Casimir effect of electromagnetic field in D-dimensional spherically symmetric cavities

TL;DR

This paper explicitly derives electromagnetic eigenmodes in D-dimensional spherical cavities, computes Casimir energies for various boundary conditions, and analyzes the resulting forces, revealing an attractive interaction between shells.

Contribution

It provides a detailed analysis of electromagnetic eigenmodes and Casimir energies in D-dimensional spherical geometries with different boundary conditions, including regularization techniques.

Findings

Eigenmodes are explicitly derived for D-dimensional spherical cavities.

Casimir energy for spherical shells is computed using cut-off and zeta regularization.

Interacting Casimir energy between shells is attractive and aligns with proximity force approximation.

Abstract

Eigenmodes of electromagnetic field with perfectly conducting or infinitely permeable conditions on the boundary of a D-dimensional spherically symmetric cavity is derived explicitly. It is shown that there are (D-2) polarizations for TE modes and one polarization for TM modes, giving rise to a total of (D-1) polarizations. In case of a D-dimensional ball, the eigenfrequencies of electromagnetic field with perfectly conducting boundary condition coincides with the eigenfrequencies of gauge one-forms with relative boundary condition; whereas the eigenfrequencies of electromagnetic field with infinitely permeable boundary condition coincides with the eigenfrequencies of gauge one-forms with absolute boundary condition. Casimir energy for a D-dimensional spherical shell configuration is computed using both cut-off regularization and zeta regularization. For a double spherical shell configuration, it is shown that the Casimir energy can be written as a sum of the single spherical shell contributions and an interacting term, and the latter is free of divergence. The interacting term always gives rise to an attractive force between the two spherical shells. Its leading term is the Casimir force acting between two parallel plates of the same area, as expected by proximity force approximation.