Charge Conjugation, Heavy Ions, e^+ e^- pairs: Was there a better way to add potentials to Dirac's free electrons?

Abstract

This is a study of a possible alternative procedure for adding a potential energy to the free electron Dirac equation. When Dirac added potentials to his free electron equation, there were two alternatives (here called D1 and D2). He chose D1 and lost charge conjugation symmetry, found Ehrenfest equations that depended on the sign of the energy of the state determining the expectation value, encountered Klein tunneling, zitterbewegung and the Klein paradox. The D1 alternative also predicted that deep potentials should pull positive energy states down into the negative energy continuum, possibly creating an unstable vacuum. Extensive experiments (1975-1997) found no evidence for this instability, but did find low energy electron-positron pairs with sharply defined energies and unusually low counting statistics. These pairs tended to disappear with higher beam currents. This paper explores the other alternative, here called D2 and finds charge conjugation symmetry preserved, Ehrenfest equations are classical, Klein tunneling is not present, unstable vacuua are forbidden, zitterbewegung is absent in the charge current density, new excitations of bound electron-positron pairs are possible in atoms, and the energies at which low energy electron-positron pair production in heavy ion scattering occurs is well described. Also all of the positive energy calculations, including those with the Coulomb potential, the hydrogen-like atom, are retained exactly the same as found in alternative D1. It might have been better if Dirac had chosen alternative D2.