Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 004, 8 pages      arXiv:0901.1644
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries

Remarks on Multi-Dimensional Conformal Mechanics

Cestmír Burdík a and Armen Nersessian b, c
a) FNSPE, Czech Technical University in Prague Trojanova 13, 120 00 Prague 2, Czech Republic
b) Artsakh State University, 5 M. Gosh Str., Stepanakert, Armenia
c) Yerevan State University, 1 A. Manoogian Str., 0025, Yerevan, Armenia

Received October 30, 2008, in final form January 10, 2009; Published online January 12, 2009

Recently, Galajinsky, Lechtenfeld and Polovnikov proposed an elegant group-theoretical transformation of the generic conformal-invariant mechanics to the free one. Considering the classical counterpart of this transformation, we relate this transformation with the Weil model of Lobachewsky space.

Key words: conformal mechanics; integrability.

pdf (182 kb)   ps (134 kb)   tex (13 kb)


  1. de Alfaro V., Fubini S., Furlan G., Conformal invariance in quantum mechanics, Nuovo Cimento A 34 (1976), 569-612.
  2. Calogero F., Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969), 2191-2196.
    Calogero F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
  3. Polychronakos A.P., Physics and mathematics of Calogero particles, J. Phys. A: Math. Gen. 39 (2006), 12793-12845, hep-th/0607033.
  4. Gurappa N., Prasanta K., Equivalence of the Calogero-Sutherland model to free harmonic oscillators, Phys. Rev. B 59 (1999), R2490-R2493, cond-mat/9710035.
    Ghosh P.K., Super-Calogero-Moser-Sutherland systems and free super-oscillators: a mapping, Nuclear Phys. B 595 (2001), 519-535, hep-th/0007208.
    Brzezinski T., Gonera C., Maslanka P., On the equivalence of the rational Calogero-Moser system to free particles, Phys. Lett. A 254 (1999), 185-196, hep-th/9810176.
  5. Galajinsky A., Lechtenfeld O., Polovnikov K., Calogero models and nonlocal conformal transformations, Phys. Lett. B 643 (2006), 221-227, hep-th/0607215.
  6. Freedman D.Z., Mende P.F., An exactly solvable N-particle system in supersymmetric quantum mechanics, Nuclear Phys. B 344 (1990), 317-343.
  7. Galajinsky A., Lechtenfeld O., Polovnikov K., N = 4 superconformal Calogero models, J. High Energy Phys. 2007 (2007), no. 11, 008, 23 pages, arXiv:0708.1075.
    Galajinsky A., Lechtenfeld O., Polovnikov K., N = 4 mechanics, WDVV equations and roots, arXiv:0802.4386.
    Lechtenfeld O., WDVV solutions from orthocentric polytopes and Veselov systems, arXiv:0805.3245.
  8. Galajinsky A.V., Remark on quantum mechanics with conformal Galilean symmetry, Phys. Rev. D 78 (2008), 087701, 3 pages, arXiv:0808.1553.
  9. Hakobyan T., Nersessian A., Lobachevsky geometry of (super)conformal mechanics, arXiv:0803.1293.
  10. Landau L.D., Lifshits E.M., Mechanics, 5th ed., Nauka, Moscow, 2004.
  11. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.
    Leemon H.I., Dynamical symmetries in a spherical geometry. II, J. Phys. A: Math. Gen. 12 (1979), 489-501.
  12. Schrödinger E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A 46 (1941), 9-16.
    Schrödinger E., Further studies on solving eigenvalue problems by factorization, Proc. Roy. Irish Acad. Sect. A 46 (1941), 183-206.
    Schrödinger E., The factorization of the hypergeometric equation, Proc. Roy. Irish Acad. Sect. A 47 (1941), 53-54, physics/9910003.
  13. Otchik V.S., Symmetry and separation of variables in the two-center Coulomb problem in three dimensional spaces of constant curvature, Dokl. Akad. Nauk BSSR 35 (1991), 420-424 (in Russian).
  14. Nersessian A., Yeghikyan V., Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems, J. Phys. A: Math. Theor. 41 (2008), 155203, 11 pages, arXiv:0710.5001.
  15. Hakobyan T., Nersessian A., Yeghikyan V., Cuboctahedric Higgs oscillator from the Calogero model, arXiv:0808.0430.
  16. Bellucci S., Krivonos S., Sutulin A., N = 4 supersymmetric 3-particles Calogero model, Nuclear Phys. B 805 (2008), 24-39, arXiv:0805.3480.
  17. Arnold V.I., Mathematical methods in classical mechanics, Nauka, Moscow, 1973.

Previous article   Next article   Contents of Volume 5 (2009)