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Quantum Spacetime and the Renormalization Group: Progress and Visions

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Progress and Visions in Quantum Theory in View of Gravity

Abstract

The quest for a consistent theory which describes the quantum microstructure of spacetime seems to require some departure from the paradigms that have been followed in the construction of quantum theories for the other fundamental interactions. In this contribution we briefly review two approaches to quantum gravity, namely, asymptotically safe quantum gravity and tensor models, based on different theoretical assumptions. Nevertheless, the main goal is to find a universal continuum limit for such theories and we explain how coarse-graining techniques should be adapted to each case. Finally, we argue that although seemingly different, such approaches might be just two sides of the same coin.

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Notes

  1. 1.

    See also the very recent results from the Event Horizon Telescope in [7], in very good agreement with GR predictions.

  2. 2.

    Of course, this assertion is restricted to path-integral approaches to quantum gravity.

  3. 3.

    This is very general and is not necessarily a cutoff in energy. As we will see in the case of tensor models, this parameter is associated to the size of the tensor being thus a dimensionless parameter.

  4. 4.

    When integrating the modes step by step in the path integral, one realizes that all terms compatible with the symmetries of the theory are generated in the effective action—apart from anomalies which we ignore in this article.

  5. 5.

    Oftenly, the asymptotically free fixed point is called Gaussian or non-interacting fixed point and the asymptotically safe one, non-Gaussian or interacting fixed point.

  6. 6.

    More precisely, it is spanned by canonically and marginally relevant couplings.

  7. 7.

    We restrict ourselves to the Euclidean path integral. The choice is for technical reasons. We should emphasize that establishing whether what is going to be presented remains valid in the Lorentzian setting is still a challenging open question.

  8. 8.

    This notation is not accurate. In fact, the metric g μν that enters as the argument of the effective average action corresponds to the expectation value of the metric g μν that appears in definition of the path integral (15). We employ the same name for both for simplicity.

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Acknowledgements

I would like to thank Astrid Eichhorn for many inspiring discussions on the topic and the organizers of the conference “Progress and Visions in Quantum Theory in View of Gravity: Bridging foundations of physics and mathematics” in Leipzig (2018) for the invitation. This work was supported by the DFG through the grant Ei/1037-1.

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, Brazil

    Antonio D. Pereira

  2. Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany

    Antonio D. Pereira

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Editors and Affiliations

  1. Department of Mathematics, University of Regensburg, Regensburg, Germany

    Felix Finster

  2. Leibniz University of Hannover, Institute for Theoretical Physics, Hannover, Germany; ZARM (Center of Applied Space Technology and Microgravity), University of Bremen, Bremen, Germany

    Domenico Giulini

  3. Institute for Theoretical Physics, Leibniz University of Hannover, Hannover, Germany

    Johannes Kleiner

  4. Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany

    Jürgen Tolksdorf

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Pereira, A.D. (2020). Quantum Spacetime and the Renormalization Group: Progress and Visions. In: Finster, F., Giulini, D., Kleiner, J., Tolksdorf, J. (eds) Progress and Visions in Quantum Theory in View of Gravity. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-38941-3_3

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