NEW TRENDS IN FIELD THEORY

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NEW TRENDS IN FIELD THEORY

K. Symanzik

To cite this version:

K. Symanzik. NEW TRENDS IN FIELD THEORY. Journal de Physique Colloques, 1973, 34 (C1),

pp.C1-117-C1-126. �10.1051/jphyscol:1973111�. �jpa-00215191�

JOURNAL DE PHYSIQUE Colloque C1, supplement au nOIO, Tome

Octobrc

page

NEW TRENDS IN FIELD THEORY K. SYMANZIK DESY, Hamburg, GFR.

This session was conceived by the Organizing C o m i t - tee to present a report on the state of quantum field theory (QFT) with emphasis on broad and promising developments, rather than a rapport or1 the papers submitted to or presented at the parallel sessions.

Due to lack of time and, more important , ^{the pre-}

sumed unfamiliarity of specifically field theoretic concepts to most of you, I s'iall in fact have to be quite selective as to the topics covered. I will only describe trends that either have a hearing on our understanding of field theory as a whole or offer seemingly far reaching aspects for applications to elementary particle physics.

The following will be the questions or trends discussed here:

0) What & field theory?

1) Does it exist?

or: constructive field theory 2) What and how to compute?

or: technical advances in renormalization theory.

3) Can one do better than perturbation theory?

or: exploiting the renormalization group.

4) What if asymptotic freedom fails?

or: conformal invariance.

5) \,'at about gravity?

or: a chance for superpropagators?

6) Parastatistics and us.

0. - WHAT S FIELD THEORY?- This we illustrate best hy reEerring to quantum electrodynamirs (QED), the theory that yields results accurate to about ten valid figures:

L = - 1 I : FU'J

QiyUa t~ - m ; ~ ₊ gauge fixing terms

4 u v u

+

eOyU A

+ counter terms is the Lagrangean formed from local fields. More generally:

Causality @ Relativity

locality, "~ahewirkung"

3 Quantum Xechanics

Local quantum fields

Then there are:

a) pgyaygean QFX, the prototype of which is QED, with a Lagrangean density

L

kinetic part (bilinear, specifies the elementary fields and perhaps particles)

+ interaction part (tri and quadrilinear terms) from which follow field equations and all that.

b) Axiomatjc-gFT: use local fields but do not mention Lagrangeans.

C)

Ay4lytic S-matrix th_e_gyy: do not mention fields or Lagrangeans, but postulate what has been derived from them (or roughly so).

All trends I will describe relate to a) except parastatistics which relates to b). There has been little recent activity in c).

1. - CONSTRUCTIVE FIELD THEORY. - The most popular theory here, .$ 4 theory has the Langrangean

1 2

~ = - a $ a $ - ! ! L + 2 - - 8 : ,+4:

2 u u 2 24 The Feynman amplitude

>

x2

... ^{> x0}

^{x ; ~ ,} for definiteness (here

+ 2n- 1

with xp, xi, the time and space arguments) allows

0 0

upon x.+zxi, all i, analytic continuation as function of

into the lower complex half plane. That

continuation at z = - i , is the Schwinger (or Euclidean Green's) function [I], S(xl, . . . xZn), which is Euclidean IO(4) invariant, real, positive, real analytic etc.. The ordinary Fourier transform (F.T.) of S (...) is the F.T. of the Feynman ampli- tude analytically continued to Euclidean momenta (ipqi, Gi) as is familiar from the Wick rotation.iJow

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973111

C1-118

K. SYMANZIK

where

2 m2 2

A(+)

Idx[i(~ + 7 4 + 6 ^{44 -} counter terms

r

I

is the Euclidean action, and ID($) an integral over

1

field function space (closely related to Feynman history integrals, the field theory analog of

Feynmanpath integrals). (1) identifies the Schwinger functions as a correlation function of the classical wave field (in four, or, for model purposes, fewer space dimensions), the denominator having the form and properties of a partition function. This makes powerful methods of classical statistical mechanics available 121 to prove the existence and derive properties of S(...) if we restrict the number of dimensions sufficiently.

Combining this with results from an analysis in real space-time it was possible to prove:

i) $4 theory in 1 time 1 space dimension: has unique vacuum, particles, an S matrix, the perturbation expansion for S(...) functions being asymptotic, etc.. . , provided f2 is sufficiently small (Glimm, Jaf f e, Spencer [3] ) .

ii) +4 theory in 1 time 2 space dimensions: has energy density bounded below (Glim, Jaffe [4]).

iii) Yukawa theory in 1 time 1 space dimension: as under ii) plus some technical results (Glim, Jaffe

theory with linear term ~4 in Lagrangean density: vacuum cannot be degenerate (Simon, Grif-

fiths161).

4

V) $

theory in 2 Euclidean dimensions: for 3 ^large

enough, <$> f 0 and +

- 6 symmetry spontaneously broken (which is the continuum analtg of spontaneous magnetization in the Ising model) (Dobrushin. Minlos

[7I )

An important technical result is: certain "Axioms"

for Schwinger functions imply certain "Axioms" to be satisfied for corresponding real time (Wightman) functions, and vice versa (Osterwalder, Schrader [a],

Glaser [9]).

