Intensely Localized Solutions of the Classical Dirac-Maxwell Field Equations

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Abstract
Intensely localized and divergence-free solutions of the classical Dirac-Maxwell field equations are obtained numerically under the assumption that the electrostatic potential is dominant. For the assumption that the electromagnetic vector potential is dominant, such solutions do not exist. The masses of the localized states are negative. However, we show the possibility of getting a positive mass by taking into account the vacuum polarization energy. Introducing this vacuum polarization effect and the electromagnetic self-energy into the classical Dirac equation we obtain the differential cross section for Compton scattering. The energies of the electron without the vacuum polarization effect take slightly different values for different forms of the spinor field in contrast to the well-known degeneracy in the Hydrogen atom.

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