QUANTUM FIELD THEORY AND STATISTICAL MECHANICS

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QUANTUM FIELD THEORY AND STATISTICAL MECHANICS

E. Brézin

To cite this version:

E. Brézin. QUANTUM FIELD THEORY AND STATISTICAL MECHANICS. Journal de Physique

Colloques, 1982, 43 (C3), pp.C3-743-C3-753. �10.1051/jphyscol:1982386�. �jpa-00221942�

JOURNAL DE PHYSIQUE

Colloque Ci, supplément cm n° 12, Tome 43, décembre 1982 page C3-743

Q U A N T U M F I E L D T H E O R Y A N D S T A T I S T I C A L M E C H A N I C S

E. B r é z i n

DPhT, Orme des Merisiers,, CEN Saalay, B.P. n° 2, 91191 Gif-sur-Yvette, France

Résumé. - Les champs de jauge sur réseau ont été étudiés de manière exhaustive par des simulations numériques. Mais dans ce rapport l'accent sera mis sur des méthodes analytiques, fréquemment inspirées de la Mécanique Statistique. Plusieurs résultats sont récents et il semble que ces méthodes soient appelées à jouer un rôle plus im- portant pour la compréhension de la Chromodynamique quantique. Néanmoins la plupart des sujets abordés dans cet exposé sont très classiques et ce rapport n'est certaine- ment pas destiné aux spécialistes.

Abstract. - Lattice gauge theories have been extensively studied by Monte-Carlo simu- lations. In this talk some analytic methods (frequently inspired by statistical me- chanics) are presented. Several results are quite recent and I believe that these methods should play an increasing role in the understanding of QCD. However most of the material is completely standard and this report should not be read by specialists.

Introduction. - Connections between quantum field theory and statistical mechanics have existed for many years. The so-called second quantized formalism, Green func- tions techniques, Feynman diagrams etc... have been widely used in both fields.

However it is only after the 70's that the two subjects became extremely similar, and it is only slightly exaggerated to say that renormalization theory is part of the theory of critical phenomena. Two basic steps led to this unification. Firstly the imaginary time (Euclidean) formalism [1] brings the quantum mechanical amplitude exp(iS/H) under the form of a Boltzmann probability weight exp-(6H), a trivial, but in practice, very important fact. Second, the existence of a renormalized quantum field theory, i.e. a theory in which the physical observables are expressed in terms inde- pendent of the intermediate regularizing length scale (or inverse cut-off), is very similar to that of a critical point in condensed matter physics [2].

These resemblances are useful in both directions. Many quantum field theory techniques, such as the renormalization group formalism, have been particularly use- ful in statistical mechanics [3] but, in spite of the fact that it is a beautiful topic (it leads to qualitative and even accurate quantitative predictions) the empha- sis, for this particle physics conference will be on the opposite direction : we shall discuss some of the ideas and of the techniques which emerged out of statistical me- chanics and led to applications to particle physics. Thus we shall be dealing mostly with analytic approaches to lattice gauge theories. Indeed the lattice formulation in 1973 by K. Wilson [4] of gauge theories is an important and natural idea which led to many developments, in particular to remarkable numerical simulations by the Monte- Carlo techniques (reviewed by C. Rebbi in this conference). It should be stated again that the elementary length scale introduced in this formulation is a powerful techni- cal device, which allows one to control the strong coupling limit, or to perform numerical simulations for instance. It has nothing to do with the deep question of a possible fundamental length scale in space-time. A major issue, a "sine qua non"

condition for the practical usefulness of lattice gauge theories was the possibility of introducing fermions in the formalism, and in particular in a numerical simulatioa This has been achieved during the last year [5] and this important break-through will, I believe, lead to many developments. (See C. Rebbi's and Martinelli's talks at this conference).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982386

JOURNAL DE P H Y S I Q U E

There a r e many important and new t o p i c s r e l a t e d t o my s u b j e c t which I should have reviewed i n t h i s t a l k . More than t h e t r a d i t i o n a l l a c k o f time argument, I t h i n k t h a t they are so important t h a t they deserve a more q u a l i f i e d e x p o s i t i o n than the one I could provide. I t i s very u n f o r t u n a t e f o r i n s t a n c e t h a t A. Polyakov has n o t been a u t h o r i z e d t o present a t t h i s conference h i s new s t a t i s t i c a l t h e o r y o f random s u r f a - ces [61, which leads t o a completely new understanding o f t h e o l d dual models, and resolves t h e i r d i f f i c u l t i e s away from 26 o r 10 dimensions (see the t a l k by A. Nevw a t t h i s conference). T h ~ i e l s e n ~ N i P o m i ~ ~ doublina theorem 171 which, a t l e a s t on a l a t t i c e , i m p l i e s t h e existence o f an equal number o f l e f t and right-handed Weyl f e r - mions, i s m a n i f e s t l y an extremely deep and important c o n s t r a i n t . The problem r a i s e d by Bhanot and Dashen [8] i n which t h e u n i v e r s a l i t y of t h e continuum l i m i t o f various l a t t i c e gauge a c t i o n s i s questioned i s important, though I b e l i e v e t h a t t h e observa- t i o n s o f S. Samuel and o f B. Grossman [ 9 ] a r e r e l e v a n t . F i n i t e temperature QCD and the issue of t h e deconfinement and c h i r a l symmetry r e s t o r a t i o n temperatures i s impor- t a n t and Monte-Carlo r e s u l t s a r e q u i t e useful [I01 b u t I had u n f o r t u n a t e l y t o i g n o r e a1 1 these t o p i c s .

