Comparative Quantizations of (2+1)-Dimensional Gravity

TL;DR

This paper compares three different approaches to quantizing (2+1)-dimensional gravity with negative cosmological constant, clarifying their relationships and how to interpret time-dependent operators across these frameworks.

Contribution

It provides a detailed comparison of three quantization methods for (2+1)-dimensional gravity and explores their interconnections and interpretations.

Findings

Established relationships among the three quantum theories.

Defined and interpreted time-dependent operators in the holonomy formulation.

Clarified the correspondence between different quantization approaches.

Abstract

We compare three approaches to the quantization of (2+1)-dimensional gravity with a negative cosmological constant: reduced phase space quantization with the York time slicing, quantization of the algebra of holonomies, and quantization of the space of classical solutions. The relationships among these quantum theories allow us to define and interpret time-dependent operators in the ``frozen time'' holonomy formulation.