Carlo Rovelli

Francesca Vidotto

On the spinfoam expansion in cosmology

We consider the technique introduced in a recent work by Ashtekar, Campiglia and Henderson, which generate a spinfoam-like sum from a Hamiltonian theory. We study the possibility of using it for finding the generalized projector of a constraint on physical states, without first deparametrising the system. We illustrate this technique in the context of a very simple example. We discuss the infinities that appear in the calculation, and argue that they can be appropriately controlled. We apply these ideas to write a spinfoam expansion for the "dipole cosmology".

I. INTRODUCTION

The spinfoam formalism [START_REF] Perez | Spin Foam Models for Quantum Gravity[END_REF] and canonical loop quantum gravity [START_REF] Rovelli | Loop space representation for quantum general relativity[END_REF][START_REF] Rovelli | Quantum Gravity[END_REF] can ideally be viewed as the covariant and canonical versions of a background-independent quantum theory of gravity, a scenario nicely realized in three dimensions [START_REF] Noui | Three dimensional loop quantum gravity: Physical scalar product and spin foam models[END_REF]. In a recent paper [START_REF] Ashtekar | Loop Quan-tum Cosmology and Spin Foams[END_REF], Ashtekar, Campiglia and Henderson find an ingenious way to implement this scenario in the context of Loop Quantum Cosmology, by introducing a technique for translating the dynamics of the quantum theory into a spinfoamlike sum-over-histories dynamics.

Here we apply this technique to the direct calculation of the projector on the kernel of a constraint, and we illustrate it the context of an extremely simple example. The exercise allows us to discuss the appearance of certain infinities. We argue that these are spurious and can be properly dealt with.

We use these results to write a spinfoam formulation for the "dipole" cosmology recently introduced in [START_REF] Rovelli | Stepping out of Homogeneity in Loop Quantum Cosmology[END_REF]. This can be done in a fully relational way; without the need of deparametrising the system using a scalar field.

A. The Ashtekar-Campiglia-Henderson expansion

The idea of the Ashtekar, Campiglia and Henderson (ACH) expansion [START_REF] Ashtekar | Loop Quan-tum Cosmology and Spin Foams[END_REF] can be resumed as follows. Consider a self-adjoint operator C on a Hilbert space H, where a discrete orthonormal basis | x is given. We are interested in the evolution operator e itC . As in the standard Feynman construction of the path integral,we write e itC = e iεC ...e iεC N times [START_REF] Perez | Spin Foam Models for Quantum Gravity[END_REF] where N is an arbitrary positive number and ε ≡ t/N. In this way, the evolution in the interval [0, t] is realized by chopping the interval in N steps and evolving one step at the time. Inserting N -1 completeness this reads

e itC = ∑ x 1 ...x N-1 e iεC | x N x N-1 |e iεC | x N-2 (2)
...

x 2 |e iεC | x 1 x 1 | e iεC
Therefore the matrix elements of the evolution operator can be written in the form

x |e itC | x = ∑ x 1 ...x N-1 P x ,x N-1 ...P x 1 ,x . ( 3 )
where

P x,x = x |e iεC | x . ( 4
)
That is, the matrix elements of the evolution operator can be expressed in terms of a sum over "histories" (x N-1 , ..., x 1 ), with an amplitude A(x N-1 , ..., x 1 ) = P x ,x N-1 ...P x 1 ,x associated at each history. Since the expression (3) is true for any N, we can equally write

x |e itC | x = lim N→∞ ∑ x 1 ...x N-1 P x ,x N-1 ...P x 1 ,x . (5)
Let us write the matrix elements of C as C xx ≡

x |C| x , and the diagonal ones as

C x ≡ C xx ≡ x |C| x .
Then, for N large enough (that is, ε small), we can write that

x |e iεC | x ∼ e iεC x ( 6
) while if x = x x |e iεC | x ∼ iεC xx , ( 7 )
up to terms of higher order in 1/N. The key idea of Ashtekar-Campiglia-Henderson is now to rearrange the sum over histories by collecting together all histories where there is a number M of "jumps". A jump being Confidential: not for distribution. Submitted to IOP Publishing for peer review 11 April 2010 a step of the history where x n+1 = x n . For each history, let N m be the position of the m-th jump. Then we have immediately

x |e itC | x = lim N→∞ N ∑ M=0 ∑ N M ...N 1 ∑ x M-1 ...x 1 iεC x x M-1 ...iεC x 1 x (e iεC x ) N-N M-1 ...(e iεC x ) N 1 (8)
where the x m 's are the M -1 intermediate values that the history takes, each lasting for N m+1 -N m steps. The only subtlety above is to keep truck of the limits of the sums over the N m 's, which are not indicated above. A moment of reflection shows that these are

∑ N M ...N 1 = N-1 ∑ N M =M N M -1 ∑ N M-1 =M-1 . . . N 2 -1 ∑ N 1 =1 (9)
the range of the sum being determined by the fact that

N < N M < N M-1 < ... < N 2 < N 1
where all the N m are different from zero. Finally, the last step of the Ashtekar-Campiglia-Henderson construction is to take the continuum limit of these sums. This can be done by noticing that the N → ∞ limit of the sums is nothing else than the definition of the Riemann integral. Thus for instance the first sum in N M converges to an integral from zero to t, and so on. This gives

N-1 ∑ N M =M ε N M -1 ∑ N M-1 =M-1 ε . . . N 2 -1 ∑ N 1 =1 ε → t 0 dt M t M 0 dt M-1 ... t 2 0 dt 1 . ( 10
)
where t m = εN m . Bringing altogether, we have

x |e itC | x = lim N→∞ N ∑ M=0 ∑ x 1 ...x M-1 C x x M-1 ...C x 1 x µ(x 1 , ..., x M-1 , t) (11)
where

µ(x 1 , ..., x M-1 , t) = i M t 0 dt M ... t 2 0 dt 1 (e iC x ) t-t M ...(e iC x ) t 1 . ( 12
)
The integrals can be performed explicitly. If all C x are different, for instance, this gives

µ(x 1 , ..., x M-1 , t) = M ∑ m=0 e itC xmxm ∏ m =m (C x m x m -C x m x m ) . ( 13
)
The expression can easily be generalized to cases where some C x 's are equal (an example is below).

B. The Projector

Suppose now that we are interested in solving the equation

C | ψ = 0 . ( 14
)
The solutions of this equation are in H if zero is a discrete eigenvalue of C (case (i)); while they are in a suitable extension K of H if zero belongs to the continuous spectrum of C (case (ii)). Following a procedure now common in quantum gravity, instead of searching directly for the | ψ states (or generalized states in K) that solve (14), we search for the operator P that maps H to the space of the solutions. In case (i), the operator is a projector which can be formally written as

P = lim T→∞ 1 2T +T -T dt e itC (15)
or, alternatively

P = lim T→∞ e -TC . ( 1 6 )
It is easy to see that these operators project on the solutions of ( 14), for instance by simply going to a basis that diagonalizes C. In case (ii), P can be written in the form

P = ∞ -∞ dt e itC . ( 1 7 )
Notice that if ( 14) is the Hamiltonian constraint of a background independent quantum theory, then the integration variable t in equations ( 15) and ( 17) is not a physical time variable: rather, it is an unphysical coordinate time, or, equivalently, a "group averaging" parameter, which will not appear in the physical transition amplitudes. We focus here on the discrete case (i) for simplicity. Inserting the ACH expansion ( 11) into (15), we have

x |P| x = lim T→∞ 1 2T +T -T dt lim N→∞ N ∑ M=0 ∑ x 1 ...x M-1 (18) C x x M-1 ...C x 1 x µ(x 1 , ..., x M-1 , t).
If we exchange the two limits, we see immediately that a problem appears: the integral of (13) in dt behaves very badly.

II. A SIMPLE EXAMPLE

Let us now compute explicitly the above expressions in a very simple case. We take H = R 2 and

C = 1 1 1 1 (19)
The solution of equation ( 14) is | ψ = 1 -1 since one can easily see that 1 1 1 1 1

-1 = 0 0 . The orthogonal matrix U = U -1 = 1 √ 2 1 1 1 -1 diagonalizes C: U C U -1 = 1 √ 2 1 1 1 -1 1 1 1 1 1 √ 2 1 1 1 -1 = 2 0 0 0 (20)
We call this matrix D. The projector on the solutions is

P = 1 2 1 -1 -1 1 . ( 2 1 )
Since zero is a discrete eigenvalue of C, we are in case (i), and P should be given by (15). To show that this is correct, let's compute the exponent. This is, by definition,

e itC = U e itD U -1 (22) = 1 √ 2 1 1 1 -1 e i2t 0 0 0 1 √ 2 1 1 1 -1 = 1 2 e i2t +1 e i2t -1 e i2t -1 e i2t +1
.

Inserting this in (15) gives

P = lim T→∞ 1 2T +T -T dt 1 2 e i2t +1 e i2t -1 e i2t -1 e i2t +1 = lim T→∞ 1 2T +T -T dt [ 1 2 e i2t C + P] ( 23
)
The first term, once integrated, gives zero in the limit T → ∞, so we are left just with the second term that gives:

P = lim T→∞ 1 2T +T -T dt 1 2 1 -1 -1 1 = 1 2 1 -1 -1 1 (24)
which is indeed (21). Let us now chose the basis | x , with x = ±1, where | +1 = 1 0 and | -1 = 0 1 for writing the sum over histories.

We consider sequences σ of state of the kind | x n = | ±1 . As in the previous section, we can write

P = lim T→∞ 1 2T +T -T dt e i t N C N (25) = lim T→∞ 1 2T +T -T dt e i2ε +1 2
e i2ε -1 2 e i2ε -1 2
e i2ε +1 2
N ( 26
)
where we have called t N = ε. We consider now the transition amplitude

±1 |P| ±1 = lim T→∞ 1 2T +T -T dt ±1 | e i2ε +1 2
e i2ε -1 2 e i2ε -1 2
e i2ε +1 2 N | ±1
We insert N -1 completeness in the product of matrices. This gives rise to a sum over the possible sequences σ, alternating between the two states | ±1 . Each time the history steps from a plus to a minus, or from a minus to a plus, we get a contribution e i2ε -1 2 , while for each step where the sign does not change we get a contribution

e i2ε +1 2 . Therefore ±1 |P| ±1 = ∑ σ lim T→∞ 1 2T +T -T dt( e i2ε +1 2 ) N-M ( e i2ε -1 2 ) M
where M is the number of times the sequence changes sign. The exponents gives terms of the form ∑ k c k e i2kε + 1, and we know that the exponents are suppressed by the integration and the limit. Thus we obtain

±1 |P| ±1 = ∑ σ lim T→∞ 1 2T +T -T dt 1 2 N-M - 1 2 M = ∑ σ lim T→∞ 1 2T +T -T dt 1 2 N (-1) M = 1 2 N ∑ σ (-1) M (27)
Now the number of changes must be even if the initial and final states are the same and odd otherwise. Therefore

±1 |P| ±1 = ± ± 1 2 N ∑ σ 1 ( 2 8 )
where N = ∑ σ 1 is the number of sequences of length N, with fixed beginning and starting point. Easily, N = 2 N-1 , so that we obtain

±1 |P| ±1 = ± ± 1 2 (29)
which is, once more, precisely the projector (21). Therefore the matrix elements of the projector can be written as sums over histories.

