QUANTUM VS CLASSICAL CALOGERO–MOSER SYSTEMS

  • Ryu Sasaki
Conference paper
Part of the NATO Science Series book series (NAII, volume 201)

Abstract

Calogero–Moser and Toda systems are best known examples of solvable many-particle dynamics on a line which are based on root systems. At the classical level, the former (C–M) is integrable for elliptic potentials (Weierstraβ β function) and their various degenerations.

Keywords

Root System Simple Root Coxeter Group Dominant Weight Algebraic Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Calogero, F. (1971) J. Math. Phys. 12, pp. 419–436.CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Sutherland, B. (1972) Phys. Rev. A5, pp. 1372–1376.ADSGoogle Scholar
  3. 3.
    Moser, J. (1975) Adv. Math. 16, pp. 197–220; Lecture Notes in Physics 38, (1975), Springer-Verlag; Calogero, F., Marchioro C., and Ragnisco, O. (1975) Lett. Nuovo Cim. 13, pp. 383–387; Calogero, F. (1975) Lett. Nuovo Cim. 13, pp. 411–416.MATHCrossRefADSGoogle Scholar
  4. 4.
    Olshanetsky, M. A. and Perelomov, A. M. (1976) Inventions Math. 37, pp. 93–108.MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Olshanetsky, M. A. and Perelomov, A. M. (1983) Phys. Rep. 94, pp. 313–404.CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Khastgir, S. P., Pocklington, A. J., and Sasaki, R. (2000) J. Phys. A33, pp. 9033–9064.ADSMathSciNetGoogle Scholar
  7. 7.
    Bordner, A. J., Corrigan, E., and Sasaki, R. (1998) Prog. Theor. Phys. 100, pp. 1107–1129; Bordner, A. J., Sasaki, R., and Takasaki, K. (1999) Prog. Theor. Phys. 101, pp. 487–518; Bordner, A. J. and Sasaki, R. (1999) Prog. Theor. Phys. 101, pp. 799–829; Khastgir, S. P., Sasaki R., and Takasaki, K. (1999) Prog. Theor. Phys. 102, pp. 749–776.CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Bordner, A. J., Corrigan, E., and Sasaki, R. (1999) Prog. Theor. Phys. 102, pp. 499–529.CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    D’Hoker, E. and Phong, D. H. (1998) Nucl. Phys. B530, pp. 537–610.CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Bordner, A. J., Manton, N. S., and Sasaki, R. (2000) Prog. Theor. Phys. 103, pp. 463–487.MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Dunkl, C. F. (1989) Trans. Amer. Math. Soc. 311, pp. 167–183; Buchstaber, V. M., Felder, G., and Veselov, A. P. (1994) Duke Math. J. 76, pp. 885–911.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Khastgir, S. P. and Sasaki, R. (2001) Phys. Lett. A279, pp. 189–193; Hurtubise, J. C. and Markman, E. (2001) Comm. Math. Phys. 223, pp. 533–552.ADSMathSciNetGoogle Scholar
  13. 13.
    Corrigan, E. and Sasaki, R. (2002) J. Phys. A35, pp. 7017–7061.ADSMathSciNetGoogle Scholar
  14. 14.
    Haldane, F. D. M. (1988) Phys. Rev. Lett. 60, pp. 635–638; Shastry, B. S. (1988) ibid 60, pp. 639–642.CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Inozemtsev, V. I. and Sasaki, R. (2001) J. Phys. A34, pp. 7621–7632.ADSMathSciNetGoogle Scholar
  16. 16.
    Inozemtsev, V. I. and Sasaki, R. (2001) Nucl. Phys. B618, pp. 689–698.CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Olshanetsky, M. A. and Perelomov, A. M. (1977) Funct. Anal. Appl. 12, pp. 121–128.CrossRefGoogle Scholar
  18. 18.
    Perelomov, A. M. (1971) Theor. Math. Phys. 6, pp. 263–282.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Wojciechowski, S. (1976) Lett. Nuouv. Cim. 18, pp. 103–107; Phys. Lett. A95, (1983) pp. 279–281.CrossRefGoogle Scholar
  20. 20.
    Lassalle, M. (1991) Acad. C. R. Sci. Paris, t. Sér. I Math. 312, pp. 425–428, 725–728, 313, pp. 579–582.MATHMathSciNetGoogle Scholar
  21. 21.
    Polykronakos, A. P. (1992) Phys. Rev. Lett. 69, pp. 703–705.CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Shastry, B. S. and Sutherland, B. (1993) Phys. Rev. Lett. 70, pp. 4029–4033.MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Brink, L., Hansson, T. H., and Vasiliev, M. A. (1992) Phys. Lett. B286, pp. 109–111; Brink, L., Hansson, T. H., Konstein, S., and Vasiliev, M. A. (1993) Nucl. Phys. B401, pp. 591–612; Brink, L., Turbiner, A., and Wyllard, N. (1998) J. Math. Phys. 39, pp. 1285–1315.ADSMathSciNetGoogle Scholar
  24. 24.
