Lifshitz formula for the Casimir force and the Gelfand-Yaglom theorem

TL;DR

This paper derives the Lifshitz formula for the Casimir force using Quantum Field Theory, modeling imperfect mirrors with background potentials and employing the Gelfand-Yaglom theorem to evaluate functional determinants related to reflection coefficients.

Contribution

It provides a QFT-based derivation of the Lifshitz formula for scalar fields, connecting the Casimir force to functional determinants and reflection coefficients at imaginary frequencies.

Findings

Derivation of Lifshitz formula via Gelfand-Yaglom theorem

Expression of vacuum energy in terms of reflection coefficients

Framework applicable to imperfect, thick plane mirrors

Abstract

We provide a Quantum Field Theory derivation of Lifshitz formula for the Casimir force due to a fluctuating real scalar field in $d+1$ dimensions. The field is coupled to two imperfect, thick, plane mirrors, which are modeled by background potentials localized on their positions. The derivation proceeds from the calculation of the vacuum energy in the Euclidean version of the system, reducing the problem to the evaluation of a functional determinant. The latter is written, via Gelfand-Yaglom's formula, in terms of functions depending on the structure of the potential describing each mirror; those functions encode the properties which are relevant to the Casimir force and are the reflection coefficients evaluated at imaginary frequencies.