Invited review: The new spin foam models and quantum gravity

 

Alejandro Perez¹

*E-mail: perez@cpt.univ-mrs.fr
¹ Centre de Physique Théorique, Campus de Luminy, 13288 Marseille, Prance. Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var; laboratoire afilié a la FRUMAM (FR 2291).

In this article, we give a systematic definition of the recently introduced spin foam models for four-dimensional quantum gravity, reviewing the main results on their semiclassical limit on fixed discretizations.

 

I. Introduction

The quantization of the gravitational interaction is a major open challenge in theoretical physics. This review presents the status of the spinfoam ap-proachto the problem. Spin foam models are defini-tions of the path integral formulation of quantum general relativity and are expected to be the co-variant counterpart of the background independent canonical quantization of general relativity known as loop quantum gravity [1-3].

This article focuses on the deñnition of the recently introduced Engle-Pereira-Rovelli-Livine (EPRL) model [4,5] and the closely related Freidel-Krasnov (FK) model [6]. An important original feature of the present paper is the explicit deriva-tion of both the Riemannian and the Lorentzian models, in terms of a notation that exhibits the cióse relationship between the two, at the algebraic level, that might signal a possible deeper relationship at the level of transition amplitudes.

We will take Plebanski's perspective in which general relativity is formulated as a constrained BF theory (for a review introducing the new models from a bottom-up perspective see Ref. [7]; for an extended version of the present review including a wide collection of related work see Ref. [8]). For that reason, it will be convenient to start this review by introducing the exact spin foam quantiza-tion of BF in the following section. In Section III, we present the EPRL model in both its Riemannian and Lorentzian versions. A unified treatment of the representation theory of the relevant gauge groups is presented in that section. In Section IV, we introduce the FK model and discuss its relationship with the EPRL model. In Section V, we describe the structure of the boundary states of these models and emphasize the relationship with the kinemati-cal Hilbert space of loop quantum gravity. In Sec-tion VI, we give a compendium of important issues (and associated references) that have been left out but which are important for future development. Finally, in Section VII, we present the recent en-couraging results of the nature of the semiclassical limit of the new models.

II. Spin foam quantization of BF theory

We will start by briefly reviewing the spin foam quantization of BF theory. This section will be the basic building block for the construction of the models of quantum gravity that are dealt with in this article. The key idea is that the quantum tran-sition amplitudes (computed in the path integral representation) of gravity can be obtained by suit-ably restricting the histories that are summed over in the spin foam representation of exactly solvable BF theory. We describe the nature of these con-straints at the end of this section.

Here, one follows the perspective of Ref. [9]. Let G be a compact group whose Lie algebra g has an invariant inner product, here denoted (), and M a d-dimensional manifold. Classical BF theory is deñned by the action

S[B,lü}= í(BAF(w)),                 (1)

M where B is a g valued (d-2)-form, w is a connection on a G principal bundle over Al. The theory has no local excitations: All the solutions of the equations of motion are locally related by gauge transforma-tions. More precisely the gauge symmetries of the action are the local G gauge transformations

¿B=[B,a],            (W = d_wa,            (2)

where a is a fl-valued 0-form, and the 'topological' gauge transformation

SB = d_wí?,            Scu = 0,                (3)

where d_w denotes the covariant exterior derivative and ri is a fl-valued 0-form. The ñrst invariance is manifest in the form of the action, while the sec-ond one is a consequence of the Bianchi identity d_uF(uj) = 0. The gauge symmetries are so vast that all the solutions to the equations of motion are locally puré gauge. The theory has only global or topological degrees of freedom.

For the time being, we assume Al to be a compact and orientable manifold. The partition func-tion, Z, is formally given by

Z=V\B\V[w\ exp(í (BAF(4). (4)

M

Formally integrating over the B field in (4), we obtain

Z = V[w] 5{F{lu)).                  (5)

The partition function Z corresponds to the 'volume' of the space of flat connections on M.

In order to give a meaning to the formal expressions above, we replace the d-dimensional manifold M with an arbitrary cellular decomposition A. We also need the notion of the associated dual 2-complex of A denoted by A*. The dual 2-complex A* is a combinatorial object deñned by a set of vertices v G A* (dual to d-cells in A) edges e G A* (dual to (d-l)-cells in A) and faces / G A* (dual to (d-2)-cells in A). In the case where A is a sim-plicial decomposition of M, the structure of both A and A* is illustrated in Figs. 1, 2 and 3 in two, three, and four dimensions, respectively.

Figure 1: On the left: A triangulation and its dual in two dimensions. On the right: The dual two complex; faces (shaded polygon) are dual to 0-simplices in 2d.

Figure 2: On the left: A triangulation and its dual in three dimensions. On the right: The dual two complex; faces (shaded wedge) are dual to 1-simplices in 3d.

 

For simplicity, we concéntrate on the case when A is a triangulation. The ñeld B is associated with Lie algebra elements B_f assigned to faces / <= A*. We can think of it as the integral of the (d-2)-form B on the (d-2)-cell dual to the face / G A*, namely

B_f =

(d-2)-cell

B.

(6)

In other words, B_f can be interpreted as the 'smearing' of the continuous (d-2)-form B on the (d-2)-cells in A. We use the one-to-one correspon-dence between faces / G A* and (d-2)-cells in A to label the discretization of the B ñeld B_f. The con-nection uj is discretized by the assignment of group elements g_e G G to edges e G A*. One can think of the group elements g_e as the holonomy of uj along e G A*, namely

9e

P

exp(- íu),

e

(7)

where the symbol "Pexp" denotes the path-order-exponential that reminds us of the relationship of the holonomy with the connection along the path eG A*.

With this, the discretized versión of the path integral (4) is

Z(A) = í Y[ dg_e Y[ dB_f e^iBf^uf

e£A*         feA*

= Yl dg_e Y[ 5{g_ei . . . g_eri), (8)

e£A*         feA*

around faces, and the second equation is the result of the B integration: It can be, thus, regarded as the analog of (5). The integration measure dB_f is the standard Lebesgue measure, while the integration in the group variables is done in terms of the invariant measure in G (which is the unique Haar measure when G is compact). For given h G G and test function F(g), the invariance property reads as follows

ídgF(g)= ídgF(g-¹)= ídgF(gh)

= ídgF(hg) (9)

The Peter-Weyl's theorem provides a useful formula or the Dirac delta distribution appearing in (8), namely

S(g) = J2d_pTr[p(g)l

(10)

where p are irreducible unitary representations of G. From the previous expression, one obtains

Z(A) = Y,

I ndge n ^áPfT^Y[pf(9i---9e)] . (n)

~r- A +               f^-A +

e£A

feA

Integration over the connection can be performed as follows. In a triangulation A, the edges e G A* bound precisely d different faces. Therefore, the 0_e's in (11) appear in d different traces. The rele-vant formula is

PLvipu--- ,Pd)

.= f dg_{e Pl}(g_e) ® p₂(g_e) ®

®_Pd{g_e). (12)

For compact G, using the invariance (and normal-ization) of the the integration measure (9), it is easy to prove that Pf_nv = {Pf_nvf is the projector onto Inv[pi <g> p₂ <8> . . . <8> p_d}. In this way the spin foam amplitudes of SO {A) BF theory reduce to

040004-3

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

Z_BF(A) = J2

n dpf n ^ptnv(pu---,Pd). (13)

/eA*

e£A*

In other words, the BF amplitude associated to a two-complex A* is simply given by the sum over of all possible assignments of irreducible represen-tations of G to faces of the number obtained by the natural contraction of the network of projectors P?_nv, according to the pattern provided deñned by the two-complex A*.

There is a nice graphical representation of the partition function of BF theory that will be very useful for some calculations. On the one hand, us-ing this graphical notation one can easily prove the discretization independence of the BF amplitudes. On the other hand, this graphical notation will sim-plify the presentation of the new spin foam models of quantum gravity that will be considered in the following sections. This useful notation was intro-duced by Oeckl [10,11] and used in Ref. [12] to give a general proof of the discretization independence of the BF partition function and the Turaev-Viro invariants for their deñnition on general cellular de-compositions.

We will present this notation in detail: The idea is to represent each representation matrix appear-ing in (11) by a line (called a wire) labeled by an irreducible representation, and integrations on the group by a box (called a cable). The traces in Eq. (11) imply that there is a wire, labeled by the representation p_f, winding around each face / <= A*. In addition, there is a cable (integration on the group) associated with each edge e G A*. As in (13), there is a projector P?_nv, which is the pro-jector in liw[_Pl <g> p₂ <g> . . . <g> p_d] associated to each edge. This will be represented by a cable with d wires, as shown in (14). Such graphical representation allows for a simple diagrammatic expression of the BF quantum amplitudes.

P1P2P3       Pd

PfnviPl' P2, P3, . . . ,Pá) =                                 (14)

a

The case of physical interest is d = 4. In such case, edges are shared by four faces; each cable has

now four wires. The cable wire diagram giving the BF amplitude is dictated by the combinatorios of the dual two complex A*. From Fig. 3, one gets

Z_BF(A) =

Cf-{f} ^Pf

n dp ^pi

fEA*

(15)

The 10 wires corresponding to the 10 faces / G A*, sharing a vértex v G A*, are connected to the neigh-boring vértices through the 5 cables (representing the projectors in (13) and Fig. 14) associated to the 5 edges e G A*, sharing the vértex v G A*.

a. SU{2) x SU{2) BF theory: a starüng point for 4d Riemannian gravity.

We now present the BF quantum amplitudes in the case G = SU(2) x SU(2). This special case is of fundamental importance in the construction of the gravity models presented in the following sections. The product form of the structure group implies the simple relationship Z_BF(SU(2) x SU(2)) = Z_Bf{SU{2))². Nevertheless, it is important for us to present this example in an explicit way as it will provide the graphical notation that is needed to introduce the gravity models in a simple manner. The spin foam representation of the BF partition function follows from expressing the projectors in (15) in the orthonormal basis of intertwiners, Le., invariant vectors in Inv[pi <g> . . . <g> p₄]. From the product form of the structure group, one has

aipzpjPA

1111

TTTT

3a 3^ 3a UJZjIÜ

3\323?,3a

3 \3 2 3 3,3 A

/TK

(16)

040004-4

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

where p_f = jj ®j+, jf and í± are half integers la-beling left and right representations of SU(2) that deñned the irreducible unitary representations of G = SU(2) x SU(2). We have used the expression of the right and left SU(2) projectors in a basis of intertwiners, namely

as a product of two SU{2) amplitudes, namely

k k k k

n

TTTT

E

k k k k

(17)

where the four-leg objects on the right hand side denote the invariant vectors spanning a basis of Inv[J! <g> . . . <g> j₄], and i is a half integer, labeling those elements. Accordingly when replacing the previous expression in (15), one gets

z_BF{¿\) = y,

nd,-d,+ 3; 3;

fGA*

and equivalently

(18)

Z_BF(A)= V TT d-d.

¿_-/          ±± 3 r 3

Cf-{f}^Pf /eA*

+

(19)

from which we finally obtain the spin foam repre-sentation of the SU(2) x SU(2) partition function

Z_BF(A)

E

C/={/}-

nd,-i 3 ¡ 3

PffeA*

e n

C_e:{e}

veA*

3a (20)

Extra remarks on jour-dimensional BF theory

The state sum (11) is generically divergent (due to the gauge freedom analogous to (3)). A regular-ized versión deñned in terms of SU¿2) x SU_q{2) was introduced by Crane and Yetter [13,14]. As in three dimensions, if an appropriate regularization of bubble divergences is provided, (11) is topolog-ically invariant and the spin foam path integral is discretization independent.

As in the three-dimensional case, BF theory can be coupled to topological defects [15] in any dimensión. In the four-dimensional case, defects are string-like [16] and can carry extra degrees of freedom, such as topological Yang-Mills ñelds [17]. The possibility that quantum gravity could be deñned directly from these simple kinds of topological the-ories has also been considered outside spin foams [18] (for which the UV problem described in the in-troduction is absent). This is attractive and should, in my view, be considered further.

It is also possible to introduce one-dimensional partióles in four-dimensional BF theory and gravity, as shown in Ref. [19].

Two-dimensional BF theory has been used as the basic theory in an attempt to deñne a manifold independent model of QFT in Ref. [20]. It is also related to gravity in two dimensions in two ways: On the one hand, it is equivalent to the so-called Jackiw-Teitelboim model [21,22], on the other hand it is related to usual 2d gravity via constraints in a way similar to the one exploited in four dimensions (see next section). The ñrst relationship has been used in the canonical quantization of the Jackiw-Teitelboim model in Ref. [23]. The second relationship has been explored in Ref. [24].

Three-dimensional BF theory and the spin foam quantization presented above are intimately related to classical and quantum gravity in three dimen-sions (for a classic reference see Ref. [25]). The state sum, as presented above, matches the quantum amplitudes first proposed by Ponzano and Regge in the 60's, based on their discovery of the asymptotic expressions of the 6j symbols [26], of-ten referred to as the Ponzano-Regge model. Di-vergences in the above formal expression require regularization. Natural regularizations are avail-able so that the model is well-defined [27-29]. For a detailed study of the divergence structure of the model, see Refs. [30-32]. The quantum de-formed version of the above amplitudes lead to the so-called Turaev-Viro model [33], which is ex-pected to correspond to the quantization of three-dimensional Riemannian gravity in the presence of a non-vanishing positive cosmological constant. For the definition of observables in the latter context, as well as in the analogue four-dimensional analog, see Ref. [34].

