Asymptotic Analysis of Ponzano-Regge Model With Non-commutative Metric Variables - Daniele Oriti, Matti Raasakka

8/13/2019 Asymptotic Analysis of Ponzano-Regge Model With Non-commutative Metric Variables - Daniele Oriti, Matti Raasak

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arXiv:1401.5819v

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[gr-qc]22Jan2

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Asymptotic analysis of Ponzano-Regge model

with non-commutative metric variables

Daniele Oriti

Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Muhlenberg 1, 14476 Potsdam, Germany

Matti Raasakka

LIPN, Institut Galilee, CNRS UMR 7030, Universite Paris 13,

Sorbonne Paris Cite, 99 av. Clement, 93430 Vil letaneuse, France

(Dated: January 24, 2014)

We apply the non-commutative Fourier transform for Lie groups to formulate the non-

commutative metric representation of the Ponzano-Regge spin foam model for 3d quantum

gravity. The non-commutative representation allows to express the amplitudes of the model

as a first order phase space path integral, whose properties we consider. In particular, westudy the asymptotic behavior of the path integral in the semi-classical limit. First, we com-

pare the stationary phase equations in the classical limit for three different non-commutative

structures corresponding to symmetric, Duflo and Freidel-Livine-Majid quantization maps.

We find that in order to unambiguously recover discrete geometric constraints for non-

commutative metric data through stationary phase method, the deformation structure of

the phase space must be accounted for in the variational calculus. When this is understood,

our results demonstrate that the non-commutative metric representation facilitates a con-

venient semi-classical analysis of the Ponzano-Regge model, which yields as the dominant

contribution to the amplitude the cosine of the Regge action in agreement with previous stud-

ies. We also consider the asymptotics of the SU(2) 6j-symbol using the non-commutative

phase space path integral for the Ponzano-Regge model, and explain the connection of our

results to the previous asymptotic results in terms of coherent states.

I. INTRODUCTION

Spin foam models have in recent years arisen to prominence as a possible candidate formulation

for the quantum theory of spacetime geometry. (See [1] for a thorough review.) Their formal-

ism derives mainly from topological quantum field theories [2], Loop Quantum Gravity [3,4] and

discrete gravity, e.g., Regge calculus [5]. On the other hand, spin foam models may also be seen

as a generalization of matrix models for 2d quantum gravity via group field theory [ 6,7]. For 3d

quantum gravity, the relation between spin foam models and canonical quantum gravity has been

fully cleared up. In particular, it is known that the Turaev-Viro model is the covariant version of

the canonical quantization (a la Witten) of 3d Riemannian gravity with a positive cosmological

constant, while the Ponzano-Regge model is the limit of the former for a vanishing cosmological

constant[8,9]. In this case, the spin foam 2-complexes have been rigorously shown to arise as his-

tories of LQG spin network states, as initially suggested in [10], while the correspondence between

Electronic address: [email protected] address: [email protected]
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LQG states and the Ponzano-Regge boundary data had been already noted in [ 11]. However, in 4d

the situation is less clear. Several different spin foam models for 4d Riemannian quantum gravity

have been proposed in the literature, such as the Barrett-Crane model[12,13], the Freidel-Krasnov

model [14], a model based on the flux representation [15], and one based on the spinor represen-

tation [16], while in the Lorentzian case the Engle-Pereira-Rovelli-Livine model[17,18] representsessentially the state of the art. (See also [19] for a review of the new 4d models.) These 4d models

differ specifically in their implementation of the necessary simplicity constraints on the underlying

topological BF theory, which should impose geometricity of the two-complex corresponding to a

discrete spacetime manifold and give rise to local degrees of freedom. Thus, a further study of the

geometric content of the different spin foam models is certainly welcome. In particular, one might

hope to recover discrete Regge gravity in the classical limit of the model, since this would imply

an acceptable imposition of the geometric constraints at least in the classical regime. Moreover,

classical general relativity can be obtained from the Regge gravity by further taking the continuum

limit, which allows for some confidence that continuum general relativity may be recovered also

from the continuum limit of the full quantum spin foam model. The Regge action is indeed known

to arise as the stationary phase solution in the 3d case in the large-spin limit for handlebodies

[20]. In 4d, Regge action was recovered asymptotically first for a single 4-simplex [21] and later

for an arbitrary triangulation with a fixed spin labeling, when both boundary and bulk spin vari-

ables are scaled to infinity [2226]. Recently, in [27, 28], an asymptotic analysis of the full 4d

partition function was given using microlocal analysis, which revealed some worrying accidental

curvature constraints on the geometry of several widely studied 4d models. This work considered

only the strict asymptotic regime of the spin variables, without further scalings of the parameters

of the theory. The work of [29,30] on the other hand dealt with the large-spin asymptotics of the

EPRL model considering also scaling in the Barbero-Immirzi parameter, with interesting results.

In particular, the analysis of [30] used also the discrete curvature as an expansion parameter and

identified an intermediate regime of large spin values (dependent on the Barbero-Immirzi parame-

ter) that, in combination with a small curvature approximation, seems to lead to the right Regge

behaviour of the amplitudes.

Classically, spin foam models, as discretizations of continuum theories, are based on a phase

space structure, which is a direct product of cotangent bundles over a Lie group that is the structure

group of the corresponding continuum principal bundle (e.g., SU(2) for 3d Riemannian gravity).

The group part of the product of cotangent bundles thus corresponds to discrete connection vari-

ables on a triangulated spatial hypersurface, while the cotangent spaces correspond to discrete

metric variables (e.g., edge vectors in 3d, or face bivectors in 4d, which correspond to discrete

tetrad variables due to the simplicity constraints). Accordingly, the geometric data of the classical

discretized model is transparently encoded in the cotangent space variables. However, when one

goes on to quantize the system to obtain the spin foam model, the cotangent space variables get

quantized to differential operators on the group. Typically (for compact Lie groups), these geomet-

ric operators possess discrete spectra, and so the transparent classical discrete geometry described

by continuous metric variables gets replaced by the quantum geometry described by discrete spin

labels. This corresponds to a representation of the states and amplitudes of the model in terms

of eigenstates of the geometric operators, the spin representation hence the name spin foams.

The quantum discreteness of geometric variables in spin foams, i.e. the use of quantum numbers as


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opposed to phase space variables, although very useful to make contact with the canonical quantum

theory, makes the amplitudes lose direct contact with classical discrete action and classical discrete

geometric variables. The use of such classical discrete geometric variables, on the other hand, has

been prevented until recently by their non-commutative nature.

