Three Dimensional Canonical Quantum Gravity

TL;DR

This paper reviews the classical and quantum aspects of three-dimensional gravity using vielbein formalism, focusing on gauge symmetries, solution classification, and explicit construction of the physical Hilbert space for specific manifolds.

Contribution

It provides a detailed analysis of the canonical quantization of 3D gravity, including explicit Hilbert space construction for torus and cylinder topologies, and discusses observer-related conceptual issues.

Findings

Explicit Hilbert space construction for torus and cylinder manifolds

Classification of Einstein solutions in dreibein formalism

Discussion of observer-related conceptual problems in quantum gravity

Abstract

General aspects of vielbein representation, ADM formulation and canonical quantization of gravity are reviewed using pure gravity in three dimensions as a toy model. The classical part focusses on the role of observers in general relativity, which will later be identified with quantum observers. A precise definition of gauge symmetries and a classification of inequivalent solutions of Einstein's equations in dreibein formalism is given as well. In the quantum part the construction of the physical Hilbert space is carried out explicitly for a torus and cylinder type space manifold, which has not been done so far. Some conceptual problems of quantum gravity are discussed from the point of view of an observer sitting inside the universe.