GRAVlTONS IN DE SITTER SPACE
Bruce ALLEN
Department of Physics and Astronomy
Tufts University
Medford, MA 02155
U.S.A.
Everyone knows that "Einstein's greatest mistake", the cosmological constant ~ , is
very close to zero 111. There have been many attempts to explain why A
must be
exactly zero 121, but none of these e f f o r t s has succeeded. In fact i t is now fashionable to believe that during the very early h i s t o ry of the universe the value of A
was quite large 131. This so-called " i n f l a t i o n a r y " epoch would have been a long period of e x p o n e n t i a l l y rapid expansion, and would elegantly explain two otherwise mysterious observational truths : the universe is uncannily f l a t ,
wave background r a d i a t i o n
and the cosmic micro-
has no r i g h t to be as i s o t r o p i c as i t is.
There are three things that we would l i k e to know. F i r s t , why is A zero today ?
Second, could A have been very big in the past ? And f i n a l l y ,
if~
was very big in
the past, what consequences would that have today ? Unfortunately this paper w i l l not
answer any of these questions, but I hope that i t w i l l nevertheless accomplish something useful. I am going to show that one of the answers that has recently been given
to the f i r s t question above - Why is A zero ? get into the technical n i t t y - g r i t t y ,
is not correct. However, before I
l e t me give you a synopsis of the kinds of ans-
wers that have been suggested to these questions.
One answer which has been given to the question - why is A
equal to zero today?-
has been that zero is the only consistent answer. Let me reveal my predJudices at once
and say that I don't believe t h i s . F i r s t of a l l ,
i t i s n ' t borne out by careful calcu-
l a t i o n . For example, someone once decided that A must be zero, because i f i t was not
zero then a certain scatteringamplitude would not be unitary. However a more careful
i n v e s t i g a t i o n showed this to be false
ked on by several people, is that i f
141. A d i f f e r e n t argument, which is being worA
is not zero then p a r t i c l e s get created out
of the vacuum, and damp the value of A to zero
151.
The problem is this : What quantities to you calculate to see i f
A r e a l l y decays (or
has to be zero from the outset) ? One may show, f or example, that a scalar p a r t i c l e
propagating in a background with
A
nonzero w i l l emit additional scalar p a r t i c l e s ,
which w i l l continue to do the same thing, and so on, ad i n f i n i t u m . Now this sounds
83
unstable. However i f you calculate the energy-momentum tensor of this process, you
find that i t only s h i f t s the value of
A
a little
bit
I6L.
Another example : you can calculate the corrections to the stress-tensor due to the
Plank-scale quantum f l u c t u a t i o n s of the vacuum. Indeed, these corrections s h i f t the
value of
~
, and one can study the semi-classical back-reaction to find out what
e f f e c t this has on the metric tensor. Do t h i s , and you find that the metric is great l y a l t e r e d . I t sounds l i k e
a tremendous physical i n s t a b i l i t y u n t i l you r e a l i z e
that the i d e n t i c a l argument implies that f l a t space with A = 0 can't be stable e i ther ! 17i. So here i t is clear that something is wrong with the argument i t s e l f ,
since l o c a l l y our spacetime is very f l a t ,
and shows no signs of decaying away be-
neath our f e e t !
One of the basic problems with these arguments is that the natural ground state
(or vacuum state) f o r de S i t t e r space is time-reversal symmetric i81. In this socalled Gibbons-Hawking vacuum state, i t is impossible f o r p a r t i c l e creation to occur,
because i f the number of p a r t i c l e s were increasing, that would break time-reversal
symmetry.
I f i t could r e a l l y be established that
A = 0 was the only consistent value,
I
don't think that i t would be a good thing . The cosmological constant is a measure of
the local energy density of the empty vacuum state. I f i t was t r u l y zero then there
would be no way to generate cosmological i n f l a t i o n , which would be very unfortunate.
(This is another reason why I am inclined to believe that there is nothing which is
inconsistent about
A
nonzero). Because A
is simply a measure of the vacuum or
l a t e n t energy, i t can change during phase t r a n s i t i o n s , and i t seems certain that i f
symmetry in gauge theories is restored at high temperatures then such phase t r a n s i tions must have taken place, as our universe cooled and expanded 191. So i t seems quite possible that A
was nonzero in the past-and this leads to our f i n a l question
above.
