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 $F^2\ne 0$. We can write $F=\psi {\gamma }_{21}\tilde{\psi }$ where $ψ\in \sec \clif^+(M)$. We can show that $\psi$ 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 $\pi$). 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 $\psi ={\psi }^{\uparrow }$ a positive parity eigenstate, $j_e=0$. Calling ${\psi }_e$ the solution corresponding to the electron, coming from $\dirac F_e =0$, we show that the NLDHE for $\psi$ such that $\psi {\gamma }_{21}\tilde{\psi }=F_e+F^{\uparrow }$ gives a linear DHE for Takabayasi angles $\pi /2$ and $3\pi /2$ with the muon mass. The Tau mass can also be obtained with additional hypothesis.

arXiv comment24 pages, KAPPROC style (Kluwer Ac. Pub. Proceedings) with named references. The Abstract to appear in the e-print archive list has been corrected. The main text is the same