A zero-curvature representation of electromagnetism and the conservation of electric charge

TL;DR

This paper presents a novel zero-curvature framework for electromagnetism in various dimensions, revealing electric charge conservation as a consequence of hidden symmetries within this geometric representation.

Contribution

It introduces a zero-curvature integral representation of electromagnetism in (D+1)-dimensional Minkowski space, highlighting hidden symmetries responsible for charge conservation.

Findings

Electromagnetic laws derived from zero-curvature equations in loop space.

Charge conservation linked to hidden symmetries in the geometric framework.

Applicable to dimensions D=1, 2, 3.

Abstract

We show that the laws of electromagnetism in $(D+1)$-dimensional Minkowski space-time $\mathcal{M}$, explicitly for $D=1$, $2$ and $3$, can be obtained from an integral representation of the zero-curvature equation in the corresponding loop space $\mathcal{L}^{(D-1)}(\mathcal{M})$. The conservation of the electric charge can be seen as the result of a hidden symmetry in this representation of the dynamical equations.