- Email: [email protected]
On the renormalization of quantum gravitation without matter
ANNALS
OF PHYSICS
104,
197-217 (1977)
On the Renormalization
of Quantum
Gravitation
Without
Matter*
P. VAN NIEUWENHUIZEN Institute
for Theoretical Stony Brook,
Physics, State University Long Island, New York
qf New York, I1794
ReceivedJune9. 1976
TheS-matrixof puregravitationisfinite at the one-looplevel.but in a trivial way dueto invarianceconsiderations which only hold in four dimensions. Theseconsiderations are not sufficientto concludethat the S-matrix is alsofinite at the two-looplevel.In order to determinewhetherthereis anylikelihoodthat extra cancellations will alsomakethe higher looppurelygravitationalS-matrixfinite, weconsidera casewhichhasseveralof thepropertiesof a two-loopcorrectionin four dimensions, a one-loopcorrectionin six dimensions. Thereare no extra cancellations and it is arguedthat puregravitationin four dimensions hasprobablyno finite S-matrixbeyondthe one-looplevel whenit is treatedasa standard particle theory.
I. INTRODUCTION
One of the fundamental problems in theoretical physics is the quantization of gravitation. The belief in quantum gauge field theory in general is strongly supported by the excellent agreement between theory and experiment in quantum electrodynamics. Since gravitation is the gauge field theory for masslessself-interacting spin two bosons with natural parity, it is tempting to apply the techniques of quantum electrodynamics to gravitation. One obtains in this way a quantum theory for gravitation which, however. is not unitary. This is a feature of non-Abelian gauge field theories, also present in Yang-Mills theories, and it is possible to find modified covariant quantization rules, either by using the path-integral formalism or by working directly with diagrams, which yield a unitary quantum field theory. In the case of Yang-Mills theories the resulting quantum theory is even renormalizdble and realistic models unifying the weak and electromagnetic interactions have been constructed. When one wants to apply the techniques of quantum Yang-Mills theory to gravitation, one must first decompose the metric into a background field (usually the flat spacemetric a,, although in this article it will be a general metric glLysatisfying the Einstein equations) and a deviation /?,“, and then treat h,,, as a normal quantum field, on the same footing as, say, the electromagnetic field A, . Admittedly, this may be a doubtful procedure in view of the close connection between the full metric *Work supportedin part by NSF Grant No. MPS-74-12208 AOl. 197 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSIi
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and the causal structure of spacetime which is needed for canonical quantization in order to define the basic commutation rules. However, it is not clear which, if any, the causal problems are when one quantizes covariantly in a path integral formalism. Consequently, we will forget for the time being that gravitons have something to do with clocks and yardsticks and build Feynman diagrams with gravitons represented by quantum operators h,, . This yields a standard particle physics theory, and investigations into the renormalizability properties of this quantum field theory have given the following results: (1) Pure gravitation, i.e., gravitation without matter sources, is not renormalizable in the sense that one can absorb the divergences in the Green’s functions by resealing of the physical parameters of the theory. However, at the one-loop level the S-matrix for any N-graviton process is finite [I] because the divergences in the N-graviton Green’s functions are polynomials in the external fields which vanish when the external gravitons have physical momenta and polarizations. (2) Coupling gravitation to such matter fields as scalars [l], photons [2], fermions [3], or Yang-Mills bosons [4], leads to a nonrenormalizable theory, as in the case of pure gravitation; however, in this case also the S-matrix diverges. Combining quantum electrodynamics with gravitation leads to a system where gravitons, photons, and fermions all couple to each other; this system turns out to be no better [5] than the coupling of gravitons to either photons or fermions. (3) When gravitons do not propagate inside loops, but are merely external classical fields, then quantum field theories which are renormalizable in flat space, stay so (to any loop order) in curved space [6]. Clearly, the only positive result in the direction of a well-defined true quantum theory of gravitation is the case of pure gravity. However, the opinions differ as to the significance of this result. Since one may show [7], using invariance arguments, that the sum of the one-loop divergences of all N-graviton S-matrices, summed over N, must be proportional to R,,R@” and R2, and since, according to the Einstein equations, R,, = R = 0 for pure gravity, some people find the finiteness of the purely gravitational S-matrix as uninteresting as the vanishing of the Dirac action (or the gravitational action for that matter) on-shell. Others believe that this finiteness is just a glimpse of the high symmetry in gravitation which heralds good news for the higher loop quantum corrections in pure gravitation. One obvious calculation which would determine whether there are extra cancellations in higher loop corrections beyond the fortuitious one-loop result is the calculation of the divergences in the S-matrix at the two-loop level. Whereas this calculation should certainly be performed, it is exceedingly hard and requires strong optimism about the final answer to sustain one in the algebraic nightmare. Instead, one might look for a calculation which is on the one hand much simpler than a two-loop calculation, but which, on the other hand, will give a more meaningful result than the one-loop case of pure gravitation in four dimensions. Such a calculation we present here: We calculate the one-loop divergences of the purely gravitational
QUANTUM
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199
S-matrix in six dimensions. As we shall show, the sum of all one-loop divergences of all N-graviton S-matrices, summed over N, is given in six dimensions by the simple expression S(div) = OL* d6x g1’2[R~yuoR(iPiURazY~](~- 6)-l. J
The factor (0 - 6)-l regularizes the divergences according to the dimensional regularization scheme, and since this nonrenormalizable divergence does not vanish automatically when R,, = R = 0, the unambiguous criterion for renormalizability is the vanishing of the coefficient (Y. In the Conclusions we argue why we believe that this calculation tells us something about two-loop finiteness of the S-matrix in four dimensions. In the next section we present the layout of this rather involved calculation, in the hope that this will give the casual reader enough information to go directly from there to the conclusions. In between, many technical problems are solved. Some of the results obtained seem to be of relevance for a future two-loop calculation in four dimensions. As already mentioned, we calculate here the sum of the divergences of all S-matrices, denoted above by S(div), because this is, paradoxically, much simpler than the sum of the divergences of a particular S-matrix. However, one can obtain the divergences of an N-graviton S-matrix by expanding S(div) into h,, and collecting terms of order h N. The formalism within which these results are obtained is an alternative to normal field theory, called the background field formalism [8]. This formalism is entirely equivalent to normal field theory, by which we mean that both give the same S-matrix [9] (but of course different Green’s functions in general). In fact, there exist calculations of particular examples which verify this equivalence explicitly [lo]. The advantages of using the background formalism are twofold. First one need not calculate the one-particle reducible diagrams in the background field method but one simply calculates loop diagrams constructed from vertices of the form h,&&gUy) h,, . The fields h are the quantum fields; they propagate in the loop and their gauge freedom must be fixed by adding a gauge fixing term. The function F is a function of the external (classical) fields g,, . By requiring that g,, satisfy R,,(g) = 0, one can replace diagrammatically each field g,“(x) by the sum of all trees which start at x and which have at their endpoints physical momenta and polorizations. In this way one calculates at once the sum of all S-matrices while only evaluating one particle irreducible diagrams. The second advantage of using the background field method is that in gauge theories one can choose the quantum gauge fixing term such that it does not break the gauge invariance in the external nonquantum fields, so that the computed counter terms, being (local) functions of the external fields only, are automatically gauge invariant objects. This reduces the form of the counter terms enormously, especially in gravitation. For more details concerning the relation between normal field theory and the background field method we refer the reader to references [9, lo]. We follow in this treatment the excellent paper by ‘t Hooft and Veltman [I],
200
P. VAN NIEUWENHUIZEN
to which we refer the reader for many of the details which we cannot again discuss here. Our conventions are as follows: the flat space metric is 6,” = SUv= 6,” = (I, 1, 1, 1) and RuvnB= a,.& + ..., while R,, = RAya,,and R = gvmR,,.
2. LAY-OUT
OF THE CALCULATION
Since the calculation we have in mind involves several logically independent steps leading up to the final result, and since each of these steps has its own technical details, we intend to give in this section a flowchart of the whole calculation while in the following short sections we discuss the individual steps in detail. The basic idea, due to ‘t Hooft and Veltman [I], is to decompose the metric into a classical background field g,, and the quantum deviation h,, about this background field. Then we rewrite the gravitational action in six dimensions in such a way that hoS (a symmetric 6 x 6 matrix) describes twenty-one scalar fields & = huB with (a/?) = i some internal indices, rather than twenty-one tensor fields h,, with (c$) Lorentz indices. Next we derive a general lemma for the one-loop divergences for a very large class of actions for scalar fields & in a classical background metric g,, , and finally we apply this lemma to the gravitational action in the form where has appear as scalar fields. The action we start from is the usual Einstein action in terms of the total gravitational field .& = g,, f !chmB (where K2 = 32~-G and G = Newton’s constant) 1 =
-2K-2
1 #x(g)112
R(f).
(1)
The dimension of K is that of a (mass)2 in six space-time dimensions. For one-loop diagrams one needs vertices with exactly two haBfields and many arbitrary external fields g,, , because the h,, fields will be the virtual fields inside the loop and the g,, fields emanate from the loop. (When g,, satisfy R,, = 0, then each field g,, represents a sum of trees with physical gravitons at the end points.) Expanding the action in ha0 about g,, and retaining the terms quadratic in h,, , we obtain the one-loop action. Since both gtiy and g,, are assumed to be tensor fields, so is h,, . Since one is dealing with a gauge theory, the usual deDonder type gauge fixing term and its accompanying ghost action are added. One then finds terms in the one-loop action of the form D,,(g) horBwhere D,(g) is the covariant derivative with respect to g,,, . By considering the connections in Dwhms as new vertices, one can consider busas a scalar field with covariant derivative auhaB. The indices $3 are from then on considered as internal indices. The kinetic term of the nonghost action is then of the form (a,h,,) P”6w(g)g~Y(ayhpO)and by doubling the number of fields h,, and defining a complex field h$ = (hg’ + ih$) 2-1/2 one can absorb the matrix P into (h$)*. Similar considerations hold for the ghost action. We have therefore brought the kinetic terms in the standard form of scalar fields but the price we had to pay was to introduce complex fields.
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WITHOUT
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201
The action then has the form (2) with ..J ‘li and A! functions of g,, and 4, =: ($:” + i$L”) the original gravitational quantum fields which are now The problem of finding the one-loop divergences of Eq. problem of finding the one-loop divergences of Eq. (2),
2-l:” where +Ji’ .= A$ are considered as scalar fields. (I) is thus reduced to the and this is solved in four
steps.
Step One. We determine the general counter term for the following action with N real fields (bi in flat space with general sources NI‘ and M. which are N N matrices
The most general term in the counter Lagrangian AY is of dimension (mass)‘j. being of the dimension of an energy density, and since 4. N, , and M have dimensions 2, I, and 2. respectively, there are quite a few possibilities. However, the action in Eq. (3) has a local gauge invariance (which is possible for scalar fields because we consider in the background field formalism N,, and M as sources independent of 4) which restricts the counter Lagrangian d.Y to five terms. whose coefficients we fix by an invariance argument. Step Tn.0. We consider N complex scalar fields in flat space (4) By decomposing +j = ($:” + i4i2’) 2-1/2, we can cast Eq. (4) into the form of Eq. (3) with 2N .i 2N matrices NU and M. The A?? belonging to Eq. (4) is. as expected, closely related to that of Eq. (3). Step Tllree. We now make the transition from flat space to curved space, but in such a way that we can use the results of Step Two. The trick is to consider metrics which are “half flat and half curved, ” i.e., conformally flat metrics g,,(x) = F(x) S,, where S,, is the flat spaceMinkovski metric. The action
can be cast in the form of Eq. (4) when g,, = F(x) S,, with matrices Jfru and A’, which are functions of NU, M, and F. Then we rewrite these counter terms in covariant form; in other words, we look for a gravitational scalar density constructed from Nl’, Ml, and g,,, which reduces to the correct counter terms when g,, = F(x) S,,, .
