The Modular Group, Operator Ordering, and Time in (2+1)-Dimensional Gravity

TL;DR

This paper explores how different operator orderings in (2+1)-dimensional quantum gravity with torus topology affect wave functions, linking them to automorphic functions and Maass Laplacians, thus addressing the problem of time and operator ambiguity.

Contribution

It demonstrates that specific operator orderings produce wave functions transforming under the modular group as automorphic functions with dynamics governed by Maass Laplacians.

Findings

Wave functions transform as automorphic functions under the modular group.

Operator ordering ambiguities lead to a family of possible wave functions.

Dynamics are described by Maass Laplacians on moduli space.

Abstract

A choice of time-slicing in classical general relativity permits the construction of time-dependent wave functions in the ``frozen time'' Chern-Simons formulation of $(2+1)$-dimensional quantum gravity. Because of operator ordering ambiguities, however, these wave functions are not unique. It is shown that when space has the topology of a torus, suitable operator orderings give rise to wave functions that transform under the modular group as automorphic functions of arbitrary weights, with dynamics determined by the corresponding Maass Laplacians on moduli space.