Asymptotic behavior of Wilson loops from Schrödinger equation on the gauge group

Abstract

Probability distribution of non-Abelian parallel transporters on the group manifold and the corresponding amplitude are investigated for quantum Yang-Mills fields. It is shown that when the Wilson area law and the Casimir scaling hold for the quantum gauge field, this amplitude can be obtained as the solution of the free Schrödinger equation on the gauge group. Solution of this equation is written in terms of the path integral and the corresponding action term is interpreted geometrically. We also note that the partition function of two-dimensional pure Yang-Mills theory living on the surface spanned on the loop solves the obtained equation.

Kramers-Moyall cumulant expansion for the probability distribution of parallel transporters in quantum gauge fields

“…The coefficients in the obtained equation can be expressed in terms of nonperturbative cumulants of the shifted curvature tensor, which are the basic objects in the method of field correlators [6,18,19]. In the limit of a gaussian-dominated QCD vacuum, where only the second-order cumulant survives, this equation reduces to the heat kernel equation, in accordance with the results of [8,9,10]. We suppose that the proposed generalization of the heat kernel equation may be helpful in finding effective action for QCD string which takes into account screening effects.…”

“…In the approximation of a gaussian-dominated QCD vacuum one assumes that the only non-zero cumulant is the secondorder cumulant [6,18,19]. In this case the equation (29) reduces to the well-known heat kernel equation, in agreement with the results of [8,9,10]:…”

“…As δ (g, g 0 (τ )) in (17) is the delta-function on group classes (15), the function p(g, τ ) is a gauge-invariant quantity. A simple calculation shows that indeed the probability distribution p(g, τ ) can be expressed in terms of Wilson loops [10]:…”

“…Thus geometric properties of the group manifold (homogeneity and absolute parallelism) are preserved and the symmetry of the equation under group transformations is explicit. The aim of this paper is to provide a theoretical ground for the phenomenological analysis of [8,9,10] and to derive the most general differential equation for the probability distribution of parallel transporters on the group manifold from the first principles. The classical loop equation [16] is first rewritten as a flow equation on the group manifold.…”

Random walks of Wilson loops in the screening regime

“…. It can be shown that p [g; C] is the Wilson loop in the regular representation of the gauge group [10,11].…”