F i n a l l y , t h i s t a l k w i l l be a survey o f some o f t h e non-perturbative, non Monte- Carlo methods i n gauge theories. The f i r s t p a r t c o n s i s t s simply o f t h e t r a d i t i o n a l connection between quantum f i e l d t h e o r y and c l a s s i c a l s t a t i s t i c a l mechanics a t a c r i t i c a l p o i n t . I n t h e second p a r t we s h a l l l o o k a t some o f t h e s t r o n g c o u p l i n g r e s u l t s . We w i l l present then t h e mean-field ideas, and t h e expansion ( i n f a c t i n powers o f t h e i n v e r s e of dimension of space-time) around t h i s p i c t u r e ;some r e c e n t r e s u l t s on t h e mass spectrum w i l l be reviewed. F i n a l l y we w i l l discuss some new and b e a u t i f u l ideas on t h e l a r g e N l i m i t o f an SU(N) gauge t h e o r y which may l e a d t o f u t u r e developments and m y j u s t i f y t h e use o f small l a t t i c e s i n numerical simula- t i o n s .

1. Quantum f i e l d theory and c l a s s i c a l s t a t i s t i c a l mechanics i n t h e neighbourhood o f a c r i t i c a l p o i n t

This connection i s so well.-known t h a t we s h a l l very b r i e f l y r e c a l l t h e main f e a t u r e s . For s i m p l i c i t y o f t h e n o t a t i o n s we s h a l l discuss i t f o r a s e l f - c o u p l e d r e a l s c a l a r f i e l d @, w i t h the a c t i o n

I n Feynman's path i n t e g r a l f o r m u l a t i o n amplitudes are given by the sum over a l l f i e l d c o n f i g u r a t i o n s i n space and t i m e

I

e x p

b

^{s { m )}

{@I

A f t e r Wick's r o t a t i o n t

*

ix4

and vacuum e x p e c t a t i o n values become s t a t i s t i c a l averages

1

A ( @ ) e x p - B H ( @ )

<O

1

^{A ( @ )}

1

^{0, =}

{'I

L'

A ( @ )

{@I

This very simple r e f o r m u l a t i o n i s a t t h e h e a r t o f many new techniques i n quantum f i e l d theory. One can i n v e n t f o r i n s t a n c e a f i c t i t i o u s time, a f i f t h dimension i n some sense, i n which t h e c o n f i g u r a t i o n s w i l l evolve and approach t h e " e q u i l i b r i u m "

prescribed by the Boltzmann weight e x p - B H ; t h i s i s the basis o f Monte-Car10 simu- l a t i o n s [ l l ] o r o f t h e r e c e n t s t o c h a s t i c approaches t o quantum theory [12]. The p o s i t i v i t y of the p r o b a b i l i t y weight may be used i n many ways ; i t leads t o

v a r i a t i o n a l c a l c u l a t i o n s o r t~ mean f i e l d ideas. Thsre :'s a whole d i c t i o n a r y o f t r a n s l a t i o n s from one f i e l d t o the o t h e r . For i n s t a n c e a change o f vacuum f o r a p o t e n t i a l v(@) o f f i g . 1, from t h e 4 = 0 t o t h e @ =

+e

vacuum i s a ( f i r s t - o r d e r )

vr01 A

phase trans1 t i o n . The a c t u a l t i m e develop- ment o f t h i s change o f uacuunl, which was s t u d i e d i n p a r t i c u l a r by Callan and Coleman [I31 i s very s i m i l a r t o the theory o f n u c l e a t i o n o f a bubble o f 1 iq u i d i n a super- cooled vapor [14]. A more extended d i c - t i o ~ r y may be found i n P a r i s i ' s c o n t r i -

-

b u t i o n [15] t o t h e 1980~ Madison Conference.

@ F i g . 1

ReguLarization and renorrnaZization

A quantum f i e l d theory cannot be constructed w i t h o u t r e g u l a r i z a t i o n o f t h e s h o r t - distance s i n g u l a r i t i e s . Many ways have been proposed i n p r a c t i c e , from simple mo- mentum c u t - o f f A t o a minimum d i s t a n c e as i n a l a t t i c e r e g u l a r i z a t i o n b u t they a l l i n v o l v e some i n t e r m e d i a t e small d i s t a n c e a, the l a t t i c e spacing o r the i n v e r s e momen- tum c u t - o f f A-1. The renormal i z e d theory ( i f t h e theory i s r e n o m a l i z a b l e ) i s t h e 1 imi t i n which S-matrix elements o r Green f u n c t i o n s a r e expressed i n terms o f physi- c a l parameters, masses, c o u p l i n g constants, momenta o r r e l a t i v e distances which are h e l d f i x e d when t h e r e g u l a r i z i n g l e n g t h a i s decreased t o zero. Therefore t h e compa- r i s o n between renormalized f i e l d theory and t h e s t a t i s t i c a l mechanics viewpoint may be expressed schematically as

w i t h a = A-' and

5

the c h a r a c t e r i s t i c c o r r e l a t i o n l e n g t h i s i d e n t i f i e d t o t h e i n v e r s e o f t h e mass. A simple example which i l l u s t r a t e s t h i s connection i s given by t h e F o u r i e r t r a n s f o r m o f t h e f r e e propagator

Renormalized f i e l d theory masses m << A

momenta p << A

Therefore i n t h e renormalized f i e l d theory c o r r e l a t i o n s extend t o i n f i n i t y (oom- pared t o a ) . This well-known and obvious f a c t i s nevertheless somewhat s u r p r i z i n g . Indeed i n t h e p r o b a b i l i t y weight

exp

- 1

d4xrapq,m+v(m)

I

S t a t i s t i c a l Mechanics near a c r i t i c a l point c o r r e l a t i o n length

6

>> a r e l a t i v e d i s t a n c e s r >> a

( i ) t h e p o t e n t i a l term [exp-( v d x ) ] does n o t couple t h e values o f 4 a t d i f f e r e n t space-time p o i n t s ; i f the k i n e t i

1

term were n o t present + ( x ) and @(y) would be inde- pendent random v a r i a b l e s and the ranqe o f c o r r e l a t i o n s would be zero.

1

²

( i i ) The k i n e t i c term [exp-(

(v@)

dx] i s q u a s i - l o c a l

.

