There have been a number of heroic attempts to
quantize gravity along the lines of other field theories [81, 55, 6, 71, 138, 54, 70, 113, 50, 51, 53, 25, 38, 7, 8
, 9
, 10
, 14
, 16
, 15
, 11
, 12
, 13
, 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.