# Some Aspects of the Canonical Analysis of Reuter-Weyer RG Improved Einstein-Hilbert Action

## TL;DR

This paper performs a canonical analysis of the RG improved Einstein-Hilbert action, revealing complex constraint structures and differences from standard General Relativity, especially when compared to Brans-Dicke theory.

## Contribution

It provides a detailed Dirac constraint analysis of the RG improved Einstein-Hilbert action, highlighting its unique constraint algebra and its distinction from Brans-Dicke theory.

## Key findings

- Constraints are second class and complex.
- The constraint algebra differs from Einstein's geometrodynamics.
- Brans-Dicke theory is inequivalent to Einstein's General Relativity.

## 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. It exhibits a Dirac's constraint algebra similar to Einstein's geometrodynamics except that the Poisson Brackets between Hamiltonian-Hamiltonian constraints is not only linear combination of the momentum constraints but also 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.