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

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

1997-08-22

arxiv.org/abs/gr-qc/9708054v2

Class.Quant.Grav. 15 (1998) 2981-3030

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

58 pages, 3 eps figures, presentation of the classical theory improved

Scientific paper

10.1088/0264-9381/15/10/008

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.

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