Non-commutative flux representation for loop quantum gravity

TL;DR

This paper introduces a novel non-commutative flux representation for loop quantum gravity, enabling a dual perspective where fluxes act multiplicatively, thus offering new insights into the theory's mathematical structure.

Contribution

It explicitly constructs a non-commutative Fourier transform for loop quantum gravity, establishing a dual flux representation previously thought impossible due to non-commutativity.

Findings

Flux operators act by *-multiplication in the dual representation

Holonomy operators act by translation in the dual space

Comparison with the U(1) case links to loop quantum cosmology

Abstract

The Hilbert space of loop quantum gravity is usually described in terms of cylindrical functionals of the gauge connection, the electric fluxes acting as non-commuting derivation operators. It has long been believed that this non-commutativity prevents a dual flux (or triad) representation of loop quantum gravity to exist. We show here, instead, that such a representation can be explicitly defined, by means of a non-commutative Fourier transform defined on the loop gravity state space. In this dual representation, flux operators act by *-multiplication and holonomy operators act by translation. We describe the gauge invariant dual states and discuss their geometrical meaning. Finally, we apply the construction to the simpler case of a U(1) gauge group and compare the resulting flux representation with the triad representation used in loop quantum cosmology.