The Casimir Effect for Conical Pistons

TL;DR

This paper calculates the Casimir force for scalar fields in conical piston geometries using zeta-function regularization, providing explicit results for various dimensions and boundary conditions.

Contribution

It introduces a method to compute Casimir forces in conical piston geometries for any dimension and boundary condition, expanding the understanding of quantum field effects in such geometries.

Findings

Explicit formulas for Casimir force in conical pistons

Results valid for any dimension and boundary conditions

Specific calculations for spherical pistons in 2 to 5 dimensions

Abstract

In this paper we utilize $\zeta$-function regularization techniques in order to compute the Casimir force for massless scalar fields subject to Dirichlet and Neumann boundary conditions in the setting of the conical piston. The piston geometry is obtained by dividing the bounded generalized cone into two regions separated by its cross section positioned at $a$ with $a\in(0,b)$ with $b>0$. We obtain expressions for the Casimir force that are valid in any dimension for both Dirichlet and Neumann boundary conditions in terms of the spectral $\zeta$-function of the piston. As a particular case, we specify the piston to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.