Representations of the holonomy algebras of gravity and nonAbelian gauge theories

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 -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 -algebra. The structure of this space is investigated and it is shown how observables labelled by "strips" arise naturally.

Hyphs and the Ashtekar–Lewandowski measure

“…Perhaps we should use a definition of the following kind: γ ∼ γ ′ iff h A (γ) = h A (γ ′ ) for all A ∈ A -maybe at least provided im γ = im γ ′ . This one is quite similar to that used originally in [1,2]. On the one hand, we expect that all the constructions made in this paper and its predecessor [6] will still go through.…”

“…Thus, the construction above is possible. Furthermore, we have τ i ≤ τ i+ 1 2 ≤ τ i,+ ≤ τ i+1,− ≤ τ i+1 and τ i < τ i+1 . 3.…”

“…But, since c j has no self-intersections and is divided according to e(τ i ) and e(τ i+ 1 2 ) (and other vertices that are not contained in im e i ), we have e i even equals c ′ k up to the parametrization.…”

“…One of the recent approaches to the quantization of gauge theories, in particular of gravity, is the investigation of generalized connections introduced by Ashtekar et al in a series of papers, see, e.g., [1,2,3]. Mathematically, there are two main ideas: First, every classical (i.e.…”

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…Such Hilbert spaces with diffeomorphism invariant states can be constructed from topological quantum field theories with defect excitations, as was suggested in [9]. The first such construction is known as the Ashtekar-Lewandowski (-Isham) representation [10][11][12][13] and involves a trivial TQFT, in which the vacuum is peaked on a geometrically totally degenerate configuration. This fact makes the construction of states describing large scale geometries extremely complicated.…”