202
P. VAN NIEUWENHUIZEN
Step Four. When the metric is conformally flat, counter terms containing the Weyl tensor C,,,, vanish because the Weyl tensor is that linear combination of R “YPD3 Rx,, and R which vanishes when g,,, = F(x) 6,“. It follows that the results of Step Three only determine the counter terms for Eq. (5) modulo the Weyl tensor. There are three additional counter terms involving C,,,, whose coefficients we again fix by calculating a few simple diagrams. At this stage we know the general counter terms for any action which can be written in the quite general form of Eq. (5). Since in particular the gravitational action and its accompanying (unitarity restoring) ghost action are of this form, we can evaluate the one-loop counter terms for pure gravitation in six dimensions by inserting the pertinent matrices N u and Ml of the nonghost and ghost action in scalar form into the general one-loop formula. For the sake of clarity we distinguish between the different kinds of derivatives we will encounter, As always, a, = ajaxu denotes the ordinary derivative and D, the gravitational covariant derivative; in addition, d, will be a Yang-Mills type covariant derivative in flat space (to be defined at the proper place) and ID, will denote a derivative which is both gravitational and Yang-Mills covariant. Real fields in flat space have sources NU and M, complex fields in general have sources JVU and JH.
3. REAL SCALARS IN FLAT SPACE
The action for N scalars I#Qin a flat six-dimensional I =
s
d”x[-iSa,+
space is given by
a,$ + $N“ au+ + #bf~l
(6)
where Nu = Nuii and M = Mij are general N x N matrices [l l] which can be cast in antisymmetric and symmetric form, respectively, by integrating partially. The action is invariant under the following local gauge transformation 4 = 4, (NW)’ = N” - Q’l + [A, N“], (M)’
= M - &,nN”
- NQ,A
(7) + [A, M],
where n(x) is an arbitrary infinitesimal antisymmetric N x N matrix. (If one allows a symmetric part in Nu one obtains the same restrictions on possible counter terms for Eq. (6) as when Nu is antisymmetric, hence NU will always be taken antisymmetric.) The transformation of Nu is the same as the matrix +FV,” where W, is a Yang-Mills field and P the generators of its gauge group, hence [l] we introduce a Yang-Mills tensor r,, and instead of M consider the quantity X Yuy = a,N” - a,N“ + [N”, NY] X=
M--NUN”
U-P”)’= r,” + [L Ywl (X)’ = x + [A, X].
‘33)
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GRAVITATION
WITHOUT
203
GRAVITY
Both Y,, and X transform as ordinary vectors (more precisely, as vectors V” contracted with TO). Any Yang-Mills invariant (any expression invariant under Eq. (7)) covariant derivative A,, only, where is a function of YUV, X, and the Yang-Mills A, acts on a matrix A as follows A,A = 3,A + [IV”, A].
(9)
This follows from the observation that one can choose the first and second derivatives for a finite gauge transformation at any given point such that N@ and a&UN”+ aVNY vanish there, in completely the same way as one can transform away in gravitation g WY- 6,” and its first derivatives. When A is an ordinary vector, then so is A,A. The most general counter term for the action in Eq. (6) is thus an invariant under Eq. (7) of dimension six, constructed from Y,” , X, A,, . Generically one can expect terms of the form AAX2, AAXY, AA Y2, X3, X2Y, XY2, and Y3. Since X and A,X are N x N matrices, symmetric N x N matrices, and Y,, and A,Y,, antisymmetric we can omit the terms of the form AAXY and X2Y. One can integrate partially with A, under the integral sign in the same way as with gravitational covariant derivatives, hence there is one term of the form AAX which we choose in the form (0,X)(0,X). A priori there are three terms [I l] of the form AA Y2 Tl = (AuyJ2
T2
= (4yvJ2
T3 = (A, yv,(Av you>
(10)
but using the cyclic identity ~~~~~~YYcm= 0
(11)
one derives T, = -2T3 . On the other hand, integrating for any matrix A
Tl partially,
and using that
[A, 74lA = [y,v 3AI
(12)
one derives Tl + T3 = [Y,, , Y,,] Y,, , and it follows that there is only one linearly independent invariant of the form (AY)2. The most general counter term is thus given by A2’
= tr[a,(A.YJ2
+ a2(Y,,,Yv,Yo,)
+ a,(X”)
-i a,(XY,,Y,,~
t u,(A,X)~I
(13)
where the trace tr is over internal indices. No invariants of the form (tr A)(tr B) are present in A9 since they correspond to disconnected loop diagrams. The five coefficients ai in Eq. (13) are now fixed by considering the five simple diagrams in Fig. 1. We write the total action as I = I(dip) + I(A9), where Z(Z) is given in Eq. (6) and I(AZ) = Jd6x AZ’, and require that the poles at II =~ 6 in the S-matrix obtained from I(Z) are cancelled by the contribution to the S-matrix from Z(A9’). The calculation is completely straightforward, but one can save some labor by calculating diagram 1C under the assumption that a,Nu = %,,a,,Nu == 0 because then terms cubic in N in (A,Yu,)2 vanish and diagram 1C yields at once the coefficient a2 . Similarly we find at once the coefficient a., from diagram 1D when
204
P. VAN
al FIG.
05
NIEUWENHUIZEN
a, and a2
a4 and a5
a3
I. The five simplediagramswhich determinethe coefficientswritten below them.
we restrict M by a,a,,M = 0, since in this case (LI,X)~ will not contribute to an MNN divergence. As a sample calculation and to define the normalization, we calculate diagram 1B. The propagator for a scalar field in n dimensions being J(2?r)-“( -i)Pe@” d,“, the S-matrix element for absorption of an“ M-particle” with momentum p and emission of an “M-particle” with momentum p’ is given by S(Mp’; Mp) = $M(-p’)
M(p) [I d6k k-2(k + P)-~] P(p - p’)
(14)
where M(p) is the Fourier transform of M(x). On the other hand, the counter term d9 = QLI~X)~ yields a contribution to the S-matrix S,2z@fp’;
Mp)
= 2iq5[P2M(-P’)
Jf(Pw4”
@(P - P’).
(15)
Using dimensional regularization [ 121,
s
d%[K2 + ITZ~]-~= i~P/~(m~)(~/~)-=r(ol - n/2)/r((w)
(16)
one obtains a, . The result of these calculations is the following set of values for the coefficients ai in Eq. (13): a, = -(l/120)
E
a$ = (l/12) E
a5 = -(l/24)
E (17)
a2 = (l/180) E
ad = (l/24) E
E = [32n3(n - 6)1-l.
An excellent and simple check on these results is obtained by considering all oneloop diagrams with six N-vertices. Since such diagrams are logarithmically divergent, their sum gives a divergent contribution of the form NwNvNDNaNTN”j d’k k”kvkDk”kTkv/k12 which is proportional to 2N6 + 3N2NL1N2N~+ 6N2NuNVN*NY + 3N@N”N”NANvNa + N~NVNANuNvNA. (We defined N2 = NUN”.) Collecting all terms in Eq. (13) with six factors N, one indeed reobtains Eq. (17). The results obtained in this section differ from the results in four dimensions
QUANTUM
GRAVITATION
WITHOUT
20.5
GRAVITY
because covariant derivatives can be present in AL?. due to the higher number of dimensions, and some tensor algebra and analysis was needed to eliminate linearly dependent terms. 4. COMPLEX
SCALARS IN FLAT
SPACE
Consider the action for N complex scalar fields $j in flat space
Decomposing $j into real fields$y’ and 4:“’ by $I = ($1” + i$!,“) 2-r’” and defining a 2N dimensional vector by $ = ($y),..., +G’, $j”‘,..., 4:‘) we can cast Eq. (18) into the form of Eq. (6) if we replace in Eq. (6) 4 by $ and define the 2N x 2N matrices NU and M by
Since 4 and $* are different fields, we cannot cast JV, and A in Eq. (18) into (anti)symmetric form and the transposed matrices .A’* and JVU,~ differ from A! and -XU. Inserting Eqs. (19) and (20) into Eq. (13) yields, after straightforward algebra, the one-loop counter term for Eq. (IS) A9
= 2 tr[a,(Ll,~&lA~,,)
+ d~~u”~J
+ a3(grrVgV$YDU) + a,(:V3
+ K&43”>(~,~~>1
(21)
where the trace tr is over the internal indices, A, is defined by d,A = ?,A + [JV~~.A] while ‘qy,” z a,./+ - i&/vu + Jl’JJ,/l/“v- ,&-U..&/‘~, (22) 3 = d&C- a,Jc - Jv-Q./v-U. The factor two in Eq. (21) is of course due to the fact that N complex fields consist of 2N real fields while the term a,MU in X would also have been found in the case of real fields if NU would not have been antisymmetrized.
5. COMPLEX
SCALARS IN CONFORMALLY
FLAT
SPACE
We now consider scalar fields in a curved background. In principle we could first treat real scalars and then generalize to complex scalars, but since we will only need the results for complex scalars, we consider directly the action
206
P. VAN
NIEUWENHUIZEN
where the index TVof the contravariant vector Mu is carefully written as a superscript and where 4, d*, and .A are gravitational scalars. It is possible to relate this action to Eq. (18) by considering a metric which is “half flat and half curved,” the conformally flat metric &” = Lw4 (24) In this case, g”” = F-T?,, and gliz = F3 so that the action becomes after partial integration Z=
s
d%[$* a,&# + 2J*(FN"
+ F-'8,F)a,cj
+ c$*(FA)$]
(25)
where we have redefined $* = $*F2. Such a redefinition of fields does not affect the one-loop divergences, see Ref. [l]. The counter terms of Eq. (23) are therefore given by Eq. (21) when we substitute in Eq. (22)
The result is thus L.v =
m,@,%J
+ a2P3) +
+ a,(9@2) + a,(&q]
a3(93)
(27)
with d;” = $9 + [
KY = a,.4 - aA + kc, 41 = Y,, , The tensors Y,, are gravitationally (V3) covariantly as
.A$ = gufl.
(28)
covariant tensors, hence we can rewrite the term tr g1~2(Y,,V*pV,U)
where indices were raised by means of gu*. Consider next 9’. Evaluating yields St' = F&M - i&A'-“ -.A$V-" - 3(a,,F)F-1.A'"u -(nF)F2]
(29) this term (30)
and noting after some algebra that in n-dimensions Z',,“, = (n/2)F,,F-l
R = (n - l)[(OF)
F-2 + (n - 6)/4(8,F)2F-3]
(31)
QUANTUM
GRAVITATION
WITHOUT
207
GRAVIn
we find that X s F-Y8
= (,,#I - D/V”
- Jy;.N@ - +R).
(32)
D, is the gravitationally covariant derivative which acts only on the Lorentz index p of Nu: D,.H” = a,NU + F;U~O. The tensor X is thus a Lorentz-scalar and we succeeded in writing also g3 and &?@ in covariant form as $3 z.z gW(X)3;
(5w&J
= gl~2(xv,,v~“).
