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8/3/2019 John Preskill- Wormholes in Spacetime and the Constants of Nature
http://slidepdf.com/reader/full/john-preskill-wormholes-in-spacetime-and-the-constants-of-nature 1/46
8/3/2019 John Preskill- Wormholes in Spacetime and the Constants of Nature
http://slidepdf.com/reader/full/john-preskill-wormholes-in-spacetime-and-the-constants-of-nature 2/46
8/3/2019 John Preskill- Wormholes in Spacetime and the Constants of Nature
http://slidepdf.com/reader/full/john-preskill-wormholes-in-spacetime-and-the-constants-of-nature 3/46
J. Preskill / Wormholes 14 3
r a n d o m l y s e l e c t e d u n i v e r s e t h e p r o b a b i l i t y i s o n e t h a t e a c h c o n s t a n t a s s u m e s i t s
s t a n d a r d v a l u e . T h u s , t h e q u a n t u m i n d e t e r m i n a c y t h a t a f f l i c t s t h e f u n d a m e n t a l
c o n s t a n t s t u r n s o u t t o b e v e r y m i l d. T h e s t a n d a r d v a l u e o f t h e c o s m o l o g i c a l c o n s t a n t
A c a n b e c a l c u l a t e d ; i t i s z e r o , a s C o l e m a n c l a i m e d . T h i s r e s u l t h o l d s q u i t eg e n e r a l l y , a n d i s n o t s e n s i ti v e t o t h e d e t a i le d p r o p e r t i e s o f s h o r t - d i s t a n c e p h y s i c s .
B u t I a r g u e t h a t t h e s t a n d a r d v a l u e s o f a ll o t h e r f u n d a m e n t a l c o n s t a n t s a r e q u i t e
s e n s i t i v e t o t h e d e t a i l s o f p h y s i c s a t th e P l a n c k s c a l e; h e n c e , n o c o n s t a n t s o t h e r t h a n
t h e c o s m o l o g i c a l c o n s t a n t c a n b e e x p l i c it ly c a lc u l a te d . T h i s c o n c l u s i o n i s ra t h e r
d i s a p p o i n t i n g , i f n o t e n t i r e l y su r p r i si n g . T o p u t i t i n a p o s i t i v e li g h t, o n e m i g h t b e
e n c o u r a g e d t h a t , b e c a u s e o f o u r i n a b i l i t y t o e x t r a c t o t h e r p r e c i s e p r e d i c t i o n s ,
C o l e m a n ' s b e a u t i f u l e x p l a n a t i o n f o r A = 0 h a s n o t b e e n f o u n d t o b e i n c o n s i s t e n t
w i t h a n y t h i n g t h a t w e k n o w a b o u t N a t u r e . I n d e e d , I w i l l a l s o p r o p o s e a m e c h a n i s m
t h a t c a n a c c o u n t n a t u r a l l y f o r t h e s u p p r e s s io n o f l a rg e w o r m h o l e s .M o s t o f t h e a n a l y s i s r e p o r t e d i n t h i s p a p e r i s p r e m i s e d o n t h e c l a i m t h a t a l l o f t h e
c o n s t a n t s o f N a t u r e o t h e r th a n t he c o s m o l o g ic a l c o n s t a n t c a n b e d e t e r m i n e d b y
r e q u i r i n g t h a t N e w t o n ' s g r a v i t a t i o n a l c o n s t a n t G m u s t a s s u m e t h e s m a l l e s t p o s s i b l e
v a l u e * . T h e p o t e n t i a l q u a n t u m i n d e t e r m i n a c y o f f u n d a m e n t a l p h y s i c s i s e v a d e d ,
b e c a u s e a r a n d o m l y s e le c te d u n i v e rs e is o v e r w h e l m i n g l y li k el y t o h a v e G a t it s
m i n i m u m . T h i s c r i t e r io n is s u ff i ci e n t t o d e t e r m i n e a l l o f th e f u n d a m e n t a l c o n s t a n t s
b e c a u s e a ll c o n t r i b u t e t o G t h r o u g h r e n o r m a l i z a t io n e f f e ct s . T h e a n a ly s i s co n s i st s o f
t w o m a i n p a r t s . I n t h e f ir s t p a r t , I s t u d y t h e e f f e c t s o f w o r m h o l e i n t e r a c t i o n s , w h i c h
w e r e n e g l e c t e d i n C o l e m a n ' s o r i g i n a l a n a l y s i s . T h e s e i n t e r a c t i o n s a r e e s s e n t i a l i ne n s u r i n g t h a t G a c t u a ll y h a s a n o n z e r o m i n i m u m . T h e y a re a ls o th e k e y t o
u n d e r s t a n d i n g w h y t h e e ff e c ts o f l a rg e w o r m h o l e s a r e s u p p r e s se d . I n t h e s e c o n d
p a r t , t h e r e n o r m a l i z a t i o n g r o u p m e t h o d i s a p p l i e d t o s t u d y t h e d e p e n d e n c e o f G o n
t h e o t h e r f u n d a m e n t a l c o n s t a n t s . I t i s b e c a u s e t h e r e n o r m a l i z a t i o n o f G i s d o m i -
n a t e d b y " n o n u n i v e r s a l " s h o r t - d i st a n c e e f fe c t s t h a t w e a r e u n a b l e in t h e e n d t o
m a k e p r e c i s e p r e d i c t i o n s a b o u t t h e v a l u e s o f t h e o t h e r c o n s t a n t s .
T h e r e m a i n d e r o f t h i s p a p e r i s o r g a n i z e d a s f o l l o w s : I n s e c t . 2 , t h e c o n n e c t i o n i s
d e r i v e d b e t w e e n w o r m h o l e s i n sp a c e ti m e a n d t he p o te n t ia l q u a n t u m i n d e t e r m i n a c y
o f t h e c o n s t a n t s o f N a t u r e . A s C o l e m a n s h o w e d , o n e c a n " i n t e g r a t e o u t " t hew o r m h o l e f l u c t u a t i o n s a n d o b t a i n a n e f f e c t iv e f ie l d t h e o r y w i t h a n e x p l ic i t s h o r t -d i s -
t a n c e c u t o f f in w h i c h w o r m h o l e f l u c t u a t i o n s n o l o n g e r o c c u r; t h e c u t o f f is t h e
c h a r a c t e r is t i c w o r m h o l e " t h i c k n e s s . " T h e e f f e ct s o f t h e w o r m h o l e s c a n t h e n b e
i n c o r p o r a t e d i n t o r e n o r m a l i z a t io n o f t h e p a r a m e t e r s o f t h e ef f e c ti v e t h e o r y . B u t
t h e r e i s a n o t h e r , m o r e s u r p ri si n g , r e m n a n t o f th e w o r m h o l e f l u c t u a t io n s i n l o w - e n e r g y
p h y s i c s . T h e e f f e c ti v e f ie l d t h e o r y is a c t u a ll y a s u p e r p o s i t io n o f m a n y " s u p e r s e l e c -
t i o n " s e c t o r s t h a t d o n o t c o m m u n i c a t e w i t h e a c h o t h e r t h r o u g h a n y l o c a l p h y s i c s .
* T h i s c r i t e r i o n w a s p r e v i o u s l y s u g g e s t e d b y G r i n s t e i n a n d W i s e [1 5]. A d i f f e r e n t c r i t e r i o n w a s
e n v i s i o n e d i n r e f s . [1 6 ,1 8 ] ; t h i s a c c o u n t s f o r s o m e o f t h e d i f f e r i n g c o n c l u s i o n s o f r e f s. [1 6 , 18 ] a n d t h e
p r e s e n t p a p e r .
8/3/2019 John Preskill- Wormholes in Spacetime and the Constants of Nature
http://slidepdf.com/reader/full/john-preskill-wormholes-in-spacetime-and-the-constants-of-nature 4/46
1 4 4 J. Preskill / Wormholes
T h e s e s e c t o r s a r e l a b e le d b y a s e t o f p a r a m e t e r s , d e n o t e d b y a , t h a t a r e p r e s u m a b l y
i n f i n i t e in n u m b e r . I n e a c h s e c to r , th e " b a r e " c o u p l i n g s t h a t s p e c i f y t h e e f fe c t iv e
f i e l d t h e o r y a t t h e w o r m h o l e s c a l e a r e f u n c t i o n s o f a ; t h i s i s t h e o r i g i n o f t h e
q u a n t u m i n d e t e r m i n a c y o f t h e f u n d a m e n t a l c o n s ta n t s .
T h e d e r i v a t i o n p r e s e n t e d h e r e o f t h is a - d e p e n d e n c e o f t h e e ff e c ti v e t h e o r y
e m p h a s i z e s t h e g e n e r a l i t y o f t h e r e su l t . I n p a r t i c u l a r , t h e a r g u m e n t d o e s n o t r e l y a t
a l l o n t h e s e m i c l a s s i c a l a p p r o x i m a t i o n i n v o k e d b y C o l e m a n . W e w i l l a l s o s e e t h a t
t h e i n t e r p r e t a t i o n o f a a s th e la b e l o f a c o h e r e n t s u p e r p o s i t io n o f " b a b y u n i v e r s e "
s t a t e s , w h i c h p l a y e d a c e n t r a l r o l e i n C o l e m a n ' s d i s c u s s i o n , i s r e a l l y p e r i p h e r a l t o
t h e a r g u m e n t , a n d m a y n o t e v e n m a k e s e n s e b e y o n d t h e s e m i c la s si ca l a p p r o x i m a -
t i o n .
I n s e ct . 3 , C o l e m a n ' s e x p l a n a t i o n f o r th e v a n i s h i n g o f t h e c o s m o l o g i c a l c o n s t a n t is
r e v i e w e d . T h e s u p e r p o s i t io n o f a - d e p e n d e n t e f f e c ti v e t h e o r ie s is s h o w n t o b ed e s c r i b e d b y a p r o b a b i l i t y d i s t ri b u t i o n t h a t i s v e r y s h a r p l y p e a k e d a t A = 0. T h is
a r g u m e n t r e q u i r e s s o m e a d d i t i o n a l a s s u m p t i o n s b e y o n d t h o s e n e e d e d i n s e c t . 2 . I n
p a r t i c u l a r , t h e a r g u m e n t s e e m s t o r e l y q u i t e h e a v i l y o n t h e v a l i d i t y o f th e e u c l i d e a n
p a t h i n te g r a l a p p r o a c h t o q u a n t u m g r a v it y * .
I n s e c t . 4 , I a r g u e t h a t t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e " a - u n i v e r s e s " s t r o n g l y
f a v o r s n o t o n l y th a t A v a n i s h, b u t a ls o t h a t N e w t o n ' s c o n s t a n t G a s s u m e t h e
s m a l l e s t p o s s i b l e ( a - d e p e n d e n t ) v a l u e * * . A g a i n , t h i s a r g u m e n t s e e m s t o m a k e
e s s e n t i a l u s e o f t h e e u c l i d e a n p a t h i n t e g r a l f o r m a l i s m . I c l a i m t h a t t h e r e q u i r e m e n t
t h a tG(a)
i s a t i t s m i n i m u m s u f f i c e s t o d e t e r m i n e a l l o f t h e a ' s , a n d h e n c e t o
d e t e r m i n e a l l o f t h e c o n s t a n t s o f N a t u r e . T h e r e s t o f t h e p a p e r f u r t h e r e x p l o r e s t h e
i m p l i c a t i o n s o f t h is c l ai m .
I f w e h o p e t o a c t u a l ly d e t e r m i n e t h e a ' s b y m i n i m i z i n g G , w e m u s t c o m p u t e t h e
f u n c t i o n G ( a ) . W e c a n i m a g i n e c a r r y i n g o u t s u c h a c a l c u l a ti o n i n t w o s ta g es . I n t h e
f i rs t s t ag e , w e i n t e g r a t e o u t t h e w o r m h o l e f l u c t u a t i o n s t o o b t a i n a n e f f e c ti v e t h e o r y
w i t h a - d e p e n d e n t b a r e c o u p l i n g s; t hi s e f fe c t iv e t h e o r y is c u t o f f a t t h e w o r m h o l e
m a s s s c a le M w . I n t h e s e c o n d s t a ge , w e a l lo w t h e c u t o f f t o " f l o a t " d o w n f r o m M w
i n t o t h e f a r i n f r a r e d ; w e t h u s o b t a i n a n e x p r e s s i o n f o r t h e r e n o r m a l i z e d G t h a t
d e s c r i b e s g r a v i t y a t l o n g d i s t a n c e s i n t e r m s o f t h e b a r e p a r a m e t e r s o f t h e e f f e c t i v e
t h e o r y . T h e f i r s t s t a g e o f t h i s c a l c u l a t i o n i s t h e s u b j e c t o f s e c t . 5 , a n d t h e s e c o n d
s t a g e i s t h e s u b j e c t o f s e c t . 6 .
I f t h e m e c h a n i s m p r o p o s e d b y C o l e m a n f o r e n s u r i n g t h a t A = 0 is to a p p l y t o
N a t u r e , a n d i f t h is m e c h a n i s m r e q u i re s G(a) t o b e a t i t s m i n i m u m , t h e n i t i s
e v i d e n t l y n e c e s s a r y f o r G(a) t o h a v e a nonzero m i n i m u m . O t h e r w i s e G w i ll w a n t t o
v a n i s h , i n c o n f l i c t w i t h o b s e r v a ti o n . B u t in t h e a p p r o x i m a t i o n a s s u m e d b y C o l e m a n ,
i n w h i c h w o r m h o l e s a r e t r e a t e d a s d i l u t e a n d t h e i n t e r a c t i o n s a m o n g w o r m h o l e s a r e
* T h e o b s e r v a t i o n t h a t A = 0 is h i g h l y " p r o b a b l e " i n e u cl i d ea n q u a n t u m g r a v i t y h a d b e e n m a d e e a r li er
b y H a w k i n g [ 2 1 ] a n d b y B a u m [ 2 2 ] . T h e y , h o w e v e r , h a d n o t a p p r e c i a t e d t h e c r u c i a l r o l e o fw o r m h o l e s i n m a k i n g A a n adjustable q u a n t i t y .
* * M o r e p r e c i s e ly , G ( a ) i s m i n i m i z e d a s a f u n c t i o n o f a o n t h a t s u r f a c e i n a - s p a c e w h e r e A(a) = O.
8/3/2019 John Preskill- Wormholes in Spacetime and the Constants of Nature
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J. Preskill / Wormholes 145
n e g l e c t e d , G i s n o t b o u n d e d a w a y f r o m z e ro . T h u s , it is n e c e s s a r y t o im p r o v e o n
t h is d i l u te a p p r o x i m a t i o n . T h e s y s t e m a t i c c o r r e c t io n s t o t h e d i lu t e a p p r o x i m a t i o n
a r e d e s c r i b e d i n s e c t . 5 . I i n d i c a t e h o w w o r m h o l e i n t e r a c t i o n s c a n g e n e r a t e a l o w e r
b o u n d o n G , a n d i l l u s t r a t e t h i s p o s s i b i l i t y w i t h a s i m p l e t o y m o d e l .T h e s t u d y o f w o r m h o l e i n t e ra c t i o n s w i ll a l so le a d m e to p r o p o s e a m e c h a n i s m
t h a t c a n a c c o u n t f o r th e s u p p r e s s i o n o f th e l o w - e n e r g y e f fe c t s o f la r g e w o r m h o l e s .
T h i s m e c h a n i s m r el ie s o n t h e o b s e r v a ti o n t h a t t he a ' s a r e d e t e r m i n e d b y t h e
r e q u i r e m e n t t h a t G ( a ) i s a t i t s m i n i m u m , a n d i t a r i s e s d u e t o t h e i n t e r a c t i o n s o f
l a rg e w o r m h o l e s w i th s m a l l w o r m h o l e s . T h e m e c h a n i s m h a s t h e u n u s u a l f e a tu r e t h a t
t h e s u p p r e s s i o n o f la rg e w o r m h o l e s c a n n o t b e u n d e r s t o o d i n t e rm s o f lo n g - d i s ta n c e
p h y s i c s a l o n e . I n t h i s s e n s e , i t m a y b e r e g a r d e d a s a v i o l a t i o n o f t h e p r i n c i p l e t h a t
s h o r t - d i s t a n c e p h y s i c s is e f f e c ti v e ly " d e c o u p l e d " f r o m l o n g - d i s t a n c e p h y s i c s .
I n s e c t. 6 , I d e s c r i b e h o w a n e f f e c t i v e th e o r y o f g r a v i t y i s r e n o r m a l i z e d a s t h e
u l t r a v i o l e t c u t o f f f l o a t s d o w n f r o m t h e w o r m h o l e s c a l e t o w a r d t h e f a r i n f r a r e d . I
n o t e t h a t s o m e n o n p e r t u r b a t i v e p r o p e r t i e s o f t h i s r e n o r m a l i z a t i o n c a n b e e x t r a c t e d
i f I m a k e t h e r a t h e r m i l d a s s u m p t i o n t h a t E i n s te i n g r a v i t y h a s n o n o n t r i v i a l
c o n t i n u u m l im i t . F r o m t h e s e p r o p e r t i e s , I t h e n i n f e r t h e c o n d i t i o n s t h a t t h e
a - d e p e n d e n t b a r e t h e o ry a t th e w o r m h o l e s c a le m u s t s a ti sf y , in o r d e r t h a t C o l e m a n ' s
d e r i v a t i o n o f A = 0 b e a p p l i c a b l e . T h e s e c o n d i t i o n s a r e f o u n d t o b e q u i t e w e a k .
I t h e n e x a m i n e w h e t h e r c a l c u l a ti o n s o f f u n d a m e n t a l c o n s t a n t s o t h e r th a n A c a n
b e c a r r i e d o u t i n p r a c t i c e . I a r g u e t h a t t h e v a l u e s o f o t h e r c o n s t a n t s c a n n o t b e
p r e c i s e l y p r e d i c t e d w i t h o u t a d e ta i le d k n o w l e d g e o f p h y s i c s a t t h e w o r m h o l e s c a l e( w h i c h i s c o m p a r a b l e t o t h e P l a n c k s c a l e ) . T h i s a r g u m e n t i s b a s e d o n s t r a i g h t f o r -
w a r d p o w e r - c o u n t i n g o f F e y n m a n d i a g r a m s , w h i ch s h o w s t h a t t h e r e n o r m a l i z a t i o n
o f G is d o m i n a t e d b y s h o r t - d i s t a n c e q u a n t u m f l u c t u a ti o n s t h a t a r e s e n s it iv e t o t h e
d e t a i l s o f P l a n c k - s c a l e p h y s i c s .
T h e a n a l y s i s o f r e n o r m a l i z a t i o n e f f e c ts l e a d s t o o n e r a t h e r s u r p r i s in g c o n c l u s io n . I
f in d t h a t i f a n e l e m e n t a r y s ca la r o r fe r m i o n h a s a n a - d e p e n d e n t b a r e m a s s t h a t c a n
b e a d j u s t e d a t t h e w o r m h o l e s c a le , t h e n t h e p h y s i c a l m a s s o f th i s p a r t i c le c a n n o t b e
f a r b e l o w t h e w o r m h o l e s c a l e , unless t h e p h y s i c a l m a s s v a n i s h e s exact ly ' . A m o n g
o t h e r c o n s e q u e n c e s , t hi s o b s e r v a t i o n im p l i e s th a t i f C o l e m a n ' s m e c h a n i s m is t o b e
c o m p a t i b l e w i th t h e k n o w n f e a t u re s o f p a r t ic l e p h y s ic s , th e n w e m u s t i n s is t t h a t t h e
l i g h t m a s s s c a l e s o f p a r t i c l e p h y s i c s ( l i k e t h e w e a k i n t e r a c t i o n s c a l e ) a r e d e t e r m i n e d
d y n a m i c a l l y . T h e w e a k i n t e r a c t i o n s c a l e m u s t n o t b e d e t e r m i n e d b y a d ju s t i n g a b a r e
m a s s p a r a m e t e r a t t h e w o r m h o l e s c a l e .
2. T h e e f f e c t s o f w o r m h o l e s
A w o r m h o l e i n s p a c e t i m e is a g r a v i t a t i o n a l q u a n t u m f l u c t u a t i o n t h a t li n k s t w o
d i s t a n t l y s e p a r a t e d s p a c e t i m e p o i n t s . H e n c e , i t is n a t u r a l t o w o n d e r w h e t h e r
w o r m h o l e s c a n i n d u c e a n a p p a r e n t v i o l a ti o n o f l o c a li ty . B u t C o l e m a n [ 8] h a s a r g u e d
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146 J . P r e s k i l l / W o r r n h o l e s
p e r s u a s i v e l y t h a t t h e p h y s ic a l e f fe c t s o f w o r m h o l e s a r e q u i t e d i f f e r e n t t h a n w e m i g h t
n a i v e l y e x p e c t . H e c o n c l u d e s t h a t , b e c a u s e o f w o r m h o l e e f f e c t s , t h e f u n d a m e n t a l
c o n s t a n t s o f N a t u r e a r e a f f li c te d w i t h a n in t r in s i c q u a n t u m i n d e t e r m i n a c y . Q u a n -
t u m g r a v i t y , t h e n , m a y t h r e a t e n o u r a b il i ty , e v e n i n p ri n c i p l e , to m a k e p r e c i s e
p r e d i c t i o n s a b o u t h o w N a t u r e b e h a ve s .
I n t h i s s e c t i o n , I w il l b r i e f l y re v i e w h o w t h i s r e m a r k a b l e c o n c l u s i o n i s re a c h e d ,
f o l l o w i n g c l o s el y a r e f o r m u l a t i o n o f C o l e m a n ' s o r i g in a l a r g u m e n t d u e t o K l e b a n o v ,
S u s s k i n d , a n d B a n k s [ 1 6 ] . I i n c l u d e t h i s s e c t i o n t o e s t a b l i s h n o t a t i o n , a n d a l s o t o
m a k e t w o i m p o r t a n t p o i n t s . F ir s t, a l t h o u g h C o l e m a n o r i g i n a ll y p r e s e n t e d h is
a r g u m e n t i n t h e c o n t e x t o f a s e m i cl a ss ic a l a p p r o x i m a t i o n , t h e v a li d i ty o f t h is
a p p r o x i m a t i o n is n o t a t a ll a n e c e s s a r y i n g r e d i e n t in t h e a r g u m e n t . T h i s p o i n t i s
i m p o r t a n t b e c a u s e w e w i l l s e e w h e n w e a p p l y C o l e m a n ' s i d e a s t o t h e c o s m o l o g i c a l
c o n s t a n t p r o b l e m t h a t t h e s e m i c l a s s i c a l a p p r o x i m a t i o n u s e d b y C o l e m a n c a n n o t b e
j u s t if i e d . S e c o n d , w h i l e t h e n o t i o n o f a " b a b y u n i v e r s e " a s s o c ia t e d w i t h a w o r m h o l e
p l a y e d a c e n t r a l r o l e i n C o l e m a n ' s d i s c u s s i o n , t h is n o t i o n i s a c t u a l l y p e r i p h e r a l t o
t h e a r g u m e n t , a n d m a y n o t m a k e s e n s e b e y o n d t h e s e m i c la s si c al a p p r o x i m a t i o n .
T h e b a s i c a s s u m p t i o n s u n d e r l y i n g C o l e m a n ' s a n a l y s is a r e t h a t w o r m h o l e s e xi st ,
a n d t h a t t h e w o r m h o l e s h a v e t w o e s se n t ia l p r o p e r ti e s : (i ) W o r m h o l e s h a v e a
c h a r a c t e r i s t i c " t h i c k n e s s " R w . T h i s t h i c k n e s s i s p r e s u m a b l y o f o r d e r t h e P l a n c k
l e n g t h M ~ , t h e c h a r a c t e ri s ti c l e n g t h s c al e o f g r a v i t a ti o n a l q u a n t u m f l u c tu a t i o n s.
W e a s s u m e t h a t w o r m h o l e s m u c h t h i c k e r th a n R w a r e r a re a n d c a n b e n e g l e c t e d * .
( i i ) W o r m h o l e s h a v e n o c h a r a c t e r i s t i c l e n g t h . W e a s s u m e t h a t t h e w o r m h o l e i s
c o m p l e t e l y i n d i f f e r e n t t o t h e s e p a r a t i o n b e t w e e n i t s t w o e n d s i n t h e b a c k g r o u n d
s p a c e t i m e ; i t t a k e s a s h o r t c u t .
