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

Abstract

In this paper the inverse problem of the correspondence between the solutions of the Dirac equation and the electromagnetic 4-potentials, is fully solved. The Dirac solutions are classified into two classes. The first one consists of degenerate Dirac solutions corresponding to an infinite number of 4-potentials while the second one consists of non-degenerate Dirac solutions corresponding to one and only one electromagnetic 4-potential. Explicit expressions for the electromagnetic 4-potentials are provided in both cases. Further, in the case of the degenerate Dirac solutions, it is proven that at least two 4-potentials are gauge inequivalent, and consequently correspond to different electromagnetic fields. This result is extremely important, because it leads to the groundbreaking conclusion that for a specific class of spinors, a fermion is possible to be in the same state under the influence of different electromagnetic fields.

Degenerate solutions to the Dirac equation for massive particles and their applications in quantum tunneling

“…( 1) is written in natural units, where both the speed of light in vacuum c and the reduced Planck constant are equal to one. Furthermore, in [1] we have shown that all solutions to the Weyl equation are degenerate, corresponding to an infinite number of electromagnetic 4potentials, explicitly calculated in Theorem 3.1 in [1]. Some very interesting properties of Weyl particles, mainly regarding their control and localization, are discussed in [2,3].…”

“…are degenerate, corresponding to an infinite number of electromagnetic 4-potentials A  , explicitly calculated in Theorem 5.4 in [1].…”

“…In addition, according to Theorem 5.4 in [1], the spinors given by Eq. (3) will also be solutions to the Dirac equation for an infinite number of 4-potentials, given by the formula b a s…”

“…In a recent article [1] we have shown that all solutions to the Dirac equation Here m is the mass of the particle and a qA  = where q is the electric charge of the particle and A  the electromagnetic 4-potential. It should also be noted that Eq.…”

“…In ref. [1] it has also been shown that, in the case of free Dirac particles, the degenerate solutions correspond to massless particles except for particle-antiparticle pairs [1]. However, the net charge of the particle -antiparticle pair is zero, and consequently the degeneracy is not particularly meaningful from a practical point of view.…”