Quantum Solitons on Quantum Chaos: Coherent Structures, Anyons, and Statistical Mechanics

  • R. K. Bullough
  • J. Timonen
Part of the NATO ASI Series book series (NSSB, volume 264)


This paper is concerned with the exact evaluation of functional integrals for the partition function Z (free energy F = -1 ℓn Z, β -1 = temperature) for integrable models like the quantum and classical sine-Gordon (s-G) models in 1+1 dimensions.1–12 These models have wide applications in physics and are generic (and important) in that sense. The classical s-G model in 1+1 dimensions
$${\phi _{xx}} - {\phi _{tt}} = {m^2}\sin \phi$$
(m > 0 is a “mass”) has soliton (kink, anti-kink and breather) solutions. In Refs 1–12 we have reported a general theory of ‘soliton statistical mechanics’ (soliton SM) in which the particle description can be seen in terms of ‘solitons’ and ‘phonons’. The situation concerning the phonons and breather solutions of models like the quantum and classical s-G models has proved unexpected and the latter part of this present report (the §4 on quantum and classical thermodynamic limits) is devoted to this problem and its actual solution.


Thermodynamic Limit Canonical Transformation Quantum Chaos Classical Statistical Mechanic Breather Solution 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Timonen, M. Stirland, D.J. Pilling, Yi Cheng and R.K. Bullough, Phys. Rev. Lett. 56:2233 (1986).MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    R.K. Bullough, D.J. Pilling and J. Timonen, J. Phys. A: Math. Gen. 19:L955 (1986).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    J. Timonen, R.K. Bullough and D.J. Pilling, Phys. Rev. B34.6525 (1986).MathSciNetADSGoogle Scholar
  4. 4.
    R.K. Bullough, D.J. Pilling and J. Timonen, ‘Soliton statistical mechanics’ in: “Solitons”, M. Lakshmanan ed. Springer-Verlag, Heidelberg (1988), pp. 250–281, and references.CrossRefGoogle Scholar
  5. 5.
    J. Timonen, Yu-zhong Chen and R.K. Bullough, Nucl. Phys. B (Proc. Suppl.) 5A:58 (1988).ADSCrossRefGoogle Scholar
  6. 6.
    R.K. Bullough, Yu-zhong Chen, S. Olafsson and J. Timonen, ‘Statistical Mechanics of the NLS Models and their Avatars’ in: “Integrable Systems and Applications” Springer Lecture Notes in Physics 342, M. Balabane, P. Lochak, and C. Sulem eds., Springer-Verlag, Heidelberg (1989), pp. 12–26.Google Scholar
  7. 7.
    R.K. Bullough, D.J. Pilling and J. Timonen, ‘Soliton statistical mechanics and the thermalisation of biological solitons’, in: “Nonlinear Coherent Structures in Physics” J. Pouget, M. Remoissenet and R. Ribotta eds., Journal de Physique Colloque C3, suppl. au n° 3, Tome 50 (mars 1989) C3–41-C3–51.Google Scholar
  8. 8.
    R.K. Bullough, J. Timonen, Yu-zhong Chen, Yi Cheng and M. Stirland, ‘Quantum and classical statistical mechanics of the integrable models’, in: “Nonlinear evolution equations: integrablility and spectral methods” A. Degasperis, A.P. Fordy and M. Lakshmanan eds., Manchester University Press, Manchester (1990). pp. 605–617.Google Scholar
  9. 9.
    R.K. Bullough and S. Olafsson, ‘Complete integrability of the integrable models: quick review’, in: IXth Intl. Congress on Math. Phys. 17–27 July, 1988, Swansea, Wales, B. Simon, A. Truman and I.M. Davies eds., Adam Hilger, Bristol (1989), pp. 329–34.Google Scholar
  10. 10.
    R.K. Bullough and S. Olafsson, ‘Algebra of Riemann-Hilbert Problems and the Integrable Models -. A Sketch’, in: “Proc. 17 Intl. Conf. on Diff. Geom. Methods in Theor. Phys.”, Allan I. Solomon ed., World Scientific, Singapore (1989), pp. 295–309.Google Scholar
  11. 11.
    R.K. Bullough, S. Olafsson, Yu-zhong Chen and J. Timonen ‘Integrability Conditions’, in: “Proc. XVIIIth Intl. Conf. on Diff. Geom. Methods in Theor. Phys.” (Lake Tahoe, July 2–9, 1989: NATO ARW Physics and Geometry) Ling-Lie Chau and Werner Nahm eds., Plenum, New York (1990) and references. To appear 1990.Google Scholar
  12. 12.
    R.K. Bullough, Yu-zhong Chen and J. Timonen, ‘Soliton statistical mechanics: thermodynamics limits for the integrable models’, in: “Proc. IV Intl. Workshop on Nonlinear and Turbulent Processes in Physics”, V.E. Zakharov, A.G. Sitenko, N.S. Erokhin and V.M. Chernousenko eds., World Scientific, Singapore (1990) and references. To appear (1991), 46pp.Google Scholar
