A biquaternionic reformulation of Maxwell's equations via Fourier analysis

TL;DR

This paper introduces a biquaternionic reformulation of Maxwell's equations using Fourier analysis, providing explicit solutions and analytical tools for complex electromagnetic problems.

Contribution

It develops a biquaternionic framework for Maxwell's equations, including a fundamental solution and inverse operator, extending previous approaches.

Findings

Explicit vectorial solutions to time-dependent Maxwell system

Characterization of the Fourier transform of fundamental solutions

Development of analytical tools for electromagnetic problems

Abstract

We analyze the parabolic Dirac operator $D \pm i\partial_t$ in a biquaternionic setting, characterizing its kernel via generalized div-curl systems and Riemann-Cauchy-type relations between the real and imaginary parts. Using Fourier transforms, we fully characterize the Fourier transform of a fundamental solution of $D \pm i\partial_t$ and construct a well-defined right inverse operator. As a key application, we derive explicit vectorial solutions to the time-dependent Maxwell system, extending prior biquaternionic approaches. These tools offer analytical efficiency for complex electromagnetic problems.