On the triviality of λϕd4 theories and the approach to the critical point in d(−) > 4 dimensions

Nuclear Physics B200 [FS4] (1982) 2 8 1 - 2 9 6 (~ North-Holland Publishing C o m p a n y

O N T H E TRIVIALITY OF k~o4 T H E O R I E S A N D THE A P P R O A C H TO THE CRITICAL POINT IN d(_>)4 D I M E N S I O N S Jiirg F R O H L I C H

lnstitut des Hautes Etudes Scientifiques, 35, route de Chartres, 91440-Bures-sur-Yvette, France Received 7 August 1981 It is shown that one- and two-component ht¢l 4 theories and non-linear g - m o d e l s in five or m o r e dimensions approach free or generalized free fields in the c o n t i n u u m (scaling) limit, and that in four dimensions the same result holds, provided there is infinite field strength renormalization, as expected. Some critical exponents for the lattice theories in five or more dimensions are shown to be m e a n field. The main tools are Symanzik's polymer representation of scalar field theories and correlation inequalities.

1. Introduction In this paper we sketch some ideas in the proofs of the results described in the abstract: triviality and approach to the critical point of h I~0[4 theories and non-linear g-models in dimension d (-) > 4. The one-component non-linear tr-model on a lattice is the usual Ising model, the two-component model is the classical rotor. Our main results are related to some prior, very beautiful results of Aizenman [ 1]; (see also [2, 3] for some previous ideas on triviality). Our methods of proof, based on refs. [4, 5], are, however, different from and complementary to his and yield some complementary information. They involve combining: (i) Symanzik's representation of scalar field theories as models of interacting random walks or "polymer chains" [4] (see [5, 6] for further developments); (ii) spin wave theory, in the form of infrared bounds [7]; (iii) Ginibre's correlation inequalities [8]. In one- and two-component h[tpl 4 t h e o r i e s and in g-models on the lattice we establish an infinite family of new inequalities for each n-point correlation (or euclidean Green) function, n = 4, 6, 8 . . . . . which express it in terms of the two-point function, i.e. the renormalized propagator, and the inverse temperature (field strength), fl, but there is no explicit dependence on bare masses and charges. Those inequalities imply that, in the continuum limit, each n -point function approaches the n-point function of a (generalized) free field with the same two-point function, provided d (>_)4. For simplicity, we only consider the four-point function of the one-component model in zero magnetic field. Details of our arguments and extensions of our results 281

282

J.

Fr6hlich / Triviality of A~p~ theories

to m o r e general models (including ones in a magnetic field) will be presented elsewhere. ,

Let Z ,a d e n o t e the d-dimensional, simple hypercubic lattice with lattice spacing a m e a s u r e d in physical units, e.g. cm, and ranging over the interval (0, 1]. A t each site j E 7/d there is attached a real-valued spin or lattice field variable, Cj, with a priori distribution I 4 1 (1) dh (~oi) = exp [-~A(pi + l/x~o i2 -- E] dcpj, where d~oi is L e b e s g u e measure, and h = h ( a ) , / x = / z ( a ) and e = e ( a ) are arbitrary functions of a with h ( a ) > 0 . If we set /z = A R 2 ,

R >0 ,

e =¼AR 4 ,

(2)

and let A tend to + oo, we obtain an Ising spin of length R. T h e H a m i l t o n function, or lattice action, of the models is defined by H(~0) = - ~ q~jq~j,,

(3)

(fi')

where Z(ir) ranges over all pairs, (if'), of nearest neighbors in Zad. T h e equilibrium state, or euclidean v a c u u m functional, of the system at inverse t e m p e r a t u r e / 3 is given by the measure* d/z~,, (~o) = Z -1 e -~H(~) I1 dA (~oj),

(4)

J

where Z is the usual partition function, and/3 =/3 (a) is s o m e positive function of a to be specified later. T h e correlation, or euclidean G r e e n functions are the m o m e n t s of dtzo,a, i.e.* (~oxl • • • ~ox.)a,a = ] ~0xl" " • ~'~. d/xt3.a(~0).

(5)

T h e distance, Ix - yl, b e t w e e n two sites, x and y, of 7/~ is m e a s u r e d in physical units, i.e. lattice u n i t s m u l t i p l i e d b y a. W e m a y choose a = a,-=2-" [cm],

n=0,1,2

.....

so that Z dm c 7/da n , for m < n. If a = 1, we shall drop the subscript a. T h e t w o - p o i n t function in m o m e n t u m space is given by

~,a --- ~. a d eik'i<~pO~Oj)13,a ,

(6)

J

where k = ( k l , • . . , kd), -- 7r/ a <~ k~ <- Ir/ a, a = 1 . . . . .

d. It is shown in ref. [7] that

0 <
(7)

• Mathematically, these quantities are defined as limits of corresponding quantities in a finite region, A, of Z~a, as A,.~Za~. The existence of the limit follows from correlation inequalities [8, 12].

