Kaluza-Klein Pistons with non-Commutative Extra Dimensions

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We calculate the scalar Casimir energy and Casimir force for a R³× N Kaluza-Klein piston setup in which the extra dimensional space N contains a non-commutative 2-sphere, SFZ. The cases to be studied are Tᵈ× SFZ and SFZ respectively as extra dimensional spaces, with Tᵈ 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 Mᵈ. A discussion on the stabilization of the extra dimensional space within the piston setup follows at the end of the article.