I t couples t h e f i e l d s @ a t neighbouring p o i n t s . I t i s manif s t i n a l a t t i c e r e g u l a r i z a t i o n i n which

a _u +a

^@

u

stands f o r (b(x*ae:)-@(x)f and nearest neighbovrs o n l y are coupled.

Therefore t h e range o f i n t e r a c t i o n s i s o f o r d e r a, and i n general f o r an a r b i - t r a r y choice of the (bare) c o u p l i n g constants t h e c o r r e l a t i o n s do no extend f u r t h e r

C3-746 JOURNAL DE PHYSIQUE

than t h e range o f i n t e r a c t i o n s . The physical requirement o f r e n o r m a l i z a t i o n

6

>> a i s p o s s i b l e o n l y i f t h e c o u p l i n g constants a r e adjusted i n o r d e r t o be i n the v i c i - n i t y o f a c r i t i c a l p o i n t ( a t which a continuous phase t r a n s i t i o n takes place w i t h a d i v e r g i n g c o r r e l a t i o n l e n g t h ) .

L e t us consider f o r example W i l s o n ' s f o r m u l a t i o n [ 4 ] o f l a t t i c e gauge t h e o r i e s

p H = - 1

ut

)

2

1

ReTr(Ux,u Ux+,J,v Ux+",p X , L ,

g B p l a q u e t t e s

One can c a l c u l a t e

5

by l o o k i n g f o r instance a t c o r r e l a t i o n f u n c t i o n s o f small Wilson loops

(5

would be simply t h e i n v e r s e o f t h e g l u e b a l l mass), and t h e r e s u l t would be f o r

5

some f u n c t i o n For an a r b i t r a r y choice o f g$.

5

remains o f t h e o r d e r o f t h e i n t e r a c t i o n r a n e a, t h e p l a q u e t t e s i z e . Thus t h e r e i s no physics

9

o u t s i d e t h e c r i t i c a l values o f gg, provided they e x i s t , a t which

5

diverges. I t i s be1 ieved, and supported by numerical simulations, t h a t f o r four-dimensional QCD E/a i s l a r g e f o r small values o f g2 only. For small i t i s a consequence o f asymptotic freedom t h a t C/a i s l a r g e . 1nd8ed t h i s f o l l o w s from t h e r e n o r m a l i z a t i o n group equa- t i o n s ; l e t us r e c a l l here simply t h a t these equations describe t r a j e c t o r i e s o f para- meters o f t h e theory which aan be m o d i f i e d w i t h o u t changing t h e physics. For i n s t a n c e i n t h e r e l a t i o n

5

= 5[gB,a] we can modify t h e l e n g t h scale a -+ a/X and change accor- d i n g l y gB gg ^-+ gB(X) so t h a t

5

remains i n v a r i a n t :

5

= ([gB(X),a/Xl. The t r a j e c t o r y gB(X) i s g i v e n by +=B[~,(x)] Adg ,B[g B I = -b o g3+0(gi). B For X l a r g e i t f o l l o w s t h a t

2

'

Therefore the r a t i o </a i s increased by a l a r g e f a c t o r X provided t h e gB(X) = Zb,llnX'

bare c o u p l i n g constant goes t o zero.

These well-known and elementary c o n s i d e r a t i o n s e x p l a i n t h e existence o f a ' s c a l i n g window" i n any numerical simulations. L e t us f o r instance sketch roughly t h e behaviour of t h e "measured" s t r i n g t e n s i o n i n a Monte-Carlo s i m u l a t i o n on a l a t t i c e ^LXLXLXL, as a f u n c t i o n o f t h e c o u p l i n g constant ; such c a l c u l a t i o n s i n i t i a t e d by M. Creutz [I61 have now been repeated by many groups.

I n t h e f i r s t r e g i o n o f Fig. 2 g i s l a r g e ;

5

i s o f o r d e r a and t h e measurements d i s p l a y o n l y l a t t i c e short-range e f ? e c t s . I n t h e t h i r d regiongB i s indeed small, b u t

5

i s order L, t h e f u l l s i z e of the l a t t i c e (which, f o r obvious reasbns, r a r e l y e x c e e d s 6 o r 8a) and t h e c a l c u l a t i o n i s showing o n l y f i n i t e si,ze e f f e c t s . I t i s o n l y I n t h e

"window" I 1 t h a t

5

i s equal t o a few l a t t i c e spacings w i t h o u t y e t any e f f e c t o f t h e f i n i t e t o t a l s i z e . I t i s remarkable t h a t asymptotic freedom ( t h e s t r a i g h t l i n e o f

t h e p i c t u r e ) i s indeed observed i n t h i s narrow window.

Universality and i r r e levant variables

Various l a t t i c e a c t i o n s may have the same p h y s i c a l content and t h e r e i s a p r i o r i no reason, o t h e r than convenience t o choose one r a t h e r than another.

For instance, i n s t e a d o f t h e s i m p l e s t p l a q u e t t e action. Wilson has found conve- n i e n t t o use i n a d d i t i o n s i x l i n k s closed c i r c u i t s [4], p i c t o r i a l l y described as

This a c t i o n has t h e same ( f o r m a l ) continuum l i m i t , the Yang-Mills a c t i o n , p l u s d i f f e r e n t c o r r e c t i o n s o f o r d e r a2. Wilson has adjusted

cl,

c2 and c3 t o improve t h e convergence t o t h e continuum l i m i t . Recently Bhanot and Dashen [ 8 ] have considered

and i n f a c t questioned t h e common b e l i e f t h a t t h i s parameter

c

should n o t a f f e c t t h e continuum l i m i t . Many o t h e r types o f m o d i f i c a t i o n s o f t h e simplest a c t i o n habe been proposed i n t h e l i t e r a t u r e . Since a l l these a c t i o n s share t h e same p r o p e r t i e s : gauge, invariance, i d e n t i c a l formal continuum l i m i t , i t would be embarassing if they had a d i f f e r e n t p h y s i c a l c o n t e n t i n t h e l o n g d i s t a n c e l i m i t . The u n i v e r s a l i t y o f t h i s con- tinuum l i m i t i s c e r t a i n l y d i f f i c u l t t o e s t a b l i s h r i g o r o u s l y ; i t c o u l d be t h a t f o r l a r g e d e v i a t i o n s from the s i m p l e s t a c t i o n one reaches a d i f f e r e n t theory, however s t r o n g arguments have been provided by Wilson's c h a r a c t e r i z a t i o n o f i r r e l e v a n t va- r i a b l e s [ 3 1 . L e t us choose a very simple example i n o r d e r t o examine t h i s concept : consider the l a t t i c e r e g u l a r i z a t i o n o f t h e k i n e t i c energy f o r a s c a l a r f i e l d , and expand o f f o r small a

