Derivative expansion for the boundary interaction terms in the Casimir effect: generalized $\delta$-potentials

TL;DR

This paper develops a local derivative expansion for boundary interaction terms in the Casimir effect, representing finite-width mirror effects with generalized delta-potentials, and computes the resulting Casimir energy.

Contribution

It introduces a derivative expansion approach to model finite-width mirror interactions in the Casimir effect using generalized delta-potentials, including quantum effects.

Findings

Explicit Casimir energy calculations for generalized delta-potentials.

Reinterpretation of certain potentials as imperfect Dirichlet boundary conditions.

Demonstration of the method's applicability to finite-width mirror models.

Abstract

We calculate the Casimir energy for scalar fields in interaction with finite-width mirrors, described by nonlocal interaction terms. These terms, which include quantum effects due to the matter fields inside the mirrors, are approximated by means of a local expansion procedure. As a result of this expansion, an effective theory for the vacuum field emerges, which can be written in terms of generalized $\delta$-potentials. We compute explicitly the Casimir energy for these potentials and show that, for some particular cases, it is possible to reinterpret them as imposing imperfect Dirichlet boundary conditions