I t f o l l o w s f r o m a s s u m p t i o n ( i ) t h a t , f o r t h e p u r p o s e o f d i s c u s s i n g p h y s i c s a t
e n e r g i e s w e l l b e l o w R w ~, w o r m h o l e s m a y b e i n t e g r a t e d o u t ; t h e ir e f f e c ts m a y b e
i n c o r p o r a t e d i n t o a n e f f ec t iv e f ie l d t h e o r y i n w h i c h w o r m h o l e f l u c t u a t i o n s n o l o n g e r
o c c u r . I n t h i s l o w - e n e r g y e f f e c ti v e t h e o r y , th e c o n t r i b u t i o n t o a q u a n t u m m e c h a n i c a l
e x p e c t a t i o n v a l u e d u e t o a w o r m h o l e t h a t c o n n e c t s t h e s p a c e t im e p o i n t s x~ a n d x 2
c a n b e r e p r e s e n t e d b y a n e x p a n s i o n i n l o c a l o p e r a t o r s a t x 1 a n d x 2, o f t h e f o r m
Y" aob(O~(X~)@b(x2). ( 2 . 1 )ab
H e r e , t h e @ ~ 's a r e a s u i t a b l e b a s is f o r t h e g a u g e - i n v a r i a n t l o c a l o p e r a t o r s , a n d A is a
r e a l s y m m e t r i c m a t r ix . B y th e a s s u m p t i o n ( ii) a b o v e , A i s i n d e p e n d e n t o f x 1 a n d x 2.
T h e b i l o c a l e x p a n s i o n e q . ( 2 . 1 ) i s u n d e r s t o o d t o b e v a l i d w h e n t h e s e p a r a t i o n
b e t w e e n x 1 a n d x 2 is la r g e c o m p a r e d t o R w .
* We will examine more closely in sect. 5 the justification of this assumption.
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J. Preski ll / Wormholes 1 4 7
A f t e r a n i n t e g r a t i o n o v e r al l p o s s ib l e p o s i ti o n s o f t h e w o r m h o l e e n d s o n t h e
b a c k g r o u n d s p a c e t im e , t h e c o n t r i b u t i o n e q. ( 2 .1 ) b e c o m e s *
C = l f d 4 X l ~ f d 4 X 2 ~ ' g 2 Z A a b ( f l a ( X I ) O b ( X 2 ) "ab
( 2 . 2 )
N o w , i n t r e a t i n g t h e e f f e c t o f a c o n f i g u r a t i o n o f N w o r m h o l e s , le t u s i m a g i n e t h a t
t h e w o r m h o l e s a r e s u f f ic i e n tl y " d i l u t e " t h a t t h e y m a y b e c o n s i d e r e d t o b e in d e p e n -
d e n t o f o n e a n o t h e r . T h e N - w o r m h o l e c o n t r i b u t io n t h e n f a c to r iz e s a n d c a n b e
e x p r e s s e d a s CN/N!, w h e r e t h e 1/N! c o m p e n s a t e s f o r o v e r c o u n t i n g o f i d e n t i c a l
w o r m h o l e s . ( I n s e c t . 5 w e w i l l s e e h o w t h i s d i l u t e a p p r o x i m a t i o n c a n b e s y s t e m a t i -
c a l l y i m p r o v e d . )
T h e c o n t r i b u t i o n d u e t o a n a r b i t r a r y w o r m h o l e c o n f i g u r a t i o n i n t h e d i l u t e l i m i tc a n n o w b e o b t a i n e d b y s u m m i n g o v e r N . T h e s u m e x p o n e n t ia t e s, a n d c a n b e
e x p r e s s e d a s a g a u s s i a n i n t e g r a l ,
e c : f ( d c ~ ) e x p ( - j y ' a . ( A 1).hab)exp(~a.fd4x~@.(x)).a,h
( z . 3 )
W e s e e th a t , i n t h e d i lu t e a p p r o x i m a t i o n , t h e p r o c e s s o f in t e g r a t in g o u t w o r m h o l e s
i n d u c e s a n c ~ - d e p e n d e n t s h i f t i n t h e l o c a l ( e u c l i d e a n ) e f f e c t i v e a c t i o n ,
8 x ( a ) = - E a , , f d 4 x ( 2 . 4 )6/
Q u a n t u m m e c h a n i c a l e x p e c t a t i o n v a lu e s in t h e e ff e c ti v e t h e o r y a r e e x p r e s s e d as
i n t e g r a l s o v e r a o f e x p e c t a t i o n v al u e s c o m p u t e d i n u n i v er s e s w i t h a - d e p e n d e n t
c o u p l i n g s * * . T h e v a r i o u s a - u n i v e r s e s a r e w e i g h t e d b y a p r o b a b i l i t y d i s t r i b u t i o n
a , b
H o w e v e r , o u r p h y s i c a l o b s e r v a t i o n s a r e a c t u a l l y c a r r i e d o u t i n a u n i v e r s e w i t h a
f i x e d v a l u e o f th e a / s , f o r w e in p r i n c ip l e c a n m e a s u r e t h e c o n s t a n t s o f N a t u r e t o
a r b i t r a r y a c c u r a c y .
T h e l o ss o f q u a n t u m c o h e r e n c e d ec l ai m e d b y H a w k i n g [ 7] m a y b e i n t e rp r e t e d a s a
l o s s o f i n f o r m a t i o n a b o u t t h e o t h e r a - u n i v e r s e s t h a t o c c u r in t h e s u p e r p o s i t i o n e q.
( 2 . 3 ) . B e c a u s e p h y s i c s i s d o n e w i t h a l l a a ' s f i x e d , t h i s l o s s o f c o h e r e n c e i s n o t
* W e n e g l e c t a s m a l l e r r o r d u e t o t h e b r e a k d o w n o f t h e b i l o c a l e x p a n s i o n f o r x 1 - x 2 .
* * T h e n u m b e r o f i n d e p e n d e n t c~'s i s t h e s a m e a s th e n u m b e r o f n o n z e r o e i g e n v a l u e s o f t he m a t r i x A i n
e q . ( 2. 1 ). " G e n e r i c a l l y " , t h i s i s t h e s a m e a s th e n u m b e r o f g a u g e - i n v a r i a n t l o c a l o p e r a t o r s : t h a t i s, i t
i s i n f i n i t e .
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148 J. Preskill / Wormholes
a c c e s s i b le t o a n y o b s e r v e r. I n s t e a d o f a f a il u r e o f q u a n t u m m e c h a n i c s o r o f l o c a li ty ,
w o r m h o l e e f f e c t s i n t ro d u c e i n t o ph y s ic s a f u n d a m e n t a l q u a n t u m i n d e t e r m i n a c y o f
t h e c o n s t a n t s o f N a t u r e . W o r m h o l e s t h r e a t e n t o i n h i b i t o u r a b i l it y to p r e d i c t w i th
c e r t a i n t y th e c o n s t a n t s o f N a t u r e o n t h e b a s is o f f u n d a m e n t a l t h e o r y .
T h a t t h e a - d i s t r i b u t i o n P ( a ) is g a u s si a n, a n d t h a t t h e s h if t o f t h e e ff e c ti v e a c t i o n
8S(a) i s l i n e a r i n a , a r e a r t if a c t s o f th e d i l u t e a p p r o x i m a t i o n t h a t w e h a v e m a d e .
C o r r e c t i o n s t o t h i s a p p r o x i m a t i o n w i l l b e d i s c u s s e d i n s e c t . 5 . B u t t h e c o n c l u s i o n
t h a t t h e e f f e c ts o f w o r m h o l e s c a n b e i n c o r p o r a t e d i n t o a n a - d e p e n d e n t r e n o r m a l iz a -
t i o n o f t h e e f f e c t i v e a c t i o n s e e m s t o b e m u c h m o r e g e n e r a l . I b e l i e v e t h a t t h i s
c o n c l u s i o n f o l lo w s f r o m j u s t t h e t w o a s s u m p t i o n s s t a te d a b o v e ( s u b je c t to s o m e
c a v e a t s t h a t w i l l b e m e n t i o n e d l a t e r ) .
T h e a b o v e r e f o r m u l a t i o n o f C o l e m a n ' s a r g u m e n t w a s p r e v io u s l y p r e s e n te d i n r e f.
[ 1 6 ] . I t d i f f e r s f r o m C o l e m a n ' s o r i g i n a l p r e s e n t a t i o n [ 8 ] i n t w o s i g n i f i c a n t r e s p e c t s .F i r s t, C o l e m a n ' s a r g u m e n t w a s c a r r i e d o u t w i t h i n t h e c o n t e x t o f t h e s em i c la s si ca l
a p p r o x i m a t i o n , i n w h i c h t h e o n l y w o r m h o l e c o n f i g u r a t i o n s th a t c o n t r i b u t e t o t h e
e u c l i d e a n p a t h i n te g r a l a r e s o lu t i o n s t o th e e u c l i d e a n e q u a t i o n s o f m o t i o n ( s t a -
t i o n a r y p o i n t s o f t h e e u c l i d e a n a c t i o n ) a n d s m a l l f l u c t u a t i o n s a b o u t s u c h s o l u t i o n s .
T h e a b o v e a r g u m e n t i s m o r e s a t is f y in g b e c a u s e it m a k e s n o r e f e r e n c e t o t h e
s e m i c l a s s ic a l a p p r o x i m a t i o n . T h i s p o i n t is i m p o r t a n t , b e c a u s e w e w il l s e e i n s ec t . 3
t h a t , i f w e a r e t o i n v o k e C o l e m a n ' s a r g u m e n t s t o s o lv e t he c o s m o l o g i c a l c o n s t a n t
p r o b l e m , t h e n t h e s e m ic la s si ca l a p p r o x i m a t i o n c a n n o t b e j u s ti f ie d . ( W e haue,so f a r ,
r e g a r d e d t h e w o r m h o l e s a s d il u te , b u t o u r c o n c l u s i o n s s ur v iv e e v e n w h e n t h e d i lu t ea p p r o x i m a t i o n i s a b a n d o n e d . )
T h e s e c o n d r e s p e c t i n w h i c h o u r
a r g u m e n t i s t h a t C o l e m a n a t t a c h e s t o
C o l e m a n i n c l u d e s i n h i s a n a l y s i s , i n
f o r m u l a t i o n d i f f e r s f r o m C o l e m a n ' s o r i g i n a l
a a n i n t e r p r e t a t i o n t h a t w a s e s c h e w e d a b o v e .
a d d i t i o n t o w o r m h o l e s a t t a c h e d t o t h e b a c k -
g r o u n d s p a c e t i m e , " s e m i w o r m h o l e s " t h a t c o n n e c t t h e b a c k g r o u n d s p a c e t i m e t o a
t i n y c l o s e d t h r e e - g e o m e t r y , o r " b a b y u n i v e r s e ." H e is th u s a b l e t o i n t e r p r e t a a s th e
l a b e l o f a c o h e r e n t s u p e r p o s i t i o n o f b a b y u n i v e r s e s t a t e s . A s w e h a v e s e e n , t h i s
i n t e r p r e t a t i o n i s n o t e s s e n t i a l i n t h e d e r i v a t i o n o f t h e m a i n r e s u l t , t h a t w o r m h o l e
e f f e c t s c a n b e a b s o r b e d i n t o a n a - d e p e n d e n t e f f e c t iv e a c t io n . T h i s is a g o o d t h in g ,
b e c a u s e i t s e e m s t h a t t h e n o t i o n o f a b a b y u n i v e r s e , o r a t l e a s t t h e c o n n e c t i o n
b e t w e e n b a b y u n i v e r se s a n d t h e a ' s a s s o c i at e d w i th w o r m h o l e p h y si c s, m a k e s s e n se
o n l y w i t h i n t h e s e m i cl a ss ic a l a p p r o x i m a t i o n .
A l t h o u g h t h is is s o m e w h a t o f f th e m a i n l i n e o f th e a r g u m e n t i n t h is p a p e r , I w il l
d e s c r i b e i n t h e re m a i n d e r o f th i s s e c ti o n h o w t h e c o n n e c t i o n b e t w e e n a ' s a n d b a b y
u n i v e r s e s c a n b e e s t a b l i sh e d . T h i s d i s c u s s i o n w i ll c l a r i f y w h y t h e s e m i c la s s ic a l
a p p r o x i m a t i o n a p p e a r s t o b e e s s e n t i a l .
L e t t h e i n d e x i l a b e l th e s t a t e s o f a b as i s f o r th e b a b y u n i v e r s e s ta t es . W e m a y
d e f i n e S ~j a s t h e a m p l i t u d e f o r t h e b a b y u n i v e r s e s t a t e i t o e v o l v e t o t h e b a b y
u n i v e r s e s t a t e j d e t e r m i n e d , f o r e x a m p l e , b y a e u c l i d e a n p a t h i n t e g r a l ( fi g. 2 ). W e
m a y a l s o c o n s i d e r t h e a m p l i t u d e f o r a s e m i w o r m h o l e to c o n n e c t b a b y u n i v e r se i t o
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J. Preskill / Wormholes
J
Fig. 2. The amplitude for bab y universe state i to evolve to bab y universe state j.
149
i
Fig. 3. The amplitude for baby universe state i to connect via a semiwormhole to the backgroundspacetime.
a b a c k g r o u n d s p a c e t i m e (f ig . 3 ); q u a n t i t i e s X j c a n b e d e f i n e d ( if t h e m a t r i x S is
i n v e r t i b l e ) s o t h a t t h is a m p l i t u d e is d e n o t e d
Y: , S , jX j .
J
I n t h is e x p r e s s io n , w e m a y t h i n k o f S a s a b a b y u n i v e r s e " p r o p a g a t o r , " a n d X j a s
t h e s tr e n g t h o f t h e " c o u p l i n g " o f b a b y u n i v e r s e j t o t h e b a c k g r o u n d s p a c e t i m e .
S o f a r t hi s is j u s t d e f i n it i o n s. B u t n o w w e m a k e t h e n o n t r i v i a l a s s u m p t i o n t h a t t h e
a m p l i t u d e f o r a w o r m h o l e t o a t t a ch t o t he b a c k g r o u n d s p a c e t im e i s de t e rm i ned b y
t h e b a b y u n i v e r s e p r o p a g a t o r a n d t he s e m i w o r m h o l e a m p l i t u d e t o b e
~ ' , X i S s jX j .ij
H e u r i s t i c a l l y , t h e n , t h e w o r m h o l e a m p l i t u d e ( fi g. 4 ) o b e y s a " c u t t i n g r u l e " o r
" c o m p l e t e n e s s r e l a ti o n , " d e n o t e d s c h e m a t i c a l l y in fi g. 5 ; i t c a n b e r e p r o d u c e d b y a
s u m o v e r b a b y u n i v er s e " i n t e r m e d i a t e s ta te s ."
Z ~ s ~ j h1,1
Fig. 4. The amplitude for a wormhole to attach to the background spacetime.
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150 J. Preskill / Wormholes
i i
F ig . 5 . T h e b a b y u n i v e r s e c o m p l e t e n e s s r e l a t io n .
I n t h e s e m i c l a s s i c a l a p p r o x i m a t i o n , s u c h a c o m p l e t e n e s s r e l a t i o n i s s a t i s f i e d ,
b e c a u s e t h e r e i s a n a t u r a l w a y t o a s s o c i a t e a b a b y u n i v e r s e w i t h a w o r m h o l e . T h e
b a b y u n i v e r s e i s t h e t h r e e - g e o m e t r y o b t a i n e d w h e n w e s li ce th e c l as s ic a l w o r m h o l e
s o l u t i o n t h r o u g h i t s m i d p o i n t [2 , 8 , 9 ]. B u t t h e r e is n o u n a m b i g u o u s w a y o f a s s o c ia t -
i n g a g e n e r a l w o r m h o l e c o n f i g u r a t i o n ( t h a t i s n o t a c l a s s i c a l s o l u t i o n ) w i t h a b a b y
u n i v e r s e , a n d I s e e n o r e a s o n t o e x p e c t t h a t t h e r e e x i s t s a b a s i s f o r t h e b a b y u n i v e r s e
s t a t e s s u c h t h a t t h e i d e n t i t y i n d i c a t e d i n f ig . 5 i s s a t i s f ie d , e x c e p t in t h e s e m i c l a s s i c a l
a p p r o x i m a t i o n .
T h i s c o m p l e t e n e s s r e l a t i o n p r o v i d e s t h e l o g ic a l c o n n e c t i o n t h a t i s n e e d e d t o r e l a te
w o r m h o l e p h y s i c s t o b a b y u n i v e r s e s . T o e s t a b l i s h t h i s r e l a t i o n s h i p , a s s u m e t h e
v a l i d i t y o f t h e c o m p l e t e n e ss r e la t io n a n d s u m o v e r w o r m h o l e a n d s e m i w o r m h o l e
c o n f i g u r a t i o n s . ( A t y p i c a l c o n f i g u r a t i o n i s s h o w n i n f i g . 6 . ) I n t h e d i l u t e a p p r o x i m a -
t i o n t h e s u m e x p o n e n t i a t e s , a n d w e h a v e
Z [ J ] : e x p ( ~ ( ~ J S , s ~ + J , i j X j + 1 2 X S i j X i ) ) , ( 2 . 6 )
w h e r e 4 i s a s o u r c e t h a t c o u p l e s t o th e b a b y u n i v e r s e i. T h i s g e n e r a t i n g f u n c t i o n c a n
b e e x p r e s s e d a s a g a u s s i a n i n t e g r a l
Z [ J ] = f ( d a ) e x p ( - l Y 'a s ( S t ) / j a d + . ~ _ . , a , ( J , + X , ) ) .i , ) i
( 2 . 7 )
N o w i t is c o n v e n i e n t t o r e d e f i n e o u r b a si s f o r t h e b a b y u n i v e r s e s ta t e s s o th a t S i jb e c o m e s 8 i j . ( T h i s i s a c o n v e n t i o n a l " r e n o r m a l i z a t i o n " o f t h e b a b y u n i v e r s e w a v e
F i g. 6 . A t y p i c a l c o n f i g u r a t i o n o f w o r m h o l e s a n d s e m i w o r m h o l e s .
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J. Preskill / Wormholes 153
Fig. 8. A connected geometry that b eco m es disconnected after wormh oles are integrated ou t. Theoperator O wh ose expectation value is to be evaluated acts on on e o f the connected components.
i s o v e r f o u r - d i m e n s i o n a l g e o m e t r i e s t h a t ar e c l o s e d a n d c o n n e c t e d * . ( T h e i n t e g r a -
t i o n o v e r f i e l d s o t h e r t h a n t h e m e t r i c h a s n o t b e e n e x p l i c i t l y i n d i c a t e d , b u t i s
u n d e r s t o o d . ) W e a s s u m e a s i n s ec t. 2 t h a t t h is p a t h i n t e g r a l in c l u d e s a s u m o v e r
c o n f i g u r a t i o n s i n w h i c h n a r r o w w o r m h o l e s l in k d is t a n t ly s e p a r a t e d p o i n t s o n a
s m o o t h b a c k g r o u n d s p ac e ti m e , a n d w e i m a g in e " i n t e g r a t i n g " o u t th e w o r m h o l e s
a n d i n c o r p o r a t i n g t h e ir e f fe c ts i n t o a l o c al e f f e ct i v e a c t i o n f o r th e b a c k g r o u n d
s p a c e t im e . B u t n o w w e n o t e t h a t th e c o n n e c t e d c l o s e d m a n i f o l d s i n c l u d e g e o m e t r ie s
i n w h i c h m a n y d i s ti n c t la r ge s m o o t h m a n i f o l d s a r e l in k e d t o g e t h e r b y n a r r o w
w o r m h o l e s . T h u s , w h e n w e in t e g ra t e o u t w o r m h o le s~ a t y p ic a l c o n n e c t e d c l o s e d
m a n i f o l d m a y y i e l d m a n y d i s c o n n e c t e d c o m p o n e n t s ( f ig . 8). I n t e g r a t in g o u t w o r m -
h o l e s , t h e n , a f f e c t s t h e e u c l i d e a n p a t h i n t e g ra l i n t w o w a y s . A s w e a l r e a d y s a w i n
s e c t . 2 , e x p e c t a t i o n v a l u e s b e c o m e e x p r e s s e d i n t h e e f f e c t i v e t h e o r y a s i n t e g r a l s o v e r
t h e c~ p a r a m e t e r s o f e x p e c t a t i o n v a l u e s c o m p u t e d i n v a r i o u s u n i v e r s es w i t h ~ - d e p e n -
d e n t c o u p l i n g s . B u t f u r th e r m o r e , t h e s u m o v e r c o n n e c t e d c l o s e d m a n i f o l d s i n th e
u n d e r l y i n g t h e o r y b e c o m e s a s u m o v e r a ll c l o se d m a n i f o l d s , i n c l u d i n g d i s c o n n e c t e d
o n e s , i n t h e e f f e c t i v e t h e o r y .
E a c h d i s c o n n e c t e d m a n i f o l d c a n b e f a c t o r e d i n t o a c o n n e c t e d c o m p o n e n t o n
w h i c h t h e o p e r a t o r (9 a c ts , a n d a r e m a i n d e r o n w h i c h n o l o c al o p e r a t o r a c t s. S i n c e
t h e s u m o v e r a ll c l o s e d f o u r - d i m e n s i o n a l g e o m e t r i e s c a n b e w r i tt e n a s t h e e x p o n e n -
t i a l o f a s u m o v e r a l l connected c l o s e d f o u r - g e o m e t r i e s , t h e e x p r e s s i o n f o r t h e
e x p e c t a t i o n v a l u e o f (9 m a y b e w r i t te n i n t h e ef f e ct iv e t h e o r y a s
. o l ) f ( d g ) e ( 3 . 2 )
w h e r e a g a i n t h e i n t e g r al s o v e r g a n d g ' d e n o t e s u m s o v e r connected c l o s e d
f o u r - g e o m e t r i e s , b u t w i t h w o r m h o l e s n o w e x c l u d e d . T h e d i s t ri b u t i o n P ( c 0 is t h a t
c o n s i d e r e d i n s ec t. 2, w h i c h w o u l d b e g a u s s i a n i n th e d i lu t e a p p r o x i m a t i o n . B u t w e
* In thi s discussion, the Hartle-Hawk ing boundary, condition [6] has been implicitly adopted. Theresults are no t sensitively dependent on this choice.
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154 J. Preskill / Wormholes
now see that the measure for the integration over (Y contains the additional factor
(3.3)
It is Coleman’s proposal that this factor is a sharply peaked function of the a’s;
since P(a) is presumably a smooth function, the factor Q(a) determines where the
support of the a-distribution lies.
To compute the factor Q(a) in eq. (3.3) Coleman makes a further hypothesis -
that the sum over connected closed four-geometries is dominated by large smooth
geometries with small curvature (in Planck units). The idea is that by integrating out
wormholes we have satiated the desire of a typical four-geometry to crinkle up on
the Planck scale. This assumption is in the same spirit as, though logically indepen-
dent of, the assumption that it makes sense to integrate out wormholes at all. Surely,
given the current status of our understanding of quantum gravity, both assumptions
must be regarded as unproved working hypotheses.
Anyway, given that the integral / (dg) e- ‘lg. al is dominated by large smooth
geometries, we may compute it by the following procedure: Recall that S[ g, a] is
the action of an effective theory with a cutoff of order M, - R;‘, the characteristic
mass scale of the wormhole fluctuations. Imagine allowing this cutoff to float down
to a mass scale M < M,; we integrate out all quantum fluctuations with wave
number k > M, and incorporate the effects of these fluctuations into a renormalized
effective action S,[g, LX]_ his renormalized effective action can be expanded in
operators that are local on the distance scale M-’ [23].
For M +Z M,, the loop corrections in the effective theory with cutoff M, and the
effects of higher dimension operators in S,, are suppressed by powers of M/M,.
Because loop corrections are small, the path integral of the effective theory can be
evaluated to good accuracy semiclassically, with the result
(3.4)
where jj is the four-geometry at which S,,,, is stationary. If this stationary point is a
large smooth geometry with volume of order R4, we improve on the semiclassical
approximation by allowing the cutoff to float down to M - R-‘. Then all fluctua-
tions at wavelengths less than R have been absorbed into the renormalization of the
effective action, and fluctuations on larger wavelengths are absent because the
volume acts as an infrared cutoff*.
l I have described the procedure in terms of a floating cutoff in order to emphasize that the action
may always be regarded as effectively local and also that infrared divergent loop integrals need not
be encountered; they arise only as R + oc or A4 + 0.
