THEORETICAL PERSPECTIVES

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Submitted on 1 Jan 1982

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THEORETICAL PERSPECTIVES

G. T Hooft

To cite this version:

G. T Hooft. THEORETICAL PERSPECTIVES. Journal de Physique Colloques, 1982, 43 (C3),

pp.C3-755-C3-764. �10.1051/jphyscol:1982387�. �jpa-00221943�

JOURNAL DE PHYSIQUE

CoZZoque C3, suppzdment

ac

n o 12, Tome 43, dEcembre 1982 page C3-755

T H E O R E T I C A L P E R S P E C T I V E S G. ' t Hooft

I n s t i t u t e fop T h e o r e t i c a l Physics, P. 0 . Box 80006,3508 !PA Utrecht, The NetherZands

When I was asked by t h e O r g a n i z i n g Committee t o speak a b o u t " T h e o r e t i c a l P e r s p e c t i v e s " t h i s l e f t me w i t h two p o s s i b i l i t i e s . One would b e t o d i s c u s s :

" P e r s p e c t i v e s i n P a r t i c l e P h y s i c s a c c o r d i n g t o s t a n d a r d t h e o r y " and t h e o t h e r :

" F u t u r e p e r s p e c t i v e s o f T h e o r e t i c a l P a r t i c l e Physics". Both a r e r a t h e r b r a v e e n t e r p r i s e s s i n c e t h e y r e q u i r e e x t r a p o l a t i o n s towards t h e f u t u r e . And s i n c e my t e l e p h o n e l i n e w i t h t h e f u t u r e h a s a b o u t a s much s t a t i c d i s t u r b a n c e a s anybody e l s e ' s my e x t r a p o l a t i o n s may w e l l b e e n t i r e l y wrong.

I. P e r s p e c t i v e s i n P a r t i c l e P h y s i c s a c c o r d i n g t o s t a n d a r d t h e o r y .

About t h i s f i r s t i n t e r p r e t a t i o n of my t i t l e I s h a l l b e b r i e f . Anything I wanted t o s a y on t h i s s u b j e c t h a s a l r e a d y been d i s c u s s e d i n much more d e t a i l i n t h e s e s s i o n s we h e a r d . The n e a r f u t u r e o f p a r t i c l e p h y s i c s l o o k s g r e a t : Z0 a t 90 GeV and !$ a t 80 GeV. I f a l l goes w e l l we w i l l l e a r n a s much from t h e s e new o b j e c t s a s we d i d from t h e J, p a r t i c l e s some y e a r s ago. One b i g d i f f e r e n c e i s t h a t t h i s t i m e

t h e o r y i s making r a t h e r p r e c i s e p r e d i c t i o n s and does n o t r u n b e h i n d t h e f a c t s l i k e i n '74. I f t h i n g s do n o t come o u t a s p r e d i c t e d we w i l l f i r s t blame t h e experimen- t a l i s t s , n o t t h e t h e o r i s t s . Our o n l y worry s h o u l d b e p e r h a p s t h a t t h e p r e d i c t i o n s w i l l b e obeyed t o o w e l l ; n o t h i n g new w i l l b e l e a r n t . R. ~ a t t o l c a l l s t h i s " S c e n a r i o Zero": o n l y minor e r r o r s i n t h e o r e t i c a l c a l c u l a t i o n s need b e c o r r e c t e d .

The Higgs p a r t i c l e

But t h e r e a r e many a s p e c t s o f p a r t i c l e p h y s i c s a b o u t which t h e o r y i s s t i l l e x t r e m e l y vague. C l e a r l y any e x p e r i m e n t a l r e s u l t s i n t h o s e a r e a s w i l l b e o f utmost importance. Going t o h i g h e r e n e r g i e s we f i n d a s t h e f i r s t u n c e r t a i n f a c t o r : t h e Higgs p a r t i c l e . I s i t t h e r e ? I s i t l i g h t ? Heavy? Composite? I f t h e Higgs p a r t i c l e i s l i g h t t h e n i t w i l l b e r e l a t i v e l y ( b u t n o t v e r y 2 ) e a s y t o d e t e c t ( s c e n a r i o a ' ) . T h i s w i l l l e a v e Theory w i t h t h e m y s t e r y o f t h e p a r a m e t e r - c o n s p i r a c y a t h i g h e r e n e r g y s c a l e s : what c a u s e d t h i s m a s s t o b e s o low? More t h e o r e t i c a l c l u e s would be n e e d e d . I f i t i s heavy t h e n t h i s would b e t h e o r e t i c a l l y more c h a l l e n g i n g . We have

where G i s t h e Fermi c o n s t a n t . I f

$

( t h e Higgs Mass) is l a r g e t h e n t h e c o u p l i n g c o n s t a n t F X must be l a r g e . Because o l a c k of a s y m p t o t i c freedom t h i s would imply t h e n e a r p r e s e n c e o f a new s t r o n g i n t e r a c t i o n a t

-

2000 G e V .

The new p o s s i b l e s t r u c t u r e a t t h e TeV range i s v e r y i l l u n d e r s t o o d . Unfortu- n a t e l y t h e p e r s p e c t i v e s h e r e a r e much l e s s b r i g h t : w i l l e x p e r i m e n t a l i s t s e v e r b e a b l e t o produce, d e t e c t and u n r a v e l t h e e x p e c t e d j e t s a t 2000 GeV ( c e n t e r o f m a s s i e n e r g y ? Can t h e y d e t e r m i n e t h e i r quantum numbers? Even a t 1 GeV t h i s seems t o b e a d i f f i c u l t job. A problem may w e l l b e t h a t c o s t s of new machines w i l l c o n t i n u e t o r i s e a s r o u g h l y t h e t h i r d power o f t h e e n e r g y , whereas t h e t h e o r i s t ' s i n t e r e s t i s c l e a r l y o n l y a l o g a r i t h m i c f u n c t i o n o f e n e r g y . I t must be t h e g r e a t e s t c h a l l e n g e f o r e x p e r i m e n t a l p a r t i c l e p h y s i c i s t s o f t h e coming decades t o s u r p r i s e t h e t h e o r i s t s h e r e .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982387

