Wightman function and vacuum fluctuations in higher dimensional brane models

TL;DR

This paper calculates the Wightman function and vacuum fluctuations for a scalar field in higher-dimensional brane models with complex boundary conditions, revealing how these quantities depend on brane positions and internal space geometry.

Contribution

It provides a general analysis of vacuum expectation values in higher-dimensional warped spacetimes with different boundary conditions, extending previous models like Randall-Sundrum.

Findings

Vacuum expectation value depends on brane position and boundary conditions.

Vacuum fluctuations vanish at the AdS horizon and boundary.

Kaluza-Klein modes significantly influence vacuum densities.

Abstract

Wightman function and vacuum expectation value of the field square are evaluated for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two codimension one parallel branes located on $(D+1)$-dimensional background spacetime $AdS_{D_1+1}\times \Sigma $ with a warped internal space $\Sigma $. The general case of different Robin coefficients on separate branes is considered. The application of the generalized Abel-Plana formula for the series over zeros of combinations of cylinder functions allows us to extract manifestly the part due to the bulk without boundaries. Unlike to the purely AdS bulk, the vacuum expectation value of the field square induced by a single brane, in addition to the distance from the brane, depends also on the position of the brane in the bulk. The brane induced part in this expectation value vanishes when the brane position tends to the AdS horizon or AdS boundary. The asymptotic behavior of the vacuum densities near the branes and at large distances is investigated. The contribution of Kaluza-Klein modes along $\Sigma $ is discussed in various limiting cases. As an example the case $\Sigma =S^1$ is considered, corresponding to the $AdS_{D+1}$ bulk with one compactified dimension. An application to the higher dimensional generalization of the Randall-Sundrum brane model with arbitrary mass terms on the branes is discussed.