nuclear physics bi70[fs1] (1980) 283-306 quantum

Nuclear Physics BI70[FS1] (1980) 283-306 © North-Holland Publishing Company

Q U A N T U M G R A V I T A T I O N A L B U B B L E S

S.W. HAWKING, D.N. PAGE I and C.N. POPE

Department of Applied Mathematics and Theoretical Physics, UniversRy of Cambridge, Silver Street, Cambridge CB3 9EW,, UK

Received 18 December 1979 (Final version received 21 March 1980)

Spacetime is expected to have a "foamlike" structure on scales of the Planck length or less with high curvatures and complicated topology. This foam can be thought of as being built out of three basic kinds of units or "gravitational bubbles", CP 2, S 2 × S 2 and K3. We investigate the propagation of particles in simple models of the first two types of bubble. The non-trivial topologies of the bubbles introduce extra singularities into the Green functions. These make large contributions to the S-matrix for scalar particles but only small contributions for spin-~ or 1 particles at energies small compared to the Planck length. These results suggest that there is no inconsistency between the spacetime foam picture and everyday observations from which spacetime appears nearly flat, because all the elementary particles we have observed have spin ½ or greater. They do, however, suggest that Higgs scalar fields, if they exist at all, are probably bound states of higher spin particles rather than being elementary fields. Further developments may enable one to calculate processes in which quantum coherence is lost and intrinsic entropy is produced.

1. In t roduct ion

The a im of this p a p e r is to expla in in more de ta i l the results ou t l ined in [1] on the

p r o p a g a t i o n of par t ic les in space t ime foam. Because the g rav i ta t iona l ac t ion is scale

d e p e n d e n t one expects large f luc tua t ions of the met r ic a n d topo logy on scales of

the P lanck length or less. On the o the r hand , space t ime appea r s to be smoo th a n d

near ly f lat f rom e v e r y d a y observa t ions . W e shall show that this m a y be because all I the e l emen ta ry par t ic les that we know have spin i or greater . Space t ime might

a p p e a r very d i f ferent if we were c o m p o s e d of e l emen ta ry scalar par t ic les .

W e a d o p t the euc l idean a p p r o a c h to q u a n t u m gravi ty [2, 3]. In this the pa th

in tegra l is eva lua ted over pos i t ive-def in i te metr ics . The v a c u u m state is de f ined by

an integral over all a sympto t i ca l l y euc l idean metrics , i.e., met r ics which a p p r o a c h

the f lat met r ic on R 4 ou ts ide some c o m p a c t set in which the topo logy m a y differ

f rom tha t of R 4. W i t h sui table fal l-off condi t ions , such metr ics can be c o m p a c t i f i e d

b y a d d i n g a po in t I a t inf in i ty a n d pe r fo rming a con fo rma l t r a n s f o r m a t i o n to m a k e

the met r ic f inite a n d su i tab ly s m o o t h there. One can therefore reduce the ques t ion

I Present address: Department of Physics, Davey Laboratory, Pennsylvania State University, Univer- sity Park, Pennsylvania 16802, USA.

283

284 S.~i:. Hawking et al. / Quantum gravitational bubbles

of the classification of the topologies of asymptotically euclidean metrics to that of compact ones. If the manifold M is simply connected (which we shall assume, see [4]), one can show that the topological sum of M with a certain number of copies of $ 2 × S 2 is diffeomorphic to a sum of copies of S2X S 2 and either K3 or anti-K3 surfaces (in the case of manifolds that admit spinor structure) or copies of CP 2 and anti-CP 2 (in the case of manifolds without spinor structures) [5]. I t seems that the number of S 2 x S 2 's that one needs to add is small and may actually be zero even

when the Euler number is very high.

One can thus regard spacetime as being made up of the three kinds of basic building blocks: S 2 x S 2, K3 and C P 2. Pursuing the analogy with foam, we shall

call these 'gravitational bubbles ' . It should be emphasized that, unlike gravitational instantons [2, 6, 7], they are not solutions of any field equations. Indeed by the positive action theorem [8], there are no asymptotically euclidean solutions of the field equations except flat space. However, these bubbles will occur as quantum fluctuations. The fact that they have non-trivial topologies means that they do not admit a time function like there is in R 4. One might, therefore, expect that Green functions defined on them would show certain acausal properties and would mix positive and negative frequencies when analytically continued to the lorentzian regime at infinity. In this respect they would be similar to the Green functions on the euclidean black hole metrics, which have periodic poles in imaginary time when continued to the lorentzian section. In fact, one might regard a gravitational bubble as a virtual black hole which appeared out of the vacuum and then disappeared

again. One would expect the foamlike structure to exist throughout all of spacetime.

However, for the purposes of interpretation it is convenient, as in ordinary particle physics, to imagine that there is an asymptotic region in which the fluctuations and

interactions are turned off and in which the fields can be regarded as propagating freely in a background metric which is nearly flat. This allows one to interpret the asymptotic in and out fields in terms of particles and to define S-matrices or more general operators which give the transformations from the initial to the final situations. In quantum gravity it is difficult to define gauge-invariant observable

quantities except in such asymptotic regions. In general it is not possible analytically to continue f rom a positive-definite

metric to a section of the complexified spacetime manifold on which the metric is real and lorentzian. This is another reason why it does not seem to make much sense to talk about quantities in the interacting region. However, in asymptotically euclidean metrics, one can analytically continue to Minkowski space at infinity and define in and out states in the normal way. In the case of zero rest mass particles, to which we shall restrict our attention, this can be done in a convenient manner, by giving the Cauchy data for the asymptotic states on null infinity ~ [9], the null cone of the infinity point I. This provides an invariant way of defining in and out states that are asymptotically plane waves [10]. One then propagates the in states to

