Electromagnetism: an intrinsic approach to Hadamard's method of descent

TL;DR

This paper introduces a geometric framework for reducing Maxwell's equations from four-dimensional spacetime to lower dimensions using descent along invariant vector fields, enhancing understanding of electromagnetic theories in various dimensions.

Contribution

It develops a systematic, geometric approach to dimensional reduction of electromagnetism based on the interplay of Lie derivatives and the Hodge star operator, offering a unified interpretation.

Findings

Provides a geometric interpretation of dimensional reduction

Derives lower-dimensional electromagnetic theories from 4D Maxwell's equations

Explores multiple descent along commuting vector fields

Abstract

We present a systematic geometric framework for the dimensional reduction of classical electromagnetism based on the concept of descent along vector fields of invariance. By exploring the interplay between the Lie derivative and the Hodge star operator, we implement descent conditions on differential forms that reduce Maxwell's equations in four-dimensional spacetime to electromagnetic theories in lower dimensions. We also consider multiple descent along pairwise commuting vector fields of invariance, yielding a finer decomposition of Maxwell's equations. Our results provide a unified and geometrically transparent interpretation of dimensional reduction, with potential applications to field theories in lower-dimensional spacetimes.