On vacuum-vacuum amplitude and Bogoliubov coefficients

TL;DR

This paper explores how virtual pairs influence the vacuum-vacuum amplitude in electromagnetic fields, emphasizing the role of Bogoliubov coefficients and phase distinctions between in- and out-solutions, even without pair creation.

Contribution

It clarifies the relationship between vacuum amplitudes, Bogoliubov coefficients, and phase definitions in electromagnetic fields without pair creation.

Findings

Virtual pairs induce a phase in vacuum-vacuum amplitude.

In- and out-solutions differ by a phase factor related to Bogoliubov coefficients.

Transition amplitude for an electron remains unity, consistent with Pauli principle.

Abstract

Even if the electromagnetic field does not create pairs, virtual pairs lead to the appearance of a phase in vacuum-vacuum amplitude. This makes it necessary to distinguish the in- and out-solutions even when it is commonly assumed that there is only one complete set of solutions as, for example, in the case of a constant magnetic field. Then in- and out-solutions differ only by a phase factor which is in essence the Bogoliubov coefficient. The propagator in terms of in- and out-states takes the same form as the one for pair creating fields. The transition amplitude for an electron to go from an initial in-state to out-state is equal to unity (in diagonal representation). This is in agreement with Pauli principal: if in the field there is an electron with given (conserved) set of quantum numbers, virtual pair cannot appear in this state. So even the phase of transition amplitude remains unaffected by the field. We show how one may redefine the phases of Bogoliubov coefficients in order to express the vacuum-vacuum amplitude through them.