Path integral representation for Wilson loops and the non-Abelian Stokes theorem

Abstract

We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in $\mathrm{SU}(N)$ gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov [Phys. Lett. B 224 131 (1989)] by using a special regularization is erroneous and predicts zero for the Wilson loop. This property is obtained by direct evaluation of path integrals for Wilson loops defined for pure $\mathrm{SU}(2)$ gauge fields and $Z(2)$ center vortices with spatial azimuthal symmetry. Further we show that both derivations given by Diakonov and Petrov for their regularized path integral, if done correctly, predict also zero for Wilson loops. Therefore, the application of their path integral representation of Wilson loops cannot give a new way to check confinement in lattice as has been declared by Diakonov and Petrov [Phys. Lett. B 242 425 (1990)]. From the path integral representation which we consider we conclude that no new non-Abelian Stokes theorem can exist for Wilson loops except the old-fashioned one derived by means of the path-ordering procedure.