Some frontiers in constructive quantum field theory and equilibrium statistical mechanics

  • Jürg Fröhlich
Main Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 80)


We present and discuss a list of important, mostly open problems in constructive quantum field theory and equilibrium statistical mechanics the solution of which requires (in rare cases : required) new ideas going beyond high - and low - temperature expansions guided by standard (super-renormalizable and infrared finite) perturbation theory about the critical points of some action or Hamilton function, beyond Peierls-type arguments and their variants and beyond spin wave theory and its rigorous counterparts. This list of problems includes higher order phase transitions, critical phenomena, long range forces, gauge theories, quantum solitons, etc.


Gauge Theory Lattice Gauge Theory Positive Type Superselection Sector Abelian Higgs Model 
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  1. I.
    Constructive Quantum Field Theory, G. Velo and A. S. Wightman, eds., Lecture Notes in Physics 25, Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
  2. II.
    Les Méthodes Mathématiques de la Théorie Quantique des Champs, F. Guerra, D. W. Robinson, R. Stora, eds., éditions du C.N.R.S., Paris, 1976.Google Scholar
  3. III.
    Proceedings of the conference on “Quantum Dynamics: Models and Mathematics”, Acta Phys. Austriaca Suppl. XVI, (1975).Google Scholar
  4. IV.
    Proceedings of the Cargése summer school in theoretical physics, 1976; to appear.Google Scholar


