The new spin foam models and quantum gravity

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 spin foam approach to the problem. Spin foam models are definitions of the path integral formulation of quantum general relativity and are expected to be the covariant counterpart of the background independent canonical quantization of general relativity known as loop quantum gravity [1][2][3].
This article concentrates on the definition 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 derivation of both the Riemannian and the Lorentzian models in terms of a notation that exhibits the close 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 where general relativity is formulated as a constrained BF theory (for a review introducing the new models from a bottom-up perspective see [7]; for an extended version of the present review including a wide collection of related work see [8]). For that reason it will be convenient to start this review by introducing the exact spin foam quantization 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 model and emphasize the relationship with the kinematical Hilbert space of loop quantum gravity. In Section VI we give a compendium of important issues (and associated references) that have been left out but which are important for future developpement. Finally, in section VII we present the resent encouraging results on the nature of the semiclassical limit of the new models.

II. SPINFOAM 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 this article is about. The key idea is that the quantum transition amplitudes (computes in the path integral representation) of gravity can be obtained by suitably restricting the histories that are summed over in the spin foam representation of exactly solvable BF theory. We describe the nature of these constraints at the end of this section.
Here one follow the perspective of [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 defined by the action where B is a g valued (d − 2)-form, ω is a connection on a G principal bundle over M. The theory has no local excitations: all solutions of the equations of motion are locally related by gauge transformations. More precisely, the gauge symmetries of the action are the local G gauge transformations where α is a g-valued 0-form, and the 'topological' gauge transformation where d ω denotes the covariant exterior derivative and η is a g-valued 0-form. The first invariance is manifest from the form of the action, while the second is a consequence of the Bianchi identity, d ω F (ω) = 0. The gauge symmetries are so vast that all the solutions to the equations of motion are locally pure gauge. The theory has only global or topological degrees of freedom. For the moment we assume M to be a compact and orientable. The partition function, Z, is formally given by Formally integrating over the B field in (4) we obtain 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 ∆. We also need the notion of the associated dual 2-complex of ∆ denoted by ∆ ⋆ . The dual 2-complex ∆ ⋆ is a combinatorial object defined by a set of vertices v ∈ ∆ ⋆ (dual to d-cells in ∆) edges e ∈ ∆ ⋆ (dual to (d−1)-cells in ∆) and faces f ∈ ∆ ⋆ (dual to (d−2)-cells in ∆). In the case where ∆ is a simplicial decomposition of M the structure of both ∆ and ∆ ⋆ is illustrated in Figures 1, 2, and 3 in two, three, and four dimensions respe1ctively. For simplicity we concentrate in the case when ∆ is a triangulation. The field B is associated with Lie algebra elements B f assigned to faces f ∈ ∆ ⋆ . We can think of it as the integral of the (d−2)-form B on the (d−2)-cell dual to the face f ∈ ∆ ⋆ , namely In other words B f can be interpreted as the 'smearing' of the continuous (d−2)-form B on the (d−2)-cells in ∆. We use the one-to-one correspondence between faces f ∈ ∆ ⋆ and (d−2)-cells in ∆ to label the discretization of the B field B f . The connection ω is discretized by the assignment of group elements g e ∈ G to edges e ∈ ∆ ⋆ . One can think of the group elements g e as the holonomy of ω along e ∈ ∆ ⋆ , namely where the symbol "P exp " denotes the path-order-exponential that reminds us of the relationship of the holonomy with the connection along the path e ∈ ∆ ⋆ . With all this the discretized version of the path integral (4) is δ(g e1 · · · g en ), (8) where U f = g e1 · · · g en denotes the holonomy 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 and test function F (g) the invariance property reads as follows dgF (g) = dgF (g −1 ) = dgF (gh) = dgF (hg) The Peter-Weyl's theorem provides a useful formula or the Dirac delta distribution appearing in (8), namely where ρ are irreducible unitary representations of G. From the previous expression one obtains Integration over the connection can be performed as follows. In a triangulation ∆, the edges e ∈ ∆ ⋆ bound precisely d different faces; therefore, the g e 's in (11) appear in d different traces. The relevant formula is P e inv (ρ 1 , · · · , ρ d ) := dg e ρ 1 (g e ) ⊗ ρ 2 (g e ) ⊗ · · · ⊗ ρ d (g e ).
For compact G it is easy to prove using the invariance (and normalization) of the the integration measure (9) that P e inv = (P e inv ) 2 is the projector onto Inv[ρ 1 ⊗ ρ 2 ⊗ · · · ⊗ ρ d ]. In this way the spin foam amplitudes of SO(4) BF theory reduce to In other words, the BF amplitude associated to a two complex ∆ ⋆ is simply given by the sum over all possible assignments of irreducible representations of G to faces of the number obtained by the natural contraction of the network of projectors P e inv according to the pattern provided defined by the two-complex ∆ ⋆ . 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, using this graphical notation one can easily prove the discretization independence of the BF amplitudes. On the other hand this graphical notation will simplify the presentation of the new spin foam models of quantum gravity that will be considered in the following sections. This useful notation was introduced by Oeckl [10,11] and used in [12] to give a general prove of the discretization independence of the BF partition function and the Turaev-Viro invariants for their definition on general cellular decompositions.
Let us try to present this notation in more detail: The idea is to represent each representation matrix appearing 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 equation (11) imply that there is a wire, labelled by the representation ρ f , winding around each face f ∈ ∆ ⋆ . In addition, there is a cable (integration on the group) associated with each edge e ∈ ∆ ⋆ . As in (13), there is a projector P e inv is the projector into Inv[ρ 1 ⊗ ρ 2 ⊗ · · · ⊗ ρ 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.
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 combinatorics of the dual two complex ∆ ⋆ . From Figure 3 one gets The 10 wires corresponding to the 10 faces f ∈ ∆ ⋆ sharing a vertex v ∈ ∆ ⋆ are connected to the neighbouring vertices through the 5 cables (representing the projectors in (13) and Figure 14) associated to the 5 edges e ∈ ∆ ⋆ sharing the vertex v ∈ ∆ ⋆ . We now present the BF quantum amplitudes in the case G = SU (2) × 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) × SU (2)) = Z BF (SU (2)) 2 . Nevertheless, it is important for us to present this example in explicit form 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, i.e. invariant vectors in Inv[ρ 1 ⊗ · · · ⊗ ρ 4 ]. From the product form of the structure group one has where ρ f = j − f ⊗ j + f , and j ± f and ι ± are half integers labelling left and right representations of SU (2) that defined the irreducible unitary representations of G = SU (2) × SU (2), and we have used the expression of the right and left SU (2) projectors in a basis of intertwiners, namely where the four-leg objects on the right hand side denote the invariant vectors spanning a basis of Inv[j 1 ⊗ · · · ⊗ j 4 ], and ι is a half integer labelling those elements. Accordingly, when replacing the previous expression in (15) one gets and equivalently from which we finally obtain the spin foam representation of the SU (2) × SU (2) partition function as a product of two SU (2) amplitudes, namely Extra remarks on four dimensional BF theory The state sum (11) is generically divergent (due to the gauge freedom analogous to (3)). A regularized version defined in terms of SU q (2) × 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 topologically invariant and the spin foam path integral is discretization independent.
As in the three dimensional case BF theory in any dimensions can be coupled to topological defects [15]. In the four dimensional case defects are string-like [16] and can carry extra degrees of freedom such as topological Yang-Mills fields [17]. The possibility that quantum gravity could be defined directly form these simple kind of topological theories has also been considered outside spin foams [18] (for which the UV problem described in the introduction is absent) is attractive and should, in my view, be considered further.
It is also possible to introduce one dimensional particles in four dimensional BF theory and gravity as shown in [19].
Two dimensional BF theory has been used as the basic theory in an attempt to define a manifold independent model of QFT in [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 first relationship has been used in the canonical quantization of the Jackiw-Teitelboim model in [23]. The second relationship has been explored in [24] Three dimensional BF theory and the spin foam quantization presented above is intimately related to classical and quantum gravity in three dimensions (for a classic reference see [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] and is often referred to as the Ponzano-Regge model. Divergences in the above formal expression require regularization. Natural regularizations are available and that the model is well defined [27][28][29]. For a detailed study of the divergence structure of the model see [30][31][32]. The quantum deformed version of the above amplitudes lead to the so called Turaev-Viro model [33] which is expected 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 analog four dimensional analog see [34].
The topological character of BF theory can be preserved by the coupling of the theory with topological defects that play the role of point particles. In the spin foam literature this has been considered form the canonical perspective in [35,36] and from the covariant perspective extensively by Freidel and Louapre [37]. These theories have been shown by Freidel and Livine to be dual, in a suitable sense, to certain non-commutative fields theories in three dimensions [38,39].
Concerning coupling BF theory with non topological matter see [40,41] for the case of fermionic matter, and [42] for gauge fields. A more radical perspective for the definition of matter in 3d gravity is taken in [43]. For three dimensional supersymmetric BF theory models see [44,45] Recursion relations for the 6j vertex amplitudes have been investigated in [46,47]. They provide a tool for studying dynamics in spin foams of 3d gravity and might be useful in higher dimensions [48].

