Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase model

In a recent contribution we identified possible points of contact between the asymptotically safe and canonical approach to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantised and one can construct the generating functional of time ordered Wightman (i.e. Feynman) or Schwinger distributions respectively from the corresponding time translation unitary group or contraction semigroup respectively as a path integral. For the unitary choice that path integral can be rewritten in terms of the Lorentzian Einstein Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantisation and a reduction term that encodes the reduction of the full phase space to the phase space of observavbles. This path integral can then be treated with the methods of asymptically safe quantum gravity in its {\it Lorentzian} version. We also exemplified the procedure using a concrete, minimalistic example namely Einstein-Klein-Gordon theory with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature which has strong impact on the convergence of ``heat'' kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.