Kohn's theorem and Galilean symmetry

TL;DR

This paper explores how Galilean symmetry and Kohn's theorem relate to the separability of charged particles in magnetic fields, using geometric frameworks to reveal hidden symmetries and system mappings.

Contribution

It demonstrates the geometric and symmetry-based conditions under which charged particle systems in magnetic fields are separable and maps these systems to harmonic oscillators.

Findings

Systems with identical charge-to-mass ratios are isometric to anisotropic harmonic oscillators.

The system's symmetry allows for a reduction to an isolated system with hidden Schrödinger symmetry.

The approach unifies the understanding of separability and symmetries via cohomological structures of the Galilei group.

Abstract

The relation between the separability of a system of charged particles in a uniform magnetic field and Galilean symmetry is revisited using Duval's "Bargmann framework". If the charge-to-mass ratios of the particles are identical, $e_a/m_a=\epsilon$ for all particles, then the Bargmann space of the magnetic system is isometric to that of an anisotropic harmonic oscillator. Assuming that the particles interact through a potential which only depends on their relative distances, the system splits into one representing the center of mass plus a decoupled internal part, and can be mapped further into an isolated system using Niederer's transformation. Conversely, the manifest Galilean boost symmetry of the isolated system can be "imported" to the oscillator and to the magnetic systems, respectively, to yield the symmetry used by Gibbons and Pope to prove the separability. For vanishing interaction potential the isolated system is free and our procedure endows all our systems with a hidden Schroedinger symmetry, augmented with independent internal rotations. All these properties follow from the cohomological structure of the Galilei group, as explained by Souriau's "d\'ecomposition barycentrique"