Representations of the holonomy algebras of gravity and non-Abelian gauge theories
Abhay Ashtekar, C. J. Isham
TL;DR
This work provides a rigorous kinematic framework for nonperturbative gauge theories and gravity by constructing the quantum holonomy algebra from loop variables and analyzing its representations via the Gel'fand spectrum. It shows that every cyclic representation is unitarily equivalent to $L^2(\Delta,d\mu)$ with holonomy operators acting by multiplication, and identifies the domain $\Delta$ of maximal ideals as the natural state space containing ${\cal A}/{\cal G}$ as a subspace. The loop transform is formalized as a generalized transform from functions on $\Delta$ to loop functions, and distributional elements of $\Delta$ are explored through strip-based constructions that yield momentum-like observables. The results provide a rigorous, gauge-invariant foundation for loop-based quantization and lay groundwork for studying nonperturbative dynamics in gravity and Yang–Mills theory, including future extension to momentum variables and full 3+1D gravity.
Abstract
Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally.