Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations

TL;DR

This paper investigates the static Maxwell-Dirac equations, proving that stationary systems with purely time-like Dirac currents must be electrically neutral and that external Coulomb fields in axially symmetric cases are necessarily magnetically charged monopoles.

Contribution

It establishes theorems showing charge neutrality in static, stationary Maxwell-Dirac systems and the necessity of magnetic monopoles for Coulomb external fields in axial symmetry.

Findings

Stationary, static Maxwell-Dirac systems are electrically neutral.

External Coulomb fields in axial symmetry imply magnetic monopoles.

Theorems hold under general assumptions.

Abstract

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -- the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be {\em asymptotically} stationary and static. Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged -- all Coulomb external sources are electrically charged magnetic monopoles.