Suppose thatGis a compact connected Lie group andP→Mis a smooth principalG-bundle. We define a “cylinder function” on the space of smooth connections onPto be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inM. Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(). Let a “generalized measure” on be a bounded linear functional on Fun(). We construct a generalized measureμ0on that is invariant under all automorphisms of the bundleP(not necessarily fixing the baseM). This result extends previous work which assumedMwas real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph withnedges embedded inMdetermines a C*-subalgebra of Fun() isomorphic toC(Gn), and the generalized measureμ0: Fun()→ restricts to the linear functional onC(Gn) given by integration against normalized Haar measure onGn. Our result implies that the group of gauge transformations acts as unitary operators onL2(), the Hilbert space completion of Fun() in the norm ‖F‖2=μ0(|F|2)1/2. Using “spin networks,” we construct explicit functions spanning the subspaceL2(/)⊆L2() consisting of vectors invariant under the action of .
Journal of Functional Analysis
Sawin, Stephen F. and Baez, John C., "Functional integration on spaces of connections" (1997). Mathematics Faculty Publications. 16.
Baez, John C., and Stephen Sawin. 1997. Functional integration on spaces of connections. Journal of Functional Analysis 150 (1), 1-26.