Yang-Mills and Segal Bargmann
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"Yang-Mills Theory and the Segal-Bargmann Transform," by B. Driver and Brian Hall.

(UCSD Preprint, July 1998 )

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The manuscript is available as a DVI file (230K).

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Abstract

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 $L^{2}$ space
over the space of connections to a \textit{holomorphic} $L^{2}$ space over
the space of complexified connections with a certain Gaussian measure. This
transform is given roughly by $e^{t\Delta _{\mathcal{A}}/2}$ followed by
analytic continuation. Here $\Delta _{\mathcal{A}}$ is the Laplacian on the
space of connections and is the Hamiltonian for the quantized theory.

On the gauge-trivial subspace, consisting of functions of the holonomy
around the spatial circle, the Segal-Bargmann transform becomes $e^{t\Delta
_{K}/2}$ followed by analytic continuation, where $\Delta _{K}$ is the
Laplacian for the structure group $K.$ This result gives a rigorous meaning
to the idea that $\Delta _{\mathcal{A}}$ reduces to $\Delta _{K}$ on
functions of the holonomy. By letting the variance of the Gaussian measure
tend to infinity we recover the standard realization of the quantized
Yang-Mills theory on a space-time cylinder, namely, $-\frac{1}{2}\Delta _{K}$
is the Hamiltonian and $L^{2}(K)$ is the Hilbert space. As a byproduct of
these considerations, we find a new one-parameter family of unitary
transforms from $L^{2}(K)$ to certain holmorphic $L^{2}$-spaces over the
complexification of $K.$ This family of transformations interpolates between
the two unitary transformations introduced in \cite{H1}.

Our work is motivated by results of Landsman and Wren \cite{LW,W1,W2,L} and
uses probabilistic techniques similar to those of Gross and Malliavin \cite
{GM}.

 

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July, 1998

08/15/2016 02:53 PM