Wilson Area Law formula on $\mathbb{R}^4$

TL;DR

This paper rigorously constructs the Yang-Mills path integral on  and derives the Wilson Area Law formula, providing a mathematical foundation for understanding quark confinement through a linear potential.

Contribution

It introduces a rigorous definition of the Yang-Mills path integral on  using an Abstract Wiener space and derives the Wilson Area Law formula via renormalization and asymptotic freedom.

Findings

Defined Yang-Mills path integral rigorously on 

Derived Wilson Area Law formula mathematically

Explained linear quark-antiquark potential

Abstract

Let $\mathfrak{g}$ be the Lie Algebra of a compact semi-simple gauge group. For a $\mathfrak{g}$-valued 1-form $A$, consider the Yang-Mills action \begin{equation} S_{{\rm YM}}(A) = \int_{\mathbb{R}^4} \left|dA + A \wedge A \right|^2\ d\omega, \nonumber \end{equation} using the Euclidean metric on $T\mathbb{R}^4$. We want to make sense of the following path integral, \begin{equation} {\rm Tr}\ \int_{A \in \mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}} \exp \left[ c\int_{S} dA\right] e^{-\frac{1}{2}S_{{\rm YM}}(A)}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space of $\mathfrak{g}$-valued 1-forms, modulo gauge transformations $\mathcal{A}_{\mathbb{R}^4, \mathfrak{g}} /\mathcal{G}$. Here, $S$ is some compact flat rectangular surface. Using an Abstract Wiener space, we can define a Yang-Mills path integral rigorously, for a compact semi-simple gauge group. Subsequently, we will then derive the Wilson area law formula from the definition, using renormalization techniques and asymptotic freedom. One of the most important applications of the Area Law formula will be to explain why the potential measured between a quark and antiquark is a linear function of its distance.