Representation Theory of Analytic Holonomy C* Algebras
Abhay Ashtekar, Jerzy Lewandowski
TL;DR
This work completes the Ashtekar–Isham program by constructing a rigorous representation theory for the holonomy $C^*$-algebra generated by Wilson loops. It provides a complete characterization of the Gel'fand spectrum $\overline{\cal A/\cal G}$ in terms of hoop-group homomorphisms to $SU(2)$, and builds a faithful, diffeomorphism-invariant cylindrical measure on this spectrum, enabling a loop representation via $L^2(\overline{\cal A/\cal G}, d\mu)$. The authors develop loop-decomposition techniques, non-linear cylindrical measures, and a robust loop transform that relates connection states to generalized knot invariants, thereby bridging connection and loop formulations of gauge theories. The framework lays a mathematically rigorous foundation for background-independent quantization of connection theories, with potential applicability to loop quantum gravity and related diffeomorphism-invariant formalisms.
Abstract
Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum theory.