Constructive Quantum Field Theory. The 1973 ''Ettore Majorana'' International School of Mathematical Physics

Lecture Notes in Physics Edited by J. Ehlers, MiJnchen, K. Hepp, Zf3rich, and H. A. WeidenmiJller, Heidelberg Managing E...

Author: G. Velo | A. S. Wightman

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Lecture Notes in Physics Edited by J. Ehlers, MiJnchen, K. Hepp, Zf3rich, and H. A. WeidenmiJller, Heidelberg Managing Editor: W. BeiglbSck, Heidelberg

25 Constructive Quantum Field Theory

The 1973 "Ettore Majorana" International School of Mathematical Physics Edited by G. Velo and A. Wightman Instituto di Fisica, A. Righi, Bologna/Italy Princeton University, Princeton, NJ/USA

Springer-Verlag Berlin - Heidelberg

New York 1973

ISBN 3-540-06608-X Springer-Verlag Berlin • Heidelberg • N e w York ISBN 0-387-06608-X Springer-Verlag N e w York • Heidelberg • Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin. Heidelberg 1973. Library of Congress Catalog Card Number 73-21055. Printed in Germany. Offsetprinting and bookbinding: Julius.Beltz, Hemsbach/Bergstr.

TABLE OF CONTENTS

LN~fRODUCTION FUNCTIONAL ANALYSIS AND PROBABILITY THEORY Michael C. Reed, Princeton University APPENDIX: SAMPLE FIELD BEHAVIOR FOR THE FREE MARKOV RANDOM FIELD Phillip Colella and Oscar E. Lanford, University of California at Berkeley EUCLIDEAN GREEN'S FUNCTIONS AND WIGHTMAN DISTRIBUTIONS Konrad Osterwalder, Harvard University PROBABILITY THEORY AND EUCLIDEAN FIELD THEORY Edward Nelson, Princeton University THE GLIMM-JAFFE #-BOUND: A MARKOV PROOF Barry Simon, Princeton University THE PARTICLE STRUCTURE OF THE WEAKLY COUPLED P(~)2 MODEL AND OTHER APPLICATIONS OF HIGH TEMPERATURE EXPANSIONS PART I: PHYSICS OF QUANTUM FIELD MODELS PART II: THE CLUSTER EXPANSION James Gllmm, Courant Institute, N.Y.U. Axthur Jaffe, Harvard University Thomas Spencer, Courant Institute, N.Y.U. BOSE FIELD THEORY AS CLASSICAL STATISTICAL MECHANICS I. THE VARIATIONAL PRINCIPLE AND THE EQUILIBRIUM EQUATIONS Francesco Guerra, University of Salerno II. THE LATTICE APPROXIMATION AND CORRELATION INEQUALITIES Lon Rosen, University! of Toronto IIl. THE CLASSICAL ISING APPROXIMATION Barry Simon, Princeton University

i 2-43

44-70

71-93

94-124 125-131

132-198 199-242

243-264 265-289 290-297

CONSTRUCTIVE MACROSCOPIC QUANTUM ELECTRODYNAMICS Klaus Hepp, E.T.H., Zurich Elliott Lieb, Mass. Inst. Tech.

298-316

PERTURBATION EXPANSION FOR THE P(~)2 SCHWINGERFUNCTIONS Jonathan Dimock, State Univ. of N.Y. at Buffalo

317-320

NONDISCRETE SPINS AND THE LEE-YANG THEOREM C~arles M. Newman, Indiana University

321-325

EUCLIDEAN FERMI FIELDS Konrad Osterwalder, Harvard University

326-331

INTRODUCTION

The present volume collects lecture notes from the session of the International School of Mathematical Physics "Ettore MaJorana" on Constructive Quantum Field Theory that took place at Erice (Sicily) July 26 to August 5, 1973. The School was a NATO Advanced Study Institute sponsored by the Italian Ministry of Public Education, the Italian Ministry of Scientific and Technological Research, and the Regional Sicilian Government. The book contains introductory material on funetionai analysis and probability theory, as well as detailed discussions of the existing state of knowledge of quantum field theory models. In the opinion of the Editors, it can serve both as a review for experts of a rapidly developing subject and as an introduction for those with only a basic knowledge of field theory. Unfortunately, the lecture notes of K. Symanzlk were not prepared in time to be published in this volume. will be published elsewhere.

We hope they

In any ease, the reader is re-

ferred to his Carg~se Lectures 1973, on related subjects, which are in course of publication.

FUNCTIONAL ANALYSIS AND PROBABILITY THEORY Michael C. Reed, Princeton University

For many years now the standard tools of functional analysis have found application in quantum mechanics and quantum field theory~ indeed the fundamental concepts of functional analysis, Hilbert and Banach spaces, bounded and unbounded operators, are the mathematical objects out of which specific models are constructedo

By "standard tools TT I mean the spectral theorem, Stone's theorem, and

various methods of proving self-adjointness and investigating the properties of specific self-adJoint operators.

It is not necessary to lecture on these topics

since many of you already know them and in any case they are readily available in functional analysis texts and in several introductory lecture series meant especially for physicists (see for example [8S or [12]).

Most of you already well-

acquainted with quantum field theory know that many different branches of mathematics have been used in attempting to understand and solve the difficult mathematical problems that are involved, among them group representation, distribution theory, several complex variables and Banach algebras.

So perhaps you

were not too surprised to learn that yet another branch of analysis, probability theory and stochastic processes is now being applied in quantum field theory. The influx of these methods has come from two sources:

first from the observa-

tion that certain field theory problems are analogous to problems in statistical mechanics; second, Irving Segal and Edward Nelson have often said that probabilistic methods are not Just tools but are generic to the problems themselves, that is, that to some extent the problems of field theory are really problems in probability theory. My purpose in these lectures then is to present an introduction to the probability theory concepts and methods which will be used by the other lecturers. We will start with the fundamentals, so you should not be angry if I say things

that you already know (though you are allowed to be impatient).

~l.

nmo~A~s

We begin with the basic definitions. < 2 , Z , ~ > where

2

is a set, Z

measure of mass one on < 2 , Z > . valued function on

2.

If

is the probability measure

x

A probability spa@e is a triple

is a o-algebra of subsets, and

is a positive

A (real) random variable is a measurable realis a random variable on

~x

~

2, the distribution of

x

on the real line given by:

x

where

of

A

is a Borel set in ~.

x, Vat(x),

We define the mean of

x, E(x), and the variance

by

and

V~(x) = f~ (~ - E(x)) 2 d~x(~)

if the integrals exist.

If

f

is a measurable function on ~

also a random variable on

then

f(x)

and

E(f(x)) = f2 f(x(W)) d~(~) = f~ f(k) d~x(k ) .

Notice that if

x E L2(~,d2)

then

L (n,d~) and

Var(x)

= (x - E(x),

x - E(x))

2 L (n,d,)

= f~ (x - E(x)) 2 d~(~)

.

is

Example la. N times.

Consider an experiment which consists of tossing an unbiased coin

If we denote the result of a toss by 0 if it is tails and by 1 if it is

heads, then the set of all outcomes = {~ = < n i > i =N l I

G

may be represented by

n.l = 0 or 1}, i.e. each point

m

represents the possible

outcome of a sequence of tosses, and each point in

~

has measure

be the random variable on

~

_i

2N "

Let

x.

l

given by

=

Ii

if

n. = 1 i

0

if

n. = 0 I

xi

Then each

xi i equal to K "

has the distribution

~x. = ½5(k) + ½5(k -i), mean ½, and variance i

We now return to our general definitions. Let

xi, i = l,...,k, be random variables

. a measure

Then the vector-valued function ~

x

on

~k

of

A

x(~) = <Xl(~),...,Xk(~) >

defines

by

~x{A] = ~{x-I[A]] = ~[ml

for each Borel set

on a probability space

in

~k.

<Xl(~),...,Xk (~) > ~ A}

The measure

~x

is called the Joint distribution

Xl~X2~...,Xk. Usually~ independence of random variables is defined in terms of Joint dis-

tributions~ but we will use the following definition in terms of ~-algebras. T

be an index set.

A family of measurable sets

[At]teT, At C ~ , is called

independent if k

for any finite subfamily, t l• E T. Et C At e

E Zt

k

A family of q-algebras

is said to be independent if each family of sets is independent.

Et, t ~ T {A t]

Finally, a family of random variables

with

where {xt]te T

is

Let

independent if the family of c-algebras the G-algebra generated by is measurable).

xt

( Zt)teT

is independent where

(i.e. the smallest ~-algebra on

~

Zt

is

so that

xt

It follows from this definition that if the random variables

Xl,X2, o..,Xk

are independent then their joint distribution ~x is just the k product measure gx = i~l gx. in ~k. This is the usual definition of indepeni deuce. If

A,B c E

and

~(B) ~ 0

we define

P(AIB) =

P(AIB)

is called the conditional probability of

are measurable sets in R

and

x, y

A

given

B.

If

are random variables with

A

and

B

~(x-l[B]} ~ 0

then we set P(yeAIxeB)

P(y eA Ix E B) that

x

is called the conditional probability that

is in

Example lb.

Zj

y

is in

A, given

B.

Let us return to example 1 and define

i = 0,1, and S~

m P(y-l(A) Ix-l(B))

j = 1,2,...,n.

Let

Zj

S~i = (~ ~ ~I

nj = i}

be the u-algebra generated by

(i.e. it Just consists of the four sets

¢, a, S~

and

S~).

for

S~

and

The a-algebras

are easily checked to be independent (this of course depends on our choice

of measure); thus the randc~ variables random variable which assigns to each

x. 1 ~

are independent.

If

c(~)

is the

the integer corresponding to the toss

on which heads first appeared or N + l if heads did not appear on any of the N

tosses then

c

has the distribution j~l ~1 ~ ( ~ . j )

+

1 ~(~.(~+l))

and

c

is

not independent of any of the x i . You can check yourself in this example that the notions of conditional probability which we have defined correspond to your intuition of conditional probability. Before leaving this elementary example let us change our point of view slightly and ask in what sense the distributions of the random variables

x. 1

determine

~°

z(J)

Let

the distribution of

xI

and

~

x2

determines

Xl,...,x N

determines

subsets of

~.

determines on ~

El' '%2". . . . ' 5 "

be the G-algebra generated by ~

Z (2)

on

Then,

Z(!), the joint distribution of

and so forth.

completely since

xI

The joint distribution of

z(N)

is just the G-algebra of all

(Notice that in this special case, since the

x.i

are indepen-

dent, we can calculate any of the joint distributions directly from the individual distributions). ~(!) C

So we have an increasing family of G-algebras

Z (2) C ... C

Z (N)

generated bylarger and larger families of random

variables and the restriction of

~

to a particular algebra is determined by the

joint distributions of the random variables which generate the algebra.

I have

reformulated this trivial example to provide some intuition about the more difficult things to

come.

Before presenting another example we make one more definition. i = I,...,N

If

xi,

are random variables, the matrix

{rij) = E((x i - Exi)(xj - Exj)) is called the covariance matrix of the variables

X.o

Notice that

l

N

g qjai~ j = E(Iz i,j=l

%(x i

so the covariance matrix is positive definite. mean then

Example 2.

Also if

xi

and

xj

have zero

Fij = (xi,xj)L2(fl,d~).

(Gaussian randcm variables).

A random variable

Gaussian random variable if its distribution

d~x(~')

for some

Exi)l 2)

-

G > 0

and

m ¢lq.

=

~x

x

is said to be a

is given by

_ (~--m)2 l 22 ~2¢~'~ e d;,.

It is easy to check that

G

and

m

are the

variance and mean respectively of

x.

A finite collection

Xl,-...,x N

of random

variables is called Gaussian if there is a symmetric positive definite matrix on A N, Q, and real numbers

ml,-.',m N

so that the joint distribution of

Xl,...,x N

is 1

(Det(Q))2 e -½(Q(~-m~'(~-m~)) d~

(~)N/2

(i.i)

It follows by integrating out variables that the joint distribution of any subset or linear combinations of the

x. l

is again Gaussian, in particular the

themselves are Gaussian random variables. explicit computation shows that Xl,...,x N

is just

Q-1

Further, a little linear algebra and

E(x i) = m i

Thus, if

x. z

and that the covariance matrix of

Xl,...,x N

is a Gaussian family their joint

distribution is completely determined by their means and covariance matrix.

In

particular, if the covariance matrix is diagonal (which is the same thing as saying that

x I - m l, ..., x N - m N

are mutually orthogonal in

L2(~,d~)), the

joint distribution is a product of one-dimensional distributions.

That is, for a

Gaussian family of random variables with means equal zero, (xi,xj)L2(Z,d~) = 0 if and only if nation of the

x. l xi

and

x. j

are independent.

is again Gaussian with mean zero, we can (by Gram-Schmidt

orthogonalization) find a set

~l,...,~N

of independent Gaussian random variables

of mean zero and variance one (this just means is a linear combination of the

Finally if

Thus, since a finite linear combi-

Xl,...,x N

II~.I12~ = l) so that each i L~(~,d~)

~i"

are random variables, the function

i z ~ixi(~) C(~ 1 .... ,0~) = I~ e

d~(~)

i Z %k i

x

is called the characteristic function of

xl,...,x ~.

Notice that if

d~_~(k) x

is

xi

given by (i.~,

(i.e., Xl,...,x N

is a Gaussian system), then

c(~ 1 .... ,%) = e

(1.2)

where

p = Q-I

i E ~.m. i 11

is just the eovariance matrix.

Z

Pij~iG j

i,j

Conversely, if

C

is given in

the form (1.2), then the uniqueness of the Fourier transform shows that the joint distribution of

Xl,...,x N

is given by (1.1) where

Q = F -1

so

Xl,...,x N

form

a Gaussian system. We remark that we have introduced real-valued random variables; we will sometimes use complex-valued random variables in which case the distributions of the random variables are just measures on measurable map from

C.

In general, a random variable is a

to another measurable space

This concludes the elementary introduction I promised.

O, and

f+ and f_ have

Then, IIEz0flI < JlEz0f+JlI + llEz0fJl

: IIf+llI ÷ llf llI : IIf;lI so EZO

is a contraction on L I. If f c L~ (,Z,d~), then for all B~

O,

-llfll~(B) ~ IB f d~ ~ Ilfll~(B) SO s

-Ilfll ~(B) < I B E~of d~ < Ilfll ~(B)

which implies that

J]EZofJI~_< J]fII. Thus EZO

is also a contraction on L~ .

(5) now follows immediately from the Riesz-Thorin theorem. To prove (6) notice that L2(g,Z0,d~) EZ02 = EZO, and L2(~,Zo,d~).

is a closed subspace of L2(C,Z,d~),

Ran EZO = L2(~,Zo,d~), so EZO

For

is a projection onto

fl,f2 e L2(~,Z,d~)

(Ezo fl' f2 ) -- (~zo fl, Ezo f2 ) : (fl' Ez o f2 )

i9

so

EZ0* = EZ0

and

EZ0

is an orthogonal projection.

Typically, conditional expectations arise by taking generated by a collection set; in this case where

[x~}~E I

E(xlZ0)

~0

to be the a-a/gebra

of random variables where

is often denoted by

I

E(xl{x~]~ei).

is some index Consider the case

I = {1,2,...,N] ; then we are conditioning with respect to an algebra

~0

generated by finitely many random variables, Xl,X2,...,x N. Given a random variaN ble x, E(xl{xi]i= l) is then a function on 2 measurable with respect to % " From this it follows that there is a Borel fhnaction ~

on Bn

so that

N

E(xl{xi)i=l)(w) = $(xI(~),...,XN(~)).

Therefore, if

B ¢~N

N

fx_l(B ) E(xI[xi}i=l)(~) d~(W) = IB ~(kl'''''kN) dgxl,...,XN (kl'''''~)

where

~Xl,...,x N

X,Xl,...,x N

is the joint distribution of

Xl,...,x N.

In the case where

is a Gaussian family the conditional expectation is especially

simple. Proposition.

Let

X,Xl,...,x N

means equal to zero and

be a Gaussian family of random variables with

Xl,...,x N

orthonormal. N

N

E(xl(xi}i__l) =

Proof:

Since

X,Xl,...,x N

Then

z i=l

(x,xi)xi

are Gaussian they are all'in

N

x - i~l (x'xi)xi of the

L2(2,Z,d~).

Further

N

xi

is orthogonal to each

xi

so

x - i~l (x'xi)xi

is independent

since the random variables are Gaussian (see ~l). Thus, by property

4, N

E(x -

Z (x,xi)xil{xi}i= i=l

N

N

1) = E(x -

Z (x,xi)xi) i=l N

= E(x) -

=0

Z (x,xi)E(xi) i=l

20

SO~

E(xl

N

N

N

{xi)i= I) = E( Z (x,xi)xi I[xi}i= 1 ) i=l N

=

7. (x,xi)x i i=l

@

We now introduce conditional probabilities as a special case of conditional expectations.

Let

x = ) ~ y_l(A ), where (Xs(W) : 1

for

y

be a random variable, A

XS

a Borel set in lq, and set

always denotes the characteristic function of the set

m e S, XS(~) = 0

if

m ~ S)o

Then

E(X

N l[xi]i:l)(m)

1

y"

S is

(A)

N

denoted by is in

A

P(y e A I [xi}i= I) given

denoted by

Xl,...,x N.

and is called the conditional probability that The corresponding

P(y ¢ A lx ! = kl, ..., xN = ~ )

bility that

y ~ A

~iven that

$(kl,...,k N)

(see above)is

and called the conditional prob a -

x I = kl, x 2 = k2, ..., xN = ~ .

We are now ready to give our precise definition of Markov process. {x(t)]

be a stochastic process indexed by

a-algebra generated by by

y

[0,~).

x(tl),... ,X(tn); Z(a,b )

Let Ztl,...,t n

Let

denote the

denotes the a-algebra generated

x(t), a < t < b; and denote the corresponding conditional expectations by

Etl,...,t n s > 0

and

E(a,b ).

[x(t)]t~(O,~ )

and random variable

(3.2)

is called a Markov process if for each

y ¢ Ll(2,Z(s,~),~),

E[o,s]y = EsY

i.e., E[o,s]y time, Z(s,~ )

is already measurable with respect to

Z s.

If

s

is the present

corresponds to future events, so the Markov condition says intui-

tively that the expectation of some event in the future given some information about the present and the past is the same as the expectation just given the present.

To illustrate that the fancy definition is Just a formal statement of

this intuitive idea, let us return briefly to the Brownianmotion constructed in ~2.

Let

r < s < t

and let

A, B, C

be Borel sets in ~.

Then

21

Prob{xt ~

A,

Xs ~ B, Xr ~ C] = E(Xx~I(A)Xx~I(B)Xx~I(c))

= I c I B ~(~,~) d~r,s(~,n) where and

~(~,~) ~r,s

is the fkmction expressing

is the joint distribution of

E xr

X

in terms of

r,s ~[I(A ) and

x s.

x

r

and

x

s

On the other hand we cal-

culated in ~2 that

Prob[x t E A, x s e B, x r E C) = I C I B (I A e " ( B - k ) 2 / 2 ( t - s )

Thus, ~(~,q) = fA e-(9- k)2/2(t " s) dk X I(A) Er's xt

is already measurable with respect to

E

× r,~

The same argument shows that if and

is a function of

dk) d~r, s(~,B)

q

only, so

Zs~ i.e.

=EX

xtl(A )

~ x~l(A)

r I < r 2 < ... < r n < s < t I < t 2 < ... < tin,

A. s R, then 1 m

E rl'r2'''''rn'S

m

Z c.X = E Z c.X i=l 1 xt-l(Ai), s i=l 1 xt~l(R ) l l

and now the dominated convergence theorem and a measure theory argument show that in fact (3.2) holds. We now return briefly to the general case. define the transition probabilities

P(s,t,k,A)

Given a Markov process by

P(s,t,k,A) = Prob(x t E A ix s = k)

•

{x t}

we

22

That is,

P(s,t,xt(~),A ) = EsXx~l(A)

For each s, t, and

s, t, and

k, P(s,t,k,')

A, P(s,t,',A)

is a probability measure on ~

is a positive Borel function on ~.

and for each

Just as in the

case of Brownian motion we can write down the joint distributions in terms of the transition probabilities and the initial distribution, for example:

~[xt ~ A, x s E B] = IB IR P(s,t,N,A)P(0,s,~,d~) d~s(~)

Let

r < s < t, then

(3.3)

~{x t e A, x s ~ R, x r ¢ B] : /B /R P(s,t,~,A)P(r,s,~,d~) d~r(~)

But, (3.3) = ~{xt ¢ A, x r e B) = fB P(r,t,~,A) d~r(~)

so by the uniqueness of the conditional expectation we have

(3.4)

P(r,t,~,A) = fR P(s,t,~,A)P(r,s,~,d~)

almost everywhere w.r.t.

~n"

Equation (3.4) is called the Chapman-Kolmogorov

equation and it holds because given a measure joint distributions must be consistent.

~

on

the family of

Conversely, given a set of transition

probabilities which satisfy (3.4) identically one can use the Kolmogorov construction exactly as we did in the case of Brownian motion to construct a Markov process with the given transition probabilities.

23

§&.

SEMI-GROUPS

In the last section we saw that studying Markov processes can in same sense be reduced to studying transition probabilities which satisfy the ChapmanKolmogorov equations.

Such a system of transition probabilities is called

stationary (and the corresponding Markov process is called homogeneous) if P(s,t,k,A) P(t,X,A)

only depends on

It - s I. That is, for each

which is a probability measure in

Borel function in

k

for

A

A

for

k

t, we have a function fixed and a positive

fixed, so that the transition probabilities are

just given by

P(s,t,k,A) = P(t-s,k,A)

.

In this case the ChalEan-Kolmogorov equations take the form:

(4.1)

P(t + s,k,A) = fir P(t,B,A)P(s,k,d~)

We now define

(4.2)

Then

(Ttf)(k) = ~

P(t,k,d~)f(~)

Tt: L~ -> L~, lITtfll~ ~ IIfll , and property (~.i) in~ediately implies that

the semigroup property holds.

Tt+sf = TsTtf = T6Tsf

Thus, in studying Markov processes with stationary transition probabilities we are led naturally to the study of an associated semi-group of contraction operators. In the lectures by Nelson semi-groups on the space

L~(~,?0,~)

will arise

directly from the conditional expectations, so it is useful to do the analogous construction here.

Each

u ¢ L~(~,ZO,~)

u(~) = f(Xo(W))

where

the map

is an isometry between

u -> f

f

can be uniquely written

is a bounded Borel function on L (,Z0,d~)

and

supp ~0 CIR.

In fact,

L~(supp ~0' d~o)

For

24

each u e L (~,Tb,d~) we define

as follows Eo

u = f(Xo(~))

>

f(xt(~))

E(f(xt(~))I~

>

O)

\ \

is clearly a linear contraction on L (,Eo,d~).

property, let A

be a Borel subset of

suIrp ~0"

To verify the semi-group

Then,

\+s×A(xO(~)) : EoXA(Xt+s(~)) EO× Xt+sl(A)

=

EOE[O,S]Xx -I(A t+s

=EÊX u s

)

-lt • xt+ s ~Aj

(the Markov property)

= EoP(s,t + S,Xs(~),A) = EoP(t,Xs(~) ,A)

(stationary trans. prob.)

= ~sP(t,Xo(~),A) ;

-- TsEO×A(Xt (~)) = Ts\~A(xO(m) ) Since linear combinations of the functions and the

\

XA(xO(~))

are dense in L~(G,ZO,~)

are bounded this proves the semi-group'property.

semi-group on L~(G,EO,~)

Of course this

is just the result of lifting the semi-group

25

on

L~(supp ~0' d~o)

spaces.

to

L~(2 Z .

'~ using the isomorphism between the two

~ 0 ~/

Our computation shows how the Markov property and the assumption of

stationary transition probabilities gives rise directly to a semi-group of contractions on

L~(~,FO,~).

I will now give a very brief sketch of the semi-group theory which you will need to know. family

[Tt]t> 0

s,t >_ O, and as

A contraction semi-group on a Banach space

t -> 0

of contraction operators so that

TO = I. for all

ITt] u e B.

B

is a one-parameter

Tt+ s = TsTt = TtT s

is said to be strongly continuous if

for all

Ttu - u --> 0

To see that the condition of strong continuity is

not trivial notice that in the case of Brownian motion the semi-group given by (~.2) is strongly continuous on the space of bounded continuous functions but is not strongly continuous on the bounded measurable functions. (Ttf)(k)

is continuous for all

t > O

even if

f

This is because

is just bounded and measura-

ble. We are interested particularly in the case real or complex functions, where ties that

(Ttf)(x) > 0

if

Tt

is assumed to have the additional proper-

f(x)>0

tion which is identically one.

B = LP(x,d~), an L p space of

and

Tt~

=~

where ~

is the fkmc-

Any such contraction semi-group on the real

valued LP(x,d~) functions can be extended uniquely to a contraction semi-group on the complex-valued LP(x,d~) functions; so from now on we will always deal with Banach spaces over the complex numbers. Let now set

Tt

be a strongly continuous semi-group and set

D(A) ~ [u e B I Lim Atu t-~ 0

exists}

and define

S

If we set

u s = fO Ttu dr, then S

%u s and

I° Tt÷r dt

A t = t-l(I - Tt).

Au = Lira Atu t-~0

for

We

u ¢ D(A).

26 S

ArUs = lr fo (Ttu - Tt+rU) dt

r

- ~ ~

-r

Thus, for all

s+r

Ttu dt

-r

u ~ B, u s e D(A) .

is in fact dense in Tt: D(A) --> D(A)

B.

! f

Ttu dt

Since

u s -> u

s

Furthermore,

if

--~--~>

as

u - T u . S

s -> 0, we see that

u e D(A), then

AtTtu = TtAtu

D(A) so

and

d-~Ttu = -ATtu = -TtAu dt

By similar techniques one can show that infinitesimal generator of

Tt

A

is closed.

and we will write

A

Tt ~ e

is called the -tA

The generators of

contraction semi-groups on Banach spaces are characterized by the Hille-Yosida theorem (see [5]) but we will only need a simple special case.

Proposition.

A closed operator

A

on a Hilbert space t'l

is the generator of a

strongly continuous semi-group of self-adjoint contractions if and only if self-adjoint and Proof:

A

is

A ~ O.

The if part follows i~nediately from the spectral theorem.

We just

define

e

-tA

u = fo

e -kt

dEku

and use the functional calculus and the dominated convergence theorem to prove the semi-group property and strong continuity, e

-~t

e

-tA

is self-adjoint because

is real valued. Conversely suppose

semi-group o n ~

Tt

and let

is a strongly continuous, A

be the generator of

Tt.

self-adJoint, If

contraction

u,v ¢ D(A), then

27

(Au,v) : Lim (~(I - ~t)u,v) t* 0

: L~ (u,~(I -~t)v) t~O

: (u,Av) so

A

is symmetric.

Further since

Tt

is a self-adjoint contraction

(Ttu,u)

is real and

(Ttu,u)S Thus

t ~ ( l - Tt)u,u) ~ 0

let

for all

k > 0; to show that

Ran (A + k) = ~ u E D(A),

or

A

II~tullll~ll ~ (u,u)

t , so

(Au,u) = l i m ~ ( ( I - Tt)u,u) ~ O. toO

Now,

is self-adjoint we need only prove that

Ker (A* + k) = {0}.

Suppose

(A* + k)v = 0

and let

Then

: -(A~tu,v) : -(~t=,A*v) = k(Ttu,v)

so

(Ttu,v) : (u,v)e kt .

contradiction unless v = 0

since

D(A)

Now, let by

A

Since

k > 0

(u,v) = O. is dense.

e -tA

and [ T t ~ a r e contractions, this is a

But

Thus

(u,v) = 0

Ker (A* + k) = (0}

w(0) : o, IIw(t)ll 2 ~ o,

stances is

e -tA

A

is self-adjoint.

w(t) = Ttu - e-tAu.

Since

for all

= -2(Aw(t),w(t)) ~ 0

t, i.eo

"-tA Tt = e

The problem which we want to investigate is this. probability space and

and

means

and

d(w(t),w(t))

w(t) = 0

u e D(A)

be the strongly continuous self-adJoint semi-group generated

as in the first part of the proof and set

we see that

for all

A > 0

is self-adJoint on

a contraction semi-group on

Suppose

L2(D,d~).

LP(2,d~)

for

is a

Under what circump # 2.

Before

28

stating and (partially) proving a theorem about this we need some.definitions. A strongly continuous, bounded semi-group

Tt

called a bounded holomorphic semi-group of angle (1)

Tt

on a Banach space . ~ 8, 0 < 8 < ~

is

if

is the restriction to the positive real axis of a family of opera-

tors

Tz, z e S 8 = {z I

larg z I < 8], so that

T u

is a holomorphic

Z

vector-valued function for all W ( z + z') = T(z)~(~') (2)

For each

eI < e, Tz

T(z)u --> u Notice that on

as

L2(G,d~)

u E~,

z e S8

and

z,z' ~ S 8.

for

is uniformly bounded in the sector

z --> 0

in

Sel

and

Ss1.

we may define

co e-ZAu = ~0 e-Zk dEku

'

re z > 0 .

Using properties of the functional calculus and the dominated convergence theorem one can easily show that (1) and (2) hold so semi-group of angle ~ on We will call all

e -tA

u e L 2 N Lp

and all

tinuous for all

e -tA

is a bounded holomorphic

L2(2,d~). an LP-contractive semi-group p e [i,~].

p < ~, we will call

If the map e "tA

if

< JJullP IIe-tAullp_

t --> e -tA

for

is strongly con-

a continuous LP-contractive semi-

group.

Theorem.

(Stein)

Let

be a finite measure space

a positive self-adJoint operator on Ca)

If

e

and

-tA

(b)

e -tA

(where k

For

Lq

f(x) ~ 0)

is the function which is identically'one)

is an LP-contractive semi-group.

Ker (e-tA ~ Lp) = {0)

(c)

if

Every LP-contractive semi-group is automatically continuous.

in

A

Then,

is positivity preserving (i.e., (e-tAf)(x) ~ 0

e-tA~ = ~

then

L2(M,d~).

(~(M) = l) and

for all

for all

p > 1

and

Ran (e "tA ~ L q)

Moreover, is dense

q < ~.

1 < p < ~, e "tA

is a bounded holomorphic semi-group in the sector

29

s(P):~zt Proof:

i~gz1~(1-

We begin by showing that

First, suppose

f g L2

and

e "tA

f ~ 0.

IF2 _ ml)}

is a contraction on all the L p spaces.

Then~

IIe-tAfIIl = (~, e'tAf) = (e-tA~_~,f) = (h,f) = IIfllI

If

f E L2

is real-valued, then we write

(f+,f) = O.

f = f+ - f.

where

f+,f_ ~ 0

and

Then Ne-tAflll < lle-tAf+II! + Ile-tAf_ll1

--llf+Ji I+ F~nally, suppose

f(X) ~ L 2

l(e-tAf)(x)I

JIfJlI --llfll

is complex-valued.

=

Then

[Re [e-i~(e-tAf(x)]} sup rational [Re [(e'tA(e'i~f))(x)]} sup rational [(e-tA(Re e-inf))(x)} sup rational

for almost all

x, where we have used the fact that

into real functions since g ~

e

-tA

e -tA

takes real functions

is positi~ty preserving.

Also, for each real

L2 ,

(e'tAg)(x) = e-tAg+ _ e'tA(g_) < e-tAg+ + e-tAg. = e-tAIg(x) I

almost everywhere.

Therefore,

l(e-tAf)(x)I ~ e-tAIf(x) l a.e.

30

which

implies that

lle-tAfIll 0. on

$, $ ~ L p.

e "t0A~ = 0

and by analyticity

is strongly continuous on

on

L2°


satisfy

Since

Ker (e "tA) = 10}

and

Lp

q

and let

Then

t > 0.

Lp

1 ~ p ~ fi then

- flip < Ile-tAf - ffl 2

is dense in

are uniformly bounded on

If

We will con-

L q, ~ = 0.

(e "tA} Now~ for

e't~ = 0 Thus

The reader can easily check that the

Lp"

So we conclude that

Then

He-tA~ - Wlp _< lle-(t÷t°)~- e-t0Allp

>

0

Ran (e "tA)

is

31

as

t -> 0

Ran e

by the holomorphicity in the interior of

(e'tOA)

-tA

is dense and the

{e "tA]


Warning.

The semi-group

is not LP-contractive°

Tt

S (p)

quoted above.

Since

are uniformly bounded~ we conclude that

Lp.

which we constructed earlier (4.2) on

T, ( ,Zo, %)

To see this one need only calculate that

IIXA(Xo(~))II Z ~ z

= ~O(A)

S ( , 0,~0) but

[I~tXA(xo(~))IILZ(~,Zo,%) = NP(t,Xo(~),A)NL1 = ~t(A) and for appropriate

A, ~t(A) > ~o(A).

If the Markov process which we con-

structed was stationary (i.e. ~t(A) = ~o(A)

for all

t

and

A) rather than

just having stationary transition probabilities, then we would have LP-con tractivity.

This is why the processes which Ed Nelson will construct will be

generalizations of the so called Ornstein-Uhlenbeck process which is stationary.

§~.

Suppose that

GENERALIZED STOCHASTIC PROCESSES

xt, t > 0

is a stochastic process of the type we have pre-

viously discussed, i.e. for each space.

is a random variable on some probability

Suppose that we write

(5.1)

where

t, xt

$(f) = ~

f

f(t)xt dt

is in some suitably nice class of functions

linear map from

E" to the random variables.

E.

Then

f --> ~(f)

This suggests the following:

is a

$2

Definition.

Let

E

probability space.

A linear map

~

from

E

s~n).

E

a

to the random variables on

is called a generalized stochastic process over

The space or

be a locally convex topological space and

E

on

<X,E,~ >.

will usually be some space of smooth functions like

S(O,~)

Given a stochastic process one can construct a generalized stochastic

process by (5.1) but the converse is not necessarily true.

Generalized stochas-

tic processes are "random variable-valued distributions" while stochastic processes are "random variable-valued functions.""

It is this more general notion of

stochastic process which arises in quantum field theory. In §2 we showed how a stochastic process can be "realized" on the set of all functions on

[0,~].

