Feynman Path Integral Formulation

216 6 Lattice Regularized Quantum Gravityhere with κ 2 ≡ 4πG. Then one can show that the combined Lagrangian (containingthe gravity part, the gravitino action and the four-fermion term) is invariant, up toterms of order (ψ) 5 , under the simultaneous transformationsδe a μ = iκ ¯εγ a ψ μδg μν = iκ ¯ε [γ μ ψ ν + γ ν ψ μ ]δψ μ = κ −1 D μ ε + 1 4 iκ (2 ¯ψ μγ a ψ b + ¯ψ a γ μ ψ b )σ ab ε ,(6.186)where ε(x) in an arbitrary Majorana spinor. The lattice action will in general notpreserve all of these symmetries, but one can hope that they will be restored in thequantum continuum limit. The difficulties one encounters in transcribing supersymmetryon a lattice are discussed, for example, in the recent review (Feo, 2003).6.17 Alternate Discrete FormulationsThe simplicial lattice formulation offers a natural way of representing gravitationaldegrees in a discrete framework by employing inherently geometric concepts suchas areas, volumes and angles. It is possible though to formulate quantum gravityon a flat hypercubic lattice, in analogy to Wilson’s discrete formulation for gaugetheories, by putting the connection centerstage. In this new set of theories the naturalvariables are then lattice versions of the spin connection and the vierbein. Also,because the spin connection variables appear from the very beginning, it is mucheasier to incorporate fermions later. Some lattice models have been based on thepure Einstein theory (Smolin, 1979; Das, Kaku and Townsend, 1979; Mannion andTaylor, 1981; Caracciolo and Pelissetto, 1988), while others attempt to incorporatehigher derivative terms (Tomboulis, 1984; Kondo, 1984).Difficult arise when attempting to put quantum gravity on a flat hypercubic latticea la Wilson, since it is not entirely clear what the gravity analogue of the Yang-Millsconnection is. In continuum formulations invariant under the Poincaré or de Sittergroup the action is invariant under a local extension of the Lorentz transformations,but not under local translations (Kibble, 1961). Local translations are replaced bydiffeomorphisms which have a different nature. One set of lattice discretizationsstarts from the action of (MacDowell and Mansouri, 1977a,b; West, 1978) whoselocal invariance group is the de Sitter group Spin(4), the covering group of SO(4).In the lattice formulation of (Smolin, 1979; Das, Kaku and Townsend, 1979) thelattice variables are gauge potentials e aμ (n) and ω μab (n) defined on lattice sites n,generating local Spin(4) matrix transformations with the aid of the de Sitter generatorsP a and M ab . The resulting lattice action reduces classically to the Einsteinaction with cosmological term in first order form in the limit of the lattice spacinga → 0; to demonstrate the quantum equivalence one needs an additional zero torsion