Feynman Path Integral Formulation

218 6 Lattice Regularized Quantum Gravityfor which the curvature transforms asU μ → Λ(n)U μ (n)Λ −1 (n + μ) , (6.194)U μ (n)U ν (n + μ)U −μ (n + μ + ν)U −ν (n + ν)→ Λ(n)U μ (n)U ν (n + μ)U −μ (n + μ + ν)U −ν (n + ν)Λ −1 (n) , (6.195)and the vierbein matrices asE μ (n) → Λ(n)E μ (n)Λ −1 (n) . (6.196)Since Λ(n) commutes with γ 5 , the expression in Eq. (6.193) is invariant. The metricis then obtained as usual byg μν (n)= 1 4 tr[E μ(n)E ν (n)] . (6.197)From the expression for the lattice curvature R abμν given above if follows immediatelythat the lattice action in the continuum limit becomesI =a44κ 2 ∑nε μνλσ ε abcd Rμνab (n)eλ c (n)e σ d (n)+O(a 6 ) , (6.198)which is the Einstein action in Cartan formI = 14κ 2 ∫d 4 xε μνλσ ε abcd Rμνab eλ c e σ d , (6.199)with the parameter κ identified with the Planck length. One can add more terms tothe action; in this theory a cosmological term can be represented byλ 0∑ε μνλσ tr[γ 5 E μ (n)E ν (n)E σ (n)E λ (n)] . (6.200)nBoth Eqs. (6.193) and (6.200) are locally SL(2,C) invariant. The functional integralis then given by∫{ }Z = ∏dB μ (n) ∏dE σ (n) exp −I(B,E) , (6.201)n,μ n,σand from it one can then compute suitable quantum averages. Here dB μ (n) is theHaar measure for SL(2,C); it is less clear how to choose the integration measureover the E σ ’s, and how it should suitably constrained, which obscures the issue ofdiffeomorphism invariance in this theory.In these theories it is possible to formulate curvature squared terms as well. Ingeneral for a hypercubic lattices the formulation of R 2 -type terms in four dimensionsinvolves constraints between the connections and the tetrads, which are a bit difficultto handle. Also there is no simple way of writing down topological invariants, which