The Dirac-Maxwell Equations with Cylindrical Symmetry

TL;DR

This paper reduces the Dirac-Maxwell equations under static cylindrical symmetry, analyzes the resulting ordinary differential equations, and finds two classes of solutions: localized Dirac fields around a wire and unbounded electrostatic potentials.

Contribution

It provides a novel analytical and numerical study of the Dirac-Maxwell system with cylindrical symmetry, revealing two distinct solution classes influenced by non-linear self-field effects.

Findings

Localized Dirac fields around a charged wire

Electrostatic potential finite on axis but unbounded at large distances

Non-linearities cause the localization of the Dirac field

Abstract

A reduction of the Dirac-Maxwell equations in the case of static cylindrical symmetry is performed. The behaviour of the resulting system of o.d.e.'s is examined analytically and numerical solutions presented. There are two classes of solutions. The first type of solution is a Dirac field surrounding a charged "wire". The Dirac field is highly localised, concentrated in cylindrical shells about the wire. A comparison with the usual linearized theory demonstrates that this localization is entirely due to the non-linearities in the equations which result from the inclusion of the "self-field". The second class of solutions have the electrostatic potential finite along the axis of symmetry but unbounded at large distances from the axis.