JOURNAL DE PHYSIQUE

Beyond

2000

TeV

A t t h e TeV (c.m.) r e g i o n Theory e n v i s a g e s a t l e a s t f o u r p o s s i b i l i t i e s : a ) . One o f t h e s e Grand U n i f i e d T h e o r i e s i s c o r r e c t 5 . Whether o r n o t i t h o l d s a l l t h e way t o t h e P l a n c k mass i s o f s e c o n d a r y i m p o r t a n c e . Most i m p o r t a n t i s t h a t t h e n t h e r e w i l l b e no s t r u c t u r e a t a l l u n t i l much h i g h e r e n e r g i e s GeV). The Higgs w i l l b e r e l a t i v e l y l i g h t (comparable w i t h t h e Z boson mass). M a t h e m a t i c a l l y n o t h i n g

seems t o b e wrong w i t h t h i s p o s s i b i l i t y . But we do n o t a t a l l u n d e r s t a n d t h e p h y s i c a l r e a s o n f o r t h e u n n a t u r a l c a n c e l l a t i o n o f p a r a m e t e r s r e q u i r e d a t t h e h i g h

GeV) mass One p o s s i b l e a t t i t u d e h e r e i s t h a t N a t u r e p r o v i d e d a p r e c e d e n t i n t h i s r e s p e c t : t h e c o s m o l o g i c a l c o u p l i n g c o n s t a n t i n E i n s f e i n ' s

L a g r a n g i a n i s a l s o u n n a t u r a l l y s m a l l . U s u a l l y we a r g u e however t h a t t h a t i s j u s t one more o f t h e s e u n u n d e r s t o o d a s p e c t s of quantum g r a v i t y b u t t h a t s u c h t h i n g s

s h o u l d n o t be a c c e p t e d i n p a r t i c l e p h y s i c s .

b ) . The supersymmetry ~ p t i o n . ~ N a t u r e may become s u p e r s y m m e t r i c beyond 2000 GeV.

A l l p a r t i c l e s come i n s u p e r m u l t i p l e t s w i t h mass s p l i t t i n g s n o t much g r e a t e r t h a n 1000 GeV. Whether t h e s e s p l i t t i n g s r e s u l t from e x p l i c i t o r s p o n t a n e o u s s u p e r - syrmnetry b r e a k i n g remains t o b e s e e n . No new s t r o n g i n t e r a c t i o n s t r u c t u r e i s t h e n needed a t o r beyond 2000 GeV, and " n a t u r a l n e s s " i s s a v e d . At l e a s t t h e s u p e r - m u l t i p l e t a r r a n g e m e n t s of p a r t i c l e s s h o u l d t h e n be u n r a v e l l e d by t h e o r i s t s and e x p e r i m e n t e r s . C e r t a i n l y many new p a r t i c l e s w i l l b e needed.

c ) . "TechnicoIor" ( o r "meta c o l o r " ) '

.

The fundamental a s s u m p t i o n h e r e i s t h a t t h e Higgs p a r t i c l e i s a two- ( o r more-) f e r m i o n c o m p o s i t e . Here a l s o t h e r e i s a

h i s t o r i c a l p r e c e d e n t : t h e Cooper p a i r i n s u p e r c o n d u c t o r s i s j u s t a " ~ i ~ g s "

article.

But f o r u n d e r s t a n d i n g fermion masses we n e e d "extended t e c h n i c o l o r " and we h e a r d a l r e a d y what t h e d i f f i c u l t i e s a r e when s u c h t h e o r i e s a r e exposed t o

he no me no-

l o g i c a l d a t a .

d ) . I l e a v e a f o u r t h o p t i o n : "hyper complexity". T h i s i s t h e e x t r e m e o p p o s i t e o f o p t i o n a. The p a r t i c l e s p e c t r u m a t and beyond 2000 GkV may b e s o r i c h t h a t n o p r e s e n t l y known t h e o r i e s a l l o w u s t o g u e s s t h e d e t a i l s . T h i s may seem t o be j u s t an

"emergency e x i t " f o r t h e o r i s t s b u t t r u l y i t i s a p o s s i b i l i t y t h a t c a n n o t b e r u l e d o u t . Some s p e c u l a t i o n s i n t h i s d i r e c t i o n c a n b e made8.

Consistency conditions for bound s t a t e t h e o r i e s

The i d e a t h a t p r e s e n t l y known "elementary" p a r t i c l e s a r e a c t u a l l y bound s t a t e s composed o f even more " e l e m e n t a r y " c o n s t i t u e n t s , h a s b e e n s p e c u l a t e d upon a l o t . A problem h e r e however i s t h a t a n e x p l a n a t i o n must b e found a s t o why t h e s e bound s t a t e s a r e s o much l i g h t e r t h a n t h e c o n s t i t u e n t mass s c a l e . One t r i e s t o f i n d a symmetry t h a t f o r b i d s a mass t e r m t o d e v e l o p . But h e r e one f i n d s r e s t r i c t i o n s . They a r e t h e s o - c a l l e d "anomaly c o n s i s t e n c y

condition^"^'^.