J. Frlihlich / Triviality of A~o] theories

283

w h e r e ( ( a ) - f l ( a ) a 2-d. H e r e M 2~,, is the l o n g - r a n g e o r d e r (spontaneous m a g n e t i z a t i o n squared), and the second t e r m on the r.h.s, of (7) is the spin wave contribution. F r o m (7) and correlation inequalities one m a y derive 0 < (¢o~p~)o,, <<-M2~,~+ c a f f ( a ) - l [ x ] 2 - a ,

d>~3 ,

(8)

for s o m e finite constant, Ca, i n d e p e n d e n t o f / 3 and a ; see [9]. F r o m (7) or (8) we conclude that (~p0~O~)O.a-~ c o n s t , as a + 0 , for all x ,

unless

(9) e.g.

0 < ~ ' ( a ) ~ < ~ " = 1,

for a l l a > 0 .

In a t h e o r y with infinite field strength renormalization, i.e. d-2 dim [~] ¢ - ~ - ,

(a -+ 0 ) ,

(lO) lira ~'(a) = 0 , a~0

as follows f r o m (8). T h e connected four-point, or Ursell function is defined by (4) ,

P w h e r e ~ a ranges over all pairings, P, of {1, 2, 3, 4}. W e can n o w state o n e of our main, new inequalities. Suppose that IXi--Xj[~,

for i ~ ] ,

for s o m e arbitrarily small, but positive g. T h e n 0 ~

(4.) , UlS,atX1 .....

X4) ~ - - ~ ( a )

2

Z z ;z',z"

(~OXp(1)¢z)lg,a

X (~z,~)xp(2))~.a(@xp(3)~)z).,a(@z,,~xp(4))O.a "~-g ( 3 ' a)

(12)

w h e r e z ranges o v e r Z d, ]z - z ' [ = Iz - z " l = a, and g(B, a) ~
.

W e n o w show that, in d (> ) 4 dimensions, lim u~,~txl (4), . . . . , x4.) = O,

(13)

a~O

p r o v i d e d I x i - x i l / > 8 > O, for i ¢ J, (i.e. at non-coinciding arguments), and M~.a ~ O.

284

Z

Fr6hlich / Triviality of h¢4a theories

We recall that some limit of all correlation functions of the h¢ 4 lattice theories, as a ~ 0, can always be constructed by using correlation inequalities and a compactness argument; see [9] and references given there. The limiting correlation functions can be analytically continued in the time variables from the euclidean region to the Minkowski region. It then follows from (13) that the connected four-point Wightman distribution in the limit a = 0 vanishes, (In fact, our inequalities can be used to show that all connected 2n-point Wightman distributions vanish, for n = 2, 3 . . . . . ) Thus the theory is a free, or generalized free field. Next, we state the renormalization conditions, which permit us to prove (13) in d t> 5 dimensions*. We choose h (a), ~ (a) and B ( a ) such that O < ( ( a ) - - t ~ ( a ) a 2-a ~< 1,

see (9),

Ma,a --- 0 ,

(14)

0 < lim (~o~x)~,, ~
<sup>a

for x = z e l , 0 < z < l , where el is the unit vector in the 1-direction of Ed. (If lim,-,o (@0(Px)a,a -----0, for all choices of A (a),/~ (a) and/3 (a) compatible with (9) and with Ma,~ ~ 0, then the field of the limiting theory vanishes identically, an uninteresting case). It is known that, in two and three dimensions, A (a), ~ ( a ) and/3(a) can be chosen such that condition (14) holds, and, for sufficiently small, bare coupling, (4)/x 1, . . . . x 4 ) # 0 . See [11-13] (also [9, 14] for some recent results). It lima-,0 u ~,a~ appears that when d t> 4 there is no complete proof of the compatibility of condition (14), although there are some partial results [2, 14] and, for d i> 5, a proof seems accessible due to the absence of infrared divergences. By (12) and the definition of ~'(a), (4) : d Ua,atXl, . . . , x 4 ) ~ - - ~ ( a ) 2 a d-4 ~. a (~xp(1)~z)13,a z;z',z" X ( ~ z ' ~ xp(2))/3,a(~ xp(3)(Pz )B,a ( ~ z " ~ xp(,t))/3, a

+ g(/3, a ) .

