Scalar cylinder-plate and cylinder-cylinder Casimir interaction in higher dimensional spacetime

摘要

We study the cylinder-plate and the cylinder-cylinder Casimir interaction inthe $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations ofmassless scalar fields. Different combinations of Dirichlet (D) and Neumann (N)boundary conditions are imposed on the two interacting objects. For thecylinder-cylinder interaction, we consider the case where one cylinder isinside the other, and the case where the two cylinders are outside each other.By computing the transition matrices of the objects and the translationmatrices that relate different coordinate systems, the explicit formulas forthe Casimir interaction energies are derived. Using perturbation technique, wecompute the small separation asymptotic expansions of the Casimir interactionenergies up to the next-to-leading order terms. The leading terms coincide withthe respective results obtained using proximity force approximation, which isof order $d^{-D+1/2}$, where $d$ is the distance between the two objects. Theresults on the next-to-leading order terms are more interesting and important.We find some universal behaviors. It is also noticed that for the case ofDirichlet-Dirichlet cylinder-plate interaction, the next-to-leading order termagrees with that obtained using derivative expansion. Hence, based on ourresults on other boundary conditions and on the cylinder-cylinder interaction,we postulate a formula for the derivative expansion to expand the Casimirinteraction energy up to the next-to-leading order terms for DD, DN, ND and NNboundary conditions, for the interaction between two curved surfaces in$(D+1)$-dimensional Minkowski spacetime. It is found that the postulate agreeswith our previous results on the sphere-sphere interactions except when $D=4$.