Quantum chaos and dynamical entropy

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion.