Abstract
We use a functional renormalization group equation tailored to the Arnowitt–Deser–Misner formulation of gravity to study the scale dependence of Newton’s coupling and the cosmological constant on a background spacetime with topology S^1 times S^d. The resulting beta functions possess a non-trivial renormalization group fixed point, which may provide the high-energy completion of the theory through the asymptotic safety mechanism. The fixed point is robust with respect to changing the parametrization of the metric fluctuations and regulator scheme. The phase diagrams show that this fixed point is connected to a classical regime through a crossover. In addition the flow may exhibit a regime of “gravitational instability”, modifying the theory in the deep infrared. Our work complements earlier studies of the gravitational renormalization group flow on a background topology S^1 times T^d (Biemans et al. Phys Rev D 95:086013, 2017, Biemans et al. arXiv:1702.06539, 2017) and establishes that the flow is essentially independent of the background topology.
Highlights
Introduction and motivationAsymptotic Safety, first suggested by Weinberg [3,4], constitutes a mechanism for constructing a consistent and predictive quantum theory for gravity within the framework of quantum field theory
We find that all cases studied in this paper admit a non-Gaussian fixed point (NGFP) suitable for Asymptotic Safety
In this work we have studied the gravitational renormalization group (RG) flow in the Arnowitt–Deser–Misner (ADM) formalism, utilizing backgrounds with a topology S1 × Sd
Summary
Asymptotic Safety, first suggested by Weinberg [3,4], constitutes a mechanism for constructing a consistent and predictive quantum theory for gravity within the framework of quantum field theory. The natural continuum analogue of the foliation structure imposed on the microscopic spacetimes studied within CDT is the Arnowitt–Deser–Misner (ADM) formulation reviewed, e.g., in [74] In this formalism spacetime is built up from a stack of spatial hypersurfaces τ on which the time-variable τ is constant. Our work is complementary to the recent investigation [1,2] in the sense that it uses a different background topology It provides a detailed analysis on how the flow is influenced by integrating over different classes of spatial fluctuations and under a change of the regulator scheme. Technical details as regards the background geometry, the structure of the flow equation, and the evaluation of the operator traces are provided in Appendix A, Appendix B, and Appendix C, respectively
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