Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity

TL;DR

This paper explores the quantum properties of a point particle in 2+1 dimensional gravity, revealing a non-commutative spacetime, discretized time, and a semi-discrete spatial structure through group-theoretic methods.

Contribution

It introduces a reduced, gauge-invariant action for a gravitating particle in 2+1 dimensions and analyzes its quantum structure using SL(2) representation theory.

Findings

Discretization of time in the quantum model

Non-commutative spacetime coordinates

Quantum dynamics governed by a discretized Klein-Gordon equation

Abstract

We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation.