This work examines the Foldy—Wouthuysen, Feynman—Gell-Mann, and self-adjoint representations of the Dirac equation for fermions with spinor wave functions. 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 non-perturbative regime with strong electromagnetic fields. Mathematical artifacts 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.