Functional Integration on Spaces of Connections

✍ Scribed by John C Baez; Stephen Sawin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 428 KB
• Volume: 150
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3108

✦ Synopsis

Suppose that G is a compact connected Lie group and P Ä M is a smooth principal G-bundle. We define a cylinder function'' on the space A of smooth connections on P to be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M. Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C\*-algebra Fun(A). Let a generalized measure'' on A be a bounded linear functional on Fun(A). We construct a generalized measure + 0 on A that is invariant under all automorphisms of the bundle P (not necessarily fixing the base M). This result extends previous work which assumed M was real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph with n edges embedded in M determines a C*-subalgebra of Fun(A) isomorphic to C(G n ), and the generalized measure + 0 : Fun(A) Ä C restricts to the linear functional on C(G n ) given by integration against normalized Haar measure on G n . Our result implies that the group G of gauge transformations acts as unitary operators on L 2 (A), the Hilbert space completion of Fun(A) in the norm &F& 2 =+ 0 (|F| 2 ) 1Â2 . Using ``spin networks,'' we construct explicit functions spanning the subspace L 2 (AÂG) L 2 (A) consisting of vectors invariant under the action of G.