Noncommuting Coordinates and Magnetic Monopoles

TL;DR

This paper investigates how noncommuting spatial coordinates emerge in quantum systems with magnetic monopoles and radial potentials, refining previous results and calculating coordinate commutators for specific potentials at fixed energy levels.

Contribution

It provides new quantum corrections to coordinate commutators in monopole systems and extends previous classical quantizations with explicit calculations for particular potentials.

Findings

Quantum corrections modify previous commutator results.

Explicit expressions for coordinate commutators at lowest energy levels.

Analysis includes harmonic oscillator and Coulomb potentials.

Abstract

The appearance of noncommuting spatial coordinates is studied in quantum systems containing a magnetic monopole and under the influence of a radial potential. We derive expressions for the commutators of the coordinates that have been restricted to the lowest energy level. Quantum corrections are found to previous results by Frenkel and Pereira based on quantizing the Dirac brackets of the classical theory. For two different potentials, the modified harmonic oscillator potential and the modified Coulomb potential, we also calculate the commutators for a projection to a fixed energy level.