So far, we have not applied the ACH expansion in the number of jumps. Let us do so now. From ( 11) and (12), we have

x |e itC | x = lim N→∞ N ∑ M=0 ∑ x 1 ...x M-1 µ(x 1 , ..., x M-1 , t) (30) where µ(x 1 , ..., x M-1 , t) = i M t 0 dt M ... t 2 0 dt 1 (e i t N ) N-M . ( 31
)
which in the large N limit will give

µ(x 1 , ..., x M-1 , t) = e it (it) M M! . ( 3 2 )
Fix the initial and final states, for instance say they are the same. Then notice that there is a single possible sequence with M steps alternating pluses and minuses if M is even, and none if M is odd. Vice-versa, if the initial and final values are different, then there is a single possible sequence if M is odd, and none if M is even. Therefore the sum over histories gives unit for even M and zero for odd M for equal boundary states, and vice-versa for different boundary states. Bringing all together, we have

+ |e itC | + = lim N→∞ e it N ∑ even M=0 (it) M M! . ( 33
)
+ |e itC | -= lim N→∞ e it N ∑ odd M=0 (it) M M! . ( 34
)
Inserting this in the definition of the projector gives

+ |P| + = lim T→∞ 1 2T T -T dt lim N→∞ e it N ∑ even M=0 (it) M M! , (35)
+ |P| -= lim T→∞ 1 2T T -T dt lim N→∞ e it N ∑ odd M=0 (it) M M! . ( 36
)
This is our key result. Let us discuss it. If we take the N limit first, we have easily

+ |P| + = lim T→∞ 1 2T T -T
dt e it cos(t) = 1 2 ; (37)

+ |P| -= lim T→∞ 1 2T T -T dt e it sin(t) = - 1 2 , (38)
which is, once again, the right result. This shows that the limits taken above, such as the expansion of the matrix elements in , the replacement of the sum with integrals and the elimination of M in the exponent in (31) are legitimate. But if we now take the T limit before the N limit in (35-36), we have a sum of divergent terms. Thus, the ACH procedure is viable in this case for computing the matrix elements of the projector, but only provided that the N limit is performed before the T limit. This may appear to be a serious problem at first sight, because the objective of the ACH expansion is to have a sum over M that can be understood as a perturbative expansion. But the individual terms of the sum over M diverge in the T limit.

III. THE REGULARIZED PROJECTOR

There is a simple way to circumvent the problem described at the end of last section. In most physical ap-plications of a highly nontrivial theory, we are not interested in the exact result, but in results obtained in some approximation scheme, such for instance a perturbation theory. Then we are only interested in solving the equations of the theory with some given margin of error. Thus, we may be content of computing an operator which "almost" projects on the solutions of ( 14). Say we fix a small number δ and search for the regularized projector

P δ = δ 2 1 δ -1 δ dt e itC ( 39
)
It is easy to see that this gives an approximate solution of ( 14), in the sense that it projects on states | ψ such that the norm of C | ψ is smaller than δ times the norm of | ψ . In the basis that diagonalizes C, we have in fact

P δ = δ 2 1 δ -1 δ dt e it 2 0 0 0 = δ 2 sin 2 δ 0 0 1 = P + O(δ). ( 40
)
Notice that the δ → 0 limit of P δ is well defined, but P δ is not analytic in δ = 0. Using the above results, we can write

+ |P δ | + = δ 2 1 δ -1 δ dt lim N→∞ e it N ∑ even M=0 (it) M M! . ( 41
)
The integral can now be easily performed, giving

+ |P δ | + = lim M max →∞ M max ∑ even M=0 i M M! c M . ( 42
)
where

c M = δ 2 1 δ -1 δ dt e it t M . ( 4 3 )
For any finite δ, the series in M is convergent. In fact, since the c M 's grow in M as (1/δ) M+1 , the sum is strongly suppressed by the factorial beyond a certain M max . See the examples of Figure 1. Pd FIG. 1: The convergence of the series (42) for a matrix element of P δ (whose exact value in the δ → 0 limit is 0.5). For δ = 0.1 (left) the sum converges to 0.58 in about 13 steps. For δ = 0.05, (right) the sum converges to 0.46 in about 25 steps.

Therefore we can compute the projector P to any given desired accuracy δ by means of a sum over his-tories where long histories do not matter.

Let us now go back to the general problem. The example shows that the ACH expansion can be viable to compute P, but spurious infinities appear if the T limit is taken before the N limit. To solve this difficulty, we can compute a regularized version of P. This is given by

x |P δ | x = ∞ ∑ M=0 ∑ x 1 ...x M-1 C x x M-1 ...C x 1 x (44) µ δ (x 1 , ..., x M-1 ),
where

µ δ (x 1 , ..., x M-1 ) = i M δ 2 1/δ -1/δ dt t 0 dt M ... t 2 0 dt 1 (e iC x ) t-t M ...(e iC x ) t 1 . ( 4 5 )
For any given value of the regulator δ, the sum over the length of the histories that defines the ACH expansion converges. This opens the possibility of computing P to any given desired accuracy δ by means of a sum over histories where long histories do not matter.

IV. DIPOLE COSMOLOGY

The key result of [START_REF] Ashtekar | Loop Quan-tum Cosmology and Spin Foams[END_REF] is the construction of a Feynmanlike sum-over-paths formulation of the quantum dynamics of a FRW cosmology, where the single histories are sequences of an (arbitrary, but) finite number of discrete steps and there is no time-parameter to keep track of the duration of each single step. This "topological" character of the sum over paths can be seen as a realization of the background independence of the theory, and represents the remnant in cosmology of the full background independence of the spinfoam theory.

This "topological" character of the final sum over paths, on the other hand, is realized in a somewhat roundabout manner in [START_REF] Ashtekar | Loop Quan-tum Cosmology and Spin Foams[END_REF]. The FRW model is coupled to a scalar field. Then, this field is taken as the independent time variable; that is, the evolution is resolved with respect to it. Finally, the intermediate scalar-field-time dependence is integrated away, leaving just a sum over histories of the purely gravitational variable, at given boundary values of both this variable and the scalar field.

More precisely, two constructions are presented in [START_REF] Ashtekar | Loop Quan-tum Cosmology and Spin Foams[END_REF][START_REF] Ashtekar | Casting Loop Quantum Cosmology in the Spin Foam Paradigm[END_REF]. In the first the system is deparametrized using the scalar field as time. In the second, the ACH expansion is directly applied to the projector on physical states. Also in this case, however, the expansion is then carried over by exploiting the peculiar deparametrizable form of the hamiltonian constraint, and the sum is over histories of just the gravitational variable, at given boundary values of both this variable and the scalar field.

Can the same technique be applied in a more straight-forward manner, without the need of singling out a time variable that can deparametrize the system? In principle, this can be done by applying the ACH expansion to the (generalized) projector P on the solutions of the Wheeler-deWitt equation of a covariant theory. This is precisely what we have done in the previous section.

Here, we apply the results of the previous sections, in a context where we do not introduce an effective scalar field and we do not deparametrize the evolution. For this, we need to work with a cosmological model with a sufficient number of degrees of freedom for the relative evolution among variables to be defined. We consider the "dipole cosmology" introduced in [START_REF] Rovelli | Stepping out of Homogeneity in Loop Quantum Cosmology[END_REF]. This system represents the lowest order of a "triangulated quantum cosmology". As shown in [START_REF] Battisti | Triangulated Loop Quantum Cosmology: Bianchi IX and inhomogeneous perturbations[END_REF], it describes a Bianchi IX closed universe, plus a small amount of inhomogeneity. The Hilbert space of the model admits a discrete basis of the form | j f , i n where (j f , i n ) are the colors of a spin network ∆ * 2 (dual to the triangulation of physical space) formed by two nodes n = 1, 2 joined by four links f = 1, ..., 4.

∆ * 2 = s s . The Hamiltonian constraint operator is C ψ(j f , ι t ) = ∑ j =0,±1 C f ι t j f ι t ψ j f + f 2 , ι t , (46)
where the coefficients C j ι t j f ι t are defined in [START_REF] Rovelli | Stepping out of Homogeneity in Loop Quantum Cosmology[END_REF]; they vanish unless f =0 for two and only two of the four j's. This Hamiltonian is not convenient for the spinfoam expansion because it lacks a diagonal term. However, it is easy to replace it with an operator that has a non-vanishing diagonal term: it suffice to write the holonomy that defines the regularization of the curvature in the spin-one representation instead than the spin-half representation [START_REF] Gaul | A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements[END_REF]. This gives

C ψ(j f , ι t ) = ∑ j =0,±1 C f ι t j f ι t ψ j f + f , ι t , (47)
where now the coefficients C j ι t j f ι t vanish unless f =0 for at least two of the four j's. In particular, they are nonvanishing for all f 's equal to zero. In this form, we can immediately apply the results of the previous section. The regularized transition amplitudes (or the physical scalar product) between two | j f , i n states will be given by

j f , i n |P δ | j f , i n = ∞ ∑ M=0 ∑ j 1 f ,i 1 n ...j M-1 f ,i M-1 n C j f ,i n j M-1 f ,i M-1 n ...C j 1 f ,i 1 n j f ,i n µ δ (j 1 f , i 1 n ...j M-1 f , i M-1 n ), (48)
where

µ δ (j 1 f , i 1 n ...j M-1 f , i M-1 n ) = i M δ 2 1/δ -1/δ dt t 0 dt M ... t 2 0 dt 1 (e iC j f ,i n ) t-t M ...(e iC j f ,in ) t 1 . ( 49
)
This expression give the matrix elements of the projection operator in terms of a sum over histories of finite length. These histories are sequences of spin networks with the same graph ∆ * 2 . The spins can change at most by one at each step. The intertwiners can change arbitrarily, compatibly with the spins, at each step. The "vertex" operator is given by the non-diagonal elements of the Hamiltonian constraint

C j f + f ,i n ,j f ,i n = j f + f , i n | C| j f , i n = C f ι t j f ι t . ( 50
)
We expect that this sum converges rapidly for finite δ.

V. CONCLUSION

The results that we have obtained are the following. First, we have introduced a simple way to regularize the spinfoam expression for the projector operator that defines the quantum dynamics in a background independent system.

