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/
Scopus

2-s2.0-105029082966
ADS NASA

2025PhyU...68.1283N
Citation: Neznamov V P "Mathematical paradoxes of Dirac equation representations" Phys. Usp. 68 1283–1288 (2025)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

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) ↓

V.I. Ritus “Finite value of the bare charge and the relation of the fine structure constant ratio for physical and bare charges to zero-point oscillations of the electromagnetic field in a vacuum” Phys. Usp. 65 468–486 (2022)
J. Gaite “The relativistic virial theorem and scale invariance” Phys. Usp. 56 919–931 (2013)
L.A. Rivlin “Photons in a waveguide (some thought experiments)” Phys. Usp. 40 291–303 (1997)
P.A. Krachkov, A.I. Mil’shtein “An operator derivation of the quasiclassical Green's function” Phys. Usp. 61 896–899 (2018)
I.N. Toptygin “Quantum description of a field in macroscopic electrodynamics and photon properties in transparent media” Phys. Usp. 60 935–947 (2017)
S.V. Vonsovskii, M.S. Svirskii “The Klein paradox and the zitterbewegung of an electron in a field with a constant scalar potential” Phys. Usp. 36 (5) 436–439 (1993)
V.I. Ritus “Duality of two-dimensional field theory and four-dimensional electrodynamics leading to finite value of the bare charge” Phys. Usp. 56 565–589 (2013)
V.S. Popov “Feynman disentangling оf noncommuting operators and group representation theory” Phys. Usp. 50 1217–1238 (2007)
L.B. Okun’ “The concept of mass (mass, energy, relativity)” Sov. Phys. Usp. 32 629–638 (1989)
S.P. Efimov “Coordinate space modification of Fock's theory. Harmonic tensors in the quantum Coulomb problem” Phys. Usp. 65 952–967 (2022)
V.S. Olkhovsky “On the multiple internal reflections of particles and phonons tunneling in one, two, or three dimensions” Phys. Usp. 57 1136–1145 (2014)
M.I. Krivoruchenko “Transitional currents of spin-1/2 particles” Phys. Usp. 37 601–606 (1994)
L.V. Prokhorov “Quantization of the electromagnetic field” Sov. Phys. Usp. 31 151–162 (1988)
V.P. Kazantsev, O.A. Zolotov et al “Interaction energy of point electric multipoles” Phys. Usp. 68 843–849 (2025)
I.M. Arbekov, S.N. Molotkov “Quantum random number generators, extraction of provably random bit sequences from Markov chain trajectories” Phys. Usp. 67 919–937 (2024)
V.I. Alshits, V.N. Lyubimov “Plasmon-polariton at the interface of a uniaxial crystal and a metal: real dispersion equation and its analysis” Phys. Usp. 66 90–102 (2023)
V.Yu. Shishkov, E.S. Andrianov et al “Relaxation of interacting open quantum systems” Phys. Usp. 62 510–523 (2019)
I.D. Novikov “Antigravitation in the Universe” Phys. Usp. 61 692–696 (2018)
A.A. Abrashkin, E.N. Pelinovsky “On the relation between Stokes drift and the Gerstner wave” Phys. Usp. 61 307–312 (2018)
V.V. Shevchenko “Localization of a stationary electromagnetic field by a planar boundary of a metamaterial” Phys. Usp. 54 1131–1142 (2011)

The list is formed automatically.