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


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).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Karasev and V. P. Maslov, “Asymptotic and Geometric Quantization,” Russian Math. Surveys 39 (6), 133–205 (1984).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. A. Berezin and M. A. Shubin, “The Schroedinger Equation,” in Mathematics and Its Applications, Soviet Series, Vol. 66 (Kluwer, 1991).Google Scholar
  3. 3.
    S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint, T. Valenzuela, J. Verdu, M. Vogel, and G. Werth, “A Planar Penning Trap,” Eur. Phys. J. D 32, 139–146 (2005).ADSCrossRefGoogle Scholar
  4. 4.
    F. Galve, P. Fernandez, and G. Werth, “Operation of a Planar Penning Trap,” Eur. Phys. J. D 40, 201–204 (2006).ADSGoogle Scholar
  5. 5.
    F. Galve and G. Werth, “Motional Frequencies in a Planar Penning Trap,” Hyperfine Interact. 174, 41–46 (2007).ADSCrossRefGoogle Scholar
  6. 6.
    J. Goldman and G. Gabrielse, “Optimized Planar Penning Traps for Quantum Information Studies,” Hyperfine Interact. 199, 279–289 (2011).ADSCrossRefGoogle Scholar
  7. 7.
    M. V. Karasev and E. M. Novikova, “Secondary Resonances in Penning Traps. Non-Lie Symmetry Algebras and Quantum States,” Russ. J. Math. Phys. 20 (3), 283–294 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. V. Karasev and E. M. Novikova, “Inserted Perturbations Generating Asymptotical Integrability,” Math. Notes 96 (6), 965–970 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. V. Karasev and E. M. Novikova, “Planar Penning Trap with Combined Resonance and Top Dynamics on Quadratic Algebra,” Russ. J. Math. Phys. 22, 463–468 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. V. Karasev, E. M. Novikova, and E. V. Vybornyi, “Non-Lie Top Tunneling and Quantum Bilocalization in Planar Penning Trap,” Math. Notes 100 (6), (2016).Google Scholar
  11. 11.
    V. P. Maslov, “An Asymptotic Expression for the Eigenfunctions of the Equation Δu + k 2 u = 0 with Boundary Conditions on Equidistant Curves and the Propagation of Electromagnetic Waves in a Waveguide,” Dokl. Akad. Nauk SSSR 123, 631–633 (1958) [Soviet Physics Dokl. 3, 1132–1135 (1959)].MathSciNetGoogle Scholar

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

Personalised recommendations