Abstract
The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood. However, recent experimental advances require universal and efficient methods to compute these forces. While several approaches have been proposed in the literature, their connection is often not entirely clear, and some of them have been introduced as purely numerical techniques. Here we present a unifying approach for the Casimir force and free energy that builds on both the Maxwell stress tensor and path integral quantization. The result is presented in terms of either bulk or surface operators that describe corresponding current fluctuations. Our surface approach yields a novel formula for the Casimir free energy. The path integral is presented both within a Lagrange and Hamiltonian formulation yielding different surface operators and expressions for the free energy that are equivalent. We compare our approaches to previously developed numerical methods and the scattering approach. The practical application of our methods is exemplified by the derivation of the Lifshitz formula.
Highlights
Laboratoire de Physique Théorique et Modèles Statistiques, CNRS UMR 8626, Université Paris-Saclay, Abstract: The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood
Other efficient approaches that have been developed before the scattering approaches include path integral quantizations where the boundary conditions at the surfaces are implemented by delta functions [23]
Numerical methods based on surface current fluctuations have been developed [32], but they rely on a full-scale numerical evaluation of matrices and their determinants, which complicates these approaches when high precision of the force is required
Summary
Consider a collection of N magneto-dielectric bodies in vacuum. In the stress-tensor (bare|r ). The second representation is less general than Equation (8) because it applies only to magneto-dielectric bodies that are (piecewise) homogeneous and isotropic 1 It expresses (x, x0 ) in the form of an integral of the surface operator M−1 defined in Equation (A44), over the union Σ =. Depending on whether we use for the kernel Kthe T-operator of Equation (8) or rather the surface operator − Mof Equation (9), Equation (19) provides us with two distinct but formally similar representations of the Casimir force, which is expressed either as an integral over the volume V occupied by the bodies or as an integral over their surfaces. An efficient numerical scheme based on surface-elements methods is described in [32], where it was used to compute the Casimir force in complex geometries, not amenable to analytical techniques
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