Uniqueness of the Representation in Homogeneous Isotropic LQC

TL;DR

This paper proves that the standard representation in homogeneous isotropic loop quantum cosmology is uniquely determined by a residual diffeomorphism-invariant state, ensuring a consistent and covariant quantum framework.

Contribution

It establishes the uniqueness of the GNS-representation in homogeneous isotropic LQC under residual diffeomorphisms, including extended algebras, and confirms covariance with prior Bianchi I results.

Findings

The standard and extended algebra representations coincide.

Residual diffeomorphisms act covariantly on the quantum algebra.

The unique invariant state leads to a consistent GNS-representation.

Abstract

We show that the standard representation of homogeneous isotropic loop quantum cosmology (LQC) is the GNS-representation that corresponds to the unique state on the reduced quantum holonomy-flux $^*$-algebra that is invariant under residual diffeomorphisms $-$ both when the standard algebra is used as well as when one uses the extended algebra proposed by Fleischhack. More precisely, we find that in both situations the GNS-Hilbert spaces coincide, and that in the Fleischhack case the additional algebra elements are just mapped to zero operators. In order for the residual diffeomorphisms to have a well-defined action on the quantum algebra, we have let them act on the fiducial cell as well as on the dynamical variables, thereby recovering covariance. Consistency with Ashtekar and Campiglia in the Bianchi I case is also shown.