{'fraction_non_alphanumeric': 0.04684289823311875, 'fraction_numerical': 0.011322649865315254, 'mean_word_length': 3.805616498464239, 'pattern_counts': {'":': 1, '<': 13, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantizable non-local gravity

It's widely recognized that general relativity emerges if we impose invariance under local translations and local Lorentz transformations. In the same manner supergravity arises when we impose invariance under local supersymmetry. In this paper we show how to treat general relativity as a common gauge theory, without introducing a metric or a tetrad field. The price to pay for such simplification is the acceptance of non-locality. At first glance the resulting theory seems renormalizable. Finally we derive Feynman vertices for such theory.


I. INTRODUCTION
Loop quantum gravity has so far achieved splendid results in the description of the kinematic of pure gravity states in terms of colored graphs, whose nodes represent quanta of volume connected by links representing quanta of area [1]. However, it is still incomplete as far as the the dynamics is concerned. A hamiltonian operator has been described by T. Thiemann [2] for the formulation of General Relativity via Ashtekar variables but, in spite of the formal definition, we have not yet been able to decipher the precise action on the states because of its complexity. In alternative, a quantum dynamics has been constructed by describing gravity as a topological theory (BF-theory) with constraints on the states [3]; in particular, he we will refer to the work by Engle Pereira and Rovelli (E.P.R.) in [4]. The amplitude obtained, sometimes referred to as E.P.R. has the correct low energy limit [11]. But so far the results are limited to a formulation via path integral over space-times of Regge type. Here we take some steps towards the derivation of the E.P.R. amplitude from the hamiltonian theory.
We start in section (II) with a standard introduction on the spin-network as functions on a compact group G and we define the approximation of a spin-network on a lattice. We perform a series of integrals on G over products of holonomies along the squares of the lattice. This allows us to calculate the expectation value of an operator that will revealed itself in section (III) as the projector on the physical states. For doing this we extend at the general case the approach adopted by Karim Noui and Alejandro Perez in [6] for the three dimensional case. In section (III) we consider BF-theory and the classical constraints that give General Relativity starting from this theory. In the quantum theory, these constraints have to be imposed on the physical states. We choose to impose them already on the kinematical states. We extract some indication for doing this from the work of Jonathan Engle, Etera Livine, Roberto Pereira, Carlo Rovelli [4] where is largely explored the case of 4-valent spin-networks. We get a projector on the physical states and we calculate a physical scalar product between the two simpler 4-valent spin-networks, obtaining the same amplitude obtained in [4] via path integral over space-times of Regge type, called E.P.R. amplitude. For another attempt to relate the spin-foam vertex with a hamiltonian operator see [10].

II. SPIN-NETWORKS
We use greek letters for spacetime indices and latin letters for space indices and follow the approach of [8]. We consider S an oriented smooth m-dimensional manifold representing space, parameterized with coordinates x a , a = 1, . . . , m. A finite collection γ of real-analytic paths γ i such that with γ i (τ ) points of S, form a graph in S if they intersect, it at all, only in their endpoints. We then call them links and call their endpoints vertices. Given a node v, we say a link γ i is outgoing from v if γ i (0) = v, and we say γ i is incoming to v if γ i (1) = v. We take a connection A of a compact group G. We can think of the holonomies along these paths as elements of G. The functionals of the form form an algebra, that we call F n (A), that is isomorphic to the algebra of all continuous complex-valued functions on G n . Given two function in this algebra, we can thus define their inner product by arXiv:1003.2056v2 [gr-qc] 31 Mar 2010 where the integral is done using normalized Haar measure on G n . Obviously if ψ ∈ F n (A) we have also ψ ∈ F k (A) if k > n. This is because we can see F n (A) as a F k (A), constant with respect to the group elements n + 1, . . . , k. In this way it is possible to extend the definition of the scalar product to arbitrary couple of function, respectively in F n (A) and F m (A), by considering both as element of F k (A) with k ≥ m, n. It is easy to see that this is independent from the choice of k.
A spin-network in S consist of: • a graph γ in S; • for each link e of γ, an irreducible representation ρ e ; • for each node v of γ, an intertwiner, that is an operator which maps where e 1 . . . e p are the links incoming to v and e 1 , . . . e q are the links outgoing from v.
A way to get a function in F n (A) from a spin-network in S is to take the holonomy along each link of γ, think of it as a group element, and write it in the representation labeling the link. Picking a basis for this representation we think of the result as a matrix with one superscript (representing the beginning of the link) and one subscript (representing the ending of the link). In addition we write the intertwiner for every node as a tensor and contract all indices.