A. The coherent states representation
In this section we introduce the coherent state representation 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 [6] that we will address in Section 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 emphasized by Livine and Speziale in [49].

Coherent states
Coherent states associated to the representation theory of a compact group have been studied by Thiemann and collaborators [50,51,[51][52][53][54][55][56][57][58][59] see also [60]. Their importance for the new spin foam models was put forward by Livine and Speziale in [49] where the emphasis is put on coherent states of intertwiners or the so-called quantum tetrahedron (see also [61]). Here we follow the presentation of [6].
In order to built coherent states for Spin(4) we start by introducing them in the case of SU (2). Starting from the representation space H j of dimension d j ≡ 2j + 1 one can write the resolution of the identity in tems of the canonical orthonormal basis |j, m as where −j ≤ m ≤ j. There exists an over complete basis |j, g ∈ H j labelled by g ∈ SU (2) such that The states |j, g ∈ H j are SU (2) coherent states defined by the action of the group on maximum weight states |j, j (themselves coherent), namely where D j mj (g) are the matrix elements of the unitary representations in the |j, m (Wigner matrices). Equation (22) follows from the orthonormality of unitary representation matrix elements, namely d j SU (2) dg |j, g j, g|, = d j where in the last equality we have used the orthonormality of the matrix elements. The decomposition of the identity (22) can be expressed as an integral on the two-sphere of directions S 2 = SU (2)/U (1) 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 (1) ⊂ SU (2). Thus one has where n ∈ S 2 is integrated with the invariant measure of the sphere. The states |j, n form (an over-complete) basis in H j . SU (2) coherent states have the usual semiclassical properties. Indeed if one considers the generators J i of su(2) one has where n i is the corresponding three dimensional unit vector for n ∈ S 2 . The fluctuations ofĴ 2 are also minimal with ∆J 2 = 2 j, where we have restored for clarity. The fluctuations go to zero in the limit → 0 and j → ∞ while j 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 R 3 of length j and of direction n. It will convenient to introduce the following graphical notation for equation (25) Finally, an important property of SU (2) coherent states stemming from the fact that is that |j, n = | 1 2 , n ⊗2j .
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 EPRLand FK modesl studied in Sections III, and IV. In the following subsection we provide an example for Spin(4) BF theory.

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 construction 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 introductions of coherent states one achieved the most difficult part of the work. In order to express the Spin(4) BF amplitude in the coherent state representation one simply inserts a resolution of the identity in the form (25) on each and every wire connecting neighbouring vertices in the expression (18) for the BF amplitudes. The result is where we have explicitly written the n ± ∈ S 2 integration variables only on a single cable. One observes that there is one n ± ∈ S 2 per each wire coming out at an edge e ∈ ∆ ⋆ ; as wires are in one-to-one correspondence with faces f ∈ ∆ ⋆ the integration variables n ± ef ∈ S 2 are labelled by an edge and face subindex. In order to get an expression 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 previous equation. One gets, where we have used the coherent states property (28), and |n ± is a simplified notation for | 1 2 , n ± . The previous equation can be finally written as where the discrete action and the indices a, b label the five edges of a given vertex. The previous expression is exactly 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 formulation of general relativity.

