TOPOLOGY CLASSES OF FLAT U(1) BUNDLES AND DIFFEOMORPHIC COVARIANT REPRESENTATIONS OF THE HEISENBERG ALGEBRA

Abstract

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