  13. 13.
    A.R. Its, A.G. Izergin and V.E. Korepin, Phys. Letts. A141:121 (1989).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    R.K. Bullough and J. Timonen, ‘Quantum and classical integrability: new approaches in Statistical Mechanics’, in: “Nonlinear Science: the Next Decade” A.R. Bishop and D.K. Campbell eds., Physica D. Nonlinear Phenomena. To appear (1991).Google Scholar
  15. 15.
    R.K. Bullough and J. Timonen, ‘Quantum groups and quantum complete integrability: theory and experiment’, in: “Proc. XlXth Intl. Conf. on Diff. Geom. Methods in Theor. Phys.” (Rapallo, Italy, June 19–24, 1990) U. Bruzzo and C. Bartocci eds., Lecture Notes in Physics, Springer-Verlag, Heidelberg. To appear (1991).Google Scholar
  16. 16.
    A. Zee ‘Semionics: a theory of high temperature superconductivity’ in: “High Temperature Superconductivity; Proc. of the Los Alamos Symp. 1989”, K.S. Bedell, D. Coffey, D.E. Keltzer, D. Pines and J.R. Schrieffer eds., Addison Wesley Publ. Co., Redwood City, Calif. (1990), pp. 248–298.Google Scholar
  17. 17.
    P.W. Anderson, ‘The Normal State of High Tc Superconductivity: A New Quantum Liquid’, in: “High Temperature Superconductivity” Ref. 16, — and references.Google Scholar
  18. 18.
    Inf. discussion with Dr. E. Courtens, IBM Zurich Rüschlikon Laboratories.Google Scholar
  19. 19.
    H.B. Thacker, Rev. Mod. Phys. 53:253 (1981).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    P.P. Kulish and E.K. Sklyanin; ‘Quantum Inverse Scattering Method’, in: “Proc. of the Tvärminne Symp. Finland, 1981”, J. Hietarinta and C. Montonen eds. Springer-Verlag, Heidelberg (1982).Google Scholar
  21. 21.
    For example E.K. Sklyanin, L.A. Takatadhyan and L.D. Faddeev, Theor. Mat. Fiz. 40:194 (1979).Google Scholar
  22. 22.
    M. Jimbo and T. Miwa ‘Infinite Dimensional Lie Algebras’, in: “Integrable Systems in Statistical Mechanics”, G.M. D’Ariano, A. Montorsi and M.G. Rasetti eds., World Scientific, Singapore (1985), pp. 105–125.Google Scholar
  23. 23.
    R.P. Feynman and A.R. Hibbs “Quantum Mechanics and Path Integrals” McGraw-Hill Book Co., New York (1965).MATHGoogle Scholar
  24. 24.
    Yu-zhong Chen, ‘Classical and quantum statistical mechanics of the 1+1 dimensional integrable models’, Ph.D. Thesis, University of Manchester, July (1989).Google Scholar
  25. 25.
    R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D11:3423 (1975).MathSciNetADSGoogle Scholar
  26. 26.
    C.N. Yang and C.P Yang, J. Math. Phys. 10:1115 (1969).ADSMATHCrossRefGoogle Scholar
  27. 27.
    C. Itzykson and J.B. Zuber “Quantum Field Theory” McGraw-Hill Book Co., New York (1980).Google Scholar
  28. 28.
    R.K. Bullough ‘Statistical Mechanics of the sine-Gordon Field: Part I’, in: “Nonlinear Phenomena in Physics” F. Claro ed., Springer-Verlag, Heidelberg (1985), pp. 70–103.CrossRefGoogle Scholar
  29. 29.
    D. Fröhlich, Lectures at XIXth Intl. Conf. on Diff. Geom. Methods in Theor. Phys. (Rapallo, Italy, June 19–24, 1990). To be published in the ‘Proceedings’ Ref. 15.Google Scholar
  30. 30.
    S.G. Chung and Yia-Chung Chang, J. Phys. A: Math. Gen. 20:2875 (1987).MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    J. Timonen and R.K. Bullough. To be published.Google Scholar
  32. 32.
    M. Fowler and X. Zotos, Phys. Rev. B 24:2634 (1981); 25:5806 (1982).MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    M. Imada, K. Hida and M. Ishikawa, J. Phys. C. 16:35 (1983).ADSCrossRefGoogle Scholar
  34. 34.
    R.K. Bullough, Y-z. Chen, J. Timonen, V. Tognetti and R. Vaia, Phys. Letts. A 145:54 (1990).MathSciNetCrossRefGoogle Scholar
  35. 35.
    Yi Cheng ‘Theory of Integrable Lattices’, Ph.D. Thesis, University of Manchester, January (1987).Google Scholar
  36. 36.
    J. Timonen and R.K. Bullough, Phys. Letts. 82A:183 (1981).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • R. K. Bullough
    • 1
  • J. Timonen
    • 2
  1. 1.Department of MathematicsUMISTManchesterUK
  2. 2.Department of PhysicsUniversity of JyväskyläJyväskyläFinland

Personalised recommendations