What this meant was that we could construct a probability

measure on the set of all functions so that the stochastic process

Et

evaluation at

x t.

t

has the same finite dimensional distributions as

this section by making an analogous construction for E*

the dual space of

E

and by

Let

F

Let

PF

={T~lg

I T(f) =0,

be the natural projection of

isomorphic to

F*

and

F*

(from ~n) of Borel set in

E~g

onto

E L g / F a.

A set

f~F~

E~Ig/F a.

A C ELg

E~alg/Fa.

The collection of cylinder sets based on

is

and define

Since

E*alg/Fa

of the form

A = pF!(B)

where F

is called a cylinder set B

is a Borel set in

will be denoted by

The smallest G-algebra containing all the

EF

spaces

Now for" f E E, T ~ E*alg, define

F

in

E

will be denoted by

ZE.

$(f)(T) = < f , T >

For each

f, ~(f)

is

is finite dimensional there is a natural notion

F

A

E

for all

based on

if

We will denote by

We first introduce a ~-algebra on

be a finite dimensional subspace of

Fa

We begin

E~alg the algebraic dual space, i.e. the set of

all everywhere defined linear functionals. E~ig.

@(f).

given by

is a function on

E* alg

for all finite dimensional sub-

~ T(f)

measurable with respect to

ZE.

Thus

33

if

~*

is a probability measure on

ized stochastic process on Now, let

~

<E*alg,% >, then

<E*alg,ZE,~*>

over

f --> ~(f)

is a general-

E.

be a generalized stochastic process over

E

on

< fl,Z,~ >

be a basis for a finite dimensional subspace

F

of

E.

and

n

let

(fi}i=l

Let

~F

n

be the joint distribution of

[~(fi)]i=l, i.e.

^

Then

~F

induces a measure

i F on E ~ I J F a by

^

~F(B) = ~F(B)

where

n

= [i{l cifil < C l " " ' e n >

and

=i

is the dual basis to

~ B]

{fi}i=l . We now define i.

~ ( A ) = ~F(OF(A))

for a cylinder set

A

in

E~g

based on

independent of the choice of basis is, if

F1 ~ F 2

then

~*F2 ~ ~ l

F.

n [fi]i=l

= ~FI"

It can be checked that and the

~

are consistent.

,

A

That

cylinder set

based on we obtain a measure on the family of cylinder sets in [~(fi ) } n-- i=l

is

Therefore, setting

~*(A) = ~(A)

bution of

~*

E* alg"

F The joint distrl-

is

-- ~F{~ ° ~ i t

i=l l i=l

Then it is easily checked that L2(E*,7_,d~). -E

d~(T)

S

extends to a unitary map of

H

onto

If we set

u(f) = s u ( f ) s "l v ( f ) -- s v ( f ) s - 1

Then

U(f)

and

V(f)

act on

h ~ L2(E*,EE,d~)

by

(~(f)h) (T) = e i < f ' T >h(T) (9.~) (V(f)h)(T) = (V(f)~)(T)h(T+f)

Conversely, given any probability measure

~

on

<E*,ZE>

define a representation of the Weyl relations over turns out that two such representations sponding measures

du I

and

d~ 2

E

on

the formulas L2(E*,EE,d~).

(9.4) It

are equivalent if and only if the corre-

are equivalent

(i.e., absolutely continuous

with respect to each other).

Remark.

It is reasonable to ask why (9.3) is the appropriate generalization of

the finite dimensional case of the Weyl relations.

Instead we could say that a

representation of the Weyl relations is a pair of families

[Uk(t)] , [Vi(t)]

of

38

strongly continuous unitary groups on a Hilbert space

H

(1)

[Uk(t),U~(s)]

= 0 =

(2)

Vk(t)U2(s ) = e

ist5 . kZu~(S)Vk(t)

(5.~)

so that:

[Vk(t),V2(S)]

Of course, given a representation of the form (9.3) we can always get a representation of the form (5.9) by choosing an orthonormal basis inner product

(',')

on

E

and defining

{fk }

in the

Uk(t ) = U(tfk) , Vk(t) = V(tfk).

How-

ever, to go frc~ (5.9) to (9.3) one must show that a certain family of infinite products

H Uk(t k) and K Vk(Sk) make sense. In fact, this can be done. Every k k representation of the form (9.9) can be continuously extended to one of the form (9-3); see [i0].

Generalized stochastic processes often arise in the following ws~v. Let (" ,-)

be a continuous inner product on a nuclear space

continuous characteristic function and thus by the Minlos probability measure

p

on

<E*,Z E >

fl '°'" 'fN

be independent vectors in

1 .z . GiGj ( f i , f j ) l,J

Then

e -~(f'f)

is a

theorem there is a

so that

e-½(f'f) = rE* e i < f ' T >

Now, let

E.

d~(T)

E.

Then

-~(z %fi z ~ifi)

-2

= e N

i

Z %~(fi)(T) i=l

= rE* e

i Z (~iki = ~N

e

dp~ifl ) ,... ,~(fn)(k~

Therefore, by our remarks in §l, q~(fl),... ,q~(fN) (i.e.

d~(fl) ,... ,~(fn)

are a Gaussian system

is a multi-variate Gaussian) with means equal to zero

$9

and covariance on

E*

and

Pij = (fi'fj)"

~(.)

For this reason

~

is called a Gaussian measure

is called the Gaussian process with mean zero and covariance

: (f,g).

The following theorem is essentially a corollary of the proof of the Minlos Theorem.

Theorem.

Let

function. tion of

E

Let E

be a nuclear space, ~ ('")0

Suppose that

Hilbert-Schmidt operator on T

(b)

E C Ran T

(c)

The map

E*, C

be a continuous inner product on

in ('")0"

(a)

a measure on

H0

C

E

is continuous on

its characteristic and

H O.

H0 Let

the compleT

be a

satisfying:

is one to one and

T'I(E)

is dense in

H0•

T-1 E

> H0

Then the support of

~

In the statement pairing between

E

and

theorem see [4], [14].

is on

(T-l)*

is continuous. (T-!)*HO * C E ~.

and

HO*

E*, not the

mean "adjoint" and "dual space" in the

inner product on

H O.

For a proof of the

To illustrate the use of this theorem we conclude with

an example.

Example.

Let

~

be the measure on

ess with means zero and covariance

S'(lqn)

corresponding to the Gaussian proc-

(f,(-A +l)-lg).

Let

P = -A + 1

and let

i

H i

denote the completion of

role of

H0

in the theorem.

S(~ n) Let

in the norm

llP'2fl12. H_I will play the l H = L2(lq~).~- Since p2 is a unitary map o f H 1

into

H 1 , T: H0 --> H0

Hilbert-Schmidt on 1

H.

is Hilbert-ScNm~.dt if

Since

T =

Thus if

TO

and only if

Top4 ' we

have

T-1

i

TO = P-2TI:~

= ~T

"2

is

and

1

(T-l) * = P-g(To1)*~.

is Hilbert-Schmidt on

conditions (a), (b), and (c), the support of

g

will be on

H

and

T

satisfies

40 1

(~-l)~l

1

i

: p-~(Tol)*~(p-~) : p-½(T~I)*H

n

Let

~2

TO = P ~P~Q-~

where P1

(5.4)

Then

=

-+l) (" -~x12

~ >~

TO

,

and Q :

(x 2 + l ) .

Suppose that

~ >o

will be Hilbert-Schmidt since it is an integral operator with kernel

[ ( ~ ) ( x and the kernel is in

L2(R 2n)

- y)](y2 + 1)-B

if (5.6) is satisfied.

This depends on the fact

that dnp ]

0.

Furthermore, it is easy to check that

the conditions (a), (b), and (c) of the theorem.

n

1

T = p-2ToP2

Thus, p

satisfies

has support on

1

[P~:-2P?Qt3f t f e L2(1Rn)} for all a

and

B

satisfying (5.4).

In the case

n = 2,

has support on the

set

(plaQ~fl f for any a > 0. d2

(. --

dXl2

+ 1)~g

~ ~2(~2)}

In particular, for almost all (with respect to is locally in

L 2.

~)

g E s'(~n),

41

The local result, i.e. that the support of n in

~

is on paths which are locally

1 2

L 2, is contained in the work of J. Cannon [16]. For further results

on the L 2 support properties of

~

see M. Reed and L. Rosen [17].

The following

appendix by P. Colella and 0. Lanford contains results on the support of terms of the lim sup properties of the sampSe paths.

~

in

42 BIBLIOGRAPHY

[i]

Breiman, L., Probability, Addison-Wesley, 1968.

[2]

Feller, W. F., An Introduction to Probability Theory and Applications, Vol. I (

[3]

), Vol. II (1971), John Wiley & Sons.

Guelfand, I. M. and N. Y. Vilenkin, Les Distributions, T. IV, Dunod, Paris, 1967.

[4]

Hida, T., Stationary Stochastic Processes, Mathematics lecture notes, Princeton Univ. Press, 1970.

[5]

Hille, E. and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc° colloq, pub. 31, 1957.

[6]

Ito, K. and H. P. McKean Jr., Diffusion Processes and Their Sample Paths, Academic Press, N. Y., 1965.

[7]

Lo&ve, M., Probability Theory, Van Nostrand, 1960.

[8]

Mathematics of Contemporary Physics, R. F. Streater, ed., Academic Press, 1972.

[9]

Minlos, R. A. , "Generalized random processes and their extension to a measure," Trudy. Moscow Mat. Ob~. 8 (1959), 497-518.

[lO]

Reed, M., "A Garding domain for quantum fields," Cc~m. Math. Phys. 1_~4 (1969) , 336-356.

[ll]

Royden, H., Real Analysis, Maemillan, 1963.

[12]

Statistical Mechanics and Quantum Field Theory, ed. By C. de Witt and R. Stora, Gordon and Breach (1971).

[13]

Stein, E., Topics in Harmonic Analysis, Annala of Mathematical Studies 63, Princeton, 1970.

43

[14]

Um~nura, Y., "Measures on infinite dimensional vector spaces," and "Carriers of continuous measures in Hilbertian norm," Pub. of Research institute of Kyoto Univ. A.1 (1965), pp. 1-54.

[15]

Varadhan, S. R. S., Stochastic Processes, lecture notes from the Courant Institute, N. Y., 1968.

[16]

Cannon, J., "Continuous sample paths in quantum field theory," to appear in Comm. Math. Phys.

[17]

Reed, M. and L. Rosen, "Global properties of the free Markov measure," to appear.

APPENDIX:

SAMPLE FIELD BEHAVIOR FOR THE

FREE MARKOV RANDOM FIELD

Phillip Colella and Oscar E. Lanford IIl Department of Mathematics, University of California Berkeley, California 94720

I.

INTRODUCTION

This appendix is concerned with the following question: denotes the Gaussian probability measure on and covariance

If

~0

S' OR 2) with mean zero

((-A+l)-if,g), what are the properties of "typical"

distributions with respect to

~0 ?

A first result in this direction

is given in the final paragraphs of Professor Reed's lectures; he shows that, if < - d ~ + I)-~T

~ > 0,

then for almost all

T E S'QR 2),

is a locally square-integrable function.

For ease of reference, we will summarize our results here in something less than their full generality:

Theorem 1.1. (a)

The set of distributions

that there exists a non-empty

open set

a signed measure is a set of

~o-measure

Alfred P. Sloan Foundation Fellow. Grant GP-15735.

UT

T

having the property

on which

T

i8 equal

zero.

Also supported in part by NSF

to

70

C

corresponding

to the

£~

vector norm on

~k

is given by

k

Thus, if

R

llcll :

sup

=

l~i~k

is large,

is also small, so

;. Icij; j=l

liB-Ill

Jak~-ll

a~

is small, so (since k

[

Jaki I are small.

R

~{Tk~2n(1-6)Z

when

large enough so that, for all

Itil ~ v ~

k

for Z~s

(i+6/2) log(lal) ,

with

wk ~

k,

~ IT1 : tl,...,Tk_ 1 : tk_ I} ~ l-exp[-£kn(l-6/2)]

the fact that the

lim k+=

We can there-

akk

i:l

fore choose

A : B-I),I}A-~11

1 ~ i < k-l.

If we do this, and if we use

are an enumeration of the numbers a e Z n and

lal > R,

we obtain from (4.3)

{l-lal (I-62/4)n} ~ {l-exp[-Zkn(l-~/2)] } = T[ k=l IaI>R a~Z n

completing the proof of the lemma and hence of the theorem.

: 0,

46 Result c) gives rather detailed information about the behavior of a typical distribution almost every

at infinity.

It says in particular that, for

T,

k}

2.3, it suffices

< cle

to show that 1

lim ~{T: l~.~

sup {'ITa/2]'A,e, B [ l o g ( l a l + 2 ) ] - ~ > I} = 0 a6~n

(2.4)

where

Ta/2(x)

By Lemma 2.4 and the translation Theorem

: T(x-a/2).

invariance

of the hypotheses

of

2.2 1 ~{T:

for any

UTa/211A

a E Z n. Hence,

,

> k [log(laI+2)] ~} ~ Cl(laI+2) -c2k

the left-hand

side of (2.4) is no larger

than

55

-c2k2 CI

lim k+~

~ a6~n

We now come to the essential of Lemma 2.4. fication measure. Their

The argument

of the argument

Sample Paths,

(X,~)

: 0.

and most difficult

step--the

we will give is a straightforward

used by Ito and McKean to construct

(See K. Ito and H.P. McKean,

For fixed on

(lal+2)

Springer-Verlag

x,y e A,

Jr.,

Diffusion

(1965)pp.

T(x) - T(y)

proof modi-

Wiener

Processes

and

12-15.)

is a Gaussian

random variable

with mean zero and variance

X(x,x)

By the hypotheses

+ X(y,y)

of Theorem

- 2X(x,y).

2.2, this variance

is no larger than

-I c2~ 2

for

(Ix-yl)

log

0 < Ix-YJ < l

~{T:

for all positive depending

only on

provided

Ix-yl ~ ~-.

Hence

again

2

IT(x)-T(y)I

y. c.

will need to complete

Here,

c4Y > y 6 ,8(Ix-yl)} < c3.1x-y I

c3, c 4

are strictly

This inequality

positive

2 (2.5)

constants

is the only property

of

~

the proof.

We now proceed by constructing, X(y) C X

that

for each positive

y,

a set

and proving

a)

^ ~(X(y)) ~ c I exp[_Csy2 ]

for all positive

b)

There exists

c6

a constant

T ~ X(y) , and all

IT(x)-T(y)I

x,y

such that

with

~< c 6 Y 6~,8(Ix-yl)

•

y

for all

y,

all

we

56

The lemma then follows For notational remainder ate.

immediately,

simplicity,

of the argument;

(of order

c2 = c5 c6

we consider

the extension

We will say that a pair

ary pair

with

x:y

of elements

if

i)

the components

of

x

are integral

2)

each component

of

y

differs

We now define j = I~2,...

of

X(Y)

T ~ X(y),

then

A

is immedi-

is an element-

of

2 -j

from the corresponding

to be the set of

,6(2-J-i).

of

for the n

multiples

either by zero or by

and some elementary

IT(x)-T(y) I > 2y~

If

x

n = 2

to arbitrary

j)

component

only

-2

pair

±2 -j-l.

T E X

such that,

(x,y)

of order

for some j,

In other words:

IT(x)-T(y) I < 2 y 6 8 ( 2 - J - l )

for all

(2 .6 ) elementary

pairs

x,y

of order

To prove a), we pick an elementary z = (Xl,Y2).

ly-z I

Then

Ix-zl

is

0

j = i~2, . . . .

pair or

x,y

2 -j-l,

(order j)

and define

and similarly

for

Thus

P{iT(x)-T(y) I > 2y~e,8(2-J-l) } ~< ~{IT(x)-T(z)[

> y~e,8(2-J-l)} (2.7)

+ ~{IT(z)-T(y) I > ¥$

,6(2-J-1)}

-c4y2(j+l)

~ 2 03 2 A

by (2.5).

To estimate

over all elementary pairs

of order

j

p(X(y)),

pairs.

we sum the right-hand

For a given

is smaller than

j,

8 ×

the number of elementary

(2J+l) 2,

so

2

~ ( X ( y ) ) ~
2-J} i:i,2 There exist of

2 -]0

x(0),

y(0),

whose components are integral multiples

such that

ix(0)_xil i

< 2

-J0

,

(0) -J0 Yi -yil < 2 ,

What we want to show is that, for

(0) (0) -J0 (i=1,2). xi -Yi I < 2

T ~ X(y),

IT(x)-T(y)I < c6Y ~a,8(Ix-Yl )

Since

Ix-yl > 2

small

e,

-J0,

and since

6

(e)

is increasing in

e

for

it suffices to show

jT(x)_T(y)i ~ c7 y6 ,8( 2 30).

(2.8)

By the triangle inequality

IT(x)-T(y)l ~ IT(x)-T(x(O))l + IT(x(O))-T(y(O))l + IT(y(O))-T(y)l.

It follows readily from (2.6) that, if

T ~ X(y),

IT(x(0))-T(y(0)) I < 4 y~ ,8(2

(Put

z (0) = ~x 1 (0) + ~y 1 (0) ;

elementary pairs of order and

IT(y)-T(y(0)) I

then

j0.)

-j 0-i

(x (0) ,z (0) ]

).

and

The estimates of

(y (0) ,z (0) ) IT(x)-T(x(G)) I

are identical; we will give only the first of

are

58

them.

Recall that

'('Ixi-xi0) l < 2

-J0

It is easy to see that we x(1),x (2),...

can construct inductively a sequence i)

x (k) , x (k+l)

such that

is an elementary pair of order

J0+k

(k=0,1,2,...) ii)

x i(k) -x i I < 2-Jo-k

It follows from ii) rational) that that, if

(i:i,2)

(and the fact that each

x (k) : x

x.l large

for sufficiently

is a dyadic k

and from i)

T ~ X(y)

IT(x(k+l))-T(x(k)) I ~ 2 y6 ~,8(2

-j0-k-1)

for all

k,

so

-j0-k-i IT(x(0))-T(x)l

Hence, for

~ 2y. k~0 6~'B(2

)"

T ~ X(Y),

IT(x)-T(y) I ~ 4 yI6c,,B(2-Jo-l) +

~ 6

2-Jo-k-l) 1

k=0 a'8( Since

lira ~(2-J-i)/6(2 -j) = 2-°" < i,

the estimate

(2.8)

follows,

completing the proof of Lemma 2.4 and

hence of Theorem 2.2. We note in the following proposition some subsidiary results which follow from the above considerations.

Proposition 2.5 a).

constants

Let

A

be a bounded set in

c 8, c 9 (depending on

A)

such that

IRn.

There exist

59

p{T:

sup x,yeA

[JT(x)-T(y) l] [6a'~(Ix-yl)] >

-c9~, 2 l} ~

c8e

Ix-y b)

In addition

is bounded.

to the hypotheses

Then for

p-almost

sup l JT<x _]

every

of Theorem 2.2, assume

X(x,x)

T

I } ~ xe[ 0,1 ]n

for all

a 6 zn;

cl0e

we then argue as in the proof of Lemma 2.3 from

Lemma 2.4. It remains to prove Proposition 2.1. We will consider only < ½; the proof for ~ = [1 is similar but slightly messier. We want to estimate:

X(0)-X(x) -

1 _ (2~)2

I

2+ 2+ -1 2+ -e . dPldP2 (Pl P2 i) (Pl i) {l-exp[i(PlXl+P2X2)]}

Using the fact that the remainder of the integrand is even in and

P2

Pl

separately, we can replace the term in braces by

{i - CoS(PlX I) cos(P2X2)} ~ {i - Cos(PlXl)} + {I - cos(P2X2)}.

60 We give the argument of the terms

on the right;

O(~o) = ~ol+2a

#(m)

for estimating

approaches

=;

2

the first is easier.

2 2 dPl(Pl+m

is continuous

for

hence,

2+

fdPldP2(Pl+P2

the contribution

,-1cp

m > 0

+l)-a =

from the second

For

m > 0

define

do(a2+l)-l(o2+l/~o2)-e

and approaches

it is bounded on

a finite

[i,~).

l)-l(p2+l)-~{l-eos(P2X2)}

;

limit as

Now:

=

1 =

21dP2(P~+l)-2--e~~

2a

) sin2(P2X2/2)

f~

2

2 . s u p ~(~) .x 2 "/_ dT(Tj ~>1

2

+x 2)

1 -T a

2

s i n (T/2)

Since 1 =dT (T 2+x2 )-2--esin2 (TI2)

f_~ approaches desired

Z

a finite

limit as

Ix21

zero, we have the

estimate.

IIIo

NON-REGULARITY

We will prove in this section a) of Theorem

Proposition Assume

approaches

version

of statement

i.i.

3.1.

Let

X

that the Fourier

measure)

an abstract

has infinite

be a distribution transform of

total mass;

X

of positive

type on

~n.

(which must be a positive

equivalently,

assume

X

is not a !

continuous

function.

Let

~

with mean zero and covariance

denote the Gaussian measure ×.

Then

g-almost

on

S (IRn)

every distribution

6!

T

has the property

TIU

such that

Proof.

~

let

open set

U C ~n

an

~i C U 2.

is a signed measure.

We will need:

Lemma 3 . 2 . Let

that there exists no non-empty

UI, U 2

Let

be b o u n d e d open sets in

be a signed measure ~n )

support

be a sequence

of finite

total variation

of continuous

functions

contained in the open ball of radius

on

with on

U2 ,

an

with

d(Ul~n\u2 )

and

about

0,

and with

~fl~m(X)Idx < =.

(3.1)

Then

lim

f~(dY)~m(X-y)

: 0

for almost all

x E U1,

m+~

where

"almost all" is to be understood in the sense of Lebesgue

measure.

Proof:

fuldx /f~ 0

and

Cs,C 9

such that

x,y e A} > l-c 8 exp[-Cpl2].(4.2)

by

(1+6) 2 = (2K2)-I

and put

q =

1 (2c9e2) 2~ .

~{T:IT(x)-T(y) 1 %

Then it follows from (4.2) that

el for all x,y

with

Ix-yl < q} > l-c8exp[-X2/2].

67 If we now choose a distance

N

Xl,...,x k

of some

such that each point of

xi,

(l+s)k

~{T: supiT(x) I > xEA Proof of Lemma 4.5.

by

~ (K3.k+c 8) exp[-12/2],

k

and recalling the definition of

~} < (K3-k+c 8) exp[-K2 ~2]

Now let A : [0,i] n.

e)

o

Then

1 ~{T: lim sup ~ (~__) , defined by ~Q.(_~)= ~._,(~ ) , where ~ = analytic

Then

(Wl) and the Bargmann-Hall-Wightman

- but not of the vectors ~ . ( ~

-

(2)

is also simple to prove°

Let ( ~ , ~ ) ~ - n

invariance

rem, we obtain a single tions ~/.(~)

that the right hand side of

Euclidean

of the Wightman Green's

function

function ~)~(_~) to ~ ) i s or Schwin~er

function.

We set ~ o = 7 ~ o = 1 and

for

x e_ -C2_n~[x {xi%xj

From the invariance

properties

mediately

derive the following

Lemma 4:

The Schwinger

geneous

group

I ~ L ~ j ~ m}

of the Wightman

functions

we im-

lemma.

functions

proper Euclidean

arguments

for all

~.(_x) are invariant

under the inhomo-

{SO~ and under permutations

of the

x I, ... x n.

Let us now introduce

the difference

variable

Schwinger

functions

S._,(~) , defined by

where ~ k vector

= Xk+l - Xk and x ~ _C~ .

valued

functions

~ : s ( ~ ) by

We also define

the real analytic

77

for × ° > o , # ~ > o ,

x~ k~-n-1.

DefinerS_ by (,~)~_ .. ( - ~ ;

Lemma 3 and the above definitions

lead immediately

Lemma 5. Let all of X o , ~ o~ , ×,o , ~,~

functions,

see

(5)

property

Then

for the Schwinger

(E2) below.

The cluster property ~' in ~ ,

to

be positive.

Lemma 5 of course yields a positivity

u).

(W4) implies that for any two vectors ~ and

and spacelike a,

lim

(~;

Lemma 6.

U(~t,~)

~' )

For 0 < Xl 0~-

f~,n ~,~

• ..

(~x,,... ~x,,

=

( ~ , ~/_~ (/Lj~')

~Xn0 , 0 ~Yl 0 ~

~ , ÷ ~ . , ... ~,.,+:~)

The lemma follows essentially

-- ~

"-" ~-ym 0 , and a =

('~x ..... ~×, ) ~ , , ~ .....

from substituting

~=

(0, ~)

5',,).

~sn( ~ ~, ~,

~(~i),

!

= Xk+l - Xk' ~ k

= Yk+l - Yk in equation

The Schwinger

functions

define distributions.

digression.

Distributions By ~ + closure.

but they also

We shall collect some simple lem-

and Laplace transforms without giving proofs.

They are either contained simple consequences

functions,

Before we investigate that aspect we have to

stop for a mathematical man on distributions

(5).

are real analytic

in refs.

[SW i, Vl i, OS 3] or they are

of things established

there.

and Laplace Transforms

we denote the open half intervals Let ~(~z)

denote the space of functions ~

~_+,given the induced topology. --

(0, _+ O0), by ~--ztheir

~z

~(~)with ~uF~$ in

) is the set of all functions,

C ~

defined on ~ e

,

on ~ e , whose derivatives

all have a continuous

tension t o ~

and are of fast decrease at infinity.

The topology

exon

~(~,)is defined by the seminorms

I H I~.,. Lemma 7. space

~up

-~

W)'O ~( ~__.,tltl

(l÷×) r~l~ ~'°c'~ I

The space ~(R÷) is isomorphic

4(~)/~(~_)

to the topological

.

The main point of this lemma is that any element restriction

to R--~ of some element

in

~(~÷)

the set of distributions in ~ / ( ~ )

is the

in {(~ ).

The dual space of {(R) / ~ ( ~ ) is the polar of ~ _ )

a distribution

quotient

3-~ ~'(~) with

~FF~-~

~e .

, which is Hence by Lemma 7

can be identified with a distribution

in ~/(R)

78

with support in R--~ .

Lemma 8. ~-~ f

For ~

{(1~+)

we define ~ by

into ~(~+) whose range is

is a continuous map of f ( ~ )

dense and whose kernel is zero. Let

T6

~'(~)

with

fines a distribution

x ~ Z c

in ~t(~+)

~.

Then by Lemma 7, T

, again denoted by T ,

also de-

and we conclude

that there are contents c and m, such that for all ~ e ~(P~+)

(6)

)T(f)

i

~

c

Ira, +

I

=

c Ill"

On the other hand we can use the Laplace transform S of T to define a distribution

in

SCx and for

~ ~

~z(~+)

)

=

4(~)

.

je -~

For x > 0 we define

T(~)~

(More precisely we first define have compact support, S(×)

,

we set

similar to

~(~)

for elements in ~ ( ~ )

then prove continuity of S

(i), and finally extend S

which

using a bound on

to all of { ( ~ + ) . )

The next theorem contains the main result of this mathematical digression. Theo[em 9: define

S

Let T as in

I sCf)

(9)

be a distribution in

(7).

$

Then for all

~

c

I~

~

l"

for some constants c, m depending on m distribution in 4 ' ( R + ) , satisfying exists a unique distribution T ~ that

~'(R) with suppq-c~+ and ~(R÷ )

only.

conversely

if

5

is a

(9) for some c, m, then there

f'(~)

with support in ~

, such

(8) holds. In our applications we will use a multivariable version of Theorem

9, which due to the nuclear theorem is easy to prove.

Let us just

introduce the necessary notations and definitions. By

~ +4n

we denote the set {~_ I ~

its closure.

For ~

~-~(~t" )

> O

we define

, k = 1 ..... ~

by

n~,

by ~~4+n

79

and we introduce a set of norms on

(lO)

I~ L~

For

_

I~- I..~_

T ~ ~l(~.)

-

~"( ~ )

~.

by

)"~ I(-~',~

c,~,~,

with s u p p T c_ R +

I

,

we can again define

S

(for-

mally) by

(Laplace transform with respect to the

~

form in the distributional sense for the S(~ ) = in

5 ~(~>

S (~ ~ ~4~

~i (~%~)

variables, Fourier transqq

variables).

For ~ { ( ~ )

again defines a distribution

We leave it as an exercise to formulate the multi-

variable version of Theorem 9 for ~

and T

defined as above.

For later use we introduce two more sub-spaces of

~(~")

for some i # k, all ~ I

1R~ ) =. where

-

~2

{*__I

o<x,

o

×.o }

4...---

Now we are prepared to study the distributional aspects of the Schwinger functions.

For ~ :

Laplace transform of

~a/n(q)

> O

version of Theorem 9 applies. Lemma i0.

, ~.(~)

, see eq.

We obtain

a) The difference variable Schwinger functions S , ( ~ )

define distributions in

~

('1~÷)

through

s.(f~ = ~f~g) s . ( g ~ d ~ F for

is just the Fourier-

(3), and the multivariable

{ e ~(R+~" )

.

Furthermore

V

S. Cf} = W~ ( ~ )

, and for some c

and m

IS,,(i:)l

"--

c ffl"

b) The Schwinger functions

m

~ n (5) define distributions in

We remark that a) implies that ~ n ( ~ )

~:(~4,)

defines a distribution in

80

.~l(~m ment

) .

In order to obtain b),

is necessary,

geometrical

argu-

see lOS 3, Si i].

In the following Schwinger

additional

an

functions

theorem we collect

derived

all the properties

of the

so far and state our main equivalence

result. Proposition

I:

The Schwinger

theory have the following Distribution

property:

(E0) 5 n d e f i n e s respect Euclidean

Positivity:

to a W i g h t m a n

For each n >/ 1

in

I" I~

Covariance:

associated

properties:

an element

to some

functions

~ / ( ~ t m)

and is continuous

-norm.

For each n > 1 and all

For all finite

with

sequences

(a, R) £ ;ZO~ ,

~=, ~,)-- fN

func-

of test

rl j ~rl

where Symmetry:

(E3)

fn (Ox)

For all permutations

~

Cluster

(5) =

Of~

(x.....

Xn ) ~-

Property: ~.-

~m

7v, (x~(,)) ..

For all n, m,

(o,~)

~

~ d

~r(m) )

~(~L

~ ) ,

~ 6

f(~)

;

TR~"

where gka is defined by g~a (x) = g ( x + ~a). Conversely, Schwinger Proof:

Schwinger functions

of

as an ordinary

and cluster

properties

they hold pointwise.

obeying

(E0) -

(E4) are the

with a unique Wightman

(E0) -

4, 5, 6, and i0.

can be defined • metry,

associated

The derivation

from Lemmas

"functions"

(E4) from the wightman

Because integral, hold

theory.

~ ~

C ~ % ~(5) ~ ,

covariance,

axioms

+

positivity,

in the distributional

sense

follows

eo symif

81

For a proof of the converse stead of

(E0) that

5,

statement

it suffices

is in the algebraic

dual of

to assume

~(~4~)

in-

and that

/

it is continuous (E0) follows of theorem &

9,

(E0) implies

for ~

We define

supp~

C

~

,

R~ ~

k/~(@)

able Wightman

(E4).

~+

in

V~

I'I ~

We now introduce

another

distribution

see N e l s o n

[Ne 2]

(w2) and the cluster conditions

(E2) and

(E3) and all the

[Jo i], p. 83.

will be discussed in

vari-

(El) we conclude

from symmetry

In the

in more detail.

(E0)-see eq. (6) for the defini-

to deal with

is a Schwartz

From

distributions,

see ref.

which appear

tion - might be difficult

There

follows

%A/ ( q )

of the difference

from the corresponding

established,

version

(in the distributional

Positivity

these arguments

The norms

Then the rest of

By the multivariable

transform

invariant

--n

(W3) finally

already

-norm.

of our theory.

(W4) follow easily

next section,

(E3).

, such that

are Lorentz

Locality

I' I m

that there exist distributions

distributions

and hence have support

other axioms

and

to be the Fourier

that the ~/~cq )

property

to some

from Lemma 8~(EI)

~)with

sense)

with respect

in constructive property

field theory.

(E0'):

normI-I S on ~(~:) and some L > O, such that

for all n and for all f k e

f(~2)

, k=l . . . .

n,

(E0 ' )

I s.(f,~&~ Our main result

Schwinger

(~!)L 7T

~

is the following

Pr0Position ' II: determine

.... f~)l

k=l

proposition.

functions

a unique W i g h t m a n

I£1~

theory

satisfying

(E0'),

(whose Schwinger

(El) -

functions

(E4) they

are). III.

RECONSTRUCTING

In this section we start satisfying belonging

(E0'), to it.

(El) -

THE WIGHTMAN

from a set of Schwinger

(E4) and reconstruct

This will prove proposition

be the vector space consisting

functions

the Wightman

~n

,

theory

If, but it will also

shed some more light on the proof of proposition Let ~

is

,

c(,+t ~)

~

for t>/O ,s>/O

Furthermore

(17

f,,(,,,_° ~ +,,,,,

and for some c, m, depending on ~

I O

in the time variables

For the following arguments

, to construct

of the S c h w i n g e r

the space variables will play

no role, so we shall drop them completely

and write

Sn-i ( ~ )

and

84 ~i~(×,~_) instead of ~ _ , ( ~ ° [ g h ) ~_

now stands

~En (~°

for the n - I time variables

Sandwiching-=H for

and

~-t~i~

between

, t>

I h)

~oj..

two vectors ~

resp., where ~o_,

and ~

we find that

0 )

-"oH E is a distribution Cauchy-Riemann

in

~

, x + x' + t, ~l and

equations

in t and s.

s

which satisfies

It follows

the

(see e.g. v l a d i m i r o v

[vl 1], p. 31) that

s,~_,C~,x+*'*~,~l~) is a d i s t r i b u t i o n z, analytic smeared

, ~I variables and a function of x + x' + 1:=

in the right half plane

in the other variables.

0, 1 . . . .

k~ the "functions"

tinuations

,~.~'~,_~'~

= s,~_,¢

in the ~

C+

= {~IWe~>o~,

For n + m - 1 = k fixed and m =

~qn+Wl_l(~'>~:~--.~')

of the same distribution

are all analytic

-~ w for

~%~_((~,w,~__')

= ~÷;v

= ~,÷m_l(~,~,~')_ _

, then the

~_(~

they all coincide.

[]~/6[

The M a l g r a n g e - Z e r n e r

o , C ~_ , z, , ~_ ) ~ c'.'~._, ~ .

