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

Hans-Juergen Matschull·Max Welling
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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.

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Hans-Juergen Matschull, Max Welling. 1998-04-21. Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity. https://doi.org/10.1088/0264-9381%2F15%2F10%2F008

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