Magnetic Charge and Photon Mass: Physical String Singularities, Dirac Condition, and Magnetic Confinement

TL;DR

This paper derives exact solutions to Proca-Maxwell equations with magnetic sources, revealing how photon mass alters magnetic charge properties and suggesting a confinement mechanism for magnetic monopoles.

Contribution

It provides the first exact solutions for massive photon magnetic charges, showing new properties and implications for magnetic confinement and Dirac quantization.

Findings

String singularities become real in magnetic fields with photon mass

Magnetic field loses rotational symmetry and depends on r and θ

Dyons carry angular momentum and satisfy Dirac condition despite photon mass

Abstract

We find exact, simple solutions to the Proca version of Maxwell's equations with magnetic sources. Several properties of these solutions differ from the usual case of magnetic charge with a massless photon: (i) the string singularities of the usual 3-vector potentials become real singularities in the magnetic fields; (ii) the different 3-vector potentials become gauge inequivalent and physically distinct solutions; (iii) the magnetic field depends on $r$ and $\theta$ and thus is no longer rotationally symmetric; (iv) a combined system of electric and magnetic charge carries a field angular momentum even when the electric and magnetic charges are located at the same place i.e. for dyons); (v) for these dyons one recovers the standard Dirac condition despite the photon being massive. We discuss the reason for this. We conclude by proposing that the string singularity in the magnetic field of an {\it isolated} magnetic charge suggests a confinement mechanism for magnetic charge, similar to the flux tube confinement of quarks in QCD.