AQV II. A new formulation of the Weyl C*-algebra

TL;DR

This paper introduces a new formulation of the Weyl C*-algebra using C*-dynamical systems, focusing on quantum configuration variables, flux operators, and their symmetries in quantum geometry.

Contribution

It presents a novel formulation of the Weyl C*-algebra in terms of C*-dynamical systems, incorporating flux groups and bisections for quantum geometry.

Findings

Defined quantum configuration variables via holonomies

Constructed flux operators as elements of flux groups

Identified a unique diffeomorphism-invariant pure state

Abstract

In this article a new formulation of the Weyl C*-algebra, which has been invented by Fleischhack, in terms of C*-dynamical systems is presented. The quantum configuration variables are given by the holonomies along paths in a graph. Functions depending on these quantum variables form the analytic holonomy C*-algebra. Each classical flux variable is quantised as an element of a flux group associated to a certain surface set and a graph. The quantised spatial diffeomorphisms are elements of the group of bisections of a finite graph system. Then different actions of the flux group associated to surfaces and the group of bisections on the analytic holonomy C*-algebra are studied. The Weyl C*-algebra for surfaces is generated by unitary operators, which implements the group-valued quantum flux operators, and certain functions depending on holonomies along paths that satisfy canonical commutation relations. Furthermore there is a unique pure state on the commutative Weyl C*-algebra for surfaces, which is a path- or graph-diffeomorphism invariant.