Hamiltonian lattice gravity

2.10 Hamiltonian lattice gravity

There is a gauge-theoretic Hamiltonian version of gravity defined on a cubic lattice, which in many aspects resembles the Lagrangian gauge formulations described earlier. It also is virtually the only discrete Hamiltonian formulation in which some progress has been achieved in the quantization (see also [155 ] for a recent review).

Renteln and Smolin [180 ] were the first to set up a continuous-time lattice discretization along the lines of Hamiltonian lattice gauge theory. Their basic configuration variables are the link holonomies $U (l)$ of the spatial Ashtekar connection along the edges. The lattice analogues of the canonically conjugate pairs $(Ai(x ),Ea (x)) a i$ are the link variables $(U (l) B,p (l)) A i$ , with Poisson brackets

B D ˆ {UA (n, ˆa),UC (m, b)} = 0, (6 ) C ˆ 1- B C {pi(n,ˆa),UA (m, b)} = − 2 δnm δˆaˆb τiA UB (n,ˆa), (7 ) ˆ {pi(n,ˆa),pj(m, b)} = δnm δˆaˆb 𝜖ijk pk(n, ˆa), (8 )

with the $SU (2)$ -generators satisfying $[τi,τj] = 2𝜖ijkτk$ . Lattice links $l = (n, ˆa)$ are labelled by a vertex $n$ and a lattice direction $ˆa$ . These relations go over to the usual continuum brackets in the limit as the lattice spacing $a$ is taken to zero. In this scheme, they wrote down discrete analogues of the seven polynomial first-class constraints, and also attempted to interpret the action of the discretized diffeomorphism and Hamiltonian constraints in terms of their geometric action on lattice Wilson loop states.