(33)
We now consider the more difficult task of rewriting the derivative terms (4U’@.,,)2 and (6,?&2 in covariant form. The latter yields upon evaluation (&@2
= g’/2(duX)(d”X)
+ (l$J’)2(X2)
+ 2(&F) F(XA,X)
(34)
where d,X = 8,X + [Jv; , X] and MU = g,,JlrV. The aim is now to rewrite this expression as a gravitational scalar density. The derivative d, in (Xd,X) can be replaced by the ordinary derivative 3, because tr(X.NUX - WXNU) = 0. Partial integration of the last term in Eq. (34) yields a term -(3,3,F)(FX2), and using Eq. (31), this term can be written as -g1/2($RX2). The other F-terms cancel. Since X is a gravitational scalar, the derivative d,, in d,X is already gravitationally covariant. Denoting by D, a derivative which is both Yang-Mills and gravitationally covariant, we have thus (6,&)2
= gl’2[(DuX)(DJ9q
- +R(x2)]
(35)
which is again a gravitational scalar density. Finally we tackle the term (8,,@‘,,)(8,@,,). All explicit F-dependence drops out not only in p,,. but also in o^,&@, , if one uses Jv; instead of NU throughout. Since s,%?u, = F( lDu%P,) - (a,F)(w,)
(36)
one has (&@u,)(&R’,,)
= g’/“(( D,w,)(DAwy
- 3( F,F)( i$F) F-y~~p)
- 2W’)(A~“ryn,Wu,)). Partially
integrating -2@,F)(A,W,)
(37)
the last term yields (omitting ~‘,y = 2g1~2F,u~F-1(WyW*)
integral signs) + &( a,F) F-‘%,(CVy,,‘%y,,)
(38)
where we used the cyclic identity in Eq. (11). Integrating the last term by parts, and using that if g,, = Fa,, the Ricci tensor is given in n dimensions by R,, = CW-lFLOF
+ (n - 2)
Fwl + (4F2)-‘I(6
- 3n) FJ,.
+ (n - 4)(F.J2 %,I (39)
leads, along with Eq. (31), to the desired result (d^,@,J(&&,) 595/1=‘4/1-I4
= g1~2(DpW‘y)(D,Wv)
+ R,,(WW”,)
- bR(tVY,,Wv)].
(40)
208
P. VAN NIEUWENHUIZEN
We have therefore shown that the one-loop counter terms of the action in Eq. (23) for the particular case of g,, = F(X) 6,” coincide with the following scalar density, provided one also puts g,, = F(x) S,, in this expression. LILT = gliz tr[aI{(UPV,,)(D~V~“)
+ R,,(V+P,)
- PR(V,,V~“)}
+ ~2w/uy~“~~/rro)+ %W) + ~4WVU”VU”) + a,{(D,X)(llP’X) - QRXZ}].
(41)
The symbol V’p was defined in Eq. (28), X in Eq. (32), Lorentz indices are lowered and raised by means of g,, and gU” while the derivative D, is both gravitatjonally and Yang-Mills covariant. For completenesswe repeat
We stress that Eq. (41) is not the general counter term for any metric g,,; only if we substitute g,, = 6,,,F in Eq. (41) does one obtain the general counter term belonging to Eq. (23) with g,, = 6,,F.
6. COMPLEX
SCALARS IN A GENERAL
CURVED SPACE
In this section we complete the counter terms, which were derived for conformally flat spaces,such that they hold for general curved spaces. Let us begin by writing down the Weyl tensor C,,,, . Its trace vanishes identically and the tensor itself vanishes when the spaceis conformally flat. In n-dimensions CLI”mJ + = R uvoo
(n - 2)HLgvo - hog,, + Rug,, - &gwl - Kn - I)@ - 2)1-k,gvo - guogv,lR.
(43)
For n = 4 one obtains a familiar result. It is clear that Eq. (41) is only determined UP to cww = 0. For example, we could have replaced R,,VuUV~, in the term with a, in Eq. (41) by an expression involving R,,,,oB and R using Eq. (43). However, we do get the most general counter terms if we add to Eq. (41) all possible, linearly independent, gravitational scalars containing at least one tensor C,,,, . Generically the list of such counter terms is given by C3,DDC2,C2X,C2V,CDDX,C[19DV,CX2,CXV,CV2
(44)
where we have omitted all terms containing either R,, or R because later on we will insert the Einstein equations R,, = R = 0. (We recall that this is the same in the background field method asin normal field theory putting all externa! gravitons on their mass shell with transverse-tracelesspolarization tensors.)
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GRAVITY
209
Since CaOYsgn;Svanishes, the terms %UIL[DX, %5X2, and @XV vanish, since they contain at least one contraction of the indices of C. Consider next the terms DDP. Repeated use of the Bianchi identity @‘BysC,aBy;6= 0 reduces these terms to C3 when R,, := R = 0. The terms C?V and CDDV (which are, by the way, not antisymmetric matrices) will vanish due to a simple theorem (called shift theorem hereafter) which we now derive. Due to the cyclic identities satisfied by the Weyl tensor (which has the same symmetry properties as the Riemann tensor) one has in graphical notation the following equality:
It follows
that one can always
bring the term C2V on the form (46)
Similarly one easily shows with Eq. (45) that the term CDDV because
can be eliminated
where we used that the commutator [D, , !3,] V,,, is proportional to RaDyBVyS and to P,,, , v,,] when R,, = 0. This leaves us with the three generic terms C”, C2X, and CV*. Using the shift theorem again, one immediately seesthat the last two terms can always be brought to the form CuBvsC31*y% and Ca~~6V/aBVvs, but perhaps less obvious is the fact that there are only two algebraically independent scalars of the form C3. To see this we note that one can always use the symmetries of the Weyl tensor to cast any scalar C3 in the form
When the contraction which starts at 01,turns left then, according to the shift theorem, there is only one invariant
u,=c
1 I 1 ICI
11____IZZ.
(49)
When. however. the contraction which starts at ai, turns right, it reproduces U, when it ends at y (use the shift theorem for the two C tensors on the right of Eq. (48)). Hence we consider the case that 01ends at 6 and 18at g. This still leaves two ways
210
P. VAN
NIEUWENHUIZEN
in which to complete the final contractions, will be called U, U2=C)
r-p7
p to y and p to E. The first possibility
a
) I\.
(50)
Using the cyclic identity, one easily derives that the second (PC) contraction is related to U, and U, by
u,=c
I,‘rl~
1 =au,i-U2.
Since in four dimensions the two-loop divergences of pure gravitation are of the form ~~(n - 4)-l@ .s.)~, it follows that the algebraic relations derived above seem to be of use for such a future two-loop calculation. At this point we have exhausted the algebraic relations between the counter terms. However, there still exists an analytical relation, the so-called Gausz-Bonnet theorem. This theorem, which holds in any even number of dimensions [l], asserts that under a small variation g,, + g,, + h,, for arbitrary g,, (not necessarily satisfying R,, = 0) and h,, , the variation of a certain scalar vanishes; in six dimensions: (52)
Hence one may omit the scalar Z from the action, since it does not contribute to the field equations (nor to boundary conditions, since g,, is arbitrary). The scalar Z itself probably diverges when g,, satisfies the Einstein equations; however, divergences in an action are quite common, e.g., energy-momentum conserving delta-functions, and do not worry us here. Expanding the product of the two E-symbols in terms of Kronecker delta functions tells us that, as far as the action is concerned,
s
d6xg112[Ul + 2U2] = 0.
From the algebraic and analytic identities
(53)
discussed above, it follows that when
R,, = R = 0, there are only three counter terms which have to be added to Eq. (41)
in order to obtain the full counter term for the action of complex scalars in a general background metric g,, AP(conf)
= g112tr[u,JJ, + a,C~~@fuvV,,o + a8ColB,,sCo~y%].
(54)
We now fix the coefficients a, and u, . Since in the next section we show that tr(X) vanishes when R,, = 0, the last term in Eq. (54) will not contribute, hence we need not fix a,. The statement that R,,(g) = 0 means, in the language of Feynman diagrams, that external gravitons (for example each of the three external gravitons
QUANTUM
GRAVITATION
WITHOUT
211
GRAVITY
q;q”-“(
C
FIG. 2. The diagrams which determine u6.
in each of the diagrams in Fig. 2) are transverse and traceless. For the S-matrix this means that expanding R,, in terms of g,, and gIIV = 6,” + K/Z,, , the fields /I,, satisfy Oh,, = 6,,h,, = aAhAv= 0. Unfortunately, when all three gravitons in Fig. 2 are on-shell (R,, = 0), then the terms in U, proportional to K3 vanish. When one of the three gravitons in Fig. 2 is off-shell, then there are three possible counter terms which might contribute: U, , RYVRwaB,,RYbB~, and RRaByaRaSyS. There are also three different terms which could come from evaluating Fig. 2:
where Iz,,(i) is the polarization tensor of graviton i, and p3 the off-shell graviton. However, also in this case the three counter terms are linearly dependent, as one may easily verify. Therefore we consider in Fig. 2 all three gravitons off-shell. There is a large number of counter terms which could in principle contribute, such as U1 and RBuV,R3, R(R,+$ etc. However, there is precisely one kind of term which can only come from U, , namely (55)
K3hamAs,a&cwy .
Only diagram 2A contributes to this term and we proceed to fix the coefficient a6 in Eq. (54) by evaluating diagram 2A. The S-matrix contribution of dZZ’ must cancel the pole at (n - 6) in the S-matrix due to the action in Eq. (23). The former stems from the term JcPx~~/~(cI,U~) in Eq. (54) and yields S,&2A)
=
~~6ic7,
lj
puih(pj)uvpvkGW6 %+
- r2 - p3).
ifjfkfi
On the other hand, the contribution from diagram 2A is found to be S(2A) =
iK3E
j-j
[p,ih(p’),,p,k]
l/15 x l/168 s6(p1 -p”
- p3)
t 57)
from which we conclude U6= -l/(90
x 168)E
(58)
with 6 given by Eq. (17). Finally, we fix the coefficient u, . In this case we put the gravitons in Fig. 3 on-shell. This means that diagram 3B vanishes since it is of the four q3*Nu”ii,,qShAA . Diagram 3A contributes both to a, , the coefficient of the invariant (D,M““)( DAVA,), and to u, . However, if we restrict the source .MJ~to satisfy a,,&.N~ =aui,Jr/-”=- 0, only u, receives contributions. One finds (omitting a factor K ~85~)
212
P. VAN NIEUWENHUIZEN
A
-
B
FIG. 3. The diagrams whichdeterminea, .
while S(3N = ~~/45~~~~~“~~‘~l~~tJlr’~p5~l~~:~~~~,~
dl.
(60)
It follows that a, = 1/(18O)C
(61)
This completes the determination of the counter terms of the following action for N complex scalar fields & in a general background metric g,,, I =
s
d6xg112tr[-a,+*
guva,4 + 24*.&-u a,+ + 4*&?4]
(62)
where the sources J+‘-wand A are not necessarily (anti)symmetric. When we are on-shell, R,, = R = 0 and (seealso Eq. (42)) A9
= .5g1i2tr[-(l/120)(D,VU”)(DAVA,) + (1/24)(XV,,V““)
+ (I/180)(Y,,V”~V,U)
- (1/24)(D,X)(WX)
+ (1/12)(X)”
- (l/ 15120)(R,,,,R”~olBR40,Y~)
+ (l/l~o)(R,",,~~"~~")l
(63)
with E = [329(n - 6)1-l. In the next section we apply this general formula to pure gravitation.
7. PURE GRAVITATION
IN SIX DIMENSIONS
The preceding sections led up to a general formula for the one-loop counter terms of N complex scalar fields in six dimensions in a general background metric g,, . In this section we will consider the system of pure gravity and cast it in the form of scalar fields. The discussion follows at most points Ref. [I] but at some points alterations are needed due to the different dimensionality of space-time; for completenesswe record all steps below. We first add a gauge fixing term to the one-loop Lagrangian and obtain the corresponding ghost Lagrangian. Subsequently we rewrite both the nonghost and the ghost action in such a way that these actions are of the form of scalar fields in a given background metric g,, . Then, in the next section, we will apply the general formula for the counter terms to these two actions. The classical action for pure gravity in six dimensions is given by Z = -21~~ 1 d6x glizR(g).
(64)
QUANTUM
GRAVITATION
WITHOUT
213
GRAVITY
The expansion of this action in quantum fields does not differ from the treatment in four dimensions. Since ultimately we will insert the classical field equations. i.e., the Einstein equations R,, = 0, we will use consistently R,,. := R -::: 0. The one-loop action is then given by -y == g’P[ -&,&@:u
+ $h,h“ - $hwhi~l + #;Uh;‘L +
4/71,1(R”LLBv
+ RnL,RJ’)/7,,]
(65)
(after dividing by a factor 2 in order to conform with Ref. [I]) where /I, =: D”(g) /I,, , == gvvh,,, and all tensor operations, including the covariant derivative (;). are performed with gU.. Adding the usual gauge fixing term h
LP = -4 g1/2[e,u(Dv( g)
/I,,”
- iD,( g)h)]”
(66)
where eoU is the vierbein field, en?,,v = gb”‘, we find the usual result 2’ + LP = g”2[-~(Dyh~4)(PUaUV)(DYhllV)
$- ihwi5( R”I”” + RL1*@@) /7,&l
where p1l3Lu= 4( gaugol + gevgN’, _ g”ag,w), Doubling the number of gravitational fields as in Section 3, taking the inverse of P (the graviton propagator in n dimensions) (p-lLrQu and redefining
g+
= guogw + &o g”” - (2/n - 2) g,,g,,
(67)
h*,.I’ = h:BPaBuu,we obtain ~2’~= g1’2[-(D.,h*CIR)(DYh,,)+ ~h*9&,( R”““” + R’“““) h,,,].
(68)
We consider the field h**B as the complex conjugates of h,, (see Ref. [l]). Rewriting D,,DYh,8 in terms of derivatives B,, , 6? which do not act on the internal indices n/3 of h,, yields the same result as in four dimensions. We thus find that
(71) where C$ and pu mean that only the part symmetric in (c&) and (pa) is to be taken. One easily findsthat the nonghost Lagrangian is characterized by (W”‘),c
= (RR(;;I -;~ RzB”),
(~~G)nuDo = (-2R”,,&)$.
(72) (73)
214
P. VAN NIEUWENHUIZEN
Turning
to the ghost Lagrangian 9’
= g1’2[q;g”vDADA~s,- T,I,*R”“Q]
(74)
one finds in the same way
(xi”): = 0;
(Jy;G>aB = 4:.
,”= -R8,,, ;
Finally we determine the quantities [19,X and D,VUu. The reader may verify the matrices Jy; in D,X provide just the necessary Christoffel symbols to make D,X a covariant object D,(XNG):;I = D,(R//
+ R,“,“)
(76)
where as we recall, D, is the usual gravitational covariant derivative. In the same way D,VU” is a covariant object, but since RaBuVi~= 0 module R,, = R = 0, we have (77)
For the ghost contribution
we get in the same way
D,(x”)p = D”(V;“),J = 0.
(78)
The terms (DJXNG)(lD~XNG) can be cast in the form R3 ogvaby using Eq. (76), integrating partially, using ~~~~~~~~~~~~ = 0 and [D,, , D,] R,,, = -R&,R,By,, -1. - R&,RmB,,A . One finds (DJ~G)(lDr’X~G) = -9u,, . (79 where we used the graphical methods of Eq. (45) and the Gausz-Bonnet theorem in Eq. (53). In the next section we will insert the explicit expressions for X, V,, , and D,XNG into the general formula for d-9.
8. ONE-LOOP
DIVERGENCES IN THE S-MATRIX IN SIX DIMENSIONS
FOR PURE GRAVITATION
Finally we are in a position to determine whether the S-matrix for pure gravitation in six dimensions is finite at the one-loop level or not. We have on the one hand derived a general formula, Eq. (63), for the one-loop divergences of the general scalar system in Eq. (62). On the other hand we have written the gravitational oneloop action in the form of Eq. (62) with definite matrices JP and A. In this section we insert these given matrices into the general formula Eq. (63) and by evaluating the traces over the internal indices, we obtain the required result. The evaluation of the traces is straightforward. All terms yielding terms of the generic form (R a..)” we use Eqs. (45) (51) and (53) to obtain d9 = ag1gl12U1,
QUANTUM
GRAVITATION
WITHOUT
GRAVITY
215
TABLE I Traces of Nonghost Contributions
TABLE II Traces of Ghost Contributions a, = 0 a2 = (v~y~,6(v”~)6~(Y~~)yb:~ = u, = -gY, as = 0 a4 = 0 a, = 0
where 01is the required coefficient. In Table I we summerize the results of the traces over the nonghost and ghost matrices. Adding them all up yields
-&+$+&-1 - -?I180
180 “, 84
+,1,11-
21
180 x 84 +&!
(80)
The factors & and -1 in front of the curly brackets account for the “dedoubling” of h,, = (h$ + ihlr28))2-112 and the fermion character of the ghost vertices. The numerators 21 and 6 in the term with 180 x 84 are the number of independent
216
P. VAN
NIEUWENHUIZEN
components in the trace over internal indices: 21 fields h,, and six ghost fields & . The total counter terms for the s-matrix is thus given by ddp = (9/l 120) g1~2(RbBysRSvPoRopBo)(32~3(n - 6))-l .
9.
031)
CONCLUSIONS
Covariantly quantized gravitation without matter sources has an S-matrix which is finite at the- one-loop level in four dimensions. However, the significance of this result is not clear because, on invariance grounds, the sum of all divergences of the sum of all S-matrices must be at the one-loop level of the form sdiv(l
- loop, 4 dim) = 1 d4x g1’2[aRuVRuY + pR2](n - 4)-l
where we used the Gauss-Bonnet theorem to express RpypoRuYoainto R RuY and R2. Since the Einstein equations imply R,, = R = 0, this expression vanisik no matter what 01 and /3 are. In order to test the divergence properties of pure gravitation without sources in a less ambiguous way, we considered quantum gravity in six dimensions; admittedly not a very physical theory, but containing some of the features of a two-loop calculation in four dimensions and much simpler than such a two-loop calculation. We found that the sum of all one-loop divergences of all S-matrices in six dimensions is of the simple, nonrenormalizable, form sdiV(l
- loop, 6 dim) = 1 d6x g1’2[yR,B,,, omuRv~a](n - 6)-l
where we used the Gauss-Bonnet theorem in six dimensions and identities. This invariant does not vanish when R,, = 0; the proof will be published elsewhere [15]. The coefficient y has to vanish cancellations if the S-matrix in six dimensions is to be finite. After tion we found
various algebraic of this statement due to nontrivial a lengthy calcula-
y = (9/l 120)(32 r3)-l
and we conclude that quantum gravity without sources in six dimensions behaves as matter-gravity interactions in four dimensions; the theory is nonrenormalizable and the S-matrix diverges. What does this result imply for the renormalization properties of pure gravitation in four dimensions? The local two-loop divergences of the S-matrix in four dimensions are of the same form as the one-loop divergences in six dimensions &iV(2 - loop, 4 dim) = (n F2 4)2
s
d4x g1’2[~R,4~oRuPT”Rv~l
where we have used the algebraic identities which were derived in section 6 and the
QUANTUM
GRAVITATION
WITHOUT
217
GRAVITY
fact that the invariant RU,,ooRuh"TRTYAP vanishes when R,, ==0 [13, 14. 151. The invariant between square brackets does not vanish when R,,. = 0 [I 51. Extra cancellations are needed in order that the coefficient E of this nonrenormalizabie counter term vanishes, and since such cancellations are absent in six dimensions, we believe that there is little likelihood that quantum gravity without matter sources will have a finite S-matrix in four dimensions. Or is, perhaps, quantum gravity in four dimensions an exceptional case and is that why our world is four dimensional? ACKNOWLEDGMENTS The author is grateful to Professors R. P. Geroch, G. ‘t Hooft, M. Veltman, and C. N. Yang for very helpful discussions. REFERENCES I. G. ‘T HOOFT AND M. VELTMAN, Ann. Inst. H. Poincart 20 (1974). 69. 3. S. DESER AND P. VAN NIEUWENHUIZEN, Phys. Rw. Lett. 32 (1974). 245: Phys.
Rev. D 10 (1974),
401. 3. S. DESER AND
P. VAN NIEUWENHUIZEN,
Lett.
Nuovo
Cimento
11 (1974).
218; Phys.
10 (1974).
Rro.
411. H.-S. TSAO, AND P. VAN NIEUWENHUIZEN, Phys. Lett. B 50 (1974). 491 : Phys. Rer. D 10 (1974), 3337. 5. M. T. GRISARU, P. VAN NIEUWENHUIZEN, AND C. C. WV, Phys. Rer;. D 12 (19X), 1813. 4. S. DESER,
6. D. Z. FREEDMAN AND E. WEINBERG, Ann. Phusics87 (1974), 354; D. 2. FREEDMAN, I. J. MUZINICH, AND E. WEINBERG, Ann. Physics 87 (1974), 959: D. Z. FREEDMAN AND SO-YOUNG PI, Ann. Physics 91 (1975), 442. 7. D. G. BOULWARE AND L. S. BROWN, Phys. Rec. 172 (1968). 1628: Ref. [8]; J. HONERKAMP, Nucl. Phys. B 48 (1972), 269; Ref. [9]. 8. B. S. DEWITT, Phys. Reu. 160 (1967), 113; 162 (1967), 1195; 162 (1967). 1239. 9. R. KALLOSH, Nucl. Phys. B 78 (1974), 293. The proof in this excellent article needs a modification since in the axial gauge the gauge fixing term (h:,e)z is not invariant under background gauge
transformations. IO. M. T. GRISARU, P. VAN NIEUWENHUIZEN, AND C. C. Wu, I I. Contraction over internal indices is always understood.
Phvs.
Rel:.
D, D12
(1975).
3203.
12. G. ‘T H~~FT AND M. VELTMAN, Nucl. Phys. 844 (1972), 189; C. G. BOLLINI AND J. J. GIAMBIAGI, Phys. Lett. 840 (1972), 566; J. F. ASHMORE, Nuouo Cimento Lett. 4 (1972), 289; G. M. CICUTA AND E. MONTALDI, Nuovo Cimento Left. 4 (1972), 329. 13. This may be proved by decomposing the Lorentz indices of Rp,,, into spinor components (CL=- 4 @ &) and using the symmetry of the Rieman tensor and R,,. -7. 0. 1 am indebted to
Dr. ‘t Hooft for pointing this out to me; see also Ref. (141. 14. See PIRANI, in “Brandeis Lectures in Theoretical Physics,” MIT Press, Cambridge, 15. P. VAN NIEUWENHUIZEN AND C. C. Wu, J. Math. Phys. 18 (1977). 182.
