On group invariant solutions to the Maxwell Dirac equations

This work constitutes a study on group invariant solutions of the Maxwell Dirac equations for a relativistic electron spinor in its own self-consistent electromagnetic field. First, the Maxwell Dirac equations are written in a gauge independent tensor form, in terms of bilinear Dirac currents and a gauge independent total four-potential. A requirement of this form is that the length of the current vector be non-zero. In this form they are amenable to the study of solutions invariant under subgroups of the Poincare group without reference to the Abelian gauge group. In particular, all subgroups of the Poincare group that generate 4 dimensional orbits by transitive action on Minkowski space, and the corresponding invariant vector fields are identifed, which will constitute invariant solutions merely if various constants satisfy a set of algebraic equations. For each such subgroup, the possibility of solutions to both the full Maxwell Dirac equations and to a classical approximation to the self-field equations is determined. Of the 19 classes of simply transitive subgroups, only one class yielded a solution.