Quantum and non-quantum anomalies in general relativity

Nuclear Physics B (Proc. Suppl,) 18B (1990) 300-301 North-Holland

300

Quantn_m and Non-Quantum Anomalies in General Relativity

Thomas Schiicker Institut fiirTheoretische Physik der Universit~it Heidelberg, Philosophenweg 16, D-6900 Heidelberg

The isometry group of a massless particle in general relativity is smaller than in special relativity. As in anomalous quantum field theories this symmetry reduction comes from a nece~ary short-distance regularization.

Many of us physicists have a tendency to think in

anomaly. The surprise here is that the isometries of a

terms of stubborn expectations. Perhaps this tendency

massless point-like particle in general relativity, G > 0,

helps understanding why so many physicist mountaineers

are a proper subgroup of those isometries in special rel-

suffer accidents.

It certainly explains the abundant

ativity, G = 0. As for quantum anomalies this symme-

use of the word anomaly in our literature, as for ex-

try breakdown is caused by a necessary short-distance

ample the high-y anomaly. In the following I concen-

regularization.

trate on those anomalies that reflect our surprise when

As warm up let us first consider the massive par-

symmetries of a classical theory do not survive in the

ticle where no such anomaly appears. In special rela-

more general theory. Such anomalies usually come with

tivity we consider the subgroup of the Poincard group

one or two other words of definite scientific consonance

leaving invariant the word line of a free massive par-

specifs, ing either

ticle. This subgroup is of course 0(3) × R with two--

- which symmetry is the victim, e.g. gauge anomaly,

-

dimensional space-like orbits under the rotation group

Lorentz anomaly, chiral anomaly,

an time-like orbits under the translation. This symme-

who is the culprit, e.g. chiral anomaly, triangle

try does not care whether the particle is point-like or

anornaly,

smeared out, i.e. has finite energy density. In general

- who was second or third to be surprised, e.g. AdlerBardeen anomaly, - in what type of theory the anomaly appears, e.g. Yang-Mills anomaly, gravitational anomaly.

relativity the gravitational field of a massive particle is the well-known Schwarzschild metric [1]. Its isometry group is again 0(3) x R. Note, however, that here the point-like limit fails to exist because of the horizon.

Finally, there is the expression quantum anomaly

Now let us examine the massless particle. In spe-

meaning that the symmetry was lost when going from

cial relativity the isometry group of the point-like par-

a classical, h = O, to a quantum field theory, h > O.

ticle is again four-dimensional, E2 x R, with two-dimen-

In contradistinction and to add to the already existing

sional space-like orbits under the Euclidean group

confusion I should like to present to you a gravitational

and light-like orbits under the translation.

0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-Ilolland)

But for

T. Sch~eker / Quantum and non-quantum anomalies

a massless particle with finite energy density the isom-

301

already in 1929 in a different context [4t:

etry group is reduced to the subgroup 0(2) x R due

dz 2 = - t ~ ( d v 2 + az 2) + dudv + l-du2 t/

to Lorentz contractions {2]. In general relativity the gravitational field of a massless particle was derived by

Its orbits under translations b'~ are light-like if and

Bonnor [3] in 1969. Suppose the particle flies on the

only if the integration constant I vankhes, i.e. in

positive z-axis with energy E. Then its metric is

space. At this point the referee will ask his sta~d~d ques-

d f 2 = - d x 2 - dy 2 + 2dudv + 2A(z, y, u)du ~

tion: So what? Gauge invaxi_ance of Y a a g - M ~ tkeoties in four dimensions ensures unitarity of t ~

where u and v are light cone coordinates:

turn theory. Quantum a n o ~ t--z

t+z "

qean-

therefore r~in the

consistency of these theode~ On the other h ~ d , the

o

symmetry reduction discussed here does not ~eem to

If the energy density of the particle is finite, say

E

have any dramatic consequence and shares ~

stalin

in a cylinder of length V~L and radius a and vanishes

of quantum anomalies of general relativity [51 and of

outside, then the function A is given by:

yang-Miils theories in higher dimensions [Oto~ Them

{

have been used extensivelyas sehction criteria ~ m#&e-

4GE

A(z, y, u)

:=

4

GE

- -

8 GE

111 r

O,

if

r<_a r>_a

andO
with G Newton's constant and r :-- V / ~ + y~. Miraculously, Bonnor's metric admits the pointlike limit L and a tending to zero independently. It

tic grounds.

In the same spirit we can ask which

alternative gravitationaltheories have fieldeffaatioRs allowing a fully symmetric point-likeparticlewithoat

mass?IS] I should like to dedicate this articleto Raymond Stora, w h o introduced m e to the ,beauty of mathematical physics.

yields an energy density E 6 ( z ) 6 ( y ) 6 ( z - t) and References

A ( z , Y, u) = 8-~_GE6(u) In r x/z

with p an arbitrary constant carrying the dimension of length. In both cases of finite and infinite energy the

[1] K. Schwarzschild, Sitzungsber., K. Preuss, Aka~ Wiss. 0916) 189.

isometry group of Bonnor's metric is only 0(2) x R. Or

[2] T. Schficker, Phys. Left. A136, (1989) 200.

in quantum field language: removing the cut-off does

[3] W.B. Bonnor, Commun. Math. Phys. 13, (1969)

not reinstate the symmetries lost through regulariza-

163. [4] G.C. McVittie, Proc. R. Soc. A, (1929) I24.

tion. In order to complete the analogy with quantum anomalies we must prove that general relativity does not admit a non-trivial vacuum solution with full Ez x R symmetry and orbits as indicated. Indeed, the most generM such solution [2] is the one that McVittie found

[5] L. Alvares-Gaum~, E. Witten, NucL Phys. B?a~4, (1984) 269. ]6] P.H. Frampton, T.W. Kephart, Phys. Rev. Left.

50, (1983) 1343. [7] J. Tolksdorf, work in progress.