Skip to main content
Log in

The quantum angular Calogero-Moser model

Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of \( {{\mathbb{R}}^n} \) to the ‘angular Calogero-Moser model’ on the sphere S n−1.We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a ‘relative angular Calogero-Moser model’, which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. M. Olshanetsky and A. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. M. Olshanetsky and A. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  5. Calogero-Moser system Scholarpedia webpage, http://www.scholarpedia.org/article/Calogero-Moser_system.

  6. T. Hakobyan, A. Nersessian and V. Yeghikyan, Cuboctahedric Higgs oscillator from the Calogero model, J. Phys. A 42 (2009) 205206 [arXiv:0808.0430] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. T. Hakobyan, S. Krivonos, O. Lechtenfeld and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374 (2010) 801 [arXiv:0908.3290] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  8. T. Hakobyan, O. Lechtenfeld, A. Nersessian and A. Saghatelian, Invariants of the spherical sector in conformal mechanics, J. Phys. A 44 (2011) 055205 [arXiv:1008.2912] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  9. T. Hakobyan, O. Lechtenfeld and A. Nersessian, The spherical sector of the Calogero model as a reduced matrix model, Nucl. Phys. B 858 (2012) 250 [arXiv:1110.5352] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. O. Lechtenfeld, A. Nersessian and V. Yeghikyan, Action-angle variables for dihedral systems on the circle, Phys. Lett. A 374 (2010) 4647 [arXiv:1005.0464] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  11. T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian and V. Yeghikyan, Action-angle variables and novel superintegrable systems, Phys. Part. Nuclei 43 (2012) 577 [INSPIRE].

    Article  ADS  Google Scholar 

  12. M.V. Feigin, Intertwining relations for spherical parts of generalized Calogero operators, Theor. Math. Phys. 135 (2003) 497.

    Article  MathSciNet  MATH  Google Scholar 

  13. J.A. Minahan and A.P. Polychronakos, Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993) 265 [hep-th/9206046] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  14. F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. F. Calogero and C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials, J. Math. Phys. 15 (1974) 1425 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. P.W. Higgs, Dynamical symmetries in a spherical geometry. 1, J. Phys. A 12 (1979) 309 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  17. H.I. Leemon, Dynamical symmetries in a spherical geometry. 2, J. Phys. A 12 (1979) 489 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. L. Brink, T. Hansson and M.A. Vasiliev, Explicit solution to the N body Calogero problem, Phys. Lett. B 286 (1992) 109 [hep-th/9206049] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  20. C.F. Dunkl and Y. Hu, Orthogonal polynomials of several variables, Cambridge University Press, Cambridge U.K. (2001).

    Book  MATH  Google Scholar 

  21. J.F. Van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Commun. Math. Phys. 188 (1997) 467 [q-alg/9609032].

    Article  ADS  MATH  Google Scholar 

  22. G.J. Heckman, A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, W. Barker and P. Sally eds., Progr. Math. 101 (1991) 181, Birkhäuser, (1991).

  23. C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994) 237.

    Article  MathSciNet  MATH  Google Scholar 

  24. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge U.K. (1990).

    Book  MATH  Google Scholar 

  25. J. Wolfes, On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974) 1420 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. M. Olshanetsky, Wave functions of quantum integrable systems, Theor. Math. Phys. 57 (1983) 1048 [Teor. Mat. Fiz. 57 (1983) 148] [INSPIRE].

  27. A. Golynski, diploma paper, Moscow State University, Moscow Russia (1999).

  28. A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett. 70 (1993) 2329 [hep-th/9210109] [INSPIRE].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Olaf Lechtenfeld.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feigin, M., Lechtenfeld, O. & Polychronakos, A.P. The quantum angular Calogero-Moser model. J. High Energ. Phys. 2013, 162 (2013). https://doi.org/10.1007/JHEP07(2013)162

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2013)162

Keywords

Navigation