Nuclear Physics B335 (1990) 155-165
North-Holland
DO WORMHOLES FIX THE CONSTANTS OF NATURE?
S .W . HAWKING
Department of Applied Mathematics and Theoretical physics, University of Cambridge,
Silver Street, Cambridge CB3 9E W, UK
Received 1 August 1989
This paper examines the claim that the wormhole effects that cause the cosmological constant
to be zero, also fix the values of all the other effective coupling constants . It is shown that the
assumption that wormholes can be replaced by effective interactions is valid in perturbation
theory, but it leads to a path integral that does not converge . Even if one ignores this difficulty,
the probability measure on the space of effective coupling constants diverges . This does not affect
the conclusion that the cosmological constant should be zero . However, to find the probability
distribution for other coupling constants, one has to introduce a cutoff in the probability
distribution . The results depend very much on the cutoff used. For one choice of cutoff at least,
the coupling constants do not have unique values, but have a gaussian probability distribution.
1. Introduction
The aim of this paper is to discuss whether wormholes introduce an extra degree
of uncertainty into physics, over and above that normally associated with quantum
mechanics [1,21. Or whether, as Coleman [3] and Preskill [4] have suggested, the
uncertainty is removed by the same mechanism that makes the cosmological
constant zero.
Wormholes [5-7] are four-dimensional positive-definite (or euclidean) metrics
that consist of narrow throats joining large, nearly flat regions of space-time . One of
the original motivations for studying them was to provide a complete quantum
treatment of gravitational collapse and black-hole evaporation . If one accepts the
"no boundary" proposal [8] for the quantum state of the universe, the class of
positive-definite metrics in the path integral, can not have any singularities or edges .
There thus has to be somewhere for the particles that fell into the hole, and the
antiparticles to the emitted particles, to go to. (In general, these two sets of particles
will be different, and so they can not just annihilate with each other .) A wormhole
leading off to another region of space-time, would seem to be the most reasonable
possibility [5] . If this is indeed the case, one would not be able to measure the part
of the quantum state that went down the wormhole. Thus there would be loss of
quantum coherence, and the final quantum state in our region of the universe would
0550-3213/90/$03 .50©Elsevier Science Publishers B.V .
(North-Holland)
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S . W. Hawking / Wormholes
be a mixed state, rather than a pure quantum state. This would represent an extra
degree of uncertainty that was introduced into physics by quantum gravity, over and
above the uncertainty normally associated with quantum theory . The entropy of the
density matrix of the final state would be a measure of this extra degree of
uncertainty.
If macroscopic wormholes occur in the formation and evaporation of black holes,
one would expect that there would also be a whole spectrum of wormholes down to
the Planck size, and maybe beyond . One might expect that such very small
wormholes would be branching off from our region of space-time all the time. So
how is it that quantum coherence seems to be conserved in normal situations? The
answer [9,10] seems to be that for microscopic wormholes, the extra degree of
uncertainty can be absorbed into an uncertainty about the values of physical
coupling constants . The argument goes as follows :
Step 1. Because Planck-size wormholes are much smaller than the scales on
which we can observe, one would not see wormholes as such. Instead, they would
appear as point interactions, in which a number of particles appeared or disappeared from our region of the universe . Energy, momentum, and gauge charges
would be conserved in these interactions, so they could be represented, at least in
perturbation theory, by the addition of gauge invariant effective interaction terms
9i(o) to the lagrangian, where 0 are the low-energy effective fields in the large
regions [5,6] . It is implicitly assumed that there is a discrete spectrum of wormhole
states labelled by the index i. This will be discussed in another paper [111.
Step 2 . The strengths of the effective interactions will depend on the amplitudes
for the wormholes to join on. This in turn will depend on what is at the other end of
the wormholes . In the dilute wormhole approximation, each wormhole is assumed to
connect two large regions, and the amplitudes are assumed to depend only on the
vertex functions B, at each end. Thus the effect of wormholes smaller than the scale
on which we can observe, can be represented by a bi-local effective addition to the
action [10]:
-
i _4`'f d4x
f
g(x) B, (x) d4Y g(Y) B;(Y)
The position independent matrix Qt l can be set to the unit matrix by a choice of the
basis of wormhole state and normalization of the vertex functions Bi . The question
of the sign of the bi-local action will be discussed later.
