Coherent State Transforms for Spaces of Connections

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 $\mathcal{H}(G^{\text{Co}})\cap L^2(G^{\text{Co}},\nu )$, where $G^{\text{Co}}$ the complexification of $G$, $\mathcal{H}(G^{\text{Co}})$ the space of holomorphic functions on $G^{\text{Co}}$, and $\nu$ an appropriate heat-kernel measure on $G^{\text{Co}}$. 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 $\mathcal{A}\mathrm{/}\mathcal{G}$ 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.