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 mu(H), the Hall transform is an isometric isomorphism hem L(2)(G, mu(H)) to H(G(C)) boolean AND L(2)(G(C), v), where G(C) the complexification of G, H(G(C)) the space of holomorphic functions on G(C), and v 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 A/g of connections module 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. (C) 1996 Academic Press, Inc.