There are vector valued functions in ~

q ~ (z, ~) :

n

, such that 4----

O

Proof:

For

(~,~)

~ ~0)

we choose a "polydisc"

k~i~...n-l} , centered at some real point C×;

)

P--{(~,~)

and containing

ll~-~I

(Note that

is always kept fixed and real. )

Taylor series expansion of 52n_i(+~i~+~ ',_~')

(~ ,×*× ° ° , ~' )

~

(~,~)~

approximation

such that supp ~

~-

P

Then the

around the point ,

(~,~') ~ P'

Now

to the 6-function 6(x - ~)-

. ~ ~

+

o

and

~x,~

'

Furthermore define

where _k =

(k I . . . .

-.811,.I Then

-- i. ks_l), _k: =~k..',~

k =~x i k i _x-"

Ik; = ~ ki , ,

%~;

~ , ~ ~,~>_ =- .[~,~>_ %-¢~,~_ ~~ " - ' ~

are vectors in ~P- , which for fixed w,]~ depend analytically on ~_ Using eq.

(13) and the fact that

~n+++_l(~ ~×+x'. ~' )

tic function we easily check that exist and satisfy equation

is a real analy-

~(~,~)_ = ~I~"~l;m~ ~e,,~]+ (x,°~ )

(19).

Lemma 12 enables us to construct an analytic extension of to a larger domain we define for

("(~') ~k

.

(×, ~_ )

For

E

~C~)

and

~(_~ )

(~,,~ t ) & D~,~ ~')

~ ~ ~+

With n + m - 1 = k fixed and n = 0, .... k, eq.

(20) yields an analy-

tic extension of S k to the domain

(~1)

q~

= ~o

[¢~_ , ~ . . ~ , ~ ' } ~ ' I ,~ , ~_' , ~

~ , (~.,~.~_

,

I

In terms of the variables a tube. S~ (~)

W~-

By the tube theorem• can be analytically --~

variables ~/< .

tt~+ iV~ = ~n ~

•

see e.g. Vladimirov

the domain

~-(~

--k

[VI i], p. 154•

extended to the envelope of holomorphy

, which is just the convex hull of As in (18) we define

C~

in the

is

86

a n d we p r o v e

Lemma 12 w i t h

Repeating extension

of

S~(~ )

q£_~ (x, ~ ) d e f i n e d

or e q u i v a l e n t l y

(A)

13:

on a d o m a i n

before.

and v e c t o r

~

O

I,,(,,N)~

I, 2,

® o .

a function

~(~,N)

,

for all t >i i, s ~ 0, ...

,

h(_s,N) )

Obviously

~

~(')

~v(t,s.O)

6 ~(0) t*s

I,

87

Suppose

now we have

N = 0, i, points

already

constructed

... L, s u c h that

are c o n t a i n e d

in

(-b(s,L),-~(~-I,L),

(25) holds.

c O+l) 2(:+t)- I

...

~(~,N)

for a l l

Then by

(23) the f o l l o w i n g

for all

[~% < ;/~

-H(Z,L),-~(hL),O,.:.q,

and

~

Y~

1 and

:

,~(I,L),

... k ( s - t , L ) )

z~-I

(-kCs-,,L),

. . . . . .

C (L+;)

Because

h(I,L),

is c o n v e x ,

2(s*t)- I

- ~
0

A simple calculation

c l o s e to ~/2_, the p o i n t s

with

ideas w h i c h

s h o u l d not do in a c o m p l e t e

~, ~, and ~,

We

)

the main

tinue neglecting

~+~

for s >~ g.

(A) a n d the e q u a t i o n

Estimating

.

in

(26) is

2_"

o 0 , k = I, 2,

integers

... n, and all n,

88

f r [(,+

(28)

s.(+++ )

and

where

~

Both

is d e f i n e d b y

+

a n d ~ im, £

5~,a(~)

×e~q.

,~

~+.

By

(" ~-~ F_. )1.+ = ~'+" + ~" "

( ×.~ )

are d e f i n e d and a n a l y t i c

(28) we h a v e

for

for z = x + iy,

=

(29) We

claim that

(30)

for

~

C (N)

a

I s ~ (~) I -~

We p r o v e just 2

-z;HH e. %lz-~,a (x~ ~') ).

( i;, 2-~ ; E'/

(30) b y

ineq. .

induction.

(27).

L

IB~ N

(~>~"

F i r s t we n o t i c e

and all n.

k = n + m - I, b y

T h e n for

(×, ~ ) ~

~L~

AS

PI+P/I - I - I+.

,

_

we can u s e the m a x i m u m that

(30) h o l d s

(30)

is

(30) for N = i,

, (×, ~, ) ~ ~(L~

2/~'(L+I)

is j u s t t h e e n v e l o p of h o l o m o r p h y --

ineq°

(29),

(~k) I~ (L+I) --k

that for N = 0,

N o w a s s u m e we h a v e v e r i f i e d

--

for N = L + i.

The next step

~

principle

of the r e g i o n

I

(see e.g. This proves

rvl

i], p.

ineq.

178)

to c o n c l u d e

(30) for a l l n and N.

is to e l i m i n a t e N f r o m the r i g h t h a n d

side of ineq.

M

(30).

In o r d e r to do that we c h o o s e and fix ~ 6 ~ ÷

N = N(~)_ so that Then by Lemma and w e m a y

~_ 6

13(B)

C (N)~ L e t N n b e s u c h that Z -N"/z ~, < 'Iz . C(N.) ~ £ ~m if l~r@ ~ ' { ~ Ir/~ for T = 1 . . . . n

set N (_~) = N n for s u c h _~ .

be unimportant.

Now take a ~

for

~-

l~e_ ~'~

, ~ =

I~1~

~;

J

The dependence

with

~ ~n

for some 1 -~ s ~ n. Then

and d e t e r m i n e

,

on n of N n w i l l

>I

89

(use t h a t W e d e f i n e N(~)

&~rc %~ × =

> ÷

for x < i. )

IS], w h e r e

i~ Z

a

for s o m e c o n s t a n t meaning

of cn m i g h t

an n - d e p e n d e n t I~

cn depending

~ ~

change

constant.

on n only.

In the f o l l o w i n g

f r o m line to line, b u t With

this c h o i c e of N

the

it w i l l a l w a y s b e

(_~) we find t h a t

for

n - N(t)/z

2

hence ~_ (30).

~

C n(u(t~/

, by Lemma

-

Now we substitute

13(B).

N (~) in ineq.

This gives

(31)

I S.,~ (~)

I 4-

(~ Symfl ® " " ® fn extends uniquely to be unitary from with

H n.

P(H) n~

onto

~nH. We shall use this mapping

to identify

P(H)n

span all of

l 2 L2(lq, B, (27[)-2e-x /2 dx), and Segal [13] extended this result to

It is well-known that the Hermite polynomials

97

arbitrary real Hilbert spaces, showing that the r(H) n span F(H).

r(H) :

Z

Consequently,

H

n=O

Thus

r(~)

is Fock space.

The space

P(H)

is intrinsically attached to the structure of

real Hilbert space.

Consequently, if

U: H -> ~

~

as a

is an orthogonal mapping of

one real Hilbert space onto another it induces a unitary mapping P(U): F(H) -> P(~). I: H -> K

On

H n , P(U)

is

U~

... ® U

( n factors).

is an iscmetric linear imbedding of one real Hilbert space into

another it induces an isometric linear imbedding H n , F(1)

If

is

I ® ... ® I

(n factors).

If

P(1): P(H) -> P(~), and on

E: H --> K

is the orthogonalpro-

jection of a real Hilbert space onto a closed linear subspace, then it induces an orthogonal projection (n factors).

If

F(E): F(~) -> F(~), and on

A: H -> ~

F(A)n , where

P(A)n: ~nH -> ~nK is given by

Halmos [ 8 ] showed that any contraction form

A = EUI

with

E, U, and

obvious imbedding, E: ~ @ H - - > K U: H @ K m >

K @ H

r(A) = r(E)r(u)r(1).

I

F(A)

E @ ... @ E

to be the direct sum of

A ® ... ® A

A: ~ -> ~

(n factors).

may be represented in the

as above (in fact, I: H - - > ~ @ ~

is the

is the obvious projection, and the operator

is defined to make the relation

out, by a non-trivialproof,

is

is any contraction (linear mapping of n o r m ~ l)

from one real Hilbert space to another, we define the

H n , F(E)

to be orthogonal).

Therefore the operator

A = EUI

valid and turns

Consequently

P(A)

is doubly Markovian, in the

sense that _< 0

(i0)

r(A)~ >

0 ,

r(A)l = 1 , Er(A)~

since

=>

F(E), r(U), and If

2toLj

F(1)

= E(~ ,

are clearly doubly Markovian.

is any doubly Markovian operator it is clearly a contraction from and from

~l

to

~l

Either by the Riesz-Thorin theorem or by a

98 I!

simple application of Holder's inequality, it follows that from

#

to

2

Theorem !

for all

P

i a a contraction

i < p _< ~.

(H~ereontractivit¥).

real Hilbert space to another.

Then

Le__tt A: H -> K P(A)

be a contraction from one

is a contraction from

~q(H)

t__oo

Lq(H)

t_~o

~P(K), fo__~r 1 ~ q ~ p ~ ~, provided that

(ll)

IIAII _~

If (ii) does not hold~ then

P(A)

~pq-1

1 •

is not a bounded operator from

We sketch here a proof which is different from the rather obscure proof [ 9] in the literature.

If

ways a contraction. so that

A = O, then

If

LP(H)

But

to

c-lA

LP(K).

theorem, we need only show that provided that (ll) holds.

is just the expectation, which is al-

A % O, then we may write

P(A) = P(c-lA)P(c).

a contraction from

P(A)

A = (e-lA)c, where

is a contraction, so that

c = NAIl,

P(c-lA)

is

Thus, to establish the first part of the

P(c)

is a contraction from

Lq(H)

to

~P(~)

It suffices to show that 1

(12)

1

(E(P(c)~)P) p < (E~ q)q

for ~ _> 0, since for any positivity preserving linear operator IIX~I _< picot.

(This is easily seen by approximating

P

we have

2 by an integral operator

with positive kernel. ) Suppose that (~,S,~)

H = H(1) • H(2 ), so that we may take the probability space

for the unit Gaussian process on

probability spaces on

H(1 )

and

H(2 ).

(~l,Sl,~l)

and

H

to be the Cartesian product of the

(G2,$2,~2)

We use the notation

E1

for the unit Gaussian processes and

E2

for the expectations on

these probability spaces, and similarly we use the notations If (~ is a positive random variable on 1

(13)

Pl(C)

(~,S,~), we claim that 1

1

llr(c)~llp -- (~,(F(c)~)P)p _< (Em(rl(C)(E2(r2(c)~)P)P)P)P

and

P2(c).

99 We may write for

P2(c).

using

r!(c) If

~

as an integral operator with kernel rl(. ,'), and similarly is a positive random variable and Pl_ + ~, = l, we have, by

Holder ' s inequality twice,

~r(c)~

8(~l,~2)Fl(~l,ql)F2(o~2,q2)~(ql,q2 ) ~ l ( ~ l ) 04~2(q2) d~l(~) d~2(~ 2)

= ffff

1 < ff (f ~(~I,~2)P' d~2(to2))P'Pl(~l,~]l) 1

(f (f £2(~2,n2)dZ(nl,q2) d~2(n2))Pd~2(~2)) p d~l(n l) d~l(ml) 1

< (If B(~l,m2 )p' d~2(m2) (4"~l(Wl))P'

1

l

(f rl(O~l,~l)(f (f r2(~2,~2)~(~l,q2 ) d~2(~2)) p dg2(~2)) p d~l(~l)) p d~l(~l)) p ,

which is just

II~IIp, times the right hand side of (13). Therefore (13) holds.

It follows at once from (13) that if the first part of the theorem holds for

H(1 )

and

H(2 ), it holds for

H(1 ) @ H(2 ).

Therefore we need only prove

it for a one dimensional real Hilbert space, for it then follows by induction for all finite dimensional real Hilbert spaces and, by approximation, for all real Hilbert spaces. Let

F

be a function on lq which is bounded below by a strictly positive

constant and which is bounded together with its derivatives up to third order. It suffices to prove (12) for ~ = F(~), where of mean

0


Let

~h

be a Gaussian random

O

that

are independent random variables on the product space

(~,S,~).

~h

Notice that

expectation is P2(c).. as above.

h

~ + ~h

h

is a Gaussian random variable

variable of mean q0 and

and variance

(~I,SI,~I).

~


is a Gaussian r%ndom variable of mean

plus the variance of Then

(~2,$2,~2), so

q~. We use the notation

O

whose

El, E2, Pl(C),

100

1

(14)

(E2(F2(c)F( g + gh))P) p

= (E2(F2(c)(F(~)

i + F'(~)~h + ½F"(@)h + o(h)))P) p i

= (E2(F(~) + cF'(~)(ph +½F"(m)h + o(hl)P) p = (E2(F(~lP + F(~)P-I(pcF'(¢%

i + ~F"(~lh) + F(~)p-2p(p-I)2 e2F'(~)2h + °(h)))~

i = (F(~) p + F(~)p-lp~F"(~)h + F(~)P "2p(p-I) c2F,(~)2h + o(h)) ~

= F(,

With

+ (½F"(¢ + ~ 2A c2 F(m) )h + o(h) . c = i

and

p = q

this gives

1 (E2F( g + q:~)q)q = F(m) + (½F"(m) + q-12 ~ ) h

(15)

which, by (ii), is greater than (14), up to terms which are

+ o(h) o(h).

By (13), we have that

llr(e)F(~ + q~)llp < ilF(m + q~)llq + o(h) ,

(16)

provided that we have already established that

Ilr(c)G(~)llp 5 IIG(~lllq ,

(17) ~here

G(x) =F(x) + ~i F"(x) + q2' I F~' ( x") 2 Let

H(x,t)

be the solution of the non-linear equation

(18)

with initial value

aH

i a2~ + ~-i (~x)2

~-t = 2 3x 2

H(x,O) = F(x).

2

H

'

Notice that (18) is equivalent to the heat

I01

equation ~H q ~t

for x

1 32H q 2 ~x 2

H q, so that (18) has a unique solution which for each

t

as a function of

has the same properties as those assumed above for the initial value

follows from the above argument that if mean

0

and variance

~

F.

It

is a Gaussian random variable of

t, then

llr(e)F(~)llp _< H(O,t) -- llF(~)llq ,

(19)

which establishes (12). Now suppose that (ii) does not hold, so that

A: H - >

is a contraction

with

(2o)

11ALL > ~ q - 1 .

By restricting

A

to the orthogonal complement of its null space, it is e n o u g h

to consider the case that position Since

A = UP

F(U) -1

has null space zero.

U: H -> K

Then

is orthogonal and

But if

and is positive. H

~

has a polar decom-

P: H -> H

is positive.

and a constant

By the spectral theorem, there is a non-zero c

with

c > /p~_'~

such that

subspace, so again it suffices to consider the case that c.

A

is a contraction, it is enough to consider the case that

A = P: H -> H subspace of

where

A

is a Gaussian random variable of mean

establish [ 9J that

r (c)ea~

eCa~e½ (i-c2) a2

lip(c)ea~llp = J [ (p-l) c 2 + 1]a 2

:

lle~llq = j q a

2 '

0

A

A > c

on that

is a scalar operator

and variance

l, we can

102

llr(c)e~llp/tfea~llq

so that

is

II.

Let

Ed

if

EUCLIDEAN Frk-~,nS

with the inner product

mean the Schwartz space. X

If

X

in

~

then

x'y = xly I + .o. + xdy d.

$

indexed by

~(f~) -> ~(f)

~

Let let

~

O(A)

~

in measure.

~(~d).

be the ~-algebra generated by the

is an arbitrary subset of E d

we let

(1)

O(A) =

n A'~A

where the intersection is over all open sets O(A)

8A

are denoted by the boundary of

~(Ed)

A

O(A).

E[-IO(A)]. A.

such that whenever

O(A), where

~(E d)

we

~(f), for

f

in

(~,~,~), and for convenience

If

A

~(f)

O(A')

$(f)

with

f

in

X.

is an open set in ~ d with

supp l E A ,

we

and if

A

,

A'

containing

A.

We also use

for the set of all random variables which are measurable

with respect to the a-algebra O(A)

Thus the

is generated by the

be a linear process over

the notation

By

which is linear and such that

X, are random variables on some probability space we assume that the a-algebra

is the real d-dimen-

is a topological vector space, a linear process

is a stochastic process

f~ -> f

a is sttffi~iently large.

be d-dimensional Euclidean space; that is, E d

sional space ~ d

over

large i f

arbitrarily

Conditional expectations with respect to

We let

Ac

denote the complement of

A Markoff field on E d a

~

and over

is a positive or integrable random variable in

is an open set in E d

(2)

is a linear process

A

E[~Io(Ac)}

then

= E(~IO(~^)}

•

We call this the Markoff property. Let E d.

10(d)

be the Euclidean group of E d, consisting of all isometries of

By a representation

a homomorphism

q ~> T(~)

T

of of

I0(d) I0(d)


(~,~,~)

we mean

into the group of automorphisms of the

103

measure algebra. then

S

We note that if

is an autunorphism of the measure algebra

acts in a natural way on the random variables.

Markoff field

~

over

~(Ed)

and

~

in

(fl,8,~) of

$

T

of

such that for all

I0(d) f

on

in

I0(d),

(3)

T(~)~(f) : ~(f o -1)

and which has the following property: plane Ed~l

A Euclidean field is a

together with a representation

the underlying probability space ~d)

S

If

p

,

is the reflection in the hyper-

then

~(~)= = ~

(4)

,

~

~

o(~ d-l)

.

Relation (3) is called Euclidean covariance and (4) is called the reflection property.

Notice that by Euclidean covariance, if the reflection property holds

for one hyperplane it holds for all hyperplanes, so that no special choice is involved. We need an assumption which guarantees that certain expectations of products of fields exist.

The following assumption is convenient, although stronger than

necessary.

(B)

For all

~(Ed)n ~__> C

f

i__nn ~(~d), ~(f)

is in

Lp

for

1 ~p

0 A

be the half-space

is the half-space

x d > O, so that

x d _< O.

Then

8A

C~ is in

is the hyperplane

O(A)

and

T(p)~

~d-i

is in

and

o(AC).

Ac We

have

= ~[

(~(~)~')~:{c~la(A c) ] ]

= ~[

(~(p)~)~{c~] o(~ d-l)

]]

(by the Markoff property)

: ~.[~.[T(o)810(~,d-l) ]~[~ I~(~.d-1)]] = ~.[(T(p)~,{~Io(Ed-l) ])~{~IO(~,d-1)]] (by Euclidean covariance)

= ~.[E[~Io~ d-l) ]~{~l0(~a-l) ]] (by the reflection property) -- ~ , l ~ { c d ~ ( ~ , d - 1 ) } l

2 _> o .

Axiom E3, syn~netry, follows immediately from the fact that randam variables commute, so that

Sn(fl,...,fn) : E ~(fl)'"~(fn)

for all permutations

7[.

: Sn(f~(1),...,f~(n))

This concludes the proof.

The 0sterwalder-Schrader theorem [ll] asserts that there is a set of tempered distributions

W

on ]Mdn, where ]Md

is d-dimensional Minkowski space)

which are the vacuum expectation values of a quantum field satisfying all of the Wightman axioms except uniqueness of the vacuum (cluster decomposition property of

W n ) , and such that the

Wn

are boundary values of hol~norphic functions

106

which agree with

Sn(Xl,...,Xn)

whenever the

xj

are distinct.

At the time of

writing these notes, there is said to be a gap in the proof, but one can confidently expect that this will soon be remedied.

For another approach to the

problem of obtaining quantum fields on Minkowski space frem Euclidean fields, not using the 0sterwalder-Schrader axioms, see [10].

The axiom E4 of 0sterwalder

and Schrader, which implies uniqueness of the vacuum, does not follow without additional assumptions on the Euclidean field, and in fact it is a question of great interest whether it always holds in the

III.

Let

~(Ed)

P(~)2

THE FREE EUCLIDEAN FIELD

be the real Schwartz space on E d, let

tive constant or merely a positive constant if Filbert space completion of

~ ( E d)

&

is the Laplace operator.

Let

$

is a linear process over

Let

Theorem 3-

~

be a strictly posiH

be the real

with respect to the inner product

,

be the unit Gaussian process on

and extend it by linearity to the complexification of Schwartz space, ~

m

d ~ 3, and let

< g , (-~ + m2)-if>

where

models.

be as above.

Then

S.

Restricted to the

~(~d).

~

is a Euclidean field satisfying

assumption (B). Proof.

Let

be an open set in E d, and let

u=

{f ~ H :

M = {f ~ H:

supp f C A

N = {f ~ E"

supp f C b A }

~=~n~ Let

f

be in

U

mud let

supp f C A }

h

, c} , ,

.

be t h e o r t h o g o n a l p r o j e c t i o n

of

f

onto

M.

We

I07

claim that

for all

h

g

is in

in

N.

To see this, observe that

M, and in particular for all C = functions

port in the interior (-A + m 2 ) - ~

A c°

of the complement of

= (-A + m2)'if

local operator, h = f

K

ed by the

as distributions

Let K

An

with

f

in

If

G(A c) = ~.

Then

K

pendent.

Now

Therefore if

G

is in

~

n ~> U(~)

quently

6n

Thus

~

is

A c°.

~

a

Therefore

be the a-algebra generat-

$ K, it is easily seen that An $ A c

4 M, so that

K

$ K.

and let O(An) = K~n $ ~' and conseG(A) = ~. ~

and

~

are inde-

is the g-algebra generated by

is a positive or integrable random variable in

~

and

~.

~,

is a Markoff field.

I0(d), we define the orthogonal operator

(~,S,~)

is a representation of of

~.

U(q)

on

~

by

-I

is an orthogonal representation of

~ ~> T(~) = F(U(~))

probability space

f = 0

-A + m 2

N.

H, let

U(~)f = f • n

Then

with compact sup-

Since

~ I ~, so that the ~-algebras

M = N • ~, so that

E[~I~} : E[~I~}. q

K

A c°.

Similarly, G(3A) = ~, and of course

We showed above that

If

K.

g

That is,

A cO, but

is indeed in

be a sequence of open sets with

= [f ~ H: supp f C A n } .

quently

h

on

is a closed linear subspace of $(f)

A.

as distributions on

supp h C A c - A c° = 8A, so that If

= < g , (-A + m 2 ) - I f >

!O(d) IO(d)

on

H, and conse-

on the underlying

Clearly,

T(n)~(f) : ~(f° n -1) , so the Euclidean covariance holds. Let

H o = [f ¢ H: supp f C]~ d'l}

and let

f

be in

H .

~O

Since

with respect to the measure

f

is in

H, its Fourier transform

is in

L2

108

dk

(i)

k2 + m2 ,

and since

supp f C E d-l, we have

}(k) = Z foiN)Pi(k d) i where

foi is some function of

since

f

p

~=

(kl,...,k d'l)

and

Pi is a polynomial.

But

is square integrable with respect to the measure (1), the polynomial

must be a constant.

Consequently, if

u(~)f

p

is the reflection in E d'l,

= f

.

Therefore

T(p)~(f)

and since

G ~ d-l) =--Ho, if (~ is in

= ~(f)

,

(~d-l)

T(~)~

That is, the reflection property holds°

: ~

then

.

Thus

$

is a Euclidean field.

Assumption (B) is obviously true for the Gaussian process We call

$

the free Euclidean field of mass

IV.

Let space

~

finite open cover

S ~ d)

of ~ d

with

8 = H 6i"

[Ai} Thus

G

(~ is additive in case for every

there exist real, (~i in

Similarly, we say that a random variable finite open cover

with the underlying probability

We say that a random variable (Ai}

on ]~d.

MULTIPLICATIVE FUNCTIONALS

be a Markoff field over

(~,S,~).

m

$.

8

~(Ai)

with

~ = E (~i"

is multiplicative in case for every

of ]~d there exist strictly positive is additive if and only if

~ = e~

8i

in

(~(Ai)

is multiplicative.

109

Theorem ~.

Le__~t ~

probability space with

E6

space

= 1.


(£,~,d~)

Then

~

and let

B

~d)

with the underlain6

be a multiplicative random variable

is a Markoff field over

~(E d)

on the probability

(£,~,# dl~) . To prove this, we need a pair of l~nmas.

If

~

and

~

~ U ~

algebras o£ measurable sets on a probability space, we let smallest complete c-algebra containing

Lez~ma i.

Le_=tt ~

an__dd B n

probability space~ with

Bn

~

and

let

~

If

A

be the

~.

be complete c-alsebras of measurable sets on a

decreasing.

Then

(AU~) =AU rq ~ n Proof.

are complete c-

B ~n

is a c-algebra of measurable sets on a probability space,

denote the von Neumann algebra of multiplication operators on

bounded random variables which are measurable with respect to

A.

~2

by

Then we need

only show that

But using

' to denote the ec~mutant, we see with a little thought that

( nn ( ~ ) ) '

-- un ( ~ o ~ ) ' =

x,~

~,

= ~ (~.' ~ ~ ' ) =

~,~(@-

~) , =

(AZu

nn

~)

'

By the double commutant theorem applied to the first and last terms, (i) holds.

Lamina 2. ~d

and let

Let A'

~


be a closed subset of

Ac

E{O'(A U A ' ) I G ( A C ) }

Proof.

Notice that

A U A'

S ~ d), le_~t A

conSainin~

= O'(A')

A'

Then

Then

.

is a closed set, since

a sequence of open sets decreasing to

~A.

be an open set in

A U An

A' D 8A.

Let

An

be

is a sequence of open

110

sets decreasing to

A U A'.

O(A U A n ) = O(A) U O(An).

Since

=

and

then, since

#i

admits partitions of unity,

Therefore, by Lemma I,

Ù(A U A') = 9 0 ( A

If C~. i

S(~ d)

U h n) = 9

(O(h) U O(An))

O(An) =

O(k) U 9

O(A) U O(A') •

are bounded random variables in

O(A)

and

O(A')

respectively,

A ~ C A c,

c~i#ilO(AC)} =

E[Z

Z

E(C~ilO(AC)}#i

= Z E[CZiIO(~A)]#i by the Markoff property, so that

E[O(A U A')Io(AC)] C O(A') .

The reverse inclusion is obvious.

Now we prove Theorem 4. tions with respect to

d~

We denote expectations and conditional expecta-

by

E

and

E(" I" }, and we denote expectations and

conditional expectations with respect to Let

A

be open in ~ d

and let

B d~

by

E6

and

~(-I" }-

Cz be a positive random variable on

O(A).

We need to show that

That is, by the Radon-Nikodym theorem there is a unique positive random variable = ~[czlO(AC)}

in

O(A e)

such that for all positive random variables

o(AC),

--

,

or equivalently

(2)

E (z~6 = E & ~

,

7

in

111

and we need to show Zhat ~ [A,Ao,A co ], where

O(~A).

is in ACO

Let

A

be any open set containing O

8A.

Now

E d.

Therefore there exist strictly positive random variables

in

O(Ao) , and

B3

in

O(A c°)

is the interior of

with

for all positive random variables variable in

B = BIB~3.

7 in o(Ac).

A c, is an open cover of B1

in

O(A), B 2

We know, by (2), that

But

B31 is a positive random

o(AC), so that

(3)

E ~TfBlp 2 = E ~TfB1B2

for all positive rand~n variables

7

in

o(AC).

Notice that (3) is also equal

to

E[~7 E(PlP2Io(AC)}] Since

%

•

is arbitrary, this means that

E[C~I~210(AC)} =~ E[~I~210(AC)} , and since

~lB2

is strictly positive this m a y b e written as

~,{o~ l~,210(i c) ]

(4)

=~.

~. { ~lB21o(Ac)}

By Lemma 2 applied to

A

and

A' = ~

O A c, both the numerator and the demoninaO

tor of (4) are in G(

), and since is in O(SA).

O(A'), which is contained in A°

O(~o).

is an arbitrary open set containing

Therefore

is in

~A, this shows that

This concludes the proof.

The proof shows that if cz is a positive random variable in

(5)

~

~!3(~lO(AC)} = E{O~Io(Ac)} EI;f310(Ao)} "

G(A)

then

112

since we may insert

~3

inside the conditional expectations of numerator and

denominator in (4). If

~

is the free Euclidean field, then

additive. Let

@(f)

for

f

in

~(~d)

is

We shall now construct a more interesting example in dimension d = 2. ~

be the free Euclidean field of mass

Fourier transform

$

m > 0

on 2 2 .

We define its

by

~(})

=

~(f)

where

f

f(k) =

is the Fourier transform of

e'ix'kf(x) dx

f, and we use the notation

~(f)

=

f

f(~)~(~)

.

Let

~K(x) :

Then 0

~K(x), for

and variance

(6)

cK2 =

x

:$K(x)n:

eiX'~(~)

"

in 2 2 , is a well-defined Gaussian random variable of mean 2

cK , where

i

~ k I ~.

eiX.k

dk

k2 + m2 "

As a consequence, :$Kn:(g) converges in

We d e n o t e t h e l i m i t

Next we assert that if A supp g C A ,

kJ : n:(g), we have

:mn:(g) ~ ~

K

oK(,) = o(^) ,

which proves (8). Since : n: (g)

~LI N ~(1R2) admits partitions of unity, it follows from (8) that

is additive.

In fact, it has the stronger property that if

finite partition of E2 (~i in

O(Ai) with

(Ai} is a

into measurable sets then there exist random variables

:qn:(g) -- E (~i' for we may take G i = : n: (gXAi ).

This

stronger property makes the analogue of Theorem 4 trivial to prove, without Le~mas 1 or 2.

It is not clear whether Theorem k will prove useful.

114

From (7) it follows that

ll:$n:(g) - :~Kn:(g)ll~ =

(9)

-

where G(x) =

We claim that for some

! (27T)2

: eix-k

> O, (9) is

0(K-g).

dk k2 + m2 °

TO prove this, it suffices to

show that

(zo)

[I: - G~II~ = o(: ~)

for some

r

with, say, 2 < r < ~, and by the Hausdorff-Young theorem it suf-

fices to show that

(11)

ii6*n

for some

s

with

~*n_-*no K

1 < s < 2.

=

- %-*n

IIs = o(:

~)

But

(~--G~:)*%:-*'"*GK÷G*(G-GK)*~ ÷ ...

÷ ~*...

*~*(~-~-K)

*'''*~K .

i

then by

Young's inequality we have that (ll) is smaller than a constant times IJG - GK]Jq, so that we need only show that

(12)

LI~ - ~Kllq = o(: ~) •

But a simple computation shows that (12) is true. By Theorem l,

(13)

IJF:

~ (:$n: (g) - :~PKn :(g))llp

\,,p-:/

_< ll:~:(g)

- :q~:(g)ll

2

115

which is

for some

O(K -E)

g > O.

But by definition of

F, the left hand side

of (13) is i I nJJ:q~ ~i_i n :(g)

Consequently,

there is an

~ > 0

and a

- :q~Kn:(g)JJp

C < ~

o

such that for all

K

and all

p,

n (141

jj: n:(g) . :$Kn:(g)[jp < (p_I)2CK-¢

If

P

is any polynomial in one variable,

(15)

P(~) = anon + ... + al~ +

we define

.

:P(m):(g) = I g(x):Pqg(x): dx

for

g

a0

in

~LI 0 ~ L ~ 2)

to be

n an:q0 :(g) + ... + alq0(g) + a 0

and similarly for that for all

K

:P(@K ):(g). and all

Again, there is an

~ > 0

and a

C O.

P If

is bounded below; that is, P ~j is Gaussian of mean

P, we have

:P($): = Q(~).

random variable

$

0

of mean

stant times the variance of

(171

and variance

is bounded below, since for a certain polynomial

leading term as

in

0

is real and in (19)

L1 N L~21

we have that

~J to the power

then by (61 there is an

Q

1

n

then

with the same

It follows that for any Gaussian :P($):

n/2.

a > 0

is bounded below by a conConsequently,

such that

:P(~K):(g ) ~ -a(log K) n + i

if

g _> 0

is

116

for all

K

(greater than 2, say), where we have put in the term

right hand side of (17) for later convenience. ~(~:

1

on the

By (16),

l:P(@):(g) - :P(~K):(g)I ~ l} n

5 ll:P(~):(g) - :P(~K):(g)ll~ ~ (p -Z)~CPK - ~ , SO that

:P(~):(g) ~ -a(log K) n

except on a set of measure at most n

(18)

(p2cK-~)P .

If we choose the value of and

E > 0

p

which minimizes (18) we find that for some

b > 0

it is less than

(19)

e

_bK ~

Consequently,

~{~:

e -:P(~):(g) ~ e a(l°g K)n} = ~[~:

:P(@):(g) ~ -a(log K) n} ~ e -bKE

K) n Let

X = e a '(l ° g

, so that

~[~:

(2o)

K = e

Then

e-:P($):(g)_> X} _< e "be ( l ° ~ a k~E/n

By (20), 7 = e-:P($):(g)

is in

~i(~,~,~), so that

8 = 7/E7

is a multiplica-

tire random variable. To summarize, let bounded below.

Then

d = 2, let ~

g ~ 0

be in

~l N ~ 2 ) ,

is a Markoff field on the probability space

where

e- f g(x):P(~(x)): dx (21)

let

B =

e" ;g(x):P(~(x)):

P

be (£,~,8d~)

117

V.

Let

A

LATTICE FIELDS

be a subset of the integral lattice points

tion defined on

Nd.

If

xEA

•

f

is a func-

A, let

(1)

~^f(x) = -2~f(x) +

Z

f(y) ,

ly-xl--i yeA

For

x,y e A

let

2d + m 2

(2)

AA(x,y) =

,

x=y

-i

Ix - yl = l

0

otherwise .

Then

(3)

If

( - A A + m2)f(x) =

m > 0

(or if

(-~A + m2)

d ~ 3

and

has an inverse on

(4)

Z AA(x,y)f(y) yeA

m ~ 0

or if

12(A).

Let

A GA

,

XCA

is finite and on

A × A

(-a^ + m2)'If(x) = z a^(x,y)f(y) ,

•

m > O) then

be such that

xEA'.

yea

Since

('~A + m2)-i

type on

A X A.

with mean on

0

We let

~A

GA

is of positive

be the Gaussian stochastic process indexed by

and covariance

GA.

