Quantization of Lorentzian 3d Gravity by Partial Gauge Fixing

TL;DR

This paper demonstrates how Lorentzian 3D gravity with a positive cosmological constant can be partially gauge fixed to an SU(2) Chern-Simons theory, enabling loop quantization and the construction of a quantum observable with a null classical counterpart.

Contribution

It shows the partial gauge fixing of Lorentzian 3D gravity to SU(2) Chern-Simons theory and constructs a quantum observable with no classical equivalent.

Findings

Successful partial gauge fixing to SU(2) Chern-Simons theory.

Quantization of the theory on a cylindrical topology.

Construction of a quantum observable with a null classical limit.

Abstract

D = 2+1 gravity with a cosmological constant has been shown by Bonzom and Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. We then review the already known quantization of the latter in the framework of Loop Quantization for the case of space being topogically a cylinder. We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity.