Derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors

TL;DR

This paper re-examines the derivation conditions of the Lifshitz-Matsubara sum formula for Casimir pressure, revealing differences between lossy and lossless models and highlighting the role of Foucault currents.

Contribution

It clarifies the validity conditions of the Lifshitz-Matsubara formula and explains the discrepancy between Drude and plasma models in Casimir physics.

Findings

The formula is valid for the Drude model but not for the lossless plasma model.

Foucault currents significantly influence the mode structure in the limit of vanishing losses.

The interpretation of the sum formula's validity depends on the material model used.

Abstract

We carefully re-examine the conditions of validity for the consistent derivation of the Lifshitz-Matsubara sum formula for the Casimir pressure between metallic plane mirrors. We recover the usual expression for the lossy Drude model, but not for the lossless plasma model. We give an interpretation of this new result in terms of the modes associated with the Foucault currents which play a role in the limit of vanishing losses, in contrast to common expectations.