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Results and problems near the interface between statistical mechanics and quantum field theory

  • Jürg Fröhlich
Quantum Field Theory Special Plenary Session
Part of the Lecture Notes in Physics book series (LNP, volume 153)

Abstract

I present a brief survey of progress in the understanding of quantum field models, lattice gauge theories and spin systems that has been achieved during the past few years — roughly since the M ∩ ϕ conference in Rome. Significant progress has occurred in the analysis of systems with abelian global or local symmetry groups, such as the Ising — and rotator models, one- and two-component \(\lambda \left| {\vec \varphi } \right|^4\)-theories and abelian lattice gauge theories, and in the mathematical foundations of renormalization group schemes for such systems. Some progress has been made with zero mass cluster expansions, with the construction of continuum gauge theories, and in surface physics.

A list of non-perturbative problems in non-abelian gauge theory and statistical physics concludes my notes.

Keywords

Gauge Theory Renormalization Group Lattice Gauge Theory Finite Size Scaling Twisted Boundary Condition 
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.

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References

  1. 1.
    J. Fröhlich, in “Mathematical Problems in Theoretical Physics”, proceedings M ⋂ ν, Rome 1977, Lecture Notes in Physics 80, Berlin-Heidelberg-New York: Springer Verlag, 1978.Google Scholar
  2. 2.
    J. Fröhlich, Commun. Math. Phys. 47, 269 (1976), and Acta Physica Austriaca, Suppl. XV, 133 (1976).Google Scholar
  3. 3.
    J. Bellissard, J. Fröhlich and B. Gidas, Commun. Math. Phys. 60, 37 (1978).Google Scholar
  4. 4.
    G. 't Hooft, Nucl. Phys. B 138, 1 (1978), and in ref. 11.Google Scholar
  5. 5.
    G. Mack and V. Petkova, Ann. Phys. (NY) 123, 442 (1979), 125, 117 (1980); and G. Mack in ref. 11.Google Scholar
  6. 6.
    D. Brydges, J. Fröhlich and E. Seiler, Nucl. Phys. B 152, 521 (1979).Google Scholar
  7. 7.
    S. Coleman, R. Jackiw and L. Susskind, Ann. Phys. (NY) 93, 267 (1975).Google Scholar
  8. 8.
    E. Witten, seminar at Les Houches Winter Advanced Study Institute, Feb. 1980, unpubl.Google Scholar
  9. 9.
    G. 't Hooft, Physica Scripts 24 (1981), Nucl. Phys. B 190, 455 (1981).Google Scholar
  10. 10.
    “Mathematical Problems in Theoretical Physics”, proceedings M ∩ ϕ, Lausanne 1979, Lecture Notes in Physics 116, Berlin-Heidelberg-New York: Springer Verlag 1980.Google Scholar
  11. 11.
    “Recent Developments in Gauge Theories”, G. 't Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P.K. Mitter, I.M. Singer and R. Stora (eds.), New York-London Plenum Press, 1980.Google Scholar
  12. 12.
    M. Göpfert and G. Mack, Proof of Confinement of Static Quarks in 3-Dimensional U(1) Lattice Gauge Theory for all Values of the Coupling Constant, Commun. Math. Phys., in press.Google Scholar
  13. 13.
    E. Seiler, “Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics”, Lecture Notes in Physics, Springer-Verlag, to appear.Google Scholar
  14. 14.
    J. Fröhlich and T. Spencer, “Massless Phases and Symmetry Restoration in Abelian Gauge Theories and Spin Systems”, Commun. Math. Phys., in press.Google Scholar
  15. 15.
    T. Bałaban, in ref. 10, and preprint, Warsaw 1981.Google Scholar
  16. 16.
    J. Fröhlich, K. Osterwalder and E. Seiler, paper on axiomatic formulation of gauge theories, in preparation.Google Scholar
  17. 17.
