The inverse Penrose transform of a solution to the Maxwell-Dirac-Weyl field equations

An explicit family of solutions to the nonlinear coupled Maxwell-Dirac-Weyl equations in Minkowski space is presented. The abstract results of Henkin and Manin (Phys. Lett. B, 95 (1980), 405–408) show that these solutions are equivalent by the Penrose transform to a coupled system of cohomology classes and a complex line bundle on ambitwistor space, the space of null lines in Minkowski space. The explicit inverse Penrose transform of this family of solutions is computed giving explicit expressions for the line bundle (transform of the vector potential), the obstruction to extension (transform of the charge), and the two cohomology classes (transform of the Dirac-Weyl coupled spinor fields).