S.W. Hawking et al. / Quantum graoitational bubbles 285

the out states using the Green functions for the fields analytically continued from the positive definite metric, weights with the action of the metric and the determinants of the operators of any fields present and integrates over all asymptotically euclidean positive-definite metrics. In the case of fields that interact only with gravity, this will give the full S-matrix amplitude with all quantum

corrections. Of course such a calculation is beyond our powers: we do not know the Green

functions in an arbitrary metric and we do not know how to perform the infinite-dimensional integral over all metrics. The procedure that is normally adopted is perturbation theory around flat space. However, this could not be expected to work for large, topologically non-trivial fluctuations and, indeed, it fails to converge even in classical general relativity when there are no topological difficulties [11]. In quantum theory the failure of perturbation theory is manifested by the fact that it is non-renormalizable: in order to make the theory finite one needs to make an infinite number of subtractions with undetermined remainders. It

is likely that this is also true in supergravity. Instead of perturbation theory we shall adopt a finite-dimensional approximation:

choose a family of metrics that depend on some finite but arbitrarily large number of parameters, integrate over the parameters with a suitable measure and take the limit as the number of parameters becomes infinite. An example of a suitable family would be the Regge calculus metrics which are built out of flat 4-simplices that are connected together with distortions or delta functions of the curvature on their boundaries [12]. However, there are other examples in which the metrics are smooth. If one performs such a finite-dimensional approximation in supergravity, one would hope that the divergences which would arise as one increased the number of parameters would cancel between the fermions and bosons.

What we shall use in this paper are only very crude finite dimensional approximations to asymptotically euclidean C P 2 and s E x S 2 metrics. The C P 2 bubbles will have 7 parameters, 4 for position and 3 for the orientation and scale, whilst the S 2 × S 2 bubbles will have 8 parameters, 4 for position and 4 for orientation and scale. However, this is sufficient to show that topologically non-trivial metrics do introduce extra acausal singularities into the Green functions. Moreover, the spin and momentum dependence of the amplitudes produced by the extra singularities seems to be given by dimensional requirements and is likely to hold in any metric with the same topology. This belief is reinforced by the fact that we get similar amplitudes for scalar particles in C P 2 and S 2 × S 2 metrics. We find that at energies small compared to the Planck value, the amplitudes are of order ( k l . k E / m 2 p ) s

where s is the spin, k I and k 2 are the in and out momenta and m p is the Planck mass. Thus the amplitudes are of order 1 for scalar particles and could give rise to an effective mass of the order of the Planck mass. The amplitudes are very small for fermions and vectors. In the case that there is more than one particle propagating in the bubble metric, the integration over the position and orientation

286 S.W. Hawking et al. / Quantum gravitational bubbles

of the bubble gives rises to an effective interaction between the particles. Scalar particles acquire an effective ~tp 4 vertex with ~ ~ - 1 and also higher vertices of the

2n 2(n--2) form q0 //mp . Fermions and vectors acquire 4-particle vertices of the form ( ~ ) 2 / m 2 and 4 4 (F~) /mp plus higher vertices which will be very small because of the powers of m p involved.

One can think of these effective vertices as arising from the exchange of very large numbers of gravitons. However, they are really non-perturbative effects and differ from ordinary graviton exchange in having different dependence on the momenta. For example, the effective 4-particle vertex produced by perturbative graviton exchange will be proportional to the square of the momenta whereas the scalar 4-particle vertex we obtain is independent of the momenta. Our effective vertices also have certain acausal properties which will be discussed elsewhere. They might transcend certain selection rules such as baryon conservation. In this way one might be able to distinguish the effective 4-fermion interaction from weak interactions which have the same low-energy form but with the Planck mass replaced by the mass of the W boson.

In sect. 2 we discuss the conformal compactification of asymptotically euclidean metrics by the addition of a point I at infinity and its light cone g, null infinity. The Cauchy data on g for the asymptotic in and out states are defined in sect. 3 for spin

1 0, ~ and 1. Sect. 4 deals with CP 2 bubbles: in subsect. 4.1 the metric and Green function are described, in subsect. 4.2 the scattering amplitudes are derived and in subsect. 4.3 they are weighted with the action and integrated over the parameters of the bubble to give the full S-matrix. Sect. 5 does the same for a certain class of S 2 × S 2 bubbles. Finally, in sect. 6 the conclusions that can be drawn are discussed and the limitations of the approximations used are pointed out.

2. Null infinity

The metric of flat space is

ds 2 -- dx~dx~ = ( d x ° ) 2 + (dx l ) 2 + (dx2) 2 + (dx3) 2 . (2.1)

The euclidean section is defined by x° ,x l , x2 ,x 3 real. The lorentzian section (Minkowski space) is defined by t, x 1, x 2, x 3 real, where t-~ - i x °. One can also define retarded and advanced null coordinates u and v by

u----t--r, c - ~ t + r ,

r = [ ( x l ) 2 + (x2) 2 + (X3)2] I/2 • (2.2)

Then the metric becomes

ds 2-- - d u d v + ¼ ( u - v) 2 dfl 2. (2.3)

S . W . Hawking et al. / Quantum gravitational bubbles 287

In order to discuss the behaviour of fields at infinity it is convenient to add a point I at infinity, and its null cone ~. The easiest way of doing this is to perform an

inversion in the sphere of radius b:

b 2 x'~' = - - x ~ (2.4)

X2 "

This brings the point I at infinity to the origin and the null infinity surface ~ to the light cone of the origin. The origin of the original spacetime and its light cone are sent to infinity. The primed space has a metric conformal to that of the original

space,

ds '2 = d x '~ dx '~ -- fll 2 d s2 , (2.5)

where

b 2 b 2 x '2 _ u'v.._..~'

~ 1 = X--~ = - - U--V---- b 2 = b 2 , ( 2 . 6 )

b 2 b 2 u ' = - - - , v ' - - - - - ( 2 . 7 )

13 u

One can define asymptotic momentum states for massless particles by their Cauchy data on the null surface ~ and propagate them into each other by using the conformally invariant Green functions either in empty space or in some background Yang-Mills field [10]. In this way one can define S-matrix elements in the presence of Yang-Mills instantons. In this paper we shall use a similar method to calculate S-matrix elements for particles propagating in asymptotically euclidean metrics.

