Abstract

Considering the scale-dependent effective spacetimes implied by the functional renormalization group in d-dimensional quantum Einstein gravity, we discuss the representation of entire evolution histories by means of a single, (d+1)-dimensional manifold furnished with a fixed (pseudo-) Riemannian structure. This “scale-spacetime” carries a natural foliation whose leaves are the ordinary spacetimes seen at a given resolution. We propose a universal form of the higher dimensional metric and discuss its properties. We show that, under precise conditions, this metric is always Ricci flat and admits a homothetic Killing vector field; if the evolving spacetimes are maximally symmetric, their (d+1)-dimensional representative has a vanishing Riemann tensor even. The non-degeneracy of the higher dimensional metric that “geometrizes” a given RG trajectory is linked to a monotonicity requirement for the running of the cosmological constant, which we test in the case of asymptotic safety.

Highlights

  • The familiar renormalization group (RG) equations of quantum field theory are formulated in a mathematical setting that is rather simple and, in a way, structureless from the geometric point of view

  • We proposed to exploit those RG-derived data, and only those, to initiate a systematic search for natural geometric structures, which can help in efficiently structuring those data and/or facilitate their physical interpretation or application.1 (2) we dealt in this paper with the nonperturbative functional RG flows of quantum Einstein gravity (QEG), i.e., quantum gravity in a metric-based formulation

  • Taking (P) for granted, we demonstrated that it is always possible to complete the specification of the (d + 1)-dimensional Riemannian geometry in such a way that it enjoys the following features: (G) The higher dimensional metric (d+1)gI J is Ricci flat: (d+1)RI J = 0

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Summary

Introduction

The familiar renormalization group (RG) equations of quantum field theory are formulated in a mathematical setting that is rather simple and, in a way, structureless from the geometric point of view. The conjectured AdS/CFT correspondence “geometrizes” RG flows by a different approach that identifies the scale variable of the RG equations with a specific coordinate on a higher dimensional (bulk) spacetime [9,10,11]. What we proposed here is a bottom-up approach that starts out from the safe harbor of a well-understood and fully general RG framework and only in a second step tries to assess whether, and under what conditions, there exist natural options for geometrizing the RG flows. This approach must be contrasted with top-down approaches like the one based on the AdS/CFT conjecture, for instance.

From Trajectories of Metrics to Higher Dimensions
Solutions of the Rescaling Type
Focusing on the Lapse Function
Distinguished Higher Dimensional Geometries
The Hubble Length as a Coordinate
Equivalent Forms of the Postulated Metric
Homothetic Killing Vector and Self-Similarity
Ricci Flatness
Strict Flatness
Asymptotic Safety
Summary and Conclusions
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