Casimir piston for massless scalar fields in three dimensions

TL;DR

This paper derives an exact formula for the Casimir force on a piston with massless scalar fields in a three-dimensional cavity, showing the force is always attractive regardless of cavity dimensions.

Contribution

It provides a new exact expression for the Casimir force in a 3D scalar field piston setup, including an alternative form for large separations.

Findings

Casimir force is always attractive for the scalar piston.

Interior alone can produce a repulsive force, but total force remains attractive.

Derived an alternative computational expression for large plate separations.

Abstract

We study the Casimir piston for massless scalar fields obeying Dirichlet boundary conditions in a three dimensional cavity with sides of arbitrary lengths $a,b$ and $c$ where $a$ is the plate separation. We obtain an exact expression for the Casimir force on the piston valid for any values of the three lengths. As in the electromagnetic case with perfect conductor conditions, we find that the Casimir force is negative (attractive) regardless of the values of $a$, $b$ and $c$. Though cases exist where the interior contributes a positive (repulsive) Casimir force, the total Casimir force on the piston is negative when the exterior contribution is included. We also obtain an alternative expression for the Casimir force that is useful computationally when the plate separation $a$ is large.