Clifford Algebra Derivation of the Characteristic Hypersurfaces of Maxwell's Equations

TL;DR

This paper presents a simplified geometric algebra approach to derive the characteristic hypersurfaces of Maxwell's equations, enabling easier analysis of electromagnetic wave fronts and their evolution.

Contribution

It introduces a Clifford algebra-based derivation of Maxwell's characteristic surfaces, simplifying the understanding of wave front propagation in electromagnetism.

Findings

Derivation of electromagnetic wave fronts using Clifford algebra

Simplified method for analyzing the evolution of electric and magnetic fields

Identification of singular characteristic surfaces in Maxwell's equations

Abstract

An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using 3D Clifford algebra (isomorphic to Pauli algebra spinorial formulation of electromagnetism). Subsequently one can more easily solve for the time evolution of both the electric and magnetic field simultaneously in terms of the fields evaluated only on a 3D hypersurface. The form of the special "characteristic" surfaces for which the time derivative of the fields can be singular are quickly deduced with little effort.