Functional Integration on Spaces of Connections
John C. Baez, Stephen Sawin
TL;DR
This work builds a rigorous framework for generalized measures on the space ${\cal A}$ of smooth connections by extending the Ashtekar–Lewandowski construction to the smooth setting using webs and tassels to capture holonomy data. A canonical uniform generalized measure $\nu$ is constructed, invariant under all bundle automorphisms ${\rm Aut}(P)$, and interpretable via a projective limit $\overline{\cal A}$; generalized measures are characterized by consistent, uniformly bounded collections on ${\cal A}_W$ for webs. The paper also provides an explicit spin-network basis for the gauge-invariant space $L^2({\cal A}/{\cal G})$, linking representation theory of $G$ to holonomy data on webs, thereby enabling concrete approximations and calculations in gauge theory and loop quantum gravity. Together, these results offer a robust, invariant functional-analytic foundation for gauge-theoretic quantum theories in the smooth category.
Abstract
Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity on the base space $M$. Using `spin networks' we construct explicit functions spanning the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge transformations.