(15)

Using (8) and (14) one sees that, for Ixe- xi[/> 8, i ~ j, ~(a) E a d(~Pxp(1)~z)B,a(~z'~xp(2))13,a(~xp(3)~z)13,a(~z"~xp(a))~,a ~" C8 <(X3, z;z',z" (16) where Y~< ranges over all z with disttcm](z, {xl . . . . . x4}) ~< 1, and C8 is a finite constant (for 8 > 0). * We work with renormalized, rather than bare fields; see sect. 7.

J. F r d h l i c h / T n v. t a. h t y.

285

o f Aq~ 4a t h e o r i e. s

Moreover, one can show that $ ( f l , a) <- C'8((a )a d-2

(17)

Finally ~(a) 2 y>

aa(~o~p,.~o~)a,a(¢,z,¢~,~,)a..,(¢~,~,¢~)a..,(~Oz,,¢~p,.,)a,~

z ;z',z"

<<-C"Z a~lz -xp<,12-~lz -

xp~3>l2-~ <- C " ' ,

(18)

z

where ~> ranges over all z with dist[cml(Z, {xl . . . . . x4})/> 1, and C", C'" are uniform constants. Inequality (18) follows from (8), (14) (combined with physical positivity) and the fact that for d/> 5

E > Iz -xl2-dlz

- y l 2-'~ -< C"' < ~ .

z

Thus, for d t> 5 and arbitrary 8 > 0 , (12) and (16)-(18) yield O >- ~.,(4) B , a l A , i_ 1 ,

.

.

. ~

x4)>~_ad-4(ks~(a)+k,)

for finite constants k8 and k', which, together with (9), proves (13). In four dimensions, the bounds (16) and (17) are still useful, but (18) must be improved. This requires a condition on the propagator (~oxCy)a.a which is stronger than (14), namely, for Ix -yl> 1 (~ox~oy)B,~ <~ k l x - yl -~ ,

(19)

for some constants e e (0, 1) (arbitrarily small) and k < oo independent of a. In the limit a = 0, inequality (19) follows from the Kiillen-Lehmann representation with e i> d - 2 , if lim~-,o (¢x~0y)a.a is euclidean invariant. By (8), (14) and (19) 0 < (g,~q~y)a.,.~<~'(a)-Vk 1-e Ix - y 1-2p-~1-,),

(20)

with p ~'--*'( 4 - 3 e ) ( 4 - 2 e ) -1 < 1, d = 4. Thus, by (19) and (20) ((a) 2 E >

ad(~Pxv,,,~Pz)t~,a

" " " (~Oz"~Pxv,4})t3,a ~

C " ' ( ( a ) ~/(z-~) •

(21)

z;z',z"

Thus, at non-coinciding arguments, lim u~.,,[xl <4>, . . . . .

x4) = O,

(22)

a.--*O

provided lim ~(a) = 0. a ---*0

(23)

286

J. Fr6hlich/ Triviality of Atp~theories

We note that \ lim (~Ox~Oy)t3,~Jlx/

a~O

y12--, +oo,

as y ~ x,

(24)

unless the fl and 3' functions in the Callan-Symanzik equation have a zero at some value g* > 0 of the renormalized coupling constant, g, defined e.g. by g =

/~(4)X-2~-4

,

where fi~4~= T, lim u~,,,(O, ~4~ x, y, z ) , x,y,z a~O

X = ~ lim <~O¢x>~.a, x a-~0

and ~ is the correlation length of the limiting theory [9]. By (8), (24) holds only if lim ~'(a) = 0 , a~0

which implies triviality, i.e. (22). We conclude that, in four dimensions, any A¢44 theory with infinite field strength renormalization, i.e. non-canonical ultraviolet dimension, is trivial. (It is generally expected that the renormalized propagator of the A¢ 4 theory in the continuum limit has (at least) logarithmic corrections to canonical short-distance behavior. This would imply that ~'(a) must tend to 0, as a -* 0.) If lim ~'(a) -- ~ ' > 0 , a~0

i.e. dim [~p] = 1,the limiting theory cannot be euclidean- and scale-invariant, unless it is a free field. This is a general theorem, due to Pohlmeyer [16] (extending the Federbush-Johnson theorem to the massless case). Thus, a non-trivial A ~ theory would have to be asymptotically free. This possibility is presumably ruled out by an inequality which sharpens (15), (see sect. 6, inequalities (46) and (47), and ref. [15]), but there is, to date, no complete proof.

e

Next, we summarize some results on critical exponents for the Ising and ~.q4 models on the lattice Z d = 7/~=l, d d I> 5. Similar results have been found independently by Aizenman [1]. Previously, Sokal already proved that the specific heat of these models in five or more dimensions is finite [17]. We fix A and ~ and study the behaviour of the correlations when/3/~flc, where Bc is the inverse of the critical temperature. (It is shown in [7] that/3c < oo, for d >/3.) We define X(fl) = Y- (~o~x)~, (susceptibility). (25) x

J. Fr6hlich / Triviality of A¢ 4 theories

287

It is shown in refs. [2, 18] that X(,8)/~

,

as , 8 , ~ ,sc .

