There have been a number of heroic attempts to quantize gravity along the lines of other field theories [80, 54, 9, 70, 137, 53, 69, 112, 50, 52, 25, 38, 10, 11, 12, 6, 16, 8, 7, 13, 14, 15, 18], and it was recognized early on that general relativity is not renormalizable. It is this technical problem of non-renormalizability which in practice has been the obstruction to performing quantum calculations with general relativity. As usually stated, the difficulty with non-renormalizable theories is that they are not predictive, since the obtention of well-defined predictions potentially requires an infinite number of divergent renormalizations.
It is not the main point of the present review to recap the techniques used when quantizing the gravitational field, nor to describe in detail its renormalizability. Rather, this review is intended to describe the modern picture of what renormalization means, and why non-renormalizable theories need not preclude making meaningful predictions. This point of view is now well-established in many areas – such as particle, nuclear, and condensed-matter physics – where non-renormalizable theories arise. In these other areas of physics predictions can be made with non-renormalizable theories (including quantum corrections) and the resulting predictions are well-verified experimentally. The key to making these predictions is to recognize that they must be made within the context of a low-energy expansion, in powers of (energy divided by some heavy scale intrinsic to the problem). Within the validity of this expansion theoretical predictions are under complete control.
The lesson for quantum gravity is clear: Non-renormalizability is not in itself an obstruction to performing predictive quantum calculations, provided the low-energy nature of these predictions in powers of , for some , is borne in mind. What plays the role of the heavy scale in the case of quantum gravity? It is tempting to identify this scale with the Planck mass , where (with denoting Newton’s constant), and in some circumstances this is the right choice. But as we shall see need not be , and for some applications might instead be the electron mass , or some other scale. One of the points of quantifying the size of quantum corrections is to identify more precisely what the important scales are for a given quantum-gravity application.
Once it is understood how to use non-renormalizable theories, the size of quantum effects can be quantified, and it becomes clear where the real problems of quantum gravity are pressing and where they are not. In particular, the low-energy expansion proves to be an extremely good approximation for all of the present experimental tests of gravity, making quantum corrections negligible for these tests. By contrast, the low-energy nature of quantum-gravity predictions implies that quantum effects are important where gravitational fields become very strong, such as inside black holes or near cosmological singularities. This is what makes the study of these situations so interesting: it is through their study that progress on the more fundamental issues of quantum gravity is likely to come.
© Max Planck Society and the author(s)