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

TL;DR

This paper derives a general equation for the probability distribution of parallel transporters in quantum gauge fields using cumulant expansion, connecting it to nonperturbative cumulants and the heat kernel in QCD.

Contribution

It introduces a novel cumulant expansion approach for gauge field transporters, linking nonperturbative effects to the probability distribution on the gauge group manifold.

Findings

Equation reduces to heat kernel in Gaussian QCD vacuum

Connects cumulant coefficients to nonperturbative curvature cumulants

Provides a general framework for probability distributions in gauge theories

Abstract

A general equation for the probability distribution of parallel transporters on the gauge group manifold is derived using the cumulant expansion theorem. This equation is shown to have a general form known as the Kramers-Moyall cumulant expansion in the theory of random walks, the coefficients of the expansion being directly related to nonperturbative cumulants of the shifted curvature tensor. In the limit of a gaussian-dominated QCD vacuum the obtained equation reduces to the well-known heat kernel equation on the group manifold.