Magnetic pseudo-differential Weyl calculus on nilpotent Lie groups

TL;DR

This paper develops a magnetic pseudo-differential Weyl calculus on nilpotent Lie groups, enabling analysis of magnetic perturbations of invariant vector fields through a novel Lie group construction and orbit-based quantization.

Contribution

It introduces a new pseudo-differential calculus on nilpotent Lie groups using a semidirect product and coadjoint orbit approach for magnetic perturbations.

Findings

Constructed an infinite-dimensional Lie group as a semidirect product.

Identified a specific coadjoint orbit for quantization.

Established a Weyl calculus for magnetic perturbations.

Abstract

We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie grup and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.