The Maxwell-Dirac Equations: Monopoles, Electric Neutrality and “Free” Fields

Recent work on the Maxwell-Dirac equations is reviewed. The system we are interested in is the full MaxwellDirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Work on the static equations (the Dirac current is purely timelike in some Lorentz frame) is examined and two interesting theorems which apply in this case are discussed. The first of these theorems asserts, under suitable conditions, 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 stationary and static exterior to some ball. The second of these theorems applies in the axially symmetric case and states, under suitable conditions, that if there are external Coulomb fields then these must necessarily be magnetically charged. Finally, some preliminary remarks about a possible new approach to the problem of “free” fields is discussed. In the full Maxwell-Dirac theory an electron cannot be free in the usual sense, it always remains dressed in its own electromagnetic field. Neglecting the interaction of the Dirac field and its electromagnetic field at large times gives rise to the infrared problem. The discussion here takes a few tentative steps towards the goal of a reasonable mathematical definition of a ‘free field’ in the Maxwell-Dirac sense. Date: October 2, 2001.