However, recently, a new mathematical tool was introduced in the context of 3d quantumgravity, which became to be called the group Fourier transform [13, 15, 3138]. This is an L2-

isometric map from functions on a Lie group to functions on the cotangent space equipped with a

(generically) non-commutative -product structure. In[39], the transform was generalized to the

non-commutative Fourier transform for all exponential Lie groups by deriving it from the canonical

symplectic structure of the cotangent bundle, and the non-commutative structure was seen to

arise from the deformation quantization of the algebra of geometric operators. Accordingly, the

non-commutative but continuous metric variables obtained through the non-commutative Fourier

transform correspond to the classical metric variables in the sense of deformation quantization.

Thus, it enables one to describe the quantum geometry of spin foam models and group field theory

[35,36] (and Loop Quantum Gravity [34,37]) by classical-like continuous metric variables.

The aim of this paper is to initiate the application of the above results in analysing the geometric

properties of spin foam models, in particular, in the classical limit ( 0). We will restrict our

consideration to the 3d Ponzano-Regge model[4042] to have a better control over the formalism in

this simpler case. However, already for the Ponzano-Regge model we discover non-trivial properties

of the metric representation related to the non-commutative structure, which elucidate aspects of

the use of non-commutative Fourier transform in the context of spin foam models. In particular,

we find that in applying the stationary phase approximation one must account for the deformation

structure of the phase space in the variational calculus in order to recover the correct geometric

constraints for the metric variables in the classical limit of the phase space path integral. Otherwise,

the classical geometric interpretation of metric boundary data depends on the ambiguous choice of

quantization map for the algebra of geometric operators, which seems problematic. Nevertheless,

once the deformed variational principle adapted to the non-commutative structure of the phase

space is employed, the non-commutative Fourier transform is seen to facilitate an unambiguous and

straightforward asymptotic analysis of the full partition function via non-commutative stationary

phase approximation.

In SectionIIwe will first outline the formalism of non-commutative Fourier transform, adapted

from [39] to the context of gravitational models. In SectionIII we introduce the Ponzano-Regge

model, seen as a discretization of the continuum 3d BF theory. In SectionIVwe then apply the

non-commutative Fourier transform to the Ponzano-Regge model to obtain a representation of

the model in terms of non-commutative metric variables, and write down an explicit expression

for the quantum amplitude for fixed metric boundary data. In Section V we further study the

classical limit of the Ponzano-Regge amplitudes for fixed metric boundary data, and find that

the results differ for different choices of non-commutative structures unless one accounts for the

deformation structure in the variational calculus. When this is taken into account the resulting

semi-classical approximation coincides with what one expects from a discrete gravity path integral.

In particular, if one considers only the partial saddle point approximation obtained by varying

the discrete connection only, one finds that the discrete path integral reduces to the one for 2nd

order Regge action, in terms of a discrete triad. In Section VI we consider in more detail the


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Ponzano-Regge amplitude with non-commutative metric boundary data for a single tetrahedron.

We recover the Regge action in the classical limit of the amplitude, and explain the connection

of our calculation to the previous studies of spin foam asymptotics in terms of coherent states.

SectionVIIsummarizes the obtained results and points to further research.

II. NON-COMMUTATIVE FOURIER TRANSFORM FORSU(2)

Our exposition of the non-commutative Fourier transform for SU(2) in this section follows [39],

adapted to the needs of quantum gravity models. Originally, a specific realization of the non-

commutative Fourier transform formalism for the group SO(3) was introduced in [31] by Freidel &

Livine, and later expanded on by Freidel & Majid[32] and Joung, Mourad & Noui[33] to the case

ofS U(2). (More abstract formulations of a similar concept have appeared also in [ 43,44].) In our

formalism this original version of the transform corresponds to a specific choice of quantization of

the algebra of geometric operators, which we will refer to as the Freidel-Livine-Majid quantizationmap, and treat it as one of the concrete examples we give of the more general formulation in

SubsectionV A.1

Let us consider the group SU(2), the Lie algebra Lie(SU(2)) =: su(2) of SU(2), and the

associated cotangent bundle TSU(2) = SU(2) su(2). As it is a cotangent bundle, TSU(2)

carries a canonical symplectic structure. This is given by the Poisson brackets

{O, O} O

XiLiO

LiOO

Xi+ kij

O

Xi

O

XjXk, (1)

where O, O C(TSU(2)) are classical observables, and Li := Li are dimensionful Lie deriva-

tives on the group with respect to a basis of right-invariant vector fields. R+ is a parameter

with dimensions [ X], which determines the physical scale associated to the group manifold via the

dimensionful Lie derivatives and the structure constants [ Li, Lj] = kij Lk. Xi are the Cartesian

coordinates on su(2).2

Let us now introduce coordinates : SU(2)\{e} su(2) = R3 on the dense subset

SU(2)\{e} =:HS U(2), where e S U(2) is the identity element, which satisfy (e) = 0 and

Lij(e) =ji . The use of coordinates onHcan be seen as a sort of one-point-decompactification

ofSU(2). We then have for the Poisson brackets of the coordinates3

{i, j} = 0 , {Xi, j} = Lij , {Xi, Xj} = kij Xk. (2)

The Poisson brackets involving i are, of course, well-defined only on H. We see that the com-

mutator {Xi, j} of the chosen canonical variables are generically deformed due to the curva-

ture of the group manifold, coinciding with the usual flat commutation relations associated with

1 In addition, another realization of the non-commutative Fourier transform for SU(2) relying on spinors was for-mulated by Dupuis, Girelli & Livine in[45], but we will not consider it here.

2 Here it seems we are giving dimensions to coordinates, which is usually a bad idea in a gravitational theory, tobe considered below. The point here is that the coordinatesXi turn out to have a geometric interpretation asdiscrete triad variables, which is exactly what one would like to give dimensions to in general relativity.

3 Strictly speaking, the coordinates are not observables of the classical system, but we may consider them definedimplicitly, since any observable may be parametrized in terms of them, and they may be approximated arbitrarilyclosely by classical observables.


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Poisson-commuting coordinates only at the identity. Moreover, let us define the deformed addi-

tion for these coordinates in the neighborhood of identity as (gh) =: (g) (h). It holds

(g)(h) =(g)+ (h)+ O(0, | ln(g)|, | ln(h)|) for any choice ofcomplying with the above men-

tioned assumptions. Indeed, the parametrization is chosen so that in the limit 0, while keeping

the coordinatesfixed, we effectively recover the flat phase space TR3

=R3

R3

su

(2)su

(2)

from TSU(2) =SU(2) su(2). This follows because keepingfixed implies a simultaneous scal-

ing of the class angles | ln(g)| of the group elements. Accordingly, the group effectively coincides

with the tangent space su(2) at the identity in this limit, and become the Euclidean Poisson-

commuting coordinates on su(2) = R3 for any initial choice of satisfying the above assumptions.

Thus, can also be thought of as a deformation parameter already at the level of the classical

phase space. For the above reasons, we will call the limit 0 the abelian limit.