One first adds a point I at infinity and performs a conformal transformation to make the metric regular there. The conformal factor can be fixed by the gauge

condition

R~,,I~'I " -- O, on ~, (2.8)

where P' is the null vector tangent to ~. This determines the conformal factor on given its value and gradient at I. Changing the value at I corresponds to rescaling the asymptotically euclidean metric and changing the gradient corresponds to translating the metric with respect to infinity. Condition (2.8) is the analogue of the gauge condition on ~ that was imposed in the Yang-Mills case [10]:

A~,P' ---- 0 , on $. (2.9)

One now defines the asymptotic momentum states in the asymptotically euclidean metric by taking their Cauchy data on ~ in the conformal gauge (2.8) to have the


same values as the data for the states in flat space in the conformal frame ds '2. One then propagates the states into each other using the Green function analytically continued from the euclidean sector.

3. The Cauchy data

we adopt relativistic normalization for the particle states. A scalar field cp in flat space will be expressed in terms of

2 ,,,b,/~, 8(0 - 0k)8(~0 - ~0k) (3.6) f = (2~r)1-~2 ou, e sin-O '

where w = k ° and (0 k, ~0k) denotes the direction of the spatial momentum k [10]. The antiparticle state I k, - ) = bt(k)l 0 ) has Cauchy data with positive frequency part

rplk,- ) =o, ~lk,- )--flO) • (3.7)

The data for the final state ( k , + [ = (Ola(k) on ~ have the negative frequency

As in ref. [10] massless complex annihilation and creation operators as

____!__1 (d3_ = (2 y/2 J ° [ a( )e ikx + - 'kx] . (3.1)

Similarly the adjoint field ~, which is the analytic continuation of the lorentzian complex conjugate ~, can be expressed as

1 f 3 2k ° [ b (k )e + ~ (x ) = (2 ~r_)a_~ j d_..~- k ik.x af(k)e-ik.x] (3.2)

The operators a t and a create and annihilate particles while b t and b do the same for antiparticles. They obey the commutation relations

[ a ( k ) , a t ( U ) ] = [ b ( k ) , b t (k ' ) ] =2k°83(_/£-_k') . (3.3)

The scalar product between fields qo I and ~2 is defined by

i f ~ , LW2 d X " . (3.4)

The one-particle state with momentum k is I k, + ) - - a t (k ) [0 ) . In the conformal frame ds '2, the positive frequency part of the Cauchy data for this state is

~ [ k , + ) = f [ 0 ) , ~11,,+ ) = 0 , (3.5)

S.W. Hawking et al. / Quantum gravitational bubbles 289

part

where

<k,+[~--O, <k,+l~=< Olg, (3.8)

2 e_,,~b21~, 8(0 -- Ok)6(q~ -- ~ , ) (3.9) g---- (27r)l/2to v, sinO

Similarly the data for < k , - [= <0[ b(k) have the negative frequency part

< k , - I~0-- <0lg, ( k , - I ~ = 0 . (3.10)

For fermion fields we shall use the 2-component notation. A 4-component upper spinor index a will correspond to a 2-component upper spinor index A and a lower index A'. Thus a Dirac field ~k '~ will be represented by the column vector

The adjoint field ~,~ will be represented by the row vector

~ = ( g a , P ' ) . (3.12)

The fermion field operators Lp and ~ can be represented by

gA' (2rr) 3/2 f - ~ [ l ~ A , [ a L ( k ) e ' k ' x + b * R ( k ) e - ' * ' x ] (3.13) 9

21/4 ( d3k [ k F ~a = (gA, ~A') = (27r)3/-----S j 2k0~ ALbrt(k)e ik'x + a*L(k)e-ik'x],

l~a'[ bL(k)e ik'~ +atR(k)e-ig 'x]) , (3.14)

where kafc A' ---- oa'4a'k~ is the spinor equivalent of the propagation vector k~,. The operators a R and a~ are the annihilation and creation operators for particles with right-handed helicity, a L and a* L are the operators for particles of left-handed helicity, and the b's are the corresponding antiparticle operators. The anti- commutators are

{ a R ( k ) , a ~ ( k ' ) } = 2k°83(_k-k ' ) , etc. (3.15)


The scalar product between two solutions tpl and Lp 2 can be expressed as

- fqT, (3.16)

= - - v ~ f ( # l A ( t 2 a , + h 2 A X l A . ) d Y . A a ' , (3.17)

where dY AA' = o~ AA' dye .

The Cauchy data for the fields ~b and ff consist of the quantities M I A, ~A,[ A' and tt,41 A, ~A'[ a, on ~, where lA[a '=o~ Aa'I~ is the spinor corresponding to the tangent vector 1 ~ on ~. In the metric ds '2 the positive frequency part of the Cauchy data for the state I k , + , R ) = a ~ ( k ) l O ) which contains a right-handed particle with momentum k are

M I a l k , + , R ) = h l O ) ,

~A. r " ' l ) = , , t ~ I) = ~,"'t'a. I ) = 0 . (3.18)

where

h = 2514b ei,ob2/u, 8 (0 -- 0k)~$(~ -- Cpk ) (3.19) (2qr)l/2tol/2u,2 sin 0

Similarly the positive frequency part of the Cauchy data for the state I k , - , L ) , a left-handed antiparticle, are ~A'I a,[~ = h 10) ; the data for I k, +, L ) , a left-handed particle, are/iaJ'A'l~ = hi0); and the data for [ k , - , R ) are t~AlA[> --- hi0) , with the other quantities being zero in each case. The data for the final state on ~ can be expressed in a similar way.

