COHERENT-STATE TRANSFORMS FOR SPACES OF CONNECTIONS

The Segal-Bargmann transform plays an important role in quantum theori es of linear fields. Recently, Hall obtained a non-linear analog of th is transform for quantum mechanics on Lie groups. Given a compact, con nected Lie group G with its normalized Haar measure mu(H), the Hall tr ansform is an isometric isomorphism hem L(2)(G, mu(H)) to H(G(C)) bool ean AND L(2)(G(C), v), where G(C) the complexification of G, H(G(C)) t he space of holomorphic functions on G(C), and v an appropriate heat-k ernel measure on G(C). We extend the Hall transform to the infinite di mensional context of non-Abelian gauge theories by replacing the Lie g roup G by (a certain extension of) the space A/g of connections module gauge transformations. The resulting ''coherent state transform'' pro vides 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 dimension s. (C) 1996 Academic Press, Inc.