Multi-plaquette solutions for discretized Ashtekar gravity

TL;DR

This paper explores a modified discretized quantum gravity model, finding new non-normalizable solutions supported on connected plaquettes through topological manipulations of lattice loops.

Contribution

It introduces a topological formulation of the discretized Hamiltonian constraint and discovers new solutions to the Wheeler-Dewitt equation in this framework.

Findings

New solutions supported on connected plaquettes

Solutions are not normalizable under the heat-kernel measure

Topological manipulations encode the Hamiltonian constraint

Abstract

A discretized version of canonical quantum gravity proposed by Loll is investigated. After slightly modifying Loll's discretized Hamiltonian constraint, we encode its action on the spin network states in terms of combinatorial topological manipulations of the lattice loops. Using this topological formulation we find new solutions to the discretized Wheeler-Dewitt equation. These solutions have their support on the connected set of plaquettes. We also show that these solutions are not normalizable with respect to the induced heat-kernel measure on $SL(2,{\bf C})$ gauge theories.