic form (18)
e−Γ1−loop = e−Γ0 DetQ0
D/a′2
= e−Γ0×Det
)]−1/2
In view of periodic boundary conditions for both oper-
ators their determinants cancel each other (their zero
modes to be eliminated because they correspond to the
conformal Killing symmetry of FRW instantons) [13].
Therefore, the contribution of the conformal mode re-
duces to the selection of instantons with a fixed time pe-
riod, effectively endowing them with a thermal nature.
As suggested in [7, 8, 16] these instantons serve as
initial conditions for inflationary universes which evolve
according to the Lorentzian version of Eq.(4) and, at late
stages, have two branches of a cosmological acceleration
with Hubble scales H2
= (m2P /B)(1±(1−2BH2)1/2). If
the initial Λ = 3H2 is a composite inflaton field decaying
at the end of inflation, then one of the branches under-
goes acceleration with H2+ = 2m
P/B. This is determined
by the new quantum gravity scale suggested in [8] – the
upper bound of the instanton Λ-range, Λmax = 3m
P /B.
To match the model with inflation and the dark energy
phenomenon, one needs B of the order of the inflation
scale in the very early Universe and B ∼ 10120 now, so
that this parameter should effectively be a growing func-
tion of time.
This picture seems to fit into string theory at its low-
energy field-theoretic level. Then, with a bounded range
of Λ, it might constrain the landscape of string vacua
[7, 8]. Moreover, string theory has a qualitative mecha-
nism to promote the constant B to the level of a mod-
uli variable indefinitely growing with the evolving size
R(t) of extra dimensions. Indeed B as a coefficient in
the overall conformal anomaly of 4-dimensional quantum
fields basically counts their number N , B ∼ N . In the
Kaluza-Klein (KK) theory and string theory the effective
4-dimensional fields arise as KK and winding modes with
the masses [17]
m2n,w =
R2 (19)
(enumerated by the KK and winding numbers), which
break their conformal symmetry. These modes remain
approximately conformally invariant as long as their
masses are much smaller than the spacetime curvature,
m2n,w ≪ H2+ ∼ m2P /N . Therefore the number of confor-
mally invariant modes changes with R. Simple estimates
show that for pure KK modes (w = 0, n ≤ N) their num-
ber grows with R as N ∼ (mPR)2/3, whereas for pure
winding modes (n = 0, w ≤ N) their number grows with
the decreasing R as N ∼ (mPα′/R)2/3. Thus, the effect
of indefinitely growing B is possible for both scenarios
with expanding or contracting extra dimensions. In the
first case this is the growing tower of superhorizon KK
modes which make the horizon scale H+ = mP
2/B ∼
mP /(mPR)
1/3 indefinitely decreasing with R → ∞. In
the second case this is the tower of superhorizon winding
modes which make this acceleration scale decrease with
the decreasing R as H+ ∼ mP (R/mPα′)1/3. This effect
is flexible enough to accommodate the present day ac-
celeration scale H0 ∼ 10−60mP (though, by the price of
fine-tuning an enormous coefficient of expansion or con-
traction of R). This gives a new dark energy mechanism
driven by the conformal anomaly and transcending the
inflationary and matter-domination stages starting with
the state of the microcanonical distribution.
To summarize, within a minimum set of assumptions
(the equipartition in the physical phase space (11)),
we seem to have the mechanism of constraining the
landscape of string vacua and get the full evolution of
the Universe as a quasi-equilibrium decay of its initial
microcanonical state. Thus, contrary to anticipations
of Sidney Coleman, “there is Nothing rather than
Something” [3], one can say that Something (rather
than Nothing) comes from Everything.
The author thanks O.Andreev, C.Deffayet, A.Kamen-
shchik, J.Khoury, H.Tye, A.Tseytlin, I.Tyutin and
B.Voronov for thought provoking discussions and espe-
cially Andrei Linde, this work having appeared as an un-
intended response to his discontent with EQG initial con-
ditions. The work was supported by the RFBR grant 05-
02-17661, the grant LSS 4401.2006.2 and SFB 375 grant
at the Ludwig-Maximilians University in Munich.
[1] J.B.Hartle and S.W.Hawking, Phys.Rev. D28, 2960
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[5] G.W.Gibbons, S.W.Hawking and M.Perry, Nucl. Phys.
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121502 (2006), hep-th/0611206.
[9] H.Firouzjahi et al, JHEP 0409, 060 (2004); S.Sarangi
and S.-H.H.Tye, hep-th/0505104; R.Brustein and S.P.de
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and reduced phase space quantization of constrained sys-
tems, gr-qc/9612003; M.Henneaux and C.Teitelboim,
Quantization of Gauge Sytems (Princeton University
Press, Princeton, 1992).
[13] A.O.Barvinsky and A.Yu.Kamenshchik, in preparation.
[14] J.B. Hartle and K. Schleich, in Quantum field theory and
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tol, 1988); K. Schleich, Phys.Rev. D 36, 2342 (1987).
[15] D. Brown and J.W. York, Jr., Phys. Rev. D 47, 1420
(1993), gr-qc/9209014.
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http://arxiv.org/abs/gr-qc/9804051
http://arxiv.org/abs/gr-qc/9812027
http://arxiv.org/abs/hep-th/0605132
http://arxiv.org/abs/hep-th/0611206
http://arxiv.org/abs/hep-th/0505104
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http://arxiv.org/abs/gr-qc/9209014
http://arxiv.org/abs/hep-th/0701201
|
0704.0086 | Clustering in a stochastic model of one-dimensional gas | Clustering in a stochastic model of one-dimensional gas
The Annals of Applied Probability
2008, Vol. 18, No. 3, 1026–1058
DOI: 10.1214/07-AAP481
c© Institute of Mathematical Statistics, 2008
CLUSTERING IN A STOCHASTIC MODEL OF
ONE-DIMENSIONAL GAS
By Vladislav V. Vysotsky1
St. Petersburg State University
We give a quantitative analysis of clustering in a stochastic model
of one-dimensional gas. At time zero, the gas consists of n identical
particles that are randomly distributed on the real line and have zero
initial speeds. Particles begin to move under the forces of mutual
attraction. When particles collide, they stick together forming a new
particle, called cluster, whose mass and speed are defined by the laws
of conservation.
We are interested in the asymptotic behavior of Kn(t) as n→∞,
where Kn(t) denotes the number of clusters at time t in the system
with n initial particles. Our main result is a functional limit theorem
for Kn(t). Its proof is based on the discovered localization property
of the aggregation process, which states that the behavior of each
particle is essentially defined by the motion of neighbor particles.
1. Introduction.
1.1. Description of the model. We give a quantitative analysis of clus-
tering in a stochastic model of one-dimensional gas. At time zero, the gas
consists of n point particles, each one of mass 1
. These particles are ran-
domly distributed on the real line and have zero initial speeds. Particles
begin to move under the forces of mutual attraction. When two or more
particles collide, they stick together forming a new particle, called cluster,
whose mass and speed are defined by the laws of mass and momentum
conservation. Between collisions, particles move according to the laws of
Newtonian mechanics.
We suppose that the force of mutual attraction does not depend on dis-
tance and equals the product of masses. This assumption is natural for
Received March 2007; revised September 2007.
1Supported in part by the Grants NSh-4222.2006.1 and DFG-RFBR 436 RUS
113/773/0-1(R).
AMS 2000 subject classifications. Primary 60K35, 82C22; secondary 60F17, 70F99.
Key words and phrases. Sticky particles, particle systems, gravitating particles, number
of clusters, aggregation, adhesion.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2008, Vol. 18, No. 3, 1026–1058. This reprint differs from the original in
pagination and typographic detail.
http://arxiv.org/abs/0704.0086v2
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/07-AAP481
http://www.imstat.org
http://www.ams.org/msc/
http://www.imstat.org
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/07-AAP481
2 V. V. VYSOTSKY
one-dimensional models because, by the Gauss law applied to flux of the
gravitational field, gravitation is proportional to the distance to the power
one minus dimension of the space. At any moment, the acceleration of a
particle is thus equal to difference of masses located to the right and to the
left of the particle.
Random initial positions of particles are usually described (see [8, 16,
25]) by the following natural models: in the uniform model, n particles are
independently and uniformly spread on [0,1]; in the Poisson model, particles
are located at points 1
S2, . . . ,
Sn, where Si is a standard exponential
random walk. In other words, particles are located at points of first n jumps
of a Poisson process with intensity n.
These two models are the most natural and interesting; let us call them
the main models of initial positions. However, we will see that behavior
of the Poisson model is essentially defined by independence of initial dis-
tances between particles rather than by the particular type of the distances’
distribution. Therefore, it is of a great mathematical interest to general-
ize the Poisson model by introducing the i.d. model, where “i.d.” stands
for “independent distances,” as follows. Particles are initially located at
S2, . . . ,
Sn, where Si is a positive random walk whose nonnegative
i.i.d. increments Xi satisfy the normalization condition EXi = 1. Note that
if we proceed to the limit as n→∞, we consider a system of total mass one,
which consists of, roughly speaking, infinitesimal particles homogeneously
spread on [0,1]; this is true for all the mentioned models of initial positions.
The mathematical interest in sticky particles systems arises mainly from
relations between these systems and some nonlinear partial differential equa-
tions originating from fluid mechanics, for example, the Burgers equation.
These equations admit interpretation in terms of sticky particles; see Gur-
batov et al. [10], Brenier and Grenier [4] or E, Rykov and Sinai [6]. Sticky
particles models are also used for numerical solving of other partial differen-
tial equations; see Chertock et al. [5] for explanations and further references.
As time goes, particles aggregate in clusters. Clusters become larger and
larger while the number of clusters decreases until they merge into a single
cluster containing all initial particles. This process of mass aggregation is
strongly connected with additive coalescence; see Bertoin [2] and Giraud [9]
for the most recent results and references.
The aggregation process resembles formation of a star from dispersed
space dust and sticky particles models indeed have relations to astrophysics.
It is appropriate to clarify these relations since they are not so direct and
cause a lot of misunderstanding.
It is known that the distribution of galaxies in the universe is very inho-
mogeneous and the regions of high density form a peculiar cellular structure.
The first attempt to understand the formation of such structures was made
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 3
in 1970 by Zeldovich. Most of the mass in the universe is believed to ex-
ist in the form of particles that practically do not collide with each other
and interact only gravitationally, for example, neutrinos. In his model, Zel-
dovich considered an initially homogeneous collisionless medium of particles
moving by pure inertia; the gravitational interaction was taken away by an
appropriate time change. He showed that singularities, that is, the thin re-
gions of very high density of particles, so called “pancakes,” appear even if
initial speeds of particles form a smooth velocity field.
Zeldovich’s approximate model, however, does not explain formation of
the cellular structure of matter. His approximation does not take into ac-
count that particles hitting a “pancake” are hampered by its strong gravita-
tional field and start oscillating inside the “pancake” instead of flying away.
Although this gravitational adhesion of collisionless particles is not precisely
the same as the real sticking, the model of sticky particles serves as a reason-
able approximation. The effect of gravitational adhesion was then analyzed
by the use of the Burgers equation; Gurbatov, Saichev and Shandarin pro-
posed it in 1984 to extend Zeldovich’s approximation, which is invalid after
formation of “pancakes.”
The model of sticky particles is directly mentioned in Gurbatov et al. [11];
a comprehensive survey of the formation of the Universe’s large-scale struc-
ture could be found in Shandarin and Zeldovich [23].
1.2. Statement of the problem and the results. In general, the problem
is to describe the process of mass aggregation. How fast is it? How large
the clusters are? Where do clusters appear most intensively, and so forth?
Numerous papers on the model (e.g., [8, 14, 16, 20, 25]) are dedicated to
probabilistic description of various properties of the aggregation process as
the number of initial particles n tends to infinity. Thus, the behavior of a
typical system consisting of a large number of particles is studied.
In this paper, we are interested in the asymptotic behavior of Kn(t), which
denotes the number of clusters at time t in the system with n initial particles.
This variable is a decreasing random step function satisfying Kn(0) = n
and Kn(t) = 1 for t ≥ T lastn , where T lastn denotes the moment of the last
collision. While calculating Kn(t), we also count initial particles that have
not experienced any collisions; in other words, Kn(t) is the total number of
particles existing at time t.
It is very important to know the behavior of Kn(t). This gives us a deep
understanding of the aggregation process since the average size of a cluster
at time t is n
Kn(t)
At first we give a short deterministic example. Suppose that particles are
located at points 1
, . . . , n
, that is, Si = i. By simple calculations, we find
that there would not be any collisions before t= 1. At the moment t= 1, all
4 V. V. VYSOTSKY
particles simultaneously stick together, hence Kn(t) = n for 0 ≤ t < 1 and
Kn(t) = 1 for t≥ 1.
However, when the initial positions are random, the aggregation process
behaves entirely differently. In [25], the author proved the following state-
ment.
Fact 1. There exists a deterministic function a(t) such that both in the
Poisson and the uniform models of initial positions, for any t≥ 0, we have
Kn(t)
P−→ a(t), n→∞.(1)
The function a(t) is continuous, a(0) = 1, and a(t) = 0 for t≥ 1. We conjec-
ture, on the basis of numerical simulations, that a(t) = 1− t2 for 0≤ t≤ 1.
The relation a(t) = 0 for t > 1 is not of a surprise because we know from
Giraud [8] that both in the Poisson and the uniform models, T lastn
P−→ 1 (the
limit constant is so “fine” due to the proper scaling of the model). Therefore,
we say that the moment t= 1 is critical ; note that this moment coincides
with the moment of the total collision in the deterministic model.
The aim of this paper is to strengthen the result of [25]. We first gen-
eralize Fact 1 and prove it for the i.d. model. We will see [relations (19)
and (27) below] that a(t) is equal to the probability of a certain event that
is expressed in terms of Xi. Also, we will prove that a(t) depends on the
common distribution of Xi as follows: a(t) = 1 on [0,
µ), where
µ := sup{y :P{Xi < y}= 0};
a(t) ∈ (0,1) on (√µ,1); and a(t) = 0 on (1,∞).
Furthermore, the recent results of the author [26] allow us to prove the
conjecture from Fact 1 that aPoiss(t) = aUnif(t) = 1− t2 for 0≤ t≤ 1. There
is an amazing contrast between the simplicity of this formula and the hard
calculations one needs to obtain it. It is remarkable that now we know the
limit function a(t) for the main models of initial positions.
Our main goal is to improve (1) by finding the next term in the asymp-
totics of Kn(t). The result is the following statement, where the standard
symbol
D−→ denotes weak convergence and D denotes the Skorohod space.
Theorem 1. In the i.d. model with continuous Xi satisfying EX
for some γ > 4, there exists a centered Gaussian process K(·) on [0,1) such
Kn(·)− na(·)√
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1)(2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 5
as n→ ∞. The process K(·) depends on the distribution of Xi. This pro-
cess satisfies K(0) = 0 and has a.s. continuous trajectories. The covariance
function R(s, t) of K(·) is continuous on [0,1)2, R(s, t)> 0 on (√µ,1)2, and
R(s, t) = 0 on [0,1)2 \ (õ,1)2.
In the uniform model, (2) holds for some centered Gaussian process KUnif(·)
on [0,1). This process satisfies KUnif(0) = 0 and has a.s. continuous trajecto-
ries. The covariance function RUnif(s, t) of KUnif(·) is continuous on [0,1)2,
and RUnif(s, t) =RPoiss(s, t)− s2t2.
Thus, the Poisson and the uniform models lead to different limit processes
KPoiss(·) and KUnif(·), although aPoiss(·) = aUnif(·).
As an immediate corollary of Theorem 1 (see Billingsley [3], Section 15),
we get
Kn(t)− na(t)√
D−→N (0, σ2(t)), n→∞(3)
for any t < 1, where σ2(t) := R(t, t). It is possible to show that in the i.d.
model, (3) holds for all t 6= 1 under the less restrictive condition EX2i <∞,
with σ2(t) = 0 for t > 1; continuity of Xi is not required.
We also study convergence of the left-hand side of (3) at the critical
moment t= 1. Apparently, the limit is not Gaussian, but this complicated
problem is related to a curious, but hardly provable conjecture on integrated
random walks. In view of this non-Gaussianity, it seems impossible to prove
any extended version of Theorem 1 that describes the weak convergence of
trajectories on the whole interval [0,1]; we refer to Section 7 for further
discussion.
We finish this subsection with a note on scaling. In our model, the masses
of particles are equal to 1
and the distances between them are of the order
. Let us rescale the i.d. model by multiplying all masses and distances
by n: the system of particles of mass one each, initially located at points
S1 − S[n/2], S2 − S[n/2], . . . , Sn − S[n/2], is called the expanding model. The
particles are shifted by S[n/2] because we want the system to expand “filling”
the whole line as n→∞ rather than only the positive half-line.
All results of our paper hold true for the expanding model. This is not
unexpected because the shift does not produce any changes and the rescaling
of masses is equivalent to the time contraction by n times while the rescaling
of distances is equivalent to the time expansion by n times. We refer the
reader to Section 2 below or to Lifshits and Shi [16] for rigorous arguments.
1.3. Organization of the paper. In Section 2 we describe a general method
which is used to study systems of sticky particles. This method is applied for
studying the i.d. model in Section 3, where we investigate some properties of
6 V. V. VYSOTSKY
the aggregation process. We will show that the aggregation process is highly
local, that is, the behavior of a particle is essentially defined by the motion
of neighbor particles. This localization property suggests that we could use
limit theorems for weakly dependent variables to prove both Fact 1 and
Theorem 1 for the i.d. model; this will be done in Section 4. Then we will
prove Theorem 1 for the uniform model in Section 5. In Section 6 we study
the number of clusters at the critical moment t = 1. Some open questions
are discussed in Section 7.
2. Method of barycenters. In this section we briefly describe the method
of barycenters, which is the main tool used to study systems of sticky par-
ticles; it is also applicable to more general models where particles could
have nonzero initial speeds and different masses. The method of barycenters
was independently introduced by E, Rykov and Sinai [6] and Martin and
Piasecki [20].
Let us start with several definitions. We always numerate particles from
left to right and identify particles with their numbers. A block of particles is
a nonempty set J ⊂ [1, n] consisting of consecutive numbers. For example,
the block (i, i+k] consists of particles i+1, . . . , i+k. Note that there are not
any relations between blocks and clusters: for example, a block’s particles
could be contained in different clusters and these clusters could even contain
particles that do not belong to the block.
It is convenient to assume that initial particles do not vanish at collisions
but continue to exist in created clusters. Then the coordinate xi,n(t) of a
particle i could be defined as the coordinate of a cluster that contains the
particle at time t. The second subscript n always indicates the number of
initial particles; we will omit this subscript as often as possible.
By xJ(t) := |J |−1
i∈J xi(t) denote the position of the barycenter of a
block J at time t. Further, define
x∗J(t) := xJ(0) +
where M
J := n
−1(n −maxj∈J) and M (L)J := n−1(minj∈J −1) are the to-
tal masses of particles located to the right and to the left of the block J ,
respectively.
A block is free from the right up to time t if, up to this time, the block’s
particles did not collide with particles initially located to the right of the
block. We similarly define blocks that are free from the left and say that a
block is free up to time t if it is both free from the right and from the left.
The next statement plays the key role in the analysis of sticky particles
systems. The barycenter of a free block moves as an imaginary particle con-
sisting of all particles of the block put together at the initial barycenter. In a
more precise and general way, we state the following.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 7
Proposition 1. If a block J is free from the right (resp. left) up to time
t, then xJ(s)≥ x∗J(s) for s ∈ [0, t] [resp. xJ(s)≤ x∗J(s)]. If a block J is free
up to time t, then xJ(s) = x
J(s) for s ∈ [0, t].
This statement could be found, for example, in Lifshits and Shi [16],
Proposition 4.1. The easy proof is based on the property of conservation of
momentum.
The moment when a particle j sticks with its right-hand side neighbor
j + 1 is called the merging time Tj,n of the particle j. In other words, Tj,n
is the first moment when particles j and j + 1 are contained in a common
cluster; here j ∈ [1, n− 1]. Proposition 4.3 from Lifshits and Shi [16], which
is stated below, gives us a way to calculate Tj,n.
Proposition 2. For every j ∈ [1, n− 1], we have
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.(4)
Thus, Tj,n is expressed by means of barycenters. Note that since
x∗(j,k](s)− x∗(l,j](s) = x(j,k](0)− x(l,j](0)−
s2,(5)
each of the equations x∗
(j,k]
(s) = x∗
(l,j]
(s) has a unique nonnegative solution.
We also mention that at the moment Tj,n appears a cluster that consists of
the particles l+1, . . . , k, where k and l are minimizers of the right-hand side
of (4).
We will prove Proposition 2 since the proof is simple and perfectly illus-
trates the sense of the method of barycenters.
Proof of Proposition 2. For any u < Tj,n, the particles j and j + 1
are contained in different clusters. Therefore, for every l < j, the block [l, j]
is free from the right up to time u, and for every k > j, the block [j + 1, k]
is free from the left. By Proposition 1,
x∗(l,j](u)≤ x(l,j](u)≤ xj(u)<xj+1(u)
≤ x(j,k](u)≤ x∗(j,k](u),
and since, by (5), the function x∗
(j,k]
(s)− x∗
(l,j]
(s) is decreasing for s≥ 0, we
conclude that
u <{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.
Taking minimum over k, l and taking supremum over u, we get Tj,n ≤
min{· · ·}.
8 V. V. VYSOTSKY
Let us prove the last inequality in the other direction. By the definition of
Tj,n, there exist an l < j and a k > j such that the blocks (l, j] and (j, k] are
free up to time Tj,n (clusters containing particles from these blocks collide
exactly at time Tj,n). In view of Proposition 1,
x∗(l,j](Tj,n) = x(l,j](Tj,n) = x(j,k](Tj,n) = x
(j,k](Tj,n);
hence Tj,n = {s≥ 0 :x∗(j,k](s) = x
(l,j]
(s)} and Tj,n ≥min{· · ·}. �
3. Study of the i.d. model. The localization property. At first, note that
Kn(t) = 1+
1{t<Ti,n}(6)
because the total number of clusters decreases by one at each moment Ti,n.
This representation plays the key role in the investigation of Kn(t). Clearly,
we need to study properties of the r.v.’s Ti,n to prove limit theorems for
Kn(t); such study will be done in this section.
3.1. The initial study. Let us simplify the representation for Tj,n from
Proposition 2. In this section we consider the i.d. model of initial positions,
where xj,n(0) =
Sj . Recall that Sj is a random walk with i.i.d. increments
{Xj}j∈Z (we will need the variables {Xj}j≤0 later).
Rewrite the initial distance between barycenters as
x(j,k](0)− x(l,j](0)
i=j+1
j − l
i=l+1
i=j+1
(Si − Sj+1) +
j − l
i=l+1
(Sj − Si) + (Sj+1 − Sj)
k−j−1
(Sj+i+1 − Sj+1) +
j − l
j−l−1
(Sj − Sj−i) +Xj+1
let us agree that
:= 0. Further, by
x(j,k](0)− x(l,j](0)
k−j−1
j+i+1
m=j+2
j − l
j−l−1
m=j−i+1
Xm +Xj+1
k−j−1
(k− j − i)Xj+i+1
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 9
j − l
j−l−1
(j − l− i)Xj−i+1 +Xj+1
and (5), we have
x∗(j,k](s)− x∗(l,j](s) =
Fk−j,j,j−l(s),
where
Fp,j,q(s) :=
(p− i)Xj+i+1
(q − i)Xj−i+1 +Xj+1 −
(for p, q ≥ 1 and j ∈ Z). Now, by Proposition 2, we get
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :Fk−j,j,j−l(s) = 0}
= min
1≤k≤n−j
1≤l≤j
{s≥ 0 :Fk,j,l(s) = 0}.(8)
Note that Fp,j,q(0) ≥ 0 for all p, j, q and Fp,j,q(s) is decreasing for s ≥ 0.
This function could be also written in the more convenient form:
Fp,j,q(s) =
(p− i)(Xj+i+1 − s2)
(q − i)(Xj−i+1 − s2) + (Xj+1 − s2).
3.2. Localization property of the aggregation process. We see that Tj,n
is a function of X2, . . . ,Xn; in other words, it is necessary to know the
distances between all n particles to find Tj,n. The aggregation process is
actually highly local, that is, the value of Tj,n is essentially defined by the
initial distances between neighbor particles {i} of j for which |j − i| is small
enough.
To make this statement rigorous, we need to introduce the following no-
tation. Let us put
j := min
1≤k,l≤M
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,M ∈N,
which is expressed in terms of the variables {Xi}|j−i|≤M only. Also, define
Tj := inf
k,l≥1
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,
10 V. V. VYSOTSKY
which is, in some sense, the merging time in an appropriate infinite system
of particles. The reader could construct such system by considering the limit
of the expanding model, see Section 1.
It is clear that
Tj ≤ Tj,n ≤ T (j∧n−j)j , j, n ∈N, j ≤ n,(10)
where by ∧ and ∨ we denote minimum and maximum, respectively, and
Tj ≤ T (M)j , j ∈ Z,M ∈N.(11)
Let us estimate the rate of the convergence of P{Tj 6= T (M)j } to zero as the
“radius of the neighborhood” M tends to infinity. We thus could “measure”
the above-mentioned locality of the aggregation process. In fact, by (10), we
have P{Tj,n 6= T (M)j } ≤ P{Tj 6= T
j } for any n ∈N, j ≤ n, andM ≤ j∧n−j.
Lemma 1. Suppose EX
i <∞ for some γ ≥ 1. Then there exists a non-
decreasing function ρ(t) such that
max(P{1{t≤Tj} 6= 1{t≤T (M)
}},P{Tj 6= T
j , T
j ≤ t})≤ ρ(t)M
1−γ(12)
for any t ∈ (0,1), j ∈ Z, and M ∈N. Moreover, for any t < 1, the left-hand
side of (12) is o(M1−γ).
Proof. Let us estimate the first probability in the left-hand side of
(12). By properties of Fk,j,l(·) and definitions of T (M)j and of Tj ,
P{1{t≤Tj} 6= 1{t≤T (M)
}}= P{Tj < t≤ T
k,l≥1
Fk,j,l(t)< 0, min
1≤k,l≤M
Fk,j,l(t)≥ 0
By (9), this expression does not depend on j, and putting j :=−1,
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)< 0,
1≤k≤M
(k− i)(Xi − t2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 11
+ min
1≤l≤M
(l− i)(X−i − t2) + (X0 − t2)≥ 0
We then compare the inequalities in the braces and obtain
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)< min
1≤k≤M
(k− i)(Xi − t2)
(Si − it2)< min
1≤k≤M
(Si − it2)
(Si − it2)< min
k∈{1,M}
(Si − it2)
Now rewrite the event in the last line as
∃k >M : 1
(Si − it2)<min
(Si − it2)
∃k >M : 1
(Si − it2)
(Si − it2)<min
(Si − it2)
Analyzing both cases 0≤ 1
i=1 (Si − it2) and 0> 1M
i=1 (Si − it2), we
conclude that the considered event implies
∃k >M : 1
(Si − it2)< 0
∃k >M :
(Si − it2)< 0
Clearly, the latter implies
{∃i≥M :Si − it2 < 0}=
hence, combining all the estimates together, we get
P{1{t≤Tj} 6= 1{t≤T (M)
}} ≤ 2P
.(13)
Note that we obtained (13) without any assumptions on the moments of Xi.
We now estimate the right-hand side of (13); recall that EXi = 1. Then
the first part of (12) immediately follows from the classical result of Baum
and Katz [1] (see their Theorem 3 and Lemma):
12 V. V. VYSOTSKY
Fact 2. If EXi = a and E|Xi|γ <∞ for some γ ≥ 1, then
= o(k1−γ), k→∞
for any ε > 0. In addition, the series
k=1P{supi≥k |Sii − a|> ε} converges
for all ε > 0 if γ = 2.
The estimation of the second probability in the left-hand side of (12) is
completely analogous, since
{Tj 6= T (M)j , T
j ≤ t}
= {Tj < T (M)j ≤ t}
1≤k,l
Fk,j,l(T
j )< 0, min
1≤k,l≤M
Fk,j,l(T
j ) = 0, T
j ≤ t
We put j :=−1, repeat the estimates, and get
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P{∃i≥M :Si − i[T
< 0, T
−1 ≤ t}
instead of (13). The right-hand side does not exceed 2P{∃i≥M : Si − it2 <
0}, hence
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P
.(14)
3.3. The distribution function of T0 in the Poisson model. It is amazing
that in the Poisson model, the distribution function of T0 could be found
explicitly. This is important because by (27) below, the limit function a(t)
equals P{T0 > t} for the i.d. model. Also, in the proof of Theorem 1 for the
uniform model, we will need aPoiss(t) = P{TPoiss0 ≥ t} to be twice differen-
tiable and have a continuous second derivative.
Lemma 2. In the Poisson model, for 0≤ t≤ 1, we have
P{T0 ≥ t}= 1− t2.(15)
In addition, for t≥ 0, n≥ 2, and 1≤ j ≤ n− 1, we have
P{Tj,n ≥ t}= et
1≤k≤j
(Si − it2)≥ 0
1≤k≤n−j
(Si − it2)≥ 0
where Si is a standard exponential random walk.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 13
Proof. We start with (16). By (8), (9) and properties of Fk,j,l(·),
P{Tj,n ≥ t}= P
1≤k≤n−j
1≤l≤j
Fk,j,l(t)≥ 0
1≤k≤n−j
(k− i)(Xj+i+1 − t2)(17)
+ min
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
In the right-hand side of the last equality, by Y denote the first minimum
and by Ỹ denote the second one.
Suppose X is a standard exponential r.v., Z is a nonnegative r.v., and
that X and Z are independent; then
P{Z ≤X}=
P{Z ≤ x}e−x dx
E1{Z≤x}e
−x dx= E
1{Z≤x}e
−x dx= Ee−Z .
Hence in view of independence of Y , Ỹ , Xj+1 we get
P{Y + Ỹ +Xj+1 − t2 ≥ 0}= EeY+Ỹ−t
EeY−t
EeỸ−t
and therefore,
P{Tj,n ≥ t}= et
P{Y +Xj+1 − t2 ≥ 0} · P{Ỹ +Xj+1 − t2 ≥ 0}.
Now, by
P{Ỹ +Xj+1 − t2 ≥ 0}
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
1≤l≤j
(l− i)(Xi+1 − t2) + l(X1 − t2)
1≤l≤j
(l− i+1)(Xi − t2)≥ 0
we conclude the proof of (16). Indeed, the expression in the last line equals
the first probability in the right-hand side of (16).
14 V. V. VYSOTSKY
Now let us prove (15). From the definition of T0 and T
0 we see that
1{t≤T (k)
} → 1{t≤T0} a.s. as k→∞; then by (16),
P{T0 ≥ t}= et
(Si − it2)≥ 0
Then we need to check that
(Si − it)≥ 0
1− te−t/2
for 0 ≤ t ≤ 1. The complicated calculations of this probability take more
then ten pages. Therefore, they were separated into independent paper [26].
Although these calculations seem to be technical, they are based on quite
original ideas. �
3.4. Some properties of the variables Ti. In this subsection we prove
several important properties of the r.v.’s Ti.
1. The sequence Ti is stationary.
Proof. This statement immediately follows from the definition of Ti and
stationarity of Xi, which are i.i.d.
2. The common distribution function of Ti is defined by
P{Ti ≥ t}= P
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)≥ 0
Proof. This formula follows from (9).
3. We have P{√µ ≤ Ti ≤ 1} = 1 while sup{y :P{Ti < y} = 0} =
µ and
inf{y :P{Ti < y} = 1} = 1; recall that µ = sup{y :P{Xi < y} = 0}. In addi-
tion, if 0<DXi <∞, then P{Ti = 1}= 0.
Proof. First, P{√µ ≤ Ti} = 1 is trivial, because both infima in (19) are
nonpositive.
Second, fix a t≥ 1 and consider P{Ti ≥ t}. Taking into account that infima
in (19) are nonpositive, we obtain
P{Ti ≥ t} ≤ P
(k− i)(Xi − t2) + (X0 − t2)≥ 0
Then by the same arguments as in (18),
P{Ti ≥ t} ≤ P
(k − i+1)(Xi − t2)≥ 0
(Si − it2)≥ 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 15
By the strong law of large numbers, this probability is zero for all t > 1.
If t= 1 and 0<DXi <∞, then
(Si − i)≥ 0
= lim
1≤k≤n
(Si − i)≥ 0
= lim
1≤k≤n
Si − i√
and from the invariance principle, we get
P{Ti ≥ 1} ≤ P
0≤s≤1
W (u)du≥ 0
It follows from the asymptotics of unilateral small deviation probabilities of
an integrated Wiener process, see (43) and (44) below, that the last expres-
sion equals zero.
Third, sup{y :P{Ti < y}= 0}=
µ and inf{y :P{Ti < y}= 1} = 1 follow
if we prove that for any t < EXi = 1, the common distribution of the i.i.d.
infima in (19) has an atom at zero. But we have
(k− i)(Xi − t2) = 0
(Si − it2) = 0
and it could be shown via the strong law of large numbers that the last
probability is strictly positive for all t < 1.
4. Suppose Xi is continuous. Then T
j and Tj,n are continuous for any
j, k,n and the common distribution of Tj could have an atom only at 1. In
addition, if EX2i <∞, then Tj are continuous.
Proof. By (7) and (8),
Tj,n = min
1≤k≤n−j
1≤l≤j
H(k, j, l),(20)
where
H(p, j, q) :=
(p− i)Xj+i+1 +
(q − i)Xj−i+1 +Xj+1
Hence Tj,n is continuous as a minimum of a finite number of continuous
r.v.’s. The T
j are also continuous because T
= Tk,2k.
16 V. V. VYSOTSKY
Now we prove the continuity of Tj . By Property 3, it only remains to
verify that P{Tj ≥ t} is continuous on [0,1). But P{T (k)j ≥ t} − P{Tj ≥ t}=
P{1{t≤Tj} 6= 1{t≤T (k)
}}, and in view of (13),
0≤t≤s
|P{T (k)j ≥ t}−P{Tj ≥ t}| ≤ sup
0≤t≤s
for every s < 1 = EXi. The last expression tends to zero by the strong law
of large numbers; then P{Tj ≥ t} is continuous on [0, s] as a uniform limit
of continuous functions P{T (k)j ≥ t}. Since s < 1 is arbitrary, P{Tj ≥ t} is
continuous on [0,1).
5. The cov(1{s≤T0},1{t≤Tk}) tends to zero as k→∞ for all s, t ∈ [0,1). If,
in addition, EX
i <∞ for some γ > 1, then for any s, t ∈ [0,1) and k ∈ N,
we have
|cov(1{s≤T0},1{t≤Tk})| ≤ 2
γ(ρ(s) + ρ(t))k1−γ .(21)
Proof. The idea is to approximate 1{s≤T0} and 1{t≤Tk} by 1{s≤T (k/2)0 }
1{t≤T (k/2)
}, respectively; here by k/2 we mean ⌈k/2⌉, where ⌈x⌉=min{m ∈
Z :m≥ x}. Note that 1{s≤T (k/2)0 }
and 1{t≤T (k/2)
} are independent because the
first is a function of {Xi}i≤k/2 while the second is a function of {Xi}i≥k/2+1.
We then have
|cov(1{s≤T0},1{t≤Tk})|
= |cov(1{s≤T0},1{t≤Tk})− cov(1{s≤T (k/2)
},1{t≤T (k/2)
≤ |E(1{s≤T0}1{t≤Tk} − 1{s≤T (k/2)
}1{t≤T (k/2)
+ |E(1{s≤T0} − 1{s≤T (k/2)
})|+ |E(1{t≤Tk} − 1{t≤T (k/2)
})|(22)
= P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
+ P{1{s≤T0} 6= 1{s≤T (k/2)0 }}+ P{1{t≤Tk} 6= 1{t≤T (k/2)k }
P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
≤ P{1{s≤T0} 6= 1{s≤T (k/2)
} ∪ 1{t≤Tk} 6= 1{t≤T (k/2)
therefore the result follows from Lemma 1.
6. The r.v.’s {Ti}i∈Z, {T (k)i }i∈Z, and {Ti,n}
i=1 are associated ; the author
owes this observation to M. A. Lifshits.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 17
Proof. Let us first recall the definition and some basic properties of as-
sociated variables. R.v.’s ξ1, . . . , ξm are associated if for any coordinate-wise
nondecreasing functions f, g :Rm →R, it is true that
cov(f(ξ1, . . . , ξm), g(ξ1, . . . , ξm))≥ 0
(assuming that the left-hand side is well defined). An infinite set of r.v.’s is
associated if any finite subset of its variables is associated.
The following sufficient conditions of association are well known; see [7].
(a) Independent variables are associated.
(b) Coordinate-wise nondecreasing functions (of finite number of argu-
ments) of associated r.v.’s are associated.
(c) If the variables ξ1,k, . . . , ξm,k are associated for every k and (ξ1,k, . . . ,
ξm,k)
D−→ (ξ1, . . . , ξm) as k→∞, then ξ1, . . . , ξm are associated.
(d) If two sets of associated variables are independent, then the union of
these sets is also associated.
Then {Ti,n}n−1i=1 are associated for every n by (a), (b) and (20). Analo-
gously, {T (k)i }i∈Z are associated for every k. Finally, since T
i → Ti a.s. as
k→∞ for every i, (c) ensures the association of {Ti}i∈Z.
7. For any s, t ∈R and k ∈ Z,
cov(1{T0≤s},1{Tk≤t})≥ 0.(23)
Proof. This inequality follows from cov(1{T0≤s},1{Tk≤t}) = cov(1{s<T0},
1{t<Tk}), the association of T0, Tk and (b).
8. If EX
i <∞ for some γ ≥ 2, then the stationary sequence min{Ti, t}
is strongly mixing for any t < 1 and its coefficients of strong mixing α(k)
satisfy α(k) = o(k2−γ).
Proof. Recall that stationary r.v.’s ξi are strongly mixing if α(k)→ 0 as
k→∞, where α(k) are the coefficients of strong mixing defined as
α(k) := sup
A∈F0−∞,B∈F
|P(AB)− P(A)P(B)|;
here F0−∞ := σ(ξ0, ξ−1, . . .) and F∞k := σ(ξk, ξk+1, . . .) are the σ-algebras of
“past” and “future,” respectively. It is readily seen that
α(k)≤ sup
0≤f,g≤1
| cov(f(ξ0, ξ−1, . . .), g(ξk, ξk+1, . . .))|,(24)
where the supremum is taken over Borel functions f, g :R∞ → [0,1].
Let us estimate α(k) in the same way we estimated the left-hand side
of (21). Fix some Borel functions f, g : R∞ → [0,1]. We approximate the
variables from the “past” T0∧ t, T−1∧ t, T−2∧ t, . . . by T (k/2)0 ∧ t, T
(k/2+1)
−1 ∧ t,
(k/2+2)
−2 ∧ t, . . . , respectively; and for the variables from the “future,” we
18 V. V. VYSOTSKY
use the analogous approximation. Now, f(T
(k/2)
0 ∧ t, T
(k/2+1)
−1 ∧ t, . . .) and
(k/2)
k ∧ t, T
(k/2+1)
k+1 ∧ t, . . .) are independent because the first is a function
of {Xi}i≤k/2 and the second is a function of {Xi}i≥k/2+1. We then argue in
the same way as in (22) to get
| cov(f(T0 ∧ t, T−1 ∧ t, . . .), g(Tk ∧ t, Tk+1 ∧ t, . . .))|
(T−i ∧ t) 6= (T (k/2+i)−i ∧ t)
(Tk+i ∧ t) 6= (T (k/2+i)k+i ∧ t)
i=k/2
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}.
Now, by the formula of total probability, we have
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}
= P{(T0 ∧ t) 6= (T (i)0 ∧ t), T
0 ≥ t}+ P{(T0 ∧ t) 6= (T
0 ∧ t), T
0 < t}
≤ P{1{t≤T0} 6= 1{t≤T (i)
}}+ P{T0 6= T
0 , T
0 ≤ t}
and combining all the estimates together, by Lemma 1 (24) and arbitrariness
of f and g, we get α(k)≤ 8
i=k/2 o(i
1−γ) = o(k2−γ) if γ > 2. For γ = 2, we
get α(k)≤ 16
i=k/2 P{inf i≥M Sii < t
2}= o(1) using the same argument and
applying (13), (14), and Fact 2 instead of Lemma 1.
3.5. The last collision. We finish this section with a statement on the
convergence of the moments of the last collision.
Proposition 3. In the i.d. model, T lastn
P−→ 1 as n→∞ if EX2i <∞.
This result is well known for the Poisson model; see Giraud [8].
Proof of Proposition 3. Let us first prove that P{T lastn ≥ t}→ 0 as
n→∞ for all t > 1. Since T lastn =max1≤j≤n−1Tj,n, we have
P{T lastn ≥ t} ≤
P{Tj,n ≥ t}.(25)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 19
By taking into account that the minima in (17) are nonpositive and by
arguing as in (18),
P{Tj,n ≥ t} ≤ P
1≤k≤j∨n−j
(k− i)(Xj+i+1 − t2) +Xj+1 − t2 ≥ 0
1≤k≤j∨n−j
(k− i+1)(Xi − t2)≥ 0
1≤k≤n/2
(Si − it2)≥ 0
We claim that (without any assumptions on the moments of Xi)
P{Tj,n ≥ t} ≤ P
i≥(t−1)/4tn
1 + t2
;(26)
recall that t > 1. Clearly, (26) follows if we check that
1≤k≤n/2
(Si − it2)≥ 0
i≥(t−1)/4tn
1 + t2
Assume the converse; then, by the nonnegativity of Si,
(Si − it2) =
(Si − it2) +
i=cn+1
(Si − it2)
(Scn − it2) +
i=cn+1
1 + t2
− it2
where c := t−1
. We estimate the last expression with
cnScn −
(cn)2
t2 − (n/2)
2 − (cn)2
2 − 1
n2 − 1/4− c
2 − 1
It is simple to check that the right-hand side is negative, thus we have a
contradiction.
Then from (25), (26) and Fact 2 it follows that P{T lastn ≥ t} =
i=1 o((cn)
−1) = o(1) for all t > 1.
Now let us prove that P{T lastn < t} → 0 as n → ∞ for all t < 1. Since
T lastn =max1≤j≤n−1Tj,n, we estimate
P{T lastn < t} ≤ P
n,n < t
P{1{t≤Tj√n,n} 6= 1{t≤T (
20 V. V. VYSOTSKY
In view of (10) and Lemma 1, the sum is
j=1 o(n
−1/2) = o(1), hence it
remains to check that the first probability in the last line tends to zero.
For a fixed n, all T
are independent because each one is a function of
{Xi}|j√n−i|≤√n/2 (to be precise, of Xj√n−√n/2+2, . . . ,Xj√n+√n/2). Thus,
n−1{T (
n/2)√
< t} ≤ P
n−1{T0 < t},
which tends to zero; indeed, P{T0 < t}< 1 by Property 3, Section 3.4. �
4. Proofs of Fact 1 and Theorem 1 for the i.d. model. Recall that the
number of clusters Kn(t) is given by (6). Our idea is to study
i=1 1{t<Ti}
instead of
i=1 1{t<Ti,n}: We thus deal with a single sequence Ti and avoid
considering the triangular array Ti,n.
Let us now prove Fact 1 for the i.d. model. We prove (1) for t 6= 1 without
any additional assumptions on Xi; for t = 1, we require EX
i < ∞. The
properties of the limit function a(t) were studied in Section 3.4, Properties
3 and 4.
Proof of Fact 1. We put
a(t) := P{T0 > t}.(27)
Let us first prove (1) for all t < 1. It is sufficient to check that
Kn(t)
1{t<Ti}
P−→ 0, n→∞.(28)
Indeed, the stationary sequence 1{t<Ti} satisfies the law of large numbers by
Property 5, Section 3.4, and the well-known result of S. N. Bernstein:
Fact 3. The law of large numbers holds for r.v.’s ξi if there exists a
sequence r(k)→ 0 such that cov(ξi, ξj)≤ r(|i− j|) for all i, j ∈N.
By (6),
Kn(t)
1{t<Ti}
(1{t<Ti,n} − 1{t<Ti}),
where we used (10) to get the nonnegativity of the right-hand side. Then
(28) immediately follows from the Chebyshev inequality provided that the
expectation of the right-hand side tends to zero. By using (10), we obtain
E(1{t<Ti,n} − 1{t<Ti})≤
(E1{t<T (i∧n−i)
} − E1{t<Ti})
P{1{t<Ti} 6= 1{t<T (i∧n−i)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 21
which is 2
i=1 o(1) = o(1) by Lemma 1. To be very precise, Lemma 1
deals with slightly different indicators, but we can estimate the considered
probability by repeating the proof of Lemma 1 word for word (or just use
Property 4, Section 3.4).
We now check that (1) holds for all t > 1. Using (26) gives E
Kn(t)
i=1 P{Ti,n > t} → 0 as n→∞ and
Kn(t)
P−→ a(t) = 0 follows from the
Chebyshev inequality.
It remains to check that (1) holds for t = 1 if EX2i < ∞ to conclude
the proof. If DXi = 0, then the situation is deterministic, this case was
described in Introduction. Here we always have Kn(1) = 1 and (1) is true.
If 0<DXi <∞, then by Property 3 from Section 3.4, we have a(1) = 0 and
P{T0 = 1} = 0; consequently, a(t) = P{T0 > t} is continuous at t= 1. Then
(1) is true for t = 1 since 0 <
Kn(1)
≤ Kn(t)
P−→ a(t) for any t ∈ (0,1) and
a(t)→ a(1) = 0 as tր 1. �
Now we prove Theorem 1 for the i.d. model. We think of D[0,1] as of a
separable metric space equipped with the Skorohod metric d, which induces
the Skorohod topology.
Proof of Theorem 1. At first, we prove (2). In view of representation
(6) for Kn(t), relation (2) follows from the relation
0≤t≤1−ε
1{t<Ti,n} −
1{t<Ti}
P−→ 0 for all ε ∈ (0,1)(29)
and the existence of a centered Gaussian process K(·) on [0,1) such that
1{t<Ti} −na(t)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(30)
Indeed, if Yn
D−→ Y and d(Yn, Y ′n)
P−→ 0 for some random elements Yn, Y ′n, Y
of the separable metric spaceD[0,1−ε], then Y ′n
D−→ Y ; recall that d(Yn, Y ′n)≤
supt∈[0,1−ε] |Yn(t)− Y ′n(t)|.
We start with (29). It is sufficient to prove that the expectation of the
left-hand side tends to zero. Since the supremum of a sum does not exceed
the sum of suprema, let us check that
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| −→ 0 for all ε ∈ (0,1).(31)
By (10), we have
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| ≤ E sup
0≤t≤1−ε
(1{t<T (i∧n−i)
} − 1{t<Ti})
22 V. V. VYSOTSKY
= P{Ti 6= T (i∧n−i)i , Ti ≤ 1− ε}
= P{Ti 6= T (i∧n−i)i , T
(i∧n−i)
i < 1− ε}
+ P{1{1−ε≤Ti} 6= 1{1−ε≤T (i∧n−i)
where the last equality was obtained via the formula of total probability.
Combining the estimates together and using Lemma 1,
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}|
≤ 2ρ(1− ε)√
(i ∧ n− i)1−γ = 4ρ(1− ε)√
i1−γ .
The last expression is O(n3/2−γ) and (31), which implies (29), follows.
Now let us prove (30). As long as
Un(t) :=−
1{t<Ti} − na(t)
1{Ti≤t} − (1− a(t))
the Un(·) is the empirical process of stationary r.v.’s Ti with the continuous
common distribution function 1− a(t). By K(·) D=−K(·), (30) is equivalent
to the existence of a centered Gaussian process K(·) on [0,1) such that
Un(·)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(32)
We will use the following result from Lin and Lu [17], Section 12 on
convergence of empirical processes. They attribute this statement to Q.-M.
Shao, who published it in 1986, in Chinese.
Fact 4. Let ξi be a sequence of stationary strongly mixing r.v.’s dis-
tributed on [0,1], and let F be the common distribution function of ξi.
Suppose F (x) = x on [0,1] (i.e., ξi are uniformly distributed) and the co-
efficients of strong mixing of the sequence F (ξi) decrease as O(k
−(2+δ))
as k → ∞ for some δ > 0. Then the empirical processes of ξi weakly con-
verge in D[0,1] to a centered Gaussian process with the covariance function
i∈Z cov(1{ξ0≤s},1{ξi≤t}).
Remark. The limit Gaussian process is a.s. continuous on [0,1]. Fact 4
also holds true if F is an arbitrary continuous distribution function.
The a.s. continuity of the limit process could be concluded by a compar-
ison of the proof from Lin and Lu [17] with the proof of Theorem 22.1 from
Billingsley [3]. The statements and the proofs of these theorems are identical,
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 23
but Lin and Lu do not state the continuity while Billingsley does. Further,
since F (ξi) is uniformly distributed on [0,1] if F is continuous, Fact 4 holds
true for every continuous F ; see the proof of Theorem 22.1 by Billingsley [3]
for explanations.
Recall that we need to prove the convergence of the empirical process of
Ti. It seems that the r.v.’s Ti are not strongly mixing; but min{Ti,1− ε} are
strongly mixing because of Property 8, Section 3.4. These variables are not
continuous and so we need to fix them. Let us fix an ε ∈ (0,1), and let αi
be i.i.d. r.v.’s independent of all Ti and, say, uniformly distributed on [0, ε];
we define T̃i := min{Ti,1− ε}+ 1{Ti≥1−ε}αi.
The stationary variables T̃i are distributed on [0,1], their common dis-
tribution function G is continuous, and the coefficients of strong mixing
of G(T̃i) decrease as o(k
2−γ). The proof of the last statement is the same
as the proof of Property 8 from Section 3.4. Indeed, approximate the vari-
ables G(T̃0),G(T̃−1), . . . from the “past” by G(T̃
(k/2)
0 ),G(T̃
(k/2+1)
−1 ), . . . where
i := min{T
i ,1− ε}+ 1{T (m)
≥1−ε}αi; use the analogous approximation
for the variables from the “future”; and then repeat word for word the ar-
guments of the previous proof.
Now, recalling that γ > 4, we see that T̃i satisfy the assumptions of Fact 4,
with the only difference that their distribution is not uniform. By Ũn(·)
denote the empirical process of T̃i; clearly, Ũn(·) coincides with the empirical
process Un(·) of Ti on [0,1− ε]. By the remark to Fact 4, we conclude that
first,
Ũn(·)
D−→ K̃(·) in D[0,1],(33)
where K̃(·) is a centered Gaussian process with the covariance function
R̃(s, t) :=
cov(1{T̃0≤s},1{T̃i≤t})
and, second, trajectories of K̃(·) are a.s. continuous on [0,1].
[There exists a simpler and more elegant proof of (33). Note that {T̃i}i∈Z
are associated as coordinate-wise nondecreasing functions of associated r.v.’s
{Ti, αi}i∈Z, see (a), (b) and (d) from Property 6, Section 3.4. Then we can
obtain (33) applying the result of Louhichi [18] on convergence of empirical
processes of stationary associated r.v.’s ξi instead of using Fact 4. This the-
orem requires only cov(F (ξ0), F (ξk)) = O(k
−(4+δ)), which could be proved
analogously to Property 5, Section 3.4. Thus we avoid the complicated esti-
mations of the strong mixing coefficients, and the proof of (33) is becomes
much simpler. The only problem is that this proof requires γ > 5.
We also note that the a.s. continuity of K̃(·) could be proved directly,
without referring to the proof of Fact 4. The arguments should be the same
as in the proof of the continuity of KUnif(·) in Section 5.]
24 V. V. VYSOTSKY
Define
R(s, t) :=
cov(1{T0≤s},1{Ti≤t}),(34)
which is, evidently, equal to R̃(s, t) on [0,1 − ε]2. Since R̃(s, t) is positive
definite and ε > 0 is arbitrary, the function R(s, t) is positive definite on
[0,1)2. Therefore, by Lifshits [15], Section 4, there exists a centered Gaussian
process K(·) on [0,1) with the covariance function R(s, t). The trajectories
of K(·) are a.s. continuous on [0,1) by K(·) D= K̃(·) on [0,1−ε], arbitrariness
of ε > 0, and the a.s. continuity of K̃(·) on [0,1].
Finally, by (33), Ũn(·) = Un(·) on [0,1− ε], K̃(·) D=K(·) on [0,1− ε], and
the a.s. continuity of K̃(·), we get (32). Since (32) implies (30), we conclude
the proof of (2).
Only the stated properties of R(s, t) remain to be proven. The continuity
of the joint distribution function of continuous variables T0 and Ti implies
that cov(1{T0≤s},1{Ti≤t}) is continuous on [0,1)
2 for every i ≥ 0. Then, in
view of (21), R(s, t) is continuous on [0,1)2 as a sum of uniformly converging
series of continuous functions.
The strict positivity of R(s, t) on (
µ,1)2 trivially follows from (34), (23)
and cov(1{T0≤s},1{T0≤t}) = a(s∨ t)(1−a(s∧ t)) > 0; the last inequality holds
by Property 3, Section 3.4. The R(s, t) = 0 on [0,1)2 \ (õ,1)2 follows from
P{Ti ≤
µ}= 0, which holds by Properties 3 and 4 from Section 3.4. �
We note that (3) holds for t 6= 1 under the less restrictive condition EX2i <
∞. For t < 1, the proof is almost the same: By (29), which is true for γ > 3/2,
we conclude that (3) holds if the stationary associated sequence 1{t<Ti}
satisfies the central limit theorem. Then we refer to the central limit theorem
for stationary associated sequences from Newman [21]; his theorem requires
only R(t, t)<∞, that is, the convergence of the right-hand side of (34). This
condition holds by (13) and Fact 2. For t > 1, relation (3) holds true with
σ2(t) = 0 because of Proposition 3.
Finally, note that the process K(·) is associated, that is, the r.v.’s
{K(t)}t∈[0,1) are associated. In fact, by (6), Property 6 from Section 3.4,
and Condition (b) from the same Property 6, the processes
Kn(·)−na(·)√
associated for every n. Then K(·) is associated by (2) and (c), Property 6.
5. Proof of Theorem 1 for the uniform model. There exists a simple
method that allows to extend results from the Poisson model to the uniform
model and vise versa. The method is based on the next statement (see
Karlin [13], Section 9.1).
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 25
Fact 5. Let Si be an exponential random walk. Then for any k ≥ 1, we
, . . . ,
= (U1,k,U2,k, . . . ,Uk,k),(35)
where Ui,k are the order statistics of k i.i.d. r.v.’s uniformly distributed on
[0,1]. Moreover, the random vector in the left-hand side of (35) is indepen-
dent of Sk+1.
Therefore, if xPoissj,n (0) =
Sj are the initial positions of particles in the
Poisson model, then for the initial positions of particles in the uniformmodel,
we have xUnifj,n (0) =
· xPoissj,n (0). By Proposition 2 and (5), we conclude
TUnifj,n = β
Poiss
j,n , βn :=
,(36)
and hence, using (6), we get
KUnifn (t) =K
Poiss
n (βnt).(37)
Note that the process KUnifn (·) and the r.v. βn are independent since val-
ues of the process are defined by xUnif1,n (0), . . . , x
n,n (0), which are mutually
independent of βn by Fact 5.
Now we prove Theorem 1 for the uniform model.
Proof of Theorem 1. Denote
Yn(t) :=
KUnifn (t)− na(t)√
, Zn(t) :=
n(a(t)− a(βnt));
we stress that Yn(·) and Zn(·) are independent.
Fix an ε ∈ (0,1). First, it follows from (2) for the Poisson model and (37)
Yn(·) +Zn(·)
D−→KPoiss(·) in D[0,1− ε].(38)
Indeed, the process Yn(·) +Zn(·) is obtained from 1√n(K
Poiss
n (·)− na(·)) by
the random time change t 7→ βnt; and since ‖βnt− t‖C[0,1−ε]
P−→ 0, we have
Yn(·) +Zn(·),
KPoissn (·)− na(·)√
P−→ 0
by the definition of the Skorohod metric d.
Second, from Fact 1, (15), and (27) it follows that aUnif(t) = aPoiss(t) =
P{TPoiss0 ≥ t}= 1− t2 for 0≤ t≤ 1, and by the central limit theorem,
Zn(t)
D−→ t2η in D[0,1− ε],(39)
26 V. V. VYSOTSKY
where η is a standard Gaussian r.v.
We claim that (38), the independence of Yn(·) and Zn(·), and (39) yield
the weak convergence of Yn(·) in D[0,1− ε]. Let us check the tightness of
Yn(·) and the convergence of their finite-dimensional distributions.
The tightness of Yn(·) in D[0,1− ε] follows from Yn(·) = (Yn(·)+Zn(·))−
Zn(·), (38), and (39). Indeed, by the Prokhorov theorem, (38) and (39) yield
that both sequences Yn(·) + Zn(·) and −Zn(·) are tight. But trajectories
of −Zn(·) are a.s. continuous because of the continuity of a(·), and the
tightness follows from the continuity of addition + :D×C →D and the fact
that under any continuous mapping, the image of a compact set is also a
compact set.
Now we study convergence of finite dimensional distributions of Yn(·).
Recall that the characteristic function of a centered Gaussian vector in Rm
is e−1/2(Ru,u), where u ∈ Rm and R is the covariance matrix of the vector.
Then (38), the independence of Yn(·) and Zn(·), and (39) yield that for the
characteristic functions of all finite-dimensional distributions of Yn(·), we
Eei(Yn(t),u) −→ e−1/2({R
Poiss(tj ,tk)−t2j t
j,k=1
,(40)
where u ∈Rm, t= (t1, . . . , tm) ∈ [0,1−ε]m, and Yn(t) := (Yn(t1), . . . , Yn(tm)).
We stress that (40) is true for every t ∈ [0,1− ε]m since the limit processes
in (38) and (39) have continuous trajectories.
We see that the matrix {RPoiss(tj , tk)− t2j t2k}mj,k=1 is positive definite for
any t = (t1, . . . , tm) ∈ [0,1− ε]m and m≥ 1 since the absolute value of the
left-hand side of (40) does not exceed one. Putting
RUnif(s, t) :=RPoiss(s, t)− s2t2,
we have {RPoiss(tj , tk)− t2j t2k}mj,k=1 = {RUnif(tj , tk)}mj,k=1; then the function
RUnif(s, t) is positive definite on [0,1)2 since ε > 0 is arbitrary. Thus, by
Lifshits [15], Section 4, RUnif(s, t) is the covariance function of some centered
Gaussian process KUnif(·) on [0,1).
Relation (2) is thus proved. Now check that KUnif(·) ∈C[0,1− ε] a.s. to
conclude the proof of Theorem 1 for the uniform model.
For this purpose, let us prove that a.s., trajectories of Yn(·) have jumps
of size 1√
only. In fact, the jumps of Yn(·) coincide with the jumps of
KUnifn (·), whose jumps are of size 1√n if and only if T
6= TUnifj2,n for
1 ≤ j1 6= j2 ≤ n − 1. By (36), we need to verify that TPoissj1,n 6= T
Poiss
for 1 ≤ j1 6= j2 ≤ n − 1. This relation follows from (20) if H(k1, j1, l1) 6=
H(k2, j2, l2) a.s. for j1 6= j2 and k1, k2, l1, l2 ≥ 1. The last a.s. nonequality is
obvious because if the equality holds true, then a certain nontrivial linear
combination of i.i.d. exponential Xi equals zero.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 27
Then there exist a.s. continuous Ỹn(·) such that supt∈[0,1−ε] |Ỹn(t)−Yn(t)| ≤
a.s.; consequently, d(Ỹn, Yn) ≤ 1√n a.s. Then by Yn(·)
D−→ KUnif(·), we
also have Ỹn(·)
D−→KUnif(·). But 1 = lim inf P{Ỹn(·) ∈C} ≤ P{KUnif(·) ∈C}
since C ⊂D is closed in the Skorohod topology, therefore, a.s., KUnif(·) is
continuous on [0,1− ε].
Since ε ∈ (0,1) is arbitrary, a.s., KUnif(·) is continuous on the whole inter-
val [0,1). The RUnif(s, t) =RPoiss(s, t)− s2t2 is continuous on [0,1)2 because
RPoiss(s, t) is. �
6. The number of clusters at the critical moment. Now we turn our
attention to the number of clusters at the critical moment t = 1. We are
interested in the behavior of
Kn(1)− na(1)√
Kn(1)√
which is the left-hand side of (3) at t = 1; here we have a(1) = 0 under
EX2i <∞, see Property 3, Section 3.4.
We do not know if this sequence is weakly convergent, but we hope that
it is. We also have a naive guess that its limit is Gaussian because the limit
in Theorem 1 is Gaussian. In view of Kn(1)≥ 1, this conjectured weak limit
is nonnegative, hence it is Gaussian if and only if it is identically equal to
zero. However, the results of this section show that the limit is nonzero, thus
our guess on Gaussianity fails.
The study of convergence of
Kn(1)√
is quite complicated. Therefore, in this
section, we consider only the Poisson model. First, let us prove the following
statement.
Proposition 4. In the Poisson model, we have limn→∞P{Kn(1) =
1}> 0.
Proof. On the one hand, Kn(1) = 1 is equivalent to T
n;Poiss ≤ 1, where
T lastn;Poiss denotes the moment of the last collision in the Poisson model. On
the other hand, a result by Giraud [8] states that in the uniform model,
n(T lastn;Unif − 1)
D−→ sup
0≤x≤1
W (y)dy −
W (y)dy
=: τ,
where
W (·) is a Brownian bridge. Now, by (36), we have T lastn;Unif = β−1n T lastn;Poiss,
hence
n(β−1n T
n;Poiss − 1)
D−→ τ.(41)
28 V. V. VYSOTSKY
But from the central limit theorem and the law of large numbers,
n(β−1n − 1) =−
Sn+1 − n√
Sn+1(
Sn+1 +
D−→ η
,(42)
where η is a standard Gaussian r.v. and Si is a standard exponential ran-
dom walk that defines initial positions of particles. Since, in view of Fact 5,
T lastn;Unif = β
n;Poiss and βn are independent, from (41), (42), and the law
of large numbers it follows that
n(T lastn;Poiss − 1)
D−→ τ − η
= τ +
where τ and η are independent. Thus,
P{Kn(1) = 1}= lim
P{T lastn;Poiss ≤ 1}= P
The main advantage of the Poisson model is that, by Lemma 2 and Prop-
erty 4, Section 3.4 we have P{Tj,n > 1}= epjpn−j , where
pk := P
1≤m≤k
(Si − ESi)≥ 0
and Si is a standard exponential random walk. We say that the sequence of
r.v.’s
i=1(Si−ESi) is an integrated random walk. In the proof of Property
3, Section 3.4, we showed that pk → 0 as k→∞. Therefore, it is reasonable
to say that pk are the unilateral small deviation probabilities of an integrated
centered random walk.
We need to obtain the asymptotics of pk → 0 to continue the study of
convergence of
Kn(1)√
. Unfortunately, the results of the rest of this section
are completely dependent on the correctness of the following conjecture.
Conjecture 1. We have pk ∼ c1k−1/4 as k→∞ for some c1 ∈ (0,∞).
Simulations show that the conjecture is true and c1 ≈ 0.36. The weaker
form pk ≍ k−1/4 of Conjecture 1 was proved by Sinai [22], but only for inte-
grated symmetric Bernoulli random walks. It also interesting to note that,
by McKean [19], the unilateral small deviation probabilities of an integrated
Wiener process have the same order as T →∞:
0≤s≤T
W (u)du≥−1
∼ c2T−1/4(43)
for some c2 ∈ (0,∞). The left-hand side of (43) is a unilateral small deviation
probability since
0≤s≤T
W (u)du≥−1
0≤s≤1
W (u)du≥−T−3/2
.(44)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 29
To be precise, McKean was interested in a more general problem, and
some calculations are required to obtain (43) from his results. Therefore,
we additionally refer to Isozaki and Watanabe [12] who state (43) explic-
itly.
By the results mentioned above, we also suppose that Conjecture 1 is
true for other integrated centered random walks that satisfy some moment
conditions.
Now we are able to prove the following result on convergence of
Kn(1)√
Proposition 5. Suppose Conjecture 1 holds true. Then in the Poisson
model, we have
Kn(1)√
= c3, sup
Kn(1)√
<∞(45)
for some c3 ∈ (0,∞); the sequence Kn(1)√n is tight and uniformly integrable;
and the limit of any weakly converging subsequence of
Kn(1)√
takes value zero
with positive probability, but is not identically equal to zero.
Numerical simulations show that
Kn(1)√
is weakly convergent and that this
convergence is quite fast. In Figure 1 we present the (empirical) distribution
function of
Kn(1)√
for n = 10,000. Since the simulations performed for n =
40,000 showed a hardly perceptible difference, this function seems to be a
good candidate for the distribution function of the conjectured limit.
Note that if we weaken Conjecture 1 to pk ≍ k−1/4, then Proposition 5
still holds true with the only difference that E
Kn(1)√
Proof of Proposition 5. We start with the convergence of the ex-
pectation. On the one hand, by (6) and Lemma 2,
Kn(1)√
pipn−i,
and on the other hand,
i−1/4(n− i)−1/4 = 1
)−1/4(
)−1/4
−→B(3/4,3/4)
as the integral sum of Beta function. Then it follows from Conjecture 1 and
standard arguments that E
Kn(1)√
converges to c3 := ec
1B(3/4,3/4) > 0.
30 V. V. VYSOTSKY
Fig. 1. The distribution function of
Kn(1)√
for n= 10,000.
Now we check the uniform boundedness of E(
Kn(1)√
)2. By (6) it is sufficient
to prove that
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1}<∞.(46)
Suppose i < j; then by using (8) and properties of Fk,j,l(·), we get
P{Ti,n > 1, Tj,n > 1}= P
1≤k≤n−i
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤j
Fk,j,l(1)> 0
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
where by (j − i)/2 we mean ⌈(j − i)/2⌉. The minima in the last expres-
sion are independent as functions of {Xm}m≤(i+j)/2 and {Xm}m≥(i+j)/2+1,
respectively; hence
P{Ti,n > 1, Tj,n > 1} ≤ P
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 31
= P{Ti,i+(j−i)/2 > 1} · P{T(j−i)/2,n−j+(j−i)/2 > 1}
= e2pip
⌈(j−i)/2⌉pn−j,
where the first equality follows from (8) and the second follows from Lemma 2.
Recalling Conjecture 1, we get
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1} ≤
i,j=1,i 6=j
e2pip
⌈|j−i|/2⌉pn−j
i,j=1,i 6=j
i−1/4⌈|j − i|/2⌉−1/2(n− j)−1/4
i,j=1,i 6=j
)−1/4∣
−1/2(
)−1/4
for some c > 0. The last expression is an integral sum converging to
x−1/4|x− y|−1/2(1− y)−1/4 dxdy,
and it is a simple exercise to check that the integral is finite. This concludes
(46).
The uniform integrability of
Kn(1)√
follows from the second relation from
(45), see Billingsley [3], Section 5, and the tightness follows from the uniform
integrability.
Finally, suppose
Kni(1)√
D−→ ξ for some subsequence ni → ∞ and some
r.v. ξ. Then Eξ = c3 > 0 by the uniform integrability and (45), and hence
ξ is not identically equal to zero. But the distribution of ξ has an atom at
zero since by Proposition 4 and properties of weak convergence,
P{ξ = 0}= lim
P{ξ ≤ ε}
≥ lim
lim sup
Kni(1)√
≥ lim
P{Kni(1) = 1}> 0. �
7. Open questions. 1. The number of clusters at the critical moment
t= 1.
Here the main question is if Conjecture 1 holds true. Even by itself, this
problem is worth studying.
But even if Conjecture 1 is true, we still do not have a proof of weak
convergence of
Kn(1)√
, it is only known that this sequence is tight. The author
32 V. V. VYSOTSKY
strongly believes, relying on numerical simulations, that the limit exists. It
would be interesting to find this conjectured limit, which should be nontrivial
by Proposition 5, in an explicit form.
2. The weak convergence of
Kn(·)−na(·)√
on the whole interval [0,1].
It is very natural to ask if it is possible to strengthen Theorem 1 by
proving the weak convergence of
Kn(·)−na(·)√
in D[0,1]. This complicated
problem returns us again to Question 1 because the weak convergence of
Kn(·)−na(·)√
in D[0,1] implies the weak convergence of
Kn(1)−na(1)√
Kn(1)√
, see
Billingsley [3], Section 15. But even if
Kn(1)√
converges, its weak limit K(1)
is not Gaussian, hence the limit process K(·), which is Gaussian on [0,1),
is no more Gaussian on [0,1]. Therefore, it is doubtful that Theorem 1 is
true in D[0,1]; at least, one should provide a proof completely different from
the presented one. Also, it is unclear how to define the finite-dimensional
distributions of the non-Gaussian K(·) on [0,1] because simulations show
that K(1) would not be independent with K(t) for t < 1.
3. The number of clusters in the warm gas.
In the presented case, initial speeds of particles are zero. This model is of-
ten called the cold gas according to its zero initial temperature. We introduce
a new model stating that initial speeds of particles are anv1, anv2, . . . , anvn,
where vi are some i.i.d. r.v.’s and an is a sequence of normalization con-
stants. This model, called the warm gas, was studied in many papers, for
example, [14, 16, 20, 25].
It is of a great interest to study the behavior of Kn(t) in the warm gas.
In [25], the author proved that in the basic case where an = 1 for all n and
Ev2i <∞, we have
Kn(t)
P−→ 0 for all t > 0. The question is to find a normal-
ization of Kn(t) leading to some nontrivial limit. Clearly, this normalization
depends on an, but it is very possible that there is an effect of phase tran-
sition similar to the one discovered by Lifshits and Shi [16]: If an are small
enough, then the gas has a low temperature and the normalization is the
same as in the cold gas. If an are big enough, as in the basic case an ≡ 1,
then the normalization and the behavior of the gas differ entirely from the
case of the cold gas.
The author believes that the localization property, which is described in
Section 3, could be helpful in a study of these questions.
It is also interesting to compare the behavior of Kn(1) in the warm and in
the cold gases; in the warm gas, the moment t= 1 plays the same “critical”
role as in the cold gas, see Lifshits and Shi [16]. The variable Kn(1) was
studied by Suidan [24], who considered the warm gas with an ≡ 1 and deter-
ministic initial positions of particles (his initial positions were 1
, . . . , n
For this case, Suidan found the distribution of Kn(1) and showed that
EKn(1)∼ logn. Recall that in the presented case, EKn(1)∼ c3
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 33
4. The number of clusters in ballistic systems of sticky particles.
A sticky particles model is called ballistic if it evolves according to the
laws introduced in Section 1, but in the absence of gravitation. Such models
are, in some sense, more natural than gravitational ones because the ba-
sic assumption that gravitation does not depend on distance is sometimes
confusing. However, an unpublished paper of Lifshits and Kuoza shows that
certain gravitational and ballistic models are tightly connected.
It seems interesting to study the number of clusters in the ballistic model.
The author does not know any results in this field.
Acknowledgments. I am grateful to my adviser Mikhail A. Lifshits for
drawing my attention into the subject and for his guidance. I also thank the
anonymous referees for carefully reading this paper and useful comments.
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Department of Probability Theory
and Mathematical Statistics
Faculty of Mathematics and Mechanics
St. Petersburg State University
Bibliotechnaya pl. 2
Stary Peterhof 198504
Russia
E-mail: [email protected]
http://www.ams.org/mathscinet-getitem?mr=2092206
http://www.ams.org/mathscinet-getitem?mr=1472736
http://www.ams.org/mathscinet-getitem?mr=2118859
http://www.ams.org/mathscinet-getitem?mr=1486580
http://www.ams.org/mathscinet-getitem?mr=1792655
http://www.ams.org/mathscinet-getitem?mr=0156389
http://www.ams.org/mathscinet-getitem?mr=1400187
http://www.ams.org/mathscinet-getitem?mr=0576267
http://www.ams.org/mathscinet-getitem?mr=1182301
http://www.ams.org/mathscinet-getitem?mr=0989562
http://www.ams.org/mathscinet-getitem?mr=1861441
http://www.ams.org/mathscinet-getitem?mr=2221711
mailto:[email protected]
Introduction
Description of the model
Statement of the problem and the results
Organization of the paper
Method of barycenters
Study of the i.d. model. The localization property
The initial study
Localization property of the aggregation process
The distribution function of T0 in the Poisson model
Some properties of the variables Ti
The last collision
Proofs of Fact 1 and Theorem 1 for the i.d. model
Proof of Theorem 1 for the uniform model
The number of clusters at the critical moment
Open questions
Acknowledgments
References
Author's addresses
|
0704.0087 | Approximate solutions to the Dirichlet problem for harmonic maps between
hyperbolic spaces | APPROXIMATE SOLUTIONS TO THE DIRICHLET PROBLEM
FOR HARMONIC MAPS BETWEEN HYPERBOLIC SPACES
DUONG MINH DUC AND TRUONG TRUNG TUYEN
Abstract. Our main result in this paper is the following: Given Hm, Hn
hyperbolic spaces of dimensional m ≥ 2 and n corresponding, and given a
Holder function f = (f1, ..., fn−1) : ∂Hm → ∂Hn between geometric bound-
aries of Hm and Hn. Then for each ǫ > 0 there exists a harmonic map
u : Hm → Hn which is continuous up to the boundary (in the sense of Eu-
clidean) and u|∂Hm = (f
1, ..., fn−1, ǫ).
1. Introduction
Let Hm and Hn are hyperbolic spaces with dimensions m ≥ 2 and n correspond-
ingly. For convenience, we use the upper-half space models for Hm and Hn. So
Hm = {(x1, ..., xm) ∈ IRm : xm > 0}, Hn = {(y1, ..., yn) ∈ IRn : yn > 0} with
metrics
d2Hm =
(xm)2
((dx1)2 + ...+ (dxm)2),
d2Hn =
(yn)2
((dy1)2 + ...+ (dyn)2).
So the tension fields of u = (y1, ..., yn) is
τα = (xm)2(∆0y
< ∇0y
α,∇0y
n >),
for 1 ≤ α ≤ n− 1 and
τn(u) = (xm)2(∆0y
α|2 − |∇0y
n|2)),
where ∇0 is the Euclidean gradient and ∆0 is the Euclidean Laplacian.
A C2 map u : Hm → Hn is called a harmonic map if τ(u)s = 0 for all s =
1, 2, ..., n. The literature about harmonic maps between Riemannian manifolds are
abundant, we refer the readers to the classical work [4].
One of the interesting problems for harmonic maps is that of the Dirichlet prob-
lem at infinity: Given ∂Hm and ∂Hn geometric boundaries of Hm and Hn, and
given a continuous map f : ∂M → ∂N (here continuity is understood in the sense
Date: October 22, 2018.
2000 Mathematics Subject Classification. 53A35.
Key words and phrases. Dirichlet problems; Harmonic functions; Hyperbolic spaces.
This work has been initiated when the second author was at Department of mathematics,
University of natural sciences, Hochiminh city, Vietnam. He would like to thank Professor Dang
Duc Trong for his many invaluable helps. He also would like to express his thankfulness to
Professor F. Helein, Professor R. Schoen, and Mr. Le Quang Nam for their generous help.
http://arxiv.org/abs/0704.0087v2
2 DUONG MINH DUC AND TRUONG TRUNG TUYEN
of Euclidean), is there a harmonic map u : Hm → Hn such that in Euclidean sense
u is continuous up to the boundary ∂Hm and takes boundary value f?
For this problem with some more requirements for the smoothness of f , there are
many results. In three papers [8], [9] and [7], Li and Tam established the existence
and uniqueness of a harmonic function u which is C1 up to the boundary and has
boundary value f , provided f is C1. But for more general types of f , according to
our knowledge, there is no answer to the existence of a solution u.
In this paper we establish the existence of approximate solutions to the Dirichlet
problem for harmonic maps between two hyperbolic spaces with prescribed bound-
ary value. More explicitly, we prove the following result
Theorem 1. Let f : Hm → Hn be a bounded uniformly continuous. Let functions
g and ϕ be as in Section 2. Assume that
t−1g(t)dt < ∞, in particular, this
condition is satisfied if f is Holder continuous. For each ǫ > 0, there exists a
harmonic map uǫ : H
m → Hn which is continuous up to the boundary ∂Hm and
u|∂Hm = (f
1, ..., fn−1, ǫ).
Our strategy for proving this result is the follows: First, we construct an initial
map, i.e., a C2 map v = (v1, ..., vn−1, vn) : Hm → Hn which has boundary value
f for any continuous map f : ∂Hm → Hn. For this step we follows the ideas in
[9], with some changes: Since the function f needs not to differentiable, we can
not take vn as in [9], and the function vn of ours is a function of one variable xm.
Then, we use this function to produce harmonic maps uǫ : H
m → Hn which takes
boundary value (f1, ..., fn−1, vn + ǫ) for every ǫ > 0.
2. Initial maps
In this part, we use the techniques in [9] to construct good initial maps v having
the map f : ∂Hm → ∂Hn as the boundary value.
Let f : IRm−1 → IRn−1 be a uniformly continuous bounded function. Let
g : Hm → (0.∞) be C2, bounded and
g(x′, xm) = 0,
uniformly in x′.
We denote by v = {f, g} : Hm → Hn the extension of f defined as follows
vα(x′, xm) =
xmfα(y′)
(|x′ − y′|2 + (xm)2)m/2
for 1 ≤ α ≤ n− 1 and
vn(x′, xm) = g(x′, xm).
By results in [9] (pp. 628-630) we have
(i) v is C2 and up to the boundary given by xm = 0 it is continuous.
(ii) If 1 ≤ α ≤ n− 1 then
xm|∇0v
α| = 0,
uniformly in x′.
Moreover, by estimates of elliptic PDEs (see Theorem 2.10 in [5]), noting that
vα is bounded, there exists constants C > 0 such that
(2.1) max{(xm)3|D3vα|, (xm)2|D2vα|, (xm)|∇0v
α|} ≤ C.
APPROXIMATE SOLUTIONS TO THE DIRICHLET PROBLEM FOR HARMONIC MAPS BETWEEN HYPERBOLIC SPACES3
We put
g(r) = sup
x′,y′∈IRm−1, |x′−y′|≤r
|f(y′)− f(x′)|,
ϕ(r) =
s2 + r2
g(s)ds.
Since g is monotone it follows that g is Lebesgue measurable. Moreover, since g is
bounded, we see that ϕ is well-defined.
Using polar coordinates with center at x′ we see that there exists a constant
C > 0 such that ∫
IRm−1
xm|f(y′)− f(x′)|
(|x′ − y′|2 + (xm)2)m/2
≤ Cϕ(xm),
for all x′ ∈ IRm−1.
Since f is uniformly continuous we see that
g(r) = 0.
Now we show that
ϕ(xm) = 0.
Indeed, for any ǫ > 0, we find δ > 0 such that
g(s) ≤ ǫ,
if 0 < s ≤ δ. So, if K = sup
s∈IR g(s) we have
ϕ(r) =
s2 + r2
g(s)ds+
s2 + r2
g(s)ds
s2 + r2
s2 + r2
= ǫ arctan(δ/r) +K(π/2− arctan(δ/r)).
Letting r → 0 we see that
lim sup
ϕ(r) ≤ ǫπ/2.
Since ǫ > 0 is arbitrary, we see that
ϕ(r) = 0.
Thus, if we put v = {f, ϕ(xm)} we see that v is an extension of f . Moreover we
have the following result
Lemma 1. Let f : ∂Hm → ∂Hn be nonconstant, uniformly continous and
bounded. Put v = {f, ϕ(xm)} as above. Then v is smooth, up to the boundary
it is continuous, v|IRm−1 = f and there exists C > 0 such that for x
m near 0 we
||τ(v)||2 ≤ C.
4 DUONG MINH DUC AND TRUONG TRUNG TUYEN
Proof. By Section 6 in [9] we have
|(xm)∇0v
α| ≤ C3|ϕ(x
where 1 ≤ α ≤ n− 1 and C3 is a positive constant.
Directly computation gives
ϕ′(r) =
s2 − r2
(s2 + r2)2
g(s)ds,
ϕ”(r) =
(s2 + r2)2
g(s)ds+
−4r(s2 − r2)
(s2 + r2)3
g(s)ds.
max{|rϕ′(r)|, |r2ϕ”(r)|} ≤ C4ϕ(r),
where C4 is a constant.
Since g is increasing, g′ exists almost everywhere and g′ ≥ 0. Using integration
by parts, noting that d
s2+r2
) = s
(s2+r2)2
, we have
ϕ′(r) =
s2 − r2
(s2 + r2)2
g(s)ds
s2 + r2
g(s)|∞0 +
s2 + r2
g′(s)ds
s2 + r2
g′(s)ds.
Differentiating the last term in above equality we get
ϕ”(r) = −
(s2 + r2)2
g′(s)ds.
Since f is nonconstant we see easily that g′ 6≡ 0 (in fact, we don’t need this
restriction since we can add g with a non-constant positive function, for example
(xm)1/2 ). So since g′ ≥ 0, it follows from above equalities that
ϕ′(r) > 0,
|rϕ”(r)| ≤ C5ϕ
′(r),
where C5 is a positive constant. Then use the formula for the tension field we are
done. �
3. Proof of Theorem 1
Proof. Fixed ǫ > 0. We define vǫ : H
m → Hn as follows:
vǫ(x) = (v
1(x), v2(x), ..., vn−1(x), ϕ(xm) + ǫ).
For each δ > 0 denote uǫ,δ : H
m k Ωδ = {x
m > δ} → Hn the harmonic map
taking value vǫ on ∂Ωδ.
By inequality (2.1) in [2] and properties of v and vǫ (see Lemma 1) we have
∆HmdHn(uǫ,δ, vǫ) ≥ −|τ(vǫ)| ≥ −C
ϕ(xm)
ϕ(xm) + ǫ
ϕ(xm),
for all x ∈ Ωδ, and here C is one constant from Lemma 1.
APPROXIMATE SOLUTIONS TO THE DIRICHLET PROBLEM FOR HARMONIC MAPS BETWEEN HYPERBOLIC SPACES5
We claim that the function
ψ(r) =
u−2ϕ(u) du ds
is well-defined for r ≥ 0. In fact, using the formula for ϕ we have
ψ(r) =
u−2ϕ du ds =
u−1(u2 + t2)−1g(t)dt du ds.
Since the integrand is non-negative, using Fubini’s theorem we have∫ r
u−1(u2 + t2)−1g(t)dt du ds =
u−1(u2 + t2)−1g(t)du dt ds
log(1 +
)g(t)dt ds
log(1 +
)g(t)ds dt
πt2 − 2 arctan( t
)t2 + r log(1 + t
g(t)dt.
Now since g(t) is bounded we have
πt2 − 2 arctan( t
g(t)dt
is convergent. Fixed r ≥ 0, near t = 0 we have
r log(1 + t
g(t) ≈ t−1g(t),
and when t→ ∞ we have
r log(1 + t
g(t) ≈ t−3g(t),
hence since g(t) is bounded and the assumption that
t−1g(t) converges, our claim
is verified.
We use the same ψ to denote the function ψ : Hm → IR defined by ψ(x) = ψ(xm)
for x = (x1, ..., xm−1, xm) ∈ Hm. Now we have ψ′(r) =
u−2ϕ du > 0 and
ψ”(r) = −r−2ϕ(r), since m ≥ 2 we have
∆Hm(−ψ(x)) = −(x
m)2[ψ”(xm)−
(m− 2)
ψ′(xm)] ≥ −(xm)2ψ”(xm) = ϕ(r).
Hence
∆Hm (dHn(uǫ,δ, vǫ)− C
ψ) ≥ 0,
for x ∈ Ωδ. Hence by maximum principle we have
dHn(uǫ,δ, vǫ) ≤ C
ψ(xm).
This bound for dHn(uǫ,δ, vǫ) is independent of δ, hence by standard arguments (see
the proof of Theorem 6.4 in [9]) we have a harmonic map uǫ : H
m → Hn which is
the subsequent limit of uǫ,δ. Moreover for all x ∈ H
m we have
dHn(uǫ, vǫ) ≤ C
ψ(xm).
6 DUONG MINH DUC AND TRUONG TRUNG TUYEN
Hence
dHn(uǫ, vǫ) = 0,
which shows that uǫ is continuous up to the boundary and takes boundary value
1, ..., xm−1, 0) = (f1, f2, ..., fn−1, ǫ). �
References
[1] Shiu-Yuen Cheng, Liouville theorem for harmonic maps, Proc. Symp. Pure Math. 36, 1980,
147–151.
[2] Wei-Yue Ding and Youde Wang, Harmonic maps of complete noncompact Riemannian man-
ifolds, Internat. J. Math. 2, 1991, 617–633.
[3] Duong Minh Duc and Alberto Verjovsky, Proper harmonic maps with Lipschitz boundary
values, preprint.
[4] James Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Ams. J.
Math. 86 (1), 1964, pp. 109–160.
[5] David Gilbarg and Neil S. Trudinger, Elliptic partial differential Equations of second order,
Springer - Verlag, Berlin - Heidelberg-New York -Tokyo, 1983.
[6] Frederic Helein, Regularite des applications faiblement harmoniques entre une surface et une
variete riemannienne, C. R. Acad. Sci. Paris 312 (1), 1991, 591–596.
[7] Peter Li and Luen-Fai Tam, The heat equation and harmonic maps of complete manifolds,
Invent. Math. 105, 1991, 1–46.
[8] Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps, Anals of
Mathematics 137, 1993, pp. 167-201.
[9] Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps II, Indiana
University Mathematics Journal 42 (2), 1993, pp. 591-635.
[10] Richard Schoen and Shing Tung Yau, Compact group actions and the topology of manifolds
with non-positive curvature, Topology 18, 1979, 361–380.
Department of Mathematics, University of natural sciences, Hochiminh city, Viet-
E-mail address: [email protected]
Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405
E-mail address: [email protected]
1. Introduction
2. Initial maps
3. Proof of Theorem ??
References
|
0704.0088 | Some new experimental photonic flame effect features | PHOTONIC FLAME EFFECT
SOME NEW EXPERIMENTAL PHOTONIC FLAME EFFECT
FEATURES
N.V.Tcherniega
P.N.Lebedev Physical Institute, RAS
Leninskii pr., 53, 119991, Moscow, Russia
[email protected]
Abstract
The results of the spectral, energetical and temporal characteristics of radiation in the
presence of the photonic flame effect are presented. Artificial opal posed on Cu plate at the
temperature of liquid nitrogen boiling point (77 K) being irradiated by nanosecond ruby laser
pulse produces long- term luminiscence with a duration till ten seconds with a finely structured
spectrum in the the antistocks part of the spectrum. Analogous visible luminescence manifesting
time delay appeared in other samples of the artificial opals posed on the same plate. In the case
of the opal infiltrated with different nonlinear liquids the threshold of the luminiscence is
reduced and the spatial disribution of the bright emmiting area on the opal surface is being
changed. In the case of the putting the frozen nonlinear liquids on the Cu plate long-term blue
bright luminiscence took place in the frozen species of the liquids. Temporal characteristics of
this luminiscence are nearly the same as in opal matrixes.
Keywords: photonic flame effect, optical luminescence, excitation, artificial opal, spectrum
1. Introduction
In [1-3] new effect called photonic flame effect was found and some its properties were
studied. This effect is determined by properties of photonic crystals.
Photonic crystals have attracted great attention since the first papers concerning such
structures [4-6]. One of the most important photonic crystals are artificial opals – self –
assembled structures composed of SiO2 spheres organizing face-centered cubic lattice. The size
of such spheres varying between 200 nm and 400 nm and defines the parameters of the face-
centered cubic lattice and the photonic bandgap. The possibility of opal infiltration with different
medium gives rise to effective processing the properties of the light propogating through the
crystal. The voids in the opal structures can be filled with semiconductors, superconductors,
ferromagnetic materials, fluorescent medium [7] and this fact gives large possibility to practical
applications of such structures for optoelectronics. The study of the linear optical properties of
the photonic band gap have been the task of many theoretical and experimental works and still
remain the task to be investigated [7,8]. The theoretical description of the electromagnetic field
inside the photonic crystal structures (obtained by transfer matrix method [9] or coupled mode
theory [10]) gives the clear picture of the transmitted and reflected spectrum, electromagnetic
field distribution inside the crystal and their dependence on the parameters of the photonic
crystal structure (values of period, number of periods, refractive index contrast). Large values of
the electromagnetic field localization in some regions lead to the expectation of the strong
enhancement of nonlinear wave-matter interaction in comparison with bulk crystals. Second
harmonic generation in different types of photonic crystals was investigated in [11,12]. Properly
chosen photonic crystal exhibits negative refraction at some conditions [13]. Some features of
the stimulated Raman scattering in one-dimensional photonic structure were considered in [14].
Fully quantum mechanical treatment of the generation of entangled photon in nonlinear photonic
crystals at the process of down-conversion was realized in [15]. Photonic band gap properties
which are demonstrated by photonic crystalls are being actively used for investigation of
photon-exciton interaction [16]. Acoustic modes excited in SiO2 balls which compose opal
photonic crystal show the effect of phonon modes quantization [17] and are the reason of
stimulated globular scattering [3]. Specific features of the acoustic wave propagation in the
photonic structures lead to the possibility of the diverging ultrasonic beam focusing into a
mailto:[email protected]
narrow focal spot with a large focal depth [18]. Optical parametric oscillations via four-wave
mixing in isotropic photonic crystals showes the possibility of the effective frequency processing
[19].
The aim of this work is to give a short review of results [1-3] and to study collective behavior
of several photonic crystals. The crystals are posed on Cu plate at the temperature of liquid
nitrogen. One of the photonic crystals is illuminated by laser pulse and the laser light is focused
on this only crystal. The phenomenon which we observe is the appearance of luminescence of
other photonic crystals. The duration of the luminescence of other crystals which are spatially
separated with the crystal illuminated by laser pulse is of the order of seconds. The appearance of
the luminescence takes place with some time delay respectively to the laser pulse. The form of
these light spots on the other crystals and their slow motions along the crystal reminds a small
flame spot. This inspired us to give the “photonic flame” name to the observed effect. In the case
of covering the surface of the Cu plate with liquid (acetone, ethanol, water) after the PFE
excitation in the opal situated on this plate blue luminescence is being seen in the frozen liquid.
The temporal characteristics of this luminescence are the sme as for single opal crystal. The
paper is organized as follows. In Sec.2 the experimental setup, laser, the photonic crystals
(artificial opals) used in the experiment are described. In Sec.3 the “photonic flame effect”
observed in the experiment is discussed. In Sec.4 perspectives and possible explanations are
presented.
2. Photonic crystals and laser used in experiment.
One of the most promising three-dimensional photonic crystals is artificial opal. Opal is a
crystal with face-centered cubic lattice consisting of the monodisperse close packed SiO2 spheres
with diameter about several hundred nanometers. Because the refractive index contrast (ratio
nSiO2/nair) is about 1,45 the complete photonic band gap does not exist but the photonic
pseudogap takes place. Empty cavities among these globules have octahedral and tetrahedral
form. It is possible to investigate both initial opals (opal matrices) and nanocomposites, in which
cavities are filled with organic or inorganic materials, for instance, semiconductors,
superconductors, ferromagnetic substances, dielectrics, displaying different types of
Fig.1. Common appearance of a globular photonic crystal, built of spherical particles (globules)
nonlinearities and so on. Filling voids of the photonic crystal with materials with different
refraction index one can effectively process the parameters of the photonic pseudogap.
Ruby laser giant pulse (λ=694.3 nm, τ=20 ns, Emax =0.3 J, spectral width of the initial light -
0.015 cm-1.) has been used as a source of excitation. Exciting light has been focused on the
material by lenses with different focal lengths (50, 90, and 150 mm). The samples of opal
crystals used had the size 3x5x5mm and were cut parallel to the plane (111) (see Fig.2) .The
angle of the incidence of the laser beam on the plane (111) varied from 0 to 600. Sample
distance from focusing system and exciting light energy were different in different runs of the
experiment. This gave possibility to make measurements for different power density at the
entrance of the sample and for different field distribution inside the sample. Opal crystals
consisting of the close-packed amorphose spheres with diameter 200 nm, 230 nm, 250 nm and
nanocomposites (opal crystals with voids filled with acetone or ethanol) were investigated.
z
θ y
[111] x
Fig.2. The scheme of illuminating the sample. Plane XY correspondes to the CU plate surface.
3.Characteristics of “photonic flame”
Opal crystals were placed on the Cu plate which was put into the cell with liquid nitrogen (see
Fig.3). The number of crystals varied from 1 to 5. The distance (d) betwen the crystals was of the
order of several centimeters (maximum value of d was 5 centimeters and was determined by the
Cu plate size). One of the crystals was illuminated by the focused laser pulse. In the case of the
reaching of the threshold visible (blue) luminescence appeared. The luminescence duration was
from 1 to 12 seconds and it looked like inhomogeneous spot changing its spatial distribution and
position on the surface of the crystal during this time.
liquid N2
opal opal
exciting light
d
opal Cu
Fig.3. Experimental setup.
Parameters of the luminescence (duration, threshold) were determined by the geometric
characteristic of the illumination and the refractive index contrast of the sample. For optimal
geometry of the excitation the power density threshold for opal crystal was 0.12 Gw/cm2, for
opal crystals filled with ethanol – 0.05 Gw/cm2, for opal crystal filled with acetone - 0,03
Gw/cm2. Typical luminescence temporal distribution measured for the part of the crystal
displaying the most intensive brightness is shown on Fig.4. The same behavior is typical for all
cases of the luminescence at these conditions of excitation, but the value of the luminescence
duration fluctuated from shoot to shoot.
a) b)
Fig.4. Temporal distribution of the visible luminiscence.
The duration of the luminiscence fluctuated from 1 till 12 seconds and demonstrated
oscillating structure. In some cases the temporal distribution had maximum at the beginning of
the luminiscence in some cases – minimum. Fig.4 a) and b) show the luminiscence of the pure
opal matrix of the nearly the same duration at the same geometrical and energetical conditions of
excitation near the threshold of excitation (0.12 Gw/cm2 ). The beginning of the mesurements
corresponds to 0.3 s delay after the laser shoot (laser pulse duration is 20 ns).
Secondary emission spectrum observed in photonic flame effect has been investigated with
the help of setup shown at the Fig.5.
12
11 10
9
8
1
3 4 2
6 7
Fig. 5.The experimental setup for PPE spectrum study. 1- ruby laser; 2- lens; 3, 4, 5 – photonic
crystals; 7 – cell with liquid nitrogen 6 – Cu plate; 8 – fiber wave guide; 9 – minipolychromator;
10 – computer; 11 – camera; 12 – computer.
Spectra of the light emitted by photonic crystal for different pumping light power density are
shown at the Fig. 6 (a and b)
200 300 400 500 600 700 800 900 1000
643I,
λ, nm
200 300 400 500 600 700 800 900 1000 1100
λ, nm
a b
Fig. 6. Secondary emission spectrum of a photonic crystal for different laser light power density:
a - I = 0.12 GW/cm2, b - I = 0.14 GW/cm2.
Spectrum consisted of the sharp lines with wavelengths: 429.0, 453.0, 489.0, 555.0, 643.0 nm,
which corresponds to the antistokes spectral range for exciting line 694.3 nm. Lines intensity in
the spectrum strongly depended on the laser pumping intensity, which was evidence of
stimulated type of the radiation emission.
In the case of several crystals placed on the Cu plate only one of them was irradiated by the
laser pulse. Luminescence took place in this crystal in the case of the threshold reaching. Bright
shining of the other crystals began with some time delay after laser shoot. The value of this delay
(and the intensity of the luminiscence) was determined by the spatial position of the crystals on
the plate. The steal screen beeing put between the crystals (in order to avoid irradiating of the
crystals by the light scattered by the crystal excited by the laser) did not stop the appearing of the
luminescence if the distance between the Cu plate and the screen was more than 0.5 mm. The
duration of the luminiscence was of the order of several seconds and temporal behavior was like
shown on Fig.4. The typical features of such distribution were existence of maximum and large
plato with near constant value of the intensity.
In order to show the role of the material of the plate used we repeated these measurements
with plates of the same size but made from steel and quarz on which opal crystals were placed
like in the previous experiments. Luminescence of the same kind in the irradiated crystal took
place but the luminescence of the other samples situated on these plates was not observed.
The effect was also determined by the angle of incidence (Fig.2). For the samples used the
value of the angle was chosen experimentally for achieving of the maximal value of the
luminescence (it worth to mention that this value differed from 0 and was about 400). Easier the
effect was excited in the unprocessed samples. In Fig.5 one can see the luminescence of the
crystals situated at the distance of about 1 centimeter from the crystal which was irradiated.
Fig.7 Visible luminiscence of the opal crystals in the case of the irradiating one of them (the
irradiated crystal can be seen by bright red light; on the left picture it was the crystal in the
center, on the right picture it was the crystal on the left). Left picture corresponds to the case
where crystals are infiltrated by acetone. Right picture corresponds to the case of the opal
crystals without infiltration.
In the case of the large laser energy (several times more the threshold) or if the crystal was
irradiated by several laser pulses the opal can be destroyed and the parts of the crystal produce
the luminiscence with the spectral and temporal properties described above (Fig.8).
Fig.8 Opal crystal is destroyed and 3 large pieces and several little pieces are going on to
produce the luminescence.
In order to clarify the role of the Cu plate surface on the energy transport between the
crystals the next experiment was realized: the pure opal matrix posed on the Cu plate was
irradiated by the ruby laser pulse and demonstrated strong luminescence lasting few seconds
with the properties described above (Fig.9)
Fig.9 Luminiscence of the single opal matrix
Next step was covering the surface of the Cu plate with the liquid (experiments were made
with water, aceton or ethanol). The thickness of the frozen liquid on the plate surface was about
1 mm. The transverse size of the frozen liquid was about 1 cm. After illuminating of the crystal
by the ruby laser pulse the luminiscence of the crystal appears the bright blue luminiscence of
the frozen species of the liquid used appears. The temporal characteristics of the luminiscence in
crystal and in the frozen liquid are approximatly the same (the luminiscence duration is about
several seconds). The luminiscence of the frozen liquid goes on in spite on the putting the screen
between the crystal and the liquid. It shows that the luminiscence of the liquid is not a reflection
of the light which is emmited by the crystal. Fig.10 shows the luminiscence of the crystal and the
frozen liquid (in this case it was water). The pictures were made with the interval of 1 second
between each other. Analogous behaviour is demonstrated by aceton and ethanol. The
luminiscence of the area covered with frozen liquid takes place even if this area is at the distance
of several cantimeters from the irradiated crystal.The explanation of the blue luminiscence of the
frozen liquid can be done in several ways and for clarifying the reasons of this luminescence
appearance it is necessary to produce additional experiments. The intensity of the laser in the
experiments is about 0.12 Gw/cm2, and the large enhancement of this field due to Mie –
resonance [20] simultaniously with the interference effect caused by the structure of the opal
matrix can lead to the extremely large field enhancement which can play an important role in this
effect.
Fig.10 Luminiscence of the opal matrix (bright round spot) and frozen liquid (large blue spot)
on the surface of the Cu plate.
4. Conclusions.
In this paper we reported about some new features of the photonic flame effect. The main
features of PFE are:
- At the excitation of the artificial opal crystal which is placed on the Cu plate at the
temperature of the liquid nitrogen by the ruby laser pulse of the nanosecond range long-
continued optical luminescence takes place in the case if the threshold of the process is
reached;
- In the case of several opal crystals being put on the Cu plate while one of them is being
irradiated bright visible luminescence occurs in all samples;
- Temporal behavior and thresholds of the luminescence have been determined. Photonic
crystals infiltrated with different nonlinear liquids and without infiltration have been
investigated. Investigated transport of the excitation between the samples spatially separated
by the length of several centimeters gives the possibility of the practical applications of
PFE;
- The blue luminiscence of the frozen liquid on the surface of the Cu plate takes place at the
precense of the photonic flame effect;
The photonic flame effect can have different explanation. Probably an essential role is played
by plasma properties. The slow transport of the excitations from the irradiated crystal to other
photonic crystals can be associated with sound waves created due to laser pulse interaction with
the sample. Exciton mechanism and surface waveguides on the surface of the Cu plate also can
play important role. It was checked that the change of the properties of the plate surface was
leading to change of the photonic flame effect. Removing the oxid layer from the plate changed
the threshold PFE. The luminescence of the frozen liquids on the surface of the Cu plate showes
the important role of the electromagnetic field enhancement due to Mie resonance and Bragg
diffraction on the photonic crystal lattice. The electromagnetic field enhancement can lead to
producing laser plasma, electron acceleration and x-ray production.
References
1. N.V.Tcherniega, A.D.Kudryavtseva, ArXiv Physics/ 0608150 (2006).
2. N.V.Tcherniega, A.D.Kudryavtseva, Journal of Russian Laser Research, V.27, N 5, стр.400-
409 (2006).
3.A.A.Esakov, V.S.Gorelik, A.D.Kudryavtseva, M.V.Tareeva and N.V.Tcherniega, SPIE
Proceedings, V 6369, 6369 OE1 - 6369 OE12, Photonic Crystals and Photonic Crystal Fibers for
Sensing Applications II; Henry H. Du, Ryan Bise; Eds, (Oct.2006).
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M. Samoilovich, Yu. A. Vlasov, Nuovo Cimento, D 17,1349 (1995).
8.A. V. Baryshev, A. A. Kaplyanskii, V. A. Kosobukin, M. F. Limonov, K. B. Samsuev,
Fiz.Tverd.Tela, 45, 434 (2003), in Russian.
9.M. Born, E. Wolf, Principles of Optics, Macmillan, New York (1964)
10.A. Yariv, Quantum Electronics, John Wiley and Sons, Inc., New York, London, Sudneu
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Akstipetrov, and G. Marovsky, Laser Physics, 14, 677 (2004).
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Appl.Phys.Letters, 87, 151111 (2005).
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Russian.
15.W. T. M. Irvine, M. J. A. de Dood, D. Bouwmeester, Phys Rev.A 72, 043815 (2005).
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|
0704.0089 | A general approach to statistical modeling of physical laws:
nonparametric regression | A general approach to statistical modeling of physical laws:
nonparametric regression
Igor Grabec∗
Faculty of Mechanical Engineering, University of Ljubljana,
Aškerčeva 6, PP 394, 1001 Ljubljana, Slovenia†
(Dated: October 26, 2018)
Abstract
Statistical modeling of experimental physical laws is based on the probability density function
of measured variables. It is expressed by experimental data via a kernel estimator. The kernel
is determined objectively by the scattering of data during calibration of experimental setup. A
physical law, which relates measured variables, is optimally extracted from experimental data by
the conditional average estimator. It is derived directly from the kernel estimator and corresponds
to a general nonparametric regression. The proposed method is demonstrated by the modeling of a
return map of noisy chaotic data. In this example, the nonparametric regression is used to predict
a future value of chaotic time series from the present one. The mean predictor error is used in
the definition of predictor quality, while the redundancy is expressed by the mean square distance
between data points. Both statistics are used in a new definition of predictor cost function. From
the minimum of the predictor cost function, a proper number of data in the model is estimated.
PACS numbers: 02.50.-r,07.05.-t,05.45.-a,89.90.+n,84.35.+i,06.20.DK
∗Also at Amanova, Kantetova 75, 1001 Ljubljana, Slovenia.
†Electronic address: [email protected]; URL: http://www.fs.uni-lj.si/lasin/
http://arxiv.org/abs/0704.0089v1
mailto:[email protected]
http://www.fs.uni-lj.si/lasin/
I. INTRODUCTION
A basic task of physical description of natural phenomena is to express relations between
experimental data about measured variables in terms of physical laws [1]. Since the corre-
sponding analytical modeling essentially depends on the intuition of the explorer performing
it, an ambiguity surrounds this basic task and there thus arises a question how this could be
avoided. This problem becomes of fundamental practical importance when developing intel-
ligent electronic systems for automatic modeling of physical laws [2]. The ambiguity could
be avoided if a unique objective method of modeling was found that would take into account
common properties of experimental observations and of transitions from experimental data
to models. The aim of this article is to show how such a method could be developed from
basic principles of probability and statistics, as well as to demonstrate an example of its
applicability.
A common property of all experimental explorations is that each experiment corresponds
to a process proceeding from preparation to execution. If we want a selected experiment
to yield any information about the phenomenon under observation, then the result of the
experiment may not be determined in advance i.e. several outcomes of the experiment must
be possible. The next common property is repeatability of experiments. Consequently, a
correct presentation of experimental observations requires the use of a distribution of ex-
perimental results and this must be related to the concept of probability. The probability
distribution is, therefore, a common basis for the description of natural properties in terms
of experimental data [3], while the transition from experimental data to an analytical ex-
pression of the corresponding probability distribution function is the crucial problem of
modeling. An objective solution of this problem represents statistical modeling of the prob-
ability distribution function by a nonparametric kernel estimator if the kernel is determined
by a calibration of the experimental setup [4, 5, 6]. For this purpose, the central theorem of
probability theory and the maximum entropy principle provide a quite general route to the
specification of the kernel function of the estimator. In this case, an experimental physical
law, which represents a relation between observed variables, can also be generally expressed
by applying the theory of optimal statistical estimators. The resulting nonparametric re-
gression is the conditional average (CA), which can be automatically extracted from the
probability density function (PDF) of experimental data in a measurement system. The
complete approach to modeling thus appears objective, independent of the intuition of the
observer and, consequently, generally applicable for automatic execution. Due to these con-
venient properties, CA is widely applicable in various fields of natural and technical sciences
A nonparametric expression of the PDF by the kernel estimator has already been proposed
by Parzen [7, 8], but weaknesses of his proposal are that the kernel function is arbitrarily
introduced, and that there is an assumption that its width should decrease to zero when the
number of data is increased to infinity. In order to avoid this weakness, we specify the kernel
function objectively by the scattering of the measurement system output during calibration
[7, 8]. The only ambiguity in the expression of the PDF is then related to the number of
experimental data, which according to Parzen’s assumption should not be limited. Since an
infinite number of experiments cannot be performed, there arises a fundamental question:
”How many experiments is it reasonable to perform in order to explore the phenomenon
properly by a given experimental setup?” Intuitively, we can conclude that it is reasonable to
repeat experiments for as long as they bring new information. However, with an increasing
number of experiments, the acquired data points become ever more concentrated in the
sample space and consequently the repetition of the experiments becomes redundant. This
is observed when distances between data points become comparable to the width of the kernel
function. This reasoning led recently to a specification of an information cost function C
[4, 5, 6, 9, 10, 11, 12, 13]. For this purpose the indeterminacy of measurements was first
expressed in terms of information entropy, which further led to definition of the experimental
information I and the redundancy R of experiments. Using these statistics, the information
cost function was expressed by the difference C = R− I. From the position of its minimum,
a proper number of experiments can then be objectively determined [4, 5, 6].
Estimation of the information cost function is related to the calculation of integrals, which
is inconvenient in a multivariate case. Therefore, another statistic, with similar properties
but more simple calculation, is sought. Since it has been shown previously that the pre-
dictor quality exhibits similar properties to the experimental information, we utilize it here
in the definition of the predictor cost function. From its minimum, a proper number of
experiments can also be estimated. If this is used as a proper number for the adaptation of
the nonparametric regression to data provided by experiments, the modeling of the corre-
sponding physical law can be performed automatically on a data acquisition system of the
experimental setup. To demonstrate this possibility, we first briefly describe the nonpara-
metric regression and then turn to the definition of the predictor quality, redundancy and
cost function. Properties of all statistics are subsequently demonstrated in the modeling of
a return map corresponding to a noisy chaotic process.
II. FUNDAMENTALS OF NONPARAMETRIC MODELING
A. Description of kernel function
Let us consider a phenomenon that can be described by just two joint variables, since the
generalization to a multivariate case is straightforward. A single result of joint measurement
is represented by the couple z = (x, y). We next assume that the phenomenon can be
characterized statistically by repetition of measurements yielding sample points zn = (xn, yn)
in the joint span of a two channel instrument S
= Sx ⊗ Sy.
Since the instruments are generally subject to stochastic disturbances, the results of
measurements are scattered even during repetition of calibration [9]. The scattering can be
described by the data provided by a series of repeated simultaneous calibrations of both
instrument channels. For this purpose, we have to perform a joint measurement on an
object representing two physical units ux and uy which we denote together by the joint unit
u = (ux, uy). The scattering of instrument outputs during calibration is characterized by the
joint PDF ψ(z|u), which we call the scattering function (SF) [2, 4, 9]. When the interaction
between both channels is negligible, the SF is given by the product ψ(z|u) = ψ(x|ux)ψ(y|uy).
Without loss of generality, we further consider a case with equal channels which are subject
to mutually independent random disturbances that do not depend on u. In such cases,
the central limit theorem of probability theory, as well as the maximum entropy principle,
suggest that we express the SF of a particular channel by the Gaussian function:
g(x− ux, σ) =
−(x− ux)
The parameters ux, σ represent the mean value and standard deviation of signal x at the cal-
ibration and can be statistically estimated from given data. The joint SF is then determined
by the product ψ(z− u) = g(x− ux, σ) g(y − uy, σ).
When reporting experimental results, experimentalists most often only specify mean val-
ues and standard deviations of variables during calibration. The maximum entropy principle
tells us that, in such cases, the Gaussian function is the best choice for SF [2, 9].
B. Nonparametric estimation of PDF pertaining to experimental data
When we perform a single measurement, we get a sample z1 = (x1, y1) that represents
the mean value of z during measurement and, therefore, we express the PDF as ψ(z−z1) =
ψ(x − x1)ψ(y − y1). When we repeat the measurements N times, we get a set of samples
{zi, 1 ≤ i ≤ N}, by which we model the joint PDF by the statistical average:
f(z) =
ψ(z− zi) (2)
that represents the kernel estimator.
Properties of the particular components x, y are described by the marginal PDFs
f(x), f(y). They are obtained from the joint PDF by integration with respect to one com-
ponent, for example:
f(x) =
f(z)dy =
ψ(x− xi). (3)
For modeling natural laws, the most important is the conditional PDF of the variable y at
a given value of x, defined as:
f(y|x) =
ψ(z− zi)
ψ(x− xj)
C. Estimation of a physical law
Distributions of joint experimental data, for example that shown in Fig. 1, often resemble
a ridge along some hypothetical line yo(x), which we want to extract from the given data
in an optimal way. For this purpose, we select from a set of joint data only those that
pertain to some selected x. These joint data generally exhibit various values of y which we
try to represent by a single value called the predictor of the variable y from a given value x.
We consider as an optimal predictor of the hypothetical yo the value yp at which the mean
square prediction error is minimal:
E[(yp − y)2|x] = min(yp). (5)
0 0.1
0.2 0.3
0.4 0.5
0.6 0.7
0.8 0.9
FIG. 1: The joint PDF f(z) utilized to demonstrate the properties of the conditional average
estimator.
Here E[. . . |x] denotes the operation of statistical averaging at given condition x. The mini-
mum satisfies the equation: dE[(yp − y)2|x]/dyp = 0 that yields as the optimal predictor yp
the conditional average:
yp(x) = E[y|x] =
y f(y|x) dy (6)
By using Eq. (4), we obtain for the conditional average the expansion:
yp(x) =
yiψ(x− xi, σ)
ψ(x− xj , σ)
yiBi(x). (7)
The coefficients of this expansion are sample values yi, while the basis functions are
Bi(x) =
ψ(x− xi, σ)
ψ(x− xj , σ)
, (8)
and satisfy the following conditions:
Bi(x) = 1 , 0 ≤ Bi(x) ≤ 1. (9)
The basis functions Bi(x) can be interpreted as a normalized measure of similarity between
the given value of x and its sample value xi. At a given x, the sample value ym contributes
most to the estimated value yp(x) whose complementary sample value xm is most similar to
The calculation of yp(x) corresponds to an associative recall of memorized items, which
is a property of an intelligence. Therefore, the estimator yp(x) could be treated as a basis
for the development of a machine intelligence based on modeling of natural laws. The
conditional average given in Eq. 7 in fact corresponds to a normalized radial basis function
neural network which is equivalent to a multilayer perceptron – the basic paradigm used in
the theory of artificial neural networks [2, 21].
III. CHARACTERISTICS OF THE MODEL
A. Predictor quality
A predictor maps the stochastic variable x to a new stochastic variable yp that gener-
ally differs from the variable y. When the variables x, y are related by some hypothetical
physical law yo(x) and the measurement noise is small, the first and second statistical mo-
ments E[y − yp], E[(y − yp)2] of the prediction error are also small. The second moment
is: E[(y − yp)2] = Var(y) + Var(yp) − 2Cov(y, yp) + [m(y) − m(yp)]2, where E,m,Var,Cov
denote statistical average, mean value, variance and covariance respectively. In the case of
statistically independent variables y and yp with equal mean values, the last two terms are
zero and we get: E[(y − yp)2] = Var(y) + Var(yp). With respect to this relation, we define
the predictor quality relatively by the formula
Q = 1− E[(y − yp)
Var(y) + Var(yp)
2Cov(y, yp)
Var(y) + Var(yp)
− [m(y)−m(yp)]
Var(y) + Var(yp)
The quality is 1 if the prediction is exact: yp = y, while it is 0 if y and yp are statistically
independent and have equal mean values. The quality Q may be negative if m(y) 6= m(yp).
For the predictor defined by the conditional average yp(x) =
y f(y|x) dy, we analytically
obtain the equalities: m(y) = m(yp) and Cov(y, yp) = Var(yp), which yield
2Var(yp)
Var(y) + Var(yp)
. (11)
From the definition of the conditional average, it follows 0 ≤ Var(yp) ≤ Var(y) and therefore
0 ≤ Q ≤ 1. This inequality need not be fulfilled exactly if CA is statistically estimated from
a finite number of samples.
With an increasing N , we generally expect that the CA statistically estimated by Eq. (7)
increasingly better represents the governing physical law and, consequently, that the corre-
sponding predictor quality Q on average increases to a certain limit value. As mentioned
previously, an unlimited increase in the number of experiments is experimentally impossible
and, consequently, there arises the question how to determine a proper number No of data
that will yield a judicious estimation of the governing law.
B. Redundancy and predictor cost function
To answer the last question, we have analyzed various experimental cases which have
shown us that, with an increasing number of experimental samples, the value of predictor
quality generally stabilizes when the distance between data points becomes similar to the
width σ of the scattering function. Therefore, it is not reasonable to surpass significantly
the corresponding number of data. This can be achieved if a ratio of σ and a proper
measure of distance δ between neighbor data points is considered. For this purpose, we
introduce δ over the mean value of minimum square distance between data points: δ2 =
E[min{(xi−xj)2+(yi−yj)2)}; i = 1 . . . N, j = 1 . . .N, ], and define a measure of redundancy
of data by the relative variable:
R = 2N
Since δ2 is comprised of two terms denoting contributions from x and y components, a
factor 2 is utilized in the nominator. The fraction 2σ2/δ2 represents an average increase of
redundancy that is assigned to the acquisition of a new data point. In order to take into
account acquisition of N data points, factor N is further used. With respect to this, we
introduce the predictor cost function by the sum:
C = R−Q+ 1
E[(y − yp)2]
Var(y) + Var(yp)
. (13)
The constant 1 is inserted in the first row in order to obtain a more simple expression
in the second row of Eq. 13. In the same way as the definition of the information cost
function given in [5, 6], the cost function is here expressed in a relative form comprised
of two terms: the first corresponds to the redundancy of experiments due to inaccurate
measurements while the second represents the influence of acquisition of information about
the phenomenon by experiments. With an increasing number of samples N , the redundancy
on average increases while the second term decreases with the decreasing error. Therefore,
the cost function C exhibits a minimum at some No that represents a proper number of data
needed for the modeling of the physical law governing the phenomenon explored. However,
the influence of the first term becomes prevailing when the distance between data points δ
becomes essentially smaller than the width σ of the scattering function.
IV. EXAMPLE
To demonstrate the properties of the CA estimator, we utilize the data generated by a
noise-corrupted chaotic return map with the span Sx = (0, 1). This example is used because
similar cases often appear in the analysis of chaotic time series [2, 14]. The basic problem
in such an analysis is to extract the return map from a given record of time series that
is influenced by additive noise of instrumental origin. In our case, we apply analytically
determined data to provide for a comparison between the original and extracted physical
law and to make feasible an objective reproduction of the complete method. The basic
governing law is here given by the logistic map:
χn+1 = 3.8χn(1− χn), (14)
while the initial value χ1 is arbitrary selected from the interval (0, 1) using a random gener-
ator. To the values of generated chaotic series, the Gaussian noise ν of zero mean value and
standard deviation σ = 0.1 is added to simulate an additive noise of measurement. The iter-
ative solution of Eq. 14 then yields a series of noise corrupted chaotic values: xn = χn + νn.
Figure 2 shows two records of such a series that were used in modeling and testing of the
proposed method.
From the series {xn ;n = 1 . . .}, the joint samples of the basic variables x, y are obtained
by treating the successive value of xn as the dependent variable: yn = xn+1. The generator
of data is thus analytically described by the rule:
xn = χn + νn
yn = xn+1, (15)
while the governing law is given by yo = 3.8 x(1−x). The sample points {xn, yn ; n = 1 . . .N}
are distributed along the corresponding parabola in the sample space. According to our
previous treatment, the standard deviation σ corresponds to the width of the instrument
scattering function ψ. The joint PDF shown in Fig. 1 is determined by the kernel estimator
0 5 10 15 20 25 30
FIG. 2: Records of the basic – (X), and the testing – (Xt) noise corrupted chaotic series.
0 0.2 0.4 0.6 0.8 1
TESTING OF CA PREDICTOR
FIG. 3: Testing of the CA predictor. Graphs represent the governing law yo and basic data y –
(top two: · · · ; ∗ ∗ ∗), test yt and predicted data yp – (middle two: + + + ; ◦ ◦ ◦ ), and prediction
error Er = yp − yt – (bottom: ♦♦♦). The upper two parabolas are displaced successively by 0.35
in the vertical direction for better visualization.
Eq. (2) using 200 data, while a reduced set of 30 data is further utilized to demonstrate
the properties of the conditional average estimator. The data obtained from the pure chaos
generator are shown by yo · ·· in the top parabola of Fig. 3, while the basic noise-corrupted
data y ∗ ∗∗ are shown by points scattered around pure data points.
The conditional average estimator is obtained by inserting data from the basic data
0 5 10 15 20 25 30
MEAN SQUARE PREDICTOR ERROR
FIG. 4: Mean square prediction error E[(y − yp)2] as a function of the data number N .
set into Eq. (7). To demonstrate its performance, we additionally generated with different
seeds of random generators a set of Nt = 60 test data {xt,i, yt,i}. Based on data xt,i from
this set, the corresponding values of yp,i are predicted by the CA estimator. The test and
predicted data are shown in Fig. 3 by the middle two sets of points (+ + + and ◦ ◦ ◦).
The prediction error Er = yp − yt, calculated from both data sets, is presented by ♦♦♦
at the bottom of Fig. 3. Relatively small differences between predicted and test points
indicate that the properties of the governing law yo(x) are properly modeled by the CA
estimator. To confirm this qualitative conclusion, we next analyze the properties of statistics
E[(ye − yt)2], Q, δ2, R, C depending on the the number of data N used in modeling. The
number of test data is kept constant Nt = 60 during calculation of these statistics. Properties
of the statistical model of the governing law depend on sets of samples utilized in modeling
and testing. To demonstrate this dependence, we repeated the modeling and testing three
times using various statistical sample sets.
The mean square predictor error E[(y − yp)2] is presented in Fig. 4 versus number of
samples N . Its value varies statistically but, on average, it decreases with the increasing
numberN . Statistical fluctuations are largest at smallN and significantly depend on samples
used in modeling. However, with the increasing N , the statistical fluctuations are ever less
pronounced. If the number of test samples Nt is much larger than the number of samples N ,
changing the testing sample set does not significantly influence the properties of estimated
statistics, which is the case in our demonstration. This is the reason why we use the value
0 5 10 15 20 25 30
PREDICTOR QUALITY
FIG. 5: Predictor quality Q as a function of the data number N .
Nt = 60.
The predictor quality Q, as determined from the prediction error, is presented in Fig. 5
versus number of samples N . For each data set the statistical fluctuations decrease with
increasing N so that qualities calculated from different data sets converge to the same limit
value. With increasing N , the curves determined from different data sets merge approxi-
mately at N ∼ 11. The quality is there ∼ 0.97 and rises to ∼ 0.98 at N = 30. At N ∼ 11,
the difference between the curves obtained from different data sets is about two orders
of magnitude smaller than the corresponding quality. With respect to these properties, we
could conjecture that in the present case about 11 data values already provide for a judicious
modeling of the governing law yo(x) by the CA predictor.
To confirm our last conjecture, we turn to the determination of the predictor cost function.
For this purpose, let us first analyze the properties of the mean square distance between data
points δ2. The corresponding graph, shown in Fig. 6, indicates that δ2 is rather monotonously
decreasing with the number of samples with the approximate dependence being ∼ 1/N .
Consequently, the corresponding redundancy R is increasing with N similarly as ∼ N2.
This conclusion is confirmed by the graph in Fig. 7.
Following the definition given by Eq. 13, we obtain from the estimated error and the
redundancy the predictor cost function C shown in Fig. 8. Its minimum is not very pro-
nounced. From various statistical data sets, we obtain the estimates of the minimal value
Co = 0.033±0.006. The corresponding number No = 10±2 confirms our previous conjecture
0 5 10 15 20 25 30
MSD BETWEEN DATA POINTS
FIG. 6: Mean square distance between data points δ2 as a function of the data number N .
0 5 10 15 20 25 30
REDUNDANCY
FIG. 7: Redundancy R as a function of the data number N .
stemming from the analysis of predictor quality.
With an increasing number of samples N , the quality Q(N) of the CA predictor exhibits
a convergence to some limit value Q∞ that characterizes hypothetical maximum quality of
proposed nonparametric statistical modeling. This limit value generally increases with the
decreasing scattering width σ. Related to this, the minimal value of cost function is dimin-
ished and takes place at a larger No ; for instance at σ = 0.005 we get Co = 0.018 ± 0.003
and No = 14± 3. However, the limit value of the quality Q∞ is less than 1 if 1/σ and N are
finite. This means that it is not possible to exactly determine the governing physical law
0 5 10 15 20 25 30
PREDICTOR COST FUNCTION
FIG. 8: Predictor cost function C as a function of the data number N .
y = yo(x) from joint data obtained by an instrument influenced by stochastic disturbances.
V. DISCUSSION
Our method of estimation of natural laws from given data can be simply generalized to
multivariate cases by substituting corresponding vectors for the variables x, y. Such modeling
has already been applied in a variety of examples stemming from physical [15], technical
[2, 16], economic [2, 16] and medical environments [2, 17, 18]. Particularly in economic
and medical environments, phenomena are often characterized by many variables that could
be either informative or disturbing. Due to the complexity of such cases, there usually
exists little or no information about a possible function that could describe the governing
law. In relation to this, researchers are faced with the problem of how to define complexity
and to reduce it by extracting informative variables from a given set [19]. Alongside mutual
information, the predictor quality could also be applied for this purpose. For instance, it has
been recently shown in the field of medicine how an analysis of predictor quality can provide
for an ordering of variables and the extraction of a set that yields an optimal predictor of
the disease healing process [17, 18]. Such an analysis makes feasible further progress towards
the origins of the treated disease.
The value of the proper number No, as defined by the minimum of predictor cost function,
could be interpreted as a measure of the complexity of an adequate predictor model. It is
important that this measure depends only on the accuracy of observation and properties of
the phenomenon represented by given experimental data.
In relation to the example demonstrated here, there emerges an important conclusion
about the description of natural phenomena by physical laws in the form y = yo(x). As
long as such a law is considered as the only basis for the description of the phenomenon,
it is not sufficient for a complete description, since no information is provided about the
properties of the sample space of joint data. Consider a well known example – the law
m = ρV that relates the mass m, the volume V and the density ρ of an object. This
law does not include the restriction m ≥ 0, and is in this aspect not complete. Similar,
but much more complex, examples are met when treating chaotic phenomena and their
strange attractors [14]. For example, the law applied here is a special case of the law
χn+1 = aχn(1 − χn), with a being a constant. Depending on the value of a and the
starting value χ1, the series {χn ;n = 1 . . .} exhibits at large values of parameter n →
∞ either a discrete or a continuous sample space. Moreover, in the continuous case, the
sample space can be comprised of disconnected intervals which could hardly be predicted
analytically. Similar, but still more cumbersome, is the situation if we consider chaotic
processes with continuous parameters. Consequently, a governing law y = yo(x) appears
incomplete for description of the phenomenon. The most outstanding deficiency is that it
does not include information about the structure of the sample space corresponding to the
observed phenomenon. This deficiency does not appear if we consider as a basis for modeling
the probability density function and estimate it nonparametrically, directly from measured
joint data. The extraction of a law that describes a relation between variables can then be
generally performed by using the conditional average estimator. However, applications of
simple parametrical laws, like m = ρV , are of tremendous importance for analytical sciences
and we do not expect that the proposed nonparametric models could substitute for them,
although they are convenient for direct applications. Consequently, the question arises of
how to find a univocal link between both paradigms of modeling.
VI. CONCLUSIONS
Our approach indicates that the objectively introduced kernel estimator provides for a
nonparametric statistical modeling of a quantitatively explored phenomenon. Since no a
priori information about the form of the governing physical law is required, the modeling
can be automatically performed by a computer in a measurement system. The proposed
predictor cost function C provides for estimating the proper number No of data needed for
the modeling. Properties of the predictor cost function resemble those of information cost
function [5, 6], but its estimation is much more simple. The properties of the extracted
model of the governing law can be quantitatively described by the predictor quality Q and
redundancy R of data from which the governing law is extracted. This law represents
the distribution of the variable y at a given value x by a single predicted value yp(x).
Such a compressed representation generally corresponds to creation of information about
the explored phenomenon [5, 6]. This is in contrast to the loss of information caused by
stochastic disturbances in signal transmission channels [20]. If the extraction of information
from observations is considered as a basis of natural intelligence [21, 22], then a system
capable of estimating a physical law from measured data autonomously must be treated as
an intelligent unit. Such an interpretation provides a common basis for a unified treatment
of experimental sciences and natural or artificial intelligence [2, 21, 22].
Acknowledgments
This work was supported by The Ministry of Higher Education, Science and Technology
of the Republic of Slovenia and EU – COST.
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[6] I. Grabec, arXiv:cs.IT/0612027 v1 5, (2006).
[7] E. Parzen, Ann. Math. Stat. 35, 1065 (1962).
[8] R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (J. Wiley and Sons,
New York, 1973), Ch. 4.
[9] J. C. G. Lesurf, Information and Measurement (Institute of Physics Publishing, Bristol, 2002).
[10] J. Risanen, Complexity, Entropy, and the Physics of Information (Addison-Wesley, 1990), ed.
W. H. Zurek, 117-125.
[11] J. Rissanen, IEEE Trans. Inf. Theory 42, 40 (1996).
[12] T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, New
York, 1991).
[13] A. N. Kolmogorov, IRE Trans. Inf. Theory IT-2, 102 (1956).
[14] F. C. Moon, Chaotic and Fractal Dynamics (John Wiley & Sons, INC. New York, 1992).
[15] S. Mandelj, I. Grabec and E. Govekar, Int. J. Bifurcation and Chaos 11, 2731 (2001).
[16] M. Thaler, I. Grabec and A. Poredoš, Physica A 35, 46 (2005).
[17] I. Grabec and D. Grošelj, Comput. Methods in Biomech. Biomed. Engin. 6, 319 (2003)
[18] I. Grabec, I. Ferkolj and D. Grošelj, Proc. of 2nd International Conference on Computational
Intelligence in Medicine and Healthcare, Lisbon, (CIMED-2005 Proceedings, ISBN: 0-86341-
520-2,IEE, 2005), ed. J. M. Fonseca, 311-316
[19] C. H. Bennett, Complexity, Entropy, and the Physics of Information (Addison-Wesley, 1990),
ed. W. H. Zurek, 137-148.
[20] C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (Univ. of Illinois
Press, Urbana, 1949).
[21] S. Haykin, Neural Networks, A Comprehensive Foundation (Mcmillan College Publishing
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[22] D. J. C. MacKay Information Theory, Inference, and Learning Algorithms (Cambridge Uni-
versity Press, Cambridge, UK, 2003)
http://arxiv.org/abs/cs/0612027
Introduction
Fundamentals of nonparametric modeling
Description of kernel function
Nonparametric estimation of PDF pertaining to experimental data
Estimation of a physical law
Characteristics of the model
Predictor quality
Redundancy and predictor cost function
Example
Discussion
Conclusions
Acknowledgments
References
|
0704.0090 | Real Options for Project Schedules (ROPS) | ASA_0.1.ps
Real Options for Project Schedules (ROPS)
Lester Ingber
Lester Ingber Research
Ashland Oregon
[email protected], [email protected]
http://www.ingber.com/
Abstract
Real Options for Project Schedules (ROPS) has three recursive sampling/optimization shells. An outer
Adaptive Simulated Annealing (ASA) optimization shell optimizes parameters of strategic Plans
containing multiple Projects containing ordered Tasks. A middle shell samples probability distributions
of durations of Tasks. An inner shell samples probability distributions of costs of Tasks. PATHTREE is
used to develop options on schedules. Algorithms used for Trading in Risk Dimensions (TRD) are
applied to develop a relative risk analysis among projects.
KEYWORDS: options; simulated annealing; risk management; copula; nonlinear; statistical
† L. Ingber, “Real Options for Project Schedules (ROPS)”, Report 2007:ROPS, Lester Ingber Research,
Ashland, OR, 2007. URL http://www.ingber.com/markets07_rops.pdf.
$Id: markets07_rops,v 1.22 2007/04/01 14:32:24 ingber Exp ingber $
Lester Ingber - 2 - Real Options for Project Schedules (ROPS)
1. Introduction
This paper is a brief description of a methodology of developing options (in the sense of financial options,
e.g., with all Greeks), to be applied in collaboration with Michael Bowman, as a first example to
scheduling a massive US Army project, Future Combat Systems (FCS) [1].
The major focus is to develop Real Options for non-financial projects, as discussed in other earlier
papers [3,4,12]. Data and some guidance on its use has been reported in a previous study of FCS [2,5].
The need for tools for fairly scheduling and pricing such a complex project has been emphasized in
Recommendations for Executive Action in a report by the U.S. General Accounting Office (GAO) on
FCS [14], and they also emphasize the need for management of FCS business plans [13].
2. Goals
A giv en Plan results in S(t), money allocated by the client/government is defined in terms of Projects
Si(t),
S(t) =
Σ Si(t)
where ai(t) may be some scheduled constraints. PATHTREE processes a probability tree developed over
the life of the plan T , divided into N nodes at times {tn}, each with mean epoch length dt [11]. Options,
including all Greeks, familiar to financial markets, are calculated for quite arbitrary nonlinear means and
variances of multiplicative noise [6,9]. This ability to process nonlinear functions in probability
distributions is essential for real-world applications.
Each Task has a range of durations, with nonzero Ai , with a disbursement of funds used, defining Si(tn).
Any Task dependent on a Task completion is slaved to its precursor(s).
We dev elop the Plan conditional probability density (CPD) in terms of differenced costs, dS,
P(S ± dS; tn + dt |S; tn)
P is modeled/cast/fit into the functional form
P(S ± dS; tn + dt |S; tn) = (2π g
2 exp(−Ldt)
(dS − fdt)2
(2g2dt2)
where f and g are nonlinear function of cost S and time t. The g2 variance function absorbs the multiple
Task cost and schedule statistical spreads, to determine P(dS, t), giving rise to the stochastic nature of
dollars spent on the Plan.
A giv en Project i with Task k has a mean duration iik , with a a mean cost Sik . The spread in dS has two
components arising from: (1) a stochastic duration around the mean duration, and (2) a stochastic spread
of mean dollars around a deterministic disbursement at a given time. Different finite-width asymmetric
distributions are used for durations and costs. For example, the distribution created for Adaptive
Simulated Annealing (ASA) [8], originally called Very Fast Simulated Re-annealing [7], is a finite-ranged
distribution with shape determined by a parameter “temperature” q. For each state (whether duration or
cost): (a) A random binary choice can be made to be higher or lower than the mean, using any ratio of
probabilities selected by the client. (b) Then, an ASA distribution is used on the chosen side. Each side
has a different q, each falling off from the mean. This is illustrated and further described in Fig. 1.
At the end of the tree at a time T (T also can be a parameter), there is a total cost at each node S(T ),
called a final “strike” in financial language. (A final strike might also appear at any node before T due to
cancellation of the Project using a particular kind of schedule alternative.) Working backwards, options
are calculated at time t0. Greeks (functional derivatives of the option) assess sensitivity to various
variables, e.g., like those discussed in previous papers [12], but here we deliver precise numbers based on
as much real-world information available.
Lester Ingber - 3 - Real Options for Project Schedules (ROPS)
-1 -0.5 0 0.5 1
ASA (q = 0.1)
1/(2 * (abs(y) + q) * log(1 + 1/q))
Fig. 1. The ASA distribution can be used to develop finite-range asymmetric distributions
from which a value can be chosen for a given state of duration or cost. (a) A random binary
distribution is selected for a lower-than or higher-than mean, using any ratio of probabilities
selected by the client. Each side of the mean has its own temperature q. Here an ASA distri-
bution is given for q = 0.1. The range can be scaled to any finite interval and the mean
placed within this range. (b) A uniform random distribution selects a value from [-1,1], and
a normalized ASA value is read off for the given state.
3. Data
The following data are used to develop Plan CPD. Each Task i has
(a) a Projected allocated cost, Ci
(b) a Projected time schedule, Ti
(c) a CPD with a statistical width of funds spent, SWSi
(d) a distribution with a statistical width of duration, SWTi
(e) a range of durations, RTi
(f) a range of costs, RSi
Expert guesses need to be provided for (c)-(f) for the prototype study.