The topological character of BF theory can be preserved by the coupling of the theory with topo-logical defects that play the role of point particles. In the spin foam literature, this has been consid-ered from the canonical perspective in Refs. [35,36] and from the covariant perspective extensively by Freidel and Louapre [37]. These theories have been proved by Freidel and Livine to be dual, in a suit-able sense, to certain non-commutative fields theo-ries in three dimensions [38,39].

Concerning coupling BF theory with non-topological matter, see Refs. [40,41] for the case of fermionic matter, and Ref. [42] for gauge fields. A more radical perspective for the definition of mat-ter in 3d gravity is taken in Ref. [43]. For three-dimensional supersymmetric BF theory models, see Refs. [44,45]

Recursion relations for the 6j vertex amplitudes have been investigated in Refs. [46,47]. They pro-vide a tool for studying dynamics in spin foams of 3d gravity and might be useful in higher dimensions [48].

i. The coherent states representation

In this section, we introduce the coherent state rep-resentation of the SU(2) and Spin(4) path integral of BF theory. This will be particularly important

for the definition of the models defined by Freidel and Krasnov in Ref. [6] that we will address in Sec-tion IV as well as in the semiclassical analysis of the new models reported in Section VII. The relevance of such representation for spin foams was first em-phasized by Livine and Speziale in Ref. [49].

a. Coherent states

Coherent states associated with the representation theory of a compact group have been studied by Thiemann and collaborators [50,51,51-59], see also Ref. [60]. Their importance for the new spin foam models was put forward by Livine and Speziale in Ref. [49], where the emphasis was put on coherent states of intertwiners or the so-called quantum tetrahedron (see also [61]). Here we follow the pre-sentationof [6].

In order to build coherent states for Spin(4), we start by introducing them in the case of SU(2). Starting from the representation space Jífj of dimensión dj = 2j + í, one can write the resolution of the identity in terms of the canonical orthonor-mal basis \j, m) as

lj= ^ |j,m)(j,m|,                    (21)

where -j < m < j. There exists an over complete basis \j,g) G Jífj, labeled by g G SU(2), such that

lj- = dj           dg\j,g){j,g\,                (22)

SU(2)

The states \j, g) G Jtfj are SU{2) coherent states deñned by the action of the group on máximum weight states \j,j) (themselves coherent), namely

\j, g) = g\j,j) = Y, \J, T)Dj_mj{g)> (23)

where D³_mj(g) are the matrix elements of the uni-tary representations in the \j, m) (Wigner matrices). Equation (22) follows from the orthonor-mality of unitary representation matrix elements, namely

040004-6

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

dj í dg\j,g)(j,g\,

SU(2)

dj Y\ \j, m)(J, m'\           dgD^j_mj{g)D^j_m,Ag)

mm>                               SU(2)

^ |j,m)(j,m|,

(24)

The above property will be of key importance in constructing effective discrete actions for spin foam models. In particular, it will play a central role in the study of the semiclassical limit of the EPRL and FK models studied in Sections III, and IV. In the following subsection, we provide an example for Spin(4) BF theory.

where in the last equality we have used the or-thonormality of the matrix elements. The decom-position of the identity (22) can be expressed as an integral on the two-sphere of directions S² = SU(2)/U(Í) by noticing that D^j_mj{g) and D^j_mj{gh) differ only by a phase for any group element h from a suitable U(í) C SU(2). Thus, one has

\j

dj f

s²

dn\j,n){j,n\,

(25)

where n G S² is integrated with the invariant mea-sure of the sphere. The states \j, n) form (an over-complete) basis in 3%. SU(2) coherent states have the usual semiclassical properties. Indeed, if one considers the generators J^¿ of su{2), one has

0',n|Jⁱ|j,n>=jnⁱ,                     (26)

where n^¿ is the corresponding three-dimensional unit vector for n G S². The fluctuations of J² are also minimal with AJ² = h²j, where we have restored h for clarity. The fluctuations go to zero in the limit h -► 0 and j -^ oo, while hj is kept constant. This kind of limit will be used often as a notion of semiclassical limit in spin foams. The state \j, n) is a semiclassical state describing a vector in R³ of length j and of direction n. It will be convenient to introduce the following graphical notation for Eq. (25)

j

s²

dn

Ó

n

0

(27)

Finally an important property of SU{2) coherent states stemming from the fact that

|j,j) = |i,i)|i,i)...|i,i) = |i,i)®2j

is that

b-,n> = |i,n>^.

b. Spin(4) BF theory: Amplitudes in the coherent state basis

Here we study the coherent states representation of the path integral for Spin(4) BF theory. The con-struction presented here can be extended to more general cases. The present case is, however, of particular importance for the study of gravity models presented in Sections III, and IV. With the intro-duction of coherent states, one achieves the most difficult part of the work. In order to express the Spin(4) BF amplitude in the coherent state rep-resentation, one simply inserts a resolution of the identity in the form (25) on each and every wire connecting neighboring vertices in the expression (18) for the BF amplitudes. The result is

Z_BF(A)= V          n d.-d.+

C_f:{f}^_Pf /£A* ³¡ ³*

J eeeA* d j- d j+ dn ef dn ef

,(29)

(28)

where we have explicitly written the n± G S² in-tegration variables only on a single cable. One observes that there is one n± G S² per each wire coming out at an edge e G A*. As wires are in one-to-one correspondence with faces / G A*, the integration variables nf_f G S² are labeled by an

7

040004-7

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

edge and face subindex. In order to get an expres-sion of the BF path integral in terms of an affective action, we restore, at this stage, the explicit group integrations represented by the boxes in the previ-ous equation. One gets

Z_BF(A)= V n d.-d.+

C/:{/}-w/eA* ^{J/ J/}

dg^fdgtf {{n-_f\{9^¡9e'f\n^f)f^3í

((ntf\(9⁺)7f9Í_fKf))^2J^             (3°)

where we have used the coherent states property (28), and |n±) is a simpliñed notation for |±,n±). The previous equation can be finally written as

Z_BF(A) = E        i! d,-d,+

C¡:{f}^P¡ /EA* ^{¡ ¡} Víee^^ái-_s^áiÍ_sdn^dn%d^íd9tí

exp^^]), where the discrete action

(31)

(32)

»£á*

with

sUg\

5

E

a<6=l

2jab\n{n_ab\g-^lg_b\n_ba), (33)

and the Índices a, 6 label the ñve edges of a given vértex. The previous expression is equal to the form (11) of the BF amplitude. In the case of the gravity models studied in what follows, the coherent state path integral representation will be the basic tool for the study of the semiclassical limit of the models and the relationship with Regge discrete formula-tion of general relativity.

ii. The relationship between gravity and BF theory

The ñeld theory described in the present section has no local degrees of freedom. It represents the sim-plest example of a topological ñeld theory in four dimensions. The interest of this theory for gravity

models stems from the fact that an action for the gravitational degrees of freedom (basically equiva-lent to general relativity in the ñrst order formu-lation) can be obtained by supplementing a 4d BF theory action with internal gauge group SL(2,C) (Lorentzian) or Spin(A) (Riemannian) with the fol-lowing set of quadratic constraints on the B-field

cuklB^B^ - e _W(T « 0,           (34)

e = cr²(l/^,A\)e_IJKLB^I¿Bf_a^Le^P'^T where

where

cr² = ±1 according to whether one is in the Riemannian or Lorentzian case. More generally a one-parameter family of gravity actions can be obtained from the imposition of the previous constraints on the following modiñed BF action

S₇(B,w)= {(*B+-B)AF(lü)),        (35)

M             7

where 7 is the Immirzi parameter. The strategy behind the deñnition of the new spin foam models for quantum gravity consists of imposing these constraints on the path integral of BF theory on the momenta J = *B + ±B conjugated to cu. In order to impose the Plebanski constraints above, it will be convenient to express the B ñeld in terms of the momenta J, namely

B

7

l__a2₇2

(J-7*J).

(36)

The imposition of the constraints (34) on the BF path integral on a ñxed discretization can be done in two different ways: by directly restricting the spin foam conñgurations (this is the EPRL ap-proach described in the following section), or by restricting the semiclassical valúes of the B ñeld in the coherent state representation of the BF path integral (this is the FK strategy described in Section IV).

III. The           Engle-Pereira-Rovelli-

Livine (EPRL) model

In this section, we introduce the Engle-Pereira-Rovelli-Livine (EPRL) model [4,5]. The section is organized as follows: The relevant representation theory is introduced in Subsection i. In Subsection ii, we present and discuss the linear simplicity constraints -classically equivalent to the Plebanski constraints-and discuss their implementation in

040004-8

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

the quantum theory. In Subsection iii, we introduce the EPRL model of Riemannian gravity. In Subsec-tion iv, we prove the validity of the quadratic Ple-banski constraints-reducing BF theory to general relativity-directly in the spin foam representation. In Subsection v, we present the coherent state rep-resentation of the Riemannian EPRL model. In Subsection vi, we describe the Lorentzian model. The material of this section will also allow us to describe the construction of the closely related (al-though derived from a different logic) Riemannian FK constructed in Ref. [6]. The idea that linear simplicity constraints are more convenient for deal-ing with the constraints that reduce BF theory to gravity was pointed out by Freidel and Krasnov in this last reference.

i. Representation theory of Spin{A) and SL(2, C) and the canonical basis

In this section, we present the representation theory of the groups Spin(A) and SL{2, C) that is neces-sary for the deñnition of the new spin foam mod-els for Riemannian and Lorentzian gravity, respec-tively. To emphasize the highly symmetric struc-ture of the two, we present them in a uniñed no-tation where a parameter a = 1 for the Riemannian sector anda = i for the Lorentzian one. The simple relationship between the two might be a hint to a possible relationship between model amplitudes in a spirit similar to the interesting link between Euclidean and Lorentzian QFT provided by Wick rotations ^l. Unitary irreducible repre-sentations 3V_p¿ of Spin{A) and SL(2, C) are la-beled by two parameters, p and k. In the case of Spin(A) = SU(2) x SU(2), the unitary irreducible representations are ñnite-dimensional and the la-bels p and k can be expressed in terms of the half integers labeling the right and left SU(2) unitary representations j±, as follows

P = j⁺+j- + í k = \j+-j-\. (37)

In the SL(2,C) case, the unitary irreducible representations are inñnite-dimensional and one has

pGR+ k G N/2.                   (38)

iSuch explicit relationship between gravity amplitudes i analytic continuation in 3d [62].

The two Casimirs are C_x = \JijJ^ij = L² + a²K² and C₂ = \*JijJ^ij = KL where Ú are the gener-ators of an arbitrary rotation subgroup and K* are the generators of the corresponding boosts. The Casimirs act on \p, k) G J^,_fc, as follows

Cí\p,k) = hk²+a²p² -l)\p,k) C₂\p,k)=pk\p,k).                          (39)

For details on the representation theory of SL(2,C), see Refs. [63-65]. The deñnition of the EPRL model requires the introduction of an (arbitrary) subgroup SU(2) C Spin(A) or SU(2) C SL(2,C), according to whether one is working in the Riemannian or in the Lorentzian sector. This subgroup corresponds to the internal gauge group of the gravitational phase space in connection variables in the time gauge (see Ref. [8] for details). Henee, in the quantum theory, the representation theory of this SU(2) subgroup will be important. This importance will soon emerge as apparent from the imposition of the constraints that define the EPRL model. The link between the unitary representations of SL(2, C) and those of SU(2) is given by the decomposition

p-í             j⁺+r

.^p,* = 0^3= 0 -^3, (40)

for the Riemannian sector, and

OO

^_fc = 0^._?                (41)

j=k

for the Lorentzian sector. As the unitary irreducible representations of the subgroup SU(2) G Spm{A) and SU(2) G SX(2, C) are essential for un-derstanding the link of the EPRL model and the operator canonical formulation of LQG, it will be convenient to express the action of the generators of the Lie algebra of the corresponding group in a basis adapted to the above equation. In order to do this, we first notice that the Lie algebra spin{A) and sl(2, C) can be characterized in terms of the generators of a rotation subgroup Ú and the remaining the Euclidean and Lorentzian sectors can be established by boost generators K\ as follows

where if_± = if¹ ± iK² and L_± = L^l±iL², respec-tively. The action of the previous generators in the basis \p, k;j, m) can be shown to be

The previous equations will be important in what follows: they will allow for the characterization of

the solutions of the quantum simplicity constraints, in both the Riemannian and Lorentzian models, in a direct manner. This concludes the review of the representation theory that is necessary for the def-inition of the EPRL model.

ii. The linear simplicity constraints

As first shown in Ref. [6], the quadratic Plebanski simplicity constraints-and more precisely in their dual version presented below (34)-are equivalent in the discrete setting to the linear constraint on each face of a given tetrahedron where the label / makes reference to a face / G A*, and where (very importantly) the subgroup SU(2) C Spm{A) (or SX(2,C)) that is necessary for the deñnition of the above constraints is chosen arbitrarily at each tetrahedron, equivalent on each edge e G A*. Such choice of the rotation subgroup is the precise analog of the time gauge in the canonical analysis of general relativity. The EPRL model is deñned by imposing the previous constraints as operator equations on the Hilbert spaces deñned by the unitary irreducible representations of the inter-nal gauge group that take part in the state-sum of BF theory. We will show in Subsection iv that the models constructed on the requirement of a suitable imposition of the linear constraints (45) satisfy the usual quadratic Plebanski constraints-that reduce BF theory to general relativity-in the path integral formulation (up to quantum corrections which are neglected in the usual semiclassical limit).

