Casimir forces for inhomogeneous planar media

Abstract

Abstract. Casimir forces arise from vacuum fluctuations. They are fully understood only for simple models, and are important in nano-and microtechnologies. We report our experience of computer algebra calculations towards the Casimir force for models involving inhomogeneous dielectrics. We describe a methodology that greatly increases confidence in any results obtained, and use this methodology to demonstrate that the analytic derivation of scalar Green's functions is at the boundary of current computer algebra technology. We further demonstrate that Lifshitz theory of electromagnetic vacuum energy can not be directly applied to calculate the Casimir stress for models of this type, and produce results that have led to alternative regularisations. Using a combination of our new computational framework and the new theory based on our results, we provide specific calculations of Casimir forces for planar dielectrics having permittivity that declines exponentially. We discuss the relative strengths and weaknesses of computer algebra systems when applied to this type of problem, and describe a combined numerical and symbolic computational framework for calculating Casimir forces for arbitrary planar models.

Stress tensor for a scalar field in a spatially varying background potential: Divergences, “renormalization”, anomalies, and Casimir forces

“…(3.2).] If we go through the fourth order, we get (4) ], (4.11) where I[g (4) ] is given in Eq. (A8c).…”

“…This is added to the numerical evaluation of the remainder, 4) and t zz + T zz wkb is shown in Fig. 5.…”

“…But the geometry still was a three-layer system: the dielectric material was spatially constant in each region. The general problem of a spatially varying medium has still not been solved [4]. (Recent papers on this topic include Refs.…”

Casimir forces in inhomogeneous media: Renormalization and the principle of virtual work

“…Later, Dzyaloshinskii et al [3,24] (DLP) introduced another homogeneous medium as the intervening material replacing the vacuum; their results have been demonstrated experimentally [9,10]. A natural next generalization is the evaluation of Casimir forces in configurations where the media are inhomogeneous [25][26][27][28][29]. However, progress in that direction has been extremely slow in the last sixty years for various reasons, of which the following two are the most significant.…”

Quantum electromagnetic stress tensor in an inhomogeneous medium

“…However, the situation is much less clear when the bodies are immersed in an inhomogeneous medium. There have been various attempts to describe Casimir forces with nonuniform dielectrics [20][21][22]. The most ambitious treatment of the inhomogeneous electromagnetic Casimir problem seems to be that of Griniasty and Leonhardt [23,24], who examine the local stress tensor and propose a specific renormalization scheme to remove the divergences that occur in such circumstances.…”