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

TL;DR

This paper constructs a rigorous mathematical framework for the abelian Yang-Mills path integral on  using an Abstract Wiener space and renormalization, and demonstrates that the Area Law does not apply in this case.

Contribution

It introduces a novel construction of an Abstract Wiener space for abelian Yang-Mills theory and applies renormalization to define the path integral rigorously.

Findings

Constructed an Abstract Wiener space for abelian Yang-Mills path integrals.

Applied lattice gauge theory renormalization techniques.

Proved the Area Law does not hold in abelian Yang-Mills theory.

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\ d\omega \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\ d\omega}\ 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.