The one-loop divergences of coupled general relativity and electrodynamics cannot be absorbed by renormalization

Deser, S.

van Nieuwenhuizen, P.

Caltech Authors - Main

VOLUME 32, NUMBER 5 PHYSICAL REVIEW LETTERS 4 FEBRUARY 1974
form A =H(a/M)(o —v, ) ", and again the equation
g(a/M, H(a/M)(o —o,), v) =0 has a continuous
solution for nearby a/M. Thus no trajectory can
have an end point on the positive imaginary axis
(except possibly at a/M = 1) and the eigenfrequen-
cy of any unstable mode must have passed through
o =0.
Finally, we observe that Carter's theorem ex-
cludes such modes. That is, Carter proves that
there are no stationary axisymmetric perturba-
tions of Kerr. But if h,.~(o)e ' ' are a family of
solutions to the time-dependent field equations,
we can show tha, t h, , (v= 0) is a, solution to the
time-independent field equations as well. We con-
clude that there are no unstable axisymmetric
modes of the Kerr geometry exterior to a black
hole.
We are indebted to James Bardeen, James
Hartle, James Ipser, Charles Sommerfield, and
Hafael Sorkin for helpful suggestions.
*Work supported in part by the National Science Foun-
dation under Contract No. NSF GP 86317.
~S. W. Hawking, Commun. Math. Phys. 25, 152 (1972).
B. Carter, Phys. Rev. Lett, 26, 831 (1971).
3W. H. Press and S. A. Teukolsky, Astrophys. J. 185,
649 (197').
S. A. Teukolsky, Astrophys. J. 185, 635 (1978).
R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8,
265 (1967).
S. Chandrasekhar and J. L. Friedman, Astrophys. J.
177, 745 (1972).
This gauge choice is analogous to a coordinate condi-
tion used to the same end in L. D. Landau and E. M. Lif-
shitz, The Classica/ Theory of Fields (Addison-Wesley,
Reading, Mass„1965), p. 98, and brought to our atten-
tion by Bafael Sorkin.
C. V. Vishveshwara, Phys. Bev. D 1, 2870 (1970).
V. de Alfaro and T. Begge, Potential Scattering
(Wiley, New York, 1965).
Nonrenormalizability of the Quantized Einstein-Maxwell System*
S. Deser and P. van Nieuwenhuizen
Department of Physics, Brandeis University, TValtham, Massachusetts 02154
(Received 7 December 1978)
The one-loop divergences of coupled general relativity and electrodynamics cannot be
absorbed by renormalization.
Although quantization of general relativity has
been discussed by many authors, it is only quite
recently that the first complete calculation, at
the one-loop level, of divergences due to gravi-
tons has been supplied by 't Hooft and Veltman. '
These authors used dimensional regularization
and a general algorithm, 2 based on the background-
field method, ~ to obtain the one-loop counterterms.
In the case of pure gravitation, the divergent
terms could be absorbed by a field renormaliza-
tion; however, this was no longer possible upon
coupling to a quantized scalar field.
While the scalar example is discouraging, one
might hope that more realistic matter sources
would keep one-loop renormalizability. We re-
port here that this is not the case for the Einstein-
Maxwell system. We also give some results for
(1) Brans-Dicke theory (nonrenormalizable ex-
cept for the singular case ~= ——,' which is equiva-
lent to Einstein theory), (2) fermion loops, and
(3) pure gravitation with a cosmological term
(formally renormalizable).
In the coupled Einstein-Maxwell Lagrangian
L—= I E+L
= —( g)'"[» '&(-g)+ '&„.I',.g"'g"-'], (1)
the fields (g&„, I'„,-=&„A,—&„A„)are written as
sums of background fields (g», F„,) and quantum
fields (~h„„f„,). The Lagrangian being invari-
ant under gravitational and electromagnetic gauge
transformations of the quantum fields h&, and f„„
we add to L the gauge-breaking terms
LEB+ LMg
= --,'v'-g[(D"I „,- -,' D&.")' +( DW„) ]2, (2)
where all tensor operations, including covariant
differentiation D, are with respect to the back-
ground metric g&, . The above choice corresponds
to the usual de Donder (harmonic) and Lorentz
gauges, which in turn give rise to a vector and
a scalar ghost with Lagrangian
g[q*"(g„sD~Dy-—R„s)its+ y+D D~y]. (3)
VOr. IJME 32, NUMBER 5 PHYSICAL RKVIKW LETTERS 4 FEBRUARY 1974
Note that the usually uncoupled electrodynamical
ghost contributes here as a result of the metric
dependence of I M~. After the Lagrangian I.+ I-~
+ I.
~
is expanded in the quantum fields about the
background, the algorithm determines the one-
loop counter Lagrangian from the coefficients of
