Generalized Measures in Gauge Theory
John C. Baez
TL;DR
This work develops generalized measures on the spaces of connections $\mathcal{A}$, gauge transformations $\mathcal{G}$, and the quotient $\mathcal{A}/\mathcal{G}$ to provide rigorous substitutes for ill-defined Lebesgue and Haar measures in gauge theory. It constructs cylinder-function algebras based on holonomies, and proves a characterization of generalized measures via consistent, uniformly bounded families $\{\mu_\phi\\}$ associated with embedded graphs, enabling explicit realizations such as the uniform measure $\mu_u$ on $\mathcal{A}$, the generalized Haar measure $\mu_H$ on $\mathcal{G}$, and the Ashtekar–Lewandowski measure $\mu_{AL}$ on $\mathcal{A}/\mathcal{G}$ (for compact $G$). The framework supports averaging measures on $\mathcal{A}$ against $\mu_H$ to enforce gauge invariance and opens routes to Hilbert-space constructions $L^2(\mathcal{A},\mu_u)$, $L^2(\mathcal{A}/\mathcal{G},\mu_{AL})$, and $L^2(\mathcal{G},\mu_H)$ with unitary actions of the symmetry groups. While not a panacea for 4D Yang–Mills, the approach provides a robust toolset for loop-based observables and potential applications in Chern–Simons theory and loop quantum gravity.
Abstract
Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure'' on the space A of connections on P and for ``Haar measure'' on the group Ga of gauge transformations. More precisely, we define algebras of ``cylinder functions'' on the spaces A, Ga, and A/Ga, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from lattice gauge theory, we characterize generalized measures on A, Ga, and A/Ga in terms of graphs embedded in M. We use this characterization to construct generalized measures on A and Ga, respectively. The ``uniform'' generalized measure on A is invariant under the group of automorphisms of P. It projects down to the generalized measure on A/Ga considered by Ashtekar and Lewandowski in the case G = SU(n). The ``generalized Haar measure'' on Ga is right- and left-invariant as well as Aut(P)-invariant. We show that averaging any generalized measure on A against generalized Haar measure gives a gauge-invariant generalized measure on A.