A Immirzi-like parameter for 3d quantum gravity

TL;DR

This paper investigates an Immirzi-like parameter in 3D quantum gravity, analyzing its impact on the canonical structure, length operator spectrum, and extending insights to 4D gravity.

Contribution

It introduces and explores an Immirzi-like ambiguity in 3D quantum gravity, detailing its effects on the canonical structure and length operator spectrum, and relates these findings to 4D gravity.

Findings

The Immirzi-like parameter modifies the Poisson brackets and constraint algebra.

The length operator spectrum is affected by the Immirzi-like ambiguity.

Topological modifications influence the canonical structure in 4D gravity.

Abstract

We study an Immirzi-like ambiguity in three-dimensional quantum gravity. It shares some features with the Immirzi parameter of four-dimensional loop quantum gravity: it does not affect the equations of motion, but modifies the Poisson brackets and the constraint algebra at the canonical level. We focus on the length operator and show how to define it through non-commuting fluxes. We compute its spectrum and show the effect of this Immirzi-like ambiguity. Finally, we extend these considerations to 4d gravity and show how the different topological modifications of the action affect the canonical structure of loop quantum gravity.