Step 3 . The bi-local action can be transformed into a sum of local additions to
the action by using the identity [10]
f
exp[z d4x g(x)
e(x) f d4Y g(Y) 0(Y)1
f
f
_
(7T
/
) -l Z daexp [- Zazlexp [ -a d4x g(x) 9(x)
I.
S. W. Hawking / Wormholes
157
This means that the path integral
Z=
fdl O]exp [ - fd x V-g LIexp [
4
-
zY_ f d 4x
g(x)ei(x)
fd y
4
g(y)Bi(y),
becomes
Z= Ida ; P(ai)Z(ai ),
where
P(ai) = exp [- z Y_ ai ai ] '
Z(ai )
= f d[O]exp [- fd x V-g (L+ Y_aiei)I
4
This can be interpreted as dividing the quantum state of the universe into
noninteracting super selection sectors labelled by the parameters ai . In each sector,
the effective lagrangian is the ordinary lagrangian L, plus an a dependent term,
La iei . The different sectors are weighted by the probability distribution P(a) . Thus
the effective interactions Oi do not have unique values of their couplings . Rather,
there is a spread of possible couplings ai. This smearing of the physical coupling
constants is the reflection for Planck-scale wormholes of the extra degree of
uncertainty introduced by black-hole evaporation . It means that even if the underlying theory is superstrings, the effective theory of quantum gravity will appear to be
unrenormalizable, with an infinite number of coupling constants that can not be
predicted, but have to be fixed by observation [2].
Coleman [3] however has suggested that the probability distributions for the
coupling constants are entirely concentrated at certain definite values, that could, in
principle, be calculated . The argument is based on a proposal for explaining the
vanishing of the cosmological constant [12], and goes as follows :
Step 4 . The probability distribution P(a) for the a parameters should be
modified by the factor Z(a) which is given by the path integral over all low energy
fields q> with the effective interactions Y_ a;Bi .
Step S. The path integral for Z(a) does not converge, because the EinsteinHilbert action is not bounded below. However, one might hope that an estimate for
Z(a) could be obtained from the saddle point in the path integral, that is, from
solutions of the euclidean field equations . If one takes the gravitational action to be
fd 4 x~(11(a)-
1
167G (a)
the saddle point will be a sphere of radius
R+O(R 2)),
3/87TGA and action -3/8G2(a)A(a) .
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S. W. Hawking / Wormholes
If one just took a single sphere, Z(a) would be exp(3/8G 2A). However, Coleman
argues that there can be many such spheres connected by wormholes. Thus
Z(a) = exp(exp(
3
)) .
8G zA
Either the single or the double exponentials blow up so rapidly, as A approaches
zero from above, that the probability distribution will be concentrated entirely at
those a for which A = 0 [3,12] .
Step 6. The argument to fix the other effective couplings takes at least two
alternative forms:
(i) Coleman's original proposal [3] was that the effective action for a single sphere
should be expanded in a power series in A . The leading term will be -3/8GA, but
there will be higher-order corrections arising from the higher powers of the curvature in the effective action :
3
+&i) +Ag(6) + . . . ,
8GA
where 6 are the directions in the a parameter space orthogonal to the direction in
which A(a) varies . The higher-order corrections to I' would not make much
difference if Z(a) = e-r. But if
Z(a) = exp(exp(- r»,
then
SZ(a)
Z(
= e-r ST .
a)
.
The factor, e -c , will be very large for A small and positive . Thus a small correction
to T will have a big effect on the probability. This would cause the probability
distribution to be concentrated entirely at the minimum of the coefficient, f(ti), in
the power series expansion of I' (always assuming that f has a minimum) .
Similarly, one would expect the probability distribution to be concentrated entirely
at the minimum of the minimum of the higher coefficients in the power series
expansion . This would lead to an infinite number of conditions on the a parameters . It is hoped that these would cause the probability distribution to be concentrated entirely at a single value of the effective couplings, a.
(ii) An alternative mechanism for fixing the effective couplings has been suggested
by Preskill [4] . If the dominant term in T is -3/GZA, one might expect that the
probability distribution would be concentrated entirely at G(a) = 0, as well as at
A(a) = 0. However, we know that G(a) =A 0, because we observe gravity. So there
S. W. Hawking / Wormholes
15 9
must be some minimum value of G(a) . One would expect that the probability
distribution would be concentrated entirely at this minimum value, and one would
hope that the minimum would occur at a single value of the effective couplings, a .