It's easy to see that a normalized state (respect to (2)) can be obtained by multiplying This holds because with i a normalized intertwiner between the representations ρ . . . σ. Hence the normalized intertwiner between only two representation is the "delta" divided by the square root of the dimensions of the representation These functions are gauge invariant. In [7] it is shown that these functions span L 2 (A/G), with A/G the space of the connections in S modulo gauge transformations. So, we have a one to one correspondence from spin networks to functions in L 2 (A/G) and we can speak about a spin-network meaning the relative function in L 2 (A/G). We consider the image in R m of a part of the graph γ contained in a open set of an atlas covered S. It is given by the x a (γ). We say that two images made with x and y coordinates are equivalent if exists a continuous, piecewise differentiable, map from R m to R m , y : x → y(x) such that y a (γ i ) = y a [x(γ i )] for every γ i ∈ γ. This take into account for the different possible choices of coordinates. A change in the coordinates (or a diffeomorphism), deform the image of the graph in R m without change its corresponding abstract graph, that is the combinatorial relation between links and nodes. The "physical scalar product" that we construct in this work depend only on abstract graphs and therefore it is diffeomorphism-invariant.
Taken an m-dimensional lattice with step , we can always approximate such an image with links of the lattice. In the limit → 0 we recover the smooth image. In this framework we can consider a minimal link of length as an element (a 1 , . . . , a m , z+) where (a 1 . . . , a m ) ∈ N m labels the starting point, z = 1, . . . , m indicates the direction and ± the orientation, with (a 1 , . . . , a z , . . . , a m , z+) = (a 1 , . . . , a z + 1, . . . , a m , z−). We indicate with U (a,...,d,e,...,m,d+) the holonomy of the link (a, . . . , d, e, . . . , m, d+). We consider the holonomy in the representation σ along one of this minimal link, we say R(U ) σ We indicate the product of the holonomies associated to the side of the square as U d+e+ (a,...,d,e,...,m) . We have the simple result graphically in figure (1), where we have used ..,d,e,...,m) .

FIG. 1:
With little work we can see that any 2 graphs related by a diffeomorphism can be related using a sequence of such integrations. Similarly, the product of two deltas along two squares with a common link, integrated in the common link, results in a delta along the rectangle (U REC ) containing the two squares So, if we consider a product of delta functions associates to all the squares of a 2-dimensional region on the lattice, and we integrate on the internal links, we obtain a delta over the holonomy along the boundary of the region. This is illustrated in figure (2) Consider now the following scalar product between two spin-networks The sum is over "indipendent" squares. By this we man the following. We write the deltas one square at a time and, when we should consider a square of which all sides have been already drawn from other squares occupied by deltas, we don't write it. We continue until we touch all the possible squares of the lattice. This procedure is largely arbitrary, depending on the sequence that we follow in considering the squares, but we are satisfied if we can calculated (9) at least for one sequence. This procedure avoid situation like this: you takes a cube and the six deltas on its faces and integrate on the twelve group elements of the links. The result is an infinitive constant. This is because, if we integrate the common links of the lower and lateral faces, we obtain a delta on the upper face. But we have already a delta on this face! In this way we have an integral of the type The formula (9) contains an integration for every link of the lattice and a delta for every independent square of the lattice. Now we note that some link appears only in the deltas and not in the initial and final states, we call them "free links". Integrating on this links and using repeatedly the result (8) we obtain where for faces we intend the closed figures formed by the links of the graph γ 1 ∪ γ 2 . Indeed if, by absurd, after this integration we would obtain deltas over closed paths whose links don't overlap the links of the graph, we would have some links of these paths that isn't in the final or initial state. But this is impossible because we have already integrated over this free links. The arbitrarily in the choice of the sequence of squares reflect an arbitrarily in the sequence of faces, following the same idea that a face doesn't appear if its contour is drawn by the other faces.
Obviously, in the calculation we obtain some "infinity": for example the integration of a delta over a path which goes to infinitive, over a free link at the infinite, gives 1. But, after this integration, remain so much integrations over now empty links at the infinitive of that canceled path, which generate an "infinite" constant, that we call "W" 1 .
