Compact dimensions and the Casimir effect: the Proca connection

TL;DR

This paper investigates how an extra compact dimension influences the Casimir effect, revealing that it introduces corrections and causes the force to depend on slab thickness, using Kaluza Klein decomposition and Lifshitz theory.

Contribution

It provides a detailed analysis of the Casimir effect in a 5D spacetime with a compact dimension, highlighting the role of the Proca sector and continuum modes.

Findings

Casimir force depends on the thickness of slabs due to extra dimension effects.

Proca continuum modes contribute to the Casimir force within Lifshitz theory.

Discrete modes are calculated using 5D formulas for piston geometry.

Abstract

We study the Casimir effect in the presence of an extra dimension compactified on a circle of radius R ($M^4\times S^1$ spacetime). Our starting point is the Kaluza Klein decomposition of the 5D Maxwell action into a massless sector containing the 4D Maxwell action and an extra massless scalar field and a Proca sector containing 4D gauge fields with masses $m_n=n/R$ where $n$ is a positive integer. An important point is that, in the presence of perfectly conducting parallel plates, the three degrees of freedom do not yield three discrete (non-penetrating) modes but two discrete modes and one continuum (penetrating) mode. The massless sector reproduces Casimir's original result and the Proca sector yields the corrections. The contribution from the Proca continuum mode is obtained within the framework of Lifshitz theory for plane parallel dielectrics whereas the discrete modes are calculated via 5D formulas for the piston geometry. An interesting manifestation of the extra compact dimension is that the Casimir force between perfectly conducting plates depends on the thicknesses of the slabs.