What kinds of effects would be associated with a large p o s i t i v e A ? Well, f i r s t
there would be classical " g r a v i t a t i o n a l " effects. Separated p a r t i c l e s , f r e e l y f a l l i n g ,
would accelerate away from one another. At a certain distance the recessional v e l o c i ty would become unity, and there would be a cosmological particle horizon Ii01. A
given observer could not see farther than this distance. In addition to these classical effects there would be quantum effects. The best known of these is the GibbonsHawking effect -a freely f a l l i n g observer would see a thermal spectrum at temperature
(I~'~'I~A
l j & r a d i a t i n g from the imaginary surface which we have Just described-
the observer's p a r t i c l e horizon i81. There are probably other i n t e r e s t i n g effects too,
but we don't know what they are yet. The subject of quantum f i e l d theory in de S i t t e r
space is s t i l l
in i t s infancy. In fact the only real results are that we know how to
construct the Fock space of states, and how to f i n d the c o r r e l a t i o n functions f o r
spin O, ½ and i I i i i .
space I121.
We also know a l i t t l e
b i t about i n t e r a c t i n g f i e l d s in de S i t t e r
84
Now l e t me t e l l you the idea which I intend to spend the rest of t h i s t a l k t r y i n g
to destroy. The idea was to show that ~
tons when A
must be zero because of properties of gravi-
is nonzero. I f we were t r y i n g to prove that ~ =
O, then, in the ab-
sence of any exact symmetry or invariance which would force ~
be the next best thing. The reason is t h i s :
to vanish, t h i s would
A is only observable through r e l a t i v i -
t y , because i t represents an otherwise completely a r b i t r a r y zero-point for measuring
the energy-density of space-time. Were i t not for general r e l a t i v i t y ,
or g r a v i t y , then
any background energy density would be completely unobservable, and i t could be set
to zero with the stroke of a pen. However in the presence of g r a v i t y , the vacuum energy density A
does become observable, f o r example through the classical effects des-
cribed above. I t would therefore be nice i f the only thing ( g r a v i t y ) that enables us
to observe
A
in the f i r s t
place would also carry with i t some quantization consis-
tency condition that would force
A
to be zero. This would be an elegant solution to
our problem : i f g r a v i t y , the only force that allows us to observe A , demands that
/~
vanish for reasons of consistency.
An argument of t h i s type has recently been made by Antoniadis, l l i o p o u l o s and
Tomaras J13J. They claim that i f one quantizes g r a v i t a t i o n a l f l u c t u a t i o n s in the presence of a background energy density A , then the r e s u l t i n g theory shows a p a r t i c u l a r kind of inconsistency called an " i n f r a - r e d divergence". Now these words can refer
to any one of several d i f f e r e n t problems. For example saying that QED has i n f r a - r e d
divergences generally refers to the fact that external lines in Feynman diagrams emit
an i n f i n i t e number of low-frequency photons 1141. However t h i s is not a real problem
because the energy carried away by t h i s process is not i n f i n i t e .
S i m i l a r l y , in the
theory of massless scalar electro-dynamics, the e f f e c t i v e action has an i n f r a - r e d
divergence, and a mass-scale must therefore be introduced into the theory. The gauge
f i e l d s thus aquire a mass, and the gauge symmetry is broken jl5J.
In the recent work by Antoniadis, l l i o p o u l o s and Tomaras I i 3 j ,
i t was claimed t h a t ,
because of an i n f r a - r e d divergence, the two-point function of g r a v i t a t i o n a l f l u c t u a tions was i n f i n i t e
this infinity
(regardless of the separation of the two points). They argued that
caused certain t r e e - l e v e l scattering amplitudes to be i n f i n i t e ,
and
thus rendered the theory of quantum g r a v i t y inconsistent unless A equaled zero.
What I am going to do in t h i s t a l k is quite straightforward. F i r s t ,
I am going to
t a l k about the graviton propagator, and explain why i t is not, in and of i t s e l f ,
a
physical object. In fact i t depends upon the choice of gauge (by which, as I w i l l
s h o r t l y explain, I mean the choice of a gauge-fixing term). What t h i s means in p r a c t i ce is that physical q u a n t i t i e s ( f o r example scattering amplitudes, or the expectation
value of the curvature tensor) depend only upon certain components of the propagator.
This dependence is Just subtle enough so that the d i f f e r e n t graviton propagators, a r i sing from d i f f e r e n t choices of gauge, give exactly the same physical r e s u l t .
