On the path integral representation for the Wilson loop and the non-Abelian Stokes theorem

TL;DR

This paper critically examines the path integral representation of Wilson loops in SU(N) gauge theory, demonstrating errors in previous derivations and reaffirming the validity of the traditional non-Abelian Stokes theorem.

Contribution

The authors derive a correct, regularization-independent path integral for Wilson loops and show that previous regularized approaches are flawed, reaffirming the traditional non-Abelian Stokes theorem.

Findings

Previous regularized path integrals predict zero for Wilson loops.

Direct evaluation confirms the correctness of the traditional non-Abelian Stokes theorem.

Erroneous regularizations cannot be used to check confinement in lattice gauge theory.

Abstract

We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in 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. B224 (1989) 131) 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 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. B242 (1990) 425). 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.