q-Deformed spin foam models of quantum gravity
q-deformed spin foam models of quantum gravity
Igor Khavkine1 and J. Daniel Christensen2
Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada
Department of Mathematics, University of Western Ontario, London, Ontario, Canada
E-mail: ikhavkin@uwo.ca and jdc@uwo.ca
November 4, 2018
Abstract
We numerically study Barrett-Crane models of Riemannian quantum grav-
ity. We have extended the existing numerical techniques to handle q-deformed
models and arbitrary space-time triangulations. We present and interpret expec-
tation values of a few selected observables for each model, including a spin-spin
correlation function which gives insight into the behaviour of the models. We
find the surprising result that, as the deformation parameter q goes to 1 through
roots of unity, the limit is discontinuous.
PACS numbers: 04.60.Pp
1 Introduction
Spin foam models were first introduced as a space-time alternative to the spin network
description of states in loop quantum gravity [3]. The most studied spin foam models
are due to Barrett and Crane [8,9]. A spin foam is a discretization of space-time where
the fundamental degrees of freedom are the areas labelling its 2-dimensional faces.
An important goal in the investigation of spin foam models is to obtain predictions
that can be compared to the large scale, classical, or semiclassical behavior of gravity.
This work continues the numerical investigation of the physical properties of spin foam
models of Riemannian quantum gravity begun in [5–7, 13]. In this paper, we extend
the computations to the q-deformed Barrett-Crane model and to larger space-time
triangulations.
The main applications of q-deformation are two-fold. On the one hand, it can act
as a regulator for divergent models, as is apparent in the link between the Ponzano-
Regge [27] and Turaev-Viro [31] models. On the other hand, Smolin [30] has argued
that q-deformation is necessary to account for a positive cosmological constant. Both
of these aspects are explored in more detail in Section 2.2. A surprising result of our
work is evidence that the limit, as the cosmological constant is taken to zero through
positive values, is discontinuous.
Large triangulations are necessary to approximate semiclassical space-times. The
possibility of obtaining numerical results from larger triangulations takes us one step
closer to that goal and increases the number of facets from which the physical proper-
ties of a spin foam model may be examined. As an example, we are able to study how
the spin-spin correlation varies with the distance between faces in the triangulation.
http://arxiv.org/abs/0704.0278v1
This paper is structured as follows. We begin in Section 2 by reviewing the basics
of q-deformation and discussing in detail its aforementioned applications. Section 3
reviews the details of the Barrett-Crane model, summarizes the necessary changes
for its q-deformation, and defines several observables associated to spin foams. In
Section 4, we review the existing numerical simulation techniques and how they need
to be generalized to handle q-deformation and larger triangulations. Section 5 presents
the results of our numerical simulations. In Section 6, we give our conclusions and list
some avenues for future research. The Appendix briefly summarizes our notational
conventions and useful formulas.
2 Deformation of su(2)
In this section, we describe the q-deformation of the Lie algebra su(2) into the algebra
suq(2) (also denoted Uq(su(2))), the representations of suq(2), and the applications
of q-deformation. The deformations of spin(4) are then obtained through the isomor-
phism spin(4) ∼= su(2)⊕ su(2).
The following is part of the general subject of quantum groups [21]. Here we shall
concentrate solely on the su(2) and spin(4) cases.
2.1 The algebra suq(2) and its representations
The Lie algebra su(2) is generated by the well known Pauli matrices σi, which obey
the commutation relations
[σ+, σ−] = 4σ3, [σ3, σ+] = 2σ+, [σ3, σ−] = −2σ−, (1)
where σ± = σ1 ± iσ2. The universal enveloping algebra of su(2) is the associative
algebra generated by σ± and σ3 subject to the above identities, with the Lie bracket
being interpreted as [A,B] = AB −BA.
The q-deformed algebra suq(2) is constructed by replacing σ3 with another gen-
erator. Formally, it is thought of as Σ = q
σ3 , where q ∈ C with the exceptions
q 6= 0, 1,−1. The Lie bracket relations are replaced by the identities
[σ+, σ−] = 4
Σ2 − Σ−2
q − q−1
, Σσ+ = qσ+Σ, Σσ− = −qσ−Σ. (2)
We can rewrite q = 1+ 2ε and think of ε as a small complex number. Then, formally
at leading order in ε, the substitution Σ = q
σ3 = 1 + εσ3 + O(ε
2) reduces the
deformed identities (2) to the standard Lie algebra relations (1). The associative
algebra generated by σ± and σ3 subject to the deformed identities (2) is the algebra
suq(2).
For generic q, that is, when q is not a root of unity, the finite-dimensional irreducible
representations of suq(2) are classified by a half-integer, j = 0, 1/2, 1, 3/2, . . . , referred
to as the spin, in direct analogy with the representations of su(2) and the theory of
angular momentum. The dimension of the representation j is 2j + 1. When q =
exp(iπ/r) is a 2rth root of unity (ROU), where r > 2 is an integer called the ROU
parameter, the representations j are still defined, but become reducible for j > (r −
2)/2. They decompose into a sum of representations with spin at most (r − 2)/2 and
so-called trace 0 ones, whose nature will be explained below.
For the purposes of this paper we are concerned only with intertwiners between
representations of suq(2), i.e., linear maps commuting with the action of the algebra,
and their (quantum) traces1.
