Magnetic monopoles, electric neutrality and the static Maxwell-Dirac equations

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) and isolated (in the sense that the fields belong to a suitable weighted Sobolev space), 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 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.