Some aspects of the canonical analysis of Reuter-Weyer RG improved Einstein-Hilbert action

S J Gionti Gabriele

doi:10.1088/1742-6596/1275/1/012048

Abstract

A canonical analysis of 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.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

Highlights

Einstein General Relativity appears to be a successful phenomenological theory at laboratory, solar system, galactic and in general at distances bigger than the Planck length l >> lP l ≡ √1 ≈ 10−33cm
This fact suggests studying the Dirac’s constraint analysis of the related Brans-Dicke theory. It exhibits a Dirac’s constraint algebra similar to Einstein’s geometrodynamics except that the Poisson Brackets between Hamiltonian-Hamiltonian constraints is linear combination of the momentum constraints and of a term note reducible to linear combination of the constraint and proportional to the extrinsic curvature. This shows that Branse-Dicke geometrodynamics is inequivalent to Einstein General Relativity geometrodynamics
A simplified Friedman Lemaitre Robertson Walker (FLRW) minisuperspace model based on the RG improved Einstein Hilbert action contains Bouncing and Emergent Universes for values of K = −1, 0, 1

Summary

Introduction

Einstein General Relativity appears to be a successful phenomenological theory at laboratory, solar system, galactic and in general at distances bigger than the Planck length l >> lP l ≡ √1 ≈ 10−33cm. G to massive bodies [1], and the recent detection of the Gravitational Waves [2], attest Einstein General Relativity is a sound classical theory. As it is even known at the popular level [3], Einstein General Relativity has an initial singularity. Stephen Weinberg [7] proposed the Asymptotic Safety conjecture He suggested that Einstein General Relativity might be defined non-perturbatively at the non-Gaussian fixed point. He himself proved that NGFP exists in 2+ dimensions [7]. There is strong evidence that the fixed point exists in the exact theory as well

Renormalization Group approach

Conclusions