Gauge Fields on a Lattice (Selected Topics)

  • G. F. De Angelis
  • D. de Falco
  • F. Guerra
  • R. Marra
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 19/1978)


Some years ago the interesting proposal has been made by Wilson [29] and others of performing the Yang-Mills [31] procedure of promoting a global symmetry in a relativistic quantum field theory to a local one after introducing a lattice ultraviolet cutoff on the globally invariant theory in its Euclidean version. This procedure gives a very convenient gauge invariant prescription for the ultraviolet regularization of gauge quantum field theories and makes possible the exploitation of the rigorous methods of modern statistical mechanics [25] for the study of these very interesting physical theories, along the lines of the general program of Quantum field Theory as Classical Statistical Mechanics [12,22,16,26].


Gauge Theory Gauge Group Gauge Transformation Gauge Field Recursive Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    R. Balian, J.M. Drouffe and C. Itzykson, Gauge Fields on a lattice. I. General outlook, Phys. Rev. 010, (1974) 3376Google Scholar
  2. R. Balian, J.M. Drouffe and C. Itzykson, Gauge Fields on a lattice. II. Gauge-invariant Ising model, ibidem Dll (1975) 2098Google Scholar
  3. R. Balian, J.M. Drouffe and C. Itzykson, Gauge Fields on a lattice. III. Strong-coupling expansions and transition points, ibidem Dll (1975) 2104.Google Scholar
  4. 2.
    A.O. Barut and R. Raczka, Theory of Groun Representations and Applications, PWN, Warsawa, 1977.Google Scholar
  5. 3.
    G.F. De Angelis and D. de Falco, Correlation Inequalities for Lattice Gauge Fields, Lettere al Nuovo Cimento 18 (1977) 536.CrossRefGoogle Scholar
  6. G.F. De Angelis, D. de Falco and F. Guerra, Scalar Quantum Electrodynamics as Classical Statistical Mechanics, Commun. Math. Phys. 57 (1977) 201.ADSCrossRefGoogle Scholar
  7. 4.
    G.F. De Angelis, D. de Falco and F. Guerra, Lattice Gauge Models in the Strong Coupling Regime, Lettere al Nuovo Cimento 19 (1977) 55.CrossRefGoogle Scholar
  8. 5.
    G.F. de Angelis, D. de Falco and F. Guerra, Note on the Abelian Higgs-Kibble on a lattice: Absence of spontaneous magnetization, Phys. Rev. D17 (1978) 1624.Google Scholar
  9. 6.
    G.F. De Angelis, D. de Falco, F. Guerra and R. Marra, Confinement as a Problem in Statistical Mechanics, Preprint 1978.Google Scholar
  10. 7.
    R.L. Dobrushin, The description of a Random Field by means of Conditional Probabilities and Conditions of its Regularity, Theory Prob. Applic. 13 (1968) 197.Google Scholar
  11. 8.
    S. Elitzur, Impossibility of spontaneously breaking local symmetries, Phys. Rev. D12 (1975) 3978.Google Scholar
  12. 9.
    J. Fröhlich, Phase Transitions, Goldstone Boson, and Topological Superselection Rules, Acta Phys. Austr. Suppl. 15 (1976) 133.Google Scholar
  13. 10.
    J. Fröhlich, B. Simon and T. Spencer, Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking, Commun. Math. Phys. 50 (1976) 79.ADSCrossRefGoogle Scholar
  14. 11.
    G. Gallavotti, F. Guerra and Miracle-Solé, A Comment to the Talk by E. Seiler, in [32].Google Scholar
  15. 12.
    J. Glimm, A. Jaffe and T. Spencer, The Particle Structure of the Weakly Coupled P(ø)2 Model and Other Applications of High Temperature Expansions, in [28].Google Scholar
  16. 13.
    J. Glimm, A. Jaffe and T. Spencer, Phase transitions for ø24 quantum fields, Commun. Math. Phys. 45 (1975) 203.MathSciNetADSCrossRefGoogle Scholar
  17. 14.
