Casimir force in brane worlds: coinciding results from Green's and Zeta function approaches

TL;DR

This paper demonstrates that Green's function and zeta function methods produce identical Casimir force results in various brane world models, correcting previous discrepancies and clarifying the role of extra dimensions.

Contribution

It proves the equivalence of Green's and Zeta function approaches in brane world scenarios, correcting earlier errors and clarifying the Casimir force's dependence on extra dimensions.

Findings

Both approaches yield the same Casimir force in brane models.

Previous discrepancies in single brane Randall-Sundrum models are corrected.

Casimir force decomposes into contributions from massless and massive modes.

Abstract

Casimir force encodes the structure of the field modes as vacuum fluctuations and so it is sensitive to the extra dimensions of brane worlds. Now, in flat spacetimes of arbitrary dimension the two standard approaches to the Casimir force, Green's function and zeta function, yield the same result, but for brane world models this was only assumed. In this work we show both approaches yield the same Casimir force in the case of Universal Extra Dimensions and Randall-Sundrum scenarios with one and two branes added by p compact dimensions. Essentially, the details of the mode eigenfunctions that enter the Casimir force in the Green's function approach get removed due to their orthogonality relations with a measure involving the right hyper-volume of the plates and this leaves just the contribution coming from the Zeta function approach. The present analysis corrects previous results showing a difference between the two approaches for the single brane Randall-Sundrum; this was due to an erroneous hyper-volume of the plates introduced by the authors when using the Green's function. For all the models we discuss here, the resulting Casimir force can be neatly expressed in terms of two four dimensional Casimir force contributions: one for the massless mode and the other for a tower of massive modes associated with the extra dimensions.