On the path integral representation for the Wilson loop 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 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.

33 pages, no figures, Latex, the explicit evaluations of the path integrals suggested by Diakonov and Petrov for the Wilson loop are added. It is shown that path integrals have been evaluated by Diakonov and Petrov incorrectly and the path integral representation for the Wilson loop is erroneous. Therefore, the path integral representation for the Wilson loop invented by Diakonov and Petrov cannot give "a new way to check confinement in lattice"

References in corpus (3)

• Potential energy of Yang-Mills vortices in three and four dimensions
• Product Integral Representations of Wilson Lines and Wilson Loops, and Non-Abelian Stokes Theorem
• Vortex solution in 2+1 dimensional Yang-Mills theory at high temperatures

Cited by in corpus (12)

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• Gauge-invariant gluon mass, infrared Abelian dominance and stability of magnetic vacuum
• Abelian Dominance in Wilson Loops
• Wilson loop and magnetic monopole through a non-Abelian Stokes theorem
• Mapping the braiding properties of the Moore-Read state
• The static quark potential from the gauge invariant Abelian decomposition
• Magnetic monopoles and center vortices as gauge-invariant topological defects simultaneously responsible for confinement
• The Static Quark Potential from the Gauge Independent Abelian Decomposition
• Magnetic Monopole Loops supported by a Meron pair as the Quark Confiner
• Abelian representation for nonabelian Wilson loops and the Non - Abelian Stokes theorem on the lattice
• Abelian Fock-Schwinger representation for the quark propagator in external gauge field
• On the evaluation of the evolution operator Z_Reg(R_2,R_1) in the Diakonov-Petrov approach to the Wilson loop