  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.
