Functional Integration on Spaces of Connections

Abstract

Suppose thatGis a compact connected Lie group andP→Mis a smooth principalG-bundle. We define a “cylinder function” on the space A 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(A). Let a “generalized measure” on A be a bounded linear functional on Fun(A). We construct a generalized measureμ0on A 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(A) isomorphic toC(Gn), and the generalized measureμ0:Fun(A)→C restricts to the linear functional onC(Gn) given by integration against normalized Haar measure onGn. Our result implies that the group G of gauge transformations acts as unitary operators onL2(A), the Hilbert space completion of Fun(A) in the norm ‖F‖2=μ0(|F|2)1/2. Using “spin networks,” we construct explicit functions spanning the subspaceL2(A/G)⊆L2(A) consisting of vectors invariant under the action of G.