L e t me e x p l a i n t h e s e . 'One l o o k s a t a l l c u r r e n t s i n t h e c o n s t i t u e n t t h e o r y ( i n p a r t i c u l a r t h o s e t o which n o gauge p a r t i c l e s a r e c o u p l e d ) and computes t h e Adler-Bell-Jackiw anomaly i n t h e i r c o n s e r v a t i o n laws. Now t h e f e r m i o n i c bound s t a t e s t h a t come o u t o f t h e s y s t e m s h o u l d r e p r o d u c e t h e same anomaly. Thus

+

T r TEa Tb Tc) = f T r TIa Tb T c l

e l e m e n t a r v f e r m i o n i c

c o n s c i t u e n c s i o u n d s t a t e s

The s i g n s r e f e r t o t h e h e l i c i t i e s o f t h e f e r m i o n i c p a r t i c l e s . Most r e s e a r c h e r s agree upon t h i s i m p o r t a n t r e s t r i c t i o n .

I f a s t a t e o c c u r s w i t h b o t h h e l i c i t i e s t h e n mass t e r m s w i l l b e p o s s i b l e t h a t l i f t t h e s e s t a t e s , p a i r w i s e , away from t h e s e t o f m a s s l e s s p a r t i c l e s . Such p a i r s do n o t c o n t r i b u t e i n t h e e q u a t i o n .

O r i g i n a l l y i t was hoped t h a t t h e r e would e x i s t s u c h a t h i n g a s a n " e f f e c t i v e L a g r a n g i a n f o r l i g h t bound s t a t e s . A c o n d i t i o n was d e r i v e d 3 ' 1 0 l a t e r c a l l e d " p e r s i s t - e n t mass c o n d i t i o n " , and i t was s u b s e q u e n t l y r e f u t e d l O . L e t me e x p l a i n t h e p h i l o s o p h y benilld t h i s . One c o u l d hope t h a t t h e effective L a g r a n g I a n would be d e t e r m i n e d

G . ' t Hooft C3-757

e s s e n t i a l l y b y s m a l l d i s t a n c e e f f e c t s i n t h e t h e o r y , a n d t h a t i t would be i n s e n s i t i v e t o phase t r a n s i t i o n s . T h e p h a s e i r t o which t h e s y s t e m w o u l d f i n a l l y condense would b e d e t e r m i n e d s o l e l y by t h e v a l u e s of t h e v a r i o u s p a r a m e t e r s i n t h i s L a g r a n g i a n ( f o r i n s t a n c e w h e t h e r o r n o t a n e f f e c t i v e p o t e n t i a l h a s o n l y one o r many d e g e n e r a t e minima). The c h a r a c t e r i s t i c numbers o f h e l i c i t y s t a t e s ( i n d i c e s ) would t h e n b e i n d e p e n d e n t o f t h e v a l u e s o f mass p a r a m e t e r s i n t h e o r i g i n a l L a g r a n g i a n :

" p e r s i s t e n t mass c o n d i t i o n " . U n f o r t u n a t e l y t h i s c a n n o t be t r u e . The two c o n d i t i o n s combined y i e l d b e a u t i f u l l y unique b u t n o n s e n s i c a l l y f r a c t i o n a l i n d i c e s . Dropping t h e p e r s i s t e n t mass c o n d i t i o n i m p l i e s d r o p p i n g t h e n o t i o n o f unambiguous e f f e c t i v e L a g r a n g i a n s and w i t h t h a t t h e hope o f b e i n g a b l e t o make p r e c i s e c a l c u l a t i o n s and p r e d i c t i o n s t h i s way.

Presumably t h e n , t h e p a i r i n g mechanism j u s t mentioned i s n o t j u s t a s m a l l d i s t a n c e e f f e c t . Without t h e s e c o n d c o n d i t i o n one h a s more f r e e d o m t o s p e c u l a t e 1 ' t h e bound s t a t e s p e c t r u m ( w i t h o u t much hope o f v e r i f y i n g s u c h s p e c u l a t i o n s by c a l c u l a t i o n s ) . B u t even s o one f i n d s t h a t t h e a l g e b r a d e s t r o y s most simple-minded

"preon", " r i s h o n " o r " s t r a t o n " models t h a t were p r o p o s e d on t h e b a s i s o f

phenomenology. As a r u l e o f thumb one may e x p e c t t h a t t h e L a g r a n g i a n needed a t t h e more " e l e m e n t a r y " l e v e l i s a b o u t a s complex a s t h e phenomenology one wi'shes t o e x p l a i n : l a r g e gauge groups and p r e o n m u l t i p l e t s seem t o b e n e c e s s a r y . I s a hypercomplex w o r l d u n a v o i d a b l e t h e n ? Who knows.

There a r e many s c a t t e r e d a r e a s i n p a r t i c l e p h y s i c s where s t a n d a r d t h e o r y g i v e s p e r s p e c t i v e s . Or d e s t r o y s them. O r d i n a r y quantum mechanics d i c t a t e s t h a t m a g n e t i c monopoles a r e q u a n t i z e d i n u n i t s o f t h e D i r a c c h a r g e :

