Subluminal and Superluminal Electromagnetic Waves and the Lepton Mass Spectrum

Abstract

Maxwell equation $\dirac F = 0$ for $F \in \sec \bwe^2 M \subset \sec \clif (M)$, where $\clif (M)$ is the Clifford bundle of differential forms, have subluminal and superluminal solutions characterized by $F^2 \neq 0$. We can write $F = ψγ_{21} \tilde ψ$ where $ψ\in \sec \clif^+(M)$. We can show that $ψ$ satisfies a non linear Dirac-Hestenes Equation (NLDHE). Under reasonable assumptions we can reduce the NLDHE to the linear Dirac-Hestenes Equation (DHE). This happens for constant values of the Takabayasi angle ($0$ or $π$). The massless Dirac equation $\dirac ψ=0$, $ψ\in \sec \clif^+ (M)$, is equivalent to a generalized Maxwell equation $\dirac F = J_{e} - γ_5 J_{m} = {\cal J}$. For $ψ= ψ^\uparrow$ a positive parity eigenstate, $j_e = 0$. Calling $ψ_e$ the solution corresponding to the electron, coming from $\dirac F_e =0$, we show that the NLDHE for $ψ$ such that $ψγ_{21} \tildeψ = F_e + F^{\uparrow}$ gives a linear DHE for Takabayasi angles $π/2$ and $3π/2$ with the muon mass. The Tau mass can also be obtained with additional hypothesis.