A

We call it the free lattice field of mass

m

A. If

A

lq#A, where

(7)

is a positive operator, the function

is finite then the #A

~A(X)

is the cardinality of

are just the coordinate functions

ux

on

A, with ,respect to the Gaussian measure

i -2l Z u xA,( x,y (det 27TGA)-2 e x,yeA

Two properties of this measure are evident.

)~ H du X xeA

First, only nearest neighbors

118

are coupled.

Second, it is ferromagnetic in the sense that the off-diagonal

terms in the exponent are all positive.

VI.

Let

P

THE INFINITE VOLUME L]ZMIT

be an even real polynomial which is bounded below.

P(~) : anon + an.2~

with m

n

even and ~d

on

then

an > Oo

n-2

Notice that if

:P(~(x)): = Q(~(x))

+ .,. +

~

where

a2~2

That is,

+ a0

is the free lattice field of mass Q

is also an even polynomial which

is bounded below, so that for the purposes of a general discussion we may ignore Wick ordering. Now let

A

be a finite subset of

~d

and consider the measure

1 (i)

NA e

where

A

A

x,yEA

is given by (5,V) and

xcA

N

A

g du x x6A

is the normalizing constant which makes

this a probability measure. We have an instance of the situation studied by Ginibre [ 2, Model 2, p. 321] for the formulation of Griffiths inequalities.

At each site

x

in

A

we have the even measure

-½(2d + ~2)Ux2 - P(~x ) (2)

e

We let

~

We let f

du

be the product of these measures on Sx

be all functions of the form

is positive, continuous, and increasing on

of such functions for the various sites

x

in

x

H A. f(lUxl )

or

sgn Uxf(lUxl), where

[0,~), we let A, and we let

S

be all products Q(S)

be the set

of all limits of polynomials with positive coefficients of such functions.

119

Q(S)

Notice that

contains

-h, where I

(3)

Z

-h : ~ J~-y1:l

u u xy

x,yCA We let

(4)

Z h = f e-h do ,

so that

N

A

=

z~ 1.

If

f

is a function on

A, we use

Ef

or

h

to denote

its expectation with respect to the measure (i), so that

Ef = < f > h

(5)

= ~ fe-h do fe-hdo

As Ginibre shows [ 2 ], we have Griffiths' second inequality: are in

Q(S)

If

f

and

Sx

are

g

then

(6)

Efg ~ EfEg

Actually, Ginibre gives the proof for the case that the functions in bounded, and so extend to be continuous on this interval is replaced by

[0,1].

[0,~], and in Ginibre's notation

However, the general statement made above

follows readily from this case. A familiar application of Griffiths' second inequality is the existence of the infinite volume limit. Xl,...,x n ~ A C A'

(7)

and consequently as

In fact, it follows readily from (6) that if

then

E ~A(Xl)-.-~A(xn) j E ~A,(Xl)'''~A,(Xn)

A

increases to

,

~d, the left hand side of (7) increases to

a limit

(8)

s(xl,...,x n) •

The same result holds if we replace the lattice

2d

by the lattice

~d

with

120 spacing

e

between nearest neighbors and include a factor

tion (1,V) of as

e -~0

A A.

For smooth functions f

on

Bd

d = 2

f

is the Laplacian

it can be shown (see [ 7]) that the limit as

the expectation values

E Sr(Xl)...$r(Xn)

in the defini-

we then have that the limit

of the difference operator (1,V) applied to

In dimension

-2

E -~0

Af .

of

exists, and is the expectation value

of what Guerra, Rosen, and Simon call the half-Dirichlet theory.

The inequality

(7) then carries over to this case, yielding the existence of the infinite volume limit.

Guerra, Rosen, and Simon [ 7] show how to establish boundedness of the

infinite volume limit, and that the limit is the Schwinger functions of a theory satisfying all of the Wightman axi~ns except possibly uniqueness of the vacuum. We conclude by showing that uniqueness of the vacuum need not hold in the lattice case.

Specifically, we consider the case

(9)

At each site

d = 2

and

P(~) = ~4 _ a~2 .

x

in

A

the measure (2) is then

(i0)

eE(Ux)du X

where

(11)

The function

E(u x) = -Ux 4 - (2 +~m 2 -

E

is an increasing function on the interval

(12)

and by choosing

b =

a

a-l-~-

large enough we may make

E(u x) ,

(13)

a)Ux2

R(~)

m

b

as large as we please.

l~xl

Sb

= co

[O,b], where

luxl > b

.

Let

121

Then

R

is a limit of functions in

Sx.

Consequently, it follows from (6)

that expectations of products of field operators decrease if we replace the measure (i0) by

(13)

eE(ux) - R(Ux)du x

But (13) is just

(14)

×[.b,b] (Ux) dux

and we have the continuous spin Ising model.

Griffiths shows [ 4] that, for

b

large enough, this model possesses long range order, and the cluster decomposition property fails.

122

NOTES

Thanks to the work of Glimm and Jaffe, constructive quantum field theory is now a large and vigorously growing subject.

We shall not review here the

origins or principal applications of the techniques discussed in these lectures, but shall confine ourselves in these notes to some comments on matters of detail.

Lecture I: F(A)

A reference for Hermite series is [16]. The fact that the

can actually be contractions into

Lp

is due to Glimm [3] - this was an

essential step in passing from box quantization to field quantization.

After

these lectures were written, a preprint by L. Gross [61 appeared which contains a beautiful, clean proof of Theorem 1.

Gross differentiates with respect to

p,

and establishes a logarithmic Sobolev inequality as equivalent to the best possible hypercontractivity result.

He then establishes the result for Bosons by

first proving it for the one degree of freedom Fermion case and applying the central limit theorem.

Lecture II:

The fact that the Markoff and reflection properties lead to

the Osterwalder-Schrader axic~s was noted in [71. for the proof which is given here.

I am grateful to Jay Rosen

Dobrushin and Minlos [11 have announced that

the uniqueness of the vacuum fails in some P(q~)2 models.

Lecture IV: theorem is [121.

A reference for von Neumann algebras and the double cc~mutant A reference for the Hausdorff-Young theorem and for Young's

inequality is [14].

The locality of

: n : , formula (8), would have been easier

to establish if we had not used a sharp momentum cutoff.

Lecture VI:

This material is a comment on the work of Guerra, Rosen, and

Simon [7]- For a discussion of the uses of Griffiths' inequalities, e.g. in proving monotonicity, see [~]. The proof given for the failure of the cluster property relies heavily on the fact that we have a fixed lattice spacing.

It

would be interesting to know whether the argument can be refined to give the Dobrushin-Minlos result [1].

123

REFERENCES

[i]

R. L. Dobrushin and R. A. Minlos, Construction of a one-dimensional quantum field via a continuous Markoff field, submitted to Functional Analysis and its Applications.

[2]

J. Ginibre, General formulation of Griffiths' inequalities, Communications in Math. Phys. 16 (1970), 310-328.

[3]

J. Glinm, Boson fields with nonlinear self-interaction in two dimensions, Communications in Math. Phys. 8 (1968), 12-25.

[4]

Robert B. Griffiths, Rigorous results for Ising ferromagnets of arbitrary spin, J. of Mathematical Physics l O (1969), 1559-1565.

[5]

Robert B. Griffiths, Phase transitions, in Statistical Mechanics and Quantum Field Theory (Les Houches 1970) ed. C. DeWitt and R. Stora, Gordon and Breach, New York (1971), 241-279.

[6]

Leonard Gross, Logarithmic Sobolev Inequalities, Cornell University preprint (1973) •

[7]

F. Guerra, L. Rosen, and B. Simon, The P(~)2 Euclidean quantum field theory as classical statistical mechanics, to appear in Annals of Mathematics.

[8]

P. R. Halmos, Normal dilations and extensions of operators, Summa Brasiliensis Math. 2_ (1950), 125-134.

[9]

Edward Nelson, The free Markoff field, J. Functional Anal. 12- (1973), 211-227.

[lO]

Edward Nelson, Construction of quantum fields from Markoff fields, J. Functional Anal. 12 (1973), 97-112.

[ll]

K. Osterwalder and R. Schrader, Axioms for Euclidean Green's functions, Communications in Math. Phys. 31, 83 (1973).

[12]

Sh~ichir~ Sakai, C*-Al~ebras and W*-A16ebras , Ergebnisse der Math. und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York, 1971.

~24

[13]

I. E. Segal, Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81 (19~6), 106-134.

[l$]

Elias Mo Stein and Guido Weiss, Introduction to Fourier AnalFsis on Euclidean Spaces, Princeton University Press, Princeton (1971).

[15]

K. Symanzik, Euclidean quantum field theory, Rend. Scuola Int. Fis. E. Fermi, XLV Corso.

[16]

G. Szeg~, Orthosonal Polvvnomi~/[s, Amer. Math. Soe. Coll. Publ. XXIII, New York (1939).

THE GLIMM-JAFFE ~-BOUND : A MARKOV PROOF Barry Simon *,% Departments of Mathematics and Physics Princeton University §I. Introduction One of the most useful estimates in the control of the thermodynamic limit for

P(~)2

is the ~-bound of Glimm-Jaffe (1972) [henceforth GJ]: ~(h)

for suitable

h

'> m 0 (Dob Min)

No: GI J a S p I

No: G r Si, Sill

R e 1.>0, 00 G1JaSp

m

Verify W i g h t m a n Axi o m s

Fr Z

Di 3 monotoni c monotonic in m 0 in Ill] GuRoSi 3,Si l G r Si N e l 5, G u R o S i 3 - - ~ G 1 J a l V , OsSeh31 G r S i , S i l l B r l , St

i

G1 J a S p I

Formulate Euclidean Axiom s

Sy Z, Nel 3, O s Sch 3 G1 Ja lll-IV, 5

Haag-Kastler Axions

G1 Ja I-II, C a Ja R o I- 3

--~

--~

Preliminary

Ja I, 2,Nel i, G I I Se 1

~

--~

----a-

--~

Figure 2.

Sp 2

--~

Physical R e p r e s entation

--~2 m02 have a m a s s

gap for small

)~c , a n d t h a t t h e m a s s

~

His

vanishes as

gap?

We

~ is increased to the crit-

gapis zero for k_> kc

> I c w e expect that neither Goldstone bosons nor a m a s s

In other words, for gap occur.

148

I. 4 Field Theory and Statistical Mechanics The equivalence of relativistic quantum field theory with statistical mechanics has a long history.

Older w o r k includes both the Landau-Ginzberg theory and

Symanzlk's p r o g r a m to construct Euclidean models.

Recent w o r k includes that of

Fisher, Wilson, Griffiths and a n u m b e r of lecturers at this conference.

W e mention

here s o m e selected aspects of this correspondence for boson quantum fields. Ideas of this nature in models with fermions have not yet proved fruitful. The Partition Function. quantum field model.

Let dq denote the Euclidean m e a s u r e for a boson

The partition function Z[J]

=

~ e ~(J) dq

is the generating function for Schwlnger functions, and has been studied in

~(q0)2

models by FrShlich [Fr 2]. A s mentioned above, the Euclidean field model has a natural approximation by a continuous spin ferromagnetic Islng lattice with nearest neighbor interaction, see for instance [Ko Wi].

The convergence of the lattice

approximation [Gr R o Si 3] and the approximation of 4

by spin I/2 Ising models

[Gr Si] sharpens this correspondence, see also [New 2]. The one point Schwlnger function

; ¢(x) dq,

which parameterizes s y m m e t r y breaking in the Goldstone

picture above, corresponds to spontaneous magnetization in the Landau-Ginzberg theory. theory,

The coupling constant corresponds

to the inverse

temperature

s t o n e p i c t u r e of t h e v a c u u m c o r r e s p o n d s body systems.

The existence

deviation from a f r e e

k / m ~ , which measures the ~ = (kT) -1.

In this way the Gold-

t o a p i c t u r e of p h a s e t r a n s i t i o n s

of a g a p i n t h e s p e c t r u m

of H,

clustering, corresponds to a finite correlation length

~ =m

and exponential -1

system. One Particle Structure.

W e define G[J}

O[J]

by

= lnZ[J} - ~ ~(J) dq

in many

,

in the m a n y body

149

and then

Gift}

is the generating function for the connected (truncated) Euclidean

Green's functions.

T h e one particle structure is displayed by an entropy principle

(Legendre transformation) 1TM[A}

= inf [ -J" A + G[J] ]

,

J or

in differential

form,

r[A]

where

J is determined by

=

-J'A+G[J]

A(x) = 6 G[J]/SJ(x) .

,

This transformation w a s intro-

duced in statistical m e c h a n i c s by D e Dominicis and Martin [De Ma], in q u a n t u m field theory by Jona-Lasinio [Jo L], and w a s developed by Syrnanzik [Sy 4]. analysis of

I" [A]

in quantum field models [GI Ja 13] m a y complement our study of

the s p e c t r u m of the Hamiltonian by expansions described below. l" [A]

The

T h e functional

generates the (amputated, one particle irreducible) vertex functions.

These

functions are directly related to the magnitude of interparticle forces, i.e. , the physical charge. B o u n d States.

In Chapter 3 w e study the presence and absence of bound states

in certain q u a n t u m field models. bound states in pure

4 ~2

models

f r o m statistical mechanics. (see below).

O u r results in Section 3.3 about the absence of depend

on m e t h o d s both f r o m field theory and

W e use high temperature expansions f r o m field theory

W e also use an idea of Lebowitz f r o m statistical m e c h a n i c s to obtain

two-particle clustering for the four point vertex function. Conversely, in Section 3.4 w e sketch a proof that bound states occur in models in a strong external field. W e

4 ~0 2

r e m a r k that in statistical mechanics, bound

state excitations appear in the transfer matrix for large values of chemical potential ~. High T e m p e r a t u r e Expansions ,.

T h e high temperature expansions in statis-

tical mechanics yield the existence of the t h e r m o d y n a m i c limit and high temperature analyticity, i.e. , the absence of phase transitions.

T h e s e Kirkwood-Salsburg or

150 Mayer-Montroll expansions converge for the c r i t i c a l t e m p e r a t u r e .

T/T

c

sufficiently large, where

T

is

c

R e l a t e d to t h e s e e x p a n s i o n s a r e the v i r i a l e x p a n s i o n s

w h i c h c o n v e r g e f o r l a r g e v a l u e s of the c h e m i c a l p o t e n t i a l ~ , and w h i c h a l s o y i e l d a n a l y ~ i c i t y ( a b s e n c e of p h a s e t r a n s i t i o n s ) , s e e [Ru]. e x p a n s i o n s p l a y an a n a l a g o u s r o l e . m~/)~

In f i e l d t h e o r y , the c l u s t e r

They converge for l a r g e i n v e r s e coupling

[G1 J a Sp I, Z] ( l a r g e T ) and f o r l a r g e e x t e r n a l f i e l d [Sp Z] ( l a r g e ~ ) .

As a

r e s u l t , the c l u s t e r e x p a n s i o n s e s t a b l i s h the e x i s t e n c e of the infinite v o l u m e l i m i t in field theory, and the existence of a single phase with a unique v a c u u m vector. T h e s e high temperature expansions do not, in general, arise f r o m K i r k w o o d Salsburg (or other) integral equations, but have a wider range of validity. W e have, however, obtained Kirkwood-Salsburg equations for the partition function Z , see Chapter 6 of Part IL

T h e s e integral equations are a useful tool in our proof of

analytic~ty of the Schwinger functions. In addition to yielding information about the v a c u u m , the high temperature expansions give us detailed ~nformation about the s p e c t r u m of the Hamiltonian

H,

e. g. , the particle structure and the presence or absence of bound states, see Chapters 2, 3 of these lectures.

W e r e m a r k that these m o r e detailed field theory

techniques m a y yield insights into statistical mechanics. L o w T e m p e r a t u r e Expansions.

T h e Peierls a r g u m e n t [Re] is the basic

proof of the existence of phase transitions at low temperatures.

T h e proof consid-

ers the energy associated with boundaries (contours) separating up spins f r o m d o w n spins.

F o r temperatures

T

sufflc~ently below

to have spins all up or all down.

T c,

it is energetically ~avorable

Gri~fiths, Dobrushln and others have modified and

extended these results, see for e x a m p l e [Dob 1-3, Gi, G r l, M L ,

M i n Sin I-2].

In particular, the contour m e t h o d s yield exponential clustering in pure phases of low temperature spin systems.

Some

continuous spin systems have been studied

[~o Or]. We r e g a r d t h e s e m e t h o d s as c o n v e r g e n t low t e m p e r a t u r e e x p a n s i o n s .

We

b e l i e v e t h a t such low t e m p e r a t u r e c o n t o u r e x p a n s i o n s e x i s t in q u a n t u m f i e l d m o d e l s .

151

T h e y should converge suffic[ently far f r o m the crit[cal point.

(Such an expansion

m a y have been used in the proof of the announced result [Dob M[n]. ) W e believe that love temperature expansions exist independent of whether multiple phases exist. In a pure phase, w e believe that they exhibit exponential clustering and thus are useful to investigate particle structure. In Figure 6 w e s h o w our conjectured region of convergence of the high t e m p e r ature (cluster) expansion and p r e s u m e d low temperature (contour) expansion in the 4 ~0 model.

F o r m o d e l s such as

k~0

4

- ~0 ,

[~ >> k ,

in which s y m m e t r y

break-

ing does not occur, the regions of convergence of the high and low temperature expansions m a y overlap.

~Im e

Convergent / Cluster 7//.. Expansion

Convergent Contour E~)ansion? Multiple phases

/

Critical Point

Figure 6.

Presumed

mo /X

convergence of cluster and contour expansions.

152

Correlation Inequalities and the Lee-Yang Theorem.

These methods yield the

convergence of the Schwinger functions for even ~(q~)Z models, [ G u R o S i 3 , Nel5], 2 2 and a unique phase for (~04 + m 0 ~0 - ~q~)Z models, ~ ~ 0, [GrSi, Si Ill.

These

methods and related developments are included in the lectures of Ouerra, Nelson, Rosen, Simon, to which w e refer the reader for further discussion.

153

I. 5. S o m e P r o b l e m s

W e discuss several open problems for ~(~)Z

In addition, p r o b l e m s closely

related to other sections are mentioned throughout the lectures. Asymptotic completeness.

In a pure ~

4

m o d e l with small coupling,

S-matrixunitary? G a n t h i s b e p r o v e d (see Chapter 3)?

is the

In m o d e l s with bound states,

does the inclusion of the bound state in and out fields for m a s s s p e c t r u m below yield asymptotic completeness?

2m

Related to these questions is the possibility of

performing a cluster expansion in asymptotic fields, as suggested formally by the LSZ

expansion of the scattering matrix or the Y a n g - F e l d m a n equations.

Asymptotic Perturbation Theory.

It is k n o w n that the Euclidean Green's functions

are asymptotic to all orders in the coupling, in the region of convergent cluster expansions, [Di 3]. W e conjecture that the S-matrix is asymptotic to its F e y n m a n perturbation series,

S = I +)~S 1 +''"

W e conjecture that the physical m a s s

Since S I ~ 0 m

, this would yield S ~ 1 .

is asymptotic in the coupling constant

expansion, m

m 0 + ~ Z m 2 + )~ 3 m 3 + " " " + ~ n m n + o(~n+l)

Cluster Expansions.

T h e high temperature expansions in Part II are based on -V 0 -V 0 -V 0 expanding the Gibbs factor e in the Gaussian m e a s u r e de into e = I+ (e -I). T h e y yield Kirkwood-Salsburg equations for Z times the Schwinger functions,

, and related expansions for Z

Z S ( X l , " " , x n) . Do these expansions generalize in

a natural w a y to yield particle structure?

W h a t is the optimal convergence d o m a i n

for these expansions ? Contour Expansions.

W e believe that a low temperature contour expansion exists

and converges, independent of whether a ~(~ ), m o d e l has an internal s y m m e t r y . W h a t is this expansion?

Does it yield the existence of the infinite v o l u m e limit, of

particles and of other properties of the m a s s spectrum.

154 Analyticity.

In P a r t II, we s h o w t h a t t h e

analytic inahalf

circle

0 < I XI < t O

'

)~P(~)Z S e h w i n g e r f u n c t i o n s a r e Re )~> 0 .

into which the Schwinger functions can be continued?

What is the c o m p l e x d o m a i n In statistical mechanics, the

L e e - Y a n g t h e o r e m is used to extend the analyticlty d o m a i n of high temperature (small ~ / m 0 2 , large m0Z/~)

expansions and of virial (large ~ ) expansions.

the Schwinger functions for ~

4_~

Icrit to co ?

real analytic in ~ , ~

except for a cut f r o m

In other words, are the Sehwinger functions real analytic in all of

Figure 6, except for a cut along the line of multiple phases? c o m p l e x analyticity?

For

I > 0 and

Haag-Doplicher-Roberts axioms. ~(~)2 m o d e l s ?

Are

The

HDR

Re~

~ 0

W h a t is the d o m a i n of

, the pressure is analytic [Sp Z].

Is duality, the missing

HDR

axiom, valid for

analysis of superselectlon sectors applies only in three

and four dimensions, but duality is still p r e s u m a b l y true for P(~)2 " Critical Points.

If a critical point exists (See Figure 5) h o w do m

behave in a neighborhood of it?

Do the m a s s ,

, (~ > , etc. ,

spontaneous magnetization, etc.,

vary with p o w e r laws (given b y critical exponents)?

For

)~ < lcrit , m

tone in n~ 0 [Gu R o Si 3]. Is the m a s s m o n o t o n e above the critical point?

is m o n o Since

C o l e m a n has s h o w n that 6 = 0 for )~(~ Z)Z models, do multiple phases exist for this m o d e l ?

Is there m o r e than one phase at the critical point for )~ 4 9 D o zero 2 4 m a s s particles occur in ~ 2 at the critical point? (We r e m a r k that zero m a s s particles do not occur in the two point function, since it is a t e m p e r e d distribution [GI Ja IV].) W h a t is the locus of multiple phases for a ~ 6 or 8

model, etc?

Do

critical manifolds exist for these m o d e l s ? Structure Analzsis.

With our control over the particle spectrum, w e have the

ingredients to carry out the particle structure analysis of Green's functions, as proposed by S y m a n z i k [Sy I]. It is also of interest to p e r f o r m a structure analysis of m o d e l s in statistical mechanics.

As a first step, one can prove the existence and

analyticity of the generating functional for one particle irreducible (IPl) Green's functions,

as

given in

[GI Ja 13] ,

These vertex parts are important in

the study of syrnrnetry breaking and of the renormalization group.

In the f o r m e r

direction, Jona-Lasinio has an effective potential which one believes gives

155

corrections to the m e a n field Goldstone picture of Section I. 3. Such potentials have by studied heuristically in [Co We].

In what sense is the m e a n field or the effective

potential m o d e l a limit of q u a n t u m field theory? A n o m a l o u s Dimensions.

A n extremely interesting circle of p r o b l e m s concerns the

m o r e refined aspects of ~(~)Z m o d e l s at the critical point. close contact with ideas of high energy theorists.

T h e s e ideas also m a k e

The short distance behavior of

4 P(~)Z and ~3(g) m o d e l s is canonical, and a rigorous proof should follow f r o m the local perturbation estimates [GI ffa IV, Fel 2]. Since these estimates hold for all k

, they hold in particular at a critical point for P(~)Z

, giving a logarithmic singu-

larity. O n the other hand, the long distance behavior at the critical point for e(~) Z models is not canonical, since

(~(x~(y))~ const,

as

Ix - Yl ~ co . Consequently,

w e do not expect that any P ~ ) Z

m o d e l w e have constructed is scale invariant.

In

fact, a scale invariant v a c u u m would ensure that scale transformations are unitarily implemented.

This wou/d ensure in turn that the long and short distance scaling

properties w e r e the same. Let us a s s u m e that a critical point exists.

T h e n w e conclude that the theory at

the critical point m u s t contain a fundamental length.

This length characterizes the

distance at which the small distance asymptotic behavior is replaced by the long distance asymptotic behavior.

Scale transformations change this length, so if a

critical point exists, there are continuously m a n y zero m a s s theories related to one another by scaling.

O n e can attempt to force scale invariance by performing an

infinite scale transformation.

D o such limits exist?

S o m e of the p r o b l e m s raised

here are unresolved for the three dimensional Ising model, and a serious effort might start with this case. T h e l~enormalization Group. a fundamental length.

A b o v e w e parameterized zero m a s s

P(~)2 theories by

A n alternative description is based on the renorrnallzation

group, which itself has intrinsic interest.

C a n the Callen-Syrnanzik equations be

used to investigate the long distance behavior of e(9))Z m o d e l s ?

156

Z.

FROM

ESTIMATES

H o w do w e o b t a i n p h y s i c a l p r o p e r t i e s bounds?

In t h i s l e c t u r e

from known cluster

two s p a c e - t i m e

For

and

states follow

for quantum field models

of disjoint regions in Euclidean phase space.

(d = Z) d = 3

from our expansions

of the one particle

These basic estimates

exp (-d/z)

dimensions

p i i n g , as i n P a r t II.

of p a r t i c l e s

we show how properties

expansions.

exhibit the decoupling

TO PHYSICS

, cluster

expansions yield space-tlme

In

decou-

, related bounds yield phase space decoupling and

4 t h e p o s i t i v i t y of ¢~3 " W e r e c a l l t h a t t h e t h e o r y of a s i n g l e t y p e o f p a r t i c l e w i t h m a s s ene rgy-momentum

m

has the

spectrum

H

H= ( %rn5 V2

e..~H=O,

divided into three disjoint parts, hyperboloid

H 2 - ~2 = m Z and the continuum

states with momentum momentum

the vacuum

P l ' P2

PR = P l - P2

P=O

P = 0

, H = 0

= p

, the one particle

H 2 u ~ 2 ~ (2m)Z

are conveniently parameterized

and the total momentum

PT = P l + P2

•

The two particle

by the relative "

The invarlant

157 mass for the two particle states is zl/Z(~I~Z - P1 ' PZ + mZ)I/Z , which for ~R=O equal s Z m

.

The mass

operator

M = (H Z - ~2)i/Z

has the corresponding s p e c t r u m

mass g a p - - - ~

fupper

0

T h e eigenspace of 0

is the v a c u u m ,

m

gap

Zm

and the eigenspace of m

is defined to be the

one particle space. In order to establish spectral properties of H

and

M

w e use estimates

proved by cluster expansions: (I) U n i f o r m v a c u u m

cluster estimates yield convergence as the v o l u m e

A-. i~z , and cluster estimates carry over to the infinite v o l u m e limit.

(Z) The limiting Schw~nger functions (for real coupling constants) satisfy the Osterwalder-Schrader

axioms,

property of the v a c u u m

and hence yield a W i g h t m a n theory.

T h e cluster

(asymptotic factorization) yields uniqueness of the v a c u u m

vector.

(3) The v a c u u m vacuum

cluster expansion bounds the exponential decay to a factorizing

and determines the m a s s gap.

obtain the upper m a s s

From

the one particle cluster expansion, w e

gap and an isolated eigenvalue

M = m

In Section Z.l w e give s o m e simple functional analysis. in Section 2.2 to establish (I) - (3) above.

.

W e apply these results

158

2.1. F u n c t i o n a l A n a l y s i s # Let

0 KH = H

and let

a dense subset of ~ Proposition > 0

, and let ~ O C ~

Z. 1.1.

for

[0, a]

Let ~ b e

be g i v e n .

Suppose that for each

~) E

, there exists X E ~0 and

such that

(z.l.1)

Then

E a be the spectral projection

(e - X , e-tH( 8 -X)) < Mse-(a+¢)t

E a $ 0 is dense in Ea~

and

0

°

IP(~) 2 m o d e l s with small coupling

~

, the

spans the subspace of energy less than y = m0(l - E) •

Let

A=

~(fl )'''~(fn) , suppt. A

in the relativistic Hilbert space

~

compact,

and let 8 A

be the vector

associated with the Euclidean function A

plan is to apply Proposition 2.I.I with ~

the dense subset of ~

. The

spanned by such

163

8A

, and with

6 0 the s u b s p a c e of 6

Since the cluster estimates volume limit

h = 1 •

time direction.

Y=m0-¢

spanned by

are uniform in

We c h o o s e

h

, t h e y c a r r y o v e r to t h e i n f i n i t e

A d to b e t h e t r a n s l a t e of A i n t h e ( E u c l i d e a n )

Thus by the vacuum cluster expansion, Theorem

2.2.1, with

,

(8A , e'dHoA) = ~ A ' A d d q

= A'dq Adq + O(MAe' dl = I(0A , 0> ]z + O(MAe'Yd)

In o t h e r w o r d s ,

i

i f OA = OA - ( ~ , 8 A ) O i s t h e c o m p o n e n t o f 0 A

(8A, e

-dH .~.

OA>
0 consider

as above, there exists an , we have

~< M A e=5~t

W e apply Proposition 2. i. 1 once again. W e choose example, and

Given

•

6 as in the previous

h E LZ(R).

164 Corollary 2.2. 5. Theorem

2. Z. 6.

T h e vectors

(Upper M a s s Gap).

coupling, the m a s s operator

in

E2m0_(~)0

M

For

has eigenvalues

span states of energy < 2 m 0 - ¢ . k~(~0)2

m o d e l s with small

0, m

and no other s p e c t r u m

[o, Zmo-E]. Proof.

Let

E = EZm0_E(I-E0),

let K0 =E~ and let K equal the union of

the Lorentz translates of }(0 " B e l o w w e obtain a cyclic vector X translation subgroup on

K 0 . B y Proposition Z. I. Z, the s p e c t r u m of

contains exactly one point (unless function converges in

S'

]~0 = [0} ). W e s h o w

m 0 , for m6

m 0 , k

}(

)t -~ 0 , using the

),

Since the free theory has one particle states

sufficiently small.

Thus

~0 # 0.

and no other spectrum in

T o complete the proof w e construct E~0(hl)~

on

the interacting theory m u s t have spectrum in a neighborhood of

[ m 0- E, m 0 + E] ,

that

M

](0 # 0 : T h e two point

to the free two point function as

dependence of the cluster bounds. w-ith m a s s

for the space

is cyclic on

]~0 " Let

U(~) ¢p(h)f~

=

X . Let

M

has the eigenvalues

0 and

[0, Z m 0- (].

h I E S(R) ,

ha(x) = h(x - a) .

hI >

0.

We show

Then

¢p(ha)f2

and

~0(I h l ( . - a ) h2(a)da)f2

~P(hl * h2)~

I d a h2(a) ~0(hla)f2

Ida hz(a) U(a~)~(hl)e Since

E

and

U(~)

commute,

E ~0(hl, ha) f~ = lies in the span of translates of CO

as

h2

ranges over

CO,

da ha(a) e-iPa Ecp(hl)f~

]Eq0(hl)f2. Since X=E~0(hl)~2

(hi*h2)~ = h1 h2

is cyclic

for

U(~)

have also used Corollary 2. 2. 5 to identigy K 0 with the span of

on

are dense in }~0" H e r e w e

Eq0(f)~ .

165

3.

BOUND

STATES

AND

RESONANCES

3. 1 I n t r o d u c t i o n An important

problem

in physics is how particles

bound states and resonances.

In atomic physics,

Coulomb

forces

and the Schr~dinger

existence

and their scattering.

similar

ideas,

is whether

of n u c l e a r

of

are atoms andmolecules:

their

Harniltonians

particle

structure

includes qualitatively

Thus a crucial physical question

quantum field model does or does not have bound states.

Do m e s o n s

s t a t e s of q u a r k s ?

namely

analysis.

but without detailed justification.

a particular

For instance:

and elementary

consequences

of a t o m i c a n d m o l e c u l a r

h a s b e e n t h e s u b j e c t of e x t e n s i v e m a t h e m a t i c a l The realm

familiar

Hamiltonian

The spectrum

form composites,

b i n d n u c l e o n s to f o r m s t a b l e n u c l e i ?

Are the p

and the

~ mesons

really

Are nucleons

~ meson

bound

resonances?

Little is k n o w n about such important questions in q u a n t u m field theory.

In

fact, no q u a n t u m field m o d e l s are k n o w n to have bound states, and heuristic calculations based on perturbation theory and the Bethe-Salpeter equation are inconclusive. In this lecture w e give a physical picture of w h e n to expect or not to expect bound states in

~ (~0)Z

m o d e l s with w e a k coupling or a strong external field. W e

prove the absence of two particle bound states in weakly coupled, pure

~04 models.

W e outline an a r g u m e n t to prove the presence of bound states in the presence of a strong external field, and certain other models. B o u n d states are eigenvalues of the m a s s operator

M,

introduced in ChapterE.

T w o particle bound states lie below the two particle continuum; getic reason would prevent their decay into free particles. states in the m a s s

otherwise no ener-

(The decay of bound

continuum m a y , however, be forbidden by additional selection

rules included in the interaction. ) O n the other hand, there is no physical interpretation of continuous m a s s

s p e c t r u m in the spectral interval [0, 2m).

H e n c e none is

believed to exist, and two particle bound states m a y occur in the "bound state interval" (m, Zm)

of the m a s s

spectrum, as illustrated in Figure 7.

t66

0

m •

Vacuum

O@,O

/

2m

'-"--~""~-.Two particle Threshold

~,

Single Particle States

PossibleTwo Particle Bound States

Figure 7.

In an even t h e o r y ,

Spectrum of the mass operator M

e. g. ,

~4

we can decompose the Hilbert space according

to w h e t h e r s t a t e s a r e e v e n o r odd u n d e r t h e s y r r n ~ e t r y e v e n n u m b e r of p a r t i c l e s IV[ h a s t h e s p e c t r u m

lie in the even subspace.

~ -~ - ~ .

R e s t r i c t e d to the odd s u b s p a c e ,

o f F i g u r e 8.

m @

@@al

5m -

-

t One particle states

Figure 8.

The resolvent of z~

for

Imz

/ O.

q u e s t i o n of r e s o n a n c e s

Possible three particle bound states

(M - z) -1 = R ( z )

resonance,

o p e r a t o r is an a n a l y t i c f u n c t i o n

It has a pole at each eigenvalue

of

concerns the analytic properties

section.

