Self-Dual Maxwell Fields from Clifford Analysis
Calum Robson
TL;DR
The paper shows that the Clifford-Cauchy-Riemann (CCR) equations, obtained by grade-by-grade decomposition of the Dirac operator in the spacetime algebra $Cl(3,1)$, encode both Maxwell and Dirac equations in a unified geometric framework. In the odd sector, CCR yield an anti-self-dual Maxwell field $F=oldsymbol{ abla} abla extbf{A}$ sourced by a divergence-free potential and gauge freedom, while in the even sector CCR reproduce the massless Dirac equation through Hestenes’ polar decomposition $oldsymbol{ extbf{φ}}= ho^{1/2}e^{IB}e^{oldsymbol{ heta}/2}$, leading to $f_{0}= ext{ln} ho$, $ extbf{f}_{2}=oldsymbol{ heta}/2$, and $ extbf{f}_{4}=IB$. A single monogenic multivector thus simultaneously defines a free Dirac field and a source-free Maxwell field, revealing a deep link between spacetime geometry and fundamental physics. The work suggests broad implications for Clifford Analysis as a foundational tool in theoretical physics and outlines future directions to include massive/charged fields, sources, and cohomological frameworks such as BRST/BV. The results highlight how the four-dimensional analogue of Cauchy–Riemann conditions governs the structure of both gauge and spinor fields in a geometrically natural setting.
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.