Two particle Quantummechanics in 2+1 Gravity using Non Commuting Coordinates

TL;DR

This paper explores the quantum mechanics of two particles in 2+1 gravity, revealing a hyperbolic momentum space, non-commuting coordinates, and a minimal distance related to the Planck length.

Contribution

It introduces a model with non-commuting coordinates on a hyperboloid momentum space, highlighting a minimal inter-particle distance in 2+1 gravity.

Findings

Momentum conjugate to relative distance is a hyperbolic angle.

Coordinates are represented by non-commuting Hermitian operators.

There exists a minimal distance between particles of half the Planck length.

Abstract

We find that the momentum conjugate to the relative distance between two gravitating particles in their center of mass frame is a hyperbolic angle. This fact strongly suggests that momentum space should be taken to be a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non commuting, Hermitian operators on this hyperboloid. We also find that there is a smallest distance between the two particles of one half times the Planck length.