Representation Theory of Analytic Holonomy C* Algebras

TL;DR

This paper develops a representation theory for abelian C* algebras generated by Wilson loops, establishing a duality between gauge classes of connections and loop invariants, with applications to quantum theories of connections.

Contribution

It introduces a novel duality framework linking gauge equivalence classes of connections to loop invariants and constructs a diffeomorphism invariant measure for quantum connection theories.

Findings

Established a one-to-one correspondence between measures and loop functions.

Constructed a diffeomorphism invariant measure for quantum states.

Demonstrated the use of Wilson loops as quantum configuration operators.

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.