The Magnetic Weyl Calculus

TL;DR

This paper develops a gauge covariant magnetic Weyl calculus that modifies the classical pseudodifferential calculus to account for variable magnetic fields, ensuring gauge invariance through a deformation based on magnetic flux.

Contribution

It introduces a new gauge covariant quantization method using magnetic canonical commutation relations and a deformation of the Moyal product based on magnetic flux.

Findings

Provides a gauge invariant pseudodifferential calculus for variable magnetic fields

Defines a deformation of the classical Moyal product incorporating magnetic flux

Ensures gauge covariance in magnetic Weyl quantization

Abstract

In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on ``the minimal coupling principle'' at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.