Boundary Terms, Long Range Effects, and Chiral Symmetry Breaking

  • G. Morchio
  • F. Strocchi
Conference paper


The aim of these lectures is to discuss the role of boundary terms in the presence of long ränge interactions (typically Coulomb-like). Significant examples are provided by gauge theories and by many-body theories with Coulomb interactions (Coulomb systems). The main point is that, in the presence of long ränge interactions, boundary terms can give rise to volume effects and there- fore be equivalent to (non negligible) externa! fields. As we will explain, this phenomenon has no counterpart in the case of short ränge interactions and this may explain why such terms have been generally regarded as irrelevant for the definition of the dynamics. One of the most interesting cases in which it has been reaiized that a boundary term (actually a four divergence term in the Lagrangean) has relevant implications and may give rise to physicai effects is the θ term in QCD [1] [2].


Symmetry Breaking Chiral Symmetry Zero Mode Boundary Term Goldstone Boson 
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  1. 1.
    R. Jackiw and C. Rebbi: Phys. Rev. Lett. 3T, 172 (1976)ADSCrossRefGoogle Scholar
  2. 2.
    C. Callan, R. Dashen and D. Gross: Phys. Lett. 63B, 334 (1976)CrossRefGoogle Scholar
  3. 3.
    J. Kogut and L. Susskind: Phys. Rev. Dil, 3594 (1975)Google Scholar
  4. 4.
    G. Morchio and F. Strocchi: J. Math. Phys. 28, 1912 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    G. Morchio and F. Strocchi: Infrared problem, Higgs phenomenon and long ränge interactions, in Fandamental Problems of Gauge Field Theory, Erice School 1985, G. Velo and A.S. Wightman dir. ( Plenum Press 1986 )Google Scholar
  6. 6.
    G. Morchio and F. Strocchi: Comm. Math. Phys. 111, 593 (1987)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    G. Morchio and F. Strocchi: Ann. Phys. (N.Y.) 170, 310 (1986)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    F. Strocchi: Long ränge dynamics and spontaneous symmetry breaking in many-body sysiems, lectures at the Workshop Fractals, Quasicrystals, Knots and Algebraic Quantum Mechanics, Maratea 1987, A. Amann et al. eds. (Kluwer Academic Publ. 1988 )Google Scholar
  9. 9.
    G. Morchio and F. Strocchi: Removal of the infrared cutoff, seizing of the vacuum and symmetry breaking in many-body and in gauge theories, invited talk at the IX Int. Conf. on Mathematical Physics, Swansea 1988, B. Simon et al. eds. (Adam Hilger Publ. 1989 )Google Scholar
  10. 10.
    R. Haag: Nuovo Cim. 25, 1078 (1962); see also W. Thirring and A. Wehrl: Comm. Math. Phys. 4, 303 (1967)MathSciNetCrossRefGoogle Scholar
  11. 11.
    G. Morchio and F. Strocchi: Comm. Math. Phys. 99, 153 (1985)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    D. Ruelle: Statistical Mechanics: Rigorous Results (W.A. Benjamin, N.Y. 1969 )MATHGoogle Scholar
  13. 13.
    C. Domb and M.S. Green eds.: Phase Transition and Critical Phenomena, Vol. I (Academic Press 1971 )Google Scholar
  14. 14.
    For a review see e.g. F. Strocchi: Elements of Quantum Mechanics of Infinite Systems ( World Scientific, Singapore 1985 )Google Scholar
  15. 15.
    G. Morchio and F. Strocchi: J. Math. Phys. 28, 622 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    J. Goldstone, A. Salam and S. Weinberg: Phys. Rev. 127, 965 (1962) J.A. Swieca: Goldstone theorem and related topics, in Cargese Lectures in Physics, Vol. 4, D. Kastler ed. ( Gordon and Breach, New York 1970 )MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    J.A. Swieca: Comm. Math. Phys. 4, 1 (1967)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    J. Schwinger: Phys. Rev. 128, 2425 (1962) J. Lowenstein and A. Swieca: Ann. Phys. (N.Y.) 68, 172 (1971)Google Scholar
