On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

Abstract

In this work we study in detail the connection between the solutions to the Dirac and Weyl equations and the associated electromagnetic four-potentials. First, it is proven that all solutions to the Weyl equation are degenerate, in the sense that they correspond to an infinite number of electromagnetic four-potentials. As far as the solutions to the Dirac equation are concerned, it is shown that they can be classified into two classes. The elements of the first class correspond to one and only one four-potential, and are called non-degenerate Dirac solutions. On the other hand, the elements of the second class correspond to an infinite number of four-potentials, and are called degenerate Dirac solutions. Further, it is proven that at least two of these four-potentials are gauge-inequivalent, corresponding to different electromagnetic fields. In order to illustrate this particularly important result we have studied the degenerate solutions to the force-free Dirac equation and shown that they correspond to massless particles. We have also provided explicit examples regarding solutions to the force-free Weyl equation and the Weyl equation for a constant magnetic field. In all cases we have calculated the infinite number of different electromagnetic fields corresponding to these solutions. Finally, we have discussed potential applications of our results in cosmology, materials science and nanoelectronics.