Time in (2+1)-Dimensional Quantum Gravity

General relativity in three spacetime dimensions is used to explore three approaches to the “problem of time” in quantum gravity: the internal Schrödinger approach with mean extrinsic curvature as a time variable, the Wheeler-DeWitt equation, and covariant canonical quantization with “evolving constants of motion.” (To appear in Proc. of the Lanczos Centenary Conference, Raleigh, NC, December 1993.) ∗email: [email protected] As Karel Kuchař has explained elsewhere in these Proceedings, the nature of time in quantum gravity is at best obscure. Straightforward attempts at quantization lead to such absurdities as vanishing Hamiltonians and undefined inner products. The problems are not merely technical, but reflect an underlying conceptual issue: time translations in general relativity are “gauge symmetries,” and do not have an obvious physical meaning. It may be that this problem cannot be resolved without a full-fledged quantum theory of gravity. But our confusion about the nature of time is itself an obstacle to the construction of such a theory. It is therefore useful to look for simpler models to explore possible approaches to the role of time in quantum gravity. General relativity in 2+1 dimensions is one such model. Like realistic (3+1)-dimensional gravity, it is a generally covariant theory of spacetime geometry. But (2+1)-dimensional general relativity has only finitely many physical degrees of freedom, reducing the problem of quantization from quantum field theory to more elementary quantum mechanics.