Does a nonzero tunneling probability imply particle production in time-independent classical electromagnetic backgrounds?

Abstract

In this paper, we probe the validity of the tunneling interpretation that is usually called forth in the literature to explain the phenomenon of particle production by time-independent classical electromagnetic backgrounds. We show that the imaginary part of the effective Lagrangian is zero for a complex scalar field quantized in a time-independent, but otherwise arbitrary, magnetic field. This result implies that no pair creation takes place in such a background. But we find that when the quantum field is decomposed into its normal modes in the presence of a spatially confined and time-independent magnetic field, there exists a nonzero tunneling probability for the effective Schr\"odinger equation. According to the tunneling interpretation, this result would imply that spatially confined magnetic fields can produce particles, thereby contradicting the result obtained from the effective Lagrangian. This lack of consistency between these two approaches calls into question the validity of attributing a nonzero tunneling probability for the effective Schr\"odinger equation to the production of particles by the time-independent electromagnetic backgrounds. The implications of our analysis are discussed.