Let us now restrict the dimensions of the space to 3 and to the case where the spin-network are constructed on the dual 1-skeleton of a triangulated manifold or a Regge space. As before, we start with a 3-dimensional real-analytic manifold S representing space. Given any triangulation of S we can choose a graph in S called the "dual 1-skeleton", having one node at the center of each 3-simplex and one link intersecting each (3 − 1)-symplex. In a 3-dimensional space we have a node for every tetrahedron and an link for every face. Given two nodes, the link which connects them correspond at the common face of the two tetrahedron. Note that all the vertices are quadrivalent. In this case the simplest transition from a normalized (truncated) initial state (the left) to a normalized final (truncated) state (the right) is represented in the figure (3) and gives, for the scalar product In figure (4)   We can now describe the amplitude (9) computed on these states using a spin-foam language. Let us start joining the initial and the final states by their external lines (the truncated parts). Returning at the triangulated manifold, this mean to identify the free face of each tetrahedron in the final states with one face of the initial tetrahedron, obtaining a 4 dimensional figure called 4-symplex, part of a Regge-spacetime. From a 4-symplex we pass at the 2complex or spin-foam, that is its dual. For construct the dual we draw a vertex in the center of the 4-symplex and, for every tetrahedron, we draw an edge which intersect its and is incident to the vertex. These edges form the boundaries of the faces of the symplex. In this case we can see the faces as marked by a couple of edges, see (6). We note that every face marked by a couple of edges corresponding to final tetrahedrons (the black upper ones) includes a link in the final spin-network. So we associate the spins of these links to these faces. In a similar way every face marked by a mixed couple includes a link in the initial state and one in the final state with the same spin. We associate these spins to these faces. Furthermore every node of the spin-networks bounds an edge of the spin-foam and so we associate the spins of the corresponding nodes to the edges. In the spin-foam formalism we associate an amplitude to the 2-complex, understood as a product of a faces amplitude, an edges amplitude and a vertex amplitude. The vertex amplitude is the part of this that is unfactorizable in terms of faces and edges. The unfactorizable part in (12) is the 15j. The spins which enter in the symbol are the six spins from the final states, the four spins from the initial state and the five spins of the spin-networks nodes. As we have seen, these 15 spins can be associated to the ten faces of the 4-symplex and to its five edges. We remain with the six factors √ dim ρ from the normalization of the non-truncated part. We can understand them as the faces amplitude in the spin-foam.
In the spin-foam framework we have a condition on the amplitude for preserving the unitarity: we image of separating an arbitrary 2-complex in two parts cutting it with a plane. The total amplitude has to be described as the product of the amplitudes of the two parts. This imposes that the face amplitude for the faces border by a link included in one of the two boundary states has to return a factor (dim ρ) n/2 if an internal face gives (dim ρ) n . This is because, dividing a 2-complex in two parts, we can divide an internal face in two parts. Every parts will be bordered by a link in a boundary state. The amplitude of the original internal face has to be the product of the amplitudes of the resulting faces bounded by a link in a boundary state. The relation (dim ρ) n = (dim ρ) n/2 · (dim ρ) n/2 ensures this. For the same reason, a face bounded by a link in the initial state and a link in the final state has to return an amplitude equal to 1. If it is not so, we can divide the 2-complex with an infinite number of planes, generating an infinite number of faces with the same character. In this way the amplitude would explode. The amplitude (12) respects this conditions with n = 1. Notice that the four faces of the final tetrahedron do not have associated deltas, because of our choice of associating deltas only to independent faces.
So far we have shown that the scalar product (9) between spin-network states can be expressed in terms of a spin-foam with vertex amplitude given be a 15j-symbol. Let us can see how this is related to quantum general relativity.

III. BF-THEORY AND GENERAL RELATIVITY
We assume that the Lie Algebra g of the group G is equipped with an invariant nondegenerate bilinear form < ·, · >. The action for BF-theory is where F (A) is the curvature of A, that is a g-valued two-form, and B is a g-valued (m − 1) form. In the 4-dimensional-SO(4) case we have where we have set abc B bc = E a and B 0c = B c . We have two first class constraints D a E a = 0 which imposes gauge invariance respect to transformations of G, and which codes the dynamic and diffeomorphisms-invariance. Riemaniann general relativity can be obtained by imposing that B has the form where IJ are indices of the so(4) algebra, β is the Immirzi parameter and * is an Hodge dual which acts in the Lie-algebra indices ( * B IJ µν = IJ KL B KL µν ). This decomposition is possible if and only if * B µν · B µν 1 + 1 for some direction n I . No sum on repeated indices is assumed. This can be rewritten as * E a · E a 1 + 1 We see that In the quantum theory we can represent E a as .
