A generalized electromagnetic theory for the mass spectrum of neutrinos

Abstract

In previous papers it was shown that solutions of Weyl equation that are eigenstates of the parity operator describe a coupled pair of a monopole anti-monopole system. These results suggest to seek a solution of the Maxwell equation \(\partial F^\infty = - g\mathcal{J}\) with a current \(\mathcal{J}\) as a source and such that the Lorentz force on the current is null. We first identify a solution where \(J_m = - \gamma ^5 \mathcal{J}\) is a spacelike field. More surprisingly we find that there exists a solution F of the free Maxwell ∂ F = 0 that is equivalent to the inhomogeneous equation for F ∞. Once this result is proved, it suggests by itself to seek more general (subluminal and even superluminal) solutions \(\mathfrak{F}\) of the free Maxwell equation equivalent to an inhomogeneous Maxwell equation for a field \(\mathfrak{F}_0\) with a current term as a source which may be a timelike or spacelike field. We exhibit one such subluminal solution, for which the Dirac-Hestenes spinor field ψ associated with the electromagnetic field \(\mathfrak{F}_0\) satisfies a Dirac-like equation for a bradyonic neutrino under the ansatz that the current is \(ce^{\lambda \gamma ^5 } g\psi \gamma ^0 \tilde \psi\) with g the quantum of magnetic charge and λ a constant to be determined in such a way that the auto-force is zero. Together with Dirac’s quantization condition this gives a quantized mass spectrum (Eq. (46)) for neutrinos, with masses of the different flavor neutrinos being of the same order of magnitude (Eq. (47)), which is in accord with recent experimental findings.