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

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.