  1. 1.
    R. Schrader, in II; Commun. math. Phys. 49, 131, (1976) and 50, 97 (1976); Phys. Rev. B14, 172, (1976).Google Scholar
  2. 2.
    J. Glimm and A. Jaffe, 1.Commun math. Phys. 52, 203, (1977), 2.Commun. math. Phys. 51, 1, (1976), 3. Phys. Rev. D 11, 2816, (1975), 4. Phys. Rev. Letters 33, 440, (1974).Google Scholar
  3. 3. 1.
    J. Fröhlich, Ann. Phys. 97, 1, (1976).Google Scholar
  4. 3. 2.
    J. Glimm and A. Jaffe, contributions to II-IV.Google Scholar
  5. 4.
    J. Fröhlich, in “Renormalization Theory”, G. Velo and A. S. Wightman, eds., Nato Adv. St. Inst. Series C, Reidel, Dordrecht-Boston, 1976.Google Scholar
  6. 5.
    J. Fröhlich and E. Seiler, Helv. Phys. Acta 49, 889, (1976).Google Scholar
  7. 6.
    J. Fröhlich and Y. M. Park, in preparation.Google Scholar
  8. 7.
    J. Fröhlich, Erice lectures 1977, to appear (G. Velo and A. S. Wightman, eds.).Google Scholar
  9. 8.
    S. Coleman, proceedings of the 1973 Int. School on Subnuclear Physics “Ettore Majorana”, A. Zichichi (ed.), Academic Press, New York, 1975.Google Scholar
  10. 9.
    A. M. Polyakov, Nuclear Physics B 20, 429, (1977), and refs. given there.Google Scholar
  11. 10.
    G. 't Hooft, Phys. Rev. D14, 3432, (1976), and refs. given there.Google Scholar
  12. 11.
    C. Callan, R. Dashen and D. Gross, Physics Letters 63B, 334, (1976).Google Scholar
  13. 12.
    R. Jackiw and C. Rebbi, Phys. Rev. Letters 37, 172, (1976).Google Scholar
  14. 13.
    C. Callan, R. Dashen and D. Gross, Princeton preprint C00-2220-115, (1977).Google Scholar
  15. 14.
    K. Wilson, Phys. Rev. D10, 2445, (1974).Google Scholar
  16. 15.
    R. Balian, J. M. Drouffé and C. Itzykson, Phys. Rev. D10, 3376, (1974), D11, 2098, (1975), D11, 2104, (1975).Google Scholar
  17. 16.
    K. Osterwalder and E. Seiler, “Gauge Field Theories on the Lattice”, Harvard preprint, 1977. See also E. Seiler's contribution to these proceedings.Google Scholar
  18. 17.
    J. Fröhlich, Commun. math Phys. 47, 269, (1976).Google Scholar
  19. 18.
    J. Fröhlich, Acta Phys. Austriaca, Suppl. XV, 133, (1976).Google Scholar
  20. 19.
    J. Béllissard, J. Fröhlich and V. Gidas, Phys. Rev. Letters 38, 619, (1977); paper in preparation.Google Scholar
  21. 20.
    R. Haag and D. Kastler, J. Math. Phys. 5, 848, (1964). S. Doplicher, R. Haag and J. Roberts, Commun. math. Phys. 23, 199, (1971), 35, 49, (1974), and refs. given there.CrossRefGoogle Scholar
  22. 21.
    J. Roberts, Commun. math. Phys. 51, 107, (1976).Google Scholar
  23. 22.
    S. Coleman, to appear in the proceedings of the 1975 Int. School on Subnuclear Physics “Ettore Majorana”, A. Zichichi (ed.).Google Scholar
  24. 23.
    J. Gervais, proceedingsof the XVIth Schladming winter school, to appear in Acta Phys. Austriaca, Suppl.Google Scholar
  25. 24.
    K. Hepp, Commun. math. Phys. 35, 265, (1974).Google Scholar
  26. 25.
    J.-P. Eckmann, “Remarks on the Classical Limit.of Quantum Field Theories”, Geneva, preprint, 1976.Google Scholar
  27. 26.
    E. H. Lieb and F. Y. Wu, in “Phase Transitions and Critical Phenomena”, C. Domb and M. S. Green (eds.)Academic Press, London-New York, 1972; (p.p. 332–490). R. J. Baxter, “Solvable Eight Vertex Model on an Arbitrary Planar Lattice”, to appear in Proc. Roy. Soc., and refs. given there. E. H. Lieb, Physica 73, 226, (1974). K. Hepp and E. H. Lieb, in I., and refs. given there.Google Scholar
  28. 27.
    J. Glimm and A. Jaffe, Physics Letters 66B, 67, (1977) and “Instantons in a U(1) Lattice Gauge Theory: A Coulomb Dipole Gas”, Harvard preprint, 1977.Google Scholar
  29. 28.
    J. Fröhlich, paper on critical phenomena, in preparation.Google Scholar
  30. 29.
    D. Buchholz, Commun. math. Phys. 52, 147, (1977).CrossRefGoogle Scholar
  31. 30.
    J. Fröhlich, “Asymptotic Condition and Structure of Scattering States in...” unpubl. report, 1974.Google Scholar
  32. 31.
    J. Fröhlich, Ann. Inst. H. Poincaré 19, 1, (1973).Google Scholar
  33. 32.
    J. Fröhlich, R. Israel, E. H. Lieb and B. Simon, to appear; see also J. Fröhlich, Bulletin A.M.S., to appear; B. Simon, to appear in the proceedings of the 1977 Haifa conference on Stat. Physics; E. H. Lieb, these proceedings.Google Scholar
  34. 33.
    J. L. Lebowitz and A. Martin-Löf, Commun. math Phys. 25, 276, (1972); J. L. Lebowitz, these proceedings, and refs. given there.Google Scholar
  35. 34.
    A. Messager, S. Miracle-Sole, C. Pfister, private communication by S. Miracle-Sole.Google Scholar
  36. 35.
    J. Bricmont, J. R. Fontaine and L. J. Landau, Louvain, preprint 1977, to appear in Commun. math. Phys.Google Scholar
  37. 36a.
    J. V. José, L. P. Kadanoff, S. Kirkpatrick and D. R. Nelson, IBM Research Report R C 6428 (27401), 1977, and refs. given there.Google Scholar
  38. 36b.
    J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181, (1973). J. M. Kosterlitz, J. Phys. C7, 1046, (1974). J. Zittartz, Z. Pnys. B23, 55, (1976), B23, 63, (1976).See also refs. 67, IV and E. Brézin and J. Zinn-Justin, Phys. Rev. B14, 3110,(1976).Google Scholar