B. The relationship between gravity and BF theory
The field theory described in the present section has no local degrees of freedom. It represents the simplest example of a topological field theory in four dimensions. The interest of this theory for gravity model stems from the fact that an action for the gravitational degrees of freedom (basically equivalent to general relativity in the first order formulation) can be obtained by supplementing a 4d BF theory action with internal gauge group SL(2, C) (Lorentzian) or Spin(4) (Riemannian) with the following set of quadratic constraints on the B-field where e ≡ σ 2 (1/4!)ǫ IJKL B IJ µν B KL ρσ ǫ µνρσ where σ 2 = ±1 according to we are in the Riemannian of Lorentzian case. More generally, a one parameter family of gravity actions can be obtained obtained from the imposition of the previous constraints on the following modified BF action where γ is the Immirzi parameter. The strategy behind the definition 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 + 1 γ B conjugate to ω. In order to impose the Plebanski constraints above it will be convenient to express the B field in terms of the momenta J, namely The imposition of the constraints (67) on the BF path integral on a fixed discretization can be done in two different ways: by directly restricting the spin foam configurations (this is the EPRL approach described in the following section), or by restricting the semiclassical values of the B field 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 Section III A. In Section III B we present and discuss the linear simplicity constraints -classically equivalent to the Plebanski constraints-and discuss their implementation in the quantum theory. In Section III C we introduce the EPRL model of Riemannian gravity. In Section III D we prove the validity of the quadratic Plebanski constraints-reducing BF theory to general relativity-directly in the spin foam representation. In Section III E we present the coherent state representation of the Riemannian EPRL model. In Section III F we describe the Lorentzian model. The material of this section will also allow us to describe the construction of the closely related (although derived from a different logic) Riemannian FK constructed in [6]. The idea that linear simplicity constraints are more convenient for dealing with the constraints that reduce BF theory to gravity was pointed out by Freidel and Krasnov in this last reference.
A. Representation theory of Spin(4) and SL(2, C) and the canonical basis In this section we present the representation theory of the groups Spin(4) and SL(2, C) that is neccesarry for the definition of the new spin foam models for Riemannian and Lorentzian gravity respectively. To emphasize the highly symmetric structure of the two we present them in a unified notation where a parameter σ = 1 for the Riemannian sector and σ = i for the Lorentzian one. The simple relationship between the two might be a hint of a possible relationship between model amplitudes in a spirit similar to the interesting link between Euclidean and Lorentzian QFT provided by Wick rotations 1 . Unitary irreducible representations H p,k of Spin(4) and SL(2, C) are labelled by two parameters p and k. In the case of Spin(4) = SU (2) × SU (2) the unitary irreducible representations are finite dimensional and the labels p and k can be expressed in terms of the half integers labelling the right and left SU (2) unitary representations j ± as follows In the SL(2, C) case the unitary irreducible representations are infinite dimensional and one has The two Casimirs are where L i are the generators of an arbitrary rotation subgroup and K i are the generators of the corresponding boosts. The Casimirs act on |p, k ∈ H p,k as follows For detail on the representation theory of SL(2, C) see [63][64][65]. The definition of the EPRL model requires the introduction of an (arbitrary) subgroup SU (2) ⊂ Spin(4) or SU (2) ⊂ 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 [8] for details). In the quantum theory, the representation theory of this SU (2) subgroup will be hence 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 for the Riemannian sector, and for the Lorentzian sector. As the unitary irreducible representations of the subgroup SU (2) ∈ Spin(4) and SU (2) ∈ SL(2, C) are essential in understanding 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(4) and sl(2, C) can be characterized in terms of the generators of a rotation subgroup L i and the remaining boost generators K i as follows where K ± = K 1 ± iK 2 and L ± = L 1 ±iL 2 respectively. The action of the previous generators in the basis |p, k; j, m can be shown to be L 3 |p, k; j, m = m|p, k; j, m , where The previous equations will be important in what follows: they will allow for the characterisation 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 definition of the EPRL model.

