C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity II

TL;DR

This paper introduces a new C*-algebra framework for quantum gravity based on holonomy-diffeomorphisms, characterizing its representations and comparing it to Loop Quantum Gravity, especially in higher dimensions.

Contribution

It defines the holonomy-diffeomorphism algebra, characterizes its separable representations, and analyzes their relation to gauge equivalence and Loop Quantum Gravity.

Findings

Separable representations correspond to measurable connections.

Unitary equivalence matches measured gauge equivalence.

Generalized connections in Loop Quantum Gravity are not in the spectrum in higher dimensions.

Abstract

We introduce the holonomy-diffeomorphism algebra, a C*-algebra generated by flows of vectorfields and the compactly supported smooth functions on a manifold. We show that the separable representations of the holonomy-diffeomorphism algebra are given by measurable connections, and that the unitary equivalence of the representations corresponds to measured gauge equivalence of the measurable connections. We compare the setup to Loop Quantum Gravity and show that the generalized connections found there are not contained in the spectrum of the holonomy-diffeomorphism algebra in dimensions higher than one. This is the second paper of two, where the prequel gives an exposition of a framework of quantum gravity based on the holonomy-diffeomorphism algebra.