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 groupGwith its normalized Haar measureμH, the Hall transform is an isometric isomorphism fromL2(G, μH) toH(GC)∩L2(GC, ν), whereGCthe complexification ofG,H(GC) the space of holomorphic functions onGC, andνan appropriate heat-kernel measure onGC. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie groupGby (a certain extension of ) the spaceA/Gof connections modulo gauge transformations. The resulting “coherent state transform” provides a holomorphic representation of the holonomyC* 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.