B. The linear simplicity constraints
As first shown in [6], the quadratic Plebanski simplicity constraints-and more precisely in their dual version presented below (67)-are equivalent in the discrete setting to the linear constraint on each face of a given tetrahedron where the label f makes reference to a face f ∈ ∆ ⋆ , and where (very importantly) the subgroup SU (2) ⊂ Spin (4) or SL(2, C) that is necessary for the definition of the above constraints is chosen arbitrarily at each tetrahedron, equivalent on each edge e ∈ ∆ ⋆ . 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 defined by imposing the previous constraints as operator equations on the Hilbert spaces defined by the unitary irreducible representations of the internal gauge group that take part in the state-sum of BF theory. We will show in Section III D 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) of previous section we can easily compute the commutator of the previous tetrahedron constraints and conclude that in fact it does not close, namely The previous commutation relations imply that the constraint algebra is not closed and cannot therefore 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: 2. The second possibility is to work in the sector where L i f = 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 representations appearing in the expression of the BF partition function so that at each tetrahedron the linear constraints (45) 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 tetrahedra is compatible with the state-space of the analogous 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 i f do not form a closed (first class) algebra in the generic case one needs to devise a weaker sense in which they are to be imposed. One possibility is to consider the Gupta-Bleuler criterion consisting in selecting a suitable class of states for which the matrix elements on D i f vanish. One notices from (43) that if we chose the subspace H j ⊂ H p,k one has The matrix elements of the linear constraints vanish in this subclass if one can chose There are two cases: 1. Case γ < 1: Following [67], in this case one restricts the representations to which amounts to choosing the maximum weight component j = p − 1 in the expansion (41). In the Riemannian case the above choice translates into j ± = (1 ± γ)j/2 for the SU(2) right and lect representations. Notice that the solutions to the simplicity constraints in the Riemannian and Lorentzian sectors look very different for γ < 1.
Simple algebra shows that condition (47) is met. There are indeed other solutions [68] of the Gupta-Bleuler criterion in this case.
2. Case γ > 1: In this case [69] one restricts the representations to which amounts to choosing the minimum weight component j = k in the expansion (41) and . For the Riemannian case we can write the solutions in terms of j ± = (γ ± 1) j 2 + γ−1 2 . Notice that for γ > 1 there is complete symmetry between the solutions of the Riemannian and Lorentzian sectors. In my opinion, this symmetry deserves further investigation as it might be an indication of a deeper connexion between the Riemannian and Lorentzian models (again such relationship is a fact in 3d gravity [62]).
Another criterion for weak imposition can be developed by studying the spectrum of the Master constraint M f = D f · D f . Strong imposition of the constraints D i f 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 [70] to express the master constraint in a in a manifestly invariant way. In order to get a gauge invariant constraint one starts from the master constraint and uses the D i f = 0 classically to write it in terms of Casimirs, namely where C 1 and C 2 are the Casimirs given in equation (39). The minimum eigenvalue condition is The minimum eigenvalue is m min = 2 γj(γ 2 − 1) for the Riemannian case and m min = γ for the Lorentzian case. The master constraint criterion works better in the Lorentzian case as pointed out in [70]. More recently, it has been shown that the constraint solution p = γj and k = j also follows naturally from a spinor formulation of the simplicity constraints [71][72][73]. The above criterion is used in the definition of the EPRL model. It is important to point out that the Riemannian case imposes strong restrictions on the allowed values of the Immirzi parameter if one wants the spin j ∈ N/2 to be arbitrary (in order to have all possible boundary states allowed in LQG). In this case the only possibilities are γ = N or γ = 1. This restriction is not natural from the viewpoint of LQG. Its relevance if any remains misterious at this stage.
Summarising, in the Lorentzian (Riemannian) EPRL model one restricts the SL(2, C) (Spin(4)) representations of BF theory to those satisfying for j ∈ N/2. From now on we denote the subset of admissible representation The admissible quantum states Ψ are elements of the subspace H j ⊂ H γj,j (i.e., minimum weight states) satisfy the constraints (45) in the following emiclassical sense: where the symbol O sc (order semiclassical) denotes a quantity that vanishes in limit → 0, j → ∞ with j =constant.
In the Riemannian case the previous equation can be written as which in turn has a simple graphical representation in terms of spin-network grasping operators, namely 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 imply the vanishing of the quadratic Plebanski constraints (67) and their fluctuations, computed in the path integral sense, in the appropriate large spin semiclassical limit.

C. Presentation of the Riemannian EPRL amplitude
Here we complete the definition of the EPRL models by imposing the linear constraints on the BF amplitudes constructed in Section (II). We will also show that the path-integral expectation value of the Plebanski constraints (67), as well as their fluctuations, vanish in a suitable semiclassical sense. This shows that the EPRL model can be considered as a lattice definition of the a quantum gravity theory.
We start with the Riemannian model for which a straightforward graphical notation is available. The first step is the translation of equation (40)-for p and k satisfying the simplicity constraints-in terms of the graphical notation introduced in Section (II). Concretely, for γ < 1 one has j ± = (1 ± γ)j/2 ∈ K γ becomes For γ > 1 we have The implementation of the linear constraints of Section (III B) consist in restricting the representations ρ f of Spin(4) appearing in the state sum amplitudes of BF theory as written in Equation (18) to the subclass ρ f ∈ K γ ⊂ Irrep(Spin(4)), defined above, while projecting to the highest weight term in (56) for γ < 1. For γ > 1 one must take the minimum weight term in (57) . The action of this projection will be denoted Y j : Explicitly, one takes the expression of the BF partition function (13) and modifies it by replacing the projector P e inv (ρ 1 , · · · , ρ 4 ) with ρ 1 , · · · ρ 4 ∈ K γ by a new object with j 1 , · · · j 4 ∈ N/2 implementing the linear constraints described in the previous section. Graphically the modification of BF theory that produces the EPRL model corresponds to the replacement P e inv (ρ 1 · · · ρ 4 ) = P e eprl (j 1 · · · j 4 ) = The previous expression is defines the EPRL model amplitude.

The spin foam 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 much more effort than the derivation of the spin foam representation for Spin(4) BF theory as we went from equation (18) to (20) in Section II. The first step is given in the following equation which follows basically from the invariance of the Haar measure (9) (in the last line we have used (17)). More presicely, the integration of the subgroup SU (2) ∈ Spin(4), represented by the green box on the right, can be absorbed by suitable redefinition 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 where the vertex amplitude (graphically represented) 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 notational simplicity. We can write the previous spin foam amplitude in another form by integrating out all the projectors (boxes) explicitly. Using, (17) we get = ι+ι−ι ι+ῑ+ ι−ῑ− ιῑ (64) thus replacing this in (61) we get where the coefficients f ι ι + ι − are the so-called fusion coefficients which appear in their graphical form already in (64), more explicitly The previous is the form of the EPRL model as derived in [5].

D. Proof of validity of the Plebanski constraints
In this section we prove that the quadratic constraints are satisfied in the sense that their path integral expectation value and fluctuation vanish in the appropriate semiclassical limit.

The quadratic Plebanski constraints
The quadratic Plebanski constraints are The constraints in this form are more suitable for the translation into the discrete formulation. More precisely, according to (6), the smooth fields B IJ µν is now associated with the discrete quantities B IJ triangles , or equivalently B IJ f as, we recall, faces f ∈ ∆ ⋆ 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 discussion) vertices v ∈ ∆ ⋆ . With all this the constraints (67) are discretized as follows: Triangle (or diagonal) constraints: for all f ∈ v, i.e., for each and every face of the 10 possible faces touching the vertex v.
Tetrahedron constraints: for all f, f ′ ∈ v such that they are dual to triangls sharing a one-simplex, i.e., belonging to the same tetrahedron out of the five possible ones.

