"YangMills Theory and the SegalBargmann 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 SegalBargmann transform to study canonically
quantized YangMills theory on a spacetime cylinder with a compact
structure group $K.$ The nonexistent Lebesgue measure on the space
of
connections is ``approximated'' by a Gaussian measure with large variance.
The SegalBargmann 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 gaugetrivial subspace, consisting of functions of the holonomy
around the spatial circle, the SegalBargmann 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
YangMills theory on a spacetime 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 oneparameter 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