From the commutation relations (42), from previous section, we can easily compute the commu-tator of the previous tetrahedron constraints and conclude that, in fact, it does not cióse, namely

[D),D*_fl] =S_ff,e%\(l + ^)L^k_f-²-K^k_e

2S_ee,eⁱ{D^k+S_ee,^eⁱÍL^k_f.(46)

k^f

The previous commutation relations imply that the constraint algebra is not closed and cannot there-fore be imposed as operator equations on the states summed over in the BF partition function in general. There are two interesting exceptions to the previous statement:

040004-10

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

1.  The ñrst one is to take 7 = ±a. This corre-sponds to the description of the model in terms of self-dual or anti-self-dual variables. Unfor-tunately, the construction of the new models is not well deñned in this case for the Lorentzian theory and leads to a trivial result in the Rie-mannian sector: SU(2) BF theory.

2.  The second possibility is to work in the sector where L) = 0. This choice leads to the Barret-Crane model [66], where the degrees of freedom of BF theory seem over constrained: Boundary states satisfying the BC constraints are a very small subset of the allowed boundary states in LQG. This is believed to be problematic if gravity is to be recovered at low energies.

The EPRL model is obtained by restricting the rep-resentations appearing in the expression of the BF partition function so that at each tetrahedron the linear constraint (45) is the strongest possible way that is compatible with the uncertainties relations stemming from (46). In addition, one would add the requirement that the state-space of the tetrahe-dra is compatible with the state-space of the anal-ogous excitation in the canonical context of LQG, so that arbitrary states in the kinematical state of LQG have non-trivial amplitudes in the model.

Due to the fact that the constraints D) do not form a closed (ñrst class) algebra in the generic case, one needs to devise a weaker sense in which they are to be imposed. One possibility is to con-sider the Gupta-Bleuler criterion consisting of se-lecting a suitable class of states for which the mate elements on D) vanish. One notices from (43) that if we chose the subspace 3% C ^,_fc, we would have

(p, k,j, q\D³_f\p, k,j, m) = 5,,_mm(l

7¿

(p,k,j,q\Df\p,k,j,m) = 6_q±ltm

x^/ܱm + l)(JTm)(l

< v

7j

7

).

The matrix elements of the linear constraints van-ish in this subclass if one chooses

7j

pk j(j + 1)

(47)

1. Case 7 < 1: Following Ref. [67], in this case one restricts the representations to

Riemannian: p = j + í,k = -fj. Lorentzian: p = ₇(j + í),k= j. (48)

which amounts to choosing the máximum weight component j = p - 1 in the expansión (41). In the Riemannian case, the above choice translates into j± = (1 ± <y)j/2 for the SU(2) right and left representations. Notice that the solutions to the simplicity constraints in the Riemannian and Lorentzian sectors look very different for 7 < 1. Simple algebra shows that condition (47) is met. There are indeed other solutions [68] to the Gupta-Bleuler criterion in this case.

2. Case 7 > 1: In this case, according to Ref. [69], one restricts the representations to

Riemannian: p Lorentzian: p =

= ₇(j + l), k = j. l(j + l),k = j.

(49)

There are two cases:

which amounts to choosing the mínimum weight component j = k in the expansión (41). For the Riemannian case, we can write the solutions in terms of j± = (7 ± 1)§ + ^i. No-tice that for 7 > 1 there is complete symme-try between the solutions of the Riemannian and Lorentzian sectors. In my opinión, this symmetry deserves further investigation as it might be an indication of a deeper connection between the Riemannian and Lorentzian models (again, such relationship is a fact in 3d gravity [62].

Another criterion for weak imposition can be devel-oped by studying the spectrum of the Master constraint Mf = D_rD_f. Strong imposition of the constraints D) would amount to looking for the kernel of the master constraint M_f. However, generically, the positive operator associated with the master constraint does not contain the zero eigenvalue in the spectrum due to the open nature of the constraint algebra (46).

It is convenient, as in Ref. [70], to express the master constraint in a manifestly invariant way. In order to get a gauge invariant constraint one starts

)

7

7

040004-11

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

from the master constraint and uses the D\

f

0

classically to write it in terms of Casimirs, namely

M_/ = (l+a²7²)C₂-2C_l7,

where C\ and C₂ are the Casimirs given in Eq. (39). The minimum eigenvalue condition is

with hj =constant. In the Riemannian case, the previous equation can be written as

[(l-7)j;-(l+7)JÍ]* = ¿?^,_sc,        (54)

which in turn has a simple graphical representa-tion in terms of spin-network grasping operators, namely

Riemannian: p Lorentzian: p =

13, k=j.

(50)

The minimum eigenvalue is m_mm = h²"fj("/² - 1) for the Riemannian case and m_mm = 7 for the Lorentzian case. The master constraint criterion works better in the Lorentzian case, as pointed out in Ref. [70]. More recently, it has been shown that the constraint solutions p = jj and k = j also follow naturally from a spinor formulation of the simplicity constraints [71-73]. The above criterion is used in the deñnition of the EPRL model.

It is important to point out that the Riemannian case imposes strong restrictions on the allowed valúes of the Immirzi parameter if one wants the spin j G N/2 to be arbitrary (in order to have all possible boundary states allowed in LQG). In this case, the only possibilities are 7 = N or 7 = 1. This restriction is not natural from the viewpoint of LQG. Its relevance, if any, remains mysterious at this stage.

Summarizing, in the Lorentzian (Riemannian) EPRL model one restricts the SL(2,C) {Spin{A)) representations of BF theory to those satisfying

P = 13 k=j

(51)

for j G N/2. From now on, we denote the subset of admissible representation

-(1+7)

k

+(1-7)

k

Gsc

(55)

3-

3>

'3-

3>

The previous equation will be of great importance in the graphical calculus that will allow us to show that the linear constraint imposed here, at the level of states, implies the vanishing of the quadratic Ple-banski constraints (34) and their fluctuations, com-puted in the path integral sense, in the appropriate large spin semiclassical limit.

iii. Presentation of the Riemannian EPRL amplitude

Here we complete the deñnition of the EPRL mod-els by imposing the linear constraints on the BF amplitudes constructed in Section II. We will also show that the path-integral expectation valué of the Plebanski constraints (34), as well as their fluctuations, vanish in a suitable semiclassical sense. This shows that the EPRL model can be considered as a lattice deñnition of the a quantum gravity theory. We start with the Riemannian model for which a straightforward graphical notation is available. The ñrst step is the translation of Eq. (40)- for p and k satisfying the simplicity constraints- in terms of the graphical notation introduced in Section II. Concretely for 7 < 1, one has j± = (1 ± ₇)j/2 G J£y and (40) becomes

JT₇ C Irrep(SX(2,C))(Irrep(S^,_ínn(4)))

(52)

The admissible quantum states * are elements of the subspace 3tfj C 3tf_liá (i.e., minimum weight states) which satisfy the constraints (45) in the fol-lowing semiclassical sense:

{Kⁱ_f-₁Lⁱ_f)^ = 0_sc,

(53)

where the symbol ff_sc (order semiclassical) denotes a quantity that vanishes in limit h -► 0, j -► 00

(1-7) For 7 > 1 we have

(l-.7)£(l + 7)§

(l+₇)£ /jk                 a.          (56)

<y-=iú (l-<)f-(l + 7)^

(7-1)

(l+₇)¿

2

13

e

a=j

(7

l)£(l+7)§ (57)

(7 -0^(1 +7) §

--

--

040004-12

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

The implementation of the linear constraints of Subsection ii consists of restricting the representa-tions _Pf oíSpin{A) (appearing in the state sum amplitudes of BF theory, as written in Eq. (18)) to the subclass _Pf G JT₇ C Irrep(Sp¿n(4)), deñned above, while projecting to the highest weight term in (56) for 7 < 1. For 7 > 1, one must take the minimum weight term in (57) . The action of this projection will be denoted ^ : ^i+_7by2,|(i-₇)_by2 - ^, graphically

for an important implication). One should simply keep in mind that green wires in the previous two equations and in the ones that follow are labeled by arbitrary spins j (which are being summed over in the expression of the amplitude (61)), while red and blue wires are labeled by j+ = (1 + j)j/2 and j~ = |1 - 7|j/2, respectively. With this, (18) is modiñed to

Wj

|7-1|

(l+7)f

(58)

Explicitly, one takes the expression of the BF par-tition function (13) and modiñes it by replacing the projector Pf_nv(pi, . . . ,p₄) with p_u . . . p₄ G Jf-, by a new object

P

eprlw11 x (% ®

¿4.) = Pfnv(pl . . . Pi)

(E)^_J4)Pl_nv(pi---pi)

(59)

with J!, . . . j₄ G N/2, implementing the linear constraints described in the previous section. Graphically, the modiñcation of BF theory that produces the EPRL model corresponds to the replacement

z*prl (A) = e n dn-7ifd (i+7)

Psex feA*

><Y[P!prlUu--- ,ji) =

e

= e n dn-7ifd (i+7)

Psex feA*

| (01)

PtnviPí . . . P4.)

Pt_vrAh---h)

(60)

on the expression (18), where we have dropped the representation labels from the figure for sim-plicity. We have done the operation (58) on each an every of the four pairs of representations. The Spin{A) integrations represented by the two boxes at the top and bottom of the previous graphical expression restore the MI Spin{A) invariance as the projection (58) breaks this latter symmetry for being based on the selection of a special subgroup SU(2) C Spin(4) in its definition (see Subsection c

The previous expression defines the EPRL model amplitude.

The spinfoam representation of the EPRL amplitude

Now we will work out the spin foam representation of the EPRL amplitude which, at this stage, will take no more effort than the derivation of the spin foam representation for Spin{A) BF theory, as we went from Eq. (18) to Eq. (20) in Section II. The first step is given in the following equation

j

x

a.

040004-13

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

^w(A)

EE_te n d |i-₇||d (1+7)

¿2

p^t3, (63)

1*4

where the vertex amplitude (graphically repre-sented) depends on the 10 spins j associated to the face-wires and the 5 intertwiners associated to the five edges (tetrahedra). As in previous equations, we have left the spin labels of wires implicit for no-tational simplicity. We can write the previous spin foam amplitude in another form by integrating out all the projectors (boxes) explicitly. Using (17), we

get

(62)

which follows, basically, from the invariance of the Haar measure (9) (in the last line, we have used (17)). More precisely, the integration of the sub-group SU(2) Spin(4), represented by the green box on the right, can be absorbed by suitable re-definition of the integration on the right and left copies of SU(2), represented by the red and blue boxes, respectively. With this, we can already write the spin foam representation of the EPRL model, namely

E

éÍ^ ^Üí

(64)

thus replacing this in (61), we get

040004-14

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

^w(a) = d_l7__lléd (₇+1)_é

l*^a(7+l)|

jf feA*                                t_e veA*

u ...;.

(65)

|l-7l^|l-₇J.. iV\ 1-7

|l_₇|W|f-7

+"|#

where the coefficients f\ _ are the so-called fusión coefficients which already appear in their graphical form in (64), more explicitly

i-₇|^

/^-(ji,-,i/i

The previous Eq. (66) is the form of the EPRL model as derived in Ref. [5].

iv. Proof of validity of the Plebanski con-straints

In this section, we prove that the quadratic con-straints are satisfied in the sense that their path integral expectation value and fluctuation vanish in the appropriate semiclassical limit.

a. The quadratic Plebanski constraints The quadratic Plebanski constraints are

nlJnKL eIJKLO tí

e t^vpa ~ 0.

(67)

JL¿ÍX ¿JO

The constraints in this form are more suitable for the translation into the discrete formulation. More

precisely, according to (6), the smooth ñelds BT are now associated with the discrete quantities B" , , or equivalently B¹/ as faces / G A* are in one-to-one correspondence to triangles in four dimensions. The constraints (67) are local constraints valid at every spacetime point. In the discrete setting, spacetime points are represented by four-simplexes or (more addapted to our discus-sion) vértices v G A*. With this, the constraints (67) are discretized as follows:

Triangle (or diagonal) constraints:

íijkl Bj

IJr>KL

0, (68)

for all / G v, i.e., for each and every face of the 10 possible faces touching the vértex v.

Tetrahedron constraints:

íIJKhB f

IJtjKL

0,

(69)

for all /, /' G v, so that they are dual to triangles sharing a one-simplex, i.e., belonging to the same tetrahedron out of the ñve possible ones.

4-simplex constraints:

r¡IJr¡KL

f

(70)

for any pair of faces fjGv that are dual to trian-

gles sharing a single point. The last constraint will require a more detailed discussion. At this point, let us point out that the constraint (70) is inter-preted as a deñnition of the four volume e_v of the four-simplex. The constraint requires such deñnition to be consistent, i.e., the true condition is

r¡IJr¡KL tIJKL&f Dj

r¡IJr¡KL

pIJpKL tllKLOfiOj»

--      --

e_v

(71)

for all ñve different possible pairs of / and / in a four simplex, and where we assume the pairs /-/ are ordered in agreement with the orientation of the complex A*.

b. The path integral expectaüon valué of the Ple-banski constraints

Here we prove that the Plebanski constraints are satisñed by the EPRL amplitudes in the path integral expectation valué sense.

