Abstract Wiener measure using abelian Yang-Mills action on R4\mathbb{R}^4

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 \nonumber \end{equation} using the standard metric on $T\mathbb{R}^4$. When we consider the Lie group $U(1)$, the Lie algebra $\mathfrak{g}$ is isomorphic to $\mathbb{R} \otimes i$, thus $A \wedge A = 0$. For some 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}\ DA, \nonumber \end{equation} whereby $DA$ is some Lebesgue type of measure on the space of $\mathfrak{g}$-valued 1-forms, modulo gauge transformations, $\mathcal{A} /\mathcal{G}$, 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, using renormalization techniques found in lattice gauge theory. We will further show that the Area Law formula do not hold in the abelian Yang-Mills theory

Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.Comment: 32 pages, 6 figure