A Jordan GNS Construction for the Holonomy-Flux *-algebra

TL;DR

This paper develops a Jordan GNS construction for the holonomy-flux *-algebra in loop quantum gravity, addressing limitations of the conventional GNS approach by utilizing a trace-based method within a Banach algebra framework.

Contribution

It introduces a novel Jordan GNS construction for the holonomy-flux *-algebra, extending the algebraic framework of loop quantum gravity beyond traditional methods.

Findings

Constructs a trace-based Jordan GNS representation.

Shows the state invariance under all inner derivations.

Discusses implications for Jordan-Schrodinger equation.

Abstract

The holonomy-flux *-algebra was recently proposed as an algebra of basic kinematical observables for loop quantum gravity. We show the conventional GNS construction breaks down when the the holonomy-flux *-algebra is allowed to be a Jordan algebra of observables. To remedy this, we give a Jordan GNS construction for the holonomy-flux *-algebra that is based on trace. This is accomplished by assuming the holonomy-flux *-algebra is an algebra of observables that is also a Banach algebra, hence a JB algebra. We show the Jordan GNS construction produces a state that is invariant under all inner derivations of the holonomy-flux *-algebra. Implications for the corresponding Jordan-Schrodinger equation are also discussed.