Kinematical uniqueness of homogeneous isotropic LQC

TL;DR

This paper demonstrates that invariance under all residual diffeomorphisms uniquely determines the standard kinematical Hilbert space in homogeneous isotropic loop quantum cosmology, using measure-theoretic and symmetry considerations.

Contribution

It extends previous invariance-based characterizations by showing that all residual diffeomorphisms uniquely select the standard LQC kinematical Hilbert space.

Findings

Haar measure on a8Ra9_{Bohr} is uniquely suitable for the inner product.

The measure must be zero on a8Ra9 in the Fleischhack case.

Standard kinematical Hilbert space is uniquely determined by residual diffeomorphism invariance.

Abstract

In a paper by Ashtekar and Campiglia, invariance under volume preserving residual diffeomorphisms has been used to single out the standard representation of the reduced holonomy-flux algebra in homogeneous loop quantum cosmology (LQC). In this paper, we use invariance under all residual diffeomorphisms to single out the standard kinematical Hilbert space of homogeneous isotropic LQC for both the standard configuration space $\mathbb{R}_{\mathrm{Bohr}}$, as well as for the Fleischhack one $\mathbb{R} \sqcup \mathbb{R}_{\mathrm{Bohr}}$. We first determine the scale invariant Radon measures on these spaces, and then show that the Haar measure on $\mathbb{R}_{\mathrm{Bohr}}$ is the only such measure for which the momentum operator is hermitian w.r.t. the corresponding inner product. In particular, the measure is forced to be identically zero on $\mathbb{R}$ in the Fleischhack case, so that for both approaches, the standard kinematical LQC-Hilbert space is singled out.