4-simplex constraints:
for any pair of faces f,f ∈ v that are dual to triangles 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 interpreted as a definition of the four volume e v of the four-simplex. The constraint requires that such definition be consistent, i.e., the true condition is for all five different possible pairs of f andf in a four simplex, and where we assume the pairs f -f are ordered in agreement with the orientation of the complex ∆ ⋆ .

The path integral expectation value of the Plebanski constraints
Here we prove that the Plebanski constraint are satisfied by the EPRL amplitudes in the path integral expectation value sense.

The triangle constraints:
We start from the simplest case: the triangle (or diagonal) constraints (68). We choose a face f ∈ v (dual to a triangle) in the cable-wire-diagram of Equation (61). This amounts to choosing a pair of wires (right and left representations) connecting two nodes in the vertex cable wire diagram. The two nodes are dual to the two tetrahedra-in the four simplex dual to the vertex-sharing the chosen triangle. From equation (36) can show that where J ± f denotes the self-dual and anti-self-dual parts of Π IJ f . The path integral expectation value of the triangle constraint is then where the double graspings on the anti-self-dual (blue) wire and the self-dual (red) wire represent the action of the Casimirs J − f · J − f and J + f · J + f on the cable-wire diagram of the corresponding vertex. Direct evaluation shows that the previous diagram is proportional to 2 j f which vanishes in the semiclassical limit → 0, j → ∞ with j =constant. We use the notation already adopted in (54) and call such quantity O sc . This concludes the proof that the triangle Plebanski constraints are satisfied in the semiclassical sense.

The tetrahedra constraints:
The proof of the validity of the tetrahedra constraints (69). In this case we also have 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 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)), and in the first and second terms on the second line we have used Equation (55) to move the graspings on self-dual wires to the corresponding anti-self-dual wires. Equation (74) follows immediately from the pervious one; the argument works in the same way for any other pair of faces. Notice that the first equality in Equation (75) implies that we can view the Plebanski constraint as applied in the frame of the tetrahedron as well as in a Lorentz invariant framework (the double grasping defines an intertwiner operator commuting with the projection P e inv represented 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 constraints in their form (71). As we show below, this last set of constraints follow from the Spin(4) gauge invariance of the EPRL node (i.e., the validity of the Gauss law) plus the validity of the tetrahedra constraints (69). Gauge invariance of the node takes the following form in graphical notation where the above equation represents the gauge invariance 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 chose an arbitrary pair f andf (where, recall,f is one of the three possible faces whose dual triangle only shares a point with the corresponding 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 pair f -f and f -f , wheref is another of the three admissible faces opposite to f . The full result follows from applying the same procedure iteratively to reach any admissible pair. It will be obvious from the treatment given below that this is possible. Thus, for a given pair of admissible faces we have where going from the first line to the second and third lines we have simply used (76) on the bottom 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 O 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.
Let us study in a bit more detail the object P e eprl (j 1 , · · · , j 4 ). We see that it is made of two ingredients. The first one is the projection to the maximum weight subspace H j for γ > 1 in the decomposition of H j + ,j − for j ± = (1 ± γ)j/2 (j ± = (γ ± 1)j/2 for γ > 1) in terms of irreducible representations of an arbitrarily chosen SU (2) subgroup of Spin(4). The second ingredient is to eliminate the dependence on the choice of subgroup by group averaging with respect to the full gauge group Spin(4). This is diagramaticaly represented in (60). However P e eprl (j 1 , · · · , j 4 ) is not a projector, namely P e eprl (j 1 , · · · , j 4 ) 2 = P e eprl (j 1 , · · · , j 4 ).
Technically this follows from (59) and the fact that i.e., the projection imposing the linear constraints (defined on the frame of a tetrahedrom or edge) and the Spin(4) (or Lorentz) group averaging-rendering the result gauge invariant-do not commute. The fact the P e eprl (j 1 , · · · , j 4 ) is not a projection operator has important consequences in the mathematical structure of the model: 1. From (61) one can immediately obtain the following expression for the EPRL amplitude This expression has the formal structure of expression (13) for BF theory. The formal similarity however is broken by the fact that P e eprl (j 1 , · · · , j 4 ) is not a projection operator. From the formal perspective is the possibility that the amplitudes be defined in term of a network of projectors (as in BF theory) might provide an interesting structure that might be of relevance in the definition of a discretization independent model. On the contrary, the failure of P e eprl (j 1 , · · · , j 4 ) to be a projector may lead, in my opinion, to difficulties in the limit where the complex ∆ is refined: the increasing of the number of edges might produce either trivial or divergent amplitudes 2 .
2. Another difficulty associated with P e eprl (j 1 , · · · , j 4 ) 2 = P e eprl (j 1 , · · · , j 4 ) is the failure of the amplitudes of the EPRL model, as defined here, to be consistent with the abstract notion of spin foams as defined in [74]. This is a point of crucial importance under current discussion in the community. The point is that the cellular decomposition ∆ has no physical meaning and is to be interpreted as a subsidiary regulating structure to be removed when computing physical quantities. Spin foams configurationa can fit in different ways on a given ∆, yet any of these different embeddings represent the same physical process (like the same gravitational field in different coordinates). Consistency requires the spin foam amplitudes to be independent of the embedding, i.e., well defined on the equivalence classes of spin foams as defined by Baez in [74] (the importance of these consistency requirements was emphasized in [75]). The amplitude (80) fails this requirement due to P e eprl (j 1 , · · · , j 4 ) 2 = P e eprl (j 1 , · · · , j 4 ).