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J. Preskill / Wormholes 155
O f c o u r s e , i t w a s a b i t d i s i n g e n u o u s t o s a y a b o v e t h a t ~ i s t h e s t a t i o n a r y p o i n t o f
S M ; w e r e a l l y w a n t g t o m i n im i z e S M . I t i s w e l l k n o w n t h a t t h e s t a t i o n a r y p o i n t s o f
t h e e u c l i d e a n a c t i o n o f g ra v i t y a r e n o t e v e n lo c a l m i n i m a ; w e c a n a l w a y s p e r f o r m a
c o n f o r m a l t r a n s f o r m a t i o n o n t h e m e t r i c t h a t l o w e r s t h e a c t i o n . I n o r d e r t h a t t h ea b o v e d i s c u s s i o n a p p l y I m u s t a s s u m e t h a t t h e r e i s s o m e c o r r e c t p r o c e d u r e f o r
d e a l i n g w i t h t h e c o n f o r m a l f lu c t u a t io n s , a n d t h a t w h e n t h is p r o c e d u r e is i n v o k e d ,
c o n f o r m a l f l u c t u a t i o n s c a n b e i n t e g r a t e d o u t a n d a b s o r b e d i n t o a r e n o r m a l i z e d
e f f e c t i v e a c t i o n .
T h e e f f e c t i v e a c t i o n S i c a n b e e x p a n d e d i n t e r m s o f l o c a l o p e r a t o r s , a n d , f o r
M << M w , o p e r a t o r s w i t h n d e r i v a t i v e s a c ti n g o n t h e m e t r i c h a v e e f f e c t s t h a t a r e
s u p p r e s s e d b y t h e f a ct o r ( M / M w ) ' . T h i s e x p a n s i o n h a s t h e f o r m *
1
S M [ g l - - 1 6v rG f d 4 x v /g ( 2 A - R + . . . ) , ( 3 . 5 )
w h e r e t e r m s w i t h m o r e t h a n t w o d e r i v a ti v e s h a v e b e e n n e g l e c te d ; t h e p a r a m e t e r s G
a n d A h a v e i m p l i c it d e p e n d e n c e o n M w , M , a n d a . A t t h e s t a t io n a r y p o i n t s o f S M ,
t h e m e t r i c o b e y s t h e e u c l i d e a n E i n s t e i n e q u a t i o n
a n d t h e a c t i o n i s
R , = A g ~ , ( 3 . 6 )
A AsM [ fdnx - 8~rG V ( 3 . 7 )
w h e r e V is t h e v o l u m e o f t h e e u c l i d e a n s p a c e t i m e . F o r A > 0 , t h e s o l u t i o n o f eq .
( 3 .6 ) w i t h m a x i m a l v o l u m e i s k n o w n t o b e a f o u r -s p h e r e w i t h
V = 2 4 ~ 2 / A 2 . ( 3 . 8 )
H e n c e t h e f a c t o r e q . ( 3 . 3 ) i s f o u n d t o b e
Q ( a ) = e x p ( e - s . [~ . , ~1) = exp [ e x p ( 3 7 r / G A ) ] , ( 3 . 9 )
f o r A > 0 . H e r e A i s t h e c o s m o l o g i c a l c o n s t a n t r e n o r m a l i z e d a t t h e s c a le M - V ~ /4 ,
w i t h V g i v e n b y e q . ( 3. 8) .
E q . ( 3 . 9 ) i s t h e s h a r p p e a k a t A = 0 f o u n d b y C o l e m a n . A i s a f u n c t i o n o f a , a n d
e q . ( 3 . 9 ) t o g e t h e r w i t h e q . ( 3 . 2 ) t e ll s u s t h a t i t i s o v e r w h e l m i n g l y l i k e l y t h a t a
u n i v e r s e s e l e c t e d a t r a n d o m w i ll h a v e a v a l u e o f a s u c h t h a t A = 0 , a s s u m i n g t h a t A
v a n i s h e s f o r s o m e c h o i c e o f a . W e n o t e t h a t t h e q u a n t i t y t h a t h as b e e n f o u n d t o
* M a s s i v e f i e l d s ( h e a v i e r t h a n M ) h a v e b e e n i n t e g r a t e d o u t , a n d l i g h t f ie l d s h a v e b e e n s e t e q u a l t o
v a l u e s t h a t m i n i m i z e S , w , s o S ~I i s e x p r e s se d a s a f u n c t i o n a l o f t h e m e t r i c o n l y.
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156 J. Preskill / Wormholes
v a n i s h i s t h e renormalized A t h a t o n e w o u l d a c t u a l l y m e a s u r e i n a l a r g e s m o o t h
u n i v e r s e ; i t i n c l u d e s t h e c o n t r i b u t i o n s f r o m t h e f lu c t u a t i o n s o f a ll q u a n t u m f ie ld s .
W e a l s o n o t e t h a t i t w a s s a f e t o n e g l e c t t h e h i g h e r d i m e n s i o n o p e r a t o r s i n e q . ( 3 . 5 ) ;
t h e s e g iv e c o n t r i b u t i o n s t o SM[g] t h a t a r e i n d e p e n d e n t o f A f o r A s m a l l, a n d d o n o tm o d i f y t h e A = 0 s i n g u l a r i t y i n e q . ( 3 . 9 ) .
( T h e p r o c e d u r e t h a t w e h a v e u se d t o c a l c ul a te Q ( a ) m a y m a k e s e n s e f o r A > 0 ,
b u t i t is il l - d e f i n e d f o r A < 0 . F o r A < 0 , t h e b e s t a p p r o a c h m a y b e t o a n a l y t i c a l l y
c o n t i n u e e q . (3 . 9) f r o m p o s i t i v e v a l u e s o f A t o n e g a t i v e v a lu e s . O u r i m p l i c i t
a s s u m p t i o n t h a t G i s n o n - n e g a t i v e w il l b e d e f e n d e d i n s e c t. 6.)
I t m a y b e w o r t h w h i l e t o c o m m e n t o n t h e u se o f " s e m i c la s s i c a l" r e a s o n i n g i n th is
d e r i v a t i o n . W e t r e a t e d g r a v i t y s e m i c l a s s ic a l ly in t h e s e n s e t h a t w e a r g u e d t h a t l o o p
e f f e c t s a re s m a l l w h e n t h e f l o a ti n g u l tr a v i o l e t c u t o f f M is m u c h l es s th a n t h e c u t o f f
M w o f t h e b a r e t h e o r y ( a n d i n fa c t c o m p a r a b l e t o th e i n f r a r e d c u t o f f V 1 /4 ). B u t w ec e r t a i n l y d i d n o t r e q u i r e t h a t t h e r e n o r m a l i z a t i o n o f S a s th e c u t o f f f l o a t s f r o m M w
t o M b e s u s c e p t i b l e t o a p e r t u r b a t i v e t r e a t m e n t . I n d e e d , it m a y b e a m i s s t a t e m e n t
t o s a y t h a t t h e p a t h i n t e g r a l i s d o m i n a t e d b y l a r g e s m o o t h g e o m e t r i e s , f o r w e d o n o t
m e a n t o e x c l u d e t h e p o s s i b il i t y th a t t h e e f fe c t iv e a c t i o n i s s t ro n g l y r e n o r m a l i z e d b y
q u a n t u m f l u c t u a t i o n s a t s c al es j u s t b e l o w M w . N o r , o f c o u r s e , h a v e w e r e q u i r e d t h a t
t h e p r o c e s s o f in t e g r a t i n g o u t w o r m h o l e s , w h i c h g e n e r a t e d t h e th e o r y w i t h c u t o f f
M w , b e a c c u r a t e l y d e s c r ib e d b y t h e s e m i c l a s s ic a l a p p r o x i m a t i o n .
T h e o b s e r v a t i o n t h a t w e m a d e i n s ec t. 2 - t h a t t h e n o t i o n o f a b a b y u n i v e r s e
m a k e s s e n s e o n l y w i th i n t he c o n t e x t o f a n u n t r u s t w o r t h y a p p r o x i m a t i o n h a s ac o n s e q u e n c e t h a t s h o u ld b e m e n t i o n e d n o w . W e h a v e n o t e d t h a t a " s u p e r s e l e c t io n
r u l e " a p p l i e s t o w o r m h o l e p h y s ic s ; u n i v e r s e s w i t h d i f fe r e n t v a lu e s o f a c a n n o t
c o m m u n i c a t e w i t h e a c h o t h e r t h r o u g h a n y l o c a l p h y s i c a l p r o c e s s . B u t C o l e m a n ' s
e x p l a n a t i o n f o r A = 0 c a n b e i n v o k e d o n l y if w e c o n s i d e r a s u p e r p o s i t i o n o f th e s e
a - u n i v e r s e s a s i n e q . ( 3 . 2 ) . O n e m i g h t o b j e c t t o s u c h a s u p e r p o s i t i o n , c i t i n g t h e
a n a l o g y w i t h t h e 0 - v a c u a o f Q C D . I t w o u l d b e p e r v e r s e to s a y t h a t t h e g r o u n d s t a t e
o f Q C D i s a s u p e r p o s i t i o n o f 0 - v a c u a . W h y s h o u l d n ' t w e t a k e t h e s a m e a t t i t u d e
t o w a r d a s u p e r p o s i t i o n o f a - u n i v e r s e s ?
T h e k e y d i s t in c t i o n b e t w e e n 0 - v a c u a a n d a - u n i v e r s e s , I s u s p e c t, is t h a t t h e as u p e r s e l e c t i o n r u l e i s a n a p p r o x i m a t e r u l e t h a t h o l d s o n l y i n t h e l o w - e n e r g y l i m i t . I t
w o u l d h a v e b e e n e a s i er to i m a g i n e t h a t t h e a s u p e r s e l e c ti o n r u l e w e r e e x a c t if t h e
i n t e r p r e t a t i o n o f a a s th e l a b e l o f a b a b y u n i v e r s e w e r e a l s o e x a ct . I f t h is
i n t e r p r e t a t i o n f ai ls , i t is h a r d t o s e e h o w a - s e c t o r s c a n b e i d e n t if ie d i n s u b - P l a n c k
s c a l e p h y s i c s . T h i s, p e r h a p s , is th e p r o p e r w a y t o i n t e r p r e t C o l e m a n ' s c l a im [ 10 ] th a t
a n a p r i o r i d i s t ri b u t i o n i n a w i th s u p p o r t a t a s in g l e p o i n t i n a - s p a c e w o u l d b e
" u n n a t u r a l . "
T h e a r g u m e n t i n t h i s s e c t i o n e v i d e n t l y r e q u i r e d s t r o n g e r a s s u m p t i o n s t h a n t h e
d i s c u s s i o n i n s e ct . 2 . H e r e w e a p p a r e n t l y n e e d e d t o ta k e m o r e s e r io u s l y th a n i n s e ct .
2 t h e p r o p o s a l t h a t e x p e c t a t i o n v a l u e s c a n b e e v a l u a t e d a s e u c l i d e a n p a t h i n t e g r a l s
( a n d t h a t a s e n s i b l e p r e s c r i p t i o n e x i s ts f o r d e a l i n g w i t h c o n f o r m a l f l u c tu a t io n s ) .
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J. Preskill / Worm holes 157
A n d w e f u r t h e r a s s u m e d t h a t , a f t e r w o r m h o l e s a r e i n t e g r a t e d o u t , t h e p a t h i n te g r a l
i s d o m i n a t e d b y l a r g e s m o o t h g e o m e t r i e s . T h a t t h e s e a s s u m p t i o n s l e a d t o t h e
e x c i t i n g c o n c l u s i o n t h a t t h e c o s m o l o g i c a l c o n s t a n t v a n i s h e s a l r e a d y p r o v i d e s s u f f i -
c i e n t m o t i v a t i o n f o r e x p l o r in g t h e i r f u r t h e r c o n s e q u e n c e s . B u t i t is o b v i o u s l y a nu r g e n t m a t t e r t o u n d e r s t a n d b e t t e r t h e j u s t i f i c a t i o n f o r th e s e a s s u m p t i o n s , i f i n d e e d
a n y c a n b e f o u n d .
4 . N e w t o n ' s c o n s t a n t
W e h a v e e m p h a s i z e d t h a t t he f a c t o r Q ( a ) in t h e m e a s u r e f o r t he i n te g r a t io n o v e r
a - u n i v e r s e s o v e r w h e l m i n g l y f a v o r s A = 0 , i f t h e re n o r m a l i z e d c o s m o l o g i c a l c o n s t a n t
c a n v a n i s h f o r s o m e v a lu e o f a . T h i s m e a s u r e f a c t o r h a s a f u r th e r c o n s e q u e n c e ;
N e w t o n ' s c o n s t a n t G w a n t s t o a s s u m e th e s m a l l e s t p o s s i b l e v a l ue , p r e f e r a b l y z e ro * .I n t h e r e m a i n d e r o f t h i s p a p e r , w e w i l l e x p l o r e t h e c o n s e q u e n c e s o f t h i s t e n d e n c y o f
G t o s e e k i ts m i n i m u m a l lo w e d v a l u e.
O n e c o n s e q u e n c e i s i m m e d i a t e : t h e r e n o r m a l i z e d G a s a fu n c t i o n o f a m u s t b e
b o u n d e d a w a y f r o m z er o
G ( a ) > / . Gmin > O . ( 4 . 1 )
O t h e r w i s e , t h e s a m e a s s u m p t i o n s t h a t l e a d t o th e s u c c e s s fu l p r e d i c t io n A = 0 w ill
a l s o l e a d t o t h e u n s u c c e s s f u l p r e d i c t i o n G = 0 . ( A g r a v i t a t i o n a l i n t e r a c t i o n is
o b s e r v e d i n o u r u n i v e r s e . )
O n e m i g h t o b j e c t , a t f i r s t , t o t h e a s s e r t i o n t h a t t h e p r o b a b i l i t y d i s t r i b u t i o n
Q ( a ) - e x p [ e x p ( 3 7 r / G A ) ] ( 4 . 2 )
is c a p a b l e o f d e t e r m i n i n g b o t h G a n d A . T h is c l a im s o u n d s s t r a ng e , b e c a u s e Q ( a ) is
a f u n c t i o n o f o n l y t h e p r o d u c t G A . T h u s , o n e m i g h t a r g u e , t h e v e r y s h a r p p e a k i n
Q ( a ) o c c u rs a t G A = 0 , a n d Q ( a ) is c o m p l e t e l y i n d i f fe r e n t t o t h e v a lu e o f G o n c e A
h a s a s s u m e d t h e v a lu e A = 0.
I n d e e d , t o e x t r a c t f r o m e q . ( 4 .2 ) a p r o b a b i l i t y d i s t r ib u t i o n o n t h e s u r f a c e G A = 0( w h e r e Q ( a ) is s i n g u la r ), w e m u s t s p e c i f y a p r e f e r r e d w a y o f a p p r o a c h i n g t h i s
s u r f a c e . I w i ll a r g u e b e l o w t h a t t h e c o r r e c t p r o c e d u r e is t o e v a l u a t e Q ( a ) o n s u r f a c e s
o f c o n s t a n t A , a n d t h e n t a k e t h e l i m i t A - -, 0 . I f A a s s u m e s a v e r y s m a l l p o s i t i v e
v a l u e , Q ( a ) s t r o n g l y f a v o r s t h a t G 1 i n c re a s e . T h i s t e n d e n c y o f G - 1 t o i n c r e a s e is
n o t s u d d e n l y l o s t w h e n A is e x a c t l y z e ro .
W e c l a im , t h e n , t h a t th e s u p p o r t o f t h e d i s t r ib u t i o n Q ( c 0 w i l l l i e a t t h a t p o i n t ( o r
t h o s e p o i n t s ) in a - s p a c e w h e r e G - l ( a ) a s s u m e s i t s m a x i m a l v a l u e o n t h e s u rf a c e
d e f i n e d b y A ( a ) = 0 . T h e r e is a n o t h e r o b j e c t i o n t h a t o n e c o u ld r a is e a g a in s t t hi s
c l a im . O n e m i g h t q u es t io n w h e t h e r t he re q u i r e m e n t t h a t G - t a s s u m e s it s m a x i m u m
* This p oint was stressed previously by Grinstein and W ise [15].
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1 5 8 J. Pres kil l / Worm holes
h a s a n y p h y s i c a l c o n t e n t . S in c e G a is a d im e n s i o n a l q u a n t i t y , th e a r g u m e n t m i g h t
g o , i t s v a l u e o f c o u r s e d e p e n d s o n t h e u n i t s i n w h i c h w e c h o o s e t o e x p r e s s i t .
N o t h i n g p r e v e n t s u s f r o m c h o o s i n g o u r u n i t o f m a s s t o b e G 1 / 2 . T h e n G - ~ is o n e,
a n d i t m a k e s n o s e n s e t o " m a x i m i z e " i t .T h i s a r g u m e n t is al s o m i s l e a d in g , f o r s e v e r a l r e a s o n s . F i r s t o f al l, t h e d i s t r i b u t i o n
Q ( c 0 q u a n t i f i e s t h e r e l a t iv e p r o b a b i l i ti e s o f different " t h e o r i e s " w i t h d i f f e r e n t
v a l u e s o f t h e c o n s t a n t s o f N a t u r e . W h e n t w o t h e o r ie s a re c o m p a r e d , w e s h o u l d
e x p r e s s t h e p h y s i c a l c o n s t a n t s o f b o t h i n t h e s a m e u n i t s ; i t i s v a l i d , f o r e x a m p l e , t o
s a y t h a t G i n t h e o r y A i s s m a l l e r t h a n G i n t h e o r y B . I n f a c t , a ll o f t h e t h e o r i e s th a t
w e a r e c o m p a r i n g s h a r e t h e s a m e a - i n d e p e n d e n t u l t r a v io l e t c u t o f f , t h e w o r m h o l e
s c a l e M w . T h u s , w h e n w e c o m p a r e d i f f e r e n t t h e o r ie s , w e m a y e x p r e s s a l l m a s s s c a le s
( l i k e G 1 /2 ) a s d i m e n s i o n l e s s q u a n t i t i e s , i n u n i t s o f M w . M o s t i m p o r t a n t , t h e
c r i t e r i o n t h a t G(~)M2w i s a t i t s m i n i m u m h a s c o n s e q u e n c e s t h a t ( i n p r i n c i p l e ) r e l a t et o e x p e r i m e n t a l l y a c c e s s i b le l o w - e n e r g y p h y s ic s . I n d e e d , w e w il l a r g u e t h a t t h i s
c r i t e r i o n a c t u a l l y f i x e s a l l o f t h e d i m e n s i o n l e s s q u a n t i t i e s t h a t s p e c i f y t h e p r o p e r t i e s
o f l o w - e n e r g y p h y s i c s .
T o s u p p o r t t h e c l a i m t h a t Q ( a ) s h o u l d b e e v a l u a t e d o n t h e s u r f a c e G A = 0 as a
A ~ 0 l im i t , w e w i ll r e g u l a te t h e v e r y si n g u la r d e p e n d e n c e o f Q ( c 0 o n A ( c 0 b y
i n t r o d u c i n g a n i n f r a r e d c u t o f f i n t o t h e c a l c u l a ti o n o f Q ( c 0 . T h e p o i n t i s t h a t A
c o n t r o l s t h e v o l u m e o f t h e fo u r - g e o m e t r y t h a t d o m i n a t e s t h e p a t h i n t e g r a l * .
I m a g i n e t h a t w e r e p e a t t h e c a l c u l a t i o n o f Q ( c 0 i n s e c t . 3 , b u t t h a t t h e c l o s e d
f o u r - g e o m e t r i e s t h a t c o n t r i b u t e t o t h e p a t h i n t e g r a l e q . ( 3 . 3 ) a r e n o w r e s t r i c t e d t o
g e o m e t r i e s w i t h v o l u m e V le ss t h a n a m a x i m u m v o l u m e V ma~; Vma~ , t hen , i s ou r
i n f r a r e d c u t o f f . W e a s s u m e a g a i n t h a t t h e p a t h i n t e g ra l is d o m i n a t e d b y t h e
f o u r - s p h e r e g e o m e t ry . T h e n w e h a v e
Q ( a ) - e x p [ e x p ( - s 0 ) ] , ( 4 .3 )
w h e r e S o i s th e m i n i m a l a c t i o n o f a f o u r - s p h e r e t h a t s a t is f ie s t h e v o l u m e c o n s t r a i n t
V~< Vmax-
F o r a f o u r - s p h e r e o f r a d i u s R , t h e a c t i o n e q. (3 .5 ) b e c o m e s * *
1S(R) = 1 6 - - - ~ # ( 2 A R 4 - 1 2 R 2 + . . . ) , ( 4 . 4 )
w h e r e ~ 2 = 8 " , 7 2 / 3 i s t h e v o l u m e o f t h e u n i t f o u r - s p h e r e . T h e m i n i m u m * * * o f S ( R )
* W e c h o o s e t o r e g u l a t e Q ( a ) b y m e a n s o f a n i n f r a r e d c u t o f f b e c a u s e s u c h a r e g u l a t o r c a n b e
f o r m u l a t e d i n t h e b a r e t h e o r y , w i t h o u t a n y r e f e r e n c e t o c ~. S i n c e i t i s o u r d e s i r e t o e x t r a c t a n
u n a m b i g u o u s f u n c t i o n o f a o n t h e s u r fa c e G A = 0 , i t s e e m s s e n s i b l e t o d e s c r i b e t h e a p p r o a c h t o t h i s
s u r f a c e i n a m a n n e r t h a t d o e s n o t i n t r o d u c e i m p l i c i t c ~ -d e p e nd e n c e.
* * F o r s u f f i c i e n t ly l a r g e R , A a n d G m a y b e t a k e n to be R - i n d e p e n d e n t c o n s t a n t s , b e c a u s e t h e y " s t o p
r u n n i n g " i n t h e i n f r a r e d l i m i t. T h i s w i l l b e e x p l a i n e d a t g r e a t e r l e n g t h i n s ec t . 6 .
* * * A s i n s e c t. 3 , t h i s m i n i m u m e x i s t s o n l y i f A > 0 . F o r A < 0 , Q ( ~ ) m a y b e d e f i n e d b y a n a l y t i c
c o n t i n u a t i o n i n A .
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160 J. Preskill / Wormholes
The condition A ( a ) = 0 determines a surface of codimension one in the a-space.
But we should expect that the condition that G ( a ) assumes its minimum on the
A (a ) = 0 surface is sufficient to determine all of the remaining a' s. A function , even
a function of many variables, generically attains its global minimum at an isolated
point, assuming that a global minimum exists.
Therefore, we have in principle recovered the ability to calculate the constants of
Nature. All of the arbitrariness that is potentially introduced by wormhole physics
can be overcome. We need only compute all of the constants of Nature as functions
of the a 's ; then we find the a tha t minimizes G on the A = 0 surface. This
determines the a's, and hence all of the other constants. In practice, this calculation
will be challenging.
The a-dependence of G, or of any coupling constant, comes from two sources,
and we can imagine carrying out the calculation of G ( a ) in two stages. First, weintegra te o ut wormholes at the wormhole scale M w, and thus obtain a "ba re " theory
with cu toff M w and a-dep enden t " ba re " parameters. Second, we allow the cutoff to
float f rom M,v to M << Mw; the effects of quantum fluctuations with wave numbe r
k satisfying M W> k > M are incorporated into a renormalization of the effective
action. The fluctuations that are integrated out in the second stage are not
wormholes, but conventional "loop" fluctuations. The G ( a ) we want to calculate is
ext rac ted from the effective action in the in frared or M --* 0 limit.
It is clear that if this G ( a ) is to have a nonvanishing minimum, then so must the
bare Newton's constant Go(a ), renormalized at the scale M W. The point is that
G = 0 is stable under renormalization; Newt on' s constant is unrenormalized when
gravity is free. (Renormalization in quantum gravity is discussed in more detail in
sect. 6.) Therefore, if the bare Newton's constant Go(a ) obtained by integrating out
wormholes, were to vanish for some a, then the physical Newton's constant G ( a )
would also vanish for that value of a. Since G ( a ) must satisfy the bound eq. (4.1),
we require the bare coupling to satisfy a similar bound,
G o (a ) >~ Go. rain > O . ( 4 . 7 )
In the dilute approximation described in sect. 2, we saw that the effective action is
linear in a. Thus G o l ( a ) is linear in a, and G o vanishes when a part icular
combination of the a's is infinite. In view of eq. (4.7), we conclude that the dilute
approximation is invalid. To understand why the gravitational interaction has a
nonvanishing strength we must go beyond the dilute approximation. The corrections
to the dilute approximation are discussed in sect. 5, with particular attention
devoted to the origin of the lower bound on G o( a )
Is it possible that the requirement that G ( a ) attains its minimum on the A ( a ) = 0
surface fails to determine all the a's? If there are undetermined a's, then there is
presumably a symmetry that prevents G from depending on those a's. But every-
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J. Preskill / Wormholes 161
t h i n g c o u p l e s t o g r a v i t y , a n d a l l o t h e r i n t e r a c t i o n s c o n t r i b u t e t o t h e r e n o r m a l i z a t i o n
o f G , s o it is h a r d t o th i n k o f s u c h a s y m m e t r y .