Let us then consider the quantization of the Poisson algebra given by the Poisson bracket

(1). In particular, we consider the algebra H generated by the operators i and Xi, modulo the

commutation relations

[i,j] = 0 , [Xi,j] =i

Lij , [Xi, Xj] =i kij Xk. (3)These relations follow from the symplectic structure ofTSU(2) in the usual way by imposing the

relation [Q(O),Q(O)] !=iQ({O, O}) with the Poisson brackets of the canonical variables, where

by Q: C(TSU(2)) H we denote the quantization map specified by linearity, the ordering of

operators, and Q(i) =:i, Q(Xi) =: Xi.

We wish to represent the abstract algebra H defined by the commutation relations (3) as op-

erators acting on a Hilbert space. There exists the canonical representation in terms of smooth

functions onH SU(2) with the L2

inner product

| := 1

3

H

dg (g)(g) , (4)

where dg is the normalized Haar measure, and the action of the canonical operators on is given by

i i , Xi iLi . (5)

However, we would like to represent our original configuration space SU(2) rather than H, and

therefore we will instead consider smooth functions on SU(2), whose restriction on H is clearly

always in C(H). Since the coordinates are well-defined only on H= SU(2)\{e}, the action of

the coordinate operators should then be understood only in a weak sense, i.e., even though strictly

speaking the action i i is not well-defined for the whole of SU(2), the inner products

|i| are, since we may write

|i| = 1

3

SU(2)

dg (g)i(g)(g) 1

3

H

dg (g)i(g)(g) (6)

for smooth, . It is easy to verify that the commutation relations are represented correctly with

this definition of the action, and the function space may be completed in the L2-norm as usual.

However, there is also a representation in terms of another function space, which is obtained

through a deformation quantization procedure applied to the operator algebra corresponding to theother factor of the cotangent bundle, su(2). (See[39] for a thorough exposition.) Notice that the

restriction ofH to the subalgebra generated by the operators Xi is isomorphic to a completion of


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the universal enveloping algebra U(su(2)) ofSU(2) due to its Lie algebra commutation relations.

A -product for functions on su(2) is uniquely specified by the restriction of the quantization

map Q on the su(2) part of the phase space via the relation f f := Q1(Q(f)Q(f)), where

f, f C(su(2)) and accordingly Q(f),Q(f) U(su(2)). One may verify that the following

action of the algebra on functions L

2

(su(2)

) constitutes another representation of the algebra:

i i

Xi, Xi Xi . (7)

The non-commutative Fourier transform acts as an intertwiner between the canonical represen-

tation in terms of square-integrable functions on SU(2) and the non-commutative dual space

L2(su(2)) of square-integrable functions on su(2) with respect to the -product. It is given by

(X)

H

dg

3 E(g, X)(g) L2(su(2)

) , L2(SU(2)) , (8)

where the integral kernel

E(g, X) eik(g)X

:=

n=0

1

n!

i

nki1(g) kin(g)Xi1 Xin (9)

is the non-commutative plane wave, and we denote k(g) := i ln(g) su(2) taken in the principal

branch of the logarithm. The inverse transform reads

(g) =

su(2)

dX

(2)3 E(g, X) (X) L2(SU(2)) , L2(su(2)

) . (10)

Let us list some important properties of the non-commutative plane waves that we will use in

the following:

E(g, X) =eik(g)X

c(g)ei(g)X , where c(g) :=E(g, 0) , (11)

E(g, X) =E(g1, X) =E(g, X) , (12)

E(adhg, X) =E(g, Ad1h X) , where adhg:= hgh

1 and AdhX :=hX h1 , (13)

E(gh,X) =E(g, X) E(h, X) , (14)su(2)

dX

(2)3 E(g, X) =(g) , (15)

(X) E(g, X) =E(g, X) (AdgX) . (16)

Notice that from (11) and (13) it follows thatc(adhg) =c(g) and (adhg) =h(g)h1 =: Adh(g).

In addition, we find that the function

(X, Y) :=

H

dg

(2)3 E(g, X)E(g, Y) (17)

acts as the delta distribution with respect to the -product, namely,su(2)

dY (X, Y) (Y) = (X) =

su(2)

dY (Y) (X, Y) (18)

or, more generally, is the integral kernel of the projection

P()(X) :=

su(2)

dY (X, Y) (Y) (19)


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onto the image L2(su(2)) of the non-commutative Fourier transform. In the following, we will

also occasionally slightly abuse notation by writing

iXi

:=

H

dg

(2)3

iE(g, Xi) (20)

for convenience, although this is not a function of the linear sum i Xi ifc(g) = 1 for someg H.III. 3D BF THEORY AND THE PONZANO-REGGE MODEL

The Ponzano-Regge model can be understood as a discretization of 3-dimensional Riemannian

BF theory. In this section, we will briefly review how it can be derived from the continuum BF

theory, while keeping track of the dimensionful physical constants, which determine the various

asymptotic limits of the theory.

Let M be a 3-dimensional base manifold to a frame bundle with the structure group SU(2).

Then the partition function of 3d BF theory on M is given by

ZMBF =

DED exp

i

2

M

tr

E F()

, (21)

where Eis an su(2)-valued triad 1-form onM, F() is the su(2)-valued curvature 2-form associ-

ated to the connection 1-form, and the trace is taken in the fundamental spin- 12 representation of

SU(2). is the reduced Planck constant and is a constant with dimensions of inverse momentum.

The connection with Riemannian gravity in three spacetime dimensions gives := 8G, where G

is the gravitational constant. Since the triad 1-form Ehas dimensions of length and the curvature

2-form Fis dimensionless, the exponential is rendered dimensionless by dividing with 8lp,

lp G being the Planck length in three dimensions. Integrating over the triad field in (21), we

get heuristically

ZMBF

D

F()

, (22)

so we see that the BF partition function is nothing but the volume of the moduli space of flat

connections onM. Generically, this is of course divergent, which (among other things) motivates

us to consider discretizations of the theory. However, since BF theory is purely topological, that

is, it does not depend on the metric structure of the base manifold, such a discretization should

not affect its essential properties.

Now, to discretize the continuum BF theory, we first choose a triangulation of the manifold

M, that is, a (homogeneous) simplicial complex homotopic to M. The dual complex of

is obtained by replacing each d-simplex in by a (3 d)-simplex and retaining the connective

relations between simplices. Then, the homotopy between and M allows us to think of , and

thus , as embedded inM. We further form a finer cellular complex by diving the tetrahedra

in along the faces of . In particular, then consists of tetrahedrat , with verticest

at their centers, each subdivided into four cubic cells. Moreover, for each tetrahedront , there

are edgestf , which correspond to half-edges off , going from the centers of the triangles

f bounding the tetrahedron to the center of the tetrahedron t. Also, for each triangle f ,

there are edges ef , which go from the center of the triangle f to the centers of the


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teifei+1

e

fei

FIG. 1: The subdivision of tetrahedra in into a finer cellular complex .

edges e bounding the triangle f. See Fig.1for an illustration of the subdivision of a single

tetrahedron in .

