Calculation of the Yang-Mills vacuum wave functional

Abstract

Working in the Schrödinger representation and A 0 = 0 gauge, an approximate Yang-Mills ground-state wave functional Ψ[ A] is constructed in the following way: we begin by constructing the vacuum wave functional Ψ 0[ A] of an Abelian gauge-field with global SU(2) symmetry, and then modify and generalize Ψ 0[ A] so that it becomes invariant under local SU(2) gauge transformations. This ansatz leads to a solution of the Schrödinger equation HΨ[ A] = ϵ 0 Ψ[ A] for the Yang-Mills vacuum, which, although approximate, may correctly describe its confinement properties. Given Ψ[ A], it is argued that the vacuum expectation values of the Wilson loop integral A( rmC) and of 't Hooft's flux-tube operator B( rmC) satisfy the Wilson-'t Hooft criteria 〈A(rmC)〉− e -area(C) , 〈B(rmC)〉− e -perimeter (C) , for the confinement phase of a gauge field. The confinement mechanism is essentially identical to the one discovered by Polyakov in 3-dimensional compact QED. The reason for the similarity is that there is an “analog-gas” approximation to fixed-time vacuum expectation values 〈 Ψ| O| Ψ〉: the analog gas in this case is a plasma of smoothed Wu-Yang monopoles.