aa r X i v : . [ g r- q c ] J u l Ponzano-Regge Model on Manifold with Torsion
T Vargas A.
Grupo de Astronom´ıa-SPACEFacultad de Ciencias F´ısicasUniversidad Nacional Mayor de San MarcosAv. Universitaria s/n. Ciudad Universitaria, Lima 1.Lima-Peru
Abstract.
The connection between angular momentum in quantum mechanics and geometricobjects is extended to manifold with torsion. First, we notice the relation betweenthe 6 j symbol and Regge’s discrete version of the action functional of Euclidean threedimensional gravity with torsion, then consider the Ponzano and Regge asymptoticformula for the Wigner 6 j symbol on this simplicial manifold with torsion. Inthis approach, a three dimensional manifold M is decomposed into a collection oftetrahedra, and it is assumed that each tetrahedron is filled in with flat space andthe torsion of M is concentrated on the edges of the tetrahedron, the length of theedge is chosen to be proportional to the length of the angular momentum vector insemiclassical limit. The Einstein-Hilbert action is then a function of the angularmomentum and the Burgers vector of dislocation, and it is given by summing theRegge action over all tetrahedra in M . We also discuss the asymptotic approximationof the partition function and their relation to the Feynmann path integral for simplicialmanifold with torsion without cosmological constant. onzano-Regge Model on Manifold with Torsion
1. Introduction
The connection between angular momentum in quantum mechanics and geometricobjects have been observed since long ago by Wigner [1]. In 1968, Ponzano and Regge [2]built a quantum gravity model in three dimensional Riemanian maniforld using theproperties of some invariants which can be obtained from the fundamental representationof the group SU (2). Noting an interesting connection between the 6 j -symbols for spinand the 3-dimensional Regge action for gravity [3], they formulated the Feynman pathintegral for three-dimensional simplicial quantum gravity in terms of a product of 6 j -symbols. Hasslacher and Perry [4] not only elucidated further details of this model andits relation to Penrose’s spin netwoks [5], but also discussed the possibility of spacetimefoam arising from this Ponzano and Regge model. At the early 90’s Turaev and Viro [6]proposed a generalization of this theory formulated in terms of quantum 6 j -symbolsand showed that this generalization provided new 3-manifold invariants. Ooguri [7]demonstrated that the Ponzano-Regge partition function is equivalent to Witten’s 2+1formulation of gravity [8] on closed orientable manifolds. Barrett and Crane [9] pointedout the relation between 6j-symbols and a discrete version of the Wheeler-de Wittequation, thereby giving further insight into the relation of Ponzano-Regge theory to3-dimensional gravity. Also in 2004, Freidel and Louapre [10] considered the Ponzano-Regge model in the context of spin foam quantization of three dimensional gravitycoupled to quantum interacting spinning particles.On the other hand, there is an alternative approach to gravitation based on theWeitzenb¨ock geometry ¸we, a manifold with torsion without curvature, which is so calledteleparallel gravity. In this theory, gravitation is attributed to torsion [12], which playsthe role of a force [13], and the curvature tensor vanishes identically. As is well known, atleast in the absence of spinor fields, teleparallel gravity is equivalent to general relativity.In this paper, relying upon this equivalence, first we will obtain the relation betweenthe 6 j symbol and Regge’s discrete version of the action functional of Euclidean threedimensional gravity with torsion without cosmological constant, then considere thePonzano and Regge asymptotic formula for the Wigner 6 j symbol on this simplicialmanifold with torsion. In this approach, a three dimensional Weitzenb¨ock manifold Mis decomposed into a collection of tetrahedra, and it is assumed that each tetrahedronis filled in with flat space and the torsion of M is concentrated on the edges of thetetrahedron, the length of the edge is chosen to be proportional to the length of theangular momentum vector in semiclassical limit. The Einstein-Hilbert action is then afunction of the angular momentum and the Burgers vector of dislocation, and it is givenby summing the Regge action over all tetrahedra in M .It is interesting to note that there is a relation between an angle ε i , which isformed by the outward normals of two faces sharing the i − th edge and the distance b i through which a vector is translated from its original position when parallel transportedaround a small loop of area Σ ∗ i . The partition function is invariant under a refinementof a simplicial decomposition, more specifically, it is invariant the 1 − − onzano-Regge Model on Manifold with Torsion b d . The resulting simplicial action is considered as theteleparallel equivalent of the simplicial Einstein’s action of general relativity. We willproceed according to the following scheme. In section 2, we review the main featuresof teleparallel gravity. In section 3, we obtain the simplicial torsion and the discreteaction. In section 4, we considere the Ponzano-Regge asymptotic formula for the Wigner6 j symbol on this simplicial manifold with torsion. Discussions and conclusions arepresented in section 5.