Acting on the holonomy state e γ 1 A and choosing n I = δ 0 I we see that the (21) and (22) can rewrite as where L I = 1 2 I KL J KL , K I = J 0I are respectively the generators of the SO(3) subgroup that leaves n I invariant and the generators of the corresponding boosts. C 1 and C 2 are the Casimir and pseudo-Casimir operators of g The constraint (24) are of second class, but we can substitute them with the first class master-constraint Substituted the (23) we have The (23) is solved, up to h corrections by where (j + , j − ) label the unitary representation of SO(4). So the (23) poses only a restriction on the representations in the spin-network states. For β > 0 the (28) imposes that for 0 < β < 1 corresponds to That is, the constraint selects the highest irreducible in the decomposition of the representation space, that we call H (j + ,j − ) , when viewed as the carrying space of a reducible representation of the SO (3) subgroup This can be implemented posing a particular form on the intertwiners k-valent in the spin-networks (3). For k = 4 this node reduces to the E.P.R.-node. Once we have imposed these constraints we can implement the dynamics. Equation (18) becomes The approximation of this condition on a lattice is It gives (remember that the delta on the group has support on the unity) the physical scalar product; that is It simplifies, as we have seen in the discussion after equation (8), to is called a projector on the physical states. At this point we can forget the lattice. If the states are the 4-valent basic graphs that we have seen in the previous section we obtain now the E.P.R. amplitude with the 15j-symbol of SO(4) which splits into two 15j-symbols of SO(3) contracted in a way given by the E.P.R. node, that is (1 + β)j ab 2 ; i + a 15j where a, b = 1, . . . , 5 are abstract labels for identify the a − th intertwiner or the contraction from the a − th and the b − th. The label f indicates the faces of the spin-foam bounded by the initial and final states. Moreover In this way we have recovered the E.P.R. spin-foam transition amplitude of reference [4]. In this framework we apparently have not need to insert a lot of complete sets between initial and final state: the only difficult for doing the calculation (33) seems to be finding the closed surfaces in the union of the graphs. But we consider < s 1 |P |s 2 >=< s 1 |P P |s 2 >= i < s 1 |P |s i >< s i |P |s 2 > .
The sum is on all the spin-networks s i , not only on those which satisfied the restrictions on the representations, and so the scalar product is incorrect. The solution can be found by inserting infinite projectors and complete sets, restricting all the sums to the states that satisfied the restrictions.
< s 1 |P |s 2 > P hys = i1,...,in < s 1 |P |s i1 >< s i1 |P |s i2 >< s i2 |P . . . P |s n−1 >< s n−1 |P |s in >< s in |P |s 2 > (36) Some "internal" scalar product will be trivially equal to one. Discarding these we obtain a sum on all the non-trivial sequence of spin-networks which connect the initial and final states. In this way we recover, calculating the single physical scalar products, the ordinary sum on spin-foams connecting the external spin-networks.

IV. CONCLUSIONS
We have shown how to reconstruct the E.P.R. spin-foam amplitude of reference [4] starting from a formulation of the canonical theory. We can add a few remarks: • The final amplitude is expressed in terms of invariant quantities of the group theory for G. Therefore it can be generalized to non compact groups. We can pass freely, for example, from SO(4) to SO(1, 3), substituting the relative invariants for the ones with those of the second. The unitary representation for the last are labeled by (n, ρ), with n positive integer and ρ real. In this case the condition (23) becomes: and fixes ρ = βn or ρ = −n/β. This makes the area spectrum discretized also in the lorentzian theory [4].
• The form of the scalar product (33) doesn't depend to the dimension of the space S. So we can in principle calculate the probability for a state to pass from a graph embedded in a N -dimensional space to a graph embedded in a N + 1 dimensional space. In this way we can study if a 3-dimensional space is statistically favored.
• The theory can be extended to gauge interactions. We can move from a 4-dimensional manifold M 4 to an 11-dimensional one with topology of M 4 × S 1 × S 2 × CP 2 . The form B becomes B IJ µ1,...,µ9 = * T β (b I ∧ b J ) µ1,...,µ9 + * L (b µ1 . . . b µ9 ) IJ where the dual * T acts on tangent space indices and * L on Lie algebra indices. The theory classically admits a solution of the form α, β, Σ = 1, . . . , 11 a, b, A = 1, . . . , 4 i, j, l, I = 5, . . . , 11 Capital letters indicate Lie SO(11) algebra indices. Indices r, s move on the 1 + 3 + 8 = 12 generators of U (1) × SU (2) × SU (3) with gauge fields Q r and Killing vectors K r . We suppose that the extra dimensions have size ∼ L P LAN CK and, at the leading order (L 0 P LAN CK ), we recover the standard model. The dual spin-networks will be of valence 11 with every link intersecting a 9-symplex. Only 4 links for nodes intersect (in a single point) a triangle in M 4 and a 7-symplex in S 1 × S 2 × CP 2 . The others 7 intersects (in a single point) a tetrahedron in M 4 and a 6-symplex in S 1 × S 2 × CP 2 . We recover the semiclassical limit putting j → ∞ for the first four and j ∼ L 2 P LAN CK for the last seven. We conjecture that, in this limit, the dominant contribution comes from intertwiners of the form i m1,...,m11 = h m1,...,m4 × k m5,...,m11 , where intertwiners h glue themselves by links j → ∞ only with intertwiners h, and intertwiners k gluing themselves by links j ∼ L 2 P L only with intertwiners k. Moreover, all the intertwiners h are also intertwiners of SO(4), because SO(4) ⊂ SO (11). It will be interesting to discover if this subset of SO(4) intertwiners corresponds or not (after solving the simplicity contraints) to the set of E.P.R. intertwiners.