5

--

040004-15

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

The tmangle constraints:

The tetrahedra constraints:

We start from the simplest case: The triangle (or diagonal) constraints (68). We choose a face f Gv (dual to a triangle) in the cable-wire-diagram of Eq. (61). This amounts to choosing a pair of wires (right and left representations) connecting two nodes in the vértex cable wire diagram. The two nodes are dual to the two tetrahedra-in the four simplex dual to the vértex-sharing the chosen triangle. Equation (36) shows that

r>IJ TjKL

tIJKL&f £>/

oc(l+₇)² Jj . J--(1-₇)²J+- J+,(72)

where jf denotes the self-dual and anti-self-dual parts of U¹/. The path integral expectation valué of the triangle constraint is then

The proof of the validity of the tetrahedra straints (69). In this case we also have

((1+₇)²J

(1+7)² (

₇)² J+ . J+) oc (73)

-(l-₇)²

0sc,

where the double graspings on the anti-self-dual (blue) wire and the self-dual (red) wire represent the action of the Casimirs Jj . Jj and Jj-Jj, on the cable-wire diagram of the corresponding vértex. Direct evaluation shows that the previous diagram is proportional to h²j_f which vanishes in the semi-classical limit h -* 0, j -* oo with hj =constant. We use the notation already adopted in (54) and cali such quantity fJ_sc. This proves that the triangle Plebanski constraints are satisñed in the semi-classical sense.

(1+7)²

-(l-₇)²

(74)

0sc-

where we have chosen an arbitrary pair of faces. In order to prove this, let us develop the term on the right. The result follows from

+ 0sc, (75)

where in the first line we have used the fact that the double grasping can be shifted through the group integration (due to gauge invariance (9)).

con-

040004-16

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

In the first and second terms on the second line, we have used Eq. (55) to move the graspings on self-dual wires to the corresponding anti-self-dual wires. Equation (75) immediately follows the pre-vious one; the argument works in the same way for any other pair of faces. Notice that the first equality in Eq. (75) implies that we can view the Plebanski constraint as applied in the frame of the tetrahedron as well as in a Lorentz invariant frame-work (the double grasping defines an intertwiner operator commuting with the projection P_i^e_nv rep-resented by the box). An analogous statement also holds for the triangle constraints (73).

The 4-simplex constraints

Now we show the validity of the four simplex con-straints in their form (71). As we will show below, this last set of constraints follow from the Spin(4) gauge invariance of the EPRL node (i.e., the va-lidity of the Gauss law) plus the validity of the tetrahedra constraints (69). Gauge invariance of the node takes the following form in graphical no-tation:

+

O, (76)

where the above equation represents the gauge in-variance under infinitesimal left SU(2) rotations. An analogous equation with insertions on the right is also valid. The validity of the previous equation can, again, be related to the invariance of the Haar measure used in the integration on the gauge group

that defines the boxes (9).

¯ Now we choose an arbitrary pair f and f (where

¯

f is one of the three possible faces whose dual tri-angle only shares a point with the one correspond-ing to f) and will show how the four volumen e_v defined by it equals the one defined by any other admissible pair. The first step is to show that we

¯             ¯

get the same result using the pairs f-f and f-f,

¯ where f is another of the three admissible faces op-posite to f. The full result follows from applying the same procedure iteratively to reach any admis-sible pair. It will be obvious from the treatment given below, that this is possible. Thus, for a given pair of admissible faces, we have

040004-17

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

e« = (l+7)² f

-(1 + 7)²

+ (l-7)²

-(1 + 7)²

(1-7)² f

+

+ jr

+ (l-₇)² fksTj^k

+ ^_sc,

(77)

where going from the ñrst line to the second and third lines we have simply used (76) on the bot-tom graspings on the right and left wires. The last line results from the validity of (69). Notice that the second terms in the second and third lines add up to ü_sc, as well as the third terms in the second and third line. There is an overall minus sign which amounts for an orientation factor. It should be clear that we can apply the same procedure to arrive at any admissible pair.

c. P_eprl is not a projector

We will study in detail the object P¡_prl{ji, ... ,Ja)-We see that it is made of two ingredients. The ñrst one is the projection to the máximum weight subspace ^ for 7 > 1 in the decomposition of .#i+ j- for j^± = (1 ± 7)j/2 (j± = (7 ± l)j/2 for

7 > 1) in terms of irreducible representations of an arbitrarily chosen SU(2) subgroup of Spin(A). The second ingredient is to eliminate the depen-dence on the choice of subgroup by group averaging with respect to the MI gauge group Spin{A). This is diagrammatically represented in (60). However, PepriÜí, . . . ,34) is not a projector, namely

P!_PriÜu . . . J4)² + PIpAh, . . . ,k)-        (78)

Technically this follows from (59) and the fact that

[PLv (Pi . . . Pi), (&ji ® . . . ® %J] = 0 (79)

i.e., the projection imposing the linear constraints (deñned on the frame of a tetrahedron or edge)

040004-18

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

and the Spm{A) (or Lorentz) group averaging- rendering the result gauge invariant-do not com-mute. The fact that the P¡_prl{ju ... ,ji)is not a projection operator has important consequences in the mathematical structure of the model:

1. Prom (61) one can immediately obtain the fol-lowing expression for the EPRL amplitude

2.

foam amplitudes to be independent of the em-bedding, i.e., well-defined on the equivalence classes of spin foams as defined by Baez in Ref. [74] (the importance of these consistency requirements was emphasized in Ref. [75]). The amplitude (80) fails this requirement due

^P¡_vrl{jl,--- ,3A?^P¡_vrl{j

d. The Warsaw proposal

eprl v

,3a)-

Zeprl(A)

E

X

II d |i-₇|id (i+₇)4 \\P!_Pri{Ju--- ,3a)- (80)

If one sees the above as difficulties, then there is a simple solution, at least in the Riemannian case. As proposed in Reí. [76,77], one can obtain a consis-tent modiñcation oí the EPRL model by replacing P¡_prl in (80) by a genuine projector P¿, graphically

This expression has the formal structure of expression (13) for BF theory. The formal sim-ilarity, however, is broken by the fact that PepriÜí, . . . ,3 a) is not a projection operator. From the formal perspective, there is the possi-bility for the amplitudes to be deñned in terms of a network of projectors (as in BF theory). This might provide an interesting structure that might be of relevance in the deñnition of a discretization independent model. On the contrary, the failure of P¡_prl{ji, . . . ,j₄) to be a projector may lead, in my opinión, to difficulties in the limit where the complex A is refined: The increasing of the number of edges might produce either trivial or divergent amplitudes ².

(81)

It is easy to check that by construction

(P¿(ji---j₄))²=P¿(ji---J4).          (82)

The variant of the EPRL model proposed in Refs. [76,77] takes then the form

Another Pí

associated

with is the

Zeprl(A)

E II dH-7lid (l+7) 3¡ /EA*

Y[PwÜu--- Ja)

3¡ '«»

n dii-7iida+7)

/GA*

difficulty

failure of the amplitudes of the EPRL model, as defined here, to be consistent with the ab-stract notion of spin foams as defined in [74]. This is a point of crucial importance under cur-rent discussion in the community. The point is that the cellular decomposition A has no physical meaning and is to be interpreted as a subsidiary regulating structure to be removed when computing physical quantities. Spin foams configurations can fit in different ways on a given A, yet any of these different em-beddings represent the same physical process (like the same gravitational field in different coordinates). Consistency requires the spin

²This is obviously not clear from the form of (80). We are extrapolating the properties of (PJj of the amplitude (80) in the large number of edges limit implied by the continuum limit.

(83)

n *i.* n

e£A*

veA*

_rl)^N for large N to those

040004-19

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

(84)

Thus, in the modiñed EPRL model, edges e G A* are assigned pairs of intertwiner quantum numbers i%_s and i%_t and an edge amplitude given by the matrix elements g^e_{í% t;} (where v_s and v_t stand for the source and target vértices of the given oriented edge). The fact that edges are not assigned a single quantum number is not really signiñcative; one could go to a basis of normalized eigenstates of P¿ and rewrite the modiñed model above as a spin foam model where edges are assigned a single (basis element) quantum number. As the nature of such basis and the quantum geometric interpreta-tion of its elements are not clear at this stage, it seems simpler to represent the amplitudes of the modiñed model in the above form.

The advantages of the modiñed model are im-portant. However, a generalization of the above modiñcation of the EPRL model in the Lorentzian case is still lacking. Notice that this modiñcation does not interfere with the results on the semiclas-sical limit (to leading order), as reviewed in Section VIL The reason for this is that the matrix elements g^e_a¡3 -+ 6_al3 in that limit [78].

v. The coherent states representation

We have written the amplitude deñning the EPRL model by constraining the state sum of BF theory. For semiclassical studies that we will review in Section VII, it is convenient to express the EPRL amplitude in terms of the coherent states basis. The importance of coherent states in spin foam models was put forward in Ref. [49] and explicitly used to re-derive the EPRL model in Ref. [79]. The coherent state technology was used by Freidel and Krasnov in [6] to introduce a new kind of spin foam models for gravity: The FK models. In some cases, the FK model is equivalent to the EPRL model. We will review this in detail in Section IV.

The coherent state representation of the EPRL model is obtained by replacing (27) in each of the intermedíate SU(2) (green) wires in the expression (61) of the EPRL amplitudes, namely

The case 7 < 1

In this case, the coherent state property (28) im-plies

where we have used, in the last line, the fact that for 7 < 1 the representations j of the sub-group SU{2) G Spin{A) are máximum weight, i.e., j = j+ +j-. Doing this at each edge, we get

040004-20

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

where the discrete action

-a.+

j¡ feA*

JS f í Y[ dj.fdnef

eeeA*

where we have explicitly written the n G S² inte-gration variables on a single cable. The expression above is very similar to the coherent states repre-sentation of Spin{A) BF theory given in Eq. (29). In fact, one gets the above expression if one starts from the expression (29) and sets n+ = n" = n_ef while dropping, for example, all the sphere inte-grations corresponding to the n+ (or equivalently n~_f). Moreover, by construction, the coherent states participating in the previous amplitude sat-isfy the linear constraints (45) in expectation valúes, namely

{j,n_ef\D)\j,n_ef)

0>e/|(l

0.

7) jy + (1 + 7) Jf ^¿b', n_ef)

(87)

Thus, the coherent states participating in the above representation of the EPRL amplitudes solve the linear simplicity constraints in the usual semiclas-sical sense. The same manipulations leading to (89) in Section II lead to a discrete effective action for the EPRL model, namely

Zlpri = J2 nd

(1-7)^ (1+7)^

(88)

II ^áJ,fdnefdg-_fdgt_f expC^^]),

e£A*

^[ff']

(89)

^ (l-₇)^.,n

ti£A

\g-]+S^v _u \g+\)

(1+7)^, n

....'¿=.1

» , (86) with

sUg\

(90)

^ 2j_o6ln(n_o6|^¹cí6|n6a),

a<6=l

and the Índices a, 6 label the ñve edges of a given vértex. The previous expression is exactly equal to the form (11) of the BF amplitude. In the case of the gravity models presented here, the coherent state path integral representation (analogous to (31)) will be the basic tool for the study of the semiclassical limit of the models and the relation-ship with Regge discrete formulation of general rel-ativity.

The case 7 > 1

The case 7 > 1 is more complicated [80]. The reason for this is that the step (85), directly lead-ing to the discrete action in the previous case, is no longer valid, as the representations of the sub-group SU{2) G Spin{A) are now minimum instead of máximum weight. However, the representations j+ = j- + j are máximum weight. We can, there-fore, insert coherent states resolution of the identity on the right representations and get:

x

040004-21

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

n 4

^^^J n 1

(91)

nd (i+₇) 2

dm

[s³]⁴

^

(1+7)

n dm¡

[S3]4

. .

.

m 3 miO-On 4

^rn2 m^Q-Q n^

where we are representing the relevant part of the diagram appearing in Eq. (85). In the last line, we have used j+ = j + j~ (i.e., máximum weight), and the graphical notation mo-on = (m|n) as it follows from our previous conventions. With this, one gets

Z

eprl

(92)

e n (1-7) íf (1+7)^

ú¡ feA*

X

II dJ,fd_{1+7)^dnefdm_efdg_efdgt_f

eeA*

xexp^J^]), where the discrete action

ST,n,J^ = E ^,n,»^]       (93)

veA*

with

= Y, [jab(l+~f)log({m_ab\g+_b\m_ba))

l<a<6<5

+Ía6(7-l)log((m_o6|_í7-₆|m6₀)) +2j₀6(log((n₀6|m_o6)) + log((m_6o|n_6o)))] .

a. Some addiüonal remarks

It is important to point out that the commuta-tion relations of basic ñelds-reflecting the simple algebraic structure of spin(4)- used here are the ones induced by the canonical analysis of BF theory presented previously. The presence of con-straints generally modiñes canonical commutation relations, in particular in the presence of second class constraints. For some investigation of the is-sue in the context of the EPRL and FK models, see Ref. [69]. In Ref. [81], it is pointed out that the presence of secondary constraints in the canonical analysis of Plebanski action should be trans-lated into additional constraints in the holonomies of the spin foam models here considered (see also Ref. [82]). A possible view is that the simplicity constraints are here imposed for all times and thus secondary constraints should be imposed automat-ically.

