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

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.

25 pages, 4 figures. Added a discussion on the renormalization process. Added some missing factors in formulas (72) and (73), App. A

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• Scalar-Mediated Quantum Forces Between Macroscopic Bodies and Interferometry
• Non-perturbative Quantum Propagators in Bounded Spaces
• Worldline approach for spinor fields in manifolds with boundaries
• Resummed heat-kernel for surface contributions: Dirichlet semitransparent boundary conditions