On the Space-Time Geometry of Quantum Systems

Details On the Space-Time Geometry of Quantum Systems On the Space-Time Geometry of Quantum Systems

1994-12-05

arxiv.org/abs/gr-qc/9412013v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

15 pages (LaTeX); figures are included via epsfig; the corresponding postscript files are uuencoded; AMS-fonts are needed

Scientific paper

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.

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