To obtain the discretized connection variables associated to the triangulation , we integrate

the connection along the edges tf and ef as

gtf := Peitf SU(2) and gef := Pe

ief SU(2) , (23)

where Pdenotes the path-ordered exponential. Thus, they are the Wilson line variables of the

connection associated to the edges or, equivalently, the parallel transports from the source to

the sink of the edges with respect to . We assume the triangulation to be piece-wise flat, and

associate frames to all simplices of . We then interpret gtfas the group element relating the

frame oft to the frame off , and similarly gefas the group element relating the frame of

f to the frame ofe . Furthermore, we integrate the triad field along the edges e as

Xe :=

e

E su(2) , (24)

where an orientation for the edge e may be chosen arbitrarily. Xeis interpreted as the vector giving

the magnitude and the direction of the edge e in the frame associated to the edge e itself.

In the case that has no boundary, a discrete version of the BF partition function (22), the

Ponzano-Regge partition function, can be written as

ZPR=

tf

dgtf

e

(He(gtf)) , (25)

where He(gtf) SU(2) are holonomies around the dual faces e obtained as products of

gtf, f e, and dgtf is again the Haar measure on SU(2). Mimicking the continuum partition

function of BF theory, the Ponzano-Regge partition function is thus an integral over the flat discreteconnections, the delta functions (He(gtf)) constraining holonomies around all dual faces to be

trivial.


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Now, we can apply the non-commutative Fourier transform to expand the delta functions in

terms of the non-commutative plane waves. This yields

ZPR =

tfdgtf

edXe

(2)3 e

c(He(gtf))

exp

i

eXe (He(gtf))

. (26)

Comparing with (21), this expression has a straightforward interpretation as a discretization of

the first order path integral of the continuum BF theory. We can clearly identify the discretized

triad variablesXe in (24) with the non-commutative metric variables defined via non-commutative

Fourier transform. We also see that, from the point of view of discretization, the form of the plane

waves and thus the choice for the quantization map is directly related to the choice of the precise

form for the discretized action and the path integral measure. In particular, the coordinate function

:SU(2) su(2) and the prefactorc: SU(2) C of the non-commutative plane wave are dictated

by the choice of the quantization map, and the coordinates specify the discretization prescription for

the curvature 2-form F(). Similar interplay between-product quantization and discretization iswell-known in the case of the first order phase space path integral formulation of ordinary quantum

mechanics [46]. Moreover, on dimensional grounds, we must identify = 8G, so that the

coordinates have the dimensions of 1F(). Therefore, the abelian limit of the non-commutative

structure of the phase space corresponds in this case also physically to the no-gravity limitG 0.

We will denote this classical deformation parameter associated with the non-commutative structure

of the phase space collectively by in the following.

IV. NON-COMMUTATIVE METRIC REPRESENTATION OF THE PONZANO-REGGE

MODEL

If the triangulated manifold has a non-trivial boundary, we may assign connection data on

the boundary by fixing the group elementsgef associated to the boundary trianglesf . Then,

the (non-normalized) Ponzano-Regge amplitude for the boundary can be written as

APR(gef|f ) =

tf

dgtf

ef

f /

dgef

e

ne1i=0

gefei+1g1tei f

ei+1

gtei fei g1efei

. (27)

The delta functions are over the holonomies around the wedges of the triangulation pictured in

grey in Fig. 1. For this purpose, the tetrahedra tei and the triangles fei sharing the edge e are

labelled by an index i = 0, . . . , ne 1 in a right-handed fashion with respect to the orientation of

the edge e and with the identification fne f0, as in Fig. 1.

Let us introduce some simplifying notation. We will choose an arbitrary spanning tree of the

dual graph to the boundary triangulation, pick an arbitrary root vertex for the tree, and label the

boundary triangles fi by i N0 in a compatible way with respect to the partial ordering

induced by the tree, so that the root has the label 0. (See Fig.2.) Moreover, we denote the set

of ordered pairs of labels associated to neighboring boundary triangles by N, and label the group

elements associated to the pair of neighboring boundary triangles (i, j) Nas illustrated in Fig.2.

The group elements gtf, f / , we will denote by a collective label hl. As we integrate over gef


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0

1

2

3

4

5

6 7

8

9

10

1112

13

14

15

16

17

18

1920

21

22

23

24 25i

j

hi hj

Kij

gji gij

FIG. 2: On the left: A portion of a rooted labelled spanning tree of the dual graph of a boundary triangulation

(solid grey edges). On the right: Boundary triangles fi, fj and the associated group elements.

for f / in (27), we obtain

APR(gij) = l

dhle/

(He(hl))(i,j)Ni


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We note that this is nothing else than the simplicial path integral for a complex with boundary,

and a fixed discrete metric on this boundary, as it can be seen by writing the explicit form of the

non-commutative plane waves, thus obtaining a formula like (26), augmented by boundary terms.

We will use this expression in the next section, to study the semi-classical limit.

Exact amplitudes for metric boundary data

By integrating over all Ye and using the property (14) for the non-commutative plane waves,

we may write (31) as

APR(Xij)

(i,j)N

dgij3

dYji(2)3

l

dhl

e/

(He(hl))

(i,j)Ni


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12

We see that the edge vectors Xij, Xji corresponding to the same edge (with opposite orientations)

in different frames of reference are identified up to a parallel transport by h1j Kji(hl)hi through

the non-commutative delta distributions (Adh1j Kji(hl)hiXji , Xij).

We wish to further integrate over the variables hi. To this aim, we employ the change of

variables Xji Adh1i Kij(hl)hjXji , i.e., we parallel transport the variablesXji to the frames ofXijto get a simple identification of the boundary variables, and move all hi-dependence to the plane

waves. We thus get

APR(Xij)

l

dhl

e/

(He(hl))

(i,j)Ni


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13

contractible along the faces of the 2-complex. Thus, given that the neighborhoods of all boundary

vertices have trivial topology, the flatness constraints impose Lij(hl) to be trivial. Accordingly, we

have

APR

(Xij

) l dhl e/ (He(hl))

i

E(hi, ijXji)

(i,j)Ni


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14

wave to express (31) as

APR(Xij) =

(i,j)N

dgij3

(i,j)Ni


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15

Variation ofYij: Similarly, variation with respect to Yij gives

(gijh1j Kji(hl)hig

1ji ) = 0 gijh

1j Kji(hl)hig

1ji = (44)

for all (i, j) N, i < j, i.e., the triviality of the connection around the dual faces e to

boundary edges e .