If A~ is a solution of the equation for the electromagnetic potential in the metric

g~v,

[--]A~, - R~,~A ~ - V~ V"A,, -- 0 , (3.20)

then it remains a solution in the metric fl2g,~. However, the Lorentz gauge condition V~A, = 0 is not maintained. For this reason we shall work with the gauge-independent field strength F~. This can be represented in 2-component form as eas~,4, B, + ea,a,~,4B where, as before, ~,4,B, represents the analytic continuation of the lorentzian complex conjugate ~a,B,. The fields ~AB and ~A'B' are symmetric and obey the equations

VAA'~AB ---- 0 = VA~'~A,B,. (3.21)

These equations are preserved under conformal transformations g~__~2g~,, if


~pAa--->f]-lep.4S and (pa,n,---->~-L~a,s,. The field operators ~Pan and ~a's' can be represented as

2- i/2 3 mAB(X)---- (2 f 2k °d~k-kakn(aR(k)eik'x+atL(k)e-ik'x ) , , (3.22)

2 - 1 / 2 3 (pA,s,(X) = (2 ~r)a/--------- ~ f 2k °d~k- fCA,l~s,(aL(k)e ' k ' x , + a~(k)e - '~ 'x ) . (3.23)

The operators aR(k ) and a~(k) are the annihilation and creation operators for right-handed photons with momentum k. They obey the commutation relations

[ aR(k),a~(k')] = 2k°Sa(k - k ' ) , etc. (3.24)

The scalar product between two solutions ~p~ and ~)2 is

- " ~ - l d ~ S n , . 2i f ")

The operator (VAA,) -~ is the inverse of the operator VAA' which relates the

potential to the field

cpa s = Vc,(A A n) c',

(3.26)

~A'B' =Vc(A 'Ac B')"

It is not gauge invariant but integration over a hypersurface makes the scalar product gauge invariant.

The Cauchy data for the electromagnetic field on ~ consist of ¢pAalAl s and ~A,B,1A'I a'. The positive frequency part of the data for the state [k, R) --a~(k)10) a r e

~AslAlB[k, R) = j [ 0 ) ,

(3.27)

~A,B,la'lB'[k, R) -- 0,

where

23/2b 2 e,Ob~/., 8 (0- 0k)8(~-- cpk ) (3.28) J = (2~r)l/2U,3 sin 0

The data for the left-handed and final states can be expressed in a similar manner.

292 S . W . H a w k i n g et a l . / Quantum gravitat ional bubbles

4. C P z

4.1. THE METRIC

We use the s tandard Fubin i -S tudy Kahler metr ic on CP 2 [13-15]. This can be

expressed in terms of four complex coordinates ~'1, ~'l, ~'2, ~2:

ds '2= o2r_ d~i d~. , i, j = 1, 2 , (4.1) a¢,a j

where

K = l °g (p '2 + ~',(1 + ~'2~2) • (4.2)

The euclidean section is given by ~. = ~. The metr ic is invar iant under a transit ive SU(3) group of isometries. Thus one can take the infinity poin t I to be at the origin and ~ to be the surface

~',(I + ~'2(2 = 0 . (4.3)

The isotropy group of the origin is the group U(2) acting in the obvious manner . The metric (4.1) is a solution of the Einstein equat ions with A = 6 / 0 '2. I t

therefore satisfies the conformal gauge condi t ion (2.8). It is convenient to in t roduce pseudo-eucl idean coordinates

~l = x O t "~- i x l t '

~ 1 ~ x O ' - - i X l t '

In these coordinates the metr ic is

~2 ~ x 2 t "1- i X 3 ' ,

__ X 2t __ i x 3t 2 - - *

(4.4)

t p 1 I po t h ds,2 __. / 9 ' 2 ~p _ x~tx p + rl~o*lvax x ) /9 ' 2 + x ' 2 /9 ' 1 + x ' 2

I is the self-dual ' t H o o f t tensor where ~

1

0 1 - 1 0 0

0 0 0 " 0 0 - 1

d x ' V d x '~, (4.5)

(4.6)

If we had pe r fo rmed a U(2) ro ta t ion on the coordinates ~l, ~2 before defining the i would have been ro ta ted into the other two ' t Hoof t coordinates x '~ by (4.4), 7/v~

i i n i 1 in (4.5) by n T/~ where is a 2 and 3 One can therefore replace ~/v~ tensors ,/v~ ,/v~. unit vector in three-dimensional space.


The conformally invariant scalar Green function is

G ( x ' , y ' ) = [4~r 2p'2(1 - L ) ] - 1 , (4.7)

_ . i i '~ "~ + + m T l ~ x y ) (4.8) L = ( p ' Z + x " y ' mrl~,~x y )(O '2 x " y ' • i i , , ,~x

(o +

It satisfies the conformally invariant equation

(l'q - ~ R ) G ( x ' , y ' ) = - 6 ( x ' , y ' ) . (4.9)

Let x '~ b e a point on ~, the light cone of the origin, i.e., x '~' =~/~av' w h e r e / ~ is a future-directed null vector with a unit timelike component and v' is the advanced time coordinate along the null geodesic generator of ~ in the d i rec t ion/~ . Then

G ( x ' , y ' ) = - 1 p,2 +y ,2 (4.10) 4~r 2 _ p,2y,2 + 2 p , 2 x , . y , + (1 + cos 2 0 ) ( x ' . y ' ) 2 '

where O is the angle between the 2-planes n;~/t~,i and /~l~y~l. For a fixed point y ' , there are two values of v' for which G is singular. In other words, the null geodesic through the origin in the d i rec t ion /~ intersects the null cone of the point y ' twice. This is in contrast to the situation in compactified flat space where a null geodesic intersects the light cone of a general point once only. It is the extra singularity of the Green function which gives rise to the interesting effects of these gravitational bubbles. If y ' is nearly on the same generator as x' , i.e., if

y'~ = ½/~w' + e ~ , (4.11)

where d" is small, the singularities in G occur for

v' = w' + O ( e ) , (4.12)

and

-- 4p '2 v' -- - w' + O ( e ) . (4.13)

(1 + COS 2 O)(/¢.e)

We shall call the first of these the direct singularity. It gives the same S-matrix as in flat space. The second, indirect, singularity will provide the non-trivial scattering amplitude. It has the opposite causal behaviour to that of the direct singularity, i.e., adding a small positive imaginary part to w' shifts the singularity in v' to negative

imaginary values.


Because CP 2 does not admit spinor structure, one cannot define a fermion Green function on it. One could introduce a generalized spin structure [16-18] by coupling the fermions to a gauge field. However the amplitudes would then be heavily damped by the action of the gauge field. We do not know the vector Green function on CP 2 although it probably has some fairly simple form.

