Some Results and Comments on Quantized Gauge Fields

  • Jürg Fröhlich
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 59)


A few basic facts concerning the geometry of classical gauge fields are summarized; in particular, it is asserted that a principal bundle with connection can be characterized uniquely by its “Wilson loops”. The quantization of gauge fields is then shown to consist of converting the Wilson loops into “random fields” on a manifold of oriented loops, a problem in “random geometry”. Other examples in random geometry are briefly sketched. A general theorem permitting to reconstruct quantized Wilson loops from a sequence of Schwinger functionals is stated, the quark-antiquark potential is introduced, and “disorder fields” are discussed in general terms. The status of the construction of quantized gauge fields in the continuum limit is indicated, and some random-geometrical arguments are applied to lattice gauge theories and used to derive estimates on the expectation of the Wilson loop, resp. the disorder field.


Gauge Group Wilson Loop Gauge Field Flux Tube Parallel Transport 
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  1. 1.
    N. Steenrod, “The Topology of Fibre Bundles,” Princeton University Press, Princeton, NJ, 1951.Google Scholar
  2. 2.
    J. Fröhlich, “On the Construction of Quantized Gauge Fields,” Proceedings of the Kaiserslautern conference on gauge theory, 1979, W. Rühl (ed.); to appear.Google Scholar
  3. 3.
    For G = (S)0(n), (S)U(n) the result was found by the author (unpublished). B. Durhuus has extended it to general, compact Lie groups, (paper to appear).Google Scholar
  4. 4.
    I.M. Singer, Commun. math. Phys. 60, 7, (1978), and these proceedings.Google Scholar
  5. 5.
    D.G. Kendall and E.F. Harding (eds.), “Stochastic Geometry,” Wiley, New York, 1977.Google Scholar
  6. 6.
    A. Connes, in “Mathematical Problems in Theoretical Physics,” G. Dell’Antonio, S. Doplicher and G. Jona-Lasinio (eds.), Lecture Notes in Physics 80, Springer-Verlag, Berlin—HeidelbergNew York, 1978.Google Scholar
  7. 7.
    B. Durhuus and J. Fröhlich, “A Connection Between v-Dimensional Yang-Mills Theory and (v-1)-Dimensional, Non-Linear a-Models,” Commun. math. Phys., to appear.Google Scholar
  8. 8.
    J. Fröhlich, “Random Geometry and Yang-Mills Theory,” to appear in the Proceedings of the “Colloquium on Random Fields,” Esztergom (Hungary) 1979; Lecture given at the “Strasbourg Rencontres,” 1979.Google Scholar
  9. 9.
    K. Wilson, Phys. Rev. D10, 2445, (1974). R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D10, 3376, (1974), D11, 2098, (1975), D11, 2104, (1975). K. Osterwalder and E. Seiler, Ann. Phys.(N.Y.) 110, 440, (1978).Google Scholar
  10. 10.
    D. Brydges, J. Fröhlich and E. Seiler, Ann. Phys. (N.Y.)121, 227 (1979); Nucl. Phys. B152, 521, (1979).ADSGoogle Scholar
  11. 11.
    M. Reed and B. Simon, “Methods of Modern Mathematical Physics,” Vol. IV, Academic Press, New York, 1978; (page 209 ).MATHGoogle Scholar
  12. 12.1
    See e.g. “Dual Theory,” M. Jacob (ed.), North Holland, Amsterdam 1974; R. Giles and C.B. Thorn, Phys. Rev. D16, 366, (1977).Google Scholar
  13. 12.2
    Y. Nambu, Phys A. Neveu, Phys. Letters 80B, 372, (1979); J.-L. Gervais and . Letters 80B, 255, (1979); F. Gliozzi, T. Regge and M.A. Virasoro, Phys. Letters 81B, 178, (1979). The lattice approach presented in §3, (II),(III) and other examples in “random geometry” are also studied in unpublished work of the author briefly summarized in 181.Google Scholar
  14. 13.
    K. Osterwalder and R. Schrader, Commun. math. Phys. 31, 83, (1973), 42, 281, (1975).MathSciNetMATHCrossRefGoogle Scholar
  15. 14.
    V. Glaser, Commun. math. Phys. 37, 257, (1974).MathSciNetADSMATHCrossRefGoogle Scholar
  16. 15.