W e h a v e e m p h a s i z e d i n t h i s s e c t i o n t h a t N e w t o n ' s c o n s t a n t G(a) m u s t o b e y a
p o s i t i v e l o w e r b o u n d ; o t h e rw i s e , o n t h e b a s i s o f t h e s a m e a s s u m p t i o n s t h a t u n d e r l ieC o l e m a n ' s e x p l a n a t i o n f o r A = 0, w e p r e d i c t G = 0 , i n c o n f li c t w i t h o b s e r v a t i o n . I n
f a c t , i t i s a l s o n e c e s s a r y f o r t h e c o e f f i c i e n t s o f h i g h e r d e r i v a t i v e t e r m s i n t h e
g r a v i t a t i o n a l e f f e c ti v e a c t i o n t o o b e y upper b o u n d s . J u s t a s t h e n e w t o n i a n g r a v i t a -
t i o n a l i n t e r a c t i o n m u s t n o t b e p e r m i t t e d t o b e a r b i t ra r i l y weak f o r a n y v a l u e o f a ,
s o t h e h i g h e r d e r i v a t i v e g r a v i t a t i o n a l i n t e r a c t i o n s m u s t n o t b e p e r m i t t e d t o b e
a r b i t r a r i l y strong f o r a n y v a l u e o f a .
I n t h e d e r i v a t i o n o f A = 0 in s e c t. 3 w e r e q u i r e d t h a t t h e c o n t r i b u t i o n f r o m h i g h e r
d e r i v a t i v e t e r m s t o t h e v a lu e o f t h e e f fe c t iv e a c ti o n a t t h e s t a t i o n a r y p o i n t b e
n e g l i g i b l e . A h i g h e r d e r i v a t i v e t e r m i n t h e a c t i o n w i t h a r b i t r a r i l y l a r g e c o e f f i c i e n tc a n u p s e t C o l e m a n ' s m e c h a n i s m b y d e s t a b il iz i n g th e s t a t io n a r y p o i n t . O r t h e
a - d i s t r i b u t i o n m a y f a v o r t h a t th i s c o e ff ic i e n t b e i n fi n it e ; t h e n t h e t h e o r y o f g r a v i t y
i n a t y p i c a l a - u n i v e r s e w i l l n o t a g r e e w i t h N e w t o n ' s t h e o r y i n t h e l o w - e n e r g y l i m i t .
T h e p a r a m e t e r s t h a t m u s t o b e y s u c h u p p e r b o u n d s a r e c o u p l in g c o n s t a n ts o f t he
r e n o r m a l i z e d e f f e c ti v e a c t i o n t h a t d e s c r i b e s g r a v i t y a t lo w e n e r g y . U n l i k e t h e lo w e r
b o u n d o n G , t h e se u p p e r b o u n d s p r o b a b l y p la c e n o re s t r ic t io n s o n t he " b a r e "
p a r a m e t e r s o f t h e e f f e c ti v e t h e o r y d e f i n e d a t t h e w o r m h o l e s c a le M w . R a t h e r , t h e
u p p e r b o u n d s a r e a n a u t o m a t i c c o n s eq u e n c e o f th e r e n o r m a l iz a t io n p r o c e d u r e a s th e
c u t o f f f l o a t s f r o m M w to M < < M w . A l l o f t h e r e n o r m a l i z e d c o u p l i n g s o f g r a v i t yo b e y u p p e r b o u n d s p r o v i d e d o n l y t h a t, a s w e e x p e c t, g ra v i t y h a s n o n o n t r iv i a l
c o n t i n u u m l i m i t * . T h i s w i l l b e e x p l a i n e d f u r t h e r i n s e c t . 6 .
5 . B e y o n d t h e d i lu t e a p p r o x i m a t i o n
U p t o n o w i n th i s p a p e r w e h a v e f o r th e m o s t p a r t t r e a t e d w o r m h o l e s a s d il u te ,
a n d h a v e i g n o r e d t he i n te r a c ti o n s a m o n g w o r m h o l e s . B u t w e h a v e al so n o t e d t h a t
t h i s d i l u t e a p p r o x i m a t i o n c a n n o t r e a l l y b e j u s t i fi e d . I n t h is s e c t i o n w e w i ll d i s c u s s
t h e c o r r e c t i o n s t o t h e d i l u t e a p p r o x i m a t i o n d u e t o i n t e r a c t i o n s .F o r t h e p u r p o s e o f t h i s d i s c u s s i o n , w e w i l l f i n d i t u s e f u l t o m a k e a d i s t i n c t i o n
b e t w e e n w o r m h o l e s a n d " i n s t a n t o n s . " I w ill u s e t h e t e r m i n s t a n t o n f o r t h e r e g i o n o f
t h e b a c k g r o u n d s p a c e t i m e w h e re a w o r m h o l e h o o k s o n; t h e t e rm i n s t a n t o n i s
a p p r o p r i a t e b e c a u s e t h i s i s a s m a l l l o c a l i z e d r e g i o n i n t h e b a c k g r o u n d ( e u c l i d e a n )
s p a c e t i m e . A w o r m h o l e c o n n e c t s t o g e t h e r tw o i n s ta n t o n s .
W e m u s t d i s t i n g u i s h t w o t y p e s o f i n t e r a c t io n s t h a t a r is e w h e n c o r r e c t i o n s t o t h e
d i lu t e a p p r o x i m a t i o n a r e c o n s id e r e d ; t he s e m a y b e c al le d " w o r m h o l e i n t e ra c t io n s "
a n d " i n s t a n t o n i n te r a c t i o n s ." W o r m h o l e i n t e ra c t io n s a re d e s c ri b e d b y v e rt ic e s a t
* A similar suggestion was m ade by W einberg [14].
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162 J . P r e s k i l l / W o r m h o l e s
Fig. 10. A w o r m h o l e i n t e r a c t i o n .
F i g . l i . A s h o r t -r a n g e i n t e r a c t i o n b e t w e e n i n s t a n t o n s a t s p a c e t i m e p o s i ti o n s x ~ a n d x ~ .
w h i c h a b r a n c h i n g o f w o r m h o l e s o c c u r s (f ig . 10 ). T h e s e i n t e r a c t io n s i n d u c e n o n -
g a u s s i a n c o r r e c t i o n s t o t h e p r o b a b i l i ty d i s t r ib u t i o n P ( a ) i n eq . (2 .5 ) a n d e q . (3 .2 ).
O f f a r g r e a t e r i n t e r e st a r e i n s t a n t o n i n t e ra c t i o n s . T h e s e o c c u r w h e n t w o i n s ta n -
t o n s c l o s e l y a p p r o a c h o n e a n o t h e r o n t h e b a c k g r o u n d s p a c e t i m e ( f i g . 1 1 ) . T h e
i n s t a n t o n i n t e r a c t i o n s i n d u c e c o r r e c t io n s to t h e a - d e p e n d e n t e f f e c ti v e a c t i o n S[g, a]
t h a t a r e n o n l i n e a r i n a . H e n c e , t h e i n s t a n t o n i n t e r a c t i o n s a l s o c o r r e c t t h e f a c t o r
Q ( a ) i n e q . ( 3 .3 ) . T h e in s t a n t o n i n t e r a c t i o n s a r e o f g r e a t e r in t e r e s t t h a n t h e
w o r m h o l e i n t e r a c t i o n s ( a t l e a s t f o r o u r p r e s e n t p u r p o s e s ) b e c a u s e , b y m o d i f y i n gQ ( a ) , t h e y c a n c h a n g e w h e r e th e s u p p o r t o f t h e a - d i s t r i b u t i o n l ie s.
I f w e a s s u m e t h a t t h e i n s t a n t o n i n t e r a c t io n s a r e o f s h o r t r a n g e, t h e n t h e e f fe c t s o f
t h e s e i n t e r a c t i o n s c a n b e s y s t e m a t i c a l ly e x p a n d e d i n p o w e r s o f a " d e n s i t y "
o f i n s t a n t o n s . T h i s e x p a n s i o n i s p r e c i se l y a n a l o g o u s t o t h e c l u s te r e x p a n s i o n o f
s t a t i s t i c a l m e c h a n i c s [ 2 4 ] . T h e c l u s t e r e x p a n s i o n , w h e n t r u n c a t e d a t a f i n i t e o r d e r ,
g i v es a c c u r a t e r e s u l t s if t h e r a n g e o f t h e i n s t a n t o n i n t e r a c ti o n s i s s m a l l c o m p a r e d t o
t h e m e a n s e p a r a t i o n b e t w e e n i n s t a n t o n s * .
A c t u a l l y , i n a t h e o r y o f g ra v i ty , w e s h o u l d n o t b e s u r p r i s e d t o f i n d t h a t i n s t a n t o n
i n t e r a c t i o n s a r e long-range. T h e i n s t a n t o n is li k e ly to b e s u r r o u n d e d b y a g ra v i ta -t i o n a l f i e l d t h a t d e c a y s l i ke a n i n v e rs e p o w e r o f t h e d i st a n c e f r o m t h e i n s t a n t o n , a n d
t h e g r a v i t a t i o n a l i n t e r a c t i o n b e t w e e n i n s t a n t o n s , t h e n , w i ll a l s o d e c a y li k e a p o w e r .
( I n d e e d , t h e r e m i g h t b e o t h e r l i g h t f i e ld s t h a t a r e e x c i t e d in t h e v i c i n i t y o f t h e
i n s t a n t o n , a n d t h e s e w o u l d a l s o i n d u c e a l o n g - r a n g e i n t e r a c t i o n . ) T h e l o n g - r a n g e
i n t e r a c t i o n s c a u s e o n e t o w o r r y a b o u t t h e v a li d i ty o f th e c l u s te r e x p a n s i o n .
H o w e v e r , a s G u p t a a n d W i s e [1 7] h a v e r e c e n t l y e m p h a s i z e d , th e l o n g - r a n g e
i n s t a n t o n i n t e r a c t i o n s a r e n o t t h e e f f e c t s t h a t w e a r e r e a ll y i n t e r e s t e d i n , b e c a u s e
t h e y d o n o t g e n e r a t e c o r r e c t i o n s t o t h e e f f e c ti v e a c t i o n S[g, a ] t h a t a r e n o n l i n e a r i n
a . I n s t e a d , t h e p h y s i c a l e ff e c ts o f t h e se l o n g - r a n g e i n s t a n t o n i n t e r a c t io n s c a n b e
* T h e c l u s t e r e x p a n s i o n f o r Q C D i n s t a n t o n s w a s d e s c r i b e d in , f o r e x a m p l e , re f . [2 5].
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J. Preskill / Worm holes 1 6 3
r e p r o d u c e d e x a c t l y i n a n e f f e c t i v e f i e ld t h e o r y w i t h a n a c t i o n l i n e a r in a ; i n th i s
e f f e c t i v e t h e o r y , t h e y a r e r e p r e s e n t e d b y e x c h a n g e s o f l i g h t f i e l d q u a n t a b e t w e e n
i n t e r a c t i o n v e r t ic e s t h a t a r e l in e a r in a . T h e t e r m s i n S[ g , a ] t h a t a r e n o n l i n e a r i n a
a r e g e n e r a t e d o n l y b y t h e g e n u i n e l y s h o r t - r a n g e i n s t a n t o n i n t e r a c ti o n s t h a t o c c u rw h e n t h e s e p a r a t i o n b e t w e e n i n s t a n t o n s is c o m p a r a b l e t o th e i n s t a n t o n s iz e. It is
s e n s i b le t o c a r r y o u t a n e x p a n s i o n o f t h e e ff e c ts o f these i n t e r a c t i o n s i n p o w e r s o f
t h e i n s t a n t o n d e n s i t y . I t w i l l b e u n d e r s t o o d , t h e n , t h a t t h e i n s t a n t o n i n t e r a c t i o n s
c o n s i d e r e d b e l o w a r e a c t u a l l y t h e s h o r t - r a n g e p o r t i o n o f i n t e r a c t i o n s t h a t m a y a l s o
i n c l u d e a l o n g - r a n g e t a i l .
I n o r d e r t o f o r m u l a t e t h e c l u s t e r e x p a n s i o n , w e m u s t f i r s t u n d e r s t a n d h o w t h e
n o t i o n o f a d e n s i t y o f in s t a n t o n s a ri se s. F o r t h is p u r p o s e , w e r e c o n s i d e r t h e d i l u te
a p p r o x i m a t i o n t h a t w a s d e s c r i b e d i n s e c t . 2 . I n s e c t . 2 w e m a d e t h e h y p o t h e s i s t h a t
t h e c o n t r i b u t i o n C o f a s in g le w o r m h o l e t o a q u a n t u m m e c h a n i c a l e x p e c t a t i o n v a lu ec a n b e e x p r e s s e d a s a b i l o c a l e x p a n s i o n i n l o c a l o p e r a t o r s , a s i n e q . ( 2 . 2 ) . I f t h e
c u r v a t u r e o f t h e b a c k g r o u n d s p a c e t i m e i s s m a ll , a n d a ll fi e ld s o n s p a c e t i m e a r e w e a k
( i n u n i t s o f t h e w o r m h o l e s c a le R ~ , ,1 ), t h e n t h e s u m o v e r l o c a l o p e r a t o r s i s
d o m i n a t e d b y t h e o p e r a t o r 1 , a n d t h e c o n t r i b u t i o n b e c o m e s
C = ~ V 2 . ( 5 . 1)
H e r e V i s th e v o l u m e o f th e b a c k g r o u n d s p a c e ti m e , a n d A i s t h e c o e f fi c i e n t o f t h e
l e a d i n g t e r m i n e q . (2 .2 ), p r e s u m a b l y o f o r d e r R ws . I n t h e s e m i c l a s s i c a l a p p r o x i m a -
t i o n d i s c u s s e d b y C o l e m a n [8] a n d G i d d i n g s a n d S t r o m i n g e r [9 ], A w o u l d b e f u r t h e rs u p p r e s s e d b y t h e s m a ll f a c t o r e - 2 S w h e r e S is th e (l a rg e ) a c t i o n o f a s e m i w o r m -
h o l e .
I n t h e d i l u t e a p p r o x i m a t i o n , t h e c o n t r i b u t i o n d u e t o N w n o n i n t e r a c t i n g w o r m -
h o l e s i s C N " / N w ! T h e s u m o v e r N w is t h u s d o m i n a t e d b y N ,,. o f o r d e r C . I n o t h e r
w o r d s , t h e n u m b e r N W o f w o r m h o l e s i n a t y p i ca l c o n f i g u r a t i o n t h a t g iv e s a
s i g n i f i c a n t c o n t r i b u t i o n t o t h e e u c l i d e a n p a t h i n t e g r a l i s
Uw - vl A V 2. (5 .2 )
I t is t h e r e f o r e p o s s i b le t o r e g a r d k a s a " d e n s i t y o f w o r m h o l e s " p e r volume squared.
O n t h e o t h e r h a n d , a c o n f i g u r a t i o n w i t h N w o r m h o l e s h a s 2 N i n s t a n t o n s ; t h e
n o t i o n o f a " d e n s i t y o f i n s t a n t o n s " p e r volume d o e s n o t a p p e a r t o m a k e s e n se in t h e
V --+ o0 l im it .
H o w e v e r , t h e a b o v e d i sc u s s io n a p p l i es t o th e c o m b i n a t o r i c s o f w o r m h o l e s o n l y
af ter t h e a i n t e g r a t i o n i n e q . ( 2 .3 ) h a s b e e n p e r f o r m e d . ( T h e e f f e c t o f t h e i n t e g r a t i o n
is t o m a t c h u p 2 N w i n s t a n t o n s w i t h N w w o r m h o l e s i n a l l p o s s ib l e w a y s . ) I n s te a d , w e
a r e i n t e r e s t e d i n s t u d y i n g p h y s i c s a t a fi x e d v a l u e o f a . T h e l e a d i n g f a c t o r i n t h e
i n t e g r a n d o f e q . (2 . 3) , a r i s in g f r o m t h e o p e r a t o r 1 i n t h e e x p a n s i o n , h a s t h e f o r m
e ( 1 / 2 ) a 2 e a a l / 2 V . ( 5 . 3 )
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164 J. Preskil l / Wormholes
( H e r e w e h a v e r e s c a l e d e~ s o t h a t t h e w o r m h o l e " p r o p a g a t o r " is o n e ; c~ is n o w
d i m e n s i o n l e s s . ) T h e t e r m i n e q . ( 5. 3) o f o r d e r V u m a y b e i n t e r p r e t e d a s t h e
c o n t r i b u t i o n a t f i x e d c~ d u e t o a n N - i n s t a n t o n c o n f i g u r a t i o n , a n d t h e s u m o v e r N i s
d o m i n a t e d b y
N / V - a A 1/2 . (5 . 4)
T h u s , a Z f / 2 m a y b e r e g a r d e d a s a d e n s i t y o f i n s t a n t o n s .
T h e r e i s a p e c u l i a r c o m p l e m e n t a r i t y a t w o r k h e r e . W e b e g a n t h i s d i s c u s s i o n b y
d e s c r i b i n g t h e p h y s i c a l e f f e c ts o f b il o c a l o b je c t s, t h e w o r m h o l e s . B u t t h e n , f o r s o u n d
p h y s i c s r e a s o n s , w e f o c u s e d a t t e n t i o n o n a s e c t o r w i th a f ix e d v a l u e o f c~. F o r c~
f i x e d , i t i s f a r m o r e a p p r o p r i a t e t o d e s c r i b e t h e p h y s i c s i n t e r m s o f l o c a l o b j e c t s , t h e
i n s t a n t o n s , i n s t e a d o f i n te r m s o f w o r m h o l e s . F u r t h e r m o r e , w h i l e th e d e n s i t y o f
w o r m h o l e s is , o f c o u r s e , c ~ - i n d e p en d e n t , th e d e n s i t y o f in s t a n t o n s d e p e n d s o n c~ a s
i n eq . ( 5 . 4 ) .
T h e i n s t a n t o n d e n s i t y a zl1/2 w i l l b e t h e e x p a n s i o n p a r a m e t e r o f o u r c l u s t e r
e x p a n s i o n , a n d t h e d i l u t e a p p r o x i m a t i o n i s a g o o d a p p r o x i m a t i o n o f aZ i1/2 s u f f i -
c i e n t l y s m a l l . T h u s , w h a t e v e r t h e v a l u e o f A ~/2, t h e d i l u t e a p p r o x i m a t i o n c a n b e
j u s t i f i e d f o r s m a l l e n o u g h a . ( A n d t h is in n o w a y r e q u i re s t h e v a l i d i ty o f a
s e m i c l a s s i c a l a p p r o x i m a t i o n ; t h a t is , w o r m h o l e e ff e ct s n e e d n o t b e d o m i n a t e d b y t h e
c o n t r i b u t i o n o f a c l as s ic a l w o r m h o l e s o l u ti o n .) O n t h e o t h e r h a n d , w h a t e v e r t h e
v a l u e o f A l /2 , t h e d i l u t e a p p r o x i m a t i o n w i ll a lw a y s b r e a k d o w n f o r c~ s u f f i c i e n t l y
large . I n p a r t i c u l a r , t h e d i l u t e a p p r o x i m a t i o n c a n n o t b e u s e d t o s t u d y t h e b e h a v i o r
o f t h e d i s t r i b u t i o n Q ( a ) f o r a s y m p t o t i c a l l y l a r g e ~.
I n t h e d i lu t e a p p r o x i m a t i o n , i t is a s s u m e d t h a t i n s t a n t o n s c a n b e f re e l y s u p e r -
p o s e d , a n d t h a t a c o n t r i b u t i o n d u e t o tw o i n s t a n t o n s f a c to r i ze s i n t o a p r o d u c t o f
o n e i n s t a n t o n c o n t r i b u t i o n s . B u t th i s a s s u m p t i o n i s e x p e c t e d t o fa il w h e n t h e
i n s t a n t o n s c l o s e ly a p p r o a c h o n e a n o t h e r . M o r e g e n e r a ll y , t h e t w o - w o r m h o l e c o n f ig -
u r a t i o n d e p i c t e d i n f i g . 1 1 , i n w h i c h i n s t a n t o n s c e n t e r e d a t x 2 a n d x ~ h a v e a s m a l l
s e p a r a t i o n , g i v e s a c o n t r i b u t i o n
f d4xl(Z d4x 2 d4 X l
a, b, a', b'
H e r e , t h e r e m a i n d e r R(b~ , (x , x ' ) i s a s m a l l c o r r e c t i o n t o t h e f a c t o r i z e d o p e r a t o r
O b ( X ) ( O h ( X ' ) w h e n ( x - x ' ) is l a rg e c o m p a r e d t o t h e c h a r a c t e ri s t ic i n s t a n t o n s iz e. ( O f
c o u r s e , t h e r e a r e a l s o c o r r e c t io n s t h a t o c c u r w h e n o t h e r p a i r s o f i n s t a n t o n s c l o s e ly
a p p r o a c h o n e a n o t h e r ; t h e se h a v e n o t b e e n i n d i c a t e d i n eq . ( 5.5 ).)
I f w e i n c l u d e t h e e f fe c t s o f a ll tw o - i n s t a n t o n i n t e r a c t io n s i n th e w o r m h o l e s u m ,
t h e i n t e g r a n d o f th e a - i n t e g r a t i o n i n eq . (2 .3 ) b e c o m e s m o d i f ie d . T h e m o d i f i c a t i o n i s
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166 J. Preskill / Wormholes
Expansions like eq. (5.9) are the basis of our earlier claim that corrections due to
instanton interactions may be expanded in powers of the instanton density. We now
see that this sta tement is true only rather schematically. The term E A (1)= linear in-a- - a IAta
a in eq. (5.9) is the density of instantons that we identified in eq. (5.4). The termshigher order in a are not really higher powers of the density; each term involves a
different combination of the a's*. But, generically, when the mean instanton
separation is comparable to the instanton size, all terms in eq. (5.9) are of roughly
the same order.
Borrowing metaphorically from the semiclassical picture, we might regard each a,
as being associated with a distinct instanton "type." Then each a, can be inter-
preted as the density of instantons of type a, and eq. (5.9) is a power series
expansion in the various densities.
There is an expansion similar to eq. (5.9) for G ol , where G o is the bare N ewton' scons tant at the wormho le scale M w. We noted in sect. 4 that, if we are to accept
Coleman's explanation for the vanishing of the cosmological constant, we must
insist that Go 1 is bounded above. Obviously, we canno t understand the origin of
this bound by considering the expansion in a to any finite order. We must somehow
sum up the series.
Before d iscussing further the upper bound on Go 1, let us notice tha t there is
another logically independent reason why we must go beyond the dilute approxima-
tion if we are to invoke Coleman's mechanism. If we compute quantum corrections
in the effective theory cut of f at the wormhole scale M w, A will be r enormalized by
an amo un t of order Mw . It would presumably require an unnatu ral fine tuning for
the a-independent bare parameter ~(0) to cancel this renormalization to high
accuracy. Therefore, if the renormalized cosmological constant is to vanish, the
a-dependent shift of / [ 0 ( a ) in eq. (5.9) must be at least of order Mw . But then, the
dens ity of instantons is of order M~, or the mean instanton separation is M w t = R w,
which is just the condition for corrections to the dilute approximation to be
important**.
We cannot escape this conclusion by appealing to supersymmetry. If supersym-
merry remains unbroken below the wormhole scale, then / [ 0 ( a ) = 0. But a bare
cosmological constant is eventually generated at a lower scale, where supersymmetry
is spontaneously broken. Again, unless the a-independent bare parameters are
carefully tuned, we expect that a large a-dependent shift of bare parameters is
required for the renormalized cosmological const ant to vanish. The a-depe nden t
shifts are small when the instantons are dilute.
Let us now consider further the origin of the bound on Newton's constant. Here
we wish to argue that it is plausible that G o l(a) , the bare coupling at the wor mhole
* Indeed, for the other coefficients in S(c~), even the linear term involves a different linear combinationof the c~'s than appears in the "instanton density."
** This argument emerged from discussions with A. Cohen and M. Wise.
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J. Pres kil l / Worm holes 1 6 7
s c a l e M w , is b o u n d e d a b o v e a s a f u n c t i o n o f c~. T h e r e n o r m a l i z a t i o n o f G - ~ b y
q u a n t u m f l u c t u a t i o n s b e l o w t h e w o r m h o l e s c a l e w i ll b e d i s c u ss e d i n s e ct . 6 .
H e u r i s t i c a l l y , o n e e x p e c ts i n s t a n t o n e f f e c ts t o " s a t u r a t e " w h e n t h e m e a n i n s t a n -
t o n s e p a r a t i o n b e c o m e s c o m p a r a b l e t o th e in s t a n t o n s iz e R w . W h e n t he i n s ta n t o n sb e c o m e d e n s e , t h e p i c tu r e o n w h i c h o u r a n a l y s is h a s b e e n b a s e d - o f w o r m h o l e s
a t t a c h i n g t o a s m o o t h b a c k g r o u n d s p a c e t i m e - m a y n o l o n ge r a p p ly . I n d e e d , th e
n o t i o n o f a n i n s t a n t o n g a s, a n d t he i n t e rp r e t a t i o n o f o u r e x p a n s i o n p a r a m e t e r a s a
d e n s i t y o f in s t a n t o n s , p r o b a b l y c e a se s to m a k e s e n s e w h e n t h e d il u te a p p r o x i m a t i o n
i s n o l o n g e r v a l i d .
I t is r e a s o n a b l e t o g u e ss t h a t, i f t h e a - d e p e n d e n t s h i ft o f a p a r a m e t e r i n t h e
e f f e c t i v e a c t i o n d o e s h a v e a m a x i m u m a s a fu n c t i o n o f c~, t h e n t h e m a x i m u m is
a t t a i n e d a t a v a l u e o f a t h a t is c o m p a r a b l e t o t h e v a l u e o f c~ f o r w h i c h t h e c l u s t e r
e x p a n s i o n b r e a k s d o w n . T h a t is , t he m a x i m u m o c c u r s w h e n t h e i n s ta n t o n d e n s i ty i so f o r d e r o n e . W e w o u l d e s t i m a t e t h en , t h a t t h e m a x i m a l s h i ft is a f a c t o r o f o r d e r o n e
t i m e s a p o w e r o f th e w o r m h o l e s c a le M w d e t e r m i n e d b y d i m e n s i o n a l a n a l y s is . F o r
e x a m p l e , f o r N e w t o n ' s c o n s t a n t w e m i g h t e x p e c t
8 [ ( 1 6 q T G o ) - 1 ] ~ < M w ( 5 . i o )
t o b e s a t i s f i e d f o r a n y a .
T o m a k e t h is d i s c u s s io n a b i t m o r e c o n c r e t e , I w ill d e s c r i b e a s i m p l e t o y m o d e l i nw h i c h t h e c l u s t e r e x p a n s i o n c a n b e s u m m e d t o a ll o r d e r s i n c~. I n t h is t o y m o d e l , w e
c a n s e e e x p l i c i t ly t h a t G o l i s a b o u n d e d f u n c t i o n o f c~. F o r t h e s a k e o f s i m p l i c i t y ,
t h is m o d e l w i ll h a v e j u s t o n e " t y p e " o f in s t a n t o n a n d j u s t o n e c ~ - p ar a m e te r ; i t is
o b v i o u s l y p o s s i b l e t o g e n e ra l iz e i t to a m o d e l w i t h m a n y c ~ ' s a n d w i t h q u a l i ta t i v e ly
s i m i l a r b e h a v i o r . T h e p o i n t o f t h e m o d e l i s t h a t e x c l u d e d v o l u m e e f f e c t s c a n b e
e x p e c t e d t o c a u s e r e p u l s iv e in t e r a c t io n s o f i n s t a n t o n s a t s m a l l s e p a r a t i o n s , a n d t h a t
s h o r t - r a n g e r e p u l s i v e i n t e r a c t i o n s c a n s i g n i f i c a n t l y a l t e r t h e c ~ - d e p e n d e n c e o f t h e
e f f e c t i v e a c t i o n a t a s y m p t o t i c a l l y la r g e c~.
F o r t h e p u r p o s e o f c h a r a c te r i z in g t h e s h o r t - r a n g e i n s t a n t o n i n t e r a c ti o n s i n th i st o y m o d e l , i m a g i n e t h a t e u c li d e a n s p a c e t i m e h a s b e e n d i v i d e d i n t o m a n y id e n t i c a l
t i n y c e l ls , e a c h w i t h a p h y s i c a l v o l u m e o f R 4,, th e v o l u m e o f a n i n s t a n t o n . A
c o n f i g u r a t i o n o f t h e in s t a n t o n g a s c a n b e d e s c r i b e d b y a s s ig n i n g t o e a c h ce ll a
n o n - n e g a t i v e i n t e g e r , t h e n u m b e r o f in s t a n t o n s t h a t o c c u p y t h e ce ll ( se e fig . 1 2). W e
w i ll l a b e l t h e c e ll s w i t h a n i n d e x i , a n d d e n o t e t h e c o r r e s p o n d i n g i n t e g e r b y n i.
N o w t h e a s s u m p t i o n o f o u r m o d e l i s t h a t i n s t a n t o n s in d is t in c t c e lls d o n o t
i n t e r a c t , b u t i n s t a n t o n s i n th e s a m e ce ll d o i n te r a c t . A n i n s t a n t o n o c c u p y i n g c e ll i
m a y e i t h e r e n c o u r a g e m o r e i n s t a n t o n s t o o c c u p y t h a t c e l l ( a t t r a c t i v e i n t e r a c t i o n ) , o r
d i s c o u r a g e a d d i t i o n a l i n s t a n t o n s ( r e p u ls i v e i n t e ra c t i o n ) .
S u p p o s e , a t f i r s t , t h a t t h e r e a r e n o i n s t a n t o n i n t e r a c t i o n s a t a l l . T h e n t h e
i n s t a n t o n s a r e n o n i n t e r a c t i n g i d e n t ic a l p a r ti c le s , a n d s u m m i n g o v e r a ll c o n f i g u r a -
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i 6 8 J . P r e s k i l l / W o r m h o l e s
~ o
o
F i g . 1 2 . E u c l i d e a n s p a c e t i m e h a s b e e n d i v i d e d in t o c e ll s; e a c h ce ll c o n t a i n s a n u m b e r o f i n s t a n t o n s t h a t
i s a s s u m e d t o b e a ( n o n - n e g a t i v e ) i n t eg e r .
t i o n s o f t h e i n s t a n t o n g a s g e n e r a t e s a n e f f e c ti v e a c t i o n 6S (a ) g i v e n b y
H e r e (9 q u a n t i f ie s t h e d e p e n d e n c e o f t h e i n s t a n t o n c o n t r i b u t i o n o n t h e b a c k g r o u n d
g e o m e t r y ; i t c a n b e e x p a n d e d i n t e r m s o f l o c a l o p e r a t o r s i n th e c e ll i . I n e q . ( 5 .1 1 )
w e h a v e r e c o v e r e d t h e r e s u l t e q . ( 2 . 4 ) t h a t w e d e r i v e d p r e v i o u s l y i n t h e d i l u t e
a p p r o x i m a t i o n .
T o s i m u l a t e i n t e r a c t i o n s , w e r e p l a c e t h e s u m o v e r n i n e q . ( 5 . 1 1 ) b y
f a, , ~ _,= 0 ~ . ( a @ , ) . ( 5 . 1 2 )
T h e n , f o r a , , > 1 , w e h a v e i n t r o d u c e d a n a t t r a c t iv e s h o r t - r a n g e n - b o d y i n t e r a c t i o n ,
a n d f o r a,, < 1 , a r e p u l s i v e n - b o d y i n t e r a c t i o n . W i t h t h i s m o d i f i c a t i o n , e q . ( 5 . 1 1 )
b e c o m e s
6 S ( a ) = - ~'~ In f (a@ i ) ,l
a"z" ( 5 . 1 3 )f ( z ) = ~_, n! "n
T o e x t r a c t t h e v a lu e o f 3Go l, w e e x p a n d 3S i n p o w e r s o f d e r i v a t i v e s o f t h e
b a c k g r o u n d m e t r ic . If t he o p e r a t o r 0 c a n b e e x p a n d e d a s
( _ 9 = c o + c l R + . . .
( w h e r e R i s t h e R i c c i s c a la r ) , t h e n w e h a v e
8 x (~ ) = f d 4 x v ~ R w ' ( S f ' ( S ) q R ( x ) + . . . ) ( 5 . 1 4 )- l n f ( S ) f ( • ) Co
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w h e r e
J. Pre skil l / Worm holes 169
S = c o ~ . ( 5 . 1 5 )
I n e q . ( 5 .1 4 ) , t h e s u m o v e r c e ll s h a s b e e n r e p l a c e d b y a v o l u m e in t e g r a l , a n d t h ev o l u m e R4w o f t h e c e l l h a s b e e n i n s e r t e d .
F r o m e q . (5 .1 4 ) w e s e e t h a t t h e s h i ft in N e w t o n ' s c o n s t a n t is g i v e n b y
8 [ ( 1 6 T r G o ) l] • f ' ( S ) c 1- f ( 8 ~ ' 4 • ( 5 . 1 6 )
(-0Rw
T h e r e a r e m a n y r e a s o n a b l e f u n c t i o n s f ( 8 ) f o r w h i c h t h i s s h i f t i s b o u n d e d ; f o r
e x a m p l e , f m a y b e a n ( e v e n ) r e a l p o l y n o m i a l w i th n o r e a l z e ro s . A s i m p l e fu n c t i o n
t h a t w o r k s i s
, 7 . ( 5 . 1 7 )( z ) = 1 + z + 5 z -
I n t h i s c a s e , o n e o r t w o i n s t a n t o n s a r e a l lo w e d t o o c c u p y a c el l w i t h o u t i n t e ra c t i n g ,
b u t a r e p u l s i v e i n t e r a c t i o n f o r b i d s t h r e e o r m o r e i n s t a n t o n s i n a s i n g l e c e l l .
C l e a r l y t h i s t o y m o d e l h a s m a n y u n r e a l i s t i c f e a t u r e s , b u t i t s e r v e s t o i l l u s t r a t e t h a t
i n s t a n t o n i n t e r a c t i o n s c a n c a u s e c ~ - d e p e n d e n t s h i f ts i n th e e f fe c t i v e a c t i o n t o b e
b o u n d e d a s f u n c t i o n s o f c ~ .
B e f o r e c o n c l u d i n g t h i s s e c t i o n , I w o u l d l i k e t o r e e x a m i n e o n e o f t h e a s s u m p t i o n su n d e r l y i n g o u r w h o l e a n a l y s is , i n th e l i g h t o f o u r o b s e r v a t i o n s a b o u t i n s t a n t o n
i n t e r a c t i o n s . T h i s a s s u m p t i o n w a s e x p li c it ly s t a te d i n s ec t. 2 - t h a t w o r m h o l e s h a v e
a c h a r a c t e r i s t i c t h ic k n e s s R w . T h e n i t m a k e s s e n s e t o i n t e g r a te o u t w o r m h o l e s , a n d
o b t a i n a n e f f e c t i v e t h e o r y w i th cu tof f Rw 1, i n w h i c h w o r m h o l e f l u c t u a t i o n s n o
l o n g e r o c c u r . T h e h e u r i s t i c p i c t u r e u n d e r l y i n g t h i s a s s u m p t i o n i s t h a t t h i c k w o r m -
h o l e s h a v e l a r g e e u c l i d e a n a c t io n , a n d s o g i ve a c o n t r i b u t i o n t o th e p a t h i n t e g ra l
t h a t i s h i g h l y s u p p r e s s e d . W e t h e r e f o r e m a k e o n l y a v e r y s m a l l e r r o r b y n e g l e c t i n g
w o r m h o l e s t h a t a r e m u c h t h i c k e r t h a n t h e c h a r a c t e r i s t i c s i z e R w .
T h i s r e a s o n i n g , h o w e v e r , m u s t b e r e g a r d e d w i t h c a u t i o n . W e m a y i m a g i n e t h a t t h ea ' s c a n b e d i v i d e d i n t o t w o s et s - p a r a m e t e r s c~ a s s o c i a t e d w i th s m a l l i n s t a n t o n s
a n d p a r a m e t e r s c% a s s o c ia t e d w i t h l a rg e in s t a n t o n s . O n t h e b a s i s o f t h e h e u ri s ti c
p i c t u r e d e s c r i b e d a b o v e , w e e x p e c t t h a t t he " p r o p a g a t o r " A L f o r la rg e w o r m h o l e s i s
e x t r e m e l y s m a l l ; t h e p r o b a b i l i t y d i s t r i b u t i o n P ( c~ ) i n e q . ( 2 .5 ) s t r o n g l y f a v o r s s m a l l
v a l u e s o f c ~L . I t is in t h i s s e n s e t h a t l a r g e a L - d e p e n d e n t s h i f ts i n t h e e f f e c t i v e a c t i o n
a r e h i g h l y i m p r o b a b l e .
B u t a s w e e m p h a s i z e d i n s e ct . 3 , i t is n o t P ( a ) b u t t h e f a c t o r
Q( a) - exp{ exp[37r/G( a )A ( a)] }( 5 . 1 8 )
t h a t d e t e r m i n e s w h e r e t h e s u p p o r t o f t h e c ~ - d i s tr ib u t io n l ie s . T h e v e r y s i n g u l a r
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17 0 J. Preskill / Wormholes
F ig . 1 3. L a rg e i n s t a n t o n s e x c l u d e s m a l l in s t a n t o n s f ro m s p a c e t i m e .
d e p e n d e n c e o f e q . (5 .1 8 ) o n a L f o r A(a)~ 0 o v e r c o m e s th e e x p ( - a ~ A O s u p p re s -
s i o n i n P(a). H e n c e , i n s p i t e o f t h e i r l a r g e e u c l i d e a n a c t i o n , t h e l a r g e i n s t a n t o n s t r y
t h e i r b e s t t o i n c r e a s e G 1 o n t h e A = 0 s u r f a c e . T h e c o n t r i b u t i o n d u e t o l a r g e
i n s t a n t o n s t o G 1 is p r e s u m a b l y m a x i m i z e d w h e n t h e l a rg e i n s t a n t o n s a r e d e n s e .
A n d s o , i n s p i t e o f t h e s tr o n g t e n d e n c y o f th e f a c t o r P ( a ) t o s u p p r e s s t h e e f fe c t s o f
t h e la r g e i n s t a n to n s , t he l ar g e i n s ta n t o n s c a n n o t b e p r e v e n t e d f r o m b e c o m i n g d e n se ,
a n d i n d u c i n g e f f e c ts th a t a r e o f o r d e r o n e .
T h i s c o n c l u s i o n i s d i s t r e s s i n g . I t a p p e a r s t h a t w o r m h o l e s o f a r b i t r a r i l y l a r g e
t h i c k n e s s w i l l c o n t r i b u t e s i g n i f i c a n t l y t o p h y s i c a l p r o c e s s e s a t l o w e n e r g y . H o w a r e
w e t o r e c o n c i l e th i s p h e n o m e n o n w i th t h e w e l l - d o c u m e n t e d s u c c e ss o f l o c a l f ie ld
t h e o r y i n d e s c r i b i n g l o w - e n e r g y p h y s i c s * ?
I b e l i e v e t h a t i n t e r a c t i o n s b e t w e e n l a r g e a n d s m a l l i n s t a n t o n s a r e r e s p o n s i b l e f o r
s u p p r e s s i n g t h e e f f e c ts o f l a rg e i n s t a n t o n s . I h a v e i n m i n d a p i c tu r e i n w h i c h s m a l l
i n s t a n t o n s c r o w d o u t t h e l a r g e o n e s ( fi g. 1 3). I f o u r g o a l is t o m a x i m i z e G 1 t h e n
c o n f i g u r a t i o n s w i t h m a n y l a rg e i n s t a n t o n s a r e in e f fi c ie n t , b e c a u s e t h e la r g e in s t a n -
t o n s e x c l u d e f r o m s p a c e t i m e a r e g io n t h a t t h e s m a l l i n s t a n t o n s w o u l d l ik e to o c c u p y .
A d e n s e g a s o f s m a l l i n s t a n t o n s g i v e s a c o n t r i b u t i o n o f o r d e r R s 2 to G O 1, w h e r e R s
i s t h e s i z e o f a s m a l l i n s t a n t o n , w h i l e a d e n s e g a s o f l a r g e i n s t a n t o n s c o n t r i b u t e s o f
o r d e r R c 2 , w h e r e R c i s th e s iz e o f a la r g e i n s t a n t o n . T h e r e f o r e , i f t h e r e i s a t r a d e - o f f
b e t w e e n l a r g e a n d s m a l l i n s t a n t o n s , t h e i n s t a n t o n g a s w il l f a v o r i n c r e a s in g t h e
a b u n d a n c e o f s m a l l i n s t a n t o n s a t th e e x p e n s e o f th e la r g e i n s t a n t o n s . T h i s e ff e c t
m i g h t p r o v i d e t h e j u s t if i c a t io n f o r i g n o r i n g t h e l a rg e w o r m h o l e s .
A v e r y c r u d e m o d e l w il l i l l u s tr a te h o w t h is m e c h a n i s m w o r k s . I m a g i n e t h a t t h e r e
a r e o n l y t w o i n s t a n t o n t y p e s , th e s m a l l o n e s w i t h s iz e R s a n d t h e l a r g e o n e s w i t h
s iz e R C" T h e n t h e a - d e p e n d e n c e o f th e b a r e N e w t o n ' s c o n s t a n t h a s t h e a p p r o x i m a t e
f o r m
ao 1 C R 2n ( s) + ( 5 . 1 9 )
H e r e n s is th e d i m e n s i o n l e s s d e n s i t y o f s m a l l i n s t a n t o n s , t h e f ra c t i o n o f t h e v o l u m e
o f s p a c e t i m e t h a t i s o c c u p i e d b y t h e i n s t a n t o n s ; n L i s t h e c o r r e s p o n d i n g d e n s i t y o f
l a r g e i n s t a n t o n s , a n d C s , C e a r e n u m e r i c a l c o n s t a n t s o f o r d e r o n e . W e w i s h t o
d e t e r m i n e a s a n d a c b y m a x i m i z i n g G o 1. B u t i f s m a l l in s t a n t o n s c a n n o t s it o n t o p
* This issue was also raised by K aplanovsky [20], and has b een discussed in refs. [16,18 .
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J. Pres kil l / Worm holes 17 1
F i g . 1 4 . I f e x c l u d e d f r o m a t t a c h i n g t o s p a c e t i m e b y a l a r g e i n s t a n t o n , a s m a l l w o r m h o l e m a y a t t a c h t o a
l a r g e w o r m h o l e i n s t e a d .
o f l a rg e i n s t a n t o n s , t h e n t h e d e n s i ti e s a re c o n s t r a i n e d b y
F / s (O ~ s ) q - H L ( @ L ) ~ 1 -. ( 5 . 2 0 )
E v i d e n t l y , t h e m a x i m u m o f G o 1 s u b j e ct t o t h e c o n s t r a i n t e q . (5 .2 0 ) is n s = 1 ,
n e = 0 , a s s u m i n g t h a t C s R s 2 > C L R L 2. T h e l ar g e i n s t a n t o n s a r e c o m p l e t e l y e li m i -
n a t e d i n f a v o r o f t h e s m a l l i n s t a n t o n s .
T h e d i s t i n c t i o n b e t w e e n i n s t a n t o n i n t e r a c t i o n s a n d w o r m h o l e i n t e r a c t i o n s i s
i m p o r t a n t f o r u n d e r s t a n d i n g w h y l a rg e a n d s m a l l i n s t a n t o n s m u t u a l l y e x c lu d e e a c h
o t h e r . W h e n e x c l u d e d f r o m s p a c e t i m e b y a l a r g e i n s t a n t o n , a s m a l l i n s t a n t o n w i l l
s l i d e u p o n t o a l a rg e w o r m h o l e (f ig . 14 ); t h a t is , t h e s m a l l w o r m h o l e w i l l a t t a c h t o
t h e l a r g e w o r m h o l e in s t e a d o f a t t a c h i n g t o th e b a c k g r o u n d s p a c e t i m e . O n e m i g h th a v e a r g u e d f r o m t h is p e r s p e c ti v e th a t t h e p h y s ic s o f s m a l l w o r m h o l e s is u n a f f e c t e d
b y l a r g e w o r m h o l e s ; t h e sm a l l w o r m h o l e f in d s t h e la r g e w o r m h o l e i n d i s ti n g u i s h a b le
f r o m t h e b a c k g r o u n d s p a c e t i m e . B u t w e h a v e a l r e a d y l e a r n e d t h a t t h e p h y s i c a l
c o n s e q u e n c e s o f w o r m h o l e i n t e r a c t i o n s a r e q u i t e d i f f e r e n t t h a n t h e c o n s e q u e n c e s o f
i n s t a n t o n i n t e r a c t io n s . T h e in t e r a c t i o n s b e tw e e n l a rg e a n d s m a ll w o r m h o l e s a l t e r t h e
d i s t r i b u t i o n P ( a ) , b u t h a v e n o e f f e c t o n t h e c r u c i a l f a c t o r Q ( a ) t h a t d e t e r m i n e s
w h a t v a l u e s o f t h e a ' s a r e f a v o re d . T h e i n s t a n t o n i n t e ra c t io n s , n o t t h e w o r m h o l e
i n t e r a c t i o n s , a f f e c t Q ( a ) , a n d i t is t h e r e f o r e th e i n s t a n t o n i n t e r a c t i o n s , n o t t h e
w o r m h o l e i n t e r a c t i o n s , t h a t m u s t a c c o u n t f o r t h e s u p p r e s si o n o f t h e e f f e ct s o f l ar g ew o r m h o l e s .
T h e s u g g e s t i o n t h a t s m a l l i n s t a n t o n s a r e r e sp o n s i b l e f o r s u p p r e s si n g t h e e ff e c t s o f
l a r g e i n s t a n t o n s s o u n d s s u r p r i s i n g a t f ir s t. It i s a c h e r i s h e d p r i n c i p l e o f p h y s i c s , t h e
d e c o u p l i n g p r i n c i p l e , t h a t l o n g - d i s t a n c e p h y s i c s is r e l a t iv e l y in s e n s i t iv e t o t h e d e t a i ls
o f s h o r t - d i s t a n c e p h y s ic s . O r d i n a r i ly , w e e x p e c t t h a t i t i s p o s s ib l e t o " i n t e g r a t e o u t "
s h o r t - w a v e l e n g t h q u a n t u m f l u c t u a t io n s a n d i n c o r p o r a t e t h e e f f e c ts o f th e s e fl u c t u a -
t i o n s i n t o t h e r e n o r m a l i z a t io n o f t h e p a r a m e t e r s o f a n e f f e c ti v e f ie l d t h e o r y t h a t
d e s c r i b e s l o n g - d i s t a n c e p h y s i c s . ( I n d e e d , w e i n v o k e d t h i s s t r a t e g y i n s e c t . 3 , a n d w i l l
d o s o a g a i n i n s e c t. 6 .) H e n c e , w e m i g h t e x p e c t t h a t t h e e f f e c t s o f la r g e i n s t a n t o n s
c a n b e u n d e r s t o o d w i t h i n t h e c o n t e x t o f a l o w - e n e r g y e f f e c ti v e f ie l d th e o r y i n w h i c h
s m a l l i n s t a n t o n s h a v e a l r e a d y b e e n i n t e g r a t e d o u t. B u t h o w c a n t h e m e c h a n i s m
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172 J. Preskill / Wormholes
d e s c r i b e d a b o v e f o r su p p r e s s in g l a rg e i n s t a n to n s , w h i c h a r is e s f r o m i n t e r a c t i o n s
w i t h s m a l l i n s t a n t o n s , b e u n d e r s t o o d i n s u c h a n e f f ec t iv e fi el d t h e o r y ?