A t order a 2 appears a n o n - r o t a t i o n a l y i n v a r i a n t o p e r a t o r o f dimension 6 (a s c a l a r f i e l d has mass dimension one). O f course i n t h e formal continuum 1 im i t a

-.

^{0 t h i s}

o p e r a t o r disappears. However the argument has t o be s l i g h t l y r e f i n e d s i n c e a p e r t u r - b a t i v e treatment o f t h i s a d d i t i o n a l d i nsion s i x o p e r a t o r y i e l d s a f a c t o r l/a2 a t each loop and t h e combined e f f e c t o f a8x 1/a2 i s n o t c l e a r . More convincing renor- m a l i z a t i o n group arguments have been i n t r o d u c e d by Wilson who c h a r a c t e r i z e d t h e

" i r r e l e v a n t operators" which have no i n f l u e n c e on t h e long d i s t a n c e l i m i t because they s c a l e as a/c r a i s e d t o some p o s i t i v e power. I n dimension f o u r a l l operators o f dimension l a r g e r than f o u r a r e i r r e l e v a n t , a t l e a s t i n some v i c i n i t y o f the o r i g i n i n c o u p l i n g constant space.

Consequently

(i) I n t h e continuum l i m i t a + 0 t h e f u l l t r a n s l a t i o n i n v a r i a n c e i s r e s t o r e d ( i n s t e a d o f j u s t the l a t t i c e t r a n s l a t i o n s ) ; r o t a t i o n symmetry ( t h e Euclidean equi- v a l e n t o f Lorentz i n v a r i a n c e ) i s a l s o r e s t o r e d .

( i i ) O n e m a y t r y t o improve t h e l a t t i c e a c t i o n , by t h e a d d i t i o n o f i r r e l e v a n t v a r i a b l e s , which would a c c e l e r a t e t h e convergence t o continuum by cancel 1 in g t h e l e a d i n g i r r e l e v a n t operators. This program, n u m e r i c a l l y explored i n i t i a l l y by Wilson, i s under systematic i n v e s t i g a t i o n by K. Symanzik and f o l l o w e r s [ 1 7 ] .

( i i i ) This p i c t u r e o f i r r e l e v a n t operators has l e d t o a conceptual r e v i s i o n o f t h e demand o f r e n o r m a l i z a b i l i t y o f qoantum f i e l d t h e o r i e s . I n a renormalizable theory, since one can g e t r i d completely o f t h e c u t - o f f , one c o u l d pretend t h a t one c o n t r o l s t h e physics a t a l l l e n g t h scales down t o zero. However we know very w e l l h h a t t h e physics i s n o t understood a t very small distances such as (and i n f a c t w e l l before)

JOURNAL DE PHYSIQUE

the Planck length. Therefore one c o u l d conclude t h a t demanding r e n o r m a l i z a b i l i t y i s more than what i s allowed by present day physics. However t h i s new way o f l o o k i n g a t continuum f i e l d t h e o r i e s shows t h a t whatever t h e o r y governs physics a t very s h o r t distances, r e n o r ~ l i z a b l e o r non-renormalizable, t h e l o n g d i s t a n c e l i m i t w i t h respect t o these unknown s h o r t distances aenerates a u t o m a t i c a l l y an e f f e c t i v e renormalizable.

theory. N. Nielsen and coworkers[l8,19lS. Shenker [201 have even gone f u r t h e r and argued t h a t Lorentz i n v a r i a n c e o r even gauge symmetry may r e s u l t o f a s i m i l a r long d i s t a n c e l i m i t o f an u n s p e c i f i e d s h o r t d i s t a n c e theory. Therefore, one can consider a cenormalizable theory such as QED, o r may be even QCD, as an e f f e c t i v e "low energy"

theory.

2. Strong coupl i n g expansions

I n t h e correspondance between f i e l d theory and s t a t i s t i c a l mechanics t h e (bare i s replaced by 0 = I / ~ T . Therefore weak and s t r o n g c o u p l i n g l i m i t s o f t h e theory coupl i n g constant plays the r o l e o f a temperature a t thermal e q u i l i b r i u m s i n c e 1/g, 2) correspond r e s p e c t i v e l y t o low and h i g h temperature l i m i t s o f t h e theory. I n t h e low temperature l i m i t , one p r o j e c t s o u t t h e v i c i n i t y o f t h e lowest energy c o n f i g u r a t i o n ; the "Hami 1 to n i an"

i s minimum i f each group element attached t o a l i n k i s equal t o the i d e n t i t y (up t o a gauge t r a n s f o r m a t i o n ) . Expanding around t h i s c o n f i g u r a t i o n ( w i t h proper gauge fiixFng) y i e l d s t h e usual p e r t u r b a t f e r theory w i t h a l a t t i c e propagator I /

