On the relativistic description of charges fermions

TL;DR

The paper addresses the asymmetry in Dirac theory by introducing a charge quantum number $\sigma=\pm 1$ to describe charged fermions with positive energy for both signs and excluding negative-energy states. It develops independent probability-amplitude fields $\psi_{\sigma}$ and $\underline{\psi}_{\overline{\sigma}}$ using a pseudo-Euclidean metric, derives Dirac equations and a corresponding Lagrangian with conserved probability, charge, and energy-momentum currents, and analyzes both stationary and non-stationary external electromagnetic fields. The approach yields explicit expressions for energy positivity $E_{\sigma}$ and a Born interpretation for the charged fields, and reveals coupling between charge sectors in non-stationary fields. This symmetric, two-charge framework generalizes Dirac theory beyond electrons/positrons and could extend to other charged fields.

Abstract

A method for describing charged relativistic Fermi fields is proposed, in which particles of opposite charges are treated equally and states with negative energy are excluded. The concept of charge quantum number is introduced. Fields of particles and antiparticles with different charge quantum numbers are associated with wave functions for which the Born interpretation as probability amplitudes is valid.