Representations of the holonomy algebras of gravity and nonAbelian gauge theories
Abstract
Holonomy algebras arise naturally in the classical description of YangMills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a nonperturbative 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 Cstar 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 Cstar algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 June 1992
 DOI:
 10.1088/02649381/9/6/004
 arXiv:
 arXiv:hepth/9202053
 Bibcode:
 1992CQGra...9.1433A
 Keywords:

 High Energy Physics  Theory
 EPrint:
 44 pages