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Casimir Force in Compact Noncommutative Extra Dimensions and Radius Stabilization

We compute the one loop Casimir energy of an interacting scalar field in a compact noncommutative space of $R^{1,d}\times T^2_θ$, where we have ordinary flat $1+d$ dimensional Minkowski space and two dimensional noncommuative torus. We find that next order correction due to the noncommutativity still contributes an attractive force and thus will have a quantum instability. However, the case of vector field in a periodic boundary condition gives repulsive force for $d>5$ and we expect a stabilized radius. This suggests a stabilization mechanism for a senario in Kaluza-Klein theory, where some of the extra dimensions are noncommutative.