  39. 37.
    G. F. DeAngelis, D. de Falco and F. Guerra, Physics Letters 68B, 255, (1977); Lettere al Nuovo Cimento, to appear. Also F. Guerra's contribution to these proceedings.Google Scholar
  40. 38.
    J. Fröhlich and Y. M. Park, Princeton preprint, 1977.Google Scholar
  41. 39.
    D. Ruelle, “Statistical Mechanics”, W. A. Benjamin, Reading-London-Amsterdam-Don Mills-Sydney-Tokyo, 1969.Google Scholar
  42. 40.
    J. Fröhlich, Commun. math. Phys. 47, 233, (1976).Google Scholar
  43. 41.
    E. Nelson, in ref. I.Google Scholar
  44. 42a.
    S. Edwards and A. Lenard, J. Math. Phys. 3, 778, (1962).Google Scholar
  45. 42b.
    S. Albeverio and R. Høegh-Krohn, Commun. math. Phys. 30, 171, (1973).Google Scholar
  46. 43.
    J. Ginibre, Commun. math. Phys. 16, 310, (1970).Google Scholar
  47. 44.
    Y. M. Park, Seoul, preprint 1977, to appear in J. Math. Phys.Google Scholar
  48. 45.
    E. Seiler, private communication and these proceedings.Google Scholar
  49. 46.
    See the contributions of Nelson, Glimm and Jaffe, Glimm, Jaffe and Spencer, Guerra, Rosen and Simon, Osterwalder to refs. I-IV and their refs. to the original arti cles. See also J. Glimm and A. Jaffe, in “Statistical Mechanics and Quantum Field Theory”, Les Houches 1970, C. de Witt and R. Stora (eds.), Gordon and Breach, New York 1971. In addition: J. Magnen and R. Sénéor, Ann. Inst. H. Poincaré 24, 95, (1976), and Harvard preprint, 1977. J. Feldman and K. Osterwalder, Ann. Phys. 97, 80, (1976). Refs. 81 and 58, 59.Google Scholar
  50. 47.
    See ref. I.Google Scholar
  51. 48.
    B. Simon, The P(ϕ)2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, N. J., 1974.Google Scholar
  52. 49.
    J. Fröhlich, in ref. II and 17.Google Scholar
  53. 50.
    F. Dunlop and C. M. Newman, Commun. math. Phys. 44, 223, (1975).CrossRefGoogle Scholar
  54. 51.
    J. Fröhlich and T. Spencer, in ref. IV.Google Scholar
  55. 52.
    K. Osterwalder and R. Schrader, Commun. math. Phys. 31, 83, (1973) and 42, 281, (1975). (See also V. Glaser, Commun. math. Phys. 37, 257, (1974)).Google Scholar
  56. 53.
    J. Glimm and A. Jaffe, in ref. IV.Google Scholar
  57. 54.
    B. Simon and R. Griffiths, Commun. math. Phys. 33, 145, (1973).Google Scholar
  58. 55.
    H. Nielsen and P. Olesen, Nuclear Physics B61, 45, (1973).Google Scholar
  59. 56.
    T. D. Lee and C. N. Yang, Phys. Rev. 87, 410, (1952).Google Scholar
  60. 57.
    J. L. Lebowitz and O. Penrose, Commun. math. Phys. 39, 165, (1974).Google Scholar
  61. 58.
    F. Guerra, L. Rosen and B. Simon, Commun. math Phys. 41, 19, (1975).CrossRefGoogle Scholar
  62. 59.
    J. Fröhlich, B. Simon and T. Spencer, Commun. math Phys. 50, 79, (1976).CrossRefGoogle Scholar
  63. 60.
    F. Dyson, E. H. Lieb and B. Simon, Phys. Rev. Letters 37, 120 (1976), and paper in preparation.CrossRefGoogle Scholar
  64. 61.
    J. Fröhlich and E. H. Lieb, Phys. Rev. Letters 38, 440, (1977); and in preparation.CrossRefGoogle Scholar
  65. 62.
    J. L. Lebowitz, Commun. math. Phys. 35, 87, (1974).CrossRefGoogle Scholar
  66. 63.
    J. Bricmont, Louvain, preprint 1977.Google Scholar
  67. 64.
    J. Rosen, Adv. Math., to appear.Google Scholar
  68. 65.
    O. McBryan and J. Rosen, Commun. math. Phys. 51, 97, (1976).CrossRefGoogle Scholar
  69. 66.
    T. Spencer, Commun. math. Phys. 39, 77, (1974).CrossRefGoogle Scholar
  70. 67.
    A. M. Polyakov, Physics Letters 59B, 79, (1975) and 59B, 82, (1975).Google Scholar
  71. 68.
    E. Brézin and J. Zinn-Justin, in IV; see also ref. 36.Google Scholar
  72. 69.
    B. M. McCoy, C. A. Tracy and T. T. Wu, Phys. Rev. Letters 38, 793, (1977). See also: D. Abrahams, Physics Letters (1977).CrossRefGoogle Scholar
  73. 70.
    O. McBryan and T. Spencer, Commun math. Phys. 53, 299, (1977).CrossRefGoogle Scholar
  74. 71.
    H. Kunz (private communication through B. Simon).Google Scholar
  75. 72.
    A. Klein and L. J. Landau, in ref. II.Google Scholar
  76. 73.
    O. McBryan and T. Spencer, private communication.Google Scholar
  77. 74.
    S. Coleman, Phys. Rev. D11, 2088, (1975).Google Scholar
  78. 75.
    E. Seiler, Commun. math. Phys. 42, 163, (1975).Google Scholar
  79. 76.
    D. Brydges, Rockefeller preprint, 1977.Google Scholar
  80. 77.
    J. Glimm, A. Jaffe and T. Spencer, Ann. Physics 101, 610 and 631, (1976).Google Scholar
  81. 78.
    E. H. Lieb and J. L. Lebowitz, Adv. Math. 9, 316, (1972); see also: E. H. Lieb, Rev. Mod. Physics 48, 553, (1976).Google Scholar
  82. 79.
    J. Glimm and A. Jaffe, private communication.Google Scholar
  83. 80.
    B. Simon, Commun math Phys. 31, 127, (1973), and ref.(to J.L.Lebowitz) given there.Google Scholar
  84. 81.
    J. Glimm, A. Jaffe and T. Spencer, Commun. math. Phys. 45, 203, (1975).Google Scholar
  85. 82.
    S. Coleman, R. Jackiw and L. Susskihd,Ann. Phys 93, 267, (1975); S. Coleman Ann. Phys. 101, 239, (1976)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrinceton

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