Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
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 Palatini form of the 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), and 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 the structure of the quantized spacetime. We find a discretization of time, and a lightcone structure emerging from the origin of our coordinate system. Inside the lightcone we have states at discrete timelike and lightlike distances from the origin, and outside a continuous set of states at spacelike distances, sarting at a minimal distance of one Plank length. Finally, we also find a discretized Klein Gordon equation for the quantized particle.Keywords
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