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 μH, the Hall transform is an isometric isomorphism from L2(G,μH) to ℋ(Gℂ) ∩ L2(Gℂ, ν), where Gℂ the complexification of G, ℋ(Gℂ) the space of holomorphic functions on Gℂ, and ν an appropriate heat-kernel measure on Gℂ. 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.
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