  19. 19.
    S. Coleman: Phys. Rev Dil, 2088 (1975) S. Mandelstam: Phys. Rev. Dil, 3026 (1975)Google Scholar
  20. 20.
    A.L. Carey and S.N.M. Ruijsenaars: Acta Appl. Math. 10, 1 (1987)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    S. Coleman, R. Jackiw and L. Susskind: Ann. Phys. (N.Y.) 93, 267 (1975)ADSCrossRefGoogle Scholar
  22. 22.
    O.E. Lanford and D. Ruelle: Comm. Math. Phys. 13, 194 (1969)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    T.W. Kibble: Proc. Int. Conf. Elementary Particles (Oxford Univ. Press 1965 ) G.S. Guralnik, C.R. Haagen and T.W. Kibble: In Advances in Particle Physics, Vol. 2, R.L. Cook and R.E. Marshak eds. ( Interscience 1968 )Google Scholar
  24. 24.
    R.F. Str iater: Goldstone ’s theorem and related topics, in Many Degrees of Freedom in Field Theory, L. Streit ed. (Plenum Press 1978 )Google Scholar
  25. 25.
    G. ’t Hooft: How instantons solve the 13(1) problem, Phys. Reports 142, 357 (1986)CrossRefGoogle Scholar
  26. 26.
    E.L. Hill: Rev. Mod. Phys. 23, 253 (1951)ADSMATHCrossRefGoogle Scholar
  27. 27.
    C. Lanczos: The Variational Principles of Mechanics (University of Toronto Press 1949)Google Scholar
  28. 28.
    L.D. Faddeev: Introduction to Functional Methods, in Methods in Field Theory, Les Houches 1975, R. Balian et al. eds. (North-Holland 1976) P. Ramond: Field Theory. A Modem Primer (Benjamin-Cummings 1981 )Google Scholar
  29. 29.
    J. Glimm and A. Jaffe: Functional Integral Methods in Quantum Field Theory, in New Developments in Quantum Field Theory and Statistical Mechanics, H. Levy et al. eds., Cargese 1976 (Plenum Press 1977 ); Quantum Physics. A Functional Integral Point of View (Springer 1987 )Google Scholar
  30. 30.
    G. ’t Hooft: Phys. Rev. Letters 37, 8 (1976)ADSCrossRefGoogle Scholar
  31. 31.
    R. Jackiw: Topological investigations of quantized gauge theories, in Relativity, Groups and Topology II, B.R. De Witt and R. Stora eds. ( North Holland 1984 )Google Scholar
  32. 32.
    G. Morchio and F. Strocchi: Infrared structures in QFT models and the 0 angle prob- Zern, invited talk at the Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Liblice, June 1989, edited by J. Niederle and J. Fisher (World Scientific)Google Scholar
  33. 33.
    G. Morchio: to be publishedGoogle Scholar
  34. 34.
    S. Coleman, R. Jackiw and L. Susskind: Ann. Phys. (N.Y.) 93, 267 (1975) S. Coleman: Ann. Phys. (N.Y.) 101, 239 (1976)ADSCrossRefGoogle Scholar
  35. 35.
    S. Coleman: In The Whys of Subnuclear Physics, A. Zichichi ed. (Plenum Press 1979 )Google Scholar
  36. 36.
    B. Schroer: Topological Methods for Gauge Theories, Schladming Lectures 1978, in Facts and Prospects of Gauge Theories, P. Urban ed. (Springer Verlag 1978 )Google Scholar
  37. 37.
    N.K. Nielsen and B. Schroer: Nucl. Phys. B127, 493 (1977)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    K. Fujikawa: Phys. Rev. D21, 2848 (1980)MathSciNetADSGoogle Scholar
  39. 39.
    A. Andrianov, L. Bonora and R. Gamboa-Saravi: Phys. Rev. D26, 2821 (1982)ADSGoogle Scholar
  40. 40.
    M.A Atiyah, V.K. Patodi and I.M. Singer: Math. Proc. Camb. Phil. Soc. 77, 43 (1975)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    M. Hortacsu, K.D. Rothe and B. Schroer: Nucl. Phys. B171, 530 (1980)MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    M. Ninominiya and C.I. Tan: Nucl. Phys. B257, 199 (1985)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • G. Morchio
    • 1
  • F. Strocchi
    • 2
    • 3
  1. 1.Dipartimento di Fisica dell’UniversitaPisaItaly
  2. 2.Accademia dei LinceiRomaItaly
  3. 3.ISAS and INFNTriesteItaly

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