The interest of all these results is twofold:

directly (vide iv) above) they contribute to the theory of phase transitions and may be expected to lead, upon further work, e.g. to strict estimates for the critical exponents of classical 2- and 3-dimen- sional systems.

indirectly they have so far borne out all heuristic conclusions one arrived at earlier using perturbation theory, whereby the confidence in such heuristic methods is strengthened. (However this success must not be overestimated: so far, only theories have been treated where only a small number of Feynman diagrams are divergent, not almost all diagrams as in any 4-dimensional theory).

2. - TECHNICAL ADVANCES ANL, RECENT DISCOVERIES IN RENORMALIZATION THEORY. - Renormalization theory is the art of extracting finite (and, hcpefully,

m

0

relevant) numbers from - ^-

^--, (or is it - 7 ) . 0 etc.

An important technical trick has been dimensional regularization [10,11] : compute in 4-E (or 1 time and 3-E space) dimensions and set

0 at the end.

For general

no ultraviolet or infrared divergences appear then, and thus Feynman integrals can be

freely manipulated. The main advantage of this method of regularization over others is that it is easy to preserve local conservation laws ("Ward identities hold. . ^") .

Next I will mention two developments that are not recent, but have been important additions to the book on non-free fields Nature may be reading, and which should still be thought about more as to their applications to particle physics.

a) N__olmal Products (Wilson [12], Zimmermann [13]).

There exist (in renormalized perturbation theory, at least) operators (here written for QED).

which are finite, local, and transform as the naive pperator product would. (Also derivatives on the constituent operators may be involved, omitted in (2) .)

b) Short distance expansions (Wilson [12] --- --- , Zimmermann 1131). Let O,(o), OB(o) two local operators of the type (2). Then there exists an (infinite) set of such operators 0

(which here must also involve deri-

Y vatives) such that

(3) Oa(x) OB(y)

;faBY(x-y)O (3) + remainder,

Y 2

where the remainder vanishes arbitrarily rapidly for

x-y

0 if in the sum on the r.h.s. sufficiently

many terms are taken. The C-number coefficients

f...(x-y) have terms like

NEW TRENDS IN FIELD THEORY C l - l l Q [(x-y)2]-... , ^(x u -y u ) r L(x-y)2]-"', kn[m 2 (x-y)'], etc. One remark is in order here: renormalizable theories

Also light cone (LC) expansions have been written down (Brandt, ~re~arata[l4], Frischman [15]) where the remainder in a sum similar (or not so similar) to (3) is supposed to vanish not only for (x-y)

0 but even for (x-Y)~

0, but this property has

not yetbeen established even in perturbation theory [16]. Nevertheless, these expansions have been used [14,15], and play an important role in, for example, deep inelastic-scattering [17].

c) Re_n_ormali_za!iliI~-_oI~!!~!!-a!!_e1ia!-~a~~e~~h~_o~ies

This is the result by 't Hooft k 8 1 , and ' ^{t Hooft and}

Veltman r191 (for a full bibliography, see Veltman's Bonn Symposium talk [20]), and has initiated the most

are great for doing calculations, in particular.

However,should renormalizability in perturbation theory be an ultimate criterion ? The fact is that so far no other efficient technique to treat four

dimensional models than perturbation expansion has been found, and that series does certainly not guarantee even mathematical existence of the model considered.

Admittedly, so far, all our intuition concerning field theory derives from QED; which appears to be descri- bable by a renormalizable Lagrangean, the Pauli term

ku,$ ^{" ' F} being conspicuously absent from it. But rather than indulge further in philosophy at this

point, let's look at some more new trends.

active trend in field theory to date, as you are

3. - THE RENORMALIZATION GROUP. - Solutions of aware. The Lagrangean is, typically,

renormalizsble theories are defined by perturbation

1 ? u v -

L

- q ^GuvGa + qa(ia

- ^m)Jla

gauge fixing terms - expansions (otherwise, by seemingly untractable IJ

gG ^{T a}

^{$ +} Faddejev Popcv ghost term (gauge- infinite systems of coupled integral equations). Such

a a E a u B

1 2 expansions can be made useful only be reordering them

dependent) + - ₂ D

+

^D'$~

^- 5 ^{$a$a - V($)}

^gauge so that under suitable circumstances they converge invariant counter terms.

Here,

G: "

B Ba - 8"~: + gf a B~B' is the analog of field

L v bc u v

strength, with fbac the Lie structure constants, T a ~

a the set of matrices corresponding to infinitesimal group transformations of the fermion multiplet, the field +a represents the Higgs scalars [21] , with D

6rSau - ^{ig tr} as ^B the covariant derivative

u r au

with trasanaiogous to T a and with V($)the most a6

general invariant quartic polynomial in +. ^{The Higgs}

scalars allow one to have spontaneous symmetry breaking treatable in renormalized perturbation theory through

<+> # 0 , just as in the sigma model but here

there are no massless scalar particles. I will not go into more of the technical details here, however.

