On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic 4-potentials

TL;DR

This paper explores the relationship between solutions to Dirac and Weyl equations and their associated electromagnetic potentials, revealing degeneracy and classification of solutions, with implications for understanding electromagnetic fields and massless particles.

Contribution

It classifies Dirac solutions into degenerate and non-degenerate types and demonstrates the existence of gauge-inequivalent potentials for degenerate solutions, providing explicit examples.

Findings

Weyl solutions are infinitely degenerate with respect to electromagnetic potentials.

Dirac solutions are classified into two classes: one-to-one and infinitely degenerate.

Degenerate solutions to the force-free Dirac equation correspond to massless particles.

Abstract

In this paper we study in detail the connection between the solutions to the Dirac and Weyl equation and the associated electromagnetic 4-potentials. First, it is proven that all solutions to the Weyl equations are degenerate, in the sense that they correspond to an infinite number of electromagnetic 4-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 4-potential, and are called non-degenerate Dirac solutions. On the other hand, the elements of the second class correspond to an infinite number of 4-potentials, and are called degenerate Dirac solutions. Further, it is proven that at least two of these 4-potentials are gauge-inequivalent, corresponding to different electromagnetic fields. In order to illustrate this particularly important result we have studied the denerate 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.