Abstract Wiener measure using abelian Yang-Mills action on $\mathbb{R}^4$

TL;DR

This work constructs a rigorous probabilistic formulation of abelian Yang–Mills theory on ${\mathbb R}^4$ by embedding the classical path integral into an Abstract Wiener space framework and employing Balaban-inspired renormalization through a Segal–Bargmann renormalization flow. By transporting to holomorphic function spaces ${\mathcal H}^2({\mathbb C}^4)$ and building a Gaussian measure on a Banach space ${\mathbb B}$, the authors define a well-posed Yang–Mills path integral for $G={\rm U}(1)$ and demonstrate that, under a renormalized coupling $c=1/\kappa$, the observable obeys an Area Law ${\rm e}^{-\frac{1}{8}\int_S d\rho}$, while with fixed $c$ the Area Law is not recovered. The framework combines Abstract Wiener space machinery, Segal–Bargmann transforms, and renormalization flow to bridge lattice and continuum perspectives, setting the stage for extending to non-abelian Yang–Mills in a follow-up work. This establishes a rigorous, renormalized probabilistic foundation for Wilson-loop-type observables in four dimensions.

Abstract

Let $\mathfrak{g}$ be the Lie algebra of a compact Lie 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\ dω\nonumber \end{equation} using the Euclidean metric on $T\mathbb{R}^4$. When we consider the Lie group ${\rm U}(1)$, the Lie algebra $\mathfrak{g}$ is isomorphic to $\mathbb{R} \otimes i$, thus $A \wedge A = 0$. For a simple closed loop $C$, we want to make sense of the following path integral, \begin{equation} \frac{1}{Z}\ \int_{A \in \mathcal{A} /\mathcal{G}} \exp \left[ \int_{C} A\right] e^{-\frac{1}{2}\int_{\mathbb{R}^4}|dA|^2\ dω}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space $\mathcal{A} /\mathcal{G}$ containing $\mathfrak{g}$-valued 1-forms modulo gauge transformations, and $Z$ is some partition function. We will construct an Abstract Wiener space for which we can define the above Yang-Mills path integral rigorously, applying renormalization techniques found in lattice gauge theory. We will further show that the Area Law formula does not hold in the abelian Yang-Mills theory.