T h e r e f o r e t h e e x i s t e n c e o f a monopole w i t h one D i r a c u n i t i s i n c o n s i s t e n t w i t h t h e e x i s t e n c e o f a f r a c t i o n a l e l e c t r i c c h a r g e , b e c a u s e t h e n t h e monopole quantum s h o u l d be m u l t i p l i e d w i t h t h e i n v e r s e o f t h i s f r a c t i o n . ' ' T h e r e f o r e I would c o n c l u d e t h a t a t l e a s t one o f t h e two famous r e c e n t S t a n f o r d e ~ ~ e r i m e n t s ' ~ " ~ must be wrong. Well, t h e r e a r e o t h e r , r a t h e r e x o t i c , ways o u t o f t h i s dilemma. One would b e t h a t t h e o b s e r v e d ( ? ) f r a c t i o n a l c h a r g e i s

also

a m a g n e t i c monopole. That i s b e c a u s e i f two p a r t i c l e s ( 1 and 2) c a r r y m a g n e t i c c h a r g e t h e n ~ i r a c ' s c o n d i t i o n s h o u l d b e r e p l a c e d b

where g. a r e t h e m a g n e t i c c h a r g e s and q . t h e e l e c t r i c c h a r g e s . n l 2 i s a n i n t e g e r . *

~ t i n d a r d t h e o r y c a n s t i l l e v e r y no6 and t h e n make s u r p r i s e d i s c o v e r i e s . One r e c e n t s u r p r i s e i s t h e d i s c o v e r y o f t h e Rubakov e f f e c t i n a monopole15. Another i s w i t t e n ' s d i s c o v e r y t h a t SU(2) gauge t h e o r i e s a r e n o t a l l o w e d t o have a n odd number o f c h i r a l f e m i o n s ^.'6 Note t h a t t h i s i s one more r e a s o n t o b e l i e v e i n t h e e x i s t e n c e o f s u c h a t h i n g a s a t o p q u a r k .

11. F u t u r e p e r s p e c t i v e s o f T h e o r e t i c a l P a r t i c l e P h y s i c s .

The t h e o r e t i c a l s i t u a t i o n today i s c l e a r e r t h a n e v e r . A number o f t h i n g s a r e now a b s o l u t e l y s t a n d a r d , w e l l known and u n d e r s t o o d :

I . We know t h e p r e s c r i p t i o n how t o b u i l d any r e n o r m a l i z a b l e (gauge-) f i e l d t h e o r y 1 7 .

*

H. Georgi p o i n t e d o u t a n o t h e r e x o t i c p o s s i b i l i t y : b o t h t h e m a g n e t i c monopole and t h e f r a c t i o n a l c h a r g e c o u l d b e c o u p l e d t o a n e n t i r e l y unknown new U(I) gauge f i e l d . We t h e n have

where i r e f e r s t o d i f f e r e n t U (I) g r o u p s .

C3-758 JOURNAL DE PHYSIQUE

2. We know how t o p e r f o r m any p e r t u r b a t i v e c a l c u l a t i o n i n s u c h t h e o r i e s . ' ' I n p r a c t i c e t h e y m y b e t e d i o u s b u t we u n d e r s t a n d t h e p r o p e r t i e s o f t h e i n t e g r a l s

i n v o l v e d . There i s even a l i t t l e more t h a n t h a t : i n s t a n t o n e f f e c t s l o o k d i f f e r e n t from p e r t u r b a t i v e e f f e c t s , b u t t h e s a d d l e p o i n t t e c h n i q u e s i n v o l v e d c o u l d b e c a l l e d p e r t u r b a t i v e i n a b r o a d e r s e n s e .

3 . The s e t of r e n o r m a l i z a b l e f i e l d t h e o r i e s i s denumerable.

O t h e r t h i n g s a r e n o t u n d e r s t o o d a t a l l and n e a r l y h o p e l e s s :

I . The v a l u e s o f masses a n d . c o u p l l n g c o n s t a n t s a r e n o t p r e d i c t e d . P r e c i s e l y b e c a u s e p e r t u r b a t i o n t h e o r y i s u n d e r s t o o d s o w e l l we know t h a t t h e r e i s n o t h e o r e t i c a l l i m i t a t i o n . Any " t h e o r y " c l a i m i n g t o u n d e r s t a n d mass- and c o u p l i n g c o n s t a n t r e l a t i o n s must be b a s e d on d u b i o u s arguments o f e s t h e t i c s . The l o g i c i s n e v e r c o n v i n c i n g . C o n s t r u c t i n g bound s t a t e t h e o r i e s seems t o b e o f l i t t l e h e l p b e c a u s e , a s s t a t e d , t h e u n d e r l y i n g t h e o r i e s a r e n o t s i m p l e r t h a n t h e phenomenology t h e y a t t e m p t t o e x p l a i n s o t h a t t h e y c o n t a i n a s many f r e e p a r a m e t e r s a s t h e r e a r e un- knowns.

2. How t o p e r f o r m r e l i a b l y n o n - p e r t u r b a t i v e c a l c u l a t i o n s . T h i s i s n o t t o s a y t h a t p r e s e n t l y p o p u l a r and w i d e l y p r a c t i c e d methods, s u c h a s t h o s e o f r e f s 19) and 20) o r t h e l a t t i c e Monte-Carlo p r o c e d u r e s 2 1 , would n o t b e o f enormous v a l u e . The p o i n t i s t h a t t h e v i n v o l v e s ~ e c u l a t i o n s t h a t one h a s t o b e l i e v e on f a i t h . Does t h e continuum l i m i t r e a l l y e x i s t and r e p r o d u c e QCD? We would l i k e t o have m a t h e m a t i c a l l y r i g o r o u s methods w i t h o u t t h e s e o r even more d u b i o u s a s s u m p t i o n s .

3 . IJe a r e s t i l l v e r y f a r away from any quantum t h e o r y o f g r a v i t y .

So t h e r e i s a wide gap between t h e c o m p l e t e l y u n d e r s t o o d and t h e a p p a r e n t l y h o p e l e s s l y i m p o s s i b l e . I f t h e r e a r e p e r s p e c t i v e s i n " T h e o r e t i c a l Theory" t h e n t h a t s h o u l d b e i n f i n d i n g l i t t l e i s l a n d s i n t h e gap t h a t some t i m e may s e r v e a s

f o u n d a t i o n s f o r a b r i d g e . I f h i s t o r y i s a g u i d e , t h e n t h e v a r i o u s u n u n d e r s t o o d problems may w e l l b e a l l c o n n e c t e d .