The physical asymptotically euclidean metric with R = 0 is

d 3 2 = ~ 2 d s ,2

• d r 2 + ' 2g(P +r2)(d02+sin20dcp2)+~r2(d~b+cosOd~p) 2, (4.14)

where

r 2 = b 4 / x , 2 , p2 = b 4 / p , 2 , (4.16)

= 4 ¢ r 2 b 2 G ( x ', O) - - 02 "~" r2 b2 (4.17)

The action of this metric is

i I 2 ~¢rp . (4.18)

4.2. THE AMPLITUDES

The amplitude for a scalar particle to propagate from the ket I f ~ with Cauchy data f on ~ to a bra (gl with data g is

= ~ G ( x , y ) V~f(x ) d~.~(x') dX~(y ' ) . (4.19) ( g l f ) o - f f g ( x ' ) V ' ' ~ ' a~J~

I f f and g correspond to states with momenta k I and k2, then in the metric (4.1) they will have the form (3.6) and (3.9) where

I ^ I f " tt p x ' ~ - - ~ k l ~ V ", y ' ~ - ~ K 2 u ,

/~1 ~ = to l - lk l ~, Jc2/~ = to2 - lk2 ~ (4.20)

Thus when k I " is not parallel to k 2 t~

(k21kl)o

[ eit°lb2/u" ~ I t2 t2 × ~ u' },~u v d u ' d v ' , (4.21)

S . W . Hawking et al. / Quantum gravitational bubbles 295

where 0 is the angle between the 2-planes i i ~1 n ~/~ and k~ t~k 2 . For a fixed value of

u', the Green function in (4.21) has singularities at v ' = 0 and v ' = -8p'2/(/~1./~2(1 +cos20)u ' ) . The first of these is the direct singularity. It is cancelled by the factor v '2 and so does not contribuie to the amplitude (in flat space the amplitude is zero if k I is not parallel to k2). To evaluate the effect of the second, indirect, singularity it is convenient to introduce new coordinates u = - b 2 / v ', v = - b2/u '. Then

(k21kl>0 = 1 uv e -i~'~ du dv . f f e 'o '2"uv+ lp21~l'l~:(1 + cos: O)

(4.22)

Consider first the u integration. Suppose to: > 0 and the contour for the u integration to be along the real axis. Then the integral will be zero if Im(v) > 0 (since/~1 and /~2 are both taken to be future directed null vectors,/~l "/~2 < 0). For Im(v) < 0 one

obtains

<k21kl>o = ip2(1 +COS z 0) ( e x p [ --ito2/~l"/~2P2(1 +COS2 0)

16~r2 J ~ 8v io,v)?

(4.23)

where the contour in v is taken below the singularity at v = 0. For to l > 0 this gives

zero. For to l < 0,

p2 +oos +oos 0)]',' 3 . (4.24)

The fact that to! < 0 and the contour in v is taken below the axis means that the incoming particle 1 is really an outgoing particle at future null infinity ~+. One also obtains a non-zero amplitude if to2 < 0, to l > 0 and the contour in v is taken above the real axis. In this case both the "incoming" particle 1 and the "outgoing" particle 2 are really incoming particles at past null infinity ~ - . The reason one gets this behaviour is that the indirect singularity in the Green function has different causality properties from that of the ordinary flat space Green function.

When k 2 is not parallel to k 1, the direct singularity in the Green function makes no contribution to the amplitude. However, when the incoming and outgoing momenta are parallel, the direct singularity gives the usual flat space amplitude

2to 11~3(k2 -- k l ) (4.25)

with the usual causality, i.e., to ~ > O, to2 > 0 and the contour in v above the real axis and the contour in u below the real axis.


4.3. AVERAGING

Suppose one has n particles propagating in the metric of a CP 2 bubble. We shall take the momenta k i of the 2n external lines all to be inward directed, i.e., ~0 i > 0 for an ingoing particle and ~0 i < 0 for an outgoing particle. To obtain the total S-matrix amplitude

( - kp+l . . . . . - k 2 n Ikl . . . . . kp ) (4.26)

(p is the number of incoming particles), one connects pairs of external lines by Green functions in all possible combinations. The crude amplitude, which is a sum of products of terms of the form (4.24) or (4.25), is weighted by e x p ( - ] ) A -1/2 where i is the action of the R = 0 metric of the bubble and A is the product of the determinants of the scalar wave operator (representing the conformal fluctuations [ 19]) and of the operators of any matter fields that may be present. For the bubbles we are considering, A will depend only on the scale. We shall discuss integration over scales below. One then integrates over the orientation, position and scale of the bubble and normalizes by dividing by the vacuum-to-vacuum amplitude obtained by integrating without any amplitude factors.

The direct flat space contributions to the amplitude (4.25) are independent of the scale, orientation and position of the bubble. The integration over these variables will cancel out with the vacuum to vacuum amplitude to give a contribution to the S-matrix which corresponds to the particles propagating in flat space without interacting. On the other hand, displacing the position of the bubble by an amount z ~ as seen from infinity will multiply the indirect contributions to the amplitude by a phase factor exp(iz~Xiki~). When integrated over all positions z ~ of the bubble this gives a factor of

where the delta function on the euclidean section 8~ = iS~a, the delta function on the Minkowski section. The corresponding integration in the vacuum to vacuum amplitude produces a factor of (2~r)484(0). This should be interpreted as the inverse of the number density N of the gravitational bubbles. Thus the net factor from integrating over positions is

The integration over the orientation angle 0 is performed with the obvious measure 2,r sin 0 d0. Finally one has the problem of integrating over the scale size p of the bubble. The measure for this is presumably of the form pa do for some value of a. Attempts to derive the measure for pure quantum gravity from

S.W. Hawking et al. / Quantum gravitational bubbles

hamiltonian techniques have produced answers like [20-22]

Hx (, )

297

(4.29)

It is unclear how to interpret such a manifestly non-convariant measure in the present context. However the non-covariant factor g00 is cancelled out by the fermions if one takes the measure for simple supergravity [23, 24]

~( Iide'+,)(y,,d~p'~)dMdNiI db °, (4.30) ~ a , p , , o

where M, N and b ° are the minimal auxiliary fields. Eliminating the auxiliary fields by gaussian integration gives the measure

IIe-4( II dea #)( II dlpa ~) . (4.31) X ~ a , ~ a , p

The gravitational part of this measure is scale invariant suggesting that the value of a above should be - 1. However, the integration over the spin-~ field ff~ ~ produces a result which is scale dependent and which would seem to raise the effective value of a above - 1. This mat ter will be discussed further elsewhere. If a > - 1, the scale dependence of the action, which is proportional to O 2, will cause the dominant contributions to the integrals to come from bubble scales of order one, i.e., the Planck length.