    J. Glimm and A. Jaffe, Fortschritte der Physik 21, 327 (1973).Google Scholar
  18. 18.
    J. Glimm and A. Jaffe, Quantum Physics, Berlin-Heidelberg-New York: Springer-Verlag, 1981.Google Scholar
  19. 19.
    Ja. G. Sinai, in “Mathematical Problems...”, Rome 1977, see ref. 1. P.M. Bleher and Ja. G. Sinai, Commun. Math. Phys. 33, 23 (1973), 45, 347 (1975).Google Scholar
  20. 20.
    E.K. Sklyanin, L.A. Takhtadzhyan and L.D. Faddeev, Theor. Math. Phys. (USSR) 40, 194 (1979); V.E. Korepin, Commun. Math. Phys. 76, 165 (1980).Google Scholar
  21. 21.
    H.B. Thacker, Reviews of Modern Physics 53, No.2, 253 (1981).Google Scholar
  22. 22.
    M. Jimbo, T Miwa and M. Sato, in ref. 10“”, proceedings M ∩ ϕ, Lausanne 1979, Lecture Notes in Physics 116, Berlin-Heidelberg-New York:Springer Verlag 1980., and refs. given there.Google Scholar
  23. 23.
    D. Buchholz and K. Fredenhagen, in in ref. 10“; these proceedings, and refs. given there.Google Scholar
  24. 24.
    Ja. G. Sinai, Theor. Prob. Appl. 21, 63 (1976). R.L. Dobrushin, proceedings of IV Int. Symp. on Information Theory, Repino 1976.Google Scholar
  25. 25.
    P. Federbush, A Mass Zero Cluster Expansion, Parts I and II, Commun. Math. Phys., to appear.Google Scholar
  26. 26.
    K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 77, 31 (1980); J. Bricmont, J.-R. Fontaine, J.L. Lebowitz and T. Spencer, Commun. Math. Phys. 78, 281 (1980), Commun. Math. Phys. 78, 363 (1980).Google Scholar
  27. 27.
    J. Fröhlich and T. Spencer, Phys. Rev. Letters _46, 1006 (1981); The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas, Commun. Math. Phys., in press.Google Scholar
  28. 28.
    K. Gawedzki and A. Kupiainen, Renormalization Group Study of a Critical Lattice Model, I and II; to appear in Commun. Math. Phys..Google Scholar
  29. 29.
    M. Aizenman, Phys. Rev. Letters 47, 1 (1981).Google Scholar
  30. 30.
    D. Brydges, J. Fröhlich and T. Spencer, The Random Walk Representation of Classical Spin Systems and Correlation Inequalities, to appear in Commun. Math. Phys.; J. Fröhlich, On the Triviality of λϕd4 Theories and the Approach to the Critical Point in \(d\mathop > \limits_{( = )} 4\)dimensions, Nucl. Phys. B, in press; D. Brydges, J. Fröhlích and A. Sokal, in preparation.Google Scholar
  31. 31.
    A.M. Polyakov, Phys. Letters, 103 B, 207 (1981), 103 B, 211 (1981).Google Scholar
  32. 32.
    E. Brézin, An Investigation of Finite Size Scaling, to appear in Journal de Physique.Google Scholar
  33. 33.
    R.L. Dobrushin, Theor. Prob. Appl. 17, 582 (1972), 18, 253 (1973); H. van Bejeren, Commun. Math. Phys. 40, 1 (1975); J. Bricmont, J.L. Lebowitz and C.E. Pfister, Commun. Math. Phys. 66, 21 (1979), 69, 267 (1979).Google Scholar
  34. 34.
    J.L. Lebowitz and C.E. Pfister, Preprint, I.H.E.S. 1981.Google Scholar
  35. 35.
    Ch. Gruber, J.L. Lebowitz and Ph. A. Martin, J. Chem. Phys., in press; B. Jancovici, Classical Coulomb Systems Near a Plane Wall, to appear in J. Stat. Phys.Google Scholar
  36. 36.
    J. Fröhlich and C. Pfister, in preparation.Google Scholar
  37. 37.
    K. Symanzik, Euclidean Quantum Field Theory. In “Local Quantum Theory”, R. Jost (ed.), New York-London: Academic Press, 1969.Google Scholar
  38. 38.
    J. Fröhlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79 (1976).Google Scholar
  39. 39.
    A. Dvoretzky, P. Erdis and S. Kakutani, Acts Sci. Math. (Szeged) 12 B, 75 (1950).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-Yvette

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