Frequency spectrum of the Casimir force: interpretation and a paradox

TL;DR

This paper reexamines the frequency spectrum of the Casimir force, revealing a paradox between oscillatory predictions and modest effects observed through alternative calculations, highlighting the need for further research.

Contribution

It generalizes the frequency spectrum analysis of the Casimir force to include imperfect reflectors and discusses the paradox arising from different computational approaches.

Findings

Oscillatory frequency dependence suggests large negative Casimir forces.

Alternative calculations using Wick rotation show only modest effects.

Recent experiments challenge the optimistic predictions based on frequency tuning.

Abstract

The frequency spectrum of the Casimir force between two plates separated by vacuum as it appears in the Lifshitz formalism is reexamined and generalised as compared to previous works to allow for imperfectly reflecting plates. As previously reported by Ford [Phys. Rev. A \textbf{48} (1993) 2962], the highly oscillatory nature of the frequency dependence of the Casimir force points to possibilities for very large and indeed negative Casimir forces if the frequency-dependent dielectric response, $\epsilon(\omega)$, of the materials could be tuned. A paradox occurs, however, because an alternative calculation of the effect of a perturbation of $\epsilon(\omega)$ involving a Wick rotation to imaginary frequencies indicate only very modest effects. A recent experiment appears to convincingly rule out the reality of Ford's optimistic predictions, although given the enormous technological promise of such frequency effects, further theoretical and experimental study is called for.