Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao, Thomas Thiemann
Coherent State Transforms for Spaces of Connections
The paper is published: J. Funct. Anal. 135, 2 (1996) 519-551

MSC:

22E70 Applications of Lie groups to physics; explicit representations, See also {81R05, 81R10}

81R30 Coherent states, See also {22E45}

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
$\mu_H$, 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.

Keywords: coherent state transform, Segal-Bargmann transform, quantum theories, quantum mechanics, Lie groups, compact, connected Lie group, Haar measure, Hall transform, holomorphic functions, heat-kind measure, non-Abelian gauge theories, real gauge fields, quantum gravity