    Ujino, H., Wadati, M., and Hikami, K. (1993) J. Phys. Soc. Jpn. 62, pp. 3035–3043; Ujino, H. and Wadati, M. (1996) J. Phys. Soc. Jpn. 65, pp. 2423–2439; Ujino, H. (1995) J. Phys. Soc. Jpn 64, pp. 2703–2706; Nishino, A., Ujino, H., and Wadati, M. Chaos Solitons Fractals (2000) 11, pp. 657–674.MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Sogo, K. (1996) J. Phys. Soc. Jpn 65, pp. 3097–3099; Gurappa, N. and Panigrahi, P. K. (1999) Phys. Rev. B59, pp. R2490–R2493.MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Ruijsenaars, S. N. M. (1999) CRM Series in Math. Phys. 1, pp. 251–352, Springer.Google Scholar
  27. 27.
    Calogero, F. (1977) Lett. Nuovo Cim. 19, pp. 505–507; Lett. Nuovo Cim. 22, (1977) pp. 251–253; Lett. Nuovo Cim. 24, (1979) pp. 601–604; J. Math. Phys. 22, (1981) pp. 919–934.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Calogero, F. and Perelomov, A. M. (1978) Commun. Math. Phys. 59, pp. 109–116.CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Heckman, G. J. (1991) in Birkhäuser, Barker, W. and Sally, P. (eds.) Basel; (1991) Inv. Math. 103, pp. 341–350.Google Scholar
  30. 30.
    Heckman, G. J. and Opdam, E. M. (1987) Comp. Math. 64, pp. 329–352; Heckman, G. J. (1987) Comp. Math. 64, pp. 353–373; Opdam, E. M. (1988) Comp. Math. 67, pp. 21–49, 191–209.MATHMathSciNetGoogle Scholar
  31. 31.
    Freedman, D. Z. and Mende, P. F. (1990) Nucl. Phys. 344, pp. 317–343.CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Dyson, F. J. (1962) J. Math. Phys. 3, pp. 140–156, 157–165, 166–175; J. Math. Phys. 3, (1962) pp. 1191–1198.CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Rühl, W. and Turbiner, A. (1995) Mod. Phys. Lett. A10, pp. 2213–2222; Haschke, O. and Rühl, W. (2000) Lecture Notes in Physics 539, pp. 118–140, Springer, Berlin.ADSGoogle Scholar
  34. 34.
    Caseiro, R., Françoise, J. P., and Sasaki, R. (2000) J. Math. Phys. 41, pp. 4679–4689.MATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Stanley, R. (1989) Adv. Math. 77, pp. 76–115; Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Lapointe, L. and Vinet, L. (1997) Adv. Math. 130, pp. 261–279; Commun. Math. Phys. 178, (1996), pp. 425–452.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Baker, T. H. and Forrester, P. J. (1997) Commun. Math. Phys. 188, pp. 175–216.MATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Awata, H., Matsuo, Y., Odake, S. and Shiraishi, J. (1995) Nucl. Phys. B449, pp. 347–374.CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Gambardella, P. J. (1975) J. Math. Phys. 16, pp. 1172–1187.CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Calogero, F. (1969) J. Math. Phys. 10, pp. 2191–2196; J. Math. Phys. 10, pp. 2197–2200.CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    Szegö, G. (1939) Orthogonal Polynomials, American Mathematical Society, New York.Google Scholar
  42. 42.
    Wolfes, J. (1974) J. Math. Phys. 15, pp. 1420–1424; Calogero, F. and Marchioro, C. (1974) J. Math. Phys. 15, pp. 1425–1430.CrossRefMathSciNetADSGoogle Scholar
  43. 43.
    Rosenbaum, M., Turbiner, A., and Capella, A. (1998) Int. J. Mod. Phys. A13, pp. 3885–3904; Gurappa, N., Khare, A., and Panigrahi, P. K. (1998) Phys. Lett. A244, pp. 467–472.ADSMathSciNetGoogle Scholar
  44. 44.
    Haschke, O. and Rühl, W. (1998) Mod. Phys. Lett. A13, pp. 3109–3122; Modern Phys. Lett. A14, (1999) pp. 937–949.ADSGoogle Scholar
  45. 45.
    Kakei, S. (1996) J. Phys. A29, pp. L619–L624; J. Phys. A30, (1997) pp. L535–L541.ADSMathSciNetGoogle Scholar
  46. 46.
    Ghosh, P. K., Khare, A., and Sivakumar, M. (1998) Phys. Rev. A58, pp. 821–825; Sukhatme, U. and Khare, A. quant-ph/9902072.ADSMathSciNetGoogle Scholar
  47. 47.
    Kuanetsov, V. B. (1996) Phys. Lett. A218, pp. 212–222.ADSGoogle Scholar
  48. 48.
    Caseiro, R., Francoise J.-P., and Sasaki, R. (2001) J. Math. Phys. 42, pp. 5329–5340.MATHCrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Sasaki, R. and Takasaki, K. (2001) J. Phys. A34, pp. 9533–9553, 10335.ADSMathSciNetGoogle Scholar
  50. 50.
    Efthimiou, C. and Spector, D. (1997) Phys. Rev. A56, pp. 208–219.ADSMathSciNetGoogle Scholar
  51. 51.
    Cherednik, I. V. (1995) Comm. Math. Phys. 169, pp. 441–461.MATHCrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Oshima, T. and Sekiguchi, H. (1995) J. Math. Sci. Univ. Tokyo 2, pp. 1–75.MATHMathSciNetGoogle Scholar
  53. 53.
    Humphreys, J. E. Cambridge Univ. Press, Cambridge 1990.Google Scholar
  54. 54.
    Krichever, I. M. (1980) Funct. Anal. Appl. 14, pp. 282–289.CrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Ryu Sasaki
    • 1
  1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

Personalised recommendations