1968.
OF PHYSICS
104,
197-217 (1977)
On the Renormalization
of Quantum
Gravitation
Without
Matter*
P. VAN NIEUWENHUIZEN Institute
for Theoretical Stony Brook,
Physics, State University Long Island, New York
qf New York, I1794
ReceivedJune9. 1976
TheS-matrixof puregravitationisfinite at the one-looplevel.but in a trivial way dueto invarianceconsiderations which only hold in four dimensions. Theseconsiderations are not sufficientto concludethat the S-matrix is alsofinite at the two-looplevel.In order to determinewhetherthereis anylikelihoodthat extra cancellations will alsomakethe higher looppurelygravitationalS-matrixfinite, weconsidera casewhichhasseveralof thepropertiesof a two-loopcorrectionin four dimensions, a one-loopcorrectionin six dimensions. Thereare no extra cancellations and it is arguedthat puregravitationin four dimensions hasprobablyno finite S-matrixbeyondthe one-looplevel whenit is treatedasa standard particle theory.
I. INTRODUCTION
One of the fundamental problems in theoretical physics is the quantization of gravitation. The belief in quantum gauge field theory in general is strongly supported by the excellent agreement between theory and experiment in quantum electrodynamics. Since gravitation is the gauge field theory for masslessself-interacting spin two bosons with natural parity, it is tempting to apply the techniques of quantum electrodynamics to gravitation. One obtains in this way a quantum theory for gravitation which, however. is not unitary. This is a feature of non-Abelian gauge field theories, also present in Yang-Mills theories, and it is possible to find modified covariant quantization rules, either by using the path-integral formalism or by working directly with diagrams, which yield a unitary quantum field theory. In the case of Yang-Mills theories the resulting quantum theory is even renormalizdble and realistic models unifying the weak and electromagnetic interactions have been constructed. When one wants to apply the techniques of quantum Yang-Mills theory to gravitation, one must first decompose the metric into a background field (usually the flat spacemetric a,, although in this article it will be a general metric glLysatisfying the Einstein equations) and a deviation /?,“, and then treat h,,, as a normal quantum field, on the same footing as, say, the electromagnetic field A, . Admittedly, this may be a doubtful procedure in view of the close connection between the full metric *Work supportedin part by NSF Grant No. MPS-74-12208 AOl. 197 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSIi
0003
-49 I6
198
P. VAN
NIEUWENHUIZEN
and the causal structure of spacetime which is needed for canonical quantization in order to define the basic commutation rules. However, it is not clear which, if any, the causal problems are when one quantizes covariantly in a path integral formalism. Consequently, we will forget for the time being that gravitons have something to do with clocks and yardsticks and build Feynman diagrams with gravitons represented by quantum operators h,, . This yields a standard particle physics theory, and investigations into the renormalizability properties of this quantum field theory have given the following results: (1) Pure gravitation, i.e., gravitation without matter sources, is not renormalizable in the sense that one can absorb the divergences in the Green’s functions by resealing of the physical parameters of the theory. However, at the one-loop level the S-matrix for any N-graviton process is finite [I] because the divergences in the N-graviton Green’s functions are polynomials in the external fields which vanish when the external gravitons have physical momenta and polarizations. (2) Coupling gravitation to such matter fields as scalars [l], photons [2], fermions [3], or Yang-Mills bosons [4], leads to a nonrenormalizable theory, as in the case of pure gravitation; however, in this case also the S-matrix diverges. Combining quantum electrodynamics with gravitation leads to a system where gravitons, photons, and fermions all couple to each other; this system turns out to be no better [5] than the coupling of gravitons to either photons or fermions. (3) When gravitons do not propagate inside loops, but are merely external classical fields, then quantum field theories which are renormalizable in flat space, stay so (to any loop order) in curved space [6]. Clearly, the only positive result in the direction of a well-defined true quantum theory of gravitation is the case of pure gravity. However, the opinions differ as to the significance of this result. Since one may show [7], using invariance arguments, that the sum of the one-loop divergences of all N-graviton S-matrices, summed over N, must be proportional to R,,R@” and R2, and since, according to the Einstein equations, R,, = R = 0 for pure gravity, some people find the finiteness of the purely gravitational S-matrix as uninteresting as the vanishing of the Dirac action (or the gravitational action for that matter) on-shell. Others believe that this finiteness is just a glimpse of the high symmetry in gravitation which heralds good news for the higher loop quantum corrections in pure gravitation. One obvious calculation which would determine whether there are extra cancellations in higher loop corrections beyond the fortuitious one-loop result is the calculation of the divergences in the S-matrix at the two-loop level. Whereas this calculation should certainly be performed, it is exceedingly hard and requires strong optimism about the final answer to sustain one in the algebraic nightmare. Instead, one might look for a calculation which is on the one hand much simpler than a two-loop calculation, but which, on the other hand, will give a more meaningful result than the one-loop case of pure gravitation in four dimensions. Such a calculation we present here: We calculate the one-loop divergences of the purely gravitational
QUANTUM
GRAVITATION
WITHOUT
GRAVITY
199
S-matrix in six dimensions. As we shall show, the sum of all one-loop divergences of all N-graviton S-matrices, summed over N, is given in six dimensions by the simple expression S(div) = OL* d6x g1’2[R~yuoR(iPiURazY~](~- 6)-l. J
The factor (0 - 6)-l regularizes the divergences according to the dimensional regularization scheme, and since this nonrenormalizable divergence does not vanish automatically when R,, = R = 0, the unambiguous criterion for renormalizability is the vanishing of the coefficient (Y. In the Conclusions we argue why we believe that this calculation tells us something about two-loop finiteness of the S-matrix in four dimensions. In the next section we present the layout of this rather involved calculation, in the hope that this will give the casual reader enough information to go directly from there to the conclusions. In between, many technical problems are solved. Some of the results obtained seem to be of relevance for a future two-loop calculation in four dimensions. As already mentioned, we calculate here the sum of the divergences of all S-matrices, denoted above by S(div), because this is, paradoxically, much simpler than the sum of the divergences of a particular S-matrix. However, one can obtain the divergences of an N-graviton S-matrix by expanding S(div) into h,, and collecting terms of order h N. The formalism within which these results are obtained is an alternative to normal field theory, called the background field formalism [8]. This formalism is entirely equivalent to normal field theory, by which we mean that both give the same S-matrix [9] (but of course different Green’s functions in general). In fact, there exist calculations of particular examples which verify this equivalence explicitly [lo]. The advantages of using the background formalism are twofold. First one need not calculate the one-particle reducible diagrams in the background field method but one simply calculates loop diagrams constructed from vertices of the form h,&&gUy) h,, . The fields h are the quantum fields; they propagate in the loop and their gauge freedom must be fixed by adding a gauge fixing term. The function F is a function of the external (classical) fields g,, . By requiring that g,, satisfy R,,(g) = 0, one can replace diagrammatically each field g,“(x) by the sum of all trees which start at x and which have at their endpoints physical momenta and polorizations. In this way one calculates at once the sum of all S-matrices while only evaluating one particle irreducible diagrams. The second advantage of using the background field method is that in gauge theories one can choose the quantum gauge fixing term such that it does not break the gauge invariance in the external nonquantum fields, so that the computed counter terms, being (local) functions of the external fields only, are automatically gauge invariant objects. This reduces the form of the counter terms enormously, especially in gravitation. For more details concerning the relation between normal field theory and the background field method we refer the reader to references [9, lo]. We follow in this treatment the excellent paper by ‘t Hooft and Veltman [I],
200
P. VAN NIEUWENHUIZEN
to which we refer the reader for many of the details which we cannot again discuss here. Our conventions are as follows: the flat space metric is 6,” = SUv= 6,” = (I, 1, 1, 1) and RuvnB= a,.& + ..., while R,, = RAya,,and R = gvmR,,.
2. LAY-OUT
OF THE CALCULATION
Since the calculation we have in mind involves several logically independent steps leading up to the final result, and since each of these steps has its own technical details, we intend to give in this section a flowchart of the whole calculation while in the following short sections we discuss the individual steps in detail. The basic idea, due to ‘t Hooft and Veltman [I], is to decompose the metric into a classical background field g,, and the quantum deviation h,, about this background field. Then we rewrite the gravitational action in six dimensions in such a way that hoS (a symmetric 6 x 6 matrix) describes twenty-one scalar fields & = huB with (a/?) = i some internal indices, rather than twenty-one tensor fields h,, with (c$) Lorentz indices. Next we derive a general lemma for the one-loop divergences for a very large class of actions for scalar fields & in a classical background metric g,, , and finally we apply this lemma to the gravitational action in the form where has appear as scalar fields. The action we start from is the usual Einstein action in terms of the total gravitational field .& = g,, f !chmB (where K2 = 32~-G and G = Newton’s constant) 1 =
-2K-2
1 #x(g)112
R(f).
(1)
The dimension of K is that of a (mass)2 in six space-time dimensions. For one-loop diagrams one needs vertices with exactly two haBfields and many arbitrary external fields g,, , because the h,, fields will be the virtual fields inside the loop and the g,, fields emanate from the loop. (When g,, satisfy R,, = 0, then each field g,, represents a sum of trees with physical gravitons at the end points.) Expanding the action in ha0 about g,, and retaining the terms quadratic in h,, , we obtain the one-loop action. Since both gtiy and g,, are assumed to be tensor fields, so is h,, . Since one is dealing with a gauge theory, the usual deDonder type gauge fixing term and its accompanying ghost action are added. One then finds terms in the one-loop action of the form D,,(g) horBwhere D,(g) is the covariant derivative with respect to g,,, . By considering the connections in Dwhms as new vertices, one can consider busas a scalar field with covariant derivative auhaB. The indices $3 are from then on considered as internal indices. The kinetic term of the nonghost action is then of the form (a,h,,) P”6w(g)g~Y(ayhpO)and by doubling the number of fields h,, and defining a complex field h$ = (hg’ + ih$) 2-1/2 one can absorb the matrix P into (h$)*. Similar considerations hold for the ghost action. We have therefore brought the kinetic terms in the standard form of scalar fields but the price we had to pay was to introduce complex fields.
QUANTUM
GRAVITATION
WITHOUT
GRAVITY
201
The action then has the form (2) with ..J ‘li and A! functions of g,, and 4, =: ($:” + i$L”) the original gravitational quantum fields which are now The problem of finding the one-loop divergences of Eq. problem of finding the one-loop divergences of Eq. (2),
2-l:” where +Ji’ .= A$ are considered as scalar fields. (I) is thus reduced to the and this is solved in four
steps.
Step One. We determine the general counter term for the following action with N real fields (bi in flat space with general sources NI‘ and M. which are N N matrices
The most general term in the counter Lagrangian AY is of dimension (mass)‘j. being of the dimension of an energy density, and since 4. N, , and M have dimensions 2, I, and 2. respectively, there are quite a few possibilities. However, the action in Eq. (3) has a local gauge invariance (which is possible for scalar fields because we consider in the background field formalism N,, and M as sources independent of 4) which restricts the counter Lagrangian d.Y to five terms. whose coefficients we fix by an invariance argument. Step Tn.0. We consider N complex scalar fields in flat space (4) By decomposing +j = ($:” + i4i2’) 2-1/2, we can cast Eq. (4) into the form of Eq. (3) with 2N .i 2N matrices NU and M. The A?? belonging to Eq. (4) is. as expected, closely related to that of Eq. (3). Step Tllree. We now make the transition from flat space to curved space, but in such a way that we can use the results of Step Two. The trick is to consider metrics which are “half flat and half curved, ” i.e., conformally flat metrics g,,(x) = F(x) S,, where S,, is the flat spaceMinkovski metric. The action
can be cast in the form of Eq. (4) when g,, = F(x) S,, with matrices Jfru and A’, which are functions of NU, M, and F. Then we rewrite these counter terms in covariant form; in other words, we look for a gravitational scalar density constructed from Nl’, Ml, and g,,, which reduces to the correct counter terms when g,, = F(x) S,,, .