This paper will examine the validity of the above steps. Steps 1 and 2 are usually
assumed without any supporting calculations . However, an explicit calculation is
given in sect. 2, for the case of a scalar field. This confirms that wormholes can
indeed be replaced by a bi-local action, at least for the calculation of low-energy
Green functions in perturbation theory . The sign of the bi-local action is that
required for the use of the identity in step 3. However, the sign also means that the
path integral does not converge, even in the case of a scalar field on a background
geometry . Thus the procedure of using the effective actions to calculate a background geometry for each set of a parameters, is suspect . However, if one is
prepared to accept it, one would indeed expect that F would diverge on a
hypersurface in a space, on which A = 0. Thus the cosmological constant will be
zero, without any uncertainty . However, to calculate the probability distributions of
the other effective coupling constants, one has to introduce a cutoff for the
divergent probability measure. Different cutoffs will give different answers . Indeed,
a natural cutoff will just give the probability distribution P(a) for all effective
couplings except the cosmological constant . Thus one can not conclude that the
effective couplings will be given unique values by wormholes.
2. The bi-local action
In this section, it will be shown that scalar field Green functions on a class of
wormhole backgrounds can be calculated approximately from a bi-local addition to
the scalar field action in flat space . In particular, the sign of the bi-local action will
be obtained . The wormhole backgrounds will be taken to be hyperspherically
symmetric, like all the specific examples considered so far. This means that they are
conformally flat. For definiteness, the conformal factor will be taken to be
b2
ds 2 = 1 +
( x-xo
2
)2
dx2
This is the wormhole solution for a conformally scalar field [13], or a Yang-Mills
field [14]. In the case of a minimally coupled scalar [7], the conformal factor will
have the same asymptotic form at infinity, and near x o , the infinity in the other
asymptotically euclidean region . The conformal factors will differ slightly in the
region of the throat, but this will just make the bi-local action slightly different.
The metric given above appears to be singular at the point xo. However, one can
see that this is really infinity in another asymptotically euclidean region, by
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S. W. Hawking / Wormholes
introducing new coordinates that are asymptotically euclidean in the other region
YV
Olv(x g - xô)
=
(x - xo)
0
+Yo
where O,,' is an orthogonal matrix. In order to study low-energy physics in the
asymptotically euclidean regions, one needs to know the Green functions for points
x 1 , x2, . . . and Y1, Y2, . . . in the two regions, far from the throat . Consider the Green
function for a point x in one asymptotic region, and a point y in the other . Since
the wormhole metric given above has R = 0, the conformally and minimally coupled
scalar fields will have the same Green functions. One can therefore calculate the
Green function using conformal invariance as
Y) = Q(A) -'
G (x,
(x
1
- x)2
2(x)-1,
where z is the image of the point y under the transformation above. For x and y
far from the wormhole ends, x 0 and yo ,
S2(x) = 1,
SZ(z)
=
b2
6
(Y - Yo)
z = xo .
2 ,
Thus
G(x,
Y)
b2
(x _ xo) 2(Y
-Yo)2
.
This is what one would have obtained from a bi-local interaction of the form
z fd4xo b ~5 (xo) fd4Yo b$(Yo) .
Note that the bi-local action has a negative sign. This is because the Green functions
are positive .
Now consider two points x 1 , x 2 and Y1, Y2 in each asymptotic region . The
four-point function will contain a term, G(x 1 , y1)G(x2, Y2), which will be given
approximately by the bi-local action
- z f d4xob 2$2 (xo)fd4Yob2$ 2 (Yo) .
In general, Green functions involving n-points in each asymptotic region will be
given by bi-local actions with vertex functions 9(x) of the form, b"(~"(x). If one
S. W. Hawking / Wormholes
16 1
takes gravitational interactions into account, one would expect that the bi-local
action would be multiplied by a factor ewhere I a n/G is the action for a
wormhole containing n scalar particles .
One can also consider higher-order corrections to the Green functions on a
wormhole background which arise because the image, .z, of the point, y, is not
exactly at xo. These will be reproduced by bi-local actions involving vertex functions containing derivatives of the scalar field . Only those vertex functions that are
scalar combinations of derivatives will survive averaging over the orthogonal matrix
O, which specifies the rotation of one asymptotically euclidean region with respect
to the other . Thus the vertex terms and the effective action will be Lorentz invariant .
