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2011

Kohn's theorem and Galilean symmetry

Zhang, P-M., Horvathy, P. A.

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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.

Kohn's theorem and Galilean symmetry · Around

Built on

W. Kohn, “Cyclotron resonance and de Haas-van Alphen oscillations of an interacting electron gas,” Phys. Rev. 123
1961
Earlier work this paper cites.
U. Niederer, “The maximal kinematical invariance group of the harmonic oscillator,” Helvetica Physica Acta
1985
Earlier work this paper cites.
C. Duval and H.-P. Künzle, “Relativistic and non-relativistic physical theories on five-dimensional spacetime,” CPT-95/PE.3295. in Semantical aspects of spacetime theories
1994
Earlier work this paper cites.
J.-M. Souriau, Structure des systèmes dynamiques
1997
Earlier work this paper cites.

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Then

F. Calogero, C. Marchioro, “Exact bound states of some n-body systems with two- and three-body forces,” J. Math. Phys. 14
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2009
Later among the works it cites.
A.V. Galajinsky “Conformal mechanics in Newton-Hooke spacetime,” Nuclear Physics B 832
Original
2010
Later among the works it cites.
G. W. Gibbons, C. N. Pope “Kohn’s Theorem, Larmor’s Equivalence Principle and the Newton-Hooke Group,” Annals of Physics 326
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2011
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C. Duval and P. A. Horvathy and L. Palla, “Conformal properties of Chern-Simons vortices in external fields.” Phys. Rev. D50
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