Patching up the monopole potential

TL;DR

This paper clarifies the correct way to define vector potentials around magnetic monopoles, emphasizing the use of multiple patches to resolve topological issues and correctly derive the Dirac quantization condition.

Contribution

It corrects previous claims by demonstrating the necessity of using two patches for the potential and extends the discussion to multiple patches with Wu-Yang type potentials.

Findings

Single patch approach is insufficient for monopole potentials

Two patches are necessary for a consistent potential definition

The Dirac quantization condition is correctly derived using two patches

Abstract

It is well known that a vector potential cannot be defined over the whole surface of a sphere around a magnetic monopole. A recent claim to the contrary is shown to have problems. It is explained however that a potential of the proposed type works if two patches are used instead of one. A general derivation of the Dirac quantization condition attempted with a single patch is corrected by introducing two patches. Further, the case of more than two patches using the original Wu-Yang type of potential is discussed in brief.