the terms bilinear in the quantum fields.
Before giving explicit results, we note that di-
mensional regularization has the virtues of avoid-
ing the usual momentum cutoff in favor of a log-
arithmically divergent dimensionless parameter
(1/e) and of eliminating tadpoles. It follows that
in our system, where there is only one dimen-
sional constant, ~, there will be no divergences
proportional either to I. itself or to a cosmolog-
ical term v 'v'-g. This means that there is no
coupling-constant renormalization.
We now list the possible gauge-invariant ingre-
dients from which the counter Lagrangian b I. is
to be constructed, namely, R, 8", F&„and D .
Both E and D must necessarily appear to an even
power (by the continuity of photon lines for E and
because the other building blocks have an even
number of indices for D). Dimension counting
requires 41. to be bilinear. in R, &", a'F„,F 8,
and D„D„. Since we are not interested in total
divergences, the only explicit D dependence is
of the form (D Egy)', (D Egg)(DATE/„), or (D"E„,)'.
The former two can be eliminated in favor of the
latter by using cyclic identities on D F&& and com-
mutator identities linking [D&, D,] to the curva-
ture. The a priori possible scalars are then the
following (absorbing ~ in E):
b,I.= (v' —g/~) (c,R„,'+ c,R'+ c, (RB ~")'+c,[TrE' ——,'(E2)']+ c,(E2)'+ c,R "~&'E BE',
+ c,R"'(E„yE„&—~ g„,E') + c,RE'+ c,(D"E„„)'j.
&I- = (v'- g/e) (137/60)R „„' (7)
and the theory is not one-loop renormalizable.
Since the coupling in question presumably does
exist in nature as a fundamental one, we are
faced with the alternatives that (a) perturbative
renormalizability is not a relevant criterion,
(b) general relativity must be replaced by a "bet-
ter" theory for quantizing, or (c) photons are not
"fundamental" fields. Possibility (a) is not very
The cs term can be eliminated in favor of (c„c,)
because of the identity4'
R 8„,'=(4R„„'-R')+a divergence
which holds in four-dimensional Riemann space.
Explicit calculation shows that because of mi-
raculous (?) cancelations, c,=c, =c, =0, which
means that F&, only appears in the combination
—
~ gq, E' and (D"E„„)', and the full
curvature 8, 8" is absent. The remaining coef-
ficients have the values c, = 9/20, c, = —29/10,
c, = 13/24, and e, = c, = 1/6. Since we are only
interested in the renormalizability question, we
may insert the field equation obeyed by the un-
perturbed background fields, namely,
1
Pv 2~Pvf D FPv OP
if AJ does not then vanish, the divergences are
not renormalizable. 2 There is effectively only
one term left since modulo the field equations
8»', T»', and R»T"'are equivalent, and 8=0.
The final result is the nonvanishing one-loop Ein-
stein-Maxwell counter Lagrangian
fruitful at our present level of understanding.
With respect to possibility (b), one hope might
lie in a Brans-Dicke-type theory, since by re-
placing e 'by a scalar field p one might avoid
the traditional nonrenormalizability of theories
with dimensional coupling constants. ' However,
direct examination of this class of theories shows
that they can always be cast, by suitable trans-
formations g -gy", p -f(p), into the unrenormal-
izable scalar-graviton system. The only excep-
tion is the singular case v = —2, which is equiv-
alent to Einstein theory. These remarks do not
rule out, perhaps, a deeper use of the Brans-
Dicke models in which the Higgs phenomenon for
p might be suitably exploited. With respect to
possibility (c), one should consider fermion cou-
plings, since presumably fermions at least are
necessary as fundamental fields. We have com-
puted the divergent part of the fermion loop con-
tribution to the graviton propagator, which for
m=0 gives a term of the form'
(6)
and the complete EI. is under investigation. Final-
ly, we consider pure gravitation with a cosmolog-
ical term, Ax 'v'-g, which involves a new dimen-
sional constant ~. A formal one-loop calculation
leads to a 4t. which may be written as propor-
tional equivalently to v'-g or R, and hence ab-
sorbed by renormalization of &/d. This is, how-
ever, not a very physical quantum theory, since
in it; the h field corresponds to a tensor-scalar
VOLUME 32, NUMBER 5 PHYSICAL RKVIEW LETTERS 4 FEBRUARY 1974
ghost admixture.
In a general vein, we remark that coupling sev-
eral matter fields in the conventional way is un-
