The Schroedinger representation on Hilbert bundles

We consider a representation of the path groupoid over the group of diffeomorphisms of the base space of a Hilbert bundle (equipped with a unitary parallel transport). The Schroedinger representation in (Weyl form) appears as a flat and measure-invariant particular case of this representation. We prove that our representation induces a representation of Mensky's path group and that generators of the representation of one-parameter groups of the path group are given by the covariant derivative plus a volume correction. We prove that, as in the case of locally compact groups, the imprimitivity systems of this representation is irreducible if and only if the diffeomorphisms group acts transitively on the manifold and the inducing representation is irreducible. In our case the inducing subgroup is the loop group and the inducing representation is the holonomy group of the parallel transport.