Self-Dual Maxwell Fields from Clifford Analysis

TL;DR

This paper extends complex analysis to higher dimensions using Clifford analysis, revealing that certain Clifford algebra equations describe self-dual Maxwell fields and massless spinors, thus linking physics and geometry.

Contribution

It decomposes Cauchy-Riemann equations in Clifford algebra and demonstrates their equivalence to Maxwell and spinor equations in spacetime algebra.

Findings

Cauchy-Riemann equations decompose into grades in Clifford algebra

Equations for Spacetime Algebra describe self-dual Maxwell fields

Links between Clifford geometry and fundamental physics established

Abstract

The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using the Geometric Algebra formalism, and show that for the Spacetime Algebra $Cl(3,1)$ these equations are the equations for a self-dual source free Maxwell field, and for a massless uncharged Spinor. This shows a deep link between fundamental physics and the Clifford geometry of Spacetime.