202
P. VAN NIEUWENHUIZEN
Step Four. When the metric is conformally flat, counter terms containing the Weyl tensor C,,,, vanish because the Weyl tensor is that linear combination of R “YPD3 Rx,, and R which vanishes when g,,, = F(x) 6,“. It follows that the results of Step Three only determine the counter terms for Eq. (5) modulo the Weyl tensor. There are three additional counter terms involving C,,,, whose coefficients we again fix by calculating a few simple diagrams. At this stage we know the general counter terms for any action which can be written in the quite general form of Eq. (5). Since in particular the gravitational action and its accompanying (unitarity restoring) ghost action are of this form, we can evaluate the one-loop counter terms for pure gravitation in six dimensions by inserting the pertinent matrices N u and Ml of the nonghost and ghost action in scalar form into the general one-loop formula. For the sake of clarity we distinguish between the different kinds of derivatives we will encounter, As always, a, = ajaxu denotes the ordinary derivative and D, the gravitational covariant derivative; in addition, d, will be a Yang-Mills type covariant derivative in flat space (to be defined at the proper place) and ID, will denote a derivative which is both gravitational and Yang-Mills covariant. Real fields in flat space have sources NU and M, complex fields in general have sources JVU and JH.
3. REAL SCALARS IN FLAT SPACE
The action for N scalars I#Qin a flat six-dimensional I =
s
d”x[-iSa,+
space is given by
a,$ + $N“ au+ + #bf~l
(6)
where Nu = Nuii and M = Mij are general N x N matrices [l l] which can be cast in antisymmetric and symmetric form, respectively, by integrating partially. The action is invariant under the following local gauge transformation 4 = 4, (NW)’ = N” - Q’l + [A, N“], (M)’
= M - &,nN”
- NQ,A
(7) + [A, M],
where n(x) is an arbitrary infinitesimal antisymmetric N x N matrix. (If one allows a symmetric part in Nu one obtains the same restrictions on possible counter terms for Eq. (6) as when Nu is antisymmetric, hence NU will always be taken antisymmetric.) The transformation of Nu is the same as the matrix +FV,” where W, is a Yang-Mills field and P the generators of its gauge group, hence [l] we introduce a Yang-Mills tensor r,, and instead of M consider the quantity X Yuy = a,N” - a,N“ + [N”, NY] X=
M--NUN”
U-P”)’= r,” + [L Ywl (X)’ = x + [A, X].
‘33)
QUANTUM
GRAVITATION
WITHOUT
203
GRAVITY
Both Y,, and X transform as ordinary vectors (more precisely, as vectors V” contracted with TO). Any Yang-Mills invariant (any expression invariant under Eq. (7)) covariant derivative A,, only, where is a function of YUV, X, and the Yang-Mills A, acts on a matrix A as follows A,A = 3,A + [IV”, A].
(9)
This follows from the observation that one can choose the first and second derivatives for a finite gauge transformation at any given point such that N@ and a&UN”+ aVNY vanish there, in completely the same way as one can transform away in gravitation g WY- 6,” and its first derivatives. When A is an ordinary vector, then so is A,A. The most general counter term for the action in Eq. (6) is thus an invariant under Eq. (7) of dimension six, constructed from Y,” , X, A,, . Generically one can expect terms of the form AAX2, AAXY, AA Y2, X3, X2Y, XY2, and Y3. Since X and A,X are N x N matrices, symmetric N x N matrices, and Y,, and A,Y,, antisymmetric we can omit the terms of the form AAXY and X2Y. One can integrate partially with A, under the integral sign in the same way as with gravitational covariant derivatives, hence there is one term of the form AAX which we choose in the form (0,X)(0,X). A priori there are three terms [I l] of the form AA Y2 Tl = (AuyJ2
T2
= (4yvJ2
T3 = (A, yv,(Av you>
(10)
but using the cyclic identity ~~~~~~YYcm= 0
(11)
one derives T, = -2T3 . On the other hand, integrating for any matrix A
Tl partially,
and using that
[A, 74lA = [y,v 3AI
(12)
one derives Tl + T3 = [Y,, , Y,,] Y,, , and it follows that there is only one linearly independent invariant of the form (AY)2. The most general counter term is thus given by A2’
= tr[a,(A.YJ2
+ a2(Y,,,Yv,Yo,)
+ a,(X”)
-i a,(XY,,Y,,~
t u,(A,X)~I
(13)
where the trace tr is over internal indices. No invariants of the form (tr A)(tr B) are present in A9 since they correspond to disconnected loop diagrams. The five coefficients ai in Eq. (13) are now fixed by considering the five simple diagrams in Fig. 1. We write the total action as I = I(dip) + I(A9), where Z(Z) is given in Eq. (6) and I(AZ) = Jd6x AZ’, and require that the poles at II =~ 6 in the S-matrix obtained from I(Z) are cancelled by the contribution to the S-matrix from Z(A9’). The calculation is completely straightforward, but one can save some labor by calculating diagram 1C under the assumption that a,Nu = %,,a,,Nu == 0 because then terms cubic in N in (A,Yu,)2 vanish and diagram 1C yields at once the coefficient a2 . Similarly we find at once the coefficient a., from diagram 1D when
204
P. VAN
al FIG.
05
NIEUWENHUIZEN
a, and a2
a4 and a5
a3
I. The five simplediagramswhich determinethe coefficientswritten below them.
we restrict M by a,a,,M = 0, since in this case (LI,X)~ will not contribute to an MNN divergence. As a sample calculation and to define the normalization, we calculate diagram 1B. The propagator for a scalar field in n dimensions being J(2?r)-“( -i)Pe@” d,“, the S-matrix element for absorption of an“ M-particle” with momentum p and emission of an “M-particle” with momentum p’ is given by S(Mp’; Mp) = $M(-p’)
M(p) [I d6k k-2(k + P)-~] P(p - p’)
(14)
where M(p) is the Fourier transform of M(x). On the other hand, the counter term d9 = QLI~X)~ yields a contribution to the S-matrix S,2z@fp’;
Mp)
= 2iq5[P2M(-P’)
Jf(Pw4”
@(P - P’).
(15)
Using dimensional regularization [ 121,
s
d%[K2 + ITZ~]-~= i~P/~(m~)(~/~)-=r(ol - n/2)/r((w)
(16)
one obtains a, . The result of these calculations is the following set of values for the coefficients ai in Eq. (13): a, = -(l/120)
E
a$ = (l/12) E
a5 = -(l/24)
E (17)
a2 = (l/180) E
ad = (l/24) E
E = [32n3(n - 6)1-l.
An excellent and simple check on these results is obtained by considering all oneloop diagrams with six N-vertices. Since such diagrams are logarithmically divergent, their sum gives a divergent contribution of the form NwNvNDNaNTN”j d’k k”kvkDk”kTkv/k12 which is proportional to 2N6 + 3N2NL1N2N~+ 6N2NuNVN*NY + 3N@N”N”NANvNa + N~NVNANuNvNA. (We defined N2 = NUN”.) Collecting all terms in Eq. (13) with six factors N, one indeed reobtains Eq. (17). The results obtained in this section differ from the results in four dimensions
QUANTUM
GRAVITATION
WITHOUT
20.5
GRAVITY
because covariant derivatives can be present in AL?. due to the higher number of dimensions, and some tensor algebra and analysis was needed to eliminate linearly dependent terms. 4. COMPLEX
SCALARS IN FLAT
SPACE
Consider the action for N complex scalar fields $j in flat space
Decomposing $j into real fields$y’ and 4:“’ by $I = ($1” + i$!,“) 2-r’” and defining a 2N dimensional vector by $ = ($y),..., +G’, $j”‘,..., 4:‘) we can cast Eq. (18) into the form of Eq. (6) if we replace in Eq. (6) 4 by $ and define the 2N x 2N matrices NU and M by
Since 4 and $* are different fields, we cannot cast JV, and A in Eq. (18) into (anti)symmetric form and the transposed matrices .A’* and JVU,~ differ from A! and -XU. Inserting Eqs. (19) and (20) into Eq. (13) yields, after straightforward algebra, the one-loop counter term for Eq. (IS) A9
= 2 tr[a,(Ll,~&lA~,,)
+ d~~u”~J
+ a3(grrVgV$YDU) + a,(:V3
+ K&43”>(~,~~>1
(21)
where the trace tr is over the internal indices, A, is defined by d,A = ?,A + [JV~~.A] while ‘qy,” z a,./+ - i&/vu + Jl’JJ,/l/“v- ,&-U..&/‘~, (22) 3 = d&C- a,Jc - Jv-Q./v-U. The factor two in Eq. (21) is of course due to the fact that N complex fields consist of 2N real fields while the term a,MU in X would also have been found in the case of real fields if NU would not have been antisymmetrized.
5. COMPLEX
SCALARS IN CONFORMALLY
FLAT
SPACE
We now consider scalar fields in a curved background. In principle we could first treat real scalars and then generalize to complex scalars, but since we will only need the results for complex scalars, we consider directly the action
206
P. VAN
NIEUWENHUIZEN
where the index TVof the contravariant vector Mu is carefully written as a superscript and where 4, d*, and .A are gravitational scalars. It is possible to relate this action to Eq. (18) by considering a metric which is “half flat and half curved,” the conformally flat metric &” = Lw4 (24) In this case, g”” = F-T?,, and gliz = F3 so that the action becomes after partial integration Z=
s
d%[$* a,&# + 2J*(FN"
+ F-'8,F)a,cj
+ c$*(FA)$]
(25)
where we have redefined $* = $*F2. Such a redefinition of fields does not affect the one-loop divergences, see Ref. [l]. The counter terms of Eq. (23) are therefore given by Eq. (21) when we substitute in Eq. (22)
The result is thus L.v =
m,@,%J
+ a2P3) +
+ a,(9@2) + a,(&q]
a3(93)
(27)
with d;” = $9 + [
KY = a,.4 - aA + kc, 41 = Y,, , The tensors Y,, are gravitationally (V3) covariantly as
.A$ = gufl.
(28)
covariant tensors, hence we can rewrite the term tr g1~2(Y,,V*pV,U)
where indices were raised by means of gu*. Consider next 9’. Evaluating yields St' = F&M - i&A'-“ -.A$V-" - 3(a,,F)F-1.A'"u -(nF)F2]
(29) this term (30)
and noting after some algebra that in n-dimensions Z',,“, = (n/2)F,,F-l
R = (n - l)[(OF)
F-2 + (n - 6)/4(8,F)2F-3]
(31)
QUANTUM
GRAVITATION
WITHOUT
207
GRAVIn
we find that X s F-Y8
= (,,#I - D/V”
- Jy;.N@ - +R).
(32)
D, is the gravitationally covariant derivative which acts only on the Lorentz index p of Nu: D,.H” = a,NU + F;U~O. The tensor X is thus a Lorentz-scalar and we succeeded in writing also g3 and &?@ in covariant form as $3 z.z gW(X)3;
(5w&J
= gl~2(xv,,v~“).