2. Teleparallel Equivalent of General Relativity
It is well known that curvature, according to general relativity, is used to geometrizethe gravitational interaction. On the other hand, teleparallelism attributes gravitationto torsion, but in this case torsion accounts for gravitation not by geometrizing theinteraction, but by acting as a force ¸pe. This means that in the teleparallel equivalentof general relativity, instead of geodesics, there are force equations quite analogous tothe Lorentz force equation of electrodynamics.A nontrivial tetrad field induces on spacetime a teleparallel structure which isdirectly related to the presence of the gravitational field. In this case, tensor and localtangent indices ‡ can be changed into each other with the use of a tetrad field h aµ . Anontrivial triad field can be used to define the linear Weitzenb¨ock connectionΓ σµν = h aσ ∂ ν h aµ , (1)a connection presenting torsion, but no curvature. It parallel transports the tetrad itself: ∇ ν h aµ ≡ ∂ ν h aµ − Γ ρµν h aρ = 0 . (2)The Weitzenb¨ock connection satisfies the relationΓ σµν = ◦ Γ σµν + K σµν , (3)where ◦ Γ σµν = 12 g σρ [ ∂ µ g ρν + ∂ ν g ρµ − ∂ ρ g µν ] (4) ‡ The greek alphabet ( µ , ν , ρ , · · · = 1 , ,
3) will be used to denote tensor indices and the latin alphabet( a , b , c , · · · = 1 , ,
3) to denote local tangent space indices, whose metric tensor is chosen to be η ab = diag(+1 , +1 , +1). Furthermore, we will use units in which c = 1. onzano-Regge Model on Manifold with Torsion g µν = η ab h aµ h bν , (5)and K σµν = 12 [ T µσν + T νσµ − T σµν ] (6)is the contorsion tensor, with T σµν = Γ σνµ − Γ σµν (7)the torsion of the Weitzenb¨ock connection.Since here we are interested in Euclidean three dimensional teleparallel gravity, wedefine the action on the manifold with torsion as S = Z d x L T . (8)Here L T is the teleparallel gravitational lagrangian given by L T = h πG (cid:20) T ρµν T ρµν + 12 T ρµν T νµρ − T ρµρ T νµν (cid:21) , (9)with h = det( h aµ ). These are all necessary information we need about the teleparallelgravity, since only the discrete version of the action (8) is required in both asymptoticapproximations of 6 j symbol and the partition function.
3. Discrete torsion and simplicial action
In order to gain further insight into the simplicial manifold with torsion we begin byreviewing the Regge calculus on which the derivation of simplicial teleparallel actionis based [14]. Similarly to the Regge construction of the simplicial manifold of generalrelativity, we assume that the usual continuous spacetime manifold with torsion can beviewed as the limit of a suitable sequence of discrete lattices composed of an increasingnumber of smaller an smaller simplices. In general, the Weitzenb¨ock manifold, whichis the stage set of teleparallel gravity, is approximated by a D -dimensional polyhedra M D . In this approach, the interior of each simplex is assumed to be flat, and this flat D -simplices are joined together at the D -hedral faces of their boundaries. The torsionturns out to be localized in the D − l between any pair of vertices serve as independent variables.To begin with, let us take a bundle of parallel dislocations (hinges) in M . Wemake the assumption that the torsion induced by the dislocations is small, so that wemay regard M as approximately euclidian. Let U be a unity vector parallel to thedislocations. We test for the presence of torsion by carrying a vector A around a smallloop of area vector S = S n , with S the area and n a unity vector normal to the surface.At the end of the test, if torsion is nonvanishing, A is found to have translated from theoriginal position, along U , by the length B = N b , where N is the number of dislocationsentangled by the loop, and b is the Burgers vector, which is a vector that gives both onzano-Regge Model on Manifold with Torsion M , the flux ofdislocation lines through the loop of area S αβ isΦ = ρ ( US ) = 12 ρ αβ S αβ , where ρ is the density of dislocation passing through the loop, and ρ αβ = ρ U αβ , with U αβ a unity antisymmetric tensor satisfying U αβ U αβ = 2. This means that we canendow the polyhedra more densely with hinges in a region