Scalar Casimir densities for cylindrically symmetric Robin boundaries

TL;DR

This paper analyzes the vacuum expectation values and Casimir forces for a massive scalar field with Robin boundary conditions between cylindrical shells, revealing how boundary parameters influence attractive or repulsive forces and surface energies.

Contribution

It provides a comprehensive calculation of vacuum densities, forces, and energies for scalar fields with general Robin boundaries in cylindrical geometries, including new formulas for surface and interference energies.

Findings

Interaction forces can be attractive or repulsive depending on boundary coefficients.

Surface divergences require renormalization, while interaction forces are finite.

The Casimir energy includes a surface part and an interference contribution.

Abstract

Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are investigated for a massive scalar field with general curvature coupling parameter in the region between two coaxial cylindrical boundaries. It is assumed that the field obeys general Robin boundary conditions on bounding surfaces. The application of a variant of the generalized Abel-Plana formula allows to extract from the expectation values the contribution from single shells and to present the interference part in terms of exponentially convergent integrals. The vacuum forces acting on the boundaries are presented as the sum of self-action and interaction terms. The first one contains well-known surface divergences and needs a further renormalization. The interaction forces between the cylindrical boundaries are finite and are attractive for special cases of Dirichlet and Neumann scalars. For the general Robin case the interaction forces can be both attractive or repulsive depending on the coefficients in the boundary conditions. The total Casimir energy is evaluated by using the zeta function regularization technique. It is shown that it contains a part which is located on bounding surfaces. The formula for the interference part of the surface energy is derived and the energy balance is discussed.