AQV IV. A new formulation of the holonomy-flux *-algebra

TL;DR

This paper introduces a new holonomy-flux cross-product *-algebra in loop quantum gravity, providing a universal algebra framework and establishing a uniqueness result for certain invariant states.

Contribution

It formulates a novel *-algebra as an abstract cross-product, expanding the algebraic structures used in loop quantum gravity research.

Findings

Defined the holonomy-flux cross-product *-algebra.

Proved a uniqueness result for a specific invariant state.

Presented new *-algebras within the cross-product framework.

Abstract

In this article the holonomy-flux *-algebra, which has been introduced by Lewandowski, Okolow, Sahlmann and Thiemann, is modificated. The new *-algebra is called the holonomy-flux cross-product *-algebra. This algebra is an abstract cross-product *-algebra. It is given by the universal algebra of the algebra of continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set, and some canonical commutator relations. There is a uniqueness result for a certain path- and graph-diffeomorphism invariant state of the holonomy-flux cross-product *-algebra. This new *-algebra is not the only *-algebra, which is generated by the algebra of certain continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set. The theory of abstract cross-product algebras allows to define different new *-algebras. Some of these algebras are presented in this article.