Coherent State Transforms for Spaces of Connections

1994/12/05 by Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf +2
Physics and Astronomy · #gr-qc #hep-th

paper · pdf

published as J.Funct.Anal. 135 (1996) 519-551 · 38 pages, latex

arxiv created 1994/12/05 · arxiv updated 2009/11/30

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 L²(G, μ_H) to \cal H(G^\Co)∩ L²(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^⋆ 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.

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