The Warsaw proposal
If one sees the above as difficulties then there is a simple solution, at least in the Riemannian case. As proposed in [76,77] one can obtain a consistent modification of the EPRL model by replacing P e eprl in (80) by a genuine projector P e w , graphically It is easy to check that by construction (P e w (j 1 · · · j 4 )) 2 = P e w (j 1 · · · j 4 ). (82) The variant of the EPRL model proposed in [76,77] takes then the form Thus in the modified EPRL model edges e ∈ ∆ ⋆ are assigned pairs of intertwiner quantum numbers ι e vs and ι e vt and an edge amplitude given by the matrix elements g e ι e vs ,ι e v t (where v s and v t stand for the source and target vertices of the given oriented edge). The fact that edges are not assigned a single quantum number is not really significative; one could go to a basis of normalized eigenstates of P e w and rewrite the modified 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 interpretation of its elements is not clear at this stage, it seems simpler to represent the amplitudes of the modified model in the above form.
The advantages of the modified model are important,; however, a generalization of the above modification of the EPRL model in the Lorentzian case is still lacking. Notice that this modification does not interfere with the results on the semiclassical limit (to leading order) as reviewed in Section VII. The reason is that the matrix elements g e αβ → δ αβ in that limit [78].

E. The coherent states representation
We have written the amplitude defining 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 [49] and explicitly used to re-derive the EPRL model in [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) where we used in the last line the fact that for γ < 1 the representations j of the subgroup SU (2) ∈ Spin(4) are maximum weight, i.e., j = j + + j − . Doing this at each edge we get where we have explicitly written the n ∈ S 2 integration variables on a single cable. The expression above is very similar to the coherent states representation of Spin(4) BF theory given in Equation (29). In fact one would get the above expression if one would start form the expression (29) and would set n + ef = n − ef = n ef while dropping for example all the sphere integrations corresponding to the n + ef (or equivalently n − ef ). Moreover, by construction the coherent states participating in the previous amplitude satisfy the linear constraints (45) in expectation values, namely Thus the coherent states participating in the above representation of the EPRL amplitudes solve the linear simplicity constraints in the usual semiclassical sense. The same manipulations leading to (89) in Section II lead to a discrete effective action for the EPRL model, namely where the discrete action and the indices a, b label the five edges of a given vertex. 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 relationship with Regge discrete formulation of general relativity.
The case γ > 1 The case γ > 1 is more complicated [80]. The reason is that the step (85) directly leading to the discrete action in the previous case is no longer valid as the representations of the subgroup SU (2) ∈ Spin(4) are now minimum instead of maximum weight. However, the representations j + = j − + j are maximum weight. We can therefore insert coherent states resolution of the identity on the right representations and get: where we are representing the relevant part of the diagram appearing in equation (84). In the last line we have used that j + = j + j − (i.e. maximum weight), and the graphical notation m n ≡ m|n as it follows from our previous conventions. With all this, one gets where the discrete action with S v j ± ,n,m [g ± ] = 1≤a<b≤5 j ab (1 + γ) log( m ab |g + ab |m ba ) + j ab (γ − 1) log( m ab |g − ab |m ba ) + +2j ab (log( n ab |m ab ) + log( m ba |n ba )) .

Some additional remarks
It is important to point out that the commutation relations of basic fields-reflecting the simple algebraic structure of spin(4)-used here is the one induced by the canonical analysis of BF theory presented previously. The presence of constraints generally modifies canonical commutation relations in particular in the presence of second class constraints. For some investigation of the issue in the context of the EPRL and FK models see [69]. In [81] it is pointed out that the presence of secondary constraints in the canonical analysis of Plebanski action should translated in additional constraints in the holonomies of the spin foam models here considered (see also [82]). A possible view is that the simplicity constraints are here imposed for all times and thus secondary constraints should be imposed automatically.
There are alternative derivations of the models presented in the previous sections. In particular one can derive them from a strict Lagrangean approach of Plebanski's action. Such viewpoint is taken in [83][84][85]. The path integral formulation of Plebansky theory using commuting B-fields was studied in [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 have a certain semiclassical input; see below). The spin foam quantization of the Holst formulation of gravity via cubulations was investigated in [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 [93].

F. Presentation of the EPRL Lorentzian model
As briefly discussed in Section III, unitary irreducible representations of SL(2, C) are infinite dimensional and labelled by a positive real number p ∈ R + and a half-integer k ∈ N/2. These representation 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 D p,k jmj ′ m ′ (g) with j ≥ k and j ≥ m ≥ −j (similarly for the primed indices) 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 version of Equation (11) in order to introduce a formal definition of the BF amplitudes, which now would involve integration of the continuous labels p f in addition of sums over discrete quantum numbers such as k, j and m. The Lorentzian version 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 = γj f the Lorentzian EPRL model becomes a state-sum model as its Riemannian relative. Using the following graphical notation the amplitude is where the boxes now represent SL(2, C) integrations with the invariant measure. The previous amplitude is equivalent to the its spin foam representation The vertex amplitude is divergent due to the presence of a redundant integration over SL(2, C), it becomes finite by dropping an arbitrary integration, i.e. removing any of the 5 boxes in the vertex expression [94].

The coherent state representation
It is immediate to obtain the coherent states representation of the Lorentzian models. As in the Riemannian case, one simply inserts resolution of the identities (22) on the intermediate SU (2) (green) wires in (97)

IV. THE FREIDEL-KRASNOV (FK) MODEL
Shortly after the appearance of the paper [4], Freidel and Krasnov [6] introduced a set of new spin foam models for four dimensional gravity using the coherent state basis of the quantum tetrahedron of Livine and Speziale [49]. The idea is to impose the linearized simplicity constraints (45) directly as a semiclassical condition on the coherent state basis. As we have seen above, coherent states are quantum states of the right and left tetrahedra 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 accept from the starting point that, due to their property of not defining set that is closed under commutation relations, the Plebansky are to be imposed semiclassically. One defines new models by restricting the set of coherent states entering in the coherent state representation of Spin(4) BF theory (29) to those that satisfy condition (45) in expectation values. They also emphasize how the model [4] corresponds indeed to the sector γ = ∞ which has been shown to be topological [95].
The case γ < 1 For γ < 1 the vertex amplitude is identical to the EPRL model. This is apparent in the coherent state expression of the EPRL model (88). Thus we have From the previous expression we conclude that the vertex amplitudes of the FK and EPRL model coincide for γ < 1 Notice however that different weights are assigned to edges in the FK model. This is due to the fact that one is restricting the Spin(4) 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 un-settled discussion concerning the measure in the path integral representation.
The case γ > 1 For the case γ > 1 the FK amplitude is given by The study of the coherent state representation of the FK model for γ > 1, and comparison with equation (91) for the EPRL model, clearly shows the difference between the two models in this regime.
where the discrete action where s(γ) = sign(1 − γ) and with the indices a, b labelling the five edges of a given vertex.