(33)
We now consider the more difficult task of rewriting the derivative terms (4U’@.,,)2 and (6,?&2 in covariant form. The latter yields upon evaluation (&@2
= g’/2(duX)(d”X)
+ (l$J’)2(X2)
+ 2(&F) F(XA,X)
(34)
where d,X = 8,X + [Jv; , X] and MU = g,,JlrV. The aim is now to rewrite this expression as a gravitational scalar density. The derivative d, in (Xd,X) can be replaced by the ordinary derivative 3, because tr(X.NUX - WXNU) = 0. Partial integration of the last term in Eq. (34) yields a term -(3,3,F)(FX2), and using Eq. (31), this term can be written as -g1/2($RX2). The other F-terms cancel. Since X is a gravitational scalar, the derivative d,, in d,X is already gravitationally covariant. Denoting by D, a derivative which is both Yang-Mills and gravitationally covariant, we have thus (6,&)2
= gl’2[(DuX)(DJ9q
- +R(x2)]
(35)
which is again a gravitational scalar density. Finally we tackle the term (8,,@‘,,)(8,@,,). All explicit F-dependence drops out not only in p,,. but also in o^,&@, , if one uses Jv; instead of NU throughout. Since s,%?u, = F( lDu%P,) - (a,F)(w,)
(36)
one has (&@u,)(&R’,,)
= g’/“(( D,w,)(DAwy
- 3( F,F)( i$F) F-y~~p)
- 2W’)(A~“ryn,Wu,)). Partially
integrating -2@,F)(A,W,)
(37)
the last term yields (omitting ~‘,y = 2g1~2F,u~F-1(WyW*)
integral signs) + &( a,F) F-‘%,(CVy,,‘%y,,)
(38)
where we used the cyclic identity in Eq. (11). Integrating the last term by parts, and using that if g,, = Fa,, the Ricci tensor is given in n dimensions by R,, = CW-lFLOF
+ (n - 2)
Fwl + (4F2)-‘I(6
- 3n) FJ,.
+ (n - 4)(F.J2 %,I (39)
leads, along with Eq. (31), to the desired result (d^,@,J(&&,) 595/1=‘4/1-I4
= g1~2(DpW‘y)(D,Wv)
+ R,,(WW”,)
- bR(tVY,,Wv)].
(40)
208
P. VAN NIEUWENHUIZEN
We have therefore shown that the one-loop counter terms of the action in Eq. (23) for the particular case of g,, = F(X) 6,” coincide with the following scalar density, provided one also puts g,, = F(x) S,, in this expression. LILT = gliz tr[aI{(UPV,,)(D~V~“)
+ R,,(V+P,)
- PR(V,,V~“)}
+ ~2w/uy~“~~/rro)+ %W) + ~4WVU”VU”) + a,{(D,X)(llP’X) - QRXZ}].
(41)
The symbol V’p was defined in Eq. (28), X in Eq. (32), Lorentz indices are lowered and raised by means of g,, and gU” while the derivative D, is both gravitatjonally and Yang-Mills covariant. For completenesswe repeat
We stress that Eq. (41) is not the general counter term for any metric g,,; only if we substitute g,, = 6,,,F in Eq. (41) does one obtain the general counter term belonging to Eq. (23) with g,, = 6,,F.
6. COMPLEX
SCALARS IN A GENERAL
CURVED SPACE
In this section we complete the counter terms, which were derived for conformally flat spaces,such that they hold for general curved spaces. Let us begin by writing down the Weyl tensor C,,,, . Its trace vanishes identically and the tensor itself vanishes when the spaceis conformally flat. In n-dimensions CLI”mJ + = R uvoo
(n - 2)HLgvo - hog,, + Rug,, - &gwl - Kn - I)@ - 2)1-k,gvo - guogv,lR.
(43)
For n = 4 one obtains a familiar result. It is clear that Eq. (41) is only determined UP to cww = 0. For example, we could have replaced R,,VuUV~, in the term with a, in Eq. (41) by an expression involving R,,,,oB and R using Eq. (43). However, we do get the most general counter terms if we add to Eq. (41) all possible, linearly independent, gravitational scalars containing at least one tensor C,,,, . Generically the list of such counter terms is given by C3,DDC2,C2X,C2V,CDDX,C[19DV,CX2,CXV,CV2
(44)
where we have omitted all terms containing either R,, or R because later on we will insert the Einstein equations R,, = R = 0. (We recall that this is the same in the background field method asin normal field theory putting all externa! gravitons on their mass shell with transverse-tracelesspolarization tensors.)
QUANTUM
GRAVITATION
WITHOUT
GRAVITY
209
Since CaOYsgn;Svanishes, the terms %UIL[DX, %5X2, and @XV vanish, since they contain at least one contraction of the indices of C. Consider next the terms DDP. Repeated use of the Bianchi identity @‘BysC,aBy;6= 0 reduces these terms to C3 when R,, := R = 0. The terms C?V and CDDV (which are, by the way, not antisymmetric matrices) will vanish due to a simple theorem (called shift theorem hereafter) which we now derive. Due to the cyclic identities satisfied by the Weyl tensor (which has the same symmetry properties as the Riemann tensor) one has in graphical notation the following equality:
It follows
that one can always
bring the term C2V on the form (46)
Similarly one easily shows with Eq. (45) that the term CDDV because
can be eliminated
where we used that the commutator [D, , !3,] V,,, is proportional to RaDyBVyS and to P,,, , v,,] when R,, = 0. This leaves us with the three generic terms C”, C2X, and CV*. Using the shift theorem again, one immediately seesthat the last two terms can always be brought to the form CuBvsC31*y% and Ca~~6V/aBVvs, but perhaps less obvious is the fact that there are only two algebraically independent scalars of the form C3. To see this we note that one can always use the symmetries of the Weyl tensor to cast any scalar C3 in the form
When the contraction which starts at 01,turns left then, according to the shift theorem, there is only one invariant
u,=c
1 I 1 ICI
11____IZZ.
(49)
When. however. the contraction which starts at ai, turns right, it reproduces U, when it ends at y (use the shift theorem for the two C tensors on the right of Eq. (48)). Hence we consider the case that 01ends at 6 and 18at g. This still leaves two ways
210
P. VAN
NIEUWENHUIZEN
in which to complete the final contractions, will be called U, U2=C)
r-p7
p to y and p to E. The first possibility
a
) I\.
(50)
Using the cyclic identity, one easily derives that the second (PC) contraction is related to U, and U, by
u,=c
I,‘rl~
1 =au,i-U2.
Since in four dimensions the two-loop divergences of pure gravitation are of the form ~~(n - 4)-l@ .s.)~, it follows that the algebraic relations derived above seem to be of use for such a future two-loop calculation. At this point we have exhausted the algebraic relations between the counter terms. However, there still exists an analytical relation, the so-called Gausz-Bonnet theorem. This theorem, which holds in any even number of dimensions [l], asserts that under a small variation g,, + g,, + h,, for arbitrary g,, (not necessarily satisfying R,, = 0) and h,, , the variation of a certain scalar vanishes; in six dimensions: (52)
Hence one may omit the scalar Z from the action, since it does not contribute to the field equations (nor to boundary conditions, since g,, is arbitrary). The scalar Z itself probably diverges when g,, satisfies the Einstein equations; however, divergences in an action are quite common, e.g., energy-momentum conserving delta-functions, and do not worry us here. Expanding the product of the two E-symbols in terms of Kronecker delta functions tells us that, as far as the action is concerned,
s
d6xg112[Ul + 2U2] = 0.
From the algebraic and analytic identities
(53)
discussed above, it follows that when
R,, = R = 0, there are only three counter terms which have to be added to Eq. (41)
in order to obtain the full counter term for the action of complex scalars in a general background metric g,, AP(conf)
= g112tr[u,JJ, + a,C~~@fuvV,,o + a8ColB,,sCo~y%].
(54)
We now fix the coefficients a, and u, . Since in the next section we show that tr(X) vanishes when R,, = 0, the last term in Eq. (54) will not contribute, hence we need not fix a,. The statement that R,,(g) = 0 means, in the language of Feynman diagrams, that external gravitons (for example each of the three external gravitons
QUANTUM
GRAVITATION
WITHOUT
211
GRAVITY
q;q”-“(
C
FIG. 2. The diagrams which determine u6.
in each of the diagrams in Fig. 2) are transverse and traceless. For the S-matrix this means that expanding R,, in terms of g,, and gIIV = 6,” + K/Z,, , the fields /I,, satisfy Oh,, = 6,,h,, = aAhAv= 0. Unfortunately, when all three gravitons in Fig. 2 are on-shell (R,, = 0), then the terms in U, proportional to K3 vanish. When one of the three gravitons in Fig. 2 is off-shell, then there are three possible counter terms which might contribute: U, , RYVRwaB,,RYbB~, and RRaByaRaSyS. There are also three different terms which could come from evaluating Fig. 2:
where Iz,,(i) is the polarization tensor of graviton i, and p3 the off-shell graviton. However, also in this case the three counter terms are linearly dependent, as one may easily verify. Therefore we consider in Fig. 2 all three gravitons off-shell. There is a large number of counter terms which could in principle contribute, such as U1 and RBuV,R3, R(R,+$ etc. However, there is precisely one kind of term which can only come from U, , namely (55)
K3hamAs,a&cwy .
Only diagram 2A contributes to this term and we proceed to fix the coefficient a6 in Eq. (54) by evaluating diagram 2A. The S-matrix contribution of dZZ’ must cancel the pole at (n - 6) in the S-matrix due to the action in Eq. (23). The former stems from the term JcPx~~/~(cI,U~) in Eq. (54) and yields S,&2A)
=
~~6ic7,
lj
puih(pj)uvpvkGW6 %+
- r2 - p3).
ifjfkfi
On the other hand, the contribution from diagram 2A is found to be S(2A) =
iK3E
j-j
[p,ih(p’),,p,k]
l/15 x l/168 s6(p1 -p”
- p3)
t 57)
from which we conclude U6= -l/(90
x 168)E
(58)
with 6 given by Eq. (17). Finally, we fix the coefficient u, . In this case we put the gravitons in Fig. 3 on-shell. This means that diagram 3B vanishes since it is of the four q3*Nu”ii,,qShAA . Diagram 3A contributes both to a, , the coefficient of the invariant (D,M““)( DAVA,), and to u, . However, if we restrict the source .MJ~to satisfy a,,&.N~ =aui,Jr/-”=- 0, only u, receives contributions. One finds (omitting a factor K ~85~)
212
P. VAN NIEUWENHUIZEN
A
-
B
FIG. 3. The diagrams whichdeterminea, .
while S(3N = ~~/45~~~~~“~~‘~l~~tJlr’~p5~l~~:~~~~,~
dl.
(60)
It follows that a, = 1/(18O)C
(61)
This completes the determination of the counter terms of the following action for N complex scalar fields & in a general background metric g,,, I =
s
d6xg112tr[-a,+*
guva,4 + 24*.&-u a,+ + 4*&?4]
(62)
where the sources J+‘-wand A are not necessarily (anti)symmetric. When we are on-shell, R,, = R = 0 and (seealso Eq. (42)) A9
= .5g1i2tr[-(l/120)(D,VU”)(DAVA,) + (1/24)(XV,,V““)
+ (I/180)(Y,,V”~V,U)
- (1/24)(D,X)(WX)
+ (1/12)(X)”
- (l/ 15120)(R,,,,R”~olBR40,Y~)
+ (l/l~o)(R,",,~~"~~")l
(63)
with E = [329(n - 6)1-l. In the next section we apply this general formula to pure gravitation.
