Feynman Path Integral Formulation

218 6 Lattice Regularized Quantum Gravity
for which the curvature transforms as
U 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 as
E 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 metric
is then obtained as usual by
g μν (n)= 1 4 tr[E μ(n)E ν (n)] . (6.197)
From the expression for the lattice curvature R ab
μν given above if follows immediately
that the lattice action in the continuum limit becomes
I =
a4
4κ 2 ∑
n
ε μνλσ ε abcd Rμν
ab (n)eλ c (n)e σ d (n)+O(a 6 ) , (6.198)
which is the Einstein action in Cartan form
I = 1
4κ 2 ∫
d 4 xε μνλσ ε abcd Rμν
ab eλ c e σ d , (6.199)
with the parameter κ identified with the Planck length. One can add more terms to
the action; in this theory a cosmological term can be represented by
λ 0∑ε μνλσ tr[γ 5 E μ (n)E ν (n)E σ (n)E λ (n)] . (6.200)
n
Both Eqs. (6.193) and (6.200) are locally SL(2,C) invariant. The functional integral
is then given by
∫
{ }
Z = ∏dB μ (n) ∏dE σ (n) exp −I(B,E) , (6.201)
n,μ n,σ
and from it one can then compute suitable quantum averages. Here dB μ (n) is the
Haar measure for SL(2,C); it is less clear how to choose the integration measure
over the E σ ’s, and how it should suitably constrained, which obscures the issue of
diffeomorphism invariance in this theory.
In these theories it is possible to formulate curvature squared terms as well. In
general for a hypercubic lattices the formulation of R 2 -type terms in four dimensions
involves constraints between the connections and the tetrads, which are a bit difficult
to handle. Also there is no simple way of writing down topological invariants, which