# Casimir Forces between Compact Objects: I. The Scalar Case - Condensed Matter > Statistical Mechanics

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Abstract: We have developed an exact, general method to compute Casimir interactionsbetween a finite number of compact objects of arbitrary shape and separation.Here, we present details of the method for a scalar field to illustrate ourapproach in its most simple form; the generalization to electromagnetic fieldsis outlined in Ref. 1. The interaction between the objects is attributed toquantum fluctuations of source distributions on their surfaces, which wedecompose in terms of multipoles. A functional integral over the effectiveaction of multipoles gives the resulting interaction. Each object-s shape andboundary conditions enter the effective action only through its scatteringmatrix. Their relative positions enter through universal translation matricesthat depend only on field type and spatial dimension. The distinction of ourmethod from the pairwise summation of two-body potentials is elucidated interms of the scattering processes between three objects. To illustrate thepower of the technique, we consider Robin boundary conditions $\phi -\lambda\partial n \phi=0$, which interpolate between Dirichlet and Neumann cases as$\lambda$ is varied. We obtain the interaction between two such spheresanalytically in a large separation expansion, and numerically for allseparations. The cases of unequal radii and unequal $\lambda$ are studied. Wefind sign changes in the force as a function of separation in certain ranges of$\lambda$ and see deviations from the proximity force approximation even atshort separations, most notably for Neumann boundary conditions.

Autor: T. Emig, N. Graham, R. L. Jaffe, M. Kardar

Fuente: https://arxiv.org/