The next thing that I w i l l do is to show that i t is indeed true that for certain
choices of gauge, the graviton propagator is indeed i n f i n i t e ,
exactly as claimed by
Antoniadis et al. However I am then going to show that there are other choices ofgau-
85
ge for which the propagator is completely f i n i t e
! Then I w i l l explain why t h i s is so.
The point w i l l be that the gauge-fixing terms of Antoniadis et al. do not completely
f i x the gauge because they s t i l l
allow a f i n i t e
number of gauge transformations. I t
is for t h i s reason that the propagator that they find is i n f i n i t e .
But t h i s i n f i n i t y
is not a real physical divergence ; i t is an a r t i f a c t of how the gauge-fixing was
done. For a better choice of gauge, the propagator is completely f i n i t e
! To make t h i s
point absolutely c l e a r , I w i l l then show that i f one calculates scattering amplitudes
in the Antoniadis et al. gauge, one s t i l l
obtains a p e r f e c t l y f i n i t e
of the fact that the graviton propagator is i n f i n i t e .
graviton propagator in t h e i r gauge is the sum of an i n f i n i t e
f a c t ) term and a f i n i t e
r e s u l t , in spite
The reason f o r t h i s is that the
(unphysical gauge-arti-
part. The i n f i n i t e term does not contribute to the scattering
amplitude of any i n t e r a c t i o n whose stress-tensor is conserved, and thus physical scatt e r i n g amplitudes remain completely f i n i t e ,
al.
contradicting the claims of Antoniadis et
Although I w i l l not show i t here, t h i s cancellation takes place at higher orders
as w e l l . The point is that for good choices of gauge, the Fadeev-Popov ghosts are w e l l behaved ; for a bad choice of gauge they introduce additional i n f r a - r e d divergences
in Just the r i g h t way to cancel those a r i s i n g from the gravitons.
So the real point of a l l t h i s technical i n v e s t i g a t i o n is t h a t , at least so f a r ,
there doesn't seem to be any i n t r i n s i c problem with
A
~ O. Of course i t ' s
entirely
possible that something else w i l l turn up in the future that w i l l render de S i t t e r
space inconsistent ; as things stand at the moment, i t seems that we have to keep on
thinking about i t .
I.
Gauge-Fixing Terms and the Choice of Gauge.
This is a straightforward subject, but one that a great many people seem to be un-
clear about. The source of most of the trouble is confusion about the r e l a t i o n s h i p
between the classical process called "choice of gauge" and i t s analogue in quantum
f i e l d theory, which is called "choice of a gauge-fixing term" in the action. Let us
begin by considering these two ideas, and the connections between them.
Suppose that hab is a small perturbation of some background metric gab. Then there
is a whole class of metric perturbations hab that represent exactly the same physical
perturbation. This is because under the i n f i n i t e s i m a l coordinate transformation
xi--)X i + Vi the metric perturbation transforms into hab +~aVb). Since coordinate
transformation does not cause any changes to p h y s i c a l l y observable q u a n t i t i e s , we can
conclude that the perturbations hab and hab +~(aVb) are gauge-equivalent. Any physical q u a n t i t y , f o r example the perturbation of the curvature induced by hab +~(aVb),
w i l l not depend upon Vb 1161.
86
For this reason, in classical perturbation and s t a b i l i t y theory, i t is very common
to "impose gauge conditions". The metric perturbations obviously l i e in equivalence
classes ; two perturbations w i l l be deemed equivalent i f f they d i f f e r by ~(aVbl for
l
l
some vector Vb. "Imposing gauge conditions" is a way to pick out one p a r t i c u l a r member from each equivalence class. For example one can impose the following conditions
on h
ab'
Vahab = 0
a
ha
= 0
(transverse),
(traceless),
tahab = 0
(i.i)
(synchronous),
to r e s t r i c t the gauge freedom. Here t a is some a r b i t r a r y vector f i e l d (usually choosen to be t i m e l i k e ) . These are not the only conditions that one could impose ; there
are c l e a r l y an i n f i n i t e number of other p o s s i b i l i t i e s .
