M. Chaichian et al., Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects, NUCL PHYS B, 611(1-3), 2001, pp. 383-402
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.