Coherent State Transforms for Spaces of 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 ${\mu }_H$, the Hall transform is an isometric isomorphism from $L^2(G,{\mu }_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, ν)$, 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 $\mathcal{A}/\mathcal{G}$ 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.

38 pages, latex

References in corpus (5)

• Projective Techniques and Functional Integration
• Representations of the holonomy algebras of gravity and non-Abelian gauge theories
• Topological Measure and Graph-Differential Geometry on the Quotient Space of Connections
• Integration on the space of Connections Modulo Gauge Transformations
• Recent Mathematical Developments in Quantum General Relativity

Cited by in corpus (2)

• Spin Network States in Gauge Theory
• Quantum Loop Representation for Fermions coupled to Einstein-Maxwell field