Now what about the quantum f i e l d theory of gravitational perturbations ? Well the
action is a scalar and i t is thus invariant under coordinate transformations, so the
perturbations hab and hab +V(aVb). have exactly the same action ; they are gauge equivalent. In t h i s situation the standard thing to do is to add to the action an a r b i t r a r i l y choosen term which is not invariant under the above transformation. This a r b i t r a r i l y choosen gauge-fixing term breaks the gauge invariance. For example i t could be,
Sgauge = ~ ( ~ a h a b ) 2
+ ~(h~ )2 + ~(tahab)2 ] d(Vol),
(1.2)
where at least one of the positive constants ( d l ~, l~ ) was nonzero. Now of course
we have made a very a r b i t r a r y choice here ; the Fadeev-Popov procedure allows us to
compensate for this choice in Just such a way that the scattering amplitudes are u l t i mately independent of i t .
Now suppose that we have determined the euclidean propagator, which we could write
for example as the path integral I171
Gabc,d,(X,X' ) = yd[hab]hab(X)hc,d,(X')exp(-S[hab]-Sgauge[hab]).
(1.3)
This propagator obviously depends upon the gauge-fixing parameters ~ l ~ and ~
. Now
suppose that we considered the divergence ~a Gabc,d' as a function o f ~ l ~ and ~" .
In general i t would not vanish. However i f ~
vanish. The reason is that i f
~
was taken to i n f i n i t y , then i t would
is very large then the f i e l d configurations in the
action which don't have ~/a hab = 0 are exponentially suppressed by the gauge-fixing
term. In the l i m i t = ( - ~
such configurations would make no contribution to the pro-
pagator. Similarly, i f ~ l
~
and ~'
were simultaneously taken to i n f i n i t y , then the
propagator would satisfy the "classical gauge conditions" (1.1) in that
a
~ Gabc,d,,
87
Gaac,d ,, and taGabc,d, would all vanish. So we can see that the classical gauge conditions are obtained in quantum theory by singular choices of gauge. This is analogous to the situation in QED. There, i f we wanted to have a transverse propagator sat i s f y i n g ~a<AaAb'> = 0 we would need to choose Landau gauge ; ie use a gauge-fixing
term
~(~A~)
Now
2 in the action and take the l i m i t ~ - ) ~
.
what gauge was used by Antoniadis et al. 1131 ? In fact there were two pos-
s i b i l i t i e s that they considered. The f i r s t one was equivalent to the three conditions
given before (1.1) where t a is a vector f i e l d orthogonal to a family of f l a t spatial
surfacer which are the standard k = 0 f o l i a t i o n of de S i t t e r space. In this choice of
gauge the propagator for gravitons can be related to that of a pair of minimally coupled scalar f i e l d s in a simple manner. Indeed in this gauge (corresponding to taking
~l~
and ~'
to i n f i n i t y in (1.2)) the propagator is i n f i n i t e . However i t was not
clear i f the reason for this was because the three gauge-fixing parameters were becoming i n f i n i t e , or i f i t was because the introduction of the vector f i e l d t a into the
action was breaking de S i t t e r invariance, or i f i t was because the gauge-conditions
did not e n t i r e l y determine the gauge. At that point in their work, Antoniadis et al,
were not themselves certain i f the infra-red divergence that they had discovered was
a gauge a r t i f a c t or not.
To resolve this uncertainty they then considered a second choice of gauge for
which the gauge-fixing term was
Sgauge =
~a(hab - 1/4 gab hCc)]2 d(Vol).
(1.4)
In this case they also found an infra-red divergence in the propagator. They then carried out a tree-level scattering-amplitude calculation, and found an i n f i n i t e result.
This, they claimed, was proof that the infra-red divergence that they had found for
two different gauge choices was t r u l y a physical effect and not merely a gauge a r t i fact.
We are going to concentrate on the second choice of gauge-fixing term (1.4) and
w i l l reach very different conclusions
than those of Antoniadis et al. We are going
to consider gauge-fixing terms of the following form, with ~ = i / 2 ,
Sgauge = ~ [ ~ a
(hab _ ~gab hCc)]2 d(Vol)
(1.5)
for a l l values of the constant ~ . We are going to prove the following three statements :
1. The graviton propagator is f i n i t e i f
does not equal one of the following
"exceptional values" (1/4, 7/10, 5/6 . . . . )
exceptional : (n2 + 3n-3) / (n 2 + 3n)
(1.6)
88
fo r n = 1, 2, 3 . . . .
If
~
has one of the exceptional values, then the propagator
is i n f i n i t e .
2. I f
~ takes one of the exceptional values, then the propagator diverges because
f o r that value of ~ the gauge-fixing term is not " s e n s i t i v e " to a gauge transformation corresponding to a p a r t i c u l a r ( f i n i t e set of) vector f i e l d s Va.
