Introduction to Lattice Gauge Theories

  • C. Itzykson
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 59)


Since 1974 when K. Wilson[1]proposed to study the large coupling behaviour of a Yang-Mills theory using a lattice to provide an ultraviolet cut-off, the literature on the subject has developped very fast. It is not however possible to claim at this moment that realistic answers have been provided to the central problem of quark confinement in quantum chromodynamics, including particle spectrum, partial conservation of the axial current... As it sometimes occur, certain aspects of questions have led to unexpected developments. The most striking is a closer relation between quantum field theory and statistical mechanics,including the behavior of disordered systems, the concept of frustration, dual transformations to describe new types of excitations... Numerical studies have been reported or are in progress. Together with various rigorous results they enable us to get a preliminary view of the phase diagram,particularly of the existence of transitions.


Gauge Theory Wilson Loop Ising Model Lattice Gauge Theory Dual Transformation 


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    It is interesting to quote the remarkable agreement between the Monte Carlo work and the analytical estimates obtained from strong coupling expansions both with an accuracy of a few percent at the present stage. Take for instance the self-dual 4-dimensional Z2 model with critical coupling βc given by tanh \({{\beta }_{c}}=\sqrt{2}-1\). From section 5 the average plaquet-te action defined as \(E=\frac{1}{6}\frac{dF}{d\beta }\) admits an expansion E = t + 4t5 - 4t7 + 60t9 + 56t11+ 1088t13 - 878t15 +... with t = tanh β. From duality it follows that E(βc+ε) + E(βc-ε) = √2 while the above series yields E(βc-ε) = 0.4819. The predicted discontinuity at the critical point is therefore ∆E = E(βc+ε) + E(βc-ε) = 0.4504 which agrees with the “observed” value reported in [4].Google Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • C. Itzykson
    • 1
  1. 1.Dph-TCEN SaclayGif-sur-YvetteFrance

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