Effective action for delta potentials: spacetime-dependent inhomogeneities and Casimir self-energy

TL;DR

This paper analyzes the vacuum fluctuations of a quantum scalar field near a thin, inhomogeneous mirror modeled by a delta potential, computing the effective action and exploring Casimir energies and particle creation.

Contribution

It introduces a perturbative evaluation of the effective action for inhomogeneous delta potentials, including nonlocal effects and applications to Casimir self-energy and dynamical Casimir effect.

Findings

Divergences are absorbed into local counterterms.

Finite part of the effective action is nonlocal in inhomogeneities.

Explicit computation for massless fields in 4D.

Abstract

We study the vacuum fluctuations of a quantum scalar field in the presence of a thin and inhomogeneous flat mirror, modeled with a delta potential. Using Heat-Kernel techniques, we evaluate the Euclidean effective action perturbatively in the inhomogeneities (nonperturbatively in the constant background). We show that the divergences can be absorbed into a local counterterm, and that the remaining finite part is in general a nonlocal functional of the inhomogeneities, which we compute explicitly for massless fields in $D=4$ dimensions. For time-independent inhomogeneities, the effective action gives the Casimir self-energy for a partially transmitting mirror. For time-dependent inhomogeneities, the Wick-rotated effective action gives the probability of particle creation due to the dynamical Casimir effect.