f f

^{(I-cos a p )}

i n s t e a d o f I / ~ ~ . _a I

u

t h e new f e a t u r e opened by t h e l a t t i c e f o r m u l a t i o n i s t h e c o n t r o l o f t h e s t r o n g c o u p l i n g l i m i t o f t h e theory. Indeed w i t h o u t l a t t i c e even i f the p o t e n t i a l energy i s l a r g e , t h e k i n e t i c energy can never be neglected since f i e l d c o n f i g u r a t i o n s w i t h r a p i d s p a t i a l v a r i a t i o n s o f t h e f i e l d may exceed t h e p o t e n t i a l energy ; t h e r e f o r e t h e f i e l d g r a d i e n t s can never be neglected unless a l a t t i c e l i m i t s a l l momenta t o an upper-bound given by t h e i n v e r s e l a t t i c e spacing ( [ p p [ s n l a ) . I t has been shown l o n g ago by Wilson [ 4 ] through a b e a u t i f u l and elementary argument using t h e h i g h tempera- t u r e expansion, t h a t i n t h e s t r o n g c o u p l i n g l i m i t the s t a t i c p o t b n t i a l between heavy quarks increases l i n e a r l y a t l a r g e distances. High temperature s e r i e s a r e c a l c u l a t e d simply by expanding exp - (BH) i n powers o f 6. A t the p r i c e o f sometimes enormous com- b i n a t o r i a l e f f o r t s any obkervable can be c a l c u l a t e d i n powers s e r i e s o f f3 :

=

Ny

a , . , ~ ~ . ~ r e e energy p l a q u e t t e energies, p l a q u e t t e - p l a q u e t t e c o r r e l a t i o n

0

f u n c t i o n s e t c

...

have been computed f o r various gauge groups and space dimensions 1201 The l o n g e s t s e r i e s have l i t t l e more than 20 terms. This method has been used f o r many years i n the area o f c r i t i c a l phenonema ; i t i s b e l i e v e d t h e r e t h a t t h e r a d i u s o f convergence o f t h e s e r i e s estimated by v a r i o u s conventional methods ( r a t i o s o f successive c o e f f i c i e n t s , Pade approximants e t c . . .) corresponds t o t h e c r i t i c a l ( i n - verse) temperature @ a t which a phase t r a n s i t i o n takes place. A naive e x t r a p o l a t i o n o f these ideas t o fo%r-dimensional non-abelian gauge t h e o r i e s i n which no t r a n s i t i o n i s expected c o u l d make one conclude t h a t t h e r a d i u s o f convergence i s i n f i n i t e . However t h i s e x p e c t a t i o n i s presumably wrong : Itzykson,Pearson andZuber [21]haveana- lyzed the Low temperature expansion o f the I s i n g model i n three-dimensions (which i s

dual t o t h e h i g h temperature expansion.of t h e 3-D Z2-gauge t h e o r y ) and found t h a t t h e l e a d i n g s i n g u l a r i t y does n o t correspond t o t h e t r a n s i t i o n p o i n t . They nave given s t r o n g i n d i c a t i o n s t h a t the same should be t r u e f o r SU(N) gauge t h e o r i e s as w e l l . Therefore one has t o e x t r a p o l a t e t h e s e r i e s by a n a l y t i c c o n t i n u a t i o n o u t s i d e t h e c i r c l e of convergence and t h e numerical accuracy may be reduced by t h e presence of these spurious s i n g u l a r i t i e s .

Extensive c a l c u l a t i o n s o f t h e p l a q u e t t e - p l a q u e t t e c o r r e l a t i o n f u n c t i o n s have l e d t o numerical estimates o f the gluebaa1 mass ( t h e lowest c o l o u r s i n g l e t , pure gauge e x c i t a t i o n ) . Hamil t o n i a n h i g h temperature s e r i e s were a1 ready c a l c u l a t e d i n 1976 by Kogut, S i n c l a i r a n d Susskind [22], and d u r i n g t h e l a s t year l o n g e r Euclidean s e r i e s have been c a l c u l a t e d by several groups [23]. O f course, s i n c e t h e continuum l i m i t

corresponds t o going t o zero, these s e r i e s i n powers o f ,6 have t o be e x t r a p o l a t e d a t = m

.

The l a t e s t values f o r an SU(2) gauge group o f t h e g l u e b a l l mass [23] i s rn(0') = ( 1 1 3 0

+

^{350) MeV} which i s c e r t a i n l y compatible w i t h t h e r e s u l t s o f an extensive

v a r i a t i o n a l Monte-Carlo r e s u l t by Berg and B i l l o i r e [24] m(O+) = (920

+

^{310) MeV.}

3. Mean f i e l d methods and beyond mean f i e l d

The simplest, and most t r a d i t i o n a l approximation, t o a system i n which t h e degrees o f freedom are coupled, i s t o replace t h e dynamics by t h a t o f independent degrees o f freedom i n some e x t e r n a l source, chosen i n t h e best p o s s i b l e way t o simulate t h e f u l l dynamics. For a gauge theory i n which t h e r e i s a group element on, every l i n k one r e - places t h e t r u e a c t i o n

H = -

~ ~ ( u u u + u +

+ h.c.) P l a q .

Ho(X) =

- 1

tr(URXQ+h.c.) l i n k s

The a r b i t r a r y matrices

xi

are then f i x e d by a v a r i a t i o n a l p r i n c i p l e t o t h e i r b e s t p o s s i b l e values. These c a l c u l a t i o n s c a r r i e d several years ago by Balian, D r o u f f e and I t z y k s o n [25] l e d t o t h e conclusion t h a t t h e r e i s a phase t r a n s i t i o n between a weak coupling, deconfined phase and a s t r o n g c o u p l i n g c o n f i n i n g phase. These r e s u l t s a r e c o r r e c t when the space-time dimension d i s l a r g e . Indeed xi may be seen as t h e c o n t r i - b u t i o n o f t h e 2(d-1) p l a q u e t t e s adjacent t o a given l i n k and f o r l a r g e d t h e sum o f these c o n t r i b u t i o n s has o n l y small s t a t i s t i c a l f l u c t u a t i o n s and may be replaced by an average s e l f - c o n s i s t e n t

x.

There i s , o f course, a more p r e c i s e way t o show t h a t mean f i e l d becomes exact a t l a r g e d [26]. The r e s u l t s o f these c a l c u l a t i o n s a r e i n

f a c t q u a l i t a t i v e l y c o r r e c t f o r d 2 5, s i n c e i n 5-D already a f i r s t order d e c o n f i n i n g t r a n s i t i o n i s a l s o observed i n Monte-Carlo s i m u l a t i o n s [271. However no such t r a n s i - t i o n i s expected i n 4-D and one has t o go beyond t h i s simple p i c t u r e . T h i s was n o t attempted f o r a l o n g time because o f a deep general p r o p e r t y o f gauge t h e o r i e s disco- vered i n 1975 by E l i t z u r [28] t h a t mean f i e l d t h e o r y apparently v i o l a t e d . E l i t z u r ' s theorem simply says t h a t no non-gauge i n v a r i a n t o p e r a t o r can a c q u i r e a non-zeroexpec- t a t i o n value. This i s t o be contrasted w i t h phase t r a n s i t i o n s i n a s p i n system f o r instance. Below t h e c r i t i c a l temperature i f we impose some e x t e r n a l f i e l d h (which breaks t h e symnetry) on a system o f f i n i t e volume V, t h e e x p e c t a t i o n value o f a s p i n a has t h e f o l 1 owing p r o p e r t y

l i m l i m <o> = 0 ; however V

'

" h + O

l i m l i m <o> # 0

.

h + O v + m

I n t h i s second sequence o f l i m i t s one discovers a broken symmetry. For a gauge theory, even i f t h e r e i s a t r a n s i t i o n t o a d e c o n f i n i n g phase, f o r a l l values o f t h e parameters E l l t z u r proved t h a t

l i m lirn <U > = l i m l i m <U > = 0 v + - h + 0

'

h + 0 v + m R ( i n which h isan i n f i n i t e s i m a l source breaking t h e gauge symmetry)

I n t h e weak c o u p l i n g phase mean f i e l d c a l c u l a t i o n s l e a d t o a non-zero e x p e c t a t i o n value f o r a l i n k and c o n t r a d i c t s E l i t z u r ' s theorem. However t h i s leads t o two puzzles ( i ) mean f i e l d theory i s q u a l i t a t i v e l y and q u a n t i t a t i v e l y exact f o r l a r g e d and E l i t z u r ' s theorem holds i n any dimension ( i i ) i n t h e weak c o u p l i n g phase i t i s o b v i o w s t h a t a l a r g e Wilson l o o p o f perimeter L behaves as <u>L w i t h i n mean f i e l d t h e o r y ; t h i s perimeter law i s indeed t h e expected behaviour o f a deconfined phase, and i t i s d i f f i c u l t t o e x p l a i n t h i s perimeter law i f <u> = 0.

The s o l u t i o n t o these paradoxes and t h e p o s s i b i l i t y o f going beyond t h i s simple mean f i e l d v a r i a t i o n a l c a l c u l a t i o n l i e s i n a r e f o r m u l a t i o n o f mean f i e l d t h e o r y [29,301. For any problem t h e r e i s an exact r e p r e s e n t a t i o n i n which the degrees o f

JOURNAL DE PHYSIQUE

freedom are decoupled b u t they a r e coupled t o a random source [311 (whose p r o b a b i l i t y measure depends upon t h e i n t e r a c t i o n s ) . Mean f i e l d p i c t u r e i s a simple saddle p o i n t approximation i n which one keeps o n l y the most l i k e l y e x t e r n a l source c o n f i g u r a t i o n . However because o f t h e gauge s y n e t r y o f t h e problem any gauge transformed o f a saddle-point i s a l s o a saddle-point. S i m i l a r l y i n a s p i n system t h e r e i s an o r b i t o f saddle-points r e l a t e d by a g l o b a l transformation, b u t since any i n f i n i t e s i m a l source l i f t s t h e degeneracy, pure s t a t e s correspond t o one s i n g l e saddle-point. Since, i n a gauge theory t h i s ( l a r g e r ) degeneracy i s n o t removed b y an e x t e r n a l i n f i n i t e s i m a l source, we do have t o keep t h e whole o r b i t o f saddle-points. Thus, summing over t h i s o r b i t i t i s immediate t o v e r i f y t h a t a l l non-gauge i n v a r i a n t e x p e c t a t i o n values va- n i s h [30]. A gauge i n v a r i a n t operator, such as a Wilson loop, i s n o t a f f e c t e d by t h i s averaging and takes indeed t h e same value as i n t h e simple one saddle-point mean f i e l d approximation. I n t h e weak c o u p l i n g l i m i t f o r instance a Wilson l o o p o f perimeter L behaves as UL and

u

i s n o t the e x p e c t a t i o n value o f a l i n k (which vanishes).

Once mean f i e l d t h e o r y i s i d e n t i f i e d as a saddle-point approximation i n an i n t e - g r a l over random e x t e r n a l sources, i t i s i n p r i n c i p l e s t r a i g h t f o r w a r d , though t h e c a l c u l a t i o n s are o f t e n tedious, t o i n c l u d e f l u c t u a t i o n s o f t h e random source around i t s m o s t l i k e l y value [32]. Successive terms i n t h i s expansion are suppressed by i n - creasing powers o f I/d ( w i t h p o s s i b l y powers o f l o g d ) . The f i r s t c o r r e c t i o n t o an SU(2) pure gauge c a l c u l a t i o n e l i m i n a t e s t h e spurious t r a n s i t i o n i n f o u r dimensions and i s a l r e a d y q u i t e c l o s e t o Monte-Carlo data. See H. W y v b j e r g ' s c o n t r i b u t i o n t o t h i s conference.

Spectrum of gauge theories with quarks i n the mean f i e l d picture

I n 1981 t h e Brussels group [331 showed t h a t i n the s t r o n g c o u p l i n g l i m i t , l a r g e N SU(N)

-

(gauge + q u a r k s ) theory, l a r g e d 1 im i t, c h i r a l s y n e t r y i s spontaneously broken, as expected from a t r u e theory o f s t r o n g i n t e r a c t i o n s . This program was then examined by several groups [34,35]. I n the work o f H. Kluberg-Stern, A . Morel and

B.

Petersson 1351 t h e assumptions o f l a r g e N was r e l a x e d and c a l c u l a t i o n s were p e r f o r - med f o r SU(2) and SU(3). They s t a r t e d from t h e f a c t t h a t when t h e bare c o u p l i n g i s very l a r g e , t h e pure gauge p a r t o f t h e a c t i o n o f quarks coupled t o gluons may be ne- g l e c t e d ( i t i s m u l t i p l i e d by a f a c t o r and t h e a c t i o n i s simply l i n e a r i n any l i n k gauge v a r i a b l e which s i t s i n between two quark f i e l d s . One can i n t e g r a t e o u t a l l t h e l i n k v a r i a b l e s because o f t h e l i n e a r i t y o f t h e a c t i o n . The r e s u l t i s an effec- t i v e a c t ' o n i n terms o f quark f i e l d s , i n v o l v i n g b i l i n e a r meson" operators

=

1

xi(r)xi(r), and mu1 t i 1 in e a r "baryon" operators (r) i=,

Q I

...

^{aN Q a}

B (r) =

1

^€i,.,.i Xi'(~)...~.N(r) ( t h e i ' s a r e c o l o u r i n d i c e s , and t h e

i l

...

^{ilJ N I l~}

a ' s a r e f l a v o i r i n d i c e s ) . This e f f e c t i v e a c t i o n i s then t r e a t e d w i t h i n a mean-field saddle-poi n t a p ~ r o x i m a t i o n , which becomes exact f o r l a r g e d. This y i e l d s a spectrum o f hadrons depending upon two parametess, t h e l a t t i c e spacing and t h e l i g h t quark mass (stranged and charmed hadrons have n o t been considered). F i t t i n g t h e m,, and t h e mp mass ( w i t h a-I = 440 MeV and ma = 8 M ~ V ) t h e authors [351 o b t a j n meeons and baryons masses. The p r e c i s e i d e n t i f i c a t i o n w i t h the t r u e world i s s l i g h t l y o b s c u r e d by t h e use o f Susskind's fermions b u t they f i n d meson s t a t e s a t 1010 and 1160 MeV which c o u l d correspond t o t h e 6 and A , mesons, and a nucleon a t 1380 MeV. Corrections i n t h e I/d expansion seem t o be r a t h e r small. The main problem i s m a n i f e s t l y t o move back c l o s e r t o t h e continuum l i m i t (which i s g; small and n o t l a r g e ) , b u t t h e present r e s u l t s are q u i t e encouraging.

4. Recent r e s u l t s concerning t h e l a r g e N - l i m i t

Many attempts have been made t o s o l v e t h e l a r g e N l i m i t o f an SU(N) -gauge theory, and one must admit t h a t , in $91

te

o f a l a r g e number o f small advances, t h e problem i s s t i l l unsolved. Some r e c e n t new Ideas % n i t ia t e d by E g o ~ t i and Kawai [37] may l e a d t o s i g n i f i c a n t improvements.

The idea o f studying t h e l a r g e N l i m i t goes back t o s t a t i s t i c a l mechanics [381, i n which a complete s o l u t i o n may be found ( t h e f i r s t 1 / ~ c o r r e c t i o n s may even be computed3 and i t i s a very u s e f u l method i n order t o understand t h e s t r u c t u r e o f t h e

t h e o r y (non-perturbative mass generation i n t h e 2-D n o n - l i n e a r a model, f o r i n s t a n c e ) [ 3 9 ] . Therefore t h e r e i s l i t t l e doubt t h a t a complete s o l u t i o n o f t h e l a r g e N - l i m i t f o r gauge t h e o r i e s would be a tremendously important way t o understand quark c o n f i n e - ment, t h e mass spectrum, e t c

...

I n a d d i t i o n ' t H o o f t has argued t h a t SU(W) might be t h e o n l y w e l l - d e f i n e d f i e l d theory (see ' t H o o f t ' s r e p o r t a t t h i s conference). For an SU(N) gauge t h e o r y t h e r e are s i m p l i f i c a t i o n s i n t h e l a r g e N - l i m i t : ' t H o o f t has shown i n 1974 [ 4 0 ] t h a t i n t h i s l i m i t a l l non-planar Feynman diagrams ( i . e . diagrams which cannot be drawn n a plane w i t h o u t l i n e s c r o s s i n g away from v e r t i c e s ) a r e suppressed

S

by powers o f I/N

.

Unfortunately, i n s p i t e o f t h e f a c t t h a t t h e r e a r e o n l y few p l a n a r diagrams

(P

i n s t e a d o f n! a t order n), i t has n o t been p o s s i b l e t o sum them (except i n a low number o f dimensions). I w i l l shorten t h e very r i c h h i s t o r y o f t h e problem, t o discuss d i r e c t l y t h e 1982 attempts.

Eguchi and Kawai 1371 have argued t h a t f o r a l a t t i c e gauge t h e o r y i n t h e large-N l i m i t one can r e p l a c e t h e i n f i n i t e volume l a t t i c e by a f i n i t e p e r i o d i c l a t t i c e I~xRxRxR, t h e s i m p l e s t being I l = 1. We consider, f o r s i m p l i c i t y t h i s reduced R = 1 mo- d e l . The o r i g i n a l model

i s replaced by t h e reduced model

i n which t h e r e a r e o n l y four N x N matrices.

A

Wilson l o o p on t h e o r i g i n a l l a t t i c ~ i s Mppea i n t o the Peducedmodel by f o r g e t t i n g i n any 1 in k v a r i a b l e Ux ( s t a r t i n g from t h e p o i n t x p o i n t i n g i n t h e d i r e c t i o n 11) t h e o r i g i n o f t h e l i n k ⁺ U

.

The authors o f r e f [ 4 0 ] proved t h a t f o r an a r b i t r a r y Wilson l o o p W, mapped by tKe pre- vious r u l e i n t o a reduced o p e r a t o r

wR

<W> true =

reduced o(>)

^'

model

The proof r e l i e s on an i d e n t i f i c a t i o n , i n t h e l a r g e N l i m i t , o f t h e equations o f motion o f t h e reduced and f u l l problems. T h e i r p r o o f i s c e r t a i n l y v a l i d t o a l l orders i n t h e s t r o n g c o u p l i n g expansion. However i t i s assumed t h a t t h e [ u ( 1 ) I 4 symmetry o f t h e reduced model, t h e very simple symmetry

uy

^{-. el0}

u

which leaves H~ i n v a r i a n t , i s unbroken i n o r d e r t o cancel unwanted extra-terms.

11

was immediate1 p o i n t e d o u t

s:

by Bhanot, H e l l e r and Neuberger [411 t h a t i n weak c o u p l i n g t h i s [ u ( I ) ] symmetry was c e r t a i n l y broken, and t h e r e f o r e t h a t t h e reduced model, as i t stands, cannot describe t h e physical r e g i o n (which i s the weak coup1 i n g l i m i t o f t h e t h e o r y ) . They proposed, i n order t o cover t h e whole range o f c o u p l i n g constants, t o use a "quenched" v e r s i o n o f the reduced model. [ T h i s concept o f quenching i s well-known i n t h e s t a t i s t i c a l mechanics o f random media ; consider a system o f p a r t i c l e s , f o r instance, an e l e c t r o n gas, i n t e r a c t i n g w i t h i m p u r i t i e s . I f these i m p u r i t i e s are mobile, they w i l l thermalize w i t h t h e e l e c t r o n gas and t h e average p h y s i c a l q u a n t i t i e s a r e obtained by a t r a c e over t h e e l e c t r o n gas and t h e i m p u r i t i e s degrees o f freedom. However i f the i m p u r i t i e s a r e frozen, t h e 'quenched" case ,the p h y s i c a l observables a r e obtained by c a l c u l a t i n g t h e i r value f o r f i x e d i m p u r i t i e s and then averaging over these i m p u r i t i e s ] . P a r i s i [421 has presented a simple v e r s i o n o f t h e quenching procedure f o r t h e reduced model o f Eguchi and Kawai. He introduces a s e t o f a r b i t r a r y phase f a c t o r s and i n s t e a d o f p e r i o d i c boundary c o n d i t i o n s on t h e u n i t c e l l o f t h e reduced model, a phase f a c t o r i s associated t o a u n i t t r a n s l a t i o n :

in the reduced model U = U x+p,v x , v

in the quenched reduced model (U ) e

'

( u ~ , ~ ) ab.

x+D, V ab

C3-752 JOURNAL DE PHYSIQUE

Wilson loops a r e then mapped, w i t h t h i s new r u l e , i n t o a reduced loop. I t s expecta- t i o n value i s c a l c u l a t e d f o r a f i x e d s e t of 4 x N phases 8; ; f i n a l l y an average i s taken over these phases w i t h a u n i f o r m d i s t r i b u t i o n on t h e u n i t c i r c l e . Again t h i s procedure i s exact i n t h e l a r g e N - l i m i t , and Gross and Kitazawa [43] have shown t h a t i t now works f o r t h e whole range o f c o u p l i n g constants. T h i s remarkable r e d u c t i o n i s s t i l l n o t a s o l u t i o n o f t h e problem, however i t may l e a d t o new developments (see I. B a r ' s c o n t r i b u t i o n t o t h i s conference).

L e t me conclude w i t h a few a d d i t i o n a l remarks on t h i s reduced problem.

( i ) The reduced model i s a l s o exact f o r f i x e d N and l a r g e dimension d:indeed, i n t h i s l i m i t , mean f i e l d t h e o r y holds and i n t h i s approximation l i n k v a r i a b l e s t a k e t h e same value a1 1 over t h e l a t t i c e . Therefore t h e e r r o r o f t h e reduced model

,

compa- r e d t o t h e t r u e S U ( N ) gauge theory, i s p r o p o r t i o n a l t o I / N ~ x I / d , which i s l e s s than 3% f o r SU(3) i n f o u r dimensions.

( i i ) T h i s r e d u c t i o n may e x p l a i n the success o f Monte-Carlo s i m u l a t i o n s on r a t h e r small l a t t i c e s . Indeed t h e arguments o f Eguchi and Kawai a r e v a l i d f o r any p e r i o d i c l a t t i c e RxRxExE. One can show [44J t h a t f o r any R t h e r e i s a breakdown o f t h i s (un- quenched) reduced model below some c r i t i c a l value g 2 ( ~ . ) o f t h i s c o u p l i n g constant.

A simple a n a l y s i s shows t h a t $ ( R ) decreases c e r t a i f f l y r a p i d l y w i t h R. For t h e pre- sent t y p i c a l Monte-Carlo s i m u l $ t i o n s i n which R i s somewhere between 4 and 8, gz(R) c o u l d be so small t h a t t h e breakdown o f t h e reduced model would be i n v i s i b l e , and t h e r e f o r e Monte-Carlo data on these small l a t t i c e s would be exact i n t h e l a r g e N l i m i t . I hope t h a t a more q u a n t i t a t i v e estimate o f g:(L) c o u l d shed some l i g h t on t h i s question.

Conclusion : Numerical s i m u l a t i o n s remain c e r t a i n l y

an

i r r e p l a c e a b l e tool i.n order t o understand t h e spectrum o f gauge t h e o r i e s . However a n a l y t i c methods, such as s t r o n g c o u p l i n g expansions, mean f i e l d expansions, l a r g e N l i m i t , e t c

...

may become i n c r e a - s i n g l y powerful.

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Discussion

M. WEINSTEIN (SLACI.- I only r a i s e t h e question because you d i d , but t h e Nielsen- Ninomiya theorem i s c a r e f u l t o exclude t h e SLAC (or long range d e r i v a t i v e ) t o which it does not apply. The only work which bears upon t h i s i s t h e claim o f Karsten and Smit which has r e c e n t l y been shown by Rabin t o be t r i v i a l l y i n c o r r e c t .