Variation ofhl in the bulk: The variations for the group elements are slightly less trivial. Tak-

ing left-invariant Lie derivatives of the exponential with respect to a group elementhl gtf

in the bulk, we obtaine/

Ye Lhlk (He(hl)) +

(i,j)Ni


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16

Variation ofgij: Taking Lie derivatives with respect to a gij of the exponential, we obtain

Yij Lgijk (gijh

1j Kji(hl)hig

1ji ) + Xij L

gijk (g

1ij ) = 0k

Ad1gij Yij D(gij)Xij = 0 = Ad

1gij

Yij+ D(gji)Xji (47)

for all i < j, where we denote (D(g))kl := Lkl(g). We see that this equation identifies

the boundary metric variables Xij with the integration variables Yij, taking into account

the orientation and the parallel transport between the frames of each vector, plus a non-

geometric deformationgiven by the matrix D(gij).4

Thus, we have obtained the constraint equations corresponding to variations of all the integration

variables. In particular, by combining the equations (47) with the boundary closure constraint

(46), we obtain

fj(i,j)N

D(gij)Xij = 0i , (48)

which gives, in general, a deformedclosure constraint for the boundary metric edge variables Xij.

In addition, from (47) alone we obtain a deformed identification

Adgij (D(gij)Xij) = Adgji (D

(gji)Xji) , (49)

naturally up to a parallel transport, of the boundary edge variables Xij and Xji . Accordingly, we

obtain for the amplitude

APR(Xij)

(i,j)N

dgij3

c(g1ij )

v

(Hv(gij))

fi

fj(i,j)N

D(gij)Xij

(i,j)Ni


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17

limit 0, || = const., do the different choices agree, in general, producing the undeformed

discrete geometric constraintsfj

(i,j)N

Xij = 0fi and AdgijXij = Adgji Xji (i, j) N (51)

for the discretized boundary metric variables Xij su(2).

Some examples

Before we go on to consider the stationary phase boundary configurations obtained through the

ordinary commutative variational calculus for some specific choices of the coordinates , and thus

the associated quantization maps, let us make a few general remarks on the apparent dependence

of the limit on this choice. As we have already emphasized above, the exact functional form of

the non-commutative plane waves, and thus the coordinate choice, is determined ultimately by the

choice of the quantization map and the-product that we thus obtain. We have found the general

expression for the plane wave as a -exponential

E(g, X) =eik(g)X

=n=0

1

n!

i

nki1(g) kin(g)Xi1 Xin. (52)

From this expression we may observe that the way the Planck constant enters into the plane wave

is very subtle. There are negative powers of () coming from the prefactor in the exponential,

while from the -monomials arise positive powers of (). The way these different contributions

go together determines the explicit functional form of the non-commutative plane wave, and ac-cordingly the behavior in the classical limit 0. Therefore, it is not too surprising that we

may eventually find different classical limits for different choices of plane waves through the appli-

cation of the ordinary stationary phase method. In particular, it is important to realize that the

non-commutative plane wave itself is purely a quantum object with an ill-defined classical limit,

and therefore has no duty to coincide with anything in this limit. For this reason, the stationary

phase solutions corresponding to different-products may also differ from each other, even though

the -product itself coincides with the pointwise product in this limit. On the contrary, in the

abelian limit 0 we also scale the coordinates k i on the group, so that ki/stay constant, since

determines the scale associated to the group manifold. Therefore, the non-commutative planewave agrees with the usual commutative plane wave in this limit. Only in the abelian limit may

one expect the different choices of non-commutative structures lead to unambiguous results, when

one applies the commutative variational calculus.

Symmetric & Duflo quantization maps:

The symmetric quantization map QS: Pol(su(2)) U(su(2)) is determined by the symmetric

operator ordering for monomials

QS(Xi1Xi2 Xin) !=

1

n! n Xi(1)Xi(2) Xi(n),


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18

where n is the group of permutations ofn elements, and extends by linearity to any completion

of the polynomial algebra Pol(su(2)). In particular, we have

Q1S (e ik(g)X) =

n=0in

n!()nki1(g) kin(g)Q1S (Xi1 Xin) = e

ik(g)X ES(g, X)

and accordingly to this quantization prescription is associated a non-commutative plane wave with

cS(g) = 1, S(g) = iln(g) su(2), where the value of the logarithm is taken in the principal

branch [39].

The Duflo quantization map QD is defined as

QD= QS j12 (x) ,

where j12 (x) is a differential operator associated to the function j : su(2) C given by

j 12 (X) := detsinh(12adX)12adX

12 .The definition is such that QDrestricts to an isomorphism from theg-invariant functions on su(2)

tog-invariant operators (i.e., Casimirs) inU(su(2)), and therefore the Duflo map can be considered

as algebraically the most natural choice for a quantization map. In the Duflo case we obtain

cD(g) = |S(g)|sin(|S(g)|)

, but the coordinates are the same D S as for the symmetric quantization

map, so the amplitudes have the same stationary phase behavior in both cases. In this respect it is

important to note that even though the Duflo factor cD(g) = |S(g)|sin(|S(g)|)

diverges for |S(g)| =,

the path integral measure is still well-behaved, since cD(g)dg= sin(|S|)

|S|

d3S remains finite.

We obtain from the equation (A1) in AppendixA for the deformation matrix as a function of

the coordinates

DSkl(S) =

|S|

sin(|S|)

cos(|S|)kl +

sin(|S|)

|S| cos(|S|)

S,kS,l

|S|2 klm

mS

. (53)

This deformation matrix has the following nice property: DSkl(k)kl =kk. This implies, in particular,

that when the edge vectors are stable under the dual connection variables, AdgijXij = Xij

k(gij) Xij , we have DS(gij)Xij =Xij , and therefore recover the undeformed closure constraints

from (48). Accordingly, classical geometric boundary data with AdgijXij =Xij , Xij = Xji andjXij = 0 satisfies the constraint equations for the symmetric quantization map. Except for thestability ansatz AdgijXij =Xij, however, there are undoubtedly other solutions to the constraint

equations that do not satisfy this stability requirement, but we have not explored the possibilities

in this general case. It is nevertheless clear that these additional solutions do not correspond to

simplicial geometries, since for them the closure constraint is again deformed.

Freidel-Livine-Majid quantization map:

We will then consider the popular choice of Freidel-Livine-Majid quantization map [ 13,15,31

36], which can be expressed in terms of the symmetric quantization map QS and a change of

parametrization : su(2) su(2) on SU(2) as[39]

QFLM= QS ,


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19

where (k) = sin1 |k||k| k. The inverse coordinate transformation

1(k) = sin |k||k| k, however, is two-

to-one: it identifies the coordinates of the antipodes g andg as

1(k(g)) =1

k(g)

2

k(g)

|k(g)|

=1(k(g)) g SU(2)\{e} .

Therefore, the coordinates 1(k) only cover the upper hemisphere SU(2)/Z2 SO(3), and theresulting non-commutative Fourier transform is applicable only to functions on S O(3).

