Anatolii I. Nikishov

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 etween 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.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.

Rindler solutions and their physical interpretation

We show that the singular behavior of Rindler solutions near horizon testifies to the currents of particles from a region arbitrarily close to the horizon. Besides, the Rindler solutions in right Rindler sector of Minkowski space can be represented as a superposition of only positive-or only negative-frequency plane waves; these states require infinite energy for their creation and possess infinite charge in a finite space interval, containing the horizon. The positive-or negative-frequency representations of Rindler solutions analytically continued to the whole Minkowski space make up a complete set of states in this space, which have, however, the aforementioned singularities. These positive (negative)-frequency states are characterized by positive (negative) total charge, the charge of the same sign in right (left) Rindler sector and by quantum number κ. But in other Lorentz invariant sectors they do not possess positive (negative)-definite charge density and have negative (positive) charge in left (right) Rindler sector. Therefore these states describe both the particle (antiparticle) and pairs, the mean number of which is given by Planck function of κ. These peculiarities make the Rindler set of solutions nonequivalent to the plane wave set and the inference on the existence of thermal currents for a Rindler observer moving in empty Minkowski space is unfounded.

Vector boson in the constant electromagnetic field

The propagator and the complete sets of in-and out-solutions of the wave equation, together with the Bogoliubov coefficients relating these solutions are obtained for the vector W-boson (with the gyromagnetic ratio g=2) in a constant electromagnetic field. When only the electric field is present, the Bogoliubov coefficients are independent of the boson polarization and are the same as for the scalar boson. For the collinear electric and magnetic fields, the Bogoliubov coefficients for states with the boson spin perpendicular to the field are again the same as in the scalar case. For the W− spin parallel (antiparallel) to the magnetic field, the Bogoliubov coefficients and the one-loop contributions to the imaginary part of the Lagrange function are obtained from the corresponding expressions for the scalar case by the substitution m2 → m2+2eH (m2 → m2-2eH). For the gyromagnetic ratio g=2, the vector boson interaction with the constant electromagnetic field is described by the functions that can be expected by comparing the scalar and Dirac particle wave functions in the constant electromagnetic field.

Energy-Momentum Tensor of Particles Created by an External Field

In Minkowski space-time the energy-momentum tensor (EMT) of particles, which arise after turning off the external field, is defined by normal ordering of out-operators. In this way the finite expression for the expectation value 〈0in| : T µν : |0 in 〉 is obtained. No regularization is needed and EMT is treated on equal footing with other observables such as current or the number of created particles. This means that an expectation value of an observable is well defined only after the process of its formation is finished. In application to particles produced by a mirror, which moves with acceleration during a finite time interval in 1+1 space-time, the value 〈0in| : T µν : |0 in 〉 does not coincide with the regularized value, 〈0in| : T µν : |0 in 〉reg although the integrals over all space of their 00-components give the same total energy of produced particles.