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: corichi@matmor.unam.mx
†Electronic address: tatjana@shi.matmor.unam.mx
‡Electronic address: zapata@matmor.unam.mx
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:corichi@matmor.unam.mx
mailto:tatjana@shi.matmor.unam.mx
mailto:zapata@matmor.unam.mx
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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