The vacuum-vacuum amplitude and Bogoliubov coefficients

We consider the problem of fixing the phases of Bogoliubov coefficients in quantum electrodynamics such that the vacuum-vacuum amplitude can be expressed via them. For a constant electric field and particles with spins of 0 and 1/2, this is done starting from the definition of these coefficients. Using the symmetry between electric and magnetic fields, we extend the result to a constant electromagnetic field. It turns out that for a constant magnetic field, it is necessary to distinguish the in- and out-states, although they differ only by a phase factor. For a spin-1 particle with a gyromagnetic of ratio g = 2, this approach fails and we reconsider the problem using the proper-time method.