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 μ_H, the Hall transform is an isometric isomorphism from L²(G,μ_H) to ℋ(G^ℂ) ∩ L²(G^ℂ, ν), where G^ℂ the complexification of G, ℋ(G^ℂ) the space of holomorphic functions on G^ℂ, and ν an appropriate heat-kernel measure on G^ℂ. 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 script A/script 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.