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 μH , the Hall transform is an isometric isomorphism from L (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 heatkernel 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 A/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. Center for Gravitational Physics and Geometry, Physics Department, The Pennsylvania State University, University Park, PA 16802-6300, USA. Institute of Theoretical Physics, University of Warsaw, 00-681 Warsaw, Poland Department of Physics, The University of California, Santa Barbara, CA 93106, USA Sector de F́ısica da U.C.E.H., Universidade do Algarve, Campus de Gambelas, 8000 Faro, Portugal