I b e l i e v e , i n f a c t, th a t t h e a - d e p e n d e n c e o f G 1 is a n exception t o t h e d e c o u p l i n g
p r in c i p le . T h e d o m i n a n t c o n t r i b u t io n t o t he d e p e n d e n c e o f G 1 o n a L c a n n o t b ea c c o u n t e d f o r i n t e r m s o f l o n g - d i s ta n c e p h y s i c s a l o n e . I t a ri se s b e c a u s e o f t h e
u n u s u a l w a y i n w h i c h s h o r t - d i s t a n c e p h y s i c s a n d l o n g - d i s t a n c e p h y s i c s a r e i n t e r -
t w i n e d h e r e . U s u a l ly , w e m a y t h i n k o f sh o r t - w a v e l e n g t h f l u c t u a t i o n s a s b e i n g
s u p e r p o s e d o n a l o n g - w a v e l e n g t h b a c k g r o u n d f ie ld c o n f i g u r a t i o n . B e c a u s e t h e
s h o r t - w a v e l e n g t h f l u c t u a t i o n s a r e i n d i f f e r e n t t o t h e l o n g - w a v e l e n g t h p h y s i c s , t h e y
i n d u c e r e n o r m a l i z a t i o n s t h a t a r e i n d e p e n d e n t o f t h e b a c k g r o u n d f i e l d ; t h a t i s w h y
d e c o u p l i n g u s u a l l y w o r k s . F r o m t h i s p e r s p e c t i v e , t h e k e y f e a t u r e o f t h e i n s t a n t o n
i n t e r a c t i o n s i s t h a t t h e y p r e v e n t s m a l l in s t a n t o n s f r o m s i tt in g o n t o p o f l ar g e
i n s t a n t o n s . T h e r e f o r e , th e e f f e c ts o f th e s m a l l in s t a n t o n s a r e n o t i n d e p e n d e n t o f t h e
c o n f i g u r a t i o n o f l a rg e in s t a n t o n s . T h e d e p e n d e n c e o f G 1 o n a L is th u s d o m i n a t e d
b y t h e e f f e c t o f a L o n t h e s m a ll in s t a n t o n s , a n d i t c a n n o t b e u n d e r s t o o d i n t e rm s o f
l o n g - d i s t a n c e p h y s i c s a l o n e . T h a t i s w h y d e c o u p l i n g f a i l s .
O n e s h o u l d a l so n o t i c e t h a t t hi s m e c h a n i s m f o r su p p r e s s i n g l ar g e in s t a n t o n s r e li es
c r u c i a l l y o n t h e c la i m t h a t t h e a ' s a r e d e t e r m i n e d b y m a x i m i z i n g G 1. T h e p o i n t i s
s i m p l y t h a t G - ~ h a s th e d i m e n s i o n s o f m a s s t o a posit ive p o w e r . T h e r e f o r e , m e r e l y
o n d i m e n s i o n a l g r o u n d s , s m a l l i n s t a n t o n s a r e m u c h m o r e e f fe c t iv e t h a n l a rg e
i n s t a n t o n s a t c a u s i n g G - ~ t o i n c re a s e.
6 . R e n o r m a l iz a t i o n in q u a n t u m g r a v i t y
I n s p i t e o f t h e i n t r i n s i c i n d e t e r m i n a c y i n t r o d u c e d b y w o r m h o l e e f f e c t s , i t i s
r e a s o n a b l e t o h o p e t h a t th e c o n s t a n t s o f N a t u r e c a n , i n p r i n c ip l e , b e c o m p u t e d . W e
m u s t c a l c u l a t e t h e d e p e n d e n c e o n t h e a ' s o f th e v a r i o u s p a r a m e t e r s t h a t c h a r a c t e r -
i ze l o w - e n e r g y p h y s i c s , a n d t h e n f i n d t h e v a l u e o f a t h a t i s o v e r w h e l m i n g l y f a v o r e d
b y t h e p r o b a b i l i t y d i s t r i b u t i o n Q ( a ) i n e q . ( 3 . 3 ) .
T o f i n d t h e f a v o r e d v a l u e o f a , it i s e n o u g h t o k n o w t h e a - d e p e n d e n c e o f tw o
q u a n t i t i e s, t h e ( r e n o r m a l i z e d ) c o s m o l o g i c a l c o n s t a n t A a n d t h e ( r e n o r m a l i z e d )
N e w t o n ' s c o n s t a n t G . A s w e h a v e e m p h a s i z e d , t h e a - d e p e n d e n c e o f th e s e q u a n t i t ie s
a r i s e s f r o m t w o s o u r c e s ( a s d o e s t h e a - d e p e n d e n c e o f a l l t h e r e n o r m a l i z e d p a r a m e -
t e r s ) , a n d w e c a n i m a g i n e c a l c u l a t i n g t h e a - d e p e n d e n c e i n t w o s t a g e s . I n t h e f i r s t
s t ag e , w e i n t e g r a t e o u t w o r m h o l e f l u c tu a t i o n s , a n d t h u s o b t a i n a n e f f e c t i v e t h e o r y
w i t h a - d e p e n d e n t b a r e c o u p l i n g s t h a t is c u t o f f at t h e w o r m h o l e s c al e M w . I n t h e
s e c o n d s t a g e , w e i n t e g r a t e o u t q u a n t u m f l u c t u a t i o n s w i t h w a v e l e n g t h s g r e a t e r t h a n
R w = M ,~ 1, to o b t a i n a n e f f e c t i v e t h e o r y t h a t is a p p r o p r i a t e f o r d e s c r i b i n g t h e
f a r - i n f r a r e d b e h a v i o r o f g r a v i ty . T h e f i r s t s t a g e w a s t h e s u b j e c t o f s e ct . 5, a n d t h e
s e c o n d s t a g e i s t h e s u b j e c t o f t h i s s e c t i o n .
I n s e c t . 5 , w e a r g u e d t h a t i t w i l l b e d i f f i c u l t t o a n a l y z e t h e f i r s t s t a g e w i t h
p r e c i s i o n , b e c a u s e t h e d i l u t e a p p r o x i m a t i o n d o e s n o t a p p l y . T h i s c o n c l u s i o n a l r e a d y
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J . Preski l l / Wormholes 173
d i s c o u r a g e s u s a b o u t t h e p r o s p e c t s fo r c a lc u l a ti n g th e a - d e p e n d e n c e o f A a n d G .
B u t e v e n i f i t is h o p e l e s s t o c a l c u la t e t h e a - d e p e n d e n c e o f t h e b a r e c o u p l i n g s
i n d u c e d a t t h e f i r s t s t a g e , o n e m i g h t s t i l l n u r t u r e t h e h o p e t h a t t h e p r e f e r r e d v a l u e s
o f a t l e a s t s o m e o f th er e n o r m a l i z e d
c o u p l i n g s c a n b e c a l c u l a t e d t o r e a s o n a b l ea c c u r a c y . T h e r e a re , p r e s u m a b l y , m a n y a ' s , a n d s o i t m a y n o t b e s e ri o u sl y w r o n g t o
a s s u m e t h a t , a s t h e a ' s v a r y , t h e b a r e c o u p l i n g s a t t h e w o r m h o l e s c a l e c a n v a r y
w i t h o u t r e s t ri c t io n * . T h e n w e c a n f o r g e t a ll a b o u t t h e o r i g in o f th e a - d e p e n d e n c e .
O u r t a s k is s i m p l y to f in d " t h e b e s t o f p o s s i b l e w o r l d s " - t h a t c h o i c e o f th e b a r e
p a r a m e t e r s a t s c a l e M W f o r w h i c h C o l e m a n ' s d i s t r ib u t i o n h a s i ts s h a r p p e a k .
U n f o r t u n a t e l y , th is p r o g r a m i n it s m o s t e x t r e m e f o r m s u g g e s te d a b o v e i s d o o m e d
t o f a il . A s w e h a v e s e e n , i n th e b e s t o f w o r l d s G v a n i s h e s a s w e ll a s A , a n d t h e r e i s
n o g r a v i t a t i o n a l i n t e r a c t io n . W e m u s t a c c e p t t h a t o u r u n i v e r s e is n o t t h e m o s t
p e r f e c t o n e , a n d a b a n d o n t h e h y p o t h e s i s t h a t w o r m h o l e e f fe c t s a l lo w t h e b a r ep a r a m e t e r s t o v a r y a r b i t r a r i l y . A m i l d e r v e r s i o n o f t h i s h y p o t h e s i s m i g h t b e
r e a s o n a b l e , h o w e v e r ; w e c a n a s s u m e t h a t w h i le G o 1 is b o u n d e d a b o v e a s a f u n c t io n
o f t h e c ~'s , a l l o t h e r b a r e c o u p l i n g s c a n b e v a r i e d w i t h o u t r e s t r i c ti o n . S u r e l y , th i s is
a t b e s t a c a r i c a t u r e o f th e a c t u a l a - d e p e n d e n c e o f th e b a r e c o u p l in g s , b u t i t m a y
s e r v e a s a f i r s t a p p r o x i m a t i o n .
T h i s p o i n t o f v ie w s u g g e s ts s o m e q u e s t i o n s t h a t w e w il l a d d r e s s b e l o w . F o r
e x a m p l e , w i ll a n u p p e r b o u n d o n G o 1 s u f fi c e t o e n s u r e t h a t t h e re n o r m a l i z e d G 1 is
a l s o b o u n d e d a b o v e , a s w e r e q u i r e f o r C o l e m a n ' s m e c h a n i s m t o b e c o n s i s t e n t w i t h
o b s e r v a t i o n , e v e n i f n o r e s t ri c ti o n s a r e p l a c e d o n o t h e r b a r e c o u p l i n g s ? A n d a ls o , d ot h e c o e f f i c i e n t s in t h e re n o r m a l i z e d g r a v i t a t i o n a l e f f e c t iv e a c t i o n o f t e r m s t h a t
i n v o l v e h i g h e r d e r i v a ti v e s o f t h e m e t r i c o b e y s u i t a b le b o u n d s e v e n a s t h e b a r e
c o u p l i n g s v a r y w i t h o u t r e s t r i c t i o n ? S u c h b o u n d s a r e a l s o r e q u i r e d f o r t h e c o n s i s -
t e n c y o f C o l e m a n ' s m e c h a n i s m .
I w ill a r g u e t h a t C o l e m a n ' s e x p l a n a t i o n f o r A - - 0 p a s s e s b o t h o f t h e c o n s i s t e n c y
t e st s p o s e d a b o v e . T h a t i s, o n l y o n e c o n d i t io n o n t h e a - d e p e n d e n c e o f th e b a s e
c o u p l i n g s - t h e u p p e r b o u n d o n G o 1 - s u f fi c e s t o j u s t i f y C o l e m a n ' s p i c t u r e o f t h e
i n f r a r e d b e h a v i o r o f g r a v i ty . T h e s e a r g u m e n t s a r e r a t h e r h e u r is t ic , b u t , I h o p e ,
c o g e n t .W h a t c a n b e s a i d a b o u t t h e c a l c u la b i l i ty , in p r a c ti c e , o f t h e c o n s t a n t s o f N a t u r e
( o t h e r t h a n t h e c o s m o l o g i c a l c o n s t a n t )? W i t h r e s p e c t t o t h is is s ue , t h e re is n o t m u c h
c a u s e f o r e n c o u r a g e m e n t . W e h o p e t o c a l cu l a te t h e v a ri o u s re n o r m a l i z e d p a r a m e t e r s
t h a t d e s c r i b e l o w - e n e r g y p h y s i c s. F o r t h is p u r p o s e , w e m u s t f i n d th e d e p e n d e n c e o f
t h e r e n o r m a l i z e d G o n t h es e re n o r m a l i z e d q u a n t it ie s . T h i s d e p e n d e n c e m i g h t h a v e
b e e n c a l c u l a b le i f t he r e n o r m a l i z a ti o n o f G w e r e d o m i n a t e d b y l o n g - w a v e l e n g th
e f f e c t s t h a t c o u l d b e n a t u r a l l y e x p r e s s e d in t e r m s o f c o u p l in g s r e n o r m a l i z e d a t lo w
e n e r g y . B u t u n f o r t u n a t e l y , t h e r e n o r m a l i z a t i o n o f G is a c t u a l ly d o m i n a t e d b y
* This means, of course, that the bare action varies "wi tho ut restri ction" within a space of reasonable
quasilocal theories.
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174 J. Preskill / Wormholes
s h o r t - w a v e l e n g t h q u a n t u m f l u c t u a t i o n s . T h e r e f o r e , t h e p r e f e r r e d v a l u e s o f t h e
c o n s t a n t s o f N a t u r e a r e a c t u a l l y s e n s i t i v e t o p h y s i c s a t t h e w o r m h o l e s c a l e M , , , a n d
c a n n o t b e c o m p u t e d b a s e d o n l o w - e n e rg y p h y s i c s a lo n e .
I n a m o r e o p t i m i s t i c l i g h t , b y f a i l i n g t o m a k e p r e d i c t i o n s b e y o n d A = 0 w e h a v ea v o i d e d m a k i n g a n y wrong p r e d i c t i o n s t h a t w o u l d c a l l C o l e m a n ' s m e c h a n i s m i n t o
q u e s t i o n . I n d e e d , b a s e d o n a d i ff e r e n t p o i n t o f v i e w th a n a d v o c a t e d h e r e , G r i n s t e i n
a n d W i s e [1 5] a n d K l e b a n o v , S u s s k in d , a n d B a n k s [1 6] r e a c h e d s o m e u n f o r t u n a t e
c o n c l u s i o n s - f o r e x a m p l e , t h a t th e p i o n h a s v a n i s h i n g m a s s . T h i s e m b a r r a s s m e n t is
e v a d e d if , a s I c la i m , t h e p r e f e r r e d v a l u e s o f th e r e n o r m a l i z e d q u a n t i t ie s a r e a c t u a l l y
d e t e r m i n e d b y s h o r t - d i s ta n c e p h y s i c s . O f c o u rs e , it r e m a i n s t o b e s e e n w h e t h e r " t h e
b e s t o f p o s s i b l e w o r l d s " i n t h e r e v i s e d s e n s e p r o p o s e d h e r e w i l l r e s e m b l e t h e
u n i v e r s e t h a t w e o b s e r v e .
L e t u s n o w c o n s i d e r i n m o r e d e t a i l t h e r e l a t i o n b e t w e e n t h e b a r e t h e o r y a t t h ew o r m h o l e s c a l e M w a n d t h e r e n o r m a l i z e d t h e o r y t h a t d e s c r i b e s p h y s i c s a t v e r y lo w
e n e r g y . I t is c o n v e n i e n t t o im a g i n e t h a t t h e r e n o r m a l i z e d t h e o r y is o b t a i n e d f r o m t h e
b a r e t h e o r y b y m e a n s o f a r e n o r m a l i z a t i o n g r o u p t r a n s f o r m a t i o n [ 23 ]. T h e b a r e
t h e o r y i s c u t o f f a t t h e m a s s s c a l e M W a n d i s q u a s i l o c a l o n t h e d i s t a n c e s c a l e M w i.
T h i s b a r e t h e o r y m i g h t b e q u i t e c o m p l i c a t e d , i n v o l v i n g m a n y d e g r e e s o f f r e e d o m
a n d i n t e r a c t i o n s . N o w w e a l lo w t h e c u t o f f t o f lo a t d o w n f r o m M w t o M << M w ; w e
i n t e g r a te o u t a l l q u a n t u m f lu c t u a ti o n s w i th w a v e n u m b e r b e t w e e n M w a n d M , a n d
w e i n c o r p o r a t e t h e e ff e c ts o f t h e se f l u c t u a ti o n s i n t o r e n o r m a l i z e d p a r a m e t e r s o f a n
e f f e c t i v e t h e o r y t h a t h a s c u t o f f M a n d i s q u a s i l o c a l o n t h e d i s t a n c e s c a l e M 1. A f t e r
M h a s d e s c e n d e d b e l o w t h e m a s s o f t h e li g h te s t m a s s i v e p a r t ic l e i n t h e o ri g in a l
t h e o r y , t h e o n l y r e m a i n i n g d e g r e e s o f f re e d o m i n t h e e f f e c ti v e t h e o r y a r e m a s s l e s s
f i e ld s , s u c h a s t h e g r a v i t o n a n d p h o t o n . I t is th e p a r a m e t e r s o f th i s e f fe c t i v e t h e o r y ,
i n t h e l i m i t M --* 0 , t h a t e n t e r i n t o C o l e m a n ' s c a l c u l a t i o n o f t h e d i s t r i b u t i o n Q ( c 0 .
T h e M ~ 0 l im i t o f th e e f fe c t iv e th e o r y i s t h e " c o n t i n u u m l i m i t " o f t h e b a r e
t h e o r y t h a t w e s t a r t e d w i t h . I t d e s c r i b e s t h e p h y s i c s o f t h e t h e o r y o n l e n g t h s c a l e s
t h a t a r e a r b i t r a r i l y l a r g e c o m p a r e d t o t h e o r i g i n a l s h o r t - d i s t a n c e c u t o f f . F u r t h e r -
m o r e , t h e o n l y s u r v i v i n g in t e r a c t i o n i n t h e M ~ 0 li m i t i s g r a v i t y .
O t h e r m a s s l e s s p a r t i c l e s ( s u c h a s t h e p h o t o n ) m a y r e m a i n c o u p l e d t o t h e g r a v i t o n ,
b u t t h e s e l f - in t e r a c t io n s o f m a t t e r h a v e b e e n i n t e g r a t e d o u t . ( F o r e x a m p l e , t h e re i s
n o e l e c t r o d y n a m i c in t e r a c t i o n b e c a u s e t h e r e a r e n o m a s s l e s s c h a r g e d p a r t ic l e s .)
T h u s , t h e M ---, 0 l i m i t is t h e c o n t i n u u m l i m i t o f a c u t o f f t h e o r y o f g r a v i t y . I n f a c t , a s
w e a l l o w M t o f l o a t t o w a r d z er o , t h e r e n o r m a l i z e d c o s m o l o g i c a l c o n s t a n t A m u s t
a l s o a p p r o a c h z e r o , a s w e d i s c u s s e d i n s e c t . 3 . S o w e a r e c o n s i d e r i n g t h e c o n t i n u u m
l i m i t o f g r a v i t y w i t h a v a n i s h i n g c o s m o l o g i c a l c o n s t a n t .
T h i s is a u s e f u l i n si g h t, b e c a u s e E i n s te i n g r a v i t y is n o n r e n o r m a l i z a b l e in p e r t u r b a -
t i o n t h e o r y , a n d i t w o u l d b e a s t u n n i n g s u r p r i s e if i t t u r n e d o u t t o b e p o s s i b l e t o
t a k e a c o n t i n u u m l im i t o f q u a n t u m g r a v i t y o t h e r t h a n f re e f ie ld t h e o ry . I n o t h e r
w o r d s , i f t h e s t r e n g t h o f g r a v i t a t i o n a l e f f e c t s i s h e l d f i x e d a t th e m a s s s c a l e M , w e
d o n o t e x p e c t t o b e a b l e t o p u s h t h e c u t o f f m a s s M o u p a r b i t r a ri l y h ig h , w h i le
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J. Pre skil l / Worm holes 175
m a i n t a i n i n g a q u a s i l o c a l t h e o r y o f g r a v i t y a t t h e s c a le M 0. I f th i s w e r e p o s s i b le , t h e n
w e c o u l d d e v i s e a n o r d i n a r y f ie l d th e o r y o f g r a v i ty t h a t m a k e s s e n s e a t a r b i t ra r i l y
s h o r t d i s t a n ce s , j u s t a s q u a n t u m c h r o m o d y n a m i c s ( p r e su m a b l y ) c a n d e s c r i b e th e
s t r o n g i n t e r a c t i o n a t a r b i t r a r i l y s h o r t d i s t a n c e s . T h e n w e w o u l d h a v e a c o n s i s t e n tf i e ld t h e o r y o f q u a n t u m g r a v it y ; th e r e w o u l d b e n o n e e d f o r s tr in g s .
T h e c o n j e c t u r e t h a t q u a n t u m g r av i ty h as n o n o n t r iv i a l c o n t i n u u m l im i t ca n b e
s t a t e d f r o m t w o d i f f e r e n t p o i n t s o f v ie w . W e m a y c o n s i d e r ( as a b o v e ) p h y s i c s a t
s c a le M t o b e f i xe d , a n d t h e n a s k w h e t h e r a b a r e t h e o r y c a n b e c o n s t r u c t e d t h a t
r e p r o d u c e s t h e p h y s i c s a t s c al e M a s t h e c u t o f f M 0 o f t h e b a r e t h e o r y a p p r o a c h e s
i n f i n i ty . A l t e r n a t iv e l y , w e m a y c o n s i d e r t h e c u t o f f M 0 o f t h e b a r e t h e o r y t o b e f ix e d ,
a n d t h e n a s k w h e t h e r a g r a v i ta t i o n a l i n t e r a c ti o n a t s ca le M c a n r e m a i n n o n t r i v i a l a s
M ~ 0 , f o r a n y c h o i c e o f t h e q u a s i l o ca l b a r e t h e o r y w i th c u t o f f M o.
T h e e f f e c t i v e t h e o r y a t sc a le M h a s a n a c t i o n S , t h a t c a n b e e x p a n d e d i n t e r m so f ( q u a s i - )l o c a l o p e r a t o r s . E a c h o p e r a t o r h a s a c o e f f ic i e n t t h a t c a n b e m a d e
d i m e n s i o n l e s s b y m u l t ip l y in g b y a n a p p r o p r i a te p o w e r o f M . T h u s , S M c a n b e
s p e c i f i e d b y a n i n f i n i t e s e t o f d i m e n s i o n l e s s c o u p l i n g s * , X % , a = 1, 2 , 3 . . . . . T h e
c o n j e c t u r e t h a t g r a v i t y h a s n o c o n t i n u u m l im i t , t h en , in t h e s e c o n d f o r m m e n t i o n e d
a b o v e , c a n b e s t a t e d : F o r a n y c h o i c e o f t he q u a s i lo c a l b a r e t h e o r y a t t h e c u t o f f M 0 ,
a l l d i m e n s i o n l e s s r e n o r m a l i z e d c o u p l i n g s ? ~ a p p r o a c h z e r o a s M - -* 0 " * . I n f a ct ,
o n e e x p e c t s t h a t f o r s m a l l n o n v a n i s h i n g M , t h e k % ' s o b e y a s o m e w h a t s t r o n g e r
i n e q u a l i t y c o n s t r a i n t ,
X% ~ X"m~, , M << M o , (6 .1 )
w h e r e t h e u p p e r b o u n d X ~ x , a d im e n s i o nl e ss f u n c t io n o f M U M , a p p r o a c h e s z e r o
f o r M o / M - - - , o o .
T h e i n e q u a l i t y e q . ( 6 . 1 ) h o l d s f o r a n y c h o i c e o f th e q u a s i lo c a l b a r e t h e o r y a t
c u t o f f s c a l e M o , e v e n a s t h e b a r e c o u p l i n g s r a n g e w i t h o u t b o u n d ; t h e r e is n o
r e n o r m a l i z e d t h e o r y a t s ca le M t h a t is d e s c e n d e d f r o m a q u a s il o c a l b a r e t h e o r y a t
s c a l e M o a n d t h a t h a s ; k ~ g r e a t e r t h a n ~ ' ~ ( M o / M ) . O n e m i g h t h a v e s u p p o s e dt h a t w e c a n s p e c i f y a n y r e n o r m a l i z e d t h e o r y w e p l e as e a t sc a le M a n d t h e n o b t a i n
t h e c o r r e s p o n d i n g b a r e t h e o r y a t s c al e M 0 b y r u n n i n g t h e r e n o r m a l i z a t i o n g r o u p
t r a n s f o r m a t i o n i n r e v e rs e . B u t t h is t y p i c a l l y fa ils , b e c a u s e t h e r e n o r m a l i z a t i o n g r o u p
f l o w r e a c h e s a b o u n d a r y o f t h e s p a c e o f q u a s i lo c a l t h e o r ie s b e f o r e t h e f l o a ti n g
c u t o f f M o is a t t a i n e d .
T h e c l a i m t h a t t h e r e n o r m a l i z e d c o u p l i n g s a r e b o u n d e d e v e n t h o u g h t h e b a r e
c o u p l i n g s a r e n o t , m a y s o u n d s u r p r i s i n g a t fi rs t , b u t t h is b e h a v i o r is j u s t t h e
* S t r i ct l y s p e a k i n g , t h e s e a r e t h e " e s s e n t i a l " c o u p l i n g s t h a t a r e n o t c h a n g e d b y l o c a l r e d e f i n i t i o n s o f
the f i e ld s [ 23 , 26 ] .
* * R e c a l l t h a t w e a r e c o n s i d e r i n g t h e i n f r a r e d b e h a v i o r o f g r a v i t y w i t h v a n i s h i n g r c n o r m a l i z e d
c o s m o l o g i c a l c o n s t a n t .