Any such intertwiner can be constructed from a small set of generators and elemen-
tary operations on them. These constructions, as well as traces, can be represented
graphically. Such graphs are called (abstract) spin networks. Their calculus is well
developed and is described in [18], whose conventions we follow throughout the paper
with one exception: we use spins (half-integers) instead of twice-spins (integers). A
brief review of our notation and conventions can be found in the Appendix.
Trace 0 representations of suq(2) are so called because the trace of an intertwiner
from such a representation to itself is always zero. Thus, they can be freely discarded,
as they do not contribute to the evaluation of q-deformed spin networks.
2.2 Applications of q-deformation
Deformation, especially with q = exp(iπ/r) a 2rth primitive ROU, is important for
spin foam models for at least two reasons. Replacing q = 1 by some ROU can act as
a regulator for a model whose partition function and observable values are otherwise
divergent. Also, suq(2) spin networks
2 naturally appear when considering a positive
cosmological constant in loop quantum gravity.
The original Ponzano-Regge model [27] attempts to express the path integral for
3-dimensional Riemannian general relativity as a sum over labelled triangulations of a
3-manifold. The edges of the triangulation are labelled by discrete lengths, identified
with spin labels of irreducible SU(2) representations. Each tetrahedron contributes a
6j-symbol factor to the summand, normalized to ensure invariance of the overall sum
under change of triangulation. Unfortunately, the Ponzano-Regge model turned out
to be divergent. Motivated by the construction of 3-manifold invariants, Turaev and
Viro were able to regularize the Ponzano-Regge model [1, 31] by replacing the SU(2)
6j-symbols with their q-deformed analogs at a ROU q. The key feature of the regu-
larization is the truncation of the summation to only the irreducible representations
of suq(2) of non-zero trace, which leaves only a finite number of terms in the model’s
partition function.
A version of the Barrett-Cranemodel, derived from a group field theory by De Pietri,
Freidel, Krasnov and Rovelli [16] (DFKR for short), was also found to be divergent.
A q-deformed version of the same model at a ROU q is similarly regularized (see Sec-
tion 3.2). Some numerical results for the regularized version of this model are given
in Section 5.2.
The argument linking q-deformation to the presence of a positive cosmological
constant is due to Smolin [29] and is given in more refined form in [30]. It is briefly
summarized as follows. Loop quantum gravity begins by writing the degrees of free-
dom of general relativity in terms of an SU(2) connection on a spatial slice and the
slice’s extrinsic curvature. A state in the Schrödinger picture, a wave function on
the space of connections, can be constructed by integrating the Chern-Simons 3-form
over the spatial slice. This state, known as the Kodama state, simultaneously satisfies
all the canonical constraints of the theory and semiclassically approximates de Sit-
1When q = 1, this notion of trace reduces up to sign to the usual trace of a linear map, but is
slightly different otherwise, cf. [10, Chapter 4].
2These are graphs embedded in a 3-manifold, labelled by representations of suq(2). They are
similar to but distinct from the abstract spin networks referred to above. See [4] for the distinction.
ter spacetime, which is a solution of the vacuum Einstein equations with a positive
cosmological constant. The requirement that the Kodama state also be invariant un-
der large gauge transformations implies discretization of the cosmological constant,
Λ ∼ 1/r, with r a positive integer. The coefficients of the Kodama state in the spin
network basis are obtained by evaluating the labelled graph, associated to a basis
state, as an abstract suq(2) spin network. Here the deformation parameter q is a
ROU, q = exp(iπ/r), where the ROU parameter r is identified with the discretization
parameter of the cosmological constant.
Given the heuristic link [4] between spin networks of loop quantum gravity and
spin foams, it is natural to q-deform a spin foam model as an attempt to account
for a positive cosmological constant. With this aim, Noui and Roche [23] have given
a q-deformed version of the Lorentzian Barrett-Crane model. The possibility of q-
deformation has been with the Riemannian Barrett-Crane model since its inception [8]
and all the necessary ingredients have been present in the literature for some time. In
the next section these details are collected in a form ready for numerical investigation.
3 Deformation of the Barrett-Crane model
Consider a triangulated 4-manifold. Let ∆n denote the set of n-dimensional simplices
of the triangulation. The dual 2-skeleton is formed by associating a dual vertex, edge
and polygonal face to each 4-simplex, tetrahedron, and triangle of the triangulation,
respectively. A spin foam is an assignment of labels, usually called spins, to the dual
faces of the dual 2-skeleton. Each dual edge has 4 spins incident on it, while each dual
vertex has 10. A spin foam model assigns amplitudes AF , AE and AV , that depend on
all the incident spins, to each dual face, edge and vertex, respectively. The amplitude
Z(F ) assigned to a spin foam F is the product of the amplitudes for individual cells of
the 2-complex, while the total amplitude Ztot assigned to a triangulation is obtained
by summing over all spin foams based on the triangulation:
Z(F ) =
AF (f)
AE(e)
AV (v), Ztot =
Z(F ). (3)
Some models, such as those based on group field theory [16,17,24], also include a sum
over triangulations in the definition of the total partition function.
3.1 Review of the undeformed model
The Riemannian Barrett-Crane model was first proposed in [8]. Its relation to the
Crane-Yetter [15] spin foam model is analogous to the relation of the Plebanski [26]
formulation of general relativity (GR) to 4-dimensional BF theory with Spin(4) as the
structure group. Both BF theory and the Crane-Yetter model are topological and the
latter is considered a quantization of the former [2]. In the Plebanski formulation, GR
is a constrained version of BF theory. Similarly, the Barrett-Crane model restricts
the spin labels summed over in the Crane-Yetter model. With this restriction, Barrett
and Crane hoped to produce a discrete model of quantum (Riemannian) GR.
