Quantum Regge Calculus of Einstein-Cartan theory

We study the Quantum Regge Calculus of Einstein-Cartan theory to describe quantum dynamics of Euclidean space-time discretized as a 4-simplices complex. Tetrad field e_\mu(x) and spin-connection field \omega_\mu(x) are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and {\it local} gauge-invariant Einstein-Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)-group valued \omega_\mu(x) fields and Dirac-matrix valued e_\mu(x) fields over 4-simplices complex.

Classical equations can be obtained by the invariance of the EC action (4) under the transfor- where δe µ and δω µ are infinitesimal variations, which can be expressed in terms of independent Dirac matrix bases γ 5 and γ µ . Therefore, for an arbitrary function θ ab , we have δS EC /δe µ = 0 and δS EC /δω µ = 0, respectively leading to Einstein equation and Cartan's structure equation FIG. 1: Assuming edge spacing a µ,ν (x) is so small that the geometry of the interior of 4-simplex and its sub-simplex (3-and 2-simplex) is approximately flat, we assign a local Lorentz frame to each 4-simplex.
On a local Lorentz manifold ξ a (x) at a space-time point "x", we sketch a closed parallelogram C P (x) lying in the 2-simplex h(x). Its edges e µ (x) and e † ν (x) = e ν (x + a ν ) are two edges of the 2-simplex h(x), and other edges (dashed lines) e † µ (x + a ν ) and e ν (x + a µ ) are parallel transports of e µ (x) and e † ν (x) along ν-and µ-directions respectively. Each 2-simplex in the 4-simplices complex has a closed parallelogram lying in it.
Group-valued gauge fields U µ (x) and U † ν (x) = U ν (x + a ν ) are respectively associated to edges e µ (x) and e † ν (x) of the 2-simplex h(x), as indicated. The fields e ρ (x + a µ ) and U ρ (x + a µ ) are associated to the third edge (x + a µ , x + a ν ) of the 2-simplex h(x).
Regularized EC action. The four-dimensional Euclidean manifold M is discretized as an ensemble of N 0 space-time points "x " and N 1 links (edges) "l µ (x)" connecting two neighboring points, which is a simplicial manifold. The way to construct a simplicial manifold depends also on the assumed topology of the manifold, which gives geometric constrains on the numbers of sub-simplices (N 0 , N 1 , · · ·, see Ref. [4]). In this Letter, analogously to the simplicial manifold adopted by Regge Calculus we consider a 4-simplices complex, whose elementary building block is a 4-simplex (pentachoron). The 4-simplex has 5 vertexes -0-simplex (a space-time point "x "), 5 "faces" -3-simplex (a tetrahedron), and each 3-simplex has 4 faces -2-simplex (a triangle), and each 2-simplex has three faces -1-simplex (an edge or a link "l µ (x)"). Different configurations of 4-simplices complex correspond to variations of relative vertex-positions {x}, edges "{l µ (x)}" and "deficit angle" around each vertex x. These configurations will be described by the configurations of dynamical fields e µ (x) and ω µ (x) (its group-valued U µ (x)) in a regularized EC-theory [11].
