Kaluza-Klein Pistons with non-Commutative Extra Dimensions

TL;DR

This paper investigates the Casimir energy and force in a Kaluza-Klein piston model with non-commutative extra dimensions, analyzing regularization, quantum corrections, and implications for extra dimension stabilization.

Contribution

It introduces a novel calculation of Casimir effects in a non-commutative geometry within Kaluza-Klein theories, including quantum corrections and stability considerations.

Findings

Casimir energy and force are computed for non-commutative extra dimensions.

Regularization methods are validated within the piston setup.

Quantum corrections suggest potential stabilization of extra dimensions.

Abstract

We calculate the scalar Casimir energy and Casimir force for a $R^3\times N$ Kaluza-Klein piston setup in which the extra dimensional space $N$ contains a non-commutative 2-sphere, $S_{FZ}$. The cases to be studied are $T^d\times S_{FZ}$ and $S_{FZ}$ respectively as extra dimensional spaces, with $T^d$ the $d$ dimensional commutative torus. The validity of the results and the regularization that the piston setup offers are examined in both cases. Finally we examine the 1-loop corrected Casimir energy for one piston chamber, due to the self interacting scalar field in the non-commutative geometry. The computation is done within some approximations. We compare this case for the same calculation done in Minkowski spacetime $M^D$. A discussion on the stabilization of the extra dimensional space within the piston setup follows at the end of the article.