Kohn's 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 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 "décomposition barycentrique"

11 pages, no figures. To appear in Physics Letters B

References in corpus (6)

• Non-relativistic conformal symmetries and Newton-Cartan structures
• The geometry of Schrödinger symmetry in non-relativistic CFT
• Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry
• Remark on quantum mechanics with N=2 Schrodinger supersymmetry
• N=2 superconformal Newton-Hooke algebra and many-body mechanics
• Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group

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• Chiral decomposition in the non-commutative Landau problem
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• Generalized Niederer's transformation for quantum Pais-Uhlenbeck oscillator
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