For O--~ 1 and momenta k 1 and k 2 small compared to the Planck value, the argument of the Bessel function J0 in (4.24) will be very small compared to one. Thus

p2 (k2 [ k, )o ~ - ~ (1 + cos 2 0 ) . (4.32)

If one considers a single particle propagating in the bubble, i.e., two external lines, this would give an effective mass

m2__ 4'n "2 fo°°pa+2e-~rP2/4dp

CO __ 2

3 fo Oae ~o ~4do (4.33)

--- ]~r(a + 1). (4.34)

This would be of order the Planck (mass) 2. However, energy-momentum conservation requires that to I and to 2 have opposite signs (we are now taking all


momenta to be ingoing). On the other hand the contours of integration seem to give a non-zero amplitude only when t~ and to 2 are of the same sign, either both ingoing or both outgoing. Thus, it seems that there may be no effect when there is only a single on-shell particle propagating in the bubble. However, the indirect singularity will be important for off-shell scalar particle internal lines. Further, it seems that more complicated bubbles can give large mass term effects even for single on-shell particles. This suggests that Higgs scalar fields either do not exist or are not elementary but are bound states of fermions.

The situation is different however when we consider two particles, i.e., four external lines. Consider a situation in which there is an incoming particle antiparticle pair with momentum k I and k 2 and an outgoing pair with momenta k 3 and k 4. One can connect line 1 with line 2 and line 3 with line 4 by Green functions to obtain a non-zero amplitude

/)4 <-k3,-k4lk,, k2)o .~ 6--~-~2 (1 + c o s 2 O) 2 • (4.35)

This amplitude will correspond to an effective h(~ff)2 interaction where

2 8 ~ 2 = - 1---~(a + 1)(a + 3) . (4.36)

In other words, ~, is large and negative. If there really were a ~(~q3) 2 term in the lagrangian it would produce a non-zero interaction also in the case when there were two incoming particles and two outgoing particles (particle-particle scattering) or two incoming and two outgoing antiparticles (antiparticle-antiparticle scattering). However, in the present case it seems that the amplitude would be zero because one cannot connect with the Green function two external lines with the same sign of to. In ordinary field theory the particle-particle and antiparticle-antiparticle amplitudes are related to the particle-antiparticle amplitude by crossing symmetry relations which are a consequence of local causality. However topologically non-trivial metric configurations such as gravitational bubbles would be expected to violate causality because they do not admit time functions whose gradient is non-zero everywhere. It is therefore not surprising that they seem to give S-matrix elements which do not have crossing symmetry.

There are also higher effective interactions which correspond to three or more particles propagating in a gravitational bubble. However, by dimensional consider- ations they will be proportional to m p 2 or higher negative powers, where m p is the Planck mass. They will be negligible, at least at low energies. The conclusion therefore is that CP 2 bubbles seem to produce an effective four-point vertex for scalar particles of the form 2,(~0q3) 2 where ~ is large and negative. In sect. 5 we shall show that S 2 x S 2 bubbles seem to give the same effect for scalars but to give only small interactions for fermion and vector particles.

S.W. Hawking et al. / Quantum gravitational bubbles

5. S 2 X S 2

299

5.1. THE METRIC

We do not know the conformally invariant Green functions in a general metric on S 2 × S 2. However, they have a simple form in a certain class of singular metrics which can be regarded as a limit of smooth metrics. One starts with the Eguchi-Hanson metric [25, 26]

( [ (a4) ] ds 2 = 1 - a41-!dr2+¼r2r4 ] dO2+s in 20d~02+ 1 - ~ ( d ~ + c o s O d ~ ) 2

(5.1)

where a ,; r < oo, 0 < 0 < or, 0 < tp < 2~r, 0 < ~k < 2~r. This metric contains a 'bolt" [7], a 2-sphere of area ~ra 2 at r = a which is the zero of the Killing vector 0/0ft . At large r, the metric tends to that of flat space but with x ~' identified with - x ~'. The metric thus is not asymptotically euclidean but is what is called asymptotically locally euclidean (ALE). In the limit that the scale parameter a tends to zero, the metric becomes that of flat space with points identified by reflection in the origin, which becomes a conical point. The scalar Green function between two points x,y is

G(x, y)= 1_.1__[ 1 1 ] (5.2) 4~r 2 ( x - y ) 2 + ( x + y 2Xo) 2 '

where we have generalized to having the conical point at x 0. In other words, the Green function is the sum of the normal expression (the first term) plus an image charge (the second term). We now pick a point z ~ x 0 and send it to infinity with the conformal factor ~2(x)= 4~r2G(x, z). This gives an asymptotically euclidean

metric

ds 2 _- ~2~t~" d x ~ d x ~ . (5.3)

This metric will contain two conical points, x+ and x _. x+ corresponds to the original conical point x o and x_ corresponds to the point at infinity in the original flat metric. The metric is singular at x+ and x_ but one could cut them out and replace them with small Eguchi-Hanson and anti- Eguchi- Hanson metrics respectively (anti-Eguchi-Hanson is Eguchi-Hanson with the opposite orientation). The metric would then have the topology of S z × S 2. Had we used two Eguchi-Hansons, the metric would have had the topology C P z # C P z and would not have had a spin structure. One can thus regard our singular metric as a limit of smooth metrics with the topology S z × S z. Because the metric is conformally flat,

300 s . w . Hawking et al. / Quantum gravitational bubbles

the conformally invariant Green function is just given by multiplying (5.2) by f~ - l (x ) f l - l (y ) . In fact it is not essential to shrink the scale a of the Eguchi-Hanson metric to zero because the Green function in this metric is known [27]. However, we would still obtain one conical point x_ when we send a point z to infinity.

