Non-commutative flux representation for loop quantum gravity
Aristide Baratin, Bianca Dittrich, Daniele Oriti, Johannes Tambornino
TL;DR
The paper addresses the longstanding tension between flux non-commutativity and a flux/triad representation in loop quantum gravity by constructing a non-commutative group Fourier transform on the LQG state space. This transform yields a dual flux representation in which flux operators act by $*$-multiplication and holonomies act by translations, while preserving the spectra of geometric operators and maintaining cylindrical consistency across graphs. It provides explicit dual actions for holonomies and fluxes, defines gauge-invariant dual states via closure constraints, and clarifies the relation to the spin-network basis, including a detailed U(1) case and implications for the semiclassical limit. The result offers a metric-like, non-commutative framework that could illuminate quantum geometry, facilitate dynamics and matter coupling, and connect LQG with non-commutative geometry and group-field-theory approaches.
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.