Localised Solutions of the Maxwell-Dirac Equations

The full ``classical" Dirac-Maxwell equations are considered under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is {\em static} and a further reduction is performed in the case of {\em spherical symmetry}. These static spherically symmetric equations are examined in some detail and a numerical solution presented. Some surprising results emerge: * Spherical symmetry necessitates the existence of a magnetic monopole. * There exists a uniquely defined solution, determined only by the demand that the solution be analytic at infinity. * The equations describe highly compact objects with an inner onion-like shell structure.