A giv en Plan must be constructed among all Tasks, specified the ordering of Tasks, e.g., obeying any
sequential constraints among Tasks.
4. Three Recursive Shell
4.1. Outer Shell
There may be several parameters in the Project, e.g., as coefficients of variables in means and variances of
different CPD. These are optimized in an outer shell using ASA [8]. This end product, including
MULTI_MIN states returned by ASA, gives the client flexibility to apply during a full Project [12]. We
may wish to minimize Cost/T , or (CostOverrun - CostInitial)/T , etc.
Lester Ingber - 4 - Real Options for Project Schedules (ROPS)
4.2. Middle Shell
To obtain the Plan CPD, an middle shell of Monte Carlo (MC) states are generated from recursive
calculations. A Weibull or some other asymmetric finite distribution might be used for Task durations.
For a giv en state in the outer middle, a MC state has durations and mean cost disbursements defined for
each Task.
4.3. Inner Shell
At each time, for each Task, the differenced cost ((Sik(t + dt) − Sik(t))) is subjected to a inner shell
stochastic variation, e.g., some asymmetric finite distribution. The net costs dSik(t) for each Project i and
Task k are added to define dS(t) for the Plan. The inner shell cost CPD is re-applied many times to get a
set of {dS} at each time.
5. Real Options
5.1. Plan Options
After the Outer MC sampling is completed, there are histograms generated of the Plan’s dS(t) and
dS(t)/S(t − dt) at each time t. The histograms are normalized at each time to give P(dS, t). At each time
t, the data representing P is “curve-fit” to the form of Eq. (0), where f and g are functions needed to get
good fits, e.g., fitting coefficients of parameters {x}
f = x f 0 + x f 1S + x f 2S
2 + . . .
g = xg0 + xg1S + xg2S
2 + . . .
At each time t, the functions f and g are fit to the function ln((P(dS, t)), which includes the prefactor
containing g and the function L which may be viewed as a Padé approximate of these polynomials.
Complex constraints as functions of Sik(t) can be easily incorporated in this approach, e.g., due to regular
reviews by funding agencies or executives. These P’s are input into PATHTREE to calculate options for a
given strategy or Plan.
5.2. Risk Management of Project Options
If some measure of risk among Projects is desired, then during the MC calculations developed for the top-
level Plan, sets of differenced costs for each Project, dSi(t) and dSi(t)/Si(t − dt), stored from each of the
Project’s Tasks. Then, histograms and Project CPDs are developed, similar to the development of the
Plan CPD. A copula analysis, coded in TRD for risk management of financial markets, are applied to
develop a relative risk analysis among these projects [10]. In such an analysis, the Project marginal CPDs
are all transformed to Gaussian spaces, where it makes sense to calculate covariances and correlations.
An audit trail back the original Project spaces permits analysis of risk dependent on the tails of the Project
CPDs.
6. Generic Applications
ROPS can be applied to any complex scheduling of tasks similar to the FCS project. The need for
government agencies to plan and monitor such large projects is becoming increasingly difficult and
necessary [15]. Many large businesses have similar projects and similar requirements to manage their
complex projects.
Lester Ingber - 5 - Real Options for Project Schedules (ROPS)
References
[1] M. Bowman and L. Ingber, ‘‘Real Options for US Army Future Combat Systems,’’ Report
2007:ROFCS, Lester Ingber Research, Ashland, OR, 2007.
[2] G.G. Brown, R.T. Grose, and R.A. Koyak, ‘‘Estimating total program cost of a long-term, high-
technology, high-risk project with task durations and costs that may increase over time,’’ Military
Operations Research 11, 41-62 (2006). [URL
http://www.nps.navy.mil/orfacpag/resumePages/papers/Brownpa/Estimating_total_
program_cost.pdf]
[3] T.E. Copeland and P.T. Keenan, ‘‘Making real options real,’’ McKinsey Quarterly 128-141 (1998).
[URL http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK98_3.pdf]
[4] G. Glaros, ‘‘Real options for defense,’’ Tr ansformation Trends June, 1-11 (2003). [URL
http://www.oft.osd.mil/library/library_files/trends_205_transforma-
tion_trends_9_june%202003_issue.pdf]
[5] R. Grose, ‘‘Cost-constrained project scheduling with task durations and costs that may increase
over time: Demonstrated with the U.S. Army future combat systems,’’ Thesis, Naval Postgraduate
School, Monterey, CA, 2004. [URL http://www.stormingmedia.us/75/7594/A759424.html]
[6] J.C. Hull, Options, Futures, and Other Derivatives, 4th Edition (Prentice Hall, Upper Saddle River,
NJ, 2000).
[7] L. Ingber, ‘‘Very fast simulated re-annealing,’’ Mathl. Comput. Modelling 12, 967-973 (1989).
[URL http://www.ingber.com/asa89_vfsr.pdf]
[8] L. Ingber, ‘‘Adaptive Simulated Annealing (ASA),’’ Global optimization C-code, Caltech Alumni
Association, Pasadena, CA, 1993. [URL http://www.ingber.com/#ASA-CODE]
[9] L. Ingber, ‘‘Statistical mechanics of portfolios of options,’’ Report 2002:SMPO, Lester Ingber
Research, Chicago, IL, 2002. [URL http://www.ingber.com/markets02_portfolio.pdf]
[10] L. Ingber, ‘‘Trading in Risk Dimensions (TRD),’’ Report 2005:TRD, Lester Ingber Research,
Ashland, OR, 2005.
[11] L. Ingber, C. Chen, R.P. Mondescu, D. Muzzall, and M. Renedo, ‘‘Probability tree algorithm for
general diffusion processes,’’ Phys. Rev. E 64, 056702-056707 (2001). [URL
http://www.ingber.com/path01_pathtree.pdf]
[12] K.J. Leslie and M.P. Michaels, ‘‘The real power of real options,’’ McKinsey Quarterly 4-22 (1997).
[http://faculty.fuqua.duke.edu/˜charvey/Teaching/BA456_2006/McK97_3.pdf]
[13] General Accounting Office, ‘‘Future Combat System Risks Underscore the Importance of
Oversight,’’ Report GAO-07-672T, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-672T]
[14] General Accounting Office, ‘‘Key Decisions to Be Made on Future Combat System,’’ Report
GAO-07-376, GAO, Washington DC, 2007. [URL http://www.gao.gov/cgi-
bin/getrpt?GAO-07-376]
[15] B. Wysocki, Jr, ‘‘Is U.S. Government ’Outsourcing Its Brain’?,’’ Wall Street Journal March 30, 1
(2007).
|
0704.0091 | Groups with finitely many conjugacy classes and their automorphisms | GROUPS WITH FINITELY MANY CONJUGACY CLASSES AND
THEIR AUTOMORPHISMS
ASHOT MINASYAN
Abstract. We combine classical methods of combinatorial group theory with
the theory of small cancellation over relatively hyperbolic groups to construct
finitely generated torsion-free groups that have only finitely many classes of
conjugate elements. Moreover, we present several results concerning embed-
dings into such groups.
As another application of these techniques, we prove that every countable
group C can be realized as a group of outer automorphisms of a group N ,
where N is a finitely generated group having Kazhdan’s property (T) and
containing exactly two conjugacy classes.
1. Introduction
We shall start with
Definition. Suppose that n ≥ 2 is an integer. We will say that a group M has the
property (nCC) if there are exactly n conjugacy classes of elements in M .
Note that a group M has (2CC) if and only if any two non-trivial elements are
conjugate in M . For two elements x, y of some group G, we shall write x
∼ y if x
and y are conjugate in G, and x
≁ y if they are not.
For a group G, denote by π(G) the set of all finite orders of elements of G. A
classical theorem of G. Higman, B. Neumann and H. Neumann ([8]) states that
every countable group G can be embedded into a countable (but infinitely gen-
erated) group M , where any two elements of the same order are conjugate and
π(M) = π(G).
For any integer n ≥ 2, take G = Z/2n−2Z and embed G into a countable group
M according to the theorem above. Then card(π(M)) = card(π(G)) = n − 1.
Since, in addition, M will always contain an element of infinite order, the theorem
of Higman-Neumann-Neumann implies that G has (nCC).
Another way to construct infinite groups with finitely many conjugacy classes
was suggested by S. Ivanov [15, Thm. 41.2], who showed for every sufficiently large
prime p there is an infinite 2-generated groupMp of exponent p possessing exactly p
conjugacy classes. The groupMp is constructed as a direct limit of word hyperbolic
groups, and, as noted in [21], it is impossible to obtain an infinite group with (2CC)
in the same manner.
In the recent paper [21] D. Osin developed a theory of small cancellation over
relatively hyperbolic groups and used it to obtain the following remarkable result:
2000 Mathematics Subject Classification. 20F65, 20E45, 20F28.
Key words and phrases. Conjugacy Classes, Relatively Hyperbolic Groups, Outer Automor-
phism Groups.
This work was supported by the Swiss National Science Foundation Grant ♯ PP002-68627.
http://arxiv.org/abs/0704.0091v2
2 ASHOT MINASYAN
Theorem 1.1 ([21], Thm. 1.1). Any countable group G can be embedded into a
2-generated group M such that any two elements of the same order are conjugate
in M and π(M) = π(G).
Applying this theorem to the group G = Z/2n−2Z one can show that for each
integer n ≥ 2 there exists a 2-generated group with (nCC). And when n = 2 we
get a 2-generated torsion-free group that has exactly two conjugacy classes.
The presence of elements of finite orders in the above constructions was impor-
tant, because if two elements have different orders, they can never be conjugate.
So, naturally, one can ask the following
Question 1. Do there exist torsion-free (finitely generated) groups with (nCC),
for any integer n ≥ 3?
Note that if G is the finitely generated group with (2CC) constructed by Osin,
then the m-th direct power Gm of G is also a finitely generated torsion-free group
which satisfies (2mCC). But what if we want to achieve a torsion-free group with
(3CC)? With this purpose one could come up with
Question 2. Suppose that G is a countable torsion-free group and x, y ∈ G are
non-conjugate. Is it possible to embed G into a groupM , which has (3CC), so that
x and y stay non-conjugate in M?
Unfortunately, the answer to Question 2 is negative as the following example
shows.
Example 1. Consider the group
(1.1) G1 = 〈a, t ‖ tat
−1 = a−1〉
which is isomorphic to the non-trivial semidirect product Z ⋊ Z. Note that G1 is
torsion-free, and t is not conjugated to t−1 in G1 because t ≁ t
−1 in the infinite
cyclic group 〈t〉 which is canonically isomorphic to the quotient of G1 by the normal
closure of a. However, if G1 is embedded into a (3CC)-group M , it is easy to see
that every element of M will be conjugated to its inverse (indeed, if y ∈ M \ {1}
and y
≁ y−1 then yǫ
∼ a−1, for some ǫ ∈ {1,−1}, hence yǫ
∼ y−ǫ – a
contradiction). In particular, t
∼ t−1.
An analog of the above example can be given for each n ≥ 3 – see Section 3. This
example shows that, in order to get a positive result, one would have to strengthen
the assumptions of Question 2.
Let G be a group. Two elements x, y ∈ G are said to be commensurable if there
exist k, l ∈ Z \ {0} such that xk is conjugate to yl. We will use the notation x
if x and y are commensurable in G. In the case when x is not commensurable with
y we will write x
6≈ y. Observe that commensurability, as well as conjugacy, defines
an equivalence relation on the set of elements of G. It is somewhat surprising that
if one replaces the words ”non-conjugate” with the words ”non-commensurable” in
Question 2, the answer becomes positive:
Corollary 1.2. Assume that G is a countable torsion-free group, n ∈ N, n ≥ 2,
and x1, . . . , xn−1 ∈ G \ {1} are pairwise non-commensurable. Then there exists a
group M and an injective homomorphism ϕ : G→M such that
1. M is torsion-free and generated by two elements;
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 3
2. M has (nCC);
3. M is 2-boundedly simple;
4. the elements ϕ(x1), . . . , ϕ(xn−1) are pairwise non-commensurable in M .
Recall that a group G is said to be k-boundedly simple if for any x, y ∈ G \ {1}
there exist l ≤ k and g1, . . . , gl ∈ G such that x = g1yg
1 · · · glyg
l in G. A group
is called boundedly simple if it is k-boundedly simple for some k ∈ N. Evidently
every boundedly simple group is simple; the converse is not true in general. For
example, the infinite alternating group A∞ is simple but not boundedly simple
because conjugation preserves the type of the decomposition of a permutation into
a product of cycles. First examples of torsion-free finitely generated boundedly
simple groups were constructed by A. Muranov (see [12, Thm. 2], [13, Thm. 1]).
Corollary 1.2 is an immediate consequence of a more general Theorem 3.5 that
will be proved in Section 3.
Applying Corollary 1.2 to the group G = F (x1, . . . , xn−1), which is free on the
set {x1, . . . , xn−1}, and its non-commensurable elements x1, . . . , xn−1, we obtain a
positive answer to Question 1:
Corollary 1.3. For every integer n ≥ 3 there exists a torsion-free 2-boundedly
simple group satisfying (nCC) and generated by two elements.
(In the case when n = 2 the above statement was obtained by Osin in [21,
Cor. 1.3].) In fact, for any (finitely generated) torsion-free group H we can set
G = H ∗ F (x1, . . . , xn−1), and then use Corollary 1.2 to embed G into a group M
enjoying the properties 1− 4 from its claim. Since there is a continuum of pairwise
non-isomorphic 2-generated torsion-free groups ([4]), and a finitely generated group
can contain at most countably many of different 2-generated subgroups, this shows
that there must be continually many pairwise non-isomorphic groups satisfying
properties 1− 3 from Corollary 1.2.
Recall that the rank rank(G) of a group G is the minimal number of elements re-
quired to generate G. In Section 4 we show how classical theory of HNN-extensions
allows to construct different embeddings into (infinitely generated) groups that have
finitely many classes of conjugate elements, and in Section 5 we use Osin’s results
(from [21]) regarding quotients of relatively hyperbolic groups to prove
Theorem 1.4. Let H be a torsion-free countable group and let M ⊳H be a non-
trivial normal subgroup. Then H can be isomorphically embedded into a torsion-free
group Q, possessing a normal subgroup N ⊳Q, such that
• Q = H ·N and H ∩N =M (hence Q/N ∼= H/M);
• N has (2CC);
• ∀ x, y ∈ Q \ {1}, x
∼ y if and only if ϕ(x)
∼ ϕ(y), where ϕ : Q → Q/N
is the natural homomorphism;
• rank(N) = 2 and rank(Q) ≤ rank(H/M) + 2.
This theorem implies that if Q/N ∼= H/M has exactly (n− 1) conjugacy classes
(e.g., if it is finite), then the group Q will have (nCC) and will not be simple
(if n ≥ 3). Thus it may be used to build (nCC)-groups in a recursive manner.
It also allows to obtain embeddings of countable torsion-free groups into (nCC)-
groups, which we could not get by using Corollary 1.2. For instance, as we saw in
Example 1, the fundamental group of the Klein bottle G1, given by (1.1), can not
be embedded into a (3CC)-group M so that t
≁ t−1. However, with 4 conjugacy
4 ASHOT MINASYAN
classes this is already possible: see Corollary 5.5 in Section 5. The idea is as
follows: the group G1 can be mapped onto Z/3Z in such a way that the images of
the elements t and t−1 are distinct. Let M be the kernel of this homomorphism.
One can apply Theorem 1.4 to the pair (G1,M) to obtain the required embedding
of G1 into a group Q. And since Z/3Z has exactly 3 conjugacy classes, the group
Q will have (4CC).
An application of Theorem 1.4 to the case when H = Z and M = 2Z⊳H also
provides an affirmative answer to a question of A. Izosov from [9, Q. 11.42], asking
whether there exists a torsion-free (3CC)-group Q that contains a normal subgroup
N of index 2.
The goal of the second part of this article is to show that every countable group
can be realized as a group of outer automorphisms of some finitely generated (2CC)-
group. This problem has some historical background: in [11] T. Matumoto proved
that every group is a group of outer automorphisms of some group (in contrast,
there are groups, e.g., Z, that are not full automorphism groups of any group); M.
Droste, M. Giraudet, R. Göbel ([7]) showed that for every group C there exists
a simple group S such that Out(S) ∼= C; I. Bumagina and D. Wise in [3] proved
that each countable group C is isomorphic to Out(N) where N is a 2-generated
subgroup of a countable C′(1/6)-group, and if, in addition, C is finitely presented
then one can choose N to be residually finite.
In Section 6 we establish a few useful statements regarding paths in the Cayley
graph of a relatively hyperbolic group G, and apply them in Section 7 to obtain
small cancellation quotients of G satisfying certain conditions. Finally, in Section
8 we prove the following
Theorem 1.5. Let C be an arbitrary countable group. Then for every non-elemen-
tary torsion-free word hyperbolic group F1 there exists a torsion-free group N sat-
isfying the following properties:
• N is a 2-generated quotient of F1;
• N has (2CC);
• Out(N) ∼= C.
The principal difference between this theorem and the result of [3] is that our
group N is torsion-free and simple. Moreover, if one applies Theorem 1.5 to the
case when F1 is a torsion-free hyperbolic group with Kazhdan’s property (T) (and
recalls that every quotient of a group with property (T) also has (T)), one will get
Corollary 1.6. For any countable group C there is a 2-generated group N such
that N has (2CC) and Kazhdan’s property (T), and Out(N) ∼= C.
The reason why Kazhdan’s property (T) is interesting in this context is the
question from [6, p. 134] which asked whether there exist groups that satisfy
property (T) and have infinite outer automorphism groups (it can be motivated
by a theorem of F. Paulin [22] which claims that the outer automorphism group is
finite for any word hyperbolic group with property (T)). Positive answers to this
question were obtained (using different methods) by Y. Ollivier and D. Wise [14],
Y. de Cornulier [5], and I. Belegradek and D. Osin [2]. Corollary 1.6 not only
shows that the group of outer automorphisms of a group N with property (T)
can be infinite, but also demonstrates that there are no restrictions whatsoever on
Out(N).
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 5
Acknowledgements. The author would like to thank D. Osin for fruitful dis-
cussions and encouragement.
2. Relatively hyperbolic groups
Assume that G is a group, {Hλ}λ∈Λ is a fixed collection of subgroups of G (called
peripheral subgroups), and X is a subset of G. The subset X is called a relative
generating set of G with respect to {Hλ}λ∈Λ if G is generated by X ∪
λ∈ΛHλ. In
this case G a quotient of the free product
F = (∗λ∈ΛHλ) ∗ F (X ),
where F (X ) is the free group with basis X . Let R be a subset of F such that the
kernel of the natural epimorphism F → G is the normal closure of R in the group
F ; then we will say that G has relative presentation
(2.1) 〈X , {Hλ}λ∈Λ ‖ R = 1, R ∈ R〉.
If the sets X and R are finite, the relative presentation (2.1) is said to be finite.
Set H =
λ∈Λ(Hλ \ {1}). A finite relative presentation (2.1) is said to satisfy a
linear relative isoperimetric inequality if there exists C > 0 such that, for every word
w in the alphabet X ∪H (for convenience, we will further assume that X−1 = X )
representing the identity in the group G, one has
f−1i R
i fi,
with equality in the group F , where Ri ∈ R, fi ∈ F , for i = 1, . . . , k, and k ≤ C‖w‖,
where ‖w‖ is the length of the word w.
The next definition is due to Osin (see [20]):
Definition. the group G is called hyperbolic relative to (the collection of peripheral
subgroups) {Hλ}λ∈Λ, if G admits a finite relative presentation (2.1) satisfying a
linear relative isoperimetric inequality.
This definition is independent of the choice of the finite generating set X and
the finite set R in (2.1) (see [20]). We would also like to note that, in general, it
does not require the group G to be finitely generated, which will be important in
this paper. The definition immediately implies the following basic facts:
Remark 2.1 ([20]). (a) Let {Hλ}λ∈Λ be an arbitrary family of groups. Then the
free product G = ∗λ∈ΛHλ will be hyperbolic relative to {Hλ}λ∈Λ.
(b) Any word hyperbolic group (in the sense of Gromov) is hyperbolic relative
to the family {{1}}, where {1} denotes the trivial subgroup.
Recall that a group H is called elementary if it has a cyclic subgroup of finite
index. Further in this section we will assume that G is a non-elementary group
hyperbolic relative to a family of proper subgroups {Hλ}λ∈Λ.
An element g ∈ G is said to be parabolic if it is conjugated to an element of Hλ
for some λ ∈ Λ. Otherwise g is said to be hyperbolic. Given a subgroup S ≤ G, we
denote by S0 the set of all hyperbolic elements of S of infinite order.
Lemma 2.2 ([17], Thm. 4.3, Cor. 1.7). For every g ∈ G0 the following conditions
hold.
6 ASHOT MINASYAN
1) The element g is contained in a unique maximal elementary subgroup EG(g)
of G, where
(2.2) EG(g) = {f ∈ G : fg
nf−1 = g±n for some n ∈ N}.
2) The group G is hyperbolic relative to the collection {Hλ}λ∈Λ ∪ {EG(g)}.
Recall that a non-trivial subgroup H ≤ G is called malnormal if for every g ∈
G \H , H ∩ gHg−1 = {1}. The next lemma is a special case of Theorem 1.4 from
[20]:
Lemma 2.3. For any λ ∈ Λ and any g /∈ Hλ, the intersection Hλ ∩ gHλg
−1 is
finite. If h ∈ G, µ ∈ Λ and µ 6= λ, then the intersection Hλ ∩ hHµh
−1 is finite. In
particular, if G is torsion-free then Hλ is malnormal (provided that Hλ 6= {1}).
Lemma 2.4 ([20], Thm. 2.40). Suppose that a group G is hyperbolic relative
to a collection of subgroups {Hλ}λ∈Λ ∪ {S1, . . . , Sm}, where S1, . . . , Sm are word
hyperbolic (in the ordinary non-relative sense). Then G is hyperbolic relative to
{Hλ}λ∈Λ.
Lemma 2.5 ([19], Cor. 1.4). Let G be a group which is hyperbolic relative to a
collection of subgroups {Hλ}λ∈Λ ∪ {K}. Suppose that K is finitely generated and
there is a monomorphism α : K → Hν for some ν ∈ Λ. Then the HNN-extension
〈G, t ‖ txt−1 = α(x), x ∈ K〉 is hyperbolic with respect to {Hλ}λ∈Λ.
In [21] Osin introduced the following notion: a subgroup S ≤ G is suitable if
there exist two elements g1, g2 ∈ S
0 such that g1
6≈ g2 and EG(g1)∩EG(g2) = {1}.
For any S ≤ G with S0 6= ∅, one sets
(2.3) EG(S) =
EG(g)
which is obviously a subgroup of G normalized by S. Note that EG(S) = {1} if the
subgroup S is suitable in G. As shown in [1, Lemma 3.3], if S is non-elementary
and S0 6= ∅ then EG(S) is the unique maximal finite subgroup of G normalized by
Lemma 2.6. Let {H}λ∈Λ be a family of groups and let F be a torsion-free non-
elementary word hyperbolic group. Then the free product G = (∗λ∈ΛHλ) ∗ F is
hyperbolic relative to {Hλ}λ∈Λ and F is a suitable subgroup of G.
Proof. Indeed, G is hyperbolic relative to {Hλ}λ∈Λ by Remark 2.1 and Lemma
2.4. Since F is non-elementary, there are elements of infinite order x, y ∈ F such
that x
6≈ y (see, for example, [16, Lemma 3.2]). Evidently, x and y are hyperbolic
elements of G that are not commensurable with each other, and the subgroups
EG(x) = EF (x) ≤ F , EG(y) = EF (y) ≤ F are cyclic (as elementary subgroups of a
torsion-free group). Hence EG(x) ∩ EG(y) = {1}, and thus F is suitable in G. �
Lemma 2.7 ([21], Lemma 2.3). Suppose that G is a group hyperbolic relative to a
family of subgroups {Hλ}λ∈Λ and S ≤ G is a suitable subgroup. Then one can find
infinitely many pairwise non-commensurable (in G) elements g1, g2, · · · ∈ S
0 such
that EG(gi) ∩ EG(gj) = {1} for all i 6= j.
The following theorem was proved by Osin in [21] using the theory of small
cancellation over relatively hyperbolic groups, and represents our main tool for
obtaining new quotients of such groups having a number of prescribed properties:
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 7
Theorem 2.8 ([21], Thm. 2.4). Let G be a torsion-free group hyperbolic relative to
a collection of subgroups {Hλ}λ∈Λ, let S be a suitable subgroup of G, and let T, U
be arbitrary finite subsets of G. Then there exist a group G1 and an epimorphism
η : G→ G1 such that:
(i) The restriction of η to
λ∈ΛHλ ∪ U is injective, and the group G1 is hy-
perbolic relative to the collection {η(Hλ)}λ∈Λ;
(ii) for every t ∈ T , we have η(t) ∈ η(S);
(iii) η(S) is a suitable subgroup of G1;
(iv) G1 is torsion-free;
(v) the kernel ker(η) of η is generated (as a normal subgroup of G) by a finite
collection of elements belonging to T · S.
We have slightly changed the original formulation of the above theorem from
[21], demanding the injectivity on V =
λ∈ΛHλ ∪ U (instead of just
λ∈ΛHλ)
and adding the last point concerning the generators of the kernel. The latter
follows from the explicit form of the relations, imposed on G (see the proof of
Thm. 2.4 in [21]), and the former – from part 2 of Lemma 5.1 in [21] and the fact
that any element from V has length (in the alphabet X ∪ H) at most N , where
N = max{|h|X∪H : h ∈ U}+ 1.
3. Groups with finitely many conjugacy classes
Lemma 3.1. Let G be a group and let x1, x2, x3, x4 ∈ G be elements of infinite order
such that x1
6≈ xi, i = 2, 3, 4. Let H = 〈G, t ‖ tx3t
−1 = x4〉 be the HNN-extension
of G with associated cyclic subgroups generated by x3 and x4. Then x1
6≈ x2.
Proof. Arguing by contradiction, assume that hxl1h
−1xm2 = 1 for some h ∈ H ,
l,m ∈ Z \ {0}. The element h has a reduced presentation of the form
h = g0t
ǫ1g1t
ǫ2 . . . tǫkgk
where g0, . . . , gk ∈ G, ǫ1, . . . , ǫk ∈ Z \ {0}, and
gj /∈ 〈x3〉 if 1 ≤ j ≤ k − 1 and ǫj > 0, ǫj+1 < 0
gj /∈ 〈x4〉 if 1 ≤ j ≤ k − 1 and ǫj < 0, ǫj+1 > 0
By the assumptions, x1
6≈ x2 hence k ≥ 1, and in the group H we have
(3.1) hxl1h
−1xm2 = g0t
ǫ1g1t
ǫ2 . . . tǫkgkx
−ǫk . . . t−ǫ2g−11 t
−ǫ1 g̃0 = 1,
where g̃0 = g
2 ∈ G. By Britton’s Lemma (see [10, IV.2]), the left hand side
in (3.1) can not be reduced, and this can happen only if gkx
k belongs to either
〈x3〉 or 〈x4〉 in G, which would contradict the assumptions. Thus the lemma is
proved. �
Definition. Suppose that G is a group and Xi ⊂ G, i ∈ I, is a family of subsets.
We shall say that Xi, i ∈ I, are independent if no element of Xi is commensurable
with an element of Xj whenever i 6= j, i, j ∈ I.
Lemma 3.2. Assume that G is a countable torsion-free group, n ∈ N, n ≥ 2, and
non-empty subsets Xi ⊂ G \ {1}, i = 1, . . . , n− 1, are independent in G. Then G
can be (isomorphically) embedded into a countable torsion-free group M in such a
way that M has (nCC) and the subsets Xi, i = 1, . . . , n− 1, remain independent in
8 ASHOT MINASYAN
Proof. For each i = 1, . . . , n− 1, fix an element xi ∈ Xi. First we embed G into a
countable torsion-free group G1 such that for each non-trivial element g ∈ G there
exist j ∈ {1, . . . , n− 1} and t ∈ G1 satisfying tgt
−1 = xj in G1, and the subsets Xi,
i = 1, . . . , n− 1, stay independent in G1.
Let g1, g2, . . . be an enumeration of all non-trivial elements of G. Set G(0) = G
and suppose that we have already constructed the group G(k), containing G, so
that for each l ∈ {1, . . . , k} there is j ∈ {1, . . . , n− 1} such that the element gl is
conjugated in G(k) to xj , and Xi, i = 1, . . . , n− 1, are independent in G(k).
Suppose, at first, that gk+1 is commensurable in G(k) with an element of Xj
for some j. Then gk+1
6≈ h for every h ∈
i=1,i6=j
Xi. Define G(k + 1) to be the
HNN-extension 〈G(k), tk+1 ‖ tk+1gk+1t
k+1 = xj〉. By Lemma 3.1 the subsets Xi,
i = 1, . . . , n− 1, will remain independent in G(k + 1).
Thus we can assume that gk+1 is not commensurable with any element from
i=1 Xi in G(k). According to the induction hypotheses one can apply Lemma 3.1
to the HNN-extension
G(k + 1) = 〈G(k), tk+1 ‖ tk+1gk+1t
k+1 = x1〉
to see that the subsets Xi ⊂ G ≤ G(k + 1), i = 1, . . . , n − 1, are independent in
G(k + 1).
Now, setG1 =
k=0G(k). EvidentlyG1 has the required properties. In the same
manner, one can embed G1 into a countable torsion-free group G2 so that each non-
trivial element of G1 will be conjugated to xi in G2, for some i ∈ {1, . . . , n − 1},
and the subsets Xi, i = 1, . . . , n− 1, continue to be independent in G2.
Proceeding like that we obtain the desired groupM =
s=1Gs. By the construc-
tion, M is a torsion-free countable group which has exactly n conjugacy classes:
[1], [x1], . . . , [xn−1]. The subsets Xi, i = 1, . . . , n− 1, are independent in M because
they are independent in Gs for each s ∈ N. �
Corollary 3.3. In Lemma 3.2 one can add that the groupM is 2-boundedly simple.
Proof. Let a torsion-free countable group G and its non-empty independent subsets
Xi, i = 1, . . . , n− 1, be as in Lemma 3.2. Let F = F (a1, . . . , an−1, b1, . . . , bn−1) be
the free group with the free generating set {a1, . . . , an−1, b1, . . . , bn−1}, and consider
the group Ḡ = G ∗ F . For each i = 1, . . . , n− 1, define
X̄i = Xi ∪ {ai, a
i } ∪ {[aj, bi] | j = 1, . . . , n− 1, j 6= i} ⊂ Ḡ,
where [aj , bi] = ajbia
i . Using the universal properties of free groups and free
products one can easily see that the subsets X̄i, i = 1, . . . , n− 1, are independent
in Ḡ.
Now we apply Lemma 3.2 to find a countable torsion-free (nCC)-group M , con-
taining Ḡ, such that X̄i, i = 1, . . . , n− 1, are independent in M . Observe that this
implies that for any given i = 1, . . . , n− 1, any two elements of X̄i are conjugate in
M . For arbitrary x, y ∈ M \ {1} there exist i, j ∈ {1, . . . , n − 1} such that x
and y
∼ aj . If i = j then x
∼ y. Otherwise, y
∼ a−1j and x
∼ [aj , bi] which
is a product of two conjugates of aj , and, hence, of y. Therefore the group M is
2-boundedly simple, and since G ≤ Ḡ ≤M , the corollary is proved. �
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 9
Below is a particular (torsion-free) case of a theorem proved by Osin in [21, Thm.
2.6]:
Lemma 3.4. Any countable torsion-free group S can be embedded into a 2-generated
group M so that S is malnormal in M and every element of M is conjugated to an
element of S in M .
Proof. Following Osin’s proof of Theorem 2.6 from [21], we see that the required
group M can be constructed as an inductive limit of relatively hyperbolic groups
G(i), i ∈ N. More precisely, one sets G(0) = S ∗ F2, where F2 is a free group of
rank 2, ξ0 = idG(0) : G(0) → G(0), and for each i ∈ N one constructs a group
G(i) and an epimorphism ξi : G(0) → G(i) so that ξi is injective on S, G(i) is
torsion-free and hyperbolic relative to {ξi(S)}, and ξi factors through ξi−1. The
group M is defined to be the direct limit of (G(i), ξi) as i → ∞, i.e., Q = G(0)/N
where N =
i∈N ker(ξi). By Lemma 2.3, ξi(S) is malnormal in G(i), hence the
image of S will also be malnormal in M . �
Theorem 3.5. Let G be a torsion-free countable group, n ∈ N, n ≥ 2, and non-
empty subsets Xi ⊂ G \ {1}, i = 1, . . . , n − 1, be independent in G. Then G can
be embedded into a 2-generated torsion-free group M which has (nCC), so that the
subsets Xi, i = 1, . . . , n− 1, stay independent in M . Moreover, one can choose M
to be 2-boundedly simple.
Proof. First, according to Corollary 3.3, we can embed the groupG into a countable
torsion-free group S such that S has (nCC) and is 2-boundedly simple, and Xi,
i = 1, . . . , n − 1, are independent in S. Second, we apply Lemma 3.4 to find the
2-generated group M from its claim. Choose any i, j ∈ {1, . . . , n − 1}, i 6= j, and
x ∈ Xi, y ∈ Xj. If x and y were commensurable inM , the malnormality of S would
imply that x and y must be commensurable in S, contradicting the construction.
Hence Xi, i = 1, . . . , n − 1, are independent in M . Since each element of M is
conjugated to an element of S, it is evident that M has (nCC), is torsion-free and
2-boundedly simple. �
Remark 3.6. A more direct proof of Theorem 3.5, not using Lemma 3.4, can be
extracted from the proof of Theorem 5.1 (see Section 5), applied to the case when
H =M .
It is easy to see that Theorem 3.5 immediately implies Corollary 1.2 that was
formulated in the Introduction. As promised, we now give a counterexample to
Question 2 (formulated in the Introduction) for any n ≥ 3.
Example 2. Let G2 = 〈a, t ‖ tat
−1 = a2〉 be the Baumslag-Solitar BS(1, 2)-group.
ThenG2 is torsion-free, and the elements t
2, t4, . . . , t2
are pairwise non-conjugate
in G2 (since this holds in the quotient of G2 by the normal closure of a). Suppose
that G2 is embedded into a group M having (nCC) so that t
2, t4, . . . , t2
pairwise non-conjugate in M . Then t2, . . . , t2
is the list of representatives of all
non-trivial conjugacy classes of M . Therefore there exist k, l ∈ {1, . . . , n− 1} such
that t
and a
. Consequently
and t2
hence k = l = n− 1 according to the assumptions. But this yields
n−1 M
∼ t2,
10 ASHOT MINASYAN
implying that t2
∼ t4, which contradicts our assumptions.
Thus G2 can not be embedded into a (nCC)-group M in such a way that
t2, . . . , t2
remain pairwise non-conjugate in M .
4. Normal subgroups with (nCC)
If M is a normal subgroup of a group H , then H naturally acts on M by con-
jugation. We shall say that this action preserves the conjugacy classes of M if for
any h ∈ H and a ∈M there exists b ∈M such that hah−1 = bab−1.
Lemma 4.1. Let G be a torsion-free group, N ⊳ G and x1, . . . , xl ∈ N \ {1}
be pairwise non-commensurable (in G) elements. Then there exists a partition
N \ {1} =
k=1Xk of N \ {1} into a (disjoint) union of G-independent subsets
X1, . . . , Xl such that xk ∈ Xk for every k ∈ {1, . . . , l}. Moreover, each subset Xk
will be invariant under conjugation by elements of G.
Proof. Since
≈ is an equivalence relation on G\{1}, one can find the corresponding
decomposition: G \ {1} =
j∈J Yj , where Yj is an equivalence class for each j ∈ J .
For each k = 1, . . . , l, there exists j(k) ∈ J such that xk ∈ Yj(k). Note that
j(k) 6= j(m) if k 6= m since xk
6≈ xm.
Denote J ′ = J \ {j(1), . . . , j(l − 1)},
X1 = Yj(1) ∩N, . . . , Xl−1 = Yj(l−1) ∩N, and Xl =
Yj ∩N.
EvidentlyN\{1} =
k=1Xk, X1, . . . , Xl are independent subsets of G and xk ∈ Xk
for each k = 1, . . . , l. The final property follows from the construction since for any
a ∈ G and j ∈ J we have aYja
−1 = Yj and aNa
−1 = N . �
Lemma 4.2. For every countable group C and each n ∈ N, n ≥ 2, there exists a
countable torsion-free group H having a normal subgroup M ⊳H such that
(i) M satisfies (nCC);
(ii) M is 2-boundedly simple;
(iii) the natural action of H on M preserves the conjugacy classes of M ;
(iv) H/M ∼= C.
Proof. Let H ′0 be the free group of infinite countable rank. Choose N
0 so that
H ′0/N
∼= C. Let F = F (x1, . . . , xn−1) denote the free group freely generated by
x1, . . . , xn−1. Define H0 = H
0 ∗ F and let N0 be the normal closure of N
0 ∪ F in
H0. Evidently, H0/N0 ∼= H
∼= C and the elements x1, . . . , xn−1 ∈ N0 \ {1} are
pairwise non-commensurable in H0.
By Lemma 4.1, one can choose a partition of N0 \ {1} into the union of H0-
independent subsets:
N0 \ {1} =
so that xk ∈ X0k for each k = 1, . . . , n− 1.
By Corollary 3.3 there exists a countable torsion-free 2-boundedly simple group
M1 with the property (nCC) containing a copy of N0, such that the subsets X0k,
k = 1, 2, . . . , n − 1, are independent in M1. Denote by H1 = H0 ∗N0 M1 the
amalgamated product of H0 and M1 along N0, and let N1 be the normal closure
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 11
of M1 in H1. Note that H1 is torsion-free as an amalgamated product of two
torsion-free groups ([10, IV.2.7]).
We need to verify that the elements x1, . . . , xn−1 are pairwise non-commen-
surable in H1. Indeed, if a ∈ X0k and b ∈ X0l, k 6= l, are conjugate in H1 then
there must exist y1, . . . , yt ∈ M1 \N0 and z1, . . . , zt−1 ∈ H0 \N0, z0, zt ∈ H0 such
z0y1 · · · zt−1ytztaz
t−1 · · · y
Suppose that t is minimal possible with this property. As conjugation by elements
of H0 preserves X0k and X0l, we can assume that z0, zt = 1. Hence
y1z1 · · · zt−1ytay
t−1 · · · z
−1 H1= 1.
By the properties of amalgamated products (see [10, Ch. IV]), the left-hand side
in this equality can not be reduced, consequently ytay
t ∈ N0 \ {1} =
k=1 X0k.
But then ytay
t ∈ X0k by the properties of M1, contradicting the minimality of t.
Thus, we have shown that xk
6≈ xl whenever k 6= l.
Assume that the groupHi = Hi−1∗Ni−1Mi, i ≥ 1, has already been constructed,
so that
0) Hi is countable and torsion-free;
1) Ni−1 ⊳Hi−1;
2) Hi−1 = H0 ·Ni−1 and H0 ∩Ni−1 = N0;
3) Mi satisfies (nCC);
4) x1, . . . , xn−1 are pairwise non-commensurable in Hi.
Let Ni be the normal closure of Mi in Hi. Because of the condition 4) and
Lemma 4.1, one can find a partition of Ni \ {1} into a union of Hi-independent
subsets:
Ni \ {1} =
so that xk ∈ Xik for each k = 1, . . . , n − 1. By Lemma 3.2 there is a countable
group a Mi+1, with (nCC), containing a copy of Ni, in which the subsets Xik,
i = 1, . . . , n− 1, remain independent. Set Hi+1 = Hi ∗Ni Mi+1. Now, it is easy to
verify that the analogs of the conditions 0)-3) hold for Hi+1 and
(4.1) Ni−1 ≤Mi ≤ Ni ≤Mi+1.
The analog of the condition 4) is true in Hi+1 by the same considerations as before
(in the case of H1).
Define the group H =
i=1Hi and its subgroup M =
i=1Ni. Observe that
the condition 0) implies that H is torsion-free, condition 1) implies that M is
normal in H , and 2) implies that H = H0 ·M and H0 ∩M = N0. Hence H/M ∼=
H0/(H0∩M) ∼= C. Applying (4.1) we getM =
i=1Mi, and thus, by the conditions
3), 4) it enjoys the property (nCC): each element of M will be a conjugate of xk
for some k ∈ {1, . . . , n− 1}. Since x1, . . . , xn−1 ∈M1 ≤M and M1 is 2-boundedly
simple, then so will be M . Finally, 4) implies that xk
≁ xl whenever k 6= l,
and, consequently, the natural action of H on M preserves its conjugacy classes.
Q.e.d. �
Lemma 4.3. Suppose that G is a group, N ⊳ G, A,B ≤ G and ϕ : A → B is
an isomorphism such that ϕ(a) ∈ aN (i.e., the canonical images of a and ϕ(a) in
12 ASHOT MINASYAN
G/N coincide) for each a ∈ A. Let L = 〈G, t ‖ tat−1 = ϕ(a), ∀ a ∈ A〉 be the
HNN-extension of G with associated subgroups A and B, and let K be the normal
closure of 〈N, t〉 in L . Then G ∩K = N .
Proof. This statement easily follows from the universal property of HNN-extensions
and is left as an exercise for the reader. �
The next lemma will allow us to construct (nCC)-groups that are not simple:
Lemma 4.4. Assume that H is a torsion-free countable group and M⊳H is a non-
trivial normal subgroup. Then H can be isomorphically embedded into a countable
torsion-free group G possessing a normal subgroup K ⊳G such that
1) G = HK and H ∩K =M ;
2) ∀ x, y ∈ G \ {1}, ϕ(x) = ϕ(y) if and only if ∃ h ∈ K such that x = hyh−1,
where ϕ : G → G/K is the natural homomorphism; in particular, K will
have (2CC);
3) ∀ x, y ∈ G \ {1}, x
∼ y if and only if ϕ(x)
∼ ϕ(y);
Proof. Choose a set of representatives Z ⊂ H of cosets of H modulo M , in such a
way that each coset is represented by a unique element from Z and 1 /∈ Z.
DefineG(0)=H andK(0) =M . Enumerate the elements ofG(0)\{1}: g1, g2, . . . .
First we embed the group G(0) into a countable torsion-free group G1, having a
normal subgroup K1 ⊳G1, such that G1 = HK1, H ∩K1 =M and for every i ≥ 0
there are ti ∈ K1 and zi ∈ Z satisfying tigit
i = zi.
Suppose that the (countable torsion-free) group G(j), j ≥ 0, and K(j) ⊳ G(j),
have already been constructed so that H ≤ G(j), G(j) = HK(j), H ∩K(j) = M
and, if j ≥ 1, then tjgjt
j = zj for some tj ∈ K(j) and zj ∈ Z. The group G(j+1),
containing G(j), is defined as the following HNN-extension:
G(j + 1) = 〈G(j), tj+1 ‖ tj+1gj+1t
j+1 = zj+1〉,
where zj+1 ∈ Z ⊂ H is the unique representative satisfying gj+1 ∈ zj+1K(j) in
G(j). Denote by K(j+1)⊳G(j+1) the normal closure of 〈K(j), tj+1〉 in G(j+1).
Evidently the group G(j + 1) is countable and torsion-free, H ≤ G(j) ≤ G(j + 1),
G(j + 1) = HK(j + 1) and H ∩K(j + 1) = H ∩K(j) =M by Lemma 4.3.
Now, it is easy to verify that the group G1 =
j=0G(j) and its normal subgroup
j=0K(j) enjoy the required properties.
In the same way we can embed G1 into a countable torsion-free group G2, that
has a normal subgroup K2⊳G2, so that G2 = HK2, H∩K2 =M and each element
of G1 \ {1} is conjugated in G2 to a corresponding element of Z. Performing such
a procedure infinitely many times we achieve the group G =
i=1Gi and a normal
subgroup K =
i=1Ki ⊳ G that satisfy the claims 1) and 2) of the lemma. It is
easy to see that the claim 2) implies 3), thus the proof is finished. �
5. Adding finite generation
Theorem 5.1. Assume that H is a countable torsion-free group and M is a non-
trivial normal subgroup of H. Let F be an arbitrary non-elementary torsion-free
word hyperbolic group. Then there exist a countable torsion-free group Q, containing
H, and a normal subgroup N ⊳Q with the following properties:
1. H is malnormal in Q;
2. Q = H ·N and N ∩H =M ;
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 13
3. N is a quotient of F ;
4. the centralizer CQ(N) of N in Q is trivial;
5. for every q ∈ Q there is z ∈ H such that q
Proof. The group Q will be constructed as a direct limit of relatively hyperbolic
groups.
Step 0 . Set G(0) = H ∗F and F (0) = F ; then G(0) is hyperbolic relative to its
subgroupH and F (0) is a suitable subgroup of G(0) by Lemma 2.6. LetN(0)⊳G(0)
be the normal closure of the subgroup 〈M,F 〉 in G(0). Evidently G(0) = H ·N(0)
and H ∩ N(0) = M . Enumerate all the elements of N(0): {g0, g1, g2, . . . }, and of
G(0): {q0, q1, q2, . . . }, in such a way that g0 = q0 = 1.
Steps 0-i . Assume the groups G(j), j = 0, . . . , i, i ≥ 0, have been already
constructed, so that
1◦. for each 1 ≤ j ≤ i there is an epimorphism ψj−1 : G(j−1) → G(j) which is
injective on (the image of) H in G(j − 1). Denote F (j) = ψj−1(F (j − 1)),
N(j) = ψj−1(N(j − 1));
2◦. G(j) is torsion-free and hyperbolic relative to (the image of) H , and F (j) ≤
G(j) is a suitable subgroup, j = 0, . . . , i;
3◦. G(j) = H ·N(j), N(j)⊳G(j) and H ∩N(j) =M , j = 0, . . . , i;
4◦. the natural image ḡj of gj in G(j) belongs to F (j), j = 0, . . . , i;
5◦. there exists zj ∈ H such that q̄j
∼ zj, j = 0, . . . , i, where q̄j is the image
of qj in G(j).
Step i+1 . Let q̂i+1 ∈ G(i), ĝi+1 ∈ N(i) be the images of qi+1 and gi+1 in G(i).
First we construct the group G(i+1/2), its normal subgroup Ki+1 and its element
ti+1 as follows.
If for some f ∈ G(i), f q̂i+1f
−1 = z ∈ H , then set G(i + 1/2) = G(i), Ki+1 =
N(i)⊳G(i + 1/2) and ti+1 = 1.
Otherwise, q̂i+1 is a hyperbolic element of infinite order in G(i). Since G(i) is
torsion-free, the elementary subgroup EG(i)(q̂i+1) is cyclic, thus EG(i)(q̂i+1) = 〈hx〉
for some h ∈ H and x ∈ N(i) (by 3◦), and q̂i+1 = (hx)
m for some m ∈ Z. Now, by
Lemma 2.2, G(i) is hyperbolic relative to {H, 〈hx〉}. Choose y ∈M so that hy 6= 1
and let G(i + 1/2) be the following HNN-extension of G(i):
G(i + 1/2) = 〈G(i), ti+1 ‖ ti+1(hx)t
i+1 = hy〉.
The group G(i+ 1/2) is torsion-free and hyperbolic relative to H by Lemma 2.5.
Let us now verify that the subgroup F (i) is suitable in G(i + 1/2). Indeed,
according to Lemma 2.7, there are two hyperbolic elements f1, f2 ∈ F (i) of infinite
order in G(i) such that fl
6≈ hx, fl
6≈ hy, l = 1, 2, and f1
6≈ f2. Then
G(i+1/2)
6≈ f2 by Lemma 3.1. It remains to check that fl is a hyperbolic element
of G(i + 1/2) for each l = 1, 2. Choose an arbitrary element w ∈ H and observe
that fl
6≈ w (since H is malnormal in G(i) by Lemma 2.3, a non-trivial power of
fl is conjugated to an element of H if and only if fl is conjugated to an element of
H in G(i), but the latter is impossible because fl is hyperbolic in G(i)). Applying
Lemma 3.1 again, we get that fl
G(i+1/2)
6≈ w for any w ∈ H . Hence f1, f2 ∈ F (i) are
hyperbolic elements of infinite order in G(i+1/2). The intersection EG(i+1/2)(f1)∩
14 ASHOT MINASYAN
EG(i+1/2)(f2) must be finite, since these groups are virtually cyclic (by Lemma 2.2),
and f1 is not commensurable with f2 in G(i+1/2). But G(i+ 1/2) is torsion-free,
therefore EG(i+1/2)(f1) ∩EG(i+1/2)(f2) = {1}. Thus F (i) is a suitable subgroup of
G(i+ 1/2).
Lemma 4.3 assures that H ∩Ki+1 =M where Ki+1 ⊳G(i + 1/2) is the normal
closure of 〈N(i), ti+1〉 in G(i + 1/2). Finally, note that
ti+1q̂i+1t
i+1 = ti+1(hx)
mt−1i+1 = (hy)
m = z ∈ H in G(i + 1/2).
Now, that the group G(i+ 1/2) has been constructed, set Ti+1 = {ĝi+1, ti+1} ⊂
Ki+1 and define G(i+ 1) as follows. Since Ti+1 ·F (i) ⊂ Ki+1 ⊳G(i+1/2), we can
apply Theorem 2.8 to find a group G(i+1) and an epimorphism ϕi : G(i+1/2) →
G(i + 1) such that ϕi is injective on H , G(i + 1) is torsion-free and hyperbolic
relative to (the image of) H , {ϕi(ĝi+1), ϕi(ti+1)} ⊂ ϕi(F (i)), ϕi(F (i)) is a suitable
subgroup of G(i + 1), and ker(ϕi) ≤ Ki+1. Denote by ψi the restriction of ϕi on
G(i). Then ψi(G(i)) = ϕi(G(i)) = G(i + 1) because G(i + 1/2) was generated by
G(i) and ti+1, and according to the construction, ti+1 ∈ ϕi(F (i)) ≤ ϕi(G(i)). Now,
after defining F (i+1) = ψi(F (i)), N(i+1) = ψi(N(i)), ḡi+1 = ϕi(ĝi+1) ∈ F (i+1)
and zi+1 = ϕi(z) ∈ H , we see that the conditions 1
◦,2◦,4◦ and 5◦ hold in the case
when j = i+1. The properties G(i+1) = H ·N(i+1) and N(i+1)⊳G(i+1) are
immediate consequences of their analogs for G(i) and N(i). Finally, observe that
ϕ−1i (H ∩N(i+ 1)) = H · ker(ϕi) ∩N(i) · ker(ϕi) =
H ∩N(i) · ker(ϕi)
· ker(ϕi)
H ∩Ki+1
· ker(ϕi) =M · ker(ϕi).
Therefore H ∩N(i+ 1) =M and the condition 3◦ holds for G(i + 1).
Let Q = G(∞) be the direct limit of the sequence (G(i), ψi) as i → ∞, and let
F (∞) and N = N(∞) be the limits of the corresponding subgroups. Then Q is
torsion-free by 2◦, N ⊳ Q, Q = H · N and H ∩N = M by 3◦. N ≤ F (∞) by 4◦,
and 5◦ implies the condition 5 from the claim.
Since F (0) ≤ N(0) we get F (∞) ≤ N . Thus N = F (∞) is a homomorphic
image of F (0) = F .
For any i, j ∈ N ∪ {∞}, i < j, we have a natural epimorphism ζij : G(i) → G(j)
such that if i < j < k then ζjk ◦ ζij = ζik. Take any g ∈ G(0). Since F = F (0)
is finitely generated, using the properties of direct limits one can show that if
w = ζ0∞(g) ∈ CQ(F (∞)) in Q, then ζ0j(g) ∈ CG(j)(F (j)) for some j ∈ N. But
CG(j)(F (j)) ≤ EG(j)(F (j)) = {1} (by formulas (2.2) and (2.3)) because F (j) is
a suitable subgroup of G(j), hence w = ζj∞
ζ0j(g)
= 1, that is, CQ(F (∞)) =
CQ(N) = {1}. This concludes the proof. �
The next statement is well-known:
Lemma 5.2. Assume G is a group and N ⊳ G is a normal subgroup such that
CG(N) ⊆ N , where CG(N) is the centralizer of N in G. Then the quotient-group
G/N embeds into the outer automorphism group Out(N).
Proof. The action of G on N by conjugation induces a natural homomorphism
ϕ from G to the automorphism group Aut(N) of N . Since ϕ(N) is exactly the
group of inner automorphisms Inn(N) of N , one can define a new homomorphism
ϕ̄ : G/N → Out(N) = Aut(N)/Inn(N) in the natural way: ϕ̄(gN) = ϕ(g)Inn(N)
for every gN ∈ G/N . It remains to check that ϕ̄ is injective, i.e., if g ∈ G \N then
ϕ̄(gN) 6= 1 in Out(N); or, equivalently, ϕ(g) /∈ Inn(N). Indeed, otherwise there
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 15
would exist a ∈ N such that ghg−1 = aha−1 for every h ∈ N , thus N 6∋ a−1g ∈
CG(N), contradicting the assumptions. Q.e.d. �
Note that for an arbitrary group N , any subgroup C ≤ Out(N) naturally acts
on the set of conjugacy classes C(N) of the group N .
Theorem 5.3. For any n ∈ N, n ≥ 2, and an arbitrary countable group C, C can
be isomorphically embedded into the outer automorphism group Out(N) of a group
N satisfying the following conditions:
• N is torsion-free;
• N is generated by two elements;
• N has (nCC) and the natural action of C on C(N) is trivial;
• N is 2-boundedly simple.
Proof. By Lemma 4.2 we can find a countable torsion-free group H and its normal
subgroup M enjoying the properties (i)-(iv) from its claim. Now, if F denotes
the free group of rank 2, we can obtain a countable torsion-free group Q together
with its normal subgroup N that satisfy the conditions 1-5 from the statement of
Theorem 5.1.
Then N is torsion-free and generated by two elements (as a quotient of F ).
Condition 2 implies that Q/N ∼= H/M ∼= C and, by 4 and Lemma 5.2, C embeds
into the group Out(N).
Using property 5, for each g ∈ N we can find u ∈ Q and z ∈ H such that
ugu−1 = z ∈ N ∩ H = M . Since Q = HN , there are h ∈ H and x ∈ N
such that u = hx. Since z, h−1zh ∈ M and the action of H on M preserves
the conjugacy classes of M , there is r ∈ M such that rh−1zhr−1 = z, hence
z = rh−1ugu−1(rh−1)−1 = rxgx−1r−1, where v = rx ∈ N . Thus for every g ∈ N
there is v ∈ N such that vgv−1 ∈ M . Evidently, this implies that N is also 2-
boundedly simple. Since M has (nCC), the number of conjugacy classes in N will
be at most n.
Suppose x1, x2 ∈ M and x1
≁ x2. Then x1
≁ x2 (by the property (iii) from
the claim of Lemma 4.2), and since H is malnormal in Q we get x1
≁ x2. Hence
≁ x2, i.e., N also enjoys (nCC).
The fact that the natural action of C on C(N) is trivial follows from the same
property for the action of H on C(M) and the malnormality of H in Q. Q.e.d. �
Now, let us proceed with the
Proof of Theorem 1.4. First we apply Lemma 4.4 to construct a group G and a
normal subgroup K ⊳ G according to its claim. Now, by Theorem 5.1, there is a
groupQ, having a normal subgroupN⊳Q such that G is malnormal in Q, Q = GN ,
G ∩ N = K, rank(N) ≤ 2 (if one takes the free group of rank 2 as F ) and every
element q ∈ Q is conjugated (in Q) to an element of G. By claim 2) of Lemma 4.4,
K has (2CC), and an argument, similar to the one used in the proof of Theorem
5.3, shows that N will also have (2CC). Consequently, rank(N) > 1 because N is
torsion-free, hence rank(N) = 2.
Since G = HK and H ∩ K = M we have Q = HKN = HN and H ∩ N =
H ∩K =M . Since Q/N ∼= H/M and N can be generated by two elements, we can
conclude that rank(Q) ≤ rank(H/M) + 2.
16 ASHOT MINASYAN
Consider arbitrary x, y ∈ Q \ {1} and suppose that ϕ(x)
∼ ϕ(y). By Theorem
5.1, there are w, z ∈ G \ {1} such that x
∼ w and y
∼ z. Therefore ϕ(w)
∼ ϕ(z),
hence the images of w and z in G/K are also conjugate. By claim 3) of Lemma
4.4, w
∼ z, implying x
∼ y. �
Theorem 1.4 provides an alternative way of obtaining torsion-free groups that
have finitely many conjugacy classes: for any countable group C we can choose
a free group H of countable rank and a normal subgroup {1} 6= M ⊳ H so that
H/M ∼= C, and then apply Theorem 1.4 to the pair (H,M) to get
Corollary 5.4. Assume that n ∈ N, n ≥ 2, and C is a countable group that
contains exactly (n− 1) distinct conjugacy classes. Then there exists a torsion-free
group Q and N ⊳Q such that
• Q/N ∼= C;
• N has (2CC) and Q has (nCC);
• rank(N) = 2 and rank(Q) ≤ rank(C) + 2.
Corollary 5.5. The group G1, given by presentation (1.1), can be isomorphically
embedded into a 2-generated torsion-free group Q satisfying (4CC) in such a way
that t
≁ t−1.
Proof. Denote by K the kernel of the homomorphism ϕ : G1 → Z3, for which
ϕ(a) = 0 and ϕ(t) = 1, where Z3 is the group of integers modulo 3. Now, apply
Theorem 1.4 to the pair (G1,K) to find the group Q, containing G1, and the normal
subgroup N ⊳ Q from its claim. Since Q/N ∼= G1/K ∼= Z3 has (3CC), the group
Q will have (4CC). We also have t
≁ t−1 because the images of t and t−1 are not
conjugate in Q/N .
Choose an element q1 ∈ Q \ N . Then q2 = q
1 ∈ N \ {1} and since N is 2-
generated and has (2CC), there is q3 ∈ N such that N = 〈q2, q3〉 in Q. As Q/N
is generated by the image of q1, the group Q will be generated by {q1, q2, q3}, and,
consequently, by {q1, q3}. Q.e.d. �
6. Combinatorics of paths in relatively hyperbolic groups
Let G be a group hyperbolic relative to a family of proper subgroups {Hλ}λ∈Λ,
and let X be a finite symmetrized relative generating set of G. Denote H =
λ∈Λ (Hλ \ {1}).
For a combinatorial path p in the Cayley graph Γ(G,X ∪H) (of G with respect
to X ∪H) p−, p+, L(p), and lab(p) will denote the initial point, the ending point,
the length (that is, the number of edges) and the label of p respectively. p−1 will
be the path obtained from p by following it in the reverse direction. Further, if Ω
is a subset of G and g ∈ 〈Ω〉 ≤ G, then |g|Ω will be used to denote the length of a
shortest word in Ω±1 representing g.
We will be using the following terminology from [20]. Suppose q is a path in
Γ(G,X ∪ H). A subpath p of q is called an Hλ-component for some λ ∈ Λ (or
simply a component) of q, if the label of p is a word in the alphabet Hλ \ {1} and
p is not contained in a bigger subpath of q with this property.
Two components p1, p2 of a path q in Γ(G,X ∪ H) are called connected if they
are Hλ-components for the same λ ∈ Λ and there exists a path c in Γ(G,X ∪ H)
connecting a vertex of p1 to a vertex of p2 such that lab(c) entirely consists of letters
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 17
from Hλ. In algebraic terms this means that all vertices of p1 and p2 belong to the
same coset gHλ for a certain g ∈ G. We can always assume c to have length at
most 1, as every nontrivial element of Hλ is included in the set of generators. An
Hλ-component p of a path q is called isolated if no other Hλ-component of q is
connected to p.
The next statement is a particular case of Lemma 2.27 from [20]; we shall for-
mulate it in a slightly more general form, as it appears in [18, Lemma 2.7]:
Lemma 6.1. Suppose that a group G is hyperbolic relative to a family of subgroups
{Hλ}λ∈Λ. Then there exists a finite subset Ω ⊆ G and a constant K ∈ N such
that the following holds. Let q be a cycle in Γ(G,X ∪H), p1, . . . , pk be a collection
of isolated components of q and g1, . . . , gk be the elements of G represented by
Lab(p1), . . . ,Lab(pk) respectively. Then g1, . . . , gk belong to the subgroup 〈Ω〉 ≤ G
and the word lengths of gi’s with respect to Ω satisfy
|gi|Ω ≤ KL(q).
Definition. Suppose that m ∈ N and Ω is a finite subset of G. Define W(Ω,m) to
be the set of all words W over the alphabet X ∪H that have the following form:
W ≡ x0h0x1h1 . . . xlhlxl+1,
where l ∈ Z, l ≥ −2 (if l = −2 then W is the empty word; if l = −1 then W ≡ x0),
hi and xi are considered as single letters and
1) xi ∈ X ∪{1}, i = 0, . . . , l+1, and for each i = 0, . . . , l, there exists λ(i) ∈ Λ
such that hi ∈ Hλ(i);
2) if λ(i) = λ(i + 1) then xi+1 /∈ Hλ(i) for each i = 0, . . . , l− 1;
3) hi /∈ {h ∈ 〈Ω〉 : |h|Ω ≤ m}, i = 0, . . . , l.
Choose the finite subset Ω ⊂ G and the constant K > 0 according to the claim
of Lemma 6.1.
Recall that a path q in Γ(G,X ∪ H) is said to be without backtracking if all of
its components are isolated.
Lemma 6.2. Let q be a path in the Cayley graph Γ(G,X ∪ H) with Lab(q) ∈
W(Ω,m) and m ≥ 5K. Then q is without backtracking.
Proof. Assume the contrary to the claim. Then one can choose a path q providing
a counterexample of the smallest possible length. Thus if p1, . . . , pl is the (consec-
utive) list of all components of q then l ≥ 2, p1 and pl must be connected Hλ′ -
components, for some λ′ ∈ Λ, the components p2, . . . , pl−1 must be isolated, and q
starts with p1 and ends with pl. Since Lab(q) ∈ W(Ω,m) we have L(q) ≤ 2l− 1.
If l = 2 then the (X ∪ {1})-letter between p1 and p2 would belong to Hλ′
contradicting the property 2) from the definition of W(Ω,m).
Therefore l ≥ 3. Since p1 and pl are connected, there exists a path v in Γ(G,X ∪
H) between (pl)− and (p1)+ with Lab(v) ∈ Hλ′ (thus we can assume that L(v) ≤ 1).
Denote by q̂ the subpath of q starting with (p1)+ and ending with (pl)−. Note that
L(q̂) = L(q)−2 ≤ 2l−3, and p2, . . . , pl−1 is the list of components of q̂, all of which
are isolated. If one of them were connected to v it would imply that it is connected
to p1 contradicting with the minimality of q. Hence the cycle o = q̂v possesses
18 ASHOT MINASYAN
k = l − 2 ≥ 1 isolated components, which represent elements h1, . . . , hk ∈ H.
Consequently, applying Lemma 6.1 one obtains that hi ∈ 〈Ω〉, i = 1, . . . , k, and
|hi|Ω ≤ KL(o) ≤ K(L(q̂) + 1) ≤ K(2l− 2).
By the condition 3) from the definition of W(Ω,m) one has |hi|Ω > m ≥ 5K for
each i = 1, . . . , k. Hence
k · 5K ≤
|hi|Ω ≤ K(2l− 2), or 5 ≤
2l − 2
which contradicts the inequality k ≥ l − 2. Q.e.d. �
Definition. Consider an arbitrary cycle o = rqr′q′ in Γ(G,X ∪ H), where Lab(q)
and Lab(q′) belong to W(Ω,m). Let p be a component of q (or q′). We will say
that p is regular if it is not an isolated component of o. If m ≥ 5K, and hence q and
q′ are without backtracking by Lemma 6.2, this means that p is either connected
to some component of q′ (respectively q), or to a component of r or r′.
Lemma 6.3. In the above notations, suppose that m ≥ 7K and denote C =
max{L(r),L(r′)}. Then
(a) if C ≤ 1 then every component of q or q′ is regular;
(b) if C ≥ 2 then each of q and q′ can have at most 4C components which are
not regular.
(c) if l is the number of components of q, then at least (l− 6C) of components
of q are connected to components of q′; and two distinct components of q
can not be connected to the same component of q′. Similarly for q′.
Proof. Assume the contrary to (a). Then one can choose a cycle o = rqr′q′ with
L(r),L(r′) ≤ 1, having at least one isolated component on q or q′, and such that
L(q) + L(q′) is minimal. Clearly the latter condition implies that each component
of q or q′ is an isolated component of o. Therefore q and q′ together contain k
distinct isolated components of o, representing elements h1, . . . , hk ∈ H, where
k ≥ 1 and k ≥ (L(q) − 1)/2 + (L(q′) − 1)/2. Applying Lemma 6.1 we obtain
hi ∈ 〈Ω〉, i = 1, . . . , k, and
|hi|Ω ≤ KL(o) ≤ K(L(q) + L(q
′) + 2).
Recall that |hi|Ω > m ≥ 7K by the property 3) from the definition of W(Ω,m).
Therefore
i=1 |hi|Ω ≥ k · 7K, implying
L(q′)
L(q)− 1
L(q′)− 1
which yields a contradiction.
Let us prove (b). Suppose that C ≥ 2 and q contains more than 4C isolated
components of o. We shall consider two cases:
Case 1. No component of q is connected to a component of q′. Then a com-
ponent of q or q′ can be regular only if it is connected to a component of r or r′.
Since, by Lemma 6.2, q and q′ are without backtracking, two distinct components
of q or q′ can not be connected to the same component of r (or r′). Hence q and
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 19
q′ together can contain at most 2C regular components. Thus the cycle o has k
isolated components, representing elements h1, . . . , hk ∈ H, where k ≥ 4C > 4 and
k ≥ (L(q)−1)/2+(L(q′)−1)/2−2C. By Lemma 6.1, hi ∈ 〈Ω〉 for each i = 1, . . . , k,
i=1 |hi|Ω ≤ K(L(q) + L(q
′) + 2C). Once again we can use the property 3)
from the definition of W(Ω,m) to achieve
L(q′)
L(q)− 1
L(q′)− 1
− 2C + 1 + 3C
L(q)− 1
L(q′)− 1
≤ 2 +
yielding a contradiction.
Case 2. The path q has at least one component which is connected to a com-
ponent of q′. Let p1, . . . , pl denote the sequence of all components of q. By part
(a), if ps and pt, 1 ≤ s ≤ t ≤ l, are connected to components of q
′, then for any
j, s ≤ j ≤ t, pj is connected to some component of q
′ (because q is without back-
tracking by Lemma 6.2). We can take s (respectively t) to be minimal (respectively
maximal) possible. Consequently p1, . . . , ps−1, pt+1, . . . , pl will contain the set of
all isolated components of o that belong to q, and none of these components will
be connected to a component of q′.
Without loss of generality we may assume that s− 1 ≥ 4C/2 = 2C. Since ps is
connected to some component p′ of q′, there exists a path v in Γ(G,X∪H) satisfying
v− = (ps)−, v+ = p
+, Lab(v) ∈ H ∪ {1}, L(v) ≤ 1. Let q̄ (respectively q̄
′) denote
the subpath of q (respectively q′) from q− to (ps)− (respectively from p
+ to q
Consider a new cycle ō = rq̄vq̄′. Reasoning as before, one can show that ō has k
isolated components, where k ≥ 2C ≥ 4 and k ≥ (L(q̄)−1)/2+(L(q̄′)−1)/2−C−1.
Now, an application of Lemma 6.1 to the cycle ō together with the property 3) from
the definition of W(Ω,m) will lead to a contradiction as before.
By the symmetry, the statement (b) of the lemma also holds for q′.
The claim (c) follows from (b) and the estimate L(r) + L(r′) ≤ 2C because if
two different components p and p̄ of q were connected to the same component of
some path in Γ(G,X ∪H), then p and p̄ would also be connected with each other,
which would contradict Lemma 6.2. �
Lemma 6.4. In the previous notations, let m ≥ 7K, C = max{L(r),L(r′)}, and
let p1, . . . , pl, p
1, . . . , p
l′ be the consecutive lists of the components of q and q
respectively If l ≥ 12max{C, 1} + 2, then there are indices s, t, s′ ∈ N such that
1 ≤ s ≤ 6C + 1, l − 6max{C, 1} ≤ t ≤ l and for every i ∈ {0, 1, . . . , t − s}, the
component ps+i of q is connected to the component p
s′+i of q
Proof. By part (c) of Lemma 6.3, there exists s ≤ 6C +1 such that the component
ps is connected to a component p
s′ for some s
′ ∈ {1, . . . , l′}. Thus there is a path
r1 between (p
s′)+ and (ps)+ with L(r1) ≤ 1. Consider a new cycle o1 = r1q1r
where q1 is the segment of q from (ps)+ to q+ = r
− and q
1 is the segment of q
from q′− = r
+ to (p
s′)+.
Observe that ps+1, . . . , pl is the list of all components of q1 and l−s ≥ l−6C−1 ≥
6max{1, C}+ 1, hence, according to part (c) of Lemma 6.3 applied to o1, there is
t ≥ l − 6max{1, C} > s such that pt is connected to p
t′ by means of a path r
where s′ + 1 ≤ t′ ≤ l′, (r′1)− = (pt)+, (r
1)+ = (p
t′)+ and L(r
1) ≤ 1. Consider
20 ASHOT MINASYAN
s′+i′ p
Figure 1.
the cycle o2 = r1q2r
2 in which q2 and q
2 are the segments of q1 and q
1 from
(ps)+ = (r1)+ to (pt)+ and from (p
t′)+ to (p
s′)+ = (r1)− respectively (Fig. 1).
Note that ps+1, . . . , pt is the list of all components of q2 and p
s′+1, . . . , p
t′ is the
list of all components of q′2
. The cycle o2 satisfies the assumptions of part (a) of
Lemma 6.3, therefore for every i ∈ {1, . . . , t − s} there exists i′ ∈ {1, . . . , t′ − s′}
such that ps+i is connected to p
s′+i′ (ps+i can not be connected to r1 [r
1] because in
this case it would be connected to ps [pt], but q is without backtracking by Lemma
6.2).
It remains to show that i′ = i for every such i. Indeed, if i′ < i for some
i ∈ {1, . . . , t − s} then one can consider the cycle o3 = r1q3r
3, where q3 and
q′3 are segments of q2 and q
2 from (q2)− = (r1)+ to (ps+i)+ and from (p
s′+i′ )+ to
(q′2)+ = (r1)− respectively, and (r
3)− = (q3)+, (r
3)+ = (q
3)−, L(r
3) ≤ 1. According
to part (a) of Lemma 6.3, each of the components ps+1, . . . , ps+i of q3 must be
connected to one of p′s′+1, . . . , p
s′+i′ . Hence, since i
′ < i, two distinct components
of q3 will be connected to the same component of q
, which is impossible by part
(c) of Lemma 6.3.
The inequality i′ > i would lead to a contradiction after an application of a
symmetric argument to q′3. Therefore i
′ = i and the lemma is proved. �
Lemma 6.5. In the above notations, let m ≥ 7K and C = max{L(r),L(r′)}. For
any positive integer d there exists a constant L = L(C, d) ∈ N such that if L(q) ≥ L
then there are d consecutive components ps, . . . , ps+d−1 of q and p
s′ , . . . , p
s′+d−1 of
q′−1, so that ps+i is connected to p
s′+i for each i = 0, . . . , d− 1.
Proof. Choose the constant L so that (L − 1)/2 ≥ 12max{C, 1} + 2 + d. Let
p1, . . . , pl be the consecutive list all components of q. Since Lab(q) ∈ W(Ω,m), we
have l ≥ (L − 1)/2 (due to the form of any word from W(Ω,m)). Thus we can
apply Lemma 6.4 to find indices s, t from its claim. By the choice of s and t, and
the estimate on l, we have t− s ≥ d+ 1, yielding the statement of the lemma. �
Corollary 6.6. Let G be a group hyperbolic relative to a family of proper subgroups
{Hλ}λ∈Λ. Suppose that a ∈ Hλ0 , for some λ0 ∈ Λ, is an element of infinite order,
and x1, x2 ∈ G \ Hλ0 . Then there exists k ∈ N such that g = a
k1x1a
k2x2 is a
hyperbolic element of infinite order in G whenever |k1|, |k2| ≥ k.
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 21
Proof. Without loss of generality we can assume that x1, x2 ∈ X , since relative
hyperbolicity does not depend on the choice of the finite relative generating set
([20, Thm. 2.34]). Choose the finite subset Ω ⊂ G and the constant K ∈ N
according to the claim of Lemma 6.1, and set m = 7K. As the order of a is infinite,
there is k ∈ N such that ak
/∈ {h ∈ 〈Ω〉 : |h|Ω ≤ m} whenever |k
′| ≥ k. Assume
that |k1|, |k2| ≥ k.
Suppose, first, that gl = 1 for some l ∈ N. Consider the cycle o = rqr′q′ in
Γ(G,X ∪ H) where q− = q+ = 1, Lab(q) ≡ (a
k1x1a
k2x2)
l ∈ W(Ω,m) (akj are
considered as single letters from the alphabet X ∪H) and r, r′, q′ are trivial paths
(consisting of a single point). Then, by part (a) of Lemma 6.3, every component of
q must be regular in o, which is impossible since q is without backtracking according
to Lemma 6.2. Hence g has infinite order in G.
Suppose, now, that there exists λ′ ∈ Λ, u ∈ Hλ′ and y ∈ G such that ygy
−1 = u.
Denote C = |y|X∪H. Since element u ∈ G has infinite order, there exists l ∈ N such
that 2l ≥ 6C+2 and ul /∈ {h ∈ 〈Ω〉 : |h|Ω ≤ m}. The equality yg
ly−1u−l = 1 gives
rise to the cycle o = rqr′q′ in Γ(G,X∪H), where r and r′ are paths of length C whose
labels represent y in G, r− = 1, q− = r+ = y, Lab(q) ≡ (a
k1x1a
k2x2)
l ∈ W(Ω,m),
r′− = q+, q
− = r
+ = y(a
k1x1a
k2x2)
ly−1 and Lab(q′) ≡ u−l ∈ W(Ω,m), L(q′) = 1.
By part (c) of Lemma 6.3, at least 2l − 6C ≥ 2 distinct components of q must
be connected to distinct components of q′, which is impossible as q′ has only one
component. The contradiction shows that g must be a hyperbolic element of G. �
Lemma 6.7. Let G be a torsion-free group hyperbolic relative to a family of proper
subgroups {Hλ}λ∈Λ, a ∈ Hλ0 \ {1}, for some λ0 ∈ Λ, and t, u ∈ G \Hλ0 . Suppose
that there exists k̂ ∈ N such that for every k ≥ k̂ the element g1 = a
ktakt−1
is commensurable with g2 = a
kuaku−1 in G. Then there are β, γ ∈ Hλ0 and
ǫ, ξ ∈ {−1, 1} such that u = γtξβ, βaβ−1 = aǫ, γ−1aγ = aǫ.
Proof. Changing the finite relative generating set X of G, if necessary, we can
assume that t, u, t−1, u−1 ∈ X . Let the finite subset Ω ⊂ G and the constant
K ∈ N be chosen according to Lemma 6.1. Define m = 7K and suppose that k is
large enough to satisfy ak /∈ {h ∈ 〈Ω〉 : |h|Ω ≤ m}.
Since g1 and g2 are commensurable, there exist l, l
′ ∈ Z\{0} and y ∈ G such that
ygl2y
−1 = gl
1 . Let C = |y|X∪H, d = 8 and L = L(C, d) be the constant from Lemma
6.5. Without loss of generality, assume that 4l ≥ L. Consider the cycle o = rqr′q′
in Γ(G,X ∪ H) such that r and r′ are paths of length C whose labels represent y
in G, r− = 1, q− = r+ = y, Lab(q) ≡ (a
kuaku−1)l ∈ W(Ω,m), L(q) = 4l, r′− = q+,
q′− = r
+ = yg
−1, Lab(q′) ≡ (aktakt−1)l
∈ W(Ω,m), L(q′) = 4l′.
Now, by Lemma 6.5, there are subpaths q̃ = p1s1p2s2p3s3p4 of q and q̃
4 of q
′−1 such that Lab(pi) ≡ a
k, Lab(p′i) ≡ a
ǫk, i = 1, 2, 3, 4, for
some ǫ ∈ {−1, 1} (which depends on the sign of l′), Lab(s1) ≡ Lab(s3) ≡ u,
Lab(s2) ≡ u
−1, Lab(s′1) ≡ Lab(s
3) ≡ t
ξ, Lab(s′2) ≡ t
−ξ, for some ξ ∈ {−1, 1}, and
pi is connected in Γ(G,X∪H) to p
i for each i = 1, 2, 3, 4. Therefore there exist paths
p̃1, p̃2, p̃3, p̃4 whose labels represent the elements α, β, γ, δ ∈ Hλ0 respectively, such
that (p̃1)− = (p1)+, (p̃1)+ = (p
1)+, (p̃2)− = (p
2)+, (p̃2)+ = (p2)+, (p̃3)− = (p3)−,
(p̃3)+ = (p
3)−, (p̃4)− = (p
4)−, (p̃4)+ = (p4)− (see Fig. 2).
The cycles s−11 p̃1s
2p̃2p
2 , s2p̃3s
p̃2 and s
3 p̃3p
3p̃4 give rise to the fol-
lowing equalities in the group G:
u = αtξaǫkβa−k, u = γtξβ and u = a−kγaǫktξδ.
22 ASHOT MINASYAN
p1 s1 p2 s2 p3 s3 p4
p′1 s
p̃1 p̃2 p̃3 p̃4
ak u ak u−1 ak u
aǫkaǫkaǫk
t−ξtξ tξ
Figure 2.
Consequently, recalling that Hλ0 is malnormal (Lemma 2.3) and that t
ξ /∈ Hλ0 , we
βakβ−1a−ǫk = t−ξγ−1αtξ ∈ Hλ0 ∩ t
−ξHλ0t
ξ = {1}, and
a−ǫkγ−1akγ = tξδβ−1t−ξ ∈ Hλ0 ∩ t
ξHλ0t
−ξ = {1}.
(6.1) βakβ−1 = aǫk and γ−1akγ = aǫk
for some β = β(k), γ = γ(k) ∈ Hλ0 and ǫ = ǫ(k), ξ = ξ(k) ∈ {−1, 1}. Note that
the proof works for any sufficiently large k, therefore we can find two mutually
prime positive integers k, k′ with the above properties such that ǫ(k) = ǫ(k′) = ǫ
and ξ(k) = ξ(k′) = ξ. Denote β′ = β(k′) and γ′ = γ(k′), then γtξβ = u = γ′tξβ′,
implying
γ−1γ′ = tξββ′
t−ξ ∈ Hλ0 ∩ t
ξHλ0t
−ξ = {1}.
Hence β′ = β, γ′ = γ,
(6.2) βak
β−1 = aǫk
and γ−1ak
γ = aǫk
It remains to observe that since k and k′ are mutually prime, the formulas (6.1)
and (6.2) together yield
βaβ−1 = aǫ and γ−1aγ = aǫ,
q.e.d. �
7. Small cancellation over relatively hyperbolic groups
Let G be a group generated by a subset A ⊆ G and let O be the set of all
words in the alphabet A±1, that are trivial in G. Then G has a presentation of the
following form:
(7.1) G = 〈A ‖ O〉.
Given a symmetrized set of words R over the alphabet A, consider the group G1
defined by
(7.2) G1 = 〈A ‖ O ∪R〉 = 〈G ‖ R〉.
During the proof of the main result of this section we use presentations (7.2) (or,
equivalently, the sets of additional relators R) that satisfy the generalized small
cancellation condition C1(ε, µ, λ, c, ρ). In the case of word hyperbolic groups this
condition was suggested by Ol’shanskii in [16], and was afterwards generalized to
relatively hyperbolic groups by Osin in [21]. For the definition and detailed theory
we refer the reader to the paper [21], as we will only use the properties, that were
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 23
already established there. The following observation is an immediate consequence
of the definition:
Remark 7.1. Let the constants εj, µj , λ, c, ρj , j = 1, 2, satisfy 0 < λ ≤ 1, 0 ≤ ε1 ≤
ε2, c ≥ 0, 0 < µ2 ≤ µ1, ρ2 ≥ ρ1 > 0. If the presentation (7.2) enjoys the condition
C1(ε2, µ2, λ, c, ρ2) then it also enjoys the condition C1(ε1, µ1, λ, c, ρ1).
We will also assume that the reader is familiar with the notion of a van Kampen
diagram over the group presentation (7.2) (see [10, Ch. V] or [15, Ch. 4]). Let ∆ be
such a diagram. A cell Π of ∆ is called an R-cell if the label of its boundary contour
∂Π (i.e., the word written on it starting with some vertex in the counter-clockwise
direction) belongs to R.
Consider a simple closed path o = rqr′q′ in a diagram ∆ over the presentation
(7.2), such that q is a subpath of the boundary cycle of an R-cell Π and q′ is a
subpath of ∂∆. Let Γ denote the subdiagram of ∆ bounded by o. Assuming that
Γ has no holes, no R-cells and L(r),L(r′) ≤ ε, it will be called an ε-contiguity
subdiagram of Π to ∂∆. The ratio L(q)/L(∂Π) will be called the contiguity degree
of Π to ∂∆ and denoted (Π,Γ, ∂∆).
A diagram is said to be reduced if it has a minimal number of R-cells among all
the diagrams with the same boundary label.
If G is a group hyperbolic relative to a family of proper subgroups {Hi}i∈I , with
a finite relative generating set X , then G is generated by the set A = X ∪
i∈I(Hi \
{1}), and the Cayley graph Γ(G,A) is a hyperbolic metric space [20, Cor. 2.54].
As for every condition of small cancellation, the main statement of the theory is
the following analogue of Greendlinger’s Lemma, claiming the existence of a cell,
large part of whose contour lies on the boundary of the van Kampen diagram.
Lemma 7.2 ([21], Cor. 4.4). Suppose that the group G is generated by a subset A
such that the Cayley graph Γ(G,A) is hyperbolic. Then for any 0 < λ ≤ 1 there is
µ0 > 0 such that for any µ ∈ (0, µ0] and c ≥ 0 there are ε0 ≥ 0 and ρ0 > 0 with the
following property.
Let the symmetrized presentation (7.2) satisfy the C1(ε0, µ, λ, c, ρ0)-condition.
Further, let ∆ be a reduced van Kampen diagram over G1 whose boundary contour
is (λ, c)-quasigeodesic in G. Then, provided ∆ has an R-cell, there exists an R-cell
Π in ∆ and an ε0-contiguity subdiagram Γ of Π to ∂∆, such that
(Π,Γ, ∂∆) > 1− 23µ.
The main application of this particular small cancellation condition is
Lemma 7.3 ([21], Lemmas 5.1 and 6.3). For any 0 < λ ≤ 1, c ≥ 0 and N > 0
there exist µ1 > 0, ε1 ≥ 0 and ρ1 > 0 such that for any symmetrized set of words
R satisfying C1(ε1, µ1, λ, c, ρ1)-condition the following hold.
1. The group G1 defined by (7.2) is hyperbolic relative to the collection of
images {η(Hi)}i∈I under the natural homomorphism η : G→ G1.
2. The restriction of η to the subset of elements having length at most N with
respect to A is injective.
3. Any element that has a finite order in G1 is an image of an element of
finite order in G.
Below is the principal lemma of this section that will later be used to prove
Theorem 1.5.
24 ASHOT MINASYAN
Lemma 7.4. Assume that G is a torsion-free group hyperbolic relative to a family
of proper subgroups {Hi}i∈I , X is a finite relative generating set of G, S is a suitable
subgroup of G and U ⊂ G is a finite subset. Suppose that i0 ∈ I, a ∈ Hi0 \ {1}
and v1, v2 ∈ G are hyperbolic elements which are not commensurable to each other.
Then there exists a word W (x, y) over the alphabet {x, y} such that the following is
true.
Denote w1 = W (a, v1) ∈ G, w2 = W (a, v2) ∈ G, and let 〈〈w2〉〉 be the normal
closure of w2 in G, G1 = G/〈〈w2〉〉 and η : G → G1 be the natural epimorphism.
• η is injective on {Hλ}λ∈Λ ∪ U and G1 is hyperbolic relative to the family
{η(Hλ)}λ∈Λ;
• η(S) is a suitable subgroup of G1;
• G1 is torsion-free;
• η(w1) 6= 1.
Proof. By Lemma 2.7 there are hyperbolic elements v3, v4 ∈ S such that vi
6≈ vj if
1 ≤ i < j ≤ 4. Then by Lemma 2.2, the group G is hyperbolic relative to the finite
collection of subgroups {Hi}i∈I ∪
j=1{EG(vj)}, and generated by the set
A = X ∪
EG(vj)
\ {1}.
Let Ω ⊂ G and K ∈ N denote the finite subset and the constant achieved after an
application of Lemma 6.1 to this new collection of peripheral subgroups.
Define m = 7K, λ = 1/3, c = 2 and N = max{|u|A : u ∈ U} + 1. Choose
µj > 0, εj ≥ 0 and ρj > 0, j = 0, 1, according to the claims of Lemmas 7.2
and 7.3. Let ε = max{ε0, ε1}, and let L = L(C, d) > 0 be the constant given by
Lemma 6.5 where C = ε0 and d = 2. Evidently there exists n ∈ N such that, for
µ = (3ε+ 11)/n, one has
0 < µ ≤ min{µ0, µ1}, 2n(1− 23µ) > L, and 2n > max{ρ0, ρ1}.
F(ε) =
h ∈ 〈Ω〉 : |h| ≤ max{K(32ε+ 70),m}
Since the subset F(ε) is finite, we can find k ∈ N such that ak
1 , v
2 /∈ F(ε)
whenever k′ ≥ k. Consider the word
W (x, y) ≡ xkykxk+1yk+1 . . . xk+n−1yk+n−1.
Let wj ∈ G be the element represented by the wordW (a, vj) in G, j = 1, 2, and let
R be the set of all cyclic shifts ofW (a, v2) and their inverses. By Lemma 2.3, Hi0 ∩
EG(v2) = {1} because G is torsion-free, hence by [21, Thm. 7.5] the presentation
(7.2) satisfies the condition C1(ε, µ, 1/3, 2, 2n), and therefore, by Remark 7.1, it
satisfies the conditions C1(ε0, µ, 1/3, 2, ρ0) and C1(ε1, µ1, 1/3, 2, ρ1).
Observe that w1 6= 1 in G because, otherwise, there would have existed a closed
path q in Γ(G,A) labelled by the word W (a, v1), and, by part (a) of Lemma 6.3,
all components of q would have been regular in the cycle o = rqr′q′ (where r, r′, q′
are trivial paths), which is obviously impossible.
Denote G1 = G/〈〈w2〉〉 and let η : G → G1 be the natural epimorphism. Then,
according to Lemma 7.3, the group G1 is is torsion-free, hyperbolic relative to
{η(Hi)}i∈I∪
j=1{η(EG(vj))} and η is injective on the set
i∈I Hi∪
j=1 EG(vj)∪
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 25
U (because the length in A of any element from this set is at most N). Since any
elementary group is word hyperbolic, G1 is also hyperbolic relative to {η(Hi)}i∈I
(by Lemma 2.4) and η(v3), η(v4) ∈ η(S) become hyperbolic elements of infinite or-
der in G1, that are not commensurable with each other (by Lemma 2.3). Therefore
EG1(η(v3)) ∩ EG2(η(v4)) = {1} (recall that these subgroups are cyclic by Lemma
2.2 and because G1 is torsion-free), and, consequently, η(S) is a suitable subgroup
of G1.
Suppose that η(w1) = 1. By van Kampen’s Lemma there exists a reduced
planar diagram ∆ over the presentation (7.2) with the wordW (a, v1) written on its
boundary. SinceW (a, v1)
6= 1, ∆ possesses at least one R-cell. It was proved in [21,
Lemma 7.1] that any path in Γ(G,A) labelled byW (a, v1) is (1/3, 2)-quasigeodesic,
hence we can apply Lemma 7.2 to find an R-cell Π of ∆ and an ε0-contiguity
subdiagram Γ (containing no R-cells) between Π and ∂∆ such that (Π,Γ, ∂∆) >
1− 23µ. Thus there exists a cycle o = rqr′q′ in Γ(G,A) such that q is labelled by a
subword of (a cyclic shift of) W (a, v2), q
′ is labelled by a subword of (a cyclic shift
of) W (a, v1)
±1, L(r),L(r′) ≤ ε0 = C and
L(q) > (1− 23µ) · L(∂Π) = (1− 23µ) · 2n > L.
In particular, Lab(q),Lab(q′) ∈ W(Ω,m). Therefore we can apply Lemma 6.5 to
find two consecutive components of q that are connected to some components of
q′. Due to the form of the word W (a, v2), one of the formers will have to be an
EG(v2)-component, but q
′ can have only EG(v1)- or Hi0 -components. This yields
a contradiction because EG(v2) 6= EG(v1) and EG(v2) 6= Hi0 . Hence η(w1) 6= 1 in
G1, and the proof is complete. �
8. Every group is a group of outer automorphisms of a (2CC)-group
Lemma 8.1. There exists a word R(x, y) over the two-letter alphabet {x, y} such
that every non-elementary torsion-free word hyperbolic group F1 has a non-elemen-
tary torsion-free word hyperbolic quotient F that is generated by two elements a, b ∈
F satisfying
(8.1) R(a, b)
6= 1, R(a−1, b−1)
= 1, R(b, a)
= 1, R(b−1, a−1)
Proof. Consider the word
R(x, y) ≡ xy101x2y102 . . . x100y200.
Denote by F (a, b) the free group with the free generators a, b. Let
R1 = {R(a, b), R(a
−1, b−1), R(b, a), R(b−1, a−1)},
and R2 be the set of all cyclic permutations of words from R
1 . It is easy to see
that the set R2 satisfies the classical small cancellation condition C
′(1/8) (see [10,
Ch. V]). Denote by Ñ the normal closure of the set
R3 = {R(a
−1, b−1), R(b, a), R(b−1, a−1)}
in F (a, b). Since the symmetrization of R3 also satisfies C
′(1/8), the group F̃ =
F (a, b)/Ñ is a torsion-free ([10, Thm. V.10.1]) word hyperbolic group (because it
has a finite presentation for which the Dehn function is linear by [10, Thm. V.4.4])
such that
R(a, b)
6= 1 but R(a−1, b−1)
= R(b, a)
= R(b−1, a−1)
26 ASHOT MINASYAN
Indeed, if the word R(a, b) were trivial in F̃ then, by Greendlinger’s Lemma [10,
Thm. V.4.4], it would contain more than a half of a relator from (the symmetriza-
tion of) R3 as a subword, which would contradict the fact that R2 enjoys C
′(1/8).
The group F̃ is non-elementary because every torsion-free elementary group is
cyclic, hence, abelian, but in any abelian group the relation R(a−1, b−1) = 1 implies
R(a, b) = 1.
Now, the free product G̃ = F̃ ∗F1 is a torsion-free hyperbolic group. Its subgroups
F̃ and F1 are non-elementary, hence, according to a theorem of Ol’shanskii [16,
Thm. 2], there exists a non-elementary torsion-free word hyperbolic group F and
a homomorphism φ : G̃ → F such that φ(F̃ ) = φ(F1) = F and φ(R(a, b)) 6= 1 in
F . Therefore F is a quotient of F1, the (φ-images of the) elements a, b generate F
and enjoy the required relations. �
We are now ready to prove Theorem 1.5.
Proof of Theorem 1.5. The argument will be similar to the one used to prove The-
orem 5.1.
First, set n = 2 and apply Lemma 4.2 to find a countable torsion-free group H
and a normal subgroup M ⊳H , where H/M ∼= C and M has (2CC) (alternatively,
one could start with a free group H ′ and M ′ ⊳H ′ such that H ′/M ′ ∼= C, and then
apply Lemma 4.4 to the pair (H ′,M ′) to obtain H and M with these properties).
Consider the word R(x, y) and the torsion-free hyperbolic group F , generated by
the elements a, b ∈ F which satisfy (8.1), given by Lemma 8.1. Denote G(−2) =
H ∗ F and let N(−2) be the normal closure of 〈M,F 〉 in G(−2), F (−2) = F ,
R(−2) = {R(a, b)} – a finite subset of F (−2). By Lemma 2.6, G(−2) will be
hyperbolic relative to the subgroup H , G(−2) = H ·N(−2), H ∩N(−2) =M and
F (−2) will be a suitable subgroup of G(−2).
The element a ∈ F (−2) will be hyperbolic in G(−2) and since the group G(−2)
is torsion-free, the maximal elementary subgroup EG(−2)(a) will be cyclic generated
by some element h−2x−2, where h−2 ∈ H , x−2 ∈ N(−2).
Choose y−2 ∈ M so that h−2y−2 6= 1. By Lemmas 2.2 and 2.5, the HNN-
extension
G(−3/2) = 〈G(−2), t−1 ‖ t−1h−2x−2t
−1 = h−2y−2〉
is hyperbolic relative to H . As in proof of Theorem 5.1, one can verify that F (−3)
is a suitable subgroup of G(−3/2), and apply Theorem 2.8 to find an epimorphism
η−2 : G(−3/2) → G(−1) such that G(−1) is a torsion-free group hyperbolic relative
to η−2(H), η−2 is injective on H ∪ R(−2) and η−2(t−1) ∈ F (−1) where F (−1) =
η−2(F (−2)) is a suitable subgroup of G(−1). Hence η−2(G(−2)) = G(−1) as
G(−3/2) was generated by G(−2) and t−1.
Denote N(−1) = η−2(N(−2)), R(−1) = η−2(R(−2)) and ψ−2 = η−2|G(−2) :
G(−2) ։ G(−1). One can show thatG(−1) = H ·N(−1) andH∩N(−1) =M using
the same arguments as in the proof of Theorem 5.1. According to the construction,
we have
η−2(t−1)η−2(a)η−2(t
−1) = η(t−1at
−1) ∈ N(−1) ∩H =M
in G(−1), therefore, since the conjugation by η−2(t−1) is an inner automorphism
of F (−1), we can assume that F (−1) is generated by a−1 and b−1, where a−1 ∈M
and R(a−1, b−1) 6= 1 in F (−1) (because η−2(R(a, b)) 6= 1 in F (−1)).
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 27
Now, if b−1 is not a hyperbolic element of G(−1), i.e., if b−1
G(−1)
∼ c for some
c ∈ H , then c ∈ N(−1)∩H =M , and since M has (2CC) we can find s−1 ∈ G(−1)
such that b−1 = s−1a−1s
−1. In this case we define G(0) = G(−1), N(0) = N(−1),
F (0) = F (−1), R(0) = R(−1), a0 = a−1, s0 = s−1 and ψ−1 = idG(−1).
Otherwise, if b−1 is hyperbolic in G(−1), then we construct the group G(0),
and an epimorphism ψ−1 : G(−1) → G(0) in an analogous way, to make sure that
η−1 is injective on H ∪ R(−1), G(0) torsion-free and hyperbolic relative to (the
image of) H , F (0) = ψ−1(F (−1)) is a suitable subgroup of G(0), G(0) = H ·N(0)
and H ∩N(0) = M where N(0) = ψ−1(N(−1)), and b0 = s0a0s
0 in G(0) where
b0 = ψ−1(b−1), a0 = ψ−1(a−1) for some s0 ∈ G(0)
Enumerate all elements of N(0): {g0, g1, g2, . . . }, and of G(0): {q0, q1, q2, . . . },
so that g0 = q0 = 1.
The groups G(j) together with N(j)⊳G(j), F (j) ≤ G(j), finite subsets R(j) ⊂
G(j), and elements aj, sj ∈ G(j), j = 1, 2, . . . , that we will construct shall satisfy
the following properties:
1◦. for each j ∈ N there is an epimorphism ψj−1 : G(j − 1) → G(j) which is
injective on H ∪R(j− 1). F (j) = ψj−1(F (j − 1)), N(j) = ψj−1(N(j − 1)),
aj = ψj−1(aj−1) ∈M , sj = ψj−1(sj−1) ∈ G(j);
2◦. G(j) is torsion-free and hyperbolic relative to (the image of) H , and F (j) ≤
G(j) is a suitable subgroup generated by aj and sjajs
3◦. G(j) = H ·N(j), N(j)⊳G(j) and H ∩N(j) =M ;
4◦. the natural image ḡj of gj in G(j) belongs to F (j);
5◦. there exists zj ∈ H such that q̄j
∼ zj, where q̄j is the image of qj in G(j);
6◦. if j ≥ 1, q̄j−1 ∈ G(j − 1) \H and for each k̂ ∈ N there is k ≥ k̂ such that
akj−1sj−1a
G(j−1)
6≈ akj−1q̄j−1a
j−1q̄
j−1, then there is a word Rj−1(x, y)
over the two-letter alphabet {x, y} which satisfies
R(j) ∋ ψj−1
Rj−1(aj−1, sj−1aj−1s
6= 1 and
Rj−1(aj−1, q̄j−1aj−1q̄
=1 in G(j).
Suppose that the groups G(0), . . . , G(i) have already been defined. The group
G(i+ 1) will be constructed in three steps.
First, assume that q̄i ∈ G(i) \ H and for each k̂ ∈ N there is k ≥ k̂ such that
aki sia
6≈ aki q̄ia
i . Observe that si /∈ H because, otherwise, one would
have F (i) ⊂ H , which is impossible as F (i) is suitable in G(i). Therefore, by
Corollary 6.6, we can suppose that k is so large that the elements v1 = a
i sia
and v2 = a
i q̄ia
i are hyperbolic in G(i). Applying Lemma 7.4 we can find a
word W (x, y) over {x, y} such that the group G(i+ 1/3) = G(i)/〈〈W (ai, v2)〉〉 and
the natural epimorphism η : G(i) → G(i + 1/3) satisfy the following: η is injective
on H ∪ R(i), G(i + 1/3) is torsion-free and hyperbolic relative to (the image of)
H , η(F (i)) ≤ G(i + 1/3) is a suitable subgroup, and η(W (ai, v1)) 6= 1. Define the
word Ri(x, y) ≡ W (x, x
kyk). Then Ri(ai, siais
i ) = W (ai, v1), Ri(ai, q̄iaiq̄
i ) =
W (ai, v2) in G(i), hence
Ri(ai, siais
6= 1 and η
Ri(ai, q̄iaiq̄
= 1 in G(i + 1/3).
28 ASHOT MINASYAN
If, on the other hand, q̄i ∈ H or there is k̂ ∈ N such that for every k ≥ k̂ one has
aki sia
≈ aki q̄ia
i , then we define G(i+ 1/3) = G(i), η : G(i) → G(i+ 1/3)
to be the identical homomorphism and Ri(x, y) to be the empty word.
Let ĝi+1 and q̂i+1 denote the images of gi+1 and qi+1 in G(i + 1/3), N̂(i) =
η(N(i)), F̂ (i) = η(F (i)) and R̂(i) = η
R(i) ∪ {Ri(ai, siais
. Then, using 3◦,
we get G(i + 1/3) = H · N̂(i) and H ∩ N̂(i) = M because ker(η) ≤ N(i) (as
ai, q̄iaiq̄
i ∈ N(i)).
Now we construct the group G(i + 2/3) in exactly the same way as the group
G(i+ 1/2) was constructed in during the proof of Theorem 5.1.
If for some f ∈ G(i + 1/3), f q̂i+1f
−1 = z ∈ H , then set G(i + 2/3) = G(i),
Ki+1 = N̂(i)⊳G(i + 2/3) and ti+1 = 1.
Otherwise, q̂i+1 is a hyperbolic element of infinite order in G(i + 1/3). Since
G(i + 1/3) is torsion-free, one has EG(i+1/3)(q̂i+1) = 〈hx〉 for some h ∈ H and
x ∈ N̂(i), and there is m ∈ Z such that q̂i+1 = (hx)
m. Now, by Lemma 2.2,
G(i + 1/3) is hyperbolic relative to {H, 〈hx〉}. Choose y ∈ M so that hy 6= 1 and
let G(i+ 2/3) be the following HNN-extension of G(i + 1/3):
G(i + 2/3) = 〈G(i + 1/3), ti+1 ‖ ti+1(hx)t
i+1 = hy〉.
The group G(i + 2/3) is torsion-free and hyperbolic relative to H by Lemma 2.5.
One can show that F̂ (i) is a suitable subgroup of G(i + 2/3) in the same way as
during the proof of Theorem 5.1. Lemma 4.3 assures that H ∩Ki+1 = M where
Ki+1 ⊳G(i+ 2/3) is the normal closure of 〈N̂(i), ti+1〉 in G(i+2/3). Finally, note
ti+1q̂i+1t
i+1 = ti+1(hx)
mt−1i+1 = (hy)
m = z ∈ H in G(i + 2/3).
Define Ti+1 = {ĝi+1, ti+1} ⊂ Ki+1. The group G(i + 1) is constructed from
G(i+2/3) as follows. Since Ti+1 · F̂ (i) ⊂ Ki+1⊳G(i+2/3), we can apply Theorem
2.8 to find a group G(i + 1) and an epimorphism ϕi : G(i + 2/3) → G(i + 1) such
that ϕi is injective on H ∪ R̂(i), G(i + 1) is torsion-free and hyperbolic relative to
(the image of) H , {ϕi(ĝi+1), ϕi(ti+1)} ⊂ ϕi(F̂ (i)), ϕi(F̂ (i)) is a suitable subgroup
of G(i + 1), and ker(ϕi) ≤ Ki+1. Denote by ψi : G(i) → G(i + 1) the composition
ϕi ◦ η. Then ψi(G(i)) = ϕi(G(i)) = G(i + 1) because G(i + 2/3) was generated
by G(i) and ti+1, and according to the construction, ti+1 ∈ ϕi(F̂ (i)) ≤ ϕi(G(i)).
Now, after defining F (i+1) = ψi(F (i)), N(i+1) = ψi(N(i)), R(i+1) = ϕi(R̂(i)),
ḡi+1 = ϕi(ĝi+1) ∈ F (i+1) and zi+1 = ϕi(z) ∈ H , we see that the conditions 1
◦ - 5◦
hold in the case when j = i+ 1, as in the proof of Theorem 5.1. The last property
6◦ follows from the way we constructed the group G(i + 1/3).
Let Q = G(∞) be the direct limit of the sequence (G(i), ψi) as i → ∞, and let
F (∞) and N = N(∞) be the limits of the corresponding subgroups. Let a∞, b∞
and s∞ be the images of a0, b0 and s0 in Q respectively. Then b∞ = s∞a∞s
∞ , Q
is torsion-free by 2◦, N ⊳Q, Q = H ·N and H ∩N =M by 3◦, N ≤ F (∞) by 4◦.
Hence Q/N ∼= H/M ∼= C.
Since F (0) ≤ N(0) we get F (∞) ≤ N . Thus N = F (∞) is a homomorphic
image of F (0) = F , and, consequently, it is a quotient of F1. By 5
◦, for any q ∈ N
there are z ∈ H and p ∈ Q such that pqp−1 = z. Consequently z ∈ H ∩ N = M .
Choose x ∈ N and h ∈ H so that p = hx. Since M has (2CC) and h−1zh ∈ M ,
there is y ∈ M such that yh−1zhy−1 = z, therefore (yx)q(yx)−1 = z ∈ M and
GROUPS WITH FINITELY MANY CONJUGACY CLASSES 29
yx ∈ MN = N . Hence each element q of N will be conjugated (in N) to an
element ofM , and since M has (2CC), therefore the group N will also have (2CC).
The property that CQ(N) = {1} can be established in the same way as in
Theorem 5.1. Therefore the natural homomorphism Q → Aut(N) is injective. It
remains to show that it is surjective, that is for every φ ∈ Aut(N) there is g ∈ Q
such that φ(x) = gxg−1 for every x ∈ N . Since all non-trivial elements of N are
conjugated, after composing φ with an inner automorphism of N , we can assume
that φ(a∞) = a∞. On the other hand, there exist q∞ ∈ N and i ∈ N such that
φ(b∞) = q∞a∞q
∞ and q∞ is the image of qi in Q. Note that s∞ /∈ H because
si ∈ G(i) \ H for every i ∈ N. This implies that H is a proper subgroup of N ,
thus q∞ /∈ H since N = F (∞) = 〈a∞, q∞a∞q
∞ 〉 ≤ Q, and a∞ ∈ H . Hence
q̄i ∈ G(i) \H .
Now we have to consider two possibilities.
Case 1: for each k̂ ∈ N there is k ≥ k̂ such that
aki sia
6≈ aki q̄ia
Then there is a word Ri(x, y) such that the property 6
◦ holds for j = i + 1. And,
since each ψj is injective on {1} ∪Rj (by 2
◦), we conclude that
Ri(a∞, s∞a∞s
∞ ) 6= 1 and Ri(a∞, q∞a∞q
∞ ) = 1 in Q,
which contradicts the injectivity of φ. Hence Case 1 is impossible.
Case 2: the assumptions of Case 1 fail. Then we can use Lemma 6.7 to find
β, γ ∈ H and ǫ, ξ ∈ {−1, 1} such that q̄i = γs
iβ, βaiβ
−1 = aǫi and γ
−1aiγ = a
in G(i). Denote by γ∞ the image γ in Q, and for any y ∈ Q let Cy be the
automorphism of N defined by Cy(x) = yxy
−1 for all x ∈ N .
If ξ = −1 then γ−1∞ a∞γ∞ = a
∞ and φ(b∞) = q∞a∞q
∞ = γ∞s
hence
Aut(N) ∋ Cs∞γ−1∞ ◦ φ :
a∞ 7→ s∞a
∞ = b
b∞ = s∞a∞s
∞ 7→ a
But N has no such automorphisms because R(a∞, b∞) 6= 1 and R(b
∞) = 1 in
N (since N is a quotient of F and 1 6= R(a0, b0) ∈ R(0) in G(0)).
Therefore ξ = 1. Similarly, ǫ = 1, as otherwise we would obtain a contradiction
with the fact that R(a−1∞ , b
∞ ) = 1 in N . Thus
Aut(N) ∋ Cγ−1∞ ◦ φ :
a∞ 7→ a∞
b∞ = s∞a∞s
∞ 7→ s∞a∞s
∞ = b∞
And since a∞ and b∞ generate N we conclude that for all x ∈ N , φ(x) = gxg
where g = γ∞ ∈ Q. Thus the natural homomorphism fromQ to Aut(N) is bijective,
implying that Out(N) = Aut(N)/Inn(N) ∼= Q/N ∼= C. Q.e.d. �
References
[1] G. Arzhantseva, A. Minasyan, D. Osin, The SQ-universality and residual properties of rela-
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http://arxiv.org/abs/math/0605553
30 ASHOT MINASYAN
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295-326.
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rithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100 pp.
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School of Mathematics, University of Southampton, Highfield, Southampton, SO17
1BJ, United Kingdom.
E-mail address: [email protected]
http://arxiv.org/abs/math/0411039
1. Introduction
2. Relatively hyperbolic groups
3. Groups with finitely many conjugacy classes
4. Normal subgroups with (nCC)
5. Adding finite generation
6. Combinatorics of paths in relatively hyperbolic groups
7. Small cancellation over relatively hyperbolic groups
8. Every group is a group of outer automorphisms of a (2CC)-group
References
|
0704.0092 | Energy density for chiral lattice fermions with chemical potential | Energy density for chiral lattice fermions with chemical potential
Christof Gattringera and Ludovit Liptakb
Institut für Physik, FB Theoretische Physik, Universität Graz,
Universitätsplatz 5, 8010 Graz, Austria
Institute of Physics, Slovak Academy of Sciences,
Dúbravská cesta 9, 845 11 Bratislava 45, Slovak Republic
We study a recently proposed formulation of overlap fermions at finite density. In particular we
compute the energy density as a function of the chemical potential and the temperature. It is shown
that overlap fermions with chemical potential approach the correct continuum behavior.
PACS numbers: 11.15.Ha, 12.38.Gc
I. INTRODUCTION
Over the last two decades lattice gauge theory was
turned into a powerful qualitative tool for analyzing
QCD. This progress is in part due to the advances in
algorithms and computer technology, but also on the con-
ceptual side important breakthroughs were made. Most
prominent among these is the correct implementation of
chiral symmetry on the lattice based on the Ginsparg-
Wilson equation for the Dirac operator [1].
An application of lattice techniques which has seen a
lot of attention in recent years, is the study of QCD at
finite temperature. The lattice implementation of the
chemical potential µ, necessary for such an analysis, is
not straightforward, however. It is well known [2], that
a naive introduction leads to µ2/a2 contributions which
diverge in the continuum limit when the lattice spacing
a is sent to zero. For more traditional formulations, such
as the Wilson or staggered Dirac operators, the problem
has been solved by introducing the chemical potential in
the same way as the 4-component of the gauge field.
A satisfactory implementation of the chemical poten-
tial should be compatible with chiral symmetry on the
lattice based on the Ginsparg-Wilson equation. When
attempting to introduce the chemical potential into the
only solution of the Ginsparg-Wilson equation know in
closed form, the overlap operator [3], a potential prob-
lem quickly surfaces: defining the sign function of a non-
hermitian matrix. In [4] Bloch and Wettig proposed a
solution based on an analytic continuation of the sign
function into the complex plane. It was shown, that the
eigenvalue spectra of this construction match the expec-
tations from random matrix theory.
In this letter we analyze the proposal [4] further
and study the energy density of free, massless overlap
fermions with chemical potential. The dependence of the
energy density on µ and the temperature T allows for a
detailed analysis of the lattice formulation at finite den-
sity. Of particular interest will be the question whether
the analytic continuation of the sign function produces
divergent µ2/a2 terms. Our study indicates the absence
of such contributions and we find that the µ and T de-
pendence of the energy density is approached correctly.
II. SETUP OF THE CALCULATION
The overlap Dirac operator D(µ) for fermions with a
chemical potential µ is given as
D(µ) =
[1− γ5 signH(µ)] ,
H(µ) = γ5 [1− aDW (µ)] . (1)
The sign function may be defined through the spectral
theorem for matrices. DW (µ) denotes the usual Wilson
Dirac operator,
DW(µ)x,y = 1
δx,y − (2)
Uj(x)δx+ĵ,y +
Uj(x− ĵ)†δx−ĵ,y
U4(x)e
µa4δx+4̂,y −
U4(x−4̂)†e−µa4δx−4̂,y .
For later use we distinguish between the lattice spacing
a in spatial direction and the temporal lattice constant
a4. Periodic boundary conditions are used in the spatial
directions, while in time direction we apply anti-periodic
boundary conditions. The chemical potential µ is cou-
pled in the usual exponential form [2].
For vanishing µ the Wilson Dirac operator is γ5-
hermitian, i.e., γ5DW (0)γ5 = DW (0)
†. This implies that
H(0) is a hermitian matrix. As soon as the chemical po-
tential µ is turned on, γ5-hermiticity no longer holds, and
H(µ) is a non-hermitian, general matrix. This fact has
two important consequences: Firstly, the eigenvalues of
H(µ) are no longer real and the sign function for a com-
plex number has to be defined in the spectral representa-
tion of signH(µ). Secondly, the spectral representation
has to be formulated using left and right eigenvectors.
This latter problem will be dealt with later when we dis-
cuss the evaluation of signH(µ). For the sign function
of a complex number we use the analytic continuation
proposed in [4] and define the sign function through the
sign of the real part
sign (x+ iy) = sign (x) . (3)
http://arxiv.org/abs/0704.0092v2
The observable we study here is the energy density
defined as
ǫ(µ) =
〈H〉 = 1
H e−β (H−µN )
= (4)
e−β(H−µN )
= − 1
∂ lnZ
Here H is the Hamiltonian of the system, N denotes the
number operator and β = 1/T is the inverse temperature
(in our units the Boltzmann constant k is set to k = 1).
The derivatives in the second line are taken such that
βµ = c = const.
The continuum result for the subtracted energy density
of free massless fermions reads (see, e.g., [5])
ǫ(µ)− ǫ(0) = µ
µ2T 2 . (5)
When working on the lattice, the inverse tempera-
ture β is given by the lattice extent in 4-direction, i.e.,
β = N4a4. Thus the derivative ∂/∂β in (4) turns into
N−14 ∂/∂a4. The partition function Z is given by the
fermion determinant detD which we write as the prod-
uct over all eigenvalues λn. We thus find
ǫ(µ) = − 1
∂ ln detD
a4µ=c
= − 1
a4µ=c
= − 1
a4µ=c
. (6)
III. EVALUATION OF THE EIGENVALUES
According to (6), for the evaluation of ǫ(µ) the eigen-
values λn of the Dirac operator D have to be computed.
This is done in three steps: First we bring the Dirac oper-
ator for free fermions to 4× 4 block-diagonal form, using
Fourier transformation. Subsequently the spectral repre-
sentation is applied to the 4× 4 blocks of H to evaluate
sign H . Finally the eigenvalues of the blocks of D are
computed and by summing over the discrete momenta
all eigenvalues are obtained.
Following this strategy, one finds for the Fourier trans-
form Ĥ of H ,
Ĥ = γ5h5 + iγ5
γνhν , (7)
h5 = 1−
[1− cos(apj)]−
[1− cos(a4(p4 − iµ))] ,
hj = − sin(apj) for j = 1, 2, 3 ,
h4 = −
sin(a4(p4 − iµ)) . (8)
The spatial momenta are given by pj = 2πkj/aN ,
where N is the number of lattice points in the spatial
directions and kj = 0, 1 ... N − 1. The momenta in time-
direction are p4 = π(2k4 + 1)/a4N4, k4 = 0, 1 ... N4 − 1.
The remaining diagonalization of Ĥ is similar to the
construction of the left- and right-eigenfunctions for the
free Dirac operator. One finds that Ĥ has two different,
doubly degenerate eigenvalues
α1 = α2 = + s , α3 = α4 = − s , s =
h2 + h25 , (9)
where h2 =
ν . The corresponding left- and right-
eigenvectors, lj and rj are given by
l1 = l
1 [Ĥ + s1] , l2 = l
2 [Ĥ + s1] ,
l3 = l
3 [Ĥ − s1] , l4 = l
4 [Ĥ − s1] ,
r1 = [Ĥ + s1]r
1 , r2 = [Ĥ + s1]r
r3 = [Ĥ − s1]r(0)3 , r4 = [Ĥ − s1]r
4 . (10)
The constant spinors l
j , r
j are (T is transposition)
1 = r
(0) T
1 = c (1, 0, 0, 0) , l
2 = r
(0) T
2 = c (0, 1, 0, 0) ,
3 = r
(0) T
3 = c (0, 0, 1, 0) , l
4 = r
(0) T
4 = c (0, 0, 0, 1) .
The constant c = (2s(s + h5))
−1/2 ensures the correct
normalization, such that the eigenvectors obey lirj = δij .
Using these eigenvectors and the spectral theorem we
find for sign Ĥ the simple result
sign Ĥ =
sign (λj) rj lj =
sign(s)
Ĥ . (12)
Plugging this back into the overlap formula (1) and di-
agonalizing the remaining 4 × 4 problem one finds two
different eigenvalues for the overlap operator at a given
momentum,
1− sign (
h2 + h25 )h5 ± i
h2 + h25
, (13)
where each of the two eigenvalues is twofold degenerate.
The momentum dependence enters through the compo-
nents hν , h5 defined in (8). In the spectral sum (6) the la-
bel n runs over all momenta and the eigenvalues at fixed
momentum as given in (13). The necessary derivative
with respect to a4 is straightforward to compute in closed
form, and the spectral sum (6) can then be summed nu-
merically. The argument of the sign function cannot be-
come purely imaginary on a finite lattice, and no δ-like
terms occur. We remark, that after taking the derivative
with respect to a4, we set a = a4 = 1, i.e., all the results
we present are in lattice units.
0.00 0.02 0.04 0.06 0.08 0.10
0.000
0.001
0.002
0.003
0.004
/ 4π
FIG. 1: The energy density ǫ(µ)−ǫ(0) as a function of µ4 (all
in lattice units). The symbols (connected to guide the eye)
are for various lattice sizes, the dashed line is the continuum
result.
IV. RESULTS
We begin the discussion of our results with Fig. 1,
where we show the subtracted energy density ǫ(µ)− ǫ(0)
as a function of µ4 for three different lattice volumes. For
those lattices all 4 sides have equal length, i.e., in the
thermodynamic limit they correspond to zero tempera-
ture. Thus, according to (5), we expect the data (sym-
bols in Fig. 1) to approach the continuum form µ4/4π2
(dashed line) as the 4-d volume is sent to infinity.
The figure clearly shows that the lattice data are pre-
dominantly linear when plotted versus µ4 and that for
small µ they approach the continuum curve when the
volume is increased. It is, however, obvious that also on
our largest lattice still a discrepancy remains for larger
µ. In particular one finds a slight curvature upwards, a
discretization effect which here, since the lattice spacing
is just the inverse lattice extension, is also a finite size
effect. Furthermore, for small µ one expects to see finite
temperature corrections according to (5).
In order to study these finite temperature corrections
systematically, we analyzed lattices with short tempo-
ral extent, i.e., lattices with non-vanishing temperature.
Fig. 2 shows the corresponding results, where we again
plot the subtracted energy density as a function of µ4.
The lattice with the shortest temporal extent, 1283×8,
which corresponds to the largest temperature, shows a
clear curvature. This curvature is due to the T 2µ2/2
term in (5), which appears as a square root when plotted
as function of µ4. The effect is visible also for the other
lattices, but becomes less pronounced as the temporal
extent is increased, i.e., the temperature T is lowered.
In order to study this effect quantitatively, we fit the
finite temperature results to the continuum form (5) plus
two terms even in µ which parameterize the cutoff effects
0.00 0.02 0.04 0.06 0.08 0.10
0.000
0.001
0.002
0.003
0.004
x 12
x 16
x 24
FIG. 2: The energy density ǫ(µ) − ǫ(0) as a function of µ4,
now for finite temperature lattices (all in lattice units).
observed in Fig. 1. The fit function is given by
2 + c4 µ
4 + c6 µ
6 + c8 µ
8 . (14)
Due to (5) the coefficient of the quadratic term should
scale with the temperature such that one expects
c2 ∼ T 2/2 = N−24 /2 . (15)
The coefficient for the quartic term should be constant,
c4 ∼ 1/4π2 = 0.02533 . (16)
The results of the fit for the data used in Fig. 2, and for
the largest lattice of Fig. 1 are given in Table 1.
The table shows that with increasing N4 the two phys-
ically significant parameters c2 and c4 approach the val-
ues expected from the continuum formula (5): c2 gets
closer to N−24 /2 as listed in the second column, and c4
approaches 1/4π2 = 0.02533. For the largest finite tem-
perature lattice 1283×24 the discrepancy is down to 9 %
/2 c2 c4 c6 c8
8 0.007812 0.010125 0.03519 0.010 -0.021
12 0.003472 0.004125 0.03178 0.023 -0.013
16 0.001953 0.002192 0.02803 0.029 -0.015
24 0.000868 0.000947 0.02587 0.025 -0.030
128 0.000030 0.000032 0.02543 0.015 0.016
TABLE I: Results of the fits to the form (14). The spatial
volume is always 1283. The temporal extension N4 is given
in the first column. In the second column we list the corre-
sponding value of N−2
/2 which is what one expects for the
fitting coefficient c2 in the third column. The coefficient c4 in
the fourth column is expected to approach the constant value
1/4π2 = 0.02533.
0.0 0.5 1.0 1.5 2.0
-0.05
Overlap , 128
Wilson , 128
FIG. 3: The ratio (ǫ(µ) − ǫ(0))/µ4 as a function of µ (in
lattice units). We compare the results for overlap to those
from Wilson fermions.
for c2, and 2 % for c4. The larger discrepancy for small
N4 can be understood as a discretization effect, since the
temporal lattice spacing a4 is related to the temporal ex-
tension through a4 = 1/N4 and thus larger N4 implies a
smaller a4. For comparison we also display the fit results
for the 1284 lattice, which corresponds to zero temper-
ature. There we find excellent agreement (less than 1%
discrepancy) for the parameter c4, governing the leading
term at T = 0. The overall picture obtained from the fit
results is that overlap fermions with chemical potential
reproduce very well both, the µ4 term, as well as the fi-
nite temperature contribution T 2µ2/2. We conclude that
the analytic continuation of the sign function does not in-
troduce lattice artifacts, such as the µ2/a2 term known
to be present in a naive implementation of the chemical
potential.
In the final step of our analysis we study the discretiza-
tion effect for larger values of µ and compare the re-
sults to the data from the standard Wilson operator. In
Fig. 3 we plot the ratio (ǫ(µ)− ǫ(0))/µ4 as a function of
µ. In the continuum at T = 0 this ratio has the value
1/4π2 = 0.02533 indicated by the horizontal line. For
small µ, up to about µ ∼ 0.7, the Wilson and overlap data
fall on top of each other. For very small µ both opera-
tors show a prominent increase which is a left-over finite
temperature effect, which for the ratio (ǫ(µ) − ǫ(0))/µ4
shows up as a 1/µ2 term. In the range between µ = 0.1
and 0.5 the data are close to the continuum value. Be-
yond 0.5 the discretization effects kick in and the overlap
and Wilson results start to differ. A comparison with
the equivalent plot in [6], where the results from various
other lattice Dirac operators were presented, shows that
the discretization effects of the overlap operator at large
µ are comparable to other formulations.
V. SUMMARY
In this article we have analyzed the energy density of
the overlap operator at finite chemical potential. Follow-
ing [4], the sign function in the overlap was implemented
through the spectral theorem using the analytic continu-
ation of the sign into the complex plane. The subtracted
energy density ǫ(µ) − ǫ(0) was analyzed for finite and
zero temperature lattices. Fits of the data show that the
expected continuum behavior is approached. No trace
of unphysical µ2/a2 terms was found. We conclude that
overlap fermions with chemical potential [4] provide both,
chiral symmetry and the correct description of fermions
at finite density.
Acknowledgments: We thank Leonard Fister,
Gabriele Jaritz, Christian Lang, Stefan Olejnik, Tilo
Wettig, and Florian Wodlei for discussions and check-
ing some of our calculations. This work is supported by
the Slovak Science and Technology Assistance Agency
under Contract No. APVT–51–005704, and the Austrian
Exchange Service ÖAD.
[1] P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649
(1982).
[2] P. Hasenfratz and F. Karsch, Phys. Lett. B 125, 308
(1983).
[3] R. Narayanan and H. Neuberger, Nucl. Phys. B 443, 305
(1995); H. Neuberger, Phys. Lett. B 417, 141 (1998).
[4] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003
(2006); J. Bloch and T. Wettig, contribution to Lattice
2006 (hep-lat/0609020).
[5] J. Kapusta, Finite temperature field theory, Cambridge
University Press, Cambridge (1989).
[6] W. Bietenholz and U. J. Wiese, Phys. Lett. B 426, 114
(1998).
http://arxiv.org/abs/hep-lat/0609020
|
0704.0093 | Aspects of Electron-Phonon Self-Energy Revealed from Angle-Resolved
Photoemission Spectroscopy | Aspects of Electron-Phonon Self-Energy Revealed from Angle-Resolved
Photoemission Spectroscopy
W.S. Lee,1 S. Johnston,2 T.P. Devereaux,2 and Z.-X. Shen1
Department of Physics, Applied Physics, and Stanford Synchrotron
Radiation Laboratory, Stanford University, Stanford, CA 94305
Department of Physics, University of Waterloo,Waterloo, Ontario, Canada N2L 3G1
(Dated: November 4, 2018)
Lattice contribution to the electronic self-energy in complex correlated oxides is a fascinating
subject that has lately stimulated lively discussions. Expectations of electron-phonon self-energy
effects for simpler materials, such as Pd and Al, have resulted in several misconceptions in strongly
correlated oxides. Here we analyze a number of arguments claiming that phonons cannot be the
origin of certain self-energy effects seen in high-Tc cuprate superconductors via angle resolved pho-
toemission experiments (ARPES), including the temperature dependence, doping dependence of the
renormalization effects, the inter-band scattering in the bilayer systems, and impurity substitution.
We show that in light of experimental evidences and detailed simulations, these arguments are not
well founded.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
The microscopic pairing mechanism of the high-Tc
superconductivity remains an unsolved question even
after twenty years of its discovery. Observations of
a kink at around 40-70 meV in the band dispersion
along the diagonal of the Brillouin zone (nodal diec-
tion) and a peak-dip-hump (PDH) structure at the zone
boundary by angle-resolved photoemission spectroscopy
(ARPES)1,2,3,4,5,6,7,8,9,10,11,12,13 have drawn a great deal
of recent attention as they may shed some light on this
problem. Although an agreement has been established
that the kink and PDH structure are signatures of the
electrons coupled to a sharp bosonic mode, it is still
strongly debated about the origin of this bosonic mode.
Influenced by the fact that the high-Tc cuprate is a doped
antiferromagnetic insulator, a common belief is that this
bosonic mode has a magnetic origin2,3,4,5,6,7,8,9. How-
ever, an alternative view is that the electron-phonon cou-
pling in such a doped-insulator can be very strong and
anomalous because of a number of unusual effects, such
as poor screening, complex structure as well as the in-
terplay with correlations. Indeed, oxygen related op-
tical phonons have been invoked to explain the tem-
perature and doping dependence of the renormalization
effects10,11,12,13,14. This idea of phonons being mainly
responsible for this low energy band renormalization ef-
fect observed by ARPES has stimulated intense debate.
There is currently no consensus whether a phonon, a set
of phonons, or a magnetic mode is the primary cause of
the band renormalization.
As mentioned, some important reasons to invoke
phonon interpretation of the ARPES data are: the
presence of an universal energy scale in all materials
at all doping10,15, particularly in the normal state of
very low Tc materials
16; the strong inferred momen-
tum dependence11,14; the existence of multiple bosonic
mode couplings12 and the decrease in the overall cou-
pling strength with increased doping, interpreted as a
screening effect, especially for phonons with eigenvectors
along the c-axis13. Yet, there is still a widespread belief
that phonons are not responsible for the kink features. In
the following sections, with a comprehensive look at all
experimental data as well as some recent simulations, we
address some of the criticisms that have been commonly
used to argue against the phonon interpretation. These
include the temperature and doping dependence of the
renormalization effects, inter-band scattering for bilayer
system, and the ARPES experiments on impurity substi-
tuted Bi2212 crystal, Bi2Sr2Ca(Cu2−xMx)O8+δ with M
= Zn or Ni. Our goal is to clarify these misconceptions
as being due to oversimplifying the effects of electron-
phonon coupling in cuprates as well as other strongly
correlated transition metal oxides.
II. ASPECTS OF THE ELECTRON-PHONON
SELF-ENERGY
A. Temperature Dependence
In the standard treatment of electron-phonon coupling
effects, the Debye temperature sets a characteristic tem-
perature scale, which is well above Tc in conventional
superconducting materials. However in the cuprates and
other low Fermi energy systems, these two energy scales
can be comparable. As a result, the temperature depen-
dence of phonon induced self-energies can be very differ-
ent from that of conventional superconductors. Accord-
ing to the ARPES measurements on Bi2212 system, the
band renormalization in the antinodal region (peak-dip-
hump structure) shows a dramatic superconductivity-
induced enhancement when the system goes through a
phase transition from the normal state to the supercon-
ducting state. It has been argued that only a mode which
emerges in the superconducting state and vanishes in
http://arxiv.org/abs/0704.0093v1
the normal state can explain this temperature-dependent
renormalization effect2,3,4,5. Phonons are thereby ex-
cluded.
The sharpness of the renormalization effects due to
electron-phonon coupling is strongly temperature depen-
dent, given by the fact that Tc of optimally-doped Bi2212
is close to 100K. To demonstrate this temperature de-
pendence, we consider the normal state (120 K) and su-
perconducting state (10 K) of a d-wave superconductor
coupled to a 36 meV B1g, 55meV oxygen A1g, and 70
meV breathing phonons14,17. The electron-phonon cou-
pling for the B1g and breathing phonons are those used
in Ref. 14. The A1g modes involve c-axis motions of
both planar and apical oxygens, and have two branches
around 55 and 80 meV. The apical electron-phonon cou-
pling, derived in Ref. 17, involves a charge-transfer from
the apex oxygen into the CuO2 plane via the Cu 4s or-
bital, the same pathway that gives rise to bi-layer split-
ting. However, for simplicity, the apical electron-phonon
coupling is treated as a constant in the calculations pre-
sented in this paper. The reason to include three modes
in the calculation was inspired by the earlier success of
the two-mode calculation11 as well as the recent discov-
ery of multiple mode coupling12,13. For this calculation,
the tight-binding band structure described in Ref. 19
has been used. The real part and imaginary part of the
self-energy Σ(k, ω) and the spectral functions A(k, ω) at
k = (0, π) are then obtained within weak coupling Eliash-
berg formalism14 and plotted in Fig. 1. Details of the
calculations are presented in the Appendix.
At high temperature, both ReΣ(k, ω) and ImΣ(k, ω)
do not exhibit a sharp renormalization feature as shown
by the dashed curves in Fig. 1 (a) and Fig. 1(b), respec-
tively. This demonstrates that the thermal broadening
effect smears out the sharp renormalization features; in
addition, broadening effects due to additional many body
effects would smear out the renormalization features fur-
ther. Thus, one should not expect to observe any sharp
renormalization features at k = (0, π) in the normal state
(∼ 100K) from the electron-phonon coupling. In the su-
perconducting state, the renormalization features of the
self-energy become much sharper, due to smaller ther-
mal broadening effect as well as the opening of a su-
perconducting gap. Consistent with the optimally-doped
Bi2212 and Bi2223 measurements2,4,11,18, the PDH struc-
ture of the spectral function at k = (0, π) emerges at low
temperature and disappears at high temperature (nor-
mal state), as illustrated in Fig. 1(c) and Fig. 1(d),
respectively. While this behavior is generally expected
for any phonon, we note that in addition, the self-energy
from electron-phonon couplings which involve momen-
tum transfers within and between anti-nodal regions of
the Fermi surface, such as the apex A1g and B1g phonons,
are greatly enhanced for all k-points due to the large
density of state enhancements in these regions via the
opening of a d-wave gap. A detailed momentum depen-
dence of the renormalization effects in the normal state
and superconducting state due to the coupling to the B1g
FIG. 1: The calculated (a) real part, ReΣ, (b) imaginary
part of the self-energy, ImΣ , and the corresponding spectral
functions, A(k,ω) in (c) normal state and (d) superconducting
state. An extra 5 meV is added to the imaginary part of the
self energy in 120K simulation to account for the thermal
broadening of the quasi-particle life time. The location for
this calculation is indicated in inset of (a) by a red dot with
a red curve representing the FS. Insets of (c) and (d) are the
data of optimally-doped Bi2223 system (Tc=110K) taken at
the normal state (120K) and superconducting state(25K)18,
respectively.
phonon has been discussed in Ref. 11 and Ref. 14. Fur-
thermore, both the dispersion kink and the PDH struc-
ture in the nodal region have been clearly observed in
the normal state when measured at a low temperature
on samples with a lower Tc
16. This lends further support
to the strongly temperature dependent renormalization
features due to electron-phonon coupling.
B. Doping Dependence
Another problematic statement against the phonon
scenario stems from the apparent strong doping depen-
dence of the kink position and strength. Based on the
wisdom for conventional good metals, phonons should
not have a strong doping dependence, either in frequency
of the mode or in strength of the coupling. This is
not necessary valid for layered, doped insulators with
strong correlation effects, such as cuprates where dop-
ing dramatically changes the metallicity and the abil-
ity of electrons to screen charge fluctuations. We first
note that many experiments on various cuprates have re-
ported strongly doping dependent anomalies for several
phonons, which implies a strongly doping dependent e-p
coupling for these phonons. For example, from inelastic
neutron scattering measurements, the breathing mode,
half-breathing mode, and the bond-stretching modes ex-
hibit prominent doping dependence of dispersion and en-
ergy renormalizations20,21. In Raman and infrared spec-
FIG. 2: The intensity plots of the (a) spectral functions with-
out resolution convolution and (b) resolution convoluted spec-
tral functions in the superconducting state (10K) along the
nodal direction, as indicated in the inset of (b) by the blue
line. Black curves are the band dispersion extracted from
the maximum positions of the momentum distribution curves,
which cut the spectral functions at a fixed energy. The MDC-
derived dispersions in (a) exhibit three sharp ”subkinks” due
to the coupling to the three phonon modes used in the model;
while in (b) the subkinks are washed out by the finite instru-
ment resolution effect leaving an apparent single kink in the
band dispersion. The white dashed line illustrates the bare
band for extracting ReΣ shown in Fig. 3 (a).
troscopy, the Fano lineshapes of phonon modes in B1g
and B1u symmetry show strong doping dependences
Furthermore, the strength of the phonon energy shift and
linewidth variation across Tc also changes strongly with
doping23.
Second, the doping dependence of the renormalization
effects to the electronic self-energy is sophisticated as in-
ferred by two recent ARPES studies. One is the observa-
tion of multiple bosonic mode couplings along the nodal
direction12. The other is the doping dependence of the
c-axis screening effect to the coupling between the elec-
tron and c−axis phonons. As proposed by Meevasana et
13,24 and Devereaux et al.17, for electron-phonon cou-
pling at long wavelengths, the screening becomes more
effective at reducing the coupling strength when the c-
axis conductivity becomes more metallic. Given these
two results plus the variation of the superconducting gap
magnitude with doping, the doping dependence of the
kink energy is highly convoluted in Bi2212 whose super-
conducting gap has an energy comparable to some of the
phonons.