of high torsion than in regionof low torsion. The closure failure is then found to be B µ = 12 ρ αβ S αβ b µ . (10)However, we know from differential geometry that, in the presence of torsion,infinitesimal parallelograms in spacetime do not close, the closure failure being equal to B µ = T µνσ S νσ . (11)By comparing the last two equations we see that T µαβ = 12 ρ αβ b µ ≡ ρ U αβ b µ . (12)On the other hand, it has already been shown t¸ that the torsion singularity takesthe form of a conical singularity. Consequently, the dislocation from the original positionthat occurs when a vector is parallel transported around a small loop encircling a givenbone is independent of the area of the loop. Furthermore, the dislocation has the nextmain characteristics: The length of the dislocation line in three-dimensions, and the areaof the triangle in four-dimensions. Therefore, there is a natural unique volume associatedwith each dislocation. To define this volume, there is a well-known procedure in whicha dual lattice is constructed for any given lattice ch,mi¸ . This involves constructingpolyhedral cells around each vertex, known in the literature as Voronoi polygon, in sucha way that the polygon around each particular vertex contains all points which arenearer to that vertex than to any other vertex. The boundary of the Voronoi polygonis always perpendicular to the edges emanating from the vertex, and each corner ofthe Voronoi polygon lies at the circuncentre of any of the simplices of the Delaunaygeometry, which shares the dislocation (bone) (see Fig. 1).By construction, the Voronoi polygon is orthogonal to the bone. If we paralleltransport a vector around the perimeter of a Voronoi polygon of area Σ ∗ d , it will traversethe flat geometry of the interior of each one of the simplices sharing the bone, and willreturn dislocated from its original position in a plane parallel to the bone by a length b µ . According to this construction, and relying on the definition (12), the torsion dueto each dislocation can be expressed by (see Fig. 2):(Torsion) = (Distance the vector is translated)(Area circumnavigated) . (13)On the other hand, it is well known that the Riemann scalar is proportional to theGauss curvature, and proportionality constant depends on the dimension D of the latticegeometry mi¸ . In similar way, we define the simplicial torsion due to each dislocation by T ( d ) µνρ = q D ( D − b ( d ) µ U ( d ) νρ Σ ∗ d ≡ √ b ( d ) µ U ( d ) νρ Σ ∗ d . (14) onzano-Regge Model on Manifold with Torsion A Figure 1.
The two-dimensional Voronoi polygon (dashed line) for a particular vertex A . Each corner of the Voronoi polygon lies at the circuncentre of any of the trianglesof the Delaunay geometry (solid line). The reason for the square root is that, as is well known from teleparallel gravity, theRiemann curvature tensor is proportional to a combination of squared torsion tensors.As we have said, the vector returns translated from its original position in a planeparallel to the hinge by a length b µ . Let us then analyze the translational group actingon it. As we know, the interior of each block is flat, so the infinitesimal translation inthese blocks is given by T ( δb ) = I − i δb a P a , (15)where I is the unity matrix, δb a are the components of an arbitrarily small three-dimensional Burgers (displacement) vector, and P a = i∂ a are the translation generators.In the presence of dislocations, and using the tetrad h aµ , this infinitesimal translationparallel to the hinge becomes T ( δb ) = I − i δb µ h aµ P a , (16)so that a finite translation will be represented by the group element T ( b ) = exp [ − i b µ h aµ P a ] . (17)On the other hand, the contour integral of the Burgers vector — which measures howmuch the infinitesimal closed contour Γ spanning a surface element d Σ ∗ αβ fails to closein the presence of hinge — by using Eq. (11), is seen to be kle¸ b µ = I Γ T µαβ d Σ ∗ αβ . (18)Therefore, the group element of translations due to torsion turns out to be T ( b ) = exp − i I Γ P a T aαβ d Σ ∗ αβ . (19) onzano-Regge Model on Manifold with Torsion 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d b A A
Figure 2.
