  The Segal-Bargmann transform plays an important role in quantum theories of
linear fields. Recently, Hall obtained a non-linear analog of this transform
for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$
with its normalized Haar measure $\mu_H$, the Hall transform is an isometric
isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$,
where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of
holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel
measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional
context of non-Abelian gauge theories by replacing the Lie group $G$ by (a
certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge
transformations. The resulting ``coherent state transform'' provides a
holomorphic representation of the holonomy $C^\star$ algebra of real gauge
fields. This representation is expected to play a key role in a
non-perturbative, canonical approach to quantum gravity in 4-dimensions.
