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

S. Deser, Infinities in Quantum Gravities, http://arxiv.org/pdf/gr-qc/9911073v1

(and references [2,4] there). This is called 'nonrenormalizability', and is the main blemish of covariant quantum gravity. (For other potential problems, see, e.g., gr-qc/0108040.) Note that quantum gravity, though nonrenormalizable in the established sense, is renormalizable in a weak sense, where infinitely many counterterms are allowed; see

J. Gomis and S. Weinberg, Are Nonrenormalizable Gauge Theories Renormalizable? http://arxiv.org/pdf/hep-th/9510087.

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:

gr-qc/9803024: Strings, loops and others: a critical survey of the present approaches to quantum gravity

gr-qc/9710008: Loop quantum gravity http://relativity.livingreviews.org/Articles/lrr-1998-1/index.html

hep-th/9709062: Introduction to superstring theory astro-ph/0304507: Update on string theory hep-th/0311044: The nature and status of string theory physics/0605105: a short review of superstring theories

gr-qc/0410049 shows how gravity derives from string theory; a more complete derivation is in section 3.7 of Polchinski's book.

Phys. Rev. Lett. 60, 2105-2108 (1988) discusses the lack of Borel summability of the S-matrix expansion for the bosonic string.

http://math.ucr.edu/home/baez/week195.html tells about the state in 2003 concerning the claims of (super)string theory to be a renormalizable quantum theory. Only the 2 loop case seems to be settled; see arXiv:hep-th/0501197 and hep-th/0211111 (especially Section 14 of the latter for the unsolved problems at 3 loops and higher).

Others treat covariant quantum gravity just as they treat nonrenormalizable effective field theories, and fare well with it. See, for example,

C.P. Burgess, Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory, Living Reviews in Relativity 7 (2004), 5. http://www.livingreviews.org/lrr-2004-5

for 1-loop corrections, and

Donoghue, J.F., and Torma, T., Power counting of loop diagrams in general relativity, Phys. Rev. D 54 (1996), 4963-4972, http://arxiv.org/abs/hep-th/9602121

for higher-loop behavior. See also

http://arxiv.org/pdf/gr-qc/9512024
http://arxiv.org/pdf/0910.4110

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

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''

The paper

D.F. Litim Fixed Points of Quantum Gravity and the Renormalisation Group http://arxiv.org/pdf/0810.3675

says on p.2: ''. It remains an interesting and open challenge to prove, or falsify, that a consistent quantum theory of gravity cannot be accommodated for within the otherwise very successful framework of local quantum field theories.''

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.
Woodard writes in the introduction:

''All of the problems that had to be solved for flat space scattering theory in the mid 20th century are being re-examined, in particular, defining observables which are infrared finite, renormalizable (at least in the sense of low energy effective field theory) and in rough agreement with the way things are measured. [...] The transformation was forced upon us by the overwhelming data in support of inflationary cosmology.''

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.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)

A theoretical physics FAQ