3.1.1 Dual vertex amplitude
All amplitudes are defined in terms of spin(4) spin networks. However, given the
isomorphism spin(4) ∼= su(2)⊕ su(2), all irreducible representations of spin(4) can be
written as tensor products of irreducible representations of su(2). The Barrett-Crane
model specifically limits itself to balanced representations, which are of the form j⊗ j,
where j is the irreducible representation of su(2) of spin j. Since the tensor product
corresponds to a juxtaposition of edges in a spin network, any spin(4) spin network
may be written as an su(2) spin network where an edge labelled j ⊗ j is replaced by
two parallel edges, each labelled j. To avoid redundancy of notation, we use a single
j instead of j ⊗ j to label spin(4) spin network edges. We then distinguish them from
su(2) networks by placing a bold dot at every vertex.
The Barrett-Crane vertex is an intertwiner between four balanced representations:
e . (4)
The graphs on the right hand side of the definition are su(2) spin networks and the
sum runs over all admissible labels e. The graphical notation and the conditions for
admissibility are defined in the Appendix.
The above expression defines the Barrett-Crane vertex in a way that breaks ro-
tational symmetry. However, it can be shown that the vertex is in fact rotation-
ally symmetric. Up to normalization, this property makes the Barrett-Crane vertex
unique [28]. The above formula defines a vertical splitting of the vertex. A ninety
degree rotation will define an analogous horizontal splitting. Both possibilities are
important in the derivation of the algorithm presented in Section 4.1.
Given a 4-simplex v of a triangulation, the corresponding vertex of the dual 2-
complex is assigned the amplitude
AV (v) =
j1,0j1,1
j1,4j1,2
j2,1 j2,4
j2,2 j2,3
. (5)
This spin network is called the 10j-symbol. The 4-simplex v is bounded by five tetra-
hedra, which correspond to the vertices of the 10j graph. The four edges incident on
a vertex correspond to the four faces of the corresponding tetrahedron; the spin labels
are assigned accordingly. The edge joining two vertices corresponds to the face shared
by corresponding tetrahedra. Evaluation of the 10j-symbol is discussed in Section 4.1.
While the crossing structure depicted above is immaterial in the undeformed case, it
is essential at nontrivial values of q. It is given here for reference.
3.1.2 Dual edge and face amplitudes
The original paper of Barrett and Crane did not specify dual edge and face amplitudes.
Three different dual edge and face amplitude assignments were considered in a previous
paper [7]. We concentrate on the same possibilities.
For the Perez-Rovelli model [25], we have
AF (f) = j , AE(e) =
j1 j2 j3 j4
. (6)
For the DFKR model [16], we have
AF (f) = j , AE(e) =
. (7)
For the Baez-Christensen model [7], we have
AF (f) = 1, AE(e) =
. (8)
The bubble diagram, when translated into su(2) spin networks, corresponds to two
bubbles (see Appendix)
. (9)
and evaluates to (2j + 1)2.
The so-called eye diagram simply counts the dimension of the space of 4-valent
intertwiners, which is also the number of admissible e-edges summed over in Equa-
tion (4). In symmetric form, it is given by
1 + min{2j, s− 2J} if positive and s is integral,
0 otherwise,
where s =
k jk, j = mink jk, and J = maxk jk.
3.2 The q-deformed model
Thanks to graphical notation, the q-deformation of the spin foam amplitudes described
above is straightforward, with only a few subtleties. The main distinction is that
q-deformed graphs are actually ribbon (framed) graphs with braiding. Thus, any
undeformed spin network has to be supplemented with information about twists and
crossings before evaluation.
In [32], Yetter generalized the Barrett-Crane 4-vertex for a q-deformed version of
spin(4). Since spin(4) ∼= su(2) ⊕ su(2), there is a two parameter family of possible
deformations of the Lie algebra, spinq,q′ (4)
∼= suq(2) ⊕ suq′(2). Yetter singles out
the one parameter family q′ = q−1, restricted to balanced representations, since it
preserves the invariance of the Barrett-Crane vertex under rotations. This family also
has especially simple curl and twist identities:
, (11)
where the left factor of j ⊗ j corresponds to suq(2) and the right one to suq−1(2), and
the 3-vertex is the obvious juxtaposition of two suq(2) and suq−1(2) 3-vertices. Once
this deformation is adopted, the ribbon structure can be ignored [32], so one only
needs to specify the crossing structure for a given spin(4) spin network to obtain a
well-defined q-evaluation.
There are three basic graphs needed to define the Barrett-Crane simplex ampli-
tudes: the bubble, the eye, and the 10j-symbol. The evaluation of the bubble graph,
Equation (9), is [2j + 1]2, where the quantum integer [2j + 1] is defined in the Ap-
pendix. Remarkably, the value of the eye diagram turns out not to depend on q and
its value is still given by Equation (10). The only exception is when q is a ROU with
parameter r. Then, the dimension of the space of 4-valent intertwiners changes to
1 + min{2j, s− 2J}
r − 1−max{2J, s− 2j}
if positive and
s is integral,
0 otherwise,
where again s =
k jk, j = mink jk, and J = maxk jk.
The 10j-symbol is the only network with a non-planar graph. Originally, it was
defined in terms of the 15j-symbol from the Crane-Yetter model. This 15j-symbol
was defined with q-deformation in mind, so its crossing and ribbon structure was fully
specified [14, Section 3]. Adapted to the 10j-graph, it can be summarized as follows:
Consider a 4-simplex. The dual 1-skeleton of the boundary has five dual vertices and
ten dual edges, and is the complete graph K5 on these five dual vertices. If we remove
one of the (non-dual) vertices from the boundary of the 4-simplex, what remains is
homeomorphic to R3. For any such homeomorphism, the embedding of K5 into R
can be projected onto a 2-dimensional plane. The crossing structure of the 10j graph
is defined by such a projection. It is illustrated in Equation (5). Although, with
crossings, the 10j graph is no longer manifestly invariant under permutations of its
vertices, it can be shown to be so.
3.3 Observables
The definition of observables in a spin foam model of quantum gravity is still open
to interpretation (see Section 6 of [7] for a brief discussion). For a fixed spin foam,
the half-integer spin labels of its faces are the fundamental variables of the model.
Practically speaking, any observable of a spin foam model should be an expectation
value of some function O(F ) of the spin labels of a spin foam F , averaged over all spin
foams with amplitudes specified by Equation (3):
〈O〉 =
O(F )Z(F )
. (13)
In this paper we choose to concentrate on a few observables representative of the
kind of quantities computable in a spin foam model. As before, fix a triangulation of
a 4-manifold, let ∆2 represent the set of its faces and let j : ∆2 → {0, 1/2, 1, . . .} be
the spin labelling. We define:
J(F ) =
⌊j(f)⌉ , (14)
(δJ)2(F ) =
(⌊j(f)⌉ − 〈J〉)
, (15)
A(F ) =
⌊j(f)⌉ ⌊j(f) + 1⌉, (16)
Cd(F ) =
f,f ′∈∆2
dist(f,f ′)=d
⌊j(f)⌉ ⌊j(f ′)⌉ − 〈J〉
〈(δJ)2〉
. (17)
where ⌊n⌉ denotes a quantum half-integer (see Appendix), | · | denotes cardinality,
dist(f, f ′) denotes the distance between faces, and Nd is a normalization factor (see
below for the definition of distance and Nd). These observables represent average spin
per face, variance of spin per face, average area per face, and spin-spin correlation as
a function of d.
The choice of observables given above is somewhat arbitrary. For instance, there
are several subtly distinct choices for the expression for (δJ)2. Fortunately, they all
yield expectation values that are nearly identical. The expression given above has the
technical advantage of falling into the class of so-called single spin observables. These
are observables whose expectation value can be directly obtained from the knowledge
of probability with which spin j occurs on any face of a spin foam. All of J , (δJ)2,
and A are single spin observables, while Cd is not.
Note that on a fixed triangulation with no other background geometry, there is
no physical notion of distance. We can, instead, define a combinatorial analog. For
any two faces f and f ′ of a given triangulation, let dist(f, f ′) be the smallest number
of face-sharing tetrahedra that connect f to f ′. Given the discrete structure of our
spacetime model, it is conceivable that this combinatorial distance, multiplied by a
fundamental unit of length, approximates some notion of distance derived from the
dynamical geometry of the spin foam model.
The correlation function Cd may be thought of as analogous to a normalized 2-
point function of quantum field theory. The d-degree of face f is the number of faces
f ′ such that dist(f, f ′) = d. If the d-degree of every face is the same, the normalization
factor Nd can be taken to be the number of terms in the sum (17), that is, the number
of face pairs separated by distance d. This choice ensures the inequality |Cd| ≤ 1. If
not all faces have the same d-degree, then the normalization factor has to be modified
Nd = |∆2|Dd, (18)
where Dd is the maximum d-degree of a face, which reduces to the simpler definition
in the case of uniform d-degree.
The choice of the q-dependent expression ⌊j⌉, instead of simply using the half-
integer j, is motivated in Section 5.1. For some q, the argument of the square root in
A(F ) may be negative or even complex. In that case, a branch choice will have to be
made. Luckily, if q = 1, q is a ROU, or q is real, the expression under the square root
is always non-negative.
4 Numerical simulation
The key development that made possible numerical simulation of variations of the
(undeformed) Barrett-Crane model [6,7] is the development by Christensen and Egan
of a fast algorithm for evaluating 10j-symbols [13]. In this section, we show how
this algorithm generalizes to the q-deformed case and discuss numerical evaluation of
observables for the previously described spin foam models.
4.1 The q-deformation of the fast 10j algorithm
The derivation of the Christensen-Egan algorithm given in [13] is contingent on the
possibility of splitting the Barrett-Crane 4-vertex as in Equation (4) and on the re-
coupling identity, Equation (43) of the Appendix. Both identities still hold in the
q-deformed case. The validity of the 4-vertex splitting was proved by Yetter [32] and
the recoupling identity is a standard part of suq(2) representation theory.
The only remaining detail of the algorithm’s generalization is the crossing structure
of the 10j graph, which was established in Section 3.2. However, its only consequence
is an extra factor from the twist implicit in the bubble diagram of Section 4 of [13],
cf. Equation (50) of the Appendix. We will not reproduce the derivation of the algo-
rithm here. However, the way in which the twist arises is schematically illustrated in
Figure 1. Note that the triviality of the twist for Yetter’s balanced representations,
Equation (11), does not apply here since the twist occurs separately in distinct suq(2)
networks.
The algorithm itself can be summarized in the following form:
{10j} = (−)2S
m1,m2
φ tr[M4M3M2M1M0]. (19)
The 10j-symbol depends on the ten spins ji,k, (i = 1, 2, k = 0, . . . , 4) specified in
Equation (5). The overall prefactor depends on the total spin S =
i,k ji,k and the
per-term prefactor is
φ = (−)m1−m2 [2m1 + 1][2m2 + 1]q
m1(m1+1)−m2(m2+1). (20)
(a) (b)
(c) (d)
Figure 1: In reference to [13], (a) corresponds to Equation (1), (b) corresponds to
Equation (2), while (c) and (d) correspond to the “ladder” and “bubble” diagrams
of Section 4, respectively. The illustrated twist introduces the explicitly q-dependent
factor into Equation (20).
The exponents of (−) and q are always integers. The Mk are matrices (not all of
the same size) of dimensions compatible with the five-fold product and trace. Their
matrix elements are
[2lk + 1](T1)
θ(j2,k−1, lk+1, j1,k) θ(j2,k+1, lk+1, j1,k+1)
, (21)
lk j2,k mi
lk+1 j2,k−1 j1,k
θ(j2,k, lk+1,mi)
. (22)
The quantum integers [n], as well as the theta θ(a, b, c) and tetrahedral Tet[· · ·] suq(2)
spin networks are defined in the Appendix.
The quantities lk and mi are spin labels (half-integers). They are constrained by
admissibility conditions (parity conditions and triangle inequalities). The parity of
each index is determined by the conditions
lk ≡ j1,k + j2,k ≡ j1,k−1 + j2,k−2, (23)
mi ≡ lk + j2,k−1, (24)
for i = 1, 2 and k = 0, . . . , 4, where ≡ denotes equivalence mod 1 and the second
subscript of j is taken mod 5. Summation bounds are determined by the triangle in-
equalities, which must be checked for each trivalent vertex introduced in the derivation
of the algorithm. They boil down to
lb3(j1,k, j2,k, j2,k−1) ≤ mi ≤ j1,k + j2,k + j2,k−1, (25)
|j1,k−1 − j2,k−2| ≤ lk ≤ j1,k−1 + j2,k−2, (26)
|j1,k − j2,k| ≤ lk ≤ j1,k + j2,k, (27)
|mi − j2,k−1| ≤ lk ≤ mi + j2,k−1, (28)
for i = 1, 2 and k = 0, . . . , 4, where we have used the notation
lb3(a, b, c) = 2max{a, b, c} − (a+ b+ c). (29)
When q = exp(iπ/r) is a ROU, extra inequalities must be taken into account to
exclude summation over reducible representations. These are
mi ≥ j1,k + j2,k + j2,k−1 − (r − 2), (30)
mi ≤ ub3(j1,k, j2,k, j2,k−1) + (r − 2), (31)
lk ≤ (r − 2)− (j1,k + j2,k), (32)
lk ≤ (r − 2)− (j1,k−1 + j2,k−2), (33)
lk ≤ (r − 2)− (m+ j2,k−1), (34)
where now
ub3(a, b, c) = 2min{a, b, c} − (a+ b+ c).
If any of the parity constraints or inequalities cannot be satisfied, the 10j-symbol
evaluates to zero.
This algorithm has been implemented and tested in the q = 1 and ROU cases,
for both j and r up to several hundreds. Unfortunately, for generic q, when Q =
max{|q|, |q|−1} > 1, the quantum integers grow exponentially as |[n]| ∼ Qn. Such a
rapid growth makes the sums involved in this algorithm numerically unstable. It is
still possible to use this algorithm with Q close to 1 or symbolically, using rational
functions of q instead of limited precision floating point numbers. Symbolic computa-
tion is, however, significantly slower (by up to a factor of 106) than its floating point
counterpart. The software library spinnet which implements these and other spin
network evaluations is available from the authors and will be described in a future
publication.
4.2 Positivity and statistical methods
The sums involved in evaluating expectation values of observables, as in Equation (13),
are very high-dimensional. For instance, a minimal triangulation of the 4-sphere (seen
as the boundary of a 5-simplex) contains 20 faces. Hence, any brute force evaluation
of an expectation value, even on such a small lattice, involves a sum over the 20-
dimensional space of half-integer spin labels.
Fortunately, in the undeformed case, the total amplitude Z(F ) for a closed spin
foam is never negative3 [5]. The proof for the q = 1 case generalizes to the ROU
case. One need only realize two facts. The first is that, in the ROU case, quantum
integers are non-negative. The second is that, for q a ROU, an suq−1(2) spin network
evaluates to the complex conjugate of the corresponding suq(2) spin network. The
3We expect the same thing to hold in Lorentzian signature [5, 12].
disjoint union of any two such spin networks evaluates to their product, the absolute
value squared of either of them, and hence is non-negative. Then, the same positivity
result follows as from Equation (1) of [5]. This positivity allows us to treat Z(F ) as
a statistical distribution and use Monte Carlo methods to extract expectation values
with much greater efficiency than brute force summation.
The main tool for evaluating expectation values is the Metropolis algorithm [20,22].
The algorithm consists of a walk on the space of spin labellings. Each step is randomly
picked from a set of elementary moves and is either accepted or rejected based on
the relative amplitudes of spin foam configurations before and after the move. An
expectation value is extracted as the average of the observable over the configurations
constituting the walk. Elementary moves for spin foam simulations are discussed in
the next section.
A Metropolis-like algorithm is possible even if individual spin foam amplitudes
Z(F ) are negative or even complex. However, if the total partition function Ztot sums
to zero, then the expectation values in Equation (13) become ill defined. Moreover, in
numerical simulations, if Ztot is even close to zero, expectation value estimates may
exhibit great loss of precision and slow convergence. In the path-integral Monte Carlo
literature, this situation is known as the sign problem [11]. Still, the sign problem
need not occur or, depending on the severity of the problem, there may be ways of
effectively dealing with it.
Independent Metropolis runs can be thought of as providing independent estimates
of a given expectation value. Thus, the error in the computed value of an observable
can be estimated through the standard deviation of the results of many independent
simulation runs [19].
4.3 Elementary moves for spin foams
The choice of elementary moves for spin foam simulations must satisfy several criteria.
Theoretically, the most important one is ergodicity. That is, any spin foam must
be able to transform into any other one through a sequence of elementary moves
which avoid configurations with zero amplitude. Practically, it is important that
these moves usually preserve admissibility. A spin foam F is called admissible if the
associated amplitude Z(F ) is non-zero. If, starting with an admissible spin foam,
most elementary moves produce an inadmissible spin foam, the simulation will spend
a lot of time rejecting such moves without any practical benefit.
As before, consider a fixed triangulation of a compact 4-manifold. The parity
conditions (23) imposed on the ji,k,
j1,k + j2,k ≡ j1,k−1 + j2,k−2, 0 ≤ k ≤ 4,
when taken together with the total spin foam amplitude (3), provide strong constraints
on admissible spin foams. One can show that a move that changes spin labels by ±1/2
(mod 1) on each face of a closed surface in the dual 2-skeleton preserves the parity
constraint. We take as the elementary moves the moves that change the spin labels
by ±1/2 on the boundaries of the dual 3-cells of the dual 3-complex; the dual 3-cells
correspond to the edges of the triangulation. If the manifold has non-trivial mod 2
homology in dimension 2, additional moves would be necessary, but for the examples
we consider the moves above suffice. From a practical point of view, extra moves
might improve the simulation’s equilibration time. For instance, in the ROU case,
parity preserving moves that change the spins from 0 to (r − 2)/2 or (r − 3)/2 were
introduced, since spins close to either admissible extreme may have large amplitudes.
This property of the Perez-Rovelli and Baez-Christensen models is illustrated in the
following section.
Unfortunately, the inequalities constraining spin labels do not have a similar geo-
metric interpretation and cannot be used to easily restrict the set of elementary moves
in advance.
5 Results
Using methods described in the previous section, we ran simulations of the three vari-
ations of the Barrett-Crane model described in Section 3 and obtained expectation
values for observables listed in Section 3.3. While previous work [7] performed simula-
tions only on the minimal triangulation of the 4-sphere, which we will refer to simply
as the minimal triangulation, we have extended the same techniques to arbitrary tri-
angulations of closed manifolds.
5.1 Discontinuity of the r → ∞ limit
The most striking result we can report is a discontinuity in the transition to the limit
r → ∞, where r, a positive integer, is the ROU parameter with q = exp(iπ/r). As
r → ∞, the deformation parameter q tends to its classical value 1. If we interpret
the cosmological constant as inversely proportional to r, Λ ∼ 1/r, this limit also
corresponds to Λ → 0, through positive values. For a fixed spin foam, the amplitudes
and observables we study tend continuously to their undeformed values as r → ∞.
However, we find that observable expectation values do not tend to their undeformed
values in the same limit, that is, 〈O〉r 9 〈O〉q=1 as r → ∞.
The discontinuity is most simply illustrated with the single spin distribution, that
is the probability of finding spin j at any spin foam face. This probability can be
estimated from the histogram of all spin labels that have occurred during a Monte
Carlo simulation. The points in Figure 2(a) show the single spin distributions for the
Baez-Christensen model with r = 50 and q = 1. The curves show the corresponding
single bubble amplitude. It is the amplitude Z(Fj) of a spin foam Fj with all spin
labels zero, except for the boundary of an elementary dual 3-cell, whose faces are all
labelled with spin j. The amplitudes and distributions are normalized as probabil-
ity distributions so their sums over j yield 1. The similarity between the points and
the continuous curves is consistent with the hypothesis that spin foams with isolated
bubbles dominate the partition function sum. The behavior of the single spin dis-
tribution for the Perez-Rovelli model is very similar, except that its peaks are much
more pronounced.
Note that the undeformed single spin distribution has a single peak at j = 0, while
the r = 50 case has two peaks, one at j = 0 and the other at j = (r − 2)/2, the
largest non-trace 0 irreducible representation. The bimodal nature of the single spin
distribution has an important impact on the large r behavior of observable expectation
values, as is most easily seen with single spin observables (Section 3.3). For instance,
if we consider the average, j̄, of the half-integers j, the large j peak would dominate
the expectation value and 〈j̄〉 would diverge linearly in r, as r → ∞. On the other
hand, since J is the average of the quantum half-integers ⌊j⌉, 〈J〉 at least approaches
a constant in the same limit. This is illustrated in Figure 2(b).
Figure 2: (a) Single spin distribution and single bubble amplitude for the Baez-
Christensen model. The distribution was obtained from 109 steps of Metropolis sim-
ulation on a triangulation with 202 faces (cf. Section 5.3). (b) Some single spin ob-
servables as functions of j, with r = 50.
However, as shown in Figure 3, this limit is not the same as the undeformed
expectation value. At the same time, as can be seen from the plot of the Perez-
Rovelli average area in the same figure, there are some observables whose large r
limits are at least very close to the undeformed values. The area observable summand
⌊j⌉ ⌊j + 1⌉ is exactly zero at both j = 0 and j = (r − 2)/2, while the spin
observable summand Jj = ⌊j⌉ is zero at j = 0 but still positive at j = (r − 2)/2,
Figure 2(b). The large j peak of the Perez-Rovelli model is very narrow and thus
the expectation value of a single spin observable is strongly influenced by its value at
j = (r − 2)/2.
The data for larger triangulations is qualitatively similar.
5.2 Regularization of the DFKR model
As expected, the ROU deformation of the DFKR model yields a finite partition func-
tion and finite expectation values. For instance, its single spin distribution for r = 40
is illustrated in Figure 4. The divergence of the amplitude for large spins in the unde-
formed, q = 1, case makes numerical simulation impossible without an artificial spin
cutoff. Thus, we do not have an undeformed analog of the single spin distribution.
For the minimal triangulation, the ROU spin distribution deviates slightly from the
single bubble amplitude close to the boundaries of admissible j. For the larger trian-
gulation, the deviation is much more pronounced and is not restricted to the edges.
This suggests that there are other significant contributions to the partition function
besides single bubble spin foams.
Note the large weight associated with spins around j = r/4. Around this value of
j, both the area Aj =
⌊j⌉ ⌊j + 1⌉ and the spin Jj = ⌊j⌉ attain their maximal values
and are proportional to r. Thus, it is natural to expect their expectation values to
grow linearly in r, which is consistent with the divergent nature of the undeformed
DFKR model. This is precisely the behavior shown in Figure 5. On the minimal
triangulation, the best linear fits for the average spin expectation value and for the
Figure 3: Observables for the Baez-Christensen (BCh) and Perez-Rovelli (PR) models
as functions of the ROU parameter r. For large r, observables do not in general tend
to their undeformed, q = 1, values; arrows show the deviation. Some observables
were scaled to fit on the graph. Data is from Metropolis simulations on the minimal
triangulation.
square root of the average spin variance are
〈J〉r = 0.146 r − 0.064, (35)
(δJ)2
= 0.014 r + 0.187. (36)
For larger triangulations, the dependence of these observables is also approximately
linear in r, with only slight variation in the effective slope.
5.3 Spin-spin correlation
The ability to work with larger lattices allows us to explore a broader range of observ-
ables. One of them is the spin-spin correlation function Cd defined in Section 3.3. In
general 〈C0〉 = 1 and 〈Cd〉 → 0 for large d. The decay of the correlation shows how
quickly the spin labels on different spin foam faces become independent. A positive
value of 〈Cd〉 indicates that, on average, any two faces distance d apart both have
spins above (or both below) the mean 〈J〉. On the other hand, a negative value of
〈Cd〉 indicates that, on average, any two faces distance d apart have one spin above
and one below the mean 〈J〉.
A small triangulation limits the maximum distance between faces. For example,
the minimal triangulation has maximum distance d = 3. Larger triangulations of
the 4-sphere were obtained by refining the minimal one by applying Pachner moves
randomly and uniformly over the whole triangulation. We restricted the Pachner
moves to those that did not decrease the number of simplices.
Figure 4: Single spin distributions and single bubble amplitudes for the DFKR model.
The distributions were obtained from 109 steps of Metropolis simulation on the mini-
mal triangulation and on a triangulation with 202 faces (cf. Section 5.3).
The largest triangulation we have used has maximum distance d = 6. Its corre-
lations for different models are shown in Figure 6 along with those from the minimal
triangulation. Correlation functions for different values of ROU parameter r (including
the q = 1 case) and other triangulations are qualitatively similar.
Notice the small negative dip for small values of d for the Perez-Rovelli and Baez-
Christensen models. As discussed in previous sections, the partition functions of these
models are dominated by spin foams with isolated bubbles. The correlation data is
consistent with this hypothesis. The values of the spins assigned to faces of the bubble
will be strongly correlated, while the values of the spins on two faces, one of which
lies on the bubble and the other does not, should be strongly anti-correlated. Since
a given face usually has fewer nearest neighbors that lie on the same bubble than
that do not, on average, the short distance correlation is expected to be negative. At
slightly larger distances, the correlation function turns positive again. This indicates
that on a larger triangulations, spin foams with several isolated bubbles contribute
strongly to the partition function. Although, with so few data points, it is difficult
to extrapolate the behavior of the correlation function to larger triangulations and
distances, its features are qualitatively similar to that of a condensed fluid, where the
density-density correlation function exhibits oscillations on the scale of the molecular
dimensions.
Note that the behavior of the DFKR correlation function is significantly different
from the other two. This is also consistent with the already observed fact that its
partition function has strong contributions from other than single or isolated bubble
spin foams.
Figure 5: Observables for the DFKR model: area 〈A〉, average spin 〈J〉, spin standard
deviation
〈(δJ)2〉. Metropolis simulation, minimal triangulation. Error bars are
smaller than the data points.
6 Conclusion
We have numerically investigated the behavior of physical observables for the Perez-
Rovelli, DFKR, and Baez-Christensen versions of the Barrett-Crane spin foam model.
Each version assigns different dual edge and face amplitudes to a spin foam, and these
choices greatly affect the behavior of the resulting model. The behavior of the models
was also greatly affected by q-deformation.
The limiting behavior of observables was found to be discontinuous in the limit of
large ROU parameter r, i.e., q = exp(iπ/r) close to its undeformed value of 1. This
result is at odds with the physical interpretation of the relation Λ ∼ 1/r between the
cosmological constant Λ and the ROU parameter. Finally, the behavior of the exam-
ined physical observables, especially of the spin-spin correlation function, indicates the
dominance of isolated bubble spin foams in the Perez-Rovelli and Baez-Christensen
partition functions, while less so for the the DFKR one.