In Fig. 2, we present the intensity plot of calculated
spectral functions demonstrating a doping dependence
of the dispersion kink in the superconducting state. The
superconducting gap size was set to be 40, 20, and 10
meV for the optimally-doped and more overdoped sys-
tems. In addition, the coupling strength of the breathing
mode, whose appreciable coupling is only for short wave-
lengths and large momentum transfers20,21, remains un-
FIG. 3: (a) The ReΣ extracted from Fig. 2(a) (dashed lines)
and Fig. 2(b) (solid lines) by subtracting a linear bare band
(dashed line in Fig. 2(a)) from the band dispersion. The ar-
rows indicate the maximum positions of the ReΣ where the
“single” apparent kink in the band dispersion is usually de-
fined. (b) Summary of the doping dependence of the apparent
kink energy and the apparent mode energy extracted by as-
suming a single mode scenario.
changed for our doping dependence simulation; while, a
filter function, ω2/(ω2+ω2sc) with different value of c-axis
screening frequency ωsc is applied to the c-axis phonons
(B1g and A1g), to simulate the doping-dependent cou-
pling strength due to the change of the c-axis screening
effect13,24. We note that although this is a simplifica-
tion, it represents the general behavior of screening con-
siderations for phonons involving small in-plane momen-
tum transfers. Full consideration of screening has been
given in Ref. 17 and Ref. 24. In addition, a component
0.005+ ω2 eV is added in the imaginary part of the self-
energy to mimic the quasiparticle life time broadening
due to electron-electron interaction.
As shown in Fig. 2(a), the coupling to multiple phonon
modes induces several “subkinks” in the dispersion. The
positions of these subkinks mostly correspond to the en-
ergies of phonons plus the maximum d-wave SC gap,
∆0, even through there is no gap along the nodal di-
rection. This is because when calculating the self-energy,
one needs to integrate the contributions from the entire
zone, which makes the electrons in the nodal region ”feel”
the presence of the gap. Furthermore, revealed by the ex-
tracted real-part of the self-energy, ReΣ (dashed curves
in Fig. 3 (a)), the dominant feature in ReΣ for the OP
case is induced by 36 meV B1g mode, while for the OD1
and OD2 case, the features of the 55 meV A1g mode and
70 meV breathing mode gradually out-weight the con-
tribution from the B1g mode. This demonstrates that
the change of the SC gap magnitude and the increasing
screening effect to these phonons because of increased
doping alters the relative strength of the subkinks caused
by different modes.
To simulate the experimental data, we convoluted the
spectral functions shown in Fig. 2(a) with a typical
ARPES instrumental resolution: 12.5 meV in energy res-
olution and 0.012 π/a in momentum resolution. As illus-
trated in Fig. 2(b) and the extracted ReΣ (solid curves
in Fig. 3(a)), the subkinks are less obvious and become
a broadened “single” kink in the dispersion which is lo-
cated at roughly the energy of the dominant phonon fea-
ture determined by the maximum position of the ReΣ
(the arrows in Fig. 3(a)). The doping dependence of
the kink position is summarized as the solid symbols in
Fig. 3(b). Assuming a single mode scenario, one can
obtain the “doping dependence” of the mode energy by
subtracting out the SC gap size. However, we note that
this extracted “apparent” mode energy does not match
any of the modes used in the model; instead, it is a av-
erage between the dominant features (open symbols in
Fig. 3(b). Clearly, since the kink energy is a sum of the
superconducting gap and a spectrum of bosonic modes,
it should not be taken as a precise measurement of the
energy of any particular bosonic mode. This casts doubts
to the analysis of the doping dependent properties of the
kink in the nodal band dispersion based on the single
bosonic mode coupling scenario7,8. More importantly,
this illustrates the complex nature of lattice effects in
these oxides.
C. Interband Scattering
Borisenko et al.8,9 observed that the scattering rate
of the bonding and antibonding bands along the nodal
direction cross each other near the energy of the Van
Hove singularity, suggesting a strong inter-band scatter-
ing between the bonding and antibonding bands. They
argued that only a mode with ”odd” symmetry, such as
spin resonance mode, can mediate such inter-band scat-
tering. The question whether phonons can induce such
inter-band scattering has also been raised by these au-
thors.
First, we note that recent high energy and momentum
resolution ARPES experiments on Bi2212 using low en-
ergy photons( <10 eV) have better resolved the bilayer
splitting at the nodal point25. However, as shown in Fig.
2 of Ref. 25, the scattering rate of the bonding and anti-
bonding band does not exhibit a crossover behavior as
reported by Borisenko at al.. The inconsistency of the
data between the two groups implies that more experi-
ments and better analysis are needed to verify whether
this inter-band scattering effect is genuine.
Second, empirically, it has been known for over 15 years
that interband electron-phonon coupling in the cuprates
is very large. The evidence comes from the strong reso-
nance profiles of many Raman active phonons, which dis-
play large intensity variations26. This is generally under-
stood as a result of strong interband coupling, whereby
phonons can be brought in and out of resonances via tun-
ing of the incident photon energy27. Since, in general,
phonons can also provide momentum to scatter electrons
along the c-axis, direct inter bi-layer scattering can occur
which involves mixing of different symmetries of phonons.
This can be viewed in a simplified way even if we first
neglect direct interband scattering and consider a bilayer
system coupled to c−axis phonons. For qz = 0, a simple
classification of c-axis modes is possible:
k,σ,α=1,2
ǫα(k)c
k,α,σ
ck,α,σ +
t⊥(k)
k,1,σck,2,σ + h.c.
k,q,σ,α=1,2,ν
gν,α(k, q)
k+q,α,σck,α,σ
aν(−q) + a
+ h.c.
, (1)
where α is the index for the electronic states of different
layers, ǫ1(k) = ǫ2(k), t⊥(k) describes the hopping of elec-
trons between two layers, and the index ν can be either
gerade or ungerade active c-axis modes, with symmetry
classification with respect to the displacement eigenvec-
tors to the inversion center of the cell, depicted in Fig.
4. After diagonalizing the first two terms by canonical
transformation, the electron-phonon coupling can be re-
cast as
(g,u)
k,q,σ
g(g,u)(k, q)
a(g,u)(q) + a
(g,u)
k+q,+,σck,(+,−),σ + c
k+q,−,σck,(−,+),σ
+ h.c.
..(2)
We have used the c+ and c− for the even and odd
linear combination of c1 and c2, and subscript g and u
for the gerade and ungerade mode, respectively. Thus
for qz = 0, where this classification is possible, ger-
ade phonons induce intra-band scattering (even chan-
nel), while the ungerade phonons mediate the inter-
band scattering process (odd channel) even without di-
rect electron-phonon coupling across the layers. Yet for
qz = π/c, the classification inverts, where gerade modes
become ungerade and vice-versa, as illustrated in Fig 4.
Thus, even in this simple case, modes at different qz con-
tribute both to intra and interband scattering, and the
net weight of the coupling appearing in the self energy is
then largely determined by the specific momentum struc-
ture g(k, q). Since the self-energy generally involves sums
over qz, and coupling directly of electrons in adjacent lay-
ers via phonons are non-negligible, clearly the inter-band
scattering phenomena can not be used to argue against
FIG. 4: The illustration of the gerade and ungerade c−axis
phonons. The eigenmode of the gerade (ungerade) phonons
is even (odd) with respect to the mirror plane between two
CuO2 layers at qz = 0, while their definition swapped at qz =
π/c. The black, grey, and white circles represent the Cu, Ca,
and O atoms, respectively.
the phonons being important to the electronic states.
We also add a remark concerning the electron-
phonon coupling derived from Raman measurements28
in YBa2Cu3O7 and Bi-2212 compared to that obtained
from ARPES. While one might naively expect the cou-
plings to be comparable from Raman and ARPES, we re-
mark that this situation is remarkably different if the cou-
pling is strongly moment dependent and whenever corre-
lations are appreciable. Since Raman measures phonons
with net zero momentum transfer and ARPES involves a
sum over all transfers, a sizeable coupling difference may
be discernable. This is specifically the case for the B1g
phonon, where scattering involving momentum transfers
across the necks of the Fermi surface near (π, 0)14, further
enhanced via correlations29, yields a strong contribution
to the electron self-energy that is absent in phonon self-
energies. Moreover, a sum rule analysis presented in Ref.
30 highlights in general how electron and phonon self-
energies may be qualitatively different in strongly corre-
lated systems.
D. ARPES Experiments on Zn and Ni substituted
Bi2212
In this section, we comment on recent experiments
about the renormalization effects in Zn and Ni substi-
tuted Bi2212 crystal31,32. The strength of sharp renor-
malizations in these substituted crystals is found to be
weakened compared to the pristine crystals. Since the
magnetic properties are expectedly modified due to the
Cu substitution by these impurities, the authors con-
cluded that the sharp renormalization effects are induced
by magnetic-related modes, not phonons.
In fact, a close examination of the data published by V.
B. Zabolotnyy et al.31 and K. Terashima et al.32 implies
that the magnetic property is not the only modification
due to the substitution by Zn and Ni. First, although
both sets of data are consistent in the antinodal region
where the strength of the band renormalization is re-
duced, they are inconsistent with each other on the kink
strength along the nodal direction. In the data set of V.
B. Zabolotnyy et al., the kink strength is weaker in the
Zn or Ni doped samples, whereas there is no detectable
change in the data set reported by K. Terashima et al..
Second, the data from K. Terashima et al. (Fig. 1(d)-
(f) in Ref. 32) suggest that the bilayer-splitting struc-
ture is much clearer in the pristine crystals than in the
Zn and Ni doped crystals. Since the authors have ruled
out the possibility of a significant doping level difference
between pristine and impurity-doped crystals, the dis-
tinct visibility variation of the bilayer structure implies
a impurity-related change in the electronic structure.
From these two observations on their data, it implies
that not only the magnetic properties could change, the
band structure and scattering behaviors could also be
affected due to these impurities. It is possible that
these changes of the electronic structures could “weaken”
the renormalizaton features observed in the ARPES
spectrum. Furthermore, we note that the strength of
electron-phonon coupling could also be modified by the
substituted impurities: this can be inferred from the
change of the Fano spectra lineshape of the B1g 340
cm−1 phonon in Raman spectral for Zn-doped YBCO33
and Th-doped YBCO34 resulting from an increase in the
phonon linewidth due to impurity scattering. Therefore,
the experiments on Ni and Zn substituted Bi2212 crys-
tals are inconclusive experiments to distinguish phonon
and magnetic modes as the origin of the renormalization
effects.
III. CONCLUSION
We have shown that the temperature and doping de-
pendence of the renormalization effects, inter-band band
scattering, and the results of Zn and Ni doped materials
can be understood in the framework of electron-phonon
coupling. On the other hand, the issues that make it
not plausible for the sharp kink being of spin origin, es-
pecially the spin resonance mode, remain: i) the nearly
constant energy scale as a function of doping in small
gap system12; ii) the multiple modes12; iii) the presence
of clear kink in the normal state4,13,16 iv) the detailed
agreement between B1g phonon based explanation of the
mode coupling as a function of momentum11,14, while
the spin resonance with tiny spectral weight (2%) is un-
likely to give an explanation for both nodal and antin-
odal renormalization; v) the accumulated evidence for
lattice polaron effect in underdoped and deeply under-
doped systems35,36. With these weaknesses of the spin
resonance interpretation, lattice effect is a more plausible
explanation of the renormalization effects. It remains a
possibility that the spin-fluctuation and other strong cor-
relation effects are also very important to determine the
electronic structure of cuprates; they likely contribute to
a smooth renormalization of the band and may be more
relevant to the higher binding energy. However, opti-
cal phonons are the most probable origin for the renor-
malization effects due to sharp modes near 40-70 meV,
which is also supported by the recent finding of STM
experiments37,38.
Acknowledgments
W.S. Lee acknowledge the support from SSRL which
is operated by the DOE Office of Basic Energy Science,
Division of Chemical Science and Material Science under
contract DE-AC02-76SF00515. T. P. Devereaux would
like to acknowledge support from NSERC, ONR grant
N00014-05-1-0127 and the A. von Humboldt Foundation.
APPENDIX A: MIGDAL-ELIASHBERG BASED
APPROACH
In the calculations presented herein, we evaluate elec-
tronic self energies and spectral functions via Migdal-
Eliashberg treatment, as discussed in Ref. 39. The
dressed Green’s function in the superconducting state is
given in Nambu notation by
Ĝ(k, ω) =
ωZ(k, ω)τ̂0 + [ǫ(k) + χ(k, ω)]τ̂3 + φ(k, ω)τ̂1
[ωZ(k, ω)]2 − [ǫ(k) + χ(k, ω)]2 − φ2(k, ω)
, (A1)
from which the spectral function follows A(k, ω) = − 1
G′′1,1(k, ω) as shown in Figs. 1c,1d, and 2. The momentum-
dependent components of the Nambu self energy are given as generalizations of those found in Ref. 39:
ωZ2(k, ω) =
|gν(k,p− k)|
[nb(Ων) + nf(Ep)][δ(ω +Ων − Ep) + δ(ω − Ων + Ep)]
+[nb(Ων) + nf (−Ep)][δ(ω − Ων − Ep) + δ(ω +Ων + Ep)]
χ2(k, ω) = −
|gν(k,p− k)|
[nb(Ων) + nf (Ep)][δ(ω +Ων − Ep)− δ(ω − Ων + Ep)]
+[nb(Ων) + nf (−Ep)][δ(ω − Ων − Ep)− δ(ω +Ων + Ep)]
φ2(k, ω) =
|gν(k,p− k)|
[nb(Ων) + nf (Ep)][δ(ω +Ων − Ep)− δ(ω − Ων + Ep)]
+[nb(Ων) + nf(−Ep)][δ(ω − Ων − Ep)− δ(ω +Ων + Ep)]
where ν denotes the phonon mode index, and nf and nb
are the Fermi and Bose occupation factors. gν(k,q) are
the corresponding electron-phonon couplings for mode
ν, given in reference14 for the B1g and breathing modes.
We choose to model the A1g coupling via a momentum
independent coupling. Further details can be found in
Ref. 17.
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|
0704.0094 | Timing and Lensing of the Colliding Bullet Clusters: barely enough time
and gravity to accelerate the bullet | arXiv:0704.0094v1 [astro-ph] 2 Apr 2007
Timing and Lensing of the Colliding Bullet Clusters: barely enough time and gravity
to accelerate the bullet
HongSheng Zhao
University of St Andrews, Scottish University Physics Alliances, KY16 9SS, UK
We present semi-analytical constraint on the amount of dark matter in the merging bullet galaxy
cluster using the classical Local Group timing arguments. We consider particle orbits in potential
models which fit the lensing data. Marginally consistent CDM models in Newtonian gravity are
found with a total mass MCDM = 1 × 10
M⊙ of Cold DM: the bullet subhalo can move with
VDM = 3000 kms
−1, and the ”bullet” X-ray gas can move with Vgas = 4200 kms
−1. These are
nearly the maximum speeds that are accelerable by the gravity of two truncated CDM halos in a
Hubble time even without the ram pressure. Consistency breaks down if one adopts higher end of
the error bars for the bullet gas speed (5000− 5400 kms−1), and the bullet gas would not be bound
by the sub-cluster halo for the Hubble time. Models with VDM ∼ 4500 kms
∼ Vgas would invoke
unrealistic large amount MCDM = 7× 10
M⊙ of CDM for a cluster containing only ∼ 10
M⊙ of
gas. Our results are generalisable beyond General Relativity, e.g., a speed of 4500 kms−1 is easily
obtained in the relativistic MONDian lensing model of Angus et al. (2007). However, MONDian
model with hot dark matter MHDM ≤ 0.6×10
M⊙ and CDM model with a halo mass ≤ 1×10
are barely consistent with lensing and velocity data.
PACS numbers: 98.10.+z, 98.62.Dm, 95.35.+d; submitted to Physical Review D, rapid publications
I. POTENTIAL FROM TIMING
Timing is a unique technique to establish the case for
dark matter halos, first and most throughly explored in
the context of the Local Group (Kahn & Woljter 1959,
Fich & Tremaine 1991, Peebles 1989, Inga & Saha 1998).
In its simplest version the Local Group consists of the
Milky Way and M31 as two isolated point masses, which
formed close to each other, moved apart due to the Hub-
ble expansion, and slowed down and moved towards each
other upto their present velocity ∼ 120 km s−1 and sepa-
ration (about 700 kpc) due to their mutual gravity. The
age of the universe sets the upper limit on the period of
this galaxy pair, hence the total mass of the pair through
Kepler’s 3rd law assuming Newtonian gravity.
Timing also finds a timely application in the pair of
merging galaxy clusters 1E0657-56 at redshift z = 0.3,
which is largely an extra-galactic grand analogy of the
M31-MW system. The sub-cluster, called the ”bullet”,
presently penetrates 400-700 kpc through the main clus-
ter with an apparent speed of ∼ 4750+710
−550 km s
−1 (Marke-
vitch 2006). The X-ray gas of the bullet (amounts to
2×1013M⊙) collides with the X-ray gas of the main clus-
ter (with the total gas up to 1014M⊙) and forms a Mach-3
cone in front of the ”bullet”. The two clusters have at
least four different centers, which are offset by 400 kpc
between the pair of X-ray gas centers and by 700 kpc
between the pair of star-light centers, which coincides
with the gravitational lensing centers and (dark matter)
potential centers (Clowe et al. 2006). The penetration
speed is unusually high, hard for standard cosmology to
explain statistically (Hayashi & White 2006), and modi-
∗Electronic address: [email protected]
fied force law has been suggested (Farrar & Rosen 2006,
Angus et al. 2007).
The timing method applies in in MONDian gravity
as well as Newtonian. Like lensing, timing is merely a
method about constraining potential distribution, and is
only indirectly related to the matter distribution. In this
Letter we model the bullet clusters as a pair of mass con-
centrations formed at high redshift, and set constraint on
their mutual force using the simple fact that their radial
oscillation period must be close to the age of universe
at z = 0.3. We check the consistency with the lensing
signal of the cluster and give interpretations in terms of
standard CDM and MOND.
First we can understand the speed of the bullet clus-
ter analytically in simplified scenarios. Approximate the
two clusters as points of fixed masses M1 and M2 on a
head-on orbit, we can apply the usual MW-M31 timing
argument. The total mass M0 = M1 + M2 is constant.
The radial orbital period is computed from
T = 2
∫ rmax
V (r)
, (1)
r3max
, Newtonian p = 2 (2)
2πrmax
, deep-MONDian, p = 1 (3)
∝ K−n/2r
max, for a K/r
p gravity, (4)
where rmax is the apocenter and is related to the present
relative velocity V (r) at separation r = 700 kpc by energy
http://arxiv.org/abs/0704.0094v1
conservation
V (r)2
= −GM0
Newtonian (5)
= V 2M (ln rmax − ln r) deep−MONDian (6)
r1−p − r1−pmax
K/(1− p) for a K/rp gravity,(7)
where VM =
ξ(GM0a0)
1/4 is the MOND cir-
cular velocity of two point masses, a0 equals one
Angstrom per square second and is the MOND
acceleration scale, and the dimensionless ξ ≡
3M1M2
∼ 0.81 ∼ 1 (cf. Mil-
grom 1994, Zhao 2007, in preparation) for a typical mass
ratio.
The predictions for simple Newtonian Keplerian grav-
ity are given in Fig. 1; the more subtle case for a MON-
Dian cluster is discussed in the final section. Setting the
orbital period T = 10Gyrs, the age of the universe at
the cluster redshift, yields presently V ∼ 3200 km s−1 in
Newtonian for a normal combined mass of M1 + M2 =
(0.7− 1)× 1015M⊙ for the clusters, which is about 7-10
times their baryonic gas content (∼ 1014M⊙) for Newto-
nian universe of Ω = 0.3 cold dark matter. In agreement
with Farrar & Rosen and Hayashi & White, the sim-
ple timing argument suggests that dark halo velocities
of 4750 km s−1, as high as the ”bullet” X-ray gas, would
require halos with unrealistically larger masses of dark
matter, ∼ 1016M⊙, an order of magnitude more than
what a universal baryon-dark ratio implies. As a sanity
check, assuming a conventional 3× 1012M⊙ Local Group
dark matter mass Fig.1 predicts the relative velocity of
∼ 100km/s for the M31-MW system at separation 700
kpc after 14 Gyrs, consistent with observation (Binney
& Tremaine 1987).
These analytical arguments, while straightforward, are
not precise given its simplifying assumptions. For one,
clusters do not form immediately at redshift infinity, and
the cluster mass and size might grow with time gradual-
lly. More important is that point mass Newtonian halo
models are far from fitting the weak lensing data of the
1E0657-56. A shallower Newtonian potential makes it
even more difficult to accelerate the bullet. On the other
hand, Angus, Shan, Zhao, Famaey (2007) show that there
are MOND-inspired potentials that fits lensing. As com-
mented in their conclusion, the same potential is deep
enough that a V = 4750 km s−1 ”bullet” is bound in an
orbit of apocenter rmax of a few Mpc, so the two clus-
ters could be accelerated by mutual gravity from a zero
velocity apocenter to 4750 km/s within the clusters’ life-
time. This line of thought was further explored by the
more systematic numerical study of Angus & McGaugh
(2007).
Our paper is a spin-off of these works and the works
of Hayashi & White and Farrar & Rosen. We emphasize
the unification of the semi-analytical timing perspective
and the lensing perspective, and aim to derive robust
constraints to the potential, without being limited to a
X-ray bullet speed
Baryon
12.5 13 13.5 14 14.5 15
Log (Combined Mass)
FIG. 1: Analytical timing-predicted dynamical mass vs. the
relative speed of two objects separated by 700 kpc after 10±
4 Gyrs (three lines in increasing order for increasing time)
assuming Keplerian potential of point masses. Three vertical
lines indicate typical Local Group Halo mass, Baryonic mass
in galaxy clusters, and most massive CDM halo masses. Three
horizontal lines indicate the error bar of the speed of the X-ray
”bullet” gas.
specific gravity theory or dark matter candidate.
Towards the completion of this work, we are made
aware by the preprint of Springel & Farrar (2007) that the
unobserved bullet DM halo could be moving slower than
its observed stripped X-ray gas. These authors, as well
as the preprint of Milosovic et al. (2007), emphasized the
effect of hydrodynamical pressure, which we will not be
able to model realisticly here. But to address the velocity
differences, instead we treat the X-ray gas as a ”bullestic
particle”. We argue that our hypothetical ballistic parti-
cle must move slow enough to be bound to vicinity of the
subhalo before the collision, but moves somewhat faster
than 4700+700
−550 km s
now, since it does not experience
ram pressure of the gas. This model follows the spirit
of classical timing models of the separation of the Large
and Small Magellanic Clouds and the Magellanic Stream
(Lin & Lynden-Bell 1982).
II. 3D POTENTIAL FROM LENSING
The weak lensing shear map of Clowe et al. (2006)
has been fitted by Angus et al. (2007) using a four-
component analytical potential each being spherical but
on different centres. For our purpose we redistribute the
minor components and simplify the potential into two
components centred on the moving centroid of galaxy
light of the main cluster with the present spatial coordi-
nates r1(t) = (−564,−176, 0) kpc and subcluster galaxy
centroid r2(t) = (145, 0, 0) kpc; the coordinate origin is
set at the present brightest point of the ”bullet” X-ray
gas; presently the cluster is at z = 0.3 or cosmic time
t = 10Gyrs. We also apply a Keplerian truncation to the
potential beyond the truncation radius rt. So the follow-
ing 3D potential is adopted for the cluster 1E0657-56 at
time t,
Φ(X,Y, Z, t) = (1800 km s−1)2φ (|r− r1|) (8)
+ (1270 km s−1)2φ (|r− r2|) ,
φ(|r − ri(t)|) = ln
|r − ri(t)|
180 kpc
+ cst, r < rt(9)
= − r̃t
|r− ri(t)|
, r ≥ rt(t) = C × t,(10)
where r̃t ≡ r
+1802
is to ensure a continuous and smooth
transition of the potential across the truncation radius rt.
The truncation rt evolves with time, since a pre-cluster
region collapses gradually after the big bang, and its
boundary and total mass grows with time till it reaches
the size of a cluster. In the interests of simplicity rather
than rigour, we use a linear model rt = C × t, where C
is a constant of the unit kpc/Gyr.
To check that the simplified potential is still consis-
tent with weak lensing data, we recompute the 3D weak
lensing convergence (Taylor et al. 2004) for sources at
distance D(0, zs) at the redshift zs,
κ(X,Y, zs) =
i=X,Y
∫ D(0,zs)
2D(z, zs)
(∂iΦ)dZ
where the integrations in square backets are the deflec-
tion angles for a source at zs, and the usrual lensing
effective distance is related to the comoving distances
by D(z, zs) = (1 + z)
−1D̃(z)
1− D̃(z)
D̃(zs)
= 587 Mpc is
for the bullet cluster z = 0.3 lensing sources at zs = 1;
the distance increases by a factor 1.3 to 1.6 for source
redshifts of 3 to infinity. Fig.2 shows the predicted κ
along the line joining the two dark centers; the result
is insensitive to the cluster truncation radius as long as
rt ≥ 1000kpc presently. The lensing model predicts a
signal in between that of the weak lensing data of Clowe
et al., and strong lensing data of Bradac et al. It is
known that these two data sets are somewhat discrepant
to each other. So the fit here is reasonable. The method
is deprojection is essentially similar to the decomoposi-
tion method of Bradac et al. whose explicit assumption
of Einsteinian gravity is however unnecessary.
The important thing here is that as far as deproject-
ing the above potential is concerned, no assumption is
needed on the gravity theory as long as light rays fol-
low geodesics, a feature built in most alternative grav-
ity theory. Similarly orbits of massive particles are also
(different) geodesics in these theories. The meaning of
potential in such theories is that the potential (scaled by
a factor 2/c2) represents metric perturbations to the flat
space-time, especially to the g00(cdt)
2 = −(1+ 2Φ
)(cdt)2
term, so the Christoffel Γi00 ∼ ∂∂XiΦ, it can be shown
–1200 –1000 –800 –600 –400 –200 0 200 400
X/kpc
FIG. 2: Predicted bullet cluster convergence (rescaled for
sources at infinity) along the line Y = 0.3X + cst connect-
ing our two potential centroids. The model predicts a lensing
signal in between that of observed weak lensing data from
sources at zs = 1 (Clowe et al, lower end of error bars) and
the united weak lensing and strong lensing (zs = 3) data
(Bradc et al. upper part of error bars); the mismatch of these
two datasets are presently unresolved.
that the geodesic equations have the same form as Ein-
steinian in the weak-field limit: d
R ≈ −(1 + v
)∇RΦ,
where R is the pair of spatial coordinates perpendicular
to the instantaneous velocity v; the pathes of light rays
are deflected twice as much by the metric perturbation
2Φ/c2 as those of low-speed particles.
III. ORBITS OF THE COLLIDING CLUSTERS
We now use this potential to predict the relative speed
of the two clusters. This is possible using the classical
timing argument, in the style of Kahn & Wolter (159),
Fich & Tremaine (1991) and Voltonen et al. (1998);
we postpone most rigourous least action models (Pee-
bles 1989, Schmoldt & Saha 1998) for later investiga-
tions since these require modeling a cosmological con-
stant and other mass concentrations along the orbital
path of the bullet clusters, which have technical issues in
non-Newtonian gravity. We trace the orbits of the two
centroids of the potentials according to the equation of
motion d
= −∇Φ(ri). We assign different relative ve-
locities presently (at z = 0.3), and integrate backward
in time and require the two centroids of the potential be
close together at a time 10 Gyrs ago. The motions are
primarily in the sky plane, but we allow for 600 km/s
relative velocity component in the line of sight. Clearly
at earlier times when t is small, the two centroids are
well-separated compared to their sizes, so they move in
the growing Keplerian potential of each other. At lat-
ter times the centroids came close and move in the cored
isothermal potential.
We shall consider models with a normal truncation
rt = C × t = 1000 kpc at time t = 10 Gyrs. We also
consider models with a very large truncation C × t =
10000 kpc. In the language of CDM, the truncation
means the virial radius of the halo. The present in-
stantaneous escape speed of the model can be com-
puted by Vesc =
−2Φ(X,Y, Z, t). We find Vesc ∼
4200− 4500 km s−1 in the central region of the shallower
potential model with a present truncation 1000 kpc. The
escape speed increases to Vesc ∼ 5700 km s−1 for models
with a present truncation 10000 kpc.
Fig. 3 shows the predicted orbits for different present
relative velocities VDM = |dr2dt −
|. Among models
with a normal truncation, we find VDM ∼ 2950 km s−1; a
model with relative velocity VDM < 2800 km s
−1 would
predict an unphysical orbital crossing at high redshift,
while models with VDM > 3000 km s
−1 would predict
that the two potential centroids were never close at high
redshift.
Larger halo velocities are only possible in models
with very large truncation. If the relative velocity is
4200 km s−1 < VDM < 4750 km s
−1 between two clus-
ter gravity centroids, then the truncation must be as big
as 10Mpc at z = 0.3.
We also track the orbit of the bullet X-ray gas cen-
troid as a tracer particle in the above bi-centric poten-
tial. We look for orbits where the bullet X-ray gas will
always be bound to one member of the binary system
since the ram pressure in a hydrodynamical collision is
unlikely to be so efficient to eject the X-ray gas out of
potential wells of both the main and sub-clusters. This
means that the bullet speed must not exceed greatly the
present instantaneous escape speed of the model, which is
∼ 4200− 4500 km s−1 in the central region of the shallow
potential of a model with a present truncation 1000 kpc.
The escape speed increases to ∼ 5700 km s−1 for models
with a present truncation 10000 kpc. The model with
normal truncation is marginally consistent with the ob-
served gas speed Vgas ∼ 4750+710−550 km s
−1. The problem
would become more severe if the potential were made
shallower by an even smaller truncation. The gas speed
is less an issue in models with larger truncation.
In short the present velocity and lensing data are eas-
ier explained with potential models of very large trun-
cation. Models with normal truncation have smaller
gravitational power, can only accelerate the subhalo to
3000 km s−1 in 10 Gyrs. Models with normal CDM trun-
cation can only accelerate the bullet X-ray gas cloud to
∼ 4200− 4400 km s−1, the escape speed, marginally con-
sistent with observations.
Above simulation results are sensitive to the present
cluster separation, but insensitive to the present direc-
tion of the velocity vector. Unmodeled effects such as
dynamical friction associated with a live halo will reduce
the predicted VDM for the same potential, but the effect
is mild since the actual collision is brief ∼ 0.1− 0.3Gyrs
and the factor exp(−M2/2) in Chandrasekhar’s formulae
Curve 1
Curve 2
Curve 3
Curve 4
Curve 9
Curve 10
Curve 11
V_DM=2850
C=100
C=100
V_DM=2950
C=1000
V_DM=4200
C=1000kpc/Gyr
V_DM=4750 km/s
–2000
–4000 –2000 0 2000 4000
X kpc
FIG. 3: The orbit of the bullet subcluster X-ray gas (red,
with present Vgas = 5400 km s
−1 for the 10 Gyrs in the past,
and pink: for the future 4 Gyrs), and the orbits of the col-
liding main cluster halo (blue dashes) and subhalo (black
dashes) in the potential (eqs. 8-10) determined by lensing
data; dashes indicate length traveled in 0.5 Gyrs steps. No
explicit assumption of gravity is needed for these calcula-
tions. Orbits with different present halo relative velocity VDM
and halo growth rate C are shown after a vertical shift for
clarity. Timing requires the present cluster relative velocity
in between 2800 kms−1 < VDM < 3000 kms
−1 for poten-
tials of normal truncation (lowest panels where the cluster
truncation grows from zero to C × 10Gyr = 1000 kpc), and
4200 kms−1 < VDM < 4750 kms
−1 for potentials with large
truncation (two upper panels where the cluster truncation
grows from zero to C × 10 Gyr = 10000 kpc).
sharply reduces dynamical friction for a supersonic body,
where M ∼ 2− 3 is the Mach number for the bullet.
IV. NEWTONIAN AND MONDIAN MEANINGS
OF THE POTENTIAL MODEL
Assuming Newtonian gravity the models with normal
truncation rt = 1Mpc at t = 10Gyrs correspond to clus-
ter (dark) masses of M1 = 0.745 × 1015M⊙ and M2 =
0.345×1015M⊙; the larger truncation rt = 10Mpc corre-
sponds to M1 = 7.45×1015M⊙ and M2 = 3.45×1015M⊙
in Newtonian. All these models fit lensing.
Interpreted in the MONDian gravity, the truncation
is due to external field effect and cosmic background
so to make the MOND potential finite hence escapable
(Famaey, Bruneton, Zhao 2007). Beyond the trun-
cation radius, MOND potential becomes nearly Kep-
lerian. The MONDian models, insensitive to trunca-
tion, would have masses only M1 = 0.66 × 1015M⊙ and
M2 = 0.16 × 1015M⊙. These masses are still higher
than their baryonic content ∼ 1014M⊙, implying the
need for, e.g., massive neutrinos; the neutrino density
is too low in galaxies to affect normal MONDian fits to
galaxy rotation curves, but is high enough to bend light
and orbits significantly on 1Mpc scale. The neutrino-to-
baryon ration, approximately 7:1 in the bullet cluster,
would be a reasonable assumption for a MONDian uni-
verse with Ωb ∼ 0.04 plus 2eV neutrinos hot dark matter
ΩHDM ∼ 0.25 ∼ 7 × Ωb (Sanders 2003, Pointecoute &
Silk 2005, Skordis et al. 2006, Angus et al. 2007). The
amount of hot dark matter inferred here is the same as
Angus et al. (2007) since their potential parameters are
fixed by the same lensing data.
V. CONCLUSION
In short a consistent set of simple lensing and dynam-
ical model of the bullet cluster is found. The present
relative speeds between galaxies of the two clusters is pre-
dicted to be VDM ∼ 2900 km s−1 in CDM and VDM ∼
4500 km s−1 in µHDM (MOND + Hot Dark Matter) if
the two clusters were born close to each other 10 Gyrs
ago; both models assume close to universal gas-DM ratio
in clusters, i.e., about (0.6 − 1) × 1015M⊙ Hot or Cold
DM. Modeling the bullet X-ray gas as ballistic particle,
we find the gas particle with speed of Vgas = 4200km/s
(at the lower end of observed speed) is bound to the
potential of the subcluster for most part of the Hubble
time for both above models, insensitive to the preference
of the law of gravity. But if future relative proper mo-
tion measurements of the subcluster galaxy speed is as
high as VDM = 4500km/s, or the gas speed is as high as
Vgas ∼ 5400 km s−1, then Newtonian models would need
to invoke unlikely 7×1015M⊙ DM halos around 1014M⊙
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[20] Angus G.W., Shan H, Zhao H., Famaey B., 2007, ApJ,
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[4] Binney, J., & Tremaine, S. 1987, Galactic Dynamics,
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[6] Clowe D., Bradac M., Gonzalez A.H., et al., 2006,astro-
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|
0704.0095 | Geometry of Locally Compact Groups of Polynomial Growth and Shape of
Large Balls | GEOMETRY OF LOCALLY COMPACT GROUPS OF
POLYNOMIAL GROWTH AND SHAPE OF LARGE BALLS.
EMMANUEL BREUILLARD
Abstract. We show that any locally compact groupG with polynomial growth
is weakly commensurable to some simply connected solvable Lie group S, the
Lie shadow of G. We then study the shape of large balls and show, generaliz-
ing work of P. Pansu, that after a suitable renormalization, they converge to
a limiting compact set, which is isometric to the unit ball for a left-invariant
subFinsler metric on the so-called graded nilshadow of S. As by-products, we
obtain asymptotics for the volume of large balls, we prove that balls are Folner
and hence that the ergodic theorem holds for all ball averages. Along the way
we also answer negatively a question of Burago and Margulis [7] on asymptotic
word metrics and recover some results of Stoll [33] of the rationality of growth
series of Heisenberg groups.
Contents
1. Introduction 1
2. Quasi-norms and the geometry of nilpotent Lie groups 12
3. The nilshadow 19
4. Periodic metrics 23
5. Reduction to the nilpotent case 27
6. The nilpotent case 31
7. Locally compact G and proofs of the main results 39
8. Coarsely geodesic distances and speed of convergence 47
9. Appendix: the Heisenberg groups 52
References 55
1. Introduction
1.1. Groups with polynomial growth. Let G be a locally compact group with
left Haar measure volG. We will assume that G is generated by a compact sym-
metric subset Ω. Classically, G is said to have polynomial growth if there exist
C > 0 and k > 0 such that for any integer n ≥ 1
volG(Ω
n) ≤ C · nk,
Date: April 2012.
http://arxiv.org/abs/0704.0095v2
2 EMMANUEL BREUILLARD
where Ωn = Ω· . . . · Ω is the n-fold product set. Another choice for Ω would only
change the constant C, but not the polynomial nature of the bound. One of the
consequences of the analysis carried out in this paper is the following theorem:
Theorem 1.1 (Volume asymptotics). Let G be a locally compact group with poly-
nomial growth and Ω a compact symmetric generating subset of G. Then there
exists c(Ω) > 0 and an integer d(G) ≥ 0 depending on G only such that the
following holds:
volG(Ω
nd(G)
= c(Ω)
This extends the main result of Pansu [27]. The integer d(G) coincides with
the exponent of growth of a naturally associated graded nilpotent Lie group, the
asymptotic cone of G, and is given by the Bass-Guivarc’h formula (4) below.
The constant c(Ω) will be interpreted as the volume of the unit ball of a sub-
Riemannian Finsler metric on this nilpotent Lie group. Theorem 1.1 is a by-
product of our study of the asymptotic behavior of periodic pseudodistances on G,
that is pseudodistances that are invariant under a co-compact subgroup of G and
satisfy a weak kind of the existence of geodesics axiom (see Definition 4.1).
Our first task is to get a better understanding of the structure of locally compact
groups of polynomial growth. Guivarc’h [21] proved that locally compact groups
of polynomial growth are amenable and unimodular and that every compactly
generated1 closed subgroup also has polynomial growth.
Guivarc’h [21] and Jenkins [15] also characterized connected Lie groups with
polynomial growth: a connected Lie group has polynomial growth if and only if
it is of type (R), that is if for all x ∈ Lie(S), ad(x) has only purely imaginary
eigenvalues. Such groups are solvable-by-compact and any connected nilpotent
Lie group is of type (R).
It is much more difficult to characterize discrete groups with polynomial growth,
and this was done in a celebrated paper of Gromov [17], proving that they are
virtually nilpotent. Losert [24] generalized Gromov’s method of proof and showed
that it applied with little modification to arbitrary locally compact groups with
polynomial growth. In particular he showed that they contain a normal compact
subgroup modulo which the quotient is a (not necessarily connected) Lie group.
We will prove the following refinement.
Theorem 1.2 (Lie shadow). Let G be a locally compact group of polynomial
growth. Then there exists a connected and simply connected solvable Lie group S
of type (R), which is weakly commensurable to G. We call such a Lie group a Lie
shadow of G.
Two locally compact groups are said to be weakly commensurable if, up to
moding out by a compact kernel, they have a common closed co-compact subgroup.
More precisely, we will show that, for some normal compact subgroupK, G/K has
1in fact it follows from the Gromov-Losert structure theory that every closed subgroup is
compactly generated.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 3
a co-compact subgroup H/K which can be embedded as a closed and co-compact
subgroup of a connected and simply connected solvable Lie group S of type (R).
We must be aware that being weakly commensurable is not an equivalence
relation among locally compact groups (unlike among finitely generated groups).
Additionally, the Lie shadow S is not unique up to isomorphism (e.g. Z3 is a
co-compact lattice in both R3 and the universal cover of the group of motions of
the plane).
We cannot replace the word solvable by the word nilpotent in the above theo-
rem. We refer the reader to Example 7.9 for an example of a connected solvable
Lie group of type (R) without compact normal subgroups, which admits no co-
compact nilpotent subgroup. In fact this is typical for Lie groups of type (R). So
in the general locally compact case (or just the Lie case) groups of polynomial
growth can be genuinely not nilpotent, unlike what happens in the discrete case.
There are important differences between the discrete case and the general case.
For example, we will show that no rate of convergence can be expected in Theorem
1.1 when G is solvable not nilpotent, while some polynomial rate always holds in
the nilpotent discrete case [9].
Theorem 1.2 will enable us to reduce most geometric questions about locally
compact groups of polynomial growth, and in particular the proof of Theorem
1.1, to the connected Lie group case. Observe also that Theorem 1.2 subsumes
Gromov’s theorem on polynomial growth, because it is not hard to see that a
co-compact lattice in a solvable Lie group of polynomial growth must be virtually
nilpotent (see Remark 7.8). Of course in the proof we make use of Gromov’s
theorem, in its generalized form for locally compact groups due to Losert. The rest
of the proof combines ideas of Y. Guivarc’h, D. Mostow and a crucial embedding
theorem of H.C. Wang. It is given in Paragraph 7.1 and is largely independent of
the rest of the paper.
1.2. Asymptotic shapes. The main part of the paper is devoted to the asymp-
totic behavior of periodic pseudodistances on G. We refer the reader to Definition
4.1 for the precise definition of this term, suffices it to say now that it is a class of
pseudodistances which contains both left-invariant word metrics on G and geodesic
metrics on G that are left-invariant under co-compact subgroup of G.
Theorem 1.2 enables us to assume that G is a co-compact subgroup of a simply
connected solvable Lie group S, and rather than looking at pseudodistances on G,
we will look at pseudodistances on S that are left-invariant under a co-compact
subgroup H. More precisely a direct consequence of Theorem 1.2 is the following:
Proposition 1.3. Let G be a locally compact group with polynomial growth and ρ
a periodic metric on G. Then (G, ρ) is (1, C)-quasi-isometric to (S, ρS) for some
finite C > 0, where S is a connected and simply connected solvable Lie group of
type (R) and ρS some periodic metric on S.
Recall that two metric spaces (X, dX ) and (Y, dY ) are called (1, C)-quasi-isometric
if there exists a map φ : X → Y such that any y ∈ Y is at distance at most C
from some element in the image of φ and if |dY (φ(x), φ(x
′)) − dX(x, x
′)| ≤ C for
all x, x′ ∈ X.
4 EMMANUEL BREUILLARD
In the case when S is Rd and H is Zd, it is a simple exercise to show that any
periodic pseudodistance is asymptotic to a norm on Rd, i.e. ρ(e, x)/ ‖x‖ → 1 as
x → ∞, where ‖x‖ = lim 1
ρ(e, nx) is a well defined norm on Rd. Burago in [6]
showed a much finer result, namely that if ρ is coarsely geodesic, then ρ(e, x)−‖x‖
is bounded when x ranges over Rd.When S is a nilpotent Lie group andH a lattice
in S, then Pansu proved in his thesis [27], that a similar result holds, namely that
ρ(e, x)/ |x| → 1 for some (unique only after a choice of a one-parameter group of
dilations) homogeneous quasi-norm |x| on the nilpotent Lie group. However, we
show in Section 8, that it is not true in general that ρ(e, x) − |x| stays bounded,
even for finitely generated nilpotent groups, thus answering a question of Burago
(see also Gromov [20]). Our main purpose here will be to extend Pansu’s result
to solvable Lie groups of polynomial growth.
As was first noticed by Guivarc’h in his thesis [21], when dealing with geometric
properties of solvable Lie groups, it is useful to consider the so-called nilshadow of
the group, a construction first introduced by Auslander and Green in [2]. Accord-
ing to this construction, it is possible to modify the Lie product on S in a natural
way, by so to speak removing the semisimple part of the action on the nilradical,
in order to turn S into a nilpotent Lie group, its nilshadow SN . The two Lie
groups have the same underlying manifold, which is diffeomorphic to Rn, only a
different Lie product. They also share the same Haar measure. This “semisimple
part” is a commutative relatively compact subgroup T (S) of automorphisms of S,
image of S under a homomorphism T : S → Aut(S). The new product g ∗ h is
defined as follows by twisting the old one g · h by means of T (S),
(1) g ∗ h := g · T (g−1)h
The two groups S and SN are easily seen to be quasi-isometric, and this is why any
locally compact group of polynomial growth G is quasi-isometric to some nilpotent
Lie group. In particular, their asymptotic cones are bi-Lipschitz. The asymptotic
cone of a nilpotent Lie group is a certain associated graded nilpotent Lie group
endowed with a left invariant geodesic distance (or Carnot group). The graded
group associated to SN will be called the graded nilshadow of S. Section 3 will be
devoted to the construction and basic properties of the nilshadow and its graded
group.
In this paper, we are dealing with a finer relation than quasi-isometry. We will
be interested in when do two left invariant (or periodic) distances are asymptotic2
(in the sense that
d1(e,g)
d2(e,g)
→ 1 when g → ∞). In particular, for every locally
compact group G with polynomial growth, we will identify its asymptotic cone
up to isometry and not only up to quasi-isometry or bi-Lipschitz equivalence (see
Corollary 1.9 below). One of our main results is the following:
2Yet a finer equivalence relation is (1, C)-quasi-isometry, i.e. being at bounded distance in
Gromov-Hausdorff metric; classifying periodic metrics up to this kind of equivalence is much
harder.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 5
Theorem 1.4 (Main theorem). Let S be a simply connected solvable Lie group
with polynomial growth. Let ρ(x, y) be periodic pseudodistance on S which is in-
variant under a co-compact subgroup H of S (see Def. 4.1). On the manifold S,
one can put a new Lie group structure, which turns S into a stratified nilpotent Lie
group, the graded nilshadow of S, and a subFinsler metric d∞(x, y) on S which is
left-invariant for this new group structure such that
ρ(e, g)
d∞(e, g)
as g → ∞ in S. Moreover every automorphism in T (H) is an isometry of d∞.
The reader who wishes to see a simple illustration of this theorem can go directly
to subsection 8.1, where we have treated in detail a specific example of periodic
metric on the universal cover of the groups of motions of the plane.
The new stratified nilpotent Lie group structure on S given by the graded
nilshadow comes with a one-parameter family of so-called homogeneous dilations
{δt}t>0. It also comes with an extra group of automorphisms, namely the image
of H under the homomorphism T . This yields automorphisms of S for both the
original group structure on S and the new graded nilshadow group structure.
Moreover the dilations {δt}t>0 are automorphisms of the graded nilshadow and
they commute with T (H).
A subFinsler metric is a geodesic distance which is defined exactly as subRie-
mannian (or Carnot-Caratheodory) metrics on Carnot groups are defined (see e.g.
[25]), except that the norm used to compute the length of horizontal paths is not
necessarily a Euclidean norm. We refer the reader to Section 2.1 for a precise
definition.
In Theorem 1.4 the subFinsler metric d∞ is left invariant for the new Lie struc-
ture on S and it is also invariant under all automorphisms in T (H) (these form
a relatively compact commutative group of automorphisms). Moreover it satisfies
the following pleasing scaling law:
d∞(δt(x), δt(y)) = td∞(x, y) ∀t > 0.
The proof of Theorem 1.4 splits in two important steps. The first is a reduction
to the nilpotent case and is performed in Section 5. Using a double averaging
of the pseudodistance ρ over both K := T (H) and S/H, we construct an asso-
ciated pseudodistance, which is periodic for the nilshadow structure on S (i.e.
left-invariant by a co-compact subgroup for this structure), and we prove that it
is asymptotic to the original ρ. This reduces the problem to nilpotent Lie groups.
The key to this reduction is the following crucial observation: that unipotent au-
tomorphisms of S induce only a sublinear distortion, forcing the metric ρ to be
asymptotically invariant under T (H). The second step of the proof assumes that
S is nilpotent. This part is dealt with in Section 6 and is essentially a reformula-
tion of the arguments used by Pansu in [27].
6 EMMANUEL BREUILLARD
Incidently, we stress the fact that the generality in which Section 6 is treated
(i.e. for general coarsely geodesic, and even asymptotically geodesic periodic met-
rics) is necessary to prove even the most basic case (i.e. word metrics) of Theorem
1.4 for non-nilpotent solvable groups. So even if we were only interested in the
asymptotics of left invariant word metrics on a solvable Lie group of polynomial
growth S, we would still need to understand the asymptotics of arbitrary coarsely
geodesic left invariant distances (and not only word metrics!) on nilpotent Lie
groups. This is because the new pseudodistance obtained by averaging, see (30),
is no longer a word metric.
The subFinsler metric d∞(e, x) in the above theorem is induced by a certain
T (H)-invariant norm on the first stratum m1 of the graded nilshadow (which
is T (H)-invariant complementary subspace of the commutator subalgebra of the
nilshadow). This norm can be described rather explicitly as follows.
Recall that we have3 a canonical map π1 : S → m1, which is a group homomor-
phism for both the nilshadow and graded nilshadow structures. Then:
{v ∈ m1, ‖v‖∞ ≤ 1} =
CvxHull
π1(h)
ρ(e, h)
, h ∈ H\F
where the right hand side is the intersection over all compact subsets F of S of
the closed convex hull of the points π1(h)/ρ(e, h) for h ∈ H\F .
Figure 1 gives an illustration of the limit shape corresponding to the word
metric on the 3-dimensional discrete Heisenberg group with standard generators.
We explain in the Appendix how one can compute explicitly the geodesics of the
limit metric and the limit shape in this example.
When S itself is nilpotent to begin with and ρ is (in restriction to H) the
word metric associated to a symmetric compact generating set Ω of H (namely
ρΩ(e, h) := inf{n ∈ N;h ∈ Ω
n}), the above norm takes the following simple form:
(2) {v ∈ m1, ‖v‖∞ ≤ 1} = CvxHull {π1(ω), ω ∈ Ω}
For instance, in the special case when H is a torsion-free finitely generated nilpo-
tent group with generating set Ω and S is its Malcev closure, the unit ball
{v ∈ m1, ‖v‖∞ ≤ 1} is a polyhedron in m1. This was Pansu’s description in
[27].
However when S is not nilpotent, and is equipped with a word metric ρΩ on
a co-compact subgroup, then the determination of the limit shape, i.e. the de-
termination of the limit norm ‖ · ‖∞ on the abelianized nilshadow, is much more
difficult. Clearly ‖ · ‖∞ is K-invariant and it is a simple observation that the unit
ball for ‖ · ‖∞ is always contained in the convex hull of the K-orbit of π1(Ω).
3The subspace m1 can be identified with the abelianized nilshadow (or abelianized graded
nilshadow) by first identifying the nilshadow with its Lie algebra via the exponential map and
then projecting modulo the commutator subalgebra. The map does not depend on the choice
involved in the construction of the nilshadow. See also Remark 3.7.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 7
Nevertheless the unit ball is typically smaller than that (unless Ω was K-invariant
to begin with).
In general it would be interesting to determine whether there exists a simple
description of the limit shape of an arbitrary word metric on a solvable Lie group
with polynomial growth. We refer the reader to Section 8 and Paragraph 8.2 for an
example of a class of word metrics on the universal cover of the group of motions
of the plane, for which we were able to compute the limit shape.
Another by-product of Theorem 1.4 is the following result.
Corollary 1.5 (Asymptotic shape). Let S be a simply connected solvable Lie
group with polynomial growth and H a co-compact subgroup. Let ρ be an H-
periodic pseudodistance on S. Then in the Hausdorff metric,
(Bρ(t)) = C,
where C is a T (H)-invariant compact neighborhood of the identity in S, Bρ(t) is
the ρ-ball of radius t in S and {δt}t>0 is a one-parameter group of dilations on S
(equipped with the graded nilshadow structure). Moreover, C = {g ∈ S, d∞(e, g) ≤ 1}
is the unit ball of the limit subFinsler metric from Theorem 1.4.
Proof. By Theorem 1.4, for every ε > 0 we have Bd∞(t−εt) ⊂ Bρ(t) ⊂ Bd∞(t+εt)
if t is large enough. Since δ 1
(Bd∞(t)) = C, for all t > 0, we are done. �
Combining this with Theorem 1.2, we also get the following corollary, of which
Theorem 1.1 is only a special case with ρ the word metric associated to the gen-
erating set Ω.
Corollary 1.6 (Volume asymptotics). Suppose that G is a locally compact group
with polynomial growth and ρ is a periodic pseudodistance on G. Let Bρ(t) be
the ρ-ball of radius t in G, i.e. Bρ(t) = {x ∈ G, ρ(e, x) ≤ t}, then there exists a
constant c(ρ) > 0 such that the following limit exists:
(3) lim
volG(Bρ(t))
td(G)
= c(ρ)
Here d(G) is the integer d(SN ), the so-called homogeneous dimension of the
nilshadow SN of a Lie shadow S of G (obtained by Theorem 1.2), and is given by
the Bass-Guivarc’h formula:
(4) d(SN ) =
dim(Ck(SN ))
where {Ck(SN )}k is the descending central series of SN .
The limit c(ρ) is equal to the volume volS(C) of the limit shape C from Corollary
1.5 once we make the right choice of Haar measure on a Lie shadow S of G. Let
us explain this choice. Recall that according to Theorem 1.2, G/K admits a co-
compact subgroup H/K which embeds co-compactly in S. Starting with a Haar
measure volG on G, we get a Haar measure on G/K after fixing the Haar measure
of K to be of total mass 1, and we may then choose a Haar measure on H/K so
that the compact quotient G/H has volume 1. Finally we choose the Haar measure
8 EMMANUEL BREUILLARD
Figure 1. The asymptotic shape of large balls in the Cayley graph
of the Heisenberg group H(Z) = 〈x, y|[x, [x, y]] = [y, [x, y]] = 1〉
viewed in exponential coordinates.
on S so that the other compact quotient S/(H/K) has volume 1. This gives the
desired Haar measure volS such that c(ρ) = volS(C).
Note that Haar measure on S is also invariant under the group of automor-
phisms T (S) and is thus left invariant for the nilshadow structure on S. It is also
left invariant for the graded nilshadow structure. In both exponential coordinates
of the first kind (on SN ) and of the second kind (as in Lemma 3.10), Haar measure
is just Lebesgue measure.
In the case of the discrete Heisenberg group of dimension 3 equipped with the
word metric given by the standard generators, it is possible to compute the con-
stant c(ρ) and the volume of the limit shape as shown in Figure 1. In this case the
volume is 31
(see the Appendix). The 5-dimensional Heisenberg group can also be
worked out and the volume of its limit shape (associated to the word metric given
by standard generators) is equal to 2009
21870
log 2
32805
. The fact that this number is
transcendental implies that the growth series of this group, i.e. the formal power
series
n≥0 |Bρ(n)|z
n is not algebraic in the sense that it is not a solution of a
polynomial equation with rational functions in C(z) as coefficients (see [33, Prop.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 9
3.3.]). This was observed by Stoll in [33] by more direct combinatorial means.
Stoll also shows there the interesting fact that the growth series can be rational
for some other choices of generating sets in the 5-dimensional Heisenberg group.
So rationality of the growth series depends on the generating set.
Another interesting feature is asymptotic invariance:
Corollary 1.7 (Asymptotic invariance). Let S be a simply connected solvable
Lie group with polynomial growth and ρ a periodic pseudodistance on S. Let ∗ be
the new Lie product on S given by the nilshadow group structure (or the graded
nilshadow group structure). Then ρ(e, g ∗ x)/ρ(e, x) → 1 as x → ∞ for every
g ∈ S.
This follows immediately from Theorem 1.4, when ∗ is the graded nilshadow
product, and from Theorem 6.2 below in the case ∗ is the nilshadow group struc-
ture.
It is worth observing that we may not in general replace ∗ by the ordinary
product on S. Indeed, let for instance S = R ⋉ R2 be the universal cover of the
group of motions of the Euclidean plane, then S, like its nilshadow R3, admits
a lattice Γ ≃ Z3. The quotient S/Γ is diffeomorphic to the 3-torus R3/Z3 and
it is easy to find Riemannian metrics on this torus so that their lift to R3 is not
invariant under rotation around the z-axis. Hence this metric, viewed on the Lie
group S will not be asymptotically invariant under left translation by elements of
S. Nevertheless, if the metric is left-invariant and not just periodic, then we have
the following corollary of the proof of Theorem 1.4.
Corollary 1.8 (Left-invariant pseudodistances are asymptotic to subFinsler met-
rics). Let S be a simply connected solvable Lie group of polynomial growth and ρ
be a periodic pseudodistance on S which is invariant under all left-translations by
elements of S (e.g. a left-invariant coarsely geodesic metric on S). Then there
is a left-invariant subFinsler metric d on S which is asymptotic to ρ in the sense
ρ(e,g)
d(e,g)
→ 1 as g → ∞.
We already mentioned above that determining the exact limit shape of a word
metric on S is a difficult task. Consequently so is the task of telling when two
distinct word metrics are asymptotic. The above statement says that in any case
every word metric on S is asymptotic to some left-invariant subFinsler metric. So
the set of possible limit shapes is no richer for word metrics than for left-invariant
subFinsler metrics.
We note that in the case of nilpotent Lie groups (where K is trivial), Theorem
1.4 shows that every periodic metric is asymptotic to a left-invariant metric. It is
still an open problem to determine whether every coarsely geodesic periodic metric
is at a bounded distance from a left-invariant metric (this is Burago’s theorem in
n, more about it below).
Theorems 1.2 and 1.4 allow us to describe the asymptotic cone of (G, ρ) for any
periodic pseudodistance ρ on any locally compact group with polynomial growth.
10 EMMANUEL BREUILLARD
Corollary 1.9 (Asymptotic cone). Let G be a locally compact group with polyno-
mial growth and ρ a periodic pseudodistance on G. Then the sequence of pointed
metric spaces {(G, 1
ρ, e)}n≥1 converges in the Gromov-Hausdorff topology. The
limit is the metric space (N, d∞, e), where N is a graded simply connected nilpo-
tent Lie group and d∞ a left invariant subFinsler metric on N . Moreover the Lie
group N is (up to isomorphism) independent of ρ. The space (N, d∞) is isometric
to “the asymptotic cone” associated to (G, ρ). This asymptotic cone is independent
of the choice of ultrafilter used to define it.
This corollary is a generalization of Pansu’s theorem ((10) in [27]). We refer
the reader to the book [18] for the definitions of the asymptotic cone and the
Gromov-Hausdorff convergence. We discuss in Section 8 the speed of convergence
(in the Gromov-Hausdorff metric) in this theorem and its corollaries about volume
growth. In particular there is a major difference between the discrete nilpotent case
and the solvable non nilpotent case. In the former, one can find a polynomial rate
of convergence [9], while in the latter no such rate exist in general (see Theorem
8.1).
1.3. Folner sets and ergodic theory. A consequence of Corollary 1.6 is that
sequences of balls with radius going to infinity are Folner sequences, namely:
Corollary 1.10. Let G be a locally compact group with polynomial growth and
ρ a periodic pseudodistance on G. Let Bρ(t) be the ρ-ball of radius t in G. Then
{Bρ(t)}t>0 form a Folner family of subsets of G namely, for any compact set F
in G, we have (∆ denotes the symmetric difference)
(5) lim
volG(FBρ(t)∆Bρ(t))
volG(Bρ(t))
Proof. Indeed FBρ(t)∆Bρ(t) ⊂ Bρ(t + c)\Bρ(t) for some c > depending on F .
Hence (5) follows from (3). �
This settles the so-called “localization problem” of Greenleaf for locally compact
groups of polynomial growth (see [16]), i.e. determining whether the powers of
a compact generating set {Ωn}n form a Folner sequence. At the same time it
implies that the ergodic theorem for G-actions holds along any sequence of balls
with radius going to infinity.
Theorem 1.11. (Ergodic Theorem) Let be given a locally compact group G with
polynomial growth together with a measurable G-space X endowed with a G-
invariant ergodic probability measure m. Let ρ be a periodic pseudodistance on
G and Bρ(t) the ρ-ball of radius t in G. Then for any p, 1 ≤ p < ∞, and any
function f ∈ Lp(X,m) we have
volG(Bρ(t))
Bρ(t)
f(gx)dg =
for m-almost every x ∈ X and also in Lp(X,m).
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 11
In fact, Corollary 1.10 above, was the “missing block” in the proof of the ergodic
theorem on groups of polynomial growth. So far and to my knowledge, Corollary
1.10 and Theorem 1.11 were known only along some subsequence of balls {Bρ(tn)}n
chosen so that (5) holds (see for instance [10] or [34]). This issue was drawn to
my attention by A. Nevo and was my initial motivation for the present work. We
refer the reader to the A. Nevo’s survey paper [26] Section 5.
It later turned out that the mere fact that balls are Folner in a given polynomial
growth locally compact group can also be derived from the fact these groups are
doubling metric spaces (which is an easier result than the precise asymptotics
vol(Ωn) ∼ cΩn
d(G) proved in this paper and only requires lower and upper bounds
of the form c1n
d(G) ≤ vol(Ωn) ≤ c2n
d(G)). This was observed by R. Tessera
[35] who rediscovered a cute argument of Colding and Minicozzi [11, Lemma 3.3.]
showing that the volume of spheres Ωn+1 \ Ωn is at most some O(n−δ) times the
volume of the ball Ωn, where δ > 0 is a positive constant depending only on the
doubling constant the word metric induced by Ω in G.
In [9], we give a better upper bound (which depends only on the nilpotency class
and not on the doubling constant) for the volume of spheres in the case of finitely
generated nilpotent groups. This is done by showing the following error term in
the asymptotics of the volume of balls: we have vol(Ωn) = cΩn
d(G)+O(nd(G)−αr ),
where αr > 0 depends only on the nilpotency class r of G. We refer the reader to
Section 8 and to the preprint [9] for more information on this. We only note here
that although the above Colding-Minicozzi-Tessera upper bound on the volume of
spheres holds generally for all locally compact groups G with polynomial growth,
unless G is nilpotent, there is no error term in general in the asymptotics of the
volume of balls. An example with arbitrarily small speed is given in §8.1.
1.4. A conjecture of Burago and Margulis. In [7] D. Burago and G. Margulis
conjectured that any two word metrics on a finitely generated group which are
asymptotic (in the sense that
ρ1(e,γ)
ρ2(e,γ)
tends to 1 at infinity) must be at a bounded
distance from one another (in the sense that |ρ1(e, γ) − ρ2(e, γ)| = O(1)). This
holds for abelian groups. An analogous result was proved by Abels and Margulis
for word metrics on reductive groups [1]. S. Krat [23] established this property
for word metrics on the Heisenberg group H3(Z). However using Theorem 1.4
(which in this particular case of finitely generated nilpotent groups is just Pansu’s
theorem [27]) we will show in Section 8.3, that there are counter-examples and
exhibit two word metrics on H3(Z) × Z which are asymptotic and yet are not at
a bounded distance. For more on this counter-example, and how to adequately
modify the conjecture of Burago and Margulis, we refer the interested reader to
1.5. Organization of the paper. Sections 2-4 are devoted to preliminaries. In
Section 2 we present the basic nilpotent theory as can be found in Guivarc’h’s
thesis [21]. In particular, a full proof of the Bass-Guivarc’h formula is given. In
Section 3, we recall the construction of the nilshadow of a solvable Lie group.
12 EMMANUEL BREUILLARD
In Section 4 we set up the axioms and basic properties of the (pseudo)distance
functions that are studied in this paper.
Sections 5-7 contain the core of the proof of the main theorems. In Section 5, we
assume that G is a simply connected solvable Lie group and reduce the problem to
the nilpotent case. In Section 6, we assume that G is a simply connected nilpotent
Lie group and prove Theorem 1.4 in this case following the strategy used by Pansu
in [27]. In Section 7, we prove Theorem 1.2 for general locally compact groups
and reduce the proof of the results of the introduction to the Lie case.
In the last section we make further comments about the speed of convergence.
In particular we give examples answering negatively the aforementioned question
of Burago and Margulis.
The Appendix is devoted to the discrete Heisenberg groups of dimension 3 and
5. We compute their limit balls, explain Figure 1, and recover the main result of
Stoll [33].
The reader who is mainly interested in the nilpotent group case can read directly
Section 6 while keeping an eye on Sections 2 and 4 for background notations and
elementary facts.
Finally, let us mention that the results and methods of this paper were largely
inspired by the works of Y. Guivarc’h [21] and P. Pansu [27].
1.6. Nota Bene. A version of this article circulated since 2007. The present ver-
sion contains essentially the same material, only the exposition has been improved
and several somewhat sketchy arguments have been replaced by full fledged proofs
(in particular in Sections 3 and 7). This delay is due to the fact that I was plan-
ning for a long time to improve Section 6 and show an error term in the volume
asymptotics of balls in nilpotent groups. E. Le Donne and I recently managed to
achieve this and it has now become an independent joint paper [9].
2. Quasi-norms and the geometry of nilpotent Lie groups
In this section, we review the necessary background material on nilpotent Lie
groups. In paragraph 2.4, we give some crucial properties of homogeneous quasi
norms and reproduce some lemmas originally due to Y. Guivarc’h which will be
used in the sequel. Meanwhile, we prove the Bass-Guivarc’h formula for the de-
gree of polynomial growth of nilpotent Lie groups, following Guivarc’h’s original
argument.
2.1. Carnot-Caratheodory metrics. Let G be a connected Lie group with Lie
algebra g and let m1 be a vector subspace of g. We denote by ‖·‖ a norm on m1.
We now recall the definition of a left-invariant Carnot-Carathéodory metric also
called subFinsler metric on G. Let x, y ∈ G. We consider all possible piecewise
smooth paths ξ : [0, 1] → G going from ξ(0) = x to ξ(1) = y. Let ξ′(u) be the
tangent vector which is pulled back to the identity by a left translation, i.e.
= ξ(u) · ξ′(u)
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 13
where ξ′(u) ∈ g and the notation ξ(u) · ξ′(u) means the image of ξ′(u) under
the differential at the identity of the left translation by the group element ξ(u).
We say that the path ξ is horizontal if the vector ξ′(u) belongs to m1 for all
u ∈ [0, 1]. We denote by H the set of piecewise smooth horizontal paths. The
Carnot-Carathéodory metric associated to the norm ‖·‖ is defined by:
d(x, y) = inf{
∥∥ξ′(u)
∥∥ du, ξ ∈ H, ξ(0) = x, ξ(1) = y}
where the infimum is taken over all piecewise smooth paths ξ : [0, 1] → N with
ξ(0) = x, ξ(1) = y that are horizontal in the sense that ξ′(u) ∈ m1 for all u. If
‖ · ‖ is a Euclidean norm, the metric d(x, y) is also called subRiemannian. In this
paper however the norm ‖ · ‖ will typically not be Euclidean (it can be polyhedral
like in the case of word metrics on finitely generated nilpotent groups) and d(x, y)
will only be subFinsler. If m1 = g, and ‖·‖ is a Euclidean (resp. arbitrary) norm
on g, then d is simply the usual left-invariant Riemannian (resp. Finsler) metric
associated to ‖·‖ .
Chow’s theorem (e.g. see [19] or [25]) tells us that d(x, y) is finite for all x and
y in G if and only if the vector subspace m1, together with all brackets of elements
of m1, generates the full Lie algebra g. If this condition is satisfied, then d is a
distance on G which induces the original topology of G.
In this paper, we will only be concerned with Carnot-Caratheodory metrics on
a simply connected nilpotent Lie group N . In the sequel, whenever we speak of a
Carnot-Carathéodory metric on N, we mean one that is associated to a norm ‖·‖
on a subspace m1 such that n = m1 ⊕ [n, n] where n = Lie(N). It is easy to check
that any such m1 generates the Lie algebra n.
Remark 2.1. Let us observe here that for such a metric d on N, we have the
following description of the unit ball for ‖·‖
{v ∈ m1, ‖v‖ ≤ 1} =
π1(x)
d(e, x)
, x ∈ N\{e}
where π1 is the linear projection from n (identified with N via exp) to m1 with
kernel [n, n]. Indeed, π1 gives rise to a homomorphism from N to the vector space
m1. And if ξ(u) is a horizontal path from e to x, then applying π1 to (6) we
get d
π1(ξ(u)) = ξ
′(u), hence π1(x) =
ξ′(u)du. Hence ‖π1(x)‖ ≤ d(e, x) with
equality if x ∈ m1.
2.2. Dilations on a nilpotent Lie group and the associated graded group.
We now focus on the case of simply connected nilpotent Lie groups. Let N be
such a group with Lie algebra n and nilpotency class r. For background about
analysis on such groups, we refer the reader to the book [12]. The exponential
map is a diffeomorphism between n and N . Most of the time, if x ∈ n, we will
abuse notation and denote the group element exp(x) simply by x. We denote
by {Cp(n)}p the central descending series for n, i.e. C
p+1(n) = [n, Cp(n)] with
C0(n) = n and Cr(n) = {0}.
14 EMMANUEL BREUILLARD
Let (mp)p≥1 be a collection of vector subspaces of n such that for each p ≥ 1,
(7) Cp−1(n) = Cp(n)⊕mp.
Then n = ⊕p≥1mp and in this decomposition, any element x in n (or N by abuse
of notation) will be written in the form
πp(x)
where πp(x) is the linear projection onto mp.
To such a decomposition is associated a one-parameter group of dilations (δt)t>0.
These are the linear endomorphisms of n defined by
δt(x) = t
for any x ∈ mp and for every p. Conversely, the one-parameter group (δt)t≥0
determines the (mp)p≥1’s since they appear as eigenspaces of each δt, t 6= 1. The
dilations δt do not preserve a priori the Lie bracket on n. This is the case if and
only if
(8) [mp,mq] ⊆ mp+q
for every p and q (where [mp,mq] is the subspace spanned by all commutators of
elements of mp with elements of mq). If (8) holds, we say that the (mp)p≥1 form a
stratification of the Lie algebra n, and that n is a stratified (or homogeneous) Lie
algebra. It is an exercise to check that (8) is equivalent to require [m1,mp] = mp+1
for all p.
If (8) does not hold, we can however consider a new Lie algebra structure on
the real vector space n by defining the new Lie bracket as [x, y]∞ = πp+q([x, y])
if x ∈ mp and y ∈ mq. This new Lie algebra n∞ is stratified and has the same
underlying vector space as n. We denote by N∞ the associated simply connected
Lie group. Moreover the (δt)t>0 form a one-parameter group of automorphisms of
n∞. In fact the original Lie bracket [x, y] on n can be deformed continuously to
[x, y]∞ through a continuous family of Lie algebra structures by setting
(9) [x, y]t = δ 1
([δtx, δty])
and letting t → +∞. Note that conversely, if the δt’s are automorphisms of n,
then [x, y] = πp+q([x, y]) for all x ∈ mp and y ∈ mq, and n = n∞.
The graded Lie algebra associated to n is by definition
gr(n) =
Cp(n)/Cp+1(n)
endowed with the Lie bracket induced from that of n. The quotient map mp →
Cp(n)/Cp+1(n) gives rise to a linear isomorphism between n and gr(n), which is
a Lie algebra isomorphism between the new Lie algebra structure n∞ and gr(n).
Hence stratified Lie algebra structures induced by a choice of supplementary sub-
spaces (mp)p≥1 as in (7) are all isomorphic to gr(n).
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 15
On N∞ the left-invariant subFinsler metrics d∞ associated to a choice of norm
on m1 are of special interest. The one-parameter group of dilations {δt}t is an
automorphism of N∞ and that
(10) d∞(δtx, δty) = td∞(x, y)
for any x, y ∈ N∞. The metric space (N∞, d∞) is called a Carnot group.
If on the other hand the simply connected nilpotent Lie groupN is not stratified,
then the group of dilations (δt)t associated to a choice of supplementary vector
subspaces mi’s as in (7) will not consist of automorphisms of N and the relation
(10) will not hold.
Note also that if we are given two different choices of supplementary subspaces
mi’s and m
i’s as in (7), then the left-invariant Carnot-Caratheodory metrics on
the corresponding stratified Lie groups are isometric if and only if (m1, ‖·‖) and
(m′1, ‖·‖
) are isometric (a linear isomorphism from m1 to m
1 that sends ‖·‖ to
extends to an isometry of the two Carnot groups).
2.3. The Campbell-Hausdorff formula. The exponential map exp : n → N
is a diffeomorphism. In the sequel, we will often abuse notation and identify N
and n without further notice. In particular, for two elements x and y of n (or
N equivalently) xy will denote their product in N , while x + y denotes the sum
in n. Let (δt)t be a one-parameter group of dilations associated to a choice of
supplementary subspaces mi’s as in (7). We denote the corresponding stratified
Lie algebra by n∞ as above and the Lie group by N∞. The product on N∞ is
denoted by x ∗ y. On N∞ the dilations (δt)t are automorphisms.
The Campbell-Hausdorff formula (see [12]) allows to give a more precise form of
the product in N. Let (ei)1≤i≤d be a basis of n adapted to the decomposition into
mi’s, that is mi = span{ej , ej ∈ mi}. Let x = x1e1 + ...+ xded the corresponding
decomposition of an element x ∈ n. Then define the degree di = deg(ei) to be the
largest j such that ei ∈ C
j−1(n). If α = (α1, ..., αd) ∈ N
d is a multi-index, then let
dα = deg(e1)α1 + ...+ deg(ed)αd.
The Campbell-Hausdorff formula yields
(11) (xy)i = xi + yi +
Cα,βx
where Cα,β are real constants and the sum is over all multi-indices α and β such
that dα + dβ ≤ deg(ei), dα ≥ 1 and dβ ≥ 1.
From (9), it is easy to give the form of the associated stratified Lie group law:
(12) (x ∗ y)i = xi + yi +
Cα,βx
where the sum is restricted to those α’s and β’s such that dα + dβ = deg(ei),
dα ≥ 1 and dβ ≥ 1.
2.4. Homogeneous quasi-norms and Guivarc’h’s theorem on polynomial
growth. Let n be a finite dimensional real nilpotent Lie algebra and consider a
decomposition
n = m1 ⊕ ...⊕mr
16 EMMANUEL BREUILLARD
by supplementary vector subspaces as in (7). Let (δt)t>0 be the one parameter
group of dilations associated to this decomposition, that is δt(x) = t
ix if x ∈ mi.
We now introduce the following definition.
Definition 2.2 (Homogeneous quasi-norm). A continuous function | · | : n → R+
is called a homogeneous quasi-norm associated to the dilations (δt)t, if it satisfies
the following properties:
(i) |x| = 0 ⇔ x = 0.
(ii) |δt(x)| = t|x| for all t > 0.
Example 2.3. (1) Quasi-norms of supremum type, i.e. |x| = maxp ‖πp(x)‖
where ‖·‖p are ordinary norms on the vector space mp and πp is the projection on
mp as above.
(2) |x| = d∞(e, x), where d∞ is a Carnot-Carathéodory metric on a stratified
nilpotent Lie group (as the relation (10) shows).
Clearly, a quasi-norm is determined by its sphere of radius 1 and two quasi-
norms (which are homogeneous with respect to the same group of dilations) are
always equivalent in the sense that
|·|1 ≤ |·|2 ≤ c |·|1
for some constant c > 0 (indeed, by continuity, | · |2 admits a maximum on the
“sphere” {|x|1 = 1}). If the two quasi-norms are homogeneous with respect to
two distinct semi-groups of dilations, then the inequalities (13) continue to hold
outside a neighborhood of 0, but may fail near 0.
Homogeneous quasi-norms satisfy the following properties:
Proposition 2.4. Let | · | be a homogeneous quasi-norm on n, then there are
constants C,C1, C2 > 0 such that
(a) |xi| ≤ C · |x|
deg(ei) if x = x1e1 + ...+ xnen in an adapted basis (ei)i.
(b) |x−1| ≤ C · |x|.
(c) |x+ y| ≤ C · (|x|+ |y|)
(d) |xy| ≤ C1(|x|+ |y|) + C2.
Properties (a), (b) and (c) are straightforward from the fact that |x| = maxp ‖πp(x)‖
is a homogeneous quasi-norm and from (13). Property (d) justifies the term “quasi-
norm” and follows from Lemma 2.5 below. It can be a problem that the constant
C1 in (d) may not be equal to 1. In fact, this is why we use the word quasi-norm
instead of just norm, because we do not require the triangle inequality axiom to
hold. However the following lemma of Guivarc’h is often a good enough remedy
to this situation. Let ‖·‖p be an arbitrary norm on the vector space mp.
Lemma 2.5. (Guivarc’h, [21] lemme II.1) Let ε > 0. Up to rescaling each ‖·‖p
into a proportional norm λp ‖·‖p (λp > 0) if necessary, the quasi-norm |x| =
maxp ‖πp(x)‖
satisfies
(14) |xy| ≤ |x|+ |y|+ ε
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 17
for all x, y ∈ N . If N is stratified with respect to (δt)t we can take ε = 0.
This lemma is crucial also for computing the coarse asymptotics of volume
growth. For the reader’s convenience, we reproduce here Guivarc’h’s argument,
which is based on the Campbell-Hausdorff formula (11).
Proof. We fix λ1 = 1 and we are going to give a condition on the λi’s so that (14)
holds. The λi’s will be taken to be smaller and smaller as i increases. We set
|x| = maxp ‖πp(x)‖
and let |x|λ = maxp ‖λpπp(x)‖
for any r-tuple of λi’s.
We want that for any index p ≤ r,
(15) λp ‖πp(xy)‖p ≤ (|x|λ + |y|λ + ε)
By (11) we have πp(xy) = πp(x) + πp(y) + Pp(x, y) where Pp is a polynomial map
into mp depending only on the πi(x) and πi(y) with i ≤ p− 1 such that
‖Pp(x, y)‖p ≤ Cp ·
l,m≥1,l+m≤p
Mp−1(x)
lMp−1(y)
where Mk(x) := maxi≤k ‖πi(x)‖
i and Cp > 0 is a constant depending on Pp and
on the norms ‖·‖i’s. Since ε > 0, when expanding the right hand side of (15) all
terms of the form |x|lλ|y|
λ with l +m ≤ p appear with some positive coefficient,
say εl,m. The terms |x|
and |y|
appear with coefficient 1 and cause no trouble
since we always have λp ‖πp(x)‖p ≤ |x|
λ and λp ‖πp(y)‖p ≤ |y|
λ. Therefore, for
(15) to hold, it is sufficient that
λpCpMp−1(x)
lMp−1(y)
m ≤ εl,m|x|
for all remaining l and m. However, clearly Mk(x) ≤ Λk · |x|λ where Λk :=
maxi≤k{1/λ
i } ≥ 1. Hence a sufficient condition for (15) to hold is
where ε = min εl,m. Since Λp−1 depends only on the first p−1 values of the λi’s, it
is obvious that such a set of conditions can be fulfilled by a suitable r-tuple λ. �
Remark 2.6. The constant C2 in Property (d) above can be taken to be 0 when N
is stratified with respect to the mi’s (i.e. the δt’s are automorphisms), as is easily
seen after changing x and y into their image under δt. And conversely, if C2 = 0
for some δt-homogeneous quasi-norm on N, then N admits a stratification. Indeed,
from (11) and (12), we see that if the δt’s are not automorphisms, then one can
find x, y ∈ N such that, when t is small enough, |δt(xy) − δt(x)δt(y)| ≥ ct
(r−1)/r
for some c > 0. However, combining Properties (c) and Property (d) with C2 = 0
above we must have |δt(xy)− δt(x)δt(y)| = O(t) near t = 0. A contradiction.
Guivarc’h’s lemma enables us to show:
Theorem 2.7. (Guivarc’h ibid.) Let Ω be a compact neighborhood of the identity
in a simply connected nilpotent Lie group N and ρΩ(x, y) = inf{n ≥ 1, x
−1y ∈ Ωn}.
18 EMMANUEL BREUILLARD
Then for any homogeneous quasi-norm | · | on N, there is a constant C > 0 such
|x| ≤ ρΩ(e, x) ≤ C|x|+ C
Proof. Since any two homogeneous quasi-norms (w.r.t the same one-parameter
group of dilations) are equivalent, it is enough to do the proof for one of them,
so we consider the quasi-norm obtained in Lemma 2.5 with the extra property
(14). The lower bound in (16) is a direct consequence of (14) and one can take
there C to be max{|x|, x ∈ Ω} + ε. For the upper bound, it suffices to show
that there is C ∈ N such that for all n ∈ N, if |x| ≤ n then x ∈ ΩCn. To
achieve this, we proceed by induction of the nilpotency length of N. The result
is clear when N is abelian. Otherwise, by induction we obtain C0 ∈ N such that
x = ω1 · ... · ωC0n · z where ωi ∈ Ω and z ∈ C
r−1(N) whenever |x| ≤ n. Hence
|z| ≤ |x|+C0n ·max |ω
i |+ εC0 · n ≤ C1n for some other constant C1 ∈ N. So we
have reduced the problem to x = z ∈ mr = C
r−1(N) which is central in N. We
have z = zn
1 where |z1| = |z|/n ≤ C1. Since Ω is a neighborhood of the identity
in N, the set U of all products of at most dim(mr) simple commutators of length
r of elements in Ω is a neighborhood of the identity in Cr−1(N) (e.g. see [19],
p113). It follows that there is a constant C2 ∈ N such that z1 is in U
C2 , hence the
product of at most C2 dim(mr) simple commutators. Then we are done because z
itself will be equal to the same product of commutators where each letter xi ∈ Ω
is replaced by xni . This last fact follows from the following lemma:
Lemma 2.8. Let G be a nilpotent group of nilpotency class r and n1, ..., nr be
positive integers. Then for any x1, ..., xr ∈ G
1 , [x
2 , [..., x
r ]...] = [x1, [x2, [..., xr]...]
n1·...·nr
To prove the lemma it suffices to use induction and the following obvious fact:
if [x, y] commutes to x and y then [xn, y] = [x, y]n. �
Finally, we obtain:
Corollary 2.9. Let Ω be a compact neighborhood of the identity in N. Then there
are positive constants C1 and C2 such that for all n ∈ N,
d ≤ volN (Ω
n) ≤ C2n
where d is given by the Bass-Guivarc’h formula:
(17) d =
i · dimmi
Proof. By Theorem 2.7, it is enough to estimate the volume of the quasi-norm
balls. By homogeneity of the quasi-norm, we have volN{x, |x| ≤ t} = t
dvolN{x, |x| ≤
1}. �
Remark 2.10. The use of Malcev’s embedding theorem allows, as Guivarc’h ob-
served, to deduce immediately that the analogous result holds for virtually nilpotent
finitely generated groups. This fact that was also proven independently by H. Bass
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 19
[3] by a direct combinatorial argument. See also Tits’ appendix to Gromov’s pa-
per [17]. In fact Guivarc’h’s Theorem 2.7 seems to have been rediscovered several
times in the past 40 years, including by Pansu in his thesis [27], the latest example
of that being [22].
3. The nilshadow
The goal of this section is to introduce the nilshadow of a simply connected
solvable Lie group G. We will assume that G has polynomial growth, although
this last assumption is not necessary for almost everything we do in this section.
The only statement which will be used afterwards in the paper (in Section 5) is
Lemma 3.12 below. The reader familiar with the nilshadow can jump directly to
the statement of this lemma and skip the forthcoming discussion.
3.1. Construction of the nilshadow. The nilshadow of G is a simply connected
nilpotent Lie group GN , which is associated to G in a natural way. This notion
was first introduced by Auslander and Green in [2] in their study of flows on
solvmanifolds. They defined it as the unipotent radical of a semi-simple splitting
of G. However, we are going to follow a different approach for its construction by
working first at the Lie algebra level. We refer the reader to the book [13] where
this approach is taken up.
Let g be a solvable real Lie algebra and n the nilradical of g.We have [g, g] ⊂ n.
If x ∈ g, we write ad(x) = ads(x) + adn(x) the Jordan decomposition of ad(x) in
GL(g). Since ad(x) ∈ Der(g), the space of derivations of g, and Der(g) is the Lie
algebra of the algebraic group Aut(g), the Jordan components ads(x) and adn(x)
also belong to Der(g). Moreover, for each x ∈ g, ads(x) sends g into n (because
so does ad(x) and ads(x) is a polynomial in ad(x)). Let h be a Cartan subalgebra
of g, namely a nilpotent self-normalizing subalgebra. Recall that the image of
a Cartan subalgebra by a surjective Lie algebra homomorphism is again a Car-
tan subalgebra. Now since g/n is abelian, it follows that h maps onto g/n, i.e.
h+ n = g. Moreover ads(x)|h = 0 if x ∈ h, because h is nilpotent.
Now pick any real vector subspace v of h in direct sum with n. Then the
following two conditions hold:
(i) v⊕ n = g .
(ii) ads(x)(y) = 0 for all x, y ∈ v.
From (i) and (ii), it follows easily that ads(x) commutes with ad(y), ads(y) and
adn(y), for all x, y in v. We have:
Lemma 3.1. The map v → Der(g) defined by x 7→ ads(x) is a Lie algebra
homomorphism.
Proof. First let us check that this map is linear. Let x, y ∈ v. By the above
ads(y) and ads(x) commute with each other (hence their sum is semi-simple) and
commute with adn(x)+adn(y). From the uniqueness of the Jordan decomposition
20 EMMANUEL BREUILLARD
it remains to check that adn(x)+adn(y) is nilpotent if x, y in v. To see this, apply
the following obvious remark twice to a = adn(x) and V = ad(n) first and then to
a = adn(y) and V = span{adn(x), ad((ad(y))
nx), n ≥ 1} : Let V be a nilpotent
subspace of GL(g) and a ∈ GL(g) nilpotent, i.e. V n = 0 and am = 0 for some
n,m ∈ N and assume [a, V ] ⊂ V. Then (a+ V )nm = 0.
The fact that this map is a Lie algebra homomorphism follows easily from the
fact that all ads(x), x ∈ v commute with one another and with [g, g] ⊂ n.
We define a new Lie bracket on g by setting:
(18) [x, y]N = [x, y]− ads(xv)(y) + ads(yv)(x)
where xv is the linear projection of x on v according to the direct sum v⊕n = g. The
Jacobi identity is checked by a straightforward computation where the following
fact is needed: ads (ads(x)(y)) = 0 for all x, y ∈ g. This holds because, as we just
saw, ads(x)(g) ⊂ n for all x ∈ g, and ads(a) = 0 if a ∈ n.
Definition 3.2. Let gN be the vector space g endowed with the new Lie algebra
structure [·, ·]N given by (18). The nilshadow GN of G is defined to be the simply
connected Lie group with Lie algebra gN .
It is easy to check that gN is a nilpotent Lie algebra. To see this, note first that
[gN , gN ]N ⊂ n, and if x ∈ gN and y ∈ n then [x, y]N = (adn(xv) + ad(xn))(y).
However, adn(xv) + ad(xn) is a nilpotent endomorphism of n as follows from the
same remark used in the proof of Lemma 3.1. Hence gN is a nilpotent.
The nilshadow Lie product on GN will be denoted by ∗ in order to distinguish
it from the original Lie product on G. In the sequel, we will often identify G (resp.
GN ) with its Lie algebra g (resp. gN ) via their respective exponential map. Since
the underlying space of gN was g itself, this gives an identification (although not
a group isomorphism) between G and GN . Then the nilshadow Lie product can
be expressed in terms of the original product as follows:
g ∗ h = g · (T (g−1)h)
Here T is the Lie group homomorphism G → Aut(G) induced by the above
choice of supplementary subspace v as follows.
(19) T (ea)(eb) = exp(eads(av)b) ∀a, b ∈ g.
In other words, T is the unique Lie group homomorphism whose differential
at the identity is the Lie algebra homomorphism deT : g → Der(g) given by
deT (a)(b) = ads(av)b, that is the composition of the map v → Der(g) from
Lemma 3.1 with the linear projection g → g/n ≃ v.
It is easy to check that this definition of the new product is compatible with
the definition of the new Lie bracket.
It can also be checked that two choices of supplementary spaces v as above yield
isomorphic Lie structures (see [13, Chap. III]). Hence by abuse of language, we
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 21
speak of the nilshadow of g, when we mean the Lie structure on G induced by a
choice of v as above.
The following example shows several of the features of a typical solvable Lie
group of polynomial growth.
Example 3.3 (Nilshadow of a semi-direct product). Let G = R ⋉φ R
n where
φt ∈ GLn(R) is some one parameter subgroup given by φt = exp(tA) = ktut where
A is some matrix in Mn(R) and A = As +Au is its Jordan decomposition, giving
rise to kt = exp(tAs) and ut = exp(tAu). The group G is diffeomorphic to R
hence simply connected. If all eigenvalues of As are purely imaginary, then G has
polynomial growth. However G is not nilpotent unless As = 0. So let us assume
that neither As nor Au is zero. Then the nilshadow GN is the semi-direct product
R⋉u R
n where ut is the unipotent part of φt.
It is easy to compute the homogeneous dimension of G (or GN) in terms of the
dimension of the Jordan blocs of Au. If nk is the number of Jordan blocks of Au
of size k, then
d(G) = 1 +
k(k + 1)
3.2. Basic properties of the nilshadow. We now list in the form of a few
lemmas some basic properties of the nilshadow.
Lemma 3.4. The image of T : G → Aut(G) is abelian and relatively compact.
Moreover T (T (g)h) = T (h) for any g, h ∈ G.
Proof. Since G has polynomial growth it is of type (R) by Guivarc’h’s theorem.
Hence all ads(x) have purely imaginary eigenvalues. It follows that K is compact.
Since T factors through the nilradical, its image is abelian. The last equality
follows from (19) and the fact that ∀x, y ∈ g, ads(ads(x)(y)) = 0. �
Lemma 3.5. T (G) also belongs to Aut(GN ) and T is a group homomorphism
GN → Aut(GN ).
Proof. The first assertion follows from (19) and the fact that deT is a derivation
of gN as one can check from (18) and the fact that ∀x, y ∈ g, ads(ads(x)(y)) = 0.
The second assertion then follows from Lemma 3.4. �
We denote by K the closure of T (G) in Aut(G) = Aut(g).
Lemma 3.6 (K-action on gN ). K preserves v and acts trivially on it. It also
preserves the ideals n and the central descending series {Ci(gN )}i of gN .
Proof. It suffices to check that ads(v) preserves n and C
i(gN ). It preserves n
because ad(x) preserves n for all x ∈ g. It preserves Ci(gN ) because it acts as a
derivation of gN as we have already checked in the proof of Lemma 3.5. �
Remark 3.7 (Well-definedness of π1). It is also easy to check from the definition
of the nilshadow bracket that the commutator subalgebra [gN , gN ] and in fact each
term of the central descending series Ci(gN ) is an ideal in g and does not depend
on the choice of supplementary subspace v used to defined the nilshadow bracket.
22 EMMANUEL BREUILLARD
In particular the projection map π1 : gN → gN/[gN , gN ] is a well defined linear
map on g = gN (i.e. independently of the choice involved in the construction of
the nilshadow Lie bracket).
Lemma 3.8 (Exponential map). The respective exponential maps exp : g → G
and expN : gN → GN coincide on n and on v.
Proof. Since the two Lie products coincide on N = exp(n), so do their exponential
map. For the second assertion, note that T (e−tv)v = v for every v ∈ v because
ads(x)(y) = 0 for all x, y ∈ ν. It follows that {e
tv}t is a one-parameter subgroup
for both Lie structures, hence it is equal to {expN (tv)}t. �
Remark 3.9 (Surjectivity of the exponential map). The exponential map is not
always a diffeomorphism, as the example of the universal cover Ẽ of the group E
of motions of the plane shows (indeed any 1-parameter subgroup of E is either a
translation subgroup or a rotation subgroup, but the rotation subgroup is compact
hence a torus, so its lift will contain the (discrete) center of E, hence will miss
every lift of a non trivial translation). In fact, it is easy to see that if g is the Lie
algebra of a solvable (non-nilpotent) Lie group of polynomial growth, then g maps
surjectively on the Lie algebra of E. Hence, for a simply connected solvable and
non-nilpotent Lie group of polynomial growth, the exponential map is never onto.
Nevertheless its image is easily seen to be dense.
However, exponential coordinates of the second kind behave nicely. Note that
[gN , gN ] ⊂ n.
Lemma 3.10 (Exponential coordinates of the second kind). Let {Ci(gN )}i≥0
be the central descending series of gN (with C
1(gN ) = [gN , gN ]) and pick linear
subspaces mi in gN such that C
i(gN ) = mi ⊕ C
i−1(gN ) for i ≥ 2. Let ℓ be a
supplementary subspace of C1(gN ) in n. Define exponential coordinates of the
second kind by setting
mr ⊕ ...⊕m2 ⊕ ℓ⊕ v → G
(ξr, ..., ξ1, v) 7→ expN (ξr) ∗ . . . ∗ expN (ξ1) ∗ expN (v)
This map is a diffeomorphism. Moreover expN (ξr) ∗ . . . ∗ expN (ξ1) ∗ expN (v) =
eξr · ... · eξ1 · ev for all choices of v ∈ v and ξi ∈ mi.
Proof. By Lemma 3.8 the exponential maps of G and GN coincide on n and on v.
Moreover g ∗ h = g · h whenever g belongs to the nilradical exp(n) of G. Hence
expN (ξr)∗. . .∗expN (ξ1)∗expN (v) = expN (ξr)·. . .·expN (ξ1)·expN (v) = e
ξr ·...·eξ1 ·ev.
The restriction of the map to n is a diffeomorphism onto exp(n), because this map
and its inverse are explicit polynomial maps (the ξi’s are coordinates of the second
kind, see the book [12]). Now the map n ⊕ v → G sending (n, v) to en · ev is a
diffeomorphism, because G is simply connected and hence the quotient group
G/ exp(n) isomorphic to a vector space and hence to exp(v). �
Lemma 3.11 (“Bi-invariant” Riemannian metric). There exists a Riemannian
metric on G which is left invariant under both Lie structures.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 23
Proof. Indeed it suffices to pick a scalar product on g which is invariant under the
compact subgroup K = T (G) ⊂ Aut(g). �
We identify K = {T (g), g ∈ G} with its image in Aut(g) under the canonical
isomorphism between Aut(G) and Aut(g). Recall that, according to Lemma 3.6,
the central descending series of gN is invariant under ads(x) for all x ∈ v and
consists of ideals of g. The same holds for n. It follows that these linear subspaces
also invariant under K. However since K is compact, its action on g is completely
reducible. Therefore we have proved:
Lemma 3.12 (K-invariant stratification of the nilshadow). Let g be the Lie algebra
of a simply connected Lie group G with polynomial growth. Let gN be the nilshadow
Lie algebra obtained from a splitting g = n⊕v as above (i.e. n is the nilradical and
v satisfies ads(x)(y) = 0 for every x, y ∈ v). Let K := {T (g), g ∈ G} ⊂ Aut(G),
where T is defined by (19). Then there is a choice of linear subspaces mi’s and ℓ
such that
(20) gN = mr ⊕ . . . m2 ⊕ ℓ⊕ v,
where each term is K-invariant, m1 := ℓ⊕ v and the central descending series of
gN satisfies C
i(gN ) = mi ⊕ C
i−1(gN ). Moreover the action on K can be read off
on the exponential coordinates of second kind in this decomposition, namely:
eξr · ... · eξ0
= k(eξr) · ... · k(eξ0) = ek(ξr) · ... · ek(ξ0)
= expN (k(ξr)) ∗ ... ∗ expN (k(ξ0))
4. Periodic metrics
In this section, unless otherwise stated, G will denote an arbitrary locally com-
pact group.
4.1. Definitions. By a pseudodistance (or metric) on a topological space X, we
mean a function ρ : X × X → R+ satisfying ρ(x, y) = ρ(y, x) and ρ(x, z) ≤
ρ(x, y) + ρ(y, z) for any triplet of points of X. However ρ(x, y) may be equal to 0
even if x 6= y.
We will require our pseudodistances to be locally bounded, meaning that the
image under ρ of any compact subset of G × G is a bounded subset of R+. To
avoid irrelevant cases (for instance ρ ≡ 0) we will also assume that ρ is proper, i.e.
the map y 7→ ρ(e, y) is a proper map, namely the preimage of a bounded set is
bounded (we do not ask that the map be continuous). When ρ is locally bounded
then it is proper if and only if y 7→ ρ(x, y) is proper for any x ∈ G.
A pseudodistance ρ on G is said to be asymptotically geodesic if for every ε > 0
there exists s > 0 such that for any x, y ∈ G one can find a sequence of points
x1 = x, x2, ..., xn = y in G such that
ρ(xi, xi+1) ≤ (1 + ε)ρ(x, y)
and ρ(xi, xi+1) ≤ s for all i = 1, ..., n − 1.
24 EMMANUEL BREUILLARD
We will consider exclusively pseudodistances on a group G that are invariant
under left translations by all elements of a fixed closed and co-compact subgroup
H of G, meaning that for all x, y ∈ G and all h ∈ H, ρ(hx, hy) = ρ(x, y).
Combining all previous axioms, we set the following definition.
Definition 4.1. Let G be a locally compact group. A pseudodistance ρ on G will
be said to be a periodic metric (or H-periodic metric) if it satisfies the following
properties:
(i) ρ is invariant under left translations by a closed co-compact subgroup H.
(ii) ρ is locally bounded and proper.
(iii) ρ is asymptotically geodesic.
Remark 4.2. The assumption that ρ is symmetric, i.e. ρ(x, y) = ρ(y, x) is here
only for the sake of simplicity, and most of what is proven in this paper can be
done without this hypothesis.
4.2. Basic properties. Let ρ be a periodic metric on G and H some co-compact
subgroup of G. The following properties are straighforward.
(1) ρ is at a bounded distance from its restriction to H. This means that if F
is a bounded fundamental domain for H in G and for an arbitrary x ∈ G, if hx
denotes the element of H such that x ∈ hxF, then |ρ(x, y)− ρ(hx, hy)| ≤ C for
some constant C > 0.
(2) ∀t > 0 there exists a compact subset Kt of G such that, ∀x, y ∈ G, ρ(x, y) ≤
t ⇒ x−1y ∈ Kt. And conversely, if K is a compact subset of G, ∃t(K) > 0 s.t.
x−1y ∈ K ⇒ ρ(x, y) ≤ t(K).
(3) If ρ(x, y) ≥ s, the xi’s in (21) can be chosen in such a way that s ≤
ρ(xi, xi+1) ≤ 2s (one can take a suitable subset of the original xi’s).
(4) The restriction of ρ to H × H is a periodic pseudodistance on H. This
means that the xi’s in (21) can be chosen in H.
(5) Conversely, given a periodic pseudodistance ρH on H, it is possible to extend
it to a periodic pseudodistance on G by setting ρ(x, y) = ρH(hx, hy) where x =
hxF for some bounded fundamental domain F for H in G.
4.3. Examples. Let us give a few examples of periodic pseudodistances.
(1) Let Γ be a finitely generated torsion free nilpotent group which is embedded
as a co-compact discrete subgroup of a simply connected nilpotent Lie group N .
Given a finite symmetric generating set S of Γ, we can consider the corresponding
word metric dS on Γ which gives rise to a periodic metric on N given by ρ(x, y) =
dS(γx, γy) where x ∈ γxF and y ∈ γyF if F is some fixed fundamental domain for
Γ in N.
(2) Another example, given in [27], is as follows. Let N/Γ be a nilmanifold with
universal cover N and fundamental group Γ. Let g be a Riemannian metric on
N/Γ. It can be lifted to the universal cover and thus gives rise to a Riemannian
metric g̃ on N . This metric is Γ-invariant, proper and locally bounded. Since Γ is
co-compact in N, it is easy to check that it is also asymptotically geodesic hence
periodic.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 25
(3) Any word metric on G. That is, if Ω is a compact symmetric generating
subset of G, let ∆Ω(x) = inf{n ≥ 1, x ∈ Ω
n}. Then define ρ(x, y) = ∆Ω(x
−1y).
Clearly ρ is a pseudodistance (although not a distance) and it is G-invariant on the
left, it is also proper, locally bounded and asymptotically geodesic, hence periodic.
(4) If G is a connected Lie group, any left invariant Riemannian metric on G.
Here again H = G and we obtain a periodic distance. Similarly, any left invariant
Carnot-Carathéodory metric on G will do.
Remark 4.3 (Berestovski’s theorem). According to a result of Berestovski [5]
every left-invariant geodesic distance on a connected Lie group is a subFinsler
metric as defined in Paragraph 2.1.
4.4. Coarse equivalence between invariant pseudodistances. The following
proposition is basic:
Proposition 4.4. Let ρ1 and ρ2 be two periodic pseudodistances on G. Then there
is a constant C > 0 such that for all x, y ∈ G
ρ2(x, y)− C ≤ ρ1(x, y) ≤ Cρ2(x, y) + C
Proof. Clearly it suffices to prove the upper bound. Let s > 0 be the number cor-
responding to the choice ε = 1 in (21) for ρ2. From 4.2 (2), there exists a compact
subset Ks in G such that ρ2(x, y) ≤ 2s ⇒ x
−1y ∈ K2s, and there is a constant
t = t(K2s) > 0 such that x
−1y ∈ K2s ⇒ ρ1(x, y) ≤ t. Let C = max{2t/s, t}, and
let x, y ∈ G. If ρ2(x, y) ≤ s then ρ1(x, y) ≤ t so the right hand side of (22) holds.
If ρ2(x, y) ≥ s then, from (21) and 4.2 (3), we get a sequence of xi’s in G from x
to y such that s ≤ ρ2(xi, xi+1) ≤ 2s and
1 ρ2(xi, xi+1) ≤ 2ρ2(x, y). It follows
that ρ1(xi, xi+1) ≤ t for all i. Hence ρ1(x, y) ≤
ρ1(xi, xi+1) ≤ Nt ≤
tρ2(x, y)
and the right hand side of (22) holds. �
In the particular case when G = N is a simply connected nilpotent Lie group,
the distance to the origin x 7→ ρ(e, x) is also coarsely equivalent to any homoge-
neous quasi-norm on N. We have,
Proposition 4.5. Suppose N is a simply connected nilpotent Lie group. Let ρ1 be
a periodic pseudodistance on N and | · | be a homogeneous quasi-norm, then there
exists C > 0 such that for all x ∈ N
|x−1y| − C ≤ ρ1(x, y) ≤ C|x
−1y|+ C
Moreover, if ρ2 is a periodic pseudodistance on the stratified nilpotent group N∞
associated to N, then again, there is a constant C > 0 such that
ρ2(e, x)− C ≤ ρ1(e, x) ≤ Cρ2(e, x) + C
The proposition follows at once from Guivarc’h’s theorem (see Corollary 2.7
above), the equivalence of homogeneous quasi-norms, and the fact that left-invariant
Carnot-Caratheodory metrics on N∞ are homogeneous quasi norms. However,
since the group structures on N and N∞ differ, (24) cannot in general be replaced
by the stronger relation (22) as simple examples show.
26 EMMANUEL BREUILLARD
The next proposition is of fundamental importance for the study of metrics on
Lie groups of polynomial growth:
Proposition 4.6. Let G be a simply connected solvable Lie group of polynomial
growth and GN its nilshadow. Let ρ and ρN be arbitrary periodic pseudodistances
on G and GN respectively. Then there is a constant C > 0 such that for all
x, y ∈ G
ρN (x, y)− C ≤ ρ(x, y) ≤ CρN (x, y) + C
Proof. According to Proposition 4.4, it is enough to show (25) for some choice of
periodic metrics on G and GN . But in Lemma 3.11 we constructed a Riemannian
metric on G which is left invariant for both G and GN . We are done. �
4.5. Right invariance under a compact subgroup. Here we verify that, given
a compact subgroup of G, any periodic metric is at bounded distance from another
periodic metric which is invariant on the right by this compact subgroup. Let K
be a compact subgroup of G and ρ a periodic pseudodistance on G. We average ρ
with the help of the normalized Haar measure on K to get:
(26) ρK(x, y) =
ρ(xk1, yk2)dk1dk2
Then the following holds:
Lemma 4.7. There is a constant C0 > 0 depending only on ρ and K such that
for all k1, k2 ∈ K and all x, y ∈ G
(27) |ρ(xk1, yk2)− ρ(x, y)| ≤ C0
Proof. From 4.2 (2), ∃t = t(K) > 0 s.t. ∀x ∈ G, ρ(x, xk) ≤ t. Applying the
triangle inequality, we are done. �
Hence we obtain:
Proposition 4.8. The pseudodistance ρK is periodic and lies at a bounded dis-
tance from ρ. In particular, as x tends to infinity in G the following limit holds
(28) lim
ρK(e, x)
ρ(e, x)
Proof. From Lemma 4.7 and 4.2 (3), it is easy to check that ρK must be asymp-
totically geodesic, and periodic. Integrating (27) we get that ρK is at a bounded
distance from ρ and (28) is obvious. �
If K is normal in G, we thus obtain a periodic metric ρK on G/K such that
ρK(p(x), p(y)) is at a bounded distance from ρ(x, y), where p is the quotient map
G→ G/K.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 27
5. Reduction to the nilpotent case
In this section, G denotes a simply connected solvable Lie group of polynomial
growth. We are going to reduce the proof of the theorems of the Introduction
to the case of a nilpotent G. This is performed by showing that any periodic
pseudodistance ρ on G is asymptotic to some associated periodic pseudodistance
ρN on the nilshadow GN . We state this in Proposition 5.1 below.
The key step in the proof is Proposition 5.2 below, which shows the asymptotic
invariance of ρ under the “semisimple part” of G. The crucial fact there is that the
displacement of a distant point under a fixed unipotent automorphism is negligible
compared to the distance from the identity (see Lemmas 5.4, 5.5), so that the
action of the semisimple part of large elements can be simply approximated by
their action by left translation.
5.1. Asymptotic invariance under a compact group of automorphisms
of G. The main result of this section is the following. Let G be a connected and
simply connected solvable Lie group with polynomial growth and GN its nilshadow
(see Section 3).
Proposition 5.1. Let H be a closed co-compact subgroup of G and ρ an H-
periodic pseudodistance (see Definition 4.1) on G. There exist a closed subset HK
containing H which is a co-compact subgroup for both G and GN , and an HK-
periodic (for both Lie structures) pseudodistance ρK such that
(29) lim
ρK(e, x)
ρ(e, x)
The closed subgroup HK will be taken to be the closure of the group generated
by all elements of the form k(h), where h belongs to H and k belongs to the
closure K in the group Aut(G) of the image of H under the homomorphism
T : G → Aut(G) introduced in Section 3. It is easy to check from the definition
of the nilshadow product (1) that this is indeed a subgroup in both G and its
nilshadow GN .
The new pseudodistance ρK is defined as follows, using a double averaging
procedure:
(30) ρK(x, y) :=
ρ(gk(x), gk(y))dkdµ(g)
Here the measure µ is the normalized Haar measure on the coset space H\HK
and dk is the normalized Haar measure on the compact group K. Recall that
all closed subgroups of S are unimodular (since they have polynomial growth by
[21][Lemme I.3.]). Hence the existence of invariant measures on the coset spaces.
An essential part of the proof of Proposition 5.1 is enclosed in the following
statement:
28 EMMANUEL BREUILLARD
Proposition 5.2. Let ρ be a periodic pseudodistance on G which is invariant
under a co-compact subgroup H. Then ρ is asymptotically invariant under the
action of K = {T (h), h ∈ H} ⊆ Aut(G). Namely, (uniformly) for all k ∈ K,
(31) lim
ρ(e, k(x))
ρ(e, x)
The proof of Proposition 5.2 splits into two steps. First we show that it is
enough to prove (31) for a dense subset of k’s. This is a consequence of the following
continuity statement:
Lemma 5.3. Let ε > 0, then there is a neighborhood U of the identity in K such
that, for all k ∈ U,
limx→∞
ρ(x, k(x))
ρ(e, x)
Then we show that the action of T (g) can be approximated by the conjugation
by g, essentially because the unipotent part of this conjugation does not move x
very much when x is far. This is the content of the following lemma:
Lemma 5.4. Let ρ be a periodic pseudodistance on G which is invariant under
a co-compact subgroup H. Then for any ε > 0, and any compact subset F in H
there is s0 > 0 such that
|ρ(e, T (h)x) − ρ(e, hx)| ≤ ερ(e, x)
for any h ∈ F and as soon as ρ(e, x) > s0.
Proof of Proposition 5.2 modulo Lemmas (5.3) and (5.4): As ρ is assumed to
be H-invariant, for every h ∈ H, we have ρ(e, h−1x)/ρ(e, x) → 1. The proof of
the proposition then follows immediately from the combination of the last two
lemmas. �
5.2. Proof of Lemmas (5.3) and (5.4). We choose K-invariant subspaces mi’s
and ℓ of the nilshadow gN of g as in Lemma 3.12 from Section 3. In particular
gN = mr ⊕ . . . ⊕m2 ⊕ ℓ⊕ v,
where each term is K-invariant, n = [gN , gN ] ⊕ l and C
i(gN ) = mi ⊕ C
i−1(gN ).
Moreover δt(x) = t
ix if x ∈ mi (here m1 = ℓ⊕ v).
We also set v(x) = maxi ‖ξi‖
i if x = expN (ξr) ∗ . . . ∗ expN (ξ0) and di = i if
i > 0 and d0 = 1. And we let |x| := maxi ‖xi‖
1/di if x = xr + . . . + x1 + x0 in the
above direct sum decomposition.
Note that | · | is a δt-homogeneous quasi-norm. Moreover, it is straightforward
to verify (using the Campbell-Hausdorff formula (12) and Proposition 2.4) that
v(x) ≤ C|x|+C for some constant C > 0. In particular ξi/|x|
di remains bounded
as |x| becomes large.
Proof of Lemma 5.3. Combining Propositions 4.5 and 4.6, there is a constant
C > 0 such that for all x, y ∈ G, ρ(x, y) ≤ C|x∗−1 ∗ y| + C. Therefore we have
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 29
reduced to prove the statement for | · | instead of ρ, namely it is enough to show
that |x∗−1 ∗ k(x)| becomes negligible compared to |x| as |x| goes to infinity and k
tends to 1.
It follows from the Campbell-Baker-Hausdorff formula (11) and (12) that, if
x, y ∈ GN and |x|, |y| are O(t), then |δ 1
(x ∗ y) − δ 1
(x) ∗ δ 1
(y)| = O(t−1/r), and
similarly |δ 1
(x1 ∗ . . . ∗ xm) − δ 1
(x1) ∗ . . . ∗ δ 1
(xm)| = Om(t
−1/r), for m elements
xi with |xi| = O(t). Hence when writing x = expN (ξr) ∗ ... ∗ expN (ξ0), and setting
t = |x|, we thus obtain that the following quantity
∣∣∣∣∣∣
(x∗−1 ∗ k(x))−
0≤i≤r
expN (−t
−diξi) ∗
0≤i≤r
expN (t
−dr−ik(ξr−i))
∣∣∣∣∣∣
is a O(t−1/r). Indeed recall from Lemma 3.12 that k(x) = expN (k(ξr)) ∗ ... ∗
expN (k(ξ0)). As x gets larger, each t
−diξi remains in a compact subset of mi.
Therefore, as k tends to the identity in K, each t−dik(ξi) becomes uniformly close
to t−diξi independently of the choice of x ∈ GN as long as t = |x| is large. The
result follows. �
Proof of Lemma 5.4. Recall that hx = h ∗ T (h)x for all x, h ∈ G (see (1).
By the triangle inequality it is enough to bound ρ(y, h ∗ y), where y = T (h)x.
From Propositions 4.5 and 4.6, ρ is comparable (up to multiplicative and additive
constants to the homogeneous quasi-norm | · |. Hence the Lemma follows from the
following:
Lemma 5.5. Let N be a simply connected nilpotent Lie group and let | · | be a
homogeneous quasi norm on N associated to some 1-parameter group of dilations
(δt)t. For any ε > 0 and any compact subset F of N, there is a constant s2 > 0
such that
|x−1gx| ≤ ε|x|
for all g ∈ F and as soon as |x| > s2.
Proof. Recall, as in the proof of the last lemma, that for any c1 > 0 there is
a c2 > 0 such that if t > 1 and x, y ∈ N are such that |x|, |y| ≤ c1t, then
(xy)− δ 1
(x) ∗ δ 1
(y)| ≤ c2t
−1/r. In particular, if we set t = |x|, then
∣∣∣δ 1
(x−1gx)− δ 1
(x)−1 ∗ δ 1
(g) ∗ δ 1
∣∣∣ ≤ c2t−1/r
On the other hand, as g remains in the compact set F, δ 1
(g) tends uniformly to
the identity when t = |x| goes to infinity, and δ 1
(x) remains in a compact set.
By continuity, we see that δ 1
(x)−1 ∗ δ 1
(g) ∗ δ 1
(x) becomes arbitrarily small as t
increases. We are done. �
30 EMMANUEL BREUILLARD
5.3. Proof of Proposition 5.1. First we prove the following continuity state-
ment:
Lemma 5.6. Let ρ be a periodic pseudodistance on G and ε > 0. Then there
exists a neighborhood of the identity U in G and s3 > 0 such that
1− ε ≤
ρ(e, gx)
ρ(e, x)
≤ 1 + ε
as soon g ∈ U and ρ(e, x) > s3.
Proof. Let ρN be a left invariant Riemannian metric on the nilshadow GN .
|ρ(e, x)− ρ(e, gx)| ≤ ρ(x, gx) ≤ ρ(x, g ∗ x) + ρ(g ∗ x, gx)
However ρ(a, b) ≤ CρN(a, b)+C for some C > 0 by Proposition 4.6. Moreover by
(1) we have gx = g ∗ T (g)x. Hence
|ρ(e, x) − ρ(e, gx)| ≤ CρN (x, g ∗ x) + CρN (x, T (g)x) + 2C
To complete the proof, we apply Lemmas 5.5 and 5.3 to the right hand side above.
We proceed with the proof of Proposition 5.1. Let L be the set of all g ∈ G
such that ρ(e, gx)/ρ(e, x) tends to 1 as x tends to infinity in G. Clearly L is a
subgroup of G. Lemma 5.6 shows that L is closed. The H-invariance of ρ insures
that L contains H. Moreover, Proposition 5.2 implies that L is invariant under K.
Consequently L contains HK , the closed subgroup generated by all k(h), k ∈ K,
h ∈ H. This, together with Proposition 5.2, grants pointwise convergence of the
integrand in (29). Convergence of the integral follows by applying Lebesgue’s
dominated convergence theorem.
The fact that ρK is invariant under left multiplication by H and invariant under
precomposition by automorphisms from K insures that ρK is invariant under ∗-
left multiplication by any element h ∈ H, where ∗ is the multiplication in the
nilshadow GN . Moreover we check that T (g) ∈ K if g ∈ HK , hence HK is a
subgroup of GN . It is clearly co-compact in GN too (if F is compact and HF = G
then H ∗ FK = G where FK is the union of all k(F ), k ∈ K).
Clearly ρK is proper and locally bounded, so in order to finish the proof, we
need only to check that ρK is asymptotically geodesic. By H-invariance of ρK and
since H is co-compact in G, it is enough to exhibit a pseudogeodesic between e
and a point x ∈ H. Let x = z1 · ... ·zn with zi ∈ H and
ρ(e, zi) ≤ (1+ε) ·ρ(e, x).
Fix a compact fundamental domain F for H in HK so that integration in (29)
over H\HK is replaced by integration over F. Then for some constant CF > 0 we
have |ρ(g, gz) − ρ(e, gz)| ≤ CF for g ∈ F and z ∈ H. Moreover, it follows from
Proposition 5.2, Lemma 5.6 and the fact that HK ⊂ L, that
(32) ρ(e, gk(z)) ≤ (1 + ε) · ρ(e, z)
for all g ∈ F, k ∈ K and as soon as z ∈ G is large enough. Fix s large enough
so that CF ≤ εs and so that (32) holds when ρ(e, z) ≥ s. As already observed in
the discussion following Definition 4.1 (property 4.2 (3)) we may take the zi’s so
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 31
that s
≤ ρ(e, zi) ≤ s. Then nCF ≤ nsε ≤ 3ερ(e, x). Finally we get for ε < 1 and
x large enough
ρK(e, zi) ≤ CFn+ (1 + ε)
2ρ(e, x)
≤ CFn+ (1 + ε)
3ρK(e, x)
≤ (1 + 10ε) · ρK(e, x)
where we have used the convergence ρK/ρ→ 1 that we just proved. �
6. The nilpotent case
In this section, we prove Theorem 1.4 and its corollaries stated in the Intro-
duction for a simply connected nilpotent Lie group. We essentially follow Pansu’s
argument from [27], although our approach differs somewhat in its presentation.
Throughout the section, the nilpotent Lie group will be denoted by N, and its Lie
algebra by n.
Let m1 be any vector subspace of n such that n = m1 ⊕ [n, n]. Let π1 the
associated linear projection of n onto m1. Let H be a closed co-compact subgroup
of N . To every H-periodic pseudodistance ρ on N we associate a norm ‖·‖0 on
m1 which is the norm whose unit ball is defined to be the closed convex hull of all
elements π1(h)/ρ(e, h) for all h ∈ H\{e}. In other words,
(33) E := {x ∈ m1, ‖x‖0 ≤ 1} = CvxHull
π1(h)
ρ(e, h)
, h ∈ H\{e}
The set E is clearly a convex subset of m1 which is symmetric around 0 (since ρ
is symmetric). To check that E is indeed the unit ball of a norm on m1 it remains
to see that E is bounded and that 0 lies in its interior. The first fact follows
immediately from (23) and Example 2.3. If 0 does not lie in the interior of E,
then E must be contained in a proper subspace of m1, contradicting the fact that
H is co-compact in N .
Taking large powers hn, we see that we can replace the set H \{e} in the above
definition by any neighborhood of infinity in H. Similarly, it is easy to see that
the following holds:
Proposition 6.1. For s > 0 let Es be the closed convex hull of all π1(x)/ρ(e, x)
with x ∈ N and ρ(e, x) > s. Then E =
s>0Es.
Proof. Since ρ is H-periodic, we have ρ(e, hn) ≤ nρ(e, h) for all n ∈ N and h ∈ H.
This shows E ⊂
s>0Es. The opposite inclusion follows easily from the fact that
ρ is at a bounded distance from its restriction to H, i.e. from 4.2 (1). �
We now choose a set of supplementary subspaces (mi) starting with m1 as in
Paragraph 2.2. This defines a new Lie product ∗ on N so that N∞ = (N, ∗) is
stratified. We can then consider the ∗-left invariant Carnot-Carathéodory metric
associated to the norm ‖·‖0 as defined in Paragraph 2.1 on the stratified nilpotent
Lie group N∞. In this section, we will prove Theorem 1.4 for nilpotent groups in
the following form:
32 EMMANUEL BREUILLARD
Theorem 6.2. Let ρ be a periodic pseudodistance on N and d∞ the Carnot-
Carathéodory metric defined above, then as x tends to infinity in N
(34) lim
ρ(e, x)
d∞(e, x)
Note that d∞ is left-invariant for the N∞ Lie product, but not the original Lie
product on N .
Before going further, let us draw some simple consequences.
(1) In Theorem 6.2 we may replace d∞(e, x) by d(e, x), where d is the left
invariant Carnot-Caratheodory metric on N (rather than N∞) defined by the
norm ‖·‖0 (as opposed to d∞ which is ∗-left invariant). Hence ρ, d and d∞ are
asymptotic. This follows from the combination of Theorem 6.2 and Remark 2.1.
(2) Observe that the choice ofm1 was arbitrary. Hence two Carnot-Carathéodory
metrics corresponding to two different choices of a supplementary subspace m1
with the same induced norm on n/[n, n], are asymptotically equivalent (i.e. their
ratio tends to 1), and in fact isometric (see Remark 2.1). Conversely, if two
Carnot-Carathéodory metrics are associated to the same supplementary subspace
m1 and are asymptotically equivalent, they must be equal. This shows that the
set of all possible norms on the quotient vector space n/[n, n] is in bijection with
the set of all classes of asymptotic equivalence of Carnot-Carathéodory metrics on
(3) As another consequence we see that if a locally bounded proper and asymp-
totically geodesic left-invariant pseudodistance on N is also homogeneous with
respect to the 1-parameter group (δt)t (i.e. ρ(e, δtx) = tρ(e, x)) then it has to be
of the form ρ(x, y) = d∞(e, x
−1y) where d∞ is a Carnot-Carathéodory metric on
6.1. Volume asymptotics. Theorem 6.2 also yields a formula for the asymptotic
volume of ρ-balls of large radius. Let us fix a Haar measure on N (for example
Lebesgue measure on n gives rise to a Haar measure on N under exp). Since d∞
is homogeneous, it is straightforward to compute the volume of a d∞-ball:
vol({x ∈ N, d∞(e, x) ≤ t}) = t
d(N)vol({x ∈ N, d∞(e, x) ≤ 1})
where d(N) =
i≥1 dim(C
i(n)) is the homogeneous dimension of N. For a pseu-
dodistance ρ as in the statement of Theorem 6.2, we can define the asymptotic vol-
ume of ρ to be the volume of the unit ball for the associated Carnot-Carathéodory
metric d∞.
AsV ol(ρ) = vol({x ∈ N, d∞(e, x) ≤ 1})
Then we obtain as an immediate corollary of Theorem 6.2:
Corollary 6.3. Let ρ be periodic pseudodistance on N. Then
td(N)
vol({x ∈ N, ρ(e, x) ≤ t}) = AsV ol(ρ) > 0
Finally, if Γ is an arbitrary finitely generated nilpotent group, we need to take
care of the torsion elements. They form a normal finite subgroup T and applying
Theorem 6.2 to Γ/T , we obtain:
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 33
Corollary 6.4. Let S be a finite symmetric generating set of Γ and Sn the ball
of radius n is the word metric ρS associated to S, then
nd(N)
#Sn = #T ·
AsV ol(ρS)
vol(N/Γ)
where N is the Malcev closure of Γ = Γ/T , the torsion free quotient of Γ, and dS
is the word pseudodistance associated to S, the projection of S in Γ.
Moreover, it is possible to be a bit more precise about AsV ol(ρS). In fact,
the norm ‖·‖0 on m1 used to define the limit Carnot-Carathéodory distance d∞
associated to ρS is a simple polyhedral norm defined by
{‖x‖0 ≤ 1} = CvxHull (π1(s), s ∈ S)
More generally the following holds. Let H be any closed, co-compact subgroup
of N. Choose a Haar measure on H so that volN (N/H) = 1. Theorem 6.2 yields:
Corollary 6.5. Let Ω be a compact symmetric (i.e. Ω = Ω−1) neighborhood of
the identity, which generates H. Let ‖·‖0 be the norm on m1 whose unit ball is
CvxHull{π1(Ω)} and let d∞ be the corresponding Carnot-Carathéodory metric on
N∞. Then we have the following limit in the Hausdorff topology
(Ωn) = {g ∈ N, d∞(e, g) ≤ 1}
volH(Ω
nd(N)
= volN ({g ∈ N, d∞(e, g) ≤ 1})
6.2. Outline of the proof. We first devise some standard lemmas about piece-
wise approximations of horizontal paths (Lemmas 6.6, 6.7, 6.10). Then it is shown
(Lemma 6.11) that the original product on N and the product in the associated
graded Lie group are asymptotic to each other, namely, if (δt)t is a 1-parameter
group of dilations of N, then after renormalization by δ 1
, the product of O(t)
elements lying in some bounded subset of N, is very close to the renormalized
product of the same elements in the graded Lie group N∞. This is why all com-
plications due to the fact that N may not be a priori graded and the δt’s may
not be automorphisms disappear when looking at the large scale geometry of the
group. Finally, we observe (Lemma 6.13), as follows from the very definition of
the unit ball E for the limit norm ‖·‖0 , that any vector in the boundary of E,
can be approximated, after renormalizing by δ 1
by some element x ∈ N lying in
a fixed annulus s(1 − ε) ≤ ρ(e, x) ≤ s(1 + ε). This enables us to assert that any
ρ-quasi geodesic gives rise, after renormalization, to a d∞-geodesic (this gives the
lower bound in Theorem 6.2). And vice-versa, that any d∞-geodesic can be ap-
proximated uniformly by some renormalized ρ-quasi geodesic (this gives the upper
bound in Theorem 6.2).
34 EMMANUEL BREUILLARD
6.3. Preliminary lemmas.
Lemma 6.6. Let G be a Lie group and let ‖·‖e be a Euclidean norm on the Lie
algebra of G and de(·, ·) the associated left invariant Riemannian metric on G.
Let K be a compact subset of G. Then there is a constant C0 = C0(de,K) > 0
such that whenever de(e, u) ≤ 1 and x, y ∈ K
|de(xu, yu)− de(x, y)| ≤ C0de(x, y)de(e, u)
Proof. The proof reduces to the case when u and x−1y are in a small neighborhood
of e. Then the inequality boils down to the following ‖[X,Y ]‖e ≤ c ‖X‖e ‖Y ‖e for
some c > 0 and every X,Y in Lie(G). �
Lemma 6.7. Let G be a Lie group, let ‖·‖ be some norm on the Lie algebra of
G and let de(·, ·) be a left invariant Riemannian metric on G. Then for every
L > 0 there is a constant C = C(de, ‖·‖ , L) > 0 with the following property.
Assume ξ1, ξ2 : [0, 1] → G are two piecewise smooth paths in the Lie group G
with ξ1(0) = ξ2(0) = e. Let ξ
i ∈ Lie(G) be the tangent vector pulled back at the
identity by a left translation of G. Assume that supt∈[0,1] ‖ξ
1(t)‖ ≤ L, and that∫ 1
‖ξ′1(t)− ξ
2(t)‖ dt ≤ ε. Then
de(ξ1(1), ξ2(1)) ≤ Cε
Proof. The function f(t) = de(ξ1(t), ξ2(t)) is piecewise smooth. For small dt we
may write, using Lemma 6.6
f(t+ dt)− f(t) ≤ de(ξ1(t)ξ
1(t)dt, ξ1(t)ξ
2(t)dt) + de(ξ1(t)ξ
2(t)dt, ξ2(t)ξ
2(t)dt)− f(t) + o(dt)
∥∥ξ′1(t)− ξ′2(t)
dt+ C0f(t)
∥∥ξ′2(t)dt
+ o(dt)
≤ ε(t)dt +C0Lf(t)dt+ o(dt)
where ε(t) = ‖ξ′1(t)− ξ
2(t)‖e . In other words,
f ′(t) ≤ ε(t) + C0Lf(t)
Since f(0) = 0, Gronwall’s lemma implies that f(1) ≤ eC0L
ε(s)e−C0Lsds ≤ Cε.
From now on, we will take G to be the stratified nilpotent Lie group N∞, and
de(·, ·) will denote a left invariant Riemannian metric on N∞ while d∞(·, ·) is a left
invariant Carnot-Caratheodory Finsler metric on N∞ associated to some norm ‖·‖
on m1.
Remark 6.8. There is c0 > 0 such that c
0 de(e, x) ≤ d∞(e, x) ≤ c0de(e, x)
r in a
neighborhood of e. Hence in the situation of the lemma we get d∞(ξ1(1), ξ2(1)) ≤
r for some other constant C1 = C1(L, d∞, de).
Lemma 6.9. Let N ∈ N and dN (x, y) be the function in N∞ defined in the
following way:
dN (x, y) = inf{
∥∥ξ′(u)
∥∥ du, ξ ∈ HPL(N), ξ(0) = x, ξ(1) = y}
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 35
where HPL(N) is the set of horizontal paths ξ which are piecewise linear with at
most N possible values for ξ′. Then we have dN → d∞ uniformly on compact
subsets of N∞.
Proof. Note that it follows from Chow’s theorem (e.g. see [25] or [19]) that there
exists K0 ∈ N such that A := supd∞(e,x)=1 dK0(e, x) < ∞. Moreover, since piece-
wise linear paths are dense in L1, it follows for example from Lemma 6.7 that for
each fixed x, dn(e, x) → d∞(e, x). We need to show that dN (e, x) → d∞(e, x) uni-
formly in x satisfying d∞(e, x) = 1. By contradiction, suppose there is a sequence
(xn)n such that d∞(e, xn) = 1 and dn(e, xn) ≥ 1 + ε0 for some ε0 > 0. We may
assume that (xn)n converges to say x. Let yn = x
−1 ∗xn and tn = d∞(e, yn). Then
dK0(e, yn) = tndK0(e, δ 1
(yn)) ≤ Atn. Thus dn(e, xn) ≤ dn(e, x) + dn(e, yn) ≤
dn(e, x) +Atn as soon as n ≥ K0. As n tends to ∞, we get a contradiction. �
This lemma prompts the following notation. For ε > 0, we let Nε ∈ N be the
first integer such that 1 ≤ dNε(e, x) ≤ 1 + ε for all x with d∞(e, x) = 1. Then we
have:
Lemma 6.10. For every x ∈ N∞ with d∞(e, x) = 1, and all ε > 0 there exists a
path ξ : [0, 1] → N∞ in HPL(Nε) with unit speed (i.e. ‖ξ
′‖ = 1) such that ξ(0) = e
and d∞(x, ξ(1)) ≤ C2ε and ξ
′ has at most one discontinuity on any subinterval of
[0, 1] of length εr/Nε.
Proof. We know that there is a path in HPL(Nε) connecting e to x with length
ℓ ≤ 1 + ε. Reparametrizing the path so that it has unit speed, we get a path
ξ0 : [0, ℓ] → N∞ in HPL(Nε) with d∞(x, ξ0(1)) = d∞(ξ0(ℓ), ξ0(1)) ≤ ε. The deriva-
tive ξ′0 is constant on at most Nε different intervals say [ui, ui+1). Let us remove
all such intervals of length ≤ εr/Nε by merging them to an adjacent interval
and let us change the value of ξ′0 on these intervals to the value on the adja-
cent interval (it doesn’t matter if we choose the interval on the left or on the
right). We obtain a new path ξ : [0, 1] → N∞ in HPL(Nε) with unit speed and
such that ξ′ has at most one discontinuity on any subinterval of [0, 1] of length
εr/Nε. Moreover
‖ξ′(t)− ξ′0(t)‖ dt ≤ ε
r. By Lemma 6.7 and Remark 6.3, we
have d∞(ξ(1), ξ0(1)) ≤ C1ε, hence
d∞(ξ(1), x) ≤ d∞(x, ξ0(1)) + d∞(ξ0(1), ξ(1)) ≤ (C1 + 1)ε
Lemma 6.11 (Piecewise horizontal approximation of paths). Let x∗y denote the
product inside the stratified Lie group N∞ and x · y the ordinary product in N .
Let n ∈ N and t ≥ n. Then for any compact subset K of N , and any x1, ..., xn
elements of K, we have
de(δ 1
(x1 · ... · xn), δ 1
(x1 ∗ ... ∗ xn)) ≤ c1
de(δ 1
(x1 ∗ ... ∗ xn), δ 1
(π1(x1) ∗ ... ∗ π1(xn))) ≤ c2
36 EMMANUEL BREUILLARD
where c1, c2 depend on K and de only.
Proof. Let ‖·‖ be a norm on the Lie algebra of N. For k = 1, ..., n let zk =
x1 · ... ·xk−1 and yk = xk+1 ∗ ...∗xn. Since all xi’s belong to K, it follows from (24)
that as soon as t ≥ n, all δ 1
(zk) and δ 1
(yk) for k = 1, ..., n remain in a bounded set
depending only on K. Comparing (12) and (11), we see that whenever y = O(1)
and δ 1
(x) = O(1), we have
∥∥∥δ 1
(xy)− δ 1
(x ∗ y)
∥∥∥ = O(
On the other hand, from (12) it is easy to verify that right ∗-multiplication by a
bounded element is Lipschitz for ‖·‖ and the Lipschitz constant is locally bounded.
It follows that there is a constant C1 > 0 (depending only on K and ‖·‖) such
that for all k ≤ n
∥∥∥δ 1
((zk · xk) ∗ yk)− δ 1
(zk ∗ xk ∗ yk)
∥∥∥ ≤ C1
∥∥∥δ 1
(zk · xk)− δ 1
(zk ∗ xk)
Applying n times the relation (35) with x = x1 · ... · xk−1 and y = xk, we finally
obtain ∥∥∥δ 1
(x1 · ... · xn)− δ 1
(x1 ∗ ... ∗ xn)
∥∥∥ = O(
) = O(
where O() depends only on K. On the other hand, using (11), it is another simple
verification to check that if x, y lie in a bounded set, then 1
de(x, y) ≤ ‖x− y‖
≤ c2de(x, y) for some constant c2 > 0. The first inequality follows.
For the second inequality, we apply Lemma 6.7 to the paths ξ1 and ξ2 starting
at e and with derivative equal on [ k
, k+1
) to nδ 1
(xk) for ξ1 and to n
π1(xk)
for ξ2.
We get
de(δ 1
(x1 ∗ ... ∗ xn), δ 1
(π1(x1) ∗ ... ∗ π1(xn)) = O(
Remark 6.12. From Remark 6.3 we see that if we replace de by d∞ in the above
lemma, we get the same result with 1
replaced by t−
Lemma 6.13 (Approximation in the abelianized group). Recall that ‖·‖0 is the
norm on m1 defined in (33). For any ε > 0, there exists s0 > 0 such that for every
s > s0 and every v ∈ m1 such that ‖v‖0 = 1, there exists h ∈ H such that
(1− ε)s ≤ ρ(e, h) ≤ (1 + ε)s
and ∥∥∥∥
π1(h)
ρ(e, h)
Proof. Let ε > 0 be fixed. Considering a finite ε-net in E, we see that there exists
a finite symmetric subset {g1, ..., gp} of H\{e} such that, if we consider the closed
convex hull of F = {fi = π1(gi)/ρ(e, gi)|i = 1, ..., p} and ‖·‖ε the associated norm
on m1, then ‖·‖0 ≤ ‖·‖ε ≤ (1 + 2ε) ‖·‖0. Up to shrinking F if necessary, we may
assume that ‖fi‖ε = 1 for all i’s. We may also assume that the fi’s generate m1 as
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 37
a vector space. The sphere {x, ‖x‖ε = 1} is a symmetric polyhedron in m1 and to
each of its facets corresponds d = dim(m1) vertices lying in F and forming a vector
basis of m1. Let f1, ..., fd, say, be such vertices for a given facet. If x ∈ m1 is of the
form x =
i=1 λifi with λi ≥ 0 for 1 ≤ i ≤ d then we see that ‖x‖ε =
i=1 λi,
because the convex hull of f1, ..., fd is precisely that facet, hence lies on the sphere
{x, ‖x‖ε = 1}.
Now let v ∈ m1, ‖v‖0 = 1, and let s > 0. The half line tv, t > 0, hits the
sphere {x, ‖x‖ε = 1} in one point. This point belongs to some facet and there
are d linearly independent elements of F, say f1, ..., fd, the vertices of that facet,
such that this point belongs to the convex hull of f1, ..., fd. The point sv then lies
in the convex cone generated by π1(g1), ..., π1(gd). Moreover, there is a constant
Cε > 0 (Cε ≤
max1≤i≤p ρ(e, gi)) such that
∥∥∥∥∥sv −
niπ1(gi)
∥∥∥∥∥
for some non-negative integers n1, ..., nd depending on s > 0. Hence
niρ(e, gi) =
∥∥∥∥∥
niπ1(gi)
∥∥∥∥∥
(‖sv‖ε + Cε)
≤ 1 + 2ε +
≤ 1 + 3ε
where the last inequality holds as soon as s > Cε/ε.
