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 $\text{cC}$ is interpreted as the time evolution operator of an observer moving along $\text{cC}$. 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 $SO(3)$ and $U(1)$-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.