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

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.

In this version we have corrected a mistake regarding the calculation of the electromagnetic fields from the electromagnetic 4-potentials. The physical interpretation of the results does not change

References in corpus (12)

• Weyl and Dirac Semimetals in Three Dimensional Solids
• Colossal Bulk Photovoltaic Effect in a Weyl Semimetal
• Weyl-Kondo Semimetal in Heavy Fermion Systems
• Current-Voltage Characteristics of Weyl Semimetal Semiconducting Devices, Veselago Lenses and Hyperbolic Dirac Phase
• Prediction of a controllable Weyl semi-metallic phase in inversion-asymmetric BiSb
• Algorithm for the solution of the Dirac equation on digital quantum computers
• The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration
• Massless Dirac-Fermions in Stable Two-Dimensional Carbon-Arsenic Monolayer
• Production of Dirac Particles in External Electromagnetic Fields
• Inverse approach to solutions of the Dirac equation for space-time dependent fields
• Elementary particles in the early Universe
• Generalized solutions of the Dirac equation, W bosons, and beta decay

Cited by in corpus (6)

• Degenerate solutions to the massless Dirac and Weyl equations and a proposed method for controlling the quantum state of Weyl particles
• On the remarkable properties of Weyl particles
• A general method for obtaining degenerate solutions to the Dirac and Weyl equations and a discussion on the experimental detection of degenerate states
• Degenerate solutions to the Dirac equation for massive particles and their applications in quantum tunneling
• A novel device for controlling the flow of information based on Weyl fermions and some interesting remarks regarding the electromagnetic interactions of high energy particles
• Degenerate wave-like solutions to the Dirac equation for massive particles