Six ways to quantize (2+1)-dimensional gravity

TL;DR

The paper surveys six quantization strategies for $2+1$-dimensional gravity, using the torus spatial topology as a tractable testbed. It systematically compares reduced phase space ADM quantization, Chern-Simons/connection representation, covariant canonical quantization, loop representation, Wheeler-DeWitt equation, and lattice methods. The comparison highlights how the resulting quantum theories differ and reveals shared conceptual challenges in quantum gravity, such as observables, constraints, and topology. Overall, the work shows that lower-dimensional gravity serves as a productive platform for evaluating the viability, implications, and interplay of diverse quantization schemes for quantum gravity.

Abstract

We do not yet know how to quantize gravity in 3+1 dimensions, but in lower dimensions we face the opposite problem: many of the approaches originally developed for (3+1)-dimensional gravity can be successfully implemented in 2+1 dimensions, but the resulting quantum theories are not all equivalent. In this talk, I discuss six such approaches --- reduced phase space ADM quantization, the Chern-Simons/connection representation, covariant canonical quantization, the loop representation, the Wheeler-DeWitt equation, and lattice methods --- for the simple case of a spacetime whose spatial topology is that of a torus. A comparison of the resulting quantum theories can provide some useful insights into the conceptual issues that underlie quantum gravity in any number of dimensions. (Talk given at the Fifth Canadian Conference on General Relativity and Relativistic Astrophysics, Waterloo, Ontario, May 1993)