The Maxwell-Dirac equations give a model of an electron in an electromagnetic (e-m) field, in which neither the Dirac or the e-m fields are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equa- tion, resulting in a nonlinear system of partial differential equations (PDE's). In this way the self-field of the electron is included.
We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its first derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter.
In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon field must be equal and opposite to the central (external) field. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die off exponentially with increasing distance from the central charge.