Uniqueness of Measures in Loop Quantum Cosmology

TL;DR

This paper demonstrates that the standard kinematical Hilbert space in homogeneous isotropic loop quantum cosmology is uniquely determined by the unitarity of extended flux translations, which singles out the Bohr measure on the configuration space.

Contribution

It shows that unitarity of flux translations uniquely determines the standard measure and Hilbert space in homogeneous isotropic LQC, extending previous results to the Fleischhack configuration space.

Findings

The Bohr measure is uniquely singled out by unitarity conditions.

The standard kinematical Hilbert space is uniquely characterized by these measure conditions.

Both the standard and Fleischhack configuration spaces are affected by this measure uniqueness.

Abstract

In a paper of Ashtekar and Campiglia, residual diffeomorphisms have been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). We show that, in the homogeneous isotropic case, unitarity of the translations w.r.t. the extended $\mathbb{R}$-action (exponentiated reduced fluxes in the standard approach) singles out the Bohr measure on both the standard quantum configuration space $\mathbb{R}_\mathrm{Bohr}$ as well as on the Fleischhack one. Thus, in both situation, the same condition singles out the standard kinematical Hilbert space of LQC.