$S$-matrix in quantum electrodynamics with external field

The radiation field being ignored, the evolution operator $S(t)$ for a state of the system is expressed in terms of solutions of the Dirac equation. Since, generally speaking, the operators of creation and annihilation of particles and antiparticles defined in the limit $t\to - \infty$ do not coincide with the corresponding operators in the limit $t\to \infty$ but are related to them by a Bogolyubov transformation, the vacuums of the initial and the final state are different as well. Study of the canonical operator corresponding to the Bogolyubov transformation enables one to refine the proof of the equivalence of the Feynman theory of positrons and the secondquantized theory.