The goal of this review has been to summarize the modern picture of quantum gravity, within which the perturbative non-renormalizability of general relativity is recognized as being a particular instance of a more general phenomena: the widespread application of non-renormalizable quantum field theories throughout many branches of physics. Regarding quantum gravity in this way shows how quantitative predictions can be made: One must simply apply the rules of effective field theories, which are known to give an accurate description of experiments in low-energy nuclear, particle, atomic, and condensed matter physics.
Thinking of general relativity as an effective theory in this way is not a new development, and underlies most approaches to quantum gravity either explicitly or implicitly. Neither is it new to calculate explicitly the behaviour of quantum fields in curved space (sometimes including the graviton). What is new (over the last few years) is proceeding beyond the qualitative statement that general relativity is an effective theory to obtain the quantitative next-to-leading predictions within a controlled semiclassical approximation. Although much of the mechanics of such calculations leans on experience obtained when calculating with quantum fields in curved space, the crucial new difference is the quantitative power-counting arguments which identify precisely which quantum effects contribute to any given order in small quantities.
What emerges from this summary is a snapshot of a work which is very much still in progress. The following loom large among the missing results:
There can be little doubt that quantum effects are extremely small in the classical systems for which gravitational measurements are possible (like the solar system), but this need not undermine the motivation for their computation. The point of such calculations is not their relevance for practical experiments (we wish!). Rather, their point is conceptual. It is only through the careful calculation of quantum effects that the theory of their size can be solidly established. In particular, any precise comparison between observations and the predictions of classical gravity is ultimately incomplete unless the quantitative size of the quantum corrections is explicitly established, as a systematic, all-orders power-counting argument would do.
Furthermore, we can always hope to get lucky, even if only theoretically. A clean understanding of how the size of quantum corrections depends on the variables (mass, size, separation, etc.) in a given system, one might hope to find larger-than-generic quantum phenomena in special systems. Even if these lie beyond the reach of present-day experimenters, they may furnish instructive theoretical laboratories within which differing approaches to quantum gravity might be more starkly compared.
In the last analysis, I hope the reader has become convinced of the utility of effective field theory techniques, and that the effective field theory point of view lifts the experimental triumphs of classical general relativity to precision tests of the leading-order implications of the quantum theory of gravity.
© Max Planck Society and the author(s)