Journal of Statistical Physics

, Volume 89, Issue 3–4, pp 633–653 | Cite as

Algebraic structure of quantum fluctuations

  • B. Momont
  • A. Verbeure
  • V. A. Zagrebnov


On the basis of the existence of second and third moments of fluctuations, we prove a theorem about the Lie-algebraic structure of fluctuation operators. This result gives insight into the quantum character of fluctuations. We illustrate the presence of a Lie algebra of fluctuation operators in a model of the anharmonic crystal, and show the dependence of the Lie-algebra structure on the fine structure of the fluctuation operator algebra. The result is also applied to construct the normal Goldstone mode in the ideal Bose gas for Bose-Einstein condensation

Key Words

Quantum fluctuations Lie-algebraic structure extremal states Goldstone theorem 


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  1. 1.
    D. Goderis, A. Verbeure, and P. Vets, Non-commutative central limits,Prob. Th. Rel. Fields 82:527–544 (1989).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Goderis, A. Verbeure, and P. Vets, Dynamics of fluctuations for quantum lattice systems,Commun. Math. Phys. 128:533–549 (1990).MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    M. Broidioi, B. Momont, and A. Verbeure, Lie algebra of anomalously scaled fluctuations,J. Math. Phys. 36:6746–6757 (1995).MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    E. Inönü and E. P. Wigner, On the contraction of groups and their representations,Proc. Nat. Acad. Sci. (US) 39:510–524 (1953).MATHCrossRefADSGoogle Scholar
  5. 5.
    E. Inönü and E. P. Wigner, Representations of the Galilei group,Nuovo Cimento 9:705–718 (1952).MATHMathSciNetGoogle Scholar
  6. 6.
    A. Verbeure and V. A. Zagrebnov, Phase transitions and algebra of fluctuation operators in an exactly soluble model of a quantum anharmonic crystal,J. Stat. Phys. 69:329–359 (1992).MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    A. Verbeure and V. A. Zagrebnov, No-go theorem for quantum structural phase transitions,J. Phys. A 28:5415–5421 (1995).MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    A. Verbeure and V. A. Zagrebnov, Dynamics of quantum fluctuations in an anharmonic crystal model,J. Stat. Phys. 79:377–393 (1995).MATHCrossRefADSGoogle Scholar
  9. 9.
    A. Car and V. A. Zagrebnov, Critical fluctuation operators for a quantum model of a ferroelectric,Physica A 212:398–414 (1994).CrossRefADSGoogle Scholar
  10. 10.
    P. Joussot and V. A. Zagrebnov, Description of quantum critical fluctuations in a ferroelectric model : Quasi-average approach, preprint CPT-Marseille 1996.Google Scholar
  11. 11.
    M. Broidioi, Abnormal quantum fluctuations, Thesis, K.U. Leuven, 1995.Google Scholar
  12. 12.
    J. Goldstone, Field theories with “superconductor” solutions,Nuovo Cimento 19:154–164 (1961).MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    J. A. Swieca, Range of forces and broken symmetries in many-body systems,Commun. Math. Phys. 4:1–7 (1967).CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    D. Kastler, D. W. Robinson, and J. A. Swieca, Currents and associated symmetries: Goldstone’s theorem,Commun. Math. Phys. 2 :108–120 (1966).MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    D. Pines,Elementary Excitations in Solids, Benjamin (1964).Google Scholar
  16. 16.
    C. Kittel,Quantum Theory of Solids, Wiley (1963).Google Scholar
  17. 17.
    A. Davidov,Théorie du Solide, Editions MIR (1980).Google Scholar
  18. 18.
    M. Broidioi and A. Verbeure, The plasmon in the one-component plasma,Helv. Phys. Acta 66:156–180 (1993).MathSciNetGoogle Scholar
  19. 19.
    M. Broidioi and A. Verbeure, Plasmon frequency for a spin-density wave model,Helv. Phys. Acta 64:1094–1112 (1991).MathSciNetGoogle Scholar
  20. 20.
    J. T. Lewis, J. V. Pulé, and V. A. Zagrebnov, The large deviation principle for the Kac distribution,Helv. Phys. Acta 61:1063–1078 (1988).MathSciNetGoogle Scholar
  21. 21.
    H. Stern, Broken symmetry, sum rules, and collective modes in many-body systems,Phys. Rev. 147:94–101 (1966).CrossRefADSGoogle Scholar
  22. 22.
    P. W. Anderson, Random-phase approximation in the theory of superconductivity,Phys. Rev. 112:1900–1916(1958).CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    M. Fannes and R. Werner, Boundary conditions for quantum lattice systems,Helv. Phys. Acta 68:635–657 (1995).MATHMathSciNetGoogle Scholar
  24. 24.
    R. H. Schonmann and N. I. Tanaka, One-dimensional caricature of phase transition,J. Stat. Phys. 61:241–252 (1990).CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  1. 1.Instituut voor Theoretische FysicaK.U. LeuvenLeuvenBelgium
  2. 2.Département de PhysiqueUniversité de la Méditerranée (Aix-Marseille II) and Centre de Physique Théorique, CNRS-LuminyMarseille Cedex 09France

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