Hyphs and the Ashtekar–Lewandowski measure

Abstract

Properties of the space A of generalized connections in the Ashtekar framework are investigated.First a construction method for new connections is given. The new parallel transports differ from the original ones only along paths that pass an initial segment of a fixed path. This is closely related to a new notion of path independence. Although we do not restrict ourselves to the immersive smooth or analytical case, any finite set of paths depends on a finite set of independent paths, a so-called hyph. This generalizes the well-known directedness of the set of smooth webs and that of analytical graphs, respectively.Due to these propositions, on the one hand, the projections from A to the lattice gauge theory are surjective and open. On the other hand, an induced Haar measure can be defined for every compact structure group irrespective of the used smoothness category for the paths. *

Representations of the Weyl Algebra in Quantum Geometry

“…The smooth case is dealt with in [8,7,27]. The facts on hyphs and the conventions are due to [14,16,19]. Let G be some arbitrary connected compact Lie group and M be some manifold.…”

“…Collect all these paths in a set γ ≥ υ. Since γ may be not a hyph again, refine, if necessary, the paths in γ further to get a hyph υ ≥ γ ≥ υ [14]. By completeness, υ contains only paths in Q.…”

On the Structure of the Space of Generalized Connections

“…As it avoids technical complications arising in more general settings, the analytic case is the most studied, and the one in which more rigorous results have been obtained. Nevertheless, important progress has been made in the case of piecewise smooth, or more general, curves [19,20,21,22].…”

Proof of a Conjecture by Lewandowski and Thiemann

“…These are paths whose parameters coincide if their images in the manifold M coincide. We will prove that those paths can always be decomposed at finitely many parameter values such that the subpaths generated this way are graph-theoretically (hence [7] measure-theoretically) independent, unless they are equal.…”