Coherent State Transforms for Spaces of Connections
Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, José Mourão, Thomas Thiemann
TL;DR
The paper develops a non-Abelian generalization of the Segal–Bargmann/Hall transform to the space of connections modulo gauge, enabling a holomorphic, gauge- and (under Baez measures) diffeomorphism-covariant representation of the holonomy C*-algebra. By leveraging projective limits over graphs and heat-kernel methods, it constructs isometric transforms that map $L^2$–states on ${\overline {\cal A}}$ to holomorphic $L^2$–spaces on the complexified configuration space ${\overline {\cal A}}^{\rm C}$ with carefully chosen measures $\nu$ or $\nu^{l}$ (and Baez measures $\mu^{(m)}$). The framework yields a coherent-state representation compatible with gauge and diffeomorphism symmetries, providing a rigorous holomorphic formulation well suited to canonical quantum gravity in four dimensions. This work thus bridges non-linear Hall transforms, loop-space geometry, and projective techniques to support a non-perturbative approach to quantum gravity with potential knot-theoretic connections.
Abstract
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 $μ_H$, the Hall transform is an isometric isomorphism from $L^2(G, μ_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, ν)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $ν$ 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.