One can exploit the conformal invariance to choose coordinate systems that are convenient for calculation. We shall define the primed frame to be one in which I

t b t t is at the origin and x_ is at infinity. The metric ds '2= d x dx~, satisfies the conformal gauge condition (2.8). A point x '~ will have an image point

$'~ = 2 x ' ~ + - x 'v . ( 5 . 4 )

In particular the point I will have an image i at 2x~.~and it will have an image light

cone ~. It will be useful also to define another conformal frame, the unprimed one, in

which I is at infinity, and x and x+ are finite points. This can be done by

x ~ = x ~ + b 2 x ' ~ / x '2 , x '~ = b 2 ( x ~ - x _ ~ ) / ( x - x _ )2. (5.5)

The unprimed metric is

ds 2 = d x ~ dx~ = ot 2 ds ,2 , (5.6)

where

a = b 2 / x '2 = ( x - x _ ) 2 / b 2 . (5.7)

In these coordinates the point x ~ has an image point

~ = (2x - x_ - x + )~ [2(x - ~+ )~x_ ~ + 2 (x - x_ )~x+ ~ - ( x + - x _ ) ~ x ~ ] .

(5.8)

The strength of the image charge at 5~ is (2x - x _ - x + )6/ (x+ - x _ )6. In particular the infinity point I has an image I at ½(x÷ ~ + x_ ~). The physical asymptotically euclidean metric in the conformal frame with R = 0 is

d s 2 - - I 1 +

The action of this metric is

( x . - x _ ) 2 ]5 d x ~ d x ~ .

( 2 x - x + - x _ ) 2 (5.9)

i = ~ ( x + - x )2. (5.10)


5.2. THE AMPLITUDES

As in the case of CP 2, the Green function has two singularities, one on the light cone of the point x ~ (the direct singularity) and the other on the light cone of the image point ~ (the indirect singularity). As before the direct singularity has the normal causal behaviour while the indirect one has the opposite causality behaviour.

The calculation of the amplitudes is most conveniently performed in the primed frame. The amplitude will consist of two parts, one arising from the direct singularity which will give the flat space result

(k2 Ikl >o = 2°q83(k2 - k , ) , (5.11)

and the other part arising from the indirect singularity which will give propagation between null infinity ~ and its image ~. This can be calculated in a simple manner by the following trick. We shall take the Cauchy data for I kl } to be of the form given in sect. 3 but multiplied by a phase factor exp(ik 1" x_ ). This will give a plane wave exp(ik I -x) in the unprimed frame. One can transform this back to the primed frame by multiplying by tx. This gives

1 b2 ( " b 2 ~ - S e x p x,~ ) ,---~k, 'x ' + ik l 'x_ . (5.12) (2~r) 3/2

One now takes the scalar product of this with the Cauchy data for the state (k 21 on ~. This gives

2ie-ik2.x e-i~b2/D' ~ ( b2 ( k 2 l k ' > ° = (2~r)2to2 f v, ov~ ( 2 x + - x , ) 2

[ib2 l 2x.x, 1) > exp (-~x+; 7 -x-~ + ik I • x_ iv'2 d v',

- ~/~2 ~v'. Let where x '~ -

(5.13)

u = - ( b 2 / v ') + b 2x + ' " ]~2

2 x ~

Then

(k2 lk )o= iq 2eip'(k'-k2) [" " ] , 16~r 2 fei,~2Uexp ~_(~k l .k2q2_l (q . l~ l ) (q . l~2) ) d___UU,u

(5.14)


where q ~ -~ ( x + ~ - x _ ~ ) a n d p ~ = ½ ( x + ~ + x _ ~ ) . Thus

2 ( k 2 l k , ~ o = - ~ e i p ' ~ k ' - k 2 ) J o ( [ ½ k l . k 2 q 2 - ( q . k , ) ( q . k 2 ) ] l / 2 ) . (5.15)

As in the case of C P 2, the causal behaviour of the indirect singularity means that

one gets the non-zero result (5.15) only if t~ l > 0, o~ 2 < 0 and the contours in u and v are taken above the real axis or ~ l < 0, o~ 2 > 0 and the contours are taken below the real axis. The first case corresponds to two incoming particles, and the second

case to two outgoing ones. It is straightforward to repeat the calculation for higher spins. The fermion

Cauchy data [k 1, R ) will produce a plane wave

2 1 / 4

(21r)3/2 exp(ik I .x + ik 1 .x+ )klA, (5.16)

in the unprimed coordinates. If one transforms this to the primed frame and multiplies by the spinor 1 "4, the tangent to ~, one obtains

where

Now

and

21/462 lJ2 ( b2 ) (2, / r )3/2.~, 2 e x p l - -~k I 'x ' + ik I .x_ l~lAl a,

^ 1/2

b 2 . ~ ' a ~ x ~ + - - .~,2

.~'~' = 2x+ '~ - x '~'

1 ^ = 2x+ '~ - i k 2 ~ v ' .

(5.17)

(5.19)

(5.20)

T h u s

b /~ 1'^ f 2\1/2 (5.21) /~lalA = 2V'U( l"ql~2"q--2K1"r2q J "

If one takes the scalar product with the Cauchy data for the state (k2 , R I on ~, one

s.w. Hawking et al. / Quantum gravitational bubbles 303

obtains

<k2]kl~°=i (wl\ l /2 q2 / ^ ^ , ^ ^ 2)1/2 -~2) 3-~27r 2(k l 'qk2"q-]kl 'k2q

ib2k l 'X ' //920)2 ) b 2 X f e x p x '2 ~ ik I .x_ - v---- 7- u-~v,2dv'

= i ( t0..../! ) 1/2 q___~2[fc.ql~2.q_½l~l.]~2q2)l/2eiP.(kt--k D W2 ] 327r2 ~ l

(5.22)

x f exp(iw2u+ 8-~[ kl.l¢2q2-2k,.qic2.q]) du J/u 2

-q2ei,'(k,-k2)Jl([ ½kl.k2q2 - kl.qk2.q]'/2) 8¢r

(5.23)

(5.24)

Here again the contours are such that particles 1 and 2 have to be both ingoing or both outgoing.

