On the space of generalized fluxes for loop quantum gravity

TL;DR

This paper investigates the structure of the space of generalized fluxes in loop quantum gravity, revealing it cannot be obtained via Fourier transform from the connection space due to non-abelian gauge group properties, and explores implications for loop quantum cosmology.

Contribution

It demonstrates the non-abelian nature prevents Fourier transform construction and characterizes the flux space as an inductive limit, with implications for cosmological models.

Findings

Flux space is an inductive limit, not a Fourier transform of connection space.

Non-abelian SU(2) gauge group prevents direct Fourier transform construction.

Insights into the Bohr compactification relevant for loop quantum cosmology.

Abstract

We show that the space of generalized fluxes - momentum space - for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections - position space - due to the non-abelianess of the gauge group SU(2). From the abelianization of SU(2), U(1)^3, we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse-graining of the underlying graphs. We comment on the applications to loop quantum cosmology, in particular, how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for LQC.