Active versus passive transformations in the presence of a magnetic field

TL;DR

This paper explores how magnetic fields affect active and passive transformations in classical and quantum mechanics, revealing differences and conditions for invariance related to canonical momentum and transformation generators.

Contribution

It clarifies the distinction between active and passive transformations in magnetic fields and identifies conditions for the existence of generators of passive transformations.

Findings

Active and passive transformations differ in magnetic fields.

Canonical momentum is not the generator of active translations.

Generators of passive transformations exist only if the magnetic field is invariant.

Abstract

It is shown that, when there is a magnetic field present, in the framework of classical or quantum mechanics, the active translations differ from the passive ones and that the canonical momentum is not the generator of them. It is also shown that an infinitesimal generator of passive translations or rotations exists only if the magnetic field is invariant under these transformations.