    Details concerning this theorem and its proof will appear in work with H. Epstein, K. Osterwalder and E. Seiler.Google Scholar
  17. 16.
    J. Fröhlich, unpublished,(1974),and preprint to appear. The relevant results can also be inferred from: V. Glaser, in “Problems of Theoretical Physics,” p. 69, “Nauka,” Moscow, 1969.Google Scholar
  18. 17.
    J. Bisognano and E. Wichmann, J. Math. Phys. 16, 985, (1975).MathSciNetADSMATHCrossRefGoogle Scholar
  19. 18.
    E. Seiler, Phys. Rev. D18, 482, (1978).ADSGoogle Scholar
  20. 19.
    J. Fröhlich, in “Les Méthodes Mathématiques de la Théorie Quantique des Champs,” F. Guerra, D.W. Robinson, R. Stora,(eds.), Editions du C.N.R.S., Paris, 1976. Commun. math. Phys. 47, 269, (1976) and 66, 223, (1979). (Results for two-and three-dimensional models go back to spring 1975 ).Google Scholar
  21. 20.
    G. ‘t Hooft, Nucl. Phys. B138, 1, (1978).ADSCrossRefGoogle Scholar
  22. 21.
    M. Jimbo, T. Miwa and M. Sato,“Holonomic Quantum Fields I, II, III,…” Publ. RIMS 14, 223, (1977), 15, 201, (1979), and Preprints, Kyoto 1977–1979.Google Scholar
  23. 22.
    G. Mack and V. Petkova,“Comparison of Lattice Gauge Theories with Gauge Group 712 and SU(2),”Preprint 1978; and “7L2 Monopoles in the Standard SU(2) Lattice Gauge Theory Model,”Preprint 1979.Google Scholar
  24. 23.
    L. Yaffe,“Confinement in SU(N) Lattice Gauge Theories,”Preprint 1979.Google Scholar
  25. 24.
    See ref. 12, in particular 121 and refs. given there; also refs. 39,40 and the contribution of J.-L. Gervais and A. Neveu to these proceedings.Google Scholar
  26. 25.
    D. Brydges, J. Fröhlich and E. Seiler,“On the Construction of Quantised Gauge Fields, II,”to appear in Commun. math. Phys., III, Preprint 1979.Google Scholar
  27. 26.
    J. Fröhlich and E. Seiler, Heiv. Phys. Acta 49, 889, (1976), J. Fröhlich, in “Renormalization Theory,” G. Velo and A.S. Wightman, (eds.), Reidel, Dordrecht-Boston, 1976.Google Scholar
  28. 27.
    J. Challifour and D. Weingarten, Preprint 1979.Google Scholar
  29. 28.
    J. Magnen and R. Sénéor, IAMP Proceedings, 1979, to appear in Springer Lecture Notes in Physics, K. Osterwalder,(ed.)Google Scholar
  30. 29.
    T. Bakaban, see ref. 28.Google Scholar
  31. 30.
    G. Benfatto, M. Cassandro, G. Gallavotti, et al, Commun. math. Phys. 59, 143, (1978), and I.H.É.S. Preprint 1978.Google Scholar
  32. 31.
    J. Glimm and A. Jaffe, Fortschr. der Physik 21, 327, (1973).MathSciNetADSCrossRefGoogle Scholar
  33. 32.
    G. F. DeAngelis, D. de Falco, F. Guerra and R. Marra, Acta Physíca Austr., Suppl. XIX, 205, (1978), and refs. given there.Google Scholar
  34. 33.
    G. Gallavotti, F. Guerra and S. Miracle-Sold, in “Mathematical Problems…,” see ref. 6.Google Scholar
  35. 34.
    R. Marra and S. Miracle-Solé, “On the Statistical Mechanics of the Gauge Invariant Ising Model,” to appear in Commun. math. Phys.Google Scholar
  36. 35.
    A. Guth,“Existence Proof of a Non-Confining Phase in Four- dimensional U(1) Lattice Gauge Theory,”Preprint 1979.Google Scholar
  37. 36.
    J. Fröhlich, Physics Letters 83B, 195, (1979).Google Scholar
  38. 37.
    J. Fröhlich and T. Spencer, unpublished. See also ref. 7 and K. Symanzik, in “Local Quantum Theory,” R. Jost,(ed.), Academic Press, New York-London, 1969; D. Brydges and P. Federbush, Commun. math. Phys. 62, 79, (1978).Google Scholar
  39. 38.
    J. Fröhlich and E.H. Lieb, Commun. math. Phys. 60, 233, (1978).ADSCrossRefGoogle Scholar
  40. 39.
    I. Bars and F. Greene,“Complete Integration of U(N) Lattice Gauge Theory in a Large N Limit,”Preprint 1979.Google Scholar
  41. 40.
    D. Förster,“Yang-Mills Theory — A String Theory in Disguise,” Preprint 1979.Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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