Now let h = g
1 · ... · g
d ∈ H. We have π1(h) =
i=1 niπ1(gi)
ρ(e, h) ≥ ‖π1(h)‖0 ≥ s− Cε ≥ s(1− ε)
Moreover
ρ(e, h) ≤
niρ(e, gi) ≤ s(1 + 3ε)
Changing ε into say ε
and for say ε < 1
, we get the desired result with s0(ε) =
max1≤i≤p ρ(e, gi). �
6.4. Proof of Theorem 6.2. We need to show that as x→ ∞ in N
1 ≤ lim
ρ(e, x)
d∞(e, x)
≤ lim
ρ(e, x)
d∞(e, x)
First note that it is enough to prove the bounds for x ∈ H. This follows from
(4.2) (1).
Let us begin with the lower bound. We fix ε > 0 and s = s(ε) as in the definition
of an asymptotically geodesic metric (see (21)). We know by 4.2 (3) and (4) that
as soon as ρ(e, x) ≥ s we may find x1, ..., xn in H with s ≤ ρ(e, xi) ≤ 2s such that
xi and
ρ(e, xi) ≤ (1 + ε)ρ(e, x). Let t = d∞(e, x), then n ≤
ρ(e, x),
hence n ≤ C
t where C is a constant depending only on ρ (see (23)). We may
38 EMMANUEL BREUILLARD
then apply Lemma 6.11 (and the remark following it) to get, as t ≥ n as soon as
s(ε) ≥ C,
d∞(δ 1
(x), δ 1
(π1(x1) ∗ ... ∗ π1(xn))) ≤ c
But for each i we have ‖π1(xi)‖0 ≤ ρ(e, xi) by definition of the norm, hence
t = d∞(e, x) ≤
‖π1(xi)‖0+ d∞(x, π1(x1) ∗ ... ∗π1(xn)) ≤ (1+ ε)ρ(e, x)+ c
Since ε was arbitrary, letting t→ ∞ we obtain
ρ(e, x)
d∞(e, x)
We now turn to the upper bound. Let t = d∞(e, x) and ε > 0. According to
Lemma 6.10, there is a horizontal piecewise linear path {ξ(u)}u∈[0,1] with unit
speed such that d∞(δ 1
(x), ξ(1)) ≤ C2ε and no interval of length ≥
contains
more than one change of direction. Let s0(ε) be given by Lemma 6.13 and assume
t > s0(ε
r)Nε/ε
r. We split [0, 1] into n subintervals of length u1, ..., un such that
ξ′ is constant equal to yi on the i-th subinterval and s0(ε
r) ≤ tui ≤ 2s0(ε
r). We
have ξ(1) = u1y1 ∗ ... ∗ unyn. Lemma 6.13 yields points xi ∈ H such that
∥∥∥∥yi −
π1(xi)
∥∥∥∥ ≤ ε
and ρ(e, xi) ∈ [(1 − ε
r)tui, (1 + ε
r)tui] (note that tui > s0(ε
r)). Let ξ be the
piecewise linear path [0, 1] → N∞ with the same discontinuities as ξ and where the
value yi is replaced by
π1(xi)
. Then according to Lemma 6.7, d∞(ξ(1), ξ(1)) ≤ Cε.
Since ρ(e, xi) ≤ 4s0(ε
r) for each i, we may apply Lemma 6.11 (and the remark
following it) and see that if y = x1 · ... · xn,
d∞(ξ(1), δ 1
(y)) ≤ c′1(ε)t
Hence d∞(δ 1
(x), δ 1
(y)) ≤ (C2+C)ε+c
1(ε)t
r and ρ(e, y) ≤
ρ(e, xi) ≤ (1+ε
while ρ(x, y) ≤ C ′td∞(e, δ 1
(x−1y)) + C ′ ≤ t(Cd∞(δ 1
(x), δ 1
(y)) + oε(1)). Hence
ρ(e, x) ≤ t+ oε(t)
Remark 6.14. In the last argument we used the fact that
∥∥∥δ 1
(xu)− δ 1
(x ∗ u)
∥∥∥ =
) if δ 1
(x) and δ 1
(u) are bounded, in order to get for y = xu,
d∞(e, δ 1
(u)) ≤ d∞(δ 1
(x), δ 1
(xu)) + d∞(δ 1
(xu), δ 1
(x ∗ u))
≤ d∞(δ 1
(x), δ 1
(y)) + o(1).
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 39
7. Locally compact G and proofs of the main results
In this section, we prove Theorem 1.2 and complete the proof of Theorem 1.4
and its corollaries. We begin with the latter.
Proof of Theorem 1.4. It is the combination of Proposition 5.1, which reduces the
problem to nilpotent Lie groups, and Theorem 6.2, which treats the nilpotent case.
It only remains to justify the last assertion that d∞ is invariant under T (H).
SinceK = T (H) stabilizesm1 (see Lemma 3.12 for the definition ofm1) and acts
by automorphisms of the nilpotent (nilshadow) structure (Lemma 3.5), given any
k ∈ K, the metric d∞(k(x), k(y)) is nothing else but the left invariant subFinsler
metric on the nilshadow associated to the norm ‖k(v)‖ for v ∈ m1 (if ‖ · ‖ denotes
the norm associated to d∞).
However, d∞ is asymptotically invariant under K, because of Proposition 5.1.
Namely d∞(e, k(x))/d∞(e, x) tends to 1 as x tends to infinity. Finally d∞(e, v) =
‖v‖ and d∞(e, k(v)) = ‖k(v)‖ for all v ∈ m1. Two asymptotic norms on a vector
space are always equal. It follows that the norms ‖ · ‖ and ‖k(·)‖ on m1 coincide.
Hence d∞(e, k(x)) = d∞(e, k(x)) for all x ∈ S as claimed. �
Proof of Corollary 1.8. First some initial remark (see also Remark 2.1). If d is
a left-invariant subFinlser metric on a simply connected nilpotent Lie group N
induced by a norm ‖ · ‖ on a supplementary subspace m1 of the commutator
subalgebra, then it follows from the very definition of subFinsler metrics (see
Paragraph 2.1) that π1 is 1-Lipschitz between the Lie group and the abelianization
of it endowed with the norm ‖·‖, namely ‖π1(x)‖ ≤ d(e, x), with equality if x ∈ m1.
From this and considering the definition of the limit norm in (33), we conclude
that ‖ · ‖ coincides with the limit norm of d. In particular Theorem 6.2 implies
that d is asymptotic to the ∗-left invariant subFinsler metric d∞ induced by the
same norm ‖ · ‖ on the graded Lie group (N∞, ∗).
We can now prove Corollary 1.8. By the above remark, the limit metric d∞
on the graded nilshadow of S is asymptotic to the subFinsler metric d induced
by the same norm ‖ · ‖ on the same (K-invariant) supplementary subspace m1
of the commutator subalgebra of the nilshadow, and which is left invariant for
the nilshadow structure on S. However, it follows from Theorem 1.4 that d∞
and the norm ‖ · ‖ are K-invariant. This implies that d is also left-invariant
with respect to the original Lie group structure of S. Indeed, by (1), we can
write d(gx, gy) = d(g ∗ (T (g)x), g ∗ (T (g)y)) = d(T (g)x, T (g)y) = d(x, y), where ∗
denotes this time the nilshadow product structure. We are done. �
Proof of Corollary 1.7. This follows immediately from Theorem 1.4, when ∗ de-
notes the graded nilshadow product. If ∗ denotes the nilshadow group structure,
then it follows from Theorem 6.2 and the remark we just made in the proof of
Corollary 1.8 (see also Remark 2.1). �
7.1. Proof of Theorem 1.2. Let G be a locally compact group of polynomial
growth. We will show that G has a compact normal subgroup K such that G/K
40 EMMANUEL BREUILLARD
contains a closed co-compact subgroup, which can be realized as a closed co-
compact subgroup of a connected and simply connected solvable Lie group of type
(R) (i.e. of polynomial growth). The proof will follow in several steps.
(a) First we show that up to moding out by a normal compact subgroup, we may
assume that G is a Lie group whose connected component of the identity has no
compact normal subgroup. Indeed, it follows from Losert’s refinement of Gromov’s
theorem ([24] Theorem 2) that there exists a normal compact subgroup K of G
such that G/K is a Lie group. So we may now assume that G is a Lie group (not
necessarily connected) of polynomial growth. The connected component G0 of G
is a connected Lie group of polynomial growth. Recall the following classical fact:
Lemma 7.1. Every connected Lie group has a unique maximal compact normal
subgroup. By uniqueness it must be a characteristic Lie subgroup.
Proof. Clearly if K1 and K2 are compact normal subgroups, then K1K2 is again
a compact normal subgroup. Considering G/K, where K is a compact normal
subgroup of maximal dimension, we may assume that G has no compact normal
subgroup of positive dimension. But every finite normal subgroup of a connected
group is central. Hence the closed group generated by all finite normal subgroups is
contained in the center ofG. The center is an abelian Lie subgroup, i.e. isomorphic
to a product of a vector space Rn, a torus Rm/Zm, a free abelian group Zk and a
finite abelian group. In such a group, there clearly is a unique maximal compact
subgroup (namely the product of the finite group and the torus). It is also normal,
and maximal in G. �
The maximal compact normal subgroup of G0 is a characteristic Lie subgroup
of of G0. It is therefore normal in G and we may mod out by it. We therefore have
shown that every locally compact (compactly generated) group with polynomial
growth admits a quotient by a compact normal subgroup, which is a Lie group G
whose connected component of the identity G0 has polynomial growth and con-
tains no compact normal subgroup. We will now show that a certain co-compact
subgroup of G has the embedding property of Theorem 1.2.
(b) Second we show that, up to passing to a co-compact subgroup, we may
assume that the connected component G0 is solvable. For this purpose, let Q be
the solvable radical of G0, namely the maximal connected normal Lie subgroup
of G0. Note that it is a characteristic subgroup of G0 and therefore normal in G.
Moreover G0/Q is a semisimple Lie group. Since G0 has polynomial growth, it
follows that G0/Q must be compact. Consider the action of G by conjugation on
G0/Q, namely the map φ : G→ Aut(G0/Q). Since G0/Q is compact semisimple,
its group of automorphisms is also a compact Lie group. In particular, the kernel
ker φ is a co-compact subgroup of G.
The connected component of the identity of Aut(G0/Q) is itself semisimple and
hence has finite center. However the image of the connected component (ker φ)0
of ker φ in G0/Q modulo Q is central. Therefore it must be trivial. We have
shown that (ker φ)0 is contained in Q and hence is solvable. Moreover (ker φ)0
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 41
has no compact normal subgroup, because otherwise its maximal normal compact
subgroup, being characteristic in (ker φ)0, would be normal in G (note that (ker φ)0
is normal in G).
Changing G into the co-compact subgroup kerφ, we can therefore assume that
G0 is solvable, of polynomial growth, and has no non trivial compact normal sub-
group. The group G/G0 is discrete, finitely generated, and has polynomial growth.
By Gromov’s theorem, it must be virtually nilpotent, in particular virtually poly-
cyclic.
(c) We finally prove the following proposition.
Proposition 7.2. Let G be a Lie group such that its connected component of the
identity G0 is solvable, admits no compact normal subgroup, and with G/G0 virtu-
ally polycyclic. Then G has a closed co-compact subgroup, which can be embedded
as a closed co-compact subgroup of a connected and simply connected solvable Lie
group.
The proof of this proposition is mainly an application of a theorem of H.C.
Wang, which is a vast generalization of Malcev’s embedding theorem for torsion
free finitely generated nilpotent groups. Wang’s theorem [36] states that any S-
group can be embedded as a closed co-compact subgroup of a simply connected real
linear solvable Lie group with only finitely many connected components. Wang
defines a S-group to be any real Lie group G, which admits a normal subgroup
A such that G/A is finitely generated abelian and A is a torsion-free nilpotent
Lie group whose connected components group is finitely generated. In particular
any S-group has a finite index (hence co-compact) subgroup which embeds as a
co-compact subgroup in a connected and simply connected solvable Lie group.
In order to prove Proposition 7.2, it therefore suffices to establish that G has a
co-compact S-group.
We first recall the following simple fact:
Lemma 7.3. Every closed subgroup F of a connected solvable Lie group S is
topologically finitely generated.
Proof. We argue by induction on the dimension of S. Clearly there is an epi-
morphism π : S → R. By induction hypothesis F ∩ ker π is topologically finitely
generated. The image of F is a subgroup of R. However every subgroup of R
contains either one or two elements, whose subgroup they generate has the same
closure as the original subgroup. We are done. �
Next we show the existence of a nilradical.
Lemma 7.4. Let G be as in Proposition 7.2. Then G has a unique maximal
normal nilpotent subgroup GN .
Proof. The subgroup generated by any two normal nilpotent subgroups of any
given group is itself nilpotent (Fitting’s lemma, see e.g. [30][5.2.8]). Let GN be
the closure of the subgroup generated by all nilpotent subgroups of G. We need
to show that GN is nilpotent. For this it is clearly enough to prove that it is
42 EMMANUEL BREUILLARD
topologically finitely generated (because any finitely generated subgroup of GN is
nilpotent by the remark we just made). Since G/G0 is virtually polycyclic, every
subgroup of it is finitely generated ([29][4.2]). Hence it is enough to prove that
GN ∩G0 is topologically finitely generated. This follows from Lemma 7.3. �
Incidently, we observe that the connected component of the identity (GN )0
coincides with the nilradicalN of G0 (it is the maximal normal nilpotent connected
subgroup of G0).
We now claim the following:
Lemma 7.5. The quotient group G/GN is virtually abelian.
The proof of this lemma is inspired by the proof of the fact, due to Malcev,
that polycyclic groups have a finite index subgroup with nilpotent commutator
subgroup (e.g. see [30][ 15.1.6]).
Proof. We will show that G has a finite index normal subgroup whose commutator
subgroup is nilpotent. This clearly implies the lemma, for this nilpotent subgroup
will be normal, hence contained in GN .
First we observe that the group G admits a finite normal series Gm ≤ Gm−1 ≤
. . . ≤ G1 = G, where each Gi is a closed normal subgroup of G such that Gi/Gi+1
is either finite, or isomorphic to either Zn, Rn or Rn/Zn. This see it pick one of the
Gi’s to be the connected component G0 and then treat G/G0 and G0 separately.
The first follows from the definition of a polycyclic group (G/G0 has a normal
polycyclic subgroup of finite index). While for G0, observe that its nilradical N is
a connected and simply connected nilpotent Lie group and it admits such a series
of characteristic subgroups (pick the central descending series), and G0/N is an
abelian connected Lie group, hence isomorphic to the direct product of a torus
n/Zn and a vector group Rn. The torus part is characteristic in G0/N , hence its
preimage in G0 is normal in G.
The group G acts by conjugation on each partial quotient Qi := Gi/Gi+1. This
yields a map G → Aut(Qi). Now note that in order to prove our lemma, it is
enough to show that for each i, there is a finite index subgroup of G whose com-
mutator subgroup maps to a nilpotent subgroup of Aut(Qi). Indeed, taking the
intersection of those finite index subgroup, we get a finite index normal subgroups
whose commutator subgroup acts nilpotently on each Qi, hence is itself nilpotent
(high enough commutators will all vanish).
Now Aut(Qi) is either finite (if Qi is finite), or isomorphic to GLn(Z) (in case
Qi is either Z
n or Rn/Zn) or to GLn(R) (when Qi ≃ R
n). The image of G in
Aut(Qi) is a solvable subgroup. However, every solvable subgroup of GLn(R)
contains a finite index subgroup, whose commutator subgroup is unipotent (hence
nilpotent). This follows from Kolchin’s theorem for example, that a connected
solvable algebraic subgroup of GLn(C) is triangularizable. We are done. �
In the sequel we assume that G/G0 is torsion-free polycyclic. It is legitimate
to do so in the proof of Proposition 7.2, because every virtually polycyclic group
has a torsion-free polycyclic subgroup of finite index (see e.g. [29][Lemma 4.6]).
We now claim the following:
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 43
Lemma 7.6. GN is torsion-free.
Proof. Since G/G0 is torsion-free, it is enough to prove that GN ∩ G0 is torsion-
free. However the set of torsion elements in GN forms a subgroup of GN (if x, y are
torsion, then xy is too because 〈x, y〉 is nilpotent). Clearly it is a characteristic
subgroup of GN . Hence its intersection with G0 is normal in G0. Taking the
closure, we obtain a nilpotent closed normal subgroup T of G0 which contains a
dense set of torsion elements. Recall that G0 has no normal compact subgroup.
From this it quickly follows that T is trivial, because first it must be discrete (the
connected component T0 is compact and normal in G0), hence finitely generated
(by Lemma 7.3), hence made of torsion elements. But a finitely generated torsion
nilpotent group is finite. Again since G0 has no compact normal subgroup, T must
be trivial, and GN is torsion-free. �
Now observe that the group of connected components of GN , namely GN/(GN )0
is finitely generated. Indeed, since G/G0 is finitely generated (as any polycyclic
group), it is enough to prove that (G0 ∩GN )/(GN )0 is finitely generated, but this
follows from the fact that G0∩GN is topologically finitely generated (Lemma 7.3).
Now we are almost done. Note thatG is topologically finitely generated (Lemma
7.3), therefore so is G/GN . By Lemma 7.5 G/GN is virtually abelian, hence has
a finite index normal subgroup isomorphic to Zn × Rm. It follows that G/GN
has a co-compact subgroup isomorphic to a free abelian group Zn+m. Hence after
changing G by a co-compact subgroup, we get that G is an extension of GN (a
torsion-free nilpotent Lie group with finitely generated group of connected com-
ponents) by a finitely generated free abelian group. Hence it is an S-group in the
terminology of Wang [36]. We apply Wang’s theorem and this ends the proof of
Proposition 7.2.
(d) We can now conclude the proof of Theorem 1.2. By (a) and (b) G has
a quotient by a compact group which admits a co-compact subgroup satisfying
the assumptions of Proposition 7.2. Hence to conclude the proof it only remains
to verify that the simply connected solvable Lie group in which a co-compact
subgroup of G/K embeds has polynomial growth (i.e. is of type (R)). But this
follows from the following lemma (see [21][Thm. I.2]).
Lemma 7.7. Let G be a locally compact group. Then G has polynomial growth if
and only if some (resp. any) co-compact subgroup of it has polynomial growth.
Proof. First one checks that G is compactly generated if and only if some (resp.
any) co-compact subgroup is. This is by the same argument which shows that
finite index subgroups of a finitely generated group are finitely generated. In
particular, if Ω is a compact symmetric generating set of G and H is a co-compact
subgroup, then there is n0 ∈ N such that Ω
n0H = G. Then H ∩ Ω3n0 generates
If G has polynomial growth and H is any compactly generated closed subgroup,
then H has polynomial growth. Indeed (see [21][Thm I.2]), if ΩH denotes a com-
pact generating set for H, and K a compact neighborhood of the identity in G,
44 EMMANUEL BREUILLARD
volG(K)volH(Ω
H) ≤ volH(KK
−1 ∩H)volG(Ω
This inequality follows by integrating over a left Haar measure of G the function
φ(x) :=
−1x)dh, where dh is a left Haar measure on H. This integral
equals the left handside of the above displayed equation, while it is pointwise
bounded by volH(xK
−1 ∩H) inside HK and by zero outside HK.
In the other direction, if H has polynomial growth, then G also has, because
one can write Ωn ⊂ ΩnHK for some compact generating set ΩH of H and some
compact neighborhood K of the identity in G (see Proposition 4.4). Then the
result follows from the following inequality
volH(ΩH)volG(Ω
HK) ≤ volH(Ω
H )volG(Ω
H K),
which itself is a direct consequence of the fact that the function
ψ(x) :=
(h−1x)dh,
where dh is a left Haar measure onH, satisfies
ψ(x)dx = volH(Ω
H )volG(Ω
on the one hand and is bounded below by volH(ΩH) for every x ∈ Ω
HK on the
other hand. �
Note that the above proof would be slightly easier if we already knew that both
G and H were unimodular, in which case G/H has an invariant measure. But
we know this only a posteriori, because the polynomial growth condition implies
unimodularity ([21]).
Similar considerations show that G has polynomial growth if and only if G/K
has polynomial growth, given any normal compact subgroup K (e.g. see [21]).
We end this paragraph with a remark and an example, which we mentioned in
the Introduction.
Remark 7.8 (Discrete subgroups are virtually nilpotent). Suppose Γ is a discrete
subgroup of a connected solvable Lie group of type (R) (i.e. of polynomial growth).
Then Γ is virtually nilpotent. Indeed, a similar argument as in Lemma 7.3 shows
that every subgroup of Γ is finitely generated. It follows that Γ is polycyclic. How-
ever Wolf [37] proved that polycyclic groups with polynomial growth are virtually
nilpotent.
Example 7.9 (A group with no nilpotent co-compact subgroup). Let G be the
connected solvable Lie group G = R ⋉ (R2 × R2), where R acts as a dense one-
parameter subgroup of SO(2,R) × SO(2,R). Then G is of type (R). It has no
compact subgroup. And it has no nilpotent co-compact subgroup. Indeed suppose H
is a closed co-compact nilpotent subgroup. Then it has a non-trivial center. Hence
there is a non identity element whose centralizer is co-compact in G. However a
simple examination of the possible centralizers of elements of G shows that none
of them is co-compact.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 45
7.2. Proof of Corollary 1.6 and Theorem 1.1. Let G be an arbitrary locally
compact group of polynomial growth and ρ a periodic pseudodistance on G.
Claim 1: Corollary 1.6 holds for a co-compact subgroup H of G, if and only
if it holds for G. By Lemma 7.7, the groups G and H are unimodular, and hence
G/H bears a G-invariant Radon measure volG/H , which is finite since H is co-
compact. Now let F be a bounded Borel fundamental domain for H inside G. And
let ρ be the periodic pseudodistance on G induced by the restriction of ρ to H, that
is ρ(x, y) := ρ(hx, hy) where hx is the unique element of H such that x ∈ hxF. By
4.2 (1) and (4), ρ and ρ are at a bounded distance from each other. In particular,
Bρ(r−C) ⊂ Bρ(r) ⊂ Bρ(r+C). Hence if the limit (3) holds for ρ, it also holds for
ρ with the same limit. However, Bρ(r) = {x ∈ G, ρ(e, hx) ≤ r} = BρH (r)F where
ρH is the restriction of ρ to H. Hence volG(Bρ(r)) = volH(BρH (r)) · volG/H(F ).
By 4.2 (4), ρH is a periodic pseudodistance on H. So the result holds for (H, ρH)
if and only if it holds for (G, ρ). Conversely, if ρ0 is a periodic pseudodistance
on H, then ρ0(x, y) := ρ0(hx, hy) is a periodic pseudodistance on G, hence again
volG(Bρ0(r)) = volH(Bρ0(r)) · volG(F ) and the result will hold for (H, ρ0) if and
only if it holds for (G, ρ0).
Claim 2: If Corollary 1.6 holds for G/K, where K is some compact normal
subgroup, then it holds for G as well. Indeed, if ρ is a periodic pseudodistance on
G, then the K-average ρK , as defined in (26), is at a bounded distance from G
according to Lemma 4.7. Now ρK induces a periodic pseudodistance ρK on G/K
and BρK (r) = BρK (r)K. Hence, volG(BρK (r)) = volG/K(BρK (r)) · volK(K). And
if the limit (3) holds for ρK , it also holds for ρK , hence for ρ too.
Thus the discussion above combined with Theorem 1.2 reduces Corollary 1.6 to
the case when G is simply connected and solvable, which was treated in Section 5
and 6. �
7.3. Proof of Proposition 1.3 and Corollary 1.9.
Proof of Proposition 1.3. We say that two metric spaces (X, dX ) and (Y, dY ) are
at a bounded distance if they are (1, C)-quasi-isometric for some finite C. This is
an equivalence relation. Now if ρ is H-periodic with H co-compact, then (G, ρ)
is at a bounded distance from (H, ρ|H). Hence we may assume that H = G, i.e.
that ρ is left invariant on G.
Now Theorem 1.2 gives the existence of a normal compact subgroup K, a co-
compact subgroup H containing K and a simply connected solvable Lie group S
such that H/K is isomorphic to a co-compact subgroup of S.
Lemma 4.7 shows that (G, ρ) is at a bounded distance from (G, ρK), where ρK
is defined as in (26). Now ρK induces a left invariant periodic metric on G/K,
and (G/K, ρK ) is clearly at a bounded distance from (G, ρK). Now by 4.2, its
restriction to H/K is at a bounded distance and is left invariant. Now we set
ρS(s1, s2) = ρ
K(h1, h2), where (given a bounded fundamental domain F for the
46 EMMANUEL BREUILLARD
left action of H/K on S) hi is the unique element of H/K such that si ∈ hiF .
Clearly then (S, ρS) is at a bounded distance from (H/K, ρ
K). We are done. �
We note that our construction of S here depends on the stabilizer of ρ in G.
Certainly not every choice of Lie shadow can be used for all periodic metrics (think
that R3 is a Lie shadow of the universal cover of the group of motions of the plane).
Perhaps a single one can be chosen for all, but we have not checked that.
Proof of Corollary 1.9. Proposition 1.3 reduces the proof to a periodic metric ρ
on a simply connected solvable Lie group S. Let d∞ the subFinsler metric on S
(left invariant for the graded nilshadow group structure SN ) as given by Theorem
1.4. Let {δt}t is the group of dilations in the graded nilshadow SN of S as defined
in Section 3. By definition of the pointed Gromov-Hausdorff topology (see [18]),
it is enough to prove the
Claim. The following quantity
ρ(s1, s2)− d∞(δ 1
(s1), δ 1
(s2))|
converges to zero as n tends to +∞ uniformly for all s1, s2 in a ball of radius O(n)
for the metric ρ.
Now this follows in three steps. First ρ is at a bounded distance from its
restriction to the (co-compact) stabilizer H of ρ (cf. 4.2 (1), 4.2 (4)). Then
for h1, h2 ∈ H, we can write ρ(h1, h2) = ρ(e, h
1 h2). However Proposition 5.1
implies the existence of another periodic distance ρK on S, which is invariant
under left translations by elements of H for both the original Lie structure and
the nilshadow Lie structure on S, such that
ρ(e,x)
ρK(e,x)
tends to 1 as x tends to
∞. Hence ρK(e, h
1 h2) = ρK(h1, h2) = ρK(e, h
1 h2), where ∗ is the nilshadow
product on S. Hence | 1
ρ(h1, h2)−
ρK(e, h
1 h2)| tends to zero uniformly as h1
and h2 vary in a ball of radius O(n) for ρ.
Finally Theorem 6.2 implies that | 1
ρK(e, h
1 h2) −
d∞(e, h
1 h2)| tends to
zero and the claim follows, as one verifies from the Campbell Hausdorff formula
by comparing (11) and (12) as we did in (35), that
|d∞(δ 1
(h1), δ 1
(h2))− d∞(e, δ 1
(h∗−11 h2)|
converges to zero.
The fact that the graded nilpotent Lie group does not depend (up to isomor-
phism) on the periodic metric ρ but only on the locally compact group G fol-
lows from Pansu’s theorem [28] that if two Carnot groups (i.e. a graded simply
connected nilpotent Lie group endowed with left-invariant subRiemannian metric
induced by a norm on a supplementary subspace to the commutator subalgebra)
are bi-Lipschitz, the underlying Lie groups must be isomorphic. This deep fact
relies on Pansu’s generalized Rademacher theorem, see [28]. Indeed, two differ-
ent periodic metrics ρ1 and ρ2 on G are quasi-isometric (see Proposition 4.4),
and hence their asymptotic cones are bi-Lipschitz (and bi-Lipschitz to any Carnot
group metric on the same graded group, by (13)). �
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 47
8. Coarsely geodesic distances and speed of convergence
Under no further assumption on the periodic pseudodistance ρ, the speed of
convergence in the volume asymptotics can be made arbitrarily small. This is
easily seen if we consider examples of the following type: define ρ(x, y) = |x−y|+
|x− y|α on R where α ∈ (0, 1). It is periodic and vol(Bρ(t)) = t− t
α + o(tα).
However, many natural examples of periodic metrics, such as word metrics or
Riemannian metrics, are in fact coarsely geodesic. A pseudodistance on G is said
to be coarsely geodesic, if there is a constant C > 0 such that any two points can
be connected by a C-coarse geodesic, that is, for any x, y ∈ G there is a map
g : [0, t] → G with t = ρ(x, y), g(0) = x and g(t) = y, such that
|ρ(g(u), g(v)) − |u− v|| ≤ C
for all u, v ∈ [0, t].
This is a stronger requirement than to say that ρ is asymptotically geodesic
(see 21). This notion is invariant under coarse isometry. In the case when G
is abelian, D. Burago [6] proved the beautiful fact that any coarsely geodesic
periodic metric on G is at a bounded distance from its asymptotic norm. In
particular volG(Bρ(t)) = c · t
d+O(td−1) in this case. In the remarkable paper [32],
M. Stoll proved that such an error term in O(td−1) holds for any finitely generated
2-step nilpotent group. Whether O(td−1) is the right error term for any finitely
generated nilpotent group remains an open question.
The example below shows on the contrary that in an arbitrary Lie group of
polynomial growth no universal error term can be expected.
Theorem 8.1. Let εn > 0 be an arbitrary sequence of positive numbers tending
to 0. Then there exists a group G of polynomial growth of degree 3 and a compact
generating set Ω in G and c > 0 such that
volG(Ω
c · n3
≤ 1− εn
holds for infinitely many n, although 1
volG(Ω
n) → 1 as n→ +∞.
The example we give below is a semi-direct product of Z by R2 and the metric
is a word metric. However, many similar examples can be constructed as soon as
the map T : G → K defined in Paragraph 5.1 in not onto. For example, one can
consider left invariant Riemannian metrics on G = R · (R2 × R2) where R acts
by via a dense one-parameter subgroup of the 2-torus S1 × S1. Incidently, this
group G is known as the Mautner group and is an example of a wild group in
representation theory.
8.1. An example with arbitrarily small speed. In this paragraph we describe
the example of Theorem 8.1. Let Gα = Z · R
2 where the action of Z is given by
the rotation Rα of angle πα, α ∈ [0, 1). The group Gα is quasi-isometric to R
and hence of polynomial growth of order 3 and it is co-compact in the analogously
defined Lie group G̃α = R ⋉ R
2. Its nilshadow is isomorphic to R3. The point is
48 EMMANUEL BREUILLARD
Figure 2. The union of the two cones, with basis the disc of radius
2, represents the limit shape of the balls Ωn in the group Z ⋉ R2,
where Z acts by an irrational rotation, with generating set Ω =
{(±1, 0, 0)} ∪ {(0, x1, x2),
x21 + x
2 ≤ 1}.
that if α is a suitably chosen Liouville number, then the balls in Gα will not be
well approximated by the limit norm balls.
Elements of Gα are written (k, x) where k ∈ Z and x ∈ R
2. Let ‖x‖
x21+x
be a Euclidean norm on R2, and let Ω be the symmetric compact generating set
given by {(±1, 0)}∪{(0, x), ‖x‖ ≤ 1}. It induces a word metric ρΩ on G. It follows
from Theorem 1.4 and the definition of the asymptotic norm that ρΩ(e, (k, x)) is
asymptotic to the norm on R3 given by ρ0(e, (k, x)) := |k| + ‖x‖0 where ‖x‖0 is
the rotation invariant norm on R2 defined by ‖x‖
(x21 + x
2). The unit ball of
‖·‖0 is the convex hull of the union of all images of the unit ball of ‖·‖ under all
rotations Rkα, k ∈ Z.
We are going to choose α as a suitable Liouville number so that (36) holds. Let
δn = (4εn)
1/3 and choose α so that the following holds for infinitely many n’s:
(37) d(kα,Z +
) ≥ 2δn
for all k ∈ Z, |k| ≤ n. This is easily seen to be possible if we choose α of the form∑
1/3ni for some suitable lacunary increasing sequence of (ni)i.
Note that, since ‖x‖0 ≥ ‖x‖ , we have ρΩ ≥ ρ0. Let Sn be the piece of R
2 defined
by Sn = {|θ| ≤ δn} where θ is the angle between the point x and the vertical axis
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 49
Re2. We claim that if x ∈ Sn, ρ0(e, (k, x)) ≤ n and n satisfies (37), then
ρΩ(e, (k, x)) ≥ |k|+ (1 +
) ‖x‖0
It follows easily from the claim that volG(Ω
n) ≤ (1− εn) · volG(Bρ0(n)). Moreover
volG(Bρ0(n)) = c · n
3 + O(n2), where c = 4π
if volG is given by the Lebesgue
measure.
Proof of claim. Here is the idea to prove the claim. To find a short path between
the identity and a point on the vertical axis, we have to rotate by a Rkα such that
kα is close to 1
, hence go up from (0, 0) to (k, 0) first, thus making the vertical
direction shorter. However if (37) holds, the vertical direction cannot be made as
short as it could after rotation by any of the Rkα with |k| ≤ n.
Note that if ρ0(e, (k, x)) ≤ n then |k| ≤ n and ρΩ(e, (k, x)) ≥ |k|+inf
‖Rkiαxi‖
where the infimum is taken over all paths x1, ..., xN such that x =
xi and all
rotations Rkiα with |ki| ≤ n. Note that if δn is small enough and (37) holds
then for every x ∈ Sn we have ‖Rkαx‖ ≥ (1 + δ
n) ‖x‖0 . On the other hand
‖x‖0 =
‖xi‖0 cos(θi) where θi is the angle between xi and the x. Hence
‖Rkiαxi‖ ≥
|θi|≤δn
‖Rkiαxi‖+
|θi|>δn
‖Rkiαxi‖
≥ (1 + δ2n)
|θi|≤δn
‖xi‖0 cos(θi) +
cos(δn)
|θi|>δn
‖xi‖0 cos(θi)
≥ (1 +
) · ‖x‖0
8.2. Limit shape for more general word metrics on solvable Lie groups
of polynomial growth. The determination of the limit shape of the word metric
in Paragraph 8.1 was possible due to the rather simple nature of the generating
set. In general, using the identity (see (1))
(38) ω1 · . . . · ωm = ω1 ∗ (T (ω1)ω2) ∗ . . . ∗ (T (ωm−1 · . . . · ω1)ωm)
it is easy to check that the unit ball of the limit norm ‖ · ‖∞ inducing the limit
subFinsler metric d∞ on the nilshadow associated to a given word metric with
generating set Ω is contained in the K-orbit of the convex hull of the projection
of Ω to the abelianized nilshadow, namely the convex hull of K · π1(Ω).
In the example of Paragraph 8.1, we even had equality between the two. How-
ever this is not the case in general. For example, the limit shape is always K-
invariant, but clearly the limit shape associated to a generating set Ω coincides
with the one associated with a conjugate gΩg−1 of it, while the convex hull of the
respective K-orbits may not be the same.
Of course if the generating set Ω is K-invariant to begin with, then Ωn = Ω∗n
and we are back in the nilpotent case, where we know that the unit ball of the
limit norm is just the convex hull of the projection of the generating set to the
abelianization. In general however it is a challenging problem to determine the
50 EMMANUEL BREUILLARD
precise asymptotic shape of a word metric on a general solvable Lie group with
polynomial growth, and there seems to be no simple description analogous to what
we have in the nilpotent case.
Even in the above example Gα = Z⋉αR
2, or in the universal cover of the group
of motions of the plane (in which Gα embeds co-compactly), it is not that simple.
In general the shape is determined by solving an optimization problem in which
one has to find the path which maximizes the coordinates of the endpoint. In
order to illustrate this, we treat without proof the following simple example.
Suppose Ω is a symmetric compact neighborhood of the identity in Gα = Z⋉αR
of the form Ω = (0,Ω0) ∪ (1,Ω1) ∪ (1,Ω1)
−1, where Ω0,Ω1 ⊂ R
2. Then the
limit shape of the word metric ρΩ associated to Ω is the solid body (rotationally
symmetric around the vertical axis as in Figure 2) made of two copies (upper and
lower) of a truncated cone with base a disc on (0,R2) of radius max{r0, r1} and
top (resp. bottom) a disc on the plane (1,R2) (resp. (−1,R2)) of radius r2, where
the radii are given by
r0 = max{‖x‖, x ∈ Ω0}, r1 =
diam(Ω1),
where diam(Ω1) is the diameter of Ω1 and r2 is given by the integral
(39) r2 =
max{πθ(Ω1)}
where πθ(Ω1) is the orthogonal projection on the x-axis of image of Ω1 ⊂ R
2 by a
rotation of angle θ around the origin. It is indeed convex (note that r2 ≤ r1).
For example if Ω1 is made of only one point, then the limit shape is the same
as in the previous paragraph and as in Figure 2, namely two copies of a cone.
However if Ω1 is made of two points {a, b}, then the upper part of the limit shape
will be a truncated cone with an upper disc of radius r2 =
‖a−b‖
(which is the
result of the computation of the above integral).
Let us briefly explain the formula (39). A path of length n reaching the highest
z-coordinate in Gα is a word of the form (1, ω1) · . . . · (1, ωn), with ωi ∈ Ω1. By
(38) this word equals
Ri−1α ωi).
Here ωi can take any value in Ω1. In order to maximize the norm of the second
coordinate, or equivalently (by rotation invariance) its x-coordinate, one has to
choose ωi ∈ Ω1 at each stage in such a way that the x-coordinate of R
α ωi is
maximized. Formula (39) now follows from the fact that {Ri−1α }1≤i≤n becomes
equidistributed in SO(2,R) as n tends to infinity.
In order to show that max{r0, r1} is the radius of the base disc and more
generally that the limit shape is no bigger than this double truncated cone, one
needs to argue further by considering all possible paths of the form (ε1, ω1) · . . . ·
(εn, ωn) where εi ∈ {0,±1} and
εi is prescribed.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 51
8.3. Bounded distance versus asymptotic metrics. In this paragraph we an-
swer a question of D. Burago and G. Margulis (see [7]). Based on the abelian case
and the reductive case (Abels-Margulis [1]), Burago and Margulis had conjectured
that every two asymptotic word metrics should be at a bounded distance. We give
below a counterexample to this. We first give an example (A) of a nilpotent Lie
group endowed with two left invariant subFinsler metrics d∞ and d
∞ that are
asymptotic to each other, i.e. d∞(e, x)/d
∞(e, x) → 1 as x → ∞ but such that
|d∞(e, x)− d
∞(e, x)| is not uniformly bounded. Then we exhibit (B) a word met-
ric that is not at a bounded distance from any homogeneous quasi-norm. Finally
these examples also yield (C) two word metrics ρ1 and ρ2 on the same finitely
generated nilpotent group which are asymptotic but not at a bounded distance.
Note that the group Gα with ρ0 and ρΩ from the last paragraph also provides
an example of asymptotic metrics which are not at a bounded distance (but this
group was not discrete).
(A) Let N = R × H3(R) where H3 is classical Heisenberg group and Γ =
Z ×H3(Z) a lattice in N . In the Lie algebra n = RV ⊕ h3 we pick two different
supplementary subspaces of [n, n] = RZ, i.e. m1 = span{V,X, Y } and m
span{V + Z,X, Y }, where h3 is the Lie algebra of H3(R) spanned by X,Y and
Z = [X,Y ].We consider the L1-norm on m1 (resp. m
1) corresponding to the basis
(V,X, Y ) (resp. (V + Z,X, Y )). Both norms induce the same norm on n/[n, n].
They give rise to left invariant Carnot-Caratheodory Finsler metrics on N , say
d∞ (resp. d
∞). We use the coordinates (v, x, y, z) = exp(vV + xX + yY + zZ).
According to Remark (2) after Theorem 6.2, d∞ and d
∞ are asymptotic. Let
us show that they are not at a bounded distance. First observe that, since V
is central, d∞(e, (v; (x, y, z))) = |v| + dH3(e, (x, y, z)) where dH3 is the Carnot-
Caratheodory Finsler metric on H3(R) defined by the standard L
1-norm on the
span{X,Y }. Similarly d′∞(e, (v; (x, y, z))) = |v| + dH3(e, (x, y, z − v))). If d∞ and
d′∞ were at a bounded distance, we would have a C > 0 such that for all t > 0
|d∞(e, (t; (0, 0, t))) − t| ≤ C
Hence |dH3(e, (0, 0, t))| ≤ C, which is a contradiction.
(B) Now let Ω = {(1; (0, 0, 1))±1 , (1; (0, 0,−1))±1 , (0; (1, 0, 0))±1 , (0; (0, 1, 0))±1}
be a generating set for Γ and ρΩ the word metric associated to it. Let | · | be
a homogeneous quasi-norm on N which is at a bounded distance from ρΩ, i.e.
|ρΩ(e, g) − |g|| is bounded. Then | · | is asymptotic to ρΩ, hence is equal to the
Carnot-Caratheodory Finsler metric d asymptotic to ρΩ and homogeneous with
respect to the same one parameter group of dilations {δt}t>0. Let m1 = {v ∈ n,
δt(v) = tv}. Then d is induced by some norm ‖·‖0 on m1, whose unit ball is
given, according to Theorem 1.4 by the convex hull of the projections to m1 of the
generators in Ω. There is a unique vector in m1 of the form V +z0Z. Its ‖·‖0-norm
is 1 and d(e, (1; (0, 0, z0))) = 1. However d(e, (v; (x, y, z))) = |v|+ dH3(e, (x, y, z −
vz0)). Since ρΩ(e, (n; (0, 0, n))) = n, we get
d(e, (n; (0, 0, n))) − ρΩ(e, (n; (0, 0, n))) = dH3(e, (0, 0, n(1 − z0)))
52 EMMANUEL BREUILLARD
If this is bounded, this forces z0 = 1. But we can repeat the same argument with
(n; (0, 0,−n)) which would force z0 = −1. A contradiction.
(C) Let now Ω2 := {(1; (0, 0, 0))
±1 , (0; (1, 0, 0))±1 , (0; (0, 1, 0))±1} and ρΩ2 the
associated word metric on Γ. Then again ρΩ and ρΩ2 are asymptotic by Theorem
6.2 because the convex hull of their projection modulo the z-coordinate coincide.
However ρΩ2 is a product metric, namely we have ρΩ2(e, (v; (x, y, z))) = |v| +
ρ(e, (x, y, z)), where ρ is the word metric on the discrete Heisenberg group H3(Z)
with standard generators {(1, 0, 0)±1, (0, 1, 0)±1}. In particular
ρΩ(e, (n; (0, 0, n))) − ρΩ2(e, (n; (0, 0, n))) = ρ(e, (0, 0, n))
which is unbounded.
Remark 8.2 (An abnormal geodesic). We refer the reader to [9] for more on
these examples. In particular we show there that ρ1 and ρ2 above are not (1, C)-
quasi-isometric for any C > 0. The key phenomenon behind this example is
the presence of an abnormal geodesic (see [25]), namely the one-parameter group
{(t; (0, 0, 0))}t .
Remark 8.3 (Speed of convergence in the nilpotent case). The slow speed phe-
nomenon in Theorem 8.1 relied crucially on the presence of a non-trivial semisim-
ple part in Gα ; this doesn’t occur in nilpotent groups. In [9], we show that for
word metrics on finitely generated nilpotent groups, the convergence in Theorem
6.2 has a polynomial speed with an error term at least as good as O(d∞(e, x)
3r ),
where r is the nilpotency class. We conjecture there that the optimal exponent is 1
This involves refining quantitatively the estimates of the above proof of Theorem
9. Appendix: the Heisenberg groups
Here we show how to compute the asymptotic shape of balls in the Heisenberg
groups H3(Z) and H5(Z) and their volume, thus giving another approach to the
main result of Stoll [33]. The leading term for the growth of H3(Z) is rational
for all generating sets (Prop. 9.1 below), whereas in H5(Z) with its standard
generating set, it is transcendental. This explains how our Figure 1 was made
(compare with the odd [22] Fig. 1).
9.1. 3-dim Heisenberg group. Let us first consider the Heisenberg group
H3(Z) = 〈a, b|[a, [a, b]] = [b, [a, b]] = 1〉 .
We see it as the lattice generated by a = exp(X) and b = exp(Y ) in the real
Heisenberg group H3(R) with Lie algebra h3 generated by X,Y and spanned by
X,Y,Z = [X,Y ]. Let ρΩ be the standard word metric on H3(Z) associated to
the generating set Ω = {a±1, b±1}. According to Theorem 1.4, the limit shape of
the n-ball Ωn in H3(Z) coincides with the unit ball C3 = {g ∈ H3(R), d∞(e, g) ≤
1} for the Carnot-Caratheodory metric d∞ induced on H3(R) by the ℓ
1-norm
‖xX + yY ‖0 = |x|+ |y| on m1 = span{X,Y } ⊂ h3.
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 53
Computing this unit ball is a rather simple task. Exchanging the roles of X
and Y , we see that C3 is invariant under the reflection z 7→ −z. Then clearly C3 is
of the form {xX + yY + zZ, with |x|+ |y| ≤ 1 and |z| ≤ z(x, y)}. Changing X to
−X and Y to −Y, we get the symmetries z(x, y) = z(−x, y) = z(x,−y) = z(y, x).
Hence when determining z(x, y), we may assume 0 ≤ y ≤ x ≤ 1, x+ y ≤ 1.
The following well known observation is crucial for computing z(x, y). If ξ(t) is a
horizontal path in H3(R) starting from id, then ξ(t) = exp(x(t)X+y(t)Y +z(t)Z),
where ξ′(t) = x(t)X + y(t)Y and z(t) is the “balayage” area of the between the
path {x(s)X + y(s)Y }0≤s≤t and the chord joining 0 to x(t)X + y(t)Y.
Therefore, z(x, y) is given by the solution to the “Dido isoperimetric problem”
(see [25]): find a path in the X,Y -plane between 0 and xX + yY of ‖·‖0-length 1
that maximizes the “balayage area”. Since ‖·‖0 is the ℓ
1-norm in the X,Y -plane,
as is well-known (see [8]), such extremal curves are given by arcs of square with
sides parallel to the X,Y -axes. There is therefore a dichotomy: the arc of square
has either 3 or 4 sides (it may have 1 or 2 sides, but these are included are limiting
cases of the previous ones).
If there are 3 sides, they have length ℓ, x and y + ℓ with y + ℓ ≤ x. Hence
1 = ℓ+ x+ y + ℓ and z(x, y) = ℓx+ 1
xy. Therefore this occurs when y ≤ 3x − 1
and we then have z(x, y) =
x(1−x)
If there are 4 sides, they have length ℓ, x+ u, y + ℓ and u, with ℓ+ y = x+ u.
Hence 1 = 2ℓ + 2u + x + y and z(x, y) = (ℓ + y)(x + u) −
. This occurs when
y ≥ 3x− 1 and we then have z(x, y) =
(1+x+y)2
Hence if 0 ≤ y ≤ x ≤ 1 and x+ y ≤ 1
(40) z(x, y) = 1y≤3x−1
x(1− x)
+ 1y>3x−1
(1 + x+ y)2
The unit ball C3 drawn in Figure 1 is the solid body C3 = {xX + yY + zZ, with
|x|+ |y| ≤ 1 and |z| ≤ z(x, y)}.
A simple calculation shows that vol(C3) =
in the Lebesgue measure dxdydz.
Since H3(Z) is easily seen to have co-volume 1 for this Haar measure on H3(R)
(actually {xX+yY +zZ, x ∈ [0, 1), y ∈ [0, 1), z ∈ [0, 1)} is a fundamental domain),
it follows that
#(Ωn)
= vol(C3) =
We thus recover a well-known result (see [4], [31] where even the full growth series
is computed and shown to be rational).
One can also determine exactly which points of the sphere ∂C3 are joined to id
by a unique geodesic horizontal path. The reader will easily check that uniqueness
fails exactly at the points (x, y,±z(x, y)) with |x| < 1
and y = 0, or |y| < 1
x = 0, or else at the points (x, y, z) with |x|+ |y| = 1 and |z| < z(x, y).
The above method also yields the following result.
54 EMMANUEL BREUILLARD
Proposition 9.1. Let Ω be any symmetric generating set for H3(Z). Then the
leading coefficient in #(Ωn) is rational, i.e.
#(Ωn)
is a rational number.
Proof. We only sketch the proof here. We can apply the method above and com-
pute r as the volume of the unit CC-ball C(Ω) of the limit CC-metric d∞ de-
fined in Theorem 1.4. Since we know what is the norm ‖·‖ in the (x, y)-plane
m1 = span 〈X,Y 〉 that generates d∞ (it is the polygonal norm given by the con-
vex hull of the points of Ω), we can compute C(Ω) explicitly. We need to know
the solution to Dido’s isoperimetric problem for ‖·‖ in m1, and as is well known
(see [8]) it is given by polygonal lines from the dual polygon rotated by 90◦. Since
the polygon defining ‖·‖ is made of rational lines (points in Ω have integer coordi-
nates), any vector with rational coordinates has rational ‖·‖-length, and the dual
polygon is also rational. The equations defining z(x, y) will therefore have only
rational coefficients, and z(x, y) will be piecewisely given by a rational quadratic
form in x and y, where the pieces are rational triangles in the (x, y)-plane. The
total volume of C(Ω) will therefore be rational. �
9.2. 5-dim Heisenberg group. The Heisenberg groupH5(Z) is the group gener-
ated by a1, b1, a2, b2,c with relations c = [a1, b1] = [a2, b2], a1 and b1 commute with
a2 and b2 and c is central. Let Ω = {a
i , b
i , i = 1, 2}. Let us describe the limit
shape of Ωn. Again, we see H5(Z) as a lattice of co-volume 1 in the group H5(R)
with Lie algebra h5 spanned by X1, Y1X2, Y2 and Z = [Xi, Yi]. By Theorem 1.4,
the limit shape is the unit ball C5 for the Carnot-Caratheodory metric on H5(R)
induced by the ℓ1-norm ‖x1X1 + y1Y1 + x2X2 + y2Y2‖0 = |x1|+ |y1|+ |x2|+ |y2|.
Since X1, Y1 commute with X2, Y2, in any piecewise linear horizontal path in
H5(R), we can swap the pieces tangent to X1 or Y1 with those tangent to X2 or
Y2 without changing the end point of the path. Therefore if ξ(t) = exp(x1(t)X1 +
y1(t)Y1 + x2(t)X2 + y2(t)Y2 + z(t)Z) is a horizontal path, then z(t) = z1(t) +
z2(t), where zi(t), i = 1, 2, is the “balayage area” of the plane curve {xi(s)Xi +
yi(s)Yi}0≤s≤t.
Since, just like for H3(Z), we know the curve maximizing this area, we can
compute the unit ball C5 explicitly. In exponential coordinates it will take the
form C5 = {exp(x1X1 + y1Y1 + x2X2 + y2Y2 + zZ), |x1|+ |y1|+ |x2|+ |y2| ≤ 1 and
|z| ≤ z(x1, y1, x2, y2)}. Then z(x1, y1, x2, y2) = sup0≤t≤1{zt(x1, y1)+ z1−t(x2, y2)},
where zt(x, y) is the maximum “balayage area” of a path of length t between 0
and xX+yY. It is easy to see that zt(x, y) = t
2z(x/t, y/t) where z is given by (40).
Hence zt is a piecewise quadratic function of t. Again z(x1, y1, x2, y2) is invariant
under changing the signs of the xi,yi’s, and swapping x and y, or else swapping
1 and 2. We may thus assume that the xi,yi’s lie in D = {0 ≤ yi ≤ xi ≤ 1 and
x1+y1+x2+y2 ≤ 1, and x2−y2 ≥ x1−y1}.We may therefore determine explicitly
the supremum z(x1, y1, x2, y2), which after some straightforward calculations takes
ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 55
on D the following form:
z(x1, y1, x2, y2) = 1Amax{d1, d2}+ 1B max{d1, c1}+ 1C max{c1, c2}
where d1 =
(1−x1− y1−x2), c1 =
(1+x1+ y1−x2− y2)
x2y2−x1y1
and d2 and c2 are obtained from d1 and c1 by swapping the indices 1 and 2. The
sets A,B and C form the following partition of D : A = D ∩ {m ≤ x1 − y1},
B = D ∩ {x1 − y1 < m < x2 − y2} and C = D ∩ {x2 − y2 ≤ m}, where m =
(1 − x1 − x2 − y1 − y2)/2.
Since C5 has such an explicit form, it is possible to compute its volume. The fact
that z(x1, y1, x2, y2) is piecewisely given by the maximum of two quadratic forms
makes the computation of the integral somewhat cumbersome but tractable. Our
equations coincide (fortunately!) with those of Stoll (appendix of [33]), where he
computed the main term of the asymptotics of #(Ωn) by a different method. Stoll
did calculate that integral and obtained
#(Ωn)
= vol(C5) =
21870
log(2)
32805
which is transcendental. It is also easy to see by this method that if we change
the generating set to Ω0 = {a
2 }, then we get a rational volume. Hence
the rationality of the growth series of H5(Z) depends on the choice of generating
set, which is Stoll’s theorem.
One advantage of our method is that it can also apply to fancier generating
sets. The case of Heisenberg groups of higher dimension with the standard gen-
erating set is analogous: the function z({xi}, {yi}) is again piecewisely defined as
the maximum of finitely many explicit quadratic forms on a linear partition of the
ℓ1-unit ball
|xi|+ |yi| ≤ 1.
Acknowledgments. I would like to thank Amos Nevo for his hospitality at the
Technion of Haifa in December 2005, where part of this work was conducted, and
for triggering my interest in this problem by showing me the possible implications
of Theorem 1.1 to Ergodic Theory. My thanks are also due to V. Losert for
pointing out an inaccuracy in my first proof of Theorem 1.2 and for his other
remarks on the manuscript. Finally I thank Y. de Cornulier, M. Duchin, E. Le
Donne, Y. Guivarc’h, A. Mohammadi, P. Pansu and R. Tessera for several useful
conversations.
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ASYMPTOTIC SHAPE OF BALLS IN GROUPS WITH POLYNOMIAL GROWTH 57
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growth, Bull. Soc. Math. France, 135(1):47–64, 2007.
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E-mail address: [email protected]
Université Paris-Sud 11, Laboratoire de Mathématiques, 91405 Orsay, France
1. Introduction
2. Quasi-norms and the geometry of nilpotent Lie groups
3. The nilshadow
4. Periodic metrics
5. Reduction to the nilpotent case
6. The nilpotent case
7. Locally compact G and proofs of the main results
8. Coarsely geodesic distances and speed of convergence
9. Appendix: the Heisenberg groups
References
|
0704.0096 | Much ado about 248 | Much ado about 248
M.C. Nucci and P.G.L. Leach∗
Dipartimento di Matematica e Informatica, Università di Perugia,
06123 Perugia, Italy
Abstract
In this note we present three representations of a 248-dimensional Lie
algebra, namely the algebra of Lie point symmetries admitted by a system
of five trivial ordinary differential equations each of order forty-four, that
admitted by a system of seven trivial ordinary differential equations each
of order twenty-eight and that admitted by one trivial ordinary differential
equation of order two hundred and forty-four.
1 Introduction
A system of n ordinary differential equations each of order M > 1,
k = fk(u
j , t), j, k = 1, n, s = 0,M − 1, (1)
has a variable number of Lie point symmetries depending upon the structure of
the functions fk. The maximal dimension D of the algebra of admitted Lie point
symmetries can be obtained by the formulæ [9]
M = 2 =⇒ D = n2 + 4n + 3 (2)
M > 2 =⇒ D = n2 +Mn + 3. (3)
Some explicit numbers are given in Table 1.
Recently the elaboration of the elements of the Lie algebra, E8, of order 248 has
been variously announced [3, 7, 13, 17, 16] in the serious popular media. The au-
thoritative source is the Atlas of Lie Groups and Representations [2] which is funded
by the National Science Foundation through the American Institute of Mathematics
[1]. The results of the E8 computation were announced in a talk at MIT by David
Vogan on Monday, March 19, 2007, and the details may be found at [15]. The Atlas
of Lie Groups and Representations is a project to make available information about
representations of semisimple Lie groups over real and p-adic fields. Of particular
importance is the problem of the unitary dual, ie the classification of all of the ir-
reducible unitary representations of a given Lie group. The goal of the Atlas of Lie
∗permanent address: School of Mathematical Sciences, Westville Campus, University of
KwaZulu-Natal, Durban 4000, Republic of South Africa
http://arxiv.org/abs/0704.0096v1
Table 1: : The maximal dimension of the algebra of admitted Lie point symmetries
for systems of equations of varying order (horizontal) and number (vertical).
M 2 3 4 5 6 7 8 9 10
1 8 7 8 9 10 11 12 13 14
2 15 13 15 17 19 21 23 25 27
3 24 21 24 27 30 33 36 39 42
4 35 31 35 39 43 47 51 55 59
5 48 43 48 53 58 63 68 73 78
6 63 57 63 69 75 81 87 93 99
7 80 73 80 87 94 101 108 115 122
8 99 91 99 107 115 123 131 139 147
9 120 111 120 129 138 147 156 165 174
10 143 133 143 153 163 173 183 193 203
Groups and Representations is to classify the unitary dual of a real Lie group, G,
by computer. A step in this direction is to compute the admissible representations
of G including their Kazhdan-Lusztig-Vogan polynomials. The computation for E8
was an important test of the technology. While the computation is an impressive
achievement, it is only a small step towards the unitary dual and should not be
ranked as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson,
Bernstein et al. (See for example [4, 5, 6, 11, 12, 14, 18, 8].) Nevertheless the result
was regarded as being suitable for a concerted campaign of publicity to heighten
awareness of Mathematics in the community at large:
“Symmetrie ist möglicherweise das erfolgreichste Prinzip der Physik überhaupt” [7].
“Un groupe de chercheurs américains et européens, parmi lesquels on trouve deux
Français, est parvenu à décoder une des structures les plus vastes de l’histoire des
mathématiques” [13].
“It may be that some day this calculation can help physicists to understand the
universe” [17].
“Eighteen mathematicians spent four years and 77 hours of supercomputer compu-
tation to describe this structure” [16].
In this note we demonstrate three representations of a Lie algebra of dimension
248. The two of us spent four hours and 77 seconds of pocket-calculator computation
to describe these three structures.
2 Three simple systems
For D = 248 formula (2) does not have integral solutions and so there is no system
of second-order ordinary differential equations of maximal symmetry possessing a
248-dimensional algebra of its Lie point symmetries1. About formula (3) the factors
of 248-3=245 are 1, 5 and 7 (49 is out of question because 492 > 245). Consequently
1Is this another instance of the intrinsically uniqueness of Classical Mechanics?
possible values of n are 1, 5 and 7. The corresponding values ofM are 244, 44 and 28,
respectively. The systems of maximal symmetry are easily obtained as one simply
puts fk = 0 ∀ k. Thus the systems we construct are the simplest representations of
the equivalence class under point transformation of systems of equations of maximal
symmetry.
Firstly we consider the following system:
k = 0, k = 1, 5. (4)
It is easy to show that this simple system admits a 248-dimensional algebra of its
Lie point symmetries since 52 + 5 · 44 + 3 = 248. The algebra is generated by the
operators
Γ1 = t
2∂t + 43t
i=1 ui∂ui ,
Γ2 = t∂t,
Γ3 = ∂t,
Γi,k = uk∂ui , k = 1, 5, i = 1, 5
Γi+5,s = t
s∂ui, s = 0, 43, i = 1, 5.
Secondly we consider the system
u(28)r = 0, r = 1, 7. (6)
This equally simple system admits a 248-dimensional algebra (72+7 · 28+ 3 = 248)
of its Lie point symmetries generated by
Γ1 = t
2∂t + 27t
j=1 uj∂uj ,
Γ2 = t∂t,
Γ3 = ∂t,
Γj,r = ur∂uj , r = 1, 7, j = 1, 7
Γj+7,n = t
n∂uj , n = 0, 27, j = 1, 7.
Thirdly and finally the scalar equation,
u(244) = 0, (8)
admits a 248-dimensional Lie algebra (12+1 ·244+3 = 248) of its point symmetries
generated by the operators
Γ1 = t
2∂t + 243tu∂u,
Γ2 = t∂t,
Γ3 = ∂t,
Γ4 = u∂u,
Γn+5 = t
n∂u, n = 0, 243.
3 Conclusion
We have demonstrated three representations of Lie algebras of dimension 248 which
is the dimension of E8. Although the algebras we present are not simple, their
method of construction is. The reason for this simplicity is that we used represen-
tations for systems of equations of maximal symmetry. We do not deny that larger
systems, be that in order or number, of less than maximal symmetry could possibly
have an algebra of dimension 248, but even on the assumption that such systems
be linear the complexity of the calculation becomes immense [10] and defeats the
purpose of the present note.
Note that we have used the simplest forms for the generators of the algebras of
the three systems, (4), (6) and (8), for our primary interest is the demonstration of
the existence of the algebras. Normally one would use combinations which reflect
subalgebraic structures. For example in the case of (8) for which the algebra is
obviously sl(250, IR) one would replace Γ2 with Γ̃2 = 2t∂t +243u∂u to underline the
subalgebraic structure {sl(2, IR)⊕ A1} ⊕s 244A1, where Γ1, Γ̃2 and Γ3 constitute a
representation of sl(2, IR), Γ4 reflects the homogeneity of the equation in the depen-
dent variable and the 244-element abelian subalgebra is composed of the solution
symmetries, so called because the coefficient functions are solutions of (8).
Acknowledgements
PGLL thanks the University of Kwazulu-Natal for its continued support.
References
[1] American Institute of Mathematics.
http://aimath.org/E8/
[2] Atlas of Lie Groups and Representations.
http://www.liegroups.org/
[3] BBC Monday, 19 March 2007, 12:28 GMT.
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
[4] Beilinson A (1983) Localization of representations of reductive Lie algebras
Proceedings of the International Congress of Mathematicians, Warsaw 699-710
[5] Beilinson A & Bernstein J (1981) Localisation de g-modules Comptes Rendus
de l’Académie des Sciences de Paris Séries I Mathématiques 292 15-18
[6] Bernstein J (1986) On the Kazhdan-Lusztig conjectures AMS Summer Research
Conference (University of California, Santa Cruz, July 1986)
[7] Der Spiegel, 19 März 2007.
http://www.spiegel.de/wissenschaft/mensch/0,1518,472569,00.html
http://aimath.org/E8/
http://www.liegroups.org/
http://news.bbc.co.uk/2/hi/science/nature/6466129.stm
http://www.spiegel.de/wissenschaft/mensch/0
[8] Gelfand S & MacPherson R (1982) Verma modules and Schubert cells: a dic-
tionary in Seminaire d’algebre Paul Dubriel et MP Malliavin (Lecture Notes in
Mathematics 925, Springer Verlag, Berlin–New York) 150
[9] González-Gascón F & González-López A (1983) Symmetries of differential equa-
tions IV Journal of Mathematical Physics 24 2006-2021
[10] Gorringe VM & Leach PGL (1988) Lie point symmetries for systems of second
order linear ordinary differential equations Quæstiones Mathematicæ 11 95-117
[11] Kazhdan D & Lusztig G (1979) Representations of Coxeter groups and Hecke
algebras Inventiones Mathematicæ 53 165184
[12] Kazhdan D & Lusztig G (1980) Schubert varieties and Poincaré duality in
Geometry of the Laplace Operator, (Proceedings of Symposium on Pure Math-
ematics 36, American Mathematical Society) 185203
[13] LEMONDE.FR avec AFP 19.03.07
http://www.lemonde.fr/web/article/0,1-0@2-3244,36-884723@51-
884724,0.html
[14] Lusztig G & Vogan D (1983) Singularities of closures of K-orbits on flag mani-
fold Inventiones Mathematicæ 71 365370
[15] http://www.liegroups.org/AIME8/technicaldetails.html
[16] NEW YORK TIMES 2007/03/20.
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
[17] The Times March 19, 2007.
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
[18] Vogan D (1983) Irreducible characters of semisimple Lie groups III: Proof of
the Kazhdan-Lusztig conjecture in the integral case Inventiones Mathematicæ
71 381417
http://www.lemonde.fr/web/article/0
http://www.liegroups.org/AIM$_$E8/technicaldetails.html
http://select.nytimes.com/gst/abstract.html?res=F40613FE3C540C738EDDAA0894DF404482
http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece
Introduction
Three simple systems
Conclusion
|
0704.0097 | Conformal Field Theory and Operator Algebras | Conformal Field Theory
and Operator Algebras
Yasuyuki Kawahigashi
Department of Mathematical Sciences
University of Tokyo, Komaba, Tokyo, 153-8914, Japan
e-mail: [email protected]
Abstract
We review recent progress in operator algebraic approach to con-
formal quantum field theory. Our emphasis is on use of representation
theory in classification theory. This is based on a series of joint works
with R. Longo.
1 Introduction
A mathematically rigorous approach to quantum field theory based on op-
erator algebras is called an algebraic quantum field theory. It has a long
history since pioneering works of Araki, Haag, Kastker. (See [22] for a gen-
eral treatment of algebraic quantum field theory.) This theory works on
Minkowski spaces on any spacetime dimension, and there have been some
recent results on curved spacetimes or even noncommutative spacetimes. In
the case of 1+1-dimensional Minkowski space with higher spacetime symme-
try, conformal symmetry, we have conformal field theory and there we have
seen many new developments in the recent years, so we survey such results
here. Our emphasis is on representation theoretic aspects of the theory and
we make various comparison with another mathematically rigorous and more
recent approach to conformal field theory, that is, theory of vertex operator
algebras.
∗Supported in part by JSPS.
http://arxiv.org/abs/0704.0097v1
Roughly speaking, a mathematical study of quantum field theory is a
study of Wightman fields, which are certain type of operator-valued distri-
butions on a spacetime with covariance with respect to a given spacetime
symmetry group. We have mathematically rigorous axioms for such Wight-
man fields, but they involve distributions and unbounded operators, so these
cause various kinds of technical difficulty. In contrast, in the algebraic quan-
tum field theory, our fundamental object is a net of von Neumann algebras
of bounded linear operators on a Hilbert space. (See [46] for general the-
ory of von Neumann algebras.) Technical problems on definition domains of
unbounded operators do not arise in this approach.
A basic idea is as follows. Suppose we have a Wightman field Φ on a
spacetime. Fix a bounded region O in the space time and consider a test
function ϕ with support contained in O. Then the pairing 〈Φ, ϕ〉 produces
an (unbounded) operator. We have many Φ and ϕ for a fixed O and obtain
many unbounded operators from such pairing. Then we consider a von Neu-
mann algebra of bounded linear operators on this Hilbert space generated
by these unbounded operators. (For example, if we have a self-adjoint un-
bounded operators, we consider its spectral projections which are obviously
all bounded. In this way, we deal with only bounded operators.) This is re-
garded as a von Neumann algebra generated by observables in the spacetime
region O. A von Neumann algebra is an algebra of bounded linear operators
which is closed under the adjoint operation and the strong operator topol-
ogy. In this way, we have a family {A(O)} of von Neumann algebras on the
same Hilbert space parameterized by spacetime regions. Since the spacetime
regions make a net with respect to the inclusion order, we call such a family a
net of von Neumann algebras. Now we forget Wightman fields and consider
only a net of von Neumann algebras. We have some expected properties for
such nets of von Neumann algebras from a physical consideration, and now
we use these properties as axioms. So our mathematical object is a net of
von Neumann algebras subject to certain set of axioms. Our mathematical
aim is to study such nets of von Neumann algebras.
2 Conformal Quantum Field Theory
We first explain formulation of full conformal quantum field theory on the
1 + 1-dimensional Minkowski space in algebraic quantum field theory. As a
spacetime region O above, it is enough to consider only open rectangles O
with edges parallel to t = ±x in (1 + 1)-dim Minkowski space. In this way,
we get a family {A(O)} of operator algebras parameterized by spacetime
regions O (rectangles). In order to realize conformal symmetry, we have to
make a partial compactification of the 1+1-dimensional Minkowski space. If
two rectangles are spacelike separated, then we have no interactions between
them even at the speed of light, so our axiom requires that the corresponding
two von Neumann algebras commute with each other. This is the locality
axiom. Since this is not our main object in this paper, we omit details of the
other axioms. See [29] for full details.
Next we briefly explain that boundary conformal field theory can be han-
dled within the same framework. Now we consider the half-space {(x, t) |
x > 0} in the 1+1-dimensional Minkowski space and only rectangles O con-
tained in this half-space. In this way, we have a similar net of von Neumann
algebras {A(O)} parameterized with rectangles in the half-space. See [38]
for full details of the axioms.
If we have a net of von Neumann algebras over the 1 + 1-dimensional
Minkowski space, we can restrict the net of von Neumann algebras to two
chiral conformal field theories on the light cones {x = ±t}. In this way, we
have two nets of von Neumann algebras on the compactified S1 as description
of two chiral conformal field theories. Since this net is our main mathematical
object in this article, we give a full set of axioms. (See [29] for details of this
“restriction” procedure.)
Now our “spacetime” is S1 and a “spacetime region” is an interval I,
which means a non-empty, non-dense open connected subset of S1. We have
a family {A(I)} of von Neumann algebras on a fixed Hilbert space H . These
von Neumann algebras are simple and such von Neumann algebras are called
factors, so the family {A(I)} satisfying the axioms below is called a net of
factors (or an irreducible local conformal net of factors, strictly speaking).
Actually, the set of intervals on S1 is not directed with respect to inclusions,
so the terminology net is not mathematically appropriate, but is widely used.
1. (isotony) For intervals I1 ⊂ I2, we have A(I1) ⊂ A(I2).
2. (locality) For intervals I1, I2 with I1∩I2 = ∅, we have [A(I1),A(I2)] = 0
3. (Möbius covariance) There exists a strongly continuous unitary repre-
sentation U of PSL(2,R) on H satisfying U(g)A(I)U(g)∗ = A(gI) for
any g ∈ PSL(2,R) and any interval I.
4. (positivity of energy) The generator of the one-parameter rotation sub-
group of U , called the conformal Hamiltonian, is positive.
5. (existence of the vacuum) There exists a unit U -invariant vector Ω in
H , called the vacuum vector, and the von Neumann algebra
I∈S1 A(I)
generated by all A(I)’s is B(H).
6. (conformal covariance) There exists a projective unitary representation
U of Diff(S1) on H extending the unitary representation of PSL(2,R)
such that for all intervals I, we have
U(g)A(I)U(g)∗ = A(gI), g ∈ Diff(S1),
U(g)AU(g)∗ = A, A ∈ A(I), g ∈ Diff(I ′),
where Diff(S1) is the group of orientation-preserving diffeomorphisms
of S1 and Diff(I ′) is the group of diffeomorphisms g of S1 with g(t) = t
for all t ∈ I.
The isotony axiom is natural because we have more test functions (or more
observables) for a larger interval. The locality axiom takes this simple form on
S1. The choice of the spacetime symmetry is not unique, and we can use the
Poincaré symmetry on the Minkowski space or the Möbius covariance on S1,
for example, but in the conformal field theory, we use conformal symmetry,
which means diffeomorphism covariance as above. This set of axioms imply
various nice conditions such as the Reeh-Schlieder property, the Bisognano-
Wichmann property and the Haag duality. See [28] and references there for
details.
In the usual situation, all the von Neumann algebras A(I) are isomorphic
to the so-called Araki-Woods type III1 factor for all nets A and all intervals
I. So each von Neumann algebra does not contain any information about
the conformal field theory, but it is the relative position of the von Neumann
algebras in the family that encodes the physical information of the theory.
(It is similar to subfactor theory of Jones where we study a relative position
of one factor in another.)
At the end of this section, we compare our formulation of conformal
quantum field theory with another mathematically rigorous approach, the-
ory of vertex operator algebras. A vertex operator algebra is an algebraic
axiomatization of Wightman fields on S1, called vertex operators. If we
have an operator valued distribution on S1, its Fourier expansion should give
countably many (possibly unbounded) operators as the Fourier coefficients.
Under the so-called state-field correspondence, any vector in the space of
“states” should give an operator-valued distribution, a quantum “field”, and
its Fourier expansion gives countably many operators. In this way, one vector
should give countably many operators on the space of these vectors. In other
words, for two vectors v, w we have countably many binary operations v(n)w,
n ∈ Z, the action of the n-th operator given by v on w. An axiomatization
of this idea gives a notion of vertex operator algebra. (See [16] for a precise
definition. There is a slightly weaker notion of a vertex algebra. See [27]
for its precise definition and related results.) In theory of vertex operator
algebra, one considers a vector space of states without an inner product and
even when we have a positive definite inner product, one considers this vec-
tor space without completion. Here in comparison to nets of factors, we are
interested in the case where we have positive definite inner products on the
spaces of states. We say that such a vertex operator algebra is unitary.
Both of one (unitary) vertex operator algebra and one net of factors
should describe one chiral conformal field theory. So unitary vertex operator
algebras and nets of factors should be in a bijective correspondence, at least
under some “nice” additional conditions, but no general theorems have been
known for such a correspondence, though there is a recent progress due to S.
Carpi and M. Weiner. However, if we have one construction or an idea on one
side, we can often “translate” it to the other side, though it can be highly non-
trivial from a technical viewpoint. Fundamental sources of constructions for
vertex operator algebras are affine Kac-Moody algebras and integral lattices.
The corresponding constructions for nets of factors have been done by A.
Wassermann [47] and his students, and Dong-Xu [12], respectively, after the
initial construction of Buchholz-Mack-Todorov [5]. If we have examples with
some nice properties, we canoften construct new examples from them, and
as such methods of constructions of vertex operator algebras, we have simple
current extensions, the coset construction, and the orbifold construction. The
simple current extensions for nets of factors are simply crossed products by
DHR-automorphisms and easy to realize. (See the next section for a notion
of DHR-endomorphisms.) The coset and orbifold constructions for nets of
factors have been studied in detail by F. Xu [50, 51, 52].
For nets of factors, we have introduced a new construction of examples
in [28] based on Longo’s notion of Q-systems [36]. Further examples have
been constructed by Xu [55] with this method. This can be translated to the
setting of vertex operator algebras, as we will see in this article later.
3 Representation Theory
An important tool to study nets of factors is a representation theory. For a
net of factors {A(I)}, all the algebras A(I) act on the initial Hilbert space
H from the beginning, but we also consider their representations on another
Hilbert space, that is, a family {πI} of representations πI : A(I) → B(K),
where K is another Hilbert space, common for all I. For I1 ⊂ I2, we must
have that the restriction of πI2 on A(I1) is equal to πI1 . The representation on
the initial Hilbert space is called the vacuum representation and plays a role of
a trivial representation. We also have to take care of the spacetime symmetry
group when we consider a representation, but this part is often automatic
(see [20]), so we now ignore it for simplicity. See [20] for a more detailed
treatment. Note that a representation of a net of factors is a counterpart of
a module over a vertex operator algebra.
Notions of irreducibility and a direct sum for such representations are
easy to formulate. Non-trivial notions are dimensions and tensor products.
Each representation {πI} is in a bijective correspondence to a certain endo-
morphism λ of an infinite dimensional operator algebra, called a Doplicher-
Haag-Roberts (DHR) endomorphism [13, 15], and we can restrict λ to a single
factor A(I) for an arbitrarily but fixed interval I. Then λ(A(I)) ⊂ A(I) is
a subfactor and we have its Jones index [26]. (See [14, 41, 43] for general
theory of subfactors.) The square root of this Jones index plays the role of
the dimension of the representation [35]. In algebraic quantum field theory,
such a dimension was called a statistical dimension, and it is analogous to
a quantum dimension in the theory of quantum groups. It is a positive real
numbers in the interval [1,∞]. We can also compose endomorphisms and
this composition gives the correct notion of tensor products. We then get a
braided tensor category as in [15].
In representation theory of a vertex operator algebra (and also a quantum
group), it sometimes happens that we have only finitely many irreducible rep-
resentations. Such finiteness is often called rationality, possibly with some
extra assumptions on some finite dimensionality. This also plays an impor-
tant role in theory of quantum invariants in low dimensional topology. In [32],
we have introduced an operator algebraic condition for such rationality for
nets of factors as follows and we called it complete rationality. We split the
circle into four intervals I1, I2, I3, I4 in this order, say, counterclockwise. Then
complete rationality is given by the finiteness of the Jones index for a subfac-
tor A(I1)∨A(I3) ⊂ (A(I2)∨A(I4))
′ where ′ means the commutant, together
with the split property. The split property is known to hold if the vacuum
character,
n=0(dimHn)q
n, is convergent for |q| < 1 by [9], so it usually
holds and is easy to verify. (Here H =
n=0Hn is the eigenspace decompo-
sition of the original Hilbert space for the positive generator of the rotation
group. So this convergence property can be verified simply by looking at
the Hilbert space, not the von Neumann algebras.) In the original definition
of complete rationality in [32], we required another condition called strong
additivity, but it was proved to be redundant by Longo-Xu [39]. We have
proved in [32] that this complete rationality implies that we have a modular
tensor category as a representation category of {A(I)}. A modular tensor
category produces a 3-dimensional topological quantum field theory. (See [45]
for general theory of topological quantum field theory.) The SU(N)k-net of
Wassermann has been shown to be completely rational by [49].
We now introduce an important notion of α-induction. For an inclusion
of nets of factors, A(I) ⊂ B(I), we have an induction procedure analogous
to the group representation. So from a representation of the smaller net A,
we would like to construct a representation of the larger net B, but what
we actually obtain is not a genuine representation of the larger net B in
general, and is something weaker called solitonic. This induction procedure
is called the α-induction and depends a choice of braiding, so we write α+
and α−. This was first defined in Longo-Rehren [37] and studied in detail
in Xu [48]. Then Böckenhauer-Evans [1] made a further study, and [2, 3]
unified this study with Ocneanu’s graphical method [42]. The intersection
of the irreducible endomorphisms appearing in the images of α+-induction
and α−-induction gives the true representation category of {B(I)} if A is
completely rational by [2, 32].
This α-induction opens an important and new connection with theory
of modular invariants. A modular tensor category produces a unitary rep-
resentation π of SL(2,Z) through its braiding as in [44], and its dimension
is the number of irreducible objects. So a completely rational net of fac-
tors produces such a unitary representation. (Note that our representation
of SL(2,Z) comes from the braiding structure, not from the action of this
group on the characters through change of variables τ 7→
aτ + b
cτ + d
, though in
all the “nice” known examples, these two representations coincide. See [30]
for a discussion on this matter.)
It has been proved in [2] that the matrix (Zλ,µ) defined by
Zλ,µ = dimHom(α
λ , α
is in the commutant of the representation π, using Ocneanu’s graphical cal-
culus [42]. Such a matrix Z is called a modular invariant, and we have only
finitely many such Z for a given π. For any completely rational net {A(I)},
any extension {B(I) ⊃ A(I)} produces such Z. Matrices Z are certainly
much easier to classify than extensions and this is a source of classification
theory in the next section.
4 Classification Theory
For a net of factors, we can naturally define a central charge and it is well-
known to take discrete values 1− 6/m(m+ 1), m = 3, 4, 5, . . . , below 1 and
all values in [1,∞) by [17, 18]. We have the Virasoro net {Virc(I)} for each
such c and it is the operator algebraic counterpart of the Virasoro vertex
operator algebra with the same c. Any net of factors {A(I)} with central
charge c is an extension of the Virasoro net with the same central charge and
it is automatically completely rational if c < 1, as shown in [28]. So we can
apply the above theory and we get the following complete classification list
for the case c < 1 as in [28].
1. The Virasoro nets {Virc(I)} with c < 1.
2. The simple current extensions of the Virasoro nets with index 2.
3. Four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
The unitary representations of SL(2,Z) for the Virasoro nets are the well-
known ones, and all the modular invariants for these have been classified by
[6]. Our result shows that each of the so-called type I modular invariants in
the classification list of [6] corresponds to a net of factors uniquely. They
are labeled with pairs of A-D2n-E6,8 Dynkin diagrams with Coxeter numbers
differing by 1. Three in (3) of the above list have been identified with coset
models, but the remaining one does not seem to be related to any other
known constructions. This is constructed with “extension by Q-system”.
Xu [55] recently applied this construction to many other coset models and
obtained infinitely many new examples based on [54], called mirror exten-
sions. Classification for the case c = 1 has been also done under some extra
assumption [7, 53].
This classification theorem also implies a classification of certain types of
vertex operator algebras as follows.
Let V be a (rational) vertex operator algebra and Wi be its irreducible
modules. We would like to classify all vertex operator algebras arising from
putting a vertex operator algebra structure on
i niWi and using the same
Virasoro element as V , where ni is multiplicity and W0 = V , n0 = 1. From
a viewpoint of tensor category, this classification problem of extensions of
a vertex operator algebras is the “same” as the classification problem of
extensions of a completely rational net of factors, as shown in [24].
So the above classification theorem of local conformal nets implies a clas-
sification theorem of extensions of the Virasoro vertex operator algebras with
c < 1 as above, and we obtain the same classification list. That is, besides
the Virasoro vertex operator algebras themselves, we have their simple cur-
rent extensions, and four exceptionals at c = 21/22, 25/26, 144/145, 154/155.
With the usual notation of L(c, h) for a module with central charge c and
conformal weight h of the Virasoro vertex operator algebras with c < 1, the
four exceptionals are listed as follows.
1. L(21/22, 0)⊕L(21/22, 8). It has 15 irreducible representations and has
two coset realizations, from SU(9)2 ⊂ (E8)2 and (E8)3 ⊂ (E8)2⊗(E8)1.
2. L(25/26, 0) ⊕ L(25/26, 10). It has 18 irreducible representations and
has a coset realization from SU(2)11 ⊂ SO(5)1 ⊗ SU(2)1.
3. L(144/145, 0)⊕L(144/145, 24)⊕L(144/145, 78)⊕ L(144/145, 189). It
has 28 irreducible representations and no coset realization has been
known.
4. L(154/155, 0)⊕L(154/155, 26)⊕L(154/155, 84)⊕ L(154/155, 203). It
has 30 irreducible representations and has a coset realization from
SU(2)29 ⊂ (G2)1 ⊗ SU(2)1.
Note that it is not obvious that the representation category of the Virasoro
net Virc and the representation category of the Virasoro vertex operator
algebra L(c, 0) are isomorphic, but as long as the two are braided tensor
category and have the same S- and T -matrices, the arguments in [28] work,
so we obtain the above classification result for vertex operator algebras.
Using the above results and more techniques, we can also completely
classify full conformal field theories within the framework algebraic quantum
field theory for the case c < 1. Full conformal field theories are given as
certain nets of factors on 1 + 1-dimensional Minkowski space. Under natu-
ral symmetry and maximality conditions, those with c < 1 are completely
labeled with the pairs of A-D-E Dynkin diagrams with the difference of
their Coxeter numbers equal to 1, as shown in [29]. We now naturally have
D2n+1, E7 as labels, unlike in the chiral case. The main difficulty in this
work lies in proving uniqueness of the structure for each modular invariant
in the Cappelli-Itzykson-Zuber list [6]. This is done through 2-cohomology
vanishing for certain tensor categories. in the spirit of [25].
Furthermore, using the above results and more techniques we can also
completely classify boundary conformal field theories for the case c < 1.
Boundary conformal field theories are given as certain nets of factors on a 1+
1-dimensional Minkowski half-space. Under a natural maximality condition,
these with c < 1 are now completely labeled with the pairs of A-D-E Dynkin
diagrams with distinguished vertices having the difference of their Coxeter
numbers equal to 1, as shown in [33] based on a general theory in [38]. The
“chiral fields” in a boundary conformal field theory should produce a net
of factors on the boundary (which is compactified to S1) as in the operator
algebraic approach. Then a general boundary conformal field theory restricts
to this boundary to produce a non-local extension of this chiral conformal
field theory on the boundary.
5 Moonshine Conjecture
The Moonshine conjecture, formulated by Conway-Norton [8], is about mys-
terious relations between finite simple groups and modular functions, since
an observation due to McKay.
Today the classification of all finite simple groups is complete and the
classification list contains 26 sporadic groups in addition to several infinite
series. The largest group among the 26 sporadic groups is called the Monster
group and its order is about 8× 1053
One the one hand, the non-trivial irreducible representation of the Mon-
ster having the smallest dimension is 196883 dimensional. On the other
hand, the following function, called j-function, has been classically studied
in algebra.
j(τ) = q−1 + 744 + 196884q +
21493760q2 + 864299970q3 + · · ·
For q = exp(2πiτ), Im τ > 0, we have modular invariance property, j(τ) =
aτ + b
cτ + d
∈ SL(2,Z), and this is the only function, up to
the constant term, satisfying this property and starting with q−1,
McKay noticed 196884 = 196883 + 1, and similar simple relations for
other coefficients of the j-function and dimensions of irreducible represen-
tations of the Monster group turned out to be true. Then Conway-Norton
[8] formulated the Moonshine conjecture roughly as follows, which has been
now proved by Borcherds [4] in 1992.
1. We have a “natural” infinite dimensional graded vector space V =
Vn with some algebraic structure having a Monster action pre-
serving the grading and each Vn is finite dimensional.
2. For any element g in the Monster, the power series
n=0(Tr g|Vn)q
is a special function called a Hauptmodul for some discrete subgroup
of SL(2,R). When g is the identity element, the series is the j-function
minus constant term 744.
For the part (1) of this conjecture, Frenkel-Lepowsky-Meurman [16] gave
a precise definition of “some algebraic structure” as a vertex operator algebra
and constructed a particular example V , which is now called the Moonshine
vertex operator algebras and denoted by V ♮.
The construction roughly goes as follows. In dimension 24, we have an
exceptional lattice Λ called the Leech lattice. Then there is a general con-
struction of a vertex operator algebra from a certain lattice, and the one for
the Leech lattice gives something very close to our final object V ♮. Then we
take a fixed point algebra under a natural action of Z/2Z arising from the
lattice symmetry, and then make a simple current extension of order 2. The
resulting vertex operator algebra is the Moonshine vertex operator algebra
V ♮. (The final step is called a twisted orbifold construction). The series
n=0(dimV
n−1 is indeed the j-function minus constant term 744.
Miyamoto [40] has a new realization of V ♮ as an extension of a tensor
power of the Virasoro vertex operator algebra with c = 1/2, L(1/2, 0)⊗48
(based on Dong-Mason-Zhu [11]). This kind of extension of a Virasoro tensor
power is called a framed vertex operator algebra as in [10].
We have given an operator algebraic counterpart of such a construction
in [31].
We realize a Leech lattice net of factors on S1 as an extension of Vir1/2
using certain Z4-code. Then we can perform the twisted orbifold construction
in the operator algebraic sense to obtain a net of factors, the Moonshine net
A♮. Theory of α-induction is used for obtaining various decompositions. We
then get a Miyamoto-type description of this construction, as an operator
algebraic counterpart of the framed vertex operator algebras. We then have
the following properties.
1. c = 24.
2. The representation theory is trivial.
3. The automorphism group is the Monster.
4. The Hauptmodul property (as above).
Outline of the proof of these four properties is as follows.
It is immediate to get c = 24. We can show complete rationality passes
to an extension (and an orbifold) in general with control over the size of the
representation category, using the Jones index. With this, we obtain (2) very
easily. Such a net is called holomorphic. Property (3) is the most difficult
part. For the Virasoro VOA L(1/2, 0), the vertex operator is indeed a well-
behaved Wightman field and smeared fields produce the Virasoro net Vir1/2.
Using this property and the fact that
g g(L(1/2, 0)
⊗48) for all g ∈ Aut(V ♮)
generate the entire Moonshine VOA V ♮, we can prove that the automorphism
group as a vertex operator algebra and the automorphism group as a net
of factors are indeed the same. Then (4) is now a trivial corollary of the
Borcherds theorem [4].
We note that the Baby Monster, the second largest among the 26 sporadic
finite simple groups, can be treated similarly with Höhn’s construction of the
shorter Moonshine super vertex operator algebra.
Still, these examples are treated with various tricks case by case. We
expect a bijective correspondence between vertex operator algebras and nets
of factors on S1 under some nice conditions. On the side of vertex operator
algebras, the most natural candidate for such a “nice” condition is the C2-
finiteness condition of Zhu [56] (with unitarity). On the operator algebraic
side, our complete rationality in [32] seems to be such a “nice” condition, but
the actual relations between the two notions are not clear at this moment.
The essential condition for complete rationality is the finiteness of the Jones
index arising from four intervals on the circle, and this finiteness somehow
has formal similarity to the finiteness appearing in the definition of the C2-
finiteness.
At the end, we list some open problems. The operator algebraic approach
has an advantage in control of representation theory, but is behind of theory
of vertex operator algebras in the theory of characters.
For a net of factors, we can naturally define a notion of a character for
each representation. But even convergence of these characters have not been
proved in general, and the modular invariance property, the counterpart of
Zhu’s result [56], is unknown, though we certainly expect it to be true. We
also expect the Verlinde identity holds, which has been proved in the context
of vertex operator algebras recently by Huang [23]. We would need an S-
matrix version of the spin-statistics theorem [21] for nets of factors.
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Introduction
Conformal Quantum Field Theory
Representation Theory
Classification Theory
Moonshine Conjecture
|
0704.0098 | Sparsely-spread CDMA - a statistical mechanics based analysis | Sparsely-spread CDMA - a statistical mechanics
based analysis
Jack Raymond and David Saad
Neural Computation Research Group, Aston University, Aston Triangle, Birmingham,
B4 7EJ
E-mail: [email protected]
Abstract.
Sparse Code Division Multiple Access (CDMA), a variation on the standard CDMA
method in which the spreading (signature) matrix contains only a relatively small number
of non-zero elements, is presented and analysed using methods of statistical physics. The
analysis provides results on the performance of maximum likelihood decoding for sparse
spreading codes in the large system limit. We present results for both cases of regular
and irregular spreading matrices for the binary additive white Gaussian noise channel
(BIAWGN) with a comparison to the canonical (dense) random spreading code.
PACS numbers: 64.60.Cn, 75.10.Nr, 84.40.Ua, 89.70.+c
AMS classification scheme numbers: 68P30,82B44,94A12,94A14
http://arxiv.org/abs/0704.0098v5
Sparsely-spread CDMA - a statistical mechanics based analysis 2
1. Background
The area of multiuser communications is one of great interest from both theoretical and
engineering perspectives [1]. Code Division Multiple Access (CDMA) is a particular
method for allowing multiple users to access channel resources in an efficient and robust
manner, and plays an important role in the current preferred standards for allocating
channel resources in wireless communications. CDMA utilises channel resources highly
efficiently by allowing many users to transmit on much of the bandwidth simultaneously,
each transmission being encoded with a user specific signature code. Disentangling the
information in the channel is possible by using the properties of these codes and much of
the focus in CDMA research is on developing efficient codes and decoding methods.
In this paper we study a variant of the original method, sparse CDMA, where the
spreading matrix contains only a relatively small number of non-zero elements as was
originally studied and motivated in [2]. While the straightforward application of sparse
CDMA techniques to uplink multiple access communication is rather limited, as it is
difficult to synchronise the sparse transmissions from the various users, the method can be
highly useful for frequency and time hopping. In frequency-hopping code division multiple
access (FH-CDMA), one repeatedly switches frequencies during radio transmission, often
to minimize the effectiveness of interception or jamming of telecommunications. At
any given time step, each user occupies a small (finite) number of the (infinite) M-ary
frequency-shift-keying (MFSK) chip/carrier pairs (with gain G, the total number of chip-
frequency pairs is MG.) Hops between available frequencies can be either random or
preplanned and take place after the transmission of data on a narrow frequency band. In
time-hopping (TH-)CDMA, a pseudo-noise sequence defines the transmission moment for
the various users, which can be viewed as sparse CDMA when used in an ultra-wideband
impulse communication system. In this case the sparse time-hopping sequences reduces
collisions between transmissions.
This study follows the seminal paper of Tanaka [3], and other recent extensions [4],
in utilising the replica analysis for randomly spread CDMA with discrete inputs, which
established many of the properties of random densely-spread CDMA with respect to
several different detectors including Maximum A Posteriori (MAP), Marginal Posterior
Maximiser (MPM) and minimum mean square-error (MMSE). Sparsely-spread CDMA
differs from the conventional CDMA, based on dense spreading sequences, in that any
user only transmits to a small number of chips (by comparison to transmission on all chips
in the case of dense CDMA). The sparse nature of this model facilitates the use of methods
from statistical physics of dilute disordered systems [5, 6] for studying the properties of
typical cases.
The feasibility of sparse CDMA for transmitting information was recently
demonstrated [2] for the case of real (Gaussian distributed) input symbols by employing a
Gaussian effective medium approximation; several results have been reported for the case of
random transmission patterns. In a separate recent study, based on the belief propagation
inference algorithm and a binary input prior distribution, sparse CDMA has also been
Sparsely-spread CDMA - a statistical mechanics based analysis 3
considered as a route to proving results in the densely spread CDMA [7]. In addition, this
study demonstrated the existence of a waterfall phenomenon comparable to the dense code
for a subset of ensembles. The waterfall phenomenon is observed in decoding techniques,
where there is a dynamical transition between two statistically distinct solutions as the
noise parameter is varied. Finally we note a number of pertinent studies concerning the
effectiveness of belief propagation as an MPM decoding method [8, 9, 10, 11], and in
combining sparse encoding (LDPC) methods with CDMA [12]. Many of these papers
however consider the extreme dilution regime – in which the number of chip contributions
is large but not O(N).
The theoretical work regarding sparsely spread CDMA remained lacking in certain
respects. As pointed out in [2], spreading codes with Poisson distributed number of non-
zero elements, per chip and across users, are systematically failing in that each user
has some probability of not contributing to any chips (transmitting no information).
Even in the “partly regular” code [7] ensemble (where each user transmits on the same
number of chips) some chips have no contributors owing to the Poisson distribution in
chip connectivity, consequently the bandwidth is not effectively utilised. We circumvent
this problem by introducing constraints to prevent this, namely taking regular signature
codes constrained such that both the number of users per chip and chips per user take
fixed integer values. Furthermore we present analytic and numerical analysis without
resort to Gaussian approximations of any quantities. Using new tools from statistical
mechanics we are able to cast greater light on the nature of the binary prior transmission
process. Notably the nature of the decoding state space and relative performance of sparse
ensembles versus dense ones across a range of noise levels; and importantly, the question
of how the coexistence of solutions found by Tanaka [3] extends to sparse ensembles,
especially close to the transition points determined for the dense ensemble.
In this paper we demonstrate the superiority of regular sparsely spread CDMA code
over densely spread codes in certain respects, for example, the anticipated bit error rate
arising in decoding is improved in the high noise regime and the solution coexistence
behaviour is less pervasive. Furthermore, to utilise belief propagation for such an ensemble
is certain to be significantly faster and less computationally demanding [13], this also has
power-consumption implications which may be important in some applications. Other
practical issues of implementation, the most basic being non-synchronisation and power
control, require detailed study and may make fully harnessing these advantages more
complex and application dependent.
The paper is organised as follows: In section 2 we will introduce the general framework
and notation used, while the methodology used for the various codes will be presented in
section 3. The main results for the various codes will be presented in section 4 followed
by concluding remarks in section 5.
Sparsely-spread CDMA - a statistical mechanics based analysis 4
i b ξ j d
1τ iτ
jτ kτ lτ K
yb yc yd yN
Figure 1. A bi-partite graph is useful for visually realising a problem. A user node i at
the bottom interacts with other variables through its set of neighbouring factor nodes (∂i)
to which it connects. The factor nodes are determined through a similar neighborhood.
The interaction at each factor (µ) is conditioned on neighbouring gain factors ξµ (the
non-zero components of s), and yµ (which is an implicit function of the noise ωµ, and
neighbouring input bits bµ and gain factors ξµ), assuming a uniform prior on the bits.
The statistical mechanics reconstruction problem associates dynamical variables τ to the
user nodes that interact through the factors. The thermodynamical equilibrium state of
this system then describes the theoretical performance of optimal detectors.
2. The model
We consider a standard model of CDMA consisting of K users transmitting in a bit
interval of N chips. We assume a model with perfect power control and synchronisation,
and consider only the single bit interval. In our case the received signal y is described by
[skbk] + ω , (1)
where the vector components describe the values for distinct chips: sk is the spreading
code for user k, bk = ±1 is the bit sent by user k (binary input symbols) and ω the noise
vector. Appropriate normalisation of the power is through the definition of the signature
matrix (s). It is possible to include a user or chip specific amplitude variation, which may
be due to fading or imperfect power control. We consider a model without these effects.
The spreading codes are sparse so that in expectation only C of the elements in vector sk
are non-zero. If, with knowledge of the signature matrix in use, we assume the signal has
been subject to additive white Gaussian channel noise of variance σ20/β, where σ
0 is the
variance of the true channel noise 〈ω2〉, we can write the posterior for the transmitted bits
τ (unknowns given the particular instance) using Bayes Theorem
P (τ |y) =
[sµk(bk − τk)] + ωµ
P (τ ) , (2)
and from this define bit error rate, mutual information, and other quantities. The
statistical mechanics approach from here is to define a Hamiltonian and partition
Sparsely-spread CDMA - a statistical mechanics based analysis 5
function from which the various statistics relating to this probability distribution may
be determined - and hence all the usual information theory measures. A suitable choice
for the Hamiltonian is
H(τ ) =
[sµk(bk − τk)] + ωµ
hkτk . (3)
We can here identify τ as the dynamical variables in the inference problem (dependence
shown explicitly). The other quenched variables (parameters), describing the instance of
the disorder, are the signature matrix (s), noise (ω) and the inputs (b). The variables
hk describe our prior beliefs about the inputs (the specific user bias), and we can assume
some simple distribution for this such as all users having the same bias hk = H . Maximal
rate transmission corresponds to unbiased bits H = 0, and this is considered throughout
the paper. The properties of such a system may be reflected in a factor (Tanner) graph,
a bipartite graph in which users and chips are represented by nodes (see figure 1).
The calculation we undertake is specific to the case of the thermodynamic limit in
which the number of chips N → ∞ whilst the load α = K/N is fixed. Note that α is
termed β in many CDMA papers, here we reserve β to mean the “inverse temperature”
in a statistical mechanics sense (which defines our prior belief for the noise level and give
rise to the corresponding MAP detector.)
In all ensembles we may identify the parameter L as the mean number of contributions
to each chip, and C as the mean number of contributions per user. As such the following
also holds
. (4)
The case in which α is greater than 1 will be called oversaturated, since more than one
bit is being transmitted per chip.
The calculations presented henceforth are specific to the case of memoryless noise,
drawn from a single distribution of mean zero and mean square σ20
Ω(ω) = P (ωµ = ω) . (5)
Defining normalised spreading codes such that
sk.sk = N , we can identify the “power
spectral density” (PSD) over a chip interval as a measure of the system noise 1/(2σ20) –
the factor two being connected with physical considerations in implementing the model.
2.1. Code Ensembles
We consider several code ensembles we call irregular, partly regular and regular, which
differ in the constraints placed on the factor and variable degree constraints of the signature
matrix s. The probability distribution
P (s) = N
δ(sµk 6= 0)− L̃)
P (L̃)
Sparsely-spread CDMA - a statistical mechanics based analysis 6
δ(sµk 6= 0)− C̃)
P (C̃)
P (sµk) , (6)
where N is a normalising constant, P (L̃) is the factor degree probability distribution of
mean L, P (C̃) is the variable degree probability distribution of mean C, and P (sµk) is the
marginal probability distribution which is common to all ensembles
P (sµk) =
δ(sµk) +
δ(sµk − ξ) . (7)
The form of (6) is then sufficient for the sparse distributions we consider in the large system
limit, and makes explicit the chip and user connectivity properties of the ensembles. The
gain factor ξ, is drawn randomly from a single distribution with zero measure at ξ = 0,
and finite moments, in any instance of a code
φ(ξ) = P (sµk = ξ|sµk 6= 0) . (8)
Unlike the dense case the details of this distribution will effect results, but only in a small
way for reasonable choices [2]. We here investigate the case of Binary Phase Shift Keying
(BPSK) which corresponds to a uniform distribution on {− 1√
}, though the analytic
results presented are applicable to any distribution of mean square = 1/L. Note that
disorder in the gain factors is not a necessity, the case ξ = 1/
L also allows decoding in
sparse ensembles.
The case where P (L̃) and P (C̃) are Poissonian distributed identifies the irregular
ensemble - where the connections between chips and users are independently distributed.
The second distribution called partly regular has P (C̃) = δC,C̃ , in which the chip
connectivity is again Poisson distributed with mean L, but each user contributes to exactly
C chips. This prevents the systematic failure inherent in the irregular ensemble since
therein an extensive number of users fail to transmit on any chips. If in addition to
the aforementioned constraint all chips receive exactly L contributions, P (L̃) = δL,L̃,
the ensemble is called regular. Regular chip connectivity amongst other things prevents
the systematic inefficiency due to leaving some chips unaccessed by any of the users.
The case of Poissonian distributions is that in which there is no global control. In
many engineering applications constraining users individually (non-Poissonian P (C̃)) is
practical, whereas coordination between users (non-Poissonian P (L̃)) is difficult. The
practicalities of implementing the different ensembles we consider are application specific:
the advantages inherent in distributing channel resources more evenly amongst users may
be lost to practical implentation problems.
3. Methodology
3.1. Spectral Efficiency Lower Bound
The inferiority of codes with Poissonian user connectivity has been pointed out previously
(e.g., in [2]), based on the understanding that codes which leave a portion of the users
Sparsely-spread CDMA - a statistical mechanics based analysis 7
disconnected cannot be optimal. Analogously we argue that codes with irregular chip
connectivity must also be inferior in that they leave a fraction of the chips (bandwidth)
unutilised, thus providing a motivation for considering fully regular codes.
In this section we show a particular case in which the regular codes are expected
to outperform any other ensemble by analysing the amount of information that can be
extracted on the sent bits by consideration of only one chip in isolation of the other chips.
This corresponds to a detector reconstructing bits based only on the value of a single chip,
and is independent of the user connectivity.
The spectral efficiency is defined as the mutual information between the received
signal and reconstructed bits per chip. In considering only a single chip (µ) we have
I(τ ; yµ) =
P (τ |yµ)
P (τ )
P0(τ ,yµ)
, (9)
where the subscript zero indicates that the true (generative), rather than model (2),
probability distribution. For brevity we consider the simplest case that the generative
and model probability distributions are the same with unbiased bits and a Gaussian noise
distribution in which case after some rearrangement
I(τ ; yµ) = L̃−
exp(−Hµ(τ µ))
τ µ exp(−Hµ(τ µ))
P0(τ µ,yµ)
, (10)
where τ µ are the bits connected to chip µ, and the chip Hamiltonian is
ξiτi + yµ
, (11)
labelling each interacting (non-zero) component on the chip by i, L̃ being the chip
connectivity.
Working from this description we wish to compare the performance of ensembles
with different chip connectivities. To do this we consider the ensemble average mutual
information by averaging the mutual information over the connectivities (L̃), load factors,
and transmitted bits. This average is complicated, however it is possible to calculate the
dominant terms in the low and high PSD limits.
In the case of low noise (PSD → ∞) we find the asymptotically dominant terms
come first from the numerator
〈log2 exp−H(τ µ)〉
/ log(2) =
2 log(2)
, (12)
which is an average over the ground state energy, and also the logarithm of the denominator
which is
exp−H(τ µ)
ξi(bi − τi)
, (13)
where yµ has been decomposed into its bit ({bi}) and noise (ω) parts, and the averages are
now over the ensembles as well as yµ. The first part of (13) gives an energy contribution
cancelling (12). We call the remaining part the average over the chip entropy, by
Sparsely-spread CDMA - a statistical mechanics based analysis 8
comparison with (10) this determines the amount of information lost in decoding. The
chip entropy term contains an indicator function counting the ground states - the average
chip entropy is zero when τ µ = bµ is the only solution. For the case of BPSK however
there may be some degeneracy in ground states with two terms in the sum being non-zero
but cancelling one another. This degeneracy has a dependence on the distribution P (L̃)
for given L. Averaging over load factors and transmitted bits we find that in the zero
noise limit
I(τ , yµ)
ξi(bi − τi)
P (L̃)
, (14)
min(p,L̃−p)
L̃− p
P (L̃)
. (15)
By numerical evaluation of this function (see results section 4.2) we find that the optimal
ensemble is in fact the regular ensemble. This is because chip entropy, when averaged over
load factors and bits is a concave function in L̃, so that the information loss is minimised
when P (L̃) = δ
. This dependency on L̃ may be a peculiarity of the detector considered,
but many other aspects of the calculation may be generalised to give a similar result.
It is possible to consider the opposite limit σ20 → ∞ perturbatively. We found that
the leading four orders in 1/σ0 were identical for all code ensembles of the same mean chip
connectivity. We would anticipate the behaviour at non-extreme PSD to fall somewhere
between these two regimes and thus for the chip regular ensemble to be atleast as good as
the chip irregular ensembles.
We note here that another reason for considering the regular code optimal amongst
sparse random codes is to consider the field term when the Hamiltonian (11) is written
in canonical form with a set of couplings ({J〈ij〉}) and user specific external fields ({hi}).
In this representation the set of external fields are in expectation aligned with the sent
bit sequence, but subject to fluctuations for each code instance. The variance of these
fluctuations may be shown to be proportional to the excess chip connectivity over the
true chip connectivity [14], which amongst all ensembles is minimised by the regular chip
ensemble. The multi-user interference is larger in irregular codes and hence information
recovery is weaker as predicted in this section.†
3.2. Replica Method Outline
We determine the static properties of our model defined in section 2, including correlations
due to the full interaction structure, we use the replica method. From the expression of
the Hamiltonian (3) we may identify a free energy and partition function as:
f = − 1
lnZ Z = Trτ exp (−βH(τ )) .
† This argument is added since published version.
Sparsely-spread CDMA - a statistical mechanics based analysis 9
To progress we make use of the anticipated self-averaging properties of the system.
The assumption being that in the large system limit any two randomly selected instances
will, with high probability, have indistinguishable statistical properties. This assumption
has firm foundation in several related problems [15], and is furthermore intuitive after
some reflection. If this assumption is true then the statists of any particular instance can
be described completely by the free energy averaged over all instances of the disorder. We
are thus interested in the quantity
F = 〈f〉 = − lim
〈lnZ〉I , (16)
where the angled brackets represent the weighted averages over I (the instances). The
entropy density may be calculated from the free energy density by use of the relation
s = β(e− f) , (17)
where e is the energy density.
To determine the free energy we must average over disorder in (16), which is a difficult
problem except in special cases. This is why we make use of the replica identity
〈lnZ〉I = lim
〈Zn〉I . (18)
We can model the system now as one of interacting replicas, where Zn is decomposed as a
product of an integer number of partition functions with conditionally independent (given
the instance of the disorder) dynamical variables. The discreteness of replicas is essential
in the first part of the calculation, but a continuation to the real numbers is required
in taking n → 0+ – this is a notorious assumption, which rigorous mathematics can not
yet justify for the general case, in spite of the progress made in recent years [16, 17, 18].
However, we shall assume validity and since the methodology for the sparse structures is
well established [19, 20, 15] we omit our particular details. The final functional form for
the free energy is determlained from
〈Zn〉 =
dP (b,σ)dP̂ (b,σ)
exp{lnN +N(G1(n) +G2(n) +G3(n))} ; (19a)
G1(n) = ln
λ2α/2
P (b,σ)
λα(b− τα)
P (L̃)
; (19b)
G2(n) =
P (b,σ)P̂ (b,σ) ; (19c)
G3(n) = α ln
(−L)C̃
P̂ (b, τ )
P (C̃)
P0(b)
; (19d)
where N is a constant due to normalising the ensembles (6). This expression may be
evaluated at the saddle point to give an expression for the free energy. In the term (19d)
Sparsely-spread CDMA - a statistical mechanics based analysis 10
we account for the cases in which the marginalised probability distribution P0(b) and
assumed marginal probability distribution (described by H) are asymmetric. In the case
of maximal rate which we will consider, the b average is trivial and H = 0. Provided
that in addition the gain factor distribution is symmetric then it is possible to remove
the b dependence in the order parameters, since the symmetry P (b,σ) = P (−b,−σ) and
P̂ (b,σ) = P̂ (−b,−σ) leaves the free energy invariant.
3.3. Replica Symmetric Equations
The concise form for our equations is attained using the assumption of replica symmetry
(RS). This amounts to the assumption that the correlations amongst replicas are all
identical, and determined by a unique shared distribution. The validity of this assumption
may be self consistently tested (section 3.5). This assumption differs from that used
by Yoshida and Tanaka [2] where the correlations are described by only a handful of
parameters rather than a distribution once RS is assumed – this approach may therefore
miss some of the detailed structure although it is easier to handle numerically. The order
parameter in our case is given by
P (b, τ ) =
dπ(x)
(1 + bταx)
; (20a)
P̂ (b, τ ) = q̂
dπ̂(x̂)
(1 + bταx) ; (20b)
where q̂ is a variational normalisation constant and π, π̂ are normalised distributions on
the interval [−1, 1]. From here onwards we may consider the case in which the bit variables
τα and gain factors ξ are gauged to b (τb → τ , ξb → ξ).
Using Laplace’s method, this gives the following expression for the (RS) free energy
at the saddle point
FRS = −
Extrπ,bπ
G1,RS(L̃)(n) + G2,RS(n) + G3,RS(C̃)(n)
where
G1,RS(n)
= − L ln 2
[dπ(xl)]
ln Tr{τl=±1}χL̃(τ ; {ξ}, ω, {x})
Ω(ω),φ(ξ)
P (L̃)
; (22a)
χL̃(τ ; {ξ}, ω, {x}) = exp
(1− τl)ξl
(1 + τlxl) ; (22b)
G2,RS(n) = − L
dπ(xc)dπ̂(x̂c) ln(1 + xx̂c) ; (22c)
G3,RS(n) = α
[dπ̂(x̂c)] ln
(1 + x̂c) +
(1− x̂c)
P (C̃)
. (22d)
Sparsely-spread CDMA - a statistical mechanics based analysis 11
and the saddle point value for ŵ (= L) has been introduced. The averages over L̃ and C̃
encapsulate the differences amongst the ensembles.
Equation (22b) describes the interaction at a single chip in the factor graph (figure 1)
of connectivity L̃. The parameter ξl and variable τ are the gain factors, and reconstructed
bits respectively, both gauged to the transmitted bit, while ω is the instance of the chip
noise.
The order variational distributions {π, π̂} are chosen so as to extremise (21). The self
consistent equations attained by the saddle point method are:
π̂(x̂) =
[dπ(xl)]
Tr{τl=±1} τL̃+1 χ̄L̃(τ ; {ξ}, {x̂})
Tr{τl=±1} χ̄L̃(τ ; {ξ}, ω, {x})
{ξ},ω
P (L̃)
(23a)
χ̄L̃(τ ; {ξ}, ω, {x}) = exp
(1− τl)ξl
(1 + τlxl) (23b)
π(x) =
[dπ̂(x̂c)] δ
c=1(1 + x̂c)−
c=1(1− x̂c)
c=1(1 + x̂c) +
c=1(1− x̂c)
P (C̃)
. (23c)
The variables P (L̃) and P (C̃) are here the excess degree distributions of the particular
ensemble (6). For regularly constrained ensembles the chip and user excesses are L − 1
and C − 1 respectively. For Poissonian distributions the excess degree distribution is the
full degree distribution.
Aside from entropy, the other quantities of interest may be determined from the
probability distribution for the overlap of reconstructed and sent variables mk = 〈τk〉,
P (m) = lim
δmk,m
, (24)
[dπ̂(x̂c)] δ
c=1(1 + x̂c)−
c=1(1− x̂c)
c=1(1 + x̂c)+
c=1(1− x̂c)
P (C̃)
. (25)
We note finally that equivalent expressions to these found with the RS assumption
may be obtained by using the cavity method [6] with the assumption of a single pure state.
This approach is a probabilistic one and hence more intuitive on some levels.
3.4. Population Dynamics
Analysis of these equations is primarily constrained by the nature of equations (23a-
23c). No exact solutions are apparent, and perturbative regimes about the ferromagnetic
solution (which is only a solution for zero noise) are difficult to handle. Consequently
we use population dynamics [21] – representing the distributions {π(x), π̂(x̂)} by finite
populations (histograms) and iterating this distribution until convergence. It is hoped,
and observed, that each histogram captures sufficient detail to describe the continuous
Sparsely-spread CDMA - a statistical mechanics based analysis 12
function and the dynamics (described below) allow convergence towards a true solution
distribution with only small corrections due to finite size effects.
To solve the equations (23a,23c) with population dynamics finite histograms
constucted from M undirected cavity magnetisations are used. Histograms approximating
each function are formed
π(x) → W = {x1, . . . , xi, . . . , xM} , (27a)
π(x̂) → Ŵ = {x̂1, . . . , x̂a, . . . , x̂M} , (27b)
with M sufficiently large to provide good resolution in the desired performance measures.
The discrete minimisation dynamics of the histograms is derived from (23a-23c).
Histogram updates are undertaken alternately, with all magnetisation in the histogram
being updated sequentially. In the update of field xa the quenched parameters {L̃, ω, ξ}
are sampled, L̃ being the chip excess degree, and L̃ magnetisations are randomly chosen
from W , defining through (23a) the update
x̂a =
Tr{τl=±1} τL̃+1 χ̄L̃(τ ; {ξ}, ω, {x})
Tr{τl=±1} χ̄L̃(τ ; {ξ}, ω, , {x})
. (28)
The update of the other histogram follows dynamics in which C̃ is sampled, C̃ being
the user excess degree, along with C̃ randomly chosen magnetisations from Ŵ , defining
through (23c) the update
c=1(1 + x̂c)−
c=1(1− x̂c)
c=1(1 + x̂c) +
c=1(1− x̂c)
. (29)
There is a strong analogy between the population dynamics algorithm and that of
message passing on a particular instance of the graph. The iteration of the histograms
implicit in (28-29) is analogous to the propagation of a population of cavity magnetisations
between factor (a) and user (i) nodes, which may be written as the self consistent equations:
x̂a→i =
Tr{τl=±1}τi exp
l∈∂ari
(1− τl)ξal
l∈∂ari
(1 + τlxl→a) ; (30a)
xi→a =
c∈∂ira
(1 + x̂c→i)−
c∈∂ira
(1− x̂c→i)
; (30b)
where Nx,x̂ are the relevant normalisations, and the abbreviation ∂y indicates the set
of nodes connected to y. In population dynamics, the notion of a particular graph
with labelled edges is absent however, and the only the distribution of the two types
of magnetisations are relevant.
3.5. Stability Analysis
To test the stability of the obtained solutions we consider both the appearance of
non-negative entropy, and a stability parameter defined through a consideration of the
Sparsely-spread CDMA - a statistical mechanics based analysis 13
fluctuation dissipation theorem. The first criteria that the entropy be non-negative is
based on the fact that physically viable solutions in discrete systems must have non-
negative entropy so that any solution found not meeting this criteria must be based on
bad premises; replica symmetry is a likely source.
The stability parameter λ is defined in connection with the cavity method for spin
glasses [22] and tests local stability of the solutions. It is equivalent to testing the local
stability of belief propagation equations as proposed in [23]. A necessary condition for
the stability of the RS solution is that the corresponding susceptibility does not diverge.
This condition ensures that fields are not strongly correlated. The spin glass susceptibility
when averaged over instances may be defined
〈τ0τd〉2c
, (31)
where d is the distance between two nodes in the factor graph, the inner average denotes
the connected correlation function between these nodes, Xd describes the typical number
of variables at distance d, and the outer average is over instances of the disorder (self-
averaging part). This quantity is not divergent provided that
λ = ln
〈τ0τd〉2c
is negative, since this indicates an asympoptically exponential decrease in the terms of (31)
and hence convergence of the sum. In the thermodynamic limit the connected correlation
function is dominated by a single direct path which may be decomposed as a chain of local
linear susceptibilities
〈τ0τd〉c ∝
(i,j)
∂xi→a
∂x̂b→i
∂x̂b→i
∂xj→b
, (33)
where (i,j) indicate the set of variables on the shortest path between nodes 0 and d in a
particular instance of the graph (30a).
This representation allows us to construct an estimation for λ numerically based
on principles similar to population dynamics [24] – the directedness and fixed structure
implicit in a particular problem is removed with the self-averaging assumption leaving a
functional description similar to (23a-23c), which may be iterated. In order to approximate
the stability parameter λ one introduces additional positive numbers in the population
dynamics histograms (27b,27a), xi → {xi, vi} and x̂a → {x̂a, v̂a} respectively. These new
values represent the relative sizes of perturbations in each magnetisation, and are updated
in parallel to (28,29) as
v̂a =
, (34)
and with similar assignments for the field update of W
. (35)
Sparsely-spread CDMA - a statistical mechanics based analysis 14
The partial derivatives are calculated from (28-29) and evaluated at the corresponding
values in the sampled population. If the final fixed point is stable against small
perturbations in the initial field then these values {v, v̂} must decay exponentially on
average. Renormalisation of {vi} and {v̂a} such that the mean is 1 after each update is
necessary. The numerical renormalisation constant for each population yields (dependent)
estimations of λ, which can be sampled at a suitable convergence time (end of the {W, Ŵ}
minimisation process).
Like population dynamics we expect behaviour to be sensitive to initialisation
conditions and finite size effects in some circumstances. In addition the estimation requires
good resolution in the histograms W and Ŵ .
4. Results
Results are presented here for the canonical case of Binary Phase Shift Keying (BPSK)
where ξl ∈ {1,−1} with equal probability. Furthermore, we assume an AWGN model for
the true noise ω (of variance σ20). For evaluation purposes we assume the channel noise
level is known precisely, so that β = 1, employing the Nishimori temperature [5]. This
guarantees that the RS solution is thermodynamically dominant. Furthermore the energy
takes a constant value at the Nishimori temperature and hence the entropy is affine to
the free energy. Where of interest we plot the comparable statistics for the Single User
Gaussian channel (SUG), and the densely spread ensemble, each with MPM detectors –
equivalent to maximum likelihood for individual bits.
For population dynamics two parallel populations (27a,27b) are initialised either
uniformly at random, or in the ferromagnetic state. These two populations are known
to converge towards the unique solution, where one exists, from opposite directions, and
so we can use their convergence as a criteria for halting the algorithm and testing for the
appearance of multiple solutions. In the case where they converge to different solutions
we can usually identify the solution converged to from the ferromagnetic initial state as
a good solution - in the sense that it reconstructs well, and that arrived at from random
initial state as a bad solution. In the equivalent belief propagation algorithm one cannot
choose initial conditions equivalent to ferromagnetic – knowing the exact solution would
of course makes the decoding redundant. We therefore expect the properties of the bad
solution to be those realisable by belief propagation (though clever algorithms may be
able to escape to the good solution under some circumstances). The stability variables
{v, v̂} were initialised independently each as the square of a value drawn from a gaussian
distribution – and tests indicated other reasonable distributions produced similar results.
Computer resources restrict the cases studied in detail to an intermediate PSD
regime, and small L. In particular, the problem at low PSD, is the Gaussian noise
average, which is poorly estimated, while at high PSD a majority of the histogram is
concentrated at magnetisations x, x̂ ≈ 1 not allowing sufficient resolution in the rest of
the histogram.
Several different measures are calculated from the converged order parameter,
Sparsely-spread CDMA - a statistical mechanics based analysis 15
indicating the performance of sparsely-spread CDMA. Using the converged histograms
for the fields we are able to determine the following quantities: free energy, energy and a
histogram for the probability distribution, from discretisations of the previously presented
equations (23a-23c). Using the probability distribution we are also able to approximate
the decoding bit error rate
dP (m)
1− sign(m)
; (36)
multi-user efficiency
MuE =
erfc−1(Pb)
; (37)
and mutual information between sent and reconstructed bits per chip, I(b; τ )/N (taking
a factorised form given the RS assumption)
MI = α
dP (m)
1 + τm
1 + τm
. (38)
The spectral efficiency is the capacity I(τ ;y) per chip, which is affine to the entropy (and
the free energy at the Nishimori temperature)
ν = α− s/ ln 2 . (39)
Negative entropy can be identified when the measured spectral efficiency exceeds the load,
and thermodynamic transition points correspond to points of coincident spectral efficiency.
Figure 2‡ demonstrates some general properties of the regular ensemble in which the
variable and factor degree connectivities are C : L = 3 : 3, respectively. Equations (23a-
23c) were iterated using population dynamics and the relevant properties were calculated
using the obtained solutions; the data presented is averaged over 100 runs and error-bars,
which are typically small, are omitted for brevity. Figure 2(a) shows the bit error rate
in regular and Poissonian codes, the inset focuses on the range where the sparse-regular
and dense cases crossover. The sparse codes demonstrate similar trends to the dense case
except the irregular code, which show weaker performance in general, and in particular at
high PSD. Detailed trends can be seen in figure 2(b) that shows the multiuser efficiency.
Codes with regular user connectivity show superior performance with respect to the dense
case at low PSD. Figure 2(c) shows similar trends in the spectral efficiency and mutual
information (shown in the inset); the effect of the disconnected (user) component is clear
in the fact that the irregular code fails to reach capacity at high noise levels. In general
it appears the chip connectivity distribution is not critical in changing the trends present,
unlike the user connectivity distribution. It was found in these cases (and all cases with
unique solutions for given PSD), that the algorithm converged to non-negative entropy
values and to a stability measure fluctuating about a value less than 0, as shown in
figure 2(d). These points would indicate the suitability of the RS assumption.
‡ This figure has been modified from the published version, the difference being that the Poissonian chip
connectivity codes have everywhere weaker performance than the dense and sparse regular code ensemble.
Sparsely-spread CDMA - a statistical mechanics based analysis 16
The outperformance of dense codes by sparse ensembles with regular user connectivity
in the low PSD regime is new to our knowledge, although Poissonian chip connectivity
is everywhere inferior to both the dense and regular sparse codes. The difference between
codes disappears rapidly with increasing (connection) density at fixed α (figure 3). This
is inline with our prediction of the regular code being a high performance ensemble in
preceeding sections.
Figure 3 indicates the effect of increasing density at fixed α in the case of the
regular code. As density is increased the statistics of the sparse codes approach that
of the dense channel in all ensembles tested. For the irregular ensemble performance
increases monotonically with density at all PSD. The rapid convergence to the dense
case performance was elsewhere observed for partly regular ensembles, and ensembles
based on a Gaussian prior input [2, 7]. At all densities for which single solutions were
found the RS assumption appeared validated in the stability parameter and entropy.
Figure 4 indicates the effect of channel load α on performance. We first explain results
for codes in which only a single solution was found (no solution coexistence). For small
values of the load a monotonic increase in the bit error rate, and capacity are observed as α
is increased with C constant, as shown in figures 4(a) and 4(b), respectively. This matches
the trend in the dense case, the dense code becoming superior in performance to the sparse
codes as PSD increases. We found that for all sparse ensembles there existed regimes with
α > 1.49 for which only a single stable solution existed, although the equivalent dense
systems are known to have two stable solutions in some range of PSD [3]. In all single
valued regimes we observed positive entropy, and a negative stability parameter. However,
in cases of large α many features became more pronounced close to the dense case solution
coexistence regime: notably the cusp in the stability parameter, gap between MI and ν
and the gradient in Pb.
4.1. Solution Coexistence Regimes
As in the case of dense CDMA [3], also here we observe a regime where two solutions, of
quite different performance, coexist. In order to investigate the regime where two solutions
coexist we investigated the states arrived at from random and ferromagnetic initial
conditions (giving bad and good solutions respectively). Separate heuristic convergence
criteria were found for the histograms, and these seemed to work well for the good
solution. For the bad solution we simply present results after a fixed number of histogram
updates (500) as all convergence criteria tested appeared either too stringent, to require
experimentally inaccessible timescales, or did not capture the asymptotic values for
important quantities like entropy. We believe 500 updates to be sufficiently conservative
to capture the properties of these solutions however.
Figure 4(a) shows the dependence of the bit error rate on the load, which is also
equivalent to L/C. There is a monotonic increase in bit error rate with the load and the
emergence and coexistence of two separate solutions above a certain point; in the case
of the 6 : 3 code the point above which the two solutions coexist is PSD = 10.23dB as
Sparsely-spread CDMA - a statistical mechanics based analysis 17
−10 −8 −6 −4 −2 0 2 4 6 8 10
Spectral Power Density [dB]
Irreg.
P. Reg.
Dense0 1 2 3 4 5 6 7 8
−10 −8 −6 −4 −2 0 2 4 6 8 10
Spectral Power Density [dB]
Irreg.
P. Reg.
Dense(b)
−10 −8 −6 −4 −2 0 2 4 6 8 10
Spectral Power Density [dB]
Irreg.
P. Reg.
Dense
2 2.5 3 3.5 4 4.5 5 5.5 6
−10 −8 −6 −4 −2 0 2 4 6 8 10
Spectral Power Density [dB]
Irreg.
P. Reg.
Reg. Type 1
Reg. Type 2
Figure 2. Performance of the sparse CDMA configuration of variable and factor degree
connectivities C : L = 3 : 3, respectively; all data presented on the basis of 100 runs, error
bars are omitted and are typically small in subfigures (a)-(c) the smoothness of the curves
being characteristic of this level (numerical accuracy was excellent only at intermediate
PSDs). (a) The bit error rate is limited by the disconnected component in the case of
irregular codes, otherwise trends match the dense case, lower bounded by the SUG. Inset
- the range where the sparse-regular and dense cases crossover.(b) Multiuser efficiency
indicates the regular user connectivity codes outperform the dense case below some PSD.
(c) The spectral efficiency [——] demonstrates similar trends, the entropy being positive.
The gap between the mutual information [· · · · · ·] and spectral efficiency (shown in the
inset) is everywhere small and especially so at small and large PSD, indicating little
information loss in the decoding process. (d) The two markers show the mean results
for the two different stability estimates in the algorithm for the regular code. There are
systematic errors at small PSD, and convergence is good only at intermediate PSD.
The lines represent the average of these quantities for each ensemble – all ensembles show
a cusp at some PSD, for 3 : 3 codes the various ensembles shows very similar trends,
indicating local stability everywhere.
Sparsely-spread CDMA - a statistical mechanics based analysis 18
−10 −8 −6 −4 −2 0 2 4 6 8 10
Spectral Power Density [dB]
Dense
0 1 2 3 4 5 6 7 8
Spectral Power Density [dB]
Dense
Figure 3. The effect of increasing density for the regular ensemble: (a) Multiuser
efficiency, (b) spectral efficiency [——] and mutual information [– – –]. Data presented on
the basis of 10 runs, error bars are omitted but of a size comparable with the smoothness
of the curves. The performance of sparse codes rapidly approaches that of the dense code
everywhere. The PSD threshold beyond which the dense code outperforms the sparse
code is fairly stable.
indicated by the vertical dotted line.
We use the regular code 6 : 3 to demonstrate the solution coexistence found above
some PSD in various codes. The onset of the bimodal distribution can be identified by
the divergence in the convergence time in the single solution regime (the time for the
ferromagnetic and random histograms to converge to a common distribution). The time
for this to occur, in a heuristically chosen statistic and accuracy, is plotted in figure 4(b).
By a naive linear regression across 3 decades we found a power law exponent of 0.59 and
a transition point of PSD = 10.23dB, but cannot provide a goodness of fit measure to
this data. This would represent the point at which at least two stable solutions co-exist.
Beyond PSD ≈ 12dB only one stable solution is found from both random and
ferromagnetic initial conditions, corresponding statistically to a continuation of the good
solution. A solution which statistically resembles a continuation of the bad solution is
occasionally arrived at from both initial conditions, this solution had a positive stability
parameter and negative entropy – so is not a viable solution. Thus we predict a second
dynamical transition in the region of 12dB, as might be guessed by comparison with the
dense case and observation of the trend in the stability parameter (see figure 4(c)).
The stability results are presented in figure 4(c). Only two stable solutions were found
in the region beyond this critical point and upto 12dB, which we infer to be viable RS
solutions (where entropy is positive). The bad solution upto 12dB has a well resolved
negative value. The good solution has a negative value in its mean, but like other near
ferromagnetic solutions investigated results are very noisy due to numerical issues relating
to histogram resolution.
Both capacity and spectral efficiency monotonically increase with the load as shown in
figure 4(d). For the 6 : 3 code we see a separation of the two solutions at PSD = 10.23dB
Sparsely-spread CDMA - a statistical mechanics based analysis 19
−4 −2 0 2 4 6 8 10 12
Spectral Power Density [dB]
6:3 (Bad)
6:3 (Good)
− PSD
Data Mean
Linear fit
Bounds
−4 −2 0 2 4 6 8 10 12
Spectral Power Density [dB]
6:3 (Bad)
6:3 (Good)
−4 −2 0 2 4 6 8 10 12
Spectral Power Density [dB]
6:3 (Bad)
6:3 (Good)
8 10 12
Figure 4. The effect of channel load α on performance for the regular ensemble. Data
presented on the basis of 10 runs, error bars omitted but characterised by the smoothness
of curves. Dashed lines indicate the dense code analogues. The vertical dotted line
indicates the point beyond which 6 : 3 random and ferromagnetic initial conditions failed
to converge to the same solution, both dynamically stable solutions are shown beyond
this point. (a) There is a monotonic increase in bit error rate with the increasing load.
(b) Investigation of the 6 : 3 code (α = 2) indicates a divergence in convergence time as
PSD → 10.23dB with exponent 0.59 based on a simple linear regression of 15 points (each
point is the mean of 10 independent runs). Beyond this point different initial conditions
give rise to one of two solutions. (c) The stability parameter was found to be negative
for all convergent solutions, indicating the suitability of RS. Where the solution is near
ferromagnetic the stability measure becomes quickly very noisy (as shown for the 5 : 3
and 6 : 3 codes). (d) As load α is increased there is a monotonic increase in capacity.
The spectral efficiency for the ’bad’ solution exceeds 2 in a small interval (equivalent to
negative entropy), similar to the behaviour reported for the dense case.
Sparsely-spread CDMA - a statistical mechanics based analysis 20
(vertical dotted line.) The dashed lines correspond to a similar behaviour observed in the
dense case (the range of interest is magnified in the inset.) A cross over in the entropy of the
two distinct solutions, near PSD ≈ 11dB, is indicative of a second order phase transition.
As in the dense case, only the solution of smallest spectral efficiency is thermodynamically
relevant at a given PSD, although the other is likely to be important in decoding dynamics.
The trends in the sparse case follow the dense case qualitatively, with the good solution
having performance only slightly worse than the corresponding solution in the dense case
(and vice versa for the bad solution).
The entropy of the bad solution becomes negative in a small interval (spectral
efficiency exceeds 2) although no local instability is observed. The static and dynamic
properties of the histograms appear to be well resolved in this region. However, the
negative entropy indicates an instability towards either a type of solution not captured
within the RS assumption, or towards some metastable configuration. We will not
speculate further, the bad solution is in any case thermodynamically subdominant in
its low and negative entropy form.
Our hypothesis is therefore that the trends in the sparse ensembles match those in
the dense ensembles within the coexistence region and RS continues to be valid for each
of two distinct positive entropy solutions. The coexistence region for the sparse codes is
however smaller than in the corresponding dense ensembles. Since our histogram updates
mirror the properties of a belief propagation algorithm on a random graph we can suspect
that the bad solution may have implications for the performance of belief propagation
decoding in the coexistence region, and that convergence problems will appear near this
region. In the user regular codes investigated the bad solution of the sparse ensemble
outperforms the bad solution of the dense ensemble, and vice-versa for the good solution.
Thus regardless of whether sparse decoding performance is good or bad, the dynamical
transition point for the dense ensemble would corresponds to a PSD beyond which dense
CDMA outperforms sparse CDMA at a particular load.
4.2. Spectral Efficiency Lower Bound Numerical Results
Finally we present figure 5, which shows the the mutual information between a single
chip and transmitted bits for sparse ensembles of differing chip connectivity in the infinite
PSD (zero noise) limit (15). This shows that in expectation a chip drawn from the
regular ensemble contains more information on the transmitted bits than a chip drawn
from any other ensemble (including the Poissonian ensemble). The difference between the
regular and Poissonian ensembles becomes relatively smaller as L increases. This appears
consistent with the replica method results found at high PSD, although regular chip
connectivity under performed by comparison with Poisson distributed chip connectivity
in the low PSD regime, which was not anticipated by the single chip approximation.
Sparsely-spread CDMA - a statistical mechanics based analysis 21
0 5 10 15 20 25
Mean Chip Connectivity, L
Poissonian
Figure 5. A PSD → ∞ limit to the expected mutual information between a single chip,
and the transmitted bits. Mutual Information is highest for regular chip connectivities,
with the Poissonian chip connectivity result also shown, the discrepancy becoming
relatively small as L increases. The inset shows the mutual information/bit decoded
(〈I(τ ; yµ)〉 /L) on a log-log plot to demonstrate an asymptotic power law behaviour and
show more detail in the cases of small L.
5. Concluding Remarks
Our results demonstrate the feasibility of sparse regular codes for use in CDMA. At
moderate PSD it seems the performance of sparse regular codes may be very good. With
the replica symmetric assumption apparently valid at practical PSD it is likely that fast
algorithms based on belief propagation may be very successful in achieving the theoretical
results. Furthermore for lower density sparse codes the problem of the coexistence regime,
which limits the performance of practical decoding methods, seems to be less pervasive
than for dense ensembles in the over saturated regime.
A direct evaluation of the properties of belief propagation may prove similar results
to those shown here. In the absence of replica symmetry breaking states it is normally
true that belief propagation performs very well. However, to make best use of the channel
resources it may be preferable to implement high load regimes in cases of high PSD, and
so overcoming the algorithmic problems arising from the solution coexistence is a challenge
of practical importance in this case.
