  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.