There are alternative derivations of the models presented in the previous sections. In particular, one can derive them from a strict Lagrangean ap-proach of Plebanski's action. Such viewpoint is taken in Refs. [83-85]. The path integral formula-tion of Plebansky theory using commuting B-ñelds was studied in Ref. [86], where it is shown that only in the appropriate semiclassical limit the amplitudes coincide with the ones presented in the previous sections (this is just another indication that the construction of the models has a certain semiclassical input; see below). The spin foam quanti-zation of the Holst formulation of gravity via cubu-lations was investigated in Ref. [87]. The simplicity constraints can also be studied from the perspective of the U(N) formulation of quantum geometry [88]. Such U(N) treatment is related to previous work [89, 90] which has been extended to a completely new perspective on quantum geometry with possible advantageous features [91,92]. For additional discussion on the simplicity constraints, see Ref.

rriA

040004-22

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

[93].

vi.

Presentation model

of the EPRL Lorentzian

As it was briefly discussed in Section III, unitary irreducible representations of SL(2, C) are infinite dimensional and labeled by a positive real number p£E+ and a half-integer k G N/2. These representations are the ones that intervene in the harmonic analysis of square integrable functions of SL(2, C) [64]. Consequently one has an explicit expression of the delta function distribution (defined on such test function), namely

where the boxes now represent SX(2,C) integra-tions with the invariant measure. The previous am-plitude is equivalent to its spin foam representation

jf <-e feA*

í-5

í-1

veA

<%)

where D

-' m+                                  -^J

k                                           j,m

(95)

p,k jmj'm

,{g) with j > k and j > m > -j (similarly fdr the primed Índices) are the matrix elements of the unitary representations p - k in the so-called canonical basis [63]. One can use the previous expression, the Lorentzian versión of Eq. (11), in order to introduce a formal definition of the BF amplitudes, which now would involve inte-gration of the continuous labels p_f, in addition to sums over discrete quantum numbers such as k, j and m. The Lorentzian versión of the EPRL model can be obtained from the imposition of the linear simplicity constraints to this formal expression. As the continuum labels p_f are restricted to p_f = jj_f, the Lorentzian EPRL model becomes a state-sum model as its Riemannian relative. Using the follow-ing graphical notation

Djmj'm'id)

the amplitude is

j,m

j',m'

(96)

The vértex amplitude is divergent due to the pres-ence of a redundant integration over SL(2,C)- It becomes finite by dropping an arbitrary integration, Le., removing any of the 5 boxes in the vértex expression [94].

a. The coherent state representation

It is immediate to obtain the coherent states representation of the Lorentzian models. As in the Riemannian case, one simply inserís resolution of the identities (22) on the intermediate SU{2) (green) wires in (97), from where it results

Z^_prl(A) = J2 It (!+7V

/ n

jf feA*

d_jsJdn_ef

eeeA*

(97)

Zeprl (A)

y n (ⁱ+^²)-?/ j

IV. The Freidel-Krasnov (FK) model

Shortly after the appearance of the paper in Ref. [4], Freidel and Krasnov [6] introduced a set of new spin foam models for four-dimensional gravity us-ing the coherent state basis of the quantum tetra-hedron of Livine and Speziale [49]. The idea is to

V

--

fe

--

040004-23

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

impose the linearized simplicity constraints (45) di-rectly as a semiclassical condition on the coherent state basis. As we have seen above, coherent states are quantum states of the right and left tetrahe-dra in BF theory which have a clear-cut semiclassical interpretation through their property (26). We have also seen that the imposition of the linear constraints (45) a la EPRL is in essence semiclassical as they are strictly valid only in the large spin limit. In the FK approach one simply accepts from the starting point that, due to their property of a non-deñning set that is closed under commutation relations, the Plebansky constraints are to be im-posed semiclassically. One defines new models by restricting the set of coherent states entering in the coherent state representation of Spin{A) BF theory (29) to those that satisfy condition (45) in expec-tation valúes. They also emphasize how the model [4] corresponds, indeed, to the sector 7 = 00 which has been shown to be topological [95].

The case 7 < 1

For 7 < 1, the vértex amplitude is identical to the EPRL model. This is apparent in the coherent state expression of the EPRL model (88). Thus, we have

^T(A)

e n dn-₇iíd (i+₇H (9^g)

is feA*

n dd+7)

e£A*

íVd^/

ft

fact that one is restricting the Spin{A) resolution of identity in the coherent basis in the previous expression, while in the EPRL model the coherent state resolution of the identity is used for SU (2) representations. This difference is important and has to do with the still unsettled discussion con-cerning the measure in the path integral representation.

The case 7 > 1

For the case 7 > 1, the FK amplitude is given by

^t(a)=e n dii-7i!da+7)

is /eA*

n dd+7)

e£A*

iV#^

(100)

The study of the coherent state representation of the FK model for 7 > 1, in comparison with Eq. (92) for the EPRL model, clearly shows the difference between the two models in this regime.

^zik=Y. n ⁽\i-^í±⁽\i+^4.

/n

is /eA*

i-₇|^§£ (1+7) -

e£A*

exp^rj^D,

j_£j_dn_efdg_efdg_ef

(101)

From the previous expression, we conclude that the where the discrete action vértex amplitudes of the FK and EPRL model coincide for 7 < 1

j ± , n L^y J

JV    r    = JV          .

v jk           v eprl

(99)

Notice, however, that different weights are assigned to edges in the FK model. This is due to the

¿-~<       (1-7)^-, n       ^J

veA

+S^V _it

(l+7)^,s(₇)n

[g⁺]\

(102)

040004-24

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

where_S(₇)=sign(l-₇) and

5

Slnlg}= Y, 2^b\n{n_ab\g-¹g_b\n_ba), (103)

a<b=1

with the Índices a, b labeling the ñve edges of a given vértex.

V. Boundary data for the new mod-els and relationship with the canonical theory

So far, we have considered cellular complexes with no boundaries. Transition amplitudes are expected to be related to the deñnition oí the physical scalar product. In order to define them, one needs to consider complexes with boundaries. Boundary states are defined on the boundary oí the dual two-complex A* that we denote dA*. The object dA* is a one-complex (a graph). According to the construction of the model (Section III), boundary states are in one-to-one correspondence with SU(2) spin networks. This comes simply from the fact that links (one-cells) l G dA* inherit the spins la-bels (unitary irreducible representations of the sub-group SU(2)) from the boundary faces while nodes (zero-cells) n G dA* inherit the intertwiner levéis from boundary edges.

At this stage, one can associate the boundary data with elements of a Hilbert space. Being in one-to-one correspondence with SU(2) spin networks, a natural possibility is to associate to them an element of the kinematical Hilbert space of LQG. More precisely, with a given colored boundary graph 7 with links labeled by spins j_e and nodes labeled by intertwiners t_n, we associate a cylindri-cal function ^₇,{_je},{_írí} G 3?²(SU(2)^N<), where here N_e denotes number of links in the graph 7. In this way, the boundary Hilbert space associated with dA* is isomorphic (if one used the natural AL measure) with the Hilbert space of LQG truncated to that fixed graph. Moreover, geometric opera-tors, such as volume and área defined in the covari-ant context, are shown to coincide with the corre-sponding operators defined in the canonical formu-lation [67,96]. Now, if cellular complexes are dual to triangulations, then the boundary spin networks can have at most four-valent nodes. This limita-tion can be easily overeóme: As in BF theory, the

EPRL amplitudes can be generalized to arbitrary complexes with boundaries given by graphs with nodes of arbitrary valence. The extensión of the model to arbitrary complexes has been first stud-ied in Refs. [97,98]. It has also been revisited in Refs. [68].

Alternatively, one can associate the boundary states with elements of ^²{Spin{A)^Nl) (in the Riemannian models)-or carefully define the ana-log of spin network states as distributions in the Lorentzian case (see Refs. [99] for some insights on the problem of defining a gauge invariant Hilbert space of graphs for non-compact gauge groups). In this case, one gets special kinds of spin network states. These are a subclass of the so-called pro-jected spin networks introduced in Refs. [100,101] in order to define a heuristic quantization of the (non-commutative and very complicated) Dirac algebra of a Lorentz connection formulation of the phase space of gravity [100,102-107]. The fact that this special subclass of projected spin networks ap-pears naturally as a boundary state of the new spin foams is shown in Ref. [108].

Due to their similarity to 7 < 1, the same relationship between boundary data and elements of the kinematical Hilbert space holds for the FK model. However, such simple relationship does not hold for the model in the case 7 > 1.

It is important to mention that the knotting properties of boundary spin networks do not seem to play a role in present definitions of transition amplitudes [109].

VI. Further developments and re-lated models

The spin foam amplitudes discussed in the previ-ous sections have been introduced by constraining the BF histories through the simplicity constraints. However, in the path integral formulation, the pres-ence of constraints has the additional effect of mod-ifying the weights with which those histories are to be summed: Second class constraints modify the path integral measure (in the spin foam context this issue was raised in Ref. [75]). As pointed out before, this question has not been completely set-tled in the spin foam community yet. The explicit modification of the formal measure in terms of con-tinuous variables for the Plebansky action was pre-

040004-25

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

sented in Ref. [110]. A systematic investigation of the measure in the spin foam context was at-tempted in Ref. [111] and [112]. As pointed out in Ref. [75], there are restrictions in the mani-fold of possibilities coming from the requirement of background independence. The simple BF mea-sure chosen in the presentation of the amplitudes in the previous sections satisfies these requirements. There are other consistent possibilities; see, for in-stance, Ref. [113] for a modified measure which re-mains extremely simple and is suggested from the structure of LQG.

An important question is the relationship be-tween the spin foam amplitudes and the canonical operator formulation. The question of whether one can reconstruct the Hamiltonian constraints out of spin foam amplitudes has been analysed in detail in three dimensions. For the study of quantum three-dimensional gravity from the BF per-spective, see Ref. [114]. We will, in fact, present this perspective in detail in the three dimensional part of this article. For the relationship with the canonical theory using variables that are natural from the Regge gravity perspective, see [115, 116]. There are generalizations of Regge variables more adapted to the interpretation of spin foams [117]. In four dimensions,the question has been investi-gated in Ref. [118] in the context of the new spin foam models. In the context of group field theo-ries, this issue is explored in Ref. [119]. Finally, spin foams can, in principle, be obtained directly from the implementation of the Dirac program us-ing path integral methods. This has been explored in Refs. [120, 121], from which a discrete path integral formulation followed [122]. The question of the relationship between covariant and canonical formulations in the discrete setting has been ana-lyzed also in Ref. [123].

By construction, all tetrahedra in the FK and EPRL models are embedded in a spacelike hyper-surface and hence have only spacelike triangles. It seems natural to ask the question of whether a more general construction allowing for timelike faces is possible. The models described in previous sections have been generalized in order to include timelike faces in the work of F. Conrady [124-126]. An ear-lier attempt to define such models in the context of the Barrett-Crane model can be found in Refs. [127].

The issue of the coupling of the new spin foam

models to matter remains to a large extend un-explored territory. Nevertheless, some results can be found in the literature. The coupling of the Barrett-Crane model (the γ → ∞ limit of the EPRL model) to Yang-Mills fields was studied in Ref. [128]. More recently, the coupling of the EPRL model to fermions has been investigated in Refs. [129, 130]. A novel possibility of unification of the gravitational and gauge fields was recently proposed in Ref. [131].

The introduction of a cosmological constant in the construction of four-dimensional spin foam models has a long history. Barrett and Crane in-troduced a vertex amplitude [132], in terms of the Crane and Yetter model [13], for BF theory with cosmological constant. The Lorentzian quantum deformed version of the previous model was stud-ied in Ref. [133]. For the new models, the coupling with a cosmological constant is explored in terms of the quantum deformation of the internal gauge symmetry in Refs. [134, 135], as well as (indepen-dently) in Ref. [136]. The asymptotics of the vertex amplitude are shown to be consistent with a cos-mological constant term in the semiclassical limit in Ref. [137].

The spin foam approach applied to quantum cos-mology has been explored in Refs. [138-143]. The spin foam formulation can also be obtained from the canonical picture provided by loop quantum cosmology (see Ref. [144] and references therein). This has been explored systematically in Refs. [145-148].

As we have discussed in the introduction of the new models, Heisenberg uncertainty principle pre-cludes the strong imposition of the Plebanski con-straints that reduce BF theory to general relativ-ity. The results of the semiclassical limit of these models seem to indicate that metric gravity should be recovered in the low energy limit. However, it seems likely that the semiclassical limit could be re-lated to certain modifications of Plebanski's formu-lation of gravity [149-153]. A simple interpretation of the new models in the context of the bi-gravity paradigm proposed in Ref. [154] could be of inter-est.

As it was already pointed out in Ref. [74], spin foams can be interpreted in close analogy to Feyn-man diagrams. Standard Feynman graphs are gen-eralized to 2-complexes and the labeling of propa-gators by momenta to the assignment of spins to

040004-26

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

faces. Finally, momentum conservation at vértices in standard feynmanology is now represented by spin-conservation at edges, ensured by the assign-ment of the corresponding intertwiners. In spin foam models, the non-trivial contení of amplitudes is contained in the vértex amplitude which, in the language of Feynman diagrams, can be interpreted as an interaction. This analogy is indeed realized in the formulation of spin foam models in terms of a group ñeld theory (GFT) [155,156].

