Unitary Equivalence of the Metric and Holonomy Formulations of 2+1 Dimensional Quantum Gravity on the Torus

TL;DR

This paper demonstrates the unitary equivalence between the metric and holonomy formulations of 2+1-dimensional quantum gravity on a torus by constructing a suitable canonical transformation and defining an appropriate inner product.

Contribution

It introduces a non-polynomial factor ordering for the canonical transformation that preserves modular properties and establishes a unitary map between the two formulations' Hilbert spaces.

Findings

The Hilbert space in the holonomy formulation is unitarily equivalent to that in the metric formulation.

A non-polynomial canonical transformation preserving modular invariance is constructed.

Gravitational theta-states from large diffeomorphisms are identified.

Abstract

Recent work on canonical transformations in quantum mechanics is applied to transform between the Moncrief metric formulation and the Witten-Carlip holonomy formulation of 2+1-dimensional quantum gravity on the torus. A non-polynomial factor ordering of the classical canonical transformation between the metric and holonomy variables is constructed which preserves their classical modular transformation properties. An extension of the definition of a unitary transformation is briefly discussed and is used to find the inner product in the holonomy variables which makes the canonical transformation unitary. This defines the Hilbert space in the Witten-Carlip formulation which is unitarily equivalent to the natural Hilbert space in the Moncrief formulation. In addition, gravitational theta-states arising from ``large'' diffeomorphisms are found in the theory.