Low temperature expansion in the Lifshitz formula

TL;DR

This paper provides a detailed analysis of the low temperature behavior of the Casimir free energy using the Lifshitz formula, employing the Abel-Plana formula to unify and extend existing results, including cases with dissipation.

Contribution

It offers a unified, detailed derivation of the low temperature expansion of the Casimir free energy, extending known results and including dissipation effects.

Findings

Re-derivation of known low temperature results

Extension beyond previous models in certain cases

Inclusion of dissipation effects in the analysis

Abstract

The low temperature expansion of the free energy in a Casimir effect setup is considered in detail. The starting point is the Lifshitz formula in Matsubara representation and the basic method is its reformulation using the Abel-Plana formula making full use of the analytic properties. This provides a unified description of specific models. We re-derive the known results and, in a number of cases, we are able to go beyond. We also discuss the cases with dissipation. It is an aim of the paper to give a coherent exposition of the topic. The paper includes the derivations and should provide a self contained representation.