Kohnʼs theorem and Galilean symmetry

Abstract

The relation between the separability of a system of charged particles in a uniform magnetic field and Galilean symmetry is revisited using Duvalʼs “Bargmann framework”. If the charge-to-mass ratios of the particles are identical, ea/ma=ϵ 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 Niedererʼs 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 Schrödinger symmetry, augmented with independent internal rotations. All these properties follow from the cohomological structure of the Galilei group, as explained by Souriauʼs “décomposition barycentrique”.