Quantum observables as magnetic pseudodifferential operators

TL;DR

This paper reviews the mathematical formulation of quantum Hamiltonians with magnetic fields using twisted Weyl quantization, focusing on the dependence on magnetic field behavior and the properties of the evolution group in magnetic pseudodifferential calculus.

Contribution

It provides a review with modified proofs emphasizing magnetic field effects and introduces a new result on the symbol of the evolution group in magnetic Moyal algebra.

Findings

The symbol of the evolution group remains in the magnetic Moyal algebra.

The approach handles unbounded magnetic fields.

The framework extends the mathematical understanding of magnetic pseudodifferential operators.

Abstract

In a series of papers we have argued that the 'basic' physical procedure of minimal coupling giving the quantum description of a Hamiltonian system interacting with a magnetic field, can be given a very satisfactory mathematical formulation as a twisted Weyl quantization \cite{MP2}. In this paper we shall present a review of some of these results with some modified proofs that allow a special focus on the dependence on the behavior of the magnetic field, having in view possible developments towards problems with unbounded magnetic fields. The main new result is contained in Theorem 2.9 and states that the the symbol of the evolution group of the self-adjoint operator defined by a real elliptic symbol of strictly positive order in a smooth bounded magnetic field is in the associated magnetic Moyal algebra, i.e. leaves invariant the space of Schwartz test functions and its dual