@MISC{Zinoviev98onwilson, author = {Yury M. Zinoviev}, title = {On Wilson Criterion}, year = {1998} }

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Abstract

U(1) gauge theory with the Villain action on a cubic lattice approximation of three- and four-dimensional torus is considered. The naturally chosen correlation functions converge to the correlation functions of the R-gauge electrodynamics on three- and four-dimensional torus as the lattice spacing approaches zero only for the special scaling. This special scaling depends on a choice of a correlation function system. Another scalings give the degenerate continuum limits. The Wilson criterion for the confinement is ambiguous. The asymptotics of the smeared Wilson loop integral for the large loop perimeters is defined by the density of the loop smearing over a torus which is transversal to the loop plane. When the initial torus radius tends to infinity the correlation functions converge to the correlation functions of the R-gauge Euclidean electrodynamics. K. Wilson [1] related the confinement problem to the study of the correlation functions for the lattice pure gauge theories. The bulk of the paper [1] is devoted to U(1) gauge theory with the periodic boundary conditions. The Wilson criterion [1] for the confinement is fulfilled if for the large closed loops Γ the correlation function 〈exp{iθ(Γ)} 〉 looks like exp{−aΣ(Γ)} where Σ(Γ) is the minimal area of a surface with boundary Γ.