It seems that any scalar polynomial in the scalar field and its covariant derivatives
can occur as a vertex function.
Earlier this year, B. Grinstein and J. Maharana [15] performed a similar calculation.
3. Convergence of the path integral
The bi-local action has a negative sign, so it appears in the path integral as a
positive exponential . This is what is required in order to introduce the a parameters
using the identity in step 3 . If the bi-local action had the opposite sign, the integral
over the a parameters would be f da e + °Z / z, which would not converge . On the
other hand, because the bi-local action is negative, the path integral will not
converge. This is true even in the case of the path integral over a scalar field on a
non-dynamic wormhole background. There will be vertex functions of the form 0"
for each n. In the case of even n, the integral f d4y 0"(y) will be positive. This
means that - f d4x (p"(x), the other part of the bi-local action, will give 0 an
effective potential that is unbounded below. Thus the path integral over q), with the
bi-local effective action, will not converge. This does not mean that scalar field
theory on non-dynamical wormhole backgrounds is not well defined . What it does
mean is that a bi-local action gives a reasonable approximation to the effect of
wormholes on low-energy Green functions, in perturbation theory. But one should
not take the bi-local action too literally . One can see this if one considers introducing the a parameters . One will then get a scalar potential which is a polynomial in
0, with a-dependent coefficients . For certain values of the a, there will be metastable
states, and decay of the false vacuum. But these obviously have no physical reality .
The moral therefore is that one can use a bi-local action to represent the effect of
wormholes in perturbation theory. But one should be wary of using the bi-local
action to calculate non-perturbatioe effects, like vacuum states .
This is even more true of the effective gravitational interactions of wormholes . It
is not clear whether there is a direct contribution of wormholes to the cosmological
constant, i .e. whether any of the vertex functions contain a constant term. This
would show up only in the pure trace contribution to linearized gravitational Green
16 2
S. W. Hawking/ Wormholes
functions in the presence of a wormhole. So far, these have not been calculated .
However, even if there is no direct wormhole contribution to the cosmological
constant, there will be indirect contributions arising from loops involving other
effective interactions . These will be cut off on the scale of the wormholes, that is, on
the scale on which the wormholes no longer appear to join on at a single point. In a
similar manner, there does not seem to be a wormhole that makes a direct
contribution to the Einstein lagrangian, R, and hence to Newton's constant. By
analogy with the case of wormholes with electromagnetic and fermion fields, one
would expect that such a wormhole would have to contain just a single graviton .
However, its effect would average to zero under rotations of the wormhole,
described by the matrix O. However, there will again be indirect contributions to
1/G from loops involving other effective interactions .
There are convergence problems with gravitational path integrals, even in the
absence of wormholes . The Einstein-Hilbert action -Jd'xVg(R/167TG-n) is
not bounded below, because conformal transformations of the metric can make the
action arbitrarily negative. Still, one might hope that the dominant contribution to
the path integral would come from metrics that were saddle points of the action,
that is they were solutions of the euclidean field equations . The spherical metric
given in sect . 1 has the lowest action of any solution of the euclidean field equations
with a given value of !l . One might therefore expect that
3
The problem of the convergence of the path integral is much worse however, if
one replaces wormholes with a bi-local action. If there were a direct wormhole
contribution to the effective cosmological constant, the path integral would contain
a factor e cvz , where V is the volume of space-time. If the constant C were negative,
the integral over a would not converge . But if C were positive, the path integral
would diverge . Even rotating the contour of the conformal factor to the imaginary
axis would not help, because in four dimensions it would leave the volume real and
positive . One might still hope that the saddle point of the effective action would give
an estimate of the path integral . However, the bi-local action would give rise to an
effective cosmological constant of value -2CV Unless this were balanced by a very
large positive cosmological constant of non-wormhole origin, the action of any
compact solution of the euclidean field equations would be positive. So it would be
suppressed, rather than enhanced, as in the case of the sphere. Even if there were a
large positive non-wormhole cosmological constant, it would not give a solution of
infinite volume, with zero effective cosmological constant . One might still use the a
identity, replace the bi-local action with a weighted sum over path integrals with an
a-dependent cosmological constant . But if gravitational path integrals can be made
sense of only by taking the saddle point, one should presumably also take the saddle
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S. W. Hawking/ Wormholes
point in the integral over a. In the case of a single exponential, this would give
3F
a+
8G2(A 0 + a FC)
2 =0,
and in the case of a double exponential
-
2
a
+
3F
8G 2 (AO +aFC ) 2
0
where A 0 is the non-wormhole contribution to the cosmological constant . In either
case, the effective cosmological constant at the saddle point will be of the order
of 11 0.