Second, we have applied this result for writing a spinfoam version of the "dipole cosmology". The main advantage of this procedure is that a scalar field to keep track of the temporal evolution is not needed. The system has enough degrees of freedom for expressing the dynamics in a fully relational way, without the need of deparameterizing it. All physical amplitudes can be expressed as "transition amplitudes" (or matrix elements of the physical scalar product) among the unconstrained spin network states. For a general discussion of the physical interpretation of these transition amplitudes, see [START_REF] Rovelli | Quantum Gravity[END_REF].

The same technique can in principle be applied to other cosmological models with enough degrees of freedom. In particular, the application of the same technique for writing a spinfoam formulation of a Bianchi I cosmology will appear elsewhere [START_REF] Henderson | Spinfoam formulation of Bianchi I and dipole quantum cosmology[END_REF].

A spinfoam expansion in cosmology has also been introduced in [START_REF] Bianchi | Towards Spinfoam Cosmology[END_REF], starting from the complete covariant theory. It would be interesting to understand if the two procedures converge.

-

We thank Ed Wilson-Ewing for many illuminating discussions, and Matteo Smerlak for a clarifying discussion on asymptotic expansions. FV acknowledges the support of the Fondazione Della Riccia.


D Oriti
email: doriti@perimeterinstitute.ca

Group field theory and simplicial quantum gravity

We present a new Group Field Theory for 4d quantum gravity. It incorporates the constraints that give gravity from BF theory, and has quantum amplitudes with the explicit form of simplicial path integrals for 1st order gravity. The geometric interpretation of the variables and of the contributions to the quantum amplitudes is manifest. This allows a direct link with other simplicial gravity approaches, like quantum Regge calculus, in the form of the amplitudes of the model, and dynamical triangulations, which we show to correspond to a simple restriction of the same.

I. INTRODUCTION AND MOTIVATION

A. GFTs, spin foams and simplicial gravity

The field of non-perturbative quantum gravity is progressing fast [START_REF]Approaches to Quantum Gravity[END_REF], in several directions. Spin foam models [START_REF] Oriti | [END_REF] are one of them, and can be understood as a covariant formulation of the dynamics of loop quantum gravity [START_REF] Rovelli | Quantum Gravity[END_REF] and as a new algebraic version of the discrete quantum gravity approach based on path integrals, as for example Regge calculus [START_REF] Williams | [END_REF] and dynamical triangulations [START_REF] Ambjorn | [END_REF]. This line of research has recently received further impetus with the introduction of new models for 4-dimensional quantum gravity in [START_REF] Freidel | [END_REF][25][26][27]. The general idea of spin foam models is to encode the kinematics of quantum gravity in discrete quantum histories given by spin foams: combinatorial 2-complexes labelled by group-theoretic data. The 2-complex is combinatorially dual to a simplicial complex, and the algebraic data are interpreted as determining a possible simplicial geometry, just as edge lengths do in traditional Regge calculus. A quantum dynamics is specified by the assignment of a probability amplitude to each spin foam, with the model being defined by a sum over both 2-complexes and algebraic data labelling them. At present the most complete definition of a spin foam model is achieved by means of the so-called group field theory formalism [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF][START_REF] Freidel | [END_REF]. Group field theories are quantum field theories over group manifolds (usually the Lorentz group) characterized by a non-local pairing of field arguments in the action, and can be seen as a generalization of matrix models [5] that have proven so useful for our understanding of 2d quantum gravity (and string theory) (and of the subsequent, but less developed, tensor models [START_REF] Gross | [END_REF]7]). The combinatorics of the field arguments in the interaction term of the GFT action matches the combinatorics of (d-2) faces of a d-simplex, with the GFT field itself interpreted as a (second) quantization of a (d-1)-simplex. The kinetic term of the action, in turn, dictates the rules for gluing two such dsimplices across a common (d-1)-simplex, and thus for the propagation of (pre-)geometric degrees of freedom from one to the next. See [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF] for details. Because of this combinatorial structure, the Feynman diagrams of a GFT are dual to d-dimensional simplicial complexes, and are themselves given by 2-complexes. The field arguments assign to these 2-complexes the same group-theoretic data that characterize spin foam models, and, most importantly, the GFT perturbative expansion in Feynman amplitudes define uniquely and completely [9] a spin foam model. This fact alone makes GFTs a very useful tool, but also leads to the suggestion that they can provide a more fundamental definition of a dynamical theory of spin networks (as a second quantized theory of spin networks, as clarified also in [START_REF] Livine | Second quantization of spin network wave functions and quantum flat 3d space[END_REF]), in particular representing the best way to investigate non-perturbative and collective properties of their quantum dynamics [START_REF] Oriti | Proceedings of Science[END_REF]. It can also be argued [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF][START_REF] Oriti | Proceedings of Science[END_REF] that GFTs represent a common framework for both the loop quantum gravity/spin foam approach and simplicial approaches, like quantum Regge calculus and (causal) dynamical triangulations, whose basic ideas and structures they incorporate.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 22 April 2010

B. Motivation

For this idea to be realized, or at least investigated in more concrete terms, it is necessary to have at one's disposal a class of group field theories whose Feynman (spin foam) amplitudes are given by path integrals for simplicial gravity actions in 1st order form (as appropriate for the type of variables appearing in them). This has been partially achieved in [17]. In this paper, we develop further, and in a sense bring to a completion, the work in [17] and present a group field theory for 4-dimensional quantum gravity, whose Feynman amplitudes are indeed simplicial gravity path integrals with a clearly identified simplicial action and with all the GFT variables possessing a clear geometric interpretation. More precisely, the model's amplitudes are path integrals for a discrete Plebanski action, i.e. a discrete BF action with constraints on the B variables that turn them into a discretization of the continuum tetrad field [START_REF] Oriti | [END_REF], augmented by quantum corrections. On the one hand, therefore, we obtain an explicit link with 4d simplicial quantum gravity, and with other simplicial approaches. On the other hand, the ability to control the imposition of the constraints in a path integral context makes the proposed GFT a nice step forward towards a spin foam formulation of 4d quantum gravity, since the correct imposition of these constraints is the declared goal of all current spin foam models [START_REF] Freidel | [END_REF][25][26][27] (we will discuss in the following both the good sides and the limitations of the model we are presenting). We achieve this by means of a generalization of the whole GFT formalism, that allows to by-pass some not entirely satisfactory (in our opinion) steps in the usual spin foam quantization procedure, when re-phrased in a Lagrangian path integral language. Let us now sketch briefly these steps and the reasons why we find them problematic.

The classical inspiration is the Plebanski formulation of gravity as a constrained BF theory [START_REF] Oriti | [END_REF]. This means that the theory one would really want to quantize in path integral (Lagrangian) language is some discrete version of the Plebanski action on a simplicial complex ∆, schematically of the form:

Z ∆ = Dg e DB f e C(B f ⊂e ) e i P f tr(B f H f (g e∈∂f ))
.

(

) 1
Here the starting point at the classical level is the action for BF theory discretized on simplicial complex. Lie algebra variables B f ∈ so(4) are associated to all the triangles f of the complex, group elements g e are associated to the tetrahedra e of the complex, and represent parallel transports of a discrete connection from a given 4-simplex to a neighboring one, across the given tetrahedron e shared by the two 4-simplices. Both set of variables are of course integrated over in the formal definition (before gauge fixing) of the partition function of the theory. The curvature associated again to each triangle is obtained from the elementary connection variables as H f = e∈∂f g e , where the product runs over the tetrahedra e sharing the triangle f . The trace is in the fundamental representation. This defines the discrete BF action. We have then simplicity constraints at the level of each tetrahedron of the simplicial complex [START_REF] Freidel | [END_REF] (see also [30]). These constraints impose the restriction on the classical discrete Lie algebra B variables constraining the set of B f variables to be determined by a discrete tetrad field [29] and thus giving gravity from BF theory. We will give the precise form of these constraints in the following. Secondary constraints arise, in the continuum, from commutators of simplicity constraints with the other constraints of the theory [33]. Similarly, additional (secondary) constraints, on top of the simplicity constraints, have been argued to be necessary also in a discrete context [31,39]. For example, in the discrete context one associated reference frames to the different tetrahedra in the simplicial complex, and then imposes simplicity constraints separately in each tetrahedron. One then expects additional constraints enforcing the correct parallel transport of such constraints from one frame to another. Any such additional constraint is to be understood as a constraint on the discrete connection variables [31,33,39], allowing in particular to determine the intrinsic (spatial) part of the connection degrees of freedom to the tetrad or Lie algebra B variables, while the remaining degrees of freedom would define the extrinsic curvature. It is not clear, at present, if and how such additional constraints are accounted for in recent spin foam models. We will see how similar restrictions on the connection, possibly related to the ones identified in [31,39], arise naturally in our model.

The usual spin foam procedure [START_REF] Oriti | [END_REF] mimics this path integral quantization but is applied at the level of quantum states directly, using methods from geometric quantization. The idea is the following. We do know the structure of states of quantum BF theory formulated as a spin foam model, the Ooguri-Crane-Yetter model, being the 4d counterpart of the Ponzano-Regge model for 3d gravity/BF), adapted to a simplicial complex: they are given by spin networks with links labeled by representations of SO(4) (in euclidean signature), with 4-valent vertices dual to tetrahedra of the corresponding simplicial complex (while their links are dual to the triangles of the same tetrahedra). We also know how to construct, starting from such states, an amplitude for a single 4-simplex that is then a function of the representations labeling its 10 triangles and the additional parameters labeling its 5 boundary tetrahedra. It is obtained by appropriate contraction of the 5 intertwining tensors associated to its 5 boundary tetrahedra. This is the main ingredient of the spin foam model defining the dynamics of the BF quantum states. The other contributions to the complete spin foam amplitude, associated to lower-dimensional simplices, i.e. triangles and tetrahedra, and usually thought of as "measure terms"can be similarly defined, but can be determined uniquely only by the requirement of triangulation independence (following from the topological character of quantum BF theory). The task for defining a spin foam dynamics for 4d gravity is then first of all to find appropriate restrictions on the SO(4) spin networks of BF theory, i.e. on their representation and intertwiner labels, that can be interpreted as the quantum analogue of the Plebanski constraints in the classical B variables of BF theory, and thus to characterize the spin network quantum states of 4d gravity. In the new spin foam models of [START_REF] Freidel | [END_REF][25][26][27], the crucial intermediate step is to re-write the same spin network states of BF theory in terms of coherent states of SO(4), whose defining parameters are interpreted as the quantum labels corresponding to the B variables. Second, one follows the steps leading to the BF spin foam amplitudes, but starting the newly defined candidate 4d gravity states, arriving at a proposal for the 4-simplex gravity amplitude. This procedure again leaves the measure terms in the spin foam model undetermined, and this time we do not expect, and thus we can not impose, any triangulation independence property. Moreover, the identification of the analogue of the discrete classical B variables, e.g. with the parameters labeling the quantum coherent states, in the quantized BF partition function or in its quantum states expressed in terms of group elements or group representations is not free of ambiguities1 . Consequently, the procedure for imposing the simplicity constraints on them, that would be rather straightforward at the path integral level, is itself ambiguous.

