A New Look at Generalized, Non-Linear σ-Models and Yang-Mills Theory

  • Jürg Fröhlich
Conference paper


First a list of recent papers on (lattice) gauge theories and non-linear o-models is presented which serves as an introduction to the subject.

Subsequently, a new, quantum mechanical Interpretation of the formalism used by Atiyah et al. and Corrigan et al. for the construction of self-dual Yang-Mills fields is attempted and criticized: Yang-Mills theory turns out to be a natural generalization of non-linear σ-models which has many conserved (Noether) currents. Confinement is linked to the presence of an “intrinsic (or parton) flavor” symmetry, at least in the case of the σ-models.


Gauge Theory Wilson Loop Lattice Gauge Theory Niels Bohr Institute Loop Observable 
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  1. [1]
    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu.I. Manin, Phys. Lett. 65 A, 185, (1978).CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    V.G. Drinfeld and Yu.I. Manin, Commun. Math. Phys. 63, 177, (1978).CrossRefMATHADSMathSciNetGoogle Scholar
  3. [3]
    E.F. Corrigan, D.B. Fairlie, S. Templeton and P. Goddard, Nucl. Phys. B 140, 31, (1978).CrossRefADSGoogle Scholar
  4. [4]
    A. D’Adda. P. Di Vecchia and M. Lüscher, Preprint, Niels Bohr Institute, 1978? and in ref. [6].Google Scholar
  5. [5]
    M. Dubois-Violette, and Y. Georgelin, Preprint, Orsay 1978.Google Scholar
  6. [6]
    E. Brézin and J.-L. Gervais (eds.), Physics Reports, to appe ar.Google Scholar
  7. [7]
    “Constructive Quantum Field Theory”, G. Velo and A.S. Wightman (eds.), Lecture Notes in Physics 25, Springer-Verlang, Berlin-Heidelberg-New York, 1973.Google Scholar
  8. [8]
    K. Osterwalder and R. Schräder, Commun. Math. Phys. 42, 281, (1975).CrossRefMATHADSGoogle Scholar
  9. [9]
    H. Eichenherr, Ph. D. thesis, Heidelberg, 1978.Google Scholar
  10. [10]
    N. Steenrod, “The Topology of Fibre Bundles”, Princeton University Press, Princeton, 1951.Google Scholar
  11. [11]
    N. Jacobson, “Lie Algebras”, Wiley, New York-London, 1962.Google Scholar
  12. [12]
    M. Gunaydin, C. Piron and H. Ruegg, Commun. Math. Phys. 61, 69, (1978); and refs. given there.CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    M. Lüscher and K. Pohlmeyer, Nucl. Phys. B 137, 46, (1978).CrossRefADSGoogle Scholar
  14. [14]
    M. Lüscher, Nucl. Phys. B 135, 1, (1978)CrossRefADSGoogle Scholar
  15. A.B. Zamolodchikov and A.B. Zamolodchikov, Nucl. Phys. B 133, 525, (1978).CrossRefADSMathSciNetGoogle Scholar
  16. [15]
    Predictions that νcrit.≥ 4, for Yang-Mills, were previously made by A. Migdal and G. Parisi. Our argument extends that in: K. Symanzik, Commun. Math. Phys. 288, (1967)Google Scholar
  17. J. Fröhlich and T. Spencer, in “New Developments in QFT and Stat. Mech.”, M. Levy and P. Mitter (eds.), Plenum, New York, 1977.Google Scholar
  18. [16]
    E. Brézin, J. Zinn-Justin and J.C. Le Guillou, Phys. Rev. D 14, 2615, (1976).CrossRefADSGoogle Scholar
  19. [17]
    J. Fröhlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79, (1976)CrossRefADSGoogle Scholar
  20. J. Fröhlich, R. Israel, E. Lieb and B. Simon, Commun. Math. Phys. 62, 1, (1978).CrossRefADSGoogle Scholar
  21. [18]
    K. Symanzik, in “Local Quantum Theory”, R. Jost (ed.), Academic Press, New York, 1969Google Scholar
  22. D. Brydges and P. Federbush, Commun. Math. Phys. 62, 79, (1978)CrossRefMATHADSMathSciNetGoogle Scholar
  23. J. Fröhlich and T. Spencer, unpublished.Google Scholar
  24. [19]
    K. Wilson, Phys. Rev. D 10, 2445, (1974)ADSGoogle Scholar
  25. R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D 11, 3376, (1974)Google Scholar
  26. R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev D 11, 2098, 2104, (1975).Google Scholar
  27. [20]
    J. Slawny, Commun. Math. Phys. 46, 75, (1976)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag/Wien 1980

Authors and Affiliations

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

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