The great attraction of non abelian gauge theories is: they allow construction of models with mass differences, analogous to the electromagnetic ones, computable in renormalized perturbation theory (It Hooft [la], Weinberg [22] ) . (If one adds, for example, to ps-ps theory electromagnetism, the resul- ting mass differences are not computable.) This has given rise to a grect activity in model building with unification of electromagnetic weak, and ultimately also strong, interactions the goal.

For this, however, 1 refer you to Weinberg's talk.

reasonably fast. For theories with sizable coupling constants, this succeeds so far only in regions either a) of very large (and essentially Euclidean) momenta or b) of very small (and essentially Euclidean) -

momenta in theories with trassless particles, depending on details of the model but never in both regions for

the same model.

n) Lar e momenta :

~onsldger Lor s~mplicity +4 theory. m is the mass and g the renormalized coupling constant. The Feynman amplitude (Zp

0).

ip x +...p x G(p,, . .pZn; m,g)

jdx . . ^.dx _zn-I ^{e I I zn 2n}

< TQ(x )...@(x 2 n ) ' connected 1

has the property r231 for X-t -,with Xp= Cr=O

where the sum of the logarithmic terms is the asymp- totic form Gas(pI.. .p

mfi ,g) of the function

2n'

Gas is the Fourier transform of a massless theory Green's function. (Attention: for this to be true, the set of p. must be "non exceptional", which amounts to the restriction that the p. must stay away from momenta configurations at which the massless theory function is infrared singular. In that case, the sum of logarithmic terms defines a function Gas which is the infrared finite part of the (at those momenta non existing) - G

as

The massless theory functions G only obey the as.

partial differential equation (PDE):

which expresses the absence of an intrinsic mass:

the m here plays the role of a normalization momentum only. Its change can be compensated by a change of coupling constant and overall normalization by a factor for each field argarnent, as expressed in the integrated form of (3):

e x p p n / T~(g(Xt))l dX1

J

1

Here, g(X) is defined by:

B(g) having the form

with bo

- 32 and bl

0 (Belakurov et al. [241).

16n In view of figure 1

"Computable"

" existence unsure"

increasing - - docreasing ^I-

FIG.l.- (Presumed) behaviour of B(g) in 44 theory and the result of integration of (3)

we see that for X

0 , g(X)

0, if 0 < g

gro such that we have "asymptotic freedom" in the infrared i.e. the small momentum behaviour could be computed reliably, since g(i)

bil (9.n and thus expansions in this parameter, as to be used on the r.h.s. of (4), become arbitrarily quickly convergent, zero coupling being the lowest approximation (and y(g)

O(g 2 ) ) . On the other hand, for 0 < g < the next zero of B(g) after g , , g(X)

gm for X

-,

which is called an ultraviolet stable fix-point [25].

Presupposing that G as (pl ,... ^p2,, m, gm) exists, an assumption that passes simple consistency tests

[23', the X -.

behaviour of the right-hand side of (4) is governed by:

=

XY('W) exp R(~(x))

I

where R(g,X)

fi

9.im R( A) satisfies:

A+= *

⁰ if only y(g) is continuous at g , . y(g,) is (the anomalous part of) the Wilson dimension of the operator

for that fix-point.

The mechanism of asymptotic suppression (here for

0

g

gw in the infrared) is the following (some-

what simplified): consider any graph and a particular (four-leg) vertex in it. Then there is an infinite set of graphs, differing from the considered one by having the vertex replaced by chains of bubbles, as follows:

This is a geometric series, which in the infrared limit sums up to:

A

1 - hP glnX 3

and yield: part of the suppression factor bf (6) (due to the simplification). Of course, for LnX

-

we are far outside the convergence radius of the

geometric series, but do not worry, the P.D.E. method

takes care of this, and also of the full combinatorics

of all such summations, in as rigorous a fashion as

NEW TRENDS IN FIELD THEORY C1-121 can be, short of proving the existence of the theory

as a whole.

The beauty and power of renormalization group arguments is: no approximations (in the sense of uncontrolled omission of graphs, for instance) are involved, only alternatives:

- Do we try a renormalizable theory, or something else?

- Does the model chosen enjoy asymptotic freedom, or not?

- Is the coupling constant small enough to be in the attraction zone of zero coupling,or not?

And in the case of non-freedom:

- Is there a gm, or not?

- Are the functions y(g) and similar ones continuous at g_, or not?

- Are certain consistency conditrons on dimensions satisfied, or not? (e.g. in $4 theory, for the massless theory to approximate the massive theory, 1 - < dim 42 < 4 is necessary).

