A mild source for the Wu-Yang magnetic monopole

TL;DR

This paper demonstrates that the Wu-Yang magnetic monopole requires a magnetic point source at the origin to satisfy Yang-Mills equations, reconciling local and global properties without violating the Bianchi identity.

Contribution

It shows that a magnetic point source is necessary for the Wu-Yang monopole to be a valid solution in Yang-Mills theory, clarifying its mathematical consistency.

Findings

Wu-Yang monopole needs a magnetic point source at the origin.

Distribution theory confirms the solution's consistency.

The monopole does not violate the Bianchi identity with the source.

Abstract

We establish that the Wu-Yang monopole needs the introduction of a magnetic point source at the origin in order for it to be a solution of the differential and integral equations for the Yang-Mills theory. That result is corroborated by the analysis through distribution theory, of the two types of magnetic fields relevant for the local and global properties of the Wu-Yang solution. The subtlety lies on the fact that with the non-vanishing magnetic point source required by the Yang-Mills integral equations, the Wu-Yang monopole configuration does not violate, in the sense of distribution theory, the differential Bianchi identity.