Asymptotically safe canonical quantum gravity: Gaussian dust matter

In a recent series of publications we have started to investigate possible points of contact between the canonical (CQG) and the asymptotically safe (ASQG) approach to quantum gravity, despite the fact that the CQG approach is exclusively for Lorentzian signature gravity while the ASQG approach is mostly for Euclidean signature gravity. Expectedly, the simplest route is via the generating functional of time ordered N-point functions which requires a Lorentzian version of the Wetterich equation and heat kernel methods employed in ASQG. In the present contribution we consider gravity coupled to Gaussian dust matter. This is a generally covariant Lorentzian signature system, which can be considered as a field theoretical implementation of the idealisation of a congruence of collision free test observers in free fall, filling the universe. The field theory version correctly accounts for geometry -- matter backreaction and thus in principle serves as a dark matter model. Moreover, the intuitive geometric interpretation selects a preferred reference frame that allows to disentangle gauge degrees of freedom from observables. The CQG treatment of this theory has already been considered in the past. For this particular matter content it is possible to formulate the quantum field theory of observables as a non-linear $\sigma$ model described by a highly non-linear conservative Hamiltonian. This allows to apply techniques from Euclidean field theory to derive the generating functional of Schwinger N-point functions which can be treated with the standard Euclidean version of the heat kernel methods employed in ASQG. The corresponding Euclidean action is closely related to Euclidean signature gravity but not identical to it despite the fact that the underlying Hamiltonian is for Lorentzian signature gravity.