M

(particles and

a cut starts at each n-particle threshold.

to the cut, is c a l l e d a r e s o n a n c e .

particle.

o£ t h e m a s s

after continuation across

as a peak in the c r o s s

Three particle threshold

Mass Spectrum on the Odd Subspace of an Even Theory.

bound states) and presumably

matrix elements)

States with an

o£ R(z)

a t h r e s h o l d cut.

The

(or suitable

A complex pole,

close

S u c h a p o l e a p p e a r s in t h e s c a t t e r i n g of p a r t i c l e s Another interpretation

of a r e s o n a n c e i s a n u n s t a b l e

T h e r e a l p a r t of t h e p o s i t i o n of t h e p o l e d e t e r m i n e s while the d i s t a n c e to the r e a l a x i s d e t e r m i n e s

the mass

the lifetime.

of t h e It is a

167

challenging question to m a k e a detailed investigation of resonances, and to determine: A r e there coupling constants for which

~(~0)Z m o d e l s have resonances?

T h e presence or absence of composite particle states depends on whether the interparticle forces are attractive or repulsive.

W e pose the related questions:

Does the mutual interaction of two particles raise or lower their energy, c o m p a r e d with the state in which they are asymptotically far apart? If the energy is raised, binding does not occur.

If the energy is lowered below the continuum, w e expect a

bound state. In Section 3. Z w e motivate our point of view on this question by perturbation theory.

In Section 3.3, w e use cluster estimates and correlation inequalities

to study the s a m e question.

In Section 3.4, w e s h o w h o w binding occurs.

Our picture of a two particle bound state is best understood in terms of the relative .-)

momentum

PR"

W e describe three kinds of forces:

attractive, repulsive and

dispersive.

T h e attractive and repulsive forces are self explanatory.

T h e disper-

sive effect arises f r o m the curvature of the m a s s hyperboloid.

A state of two free

particles, with

and in g e n e r a l , f o r

~T = O,

has a t o t a l e n e r g y

(4m Z + ~ 2 ) l / Z ,

s m a l l m o m e n t u m , a two p a r t i c l e s t a t e has e n e r g y

2m + O(p~ + ~ T ).

This raising

of the e n e r g y a w a y f r o m z e r o m o m e n t u m is what we c a l l the d i s p e r s i v e f o r c e ,

For

bound s t a t e s to o c c u r , the a t t r a c t i v e f o r c e m u s t d o m i n a t e the r e p u l s i v e and d i s p e r sire forces. W e introduce a p a r a m e t e r packet.

For a momentum

6 to m e a s u r e the spread of the bound state w a v e

space distribution concentrated in

a configuration space spreading of order

8-I.

I{RI ~ 6,

w e have

F o r w e a k coupling, w e expect

increased spreading in configuration space, as a bound state g r o w s in size and disappears into the continuum.

Thus w e expect

8 -~ 0

as

)% -~ 0.

T h e binding forces

have characteristic dependences on 8 and k : T h e dispersive effect is

0(8 2 ) . In

p (~0)2 models, w e find in perturbation theory i:hat attractive and repulsive effects are

O(6),

times the appropriate dimensionless coupling constants

discuss the balance of these forces in Section 3.4.

2 kj/m 0 . We

168 3.2 For

a

k~04

F o r m a l Perturbation T h e o r y

interaction,

given by the Feynman

the first order shift i n the two particle energy is

diagram

X which is positive for contributions,

k > 0.

In second order,

a second order mass

we find the shift has two sorts

shift with the disconnected

Feynman

of

diagrams

Q +

O and a second order attractive

' ( n e g a t i v e ) c o n t r i b u t i o n o£ t h e f o r m A

+

The first order repulsive particle

bound states to

shift dominates

for small

occur in weakly coupled

)~.

T h u s we do n o t e x p e c t t w o

4 ~02 m o d e l s ,

and we establish

this

i n S e c t i o n 3.3o We remark mass

that the mass

renormalization

Of c o u r s e , respect

to

of single particle

to second order, nm 2.

by considering

shift diagrams

we measure

states,

forces

We do not include vacuum energy shifts,

perturbations

o£ t h e e x a c t

the second order

i. e. , t h e s h i f t f r o m

our n-particle

m0

to

m 2.

(energy shifts) with

since they are eliminated

( c o u p l i n g X) g r o u n d s t a t e °

If we consider three particle interactions, form

above represent

in lowest order,

diagrams

of t h e

169

give an a t t r a c t i v e t h r e e

body f o r c e .

H o w e v e r the d i a g r a m

X g i v e s a r e p u l s i v e e f f e c t in the two p a r t i c l e s u b s y s t e m s .

S i n c e the two body f o r c e is

f i r s t o r d e r , and the t h r e e body f o r c e is s e c o n d o r d e r , we e x p e c t the r e p u l s i v e f o r c e to d o m i n a t e at s m a l l coupling. A t h r e e p a r t i c l e u n s t a b l e s t a t e ( r e s o n a n c e ) is possible. With a

~p3

i n t e r a c t i o n , the l o w e s t o r d e r two body f o r c e is a t t r a c t i v e

I

1

I l Similarly,

n body f o r c e s in l o w e s t o r d e r a r e a t t r a c t i v e .

F o r i n s t a n c e , in t h i r d

o r d e r we h a v e d i a g r a m s of the f o r m

J

l

I

l

T h e s e a t t r a c t i v e f o r c e s c o m p l e m e n t the a t t r a c t i v e f o r c e s in two body s u b s y s t e m s , i.e.,

in the t h r e e body c a s e ,

l

l

Thus we e x p e c t two p a r t i c l e bound s t a t e s , and bound s t a t e s of t h r e e or m o r e p a r t i c l e s if a s e l e c t i o n r u l e p r e v e n t s t h e i r decay.

O t h e r w i s e , the a t t r a c t i v e m a n y body

f o r c e s should y i e l d m a n y body r e s o n a n c e s . Of c o u r s e , a p u r e f r o m below.

¢P3 t h e o r y does not e x i s t , b e c a u s e the e n e r g y is unbounded

H o w e v e r , if the

¢p3 t e r m in an i n t e r a c t i o n has a c o e f f i c i e n t m u c h

l a r g e r than the o t h e r coupling c o n s t a n t s , we e x p e c t that the ¢p3 e f f e c t s w i l l d o m i n ate.

Thus the above q u a l i t a t i v e d i s c u s s i o n a p p l i e s to the

k l~°3 + kZP(~P)2

model,

170

where

k I >> X z.

In this case w e expect bound states, and in particular, two

particle bound states. Closely related is the case of a the

P(~)Z - ~

locally by X~

model.

~ (~0)2 m o d e l in an external field, i. e. ,

B y the transformation

~0 -* ~ + const.

(implemented

exp (i ~ TT) ) w e can eliminate the external field. F o r instance, the

- ~

m o d e l i s transformed into a k~04 + a ~ 3 + b ~ 2 Z 4ka 3 + am = ~. T h e m a s s t e r m b also grows with ~,

model, w h e r e but by scaling it can be

reduced to unity. T h u s w e conjecture: B o u n d states exist in the ~04 m o d e l with a strong external field,

~ >>

)~.

A similar analysis applies to an arbitrary

)~P(~)2 - ~ )

model.

ing a w a y the external field, w e add to P a lower degree polynomial. the dominant coefficients have degree Z and 3.

TransformFor

~ large,

T h e degree 2 t e r m gives a m a s s

shift, while the degree 3 t e r m yields an attractive potential in lowest order. w e conjecture: B o u n d states exist in

Thus

) ~ (~0)2 m o d e l s in external fields with

~>>k. Question: t h e case.

D o bound states occur in Y2

models?

W e conjecture that this is

1"71

3.3

O n the A b s e n c e of B o u n d States zt k~02

We c o n s i d e r the w e a k l y c o u p l e d

model,

and we prove that two particle

bound states do not occur.

Theorem Then the mass

3. 3. 1. operator

bound state interval

From

)Jm 0

Let

be sufficiently s m a l l i n the

M = (H 2 - p g ) l / Z

can be unitar.ily implemented, ]~e' ~ o

depends on three facts:

in the two particle

w e infer that the s y m m e t r y

and that the Hilbert space each [nvariant under

Cluster expansions

consideration of the two point function for Second, an inequality possibility that m a s s function.

model.

(m, 2m).

the uniqueness of the v a c u u m ,

and odd subspaces

h a s no s p e c t r u m

)~0 4

that Lebowitz

~

U(a,A)

~0 -* -~0

decomposes

into even

and ~0. O u r t h e o r e m

[GI Ja Sp I] reduce the p r o b l e m to the ~o ' and the four point function for ]$ e"

[ L e b 2] proved for Ising m o d e l s

spectrum in the interval

(O,2m)

Finally, cluster bounds exclude m a s s

excludes the

occurs in the four point

s p e c t r u m in the interval

(m, Zm)

in the two point function. T h e condition of w e a k coupling in T h e o r e m exponential decay

e -Yd,

in the error t e r m of the two particle cluster expansion.

W e s h o w in [Ol J a Sp 1] t h a t Theorem

odd subspace, Theorem k / m 02

y-~ 3 m 0

3. 3. 1 w e r e q u i r e t h a t

More generally,

3. 3. I concerns the rate y, of

k/m 0

and

m -~ m O

as

k/m~

be s u f f i c i e n t l y s m a l l t o e n s u r e

we obtain for even

P(~0)2

models a larger mass

3. 3. Z.

Consider an even

)~P(~0)2

model.

Y > 3m 0 - E ,

Given

for the rate

tial decay for the error in the two particle cluster expansion. no spectrum in the interval

A

dq

In y > Zm. gap on the

as s u g g e s t e d in F i g u r e 8 above.

be sufficiently small to ensure

Let

-~ 0 .

let

y of exponenM~

~o

has

(m, 3 m 0 - E) •

be the Euclidean m e a s u r e

of the Euclidean field ~

Then

E > 0,

let

for the

),~04

model,

and for a function

172

(A>

For the

Proposition 3. 3. 3.

=- ~ A d q

)~q4

model,

- (~(x3)#(x4)) (3.3. l)

+ = 0 ,

this inequality states that the connected

f o u r p o i n t f u n c t i o n is n e g a t i v e .

T h i s b o u n d i s s p e c i a l to

I n f a c t t h e p h i l o s o p h y of S e c t i o n 3 . 2 s u g g e s t s t h e p r e s e n c e states

in

6 4 ¢p - ¢p

4 ~0

of two p a r t i c l e

models. bound

models.

The key inequality due to Lebowitz concerns Lndependent spin variables ~Yi = + I

for a ferromagnetic [sing model.

H([)

where

Jij ~

0.

=

-

For a function

The energy of a spin configuration

~

C;I ~j

f(~),

let

i< j Jij

completing the proof. ary to this point.

0 ,

[0, 3 m 0 ~ ¢]

s m a l l to e n s u r e

[2 and

dense in

E~

and we let

in an even

a decay rate

Proposition 3.3.5. vector

[G1 J a S p 1] w h i c h f o l l o w s f r o m t h e t w o p a r t i c l e E

be the spectral

k~(cp) Z

• = 3m 0 - ¢

model.

projection

We a s s u m e

in the two particle

for the energy

k/m z cluster

sufficiently

expansion.

With the above assumptions, linear combinations of the

etHE@(fl,f2)

are dense in

E ~ e. Also the vectors

E~0(f)[2 are

O

W e r e m a r k that in [GI ffa Sp I] w e prove a w e a k e r result for that vectors

cluster

etHE~(f)[2

span

E~

E M o,

. A simple modification of T h e o r e m

namely 4.2,

O

[GI Ja Sp I] can be used to bring first degree polynomials in the n-particle cluster expansion to time zero.

This yields Proposition 3. 3. 5, for

P r o o f of t h e T h e o r e m s . interval

( m , Zm)°

Suppose that

By Lorentz

invariance,

M ~ Me

has mass

proving Theorem

of v e c t o r s

3.3. 1 on

Finally we show

e

tH

spectrum

there is a nonzero vector

corresponding to that spectral interval a n d w i t h energy i s a l i m i t of s u m s

n = Z.

ES(fl,fZ).

in the

~ E M

e

< 2 m . B y P r o p o s l t i o n 3.3.5,

B y Corollary 3. 3.4,

~ = 0,

Me .

M~ E~4

has only one point in its spectrum, n a m e l y

m.

O

B y Proposition 3.3. 5, the vectors

E 0 = [F_~(f)[2] span

E M O . W e let }~ be the

closure of the union of Lorentz translates of }~0 " O u r assertion then follows by Proposition Z. I.Z.

Theorems

3.3. Z and 3.3. 1 then follow by Lorentz invariance.

175

3.4

O n the P r e s e n c e

of B o u n d States

T h e ideas of Section 3.2 suggest the presence of bound states in certain p(q0)2

models.

We

give

two methods to establish the existence of m a s s

t r u m in the two particle bound state interval

(m, Zm).

spec-

A s w e m e n t i o n e d above,

there is no physical interpretation of continuous s p e c t r u m in this interval, existence of s p e c t r u m p r e s u m a b l y ensures the existence of eigenvalues,

so the

i.e. ,

bound states. Variational Method.

state wave function

O,

The first m e t h o d is to choose an approximate bound

with t h e p r o p e r t i e s :

orthogonal to the vacuum and one particle M g H,

we may replace

(3.4. 1)

the bound on ( 0, HS>

(i)

states;

and

.

by the time zero field

= 0 and the canonical c o m m u t a t i o n relations.

[H, a*(f)]

,

is a time zero creation operator.

Vgith this variational method, w e eliminate H~

(fz)~

~

from

in (3.4.2). ( @, HO>

by using

For instance,

where

then

= (f,~OLz + (e,a*(~f) a(F)n) (3.4.3)

+ ( a*(f)~, [HI,a*(f)]~>

z,1/z ~ = (~Z + m0J

176

We estimate

vacuum

expectation

W

=

values

of Wick ordered

a (Xl) . ' '

a(Xn)W(X) d x

monomials

,

by the cluster expansion [GI Ja Sp i]. In fact, before estimation, w e expand < f~,Wf~) ing

using integration by parts, to isolate low order dependence in the coupl-

). , see Chapter 4.

F o r instance, in second order, w e obtain a second order

mass-shift correction to

( f, ~ f > L z .

In this m a n n e r , w e need not calculate the physical m a s s can obtain explicitly the relevant low order corrections to that m

is asymptotic to m 0. ) F u r t h e r m o r e ,

give m o m e n t u m

localization

w e explained that

6 -* 0

O(6) , n a m e l y

as

~ -* 0 . )

=

which exhibits the m o m e n t u m

sketch

our

f(~) = gllZh(~/8).

(In Section 3. i

Then

dispersion about

,

~ = 0

of the single particle state.

X(~O6 - ~04)

(e,HS>

= (@,[a

and integrate by parts. in )..

(~f)a

m z X 2 llflI2+0(k282).

interaction.

8 = a*(f)Zf~ - ( f2, a*(f)ZQ> f~

which satisfies (i), (ii) above.

(Here w e a s s u m e f is scaled to

mollfll z+o(~ z)

proof for the

m 0.

exactly, but w e

let us a s s u m e that

Similarly, the second order m a s s correctionwlll equal We

m

We

take

with Iif11% =

z-I14

W e study

(f)+a*(f)a*(~f)}~>

+

,

W e isolate, in closed form, all t e r m s of degree 0, 1 or 2

The m a s s t e r m s have the f o r m 2[m 0 + kZm2

+ O(82+k3+62k2)}

T h e attractive contribution f r o m diagrams of the f o r m

X

177

l o w e r s t h e e n e r g y by

-O(Sk).

We c h o o s e

Then for small

8 = k I+E.

d o m i n a t e s the d i s p e r s i v e O(k28 + k 3) ~ O(k 3) .

effect

Other contributions are k,

O(8 2) = 0 0 , 2+2¢)

k(~06_ 4)

Similar arguments should hold for

8 = k 2+E .

or higher order.

in e n e r g y

-O(5 %) = -O(X2+¢)

and the r e p u l s l v e e f f e c t s result from a variant

This completes our sketch of the proof that bound state

spectrum exists in the weakly coupled

O(k28).

the d e c r e a s e

T h e o p e r a t o r p a r t s of t h e s e e s t i m a t e s

of the cluster expansion.

tion is

O(k25)

%

model.

6. In this case, however, the attrac-

W e m u s t therefore isolate the fourth order m a s s shift and w e set

F o r the interaction

k 3 + )6 4 , w e m u s t orthogonalize

8 to the one

particle states (at least to third order in ~.). W e would then isolate the fourth order m a s s

renormalization and take

6 =)Z+(.

W e thank B. S i m o n for ohserving

that an even theory is technically simpler. Cluster Method.

(@,e-tH8>

In an even

[~(¢0)2

model, for e of the f o r m

= (~(gl)¢(ga)~tg3)¢(g4)>C

+ ( ¢(gl)¢(g3)> ( #(gz)~(g4)>

+ (~(gl)~(g4)) where

('>C

denotes the connected (truncated) part.

the two particle decay

(3.4.4)

O(e "2mr) ,

(3.3.2),

({(gz)~(g3)) ( 8, e-tH 8 >

Thus

exhibits

unless

( ~ ( g l ) . . . ~(g4)>C m O(e - Z ( m - ~ ) t )

Using the Bethe-Salpeter equation, w e can isolate in decaying part of

(~(gl).L. ~(g4))C,

k( 6.

4)

4 given by (positive) ~0

propose using cluster expansions to estimate t~he errors. would establish the existence of m a s s s p e c t r u m on This proposed calculation appears interesting.

~e

However,

models a slowly contributions.

The inequality (3.4.4) in the interval

(0, 2m-E].

unlike the variational

proof above, w e presently have no error estimates using this method.

Conversely,

w e r e m a r k that the existence of two-particle bound state spectrum in a weakly coupled even

(3.4.4).

f~(~0)z

model

We

(as established by the variational method)

ensures

178

4.

PHASE

SPACE

LOCALIZATION

AND

RENORMALIZATION

4 4. i Results for ~0 3

In a series

of related papers,

we

have given convergent

[GI Ja Sp I, Z] and convergent upper bounds

expansions

[GI Ja IV, 8] for q u a n t u m field models.

These expansions and bounds deal with the p r o b l e m of r e m o v i n g n a m e l y in taking infinite v o l u m e limits in phase space. dealt with the

A -+ R 2

limit.

H o w e v e r the

~ -~ oo

~04: Let

the choice Let

limit in

YZ

and in higher

for both physics and

describe the results C, and let d ~ denote

C = (- A + m ~ ) - I V(A,~)

denote the Euclidean action, the s u m of the V C.

4 ~0

interaction

VI

Then

VI =

VC

In this section w e

d ~ C be the Gaussian m e a s u r e with covariance

and the counterterms

and

A,

w e hope these ultraviolet p r o b l e m s will be the focus of increasing

attention in constructive field theory. for

K,

M o s t of this conference has

dimensional m o d e l s presents the m o s t challenging problems, for mathematics;

cutoffs

k

[

:14:

dx

A~"R 3

are the Green's function counterterms given in second and third order

perturbation theory.

T h e partition function for the action

Z(A,~)

V = VI+ VC ,

namely

= f e -V(A'~) d )

contains the ultraviolet divergent counterterms.

Theorem

4. I. 1 [GI Ja 8].

(4. 1.1)

Z(A,~)

uniformly in We ally by

For

;~ . F o r

n o w let

~

0 ~ )%

e c~IAI)

k bounded, (4. i. i) is u n i f o r m in k

H(If)

denote the renormalized

~04 J

also.

Hamiltonian,

defined f o r m -

179

H(I;) = H 0 + k Is~c R 2

6m~,

Here order

EZ

and

perturbation

from the Green's Corollary

tE 3

:q4 : d~x - ~I 5 m 2Z '[ :~02 : d~x - E 2 - E 3

are the Hamiltonian

theory.

(These counterterms

function counterterms, 4. 1 . 2 .

constant proportional

see

The Hamiltonian

to t h e v o l u m e

(4. l . Z )

0

The corollary

counterterms

in second and third

differ by a constant and a transient

[G1 J a 8 ] . )

H(U)

is bounded from below by a

Ils I ,

H(~) + O([b [)

a

follows from the theorem

and the fact that

( f 2 ( U , n ) , f~0> # 0 .

In fact

(f~0,e-tH(U,n) Q0 )

where

E(%/,K)

=

e-tE(U,K)-A(U,K) + T(U,K,t)

is the partially renormalized v a c u u m energy, vanishing in second

and third order, and convergent as e-A(If'K) = l[ 2 alization constant, bounded as As

and

II; I ~ c o ,

K -~ co

for fixed volume.

The

constant

o(1)

as

the constants

t.

Also

T(ls , n , t)

is a transient

that is

t-~ co. E(ls,~t) ,

A ( U , ~.)

and

The second order, i.e. , the ultraviolet divergent, part of

T ( l s , ~, t) A(ls ,~)

diverge.

has been

Z(A, K).

These results have been extended by Joel ~'eldrnan [Fel 2], w h o proved Theorem

(4. 1 . 3 )

converge

4. 1 . 3 .

The finite volume partition function

Z(A,n)S(A,~;

as

~ -, co.

A

in

is the logarithmically divergent w a v e function r e n o r m -

i n d e p e n d e n t of

n ~ co

cancelled in

,

Z(A,~)

fl ..... fn ) = ~'~(fl )''" ~(fn )e-V(A'n)

The limits are continuous in

Izc. lA; fi, "", fn l

k

and satisfy

llf ll eO(IAII)

and d~

180

for a S c h w a r t z space n o r m

t[" II.

:From continuity in )~ and fixed and

~

Z(A) = 1

h

k = 0,

we conclude that for

h

sufficiently small,

Z(A) Thus for

for

fixed and

I/Z

>

>, small, the approximate Schwinger functions

S(A,fl. . . . . fn)

do n o t v a n i s h i d e n t i c a l l y a n d

IS(A; fl. . . . . fn)I ~ n! ~

(4.1.5)

Ilfi[[ .

i

C o r o l l a r y 4. 1 . 4 are the moments

[ F e Z].

For small

of a u n i q u e m e a s u r e dq

=

lira

on

X,

volume

g '(R3),

h

Schwinger functions

namely

Z ( h , n ) -1 e - V ( h ' n )

d~

K->oo

=

lim ~->oo

dq^~

K

T h e corollary is based on a study of the perturbation of

Z

in a n external

Euclidean field, n a m e l y on the study of the generating functional

Z(h)

= ~ e ¢(h)

for the (disconnected) Schwinger functions.

dqA

This functional w a s studied in

~(~0)2

by Fr~hlich [ F r 2]; see also [Ol J a 13]. Of c o u r s e ,

functions theory.

we a r e i n t e r e s t e d in the

S(A; • )

and of m e a s u r e s

h -~ R 3

dqh,

l i m i t of t h e s e S c h w i n g e r

in order to obtain the full relativistic

W e conjecture that the Kirkwood-Salsburg equations of Part II can be

generalized to

4 ~03

and yield the limit.

In fact the local estimates of T h e o r e m 4 . 1 . 3

and Corollary 4. i. 4 are exactly the type of local estimates which the cluster expansion for small

P(~)2

uses as input. W e conjecture that

Z(A) ~ exp(-OIA I)

k • W e expect that such estimates lead to the W i g h t m a n axioms for

for ~0 4 3

181

4. 2_. E l e m e n t a r y Expansion Steps

T h e proof of the estimates for

~(~0)z,

as well as those for

4 ¢P3' results

f r o m the use of four.elementary identities and bounds concerning the non-Gaussian measure e - V ( A ' n) d ~ c

(4. 2. I) T h e four steps are I. C h a n g e of c o v a r i a n c e

C.

II. C h a n g e of e x p o n e n t Ill. W i c k o r d e r i n g IV.

Integration

V.

bound.

by parts.

The four steps are combined to yield expansions the construction the desired

is generally

property

or bomlds.

T h e d i f f i c u l t p a r t of

t h e q u e s t i o n of h o w to c o m b i n e t h e s e s t e p s to i s o l a t e

of t h e m o d e l ,

at the same time to ensure

convergence.

We

use three expansion techniques: a)

Explicit expansions.

expansion,

We prescribe

as the expansion for

b) N e u m a n n

series.

b y e x p l i c i t e x p a n s i o n of

ZS

definite elementary

i n P a r t II.

The Kirkwood-Salsburg Z,

s t e p s to y i e l d a n

e q u a t i o n s of P a r t II, o b t a i n e d

yield a Neuxnann series

(I - }~)-I = ~ n }~ n

for

their solution. c)

Inductively defined expansions.

(integral)

in our expansion,

There

is considerable

the inverse

freedom

l e a v e t h e w i d e s t l a t t i t u d e of c h o i c e ,

the most detailed information

In addition,

in closed form,

the inductive expansions

about our models,

term

terms.

i n t h e d e f i n i t i o n of o u r e x p a n s i o n s

an expansion expressible

of a n o p e r a t o r .

for each possible

r u l e s t o e x p a n d i t i n t o a s u m , of s i m i l a r

The inductively defined expansions are not tied to recovering

We prescribe

and bounds. since they

o r to o b t a i n i n g

and bounds yield

i n c l u d i n g t h e p o s i t i v i t y of

4 q~3

[GI Ja 8] and the ¢p-bounds for all couplings [GI Ja IV]. These expansions and bounds are not tied to the use of particular boundary conditions on the covariance

182 C, b u t h a v e m o r e g e n e r a l v a l i d i t y . We now g i v e the e l e m e n t a r y

steps;

the f i r s t two s t e p s a r e m e r e l y t h e f u n d a m -

e n t a l t h e o r e m of c a l c u l u s : I.

C h a n g e of C o v a r i a n c e .

polating covariances,

Let

C

= aC 1 + (1-~)C 0

be a f a m i l y of i n t e r -

and let 1

d~Cl

:

d~Co +

f

d d#ca d~ ~-~ l

1

= d ~ C o + ~(CI-C0).

dad~c~

Ai

T h i s f o r m u l a h a s b e e n u s e d to d e a l w i t h the i n f i n i t e v o l u m e l i m i t , s e e P a r t lI.

It is established by integration by parts on function space [Di GI]; see also the proof of iV below. II.

W e do not use Step [ in this chapter.

C h a n g e of V .

Let

Va

interpolating Euclidean actions.

~ £[0,1]

b e a d i f f e r e n t i a b l e f a m i l y of

Then 1

e

-Vl

= e

-Vo

~0

+

d -V ~-~ e t'

da

(4. z. z)

v°

1%

= e

~---'~-e

do~

0 W e use this identity to lower an upper m o m e n t u m positiv[ty proofs for

e (~0)Z

cutoff in the action V , in the

[See G1 Ja 7, iV] and

4

[GI Ja 8], and we call

this formula the perturbation or Duhamel identity. I t e r a t i n g (4. 2. 2) l e a d s to the u n r e n o r m a l i z e d p e r t u r b a t i o n s e r i e s .

ultraviolet cutoff, this series diverges because of the in n th order.

O(n~ Z)

diagrams arising

For example, with one degree of freedom,

e_qZ_k q4 dq

#

co ~ n=O

With a n

(_x)n q4n n: ~ e'q2 dq

183

since the series on the right side diverges.

It is therefore

necessary

to truncate

perturbation theory, for which w e use step III below. III.

Wick Bound.

4 :~X:

For

(4. Z. 3)

e

we have

eO(log z ~IIAI

d=Z

eO(~ZllAI

d=3

-V(A,~)

This bound follows by integrating :~4:

=

(~

over the space time volume

- 3c )2 - 6cZ

-6c

2

A . Here

I O(log~) c~

= [ ~ ( x ) z d~

=

d = 2

C~(x,x) d = 3

O (K)

This W i c k b o u n d is used to raise the lower m o m e n t u m O u r expansions t e r m i n a t e

V(A,n,O). IV.

Integration

cutoff

p in the exponent

if n = p.

by P a r t s . 5F

We use this integration renormallzation other forms

by parts formula to exhibit the cancellation

counterterms

of ( 4 . 2 . 4 ) ,

V 1 - V0

o f (4. 2. Z).

I n [G1 ffa IV, 8] w e u s e

called there the pull through and contraction

It is easy to establish

(4. Z. 4) b y s t u d y i n g f i n i t e d i m e n s i o n a l

to the function space integral. to

in

We choose a Gaussian

measure

d~ C , d~ N

of t h e d i v e r g e n t

=

N exp

-g

-= V N d q N

,

.~ . 1.,j

qj

dqk

formulas.

approximations

d~ N

converging

184 where C..1j [s the covariance m a t r i x integration by parts then yields

~ %1 qjFIqld0~ =

and

N

-;F(q)--

is a normalizationconstant.

5v N

Ordinary

dqN =; 8,F(R),,, d~ N 5 qi

qi

Inverting C,

5F

w h i c h c o n v e r g e s to (4.2.4) as

d ~ N -~ d ~ C .

For Wick ordered monomials,

w e obtain similarly

. . . . ;:~(X 1) . ~(Xn' :F(~)d~ G =.; :#(x 2)

A s an e x a m p l e ,

d{ N

~'(Xn): ; o~x' 1 C"'Xl,Xl},, ~~ F

w e integrate b y parts one ~(x)

d~ G

factor in a simple expresskon,

J" :}4(x):e -J':#4:dz d~ C = -4~dxdyJ':~3(x):C(x-y):,3(y):e -S:#4:dz d~ C

F u r t h e r integration by parts yields

~4, 2. ~

;:~%:e4:~:dz

d0c = ~ ; ~ d y C~x-y,~ ;e4:0~:dz d0~ + other terms .

185

4. 3.

Synthesis of the E l e m e n t a r y Steps

W e have two basic a i m s in combining the e l e m e n t a r y expansion steps.

First,

w e desire convergent expansions in a given space-time or phase space volume. Second, w e desire polynomial decoupling of different localization regions. w e present the v a c u u m case, m o m e n t u m

cluster expansion for

localization is u n n e c e s s a r y

~(~)2

In Partll

m o d e l s in full detail.

In that

(no ultraviolet divergences occur) and

our localization regions are unions of unit lattice squares in space time.

W i t h no

cutoff, distant regions decouple exponentially, and the d e c a y rate determines the physical m a s s .

In this section w e present the basic ideas of phase space localiza4 ~03

tion w h i c h w e used to deal with the ultraviolet divergent yield the results s u m m a r i z e d

in Section 4. I.

model, a n d w h i c h

F o r s m o o t h cutoffs in m o m e n t u m

space, w e obtain polynomial decoupling. F o r simplicity w e discuss the partition function Z . W e fix the v o l u m e and investigate h o w

Z

depends on the ultraviolet cutoff ){. In order to truncate

the perturbation expansions, w e introduce a lower cutoff p ensure that

V(~{,p)

h

into the action

is b o u n d e d f r o m below, w e introduce the m o m e n t u m

in a s y m m e t r i c fashion:

each m o m e n t u m

c o m p o n e n t in

V(;{,p)

V.

To

cutoffs

lies in the interval

[p, *.]. W e p e r f o r m our expansions o n integrals of the f o r m

(4.3.1) where

~ R(¢)e -v(n'p) d~ R

i s a p o l y n o m i a l f u n c t i o n of { .

a s u m of s i m i l a r t e r m s . to lower

A t the start of the expansion, p = K,

~R

i.e. ,

d~ . W e

Each expansion step replaces

"We u s e a h i g h m o m e n t u m

K , and w e use a low m o m e n t u m

V(K,p) = 0 ,

p = 0

and

,

(perturbation)

( 4 . 3 . 1) b y

expansion step

(truncation) expansion step to raise

p.

K = K 0 . T h e expansion terminates w h e n

and (4.3. I) is r e d u c e d to a s u m of G a u s s i a n integrals

estimate this s u m uniformly in K 0 .

T h e rules for alternating the expansion steps are s o m e w h a t complicated.

The

m a i n idea is to obtain a small contribution f r o m each high energy vertex in R , by

i86

performing explicit renormallzation cancellations.

W e avoid the

(n!)r

number

of

t e r m s which would arise f r o m iterating (4.2 2), by truncating the perturbation expansion. T h e high m o m e n t u m V 1 = V(~',p),

expansion.

V 0 = V(K_,p).

W e use Step II to lower

T h e first t e r m in (4.2.2) has the desired form.

second t e r m has the s a m e upper cutoff Since

6V

has a lower m o m e n t u m

convergence factor

-K (

~ , taking

and a n e w vertex

6V = dV

cutoff at K_, w e desire that 8 V

to our final estimates.

We

/d~

The

in R.

contributes a

obtain the proof of this fact

only after performing the renormalization cancellation of the dive rgent counterterms 8V C

in 8V.

6V o We

Using Step IV, w e integrate by parts

also integrate any n e w

the exponent.

~4

part of

(and w e cancel) the ultraviolet diver-

F o r instance, in (4. Z. 5), w e displayed the second order v a c u u m

energy contribution.

T h e third order v a c u u m

a m o n g the "other t e r m s " in (4.2. 5).

and the m a s s

The vacuum

exactly with the correspondlng counterterm in diagram,

n a m e l y the

V I p r o d u c e d in I~ as a result of differentiating

W e thus obtain in closed f o r m

gent part of 8V.

6 V l,

energy contributions cancel

8V C . The mass

after cancellation, leaves a r e m a i n d e r

t e r m s " f r o m this procedure are convergent

counterterms occur

renormalization

O(K--(). T h e remaining "other

and so contribute

Kj (

to the final

e stimate. In this m a n n e r ,

Steps II and IV c o m b i n e to yield one order

a renormalized perturbation expansion.

B e c a u s e of the large n u m b e r

it is necessary to truncate this expansion after introducing

The low m o m e n t u m

expansion.

T h e W i c k bound is an

in space-time cubes

A

Leo

~-6

8V) in

of terms,

vertices

8V.

W e truncate the perturbation series by

raising the lower cutoff in the exponent. p-.

(i. e. , one

We

use Step III to raise

p

from

9_ to

estimate on path space, so w e expect to apply it

on which (4. Z. 3) remains bounded,

[zl,,i

~

i.e. , cubes for w h i c h

o(I)

This restriction m e a n s that the localization length

L = IA 1I/3

satisfies

187

(4.3. z)

L ~

O(~-'z/3)

and defines our phase space localization.