3. The scattering amplitude is f i n i t e and independent of the value of ~ .
While the gauge f i x i n g term that we consider has ~ = 1/2 and not ~ - ~ ( ~
ults apply equally well to the
~=~
because f o r the exceptional values of ~
the gauge-fixing-term f a i l s to f i x the gau-
ge f o r any value of ~ . In other words, when ~
(regardless of ~
, our res-
gauge of Antoniadis et a l. The reason why is
takes one of the exceptional values
) then there e x i s t certain perturbations hab which are pure gauge
hab = ~(aVb) ~ 0 and such that Sgauge~(aVbl] vanishes. Thus our conclusion w i l l
be that Antoniadis et a l . found an i n f r a - r e d divergence only because they had the bad
%
i
luck to choose an i n e f f e c t i v e gauge-fixing term, and not because the graviton propagator in de S i t t e r space has any i n t r i n s i c physical i n f r a - r e d divergence. For most
choices of gauge the propagator would have been completely f i n i t e .
2.
How to find the Graviton Propagator.
The basic idea of this section is to find a form f o r the graviton propagator which
w i l l make i t easy f o r us to see how i t depends upon the choice of a gauge-fixing term.
For this purpose i t turns out to be very convenient to represent the propagator as a
mode sum. Such mode sums are very f a m i l i a r in the context of Lorentzian space-time
calculations of ( f o r example) the commutator and symmetric functions f or a scalar
f i e l d . Here we are doing something s l i g h t l y less f a m i l i a r - a Euclidean mode sum.
The point is this : in the Hawking-Gibbons vacuum state, which is de S i t t e r i n v a r i a n t ,
the two-point function only depends upon the geodesic distance between the two-points.
I t is also an a n a l y t i c function f o r spacelike-separated points. Therefore i f we can
find this function f o r spacelike separations, i t s analytic continuation to t i m e l i k e
separation w i l l y i e l d a l l information about the physically i n t e r e s t i n g function,
which is the Lorentzian two-point
function.
Thus we w i l l look fo r the two-point function only f o r spacelike separated points.
One way to do this is to carry out the c a l c u l a t i o n on a Euclidean (++++) metric 4sphere of radius a, whose scalar curvature R has the same constant value as the curvature of the physical Lorentzian de S i t t e r space R = 4 A . On this four-sphere the
distance between two points is always p o s i t i v e , so that spacelike separation is the
only p o s s i b i l i t y . I t can be e a s i l y shown 1181 that f or such spacelike separation the
two-point function on the sphere, considered Just as a function of geodesic distance,
89
is exactly the same as the Lorentzian two-point function for spatial separations.
For that reason, from this moment on, we w i l l perform all calculations on a foursphere of radius a and volume 81T2 a4/3. The cosmological constant is then A = 3a-{
To find the two-point function, we need to know the quadratic part of the gravitational action for a small metric fluctuation hab. This has been calculated in many
places
J191. When we add to i t the gauge-fixing term previously given (1.5), we find
that the total gauged-fixed action is
S +
Sgauge = ( 6 4 ~ G ) - 1 S
habWabCdhcd d(Vol) •
Here the second-order d i f f e r e n t i a l
(2.1)
operator Wabcd is given by
A
Wabcd = ([l-2~2]gabgcd -gc(agb)d ) ~
+ (2gc(agb) d + gabgcd) ~
(2.1)
+ (2~-Z)(gab~(c~d)
+ gcd ~ ( a ~ b )
)
where ~ is the gauge-fixing parameter. Now the propagator is defined by the different i a l equation
Wabcd
GCda'b '
(X,X')
= ~(a
a'
~ b)
b'
'
(2.3)
together with the boundary conditions that Gaba'b' depend only upon the distance from
X to X' (in the sen6e of 1181) and that i t only be singular i f X = X'.
This equation can be solved in many ways. One method that we have already exploited is to actually perform the path integral (1.3). This is done by choosing lO"coordinates" in the space of all metric perturbations hab, and then integrating all of
them from - ~ t o
~
120J. A simpler method w i l l be followed here ; i t involves using
an ansatz which is J u s t i f i e d by the previous method.
