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Russian Journal of Mathematical Physics

, Volume 23, Issue 4, pp 484–490 | Cite as

Quantization due to breaking the commutativity of symmetries. Wobbling oscillator and anharmonic Penning trap

  • M. V. KarasevEmail author
Article
  • 37 Downloads

Abstract

We discuss two examples of classical mechanical systems which can become quantum either because of degeneracy of an integral of motion or because of tuning parameters at resonance. In both examples, the commutativity of the symmetry algebra is breaking, and noncommutative symmetries arise. Over the new noncommutative algebra, the system can reveal its quantum behavior including the tunneling effect. The important role is played by the creation-annihilation regime for the perturbation or anharmonism. Activation of this regime sometimes needs in an additional resonance deformation (Cartan subalgebra breaking).

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Laboratory for Mathematical Methods in Natural ScienceNational Research University “Higher School of Economics”MoscowRussia

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