Classical Mechanics in Noncommutative Spaces: Confinement and More

TL;DR

This paper explores the effects of noncommutative geometry on classical particle dynamics, revealing phenomena like velocity bounds and bounded motion, with implications for understanding physics in curved momentum spaces.

Contribution

It introduces a semi-classical framework for noncommutative spaces using Poisson brackets, linking phase space geometry to novel physical effects.

Findings

Velocity of free particles can be bounded.

Bounded motion occurs under repulsive forces.

Effective monopole potential describes corrections in Kepler problem.

Abstract

We consider a semi-classical approximation to the dynamics of a point particle in a noncommutative space. In this approximation, the noncommutativity of space coordinates is described by a Poisson bracket. For linear Poisson brackets, the corresponding phase space is given by the cotangent bundle of a Lie group, with the Lie group playing the role of a curved momentum space. We show that the curvature of the momentum space may lead to rather unexpected physical phenomena such as an upper bound on the velocity of a free nonrelativistic particle, bounded motion for repulsive central force, and no-fall-into-the-centre for attractive Coulomb potential. We also consider a superintegrable Hamiltonian for the Kepler problem in $3$-space with $su(2)$ noncommutativity. The leading correction to the equations of motion due to noncommutativity is shown to be described by an effective monopole potential.