The Dirac-Hestenes Equation for Spherical Symmetric Potentials in the Spherical and Cartesian Gauges

TL;DR

This paper demonstrates how to solve the Dirac-Hestenes equation with spherically symmetric potentials in Minkowski spacetime using Clifford bundle formalism, showing equivalent complexity in Cartesian and spherical gauges.

Contribution

It provides a straightforward method for solving the Dirac-Hestenes equation with spherical symmetry in different gauges using Clifford bundle formalism.

Findings

Solutions in both gauges have the same mathematical difficulty.

Separation of variables is effective for spherically symmetric potentials.

Clifford bundle formalism simplifies the Dirac-Hestenes equation analysis.

Abstract

In this paper using the apparatus of the Clifford bundle formalism we show how straightforwardly solve in Minkowski spacetime the Dirac-Hestenes equation -- which is an appropriate representative in the Clifford bundle of differential forms of the usual Dirac equation -- by separation of variables for the case of a potential having spherical symmetry in the Cartesian and spherical gauges. We show that contrary to what is expected at a first sight, the solution of the DHE in both gauges has exactly the same mathematical difficulty.