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Mathematical Paradoxes of Dirac Equation Representations

V. P. Neznamov

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.

Mathematical Paradoxes of Dirac Equation Representations

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.
Paper Structure (8 sections, 32 equations, 2 figures)

This paper contains 8 sections, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Perturbation theory in Foldy-Wouthuysen representation
  3. Nonperturbative quantum electrodynamics and representations of Dirac equation
  4. Standard quantum electrodynamics in fields of hydrogen-like ions with large charge number Z
  5. Representations of Feynman-Gell-Mann and Foldy-Wouthuysen
  6. Dirac equation with nonrelativistic Hamiltonian in Foldy-Wouthuysen representation
  7. Conclusions
  8. Acknowledgements

Figures (2)

  • Figure 1: Lower energy levels of hydrogen-like ion as a function of nuclear charge number $Z$.
  • Figure 2: Energy spectrum of a) equation for electrons (\ref{['eq24']}) and b) equation for positrons (\ref{['eq25']}).