Other practical issues in implementation are certainly significant. Similar to the case
of dense CDMA there are considerable problems relating to multipath, fading and power
control, in fact it is known that these effects are more disruptive for the sparse codes,
especially regular codes. However, certain situations such as broadcasting (one to many)
channels and downlink CDMA, where synchronisation can be assumed, may be practical
points for future implementation. There are practical advantages of the sparse case over
dense and orthogonal codes in some regimes. The sparse CDMA method is likely to be
particularly useful in frequency-hopping and time-hopping code division multiple access
(FH and TH -CDMA) applications where the effect of these practical limitations is less
Sparsely-spread CDMA - a statistical mechanics based analysis 22
emphasised.
Extensions based on our method to cases without power control or synchronisation
have been attempted and are quite difficult. A consideration of priors on the inputs, in
particular the effects when sparse CDMA is combined with some encoding method may
also be interesting.
Acknowledgments
Support from EVERGROW, IP No. 1935 in FP6 of the EU is gratefully acknowledged.
DS would like to thank Ido Kanter for helpful discussions.
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Background
The model
Code Ensembles
Methodology
Spectral Efficiency Lower Bound
Replica Method Outline
Replica Symmetric Equations
Population Dynamics
Stability Analysis
Results
Solution Coexistence Regimes
Spectral Efficiency Lower Bound Numerical Results
Concluding Remarks
|
0704.0099 | On Ando's inequalities for convex and concave functions | On Ando’s inequalities for convex and concave
functions
Koenraad M.R. Audenaert
Institute for Mathematical Sciences, Imperial College London,
53 Prince’s Gate, London SW7 2PG, United Kingdom
Jaspal Singh Aujla
Department of Applied Mathematics, National Institute of Technology,
Jalandhar 144011, Punjab, India
Abstract
For positive semidefinite matrices A and B, Ando and Zhan proved the inequalities
|||f(A) + f(B)||| ≥ |||f(A + B)||| and |||g(A) + g(B)||| ≤ |||g(A + B)|||, for any
unitarily invariant norm, and for any non-negative operator monotone f on [0,∞)
with inverse function g. These inequalities have very recently been generalised to
non-negative concave functions f and non-negative convex functions g, by Bourin
and Uchiyama, and Kosem, respectively.
In this paper we consider the related question whether the inequalities |||f(A)−
f(B)||| ≤ |||f(|A−B|)|||, and |||g(A)− g(B)||| ≥ |||g(|A−B|)|||, obtained by Ando,
for operator monotone f with inverse g, also have a similar generalisation to non-
negative concave f and convex g. We answer exactly this question, in the negative
for general matrices, and affirmatively in the special case when A ≥ ||B||.
In the course of this work, we introduce the novel notion of Y -dominated majori-
sation between the spectra of two Hermitian matrices, where Y is itself a Hermitian
matrix, and prove a certain property of this relation that allows to strengthen the
results of Bourin-Uchiyama and Kosem, mentioned above.
Key words: Matrix norm inequality, Convex functions, Majorization.
1991 MSC: 15A60
Email addresses: [email protected] (Koenraad M.R. Audenaert),
[email protected] (Jaspal Singh Aujla).
Preprint submitted to Elsevier 1 November 2018, 20:38
http://arxiv.org/abs/0704.0099v1
1 Introduction
In [1], Ando proved the following inequalities for positive semidefinite (PSD)
matrices A, B, and any unitarily invariant (UI) norm. For any non-negative
operator monotone function f(t) on [0,∞):
|||f(A)− f(B)||| ≤ |||f(|A−B|)|||, (1)
and, when f(0) = 0 and f(∞) = ∞, and g is the inverse function of f ,
|||g(A)− g(B)||| ≥ |||g(|A−B|)|||. (2)
In a later paper [2], Ando and Zhan proved the related inequalities (with the
same conditions on f and g)
|||f(A) + f(B)||| ≥ |||f(A+B)|||, (3)
|||g(A) + g(B)||| ≤ |||g(A+B)|||. (4)
The conditions on f are satisfied by every operator concave function f with
f(0) = 0.
Inequality (4) was generalised by Kosem [7] to non-negative convex functions
g on [0,∞), with g(0) = 0. Inequality (3) was generalised very recently to any
non-negative concave function on [0,∞) by Bourin and Uchiyama [5], who
also gave a simpler proof of Kosem’s result.
In the same spirit, we consider the question whether inequalities (1) and (2)
can also be generalised to non-negative concave f and convex g, respectively.
After introducing the necessary prerequisites in Section 2, we present our main
results concerning this question in Section 3. Regrettably, most of our results
are negative answers, and we give counterexamples to this generalisation. The
answer is even negative for the special case A ≥ B, although the apparent
hardness of finding counterexamples had led us temporarily into believing
that in that case the generalisation might actually hold.
All is not bad news, however. In Section 4 we answer the question affirmatively
in the special case when A ≥ ||B||. In Section 5, we introduce the novel notion
of Y -dominated majorisation between the spectra of two Hermitian matrices,
where Y is itself a Hermitian matrix. We prove a certain property of this
relation, namely Proposition 3, which we subsequently use, first in a rather
destructive fashion. To wit, the Proposition has been instrumental in finally
discovering a counterexample to the generalisation of (1) for A ≥ B; this will
be reported in Section 6. On the more constructive side, the Proposition also
allows to strengthen the results of Bourin-Uchiyama and Kosem mentioned
above. This is the topic of the final Section, along with a few other applications.
2 Preliminaries
In this Section, we introduce the notations and necessary prerequisites; a more
detailed exposition can be found, e.g. in [4]. We will use the abbreviations LHS
and RHS for left-hand side and right-hand side, respectively.
We denote the vector of diagonal entries of a matrix A by Diag(A).
We denote the absolute value by | · |, both for scalars and for matrices. For
matrices this is defined as |A| := (A∗A)1/2. Similarly, we denote the positive
part of a real scalar or Hermitian matrix by (·)+, and define it by A+ :=
(A+ |A|)/2.
In this paper, we are mainly concerned with monotonously increasing convex
and concave functions from R to R. Kosem noted in [7] that any such function
can be approximated by a sum of angle functions x 7→ ax+ b(x−x0)
+, where
a ≥ 0, and b > 0 for a convex angle function (b < 0 for a concave one).
We are also concerned with the unitarily invariant (UI) matrix norms, which
we denote by ||| · |||, and which are defined in terms of the singular values
σj(·) of the matrix only. We adopt the customary convention that the singular
values are sorted in non-increasing order: σ1 ≥ σ2 ≥ . . . ≥ σd. Special cases
of these norms are the operator norm || · ||, which is just equal to the largest
singular value σ1(·), and the Ky Fan norms || · ||(k), which are defined as the
sum of the k largest singular values:
||A||(k) :=
σj(A).
The famous Ky Fan dominance theorem states that a matrix B dominates
another matrix A in all UI norms if and only if it does so in all Ky Fan norms.
The latter set of relations can be written as a weak majorisation relation
between the vectors of singular values of A and B:
σ(A) ≺w σ(B) :
σj(A) ≤
σj(B), ∀k.
For PSD matrices, the above domination relation translates to a weak majori-
sation between the vectors of eigenvalues: λ(A) ≺w λ(B).
Weyl’s monotonicity Theorem ([4], Corollary III.2.3) states that
k(A) ≤ λ
k(A +B), ∀k,
for Hermitian A and positive semi-definite B. Here, λ↓(A) denotes the (real)
vector of eigenvalues of A sorted in non-increasing order.
Finally, we refer the reader to Chapter 2 of [6] for an exposition of a number
of important functional analytic properties of eigenvalues and corresponding
eigenspaces of a Hermitian matrix, which we will need in the proof of Propo-
sition 2.
3 Main Results
The question we start with is about the straightforward generalisation of in-
equality (2) to non-negative convex functions.
Question 1 For all A,B,≥ 0, for all UI norms, and for non-negative convex
functions g on [0,∞) with g(0) = 0, does the inequality |||g(A) − g(B)||| ≥
|||g(|A− B|)||| hold?
The answer to this question is negative, as shown by the following counterex-
ample. We consider the convex angle function g(x) = x + (x − 1)+ and the
operator norm. For the 2× 2 PSD matrices
0.9 0
0 0.6
, B =
0.8 0.5
0.5 0.4
, (5)
the eigenvalues of g(|A− B|) are 0.65249 and 0.35249, while those of g(A)−
g(B) are 0.65010 and −0.48862. Thus, ||g(|A − B|)||∞ = 0.65249, which is
larger than ||g(A)− g(B)||∞ = 0.65010. ✷
Under the additional restriction A ≥ B, the absolute value in the argument
of g in the right-hand side vanishes, leading to a simplified statement, and a
second question, with better hopes for success. Introducing the matrix ∆ =
A− B,
Question 2 For all B,∆ ≥ 0, for all UI norms, and for non-negative convex
functions g on [0,∞) with g(0) = 0, does the inequality |||g(B+∆)−g(B)||| ≥
|||g(∆)||| hold?
This restricted case also turns out to have a negative answer. Counterexam-
ples, however, were much harder to find, and required a reduction of the prob-
lem based on certain results about a novel majorisation-like relation, which
we call the Y -dominated majorisation. This will be the subject of Sections 5
and 6, where a number of Propositions of independent interest are proven.
It is also very reasonable to ask:
Question 3 For all B,∆ ≥ 0, for all UI norms, and for non-negative concave
functions f on [0,∞), does the inequality |||f(B + ∆) − f(B)||| ≤ |||f(∆)|||
hold?
In fact, if this were true, a positive answer to Question 2 would easily follow,
using the same reasoning that was used in [5] to derive the generalisation of
(3) from the generalisation of (4).
Again, this statement is false, as the following counterexample shows. Consider
the concave angle function f(x) = min(x, 1) = x − (x − 1)+, and the 3 × 3
PSD matrices
0.701816 0.317887 0.198910
0.317887 1.014950 −0.093826
0.198910 −0.093826 0.274236
0.192713 0 0
0 0.446505 0
0 0 0.455416
One gets
||f(∆)||∞ = 0.455416
while
||f(B +∆)− f(B)||∞ = 0.455776.
In Section 4, we consider an even more restricted special case, in which the
inequalities (1) and (2) finally do hold. We actually prove that a stronger
relationship holds in this special case:
Proposition 1 For non-negative, monotonously increasing and concave func-
tions g, and A,B ≥ 0 such that A ≥ ||B||, we have
λ↓(g(A− B)) ≥ λ↓(g(A)− g(B)). (6)
An easy Corollary is the corresponding statement for monotonously increasing
convex functions.
Corollary 1 Let f be a non-negative convex function on [0,∞) with f(0) = 0.
Let A,B ≥ 0 such that A ≥ ||B||. Then
λ↓(f(A− B)) ≤ λ↓(f(A)− f(B)). (7)
Proof. Let f = g−1, with g satisfying the conditions of Proposition 1. Replace
in (6) A by f(A) and B by f(B), yielding
λ↓(g(f(A)− f(B))) ≥ λ↓(A− B).
Applying the function f on both sides does not change the ordering, because
of monotonicity of f , and yields validity of inequality (7). ✷
These two results obviously imply the corresponding majorisation relations,
and by Ky Fan dominance, relations in any UI norm.
4 Proof of Proposition 1
We want to prove inequality (6):
λ↓(g(A)− g(B)) ≤ λ↓(g(A−B)),
for A,B ≥ 0, A ≥ ||B||, and concave, monotonously increasing and non-
negative g.
W.l.o.g. we will assume ||B|| = 1, since any other value can be absorbed in
the definition of g.
It is immediately clear that if (6) holds for g that in addition satisfy g(0) = 0,
then it must also hold without that constraint, i.e. for functions g(x)+ c, with
c ≥ 0. This is because the additional constant c drops out in the LHS, while
λ↓(g(A−B) + c) ≥ λ↓(g(A− B)).
Furthermore, (6) remains valid when replacing g(x) with ag(x), for a > 0.
Thus, w.l.o.g. we can assume g(0) = 0 and g(1) = 1. Together with concavity
of g, this implies that, for 0 ≤ x ≤ 1, g(x) ≥ x, while for x ≥ 1, the derivative
g′(x) ≤ 1.
Since 0 ≤ B ≤ 11, and for 0 ≤ x ≤ 1, g(x) ≥ x holds, we have g(B) ≥ B,
or −g(B) ≤ −B. By Weyl monotonicity, this implies λ↓(g(A) − g(B)) ≤
λ↓(g(A)−B). Thus, statement (6) would be implied by the stronger statement
λ↓(g(A)−B) ≤ λ↓(g(A− B)). (8)
Now note that the argument of g in the LHS, A, is never below 1. Thus, in
principle, we could replace g(x) in the LHS by another function h(x) defined
h(x) =
g(x), if x ≥ 1
x, otherwise.
If we also do that in the RHS, we get a stronger statement than (8). Indeed,
h(x) ≤ g(x) for x ≥ 0 and A − B ≥ 0, and therefore h(A − B) ≤ g(A − B)
holds. By Weyl monotonicity again, we see that (8) is implied by
λ↓(h(A)−B) ≤ λ↓(h(A− B)). (10)
The importance of this move is that h(x) is still a monotonously increasing
and concave function (because g′(x) ≤ 1 for x ≥ 1), but now has gradient
h′(x) ≤ 1 for x ≥ 0.
Defining C = A−B, which is positive semi-definite, we now have to show the
inequality
k(h(C +B)− B) ≤ λ
k(h(C)) = h(λ
k(C)),
for every k. Fixing k, and introducing the shorthand x0 = λ
k(C), we can
exploit concavity of h to upper bound it as h(x) ≤ a(x − x0) + h(x0), where
a = h′(x0) ≤ 1. Again by Weyl monotonicity, we find
k(h(C +B)− B)≤λ
k(a(C +B − x0) + h(x0)− B)
k(aC + (a− 1)B − ax0 + h(x0))
k(aC)− ax0 + h(x0) = h(x0),
where in the second line we could remove the term (a−1)B because it is nega-
tive. This being true for all k, we have proved (10) and all previous statements
that follow from it, including the statement of the Theorem. ✷
5 On Y -dominated Majorisation
To answer Question 2, we have to consider the property that a convex function
f satisfies
λ(f(∆)) ≺w λ(f(B +∆)− f(B)) (11)
for all PSD B and ∆, which is equivalent to the statement
λ(f(A− B)) ≺w λ(f(A)− f(B)) (12)
for all A ≥ B ≥ 0.
The monotone convex angle functions x 7→ ax + (x − 1)+ (a ≥ 0) already
have proven their valour as a testing ground for similar statements, in Section
3. Numerical experiments using angle functions for inequality (11) did not
directly lead to any counterexamples, however. This temporarily increased
our belief that the inequality might actually hold, and led us to investigate,
as an initial step towards a “proof”, whether the inequality
j(aY +B) ≤
j(aY + C)
might be true for all a ≥ 0, where B = f(Y ) and C = f(X + Y )− f(X), and
f(x) = (x−1)+. The crucial observation is now that if this holds for all a ≥ 0,
then, actually, a much stronger relationship than just majorisation must hold
between aY +B and aY + C.
To describe this phenomenon, we’ll consider a somewhat broader setting. Let
G and C be Hermitian matrices, and let f1 and f2 be monotonously increasing
real functions on R. Suppose that for all a ≥ 0, the following holds:
j(aA +B) ≤
j(aA+ C), (13)
with A = f1(G) and B = f2(G).
It is easily seen that if (13) holds for a certain value of a, it also holds for all
smaller positive values. Let b be a scalar such that 0 ≤ b < a. Because both
A and B exhibit their eigenvalues as diagonal elements in the eigenbasis of G,
and both in non-increasing order, we get
j(aA +B) =
j (bA+B) + (a− b)
j (A).
On the other hand, for aA + C we only have the subadditivity inequality
j (aA+ C) ≤
j(bA + C) + (a− b)
j(A).
As a consequence, we obtain that, indeed,
j (bA+B) ≤
j(bA + C)
follows from (13).
We are therefore led to consider what happens when a tends to infinity, because
that value dominates all others. Subtracting
j=1 λ
j(aA) from both sides, and
substituting a = 1/t, we obtain
j(A + tB)− λ
j(A)) ≤
j(A+ tC)− λ
j(A)).
In the limit of t going to 0, this yields a comparison between derivatives:
j(A + tB) ≤
j(A+ tC). (14)
We will show below that the derivatives ∂
j (A + tC) are the diagonal
elements of C in a certain basis depending on G and C. Let us first introduce
the vector δ(C;A) whose entries satisfy the following relation:
δj(C;A) :=
j(A+ tC). (15)
With this notation, relation (14) becomes
δj(B;G) ≤
δj(C;G).
That is, the entries of δ(B;G) are “majorised” by those of δ(C;G). How-
ever, this is a much stronger relation than ordinary majorisation, since the
rearrangement of the entries in decreasing order is absent.
Introducing the symbol ≺dw for weak majorisation with missing rearrange-
ment:
a ≺dw b ⇐⇒
bj , (16)
relation (14) is expressed as
δ(B;G) ≺dw δ(C;G). (17)
To justify these notations, we now show:
Proposition 2 Let A and C be Hermitian matrices. With δ(C;A) defined by
(15), the entries of the vector δ(C;A) are the diagonal entries of C in a certain
basis in which A is diagonal and its diagonal entries appear sorted in non-
increasing order. When all eigenvalues of A are simple (i.e. have multiplicity
1), this basis is just the eigenbasis of A and does not depend on C.
Proof. There are three cases to consider, according to whether A is non-
degenerate, A + tC has an accidental degeneracy at t = 0, or A + tC is
permanently degenerate.
1. The most important case is when all eigenvalues of A are simple, i.e. when
they have multiplicity 1. We then show that the derivative is given by
j(A+ tC) = Tr[Pk(A) C],
where Pk(A) denotes the projector on the subspace spanned by the k eigen-
vectors of A corresponding to its k largest eigenvalues.
By the simplicity of the eigenvalues of A, the eigenvalues of A + tC are also
simple for small enough values of t. This follows easily fromWeyl’s inequalities:
j(A) + λ
n(tC) ≤ λ
j (A+ tC) ≤ λ
j(A) + λ
1(tC);
thus if t||C|| is strictly less than one half the minimal difference between
all pairs of eigenvalues of A, the difference between all pairs of eigenvalues of
A+tC is bounded away from 0. Therefore, for small enough t, every eigenvalue
of A+ tC has a unique eigenvector, and as a result Pk(A+ tC) is well-defined
as the sum of the projectors on the eigenvectors pertaining to the k largest
eigenvalues.
It is well-known that the eigenvalues of A+tC as functions of the real variable
t can be so ordered that they are analytic functions of t (see [6], Chapter 2),
and hence continuous. This implies that the k-th largest eigenvalue of A+ tC
is also a continuous function of t, for any k.
If, furthermore, an eigenvalue λ(t) of A + tC is simple in an interval of t,
then the projector P (t) on the eigenvector x(t) associated to it (with P (t) =
x(t)x(t)∗) is also analytic, and therefore continuous in t on this interval. We
conclude that Pk(A+ tC) is analytic in t, and therefore differentiable.
By the maximality of Pk(A) in the variational characterisation
j(A) = max
Tr[AQk] = Tr[APk(A)],
where Qk runs over all rank-k projectors, we have
Tr[APk(A + tC)] = 0,
which implies
j (A+ tC)
Tr[(A+ tC)Pk(A+ tC)]
Tr[APk(A+ tC)] +
Tr[(A+ tC)Pk(A)]
=Tr[C Pk(A)].
Let U be the unitary that diagonalises A, i.e. UAU∗ = Λ↓(A). Then
Tr[C Pk(A)] =
(UCU∗)jj,
and the statement of the Proposition follows.
2. When A has degenerate eigenvalues, the situation becomes somewhat more
complicated, but there are no really significant changes. There is no longer
a unique eigenbasis of A, so that Pk(A) is not well-defined for all k. We will
first consider the case where C is such that it removes the degeneracy of the
eigenvalues of A in A+ tC for small enough positive t. In that case Pk(A+ tC)
will be uniquely defined for all positive t less than some value t0, which is the
smallest positive t for which A + tC has an accidental degeneracy (which is
what also happens at t = 0).
This occurs, for instance, when C has simple eigenvalues. Indeed, by analyt-
icity of the eigenvalues of A + tC in t, degeneracy is either accidental (for
isolated values of t) or permanent (for all values of t). Since all eigenvalues
are simple for large enough t, they have to remain simple for all values of t
except possibly for some isolated values, such as t = 0, in this case. Let t0
be the smallest positive such value, then A + tC has simple eigenvalues for
0 < t < t0.
We can therefore define Pk(A) in a unique way as the limit limt→0 Pk(A +
tC). This is an allowed choice because of the continuity of the eigenvalues:
j=0 λ
k(A) = Tr[limt→0 Pk(A + tC) A]. Using the same argument as in the
previous case, we obtain δ(C;A) := Tr[limt→0 Pk(A+ tC) C].
Let λl be the eigenvalues of A (multiplicity not counted), and Ql the projec-
tions onto the corresponding eigenspaces of A (with Q∗l the corresponding in-
clusion operators); the rank of Ql equals the multiplicity of λl, which we denote
by ml. To obtain δ(C;A), we first construct the diagonal blocks Cl := QlCQ
(of size ml), then take the eigenvalues λ
↓(Cl) in non-increasing order of each
block, and then concatenate the obtained sequences of eigenvalues:
δ(C;A) := (λ↓(C1), . . . , λ
↓(Cm)).
If all eigenvalues of A are distinct, this reduces to the vector of diagonal
elements of C in the eigenbasis of A that we encountered in case 1.
For example, if λ↓(A) = (5, 5, 3, 1), then δ(C;A) = (λ
1(C1), λ
2(C1), C33, C44),
where C1 =
C11 C12
C21 C22
and all entries of C are taken in the eigenbasis of A.
Let U be a unitary (which, in this case, is not unique) that diagonalises A as
UAU∗ = Λ↓, and take the diagonal blocks Cl of UCU
∗, as above. Each block
can be diagonalised using a unitary Vl. Together with U we obtain the total
basis rotation W := U(
l Vl). By construction,
l Vl leaves Λ invariant, and
resolves the ambiguity in U . We obtain that δ(C;A) is the vector of diagonal
entries of C in the basis obtained by applying the unitary W .
3. Finally, we look at the case when A + tC is permanently degenerate, i.e.
when it has degenerate eigenvalues for all values of t. W.l.o.g. we just have to
look at t in an interval [0, t0), where t0 is the smallest positive value for which
A + tC has an accidental degeneracy. Let us denote by λj(t) the eigenvalues
of A+ tC in non-increasing order, multiplicity mj not counted, and by Pj(t)
the projectors on the corresponding eigenspaces. In that case Pk(A + tC) is
only well-defined if there is a j′ such that k = m1 +m2 + . . . +mj′; then we
have Pk(A + tC) = P1(t) + P2(t) + . . .+ Pj′(t).
If there is no such j′, let j′ be the largest integer such that k > m1 + m2 +
. . .+mj′ =: k
′. Thus 0 < k − k′ < mj′+1. Then we have
j (A+ tC)
miλi(t) + (k − k
′)λj′+1(t)
=Tr[(A + tC) (P1(t) + . . .+ Pj′(t) +
k − k′
mj′+1
Pj′+1(t))]
=Tr[(A + tC) (
k − k′
mj′+1
Pk′+mj′+1(A+ tC) + (1−
k − k′
mj′+1
)Pk′(A+ tC))].
Thus, if we define α := (k − k′)/mj′+1,
j=1 λ
j(A + tC) interpolates linearly
between
j=1 λ
j(A+ tC) and
∑k′+mj′+1
j=1 λ
j(A + tC) with parameter α.
Proceeding in the same way as in the two previous cases, we obtain for the
derivative
j (A+ tC) = Tr[C(αPk′+mj′+1(A) + (1− α)Pk′(A))],
where the Pk(A) have to be replaced with the limits limt→0 Pk(A + tC) if in
addition there are accidental degeneracies at t = 0. Let us consider the entries
of C again as before, in an eigenbasis of A in which the eigenvalues of A appear
on the diagonal, in non-increasing order. We get
δ(C;A)k = (1− α)
Cii + α
k′+mj′+1
Cii. (18)
Because of the permanent degeneracy, an eigenbasis is determined up to “lo-
cal” rotations within the various eigenspaces. We consider a partitioning of
C in such an eigenbasis corresponding to these eigenspaces. That is, in C we
can single out diagonal blocks, each of which corresponds to the eigenspace of
eigenvalue λj . We can use our freedom to choose the local rotations to make
all diagonal elements of C equal within each diagonal block. This allows us to
get rid of the interpolation in (18), and we finally obtain that, again,
δ(C;A)k =
with the entries of C taken in the eigenbasis that we have just chosen. ✷
The upshot of this Proposition is that there exists a unitary U such that
UAU∗ = Λ↓(A) and δ(C;A) = Diag(UCU∗). In the generic case that all
λi(A) are distinct, U is unique and does not depend on C.
A number of easy consequences follow immediately from this Proposition:
Corollary 2 Let G and C be Hermitian matrices, f be any monotonously
increasing real function on R, and g any strictly increasing real function on
R, then
(i) δ(f(G);G) = f(λ↓(G)).
(ii) δ(C;G) obeys Schur’s majorisation Theorem: δ(C;G) ≺ λ↓(C).
(iii) δ(C;G) + aλ↓(f(G)) = δ(C + af(G);G), ∀a ≥ 0.
(iv) δ(C; f(A)) = δ(C;A).
Along with the previously demonstrated equivalence of (13) with (17), the
Corollary immediately leads to the following Proposition:
Proposition 3 For Hermitian G,C, monotonously increasing real functions
f1, f2 on R, and A = f1(G), B = f2(G), the following are equivalent:
λ(aA+B) ≺w λ(aA+ C), ∀a ≥ 0 (19)
δ(B;G)≺dw δ(C;G) (20)
δ(aA+ B;G)≺dw δ(aA+ C;G), ∀a ≥ 0. (21)
Proof.
(19) implies (20): This is just Proposition 2.
(20) implies (21): Add aλ↓(A) to both sides and invoke statement (iii) of the
Corollary.
(21) implies (19): By statement (i) of the Corollary, the LHS of (21) is equal
to λ↓(aA+B), while, by statement (ii) of the Corollary, its RHS is majorised
by λ(aA+ C). ✷
6 Counterexample to Question 2
If the answer to Question 2 is to be affirmative, it should at least hold for all
angle functions f(x) = ax+ b(x−x0)
+. By Proposition 3 this is equivalent to
the statement
δ((Y − 11)+; Y ) ≺dw δ((X + Y − 11)
+ − (X − 11)+; Y ).
Consider the 3× 3 matrices
0.35614 −0.053243 0.10116
−0.053243 0.87456 0.40559
0.10116 0.40559 0.82474
0.53642 0 0
0 0.42018 0
0 0 0.094866
The eigenbasis of Y is therefore the standard basis. Then δ((Y − 11)+; Y ) =
(0, 0, 0) and
(X + Y − 11)+ − (X − 11)+ =
−0.00018194 0.00052449 −0.0016345
0.00052449 0.2573 0.12368
−0.0016345 0.12368 0.04
so that δ((X+Y −11)+−(X−11)+; Y ) = (−0.00018194, 0.2573, 0.04). The first
entry is negative, violating the ≺dw relation, and thereby answering Question
2 in the negative. ✷
7 Further Applications of Y -dominated majorisation
One issue we had to address during our attempts at giving a positive answer
to Question 2 dealt with the possibility of reducing the question for convex
functions to convex angle functions. One way of doing so would have been
possible if the set of (monotonously increasing and convex) functions satisfying
(11) were closed under addition. While we were unable to prove this particular
statement (which is most likely false, anyway), Proposition 3 enables us to
prove the corresponding statement for the relation
δ(f(Y ); Y ) ≺dw δ(f(X + Y )− f(X); Y ). (22)
Proposition 4 Let all the eigenvalues of Y be distinct. Let f and g be func-
tions from R to R satisfying (22). Then f + g also satisfies (22).
Proof. By the assumption on the eigenvalues of Y , δ(A; Y ) equals Diag(A) in
a basis only depending on Y and is therefore a linear function of A. We can
therefore add up the inequalities (22) for f and g and obtain the corresponding
inequalities for f + g. ✷
A second application of Proposition 3 is a strengthening of the following Propo-
sition, which we also obtained in the course of our attempts at positively
answering Question 2.
Proposition 5 For X, Y ≥ 0 and ga(x) = ax+
, with a ≥ 0, the following
majorisation statement holds:
λ(ga(Y )) ≺w λ(ga(X + Y )− ga(X)).
Proof. From the proof of Lemma X.1.4 in [4], we have, for X, Y ≥ 0,
j ((X + 11)
−1 − (X + Y + 11)−1) ≤ λ
j(11 − (Y + 11)
Defining the function f(x) = x
= 1− (x+ 1)−1, this turns into:
j(f(X + Y )− f(X)) ≤ λ
j(f(Y )).
This implies the majorisation statement
j(f(X + Y )− f(X)) ≤
j(f(Y )). (23)
We want to prove a somewhat similar statement for the function ga(x). Since
both f and ga are monotonously increasing over R
+, and noting that ga(x) =
(a+ 1)x− f(x), we have
j(ga(Y ))= ga(λ
j (Y )) = (a+ 1)λ
j(Y )− f(λ
j(Y ))
j(f(Y ))= f(λ
j(Y )),
so that
j(ga(Y )) = (a+ 1)λ
j(Y )− λ
j(f(Y )).
This implies in particular
j (ga(Y )) = (a+ 1)
j (Y )−
j(f(Y ))
≤ (a+ 1)
j (Y )−
j(f(X + Y )− f(X)),
where we have inserted (23). Exploiting the well-known relation ([4], Th.
III.4.1)
j(A+B) ≤
j (A) +
j(B),
for A = (a+ 1)Y − f(X + Y ) + f(X) and B = f(X + Y )− f(X) then yields
j (ga(Y ))≤
j((a+ 1)Y − f(X + Y ) + f(X))
j(ga(X + Y )− ga(X)).
Proposition 3, with A = G = Y , B = f(Y ), C = f(X + Y ) − f(X), where
f(x) = x2/(x+ 1), then yields the following strengthening of Proposition 5:
Proposition 6 For X, Y ≥ 0, and ga(x) = ax+
, with a ≥ 0,
δ(ga(Y ); Y ) ≺dw δ(ga(X + Y )− ga(X); Y ).
Here we noted that ga(X + Y )− ga(X) = aY + f(X + Y )− f(X).
To end this Section, we present a third application of Proposition 3, namely
to the results of Kosem and Bourin-Uchiyama. Consider first inequality (3),
which holds for all non-negative concave functions f(x). In particular, it holds
for all functions f = ax+ f0(x), where f0 is non-negative concave, and a ≥ 0.
Inserting this in the eigenvalue-majorisation form of inequality (3), we get the
(A+B)-dominated majorisation relation
λ(a(A+B) + f0(A+B)) ≺w λ(a(A +B) + f0(A) + f0(B)),
for A,B ≥ 0. Proposition 3 then immediately yields the stronger form
δ(f(A+B);A+B) ≺dw δ(f(A) + f(B);A+B), (24)
for all non-negative concave functions f . The strengthening of inequality (4)
is performed in a completely identical way and yields the reversed inequality
of (24) for non-negative convex functions g such that g(0) = 0.
Acknowledgements
JSA thanks Professor Moin Uddin, Director of his institute for encourage-
ment and supporting his visit to attend the conference at Nova Southeastern
University, Fort Lauderdale, Florida, USA, which lead to his introduction to
Koenraad M.R. Audenaert and the completion of this work.
KA thanks the Institute for Mathematical Sciences, Imperial College London,
for support. His work is part of the QIP-IRC (www.qipirc.org) supported by
EPSRC (GR/S82176/0).
References
[1] T. Ando, “Comparison of norms |||f(A)− f(B)||| and |||f(|A−B|)|||,” Math. Z.
197, 403–409 (1988).
[2] T. Ando and X. Zhan, “Norm inequalities related to operator monotone
functions,” Math. Ann. 315, 771–780 (1999).
[3] J.S. Aujla and F.C. Silva, “Weak majorization inequalities and convex functions,”
Lin. Alg. Appl. 369, 217–233 (2003).
[4] R. Bhatia, Matrix Analysis, Springer, Heidelberg (1997).
[5] J.-C. Bourin and M. Uchiyama, “A matrix subadditivity inequality for f(A+B)
and f(A) + f(B),” Arxiv.org E-print math.FA/0702475 (2007).
[6] T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition,
Classics in Mathematics, Springer-Verlag, Berlin (1995).
[7] T. Kosem, “Inequalities between ||f(A + B)|| and ||f(A) + f(B)||,” Lin. Alg.
Appl. 418, 153–160 (2006).
http://arxiv.org/abs/math/0702475
Introduction
Preliminaries
Main Results
Proof of Proposition 1
On Y-dominated Majorisation
Counterexample to Question 2
Further Applications of Y-dominated majorisation
Acknowledgements
References
|
0704.0100 | Topology Change of Black Holes | Topology Change of Black Holes
Daisuke Ida
and Masaru Siino
1Department of Physics, Gakushuin University, Tokyo 171-8588, Japan.
2Department of Physics, Tokyo Institute of Technology, Tokyo 152-8550, Japan.
(April 1, 2007)
Abstract.
The topological structure of the event horizon has been investigated in terms of the
Morse theory. The elementary process of topological evolution can be understood
as a handle attachment. It has been found that there are certain constraints on
the nature of black hole topological evolution: (i) There are n kinds of handle
attachments in (n+1)-dimensional black hole space-times. (ii) Handles are further
classified as either of black or white type, and only black handles appear in real
black hole space-times. (iii) The spatial section of an exterior of the black hole
region is always connected. As a corollary, it is shown that the formation of a
black hole with an Sn−2 × S1 horizon from that with an Sn−1 horizon must be
non-axisymmetric in asymptotically flat space-times.
1. Introduction
Black holes in space-times of greater than or equal to five dimensions have rich
topological structure. According to the well-known results of Hawking concerning
the topology of black holes in four-dimensional space-time, the apparent horizon
or the spatial section of the stationary event horizon is necessarily diffeomorphic
to a 2-sphere. [1, 2] This follows from the fact that the total curvature, which is
the integral of the intrinsic scalar curvature over the horizon, is positive under the
dominant energy condition and from the Gauss-Bonnet theorem. Alternative and
improved proofs of Hawking’s theorem have been given by several authors. [3, 4, 5, 6]
However in higher dimensional space-times, an apparent horizon or the spatial
section of the stationary event horizon may not be a topological sphere, [7, 8, 9, 10]
because the Gauss-Bonnet theorem does not hold in such cases. Nevertheless,
the positivity of the total curvature of the horizon still holds. This puts certain
topological restrictions on the black hole topology, though they are rather weak. For
example, the apparent horizon in five-dimensional space-time can consist of finitely
many connected sums of copies of S3/Γ and copies of S2 × S1. In fact, exact
solutions representing a black hole space-time possessing a horizon of nonspherical
topology have recently been found in five-dimensional general relativity. When such
black holes with nontrivial topologies are regarded as being formed in the course
of gravitational collapse, questions regarding the evolution of the topology of black
holes naturally arise. Our purpose here is to understand the time evolution of the
topology of event horizons in a general setting. The relation between the crease set,
where the event horizon is nondifferentiable, and the topology of the event horizon
is studied in Refs. [11, 12, 13] for four-dimensional space-times. In the present
work, we carry out a systematic investigation and find useful rules to determine
admissible processes of topological evolution for time slicing of a black hole.
Our approach is to utilize the Morse theory [14, 15] in differential topology. The
Morse theory is useful for the purpose of understanding the topology of smooth
http://arxiv.org/abs/0704.0100v3
manifolds. The basic tool used in this approach is a smooth function on a differ-
entiable manifold. The event horizon, however, is not a differentiable manifold but
has a wedge-like structure at the past endpoints of the null geodesic generators
of the horizon. For this reason, we first smooth the wedge. Then, the smooth
time function which is assumed to exist plays the role of the Morse function on the
smoothed event horizon. According to the Morse theory, the topological evolution
of the event horizon can then be decomposed into elementary processes called “han-
dle attachments.” In such a process, starting with a spherical horizon, one adds
several handles, each characterized by the index of the critical points of the Morse
function, which is an integer ranging from 0 to n (the dimension of the smoothed
horizon as a differentiable manifold).
The purpose of the present article is to show that there are several constraints
on the handle attachments for real black hole space-times.
2. The Morse theory for event horizons
Let M be an (n+1)-dimensional asymptotically flat space-time. We require the
existence of a global time function t : M → R that is smooth and has an everywhere
time-like and future-pointing gradient. The event horizon H is defined as the
boundary of the causal past of the future null infinity H = ∂J−(I +). [2] We treat
the event horizon defined with respect to a single asymptotic end, unless otherwise
stated. In other words, the future null infinity, I +, is assumed to be connected.
The black hole region B is defined as the interior region of H , specifically, as B =
M \ J−(I +), and the exterior region E of the black hole region is its complement,
E = int(J−(I +)). We refer to the intersection of the black hole region and the
time slice Σ(t0) = {t = t0} as the black hole B(t0) = B ∩ Σ(t0) at time t = t0.
The exterior region at time t = t0 is, accordingly, written E (t0) = E ∩ Σ(t0).
One of most basic properties of the event horizon is that it is generated by null
geodesics without future endpoints. In general, the event horizon is not smoothly
imbedded into the space-time manifold M , but it has a wedge-like structure at
the past endpoints of the null geodesic generators, where distinct null geodesic
generators intersect. We call the set of past endpoints of null geodesic generators
of H , from which two or more null geodesic generators emanate, the crease set
S. [11, 12] When no crease set S exists between the time slices t = t1 and t = t2, the
null geodesic generators of H naturally define a diffeomorphism ∂B(t1) ≈ ∂B(t2).
Hence, the topological evolution of a black hole can take place only when the time
slice intersects the crease set S. Of course, the event horizon itself is a gauge-
independent object. Nevertheless, we often understand the dynamics of space-time
by scanning it along time slices. Thus, the topological evolution of a black hole
depends on the time function.
It is expected that Morse theory [14] provides useful techniques to analyze such
a process of topological evolution. Because the Morse theory is concerned with
functions on smooth manifolds, we first regularize H around the crease set S. The
event horizon is not necessarily smooth, even on H \ S, in the case that the future
null infinity I + has a pathological structure. [16] Here it is assumed that H is
smooth on H \ S. Then, small deformations of H near the crease set S will make
H a smooth hypersurface H̃ in M , while B̃(t0) remains deformed in such a manner
that ∂B̃(t0) = H̃ ∪ Σ(t0) holds and B̃(t0) remains homeomorphic to the original
black hole for all t0 ∈ R. This deformation is assumed to be such that the time
Figure 1. An example in which no smoothing procedure makes
t| eH a Morse function on H̃ . Here, the intersection of the crease set
S of the event horizon and t = t0 hypersurface has an accumulation
point.
function t| eH , which is the restriction of t on H̃, gives a Morse function on H̃ that
has only nondegenerate critical points, where the gradient of t| eH defined on H̃
becomes zero and where also the Hessian matrix (∂i∂jt| eH) of t| eH is nondegenerate.
Though this assumption should hold for a wide class of systems, it does not always
hold. Figure 1 gives an example for which no smoothing procedure makes the
induced time function t| eH a Morse function on H̃ , because the intersection of the
crease set S of the event horizon and the t = t0 hypersurface has an accumulation
point. It is highly nontrivial to determine whether such a smoothing procedure is
generically possible. It is, however, not easy nor the primary purpose of this article
to assertain the realm of validity of the assumption, and therefore we make this
assumption without inquiring into its validity.
According to the Morse Lemma, there is a local coordinate system {x1, · · · , xn}
on H̃ in the neighborhood of the critical point p ∈ H̃ such that the restriction t| eH
of the time function t on H̃ takes the form
t| eH(x
1, · · · , xn) = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2.
The integer λ, ranging from 0 to n, is called the index of the critical point p. The
topology of the black hole B̃(t) changes when Σ(t) pass through critical points,
or equivalently, when the time function t takes critical values. This implies that
critical points appears only near the crease set S.
The gradient-like vector field for t| eH is defined to be the tangent vector field X
on H̃ such that Xt| eH > 0 holds on H̃ , except for critical points, and has the form
X = −2x1
− · · · − 2xλ
+ 2xλ+1
∂xλ+1
+ · · ·+ 2xn
near the critical point of index λ, in terms of the standard coordinate system
appearing in the Morse Lemma. We choose a gradient-like vector field X such that
it coincides with the future-directed tangent vector field of null geodesic generators
of H , except in a small neighborhood of the crease set S (Fig. 2). The effect of
a critical point p of index λ is equivalent to the attachment of a λ-handle. [14, 15]
The handlebody is just a topological n-disk Dn ≈ In (I = [0, 1]), but it is regarded
as the product space Dn ≈ Dλ×Dn−λ (Fig. 3). The λ-handle attachment to an n-
dimensional manifold N with a boundary consists of the set hλ = (Dλ ×Dn−λ, f),
where the attaching map f induces the imbedding of ∂Dλ × Dn−λ ⊂ ∂Dn into
∂N (Fig. 4). The new manifold obtained through the λ-handle attachment to N is
Figure 2. The smoothing procedure of the event horizon H . The
gradient-like vector field on H̃ can be constructed through a slight
deformation of the null geodesic generators of H . Here, the effect
of the crease set S has been replaced by that of the critical points
p1, p2 and p3.
Figure 3. The local structure around the critical point p of index
λ. It can be seen that H̃t(p)+ǫ is homeomorphic to H̃t(p)−ǫ with a
λ-handle attached.
given by
N ∪ hλ = N ∪ (Dλ ×Dn−λ)/(x ∼ f(x)), (x ∈ ∂Dλ ×Dn−λ).
Let us denote by H̃t0 the t ≤ t0 part of H̃. Then, H̃t(p)+ǫ (ǫ > 0) just above the
critical point p of index λ is homeomorphic (in fact diffeomorphic, taking account
of the smoothing procedure) to that just below p, H̃t(p)−ǫ attached with a λ-handle,
H̃t(p)+ǫ ≈ H̃t(p)−ǫ ∪ h
if there are no other critical points satisfying t(p)−ǫ ≤ t ≤ t(p)+ǫ. The handlebody
itself is denoted by hλ as well.
Let us consider several examples. The 0-handle attachment does not need an
attaching map f . It simply corresponds to the emergence of the (n − 1)-sphere
Sn−1 ≈ ∂Dn as a black hole horizon ∂B(t). A typical example is the creation of a
black hole (Fig. 5): A black hole always emerges as 0-handle attachment. The other
Figure 4. The attachment of a 1-handle and a 2-handle to a
3-manifold N creates a new 3-manifold N ∪ h1 ∪ h2.
Figure 5. The emergence of a black hole through a 0-handle attachment.
Figure 6. The emergence of a bubble in the black hole region by
0-handle attachment, which does not occur in the real black hole
space-times.
Figure 7. The collision of a pair of black holes, creating a single
black hole, is realized through 1-handle attachment.
possiblity is the creation of a bubble that is subset of J−(I +) in a black hole region
(Fig. 6). One might think that this corresponds to wormhole creation between the
internal and external regions of the event horizon. Although in the framework of the
standard Morse theory on H̃ , these two examples are indistinguishable, we below
see that the latter process is in fact impossible.
Next, we consider 1-handle attachment. A typical example is the collision of two
black holes. A 1-handle serves as a bridge connecting black holes, or it corresponds
to taking the connected sum of each component of multiple black holes (Fig. 7).
Figure 8. The bifurcation of one black hole into two is represeted
by an (n − 1)-handle attachment. This, however, never occurs in
real black hole space-times.
Figure 9. The structure of λ-handle. The core Dλ × {0} corre-
sponds to the stable submanifold with respect to the flow gener-
ated by the gradient-like vector field, and the co-core {0} ×Dn−λ
corresponds to the unstable submanifold.
The time reversal of the collision of black holes consists of the bifurcation of one
black hole into two. This would be realized through an (n− 1)-handle attachment,
if such a process were possible (Fig. 8). It is, however, well known that such a
process is forbidden. [2] In general, the time reversal of the λ-handle attachment
corresponds to (n− λ)-handle attachment.
Before discussing general cases, let us consider the structure of a handlebody.
Recall that a λ-handle consists of the product space Dλ × Dn−λ. The subset
Dλ × {0} ⊂ Dλ × Dn−λ is called the core of the handlebody, and {0} ×Dn−λ ⊂
Dλ ×Dn−λ is called the co-core. The core and co-core intersect transversely at a
point. This point can be regarded as a critical point p.
Let us refer to the subset Ws(p) of H̃
(1) Ws(p) = {q ∈ M | lim
expq tX = p}
which consists of points that converge to p along the flow generated by the gradient-
like vector field X , as the stable manifold with respect to the critical point p. The
stable manifold Ws(p) is homeomorphic to R
λ if the index of p is given by λ. [17]
Similarly, let us refer to the subset Wu(p) ⊂ H̃ consisting of points which converge
to p along the flow generated by (−X) as the unstable manifold with respect to p.
For the unstable manifold, Wu(p) ≈ R
n−λ holds. The portions of the stable and
unstable manifolds in the handlebody can be regarded as corresponding to the core
and co-core, respectively.
The effect of smoothing the event horizonH to H̃ is to deform the null vector field
generating H into a gradient-like vector field X . The primary difference between
the null geodesic generators and the flow generated by X is that the former does
not have future endpoints, but the latter can. Thus, there are admissible and
inadmissible processes for the smoothed manifold H̃. An admissible process is
given by H̃ , which is obtained from an in priciple realizable event horizon, while an
inadmissible one is constructed from a spurious event horizon, i.e., one that consists
of the null hypersurface containing null geodesic generators with a future endpoint.
3. The structure of the critical points
The spatial topology of a black hole changes only when the time function takes a
critical value. The time evolution of the black hole topology can be understood by
considering its local structure around critical points. To determine whether a given
topological change is admissible or inadmissible, it is not sufficient to consider only
the intrinsic structure of the event horizon. Rather, it is required to take account
of its imbedding structure relative to the space-time.
In a time slice, any point separate from H̃ belongs to either of the black hole
or the exterior of the black hole region. It is useful to consider the local behavior
of the black hole region or the exterior region near the critical point p. Let us call
the exterior E of the black hole region simply the exterior region, for brevity. The
exterior region is slightly deformed by the smoothing procedure. The deformed
exterior region is denoted by Ẽ , and the deformed exterior region at the time t by
Ẽ (t) = Ẽ ∩ Σ(t) = Σ(t) \ B̃(t).(2)
The 0-handle is placed at some t ≥ t(p). Such an attachment describes the emer-
gence of the black hole region at the critical point p and its expansion with time.
The emergence of a bubble, which consists of a part of J−(I +), in the background
of the black hole region would also be described by a 0-handle attachment. This,
however, never occurs, as we explain below in detail. Hence, a 0-handle attachment
always describes the creation of a black hole homeomorphic to the n-disk.
An n-handle attachment corresponds to the time reversal of a 0-handle attache-
ment. This process, however, never occurs in real black hole space-time. An n-
handle is defined for t ≤ t(p), which means that it terminates at the critical point
p. The crease set is isolated into critical points during the course of the smoothing
procedure. The gradient-like vector field, which can be regarded as being tangent
to the generator of the deformed event horizon H̃ , may have several inward (con-
verging) directions at the critical point due to this smoothing procedure, while the
original null generator of the event horizon does not have an inward direction at
the crease set. In the case of the n-handle, all the directions become inward at the
critical point. This implies that the null generators of the event horizon H must
have future endpoints at the critical point, which is, of course, impossible. It is
thus seen that an n-handle attachment never occurs in real black hole space-times.
The remaining cases are λ-handle attachments for 1 ≤ λ ≤ n − 1. In these
cases, the λ-handle lies on either side of the critical point p both in the future
[t > t(p)] and in the past [t < t(p)]. Then, we consider the case in which the handle
exists during the sufficiently small time interval t ∈ [t(p) − δ, t(p) + δ] (δ > 0), to
understand the topological change of the black hole region at the critical point p.
Figure 10. The neighborhood U of p is separated by hλ into the
future region, U+, and the past region, U−.
First, we introduce a coordinate system {t, xi} (i = 1, · · · , n) in the neighborhood
U of p, where t is a given function of time, and {xi} is the extension over U of
the cannonical coordinate appearing in the Morse Lemma such that each curve
(x1, · · · , xn) = [const] is timelike in U . We assume that U is the solid cylinder
given by t ∈ [t(p)−δ, t(p)+δ],
(xi)2 ≤ δ. In this coordinate system, the λ-handle
hλ is given by the saddle surface
t = t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , which is an acausal set if the constant δ is taken sufficiently small, since hλ
is tangent to the space-like hypersurface t = t(p) at p. Therefore, hλ separates
U into two open subsets, the future and past regions U+ and U− of U , where
U+ and U− are the subsets lying chronological future and past, respectively, of
hλ: U± = I±(hλ) ∩ U . Explicitly, the future and past regions U± are the regions
satisfying
t ≷ t(p)− (x1)2 − · · · − (xλ)2 + (xλ+1)2 + · · ·+ (xn)2
in U , respectively (Fig. 10).
Because the λ-handle is a subset of the black hole boundary H̃ , one of U± is
contained in the black hole region, B̃, and the other in the exterior region, Ẽ .
However, the future region U+ of U is always included in the black hole region, i.e.
U+ ⊂ B̃, and hence we have U− ⊂ Ẽ , since the horizon is the boundary of the past
set, J−(I +). Therefore, the black hole region B̃(t(p) − ǫ) ∩ U in U at the time
t = t(p)− ǫ just before the critical time is given by
(x1)2 + · · ·+ (xλ)2 > (xλ+1)2 + · · ·+ (xn)2 + ǫ,
which is homotopic to the (λ − 1)-sphere Sλ−1. (For λ = 1, S0 simply consists of
two points.) Similarly, B̃(t(p) + ǫ) ∩ U just after the critical time is given by
(x1)2 + · · ·+ (xλ)2 + ǫ > (xλ+1)2 + · · ·+ (xn)2,
which is homotopic to the n-disk. In this way, the black hole region restricted to
the small neighborhood of the critical point p is initially homotopic to a sphere.
Then, the internal region of the sphere is filled up at the critical time t = t(p) and
eventually becomes homotopically trivial. The exterior region, Ẽ (t) ∩ U , in U is
initially homotopic to an n-disk for t = t(p) − ǫ. Then, its (n − λ)-dimensional
direction is penetrated by the black hole region at t = t(p), and thus it becomes
homotopic to an (n− λ− 1)-sphere Sn−λ−1 for t = t(p) + ǫ. If the spurious event
horizon is also taken into account, the future region U+ might be a subset of Ẽ , and
therefore the past region U− might be a subset of B̃. Then, the black hole region in
the λ-handle might be homotopic to an n-disk initially and become homotopic to an
(n−λ−1)-sphere finally, and vice versa for the exterior region. Let us refer to such
a topological change of the black hole region B̃(t)∩U from a region homotopic to a
sphere to a region homotopic to a disk as a black handle attachment, and that from
a region homotopic to a disk to a region homotopic to the sphere as a white handle
attachment. The above observation shows that only a black handle attachment
occurs if a sufficiently small neighborhood of the critical point is considered. For
example, a collision of black holes corresponds to a black 1-handle attachment, while
the bifurcation of a black hole corresponds to a white (n − 1)-handle attachment
in the sense that the homotopy type of the exterior region Ẽ (t) ∩ U changes from
that of Sn−2 to that of Dn. This local argument also elucidates te reason that a
black hole collision is admissible while a black hole bifurcation, which is its time
reversal, is inadmissible. We also note that the effect of time reversal is to convert
a black λ-handle attachment into a white (n− λ)-handle attachment.
It is appropriate to refer to the 0-handle attachment corresponding to the cre-
ation of a black hole as a black 0-handle attachment. Then, the proposition above
also applies to a 0-handle attachment.
4. Connectedness of the exterior region
There also exist processes that are unrealizable due to global conditions. Let
us, for a moment, consider the event horizon in maximally extended Schwarzschild
space-time. Though we are interested in the event horizon defined with respect to
a specific asymptotic end, for the purpose of explanation, we examine the event
horizon defined with respect to a pair of asymptotic ends in Schwarzschild space-
time (Fig. 11).
Let I +1 and I
2 be the pair of future null infinities of the maximally extended
Schwarzschild space-time. The event horizon here is defined by H = ∂J−(I +1 ∪
2 ), which is nondifferentiable at the bifurcate horizon F = ∂J
−(I +1 )∩∂J
−(I +2 ).
Let t be a global time function and χ be a global radial coordinate function such
that each two-surface t, χ = [const] is invariant under the SO(3) isometry. These
coordinates are chosen such that the bifurcation surface F is located at t = χ = 0
and the event horizon H is determined by t = |χ| around F . The smoothed event
horizon H̃ is also taken to be invariant under the SO(3) isometry. Due to the
symmetry of the configuration, the time function t has critical points of degenerate
type. In fact, any point on bifurcate horizon F is critical. Here, we are not interested
in such a nongeneric situation. Instead, we consider a slightly different time slicing
determined by the new time function
t′ = t+ ǫ sin2
where ǫ > 0 is a sufficiently small positive constant and ϑ, which satisfies 0 ≤ ϑ ≤ π,
is the usual polar coordinate of the 2-sphere. Then, there appears only a pair of
isolated critical points at the north pole (ϑ = 0) and the south pole (ϑ = π) on
the bifurcate horizon F , and the time function t′ becomes the Morse function on
H̃. At the time t′ = 0, the black hole appears at the north pole. This is the
0-handle attachment. The black hole formed there grows into a geometrically thick
spherical shell with a hole at the south pole, which is nevertheless a topological
3-disk. At the time t′ = ǫ, the puncture at the south pole is filled, and the black
hole region becomes topologically S2 × [0, 1]. The deformed event horizon H̃ splits
into a disjoint union of a pair of 2-spheres. This is the 2-handle attachment.
This kind of 2-handle attachment occurs because the event horizon is defined
with respect to the two asymptotic ends, which is in general inadmissible if the
future null infinity is connected, as we assume from this point. To understand the
above statement, it should be noted that there is no process through which the
several connected components of the exterior region Ẽ (t) = Ẽ ∩ Σ(t) at time t
merge together at a later time because such a handle attachment is not admissible.
It is also seen that no connected component of Ẽ (t) disappears, because possi-
ble n-handle attachments are inadmissible. These facts imply that the number of
connected components of the exterior region Ẽ (t) cannot decrease with the time
function t. On the other hand, there is only one connected component of the exte-
rior region Ẽ (t) for sufficiently large t, because of the connectedness of I +. This
observation shows that the exterior region Ẽ (t) remains connected in any process.
The only possible process through which the number of connected components
of the exterior region Ẽ (t) changes is an (n− 1)-handle attachment, as constructed
above in the Schwarzschild space-time. This is because the subset Dλ × ∂Dn−λ of
H2 H1
t' = t' (p)
B (t'(p))
II 12
Figure 11. The figure on the left is a conformal diagram of the
maximally extended Schwarzschild space-time. The structure of
the event horizon defined with respect to the two asymptotic ends
is depicted on the right, with one dimension omitted. The shaded
region represents the black hole region at the critical time t = t(p).
This corresponds to the 2-handle attachment, where the exterior
region is separated into a pair of connected components.
the boundary of the λ-handle
∂hλ ≈ (∂Dλ ×Dn−λ) ∪ (Dλ × ∂Dn−λ),
namely the part of ∂hλ which is the complement of the preimage of the attaching
f : ∂hλ ⊃ ∂Dλ ×Dn−λ → H̃t,
is disconnected only when λ = n − 1. In this case, the homotopy type of the
exterior region Ẽ (t) changes from that of an n-disk to that of S0, namely two points.
Note, however, that this does not imply that the exterior region Ẽ (t) is always
separated into two disconnected parts through the (n− 1)-handle attachment. For
example, a transition from the black ring horizon ≈ Sn−2 × S1 to the spherical
black hole horizon ≈ Sn−1 is realized through a black (n− 1)-handle attachment,
which pinches the longitude {a point}×S1 ⊂ Sn−2 ×S1 into a point. The exterior
region Ẽ (t) remains connected all the while. Thus, there are both admissible and
inadmissible processes for (n−1)-handle attachments. An (n−1)-handle attachment
is inadmissible if it separates the exterior region Ẽ (t).
5. Concluding remarks
The arguments given in this paper are summerized by the following rules. As-
sume that (i) an (n+ 1)-dimensional space-time M is asymptotically flat and the
future null infinity I + is connected, or the event horizon H = ∂J−(I +) is defined
with respect to a single asymptotic end, (ii) the space-time M admits a smooth
global time function t, (iii) the event horizon H can be deformed so that the black
hole B̃(t) deformed accordingly at each time t is smooth and homeomorphic to orig-
inal one B(t) at each time t and the time function t becomes the Morse function
on H̃. Then, the topological evolution of the event horizon can be regarded as a
λ-handle attachment (0 ≤ λ ≤ n) subject to the following rules:
(1) The n-handle attachment is inadmissible.
(2) Only the black λ-handle attachment (0 ≤ λ ≤ n − 1), where the black
hole region in the neighborhood of the critical point varies from the region
homotopic to the sphere Sλ−1 (regarded as the empty set for λ = 0) to the
n-disk Dn, is admissible.
(3) The (n − 1)-handle attachment which separates the spatial section of the
exterior region of the black hole is inadmissible.
The first rule simply states that no connected component of a black hole disap-
pears. It also implies that if a bubble of the exterior region forms within the black
hole region, it does not vanish.
The second rule is concerned with the imbedding structure of the event horizon
relative to the space-time manifold. The neighborhood of the critical point is sep-
arated into two regions by the event horizon. One changes homotopically from a
sphere to a disk and the other from a disk to a sphere. We call it a black handle at-
tachment when the former corresponds to the black hole region and a white handle
attachment otherwise. Then, the second rule states that a white handle attach-
ment never occurs. The reverse process, in which a black hole region homotopically
changes from a disk to a sphere, is ruled out. A white 0-handle attachment, which
Figure 12. Black ring formation from a spherical black hole must
be non-axisymmetric in real black hole space-times.
describes the emergence of the exterior region, is also forbidden. This gives an-
other reason for the well-known result that a black hole cannot bifurcate, because
it corresponds to a white (n− 1)-handle attachment.
The second rule applies to more general situations. For example, let us consider
the topological evolution of the event horizon from Sn−1 to Sn−2 × S1 in (n+ 1)-
dimensional space-time (n ≥ 3). When it is realized with a single critical point, it
corresponds to a 1-handle attachment. Here, one might expect two possibilities if
the second rule is not considered. One possibility is that the 1-handle is attached in
the exterior region of the black hole. This is locally equivalent to the merging of a
pair of black holes, where these two black holes are connected elsewhere irrelevant.
The other possibility is that it is attached from the inside such that the 1-handle
pierces the black hole region. In asymptotically flat space-times, only the latter
includes axisymmetric configurations such that a spherical black hole is pinched
out along the symmetric axis; here the axisymmetric configuration is such that
the space-time possesses the SO(n − 1) isometry and the time slicing respects
this symmetry. However, this latter possibility corresponds to a white 1-handle
attachment, which is impossible, and only the former, which corresponds to a black
1-handle attachment, is possible. In particular, a transition from a spherical event
horizon (≈ Sn−1) to a black ring horizon (≈ Sn−2×S1) in asymptotically flat space-
times is always non-axisymmetric in the sense that such a configuration cannot
possess SO(n− 1) symmetry (Fig. 12).
While the apparent horizon must be diffeomorphic to a two-sphere in four-
dimensional space-times under the dominant energy condition, a torus event hori-
zon may appear, even under the dominant energy condition, via a black 1-handle
attachment to the spherical horizon. More generally, an event horizon with an
arbitrary number of genura may be formed by several black 1-handle attachments.
The third rule is not directly determined by the local structure of the critical
point. It states that the exterior region E (t) = E ∩ Σ(t) at each time is always
connected under the assumption that I + is connected. Thus, the possibility that
there forms a bubble of the exterior region inside the black hole horizon is ruled
out. It should, however, be noted that such a process is possible if I + consists
of several connected components. This may also be related to the topological
censorship theorem. [19] The topological censorship theorem states that all causal
curves from I − to I + are homotopic under the null energy condition. This also
forbids the formation of a bubble of the exterior region inside the black hole, because
otherwise there would be two nonhomotopic causal curves from I − to I +, one
passing inside the horizon and the other outside. Our argument, however, does not
depend on energy conditions.
References
[1] S. W. Hawking, Commun. Math. Phys. 25 (1972), 152.
[2] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-times (London, Cam-
bridge University Press, 1973).
[3] D. Gannon, Gen. Relat. Gravit. 7 (1974), 219.
[4] P. T. Chrusciel and R. M. Wald, Class. Quantum Grav. 11 (1994), L147; gr-qc/9410004.
[5] T. Jacobson and S. Venkataramani, Class. Quantum Grav. 12 (1995), 1055; gr-qc/9410023.
[6] S. F. Browdy and G. J. Galloway, J. Math. Phys. 36 (1995), 4952.
[7] M. l. Cai and G. J. Galloway, Class. Quantum Grav. 18 (2001), 2707; hep-th/0102149.
[8] C. Helfgott, Y. Oz and Y. Yanay, JHEP 0602 (2006), 025; hep-th/0509013.
[9] G. J. Galloway and R. Schoen, Commun. Math. Phys. 266 (2006), 571; gr-qc/0509107.
[10] G. J. Galloway, gr-qc/0608118.
[11] M. Siino, Phys. Rev. D 58 (1998), 104016; gr-qc/9701003.
[12] M. Siino and T. Koike, arXiv:gr-qc/0405056.
[13] S. L. Shapiro, S. A. Teukolsky and J. Winicour, Phys. Rev. D 52 (1995), 6982.
[14] J. Milnor, Lectures on h-cobordism theorem (Princeton, Princeton University Press, 1965).
[15] I. Tamura, The differential topology (Tokyo, Iwanami Shoten Publishers, 1978).
[16] P. T. Chrusciel and G. J. Galloway, Commun. Math. Phys. 193 (1998), 449 gr-qc/9611032.
[17] S. Smale, Ann. of Math. 74 (1961), 199.
[18] R. Emparan and H. S. Reall, Phys. Rev. Lett. 88 (2002), 101101; hep-th/0110260.
[19] J. L. Friedman, K. Schleich and D. M. Witt, Phys. Rev. Lett. 71 (1993), 1486 [Errata; 75
(1995), 1872]; gr-qc/9305017.
http://arxiv.org/abs/gr-qc/9410004
http://arxiv.org/abs/gr-qc/9410023
http://arxiv.org/abs/hep-th/0102149
http://arxiv.org/abs/hep-th/0509013
http://arxiv.org/abs/gr-qc/0509107
http://arxiv.org/abs/gr-qc/0608118
http://arxiv.org/abs/gr-qc/9701003
http://arxiv.org/abs/gr-qc/0405056
http://arxiv.org/abs/gr-qc/9611032
http://arxiv.org/abs/hep-th/0110260
http://arxiv.org/abs/gr-qc/9305017
1. Introduction
2. The Morse theory for event horizons
3. The structure of the critical points
4. Connectedness of the exterior region
5. Concluding remarks
References
|
0704.0101 | The birth of string theory | The birth of string theory
Paolo Di Vecchia1
Nordita, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark [email protected]
Summary. In this contribution we go through the developments that in the years
from 1968 to about 1974 led from the Veneziano model to the bosonic string theory.
They include the construction of the N-point amplitude for scalar particles, its
factorization through the introduction of an infinite number of oscillators and the
proof that the physical subspace was a positive definite Hilbert space. We also discuss
the zero slope limit and the calculation of loop diagrams. Lastly, we describe how
it finally was recognized that a quantum relativistic string theory was the theory
underlying the Veneziano model.
1 Introduction
The sixties was a period in which strong interacting processes were studied
in detail using the newly constructed accelerators at Cern and other places.
Many new hadronic states were found that appeared as resonant peaks in var-
ious cross sections and hadronic cross sections were measured with increasing
accuracy. In general, the experimental data for strongly interacting processes
were rather well understood in terms of resonance exchanges in the direct
channel at low energy and by the exchange of Regge poles in the transverse
channel at higher energy. Field theory that had been very successful in de-
scribing QED seemed useless for strong interactions given the big number of
hadrons to accomodate in a Lagrangian and the strength of the pion-nucleon
coupling constant that did not allow perturbative calculations. The only do-
main in which field theoretical techniques were successfully used was current
algebra. Here, assuming that strong interactions were described by an almost
chiral invariant Lagrangian, that chiral symmetry was spontaneously broken
and that the pion was the corresponding Goldstone boson, field theoretical
methods gave rather good predictions for scattering amplitudes involving pi-
ons at very low energy. Going to higher energy was, however, not possible
with these methods.
Because of this, many people started to think that field theory was use-
less to describe strong interactions and tried to describe strong interacting
http://arxiv.org/abs/0704.0101v1
2 Paolo Di Vecchia
processes with alternative and more phenomenological methods. The basic
ingredients for describing the experimental data were at low energy the ex-
change of resonances in the direct channel and at higher energy the exchange
of Regge poles in the transverse channel. Sum rules for strongly interacting
processes were saturated in this way and one found good agreement with the
experimental data that came from the newly constructed accelerators. Be-
cause of these successes and of the problems that field theory encountered to
describe the data, it was proposed to construct directly the S matrix without
passing through a Lagrangian. The S matrix was supposed to be constructed
from the properties that it should satisfy, but there was no clear procedure on
how to implement this construction1. The word “bootstrap” was often used
as the way to construct the S matrix, but it did not help very much to get an
S matrix for the strongly interacting processes.
One of the basic ideas that led to the construction of an S matrix was
that it should include resonances at low energy and at the same time give
Regge behaviour at high energy. But the two contributions of the resonances
and of the Regge poles should not be added because this would imply double
counting. This was called Dolen, Horn and Schmidt duality [2]. Another idea
that helped in the construction of an S matrix was planar duality [3] that
was visualized by associating to a certain process a duality diagram, shown in
Fig. (1), where each meson was described by two lines representing the quark
and the antiquark. Finally, also the requirement of crossing symmetry played
a very important role.
Fig. 1. Duality diagram for the scattering of four mesons
Starting from these ideas Veneziano [4] was able to construct an S matrix
for the scattering of four mesons that, at the same time, had an infinite number
of zero width resonances lying on linearly rising Regge trajectories and Regge
behaviour at high energy. Veneziano originally constructed the model for the
1 For a discussion of S matrix theory see Ref.s [1]
The birth of string theory 3
process ππ → πω, but it was immediately extended to the scattering of four
scalar particles.
In the case of four identical scalar particles, the crossing symmetric scat-
tering amplitude found by Veneziano consists of a sum of three terms:
A(s, t, u) = A(s, t) +A(s, u) +A(t, u) (1)
where
A(s, t) =
Γ (−α(s))Γ (−α(t))
Γ (−α(s)− α(t))
dxx−α(s)−1(1− x)−α(t)−1 (2)
with linearly rising Regge trajectories
α(s) = α0 + α
′s (3)
This was a very important property to implement in a model because it was
in agreement with the experimental data in a wide range of energies. s, t and
u are the Mandelstam variables:
s = −(p1 + p2)2 , t = −(p3 + p2)2 , u = −(p1 + p3)2 (4)
The three terms in Eq. (1) correspond to the three orderings of the four parti-
cles that are not related by a cyclic or anticyclic 2 permutation of the external
legs. They correspond, respectively, to the three permutations: (1234), (1243)
and (1324) of the four external legs. They have only simple pole singularities.
The first one has only poles in the s and t channels, the second only in the s
and u channels and the third only in the t and u channels. This property fol-
lows directly from the duality diagram that is associated to each inequivalent
permutation of the external legs. In fact, at that time one used to associate
to each of the three inequivalent permutations a duality diagram where each
particle was drawn as consisting of two lines that rappresented the quark and
antiquark making up a meson. Furthermore, the diagram was supposed to
have only poles singularities in the planar channels which are those involving
adjacent external lines. This means that, for instance, the duality diagram
corresponding to the permutation (1234) has only poles in the s and t chan-
nels as one can see by deforming the diagram in the plane in the two possible
ways shown in figure (2).
This was a very important property of the duality diagram that makes
it qualitatively different from a Feynman diagram in field theory where each
diagram has only a pole in one of the three s, t and u channels and not
simultaneously in two of them. If we accept the idea that each term of the
sum in Eq. (1) is described by a duality diagram, then it is clear that we
2 An anticyclic permutation corresponding, for instance, to the ordering (1234) is
obtained by taking the reverse of the original ordering (4321) and then performing
a cyclic permutation.
4 Paolo Di Vecchia
Fig. 2. The duality diagram contains both s and t channel poles
do not need to add terms corresponding to equivalent diagrams because the
corresponding duality diagram is the same and has the same singularities. It
is now clear that it was in some way implicit in this picture the fact that the
Veneziano model corresponds to the scattering of relativistic strings. But at
that time the connection was not obvious at all. The only S matrix property
that the Veneziano model failed to satisfy was the unitarity of the S matrix.
because it contained only zero width resonances and did not have the various
cuts required by unitarity. We will see how this property will be implemented.
Immediately after the formulation of the Veneziano model, Virasoro [5]
proposed another crossing symmetric four-point amplitude for scalar particles
that consisted of a unique piece given by:
A(s, t, u) ∼
Γ (−α(u)
)Γ (−α(s)
)Γ (−α(t)
Γ (1 +
)Γ (1 +
)Γ (1 +
where
α(s) = α0 + α
′s (6)
The model had poles in all three s, t and u channels and could not be written
as sum of three terms having poles only in planar diagrams. In conclusion,
the Veneziano model satisfies the principle of planar duality being a crossing
symmetric combination of three contributions each having poles only in the
planar channels. On the other hand, the Virasoro model consists of a unique
crossing symmetric term having poles in both planar and non-planar channels.
The attempts to construct consistent models that were in good agreement
with the strong interaction phenomenology of the sixties boosted enormously
the activity in this research field. The generalization of the Veneziano model to
the scattering ofN scalar particles was built, an operator formalism consisting
of an infinite number of harmonic oscillators was constructed and the complete
spectrum of mesons was determined. It turned out that the degeneracy of
states grew up exponentially with the mass. It was also found that the N
point amplitude had states with negative norm (ghosts) unless the intercept
of the Regge trajectory was α0 = 1 [6]. In this case it turned out that the
model was free of ghosts but the lowest state was a tachyon. The model was
called in the literature the “dual resonance model”.
The birth of string theory 5
The model was not unitary because all the states were zero width reso-
nances and the various cuts required by unitarity were absent. The unitarity
was implemented in a perturbative way by adding loop diagrams obtained by
sewing some of the external legs together after the insertion of a propagator.
The multiloop amplitudes showed a structure of Riemann surfaces. This be-
came obvious only later when the dual resonance model was recognized to
correspond to scattering of strings.
But the main problem was that the model had a tachyon if α0 = 1 or had
ghosts for other values of α0 and was not in agreement with the experimental
data: α0 was not equal to about
as required by experiments for the ρ
Regge trajectory and the external scalar particles did not behave as pions
satisfying the current algebra requirements. Many attempts were made to
construct more realistic dual resonance models, but the main result of these
attempts was the construction of the Neveu-Schwarz [7] and the Ramond [8]
models, respectively, for mesons and fermions. They were constructed as two
independent models and only later were recognized to be two sectors of the
same model. The Neveu-Schwarz model still contained a tachyon that only in
1976 through the GSO projection was eliminated from the physical spectrum.
Furthermore, it was not properly describing the properties of the physical
pions.
Actually a model describing ππ scattering in a rather satisfactory way
was proposed by Lovelace and Shapiro [9] 3. According to this model the
three isospin amplitudes for pion-pion scattering are given by:
[A(s, t) +A(s, u)]− 1
A(t, u)
A1 = A(s, t)−A(s, u) A2 = A(t, u) (7)
where
A(s, t) = β
Γ (1− α(s))Γ (1 − α(t))
Γ (1− α(t)− α(s))
; α(s) = α0 + α
′s (8)
The amplitudes in eq.(7) provide a model for ππ scattering with linearly rising
Regge trajectories containing three parameters: the intercept of the ρ Regge
trajectory α0, the Regge slope α
′ and β. The first two can be determined by
imposing the Adler’s self-consistency condition, that requires the vanishing of
the amplitude when s = t = u = m2π and one of the pions is massless, and the
fact that the Regge trajectory must give the spin of the ρ meson that is equal
to 1 when
s is equal to the mass of the ρ meson mρ. These two conditions
determine the Regge trajectory to be:
α(s) =
s−m2π
m2ρ −mπ2
= 0.48 + 0.885s (9)
3 See also Ref. [10].
6 Paolo Di Vecchia
Having fixed the parameters of the Regge trajectory the model predicts the
masses and the couplings of the resonances that decay in ππ in terms of a
unique parameter β. The values obtained are in reasonable agreement with
the experiments. Moreover, one can compute the ππ scattering lenghts:
a0 = 0.395β a2 = −0.103β (10)
and one finds that their ratio is within 10% of the current algebra ratio given
by a0/a2 = −7/2. The amplitude in eq.(8) has exactly the same form as that
for four tachyons of the Neveu-Schwarz model with the only apparently minor
difference that α0 = 1/2 (for mπ = 0) instead of 1 as in the Neveu-Schwarz
model. This difference, however, implies that the critical space-time dimension
of this model is d = 4 4 and not d = 10 as in the Neveu-Schwarz model. In
conclusion this model seems to be a perfectly reasonable model for describing
low-energy ππ scattering. The problem is, however, that nobody has been able
to generalize it to the multipion scattering and therefore to get the complete
meson spectrum.
As we have seen the S matrix of the dual resonance model was constructed
using ideas and tools of hadron phenomenology of the end of the sixties.
Although it did not seem possible to write a realistic dual resonance model
describing the pions , it was nevertheless such a source of fascination for those
who actively worked in this field at that time for its beautiful internal structure
and consistency that a lot of energy was used to investigate its properties and
for understanding its basic structure. It turned out with great surprise that
the underlying structure was that of a quantum relativistic string.
The aim of this contribution is to explain the logic of the work that was
done in the years from 1968 to 1974 5 in order to uncover the deep properties of
this model that appeared from the beginning to be so beautiful and consistent
to deserve an intensive study.
This seems to me a very good way of celebrating the 65th anniversary of
Gabriele who is the person who started and also contributed to develop the
whole thing with his deep physical intuition.
2 Construction of the N -point amplitude
We have seen that the construction of the four-point amplitude is not sufficient
to get information on the full hadronic spectrum because it contains only
those hadrons that couple to two ground state mesons and does not see those
intermediate states which only couple to three or to an higher number of
ground state mesons [12]. Therefore, it was very important to construct the
N -point amplitude involving identical scalar particles. The construction of
4 This can be checked by computing the coupling of the spinless particle at the
level α(s) = 2 and seeing that it vanishes for d = 4.
5 Reviews from this period can be found in Ref. [11]
The birth of string theory 7
the N -point amplitude was done in Ref. [13] (extending the work of Ref. [14])
by requiring the same principles that have led to the construction of the
Veneziano model, namely the fact that the axioms of S-matrix theory be
satisfied by an infinite number of zero width resonances lying on linearly
rising Regge trajectories and planar duality.
The fully crossing symmetric scattering amplitude of N identical scalar
particles is given by a sum of terms corresponding to the inequivalent permu-
tations of the external legs:
An (11)
Also in this case two permutations of the external legs are inequivalent if they
are not related by a cyclic or anticyclic permutation. Np is the number of
inequivalent permutations of the external legs and is equal to Np =
(N−1)!
and each term has only simple pole singularities in the planar channels. Each
planar channel is described by two indices (i, j), to mean that it includes the
legs i, i+ 1, i+ 2 . . . j − 1, j, by the Mandelstam variable
sij = −(pi + pi+1 + . . .+ pj)2 (12)
and by an additional variable uij whose role will become clear soon. It is
clear that the channels (ij) and (j + 1, i− 1) 6 are identical and they should
be counted only once. In the case of N identical scalar particles the number
of planar channels is equal to
N(N−3)
. This can be obtained as follows. The
independent planar diagrams involving the particle 1 are of the type (1, i)
where i = 2 . . .N − 2. Their number is N − 3. This is also the number of
planar diagrams involving the particle 2 and not the 1. The number of planar
diagrams involving the particle 3 and not the particles 1 and 2 is equal to
N − 4. In general the number of planar diagrams involving the particle i and
not the previous ones from 1 to i-1 is equal to N − 1− i. This means that the
total number of planar diagram is equal to:
2(N − 3) +
(N − 1− i) = 2(N − 3) +
= 2(N − 3) +
(N − 4)(N − 3)
N(N − 3)
If one writes down the duality diagram corresponding to a certain planar
ordering of the external particles, it is easy to see that the diagram can have
simultaneous pole singularities only in N − 3 channels. The channels that
allow simultaneous pole singularities are called compatible channels, the other
6 This channel includes the particles (j + 1, . . . , N, 1, . . . i− 1).
8 Paolo Di Vecchia
are called incompatible. Two channels (i,j) and (h,k) are incompatible if the
following inequalities are satisfied:
i ≤ h ≤ j ; j + 1 ≤ k ≤ i− 1 (14)
The aim is to construct the scattering amplitude for each inequivalent per-
mutation of the external legs that has only pole singularities in the
N(N−3)
planar channels. We have also to impose that the amplitude has simultaneous
poles only in N − 3 compatible channels. In order to gain intuition on how to
proceed we rewrite the four-point amplitude in Eq. (2) as follows:
A(s, t) =
du23 u
−α(s12)−1
−α(s23)−1
23 δ(u12 + u23 − 1) (15)
where u12 and u23 are the variables corresponding to the two planar chan-
nels (12) and (23) and the cancellation of simultaneous poles in incompatible
channels is provided by the δ-function which forbids u12 and u23 to vanish
simultaneously.
We will now extend this procedure to the N -point amplitude. But for the
sake of clarity let us start with the case of N = 5 [14]. In this case we have 5
planar channels described by u12, u13, u23, u24 and u34. Since we have only two
compatible channels only two of the previous five variables are independent.
We can choose them to be u12 and u13. In order to determine the depen-
dence of the other three variables on the two independent ones, we exclude
simultaneous poles in incompatible channels. This can be done by imposing
relations that prevent variables corresponding to incompatible channels to
vanish simultaneously. A sufficient condition for excluding simultaneous poles
in incompatible channels is to impose the conditions:
uP = 1−
uP̄ (16)
where the product is over the variables P̄ corresponding to channels that
are incompatible with P . In the case of the five-point amplitude we get the
following relations:
u23 = 1− u34u12 ; u24 = 1− u13u12
u13 = 1− u34u24 ; u34 = 1− u23u13 ; u12 = 1− u24u23 (17)
Solving them in terms of the two independent ones we get:
u23 =
1− u12
1− u12u13
; u34 =
1− u13
1− u12u13
; u24 = 1− u12u13 (18)
In analogy with what we have done for the four-point amplitude in Eq. (15)
we write the five-point amplitude as follows:
The birth of string theory 9
du34u
−α(s12)−1
−α(s13)−1
×u−α(s24)−124 u
−α(s23)−1
−α(s34)−1
δ(u23 + u12u34 − 1)δ(u24 + u12u13 − 1)δ(u34 + u13u23 − 1) (19)
Performing the integral over the variables u23, u24 and u34 we get:
du13u
−α(s12)−1
−α(s13)−1
× (1− u12)−α(s23)−1(1 − u13)−α(s13)−1(1− u12u13)−α(s24)+α(s23)+α(s34)(20)
We have implicitly assumed that the Regge trajectory is the same in all chan-
nels and that the external scalar particles have the same common mass m
and are the lowest lying states on the Regge trajectory. This means that their
mass is given by:
α0 − α′p2i = 0 ; p2i ≡ −m2 (21)
Using then the relation:
α(s23) + α(s34)− α(s24) = 2α′p2 · p4 (22)
we can rewrite Eq. (20) as follows:
−α(s2)−1
−α(s3)−1
3 (1− u2)−α(s23)−1×
× (1 − u3)−α(s34)−1
(1− xij)2α
′pi·pj (23)
where
si ≡ s1i , ui ≡ u1i ; i = 2, 3 ; xij = uiui+1 . . . uj−1. (24)
We are now ready to construct the N -point function [13]. In analogy with
what has been done for the four and five-point amplitudes we can write the
N -point amplitude as follows:
. . .
−α(sP )−1
δ(uQ − 1 +
uQ̄) (25)
10 Paolo Di Vecchia
where the first product is over the
N(N−3)
variables corresponding to all
planar channels, while the second one is over the
(N−3)(N−2)
independent
δ-functions. The product in the δ-function is defined in Eq. (16).
The solution of all the non-independent linear relations imposed by the
δ-functions is given by
uij =
(1 − xij)(1− xi−1,j+1)
(1− xi−1,j)(1 − xi,j+1)
where the variables xij are given in Eq. (24). Eliminating the δ-function from
Eq. (25) one gets:
−α(si)−1
i (1 − ui)
−α(si,i+1)−1
j=i+2
(1− xij)−γij(27)
where
γij = α(sij) + α(si+1;j−1)− α(si;j−1)− α(si+1;j) ; j ≥ i+ 2 (28)
It is easy to see that
α(si,i+1) = −α0 − 2α′pi · pi+1 ; γij = −2α′pi · pj ; j ≥ i+ 2 (29)
Inserting them in Eq. (27) we get:
−α(si)−1
i (1− ui)
j=i+1
(1− xij)2α
′pi·pj (30)
This is the form of the N -point amplitude that was originally constructed.
Then Koba and Nielsen [15] put it in the form that is more known nowadays.
They constructed it using the following rules. They associated a real variable
zi to each leg i. Then they associated to each channel (i, j) an anharmonic
ratio constructed from the variables zi, zi−1, zj, zj+1 in the following way
(zi, zi+1, zj, zj+1)
−α(sij)−1 =
(zi − zj)(zi−1 − zj+1)
(zi−1 − zj)(zi − zj+1)
]−α(sij)−1
and finally they gave the following expression for the N -point amplitude:
dV (z)
(i,j)
(zi, zi+1, zj, zj+1)
−α(sij)−1 (32)
where
dV (z) =
1 [θ(zi − zi+1)dzi]
i=1(zi − zi+2)dVabc
; dVabc =
dzadzbdzc
(zb − za)(zc − zb)(za − zc)
The birth of string theory 11
and the variables zi are integrated along the real axis in a cyclically ordered
way: z1 ≥ z2 . . . ≥ zN with a, b, c arbitrarily chosen.
The integrand of the N -point amplitude is invariant under projective
transformations acting on the leg variables zi:
αzi + β
γzi + δ
; i = 1 . . .N ; αδ − βγ = 1 (34)
This is because both the anharmonic ratio in Eq. (31) and the measure dVabc
are invariant under a projective transformation. Since a projective transfor-
mation depends on three real parameters, then the integrand of the N -point
amplitude depends only on N − 3 variables zi. In order to avoid infinities, one
has then to divide the integration volume with the factor dVabc that is also
invariant under the projective transformations. The fact that the integrand
depends only on N − 3 variables is in agreement with the fact that N − 3 is
also the maximal number of simultaneous poles allowed in the amplitude.
It is convenient to write the N -point amplitude in a form that involves the
scalar product of the external momenta rather than the Regge trajectories.
We distinguish three kinds of channels. The first one is when the particles
i and j of the channel (i, j) are separated by at least two particles. In this
case the channels that contribute to the exponent of the factor (zi − zj) are
the channels (i, j) with exponent equal to −α(sij) − 1, (i + 1, j − 1) with
exponent −α(si+1,j−1)− 1, (i+1, j) with exponent α(si+1,j)+1 and (i, j− 1)
with exponent α(si,j−1) + 1. Adding these four contributions one gets for the
channels where i and j are separated by at least two particles
− α(sij)− α(si+1,j−1) + α(si+1,j) + α(si,j−1) = 2α′pi · pj (35)
The second one comes from the channels that are separated by only one
particle. In this case only three of the previous four channels contribute. For
instance if j = i+2 the channel (i+1, j− 1) consists of only one particle and
therefore should not be included. This means that we would get:
− α(si;i+2)− 1 + α(s1+1;i+2) + 1 + α(si;i+1 + 1) = 1 + 2α′pi · pi+2 (36)
Finally the third one that comes from the channels whose particles are adja-
cent, gets only contribution from:
− α(si;i+1)− 1 = α0 − 1 + 2α′pi · pi+1 (37)
Putting all these three terms together in Eq. (32) and remembering the factor
in the denominator in the first equation of (33) we get:
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
(zi − zj)2α
′pi·pj(38)
A convenient choice for the three variables to keep fixed is:
12 Paolo Di Vecchia
za = z1 = ∞ ; zb = z2 = 1 ; zc = zN = 0 (39)
With this choice the previous equation becomes:
dziθ(zi − zi+1)
(zi − zi+1)α0−1×
j=i+1
(zi − zj)2α
′pi·pj (40)
We now want to show that this amplitude is identical to the one given in Eq.
(30). This can be done by performing the following change of variables:
; i = 2, 3 . . .N − 2 (41)
that implies
zi = u2u3 . . . ui−1 ; i = 3, 4 . . .N − 1 (42)
Taking into account that the Jacobian is equal to:
uN−2−ii (43)
using the following two relations:
(zi − zi+1)α0−1 =
(N−1−i)α0−1
(1− ui)α0−1 (44)
j=i+1
(zj − zi)2α
′pi·pj =
j=i+1
(1− xij)2α
′pi·pj
−α(si)−(N−i−1)α0
i (45)
and the conservation of momentum
pi = 0 (46)
together with Eq. (21), one can easily see that Eq.s (30) and (40) are equal.
The birth of string theory 13
The N -point amplitude that we have constructed in this section corre-
sponds to the scattering of N spinless particles with no internal degrees of
freedom. On the other hand it was known that the mesons were classified
according to multiplets of an SU(3) flavour symmetry. This was implemented
by Chan and Paton [16] by multiplying the N -point amplitude with a factor,
called Chan-Paton factor, given by
Tr(λa1λa2 . . . λaN ) (47)
where the λ’s are matrices of a unitary group in the fundamental representa-
tion. Including the Chan-Paton factors the total scattering amplitude is given
Tr(λa1λa2 . . . λaN )BN (p1, p2, . . . pN ) (48)
where the sum is extended to the (N − 1)! permutations of the external legs,
that are not related by a cyclic permutations. Originally when the dual reso-
nance model was supposed to describe strongly interacting mesons, this factor
was introduced to represent their flavour degrees of freedom. Nowadays the
interpretation is different and the Chan-Paton factor represents the colour
degrees of freedom of the gauge bosons and the other massive particles of the
spectrum.
The N -point amplitude BN that we have constructed in this section con-
tains only simple pole singularities in all possible planar channels. They cor-
respond to zero width resonances located at non-negative integer values n of
the Regge trajectory α(M2) = n. The lowest state located at α(m2) = 0 cor-
responds to the particles on the external legs of BN . The spectrum of excited
particles can be obtained by factorizing the N -point amplitude in the most
general channel with any number of particles. This was done in Ref.s [17] and
[18] finding a spectrum of states rising exponentially with the mass M . Being
the model relativistic invariant it was found that many states obtained by
factorizing the N -point amplitude were ”ghosts”, namely states with negative
norm as one finds in QED when one quantizes the electromagnetic field in a
covariant gauge. The consistency of the model requires the existence of rela-
tions satisfied by the scattering amplitudes that are similar to those obtained
through gauge invariance in QED. If the model is consistent they must decou-
ple the negative norm states leaving us with a physical spectrum of positive
norm states. In order to study in a simple way these issues, we discuss in the
next section the operator formalism introduced already in 1969 [19, 20, 21].
Before concluding this section let us go back to the non-planar four-point
amplitude in Eq. (5) and discuss its generalization to an N -point amplitude.
Using the technique of the electrostatic analogue on the sphere instead of on
the disk Shapiro [22] was able to obtain a N -point amplitude that reduces
to the four-point amplitude in Eq. (5) with intercept α0 = 2. The N -point
amplitude found in Ref. [22] is:
14 Paolo Di Vecchia
i=1 d
dVabc
|zi − zj |α
′pi·pj (49)
where
dVabc =
d2zad
|za − zb|2|za − zc|2|zb − zc|2
The integral in Eq. (49) is performed in the entire complex plane.
3 Operator formalism and factorization
The factorization properties of the dual resonance model were first studied by
factorizing by brute force the N-point amplitude at the various poles [17, 18].
The number of terms that factorize the residue of the pole at α(s) = n,
increases rapidly with the value of n. In order to find their degeneracy it turned
out to be convenient to first rewrite the N-point amplitude in an operator
formalism. In this section we introduce the operator formalism and we rewrite
the N -point amplitude derived in the previous section in this formalism.
The key idea [19, 20, 21] is to introduce an infinite set of harmonic oscil-
lators and a position and momentum operators 7 which satisfy the following
commutation relations:
[anµ, a
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (51)
where ηµν is the flat Minkowski metric that we take to be ηµν = (−1, 1, . . . 1).
A state with momentum p is constructed in terms of a state with zero mo-
mentum as follows:
p̂|p〉 ≡ p̂eip·q̂|0〉 = p|p〉 ; p̂ |0〉 = 0 (52)
normalized as 8
〈p|p′〉 = (2π)dδ(d)(p+ p′) (53)
In order to avoid minus signs we use the convention that
〈p| = 〈0|eip·q̂ (54)
A complete and orthonormal basis of vectors in the harmonic oscillator space
is given by
|λ1, λ2, . . . λi; p〉 =
(a†µn;n)
λn;µn
λn,µn !
eipq̂|0, 0〉 (55)
7 Actually the position and momentum operators were introduced in Ref. [23].
8 Although we now use an arbitrary d we want to remind you that all original
calculations were done for d = 4.
The birth of string theory 15
where the first |0〉 corresponds to the one annihilated by all annihilation op-
erators and the second one to the state of zero momentum:
aµn;n|0, 0〉 = p̂|0, 0〉 = 0 (56)
Notice that Lorentz invariance forces to introduce also oscillators that create
states with negative norm due to the minus sign in the flat Minkowski metric.
This implies that the space spanned by the states in Eq. (55) is not positive
definite. This is, however, not allowed in a quantum theory and therefore if
the dual resonance model is a consistent quantum-relavistic theory we expect
the presence of relations of the kind of those provided by gauge invariance in
Let us introduce the Fubini-Veneziano [23] operator:
Qµ(z) = Q
µ (z) +Q
µ (z) +Q
µ (z) (57)
where
Q(+) = i
z−n ; Q(−) = −i
Q(0) = q̂ − 2iα′p̂ log z (58)
In terms of Q we introduce the vertex operator corresponding to the external
leg with momentum p:
V (z; p) =: eip·Q(z) :≡ eip·Q
(−)(z)eipq̂e+2α
′p̂·p log zeip·Q
(+)(z) (59)
and compute the following vacuum expectation value:
〈0, 0|
V (zi, pi)|0, 0〉 (60)
It can be easily computed using the Baker-Haussdorf relation
eAeB = eBeAe[A,B] (61)
that is valid if the commutator, as in our case, [A,B] is a c-number. In our
case the commutation relations to be used are:
[Q(+)(z), Q(−)(w)] = −2α′ log
and the second one in Eq. (51). Using them one gets:
V (z; p)V (w; k) =: V (z; p)V (w; k) : (z − w)2α
′p·k (63)
16 Paolo Di Vecchia
〈0, 0|
V (zi, pi)|0, 0〉 =
(zi − zj)2α
′pi·pj (2π)dδ(d)(
pi) (64)
where the normal ordering requires that all creation operators be put on the
left of the annihilation one and the momentum operator p̂ be put on the right
of the position operator q̂. This means that
(2π)dδ(d)(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
(zi − zi+1)α0−1
× 〈0, 0|
V (zi, pi)|0, 0〉 (65)
By choosing the three variables za, zb and zc as in Eq. (39) we can rewrite the
previous equation as follows:
(2π)dδ(d)(
pi)BN =
θ(zi − zi+1)×
(zi − zi+1)α0−1
〈0, p1|
V (zi; pi)|0, pN 〉 (66)
where we have taken z2 = 1 and we have defined (α0 ≡ α′p2i ; i = 1 . . .N) :
V (zN ; pN)|0, 0〉 ≡ |0; pN〉 ; 〈0; 0| lim
z2α01 V (z1; p1) = 〈0, p1| (67)
Before proceeding to factorize the N -point amplitude let us study the prop-
erties under the projective group of the operators that we have introduced.
We have already seen that the projective group leaves the integrand of the
Koba-Nielsen representation of the N -point amplitude invariant. The projec-
tive group has three generators L0, L1 and L−1 corresponding respectively to
dilatations, inversions and translations. Assuming that the Fubini-Veneziano
fields Q(z) transforms as a field with weight 0 (as a scalar) we can immedi-
ately write the commutation relations that Q(z) must satisfy. This means in
fact that, under a projective transformation, Q(z) transforms as follows:
Q(z) → QT (z) = Q
αz + β
γz + δ
; αδ − βγ = 1 (68)
Expanding for small values of the parameters we get:
QT (z) = Q(z) + (ǫ1 + ǫ2z + ǫ3z
dQ(z)
+ o(ǫ2) (69)
The birth of string theory 17
This means that the three generators of the projective group must satisfy the
following commutation relations with Q(z):
[L0, Q(z)] = z
; [L−1, Q(z)] =
; [L1, Q(z)] = z
They are given by the following expressions in terms of the harmonic oscilla-
tors:
L0 = α
′p̂2 +
na†n · an ; L1 =
2α′p̂ · a1 +
n(n+ 1)an+1 · a†n (71)
L−1 = L
2α′p̂ · a†1 +
n(n+ 1)a
n+1 · an (72)
They annihilate the vacuum
L0|0, 0〉 = L1|0, 0〉 = L−1|0, 0〉 = 0 (73)
that is therefore called the projective invariant vacuum, and satisfy the algebra
that is called Gliozzi algebra [24]9:
[L0, L1] = −L1 ; [L0, L−1] = L−1 ; [L1, L−1] = 2L0 (74)
The vertex operator with momentum p is a projective field with weight equal
to α0 = α
′p2. It transforms in fact as follows under the projective group:
[Ln, V (z, p)] = z
n+1 dV (z, p)
+ α0(n+ 1)z
nV (z, p) ; n = 0,±1 (75)
or in finite form as follows:
UV (z, p)U−1 =
(γz + δ)2α0
αz + β
γz + δ
where U is the generator of an arbitrary finite projective transformation.
Since U leaves the vacuum invariant, by using Eq. (76) it is easy to show
that:
〈0, 0|
V (z′i, p)|0, 0〉 =
(γzi + δ)
2α0〈0, 0|
V (zi, p)|0, 0〉 (77)
that together with the following equation:
(z′i − z′i+1)α0−1 =
(zi − zi+1)α0−1
(γzi + δ)
−2α0(78)
9 See also Ref. [25].
18 Paolo Di Vecchia
implies that the integrand of the N -point amplitude in Eq. (65) is invariant
under projective transformations.
We are now ready to factorize the N -point amplitude and find the spec-
trum of mesons.
From Eq.s (75) and (76) it is easy to derive the transformation of the
vertex operator under a finite dilatation:
zL0V (1, p)z−L0 = V (z, p)zα0 (79)
Changing the integration variables as follows:
; i = 2, 3 . . .N − 2 ; det
= z3z4 . . . zN−2 (80)
where the last term is the jacobian of the trasformation from zi to xi, we get
from Eq.(66) the following expression:
AN ≡ 〈0, p1|V (1, p2)DV (1, p3) . . .DV (1, pN−1)|0, pN〉 (81)
where the propagator D is equal to:
dxxL0−1−α0(1− x)α0−1 = Γ (L0 − α0)Γ (α0)
Γ (L0)
AN = (2π)
dδ(d)
BN (83)
The factorization properties of the amplitude can be studied by inserting in
the channel (1,M) or equivalently in the channel (M +1, N) described by the
Mandelstam variable
s = −(p1 + p2 + . . . pM )2 = −(pM+1 + pM+2 . . .+ pN )2 ≡ −P 2 (84)
the complete set of states given in Eq. (55):
〈p(1,M)|λ, P 〉〈λ, P |D|µ, P 〉〈µ, P |p(M+1,N)〉 (85)
where
〈p(1,M)| = 〈0, p1|V (1, p2)DV (1, p3) . . . V (1, pM ) (86)
|p(M+1,N)〉 = V (1, pM+1)D . . . V (1, pN−1)|pN , 0〉 (87)
Introducing the quantity:
The birth of string theory 19
na†n · an (88)
it is possible to rewrite
〈λ, P |D|µ, P 〉 =
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉 (89)
where s is the variable defined in Eq. (84). Using this equation we can rewrite
Eq. (85) as follows
〈p(1,M)|λ, P 〉
〈λ, P |
(−1)m
α0 − 1
R+m− α(s)
|µ, P 〉〈µ, P |p(M+1,N)〉(90)
This expression shows that amplitude AN has a pole in the channel (1,M)
when α(s) is equal to an integer n ≥ 0 and the states |λ〉 that contribute to
its residue are those satisfying the relation:
R|λ〉 = (n−m)|λ〉 ; m = 0, 1 . . . n (91)
The number of independent states |λ〉 contributing to the residue gives the
degeneracy of states for each level n.
Because of manifest relativistic invariance the space spanned by the com-
plete system of states in Eq. (55) contains states with negative norm corre-
sponding to those states having an odd number of oscillators with timelike
directions (see Eq. (51)). This is not consistent in a quantum theory where
the states of a system must span a positive definite Hilbert space. This means
that there must exist a number of relations satisfied by the external states that
decouple a number of states leaving with a positive definite Hilbert space. In
order to find these relations we rewrite the state in Eq. (87) going back to the
Koba-Nielsen variables:
|p(1,M)〉 =
dziθ(zi − zi+1)]
(zi − zi+1)α0−1×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1)|0, pM 〉 (92)
Let us consider the operator U(α) that generate the projective transformation
that leaves the points z = 0, 1 invariant:
1− α(z − 1) = z + α(z
2 − z) + o(α2) (93)
From the transformation properties of the vertex operators in Eq. (76) it
is easy to see that the previous transformation leaves the state in Eq. (92)
invariant:
20 Paolo Di Vecchia
U(α)|p(1,M)〉 = |p(1,M)〉 (94)
This means that the generator of the previous transformation annihilates the
state in Eq. (92):
W1|p(1,M)〉 = 0 ; W1 = L1 − L0 (95)
The explicit form of W1 follows from the infinitesimal form of the transforma-
tion in Eq. (93). This condition that is of the same kind of the relations that
on shell amplitudes with the emission of photons satisfy as a consequence of
gauge invariance, implies that the residue at the pole in Eq. (90) can be fac-
torized with a smaller number of states. It turns out, however, that a detailed
analysis of the spectrum shows that negative norm states are still present.
This can be qualitatively understood as follows. Due to the Lorentz metric
we have a negative norm component for each oscillator. In order to be able
to decouple all negative norm states we need to have a gauge condition of
the type as in Eq. (95) for each oscillator. But the number of oscillators is
infinite and, therefore, we need an infinite number of conditions of the type
as in Eq. (95). It was found in Ref. [6] that, if we take α0 = 1, then one can
easily construct an infinite number of operators that leave the state in Eq.
(92) invariant. In the next section we will concentrate on this case.
4 The case α0 = 1
If we take α0 = 1 many of the formulae given in the previous section simplify.
The N -point amplitude in Eq. (38) becomes:
1 dziθ(zi − zi+1)
dVabc
(zi − zj)2α
′pi·pj (96)
that can be rewritten in the operator formalism as follows:
(2π)4δ(
pi)BN =
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
V (zi, pi)|0, 0〉 (97)
By choosing z1 = ∞, z2 = 1 and zN = 0 it becomes
(2π)4δ(
pi)BN =
θ(zi − zi+1)〈0, p1|
V (zi; pi)|0, pN 〉 (98)
The birth of string theory 21
where
V (zN ; pN )|0, 0〉 ≡ |0; pN 〉 ; 〈0; 0| lim
z21V (z1; p1) = 〈0, p1| (99)
Eq. (81) is as before, but now the propagator becomes:
dxxL0−2 =
L0 − 1
(100)
This means that Eq. (89) becomes:
〈λ, P |D|µ, P 〉 = 〈λ, P | 1
L0 − 1
|µ, P 〉 (101)
and Eq. (90) has the simpler form:
〈p(1,M)|λ, P 〉〈λ, P |
R − α(s)
|λ, P 〉〈λ, P |p(M+1,N)〉 (102)
BN has a pole in the channel (1,M) when α(s) is equal to an integer n ≥ 0 and
the states |λ〉 that contribute to its residue are those satisfying the relation:
R|λ〉 = n|λ〉 (103)
Their number gives the degeneracy of the states contributing to the pole at
α(s) = n. The N -point amplitude can be written as:
BN = 〈p(1,M)|D|p(M+1,N)〉 (104)
where
|p(1,M)〉 =
∫ M−1
[dziθ(zi − zi+1)]×
× V (1, p1)V (z2, p2) . . . V (zM−1, pM−1|0, pM 〉 (105)
Using Eq. (79) and changing variables from zi, i = 2 . . .M−1 to xi = zi+1zi , i =
1 . . .M − 2 with z1 = 1 we can rewrite the previous equation as follows:
|p(1,M)〉 = V (1, p1)DV (1, p2) . . .DV (1, pM−1)|0, pM 〉 (106)
where the propagator D is defined in Eq. (100).
We want now to show that the state in Eq.s (105) and (106) is not only
annihilated by the operator in Eq. (95), but, if α0 = 1 [6], by an infinite set
of operators whose lowest one is the one in Eq. (95). We will derive this by
using the formalism developed in Ref. [26] and we will follow closely their
derivation.
Starting from Eq.s (70) Fubini and Veneziano realized that the generators
of the projective group acting on a function of z are given by:
22 Paolo Di Vecchia
L0 = −z
; L−1 = −
; L1 = −z2
(107)
They generalized the previous generators to an arbitrary conformal transfor-
mation by introducing the following operators, called Virasoro operators:
Ln = −zn+1
(108)
that satisfy the algebra:
[Ln, Lm] = (n−m)Ln+m (109)
that does not contain the term with the central charge! They also showed that
the Virasoro operators satisfy the following commutation relations with the
vertex operator:
[Ln, V (z, p)] =
zn+1V (z, p)
(110)
More in general actually they define an operator Lf corresponding to an
arbitrary function f(ξ) and Lf = Ln if we choose f(ξ) = ξ
n. In this case the
commutation relation in Eq. (110) becomes:
[Lf , V (z, p)] =
(zf(z)V (z, p)) (111)
By introducing the variable:
ξf(ξ)
(112)
where A is an arbitrary constant, one can rewrite Eq. (111) in the following
form:
[Lf , zf(z)V (z, p)] =
(zf(z)V (z, p)) (113)
This implies that, under an arbitrary conformal transformation z → f(z),
generated by U = eαLf , the vertex operator transforms as:
eαLfV (z, p) zf(z) e−αLf = V (z′, p)z′f(z′) (114)
where the parameter α is given by:
ξf(ξ)
(115)
On the other hand, this equation implies:
zf(z)
z′f(z′)
(116)
The birth of string theory 23
that, inserted in Eq. (114), implies that the quantity V (z, p) dz is left invariant
by the transformation z → f(z):
eαLfV (z, p)dze−αLf = V (z′, p)dz′ (117)
Let us now act with the previous conformal transformation on the state in
Eq. (105). We get:
eαLf |p(1,M)〉 =
[dziθ(zi − zi+1)] eαLfV (1, p1)e−αLf×
×eαLfV (z2, p2)e−αLf . . . . . . eαLfV (zM−1, pM−1)e−αLf eαLf |0, pM 〉 =
θ(zi − zi+1)× eαLfV (1, p1)e−αLf×
× V (z′2, p2)dz′2 . . . V (z′M−1, pM−1)dz′M−1eαLf |0, pM 〉 (118)
where we have used Eq. (117). The previous transformation leaves the state
invariant if both z = 0 and z = 1 are fixed points of the conformal transfor-
mation. This happens if the denominator in Eq. (115) vanishes when ξ = 0, 1.
This requires the following conditions:
f(1) = 0 ; lim
ξf(ξ) = 0 (119)
Expanding ξ near the poinr ξ = 1 we can determine the relation between z
and z′ near z = z′ = 1. We get:
ze−αf
1− z + ze−αf ′(1)
(120)
and from it we can determine the conformal factor:
(1 − z + ze−αf ′(1))2
→ eαf
′(1) (121)
in the limit z → 1. Proceeding in the same near the point z = z′ = 0 we get:
zf(0)eαf(0)
f(0) + zf ′(0)(1− eαf(0)
→ zeαf(0) (122)
in the limit z → 0. This means that Eq. (118) becomes
eα(Lf−f
′(1)−f(0))|p(1,M)〉 = |p(1,M)〉 (123)
A choice of f that satisfies Eq.s (119) is the following:
24 Paolo Di Vecchia
f(ξ) = ξn − 1 (124)
that gives the following gauge operator:
Wn = Ln − L0 − (n− 1) (125)
that annihilates the state in Eq. (105):
Wn|p1...M 〉 = 0 ; n = 1 . . .∞ (126)
These are the Virasoro conditions found in Ref. [6]. There is one condition for
each negative norm oscillator and, therefore, in this case there is the possibility
that the physical subspace is positive definite. An alternative more direct
derivation of Eq. (126) can be obtained by acting with Wn on the state in Eq.
(106) and using the following identities:
WnV (1, p) = V (1, p)(Wn + n) ; (Wn + n)D = [L0 + n− 1]−1Wn (127)
The second equation is a consequence of the following equation:
L0 = xL0+nLn (128)
Eq.s (127) imply
WnV (1, p)D = V (1, p)[L0 + n− 1]−1Wn (129)
This shows that the operator Wn goes unchanged through all the product
of terms V D until it arrives in front of the term V (1, pM−1)|0, pM 〉. Going
through the vertex operator it becomes Ln − L0 + 1 that then annihilate the
state
(Ln − L0 + 1)|pM , 0〉 = 0 (130)
This proves Eq. (126).
Using the representation of the Virasoro operators given in Eq. (108) Fu-
bini and Veneziano showed that they satisfy the algebra given in eq. (109)
without the central charge. The presence of the central charge was recognized
by Joe Weis10 in 1970 and never published. Unlike Fubini and Veneziano [26]
he used the expression of the Ln operators in terms of the harmonic oscillators:
2α′np̂ · an +
m(n+m)an+m · am+
m(n−m)am−n · am ;n ≥ 0 Ln = L†n (131)
10 See noted added in proof in Ref. [26].
The birth of string theory 25
He got the following algebra:
[Ln, Lm] = (n−m)Ln+m +
n(n2 − 1)δn+m;0 (132)
where d is the dimension of the Minkowski space-time. We write here d for the
dimension of the Minkowski space, but we want to remind you that almost
everybody working in a model for mesons at that time took for granted that
the dimension of the space-time was d = 4. As far as I remember the first
paper where a dimension d 6= 4 was introduced was Ref. [27] where it was
shown that the unitarity violating cuts in the non-planar loop become poles
that were consistent with unitarity if d = 26.
In the last part of this section we will generalize the factorization procedure
to the Shapiro-Virasoro model whose N -point amplitude is given in Eq. (49).
In this case we must introduce two sets of harmonic oscillators commuting
with each other and only one set of zero modes satisfying the algebra [28] :
[anµ, a
mν ] = [ãnµ, ã
mν ] = ηµνδnm ; [q̂µ, p̂ν ] = iηµν (133)
In terms of them we can introduce the Fubini-Veneziano operator
Q(z, z̄) = q̂ − 2α′p̂ log(zz̄) + i
−n − a†nzn
ãnz̄
−n − ã†nz̄n
(134)
We can then introduce the vertex operator:
V (z, z̄; p) =: eip·Q(z,z̄) : (135)
and write the N -point amplitude in Eq. (95) in the following factorized form:
i=1 d
dVabc
V (zi, z̄i, pi))
|0〉 =
= (2π)4δ(4)(
i=1 d
dVabc
|zi − zj|α
′pi·pj (136)
where the radial ordered product is given by
V (zi, z̄i, pi))
V (zi, z̄i, pi))
θ(|zi| − |zi+1|) + . . . (137)
26 Paolo Di Vecchia
and the dots indicate a sum over all permutations of the vertex operators.
By fixing z1 = ∞, z2 = 1, zN = 0 we can rewrite the previous expression
as follows:
∫ N−1
d2zi〈0, p1|R
V (zi, z̄i, pi))
|0, pN〉 (138)
For the sake of simplicity let us consider the term corresponding to the per-
mutation 1, 2, . . .N . In this case the Koba-Nielsen variables are ordered in
such a way that |zi| ≥ |zi+1| for i = 1, . . .N −1. We can then use the formula:
V (zi, z̄i, pi)) = z
L̃0−1
i V (1, 1, pi)z
i (139)
and change variables:
; |wi| ≤ 1 (140)
to rewrite Eq. (138) as follows:
〈0, p1|V (1, 1, pi1)DV (1, 1, p2)D . . . V (1, 1, pN−1)|0, pN 〉 (141)
where
wL0−1w̄L̃0−1 =
L0 + L̃0 − 2
· sinπ(L0 − L̃0)
L0 − L̃0
(142)
We can now follow the same procedure for all permutations arriving at the
following expression:
〈0, p1|P [V (1, 1, p2)DV (1, 1, p3)D . . . V (1, 1, pN−1)]|0, pN〉 (143)
where P means a sum of all permutations of the particles.
If we want to consider the factorization of the amplitude on the pole at
s = −(p1 + . . . pM )2 we get only the following contribution:
〈p(1...M)|D|p(M+1...N)〉 (144)
where
|p(M+1...N)〉 = P [V (1, 1, pM+1)D . . . V (1, 1, pN−1]|0, pN 〉 (145)
〈p(1...M)| = 〈0, p1|P [V (1, 1, p2)D . . . V (1, 1, pM)] (146)
The amplitude is factorized by introducing a complete set of states and rewrit-
ing Eq. (141) as follows:
The birth of string theory 27
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δL0,L̃0|λ, λ̃〉
L0 + L̃0 − 2
〈λ, λ̃|p(M+1,...N)〉 (147)
By writing
p̂2 +R ; L̃0 =
p̂2 + R̃ (148)
na†n · an ; R̃ =
nã†n · ãn (149)
we can rewrite Eq. (147) as follows
〈p1...M |λ, λ̃〉
2π〈λ, λ̃|δR,R̃|λ, λ̃〉
R + R̃− α(s)
〈λ, λ̃|p(M+1,...N)〉 (150)
We see that the amplitude for the Shapiro-Virasoro model has simple poles
only for even integer values of αSV (s) = 2 +
s = 2n ≥ 0 and the residue at
the poles factorizes in a sum with a finite number of terms. Notice that the
Regge trajectory of the Shapiro-Virasoro model has double intercept and half
slope of that of the generalized Veneziano model.
5 Physical states and their vertex operators
In the previous section, we have seen that the residue at the poles of the N -
point amplitudes factorizes in a sum of a finite number of terms. We have also
seen that some of these terms, due to the Lorentz metric, correspond to states
with negative norm. We have also derived a number of ”Ward identities” given
in Eq. (126) that imply that some of the terms of the residue decouple. The
question to be answered now is: Is the space spanned by the physical states
a positive norm Hilbert space? In order to answer this question we need first
to find the conditions that characterize the on shell physical states |λ, P 〉
and then to determine which are the states that contribute to the residue
of the pole at α(s = −P 2) = n. In other words, we have to find a way of
characterizing the physical states and of eliminating the spurious states that
decouple in Eq. (102) as a consequence of Eq.s (126). A state |λ.P 〉 contributes
at the residue of the pole in Eq.(102) for α(s = −P 2) = n if it is on shell,
namely if it satisfies the following equations:
R|λ, P 〉 = n|λ, P 〉 ; α(−P 2) = 1− α′P 2 = n (151)
that can be written in a unique equation:
28 Paolo Di Vecchia
(L0 − 1)|λ, P 〉 = 0 (152)
Because of Eq. (126) we also know that a state of the type:
|s, P 〉 = W †m|µ, P 〉 (153)
is not going to contribute to the residue of the pole. We call it a spurious or
unphysical state. We start constructing the subspace of spurious states that
are on shell at the level n. Let us consider the set of orthogonal states |µ, P 〉
such that
R|µ, P 〉 = nµ|µ, P 〉 ; L0|µ, P 〉 = (1−m)|µ, P 〉 ; 1− α′P 2 = n (154)
where
m = n+ nµ (155)
In terms of these states we can construct the most general spurious state that
is on shell at the level n. It is given by
|s, P 〉 = W †m|µ, P 〉 ; (L0 − 1)|s, P 〉 = 0 (156)
per any positive integer m. Using Eq. (154), eq. (156) becomes:
|s, P 〉 = L†m|µ, P 〉 (157)
where |µ, P 〉 is an arbitrary state satisfying Eq.s (154).
A physical state |λ, P 〉 is defined as the one that is orthogonal to all spuri-
ous states appearing at a certain level n. This means that it must satisfy the
following equation:
〈λ.P |L†ℓ |µ, P 〉 = 0 (158)
for any state |µ, P 〉 satisfying Eq.s (154). In conclusion, the on shell physical
states at the level n are characterized by the fact that they satisfy the following
conditions:
Lm|λ, P 〉 = (L0 − 1)|λ, P 〉 = 0 ; 1− α′P 2 = n (159)
These conditions characterizing the physical subspace were first found by Del
Giudice and Di Vecchia [28] where the analysis described here was done.
In order to find the physical subspace one starts writing the most general
on shell state contributing to the residue of the pole at level n in Eq. (154).
Then one imposes Eq.s (159) and determines the states that span the physical
subspace. Actually, among these states one finds also a set of zero norm states
that are physical and spurious at the same time. Those states are of the form
given in Eq. (157), but also satisfy Eq.s (159). It is easy to see that they are
not really physical because they are not contributing to the residue of the pole
The birth of string theory 29
at the level n. This follows from the form of the unit operator given in the
space of the physical states by:
norm 6=0
|λ, P 〉〈λ, P |+
[|λ0, P 〉〈µ0, P |+ |µ0, P 〉〈λ0, P |] (160)
where |λ0, P 〉 is a zero norm physical and spurious state and |µ0, P 〉 its con-
jugate state. A conjugate state of a zero norm state is obtained by changing
the sign of the oscillators with timelike direction. Since |λ0, P 〉 is a spurious
state when we insert the unit operator, given in Eq. (160), in Eq. (102) we
see that the zero norm states never contribute to the residue because their
contribution is annihilated either from the state 〈p(1,M)| or from the state
|p(M+1,N)〉. In conclusion, the physical subspace contains only the states in
the first term in the r.h.s. of Eq. (160).
Let us analyze the first two excited levels. The first excited level corre-
sponds to a massless gauge field. It is spanned by the states ǫµa
1µ|0, P 〉. In
this case the only condition that we must impose is:
1µ|0, P 〉 = 0 =⇒ P · ǫ = 0 (161)
Choosing a frame of reference where the momentum of the photon is given by
Pµ ≡ (P, 0....0, P ) , Eq. (161) implies that the only physical states are:
1i |0, P 〉+ ǫ(a
1;0 − a
1;d−1)|0, P 〉 ; i = 1 . . . d− 2 (162)
where ǫi and ǫ are arbitrary parameters. The state in Eq. (162) is the most
general state of the level N = 1 satisfying the conditions in Eq. (159). The
first state in eq. (162) has positive norm, while the second one has zero norm
that is orthogonal to all other physical states since it can be written as follows:
1;0 − a
1;D−1)|0, P 〉 = L
1|0, P 〉 (163)
in the frame of reference where Pµ ≡ (P, ...0, P ). Because of the previous
property it is decoupled from the physical states together with its conjugate:
1,0 + a
1,d−1)|0, P 〉 (164)
In conclusion, we are left only with the transverse d− 2 states corresponding
to the physical degrees of freedom of a massless spin 1 state. At the next level
n = 2 the most general state is given by:
[αµνa
1,ν + β
2,µ]|0, P 〉 (165)
If we work in the center of mass frame where Pµ = (M,0) we get the following
most general physical state:
|Phys >= αij [a†1,ia
1,j −
(d− 1)
1,k]|0, P 〉+
30 Paolo Di Vecchia
+βi[a
2,i + a
1,i]|0, P >〉+
1,i +
1,0 − 2a
|0, P 〉 (166)
where the indices i, j run over the d− 1 space components. The first term in
(166) corresponds to a spin 2 in (d− 1) dimensional space and has a positive
norm being made with space indices. The second term has zero norm and is
orthogonal to the other physical states since it can be written as L+1 a
1,i|0, P 〉.
Therefore it must be eliminated from the physical spectrum together with its
conjugate, as explained above. Finally, the last state in (166) is spinless and
has a norm given by:
2(d− 1)(26− d) (167)
If d < 26 it corresponds to a physical spin zero particle with positive norm. If
d > 26 it is a ghost. Finally, if d = 26 it has a zero norm and is also orthogonal
to the other physical states since it can be written in the form:
2 + 3L
1 )|0 > (168)
It does not belong, therefore, to the physical spectrum. The analysis of this
level was done in Ref. [29] with d = 4. This did not allow the authors of
Ref. [29] to see that there was a critical dimension.
The analysis of the physical states can be easily extended [28] to the
Shapiro-Virasoro model. In this case the physical conditions given in Eq. (159)
for the open string, become [28]:
Lm|λ, λ̃〉 = L̃m|λ, λ̃〉 = (L0 − 1)|λ, λ̃〉 = (L̃0 − 1)|λ, λ̃〉 = 0 (169)
for any positive integer m. It can be easily seen from the previous equations
that the lowest state of the Shapiro-Virasoro model is the vacuum |0a, 0ã, p〉
corresponding to a tachyon with mass α′p2 = 4, while the next level described
by the state a
1ν |0a, 0ã, p〉 contains massless states corresponding to the
graviton, a dilaton and a two-index antisymmetric tensor Bµν .
Having characterized the physical subspace one can go on and construct
a N -point scattering amplitude involving arbitrary physical states. This was
done by Campagna, Fubini, Napolitano and Sciuto [30] where the vertex oper-
ator for an arbitrary physical state was constructed in analogy with what has
been done for the ground tachyonic state. They associated to each physical
state |α, P 〉 a vertex operator Vα(z, P ) that is a conformal field with conformal
dimension equal to 1:
[Ln, Vα(z, p)] =
zn+1Vα(z, p)
(170)
and reproduces the corresponding state acting on the vacuum as follows:
Vα(z; p)|0, 0〉 ≡ |α; p〉 ; 〈0; 0| lim
z2Vα(z; p) = 〈α, p| (171)
The birth of string theory 31
It satisfies, in addition, the hermiticity relation:
V †α (z, P ) = Vα(
,−P )(−1)α(−P
2) (172)
An excited vertex that will play an important role in the next section is the
one associated to the massless gauge field. It is given by:
Vǫ(z, k) ≡ ǫ ·
dQ(z)
eik·Q(z) ; k · ǫ = k2 = 0 (173)
Because of the last two conditions in Eq. (173) the normal order is not neces-
sary. It is convenient to give the expression of
dQ(z)
in terms of the harmonic
oscillators:
P (z) ≡ dQ(z)
−n−1 (174)
It is a conformal field with conformal dimension equal to 1. The rescaled
oscillators αn are given by:
nan ; α−n =
na†n ; n > 0 ; α0 =
2α′p̂ (175)
In terms of the vertex operators previously introduced the most general
amplitude involving arbitrary physical states is given by [30]:
(2π)4δ(
1 dziθ(zi − zi+1)
dVabc
〈0, 0|
Vαi(zi, pi)|0, 0〉(176)
In the case of the Shapiro-Virasoro model the tachyon vertex operator is
given in Eq. (135). By rewriting Eq. (134) as follows:
Q(z, z̄) = Q(z) + Q̃(z̄) (177)
where
Q(z) =
q̂ − 2α′p̂ log(z) + i
−n − a†nzn
(178)
Q̃(z̄) =
q̂ − 2α′p̂ log(z̄) + i
ãnz̄
−n − ã†nz̄n
(179)
we can write the tachyon vertex operator in the following way:
V (z, z̄, p) =: eip·Q(z)eip·Q̃(z̄) : (180)
32 Paolo Di Vecchia
This shows that the vertex operator corresponding to the tachyon of the
Shapiro-Virasoro model can be written as the product of two vertex oper-
ators corresponding each to the tachyon of the generalized Veneziano model.
Analogously the vertex operator corresponding to an arbitrary physical
state of the Shapiro-Virasoro model can always be written as a product of
two vertex operators of the generalized Veneziano model:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (181)
The first one contains only the oscillators αn, while the second one only the
oscillators α̃n. They both contain only half of the total momentum p and
the same zero modes p̂ and q̂. The two vertex operators of the generalized
Veneziano model are both conformal fields with conformal dimension equal
to 1. If they correspond to physical states at the level 2n, they satisfy the
following relation (n = ñ):
+ n = 1 (182)
They lie on the following Regge trajectory:
p2 ≡ αSV (−p2) = 2n (183)
as we have already seen by factorizing the amplitude in Eq. (150).
6 The DDF states and absence of ghosts
In the previous section we have derived the equations that characterize the
physical states and their corresponding vertex operators. In this section we
will explicitly construct an infinite number of orthonormal physical states with
positive norm.
The starting point is the DDF operator introduced by Del Giudice, Di Vec-
chia and Fubini [31] and defined in terms of the vertex operator corresponding
to the massless gauge field introduced in eq. (173):
Ai,n =
i Pµ(z)e
ik·Q(z) (184)
where the index i runs over the d−2 transverse directions, that are orthogonal
to the momentum k. We have also taken
= 1. Because of the log z term
appearing in the zero mode part of the exponential, the integral in Eq. (184),
that is performed around the origin z = 0, is well defined only if we constrain
the momentum of the state, on which Ai,n acts, to satisfy the relation:
2α′p · k = n (185)
The birth of string theory 33
where n is a non-vanishing integer.
The operator in Eq. (184) will generate physical states because it com-
mutes with the gauge operators Lm:
[Lm, An;i] = 0 (186)
since the vertex operator transforms as a primary field with conformal dimen-
sion equal to 1 as it follows from Eq. (170).
On the other hand it also satisfies the algebra of the harmonic oscillator
as we are now going to show. From Eq. (184) we get:
[An,i, Am,j] = −
dzǫi · P (z)eik·Q(ζ)ǫj · P (ζ)eik
′ ·Q(ζ) (187)
where
2α′p · k = n ; 2α′p · k′ = m (188)
and k and k′ are supposed to be in the same direction, namely
kµ = nk̂µ ; k
µ = mk̂µ (189)
2α′p · k̂ = 1 (190)
Finally the polarizations are normalized as:
ǫi · ǫj = δij (191)
Since k̂ · ǫi = k̂ · ǫj = k̂2 = 0 a singularity for z = ζ can appear only from the
contraction of the two terms P (ζ) and P ((z) that is given by:
〈0, 0|ǫi · P (z)ǫj · P (ζ)|0, 0〉 = −
2α′δij
(z − ζ)2
(192)
Inserting it in Eq. (187) we get:
[An,i, Am,j ] = δij in
dζk̂ · P (ζ)e−i(n+m))k̂·Q(ζ) =
= inδijδn+m;0
dζk̂ · P (ζ) (193)
where we have used the fact that the integrand is a total derivative and
therefore one gets a vanishing contribution unless n + m = 0. If n + m = 0
from Eq.s (174) and (190) we get:
[An,i, Am,j ] = nδijδn+m;0 ; i, j = 1 . . . d− 2 (194)
34 Paolo Di Vecchia
Eq. (194) shows that the DDF operators satisfy the harmonic oscillator alge-
In terms of this infinite set of transverse oscillators we can construct an
orthonormal set of states:
|i1, N1; i2, N2; . . . im, Nm〉 =
Aik,−Nk√
|0, p〉 (195)
where λh is the multiplicity of the operator Aih,−Nh in the product in Eq.
(195) and the momentum of the state in Eq. (195) is given by
P = p+
k̂Ni (196)
They were constructed in four dimensions where they were not a complete
system of states 11 and it took some time to realize that in fact they were a
complete system of states if d = 26 [32, 33] 12. Brower [32] and Goddard and
Thorn [33] showed also that the dual resonance model was ghost free for any
dimension d ≤ 26. In d = 26 this follows from the fact that the DDF operators
obviously span a positive definite Hilbert space (See Eq. (194)). For d < 26
there are extra states called Brower states [32]. The first of these states is
the last state in Eq. (166) that becomes a zero norm state for d = 26. But
also for d < 26 there is no negative norm state among the physical states.
The proof of the no-ghost theorem in the case α0 = 1 is a very important
step because it shows that the dual resonance model constructed generalizing
the four-point Veneziano formula, is a fully consistent quantum-relativistic
theory! This is not quite true because, when the intercept α0 = 1, the lowest
state of the spectrum corresponding to the pole in the N -point amplitude for
α(s) = 0, is a tachyon with mass m2 = − 1
. A lot of effort was then made
to construct a model without tachyon and with a meson spectrum consistent
with the experimental data. The only reasonably consistent models that came
out from these attempts, were the Neveu-Schwarz [7] for mesons and the
Ramond model [8] for fermions that only later were recognized to be part of a
unique model that nowadays is called the Neveu-Schwarz-Ramond model. But
this model was not really more consistent than the original dual resonance
11 Because of this Fubini did not want to publish our result, but then he went to a
meeting in Israel in spring 1971 giving a talk on our work where he found that
the audience was very interested in our result and when he came back to MIT we
decided to publish our result.
12 I still remember Charles Thorn coming into my office at Cern and telling me:
Paolo, do you know that your DDF states are complete if d = 26? I quickly redid
the analysis done in Ref. [29] with an arbitrary value of the space-time dimension
obtaining Eq.s (166) and (167) that show that the spinless state at the level
α(s) = 2 is decoupled if d = 26. I strongly regretted not to have used an arbitrary
space-time dimension d in the analysis of Ref. [29] .
The birth of string theory 35
model because it still had a tachyon with mass m2 = − 1
. The tachyon
was eliminated from the spectrum only in 1976 through the GSO projection
proposed by Gliozzi, Scherk and Olive [34].
Having realized that, at least for the critical value of the space-time dimen-
sion d = 26, the physical states are described by the DDF states having only
d− 2 = 24 independent components, open the way to Brink and Nielsen [35]
to compute the value α0 = 1 of the Regge trajectory with a very physical ar-
gument. They related the intercept of the Regge trajectory to the zero point
energy of a system with an infinite number of oscillators having only d − 2
independent components:
α0 = −
n (197)
This quantity is obviously infinite and, in order to make sense of it, they in-
troduced a cutoff on the frequencies of the harmonic oscillators obtaining an
infinite term that they eliminated by renormalizing the speed of light and a
finite universal constant term that gave the intercept of the Regge trajectory.
Instead of following their original approach we discuss here an alternative ap-
proach due to Gliozzi [36] that uses the ζ-function regularization. He rewrites
Eq. (197) as follows:
α0 = −
n = −
n−s = −
ζR(−1) = 1 (198)
where in the last equation we have used the identity ζR(−1) = − 112 and we
have put d = 26. Since the Shapiro-Virasoro model has two sets of trans-
verse harmonic oscillators it is obvious that its intercept is twice that of the
generalized Veneziano model.
Using the rules discussed in the previous section we can construct the
vertex operator corresponding to the state in Eq. (195). It is given by:
V(i;Ni)(z, P ) =
dziǫi · P (zi)eiNik̂·Q(zi) : eip·Q(z) : (199)
where the integral on the variable zi is evaluated along a curve of the complex
plane zi containing the point z. The singularity of the integrand for zi = z is
a pole provided that the following condition is satisfied.
2α′p · k̂ = 1 (200)
The last vertex in Eq. (199) is the vertex operator corresponding to the ground
tachyonic state given in Eq. (59) with α′p2 = 1.
Using the general form of the vertex one can compute the three-point
amplitude involving three arbitrary DDF vertex operators. This calculation
36 Paolo Di Vecchia
has been performed in Ref. [37] and since the vertex operators are conformal
fields with dimension equal to 1 one gets:
〈0, 0|V
(z1, P1)V(i(2)
(z2, P2)V(i(3)
(z3, P3)|0, 0〉 =
(z1 − z2)(z1 − z3)(z2 − z3)
(201)
where the explicit form of the coefficient C123 is given by:
C123 = 1〈0, 0|2〈0, 0|3〈0, 0|e
r.s=1
n,m=1
−n;iN
−m;i+
−n;i×
× eτ0
(α′Π2r−1)|N (1)k1 , i
〉1|N (2)k2 , i
〉2|N (3)k3 , i
〉3 (202)
where
N rsnm = −N rnNsm
nmα1α2α3
nαs +mαr
; N rn =
Γ (−nαr+1
αrn!Γ (1− nαr+1αr − n)
(203)
Π = Pr+1αr − Prαr+1 ; r = 1, 2, 3 (204)
Π is independent on the value of r chosen as a consequence of the equations:
Pr = 0 (205)
7 The zero slope limit
In the introduction we have seen that the dual resonance model has been
constructed using rules that are different from those used in field theory.
For instance, we have seen that planar duality implies that the amplitude
corresponding to a certain duality diagram, contains poles in both s and t
channels, while the amplitude corresponding to a Feynman diagram in field
theory contains only a pole in one of the two channels. Furthermore, the
scattering amplitude in the dual resonance model contains an infinite number
of resonant states that, at high energy, average out to give Regge behaviour.
Also this property is not observed in field theory. The question that was
natural to ask, was then: is there any relation between the dual resonance
model and field theory? It turned out, to the surprise of many, that the dual
resonance model was not in contradiction with field theory, but was instead
an extension of a certain number of field theories. We will see that the limit in
The birth of string theory 37
which a field theory is obtained from the dual resonance model corresponds
to taking the slope of the Regge trajectory α′ to zero.
Let us consider the scattering amplitude of four ground state particles in
Eq. (1) that we rewrite here with the correct normalization factor:
A(s, t, u) = C0N
0 (A(s, t) +A(s, u) +A(t, u)) (206)
where
2g(2α′)
4 (207)
is the correct normalization factor for each external leg, g is the dimensionless
open string coupling constant that we have constantly ignored in the previous
sections and C0 is determined by the following relation:
′ = 1 (208)
that is obtained by requiring the factorization of the amplitude at the pole
corresponding to the ground state particle whose mass is given in Eq. (21).
Using Eq. (21) in order to rewrite the intercept of the Regge trajectory in
terms of the mass of the ground state particle m2 and the following relation
satisfied by the Γ - function:
Γ (1 + z) = zΓ (z) (209)
we can easily perform the limit for α′ → 0 of A(s, t) obtaining:
A(s, t) =
m2 − s
m2 − s
(210)
Performing the same limit on the other two planar amplitudes we get the
following expression for the total amplitude in Eq. (206):
A(s, t, u) =
2g(2α′)
(α′)2
m2 − s
m2 − s
m2 − u
(211)
By introducing the coupling constant:
g3 = 4g(2α
4 (212)
Eq. (211) becomes
A(s, t, u) = g23
m2 − s
m2 − s
m2 − u
(213)
that is equal to the sum of the tree diagrams for the scattering of four particles
with mass m of Φ3 theory with coupling constant equal to g3. We have shown
that, by keeping g3 fixed in the limit α
′ → 0, the scattering amplitude of four
38 Paolo Di Vecchia
ground state particles of the dual resonance model is equal to the tree diagrams
of Φ3 theory. This proof can be extended to the scattering of N ground state
particles recovering also in this case the tree diagrams of Φ3 theory. It is also
valid for loop diagrams that we will discuss in the next section. In conclusion,
the dual resonance model reduces in the zero slope limit to Φ3 theory. The
proof that we have presented here is due to J. Scherk [38] 13
A more interesting case to study is the one with intercept α0 = 1. We will
see that, in this case, one will obtain the tree diagrams of Yang-Mills theory,
as shown by Neveu and Scherk [40] 14.
Let us consider the three-point amplitude involving three massless gauge
particles described by the vertex operator in Eq. (173). It is given by the sum
of two planar diagrams. The first one corresponding to the ordering (123) is
given by:
3Tr (λa1λa2λa3)
〈0, 0|Vǫ1(z1, p1)Vǫ2(z2, p2)Vǫ3(z3, p3)|0, 0〉
[(z1 − z2)(z2 − z3)(z1 − z3)]−1
(214)
Using momentum conservation p1+ p2+ p3 = 0 and the mass shell conditions
p2i = pi · ǫi = 0 one can rewrite the previous equation as follows:
0Tr(λ
a1λa2λa3)
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (215)
The second contribution comes from the ordering 132 that can be obtained
from the previous one by the substitution
Tr(λa1λa2λa3) → −Tr(λa1λa3λa2) (216)
Summing the two contributions one gets
oTr(λ
a1 [λa2 , λa3 ])
× [(ǫ1 · ǫ2)(p1 · ǫ3) + (ǫ1 · ǫ3)(p3 · ǫ2) + (ǫ2 · ǫ3)(p2 · ǫ1)] (217)
The factor
N0 = 2g(2α
′)(d−2)/4 (218)
is the correct normalization factor for each vertex operator if we normalize
the generators of the Chan-Paton group as follows:
δij (219)
13 See also Ref. [39].
14 See also Ref. [41].
The birth of string theory 39
It is related to C0 through the relation
′ = 2 (220)
g is the dimensionless open string coupling constant. Notice that Eq.s (218)
and (220) differ from Eq.s (207) and (208) because of the presence of the
Chan-Paton factors that we did not include in the case of Φ3 theory.