The GFT formulation resolves, by deñnition, the two fundamental conceptual problems of the spin foam approach: Diffeomorphism gauge symmetry and discretization dependence. The difficulties are shifted to the question of the physical role of A and the convergence of the corresponding perturbative series.

This idea has been studied in more detail in three dimensions. In Ref. [157], the scaling properties of the modiñcation of the Boulatov group ñeld theory introduced in Ref. [158] were studied in detail. In a further modiñcation of the previous model (known as colored tensor models [159], new tech-niques based on a suitable Í/N expansión imply that amplitudes are dominated by spherical topol-ogy [160]. Moreover, it seems possible that the con-tinuum limit might be critical as in certain matrix models [161-165]. However, it is not yet clear if there is a sense in which these models correspond to a physical theory. The naive interpretation of the models is that they correspond to a formulation of 3d quantum gravity including a dynamical topology.

VII. Results on the semiclassical limit of EPRL-FK models

Having introduced the relevant spin foam models in the previous sections, we now present the results of the large spin asymptotics of the spin foam amplitudes suggesting that on a fixed discretization the semiclassical limit of the EPRL-FK models is given by Regge's discrete formulation of general relativity [80, 166].

The semiclassical limit of spin foams is based on the study of the the large spin limit asymptotic be-havior of coherent state spin foam amplitudes. The notion of large spin can be defined by the rescaling of quantum numbers and Planck constant accord-

ing to j -► Xj and h -► ft/A and taking A >> 1. In this limit, the quantum geometry approximates the classical one when tested with suitable states (e.g., coherent states). However, the geometry remains discrete during this limiting process as the limit is taken on a ñxed regulating cellular structure. That is why one usually makes a clear distinction be-tween semiclassical limit and the continuum limit. In the semiclassical analysis presented here, one can only hope to make contact with discrete formula-tions of classical gravity. Henee, the importance of Regge calculus in the discussion of this section.

The key technical ingredient in this analysis is the representation of spin foam amplitudes in terms of the coherent state basis introduced in Subsec-tion i. Here we follow Refs. [80, 166-169]. The idea of using coherent states and discrete effective actions for the study of the large spin asymptotics of spin foam amplitudes was put forward in Refs. [170,171]. The study of the large spin asymptotics has a long tradition in the context of quantum gravity, dating back to the study of Ponzano-Regge [26]. More directly related to our discussion, here are the earlyworks [172,173]. The key idea is to use asymp-totic stationary phase methods for the amplitudes written in terms of the discrete actions presented in the previous section.

In this section, we review the results of the analysis of the large spin asymptotics of the EPRL vértex amplitude for both the Riemannian and Lorentz-tian models. We follow the notation and terminol-ogy of Ref. [80] and related papers.

b. SU(2) 15j-symbol asymptoücs

As SU(2) BF theory is quite relevant for the con-struction of the EPRL-FK models, the study of the large spin asymptotics of the SU(2) vértex amplitude is a key ingredient in the analysis of [80]. The coherent state vértex amplitude is

15j(j,n)                                              (104)

5 = Y[dg_a Y[ (nab\ga¹9b\n_ba)^2jab,

a=1         1<a<6<5

which depends on 10 spins j_ab and 20 normáis n_ab + n_ba. The previous amplitude can be ex-pressed as

040004-27

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

15j(j,n)

r [[dg_a Yl exP ^siM

-¹           Ka<b<5

(105)

a=í

Sj,_a\g]= J2 ZJablninablg-^blnba), (106)

a<b=l

and the Índices a, b label the ñve edges of a given vértex. The previous expression is equal to the form (11) of the BF amplitude. In the case of the EPRL model studied in Section III, the coherent state representation-see Eqs. 88, 92, and 97- is the basic tool for the study of the semiclassical limit of the models and the relationship with Regge discrete formulation of general relativity.

In order to study the asymptotics of (105), one needs to use extended stationary phase methods due to the fact that the action (106) is complex (see Refs. [170,171]). The basic idea is that, in addition to stationarity, one requires real part of the action to be maximal. Points satisfying these two conditions are called cmtical points. As the real part of the action is negative deñnite, the action at critical points is purely imaginary.

Notice that the action (106) depends parametri-cally on 10 spins j and 20 normáis n. These pa-rameters define the so-called boundary data for the four simplex v G A*. Thus, there is an action principie for every given boundary data. The number of critical points and their properties depend on these boundary data, henee the asymptotics of the vértex amplitude is a function of the boundary data. Dif-ferent cases are studied in detail in Ref. [80]. Here we present their results in the special case where the boundary data describe a non-degenerate Regge ge-ometry for the boundary of a four simplex. These data are referred to as Regge-like, and satisfy the gluing constraints. For such boundary data, the action (106) has exactly two critical points, leading to the asymptotic formula

15j(Aj,n)

1

_\12

U₊exp(í^Aj_o6ef

a<b

+N-exp(-iY^*Jab&ab)\,

a<b

where Q_ab are the appropriate dihedral angles de-fined by the four-simplex geometry. Finally, the N_± are constants that do not scale with A.

c. The Riemanman EPRL vértex asymptotics

The previous result, together with the fact that the EPRL amplitude for 7 < 1 is a product of SU (2) amplitudes with the same n in the coherent state representation (88), implies the asymptotic formula for the vértex amplitude to be given by the unbal-anced square of the above formula [167], namely

1 í^p¹ 1

A"v^rl ~TñN+e

+N_ e

^abQab

(I+7)

x N+ e

£ Aj atef,,

+N_ e

One can write the previous expression as

Áeprl

1

\Í2

S^B

Nlé

- S

-¡ Regg

2N₊N_

where

a<b

'] . (108)

(109)

is the Regge-like action for A₇j_o6 = A_ab, the ten tri-angle áreas (according to the LQG área spectrum [1,2]). Remarkably, the above asymptotic formula is also valid for the case 7 > 1 [80]. The first term in the vértex asymptotics is in essence the expected one: It is the analog of the 6j symbol asymptotics in three-dimensional spin foams. Due to their ex-plicit dependence on the Immirzi parameter, the last two terms are somewhat strange from the the-oretical point of view of the continuum field. How-ever, this seems to be a peculiarity of the Rieman-nian theory alone, as shown by the results of Ref. [166] for the Lorentzian models. Non-geometric configurations are exponentially suppressed

2

a<b

040004-28

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

d. Lorentzian EPRL model

To each solution, one can associate a second so-lution corresponding to a parity related 4-simplex and, consequently, the asymptotic formula has two terms. It is given, up to a global sign, by the ex-pression

Av^Pr' ~ Tl2 ^W+ e^XP ^ÍX1^iab<S>ab

a<6

+JV_exp j _¿A₇^j_o6e^ J ,        (110)

V          a<b            / \

where N_± are constants that do not scale. Non-geometric conñgurations are exponentially sup-pressed

In Ref. [171], Preidel and Conrady gave a de-tailed description of the coherent state representa-tion of the various spin foam models described so far. In particular, they provided the deñnition of the effective discrete actions associated to each case which we presented in (101). This provides the ba-sic elements for setting up the asymptotic analysis presented in Ref. [170] (the ñrst results of the semi-classical limit of the new spin foam models). This is similar to the studies of the asymptotic of the vértex amplitude reviewed above, but more general in the sense that the semiclassical limit of a full spin foam conñguration (involving many vértices) is studied. The result is technically more complex as one studies now critical points of the action associated to a colored complex which, in addition of depending on group variables g, depends on the coherent state parameters n. The authors of Ref. [170] write Eq. (101) in the following way:

^zh=Y, n v^d(i₊,)¥^ü/uni)

where

Wa*07)                                        (112)

= / II d |l-₇|£|/d (l+₇)£|/dne/^e7^e⁺/

e£A*

xexp^;^]).

They show that those solutions of the equations of motion of the effective discrete action that are non-geometric (Le., the contrary of Regge-like) are not critical and henee exponentially suppressed in the scaling j_f -► Xj_f with A >> 1. If conñgurations are geometric (Le., Regge-like), one has two kinds of contributions to the amplitude asymptotics: Those coming from degenerate and non-degenerate conñgurations. If one (by hand) restriets to the non-degenerate conñgurations, then one has

WA*Uf) ~ ^(33n_e-6n"-4n_/)

xexp(iXS^_gJA\j_f)),          (113)

where n_e, n_v, and n_f denote the number of edges, vértices, and faces in the two complex A*, respectó vely. There are recent works by M. Han in which asymptotics of general simplicial geometry amplitudes are studied in the context of the EPRL model [174,175].

The problem of computing the two-point func-tion and higher correlation functions in the context of spin foam has received a lot of attention recently. The framework for the deñnition of the correlation functions in the background indepen-dent setting has been generally discussed by Rov-elli in Ref. [176], and correspods to a special ap-plication of a more general proposal investigated by Oeckl [177-184]. It was then applied to the Barrett-Crane model in Refs. [185-187], where it was discovered that certain components of the two-point function could not yield the expected result compatible with Regge gravity in the semiclassical limit. This was used as the main motivation for the weakening of the imposition of the Pleban-ski constraints, leading to the new models. Soon thereafter, it was argued that the difficulties of the Barrett-Crane model where indeed absent in the EPRL model [188]. The two-point function for the EPRL model was calculated in Ref. [189] and it was shown to produce a result in agreement with that of Regge calculus [190,191], in the limit 7 -► 0.

The fact that, for the new model, the double scaling limit 7 -► 0 and j -* 00 with ₇j= constant defines the appropriate regime where the fluctuation behave as in Regge gravity (in the leading order) has been further clarified in Ref. [192]. This in-dicates that the quantum fluctuations in the new models are more general than simply metric fluc-

040004-29

Papers in Physics, vol. 4, art. 040004 (2012) / A. Perez

tuations. The fact that the new models are not metric at all scales should not be surprising as we know that the Plebanski constraints that produce metric general relativity out of BF theory have been implemented only semiclassically (in the large spin limit). At the deep Planckian regime, fluctuations are more general than metric. However, it is not clear at this stage why this is controlled by the Im-mirzi parameter.

All the previous calculations involve a complex with a single four-simplex. The first computation involving more than one simplex was performed in Refs. [187, 193], for the case of the Barrett-Crane model. Certain peculiar properties were found and it is not clear at this stage whether these issues remain in the EPRL model. Higher order correlation functions have been computed in Ref. [194], the results are in agreement with Regge gravity in the γ → 0 limit.

VIII. Acknowledgements

I would like to thank the help from many people in the field that have helped me in various ways. I am grateful to Eugenio Bianchi, Carlo Rovelli and Si-mone Speziale for the many discussions on aspects and details of the recent literature. Many detailed calculations that contributed to the presentation of the new models in this review were done in collab-oration with Mercedes Vel´azquez to whom I would like to express my gratitude. I would also like to thank You Ding, Florian Conrady, Laurent Freidel, Muxin Han, Merced Montesinos for their help and valuable interaction.

 

[1] C Rovelli, Quantum Gravity, Cambridge Uni-versity Press, Cambridge (UK) (2004), Pag. 480.

[2] T Thiemann, Modern canonical quantum general relativity, Cambridge University Press, Cambridge (UK) (2007), Pag. 819.

[3] A Ashtekar, J Lewandowski, Background in-dependent quantum gravity: A status report, Class. Quant. Grav. 21, R53 (2004).

[4] J Engle, R Pereira, C Rovelli, The loop-quantum-gravity vertex-amplitude, Phys. Rev. Lett. 99, 161301 (2007).

[5] J Engle, E Livine, R Pereira, C Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys. B 799, 136 (2008).

[6] L Freidel, K Krasnov, A new spin foam model for 4d gravity, Class. Quant. Grav. 25, 125018 (2008).

[7] C Rovelli, Zakopane lectures on loop gravity, arXiv:1102.3660 (2011).

[8] A Perez, The spin foam approach to quantum gravity, Liv. Rev. Rel. (in press).

[9] J C Baez, An introduction to spin foam mod-els of quantum gravity and bf theory, Lect. Notes Phys. 543, 25 (2000).

[10] R Oeckl, Discrete gauge theory: From lattices to TQFT, Imperial College Press, London (UK) (2005), Pag. 202.

[11] R Oeckl, H Pfeiffer, The dual of pure non-Abelian lattice gauge theory as a spin foam model, Nucl. Phys. B 598, 400 (2001).

[12] F Girelli, R Oeckl, A Perez, Spin foam dia-grammatics and topological invariance, Class. Quant. Grav. 19, 1093 (2002).

[13] D Yetter L Crane, A Categorical construc-tion of 4-D topological quantum field theories, In: Quantum Topology, Eds. L Kaufmann, R Baadhio, Pag. 120, World Scientific, Singa-pore (1993).

[14] D N Yetter, L Crane, L Kauffman, State-sum invariants of 4-manifolds, J. Knot Theor. Ramif. 6, 177 (1997).

[15] J C Baez, A Perez, Quantization of strings and branes coupled to BF theory, Adv. Theor. Math. Phys. 11, 3 (2007).

[16] W J Fairbairn, A Perez, Extended matter coupled to BF theory, Phys. Rev. D, 78, 024013 (2008).