Unfortunately, even if asymptotic freedom (in the UV, say) applies, it leads to direct consequences for large momenta behaviour only in the "deep Euclidean region" [26:, which is essentially the one where no square of a partial sum of momenta is zero. (In that case, we have to deal with a G rather than a G as

as as

explained earlier, and the transformation laws of the G differ from the ones (4), of Gas and have been

as

detezned only in the simplest cases !23] _{) .} Let us return to 6 4 theory. For g < 0, and [g 1

not too large, see figure 1, g

0 is formally the UV stable fix-point, and the asymptotic expansion

[27] based on the form of the right hand side of (4) has all desirable properties. However, for g

negative, the b4 model cannot be stable (Coleman [28]).

When learning about canonical behaviour in such circumstances, however, G.Parisi to explain Bjorken scaling by such a model, since the light-cone expansion coefficients are, in the Bjorken limit, essentially in their deep-Euclidean region.

Clearly the question arises whether there are

"good" models (as far as one can say) for which asymptotic freedom holds. It was posedC3~]at the conference on Yang Mills theories, Marseille, June 1972, and there also answered by t'Hooft, who announced that Yang-Mills theories (i.e. SU(2) non

abelian gauge fields ) even with not too many fer- mions, do yield asymptotic freedom.

3nfortunately. being busy with gravity, he did not publish this result until recently. A few months ago, it was found independently of t'Hooft by Polit~er[~~] and Gross and WilczekC3q.

In non abelian gauge theory,(see sect.2c), the expansion parameter is g 2 , and thus the PDE reads

(in Landau gauge) 2

-&

B(g2)-&2

2~(g

Gas(---;m.g)=O

a m a g

where (32, 33)

2 1 1 8 4

B(g ) = -(?

- c2)

O(G 6 )

with c >O for a = abelian gauge group, e.g. c =N for SU(N), and c >O if there are fermions, e.g. c = 1 for the defining representation of

2

+

SU(N). c >O is due to the one-loop graphs invol-

1

ving the three-and/or four-meson coupling, absent in QED.

Coleman and in an elegant paper, have shown that no renormalizable theory not involving non abelian gauge fields (and not subject to the

-

instability objection mentioned above) can be asymp totically free.

Gross and Wilczck proposed to use non abelian gauge theories for the strong interactions to explain Bjorken scaling. They also showed that there are models including Higgs scalars and still having asymptotic freedom; however these models must employ very large gauge groups having other undvsirahle features that makes them unappealing.

Politzer b4 proposed to speculate, for the spon- raneous gauge symmetry breaking, on the Nambu-Jona Lasinio mechanism pq where chiral invariance is broken spontaneously in a four-fermion coupling theory, without elementary scalar field, in con- trast to the 0-model. Politzer, however, also points out unsatisfactory features of the Nambu- Jona-Lasinio calculation. Weinberg has proposed to accept unbroken gauge symetry, as he discussed in his talk here.

There is a new trend 138' 393 to set up detailed models in which the Nambu-Jona-Lasinio mechanism is supposed to take place; however, so far no systematic computation procedure to do this has been found .

(cp.the closely related bound-state problem : it also is notoriously untractable in perturbation theory).

In any case, one expects that the spontaneous- symmetry-breaking mechanism has no effect on the deep-Euclidean behaviour. This leads for Bjorken scaling, as mentioned by Gatto and Weinberg in

10

t h e i r t a l k s , t o t h e f o l l o w i n g asymptotic behaviour of t h e moments of t h e s t r u c t u r e f u n c t i o n . i n t h e c a s e of asymptotic freedom [35, 401 .

xn-' (7)

q2+- where c i s computable and model dependent, w h i l e t h e forward m a t r i x element of t h e s p i n n , o p e r a t o r 0 o c c u r i n g i n t h e L C expansion, of c o u r s e i s n o t computable.

A s i d e remark : asymptotic freedom does n o t s o l v e t h e r i d d l e of t h e f i n a l s t a t e i n d e e p - i n e l a s t i c s c a t t e r i n g i f p a r t o n s ( h e r e , t h e elementary f i e l d s a p p e a r i n g i n t h e Lagrangean) c a r r y quark quantum

7

91 ncmbers, b u t o n l y makes i t more a c u t e : P a r i s i l - .

a r g u e s t h a t under asymptotic freedom c o n d i t i o n s ( i n t h e model h e considered) t h e main c o n t r i b u t i o n s do come from lowest-order diagrams, which should i n t h e f i n a l s t a t e g i v e r i s e t o w e l l s e p a r a t e d j e t s of p a r t i c l e s ( i n t h e s e n s e of Polyakov C4I1), car- r y i n g t h e quantum numbers of t h e elementary f i e l d s , and t h e argument l e a d i n g t o asymptotic freedom r e - q u i r e s t o i n t e r p r e t t h e Lagrangean q u i t e l i t e r a l l y w i t h no escape mechanism n o t embodied i n i t allowed.

One may s u s p e c t t h a t t h e c o n c l u s i o n h o l d s a l s o i n t h e non a b e l i a n gauge models.