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176 J. Preskill /Worrnholes
( n o n p e r t u r b a t i v e ) " L a n d a u g h o s t " p h e n o m e n o n . T h e L a n d a u g h o s t is f a m i l ia r in ,
f o r e x a m p l e , Q E D a n d 0 4 t h e o r y i n f o u r d i m e n s i o n s , t w o f i e ld t h e o r ie s t h a t , t h o u g h
p e r t u r b a t i v e l y r e n o r m a l i z a b l e , a r e n o t e x p e c t e d t o h a v e n o n t r i v i a l c o n t i n u u m l i m i t s
[ 2 3 ,2 7 ]. I t is c h a r a c t e r is t i c o f t h is p h e n o m e n o n t h a t a r e n o r m a l i z e d c o u p l i n g )tM ,
w h e n e v o l v e d b y m e a n s o f t h e r e n o r m a l i z a t i o n g r o u p t o a l a r g e r r e n o r m a l i z a t i o n
s c a l e , a t t a i n s a n i n f i n i t e v a l u e a t a f i n i t e s c a l e M ' > M . E q u i v a l e n t l y , L a n d a u g h o s t
b e h a v i o r m e a n s t h a t r e n o r m a l i z e d c o u p l i n g s a t s ca l e M o b e y t h e i n e q u a l i ty e q . ( 6.1 ),
e v e n a s b a r e c o u p l i n g s a t t h e c u t o f f M 0 v a r y w i t h o u t r e s tr ic t io n . P h y s i c a ll y , v e r y
s t r o n g c h a r g e s c r e e n in g e f f e c ts k e e p t h e r e n o r m a l i z e d c o u p l i n g f i n it e e v e n i f t h e
b a r e c o u p l i n g i s i n f i n i t e .
W e c o n c l u d e , t h e n , th a t i f n o q u a n t u m f ie ld t h e o r y o f g r a v i t y h a s a c o n t i n u u m
l i m i t , t h e n a l l g r a v i t a t i o n a l e f f e c t s m u s t b e c o m e w e a k i n th e M - ~ 0 li m i t, ir r e s p e c -
t i v e o f t h e b a r e t h e o r y d e f i n e d a t s c a l e M 0 >> M ; a l l d i m e n s i o n l e s s c o u p l i n g s ) t%a p p r o a c h z e r o . B e c a u s e a l l d i m e n s i o n l e s s c o u p l i n g s a r e s m a l l , t h e r e n o r m a l i z a t i o n
g r o u p f l o w c a n b e r e li a b l y c o m p u t e d f o r M << M 0 in p e r t u r b a t i o n t h e o r y . T h i s
p e r t u r b a t i v e a n a l y s i s a l lo w s u s t o j u s t i fy n e g l e c t in g t e r m s i n t h e e f f e c ti v e a c t i o n t h a t
i n v o l v e h i g h e r d e r i v a t i v e s o f t h e m e t r i c i n t h e d e r i v a t i o n i n s e c t . 3 . ( T h e p e r t u r b a t i v e
f l o w e q u a t i o n s w i l l b e d i s c u s s e d f u r t h e r b e l o w . ) T h u s , C o l e m a n ' s e x p l a n a t i o n f o r
A = 0 h a s p a s s e d a n o n t r iv i a l c o n s i s t e n c y t e s t; w e n e e d n o t w o r r y a b o u t h i g h e r-
d e r i v a t i v e i n t e r a c t io n s u p s e t t i n g h is m e c h a n i s m , r e g a r d le s s o f t h e a - d e p e n d e n c e o f
t h e b a r e h i g h e r - d e r i v a t i v e c o u p l i n g s .
L i k e t h e o t h e r d i m e n s i o n l e s s c o u p l i n g s , GM M2 o b e y s t h e i n e q u a l i t y e q . ( 6 . 1 ) .
P e r t u r b a t i v e p o w e r - c o u n t i n g a s d e s cr ib e d b e l o w t h e n s h o w s t h a t G M s t o p s ru n n i n g
a s M - , 0 ; i t a p p r o a c h e s t h e l im i t G , t h e r e n o r m a l i z e d N e w t o n ' s c o n s t a n t . T h e r e f o r e
G o b e y s a n i n e q u a l i ty
G < C mo 2 , ( 6 . 2 )
w h e r e C is a c o n s t a n t , p r e s u m a b l y o f o r d e r o n e , i n d e p e n d e n t o f t h e b a r e t h e o r y a t
t h e c u t o f f M 0. C o r r e s p o n d i n g l y , t h e c o e f f i c i e n t G 1 o f R i n t h e r e n o r m a l i z e d
e f f e c t iv e a c t i o n e q . (3 .5 ) is b o u n d e d f r o m be lo w . I n p a r t i c u l a r , t h e n , G 1 i s p o s i t i v e ,
a s w e a s s u m e d i n s e c t . 3 .
B u t t h e r e i s n o l o we r b o u n d o n G t h a t w e c a n i nf e r f r o m a re n o r m a l i z a ti o n g r o u p
a r g u m e n t a l o n e . I n d e e d , a l l g r a v i t a t io n a l i n t e r a c t io n s a r e p r o p o r t i o n a l t o a p o s i t i v e
p o w e r o f G , a n d g r a v i t y b e c o m e s a f r ee t h e o r y f o r G = 0 . E v i d e n t ly , t h e n , t h e r e is n o
r e n o r m a l i z a t i o n f o r G = 0 , a n d G = 0 is a f ix e d p o i n t o f t h e r e n o r m a l i z a t io n g r o u p .
I f th e b a r e N e w t o n ' s c o n s t a n t G O v a n i s h es , t h e n s o d o e s t h e r e n o r m a l i z e d c o u p l i n g
G . W e a r r i v e a g a i n a t t h e c o n c l u s i o n a n n u n c i a t e d e a r l i e r ; C o l e m a n ' s m e c h a n i s m
f a v o r s t h e l o w e s t p o s s ib l e v a lu e o f G , a n d h e n c e i s i n c o m p a t i b l e w i t h o b s e r v a t i o n
u n l e s s t h e b a r e p a r a m e t e r G O is b o u n d e d a w a y f r o m z e ro .
M a y w e a l s o s a y th a t a lo w e r b o u n d o n t h e b a r e N e w t o n ' s c o n s t a n t G O>/G0,min > 0
s u f f ic e s t o e n s u r e t h a t t h e r e n o r m a l i z e d G d o e s n o t v a n i s h ? T h e i ss u e is w h e t h e r
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G M 1 c a n b e i n f i n i t e l y r e n o r m a l i z e d a s M f l o a ts f r o m M o d o w n t o ze r o , o r , i n o t h e r
w o r d s , w h e t h e r t h e m e t r i c c a n u n d e r g o a n i n f i n i t e f i e l d r e n o r m a l i z a t i o n . S i n c e t h e
r e n o r m a l i z a t i o n f o r M << M 0 c a n b e s t u d i e d p e r t u r b a t i v e l y , a n d w e k n o w t h a t G M
s t o p s r u n n i n g , t h e o n l y q u e s t i o n i s w h e t h e ri n f i n i t e
h i g h e r - d e r i v a t i v e b a r e c o u p l i n g sc a n i n d u c e a n i n f i n it e f ie l d r e n o r m a l i z a t i o n a t M - M 0. T h i s q u e s t i o n i s h ig h l y
n o n p e r t u r b a t i v e a n d is h a r d t o a n s w e r d e f in i ti v e ly , b u t I b e l ie v e th a t t h e s a m e
s c r e e n i n g e f f e c t s t h a t u n d e r l i e t h e i n e q u a l i ti e s e q . ( 6 .1 ) w i ll a ls o p r e v e n t i n f i n i t e f i e ld
r e n o r m a l i z a t i o n , e v e n i f s o m e b a r e c o u p l i n g s a r e i n fi n it e . T h e p o i n t is t h a t i n fi n i te
b a r e c o u p l i n g s r u n i n f in i te l y q u i c k l y a s th e c u t o f f f l o a ts , a n d s o a lw a y s i n d u c e f in i te
e f f e c ts . T h e r e f o r e , i f th e b a r e N e w t o n ' s c o n s t a n t i s G o << M o 2 , t h e n t h e r e n o r m a l -
i z e d c o u p l i n g i s G - G 0, r e g a r d l e s s o f th e v a l u e s o f th e o t h e r b a r e c o u p l i n g s . ( F o r
G o > 3 4 o 2, t h e r e n o r m a l i z e d c o u p l i n g i s G - M 0 2, i n a c c o r d w i t h t h e b o u n d e q .
( 6 . 2 ) . )I f t h e b a r e c o u p l i n g G o h a s a m i n i m u m a s a f u n c t i o n o f a , o n e e x p e c t s o n
d i m e n s i o n a l g r o u n d s t h a t t he m i n i m u m o c c u r s f o r G o - 3 4 o - 2 ; t h e n t h e m i n i m a l
r e n o r m a l i z e d c o u p l i n g is a ls o G - M 0 2. T h i s m i n i m a l r e n o r m a l i z e d c o u p l i n g s h o u l d
c o i n c i d e w i t h t h e o b s e r v e d N e w t o n ' s c o n s t a n t . I t is t h e r e f o r e re a s o n a b l e t o id e n t i f y
t h e w o r m h o l e s c a le M w = M o w i th t h e P l a n c k s c a le M v - 1 0 1 9 G e V .
W h i l e t h e a b o v e d is c u s s i o n s u g g e s ts t h a t t h e o th e r b a r e c o u p l i n g s d o n o t r e n o r -
m a l i z e G o b y a n e n o r m o u s f a c to r , e a c h c o u p l i n g d o e s g i ve a c o n t r i b u t i o n , h o w e v e r
s m a l l , t o t h e r e n o r m a l i z a t i o n o f G o , b e c a u s e a ll d e g r e e s o f f r e e d o m i n t h e b a r e
t h e o r y c o u p l e t o g r a v i t y . I n o t h e r w o r d s , t h e r e n o r m a l i z e d c o u p l i n g G i s a f u n c t i o no f a ll t h e b a r e p a r a m e t e r s . T h e r e f o r e , t h e r e q u i r e m e n t t h a t G a t t a i n i ts m i n i m u m
v a l u e i s e x p e c t e d t o c o m p l e t e l y d e t e r m i n e t h e b a r e t h e o r y , a n d h e n c e t o c o m p l e t e l y
d e t e r m i n e a ll lo w - e n e r g y p h y s ic s . Q u a n t u m i n d e t e r m i n a c y o f t he c o n s t a n t s o f
N a t u r e i s t h u s , i n p r i n c i p l e , a v o i d e d .
O f c o u r s e , t h is a n a l y s is is p r e m i s e d o n t h e a s s u m p t i o n t h a t q u a n t u m g r a v i t y h a s
n o n o n t r i v i a l c o n t i n u u m l i m i t . I t i s a t l e a s t a p o s s i b i l i t y t h a t t h e r e n o r m a l i z a t i o n
g r o u p e q u a t i o n s o f q u a n t u m g r a v i t y h a v e a n o n t r i v i a l fi xe d p o in t . T h e n a n o n t r i v i a l
c o n t i n u u m t h e o r y c o u ld b e f o r m u l a t e d * . V a r io u s d i m e n s i o n le s s c o u p l i n g s w o u l d b e
n o n v a n i s h i n g a t th e f ix e d p o n t , a n d h i g h e r - d e r i v a ti v e i n t e r a c ti o n s w o u l d h a v ei m p o r t a n t e f f e c t s i n t h e c o n t i n u u m t h e o r y . T h e r e w o u l d t h e n b e a s e r io u s d a n g e r
t h a t t h e s e i n t e r a c t io n s w o u l d d e s t a b il iz e t h e p e a k a t A = 0 th a t C o l e m a n f o u n d i n
t h e d i s t r i b u t i o n Q ( a ) . B u t i f , a s o n e e x p e c t s , t h e r e i s n o n o n t r i v i a l c o n t i n u u m l i m i t
o f q u a n t u m g r a v it y , t h e n o u r a r g u m e n t s b u t t r e ss C o l e m a n ' s p r e d ic t io n t h a t A = 0 .
T o f u r t h e r s u p p o r t t h e s e c l a i m s , w e s h o u l d n o w d i s c u s s s o m e w h a t m o r e e x p l i c i t l y
t h e r e n o r m a l i z a t i o n g r o u p e q u a t i o n s o f q u a n t u m g r a v i t y [2 6]. T h e s e e q u a t i o n s
d e s c r i b e h o w t h e e f fe c t iv e a c t i o n S M f l o w s a s t h e f l o a t i n g c u t o f f M i s l o w e r e d . T h e
* T h e p o s s i b l e e x i s t e n c e o f a n o n t r i v i a l f i x e d p o i n t h a s b e e n d i s c u s s e d ( f ro m q u i t e d i f f e r e n t p o i n t s o f
v i e w ) b y W e i n b e rg [ 2 6 ] a n d b y A n t o n i a d i s a n d T o m b o u l i s [ 2 8 ] . T h e c o n c l u s i o n s o f r ef . [ 2 8 ] h a v e
b e e n c r i t i c i z e d b y J o h n s t o n [ 2 9 ] .
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178 J. Preskill / Wormholes
g r e a t u t i l i t y o f t h e s e e q u a t i o n s d e r i v e s f r o m t h e o b s e r v a t i o n t h a t , i n t h e i n f r a r e d
l i m i t M ---, O, S M e x h i b it s " u n i v e r s a l " b e h a v i o r th a t d e p e n d s o n l y r a t h e r w e a k l y o n
t h e o r i g i n a l b a r e a c t i o n SMo.
I f t h e b a r e t h e o r ySM o
d e p e n d s u p o n a n in f i n it e n u m b e r o f a d ju s t a b l e b a r ep a r a m e t e r s , o n e m i g h t w o r r y t h a t t h e th e o r y h a s n o p r e d i c t iv e p o w e r . B u t a s W i l s o n
e m p h a s i z e d [2 3], p h y s i c s a t t h e e n e r g y s c a l e M << M 0 c a n b e p r e d i c t e d , b e c a u s e , u p
t o c o r r e c t i o n s s u p p r e s s e d b y p o w e r s o f M / M o, S M c a n b e e x p r e s s e d i n t e r m s o f a
s m a l l n u m b e r o f r e n o r m a l i z e d p a r a m e t e r s . I n d e e d , e v e n t h e p o w e r c o r r e c ti o n s c a n
b e s y s t e m a t i c a l l y t a k e n i n t o a c c o u n t ; t o a n y f i n i t e o r d e r i n M / M o , S M c a n b e
e x p r e s s e d i n t e r m s o f a f i n it e n u m b e r o f r e n o r m a l i z e d q u a n t i t ie s .
I n d i s c u s s i n g C o l e m a n ' s m e c h a n i s m , w e c o n s i d e r t h e v a l u e o f t h e r e n o r m a l i z e d
e u c l i d e a n a c t i o n a t i t s s t a t i o n a r y p o i n t . I n t h i s c o n t e x t , t h e e x p a n s i o n o f S M in
p o w e r s o f M b e c o m e s a n e x p a n s i o n o f t h e s t a t i o n a r y v a lu e o f t h e a c t i o n in p o w e r so f R 1, w h e r e R i s t h e " r a d i u s " o f th e c la s s ic a l s o l u t i o n . F o r e x a m p l e , th e a c t i o n
S M [ g ] o f e q . (3 .5 ), w h e n e v a l u a t e d f o r a f o u r - s p h e r e o f r a d iu s R , b e c o m e s
S ( R ) - 1 6 ~ r G ( 2 A R a - 1 2 R 2 + A 4 + A 6 R 2 + . . . ) , ( 6 . 3 )
whe re ~ ' 2 = 8 q r 2 / 3 i s t h e v o l u m e o f a u n i t f o u r - s p h e r e . H e r e A 4 , A 6 . . . . a r i s e f r o m
t e r m s i n t h e a c t i o n i n v o l v i n g 4 , 6 . . . . d e r i v a t i v e s o f t h e m e t r ic . B e c a u s e o f r e n o r m a l -
i z a t i o n e f f e c ts , t h e p a r a m e t e r s A , G , A 4, A 6 . . . . . a r e a ll i m p l i ci tl y d e p e n d e n t o n t h er a d i u s R .
N a i v e l y , e q . ( 6 . 3 ) s h o w s " u n i v e r s a l " b e h a v i o r i n t h a t S ( R ) c a n b e e x p r e s s e d t o
o r d e r R -2m i n t e r m s o f m + 3 r e n o r m a l i z e d p a r a m e t e r s . T h i s s t a t e m e n t is n a i v e
b e c a u s e w e h a v e i g n o r e d t h e d e p e n d e n c e o f t h e r e n o r m a l i z e d p a r a m e t e r s o n
R - M 1. F o r e x a m p l e , w e m a y w is h t o c l a im t h a t S ( R ) u p t o o r d e r R ° c a n b e
e x p r e s s e d i n t e r m s o f o n l y th e re n o r m a l i z e d A a n d G . W e m u s t a r g u e , t h e n , t h a t as
t h e b a r e p a r a m e t e r s o f SM , v a r y o n t h e s u r f a c e w h e r e A M a n d G M h a v e s p e c i f i e d
v a l u e s , A 4. M v a r ie s b y a n a m o u n t o f o r d e r o n e ( i n d e p e n d e n t o f M ) . W e c a n a n a l y z e
t h is p r o b l e m i n p e r t u r b a t io n t h e o r y u si ng e l e m e n t a r y a r g u m e n t s b a s e d o n d i m e n -s i o n a l a n a l y s i s a n d p o w e r - c o u n t i n g o f F e y n m a n d i a g r a m s .
( T h e r e i s a t e c h n i c a l p o i n t t h a t w e s h o u l d c o m m e n t o n h e r e * . A l l te r m s i n S M
t h a t i n v o l v e f o u r d e r i v a t i v e s o f th e m e t r i c a r e a c t u a l l y " i n e s s e n t i a l " [2 6, 1 4]; t h e s e
t e r m s a r e e i t h e r to p o l o g i c a l a n d c a n b e i g n o r e d in p e r t u r b a t i o n t h e o r y o r c a n b e
e l i m i n a t e d f r o m t h e a c t io n b y a lo c a l r e d e f i n i ti o n o f t h e m e t r i c g v O r d i n a r i ly , i t is
a d v i s a b l e t o e l i m i n a t e i n e s s e n t i a l c o u p l i n g s w h e n r e n o r m a l i z a t i o n g r o u p e q u a t i o n s
a r e d e r i v e d , b u t w e w ill f in d i t c o n v e n i e n t to c o n s i d e r t h e r e n o r m a l i z a t i o n o f A 4 . I n
p a r t , t h is is b e c a u s e w e a r e u l ti m a t e l y i n te r e s t e d in m i n i m i z i n g S ( R ) w i t h r e s p e c t to
R a n d t h e a - p a r a m e t e r s , s u b j ec t t o a n i n f r a r e d c u t o f f . T h e r e d e f i n i t io n o f t h e m e t r i c
* T h i s p o i n t w a s d i s c u s s e d b y G r i n s t e i n a n d W i s e [ 15 ].
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J. Pres kil l / Worm holes 17 9
t h a t e l i m i n a t e s A 4 p r e s e r v e s t h e v a l u e o f t h e a c t i o n a t i t s m i n i m u m , b u t m o d i f i e s
t h e ( i n v a r i a n t ) r a d i u s o f t h e f o u r - s p h e r e s o lu t io n . F u r t h e r m o r e , t h is r e d e f i n i t io n
d e p e n d s o n c~; i f w e c h a n g e t h e b a r e p a r a m e t e r s , a c h a n g e i n A 4 i s t y p i c a l l y
i n d u c e d , a n d w e m u s t r e d e f i n e t h e fi e l d s a g a i n t o e l i m i n a t e i t. T h u s , i f w e w e r e t oe l i m i n a t e A 4 a n d then m i n i m i z e t h e a c t i o n w i t h r e s p e c t t o a , w e w o u l d b e
m i n i m i z i n g s u b j e c t t o a n a-dependent i n f r a r e d c u t o f f . I t s e e m s l e s s c o n f u s i n g t o
k e e p t h e i n f r a r e d c u t o f f fi x ed a n d a - i n d e p e n d e n t ; w e t h e n lo s e t h e fr e e d o m t o
e l i m i n a t e A 4 . )
R o u g h l y s p e a k i n g , S ~t c a n b e o b t a i n e d f r o m SMo i n p e r t u r b a t i o n t h e o r y b y
e v a l u a t i n g F e y n m a n d i a g r a m s t h a t a r e c u t o f f in t h e u l t r av i o l e t a t M 0 a n d i n t h e
i n f r a r e d a t M . T h e s e d i a g r a m s c a n b e d i v i d e d , a c c o r d i n g t o t h e i r s e n s i t i v i t y t o t h e
u l t r a v i o l e t c u t o f f , i n t o t h r e e c l a s s e s - p o w e r - d i v e r g e n t , l o g - d i v e r g e n t , a n d c o n v e r -
g e n t .P o w e r - d i v e r g e n t d ia g r a m s a r e o f o r d e r M 0D w h e r e D > 0 is t h e s u p e r f i c i a l d e g r e e
o f d i v e r g e n c e o f t h e d ia g r am . T h e s e d i a g ra m s a re " n o n u n i v e r s a l " c o n t r i b u t i o n s t h a t
a r e d o m i n a t e d b y q u a n t u m f lu c t u a ti o n s w it h w a v e n u m b e r o f o r d e r M 0 a n d t h a t a r e
s e n s i t i v e t o t h e d e t a i l e d f o r m o f t h e b a r e a c t i o n SMo. B e c a u s e o f t h e p o w e r
d i v e r g e n c e s , t h e r e n o r m a l i z a t i o n g r o u p f l o w is v e r y c o m p l e x fo r M - M o ; m a n y
h i g h e r d i m e n s i o n o p e r a t o r s i n S M c o n t r i b u t e t o t h e r e n o r m a l i z a t i o n o f l o w e r
d i m e n s i o n o p e r a t o r s . B u t , i n p e r t u r b a t i o n t h e o r y , t h e p o w e r - d i v e r g e n t d i a g ra m s g i ve
a d e c r e a s i n g l y i m p o r t a n t c o n t r i b u t i o n t o t h e fl o w as M d e c r ea s e s.
O f m u c h g r e a t e r i n te r e s t ar e th e l o g - d iv e r g e n t d i a g r a m s t h a t a r e p r o p o r t i o n a l t o ap o w e r o f l n ( M 0 / M ). T h e s e d i a g r a m s r e c e iv e a si g n i fi c a n t c o n t r i b u t i o n f r o m q u a n -
t u m f l u c t u a t i o n s o v e r a b r o a d r a n g e o f l e n g th s c a le s. U n l i k e p o w e r d i v e rg e n c e s , th e
l o g d i v e r g e n c e s a r e u n i v e r s al ; t h e c o e f f i c ie n t s o f le a d i n g a n d n o n l e a d i n g p o w e r s o f
l n ( M o / M ) a r e c a l c u l a b l e i n t e r m s o f l o w - e n e r g y p h y s i c s a l o n e a n d a r e in s e n s i t i v e t o
t h e d e t a i l s o f p h y s i c s a t t h e c u t o f f .
F i n a l l y , t h e r e a r e c o n v e r g e n t d i a g r a m s . T h e s e t o o a r e c a l c u l a b l e i n t e r m s o f
l o w - e n e r g y p h y s i c s a l o n e . T h e i r s e n s it i v it y to t h e c u t o f f is o f o r d e r M ~ , w h e r e D < 0
i s th e s u p e r f i c i a l d e g r e e o f d i v e r g e n c e o f t h e d i a g r a m . I f w e a r e i n t e r e s t e d i n t h e
f l o w o f t h e r e n o r m a l i z e d e f f e c t i v e a c t i o n S M f o r M << M o , w e n e e d c o n s i d e r o n l yt h e l o g - d i v e r g e n t a n d c o n v e r g e n t d i a g r am s , f o r th e r e n o r m a l i z a t i o n e f f e ct s i n d u c e d
b y t h e p o w e r - d i v e r g e n t d i a g r a m s a r e n o t s e n s i t i v e t o M .
N o w c o n s i d e r t h e i m p l i c a t i o n s o f t h i s c l a s s i f i c a t i o n f o r t h e r e n o r m a l i z a t i o n o f
q u a n t u m g r a v i t y . F i r s t w e c o n s i d e r p u r e g r a v i t y , u n c o u p l e d t o m a t t e r , w h i c h w e
h a v e a r g u e d is a d e q u a t e f o r s t u d y i n g t h e a s y m p t o t i c M -+ 0 b e h a v i o r o f t h e e f f e c ti v e
a c t i o n S M . T h e n w e w i ll d i sc u s s h o w o u r c o n c l u s i o n s a r e m o d i f i e d w h e n g r a v i t y is
c o u p l e d t o m a t t e r w i t h r e n o r m a l i z a b l e o r s u p e r r e n o r m a l i z a b l e i n t e r a c t i o n s .
