Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects

Abstract After discussing the peculiarities of quantum systems on noncommutative (NC) spaces with nontrivial topology and the operator representation of the ★-product on them, we consider the Aharonov–Bohm and Casimir effects for such spaces. For the case of the Aharonov–Bohm effect, we have obtained an explicit expression for the shift of the phase, which is gauge invariant in the NC sense. The Casimir energy of a field theory on a NC cylinder is divergent, but it becomes finite on a torus, when the dimensionless parameter of noncommutativity is a rational number. The latter corresponds to a well-defined physical picture. Certain distinctions from other treatments based on a different way of taking the noncommutativity into account are also discussed.