One new result is the following: in five or more dimensions, for the Ising model or the X~ 4 lattice theory, X ( , 8 ) - ('8c-'8) -a ,

as '8/~'8~,

(26)

i.e. the critical exponent, y, of X(,8) takes its m e a n f i e l d v a l u e y = 1. (See also [1].) The proof involves deriving a lower and an upper bound, ocX(,8)2, for (0/a,8)X ('8), by using the Lebowitz inequality and the new inequality (15), or related inequalities in [1, 14]. Combining these inequalities with an argument, due to Glimm and Jaffe [2b], one obtains (under slightly more restrictive hypotheses on the coupling constants) 0

a~ ~ (,8)-2 ~ Z (,8),

(27)

for ,8 <,8c and ,8c-,8 small. Here ~:(,8) is the correlation length (inverse physical mass), and Z('8) is the strength of the one-particle pole in (~#x~y)~, at zero spatial momentum. We conjecture that

Z('8)~>Zo>O,

(28)

for '8 < '8~, in d/> 5 dimensions. This would imply

~(,8 ) ~ (,8c- ,8 ) -1/2 , which is the mean field result. Conjecture (28) is a stronger form of the conjecture that r/=0,

for d > 1 5 ,

(29)

where 7/is the critical exponent defined by (~#x~y)t3 Ixl~ooconst • Ix - yl 2-d-'~ • In ref. [15] we propose a strategy for proving (29), but there is no complete proof. The proofs of (26) and (27) are straightforward consequences of the lower and upper bounds on u~4) and are given in [15].

,

We now describe some of the ideas which go into the proof of the basic lower bound (12) on u~4). We start by recalling the random walk representation of the

288

J. Fr6hlich / Triviality of A~O4dtheories

correlation functions of a lattice )t¢ 4 theory or an Ising model d e v e l o p e d in [5, 6] which is inspired by Symanzik's work [4]. Without loss of generality we m a y set a = 1. W e define fd~(s) ds, if n = O,

|

dp.(s) = ~ sn-1 [ ~

XCO,oo)(S)ds,

if n = 1, 2, 3 . . . . .

where Xt0,oo)is the characteristic function of the positive half axis. Let to be a r a n d o m walk starting at some site x in l_d and ending at another site y. Let nj(to) be the n u m b e r of visits of to at some site L W e define dpo, (t)----1-I dp,,~`o)(tfl. J

(30)

Let Ito[ be the total n u m b e r of nearest-neighbour steps m a d e by to. W e note that dP(to; t) =/3 I`°1l-I e -~2at3+"2)t~dp,,,o)(ti) i is the r a n d o m walk analogue of the Wiener measure conditioned on r a n d o m walks with killing rate m starting at x and ending at y. T h e variable tj is the total a m o u n t of time spent by to at site j. W e now define a t - d e p e n d e n t partition function

Z ( t ) = f e-tJi4~)l- I g(~p2 +2ti) d~oi, ,/

(31)

i

where g(¢2)= dA (~o)/d¢, see (1). F u r t h e r m o r e

z(t)-~Z(t)>o.

(32)

Z

Then (~Px~Py)o=

~

/3l~'tI

dpo,(t)z(t).

(33)

`o:x~y

If A = O, ~ = - ( 2 d B + m 2) we find

z (t) = [I e-~2aa+'~)tJ, i

so that

(~Pxq~r)t3,~=o= ~

f dP(to; t) = (--/3A +rn2)~ 1 ,

t o : x --~ y •

where A is the finite difference laplacian. As expected, this is the two-point function of the free (gaussian) lattice field with mass m/~/-~. For the four-point function one finds

f

/3 I'°1t+1'°21 dpo, l(t 1) do,o2(t2)[z (t I + t 2) - z ( t l ) z ( t 2 ) ] ,

.... tobto

2

(34)

J. Fr6hlich

/ T r t v. t a• h t y.

289

o f Aq~ a4 t h e o r t e. s

P

where Y'-,,~,o,2ranges over all walks 0)1 and 0)2, with 0)1 : xp(1)-~ Xp(2), 0)2 : Xp(3) -* Xp(4). Formulas (31)-(34), along with m a n y applications, can be found in [5]. W e now define (35)

~(t) = II eXt~z(t) . i

If we write In ~(t 1 + t 2) as an integral over derivatives in t 1 and t 2 and use Ginibre's inequality [8] to estimate the integrand, we obtain [15] ~(t I + t 2)/> ~(tl)~(t2).

