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 conceptual problems 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.
The conventional approach to quantum gravity is to leave the framework of local quantum field theory; examples of this kind are string theory and loop quantum gravity. A different logical possibility, explored by our group, is to stay within the framework of quantum field theory but to go beyond perturbation theory in an essential way. Wilson’s generalized theory of renormalization amounts to solving a field theory or 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 “effective average action”.
Approximate nonperturbative solutions of the flow equation for the effective average action have been obtained which possess a non-Gaussian fixed point, a simultaneous zero of all beta-functions. If this fixed point also exists in the full theory, then a mathematically consistent quantum field theory of gravity, Quantum Einstein Gravity or “QEG”, can be defined nonperturbatively. It could be predictive at arbitrarily small distances. Such a scenario would have potential implications for the cosmology of the very early universe and the structure of black holes, perhaps also for the cosmology of the present universe and the dark-matter problem. There are first results indicating that the spacetimes of QEG have a fractal structure with a scale-dependent (spectral) dimension which equals 4 on macroscopic, and 2 on microscopic length scales.
A comprehensive 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.