A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity

TL;DR

This paper demonstrates that the non-Abelian characteristics of geometric fluxes in loop quantum gravity stem from fundamental continuum commutation relations and the Gauss law, simplifying previous approaches and clarifying the origin of geometric operator discreteness.

Contribution

It provides a simplified derivation showing the non-Abelian nature of fluxes directly from continuum relations and Gauss law, generalizing to other gauge theories.

Findings

Non-Abelian fluxes follow from continuum commutation relations.

Simplification of previous formulations of quantum geometry.

Clarification of the origin of geometric operator discreteness.

Abstract

We show that the non-Abelian nature of geometric fluxes---the corner-stone in the definition of quantum geometry in the framework of loop quantum gravity (LQG)---follows directly form the continuum canonical commutations relations of gravity in connection variables and the validity of the Gauss law. The present treatment simplifies previous formulations and thus identifies more clearly the root of the discreteness of geometric operators in LQG. Our statement generalizes to arbitrary gauge theories and relies only on the validity of the Gauss law.