The Weyl–Heisenberg group on the noncommutative two-torus: a zoo of representations

Abstract

In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl–Heisenberg group on the two-torus are constructed. This analysis extends the well-known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no observable effect of noncommutativity is implied.