Yang–Mills Theory and the Segal–Bargmann Transform

We use a variant of the Segal–Bargmann transform to study canonically quantized Yang–Mills theory on a space-time cylinder with a compact structure group K. The non-existent Lebesgue measure on the space of connections is “approximated” by a Gaussian measure with large variance. The Segal–Bargmann transform is then a unitary map from the L2 space over the space of connections to a holomorphicL2 space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} followed by analytic continuation. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory.