Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness

TL;DR

This paper explores how quantized particles in 2+1 dimensional gravity inherently reside on a discrete space-time lattice, with the lattice structure depending on the topology of energy-momentum space, affecting particle quantization.

Contribution

It demonstrates that particles in 2+1D quantum gravity are confined to a space-time lattice characterized by three integers, with the lattice structure influenced by the topology of energy-momentum space.

Findings

Particles live on a space-time lattice with three integer coordinates.

An $S_2\times S_1$ topology allows first quantization of Dirac particles.

An $S_3$ topology also forms a lattice but prevents first quantization.

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 $S_2\times S_1$ 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.