To illustrate how to construct a regularized EC theory describing dynamics of 4-simplices complex, we consider a 2-simplex (triangle) h(x) (see Fig. 1). The fundamental tetrad field e µ (x) and undergoes its parallel transport to e ν (x + a µ ) [e µ (x + a ν )] along the µ [ν]-direction for an edge spacing a µ (x) [a ν (x)], following the discretized Cartan equation and µ ↔ ν. The parallel transports e a ν (x + a µ ) and e a µ (x + a ν ) are neither independent fields, nor assigned to any edges of the 4-simplices complex. They are related to e µ (x) and ω µ (x) fields assigned to edges of the 2-simplex h(x) by the Cartan equation (9). Because of torsion-free, e µ (x), e ν (x) and their parallel transports e µ (x + a ν ), e ν (x + a µ ) form a closed parallelogram C P (x) (Fig. 1). Otherwise this would means the curved space-time could not be approximated locally by a flat space-time [12]. We define ω µ (x + a ν ) and ω ν (x + a µ ) by using the discretized equation for curvature (3), and µ ↔ ν. For zero curvature case, analogously to (9), parallel transportsω ab and µ ↔ ν. The difference ("deficit angle") between ω ab ν (x + a µ ) andω ab ν (x + a µ ) is the curvature a µ R ab µν (x). Instead of ω µ (x) field, we assign a group-valued field U µ (x) to each 1-simplex of 4-simplices complex. For example, at edges (x, µ) and (x, ν) of the 2-simplex h(x) (µ = ν see Fig. 1), we define SO(4) group-valued spin-connection fields, which take value of fundamental representation of the compact group SO(4), and their local gauge transformations, and µ ↔ ν in accordance with (2). Actually, these group-valued fields (12) can be viewed as unitary operators for finite parallel transportations. Eq. (9) can be generalized to and µ ↔ ν. Eq. (17) characterizes relative angles θ µν (x) between two neighboring edges e µ (x) and e ν (x) (see Fig. 1). In the naive continuum limit: agω µ ≪ 1 (small coupling or weak-field), indicating that the wavelengths of weak and slow-varying fields ω µ (x) are much larger than the edge spacing a µ,ν , we have where O(a 3 ) indicates high-order powers of agω µ .
Using the tetrad fields e µ (x) to construct coordinate and Lorentz scalars so as to obtain a regularized EC action preserving the diffeomorphism and local gauge-invariance, we define the smallest holonomy along closed triangle path of 2-simplex: whose orientation is anti-clock-like, and X † h (e, U ) is clock-like (see Fig. 1). We have following two possibilities for the vertex-field v νµ (x). The first v µν (x) = e µν (x)γ 5 : where h is the sum over all 2-simplices h(x). In the limit: agω µ ≪ 1, Eq. (20) becomes We define a 4-d volume element V (x) = h(x) S 2 h (x) around the vertex x. The interior of 4-simplex is approximately flat, leading to and Eq. (22) approaches to S P (e, ω) (5) with an effective Newton constant G eff = gG/4. The second v µν (x) = e µν (x): where the real parameter γ = iγ. Analogously, in the limit: agω µ ≪ 1, Eq. (24) approaches to S H (e, ω) (6), Under the gauge transformation (1), The diffeomorphism and local gauge-invariant regularized EC action is then given by Considering the following diffeomorphism and local gauge-invariant holonomies along a large loop C on the Euclidean manifold M where P C is the path-ordering and "Tr" denotes the trace over spinor space, we attempt to regularize these holonomies on the 4-simplices complex. Suppose that an orientating closed path C passes space-time points x 1 , x 2 , x 3 , · · ·, x N = x 1 and edges connecting between neighboring points in the 4-simplices complex. At each point x i two tetrad fields e µ (x i ) and e µ ′ (x i ) (µ = µ ′ ) respectively orientating path incoming to (i − 1 → i) and outgoing from (i → i + 1) the point x i , we have the vertex-field v µµ ′ (x i ) defined by Eqs. (21,24). Link fields U µ (x i ) are defined on edges lying in the loop C, recalling the relationship U µ (x i ) = U −µ (x i+1 ) = U † µ (x i+1 ), we can write the regularization of the holonomies (28) as follows, preserving diffeomorphism and local gauge-invariances. Eq. (29) is consistent with Eq. (19).
Euclidean partition function. The partition function Z EC and effective action A eff EC are with the diffeomorphism and local gauge-invariant measure where x,µ indicates the product of overall edges, dU µ (x) is the Haar measure of compact gauge group SO(4) or SU (2), and de µ (x) is the measure of Dirac-matrix valued field e µ (x) = a e a µ (x)γ a , determined by the functional measure de a µ (x) of the bosonic field e a µ (x). It should be mentioned that the measure (31) is just a lattice form of the standard DeWitt functional measure [13] In the action (20,24), X h (v, U ) (19) contains the quadric term of e µ (x)-field associated to each edge (x, µ), the partition function Z EC (30) and v.e.v. (32) are converge.