To use the simplest method, i t is only necessary to have an orthogonal expansion
of the delta function appearing on the right hand side of (2.3). This orthogonal
expansion is
~(aa'~
b)
b' = S h ~ , ( X ) h ~ ' b ' ( x ' ) + ~ V~,(X)v~'b'(x')+n~ 2 W~b(X)w~'bix')
n=O au
n=l au
,
+ =
a
b (X)X n b(x') "
Here the tensor f i e l d s hab Vab
n , wab and~ ab are all eigenfunctions of I ~
and they
n '
n
form a complete set for the representation of any symmetric rank-two tensor. What this
means is that any such tensor,
Qab = O ~ n
Qab' can be represented as a sum of the form
hnab +~iPnV~b +~2~nW~b +o~n](~b
(2.5)
90
f o r a unique set of constant c o e f f i c i e n t s
{~.j#,,~m,~.~._
I f the eigenfunc-
tions are appropriately normalized, so that
= Sh
b(X)hab(x)dV = . . .
,
(2.6)
O:
then i t is easy to show from equations (2.5) and (2.6) that the delta function defined by (2.4) satisfies
b'
Qa'b'(x') =
Qab(x) ~ a l a ( X , X ' ) ~ b ) (X,X') dVx
(2.7)
for any symmetric tensor Qab"
The nice thing about t h i s method is that we don't have to e x p l i c i t l y construct
ab
ab
any of the eigenfunctions hn . . . . . ~ n " We w i l l only need basic information about
their eigenvalues and m u l t i p l i c i t y which can be found in many articles 1211 and which
can be obtained entirely from the group representation theory of S0(5). The basic
facts are as follows. The details, including the normalizations and eigenvalues of
these modes, can be found i:n reference 1201. The ten degrees of freedom in the symmet r i c tensor Qab (2.5) are divided among the different modes. There are five degrees
of freedom in the tensor modes hab which are transverse and traceless (TT)
n '
0 = hab
ab
n gab - ~ahn
r~hab =
n
~(~)
_
~(2) ham
-,n
n
(2.8)
A (n 2 + 7n + 8)
3
There are three degrees of freedom in the modes Vab which are the symmetrized derivan
t i v e s of transverse vectors.
[½
a
+^>] v(a~nb)
0 :~afn
(2.9)
[]vab = ~X(~)+ ~^ ) vab
n
n
)~(1)= _--~(n2 +
n
3
5n
+ 3)
There is one degree of freedom in the modes W
ab which are the traceless derivatives
n
of longitudinal vectors
91
Wab = IX(On) (3/4 ~(0n) +A)] -I/2 (~avb-¼gab I-l)~n
(2.10)
[] wab
: (X(On)+8/3A) Wab
n
n "
Finally there is one degree of freedom in the pure-trace scalar modes ~ab
n"
~b = 1/2 gab ~n
F]'xab
: X(On)Xabn
(2.11)
~(On) = -~--n (n+3)
3
From a detailed examination of these modes and t h e i r normalizations (for which see
1201) i t can be shown that
~a~Zb
wab
n
:
~ ( 0 ) 3 4 ~ (0) +A) ~n
zl n ( / ,, n
(2.12)
(gab
d + gcd~a~b ) XCdn
n + 2~(On)~n b .
= 2[~ (0)
n (3/4% (0)
n +/~ )]½ Wab
This is all that we w i l l need to know about the mode functions.
As we said e a r l i e r , the path integral calculation of the propagator 1201 J u s t i f i e s
our use of the following ansatz for the propagator :
Gaba'b'(x'x')
=~
I ~ n h~b(X) ha'b'(X')n
0
+
~n V~b(x) va'b'(X')n +
~n W~ b(X) W~ 'b'(x') +
~n
~ab(x ) ~ a ' b ' ( x , )
n " " n
rn LnEab(x)wa'biX')
+n
(2.13)
+
b(x)
(X,)] 1
Applying the wave operator Wabcd to the propagator (2.13) and demanding that i t yield
the delta function (2.4) gives linear equations that uniquely determine the coefficients ~n . . . . . ~'n to be
~n
= K(-~(n2) + 2 / 3 • ) - i
n = K ( - ~ ( ~ ) - ~ )-1
~n
for n > 0
for n >~ I, else zero.
= K(-~(On)(1-~)+A)-2[(2~2-E-1/4)~(On)-A/21
for n>2, el se zero.
(2.14)
92
~n
:
K(1/4)k(On) + A / 2 ) ( ) ~ (0)
n ( -1~ )
=
K()k(On)(1-~) +A)-2(~-1/2)(,.n)~(O)(3/4)k(~ ) + A ) ) ½ f o r n>~2, else zero.
