Magnetic fields and factored two-spheres

TL;DR

This paper develops a quantum mechanical framework for a magnetic monopole on a 3-ball with a tiled surface, analyzing spectral properties, quantization, and Casimir energies, with numerical methods for Barnes zeta-function derivatives.

Contribution

It introduces a formalism for quantum mechanics on quotient spaces with magnetic monopoles, including spectral analysis and Casimir energy calculations, supported by numerical techniques.

Findings

Monopole charge is quantized as expected.

Heat-kernels and zeta-functions are explicitly evaluated.

Casimir energies are computed for the system.

Abstract

A magnetic monopole is placed at the centre of a 3-ball whose surface, S, is tiled by the symmetry group, G, of a regular solid. The quantum mechanics on the two-dimensional quotient, S/G, is developed and the monopole charge is found to be quantized in the expected manner. The heat-kernels and zeta-functions are evaluated and the Casimir energies computed as examples of the formalism. Numerical approaches to the calculation of the derivative of the Barnes zeta-function are presented.