Dirac/Rarita-Schwinger plus Maxwell theories in $\mathbb{R} \times S^3$ spacetime in the Hopf coordinates

TL;DR

This paper analyzes Dirac-Maxwell and Rarita-Schwinger-Maxwell equations in a curved spacetime using Hopf coordinates, revealing variable separation, solutions, and a link between Dirac current and Hopf invariant.

Contribution

It introduces a method to separate variables and solve these equations analytically and numerically in $ ext{R} imes S^3$ spacetime, highlighting the relation to topological invariants.

Findings

Successful separation of variables in Hopf coordinates

Analytic and numerical solutions obtained

Dirac current linked to Hopf invariant

Abstract

We consider the sets of Dirac-Maxwell and Rarita-Schwinger-Maxwell equations in $\mathbb{R} \times S^3$ spacetime. Using the Hopf coordinates, we show that these equations allow separation of variables and obtain the corresponding analytic and numerical solutions. It is also demonstrated that the current of the Dirac field is related to the Hopf invariant on the $S^3 \rightarrow S^2$ fibration.