Compact dimensions and the Casimir effect: the Proca connection

Understand

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.