The parallel transport of a vector A along the perimeter of a Voronoipolygon (dashed line) around the dislocation line in three-dimension. The vectorreturns translated in a plane parallel to the dislocation d by the length b . The D -volume Ω d associated with each dislocation, as described above, isproportional to the product of Σ d , the D − ∗ d , the area of the Voronoi polygon mi¸ :Ω d ≡ D ( D −
1) Σ d Σ ∗ d = 13 Σ d Σ ∗ d . (20)The invariant volume element h d x , therefore, is represented by Ω d , and we have thefollowing relation, Z h d x = ⇒ X dis Ω d = 13 X dis Σ d Σ ∗ d , (21)where the sum is made over all dislocations. We are ready then for constructing thesimplicial action. Let us take the lagrangian (9) of teleparallel gravity, whose termsare proportional to the square of the torsion tensor, and substitute torsion as given byEq. (14). For the first term, we obtain T µνρ ( d ) T ( d ) µνρ = 6 ∗ d ! b µ ( d ) b ( d ) µ . (22)Writing the other two terms in a similar way, the simplicial teleparallel action will be S T = 116 πG X dis b d Σ ∗ d ! l d , (23)where b d denotes the projected Burgers vector parallel to the hinge, and l d is thedislocation line, or hinge.
4. Ponzano-Regge model
Ponzano and Regge [2] studied a model in which the simplicial bloks of three-dimensionalRiemanian manifold are 3-dimensional tetrahedra. Each edge of a tetrahedron is labelled onzano-Regge Model on Manifold with Torsion j j j j j j Figure 3.
Geometric representation of 6 j -coefficients. by a half integer j , corresponding to the (2 j +1)-dimensional fundamental representationof the group SU (2), such that q j ( j + 1)¯ h ≈ ( j + )¯ h for large j is the descritized lengthof that edge. A tetrahedron with four vertices, six edges and four triangular faces is thenthe natural geometric representation of the recoupling coefficients between four angularmomenta (see Fig. 3). There is thus a one-to-one correspondence betwen the numberof edges of tetrahedron and the number of arguments of the 6 j -symbol, namely l i = ( j i + 12 )¯ h , i = 1 , , ..., . (24)These lengths must satisfy the triangle inequalities corresponding to the triangularfaces of the tetrahedron, then 6 j -symbol is only defined when the values of the angularmomenta triades which correspond to the edge lengths around a face astisfy the triangleinequalities, | j − j | ≤ j ≤ j + j . This implies that the edge lenghts around a facesatisfy the triangular inequalities, up to additional terms of ± : though four momentatriads satisfy triangle inequalities, the same triades shifted by 1 / j symbol is said to be classically forbidden, and it is exponentially suppressedat large j i . Also, j , j , j are required to satisfy j + j + j =integer, for each face.These inequalities for the angular momentum guarantees that the edges l , l , l oftetrahedron form a closed triangle af non-zero surface area: | l − l | < l < l + l . Ifthese inequalities are not satisfied, the value of the 6 j -symbol is defined to be zero.Ponzano and Regge main porpuse was to evaluate the asymptotic limit of 6 j -symbolwhen all six arguments j , j , ...... , j are large or when j i lie in the classically allowedregion, and the asymptotic formula obtained was j j j j j j ≃ vuut ¯ h πV ( j ) cos " h X i =1 ( j i + 12 ) ε i + π , (25) onzano-Regge Model on Manifold with Torsion j j j j j j Figure 4. ε i is the angle between the outward normals of the tetrahedral faces whichhave the edge j thi in common. where ε i is the interior angle between the outward normals of the tetrahedral facessharing the l thi edge (see Fig. 4).Because we are interested in the large j i limit, the shifts by 1 / V ( j ) can be found from the Cayley formula: V ( j ) = 1288 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j j j j j j j j j j j j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (26)In order to obtain a meaninful non-imaginary result from equation (25) the tetrahedralvolume must be real. It means that, tetrahedra with non-negative V can be embeddedin a three-dimensional Euclidean space, while the same can not be done with tetrahedrawith negative V . However, the V < S R = 18 πG X hinge ε h l h , (27) onzano-Regge Model on Manifold with Torsion ε h is the deficit angle associated to each hinge, which is directly related to thecurvature of spacetime, and for the tetrahedron using (24) it is given by S R = 18 πG X i =1 l i ε i = 18 πG X i =1 ( j i + 12 ) ε i , (28)which means that, the gravitational contribution at each edge of the tetrahedron is l i ε i = ( j i + ) ε i . Therefore, the semiclasssical approximation (25) is equal to the cosineof the Regge action for a single tetrahedron up to a constant factor and a phase shift:cos (cid:16) S R ¯ h + π (cid:17) .We now turn to the Ponzano-Regge model on manifold with torsion. In a previoussection we have seen that the usual continuous spacetime manifold with torsion can beviewed as the limit of a suitable sequence of discrete lattices composed of an increasingnumber of smaller an smaller simplices, and in a vacuum three-dimensional case, thetorsion tensor is localized in one-dimensional dislocation line l i , called hinges. Also wehave demostrated that when torsion is present, it is detected a dislocation parallel tothis hinge, and this dislocation is measured by the Burgers vector b d . From these set of l i , let us choose six of them in such a way that they must satisfy triangle inequalities.More precisely, let l , l , l , l , l , l be non-negative integers and an unordered triades ofthis family of dislocation lines written as ( l i , l j , l k ) with i, j, k = 1 , , ..., i = j = k ,is said to be admissible if the triangular inequalities | l j − l k | < l i < l j + l k are met.These admissible l i are the edge lengths of the tetrahedron and also they completelycharacterizes it in Euclidean 3-space: the triads ( l , l , l ), ( l , l , l ), ( l , l , l ) and( l , l , l ) form the four faces of the tetrahedron and the following determinant V ( l ) = 1288 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l l l l l l l l l l l l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (29)must be positive.Then, the simplicial teleparallel action (23) for the tetrahedron reduces to S T = 116 πG X i =1 l i b i Σ ∗ i ! , (30)where l i , b i are the edge length and the closure failure or gap at the edge correspondingly.Σ ∗ i is the area of a Voronoi polygon which is orthogonal to the edge. As mentioned inthe introduction of this section, the tetrahedron is the natural geometric representationof the 6 j -symbol, and there is one-to-one correspondence between the number oftetrahedral edges and the number of arguments of the 6 j -symbol given by (24). Then,the Regge action may be re-expressed as the sum of the gravitational contribution fromeach edge of the tetrahedron: S T = 116 πG X i =1 ( j i + 12 ) b i Σ ∗ i ! . (31) onzano-Regge Model on Manifold with Torsion N internal edges, then the action would bewritten as S T = 116 πG N X i =1 ( j i + 12 ) b i Σ ∗ i ! . (32)The discrete action is then a function of the angular momentum, the Burgers vector ofdislocation and the area of Voronoi polygon.Consequently, in this approach we can approximate a three dimensional manifoldwith torsion M by a large collection of tetrahedra glued together. If we assume thateach tetrahedron is filled in with flat space and the torsion of M is concentrated on theedges of the tetrahedron, and since the length of the edge is chosen to be proportionalto the length of the angular momentum vector in semi classical limit, a metric g µν whichis related to triad on M by (5) is specified once the length ( j i + ) of each edge is fixed.The Euclidean Einstein-Hilbert action given by (8) is then a function of the angularmomentum on the edges and is given by summing the simplicial action (31) over all thetetrahedra in M .Having identified the edges l i of the tetrahedron where the torsion is concentratedwith the angular momenta, the asymptotic form of 6 j symbol for large values of j i isgiven by j j j j j j ≃ vuut ¯ h πV ( j ) cos " πG ¯ h X i =1 ( j i + 12 ) b i Σ ∗ i ! + π . (33)This is the Ponzano and Regge asymptotic formula for the Wigner 6 j symbol onsimplicial manifold with torsion. Here V ( j ) is the three dimensional volume of thetetrahedron and b i is the Burgers vector which gives both the length and directionof the closure failure or gap for every dislocation in the tetrahedron corresponding tothe edge j i + . Σ ∗ i is the area of a Voronoi polygon which is located in the planeperpendicular to a edge of tetrahedron.Comparing right hand side of the asymptotic equations (25) and (33), we obtaina relation between the angle ε i , which is formed by the outward normals of two facessharing the i − th edge and the distance b i through which a vector is translated from itsoriginal position when parallel transported around a small loop of area Σ ∗ i perpendicularto l i .In this way, in closed three dimensional Euclidean manifold with torsion M wehave introduced a tetrahedral triangulation, and have associated 6 j symbol to eachtetrahedron. Then, the set of vertices, edges, faces and tetrahedra with various6 j symbols associated to the tetrahedra defines the simplicial decomposition of M .Following Ponzano and Regge, we define a partition function by summing over allpossible edge lengths simillar to Regge calculus and by taking the product of the 6 j symbols over all fixed number of tetrahedra and connectivity of the simplicial manifold: Z P R ( M ) = lim L →∞ X j ≤ L Y vertices Λ( L ) − Y edges (2 j +1) Y tetrahedra ( − P i j i j j j j j j (34) onzano-Regge Model on Manifold with Torsion L is a non-negative integer or half-integer cut off and the factor Λ( L ) − per eachvertex was introduced to regulate divergences and is defined asΛ( L ) = X p =0 , , ,...,L (2 p + 1) (35)which behaves as Λ( L ) ∼ L in the limit L −→ ∞ .An important characteristic of this partition function is that it is independent ofthe way the interior of the