We introduce the basic concepts and the main underlying philosophy of the Asymptotic Safe approach to Quantum Gravity.
A Hamiltonian Analysis of the Reuter-Weyer Renormalization Group (RG) improved action of the Einstein-Hilbert functional
is performed. The gravitational and cosmological constants as function of the
space-time coordinates are treated as external non-geometrical fields. Dirac's
constraint analysis is performed, in the general case, up to secondary
constraints. The constraints are second class and, in general, the problem
appears to be technically complicated. This fact suggests studying the Dirac's
constraint analysis of the related Brans-Dicke theory, which shows the Poisson
Brackets between Hamiltonian-Hamiltonian constraints do not close.
The equivalence between the Jordan frame and the Einstein frame is introduced and discussed in light of this last result.
A simplified FLRW minisuperspace model based on the RG improved Einstein Hilbert action
contains Bouncing and Emergent Universes for values of K=-1, 0, 1.