Green Functions, Determinants and Induced Actions in Gauge Theories

  • K. D. Rothe
  • B. Schroer
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 55)


In a theory of quarks and gluons for which the Lagrangian depends bilinearly on the matter fields, important physical properties only become exposed after integration over the matter fields. The conventional integration rules for this integration are well-known [1]:


Correlation Function Gauge Theory Dirac Equation Dirac Operator Zero Mode 


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References and Footnotes

  1. 1.
    A. Salam and P.T. Matthews, Phys. Rev. 90, 690 (1953).MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    The notion of induced action was introuduced in connection with the functional representation of correlation functions in the Schwinger model: N.K. Nelsen and B. Schroer, Nucl. Phys. Bl20, 62 (1977); N.K. Nielsen and B. Schroer, Phys. Lett. 66B, 373 (1977).Google Scholar
  3. 3.
    M. Hortacsu, K.D. Rothe and B. Schroer, FUB/HEP March 79, to be published in Phys. Rev. D (Nov. 79).Google Scholar
  4. 4.
    N.K. Nielsen, K.D. Rothe and B. Schroer, FUB/HEP May 79/ to be published in Nucl. Phys. B. See also C. Sorensen and G.H. Thomas, ANL-HEP-PR-79-13; The method of these authors does not yield however the zero mode contribution to the determinant.Google Scholar
  5. 5.
    V. Kurak, B. Schroer and J.A. Swieca, Nucl. Phys. B134, 61 (1978); S. Coleman, Ann. Phys. N.Y. 101, 239 ( 1976 ). See also J.A. Swiecafs lectures at this school.ADSCrossRefGoogle Scholar
  6. 6.
    L.V. Belvedere, K.D. Rothe, B. Schroer and J.A. Swieca, Nucl. Phys. Bl53, 112 (1979).Google Scholar
  7. 7.
    H.J. Rothe, K.D. Rothe and J.A. Swieca, Phys. Rev. D19, 3020 (1979).ADSGoogle Scholar
  8. 8.
    This argument was used by Coleman in a private discussion with one of the authoers (B.S.)Google Scholar
  9. 9.
    M. Lüscher, Phys. Lett. 78B, 465 (1978). A. D’Adda, P. Di Vecchia and M. Lüscher, Nucl.Phys. B146, 63 (1978).Google Scholar
  10. 10.
    M. Karowski and P. Weisz, Nucl. Phys. B139, 455 (1978).MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    A.S. Schwarz, Comm. Math. Phys. 6£f 233 (1979).Google Scholar
  12. 12.
    R. Jackiw and C. Rebbi, Phys. Rev. D14, 517 (1976).MathSciNetADSGoogle Scholar
  13. 13.
    N.K. Nielsen and B. Schroer, Nucl. Phys. B127, 493 (1977).MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    R.R. Seeley, Proc. Symp. Pure and Appl. Math. 10, 288 (1967).Google Scholar
  15. 15.
    B. DeWitt, Dynamical Theory of Groups and Fields, Gordon and Breach 1965.Google Scholar
  16. 16.
    P.B. Gilkey, Proc. Symp. Pure Math. 27 (1973), and J. Diff. Geom. 10, 601 (1975).Google Scholar
  17. 17.
    See also an unpublished paper by N.K. Nielsen, Nordita 78/24.Google Scholar
  18. 18.
    See the lectures of H. Leutwyler at this school.Google Scholar
  19. 19.
    H. Hogreve, R. Schräder and R. Seiler, Nucl. Phys. B142, 525, (1978).ADSCrossRefGoogle Scholar
  20. 20.
    S. Hawking, Comm. Math. Phys. 55, 133 (1977).MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    B. Schroer, Acta Austriaca, Suppl. XIX, 155 (1978).Google Scholar
  22. 22.
    M.F. Atiyah, N.J. Hitchin, V.C. Drinfeld and Yu.I. Manin, Phys. Lett. 65A, 185 (1978); E.F, Corrigan, D.B, Farlie, P. Goddard and S. Templeton, Nucl. Phys. B140, 31 (1978). N.H. Christ, E.J. Weinberg and N.K. Stanton, Phys. Rev. D18, 2013 (1978).Google Scholar
  23. 23.
    L.S. Brown and D.B. Creamer, Phys. Rev. D18, 3695 (1978).ADSGoogle Scholar
  24. 24.
    B. Berg and M. Lüscher, DESY 79/40.Google Scholar
  25. 25.
    B. Berg and M. Lüscher, DESY 79/17.Google Scholar
  26. 26.
    This simplification for the computation of the determinant of the constrained Dirac operator in terms of the unconstrained determinant divided by the Dirac operator parallel to the constraint was suggested to us by M.F. Atiyah.Google Scholar
  27. 27.
    A. Patrascioiu, Phys. Rev. D20, 491 (1979).ADSCrossRefGoogle Scholar
  28. 28.
    M.F. Atiyah and R. Bott., “The Index Theorem for Manifolds with Boundary”, Differential Analysis (Bombay Colloquium) Oxford, 1964.Google Scholar
  29. 29.
    M.F. Atiyah, V.K. Patodi and I.M. Singer, Math. Proc, Camb. Phil. Soc. 11, 43 (1975); Math. Proc. Camb. Phil. Soc. 78, 405 (1975).Google Scholar
  30. 30.
    J. Milnor, Enseignement Mathematique 9, 198 (1963).MathSciNetADSMATHGoogle Scholar
  31. 31.
    J.F. Schonfeld, Minnesota preprint 1979.Google Scholar
  32. 32.
    P. D’Adda, R. Horsley and P. Di Vecchia, Phys. Lett. 76B, 298 (1978).Google Scholar
  33. 33.
    We thank N.K. Nielsen for bringing this controversy to our attention.Google Scholar
  34. 34.
    J. Lowenstein and J.A. Swieca, Ann. of Phys. N.Y. 68, 172 (1971).MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    J. Schwinger, Phys. Rev. 128, 2425 (1962); Theoretical Physics (IAEA Vienna, 1963 ) 88.Google Scholar
  36. 36.
    K.D. Rothe and J.A. Swieca, Phys. Rev. D15, 541 (1977).ADSGoogle Scholar
  37. 37.
    K.D. Rothe and J.A. Swieca, Ann. of Phys. N.Y. 117, 382 (1979).MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Subsequent to our paper, ref.(6), P. Mitra and Probir Roy reconsidered the torus solutions interpreting them as QCD2 solutions in the broken phase (Bombay preprints I-III to appear in Phys. Rev. D); see also Patrascioiu, Phys. Rev. D15, 3592 (1977). Their interpretation is wrong.Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • K. D. Rothe
    • 1
  • B. Schroer
    • 1
  1. 1.Institut für Theoretische PhysikFreie Universität BerlinBerlinDeutschland

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