Mathematical paradoxes of Dirac equation representations

V.P. Neznamov^†
Russian Federal Nuclear Center - All-Russian Research Institute of Experimental Physics, prosp. Mira 37, Sarov, Nizhny Novgorod region, 607188, Russian Federation

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 contradict 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.

Keywords: Dirac equation representations, quantum electrodynamics, fermion vacuum, positive and negative energy states, mathematical paradoxes of the theory
PACS: 03.65.Pm, 11.10.St, 12.20.−m (all)
DOI: 10.3367/UFNe.2025.11.040057
URL: https://ufn.ru/en/articles/2025/12/h/
Citation: Neznamov V P "Mathematical paradoxes of Dirac equation representations" Phys. Usp. 68 1283–1288 (2025)

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Received: 2nd, June 2025, revised: 6th, November 2025, accepted: 11th, November 2025

Оригинал: Незнамов В П «Математические парадоксы представлений уравнения Дирака» УФН 195 1356–1361 (2025); DOI: 10.3367/UFNr.2025.11.040057

References (28) ↓ Similar articles (20)

Dirac P A M The Principles Of Quantum Mechanics (Oxford: The Univ. Press, 1930)
Stuekelberg E C G Helv. Phys. Acta 14 L32 (1941); Stuekelberg E C G Helv. Phys. Acta 14 588 (1941)
Feynman R P Phys. Rev. 76 749 (1949)
Feynman R P Phys. Rev. 76 769 (1949)
Foldy L L, Wouthuysen S A Phys. Rev. 78 29 (1950)
Feynman R P, Gell-Mann M Phys. Rev. 109 193 (1958)
Zel’dovich Ya B, Popov V S Usp. Fiz. Nauk 105 403 (1971); Zel’dovich Ya B, Popov V S Sov. Phys. Usp. 14 673 (1971)
Neznamov V P, Safronov I I Zh. Eksp. Teor. Fiz. 155 792 (2019); Neznamov V P, Safronov I I J. Exp. Theor. Phys. 128 672 (2019); Neznamov V P, Safronov I I arXiv:1907.03579
Neznamov V P Fiz. Elem. Chast. At. Yad. 37 152 (2006); Neznamov V P Phys. Part. Nucl. 37 86 (2006); Neznamov V P hep-th/0411050
Neznamov V P Fiz. Elem. Chast. At. Yad. 43 70 (2012); Neznamov V P Phys. Part. Nucl. 43 36 (2012); Neznamov V P arXiv:1107.0693
Brown L M Phys. Rev. 111 957 (1958)
Neznamov V P, Shemarulin V E Int. J. Mod. Phys. A 36 2150086 (2021); Neznamov V P, Shemarulin V E arXiv:2108.04664
Neznamov V P Int. J. Mod. Phys. A 36 2150173 (2021); Neznamov V P arXiv:2110.03530
Neznamov V P Int. J. Mod. Phys. A 40 2550049 (2025)
Neznamov V P Int. J. Mod. Phys. A 40 2550074 (2025)
Neznamov V P, Silenko A J J. Math. Phys. 50 122302 (2009); Neznamov V P, Silenko A J arXiv:0906.2069
B’erken D D, Drell S D Relyativistskaya Kvantovaya Teoriya Vol. 1 (M.: Nauka, 1978); Per. s angl. yaz., Bjorken J D, Drell S D Relativistic Quantum Mechanics (New York: McGraw-Hill, 1964)
Greiner W, Reinhardt J Quantum Electrodynamics (Berlin: Springer, 2002)
Pomeranchuk I, Smorodinsky J J. Phys. USSR 9 97 (1945)
Gershtein S S, Zel’dovich Ya B Zh. Eksp. Teor. Fiz. 57 654 (1969); Gershtein S S, Zel’dovich Ya B Sov. Phys. JETP 30 358 (1970)
Pieper W, Greiner W Z. Phys. 218 327 (1969)
Zel’dovich Ya B, Popov V S Usp. Fiz. Nauk 105 403 (1971); Zel’dovich Ya B, Popov V S Sov. Phys. Usp. 14 673 (1972)
O’Connell R F "Rotation and spin in physics" General Relativity And John Archibald Wheeler (Astrophysics and Space Science Library) Vol. 367 (Eds I Ciufolini, R A Matzner) (Dordrecht: Springer, 2010) p. 325
O’Connell R F Mod. Phys. Lett. A 26 469 (2011)
Neznamov V P FIZMAT 2 (2) 94 (2024)
Neznamov V P Int. J. Mod. Phys. A 40 2550104 (2025)
Dirac P A M Proc. R. Soc. London A 322 435 (1971)
Dirac P A M Proc. R. Soc. London A 328 1 (1972)