Flux formulation of loop quantum gravity: Classical framework

TL;DR

This paper develops a classical framework for a new loop quantum gravity formulation based on integrated fluxes and Wilson surface operators, enabling better coarse graining and curvature encoding.

Contribution

It introduces a classical phase space with integrated fluxes and a closed Poisson algebra, advancing the mathematical foundation of the new quantum gravity representation.

Findings

Integrated fluxes are key in encoding curvature and torsion.

The continuum phase space is constructed as a modified projective limit.

Poisson brackets of fluxes can intersect, maintaining algebra closure.

Abstract

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.