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.
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