(36)

By combining (34)-(36) we obtain the lower b o u n d

u(f(xl,.,

., x,)~>E

E P /31°"l+l~211 doo,l(t 1) z ( t 1) d p ~ ( t 2) z ( t 2)

P

o.~1 ,¢o 2

× [ e x p ( - 2 A ~ / t ~ t ~ ) - 1] .

(37)

N o t e that if j~ 0), t ii = 0, i.e. the r.h.s, of (37) vanishes, unless 0)x c~ 0)2 # ~ .

(38)

If z ( t ) = l q j e x p [-2dfltj] a term on the r.h.s, of (37) corresponding to a given P is proportional to the probability that a standard r a n d o m walk 0)1:Xp(1)-* Xe(2) and a walk 0)2 : xp(z)~ xP(3) intersect. It is well known that, in the scaling limit (a ~ 0), that probability vanishes in four or m o r e dimensions [19]. This fact together with representation (37) represent the basic intuition behind the proof of the vanishing of u ~,~(4)in the limit a ~ 0. In order to m a k e it precise and derive the lower b o u n d in (12) we must resum the r.h.s, of (37). F r o m (37) and (38) we obtain

u~,'~(xl . . . . . x,) > / - E E ~ t~l'°'l+l'°21x({0)l, 0)=: 0)1 ~ 0)2 ~ @}) P ×

~I,~2

I dp~°l(tl)dP'°~(tZ)z(tl)z(t2)"

By the exclusion-inclusion principle, ,I(({0)1, 0 ) 2 : 0 ) 1 0 09 2 ~ (~)})

= ~ (--1)"-1 n=l n !

E

X({0)l,0)2:o)ln0)29{zl

.....

z~})

Zl,...,z n Zi T~ Z i

2m+1 (__1) n-1

<~ Y~ n=l

for rn -- O, 1, 2 . . . . .

E FI !

z l,...,zn zi#z i

X({0)h0)2:0)ln0)2~{Zl

.....

Z,,}),

(39)

J. Fr6hlich/

290

Trivialityof .~o4 theories

If we set m = 0, we obtain U(4)(X ' .....

~',

X4)~__~.~

r,

/310,,1+1,o21Idpo,~(tl)z(t,)f

dp,~(t2)z(t2) "

o21,~o2

P z~7¢ a

(40) Next, if

ZI~ {X1, X2, X3,

X4}, we decompose wi, i = 1, 2, into two paths

o9'1 :xp~l)~z,

oJ~: z'->Xp<2),

with

(zz')~wl,

W 2 " Xp(3)'-)' Z ,

W~ : Z " " ~ Xp(4) ,

with (zz") 6 oJ2,

(41) t¢ O j t~ , and then sum independently over w~, w~, z', z t¢ , o91, In this summation we overcount the terms on the r.h.s, of (40) and therefore obtain a further lower bound on u~4). (A separate, but straightforward analysis is required when z ~ {xl . . . . . x4}, but the corresponding contribution vanishes when a ~ 0.) We now use the following identity: Let w': x ~ z, w": z'-> y, with (zz') nearest neighbours. Then

fll°"l+l°'"l+11 dp,o,o(zz')o,~,,(t)z(t) =/3 Y flt,o'l+l,o"t[ dp,o,(t')

dpo,,,(t")z(t'+t")

J

¢.o',to"

= fl(¢x,;z)o(,pz,¢y)~

+fl E fll°"l+l°Y'tf dp~'(t')dP,o"(t")[z(t'+t")-z(t')z(t")],

(42)

d

~o',co"

where oj'o (zz')ooy' is the path obtained by composing t-dependent two-point function [5]

w', (zz')

and w". We define a

+2t]) d~oj = z ( t ) -1

Y. fll'°t f dp,o(s)z(s+t). ¢o:x--~.y

(43)

4

Using (43), the second term on the r.h.s, of (42) can be resummed to yield fl y~ fll,o'H,o"tf dp,o,(t') ~',to"

J

dp~,,(t")[z(t'+t")-z(t')z(t")]

= fl ~ fll'°'I I dpo,,(t')z(t')[(q~,q~x)t3(t')-

(q~,~o~)~].

By Ginibre's inequality [8] (~,~o~)o (t') ~ (q~z,~0~)a ,

(44)

for t' t> 0. Thus the second term on the r.h.s, of (42) is negative. (For more details, see [5, 14, 15].)

