Subluminal and Superluminal Electromagnetic Waves and the Lepton Mass Spectrum

Abstract

Maxwell equation ∂F = 0 for F ∊ sec ^² M ⊂ sec Cℓ(M), where Cℓ(M) is the Clifford bundle of differential forms, have subluminal and superluminal solutions characterized by F ² ≠ 0. We can write F = ψγ ₂₁ ψγ where ψ ∊ sec Cℓ⁺ (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 ∂ψ 0, ψ ∊ sec Cℓ⁺(M), is equivalent to a generalized Maxwell equation ∂F = J _e − γ ₅ J _m = J. For ψ = ψ ^↑ a positive parity eigenstate, J _e = 0. Calling ψ _e the solution corresponding to the electron, coming from ∂F _e = 0, we show that the NLDHE for ψ such that ψγ ₂₁ ψγ = F _e + F ^↑ 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.