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Is There a C-Function in 4D Quantum Einstein Gravity?

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Quantum Mathematical Physics

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Abstract

We describe a functional renormalization group-based method to search for ‘C-like’ functions with properties similar to that in 2D conformal field theory. It exploits the mode counting properties of the effective average action and is particularly suited for theories including quantized gravity. The viability of the approach is demonstrated explicitly in a truncation of 4 dimensional Quantum Einstein Gravity, i.e. asymptotically safe metric gravity.

Mathematics Subject Classification (2010). Primary 81T06; Secondary 81Q06.

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Notes

  1. 1.

    Our conventions are as follows. We use dimensionless coordinates, [x μ] = 0. Then \([\text{d}s^{2}] = -2\) implies that all components of the various metrics have \([\hat{g}_{\mu \nu }] = [\bar{g}_{\mu \nu }] = [g_{\mu \nu }] = -2\), and likewise for the fluctuations: \([\hat{h}_{\mu \nu }] = [h_{\mu \nu }] = -2\).

  2. 2.

    We use the notation \(c^{[\varphi ]}\varphi \equiv \{ c^{[\varphi _{i}]}\varphi _{i}\}\) for the set in which each field is rescaled according to its individual canonical dimension.

  3. 3.

    Here one should also switch from k to the manifestly dimensionless ‘RG time’ \(t \equiv \ln (k) + \text{const}\), but we shall not indicate this notationally.

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Acknowledgements

M. R. would like to thank the organizers of Quantum Mathematical Physics for their hospitality at Regensburg and for a particularly stimulating conference.

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Authors and Affiliations

  1. Institute of Physics, University of Mainz, Staudingerweg 7, D-55099, Mainz, Germany

    Daniel Becker & Martin Reuter

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  1. Daniel Becker
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Correspondence to Daniel Becker .

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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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Becker, D., Reuter, M. (2016). Is There a C-Function in 4D Quantum Einstein Gravity?. 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_2

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