manifold is triangulated and it is invariant under a refinementof a simplicial decomposition, more specifically, it is invariant under the following 1 − − Z P R ( M ). Thus the partition functiondefines a topological invariant of the manifold M and it is expected to be a topologiaclfield theory [22].Let us remark that the sum of contributions to S T in (23) from all tetrahedra ina tesselation approaches a value proportional to the action of teleparallel gravity, S in(8), provided the number of edges and vertices in the simplical manifold becomes verylarge: lim N →∞ N X j = i ( j i + 12 ) b i Σ ∗ i ! ≃ πGS = Z d x L T . (36)Comparing (34) and (36) we see that the partition function Z P R ( M ) can be interpretedas the path integral formulation of three dimensional Euclidean teleparalled gravity ona lattice. As it was remarked in [23], a given 6 j symbol is proportional to the pathintegral amplitude for the associated tetrahedron, so the product of all 6 j symbols isequivalent to the path integral amplitude for a given simplicial geometry: If in the sumover edges the large values dominate, then the sum over j i is replaced by an integraland the asymptotic value for 6 j symbols is used, Z P R ( M ) contains a term proportionalto Z Y i dj i (2 j i + 1) Y tetrahedra p vuut ¯ h πV p ( e iS T + e − iS T ) = Z Y i dµ ( j i ) e iS T (37)which looks like a Feynmann path integral with the Regge action in three dimension, andwith other terms contributing to the measure of the integral. However this interpretationsuffer from two problems: The integrals is over the form e iS T rather than e − S T althoughwe are considering Euclidean three dimensional spacetime, and the justification ofassociating various interference terms to measure by hand. Despite these difficulties,it seems natural to expect that this asymptotic approximation of the partition functionleads to Feynmann path integral for simplicail manifold with torsion and the 6 j symbolappears to be related to semiclassical quantum gravity. onzano-Regge Model on Manifold with Torsion
5. Final Remarks
In this paper we have considered the connection between angular momentum in quantummechanics and geometric objects, namely the relation between angular momentum andtetrahedra on manifold with torsion without the cosmological term. First, we noticedthe relation between the 6 j symbol and Regge’s discrete version of the action functionalof Euclidean three dimensional gravity with torsion given by (31). Then we consideredthe Ponzano and Regge asymptotic formula for the Wigner 6 j symbol on this simplicialmanifold with torsion (33). Let us remark that in this approach, a three dimensionalmanifold M is decomposed into a collection of tetrahedra, and it is assumed that eachtetrahedron is filled in with flat space and the torsion of M is concentrated on the edgesof the tetrahedron, the length of the edge is chosen to be proportional to the lengthof the angular momentum vector in semiclassical limit. The Einstein-Hilbert action onthis manifold is then a function of the angular momentum and the Burgers vector ofdislocation, and it is given by summing the Regge action over all tetrahedra in M (32).Following Ponzano and Regge [2], we defined a partition function (34) by summingover all possible edge lengths simillar to Regge calculus and by taking the product ofthe 6 j symbols over all fixed number of tetrahedra and connectivity of the simplicialmanifold, and in order to regulate the divergences by a cut off we have introduced a non-negative integer or half-integer parameter L . Although the partition function is finite insimple examples, it diverges in general cases since the set of irreducible representationsof SU (2) is infinite and the partition function is often a sum of an infinite number ofterms, as explained in [24]. A regularization of this infinite sum and its relation to thecosmological constant was provided by the Turaev − Viro model [6] and [25] by replacingthe Lie group SU (2) by its quantum deformation U q ( sl ( C )), which has only a finitenumber of representation. This q-deformed Ponzano Regge model on manifold withtorsion will be given elsewhere.The asymptotic approximation of the partition function can be interpreted as thepath integral formulation of three dimensional Euclidean teleparalled gravity on a lattice,since a given 6 j symbol is proportional to the path integral amplitude for the associatedtetrahedron, so the product of all 6 j symbols is equivalent to the path integral amplitudefor a given simplicial geometry. This interpretation is possible because in the sumover edges only the large values dominate and the number of edges and vertices in thesimplical manifold is very large. Consiquently, equation (37) looks like a Feynmann pathintegral for simplicail manifold with torsion with the Regge action in three dimensionand the 6 j symbol appears to be related to semiclassical quantum gravity. References [1] Wigner E P
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