"Yang-Mills Theory and the Segal-Bargmann Transform," by B. Driver
and Brian Hall.
(UCSD Preprint, July 1998 )
The manuscript is available as a DVI
file (230K).
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}.
July, 1998
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