The l a t t i c e . UniversaZity

The l a t t i c e h a s b e e n d i s c u s s e d i n s e v e r a l o t h e r s e s s i o n s a t t h i s c o n f e r e n c e 2 1 . Here I m a i n l y wish t o s t r e s s t h a t t h e r e i s a f u n d a m e n t a l u n d e r l y i n g a s s u m p t i o n when t h e s e methods a r e a p p l i e d t o quantum f i e l d t h e o r y . A l a t t i c e L a g r a n g i a n e s s e n t i a l l y o f t h e form

E = I ~r

[wu'u'] ,

3

J a q u e t t e s

where U a r e u n i t a r y m a t r i x v a r i a b l e s d e f i n e d on t h e l a t t i c e l i n k s , i s c o n s i d e r e d on a l a t t i c e w i t h l a t t i c e s p a c i n g s o f l e n g t h a . I n t h e "continuum l i m i t " i t s h o u l d r e p r o d u c e t h e L a g r a n g i a n

I a l +

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4 G,,"

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i f t h e i d e n t i f i c a t i o n

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^(x)]

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i s made. The t h e o r y i s " r e g u l a r i z e d " w i t h a c u t - o f f M

"

] / a .

Now a c c o r d i n g 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 t h e l i m i t a C o ; g + 0 must be t a k e n s u c h t h a t

G. ' t Hooft

w i t h A (= l a t t i c e lambda p a r a m e t e r ) k e p t f i x e d . Does t h i s limit e x i s t ? I f

p e r t u r b a t i o n e x p a n s i o n i s performed w i t h r e s p e c t t o g 2 , o r r a t h e r a r u n n i n g c o u p l i n g c o n s t a n t g 2 ( u ) t h e n e v e r y t h i n g s e e m t o b e i n o r d e r . But t h i s e x p a n s i o n i s

f u n d a m e n t a l l y d i v e r g e n t . We do n o t know w h e t h e r t h e e x p a n s i o n u n i q u e l y d e t e r m i n e s t h e t h e o r y . So maybe t h e l i m i t does n o t e x i s t , o r maybe i t depends on t h e d e t a i l s of t h e l a t t i c e ( c u b i c , t r i a n g u l a r , body-centered c u b e s , e t c ) . The u n i q u e n e s s o f t h e l i m i t ( f o r s u b t r a c t e d , g a u g e - i n v a r i a n t G r e e n ' s f u n c t i o n s ) i s c a l l e d " u n i v e r s a l i t y "

and i f u n i v e r s a l i t y i n d e e d h o l d s t h e n t h e l a t t i c e L a g r a n g i a n i s a good way t o d e f i n e t h e p o p u l a r t h e o r y c a l l e d "quantum-chromodynamicsl'. I f i t h o l d s t h e n i t s h o u l d b e p o s s i b l e t o p r o v e t h i s . And t h e n i t s h o u l d and p r o b a b l y w i l l b e p o s s i b l e t o c a l c u l a t e h a d r o n i c f e a t u r e s w i t h g r e a t a c c u r a c y .

But what i f u n i v e r s a l i t y does n o t h o l d ? I f d i f f e r e n t l a t t i c e s g i v e d i f f e r e n t p r o t o n m a s s - t o - s t r i n g c o n s t a n t r a t i o s (I am n o t t a l k i n g about 1% b u t a b o u t

d i s c r e p e n c i e s o f t h e o r d e r o f l o o % ) ? T h i s may s t i l l b e t h e c a s e b e c a u s e u n i v e r s a l i t y h a s n o t y e t been checked a c c u r a t e l y ( a l t h o u g h a t t e m p t s a r e u n d e r way22. T h i s would be a h i g h l y i n t e r e s t i n g s i t u a t i o n b e c a u s e i t would imply t h a t h a d r o n i c p r o p e r t i e s a r e s t i l l d e t e r m i n e d by a k i n d of p h y s i c s t h a t we do n o t u n d e r s t a n d . I come b a c k t o t h i s l a t e r .

Quenching

An i n t r i g u i n g o b s e r v a t i o n h a s been made by Eguchi and ~ a w a i ' ~ , l a t e r c r i t i c i s e d and r e f i n e d , a l s o on t h e b a s i s of e a r l i e r work24. Suppose one c o n s i d e r s t h e

e x p e c t a t i o n v a l u e of a Wilson l o o p i n a c l o s e d box. Now we c o u l d d i v i d e t h e b o x i n t o s m a l l e r boxes w i t h a d d i t i o n a l p e r i o d i c boundary c o n d i t i o n s . T h e Wilson l o o p e x p r e s s i o n t h e n compares w i t h an e x p r e s s i o n f o r a l o o p f o l d e d up i n s i d e one o f t h e s m a l l e r boxes. But i t s t o t a l a r e a may n o t have changed. And i n SU(N) t h e o r i e s , w i t h N l a r g e , t h e s u r f a c e spanned by t h e l o o p does n o t i n t e r a c t w i t h i t s e l f . So one may e x p e c t t h a t t h e Wilson l o o p e x p r e s s i o n w i l l n o t have a l t e r e d . T h i s i s what Eguchi and Kawai f i n d , working on a l a t t i c e . They r e d u c e t h e s y s t e m t o t h e extreme:

a Ix lx lx l l a t t i c e .

A t f i r s t s i g h t t h i s l o o k s l i k e an enormous s i m p l i f i c a t i o n , b r i n g i n g t h e e x a c t s o l u t i o n of SU(-) gauge t h e o r i e s w i t h i n range. U n f o r t u n a t e l y t h a t i s p r o b a b l y j u s t an i l l u s i o n . F o r one t h i n g : one s h o u l d n o t f o l d t h e T J i l s o n ' l o o p t o o many t i m e s b e c a u s e t h e n t h e number o f i n t e r s e c t i o n s might b e a t t h e damping 1 1 ~ 2 f a c t o r f o r t h e i n t e r a c t i o n s . I n a d i f f e r e n t f o r m a l i s m i t was found t h a t t h e energy-momentum

o p e r a t o r P,, c a n b e r e w r i t t e n a s a n NxN m a t r i x , which a f t e r d i a g o n a l i z a t i o n r e a d s :

But i n a volume V = (Ax) 4 we want a s e t of e i g e n v a l u e s s e p a r a t e d by d i s t a n c e s Ap = h/Ax. So t h e number of e i g e n v a l u e s r e q u i r e d i n c r e a s e s w i t h ( 1 1 ~ ~ ) ~ ^a V. Thus t h e u n f o l d e d volume V c a n n o t become l a r g e r t h a n N. A p p a r e n t l y a l l space-time dependent s t r u c t u r e o f t h e t h e o r y h a s been t r a d e d f o r an N dependent s t r u c t u r e . F o r computing l a r g e Wilson l o o p s one r e a l l y n e e d s tremendously l a r g e N v a l u e s , i f quenching i s used. These l i m i t s may now b e v e r y h a r d t o perform. So a l t h o u g h t h e s e q u e n c h i n g methods g i v e i n t e r e s t i n g r e l a t i o n s between l a r g e N and l a r g e V e f f e c t s , i t i s q u i t e p r e p o s t e r o u s t o s u g g e s t t h a t t h e N +

-

t h e o r y i n t h e continuum i s made any e a s i e r t h i s way25.

As s t a t e d e a r l i e r , quantum chromodynamics i s p e r h a p s n o t a p e r f e c t t h e o r y , i f u n i v e r s a l i t y does n o t h o l d . I would l i k e t o p u t t h i s s i t u a t i o n i n a h i s t o r i c a l p e r s p e c t i v e .

JOURNAL DE PHYSIQUE

1. I n t h e b e g i n n i n g t h e r e was Lagrange f i e l d t h e o r y . Any L a g r a n g i a n was a s good any o t h e r , a s f a r a s t r e e d i a g r a m s were c o n c e r n e d . I f t h e c o u p l i n g c o n s t a n t s a r e s m a l l enough t h e n s u c h a t h e o r y i s r e a s o n a b l y a c c u r a t e . An example i s t h e 4-fermion t h e o r y f o r t h e weak i n t e r a c t i o n s .

2 . I f t h e c o u p l i n g i s a s l a r g e a s 11137 t h e n h i g h e r o r d e r e f f e c t s become w o r t h w h i l e . Now we w i s h t h a t s u c h t h e o r i e s a r e r e n o r m a l i z a b l e . Gauge t h e o r i e s f o r weak and e l e c t r o m a g n e t i c i n t e r a c t i o n s were i n d e e d shown t o s a t i s f y t h i s r e q u i r e m e n t . A v e r y i m p o r t a n t r e s u l t was o b t a i n e d by Cornwall e t a l l i n 1973: e r e n o r m a l i z a b l e f i e l d t h e o r i e s a r e of t h e gauge t h e o r y t y p e 2 6 .

3 . A t s t i l l s t r o n g e r c o u p l i n g s o n e might worry of f e a t u r e s such a s t h e Landau g h o s t . A t h i g h e n e r g i e s accumulated i n t e r a c t i o n s blow up and c e a s e t o make s e n s e . T h i s i s why i t i s s o i m p o r t a n t t h a t some gauge t h e o r i e s , i n p a r t i c u l a r QCD, a r e a s y m p t o t i c a l l y

free:

a t h i g h e n e r g i e s t h e a c c u m u l a t e d i n t e r a c t i o n s c a n c e l .

And now p e r h a p s we r e a c h s t a g e 4: a s y m p t o t i c freedom i s s t i l l n o t good enough i f we c a n n o t p r o v e u n i v e r s a l i t y . I n t h a t c a s e t h e f u t u r e may show " a s y m p t o t i c u n i v e r s a l i t y "

o r " a s y m p t o t i c convergence" meaning t h a t t h e r a t e o f convergence o f p e r t u r b a t i o n e x p a n s i o n grows r a p i d l y w i t h e n e r g y . 2 7 S t a g e 5 , s t i l l f u r t h e r i n t h e f u t u r e , may f i n a l l y g i v e us a c o n v e r g e n t f i e l d t h e o r y . T h i s i s a f i e l d t h e o r y t h a t i s

m a t h e m a t i c a l l y a s r i g o r o u s a s p r e s e n t l y known " c o n s t r u c t i v e f i e l d t h e o r i e s " i n 2 o r 3 space-time d i m e n s i o n s z 8 .

