Quantum Geometry II : The Mathematics of Loop Quantum Gravity Three dimensional quantum gravity

TL;DR

This paper introduces the mathematical framework of three-dimensional loop quantum gravity, aiming to bridge general relativity and quantum mechanics through a geometric quantisation approach.

Contribution

It provides a mathematical perspective on loop quantum gravity, focusing on the three-dimensional case, for postgraduate students in physics and mathematics.

Findings

Mathematical formulation of 3D loop quantum gravity

Clarification of geometric quantisation in gravity

Educational resource for postgraduate students

Abstract

Loop quantum gravity is a physical theory which aims at unifying general relativity and quantum mechanics. It takes general relativity very seriously and modifies it via a quantisation. General relativity describes gravity in terms of geometry. Therefore, quantising such theory must be equivalent to quantising geometry and that is what loop quantum gravity does. This sounds like a mathematical task as well. This is why in this paper we will present the mathematics of loop quantum gravity. We will do it from a mathematician point of view. This paper is intended to be an introduction to loop quantum gravity for postgraduate students of physics and mathematics. In this work we will restrict ourselves to the three dimensional case.