[17] M Montesinos, A Perez, Two-dimensional topological field theories coupled to four-dimensional BF theory, Phys. Rev. D 77, 104020 (2008).

[18] G 't Hooft, A locally finite model for gravity, Found. Phys. 38, 733 (2008).

[19] L Freidel, J Kowalski-Glikman, A Starodubt-sev, Particles as Wilson lines of gravitational field, Phys. Rev. D 74, 084002 (2006).

[20] E R Livine, A Perez, C Rovelli, 2D manifold-independent spinfoam theory, Class. Quant. Grav. 20, 4425 (2003).

[21] R Jackiw, Liouville field theory: A two-dimensional model for gravity? In: Quantum theory of gravity, Eds. S M Christensen, B S DeWitt, Pag. 403, Adam Hilger Ltd., Bristol (1984).

[22] C Teitelboim, The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly, In: Quantum theory of gravity, Eds. S M Christensen, B S DeWitt, Pag. 403, Adam Hilger Ltd., Bristol (1984).

[23] C P Constantinidis, O Piguet, A Perez, Quantization of the Jackiw-Teitelboim model, Phys. Rev. D 79, 084007 (2009).

[24] D Oriti, C Rovelli, S Speziale, Spinfoam 2d quantum gravity and discrete bundles, Class. Quant. Grav. 22, 85 (2005).

[25] S Carlip, Quantum gravity in 2+1 dimen-sions, Cambridge University Press, Cambridge (UK) (1998), Pag. 276.

[26] T Regge, G Ponzano, Semiclassical limit of Racah coeficients, In: Spectroscopy and group theoretical methods in physics, Eds. F Block et al., North-Holland, Amsterdam (1968).

[27] J W Barrett, I Naish-Guzman, The Ponzano-Regge model, Class. Quant. Grav. 26, 155014 (2009).

[28] K Noui, A Perez, Three dimensional loop quantum gravity: Physical scalar product and spin foam models, Class. Quant. Grav. 22, 1739 (2005).

[29] L Freidel, D Louapre, Diffeomorphisms and spin foam models, Nucl. Phys. B 662, 279 (2003).

[30] V Bonzom, M Smerlak, Bubble divergences from cellular cohomology, Lett. Math. Phys. 93, 295 (2010).

[31] V Bonzom, M Smerlak, Bubble divergences from twisted cohomology, arXiv:1008.1476 (2010).

[32] V Bonzom, M Smerlak, Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincare 13, 185 (2012).

[33] O Y Viro, V G Turaev, Statesum invari-ants of 3-manifolds and quantum 6j-symbols, Topology 31, 865 (1992).

[34] J W Barrett, J M Garcia-Islas, J F Martins, Observables in the Turaev-Viro and Crane-Yetter models, J. Math. Phys. 48, 093508 (2007).

[35] K Noui, A Perez, Observability and geome-try in three dimensional quantum gravity, In: Quantum theory and symmetries, Eds. P C Argyres et al., Pag. 641, World Scientific, Sin-gapore (2004).

[36] K Noui, A Perez, Three dimensional loop quantum gravity: Coupling to point particles, Class. Quant. Grav. 22, 4489 (2005).

[37] L Freidel, D Louapre, Ponzano-Regge model revisited. I: Gauge fixing, observables and in-teracting spinning particles, Class. Quant. Grav. 21, 5685 (2004).

[38] L Freidel, E R Livine, Ponzano-Regge model revisited. III: Feynman diagrams and effec-tive field theory, Class. Quant. Grav. 23, 2021 (2006).

[39] L Freidel, E R Livine, Effective 3d quantum gravity and non-commutative quantum field theory, Phys. Rev. Lett. 96, 221301 (2006).

[40] W J Fairbairn,  Fermions in threedimensional spinfoam quantum gravity, Gen. Rel. Grav. 39, 427 (2007).

[41] R J Dowdall, W J Fairbairn, Observables in 3d spinfoam quantum gravity with fermions, Gen. Rel. Grav. 43, 1263 (2011).

[42] S Speziale, Coupling gauge theory to spin-foam 3d quantum gravity, Class. Quant. Grav. 24, 5139 (2007).

[43] W J Fairbairn, E R Livine, 3d spinfoam quantum gravity: Matter as a phase of the group field theory, Class. Quant. Grav. 24, 5277 (2007).

[44] E R Livine, R Oeckl, Three-dimensional quantum supergravity and supersymmetric spin foam models, Adv. Theor. Math. Phys. 7, 951 (2004).

[45] V Baccetti, E R Livine, J P Ryan, The par-ticle interpretation of N = 1 supersymmetric spin foams, Class. Quant. Grav. 27, 225022 (2010).

[46] V Bonzom, E R Livine, Yet another recursion relation for the 6j-symbol, arXiv:1103.3415 (2011).

[47] M Dupuis, E R Livine, The 6j-symbol: Re-cursion, correlations and asymptotics, Class. Quant. Grav. 27, 135003 (2010).

[48] V Bonzom, E R Livine, S Speziale, Recur-rence relations for spin foam vertices, Class. Quant. Grav. 27, 125002 (2010).

[49] E R Livine, S Speziale, A new spinfoam ver-tex for quantum gravity, Phys. Rev. D 76, 084028 (2007).

[50] T Thiemann, Coherent states on graphs, Prepared for 9th Marcel Grossmann Meet-ing on Recent Developments in Theoretical and Experimental General Relativity, Grav-itation and Relativistic Field Theories (MG 9), Rome (Italy), 2-9 July (2000).

[51] T Thiemann, Gauge field theory coherent states (gcs). i: General properties, Class. Quant. Grav. 18, 2025 (2001).

[52] H Sahlmann, T Thiemann, O Winkler, Co-herent states for canonical quantum general relativity and the infinite tensor product ex-tension, Nucl. Phys. B 606, 401 (2001).

[53] T Thiemann, O Winkler, Gauge field theory coherent states (GCS) 2. Peakedness proper-ties, Class. Quant. Grav. 18, 2561 (2001).

[54] T Thiemann, O Winkler, Gauge field theory coherent states (GCS) 3. Ehrenfest theorems, Class. Quant. Grav. 18, 4629 (2001).

[55] T Thiemann, O Winkler, Gauge field theory coherent states (GCS) 4. Infinite tensor prod-uct and thermodynamical limit, Class. Quant. Grav. 18, 4997 (2001).

[56] T Thiemann, Complexifier coherent states for quantum general relativity, Class. Quant. Grav. 23, 2063 (2006).

[57] B Bahr, T Thiemann, Gauge-invariant co-herent states for Loop Quantum Gravity. I. Abelian gauge groups, Class. Quant. Grav. 26, 045011 (2009).

[58] B Bahr, T Thiemann, Gauge-invariant co-herent states for loop quantum gravity. II. Non-Abelian gauge groups, Class. Quant. Grav. 26, 045012 (2009).

[59] C Flori, T Thiemann, Semiclassical analysis of the Loop Quantum Gravity volume opera-tor. I. Flux Coherent States, arXiv:0812.1537 (2008).

[60] E Bianchi, E Magliaro, C Perini, Coher-ent spin-networks, Phys. Rev. D 82, 024012 (2010).

[61] F Conrady, L Freidel, Quantum geometry from phase space reduction, J. Math. Phys. 50, 123510 (2009).

[62] E Buffenoir, P Roche, Harmonic analysis on the quantum Lorentz group, Commun. Math. Phys. 207, 499 (1999).

[63] W Ruhl, The Lorentz group and harmonic analysis, W. A. Benjamin Inc., New York (1970).

[64] I M Gelfand, Generalized Functions, Aca-demic Press, New York (1966), Vol. 5.

[65] I M Gelfand, R A Minlos, Z Ya Shapiro, Representations of the rotation and Lorentz groups and their applications, Pergamon Press, Oxford (1963).

[66] J W Barrett, L Crane, Relativistic spin net-works and quantum gravity, J. Math. Phys. 39, 3296 (1998).

[67] Y Ding, C Rovelli, The volume operator in covariant quantum gravity, Class. Quant. Grav. 27, 165003 (2010).

[68] Y Ding, M Han, C Rovelli, Generalized Spin-foams, Phys. Rev. D 83, 124020 (2011).

[69] S Alexandrov, The new vertices and canonical quantization, Phys. Rev. D 82, 024024 (2010).

[70] C Rovelli, S Speziale, Lorentz covariance of loop quantum gravity, Phys. Rev. D 83, 104029 (2011).

[71] W M Wieland, Twistorial phase space for complex Ashtekar variables, Class. Quant. Grav. 29, 045007 (2012)..

[72] M Dupuis, L Freidel, E R Livine, S Speziale, Holomorphic Lorentzian simplicity constraints, arXiv:1107.5274 (2011).

[73] E R Livine, S Speziale, J Tambornino, Twistor Networks and Covariant Twisted Ge-ometries, Phys. Rev. D 85, 064002 (2012).

[74] J C Baez, Spin foam models, Class. Quant. Grav. 15, 1827 (1998).

[75] M Bojowald, A Perez, Spin foam quantiza-tion and anomalies, Gen. Rel. Grav. 42, 877 (2010).

[76] B Bahr, F Hellmann, W Kaminski, M Kisielowski, J Lewandowski, Operator spin foam models, Class. Quant. Grav. 28, 105003 (2011).

[77] W Kaminski, M Kisielowski, J Lewandowski, The EPRL intertwiners and corrected parti-tion function, Class. Quant. Grav. 27, 165020 (2010).

[78] E Alesci, E Bianchi, E Magliaro, C Perini, Asymptotics of LQG fusion coefficients, Class. Quant. Grav. 27, 095016 (2010).

[79] E R. Livine, S Speziale, Consistently solv-ing the simplicity constraints for spinfoam quantum gravity, Europhys. Lett. 81, 50004 (2008).

[80] J W Barrett, R J Dowdall, W J Fairbairn, H Gomes, F Hellmann, Asymptotic analysis of the EPRL four-simplex amplitude, J. Math. Phys. 50, 112504 (2009).

[81] S Alexandrov, Simplicity and closure con-straints in spin foam models of gravity, Phys. Rev. D 78, 044033 (2008).

[82] S Alexandrov, Spin foam model from canonical quantization, Phys. Rev. D 77, 024009 (2008).

[83] V Bonzom, Spin foam models for quantum gravity from lattice path integrals, Phys. Rev. D 80, 064028 (2009).

[84] V Bonzom, From lattice BF gauge theory to area-angle Regge calculus, Class. Quant. Grav. 26, 155020 (2009).

[85] V Bonzom, E R Livine, A Lagrangian ap-proach to the Barrett-Crane spin foam model, Phys. Rev. D 79, 064034 (2009).

[86] M Han, T Thiemann, Commuting simplic-ity and closure constraints for 4D spin foam models, arXiv:1010.5444 (2010).

[87] A Baratin, C Flori, T Thiemann, The Holst spin foam model via cubulations, arXiv:0812.4055 (2008).

[88] M Dupuis, E R Livine, Revisiting the sim-plicity constraints and coherent intertwiners, Class. Quant. Grav. 28, 085001 (2011).

[89] L Freidel, E R Livine, U(N) Coherent States for Loop Quantum Gravity, J. Math. Phys. 52, 052502 (2011).

[90] L Freidel, E R Livine, The fine structure of SU(2) intertwiners from U(N) representa-tions, J. Math. Phys. 51, 082502 (2010).

[91] E F Borja, L Freidel, I Garay, E R Livine, U(N) tools for loop quantum gravity: The re-turn of the spinor, Class. Quant. Grav. 28, 055005 (2011).

[92] E R Livine, J Tambornino, Spinor repre-sentation for loop quantum gravity, J. Math. Phys. 53, 012503 (2012).

[93] B Dittrich, J P Ryan, Simplicity in simplicial phase space, Phys. Rev. D 82, 064026 (2010).

[94] J Engle, R Pereira, Regularization and finite-ness of the Lorentzian LQG vertices, Phys. Rev. D 79, 084034 (2009).

[95] L Liu, M Montesinos, A Perez, A topological limit of gravity admitting an SU(2) connec-tion formulation, Phys. Rev. D 81, 064033 (2010).

[96] Y Ding, C Rovelli, Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory, Class. Quant. Grav. 27, 205003 (2010).

[97] W Kaminski, J Lewandowski, T Pawlowski, Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC, Class. Quant. Grav. 26, 245016 (2009).

[98] W Kaminski, M Kisielowski, J Lewandowski, Spin-foams for all loop quantum gravity, Class. Quant. Grav. 27, 095006 (2010).

[99] L Freidel, E R Livine, Spin networks for non-compact groups, J. Math. Phys. 44, 1322 (2003).

[100] S Alexandrov, E R Livine, SU(2) loop quantum gravity seen from covariant theory, Phys. Rev. D 67, 044009 (2003).

[101] E R Livine, Projected spin networks for Lorentz connection: Linking spin foams and loop gravity, Class. Quant. Grav. 19, 5525 (2002).

[102] S Alexandrov, E Buffenoir, P Roche, Pleban-ski theory and covariant canonical formulation, Class. Quant. Grav. 24, 2809 (2007).

[103] S Alexandrov, Reality conditions for Ashtekar gravity from Lorentz- covariant for-mulation, Class. Quant. Grav. 23, 1837 (2006).

