Abstract
A new class of exact solutions of the Dirac equation with external electromagnetic fields is derived by assuming a set of field-dependent solution matrices which obey an algebra isomorphic to the Pauli matrices. The method of exact solution may be applied to any field having a four-vector potential Aμ depending only on kμxμ, but for which the field tensor and initial electron momentum are such that AμAμ, Aμpμ, (σμνFμν)2, and (σμνF′μν)2 are independent of kμxμ. Exact solutions for a circularly polarized propagating electromagnetic wave in an isotropic medium, for a screw symmetric static magnetic field, and for a rotating uniform electric field are given in terms of the roots of a quartic equation. A class of solutions is given explicitly in the weak field limit to lowest order in eA/mc2. The vacuum limit of the solution of a wave propagating in a medium is shown to be the Volkov solution.
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