By using the commutation relations:
[λa, λb] = ifabcλc (221)
and the previous normalization factors we get for the three-gluon amplitude:
igYMf
a1a2a3 [(ǫ1 · ǫ2)((p1 − p2) · ǫ3 +
+(ǫ1 · ǫ3)((p3 − p1) · ǫ2) + (ǫ2 · ǫ3)((p2 − p3) · ǫ1)] (222)
that is equal to the 3-gluon vertex that one obtains from the Yang-Mills action
LYM = −
F aαβF
a , F
αβ = ∂αA
β − ∂βAaα + gYMfabcAbαAcβ (223)
where
gYM = 2g(2α
4 (224)
The previous procedure can be extended to the scattering of N gluons finding
the same result that one gets from the tree diagrams of Yang-Mills theory.
In the next section, we will discuss the loop diagrams. Also, in this case one
finds that the h-loop diagrams involving N external gluons reproduces in the
zero slope limit the sum of the h-loop diagrams with N external gluons of
Yang-Mills theory.
We conclude this section mentioning that one can also take the zero slope
limit of a scattering amplitude involving three and four gravitons obtaining
agreement with what one gets from the Einstein Lagrangian of general rela-
tivity. This has been shown by Yoneya [43].
8 Loop diagrams
The N -point amplitude previously constructed satisfies all the axioms of S-
matrix theory except unitarity because its only singularities are simple poles
corresponding to zero width resonances lying on the real axis of the Mandel-
stam variables and does not contain the various cuts required by unitarity [1].
15 The determination of the previous normalization factors can be found in the
Appendix of Ref. [42].
40 Paolo Di Vecchia
In order to eliminate this problem it was proposed already in the early days of
dual theories to assume, in analogy with what happens for instance in pertur-
bative field theory, that the N -point amplitude was only the lowest order (the
tree diagram) of a perturbative expansion and, in order to implement unitar-
ity, it was necessary to include loop diagrams. Then, the one-loop diagrams
were constructed from the propagator and vertices that we have introduced
in the previous sections [44]. The planar one-loop amplitude with M external
particles was computed by starting from a (M + 2)-point tree amplitude and
then by sewing two external legs together after the insertion of a propagator
D given in Eq. (100). In this way one gets:
(2α′)d/2(2π)d
〈P, λ|V (1, p1)DV (1, p2) . . . V (1, pN)D|P, λ〉 (225)
where the sum over λ corresponds to the trace in the space of the harmonic os-
cillators and the integral in ddP corresponds to integrate over the momentum
circulating in the loop. The previous expression for the one-loop amplitude
cannot be quite correct because all states of the space generated by the oscil-
lators in Eq. (51) are circulating in the loop, while we know that we should
include only the physical ones. This was achieved first by cancelling by hand
the time and one of the space components of the harmonic oscillators reducing
the degrees of freedom of each oscillator from d to d − 2 as suggested by the
DDF operators at least for d = 26. This procedure was then shown to be cor-
rect by Brink and Olive [45]. They constructed the operator that projects over
the physical states and, by inserting it in the loop, showed that the reduction
of the degrees of freedom of the oscillators from d to d− 2 was indeed correct.
This was, at that time, the only procedure available to let only the physical
states circulate in the loop because the BRST procedure was discovered a bit
later also in the framework of the gauge field theories!
To be more explicit let us compute the trace in Eq. (225) adding also the
Chan-Paton factor. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
τd/2+1
[f1(k)]
12 (2π)M×
dνM−1 . . .
dν2 τ
eG(νji)
]2α′pi·pj
; k ≡ e−πτ(226)
where νji ≡ νj − νi,
G(ν) = log
ie−πν
2τ Θ1(iντ |iτ)
f31 (k)
; f1(k) = k
(1− k2n) (227)
The birth of string theory 41
Θ1(ν|iτ) = −2k1/4 sinπν
1− e2iπνk2n
1− e−2iπνk2n
(1− k2n)(228)
Finally the normalization factor N0 is given in Eq. (218). We have performed
the calculation for an arbitrary value of the space-time dimension d. However,
in this way one gets also the extra factor of k
12 appearing in the first line
of Eq. (226) that implies that our calculation is actually only consistent if
d = 26. In fact, the presence of this factor does not allow one to rewrite the
amplitude, originally obtained in the Reggeon sector, in the Pomeron sector
as explained below. In the following we neglect this extra factor, implicitly
assuming that d = 26, but, on the other hand, still keeping an arbitrary d.
Using the relations:
f1(k) =
tf1(q) ; Θ1(iντ |iτ) = iΘ1(ν|it)t1/2eπν
2/t (229)
where t = 1
and q ≡ e−πt, we can rewrite the one-loop planar diagram in the
Pomeron channel. We get:
(2π)dδ(d)
NTr(λa1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2π)M×
dνM−1 . . .
−Θ1(νji|it)
f31 (q)
]2α′pi·pj
(230)
Notice that, by factorizing the planar loop in the Pomeron channel, one con-
structed for the first time what we now call the boundary state [46] 16. This
can be easily seen in the way that we are now going to describe. First of all,
notice that the last quantity in Eq. (230) can be written as follows:
Θ1(νji|it)
f31 (q)
]2α′pi·pj
−2 sin(πνji)
1− q2ne2πiνji
1− q2ne−2πiνji
(1− q2n)2
]2α′pi·pj
(231)
This equation can be rewritten as follows:
〈p = 0|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0〉
Tr (〈p = 0|q2N |p = 0〉)
; R =
na†n · an (232)
16 See also the first paper in Ref. [47].
42 Paolo Di Vecchia
where the trace is taken only over the non-zero modes and momentum con-
servation has been used. It must also be stressed that the normal ordering of
the vertex operators in the previous equation is such that the zero modes are
taken to be both in the same exponential instead of being ordered as in Eq.
(59). By bringing all annihilation operators on the left of the creation ones,
from the expression in Eq. (232) one gets (zi ≡ e2πiνi):
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
n=1 Tr
n·ane
2α′pj ·
znj e
2α′pi· an√
Tr (〈p = 0|q2N |p = 0〉)
(233)
The trace can be computed by using the completeness relation involving co-
herent states |f〉 = efa† |0〉:
e−|f |
|f〉〈f | = 1 (234)
Inserting the previous identity operator in Eq. (233) one gets after some cal-
culation:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
i.j=1
−2α′pi·pje2πinνji q
n(1−q2n) (235)
Expanding the denominator in the last exponent and performing the sum over
n one gets:
(2π)dδ(d)
(−2 sinπνji)2α
′pi·pj×
2α′pi·pj
log(1−e2πiνji q2(m+1)) (236)
that is equal to the last line of Eq. (231) apart from the δ-function for mo-
mentum conservation. In conclusion, we have shown that Eq.s (231) and (232)
are equal.
Using Eq. (231) we can rewrite Eq. (230) as follows:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt[f1(q)]
2−d(2πi)M
dνM−1 . . .
The birth of string theory 43
. . .
λ〈p = 0, λ|q2R
i=1 : e
ipi·Q(e2iπνi ) : |p = 0, λ〉
λ〈p = 0, λ|q2N |p = 0, λ〉
(237)
where the sum over any state |λ〉 corresponds to taking the trace over the
non-zero modes. If d = 26 we can rewrite Eq. (237) in a simpler form:
NNM0 Tr(λ
a1 . . . λaM )
(8π2α′)d/2
dt (2πi)M
dνM−1 . . .
〈p = 0, λ|q2R−2
: eipi·Q(e
2iπνi ) : |p = 0, λ〉 (238)
The previous equation contains the factor
dtq2R−2 that is like the propa-
gator of the Shapiro-Virasoro model, but with only one set of oscillators as
in the generalized Veneziano model. In the following we will rewrite it com-
pletely with the formalism of the Shapiro-Virasoro model. This can be done
by introducing the Pomeron propagator:
dt q2N−2 =
D̂ ; D̂ ≡ α
zL0−1z̄L̃0−1; |z| ≡ q = e−πt(239)
and rewriting the planar loop in the following compact form:
〈B0|D̂|BM 〉 ; |B0〉 ≡
n |p = 0, 0a, 0ã〉 (240)
where |B0〉 is the boundary state without any Reggeon on it,
Td−1 =
2(d−10)/4
α′)−d/2−1 (241)
and |BM 〉 is instead the one with M Reggeons given by:
|BM 〉 = NM0 Tr(λa1 . . . λaM )(2πi)M
dνM−1 . . .
: eipi·Q(e
2iπνi ) : |B0〉 (242)
We want to stress once more that the normal ordering in the previous equa-
tion is defined by taking the zero modes in the same exponential. Both the
boundary states and the propagator are now states of the Shapiro-Virasoro
model. This means that we have rewritten the one-loop planar diagram, where
the states of the generalized Veneziano model circulate in the loop, as a tree
44 Paolo Di Vecchia
diagram of the Shapiro-Virasoro model involving two boundary states and a
propagator. This is what nowadays is called open/closed string duality.
Besides the one-loop planar diagram in Eq. (225), that is nowadays called
the annulus diagram, also the non-planar and the non-orientable diagrams
were constructed and studied. In particular the non-planar one, that is ob-
tained as the planar one in Eq. (225) but with two propagators multiplied
with the twist operator
Ω = eL−1(−1)R , (243)
had unitarity violating cuts that disappeared [27] if the dimension of the
space-time d = 26, leaving behind additional pole singularities. The explicit
form of the non-planar loop can be obtained following the same steps done
for the planar loop. One gets for the non-planar loop the following amplitude:
〈BR|D̂|BM 〉 (244)
where now both boundary states contain, respectively, R and M Reggeon
states. The additional poles found in the non-planar loop were called Pomerons
because they occur in the Pomeron sector, that today is called the closed string
channel, to distinguish them from the Reggeons that instead occur in the
Reggeon sector, that today is called the open string sector of the planar and
non-planar loop diagrams. At that time in fact, the states of the generalized
Veneziano models were called Reggeons, while the additional ones appearing
in the non-planar loop were called Pomerons. The Reggeons correspond nowa-
days to open string states, while the Pomerons to closed string states. These
things are obvious now, but at that time it took a while to show that the
additional states appearing in the Pomeron sector have to be identified with
those of the Shapiro-Virasoro model. The proof that the spectrum was the
same came rather early. This was obtained by factorizing the non-planar dia-
gram in the Pomeron channel [46] as we have done in Eq. (244). It was found
that the states of the Pomeron channel lie on a linear Regge trajectory that
has double intercept and half slope of the one of the Reggeons. This follows
immediately from the propagator D̂ in Eq. (239) that has poles for values of
the momentum of the Pomeron exchanged given by:
p2 = 2n (245)
that are exactly the values of the masses of the states of the Shapiro-Virasoro
model [48], while the Reggeon propagator in Eq. (100) has poles for values of
momentum equal to:
1− α′p2 = n (246)
However, it was still not clear that the Pomeron states interact among them-
selves as the states of the Shapiro-Virasoro model. To show this it was first
The birth of string theory 45
necessary to construct tree amplitudes containing both states of the general-
ized Veneziano model and of the Shapiro-Virasoro model [49]. They reduced
to the amplitudes of the generalized Veneziano (Shapiro-Virasoro) model if
we have only external states of the generalized Veneziano (Shapiro-Virasoro)
model. Those amplitudes are called today disk amplitudes containing both
open and closed string states. They were constructed [49] by using for the
Reggeon states the vertex operators that we have discussed in Sect. (5) in-
volving one set of harmonic oscillators and for the Pomeron states the vertex
operators given in Eq. (181) that we rewrite here:
Vα,β(z, z̄, p) = Vα(z,
)Vβ(z̄,
) (247)
because now both component vertices contain the same set of harmonic os-
cillators as in the generalized Veneziano model. Furthermore, each of the two
vertices is separately normal ordered, but their product is nor normal ordered.
The amplitude involving both kinds of states is then constructed by taking
the product of all vertices between the projective invariant vacuum and inte-
grating the Reggeons on the real axis in an ordered way and the Pomerons in
the upper half plane, as one does for a disk amplitude.
We have mentioned above that the two vertices are separately normal
ordered, but their product is not normal ordered. When we normal order
them we get, for instance for the tachyon of the Pomeron sector, a factor
(z − z̄)α′p2/2 that describes the Reggeon-Pomeron transition. This implies a
direct coupling [51] between the U(1) part of gauge field and the two-index
antisymmetric field Bµν , called Kalb-Ramond field [50], of the Pomeron sector,
that makes the gauge field massive [51].
It was then shown that, by factorizing the non-planal loop in the Pomeron
channel, one reproduced the scattering amplitude containing one state of
the Shapiro-Virasoro and a number of states of the generalized Veneziano
model [52]. If we have also external states belonging to the generalized
Shapiro-Virasoro model, then by factorizing the non-planar one loop ampli-
tude in the pure Pomeron channel, one would obtain the tree amplitudes of
the Shapiro-Virasoro model [52].
All this implies that the generalized Veneziano model and the Shapiro-
Virasoro model are not two independent models, but they are part of the
same and unique model. In fact, if one started with the generalized Veneziano
model and added loop diagrams to implement unitarity, one found the ap-
pearence in the non-planar loop of additional states that had the same mass
and interaction of those of the Shapiro-Virasoro model.
The planar diagram, written in Eq. (230) in the closed string channel, is
divergent for large values of t. This divergence was recognized to be due to
exchange, in the Pomeron channel, of the tachyon of the Shapiro-Virasoro
model and of the dilaton [47]. They correspond, respectively, to the first two
terms of the expansion:
[f1(q)]
−24 = e2πt + 24 +O
e−2πt
(248)
46 Paolo Di Vecchia
The first one could be cancelled by an analytic continuation, while the second
one could be eliminated through a renormalization of the slope of the Regge
trajectory α′ [47].
We conclude the discussion of the one-loop diagrams by mentioning
that the one-loop diagram for the Shapiro-Virasoro model was computed by
Shapiro [53] who also found that the integrand was modular invariant.
The computation of multiloop diagrams requires a more advanced tech-
nology that was also developed in the early days of the dual resonance model
few years before the discovery of its connection to string theory. In order to
compute multiloop diagrams one needs first to construct an object that was
called the N -Reggeon vertex and that has the properties of containing N sets
of harmonic oscillators, one for each external leg, and is such that, when we
saturate it with N physical states, we get the corresponding N -point ampli-
tude. In the following we will discuss how to determine the N -Reggeon vertex.
The first step toward the N -Reggeon vertex is the Sciuto-Della Selva-
Saito [54] vertex that includes two sets of harmonic oscillators that we denote
with the indices 1 and 2. It is equal to:
VSDS = 2〈x = 0, 0| : exp
dzX ′2(z) ·X1(1− z)
: (249)
where X is the quantity that we have called Q in Eq. (57) and the prime
denotes a derivative with respect to z. It satisfies the important property
of giving the vertex operator Vα(z = 1) of an arbitrary state |α〉 when we
saturate it with the corresponding state:
VSDS |α〉2 = Vα(z = 1) (250)
A shortcoming of this vertex is that it is not invariant under a cyclic permu-
tation of the three legs. A cyclic symmetric vertex has been constructed by
Caneschi, Schwimmer and Veneziano [55] by inserting the twist operator in
Eq. (243). But the 3-Reggeon vertex is not enough if we want to compute an
arbitrary multiloop amplitude. We must generalize it to an arbitrary number
of external legs. Such a vertex, that can be obtained from the one in Eq. (249)
with a very direct procedure, or that can also be obtained by sewing together
three-Reggeon vertices, has been written in its final form by Lovelace [56] 17.
Here we do not derive it, but we give directly its expression written in Ref. [56]:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
i,j=1
n,m=0
a(i)n Dnm(ΓV
i Vj) a
(251)
17 See also Ref. [57]. Earlier papers on the N-Reggeon can be found in Ref.s [58].
The birth of string theory 47
where a
0 ≡ αi0 =
2α′p̂i is the momentum of particle i and the infinite
matrix:
Dnm(γ) =
∂mz [γ(z)]
n|z=0 ; n,m = 1.. : D00(γ) = − log |
AD −BC
Dn0 =
)n ; D0n =
)n ; γ(z) =
Az +B
Cz +D
(252)
is a ”representation” of the projective group corresponding to the conformal
weight ∆ = 0, that satisfies the eqs.:
Dnm(γ1γ2) =
Dnl(γ1)Dlm(γ2) +Dn0(γ1)δ0m +D0m(γ2)δn0 (253)
Dnm(γ) = Dmn(Γγ
−1Γ ) Γ (z) =
(254)
Finally Vi is a projective transformation that maps 0, 1 and ∞ into zi−1, zi
and zi+1.
The previous vertex can be written in a more elegant form as follows:
VN,0 =
i=1 dzi
dVabc
i=1[V
i (0)]
[i<x = 0, Oa|] δ(
dz∂X(i)(z)p̂i logV
i (z)
i,j=1
dy∂X(i)(z) log[Vi(z)− Vj(y)]∂X(j)(y)
(255)
where the quantities X(i) are what we called Q, namely the Fubini-Veneziano
field, in the previous sections. The N -Reggeon vertex that satisfies the impor-
tant property of giving the scattering amplitude of N physical particle when
we saturate it with their corresponding states, is the fundamental object for
computing the multiloop amplitudes. In fact, if we want to compute a M -loop
amplitude withN external states, we need to start from the (N+2M)-Reggeon
vertex and then we have to sew the M pairs together after having inserted a
propagator D. In this way we obtain an amplitude that is not only integrated
over the punctures zi (i = 1 . . . N) of the N external states, but also over
the additional 3h− 3 moduli corresponding to the punctures variables of the
48 Paolo Di Vecchia
states that we sew together and the integration variable of the M propaga-
tors. h is the number of loops. The multiloop amplitudes have been obtained
in this way already in 1970 [59, 60, 61] and, through the sewing procedure,
one obtained functions, as the period matrix, the abelian differentials, the
prime form, etc., that are well defined on Riemann surface! The only thing
that was missing, was the correct measure of integrations over the 3h−3 vari-
ables because it was technically not possible to let only the physical states to
circulate in the loops. This problem was solved only much later [62, 63] when
a BRST invariant formulation of string theory and the light-cone functional
integral could be used for computing multiloops. They are two very different
approaches that, however, gave the same result. For the sake of completeness
we write here the planar h-loop amplitude involving M tachyons:
M (p1, . . . , pM ) = N
h Tr(λa1 · · ·λaM ) Ch
2gs (2α
(d−2)/4
[dm]Mh
G(h)(zi, zj)
V ′i (0)V
j (0)
2α′pi·pj
, (256)
where Nh Tr(λa1 · · ·λaM ) is the appropriate U(N) Chan-Paton factor, g is
the dimensionless open string coupling constant, Ch is a normalization factor
given by
(2π)dh
g2h−2s
(2α′)d/2
, (257)
and G(h) is the h-loop bosonic Green function
G(h)(zi, zj) = logE(h)(zi, zj)−
ωµ (2πImτµν)
ων , (258)
with E(h)(zi, zj) being the prime form, ω
µ (µ = 1, . . . , h) the abelian differen-
tials and τµν the period matrix. All these objects, as well as the measure on
moduli space [dm]Mh , can be explicitly written in the Schottky parametrization
of the Riemann surface, and their expressions for arbitrary h can be found for
example in Ref. [64]. It is given by
[dm]Mh =
dVabc
V ′i (0)
dkµ dξµ dηµ
k2µ (ξµ − ηµ)2
(1− kµ)2
(259)
× [det (−iτµν)]−d/2
(1− knα)−d
(1− knα)2
where kµ are the multipliers, ξµ and ηµ are the fixed points of the generators
of the Schottky group,
The birth of string theory 49
9 From dual models to string theory
The approach presented in the previous sections is a real bottom-up ap-
proach. The experimental data were the driving force in the construction of
the Veneziano model and of its generalization to N external legs. The rest of
the work that we have described above consisted in deriving its properties. The
result is, except for a tachyon, a fully consistent quantum-relativistic model
that was a source of fascination for those who worked in the field. Although
the model grew out of S-matrix theory where the scattering amplitude is the
only observable object, while the action or the Lagrangian have not a central
role, some people nevertheless started to investigate what was the underly-
ing microscopic structure that gave rise to such a consistent and beautiful
model. It turned out, as we know today, that this underlying structure is that
of a quantum-relativistic string. However, the process of connecting the dual
resonance model (actually two of them the generalized Veneziano and the
Shapiro-Virasoro model) to string theory took several years from the origi-
nal idea to a complete and convincing proof of the conjecture. The original
conjecture was independently formulated by Nambu [20, 65], Nielsen [66] and
Susskind [21] 18. If we look at it in retrospective, it was at that time a fantastic
idea that shows the enormous physical intuition of those who formulated it.
On the other hand, it took several years to digest it before one was able to
derive from it all the deep features of the dual resonance model. Because of
this, the idea that the underlying structure was that of a relativistic string,
did not really influence most of the research in the field up to 1973. Let me
try to explain why.
A common feature of the work of Ref.s [20, 66, 21] is the suggestion that
the infinite number of oscillators, that one got through the factorization of
the dual resonance model, naturally comes out from a two-dimensional free
Lagrangian for the coordinate Xµ(τ, σ) of a one-dimensional string, that is an
obvious generalization of the Lagrangian that one writes for the coordinate
Xµ(τ) of a pointlike object in the proper-time gauge:
=⇒ L ∼
(260)
Being this theory conformal invariant the Virasoro operators were also con-
structed together with their algebra. In this very first formulation, however,
the Virasoro generators Ln were just the generators associated to the confor-
mal symmetry of the string world-sheet Lagrangian given in Eq. (260) as in
any conformal field theory. It was not clear at all why they should imply the
gauge conditions found by Virasoro or, in modern terms, why they should be
zero classically. The basic ingredient to solve this problem was provided by
Nambu [65] and Goto [68] who wrote the non-linear Lagrangian proportional
18 See also Ref. [67].
50 Paolo Di Vecchia
to the area spanned by the string in the external target space. They proceeded
in analogy with the point particle and wrote the following action:
−dσµνdσµν (261)
where
dσµν =
dζα ∧ dζβ = ∂Xµ
ǫαβdσdτ (262)
Xµ(σ, τ) is the string coordinate and ζ
0 = τ and ζ1 = σ are the coordinates of
the string worldsheet. ǫαβ is an antisymmetric tensor with ǫ01 = 1. Inserting
eq. (262) in (261) and fixing the proportionality constant one gets the Nambu-
Goto action [65, 68]:
S = −cT
(Ẋ ·X ′)2 − Ẋ2X ′2 (263)
where
Ẋµ ≡ ∂X
µ ≡ ∂X
(264)
and T ≡ 1
is the string tension, that replaces the mass appearing in the
case of a point particle. In this formulation, the string Lagrangian is invariant
under any reparametrization of the world-sheet coordinates σ and τ and not
only under the conformal transformations. This, in fact, implies that the two-
dimensional world-sheet energy-momentum tensor of the string is actually zero
as we will show later on. But it took still a few years to connect the Nambu-
Goto action to the properties of the dual resonance model. In the meantime
an analogue model was formulated [69] that reproduced the tree and loop
amplitudes of the generalized Veneziano model. This approach anticipated
by several years the path integral derivation of dual amplitudes. It was very
closely related to the functional integral formulation of Ref.s [70].
However, one needed to wait until 1973 with the paper of Goddard,
Goldstone, Rebbi and Thorn [71], where the Nambu-Goto action was cor-
rectly treated, all its consequences were derived and it became completely
clear that the structure underlying the dual resonance model was that of a
quantum-relativistic string. The equation of motion for the string were de-
rived from the action in Eq. (263) by imposing δS = 0 for variations such
that δXµ(τi) = δX
µ(τf ) = 0. One gets:
∂X ′µ
δXµ +
∂X ′µ
δXµ|σ=πσ=0
(265)
where L is the Lagrangian in Eq. (263). Since δXµ is arbitrary, from eq. (265)
one gets the Euler-Lagrange equation of motion
The birth of string theory 51
∂X ′µ
= 0 (266)
and the boundary conditions
∂X ′µ
= 0 or δXµ = 0 at σ = 0, π (267)
for an open string and
Xµ(τ, 0) = Xµ(τ, π) (268)
for a closed string. In the case of an open string, the first kind of boundary
condition in Eq.(267) corresponds to Neumann boundary conditions, while
the second one to Dirichlet boundary conditions. Only the Neumann bound-
ary conditions preserve the translation invariance of the theory and, there-
fore, they were mostly used in the early days of string theory. It must be
stressed, however, that Dirichlet boundary conditions were already discussed
and used in the early days of string theory for constructing models with off-
shell states [72].
From Eq. (263) one can compute the momentum density along the string:
≡ Pµ = cT
′2 −X ′µ(Ẋ ·X ′)
(Ẋ ·X ′)2 − Ẋ2X ′2
(269)
and obtain the following constraints between the dynamical variables Xµ and
c2T 2x′
+ P 2 = x′ · P = 0 (270)
They are a consequence of the reparametrization invariance of the string La-
grangian. Because of this one can choose the orthonormal gauge specified by
the conditions:
Ẋ2 +X ′
= Ẋ ·X ′ = 0 (271)
that nowadays is called conformal gauge. In this gauge eq. (269) becomes:
Pµ = cT Ẋµ
∂X ′µ
= −cTX ′µ (272)
and therefore the eq. of motion in eq.(266) becomes:
Ẍµ −X ′′µ = 0 (273)
while the boundary condition in eq.(267) becomes:
X ′µ(σ = 0, π) = 0 (274)
52 Paolo Di Vecchia
The most general solution of the eq. of motion and of the boundary conditions
can be written as follows:
Xµ(τ, σ) = qµ + 2α′pµτ + i
[aµne
−inτ − a+µn einτ ]
cosnσ√
(275)
for an open string and
Xµ(τ, σ) = qµ + 2α′pµτ +
[ãµne
−2in(τ+σ) − ã+µn e2in(τ+σ)]
[aµne
−2in(τ−σ) − a+µn e2in(τ−σ)]
(276)
for a closed string. This procedure really shows that, starting from the
Nambu-Goto action, one can choose a gauge (the orthonormal or confor-
mal gauge) where the equation of motion of the string becomes the two-
dimensional D’Alembert equation in Eq. (273). Furthermore, the invariance
under reparametrization of the Nambu-Goto action implies that the two-
dimensional energy-momentum tensor is identically zero at the classical level
(See Eq. (271)).
As the Lorentz gauge in QED the orthonormal gauge does not fix com-
pletely the gauge. We can still perform reparametrizations that leave in the
conformal gauge: they are conformal transformatiuons. Introducing the vari-
able z = eiτ the generators of the conformal transformations for the open
string can be written as follows:
dzzn+1
αn−m · αm = 0 (277)
where
αµn =
naµn if n > 0√
2α′pµ if n = 0√
na†µn if n < 0
(278)
They are zero as a consequence of Eq.s (270) that in the conformal gauge
become Eq.s (271). In the case of a closed string we get instead:
L̃n =
dzzn+1
= 0 (279)
dz̄z̄n+1
= 0 (280)
The birth of string theory 53
In terms of the harmonic oscillators introduced in eq. (276) we get
αm · αn−m = 0 ; L̃n =
α̃m · α̃n−m = 0 (281)
where for the non-zero modes we have used the convention in (278), while the
zero mode is given by:
0 = α̃
(282)
In conclusion, the fact that we have reparametrization invariance implies that
the Virasoro generators are classically identically zero. When we quantize the
theory one cannot and also does not need to impose that they are vanishing at
the operator level. They are imposed as conditions characterizing the physical
states.
〈Phys′|Ln|Phys〉 = 〈Phys′|(L0 − 1)|Phys〉 = 0 ; n 6= 0 (283)
These equations are satisfied if we require:
Ln|Phys >= (L0 − 1)|Phys >= 0 (284)
The extra factor −1 in the previous equations comes from the normal ordering
as explained in Eq. (198).
The authors of Ref. [71] further specified the gauge by fixing it completely.
They introduced the light-cone gauge specified by imposing the condition:
X+ = 2α′p+τ (285)
where
X0 ±Xd−1√
X0 ±Xd−1√
(286)
In this gauge the only physical degrees of freedom are the transverse ones.
In fact the components along the directions 0 and d − 1 can be expressed in
terms of the transverse ones by inserting Eq. (285) in the constraints in Eq.
(271) and getting:
Ẋ− =
4α′p+
(Ẋ2i +X
i ) X
2α′p+
Ẋi ·X ′i (287)
that up to a constant of integration determine completely X− as a function
of X i. In terms of oscillators we get
α+n = 0 ;
2α′α−n =
αin−mα
m n 6= 0 (288)
54 Paolo Di Vecchia
for an open string and
α+n = α̃
n = 0 n 6= 0 (289)
together with
2α′α−n =
αin−mα
2α′α̃−n =
α̃in−mα̃
m (290)
in the case of a closed string.
This shows that the physical states are described only by the transverse
oscillators having only d − 2 components. Those transverse oscillators corre-
spond to the transverse DDF operators that we have discussed in Section 6.
The authors of Ref. [71] also constructed the Lorentz generators only in terms
of the transverse oscillators and they showed that they satisfy the correct
Lorentz algebra only if the space-time dimension is d = 26. In this way the
spectrum of the dual resonance model was completely reproduced starting
from the Nambu-Goto action if d = 26! On the other hand, the choice of
d = 26 is a necessity if we want to keep Lorentz invariance!
Immediately after this, the interaction was also included either by adding
a term describing the interaction of the string with an external gauge field [73]
or by using a functional formalism [74, 75].
In the following we will give some detail only of the first approach for the
case of an open string. A way to describe the string interaction is by adding
to the free string action an additional term that describes the interaction of
the string with an external field.
SINT =
dDyΦL(y)JL(y) (291)
where ΦL(y) is the external field and JL is the current generated by the string.
The index L stands for possible Lorentz indices that are saturated in order to
have a Lorentz invariant action.
In the case of a point particle, such an interaction term will not give any
information on the self-interaction of a particle.
In the case of a string, instead, we will see that SINT will describe the
interaction among strings because the external fields that can consistently
interact with a string are only those that correspond to the various states of
the string, as it will become clear in the discussion below.
This is a consequence of the fact that, for the sake of consistency, we must
put the following restrictions on SINT :
• It must be a well defined operator in the space spanned by the string
oscillators.
The birth of string theory 55
• It must preserve the invariances of the free string theory. In particular, in
the ”conformal gauge” it must be conformal invariant.
• In the case of an open string, the interaction occurs at the end point of
a string (say at σ = 0). This follows from the fact that two open strings
interact attaching to each other at the end points.
The simplest scalar current generated by the motion of a string can be written
as follows
J(y) =
dσδ(σ)δ(d)[yµ − xµ(τ, σ)] (292)
where δ(σ) has been introduced because the interaction occurs at the end of
the string. For the sake of simplicity we omit to write a coupling constant g
in (292).
Inserting (292) in (291) and using for the scalar external field Φ(y) = eik·y
a plane wave, we get the following interaction:
SINT =
dτ : eik·X(τ,0) : (293)
where the normal ordering has been introduced in order to have a well defined
operator. The invariance of (293) under a conformal transformation τ → w(τ)
requires the following identity:
SINT =
dτ : eik·X(τ,0) : =
dw : eik·X(w,0) : (294)
or, in other words, that
: eik·X(τ,0) :=⇒ w′(τ) : eik·X(w,0) : (295)
This means that the integrand in Eq. (294) must be a conformal field with
conformal dimension equal to one and this happens only if α′k2 = 1. The
external field corresponds then to the tachyonic lowest state of the open string.
Another simple current generated by the string is given by:
Jµ(y) =
dσδ(σ)Ẋµ(τ, σ)δ
(d)(y −X(τ, σ)) (296)
Inserting (296) in (291) we get
SINT =
dτẊµ(τ, 0)ǫ
µeik·X(τ,0) (297)
if we use a plane wave for Φµ(y) = ǫµe
ik·y. The vertex operator in eq. (297)
is conformal invariant only if
k2 = ǫ · k = 0 (298)
56 Paolo Di Vecchia
and, therefore, the external vector must be the massless photon state of the
string. We can generalize this procedure to an arbitrary external field and
the result is that we can only use external fields that correspond to on shell
physical states of the string.
This procedure has been extended in Ref. [73] to the case of external
gravitons by introducing in the Nambu-Goto action a target space metric and
obtaining the vertex operator for the graviton that is a massless state in the
closed string theory. Remember that, at that time, this could have been done
only with the Nambu-Goto action because the σ-model action was introduced
only in 1976 first for the point particle [76] and then for the string [77]. As
in the case of the photon it turned out that the external field corresponding
to the graviton was required to be on shell. This condition is the precursor of
the equations of motion that one obtains from the σ-model action requiring
the vanishing of the β-function [78].
One can then compute the probability amplitude for the emission of a
number of string states corresponding to the various external fields, from an
initial string state to a final one. This amplitude gives precisely the N -point
amplitude that we discussed in the previous sections [73]. In particular, one
learns that, in the case of the open string, the Fubini-Veneziano field is just
the string coordinate computed at σ = 0:
Qµ(z) ≡ Xµ(z, σ = 0) ; z = eiτ (299)
In the case of a closed string we get instead:
Qµ(z, z̄) ≡ Xµ(z, z̄) ; z = e2i(τ−σ) , z̄ = e2i(τ+σ) (300)
Finally, let me mention that with the functional approach Mandelstam [74]
and Cremmer and Gervais [79] computed the interaction between three arbi-
trary physical string states and reproduced in this way the coupling of three
DDF states given in Eq. (202) and obtained in Ref. [37] by using the operator
formalism. At this point it was completely clear that the structure underlying
the generalized Veneziano model was that of an open relativistic string, while
that underlying the Shapiro-Virasoro model was that of a closed relativistic
string. Furthermore, these two theories are not independent because, if one
starts from an open string theory, one gets automatically closed strings by
loop corrections.
10 Conclusions
In this contribution, we have gone through the developments that led from
the construction of the dual resonance model to the bosonic string theory
trying as much as possible to include all the necessary technical details. This
is because we believe that they are not only important from an historical point
of view, but are also still part of the formalism that one uses today in many
The birth of string theory 57
string calculations. We have tried to be as complete and objective as possible,
but it could very well be that some of those who participated in the research
of these years, will not agree with some or even many of the statements we
made. We apologize to those we have forgotten to mention or we have not
mentioned as they would have liked.
Finally, after having gone through the developments of these years, my
thoughts go to Sergio Fubini who shared with me and Gabriele many of the
ideas described here and who is deeply missed, and to my friends from Flo-
rence, Naples and Turin for a pleasant collaboration in many papers discussed
here.
Acknowledgments
I thank R. Marotta and I. Pesando for a critical reading of the manuscript.
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The birth of string theory
Paolo Di Vecchia
|
0704.0102 | Duality and Tameness | arXiv:0704.0102v1 [math.AC] 1 Apr 2007
DUALITY AND TAMENESS
MARC CHARDIN, STEVEN DALE CUTKOSKY, JÜRGEN HERZOG AND HEMA SRINIVASAN
Abstract. We prove a duality theorem for certain graded algebras and show by various
examples different kinds of failure of tameness of local cohomology.
Introduction
The purpose of this paper is to construct examples of strange behavior of local coho-
mology. In these constructions we follow a strategy that was already used in [CH] and
which relates, via a spectral sequence introduced in [HR], the local cohomology for the
two distinguished bigraded prime ideals in a standard bigraded algebra.
In the first part we consider algebras with rather general gradings and deduce a similar
spectral sequence in this more general situation. A typical example of such an algebra
is the Rees algebra of a graded ideal. The proof for the spectral sequence given here is
simpler than that of the corresponding spectral sequence in [HR].
In the second part of this paper we construct examples of standard graded rings A,
which are algebras over a field K, such that the function
(1) j 7→ dimK(H iA+(A)−j)
is an interesting function for j ≫ 0. In our examples, this dimension will be finite for all
Suppose that A0 is a Noetherian local ring, A =
j≥0Aj is a standard graded ring and
set A+ :=
j>0Aj. Let M be a finitely generated graded A-module and F := M̃ be the
sheafification of M on Y = Proj(A). We then have graded A-module isomorphisms
H i+1A+ (M)
H i(Y,F(n))
for i ≥ 1, and a similar expression for i = 0 and 1.
By Serre vanishing, H iA+(M)j = 0 for all i and j ≫ 0. However, the asymptotic
behaviour of H iA+(M)−j for j ≫ 0 is much more mysterious.
In the case when A0 = K is a field, the function (1) is in fact a polynomial for large
enough j. The proof is a consequence of graded local duality ([BS, 13.4.6] or [BH, 3.6.19])
or follows from Serre duality on a projective variety.
For more general A0, HA+(M)−j are finitely generated A0 modules, but need not have
finite length.
The following problem was proposed by Brodmann and Hellus [BrHe].
The second author was partially supported by NSF.
http://arxiv.org/abs/0704.0102v1
Tameness problem: Are the local cohomology modules H iA+(M) tame? That is, is it
true that either
{H iA+(M)j 6= 0, ∀j ≪ 0} or {H
A+(M)j = 0, ∀j ≪ 0}?
The problem has a positive solution for A0 of small dimension (some of the references
are Brodmann [Br], Brodmann and Hellus [BrHe], Lim [L], Rotthaus and Sega [RS]).
Theorem 0.1 ([BrHe]). If dim(A0) ≤ 2, then M is tame.
However, it has recently been shown by two of the authors that tameness can fail if
dim(A0) = 3.
Theorem 0.2 ([CH]). There are examples with dim(A0) = 3 where M is not tame.
The statement of this example is reproduced in Theorem 3.1 of this paper. The function
(1) is periodic for large j. Specifically, the function (1) is 2 for large even j and is 0 for
large odd j.
In Theorem 3.3 we construct an example of failure of tameness of local cohomology
which is not periodic, and is not even a quasi polynomial (in −j) for large j. Specifically,
we have for j > 0,
dimK(H
(A)−j) =
1 if j ≡ 0 (mod) (p+ 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where the characteristic of K is p. We have pt ≡ −1 (mod) (p + 1) for all odd t ≥ 0.
We also give an example (Theorem 3.5) of failure of tameness where (1) is a quasi
polynomial with linear growth in even degree and is 0 in odd degree.
In Theorem 3.6, we give a tame example, but we have
dimK(H
(A)−j)
so (1) is far from being a quasi polynomial in −j for large j.
While the example of [CH] is for M = ωA, where ωA is the canonical module of A, the
examples of the paper are all for M = A. This allows us to easily reinterpret our examples
as Rees algebras in Section 4, and thus we have examples of Rees algebras over local rings
for which the above failure of tameness holds.
In the final section, Section 5, we give an analysis of the explicit and implicit role of
bigraded duality in the construction of the examples, and some comments on how it effects
the geometry of the constructions.
1. Duality for polynomial rings in two sets of variables
Let K be any commutative ring (with unit). In later applications K will be mostly
a field. Furthermore let S = K[x1, . . . , xm, y1, . . . , yn], P = (x1, . . . , xm) and Q =
(y1, . . . , yn).
The homology of the Čech complex CP ( ) (resp. CQ( )) will be denoted by HP ( ) (resp.
HQ( )). Notice that for any commutative ring K, this homology is the local cohomology
supported in P (resp. Q), as P and Q are generated by a regular sequences.
Assume that S is Γ-graded for some abelian group Γ, and that deg(a) = 0 for a ∈ K.
If xsyp ∈ R, deg(xsyp) = l(s) + l′(p) with l(s) :=
i si deg(xi) and l
′(p) :=
j pj deg(yj).
Definition 1.1. Let I ⊂ S be a Γ-graded ideal. The Γ-grading of S is I-sharp if H iI(S)γ
is a finitely generated K-module, for every i and γ ∈ Γ.
Lemma 1.2. The following conditions are equivalent:
(i) the Γ-grading of S is P -sharp.
(ii) the Γ-grading of S is Q-sharp.
(iii) for all γ ∈ Γ, |{(α, β) : α ≥ 0, β ≥ 0, l(α) = γ + l′(β)}| < ∞.
Note that if K is Noetherian, M is a finitely generated Γ-graded S-module, and the
Γ-grading of S is I-sharp, then H iI(M)γ is a finite K-module, for every i and γ ∈ Γ. This
follows from the converging Γ-graded spectral sequence Hp−q(H
I (F)) ⇒ H
I (M), where F
is a Γ-graded free S-resolution of M with Fi finite for every i.
We will assume from now on that the Γ-grading of S is P -sharp (equivalently Q-
sharp). Set σ = deg(x1 · · · xmy1 · · · yn), and if N is a Γ-graded module, then let N∨ =
HomS(N,S(−σ)) and N∗ = ∗HomK(N,K) where the Γ-grading of N∗ is given by (N∗)γ =
HomK(N−γ ,K). More generally, we always denote the graded K-dual of a graded mod-
ule N (over what graded K-algebra soever) by N∗. Finally we denote by ϕαβ the map
S(−a) → S(−b) induced by multiplication by xαyβ where a = deg xα and b = − deg yβ.
Lemma 1.3. HmP (ϕαβ)γ
∼= HnQ(ϕ∨αβ)∗.
Proof. The free K-moduleHmP (S)γ is generated by the elements x
−s−1yp with s, p ≥ 0 and
−l(s)− l(1) + l′(p) = γ, and HnQ(S)γ′ is generated by the elements xty−q−1 with t, q ≥ 0
and l(t)− l′(q)− l′(1) = γ′.
Let dγ : H
P (S)γ → (HnQ(S∨)∗)γ = HnQ(S)−γ−σ be the K-linear map defined by
−s−1yp)(xty−q−1) =
1, if s = t and p = q,
0, else.
Then dγ is an isomorphism (because the Γ-grading of R is Q-sharp) and there is a com-
mutative diagram
HmP (S)γ−a
(ϕαβ)γ−−−−−−−→ HmP (S)γ−b
y dγ−b
(HnQ(S)−γ+a−σ)
−−−−−−−−−→ (HnQ(S)−γ+b−σ)∗.
The assertion follows. �
As an immediate consequence we obtain
Corollary 1.4. (a) Let f ∈ S be an homogeneous element of degree a−b, and ϕ : S(−a) →
S(−b) the graded degree zero map induced by multiplication with f . Then
HmP (ϕ) ≃ HnQ(ϕ∨)∗.
(b) Let F be a Γ-graded complex of finitely generated free S-modules. Then
(i) H iP (F) = 0 for i 6= m and H
Q(F) = 0 for j 6= n,
(ii) HmP (F) ≃ HnQ((F)∨)∗.
As the main result of this section we have
Theorem 1.5. Assume that K is Noetherian, the Γ-grading of S is P -sharp (equivalently
Q-sharp) and M is a finitely generated Γ-graded S-module. Set ωS/K := S(−σ). Let F be
a minimal Γ-graded S-resolution of M . Then,
(a) For all i, there is a functorial isomorphism
H iP (M) ≃ Hm−i(HmP (F)).
(b) There is a convergent Γ-graded spectral sequence,
H iQ(Ext
S(M,ωS/K)) ⇒ H
i+j−n(HmP (F)
In particular, if K is a field, there is a convergent Γ-graded spectral sequence,
H iQ(Ext
S(M,ωS)) ⇒ H
dimS−(i+j)
P (M)
Proof. Claim (a) is an immediate consequence of Corollary 1.4 via the Γ-graded spec-
tral sequence Hp−i(H
P (F)) ⇒ H iP (M). For (b), the two spectral sequences arising from
the double complex CQF∨ have as second terms respectively ′Eij2 = H iQ(Ext
S(M,ωS/K)),
2 = 0 for i 6= n and ′′E
2 = H
j(HnQ(F
∨)) ≃ Hj(HmP (F)∗). If further K is a field,
Hj(HmP (F)
∗) ≃ (Hj(HmP (F)))∗ ≃ H
P (F)
Corollary 1.6. Under the hypotheses of the theorem, if K is a field, then for any γ ∈ Γ,
there are convergent spectral sequences of finite dimensional K-vector spaces
H iQ(Ext
S(M,ωR))γ ⇒ H
dimS−(i+j)
P (M)−γ ,
H iP (Ext
S(M,ωR))γ ⇒ H
dimS−(i+j)
Q (M)−γ .
We now consider the special case that Γ = Z2, S := K[x1, . . . , xm, y1, . . . , yn] with
deg(xi) = (1, 0) and deg(yj) = (dj , 1) with dj ≥ 0. Set T := K[x1, . . . , xm] and let M be
a Γ-graded S-module. We view M as a Z-graded module by defining Mk =
j M(j,k).
Observe that each Mk itself is a graded T -module with (Mk)j = M(j,k) for all j. We also
note that H iP (M)k
∼= H iP0(Mk), as can been seen from the definition of local cohomology
using the Čech complex. Here P0 = (x1, . . . , xm) is the graded maximal ideal of T .
Corollary 1.7. With the notation introduced, let s := dimS = m + n and d := dimM .
(a) H0P (Ext
S (M,ωS))
∼= HdQ(M)∗ for any k,
(b) there is an exact sequence
0→H1P (Exts−dS (M,ωS))→H
Q (M)
∗→H0P (Exts−d+1S (M,ωS)).
(c) Let i ≥ 2. If ExtjS(M,ωS) is annihilated by a power of P for all s−d < j < s−d+i,
then there is an exact sequence
Exts−d+i−1S (M,ωS)→H
P (Ext
S (M,ωS))→H
Q (M)
∗→H0P (Exts−d+iS (M,ωS)).
In particular, if Ext
S(M,ωS) has finite length for all s − d < j ≤ s − d+ i0, for some
integer i0, then
H iP0(Ext
S (M,ωS)k)
∼= (Hd−iQ (M)−k)
∗ for all i ≤ i0 and k ≫ 0.
Consequently, if M is a generalized Cohen-Macaulay module (i.e. Exts−iS (M,ωS) has
finite length for all i 6= d), and if we set N = Exts−dS (M,ωS), then
H iP0(Nk)
∼= (Hd−iQ (M)−k)
∗ for all i and all k ≫ 0.
Proof. (a), (b) and (c) are direct consequences of Corollary 1.6. For the application, notice
that if γ = (ℓ, k) ∈ Γ with k ≫ 0 one has ExtjS(M,ωS)γ = 0 for all s− d < j ≤ s− d+ i0.
Therefore, for such γ, the desired conclusion follows. �
A typical example to which this situation applies is the Rees algebra of a graded ideal
I in the standard graded polynomial ring T = K[x1, . . . , xm]. Say, I is generated be the
homogeneous polynomials f1, . . . , fn with deg fj = dj for j = 1, . . . , n. Then the Rees
algebra R(I) ⊂ T [t] is generated the elements fjt. If we set deg fjt = (dj , 1) for all j
and deg xi = (1, 0) for all i, then R(I) becomes a Γ-graded S-module via the K-algebra
homomorphism S → R(I) with xi 7→ xi and yj 7→ fjt.
According to this definition we have R(I)k = Ik for all k.
Since dimR(I) = m+1, the module ωR(I) = Extn−1S (R(I), ωS) is the canonical module
of R(I) (in the sense of [HK, 5. Vortrag]). Recall that if a ring R is a finite S-module
of dimension m + 1, the natural finite map R→Hom(ωR, ωR) ∼= Extn−1S (ωR, ωS) is an
isomorphism if and only if R is S2. Thus in combination with Corollary 1.7 we obtain
Corollary 1.8. Let R := R(I). Suppose that Rp is Cohen-Macaulay for all p 6= (m, R+)
where m = (x1, . . . , xm) and R+ =
k>0 I
ktk. Then
H im(I
k) ∼= (Hm+1−iR+ (ωR)−k)
∗ for all i and all k ≫ 0.
Proof. Since ωR localizes, the conditions imply that (ωR)p is Cohen-Macaulay for all p 6=
(m, R+). Hence the natural into map R→R′ := Extn−1S (ωR, ωS) has a cokernel of finite
length. In particular, R′k = Rk = I
k for k ≫ 0. Thus Corollary 1.7 applied to M = ωR
gives the desired conclusion. �
Remark 1.9. Let R := R(I). If the cokernel of R→Hom(ωR, ωR) is annihilated by a
power of R+ (in other words, the blow-up is S2, as a projective scheme over Spec(T )),
then R′k = I
k for k ≫ 0 and therefore one has an exact sequence
0→H0m(T/Ik)→(HmR+(ωR)−k)
∗→H0m(ExtnS(ωR, ωS)k)→H1m(T/Ik)→(Hm−1R+ (ωR)−k)
for such a k.
2. A method of constructing examples
Suppose that R =
i,j≥0Rij is a standard bigraded algebra over a ring K = R00.
Define Ri =
j≥0Rij and Rj =
i≥0Rij . Define ideals P =
i and Q =
j>0Rj
in R. Suppose that M =
ij∈ZMij is a finitely generated, bigraded R-module. Define
M i =
j∈ZMij and Mj =
i∈ZMij. M
i is a graded R0-module and Mj is a graded
R0-module. Let Q0 = R01R
0, so that Q = Q0R. Let P0 = R10R0 so that P = R10R. We
have K module isomorphisms
H lQ(M)m,n
∼= H lQ0(M
for m,n ∈ Z. Let M̃m be the sheafification of the graded R0-module Mm on Proj(R0).
We have K module isomorphisms
H lQ0(M
m)n ∼= H l−1(Proj(R0), M̃m(n))
for l ≥ 2 and exact sequences
0 → H0Q0(M
m)n → (Rm)n = Rm,n → H0(Proj(R0), M̃m(n)) → H1Q0(M
m)n → 0.
We have similar formulas for the calculation of H lP (M).
Now assume that X is a projective scheme over K and F1 and F2 are very ample line
bundles on X. Let
Rm,n = Γ(X,F⊗m1 ⊗F
We require that R =
m,n≥0Rm,n be a standard bigraded K-algebra. We have
X ∼= Proj(R0) ∼= Proj(R0).
The sheafification of the graded R0-module Rm on X is R̃m = F⊗m1 , and the sheafification
of the graded R0-module Rn on X is R̃n ∼= F⊗n2 (Exercise II.5.9 [Ha]).
For l ≥ 2 we have bigraded isomorphisms
H lQ(R)
H lQ0(R
m)n ∼=
m≥0,n∈Z
H l−1(X,F⊗m1 ⊗F
Viewing R as a graded R0 algebra, we thus have graded isomorphisms
(2) H lQ(R)n
H l−1(X,F⊗m1 ⊗F
for l ≥ 2 and n ∈ Z. Let d = dim(R) = dim(X) + 2.
We now further assume that K is an algebraically closed field and X is a nonsingular
K variety. Let
V = P(F1 ⊕F2),
a projective space bundle over X with projection π : V → X. Since F1 ⊕ F2 is an ample
bundle on X, OV (1) is ample on V . Since
Γ(V,OV (t))
Γ(V,OV (t)) ∼= Γ(X,St(F1 ⊕F2)) ∼=
i+j=t
and R is generated in degree 1 with respect to this grading, OV (1) is very ample on V and
R is the homogeneous coordinate ring of the nonsingular projective variety V , so that R
is generalized Cohen Macaulay (all local cohomology modules H iR+(R) of R with respect
to the maximal bigraded ideal R+ of R have finite length for i < d). We further have that
V is projectively normal by this embedding (Exercise II.5.14 [Ha]) so that R is normal.
3. Strange behavior of local cohomology
In [CH], we constructed the following example of failure of tameness of local cohomology.
In the example, R0 has dimension 3, which is the lowest possible for failure of tameness
[Br].
Theorem 3.1. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j ≫ 0,
dimK(H
Q(ωR)−j) =
2 if j is even,
0 if j is odd,
where ωR is the canonical module of R, Q =
n>0Rn.
We first show that the above theorem is also true for the local cohomology of R.
Theorem 3.2. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j > 0,
dimK(H
Q(R)−j) =
2 if j is even,
0 if j is odd,
where Q =
n>0Rn.
Proof. We compute this directly for the R of Theorem 3.1 from (2) and the calculations of
[CH]. Translating from the notation of this paper to the notation of [CH], we have X = S
is an Abelian surface, F1 = OS(r2laH) and F2 = OS(r2(D + alH)).
By (2) of this paper, for n ∈ N, we have
dimK(H
Q(R)n) =
m≥0 h
1(X,F⊗m1 ⊗F
m≥0 h
1(S,OS((m+ n)r2alH + nr2D)).
Formula (1) of [CH] tells us that for m,n ∈ Z,
(3) h1(S,OS(mH + nD)) =
2 if m = 0 and n is even,
0 otherwise.
Thus for n < 0, we have
dimK(H
Q(R)n) =
2 if n is even,
0 if n is odd,
giving the conclusions of the theorem. �
The following example shows non periodic failure of tameness.
Theorem 3.3. Suppose that p is a prime number such that p ≡ 2 (mod) 3 and p ≥ 11.
Then there exists a normal standard graded K-algebra R0 over a field K of characteristic
p with dim(R0) = 4, and a normal standard graded R0-algebra R with dim(R) = 5 such
that for j > 0,
dimK(H
Q(R)−j) =
1 if j ≡ 0 (mod) (p+ 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where Q =
n>0Rn. We have p
t ≡ −1(mod)(p + 1) for all odd t ≥ 0.
To establish this, we need the following simple lemma.
Lemma 3.4. Suppose that C is a non singular curve of genus g over an algebraically closed
field K, and M, N are line bundles on C. If deg(M) ≥ 2(2g+1) and deg(N ) ≥ 2(2g+1),
then the natural map
Γ(C,M) ⊗ Γ(C,N ) → Γ(C,M⊗N )
is a surjection.
Proof. If L is a line bundle on C, then H1(C,L) = 0 if deg(L) > 2g − 2 and L is very
ample if deg(L) ≥ 2g + 1 (Chapter IV, Section 3 [Ha]).
Suppose that L is very ample and G is another line bundle on C. If deg(G) > 2g − 2−
deg(L), then G is 2-regular for L (Lecture 14, [M1]). Thus if deg(G) > 2g − 2 + deg(L),
Γ(C,G) ⊗ Γ(C,L) → Γ(C,G ⊗ L)
is a surjection by Castelnuovo’s Proposition, Lecture 14, page 99 [M1].
We now apply the above to prove the lemma. Write M ∼= A⊗q ⊗ B where A is a
line bundle such that deg(A) = 2g + 1, and 2g + 1 ≤ deg(B) < 2(2g + 1). deg(N ) >
2g − 2 + deg(A). Thus there exists a surjection
Γ(C,N ) ⊗ Γ(C,A) → Γ(C,A⊗N ).
We iterate to get surjections
Γ(C,A⊗i ⊗N )⊗ Γ(C,A) → Γ(C,A⊗(i+1) ⊗N )
for i ≤ q, and a surjection
Γ(C,A⊗q ⊗N )⊗ Γ(C,B) → Γ(C,M⊗N ).
We now prove Theorem 3.3. For the construction, we start with an example from
Section 6 of [CS]. There exists an algebraically closed field K of characteristic p, a curve
C of genus 2 over K, a point q ∈ C and a line bundle M on C of degree 0, such that for
n ≥ 0,
H1(C,OC (q)⊗M⊗n) =
1 if n = pt for some t ≥ 0,
0 otherwise.
Further, H1(C,OC (2q)⊗M⊗n) = 0 for all n > 0.
Let a = p+ 1. Let E be an elliptic curve over K, and let T = E × E, with projections
πi : T → E. Let b ∈ E be a point and let A = π∗1(OE(b)) ⊗ π∗2(OE(b)). Let X = T × C,
with projections ϕ1 : X → T , ϕ2 : X → C. Let L = OC(q). Let
F1 = ϕ∗1(A)⊗a ⊗ ϕ∗2(L)⊗a,
F2 = ϕ∗1(A)⊗(1+a) ⊗ ϕ∗2(L⊗(1+a) ⊗M−1).
For m,n ≥ 0, we have
Γ(X,F⊗m1 ⊗F
2 ) = Γ(T,A⊗(ma+n(1+a)))⊗ Γ(C,L⊗(ma+n(1+a)) ⊗M−⊗n)
= Γ(T,A⊗a)⊗m ⊗ Γ(T,A⊗(1+a))⊗n ⊗ Γ(C,La)⊗m ⊗ Γ(C,L⊗(1+a) ⊗M−1)⊗n
= Γ(X,F1)⊗m ⊗ Γ(X,F2)⊗n
by the Künneth formula (IV of Lecture 11 [M1]) and Lemma 3.4.
Let Rm,n = Γ(X,F⊗m1 ⊗F
2 ). R =
m,n≥0Rm,n is a standard bigraded K-algebra by
(4). Thus (2) holds.
By the Riemann Roch Theorem, we compute,
(5) h0(C,L⊗r ⊗M−⊗s) = h1(C,L⊗r ⊗M−⊗s) + r − 1,
and for s < 0,
(6) h1(C,L⊗r ⊗M−⊗s) =
1− r r < 0,
1 r = 0, s < 0,
1 r = 1, s = −pt, for some t ∈ N,
0 r = 1, s 6= −pt for some t ∈ N,
0 r = 2, s < 0,
0 r ≥ 3.
We further have
(7) h1(T,A⊗r) =
0 r 6= 0,
2 r = 0,
(8) h0(T,A⊗r) =
0 r < 0,
1 r = 0,
r2 r > 0.
By (2), for n ∈ Z, we have
dimK(H
Q(R)n) =
h1(X,F⊗m1 ⊗F
By the Künneth formula,
H1(X,F⊗m1 ⊗F
∼= H0(T,A⊗(ma+n(1+a)))⊗H1(C,L⊗(ma+n(1+a)) ⊗M−⊗n)
⊕H1(T,A⊗(ma+n(1+a)))⊗H0(C,L⊗(ma+n(1+a)) ⊗M−⊗n).
Thus by (5) - (8), we have for j > 0,
dimK(H
Q(R)−j) =
1 j ≡ 0 (mod) a,
1 j = pt for some odd t ∈ N,
0 otherwise.
and we have the conclusions of Theorem 3.3.
Theorem 3.5 gives an example of failure of tameness of local cohomology with larger
growth.
Theorem 3.5. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 over K with dim(R0) = 4, and a normal standard graded
R0-algebra R with dim(R) = 5 such that for j > 0,
dimK(H
Q(R)−j) =
6j if j is even,
0 if j is odd,
where Q =
n>0Rn.
Proof. Let E be an elliptic curve over K, and let q ∈ E be a point. Let L = OE(3q). By
Proposition IV.4.6 [Ha], L is very ample on E, and
(9) ⊕n≥0Γ(E,L⊗n)
is generated in degree 1 as a K-algebra. For n ∈ N,
(10) h0(C,L⊗n) =
0 n < 0,
1 n = 0,
3n n > 0.
(11) h1(C,L⊗n) =
−3n n < 0,
1 n = 0,
0 n > 0.
Let X = E3, with the three canonical projections πi : X → E. Define
F1 = π∗1(L⊗2)⊗ π∗2(L⊗2)⊗ π∗3(L⊗2)
F2 = π∗1(L)⊗ π∗2(L)⊗ π∗3(L⊗2).
Rm,n = Γ(X,F⊗m1 ⊗F
m,n≥0
Rm,n.
By (9) and the Künneth formula, R is standard bigraded. By (2), the fact that ωX ∼= OX
and Serre duality,
dimK(H
Q(R)−j) =
h2(X,F⊗m1 ⊗F
2 ) =
h1(X,F⊗m1 ⊗F
for j ∈ Z.
Now by (10), (11) and the Künneth formula, we have that for n > 0,
h1(X,F⊗m1 ⊗F
2 ) =
0 if 2m+ n 6= 0,
2h0(X,L⊗n) if 2m+ n = 0.
Thus the conclusions of Theorem 3.5 hold. �
The following theorem gives an example of tame, but still rather strange local cohomol-
ogy. Let [x] be the greatest integer in a real number x.
Theorem 3.6. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j > 0,
dimK(H
Q(R)−j) = 162
dimK(H
Q(R)−j)
where Q =
n>0Rn.
Proof. We use the method of Example 1.6 [Cu]. Let E be an elliptic curve over an
algebraically closed field K, and let p ∈ E be a point. Let X = E × E with projections
πi : X → E. Let C1 = π∗1(p), C2 = π∗2(p) and
∆ = {(q, q) | q ∈ E}
be the diagonal of X. We compute (as in [Cu]) that
(12) (C21 ) = (C
2 ) = (∆
2) = 0
(13) (∆ · C1) = (∆ · C2) = (C1 · C2) = 1.
If N is an ample line bundle on X, then
(14) H i(X,N ) = 0 for i > 0
by the vanishing theorem of Section 16 [M2].
Suppose that L is a very ample line bundle on X, and M is a numerically effective (nef)
line bundle. Then M is 3 regular for L, so that
Γ(X,M⊗L⊗n)⊗ Γ(X,L) → Γ(X,M⊗L⊗(n+1))
is a surjection if n ≥ 3. C1 + 2C2 is an ample divisor by the Moishezon Nakai criterion
(Theorem V.1.10 [Ha]), so that 3(C1+2C2) is very ample by Lefschetz’s theorem (Theorem,
Section 17 [M2]). Let
F1 = OX(9(C1 + 2C2)).
Then OX is 3 regular for OX(3(C1 + 2C2)), so we have surjections
Γ(X,F⊗n1 )⊗ Γ(X,F1) → Γ(X,F
⊗(n+1)
for all n ≥ 1.
∆+C2 is ample by the Moishezon Nakai criterion. Let D = 3(∆+C2). D is very ample
by Lefschetz’s theorem, and thus OX(D)⊗F1 is very ample. Let
F2 = OX(3D)⊗F⊗31 .
OX is 3 regular for OX(D)⊗F1, so we have surjections
Γ(X,F2⊗n)⊗ Γ(X,F2) → Γ(X,F⊗(n+1)2 )
for all n ≥ 1.
for n > 0 and m ≥ 0, we have
F⊗m1 ⊗F
∼= OX(3nD)⊗F⊗(m+3n)1 .
Since D is nef, it is 3 regular for F1, and we have a surjection for all m ≥ 0, n > 0,
Γ(X,F⊗m1 ⊗F
2 )⊗ Γ(X,F1) → Γ(X,F
⊗(m+1)
Rm,n = Γ(X,F⊗m1 ⊗F
We have shown that ⊕m,n≥0Rm,n is a standard bigraded K-algebra. Thus (2) holds.
For m,n ∈ Z, let G = F⊗m1 ⊗ F
2 . As in Example 1.6 [Cu], and by (14) and Serre
Duality (ωX ∼= OX since X is an Abelian variety), we deduce that
1. (G2) > 0 and (G · F1) > 0 imply G is ample and h1(X,G) = h2(X,G) = 0.
2. (G2) < 0 implies h0(X,G) = h2(X,G) = 0.
3. (G2) > 0 and (G · F1) < 0 imply G−1 is ample and h0(X,G) = h1(X,G) = 0.
Let τ2 = −4−
and τ1 = −4 +
Using (12) and (13), we compute
(F21 ) = 2 · 162, (F2)2 = 31 · 162, (F1 · F2) = 8 · 162.
We have
(G2) = 324(m2 + 8mn+ 31
= 324(m− τ1n)(m− τ2n).
(G · F1) = 324(m + 4n).
Since τ2 < −4 < τ1 < 0, for n < 0 and m ∈ Z, we have
1. m > τ2n if and only if G2 > 0 and G · F1 > 0
2. τ1n < m < τ2n if and only if (G2) < 0
3. m < τ1n if and only if (G2) > 0 and (G · F1) < 0.
By the Riemann Roch Theorem for an Abelian surface (Section 16 [M2]),
χ(G) = 1
(G2).
Thus for m ∈ Z and n < 0,
h1(X,G) =
(G2) = −162(m2 + 8mn+ 31
n2) if τ1n < m < τ2n,
0 otherwise.
For n ∈ Z, let σ(n) = dimK(H2Q(Rn)). By (2),
σ(n) =
h1(X,F⊗m1 ⊗F
For n < 0, we have
σ(n) = −162(
τ1n<m<τ2n
(m2 + 8mn+
n2)).
Setting r = m+ 4n, we have
σ(n) = −162(
n<r<−
(r2 − 1
= −324
r=1 (r
2 − 1
n2) + 81n2
= −324
+ 81n2
= 162
We thus have the conclusions of the theorem. �
4. Strange examples of Rees Algebras
Let notation and assumptions be as in Section 2. Since F1 is ample, there exists l > 0
such that Γ(X,F⊗l1 ⊗ F
2 ) 6= 0. Thus we have an embedding F2 ⊗ F
1 ⊂ OX . Let
A = F2 ⊗F−l1 , which we have embedded as an ideal sheaf of X. For j ≥ 0 and i ≥ jl, let
Tij = Γ(X,F⊗i1 ⊗A⊗j) = Ri−jl,j.
For j ≥ 0, let Tj =
i≥jl Tij and T =
j≥0 Tj . Let B =
j>0 Tj. R
∼= T as graded rings
over R0 ∼= T0, although they have different bigraded structures. Thus for all i, j we have
(15) H iB(T )j
∼= H iQ(R)j .
T1 is a homogeneous ideal of T0, and T is the Rees algebra of T1. Thus all of the exam-
ples of Section 3 can be interpreted as Rees algebras over normal rings T0 with isolated
singularities.
We thus obtain the following theorems from Theorems 3.2 - 3.6. Theorems 4.1, 4.2 and
4.3 give examples of Rees algebras with non tame local cohomology.
Theorem 4.1. Suppose that K is an algebraically closed field. Then there exists a normal,
standard graded K algebra T0 with dim(T0) = 3 and a graded ideal A ⊂ T0 such that the
Rees algebra T = T0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) =
2 if j is even,
0 if j is odd.
where B is the graded ideal AtT of T .
Theorem 4.2. Suppose that p is a prime number such that p ≡ 2(mod)3 and p ≥ 11.
Then there exists a normal standard graded K-algebra T0 over a field K of characteristic
p with dim(T0) = 4, and a graded ideal A ⊂ T0 such that the Rees algebra T = T0[At] of
A is normal, and for j > 0,
dimK(H
Q(T )−j) =
1 if j ≡ 0(mod)(p + 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where B is the graded ideal AtT of T . We have pt ≡ −1(mod)(p + 1) for all odd t ≥ 0.
Theorem 4.3. Suppose that K is an algebraically closed field. Then there exists a normal,
standard graded K-algebra T0 with dim(T0) = 4 and a graded ideal A ⊂ T0 such that the
Rees algebra T = R0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) =
6j if j is even,
0 if j is odd,
where B is the graded ideal AtT of T .
Theorem 4.4. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K algebra T0 with dim(T0) = 3, and a graded ideal A ⊂ T0 such that the
Rees algebra T = T0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) = 162
dimK(H
B(T )−j)
where B is the graded ideal AtT of T .
By localizing at the graded maximal ideal of T0, we obtain examples of Rees algebras
of local rings with strange local cohomology.
In all of these examples, T0 is generalized Cohen Macaulay, but is not Cohen Macaulay.
This follows since in all of these examples,
H2P0(R0)0
∼= H1(X,OX ) 6= 0.
5. Local duality in the examples
The example of [CH], giving failure of tameness of local cohomology, is stated in The-
orem 3.1 of this paper. The proof of [CH] uses the bigraded local duality theorem of
[HR], which now follows from the much more general bigraded local duality theorem, The-
orem 1.5 and Corollary 1.7 of this paper, to conclude that in our situation, where R is
generalized Cohen Macaulay,
(16) (Hd−iQ (ωR)−j)
∗ ∼= H iP (R)j
for j ≫ 0.
In [CH], the formula
H iP (R)j
∼= H iP0(Rj)
i−1(X, R̃j(m))
i−1(X,F⊗m1 ⊗F
for i ≥ 2 and j ≥ 0 is then used with formula (1) of [CH] ((3) of this paper) to prove
Theorem 3.1.
In Section 2 we derive (2) from which we directly compute the local cohomology in the
examples of this paper. We make essential use of Serre duality on X in computing the
examples.
In this section, we show how (16) can be obtained directly from the geometry of X and
V , and how this formula can be directly interpreted as Serre duality on X.
Let notation be as in Section 2, so that K is an algebraically closed field, F1 and F2
are very ample line bundles on the nonsingular variety X.
Let ωR be the dualizing module of R, and let ωX be the canonical bundle of X (which
is a dualizing sheaf on X). For a K module W , let W ′ = HomK(W,K).
Lemma 5.1. We have that
(ωR)ij =
Γ(X,F⊗i1 ⊗F
2 ⊗ ωX) if i ≥ 1 and j ≥ 1
0 otherwise.
Set (ωR)
j∈Z(ωR)i,j , a graded R
0 module. The sheafification of (ωR)
i on X is
(18) (̃ωR)i =
F⊗i1 ⊗ ωX if i ≥ 1
0 if i ≤ 0.
Set (ωR)j =
i∈Z(ωR)i,j , a graded R0 module. The sheafification of (ωR)j on X is
(19) (̃ωR)j =
F⊗j2 ⊗ ωX if j ≥ 1
0 if j ≤ 0.
Proof. Give R the grading where the elements of degree e in R are [R]e =
i+j=eRij.
We have realized R (with this grading) as the coordinate ring of the projective embed-
ding of V = P(F1 ⊕F2) by the very ample divisor OV (1), with projection π : V → X.
Let ωV be the canonical line bundle on V . We first calculate ωV . Let f be a fiber of
the map π : V → X. By adjunction, we have that (f · ωV ) = −2. Since
Pic(V ) ∼= ZOV (1) ⊕ π∗(Pic(X)),
we see that there exists a line bundle G on X such that
ωV ∼= OV (−2)⊗ π∗(G).
The natural split exact sequence
(20) 0 → F2 → F1 ⊕F2 → F1 → 0
determines a section X0 of X, such that π∗ of the exact sequence
0 → OV (1)⊗OV (−X0) → OV (1) → OV (1) ⊗OX0 → 0
is (20) (Proposition II.7.12 [Ha]). Thus
OV (1) ⊗OV (−X0) ∼= π∗(F2)
OV (1)⊗OX0 ∼= F1.
By adjunction, we have that the canonical line bundle of X0 is
∼= ωV ⊗OV (X0)⊗OX0 .
Putting the above together, we see that
G ∼= F1 ⊗F2 ⊗ ωX .
ωV ∼= OV (−2)⊗ π∗(F1 ⊗F2 ⊗ ωX).
We realize R as a bigraded quotient of a bigraded polynomial ring
S = K[x1, . . . , xm, y1, . . . , yn],
with deg(xi) = (1, 0) for all i and deg(yj) = (0, 1) for all j. Viewing S as a graded K-
algebra with the grading determined by d(xi) = d(yj) = 1 for all i, j, we have a projective
embedding V ⊂ P = Proj(S). Since V is nonsingular, we see from Section III.7 of [Ha]
ωV ∼= ExtrP(OV ,Op(−e))
where e = m+ n is the dimension of S, and r = e− dim(R). ωR is defined as
ωR = *Ext
S(R,S(−e)) ∼=
ExtrP(OV ,OP(m− e)).
For m ≫ 0,
Γ(P, ExtrP(OV ,Op(m− e))) ∼= ExtrP(OV ,OP(m− e))
(by Proposition III.6.9 [Ha]). Thus ωR and
Γ∗(ωV ) =
Γ(V, ωV (m))
are isomorphic in high degree. Since both modules have depth≥ 2 at the maximal bigraded
ideal of R, we see that
ωR ∼= Γ∗(ωV ).
m∈Z Γ(V, ωV (m))
m∈Z Γ(V,OV (m− 2)⊗ π∗(F1 ⊗F2 ⊗ ωX)).
Since a fiber f of π satisfies (f ·OV (m− 2)⊗π∗(F1⊗F2)) < 0 if m < 2, we see that (with
this grading) [ωR]m = 0 if m < 2 and For m ≥ 2, we have
[ωR]m = Γ(X,S
m−2(F1 ⊕F2)⊗F1 ⊗F2)
i+j=m−2 Γ(X,F
⊗(i+1)
⊗(j+1)
2 ⊗ ωX).
The conclusions of the lemma now follow. �
Suppose that 2 ≤ i ≤ d − 2. Since F1 and F2 are ample, and d − (i + 1) > 0, there
exists a natural number n0 such that
(21) Hd−(i+1)(X,F⊗m1 ⊗Fn2 ⊗ ωX) = 0
for n ≥ n0 and all m ≥ 0.
By (18), we have graded isomorphisms
(22) H iQ(ωR)n
H i−1(X,F⊗m1 ⊗F
2 ⊗ ωX)
for n ∈ Z.
By Serre duality,
(23) H iQ(ωR)n
(Hd−i−1(X,F−⊗m1 ⊗F
By (21), there exists n0 such that
(24) H iQ(ωR)−n
(Hd−i−1(X,F−⊗m1 ⊗F
for n ≥ n0.
Now apply the functor L∗ = HomK(L,K) on graded R0-modules, with the grading
(L∗)i = HomK(L−i,K)
to (24), and compare with (17), to obtain
(25) Hd−iP (R)n
∼= (H iQ(ωR)−n)∗
for n ≥ n0, from which (16) immediately follows.
We can now verify that Theorem 3.1 is in fact true for all j > 0, using (22) and (3).
We finally comment that an alternate proof of Theorem 3.2 for j ≫ 0 is obtained
from Theorem 3.1, Formulas (2) and (22), the fact that X is an Abelian variety so that
ωX ∼= OX , and the observation that
h1(X,F−⊗n2 ) = h
1(X,F⊗n2 ) = 0
for n > 0.
References
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6(1980), 61–69.
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schemes, J. Pure and Appl. Alg. 172 (2002), 165–182.
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Joint International Meeting of the American Mathematical Society and the Indian Mathematical
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vanced Mathematics 39, Cambridge University Press, 1998.
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149 -156.
[CH] Cutkosky, S.D. and Herzog, J., Failure of tameness of Local Cohomology, to appear in Journal
of Pure and Applied Algebra.
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New York, Heidelberg, Berlin (1995).
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Mathematics 238, Springer, 1971.
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E-mail address: [email protected]
Dale Cutkosky, Mathematics Department, 202 Mathematical Sciences Bldg, University
of Missouri, Columbia, MO 65211 USA
E-mail address: [email protected]
Jürgen Herzog, Fachbereich Mathematik und Informatik, Universität Duisburg-Essen,
Campus Essen, 45117 Essen, Germany
E-mail address: [email protected]
Hema Srinivasan, Mathematics Department, 202 Mathematical Sciences Bldg, University
of Missouri, Columbia, MO 65211 USA
E-mail address: [email protected] <[email protected]>
|
0704.0103 | Generalized regularly discontinuous solutions of the Einstein equations | Generalized regularly discontinuous solutions
of the Einstein equations
Gianluca Gemelli
L.S. B. Pascal, V. P. Nenni 48, Pomezia (Roma), Italy.
e-mail: [email protected]
Abstract
The physical consistency of the match of piecewise-C0 metrics is
discussed. The mathematical theory of gravitational discontinuity hy-
persurfaces is generalized to cover the match of regularly discontinuous
metrics. The mean-value differential geometry framework on a hy-
persurface is introduced, and corresponding compatibility conditions
are deduced. Examples of generalized boundary layers, gravitational
shock waves and thin shells are studied.
Submitted to Int. J. Theor. Phys.
1 Introduction
Is it possible to define weak solutions of the Einstein equations of class
piecewise-C0, i.e. to generalize the compatibility conditions which replace
the field equations on a singular hypersurface to the case when the metric is
regularly discontinuous?
To reach this goal would probably mean to define the most general class
of regularly discontinuous weak solutions of the Einstein equations. It seems
that this problem was never studied before in the literature. But, before we
proceed, we need to discuss whether we are talking of something mathemat-
ically and physically consistent or not.
A fundamental concept of Riemannian geometry is that at any point of
a submanifold there are coordinate choices for which the metric reduces to
http://arxiv.org/abs/0704.0103v1
the Minkowski flat metric. Clearly, if this choice is made on both sides of
the discontinuity surface, any ”jump” in the metric disappears. Thus, the
metric discontinuity appears as a coordinate dependent concept, which is
neither geometrically (nor physically) acceptable in the context of General
Relativity.
But we also have to consider that regularity of the global coordinates plays
an important role in our approach, which is that of [1] and of the literature
cited therein. In particular, since here the spacetime is only C0, we are led to
considering (C0, piecewise C1) coordinate transformations. If the metric is
discontinuous in some globally C0 chart, it is in general impossible to obtain
the vanishing of the metric jump on both sides of a hypersurface with a C0
coordinate transformation (see section 2). Moeover in the following we are
led in a natural way to considering C1 coordinate transformations; the metric
discontinuity is a tensor with respect to such coordinate changes!
In other words the jump of the metric has a precise mathematical mean-
ing, if we consider it in connection with global regular coordinates.
In a well consolidated procedure, the assumption of continuity for the
metric across a gravitational interface is usually taken for granted; however
it follows from the limiting process of the thin sandwich modelization, in
consequence of the hypothesis that the external derivatives of the metric
are bounded [2]. Yet in this paper we are going to see that, even removing
the assumption of continuity, it is still possible to define a generalized inner
geometry of the discontinuity hypersurface; one thus can consistently find
a corresponding generalized set of compatibility conditions, which obviously
reduces to the usual ones when the continuity hypothesis is restored.
Yet, which are the physical motivations to move to such generalization?
Actually gravitational shock waves and thin shells are usually defined by the
presence of singular curvature with a “delta” component concentrated on a
hypersurface, situation which is well cast within the classic C0 piecewise-C1
match of metrics [1, 3].
We were originally led to consider solutions of class piecewise-C0, as pos-
sible generalizations of shock waves and thin shells, by the sake of mathe-
matical completeness, with the idea that phisical interpretation would follow.
We actually found a reacher framework than the usual one, with some in-
teresting new features (and even some rather undesiderable ones), which we
display in this paper.
There are two main theories in the literature for solutions of class C0
piecewise-C1, i.e. that in terms of the second fundamental form (heuristic
theory, see e.g. [4, 5]) and that in terms of the curvature tensor-distribution
(axiomatic theory, see e.g. [6, 1]); such are equivalent through appropiate
extensions (for a self-contained overview see e.g. [1]).
The axiomatic theory appears to be inappropriate to the study of general-
ized solutions, since the theory of distributions is basically linear. Even if we
could in principle replace the discontinuous metric with its associated distri-
bution gD, then it would be impossible to define, for example, replacements
for the Christoffel symbols, since this would involve product of distributions,
which, as it is generally believed, is impossible to define. In fact it was proved
by Schwarz [7] that, under reasonable hypothesis, there can be no definition
of commutative and associative operation on distributions which reduce to
ordinary multiplication on integrable distributions (say on regular functions);
thus in a word it is impossible to define product of distributions.
Or is it? Colombeau [8, 9, 10] developed a theory which apparently con-
tradicts Schwarz’s result. He introduced a very broad space of generalized
functions, which extends the usual space of distributions, a subspace of which
corresponds, in a certain sense (the correspondance is not 1 to 1), to usual
distributions. Colombeau’s formalism permits multiplication of generalized
functions; but the contradiction with the impossibility theorem is only ap-
parent, in fact Schwarz’s hypothesis are violated, since the operation does
not coincide with ordinary multiplication on regular functions nor with mul-
tiplication of a regular function times a distribution (although it does at least
for C∞ functions).
Such theory, however, does not fit in a natural way in general relativity,
since it is impossible to define covariantly invariant geometrical objects; in
fact Colombeau’s space is not invariant for smooth coordinate transforma-
tions, unless they are linear. Such difficulty, however, seems to have been
overcome in subsequent adjustments of the theory, with the introduction of
a richer mathematical framework [11, 12], so that the generalized functions
current apparatus can be used in general relativity, and indeed it has been
applied at least to the calculation of singular curvatures of the spacetimes
of Kerr [13], Reissner-Nordstrom [14], and so-called cosmic-string spacetime
[15]. In such literature Colombeau’s theory is adapted to the handling of
curvature when the metric has a singularity in the sense of functions, i.e. the
ordinary curvature would explode, at a singular event-point or at a singular
worldline. There seems to be no particular reason to forbid Colombeau’s
method also for defining the match of piecewise-C0 regularly discontinuous
metrics at a singular hypersurface; however, as far as the author is aware, no
attempt has been made yet to use it in this framework.
The direct method we will introduce in the following sections, however,
is so conceptually simple that we prefer not to experiment with Colombeau’s
generalized functions, which would instead mean introducing a far more com-
plicated and unfamiliar mathematical apparatus.
In this paper in fact we propose a new, generalized theory for regularly
discontinuous solutions, covering also the match of piecewise-C0 metrics. Our
theory is heuristic, as it is constructed in a way similar to the heuristic the-
ory of C0, piecewise-C1 solutions originated from the studies of Israel, but
we completely avoid the traditional or projectional Gauss-Codazzi framework
(which either does not include the lightlike case [4, 5], or needs a special adap-
tation for it [16, 1]) and introduce what we called “mean-value differential
geometry” framework, instead (see section 3). This is conceptually very sim-
ple, and permits to construct in a natural way a generalized theory, where
the main role (which used to be that of the jump of the secund fundamental
form) is here played by the jump of the Christoffel symbols. The theory is an
extension of the theory of gravitational discontinuity hypersurfaces we have
studied in [1], to which it reduces when the metric is C0. Even if we should
restrict to C0 solution, by adding the traditional assumption of continuity for
the metric, our theory would undoubtedly have at least the good qualities of
not needing the timelike and the lightlike case to be distinguished (different
from usual heuristic theory), and of just requiring C0 continuity for the co-
ordinates (different from the axiomatic theory). Moreover, it is completely
cast in the framework of general coordinates of the ambient (glued) space-
time, with no use of parametric equations of the hypersurface, nor of inner
coordinates and holonomic 3-basis, which could be considered a good quality
in some applications as well.
Piecewise-C0 weak solutions of the Einstein equations, as far as the au-
thor is aware, have never been considered previously in the literature. They
generalize the corresponding C0 solutions, as examples in this paper show;
however there is more. Apparently in fact the theory allows the propagation
of free gravitational discontinuity at lower speed than the speed of light (sec-
tion 8); or rather, we still have no general proof that the absence of stress
energy concentrated on Σ should, in the timelike case, necessarily imply the
degeneracy of a generalized solution to a boundary layer, although it does
at least for a wide class of spherical matchs (see section 6). Moreover, non-
simmetric stress-energy is allowed on the hypersurface (section 9), like e.g. in
Einstein-Cartan dynamics. This possible link to classical unification theories
is surprising, since in our framework we have nothing similar to Einstein-
Cartan torsion. We therefore see a lot of space for future investigation.
2 Discontinuous metrics
Let us suppose V4 an oriented differentiable manifold of dimension 4, of class
(C0, piecewise C2), provided with a strictly hyperbolic metric of signature
–+++ and class piecewise-C0. Let Ω ⊂ V4 be an open connected subset with
compact closure. Let units be chosen in order to have the speed of light in
empty space c ≡ 1. Greek indices run from 0 to 3.
Let Σ ⊂ Ω be a regular hypersuperface of equation f(x) = 0; let Ω+ and
Ω− denote the subdomains distinguished by the sign of f . We suppose the
metric and its first and second partial derivatives to be regularly discontin-
uous on Σ in all charts of class C0(Ω). Let f ∈ C0(Ω) ∩ C2(Ω\Σ), and let
second and third derivatives of f be regularly discontinuous on Σ. Finally,
let ℓα ≡ ∂αf denote the gradient of f .
Let the metric be a solution of the ordinary Einstein equations on each of
the two domains Ω+ and Ω−. In this situation Σ is the interface between two
general relativistic spacetimes and it is called a (generalized) gravitational
discontinuity hypersurface.
In the following we will develope a theory to justify the introduction of
suitable generalized compatibility conditions to replace the Einstein equa-
tions on Σ (section 5); if such conditions are satisfied the match across the
generalized gravitational hypersurface Σ will be called a generalized regularly
discontinuous solution of the Einstein equations.
Now let us briefly recall some basics notions on regularly discontinuous
fields, which we will use as tools. In any case, for notation and terminology
we refer to [1].
A field ϕ is said to be regularly discontinuous on Σ if its restrictions to
the two subdomains Ω+ and Ω− both have a finite limit for f −→ 0; we
denote such limits by ϕ+ and ϕ−, respectively.
In this case the jump [ϕ] across Σ and its arithmetic mean value ϕ are
well defined on the hypersurface:
[ϕ] = ϕ+ − ϕ−
ϕ = (1/2)(ϕ+ + ϕ−)
If ϕ is continuous across Σ, we obviously have: [ϕ] = 0, ϕ = ϕ. We also have
the converse formulae:
ϕ+ = ϕ+ (1/2)[ϕ]
ϕ− = ϕ− (1/2)[ϕ].
As for the product of two functions ϕ and ψ, we have:
[ϕψ] = [ϕ]ψ + ϕ[ψ]
ϕψ = ϕψ + (1/4)[ϕ][ψ]
If a field ϕ is regularly discontinuous on Σ, its jump [ϕ] is sometimes called
its discontinuity of order 0.
The jump of a regularly discontinuous function has support on Σ, but in
general, the partial derivative of the jump is well defined as the jump of the
derivative of the function (see [17, 18]). In particular, the derivative of the
jump of a continuous field is not null, unless the field is also C1.
Similarly, we define the partial derivative of the mean value as the mean
value of the partial derivative. We can also use regular prolongations to
extend, in a sense, the definition of ϕ and [ϕ] to the whole domain Ω. Thus
they can be regarded as regular and derivable fields in Ω, but their values
(and those of their derivatives) are well defined only on Σ, while in Ω\Σ
they depend on the choice of the prolongation. For details on the method of
regular prolongations see e.g. [17, 18].
We moreover define the covariant derivative of a field with support on Σ
by means of the mean value Γβρ
σ of the Christoffel symbols. For the jump of a
regularly discontinuous vector, for example, with this definition one has that
the jump of the covariant derivative is different than the covariant derivative
of the jump. Thus, by definition, we have:
β] = ∂α[V
β] + Γασ
β[V σ] (4)
and in consequence of (3):
β] = [∇αV
β]− [Γασ
, (5)
and similarly for the jump of any regularly discontinuous tensor.
Since the spacetime is only C0, we are led to considering (C0, piecewise
C1) coordinate transformations, with regularly discontinuous first deriva-
tives; the metric discontinuity [gαβ] is not a tensor with respect to such
changes of coordinates. In fact we have:
[gαβ] = [gα′β′]
+ qαβ′
+ qα′β
where:
qα′β =
[gα′β′ ]
+ ḡα′β′
We therefore can simulate all (C0, piecewise C1) coordinate changes by com-
bining C1 changes with metric gauge changes:
[gαβ ]←→ [gαβ] + qαβ′
+ qα′β
which generalize usual gravitational gauge changes of the theory of (C0, piece-
wise C1) solutions [1].
Is it always possible to make [gαβ] vanish with an appropriate C
0 trans-
formation? Clearly the answer is negative. In fact it suffices to consider the
case when [gαβ] and ḡαβ are both definite positive in a given chart to see that
the equation obtained from (6) by replacing the first hand side with 0 has no
solution for [∂xα
/∂xα] and ∂xα
/∂xα. Thus the set of effective generalized
gravitational discontinuity hypersurfaces is non empty.
Moreover it will occur in many applications to have ℓα ∈ C
0. Therefore
it will be often desiderable to work in the framework of (C1, piecewise C2)
coordinate transformations, which preserve such condition. The metric dis-
continuity is a tensor with respect to such changes of coordinates, but the
jump of the Christoffel symbols, which appear to play a main role in the
following, is not; we have in fact:
σ] = [Γα′β′
∂xα∂xβ
If the coordinates are C0 and so is the form ℓα we can write:
∂xα∂xβ
= ℓαℓβ∂
where ∂2 denotes the weak discontinuity of order 2 (see e.g. [17, 18]). Thus on
Σ we can generate all (C1, piecewise C2) transformations for [Γ] combining
C2 transformations (with respect to which Γ is a tensor) and Christoffel
gauges transformations, i.e. of the kind:
σ]↔ [Γαβ
σ] + ℓαℓβQ
σ (11)
with some analogy with the case of C0 metrics (where the main role is played
by the first order metric discontinuity ∂g, see [1] section 3).
In any case neither the mean value of the metric g or its jump [g] now
have null covariant derivatives. Consider in fact the identity ∇αgβρ = 0 in
the domain Ω+; from the limit f −→ 0+, on Σ we have:
βρ − (Γαβ
ν)+g+νρ − (Γαρ
ν)+g+βν = 0 (12)
Here, with obvious meaning of the symbols, we denote: g+βρ = (gβρ)
+, gβρ =
gβρ, etc. Consequently on Σ, from (2)1 we have:
∂αgβρ + (1/2)∂α[gβρ]− Γαβ
νgνρ − Γαρ
νgνβ+
−(1/2)([Γαβ
ν ]gνρ + Γαβ
ν [gρν ] + [Γαρ
ν ]gνβ + Γαρ
ν [gβν ])+
−(1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) = 0
Similarly, from the limit f −→ 0− and from (2)2 we also have on Σ:
∂αgβρ − (1/2)∂α[gβρ]− Γαβ
νgνρ − Γαρ
νgνβ+
+(1/2)([Γαβ
ν ]gνρ + Γαβ
ν [gρν ] + [Γαρ
ν ]gνβ + Γαρ
ν [gβν ])+
−(1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) = 0
From the sum of expressions (13) and (14) we thus have:
∇αgβρ = (1/4)([Γαβ
ν ][gνρ] + [Γαρ
ν ][gβν ]) (15)
and, from difference:
∂α[gβρ] = [Γαβρ] + [Γαρβ ] (16)
From (16), (3), and from the definition of covariant derivative over Σ, we
then have:
∇α[gβρ] = [Γαβ
ν ]gνρ + [Γαρ
ν ]gβν (17)
As for the jump and the mean value of the Christoffel symbols we have, from
ν = (1/2){gνσ(∂αgβσ + ∂βgσα − ∂σgαβ)+
+(1/4)[gνσ](∂α[gβσ] + ∂β[gσα]− ∂σ[gαβ ])}
ν ] = (1/2){gνσ(∂α[gβσ] + ∂β[gσα]− ∂σ[gαβ ])+
+[gνσ](∂αgβσ + ∂βgσα − ∂σgαβ)
or, from (15) and (17):
ν ]gνρ = (1/2)(∇α[gβρ] +∇β[gρα]−∇ρ[gαβ])
ν ][gνρ] = 2(∇αgβρ +∇βgρα −∇ρgαβ)
3 Mean-value geometry on a hypersurface
Let us consider a 4-vector V α, regularly discontinuous on Σ, the jump and the
mean value of which will work as a prototype of vectors with Σ as support.
We have, by definition:
[∇β∇αV
σ] = ∇β[∇αV
σ]− [Γβα
ν ]∇νV σ + [∇βν
σ]∇αV ν (21)
where [∇αV
σ] = ∇α[V
σ]+ [Γαν
and where, again by definition, we have:
∇νV σ =
{∂ν(V
+)σ + (Γ+)νλ
σ(V +)λ + ∂ν(V
−)σ + (Γ−)νλ
σ(V −)λ} (22)
Thus, from (2) we have:
∇νV σ = ∇νV σ + (1/4)[Γνλ
σ][V λ], (23)
which, incidentally, is the same result we could get from the formal applica-
tion of (3), wich actually can be applied to covariant derivatives, provided
one interpretes ∇ = ∇. We therefore have:
[∇α∇βV
σ] = ∇α∇β[V
σ] +∇β[Γαν
+ [Γαν
σ]∇βV
−[Γβα
ν ]∇νV
− (1/4)[Γβαν][Γνλ
σ][V λ]+
+[Γβν
σ]∇αV
+ (1/4)[Γβν
σ][Γαλ
ν ][V λ]
and thus, by antisymmetrization:
[∇[β∇α]V
σ] = ∇[β∇α][V
σ] +∇[β[Γα]νσ ]V
[Γν[β
σ][Γα]λ
ν ][V λ] (25)
Now, from the Ricci identity we have: [∇[β∇α]V
σ] = [Rαβρ
σV ρ] and then, by
[∇[β∇α]V
σ] = [Rαβρ
+Rαβρ
σ[V ρ], (26)
and thus from a well known identity which follows from (3) as a consequence
our definition (5) for the covariant derivative on Σ, i.e. (see [1]):
[Rαβρ
σ] = ∇β[Γαρ
σ]−∇α[Γβρ
σ] (27)
we have that the commutator of the covariant derivatives of the jump of a
generic regularly discontinuous vector obeys the following Ricci-like formula:
∇[βα][V
(1/2)Rαβρ
σ − (1/4)[Γν[β
σ][Γα]ρ
[V ρ]. (28)
Not surprisingly, working in a similar way starting from ∇β∇αV σ and anti-
symmetrizing, we find again:
∇[βα]V
(1/2)Rαβρ
σ − (1/4)[Γν[β
σ][Γα]ρ
; (29)
in fact any given field with support on Σ can be considered, by the help of
suitable prolongations, as the jump (or as the mean value of) some regularly
discontinuous field. Thus, for any vector V with support on Σ, we can
introduce the following mean-value geometry Ricci-like formula on Σ:
(∇[β∇α])V
σ = (RΣ)αβρ
σV ρ; (30)
where we have introduced the mean-value geometry curvature (RΣ), defined
by the following mean-value geometry first Gauss-Codazzi identity:
(RΣ)αβρ
σ = Rαβρ
σ − (1/4)([Γβν
σ][Γαρ
ν ]− [Γαν
σ][Γβρ
ν ]) (31)
Notice that, for the sake of simplicity, we have introduced a slight abuse
of notation, since in [1] and [16] the same symbol RΣ instead denotes the
inner curvature defined with the help of projections. Actually anything goes
like in [1] section 4 with the Gauss-Codazzi framework, with the difference
that here we don’t have to make projections, which would involve product
times a discontinuous tangent metric. Moreover here we don’t even have
to distinguish between the cases of Σ timelike or lightlike. In other words
our mean-value differential geometry on a hypersurface is a very simple, in
conceptual terms, analogue of the Gauss-Codazzi apparatus.
Thus, with the heuristic theory of [1] section 6 (see also [4] for the timelike
case) in mind as a prototype, we expect the jump of the Christoffel symbols to
play the main role, in place of the secund fundamental form, in the definition
of compatibility conditions for very weak solutions of the Einstein equations.
Indeed, this happens, as it will be shown in the following.
4 Complex mean-value formalism
The metric being dicontinuous on Σ, we are missing the fundamental tool
to rise and lower indices, and to construct curvature in the traditional way.
This is the reason why sometimes one is tempted to introduce some hybrid
metric object on Σ to replace the metric, even in the (C0, piecewise C1) case
(see e.g. [5]). It is reassuring to find out that the framework of the preceeding
section can be confirmed by such a kind of approach.
It would be desiderable to simply replace g with g on Σ, but it is easy to
check that g has not the necessary algebraic requisites; in particular we have
αρ 6= δβ
ρ. Consider instead:
g̃αβ = gαβ + i(1/2)[gαβ], g̃
αβ = gαβ − i(1/2)[gαβ] (32)
where i is the imaginary unit (i.e. we have i2 = −1). It is easy to check,
with the help of (3), that we have:
g̃αβ g̃
αρ = δα
ρ + i[gαβ ]g
αρ (33)
i.e., in particular: ℜ(g̃αβ g̃
αρ) = δβ
ρ. For the sake of brevity in the following
we will denote A ≈ B the relation ℜ(A) = ℜ(B). Thus the pair g̃αβ and
g̃αβ is a good candidate replacement for the metric on Σ, for the purposes of
rising and lowering indices. Now, similar to (32) let us introduce:
Γ̃αβν = Γαβν + i(1/2)[Γαβν ], Γ̃αβ
σ = Γαβ
σ − i(1/2)[Γαβ
σ] (34)
so that we have: Γ̃αβ
σ ≈ Γ̃αβν g̃
σν and conversely: Γ̃αβν ≈ Γ̃αβ
σg̃νσ. Let us
now introduce the differential operator ∇̃ on Σ, which makes use of Γ̃ in
place of Γ. As we could expect we have:
∇̃ρg̃αβ ≈ 0, ∇̃ρg̃
αβ ≈ 0 (35)
which is the replacement on Σ for the covariant conservation of the metric
tensor.
Now let us construct on Σ the complex curvature tensor R̃ in the familiar
way, but with Γ̃ in place of the ordinary Christoffel symbols (which are
undefined on Σ):
R̃αβρ
σ = ∂βΓ̃αρ
σ − ∂αΓ̃βρ
σ + Γ̃βµ
σΓ̃αρ
µ − Γ̃αµ
σΓ̃βρ
µ (36)
We rather unespectedly find out that
R̃αβρ
σ = (RΣ)αβρ
σ + i(1/2)[Rαβρ
σ] (37)
i.e. in particular we have: R̃αβρ
σ ≈ (RΣ)αβρ
σ, where RΣ is given by (31).
This is just another reason for identifying RΣ as the replacement for the
curvature tensor of Σ, which is the first step of our path to the generalized
compatibility conditions.
5 Generalized compatibility conditions
Let us now consider limit f → 0+ of the curvature tensor of the subdomain
Ω+; by (2) we have:
(Rαβρ
σ)+ = Rαβρ
σ + (1/2)[Rαβρ
σ] (38)
and, by (27):
(Rαβρ
σ)+ = Rαβρ
σ +∇[β[Γα]ρ
σ] (39)
We also have, by (31):
(Rαβρ
σ)+ = (RΣ)αβρ
σ +∇[β[Γα]ρ
σ] + [Γν[β
σ][Γα]ρ
ν ] (40)
We see that R and RΣ only differ by terms proportional to [Γ], and not
involving derivatives of it. Thus, in view of neglecting these tems, in the
following we will consider R instead ofRΣ; this simply avoids the introduction
of the symbol “ ∼= ”, with the meaning of equality but for terms not involving
derivatives of [Γ] (which here replaces the second fundamental form K) as in
[1] section 6. Then for the Ricci tensor Rβρ = Rαβρ
α we have:
(Rβρ)
+ = Rβρ + (1/2)∇µ
µ[Γνρ
ν ]− [Γβρ
and for the curvature scalar R = Rα
R+ = R + (1/2)∇µ
µν ]− [Γν
Now, to construct the Einstein tensor G+ we have to remember that, since
the metric is also discontinuous:
(gαβ)
+ = gαβ + (1/2)[gαβ] (43)
so that we have:
(Gβρ)
+ = Gβρ + (1/2)∇µ
µ − (1/8)[gβρ]
µν ]− [Γν
where we have denoted, for the sake of brevity:
µ[Γνρ
ν ]− [Γβρ
µ]− (1/2)gβρ
µν ]− [Γν
Let us fix a coordinate chart and consider a generic (for the moment) regular
prolongation for G, so that its mean value is defined in the whole Ω. Now
consider the Riemann 4-volume integral of G+ over the domain Ω+; from
the Green theorem we obtain (for the general definition of integral on a
hypersurface see [6] p. 6):
Gβρ =
Gβρ + (1/2)
ℓ+µHβρ
µ − (1/8)
ℓ+µ [gβρ]
µν ]− [Γν
The analogous formula for Ω− involves −ℓ− as the outgoing normal vector
and the metric g−αβ = gαβ − (1/2)[gαβ], so we have:
Gβρ =
Gβρ + (1/2)
ℓ−µHβρ
µ + (1/8)
ℓ−µ [gβρ]
µν ]− [Γν
and consequently we have:
Gβρ =
Gβρ +
ℓµHβρ
µ (48)
Thus reasons similar to those of the heuristic theory (see [4] and [1] section
6) lead to the reasonable hypothesis that G remain bounded in the neigh-
bourhood of Σ, for any admissible prolongation, so that from the volume
integral of the Einstein equations, with the presence of an eventual source
term concentrated on Σ:
Gαβ = −χ
Tαβ − χ
T̆αβ (49)
where χ denotes the gravitational constant, we conclude that
ℓµHβρ
µ = −χ
T̆βρ (50)
which is our heuristic reason for considering the following set of general-
ized compatibility conditions to hold on Σ as a replacement for the Einstein
equations:
ℓµHβρ
µ = −χT̆βρ (51)
Here T̆ represents the stress-energy content of the hypersurface.
In the simpler case ℓα ∈ C
0, it is very easy to check that the object ℓµHβρ
is gauge-invariant in the sense of (11), as we could hope.
Turning now to the comparison with the C0 case, we see from eq.s (71)
and (85) of [1] that our generalized conditions (51) are formally identical to
ordinary compatibility conditions [eq. (110) of the same paper], if expressed
in terms of [Γ] (which in the general case is a function of the jump of the
metric [g] as well as of its weak discontinuity ∂g). Therefore it is clear
that generalized compatibility conditions reduce to ordinary ones in case the
metric is continuous, i.e. in case [gαβ] = 0.
In particular, let us suppose g ∈(C0, piecewise C1) and f ∈ C0(Ω); let
us moreover suppose (ℓ · ℓ) > 0, i.e. Σ timelike. By definition of Christoffel
symbols, and from (11) of [1], we have:
σ] = (ℓ · ℓ)−1/2(NβGρ
σ +NρGβ
σ −NσGβρ) + (ℓ · ℓ)
1/2NβNρQ
σ (52)
Q is a vector which can be set to zero with a suitable gauge choice; it plays
no role in (51), as one would expect, in fact we have:
ℓµ[Γβρ
µ] = −Gβρ + (ℓ · ℓ)Nβρ(Q ·N)
ℓβ[Γνρ
ν ] = NβNρGν
ν + (ℓ · ℓ)NβNρ(Q ·N)
ℓµ[Γν
µν ] = Gν
ν + (ℓ · ℓ)(Q ·N)
ℓµ[Γν
νµ] = −Gν
ν + (ℓ · ℓ)(Q ·N)
and, since g = g = h(N) +N ⊗N , we have from (45):
ℓµHβρ
µ = Gβρ − h(N)βρGν
ν (54)
i.e., according to (88) of [1]:
ℓµHβρ
µ = Hβρ (55)
as expected. Now let us instead suppose (ℓ · ℓ) = 0, i.e. Σ lightlike. Let
u ∈ C0(Ω) be a given auxiliary reference frame. From eq.s (21) and (16) of
[1] we have:
σ] = (u ·ℓ)−1(−LβF(u)ρ
σ−LρF(u)β
σ+LσF(u)βρ)+(u ·ℓ)
2LβLρQ̂
σ (56)
and consequently, from (18) and (19) of the same paper:
ℓµ[Γβρ
µ] = LβB(u, n)ρ + LρB(u, n)β − (u · ℓ)
3LβLρ(Q̂ · L)
ℓβ[Γνρ
ν ] = LβLρG(u, n)ν
ν − (u · ℓ)3LβLρ(Q̂ · L)
ℓµ[Γν
µν ] = ℓµ[Γν
νµ] = 0
We therefore have:
ℓµHβρ
µ = G(u, n)ν
νLβLρ − LβB(u, n)ρ − LρB(u, n)β (58)
i.e. again, according to (83) of [1], we have: ℓµHβρ
µ = Hβρ, as expected.
Therefore the set (51) of compatibility conditions, together with ordinary
Einstein Equations to hold on each side of the discontinuity hypersurface,
defines the class of generalized regularly discontinuous solutions of the Ein-
stein equations. And in case [g] = 0, i.e. for continuous metric, from such
conditions we recover the ordinary compatibility conditions for regularly dis-
continuous weak solutions.
However, in the generic case we have some differences, as we are going to
show in the following.
6 A class of spherical boundary layers
Let us consider the match of two piecewise-C0 regularly discontinuous spher-
ical solutions of the vacuum Einstein equations, of the form
ds2 = −a(r, t)dt2 + b(r, t)dr2 + r2dΩ2 (59)
with dΩ2 = dθ2 + sin2 θdϕ2, across a spherical admissible gravitational dis-
continuity hypersurface Σ of equation r = ρ(t), with ρ(t) ∈ C1. Therefore
the form ℓα = δα
r − ρ̇δα
t is continuous (while ℓβ = gβαℓα in general is not).
We suppose globally C0 coordinates, the same form of the metric in both
domains Ω+ and Ω−, and the identification t+ = t−, r+ = r−, θ+ = θ−,
ϕ+ = ϕ− on Σ. Leta, b > 0 and let a, b ∈ piecewise-C0 be regularly dis-
continuous on Σ and with regularly discontinuous first derivatives. Let us
denote by a dot the partial derivative with respect to t, and by a prime that
with respect to r. Let moreover condition a− bρ̇ > 0, i.e. (ℓ · ℓ) > 0, hold on
both sides on Σ.
We have:
[gαβ ] = −[a]δα
t + [b]δα
r (60)
Now let us define the match as a generalized regularly discontinuous solution
by (51), with T̆ = 0, i.e. in the absence of stress-energy concentrated on Σ.
In this case our compatibility conditions reduce to:
ℓβ[Γµρ
µ]− ℓµ[Γβρ
µ] = 0 (61)
which, for a match of metrics of the kind (59), are equivalent to the following
system:
ρ̇[ḃb−1] + [a′b−1] = 0
ρ̇[ḃa−1] + [a′a−1] = 0
ρ̇[a′a−1] + [ȧa−1] = 0
ρ̇[b′b−1] + [ḃb−1] = 0
[b−1] = 0
i.e. we have [b] = 0 and consequently:
ρ̇[ḃ] + [a′] = 0
ρ̇[ḃa−1] + [a′a−1] = 0
ρ̇[a′a−1] + [ȧa−1] = 0
ρ̇[b′] + [ḃ] = 0
and from (3):
ρ̇[ḃ] + [a′] = 0
(ρ̇ḃ+ a′)[a−1] = 0
(ρ̇ a′ + ȧ)[a−1] + (ρ̇[a′] + [ȧ])a−1 = 0
ρ̇[b′] + [ḃ] = 0
Now if we had both ρ̇[ḃ] + [a′] = 0 and ρ̇ḃ + a′ = 0, by (2) we would have
ρ̇ḃ + a′ = 0 on both sides of the hypersurface. We discard for the moment
this singular situation, and from (64)2 we conclude that [a
−1] = 0.
Thus in this case our generalized compatibility conditions imply [a] =
[b] = 0, i.e. they force the match to be C0, piecewise-C1.
In [1] we have already studied some examples of C0, piecewise-C1 matchs
of metrics of the kind (59) at a hypersurface of constant radius r = rb, with
ℓα = δα
r. Namely, we have considered: external Schwarzschild - internal
Schwarzschild; external Schwarzschild - Tolman VI; external Schwarzschild -
Tolman V. Such matchs obviously have ℓα ∈ C
0; moreover condition ρ̇ ḃ+a′ 6=
0 reduce in this case to a′ 6= 0, which is obviously satisfied. In each case we
have verified that condition [a] = [b] = 0 imply ∂a = 0 (where ∂ denotes first
order discontinuity), which then define the match as a boundary layer [1] (it
actually also imply ∂b = 0, as one can verify). Such is a general result, since
for a metric of the kind (59) the completely temporal and radial components
of the Einstein tensor are independent from the second derivatives of the
metric:
Gtt = −a(b
′r + b2 − b)/r2b2
Grr = −(a
′r − ab+ a)/ar2
so that the corresponding vacuum Einstein equations reduce to:
b′r + b2 − b = 0,
a′r − ab+ a = 0.
Now, since in the match of (59) vacuum solutions equations (66) are satisfied
on each side of the interface Σ, their jump is in particular null, and from (3)
we have:
′] + (2b− 1)[b] = 0
′]− a[b]− (b+ 1)[a] = 0
from which it clearly follows that conditions [a] = [b] = 0 imply [a′] = [b′] = 0,
i.e. ∂a = ∂b = 0.
Summarizing, for the match of two piecewise-C0 regularly discontinuous
spherical solutions, in the above hypothesis, generalized compatibility con-
ditions (51) imply [a] = [b] = 0 i.e. they force the match to be C0. On
the other hand conditions [a] = [b] = 0 imply that Σ is a boundary layer.
Therefore for such spherical matchs generalized compatibility conditions (51)
are necessary and sufficient for the match to be a boundary layer.
7 Generalized gravitational shock waves
Let us consider the match of two plane wave metrics of the form
ds2 = −2dξdη + F (ξ)2dx2 +G(ξ)2dy2 (68)
across a hypersurface Σ of equation ξ = 0. Here ξ and η are the two null
coordinates. We suppose continuously matching coordinates and F,G reg-
ularly discontinuous, together with their first and second derivatives. The
gradient vector of Σ is the continuous characteristic (on each side of Σ) vector
ℓα = δα
Generalized compatibility conditions (51) in the case T̆ = 0 (i.e. no stress-
energy concentrated on the hypersurface) reduce to the following single scalar
equation:
[F−1F ′ +G−1G′] = 0 (69)
which characterize the generalized gravitational shock wave. Let us now
study compatibility of (69) with the Einstein Equations. Einstein vacuum
equations also reduce to a single scalar equation:
F−1F ′′ +G−1G′′ = 0 (70)
which we suppose to hold on each side of the hypersurface Σ; thus replacing
F+ and G+ by their expressions in terms of F , [F ], G and [G] according to
(2) gives rise to the following couple scalar conditions:
(2F ′′ + [F ′′])(2G+ [G]) + (2G′′ + [G′′])(2F + [F ]) = 0 (71)
(2F ′′ − [F ′′])(2G− [G]) + (2G′′ − [G′′])(2F − [F ]) = 0 (72)
Equations (71)-(72) are compatible with (69), i.e. the three equations set
can be solved algebraically with respect to F , [G] and to any member of the
pair (F , G), and the solution is not necessarily trivial.
Finally let us notice that, if the additional condition [F ] = [G] = 0 holds,
i.e. if the solution is C0, condition (69) reduces to F−1[F ′]+G−1[G′] = 0 i.e.:
F−1∂F +G−1∂G = 0 (73)
which is the analogous condition for the ordinary shock wave, according to
[1] section 10.5.
8 Slow generalized gravitational waves
Let us start trying to match two vacuum solutions of the kind (68) across
the timelike (on both sides) hypersurface Σ of equation ξ = ζ . Again we
suppose continuously matching coordinates, F,G regularly discontinuous to-
gether with their first and second derivatives, and T̆ = 0. This times gener-
alized compatibility conditions include (69) and the following two additional
scalar conditions:
[FF ′] = 0, [GG′] = 0 (74)
i.e., in terms of F , [F ], G and [G], according to (3):
F [F ′] + [F ]F ′ = 0 (75)
G[G′] + [G]G′ = 0 (76)
It is easy to check that the system (75)-(76) is not compatible with (71)-(72),
in the sense that the whole system does not admit non-trivial solutions for
F , [F ], G and [G].
On the other hand we have proved in section 6 that a wide class of gen-
eralized spherical matchs at a hypersurface of constant radius necessarily
degenerate to a C0 match.
Other examples of degeneracy have not been included in the paper for
the sake of brevity, but at least it seems to be a hard task to construct a non-
trivial generalized match across a timelike (on each side) hypersurface, with
no stress-energy content. Such difficulty is certainly not a proof that this is
an impossible task, but it makes us wonder whether such a solution should
necessarily degenerate to a boundary layer, just like it happens for ordinary
C0 solutions (see e.g. [1]). This would forbid the existence of generalized
solutions which propagate at a speed slower than light. Such would be a
desiderable prohibition under certain respect, since one could expect that
gravitational interactions in vacuo must necessarily propagate at the speed
of light also in a generalized theory.
In general terms, since for generalized solutions the metric is discontinu-
ous, a hypersurface can in principle have different signatures on the different
sides; for this reason we cannot simply distinguish between the timelike and
the lightlike case, as for usual C0 solutions. We should rather distinguish
between three cases: timelike-timelike, timelike-lightlike (or conversely) and
lightlike-lightlike.
In any case it is legitimate to expect that, at least in the timelike-timelike
case, similar to the timelike case of (C0, piecewise C1) solutions, absence of
stress-energy concentrated on Σ should imply the solution to degenerate to
a boundary layer [1].
Unfortunately for generalized solutions we still have no proof that absence
of stress energy concentrated on Σ does necessarily imply the degeneracy of
the solution to a boundary layer.
Therefore, although the examples considered in this paper seem to sug-
gest that such property could hold true also in the generalized case, for the
moment such result is still a conjecture; we thus have to admit that the the-
ory in principle allows propagation of generalized gravitational shock waves
at lower speed than the speed of light. We would call such waves “slow gen-
eralized gravitational shock waves”. It would be reasonable to forbid this
situation as unphysical, but for now this can only be done ad hoc, by means
of a corresponding additional hypothesis.
9 Non-symmetric stress-energy
Notice that ℓµHβρ
µ is not necessarily symmetric; from identity:
ν = (1/2)g−1∂αg (77)
where g denotes the determinant of the contravariant metric, we have:
ℓµH[βρ]
µ = (1/4)(ℓβ[g
−1∂ρg]− ℓρ[g
−1∂βg]) (78)
Thus the generalized scheme allows in principle the presence of non sym-
metric stress-energy on the discontinuity hypersurface. We will display non-
trivial examples of non-symmetry in the following section. Notice that the
right hand side of (78) is identically null in case g ∈ C0 and ℓα ∈ C
0, since
in this case we have [g−1∂βg] = ℓβg
−1∂g.
A non-symmetric Einstein tensor is a feature of Einstein-Cartan theory of
gravitation (see [19], see also [3] section 7.2), where it is due to the presence
of torsion in the non-symmetric connection used to construct generalized
curvature. Thus the generalized theory can be interpreted, at least to some
extent, as introducing a torsion equivalent tool on the shell only, even if there
are no geometrical objects in our theory which can be directly interpretated
as torsion. However, Einstein-Cartan theory also has a spin - angular mo-
mentum field equation in addition to the Einstein equations, which here is
missing.
In the literature, compatibility conditions for C0 solutions of boundary
layers [20], and recently of shock waves and thin shells [21], have been studied
also in the framework of Einstein-Cartan theory; actually this can lead to
non-symmetric stress-energy on the shell. But in that theory this feature
is inherited from the ambient spacetime, which is not here: non-symmetric
stress-energy arises on the shell only, in consequence of the theory. This
interesting feauture probably is worth investigating.
10 Generalized thin shells
Now let us consider a more general form of the spherical metric:
ds2 = −a(r, t)dt2 + b(r, t)dr2 + c(r, t)dΩ2 (79)
Let us consider a match of two spherical solutions of the Einstein equations
across a timelike (on each side) hypersurface of equation r = ρ(t). Again we
suppose ρ(t) ∈ C1 and therefore ℓα = δα
r − ρ̇δα
t ∈ C0. Let the coordinates
be C0 globally, and let the metric have the same form (79) in both domains
Ω+ and Ω−, with the identification t+ = t−, r+ = r−, θ+ = θ−, ϕ+ = ϕ− on
Let moreover a, b, c > 0 and let a, b, c ∈ piecewise-C0 be regularly dis-
continuous on Σ and with regularly discontinuous first derivatives. Again we
denote by a dot the partial derivative with respect to t, and by a prime that
with respect to r.
In this case for the left hand side of the generalized compatibility condi-
tions ℓµHβρ
µ we obtain:
ℓµHβρ
µ = −
[a′b−1/2] + ρ̇[ḃb−1/2 + ċc−1]
+([a′a−1/2 + c′c−1] + ρ̇[ḃa−1/2])δβ
+([ȧa−1/2 + ċc−1] + ρ̇[a′a−1/2])δβ
−([ḃb−1/2] + ρ̇[b′b−1/2 + c′c−1])δβ
+([c′b−1/2] + ρ̇[ċa−1/2])(δβ
θ + sin2 θδβ
[b−1(a′a−1/2 + c′c−1)] + ρ̇[a−1(ḃb−1/2 + ċc−1)]
where, obviously:
gβρ = −aδβ
t + bδβ
r + c(δβ
θ + sin2 θδβ
ϕ) (81)
We now mean to interpret (80) as the matter-energy of a thin shell. Let
us first get back to the particular case ρ̇ = 0 (static shell) and ȧ = ḃ =
ċ = 0, to make the interpretation simpler by eliminating the non-symmetric
component; rearranging some terms we in fact obtain:
ℓµHβρ
µ = (−[a′b−1/2] + ac−1[c′b−1/2])δβ
+([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δβ
c−1[c′b−1/2]− [b−1(a′a−1/2 + c′c−1)]
This can be interpretated as a perfect isotropic magneto-fluid thin shell with
infinite conductivity, i.e. we can solve the compatibility conditions by con-
sidering the following stress-energy as the right hand side:
T̆αβ = (ρ0 + p+ µh
2)UαUβ + (p+ (1/2)µh
2)gαβ − µhαhβ (83)
where ρ0 is the proper density, h the magnetic field and µ the magnetic
permeability [22, 23, 24, 25, 6]; here we define h2 = hαhβg
αβ . In fact it
suffices to define the following 4-velocity vector:
a− (1/4)[a2]a−1δα
t = (a−1)1/2δα
t (84)
which by construction is unitary on Σ, in the following sense: UαUβg
αβ = −1,
and the following magneto-hydrodynamical variables:
ρ0 = χ
−1a−1[a′b−1]/2 + χ−1c−1(b b−1/2− aa−1 + 1)[c′b−1]/2+
−χ−1[b−1](a′a−1/2 + c′c−1)− (3/2)χ−1b−1[a′a−1/2 + c′c−1]
p = χ−1c−1(bb−1/2− 1)[c′b−1/2] + (1/2)χ−1b−1[a′a−1/2 + c′c−1]+
+χ−1[b−1](a′a−1/2 + c′c−1)
hα = ±
b−1χ−1([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δα
to match (82) and (83) via ℓµHβρ
µ = −χT̆βρ. If [a] = [b] = [c] = 0 then the
generalized shell (85) degenerates to the C0 magnetohydrodynamical shell
considered in [1] section 10.1, in the particular case ρ̇ = 0.
The slightly more general case of ȧ = ḃ = ċ = 0, but ρ̇ 6= 0, displays
non-symmetric terms in (80); however it is not difficult to see that the per-
fect magnetofluid interpretation still holds, provided such additional non-
symmetric terms are interpreted, or neglected. In fact in this case we have:
ℓµHβρ
µ = (−[a′b−1/2] + a c−1[c′b−1/2])δβ
+([a′a−1/2 + c′c−1]− bc−1[c′b−1/2])δβ
+(1/2)ρ̇[a′a−1/2− b′b−1/2− c′c−1](δβ
t + δβ
+(1/2)ρ̇[a′a−1/2 + b′b−1/2 + c′c−1](δβ
t − δβ
c−1[c′b−1/2]− [b−1(a′a−1/2 + c′c−1)]
Now let us consider, for the sake of brevity, the following quantities:
ρ̇2[ a
]2b−1 − (a
]− [ a
])2a−1
]− [ a
]− [ a
and let us suppose that inequality α < 0 holds, which is necessary for the
physical interpretation. In fact in this case the following vector:
]− [ a
t + 1
ρ̇[ a
]− [ a
is a unit timelike vector on Σ, in the sense that UαUβg
αβ = −1. Rearranging
terms, (86) now reads:
ℓµHβρ
µ = αUβUρ + βδβ
r + 1
ρ̇[ a
t − δβ
c−1[ c
]− [b−1( a
which can be matched via ℓµHβρ
µ = −χT̆βρ with a stress-energy tensor of
the following kind:
T̆αβ = (ρ0 + p+ µh
2)UαUβ + (p+ (1/2)µh
2)gαβ − µhαhβ + Aαβ (91)
where A denotes the anti-symmetric term. We have:
ρ0 = −χ
−1α + χ−1 1
]− χ−1[b−1( a
)]− 1
p = −χ−1 1
] + χ−1[b−1( a
)]− 1
µh2 = χ−1βb−1
while the anti-symmetric term A reads:
Aαβ = −χ
t − δβ
r) (93)
The interpretation of such term is still missing; alternatively it could be
neglegted by adding the further hypothesys:
= 0 (94)
which is equivalent to [g−1g′] = 0, as we could expect from (78).
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Introduction
Discontinuous metrics
Mean-value geometry on a hypersurface
Complex mean-value formalism
Generalized compatibility conditions
A class of spherical boundary layers
Generalized gravitational shock waves
Slow generalized gravitational waves
Non-symmetric stress-energy
Generalized thin shells
|
0704.0104 | A geometric realization of sl(6,C) | A GEOMETRIC REALIZATION OF sl(6,C)
GIOVANNI GAIFFI, MICHELE GRASSI
Abstract. Given an orientable weakly self-dual manifold X of rank two, we
build a geometric realization of the Lie algebra sl(6,C) as a naturally defined
algebra LC of endomorphisms of the space of differential forms of X. We pro-
vide an explicit description of Serre generators in terms of natural generators
of LC. This construction gives a bundle on X which is related to the search
for a natural Gauge theory on X. We consider this paper as a first step in the
study of a rich and interesting algebraic structure.
1. Introduction
This paper is a step in a broader program, which aims at finding a geomet-
ric counterpart to the Mirror Symmetry phaenomenon, and possibly a geometric
language in which to formulate a physical theory interpolating between different
σ-models. While we direct the reader to [G2],[G3] for more details, we list here
only some aspects of this theory to put the present work into context.
In the Strominger-Yau-Zaslow approach to Mirror Symmetry you have that two
mirror dual Calabi-Yaus should posses (in some limiting sense) semi-flat special
lagrangian torus fibrations f : M → B, f̂ : M̂ → B which have as fibres flat tori
which are dual in the metric sense (see [SYZ], and [G2] for the terminology and the
definitions). As it is widely known, the major drawback of this approach is that it
is very difficult to build special lagrangian tori fibrations. Usually this construction
can be carried out only when the dual Calabi-Yau manifolds are actually hyper-
kahler, and the special lagrangian tori can be viewed as complex submanifolds (with
respect to a rotated complex structure), so that the methods of complex algebraic
geometry can be put to work.
When you do have the fibrations, then the idea is to construct the mirror map as
a sort of Fourier-Mukai transform (see for example [BMP]). This Fourier-Mukai
transform is a correspondence induced by pull-back and push forward from the
space X = M ×B M̂ . In the hyperkähler case this space is a complex manifold,
while in the general case (for example for Mirror Symmetry for Calabi-Yau three-
folds) it is just a real manifold of (real) dimension 3 · dimC(M).
Background. The notion of (Weakly) self-dual manifold (cf. [G2]) was con-
ceived in the first place to isolate the geometric aspects of the X above which are
needed to obtain Mirror Symmetry betweenM and M̂ . We reproduce here the def-
inition for the reader, while referring to [G2] and [G3] for all the remarks, examples
and observations:
Definition 1.1. A weakly self-dual manifold (WSD manifold for brevity) is given
by a smooth manifold X, together with two smooth 2-forms ω1, ω2 a Riemannian
metric and a third smooth 2-form ωD (the dualizing form) on it, which satisfy the
following conditions:
1) dω1 = dω2 = dωD = 0 and the distribution ω
1 + ω
2 is integrable.
2) For all p ∈ X there exist an orthogonal basis dx1, .., dxm, dy
1 , ..., dy
m, dy
1 , ..., dy
Date: October 24, 2006.
http://arxiv.org/abs/0704.0104v1
2 GIOVANNI GAIFFI, MICHELE GRASSI
dz1, ..., dzc, dw1, ..., dwc of T
pX such that the dx1, .., dxm, dy
1 , ..., dy
m, dy
1 , ..., dy
are orthonormal and
(ω1)p =
dxi ∧ dy
i , (ω2)p =
dxi ∧ dy
i , (ωD)p =
dy1i ∧ dy
dzi ∧ dwi
Any orthogonal basis of TpX dual to a basis of 1- forms as above is said to be
adapted to the structure, or standard. The number m is the rank of the structure.
For a more intrinsic definition of WSD manifolds the reader should refer to [G2].
Here we have chosen the quickest way to introduce them.
When the forms ω1, ω2, ωD are covariant constant with respect to the Levi-Civita
connection, we speak of 2-Kähler manifolds. An example of these comes from mirror
symmetry for abelian varieties.
Remark 1.2. The form ωD is symplectic once restricted to ω
1 + ω
2. We have
therefore that ω
dim(X)−m
6= 0.
Definition 1.3. 1) A WSD manifold is nondegenerate if dim(ω01 ∩ ω
2)p = 0 at all
points (equivalently if its dimension is 3 times the rank).
2) A WSD manifold is self-dual (SD manifold for brevity) if all the leaves of the
distribution ω01 + ω
2 have volume one (with respect to the volume form induced by
the metric)
Using Self dual manifolds, you can give a first näıve geometric definition of Mir-
ror Symmetry as follows:
Two Calabi-Yau manifolds with B-field (M,BM ) and (M̂,BM̂ ) are mirror dual
if there is a Self-dual manifold X together with surjections π : X → M and
π̂ : X → M̂ such that:
a) π∗(ωM ) = ω1, π̂
) = ω2.
b) The leaves of ω⊥1 are the fibres of π̂
c) The leaves of ω⊥2 are the fibres of π
d) The induced B-fields on M and M̂ are the ones given.
Here make their first appearence the B-fields BM and BM̂ , which are flat unitary
gerbes on M and M̂ respectively, and which are not relevant for the discussions of
this paper. In [G2] it was shown that this picture works well in the case of elliptic
curves, and for some other flat situations.
Physical motivation. One of the reasons to introduce SD manifolds however
was to get rid of special lagrangian fibrations, which are so difficult to construct,
and to be able to attack the problem of Mirror Symmetry also when these fibra-
tions are not expected to exist. In this more general context one expects that the
Mirror Symmetry phaenomenon will not be obtained directly from fibrations of
a SD manifold to the dual Calabi-Yaus, but via a more sophisticated procedure,
which involves a Gromov-Hausdorff type of limit. In [G3] it was shown that for
the family of anticanonical divisors in complex projective space one can build a
(real) two-dimensional family of WSD manifolds, which degenerate in a normalized
Gromov-Hausdorff sense to the correct limits of the mirror dual Calabi-Yaus. The
picture is the following:
A GEOMETRIC REALIZATION OF sl(6,C) 3
MB MAS
where MA and MB are the large Kähler and large complex structure limits of M
and M̂ respectively. To be precise, the manifolds which come out of the costruc-
tion of [G3] are 11 dimensional (degenerate) Weakly self-dual manifolds or rank 3.
Dimension 11 is very appealing in this context from a physical point of view, and
it brings us to the motivation for the present work.
The point of view of [G3] is very different from the current one in the main liter-
ature on mathematical Mirror Symmetry: instead of considering the fibre product
M×B M̂ (when it exists) as a device for proving Mirror Symmetry for Calabi-Yaus,
the limiting Calabi-Yaus of Mirror Symmetry are seen as very special limits of a
family of Self-Dual manifolds, which are the main objects of study. This is actually
more in line with what can be found in the physical literature, where the σ-models
defining the string theories from which Mirror Symmetry originates are seen as
just ”phases” of a unique theory, which is not necessarily in the form of a σ-model
but could very likely be similar to a quantized Gauge theory on an 11-dimensional
manifold. To make this circle of ideas more concrete (and hence more verifiable) at
the end of [G3] it is suggested that one should try to build a natural gauge theory
on Self-dual manifolds: the hope is that once quantized this gauge theory might in-
terpolate between the σ-models associated to the Calabi-Yau’s, and as a byproduct
prove Mirror Symmetry for them. Of course one can always put a gauge bundle on
the Self-dual manifolds ”artificially”, but a natural bundle which depends only on
the geometric structure would be much more appealing. We ignore here the issue
of which action to put on the theory, but it too should be a natural geometric one.
Finally, on [GG] we analyzed the situation for rank three WSD manifolds, and we
found that in this case the corresponding natural bundle is formed by complex Lie
superalgebras. We were able to find a geometrically motivated real form, and to
split it into simple factors. The results of [GG] confirm the suspicion that on a WSD
manifold of high enough rank there could be enough natural algebraic bundles of
operators to build interesting gauge theories.
The construction of LC. From a physical point of view the case of Calabi-
Yau threefolds (i.e. rank three WSD manifolds) or fourfolds (i.e. rank four WSD
manifolds) would be the most interesting one to start with. However, its technical
difficulty convinced us to start more modestly from the case of Calabi-Yau two-
folds (i.e. K3 surfaces) which correspond to rank two Self-dual manifolds. We also
considered only orientable nondegenerate Self-dual manifolds of rank two, hence of
dimension 6. This could be considered a proof of concept from a physicist’s point
of view, however Mirror Symmetry for K3’s is in itself very interesting mathemati-
cally, so we hope that our results could have some useful geometric consequences.
The rank three case is treated in our subsequent [GG], as mentioned in the previous
section of this introduction. The main result of the present paper is the following
4 GIOVANNI GAIFFI, MICHELE GRASSI
(which is a geometric restatement of Theorem 5.11):
The Lie algebra sl(6,C) acts via canonical operators (depending only on the geo-
metric structure) on the smooth differential forms of any orientable nondegenerate
WSD manifold of rank 2.
This action generalizes naturally the action of sl(2,C) on smooth differential
forms of any almost Kähler manifold, and is induced by a bundle action on the
exterior power of the cotangent bundle.
Recall that a Weakly self-dual manifold is a Riemannian manifold with three
”compatible” closed differential forms. We will build a Lie algebra of pointwise
operators on complex differential forms on X , as smooth sections of a bundle of Lie
algebras of operators on the complexified cotangent bundle of X . To start, one can
define the following operators:
Definition 1.4. For φ ∈ Ω∗
L0(φ) = ωD ∧ φ, L1(φ) = −ω2 ∧ φ, L2(φ) = ω1 ∧ φ
One can notice immediately the strong resemblance of the operators above with
the Lefschetz operator of Kähler geometry. Indeed, one can elaborate on this simi-
larity, and use the metric to define the adjoints Λj = L
j (using a pointwise proce-
dure, as in the almost Kähler case).
Simply using the Lj and the Λj , one can show that the algebra generated is iso-
morphic to SL(4,C) ([G2]). However, there are other natural differential forms on
a WSD manifold (which do not have a counterpart in the Kähler case), namely the
volume forms of the distributions ω⊥1 , ω
2 , ω
D of vectors which contract to zero
with the forms ω1, ω2 and ωD respectively. If one calls V0, V1, V2 the corresponding
wedge operators, and A0, A1, A2 their adjoints, the complexity of the calculations
to describe the generated Lie algebra grows a lot. We called L the algebra generated
by the Lj, Vj and their adjoints, and LC its complexification. To study LC we intro-
duced an operator J , which is a complex structure on each of the two-dimensional
distributions mentioned above and generates a group isomorphic to SO(2,R) (recall
that we are in the ”hyperkahler” case, corresponding to Mirror Symmetry for K3’s,
so an ”extra” complex structure shouldn’t be surprising; moreover the holonomy of
a WSD manifold in which all ω1, ω2, ωD are invariant is actually always included in
the group generated by J). One checks that all the operators introduced commute
with it:
∀j [Lj , J ] = [Λj , J ] = [Vj , J ] = [Aj , J ] = 0
and therefore one can try to decompose Λ∗T ∗
X with respect to J and then use
Shur’s Lemma to reduce to the study of the operators on the isotypical components.
One should mention that in the (very) good cases (for instance 2-Kähler manifolds)
the operators above are all covariant constant with respect to the metric connec-
tion, and define an action on the cohomology of X much in the same way as in the
Kähler setting the operators L and Λ do (due to Hodge-type identities). We don’t
explore this aspect here, although it may be relevant to the (homological) mirror
map construction.
Coming back to the construction, we point out the inclusion of the Lie algebra LC
inside a copy of the Clifford algebra Cl6,6.
Using this Clifford algebra one can identify ”degree two” or ”quadratic” operators
(in a way similar to the ones involved in the Spinor representations on standard
Spin manifolds) and among these the SO(2,R)-invariant ones. A posteriori, it turns
out that the operators of LC⊕ < J > are all the J-invariant operators of ”degree
A GEOMETRIC REALIZATION OF sl(6,C) 5
two”, and this strengthens the rationale in our selection of natural operators.
As a last step one finds that inside Λ∗T ∗X there is an SO(2,R)-isotypical compo-
nent of dimension 6, and by direct computation we prove that indeed the operators
restricted to this sub-representation determine a copy of sl(6,C) (with the defin-
ing representation). Using the bound on the dimension of LC obtained computing
”quadratic” invariants, one then shows that the representation on this isotypical
component is faithful. This provides as a byproduct a method for giving presenta-
tion of standard Serre generators of LC, explicitely written in terms of the natural
geometrical generators.
2. Basic operators
In this section we fix a point p in the WSD manifold X . The WSD structure
splits the cotangent space as T ∗pX =W0⊕W1⊕W2 where theWj are three mutually
orthogonal canonical distributions defined as:
W0 = {φ ∈ T
pX | φ ∧ ω
1 = φ ∧ ω
2 = 0}
W1 = {φ ∈ T
pX | φ ∧ ω
1 = φ ∧ ω
D = 0}
W2 = {φ ∈ T
pX | φ ∧ ω
2 = φ ∧ ω
D = 0}
The WSD structure also determines canonical pairwise linear identifications
among W0,W1 and W2, so that one can also write T
pX = W0 ⊗R R
3 or more
simply
T ∗pX =W ⊗R R
where W =W0 ∼=W1 ∼=W2.
Let us now come back to the canonical operators Lj mentioned in the introduction:
Definition 1.4 For φ ∈ Ω∗
L0(φ) = ωD ∧ φ, L1(φ) = −ω2 ∧ φ, L2(φ) = ω1 ∧ φ
We now choose a (non-canonical) orthonormal basis γ1, γ2 forW0, and this together
with the standard identifications of the Wj determines an orthonormal basis for
T ∗pX , which we write as {vij = γi ⊗ ej | i = 1, 2, j = 0, 1, 2}. We remark that
the vij are an adapted coframe for the WSD structure, and therefore we have the
explicit expressions:
ω1 = v10 ∧ v11 + v20 ∧ v21
ω2 = v10 ∧ v12 + v20 ∧ v22
ωD = v11 ∧ v12 + v21 ∧ v22
A different choice of the γ1, γ2 would be related to the previous one by an element in
O(2,R) or, taking into account the orientability ofX mentioned in the Introduction,
an element of SO(2,R). The Lie algebra of the group SO(2,R) expressing the
change from one oriented adapted basis to another is generated (point by point) by
the global operator J :
Definition 2.1. The operator J ∈ EndR(Ω
∗(X)) is induced by its pointwise action
on the Λ∗T ∗pX for varying p ∈ X, defined in terms of the standard basis vij as
J(v1j) = v2j , J(v2j) = −v1j for j ∈ {0, 1, 2}
and J(v ∧ w) = J(v) ∧ w + v ∧ J(w) for v, w ∈ Λ∗T ∗pX
Remark 2.2. As J commutes with itself, it is well defined, independently of the
choice of an oriented adapted basis.
