Uniqueness of the Representation in Homogeneous Isotropic LQC

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 ${}^{\ast }$-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.

arXiv comment9 pages; Bianchi I case added in Section V; gamma factors corrected in equations (23) and (24); characters defined at the end of Section II; technical oversight corrected in Section IV; Acknowledgements updated; entry added to the bibliography

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