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arXiv:2602.23384

Mathematical Paradoxes of Dirac Equation Representations

V. P. Neznamov

hep-thhep-phquant-phSubmitted Feb 15, 2026• Announced Feb 17, 2026
arXivHTMLPDF

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.

Journal reference
Phys. Usp. 68 (2025) 1283-1288
DOI
10.3367/UFNe.2025.11.040057
Comments
12 pages, 2 figures

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This paper re-examines the key aspects of Dirac's derivation of his relativistic equation for the electron in order advance our understanding of the nature of quantum field theory. Dirac's derivation, the paper argues, follows the key principles behind Heisenberg's discovery of quantum mechanics, which, the paper also argues, transformed the nature of both theoretical and experimental physics vis-a-vis classical physics and relativity. However, the limit theory (a crucial consideration for both Dirac and Heisenberg) in the case of Dirac's theory was quantum mechanics, specifically, Schrodinger's equation, while in the case of quantum mechanics, in Heisenberg's version, the limit theory was classical mechanics. Dirac had to find a new equation, Dirac's equation, along with a new type of quantum variables, while Heisenberg, to find new theory, was able to use the equations of classical physics, applied to different, quantum-mechanical variables. In this respect, Dirac's task was more similar to that of Schrodinger in his work on his version of quantum mechanics. Dirac's equation reflects a more complex character of quantum electrodynamics or quantum field theory in general and of the corresponding (high-energy) experimental quantum physics vis-a-vis that of quantum mechanics and the (low-energy) experimental quantum physics. The final section examines this greater complexity and its implications for fundamental physics.
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