Whatever t h e answer t o t h e f i n a l - s t a t e problem i s : t h e p r e c i s e p r e d i c t i o n ( 7 ) of asymptotic f r e e - dom i s i n e s c a p a b l e , and i f i t i s v i o l a t e d , t h a t freedom does n o t apply. Another c a s e where asympto- t i c freedom l e a d s t o a p r e c i s e p r e d i c t i o n d i r e c t l y

+ -

i s e e - a n n i h i l a t i o n , t r e a t e d by P a r i s i r291(in t h e g <O model, however), Appelquist and G i o r g i k21

and Zeet433 :

R(s) E ~ ( e + e -

hadrons)= X.

[ I + - ⁺ --- ^](a)

o(ece-+ u+ 1.1-) s p i n -

t h e c o n s t being model dependent, and - 4 i n a simple model d i s c u s s e d by Zee. 9

That R(s) should go t o a c o n s t a n t a s y m p t o t i c a l l y r e s t s on more g e n e r a l grounds t h a n a s y m p t o t i c f r e e - dom, s i n c e t h e Wilson dimension of e l e c t r o m a g n e t i c c u r r e n t i s t h r e e g e n e r a l l y . I f R(s) does n o t go t o a c o n s t a n t , t h i s would mean t h a t t h e simple Gell- Mann-Low f i x - p o i n t p i c t u r e does n o t a p p l y h e r e . From t h e t h e o r e t i c a l p o i n t of view, t h e r e i s no r e a s o n why i t must, d i s r e g a r d i n g , however, t h e e m p i r i c a l phenomenon of a p p a r e n t Bjorken s c a l i n g .

For o r d i n a r y s c a t t e r i n g amplitudes and form f a c t o r s , a s y m p t o t i c freedom makes d i r e c t l y no pre- d i c t i o n . The p o i n t i s t h a t i n t h e t h e o r i e s of in- t e r e s t hereby, one runs i n t o i n f r a r e d d i v e r g e n c e s

i f one a t t e m p t s t o r e s t r i c t t h e a p p r o p r i a t e zero- mass-theory f u n c t i o n G t o t h e zero-mass s h e l l

a s

p . = 0. The s t r u c t u r e of t h a t i n f r a r e d s i n g u l a r i t y must be analyzed f i r s t b e f o r e asymptotic freedom can be brought i n t o p l a y ( i f a t a l l ) . The deep connection between i n f r a r e d s i n g u l a r i t i e s and large-energy prohlems was f i r s t observed by Appel- q u i s t and PrimacJ!470nl>- v e r y few, and p h y s i c a l l y n o t r e a l l y i n t e r e s t i n g c a s e s of such s i n g u l a r i t i e s have s o f a r been analyzed. Recently, Mueller c6J

has taken a n o r i g i n a l l o o k a t t h i s problem.

As a n i l l u s t r a t i o n of t h e d i f f i c u l t y , l e t me mention t h e form f a c t o r i n a quark-gluon model, where t h e summation of l e a d i n g l o g a r i t h m s g i v e s @71

The c h a r a c t e r i s t i c f e a t u r e i s t h a t w i t h each power of t h e expansion parameter, g 2 , t h e r e go two powers of l o g a r i t h m , r a t h e r t h a n one a s f o r i n s t a n c e i n leading-logarithm summation of t h e form f a c t o r i n ps-ps t h e o r y [441

Here asymptotic freedom would have helped b u t does n o t apply; i n c a s e (9) one does n o t know t h e s t r u c t u r e of t h e f u n c t i o n d e f i n e d by s u m m i n g s l o g a r i t h m s i n terms, e . g . , of a t r a n s f o r m a t i o n law of t h e type ( 4 ) , which g e t s o n l y s l i g h t l y mo- d i f i e d , i n t h e exceptional-momenta c a s e s , f o r t h e f u n c t i o n s G _{a s} s o f a r t r e a t e d [451.

b)Small momenta : As e x p l a i n e d a l r e a d y (4) g i v e s

i n massless @'theory t h e p r e c i s e behaviour a s

A+ 0 , due t o asymptotic freedom i n t h e i n f r a r e d

(provided o< g< g

G e n e r a l l y , a l l m a s s l e s s one

(dimensionless) coupling c o n s t a n t t h e o r i e s t h a t

a r e n o t a s y m p t o t i c a l l y f r e e i n t h e u l t r a v i o l e t

a r e s o i n t h e i n f r a r e d ( f o r t h e c o u p l i n g c o n s t a n t

s u f f i c i e n t l y s m a l l ) . One i s seldom, however, i n -

t e r e s t e d i n a t r u l y m a s s l e s s t h e o r y b u t r a t h e r

more i n t h e o r i e s t h a t have m a s s l e s s a s w e l l a s

massive p a r t i c l e s . I n t h i s c a s e , s c a l i n g momenta

down i s n o t p o s s i b l e because of t h e f i x e d p h y s i c a l

masses. N e v e r t h e l e s s , i n t h e c a s e of m a s s l e s s

s c a l a r p a r t i c l e s i n 0' t y p e s e l f coupling, one can

prove t h a t t h e i n f r a r e d s i n g u l a r i t i e s a r e n o t a f -

f e c t e d by presence a l s o of massive p a r t i c l e s . Re-

c a l l t h a t Weinberg proposes t o c o n s i d e r m a s s l e s s

v e c t o r bosons, due t o unbroken gauge symmetries,

and hopes t h e i n f r a r e d phenomena t h e n t o b e ex-

NEW TRENDS IN FIELD THEORY C1-123 pected to have some good effects. Let us again dis-

regard massive particles at the moment, and try to start with the renormalization group.

