Observables, gauge invariance, and time in (2+1)-dimensional quantum gravity

Two formulations of quantum gravity in 2+1 dimensions have been proposed: one based on Arnowitt-Deser-Misner variables and York's "extrinsic time," the other on diffeomorphism-invariant ISO(2,1) holonomy variables. In the former approach, the Hamiltonian is nonzero, and states are time dependent; in the latter, the Hamiltonian vanishes, and states are time independent but manifestly gauge invariant. This paper compares the resulting quantum theories in order to explore the role of time in quantum gravity. It is shown that the two theories are exactly equivalent for simple spatial topologies, and that gauge-invariant "time"-dependent operators can be constructed for arbitrary topologies.