Using the chosen (orthonormal) basis, one can define corresponding (non canon-
ical) wedge and contraction operators:
6 GIOVANNI GAIFFI, MICHELE GRASSI
Definition 2.3. Let i ∈ {1, 2} and j ∈ {0, 1, 2}. The operators Eij and Iij are
respectively the wedge and the contraction operator with the form vij on
(defined using the given basis); we use the notation ∂
to indicate the element of
TpX dual to vij ∈ T
Eij(φ) = vij ∧ φ, Iij(φ) =
Proposition 2.4. The operators Eij , Iij satisfy the following relations:
∀i, j, k, l EijEkl = −EklEij , IijIkl = −IklIij
∀i, j EijIij + IijEij = Id
∀(i, j) 6= (k, l) EijIkl = −IklEij
∀i, j E∗ij = Iij , I
ij = Eij
where ∗ is adjunction with respect to the metric.
Proof The proof is a simple direct verification, which we omit. �
It is then immediate to verify that:
Proposition 2.5. J can be expressed as
(E2jI1j − E1jI2j)
on the whole
T ∗pX. From this expression and the previous proposition one ob-
tains that J∗ = −J , i.e. for every p the Lie algebra generated by J is a subalgebra
of o(
T ∗pX) isomorphic to so(2,R)
∼= R. Moreover, the exponential images in-
side AutR(Ω
∗(X))of the operators of type tJ for t ∈ R form a group isomorphic to
SO(2,R) ∼= S1, as this isomorphism holds for the (faithful) restriction of the group
action to T ∗pX.
Using the (non canonical) operators Eij we can obtain simple expressions for the
pointwise action of the other canonical operators, the volume forms Vj :
Definition 2.6. For φ ∈
T ∗pX,
V0(φ) = E10E20(φ), V1(φ) = E11E21(φ), V2(φ) = E12E22(φ)
Remember however that the operators Vj do not depend on the choice of a basis,
as they are simply multiplication by the volume forms of the spaces Wj .
We use the vij also as a orthonormal basis for the complexified space T
p ⊗R C
(with respect to the induced hermitian inner product). We indicate with the same
symbols Vj the complexified operators acting on the spaces
T ∗pX .
The riemannian metric induces a Riemannian metric on T ∗pX and on the space
T ∗pX .
Definition 2.7. For j ∈ {0, 1, 2}
Λj = L
j , Aj = V
By construction the canonical operators Lj , Vj ,Λj, Aj on
T ∗pX are the point-
wise restrictions of corresponding global operators on smooth differential forms,
which we indicate with the same symbols: for j ∈ {0, 1, 2},
Lj , Vj ,Λj , Aj : Ω
∗(X) → Ω∗(X)
Summing up:
A GEOMETRIC REALIZATION OF sl(6,C) 7
Definition 2.8. The ∗-Lie algebra L is the ∗-Lie subalgebra of EndR (Ω
∗(X)) gen-
erated by the operators
{Lj, Vj ,Λj, Aj | for j = 0, 1, 2}
The ∗ operator on L is induced by the adjoint with respect to the Riemannian
metric. The ∗-Lie algebra LC is L ⊗ C, and is in a natural way a ∗-Lie subalgebra
of EndC (Ω
(X)). The ∗ operator on LC is induced by the adjoint with respect to
the induced Hermitian metric.
The canonical splitting T ∗pX = W0 ⊕ W1 ⊕ W2 together with the canonical
identifications W0 ∼=W1 ∼=W2 induce an action of the symmetric group S3, which
propagates to
T ∗X and to its C∞ sections. At every point, the action can be
written explicitly in terms of the basis as
σ(vij) = viσ(j)
The induced action on endomorphisms via conjugation, σ(φ) = σ◦φ◦σ−1, preserves
LC. Indeed, one can check directly using the basis vij at every point that for σ ∈ S3
σ(Vj) = Vσ(j), σ(Lj) = ǫ(σ)Lσ(j)
Since S3 acts on LC by conjugation with unitary operators, its action commutes
with adjunction (the ∗ operator), and therefore
σ(Aj) = Aσ(j), σ(Λj) = ǫ(σ)Λσ(j)
Moreover, one also has that σ(J) = J which means that the action of S3 commutes
with that of so(2,R).
3. The action of so(2,R)
When one deals with mirror simmetry for 2-Kähler manifolds (see the Introduc-
tion), the WSD manifolds which arise have the property that the forms ω1, ω2 and
ωD are covariant constant with respect to the metric. In this case, the maximal
possible holonomy of the WSD manifold X is included in the so(2,R) generated
by the operator J . We will show now that J commutes with LC. Our proof will
be strictly algebraic, so that the commutativity between so(2,R) and LC will hold
also on WSD manifolds for which the holonomy is more general.
Definition 3.1. Given n ∈ Z, we indicate with Vn the one dimensional complex
representation of SO(2,R) ∼= S1 ∼= R/Z given by the character:
θ → e2πınθ
Proposition 3.2. Under the SO(2,R) representation induced by the operator J ,
for any p ∈ X :
1) The space
Xp) splits as
V ⊕31
8 GIOVANNI GAIFFI, MICHELE GRASSI
2) The whole space
Xp) splits according to the following picture:
Xp) = V0
Xp) = V
V ⊕31
Xp) = V
V ⊕90
V ⊕32
Xp) = V−3
V ⊕91
Xp) = V
V ⊕90
V ⊕32
Xp) = V
V ⊕31
Xp) = V0
Proof 1) The space T ∗
Xp is a direct sum of the three Wj , and each one of these
is the standard two dimensional real representation of so(2,R). We therefore diag-
onalize the representation introducing a new basis for each Wj =< v1j , v2j >:
wj = v1j + ı v2j , wj = v1j − ıv2j
From the definition of J , one has then for every j ∈ {0, 1, 2}
J(wj) = −ıwj , J(wj) = ıwj
Therefore one has for every j ∈ {0, 1, 2}
< wj >∼= V−1, < wj >∼= V1
2) To prove the general case, we use the fact that the operator J determines an
almost complex structure on the manifold X , compatible with the metric. From
this, following standard arguments, the complex differential forms and also the
elements of
Xy for any y ∈ Y can be divided according to their type:
p+q=n
In the notation adopted in the proof of the first statement, one has
Xy =< wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq | i1, ..., jq ∈ {0, 1, 2} >
From the definition of the action of J one has therefore that for any p, q
Xy ∼= V
with k =
from which the second statement of the proposition can be esily
deduced. �
Theorem 3.3. The operators Lj, Vj for j ∈ {0, 1, 2} commute with the generator
J of so(2,R).
A GEOMETRIC REALIZATION OF sl(6,C) 9
Proof We prove the statements by a direct computation using the basis vij ;
moreover, using the action of S3 (which permutes the Lj, Vj and fixes J), it is
enough to prove the commutativity for L0 and V0. It useful to rewrite ω0 (and
hence L0 which is wedge with ω0) in terms of the basis generated by the wj :
ω0 = v11 ∧ v12 + v21 ∧ v22 =
(w1 ∧ w2 − w2 ∧ w1)
and then:
[J, L0](wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq ) =
(w1 ∧ w2 − w2 ∧ w1)
wi1 ∧ · · · ∧wip ∧ wj1 ∧ · · · ∧ wjq
(w1 ∧ w2 − w2 ∧ w1)
wi1 ∧ · · · ∧wip ∧ wj1 ∧ · · · ∧ wjq
(w1 ∧ w2 − w2 ∧w1) ∧ J(wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq )
Therefore the result follows from the fact that
(w1 ∧w2 − w2 ∧ w1) = 0
as wj and wk have opposite weight with respect to J for any j, k.
Similarly, [J, V0] = 0 follows from the fact that for any α
V0(α) = v10 ∧ v20 ∧ α =
w0 ∧ w0 ∧ α
From the previous theorem one obtains the following corollary, which holds on any
WSD manifold (not necessarily 2-Kähler ):
Corollary 3.4. The algebra LC commutes with the action of so(2,R) induced by
Proof We already know that [J, Lj] = [J, Vj ] = 0 for j ∈ {0, 1, 2}. The corre-
sponding commutation relations for the adjoint generators Λj , Aj of LC follow from
the fact that J∗ = −J , as noticed in Proposition 2.5. �
Remark 3.5. From Schurs’s lemma it follows that the columns of the diagram of
Proposition 3.2 are preserved by the action of LC.
4. An irreducible representation of LC
Looking at the table in Proposition 3.2 we notice that the second column from
the left is a representation of LC (by Remark 3.5) of dimension 6:
V ∼= V
−2 =< w0 ∧ w1, w0 ∧ w2, w1 ∧w2, w0 ∧ w1 ∧w2 ∧w0,
w0 ∧w1 ∧w2 ∧ w1, w0 ∧ w1 ∧ w2 ∧ w2 >
In this section we will compute explicitely this representation.
Using the above described basis, it is not difficult to compute the matrices by
hand:
Proposition 4.1. Indicating with β the ordered basis for V indicated above, the
matrices for the (restrictions to V of) the generators of LC are the following:
Mβ(L0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0
, Mβ(Λ0) =
0 0 0 0 −2 0
0 0 0 0 0 −2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
10 GIOVANNI GAIFFI, MICHELE GRASSI
Mβ(L1) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 −
0 0 0
, Mβ(Λ1) =
0 0 0 2 0 0
0 0 0 0 0 0
0 0 0 0 0 −2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(L2) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0
0 0 0 0 0 0
, Mβ(Λ2) =
0 0 0 0 0 0
0 0 0 2 0 0
0 0 0 0 2 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
, Mβ(A0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 −2ı 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V1) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
, Mβ(A1) =
0 0 0 0 0 0
0 0 0 0 2ı 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V2) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
, Mβ(A2) =
0 0 0 0 0 −2ı
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Proof Direct computation using the basis generated by the wj . �
Corollary 4.2. The algebra generated by the restriction of LC to V is isomorphic
to sl(6,C), with V its natural representation.
One can sum up the computations above in the following theorem:
Theorem 4.3. There is an exact sequence of Lie algebras
0 → K → LC → sl(6,C) → 0
given by the restriction to V .
In the next section we will prove that K = {0}, and therefore the representation
V is faithful and LC ∼= sl(6,C).
5. Quadratic invariants
We begin by showing that the action of Lie algebra LC is induced by a (non-
canonical) Clifford algebra representation. We use for simplicity the canonical
identification T ∗∗Xp ∼= TXp without further comment, so that if {vij} is a basis
for T ∗pX , then {
} is the corresponding dual basis for TpX .
Definition 5.1. For p ∈ X, the Clifford algebra Cp is
Cp = Cl(TpX ⊕ T
pX, q)
A GEOMETRIC REALIZATION OF sl(6,C) 11
with the quadratic form q induced by the metric
∀i, j, h, k < vij , vhk >= 0
∀i, j, h, k < ∂
∀(i, j) 6= (h, k) < vij ,
∀i, j < vij ,
>= − 1
Remark 5.2. The Clifford algebras Cp for varying p define a Clifford bundle C
on X, as the definition of Cp is independent on the choice of a basis. Indeed, the
quadratic form used to define it is simply induced by − 1
times the natural bilinear
pairing TpX ⊗ T
pX → R.
Proposition 5.3. The Clifford algebra Cp has a canonical representation ρp on
T ∗pX, induced by the operators Eij and Iij via the map
ρp(vij) = Eij , ρp
= Iij
Proof The Clifford relations
φψ + ψφ = −2 < φ,ψ >
are precisely the content of Proposition 2.4. The representation is canonical, even if
the operators Eij and Iij are not, because it can be defined in a basis independent
way as
ρp(v)(α) = v ∧ α, ρp
Abusing slightly the notation, we will identify Cp with its (faithful) image inside
T ∗pX
, and we will omit any reference to the map ρp. Actually, as the
representation above is a real analogue of the Spinor representation, it is easy to
check that the map ρp is an isomorphism of associative algebras. One then has:
Definition 5.4. The linear subspace C2p of Cp is the image of the natural map
(TpX ⊕ T
pX) → Cp. The linear subspace C
p of Cp is the subspace generated by
Recall that C2p is a Lie subalgebra of Cp (with the commutator bracket).
Proposition 5.5. The Lie algebra Lp and the operator J sit inside C
p for all
p ∈ X.
Proof The operators Lj , the Λj, the Vj and the Aj lie inside C
p by Propo-
sition 2.4 and the fact that ω1, ω,ωD lie in
T ∗pX . The operator J lies inside
C2p ⊕ C
p by Proposition 2.5. By definition the elements C
p are commutators, and
therefore have trace zero in any representation, and hence also in the ρp. Moreover,
again by inspection all the generators of Lp have trace zero once represented via
ρp (they are nilpotent), and therefore they must lie inside C
p. The operator J is
in the Lie algebra of the isometry group, and therefore it too has trace zero and
hence sits inside C2p . As C
p is closed under the commutator bracket of Cp, and
this commutator coincides with the composition bracket of operators, we have the
conclusion. �
Remark 5.6. Giving degree 1 to the operators Eij and degree −1 to the opera-
tors Iij , we induce a Z-degree on Cp. This degree coincides with the degree of the
operators induced from the grading on the forms from
T ∗X.
12 GIOVANNI GAIFFI, MICHELE GRASSI
Remark 5.7. For any p ∈ X, the Clifford algebra Cp is isomorphic to Cl6,6, as
the metric used to define it has signature (6, 6). The previous proposition therefore
shows that Lp is a Lie subalgebra of Cl
∼= spin(6, 6) = so(6, 6), generated by
smooth global sections of the Clifford bundle C.
The operator J acts on all of Cp by adjunction with respect to the commutator
bracket, and sends its quadratic part C2p to itself from Proposition 5.5.
We will show that the space of J-invariants inside C2p (the “quadratic” J-invariants)
coincides with LC. To describe it explicitely, let us introduce the following notation:
Definition 5.8.
Ewj = E1j + ıE2j , Ewj = E1j − ıE2j
Iwj = I1j − ıI2j , Iwj = I1j + ıI2j
Lemma 5.9. The adjoint action of the operator J on Ewj , Iwj , Ewj , Iwj is:
[J,Ewj ] = −ıEwj , [J, Iwj ] = ıIwj
[J,Ewj ] = ıEwj , [J, Iwj ] = −ıIwj
Proof It is enough to consider the corresponding J-weights of the wj , wj . �
Proposition 5.10. The following 36 operators provide a linear basis for the qua-
dratic J-invariants:
(1) [Ew0 , Ew1 ], [Ew0 , Ew2 ], [Ew1 , Ew2 ], [Ew1 , Ew0 ], [Ew2 , Ew0 ], [Ew2 , Ew1 ]
(2) [Iw0 , Iw1 ], [Iw0 , Iw2 ], [Iw1 , Iw2 ], [Iw1 , Iw0 ], [Iw2 , Iw0 ], [Iw2 , Iw1 ]
(3) [Ew0 , Ew0 ], [Ew1 , Ew1 ], [Ew2 , Ew2 ]
(4) [Iw0 , Iw0 ], [Iw1 , Iw1 ], [Iw2 , Iw2 ]
(5) [Ew0 , Iw1 ], [Ew0 , Iw2 ], [Ew1 , Iw0 ], [Ew1 , Iw2 ], [Ew2 , Iw0 ], [Ew2 , Iw1 ]
(6) [Ew0 , Iw1 ], [Ew0 , Iw2 ], [Ew1 , Iw0 ], [Ew1 , Iw2 ], [Ew2 , Iw0 ], [Ew2 , Iw1 ]
(7) [Ew0 , Iw0 ], [Ew1 , Iw1 ], [Ew2 , Iw2 ], [Ew0 , Iw0 ], [Ew1 , Iw1 ], [Ew2 , Iw2 ]
Proof The J-weight of a bracket of J-homogeneous operators is the sum of
the respective weights. The quadratic ”monomials” (with respect to the bracket)
in the Ewj , Iwj , Ewj , Iwj are all J-homogeneous, and therefore to find a basis of
J-invariant quadratic operators it is enough to identify the J-invariant quadratic
monomials. To be J-invariant means simply to have weight zero, and the compu-
tation of the J-weight of the quadratic mononials follows immediately from those
of Ewj , Iwj , Ewj , Iwj , which are respectively −ı, ı, ı,−ı. �
We end this section with the following:
Theorem 5.11. In the exact sequence of Theorem 4.3 the kernel K is equal to {0}.
The algebra LC is therefore isomorphic to sl(6,C).
Proof Since LC is included in the Lie algebra of quadratic invariants, it is
enough to show that J 6∈ LC, as from this and the previous proposition it follows
that dimC(LC) ≤ 35. As LC maps surjectively to sl(6,C) which has dimension 35,
the kernel must be zero. When restricted to the subrepresentation V , the generators
of LC have all trace zero by inspection of their matrices. However, by definition
of V , J restricted to it is multiplication by −2ı, and has therefore trace equal to
−12ı. �
Corollary 5.12. The Lie algebra LC⊕ < J > equals the Lie algebra of quadratic
invariants inside C2p .
A GEOMETRIC REALIZATION OF sl(6,C) 13
6. A geometric presentation of Serre generators
In this section, to gain a better geometric understanding of the representation
LC of sl(6,C), we explore in greater detail its relation to the geometric structure of
a WSD manifold. In particular, we give a presentation of a natural choice of Cartan
subalgebra and Serre generators in terms on the geometric generators Lj ,Λj, Vj , Aj .
The Lj operators are similar in nature to the Lefschetz operators of a Kähler
manifold. This analogy is what provided the initial interest in the algebraic struc-
ture of LC. Similarly to the corresponding standard construction of a representation
of sl(2,C), we define
Definition 6.1. For j ∈ {0, 1, 2}
Hj = [Lj ,Λj]
These operators are self-adjoint, as L∗j = Λj by definition. As in the context of
Kählerian geometry, for every j the algebra < Lj ,Λj, Hj > turns out to be a copy
of sl(2,C). Moreover, the following proposition shows that the operators Hj are
semisimple on the whole algebra LC, and therefore generate a toral subalgebra of
Proposition 6.2. The geometric operators Hj generate a toral subalgebra of LC,
and the following relations hold: for j 6= k ∈ {0, 1, 2}
(1) [Hj , Lj] = 2Lj, [Hj ,Λj] = −2Λj
(2) [Hj , Lk] = Lk, [Hj ,Λk] = −Λk
(3) [Hj , Vj ] = 0, [Hj , Aj ] = 0
(4) [Hj , Vk] = 2Vk, [Hj , Ak] = −2Ak
Proof In view of Theorem 5.11, at this point the quickest method of proof of
this proposition is to refer to the explicit matrices of the (faithful) restriction of LC
to V . �
The whole algebra LC splits into a direct sum of weight spaces with respect to
< H0, H1, H2 >, as this subalgebra is toral. The weight of L0 with respect to the
basis dual to H0, H1, H2 is:
αL0 = (αL0(H0), αL0(H1), αL0(H2)) = (2, 1, 1)
The full list is:
αL0 = (2, 1, 1), αΛ0 = −αL0
αL1 = (1, 2, 1), αΛ1 = −αL1
αL2 = (1, 1, 2), αΛ2 = −αL2
αV0 = (0, 2, 2), αA0 = −αV0
αV1 = (2, 0, 2), αA1 = −αV1
αV2 = (2, 2, 0), αA2 = −αV2
To find a natural geometric expression for two ad-semisimple elements which com-
plete < H0, H1, H2 > to a Cartan subalgebra we look at the generators Vj and Aj .
However, it turns out that the natural candidates [Vj , Aj ] already lie in the algebra
< H0, H1, H2 >. We instead build the new operators by ”subtracting” from the Vj
their weight αVj :
Definition 6.3. We define
S0 = ı[[[V0,Λ1],Λ2], L0]
S1 = ı[[[V1,Λ2],Λ0], L1]
S2 = ı[[[V2,Λ0],Λ1], L2]
14 GIOVANNI GAIFFI, MICHELE GRASSI
and denote by H the Lie algebra (over C):
H =< H0, H1, H2, S0, S1, S2 >
The coefficients ı which appear in the formulas above are dictated by the fact
that with this choice the (diagonal) matrices of the Sj restricted to V have integer
entries.
Proposition 6.4. The algebra H is a Cartan subalgebra of LC. More precisely,
the following are the diagonals of the operators H0, ..., S2 once restricted to V
, H1 :
, H2 :
, S0 :
, S1 :
, S2 :
Proof The computation of the matrices above shows that, once restricted to
V , the algebra H spans the space of diagonal matrices of trace zero in the given
basis. �
Remark 6.5. The computation above shows also that operators S0, S1, S2 safisfy
the relation
S0 + S1 + S2 = 0
Even if from the previous proposition we know that H is maximal toral inside LC,
the natural geometric generators Lj ,Λj are not eigenvectors for the adjoint action
of the Sk. At this point however it is possible to single out in natural geometric
terms operators of LC which have ”pure” weight with respect to the algebra H and
which contain in their linear span the Lj,Λj :
Definition 6.6. For j ∈ {0, 1, 2}
L1j = −2Lj + [Sj , Lj], L2j = 2Lj + [Sj , Lj]
Λ1j = −2Λj − [Sj ,Λj], Λ2j = 2Λj − [Sj ,Λj]
Proposition 6.7. Indicating with ehk the 6× 6 matrix with a 1 in position k (row)
and h (column) and zero otherwise, the matrices of the operators Lij and Λij re-
stricted on V are:
L10 = 2e
6 L11 = −2e
4 L12 = −2e
L20 = −2e
5 L21 = −2e
6 L22 = 2e
Λ10 = 8e
2 Λ11 = −8e
1 Λ12 = −8e
Λ20 = −8e
1 Λ21 = −8e
3 Λ22 = 8e
Corollary 6.8. We have the following relations for the operators of LC restricted
to V :
[Hk, Lij ] = (1 + δkj)Lij , [Hk,Λij ] = −(1 + δkj)Λij
[Sk, Lij ] = (−1)
i+1(1− 3δkj)Lij , [Sk,Λij ] = (−1)
i(1− 3δkj)Λij
[Sk, Vj ] = 0, [Sk, Aj ] = 0
Guided by all the explicit computations of the action on the isotypical component
V = V ⊕6
−2 made up to this point, we now define in terms of the natural geometric
operators a set of Serre generators for the algebra LC.
A GEOMETRIC REALIZATION OF sl(6,C) 15
Definition 6.9.
[L20, A1] f1 =
[V1,Λ20]
[L22, A0] f2 =
[V0,Λ22]
e3 = V0 f3 = A0
[L12, A0] f4 =
[V0,Λ12]
[L10, A1] f5 =
[V1,Λ10]
Moreover, for all i ∈ {1, .., 5} we define hi = [ei, fi].
As the ei have by construction associated matrix e
i+1 once restricted to V and
the fi are their respective adjoints, one gets:
Proposition 6.10. The operators ei, fj ,hk satisfy the Serre relations for sl(6,C)
and the hi span the Cartan subalgebra H:
(H1 −H2 − S1 − S2)
(H0 −H1 + S2)
(−H0 +H1 +H2)
(H0 −H1 − S2)
(H1 −H2 + S1 + S2)
It would be interesting as a last remark to identify in the list of quadratic in-
variants the geometric operators Lij ,Λij , Vj , Aj , the algebra H and the so(2,R)
generator J . To do this one could of course use the explicit matrices for the qua-
dratic invariants once restricted to V , which are not difficult to compute. One can
however get very quickly a qualitative picture by using the notion of multidegree
which we now introduce.
The decomposition T ∗X = W0 ⊕ W1 ⊕ W2 induces naturally a multi-degree on
X with values in Z3, which we indicate with mdeg. This follows from the
equation
p+q+r=n
(W0 ⊗ C)⊕
(W1 ⊗ C)⊕
(W2 ⊗ C)
We notice furthermore that the (complexified) decomposition above is preserved by
the operator J , and therefore mdeg commutes with the action of so(2,R).
Proposition 6.11. The operators Lj, Vj ,Λj , Aj , Hj , Sj are mdeg-homogeneous,
with multi-degrees:
mdeg(L0) = (0, 1, 1) mdeg(L1) = (1, 0, 1) mdeg(L2) = (1, 1, 0)
mdeg(Λ0) = (0,−1,−1) mdeg(Λ1) = (−1, 0,−1) mdeg(Λ2) = (−1,−1, 0)
mdeg(V0) = (2, 0, 0) mdeg(V1) = (0, 2, 0) mdeg(V2) = (0, 0, 2)
mdeg(A0) = (−2, 0, 0) mdeg(A1) = (0,−2, 0) mdeg(A2) = (0, 0,−2)
mdeg(H0) = (0, 0, 0) mdeg(H1) = (0, 0, 0) mdeg(H2) = (0, 0, 0)
mdeg(S0) = (0, 0, 0) mdeg(S1) = (0, 0, 0) mdeg(S2) = (0, 0, 0)
Proof The values for mdeg for the Lj and the Vj follow immediately from mdeg
of the corresponding forms and the dual (contraction) operators have opposite value
16 GIOVANNI GAIFFI, MICHELE GRASSI
of mdeg. The remaing values can be computed using the additivity of mdeg with
respect to the bracket. �
Proposition 6.12. Let {j, k, l} = {0, 1, 2}. Then
Span (L1j, L2j) = Span ([Ewk , Ewl ], [Ewl , Ewk ])
Span (Λ1j ,Λ2j) = Span ([Iwk , Iwl ], [Iwl , Iwk ])
Span (Vj) = Span
[Ewj , Ewj ]
Span (Aj) = Span
[Iwj , Iwj ]
H⊕ Span (J) =
Span ([Ewm , Iwm ], [Ewm , Iwm ])
Proof The mdeg of the Lij is the same of the corresponding Lj, and sim-
ilarly for their adjoints. The mdegs of the quadratic monomials are immedi-
ately computed as they are the sum of those of their components. For example,
mdeg(Ew0) = mdeg(Ew0) = (1, 0, 0) , mdeg(Ew1) = mdeg(Ew1) = (0, 1, 0) and
therefore mdeg([Ew0 , Ew1 ] = (1, 1, 0), equal to that of L12 and L22. �
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http://arxiv.org/abs/math/0006154
http://arxiv.org/abs/math/0202016
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http://arxiv.org/abs/math/0011041
http://arxiv.org/abs/hep-th/9606040
1. Introduction
2. Basic operators
3. The action of so(2,R)
4. An irreducible representation of LC
5. Quadratic invariants
6. A geometric presentation of Serre generators
References
|
0704.0105 | Rigid subsets of symplectic manifolds | 8 Rigid subsets of symplectic manifolds
Michael Entova and Leonid Polterovichb
August 31, 2018
Abstract
We show that there is an hierarchy of intersection rigidity proper-
ties of sets in a closed symplectic manifold: some sets cannot be dis-
placed by symplectomorphisms from more sets than the others. We
also find new examples of rigidity of intersections involving, in par-
ticular, specific fibers of moment maps of Hamiltonian torus actions,
monotone Lagrangian submanifolds (following the works of P.Albers
and P.Biran-O.Cornea), as well as certain, possibly singular, sets de-
fined in terms of Poisson-commutative subalgebras of smooth func-
tions. In addition, we get some geometric obstructions to semi-simpli-
city of the quantum homology of symplectic manifolds. The proofs are
based on the Floer-theoretical machinery of partial symplectic quasi-
states.
aPartially supported by E. and J. Bishop Research Fund and by the Israel Science
Foundation grant # 881/06.
bPartially supported by the Israel Science Foundation grant # 11/03.
http://arxiv.org/abs/0704.0105v2
Contents
1 Introduction and main results 3
1.1 Many facets of displaceability . . . . . . . . . . . . . . . . . . 3
1.2 Preliminaries on quantum homology . . . . . . . . . . . . . . . 8
1.3 An hierarchy of rigid subsets within Floer theory . . . . . . . 10
1.4 Hamiltonian torus actions . . . . . . . . . . . . . . . . . . . . 14
1.5 Super(heavy) monotone Lagrangian submanifolds . . . . . . . 19
1.6 An effect of semi-simplicity . . . . . . . . . . . . . . . . . . . . 23
1.7 Discussion and open questions . . . . . . . . . . . . . . . . . . 27
1.7.1 Strong displaceability beyond Floer theory? . . . . . . 27
1.7.2 Heavy fibers of Poisson-commutative subspaces . . . . 28
2 Detecting stable displaceability 32
3 Preliminaries on Hamiltonian Floer theory 33
3.1 Valuation on QH∗(M) . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Hamiltonian Floer theory . . . . . . . . . . . . . . . . . . . . 34
3.3 Conley-Zehnder and Maslov indices . . . . . . . . . . . . . . . 36
3.4 Spectral numbers . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Partial symplectic quasi-states . . . . . . . . . . . . . . . . . . 44
4 Basic properties of (super)heavy sets 45
5 Products of (super)heavy sets 48
5.1 Product formula for spectral invariants . . . . . . . . . . . . . 48
5.2 Decorated Z2-graded complexes . . . . . . . . . . . . . . . . . 49
5.3 Reduced Floer and Quantum homology . . . . . . . . . . . . . 50
5.4 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Proof of algebraic Theorem 5.2 . . . . . . . . . . . . . . . . . 52
6 Stable non-displaceability of heavy sets 57
7 Analyzing stable stems 59
8 Monotone Lagrangian submanifolds 61
9 Rigidity of special fibers of Hamiltonian actions 66
9.1 Calabi and mixed action-Maslov . . . . . . . . . . . . . . . . . 76
1 Introduction and main results
1.1 Many facets of displaceability
A well-studied and easy to visualize rigidity property of subsets of a symplec-
tic manifold (M,ω) is the rigidity of intersections: a subset X ⊂ M cannot
be displaced from the closure of a subset Y ⊂ M by a compactly supported
Hamiltonian isotopy:
φ(X) ∩ Y 6= ∅ ∀φ ∈ Ham(M) .
We say in such a case that X cannot be displaced from Y . If X cannot be
displaced from itself we call it non-displaceable. These properties become
especially interesting and purely symplectic when X can be displaced from
itself or from Y by a (compactly supported) smooth isotopy.
One of the main themes of the present paper is that “some non-displace-
able sets are more rigid than others.” To explain this, we need the following
ramifications of the notion of a non-displaceable set:
Strong non-displaceability: A subset X ⊂ M is called strongly non-
displaceable if one cannot displace it by any (not necessarily Hamiltonian)
symplectomorphism of (M,ω).
Stable non-displaceability: Consider T ∗S1 = R × S1 with the coordi-
nates (r, θ) and the symplectic form dr ∧ dθ. We say that X ⊂ M is stably
non-displaceable if X × {r = 0} is non-displaceable in M × T ∗S1 equipped
with the split symplectic form ω̄ = ω ⊕ (dr ∧ dθ). Let us mention that de-
tecting stably non-displaceable subsets is useful for studying geometry and
dynamics of Hamiltonian flows (see for instance [50] for their role in Hofer’s
geometry and [51] for their appearance in the context of kick stability in
Hamiltonian dynamics).
Formally speaking, the properties of strong and stable non-displaceability
are mutually independent and both are strictly stronger than displaceability.
In the present paper we refine the machinery of partial symplectic quasi-
states introduced in [23] and get new examples of stably non-displaceable
sets, including certain fibers of moment maps of Hamiltonian torus actions
as well as monotone Lagrangian submanifolds discussed by Albers [2] and
Biran-Cornea [15]. Further, we address the following question: given the class
of stably non-displaceable sets, can one distinguish those of them which are
also strongly non-displaceable by means of the Floer theory? Or, other way
around, what are the Floer-homological features of stably non-displaceable
but strongly displaceable sets? Toy examples are given by the equator of the
symplectic two-sphere and by the meridian on a symplectic two-torus. Both
are stably non-displaceable since their Lagrangian Floer homologies are non-
trivial. On the other hand, the equator is strongly non-displaceable, while
the meridian is strongly displaceable by a non-Hamiltonian shift. Later on we
shall explain the difference between these two examples from the viewpoint
of Hamiltonian Floer homology and present various generalizations.
The question on Floer-homological characterization of (strongly) non-displa-
ceable but stably displaceable sets is totally open, see Section 1.7.1 below for
an example involving Gromov’s packing theorem and discussion.
Leaving Floer-theoretical considerations for the next section, let us outline
(in parts, informally) the general scheme of our results: Given a symplectic
manifold (M,ω), we shall define (in the language of the Floer theory) two
collections of closed subsets of M , heavy subsets and superheavy subsets.
Every superheavy subset is heavy, but, in general, not vice versa. Formally
speaking, the hierarchy heavy-superheavy depends in a delicate way on the
choice of an idempotent in the quantum homology ring ofM . This and other
nuances will be ignored in this outline. The key properties of these collections
are as follows (see Theorems 1.2 and 1.5 below):
Invariance: Both collections are invariant under the group of all symplec-
tomorphisms of M .
Stable non-displaceability: Every heavy subset is stably non-displace-
able.
Intersections: Every superheavy subset intersects every heavy subset. In
particular, superheavy subsets are strongly non-displaceable. In contrast to
this, heavy subsets can be mutually disjoint and strongly displaceable.
Products: Product of any two (super)heavy subsets is (super)heavy.
What is inside the collections? The collections of heavy and superheavy
sets include the following examples:
Stable stems: Let A ⊂ C∞(M) be a finite-dimensional Poisson-commuta-
tive subspace (i.e. any two functions from A commute with respect to the
Poisson brackets). Let Φ :M → A∗ be the moment map: 〈Φ(x), F 〉 = F (x).
A non-empty fiber Φ−1(p), p ∈ A∗, is called a stem of A (see [23]) if all
non-empty fibers Φ−1(q) with q 6= p are displaceable and a stable stem if
they are stably displaceable. If a subset of M is a (stable) stem of a finite-
dimensional Poisson-commutative subspace of C∞(M), it will be called just
a (stable) stem. Clearly, any stem is a stable stem. The collection of
superheavy subsets includes all stable stems (see Theorem 1.6 below).
One readily shows that a direct product of stable stems is a stable stem and
that the image of a stable stem under any symplectomorphism is again a
stable stem.
The following example of a stable stem is borrowed (with a minor mod-
ification) from [23]: Let X ⊂ M be a closed subset whose complement is
a finite disjoint union of stably displaceable sets. Then X is a stable stem.
For instance, the codimension-1 skeleton of a sufficiently fine triangulation of
any closed symplectic manifold is a stable stem. Another example is given by
the equator of S2: it divides the sphere into two displaceable open discs and
hence is a stable stem. By taking products, one can get more sophisticated
examples of stable stems. Already the product of equators of the two-spheres
gives rise to a Lagrangian Clifford torus in S2× . . .×S2. To prove its rigidity
properties (such as stable non-displaceability) one has to use non-trivial sym-
plectic tools such as Lagrangian Floer homology, see e.g. [44]. Products of
the 1-skeletons of fine triangulations of the two-spheres can be considered as
singular Lagrangian submanifolds, an object which is currently out of reach
of the Lagrangian Floer theory.
Another example of stable stems comes from Hamiltonian torus actions.
Consider an effective Hamiltonian action ϕ : Tk → Ham(M) with the mo-
ment map Φ = (Φ1, . . . ,Φk) : M → R
k. Assume that Φi is a normalized
Hamiltonian, that is
Φi = 0 for all i = 1, . . . , k. A torus action is called
compressible if the image of the homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)),
induced by the action ϕ, is a finite group. One can show that for compressible
actions the fiber Φ−1(0) is a stable stem (see Theorem 1.7 below).
Special fibers of Hamiltonian torus actions: Consider an effective
Hamiltonian torus action ϕ on a spherically monotone symplectic manifold.
Let I : π1(Ham(M)) → R be the mixed action-Maslov homomorphism intro-
duced in [49]. Since the target space Rk of the moment map Φ is naturally
identified with Hom(π1(T
k),R), the pull back pspec := −ϕ
♯I of the mixed
action-Maslov homomorphism with the reversed sign can be considered as a
point of Rk. The preimage Φ−1(pspec) is called the special fiber of the action.
We shall see below that the special fiber is always non-empty. For monotone
symplectic toric manifolds (that is when 2k = dimM) the special fiber is a
monotone Lagrangian torus. Note that when the action is compressible we
have pspec = 0 and therefore the special fiber is a stable stem according to the
previous example. It is unknown whether the latter property persists for gen-
eral non-compressible actions. Thus in what follows we treat stable stems
and special fibers as separate examples. The collection of superheavy
subsets includes all special fibers (see Theorem 1.9 below).
For instance, consider CP 2 and the Lagrangian Clifford torus in it (i.e.
the torus {[z0 : z1 : z2] ∈ CP
2 | |z0| = |z1| = |z2|}). Take the standard
Hamiltonian T2-action on CP 2 preserving the Clifford torus. It has three
global fixed points away from the Clifford torus. Make an equivariant sym-
plectic blow-up, M , of CP 2 at k of these fixed points, 0 ≤ k ≤ 3, so that
the obtained symplectic manifold is spherically monotone. The torus action
lifts to a Hamiltonian action on M . One can show that its special fiber is
the proper transform of the Clifford torus.
Monotone Lagrangian submanifolds: Let (M2n, ω) be a spherically
monotone symplectic manifold, and let L ⊂ M be a closed monotone La-
grangian submanifold with the minimal Maslov number NL ≥ 2. We say
that L satisfies the Albers condition [2] if the image of the natural morphism
H∗(L;Z2) → H∗(M ;Z2) contains a non-zero element S with
deg S > dimL+ 1−NL .
The collection of heavy sets includes all closed monotone Lagran-
gian submanifolds satisfying the Albers condition (see Theorem 1.15
below).
Specific examples include the meridian on T2, RP n ⊂ CP n and all La-
grangian spheres in complex projective hypersurfaces of degree d in CP n+1
with n > 2d − 3. In the case when the fundamental class [L] of L divides
a non-trivial idempotent in the quantum homology algebra of M , L is, in
fact, superheavy (see Theorem 1.18 below). For instance, this is the case
for RP n ⊂ CP n. Furthermore, a version of superheaviness holds for any
Lagrangian sphere in the complex quadric of even (complex) dimension.
However, there exist examples of heavy, but not superheavy, Lagrangian
submanifolds: For instance, the meridian of the 2-torus is strongly displa-
ceable by a (non-Hamiltonian!) shift and hence is not superheavy. Another
example of heavy but not superheavy Lagrangian submanifold is the sphere
arising as the real part of the Fermat hypersurface
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
with even d ≥ 4 and n > 2d− 3. We refer to Section 1.5 for more details on
(super)heavy monotone Lagrangian submanifolds.
Motivation: Our motivation for the selection of examples appearing in the
list above is as follows. Stable stems provide a playground for studying
symplectic rigidity of singular subsets. In particular, no visible analogue of
the conventional Lagrangian Floer homology technique is applicable to them.
Detecting (stable) non-displaceability of Lagrangian submanifolds via La-
grangian Floer homology is one of the central themes of symplectic topology.
In contrast to this, detecting strong non-displaceabilty has at the moment the
status of art rather than science. That’s why we were intrigued by Albers’
observation that monotone Lagrangian submanifolds satisfying his condition
are in some situations strongly non-displaceable. In the present work we tried
to digest Albers’ results [2] and look at them from the viewpoint of theory
of partial symplectic quasi-states developed in [23]. In addition, our result
on superheaviness of the Lagrangian anti-diagonal in S2 × S2 allows us to
detect an “exotic” monotone Lagrangian torus in this symplectic manifold:
this torus does not intersect the anti-diagonal, and hence is not heavy in
contrast to the standard Clifford torus, see Example 1.20 below.
In [23] we proved a theorem which roughly speaking states that every
(singular) coisotropic foliation has at least one non-displaceable fiber. How-
ever, our proof is non-constructive and does not tell us which specific fibers
are non-displaceable. The notion of the special fiber arose as an attempt to
solve this problem for Hamiltonian circle actions.
Let us mention also that the product property enables us to produce even
more examples of (super)heavy subsets by taking products of the subsets
appearing in the list.
A few comments on the methods involved into our study of heavy and su-
perheavy subsets are in order. These collections are defined in terms of
partial symplectic quasi-states which were introduced in [23]. These are cer-
tain real-valued functionals on C∞(M) with rich algebraic properties which
are constructed by means of the Hamiltonian Floer theory and which conve-
niently encode a part of the information contained in this theory. In general,
the definition of a partial symplectic quasi-state involves the choice of an
idempotent element in the commutative part QH•(M) of the quantum ho-
mology algebra of M . Though the default choice is just the unity of the
algebra, there exist some other meaningful choices, in particular in the case
when QH•(M) is semi-simple. This gives rise to another theme discussed in
this paper: “visible” topological obstructions to semi-simplicity (see Corol-
lary 1.24 and Theorem 1.25 below). For instance, we shall show that if a
monotone symplectic manifold M contains “too many” disjoint monotone
Lagrangian spheres whose minimal Maslov numbers exceed n+ 1, the quan-
tum homology QH•(M) cannot be semi-simple.
Let us pass to the precise set-up. For the reader’s convenience, the ma-
terial presented in this brief outline will be repeated in parts in the next
sections in a less compressed form.
1.2 Preliminaries on quantum homology
The Novikov Ring: Let F denote a base field which in our case will be
either C or Z2, and let Γ ⊂ R be a countable subgroup (with respect to the
addition). Let s, q be formal variables. Define a field KΓ whose elements are
generalized Laurent series in s of the following form:
KΓ :=
θ, zθ ∈ F , ♯
θ > c | zθ 6= 0
<∞, ∀c ∈ R
Define a ring ΛΓ := KΓ[q, q
−1] as the ring of polynomials in q, q−1 with
coefficients in KΓ. We turn ΛΓ into a graded ring by setting the degree of s
to be 0 and the degree of q to be 2.
The ring ΛΓ serves as an abstract model of the Novikov ring associated to
a symplectic manifold. Let (M,ω) be a closed connected symplectic manifold.
Denote by HS2 (M) the subgroup of spherical homology classes in the integral
homology group H2(M ;Z). Abusing the notation we will write ω(A), c1(A)
for the results of evaluation of the cohomology classes [ω] and c1(M) on
A ∈ H2(M ;Z). Set
π̄2(M) := H
2 (M)/ ∼,
where by definition
A ∼ B iff ω(A) = ω(B) and c1(A) = c1(B).
Denote by Γ(M,ω) := [ω](HS2 (M)) ⊂ R the subgroup of periods of the
symplectic form on M on spherical homology classes. By definition, the
Novikov ring of a symplectic manifold (M,ω) is ΛΓ(M,ω). In what follows,
when (M,ω) is fixed, we abbreviate and write Γ, K and Λ instead of Γ(M,ω),
KΓ(M,ω) and ΛΓ(M,ω) respectively.
Quantum homology: Set 2n = dimM . The quantum homology QH∗(M)
is defined as follows. First, it is a graded module over Λ given by
QH∗(M) := H∗(M ;F)⊗F Λ,
with the grading defined by the gradings on H∗(M ;F) and Λ:
deg (a⊗ zsθqk) := deg (a) + 2k .
Second, and most important, QH∗(M) is equipped with a quantum prod-
uct: if a ∈ Hk(M ;F), b ∈ Hl(M ;F), their quantum product is a class
a ∗ b ∈ QHk+l−2n(M), defined by
a ∗ b =
A∈π̄2(M)
(a ∗ b)A ⊗ s
−ω(A)q−c1(A),
where (a ∗ b)A ∈ Hk+l−2n+2c1(A)(M) is defined by the requirement
(a ∗ b)A ◦ c = GW
A (a, b, c) ∀c ∈ H∗(M ;F).
Here ◦ stands for the intersection index and GWFA (a, b, c) ∈ F denotes the
Gromov-Witten invariant which, roughly speaking, counts the number of
pseudo-holomorphic spheres inM in the class A that meet cycles representing
a, b, c ∈ H∗(M ;F) (see [55], [56], [41] for the precise definition).
Extending this definition by Λ-linearity to the whole QH∗(M) one gets
a correctly defined graded-commutative associative product operation ∗ on
QH∗(M) which is a deformation of the classical ∩-product in singular ho-
mology [37], [41], [55], [56], [69]. The quantum homology algebra QH∗(M)
is a ring whose unity is the fundamental class [M ] and which is a module
of finite rank over Λ. If a, b ∈ QH∗(M) have graded degrees deg (a), deg (b)
deg (a ∗ b) = deg (a) + deg (b)− 2n. (1)
We will be mostly interested in the commutative part of the quantum
homology ring (which in the case F = Z2 is, of course, the whole quantum
homology ring). For this purpose we introduce the following notation:
We denote by QH•(M) the whole quantum homology QH∗(M) if
F = Z2 and the even-degree part of QH∗(M) if F = C.
In general, given a topological space X, we denote by H•(X ;F) the
whole singular homology group H∗(X ;F) if F = Z2 and the even-
degree part of H∗(X ;F) if F = C.
Thus, in our notation the ring QH•(M) = H•(M ;F)⊗F Λ is always a com-
mutative subring with unity of QH∗(M) and a module of finite rank over Λ.
We will identify Λ with a subring of QH•(M) by λ 7→ [M ]⊗ λ.
1.3 An hierarchy of rigid subsets within Floer theory
Fix a non-zero idempotent a ∈ QH2n(M) (by obvious grading considera-
tions the degree of every idempotent equals 2n). We shall deal with spectral
invariants c(a,H), where H = Ht : M → R, t ∈ R, is a smooth time-
dependent and 1-periodic in time Hamiltonian function on M , or c(a, φH),
where φH is an element of the universal cover H̃am (M) of Ham(M) rep-
resented by an identity-based path given by the time-1 Hamiltonian flow
generated by H . If H is normalized, meaning that
dimM/2 = 0 for all
t, then c(a,H) = c(a, φH). These invariants, which nowadays are standard
objects of the Floer theory, were introduced in [45] (cf. [59] in the aspherical
case; also see [42],[43] for an earlier version of the construction and [22] for a
summary of definitions and results in the monotone case).
Disclaimer: Throughout the paper we tacitly assume that (M,ω) (as well
as (M ×T2, ω̄), when we speak of stable displaceability) belongs to the class
S of closed symplectic manifolds for which the spectral invariants are well
defined and enjoy the standard list of properties (see e.g. [41, Theorem
12.4.4]). For instance, S contains all symplectically aspherical and spherically
monotone manifolds. Furthermore, S contains all symplectic manifolds M2n
for which, on one hand, either c1 = 0 or the minimal Chern number (on
HS2 (M)) is at least n − 1 and, on the other hand, [ω](H
2 (M)) is a discrete
subgroup of R (cf. [64]). The general belief is that the class S includes all
symplectic manifolds.
Define a functional ζ : C∞(M) → R by
ζ(H) := lim
c(a, lH)
It is shown in [23] that the functional ζ has some very special algebraic
properties (see Theorem 3.6) which form the axioms of a partial symplectic
quasi-state introduced in [23]. The next definition is motivated in part by
the work of Albers [2].
Definition 1.1. A closed subset X ⊂ M is called heavy (with respect to ζ
or with respect to a used to define ζ) if
ζ(H) ≥ inf
H ∀H ∈ C∞(M) , (3)
and is called superheavy (with respect to ζ or a) if
ζ(H) ≤ sup
H ∀H ∈ C∞(M) . (4)
The default choice of an idempotent a is the unity [M ] ∈ QH∗(M). In this
case, as we shall see below, the collections of heavy and superheavy sets
satisfy the properties listed in Section 1.1 and include the examples therein.
In view of potential applications (including geometric obstructions to semi-
simplicity of the quantum homology), we shall work, whenever possible, with
general idempotents.
The asymmetry between supX H and infX H is related to the fact that
the spectral numbers satisfy a triangle inequality c(a ∗ b, φFφG) ≤ c(a, φF ) +
c(b, φG), while there may not be a suitable inequality “in the opposite direc-
tion”. In the case when such an “opposite” inequality exists (e.g. when a = b
is an idempotent and ζ defined by it is a genuine symplectic quasi-state – see
Section 1.6 below) the symmetry between supX H and infX H gets restored
and the classes of heavy and superheavy sets coincide.
Let us emphasize that the notion of (super)heaviness depends on the
choice of a coefficient ring for the Floer theory. In this paper the coefficients
for the Floer theory will be either Z2 or C depending on the situation. Unless
otherwise stated, our results on (super)heavy subsets are valid for any choice
the coefficients.
The group Symp (M) of all symplectomorphisms of M acts naturally on
H∗(M ;F) and hence on QH∗(M) = H∗(M ;F) ⊗F Λ. Clearly, the identity
component Symp0(M) of Symp (M) acts trivially on QH∗(M) and hence for
any idempotent a ∈ QH∗(M) the corresponding ζ is Symp0(M)-invariant.
Thus the image of a (super)heavy set under an element of Symp0(M) is again
a (super)heavy set with respect to the same idempotent a. If a is invariant
under the action of the whole Symp (M) (for instance, if a = [M ]) the classes
of heavy and superheavy sets with respect to a are invariant under the action
of the whole Symp (M) in agreement with the invariance property presented
in Section 1.1 above.
Let us mention also that the collections of (super)heavy sets enjoy a
stability property under inclusions: If X, Y , X ⊂ Y , are closed subsets of M
and X is heavy (respectively, superheavy) with respect to an idempotent a
then Y is also heavy (respectively, superheavy) with respect to the same a.
We are ready now to formulate the main results of the present section.
Theorem 1.2. Assume a and ζ are fixed. Then
(i) Every superheavy set is heavy, but, in general, not vice versa.
(ii) Every heavy subset is stably non-displaceable.
(iii) Every superheavy set intersects every heavy set. In particular, a super-
heavy set cannot be displaced by a symplectic (not necessarily Hamil-
tonian) isotopy and if the idempotent a is invariant under the symplec-
tomorphism group of (M,ω) (e.g. if a = [M ]), every superheavy set is
strongly non-displaceable.
The following theorem discusses the relation between heaviness/super-
heaviness properties with respect to different idempotents. In particular, it
shows that [M ] plays a special role among all the other non-zero idempotents
in QH∗(M).
Theorem 1.3. Assume a is a non-zero idempotent in the quantum homology.
(i) Every set that is superheavy with respect to [M ] is also superheavy with
respect to a.
(ii) Every set that is heavy with respect to a is also heavy with respect to
[M ].
(iii) Assume that the idempotent a is a sum of non-zero idempotents
e1, . . . , el and assume that a closed subset X ⊂ M is heavy with re-
spect to a. Then X is heavy with respect to ei for at least one i.
The next proposition shows that, in general, the heaviness of a set does
depend on the choice of an idempotent in the quantum homology.
Proposition 1.4. Consider the torus T2n equipped with the standard sym-
plectic structure ω = dp∧dq. Let M2n = T2n♯CP n be a symplectic blow-up of
T2n at one point (the blow up is performed in a small ball around the point).
Assume that the Lagrangian torus L ⊂ T2n given by q = 0 does not intersect
the ball in T2n, where the blow up was performed.
Then the proper transform of L (identified with L) is a Lagrangian sub-
manifold of M , which is not heavy with respect to some non-zero idempotent
a ∈ QH∗(M) but heavy with respect to [M ]. (Here we work with F = Z2).
Next, consider direct products of (super)heavy sets. We start with the fol-
lowing convention on tensor products. Let Γi, i = 1, 2, be two countable
subgroups of R. Let Ei be a module over KΓi. We put
E1⊗̂KE2 =
E1 ⊗KΓ1 KΓ1+Γ2
⊗KΓ1+Γ2
E2 ⊗KΓ2 KΓ1+Γ2
. (5)
If E1, E2 are also rings we automatically assume that the middle tensor prod-
uct is the tensor product of rings. In simple words, we extend both modules
to KΓ1+Γ2-modules and consider the usual tensor product over KΓ1+Γ2 .
Given two symplectic manifolds, (M1, ω1) and (M2, ω2), note that the
subgroups of periods of the symplectic forms satisfy
Γ(M1 ×M2, ω1 ⊕ ω2) = Γ(M1, ω1) + Γ(M2, ω2) .
Furthermore, due to the Künneth formula for quantum homology (see e.g.
[41, Exercise 11.1.15] for the statement in the monotone case; the general
case in our algebraic setup can be treated similarly) there exists a natural
ring monomorphism linear over KΓ1+Γ2
QH2n1(M1)⊗̂KQH2n2(M2) →֒ QH2n1+2n2(M1 ×M2) ,
We shall fix a pair of idempotents ai ∈ QH∗(Mi), i = 1, 2. The notions
of (super)heaviness in M1,M2 and M1 ×M2 are understood in the sense of
idempotents a1, a2 and a1 ⊗ a2 respectively.
Theorem 1.5. Assume that Xi is a heavy (resp. superheavy) subset of Mi
with respect to some idempotent ai, i = 1, 2. Then the product X1 × X2
is a heavy (resp. superheavy) subset of M with respect to the idempotent
a1 ⊗ a2 ∈ QH•(M1 ×M2).
An important class of superheavy sets is given by stable stems introduced
and illustrated in Section 1.1.
Theorem 1.6. Every stable stem is a superheavy subset with respect to any
non-zero idempotent a ∈ QH∗(M). In particular, it is strongly and stably
non-displaceable.
In the next section we present an example of stable stems coming from Hamil-
tonian torus actions.
1.4 Hamiltonian torus actions
Fibers of the moment maps of Hamiltonian torus actions form an interesting
playground for testing the various notions of displaceability and heaviness
introduced above. Throughout the paper we deal with effective actions only,
that is we assume that the map ϕ : Tk → Ham(M) defining the action
is a monomorphism. Furthermore, we assume that the moment map Φ =
(Φ1, . . . ,Φk) : M → R
k of the action is normalized: Φi is a normalized
Hamiltonian for all i = 1, . . . , k. By the Atiyah-Guillemin-Sternberg theorem
[6], [30], the image ∆ = Φ(M) of Φ is a k-dimensional convex polytope,
called the moment polytope. The subsets Φ−1(p), p ∈ ∆, are called fibers of
the moment map. A torus action is called compressible if the image of the
homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)), induced by the action ϕ, is a
finite group.
Theorem 1.7. Assume that (M,ω) is equipped with a compressible Hamilto-
nian Tk-action with moment map Φ and moment polytope ∆. Let Y ⊂ ∆ be
any closed convex subset which does not contain 0. Then the subset Φ−1(Y )
is stably displaceable. In particular, the fiber Φ−1(0) is a stable stem.
Note that for symplectic toric manifolds, that is when 2k = dimM , the point
0 is the barycenter of the moment polytope with respect to the Lebesgue
measure. This follows from our assumption on the normalization of the
moment map.
Theorems 1.6 and 1.7 imply that the fiber Φ−1(0) of a compressible torus
action is stably non-displaceable, and thus we get the complete description
of stably displaceable fibers for such actions.
In the case when the action is not compressible, the question of the com-
plete description of stably non-displaceable fibers remains open. We make a
partial progress in this direction by presenting at least one such fiber, called
the special fiber, explicitly in the case when (M,ω) is spherically monotone:
[ω]|HS2 (M)
= κ c1(TM)|HS2 (M)
, κ > 0 .
The special fiber can be described via the mixed action-Maslov homomor-
phism introduced in [49]: Let (M2n, ω) be a spherically monotone symplectic
manifold, and let {ft}, t ∈ [0, 1], be any loop of Hamiltonian diffeomorphisms,
with f0 = f1 = 1, generated by a 1-periodic normalized Hamiltonian func-
tion F (x, t). The orbits of any Hamiltonian loop are contractible due to the
standard Floer theory1. Pick any point x ∈ M and any disc u : D2 → M
spanning the orbit γ = {ftx}. Define the action
2 of the orbit by
AF (γ, u) :=
F (γ(t), t)dt−
u∗ω .
Trivialize the symplectic vector bundle u∗(TM) over D2 and denote by
mF (γ, u) the Maslov index of the loop of symplectic matrices corresponding
to {ft∗} with respect to the chosen trivialization. One readily checks that,
in view of the spherical monotonicity, the quantity
I(F ) := −AF (γ, u)−
mF (γ, u)
does not depend on the choice of the point x and the disc u, and is invariant
under homotopies of the Hamiltonian loop {ft}. In fact, I is a well defined
homomorphism from π1(Ham(M)) to R (see [49], [68]).
Assume again that ϕ : Tk → Ham(M,ω) is a Hamiltonian torus ac-
tion. Write ϕ♯ for the induced homomorphism of the fundamental groups.
Since the target space Rk of the moment map Φ is naturally identified with
Hom(π1(T
k),R), the pull back −ϕ∗♯I of the mixed action-Maslov homomor-
phism with the reversed sign can be considered as a point of Rk. We call
it a special point and denote by pspec. The preimage Φ
−1(pspec) is called the
special fiber of the moment map. In the case k = 1, when Φ is a real-valued
function on M , we will call pspec the special value of Φ.
1The Floer theory guarantees the existence of at least one contractible periodic orbit –
this is not obvious a priori if {ft} is not an autonomous flow. Since all the orbits of {ft}
are homotopic, all of them are contractible.
2Note that our action functional and the one in [49] are of opposite signs.
If k = n and M is a symplectic toric manifold, then pspec can be defined
in purely combinatorial terms involving only the polytope ∆. Namely, pick
a vertex x of ∆. Since ∆ in this case is a Delzant polytope [20], there is a
unique (up to a permutation) choice of vectors v1, . . . ,vn which
• originate at x;
• span the n rays containing the edges of ∆ adjacent to x;
• form a basis of Zn over Z.
Proposition 1.8.
pspec = x+ κ
vi. (6)
Proof. The vertices of the moment polytope are in one-to-one correspondence
with the fixed points of the action. Let x ∈ M be the fixed point corre-
sponding to the vertex x = (x1, . . . ,xn). Then the vectors vj = (v
j , . . . , v
j = 1, . . . , n, are simply the weights of the isotropy Tn-action on TxM . Since
the definition of the mixed action-Maslov invariant of a Hamiltonian circle
action does not depend on the choice of a 1-periodic orbit and a disc span-
ning it, let us compute all Ii, l = 1, . . . , n, using the constant periodic orbit
concentrated at the fixed point x and the constant disc u spanning it. Clearly,
AΦi(x, u) = Φi(x) = xi and mΦi(x, u) = 2
vij ∀i = 1, . . . , n,
which readily yields formula (6).
E.Shelukhin pointed out to us that by summing up equations (6) over all the
vertices x(1), . . . ,x(m) ∈ Rn of the moment polytope, one readily gets that
pspec =
Theorem 1.9. Assume M2n is a spherically monotone symplectic manifold
equipped with a Hamiltonian Tk-action. Then the special fiber of the moment
map is superheavy with respect to any (non-zero) idempotent a ∈ QH2n(M).
In particular, it is stably and strongly non-displaceable.
Let us mention that, in particular, the special fiber is non-empty and so
pspec ∈ ∆. Moreover pspec is an interior point of ∆ – otherwise Φ
−1(pspec) is
isotropic of dimension < n and hence displaceable (see e.g. [9]).
Remark 1.10. If dimM = 2dimTk (that is we deal with a symplectic toric
manifold), the special fiber, say L, is a Lagrangian torus. In fact, this torus
is monotone: for every D ∈ π2(M,L) we have
ω = κ ·mL(D) ,
where mL stands for the Maslov class of L. This is an immediate consequence
of the definitions.
Remark 1.11. Note that when M is spherically monotone and the action is
compressible Theorems 1.7 and 1.9 match each other: in this case pspec = 0
and therefore the special fiber is a stable stem by Theorem 1.7. It is unknown
whether this property persists for the special fibers of non-compressible ac-
tions.
Example 1.12. Let M be the monotone symplectic blow up of CP 2 at k
points (0 ≤ k ≤ 3) which is equivariant with respect to the standard T2-
action and which is performed away from the Clifford torus in CP 2. Since
the blow-up is equivariant, M comes equipped with a Hamiltonian T2-action
extending the T2-action on CP 2. The Clifford torus is a fiber of the moment
map of the T2-action on CP 2. Let L ⊂M be the Lagrangian torus which is
the proper transform of the Clifford torus under the blow-up – it is a fiber of
the moment map of the T2-action on M . Using Proposition 1.8 it is easy to
see that L is the special fiber of M . According to Theorem 1.9, it is stably
and strongly non-displaceable. In fact, it is a stem: the displaceability of
all the other fibers was checked for k = 0 in [10], for k = 1 in [23] and for
k = 2, 3 in [40].
We refer to Section 1.7.2 for further discussion of related problems and very
recent advances.
Digression: Calabi vs. action-Maslov. The method used to prove
Theorem 1.9 also allows to prove the following result involving the mixed
action-Maslov homomorphism. Denote by vol (M) the symplectic volume of
M . Consider the function µ : H̃am (M) → R defined by
µ(φH) := −vol (M) lim
c(a, φlH)/l.
In the case when a is the unity in a field that is a direct summand in the
decomposition of the K-algebra QH2n(M,ω), as an algebra, into a direct
sum of subalgebras, µ is a homogeneous quasi-morphism on H̃am (M) called
Calabi quasi-morphism [22],[24],[46]; in the general case it has weaker prop-
erties [23]. With this language the functional ζ (on normalized functions) is
induced (up to a constant factor) by the pull-back of µ to the Lie algebra of
H̃am (M).
Following P.Seidel we described in [22] the restriction of µ (in fact, for any
spherically monotoneM) on π1(Ham(M)) ⊂ H̃am (M) in terms of the Seidel
homomorphism π1(Ham(M)) → QH
∗ (M), where QH
∗ (M) denotes the
group of invertible elements in the ring QH∗(M). Here we give an alternative
description of µ|π1(Ham(M)) in terms of the mixed action-Maslov homomor-
phism I which, in turn, also provides certain information about the Seidel
homomorphism.
Theorem 1.13. Assume M is spherically monotone and let µ be defined as
above for some non-zero idempotent a ∈ QH∗(M). Then
µ|π1(Ham(M)) = vol (M) · I.
Note that, in particular, µ|π1(Ham(M)) does not depend on a used to de-
fine µ. The theorem also implies that µ descends to a quasi-morphism on
Ham(M) if and only if I : π1(Ham(M)) → R vanishes identically (since µ
descends to a quasi-morphism on Ham(M) if and only if µ|π1(Ham(M)) ≡ 0 –
see e.g. [22], Prop. 3.4). The proof of the theorem is given in Section 9.1.
Let us mention also that, interestingly enough, the homomorphism I
coincides with the restriction to π1(Ham(M)) of yet another quasi-morphism
on H̃am (M) constructed by P.Py (see [52, 53]).
Digression: Action-Maslov homomorphism and Futaki invari-
ant. This remark grew from an observation pointed out to us by Chris
Woodward – we are grateful to him for that. Assume that our symplectic
manifoldM is complex Kähler (i.e. the symplectic structure onM is induced
by the Kähler one) and Fano (by this we mean here that [ω] = c1). Assume
also that a Hamiltonian S1-action {ft} preserves the Kähler metric and the
complex structure. For instance, if M2n is a symplectic toric manifold it can
be equipped canonically with a complex structure and a Kähler metric invari-
ant under the Tn-action on M , hence under the action of any S1-subgroup
{ft} of T
Let V be the Hamiltonian vector field generating the Hamiltonian flow
{ft}. Since {ft} preserves the complex structure, one can associate to V its
Futaki invariant F(V ) ∈ C [29]. It has been checked by E.Shelukhin [63]
that, up to a universal constant factor, this Futaki invariant is equal to the
value of the mixed action-Maslov homomorphism on the loop {ft}:
F(V ) = const · I({ft}).
Note that if such an M admits a Kähler-Einstein metric then the Futaki
invariant has to vanish [29] – thus if I({ft}) 6= 0 the manifold does not admit
a Kähler-Einstein metric. Moreover, if M2n is toric the opposite is also true:
if the Futaki invariant vanishes for any V generating a subgroup of the torus
Tn acting onM thenM admits a Kähler-Einstein metric – this follows from a
theorem by Wang and Zhu [67], combined with a previous result of Mabuchi
[38]. In terms of the moment polytope, the vanishing of the Futaki invariant,
and accordingly the existence of a Kähler-Einstein metric, on a Kähler Fano
toric manifold means precisely that the special point of the polytope coincides
with the barycenter.
1.5 Super(heavy) monotone Lagrangian submanifolds
Let (M2n, ω) be a closed spherically monotone symplectic manifold with [ω] =
κ · c1(TM) on π2(M), κ > 0. Let L ⊂ M be a closed monotone Lagrangian
submanifold with the minimal Maslov number NL ≥ 2. As usually, we put
NL = +∞ if π2(M,L) = 0. As before, we work with the basic field F which
is either Z2 or C. In the case F = C, we assume that L is relatively spin, that
is L is orientable and the 2nd Stiefel-Whitney class of L is the restriction of
some integral cohomology class of M .
Disclaimer: In the case F = C the results of this section are conditional:
We take for granted that Proposition 8.1 below, which was proved by Biran
and Cornea [15] for homologies with Z2-coefficients, extends to homologies
with C-coefficients. In each of the specific examples below we will explicitly
state which F we are using and whenever we use F = C we assume that L
is relatively spin.
Denote by j the natural morphism j : H•(L;F) → H•(M ;F). We say that
L satisfies the Albers condition [2] if there exists an element S ∈ H•(L;F) so
that j(S) 6= 0 and
deg S > dimL+ 1−NL .
We shall refer to such S as to an Albers element of L.
Example 1.14. Assume [L] ∈ H•(L;F) and j([L]) ∈ H•(M ;F) is non-zero.
This means precisely that [L] is an Albers element of L.
A closed monotone Lagrangian submanifold L which satisfies this con-
dition (and whose minimal Maslov number is greater than 1) will be called
homologically non-trivial in M .
Theorem 1.15. Let L be a closed monotone Lagrangian submanifold satisfy-
ing the Albers condition. Then L is heavy with respect to [M ]. In particular,
any homologically non-trivial Lagrangian submanifold is heavy with respect
to [M ].
Example 1.16. Assume that π2(M,L) = 0. Then the homology class of a
point is an Albers element of L, and hence L is heavy. Note that in this
case heaviness cannot be improved to superheaviness: the meridian on the
two-torus is heavy but not superheavy. Here we took F = Z2.
Example 1.17 (Lagrangian spheres in Fermat hypersurfaces). More exam-
ples of heavy (but not necessarily superheavy) monotone Lagrangian sub-
manifolds can be constructed as follows3.
Let M ⊂ CP n+1 be a smooth complex hypersurface of degree d. The
pull-back of the standard symplectic structure from CP n+1 turns M into a
symplectic manifold (of real dimension 2n). If d ≥ 2, then, as it is explained,
for instance, in [12],M contains a Lagrangian sphere: M can be included into
a family of algebraic hypersurfaces of CP n+1 with quadratic degenerations at
isolated points and the vanishing cycle of such a degeneration can be realized
by a Lagrangian sphere following [5], [21], [60], [61], [62].
Let M ⊂ CP n+1 be a projective hypersurface of degree d, 2 ≤ d < n+ 2.
The minimal Chern number of M equals N := n+2− d > 0. Let Ln ⊂ M2n
be a simply connected Lagrangian submanifold (for instance, a Lagrangian
sphere).
First, consider the case when n is even, L is relatively spin and the Euler
characteristics of L does not vanish (this is the case for a sphere). Then the
3We thank P.Biran for his indispensable help with these examples.
homology class j([L]) ∈ Hn(M ;Z) is non-zero: its self-intersection number
in M up to the sign equals the Euler characteristic. Thus [L] is an Albers
element. (Here we use F = C). In view of Theorem 1.15, L is heavy with
respect to [M ].
Second, suppose that n is of arbitrary parity but n > 2d − 3, and no
restriction on the Euler characteristics of L is assumed anymore. This yields
NL = 2N > n+ 1 and thus L satisfies the Albers condition with the class of
a point P as an Albers element. Thus L is heavy with respect to [M ] – here
we use F = Z2.
Finally, fix n ≥ 3 and an even number d such that 4 ≤ d < n+2. Consider
a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
n+1 .
Its real part L := M ∩ RP n+1 lies in the affine chart z0 6= 0 and is given by
the equation
xd1 + . . .+ x
n+1 = 1,
where xj := Re(zj/z0) . Since d is even, L is an n-dimensional sphere. As
it was explained above, L is heavy with respect to [M ] if either n is even
(and F = C) or n > 2d − 3 (and F = Z2). However, in either case L is not
superheavy with respect to [M ]. Indeed, let Σd ≈ Zd be the group of complex
roots of unity. Given a vector α = (α1, . . . , αn) ∈ (Σd)
n+1, denote by fα the
symplectomorphism of M given by
fα(z0 : z1 : . . . : zn+1) = (z0 : α1z1 : . . . : αn+1zn+1) . (7)
If all αj ∈ C\R, then αjx /∈ R whenever x ∈ R\{0}, and thus fα(L)∩L = ∅.
Therefore L is strongly displaceable and the claim follows from the part (iii)
of Theorem 1.2.
The next result gives a user-friendly sufficient condition of superheaviness.
Theorem 1.18. Assume L is homologically non-trivial in M and assume
a ∈ QH2n(M) is a non-zero idempotent divisible by j([L]) in QH•(M), that
is a ∈ j([L]) ∗QH•(M). Then L is superheavy with respect to a.
The homological non-triviality of L in the hypothesis of the theorem means
just that [L] is an Albers element of L (see Example 1.14). In fact, the
theorem can be generalized to the cases when L has other Albers elements –
see Remark 8.3 (ii).
Example 1.19 (Lagrangian spheres in quadrics). Here we work with F = C.
Let M be the real part of the Fermat quadric M = {−z20 +
j=1 z
j = 0}.
Assume that n is even and L is a simply connected Lagrangian submanifold
with non-vanishing Euler characteristic (e.g. a Lagrangian sphere). Under
this assumption, [L] ∈ H•(L) and j([L]) 6= 0, since L has non-vanishing self-
intersection. Denote by p ∈ H∗(M ;F) the class of a point. The quantum
homology ring of M was described by Beauville in [8]. In particular, p ∗ p =
w−2[M ], where w = sκnqn. Thus
a± :=
[M ]± pw
are idempotents. One can show that j([L]) divides a− and hence L is a−-
superheavy. Since a− is invariant under the action of Symp(M), the manifold
L is strongly non-displaceable.
For simplicity, we present the calculation in the case n = 2 – the general
case is absolutely analogous. The 2-dimensional quadric is symplectomorphic
to (S2 × S2, ω ⊕ ω). Denote by A and B the classes of [S2] × [point] and
[point] × [S2] respectively. Since the symplectic form vanishes on j([L]) we
get that j([L]) = l(B − A) with l 6= 0. It is known that A ∗ B = p and
B ∗B = w−1[M ]. Thus j([L]) ∗ 1
wB = a−, that is j([L]) divides a−.
In particular, the Lagrangian anti-diagonal
∆ := {(x, y) ∈ S2 × S2 : x = −y} ,
which is diffeomorphic to the 2-sphere, is superheavy with respect to a−. It is
unknown whether ∆ is super-heavy with respect to a+. Further information
on superheavy Lagrangian submanifolds in the quadrics can be extracted
from [15].
Example 1.20 (A non-heavy monotone Lagrangian torus in S2 × S2). Con-
sider the quadric M = S2 × S2 from the previous example. We will think of
S2 as of the unit sphere in R3 whose symplectic form is the area form divided
by 4π. We will work again with F = C. Interestingly enough, such an M
contains a monotone Lagrangian torus that is not heavy with respect to a−.
Namely, consider a submanifold K given by equations4
K = {(x, y) ∈ S2 × S2 : x1y1 + x2y2 + x3y3 = −
, x3 + y3 = 0} .
4We thank Frol Zapolsky for his help with calculations in this example.
One readily checks that K is a monotone Lagrangian torus with NK = 2
which represents a zero element inH2(M ;F) (both with F = C and F = Z2).
Thus H•(K;F) does not contain any Albers element. Furthermore, K is
disjoint from the Lagrangian anti-diagonal ∆ and hence is not heavy with
respect to a− since, as it was shown above, ∆ is superheavy with respect to
a−. In particular, K is an exotic monotone torus: it is not symplectomorphic
to the Clifford torus which is a stem and hence a−-superheavy. A further
study of exotic tori in products of spheres is currently being carried out by
Y.Chekanov and F.Schlenk.
It is an interesting problem to understand whether K is superheavy with
respect to a+, or at least non-displaceable. Identify M \ {the diagonal} with
the unit co-ball bundle of the 2-sphere. After such an identification ∆ corre-
sponds to the zero section, while K corresponds to a monotone Lagrangian
torus, say K ′. Interestingly enough, the Lagrangian Floer homology of K ′
in T ∗S2 (with F = Z2) does not vanish as was shown by Albers and Frauen-
felder in [3], and thus K is not displaceable in M \ {the diagonal}. Thus
the question on (non)-displaceability of K is related to understanding of the
effect of the compactification of the unit co-ball bundle to S2 × S2.
The proofs of theorems above are based on spectral estimates due to
Albers [2] and Biran-Cornea [15]. Furthermore, the results above admit
various generalizations in the framework of Biran-Cornea theory of quantum
invariants for monotone Lagrangian submanifolds, see [15] and the discussion
in Section 8 below.
1.6 An effect of semi-simplicity
Recall that a commutative (finite-dimensional) algebra Q over a field A is
called semi-simple if it splits into a direct sum of fields as follows: Q =
Q1 ⊕ . . .⊕Qd , where
• each Qi ⊂ Q is a finite-dimensional linear subspace over A;
• each Qi is a field with respect to the induced ring structure;
• the multiplication in Q respects the splitting:
(a1, . . . , ad) · (b1, . . . , bd) = (a1b1, . . . , adbd).
A classical theorem of Wedderburn (see e.g. [66], §96) implies that the semi-
simplicity is equivalent to the absence of nilpotents in the algebra.
Remark 1.21. Assume that the K-algebra QH2n(M,ω) splits, as an algebra,
into a direct sum of two algebras, at least one of which is a field, and let e
be the unity in that field. In particular, this is the case when QH2n(M,ω) =
Q1⊕ . . .⊕Qd is semi-simple and e is the unity in one of the fields Qi. A slight
generalization of the argument in [23, 46] (see [24], the remark on pp. 56-57)
shows that the partial quasi-state ζ(e, ·) associated to e is R-homogeneous
(and not just R+-homogeneous as in the general case).
This immediately yields that every set which is heavy with respect to e is
automatically superheavy with respect to e.
In fact, in this situation ζ is a genuine symplectic quasi-state in the sense of
[23] and, in particular, a topological quasi-state in the sense of Aarnes [1] (see
[23] for details). In [1] Aarnes proved an analogue of the Riesz representation
theorem for topological quasi-states which generalizes the correspondence
between genuine states (that is positive linear functionals on C(M)) and
measures. The object τζ corresponding to a quasi-state ζ is called a quasi-
measure (or a topological measure). With this language in place, the sets that
are (super)heavy with respect to ζ are nothing else but the closed sets of the
full quasi-measure τζ . Any two such sets have to intersect for the following
basic reason: any quasi-measure is finitely additive on disjoint closed subsets
and therefore if two closed subsets of M of the full quasi-measure do not
intersect, the quasi-measure of their union must be greater than the total
quasi-measure of M , which is impossible.
Example 1.22. In this example we again assume that F = Z2. Let M =
CP n be equipped with the Fubini-Study symplectic structure ω, normalized
so that [ω] = c1, and let A ∈ H2n−2(M) be the homology class of the hyper-
plane. One readily verifies the following K-algebra isomorphism
QH2n(M) ∼= K[X ]/〈X
n+1 − u−1〉,
where
K = Z2[[u] = {zku
k + zk−1u
k−1 + . . . , zi ∈ Z2 ∀i}
is the field of Laurent-type series in u := sn+1 with coefficients in Z2 and
X = qA. Since no root of degree 2 or more of u−1 is contained in K, the
polynomial P is irreducible over K for any n (see e.g. [34], Theorem 9.1) and
therefore QH2n(M) is a field. Hence the collections of heavy and superheavy
sets with respect to the fundamental class coincide.
We claim that L := RP n ⊂ CP n is superheavy. The case n = 1 cor-
responds to the equator of the sphere, which is known to be a stable stem.
For n ≥ 2, note that NL = n + 1 and S = [RP
2] is an Albers element of L.
Therefore, L is [M ]-heavy by Theorem 1.15, and hence superheavy.
The next result follows directly from Theorem 1.3 (iii) and Remark 1.21:
Theorem 1.23. Assume that QH2n(M) is semi-simple and splits into a
direct sum of d fields whose unities will be denoted by e1, . . . , ed. Assume that
a closed subset X ⊂M is heavy with respect to a non-zero idempotent a – as
one can easily see, such an idempotent has to be of the form a = ej1+ . . .+ejl
for some 1 ≤ j1 < . . . < jl ≤ d. Then X is superheavy with respect to some
eji, 1 ≤ i ≤ l.
The theorem yields the following geometric characterization of non-semi-
simplicity of QH2n(M). Namely, define the symplectic Torelli group as the
group of all symplectomorphisms of M which induce the identity map on
H•(M ;F). For instance, this group contains Symp0(M). Note that any ele-
ment of the symplectic Torelli group acts trivially on the quantum homology
of M and hence maps sets (super)heavy with respect to an idempotent a to
sets (super)heavy with respect to a.
Now Theorem 1.23 readily implies the following
Corollary 1.24. Assume that (M,ω) contains a closed subset X which is
heavy with respect to a non-zero idempotent and displaceable by a symplec-
tomorphism from the symplectic Torelli group. Then QH2n(M) is not semi-
simple.
The simplest examples are provided by sets of the form X×{a meridian}
in M × T2 with a heavy X .
Another result in the same vein is as follows5. Given a set Y of positive
integers, put βY (M) =
i∈Y βi(M), where βi(M) stands for the i-th Betti
number of M over F .
5In the case F = C, Theorem 1.25 is conditional, see the disclaimer in the previous
section.
Theorem 1.25. Assume that either of the following (not mutually excluding)
conditions holds:
(a) M contains m > βY (M) + 1 pair-wise disjoint closed monotone La-
grangian submanifolds whose minimal Maslov numbers are greater than n+1
and belong to a set Y of positive integers.
(b) M contains pair-wise disjoint homologically non-trivial Lagrangian sub-
manifolds6 whose fundamental classes, viewed as (non-zero) elements of
H•(M ;F), are linearly dependent over F .
(In the case F = C assume that all the Lagrangian submanifolds above are
also relatively spin.)
Then QH2n(M) is not semi-simple.
The proof is given in Section 8.
Example 1.26. For instance, if all the Lagrangian submanifolds from part
(a) of the theorem are simply connected, their minimal Maslov numbers are
equal to 2N , so that the set Y consists of one element: Y = {2N}. Thus
if 2N > n + 1 and QH2n(M) is semi-simple, M cannot contain more than
β2N(M)+ 1 pair-wise disjoint simply-connected Lagrangians (provided all of
them are relatively spin if we work with F = C).
Example 1.27. Set F = C. Fix n ≥ 11 and an even number d such that
6 ≤ d < (n + 3)/2. Consider a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
As we already saw in Example 1.17, the manifold L := M ∩ RP n+1 is an
n-dimensional Lagrangian sphere. Consider the images fα(L), where sym-
plectomorphisms fα are defined by (7). Note that, as long as αj/βj 6= ±1
for all j, the Lagrangian spheres fα(L) and fβ(L) are disjoint. Using this
observation, it is easy to find d/2 disjoint Lagrangian spheres in M .
The minimal Chern number N ofM equals n+2−d, and so 2N lies in the
interval [n+2, 2n−4]. In this case β2N(M) = 1 (see e.g. [31]). Since d/2 > 2,
we conclude from the previous example that QH2n(M) is not semi-simple.
This conclusion agrees with the computation of QH∗(M) by Beauville [8].
6See Example 1.14 for the definition. As in that example we again assume that all our
Lagrangian submanifolds are closed, monotone and have minimal Maslov number greater
than 1.
It would be interesting to find examples of symplectic manifolds where the
quantum homology is not known a priori and where the above theorems are
applicable. Let us mention that different obstructions to the semi-simplicity
of QH•(M) coming from Lagrangian submanifolds were recently found by
Biran and Cornea [14].
1.7 Discussion and open questions
1.7.1 Strong displaceability beyond Floer theory?
Clearly, displaceability implies stable displaceability. The converse is not
true, as the next example shows:
Example 1.28. Consider the complex projective space CP n equipped with
the Fubini-Study symplectic form (in our normalization the area of a line
equals 1). Identify CP n with the symplectic cut of the Euclidean ball B(1) ⊂
Cn (that is the boundary of B(1) is collapsed to CP n−1 along the fibers of
the Hopf fibration, see [36]), where B(r) := {π|z|2 ≤ r}. Then B(r) ⊂ CP n
(i) displaceable for r < 1/2;
(ii) strongly non-displaceable but stably displaceable for r ∈ [1/2, n/n+1);
(iii) strongly and stably non-displaceable for r ≥ n/n+ 1.
It is instructive to analyze the techniques involved in the proofs: The strong
non-displaceability result in (ii) is an immediate consequence of Gromov’s
packing-by-two-balls theorem, which is proved via the J-holomorphic variant
of the theorem which states that there exists a J-holomorphic line in CP n
passing through any two points. In the case (iii) the ball B(r) contains the
Clifford torus, which is stably non-displaceable. This follows either from the
fact that the Clifford torus is a stem (see [10]), or from non-vanishing of its
Lagrangian Floer homology [16].
The displaceability of B(r) in (i) follows from the explicit construction
of the two balls packing (see [33]). The stable displaceability in (ii) is a
direct consequence of Theorem 1.7 above: Indeed, consider the standard Tn-
action on CP n. The normalized moment polytope ∆ ⊂ Rn has the form
∆ = ∆stand + w where ∆stand is the standard simplex {ρi ≥ 0,
ρi ≤ 1} in
Rn, where (ρ1, . . . , ρn) denote coordinates in R
n, and w = − 1
(1, . . . , 1).
Note that the ball B(r) equals to Φ−1(∆r) where ∆r := r ·∆stand + w. Note
that ∆r does not contain the origin exactly when r ≤
which yields the
stable displaceability in (ii) above.
A mysterious feature of Example 1.28 is as follows. On the one hand, we
believe in the following general empiric principle: whenever one can establish
the non-displaceability of a subset by means of the Floer homology theory,
one gets for free the stable non-displaceability. On the other hand, we be-
lieve, following a philosophical explanation provided by Biran, that Gromov’s
packing-by-two-balls theorem may be extracted from some “operations” in
Floer homology. Example 1.28 shows that at least one of these beliefs is
wrong. It would be interesting to clarify this issue.
1.7.2 Heavy fibers of Poisson-commutative subspaces
It was shown in [23] that for any finite-dimensional Poisson-commutative
subspace A ⊂ C∞(M) at least one of the fibers of its moment map Φ has to
be non-displaceable.
Question. Is it true that at least one fiber of Φ has to be heavy (with respect
to some non-zero idempotent a ∈ QH∗(M))?
It is easy to construct an example of A whose moment map Φ has no
superheavy fibers: take T2 with the coordinates p, q mod 1 on it and take A
to be the set of all smooth functions depending only on p – the corresponding
Φ defines the fibration of T2 by meridians none of which is superheavy.
Here is another question which concerns fibers of symplectic toric man-
ifolds, i.e. fibers of a moment map Φ of an effective Hamiltonian Tn-action
on (M2n, ω). Assume M is (spherically) monotone. Theorem 1.9 shows that
in such a case the special fiber ofM is superheavy, hence stably and strongly
non-displaceable. In all the examples where it has been checked this turns
out to be the only non-displaceable fiber of M .
Question. Is the special fiber for a monotone symplectic toric M always
a stem? In particular, is it the only non-displaceable fiber of the moment
In the monotone case the special fiber is clearly the only heavy fiber of
the moment map, because it is superheavy and any other heavy fiber would
have had to intersect it. On the other hand, if we consider a Hamiltonian Tk-
action on M2n with k < n there can be more than one non-displaceable fiber
of the moment map – for instance, because of purely topological obstructions:
the simplest Hamiltonian T1-action on CP 2 provides such an example. In
the case of monotone symplectic toric manifolds of dimension bigger than 4
the question above is absolutely open.
After the first draft of this paper appeared, a remarkable progress in this
direction has been achieved in the works by Cho [17] and Fukaya, Oh, Ohta
and Ono [28]: In particular, it turns out that a non-monotone symplectic
toric manifold can have more than one non-displaceable fiber – this happens
already for certain equivariant blowups of CP 2.
Organization of the paper:
In Section 2 we prove Theorem 1.7 which in particular states that the
special fiber of a compressible torus action is a stable stem.
In Section 3 we sum up various preliminaries from Floer theory including
basic properties of spectral invariants and partial symplectic quasi-states. In
addition we spell out a useful property of the Conley-Zehnder index: it is a
quasi-morphism on the universal cover of the symplectic group (see Propo-
sition 3.5). For completeness we extract a proof of this property from [54];
alternatively, one can use the results of [19].
In Section 4 we prove parts (i) and (iii) of Theorem 1.2 and Theorem 1.3
on basic properties of (super)heavy sets.
In Section 5 we prove Theorem 1.5 on products of (super)heavy sets. Our
approach is based on a quite general product formula for spectral invariants
(Theorem 5.1), which is proved by a fairly lengthy algebraic argument.
In Section 6 we prove Theorem 1.2 (ii) on stable non-displaceability of
heavy subsets. The argument involves a “baby version” of the above-men-
tioned product formula.
In Section 7 we prove superheaviness of stable stems.
In Section 8 we bring together the proofs of various results related to
(super)heaviness of monotone Lagrangian submanifolds satisfying the Albers
condition, including Theorems 1.15, 1.18, 1.25 and Proposition 1.4.
In Section 9 we prove Theorem 1.9 on superheaviness of special fibers of
Hamiltonian torus actions on monotone symplectic manifolds. The proof is
quite involved. In fact, two tricks enabled us to shorten our original argu-
ment: First, we use the Fourier transform on the space of rapidly decaying
functions on the Lie coalgebra of the torus in order to reduce the problem
to the case of Hamiltonian circle actions. Second, we systematically use the
quasi-morphism property of the Conley-Zehnder index for asymptotic calcu-
lations with Hamiltonian spectral invariants. Finally, in Section 9.1 we prove
Theorem 1.13.
Figure 1 sums up the hierarchy of the non-displaceability properties dis-
cussed above.
������������������������
MHierarchy of non−displaceability properties of a closed subset of
Heavy
aidempotent
Superheavy
idempotent a
wrt a non−zero wrt a non−zero
(3) (4) (5) (6)
action on a spherically
torus action on a (not
a compressible Hamiltonian
necessarily monotone)
Special fiber of
monotone M
Always true.
True under certain conditions (see below)
? Question (under certain conditions − see below)
Monotone
Lagrangian
submanifold L
(14) (15)
(17) (18)
(21) (22)
(16b)
Product of codimension−1
skeletons of fine
triangulations
Strongly
non−displaceable
a Hamiltonian torus
Non−displaceable
a symplectic isotopy
Non−displaceable by
wrt [M]
Heavy
(16a)
Superheavy
wrt [M]
Stable stem
Stably
non−displaceable
Zero fiber of
Figure 1: Hierarchy of non-displaceability properties
(1),(2),(6),(19) - Trivial.
(3) True if a is invariant under the action of the whole group Symp (M) –
Theorem 1.2, part (iii).
(4), (9) Theorem 1.2, part (iii).
(5) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(7a) True if the algebra QH2n(M) splits, as an algebra, into a direct sum of
two algebras, at least one of which is a field, and a is the unity element in
that field – see Remark 1.21.
(7b), (16b) Theorem 1.2, part (i).
(8) Theorem 1.2, part (ii).
(10) Theorem 1.18 (see the assumptions on L there).
(11) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(12) Theorem 1.3, part (i).
(13) Theorem 1.3, part (ii).
(14) Theorem 1.18 (see the assumptions on L there) with a = [M ] – i.e. j(L)
is invertible in QH•(M).
(15) L satisfies the Albers condition – see Theorem 1.15.
(16a) True if QH2n(M) is a field – see Remark 1.21.
(17) Theorem 1.6.
(18) Theorem 1.9.
(20) Theorem 1.7.
(21) Is the special fiber for a monotone symplectic toric M always a stem?
See Section 1.7.2.
(22) True if M is spherically monotone and the torus action is compressible
– see Remark 1.11.
(23) See [23].
2 Detecting stable displaceability
For detecting stable displaceability of a subset of a symplectic manifold we
shall use the following result (cf. [48, Chapter 6]).
Theorem 2.1. Let X be a closed subset of a closed symplectic manifold
(M,ω). Assume that there exists a contractible loop of Hamiltonian diffeo-
morphisms of (M,ω) generated by a normalized time-periodic Hamiltonian
Ht(x) so that Ht(x) 6= 0 for all t ∈ [0, 1] and x ∈ X. Then X is stably
displaceable.
Proof. Denote by ht the Hamiltonian loop generated by H . Let h
t be its
homotopy to the constant loop: h
t = ht and h
t = 1. Write H
(s)(x, t) for
the corresponding normalized Hamiltonians. Consider the family of diffeo-
morphisms Ψs of M × T
∗S1 given by
Ψs(x, r, θ) = (h
θ x, r −H
(s)(h
θ x, θ), θ) .
One readily checks that Ψs, s ∈ [0, 1], is a Hamiltonian isotopy (not com-
pactly supported). We claim that Ψ1 displaces Y := X × {r = 0}. Indeed,
if Ψ1(x, 0, θ) ∈ Y we have hθx ∈ X and Hθ(hθx) = 0 which contradicts the
assumption of the theorem. This completes the proof.
Proof of Theorem 1.7: Choose a linear functional F : Rk → R with
rational coefficients which is strictly positive on Y . Then for some suffi-
ciently large positive integer N the Hamiltonian H := NΦ∗F generates a
contractible Hamiltonian circle action on M and H is strictly positive on
X := Φ−1(Y ). Thus X is stably displaceable in view of the previous theo-
3 Preliminaries on Hamiltonian Floer theory
3.1 Valuation on QH∗(M)
Define a function ν : K → Γ by
θ) = max{ θ | zθ 6= 0} .
The convention is that ν(0) = −∞. In algebraic terms, exp ν is a non-
Archimedean absolute value on K.
The function ν admits a natural extension to Λ and then to QH∗(M) –
abusing the notation we will denote all of them by ν. Namely, any element
of λ ∈ Λ can be uniquely represented as λ =
θ uθs
θ, where each uθ belongs
to F [q, q−1], and any non-zero a ∈ QH∗(M) can be uniquely represented as
i λibi, 0 6= λi ∈ Λ, 0 6= bi ∈ H∗(M ;F). Define
ν(λ) := max
θ | uθ 6= 0
ν(a) := max
ν(λi).
3.2 Hamiltonian Floer theory
We briefly recall the notation and conventions for the setup of the Hamilto-
nian Floer theory that will be used in the proofs.
Let L be the space of all smooth contractible loops γ : S1 = R/Z → M .
We will view such a γ as a 1-periodic map γ : R → M . Let D2 be the
standard unit disk in R2. Consider a covering L̃ of L whose elements are
equivalence classes of pairs (γ, u), where γ ∈ L, u : D2 → M , u|∂D2 = γ (i.e.
u(e2π
−1t) = γ(t)), is a (piecewise smooth) disk spanning γ in M and the
equivalence relation is defined as follows: (γ1, u1) ∼ (γ2, u2) if and only if γ1 =
γ2 and the 2-sphere u1#(−u2) vanishes in H
2 (M). The equivalence class of
a pair (γ, u) will be denoted by [γ, u]. The group of deck transformations of
the covering L̃ → L can be naturally identified with HS2 (M). An element
A ∈ HS2 (M) acts by the transformation
A([γ, u]) = [γ, u#(−A)]. (8)
Let F :M× [0, 1] → R be a Hamiltonian function (which is time-periodic
as we always assume). Set Ft := F (·, t). We will denote by ft the Hamiltonian
flow generated by F , meaning the flow of the time-dependent Hamiltonian
vector field Xt defined by the formula
ω(·, Xt) = dFt(·) ∀t.
(Note our sign convention!)
Let PF ⊂ L be the set of all contractible 1-periodic orbits of the Hamilto-
nian flow generated by F , i.e. the set of all γ ∈ L such that γ(t) = ft(γ(0)).
Denote by P̃F the full lift of PF to L̃.
Denote by Fix (F ) the set of those fixed points of f that are endpoints of
contractible periodic orbits of the flow:
Fix (F ) := {x ∈M | ∃γ ∈ PF , x = γ(0)}.
We say that F is regular if for any x ∈ Fix (F ) the map dxf : TxM → TxM
does not have eigenvalue 1.
Recall that the action functional is defined on L̃ by the formula:
AF ([γ, u]) =
F (γ(t), t)dt−
Note that
AF (Ay) = AF (y) + ω(A) (9)
for all y ∈ L̃ and A ∈ HS2 (M).
For a regular Hamiltonian F define a vector space C(F ) over F as the
set of all formal sums
λiyi, λi ∈ Λ, yi ∈ P̃F ,
modulo the relations
Ay = s−ω(A)q−c1(A)y,
for all y ∈ P̃F , A ∈ H
2 (M). The grading on Λ together with the Conley-
Zehnder index on elements of P̃F (see Section 3.3) defines a Z-grading on
C(F ). We will denote the i-th graded component by Ci(F ).
Given a loop {Jt}, t ∈ S
1, of ω-compatible almost complex structures,
define a Riemannian metric on L by
(ξ1, ξ2) =
ω(ξ1(t), Jtξ2(t))dt,
where ξ1, ξ2 ∈ TγL. Lift this metric to L̃ and consider the negative gradient
flow of the action functional AF . For a generic choice of the Hamiltonian
F and the loop {Jt} (such a pair (F, J) is called regular) the count of iso-
lated gradient trajectories connecting critical points of AF gives rise in the
standard way [26], [32], [58] to a Morse-type differential
d : C(F ) → C(F ), d2 = 0. (10)
The differential d is Λ-linear and has the graded degree −1. It strictly de-
creases the action. The homology, defined by d, is called the Floer homology
and will be denoted by HF∗(F, J). It is a Λ-module. Different choices of a
regular pair (F, J) lead to natural isomorphisms between the Floer homology
groups.
The following proposition summarizes a few basic algebraic properties of
Floer complexes and Floer homology that will be important for us further.
The proof is straightforward and we omit it.
Proposition 3.1.
1) Each Ci(F ) and each HFi(F, J), i ∈ Z, is a finite-dimensional vector
space over K.
2) Multiplication by q defines isomorphisms Ci(F ) → Ci+2(F ) and
HFi(F, J) → HFi+2(F, J) of K-vector spaces.
3) For each i ∈ Z there exists a basis of Ci(F ) over K consisting of the
elements of the form ql[γ, u], with [γ, u] ∈ P̃F .
4) A finite collection of elements of the form ql[γ, u], [γ, u] ∈ P̃F , lying
in C0(F ) ∪C1(F ) is a basis of the vector space C0(F )⊕C1(F ) over the field
K if and only if it is a basis of the module C(F ) over the ring Λ.
3.3 Conley-Zehnder and Maslov indices
In this section we briefly outline the definition and recall the relevant proper-
ties of the Conley-Zehnder index referring to [54, 58, 57] for details. In par-
ticular, we show that the Conley-Zehnder index is a quasi-morphism on the
universal cover S̃p (2k) of the symplectic group Sp(2k) (see Proposition 3.5
below), a fact which will be useful for asymptotic calculations with Floer
homology in the next sections. There are several routes leading to this fact,
which is quite natural since all homogeneous quasi-morphisms on S̃p (2k) are
proportional, and hence the same quasi-morphism admits quite dissimilar
definitions [7]. We extract the quasi-morphism property from the paper of
Robbin and Salamon [54] by bringing together several statements contained
therein7.
The Conley-Zehnder index assigns to each [γ, u] ∈ P̃F a number. Orig-
inally the Conley-Zehnder index was defined only for regular Hamiltonians
[18] – in this case it is integer-valued and gives rise to a grading of the ho-
mology groups in Floer theory. Later the definition was extended in different
ways by different authors to arbitrary Hamiltonians. We will use such an ex-
tension introduced in [54] (also see [57, 58]). In this case the Conley-Zehnder
index may take also half-integer values.
Let k be a natural number. Consider the symplectic vector space R2k
with a symplectic form ω2k on it. Denote by p = (p1, . . . , pk), q = (q1, . . . , qk)
the corresponding Darboux coordinates on the vector space R2k.
7We thank V.L. Ginzburg for stimulating discussions on the material of this section.
Robbin-Salamon index of Lagrangian paths: Let V ⊂ R2k be a
Lagrangian subspace. Consider the Grassmannian Lagr (k) of all Lagrangian
subspaces in R2k and consider the hypersurface ΣV ⊂ Lagr (k) formed by all
the Lagrangian subspaces that are not transversal to V . To such a V and
to any smooth path {Lt}, 0 ≤ t ≤ 1, in Lagr (k) Robbin and Salamon [54]
associate an index, which may take integer or half-integer values and which
we will denote by RS({Lt}, V ). The definition of the index can be outlined
as follows.
A number t ∈ [0, 1] is called a crossing if Lt ∈ ΣV . To each crossing t one
associates a certain quadratic form Qt on the space L(t) ∩ V – see [54] for
the precise definition. The crossing t is called regular if the quadratic form
Qt is non-degenerate. The index of such a regular crossing t is defined as the
signature of Qt if 0 < t < 1 and as half of the signature of Qt if t = 0, 1.
One can show that regular crossings are isolated. For a path {Lt} with only
regular crossings the index RS({Lt}, V ) is defined as the sum of the indices
of its crossings. An arbitrary path can be perturbed, keeping the endpoints
fixed, into a path with only regular crossings and the index of the perturbed
path does not depend on the perturbation – in fact, it depends only on the
fixed endpoints homotopy class of the path. Moreover, it is additive with
respect to the concatenation of paths and satisfies the naturality property:
RS({ALt}, AV ) = RS({Lt}, V ) for any symplectic matrix A.
Indices of paths in Sp (2k): Consider the group Sp (2k) of symplectic
2k × 2k-matrices. Denote by S̃p (2k) its universal cover. One can use the
index RS in order to define two indices on the space of smooth paths in
Sp (2k).
The first index, denoted by Ind2k, is defined as follows. Fix a Lagrangian
subspace V ⊂ R2k. For each smooth path {At}, 0 ≤ t ≤ 1, in Sp (2k) define
Ind2k ({At}, V ) as
Ind2k ({At}, V ) := RS({AtV }, V ).
The naturality of the RS index implies that
RS({BAtB
−1(BV )}, BV ) = RS({BAtV )}, BV ) =
= RS({AtV )}, V ) for any B ∈ Sp (2k)
and thus we get the following naturality condition for Ind2k:
Ind2k ({BAtB
−1}, BV ) = Ind2k ({At}, V ) for any B ∈ Sp (2k). (11)
The second index, which we will call the Conley-Zehnder index of a matrix
path and which will be denoted by CZmatr, is defined as follows. For each
A ∈ Sp (2k) denote by GrA the graph of A which is a Lagrangian subspace
of the symplectic vector space R4k = R2k×R2k equipped with the symplectic
structure ω4k = −ω2k ⊕ ω2k. Denote by ∆ the diagonal in R
4k = R2k × R2k
– it is a Lagrangian subspace with respect to ω4k. Now for any smooth path
{At}, 0 ≤ t ≤ 1, in Sp (2k) define CZmatr as
CZmatr({At}) := RS({GrAt},∆).
Equivalently, one can define CZmatr({At}) similarly to the index RS by look-
ing at the intersections of {A(t)} with the hypersurface Σ ⊂ Sp (2k) formed
by all the symplectic 2k× 2k-matrices with eigenvalue 1 and translating the
notions of a regular crossing and the corresponding quadratic form to this
setup.
Both indices Ind2k ({At}, V ) and CZmatr({At}) depend only on the fixed
endpoints homotopy class of the path {At} and are additive with respect to
the concatenation of paths in Sp (2k). The relation between the two indices
is as follows. Denote by I2k the 2k × 2k identity matrix. Given a smooth
path {At}, 0 ≤ t ≤ 1, in Sp (2k), set Ât := I2k ⊕ At ∈ Sp (4k). Then
CZmatr({At}) = Ind4k({Ât},∆). (12)
Remark 3.2. Note that near each W ∈ ΣV there exists a local coordinate
chart (on Lagr (k)) in which ΣV can be defined by an algebraic equation of
degree bounded from above by a constant C depending only on k and W .
Moreover, since for any two V, V ′ ∈ Lagr (k) there exists a diffeomorphism of
Lagr (k) mapping ΣV into ΣV ′ we can assume that C = C(k) is independent
of W and depends only on k. Therefore for any V , for any point W ∈ ΣV
and for any sufficiently small open neighborhood UW of W in Lagr (k) the
number of connected components of UW \(UW ∩ΣV ) is bounded by a constant
depending only on k.
Using these observations and the fact that regular crossings are isolated it
is easy to show that there exists a constant C(k), depending only on k, such
that for any Lagrangian subspace V ⊂ R2k and any path {At} ⊂ Sp (2k),
0 ≤ t ≤ 1, there exists a δ > 0 such that for any smooth path {A′t} ⊂ Sp (2k),
0 ≤ t ≤ 1, which is δ-close to {At} in the C
0-metric, one has
|Ind2k({At}, V )− Ind2k({A
t}, V | < C(k),
|CZmatr({At})− CZmatr({A
t}| < C(k).
Leray theorem on the index Ind2k: The following result follows from
Theorem 5.1 in [54] which Robbin and Salamon credit to Leray [35], p.52.
Denote by L the Lagrangian (q1, . . . , qk)-coordinate plane in R
2k. Any sym-
plectic matrix S ∈ Sp (2k) can be decomposed into k × k blocks as
where the blocks satisfy, in particular, the condition that
EF T − FET = 0. (13)
If SL ∩ L = 0 then the k × k-matrix F is invertible and multiplying (13)
by F−1 on the left and (F T )−1 = (F−1)T on the right, we get that F−1E −
ET (F−1)T = 0. Therefore the matrix QS := F
−1E is symmetric.
Theorem 3.3 ([54], Theorem 5.1; [35], p.52). Assume {At}, {Bt}, 0 ≤ t ≤ 1,
are two smooth paths in Sp (2k), such that A0 = B0 = I2k and A1L ∩ L = 0,
B1L ∩ L = 0, A1B1L ∩ L = 0. Then
Ind2k({AtBt}, L) = Ind2k({At}, L) + Ind2k({Bt}, L) +
sign (QA1 +QB1),
where sign (QA1 +QB1) is the signature of the quadratic form defined by the
symmetric k × k-matrix QA1 +QB1.
Corollary 3.4. Let V be any Lagrangian subspace of R2k. Then there exists
a positive constant C, depending only on k, such that for any smooth paths
{Xt}, {Yt}, 0 ≤ t ≤ 1, in Sp (2k), such that X0 = Y0 = I2k (there are no
assumptions on X1, Y1!),
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C.
Proof. We will write C1, C2, . . . for (possibly different) positive constants de-
pending only on k.
Pick a map Ψ ∈ Sp (2k) such that ΨV = L. Denote At = ΨXtΨ
Bt = ΨYtΨ
−1. Note that the paths {At}, {Bt} are based at the identity.
Using the naturality property (11) of Ind2k we get
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({ΨXtYtΨ
−1},ΨV )− Ind2k({ΨXtΨ
−1},ΨV )−
−Ind2k({ΨYtΨ
−1},ΨV )| =
= |Ind2k({(ΨXtΨ
−1)(ΨYtΨ
−1)}, L)− Ind2k({ΨXtΨ
−1}, L)−
−Ind2k({ΨYtΨ
−1}, L)| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|.
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|. (14)
Further on, Remark 3.2 implies that we can find sufficiently C0-close identity-
based perturbations {A′t}, {B
t} of {At}, {Bt} such that
A′1L ∩ L = 0, B
1L ∩ L = 0, A
1L ∩ L = 0. (15)
|Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|−
−|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C1, (16)
for some C1. On the other hand, since the three identity-based paths {A
{B′t}, {A
t}, satisfy the conditions (15), we can apply to them Theorem 3.3.
Hence there exists C2 such that
|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C2.
Combining it with (14) and (16) we get that there exists C3 such that
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C3,
which finishes the proof.
Conley-Zehnder index as a quasi-morphism: Recall that 2n = dimM .
Restricting CZmatr to the identity-based paths in Sp (2n) one gets a function
on S̃p (2n) that will be still denoted by CZmatr.
Proposition 3.5 (cf. [19]). The function CZmatr : S̃p (2n) → R is a quasi-
morphism. It means that there exists a constant C > 0 such that
|CZmatr(ab)− CZmatr(a)− CZmatr(b)| ≤ C ∀a, b ∈ S̃p (2n).
Proof. Represent a and b by identity-based paths {At}, {Bt}, 0 ≤ t ≤ 1, in
Sp (2n). Then use (12) and apply Corollary 3.4 for k = 2n, V = ∆ to {Ât},
{B̂t} in Sp (4n).
Maslov index of symplectic loops: The Conley-Zehnder index for
identity-based loops in Sp (2n) is called the Maslov index of a loop. Its
original definition, going back to [4], is the following: it is the intersection
number of an identity-based loop with the stratified hypersurface Σ whose
principal stratum is equipped with a certain co-orientation. Note that we do
not divide the intersection number by 2 and thus in our case the Maslov index
takes only even values; for instance, the Maslov index of a counterclockwise
2π-twist of the standard symplectic R2 is 2. We denote the Maslov index of
a loop {B(t)} by Maslov ({B(t)}).
Conley-Zehnder and Maslov indices of periodic orbits: The Con-
ley-Zehnder index for periodic orbits is defined by means of the Conley-
Zehnder index for matrix paths as follows. Given [γ, u] ∈ P̃F , build an
identity-based path {A(t)} in Sp (2n) as follows: take a symplectic trivial-
ization of the bundle u∗(TM) over D2 and use the trivialization to identify
the linearized flow dγ(0)ft, 0 ≤ t ≤ 1, along γ with a symplectic matrix
{A(t)}. Then the Conley-Zehnder index CZF ([γ, u]) is defined as
CZF ([γ, u]) := n− CZmatr ({A(t)}). (17)
With such a normalization of CZF for any sufficiently C
2-small autonomous
Morse Hamiltonian F , the Conley-Zehnder index of an element of P̃F , rep-
resented by a pair [x, u] consisting of a critical point x of F (viewed as a
constant path in M) and the trivial disk u, is equal to the Morse index of
x. Note that with such a normalization CZF (Sy) = CZF (y)+2
c1(M) for
every y ∈ P̃F and S ∈ H
2 (M).
Similarly, if the time-1 flow generated by F defines a loop in Ham(M) then
to each [γ, u] ∈ P̃F one can associate its Maslov index. Namely, trivialize the
bundle u∗(TM) over D2 and identify the linearized flow {dxft} along γ with
an identity-based loop of symplectic 2n × 2n-matrices. Define the Maslov
index mF ([γ, u]) as the Maslov index for the loop of symplectic matrices.
Under the action of HS2 (M) on P̃F the Maslov index changes as follows:
mF (S · [γ, u]) = mF ([γ, u])− 2
c1(M), S ∈ H
2 (M).
Let us make the following remark. Assume γ ∈ PF and assume that a
symplectic trivialization of the bundle γ∗(TM) over S1 identifies {dγ(0)ft}
with an identity-based path {A(t)} of symplectic matrices. Assume there
is another symplectic trivialization of the same bundle, coinciding with the
first one at γ(0), and denote by {B(t)} the identity-based loop of transition
matrices from the first symplectic trivialization to the second one. Use the
second trivialization to identify {dγ(0)ft} with an identity-based path {A
′(t)}.
CZmatr ({A
′(t)}) = CZmatr ({A(t)}) +Maslov ({B(t)}), (18)
and if {A(t)} is a loop then so is {A′(t)} and
Maslov ({A′(t)}) = Maslov ({A(t)}) +Maslov ({B(t)}). (19)
3.4 Spectral numbers
Given the algebraic setup as above, the construction of the Piunikhin-Sala-
mon-Schwarz (PSS) isomorphism [47] yields a Λ-linear isomorphism (PSS-
isomorphism) φM : QH∗(M) → HF∗(F, J) which preserves the grading and
which is actually a ring isomorphism (the pair-of-pants product defines a ring
structure on HF∗(F, J)).
Using the PSS-isomorphism one defines the spectral numbers c(a, F ),
where 0 6= a ∈ QH∗(M), in the usual way [45]. Namely, the action functional
AF defines a filtration on C(F ) which induces a filtration HF
∗ (F, J), α ∈ R,
on HF∗(F, J), with HF
∗ (F, J) ⊂ HF
∗ (F, J) as long as α < β. Then
c(a, F ) := inf {α | φM(a) ∈ HF
∗ (F, J)}.
Such spectral number is finite and well-defined (does not depend on J). Here
is a brief account of the relevant properties of spectral numbers – for details
see [45] (see also [65, 42, 59, 43] for earlier versions of this theory).
(Spectrality) c(a,H) ∈ spec (H), where the spectrum spec (H) of H is
defined as the set of critical values of the action functional AH , i.e.
spec (H) := AH(P̃H) ⊂ R.
(Quantum homology shift property) c(λa,H) = c(a,H) + ν(λ) for all
λ ∈ Λ, where ν is the valuation defined in Section 3.1.
(Hamiltonian shift property) c(a,H + λ(t)) = c(a,H) +
λ(t) dt for
any Hamiltonian H and function λ : S1 → R.
(Monotonicity) If H1 ≤ H2, then c(a,H1) ≤ c(a,H2).
(Lipschitz property) The map H 7→ c(a,H) is Lipschitz on the space of
(time-dependent) Hamiltonians H : M × S1 → R with respect to the
C0-norm.
(Symplectic invariance) c(a, φ∗H) = c(a,H) for every φ ∈ Symp0(M),
H ∈ C∞(M); more generally, Symp (M) acts on H∗(M ;F), and hence
on QH∗(M), and c(a, φ
∗H) = c(φ∗a,H) for any φ ∈ Symp (M).
(Normalization) c(a, 0) = ν(a) for every a ∈ QH∗(M).
(Homotopy invariance) c(a,H1) = c(a,H2) for any normalized H1, H2
generating the same φ ∈ H̃am (M). Thus one can define c(a, φ) for any
φ ∈ H̃am (M) as c(a,H) for any normalized H generating φ.
(Triangle inequality) c(a ∗ b, φψ) ≤ c(a, φ) + c(b, ψ).
The commutative ring QH•(M) admits a K-bilinear and K-valued form Ω
on QH•(M) which associates to a pair of quantum homology classes a, b ∈
QH•(M) the coefficient (belonging to K) at the class [point] = [point] · q
a point in their quantum product a ∗ b ∈ QH•(M) (the Frobenius structure).
Let τ : K → F be the map sending each series
θ∈Γ zθs
θ, zθ ∈ F , to its free
term z0. Define a non-degenerate F -valued F -linear pairing on QH•(M) by
Π(a, b) := τΩ(a, b) = τΩ(a ∗ b, [M ]) . (20)
Note that Π is symmetric and
Π(a ∗ b, c) = Π(a, b ∗ c) ∀a, b, c ∈ QH•(M). (21)
With this notion at hand, we can present another important property of
spectral numbers:
(Poincaré duality) c(b, φ) = − infa∈Υ(b) c(a, φ
−1) for all b ∈ QH•(M)\{0}
and φ. Here Υ(b) denotes the set of all a ∈ QH•(M) with Π(a, b) 6= 0.
The Poincaré duality can be extracted from [47] (cf. [22]) – for a proof see
[46].
The next property is an immediate consequence of the definitions (see [22]
for a discussion in the monotone case):
(Characteristic exponent property) Given 0 6= λ ∈ F , a, b ∈ QH∗(M),
a, b, a + b 6= 0, and a (time-dependent) Hamiltonian H , one has
c(λ · a,H) = c(a,H) and c(a+ b,H) ≤ max(c(a,H), c(b,H)).
3.5 Partial symplectic quasi-states
Given a non-zero idempotent a ∈ QH2n(M) and a time-independent Hamil-
tonian H :M → R, define
ζ(a,H) := lim
c(a, lH)
. (22)
When a is fixed, we shall often abbreviate ζ(H) instead of ζ(a,H). The limit
in the formula (22) always exists and thus the functional ζ : C∞(M) → R is
well-defined. The functional ζ on C∞(M) is Lipschitz with respect to the C0-
norm ‖H‖ = maxM |H| and therefore extends to a functional ζ : C(M) → R,
where C(M) is the space of all continuous functions on M . These facts were
proved in [23] in the case a = [M ] but the proofs actually go through for any
non-zero idempotent a ∈ QH2n(M).
Here we will list the properties of ζ for such an M . Again, these proper-
ties were proved in [23] in the case a = [M ] but the proof goes through for
any non-zero idempotent a ∈ QH2n(M). The additivity with respect to con-
stants property was not explicitly listed in [23] but follows immediately from
the definition of ζ and the Hamiltonian shift property of spectral numbers.
The triangle inequality follows readily from the definition of ζ and from the
triangle inequality for the spectral numbers.
Theorem 3.6. The functional ζ : C(M) → R satisfies the following prop-
erties:
Semi-homogeneity: ζ(αF ) = αζ(F ) for any F and any α ∈ R≥0.
Triangle inequality: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 then ζ(F1 + F2) ≤
ζ(F1) + ζ(F2).
Partial additivity and vanishing: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 and the
support of F2 is displaceable, then ζ(F1 + F2) = ζ(F1); in particular, if the
support of F ∈ C(M) is displaceable, ζ(F ) = 0.
Additivity with respect to constants and normalization: ζ(F +α) = ζ(F )+α
for any F and any α ∈ R. In particular, ζ(1) = 1.
Monotonicity: ζ(F ) ≤ ζ(G) for F ≤ G.
Symplectic invariance: ζ(F ) = ζ(F ◦ f) for every symplectic diffeomorphism
f ∈ Symp0 (M).
Characteristic exponent property: ζ(a1+a2, F ) ≤ max(ζ(a1, F ), ζ(a2, F )) for
each pair of non-zero idempotents a1, a2 with a1 ∗ a2 = 0, a1+ a2 6= 0 (in this
case a1 + a2 is also a non-zero idempotent), and for all F ∈ C(M) .
We will call the functional ζ : C(M) → R satisfying all the properties
listed in Theorem 3.6 a partial symplectic quasi-state.
4 Basic properties of (super)heavy sets
In this section we prove parts (i) and (iii) of Theorem 1.2, as well as The-
orem 1.3. We shall use that a partial symplectic quasi-state ζ extends by
continuity in the uniform norm to a monotone functional on the space of
continuous functions C(M), see Section 3.5 above. In particular, one can
use continuous functions instead of the smooth ones in the definition of (su-
per)heaviness in formulae (3) and (4).
Assume a partial quasi-state ζ defined by a non-zero idempotent is fixed
and we consider heaviness and superheaviness with respect to ζ . We start
with the following elementary
Proposition 4.1. A closed subset X ⊂ M is heavy if and only if for every
H ∈ C∞(M) with H|X = 0, H ≤ 0 one has ζ(H) = 0. A closed subset
X ⊂ M is superheavy if and only if for every H ∈ C∞(M) with H|X = 0,
H ≥ 0 one has ζ(H) = 0.
Proof. The “only if” parts follow readily from the monotonicity property of
ζ . Let us prove the “if” part in the “heavy case” – the “superheavy” case is
similar. Take a function H on M and put
F = min(H − inf
H, 0) .
Note that F |X = 0 and F ≤ 0. Thus ζ(F ) = 0 by the assumption of the
proposition. Thus
0 = ζ(F ) ≤ ζ(H − inf
H) = ζ(H)− inf
which yields heaviness of X .
The following proposition proves part (i) of Theorem 1.2.
Proposition 4.2. Every superheavy set is heavy.
Proof. Let X ⊂ M be a superheavy subset. Assume that H|X = 0, H ≤ 0.
By the triangle inequality for ζ we have ζ(H) + ζ(−H) ≥ 0. Note that
−H|X = 0, −H ≥ 0. Superheaviness yields ζ(−H) = 0, so ζ(H) ≥ 0. But
by monotonicity ζ(H) ≤ 0. Thus ζ(H) = 0 and the claim follows from
Proposition 4.1.
Superheavy sets have the following user-friendly property.
Proposition 4.3. Let X ⊂ M be a superheavy set. Then for every α ∈ R
and H ∈ C∞(M) with H|X ≡ α one has ζ(H) = α.
Proof. Since ζ(H + α) = ζ(H) + α it suffices to prove the proposition for
α = 0. Take any function H with H|X = 0. Since X is superheavy and, by
Proposition 4.2, also heavy, we have
0 = ζ(−|H|) ≤ ζ(H) ≤ ζ(|H|) = 0 ,
which yields ζ(H) = 0.
As an immediate consequence we get part (iii) of Theorem 1.2.
Proposition 4.4. Every superheavy set intersects with every heavy set.
Proof. Let X be a superheavy set and Y be a heavy set. Assume on the
contrary that X ∩ Y = ∅. Take a function H ≤ 0 with H|Y ≡ 0 and
H|X ≡ −1. Then ζ(H) = −1 by Proposition 4.3. On the other hand,
ζ(H) = 0 since Y is heavy, and we get a contradiction.
Note that two heavy sets do not necessarily intersect each other: a meridian
of T2 is heavy (see Corollary 6.4 below), while two meridians can be disjoint.
Proof of Theorem 1.3 (i) and (ii): The triangle inequality yields
c(a,H) = c(a ∗ [M ], 0 +H) ≤ c(a, 0) + c([M ], H) = ν(a) + c([M ], H).
Passing to the partial quasi-states ζ(a,H) and ζ([M ], H) we get:
ζ(a,H) = lim
c(a, kH)/k ≤
≤ lim
(ν(a) + c([M ], kH))/k = lim
c([M ], kH)/k = ζ([M ], H).
The result now follows from the definition of heavy and superheavy sets (see
Definition 1.1).
Proof of Theorem 1.3 (iii): By the characteristic exponent property of
spectral invariants,
ζ(a, F ) ≤ max
i=1,...,l
ζ(ei, F ) ∀F ∈ C
∞(M) . (23)
Choose a sequence of functions Gj ∈ C
∞(M), j → +∞, with the fol-
lowing properties: Gk ≤ Gj for k > j, Gj = 0 on X , Gj ≤ 0 and for
every function F ≤ 0 which vanishes on an open neighborhood of X there
exists j so that Gj ≤ F (existence of such a sequence can be checked easily).
In view of inequality (23), we have that for every j there exists i so that
ζ(a,Gj) ≤ ζ(ei, Gj). Passing, if necessary, to a subsequence Gjk , jk → +∞,
we can assume without loss of generality that i is the same for all j. In view
of heaviness of X with respect to a, we have that ζ(a,Gj) = 0. Therefore
ζ(ei, Gj) ≥ 0.
Choose any function F ≤ 0 onM which vanishes on an open neighborhood
of X . Then there exists j large enough so that F ≥ Gj. By monotonicity
combined with the previous estimate we have
0 ≥ ζ(ei, F ) ≥ ζ(ei, Gj) ≥ 0 ,
which yields ζ(ei, F ) = 0.
Now let F be any continuous function on M that vanishes on X . Take
a sequence of continuous functions Fj , converging to F in the C
0-norm, so
that each Fj vanishes on an open neighborhood of X . Then ζ(ei, Fj) =
limj→+∞ ζ(ei, Fj) = 0, because ζ(ei, ·) is Lipschitz with respect to the C
norm. The heaviness ofX with respect to ei now follows from Proposition 4.1.
This finishes the proof of the theorem.
5 Products of (super)heavy sets
In this section we prove Theorem 1.5 on products of (super)heavy subsets.
5.1 Product formula for spectral invariants
The proof of Theorem 1.5 is based on the following general result.
Theorem 5.1. For every pair of time-dependent Hamiltonians G1, G2 onM1
and M2, and all non-zero a1 ∈ QHi1(M1), a2 ∈ QHi2(M2) we have
c(a1 ⊗ a2, G1(z1, t) + G2(z2, t)) = c(a1, G1) + c(a1, G2) .
Here G1(z1, t) +G2(z2, t) is a time-dependent Hamiltonian on M1 ×M2.
Let us deduce Theorem 1.5 from Theorem 5.1.
Proof of Theorem 1.5: We show that the product of superheavy sets is
superheavy (the proof for heavy sets goes without any changes). We denote
by ζ1, ζ2 and ζ the partial quasi-states on M1,M2 and M := M1 ×M2 as-
sociated to the idempotents a1, a2 and a1 ⊗ a2 respectively. Let Xi ⊂ Mi,
i = 1, 2, be a superheavy set. By Proposition 4.1 it suffices to show that if a
non-negative function G ∈ C∞(M) vanishes on some neighborhood, say U ,
of X := X1 × X2 then ζ(G) = 0. (Since ζ is Lipschitz with respect to the
C0-norm this would imply that ζ(G) = 0 for any non-negative G ∈ C(M)
that vanishes on X). Put K := maxM G. Choose neighborhoods Ui of Xi so
that U1 ×U2 ⊂ U . Choose non-negative functions Gi on Mi which vanish on
Xi and such that Gi(z) > K for all z ∈Mi \Ui. Observe that G ≤ G1 +G2.
But, in view of Theorem 5.1 and superheaviness of Xi, we have
ζ(G1 +G2) = ζ1(G1) + ζ2(G2) = 0 .
By monotonicity
0 ≤ ζ(G) ≤ ζ(G1 +G2) = 0 ,
and thus ζ(G) = 0.
It remains to prove Theorem 5.1. Note that the left-hand side of the equality
stated in the theorem does not exceed the right-hand side: this is an imme-
diate consequence of the triangle inequality for spectral invariants. However,
we were unable to use this observation for proving the theorem. Our ap-
proach is based on a rather lengthy algebraic analysis which enables us to
calculate separately the left and the right-hand sides “on the chain level”. A
simple inspection of the results of this calculation yields the desired equality.
5.2 Decorated Z2-graded complexes
A Z2-complex is a Z2-graded finite-dimensional vector space V over a field
K equipped with a K-linear differential ∂ : V → V satisfying ∂2 = 0 and
shifting the grading. A decorated complex over K = KΓ includes the following
data:
• a countable subgroup Γ ⊂ R;
• a Z2-graded complex (V, d) over KΓ;
• a preferred basis x1, . . . , xn of V ;
• a function F : {x1, . . . , xn} → R (called the filter) which extends to V
λjxj) = max{ν(λj) + F (xj)
∣∣∣ λj 6= 0},
and satisfies F (dv) < F (v) for all v ∈ V \ {0}. The convention is that
F (0) = −∞. Here ν is the valuation defined in Section 3.1 above.
We shall use the notation
V := (V, {xi}i=1,...,n, F, d,Γ)
for a decorated complex.
The ⊗̂K-tensor product V = V1⊗̂KV2 of decorated complexes
Vi = (Vi, {x
j }j=1,...,ni, Fi, di,Γi) , i = 1, 2
is defined as follows. Consider the space V = V1⊗̂KV2 (see formula (5) above)
with the natural Z2-grading. Define the differential d on V by
d(x⊗ y) = d1x⊗ y + (−1)
deg xx⊗ d2y .
The preferred basis in V is given by {xpq := x
p ⊗ x
q } and the filter F is
defined by
F (xpq) = F1(x
p ) + F2(x
Finally, we put V := (V, {xpq}, F, d,Γ1 + Γ2) .
The (Z2-graded) homology of decorated complexes are denoted by H∗(V)
– they are K-vector spaces. By the Künneth formula, H(V1⊗̂KV2) =
H(V1)⊗̂KH(V2).
Next we define spectral invariants associated to a decorated complex V :=
(V, {xpq}, F, d) . Namely, for a ∈ H(V) put
c(a) := inf{F (v) | a = [v], v ∈ Ker d} .
We shall see below that c(a) > −∞ for each a 6= 0.
The purpose of this algebraic digression is to state the following result:
Theorem 5.2. For any two decorated complexes V1,V2
c(a1 ⊗ a2) = c(a1) + c(a2) ∀a1 ∈ H(V1), a2 ∈ H(V2)
5.3 Reduced Floer and Quantum homology
The 2-periodicity of the Floer complex and Floer homology defined by the
multiplication by q (see Proposition 3.1 above) allows to encode their al-
gebraic structure in a decorated Z2-complex. Consider a regular pair (G, J)
consisting of a Hamiltonian function and a compatible almost-complex struc-
ture on M (both, in general, are time-dependent). Let (C∗(G), dG,J) be the
corresponding Floer complex. Let us associate to it a Z2-complex: a Z2-
graded vector space VG over KΓ, defined as
VG := C0(G)⊕ C1(G),
with the obvious Z2-grading, and a differential ∂G,J : VG → VG, defined as
the direct sum of dG,J : C1(G) → C0(G) and qdG,J : C0(G) → C1(G). One
readily checks that this is indeed a Z2-complex because dG,J : C(G) → C(G)
is ΛΓ-linear. We will call (VG, ∂G,J) the Z2-complex associated to (G, J).
Note that the cycles and the boundaries of (VG, ∂G) having Z2-degree
i ∈ {0, 1} in VG coincide, respectively, with the cycles and the boundaries
having Z-degree i of (C(G), dG,J). Therefore the Floer homology HFi(G, J)
is isomorphic, as a vector space over KΓ, to the i-th degree component of the
homology of the complex (VG, ∂G,J).
The Z2-complex (VG, ∂G,J) carries a structure of the decorated complex
VG,J as follows. Let γi(t), i = 1, . . . , m, be the collection of all contractible
1-periodic orbits of the Hamiltonian flow generated by G. Choose disc ui
in M spanning γi. For each i there exists unique integer, say ri, so that
the Conley-Zehnder index of the element xi := q
ri · [γi, ui] lies in the set
{0, 1}. Clearly, the collection {xi} forms a basis of VG over KΓ. We shall
consider it as a preferred basis. Note that the preferred basis is unique up to
multiplication of xi’s by elements of the form s
αi , αi ∈ Γ. Finally, the action
functional associated to G defines a filtration on VG.
The homology of (VG, ∂G,J) can be canonically identified via the PSS-
isomorphism with the object which we call reduced quantum homology:
QHred(M) := QH0(M)⊕QH1(M) .
We call this isomorphism the reduced PSS-isomorphism and denote it by ψG,J .
Note that we have a natural projection p : QH∗(M) → QHred(M) which
sends any degree homogeneous element a to aqr with deg a + 2r ∈ {0, 1}.
With this notation, the usual Floer-homological spectral invariant c(a,G)
coincides with the spectral invariant c(p(a)) of the decorated complex VG,J .
5.4 Proof of Theorem 5.1
By the Lipschitz property of spectral numbers it is enough to consider the
case when G1 and G2 belong to regular pairs (Gi, Ji), i = 1, 2. Set
G(z1, z2, t) := G1(z1, t) +G(z2, t)
and J := J1 × J2. Then (G, J) is also a regular pair. Put Γi = Γ(Mi, ωi). It
is straightforward to see that the decorated complex VG,J is the ⊗̂K-tensor
product of the decorated complexes VGi,Ji for i = 1, 2.
Put (M,ω) = (M1×M2, ω1⊕ω2). An obvious modification of the Künneth
formula for quantum homology (see e.g. [41, Exercise 11.1.15] for the state-
ment in the monotone case) yields a natural monomorphism
ı : QHi1(M1, ω1)⊗̂KQHi2(M1, ω1) → QHi1+i2(M,ω) .
Since in our setting quantum homologies are 2-periodic, the collection of these
isomorphisms for all pairs (i1, i2) from the set {0, 1} induces an isomorphism
j : QHred(M1)⊗̂KQHred(M2) → QHred(M) .
It has the following properties: First, given two elements a1 ∈ QHi1(M1, ω1)
and a2 ∈ QHi2(M2, ω2) we have that
p(a1)⊗ p(a2) = p(a1 ⊗ a2) .
Second, the following diagram commutes:
H(VG1, ∂G1,J1)⊗̂KH(VG2 , ∂G2,J2)
ψG1,J1⊗ψG2,J2
H(VG, ∂G,J)
QHred(M1)⊗̂KQHred(M2)
// QHred(M)
Here k is the isomorphism coming from the Künneth formula for Z2-comple-
xes, and ψGi,Ji, ψG,J stand for the reduced PSS-isomorphisms. It follows that
the definition of c(ai, Gi), c(a1⊗a2, G) matches the definition of c(p(ai)) and
c(p(a1)⊗ p(a2)). By Theorem 5.2 we get that
c(a1⊗a2, G) = c(p(a1)⊗p(a2)) = c(p(a1))+ c(p(a2)) = c(a1, G1)+ c(a2, G2) .
This proves Theorem 5.1 modulo Theorem 5.2.
5.5 Proof of algebraic Theorem 5.2
A decorated complex is called generic if F (xi) − F (xj) /∈ Γ for all i 6= j
(recall that under our assumptions Γ, the group of periods of the symplectic
form ω over π2(M), is a countable subgroup of R). We start from some
auxiliary facts from linear algebra. Let V := (V, {xi}i=1,...,n, F, d,Γ) be a
generic decorated complex. We recall once again that for brevity we write K
instead of KΓ wherever it is clear what Γ is taken.
An element x ∈ V is called normalized if
x = xp +
i 6=p
λixi , λi ∈ K, F (xp) > max
i 6=p
F (λixi) .
We shall use the notation x = xp+o(xp). In generic complexes, every element
x 6= 0 can be uniquely written as x = λ(xp+o(xp)) for some p = 1, . . . , n and
λ ∈ K. A system of vectors e1, . . . , em in V is called normal if every ei has
the form ei = xji +o(xji) for ji ∈ {1, . . . , n} and the numbers ji are pair-wise
distinct.
Lemma 5.3. Let e1, . . . , em be a normal system. Then
λiei) = max
F (λiei) .
Proof. We prove the result using induction in m. For m = 1 the statement is
obvious. Let’s check the induction step m− 1 → m. Observe that it suffices
to check that
F (e1 +
λiei) ≥ F (e1) . (24)
Then obviously
λiei) ≥ max
F (λiei) ,
while the reversed inequality is an immediate consequence of the definitions.
By the induction step,
λiei) = max
i=2,...,n
F (λiei) .
In view of the genericity, the maximum at the right hand side can be uniquely
written as F (λi0xi0). Without loss of generality we shall assume that ei =
xi + o(xi) and i0 = 2.
λ−12 λiei = x2 + o(x2) .
Write
e1 = x1 + αx2 +X, v = x2 + βx1 + Y,
where α, β ∈ K and X, Y ∈ SpanK(x3, . . . , xn). Note that F (x1) > F (αx2),
F (x2) > F (βx1), which yields
ν(α) < F (x1)− F (x2) < −ν(β) = ν(β
−1) . (25)
In particular, ν(α) < ν(β−1). Note that
e1 + λ2v = (1 + λ2β)x1 + (α + λ2)x2 + Z, Z ∈ SpanK(x3, . . . , xn) .
F (e1 + λ2v) ≥ max(ν(1 + λ2β) + F (x1), ν(α + λ2) + F (x2)) .
If ν(1 + λ2β) ≥ 0 we have F (e1 + λ2v) ≥ F (x1) = F (e1) and inequality (24)
follows. Assume that ν(1+λ2β) < 0 = ν(1). Then ν(λ2β) = 0 = ν(λ2)+ν(β),
and hence ν(λ2) = ν(β
−1) 6= ν(α). Thus
ν(α + λ2) ≥ ν(λ2) = −ν(β) .
Combining this inequality with (25) we get that
F (e1 + λ2v) ≥ ν(α + λ2) + F (x1) + (F (x2)− F (x1))
≥ F (x1) + (ν(α + λ2) + ν(β)) ≥ F (x1) = F (e1) .
This completes the proof of inequality (24), and hence of the lemma.
It readily follows from the lemma that every normal system is linearly inde-
pendent.
Lemma 5.4. Every subspace L ⊂ V has a normal basis.
Proof. We use induction over m = dimK L. The case m = 1 is obvious,
so let us handle the induction step m − 1 → m. It suffices to show the
following: Let e1, . . . , em−1 be a normal basis in a subspace L
′, and let v /∈ L′
be any vector. Put L = SpanK(L
′ ∪ {v}). Then there exists em ∈ L so that
e1, . . . , em is a normal basis. Indeed, assume without loss of generality that
for all i = 1, . . . , m−1 one has ei = xi+ o(xi). Put W = SpanK(xm, . . . , xn).
We claim that L′ ∩W = {0}. Indeed, otherwise
λ1e1 + . . .+ λm−1em−1 = λmxm + . . .+ λnxn
where the linear combinations in the right and the left-hand sides are non-
trivial. Apply F to both sides of this equality. By Lemma 5.3
F (λ1e1 + . . .+ λm−1em−1) = F (xp) mod Γ, where 1 ≤ p ≤ m− 1 ,
while
F (λmxm + . . .+ λnxn) = F (xq) mod Γ, where q ≥ m .
This contradicts the genericity of our decorated complex, and the claim fol-
lows. Since dimL′+dimW = dimV , we have that V = L′⊕W . Decompose v
as u+w with u ∈ L′, w ∈ W , and note that w ∈ L. Note that e1, . . . , em−1, w
are linearly independent. Furthermore, w = λ(xp + o(xp)) for some p ≥ m.
Put em = λ
−1w. The vectors e1, . . . , em form a normal basis in L.
The same proof shows that if L1 ⊂ L2 are subspaces of V , every normal basis
in L1 extends to a normal basis in L2.
Now we turn to the analysis of the differential d. Choose a normal basis
g1, . . . , gq in Im d, and extend it to a normal basis g1, . . . , gq, h1, . . . , hp in
Ker d. Note that each of these p + q vectors has the form xj + o(xj) with
distinct j. Let us assume without loss of generality that the remaining n−p−q
elements of the preferred basis in V are x1, . . . , xq, and
gi = xi+q + o(xi+q), hj = xj+2q + o(xj+2q) .
