Mathematical Paradoxes of Dirac Equation Representations

TL;DR

This paper analyzes different mathematical representations of the Dirac equation, identifying paradoxes and proposing that using only positive energy states resolves these issues in quantum electrodynamics.

Contribution

It uncovers mathematical paradoxes in Dirac equation representations and suggests a resolution by restricting to positive energy amplitude states.

Findings

Mathematical artifacts contradict physical premises

Paradoxes are resolved with positive energy states

Analysis covers perturbative and nonperturbative QED

Abstract

This paper examines the Foldy-Wouthuysen and Feynman-Gell-Mann representations of the Dirac equation. The analysis is conducted for electrons and positrons interacting with electromagnetic fields. Versions of quantum electrodynamics are considered both within the scope of perturbation theory and in the nonperturbative case with strong electromagnetic fields. Mathematical artifacts that contradicting the physical premises of the theory are identified in the studied representations of the Dirac equation. These mathematical paradoxes are resolved if the theory only employs amplitude states (real and virtual) with positive energies.