My own work now p o i n t s t o w a r d s a c o n j e c t ~ r e ~ ~ : SU(N -+ m) gauge t h e o r i e s may be m a t h e m a t i c a l l y w e l i d e f i n e d . U n i v e r s a l i t y h o l d s i n t h e N + l i m i t . As y e t we c l a i m t h e f o l l o w i n g r e s u l t 2 9 : i n a v a r i a n t of SU(m)gauge t h e o r y w i t h maqsive g l u o n s o n l y ( a l l masses

>

^{m )} and weak enough c o u p l i n g c o n s t a n t s ,$N (gcr't)2 , t h e Dyson-Schwinger e q u a t i 8 n s have a unique s o l u t i o n t h a t can be o b t a i n e d t h r o u g h a n i t e r a t i o n p r o c e d u r e . So we do have a " c o n s t r u c t i v e f i e l d t h e o r y i n f o u r dimensions"

a l t h o u g h i t i s s t i l l i n an u n p h y s i c a l l i m i t : t h e number o f " c o l o r s " i s s t r i c t l y i n f i n i t e . Two t e c h n i c a l p r o c e d u r e s a r e u s e d :

one i s t h e s k e l e t o n e x p a n s i o n . We w r i t e c l o s e d b l o b s r e p r e s e n t i n g t h e sum o f diagrams f o r t h e d r e s s e d p r o p a g a t o r s , d r e s s e d i r r e d u c i b l e 3-point v e r t i c e s and d r e s s e d i r r e d u c i b l e 4-point v e r t i c e s ( F i g u r e 1) c a l l e d e l e m e n t a r y Green's f u n c t i o n s .

G. ' t Hooft

The h i g h e r ("composite") Green's f u n c t i o n s such a s t h e i r r e d u c i b l e 5-point function, a r e expressed as sums of diagrams c o n t a i n i n g only d r e s s e d p r o p a g a t o r s , 3-point and 4-point v e r t i c e s . Because N + a l l t h e s e diagrams a r e p l a n a r . A c r u c i a l theorem now i s t h a t i f t h e elementary Green's f u n c t i o n s remain small enough

( g 2 ~

<

(gcrit)2) then t h e sum of t h e "skeleton graphs" f o r t h e composite 5 p o i n t f u n c t i o n converges. Powerful mathematical theorems of r e f . ^O) can be a p p l i e d h e r e . Secondly we use d i f f e r e n c e e q u a t i o n s (Figure 2 ) .

The d i f f e r e n c e between t h e v a l u e s f o r elementary Green f u n c t i o n s a t d i f f e r e n t e x t e r n a l momenta i s given by a Green f u n c t i o n with one e x t r a e x t e r n a l l e g . A f t e r d i f f e r e n t i a t i n g a few times t h i s way we g e t f i v e l e g s and f o r t h o s e diagrams again t h e s k e l e t o n expansion can be w r i t t e n down.

The s e t ( d i f f e r e n c e e q u a t i o n s + s k e l e t o n expansions) determines t h e Green f u n c t i o n s completely, up t o " i n t e g r a t i o n c o n s t a n t s " which r e p l a c e t h e b a r e parameters of t h e theory. No bare parameters o c c u r e x p l i c i t l y . O n e can s o l v e t h e s e e q u a t i o n s by an i t e r a t i o n procedure which under t h e c o n d i t i o n s I mentioned can be shown t o converge! It was important t o use d i f f e r e n c e e q u a t i o n s , n o t d i f f e r e n t i a l e q u a t i o n s i n o r d e r t o avoid some bothersome i n f r a r e d problems. At some p o i n t t h e d i f f e r e n c e equation looks l i k e t h e o r d i n a r y r e n o r m a l i z a t i o n group e q u a t i o n i n Gell- Mann-Low form:

I t i s i n s t r u c t i v e t o observe t h a t i f i n t h i s e q u a t i o n we i t e r a t i v e l y determine f3(g2) from g 2 ( 1 p l ) and then i n t e g r a t e t o r e o b t a i n g 2 ( 1 p l ) , then t h i s i t e r a t i o n converges i f we s t a r t w i t h i n c e r t a i n l i m i t s . We should mention t h a t t h e above d e r i v a t i o n s a r e c e r t a i n l y r e l a t e d t o , o r r a t h e r mathematically p r e c i s e

implementations o f , more i n t u i t i v e and p r a c t i c a l arguments p r e s e n t e d i n r e f 3 1 .

Science Fiction

The a u t h o r s p e c u l a t e d t h a t t h e only theory f o r the world t h a t i s indeed mathematically unique, with p r e d i c t i v e power, i s SU(N) with N -+ a s t h e energy E One can w e l l imagine such a system. A t low e n e r g i e s we have SU(3) of c o l o r . A t some energy s c a l e t h i s t u r n s o u t t o be SU(4) broken down t o SU(3) by some Higgs mechanism. The s t o r y r e p e a t s e n d l e s s l y a t h i g h e r e n e r g i e s . This way one can hope t o o b t a i n an I / N expansion t h a t i s " a s y m p t o t i c a l l y f r e e " , with N i n c r e a s i n g l i n e a r l y with energy. In t h a t c a s e t h e r e a r e some hopes t h a t t h e 1 / N expansion becomes Bore1 s u m a b l e . Indeed such a theory would have an immensely complex s t r u c t u r e a t h i g h e r e n e r g i e s so it would r e p r e s e n t o p t i o n d.

JOURNAL DE PHYSIQUE

N e e d l e s s t o s a y t h a t t h e s e w e r e p h a n t a s i e s more l i k e l y t o b e wrong t h a n r i g h t . The main message i s t h a t we s h o u l d t r y t o f i n d b e t t e r model b u i l d i n g methods t h a n j u s t " t a k e a gauge t h e o r y . . . " , a n d i n d e e d t h e r e s h o u l d b e c o m p e l l i n g m a t h e m a t i c a l a r g u m e n t s f o r them. P r e s e n t l y p r o p o s e d t h e o r i e s a r e o f t e n b a s e d j u s t o n a r g u m e n t s o f e s t h e t i c s a n d t h o s e h a v e h i s t o r i c a l l y a l w a y s shown t o b e m i s l e a d i n g . Only on h i n d s i g h t , y e s , s u c c e s s f u l t h e o r i e s o f N a t u r e a l w a y s t u r n o u t t o b e b e a u t i f u l , b u t o f t e n i n some u n p r e d i c t a b l e way. I n c o n c l u s i o n 3 2 : we s h o u l d n o t p r i m a r i l y a t t e m p t t o u n i f y p a r t i c l e s a n d f i e l d s . W e s h o u l d u n i f y knowledge.

