Localised Solutions of the Maxwell-Dirac Equations

TL;DR

This paper investigates static, spherically symmetric solutions to the Maxwell-Dirac equations, revealing the necessity of magnetic monopoles and describing compact objects with layered structures.

Contribution

It provides a detailed analysis of simplified Maxwell-Dirac equations under symmetry assumptions, including numerical solutions and novel insights into monopole existence.

Findings

Spherical symmetry requires magnetic monopoles.

Unique analytic solutions determined by boundary conditions.

Solutions describe compact, layered objects with onion-like shells.

Abstract

The full ``classical" Dirac-Maxwell equations are considered under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is {\em static} and a further reduction is performed in the case of {\em spherical symmetry}. These static spherically symmetric equations are examined in some detail and a numerical solution presented. Some surprising results emerge: * Spherical symmetry necessitates the existence of a magnetic monopole. * There exists a uniquely defined solution, determined only by the demand that the solution be analytic at infinity. * The equations describe highly compact objects with an inner onion-like shell structure.