J. Fr6hlich/ Trivialityof h¢~ theories

291

If we insert (42) on the r.h.s, of (40), using (41) and (44), we finally obtain u~(xl

.....

x 4 ) / > - ~ E Y (u,~,.u,~)~
where ~(/3) takes into account the correction required when z E {xl, completes our sketch of the proof of (12).

X2, X3, X4}.

This

.

T h e r e are various refinements of inequality (12) which we believe might be important for the analysis of mean-field behaviour in d/> 5 dimensions and of the four-dimensional kq~4 theory. (I) By using the upper bound (39) for X({0)h 0)2 : 0)1C~ 0)Z ~ (1)}) we obtain a lower bound on u (4) for each choice of rn = 0, 1, 2 . . . . . Given rn, that lower bound can actually be expressed as a sum over all "skeleton diagrams" (---Feynman diagrams without any self-energy subdiagrams) of order 2rn + 1, computed according to the following " F e y n m a n rules": (i) Each propagator is given by the full two-point function, ((px~py)~. (ii) Each vertex corresponds to

-/3 2

E

G,:G~:G~,G,,:,,,

Xl, • . . , x4

where Xl, • • •, x4 are arguments of propagators attached to the given vertex which is localized at z, and z ,p Z It are nearest neighbours of z. These lower bounds appear to be useful in a refined analysis of the lattice h(p 4 theory or the Ising model in d >t 5 dimensions (with a = 1), for which all skeleton diagrams are infrared convergent, because the renormalized propagator is square summable; see (8). Odd order lower bounds and even order upper bounds on u~4~ in terms of skeleton diagrams with vertices proportional to the coupling constant k have previously been proven in [14]. For applications, see [14, 15]. (II) We now outline another refinement of the basic lower bound (15) on u~4) which is potentially useful to complete the analysis of the four-dimensional kq~4 theory and to show that it is trivial even if lima,0 ~'(a)> 0. We reinstall the lattice spacing, a. First, we recall the lower bound (40) on u~.a, i.e. (4)

UO,atXl . . . .

P

, X4) ~>--Y 2 P z~da

Y

fll~"l+L°'211 dP'~'(t')z(tl) I dP°'2(t2)z(t2)"

¢oI ,to 2 o) 1t.'-~t~t}2D 2

We orient 0)1 to be directed from xp(x) to Xp(2)oGiven some fixed path 0)2, choose z to be the last point at which (.01 intersects w2. Adhering to the notations introduced in

J. Fr6hlich/ Triviality of A¢4 theories

292 (41) we then obtain (4) / Ul3,a~Xl . . . . .

X4) ~ - - ~ ~'~' ~

~ I,o~I+I~,~I+I,o21

~

P ZIZ' oJi,w~

X oJ2:XP(3)"~Xp(4)

I

×X({wT, ~2:(zz')oo~Tnw2 = {z}}) I dp~,i (t') dpo, i(t")z(t'+t")

× I dp~(t2)z $

(45)

(t2) •

We now appeal to an argument used in [5] (new proof of Lieb's inequality [20]) to show that the r.h.s, of (45) can be resummed over oJ~ and w7 to yield

U(O4)a(X~. . . . . X.) >1--fl X P

Y

y filial

zeZ a

z':l~'-zl=~

¢o2

~(,o2) jf dpo,2(t2)z(t2), × (q~x...~Oz)O.a(~z'~Pxv(2)?t3.a

(46)

~(~o2) is the two-point function of the theory with Dirchlet boundary where ~o~q~y~t3,~ conditions along the walk ~o2, i.e. ~(,o2) = lim '/ \ ~ O x'~ Oakt('°2)), CxC,y?~,a y 'l O , t~OO

with t(~'~) = { t , O,

i f ] 6 w2, otherwise.

We expect that, for IXp(2)- z'l ~>e > 0 (measured in cm), (~0~,~~,2, )~g) ~
(47)

for some constant K. which is independent of a and finite for all e > 0, and for some x(w2) which is positive for almost all w2 [with respect to the weight fll~'21Jdp~(tZ)z(t2)]. Although we have no proof of (47) the following heuristic considerations suggest that it ought to be true. (a) If lima-,0~'(a)>0 we expect that (q~xCy)~f behaves qualitatively like (-A (a~'2))~lr, where A~'2) is the finite difference laplacian on 7/a with zero Dirichlet data imposed on the set of sites visited by ~o2. (b) In the scaling limit (a -, 0) the Hausdorff dimension of a typical path is 2. We therefore expect that "(~°2)

( a (¢02)~-1

(log a - l ) -~('°2) , with x (~o2)> 0, almost surely. This behaviour is expected, since Iz' - xp(2)[ ~>e > 0 and dist (z', oJ2) = a ~ 0. (Our considerations are motivated by the analogy with brownian motion [19].)