7. PURE GRAVITATION
IN SIX DIMENSIONS
The preceding sections led up to a general formula for the one-loop counter terms of N complex scalar fields in six dimensions in a general background metric g,, . In this section we will consider the system of pure gravity and cast it in the form of scalar fields. The discussion follows at most points Ref. [I] but at some points alterations are needed due to the different dimensionality of space-time; for completenesswe record all steps below. We first add a gauge fixing term to the one-loop Lagrangian and obtain the corresponding ghost Lagrangian. Subsequently we rewrite both the nonghost and the ghost action in such a way that these actions are of the form of scalar fields in a given background metric g,, . Then, in the next section, we will apply the general formula for the counter terms to these two actions. The classical action for pure gravity in six dimensions is given by Z = -21~~ 1 d6x glizR(g).
(64)
QUANTUM
GRAVITATION
WITHOUT
213
GRAVITY
The expansion of this action in quantum fields does not differ from the treatment in four dimensions. Since ultimately we will insert the classical field equations. i.e., the Einstein equations R,, = 0, we will use consistently R,,. := R -::: 0. The one-loop action is then given by -y == g’P[ -&,&@:u
+ $h,h“ - $hwhi~l + #;Uh;‘L +
4/71,1(R”LLBv
+ RnL,RJ’)/7,,]
(65)
(after dividing by a factor 2 in order to conform with Ref. [I]) where /I, =: D”(g) /I,, , == gvvh,,, and all tensor operations, including the covariant derivative (;). are performed with gU.. Adding the usual gauge fixing term h
LP = -4 g1/2[e,u(Dv( g)
/I,,”
- iD,( g)h)]”
(66)
where eoU is the vierbein field, en?,,v = gb”‘, we find the usual result 2’ + LP = g”2[-~(Dyh~4)(PUaUV)(DYhllV)
$- ihwi5( R”I”” + RL1*@@) /7,&l
where p1l3Lu= 4( gaugol + gevgN’, _ g”ag,w), Doubling the number of gravitational fields as in Section 3, taking the inverse of P (the graviton propagator in n dimensions) (p-lLrQu and redefining
g+
= guogw + &o g”” - (2/n - 2) g,,g,,
(67)
h*,.I’ = h:BPaBuu,we obtain ~2’~= g1’2[-(D.,h*CIR)(DYh,,)+ ~h*9&,( R”““” + R’“““) h,,,].
(68)
We consider the field h**B as the complex conjugates of h,, (see Ref. [l]). Rewriting D,,DYh,8 in terms of derivatives B,, , 6? which do not act on the internal indices n/3 of h,, yields the same result as in four dimensions. We thus find that
(71) where C$ and pu mean that only the part symmetric in (c&) and (pa) is to be taken. One easily findsthat the nonghost Lagrangian is characterized by (W”‘),c
= (RR(;;I -;~ RzB”),
(~~G)nuDo = (-2R”,,&)$.
(72) (73)
214
P. VAN NIEUWENHUIZEN
Turning
to the ghost Lagrangian 9’
= g1’2[q;g”vDADA~s,- T,I,*R”“Q]
(74)
one finds in the same way
(xi”): = 0;
(Jy;G>aB = 4:.
Finally we determine the quantities [19,X and D,VUu. The reader may verify the matrices Jy; in D,X provide just the necessary Christoffel symbols to make D,X a covariant object D,(XNG):;I = D,(R//
+ R,“,“)
(76)
where as we recall, D, is the usual gravitational covariant derivative. In the same way D,VU” is a covariant object, but since RaBuVi~= 0 module R,, = R = 0, we have (77)
For the ghost contribution
we get in the same way
D,(x”)p = D”(V;“),J = 0.
(78)
The terms (DJXNG)(lD~XNG) can be cast in the form R3 ogvaby using Eq. (76), integrating partially, using ~~~~~~~~~~~~ = 0 and [D,, , D,] R,,, = -R&,R,By,, -1. - R&,RmB,,A . One finds (DJ~G)(lDr’X~G) = -9u,, . (79 where we used the graphical methods of Eq. (45) and the Gausz-Bonnet theorem in Eq. (53). In the next section we will insert the explicit expressions for X, V,, , and D,XNG into the general formula for d-9.
8. ONE-LOOP
DIVERGENCES IN THE S-MATRIX IN SIX DIMENSIONS
FOR PURE GRAVITATION
Finally we are in a position to determine whether the S-matrix for pure gravitation in six dimensions is finite at the one-loop level or not. We have on the one hand derived a general formula, Eq. (63), for the one-loop divergences of the general scalar system in Eq. (62). On the other hand we have written the gravitational oneloop action in the form of Eq. (62) with definite matrices JP and A. In this section we insert these given matrices into the general formula Eq. (63) and by evaluating the traces over the internal indices, we obtain the required result. The evaluation of the traces is straightforward. All terms yielding terms of the generic form (R a..)” we use Eqs. (45) (51) and (53) to obtain d9 = ag1gl12U1,
QUANTUM
GRAVITATION
WITHOUT
GRAVITY
215
TABLE I Traces of Nonghost Contributions
TABLE II Traces of Ghost Contributions a, = 0 a2 = (v~y~,6(v”~)6~(Y~~)yb:~ = u, = -gY, as = 0 a4 = 0 a, = 0
where 01is the required coefficient. In Table I we summerize the results of the traces over the nonghost and ghost matrices. Adding them all up yields
-&+$+&-1 - -?I180
180 “, 84
+,1,11-
21
180 x 84 +&!
(80)
The factors & and -1 in front of the curly brackets account for the “dedoubling” of h,, = (h$ + ihlr28))2-112 and the fermion character of the ghost vertices. The numerators 21 and 6 in the term with 180 x 84 are the number of independent
216
P. VAN
NIEUWENHUIZEN
components in the trace over internal indices: 21 fields h,, and six ghost fields & . The total counter terms for the s-matrix is thus given by ddp = (9/l 120) g1~2(RbBysRSvPoRopBo)(32~3(n - 6))-l .
9.
031)
CONCLUSIONS
Covariantly quantized gravitation without matter sources has an S-matrix which is finite at the- one-loop level in four dimensions. However, the significance of this result is not clear because, on invariance grounds, the sum of all divergences of the sum of all S-matrices must be at the one-loop level of the form sdiv(l
- loop, 4 dim) = 1 d4x g1’2[aRuVRuY + pR2](n - 4)-l
where we used the Gauss-Bonnet theorem to express RpypoRuYoainto R RuY and R2. Since the Einstein equations imply R,, = R = 0, this expression vanisik no matter what 01 and /3 are. In order to test the divergence properties of pure gravitation without sources in a less ambiguous way, we considered quantum gravity in six dimensions; admittedly not a very physical theory, but containing some of the features of a two-loop calculation in four dimensions and much simpler than such a two-loop calculation. We found that the sum of all one-loop divergences of all S-matrices in six dimensions is of the simple, nonrenormalizable, form sdiV(l
- loop, 6 dim) = 1 d6x g1’2[yR,B,,, omuRv~a](n - 6)-l
where we used the Gauss-Bonnet theorem in six dimensions and identities. This invariant does not vanish when R,, = 0; the proof will be published elsewhere [15]. The coefficient y has to vanish cancellations if the S-matrix in six dimensions is to be finite. After tion we found
various algebraic of this statement due to nontrivial a lengthy calcula-
y = (9/l 120)(32 r3)-l
and we conclude that quantum gravity without sources in six dimensions behaves as matter-gravity interactions in four dimensions; the theory is nonrenormalizable and the S-matrix diverges. What does this result imply for the renormalization properties of pure gravitation in four dimensions? The local two-loop divergences of the S-matrix in four dimensions are of the same form as the one-loop divergences in six dimensions &iV(2 - loop, 4 dim) = (n F2 4)2
s
d4x g1’2[~R,4~oRuPT”Rv~l
where we have used the algebraic identities which were derived in section 6 and the
QUANTUM
GRAVITATION
WITHOUT
217
GRAVITY
fact that the invariant RU,,ooRuh"TRTYAP vanishes when R,, ==0 [13, 14. 151. The invariant between square brackets does not vanish when R,,. = 0 [I 51. Extra cancellations are needed in order that the coefficient E of this nonrenormalizabie counter term vanishes, and since such cancellations are absent in six dimensions, we believe that there is little likelihood that quantum gravity without matter sources will have a finite S-matrix in four dimensions. Or is, perhaps, quantum gravity in four dimensions an exceptional case and is that why our world is four dimensional? ACKNOWLEDGMENTS The author is grateful to Professors R. P. Geroch, G. ‘t Hooft, M. Veltman, and C. N. Yang for very helpful discussions. REFERENCES I. G. ‘T HOOFT AND M. VELTMAN, Ann. Inst. H. Poincart 20 (1974). 69. 3. S. DESER AND P. VAN NIEUWENHUIZEN, Phys. Rw. Lett. 32 (1974). 245: Phys.
Rev. D 10 (1974),
401. 3. S. DESER AND
P. VAN NIEUWENHUIZEN,
Lett.
Nuovo
Cimento
11 (1974).
218; Phys.
10 (1974).
Rro.
411. H.-S. TSAO, AND P. VAN NIEUWENHUIZEN, Phys. Lett. B 50 (1974). 491 : Phys. Rer. D 10 (1974), 3337. 5. M. T. GRISARU, P. VAN NIEUWENHUIZEN, AND C. C. WV, Phys. Rer;. D 12 (19X), 1813. 4. S. DESER,
6. D. Z. FREEDMAN AND E. WEINBERG, Ann. Phusics87 (1974), 354; D. 2. FREEDMAN, I. J. MUZINICH, AND E. WEINBERG, Ann. Physics 87 (1974), 959: D. Z. FREEDMAN AND SO-YOUNG PI, Ann. Physics 91 (1975), 442. 7. D. G. BOULWARE AND L. S. BROWN, Phys. Rec. 172 (1968). 1628: Ref. [8]; J. HONERKAMP, Nucl. Phys. B 48 (1972), 269; Ref. [9]. 8. B. S. DEWITT, Phys. Reu. 160 (1967), 113; 162 (1967), 1195; 162 (1967). 1239. 9. R. KALLOSH, Nucl. Phys. B 78 (1974), 293. The proof in this excellent article needs a modification since in the axial gauge the gauge fixing term (h:,e)z is not invariant under background gauge
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Phvs.
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3203.
12. G. ‘T H~~FT AND M. VELTMAN, Nucl. Phys. 844 (1972), 189; C. G. BOLLINI AND J. J. GIAMBIAGI, Phys. Lett. 840 (1972), 566; J. F. ASHMORE, Nuouo Cimento Lett. 4 (1972), 289; G. M. CICUTA AND E. MONTALDI, Nuovo Cimento Left. 4 (1972), 329. 13. This may be proved by decomposing the Lorentz indices of Rp,,, into spinor components (CL=- 4 @ &) and using the symmetry of the Rieman tensor and R,,. -7. 0. 1 am indebted to
Dr. ‘t Hooft for pointing this out to me; see also Ref. (141. 14. See PIRANI, in “Brandeis Lectures in Theoretical Physics,” MIT Press, Cambridge, 15. P. VAN NIEUWENHUIZEN AND C. C. Wu, J. Math. Phys. 18 (1977). 182.
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