Heun-type solutions for the Dirac particle on the curved background of Minkowski space-times

TL;DR

This paper derives Heun-type solutions for the Dirac equation in Minkowski space-time near a light cone, revealing energy spectra and connections to Schrödinger-like equations, applicable also to spinless particles in magnetic fields.

Contribution

It introduces a novel reduction of the Dirac equation to Heun-type equations in curved space-time, providing new analytical solutions and insights into particle behavior in such backgrounds.

Findings

Derivation of Heun-type equations for the Dirac particle

Energy spectra obtained from the solutions

Extension to spinless particles in magnetic fields

Abstract

In the present paper, we study the Dirac equation in the background of Minkowski space-time on a light cone. With the help of the coupling of the radial parts, the system of 4 equations is reduced to two different second-order differential equations, which coincide with a particle in present potentials. It turns out that the central equation reduces to Heun-type equations and provides us with energy spectra. Also, it is shown that the Schr\"odinger equation like with a combination of the external field is related to a second-order differential equation. In another way to solve the problem, results are valid for a spinless charged particle in the context of the magnetic field. Keywords: Heun function, Dirac equation, curved space-time, light cone