A spherically symmetric bound state of the coupled Maxwell-Dirac equations with self-interaction alone

Abstract

Stationary, static, spherically symmetric solutions of the Maxwell-Dirac system, treated as classical fields, have been found which are localised and normalisable. The solutions apply to any bound energy eigenvalue in the range 0 < E < m, where m is the bare mass in the Dirac equation. A point charge of any magnitude and either sign may be placed at the origin and the solutions remain well behaved and bound. However, no such central charge is necessary to result in a bound solution. As found previously by Radford, the magnetic flux density is equal to that of a monopole at the origin. However, no monopole is present, the magnetic flux being a result of the dipole moment distribution of the Dirac field. The Dirac field magnetic dipole moment is aligned with the magnetic flux density and so the resulting magnetic self-energy is negative. It is this which results in the states being bound (E < m). The case which omits any central point charge is therefore a self-sustaining bound state solution of the Maxwell-Dirac system which is localised, normalisable, and requires no arbitrarily added “external” features (i.e., it is a soliton). As far as the author is aware, this is the first time that such an exact solution with a positive energy eigenvalue has been reported. However, the solution is not unique since the energy eigenvalue is arbitrary within the range 0 < E < m. The stability of the solution has not been addressed.