Casimir effect for scalar fields under Robin boundary conditions on plates

TL;DR

This paper investigates the Casimir effect for scalar fields with Robin boundary conditions on parallel plates, deriving energy expressions and exploring conditions for repulsive or attractive forces that could stabilize plate separation.

Contribution

It provides a generalized analysis of the Casimir effect under Robin boundary conditions, including energy decomposition and conditions for force sign changes.

Findings

Casimir energy depends on boundary parameters and plate separation.

Existence of parameter regions with repulsive and attractive Casimir forces.

Potential for vacuum force-based stabilization of plate distance.

Abstract

We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions $(1+\beta_{m}n^{\mu}\partial_{\mu})\phi =0$ at $x=a_{m}$ on one ($m=1$) and two ($m=1,2$) parallel plates at a distance $a\equiv a_{2}-a_{1}$ from each other. Making use of the generalized Abel-Plana formula previously established by one of the authors \cite{Sahrev}, the Casimir energy densities are obtained as functions of $\beta_{1}$ and of $\beta_{1}$,$\beta_{2}$,$a$, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is provided. The possibility of finding a vanishing energy for particular parameter choices is shown, and the existence of a minimum to the surface part is also observed. We show that there is a region in the space of parameters defining the boundary conditions in which the Casimir forces are repulsive for small distances and attractive for large distances. This yields to an interesting possibility for stabilizing the distance between the plates by using the vacuum forces.