Borel quantum kinematics of rank k on smooth manifolds

The authors propose a model for quantizing a nonrelativistic physical system that admits a finite number of internal degrees of freedom. They assume that the configuration space of the given system carries the structure of a smooth manifold M. The kinematics of the given system is assumed to be describable by a flow model. Under certain technical restrictions momentum and position operators are seen to act on a Hilbert space of square-integrable sections of a Hermitian vector bundle over M. They determine their necessary shape up to isometric isomorphism and distinguish two types. In particular, they show that the number of inequivalent quantizations depends on the topology of the underlying configuration space.