3". Friihlich / Triviality of A~o4atheories

293

The renormalization condition (19) and inequalities (15), (46) and (47) would imply [under suitable assumptions on the behaviour of x (w2)] that, in four dimensions, lim uo.~t.xa, ~4) • • •, x4) = 0, a-~0

at non-coinciding arguments, even if lima-,o ~'(a)> 0. The conjecture that J¢(£02) is positive, almost surely, is closely related to a conjecture concerning intersection properties of two random walks, £01 and w2, with statistical weight given by /31,o,I d o o , , ( t ) z ( t ) ,

i = 1, 2 ,

[see (30)-(32), sect. 5] on a four-dimensional lattice. As noted in sect. 5, one obtains from (37) and (38) the lower bound

u~ ~(xl . . . . . x , ) / > - X E P /31~'l+t~lX({£01,£02:£01~,o2 ~ qb}) P ¢ a 1~to2

x

I dP"(tl) dP'°~(t2)z(tl)z(t2)"

One may now insert the identity [used in the proof of (39)] X ( { ¢ ' O l , 0 ) 2 : O ) 1 ('~ £02 ~;~ ( ~ } ) = E ,I(({£01, £02" £01 f'~ £02 ~ z

z})N(£01,

£02) - 1 ,

where N ( o 1 , ¢oz) is the total number of intersections of wl and ¢o2, given that wl c~ £0z # q). The problem is then to show that in four dimensions N(£01, £02)diverges to + oo, almost surely, as a ~ 0. Although this is very plausible, we have not found a proof of it, yet. (See also [1].) o

We conclude by describing some related problems and results (in particular concerning the self-avoiding random walk). (a) An alternative way of constructing the continuum limit of a lattice field theory consists of keeping the lattice spacing, a, fixed and analyzing the s c a l i n g limit. One then works with bare fields, (po, introduces a scale parameter O = 1, 2, 3 . . . . and chooses functions/3 (O) and a (0) with the following properties: (i) / 3 ( 0 ) 7 / 3 c , as O/~oo, (e.g. in such a way that the scaled correlation length, 0-1~:(/3 (0)), remains bounded); (ii) hmo-.oo " o o a ( O) 2(q~sxq~oy)t3(o)-G ( x - y ) exists. From the infrared bound, i.e. 0 0 (~o~ y)t3 ~
- yl2-d,

for/3 ~ 3, we conclude that a (0) >>-0 ~e-2)/2 .

(48)

294

J. Fr6hlich / Triviality of A~p4atheories 0

•

•

By setting a = O-1, ~0x = ~ (0)~o0x, this approach is seen to be completely equivalent to the one used in this note, where the lattice spacing a tends to 0, distances are kept fixed in physical units, and the renormalized field, ~, is used. See, e.g. ref. [9]. (Our results then yield triviality of the scaling limit,0 ~ oo, in d~>_~4 dimensions). (b) If in formulas (33) and (34) one sets z(t) =- za.e(t) = [I e -e'-a'~ , J one obtains the Edwards model of the self-suppressing random walk [21] on the lattice. The function G o ( x ' y)__

~

/31,o1[ dpo,(t)zx.~(t)

oJ:x~y

d

is a weighted sum over all random walks with self-suppression starting at x and ending at y. It corresponds to the two-point function (~ox~oy)0 of t h e / ~ 4 theory. We also define G~4) (xl,

yl;x2, y2)--- E

/31~1Ho~11[dp,ol(tl)zA,~(t 1)

o) 1 :Xl---l.yl oJ2:x2--~ y2

d

x dpo,2(t2)zx,e(t2)[e -2xxj'~'~- 1].

(49)

We note that the integrand on the r.h.s, of (49) vanishes, unless ~o~c~o2~ (~. Obviously the function G~, more precisely ~

t-,(4) / L I B ~Xp(1), X p ( 2 ) ; X p ( 3 ) , X p ( 4 ) ) ,

P

is the analogue of u~ ~. Since Zx.,(tl)zx.£(t2)[e -2xx''}t~ -- 1] = zx,,(t 1 +

t 2) -

zx,t2(tl)zx,,(t2) ,

inequality (37) is saturated in the present model. Therefore, all inequalities previously derived for the connected four-point function, u~ ), of the lattice A~04theory extend to G~4). (The proof follows directly from (37), (39) and the repulsive character of the interaction between different random walks.) These inequalities are useful in trying to show that in five or more dimensions, in the scaling limit, two selfsuppressing random walks with non-coinciding starting and ending points never intersect with probability 1. This would follow by showing that G~4) vanishes in the scaling limit. As remarked already, G~4) satisfies the upper and lower bound in (12). The obstruction against completing the proof that G ~) vanishes in the scaling limit is that one does not know for this model that G o (x, y) satisfies a spin wave upper bound of the form of (8). For fixed A and ~:, let B~ be the supremum over all values of/3 for which Go(x, y) has exponential decrease. We conjecture that in three or more dimensions Goo(x, y) ~ 0.

