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
Summary
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.
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