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

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 expl icit 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 diverge nt, but it becomes finite on a torus, when the dimensionless parameter of n oncommutativity is a rational number. The latter corresponds to a well-defi ned physical picture. Certain distinctions from other treatments based on a different way of taking the noncommutativity into account are also discuss ed. (C) 2001 Elsevier Science B.V. All rights reserved.