Abstract
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.
Highlights
Segal [1, 2] and Bargmann [3] introduced an integral transform that led to a holomorphic representation of quantum states of linear, Hermitian, Bose fields. (For a review of the holomorphic or, coherent-state representation, see Klauder [4].) The purpose of this paper is to extend that construction to non-Abelian gauge fields and, in particular, to general relativity
The key idea is to combine two ingredients: (i) A non-linear analog of the Segal-Bargmann transform due to Hall [5] for a system whose configuration space is a compact, connected Lie group; and, (ii) A calculus on the space of connections modulo gauge transformations based on projective techniques [6,7,8,9,10,11,12,13,14,15]
In theories of connections, the classical configuration space is given by AÂG, where A is the space of connections on a principal fibre bundle P(7, G) over a (``spatial'') manifold 7, and G is the group of vertical automorphisms of P
Summary
Segal [1, 2] and Bargmann [3] introduced an integral transform that led to a holomorphic representation of quantum states of linear, Hermitian, Bose fields. (For a review of the holomorphic or, coherent-state representation, see Klauder [4].) The purpose of this paper is to extend that construction to non-Abelian gauge fields and, in particular, to general relativity. All these measures project down unambiguously to AÂG.
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