V. BOUNDARY DATA FOR THE NEW MODELS AND RELATIONSHIP WITH THE CANONICAL THEORY
So far we have considered cellular complexes with no boundary. Transition amplitudes are expected to be related to the definition of the physical scalar product. In order to define them one needs to consider complexes with boundaries. Boundary states are defined on the boundary of the dual two-complex ∆ ⋆ that we denote ∂∆ ⋆ . The object ∂∆ ⋆ 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 ∈ ∂∆ ⋆ inherit the spins labels (unitary irreducible representations of the subgroup SU (2)) of the boundary faces while nodes (zero-cells) n ∈ ∂∆ ⋆ inherit the intertwiner levels of 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 coloured boundary graph γ with links labelled by spins j ℓ and nodes labelled by interwiners ι n we associate a cylindrical function Ψ γ,{j ℓ },{ιn} ∈ L 2 (SU (2) N ℓ ), where here N ℓ denotes number of links in the graph γ. In this way, the boundary Hilbert space associated with ∂∆ ⋆ is isomorphic (if one used the natural AL measure) with the Hilbert space of LQG truncated to that fixed graph. Moreover, geometric operators such as volume and area defined in the covariant context are shown to coincide with the corresponding operators defined in the canonical formulation [67,96]. Now, if cellular complexes are dual to triangulations then the boundary spin networks can have at most four valent nodes. This limitation can be easily overcome: as in BF theory the EPRL amplitudes can be generalized to arbitrary complexes with boundaries given by graphs with nodes of arbitrary valence. The extension of the model to arbitrary complexes has been first studied in [97,98], it has also been revisited in [68].
Alternatively, one can associate the boundary states with elements of L 2 (Spin(4) N ℓ ) (in the Riemannian models)or carefully define the analog of spin network states as distributions in the Lorentzian case (see [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 kind of spin network states that are a subclass of the so-called projected spin networks introduced in [100,101] in order to define an heuristic quantization of the (non-commutative and very complicated) Dirac algebra of a Lorentz connection formulation of the phase space of gravity [100,[102][103][104][105][106][107]. The fact that these special subclass of projected spin networks appear naturally as boundary states of the new spin foams is shown in [108].
Due to their similarity for γ < 1 the same relationship between boundary data and elements of the kinematical Hilbert space hold for the FK model. However, the such simple relationship does not hold for the model in the case γ > 1.
It is important to mention that the knotting properties of boundary spin network do not seem to play a role in present definitions of transition amplitudes [109].

VI. FURTHER DEVELOPMENTS AND RELATED MODELS
The spin foam amplitudes discussed in the previous sections have been introduced by constraining the BF histories through the simplicity constraints. However, in the path integral formulation, the presence of constraints has the additional effect of modifying 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 [75]). As pointed out before, this question has not been completely settled yet in the spin foam community. The explicit modification of the formal measure in terms of continuous variables for the Plebansky action was presented in [110]. A systematic investigation of the measure in the spin foam context was attempted in [111] and [112]. As pointed out in [75], there are restrictions in the manifold of possibilities coming from the requirement of background independence. The simple BF measure chosen in the presentation of the amplitudes in the previous sections satisfy these requirements. There are other consistent possibilities; see for instance [113] for a modified measure which remains extremely simple and is suggested from the structure of LQG.
An important question is the relationship between the spin foam amplitudes and the canonical operator formulation. The question of wether 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 perspective see [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 moro adapted to the interpretation of spin foams [117]. In four dimensions the question has been investigated in [118] in the context of the new spin foam models. In the context of group field theories this issue is explored in [119]. Finally, spin foams can in principle be obtained directly from the implementation of the Dirac program using path integral methods this avenue has been explored in [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 analyzed also in [123].
By construction all tetrahedra in the FK and EPRL models are embedded in a spacelike hypersurface and hence have only spacelike triangles. It seem 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][125][126]. An earlier attempt to define such models in the context of the Barrett-Crane model can be found in [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 [128]. More recently the coupling of the EPRL model to fermions has been investigated in [129,130]. A novel possibility of unification of the gravitational and gauge fields was recently proposed in [131].
The introduction of a cosmological constant in the construction of four dimensional spin foam models has a long history. Barrett and Crane introduced 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 studied in [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 [134,135] as well as (independently) in [136]. The asymptotics of the vertex amplitude are shown to be consistent with a cosmological constant term in the semiclassical limit in [137].
As we have discussed in the introduction of the new models, Heisenberg uncertainty principle precludes the strong imposition of the Plebanski constraints that reduce BF theory to general relativity. The results on the semiclassical limit of these models seem to indicate that metric gravity should be recovered in the low energy limit. However, its seems likely that the semiclassical limit could be related to certain modifications of Plebanski's formulation of gravity [149][150][151][152][153]. A simple interpretation of the new models in the context of the bi-gravity paradigm proposed in [154] could be of interest.
As already pointed out in [74] spin foams can be interpreted in close analogy to Feynman diagrams. Standard Feynman graphs are generalized to 2-complexes and the labeling of propagators by momenta to the assignment of spins to faces. Finally, momentum conservation at vertices in standard feynmanology is now represented by spin-conservation at edges, ensured by the assignment of the corresponding intertwiners. In spin foam models the non-trivial content of amplitudes is contained in the vertex 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 field theory (GFT) [155,156].
The GFT formulation resolves by definition 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 λ and the convergence of the corresponding perturbative series.
In three dimensions this idea has been studied in more detail. In [157] scaling properties of the modification of the Boulatov group field theory introduced in [158] was studied in detail. In a further modification of the previous model (known as coloured tensor models [159]) new techniques based on a suitable 1/N expansion imply that amplitudes are dominated by spherical topology [160]; moreover, it seem possible that the continuum limit might be critical as in certain matrix models [161][162][163][164][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 on 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 behaviour of coherent state spin foam amplitudes. The notion of large spin can be defined by the rescaling of quantum numbers and Planck constant according to j → λj and → /λ and taking λ >> 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 fixed regulating cellular structure. That is why one usually makes a clear distinction between semiclassical limit and the continuum limit. In the semiclassical analysis presented here one can only hope to make contact with discrete formulations of classical gravity; hence 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 Section II A. Here we follow [80,[166][167][168][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 [170,171]. The study of the large spin asymptotics has a long tradition in the context of quantum gravity dating back to the studied of Ponzano-Regge [26]. More directly related to our discussion here are the early works [172,173]. The key idea is to use asymptotic 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 vertex amplitude for both the Riemannian and Lorentztian models. We follow the notation and terminology of [80] and related papers.

