Quantum Theories on Noncommutative Spaces with Nontrivial Topology: Aharonov-Bohm and Casimir Effects

TL;DR

This paper explores quantum phenomena like the Aharonov-Bohm and Casimir effects on noncommutative spaces with complex topology, providing explicit formulas and analyzing divergences and finiteness conditions.

Contribution

It introduces explicit gauge-invariant phase shifts for the Aharonov-Bohm effect and analyzes Casimir energy behavior on noncommutative geometries with different topologies.

Findings

Explicit gauge-invariant phase shift derived for NC Aharonov-Bohm effect

Casimir energy divergent on NC cylinder, finite on NC torus with rational noncommutativity

Distinct treatment of noncommutativity compared to other approaches

Abstract

After discussing the peculiarities of quantum systems on noncommutative (NC) spaces with non-trivial topology and the operator representation of the $\star$-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, while 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.