39,506 research outputs found
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
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
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