Subluminal and Superluminal Electromagnetic Waves and the Lepton Mass Spectrum

TL;DR

This paper explores solutions to Maxwell's equations that include subluminal and superluminal electromagnetic waves, linking these solutions to the Dirac equation and the lepton mass spectrum, offering a novel theoretical perspective on particle masses.

Contribution

It introduces a framework connecting electromagnetic wave solutions to the Dirac-Hestenes equation and derives lepton masses from these solutions under specific conditions.

Findings

Subluminal and superluminal solutions characterized by F^2 ≠ 0.

Reduction of nonlinear Dirac-Hestenes equation to linear form for specific angles.

Derivation of muon and tau masses from electromagnetic wave solutions.

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 $F = \psi \gamma_{21} \tilde \psi$ where $\psi \in \sec \clif^+(M)$. 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 $\dirac \psi =0$, $\psi \in \sec \clif^+ (M)$, is equivalent to a generalized Maxwell equation $\dirac F = J_{e} - \gamma_5 J_{m} = {\cal J}$. 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.