Non-existence theorems and solutions of the Wu-Yang monopole equation

TL;DR

This paper investigates the classical SU(2) Wu-Yang monopole problem, providing new analytical methods, non-existence proofs for certain solutions, and exploring physical interpretations and generalizations of monopoles.

Contribution

It introduces a variational framework, new solution techniques including Padé approximations, and proves non-existence of smooth finite energy solutions, advancing understanding of Wu-Yang monopoles.

Findings

Non-existence of smooth finite energy solutions.

New solution methods using Padé approximations.

Interpretation of monopoles as magnetic dipoles.

Abstract

We consider the dynamics of the classical $SU(2)$ Wu-Yang monopole problem and show a set of new directions for its analysis starting from a variational setting. This allows us to give a new interpretation of the monopole charge as a string in a centrifugal potential. The field equations are solved using both standard power series, as well as a new Pad\'e approach. Furthermore, we discuss non-existence, and give a proof that rule out even singular non-trivial smooth finite energy solutions. Due to its non-vanishing chromomagnetic fields, the monopole can be interpreted as a collection of magnetic dipoles. Along the same lines, a non-smooth glue-ball solution is discussed that may serve as a toy model of the proton or a radiation-dominated cosmological expansion. Finally, we discuss generalizations of the Wu-Yang monopole.