Towards a Geometrization of Renormalization Group Histories in Asymptotic Safety

Understand

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-space-time" 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.