Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. This will be reviewed in chapter 2 after a general overview in the introductory chapter 1. Then we discuss various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so–called Einstein–Hilbert truncation in chapter 3, where only the cosmological constant and Newton’s constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences. We discuss some possible phenomenological consequences leading to modified dispersion relations and show connections to phenomenological models where Lorentz invariance is either broken or deformed. We then apply the Renormalization Group to higher derivative gravity in chapter 4. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four–derivative terms, while Newton’s constant and the cosmological constant have a nontrivial fixed point. In the case of actions that are polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV–attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three–dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme–dependent and scheme–independent results, and provide several examples of the fact that only dimensionless couplings can have “universal” behavior.