likely to lead to miraculous cancelations, since
all R»' terms obtained so far have the same
sign. Also, if one starts with an original gravi-
tational Lagrangian with R&,' terms to absorb
the unrenormalizable AI., these terms wiII in
turn generate higher powers of R at the one-loop
level (in addition to implying a very unpleasant
higher-order equation for the quantum fields h&,),
so that a Lagrangian which is a finite polynomial
in the curvature is unlikeIy to help.
As for two-loop calculations, they present enor-
mous technical complications. It would be most
important if one could at least discover whether
pure gravity involves terms hL- Tr[(R„&")']
there, since these are presumably a sign of non-
renormalizability.
Detailed results wi11 be published elsewhere.
Note added. —The coupling of fermions to grav-
itation has now been analyzed; the one-loop diver-
gences include a, term proportional to (+,y g)'.
Hence this system is also nonrenormalizable.
*Work supported in part by the U.S. Air Force Office
of Scientific Research, the National Science Foundation,
and the Nederlands Ministerie van Onderwijs en Weten-
schappen.
~G. 't Hooft and M. Veltman, CERN Report No. TH
1728, 1978 (unpublished).
G. 't Hooft, CEHN Report No. TH 1692, 1978 (unpub-
lished) .
B. S. DeWitt, Phys. Rev. 162, 1195, 1289 (1967),
and in Relativity, Groups and Topology (Gordon and
Breach, New York, 1964).
4B„Bach, Math. Z. 9, 110 (1921); C. Lanczos, Ann.
Math. 39, 842 (1938).
The one-loop renormalizability of pure gravitation is
an immediate consequence of this identity, the generic
form DL= [(-g)' '/ej(n &R» + n2R + n&R~»~), and
the field equations 8» =0.
6This idea has also been proposed by S. Weinberg,
private communication.
VScale invariance of massless fields of spin +&, 1 nec-
essarily leads to this combination of curvatures, which
is just the square of the (scale-invariant) Weyl tensor
C„~s". Indeed, the pure photon contribution to Eq. (4)
is just 4 times EL of Eq. (8), which result was also
found by D. M. Capper, M. J. Duff, and L. Halpern, In-
ternational Centre for Theoretical Physics Report No.
IC/72/190, 1972 (unpublished).
An initial photon-graviton Lagrangian of the Weyl
type (purely quadratic in R) does not involve dimension-
al constants and so is reproduced by AL; however, the
graviton propagator behaves as P 4.
Observation of Structure in pp and pd Total Cross Sections below 1.1 GeV/ce
A. S. Carroll, I-H. Chiang, T. F. Kycia, K. K. Li,
P. 0. Mazur, D. N. Michael, 'f P. Mockett, f. D. C. Rahm, and R. Rubinstein
Bxookhaven National Laboratory, Upton, Nese Fork ll 973
(Received 1 November 1978)
Total cross sections of PP and Pd have been mea, sured between 260 and 1060 Mev/c,
with high statistical precision. Structures are observed in both cross sections at about
the same momenta. For pp, the central mass is 1922+2 MeV/c2, and a fit to the data
with a simple Brett-Wigner resonance plus background gives & =9+4 MeV/c2. The data
suggest that the structures are in the isospin-1 state.
There have been many claims' "of narrow
meson resonances with masses in the range 1900
to 2000 MeV/c'; the observed widths varied from
-10 to 200 MeV/c'. However, many other simi-
lar experiments have not observed such reso-
nances and others have directly challenged the
basis of such claims. " -" Searches for meson
resonances in this mass region have been made
in missing-mass experiments, multipion mass
spectra, and pp-formation experiments. A re-
view of the subject is given by Diebold, ' where
the conclusion is that no narrow resonances in
this mass region have been confirmed.
Total-cross-section measurements have in
the past been very successful in searching for
new resonances, because even though they are
not intrinsically sensitive, they can be carried
out to a very high statistical accuracy and with
small point-to-point systematic errors. In an
attempt to clarify the problem of the possible
existence of meson resonances in the mass re-
gion around 1950 MeV/c, we have measured
Pp and P'd total cross sections using antiprotons
of momenta 360 to 1050 MeV/c (center-of-mass
247


Nonrenormalizability of the Quantized Einstein-Maxwell System

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Nonrenormalizability of the Quantized Einstein-Maxwell System

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