(50)

•

.

,

4

•

J. Fr6hlich / Trtvtahty of A~od theortes

295

If (50) t u r n e d o u t to be true, as expected, t h e n the f u n c t i o n ~ (0) i n t r o d u c e d a b o v e w o u l d satisfy i n e q u a l i t y (48), a n d it w o u l d t h e n follow f r o m (12) that G ~4) (at n o n - c o i n c i d i n g a r g u m e n t s ) vanishes in the scaling limit, (i.e. two self-suppressing r a n d o m walks do n o t intersect), in five or m o r e d i m e n s i o n s . I a m deeply i n d e b t e d to D. Brydges a n d T. S p e n c e r for g e n e r o u s l y c o n t r i b u t i n g i m p o r t a n t ideas u n d e r l y i n g m y results. I t h a n k t h e m a n d A. Sokal for all they have t a u g h t m e a n d for a most p l e a s a n t c o l l a b o r a t i o n . M. A i z e n m a n deserves t h a n k s for h a v i n g described his results [1] to m e prior to p u b l i c a t i o n a n d for very v a l u a b l e discussions. I also t h a n k K. S y m a n z i k (the o r i g i n a t o r of m u c h in the a p p r o a c h used here) for an i n t e r e s t i n g discussion a n d e n c o u r a g i n g c o m m e n t s .

Note added in proof It is easy to check that h y p e r s c a l i n g (i.e. n o n - t r i v i a l i t y of the scaling limit) a n d i n e q u a l i t y (12) imply that 1

r/~<2-5d,

d=2,3,4.

References [1] M. Aizenman, Phys. Rev. Lett. 47 (1981) 1 [2] J. Glimm and A. Jaffe, (a) Comm. Math. Phys. 51 (1976) 1; (b) Comm. Math. Phys. 52 (1977) 203; (c) Phys. Rev. D10 (1974) 536 [3] C. Newman, Phys. Lett. 83B (1979) 63; R. Schrader, Comm. Math. Phys. 49 (1976) 131 [4] K. Symanzik, Euclidean quantum field theory, in Local quantum theory, ed. R. Jost (Academic Press, New York, London, 1969) [5] D. Brydges, J. Fr6hlich and T. Spencer, The random walk representation of classical spin systems and correlation inequalities, Comm. Math. Phys., to be published [6] D. Brydges and P. Federbusch, Comm. Math. Phys. 62 (1978) 79 [7] J. Fr6hlich, B. Simon and T. Spencer, Comm. Math. Phys. 50 (1976) 79 [8] J. Ginibre, Comm. Math. Phys. 16 (1970) 310 [9] A. Sokal, An alternate constructive approach to the ~04 quantum field theory, and a possible destructive approach to ~04,Princeton preprint (Jan., 1981) [10] J.L. Lebowitz, Comm. Math. Phys. 35 (1974) 87 [11 ] Constructive quantum field theory, ed. G. Velo and A.S. Wightman, Lecture notes in physics,vol. 25 (Springer-Verlag, Berlin, Heidelberg, New York, 1973) (in particular, contributions of J. Glimm, A. Jaffe and T. Spencer, and references therein) [12] B. Simon, The P(4~)2 euclidean (quantum) field theory, Princeton Series in Physics (Princeton University Press, 1974) [13] J. Glimm and A. Jaffe, Fortschr. Phys. 21 (1973) 327; J. Feldman and K. Osterwalder, Ann. of Phys. 97 (1976) 80; J. Magnen and R. S6n6or, Ann. Inst. H. Poincar6 24 (1976) 95 [14] D. Brydges, J. Fr6hlich and A. Sokal, in preparation [15] J. Fr6hlich, in preparation [16] K. Pohlmeyer, Comm. Math. Phys. 12 (1969) 204 [17] A. Sokal, Phys. Lett. 71A (1979) 451

296 [18] [19] [20] [21]

J. Fr6hlich / Triviality of A~o~theories B. Simon, Comm. Math. Phys. 77 (1980) 111 A. Dvoretzky, P. Erd6s and S. Kakutani, Acta Sci. Math. (Szeged) 12B (1950) 75 E.H. Lieb, Comm. Math. Phys. 77 (1980) 127 S.F. Edwards, Proc. Phys. Soc. London 85 (1965) 613; M.J. Westwater, Comm. Math. Phys. 72 (1980) 131</sup>