Multiple scattering methods in Casimir calculations

Abstract

Multiple scattering formulations have been employed for more than 30 years as a method of studying the quantum vacuum or Casimir interactions between distinct bodies. Here we review the method in the simple context of δ-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) After applying the method to rederive the Casimir force between two semitransparent plates and the Casimir self-stress on a semitransparent sphere, we obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. Simplifications occur for weak and strong coupling. In particular, after performing a power series expansion in the ratio of the radii of the objects to the separation between them, we are able to sum the weak-coupling expansions exactly to obtain explicit closed forms for the Casimir interaction energy. The same can be done for the interaction of a weak-coupling sphere or cylinder with a Dirichlet plane. We show that the proximity force approximation (PFA), which becomes the proximity force theorem when the objects are almost touching, is very poor for finite separations.

Casimir effect between two spheres at small separations

“…The formula for the zero temperature Casimir energy can be derived using the multiple scattering or functional determinant approach [1][2][3][4][5][6][7][8][9] or the mode summation approach [22,23]. For a scalar field ϕ, it is given by…”

Fluctuating volume-current formulation of electromagnetic fluctuations in inhomogeneous media: Incandescence and luminescence in arbitrary geometries

“…Here, the unknowns are volume currents within objects rather than surface currents as in FSC, and can therefore easily handle more complex structures, including inhomogeneous bodies with temperature gradients or spatially varying permittivities. In contrast to recently developed scattering-matrix methods, [28][29][30][31][32][33][34][35][36][37][38][39][40] the FVC and FSC methods do not require a separate basis of incoming/outgoing wave solutions to be selected (a potentially difficult task in geometries involving interleaved objects or complex structures favoring nonuniform spatial resolution), although VIE can be used to compute the scattering matrix if desired. We show that regardless of which quantity is computed, the final expressions for power and momentum transfer are based on simple trace formulas involving well-studied VIE and current-current correlation matrices that encode the spectral properties of fluctuating sources.…”

Casimir torque between two inhomogeneous semi-transparent concentric cylinders

“…The interpretation of the Casimir effect in terms of the radiation pressure associated with zero-point energy was another method for these kind of studies which is reviewed in [13] and used by Matloob [14]. Finally, there is the formalism of multiple scattering method proposed by Milton [15] and Rahi [16] which it is possible to use for different geometries; we used it in our study here.…”

“…In this paper we calculate the Casimir energy and torque between two inhomogeneous semi-transparent concentric cylinders using the multiple scattering method [15,16] with a massless scalar field in the presence of the Dirichlet boundary condition. We study the quantum fluctuation induced by Casimir energy to find the equilibrium points and maximum torque between the above geometries both in weak and strong coupling limits.…”