The FLM quantization map yields

Q1FLM(e ik(g)X) =e

i

sin |k(g)||k(g)| k(g)X EFLM(g, X) .

Accordingly, it leads to a form of the non-commutative plane wave with cFLM(g) = 1, FLM(g) =1sin |k(g)||k(g)| k(g) =

i2 tr 12

(gk)k su(2), where tr 12

denotes the trace in the fundamental spin- 12representation of SU(2). Due to the linearity of the trace, it is straightforward to calculate the

deformation matrix

DFLMkl (g) = 12tr 1

2(g)kl+ i

2tr 1

2(gj)jkl 1 2|FLM(g)|2 kl jFLM(g)jkl . (54)

Thus, according to our general description above, the classical discrete geometricity constraints are

satisfied by the deformed boundary metric variables

DFLM(gij)Xij =

1 2|FLM(gij)|2 Xij (FLM(gij) Xij) . (55)

We have not solved these constraints explicitly, which would generically impose relations between

the stationary phase boundary connection gij and the given boundary metric data Xij . However,

one can easily confirm that data corresponding to generic classical discrete geometries does not

satisfy the constraints, and therefore the geometry resulting from the constraints does not, ingeneral, describe discrete geometries. In fact, the deformed and the undeformed closure constraints

are compatible only for gij , or equivalently, in the abelian limit. Therefore, we conclude that

the non-commutative metric boundary variables do not have a classical geometric interpretation

in the case of FLM quantization map outside the abelian approximation, when one studies the

commutative variation of the action.

B. Stationary phase approximation via non-commutative variational method

We emphasize that in the above variation of the amplitude we did not take into account thedeformation of phase space structure, which appeared crucial for obtaining the correct classical

equations of motion in [38] in the case of quantum mechanics of a point particle on SO(3). This

could be guessed to be the origin of the discrepancies between the amplitudes corresponding to

different choices of quantization maps in the semi-classical limit. Indeed, we may define the non-

commutative variation Sof the action Sin the amplitude viaeiS+O(2) e

iS e

iS , where the

-product acts on the fixed boundary variables Xij , O(2) refers to terms higher than first order

in the variations, and S is the varied action. It is easy to see that the non-commutative variation

so defined undeforms the above identification (47) ofXij and Yij (up to orientation and parallel

transport), simply because we have

E(g1, X) E(geiZ, X) =E(g1, X) E(g, X) =1

E(eiZ, X) =e i(ZX)+O(2) (56)


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20

for any Z su(2) and R implementing the variation of g. Then, all the above results for

variations remain the same by requiring the non-commutative variation Sof the action to van-

ish except for Eq. (47), which becomes undeformed, i.e., we obtain the geometric identification

AdgijXij = Yij = AdgjiXji . Thus, the non-geometric deformation of the constraints does not

appear, and we recover exactly the simplicial geometry relations for the boundary metric variables,regardless of the choice of a quantization map.

To begin with, let us consider the partially off-shell amplitude, where we only substitute the

identifications AdgijXij =Yij = Adgji Xji arising from the variations of the boundary connection

gij . The substitution is done, again, by multiplying the amplitude by -delta functions imposing

the identities, and integrating over Yij. We also integrate over Xji for i < j in order to explicitly

impose the identifications AdgijXij = AdgjiXji on the boundary variables. Using the properties

of the non-commutative plane waves, we find from (42) through a simple substitution

APR(Xij) (i,j)Ni


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21

h1j Kji(hl)hi exp(iji nji ) in the spin-12 representation, where e, ji [0, ] are now the class

angles of the group elements and ne, nji S2 unit vectors. Then, we may write

Ye S(He(hl)) = |Ye|e

Ye|Ye|

ne

, Xij (h

1j Kji(hl)hi) = |Xij|

ji

Xij|Xij |

nji

. (58)

Considering then variations in the unit vectors neand nji, it is immediate to find that the stationary

phase of the amplitude is given by ne = Ye|Ye|

ne and nji = Xij|Xij |

, the signs corresponding

to the two opposite orientations of the edge vectors or, equivalently, the dual faces. Now, if a

configuration of edge vectors satisfies the constraints for a given discrete connection, it does so also

for the oppositely oriented configuration obtained by reversing the orientations of all the dual faces.

For the oppositely oriented configuration the holonomies around dual faces are also inverted, which

gives opposite signs for ne and nji with respect to the original configuration. Therefore, choosing

ne and nji to have positive signs for one of the orientations and thus negative signs for the other,

we may further write for the Ponzano-Regge amplitude in the semi-classical limit

APR(Xij)

l

dhl

e/

dYe

cos

ie/

|Ye|e+

(i,j)Ni


22/32


23/32

23

vectors nij := k(Gij)/|k(Gij )| S2, such that Gij exp(iij nij ) in the spin-

12 representation.

Adopting again the symmetric quantization map, we then have for the exponent

(i,j)Ni


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24

a discrete BF path integral in terms of the standard classical action. The choice of quantization

map affects the exact form of such discrete classical action, but in a rather minor way. In fact,

it leads to a discretization of the continuum curvature in terms of either the holonomy of the

connection (a group element), for the FLM map, or its logarithm (a Lie algebra element), for the

Duflo map. In general, the analysis appears to be more straightforward, than the one based on thespin (or coherent state) representation, as one would expect from a straightforward path integral

representation. We will exemplify this comparison in the simple case of a triangulation formed by

a single tetrahedron.

VI. SEMI-CLASSICAL LIMIT FOR A TETRAHEDRON

To conclude our asymptotic analysis, we consider in this section the relation of the classical limit

of Ponzano-Regge amplitudes formulated in terms of non-commutative boundary metric variables,

as discussed above, to the usual formulation of spin foam asymptotics as the large spin limit inthe spin representation. We will restrict our treatment to the case of a single tetrahedron, since

in this case the asymptotics for the Ponzano-Regge amplitude is simple and well-known in the

literature. In particular, it has been found that (for non-degenerate boundary data) the amplitude

of a tetrahedron is approximated in the large spin limit by the cosine of the Regge action [ 20].

We derive this result from the asymptotic behavior obtained through the non-commutative phase

space path integral, and thus establish a firm connection between the two asymptotic analyses.

