Semiclassical Dynamics and Magnetic Weyl Calculus

TL;DR

This paper develops a magnetic Weyl calculus to rigorously derive semiclassical equations of motion for crystalline solids in magnetic fields, addressing the mathematical challenges in modeling phenomena like the quantum Hall effect.

Contribution

It introduces a systematic magnetic Weyl calculus with a semiclassical parameter, extending semiclassical analysis to magnetic pseudodifferential operators for the first time.

Findings

Established properties of magnetic pseudodifferential operators

Derived semiclassical equations of motion in magnetic fields

Provided mathematical framework for quantum Hall effect analysis

Abstract

Weyl quantization and related semiclassical techniques can be used to study conduction properties of crystalline solids subjected to slowly-varying, external electromagnetic fields. The case where the external magnetic field is constant, is not covered by existing theory as proofs involving usual Weyl calculus break down. This is the regime of the so-called quantum Hall effect where quantization of transverse conductance is observed. To rigorously derive semiclassical equations of motion, one needs to systematically develop a magnetic Weyl calculus which contains a semiclassical parameter. Mathematically, the operators involved in the analysis are magnetic pseudodifferential operators, a topic which by itself is of interest for the mathematics and mathematical physics community alike. Hence, we will devote two additional chapters to further understanding of properties of those operators.