When re-phrased in terms of a path integral quantization, the same spin foam procedure consists of four basic steps: 1) start from the BF path integral discretized on a simplicial complex, with the B field discretized in terms of Lie algebra elements B t associated to the triangles of the complex, and the connection degrees of freedom encoded in group elements (to be thought of as parallel transports of the same connection) associated to the edges of the dual 2-complex and thus in 1-1 correspondence with the tetrahedra of the triangulation; 2) integrate out the B t variables so that one is left with a quantum amplitude function of group elements only, and given by a delta function for each triangle (dual face) imposing flatness of the corresponding holonomy (this is the entire content of BF theory); 3) expand the delta function in irreducible representations of the Lorentz group obtaining a re-formulation of the original discrete path integral in terms of group elements and representations of the same group; 4) identify in this reformulation variables that can be argued to correspond to the original Lie algebra variables B t and impose on them an analogue of the Plebanski constraints on the B field, modifying the measure on such variables in the partition function. This last step leaves us with a candidate partition function for 4d quantum gravity, that can then be put in a pure spin foam form by integrating out the group variables. In the most recent spin foam models [START_REF] Freidel | [END_REF][25][26][27], for example, a basis of coherent states for the Lorentz group is used for expressing the amplitudes in the representation picture, i.e. after the expansion in step 3), their characterizing parameters are identified with the discrete B variables, and the simplicity (Plebanski) constraints are imposed on them to give the final model. The initial BF path integral with group elements and Lie algebra variables does not play an essential role and one could have as well started directly from a definition of BF theory as a product of delta functions on the group, i.e. from the result of step 2)2 .

The corresponding group field theory construction of the same spin foam models follows an analogous path (see [START_REF] Oriti | [END_REF] and, for the new models, [START_REF] Freidel | [END_REF]). One starts from a group field theory action designed to produce Feynman amplitudes given by products of delta functions. The action for such model (for φ ∈ R) is [16]:

S[φ] = 1 2 dg 1 .
..dg 4 φ(g 1 , g 2 , g 3 , g 4 )φ(g 4 , g 3 , g 2 , g 1 ) + -λ 5! dg 1 ...dg 10 [φ(g 1 , g 2 , g 3 , g 4 ) φ(g 4 , g 5 , g 6 , g 7 ) φ(g 7 , g 3 , g 8 , g 9 ) φ(g 9 , g 6 , g 2 , g 10 ) φ(g 10 , g 8 , g 5 , g

1 )] . ( 2
)
The field is assumed to be symmetric under the diagonal (right) action of the Lorentz group G φ(g i ) = φ(g i g) ∀g ∈ G. This symmetry requirement is ultimately responsible for the form of the quantum amplitudes. It introduces a gauge connection on the dual links of the simplicial complex and imposes flatness inside each 4-simplex, which, combined with the trivial gluing condition imposed by the kinetic term, results in imposing flatness on each dual face (triangle). Starting from the above, the crucial step is the insertion of appropriate constraints in the GFT action, in the above group picture (e.g. for the Barrett-Crane model [START_REF] Oriti | [END_REF]) or, in the representation picture, following step 3) and 4) above. For example, in the new models [START_REF] Freidel | [END_REF] one can expand the GFT kinetic term in group representations, and then in coherent states of G, and finally constrain the corresponding parameters as the simplicial geometry of a tetrahedron (corresponding to the GFT field), when expressed in terms of bivectors B t suggests [27]. Once more, there is no role for the Lie algebra variables, that would directly correspond to the discrete B field.

There are reasons to be dissatisfied with this procedure, and to look for alternatives, in particular to look for a proper Lagrangian formulation of a 4d spin foam model for gravity.

We have already mentioned the difficulty in specifying uniquely the measure terms in the usual spin foam construction. A proper Lagrangian path integral derivation starting from BF theory (whose measure is unambiguous) could provide a prescription for these measure terms.

In a Lagrangian/path integral context, the two sets of Lie algebra and group variables are classical and independent of each other. Any relation between the two should arise in some (dominant) configurations, satisfying the discrete equations of motion. We want to reproduce exactly this feature, with a corresponding "doubling"of independent geometric variables, at the GFT level, with our generalised GFT formalism. Alternatively, the relation between the two should be in terms of some classical Fourier transform mapping one set of variables to the other. We will discuss briefly this possibility in the following. On the other hand, the usual spin foam quantization introduces a quantum analogue of the B variables in terms of the Lie algebra generators acting on function of the group variables, and thus lives from the start at the quantum (canonical) level, outside the path integral context.

Once an appropriate discretization of the simplicity constraints has been chosen, they are imposed as delta function restriction in the path integral measure on the classical independent B variables. This means that, contrary to steps 3) and 4) above, there is no ambiguity in how to impose the simplicity constraints, which is the point of debate in the usual spin foam construction. Furthermore, this also means that they have to be imposed before performing the integration over these variables, i.e. at the classical level, as they drastically affect exactly this integration, contrary to the usual spin foam procedure.

In this work, we try to overcome the pitfalls of the usual procedure when seen from a Lagrangian path integral point of view3 . We also make direct contact with simplicial gravity at the level of the resulting partition function (and not only as a guideline for the construction). We work in (and develop further) a generalised group field theory framework in which both group variables (representing discrete connection variables) and Lie algebra elements (representing discrete B variables) are present. We obtain an explicit simplicial gravity path integral for 4d gravity in 1st order variables, with BF-like action and the simplicity constraints manifestly implemented on the Lie algebra variables. Moreover, thanks to the presence of both sets of variables, we can keep under control and manifest the simplicial geometric picture at all stages in the construction. This also allows us to establish a direct link with other simplicial gravity approaches like quantum Regge calculus, in the very form of the quantum amplitudes of the model, and dynamical triangulations, which we can show to correspond to a simple restriction of the same.

The new model can be understood as a refinement of the class of models introduced in [17]. More details on the new 4d gravity model, together with the construction and thorough analysis of the simpler 3d gravity model can instead be found in [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]. This work can be also understood in the context of the line of research on the issue of causality in spin foam quantum gravity and GFT, and on the construction of a unified GFT framework for loop quantum gravity and simplicial quantum gravity, developed in [17][18][19]21].

II. NEW GFT MODEL FOR 4D QUANTUM GRAVITY

A. Definition of the model

As it is usually the case in group field theory [START_REF] Oriti | [END_REF], our model, defining a theory in d spacetime dimensions, is based on a (real) field φ(z 1 , z 2 , ..., z d ) of d variables, interpreted as (pre-)geometric data, and by a classical action for this field of the general form:

S(φ) = dz i φ(z i ) (Kφ) (z i ) - λ d + 1! dz ij φ(z 1j )φ(z 2j )...φ(z (d+1)j ) V(z ij ) ( 3 )
with a local quadratic term (the arguments of the two fields in it are identified) defined by some (differential) operator K and a combinatorially non-local interaction term, defined by the operator V, which couples in a peculiarly non-local way the arguments of d + 1 fields. The combinatorics of field arguments in the action, in particular, in the interaction term, is best understood in terms of a diagram (representing the d=4 case).

! !

FIG. 1: Kinetic and vertex term of 4d group field theory

In the vertex term, in particular, each of the 4 arguments of each field is identified with one and only one argument of another field. Interpreting each field as a tetrahedron, with its 4 arguments corresponding to the 4 triangles in the boundary of the same tetrahedron, the above identification corresponds to the gluing of two tetrahedra along a common triangle, and the combinatorics of arguments in the GFT vertex is that of a 4-simplex [START_REF] Oriti | [END_REF] with 5 tetrahedra glued along triangles. The kinetic term, on the other hand, represent the gluing of two 4-simplices along a common tetrahedron obtained by identification of the corresponding boundary triangles. Within this interpretation, the indices i in the kinetic term above identify the variables associated to the 4 triangles in the tetrahedron represented by the field, while, in the vertex term, the variables of the field are labeled by two indices ij each corresponding to a tetrahedron in the 4-simplex, so that the pair of indices corresponds the their common triangle, with the indices ij and ji associated to the same triangle seen in the two tetrahedra i and j respectively.

To be more specific, the field of our model

4 φ(x 1 , b + 1 ; x 2 , b + 2 ; x 3 , b + 3 ; x 4 , b + 4
) lives on the cartesian product of 4 copies of S 3 × su(2). For our present concerns (but see the discussion in [17]), we can restrict our attention to real fields. In turn, this field is obtained from a generic (real) field ϕ(g 1 , B 1 ; ...

; g 4 , B 4 ) = ϕ(g 1 , (b + 1 , b - 1 ); ...; g 4 , (b + 4 , b - 4
)) on (Spin(4) × so(4)) 4 by application of the two maps P B and P h , to be defined below: φ = P B P h ϕ. In terms of these maps, as we are about to see in more details, one identifies the domain of dependence of φ, "S 3 ", as the quotient of the domain of ϕ, "Spin(4)", by the diagonal SU(2) subgroup, and the su(2) algebra on which φ depends as the self-dual part of the Spin(4) algebra on which ϕ depends.

The action defining the classical dynamics of the model is then given by:

S = 1 2 Dx i Db + i φ(x i ; b + i ) K m (x i , b + i ) φ(x i ; b + i ) - λ 5! Dx ij Db + ij [P g φ](x 1j ; b + 1j )..[P g φ](x 5j ; b + 5j ) V(x ij , b ij ) = = 1 2 Dx i Db + i [P B P h ϕ](x i ; b + i ) i S 3 i + |b + i | 2 - m 2 4 [P B P h ϕ](x i ; b + i ) + - λ 5! Dx ij Db + ij [P g P B P h ϕ](x 1j ; b + 1j )..[P g P B P h ϕ](x 5j ; b + 5j ) V(x ij , b ij ) = = 1 2 Dg i Dg i Db + i Db - i D b+ i D b- i Dh i D hi DN D Ñ i δ b - i + Nb + i N -1 δ( i b + i ) i δ b- i + Ñb + i Ñ -1 δ( i b+ i ) ϕ(g i h i ; (b + i , b - i )) i S 3 i + | b+ i | 2 - m 2 4 δ(g i g-1 i )δ(b + i -b+ i ) ϕ(g i hi ; ( b+ i , b- i )) + - λ 5! Dx ij Db + ij [P g P B P h ϕ](x 1j ; b + 1j )..[P g P B P h ϕ](x 5j ; b + 5j ) = = 1 2 Dg i Dg i DB i D Bi C(B i ) C( Bi )ϕ(g i ; B i ) i Dh i Spin(4) i + 1 2 | Bi | 2 - m 2 4 δ(g i h i g-1 i )δ(b + i -b+ i ) ϕ(g i ; Bi ) + - λ 5! Dx ij Db + ij [P g P B P h ϕ](x 1j ; b + 1j )..[P g P B P h ϕ](x 5j ; b + 5j )V(x ij , b ij )
with the following ingredients:

• the measures of integration Dg, Dh and Dx, as the standard Haar measures on Spin(4) and SU(2) respectively, and the corresponding induced measure on the homogeneous space S 3 Spin(4)/SU(2), DB and Db are instead the standard Lebesgue measure on the Lie algebra considered as a vector space.

• the map P B , acting on functions on so(4) 4 as

P B F (B 1 , .., B 4 ) = P B F ((b + 1 , b - 1 ), ..., (b + 4 , b - 4 )) = i Db - i S 3 DN D Ñ i δ b - i + Nb + i N -1 δ( i b + i ) F ((b + 1 , b - 1 ), ..., (b + 4 , b - 4 )) = i Db - i C(B i )F (B 1 ; ....; B 4
) constrains the four Lie algebra variables associated to the field ϕ imposing the simplicity constraints; its geometric meaning will be discussed below.

• the projector P h maps ϕ to a function on the homogeneous space Spin(4)/SU(2) S 3 4 , and can be obtained explicitly by group averaging P h ϕ(g 1 , B 1 ; ...; g 4 , B 4 ) = dh 1 ..dh 4 ϕ(g 1 h 1 , B 1 ; ...; g 4 h 4 , B 4 ).

• the kinetic operator is given by:

K(g i , B i ) = 4 i=1 B 2 i + Gi - m 2 4 ( 4
)
with Gi the Laplacian operator on the i-th copy of the group manifold G = Spin(4) or on the homogeneous space S 3 , the square |B| 2 (or

|b + i | 2
) is taken using the fundamental Killing form on the Lie algebra, and m 2 is an arbitrary positive constant.

• the projector P g imposes invariance under the diagonal action of the group Spin(4) on the group and Lie algebra variables: P g ϕ(g 1 , B 1 ; ...; g 4 , B 4 ) = dg ϕ(g 1 g, g -1 B 1 g; ...; g 4 g, g -1 B 4 g).

• the vertex or interaction operator V is given by

V(g ij , B ij ) = 5 i =j=1 δ(g ij g -1 ji )δ(B ij -B ji ). (5)
The four variables appearing in each field, interpreted as associated to the four triangles in the boundary of the tetrahedron i corresponding to it, are represented in the figure below.

Let us now explain motivation and geometric content of all these ingredients (for more details, see [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]).

The key to the geometric interpretation of the various ingredients and of the whole model is the geometric interpretation of the field ϕ(g i , B i ) itself. We can think of it as the second quantization of a tetrahedron whose geometry is characterized by the four pairs of 1st order variables g i representing elementary parallel transports of a Lorentz connection along paths dual to the triangles of the tetrahedron, and B i representing Lie algebra variables (or bivectors) associated to the same triangles [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF]. In addition, we assume that a reference frame is associated to a tetrahedron and, thus, to each field ϕ, and that the B variables) are expressed in this frame.

Let us now look at the geometric content of the interaction term. First of all, as in other GFT models, the combinatorics of field arguments is chosen to respect the combinatorics of triangles in a 4-simplex. It is this feature that characterizes GFTs as peculiarly non-local field theories. Second, the symmetry imposed by the projector P g amounts to the requirement that the reference frame associated to each tetrahedron should be arbitrary. In other words, one should be free to relate this frame by means of an arbitrary parallel transport to some given, and equally arbitrary, frame associated to the 4-simplex as a whole. This parallel transport is effected by the group element g integrated over in the projection P g , that can be thought of as associated to (half) dual edges in the 2-complex dual

!" $ % & '( ') '* '+ '+ '( ') '* !" $ % & !" $ % & !" $ % & !" $ % &
FIG. 2: The variables associated to the four triangles of each tetrahedron, a group element, g, and a Lie algebra element B.

to the triangulation to which the 4-simplex belongs. One can do the same for all the tetrahedra/fields in the vertex term. This means that their different variables all come from a single set of triangle variables for the 4-simplex they belong to, then parallel transported to different locations (the boundary tetrahedra) in the simplex. This symmetry imposition amounts to a relaxation of the usual symmetry imposed on the group variables alone in the usual GFTs (and interpreted as imposing the closure constraint of BF-like theories). The net result is that the field is not invariant anymore under the diagonal action of the group, when seen as a function of the group elements only, but it becomes covariant, and this translates in representation space as a covariance, as opposed to invariance, of the tensors between the 4 representations associated to the triangles of the tetrahedron. The need to relax this invariance to a covariance requirement has been emphasized also, from a canonical perspective, by Alexandrov [31]. By a change of variables one can move the group variables defining the projection operators P g into the vertex function V, then defined on non-projected fields φ's (or ϕ). The result is a vertex function given by:

V(g ij , B ij ) = 5 i=1 dg i 5 i =j=1 δ(g ij g i g -1 j g -1 ji ) 5 i =j=1 δ(B ij -g -1 i g j B ji (g -1 i g j ) -1 ). (6)
The geometric meaning is now transparent. The delta functions on the group impose that the parallel transport g ij g i g -1 j g -1 ji of the Lorentz connection along the boundary of the wedge (portion of the dual face inside each 4simplex) associated to the triangle ij shared by the tetrahedra labelled i and j and living inside the 4-simplex that contains both these tetrahedra, is flat. This is consistent with our piecewise-flat context. The deltas on the Lie algebra impose that the Lie algebra variables (discrete B field) associated to the same triangle in two different tetrahedra are identified only after parallel transport from the center of one tetrahedron (where they are originally defined) to the center of the 4-simplex, i.e. to the single reference frame in which they are in fact the same, and back to the other tetrahedron, with this parallel transport effected by the group elements g i5 . The variables associated to each wedge are shown in the figure below. We see therefore that the interaction term of the GFT model enforces a symmetry requirements (local Lorentz invariance) and the trivial kinematical geometry dictated by a piecewise-flat setting. Notice that, because of the projectors P B and P h acting on the field ϕ, the variables B ij entering the vertex term will be constrained to be simple, and we will also have g ij ∈ S 3 .

Let us then give a look at the kinetic term. This is given by the product of four Klein-Gordon operators acting on the group manifold G with a variable mass term given by B 2 plus an arbitrary constant mass square shift, one for each triangle in the tetrahedron the field refers to. This choice [17], first of all, relaxes the identification between the discrete bivector B associated to the triangle and the Lie algebra generator J seen as an operator acting on functions of the group, and thus between its modulus square and the (1st) Casimir of the algebra, which in turn is defined up to an arbitrary constant shift. This specific choice is the simplest Lorentz invariant one achieving the above. Obviously, others could be considered. On the one hand, in the present context this step allows to deal with both B and g variables at the same time, it leads to a nice simplicial path integral, as we are going to see, and keeps the simplicial

! " # "# #" ! ! ! $ $ #" "#
FIG. 3: The variables associated to each wedge in the (dual of the) simplicial complex: two group elements gi and gj representing parallel transports from the center of the 4-simplex to the center of the two tetrahedra i and j sharing the triangle ij dual to the wedge; two group elements gij and gji (arguments of two fields corresponding to the tetrahedra sharing the same triangle) representing parallel transports from the center of the two tetrahedra to the center of the triangle; the two Lie algebra variables Bij and Bji representing the B variable associated to the triangle ij seen in the reference frame of the tetrahedron i and of the tetrahedron j, respectively. geometry of the model fully manifest. On the other hand, we may suspect already at this stage that it will lead us away from the dynamics of BF theory, even in absence of simplicity constraints, since the conjugate nature of these variables (characteristic of BF theory) is then not imposed in our model.

The additional projectors P h lead to a Klein-Gordon operator on the homogeneous space S 3 . It results in the amplitudes depending on a single angle parameter on the group manifold. It is mainly motivated by the specific discrete gravity action this leads to. However, it also makes the same amplitudes, when expanded in representations of the Lorentz group, involve only class 1 representations, as in the Barrett-Crane model and in the other new spin foam models [START_REF] Freidel | [END_REF]25,26]. We know that this represents partial imposition of the simplicity constraints in terms of the connection variables, following their (partial) imposition on the Lie algebra generators identified with the discrete B variables in a geometric quantization of BF theory . Here, on the other hand, the same constraints are fully implemented on the Lie algebra variables B. Therefore, we could expect this additional restriction to be redundant, to some extent. We will discuss later on to what extent this is in fact true.

The simplicity constraints on the B variables are imposed by the factor C(B i ) that is inserted for each field and thus for each tetrahedron in the triangulation, separately in each 4-simplex to which the tetrahedron belongs. In fact, the GFT kinetic term defines the gluing of two 4-simplices in the 4d triangulation associated to each Feynman diagram (see [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF]) across a common tetrahedron, represented by the two fields in the GFT kinetic term (same tetrahedron seen in the two different 4-simplices). The simplicity constraints amount to the imposition that the four bivectors associated to the four triangles f of the tetrahedron t all belong to the same hypersurface (in flat space), i.e. they are all normal to the same unit 4-vector, interpreted as the normal to the tetrahedron [START_REF] Freidel | [END_REF][25][26][27]31]: ∃N t ∈ S 3 / B IJ f n tJ = 0 ∀f ⊂ t. In this case, the bivectors are interpreted as defining the so-called "area bivectors"A f = e 1 ∧ e 2 formed by the wedge product of two edge (tetrad) vectors e 1,2 of the triangle f . A dual version of the same constraints is the requirement that the four bivectors are such that their duals are normal to the same unit normal vector and thus belong to the same hypersurface: ∃N t ∈ S 3 / ( * B f ) IJ N tJ = 0 ∀f ⊂ t. In this case, they are interpreted as dual to the same area bivectors: B f = * A f 6 . To the above constraints, it is necessary to add the so-called closure constraint, imposing that the four bivectors associated to the triangles in a given tetrahedron sum to zero, i.e. that the tetrahedron "closes". Only when all these conditions are satisfied it is possible to invert the set of bivectors associated to the triangles of the simplicial complex for a set of tetrad vectors associated to the edges of the same and specifying a simplicial geometry. It is possible to show, moreover, that these set of constraints represents the discrete version of the continuum Plebanski constraints. The constraining factor C(B i ), brought in the action by the mentioned map P B , has thus the form:

C(B i ) = S 3 SU(2) dN t 4 i=1 δ (3) (b - i ∓ N t b + i ) δ( i b + i ) = S 3 SU(2) dN t 4 i=1 δ (3) (b - i ∓ N t b + i N -1 t ) δ( i b + i ) = = S 3 SU(2) dN t 4 i=1 su(2) db i δ(B i -(b i , ±N t b i N -1 t ))δ( i b + i ) ( 7 )
where we have written the simplicity constraints using the selfdual/anti-selfdual decomposition of the Lie algebra elements B i 's, and the identification of elements of the homogeneous space S 3 with the Spin(4) elements mapping the reference unit vector (1, 0, 0, 0) into them, and further parametrized them as SU (2) × SU (2) elements as N t = (N + t , N - t ) = (1, N t ) at the cost of a slight abuse of notation (see [START_REF] Freidel | [END_REF]). The two possible signs ∓ correspond to the first (second) way of imposing the simplicity constraints, respectively. The arbitrariness of the normal vector N t is enforced by the integration over S 3 . In the last line, we have defined the simple bivectors b i = (b i , ±b i ) in selfdual/anti-selfdual decomposition, with similar abuse of notation, and with the same sign ambiguity.

