The Casimir effect as a sum-over-modes in dissipative systems

TL;DR

This paper demonstrates that the sum-over-modes approach within an open-system framework can derive the Lifshitz formula for Casimir free energy, generalizing to arbitrary geometries and addressing complex mode sums.

Contribution

It establishes a general method to derive the Lifshitz formula from a sum-over-modes perspective in dissipative systems, extending previous approaches.

Findings

Derivation of Lifshitz formula from sum-over-modes in dissipative systems

Generalization to arbitrary geometries using Ford, Lewis, & O'Connell's formula

Identification of necessary modifications for complex mode sums

Abstract

The aim of this paper is to show that within the open-system framework the sum-over-modes approach \'a la Casimir leads to the Lifshitz formula for the Casimir free energy. A general result applicable to arbitrary geometries is obtained through the use of Ford, Lewis, & O'Connell's remarkable formula. Additionally, we address the possibility for obtaining the Casimir energy as a sum over complex "modes". We show in this case that the standard sum-over-modes formula must be suitably generalized to avert unphysical complex energies. Finally, we apply our results to several standard examples.