W e h a v e a r g u e d t h a t t h e f lo w o f SM f o r M s m a ll is d e t e r m i n e d b y l o g - d i v e r g e n t
a n d c o n v e r g e n t d i a g r a m s . B u t i n a t h e o r y li ke p u r e g r a v i t y w i t h A = 0 , i n w h i c h a l l
c o u p l i n g s a r e o f n o n r e n o r m a l i z a b l e t y p e , a l l r e n o r m a l i z a t i o n s i n d u c e d b y s u c h
d i a g r a m s a r e r e n o r m a l i z a t io n s o f h ig h e r d im e n s i o n a l c o u p l i n g s b y l o w e r d i m e n -
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F i g . 1 5 . A o n e - l o o p d i a g ra m t h a t c o n t r i b u t e s t o t h e r e n o rm a l i z a t i o n o f t h e g r a v i t a t i o n a l e f f e c t iv e a c t i o n .
A l l l i n e s r e p re s e n t g r a v i t o n s .
s i o n a l c o u p l i n g s . T h i s c o n c l u s i o n f o l l o w s d i r e c tl y f r o m s i m p l e d i m e n s i o n a l a n a l y s is .
C o n s i d e r , f o r e x a m p l e , t h e lo g a r i t h m i c r e n o r m a l i z a t io n s . I f w e s c al e G - 1 o u t o f t h e
a c t i o n a s in e q . (6 .3 ), th e n G is a l o o p - c o u n t i n g p a r a m e t e r ; a n L - l o o p d i a g r a m is o f
o r d e r G L - 1 . S i n c e al l o t h e r c o u p l i n g s h a v e d i m e n s i o n s o f m a s s t o a n e g a t i v e p o w e r ,
i t f o l lo w s i m m e d i a t e l y o n d i m e n s i o n a l g r o u n d s ( f o r A = 0 ) th a t t h e r e i s n o l o g a r i th -
m i c r e n o r m a l i z a t i o n o f G - 1 . S im i l a rl y , a l o g a r it h m i c r e n o r m a l i z a t io n o f A 4 oc cur s
o n l y i n o n e - l o o p o r d e r , a n d h a s t h e f o r m
(~A 4
1 6 ~ G- b 4 l n ( M o /M ) , ( 6 . 4 )
w h e r e b 4 is a n u m e r i c a l c o n s t a n t . ( T h i s r e n o r m a l i z a t i o n i s g e n e r a t e d b y t h e d i a g r a m
i n f i g . 1 5 , f r o m w h i c h t h e v a l u e o f b4 c a n b e e x t r a c t e d . ) E a c h c o u p l i n g A 2 , r e c e i v e s
a lo g a r i t h m i c r e n o r m a l i z a t io n o n l y u p t o (n - 1 ) - lo o p o r d e r in p e r t u r b a t i o n t h e o r y
[ 2 6 ] . F o r e x a m p l e , A 6 r e ce iv e s t w o - l o o p r e n o r m a l i z a t io n o f o r d e r G 2 a n d a o n e - l o o pr e n o r m a l i z a t i o n o f o r d e r G A a .
W e s ee , t h e r e f o r e , t h a t t h e i n f ra r e d - s e n s it i v e r e n o r m a l i z a t i o n o f e a c h c o u p l i n g i n
t h e e f f e c t i v e a c t i o n i s d e t e r m i n e d b y r e n o r m a l i z e d c o u p l i n g s o f l o w e r d i m e n s i o n .
O n c e w e h a v e f i x e d t h e v a l u e o f A , G , a n d A 2m f o r m < n , t h e s e n s i t i v it y o f t h e
r e n o r m a l i z e d A 2 n t o t h e b a r e p a r a m e t e r s i n v o l v e s o n l y t h e n o n u n i v e r s a l p o w e r -
d i v e r g e n t r e n o r m a l i z a t i o n s . I n p e r t u r b a t i o n t h e o r y , G M o a n d A 2 m M o2 " 2 a r e
r e g a r d e d a s s m a ll , a n d t h e p o w e r - d i v e r g e n t r e n o r m a l i z a t io n o f A 2n is b o u n d e d b y a
p o w e r o f M 0 d e t e r m i n e d b y d i m e n s i o n a l a n a l y s is
( S A 2 n ) . . . . n i . . . . 1 ~ C M ° 2 n + 2 . ( 6 . 5 )
O u r e a r l ie r d i s c u s s io n o f n o n p e r t u r b a t i v e e f f e c t s t h e n s u g g e s ts t h a t a n i n e q u a l i ty o f
t h is f o r m h o l d s e v e n b e y o n d t h e d o m a i n o f v a li d it y o f p e r t u r b a t io n t h e o r y * .
S p e c i f i c a l l y , t h e r e n o r m a l i z a t i o n o f t h e c o s m o l o g i c a l c o n s t a n t A i s d u e t o
p o w e r - d i v e r g e n t d i a g r a m s o n ly . T h e se d i a g ra m s r e n o r m a l i ze A / 8 ~ r G b y a n a m o u n t
o f o r d e r M 4 , a n d A / 8 ~ r G a p p r o a c h e s a f i n i t e l i m i t a s t h e f l o a t i n g c u t o f f M
a p p r o a c h e s z e r o . S im i l a rl y , i n p u r e g r a v i t y w i t h v a n i s h i n g r e n o r m a l i z e d A , G 1 is
r e n o r m a l i z e d b y p o w e r - d i v e r g e n t d i a g r a m s o n l y , a n d i t t o o a p p r o a c h e s a f in i te li m i t
* A related suggestion was m ade by Weinberg [14].
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J. Preskill / Wormholes" 181
f o r M ~ 0 . W e s a y th a t A a n d G s t o p r u n n i n g i n t h e f a r i n f r a r e d . O n t h e o t h e r
han d , A 4 d o e s n o t s t o p r u n n i n g ; i t r e c e i v e s t h e l o g a r i t h m i c r e n o r m a l i z a t i o n e q .
( 6 .4 ) . T h i s l o g a r i t h m i c r e n o r m a l i z a t i o n d i v e r g e s in t h e i n f r a r e d l im i t M --+ 0 , b u t t h e
c o e f f i c i e n t o f t h e l o g a r i t h m i s d e t e r m i n e d b y t h e r e n o r m a l i z e d G . W h e n A a n d Ga r e h e l d f i x e d , t h e n , S(R) i n e q. (6 . 3) v a r i e s a s a f u n c t i o n o f t h e a ' s b y a n a m o u n t
o f o r d e r o n e ; i t is a b o u n d e d f u n c t i o n o f b o t h a a n d R . T h e r e f o r e , t h e a - d e p e n -
d e n c e t h a t a r i s e s f r o m h i g h e r - d e r i v a t i v e o p e r a t o r s c a n b e n e g l e c t e d i n t h e R --+
l i m i t , j u s t a s w e a s s u m e d i n s e ct . 3.
T h e p o w e r - c o u n t i n g d e s c r i b e d a b o v e a p p l i e s t o p u r e g r a v i t y , a n d h e n c e t o t h e
r e n o r m a l i z a t i o n g r o u p f lo w o f t h e g r a v i t a t io n a l e f f e ct i v e a c t i o n S M in t h e f a r
i n f r a r e d , a f t e r a ll m a t t e r h a s b e e n i n t e g r a t e d o u t . I t is a ls o o f i n t e r e s t t o c o n s i d e r t h e
f l o w o f S M a t i n t e r m e d i a t e l e n g t h s ca le s, w h e r e t h e c o u p l i n g o f g r a v i ty t o m a t t e r
c a n n o t b e n e g l e c te d . B e c a u s e t h e m a t t e r a c t io n c a n c o n t a i n t e r m s o f re n o r m a l i z a b l eo r s u p e r r e n o r m a l i z a b l e t y p e , w e m u s t m o d i f y t h e e a r li e r a n a ly s is , w h i c h a p p l i e d t o a
t h e o r y i n w h i c h a l l i n t e r a c t i o n s a r e o f n o n r e n o r m a l i z a b l e t y p e .
O f p a r t i c u l a r i n t e r e s t is th e d e p e n d e n c e o f t h e r e n o r m a l i z a t i o n o f G o n t h e m a s se s
o f m a t t e r p a r t i c l e s , a n d t h is is t h e o n l y e f f e c t o f t h e m a t t e r t h a t w e w i ll d is c u s s h e r e
i n a n y d e t a i l . I n s o f a r a s t h e m a s s - d e p e n d e n t r e n o r m a l i z a t i o n o f G i s c a l c u l a b l e , w e
m i g h t h o p e t o r e a c h n o n t r i v ia l c o n c l u s i o n s a b o u t t h e m a s s es o f e l e m e n t a r y p a r t ic l e s
f o r m t h e r e q u i r e m e n t th a t G a a t t a i n s it s l a r g e s t p o s s i b l e v a l u e.
D i m e n s i o n a l l y , a c a l c u l ab l e o r d e r - m 2 l o g a r i th m i c r e n o r m a l i z a t i o n o f G 1 is
p o s s i b l e i n o n e - l o o p o r d e r , a n d i n d e e d , o n e f i n d s t h a t t h i s r e n o r m a l i z a t i o n isg e n e r a t e d b y t h e d i a g r a m i n f i g . 1 6 . C a l c u l a t i o n o f t h i s d i a g r a m [ 3 0 ] s h o w s t h a t t h e
l o g a r i t h m i c r e n o r m a l i z a t i o n i s
1 ) 1 m21n(Mo2/mZ)+O(m Z)+O(Gm 4 ) ( 6 . 6 )6 ~ 192~r 2
w h e r e m i s t h e m a s s o f either a f r e e ( D i r a c ) f e r m i o n o r o f a f r e e m i n i m a l l y c o u p l e d
( r e a l ) s c a l a r . T h e o r d e r - m 2 c o r r e c t i o n i n e q . (6 . 6 ) is a n o n u n i v e r s a l e f f e c t ( w h i c h c a n
b e a b s o r b e d b y s h i ft in g M 2 i n t h e l o g a r it h m ) , a n d t h e o r d e r - G m 4 c o r r e c t i o n is d u e
t o d i a g r a m s w i t h t w o o r m o r e l o o p s . O f c o u r s e, if w e i n c l u d e t h e c o n t r i b u t i o n s d u e
t o r e n o r m a l i z a b l e i n t e r a c ti o n s , t h e l o g a r i th m i c re n o r m a l i z a t i o n o f G 1 is m o d i f i e d .
T h e i n t e r a c t i o n s g e n e r a t e a d d i t io n a l l o g a r i t h m s i n e a c h o r d e r o f p e r t u r b a t i o n
t h e o r y . T h e s e l o g a r i t h m s c a n b e s u m m e d , a n d t h e y h a v e t h e e f f e c t o f c a u s i n g m 2 t o
Fig. 16. A one-lo op diagram that induce s a logarithmi c renormaliza tion of G. The solid linc repres ents a
massil~e matt er particle; it may be either a scalar or a spin - ½ fermion.
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182 J. Preskill / Wormholes
b e c o m e a s c a l e - d e p e n d e n t r u n n i n g p a r a m e t e r . I f t h e m a t t e r p a r ti c le t h a t i n d u c e s
t h i s r e n o r m a l i z a t i o n is a c o m p o s i t e o b j e c t ( li k e th e p i o n ) , t h e n t h e c u t o f f M 0 i n t h e
l o g a r i t h m i s n o t t h e w o r m h o l e s c a l e , b u t r a t h e r t h e m a s s s c a l e ( l i k e 4 ~ r f = ) b e l o w
w h i c h t h e p a r t i c l e b e h a v e s l ik e a n e f f e c t iv e l y e l e m e n t a r y o b j e c t.
T h e p h y s i c a l m a s s m o f t h e m a t t e r p a r ti c l e a p p e a r s i n th e a r g u m e n t o f th e
l o g a r i t h m i n e q . ( 6.6 ). T h i s is b e c a u s e t h e m a t t e r l o o p c o n t i n u e s t o c o n t r i b u t e t o t h e
r e n o r m a l i z a t i o n o f G 1 a s t h e c u t o f f M f l o a t s d o w n u n t il M - m , a t w h i c h p o i n t
w e i n t e g r a t e t h e m a t t e r p a r t i c le o u t o f t h e e f f e c ti v e a c t io n , a n d i t c e a s e s t o
c o n t r i b u t e t o l o o p d i a g r a m s . I n s p i t e o f t h e l o g a r i t h m , t h o u g h , t h e r e n o r m a l i z a t i o n
e q . (6 .6 ) a c t u a l l y v a n is h e s in t h e l im i t m 2 ~ 0 . T h i s o b s e r v a t i o n c o r r o b o r a t e s o u r
e a r l i e r c o n c l u s i o n t h a t G 1 s t o p s ru n n i n g i n t h e i n f ra r e d . L o n g - w a v e l e n g t h fl u c t u a -
t i o n s o f li g h t m a t t e r p a r t i c le s , li k e l o n g - w a v e l e n g t h g r a v i t o n f l u c t u a t i o n s , c o n t r i b u t e
v e r y l i t t l e t o t h e r e n o r m a l i z a t i o n o f G .T h e sign o f t h e c a l c u l a b l e l o g a r i t h m i c r e n o r m a l i z a t i o n i n e q . ( 6 . 6 ) i s s u c h a s t o
d e c r e a s e G 1. T h e c r it e r io n o f m a x i m i z i n g G - l , t h e n , f a v o r s l ighter m a t t e r p a r t i c l e s .
W e o u g h t n o t t o a c c e p t t h i s c o n c l u s i o n t o o r e a d i l y , h o w e v e r . B e c a u s e t h e c a l c u l a b l e
i n f r a r e d r e n o r m a l i z a t i o n is sm a l l f o r s m a l l m z, o n e m u s t w o r r y a b o u t w h e t h e r t h is
e f f e c t is s w a m p e d b y n o n u n i v e r s a l c o n t r i b u t i o n s .
T h e f o r m o f eq . ( 6.6 ) s h o u l d b e c o n t r a s t e d w i th t h e l o g a r i t h m i c m a s s - d e p e n d e n t
o n e - l o o p r e n o r m a l i z a t io n o f A4; f o r a m i n i m a l l y c o u p l e d r e a l s c a l a r t h i s i s
8 A 4 2 9
16~rG 30( 6 . 7 )
A s n o t e d b y G r i n s t e i n a n d W i s e [ 1 5 ] a n d K l e b a n o v , S u s s k i n d , a n d B a n k s [ 1 6 ] , t h e n ,
i f t h e c o r r e c t c r i t e r i o n t o d e t e r m i n e t h e c ~'s w e r e t o m i n i m i z e A 4 r a t h e r t h a n t o
m a x i m i z e G - l , t h e n e q . ( 6.7 ) w o u l d f a v o r t h a t m 2 a p p r o a c h z e ro . B u t fu r t h e r m o r e ,
b e c a u s e t h e m a s s - d e p e n d e n t r e n o r m a l i z a t i o n o f A 4 b e c o m e s infinite f o r m 2 ~ 0 , w e
c a n b e c o n f i d e n t t h a t t h e c a l c u l a b l e d e p e n d e n c e o f A 4 on t h e p h y s i c a l m a s s m 2
d o m i n a t e s n o n u n i v e r s a l e f f e c ts f o r rn Z ~ 0 . W e w o u l d c o n c l u d e , t h e r e f o r e , th a t t h e
p i o n m a s s m , , i s e x a c t l y z e r o , i f m ~ = 0 c a n b e a c h i e v e d f o r a n y v a l u e o f t h e c ~ 's . ( I t
d o e s n o t e v e n m a t t e r t h a t , f o r t h e p i o n , M 0 i n e q . (6 .7 ) is o f o r d e r 4 ~rf~ , r a t h e r t h a n
t h e w o r m h o l e s c a l e . )
T h i s u n p l e a s a n t c o n c l u s i o n c a n b e a v o i d e d if , a s w e h a v e p r o p o s e d , t h e c o r re c t
c r i t e r io n t o d e t e r m i n e t h e a ' s is a c t u a l l y th a t G - 1 i s a t is m a x i m u m . T h e n b a r e
p a r a m e t e r s a t t h e w o r m h o l e s c a le m u s t b e a d j u s t e d s o a s to m a x i m i z e t h e s u m o f t h e
c a l c u l a b l e i n f r a r e d r e n o r m a l i z a t io n o f G 1 i n eq . ( 6.5 ) a n d o f th e u n c a l c u l a b l e ( b u t
n o n - n e g l i g i b l e ) n o n u n i v e r s a l c o n t r i b u t i o n s t o th e r e n o r m a l i z a t i o n o f G 1 . T h e
n a t u r e o f t h e n o n u n i v e r s a l c o n t r i b u t i o n s d e p e n d s o n w h e t h e r t h e li g ht p a r t ic l e s h a v e
b a r e m a s s e s o r a c q u i r e t h e i r m a s s e s f r o m r e n o r m a l i z a b l e i n t e r a c t i o n s .
C o n s i d e r , f o r e x a m p l e , a n e l e m e n t a r y s c a l a r w i t h a n a d j u s t a b l e b a r e m a s s . In t h is
c a s e w e can a r g u e t h a t i f t h e b a r e m a s s m 0 i s c h o s e n t o m a x i m i z e G 1, t h e n t h e
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J . P r e s k i l l / W o r m h o l e s 183
r e n o r m a l i z e d s c a l a r m a s s m u s t e i t h e r v a n i s h o r b e o f t h e o r d e r o f t h e c u t o f f M o. I f
G - 1 is s t a t i o n a r y a s a f u n c t i o n o f m 2 , w e m u s t h a v e
0
0 = -~ m ~ ( 1 6 7 rG ) 1
0 1 ( o 2 )a m 2 ( 1 6 ~ rG o ) - 1 1 92 ~ r2 l n ( M ( 2 / m 2) -~m~ )m + . . . , ( 6 . 8 )
w h e r e G o i n c l u d e s a ll n o n u n i v e r s a l m a s s - d e p e n d e n t c o n t r i b u t i o n s t o t h e r e n o r m a l -
i z a t i o n o f G . B u t b e c a u s e w e e x p e c t t h a t
0- - m 2 ~ 1 ,Om~
3 1
3m2o ( 1 6 ~ r G o ) - i Z 1 9 2~ r2 , ( 6 . 9 )
e q . ( 6 . 8 ) c a n b e s a t i s f i e d o n l y f o r
In ( Mo2/m 2 ) - 1 . ( 6 . 1 0 )
T h u s , w h i l e 1 6 7 rG o 1 m i g h t b e a c o m p l i c a t e d f u n c t i o n o f m ~ t h a t is m a x i m i z e d a t a
n o n t r i v i a l v a l u e o f m o , th is m a x i m u m w i ll b e d e s t a b i l iz e d b y t h e c a l c u l a b l e
i n f r a r e d - s e n s i t i v e r e n o r m a l i z a t i o n o f G , i f m 2 << M o .
T h e s i t u a t i o n i s q u i t e d i f f e r e n t , h o w e v e r , i n t h e m o r e r e a l i s t i c c a s e o f a p a r t i c l e
t h a t h a s n o b a r e m a s s a n d a c q u i r e s i t s p h y s i c a l m a s s f r o m a r e n o r m a l i z a b l e
i n t e r a c t i o n , l ik e a Y u k a w a c o u p l i n g . I n th i s c a s e , w e re q u i r e G 1 t o b e s t a t i o n a r y a s
a f u n c t i o n o f t h e b a r e Y u k a w a c o u p l i n g X o , o r
0 = ~ o ( 1 6 rr G o ) 1 l n ( M ( Z / m 2) +1 92 Tr2 ~ o m . . . . ( 6 . 1 1 )
B u t n o w w e e x p e c t
0 102% (16~ ' G° ) -1 < (16~v2)~M o2 (6 .1 2 )
d u e t o d i a g r a m s l i k e f i g . 1 7 , a n d
0Oq~-~O m 2 ~ O 2 , ( 6 . 1 3 )
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184 J. Pre skill / Wormholes
Fig. i7. A two-loop diagram that contributes to the renormalization G. The solid line represents a spin~ fermion, and the dotted line represents a scalar.
where v is an expectation value that is determined by other, independent bare
parameter s*. The maximum of Go I will not be destabilized by the infrared
renorma lizat ion of G, then, provided only that v 2 << Mo .
We see, then, that if one accepts Coleman's explanation for the vanishing of the
cosmological constant, then one must also find unacceptable any elementary particle
that is light compared to the wormhole scale and that has a bare mass that can be
adjusted at the wormhole scale, unless the physical mass of the particle vanishes
identically**. This argument has nothing to do with the quadratically divergent
renormalization of the mass of an elementary scalar. In fact, the same conclusion
applies to a light elementary fermion with an adjustable bare mass, even though the
fermion mass is protected from any power-divergent renormalization by approxi-
mate chiral symmetry.
If Coleman's mechanism is to apply to Nature, we must require therefore that all
mass scales of low-energy physics are determined dynamically, rather than by
adjusting bare mass parameters in the action Sgo at the wormhole scale. This
conclusion is actually unsurprising. If the bare parameters are to be determined by
the in f rared- insensi t ive renormalization of G, it would evidently require a miracle
for the bare Higgs mass, for example, to be fine-tuned just so as to ensure that the
weak-interaction mass scale is far below the Planck scale. But if all low-energy mass
scales are determined dynamically, then it seems at least conceivable that the
low-energy physics that we observe can be accounted for (to excellent accuracy) by
maximizing the infrared-insensitive nonuniversal renormalization of G-~.
The curious and rather surprising thing that we have found is that a light
elementary particle with an adjustable bare mass is compatible with Coleman's
mechanism if its physical mass is exactly zero. This observation, unfortunately, does
not help us to understand why the weak-interaction scale is so small compared to
the Planck scale, but it might have other interesting implications for low-energy
physics.
In the end, we have been unable to extract from wormhole physics any quantita-
tive conclusion other than A = 0. Indeed, it seems unlikely that any other precise
* The logarithmic renormalization of ?, has been ignored here.* * I am assuming that the a-dependence of the bare mass is such that the physical mass vanishes for
some value of a. Note also that, in order to simplify the discussion, I have assumed that elementary"scalars are minimally coupled.
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J. Preskill / Wormholes 18 5
predictions can be made without a detailed understanding of Planck-scale physics.
This is disappointing. On the other hand, we have probed carefully for fatal flaws in
Coleman's beautiful explanation for the vanishing of the cosmological constant, and
have found none. This is encouraging.
This research began in collaboration with Andy Cohen and Mark Wise; I am very
grateful to both of them for many enlightening discussions. I have also benefited
from conversations about wormholes with Jim Cline, Sidney Coleman, Gerry
Gilbert, Stephen Hawking, Igor Klebanov, and Frank Wilczek.
N o t e s A d d e d
There is an interesting exception to the general feature that the dependence of therenormalized Newton's constant G on other renormalized parameters cannot be
computed. The dependence of G on the vacuum angle 0 of quantum chromodynam-
ics i s determined by low-energy physics [31]. This exception arises because the
0-dependence of G is entirely due to nonperturbative strong-interaction effects that
are extremely weak at very short distances.
More careful consideration indicates that the excluded volume interaction be-
tween large and small instantons, discussed in sect. 5, does not by itself suffice to
suppress the effects of large wormholes. Rather, these interactions induce a very
strong renormalization of the relation between the a-parameters of the largewormholes and the density of large instantons. I am indebted to Joe Polchinski for
an enlightening discussion of this point. (See ref. [32].) If the large wormhole
problem is to be evaded within the framework described in this paper, it will
presumably be necessary to invoke a more subtle interaction between large and
small instantons. Other recent discussions of the large wormhole problem may be
found in ref. [33].
R e f e r e n c e s
[1] S.W. H aw king , Phys . Lett . B 195 (1987) 337: Phys . Rev. D3 7 (1988) 904
[2] G.V. Lavre las hv i l i , V. Rubako v and P .G. Tinyakov , JET P Let t . 46 (1987) 167
[3] S.B. G idd ing s and A. Strominger, N ucl. Phys . B30 6 (1988) 890;
K. Lee, Phys. Rev. Lett. 61 (1988) 263
[4] S .W. Hawking , in Rela t iv i ty , g roups , an d topo logy I I , ed . B . De W it t and R. S to ra (N or th -H ol land ,
Am s te r d a m , 1 9 8 3 )
[5] J . Hartle , in H igh ene rgy physics 1985, ed. M. Bow ick and F. G iirsey (Wor ld Scientif ic , Singapore~
1986)
[6] J . Hartle and S.W. Hawking, Phys . Rev. D28 (1983) 2960
[7] S .W. Haw king , D .N. P age and C.N . Pope , Nucl . Phys . B170 (1980) 283 ;
S .W. Haw king , Com mu n. Math . Phys . 87 (1982) 395 : N ucl . Phys . B244 (1984) 135
[8] S. Co lem an , Nu cl. Phys . B3 07 (1988) 867[9] S.B. Gid din gs a nd A . Strominger, N ucl. Phys . B3 07 (1988) 854
[10] S. Coleman, Nucl. Phys . B310 (1988) 643
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