Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole

TL;DR

This paper develops a group-theoretic quantization method for a particle on a sphere with a magnetic monopole, deriving energy spectra and quantization conditions through algebraic and symmetry considerations.

Contribution

It introduces a gauge-invariant algebraic approach to quantize a particle on a sphere with magnetic flux, recovering known spectra and Dirac's quantization condition.

Findings

Algebraic classification of quantizations and spectra.

Derivation of Dirac quantization condition from Casimir invariants.

Connection between algebraic approach and line bundle topology.

Abstract

The problem of quantizing a particle on a 2-sphere has been treated by numerous approaches, including Isham's global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham's scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.