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Kinematical Foundations of Loop Quantum Cosmology

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Quantum Mathematical Physics

Abstract

First, we review the C ∗-algebraic foundations of loop quantization, in particular, the construction of quantum configuration spaces and the implementation of symmetries. Then, we apply these results to loop quantum gravity, focusing on the space of generalized connections and on measures thereon. Finally, we study the realm of homogeneous isotropic loop quantum cosmology: once viewed as the loop quantization of classical cosmology, once seen as the symmetric sector of loop quantum gravity. It will turn out that both theories differ, i.e., quantization and symmetry reduction do not commute. Moreover, we will present a uniqueness result for kinematical measures. These last two key results have originally been due to Hanusch; here, we give drastically simplified and direct proofs.

Mathematics Subject Classification (2010). Primary 46L60; Secondary 58D19, 46L65, 81T05, 83C45, 83F05

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Notes

  1. 1.

    One may assume w.l.o.g. that the action is effective although we will never use this.

  2. 2.

    Parts of Proposition 1 originate in [47].

  3. 3.

    The kernel of \(\mathfrak{A}\) is defined by \(\bigcap _{a\in \mathfrak{A}}a^{-1}(0)\). In particular, each unital \(\mathfrak{A}\) has empty kernel. Throughout the whole article, any \(\mathfrak{A}\) will have empty kernel – or we will assume that.

  4. 4.

    If the natural mappings ι i are injective, this just means that \(\overline{\sigma }\) extends \(\sigma\).

  5. 5.

    To explain the term “restriction algebra”, assume that \(\sigma\) is injective, whence \(\mathcal{X}_{1}\) can be considered as a subset of \(\mathcal{X}_{2}\). Then \(\sigma ^{{\ast}}\mathfrak{A}_{2}\) consists just of the restrictions of the functions in \(\mathfrak{A}_{2} \subseteq \ell^{\infty }(\mathcal{X}_{2})\) to the domain \(\mathcal{X}_{1}\). In order to avoid conflicts with the different notion of pull-back C ∗-algebras, we will use the notion “restriction algebra” also in the case where \(\sigma\) is not injective.

  6. 6.

    Note that later we will refrain from writing \(\sigma\) in the case of subspaces of invariant elements.

  7. 7.

    Alternatively, one can also consider the space of all Riemannian 3-metrics modulo diffeomorphisms. This leads to the so-called superspace. However, its mathematical structure is rather complicated [36].

  8. 8.

    There are spin structures, hence spin bundles, as \(\Sigma \) is orientable [43].

  9. 9.

    Note that we do not require ν to even be a local section in the bundle sense. In fact, ν need not be continuous; this is referred to by noting “set-theoretic” [26].

  10. 10.

    We assume to have fixedly chosen G as a Lie subgroup of some U(n).

  11. 11.

    For this, one has to identify paths that coincide up to their parametrization.

  12. 12.

    The results of Sects. 4.7 and 4.8 can, in principle, be found in the very voluminous and interesting thesis [38] by Hanusch. Here, however, we have drastically simplified and streamlined notations, statement presentations and proofs.

  13. 13.

    In order not to overload the notation, we refrain from writing “paths modulo reparametrization”, here and in the following. Indeed, it should be clear that and how the action of S transfers to these equivalence classes.

  14. 14.

    This way, rotation in direction n with length t equals rotation in direction − n with length − t. The homomorphy property of b together with \(T_{-n} = T_{n}^{-1}\) shows that \(\overline{A}_{\mathbf{b}}(\gamma )\) is well defined.

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Acknowledgements

The author is very grateful to the organizers of the Regensburg conference on quantum mathematical physics for the kind invitation. The author is also very grateful to Maximilian Hanusch for numerous discussions and helpful comments on a draft version of this article.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Paderborn, 33095, Paderborn, Germany

    Christian Fleischhack

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  1. Christian Fleischhack
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Correspondence to Christian Fleischhack .

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Editors and Affiliations

  1. Universität Regensburg, Fakultät für Mathematik, Regensburg, Germany

    Felix Finster

  2. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Johannes Kleiner

  3. Regensburg, Germany, Fakultät für Mathematik Universität Rege, Regensburg, Germany

    Christian Röken

  4. Leipzig, Germany, MPI für Mathematik in den Naturwissensch, Leipzig, Germany

    Jürgen Tolksdorf

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Fleischhack, C. (2016). Kinematical Foundations of Loop Quantum Cosmology. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds) Quantum Mathematical Physics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26902-3_11

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