Abstract
This paper focuses on a phase transition from the asymptotic safety
approach of renormalizing the quantum gravity (QG) to a more granular
approach of the loop quantum gravity (LQG) and then merging it with the
Regge calculus for deriving the spin-(2) graviton. From loop-(2)
onwards, the higher derivative curvatures make the momentum go to
infinity which assaults a problem in renormalizing the QG. If the
Einstein-Hilbert (E-H) action, is computed, and a localized path
integral (or partition functions) is defined over a curved space, then
that action is shown to be associated with the higher order dimension in
a more compactified way, resulting in an infinite winding numbers being
accompanied through the exponentiality coefficients of the partition
integrals in the loop expansions of the second order term onwards. Based
on that localization principle, the entire path integral got collapsed
to discrete points that if corresponds the aforesaid actions, results in
negating the divergences’ with an implied bijections’ and reverse
bijections’ of a diffeormorphism of a continuous differentiable
functional domains. If those domains are being attributed to the spatial
constraints, Hamiltonian constraints and Master constraints then,
through Ashtekar’s variables, it can be modestly shown that the behavior
of quantum origin of asymptotic safety is similar to the LQG granules of
spinfoam spacetime. Then, we will proceed with the triangulation of the
entangled-points that results in the inclusion of Regge poles via the
quantum number (+2,-2,0) as the produced variables of the spin-(2)
graviton and spin-(0) dilaton.