,

O n the other hand, the uncertainty

principle requires that

(4.3.3)

O(~.'I)

~

L

O(p_ 1)-

for the localization to be proper, i.e. , for the spreading packet (due to m o m e n t u m

(4. 3. 3) a r e c o m p a t i b l e . renormallzability.

localization) to be less than

L.

of the wave

We note t h a t (4.3. Z) -

T h i s c o m p a t i b i l i t y is a c t u a l l y a n o t h e r a s p e c t of s u p e r -

F o r the

4

model,

(4. 3.2) would be r e p l a c e d by

0(~--I),

L

for which our estimates are borderline. O u r analysis has s h o w n that w e m u s t treat separately cubes space-tlme cover

4.

(Also s o m e

A's tend to zero as

deal with upper and lower cutoff functions Furthermore, low m o m e n t u m V(~,p),

part of

are less than

which are functions of A .

p-.

4 ~P4' however,

t e r m s and by

part equals

T h e cross t e r m s in 6 V

in order to raise the lower cutoff in the exponent.

procedure has s o m e complications,

divergences,

}to -~ oo . ) T h u s w e actually

8 V = V(}t, 9_) - V(}t, p-) . This low m o m e n t u m

cross t e r m s by the low m o m e n t u m

In

belonging to a

w e r e m a r k that the W i c k bound (4.2.3) deals only with the pure

i.e., all m o m e n t a

be r e m o v e d

}t(A), p(A)

&

V(}t, p-) ,

must

W e dominate the

the n e w exponent.

but poses no essential difficulty, see [OiJa 8].

our biggest challenge.

Finally w e r e m a r k that the relevant distance

p a r a m e t e r that w e m u s t use with phase space 1ocallzation is the scaled distance

d

= Euclidean distance

For smooth momentum

×

lower m o m e n t u m

d,

cutoff.

cutoffs, scaling standard estimates gives

correlations between different phase cells with proper localization. n > 3

This

it is such cross t e r m s which yield the charge renormalization

Independence of P h a s e Cells.

for

also

O(d -n)

decay of

A n y such decay

is sufficient to control distance factors in s u m s over phase cells (whose

188

d i a m e t e r goes to zero as

~0 -~ co) . W e

remark

that in the llmLting t h e o r y without

ultraviolet cutoff, w e expect to r e c o v e r exponential decoupling a n d a m a s s

gap.

189

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Harvard thesis.

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model,

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The

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Renormalization of the Hilbert space in the m a s s shift model, J. Math. Phys. 13, 918-927 (1972).

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Phys. 31, IZ7-136 (1973).

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Supported in part by the National Science Foundation, Grant N S F - G P - Z 4 0 0 3 . t

Supported in part by the National Science Foundation, Grant N S F - G P - 4 0 3 5 4 X .

$

Supported in part by the Alfred P. Sloan Foundation.

THE PARTICLE

STRUCTURE OF THE WEAKLY

COUPLED P(¢)2 MODEL AND OTHER APPLICATIONS OF HIGH TEMPERATURE Part II:

EXPANSIONS

The cluster expansion #

James Glimm* Courant Institute, New York, N.Y.

NYU

Arthur Jaffe % Harvard University Cambridge, Mass.

Thomas Spencer Courant Institute, New York, N.Y.

Contents 1.

Introduction

2.

The main results

3.

The cluster expansion

4.

Clustering

5.

Convergence:

6.

An equation of Kirkwood

7.

Covariance

8.

Derivatives

9.

Gaussian

10.

and analyticity:

Proof of the main results

The main ideas Salsburg type

operators of covariance

operators

integrals

Convergence:

The proof completed

Supported in part by the National

Science Foundation,

Grant

Science Foundation,

Grant

NSF-GP-24003. %Supported

in part by the National

NSF-GP- 40354X

%

Supported in part by t-he Alfred P. S!oan Foundation.

NYU

200

§l.

INTRODUCTION

From the point of view of physics,

there are essentially two pro-

blems in the study of quantum field models. find the phenomena of interest to physics states, resonances, broken symmetries, expansions)

The first problem is to

(e.g. particles,

bound

and asymptotic power series

in the two and three dimensional quantum field models. 1

This problem includes these phenomena,

obtaining the mathematical structure underlying

and determining qualitatively which interactions

and/or parameter values give rise to these phenomena and which do not. We believe this problem is feasible at the present time; indeed the initial progress is the subject of the present lectures.

The second

problem is the construction of nontrivial fields in four space time dimensions.

Here it seems that some new ideas are needed to supple-

ment those which have already been used in the construction of quantum fields. These two problems are also interesting from the point of view of mathematics.

In addition we mention a third problem, which is to

place the solutions framework.

already constructed in a more general conceptual

In particular,

of independent variables,

elliptic equations with an infinite number Markov fields, non Gaussian integrals

over

functions of several variables, and C*-algebras arise naturally in the construction of quantum fields. directions

The quantum field results may suggest

for further development of these theories.

In Part II of these lectures, we give in detail a high temperature (cluster) expansion,

in a form which yields an exponential

property for the Schwinger functions,

cluster

as well as analyticity in the

coupling constant ~, for k in a bounded sector 0 < Ikl < ~o '

-~/2 < arg ~ < ~/2 ,

for the P(@)2 quantum field theory.

Other applications

of expansion to the study of quantum field models part I of these lectures;

of this type

were developed in

see also [4,5,6,14].

The expansion itself is related to the virial and cluster expansions [13, Chapter 4] of statistical mechanics. statistical mechanics for some time.

The analogy between

and P(@) quantum field theories has been known

For a discussion of the relation of statistical

mechanics to quantum field theory, see the lectures of Guerra, Hose

1For three space time dimensions, it is still necessary to complete the construction of the quantum fields. However enough progress has been made to eliminate any serious doubt as to their eventual existence.

201

and Simon,

as well as Part I of these lectures.

These lectures coupling.

develop a cluster expansion in the case of small

We believe expansion

techniques

meter values which are sufficiently

far from the critical point.

belief is based partly on low temperature Pelerls

and by Minlos

as work of Dobrushin Spencer

will apply for any paraexpansions

Our

developed by

and Sinai [10] in statistical

mechanics

and Minlos mentioned in part I.

as well

Furthermore

[14] has extended the expansion presented here to the case of

large external Within

field.

their region of convergence

the most detailed information the main information

expansion methods

available.

In fact these methods

known about the energy spectrum,

level E = 0 of the vacuum.

Correlation

theorem and other analyticity

methods

inequalities,

complement

the Lee-Yang

these techniques

and

See [7,10,14]

of this series.

For the cluster expansion, mechanics

yield

abo~e the energy

often yield results outside the region of convergence. and other lectures

tend to give

the relevant

is the following expansion

formula from statistical

for the density

of the Gibbs

ensemble : -BV(xi-xj ) (I.i)

I ] e i<J

[i =

I I i<J

= [ r

(e-BV(xi-xj ) +

I f

(e-Bv(xi-xJ ) - I)

(i,J)el"

Here F is a set of distinct unordered pairs and the sum extends interaction e -BV(xi-xj),

] - i)

over all such graphs.

(i,J), i.e. Mayer graphs, This formula expresses

the

between the distinct i and J (i~J) particles, as a sum of a zero interaction

term, l, for which there

-BV(xi-x j ) is no coupling between the particles which is small at high temperatures

and a perturbation kT = B-l°

e

Heuristically,

-i the role

of the Gibbs density in the P(¢)2 field theory is replaced by the measure (1.2)

e

~-~2 d e ( x ) xeR

Here

.

202

(1.3)

~o(X)

=

21--(V¢(x)2 + m2¢(x) 2)

and the formal expression

-f~o(X)dx (1.4)

e

denotes

x-~R2 de(x)

the Gaussian measure

(1.5)

C~

=

on ~'(R 2) with mean zero and covariance

(-A + m2)-lo-

We also use the notation de C for the Gaussian measure with covarlance C, so that

(1°4) =dec

.

density for transverse the nonlinear

We recognize

vibrations

restoring F

=

of an elastic membrane

Gibbs

subject

to

force

-m2¢(x) O

after integration

(1.2) as the canonical

-

kP'

(¢(x))

over the momentum variables

E(x).

In (1.2), the coupling between distinct points comes entirely from the V¢ term i n t o ( X ) e -SV in (1.1). (1.1).

and so e -/V¢2 in (1.2) plays the role of

Our cluster expansion is constructed

In the completely f~o(X)dx

=

is replaced by zero.

decoupled

The resulting

ultralocal

theory [9,12] seems to

to the theory defined by (1.2).

and control the singularity a lattice structure

in the spirit of

the Laplacian in

½]¢(x)(-A+m~)¢(x)dx

be very singular relative decoupled theories

theory,

of the difference

in two separate steps.

We reduce

between the coupled and

The first step introduces

into R 2 and into the expansion generalizing

We do not introduce

the lattice into the measure

resulting expansion

is an expansion for the continuum infinite

(1.2) and so the

P(¢)2 theory, rather than for a lattice approximation Let F denote a set of lattice

(1.1). volume

to this theory.

lines, Joining nearest neighbor

lattice

points in Z 2, let A r be the Laplace operator with Dirichlet 2 boundary conditions on F, and let (1.6)

Cr

=

(-AF + m~ )-I

2From the point of view of this discussion, Neumann data might seem more appropriate, but we use Dirichlet data because it is technically easier.

203 Then d¢CF plays the role of the decoupled measure, with decoupling along the curve F. 3 In summary, the lattice structure gives discrete variables in the sum and product

r (i,J)er in (I.I), even when this formula is applied to the continuum P(¢)2 model. The second step regularizes

the differences

corresponding

to

e -Sv(xi-xj)- i in (i.i). The difference between two Gaussian measures can be expressed as 1 O

where C(s)

=

sC 1 + (I-s)C 2 .

This formula applies equally well to the non Gaussian measures

e - k i P (¢ (x)) dx de C of interest to us. There is a simple formula for the evaluation of d d-~ dec(s)' namely [i] (1.7)

d I Fd¢c( s) d--s

=

1 I (C'(s)-A¢)F dec(s)

where

(z.8)

C'(s).AcF =

=

~2 d i C(s,x,y) ~¢(x)6¢(y) F dxdy ds

f [CI(X'Y)

-C2(x'Y)]

62

~¢(x)6¢(y) F dxdy

and C(s,x,y) is the kernel of C(s). The proof of this formula contains an integration by parts in 3In comparison with (i.i), F should be replaced by (Z2) * % F, where (Z2) * is the set of all nearest neighbor bonds in Z 2.

204 the function space integral.

In the case of interest to us, C 1

and C 2 = CF2, and the integration by parts regularizes 4 d d-~d#c(s) "

4Forma!!y, d

d~c(s) = J(¢)d~c(s) , where the Jacobean J is

j(~) = ½ f:~(x)[C(s)-iC'(s)C(s)-l](x,y)~(y):dx

dy

= -51 I:~(x)[~d c-l(s)](x,y)#(y):dx dy Since

CFI

and

CF2

are mutually singular,

singularity in J(#) should Be anticipated.

for

F 1 ~ F2,

some

205

§2. For small values ty on the Schwinger

THE MAIN RESULTS

of I/m~, we prove an exponential

functions.

We prove the cluster property

finite volume with bounds independent results,

it follows

the infinite functions

of the volume.

easily that the Schwlnger

of the boundary

Osterwalder-Schrader

conditions.)

field theory is constructed

See [6].

theorem,

Using the volume

functions,

axioms

strlctly posltive.

We also show that the Schwinger functions

and the physical mass is

0 0

r ~ Zr(A). Theorem 6.1. let

m0

Let

is the unique solution satisfies

Ikl £ ~ ,

be sufficiently the bounds

By the dominated Thus for small

large.

in

~

Re k L 0 Then

of equation

on

(6.1).

is small and defined above

Moreover

PA

convergence

theorem,

Z~A(A) + 1

as

c + 0.

~,

restriction

restriction

> 0

PA

(6.2)°

1 - < I z ~ (A)I F This

where

Z(A) ~ 0, and

fixes e

in

the these

proof of Prop.0sition IZ~x(A~X)/Z(A)I

c

in

Theorem

6.1,

and

in

fact

is

the

only

lectures.

5.2, assuming Theorem 6.1. -- IZx*(A)/Z(A)I

1

and

and C e ~

6 > 0 , there is a constant such that

|C(J)aLq_(AJI×Aj2) --< K4mo2/q exp(-m 0(1-6)d(j)) Proof.

Because of Proposition 7.i, we may assume

d(J) > 0, necessarily property choosing

(c). ~

d(J) ~ l,

The factor d(J) = 0

For

and the required bound follows from

m0 2/q in the proposition comes from

in (c) smaller than the

consider the case

C = C~.

6

in the proposition. Next

(i.e. equal or adjacent squares).

We have

|C(x'Y)gLq(AJlXAJ2 ) ~ K3JC(''0)ILq (R2) K4(q)mo 2/q The last inequality follows from the fact that (-A+l)-l(x,0) e Lq for q e [1,~) and from the identity (-A+m~)-l(x,0) = (-A+l)-l(m0x,0) Remark.

The differentiated covariance

bound of Proposition

8FC also satisfies the

7.1, as we can see from (7.3) and the inequality

226

0 ! ~ Hence

~rC

Local regularity and basic

the bound of Proposition

of the covariance

C e~

formal properties

integrals

obtain fractional than

(s b + (l-Sb)J b) = l-J b ! 1 .

also satisfies

definition Gaussian

d

is used to justify

(e.g. integration

and it is used in the bound derivatives

of

7.2.

C e ~

IZI ~ e OIAI .

, with bounds

the

by parts)

of

We

slightly w e a k e r

(e). Proposition

independent

of

Proof.

7.3.

C 6 ~loc "~ 2,5

for

0 < 6 < 1/2 with bounds -'

C .

We start with the observation

that

o !-a !-A r Consequently one.

COI/2cFc~ I/2

is a bounded operator,

By convex combinations,

|B| ~ 1

with norm at most

also, where

B = C~ !/2 C C~ I/2 Let

-(1/4)+~ ~/2

A = C~ for

~ e C~(R2).

calculate (7.6)

~C

We use the fact that A A*

the Hilbert

|(-A+m~) -C+I/4

=

Tr A*A B2A*A

< |BN 2 Tr (A'A) 2

|A*AaH2S

Ei=l ¢(xi) des(F)

Let # ( F ) be the set of all partitions and (1.7), (i0.I) equals (10.2) where

0})

~_ e~p (o [~ d(J,~-)] exp ( o l r l )

242

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III

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BOSE FIELD THEORY AS CLASSICAL STATISTICAL MECHANICS. I. THE VARIATIONAL PRINCIPLE AND THE EQUILIBRIUM EQUATIONS FRAN~ESCO GUERRA I n s t i t u t e of Physics , University of Salerno, Salerno, I t a l y

1. INTRODUCTION In the l a s t two years new powerful methods have been exploited f o r the advancement of the constructive quantum f i e l d theory program, ~i0,II,43J

of Glimm and Jaffa

and t h e i r followers. These new methods rely on ideas from Euclidean f i e l d theory 23,35~

and use p r o b a b i l i s t i c techniques and concepts. They have been mostly

advocated by Nelson [24,25,26], following e a r l i e r proposals by Symanzik L42]. In p a r t i c u l a r Nelson isolated the crucial Markov property of the Euclidean f i e l d s , which plays a very important role for the construction of the Euclidean theory and i t s physical i n t e r p r e t a t i o n . One of the most a t t r a c t i v e features of the Euclidean-Markov f i e l d theory for Bosons is that a l l physical quantities are expressed by means of commutative f i e l d s . Moreover the vacuum expectation values for interacting f i e l d s have a remarkable s i m i l a r i t y with the expectation values in Gibbsian ensembles of c l a s s i cal s t a t i s t i c a l mechanics. This s i m i l a r i t y , very well known since many years, suggests the p o s s i b i l i t y to e x p l o i t the modern techniques of rigorous s t a t i s t i c a l for the study of constructive f i e l d theory. Such a program

mechanics ~3,141

has been advocated by Guerra, Rosen and Simon ~17,18,19~, with f u r t h e r developments by Nelson ~ 8 ] , Simon ~6,37]

and G r i f f i t h s and Simon ~9,40 I .

In these lectures we present part of this program, dealing mainly with the variational p r i n c i p l e for the entropy and the equilibrium equations for i n f i nite volume systems of the type proposed by Dobrushin, Lanford and Ruelle in s t a t i s t i c a l mechanics. Our main concern w i l l be to provide a characterization of the i n f i n i t e volume states associated to a given i n t e r a c t i o n , independently of l i m i t i n g procedures on volume cut o f f theories. In the following lectures Lon Rosen 131]

and Barry Simon 13~ w i l l present the other parts of the program

and i t s developments, mainly the l a t t i c e approximation, the correlation inequalities with applications and the Lee-Yang theorem with i t s important consequences. For other applications of s t a t i s t i c a l mechanics ideas to constructive f i e l d theory we refer to the talks by Nelson [27]. The powerful techniques of ~Postal Address : I s t i t u t o di Fisica d e l l ' U n i v e r s i t ~ Via Vernieri 42, 84100, Salerno, I t a l y .

244

contour expansion ~ 3 ] , also related to s t a t i s t i c a l mechanics ideas, with t h e i r important applications to the i n f i n i t e

volume l i m i t and the p a r t i c l e spectrum are

presented in the lectures by Glimm [8] and Jaffe I20]. The content of these lectures is the f o l l o w i n g . In Sections 2 and 3 we review the basic properties of the free and i n t e r a c t i n g Euclidean-Markov f i e l d , in order to f i x notations and introduce the motivations of the s t a t i s t i c a l

mecha-

nics analogy which w i l l be exploited in the f o l l o w i n g Sections.ln Section 4 we introduce the i n f i n i t e

volume l i m i t of the pressure associated to a given

i n t e r a c t i o n . The structure of the i n f i n i t e

volume states and t h e i r entropy density

are discussed in Section 5 . F i n a l l y in Section 6 we introduce the v a r i a t i o n a l p r i n c i p l e f o r the entropy density and in Section 7 the e q u i l i b r i u m equations of the Dobrushin-Lanford-Ruelle type. For the basic concepts of p r o b a b i l i t y theory and stochastic processes

we refer to Reed's talks ~0] and to [7]. 2. THE FREE EUCLIDEAN-MARKOVFIELD. We consider f i e l d s on the Euclidean space R~. The physical case is

dL=~

(three

space-one time dimensions). The harmonic and anharmonic o s c i l l a t o r s correspond to 0[=C , the l~(u~)z theory to &=;L and t h e ~

theory[12] to ~[:3 .

F i r s t of a l l we introduce the Sobolev H i l b e r t space N of real temperate distributions

F on R~ , with symmetric scalar product

where < , >

is the usual Lebesgue scalar product on Fourier transforms, ~

the Laplacian in d. dimensions and mz

is

is a p o s i t i v e constant.

D e f i n i t i o n i . The free Euclidean-Markov f i e l d is the real Gaussian random f i e l d ~(~) , indexed by W and defined by the expectations

We call ( O , Z , p ) ~(F) are represented as still

the underlying p r o b a b i l i t y space, then the f i e l d s LP(O,Z,I~)

functions on (~ , L-~J:~=o

, which we

call .~o(F), in such a way that the expectations can be expressed as i n t e g r a l s

We assume that ~

is the smallest G--algebra with respect to which a l l

f i e l d s ~(#) , F(~( , are measurable. Due to the Euclidean invariance of the scalar product in fq , the f u l l Euclidean group E ( d ) ( i n c l u d i n g r e f l e c t i o n s ) can be represented in the natural way as a group of measure preserving automorphisms of the -algebra ~ . To each closed region A

of F~& we associate the s u b - ~ - a l g e b r a >-A of ~_-

245

generated by f i e l d s ~(#) with

s u p p i % A . We call

EA

the conditional expectation

with respect to ~-A" Proposition 2.

Let ~T be a smooth (d-l)-dimensional closed manifold dividing Ri

in two closed regions A and B , such that A U B : Ra and AMB = IT

E~ =

EA

E~

,then

.

In order to describe the connection with the Hamiltonian theory, let us introduce the Sobolev Hilbert space F of real temperate distributions in

R~'i

with scalar product

whereA is the (d-1)-dimensional Laplancian. Then the time zero physical free field of mass 4~ in

d-dimensional space-time is the real Gaussian random process ~£(#) ,

indexed by F and defined by the expectations

If CQ,~,~)

is the underlying minimal p r o b a b i l i t y space, then the

~o~ space is represented as ~_z(~,~,~) and the ~oK function - C L o ~

on Q .

Let us now define, for ~-C R Jr: "

vacuum corresponds to the

Lv((~,E,~.)

->

, the operators ]'~

LC[O,z,V )

,

~.~p_~oo,

such that ]'t..C'L~, =

~0o

,

where ~ is the function q : - ~

on 6) and ~'~ is, for

gcF , the distribution

in R4 belonging to N and defined by

R The operators ~

provide the connection between the Hamiltonian theory

and the Euclidean-Markov structure. By means of ~e t r i c a l l y into L~((2,Z, F) with image L~(O,F~,F')

/~(Q,Z,~) is embedded isome, where Ze is the sub-O'-algebra

of 5" generated by the fields with test functions having support on the hyperplane The physical free Hamiltonian No can be expressed in the form e

:

,

or equivalently -~Wo where ~c

and ~

are

space functions.

The well known hypercontractivity [41] of the free Hamiltonian can be

246

expressed in the best Nelson form [26] as

with

~;0

, i ~

~~ ~ ~

, where

II

11~ denotes the norm of a map from

At the level of the Euclidean-Markov theory the following general version of the hypercontractive property can be proved [18]. Theoremj. D~ (Basic hypercontractive estimate). Let A~ andA2 be two regions in R& separated by a distance ~ o I f ~ L with respect to Ai

provided

and A2

(¢~-~)6~-£)

and ~2 are two Q space functions measurable

respectively, then the following estimate holds

~ G(~>

, where

~/e)=

O(~d'~e -2~"')"

This estimate is an improvement with respect to plain H~Ider's i n e q u a l i t y , corresponding to ~(~)=~ , because we can take ~ provided ~

and ~

as near to

i

as we l i k e

is large enough. On the other hand, should distant regions be stochasti-

c a l l y independent, we would have

But the free measure couples d i f f e r e n t regions of ~

, in fact

E-A~ EAaz/: ~"

Therefore the hypercontractive estimate t e l l s us that stochastic independence is almost realized as ~ - ~ .

Thus we speak of exponential decoupling of distant regions

and we expect a kind of thermodynamic behaviour for the local interacting theories. We conclude this section with the following checkerboard estimate, stated for the two-dimensional case. Theorem 4. ~8] (Checkerboard estimate). Consider two orthogonal systems of p a r a l l e l lines at distances ~ + ~ , partitioning Ra in squares of side ~+2c . Let A~ , : £,..,a~

, be ~

d i s t i n c t squares concentric to squares of the p a r t i t i o n and

with sides of length ~ p a r a l l e l to the lines of the p a r t i t i o n . I f ~ are measurable then the following estimate holds

II E.. F.

~-A~

II II = .. II f.

Remark. I f both ~ and a become very large then ~ Z stochastically independent.

, thus the regions A~ become

3. THE INTERACTING THEORY When the interaction is turned on i t is expected that the interacting f i e l d s ~ equal to the free f i e l d s

~

are

as functions on (~ space, but there is a change in the

measure, so that the expectations of the interacting f i e l d s ~

are given by

247

where ~

is a new measure depending on the i n t e r a c t i o n ,

For the two-dimensional case the interaction is specified by a polynomial I)(X) with real coefficients, bounded below and (without loss of generality) normalized to P ( O ) - O . To each compact region A

of ~Zwe associate the Euclidean action

defi ned by

U^ = where the local l i m i t is obtained through the removal of an u l t r a v i o l e t cutoff (see [18]) and the normal Wick product can be introduced in a purely stochastic fashion as explained in Nelson talks [27] . The main properties of the Euclidean action are summarized in the following theorem. Theorem 5. (Properties of the Euclidean action U^ ). a)

U^ E ~ ( Q , Z A , ~ )

,

~ - = < ~ o ,

e_2H~

p. ° > ,

which is at the basis o f [15,16] . On the o t h e r hand, formal reasoning, based on the a n a l y t i c c o n t i n u a t i o n of the well known Gell-Mann-Low formula, suggests that f o r the volume c u t o f f i n t e r -

acting theory we must take as new measure

248

which is s t i l l

a Markov measure ~27,18] even though i t is not covariant. The f u l l y

covariant interacting measure ~

must be obtained taking the l i m i t as A-~ Rz

of

~A in a suitable sense. The volume cut off vacuum expectation values of products of interacting fields are given by

This expression is the starting point of the s t a t i s t i c a l mechanics analogy, exploited in these lectures and summarized in the following table. (~ space

Configuration space

fields

Basic observables

(p(~)

Free expectations

e-U^ Z^ : JQe-U^dM.

Gibbsian factor Partition function Gibbs expectations in A

= IAI -~ .2o# Z,

Pressure Correlation functions

....

=

family {~c^~ of positive, normalized

State

consistent densities on C~ Entropy This analogy can be further deepened i f we go to the l a t t i c e approximation ~8] as explained in the talks

by Nelson ~

, Rosen [3~ and Simon ~8].

In this case a kind of nearest neighbor interaction can be extracted from the free measure, so that d i s t i n c t regions become stochastically independent but coupled through this interaction. In this approximation the free theory is represented as an array of Gaussian spins with nearest neighbor ferromagnetic interaction, when the interaction ~

is turned on then only the distributions of the single spins

are affected but not the nearest neighbor coupling. From this point of view, in the l a t t i c e approximation the local interaction Ua

acts like a kind of chemical

potential. In the rest of these lectures we consider the problem of characterizing the states of the i n f i n i t e volume system in terms of the interaction, using s t a t i s t i c a l mechanics ideas. We follow two main lines of development connected with the variational principle for the entropy density, as introduced by Ruelle [32,33] , and the equilibrium equations of the type considered by Dobrushin •3,4,5] and Lanford and Ruelle [21] . We show also the connections between the two lines of

249

development. 4. THE PRESSURE Since in the following we need f i e l d s in a region with zero boundary conditions, we begin this section presenting some basic facts related to the general conditioning theory for which we refer to ~8,19]. Consider a closed region A of

R~ , l e t WA be the subspace of ~ made

of distributions with support on A , em the orthogonal projection on WA and

EA

the conditional expectation with respect to the sub-C-algebraZA of 7- generated by fields~(F) with fe~A

, as in Section 2. We call

space of the random f i e l d ~(F)

IOA,ZA, ~^) the

probability

, with FEN A

For the free Markov f i e l d

~(F) , FE ~ , we can write

~(~)=q~((~-eA)f) + and define the two independent fields

~(e~f)

~

and ~A through

~(~)=~o((~-eA)~), SoA(F)=so(e~F), FEN. Obviously ~X(F) is zero i f

FEWA

, therefore we call

~

the f i e l d obtained

from ~ by conditioning i t to be zero on the region A . Let us call the probability space of ~

~(~)=@~(F).~A(F) and

EA~ = ~

~

,

0 0 o (Q~,Z.A, ~A)

, then we have

O=~xQ~

0

, >- = Z ~ x Z A , ~ = ~ x p m

,

~.

Let A' be the closure of the complement in R& of the compact regionA , then for f~ WA

we call

~(F)

the D i r i c h l e t f i e l d in A

with zero boundary

conditions on ~A , the boundary of A . As a consequence of Markov property i f

~E ~A

then

~(F)=

~(F)

For a two-dimensional f i e l d theory we introduce the Euclidean action in A with zero boundary conditions on ~A , defined by

Using the properties of Wick ordering i t can be shown ~8] that

Given a normalized interaction I )

, l e t us introduce the following

definitions. Definitions 6 Partition function

Z A -

~

e-U~

250

Dirichlet partition function

ZA =

°

I

e- m (;IF :

e -UA° c[~;,

,

Pressure

p~ - I A I'~ //~ Z ,~ .

Dirichlet pressure

m Im ° : IA l'~z~#~Z~.

and Io~

lo:

°

Our main objective of this section is to investigate the behaviour of as A-> oo

Lemma 7. For the partition functions we have a) ~L ~ Z ~ b) I f

.< Z ^

A=A~.UAz

and AL(]Az

has zero measure, then

Z~ ~ Z A~ ° Z A2 ° i f Ai and Az are disjoint then the equality holds. c) logZ^ and log Z~

are convex functions of I) for fixed A , i.e. for two inter-

action polynomials Pi

for

0-~- A i ~ -i ,

and ~ we have

A ~ ÷ z \ z = dL.

Proof. a) We use Jensen's inequality

f~A

~

J~

.. ~

and the relation

,

I ,, II,.,,, o& I"-,~, o Q^,

for f ~ : ~

,

= o

following from the properties of Wick ordering and the normalization of 1) , to write

Z^

o,~, Gt,,

~

a,t,.

4h,,.

4.p^, 4 p,,, ~ a:,

= Z^

>1 e

4p;, =

= d..

b) We can write

,p;,0

=

~o4#

+

% : 0,

*

So~,

where the three fields on the right hand side are independent and ¢pedescribes the degrees of freedom associated to the intersection of A~ andAz (thus ~ : 0

if

A±

andAzare disjoint). Therefore

z~ =

~, e-

° = ~X~xQo~ aP A'

where we have used Jensen's inequality and

~ e- U~

4p~ ~P~i ~A'~ >I

251

I f A~ and Az are disjoint then we have directly 0

and therefore the equality in

0

b) .

c) Follows t r i v i a l l y from H61der's inequality. Let us now introduce the i n f i n i t e

volume energy d e n s i t y ~

~15,16],

defi ned by

~

:

z~.

-

E.,/.e

=

.,.

-E.el~

where E~ is the ground state energy of the volume cut off Hamiltonian ~l introduced in Section 3. Lemma 8.

ib~ ~ ~

~ o(~ .

Proof. The f i r s t bound follows from a)

A{~)

rectangles

of Lemma7. The second bound is obvious for

of sides ~ and 4# , since by the Feynman-Kac-Nelson formula

we have

_Yc E~

~ H~~ °

Z A (~,/~)

4.cz o , e -

=

>

~ e

,~.~o~,~ -< e

More general regions can be handled through a limiting procedure as explained in[18]. Now we can take the thermodynamic l i m i t for rectangles / ~ ( ~ ' ) . Theorem 9. a)

~A(.~,~)is

monotone increasing in ~ and ~T and

4,~ b) For fixed

log

0

Z AIS,+.)

is superadditive in ~ , the following l i m i t exists

~;~ ~,~-~

o

/PA(~,~)

and is equal to

c)

o.

o~°~

~

d) (:,4=, and o ~ Proof.

o~

•

are convex functions of ~

a) For any } ~

.

we have ,e

therefore

f:

ZAIe,~)

~ ~(~,l) ~,~-,~ b) The s u p e r a d d i t i v i t y of ZA~2,~}

>I ~ A ~ , ~ )

log

o

ZA(t,t )

= d~

"

follows from part

ZA(~,Iz~ }

for

b)

of Lemma 7 , in fact

~-= ~ i ' / ' ~ z

then, by a standard argument using s u p e r a d d i t i v i t y , we have o

c) and d) follow from Lemma 8 and Lemma 7 (part

c).

The following theorem, whose proof can be found in I19], establishes the convergence of the pressures for more general regions and the equality of the l i m i t s . Theorem I0. As A-> om

(Van Hove)

In this way we have introduced the f i r s t basic thermodynamical quantity, the pressure o I ~ 1~) associated to a given interaction. 5. STATES AND ENTROPY. The basic objectives of constructive Euclidean f i e l d theory are to prove the existence of states associated to a given interaction and to study t h e i r physical properties. In general a state w i l l be given by some p r o b a b i l i t y measure ~ and the problem arises to see how ~

on Q space

is related to the free measure ~- and the

interaction F i r s t of a l l l e t us remark that i f

~

is translation i n v a r i a n t then i t

cannot be absolutely continuous with respect to the free measure unless i t is t r i v i a l . In fact, by the ergodicity of the translation group on ~:f&~,

with

FE L~CQ, Z , ~ )

~(Q,Z,

~)

, the r e l a t i o n

, would imply F=~ . This is the Euclidean ver-

sion of Haag's theorem. We can also look at i t from the point of view of Van Hove phenomenon ~ 5 ] . Consider a volume cut o f f two-dimensional interacting theory. Then the expectation value of a general observable A

can be expressed as

where we have introduced the Q space normalized wave function of the approximate vacuum !

~H^ = Z . .

I

e -~u^

,

% e L~(Q,Z,~

)

~ ~ ? < ~ .

Then using the same techniques of ~5] and ~6] we can prove the following Euclidean version of the Van Hove phenomenon. Theorem I i .

In the l i m i t A->R z the Van Hove phenomenon holds, in the sense that

253

%

--) O

, weakly in

Lz{Q,Z,

~

.

In general we have that IIL~AII~ tends to zero i f I-A,~)

and normalized

b) the family I ~ ] is compatible in the sense that ~^ ~^, = ~^ for regions A , A ~ such that A~A' , where E^ is the conditional expectation with respect to ~'m . We say that the state is

~-smooth for some ~>i i f in addition we have

~m ~ L~( Q, Z A , ~.) for each A By definition the physical expectation E(~}~A) of an observable A >-A measurable, in the state F is given by =

ECA^)

I t is clear that the family ~'-algebra containing all ~^

~Î

=

, which is

mF^ a t , .

defines a unique measure ~

on the smallest

for A compact.

The similarity with the corresponding definition in statistical mechanics [33] is evident. In analogy with statistical mechanics [33] and information theory, we now introduce the concept of entropy for these states.

254 D e f i n i t i o n 13. Let F be a )~-smooth state. For each compact region A the entropy of F in

A

is given by

when J~ is fixed we write simply ~ ( A ) in place of

~

~4: )

Theorem 14. The following inequalities hold a) (boundedness)

-o0

C) (weak subadditivity)

~(A)

< ~(A

~(A')

b) (decrease)

~ i=-( ~

~-

if

) ~ O

if

A = z~i / ~

,5(At)

A ~A'

,

, then

*" 209. IIt^~.. Fa " Ib_"

Proof. We use the elementary inequality

where ~

,

,.~(A)

- log sc ~ C ~ - £

is any probability measure and F

and Jensen's i n e q u a l i t y

is a non-negative L~ function,

a) By Jensen's inequality

On the other hand

b) Using the compatibility condition on }

we have

= 5f^, %(#^~;,*)a> ~ % f ¢ ~ > : o . c) As in the proof of

b)

we have

m

"-~L

This theorem is very s i m i l a r to analogous results in s t a t i s t i c a l mechanics /33] The main difference is in the correction term given by the logarithm in the weak s u b a d d i t i v i t y condition. This term is c l e a r l y related to the lack of stochastic independence of d i s j o i n t regions. As in s t a t i s t i c a l mechanics we are interested in the l i m i t of the entropy density

JAI"± ~ ( A )

Definition 15. tants ei 0

4

+

~z(~)

,where

I~ I -~- J~llh, I1~ .

