Renormalization in quantum gravity
Renormalization of QFTs is needed to make the coefficients in the loop expansion (i.e., the expansion in powers of Planck's number hbar) of the S-matrix well-defined.
Canonical quantum gravity is the theory obtained by writing down the Einstein-Hilbert action in a (3+1)-dimensional splitting (ADM formalism) and either fixing coordinates and solving the constraints (reduced phase space quantization) or quantizing using Dirac's approach to constrained systems (Dirac quantization).
Covariant quantum gravity is the theory obtained as follows: Write down the classical Hilbert action for general relativity, look at the corresponding functional integral defined perturbatively as for QED or QCD, and try to compute S-matrix elements using the usual renormalization prescriptions for the integrals corresponding to the various Feynman diagrams.
Quantum field theories are nowadays almost always defined in the covariant way; the covariant approach has the advantage of being manifestly invariant under the full symmetry group. (The canonical approach to scalar QED fails in certain versions to preserve Poincar'e symmetries, due to term ordering problems; see gr-qc/9403065.) On the other hand, the canonical approach is intrinsically nonperturbative, while the covariant approach needs extra tricks (renormalization group enhancements) to get partial nonperturbative results.
Covariant quantum gravity only works in the traditional way up to 1 loop (and together with matter not even then); at higher loops (i.e., for corrections of higher order in the Planck constant hbar) one needs more and more counterterms to make the resulting combination of integrals finite. See
Most researchers in quantum gravity want a renormalizable theory in the strong sense (so that finitely many counterterms suffice); then covariant quantum gravity is out, and people look for fancy alternatives (loop quantum gravity, superstring theory, etc.). However, these theories have their own difficulties. Some online references are:
Others treat covariant quantum gravity just as they treat nonrenormalizable effective field theories, and fare well with it. See, for example,
Section 4.1 of the paper by Burgess discussed recent computational
studies showing that covariant quantum gravity regarded as an effective
field theory predicts quantitative leading quantum corrections to the
Schwarzschild, Kerr-Newman, and Reisner-Nordstroem metrics.
Only a few new parameters arise at each loop order, in particular only
one (the coefficient of curvature^2) at one loop.
In particular, at one loop, Newton's constant of gravitation becomes
a running coupling constant with
The paper
My bet is that the canonical approach will win the race!
There is a recent survey of canonical quantum gravity and its
confrontation with exciting experimental data:
R.P. Woodard, Perturbative Quantum Gravity Comes of Age,
Int. J. Modern Physics D 23 (2014), 1430020.
See also the blog
Effective field theory treatment of gravity (very recommendable),
two blog posts by Jacques Distler, from
September 1, 2005 and from
April 26, 2007.
The discussion following
this thread in PhysicsForums
contains some interesting updates from 2016. See also the discussion
starting
here.
G(r) = G - 167/30pi G^2/r^2 + ...
in terms of a renormalization length scale r.
Here is a quote from Section 4.1:
''Numerically, the quantum corrections are so miniscule as to be
unobservable within the solar system for the forseeable future.
Clearly the quantum-gravitational correction is numerically extremely
small when evaluated for garden-variety gravitational fields in the
solar system, and would remain so right down to the event horizon even
if the sun were a black hole. At face value it is only for separations
comparable to the Planck length that quantum gravity effects become
important. To the extent that these estimates carry over to quantum
effects right down to the event horizon on curved black hole
geometries (more about this below) this makes quantum corrections
irrelevant for physics outside of the event horizon, unless the
black hole mass is as small as the Planck mass''
Woodard writes in the introduction: