Relativistic dynamical inversion in manifestly covariant form

TL;DR

This paper introduces a covariant form of the relativistic dynamical inversion technique for the Dirac equation, enabling easier coordinate transformations and the derivation of new analytical solutions in magnetic and electric field configurations.

Contribution

It presents a covariant formulation of the relativistic dynamical inversion method and constructs new exact solutions to the Dirac equation in various electromagnetic fields.

Findings

New covariant form of the dynamical inversion technique.

Exact solutions for Dirac electron in magnetic and electric fields.

Insights into spinor geometry and spin-orbit coupling.

Abstract

The Relativistic Dynamical Inversion technique, a novel tool for finding analytical solutions to the Dirac equation, is written in explicitly covariant form. It is then shown how the technique can be used to make a change from Cartesian to spherical coordinates of a given Dirac spinor. Moreover the most remarkable feature of the new method, which is the ease of performing non-trivial change of reference frames, is demonstrated. Such a feature constitutes a potentially powerful tool for finding novel solutions to the Dirac equation. Furthermore, a whole family of normalizable analytic solutions to the Dirac equation is constructed. More specifically, we find exact solutions for the case of a Dirac electron in the presence of a magnetic field as well as a novel solution comprising of a combination of a spherically symmetric electric field and magnetic fields. These solutions shed light on the possibility of separating the positive and negative energy parts of localized Dirac spinors in the presence as well as in the absence of magnetic fields. The presented solutions provide an illustration of the connection between the geometrical properties of the spinor and spin-orbit coupling for normalizable spinorial wave functions.