For a single tetrahedron the Ponzano-Regge amplitude with boundary connection data reads

APR(gij) = i dhii,ji


25/32

25

for a tetrahedron in non-commutative boundary metric variables. As before, by studying the

non-commutative variations of the action

SPR=i,jii Xij j


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26

the spin basis, which is given by the SU(2) Wigner D-matrices Djkl(g). We find for the Ponzano-

Regge amplitude in the spin basis

APR(jij; kij , lij) = i,j

dgij Djijkij lij

(gij)

APR(gij) = j12 j13 j14

j23 j24 j34 i


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27

it is easy to show thatDjmn(g) constitute an over-complete basis of functions on SU(2), as we have

(g1g) =j

(2j+ 1)3S2

dm

4

S2

dn

4 Djmn(g)D

jmn(g

) . (76)

Similarly, for the S U(2) character function we may write

j(g) = tr(Dj(g)) = (2j+ 1)

S2

dm

4Djmm(g) . (77)

In the following, without a serious danger of confusion, we will denote the spin- 12 representation

Wigner matrices and coherent states simply by D12 (g) =: g and |12 , m=:|m, respectively, so in

our notation Djmn(g) m|g|n2j . Using (77), the 6j-symbol (72) may be re-expressed as

j12 j13 j14

j23 j24 j34

2=

i,j(2jij+ 1)

2

dmij

4

dnij4

APR(jij ; mij, nij) . (78)

This is simply the equation (74) transformed into the coherent state basis. Here, APR(jij ; mij, nij)

is indeed the Ponzano-Regge amplitude for a single tetrahedron with the coherent state labels as

boundary metric data, which may be written as

APR(jij ; mij , nij) =

i,j

dgij Djijmij nij

(gij)

APR(gij)=

j12 j13 j14

j23 j24 j34

i,ji


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28

we obtain by a straightforward calculation n= Adgm and 2jm = 1X. Note that, since we had

already understood the Xvariables as the classical discrete BF variables, this result confirms that

the coherent state variables acquire the correct geometric interpretation in the classical limit.

Then, from (69) and 2jijmij = Xij , nij = Adgijmij , we have for the classical limit of (80)

the expression

APR(jij; mij, nij)

i,j

dgij

v

(Hv(gij))

i

(j>i

jijmijj


29/32

29

for the boundary data in the classical limit, when we apply the ordinary commutative variational

calculus to find the stationary phase solutions. Furthermore, in this case, the constraints that arise

as the classical equations of motion generically do not correspond to discrete geometries, since

the edge vectors in the constraint equations are deformed due to the non-linearity of the group

manifold. We verified our observation by considering as explicit examples the non-commutativestructures that arise from symmetric, Duflo and Freidel-Livine-Majid quantization maps.

Accordingly, we were led to consider a non-commutativevariational method to extract the sta-

tionary phase behavior, which was motivated by the fact that the amplitude for non-commutative

metric boundary data acts as the integral kernel in the propagator with respect to the correspond-

ing-product, and not the commutative product. We showed that the non-commutative variations

produce the correct geometric constraints for the discrete metric boundary data in the classical

limit. Thus, we concluded that only by taking into account the deformation of phase space struc-

ture in studying the variations, we find the undeformed and unambiguous geometric constraints,

independent on the choice of the quantization map.

Finally, we considered the asymptotics of theSU(2) 6j-symbol, which is related to the Ponzano-

Regge amplitude for a tetrahedra with fixed quantized edge lengths. We found the Regge action,

previously recovered in the large spin limit of the 6j-symbol, in the classical limit. Our calculations

thus not only verify the previous results, but also allows for a better understanding of them due

to the clear-cut connection to the phase space of classical discretized 3d gravity. This concrete

example also illustrates the use of the non-commutative path integral as a computational tool.

There are several conclusions and further directions of research pointed to by our results. First

and foremost, we have seen that the non-commutative metric representation obtained through

the non-commutative Fourier transform facilitates a full asymptotic analysis of spin foam models,

when proper care is taken in applying variational methods to the first order path integral. In

particular, by studying the non-commutative variations one may recover the classical geometric

constraints for all cases of non-commutative structures. The need for a non-commutative variational

method requires further analysis, and must be taken into account in any future application of the

non-commutative methods to spin foam models. Our consideration of the 6j-symbol asymptotics

further illustrates the usefulness of the non-commutative methods. As the non-commutative Fourier

transform formalism has recently been extended to all exponential Lie groups [39], in particular the

double-cover S L(2,C) of the Lorentz group, we look forward to extending the asymptotic analysis

to the 4d spin foam models in future work, now equipped with this improved understanding.

Acknowledgments

We would like to thank A. Baratin for several useful discussions on the non-commutative Fourier

transform and spin foam models. We also thank C. Guedes, F. Hellmann and W. Kaminski for

several discussions. This work was supported by the A. von Humboldt Stiftung, through a Sofja

Kovalevskaja Prize, which is gratefully acknowledged. The work of M. Raasakka was partially

supported by Emil Aaltonen Foundation.


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30

Appendix A: Infinitesimal Baker-Campbell-Hausdorff formula

In order to calculate the deformation matrix (53) for the symmetric quantization map, we need

to compute Lie derivatives LkS,l of the coordinates S= iln(g) onS U(2). To do this, we derive

the explicit form of the Baker-Campbell-Hausdorff formula B(k, k) in eikeik eiB(k,k

) for

the case, when one of the arguments is infinitesimal. We may write

cos |B(k, k)| + isin |B(k, k)|

|B(k, k)| B(k, k) =

cos |k| + i

sin |k|

|k| k

cos |k| + i

sin |k|

|k| k

=

cos |k| cos |k|

sin |k|

|k|

sin |k|

|k| (k k)

+ i

cos |k|

sin |k|

|k| k+ cos |k|

sin |k|

|k| k

sin |k|

|k|

sin |k|

|k| (k k)

,

where by we denote the cross-product in R3, from which one can extract

cos |B(k, k)| = cos |k| cos |k| sin |k|

|k|

sin |k|

|k| (k k)

sin |B(k, k)|

|B(k, k)| B(k, k) = cos |k|

sin |k|

|k| k+ cos |k|

sin |k|

|k| k

sin |k|

|k|

sin |k|

|k| (k k) .

From these identities it is not too difficult to find a closed form for the Baker-Campbell-Hausdorff

formula for SU(2) in terms of elementary functions [39]. However, we will only need to consider

the special case k = tek, where t > 0 is an expansion parameter, and ek, k = 1, 2, 3, are or-

thonormal basis vectors in R3. Then we obtain the deformation matrix as DSkl(g) =LkS,l(g) =

ddtB(k(g), tek)lt=0, i.e., it is the t-linear term in B (k(g), tek)l. From above we have

sin |B(k,tek)|

|B(k,tek)| B(k,tek)l=

sin |k|

|k| kl+ t (cos |k|kl (k ek)l) + O(t

2) ,

from which we may deduce

|B(k,tek)| = |k| + tkk|k|

+ O(t2)

1

sin |B(k,tek)|=

1

sin |k|

1 t

cos |k|

sin |k|

kk|k|

+ O(t2) .

Using these formulae, we get

B(k,tek)l = kl+ t |k|

sin |k|

cos |k|kl+

sin |k|

|k| cos |k|

kkkl|k|2

mkl km

+ O(t2) ,

and so

DSkl(g) = |k(g)|

sin |k(g)|

cos |k(g)|kl +

sin |k(g)|

|k(g)| cos |k(g)|

kk(g)kl(g)

|k(g)|2 mkl km(g)

. (A1)

[1] A. Perez, Living Rev. Relativity 16, 3 (2013), arXiv:1205.2019.[2] J. Baez, Lect. Notes Phys.543, 25 (2000), arXiv:gr-qc/9905087.