    J. Glimm and A. Jaffe, Quark Trapping for Lattice U(1) Gauge Fields, Phys. Lett. B66 (1977) 67.Google Scholar
  18. J. Glimm and A. Jaffe, Instantons in a U(1) lattice gauge theory: a Coulomb dipole gas, Commun. Math. Phys. 56 (1977) 195.MathSciNetADSCrossRefGoogle Scholar
  19. J. Glimm and A. Jaffe, A Droplet Model for Quark Confinement, The Rockefeller University Preprint, 1978, and references quoted there.Google Scholar
  20. 15.
    F. Guerra, G. Immirzi and R. Marra, Strong Coupling Expansions for Lattice Yang-Mills Fields, Preprint 1978, and paper in preparation.Google Scholar
  21. 16.
    F. Guerra, L. Rosen and B. Simon, The P(ø)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics, Annals of Math. 101 (1975) 111.MathSciNetCrossRefGoogle Scholar
  22. 17.
    J.M. Drouffe and C. Itzykson, Lattice Gauge Fields, Phys. Rep. 38C (1978) 133.MathSciNetCrossRefGoogle Scholar
  23. 18.
    C.P. Korthals Altes, Duality for Z(N) Gauge Theories, Preprint CPT-CNRS 78/o. 1003, Marseille 1978.Google Scholar
  24. 19.
    O. Lanford and D. Ruelle, Observables at Infinity and States with Short Range Correlations in Statistical Mechanics, Commun. Math. Phys. 13 (1969) 194.MathSciNetADSCrossRefGoogle Scholar
  25. 20.
    M. Löscher, Absence of spontaneous gauge symmetry breaking in Hamiltonian Lattice gauge theories, Preprint 1977.Google Scholar
  26. 21.
    S. Mandelstam, Vortices and quark confinement in non-Abelian gauge theories,Phys. Rep. 23C (1976).Google Scholar
  27. 22.
    E. Nelson, Probability theory and Euclidean field theory, in [28].Google Scholar
  28. 23.
    K. Osterwalder, Yang-Mills Fields on the Lattice (lecture delivered at the 1976 Cargèse Summer School), Preprint, 1976.Google Scholar
  29. K. Osterwalder and E. Seiler, Gauge Field Theories on a Lattice, Annals of Phys. 110 (1978) 440 and earlier references quoted there.Google Scholar
  30. 24.
    A.M. Polyakov, Quark confinement and topology of gauge theories, Nucl. Phys. B120 (1977) 429.Google Scholar
  31. 25.
    D. Ruelle, Statistical Mechanics, Benjamin, New York, 1969.Google Scholar
  32. 26.
    B. Simon, The P(ø)2 Euclidean (quantum) field theory, Princeton University Press, Princeton 1974.Google Scholar
  33. 27.
    F. Strocchi, Spontaneous Symmetry Breaking in Local. Gauge Quantum Field Theory; the Higgs Mechanism, Commun. Math. Phys. 56 (1977) 57.MathSciNetADSMATHCrossRefGoogle Scholar
  34. 28.
    G. Velo and A.S. Wightman (eds), Constructive quantum field theory, Berlin-Heidelberg-New York, 1973.Google Scholar
  35. 29.
    K.G. Wilson, Confinement of Quarks, Phys. Rev. D10 (1974) 2445.Google Scholar
  36. 30.
    F.J. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, J. Math. Phys. 12 (1971) 2259.Google Scholar
  37. 31.
    C.N. Yang and R.L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96 (1954) 191.MathSciNetCrossRefGoogle Scholar
  38. 32.
    G. Dell’Antonio, S. Doplicher and G.Jona-Lasinio (eds), Mathematical Problems in Theoretical Physics, Springer Verlag, Berlin, Heidelberg (1978).Google Scholar
  39. 33.
    R. Marra and S. Miracle-Sole, Preprint 1978.Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • G. F. De Angelis
    • 1
    • 2
    • 3
  • D. de Falco
    • 1
    • 2
    • 3
  • F. Guerra
    • 1
    • 2
    • 3
  • R. Marra
    • 1
    • 2
    • 3
  1. 1.Institute of PhysicsUniversity of SalernoItaly
  2. 2.U.E.R. Scientifique de LuminyUniv.of Aix-Marseille IIFrance
  3. 3.Centre de Physique TheoriqueCNRS MarseilleFrance

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