**Asymptotic Safety: Renormalization of Quantum Einstein Gravity**

As the quantization of the Yang-Mills theories underlying the standard model of the electromagnetic, weak, and strong interactions is well understood by now, it remains the challenge of constructing a corresponding theory of Nature’s fourth fundamental interaction, namely gravity. Since the gravitational field is closely linked to the structure of spacetime, any attempt at quantizing gravity has to overcome a number of conceptual problems which are not present in conventional quantum field theories. Indeed, a straightforward quantization of General Relativity leads to a perturbatively nonrenormalizable theory with very limited predictive power.

Conventional approaches to quantum gravity typically leave the framework of local Quantum Field Theory; well known examples of this kind include string theory and loop quantum gravity. A different logical possibility, originally hypothesized by S. Weinberg, and explored in our group, is to stay within the framework of Quantum Field Theory but to go beyond perturbation theory in an essential way.

The underlying conjecture about gravity's "Asymptotic Safety" can be formulated abstractly on the basis of K. Wilson's generalized theory of renormalization. Within the latter, "renormalization" amounts to solving a quantum field theory or a statistical-mechanics system by a systematic coarse graining of the dynamics, i.e., by integrating out fluctuations of continuously increasing wavelength.

As a specific implementation of this idea, applicable to continuum systems, we use the method of the Functional Renormalization Group Equations. It leads to a mathematically concrete formulation of Weinberg's idea, thus allowing for an explicit exploration and assessment of the Asymptotic Safety conjecture, i.e., the renormalizability of gravity at the exact (non-perturbative) level.

In the late 1990s, our group obtained the first approximate, yet nonperturbative renormalization group flows for gravity. Remarkably, they were found to possess a non-Gaussian fixed point, the very hallmark of Asymptotic Safety. If this fixed point also exists in the full theory, and all available evidence does indeed point in this direction, then a mathematically consistent quantum field theory of gravity, "Quantum Einstein Gravity" or “QEG”, can be defined nonperturbatively. It would be predictive at arbitrarily small distances even.

Such a scenario has far reaching implications for the cosmology of the very early universe and the structure of black holes, for instance, and perhaps also for the cosmology of the present universe and the dark-matter problem. We also obtained first results indicating that the spacetimes of QEG have a fractal structure, displaying a scale-dependent (spectral) dimension that equals 4 on macroscopic, and 2 on microscopic length scales.

In the meantime, our original results have been confirmed and considerably extended, also by many other research groups.

An introduction to the Asymptotic Safety scenario in Quantum Einstein Gravity can be found in

M. Niedermaier and M. Reuter: Living Rev. Relativity 9 (2006) 5.

For a comprehensive monograph on this topic we refer to:

M. Reuter and F. Saueressig, Quantum Gravity and the Functional Renormalization Group:

The Road towards Asymptotic Safety, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2019.

For further details see also my corresponding publications in INSPIRE.