If asymptotic freedom holds in the ultra-riolet it does not hold in the infrared. What then ? General- ly, what is the alternative to asymptotic freedom?

4. CONFORMAL INVARIANCE :

If the relevant Gell-Mann-Low fix-point is not at the origin, the limiting theory G (---' ,m9gm) is a non one. Schroer kOJ has shown that in the 0' theory case that theory should be con- fcrmal invariant. The argument holds for all but gauge theories. In this case one can argue only for conformal invariance up to a gauge transforma- tion f'il . It appears that there is no choice of family of gauges such that the gauge change by conformal transformation does not result in great complication b2 I.

The vacuum expectation values of products of gauge invariant quantities are conformal invariant but not very helpful - e.g. in conformal invariant Q.E.D., the vacuum expectation value of any product of field strengths alone vanishes.

I will disregard gacge theories now, except men- tioning that the trend, initiated by Adler's b a ) proposal of computing the fine structure constant on the basis of the eigenvalue condition 6(e2)=0, and related matter have been described by K. John- son at the N.A.L. conference.

A short reminde: : the transformations characte- rising the conformal group [531 are

translations xu+ xp + ' ' a Lorentz transformations xp

A'\, xp scale transformations xu

special conformal transformations

the latter obtainable also by the sequence :

diagrams, in the sense of Dyson's skeletoa expan- sion, the constants appearing in (lo), and in the propagator G(x, y)

[- (x-~)' + i~] -d (11) are the only (not all independent) parameters spe- cifying the theory.

To construct a conformal invariant theory (in strictly four dimensions) two approaches have been proposed :

a) The Migdal-Pari~i-PelitiC~~' 56 1 ^bootstrap

approach : one inserts (10) and (11) in the Dyson equations (the infinite system of integral equa- tions for Green's functions) in skeleton-expanded form and carries out the integrals in a conformal invariant way. (Hereby it is advantageous to work in Euclidean space, the information one is after being embodied in the Euclidean equations just as in the original Dyson equations). There result algebraic relations that determine, in principle, the constants in(l0) and (1 1). Although the skele- ton expansion has formally all desirable proper- ties E 9 j , the occurrence of infinite sums in the relations just mentioned poses a seemingly unsur- mountable problem,

b) Mack's group theoretical approach p91

Here in the coupled Dyson equations, not developped into skeleton sums, a conformal partial wave de- composition is made. Such decomposition was first suggested by Migdal and Polyakov EO], and was ge- neralized and put on more rigorous group theoreti- cal basis by Mack, and is related to work by Gatto and collaborators. This allows to diagonalise the Bethe Salpeter equation and to solve, in a way, all Dyson equations with, however, the crossing symmetry constraint in the form of linear inte- gral equations. The advantage of this approach is that no infinite sums are involved: the formula --

tion obtained, though not yet tested on its pro- mise, is of high esthetic appeal and so different

from the relatively conventional bootstrap one that it merits closest attention. It has already

U yielded an almost trivial derivation of the con-

inversion xp

- Z-- , translation, and inversion.

x2 formal invariant Wilson expansion, obtained befoqe

Conformal invariance puts restrictions on the form with rather more labor bv the Roma-Frascati eroub. - . ₆ 0

of the Green's functions of a theory possessing

A cautionary remark : conformal invariants are that invariance; in particular, Polyakov b41 ^has

only the G (---' ,m, gm) functions; the properties shown that a three point function in coordinate as

of the infrared finite Darts GI of tilese functions space must then have the simple form:

~(x,y,z)=const [-(x-y)' +id E(y-z) +i~]-~2

as

in the case of exceptional momenta have to be

worked out anew and may show no resemblance to

[-(z-x)~+~E]

(10) conformal invariance. In simpler words, conformal

x-ritten for scalar fields only. In a theory where invariance can be asserted to hold asymptotically

the three-point function is the building block of only in the deep Euclidean region; in particular,

r e s t r i c t i o n s t o C1p=O, o r ( c ' ~ ) =O ( 1' a par- i n t e r n a l c o n s i s t e n c y and a t most a f i n i t e number of t i a l sum) a r e n o t conformal i n v a r i a n t .

One simple d i r e c t consequence of conformal inva- r i a n c e should be mentioned : t h e t w i s t of t h e s p i n n - o p e r a t o r s appearing i n t h e l i g h t - c o n e expan- s i o n s , which i s t h e i r Wilson dimension minus n , i s n e c e s s a r i l y >2, except f o r n=O, t h e Schwinger term o p e r a t o r . Tbis f o l l o w s from work of Riihl L6'j a s was pointed o u t by Mack c62aI.

