AQV V. The localised holonomy-flux cross-product *-algebra

TL;DR

This paper introduces a new *-algebra called the localised holonomy-flux cross-product *-algebra for Loop Quantum Gravity, aiming to incorporate quantum constraints and observables into a comprehensive physical algebra.

Contribution

It proposes a novel *-algebra structure for Loop Quantum Gravity that includes quantum constraints and observables, advancing the algebraic formulation of the theory.

Findings

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

Suggested a physical *-algebra containing key quantum constraints

Provided a framework for incorporating complete observables

Abstract

In the project AQV the issue of quantum constraints, KMS-states and algebras of quantum configuration and momentum variables in Loop Quantum Gravity has been argued. There a physical algebra has been required to contain complete observables and the quantum constraints, or at least the quantum constraints are affilliated with this algebra. In this context a first conjecture for a physical algebra is presented in this article. A new *-algebra for LGQ, which is called the localised holonomy-flux cross-product *-algebra, is studied. A suggestion for a physical *-algebra, which contains the localised holonomy-flux cross-product *-algebra, a modified quantum Hamilton constraint, a localised quantum diffeomorphism constraint and even a modified quantum Master constraint, is given.