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.

复函数的研究基于对满足 Cauchy-Riemann 方程的全态函数的研究。Clifford 分析相对较新的领域使我们能够将复杂分析的许多结果扩展到更高的维度。在本文中，我使用几何代数形式将一般 Clifford 代数的 Cauchy-Riemann 方程分解为等级，并表明对于时空代数 Cl（3， 1），这些方程是自对偶无源麦克斯韦场和无质量不带电荷的旋量的方程。这表明了基础物理学与时空的 Clifford 几何之间的深刻联系。