Topology classes of flat U(1) bundles and diffeomorphic covariant representations of the Heisenberg algebra

The general construction of self-adjoint configuration space representation s of the Heisenberg algebra over an arbitrary manifold is considered. All s uch inequivalent representations are parametrized in terms of the topology classes module integer holonomies of flat; U(1) bundles over the configurat ion space manifold. In the case of Riemannian manifolds, these representati ons are also manifestly diffeomorphic covariant. The general discussion, il lustrated by some simple examples in nonrelativistic quantum mechanics, is of particular relevance to systems whose configuration space is parametrize d by curvilinear coordinates or is not simply connected, which thus include for instance the modular spaces of theories of non-Abelian gauge fields an d gravity.