SU(2) 15j-symbol asymptotics
As SU (2) BF theory is quite relevant for the construction of the EPRL-FK models, the study of the large spin asymptotics of the SU (2) vertex amplitude is a key ingredient in the analysis of [80]. The coherent state vertex amplitude is which depends on 10 spins j ab and 20 normals n ab = n ba . The previous amplitude can be expressed as and the indices a, b label the five edges of a given vertex. The previous expression is exactly equal to the form (11) of the BF amplitude. In the case of the EPRL model studied in Sections III the coherent state representation-see equations 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 the the action (106) is complex (see [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 critical points. As the real part of the action is negative definite, the action at critical points is purely imaginary.
Notice that the action (106) depends parametrically on 10 spins j and 20 normals n. These parameters define the so-called boundary data for the four simplex v ∈ ∆ ⋆ . Thus, there is an action principle for every given boundary data. The number of critical points and their properties depend on these boundary data, hence the asymptotics of the vertex amplitude is a function of the boundary data. Different cases are studied in detail in [80], here we present their results in the special case where the boundary data describe a non-degenerate Regge geometry 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(λj, n) ∼ 1 λ 12 N + exp(i a<b λj ab Θ E ab ) + N − exp(−i a<b λj ab Θ E ab ) , where Θ ab the appropriate diahedral angles defined by the four simplex geometry; finally the N ± are constants that do not scale with λ.

The Riemannian EPRL vertex asymptotics
The previous result together with the fact that the EPRL amplitude for γ < 1 is a product of SU (2) amplitudes with the same n in the coherent state representation (88) implies the asymptotic formula for the vertex amplitude to be given by the unbalanced square of the above formula [167], namely One can write the previous expression as where is the Regge like action for λγj ab = A ab the ten triangle areas (according to the LQG area spectrum [1,2]). Remarkably, the above asymptotic formula is also valid for the case γ > 1 [80]. The first term in the vertex asymptotics is in essence the expected one: it is the analog of the 6j symbol asymptotics in three dimensional spin foams. Due to their explicit dependence on the Immirzi parameter, the last two terms are somewhat strange from the point of view of the continuum field theoretical view point. However, this seems to be a peculiarity of the Riemannian theory alone as the results of [166] for the Lorentzian models show. Non geometric configurations are exponentially surpressed

Lorentzian EPRL model
To each solution one can associate a second solution 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 expression where N ± are constants that do not scale. Non geometric configurations are exponentially surpressed In [171] Freidel and Conrady gave a detailed description of the coherent state representation of the various spin foam models described so far. In particular they provided the definition of the effective discrete actions associated to each case which we presented in (101). This provides the basic elements for setting up the asymptotic analysis presented in [170] (the first results on the semiclassical limit of the new spin foam models) which is similar to the studies of the asymptotic of the vertex amplitude reviewed above but more general in the sense that the semiclassical limit of a full spin foam configuration (involving many vertices) is studied. The result is technically more complex as one studies now critical points of the action associated to a coloured complex which in addition of depending on group variables g it depends on the coherent state parameters n. The authors of [170] write Equation (101) in the following way where They show that those solutions of the equations of motion of the effective discrete action that are non geometric (i.e. the contrary of Regge like) are not critical and hence exponentially suppressed in the scaling j f → λj f with λ >> 1. If configurations are geometric (i.e. Regge like) one has two kind of contributions to the amplitude assymptotics: those coming from degenerate and non-degenerate configurations. If one (by hand) restricts to the non-degenerate configurations then one has where n e , n v , and n f denote the number of edges, vertices, and faces in the two complex ∆ ⋆ respectively. There are recent works by M. Han where asymptotics of general simplicial geometry amplitudes are studied in the context of the EPRL model [174,175]. The problem of computing the two point function and higher correlation functions in the context of spin foam has received lots of attention recently. The framework for the definition of the correlation functions in the background independent setting has been generally discussed by Rovelli in [176] and correspods to a special application of a more general proposal investigated by Oeckl [177][178][179][180][181][182][183][184]. It was then applied to the Barrett-Crane model in [185][186][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 of the weakening of the imposition of the Plebanski 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 [189] and it was shown to produce a result in agreement with that of Regge calculus [190,191] in the limit γ → 0.
The fact that, for the new model, the double scaling limit γ → 0 and j → ∞ with γj=constant defines the appropriate regime where the fluctuation behave as in Regge gravity (in the leading order) has been further clarified in [192]. This indicates that the quantum fluctuations in the new models are more general than simply metric fluctuations. The fact the 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 has been implemented only semiclassically (in the large spin limit). At the deep Planckian regime fluctuations are more general than metric. However, it not clear at this stage why this is controlled by the Immirzi parameter.
All the previous calculations involve a complex with a single four-simplex. The first computation involving more than one simplex was performed in [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 [194], the results are in agreement with Regge gravity in the γ → 0 limit.

VIII. ACKNOWLEDGEMENTS
I would like to thank the help of many people in the field that have helped me in various ways. I am grateful to Eugenio Bianchi, Carlo Rovelli and Simone Speziale for the many 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 where done in collaboration with Mercedes Velázquez 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 help and valuable interaction.