Monopole Vector Spherical Harmonics

TL;DR

This paper constructs and analyzes eigenfunctions of total angular momentum for a charged vector field in the presence of a magnetic monopole, revealing unique properties and limitations of these functions.

Contribution

It introduces a method to construct these eigenfunctions and identifies special cases where standard methods fail, providing new insights into monopole vector harmonics.

Findings

Eigenfunctions constructed for most angular momenta

Standard construction fails for minimum angular momentum harmonics

Minimum angular momentum harmonics have vanishing covariant curl and divergence

Abstract

Eigenfunctions of total angular momentum for a charged vector field interacting with a magnetic monopole are constructed and their properties studied. In general, these eigenfunctions can be obtained by applying vector operators to the monopole spherical harmonics in a manner similar to that often used for the construction of the ordinary vector spherical harmonics. This construction fails for the harmonics with the minimum allowed angular momentum. These latter form a set of vector fields with vanishing covariant curl and covariant divergence, whose number can be determined by an index theorem.