Asymptotically safe -- canonical quantum gravity junction

TL;DR

This paper explores the conceptual connections between canonical quantum gravity and asymptotically safe quantum gravity, aiming to reconcile differences and foster communication between the two approaches using path integral and flow equation formalisms.

Contribution

It provides a framework to compare CQG and ASQG by focusing on their path integral formulations and effective actions, emphasizing conceptual rather than technical differences.

Findings

Path integral formulation links CQG and ASQG.

Effective actions can be analyzed via Wetterich flow equations.

Conceptual bridges are identified between the two approaches.

Abstract

The canonical (CQG) and asymptotically safe (ASQG) approach to quantum gravity share to be both non-perturbative programmes. However, apart from that they seem to differ in several aspects such as: 1. Signature: CQG is Lorentzian while ASQG is mostly Euclidian. 2. Background Independence (BI): CQG is manifesly BI while ASQG is apparently not. 3. Truncations: CQG is apparently free of truncations while ASQG makes heavy use of them. The purpose of the present work is to either overcome actual differences or to explain why apparent differences are actually absent. Thereby we intend to enhance the contact and communication between the two communities. The focus of this contribution is on conceptual issues rather than deep technical details such has high order truncations. On the other hand the paper tries to be self-contained in order to be useful to researchers from both communities. The point of contact is the path integral formulation of Lorentzian CQG in its reduced phase space formulation which yields the formal generating functional of physical (i.e. gauge invariant) either Schwinger or Feynman N-point functions for (relational) observables. The corresponding effective actions of these generating functionals can then be subjected to the ASQG Wetterich type flow equations which serve in particular to find the rigorous generating fuctionals via the inverse Legendre transform of the fixed pointed effective action.