One can perform a very similar calculation for spin-1 fields. The main difference is that there is now a factor of (kl~lA) 2 instead of klAl ~. The result of this is that the amplitude is a Bessel function of order 2:

q 2 1 (k2 [ k l ) 0 = _ __eiv'(k,--k2)j [[ ikl"k2q 2 - ka'qk2"q] I/2)

87r 2~ (5.25)

5.3. AVERAGING

The averaging procedure to obtain the physical S-matrix amplitude is similar to that for CP 2. Averaging over the pos i t ionp" of the centre of the bubble produces a delta function which ensures conservation of energy and momentum. Integrating over the orientation of q~ ensures angular momentum conservation while the integral over the magnitude of q a, which determines the scale of the bubble, will be dominated by q ~ 1, the Planck length, provided that the effective value of the constant a in the measure is greater than - 1. If this is the case, the arguments of the Bessel functions will be very small for particle energies small compared to the Planck length. For scalar particles this will give results similar to those for CP 2, namely an effective h~p 4 interaction with h large and negative. However the

1 2s ! amplitudes will be small for spin-½ and 1 because J 2 , ( x ) ~ ( 5 x ) / (2 s ) . for x<< 1. In the case of one spin-½ particle propagating in a bubble, energy momen tum

conservation would require k I -- - k 2. In this case the averaging over the direction of q~ produces zero. One would also expect the answer to be zero because the choice of contours seems to require that both particle lines are ingoing or both outgoing. In fact no kind of $ 2 × S 2 bubble can produce an effective mass term

304 S .W. Hawking et al. / Quantum gravitational bubbles

because the Green functions conserve helicity since the index of the Dirac operator is zero. K3 bubbles would give a fermion mass term but only if there were only one species of fermion and it was either Weyl or Majorana.

In the case of two spin-½ particles it is difficult to obtain an analytic expression for the averaging over the directions of q~'. If one has forward scattering, i.e., if

k 3 = - k I and k 4 = - k 2, then averaging over q~ gives

(kl, k21kl, k2~_(a+ 5)(a+ 3)(a+ l) kl.k2 " 432 ~r 5

(5.26)

Such an amplitude would arise from an effective interaction term of the form ( ~ k ) 2. When one converts from geometrical units in which G = 1 to conventional units, this becomes (q~q~)2/mp 2 where mp is the Planck mass. One should compare this with the effective interaction of the form (~q,)2/m2 v arising from the exchange of a heavy vector particle of mass m v which occurs in unified theories of the strong, weak and electromagnetic forces. Presumably, the gravitational interaction will be much weaker because mp>>m v. However, it might be distinguishable because it might produce entropy increase and a final situation that was described by a density matrix rather than a pure state. This seems to happen with macro- scopic black holes [28, 29] and one could regard a gravitational bubble as a virtual black hole which appeared and disappeared again. One might expect a baryon to decay gravitationally by two of the three quarks falling into the virtual black hole and emerging as a lepton and an antiquark. This would give a lifetime of the order of 4 5 1050 m p / m ~ years. By comparison the lifetime for the proton to decay in the

mx/m ~ ~ years if m x ~ GeV. grand unified theories [30] is of the order of 4 5 1030 1014

However, the lifetimes would be of the same order if the grand unification did not take place until the Planck energy.

In the case of a single spin-1 particle propagating in a gravitational bubble, the averaging over q~ gives

(k2lk, ~ = ( a + 3 ) ( a + 1) k .k2 " (5.27) 144~r 3

This will be zero for a single particle on-mass-shell and one might expect it to be zero anyway because of the contours. However, one might obtain a non-zero result for photons off-mass-shell which would correspond to an interaction term of the form ( F ~ ) 2. Such a term would contribute to the vacuum polarization and charge renormalization. This will be discussed further elsewhere.

For two spin-I particles, averaging over qZ gives

(a + 7)(a + 5)(a + 3)(a + 1) ( - k3, - k 4 l k l , k 2 ) = 384~r6

×(kl'k2ka'k4+kl'k3k2"k4+kl'k4k2"k3). (5.28)


This would correspond to an effective interaction of the form (F~) 4 - ( F ~ . F ~ ) 2. 4 Such an When one converted to conventional units this would be divided by mp.

interaction would be completely negligible compared to vacuum polarization effects of order 4 4 (F~) ~me, where m e is the electron mass.

6. Conclusion

We have shown that topologically non-trivial gravitational bubbles introduce extra singularities into the Green function which produce S-matrix amplitudes that are not obtainable from perturbation theory. However, the effects seem to be small for spin-½ and 1 particles at energies small compared to the Planck mass. Had this not been the case, the picture of spacetime foam would have been inconsistent with everyday observations from which spacetime appears to be practically flat on normal length scales. Of course the calculations in this paper do not fully establish the consistency of the foam picture because we have considered only individual bubbles in asymptotically euclidean space. There might be a qualitative change when all the bubbles were packed together to form the spacetime foam. We think this unlikely, however, because the dependence on the spin seems to be given by dimensional requirements and not to involve the precise details of the metric.

The finite-dimensional approximations to gravitational bubbles that we have considered are special and untypical in at least two respects. First, they have zero or self-dual Weyl tensors. This means that they obey Huygens' principle [ 19] which can be formulated in this context as saying that the euclidean Green functions contain no terms which depend logarithmically on the separation of the points. It is this which makes the Green functions simple and which makes the whole calculation fairly straightforward. In a generic bubble metric, however, the Green functions will contain logarithmic terms and will have branch cuts. This should not affect the dependence of the amplitudes on the spin but it might introduce new effects which could be interpreted as the absorption or emission of particles by the bubble or virtual black hole.

The second respect in which the bubble metrics we have used are not fully general is that they correspond to zero total energy momentum because the infinity point I is regular and does not exhibit the conformal curvature singularity that occurs for example in the conformal compactification of the Schwarzschild solution. Integrating over metrics with asymptotic behaviour corresponding to different values of the energy momentum should ensure that the energy momentum of particles emitted by the virtual black holes is balanced by that of particles absorbed. In this way one should be able to calculate the superscattering operator $ [29] which takes density matrices describing the original situation to density matrices describing the final one.

306 S .W. Hawking et al. / Quantum gravitational bubbles

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