+/~) - 2
f o r n >~0
(2.14)
Here the constant K = 64~G where G is Newton's constant. With the propagator now
determined by (2.13) and (2.14), we can discuss i t s i n f r a - r e d behavior.
3.
I n f r a - r e d Behavior of the Graviton Propagator.
We begin the discussion of the i n f r a - r e d behavior of the graviton propagator by
asserting that the propagator is f i n i t e
the c o e f f i c i e n t s
~n .....
(I n are f i n i t e .
f o r separated points i f and only i f a l l of
This is indeed the case, provided t h a t the
g a u g e - f i x i n g parameter ~ does not have one of the values
exceptional =
f o r n = 1,2 . . . . .
The f i r s t
i +
~/~(~)
-
n2+3n-3
n(n+3)
(3.1)
{m . Now l e t us prove our assertion.
terms in the mode sum f o r the propagator, corresponding to ~nhnhn' and
~ n V n ~ , have been evaluated by A l l e n and Turyn 1221 and shown to be completely f i n i te. This leaves the f i n a l
for different
three terms, which can be related to the scalar propagator,
values of the scalar mass. Thus to understand the i n f r a - r e d behavior of
the graviton propagator, a l l we have to do is understand the scalar case.
Here the s i t u a t i o n is very simple. For two points X and X', separated by a geodesic distance
j(~(X,X'),
G(m2,jL~) = ~
~n(X)~n (
-rl
n
the massive scalar propagator is I18,20,221,
X'
)
P(3/2 + V)~(3/2-V)
16~r 2 a
F(3/2+V,3/2_V;2;cos2~W,/2a)).
(3.2)
The r i g h t hand s i d e of t h i s e q u a t i o n , and hence the mode sum, is c o m p l e t e l y f i n i t e
provided t h a t 3/2-V is a not a n o n p o s i t i v e i n t e g e r . Since V= (9/4 - a2m2) ½, t h i s means
the p r o p a g a t o r is f i n i t e
m2 = _
provided t h a t m2 does not take one of the ( n e g a t i v e ) v a l u e s
i n(n+3)
a2
f o r n = O, 1 2,
'
....
(3.3)
But these are e x a c t l y the values of m2 f o r which the d e n o m i n a t o r - ~ ( ~ ) + m2 in the
mode sum vanishes ! Exactly the same analysis applies to the " s c a l a r " parts of the
g r a v i t o n propagator. We have thus proved t h a t p r o v i d e d t h a t i f ~ i s
not given one of the
"exceptional" values given above ( 3 . 1 ) , the propagator is completely f i n i t e .
will
now
do is to show why t h i s i s .
What we
93
4.
How Can The Gauge-Fixing Term Fail ?
The infra-red divergence that occurs in the propagator for the exceptional values
of ~ can be e a s i l y understood. Imagine expressing the propagator as a path i n t e g r a l ,
or average, over a l l f i e l d configurations. I f the gauge-fixing term was not present,
then this integral would y i e l d i n f i n i t y ,
because i t would include an i n f i n i t e number
of gauge-equivalent f i e l d configurations which had the same value of the action. The
purpose of the gauge f i x i n g term is to make the integral converge by giving gaugeequivalent f i e l d configurations d i f f e r e n t values of the action. Thus the gauge-fixing
term " f a i l s to do i t s duty" i f there e x i s t a d i s t i n c t pair of configurations which
are physically gauge equivalent and which have the same value of the gauge-fixed
action. Let us now show that this is exactly what happens i f
~ is given one of the
"exceptional" values.
We can write the gauge f i x i n g term (1.5) in the following form, a f t e r integrating
by parts.
Sgauge :
_=(~(hab _ ~gab he)~a
Vce
(hbc -
gbc h~) d(Vol).
(4.1)
Now consider the following gauge transformation : hab-~h ab +~(avb) where vb=v b~n
for the scalar mode ~ n ' and n > i . I t is easy to v e r i f y that for n > O, ~/(avb)
is nonzero, and
Sgauge[~(avb)] : 2 W [ ~ ( ~ ) ( I _ ~ ) + A ] 2 : 2 ~ A
Thus, i f
~
2 n2(n+3)2[~_
9
n2
+3n - 3]2
n(n+3)
"
(4.2)
takes on one of the exceptional values - say the n'th exceptional value-
then the gauge-fixing term f a i l s to be sensitive to the gauge transformation
hab")hab + ~ a V b
~ n induced by the n'th scalar mode~ because the r . h . s , vanishes!
