A Calculus for Magnetic Pseudodifferential Super Operators

TL;DR

This paper introduces a gauge-covariant magnetic pseudodifferential calculus for super operators, extending classical theories to include magnetic fields and non-commutative geometries, with applications to Liouville super operators.

Contribution

It develops a new magnetic pseudodifferential calculus for super operators that incorporates gauge covariance and non-commutative structures, advancing the mathematical framework for magnetic quantum systems.

Findings

Established a gauge-covariant calculus for magnetic super operators.

Defined magnetic Weyl and semi-super products for operator composition.

Connected the calculus to non-commutative torus pseudodifferential theory.

Abstract

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a H\"ormander symbol. An important example is Liouville super operators defined in terms of a magnetic pseudodifferential operator. Our work combines ideas from magnetic Weyl calculus developed in [MP04, IMP07, Lei11] and the pseudodifferential calculus on the non-commutative torus from [HLP18a, HLP18b]. Thus, our calculus is inherently gauge-covariant, which means all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.