Towards a Rigorous Derivation of Semiclassical Dynamics for Semigroups Generated by Magnetic Pseudodifferential Super Operators

One way to formulate a semiclassical limit is via an Egorov-type theorem, which compares the quantum mechanically evolved Weyl quantization of a suitable function on phase space with the Weyl quantization of the function, evolved with the corresponding classical hamiltonian flow. In physics parlance, the Weyl quantization of a suitable function defines a pseudodifferential operator with symmetric operator ordering.

As soon as one wants to include magnetic fields and effects of dissipation and such, the quantum dynamics are generated by magnetic Lindblad super operators. They are called super operators, because they are operators acting on other operators such as trace-class or, more generally, p-Schatten class operators. The purpose of the talk is to report on progress proving an Egorov-type theorem for magnetic pseudodifferential super operators, which are potentially more general than Lindblad-type operators. All we ask is that their “magnetic super Weyl quantization” generates a semigroup. We will contrast and compare our results with a recently published work by Galkowski, Huang and Zworski on that topic.