4. The divergence of the probability measure
Suppose, as one often does, one ignores problems about the convergence of the
path integral. Then, as described in sect. l, there will be a probability measure on
the space of the a parameters
tt(a)=P(a)Z(a),
where P(a) = exp[Y- - Za;a;] and Z(a) = exp [ -l(a)] or exp[exp [- l(a)]]. If
r_
3
8G2(a)A(a)
and GA vanishes on some surface K in a space, the measure y(a) will diverge .
That is to say, the total measure of a space will be infinite.
The total measure of the part of a space for which GZA > e > 0 may well be finite .
In this case, one could say that
G2A =0,
with probability one. Since we observe that G =# 0, one could deduce that
A=0 .
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S. W. Hawking / Wormholes
However, with such a badly divergent probability measure, this is about the only
conclusion one could draw . To go further, and to try to argue as in sect. 1, that the
probability measure is concentrated entirely at a certain point in a space, one has to
introduce some cutoff in the probability measure. One then takes the limit as the
cutoff is removed . The trouble is, different ways of cutting off the probability
measure will give different results . And it hard to see why one cut-off procedure
should be preferred to another .
One can cut off the probability measure by introducing a function F on a, which
is zero on the surface K where 1/F = 0, and which is positive for small negative
1/F. One then cuts the region 0 < F < e out of a space . One would expect the
probability measure on the rest of a space to be finite, and therefore to give a
well-defined probability distribution for the effective coupling constants . If Z(a) is
given by a double exponential, the probability distribution will be highly concentrated near the minimum of F on the surface, F = e . Thus, in the limit e tends to
zero, the probability would be concentrated entirely at a single point of a space . But
the point will depend on the choice of the function F, and different choices will give
different results . For example, Coleman's procedure [3] is equivalent to choosing
F= A . On the other hand, Preskill [4] has suggested using a cutoff on the volume of
space-time . This would be equivalent to using
F=G2A2 .
But if you minimise G 2A for fixed G2A2, you would drive G to zero and A to a
non-zero value, if G can be zero anywhere in a space . This is not what one wants .
One therefore has to suppose that G is bounded away from zero, at least in the
region of a space in which the bi-local action is a reasonable approximation for
wormholes .
It seems therefore that one can get different results by different methods of
cutting off the divergence in the probability measure . There does not seem to be a
unique preferred cutoff. A possible candidate would be to use F or Z(a) themselves
to define the cutoff ; for example, F= -1/F. This would lead to A =0, but the
other effective couplings would be distributed with the probability distribution
P(a). In this case, wormholes would have introduced an extra degree of uncertainty
into physics. This uncertainty would reflect the fact that we can observe only our
large region of the universe, and not the major part of space-time, which is down a
wormhole, beyond our ken.
References
[1]
S.W . Hawking, Commun . Math . Phys. 87 (1982) 395
[2] S.W . Hawking and R. Laflamme, Phys. Lett . B209 (1988) 39
[3] S. Coleman, Nuel . Phys. B310 (1988) 643
S . W. Hawking / Wormholes
[4] J. Preskill, Nucl . Phys . B323 (1989) 141
[5] S.W . Hawking, Phys . Lett. B195 (1987) 337
[6] S.W . Hawking, Phys . Rev. D37 (1988) 904
[7] S.B. Giddings and A. Strominger, Nucl . Phys . B306 (1988) 890
[8] J.B . Hartle and S.W . Hawking, Phys. Rev. D28 (1983) 2960
[9] S. Coleman, Nucl . Phys . B307 (1988) 864
[10] 1. Klebanov, L. Susskind and T. Banks, Nucl. Phys . B317 (1989) 665
[11] S.W . Hawking and D. Page, in preparation
[12] S.W . Hawking, Phys . Lett. B134 (1984) 402
[13] J.J . Halliwell and R. Laflamme, Santa Barbara ITP preprint NSF-ITP-89-41 (1989)
[14] A. Hosoya and W. Ogura, Phys . Lett . B225 (1989) 117
[15] B. Grinstein and J. Maharana, Fermilab preprint FERMILAB-PUB-89/121-T (1989)
16 5