REFERENCES

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2. VELTMAN M . , Phys. L e t t . (1980) 95.

3. HOOFT G. ' t . , i n "Recent Developments i n Gauge t h e o r i e s " , C a r g z s e 1979, G. ' t H o o f t e t a 1 e d s . (Plenum, New York a n d London) p. 135.

4 . VELTMAN M., A c t a Phys. P o l . B12 (1981) 437.

5 . PAT1 J.C., a n d A. S A L W ~ h ~ c R e v . D8(1973) 1240.

GEORGI H., a n d GLASHOW S.L., P h y s . ~ e c ~ e t t . z ( 1 9 7 4 ) 438.

GEORGI H., QUINN H. and WEINBERG S . , P h y s . R e v . L e t t .

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B a s e l , S t u t t g a r t , 1981 6. DIIIOPOULOS S. a n d GEORGI H., H a r v a r d p r e p r i n t HUTP-82/AO 46, J u l y 1982.

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G. ' t Hooft

18. HOOFT G. ' t a n d VELTMAN M., "DIAGRAMi?AR", CERN r e p o r t 73-9, r e p r i n t e d i n

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Discussion

E.

BREZIN (Saclay).- The large-N l i m i t always assumes that N goes t o i n f i n i t y faster than a l l other parameters such as the uolwne. This takes p h c e

for

i n s t m c e i n the

O I N )

-GmodeZ i n the master fieZd approach, however it does yield the correct conti- nuum solution.

G .

' T HO0FT.- The question r e f e r s t o my comnent on t h e Eguchi-Kawai theorem. I e n t i - r e l y agree.

I

simply conclude t h a t t h i s f a c t implies t h a t when large Wilson loops a r e considered one has t o choose between large

N

o r large

V

so t h a t the

E-K

theorem may be not so powerful in practice f o r computational purposes.

C. CALLAN

(Princeton Univ.) .- Some people (Tornboulis, i particular)

have proposed

t o m e the sicknesses of quantum g m v i t g by adding an

R'

t e r n t o the Einstein action thereby improving u l t r a v i o l e t convergence. What do you think of these schemes

?

G. _{' T} H0OFT.- This i s one of many theories i n which space-time is kept continuous.

My

problem i s then always the same

:

what does the system look l i k e a t distance scales much smaller than t h e Planck mass inverse

?

I t does not seem t o make sense.

Perturbation-ion may be convergent term by term

b u t

diverges when summed.

JOURNAL DE PHYSIQUE

C.

TTZYXSON (Saclay).- This i s a questwn,not a conunent. WouZd your conjecture o r theorem o f t h e consistency o f t h e large N-gauge theory imply t h a t f i n i t e N-gauge t h e o r i e s are i n c o n s i s t e n t

?

G.

'T H0OFT.- I wish

I

knew. The problem i s t h a t t h e s k e l e t o n expans.ion f o r t h e 5- p o i n t f u n c t i o n i s guaranteed t o diverge. E i t h e r one w i l l f i n d Bore1 and i n s t a n t o n methods t o improve p e r t u r b a t i o n theory o r prove t h a t i t cannot be done.

J .

ILIOPOULOS (ENS).- How are t h e

2-,3- and 4-

point bubbles t o be detomnined

?

Are they a r b i t r a r y parameters of t h e theory

?

G. ' T HO0FT.- The a r b i t r a r y parameters come i n t h e boundary c o n d i t i o n s a t i n f i n i t e (Euclidean) energy. The 2-, 3- and 4- p o i n t f u n c t i o n s come from i t e r a t i v e l y i n t e g r a - t 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 - l i k e equations.

M.

CAHILL (Univ. o f New Me3sicoJ.- Are you p e s s i m i s t i c about supergravity

?

And i f so, why

?

G. ' T H9OFT.- The answer i s t h e same as I gave t o Callan. P e r t u r b a t i o n expansion may be f i n i t e term by term, b u t diverges beyond t h e Planck-length.

E.

BREZIN (Sackzyl.- Are your i n t e g r a l equations t h e large N-limit o f t h e Migdal- Polyakov bootstrap

?

G. ' T HI)OFT.- They a r e s i m i l a r b u t I t r y t o prove r i g o r o u s l y t h a t they converge (under l i m i t e d c o n d i t i o n s ) . They seem t o r e l y on Pad6 techniques.

L.

MAIANI (Univ. of Romal.- There are i n d i c a t i o n s t h a t some extended super-synrmetric t h e o r i e s are f i n i t e . Could you make some conunent whether t h e s e t h e o r i e s could provi- de an a l t e r n a t i v e t o the SU(hl=-l theory

?

G.

' T HO0FT.- They would be f i n i t e o r d e r by order. F o r p e r t u r b a t i o n expansion i t s e l f t o converge you need p l a n a r i t y . I n t h e equations I use a l l i n t e g r a l s converge, b u t t h e i n d i c a t i o n s you mentioned (!4andelstam1s t a l k a t t h i s conference) c o u l d i m p l y t h a t t h e N + . . l i m i t o f such t h e o r i e s perhaps converges w i t h i n some r a d i u s o f conver- gence w i t h o u t c e q u i r i n g , as I do.asymptotic freedom.