Here we use that, by the dimension theorem, n = p+ 2q. Note that
x1, . . . , xq, g1, . . . , gq, h1, . . . , hp
is a normal system, and hence a basis in V . We call such a basis a spectral
basis of the decorated complex V.
Note that [h1], . . . , [hp] is a basis in the homology H(V). Consider any
homology class a =
λi[hi]. Every element v ∈ V with a = [v] can be
written as v =
λihi +
αjgj. Thus, by Lemma 5.3, F (v) ≥ maxi F (λihi)
and hence
c(a) = max
F (λihi) . (26)
This proves in particular that the spectral invariants are finite provided a 6= 0.
For finite sets A = {v1, . . . , vs} and B = {w1, . . . , ws} we write A⊗B for the
finite set {vi ⊗ wj}.
Assume now that V1,V2 are generic decorated complexes. We say that they
are in general position if their tensor product V = V1⊗̂KV2 is generic. Let
Bi = {x
1 , . . . , x
1 , . . . , g
1 , . . . , h
}, i = 1, 2
be a spectral basis in Vi. Obviously, B1 ⊗ B2 is a normal basis in V1⊗̂KV2.
We shall denote by d1, d2, d the differentials and by F1, F2, F the filters in
V1,V2 and V respectively. Put Gi = {g
1 , . . . , g
qi }, Hi = {h
1 , . . . , h
and K = G1 ⊗ B2 ∪B1 ⊗G2. Observe that
Im d ⊂W := Span(K) .
Take any two classes
j ] ∈ H(Vi) , i = 1, 2.
Suppose that a1 ⊗ a2 = [v]. Then v is of the form
λ(1)m λ
m ⊗ h
l + w
where w must lie in W . Observe that (H1 ⊗H2) ∩K = ∅. By Lemma 5.3,
F (v) ≥ max
F (λ(1)m λ
m ⊗ h
l ) ,
and hence
c(a1 ⊗ a2) = max
F (λ(1)m λ
m ⊗ h
= max
m ) + F2(λ
= max
m ) + max
l ) = c(a1) + c(a2) .
In the last equality we used (26). This completes the proof of Theorem 5.2
for decorated complexes in general position.
It remains to remove the general position assumption. This will be done
with the help of the following lemma. We shall work with a family of deco-
rated complexes
V := (V, {xi}i=1,...,n, F, d,Γ)
which have exactly the same data (preferred basis, grading, differential and
Γ) with the exception of the filter F which will be allowed to vary in the class
of filters. The corresponding spectral invariants will be denoted by c(a, F ).
Lemma 5.5.
(i) If filters F, F ′ satisfy F (xi) ≤ F
′(xi) for all i = 1, . . . , n, then c(a, F ) ≤
c(a, F ′) for all non-zero classes a ∈ H(V).
(ii) If F is a filter and θ ∈ R, then F + θ is again a filter and c(a, F + θ) =
c(a, F ) + θ for all non-zero classes a ∈ H(V).
The proof is obvious and we omit it. It follows that for any two filters F, F ′
|c(a, F )− c(a, F ′)| ≤ ||F − F ′||C0 ∀a ∈ H(V) \ {0} .
Assume now that V1,V2 are decorated complexes. Denote by F1, F2 their
filters. Fix ǫ > 0. By a small perturbation of the filters we get new filters,
F ′1 and F
2, on our complexes so that the complexes become generic and in
general position, and furthermore
||F1 − F
1||C0 ≤ ǫ , ||F2 − F
2||C0 ≤ ǫ .
Given homology classes ai ∈ H(Vi) we have
|c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2)| ≤
|c(a1, F
1) + c(a2, F
2)− c(a1 ⊗ a2, F
1 + F
2)|+ 4ǫ = 4ǫ .
Here we used that Theorem 5.2 is already proved for generic complexes in
general position. Since ǫ > 0 is arbitrary, we get that
c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2) = 0 ,
which completes the proof of Theorem 5.2 in full generality.
6 Stable non-displaceability of heavy sets
In this section we prove part (ii) of Theorem 1.2.
Proposition 6.1. Every heavy subset is stably non-displaceable.
For the proof we shall need the following auxiliary statement. Given R > 0,
consider the torus T2R obtained as the quotient of the cylinder T
∗S1 = R(r)×
S1 (θ mod 1) by the shift (r, θ) 7→ (r + R, θ). For α > 0 define the function
Fα(r, θ) := αf(r) on T
R, where f(r) is any R-periodic function having only
two non-degenerate critical points on [0, R]: a maximum point at r = 0 with
f(0) = 1, and a minimum point at r = R/2, f(R/2) =: −β < 0. We denote
by [T ] the fundamental class of T2R. We work with the symplectic form dr∧dθ
on T2R.
Lemma 6.2. c([T ], Fα) = α.
Proof. Note that the contractible closed orbits of period 1 of the Hamiltonian
flow generated by Fα are fixed points forming circles S+ = {r = 0} and
S− = {r = R/2}. The actions of the fixed points on S± equal respectively to
α and −αβ, and thus the spectral invariants of Fα lie in the set {α,−αβ}.
Recall from [59] that c([T ], Fα) > c([point], Fα). Thus c([T ], Fα) = α.
Lemma 6.3. Let H ∈ C∞(M) so that H−1(maxH) is displaceable. Then
ζ(H) < maxH.
Proof. Choose ǫ > 0 so that the set
H−1((maxH − ǫ,maxH ])
is displaceable. Choose a real-valued cut-off function ρ : R → [0, 1] which
equals 1 near maxH and which is supported in (maxH−ǫ,maxH+ǫ). Thus
ρ(H) is supported in H−1((maxH − ǫ; maxH ]) and ζ(ρ(H)) = 0. Since H
and ρ(H) Poisson-commute, the vanishing and the monotonicity axioms yield
ζ(H) = ζ(ρ(H)) + ζ(H − ρ(H)) ≤ max(H − ρ(H)) < maxH .
Proof of Proposition 6.1: It suffices to show that for every R > 0 the set
Y := X × {r = 0} ⊂M ′ :=M × T2R
is non-displaceable. Assume on the contrary that Y is displaceable. Choose
a function H on M with H ≤ 0, H−1(0) = X . Put
H ′ = H + F1 = H + f(r) :M
′ → R.
Assume that the partial quasi-state ζ on M is associated to some non-zero
idempotent a ∈ QH∗(M) by means of (2). Denote by ζ
′ the quasi-state on
M ′ associated to a⊗ T . Note that
Y = (H ′)−1(maxH ′) , where maxH ′ = 1 ,
while Theorem 5.1 and Lemma 6.2 imply that
ζ ′(H ′) = ζ(H) + 1 .
By Lemma 6.3 ζ ′(H ′) < 1 and so ζ(H) < 0. In view of Proposition 4.1, we
get a contradiction with the heaviness of X .
Lemma 6.2 also yields a simple proof of the following result which also follows
from Corollary 1.15:
Corollary 6.4. Any meridian of T2 is heavy (with respect to the fundamental
class [T ]).
Proof. In the notation as above identify T2 with T21 for R = 1. Since any
two meridians of T2 can be mapped into each other by a symplectic isotopy
and since such an isotopy preserves heaviness, it suffices to prove that the
meridian S := S+ = {r = 0} (see the proof of Lemma 6.2) is heavy.
Let H : T2 → R be a Hamiltonian and let us show that ζ(H) ≥ infSH ,
where ζ is defined using [T ]. Shifting H , if necessary, by a constant, we may
assume without loss of generality that infSH = 1. Pick f = f(r) : T
2 → R
as in the definition of Fα so that F1 = f ≤ H on T
2 (note that f equals 1 on
S). Then Lemma 6.2 yields
ζ(H) ≥ ζ(F1) = 1 = inf
7 Analyzing stable stems
Proof of Theorem 1.6: Assume that A is a Poisson-commutative subspace
of C∞(M), Φ : M → A∗ its moment map with the image ∆, and let X =
Φ−1(p) be a stable stem of A.
Take any functionH ∈ C∞(A∗) with H ≥ 0 andH(p) = 0. We claim that
ζ(Φ∗H) = 0. By an arbitrarily small C0-perturbation of H we can assume
that H = 0 in a small neighborhood, say U , of p. Choose an open covering
U0, U1, . . . , UN of ∆ so that U0 = U , and all Φ
−1(Ui) are stably displaceable
for i ≥ 1 (it exists by the definition of a stem). Let ρi : ∆ → R, i = 0, . . . , N ,
be a partition of unity subordinated to the covering {Ui}.
Take the two-torus T2R as in Section 6. Choose R > 0 large enough so that
Φ−1(Ui)× {r = const} is displaceable in M × T
R for all i ≥ 1. Choose now
a sufficiently fine covering Vj , j = 1, . . . , K, of the torus T
R by sufficiently
thin annuli {|r − rj | < δ} so that the sets Φ
−1(Ui) × Vj are displaceable in
M × T2R for all i ≥ 1 and all j. Let ̺j = ̺j(r), j = 1, . . . , K, be a partition
of unity subordinated to the covering {Vj}.
Denote by ζ ′ the partial quasi-state corresponding to a⊗T . Put F (r, θ) =
cos(2πr/R). Write
Φ∗H + F =
(Φ∗H + F ) · Φ∗ρi · ̺j =
Φ∗(Hρ0) + F · Φ
∗ρ0 +
(Φ∗H + F ) · Φ∗ρi · ̺j .
Note that Hρ0 = 0 and F · Φ
∗ρ0 ≤ 1. Applying partial quasi-additivity
and monotonicity we get that
ζ ′(Φ∗H + F ) = ζ ′(F · Φ∗ρ0) ≤ 1.
By Lemma 6.2 and the product formula (Theorem 5.1 above) we have
ζ ′(Φ∗H + F ) = ζ(Φ∗H) + 1 ≤ 1
and hence ζ(Φ∗H) ≤ 0. On the other hand, ζ(Φ∗H) ≥ 0 since H ≥ 0. Thus
ζ(Φ∗H) = 0 and the claim follows.
Further, given any function G on M with G ≥ 0 and G|X = 0, one can
find a function H on A∗ with H(p) = 0 so that G ≤ Φ∗H . By monotonicity
and the claim above
0 ≤ ζ(G) ≤ ζ(Φ∗H) = 0 ,
and hence ζ(G) = 0. Thus X is superheavy.
8 Monotone Lagrangian submanifolds
The main tool of proving (super)heaviness of monotone Lagrangian subman-
ifolds satisfying the Albers condition is the spectral estimate in Proposi-
tion 8.1(iii) below, which originated in the work by Albers [2]. Later on
Biran and Cornea pointed out a mistake in [2], and suggested a correction
together with a far reaching generalization in [15]. Let us mention that the
original Albers estimate was used in the first version of the present paper. We
thank Biran and Cornea for informing us about the mistake, explaining to
us their approach and helping us to correct a number of our results affected
by this mistake.
The main ingredient of Biran-Cornea techniques which is needed for our
purposes is the following result. Let (M,ω) be a closed monotone symplectic
manifolds with [ω] = κ·c1(M), κ > 0. WriteN for the minimal Chern number
of (M,ω). Let Ln ⊂M2n be a closed monotone Lagrangian submanifold with
the minimal Maslov number NL ≥ 2.
We shall treat slightly differently the cases when NL is even and odd. Let
us mention that for orientable L, NL is automatically even. Thus, due to
our convention, when NL is odd we work with the basic field F = Z2. Let
Γ = κN · Z be the group of periods of M . Recall that the quantum ring has
the form QH∗(M) = H∗(M ;F) ⊗F Λ, where the Novikov ring Λ is defined
as Λ = KΓ[q, q
−1] . Put Γ′ = (κN/2) · Z. Consider an extended Novikov ring
Λ′ := KΓ′ [q
2 , q−
2 ]. Define now QH ′∗(M) as QH∗(M) if NL is even, and as
H∗(M,Z2)⊗Z2 Λ
′ if NL is odd. In the latter case QH
∗(M) is an extension of
QH∗(M), and we shall consider without special mentioning QH∗(M), Λ, KΓ
as subrings of QH ′∗(M), Λ
′, KΓ′. The grading of QH
∗(M) is determined by
the condition deg q
2 = 1. As before, we shall use notation QH ′•(M), where
• = “even” when F = C and • = ∗ when F = Z2.
Note that the spectral invariants (and hence partial symplectic quasi-
states) are well-defined over the extended ring, and furthermore, their values
and properties, by tautological reasons, do not alter under such an extension
(cf. a discussion in [15], Section 5.4). Put w := sκNL/2qNL/2. Recall that j
stands for the natural morphism H•(L;F) → H•(M ;F).
Proposition 8.1. Assume that k > n+1−NL. If F = C assume in addition
that k is even. Then there exists a canonical homomorphism jq : Hk(L;F) →
QH ′k(M) with the following properties
8The letter “q” in jq stands for quantum.
(i) jq(x) = j(x) + w−1y, where y is a polynomial in w−1 with coefficients
in H•(M ;F);
(ii) jq([L]) = j([L]);
(iii) If jq(x) 6= 0 then c(jq(x), H) ≤ supLH for every H ∈ C
∞(M).
In particular, if S is an Albers element of L, we have jq(S) = j(S)+O(w−1) 6=
This proposition was proved by Biran and Cornea in [15] in the case
F = Z2: The map j
q is essentially the map iL appearing in Theorem A(iii)
in [15]. Proposition 8.1(i) above is a combination of Theorem A(iii) and
Proposition 4.5.1(i) in [15]. Our variable w corresponds to the variable t−1 in
[15], while our sNκqN corresponds to the variable s−1 in Section 2.1.2 of [15].
After such an adjustment of the notation, the formula w := sκNL/2qNL/2 above
can be extracted from Section 2.1.2 of [15]. For Proposition 8.1(ii) above
see Remark 5.3.2.a in [15]. Proposition 8.1(iii) above follows from Lemma
5.3.1(ii) in [15]. Finally, let us repeat the disclaimer made in Section 1.5: we
take for granted that Proposition 8.1 remains valid for the case F = C.
Let us pass to the proofs of our results on (super)-heaviness of monotone
Lagrangian submanifolds. We start with the following remark. Let S be an
Albers element of L. The Poincaré duality property of spectral invariants
(see Section 3.4 above) extends verbatim to the case of the ring QH ′(M)
with the following modification: When NL is odd, the pairing Π introduced
in Section 3.4 extends in the obvious way to a non-degenerate F -valued
pairing on QH ′•(M) which we still denote by Π. Applying Poincaré duality
and substituting H := −F into Proposition 8.1 (iii) above we get that for
every function F ∈ C∞(M)
c(T, F ) ≥ inf
F ∀T ∈ QH ′•(M) with Π(T, j
q(S)) 6= 0.
In particular, given a non-zero idempotent a ∈ QH ′•(M) and a class b ∈
QH ′•(M), so that Π(a∗b, j
q(S)) 6= 0, we get, using the normalization property
of spectral invariants, that
c(a, F ) + ν(b) ≥ c(a ∗ b, F ) ≥ inf
F ∀F ∈ C∞(M) . (27)
Therefore, applying (27) to kF for k ∈ N, dividing by k and passing to the
limit as k → +∞, we get that for the partial quasi-state ζ , defined by a,
ζ(F ) ≥ inf
F ∀F ∈ C∞(M),
meaning that L is heavy with respect to a.
Proof of Theorem 1.15: Let S be an Albers element of L. Let T ∈
H•(M ;F) be any singular homology class such that T ◦ j(S) 6= 0. Thus,
applying Proposition 8.1 (i) we see that Π([M ]∗T, jq(S)) = Π(T, jq(S)) 6= 0,
and hence inequality (27), applied to a = [M ], b = T , yields that L is heavy
with respect to [M ].
Let us pass to the proof of Theorem 1.25 on the effect of semi-simplicity
of the quantum homology. It readily follows from the next more general
statement. Let L1, . . . , Lm be Lagrangian submanifolds satisfying the Albers
condition. Let Si be any Albers element of Li. Denote by Zi = j
q(Si) ∈
QH ′•(M) its image under the inclusion morphism from Proposition 8.1 above.
Theorem 8.2. Given such L1, . . . , Lm and Z1, . . . , Zm, assume, in addition,
that QH2n(M) is semi-simple and the Lagrangian submanifolds L1, . . . , Lm
are pair-wise disjoint. Then the classes Z1, . . . , Zm are linearly independent
over KΓ′.
Proof. Arguing by contradiction, assume that
Z1 = α2Z2 + . . .+ αmZm (28)
for some α2, . . . , αm ∈ KΓ′ . Since QH2n(M) is semi-simple, it decomposes
into a direct sum of fields with unities e1, . . . , ed. Since the pairing Π (on
QH ′•(M ;F)) is non-degenerate, there exists T ∈ QH
•(M ;F) such that
Π(T, Z1) 6= 0. (29)
Let us write T as
T = [M ] ∗ T =
ei ∗ T. (30)
Equations (29), (30) imply that there exists l ∈ [1, d] such that
Π(el ∗ T, Z1) 6= 0 . (31)
Then (28) implies that there exists r ∈ [2, m] such that
Π(el ∗ T, αrZr) 6= 0.
Using (21) (for Π on QH ′•(M ;F)) we can rewrite the last equation as
Π(el ∗ αrT, Zr) 6= 0. (32)
Applying now formula (27) for S = Z1 ∈ H•(L1;F), a = el, b = T , and also
for S = Zr ∈ H•(Lr;F), a = el, b = αrT , we conclude that both L1 and Lr
are heavy with respect to el. Thus they are superheavy with respect to el,
because el is the unity in a field factor of QH2n(M) (see Section 1.6). Hence
they must intersect – in contradiction to the assumption of the theorem. This
finishes the proof of the first part of the theorem.
Proof of Theorem 1.25(a): Assume that L1, . . . , Lm are pair-wise disjoint
Lagrangian submanifolds satisfying the condition (a) from the formulation
of the theorem. Denote by Ni the minimal Maslov number of Li. Since
Ni > n + 1, the class of a point from H0(Li;F) is an Albers element for Li.
Let Zi ∈ QH
0(M) be its image under the Biran-Cornea inclusion morphism
associated to Li. Note that by Proposition 8.1(i) Zi = p + aiw
i , where
wi = s
κNi/2qNi/2, ai ∈ HNi(M ;F) and p ∈ H0(M ;F) is the homology class of
a point. Observe that degwi = Ni > n + 1, and hence the expression for Zi
cannot contain terms in w−1i of order two and higher, since HkNi(M ;F) = 0
for k ≥ 2.
Recall now that all Ni’s lie in some set Y of positive integers. Let W ⊂
QH ′0(M) be the span over KΓ′ of
H0(M ;F)⊕
s−κE/2q−E/2 ·HE(M ;F) .
Note that
dimKΓ′ W = βY (M) + 1 < m .
Thus the elements Zi, i = 1, . . . , m, are linearly dependent over KΓ′ . By
Theorem 8.2, QH2n(M) is not semi-simple.
Proof of Theorem 1.25(b): Assume that L1, . . . , Lm are pair-wise disjoint
homologically non-trivial Lagrangian submanifolds. By Proposition 8.1(ii)
jq([Li]) = j([Li]) for every i = 1, . . . , m. Since the classes j([Li]) are linearly
dependent, Theorem 8.2 implies that QH2n(M) is not semi-simple.
Proof of Theorem 1.18: Combining Proposition 8.1 (ii) and (iii) we get
that for any H ∈ C∞(M)
c(j([L]), H) ≤ sup
H ∀H ∈ C∞(M) .
By the hypothesis of the theorem, we can write j([L]) ∗ b = a for some b.
c(a,H) = c(j([L]) ∗ b,H) ≤ c(j([L]), H) + c(b, 0) .
c(a,H) ≤ sup
H + c(b, 0) .
Applying this inequality to E · H with E > 0, dividing by E and passing
to the limit as E → +∞ we get that ζ(H) ≤ supLH for all H . Thus L is
superheavy.
Remark 8.3. The results above admit the following generalizations in the
framework of the Biran-Cornea theory. The main object of this theory is the
quantum homology ring QH∗(L) of a monotone Lagrangian submanifold,
which is isomorphic to the Lagrangian Floer homology HF∗(L, L) of L up to
a shift of the grading.
(i) If QH∗(L) does not vanish then L is heavy (see Remark 1.2.9b in [15]).
In fact, it follows from [15] that if L satisfies the Albers condition,
QH∗(L) does not vanish.
(ii) The map jq of the Proposition 8.1 above is a footprint of the quan-
tum inclusion map iL : QH∗(L) → QH
∗(M) constructed in [15]. The
analogue of the action estimate in item (iii) of the proposition is ob-
tained in [15] for the classes iL(x) for elements x ∈ QH∗(L) of a certain
special form, yielding the following generalization of Theorem 1.18: for
these special classes x ∈ QH∗(L) the condition that the class iL(x)
does not vanish and divides a non-trivial idempotent a implies that L
is superheavy with respect to a. This enables, for instance, to general-
ize Example 1.19 on Lagrangian spheres in quadrics above to the case
when dimL is odd.
(iii) In [15] one can find another action estimate which comes from the
QH∗(M)-module structure on QH∗(L), which yields more results on
(super)heaviness of monotone Lagrangian submanifolds.
Proof of Proposition 1.4: The quantum homology QH2n(M) splits as an
algebra over K into a direct sum of two algebras one of which is a field. This
was proved by McDuff for the field F = C (see [39] and [24, Section 7]), but
the proof goes through for the case F = Z2 as well. Denote the unity of the
field by a. It is a non-zero idempotent in QH2n(M). As we already pointed
out in Remark 1.21, such an idempotent a defines a genuine symplectic quasi-
state and therefore the classes of heavy and superheavy sets with respect to
a coincide.
By Theorem 1.2, the Lagrangian torus L ⊂ M cannot be superheavy
with respect to a, since it can be displaced from itself by a symplectic (non-
Hamiltonian) isotopy. Indeed, take an obvious symplectic isotopy φt of T
that displaces L (a parallel shift) and compose it with a Hamiltonian isotopy
ψt so that for every t we have that ψt is constant on φt(L) and ψtφt is identity
on the ball where the blow up of T2n was performed. Clearly, the resulting
symplectic isotopy ψtφt extends to a symplectic isotopy of M that displaces
On the other hand, NL ≥ 2 because in this case NL = 2N , where N ≥ 1
is the minimal Chern number of M . Finally, note that L represents a non-
trivial homology class in Hn(M ;Z2). Therefore we can apply Theorem 1.15
and get that L is heavy with respect to [M ].
9 Rigidity of special fibers of Hamiltonian ac-
tions
In this section we prove Theorem 1.9. Denote the special fiber of Φ by
L := Φ−1(pspec).
Reduction to the case of T1-actions: First, we claim that it is enough
to prove the theorem for Hamiltonian T1-actions and the general case will
follow from it. Indeed, assume this is proved. The superheaviness of the
special fiber immediately yields that for any function H̄ : R → R
ζ(Φ∗H̄) = H̄(pspec), (33)
where Φ :M → R is the moment map of the T1-action.
Let us turn to the multi-dimensional situation and let Φ : M → Rk
be the normalized moment map of a Hamiltonian Tk-action on M . For a
v ∈ Rk denote by Φv(x) = 〈v,Φ(x)〉, where 〈·, ·〉 is the standard Euclidean
inner product on Rk. Note that if v ∈ Zk the function Φv is the normalized
moment map of a Hamiltonian circle action and its special value is 〈v, pspec〉.
Thus by (33)
K) = K(〈v, pspec〉) ∀K ∈ C
∞(R) . (34)
By homogeneity of ζ , equality (34) holds for all v ∈ Qk, and by continuity
for all v ∈ Rk.
Observe that for each pair of smooth functions P,Q ∈ C∞(R) and for each
pair of vectors v,w ∈ Rk the functions Φ∗
P and Φ∗
Q Poisson-commute on
M . Thus the triangle inequality for the spectral numbers (see Section 3.4)
yields
P + Φ∗
Q) ≤ ζ(Φ∗
P ) + ζ(Φ∗
Q) . (35)
Since M is compact, it suffices to assume that the function H̄ ∈ C∞(Rk) on
Rk is compactly supported. By the inverse Fourier transform we can write
H̄(p) =
sin〈v, p〉 · F (v) + cos〈v, p〉 ·G(v)
for some rapidly (say, faster than (|p| + 1)−N for any N ∈ N) decaying
functions F and G on Rk. For every v ∈ Rk define a function Kv ∈ C
Kv(s) := sin s · F (v) + cos s ·G(v) .
Observe that
Φ∗H̄ =
Kv dv .
Denote by B(R) the Euclidean ball of radius R in Rk with the center at the
origin. Put
H̄R(p) =
Kv(〈v, p〉) dv, p ∈ R
Since the functions F and G are rapidly decaying, we get that
||H̄R − H̄||C0(Rk) → 0 as R → ∞ . (36)
We claim that for every R
ζ(Φ∗H̄R) ≤ H̄R(pspec) . (37)
Indeed, for ǫ > 0 introduce the integral sum
H̄R,ε(p) =
v∈ ε·Zk∩B(R)
εk ·Kv(〈v, p〉) .
Φ∗H̄R,ε =
v∈ ε·Zk∩B(R)
εk · Φ∗
Applying repeatedly (35) and (34) we get that
ζ(Φ∗H̄R,ε) ≤ H̄R,ε(pspec) .
Note now that for fixed R the family H̄R,ǫ converges to H̄R as ε → 0 in
the uniform norm on C0(Rk). Using that ζ is Lipschitz with respect to the
uniform norm on C0(M) we readily get the inequality (37).
Combining the fact that ζ is Lipschitz with (36) and (37) we get that
ζ(Φ∗H̄) = lim
ζ(Φ∗H̄R) ≤ lim
H̄R(pspec) = H̄(pspec) .
Now, assume that H̄ ≥ 0 and H̄(pspec) = 0. We just have proved that
ζ(Φ∗H̄) ≤ 0, and hence ζ(H) = 0, which immediately yields the desired su-
perheaviness of the special fiber. This completes the reduction of the general
case to the 1-dimensional case.
From now on we will consider only the case of an effective Hamil-
tonian T1-action on M with a moment map Φ :M → R. Its moment
polytope ∆ is a closed interval in R and pspec = −I(Φ) ∈ R.
Reduction to the case of a strictly convex function: We claim
that it is enough to show the following proposition:
Proposition 9.1. Assume H̄ : R → R is a strictly convex smooth function
reaching its minimum at pspec. Set H := Φ
∗H̄. Then ζ(H) = H̄(pspec).
Postponing the proof of the proposition for a moment let us show that it
implies the theorem. Indeed, let F : M → R be a Hamiltonian on M . In
order to show the superheaviness of L = Φ−1(pspec) we need to show that
ζ(F ) ≤ supL F . Pick a very steep strictly convex function H̄ : R → R with
the minimum value supL F reached at pspec and such that Φ
∗H̄ =: H ≥ F
everywhere on M . Then using Proposition 9.1 and the monotonicity of ζ we
ζ(F ) ≤ ζ(H) = H̄(pspec) = sup
yielding the claim.
Preparations for the proof of Proposition 9.1: Given a (time-
dependent, not necessarily regular) Hamiltonian G, we associate to every
pair [γ, u] ∈ P̃G a number
DG([γ, u]) := AG([γ, u])−
· CZG([γ, u]).
(Recall that we defined the Conley-Zehnder index for all Hamiltonians and
not only the regular ones – see Section 3.3). The number DG([γ, u]) is in-
variant under a change of the spanning disc u – an addition of a sphere
jS ∈ HS2 (M) to the disc u changes both AG([γ, u]) and κ/2 · CZG([γ, u]) by
the same number. Thus we can write DG([γ, u]) = DG(γ).
Given [γ, u] ∈ P̃G and l ∈ N define γ
(l) and u(l) as the compositions
of γ and u with the map z → zl on the unit disc D2 ⊂ C (here z is a
complex coordinate on C). Denote by t 7→ gt the time-t flow of G and by
G(l) :M × R → R the Hamiltonian whose time-t flow is t 7→ (gt)
l and which
is defined by
G(l) := G♯ . . . ♯G (l times),
where G♯K(x, t) := G(x, t) +K(g−1t x, t) for any K :M × R → R.
Proposition 9.2. There exists a constant C > 0, depending only on n, with
the following property. Given a 1-periodic orbit γ ∈ PG of the flow t 7→ gt
generated by G, assume that γ(l) is a 1-periodic orbit of the flow t 7→ glt
generated by G(l), and therefore for any u such that [γ, u] ∈ P̃G we have
[γ(l), u(l)] ∈ P̃G(l). Then
|DG(l)([γ
(l), u(l)])− lDG([γ, u])| ≤ l · C.
Proof. The action term in DG gets multiplied by l as we pass from G to G
As for the Conley-Zehnder term, the quasi-morphism property of the Conley-
Zehnder index (see Proposition 3.5) implies that there exists a constant C > 0
(depending only on n) such that
|lCZG[γ, u]− CZG(l)([γ
(l), u(l)])| ≤ C.
This immediately proves the proposition.
Proposition 9.3. Let G :M × [0, 1] → R be a Hamiltonian as above. Then
one can choose ǫ > 0, depending on G, and a constant Cn > 0, depending
only on n = dimM/2, so that any function F : M × [0, 1] → R which is ǫ-
close to G in a C∞-metric on C∞(M×[0, 1]) satisfies the following condition:
for every γ0 ∈ PF there exists γ ∈ PG such that the difference between DF (γ0)
and DG(γ) is bounded by Cn.
Proof. Denote the flow of G by gt (as before) and the flow of F by ft. We
will view time-1 periodic trajectories of these flows both as maps of [0, 1] to
M having the same value at 0 and 1 and as maps from S1 to M .
First, consider the fibration D2×M →M and, slightly abusing notation,
denote the natural pullback of ω again by ω. Second, look at the fibration
pr : D2 ×M → D2. Denote by V ert the vertical bundle over D2 ×M formed
by the tangent spaces to the fibers of pr. For each loop σ : S1 →M define by
σ̂ : S1 → D2 ×M the map σ̂(t) := (t, γ(t)). The bundles σ∗TM and σ̂∗V ert
over S1 coincide. Similarly for each w : D2 →M denote by ŵ : D2 → D2×M
the map ŵ(z) := (z, w(z)).
There exists δ > 0, depending on G, such that for each γ ∈ PG a tubular
δ-neighborhood of the image of γ̂ in S1 ×M ⊂ D2 ×M , denoted by Ubγ, has
the following properties:
• there exists a 1-form λ on Ubγ satisfying dλ = ω;
• V ert admits a trivialization over Ubγ .
Given an ǫ > 0, we can choose F sufficiently C∞-close to G so that the
paths t 7→ ft and t 7→ gt in Ham(M) are arbitrarily C
∞-close and therefore
• for every x ∈ Fix (F ) there exists y ∈ Fix (G) which is ǫ-close to x
(think of the fixed points as points of intersection of the graph of a
diffeomorphism with the diagonal);
• the C∞-distance between the maps γ0 : t 7→ ft(x) and γ : t 7→ gt(y)
from [0, 1] to M is bounded by ǫ and the image of γ̂0 lies in Ubγ.
Pick a map u0 : D
2 → M , u|∂D2 = γ0. Since γ0 and γ are C
∞-close one
can enlarge D2 to a bigger disc D21 ⊃ D
2 and find a smooth map u : D21 →M
so that
• u|∂D21
= u0;
• u(D21 \ D
2) ⊂ Ubγ.
Rescaling D21 we may assume without loss of generality that [γ, u] ∈ PG.
Trivialize the vector bundles γ∗0TM and γ
∗TM so that the trivializations
extend to a trivialization of u∗TM over D21 (and hence of u
0TM over D
Using the trivializations we can identify the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
with some identity-based paths of symplectic matrices A(t), B(t). Fixing a
small ǫ as above, we can also assume that F is chosen so C∞-close to G that,
in addition to all of the above, the C∞-distance between the paths t 7→ A(t)
and t 7→ B(t) in Sp (2n) is bounded by ǫ (for instance, make sure first that
the matrix paths obtained by writing the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
using some trivialization of V ert over Ubγ are close enough – then the matrix
paths t 7→ A(t) and t 7→ B(t) will also be close enough).
We claim that by choosing ǫ sufficiently small in the construction above we
can bound the difference between DF ([γ0, u0]) and DG([γ, u]) by a quantity
depending only on dimM .
Indeed, the difference |
F (γ0(t), t)dt −
G(γ(t))dt| is bounded by a
quantity depending only on some universal constants and ǫ, because γ0 is
ǫ-close to γ and F is ǫ-close to G with respect to the C∞-metrics. It can be
made arbitrarily small by choosing a sufficiently small ǫ. The difference
u∗0ω −
u∗ω| = |
û∗0ω −
û∗ω|
is bounded by the difference |
γ̂∗0λ −
γ̂∗λ|. Since, γ0 and γ are ǫ-close
in the C∞-metric the later difference can be made less than 1 if we choose
a sufficiently small ǫ. Thus we have shown that by choosing a sufficiently
small ǫ we can bound |AF ([γ0, u0])−AG([γ, u])| by 1.
Now, as far as the Conley-Zehnder indices are concerned, our choice
of the trivializations means that the difference between CZF ([γ0, u0]) and
CZG([γ, u]) is just the difference between the Conley-Zehnder indices for the
matrix paths t 7→ A(t) and t 7→ B(t). But the latter paths in Sp (2n) are
ǫ-close in the C∞-sense, hence represent close elements of S̃p (2n) and if ǫ
was chosen sufficiently small, then, as we mentioned in Section 3.3, their
Conley-Zehnder indices differ at most by a constant depending only on n.
This finishes the proof of the claim and the proposition.
Plan of the proof of Proposition 9.1: We assume now that H̄ is
a fixed strictly convex function on R. Our calculations will feature E as a
large parameter. For quantities α, β depending on E we will write α � β if
α ≤ β+const holds for large enough E, where const depends only on (M,ω),
Φ and H̄ , and in particular does not depend on E. We will write α ≈ β if
α � β and β � α. Using this language the proposition can be restated as
c(a, EH) ≈ EH̄(pspec). (38)
In general, 1-periodic orbits of the flow of EH are not isolated and there-
fore the Hamiltonian is not regular. Let F be a regular (time-periodic)
perturbation of EH .
By the spectrality axiom, the spectral number c(a, F ) for a ∈ QH2n(M)
equals AF ([γ0, u0]) for some pair [γ0, u0] ∈ P̃F with CZF ([γ0, u0]) = 2n. Thus
c(a, F ) ≈ DF (γ0). Combining this with Proposition 9.3 we get that for some
γ ∈ PEH
EH̄(pspec) � c(a, EH) ≈ c(a, F ) ≈ DF (γ0) ≈ DEH(γ) . (39)
Thus it would be enough to show that
DEH(γ) � EH̄(pspec) for all γ ∈ PEH , (40)
which together with (39) would imply (38).
Inequality (40) will be proved in the following way. Note that each γ ∈
PEH lies in Φ
−1(p) for some p ∈ ∆. We will show that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p). (41)
Note that (41) implies (40). Indeed, since H̄ is strictly convex and reaches
its minimum at pspec, it follows from (41) that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p) ≤ EH̄(pspec),
which is true for any γ ∈ PEH thus yielding (40).
Proof of (41): Let the T1-action on M be given by a loop of sym-
plectomorphisms {φt}, t ∈ R, φt = φt+1. The flow of EH has the form
htx = φEH̄′(Φ(x))tx.
We view γ as a map γ : [0, 1] → M satisfying γ(0) = γ(1). Denote
x := γ(0). The curve γ lies in Φ−1(p).
Denote N := γ([0, 1]). This is the T1-orbit of x and it is either a point or
a circle.
In the first case γ is a constant trajectory concentrated at a fixed point
N ∈M of the action. Using this constant curve γ together with the constant
disc u spanning for the definitions of I(Φ) and DEH(γ) one gets
pspec − p = mΦ(γ, u) · κ/2,
DEH(γ) = EH̄(p)− κ/2 · CZEH([γ, u]).
Thus proving (41) reduces in this case to proving
−CZEH([γ, u]) ≈ EH̄
′(p) ·mΦ(γ, u).
Let us fix a symplectic basis of TNM and view each differential dNφt as a
symplectic matrix A(t), so that {A(t)} is an identity-based loop in Sp (2n).
−CZEH([γ, u]) ≈ CZmatr({A(EH̄
′(p)t)}),
while
EH̄ ′(p) ·mΦ(γ, u) ≈ EH̄
′(p)Maslov({A(t)}).
Thus we need to prove
CZmatr({A(EH̄
′(p)t)}) ≈ EH̄ ′(p)Maslov({A(t)}),
which follows easily from the definitions of the Conley-Zehnder index and
the Maslov class.
Thus from now on we will assume that N is a circle. Take any point
x ∈ N . The stabilizer of x under the T1-action is a finite cyclic group of
order k ∈ N. Thus the orbit of the T1-action turns k times along N . Since γ
is a non-constant closed orbit of the Hamiltonian flow generated by EΦ∗H̄ ,
it turns r times along N with r ∈ Z \ {0}. This implies that EH̄ ′(p) = r/k.
We claim that without loss of generality we may assume that l := r/k is an
integer.
Indeed, we can always pass to γ(k) ∈ PkEH , so that (kEH̄)
′(p) ∈ Z, and
if we can prove the proposition for γ(k), then
DkEH(γ
(k)) ≈ kEH̄(p) + kEH̄ ′(p)(pspec − p).
Applying Proposition 9.2 we get
kDEH(γ) ≈ kEH̄(p) + kEH̄
′(p)(pspec − p) + k · const ,
and hence
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p),
proving the claim for the original γ.
From now on we assume that l := EH̄ ′(p) ∈ Z\{0} and that [γ, u] ∈ P̃lΦ.
Consider the Hamiltonian vector field X := sgradΦ at a point x ∈ N . Since
N is a non-constant orbit we get X 6= 0. Then V = Tx(Φ
−1(p)) is the skew-
orthogonal complement to X . Choose a T1-invariant ω-compatible almost
complex structure J in a neighborhood of N . Together ω and J define a T1-
invariant Riemannian metric g. Decompose the tangent bundle TM along
N as follows. Put Z = Span(JX,X) and set W to be the g-orthogonal
complement to X in V . Thus we have a T1-invariant decomposition
TxM = W ⊕ Z , x ∈ N . (42)
Furthermore, W and Z carry canonical symplectic forms. Thus W and Z
define symplectic (and hence trivial) subbundles of TM over N . They induce
trivial subbundles of the bundle γ∗TM over S1.
We calculate
dht(x)ξ = dφEH′(Φ(x))t(x)ξ + EH
′′(Φ(x)) · dΦ(ξ) ·X . (43)
We consider two trivializations of the bundle γ∗TM over S1. The first trivi-
alization is defined by means of sections invariant under the T1-action. The
second one is chosen in such a way that it extends to a trivialization of u∗TM
over D2. Using these trivializations we can identify dht(x), respectively, with
two identity-based paths {Ct}, {C
t} of symplectic matrices. The decompo-
sition (42) induces a split
Ct = 1⊕ Bt .
We claim that |CZmatr({Bt})| is bounded by a constant independent of E.
Indeed, observe that in the basis (X, JX) of Z
1 b12(t)
Denote by L the line spanned by X = (1, 0). Perturb {Bt} to a path {B
RδtBt}, where Rt is the rotation by angle t, and δ > 0 is small enough.
Observe thatB′(t)L∩L = {0} for t > 0. It follows readily from the definitions
that |CZmatr(B
t)| and |CZmatr(Rδt)| do not exceed 2. Thus by the quasi-
morphism property of the Conley-Zehnder index (see Proposition 3.5) we
have that |CZmatr({Bt})| is bounded by a constant independent of E, which
yields the claim. Therefore
CZmatr ({Ct}) ≈ 0 .
On the other hand, by formula (18)
CZmatr ({C
t}) = CZmatr ({Ct}) +mlΦ([γ, u]) .
CZEH([γ, u]) := n− CZmatr ({C
t}) ≈ −mlΦ([γ, u]). (44)
Since the periodic trajectory γ lies inside Φ−1(p), we get
AEH([γ, u]) =
EH(γ(t))dt−
u∗ω = EH̄(p)−
u∗ω. (45)
Using (45) and (44) the precise equality
DEH([γ, u]) = AEH([γ, u])−
· CZEH([γ, u])
can be turned into an asymptotic inequality
DEH([γ, u]) ≈ EH̄(p)−
u∗ω +
mlΦ([γ, u]). (46)
Since the periodic trajectory γ lies inside Φ−1(p), we have
AlΦ([γ, u]) =
lΦ(γ(t))dt−
u∗ω = lp−
u∗ω. (47)
Adding and subtracting lp from the right-hand side of (46) and using (47)
we get
DEH(γ) = DEH([γ, u]) ≈
EH̄(p)− lp)
mlΦ([γ, u])
EH̄(p)− lp
AlΦ([γ, u])+
mlΦ([γ, u])
EH̄(p)− lp
− I(lΦ) =
= EH̄(p) + l(−I(Φ)− p) = EH̄(p) + l(pspec − p) .
Recalling that l = EH ′(p), we finally obtain that
DEH(γ) = EH̄(p) + EH
′(p)(pspec − p),
which is precisely the equation (41) that we wanted to get. This finishes the
proof of Proposition 9.1 and Theorem 1.9.
9.1 Calabi and mixed action-Maslov
Proof of Theorem 1.13.
Assume H :M × [0, 1] → R is a normalized Hamiltonian which generates
a loop in Ham(M) representing a class α ∈ π1(Ham(M)) ⊂ H̃am (M). Then
H(l) is also normalized and generates a loop representing αl. Let us compute
µ(α) = −vol (M) · liml→+∞ c(a,H
(l))/l.
Arguing as in the proof of (39) we get that there exists a constant C >
0 such that for each l ∈ N there exists γ ∈ PH(l) for which |c(a,H
(l)) −
DH(l)(γ)| ≤ C. But, as it follows from the definitions and from the fact that
I is a homomorphism, DH(l)(γ) does not depend on γ and equals −I(α
−lI(α). This immediately implies that µ(α) = vol (M) · I(α).
Acknowledgements. The origins of this paper lie in our joint work with
P.Biran on the paper [10] – we thank him for fruitful collaboration at an
early stage of this project, as well as for his crucial help with Example 1.17
on Lagrangian spheres in projective hypersurfaces. We also thank him and
O.Cornea for pointing out to us a mistake in the original version of this paper
and helping us with the correction (see Section 8). We thank F. Zapolsky
for his help with the “exotic” monotone Lagrangian torus in S2 × S2 dis-
cussed in Example 1.20. We thank C. Woodward for pointing out to us the
link between the special point in the moment polytope of a symplectic toric
manifold and the Futaki invariant, and E. Shelukhin for useful discussions
on this issue. We are also grateful to V.L. Ginzburg, Y. Karshon, Y. Long,
D. McDuff, M. Pinsonnault, D. Salamon and M. Sodin for useful discussions
and communications. We thank K. Fukaya, H. Ohta and K. Ono, the orga-
nizers of the Conference on Symplectic Topology in Kyoto (February 2006),
M. Harada, Y. Karshon, M. Masuda and T. Panov, the organizers of the Con-
ference on Toric Topology in Osaka (May 2006), O. Cornea, V.L. Ginzburg,
E. Kerman and F. Lalonde, the organizers of the Workshop on Floer theory
(Banff, 2007), and A. Fathi, Y.-G. Oh and C. Viterbo, the organizers of the
AMS Summer Conference on Symplectic Topology and Measure-Preserving
Dynamical Systems (Snowbird, July 2007), for giving us an opportunity to
present a preliminary version of this work and for the superb job they did
in organizing these conferences. Finally, we thank an anonymous referee for
helpful comments and corrections.
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Michael Entov Leonid Polterovich
Department of Mathematics School of Mathematical Sciences
Technion Tel Aviv University
Haifa 32000, Israel Tel Aviv 69978, Israel
[email protected] [email protected]
http://arxiv.org/abs/0709.1127
Introduction and main results
Many facets of displaceability
Preliminaries on quantum homology
An hierarchy of rigid subsets within Floer theory
Hamiltonian torus actions
Super(heavy) monotone Lagrangian submanifolds
An effect of semi-simplicity
Discussion and open questions
Strong displaceability beyond Floer theory?
Heavy fibers of Poisson-commutative subspaces
Detecting stable displaceability
Preliminaries on Hamiltonian Floer theory
Valuation on QH* (M)
Hamiltonian Floer theory
Conley-Zehnder and Maslov indices
Spectral numbers
Partial symplectic quasi-states
Basic properties of (super)heavy sets
Products of (super)heavy sets
Product formula for spectral invariants
Decorated Z2-graded complexes
Reduced Floer and Quantum homology
Proof of Theorem 5.1
Proof of algebraic Theorem 5.2
Stable non-displaceability of heavy sets
Analyzing stable stems
Monotone Lagrangian submanifolds
Rigidity of special fibers of Hamiltonian actions
Calabi and mixed action-Maslov
|
0704.0106 | Multiple Parton Scattering in Nuclei: Quark-quark Scattering | Multiple Parton Scattering in Nuclei:
Quark-quark Scattering
Andreas Schäfer, a Xin-Nian Wang b and Ben-Wei Zhang c,d,a
aInstitut für Theoretische Physik, Universität Regensburg
D-93040 Regensburg, Germany
bNuclear Science Division, MS 70R0319
Lawrence Berkeley National Laboratory, Berkeley, CA 94720
cCyclotron Institute and Physics Department, Texas A&M University
College Station, Texas 77843-3366
dInstitute of Particle Physics, Central China Normal University
Wuhan 430079, China
Abstract
Modifications to quark and antiquark fragmentation functions due to quark-quark
(antiquark) double scattering in nuclear medium are studied systematically up to
order O(α2s) in deeply inelastic scattering (DIS) off nuclear targets. At the order
O(α2s), twist-four contributions from quark-quark (antiquark) rescattering also ex-
hibit the Landau-Pomeranchuck-Midgal (LPM) interference feature similar to gluon
bremsstrahlung induced by multiple parton scattering. Compared to quark-gluon
scattering, the modification, which is dominated by t-channel quark-quark (anti-
quark) scattering, is only smaller by a factor of CF /CA = 4/9 times the ratio of
quark and gluon distributions in the medium. Such a modification is not negli-
gible for realistic kinematics and finite medium size. The modifications to quark
(antiquark) fragmentation functions from quark-antiquark annihilation processes
are shown to be determined by the antiquark (quark) distribution density in the
medium. The asymmetry in quark and antiquark distributions in nuclei will lead
to different modifications of quark and antiquark fragmentation functions inside
a nucleus, which qualitatively explains the experimentally observed flavor depen-
dence of the leading hadron suppression in semi-inclusive DIS off nuclear targets.
The quark-antiquark annihilation processes also mix quark and gluon fragmentation
functions in the large fractional momentum region, leading to a flavor dependence
of jet quenching in heavy-ion collisions.
Key words: Jet quenching, modified fragmentation, parton energy loss.
PACS: 24.85.+p, 12.38.Bx, 13.87.Ce, 13.60.-r
Preprint submitted to Elsevier 30 October 2018
http://arxiv.org/abs/0704.0106v2
1 Introduction
Multiple parton scattering in a dense medium can be used as a useful tool to study
properties of both hot and cold nuclear matter. The success of such an approach has been
demonstrated by the discovery of strong jet quenching phenomena in central Au + Au
collisions at the Relativistic Heavy-ion Collider (RHIC) [1,2,3] and their implications
on the formation of a strongly coupled quark-gluon plasma at RHIC [4,5]. However,
for a convincing phenomenological study of the existing and future experimental data,
a unified description of all medium effects in hard processes involving nuclei, such as
electron-nucleus (e + A), hadron-nucleus (h + A) and nucleus-nucleus collisions (A +
A) has to be developed [6,7]. This must include the physics of transverse momentum
broadening [8], strong nuclear enhancement in DIS [9] and Drell-Yan production [10,11],
nuclear shadowing [12], and parton energy loss due to gluon radiation induced by multiple
scattering [13,14,15,16,17,18,19].
There exist many different frameworks in the literature to describe multiple scattering
in a nuclear medium [20,21,22]. Among them the twist expansion approach is based on
the generalized factorization in perturbative QCD as initially developed by Luo, Qiu
and Sterman (LQS) [23]. In the LQS formalism, multiple scattering processes generally
involve high-twist multiple-parton correlations in analogy to the parton distribution op-
erators in leading twist processes. Though the corresponding higher twist corrections are
suppressed by powers of 1/Q2, they are enhanced at least by a factor of A1/3 due to
multiple scattering in a large nucleus. This framework has been applied recently to study
medium modification of the fragmentation functions as the leading parton propagates
through the medium [18,19]. Because of the non-Abelian Landau-Pomeranchuck-Midgal
interference in the gluon bremsstrahlung induced by multiple parton scattering in nuclei,
the higher-twist nuclear modifications to the fragmentation functions are in fact enhanced
by A2/3, quadratic in the nuclear size [18,19]. Phenomenological study of parton energy
loss and nuclear modification of the fragmentation functions in cold nuclear matter [24]
gives a good description of the nuclear modification of the leading hadron spectra in semi-
inclusive deeply inelastic lepton-nucleus scattering observed by the HERMES experiment
[25,26]. The same framework also gives a compelling explanation for the suppression of
large transverse momentum hadrons discovered at RHIC [27].
The emphasis of recent studies of medium modification of fragmentation functions has
been on radiative parton energy loss induced by multiple scattering with gluons. Such
processes indeed are dominant relative to multiple scattering with quarks because of the
abundance of soft gluons in either cold nuclei or hot dense matter produced in heavy-ion
collisions. Since gluon bremsstrahlung induced by scattering with medium gluons is the
same for quarks and anti-quarks, one also expects the energy loss and fragmentation
modification to be identical for quarks and anti-quarks. However, in a medium with fi-
nite baryon density such as cold nuclei and the forward region of heavy-ion collisions, the
difference between quark and anti-quark distributions in the medium should lead to differ-
ent energy loss and modified fragmentation functions for quarks and antiquarks through
quark-antiquark annihilation processes. To study such an asymmetry, one must consider
systematically all possible quark-quark and quark-antiquark scattering processes, which
will be the focus of this paper.
In this study we will calculate the modifications of quark and antiquark fragmentation
functions (FF) due to quark-quark (antiquark) double scattering in a nuclear medium,
working within the LQS framework for generalized factorization in perturbative QCD.
For a complete description of nuclear modification of the single inclusive hadron spec-
tra, one still have to consider medium modification of gluon fragmentation functions in
addition to modified quark fragmentation function due to quark-gluon scattering [18].
The theoretical results presented in this paper will be a second step toward a complete
description of medium modified fragmentation functions. However, one can already find
that quark-quark (antiquark) double scattering will give different corrections to quark
and antiquark FF, depending on antiquark and quark density of the medium, respec-
tively. This difference between modified quark and antiquark FF may shed light on the
interesting observation by the HERMES experiment [25,26] of a large difference between
nuclear suppression of the leading proton and antiproton spectra in semi-inclusive DIS
off large nuclei. Such a picture of quark-quark (antiquark) scattering can provide a com-
peting mechanism for the experimentally observed phenomenon in addition to possible
absorption of final state hadrons inside nuclear matter [28,29].
The paper is organized as follows. In the next section we will present the general for-
malism of our calculation including the generalized factorization of twist-4 processes.
In Section III we will illustrate the procedure of calculating the hard partonic parts of
quark-quark double scattering in nuclei. In Section IV we will discuss the modifications
to quark and antiquark fragmentation functions due to quark-quark (antiquark) dou-
ble scattering in nuclei. In Section V, we will focus on the flavor dependent part of the
medium modification to the quark FF’s due to quark-antiquark annihilation and we will
discuss the implications for the flavor dependence of the leading hadron spectra in both
DIS off a nucleus and heavy-ion collisions. We will summarize our work in Section VI. In
the Appendix A-1, we collect the complete results for the hard partonic parts for different
cut diagrams of quark-quark (antiquark) double rescattering in nuclei. We also provide
an alternative calculation of the hard parts of the central-cut diagrams in Appendix A-3
through elastic quark-quark scattering or quark-antiquark annihilation as a cross check.
2 General formalism
In order to study quark and antiquark FF’s in semi-inclusive deeply inelastic lepton-
nucleus scattering, we consider the following processes,
e(L1) + A(p) −→ e(L2) + h(ℓh) +X ,
Ap Ap
Fig. 1. Lowest order and leading-twist contribution to semi-inclusive DIS.
where L1 and L2 are the four momenta of the incoming and outgoing leptons, and ℓh
is the observed hadron momentum. The differential cross section for the semi-inclusive
process can be expressed as
EL2Eℓh
dσhDIS
d3L2d3ℓh
LµνEℓh
dW µν
, (1)
where p = [p+, 0, 0⊥] is the momentum per nucleon in the nucleus, q = L2 − L1 =
[−Q2/2q−, q−, 0⊥] the momentum transfer carried by the virtual photon, s = (p + L1)2
the lepton-nucleon center-of-mass energy and αEM is the electromagnetic (EM) coupling
constant. The leptonic tensor is given by Lµν = 1/2Tr(γ · L1γµγ · L2γν) while the semi-
inclusive hadronic tensor is defined as,
〈A|Jµ(0)|X, h〉〈X, h|Jν(0)|A〉2πδ4(q + p− pX − ℓh) (2)
where
X runs over all possible final states and Jµ =
q eqψ̄qγµψq is the hadronic EM
current.
Assuming collinear factorization in the parton model, the leading-twist contribution to
the semi-inclusive cross section can be factorized into a product of parton distributions,
parton fragmentation functions and the partonic cross section. Including all leading log
radiative corrections, the lowest order contribution [O(α0s)] from a single hard γ∗ + q
scattering, as illustrated in Fig. 1, can be written as
dW Sµν
dxfAq (x, µ
µν (x, p, q)Dq→h(zh, µ
2) ; (3)
H(0)µν (x, p, q) =
Tr(γ · pγµγ · (q + xp)γν)
2p · q
δ(x− xB) , (4)
where the momentum fraction carried by the hadron is defined as zh = ℓ
−, xB =
Q2/2p+q− is the Bjorken scaling variable, µ2I and µ
2 are the factorization scales for
the initial quark distributions fAq (x, µ
I) in a nucleus and the fragmentation functions
in vacuum Dq→h(zh, µ
2), respectively. The renormalized quark fragmentation function
Dq→h(zh, µ
2) satisfies the DGLAP QCD evolution equations [30]:
∂Dq→h(zh, µ
∂ lnµ2
γq→qg(z)Dq→h(zh/z, µ
+ γq→gq(z)Dg→h(zh/z, µ
; (5)
∂Dg→h(zh, µ
∂ lnµ2
γg→qq̄(z)Dq→h(zh/z, µ
+ γg→gg(z)Dg→h(zh/z, µ
, (6)
where γa→bc(z) denotes the splitting functions of the corresponding radiative processes
[31,32].
In DIS off a nuclear target, the propagating quark will experience additional scatterings
with other partons from the nucleus. The rescatterings may induce additional parton
(quark or gluon) radiation and cause the leading quark to lose energy. Such induced
radiation will effectively give rise to additional terms in the DGLAP evolution equation
leading to a modification of the fragmentation functions in a medium. These are power-
suppressed higher-twist corrections and they involve higher-twist parton matrix elements.
We will only consider those contributions that involve two-parton correlations from two
different nucleons inside the nucleus. They are proportional to the thickness of the nucleus
[18,23,33] and thus are enhanced by a nuclear factor A1/3 as compared to two-parton
correlations in a nucleon. As in previous studies [18,19], we will limit our study to such
double scattering processes in a nuclear medium. These are twist-four processes and give
leading contributions to the nuclear effects. The contributions of higher twist processes or
contributions not enhanced by the nuclear medium will be neglected for the time being.
When considering double scattering with nuclear enhancement, a very important process
is quark-gluon double scattering as illustrated in Fig. 2. Such processes give the dominant
contribution to the leading quark energy loss and have been studied in detailed in Refs.
[18,19]. The modification to the vacuum quark fragmentation function from quark-gluon
scattering is,
qg→qg
q→h (zh)=
α2sCA
Dq→h(zh/z)
1 + z2
(1− z)+
TAqg(x, xL)
fAq (x)
+ δ(z − 1)
∆TAqg(x, ℓ
fAq (x)
+Dg→h(zh/z)
1 + (1− z)2
TAqg(x, xL)
fAq (x)
, (7)
Fig. 2. A typical diagram for quark-gluon double scattering with three possible cuts [central(C),
left(L) and right(R)].
where the +function is defined as
F (z)
(1 − z)+
F (z)− F (1)
for any F (z) that is sufficiently smooth at z = 1 and the twist-four quark-gluon correla-
tion function,
TAqg(x, xL) =
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
×〈A|ψ̄q(0)
F +σ (y
+σ(y−1 )ψq(y
−)|A〉θ(−y−2 )θ(y− − y−1 ), (9)
has explicit interference included. The matrix element in the virtual correction [the term
with δ(z − 1)] is defined as
∆TAqg(x, ℓ
T ) ≡
2TAqg(x, xL)|z=1 − (1 + z2)TAqg(x, xL)
. (10)
Since TAqg(x, xL)/f
q (x) is proportional to gluon distribution and independent of the flavor
of the leading quark, the suppression of the hadron spectrum caused by quark-gluon or
antiquark-gluon scattering should be proportional to the gluon density of the medium
and is identical for quark and antiquark fragmentation. It was shown in Ref. [24] that
such modification of parton fragmentation functions by quark-gluon double scattering
and gluon bremsstrahlung in a nuclear medium describes very well the recent HERMES
data [25] on semi-inclusive DIS off nuclear targets.
Fig. 3. Diagram for leading order quark-antiquark annihilation with three possible cuts [cen-
tral(C), left(L) and right(R)].
Fig. 4. A typical diagram for next-to-leading order correction to quark-antiquark annihilation
with three possible cuts [central(C), left(L) and right(R)].
In this paper, we will consider quark-quark (antiquark) double scattering such as the
process shown in Fig. 3 and its radiative corrections at order O(α2s) in Fig. 4. The
contributions of quark-quark double scattering is proportional to the quark density in
a nucleon, while the contribution of quark-gluon double scattering is proportional to
the gluon density in a nucleon; and the gluon density is generally larger than the quark
density in a nucleon at small momentum fraction. However, as pointed out in earlier works
[18], quark-quark double scattering mixes quark and gluon fragmentation functions and
therefore gives rise to new nuclear effects. The annihilation processes as shown in Figs. 3
and 4 will lead to different modifications of quark and antiquark fragmentation functions
in a medium with finite baryon density (or valence quarks). Such differences will in
turn lead to flavor dependence of the nuclear modification of leading hadron spectra as
observed in HERMES experiment [25,26].
Quark-quark double scattering as well as quark-gluon double scattering are twist-4 pro-
cesses. We will apply the same generalized factorization procedure for twist-4 processes
as developed by LQS [23] for semi-inclusive processes in DIS. In general, the twist-four
contributions can be expressed as the convolution of partonic hard parts and two-parton
correlation matrix elements. In this framework, contributions from double quark-quark
scattering in any order of αs, e.g., the quark-antiquark annihilation process as illustrated
in Fig. 4, can be written in the following form,
dWDµν
p+dy−
dy−1 dy
−, y−1 , y
2 , p, q, zh)
×〈A|ψ̄q(0)
−)ψ̄q(y
2 )|A〉. (11)
Here we have neglected transverse momenta of all quarks in the hard partonic part.
Transverse momentum dependent contributions are higher twist and are suppressed by
〈k2⊥〉/Q2, Therefore, all quarks’ momenta are assumed collinear, k2 = x2p and k3 = x3p.
−, y−1 , y
2 , p, q, zh) is the Fourier transform of the partonic hard part H̃µν(x, x1, x2, p, q, zh)
in momentum space,
−, y−1 , y
2 , p, q, zh) =
eix1p
+y−+ix2p
+i(x−x1−x2)p
×H̃Dµν(x, x1, x2, p, q, zh)
dxH(0)µν (x, p, q)H
(y−, y−1 , y
2 , x, p, q, zh) , (12)
where, in collinear approximation, the hard partonic part H(0)µν (x, p, q) [Eq. (4)] in the
leading twist without multiple parton scattering can be factorized out of the high-twist
hard part H̃Dµν(x, x1, x2, p, q, zh). The momentum fractions x, x1, and x2 are fixed by
δ-functions of the on-shell conditions of the final state partons and poles of parton prop-
agators in the partonic hard part. The phase factors in H
−, y−1 , y
2 , p, q, zh) can then
be factored out, which in turn will be combined with the partonic fields in Eq. (11)
to give twist-four partonic matrix elements or two-parton correlations. The quark-quark
double scattering corrections in Eq. (11) can then be factorized as the convolution of
fragmentation functions, twist-four partonic matrix elements and the partonic hard scat-
tering cross sections. For scatterings (versus the annihilation) with quarks (antiquarks),
a summation over the flavor of the secondary quarks (antiquarks) should be included
in two-quark correlation matrix elements and both t, u channels and their interferences
should be considered for scattering of identical quarks in the hard partonic parts.
After factorization, we then define the twist-four correction to the leading twist quark
fragmentation function in the same form [Eq. (3)],
dWDµν
dxfAq (x)H
µν (x, p, q)∆Dq→h(zh) . (13)
3 Quark-quark double scattering processes
In this section we will discuss the calculation of the hard part of quark-quark double
scattering in detail. The lowest order process of quark-quark (antiquark) double scattering
in nuclei is quark-antiquark annihilation (or quark-gluon conversion) as shown in Fig. 3.
The hard partonic parts from the three cut diagrams in this figure are [18]:
0,C(y
−, y−1 , y
2 , x, p, q, zh)=Dg→h(zh)
×θ(−y−2 )θ(y− − y−1 ) , (14)
0,L(y
−, y−1 , y
2 , x, p, q, zh)=Dq→h(zh)
×θ(y−1 − y−2 )θ(y− − y−1 ) , (15)
0,R(y
−, y−1 , y
2 , x, p, q, zh)=Dq→h(zh)
×θ(−y−2 )θ(y−2 − y−1 ) . (16)
The main focus of this paper is about contributions from the next-leading order correc-
tions to the above lowest order process. There is a total of 12 diagrams for real corrections
at one-loop level as illustrated in Fig. 5 to Fig. 16 in the Appendix A-1, each having up
to three different cuts. In this section, we demonstrate the calculation of the hard parts
from the quark-antiquark annihilation in Fig. 4 in detail as an example. We will list the
complete results of all diagrams in Appendix A.
One can write down the hard partonic part of the central-cut diagram of Fig. 4 (Fig. 5
in Appendix A-1) according to the conventional Feynman rule,
C µν(y
−, y−1 , y
2 , p, q, zh)=
Dg→h(
eix1p
+y−+ix2p
×ei(x−x1−x2)p+y
(2π)4
γµĤγν
2πδ+(ℓ
2)2πδ+(ℓ
g) δ(1− z −
γ · (q + x1p)
(q + x1p)2 − iǫ
γ · (q + x1p− ℓ)
(q + x1p− ℓ)2 − iǫ
γ · (q + xp− ℓ)
(q + xp− ℓ)2 + iǫ
γ · (q + xp)
(q + xp)2 + iǫ
εαρ(ℓ) εβσ(ℓg) , (17)
where δ+ is a Dirac delta-function with only the positive solution in its functional variable,
εαρ(ℓ) = −gαρ + (nαℓρ + nρℓα)/n · ℓ is the polarization tensor of a gluon propagator in
an axial gauge (n · A = 0) with n = [1, 0−,~0⊥], ℓ and ℓg = q + (x1 + x2)p − ℓ are the
4-momenta carried by the two final gluons respectively. The fragmenting gluon carries
a fraction, z = ℓ−g /q
−, of the initial quark’s longitudinal momentum (the large minus
component).
To simplify the calculation in the case of small transverse momentum ℓT ≪ q−, p+, we
can apply the collinear approximation to complete the trace of the product of γ-matrices,
Ĥ ≈ γ · ℓq
γ · ℓqĤ
. (18)
According to the convention in Eqs. (11) and (12), contributions from quark-quark double
scattering in the nuclear medium to the semi-inclusive hadronic tensor in DIS off a nucleus
can be expressed in the general factorized form:
dWDqq̄,µν
dxH(0)µν (x, p, q)
p+dy−
dy−1 dy
(y−, y−1 , y
2 , x, p, q, zh)
× 〈A|ψ̄q(0)
ψq̄(y
−)ψ̄q(y
ψq̄(y
2 )|A〉 . (19)
After carrying out the momentum integration in x, x1, x2 and ℓ
± in Eq. (17) with the
help of contour integration and δ-functions, one obtains the hard partonic part, H
the rescattering for the central-cut diagram in Fig. 4 (Fig. 5) as
5,C(y
−, y−1 , y
2 , x, p, q, zh) =
α2sxB
Dg→h(zh/z)
× 2(1 + z
z(1− z)
I5,C(y
−, y−1 , y
2 , x, xL, p) , (20)
I5,C(y
−, y−1 , y
2 , x, xL, p) = e
i(x+xL)p
+y−θ(−y−2 )θ(y− − y−1 )
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (21)
where the momentum fractions xL is defined as
2p+q−z(1− z)
. (22)
Note that the function I5,C(y
−, y−1 , y
2 , x, xL, p) contains only phase factors. One can
combine these phase factors with the matrix elements of the quark fields to define a
special two-quark correlation function
A(5,C)
qq̄ (x, xL) =
p+dy−
dy−1 dy
2 〈A|ψ̄q(0)
ψq̄(y
−)ψ̄q(y
ψq̄(y
2 )|A〉
× I5,C(y−, y−1 , y−2 , x, xL, p) . (23)
The contribution from quark-antiquark annihilation in the central-cut diagram in Fig. 4
to the hadronic tensor can then be expressed as
dWDqq̄,µν
dxH(0)µν (x, p, q)
α2sxB
Dg→h(
2(1 + z2)
z(1 − z)
A(5,C)
qq̄ (x, xL). (24)
Contributions from all quark-quark (antiquark) double scattering processes can be cast
in the above factorized form.
The structure of the phase factors in I5,C(y
−, y−1 , y
2 , x, xL, p) is exactly the same as for
gluon bremsstrahlung induced by quark-gluon scattering as studied in Ref. [18,19]. It
resembles the cross section of dipole scattering and represents contributions from two
different processes and their interferences. It contains essentially four terms,
I5,C(y
−, y−1 , y
2 , x, xL, p) = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×[1 + e−ixLp+(y−+y
) − e−ixLp+y
2 − e−ixLp+(y−−y
)] . (25)
The first term corresponds to the so-called hard-soft processes where the gluon emission
is induced by the hard scattering between the virtual photon γ∗ and the initial quark
with momentum (x + xL)p. The quark then becomes on-shell before it annihilates with
a soft antiquark from the nucleus that carries zero momentum and converts into a real
gluon in the final state. The second term corresponds to a process in which the initial
quark with momentum xp is on-shell after the first hard γ∗-quark scattering. It then
annihilates with another antiquark and produces two final gluons in the final state. In
this process, the antiquark carries finite (hard) momentum xLp. Therefore one often refers
to this process as double-hard scattering as compared to the first process in which the
antiquark carries zero momentum. Set aside the change of flavors in the initial and final
states, the double-hard scattering corresponds essentially to two-parton elastic scattering
with finite momentum and energy transfer. This is in contrast to the hard-soft scattering
which is essentially the final state radiation of the γ∗-quark scattering and the total
energy and momentum of the two final state gluons all come from the initial quark. The
corresponding matrix elements of the two-quark correlation functions from these first two
terms are called ‘diagonal’ elements.
The third and fourth terms with negative signs in I5,C(y
−, y−1 , y
2 , x, xL, p) are interfer-
ences between hard-soft and double hard processes. The corresponding matrix elements
are called ‘off-diagonal’. The cancellation between the two diagonal and off-diagonal
terms essentially gives rise to the destructive interference which is very similar to the
Landau-Pomeranchuk-Migdal (LPM) interference in gluon bremsstrahlung induced by
quark-gluon double scattering [18,19]. One can similarly define the formation time of the
parton (quark or gluon) emission as
. (26)
In the limit of collinear emission (xL → 0) or when the formation time of the parton
emission, τf , is much larger than the nuclear size, the effective matrix element vanishes
because
I5,C(y
−, y−1 , y
2 , x, xL, p)|xL=0 → 0 , (27)
when the hard-soft and double hard processes have complete destructive interference.
We should note that in the central-cut diagram of Fig. 4, the final state partons are two
gluons. Therefore, in Eq. (20) the gluon fragmentation function in vacuum Dg→h(zh/z)
enters. If the other gluon (close to the γ∗-quark interaction) fragments, the contribution
to the semi-inclusive hadronic tensor is similar except that the corresponding effective
“splitting function” should be replaced by
1 + z2
z(1 − z)
→ 1 + (1− z)
z(1 − z)
. (28)
As we will show in Appendix A-1, the two gluons in the quark-antiquark annihilation
processes (central-cut diagrams) are symmetric when contributions from all possible an-
nihilation processes and their interferences are summed. Therefore, one can simply mul-
tiply the final results by a factor of 2 to take into account the hadronization of the second
final-state gluon.
In addition to the central-cut diagram, one should also take into account the asymmetrical-
cut diagrams in Fig. 4, which represent interference between gluon emission from single
and triple scattering. The hard partonic parts are mainly the same as for the central-cut
diagram. The only differences are in the phase factors and the fragmentation functions
since the fragmenting parton can be the final-state quark or gluon. These hard parts can
be calculated following a similar procedure and one gets,
5,L(R)(y
−, y−1 , y
2 , x, p, q, zh) =
α2sxB
Dq→h(
2(1 + z2)
z(1 − z)
× I5,L(R)(y−, y−1 , y−2 , x, xL, p) , (29)
I5,L(y
−, y−1 , y
2 , x, xL, p) =−ei(x+xL)p
+y−(1− e−ixLp+(y−−y
×θ(y−1 − y−2 )θ(y− − y−1 ) , (30)
I5,R(y
−, y−1 , y
2 , x, xL, p) =−ei(x+xL)p
+y−(1− e−ixLp+y
×θ(−y−2 )θ(y−2 − y−1 ) . (31)
In the asymmetrical cut diagrams, the above contributions come from the fragmentation
of the final-state quark. Therefore, quark fragmentation function Dq→h(zh/z) enters this
contribution. For gluon fragmentation into the observed hadron in this asymmetrical-cut
diagrams, the contribution can be obtained by simply replacing the quark fragmentation
function by the gluon fragmentation function Dg→h(zh/z) and replacing z by 1 − z.
Summing the contributions from three different cut diagrams of Fig. 4, we can observe
further examples of mixing (or conversion) of quark and gluon fragmentation functions.
This medium-induced mixing was first observed by Wang and Guo [18] and is a unique
feature of quark-quark (antiquark) double scattering among all multiple parton scattering
processes.
With the same procedure we can calculate contributions from all other cut diagrams
of quark-quark (antiquark) double scattering at order O(α2s), which are listed in Ap-
pendix A-1. There are three types of processes: two annihilation processes, qq̄ → gg
(central-cut diagrams in Figs. 5, 6, 7, 8 and 9), qq̄ → qiq̄i (central-cut diagram in Fig. 10)
and quark-quark (antiquark) scattering, qqi(q̄i) → qqi(q̄i) (central-cut diagram in Fig. 11).
One also has to consider the interference of s and t-channel amplitude for annihilation
into an identical quark pair, qq̄ → qq̄ (central-cut diagrams in Figs. 12 and 13) and the
interference between t and u channels of identical quark scattering qq → qq (central-cut
diagram in Fig. 14).
Contributions from left and right-cut diagrams correspond to interference between the
amplitude of gluon radiation from single γ∗-quark scattering and triple quark scattering.
The amplitudes of gluon radiation via triple quark scattering essentially come from ra-
diative corrections to the left and right-cut diagrams of the lowest-order quark-antiquark
annihilation in Fig. 3 (as shown in left and right-cut diagrams in Figs. 5, 6, 8, 9, 12,
13, 15 and 16). Two other triple quark scatterings with gluon radiation, shown as the
left and right-cut diagrams in Figs. 11 and 14, correspond to the case where one of the
final state quarks, after quark-quark scattering, annihilates with another antiquark and
converts into a final state gluon.
4 Modified Fragmentation Functions
In order to simplify the contributions from quark-quark (antiquark) scattering (annihi-
lation), one can first organize the results of the hard parts in terms of contributions from
central, left or right-cut diagrams, which are associated by contour integrals with specific
products of θ-functions,
= HDC θ(−y−2 )θ(y− − y−1 )−HDL θ(y−1 − y−2 )θ(y− − y−1 )
−HDR θ(−y−2 )θ(y−2 − y−1 ) . (32)
These θ-functions provide a space-time ordering of the parton correlation and will restrict
the integration range along the light-cone. For contributions from central, left and right-
cut diagrams that have identical hard partonic parts, H
C = H
L = H
R , they will
have a common combination of θ-functions that produces a path-ordered integral,
dy−2 =−
dy−1 dy
θ(−y−2 )θ(y− − y−1 )− θ(−y−2 )θ(y−2 − y−1 )
−θ(y− − y−1 )θ(y−1 − y−2 )
that is limited only by the spatial-spread y− of the first parton along the light-cone
coordinate. For a high-energy parton that carries momentum fraction xp+, y− ∼ 1/xp+
should be very small. Those contributions that are proportional to the above path-ordered
integral are referred to as contact contributions (or contact interactions).
Similarly, y−1 − y−2 is the spatial spread of the second parton and can only be limited
by the spatial size of its host nucleon even for small value of momentum fraction. The
spatial position of its host nucleon, y−1 + y
2 , however, can be anywhere within the nu-
cleus. Therefore, any contributions from double parton scattering that have unrestricted
integration over y−1 and y
2 should be proportional to the nuclear size of the target A
and therefore are nuclear enhanced. In this paper, we will only keep the nuclear enhanced
contributions and neglect the contact contributions. This will greatly simplify the final
results for double parton scattering.
4.1 qq̄ → g annihilation
For the lowest order of quark-antiquark annihilation in Eqs. (14)-(16), the hard parts
from the three cut diagrams are almost the same except for the parton fragmentation
functions. The central-cut diagram is proportional to the gluon fragmentation function
while the left and right-cut diagrams are proportional to quark fragmentation functions.
Rearranging the contributions from the three cut diagrams and neglecting the contact
term that is proportional to the path-ordered integral as in Eq. (33), the total contribution
can be written as
dWD(0)µν
qq̄ (x, 0)
H(0)µν (x, p, q)
× [Dg→h(zh)−Dq→h(zh)] . (34)
According to our definition in Eq. (13) of the twist-four correction to the quark fragmen-
tation functions, the modification to the quark fragmentation function from the lowest
order quark-antiquark annihilation is then,
(qq̄→g)
q→h (zh) =
[Dg→h(zh)−Dq→h(zh)]
qq̄ (x, 0)
fAq (x)
. (35)
Here the effective quark-antiquark correlation function T
qq̄ (x, 0) is defined as,
qq̄i (x, xL)≡
p+dy−
dy−1 dy
ixp+y−−ixLp
)θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (36)
with the antiquark q̄i carrying momentum fraction xL. This two-parton correlation func-
tion is always associated with double-hard rescattering processes. Similarly, we define
three other quark-antiquark correlation matrix elements
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (37)
A(I−L)
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−−ixLp
+(y−−y
)θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (38)
A(I−R)
qq̄i (x, xL)≡
p+dy−
dy−1 dy
i(x+xL)p
+y−−ixLp
2 θ(y− − y−1 )
× θ(−y−2 )〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (39)
that are associated with hard-soft rescattering and interference between double hard
and hard-soft rescattering. In the first parton correlation T
qq̄i (x, xL), the antiquark q̄i
carries momentum fraction xL while the initial quark has the momentum fraction x. The
two-parton correlation T
qq̄i (x, xL) corresponds to the case when the leading quark has
x+ xL but the antiquark carries zero momentum. The two interference matrix elements
are approximately the same for small value of xL and will be denoted as T
qq̄i (x, xL).
4.2 qq̄ → qiq̄i annihilation
Contributions from the next-to-leading order quark-antiquark annihilation or quark-
quark (antiquark) scattering are more complicated since they involve many real and vir-
tual corrections. The simplest real correction comes from qq̄ → qiq̄i annihilation (qi 6= q)
[Fig. 10 and Eqs. (A-25) and (A-26)] which has only a central-cut diagram,
(qq̄→qiq̄i)
q→h (zh) =
α2sxB
[z2 + (1− z)2]
qi 6=q
[Dqi→h(zh/z) +Dq̄i→h(zh/z)]
qq̄ (x, xL)
fAq (x)
. (40)
This kind of qq̄ annihilation is truly a hard processes and thus requires the second an-
tiquark to carry finite initial momentum fraction xL. Furthermore, there are no other
interfering processes.
4.3 qqi(q̄i) → qqi(q̄i) scattering
Contributions from non-identical quark-quark scattering qq̄i → qq̄i (qi 6= q) are a little
complicated because they involve all three cut diagrams (central, left and right) [Eqs. (A-
28)-(A-32)]. One can factor out the θ-functions in the hard parts according to Eq. (32)
and re-organize the phase factors in each cut diagram,
I11,C = e
i(x+xL)p
+y−(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
+(y−+y
I11,L = e
i(x+xL)p
+y−(1− e−ixLp+(y−−y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
2 ] ;
I11,R = e
i(x+xL)p
+y−(1− e−ixLp+y
= ei(x+xL)p
+y−[1− e−ixLp+y
2 − e−ixLp+(y−−y
) + e−ixLp
+(y−−y
)] , (41)
such that the first three terms in each amplitude are the same. These three common phase
factors will give rise to a contact contribution for all similar hard parts from the three
cut diagrams, which we will neglect since they are not nuclear enhanced. The remaining
part will have the following phase factors,
I11= e
i(x+xL)p
+y−[θ(−y−2 )θ(y− − y−1 )e−ixLp
+(y−+y
−θ(y−1 − y−2 )θ(y− − y−1 )e−ixLp
2 − θ(−y−2 )θ(y−2 − y−1 )e−ixLp
+(y−−y
)] . (42)
Note that the phase factors of the last two terms in the above equation give identi-
cal contributions to the matrix elements when intergated over y−1 and y
2 as they dif-
fer only by the substitution y−2 ↔ y−1 − y−. One therefore can combine them with
θ(−y−2 )θ(y− − y−1 )e−ixLp
+(y−−y
) to form another contact contribution (path-ordered)
which can be neglected. The final effective phase factor is then
I11 = e
ixp+y−−ixLp
)(1− eixLp+y
2 ) . (43)
Using the above effective phase factor, one can obtain the effective modification to the
quark fragmentation function due to quark-antiquark scattering, qq̄i → qq̄i,
(qq̄i→qq̄i)
q→h (zh) =
α2sxB
q̄i 6=q̄
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dg→h(zh/z)
1 + (1− z)2
A(HI)
qq̄i (x, xL)
fAq (x)
+ [Dq̄i→h(zh/z))−Dg→h(zh/z)]
1 + (1− z)2
A(HS)
qq̄i (x, xL)
fAq (x)
α2sxB
q̄i 6=q̄
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dq̄i→h(zh/z)
1 + (1− z)2
A(HI)
qq̄i (x, xL)
fAq (x)
+ [Dq̄i→h(zh/z)−Dg→h(zh/z)]
1 + (1− z)2
A(SI)
qq̄i (x, xL)
fAq (x)
, (44)
where three types of two-parton correlations are defined:
A(HI)
qq̄i (x, xL)≡T
qq̄i (x, xL)− T
qq̄i (x, xL)
p+dy−
dy−1 dy
ixp+y−−ixLp
)(1− eixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉θ(−y−2 )θ(y− − y−1 ) , (45)
A(SI)
qq̄i (x, xL)≡T
qq̄i (x, xL)− T
qq̄i (x, xL)
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉θ(−y−2 )θ(y− − y−1 ) , (46)
A(HS)
qq̄i (x, xL)≡T
A(HI)
qq̄i (x, xL) + T
A(SI)
qq̄i (x, xL)
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
× (1− e−ixLp+(y−−y
))θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 . (47)
One can similarly obtain the modification of quark fragmentation from non-identical
quark-quark scattering by replacing q̄i → qi in Eq. (44),
(qqi→qqi)
q→h (zh) =
α2sxB
qi 6=q
Dq→h(zh/z)
1 + z2
(1− z)2
+ Dqi→h(zh/z)
1 + (1− z)2
TA(HI)qqi (x, xL)
fAq (x)
+ [Dqi→h(zh/z)−Dg→h(zh/z)]
1 + (1− z)2
TA(SI)qqi (x, xL)
fAq (x)
. (48)
The two-quark correlations, TA(HI)qqi (x, xL) and T
A(SI)
(x, xL) can be obtained from T
A(HI)
qq̄i (x, xL)
and T
A(SI)
qq̄i (x, xL), respectively, by making the replacements ψqi(y2) → ψ̄qi(y2) and ψ̄qi(y1) →
ψqi(y1) in Eqs. (45) and (46),
TA(HI)qqi (x, xL)≡
p+dy−
dy−1 dy
ixp+y−−ixLp
)(1− eixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
1 )|A〉θ(−y−2 )θ(y− − y−1 ) , (49)
TA(SI)qqi (x, xL) =
p+dy−
dy−1 dy
i(x+xL)p
+y−(1− e−ixLp+y
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
1 )|A〉θ(−y−2 )θ(y− − y−1 ) , (50)
and TA(HS)qqi (x, xL) = T
A(HI)
(x, xL) + T
A(SI)
(x, xL).
Note that the contribution from fragmentation of quark qi or antiquark q̄i only comes from
the central-cut diagram. This contribution is positive and is proportional to T
A(HI)
qq̄i (x, xL)+
A(SI)
qq̄i (x, xL), containing all four terms: hard-soft, double-hard and both interference
terms . The gluon fragmentation comes only from the single-triple interferences (left and
right-cut diagrams). Its contribution is therefore negative and partially cancels the pro-
duction of qi(q̄i) from the hard-soft rescattering. The cancellation is not complete since
the gluon and quark fragmentation functions are different. The structure of this hard-soft
rescattering (quark plus gluon) is very similar to the lowest order result of qq̄ → g in
Eq. (35). It contributes to the modification of the effective fragmentation function but
does not contribute to the energy loss. The energy loss of the leading quark comes only
from double-hard rescattering, since the leading quark fragmentation comes both from
the central-cut and single-triple interferences, and the single-triple interference terms can-
cel the effect of hard-soft scattering for the leading fragmentation. Its net contribution is
therefore proportional to T
A(HI)
qq̄i(q̄i)
. Since the double-hard rescattering amounts to elastic
qqi(q̄i) scattering, the effective energy loss is essentially elastic energy loss as shown in
Ref. [34]. There is, however, LPM suppression due to partial cancellation by single-triple
interference contributions. For long formation time, 1/xLp
+ ≫ RA, the cancellation is
complete. Therefore, LPM interference effectively imposes the lower limit xL ≥ 1/p+RA
on the fractional momentum carried by the second quark (antiquark).
4.4 qq → qq scattering
For identical quark-quark scattering, qq → qq, one has to include both t and u-channels,
their interferences, and the related single-triple interference contributions. Using the same
technique to identify and neglect the contact contributions, one can find the correspond-
ing modification to the quark fragmentation function from Eqs. (48) and (A-45)-(A-49),
(qq→qq)
q→h (zh) =
α2sxB
TA(HS)qq (x, xL)
fAq (x)
× [Dq→h(zh/z))−Dg→h(zh/z)]
1 + (1− z)2
z(1 − z)
Dq→h(zh/z)
1 + z2
(1− z)2
z(1 − z)
+Dg→h(zh/z)
1 + (1− z)2
z(1 − z)
TA(HI)qq (x, xL)
fAq (x)
α2sxB
TA(SI)qq (x, xL)
fAq (x)
P (s)qq→qq(z)[Dq→h(zh/z)
−Dg→h(zh/z)] +Dq→h(zh/z)Pqq→qq(z)
TA(HI)qq (x, xL)
fAq (x)
, (51)
where the effective splitting functions are defined as
P (s)qq→qq(z) =
1 + (1− z)2
z(1 − z)
, (52)
Pqq→qq(z) =
1 + (1− z)2
1 + z2
(1− z)2
z(1− z)
. (53)
4.5 qq̄ → qq̄, gg annihilation
The most complicated twist-four processes involving four quark field operators are quark-
antiquark annihilation into two gluons or an identical quark-antiquark pair. We have to
consider them together since they have similar single-triple interference processes and
they involve the same kind of quark-antiquark correlation matrix elements, T
qq̄ (x, xL),
(i = HI, SI,HS).
For notation purpose, we first factor out the common factor (CF/Nc)α
sxB/Q
2/fAq (x) and
the integration over ℓT and z and define
(qq̄→gg,qq̄)
q→h (zh) ≡
α2sxB
Q2fAq (x)
(qq̄→gg,qq̄)
q→h (zh, z, x, xL). (54)
After rearranging the phase factors and identifying (by combining central, left and right
cut diagrams) and neglecting contact contributions we can list in the following the twist-
four corrections to the quark fragmentation from the hard partonic parts of each cut
diagram (see Appendix A):
Fig. 5 (t-channel qq̄ → gg),
(qq̄→gg,qq̄)
q→h(5) =Dg→h(zh/z)2CF
1 + (1− z)2
z(1 − z)
1 + z2
z(1− z)
A(HI)
qq̄ (x, xL)
+ [Dg→h(zh/z)−Dq→h(zh/z)] 2CF
1 + z2
z(1− z)
A(SI)
qq̄ (x, xL) ; (55)
Fig. 6 (interference between u and t-channel of qq̄ → gg),
(qq̄→gg,qq̄)
q→h(6) =Dg→h(zh/z)
−4(CF − CA/2)
z(1 − z)
A(HI)
qq̄ (x, xL)
+ [Dg→h(zh/z)−Dq→h(zh/z)]
−2(CF − CA/2)
z(1 − z)
A(SI)
qq̄ (x, xL) ; (56)
Fig. 7 (s-channel of qq̄ → gg),
(qq̄→gg,qq̄)
q→h(7) = Dg→h(zh/z)4CA
(1− z + z2)2
z(1 − z)
qq̄ (x, xL) ; (57)
Figs. 8 and 9 (interference of s and t-channel qq̄ → gg),
(qq̄→gg,qq̄)
q→h(8+9) =Dg→h(zh/z)(−2CA)
1 + z3
z(1− z)
1 + (1− z)3
z(1 − z)
×TA(HI)qq̄ (x, xL) + CA
Dq→h(zh/z)
1 + z3
z(1 − z)
+Dg→h(zh/z)
1 + (1− z)3
z(1 − z)
× [TA(I2)qq̄ (x, xL)− T
qq̄ (x, xL)] ; (58)
Fig. 10 (s-channel of qq̄ → qq̄),
(qq̄→gg,qq̄)
q→h(10) = [Dq→h(zh/z) +Dq̄→h(zh/z)] [z
2 + (1− z)2]
×TA(H)qq̄ (x, xL) , (59)
Fig. 11 (t-channel of qq̄ → qq̄), similar to Eq. (44),
(qq̄→gg,qq̄)
q→h(11) =Dq→h(zh/z)
1 + z2
(1 − z)2
A(HI)
qq̄ (x, xL)
+Dq̄→h(zh/z)
1 + (1− z)2
A(HS)
qq̄ (x, xL)
−Dg→h(zh/z)
1 + (1− z)2
A(SI)
qq̄ (x, xL) ; (60)
Figs. 12 and 13 (interference between s and t-channel qq̄ → qq̄),
(qq̄→gg,qq̄)
q→h(12+13) =−4(CF − CA/2)
Dq→h(zh/z)
1 − z
+ Dq̄→h(zh/z)
(1− z)2
A(HI)
qq̄ (x, xL)
+2(CF − CA/2)
Dq→h(zh/z)
+Dg→h(zh/z)
(1− z)2
× [TA(I2)qq̄ (x, xL)− T
qq̄ (x, xL)] ; (61)
Figs. 15 and 16 (two additional single-triple interference diagrams),
(qq̄→gg,qq̄)
q→h(15+16) =−2CF
Dq→h(zh/z)
1 + z2
+ Dg→h(zh/z)
1 + (1− z)2
A(I2)
qq̄ (x, xL) . (62)
Most processes involve both TA(HI)(x, xL) for double-hard rescattering with interference
and TA(SI)(x, xL) for hard-soft rescattering with interference. All the s-channel (Figs. 7
and 10) processes involve double-hard scattering only. Therefore, they depend only on the
qq̄ (x, xL) = T
A(HI)
qq̄ (x, xL) + T
qq̄ (x, xL). For interference between single and triple
scattering (left and right-cut diagrams in Figs. 8, 9, 12 13, 15 and 16), where a hard
rescattering with the second quark (antiquark) follows a soft rescattering with the third
antiquark (quark), only interference matrix elements, T
qq̄ (x, xL) and T
A(I2)
qq̄ (x, xL), are
involved. Here,
A(I2)
qq̄ (x, xL)≡
p+dy−
dy−1 dy
ixp+y−+ixLp
×〈A|ψ̄q(0)
−)ψ̄q(y
2 )|A〉θ(−y−2 )θ(y− − y−1 )
p+dy−
dy−1 dy
ixp+y−+ixLp
+(y−−y
×〈A|ψ̄q(0)
−)ψ̄q(y
2 )|A〉θ(−y−2 )θ(y− − y−1 ) , (63)
is a new type of interference matrix elements that is only associated with this type of
single-triple interference processes. One can categorize the above contributions according
to the associated two-quark correlation matrix elements and rewrite the above contribu-
tions as,
qq̄→qq̄,gg
q→h(HI) = T
A(HI)
qq̄ (x, xL)[Dg→h(zh/z)Pqq̄→gg(z) +Dq→h(zh/z)Pqq̄→qq̄(z)
+Dq̄→h(zh/z)Pqq̄→qq̄(1− z)] (64)
qq̄→qq̄,gg
q→h(SI) = T
A(SI)
qq̄ (x, xL)
z(1 − z)
+ 2CF
1 + (1− z)2
×Dg→h(zh/z)−Dq→h(zh/z)
z(1− z)
+ 2CF
+ Dq̄→h(zh/z)
1 + (1− z)2
A(SI)
qq̄ (x, xL)
[Dq→h(zh/z)−Dg→h(zh/z)]
P (s)qq→qq(z)
− 2CF
1 + z2
z(1− z)
+ [Dq̄→h(zh/z)−Dq→h(zh/z)]
1 + (1− z)2
qq̄→qq̄,gg
q→h(I) = T
qq̄ (x, xL)
4(1− z + z2)2 − 1
z(1 − z)
− 2CF
(1− z)2
×Dg→h(zh/z) + [z2 + (1− z)2]Dq̄→h(zh/z)]
+ Dq→h(zh/z)
z2 + (1− z)2 −
z(1 − z)
− 2CF
A(I2)
qq̄ (x, xL)
Dq→h(zh/z)
z(1 − z)
− 2CF
+ Dg→h(zh/z)
z(1 − z)
− 2CF
, (66)
where P (s)qq→qq(z) is given in Eq. (52) and the effective splitting functions for qq̄ → gg and
qq̄ → qq̄ are defined as
Pqq̄→gg(z) = 2CF
z2 + (1− z)2
z(1− z)
− 2CA[z2 + (1− z)2] ; (67)
Pqq̄→qq̄(z) = z
2 + (1− z)2 +
1 + z2
(1− z)2
, (68)
which come from the complete matrix elements of qq̄ → gg and qq̄ → qq̄ scattering (see
Appendix A-3). Again, double-hard rescattering corresponds to the elastic scattering of
the leading quark with another antiquark in the medium and the interference contribu-
tions. The structure of the hard-soft rescattering contribution we identify above shows
the same kind of gluon-quark (or quark-antiquark) mixing in the fragmentation functions
and does not contribute to the energy loss of the leading quark. The unique contributions
in the qq̄ → qq̄, gg processes are the interference-only contributions. They mainly come
from single-triple interference processes in the multiple parton scattering.
5 Modification due to quark-gluon mixing
We have so far cast the modification of the quark fragmentation function due to quark-
quark (antiquark) scattering (or annihilation) in a form similar to the DGLAP evolution
equation in vacuum. In fact, one can also view the evolution of fragmentation functions in
vacuum as modification due to final-state gluon radiation. In both cases, the modification
at large zh is mainly determined by the singular behavior of the splitting functions for
z → 1, whereas the modifications at mall zh is dominated by the singular behavior of the
splitting function for z → 0.
Let us first focus on the modification at large zh. A careful examination of the contribu-
tions from all possible processes shows that the dominant modification to the effective
quark fragmentation function comes from the t-channel of double hard quark-quark scat-
tering processes,
∆Dq→h(zh)∼
α2sxB
Dq→h(
TA(HI)qqi (x, xL)
fAq (x)
× 1 + z
(1 − z)2+
+δ(1− z)∆qi(ℓ2T )
Dq→h(
A(HI)
(x, xL)
fAq (x)
×z(1 + z
(1 − z)+
+ δ(1− z)∆qi(ℓ2T )
, (69)
where the summation is over all possible quark and antiquark flavors including qi = q, q̄
and ∆qi(ℓ
T ) represents the contribution from virtual corrections. We have expressed the
modification in a form that it is proportional to the matrix elements xLT
A(HI)
(x, xL)/f
q (x) ∼
A1/3xLf
(xL) as compared to the modification from quark-gluon scattering where the
corresponding matrix element [Eq. (9)] is TA(HI)qg (x, xL)/f
q (x) ∼ A1/3xLGN(xL). Here,
fNqi (x) and G
N(x) are quark and gluon distributions, respectively, in a nucleon. This
leading contribution to the modification from quark-quark scattering is very similar in
form to that from quark-gluon scattering [see Eq. (7)]. However, it is smaller due to the
different color factors CF/CA = 4/9 and the different quark and gluon distributions,
fNqi (xL) and G
N(xL) in a nucleon. Because of LPM intereference, small angle scattering
with long formation time τf = 1/xLp
+ is suppressed, leading to a minimum value of
xL ≥ xA = 1/mNRA = 0.043 for a Kr target. For this value of xL, the ratio
fNqi (xL, Q
GN(xL, Q2)
≥ 1.40/1.85 ∼ 0.75, (70)
at Q2 = 2 GeV2 according the CTEG4HJ parameterization [35]. Therefore, one has to
include the effect of quark-quark scattering for a complete calculation of the total quark
energy loss and medium modification of quark fragmentation functions.
In a weakly coupled and fully equilibrated quark-gluon plasma, quark to gluon number
density ratio is ρq/ρg = nf (3/2)Nc/(N
c − 1) = 9nf/16. An asymptotically energetic
jet in an infinitely large medium actually probes the small x = 〈q2T 〉/2ET regime, where
quark-antiquark pairs and gluons are predominantly generated by thermal gluons through
pQCD evolution. In this ideal scenario one expects Nq/Ng ∼ 1/4CA = 1/12 and therefore
can neglect quark-quark scattering. The modification of quark fragmentation function will
be dominated by quark-gluon rescattering. However, for moderate jet energy E ≈ 20 GeV
and a finite medium L ∼ 5 fm, parton distributions in a quark-gluon plamsa are close to
the thermal distribution. In particular, if quark and gluon production is dominated by
non-perturbative pair production from strong color fields in the initial stage of heavy-ion
collisions [36], the quark to gluon ratio is comparable to the equilibrium value. In this
case, we should take into account the medium modification of the quark fragmentation
functions by quark-quark scattering.
An important double hard process in quark-quark (antiquark) scattering is qq̄ → gg
[Eq. (64)]. In this process, the annihilation converts the initial quark into two final gluons
that subsequently fragment into hadrons. This will lead to suppression of the leading
hadrons not only because of energy loss (energy carried away by the other gluon) but
also due to the softer behavior of gluon fragmentation functions at large zh. Even though
the leading behavior of the effective splitting function [Eq. (67)]
Pqq̄→gg(z) ≈ 2CF
z2 + (1− z)2
z(1− z)
is not as dominating as that of t-channel quark-quark scattering, it is enhanced by a color
factor 2CF = 8/3. One expects this to make a significant contribution to the medium
modification at intermediate zh.
In high-energy heavy-ion collisions, the ratios of initial production rates for valence
quarks, gluons and antiquarks vary with the transverse momentum pT . Gluon production
rate dominates at low pT while the fraction of valence quark jets increases at large pT .
Quarks are more likely to fragment into protons than antiprotons, while gluons fragment
into protons and antiprotons with equal probabilities. Therefore, the ratio of large pT
antiproton and proton yields in p + p collisions is smaller than 1 and decreases with pT
as the fraction of valence quark jets increases. Since gluons are expected to lose more
energy than quark jets, one would naively expect to see the antiproton to proton ratio
p̄/p becomes smaller due to jet quenching. However, if the quark-gluon conversion due
to qq̄ → gg becomes important, one would expect that the fractions of quark and gluon
jets are modified toward their equilibrium values. The final p̄/p ratio could be larger than
or comparable to that in p+ p collisions. Such a scenario of quark-gluon conversion was
recently considered in Ref. [37] via a master rate equation.
The mixing between quark and gluon jets also happens at the lowest order of quark-
antiquark annihilation as shown in Fig. 3. At NLO, all hard-soft quark-quark (antiquark)
scattering processes have this kind of mixing between quark and gluon fragmentation
functions. Their contributions generally have the form,
α2sxB
Dqi→h(
)−Dg→h(
×Pqqi→qqi(z)
TA(SI)qqi (x, xL)
fAq (x)
, (72)
where again the summation over the quark flavor includes qi = q, q̄. This mixing does
not occur on the probability but rather on on the amplitude level since it involves in-
terferences between single and triple scattering. Therefore, this contribution depends on
the difference between gluon and quark fragmentation functions [Eq. (35)] and can be
positive or negative in different region of zh. Nevertheless, they contribute to the mod-
ification of the effective quark fragmentation function and the flavor dependence of the
final hadron spectra.
6 Flavor dependence of the medium modified fragmentation
Summing all contributions to quark-quark (antiquark) double scattering as listed in Sec-
tion 4, we can express the total twist-four correction up to O(α2s) to the quark fragmen-
tation function as
∆Dq→h(zh)=
2 [Dg→h(zh)−Dq→h(zh)]
qq̄ (x, 0)
fAq (x)
a,b,i
Db→h(zh/z)P
qa→b(z)
TA(i)qa (x, xL)
fAq (x)
, (73)
where the summation is over all possible q+a→ b+X processes and all different matrix
elements TA(i)qa (x, xL) (i = HI, SI, I, I2), which will be four basic matrix elements we will
use. The effective splitting functions P
qa→b(z) are listed in Appendix A-2. One should also
include virtual corrections which can be constructed from the real corrections through
unitarity constraints [18].
Similarly, we can also write down the twist-four corrections to antiquark fragmentation
in a nuclear medium,
∆Dq̄→h(zh)=
2 [Dg→h(zh)−Dq̄→h(zh)]
q̄q (x, 0)
fAq̄ (x)
a,b,i
Db→h(zh/z)P
q̄a→b(z)
q̄a (x, xL)
fAq̄ (x)
, (74)
where the matrix elements T
q̄a (x, xL) and the effective splitting functions P
q̄a→b(z) can
be obtained from the corresponding ones for quarks. Given a model for the two-quark
correlation functions, one will be able to use the above expressions to numerically evaluate
twist-four corrections to the quark (antiquark) fragmentation functions. In this paper,
we will instead give a qualitative estimate of the flavor dependence of the correction in
DIS off a large nucleus.
For the purpose of a qualitative estimate, one can assume that all the twist-four two-quark
correlation functions can be factorized, as has been done in Refs. [18,19,23,33],
p+dy−
dy−1 dy
+y−+ix2p
)θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉
fAq (x1) f
(x2) , (75)
p+dy−
dy−1 dy
+y−+ix2p
)±ixLp
2 θ(−y−2 )θ(y− − y−1 )
×〈A|ψ̄q(0)
ψq̄(y
−)ψ̄qi(y
ψqi(y
2 )|A〉
fAq (x1) f
(x2)e
A , (76)
where xA = 1/mNRA, mN is the nucleon mass, RA the nucleus size, f
(x2) is the
antiquark distribution in a nucleon and C is assumed to be a constant, parameterizing
the strength of two-parton correlations inside a nucleus. The integration over the position
of the antiquark (y−1 +y
2 )/2 in the twist-four two-quark correlation matrix elements gives
rise to the nuclear enhancement factor 1/xA = mNRA = 0.21A
We should note that we set kT = 0 for the collinear expansion. As a consequence, the
secondary quark field in the twist-four parton matrix elements will carry zero momentum
in the soft-hard process. Finite intrinsic transverse momentum leads to higher-twist cor-
rections. If a subset of the higher-twist terms in the collinear expansion can be resummed
to restore the phase factors such as eixT p
+y−, where xT ≡ 〈k2T 〉/2p+q−z(1 − z), the soft
quark fields in the parton matrix elements will carry a finite fractional momentum xT .
Under such an assumption of factorization, one can obtain all the two-quark correlation
matrix elements:
A(HI)
qq̄i (x, xL)≈
fAq (x) f
(xL + xT )[1− e−x
A], (77)
A(SI)
qq̄i (x, xL)≈
fAq (x+ xL) f
(xT )[1− e−x
A], (78)
qq̄i (x, xL)≈T
A(I2)
qq̄i (x, xL) ≈
fAq (x+ xL) f
(xT )e
fAq (x) f
(xL + xT )e
A. (79)
In the last approximation, we have assumed xL ∼ xT ≪ x. Similarly, one can obtain
TA(i)qqi (x, xL), T
q̄qi (x, xL) and T
q̄q̄i (x, xL). With these forms of two-quark correlation
matrix elements, we can estimate the flavor dependence of the nuclear modification to
the quark (antiquark) fragmentation functions.
The lowest order corrections [O(αs)] are very simple
q→h (zh) ∝ CA1/3[Dg→h(zh)−Dq→h(zh)]fNq̄ (xT ) , (80)
q̄→h̄
(zh) ∝ CA1/3[Dg→h̄(zh)−Dq̄→h̄(zh)]fNq (xT ) . (81)
We consider the dominant contribution from the fragmentation of a quark (antiquark)
which is one of the valence quarks (antiquarks) of the final particle h (antiparticle h̄).
The gluon fragmentation functions into h and h̄ are the same. For large zh, the gluon
fragmentation function is always softer than the valence quark (antiquark) fragmenta-
tion [38]. Therefore, the lowest order twist-four corrections are always negative for large
zh, leading to a suppression of the valence quark (antiquark) fragmentation function,
Dqv→h(zh) [Dq̄v→h̄(zh)]. Consider those quarks that are also valence quarks of a nucleon:
n = udd
p = uud , p̄ = ūūd̄ , (82)
K+ = us̄ ,K− = ūs . (83)
π+, π0, π− = ud̄ , (uū − dd̄ )/
2 , dū . (84)
One can find the following flavor dependence of the lowest order twist-four corrections
to the quark (antiquark) fragmentation functions,
q̄v→h̄
−|∆D(LO)
q̄v→h̄
(zh)|
−|∆D(LO)qv→h(zh)|
fNqv (xT )
fq̄v(xT )
> 1 , (85)
q̄v→h̄
1 + ∆D
q̄v→h̄
(zh)/Dq̄→h̄(zh)
1 + ∆D
(zh)/Dq→h(zh)
< 1 , (86)
where R
is the corresponding leading order suppression of the fragmentation function
at large zh for proton (anti-proton) and K
+ (K−). Since pions contain both valence
quark and antiquark, the suppression factors should be similar for all pions. For xT ≥
0.043, u(x)/ū(x) ≥ 3 and d(x)/d̄(x) ≥ 2 [35]. Therefore, the modification of antiquark
fragmentation functions due to quark-antiquark annihilation is significantly larger than
that of a quark.
The flavor dependence of the NLO results are more complicated since they involve scatter-
ing with both quarks and antiquarks in the medium. One can observe first that effective
splitting functions (or quark-quark scattering cross section) are the same for the t-channel
qq′ → qq′ and qq̄′ → qq̄′ (q′ 6= q) scatterings,
qq′→b(z) = P
q̄q′→b(z) = P
qq̄′→b(z) = P
q̄q̄′→b(z) . (87)
For identical quark-quark scattering or quark-antiquark annihilation, one can separate
the qq̄ annihilation splitting functions (or cross sections) into singlet and non-singlet
contributions by singling out the t-channel contributions,
qq̄→b(z)≡P
qq→b(z) + ∆P
qq̄→b(z), (88)
q̄q→b(z)≡P
q̄q̄→b(z) + ∆P
q̄q→b(z). (89)
These singlet contributions to the modified fragmentation functions are,
S(NLO)
q→h (zh)∝
b,q′,i
Db→h ⊗ P (i)qq′→b(zh)
×[fNq′ (xT ) + fNq̄′ (xT )]C(i) , (90)
S(NLO)
q̄→h̄
(zh)∝
b,q′,i
Db̄→h̄ ⊗ P
q̄q̄′→b̄
×[fNq′ (xT ) + fNq̄′ (xT )]C(i) , (91)
where the summation over q′ now includes q′=q and C(i)(xL) are flavor-independent
functions determined from Eqs. (77)-(79),
C(HI) =C(SI) = C(xL)(1− e−x
C(I) =C(I2) = C(xL)e
A , (92)
and C(xL) is a common coefficient that is a function of xL. Using P
q̄q̄→b̄
(z) = P
qq→b(z)
, one can conclude that the singlet contributions to the modified quark and antiquark
fragmentation functions are the same, ∆D
S(NLO)
q→h (zh) = ∆D
S(NLO)
q̄→h̄
(zh).
The non-singlet contributions, mainly from s-channel and s-t interferences, are,
N(NLO)
q→h (zh)∝
Db→h ⊗∆PN(i)qq̄→b(zh)fNq̄ (xT )C(i) , (93)
N(NLO)
q̄→h̄
(zh)∝
Db̄→h̄ ⊗∆P
q̄q→b̄
(zh)f
q (xT )C
(i) , (94)
where again ∆P
qq̄→b(z) = ∆P
q̄q→b̄
(z) due to crossing symmetry. We have listed all non-
vanishing nonsinglet splitting functions ∆P
qq̄→b(z) in Appendix A-2.
We again consider the limit zh → 1. In this region the convolution in the modified
fragmentation function is dominated by the large z → 1 behavior of the effective split-
ting functions. From the listed ∆P
qq̄→b(z) in Appendix A-2, we can obtain the leading
contributions,
C(i)∆P
qq̄→q(z)≈−4CF
C(xL)
C(i)∆P
qq̄→g(z)≈ 2
2CF + CF (1− e−x
A) + CAe
] C(xL)
, (95)
where we have also neglected terms proportional to 1/Nc. All ∆P
qq̄→q̄(z) are non-leading
in the limit z → 1 and therefore can be neglected. With these leading contributions,
the non-singlet modification to the quark and antiquark fragmentation functions can be
estimated as
N(NLO)
q→h (zh)∝
C(xL)
(1− z)+
CF (1− e−x
A) + CAe
+ δ(1− z)∆1(ℓT )
−Dq→h
C(xL)
(1− z)+
+ δ(1− z)∆2(ℓT )
fNq̄ (xT ) , (96)
N(NLO)
q̄→h̄
(zh)∝
Dg→h̄
C(xL)
(1− z)+
CF (1− e−x
A) + CAe
+ δ(1− z)∆1(ℓT )
Dg→h̄
−Dq̄→h̄
C(xL)
(1− z)+
+ δ(1− z)∆2(ℓT )
fNq (xT ) , (97)
where ∆1(ℓT ) and ∆2(ℓT ) are from virtual corrections,
∆1(ℓT )=
CFC(xL)|z=1 − [CF (1− e−x
+ CAe
A ]C(xL)
, (98)
∆2(ℓT )=
2CF [C(xL)|z=1 − C(xL)] . (99)
Because of momentum conservation, C(xL) = 0 when xL → ∞ for z = 1. Therefore,
the above virtual corrections are always negative. At large zh, these virtual corrections
dominate over the real ones.
There are two kinds of non-singlet contributions in the expressions given above. One that
is proportional to gluon fragmentation functions is due to quark-antiquark annihilation
into gluons which then fragment. The fragmenting gluon not only carries less energy than
the initial quark but also has a softer fragmentation function, leading to suppression of
the final leading hadrons. The second type of contributions is proportional to Dg→h(zh)−
Dq→h(zh) and therefore mixes quark and gluon fragmentation functions, similarly as the
lowest order quark-antiquark annihilation processes [see Eqs. (80) and (81)]. Since a gluon
fragmentation function is softer than a quark one, the real corrections from this type of
processes are positive for small zh and negative for large zh. The virtual corrections
have just the opposite behavior. Therefore, the second type of contributions will reduce
the total net modification. For intermediate values of zh where 2Dg→h(zh) > Dq→h(zh),
the net effect is still the suppression of the effective fragmentation functions for leading
hadrons.
Since fNq (xT ) > f
q̄ (xT ), we can conclude that the LO and NLO combined non-singlet
suppression for antiquark fragmentation into valence hadrons is larger than that for quark
fragmentation into valence hadrons. This qualitatively explains the flavor dependence of
nuclear suppression of leading hadrons in DIS off heavy nuclear targets as measured by
the HERMES experiment [25,26]. The ratio of differential semi-inclusive cross sections
for nucleus and deuteron targets were used to study the nuclear suppression of the frag-
mentation functions. It was observed that suppression of leading anti-proton is stronger
than for leading proton and K− suppression is stronger than K+. In the valence quark
fragmentation picture, the leading proton (K+) is produced mainly from u, d (u) quark
fragmentation while anti-protons come primarily from ū, d̄ (ū) fragmentation. Therefore,
HERMES data are consistent with stronger suppression of antiquark fragmentation.
Since gluon bremsstrahlung and the singlet qqi(q̄i) scattering also suppress quark and
antiquark fragmentation, but independently of quark flavor, one has to include all the
processes in order to have a complete and quantitative numerical evaluation of the flavor
dependence of the nuclear modification of the quark fragmentation functions. Further-
more, the NLO contributions are proportional to αs ln(Q
2)/2π. They are as important
as the lowest order correction for large values of Q2. In principle, one should resum these
higher order corrections via solving a set of coupled DGLAP evolution equations, in-
cluding medium modification for gluon fragmentation functions. The contributions from
quark-quark (antiquark) scattering derived in this paper will be an important part of the
complete dscription. Detailed numerical study of the effect of quark-quark (antiquark)
scattering will be possible only after the completion of this complete description in the
future.
7 Summary
Utilizing the generalized factorization framework for twist-four processes we have stud-
ied the nuclear modification of quark and antiquark fragmentation functions (FF) due
to quark-quark (antiquark) double scattering in dense nuclear matter up to order O(α2s).
We calculated and analyzed the complete set of all possible cut diagrams. The results
can be categorized into contributions from double-hard, hard-soft processes and their
interferences. The double-hard rescatterings correspond to elastic scattering of the lead-
ing quark with another medium quark. It requires the second quark to carry a finite
fractional momentum xL. Therefore, the energy loss of the leading quark through such
processes can be identified as elastic energy loss at order O(α2s). The quark energy loss
and modification of quark fragmentation functions are dominated by the t-channel of
quark-quark (antiquark) scattering and are shown to be similar to that caused by quark-
gluon scattering. The contribution from quark-quark scattering is smaller than that from
quark-gluon scattering by a factor of CF/CA times the ratio of quark and gluon distribu-
tion functions in the medium. We have shown that such contributions are not negligible
for realistic kinematics and finite medium size. The soft-hard rescatterings mix gluon and
quark scattering, in the same way as the lowest order qq̄ → g processes. Such processes
modifies the final hadron spectra or effective fragmentation functions but do not con-
tribute to energy loss of the leading quark. For qq̄ → qq̄, gg processes, there also exist
pure interference contributions mainly coming from single-triple-scattering interference.
With a simple model of a factorized two-quark correlation functions, we further investi-
gated the flavor dependence of the medium modified quark fragmentation functions in a
large nucleus. We identified the flavor dependent part of the modification and find that
the nuclear modification for an antiquark fragmentation into a valence hadron is larger
than that of a quark. This offers an qualitative explanation for the flavor dependence of
the leading hadron suppression in semi-inclusive DIS off nuclear targets as observed by
the HERMES experiment [25,26].
Acknowledgements
The authors thank Jian-Wei Qiu and Enke Wang for helpful discussion. This work
was supported by NSFC under project No. 10405011, by MOE of China under project
IRT0624, by Alexander von Humboldt Foundation, by BMBF, by the Director, Office
of Energy Research, Office of High Energy and Nuclear Physics, Divisions of Nuclear
Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231,
and by the US NSF under Grant No. PHY-0457265, the Welch Foundation under Grant
No. A-1358.
A-1 Hard partonic parts for quark-quark double scattering
In Section 3 we have discussed the calculation of the hard part of one example cut-diagram
(Fig. 5) in detail. In this appendix we list the results for all possible real corrections to
quark-quark (antiquark) double scattering in the next-to-leading order O(α2s). There are
a total of 12 diagrams as illustrated in Figs. 5-16. For the purpose of abbreviation, we
will suppress the variables in the notations of partonic hard parts
D ≡ HD(y−, y−1 , y−2 , x, p, q, zh) , (A-1)
and phase factor functions
I ≡ I(y−, y−1 , y−2 , x, , xL, p) . (A-2)
We first consider all qq̄ → gg annihilation diagrams with different possible cuts. The
contributions of Fig. 5 are:
5,C =
α2sxB
I5,CDg→h(zh/z)
1 + z2
z(1 − z)
1 + (1− z)2
z(1 − z)
, (A-3)
Fig. 5. The t-channel of qq̄ → gg annihilation diagram with three possible cuts, central(C),
left(L) and right(R).
Fig. 6. The interference between t and u-channel of qq̄ → gg annihilation.
I5,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-4)
5,L(R) =
α2sxB
I5,L(R)
Dq→h(zh/z)2
1 + z2
z(1 − z)
+Dg→h(zh/z)2
1 + (1− z)2
z(1 − z)
, (A-5)
I5,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (A-6)
I5,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−(1− e−ixLp+y
2 ) . (A-7)
Here we have included the fragmentation of both final-state partons.
The contributions from Fig. 6 are:
Fig. 7. The s-channel of qq̄ → gg annihilation diagram with only a central-cut.
6,C =
α2sxB
I6,C 2Dg→h(zh/z)
(1− z)z
CF (CF − CA/2)
, (A-8)
6,L(R) =
α2sxB
I6,L(R) [Dg→h(zh/z) +Dq→h(zh/z)]
(1− z)z
CF (CF − CA/2)
, (A-9)
I6,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-10)
I6,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) . (A-11)
I6,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−(1− e−ixLp+y
2 ) , (A-12)
Note that the central-cut diagram in Fig. 6 corresponds to the interference between t and
u-channel of the qq̄ → gg annihilation processes in Fig. 5. Since the splitting function is
symmetric in z and 1 − z, a factor of 2 comes from the fragmentation of both gluons in
the central-cut diagram.
The s-channel of qq̄ → gg is shown in Fig. 7 which has only one central-cut. Its contri-
bution to the partonic hard part is,
7,C =
α2sxB
I7,C 2Dg→h(zh/z)
2(z2 − z + 1)2
z(1 − z)
, (A-13)
I7,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y−
)e−ixLp
2 . (A-14)
Note that the splitting function 2(z2 − z + 1)2/z(1 − z) = 2[1 − z(1 − z)]2/z(1 − z) is
symmetric in z and 1− z. Therefore, fragmentation of the two final gluons gives rise to
the factor of 2 in front of the gluon fragmentation function.
Fig. 8. The interference between t and s-channel of qq̄ → gg annihilation.
Fig. 9. The complex conjugate of Fig. 8.
The interferences between t and s-channel of qq̄ → gg processes are shown in Figs. 8 and
9. There are only two possible cuts in these diagrams. The contributions from Fig. 8 are:
8,C =
α2sxB
I8,C Dg→h(zh/z)
1 + z3
z(1 − z)
1 + (1− z)3
z(1 − z)
, (A-15)
α2sxB
Dq→h(zh/z)2
1 + z3
z(1− z)
+ Dg→h(zh/z)2
1 + (1− z)3
z(1 − z)
, (A-16)
I8,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )e−ixLp
+(y−−y−
) , (A-17)
I8,L = θ(y
1 − y−2 )θ(y− − y−1 )ei(x+xL)p
×(e−ixLp+(y−−y
) − e−ixLp+(y−−y
)) . (A-18)
Contributions from Fig. 9, which are just the complex conjugate of Fig. 8, are:
9,C =
α2sxB
I9,C Dg→h(zh/z)
1 + z3
z(1 − z)
1 + (1− z)3
z(1 − z)
, (A-19)
9,R =
α2sxB
Dq→h(zh/z)2
1 + z3
z(1 − z)
+ Dg→h(zh/z)2
1 + (1− z)3
z(1 − z)
, (A-20)
I9,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+(y−−y
))e−ixLp
2 , (A-21)
I9,R = θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
×(1− e−ixLp+(y
))e−ixLp
1 . (A-22)
One can collect all contributions of the double hard qq̄ → gg processes from the central-
cut diagrams, which should have the common phase factor
ĪC = θ(−y−2 )θ(y− − y−1 )eixp
+y−e−ixLp
) , (A-23)
and obtain the total effective splitting function in the hard partonic part,
Pqq̄→gg(z) =
z(1 − z)
{C2F [1 + z2 + 1 + (1− z)2]− 2CF (CF − CA/2)
+2CFCA(1− z + z2)2 − CFCA[1 + z3 + 1 + (1− z)3]}
z2 + (1− z)2
z(1− z)
− 2CA[z2 + (1− z)2]
. (A-24)
We will find later in Appendix A-3 that the above result can also be obtained from the
total matrix elements squared for qq̄ → gg annihilation.
We now consider the annihilation processes qq̄ → qiq̄i with qi 6= q. There is only the
s-channel process with one central-cut diagram as shown in Fig. 10. Its contribution to
the hard part is
10,C =
α2sxB
I10,C
qi 6=q
[Dqi→h(zh/z) +Dq̄i→h(zh/z)]
Fig. 10. s-channel qq̄ → qiq̄i annihilation.
Fig. 11. t-channel qqi(q̄i) → qqi(q̄i) scattering.
×[z2 + (1− z)2]
, (A-25)
I10,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y
)e−ixLp
2 . (A-26)
Here we define the effective splitting function for qq̄ → qiq̄i annihilation as,
Pqq̄→qiq̄i(z) =
[z2 + (1− z)2] . (A-27)
Similarly, for qq̄i → qq̄i scattering with qi 6= q, there is only the t-channel as shown in
Fig. 11. There are, however, three cut diagrams. Their contributions to the partonic hard
part are:
11,C =
α2sxB
I11,C
Dq→h(zh/z)
1 + z2
(1− z)2
+Dq̄i→h(zh/z)
1 + (1− z)2
, (A-28)
11,L(R) =
α2sxB
I11,L(R)
Dq→h(zh/z)
1 + z2
(1− z)2
+Dg→h(zh/z)
1 + (1− z)2
, (A-29)
I11,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-30)
I11,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (A-31)
I11,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−(1− e−ixLp+y
2 ) . (A-32)
The twist-four two-parton correlation matrix element associated with the above quark-
antiquark scattering is the quark-antiquark correlator,
TAqq̄i(x, xL)∝ e
ixp+y−−ixLp
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
2 )|A〉 , (A-33)
and one should sum over all possible qi 6= q flavors. Note that in the above matrix element,
the momentum flow for the antiquark (q̄i) is opposite to that of the quark (q) fields.
For quark-quark scattering, qqi → qqi, the hard part is essentially the same. The only
difference is the associated matrix element for the quark-quark correlator which is ob-
tained from that of the quark-antiquark correlator via the exchange ψqi(y2) → ψ̄qi(y2)
and ψ̄qi(y1) → ψqi(y1),
TAqqi(x, xL)∝ e
ixp+y−+ixLp
×〈A|ψ̄q(0)
−)ψ̄qi(y
ψqi(y
1 )|A〉 . (A-34)
Note that the momentum flows of the two quarks (q and qi) point in the same direction.
The effective splitting function of this scattering process is defined through the fragmen-
tation of the quark in the central-cut diagram,
Pqqi(q̄i)→qqi(q̄i)(z) =
1 + z2
(1− z)2
. (A-35)
For annihilation qq̄ → qq̄ into identical quark and antiquark pairs, in addition to the
s-channel (Fig. 10 for qi = q) and t-channel (Fig. 11 for qi = q̄), one has also to consider
the interference between s and t-channel amplitudes as shown in Figs. 12 and 13, each
having two cuts. Their contributions to the hard partonic parts are, respectively:
Fig. 12. Interference between s and t-channel of qq̄ → qq̄ scattering
Fig. 13. The complex conjugate of Fig. 12.
12,C =
α2sxB
I12,C
Dq→h(zh/z)
(1− z)
+Dq̄→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-36)
12,L =
α2sxB
I12,L
Dq→h(zh/z)
(1− z)
+Dg→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-37)
I12,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+y
2 )e−ixLp
+(y−−y−
) , (A-38)
I12,L = θ(y
1 − y−2 )θ(y− − y−1 )ei(x+xL)p
×(e−ixLp+(y−−y
) − e−ixLp+(y−−y
)) ; (A-39)
Fig. 14. The interference between t and u-channel of identical quark-quark scattering qq → qq.
13,C =
α2sxB
I13,C
Dq→h(zh/z)
(1− z)
+Dq̄→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-40)
13,R =
α2sxB
I13,R
Dq→h(zh/z)
(1− z)
+Dg→h(zh/z)
2(1− z)2
CF (CF − CA/2)
, (A-41)
I13,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
×(1− e−ixLp+(y−−y
))e−ixLp
2 , (A-42)
I13,R = θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
×(e−ixLp+y
1 − e−ixLp+y
2 ) . (A-43)
One can again collect contributions from the central-cut diagrams of the double scattering
processes in Figs. 10, 11 12 and 13 and obtain the total effective splitting function for
qq̄ → qq̄,
Pqq̄→qq̄(z) =
[z2 + (1− z)2] +
1 + z2
(1− z)2
CF (CF − CA/2)
z2 + (1− z)2 +
1 + z2
(1− z)2
. (A-44)
Here we have used CF − CA/2 = −1/2Nc. For antiquark fragmentation, Pqq̄→q̄q(z) =
Pqq̄→qq̄(1 − z). One can also obtain the above result from qq̄ → qq̄ scattering matrix
squared as shown in Appendix A-3.
Similarly, for scattering of identical quarks qq → qq, one should set qi = q in Fig. 11[in
Eq. (A-28)]. In addition, one should also also include interference between t and u-channel
of the scattering as shown in Fig. 14. The contributions from such interference diagram
14,C =
α2sxB
I14,C
× 2Dq→h(zh/z)
z(1 − z)
CF (CF − CA/2)
, (A-45)
14,L(R) =
α2sxB
I14,L(R)
× [Dq→h(zh/z) +Dg→h(zh/z)]
z(1− z)
CF (CF − CA/2)
, (A-46)
I14,C = θ(−y−2 )θ(y− − y−1 )ei(x+xL)p
× (1− e−ixLp+y
2 )(1− e−ixLp+(y−−y
)) , (A-47)
I14,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−(1− e−ixLp+(y−−y
)) , (A-48)
I14,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−(1− e−ixLp+y
2 ) . (A-49)
Note again that the fragmentation of both quarks contributes to the factor 2 in Eq. (A-
45) since the splitting function is symmetric in z and 1 − z. The twist-four two-quark
correlation matrix element associated with qq → qq scattering is TAqq(x, xL) as compared
to TAqq̄(x, xL) for quark-antiquark annihilation processes.
We can sum contributions from the double hard scattering in all the central-cut diagrams
in Figs. 11 and 14 and obtain the total effective splitting function for qq → qq processes,
Pqq→qq(z) =
1 + z2
(1− z)2
1 + (1− z)2
CF (CF − CA/2)
z(1 − z)
1 + z2
(1− z)2
1 + (1− z)2
z(1 − z)
. (A-50)
There are two remaining cut diagrams that contribute to the quark-antiquark annihilation
at the order of O(α2s) as shown in Figs. 15 and 16. Their contributions are:
15,L =
α2sxB
I15,L
Dq→h(zh/z)2
1 + z2
+Dg→h(zh/z)2
1 + (1− z)2
, (A-51)
I15,L =−θ(y−1 − y−2 )θ(y− − y−1 )ei(x+xL)p
+y−e−ixLp
+(y−−y
) , (A-52)
Fig. 15. Interference between final-state gluon radiation from single and triple-quark scattering.
Fig. 16. The complex conjugate of Fig. 15.
16,R =
α2sxB
I16,R
Dq→h(zh/z)2
1 + z2
+Dg→h(zh/z)2
1 + (1− z)2
, (A-53)
I16,R =−θ(−y−2 )θ(y−2 − y−1 )ei(x+xL)p
+y−e−ixLp
1 . (A-54)
A-2 Effective splitting functions
In this Appendix, we list the effective splitting functions associated with each process
qa→ b and the double-hard (HI), hard-soft (SI) or their interferences (I, I2) according
to Eq. (73).
qqi(q̄i)→qi(q̄i)
(z) =
1 + (1− z)2
qqi(q̄i)→q
(z) =
1 + z2
(1− z)2
qqi(q̄i)→qi(q̄i)
(z) =
1 + (1− z)2
qqi(q̄i)→g
(z) = −1 + (1− z)
(A-55)
qq̄→qi(z) =P
qq̄→q̄i(z) = z
2 + (1− z)2,
qq̄→qi(z) =P
qq̄→q̄i(z) = z
2 + (1− z)2, (A-56)
P (HI)qq→q(z) =
1 + (1− z)2
1 + z2
(1− z)2
z(1 − z)
P (SI)qq→g(z) =−P (SI)qq→q(z) , (A-57)
P (SI)qq→q(z) =
1 + (1− z)2
z(1 − z)
qq̄→q(z) = z
2 + (1− z)2 + 1 + z
(1− z)2
qq̄→q̄(z) =P
qq̄→q(1− z) ,
qq̄→g(z) = 2CF
z2 + (1− z)2
z(1 − z)
− 2CA[z2 + (1− z)2], (A-58)
qq̄→q(z) =−
z(1 − z)
+ 2CF
qq̄→q̄(z) =
1 + (1− z)2
qq̄→g(z) =
z(1 − z)
+ 2CF
− 1 + (1− z)
(A-59)
qq̄→q(z) = z
2 + (1− z)2 −
z(1 − z)
− 2CF
qq̄→q̄(z) = z
2 + (1− z)2 ,
qq̄→g(z) =CA
4(1− z + z2)2 − 1
z(1 − z)
− 2CF
(1− z)2
, (A-60)
qq̄→q(z) =
z(1 − z)
− 2CF
qq̄→g(z) =
z(1 − z)
− 2CF
. (A-61)
The non-singlet splitting functions for qq̄ → b, defined as
qq̄→b(z) ≡ P
qq̄→b(z)− P
qq→b(z), (A-62)
are listed as below:
N(HI)
qq̄→qi(q̄i)
(z) =P
qq̄→qi(q̄i)
(z), ∆P
qq̄→qi(q̄i)
(z) = P
qq̄→qi(q̄i)
(z), (A-63)
N(HI)
qq̄→q (z) =−
(1− z2)(1 + z2 + (1− z)2)
1 + z3
z(1− z)
N(HI)
qq̄→q̄ (z) =P
qq̄→q̄(z), ∆P
N(HI)
qq̄→g (z) = P
qq̄→g(z), (A-64)
N(SI)
qq̄→q (z) =−
1 + z2
1 + (1− z)2
N(SI)
qq̄→q̄ (z) =P
qq̄→q̄(z)
N(SI)
qq̄→g (z) = 2CF
1 + z2
z(1 − z)
1 + (1− z)2
(A-65)
qq̄→b(z) =P
qq̄→b(z), ∆P
N(I2)
qq̄→b (z) = P
qq̄→b(z) (b = q, q̄, g) (A-66)
A-3 Alternative calculations of central-cut diagrams
As a cross-check of the hard partonic parts calculated from different cut diagrams in
Appendix A-1, we provide an alternative calculation of all the central-cut diagrams,
which correspond to quark-quark (antiquark) scattering.
Considering a parton (a) with momentum q scattering with another parton (b) that
carries a fractional momentum xp, a(q) + b(xp) → c(ℓ) + d(p′), the cross section can be
written as
dσab =
|M |2ab→cd(t̂/ŝ, û/ŝ)
(2π)32ℓ0
2πδ[(p+ q − ℓ)2]
(4π)2
|M |2ab→cd(t̂/ŝ, û/ŝ)
z(1− z)
dℓ2T δ
, (A-67)
where q = [0, q−, 0] and p = [xp+, 0, 0] are momenta of the initial partons and
, zq−, ~ℓT
(A-68)
is the momentum of one of the final partons. With the given kinematics, the on-shell
condition in the cross section can be recast as
(xp + q − ℓ)2 = 2(1− z)xp+q−
1− xL
, xL =
2z(1− z)p+q−
. (A-69)
The Mandelstam variables of the collision are,
ŝ=(q + xp)2 = 2xp+q− =
z(1 − z)
, û = (ℓ− xp)2 = −zŝ,
t̂=(ℓ− q)2 = −(1 − z)
ŝ = −(1− z)ŝ, (A-70)
where we have used the on-shell condition x = xL.
With Eq. (A-67) and parton distribution functions fNb (x), one can obtain the parton-
nucleon cross section,
dσaN =
dσabf
b (x)dx
fNb (xL)xL|M |2ab→cd(t̂/ŝ, û/ŝ)
z(1− z)
fNb (xL)
C0Pab→cd(z)dz
, (A-71)
where s = 2p+q− is the center-of-mass energy for aN collision, C0 is some common color
factor in the scattering matrix elements and
Pab→cd(z) = (1/C0)|M |2ab→cd(t̂/ŝ, û/ŝ) (A-72)
is what we have defined as the effective splitting function for the corresponding processes.
One can therefore easily obtain these effective splitting functions from the corresponding
matrix elements for elementary parton-parton scattering [39]. We will list them in the fol-
lowing. A common color factor for all quark-quark(antiquark) scattering is C0 = CF/Nc.
qq̄ → qiq̄i annihilation:
|M |2qq̄→qiq̄i =
t̂2 + û2
Pqq̄→qiq̄i(z) = z
2 + (1− z)2 . (A-73)
qq̄ → qq̄ annihilation:
|M |2qq̄→qq̄ =
û2 + ŝ2
û2 + t̂2
Pqq̄→qq̄(z) =
1 + z2
(1− z)2
+ z2 + (1− z)2 +
. (A-74)
qq̄ → gg annihilation:
|M |2qq̄→gg =
− 2CA
û2 + t̂2
Pqq̄→gg(z) = 2CF
z2 + (1− z)2
z(1− z)
− 2CA(z2 + (1− z)2) . (A-75)
qqi(q̄i) → qqi(q̄i) scattering:
|M |2qqi(q̄i)→qqi(q̄i)=
û2 + ŝ2
Pqqi(q̄i)→qqi(q̄i)(z) =
1 + z2
(1− z)2
. (A-76)
qq → qq scattering:
|M |2qq→qq =
û2 + ŝ2
ŝ2 + t̂2
Pqq→qq(z) =
1 + z2
(1− z)2
1 + (1− z)2
z(1− z)
. (A-77)
For quark-gluon Compton scattering, the relevant gluon distribution function is xLGN (xL).
One can therefore rewrite contribution from qg → qg to Eq. (A-71) as,
dσqN = xLGN(xL)πα
sz(1 − z)|M |2qg→qg(t̂/ŝ, û/ŝ)dz
≡xLGN(xL)πα2s
Pqg→qg(z)dz
. (A-78)
We have then for qg → qg scattering,
|M |2qg→qg =
ŝ2 + û2
û2 + ŝ2
Pqg→qg(z) = z(1 − z)
1 + z2
(1− z)2
1 + z2
. (A-79)
Comparing this result with that in Ref. [18] for the quark-gluon rescattering, we can see
that they agree in the limit 1 − z → 0. This is a consequence of the collinear approxi-
mation employed in Ref. [18] in the calculation of the hard partonic part in quark-gluon
rescattering.
We can also extend this calculation to the case of gluon-nucleon scattering. One can use
Eq. (A-71) to define the splitting function for gq → gq scattering,
|M |2gq→gq =
ŝ2 + t̂2
t̂2 + ŝ2
Pgq→gq(z) = z(1 − z)
1 + (1− z)2
1 + (1− z)2
(1− z)
. (A-80)
Here for gluon-parton scattering, there is no common color factor.
gg → qq̄ annihilation,
|M |2gg→qq̄ =
t̂2 + û2
t̂2 + û2
Pgg→qq̄(z) = z(1 − z)
z2 + (1− z)2
z(1− z)
[z2 + (1− z)2]
. (A-81)
gg → gg scattering
|M |2gg→gg =2
3− t̂û
− ûŝ
− t̂ŝ
Pgg→gg(z) = 2
(1− z + z2)3
z(1− z)
. (A-82)
One can use this technique to extend the study of modified fragmentation functions to
propagating gluons. Since the modification is dominated by quark-gluon and gluon-gluon
scattering, comparing the effective splitting functions,
Pqg→qg(z)≈
, (A-83)
Pgg→gg(z)≈
, (A-84)
in the limit z → 1, one can conclude that a gluon’s radiative energy loss is larger than
a quark by a factor of Nc/CF = CA/CF = 9/4. We will leave the complete derivation of
medium modification of gluon fragmentations to a future publication.
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Introduction
General formalism
Quark-quark double scattering processes
Modified Fragmentation Functions
q"7016q g annihilation
q"7016q qi"7016qi annihilation
qqi("7016qi) qqi("7016qi) scattering
qqqq scattering
q"7016q q"7016q, gg annihilation
Modification due to quark-gluon mixing
Flavor dependence of the medium modified fragmentation
Summary
Hard partonic parts for quark-quark double scattering
Effective splitting functions
Alternative calculations of central-cut diagrams
References
|