[104] S Alexandrov, Hilbert space structure of covariant loop quantum gravity, Phys. Rev. D 66, 024028 (2002).

[105] S Alexandrov, Choice of connection in loop quantum gravity, Phys. Rev. D 65, 024011 (2002).

[106] S Alexandrov, SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter, Class. Quant. Grav. 17, 4255 (2000).

[107] S Alexandrov, I Grigentch, D Vassilevich, SU(2)-invariant reduction of the 3+1 dimensional Ashtekar's gravity, Class. Quant. Grav. 15, 573 (1998).

[108] M Dupuis, E R Livine, Lifting SU(2) spin networks to projected spin networks, Phys. Rev. D 82, 064044 (2010).

[109] B Bahr, On knottings in the physical Hilbert space of LQG as given by the EPRL model, Class. Quant. Grav. 28, 045002 (2011).

[110] E Buffenoir, M Henneaux, K Noui, Ph Roche, Hamiltonian analysis of Plebanski theory, Class. Quant. Grav. 21, 5203 (2004).

[111] J Engle, M Han, T Thiemann, Canonical path integral measures for Holst and Pleban-ski gravity. I. Reduced Phase Space Deriva-tion, Class. Quant. Grav. 27, 245014 (2010).

[112] M Han, Canonical path-integral measures for Holst and Plebanski gravity. II. Gauge in-variance and physical inner product, Class. Quant. Grav. 27, 245015 (2010).

[113] E Bianchi, D Regoli, C Rovelli, Face am-plitude of spinfoam quantum gravity, Class. Quant. Grav. 27, 185009 (2010).

[114] K Noui, A Perez, Three dimensional loop quantum gravity: Physical scalar product and spin foam models, Class. Quant. Grav. 22, 1739 (2005).

[115] V Bonzom, L Freidel, The Hamiltonian con-straint in 3d Riemannian loop quantum grav-ity, Class. Quant. Grav. 28, 195006 (2011).

[116] V Bonzom, A taste of Hamiltonian constraint in spin foam models, arXiv:1101.1615 (2011).

[117] B Dittrich, S Speziale, Area-angle variables for general relativity, New J. Phys. 10, 083006 (2008).

[118] E Alesci, K Noui, F Sardelli, Spin-foam mod-els and the physical scalar product, Phys. Rev. D 78, 104009 (2008).

[119] E R Livine, D Oriti, J P Ryan, Effective Hamiltonian constraint from group field the-ory, Class. Quant. Grav. 28, 245010 (2011).

[120] M Han, T Thiemann, On the relation be-tween operator constraint, master constraint, reduced phase space, and path integral quan-tisation, Class. Quant. Grav. 27, 225019 (2010).

[121] M Han, T Thiemann, On the relation between rigging inner product and master constraint direct integral decomposition, J. Math. Phys. 51, 092501 (2010).

[122] M Han, A path-integral for the master con-straint of loop quantum gravity, Class. Quant. Grav. 27, 215009 (2010).

[123] B Dittrich, P A Hohn, From covariant to canonical formulations of discrete gravity, Class. Quant. Grav. 27, 155001 (2010).

[124] F Conrady, J Hnybida, Unitary irreducible representations of SL(2,C) in discrete and continuous SU(1,1) bases, J. Math. Phys. 52, 012501 (2011).

[125] F Conrady, Spin foams with timelike surfaces, Class. Quant. Grav. 27, 155014 (2010).

[126] F Conrady, J Hnybida, A spin foam model for general Lorentzian 4-geometries, Class. Quant. Grav. 27, 185011 (2010).

[127] A Perez, C Rovelli, 3+1 spinfoam model of quantum gravity with spacelike and timelike components, Phys. Rev. D 64, 064002 (2001).

[128] D Oriti, H Pfeiffer, A spin foam model for pure gauge theory coupled to quantum grav-ity, Phys. Rev. D 66, 124010 (2002).

[129] M Han, C Rovelli, Spinfoam fermions: PCT symmetry, Dirac determinant, and correla-tion functions, arXiv:1101.3264 (2011).

[130] E Bianchi et al., Spinfoam fermions, arXiv:1012.4719 (2010).

[131] S Alexander, A Marciano, R A Tacchi, To-wards a Spin-foam unification of gravity, Yang-Mills interactions and matter fields, arXiv:1105.3480 (2011).

[132] J W Barrett, L Crane, A lorentzian signature model for quantum general relativity, Class. Quant. Grav. 17, 3101 (2000).

[133] K Noui, P Roche, Cosmological deforma-tion of Lorentzian spin foam models, Class. Quant. Grav. 20, 3175 (2003).

[134] Y Ding, M Han, On the asymptotics of quantum group spinfoam model, arXiv:1103.1597 (2011).

[135] M Han, 4-dimensional spin-foam model with quantum Lorentz group, J. Math. Phys. 52, 072501 (2011).

[136] W J Fairbairn, C Meusburger, Quantum de-formation of two four-dimensional spin foam models, J. Math. Phys. 53, 022501 (2012).

[137] M Han, Cosmological constant in LQG vertex amplitude, arXiv:1105.2212 (2011).

[138] E Bianchi, T Krajewski, C Rovelli, F Vi-dotto, Cosmological constant in spinfoam cosmology, Phys. Rev. D 83, 104015 (2011).

[139] F Vidotto, Spinfoam Cosmology: quantum cosmology from the full theory, arXiv:1011.4705 (2010).

[140] A Henderson, C Rovelli, F Vidotto, E Wilson-Ewing, Local spinfoam expansion in loop quantum cosmology, Class. Quant. Grav. 28, 025003 (2011).

[141] E Bianchi, C Rovelli, F Vidotto, Towards spinfoam cosmology, Phys. Rev. D 82, 084035 (2010).

[142] C Rovelli, F Vidotto, On the spinfoam expan-sion in cosmology, Class. Quant. Grav. 27, 145005 (2010).

[143] C Rovelli, F Vidotto, Stepping out of Homo-geneity in Loop Quantum Cosmology, Class. Quant. Grav. 25, 225024 (2008).

[144] M Bojowald, Loop quantum cosmology, Liv. Rev. Rel. 8, 11 (2005).

[145] A Ashtekar, M Campiglia, A Henderson, Path integrals and the WKB approximation in loop quantum cosmology, Phys. Rev. D 82, 124043 (2010).

[146] A Ashtekar, M Campiglia, A Henderson, Casting loop quantum cosmology in the spin foam paradigm, Class. Quant. Grav. 27, 135020 (2010).

[147] A Ashtekar, Ml Campiglia, A Henderson, Loop quantum cosmology and spin foams, Phys. Lett. B 681, 347 (2009).

[148] M Campiglia, A Henderson, W Nelson, Ver-tex expansion for the Bianchi I model, Phys. Rev. D 82, 064036 (2010).

[149] K Krasnov, Renormalizable non-metric quantum gravity? arXiv:hep-th/0611182 (2006).

[150] K Krasnov, On deformations of Ashtekar's constraint algebra, Phys. Rev. Lett. 100, 081102 (2008).

[151] K Krasnov, Plebanski gravity without the simplicity constraints, Class. Quant. Grav. 26, 055002 (2009).

[152] K Krasnov, Gravity as BF theory plus poten-tial, Int. J. Mod. Phys. A 24, 2776 (2009).

[153] K Krasnov, Metric Lagrangians with two propagating degrees of freedom, Europhys. Lett. 89, 30002 (2010).

[154] S Speziale, Bi-metric theory of gravity from the non-chiral Plebanski action, Phys. Rev. D 82, 064003 (2010).

[155] M P Reisenberger, C Rovelli, Spacetime as a feynman diagram: the connection formula-tion, Class. Quant. Grav. 18, 121 (2001).

[156] M P Reisenberger, C Rovelli, Spin foams as feynman diagrams, In: 2001, a relativistic spacetime odyssey. Eds. I Ciufolini, D Do-minici, L Lusanna, Pag. 431, World Scientific, Singapore (2003).

[157] J Magnen, K Noui, V Rivasseau, M Smerlak, Scaling behaviour of three-dimensional group field theory, Class. Quant. Grav. 26, 185012 (2009).

[158] L Freidel, D Louapre, Non-perturbative sum-mation over 3D discrete topologies, Phys. Rev. D 68, 104004 (2003).

[159] R Gurau, Colored Group Field Theory, Com-mun. Math. Phys. 304, 69 (2011).

[160] R Gurau, The 1/N expansion of colored tensor models, Ann. Henri Poincare 12, 829 (2011).

[161] R Gurau, A generalization of the Virasoro algebra to arbitrary dimensions, Nucl. Phys. B 852, 592 (2011).

[162] V Bonzom, R Gurau, A Riello, V Rivasseau, Critical behavior of colored tensor models in the large N limit, Nucl. Phys. B 853, 174 (2011).

[163] R Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincare 13, 399 (2011).

[164] R Gurau, V Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, Europhys. Lett 95, 50004 (2011).

[165] J P Ryan, Tensor models and embedded Rie-mann surfaces, Phys. Rev. D 85, 024010 (2012).

[166] J W Barrett, R J Dowdall, W J Fairbairn, F Hellmann, R Pereira, Lorentzian spin foam amplitudes: graphical calculus and asymp-totics, Class. Quant. Grav. 27, 165009 (2010).

[167] J W Barrett, R J Dowdall, W J Fairbairn, H Gomes, F Hellmann, A Summary of the asymptotic analysis for the EPRL amplitude, In: AIP Conf. Proc. 1196, Pag. 36, (2009).

[168] J W Barrett, W J Fairbairn, F Hellmann, Quantum gravity asymptotics from the SU(2) 15j symbol, Int. J. Mod. Phys. A 25, 2897 (2010).

[169] J W Barrett et al., Asymptotics of 4d spin foam models, Gen. Relat. Gravit. 43, 2421 (2011).

[170] F Conrady, L Freidel, On the semiclassical limit of 4d spin foam models, Phys. Rev. D 78, 104023 (2008).

[171] F Conrady, L Freidel, Path integral repre-sentation of spin foam models of 4d gravity, Class. Quant. Grav. 25, 245010 (2008).

[172] J W Barrett, Ch M Steele, Asymptotics of relativistic spin networks, Class. Quant. Grav. 20, 1341 (2003).

[173] J W Barrett, R M Williams, The asymp-totics of an amplitude for the 4-simplex, Adv. Theor. Math. Phys. 3, 209 (1999).

[174] M Han, M Zhang, Asymptotics of spinfoam amplitude on simplicial manifold: Euclidean theory, arXiv:1109.0500 (2011).

[175] M Han, M Zhang, Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory, arXiv:1109.0499 (2011).

[176] C Rovelli, Graviton propagator from background-independent quantum gravity, Phys. Rev. Lett. 97, 151301 (2006).

[177] R Oeckl, Affine holomorphic quantization, arXiv:1104.5527 (2011).

[178] R Oeckl, Observables in the general bound-ary formulation, In: Quantum field theory and gravity, Eds. F Finster et al., Pag. 137, Birkh¨auser, Basel, (2012).

[179] R Oeckl, Holomorphic quantization of linear field theory in the general boundary formula-tion, arXiv:1009.5615 (2010).

[180] D Colosi, Robert Oeckl, On unitary evolution in quantum field theory in curved spacetime, Open Nucl. Part. Phys. J. 4, 13 (2011).

[181] D Colosi, Robert Oeckl, States and amplitudes for finite regions in a two-dimensional Euclidean quantum field theory, J. Geom. Phys. 59, 764 (2009).

[182] D Colosi, R Oeckl, Spatially asymptotic S-matrix from general boundary formulation, Phys. Rev. D 78, 025020 (2008).

[183] D Colosi, R Oeckl, S-matrix at spatial infinity, Phys. Lett. B 665, 310 (2008).

[184] R Oeckl, Probabilites in the general boundary formulation, J. Phys. Conf. Ser. 67, 012049 (2007).

[185] E Alesci, C Rovelli, The complete LQG prop-agator. II. Asymptotic behavior of the vertex, Phys. Rev. D 77, 044024 (2008).

[186] E Alesci, C Rovelli, The complete LQG prop-agator. I. Difficulties with the Barrett-Crane vertex, Phys. Rev. D 76, 104012 (2007).

[187] E Bianchi, L Modesto, C Rovelli, S Speziale, Graviton propagator in loop quantum gravity, Class. Quant. Grav. 23, 6989 (2006).

[188] E Alesci, E Bianchi, C Rovelli, LQG propagator: III. The new vertex, Class. Quant. Grav. 26, 215001 (2009).

[189] E Bianchi, E Magliaro, C Perini, LQG propagator from the new spin foams, Nucl. Phys. B 822, 245 (2009).

[190] E Bianchi, A Satz, Semiclassical regime of Regge calculus and spin foams, Nucl. Phys. B 808, 546 (2009).

[191] E Magliaro, C Perini, Comparing LQG with the linearized theory, Int. J. Mod. Phys. A 23, 1200 (2008).

[192] E Magliaro, C Perini, Regge gravity from spinfoams, arXiv:1105.0216 (2011).

[193] D Mamone, C Rovelli, Second-order amplitudes in loop quantum gravity, Class. Quant. Grav. 26, 245013 (2009).

[194] C Rovelli, M Zhang, Euclidean three-point function in loop and perturbative gravity, Class. Quant. Grav. 28, 175010 (2011).