Notice that, due to the delta functions imposing the closure constraint, a (straightforward) regularization of the action and, later, of the amplitudes of the model will be needed.

Notice also that the anti-sefdual variables b - i of the field ϕ are not coupled among different fields, but are instead fixed completely as functions of the b + i by the map P B (simplicity constraints) in each field separately.

B. Feynman amplitudes: discrete gravity path integrals

The quantization of the model is obtained by the partition function defined as a perturbative expansion in Feynman diagrams in the coupling constant λ:

Z = Dφ e iS[φ] = Γ λ VΓ sym(Γ) Z Γ (8)
where V Γ is the number of vertices in the Feynman diagram Γ (see [START_REF] Oriti | [END_REF][START_REF] Oriti | Quantum Gravity[END_REF]), sym(Γ) is the order of automorphisms of the diagram/complex, and Z Γ is the corresponding Feynman amplitude, whose construction is detailed in [17], for this general class of models, and in [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF] for this specific one and its 3d counterpart. The key ingredient is the non-trivial propagator, the inverse of the kinetic term 4. Being this a product of Klein-Gordon operators on the homogeneous space S 3 Spin(4)/SU (2) thanks to the projection operators P h , its inverse is taken to be the product of Feynman propagator on the same homogeneous space, in turn equal to the propagators on the group SU (2), due to the isomorphism between the two spaces, with variable mass given by b 2 i -m 2 4 . From now on we set m 2 = 1 because this simplifies the resulting formulae. They can also be written in terms of the analogous propagators on the full Spin(4) group acted upon by the same projectors P h (see [32]). The resulting schematic form of the amplitudes is:

Z Γ = (ev) Spin(4) dg ve (ef ) su(2) db + ef su(2) db - ef d b- ef (ev) C(B f ⊂e;v ) f A[B ef , g ve ]. (9)
The amplitudes factorize per dual face, apart from the measure factor C imposing the simplicity constraints, which is associated to each dual edge e (tetrahedron) for each vertex v (4-simplex) it touches (belongs to). They depend on one su(2) Lie algebra variable b + ef for each tetrahedron e sharing the triangle dual to the face f , two similar (anti-selfdual) Lie algebra elements associated to the same tetrahedron-triangle pair, one for each 4-simplex v sharing it, and one group element g ev = (g ve ) -1 for each half dual edge incident to a given vertex v. Recall that all the anti-selfdual Lie algebra elements are fully fixed as functions of the self-dual ones, by the constraints C. Equivalently, one can think of a single so(4) Lie algebra element B ef ;v = (b + ef , b - ef ) v associated to each dual face (triangle) in each tetrahedron sharing it, within each one of the two 4-simplices v to which the tetrahedron belongs, and with only the selfdual components of these Lie algebra elements being identified from one tetrahedron to the next, and across 4-simplices. The full holonomy around the dual face is given by H = e∈∂f g e (having chosen an arbitrary ordering of the edges in the boundary of the face), where we have defined g e = g ve g ev . They have the generic form:

A[B ef , g ev ] = µ(H f , B ef ) W(g ev , B ef ) e iS f R [B ef ,H f ] e iS f c [B ef ,H f ] . (10)
We now discuss the various contributions to these amplitudes.

The first term to notice is the action term

S f R (B ef , H f ) = |B ef ||[θ f (H f )]|. |B ef |
is the modulus of a Lie algebra variable associated to the dual face f , and thus to the corresponding triangle in ∆, in the reference frame of one of the tetrahedra sharing it. θ f (H f ) is the distance on the homogeneous space S 3 defined by the group element H f , which computes the simplicial curvature associated to the dual triangle. The holonomy is defined starting from and ending to the same tetrahedron in whose frame we defined B ef ; the choice of the tetrahedron, and thus of the frame, is immaterial [25,26]. The notation [..] indicates that the amplitude depends only on the equivalence class of distance angles corresponding to the same group element H f ; this is a result of the periodic boundary conditions imposed on the propagators on the compact manifold S 3 . It is realized in practice by defining this distance to be θ ± 2πn, with θ ∈ [0, 2π] and n ∈ N identifying each of the possible geodesics in S 3 on which the distance is computed, and summing over n, thus adding the variable n to the configuration variables on which the amplitudes depend (see [17,32] for details). Taking into account the product over dual faces in (9), this gives an action term associated to the whole triangulation ∆ dual to Γ given by: S

R = f S f R (B f , H f ) = f |B f ||[θ f (H f )]|.
This action term characterizes the Feynman amplitudes of the model as a simplicial gravity path integral. However, it is the presence of the terms C e and W f that fully characterizes the geometric content of this action and thus the dynamics of the theory.

The simplicity constraints C ev (B f ⊂e ), as discussed above, impose the geometric restriction that the set of B variables associated to the triangles of ∆ are either identified with or dual to the area bivectors for the same triangles [START_REF] Freidel | [END_REF][25][26][27]. These constraints ensure that the set of B f variables can be put in bijective correspondence with a set of tetrad vectors E I l associated with the links of the triangulation ∆, and thus with a unique (up to local Lorentz transformations) simplicial geometry [29]. In practice, they impose that each B variable is of the form (b f , ±N e b f N -1 e ) for an arbitrary vector N e common to all the bivectors associated to the same tetrahedron e, and that tetrahedra close in the hypersurface orthogonal to N e . It can be rephrased by saying that the Lie algebra element B f is, up to the common Lorentz rotation N e , the generator of a U (1) subgroup of the diagonal SU (2) or of a corresponding anti-diagonal one, depending on the sign chosen. The vector N e can be set to the form (1, 0, 0, 0) by appropriate gauge choice in each tetrahedron (see [START_REF] Freidel | [END_REF]25,26]).

The contributions W(g ve , B ef ) are delta functions on the Lie algebra. Choosing an ordering 1, ..., N of the N edges in the boundary of the dual face, fixing an arbitrary edge as the reference one and labelling it by 1 (with N + 1 ≡ 1), and defining g e e = g e v g ve , they have the form:

W = N ē=1 δ(B ē -g 1N ..g ē+1ē ..g 21 B 1 ) = δ(B N -g 21 B 1 )...δ(B 1 -g 1N ...g 21 B 1 )
where we simplified the notation B ef → B e as we are dealing with a single dual face.

These impose the obvious geometric requirement that the Lie algebra variables appearing in the different tetrahedra but corresponding to the same triangle, being the same fundamental Lie algebra element just expressed in different reference frames, can be obtained all from a single arbitrary one (here B 1 ) by parallel transports of this across the various simplices sharing the same triangle. Recall that the variables B ef are integrated over in the discrete partition function. If this was a free integration, then one could simply use the above deltas to eliminate all the redundant B variables, leaving in the end a single one per dual face. This is what happens in the 3d case [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]. However, the simplicity constraints C ev impose restrictions on these integrals, and these restrictions, together with the above deltas, imply further restrictions on the parallel transport (connection) variables g ev . We leave a complete analysis of these restrictions to further work, but it is clear that they impose a consistency or compatibility requirement between simplicity constraints and gauge invariance, thus parallel transport 7 .

A careful analysis of the constraints W, and of their interplay with the simplicity constraints C, has been performed in [37], after a first version of the present work had appeared. The analysis confirms the above and goes much further, in that it shows that these compatibility conditions can be enough to solve completely the connection degrees of freedom in terms of the tetrad or bivector ones, and thus go from a 1st order formulation to a 2nd order one, which is fully equivalent to ordinary Regge calculus at the classical level. Moreover, the restriction on the connection degrees of freedom is such that the projection P h we have imposed on our fields, that leads to our quantum amplitudes to depend only on the S 3 component of the group elements, with the consequences we have discussed, become un-necessary. In fact, they would result in the component of the holonomy along (b f , b f ) to be trivialized altogether, obtaining the same form and geometric interpretation even when working with generic Spin(4) group elements. However, at the same time the results of [37] suggest that the correct way of imposing the simplicity constraints may differ slightly but crucially from the one we adopted here. Having imposed the simplicity constraints at the level of each field ϕ independently, by means of the map P B , we end up with two normal vectors N ev ∈ S 3 associated to the same tetrahedron e one for each 4-simplex v, which our model treats as completely independent from each other. This has one important consequence. The restriction on the connection implied by the requirement that the two normals to two tetrahedra sharing the same triangle (either within the same 4-simplex or in two adjacent ones), both lie in the plane orthogonal to the area bivector associated to that triangle [37], is missing in our formulation. This requirement is important for the complete inversion of the connection variables as function of the bivector variables, thus for the definition of geometric dihedral angles from them, and it is not clear, at this stage, how much of this inversion one can still perform given our weaker form of the constraints. Notice also that this weaker form of the constraints is the same imposed in the Barrett-Crane model (within a different formalism), and possible the origin of the problems faced by it (see also [36]). How the stronger form of constraints can be imposed in a GFT context is the subject of work in progress [START_REF] Baratin | A new group field theory for 4d quantum gravity[END_REF].

Of particular interest is the last of these deltas:

δ(B 1 -g 1N ..g 21 B 1 ) = δ(B 1 -H f B 1 ). ( 11
)
This imposes on the face holonomy the condition H f = e iθ1 Be + iθ2 * Be , for arbitrary angles θ 1,2 ; in other words, it imposes that H f lives in the U (1) × U (1) Cartan subgroup of Spin( 4) and, moreover, that this is aligned with the subgroup generated by the two commuting Lie algebra elements Be = Be |Be| and * Be . The geometric content of this condition is revealed by taking the simplicity constraints into account as well. First of all, notice that the presence of this delta function reduces the gauge invariance of the Feynman amplitude Z Γ from the invariance under

B ef → GB ef G -1 , H f → ḠH f
Ḡ-1 for arbitrary G, Ḡ8 , that could be deduced from the action term alone and the simplicity constraints (we will see that also the other terms µ and S c would allow for this large symmetry) to the smaller

B ef → GB ef G -1 , H f → GH f G -1
for the same group element G. This is indeed the symmetry of BF theory (see, for example, [25,26]). Consider now the simplicity constraints with negative sign

B ef = (b ef , -n e b ef n -1
e ) implying that B ef is an area bivector A f . This can be re-written as B ef = N e * b ef for the group element N e = (1, n e ), with n e ∈ SU (2) [START_REF] Freidel | [END_REF], and b ef = (b ef , b ef ) in selfdual/anti-selfdual notation. This implies (simplifying again the notation) that

H f = e iθ1 Bf + iθ2 * Bf = N e e iθ1 * bf + iθ2 bf N -1 e = N e h f N -1 e with h f = e iθ1 * bf + iθ2 bf .
Let us then take into account the symmetry noticed above. The amplitudes, and in particular the action term depend on H f and B f only up to their simultaneous rotation by an arbitrary group element G. Let us choose this as G = N e . Then it results that the amplitudes are the same as those computed from b f and h f . More precisely, the action S R depends only on the modulus of b f ,

|b f | = |B f | = |A f |
and on the distance on S 3 corresponding to the holonomy h f . But h f = e iθ1 * bf + iθ2 bf , with b f = (b f , b f ) = A f , thus with e iθ2 bf belonging to the diagonal SU (2) subgroup of Spin(4). Therefore we conclude that the angle measuring the distance on S 3 and entering the action S R is the component of h f along * b f = * A f , or of H f along B f , i.e. θ f (h f ) = ±θ 1 . We can therefore interpret the contribution of each dual face to the simplicial action as

|B f ||[θ f ]| = tr(B f F f ) = tr(B f ln H f ) (the
trace is in the Lie algebra) with the additional restriction (that can be included in the measure of integration over the B and g variables by means of a simple Heaviside function) tr(B f F f ) > 0. This same restriction has been argued for as a (pre-)causality condition needed to define causal transition amplitudes for BF theory and gravity in [20]. The simplicity constraints imply that each B variable comes from an appropriate set of discrete tetrad variables. In turn, this implies that the simplicial action S R can be understood schematically as S R (E, g) = f tr( * A f (E)F f (g)), i.e. as a simplicial action for 4d gravity of a 1st order Regge type [17,24,28], in turn a discrete version of the Palatini action for 4d gravity. Equivalently, this shows our model to be a simplicial gravity path integral for a discrete Plebanski formulation of 4d gravity as a constrained BF theory.

Finally, let us mention the last two contributions to the Feynman amplitudes Z Γ . The first is a measure term µ(B ef , H f ) which is a real function of |B f | and H f only (with parametric dependence on the number of N dual edges/vertices in the dual face f ). The second is an additional contribution S c to the classical action S R that we can interpret as a quantum correction to the same. It has the same dependence on the basic variables as µ(B, H). These two terms are identified [17] as the modulus and phase, respectively, of the complex function:

ν(H, B, N ) ∝ -i (N -1)! 1 sin(θ(H f )) |[θ(H f )]| |B f | N -1 N -2 K=0 (N + K -2)! K!(N -K -2)! i 2|B f | |[ θ(H f ) ]| K
Their explicit expressions and asymptotic form for |B f | → ∞, H f = I are reported and discussed in [17,[START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]. This analysis, which is the analogue of the semi-classical expansion j → ∞ performed in usual spin foam models, is easily done for arbitrary triangulation thanks to the fact that the full amplitude, modulo the various constraints (whose form is of course unmodified in the asymptotic regime), i.e. ν(B, H, N )e iSR(B,H) , is given by an Hankel function, whose asymptotic formulae can be then used. The end result is that indeed the terms S c give subdominant large scale corrections (of 1/R n type) in this regime to the dominant Regge term.

In the end, the aim of obtaining a GFT model reproducing the Plebanski formulation of gravity (1) in a simplicial setting, and thus (implicitly 9 ) a new promising spin foam model for 4d quantum gravity is realized.

III. DISCUSSION

A. Features of the model

In the 3d case, and in 4d in absence of the simplicity constraints), the static-ultralocal truncation of the generalised GFT models reproduces the usual 3d gravity spin foam model, i.e. the Ponzano-Regge model, and the analogous spin foam model for 4d BF theory [17,[START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]. This amounts to maintaining the same field varibles, symmetries and interaction, but dropping the derivative (and variable mass) terms in the kinetic term, being left with simple delta functions. One can consider here a similar truncation, and analyse the corresponding Feynman amplitudes.

Because of the projections P h , and because of the constraints on the connection following the imposition of simplicity on the bivectors B, that we discuss below, the amplitudes will not force the dual face holonomies to be flat, as in BF. The simplicity constraints C(B) will still restrict the Lie algebra elements to be simple, and, in conjunction with the symmetry projectors P g will still imply further restrictions W on the connection variables ensuring compatibility between simplicity conditions and parallel transport. Because of the projections P h the resulting model can be expected to be similar to the Barrett-Crane spin foam model, also due to the weak form of the simplicity constraints we have chosen. However, because of the mentioned compatibility restrictions, thus, because of the modified symmetry requirement P g , we may still obtain a different model. In particular, we expect a stronger correlations among 4simplices, effected by the matching of B variables (and not only their modulus) across them, even if the connection variables would turn out to be not sufficiently constrained to be geometric.

The compatibility conditions W(B, g) are another aspect of the new model that need further study. First of all, one needs to clarify which restrictions they impose, exactly, on connection degrees of freedom, as we have discussed. Most important, maybe, is a clearer understanding of their geometric meaning and of their interpretation from the point of view of the canonical quantization of Plebanski action. One could in fact conjecture a relation between these constraints and the secondary second class constraints identified in [33] and whose importance for the spin foam quantization has been emphasized in [31]. In particular, in [31] it is argued that the presence of these secondary second class constraints is a necessary consequence of the simplicity conditions when considered together with the relaxed Lorentz covariance of the intertwiners. Moreover, [31] argues that these constraints would be essential for a proper integration over the connection variables by modifying the path integral measure with respect to the one in BF theory. Both these origin, features and consequences are shared by our constraints W(B, g), thus supporting our conjecture. In fact, they are (at least in the stronger form of [37]) equivalent in terms of resulting restrictions on the connection to the discrete gluing conditions ("edge simplicity") identified and studied in [39].

The quantum corrections S c to the classical Regge action S R deserve more study as well. Three features can be already noticed. First of all, they follow directly from the choice of GFT action, that therefore somehow fixes not only which terms will appear in the large distance limit or in any other approximate regime of the amplitudes, but also their relative coefficients. This is also true for the corrections appearing in the j → ∞, H f ≈ Id limit of the simplicial action defined from coherent states in [29]. Second, this approximation would be the analogue of the approximation |B f | → ∞, H f ≈ Id of our full action S R + S c . In order to analyze this regime, however, a more careful study than the one in [17], that effectively applies to the regime |B f |θ f (H f ) >> 1 only, is needed. Third, it can be shown [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF] that these corrections originate directly from the gluing of 4-simplices, i.e. from the off-shell propagation of B and g variables across them. In fact, the amplitude for a single 4-simplex with boundary has a similar simplicial path integral form, depends exactly on the variables characterizing BF/Plebanski theory on a manifold with boundary, with fixed-B boundary conditions, but involves the action S R only [START_REF] Oriti | Encoding simplicial geometry in group field theories[END_REF]. More generally, it is not clear, at this stage of development, whether the presence of the quantum corrections S c , and in general the precise form of the measure ν is to be considered correct from the geometric point of view. It is possible that the GFT model itself will have to be amended in order to eliminate the corrections S c altogether and thus simplify the measure ν.

An interesting extension of the model would be instead one including the Immirzi parameter in the quantum amplitudes, and thus giving a quantization of (the simplicial version of) the Holst action. This parameter, crucial for the LQG formalism, enters prominently in the new spin foam models [START_REF] Freidel | [END_REF]34], and affects significantly both their kinematics and dynamics. One would expect the Immirzi parameter to be straightforwardly introduced by modifying the simplicity map P B only. This generalized map involving the Immirzi parameter is easy to introduce, indeed, but this modification does not seem to be enough to achieve the expected form of the action in our simplicial path integral. The reason seems to be that a modification of the map P h seems needed as well, and in turn this can be understood as due to our weaker imposition of the simplicity constraints. We stress again that, indeed, with a stronger imposition [37] of the same, the maps P h could be avoided altogether and the connection would be completely fixed as a function of the B's. What is less clear at this stage is what should substitute the projections P h or, in other words, what restrictions on the connection degrees of freedom does the inclusion of the Immirzi parameter imply, given our weaker constraints. This is actually rather unclear also in the usual spin foam models [START_REF] Freidel | [END_REF]34,35].

While all of the above concerns the analysis or improvement of our model within the same generalized GFT formalism, an altogether different line of development is suggested by several of its features.

This new direction [START_REF] Baratin | The non-commutative representation for group field theories and simplicial gravity[END_REF][START_REF] Baratin | A new group field theory for 4d quantum gravity[END_REF] stems from the idea that, instead of relaxing the conjugate nature (at the classical level) of B and g variables in the definition of the model, treating them on equal footing as arguments of the GFT field, as we do here, one could instead take this conjugacy relation as the starting point for introducing the B variables in the GFT formalism. This would mean to map the usual group field theories in which the field is a function of group elements into non-commutative and non-local field theories on several copies of the corresponding Lie algebra, using a generalized (non-commutative) Fourier transform, of the type developed in [START_REF] Freidel | [END_REF]. A first motivation for doing so is that, in the model we have presented, we are dealing with the Lie algebra of so(4) (and its subalgebras) as an ordinary vector space, thus neglecting its non-commutative nature; this would lead to suspect that, in doing so, we are neglecting some relevant information that should instead be incorporated i a correct model using the B variables as arguments of the GFT field. More precisely, The non-commutative nature of the algebra would result in a non trivial star product for functions on it, when seen as functions on R 3 (for su( 2)) or R 6 (for so(4)), as we do here. Moreover, the use of a non-trivial star product for the products of fields in the GFT action would result in a different composition rule for vertex amplitudes with propagators in the construction of the Feynman amplitudes of the model, and we have already noticed above how the rather puzzling quantum corrections S c to the BF action appearing in the model we have presented are the result of the way our individual vertex (4-simplex) amplitudes (itself given by a simple BF action) compose. Also, notice that the maps P B could be turned into true projections, in a non-commutative setting, and one could avoid any divergence resulting from them, in particular from the use of an ordinary delta function to impose the closure constraint; this would be due to the fact that the non-commutative delta function on the Lie algebra [START_REF] Freidel | [END_REF] is a perfectly regular function when seen as a function on R 3 (for su(2)) or R 6 (for so(4)). The step to a non-commutative setting could then be motivated from the point of view of regularization only, even in absence of the other mentioned motivations. Finally, we mention another hint for the need to move to a non-commutative setting, arising from the analysis of the model we have presented. It has recently been shown, both in 3d and in 4d [43], that effective scalar field theories on flat non-commutative spaces emerge naturally from group field theories, with the group manifold underlying the GFT providing the momentum degrees of freedom of the effective matter field, while the non-commutative position variables are expected to be related to the conjugate B variables. A similar analysis performed using the generalized GFT formalism we have presented in this paper shows [START_REF] Di Mare | Emergent non-commutative matter from a quantum geometric background[END_REF] that, while everything works fine even for these generalised GFT models as long as one focuses on the group sector, the expected matter field theory fails to emerge if one focuses instead on the Lie algebra sector of the same GFT models. This failure can be indeed traced back to having neglected the non-commutative structure of the same Lie algebra at the level of the fundamental GFT, i.e. in the definition of the generalised formalism itself.