[3] C. Rovelli,Quantum Gravity (Cambridge University Press, Cambridge, UK, 2004).


31/32

31

[4] T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cam-

bridge, UK, 2007).

[5] T. Regge and R. Williams, J. Math. Phys.41, 3964 (2000), arXiv:gr-qc/0012035.

[6] D. Oriti, in Foundations of Space and Time: Reflections on Quantum Gravity, edited by J. Muru-

gan, A. Weltman, and G. F. Ellis (Cambridge University Press, Cambridge, UK, 2011), pp. 257320,

arXiv:1110.5606.[7] L. Freidel, Int. J. Theor. Phys.44, 1769 (2005), arXiv:hep-th/0505016.

[8] K. Noui and A. Perez, Class. Quant. Grav.22, 1739 (2005), arXiv:gr-qc/0402110.

[9] S. Alexandrov, M. Geiller, and K. Noui, SIGMA 8, 055 (2012), arXiv:1112.1961.

[10] M. Reisenberger and C. Rovelli, Phys. Rev. D56, 3490 (1997), arXiv:gr-qc/9612035.

[11] C. Rovelli, Phys. Rev.D48, 2702 (1993), arXiv:hep-th/9304164.

[12] J. Barrett and L. Crane, J. Math. Phys.39, 3296 (1998), arXiv:gr-qc/9709028.

[13] A. Baratin and D. Oriti, New J. Phys. 13, 125011 (2011), arXiv:1108.1178.

[14] L. Freidel and K. Krasnov, Class. Quant. Grav.25, 125018 (2008), arXiv:0708.1595.

[15] A. Baratin and D. Oriti, Phys. Rev.D85, 044003 (2012), arXiv:1111.5842.

[16] M. Dupuis and E. Livine, Class. Quant. Grav.28, 215022 (2011), arXiv:1104.3683.[17] J. Engle, R. Pereira, and C. Rovelli, Phys. Rev. Lett.99, 161301 (2007), arXiv:0705.2388.

[18] J. Engle, E. Livine, R. Pereira, and C. Rovelli, Nucl. Phys.B799, 136 (2008), arXiv:0711.0146.

[19] A. Perez, Papers Phys.4, 040004 (2012), arXiv:1205.0911.

[20] R. Dowdall, H. Gomes, and F. Hellmann, J. Phys.A43, 115203 (2010), arXiv:0909.2027.

[21] J. Barrett, R. Dowdall, W. Fairbairn, F. Hellmann, and R. Pereira, Class. Quant. Grav. 27, 165009

(2010), arXiv:0907.2440.

[22] F. Conrady and L. Freidel, Phys. Rev. D78, 104023 (2008), arXiv:0809.2280.

[23] M. Han and M. Zhang, Class. Quant. Grav. 29, 165004 (2012), arXiv:1109.0500.

[24] M. Han and M. Zhang, Class. Quant. Grav. 30, 165012 (2013), arXiv:1109.0499.

[25] M. Han and T. Kra jewski, Class. Quant. Grav.31, 015009 (2013), arXiv:1304.5626.

[26] M. Han, Class. Quant. Grav.31, 015004 (2013), arXiv:1304.5627.

[27] F. Hellmann and W. Kaminski (2012), arXiv:1210.5276.

[28] F. Hellmann and W. Kaminski, JHEP1310, 165 (2013), arXiv:1307.1679.

[29] E. Magliaro and C. Perini, Int. J. Mod. Phys. D22, 1 (2013), arXiv:1105.0216.

[30] M. Han, Phys. Rev.D88, 044051 (2013), arXiv:1304.5628.

[31] L. Freidel and E. Livine, Phys. Rev. Lett.96, 221301 (2006), arXiv:hep-th/0512113.

[32] L. Freidel and S. Majid, Class. Quant. Grav.25, 045006 (2008), arXiv:hep-th/0601004.

[33] E. Joung, J. Mourad, and K. Noui, J. Math. Phys. 50, 052503 (2009), arXiv:0806.4121.

[34] A. Baratin, B. Dittrich, D. Oriti, and J. Tambornino, Class. Quant. Grav. 28, 175011 (2011),

arXiv:1004.3450.

[35] A. Baratin and D. Oriti, Phys. Rev. Lett. 105, 221302 (2010), arXiv:1002.4723.[36] A. Baratin, F. Girelli, and D. Oriti, Phys. Rev.D83, 104051 (2011), arXiv:1101.0590.

[37] B. Dittrich, C. Guedes, and D. Oriti, Class. Quant. Grav.30, 055008 (2013), arXiv:1205.6166.

[38] D. Oriti and M. Raasakka, Phys. Rev.D84, 025003 (2011), arXiv:1103.2098.

[39] C. Guedes, D. Oriti, and M. Raasakka, J. Math. Phys.54, 083508 (2013), arXiv:1301.7750.

[40] G. Ponzano and T. Regge, inSpectroscopic and group theoretical methods in physics, edited by F. Bloch

(North-Holland Publ. Co., Amsterdam, 1968), pp. 158.

[41] D. Boulatov, Mod. Phys. Lett.A7, 1629 (1992), arXiv:hep-th/9202074.

[42] J. Barrett and I. Naish-Guzman, Class. Quant. Grav.26, 155014 (2009), arXiv:0803.3319.

[43] B. Schroers, in Quantization of Singular Symplectic Quotients, edited by N. Landsman, M. Pflaum,

and M. Schlichenmaier (Springer, 2001), vol. 198 of Progress in Mathematics, pp. 307327,arXiv:math/0006228.

[44] S. Majid and B. Schroers, J. Phys.A42, 425402 (2009), arXiv:0806.2587.


32/32

32

[45] M. Dupuis, F. Girelli, and E. Livine, Phys. Rev.D86, 105034 (2012), arXiv:1107.5693.

[46] M. Chaichian and A. Demichev, Path Integrals in Physics, vol. I (Institute of Physics Publishing,

London, 2001).

[47] M. Caselle, A. DAdda, and L. Magnea, Phys. Lett. B232, 457 (1989).

[48] N. Kawamoto and H. Nielsen, Phys. Rev. D43, 1150 (1991).

[49] B. Bahr and B. Dittrich, Class. Quant. Grav.26, 225011 (2009), arXiv:0905.1670.[50] A. Perelomov,Generalized Coherent States and Their Applications(Springer-Verlag, Heidelberg, 1986).

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Asymptotic Analysis of Ponzano-Regge Model With Non-commutative Metric Variables - Daniele Oriti, Matti Raasakka