Quantisation of 2+1 gravity for genus 2

TL;DR

This paper simplifies the algebra of observables for 2+1 gravity on a genus 2 surface, introduces a Hilbert space framework, and explores its representations via a deformed Euler-Gamma function influenced by physical constants.

Contribution

It provides a simplified algebraic structure for genus 2 2+1 gravity and develops a novel representation approach using a deformed special function.

Findings

Algebra of observables reduced to a simple form for genus 2

Hilbert space of state vectors constructed

Representations analyzed with a deformation parameter

Abstract

In [1,2] we established and discussed the algebra of observables for $2+1$ gravity at both the classical and quantum level, and gave a systematic discussion of the reduction of the expected number of independent observables to $6g - 6 (g > 1)$. In this paper the algebra of observables for the case $g=2$ is reduced to a very simple form. A Hilbert space of state vectors is defined and its representations are discussed using a deformation of the Euler-Gamma function. The deformation parameter $\th$ depends on the cosmological and Planck's constants.