Sourceless Maxwell and Dirac equations via Clifford Analysis
The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The complex numbers are a subset [the even subalgebra] of $Cl(2)$, and therefore we can ask whether there are analogues for the Cauchy-Riemann equations for other Clifford algebras. This has been extensively explored under the name of Clifford Analysis. Here I explicitly 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 Electromagnetic field, and for a massless uncharged Spinor.