This is the source of the infra-red divergence that occurs for the exceptional values
of ~ .
We w i l l now show that this infra-red divergence, should i t happen to arise be-
cause of a bad choice of ~ ,
is a harmless gauge a r t i f a c t and makes no contribution
to scattering amplitudes.
5.
The Infra-red Divergence is a Gauge-Artifact.
Consider the tree level scattering process where two matter f i e l d s , which we deno-
te
~ ,
i n t e r a c t by exchanging a graviton. Here ~
Just a scalar f i e l d . Schematically this looks l i k e :
could be any kind of matter, not
94
The amplitude for this process is determined by the stress tensor Tab of the matter.
I t is
A = I ' ] T a b ( x ) G a b c , d , ( X , X ' ) TC'd'(x')dVdV'
,
(5.1)
where dV denotes the invariant four-volume element g J ~ ) d 4 X
at the point X, and
dV' denotes the same thing at X'. Let us assume only that ~7a Tab = 0 ; ie that the
operator Tab , which is quadratic in the f i e l d ~ , is conserved. This is true even
in the presence of trace anomalies, for the renormalized operator, provided that i t
is a matrix element between physical (on-shell) states
1231. We w i l l show that this
amplitude is f i n i t e regardless of the value of the gauge-fixing parameter ~ , and in
p a r t i c u l a r for the "exceptional" values of ~ , for which Gabc,d, contains infra-red
divergences.
The amplitudes A is a sum of f i v e terms arising from the propagator (2.13). The
first
two terms are independent of ~ .
The f i n a l three terms, upon integration bypart~
can be expressed as
A~ + As + A~
: ~IT(X)p~(X,X')
T(X') dVdV'
(5.2)
where T(X) = Ta a(x) is the trace of the stress tensor. The function
p~(X,X')
is
of the form
~(X,X') = CI + O2(E-1/4)-2 ~I(X)~I(X')
+ C3 n~_-2 ~n(X)~n(X')
:
X(O)+ 4
n
Here C1, C2 and C3 are nonzero constants. What matter is that there appears to be a
single term in the amplitude that depends upon ~ .Howeverfrom gauge-invariance we
known that the amplitude can not depend upon ~ at a l l ! We w i l l now show that the
second term above contributes nothing, even in the l i m i t
~1/4
!
1241.
The reason why is simple : the mode(s) ~)I(X) obey ~ / a V b 9 1 = - 31~ g a b ~ 1
Thus replacing Ta~la by T a b V a V b ~ l , and integrating by parts 1251 to get
(~a Tab)~b~l,_ we see that the ~,-dependent term vanishes as long as the stress
tensor is conserved. What this means is that even in those cases where the two-point
function has an infra-red divergence, the scattering amplitude is f i n i t e . This shows
95
that in those cases where i t occurs, i n f r a - r e d divergence is a h a r m l e s s g a u g e a r t i f a c t .
6.
Conclusion.
What has been shown in t h i s t a l k is that the graviton propagator in de S i t t e r spa-
ce is OK. I f one makes a bad choice of gauge ( - f i x i n g term) then the propagator is
i n f r a - r e d divergent. However t h i s is not a problem. You can e i t h e r make a better choice of gauge (of which there are an i n f i n i t e
pletely finite,
number), f o r which the propagator is com-
or else you can go r i g h t ahead and use the i n f r a - r e d divergent one.
We demonstrated that i t doesn't matter. Gauge-invariance is the o v e r - r i d i n g p r i n c i p l e ,
and i t ensures that even i f the propagator has an i n f r a - r e d divergence, the physical
scattering amplitudes are f i n i t e .
A more detailed discussion of these points can also be found in an e a r l i e r published paper 1201. The complete closed form f o r the graviton propagator with ~ = 1/2
has also been found 1221. F i n a l l y a closed form in the de S i t t e r - n o n - i n v a r i a n t gauge
(I.I)
has
been
recently obtained 126i. This form applies to any s p a t i a l l y - f l a t
Robertson-Walker model.
Acknowledgements.
I would l i k e to thank S. Coleman, J. l l i o p o u l o s a n d M . T u r y n f o r h e l p f u l discussions.
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with the same eigenvalue. I f the four-sphere is X2 + . . . + X2 = 1 then the modes
~i are proportional to the i ' t h coordinate Xi.
1
5
25 T~e boundary terms can be shown to vanish in the Lorentzian spacetime case
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(submitted to Nucl. Phys.)