=

as ~ - ~ o

, i f we can show that in the same l i m i t

, then we have

~(#) _< ~_~

l~j-i~(A)

and therefore the existence of the following l i m i t is proved

S~F) : Now for large ~

~ ~->

we have ~ - > 4

I A I -~ , . ~ ( A )

.

, therefore by interpolation

But by weak temperedness logll~^]l~increases at most as exp [ ~ ) hand we have ~2-C ~ p ( - c ~

~')

, with

~o as ~ - ~

To complete the proof of the theoremwe must show the a f f i n i t y of S(F3 . I f we put

4, :

-\'- ,%,

,S,,

),

then we find as in the s t a t i s t i c a l mechanics case [33]

where the f i r s t i n e q u a l i t y follows from the monotonicity of log.# from Jensen's i n e q u a l i t y . Dividing by IAI

and taking A - > ~

and the second

we obtain the a f f i -

nity relation. 6. THE VARIATIONAL PRINCIPLE In s t a t i s t i c a l mechanics, a variational p r i n c i p l e for the entropy density ~2,33] provides a very elegant characterization of the i n f i n i t e volume equilibrium states associated to a given interaction. In this section we introduce s i m i l a r ideas in Euclidean f i e l d theory. The v a r i a t i o n a l p r i n c i p l e involves three quantities : the pressure, ~(p)

, which depends only on the interaction P ,

introduced in Section 4 ; the

entropy density,S(F) , which depends only on the state and f i n a l l y the mean i n t e r a c t i o n ,

~(#,p)

F

, introduced in Section 5;

, which depends on both the interaction

I ~ and the state # . A complete discussion of the v a r i a t i o n a l principle should involve the following steps. a) Isolate a class ~}" of states

~

, which is expected to contain a l l states of

physical i n t e r e s t , in p a r t i c u l a r those associated to a given i n t e r a t i o n I > . b) Prove that for a l l states in ~

the following inequality holds

257

S(f)-

~(~,P)

--~ o ~ ( 1

p)

(Gibbs v a r i a t i o n a l i n e q u a l i t y ) .

c) Prove that ~,p(S(#)-f(~,]P))

= ~('j>)

(Gibbs v a r i a t i o n a l p r i n c i p l e ) ,

where the supremum must be taken with respect to a l l # E ~ I ~ . d) Prove the existence of a state #~ (or a family of states) such that

s(G)- ~C~p,P)=~CP). The states ~

are, by d e f i n i t i o n the e q u i l i b r i u m states associated to the given

i n t e r a c t i o n Ip . e) Prove that t h i s notion of e q u i l i b r i u m states agrees with the d e f i n i t i o n given by means of the e q u i l i b r i u m equations of the Dobrushin, Lanford, Ruelle type discussed in the next section. We t a k e ~

to be the class of t r a n s l a t i o n i n v a r i a n t , weakly tempered

states. In this section we prove

b) and c)

the available p a r t i a l results about problem

and in the next section we w i l l give e) . Unfortunately, at the present

stage of development of the theory, i t is not possible to answer little

is known about the i n f i n i t e

d) , since very

volume states as measures in general. Therefore

our results are f a r from d e f i n i t i v e , but they strongly support the idea that a v a r i a t i o n a l p r i n c i p l e can be used to characterize the i n f i n i t e volume e q u i l i b r i u m states in Euclidean f i e l d theory. D e f i n i t i o n 18. Given a~-smooth i n t e r a c t i o n polynomial 1) region A by

Proposition 19.

(~7£)

t r a n s l a t i o n i n v a r i a n t state # and an

, we define the mean i n t e r a c t i o n associated to the compact

e^(f,P) has a value,

~(F, P I

, independent of A .

Proof. I f A i

and A~ are two regions with i n t e r s e c t i o n of zero Lebesgue measure,

then, f o r

A = A± UA~

, we have

where we have used the l o c a l i t y of U^ and the c o m p a t i b i l i t y condition f o r Since A ~ ~ that

~^

The step

is continuous in A , using t r a n s l a t i o n invariance, i t e a s i l y follows

is indipendent of b)

A

of the previous discussion is now straightforward.

Theorem 20. (Gibbs v a r i a t i o n a l i n e q u a l i t y ) . For any semibounded i n t e r a c t i o n polynomial

P and any weakly tempered t r a n s l a t i o n i n v a r i a n t state F , we have

s{;}-

~(¢,P)

-~ ~ o ( t " }

.

258

Proof. By Jensen's i n e q u a l i t y we have

dividing bylAl and taking A-~o~ we conclude the proof. Let us now consider a class of non t r i v i a l

t r a n s l a t i o n i n v a r i a n t smooth states, very

near to e q u i l i b r i u m states for a given i n t e r a c t i o n , defined as follows. Consider two orthogonal families of p a r a l l e l l i n e s at distance ~ in squares of side ~

, A± ~Az~ . . . .

Call o-

, p a r t i t i o n i n g the plane

~z

the region of R~ made by these

l i n e s . Then the free Markov f i e l d can be w r i t t e n as a countable sum of independent fields

Here ~

describes the degrees of freedom on o- and is given by

qp~('~c}= qp(e~F) ciated to ~

, where e~ is the projection in H on the subspace IVy- asso-

as explained in Section 4. On the other hand each

~2

is defined as

and describes the residual degrees of freedom in each square A ; . Recall t h a t , as in Section 4 , At~ is the closed complement of A~ , therefore

~Z is the D i r i c h l e t

f i e l d in A~ and is zero outside A: . For the p r o b a b i l i t y space

~Q,>-,~)

of

the free Markov f i e l d we have

Q :

Q~ x Q:[ x Q~ x . .

,

Z=

Z_~-x Z _ ~ . x Z ~ x . .

,

in terms of the probability spaces ( ~ . , Z~, ~ ) associated to ~

and ~

Consider now the state

where

and

(Q~, m~ , t~i)

respectively, {=~.z,.. ~: given by the measure

OL~ = Z ~ ~ e -U~ (L~ R z . Using the

Markov property then i t is easy to verify that kc^, is P-Gibbsian in every region A

contained in A'

, therefore

~

w i l l be P-Gibbsian in general.

In classical s t a t i s t i c a l mechanics of l a t t i c e systems the equilibrium equations are very powerful, since they allow to find all properties of the equilibrium states [3,4,5] . But in Euclidean f i e l d theory the situation is similar to the case of classical continuous systems studied by Ruelle [34] . In fact i t is very easy to see that the equilibrium equations of Definition 26 are not s u f f i c i e n t to characterize completely the "physical" states, but must be supplemented by some restriction on the allowed behaviour of states at i n f i n i t y , otherwise unphysical spurious states can appear as solutions of the equilibrium equations. A similar situation is found in quantum s t a t i s t i c a l mechanics in the study of states satisfying the KMS conditions [22]. We give an example of the occurrence of spurious solutions in Euclidean f i e l d theory and then propose that the right boundary condition on the states is weak temperedness. For our example we consider the case of zero interaction for the onedimensional Markov f i e l d on the real l i n e , characterized by the free covariance

The same method works for a linear or quadratic interaction and in several dimensions.

262

Proposition 27. For any interval [~,h] consider the system of densities

bq(b))

, cze zb

f o r real

C . This system is normalized and consistent, moreover i t is Gibbsian

for zero interaction. Proof. A simple computation. Since the "physical" state for zero interaction must be the free state (corresponding to c = O ) we see t h a t a l l cases c~o must be considered spurious even though they s a t i s f y the equilibrium equations.

On the other hand by e x p l i c i t compu-

tation we have t cZe 2b therefore only the case c=o

gives a weak tempered state.

The example strongly suggests the idea that weak temperedness is the r i g h t boundary condition to be imposed to l~-Gibbsian states in order to characterize completely the equilibrium states associated to a given interaction. Further support comes from the following ( p a r t i a l ) discussion about the possible equivalence of the two notions of equilibrium state given by the variational p r i n c i p l e and the l~-Gibb sian condition. F i r s t of a l l l e t us remark that f o r a P-Gibbsian state we have the following form of the densities

e-U^G^ where

~-~^

= E~^ (~Ua)

,

is the conditional p a r t i t i o n function (see

Proposition 28. For any Gibbsian state IAI'~A(~) where

~

=

f ^ ( G 1 ~) ÷

= lAl'l~o~ 7~^

jAI-Z g ~ ^ ( # )

+ E({~,~,'lO"~, ~ ) ,

is the conditional pressure.

Proof. We have

Multiplying by

-lal-tfa

and integrating we get the proposition.

Now i t is e a s i l y proved, using the properties of the entropy and the basic hypercontractive estimate, that f o r any weakly, tempered, translation i n v a r i a n t state we have

IAl'i~^(f-)->

0

as A->~

(see [18] ). The conclusion is that

for any weakly tempered, translation i n v a r i a n t , Gibbsian state we have

Therefore i f i t is possible to prove that the conditional pressure ,~.~^ converges to oloo in a suitable sense as

A-> o~ , then the following conjecture would be

263

veri lied. Conjecture 29. Every weakly tempered, translation invariant, ~-Gibbsian state satisfies the variational equality. We would like to conclude this section by referring to recent announced results of Dobrushin and Minlos [6], which show that a complete control of the equilibrium equations can be extremely important for the actual construction of the theory and the verification of the conventional wisdom ~3,20,1] about dynamical i n s t a b i l i t y and broken symmetry. Theorem 30.(Dobrushin and Minlos, announced in [6] ). For all polynomials ID bounded below, the set of all Euclidean invariant, P-Gibbsian states ~ the integrals F~ ~{Rz~

~)~d~

, for which

exist for any integer ~ and depend continuously on

, is a non empty convex set VqCP) . The extremal elements of ~ ( P )

are ergodic with respect to the Euclidean translations and ~(P) is a Choquet simplex, i.e. any element of ~{P) can be represented as the baricenter of some probability measure on the extremal elements of ~q~P) Let for all

P~X)=~p(x),

A~A{P)

~0

, then there is a A~P)>O such that

the set ~q{P) consists of exactly one element. I f P

then there is a A ' ( P ) < ~

such that Vq(P) for

.~ ~ A'(P)

is even

contains at

least two different extremal states.

REFERENCES 11] R. BAUMEL , Princeton University Thesis, 1973. I2] W. CRUTCHFIELD, Princeton University Senior Thesis, 1973. L3] R.L.DOBRUSHIN, Gibbsian Random Fields for Lattice Systems with Pairwise Interactions, Funct. Anal. Applic. 2 (1968) 292. [4] R.L. DOBRUSHIN, The Problem of Uniqueness of a Gibbsian Random Field and the Problem of Phase Transitions, Funct. Anal. Applic. 2 (1968) 302. [5] R.L. DOBRUSHIN, Gibbsian Random Fields, The General Case, Funct. Anal. Applic. 3 (1969) 22. [6] R.L. DOBRUSHIN and R.A. MINLOS, Construction of a One Dimensional Quantum Field Via a Continuous Markov Field, Moscow Preprint, 1973. [71 I . I . GIKHMAN and A.V. SKOROKHOD, Introduction to the Theory of Random Processes, W.B. Saunders Co., Philadelphia, 1969. ~] J. GLIMM, These Proceedings. 191 J. GLIMMand A. JAFFE, The A ( ~ Quantum Field Theory Without Cutoffs. I l l . The Physical Vacuum, Acta Math. 125 (1970) 203. ilOIJ. GLIMMand A. JAFFE, Quantum Field Theory Models, in Statistical Mechanics and Quantum Field Theory, Les Houches 1970, C. DE WITT, R. STORA, Editors, Gordon and Breach, New York, 1971. I111J. GLIMMand A. JAFFE, Boson Quantum Field Models, in Mathematics of Contemporary Physics, R. STREATER, Editor, Academic Press, New York, 1972. ~2IJ. GLIMMand A. JAFFE, Positivity of the ~ Hamiltonian,Fort. der Physik, 21 (1973) 327. ~ J . GLIMM, A. JAFFE and T. SPENCER, The Wightman Axioms and Particle Structure in the P¢~)z Quantum Field Model, New York University Preprint, 1973. ~ R . GRIFFITHS, Rigorous Results and Theorems, in Phase Transitions and Critical Phenomena, vol. I ved. C. Comb and M.S. Green, Academic Press, London and

264

New York, 1972. [15] F. GUERRA, Uniqueness of the Vacuum Energy Density and Van Hove Phenomenon in the I n f i n i t e Volume Limit for Two-Dimensional Self-Coupled Bose Fields, Phys. Rev. Lett. 28 (1972) 1213. [16] F. GUERRA, L. ROSENand B. SIMON, Nelson's Symmetry and the I n f i n i t e Volume Behavior of the Vacuum in P(~}~ , Commun.math.Phys. 27 (1972) I0. [17:1F. GUERP~A, L. ROSENand B. SIMON, Statistical Mechanics Results in the P@~ Quantum Field Theory, Phys.Lett. 44B (1973) 102. [1~]F. GUERRA, L. ROSENand B. SIMON, The ~(~)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. Math., to appear. 9] Fo GUERRA, L. ROSENand B. SIMON, in preparation. o] A. JAFFE, These Proceedings. 110. LANFORD and D. RUELLE, Observables at I n f i n i t y and States with Short Range Correlations in Statistical Mechanics, Commun.math.Phys. 13 (1969) 194. ~ R . P . MOYA, Equilibrium States for the I n f i n i t e Free Bose Gas, University of London, Preprint, 1973. ~ T . NAKANO, Quantum Field Theory in Terms of Euclidean Parameters, Prog. Theor. Phys. 21 (1959) 241. ~ E . NELSON, Quantum Fields and Markoff Fields, in Proceedings of Summer Institute of Partial Differential Equations, Berkeley 1971, Amer. Math. Soc. Providence, 1973. ~ E . NELSON, Construction of Quantum Fields from Markoff Fields, J.Funct.Anal. 12 (1973) 97. 6]E. NELSON, The Free Markoff Field, J.Funct.Anal. 12 (1973) 211. E. NELSON, These Proceedings. ~ E . NELSON, in preparation. ~ C . NEWMAN,The Construction of Stationary Two-Dimensional Markoff Fields with an Application to Quantum Field Theory, J.Funct.Anal., to appear. ~ M . REED, These Proceedings. ~L. ROSEN, These Proceedings. ~ D . RUELLE, A Variational Formulation of Equilibrium Statistical Mechanics and the Gibbs Phase Rule, Commun.math.Phys. 5 (1967) 324. ~3]D. RUELLE, Statistical Mechanics, Benjamin, New York, 1969. ~ D . RUELLE, Superstable Interactions in Classical Statistical Mechanics, Commun.math.Phys. 18 (1970) 127. ~J. SCHWINGER, On the Euclidean Structure of Relativistic Field Theory, Proc.Nat.Acad.Sci. 44 (1958) 956. ~ B . SIMON, Correlation Inequalities and the Mass Gap in P(~)~ . I. Domination by the Two Point Function, Commun.math.Phys. 31 (1973) 127. ~71B. SIMON, Correlation Inequalities and the Mass Gap in P(@)~ . I I . Uniqueness of the Vacuum for a Class of Strongly Coupled Theories, Toulon University Preprint, 1973. [3~B. SIMON, These Proceedings. [39]B. SIMON and R. G r i f f i t h s , Griffiths-Hurst-Sherman Inequalities and a Lee-Yang Theorem for the (Vv)z Field Theory, Phys. Rev. Lett. 30 (1973) 931. ~O]B. SIMON and R. GRIFFITHS, The ( ~ ) z Field Theory as a Classical Ising Model, Commun.math.Phys., to appear. ~IIB. SIMON and R. HOEGH-KROHN,Hypercontractive Semigroups and Two Dimensional Self-Coupled Bose Fields, J. Funct.Anal. 9 (1972) 121. ~2JK. SYMANZIK, Euclidean Quantum Field Theory, in Local Quantum Theory, R. JOST, Editor, Academic Press, New York, 1969. ~3]A.S. WIGHTMAN,Constructive Field Theory. Intro4uction to the Problems, Coral Gables Lectures, 1972.

f!

~

BOSE F I E L D THEORY AS C L A S S I C A L STATISTICAL MECHANICS. II. THE L A T T I C E A P P R O X I M A T I O N AND C O R R E L A T I O N I N E Q U A L I T I E S

Lon Rosen* Mathematics Department U n i v e r s i t y of Toronto Toronto, C a n a d a MSS IAI

T h e s e lectures are devoted to the idea that nothing but a m o d e l of classical statistical mechanics.

P(~12

is

Francesco

Guerra has a l r e a d y d e s c r i b e d to you the v a r i a t i o n a l p r i n c i p l e and equilibrium equations, inequalities.

and I wish now to discuss the role of c o r r e l a t i o n

My lectures consist of three parts:

i.

The lattice a p p r o x i m a t i o n

2.

Proof of c o r r e l a t i o n inequalities

3.

Applications

The purpose of the lattice a p p r o x i m a t i o n is to exhibit the ferromagnetic n a t u r e of E u c l i d e a n Bose field theories,

this being the

critical i n g r e d i e n t in the proof of c o r r e l a t i o n inequalities.

It turns

out that it is the free theory w h i c h determines the f e r r o m a g n e t i c properties and so these results are e s s e n t i a l l y m o d e l independent. they depend on the number of space dimensions,

Nor do

at least on a formal level.

The c o r r e l a t i o n inequalities that have been e s t a b l i s h e d in Bose field theories are of types G-I

and

G-If

respectively

G-I, G-II, FKG,

refer to Griffiths'

and

GHS

Hurst and Sherman

Here

first and second i n e q u a l i t i e s

[7,4], FKG to Fortuin, K a s t e l e y n and G i n i b r e

GHS to Griffiths,

.

[9] .

[2], and

In the statistical m e c h a n i c s

* Research p a r t i a l l y supported by USNSF under grant GP39048.

266

context,

the first three types

Ising models

(in particular,

single spin distributions) only to "classical actions).

(G-I, G-II, FKG)

continuous

hold for quite general

spins and fairly a r b i t r a r y

while the GHS inequalities

seem to apply

1 (i.e. spin [ ferromagnetic

Ising models"

A further approximation

(the "classical

pair inter-

Ising approximation")

is needed in order to bring GHS to Bose field theories and this will be the subject of Barry Simon's

lectures.

The m a i n reference Guerra-Rosen-Simon

[I0].

for these lectures will be the paper of

Since it will not be possible

supply all of the technical details by referring

freely to

found in Ed Nelson's exclusively

in the time available,

[10] I can concentrate

of the relevant material

to correlation

inequalities,

lectures

Much

that I quote can be

Since these lectures are devoted the reader should refer to the

lectures of Guerra and Jaffe for the general setting and to Jaffe's

I hope that

on the main ideas.

and some of the results

lectures.

for m e to

statistical m e c h a n i c s

for a comprehensive

survey of recent

results. i.

The Lattice A p p r o x i m a t i o n ~(x)

described

denotes

in Nelson's

measure on Q-space mation

the Euclidean Bose field over ~ _ I ( I R d)

lectures,

(see also

the Laplacian

A

d~

is the corresponding

[13,10]).

is a type of ultraviolet

is replaced by the lattice

and

cutoff

Briefly,

finite difference operator. subspace of Q-space,

By restricting

we obtain,

space

with spacing

in the Euclidean propagator

free

the lattice approx~-

in which Euclidean

~Z d = {nSln~Z d}

, as

~>0

is approximated

IR

d

and by a

to a finite dimensional

instead of the full measure

dZ , a

Gaussian

d~ ~ (2~)-N/21BII/2e-~"~ dNq in terms of a finite number of variables with a lattice point.

Here

B

ql,...,q N

(1. l] each associated

is a positive definite

N×N

matrix

267 which is ferromagnetic nonpositive,

and

in the sense that its off-diagonal

Formally,

-4

(6>0)

=

is positive

diagonal"

property

can be understood

the free field measure

d~ Now

are

IBI = det B .

The ferromagnetic follows.

elements

const,

as

is just the Gaussian

e -~(~' (-~+m2)~)d~

"on-diagonal"

as is evident

intuitively

(1.2)

and negative

"infinitesimally

from its finite difference

off-

approximation

:

(-A6f) (n6)

=

6-212df(nS)

-

[

f(n'~)]

(1.3)

In' nl= where we norm sum over

Zd

by

n' ~ Z d

takes place over the

To convert transform

T6 =

~16]

from

~2 (Z d)

to

nearest

A =

(-A6+m2)

~ a(n-n')h(n') n'

a(n)

as an operator

,

of

n .

we introduce

where

on

=

its image

on

by means

operator

Inl = 1

(1.5)

otherwise.

L 2 (Td)

is multiplication

by

By a simple computation (2~/6)d/2~6(k)

of

n = 0

-6 -2

A

~2 (Z d)

(1.4)

where

0

(2z/6)d/2a~(k)

neighbours

( 6 ~d/2 (n)e_ik.n6 [~-~J n!zd h

=

m 2 + 2d6 -2

Therefore

so that the

theorem,

L 2(T6 d)

(1.3), then we see that it is a convolution (Ah) (n) =

IndJ

:

A h6(k )

If we regard

2d

this remark to a rigorous

the Fourier [-~/6,

I (nl ..... nd) I = Inl] +...+

=

d 26-2(d - [ cos(6ki)) i=1'

-= P6 (k) 2

+ m2

268

Note that as

6 + 0 ,

p~(k) + p(k) = (k2+m 2) i,/2

We wish the cutoff field C = ~-dA-~

-= I ~ (n) ~8 (n')dp

Accordingly, Definitions of the field

i.i.

Cnn' =

(2z) - d [ d eik" (n-n')~p6(k)-2dk" ~T8

=

~(x)

by

~

is defined in terms

and the

]'16(k) dk

p(k) 's

cancel with the

in the definition of the inner product, yielding

Wick powers of

$6(n)

: ~6(n) 2

:

(1.6).)

are defined in the usual way, e.g.,

=

¢8(n)

(i .7)

2 - C00

and smeared powers by r :~5:(g)

[ dSd:~6 (n)r:g(nS) nEZ

=

g E Co( IR d )

where

For the with space cutoff

P($)d A

theory the (smeared) Schwinger functions

and lattice cutoff

~

are defined by _U A

SA,6(hl,o..,h r)

Here

A

where

=

is a bounded region in

UA, ~ P

f~8 (hl) . . . ~ (hr) e le_UA,~d~

]R d =

'

(1.8)

hl,...,h r ~ C~(]R d)-0

[ 6d : P(¢6(n)} n~eA8

is a semibounded polynomial,

'Sd~

and

A6 =

, and

:

A n (~Z d)

denotes

269

the lattice

points

enclosed

by

A .

What have these definitions the integrands namely

in

(1.8)

qn ~ ~ ( n )

for

supp hj c A).

accomplished?

involve only a finite number of field variables, n~ ~ A 6

(where we assume for convenience

Thus by the definition

the numerator

in

In the first place

(1.8) reduces

of

d~

(see Reed's

that

lectures),

to a sum of terms of the form

const. I qn l...qn r e

_z6d :P (qn) : d~A,~(q)

(1.9)

where the Gaussian measure i/2

d~A,~(q)

Here

N =

IA61

,

=

(2v)-N/21CAI-

e

the number of points

covariance

matrix

of the

restricted

to the indices

q's in

1

~I

_2--q'CA q dNq

in

A6 ,

(1.10)

and

with entries defined A

Similarly

CA by

is the

NxN

(1.6), but

for the denominator

in

(1.8). Secondly,

this approximation

sense that the interaction which each term involves

is locality

is approximated

the field

qn

by a sum

matrix

CX I

which occurs

infinite matrix

C

entries defined

by

equality

Theorem

where

for our purposes

in the Gaussian

has a particularly (1.5).

in

lattice

point.

cutoff procedure.

is the structure

of the

density.

simple

Now of course

in the

[~d:p(qn) :

at only a single

This is not the case for the usual ultraviolet But m o s t critical

preservin~

By construction, the --i inverse, C = ~dA with

(CA) -I ~

(C-I)A = ~dA A , but

almost holds:

1.2 B~A

[i0]

If

A c IR d

is a positive

that is "concentrated

is bounded,

semi-definite

on the boundary

then

CA I

=

~d (AA-B~A)

matrix with nonnegative 3A~ ".

elements

270 By this we mean the following: .ext = 6zd\A~ A~

points we let

• .. int ~A~ = A~\A 6 .

and R

if

Bnn , = 0

A~

of lattice

A~in,6 e A ~

if

In-n'I = i},

.int = { n ~ , A6

A matrix

unless

given a set

B

is said to be c o n c e n t r a t e d

n6,n6' ~ R

.

on a set

The proof of the theorem

is based

on the r e p r e s e n t a t i o n i

A_-I

A

e _2--q "U A q

where

qR

=

const,

stands for the variables

nearest neighbours, integrated

lim R-~oo

i I R ~d A R -IR61/2 IARI /ale-2--q " R~ dq R\A (2~)

out in

R6 .

Since

(I.ii) the elements of is concentrated

AR

on

off-diagonal

elements.

analogy with the Ising ferromagnet boundary variables,

, like

This is the M a r k o v property

B~A

The covariance

correspond

if we take

B.C.

This statement

remain unchanged;

A we see that CA .I CA ferromagnetic in

Moreover,

except for the

in the lattice setting. of the matrix

to different

B~A = 0 ,

B~A ?

Different

choices of boundary

condition.

On the other

we obtain the lattice theory with Dirichlet

is justified by Theorem 1.3 below

to the operator

(see also

[10,

that the finite d i f f e r e n c e

(-AA+m2)

with Dirichlet

just

AA

since the effect is to ignore the variables

A~ .

We obtain free B.C. by "integrating

B°C. on

in

.ext A~

~A

is

immediately outside

out" the variables

and Dirichlet B.C. by setting the variables simple expression

are

A , links only nearest neighbours.

and is based on the observation

approximation

A int

qR\A

(1.6) of Theorem 1.2 is .that of free Be C.

hand,

#IV.3])

We call

[ 14].

What is the significance choices of

links only

on

From the theorem and the definition of has nonpositive

AR

it is clear that when the variables

CAI-6dAA

consequently,

on

(i.ii

in

A~ ext

equal to zero.

for the Dirichlet covariance on the lattice

-I

=

A

(in)

The

271 shows that Dirichlet B.C.

B.C.

are in some sense more natural

Note also that free

the covariance obtained I(C~) "I_

operator

B.C.

by restricting

The usefulness in

(as has been emphasized

already

choices

are obtained whereas

A~

in Glimm's

the other types of B~A

measure

D d~A, a

There are two logical

• 2.

or with respect

to

possibilities

"qn"

and similarly natural

=

to

=

for higher

d~,6

2 _

Jf

qn

dPA,6

• 2.

in

êxt -6

We shall not have

obtained

5y more general

about the non-interacting

Let us denote

paying

some

the free Dirichlet

z _

,

n~ e A)

e.g.

D

Jq~ d~A,6

powers.

measure with space cutoff

A

2 _

=

qn

(C~)nnA

,

,

(1.14)

=

2 _ c qn

The first choice P(~)d

O0

(1.i5)

'

is perhaps more Dirichlet lattice

:

I

exp-

(assuming

(1.7), i.e.,

~

qn

(i.13)

of Wick ordering:

q~ d~A,6

as in

or

for the definition

and gives rise to what we call the

=

from those

lies in the

(ad/2=)N/21AAl~e-z/'q'6dAAq dNq

we can order with respect

"qn'D,A

B.C.

the interaction,

of the Wick dots.

=

are

-

and now I wish to include

by

B.C.

operator

lectures)•

B.C.

So far I have only been talking

care to the meaning

Dirichlet

of Dirichlet

are decoupled

of the boundary matrix

free theory

by restricting

the inverse of the covariance

= C-I~£2(A6)I

to employ

As

IcA = C~£2(A~))

fact that the variables

occasion

on

than free

~ 6d :P(qn ') :D,A d~A,~

n6eA

whereas we call the second choice

I °

the half-Dirichlet

;

(1.16)

lattice measure:

272

exp HD

d~A, ~

The advantage orderin9

~d :p(qn) : d~A, ~

I °

=

of the half-Dirichlet

does not change with

A

(1.17)

state is that the d e f i n i t i o n (see

(1.14) and

(1.15)).

We shall

return to this point in the discussion of the infinite volume §3.

The lattice Schwinger

functions

are defined

of Wick

limit in

in the obvious way,

e.g.

s°A,6(hl ..... h r)

=

J~6(hl) ...~6(hr)dU~, ~

(1.18)

co

where

h. ~ C (A) 3 0 Let me restate the essential properties

in the following way: Dirichlet,

whether we consider

or m o r e general B.C.,

arising

from the polynomial

which determines P(~)d

spin.

type that are nearest

of spins by perturbing

From this point of view,

the basic properties

The Gibbs factor spins but

the d i s t r i b u t i o n

it is the free theory

of the model.

This r e a l i z a t i o n

of

leads to the c o r r e l a t i o n

of the next section.

In closing "approximation"

this section I ought to justify the word

in the title.

can converge,

We need something

and so we take d < 2

ultraviolet divergences. 6 + 0

half-

is local and does not couple

theory as an Ising ferrqma~net

inequalities

as

P

the interactions

of each uncoupled

theories

of ferroma@netic

except possibly for the boundary variables.

only mediates

the

free, Dirichlet,

the free lattice theory is an array of

G a u s s i a n spins with interactions neighbours

of the lattice theories

The convergence

is then just a v a r i a t i o n

to which the lattice

in order to soften the of the theory w i t h free B.C.

of the standard

semi-boundedness

proof and removal of an ultraviolet

In

particular,

for

any

p< ~

objects

like

~d(h)

so that by Holder's

cutoff (see Nelson's lectures). -UA,6 and e converge in LP(Q,d~)

inequality,

the smeared Schwinger

273

functions converge.

In the case of D i r i c h l e t and h a l f - D i r l c h l e t B.C.

some further a r g u m e n t s are r e q u i r e d and we m u s t impose some m i l d r e g u l a r i t y c o n d i t i o n s on A The issue for h a l f - D i r i c h l e t B.C. is to -UA show that e ~ LP(Q,d~ ) in spite of the W i c k ordering being "wrong" In two d i m e n s i o n s this is so b e c a u s e the d i f f e r e n c e s in W i c k o r d e r i n g involve c o e f f i c i e n t s w i t h at m o s t logarithmic singularities. see

[i0].

For d e t a i l s

Thus:

T h e o r e m 1.3.

[i0]

Suppose

polynomial.

T h e n as

d = 1

6 + 0

or

2

and let

P

be a s e m i b o u n d e d

the smeared Schwinger functions w i t h free,

D i r i c h l e t or h a l f - D i r i c h l e t B.C. converge to the c o r r e s p o n d i n g Schwinger functions of the c o n t i n u u m theories.

In principle, inequalities

these techniques and the attendant c o r r e l a t i o n

should hold when

counterterms violet cutoff

d e p e n d i n g on 6 + 0 .

~

d > 2

Of course one m u s t add a p p r o p r i a t e

and then prove c o n v e r g e n c e as the ultra-

We expect that w h e n

d > 2

h a l f - D i r i c h l e t states

will not exist since the r e q u i r e d counterterms will be

2.

A

dependent.

C o r r e l a t i o n Inequalities

I w i s h next to outline the proof of G-I and G-II for the lattice model.

By virtue of the a p p r o x i m a t i o n theorem

(Theorem 1.31

these i n e q u a l i t i e s extend i m m e d i a t e l y to the spatially cutoff c o n t i n u u m P(~)2

theory and to the infinite volume

P(~)2

convergence of the Schwinger functions is known. d i r e c t l y from G i n i b r e ' s

theories for w h i c h the Our proof is adapted

[4] elegant a n a l y s i s of Griffiths'

G-I was first p r o v e d for the special case

(~)2

inequalities.

by S y m a n z i k

[17] using

somewhat d i f f e r e n t methods. M o t i v a t e d by the lattice m e a s u r e s of the p r e v i o u s section, we define a f e r r o m a g n e t i c m e a s u r e ' on

IR N

to be of the form

274

d~(q)

where

A

is an

diagonal

NXN

elements

functions write

=

on

~

(0, ~)

functions fi ( ~ i

"

e(q) = 1

IR N

or

il,...,i N

continuous ~

is even.

f(qi ) = e(qi)g(lqi ])

sgn q .

We let ~

in ~

non-negative

fl(ql)

integers.

Note also that ~

measure on

if

f,g ~

,

is an even ferromagnetic

~,T

Typically, a polynomial,

section.

H

> --

< fg > ~

(2.1) -

Fi

~ g >p

~

part of the free m e a s u r e

inequalities).

(2.2)

e

_~Pi (qi)

A0

(after the

This does not work.

inequalities

is to regard

and to think of the coupling tera~s in

Thus we shall expand the'off-dia@onal

A~ = A 0 + 8(A-A 0)

0

is of the form

is to expand this exponential

correct point of view for correlation

the interaction.

< f ~

as in the lattice field theories of the previous

The temptation

Let

and

0

the "interaction"

standard proofs of Griffiths'

Gaussian.

IR N ,

then

P. l

where

is the m o n o m i a l

Theorem

~

where

..° fN(qn )

We shall prove:

If

be

be the set of

is closed under addition and multiplication.

2.1.

We

bounded function on

that are sums of products

with

off-

Following Ginibre we let ~ i

polynomially

An example of a function

iI iN ql "'" qN

bounded,

is even we say that

of the form

increasing,

and

on

Fi

qi

matrix with n o n - p o s i t i v e

are positive,

If (q) d~ (q) /Id~ (q)

is a positive,

IR+ =

Fi

If each

the set of functions of g

-.o FN(qN) e-q'Aq d N q

positive-definite

and the

~.

p

FI (ql)

be the diagonal part of interpolate

between

A0

A and

The Fi

d~

as as

part of the

and let A

for

8 ~ [0,I]

Define d~8(q)

=

F 1 ... F N

e~q'Asq

dNq

.

