The Casimir effect for pistons with transmittal boundary conditions

TL;DR

This paper analyzes the Casimir effect for pistons with transmittal boundary conditions on product manifolds, calculating the Casimir energy and force using spectral zeta function regularization, with explicit results for a spherical manifold.

Contribution

It introduces a method to compute Casimir forces for pistons with transmittal boundary conditions on product manifolds, including explicit calculations for spherical geometries.

Findings

Casimir energy computed via spectral zeta function

Explicit Casimir force results for spherical pistons

Method applicable to various manifold geometries

Abstract

This work focuses on the analysis of the Casimir effect for pistons subject to transmittal boundary conditions. In particular we consider, as piston configuration, a direct product manifold of the type $I\times N$ where $I$ is a closed interval of the real line and $N$ is a smooth compact Riemannian manifold. By utilizing the spectral zeta function regularization technique, we compute the Casimir energy of the system and the Casimir force acting on the piston. Explicit results for the force are provided when the manifold $N$ is a $d$-dimensional ball.