AQV III. The holonomy-flux cross-product C*-algebra

TL;DR

This paper introduces a new C*-algebra in Loop Quantum Gravity based on holonomies and flux operators, utilizing cross-product theory to analyze surface-preserving symmetries and extend the algebra with graph-diffeomorphisms.

Contribution

It constructs a novel holonomy-flux cross-product C*-algebra and explores its states and symmetries, advancing the mathematical framework of Loop Quantum Gravity.

Findings

Defined the holonomy-flux cross-product C*-algebra for a surface set

Analyzed surface-preserving path- and graph-diffeomorphism-invariant states

Extended the algebra to include graph-diffeomorphisms and related operators

Abstract

In this article a new C*-algebra derived from the basic quantum variables: holonomies along paths and group-valued quantum flux operators in the framework of Loop Quantum Gravity is constructed. This development is based on the theory of cross-products and C*-dynamical systems. The author has presented a set of actions of the flux group associated to a surface set on the analytic holonomy C*-algebra, which define C*-dynamical systems. These objects are used to define the holonomy-flux cross-product C*-algebra associated to a surface set. Furthermore surface-preserving path- and graph-diffeomorphism-invariant states of the new C*-algebra are analysed. Finally the holonomy-flux cross-product C*-algebra is extended such that the graph-diffeomorphisms generate among other operators the holonomy-flux-graph-diffeomorphism cross-product C*-algebra associated to a surface set.