Quantization of point particles in (2+1)-dimensional gravity and spacetime discreteness

Abstract

By investigating the canonical commutation rules for gravitating quantized particles in a 2+1 dimensional world it is found that these particles live on a space-time lattice. The space-time lattice points can be characterized by three integers. Various representations are possible, the details depending on the topology chosen for energy-momentum space. We find that an $S_2imes S_1$ topology yields a physically most interesting lattice within which first quantization of Dirac particles is possible. An $S_3$ topology also gives a lattice, but does not allow first quantized particles.