Frequency spectrum of the Casimir force: interpretation and a paradox

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 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, ǫ(ω), of the materials could be tuned. A paradox occurs, however, because an alternative calculation of the effect of a perturbation of ǫ(ω) 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. In an interesting paper [1], Ford analysed the frequency spectrum of the Casimir pressure as it appears when read directly out of Lifshitz’ celebrated formula [2]. His calculations extended a previous study of the spectrum of the Casimir effect for a massless scalar field [3] and subsequent analysis of the electromagnetic vacuum stress tensor by Hacyan et al. [4]. In [1], the classical Casimir set-up is considered, where two perfectly reflecting metal plates of infinite transverse size are separated by a vacuum-filled gap of width a. For this system, the pressure between the plates was found by Casimir [9] to be PC(a) = − ~cπ 240a4 . (1) Ford’s puzzling finding was that if the pressure is expressed as an integral over all frequencies of the zero-point oscillations of the electromagnetic field in the cavity, the integrand is wildly oscillating and discontinuous as a function of frequency and the integral a sum of almost exactly cancelling positive and negative contributions, each of which far larger in magnitude than the measurable pressure itself. By a suitable cutoff procedure, however, the integral is calculable and the result correct. Similar considerations were later performed for a sphere and plate set-up [5, 6] and the electromagnetic stress tensor in a cavity [7]. An extension of Ford’s work on two ideally conducting plates was recently presented by Lang [8]. The unruly behaviour of the Casimir force as a function of real frequencies was recently treated for numerical purposes [10] and the same oscillatory behaviour was found. As a consequence these latter authors like most before performed the Wick rotation to imaginary time (and imaginary frequencies), which is legal when the permittivity is assumed causal. As expressed for imaginary frequencies, the Lifshitz expression is much more well behaved and rather than complex and strongly oscillating, the frequency integrand is real, nicely peaked and exponentially decreasing at high imaginary frequency. The importance of good optical data for the precise calculation of Casimir forces has been emphasised in a number of recent efforts [11–14], but these have all employed a Wick rotated formalism. Ford suggested that if the frequency response of the plate materials could be tuned, for example if a material could be found which is transparent for all but a small band of frequencies in which it was a good reflector, Casimir forces much larger than that between perfect conductors could be observed and by changing the reflection band the force could be changed from attractive to repulsive. Despite the potentially enormous technological