Casimir effect for a $D$-dimensional sphere

TL;DR

This paper calculates the Casimir force on a D-dimensional sphere for a massless scalar field, revealing how the force varies with dimension, including its vanishing points and singularities at specific dimensions.

Contribution

It introduces a straightforward Green's function method to analyze the Casimir effect across continuous dimensions, including negative and non-integer values.

Findings

Casimir force vanishes as D approaches +infinity (non-even integers)

Force vanishes at negative even integers

Force exhibits simple poles at positive even integers

Abstract

The Casimir force on a $D$-dimensional sphere due to the confinement of a massless scalar field is computed as a function of $D$, where $D$ is a continuous variable that ranges from $-\infty$ to $\infty$. The dependence of the force on the dimension is obtained using a simple and straightforward Green's function technique. We find that the Casimir force vanishes as $D\to +\infty$ ($D$ non-even integer) and also vanishes when $D$ is a negative even integer. The force has simple poles at positive even integer values of $D$.