On the space-time geometry of quantum systems

We describe the time evolution of quantum systems in a classical background space-time by means of a covariant derivative in an infinite dimensional vector bundle. The corresponding parallel transport operator along a timelike curve \cC is interpreted as the time evolution operator of an observer moving along \cC. The holonomy group of the connection, which can be interpreted as a group of local symmetry transformations, and the set of observables have to satisfy certain consistency conditions. Two examples related to local \mbox{SO}(3) and \mbox{U}(1)-symmetries, respectively, are discussed in detail. The theory developed in this paper may also be useful to analyze situations where the underlying space-time manifold has closed timelike curves.