Some questions raised by these results deserve attention. For instance, it is not
known whether the same q → 1 limit behavior will be observed when q is taken
through non-ROU values. While calculations with max{|q|, |q|−1} > 1 are numerically
unstable, they should still be possible for |q| ∼ 1.
Another important project is to perform a more extensive study of the effects of
triangulation size in order to better understand the semi-classical limit.
Finally, all of this work should also be carried out for the Lorentzian models, which
are physically much more interesting but computationally much more difficult.
These and other questions will be the subject of future investigations.
Figure 6: Spin-spin correlation functions for the Baez-Christensen (BCh), Perez-
Rovelli (PR) and DFKR models, on the minimal triangulation (6 vertices, 15 edges, 20
faces, 15 tetrahedra, and 6 4-simplices) as well as a larger triangulation (23 vertices,
103 edges, 202 faces, 200 tetrahedra, and 80 4-simplices). ROU parameter r = 10.
Acknowledgements
The authors would like to thank Wade Cherrington for helpful discussions. The first
author was supported by NSERC and FQRNT postgraduate scholarships and the
second author by an NSERC grant. Computational resources for this project were
provided by SHARCNET.
A Spin network notation and conventions
Quantum integers are a q-deformation of integers. For an integer n, the corresponding
quantum integer is denoted by [n] and is given by
[n] =
qn − q−n
q − q−1
. (37)
In the limit q → 1, we recover the regular integers, [n] → n. Note that [n] is invariant
under the transformation q 7→ q−1. When q = exp(iπ/r) is a root of unity (ROU), for
some integer r > 1, an equivalent definition is
[n] =
sin(nπ/r)
sin(π/r)
. (38)
This expression is non-negative in the range 0 ≤ n ≤ r. Quantum factorials are
defined as
[n]! = [1][2] · · · [n]. (39)
In many cases, q-deformed spin network evaluations can be obtained from their unde-
formed counterparts by simply replacing factorials with quantum factorials. For con-
venience, when dealing with half-integral spins, we also define quantum half-integers
⌊j⌉ =
when j is a half-integer.
Abstract suq(2) spin networks can be approached from two different directions.
They can represent contractions and compositions of suq(2)-invariant tensors and in-
tertwiners [10]. At the same time, they can represent traces of tangles evaluated
according to the rules of the Kauffman bracket [18]. Either way, the computations
turn out to be the same. We present here formulas for the evaluation of a few spin
networks of interest.
The single bubble network evaluates to what is sometimes called the superdimension
of the spin-j representation:
j = (−)2j [2j + 1]. (41)
(As in the rest of the paper, the spin labels are half-integers.)
Up to a constant, there is a unique 3-valent vertex (corresponding to the Clebsch-
Gordan intertwiner) whose normalization is fixed up to sign by the value of the θ-
network :
θ(a, b, c) =
(−)s[s+ 1]![s− 2a]![s− 2b]![s− 2c]!
[2a]![2b]![2c]!
, (42)
where s = a+ b + c. The θ-network is non-vanishing, together with the three-vertex
itself, if and only if s is an integer and the triangle inequalities are satisfied: a ≤ b+ c,
b ≤ c+ a, and c ≤ a+ b. In addition, when q is a ROU, one extra inequality must be
satisfied: s ≤ r − 2. The triple (a, b, c) of spin labels is called admissible if θ(a, b, c) is
non-zero.
The recoupling identity gives the transformation between different bases for the
linear space of 4-valent tangles (or intertwiners):
(−)2e[2e+ 1]Tet
a b e
c d f
θ(a, d, e) θ(c, b, e)
e , (43)
where the sum is over all admissible labels e and the value of the tetrahedral network
a b e
c d f
m≤S≤M
(−)S [S + 1]!
i[S − ai]!
j [bj − S]!
, (44)
where
[bj − ai]! E ! = [2A]![2B]![2C]![2D]![2E]![2F ]! (45)
a1 = (a+ d+ e) b1 = (b+ d+ e+ f) (46)
a2 = (b + c+ e) b2 = (a+ c+ e+ f) (47)
a3 = (a+ b+ f) b3 = (a+ b+ c+ d) (48)
a4 = (c+ d+ f) m = max{ai} M = min{bj}. (49)
Due to parity constraints, the ai, bj, m, M , and S are all integers.
Since the three-vertex is unique up to scale, its composition with with a braiding
applied to two incoming legs yields a multiplicative factor:
= (−)a+b−cqa(a+1)+b(b+1)−c(c+1)
. (50)
Note that the above braiding factor is not invariant under the transformation q 7→ q−1,
while the bubble, tetrahedral and θ-networks are all invariant under this transforma-
tion, by virtue of their expressions in terms of quantum integers.
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	Introduction
	Deformation of su(2)
	The algebra suq(2) and its representations
	Applications of q-deformation
	Deformation of the Barrett-Crane model
	Review of the undeformed model
	Dual vertex amplitude
	Dual edge and face amplitudes
	The q-deformed model
	Observables
	Numerical simulation
	The q-deformation of the fast 10j algorithm
	Positivity and statistical methods
	Elementary moves for spin foams
	Results
	Discontinuity of the r limit
	Regularization of the DFKR model
	Spin-spin correlation
	Conclusion
	Spin network notation and conventions
