Thomas Thiemann and Renata Ferrero: Asymptotically safe – canonical quantum gravity junction

The asymptotically safe (ASQG) and the canonical (CQG) approach to quantum gravity share to be both non-perturbative programmes. However, apart from that they have been developed to a large extent independently of each other. Our aim is to overcome actual differences, to explain why apparent differences are actually absent and to enhance the contact between the two communities.

The first part of the talk is devoted to conceptual issues and to establish the inteface between the two programmes. The idea is to start from the path integral formulation of Lorentzian CQG in its reduced phase space formulation which in principle yields the generating functional of physical (i.e. gauge invariant) either Schwinger or Feynman N-point functions for (relational) observables. The corresponding effective average actions of these generating functionals can then be subjected to the ASQG Wetterich flow equation. Particular care is needed due to the necessary switch to Lorentzian signature which has strong impact on the convergence of “heat” kernel time integrals which requires different cut-off functions than in the Euclidian version.

In the second part of the talk we exemplify the procedure using a concrete example, namely Einstein-Hilbert-Klein-Gordon theory. We derive and solve the resulting Lorentzian flow equations in that truncation. We compute the flow of Newton’s constant and the cosmological constant and we find an attractive UV fixed point. A new feature of the Lorentzian flow is that the couplings become generically complex valued. However, only the k = 0 values of the couplings have physical meaning. We numerically prove the existence of so-called admissible trajectories. These 1. allow to integrate down the flow all the way to k = 0, 2. reach real valued dimensional couplings in the IR and 3. reach the, generically complex valued, UV fixed point of the dimensionless couplings.