A calculus for magnetic pseudodifferential super operators

Abstract

This work develops a magnetic pseudodifferential calculus for super operators Op A( F); these map operators onto operators (as opposed to L p functions onto L q functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators [Formula: see text] defined in terms of a magnetic pseudodifferential operator op A( h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of Op A( F) are determined by properties of the magnetic field B = d A 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.