At the same time, we think we should expect several features of the model we have presented to show up also in any new model using the B variables and constructed by means of non-commutative tools. For example, the way the simplicial geometry is implemented at the level of the action and in particular the way the simplicity constraints are implemented, the interplay between simplicity constraints and gauge invariance, and the consequent compatibility relations encoded in the contributions W to the feynman amplitudes.

B. The dynamical triangulations sector

Finally, let us conclude by showing a direct link between our new GFT model and the dynamical triangulations approach. This is made possible by the manifest simplicial geometric meaning of variables and amplitudes, that we can now put to use. We do not consider any additional causality restriction that one could impose on the model in order to obtain a restricted class of Feynman diagrams/triangulations summed over. These restrictions are crucial for the dynamical triangulations approach in its most modern form, called indeed causal dynamical triangulations [START_REF] Ambjorn | [END_REF]. Unfortunately, a field theoretic understanding of them in 4d in terms of some generalization of matrix models, and thus at the GFT level, is lacking, having been only recently achieved in the 2d case [START_REF] Benedetti | [END_REF]. Consequently, we can make a link only with the "old"dynamical triangulations approach: we obtain a model defined as a sum over equilateral triangulations (of arbitrary topology) weighted by a simplicial gravity action (Regge action plus corrections). The 4d DT approach is based on a 2nd order formulation of gravity where both area of triangles and dihedral angles between (d-1)-simplices are functions of fixed edge lengths. Our GFT model (4) corresponds to a 1st order formulation where areas are given by |B f | and dihedral angles are encoded in the connection variables g e . Therefore, in order to reproduce a DT setting we have to: a) kill all connection degrees of freedom, in turn fixing the local Lorentz invariance, and do so in such a way as to obtain dihedral angles in each 4-simplex corresponding to the values φ obtained from the edge lengths in the equilateral case; b) kill "direction"degrees of freedom in the B's and fix their modulus to a constant |B f | equal to the value of the area of triangle f again in the equilateral case. Note that clearly this means having taken already into account, in both the B and g variables, the simplicity constraints. A candidate model for a GFT formulation of dynamical triangulations10 (in the Riemannian case), or for the DT subsector of our GFT model, is: [START_REF]Approaches to Quantum Gravity[END_REF]. From this action, as it can easily be verified, one derives Feynman amplitudes of the same type of (9) without the constraining factors C and W nor any integration over the group or Lie algebra variables. The area of every triangle will be |B| and the angle of holonomy will be θ f = N f φ, with N f the number of 4-simplices sharing the triangle f . These parameters |B| and φ can be of course chosen to match those computed from the edge lengths of an equilateral triangulation. The above model still defines a group field theory, but the group variables g i have the only effect of ensuring the gluing of simplices in the construction of the Feynman diagrams/amplitudes, and then disappear from the same (in the bulk, while they will still label boundary states whose kinematical variables, however, will not have any dynamics). One obtains therefore a sum over equilateral triangulations weighted by an amplitude involving the Regge action and depending on the above two parameters. When taken together with the weighting factor depending on the coupling constant λ in (8) one notices that the GFT coupling constant can be interpreted as the exponential of (i times) the cosmological constant, that ends up multiplying the number of 4-simplices in the triangulation, matching exactly the usual DT construction. A Wick rotation can then easily devised, as in the DT approach to give a euclideanized partition function, suitable for numerical analysis. The main differences from the usual dynamical triangulations approach are: a) the Regge term is augmented by the quantum corrections term S c , function of the same parameters, also resulting from b) the measure term ν depending on the number of simplices per triangle. These terms have to be thoroughly analyzed, but it is a rather general result in the DT approach that its main features do not depend heavily on the details of the measure or of the action chosen, being rather dictated by entropic factors [START_REF] Ambjorn | [END_REF]. c) The holonomy angle θ still enters the amplitudes through its equivalence class [θ f ] (in fact, the expression for the deficit angle in terms of the holonomy angle obtained from dihedral angles would be f = 2π -θ f ). This is a direct consequence of our group-theoretic framework, with fields and propagators defined on a compact group manifold. This model will be analyzed elsewhere.

S = 1 2 Dg φ(g i )K |B| (g)φ(g i ) + λ 5! Dg φ(g 1j )...φ(g 5j ) V h (g) for K |B| (g i ) = 4 i=1 |B| 2 + Gi -1 4 , with fixed |B| and V h (g) = 5 i =j=1 δ(g ij h g -1 ji ) with fixed h = e iφ ∈ U (
However, it already allows us to consider two scenarios, concerning the relation between GFTs and the dynamical triangulations approach.

I) The reduced GFT model matching the dynamical triangulations amplitudes may be still too general to achieve a good continuum limit. This would be the perspective coming from the dynamical triangulations side. In fact, first of all, it lacks the causality restrictions we mentioned above and that proved so crucial in recent developments: it still sums over different topologies and it does not incorporate any foliation structure for the triangulations of trivial topology. Second, it does include, in the configurations summed over, pseudo-manifolds, i.e. manifolds with conical singularities, whose characterizing features are also not understood at the field theoretic level. In this situation, with trivial amplitudes associated to the simplicial complexes, entropic considerations dominate and the model is likely too pathological.

However, a second perspective is possible, coming from our GFT approach. II) One could argue that the model is already too restricted, and that its pathologies are possibly cured within the full model instead. In particular, the presence of non-trivial amplitudes is crucial for both symmetry and renormalization considerations. The restricted model implies a trivialization of the gauge symmetries of the full one, with a consequent trivialization of the Ward identities, that lead to non-trivial relations among amplitudes associated to different Feynman diagrams (simplicial complexes). Complexes that would not be counted twice in the full sum, because of symmetries, are now distinguished in terms of their amplitudes depending on pure combinatorics. Moreover, the non-trivial amplitudes can be crucial in implementing a good renormalization procedure, and in devising suitable approximate regimes in which manifold-like configurations dominate with respect to pseudo-manifold ones. Recent results in 3d confirm this possibility [START_REF] Freidel | [END_REF].

IV. CONCLUSIONS

We have presented a new GFT/spin foam model for 4d quantum gravity. It is based on a recent extension of the GFT formalism, in which the field depends on both group and Lie algebra variables, representing the discrete analogue of the variables in a BF-like formulation of gravity. The new model allows a straightforward implementation of the simplicity constraints that give gravity from BF theory, and has quantum amplitudes with the explicit form of simplicial path integrals for gravity. In doing so it sidesteps some ambiguous aspects of the usual spin foam quantization procedure. The geometric interpretation of the variables and of the different contributions to the quantum amplitudes is also made manifest. Moreover, thanks to the links with other discrete approaches to quantum gravity, that we have explicitly shown, its usefulness and interest may well extend beyond the spin foam or loop quantum gravity framework. While there are still several aspects of the new model that should be analyzed further, and interesting variations of the construction leading to it that can be considered, we believe that the new model represents an altogether new approach to the spin foam quantization of gravity and opens up a whole landscape possible new developments, some of which we have discussed in some detail.

For example, as the parameters labeling the coherent states correspond to the mean value only of the operators corresponding to the B variables, when computed in the same coherent states, one could argue that they can be identified with the same B variables only in some semi-classical approximation, and not in the fundamental theory; consequently, the Plebanski constraints should not be imposed on these parameters, as done in the new spin foam models, strongly.

The model presented in[START_REF] Freidel | [END_REF] was recently given a simplicial path integral form in[28], used in[29] to study its semiclassical limit. The resulting simplicial action is a non-standard BF-like action involving explicitly discrete bivectors B which are, however, defined in term of the group representations and coherent state parameters mentioned above, and not introduced as independent Lie algebra elements and later identified with the above in some approximation, as one would expect. So, the model can still be understood in the context outlined above, and thus subject to the same criticism. Concerning this last model as well as the problems in interpreting it in a path integral context, see the recent[38]

Some of these motivations, and similar technical issues, are shared by the work[30] where a Lagrangian approach to a

4d gravity spin foam model on a given simplicial complex is presented. Along the same line of reasoning, see the recent results of[37,38], appeared after the completion of this work, and that we will discuss more in the following, as on the one hand they subtantiate further the above criticisms, and on the other hand clarify better some of our own results.

We deal here with the Riemannian version of the new model.

This piecewise flat geometric requirements have been also nicely discussed in[25,26], even though they do not seem to be fully implemented in the quantization and thus in the model presented there.

Note that, in absence of the Immirzi parameter, the two new spin foam models[START_REF] Freidel | [END_REF][25][26][27], based on imposing the simplicity constraints on coherent states of the Lorentz group, are distinguished exactly by the above choice.

Analogous compatibility conditions were noticed in[30].

This follows from the fact that each propagator contributing in building up the Feynman amplitude is invariant under the conjugate action of the group G on the homogeneous space S 3 and that it depends on each B only through its modulus.

The re-writing of our quantum amplitudes in pure spin foam form, i.e. as a function of group representations only, can be straightforwardly obtained. This has been done for the simpler but analogous models in[17]. While such re-writing can be useful, it does not modify the content of the model.

It is also possible that restriction to the dynamical triangulations sector of our full GFT model is achieved dynamically, without an explicit restriction of the initial model[17]

Acknowledgements

We thank T. Tlas, with whom part of this line of research has been developed, for many useful discussions, criticisms and suggestions. Special thanks are due to V. Bonzom and M. Smerlak, and to S. Alexandrov, for their interest, insightful criticisms and help. We also thank A. Baratin, B. Dittrich, J. Engle, E. Livine, K. Noui and C. Rovelli for comments on this work. Financial support from the A. Von Humboldt Foundation through a Sofja Kovalevskaja Prize is gratefully acknowledged.