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Abstract Wiener measure using abelian Yang-Mills action on
Let be the Lie algebra of a compact Lie group. For a
-valued 1-form , 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
. When we consider the Lie group , the Lie algebra
is isomorphic to , thus .
For some simple closed loop , 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 is some Lebesgue type of measure on the
space of -valued 1-forms, modulo gauge transformations,
, and 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
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