2+1 Quantum Gravity with Barbero-Immirzi like parameter on Toric Spatial Foliation

TL;DR

This paper explores 2+1 dimensional quantum gravity with a Barbero-Immirzi-like parameter on a toric spatial topology, analyzing its phase space, quantization, and how the parameter influences the finite-dimensional Hilbert space.

Contribution

It provides a detailed phase space analysis and explicit wave function construction for 2+1 gravity with a Barbero-Immirzi-like parameter on a torus topology, revealing the parameter's effect on the Hilbert space dimension.

Findings

The physical Hilbert space is finite-dimensional.

The Hilbert space dimension depends on the Barbero-Immirzi-like parameter.

Explicit wave functions are constructed for the theory.

Abstract

We consider gravity in 2+1 space-time dimensions, with negative cosmological constant and a `Barbero-Immirzi' (B-I) like parameter, when the space-time topology is of the form $ T^2 \times \mathbbm{R}$. The phase space structure, both in covariant and canonical framework is analyzed. Full quantization of the theory in the 'constrain first' approach reveals a finite dimensional physical Hilbert space. An explicit construction of wave functions is presented. The dimension of the Hilbert space is found to depend on the `Barbero-Immirzi' like parameter in an interesting fashion. Comparative study of this parameter in light of some of the recent findings in literarure for similar theories is presented.