Asymptotic Safety-30 Years After - 2009

In Weinberg’s asymptotic safety approach to quantum gravity, one has a finite dimensional critical surface for a UV stable fixed point to generate a theory of quantum gravity with a finite number of physical parameters. The task is to demonstrate how this fixed point behavior actually arises. We argue that, in a recently formulated extension of Feynman’s original formulation of the theory, which we have called resummed quantum gravity, we recover this fixed-point UV behavior from an exact re-arrangement of the respective perturbative series. We argue that the results we obtain are consistent both with the exact field space Wilsonian renormalization group results of Reuter and Bonanno and with recent Hopf-algebraic Dyson-Schwinger renormalization theory results of Kreimer. We calculate the first "first principles" predictions of the respective dimensionless gravitational and cosmological constants and argue that they support the Planck scale cosmology advocated by Bonanno and Reuter as well. Comments on the prospects for actually predicting the currently observed value of the cosmological constant are also given.

The fixed point structure of the renormalization flow in Einstein gravity and higher derivative gravity is investigated in terms of the background effective action. Using a covariant operator cutoff that keeps track of powerlike divergences and the transversal-traceless decomposition a construction is proposed that renders the {\it regularized} one-loop effective action gauge independent on-shell. In combination with a `Wilsonian' matching condition nontrivial strictly positive fixed points for the dimensionless Newton constant $g$ and the cosmological constant $\lambda$ can then be identified already in one loop perturbation theory. The renormalization flow is asymptotically safe with respect to the nontrivial fixed points in both cases. In Einstein gravity a residual gauge dependence of the fixed points is unavoidable while in higher derivative gravity both the fixed point and the flow equations are universal. Along this flow spectral positivity of the Hessians can be satisfied, evading the traditional positivity problems. Dependence on $O(10)$ initial data is erased to accuracy $10^{-5}$ after $O(10)$ units of the renormalization mass scale and the flow settles on a $\lambda(g)$ orbit.