This i m p l i e s t h a t t h e l i g h t cone s i n g u l a r i t y of a product of two c u r r e n t s cannot be s t r o n g e r than c a n o n i c a l , except f o r t h e s c a l a r Schwinger terms c o n t r i b u t i o n .

5. A CHANCE FOR SUPERPROPAGATOR ?

6 -

-

' T Hooft and vkltmank have shown t h a t quantized E i n s t e i n g r a v i t a t i o n becomes n o t r e n o r m a l i z a b l e i n p e r t u r b a t i o n t h e o r y , i n t h e p r e s e n c e of a quantized s c a l a r f i e l d . Another p o s s i b l e approach i s t o f o l l o w

Isham, Salam, and S t r a t h d e e who propose t h a t one u s e s superpropagator methods. The proposal i n v o l v e s

t h e s t e p s :

a) p a r a m e t r i s e t h e g r a v i t a t i o n a l f i e l d such t h a t e x p o n e n t i a l s ( l i k e ekd " ( X ) ) appear i n t h e Lagran-

g i a n , (@,*being a m a t r i x ) .

b) make a p e r t u r b a t i o n expansion whereby t h e ex- p o n e n t i a l ~ a r e k e p t i n t a c t . The r e l e v a n t bare va- cuum e x p e c t a t i o n v a l u e s then f o r m a l l y come o u t t o be products of "superpropagators"

T h e a u t h o r s hope t h a t by a s u i t a b l e choice of gau- g e , a l l t h e s e superpropagators (whose d i r e c t evalu- a t i o n l e a d s t o ambiguous r e s u l t s ) can then be d e f i - ned such a s t o be f u n c t i o n s s t r o n g l y v a n i s h i n g a t c o i n c i d i n g argument ( h e r e x * y ) i n Euclidean s p a c e ,

such t h a t a l s o products of such p r o p a g a t o r s a r e am- b i g u i t y f r e e .

c ) c o n t i n u e a n a l y t i c a l l y t o Minkowski space ( i . e . r e a l t i m e ) .

Whether or n o t s t e p b) goes through i s open. To s t e p c ) , t h e work of Osterwaaer and Schrader r8] ^,

mentioned e a r l i e r , would be r e l e v a n t . The a u t h o r s b e l i e v e t h a t g r a v i t y may b e t h e only theory s t i l l meaningful, and h o p e f u l l y t r a c t a b l e , f o r superpro- p a g a t o r s , i n view of u l t i m a t e l y u n s u c c e s s f u l a t - t e m p t ~ [ ~ ~ ] t o t r e a t e.g. c h i r a l Lagrangeans of t h e type introduced phenomenologically by Weinberg sa- t i s f a c t o r i l y by t h i s method.

While t h e q u a n t i z a t i o n of g r a v i t y i s n o t a p r e s - s i n g problem, t h e a t t e m p t t o t r e a t "nonrenormali- zable" Lagrangeans by any method t h a t has a t l e a s t

a m b i g u i t i e s d e s e r v e s a t t e n t i o n . And, of c o u i s e , we hope t h a t someday man w i l l know how t o q u a n t i z e gra- v i t y .

6. - - PARASTATISTICS

Haag r e p o r t e d on t h e r e s u l t s of a r e c e n t l y con- cluded s e r i e s of papers i n a l g e b r a i c quantum f i e l d theory. They r e l a t e t o s u p e r s e l e c t i o n r u l e s and p a r a s t a t i s t i c s , and a r e o b t a i n e d by a deep, and d i f f i c u l t , mathematical a n a l y s i s . I can g i v e o n l y a h i g h l y s i m p l i f i e d v e r s i o n h e r e .

The QFT p r i n c i p l e s adopted a r e l o c a l i t y ( i n t h e a b s t r a c t v e r s i o n of l o c a l o b s e r v a b l e s ) , u n i t a r i t y , Poincard i n v a r i a n c e , and p o s i t i v e m e t r i c of t h e s t a t e space. Let such t h e o r y p o s s e s s s t r i c t l y conserved "charges" ( l i k e t h e e l e c t r o m a g n e t i c one, however, gauge v e c t o r bosons a r e s o f a r n o t consi- dered i n t h i s work; a b e t t e r example i s i s o s p i n imagining i t t o be e x a c t l y conserved; s p i n i n t e - g r i t y o r h a l f - i n t e g r i t y i s included h u t t i r n s o u t t o be a f u n c t i o n of t h e o t h e r charges i n a way t h a t e x p r e s s e s t h e connection between s p i n and s t a t i s t i c s ) then t o each charge quantum number, which i s a maximal compatible s e t of v a l u e s f o r t h e c h a r g e s ,

t h e r e belongs p r e c i s e l y one s t a t i s t i c s parameter which i s a p a i r

d) w i t h

p l u s o r minus, and d a p o s i t i v e i n t e g e r . E = I c h a r a c t e r i s e s parabosons,

&=-I parafermions. d i s t h e o r d e r of t h e p a r a s t a - t i s t i c s , i . e .

f l : wave f u n c t i o n s can be a n t i - symmetric r e s p e c t i v e l y symmetric i n a t most d arguments. The s t a t i s t i c s parameter i s t h e same f o r a l l p a r t i c l e s w i t h t h e same charge quantum numbers.

Furthermore, t h e r e i s ( i n many c a s e s , a t l e a s t ) a gauge group of f i r s t kind a s s o c i a t e d w i t h t h e s t r u c t u r e of t h e charge quantum numbers such t h a t t h e l a t t e r a r e t h e l a b e l s of t h e i r r e d u c i b l e r e - p r e s e n t a t i o n s of t h a t group, whereby t h e s t a t i s t i c s parameter d i s t h e dimension of t h e p a r t i c u l a r r e - p r e s e n t a t i o n . I n s e v e r a l - p a r t i c l e s t a t e s represen- t a t i o n s combine i n t h e way f a m i l i a r from t h e vec- t o r a d d i t i o n of a n g u l a r momenta, and charge conju- g a t i o n corresponds t o t h e t r a n s i t i o n t o t h e conju- g a t e complex r e p r e s e n t a t i o n .

P a r a s t a t i s t i c s i s t h e same a s o r d i n a r y (namely, Bose o r Fermi) s t a t i s t i c s of p a r t i c l e s t h a t have unobservable ("hidden") d e g r e e s of freedom. Haag observes t h a t h i s t o r i c a l l y a l l d e g r e e s of freedom t h a t were n e c e s s a r y t o c l a s d i f y s t a t e s were found t o be a s s o c i a t e d w i t h o b s e r v a b l e a t t r i b u t e s , e.g.

l i k e e l e c t r o n s p i n i n t h e atomic h u l l , o r e l e c t r i c

NEW

TRENDS IN FIELD THEORY C1-125 charge i n c a s e of i s o s p i n , i . e . t h e p h y s i c a l d i s t i n -

g u i s h a b i l i t y of proton and n e u t r o n . So maybe n a t u r e does n o t l i k e p a r a s t a t i s t i c s .

I n t h i s s e t t i n g i t would be impossible t o have a OFT w i n g quark f i e l d s b u t t o exclude t h e occu- r e n c e of p h y s i c a l s t a t e s w i t h non zero t r i a l i t y .

(Thus, t h e proposal of K. Johnson t o exclude f r e e - l y o c c u r r i n g quarks due t o a p o t e n t i a l i n c r e a s i n g w i t h d i s t a n c e v i o l a t e s b a s i c propei-ties (namely,

t h e c l u s t e r p r o p e r t y ) d e r i v e d from l o c a l i t y and t h e p o s i t i v i t y of the energy ( n o t e , however, t h e l i m i t e d g e n e r a l i t y of t h e a n a l y s i s c a r r i e d o u t s o f a r . )

CONCLUSION :

The t r e n d i s t o t a k e QFT, down t o t h e u s e of s p e c i f i c r e n o r m a l i s a b l e Lagrangeans s e r i o u s l y . To g e t i n f o r m a t i o n from such Lagraneeans r e l i a b l y , one i s developing t e c h n i q u e s , r e s t i n g e . g . on r e s h u f f l i n g of t h e p e r t u r b a t i o n s e r i e s , t h a t do n o t amount t o n e g l e c t i n g some ( o r most) c l a s s e s of Feynman graphs u n l e s s t h e i r n e g l i g i b i l i t y i n t h e problem c o n s i d e r e d , i s proven.

Nonabelian gauge t h e o r i e s have g i v e n and w i l l c o n t i n u e t o g i v e , a s t r o n g impetus t o Lagrangean f i e l d t h e o r y , s i n c e i n c o n j o n c t i o n w i t h sponta- neous symmetry breaking they appear t o be a b l e t o d e s c r i b e a wide r a n g e of phenomenological s i t u a - t i o n s . Before t h e i r advent t h e r e n o r m a l i z a b l e Lagrangeans a v a i l a b l e were too few, and too p o o r l y s t r u c t u r e d , t o make t h e i d e a a t t r a c t i v e t h a t a l l i n t e r a c t i o n s , e x c e p t t h e g r a v i t a t i o n a l one, might be d e s c r i b e d (with r e a s o n a b l y c l o s e r e l a t i o n b e t - ween observed p a r t i c l e s and elementary f i e l d s ) by

a r e n o r m a l i z a b l e Lagrangean.

Of c o u r s e , we have no g u a r a n t e e e i t h e r t h a t t h i s happens, and i f no workable Lagrangean model can be found, we may have t o go back t o axiomatic the0 r y . The l a t t e r b e i n g known t o pose mathematical problems of h i g h e s t d e g r e e of d i f f i c u l t y , i t i s r e a s s u r i n g t h a t by perseverence and e r u d i t i o n new and d e f i n i t i v e r e s u l t s can t h e r e s t i l l be o b t a i n e d . Before we g i v e up Lagrangean quantum f i e l d t h e o r y , however, l e t u s have some more r e s u l t s a l s o t h e r e .

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