Analogously to Eq. (7), the local gauge-invariance of the partition function (30) (δZ EC = 0) leads to which becomes "averaged" Einstein equation δA EC /δe µ + h.c. = 0, and Eq. (34) is "averaged" torsion-free Cartan equation (8), which actually shows the impossibility of spontaneous breaking of local gauge symmetry. This should not be surprised, since the torsion-free (8) is a necessary condition to have a local Lorentz frame, therefore a local gauge-invariance.

The regularized EC theory (27) can be separated into left-and right-handed parts by replacing
. In addition, we can generalize the link field U µ (x) to be all irreducible representations U j µ (x) of the gauge group SO(4). The regularized EC action (27) should be a sum over all representations j ≡ j L,R = 1/2, 3/2, · · ·, and the measure (31) should include all representations of gauge group.
Some calculations in 2-dimensional case. We consider a 2-simplices complex, i.e., random simplicial surface, whose elementary building block is a triangle h(x) (see Fig. 1). In this case, local gauge transformations (13,26) can be made so that all fields v µρ (x + a µ )U ρ (x + a µ )v ρν (x + a ν ) = 1 in Eq. (19), as if we choose a particular gauge. The partition function (30) can be calculated by integrating over e µ (x)-and U µ (x)-fields, using the Cayley-Hamilton formula for a determinant [14] and the properties of invariant Haar measure: where d j = n j L n j R ( n j L ,j R = 2j L,R + 1), the dimension of irreducible representations j = (j L , j R ) of SU L (2) ⊗ SU R (2). We obtain the entropy S = ln Z EC where is the sum over all 2-simplices, degrees of freedom of gauge group representations and Dirac spinors. The 2-dimensional surface where N h is the total number of 2-simplices and P a averaged area of 2-simplices. The free energy F = − 1 β ln Z EC , where the inverse "temperature" β = 1/g 2 , see Eqs. (20,24). Selecting fundamental representation d j = 4, we obtain S = S surf /(g 2 γa 2 ) and F = −S surf /(γa 2 ) .
In the same way, we calculate the average of regularized EC action A EC (37), in the strong coupling (field) limit g ≫ 1 or gaω µ ∼ O(1), which implies that ω µ field's wavelength is comparable to the Planck length a, The average (41) of regularized EC action has discrete values corresponding to the fundamental state d j = 4 and excitation states d j = 16.
Using the convexity inequality e −A j EC ≥ e − A j EC , we have Using Eqs. (39,40), we obtain and averaged area of a 2-simplex implying that the Planck length is minimal separation between two space-time points [15].
Some remarks. Although the regularized EC action (27) approaches to the EC action (4) in the "naive continuous limit" agω µ ≪ 1, the regularized EC theory is physically sensible, provided it has a non-trivial continuum limit. It is crucial, on the basis of non-perturbative methods and renormalization group invariance, to find: (1) the scaling invariant regimes (ultraviolet fix points) g c , where phase transition takes place and physical correlation length ξ is much larger than the Planck length a; (2) β-function β(g) and renormalization-group invariant equation ξ = const. a exp g dg ′ /β(g ′ ); (3) all relevant and renormalizable operators (one-particle irreducible (1PI) functions) with effective dimension-4 in these regimes to obtain effective low-energy theories. One may add by hand the cosmological Λ-term λ 4·4! ǫ µνρσ x tr[e µ e ν e ρ e σ ] + h.c., where λ = Λa 2 , into the regularized EC action (27). However, 1PI functions A eff EC (30) effectively contain this dimensional operator, which is related to the truncated Green function A EC A EC . It is then a question what is the scaling property of this operator in terms of ξ −2 , where inverse correlation length ξ −1 gives the mass scale of low-energy excitations of the theory.
One can consider the following regularized fermion action, where fermion fields ψ(x) and ψ(x + a µ ) are defined at two neighboring points (vertexes) of 4simplices complex, fields U µ (x) and e µ (x) are added to preserve local gauge and diffeomorphism invariances, and xµ is the sum over all edges (1-simplices) of 4-simplices complex. This bilinear fermion action (45) introduces a non-vanishing torsion field [16,17]. We need to study whether the regularized EC action (27) with fermion action (45) can be effectively written in form of a torsionfree part and four fermion interactions, as the EC theory in continuum. In addition, the bilinear