### 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
of the spatial Ashtekar connection along the edges. The lattice analogues of the canonically
conjugate pairs are the link variables , with Poisson brackets

with the -generators satisfying . Lattice links are labelled by a vertex
and a lattice direction . These relations go over to the usual continuum brackets in the limit as the
lattice spacing 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.