Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity

Abstract

These lecture notes give an introduction to the functional renormalisation group and asymptotically safe quantum gravity. We cover the basics of generating functionals in quantum field theories and derive the Wetterich equation. The anharmonic oscillator is presented as a simple example and compared to perturbation theory with resummation methods. A special focus is set on the implementation of symmetries in the functional renormalisation group. The second half of these notes is dedicated to quantum gravity, explaining the failure of perturbative gravity and how these issues are addressed in the non-perturbative approach. After detailing the implementation of the diffeomorphism symmetry, we present a simple computation in the Einstein-Hilbert truncation and give a short outlook on the state-of-the-art in the field. These notes are based on lectures given at the XV Modave Summer School in Mathematical Physics.

Finite Quantum Gravity Amplitudes: No Strings Attached

“…A first principle computation deriving the pole structure underlying this work from the Wetterich equation for gravity [48] then requires the full momentum dependence of the renormalized graviton propagator and low-order vertices. The form factor program [49][50][51] and the momentum dependence studied within the vertex expansion [52][53][54][55][56][57][58] constitute first steps in this direction. Our results then provide an important proof of principle that the quantum effective action comes with sufficient freedom to accommodate the physics requirements of asymptotic safety.…”

Graviton-mediated scattering amplitudes from the quantum effective action

“…Therefore, it is commonly accepted that GR is an effective field theory valid up to the Planck scale M Pl 10 19 GeV [16][17][18][19]. By now, there is a significant number of proposals on how to complete GR into a fundamental theory valid on all scales with String Theory [20,21], Loop Quantum Gravity [22][23][24], Asymptotic Safety [25][26][27][28][29][30][31], Causal Dynamical Triangulations [32,33], Causal Set Theory [34,35], Group Field Theory [36,37], infinite derivative gravity [38][39][40][41], and weakly non-local gravity [42] constituting only some selected examples.…”

The phase diagram of the multi-matrix model with ABAB interaction from functional renormalization

“…At its core lies the implementation of the Wilsonian idea of a coarse graining operation which progressively eliminates short scale fluctuations. Indeed, all explicit calculations within truncated RG flows find evidence for the existence of such a fixed point, defining the Reuter universality class, providing compelling indications for asymptotic safety in Euclidean gravity, see, e.g., [47][48][49][50][51][52][53] for recent reviews and introductions. Rephrased in a lattice-like language, such a universality class enables one to take a universal continuum limit.…”