(2.3)

275

An important role is played by: Lemma 2.2.

For

~

If(q)du8

E(8)

an even ferromagnetic measure and

.

Then the Taylor series for

negative coefficients Proof.

The

nth

E(8)

and radius of convergence

derivative

E (n) (0)

f ¢ ~, at

define

greater than

=

has non-

8 = 0

(_Aij)qiqj

1 .

n f(q)d~ ° .

J Since each Because

A.. < 0 13 --

dp °

for

i ¢ j , the integrand

factors and is even,

E(n) (0) > 0 .

~gdp 0 ~ 0

is a function for any

in hence

g

Now an analytic function with nonnegative

Taylor

w

coefficients

at the origin must have its nearest singularity

positive real axis, of

and

E(8)

is clearly analytic

on the

in a n e i g h b o u r h o o d

[0,i] In his general analysis of Griffiths'

[4] isolated a condition

(Q3)

satisfy to yield correlation

inequalities,

that the measure and observables

inequalities.

In our case,

essentially

proved by Ginibre by the following argument:

Lemma 2.3.

Let

fl,...,fn

~/

.

d~ 0

be the even m e a s u r e given by

Because

(2.3)

(Q3)

d~ 0

factors we first reduce

was

signs

~ 0

(2.4)

should

and let

Then for any choice of the plus or minus

IIi~ln (fi (q) ± fi(P))d~0 (q) dp0(p)

Proof.

Ginibre

(2.4)

to the case

N = 1

by repeated use of the identity hl (ql)h2(q2)

± hl (pl)h2(P2)

Secondly,

we think of

function

fi ~ ~

=

1 ~(hl (ql)+h1(pl)l (h2(q2)±h2(p2))

IR as the product

has the form

space

fi = ei(~)gi (q)

{+l} x IR+ for

so that a

~ ¢ {-+i},

q ¢ 3R+

276 where

ci(~)

function

H 1

on

or

IR+

ei(~)

= ~

gi

Then the left side of

ff

is a positive

increasing

(2.4) becomes

n d~0(q)d~0(p)

[ K lei(~)gi(q)+Ei(T)gi(p) O=+1 i=l T=-+I

lq+]~+ A further becomes

and

application

a sum of

of

2n

(2.5)

separates

(2.6)

1

out the variables

so that

(2.6)

terms of the form

(2.7)

I~+I d ~ ° ( q ) d ~ ° ( P ) [ { g i ( q ) + - g i ( P ) ~

But each of the factors

in

are increasing

functions.

By the change

of variables

even number of minus nonnegative

2.1.

is nonnegative

Consider, qp

signs.

since each

Proof of Theorem

(2.7)

because

for example,

e.

gi

the second factor:

we need only consider

the case of an

But in this case the integrand

gi

and

is clearly

is increasing.

(2.1) follows

immediately

from Lemma

2.2 since

oo

E(1)

= • E (n) (0)/n! > 0 .

As for

(2.2) we must

show that

0 2

2

d~

~ fg > ~

= fief(q) f(p)l 0

be even. and

to ask how essential

is the assumption

It is not hard to give counter-

(2.2) when

~

is not even;

e.g.

take

277

Fi(x ) = e -ax ,

a > 0

and

f

and

g

it is easy to see that (2.1) holds if assumption that

Fi = Si ~ fi,k

and, by Lemma 2.3,

where

mono~ials. Fi Gi

where

Gj

is even and

=

F's

fi,k e~i

;

satisfy

nk

n

j~IFj (qj)Fj (pj)

even is replaced 5y the

(2.2) holds if products of the

n

On the other hand,

j=l ~ Gj (qj)Gj(pj) k7 i=l~ (gk'i(q) -+ gk,i (p)~

is even and

gk,i ~

"

In particular, we obtain this result by expanding the exponentials

e Qi

and appealing to the arguments of Lemmas 2.2 and

2.3: Corollary 2~4. Fi(qi) and

=

Qi

e

Let

~

be a ferromagnetic measure on

-Pi(qi)+Qi(q i)

where

is an odd polynomial with

and increasing on

~+

.

Pi

]R N

with

is an even semibounded polynomial

deg Qi < deg Pi

that is positive

Then the correlation inequalities

[2.1) and

(2.2) are valid. As I mentioned before, these correlation inequalities, case of observables

f,g

that are polynomials in

q , go over from the

lattice to the continuous models by virtue of Theorem 1.3. when

For instance

d = i:

Theorem 2.'5. Consider the P = Pe - P0 increasing on

where IR+

Pe

P(~)I

is even and

Euclidean Markov theory with PQ

is odd and positive and

and with Schwinger functions

-r^Pj(~(s)Ids Idv~(t l)...~(t r)e SA(t I ..... t r)

= Id~ e-]AP(#(s)Ids

in the

278

where

d~

is the free m e a s u r e w i t h

periodic

B.C.

on

~A

SA(t I ..... t r) ~ 0

where

free,

Dirichlet,

A c IR is a finite

Neumann,

interval.

or Then

and

S A ( t l , . . . , t r + s) ~ S A ( t l , - . ° , t r) S A ( t r + l , . . ° , t r + s)

N o t e t h a t in the case necessary

and t h a t

tl,...,t r

without

Wick order

and as a r e s u l t

d = 1 , Wick ordering

S A ( t l , . . . , t r)

is a w e l l - d e f i n e d

the n e e d for s m e a r i n g .

for w h e n

Hermite polynomial)

is n o t p o s i t i v e

2.6.

Consider

f u n c t i o n of

d = 2 , we m u s t

the o n l y o d d t e r m that w e can a l l o w

is the l i n e a r term;

Theorem

When

is n o t

the

r > 1 ,

P(~)

:q~:

(essentially

increasing

in

in

the

P

rth

qn ~ 0

Euclidean Markov

theory with

2

P(X)

= P

e

(X) - ~X

where

be the c o r r e s p o n d i n g [i0])

region

B.C.

Then

A

P

e

is e v e n and

Schwinger

where

~

functions

denotes

S~ A ( x I ,... ,x r) _> 0

freer

I > 0 -"

Let

S Aa (X 1 , • • . ,x r)

in the b o u n d e d Dirichlet

regular

(see

or h a l f - D i r i c h l e t

and

S A ( X l , . . - , X r + s) ~ S A ( X l , . . - , x r) S A ( X r + 1 .... ,Xr+ s)

Actually inequalities

when

odd in T h e o r e m e.g.,

by

~ + -#

I < 0

covariance

in T h e o r e m

we r e t a i n

2.6 or w h e n

the c o r r e l a t i o n

ai > 0

for

i

2.5, b u t t h e r e m a y be a sign change;

(-i) r S A ( x I .... ,x r) ~ 0 .

correlation

inequalities

one is for

:~2: = ~2 _ ~

It w o u l d be d e s i r a b l e

to h a v e

i n v o l v i n g W i c k p o w e r s b u t the o n l y o b v i o u s since the W i c k r e n o r m a l i z a t i o n

is a

constant: Corollary I > 0

and

2.7. B.C.

Consider ~

the

(= free,

P(#)2

theory with

Dirichlet,

P = P e - Ix

or h a l f - D i r i c h l e t ) .

,

Then

279

A

~:#2(x):

> A "

the interacting m e a s u r e

of F o r t u i n - K a s t e l e y n - G i n i b r e (for details see

[i0]).

P .

p r e s s e d in terms of the cone

of increasin~ functions o_ffth___e

fields; where

I

consists of elements of the form

hl,...,h r > 0 -

are in

C~(IR 2)

and

F(x) ~

F(y)

IR r .

T h e o r e m 2.8.

Let

let

be bounded.

A c ~ 2

where Remark.

~

P

F(~(hl),...,#(hr)) F : ]R r ÷ IR

if

For technical reasons we also assume that n o m i a l l y b o u n d e d on

inequalities are ex-

is in-

0

c r e a s i n g in the sense that

the fields,

FKG

There is no

r e s t r i c t i o n on the odd terms of I

The

[2], and so we

F

xi ~ Yi

'

i = 1 .... ,r

is continuous

and poly-

Then:

be an arbitrary If

(semibounded)

F, G ~ I

p o l y n o m i a l and

are i n c r e a s i n g functions of

then the truncated e x p e c t a t i o n A , T ~ 0

denotes free, Dirichlet,

or h a l f - D i r i c h l e t

B.C.

The inequalities of Theorems 2.6 and 2.8 extend to the

infinite volume theories once it is known that the various expectations involved converge as

A + =

(see [6], Nelson's lectures and

C o r o l l a r y 3.5 in the next section for results along these lines). It should be noted that both

G-II

and

FKG

are both

statements of p o s i t i v e correlation but for somewhat d i f f e r e n t classes of observables.

The appropriate class for the

P(9)

Griffiths' 2

inequalities is the cone of polynomials negative test functions. #(hl)~(h z) f [

since

in the fields w i t h non-

But clearly a p r o d u c t like

xl 2x

is not an increasing function on

In order to produce some nontrivial examples of o b s e r v a b l e s we introduce, f o l l o w i n g Simon

[15], the analogues

p(f)

occupation number v a r i a b l e s used in the lattice gas case the cutoff function

Y(x) = x

if

Ixl ~ 1 ,

in

m 2 I

of the [2].

Y(x) = sgn x

if

Define Ixl > 1 .

280

Definitions 2.9. ~A =

~(f) and

~ P(fi ) ' iEA

test functions in

= Y(~(f)) ZA =

,

p(f) = ½(I + oif))

Z p(fi ) , i~A

C ~(IR 2) 0

and

A

where

f, f. 1

,

are n o n n e ~ a t i v e

is a finite index set.

Note that the o c c u p a t i o n number variable linear f u n c t i o n of the field taking values b e t w e e n

p(f) 0

is a non-

and

1 .

It

is easy to v e r i f y that: Lemma 2.10.

The f o l l o w i n g are all in

I : ~(f), ~(f),

p(f),

~(f)

- o(f),

hA' ZA' ZA - ~A " 3. A p p l i c a t i o n s Some of the traditional applications of c o r r e l a t i o n inequalities in s t a t i s t i c a l m e c h a n i c s are as follows (i)

[8, 14]:

m o n o t o n i c c o n v e r g e n c e of c o r r e l a t i o n functions in the i n f i n i t e volume limit;

(ii)

bounds on c o r r e l a t i o n s in terms of the two point f u n c t i o n

[ii];

(iii) m o n o t o n i c b e h a v i o u r of correlation lengths as the i n t e r a c t i o n is made more ferromagnetic; (iv)

p e r s i s t e n c e of phase transitions if an i n t e r a c t i o n is m a d e m o r e ferromagnetic. We now discuss the field theoretic translations of these

statements.

For

efficient of

#2

the bare mass.

P(~)

2

"more ferromagnetic" means that the co-

has been decreased;

this amounts to a d e c r e a s e in

The analogue of c o r r e l a t i o n length is

-i mphy s

The occurrence of a phase transition or spontaneous m a g n e t i z a t i o n means that

lim > 0 ~+0 + ~

where

in the infinite volume theory w i t h The Griffiths' monotonicity statements

denotes e x p e c t a t i o n

P(x) = Pe(X)

- ~x .

inequalities lead i m m e d i a t e l y to for the Schwinger functions,

we collect in the following lemma:

some of w h i c h

281 Lemma

3.1.

Schwinger

(i) Consider functions

coefficients (ii) the

functions

S~(tl,...,t r)

P

Schwinger

= Pe(X)

Proof.

The

functions

of the

coefficients.

Then

are d e c r e a s i n g

A .

the spatially cutoff

+ a x2 + a x , 1

theory with 2 < 0 , as in Theorem 2.6.

a I

P(~)

Then the

--

o

Schwinger functions and

are decreasing

_free ~A (tl,...,t r)

functions

2

a1

theory of Theorem 2.5.

is even with nonnegative

of the interval

(iii) Consider P(x)

P(~)I

al,...,a2n

Suppose that P(~)2

the

SA(Xl,...,x

r)

are decreasing

functions

of

a2 All the assertions

we prove the m o n o t o n i c i t y S~(Xl,...,x r ) ~

are proved in the same way; in

a

is differentiable

(x I ..... x r) = 2

of

2

(iii).

in

dx[ A

..

A

• ~] which is nonpositive

by Corollary

We turn to application in which monotone

convergence

For the other cases which convergence

i. (i) and describe

of the Schwinger

(high temperature

the circumstances

functions

and large magnetic

is known. field)

in

has been proved see the lectures of Glimm and Jaffe.

1. First notice how much better the situation

is in one d i m e n s i o n

than in two, owing to our lack of control over Wick powers higher than two.

We conulude

immediately f r o m Lemma 3.1(ii)

Schwinger functions monotonically

to an infinite volume

with nonnegative matrix method vergence

~A-free (which are nonnegative

coefficients•

[10] that there is

for general

P.

that the by

i

G-I) decrease

limit, at least when

Actually,

P(~)

P

is even

we know by the transfer

(not necessarily monotonic)

con-

282

2. As for Corollary sfree

3.2.

converge

A

as

d = 2

with

free B.C.

there

For the

(~2)

theory,

monotonically

downward

the Schwinger

result:

functions

to an infinite

volume

limit

IAI ~

3. The

above two c o n v e r g e n c e

from the Griffiths' happily, Theorem

been 3.3

Dirichlet

salvaged

Nelson's

were

all that

The general

Let

P = P

functions

- Xx

e

S~ D-

GRS

P(~)

has,

2

that

, X > 0 o

converge

could deduce

case of

by N e l s o n who d i s c o v e r e d

(Nelson).

volume

results

inequalities.

Schwinger

an i n f i n i t e

his

2

is only this m e a g r e

Then the half-

monotonically

upward

to

limit. argument

is b a s e d

on

G-II

as he has d e s c r i b e d

in

lectures.

4. It is doubtful or D i r i c h l e t

that there

B.C. w h e n

deg P > 4 .

is that the d e f i n i t i o n However, occurs

term

can control by Lemma Theorem

3.4.

a > 0, ~ ~ 0 ly upward

the

bound

the S c h w i n g e r

to an i n f i n i t e

of o b t a i n i n g

volume

[i0] on the basis

Recently

Frohlich

[3]

in T h e o r e m

P(x) SD A

w h i c h we

= ax ~ + b x 2 - ~x

converge

on the Schwinger

with non-coincident

It is s u f f i c i e n t

It is here

enters,

G-II we deduce

s~D ! s A

term)

monotonical-

3.4 and in N e l s o n ' s

of the G l i m m - J a f f e

case of free B.C. for using

(1.14)).

is:

functions

(see also Simon's

arguments.

(see

limit.

functions

noted in

A

B.C°

in Wick o r d e r i n g

constant

theory w i t h

an upper b o u n d

for the S c h w i n g e r

for general

2

for free

with D i r i c h l e t

with

the change

The result

P(~)

Part of the a r g u m e n t consists

~

changes

(and a trivial

3.1(iii).

Consider Then

case of

results

The d i f f i c u l t y

of Wick o r d e r i n g

for the special

as a q u a d r a t i c

are any m o n o t o n i c i t y

to b o u n d

that the sign of

functions. arguments

linear b o u n d

lectures)

obtained the

B~A

via the lattice

Theorem

SA

A was

[5]. bounds in the

of T h e o r e m

approximation

1.2 that

,

283

5. F r o h l i c h ' s basic results are best expressed in terms of the g e n e r a t i n g functional functions: = D

Let

with

J~(f)

=

~ = free or

~e~(f) ~ A HD

with

for the S c h w i n g e r

- Ix , e c < 0 , and let A

P = ax ~ + bx z + cx ,

P = P

-

region.

Then there are constants

and for any

f

w i t h supp f c A

cI

I > 0 , or be a b o u n d e d 0

and

c

such that for any

2

A

, 0

2 II f II )

IJA(f) I < c I exp(c --

Moreover,

2

(3.1)

2

in the situations of Theorems 3.3 and 3.4 J~(f) = lim JA(f)

exists and is continuous Theorems

(3.2)

in

f

in

3.3 and 3.4, the bound

L2

norm.

(3.2)

follows

from

(3.1), and Vitali's Theorem.

By a p p r o x i m a t i n g by exponentials

(the Fourier t r a n s f o r m

theorem) we can deduce the following convergence result from and

(3.2):

C o r o l l a r y 3.5. F ~r

(3.1)

C o n s i d e r the situations of Theorems 3.3 or 3.4.

be a continuous, ,

e x p o n e n t i a l l y b o u n d e d function on

iF(x I .... Xr) I ~ exp(c ~Ixil)

hl,..,h r

in

Let

L 2 ( m 2) ' < F ( ~ ( h l ) '

Then for r e a l - v a l u e d

"''' ~(hr)

Thus we see that the Griffiths and

>A

converges as

FKG

A +

inequalities

transfer to the infinite volume limit in these cases. 6.

Before leaving a p p l i c a t i o n

(i), I wish to m e n t i o n one more result

of this type due to A l b e v e r i o and H o e g h - K r o h n

[i].

They c o n s i d e r the

(e~) 2

model with i n t e r a c t i o n U A = I d ~ ( ~ ) I A d X : e ~ ( x ) :

dg(~)

is a p o s i t i v e m e a s u r e of compact support on

If, moreover,

d~

is even then

UA

= exp(-

(-4/~,

4//~)

contains only even powers;

indeed, B e c a u s e :exp(~#h(X)):

where

½=IIhllil)exp¢~+hCX))

.

284

where

~hix) = I h(x-y)~(y)dy

follows

that

UA, h

2n ~h

as in Corollary

are decreasing volume

an ultraviolet

of

it

cutoff,

cutoff is a power series

with nonnegative

3.2, that the

functions

coefficients.

(nonnegative)

A , and thus converge

in

We con-

Schwinger

functions

in the infinite

limit° Application

after Lebowitz' FKG

h > 0

with an ultraviolet

ordinary even powers clude,

,

(ii) is due to Simon

proof that,

inequalities,

[15] and is patterned

for Ising spin systems satisfying

a spin-spin correlation

of any order can be

dominated by the two point spin correlation estimate

is expressed

Defn.

2.9; A, B

Lemma

3.6.

= ~1

In our case the

of Lemma 2.10

[15]

Let denote the expectation

~B>T

FKG

< ~A --

inequalities 7B>T

=

7~ 7, T icA j~B

to the increasing

~ ~ T icA j~B

i 0

be fixed for

Z(h I ..... hn) =

Then

Z # 0

if each

Griffiths

1 .< i < j ~< n

[ ~I =±i "" . . , O n = ± l

and

let

exp(~aij~i~ j + [hi~i)

(i)

h i ~ ~ = {hlReh > 0} U {h = 0} .

(1970) proposed a very simple and beautiful way of extending

Theorem i' to more complex situations. ferromagnet, i.e. each spin

s

As a typical case, consider a spin 1

can take the values

0,+2

with equal probability.

We thus seek a zero theorem for the function ~(h I ..... hn) =

[ exp(~aijsis j + [hisi) si=±2,0

(2)

Griffiths suggests first looking at a two spin, spin 1/2 ferromagnet with a12 = 1/2 £n z .

Thus:

prob (s = °l + 02 = +2) = /2/Normalization = prob (s = -2) prob (S = O ) = That is,

s

/f/Normalization

In particular

i j aij(oil+Oi2)(Ojl+oj2)

+ 11/2 £n z OilOjl) i

exp/[h (~ _-~. )~ \i i il 12 /

(3)

292

by replacing the sum over

oil = ±i,

oiz = ±i

doing the sum over the other degrees explicitly.

hie

~

§2.

The Improved DeMoivre-Laplace Limit Theorem

by a sum over

°i/ + ~i2 = ±2,0

On account of

(3),

Z ~ 0

if

It is now clear how to go about trying to prove a Lee-Yang theorem for P(~)2 "

First approximate

P(~)2

by the lattice approximation, i.e. by Ising

ferromagnets with pair interactions and single spin distributions deg Q = deg P

and

Q

need only obtain

is even if e-Q(q)dq

P

e-Q(q)dq

is even (which we will suppose).

where

Thus we

as the output probability distribution for the total

spin of an Ising ferromagnet with spin-i/2 spins and pair distributions.

More

accurately, we need only obtain it as the limit of suitably rescaled output distributions.

This is because the Lee-Yang theorem in the form of Theorem I' is

preserved under limits on account of the following consequence of the argument principle:

If

f (z) is a sequence of functions analytic and non-zero in a n D ~ C and if f ÷ f uniformly on compacts of D , then f n is either identically zero or non-vanishing in D . To apply this limit theorem, connected region

one needs uniform bounds on the approximating distributions as well as pointwise convergence.

Below we will only prove pointwise convergence; the extra bounds

(which require higher order terms in Stirling's formula) can be found in SG. Consider first the case take

N

is just

deg Q = 2 .

uncoupled spin-i/2 spins.

It is easy to handle this case, for

Then the probability that

s = Es i

2-N/\(N+~)/2]N ~ where ( a)b is a binomial coefficient.

is

The DeMoivre-Laplace

limit theorem asserts that the binomial distribution for large

N

looks like a

Gaussian, explicitly: (N~--U) ..,DN exp (-~2/2N) for a suitable constant

DN

What (4) means is that if

(4) s

is fixed then

N

- (N+UN(S)) -~ DN i 2 as

and

N -~ =

where

~N(S)

exp (-s2/2)

is defined by

[x] = greatest integer less than

x .

To prove (4), one needs Stirling's

formula: log nl-~- n log n from which log ( N ~ )

-- % - Nh(]//N)

293

with

CN

a suitable constant and h(x) = 1/2[(l+x)log(l+x)

For

x

small,

+ (1-x)log(l-x)]

h(x) = 1/2 x 2 + 1/12 x 4 + 0(x 6) .

log

..~ cN - N ( 1 / 2 ( ~ / N ) 2 + o ( ~ / N )

Thus for

~) = c N -

s = ~/~

fixed,

1/2 s 2 + o ( 1 / N )

from which (4) follows. Next consider

deg Q = 4 ; in fact suppose

modify the Gaussian behavior above to get out

Q(q) = q4 .

It is clear how to

q~ ; just cancel the Ganssian and

re-s cale ; i.e. (N~+~> For if we fix

e~2/2N

DN e-~4/12N 3

(5)

s = ~/N3/4:

log IN-!~-I e ~2/2N --- % \--/

- N [I/12(~/N)4 + 0(~/N) 6]

= ~ But ( N 2 ) e B 2 / 2 N Ising magnet

- 1/12 s 4 + 0(I/N ½)

is the unnormalized probability distribution for an with energy

argument works for any

H = - i/2N(Zsi)2 which is ferromagnetic.

Q(q) = aq 4 + bq 2

with

N

spin-i/2

A similar

a > 0 .

We are thus able to conclude the following basic theorem from SG: Theorem 2

(Lee-Yang theorem for (~4) 2)

Let

0 .

Moreover,

=A(B) it is

clear that

Re ~ ) Let

fA(~) = exp(~A(~))

compacts of convergence

theorem,

vanishing there and for all

~

Theorem 3

.

{~[Re ~ > 0}

with ~ (~)

~A(B) -~ e (~)

We thus see that and converging

convergent on f

~ aA(Re @

]fA(~)[

is uniformly bounded on

~ ~ ~

and thus, by the Vitali

for

{B[Re ~ > 0} .

is not identically

Re ~ > 0 .

In particular,

fA(~)

zero, it is never zero, so

are non=A(~) + ~

s~p d2~A/d~2 value

d2~A d~ 2

to the entire right half-plane

< = , for any fixed
A for the

~ > 0 .

ax 4 + bx 2 - ~x

and

In terms of the theory:

i

[A[ [A - AA]

s~p d2ah/d~2 < ~ ,

there is a

D with

[A- At] ~< D[A[ Now the two-point

(~)

on compacts of the right half-plane.

Dirichlet state expectation

so by t h e Lee-Yang r e s u l t

Since the

We summarize by:

has an analytic continuation

uniformly

.

(6)

truncated Schwinger function

sT(x-y) = < ¢ ( x ) ¢ ( y ) > - ® is positive and monotone decreasing

as

[x-y] ÷ ~ •

If the limit is some

C > 0 ,

then

[ (6) and (7)

- . CIA[ 2 .

are not directly contradictory

further argument

(7)

although they clearly almost are and a T S2 -~ 0 as

[Simon (1973b)] show they are: we conclude that

295

Ix-Y] ÷ ~ •

But then, by a result of Simon (1973a) (descrlbed~in Rosen's lectures),

all the truncated Schwinger functions go to Theorem 4

In the infinite volume

P(¢)2

0 .

We summarize:

Dirichlet state with

P(x) = ax4 + bx 2 _ ~x

(B @ 0) , the Schwinger functions obey clustering, i.e. f ¢(fl ~

~t)'"¢(fk(~)St)¢(fk+l~80)'"¢(fn(~80)d~+

If d9 ¢(fk+l~)80)...¢(fn~)~0)]

§4

as

If d~ ¢ ( f 1 0

~0)...¢(fk~80)]

t + ~ .

The Wi~htman Axioms The status of the basic Euclidean objects for Dirichlet and half-Dirichlet

states when

P(x) = ax~ + bx 2 - Bx

(~ @ 0)

are given by the following table:

TABLE Dirichlet (A ÷ ~) Schwinger Functions

Yes (2)

Yes (1)

(A + ~) Pressure

Yes (2)

Yes (3)

?

(~ < ~) Transfer Matrix

where

Half-Dirichle t

Yes (3)

T (A = ~) S 2 + 0

Yes (4)

Yes (4)

(A = ®) OS Axioms

Yes

Yes

(1) = Nelson (these lectures), (2) = Guerra et el. (1973) (3) = Guerra e t a l .

(1974), (4) = Simon (1973a).

If the Osterwalder-Schrader (1973) reconstruction

theorem is valid ~there is presently a gap in the proof), all the Wightman axioms hold for the infinite volume

D

and

HD

theories.

And in any event, on account

of the existence of a transfer matrix, the Wightman axioms do hold for the

HD

theory [see Simon (1974)].

§5.

~

and Dymanical Instability The field theoretic analog of a phase transition is the notion of dynmaical

instability (Wightman (1969)), i.e. the existence of more than one infinite volume theory associated to a fixed interaction by some mechanism for associating infinite volume theories to interactions, e.g. the DLR equations described in Guerra's lectures.

The expe=ted picture for

Jaffe's lecture.

¢4 + b¢2 _ ~¢

theories has been described in

In this section we want to supplement the picture given by Jaffe

explaining the connection between dynamical instability and the Fock space energy per unit volume, a is Just the

, of Guerra (1972).

pressure.

As Guerra explained in his lectures)

=~

There is an old idea in field t~eory associated wit~ the

296 name of Bogoliubov that dynamical instability is present precisely when 4+b~2_B ~

is discontinuous in

~ .

In statistical mechanical language,

Bogoliubov is saying that the phase transition is first order and has the field as long range order parameter.

This picture is supported by the following which

combines results from SG and Simon (1973b): Theorem 5 let

< >

Fix

b

and let

uoo(~) denote the pressure for the

~4 + b~2 _ ~

denote the infinite volume (Dirichlet) state for this theory.

and

Consider

the statements: (A)

Then

There is a mass gap in the

< >

~=0 ~ = 0.

(B)

~(~)

(C)

The "magnetization"

is differentlable at

(D)

There is a unique vacuum in the

theory.

is continuous at < >B=0

~ = 0 .

theory.

(A) => (B) (C) => (D).

Remarks I.

We emphasize that

(A) is a statement about the

< >~=0

theory and not

its decomposition into unique vacuums. 2.

Suppose the picture described in Jaffe's lectures holds.

a critical value b < b for

b

When

c

b > b

Then there is

we expect (A) to hold so (B),(C) hold.

c to fail for the following reason:

When

, we expect (D) The Wightman theories c B = 0 with unique vacuum (there should be two such theorlesf) have

@ 0 .

But by

= 0 o

~ ~ -4

Thus the

fails so do (A),(B).

symmetry in the Dirichlet B.C. theories the value of

~

so

But b y s y m m e t r y

$2,~> 0 + d 2 >~ c 2

> 0 >~ c . = 0 .

as

¢ - $

x-y + 0 .

By symmetry

m . Then

See SG for details. ST (x-y) -~ c 2 > 0 2,!a=O

T , S2,]a=O = S2,]a=O .

By

But by Theorem 4,

S T2,~ >0 ÷ 0

> 0 ~< c

so

GII,

as

S2,V> 0 >. S2,~= ° so

is not continuous at

For details see Simon (1973b).

References

ASANO, T. (1970) :J. Phys. Soc. Jap. 29, 350. GRIFFITHS, R. (1970): J. Math. Phys. iO, 1559. GUERRA, F. (1972): Phys. Rev. Lett. 28, 1213. GUERRA, F., ROSEN, L. SIMON, B. (1973): The P(~)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. Math., to appear GUERRA, F., ROSEN, L., SIMON, B. (1974): Boundary Conditions for the P(~)2 Euclidean Field Theory, in preparation. LEBOWITZ, J., PENROSE, O. (1968):

Commun. Math. Phys. ii, 99.

LEE, T.D., YANG, C.N. (1952): Phys. Rev. 87, 410. NEWMAN, C. (1973): Zeroes of the Partition Function for Generalized Islng Systems, N.Y.U. Preprint. OSTERWALDER, K., SCHRADER, R. (1973): Commun. Math Phys. 31, 83. RUELLE, D. (1969) : Statistical Mechanics, Benjamin, New York. SIMON, B. (1973a): Commun. Math. Phys. 31, 127. SIMON, B. (1973b): Correlation Inequalities and the Mass Gap in P(~)2 , If. Uniqueness of the Vacuum for a Class of Strongly Coupled Theories, Ann. Math., to appear. SIMON, B. (1974): Th e P(~)2 Euclidean Quantum Fie!d Theory , Princeton Series in Physlcs, Princeton University Press. SIMON, B. GRIFFITHS, R. (1973): The (~%)2 Field Theory as a Classical Ising Model, Commun. Math. Phys., to appear. WIGHTMAN, A.S. (1969): Phys. Today 2 2 53-58. YANG, C.N., LEE, T.D. (1952): Phys Rev. 8 7 404.

CONSTRUCTIVE MACROSCOPIC

QUANTUM ELECTRODYNAMICS

Elliott H. Lieb

Klaus Hepp Department of Physics, CH-8049 ZOrich,

E.T.H.

Department of Physics, M.i.T. Cambridge, Mass.

Schweiz

02139, U.S.A.

§i. Introduction

After ten days of difficult lectures the audience and the lecturers need some holidays.

I have chosen the subject of this last talk

half for your recreation,

half for exposing you to some new and exotic

aspects of the quantum world of infinitely many degrees of freedom, where there are many interesting problems

in mathematical physics.

My lecture will be centered around the quantum electrodynamics laser in the thermodynamic for some time.

It is hard to give fair references

a continuous transition to applied physics. laser theory is the book by Haken ~HI] in

[AI]

, [G3]

of the

limit, on which E.H.Lieb and I have worked in this field with

A good starting point on

as well as various contributions

, [KI]. The statistical theory of instabilities

in

stationary nonequilibrium systems is treated in [GI] and [G4] with many references to older contributions. phase transitions given in [ H ~

, with great emphasis on the nonlinear analysis of the

Heisenberg equations of motion. ified

A general approach to nonequilibrium

in mean field models with linear dissipation has been

In this lecture I shall presenta

approach to these problems,

simpl-

using only linear functional ana-

lysis and working in the Schr~dinger picture.

By this method one can

easily incorporate the unbounded boson operators of the quantized radiation field, and one sees better the analogy to the usual treatment of the classical

limit in quantum mechanics

[HS]

, [MI]

299

§2.

H e u r i s t i c D i s c u s s i o n of the Laser

The Dicke H a k e n Lax model of the 1-mode 2-level h o m o g e n e o u s l y b r o a d e n e d laser starts from the following a p p r o x i m a t i o n to the H a m i l t o nian of q u a n t u m e l e o t r o d y n a m i c s

H = ~ Z Y a * amm m + £~.~3~.i n + V-~2~,=, ~ a m ( ~ m n S : Here the

am

~mn

)

+

h.c.{ . (2.1)

are c ~ e a t i o n and a n n i h i l a t i o n operators for the d i s c r e t e

set of photon modes of energy and

+ ~nSn

Ym

are c o u p l i n g constants

i n t e r a c t i o n of the mode

m

of a cavity of volume

V o The

~mn

for the rotating and c o u n t e r - r o t a t i n g th atom. We assume f i n i t e l y

w i t h the n

many atoms~ N , in the cavity and shall later take passage to the t h e r m o d y n a m i c

limit.

N = V

in the

The atoms have two states with

fermion c r e a t i o n and a n n i h i l a t i o n operators b ~ and b~ for the up+n -n per and lower level and no t r a n s l a t i o n a l degrees of freedom. Then

S+

* n = b+nb-n

satisfy

'

_ = (Sn)

SU 2

n]

,

'

S3

* , n : (b+nb+n - b-nb-n)/2

(2.29

commutation relations

: +

n

'

n' S

: 2S

,

, S

= 0

for

m @ n

Little is known in general about the system d e s c r i b e d by except for t h e r m o d y n a m i c

(2.3)

(2.1),

s t a b i l i t y with hard cores and i n s t a b i l i t y

without a s u f f i c i e n t l y strong r e p u l s i o n at short distances

[H3]

For f i n i t e l y many modes, an e q u i l i b r i u m phase t r a n s i t i o n f r o m a normally r a d i a t i n g to a superradiant phase can be established

In these lectures we are interested

[H2].

in the n o n e q u i l i b r i u m beha-

viour of the system, and we shall restrict ourselves to one mode and, for notational c o n v e n i e n c e ,

H NS

:

Ya

a +

£S

+

to the r o t a t i n g wave a p p r o x i m a t i o n

~N-Y2(

S a +

where only the total spin o p e r a t o r s i SN

:

M ~- S i n

a SN ) ~

enter

:

(2.4)

:

(2.5)

300 The total Hamiltonian

HN

of the laser cavity with

N~V

atoms and

photons coupled to atomic pumping devices and photonic loss mechanisms is of the form later,

(4.1),

HN = H

+ H RN , where the reservoir part will be given

(4.6), once we have acquired a qualitative

of the laser action.

Since the

S Ni

it is natural to consider the Heisenberg five operators aN(t)

= -iYaN(t)

SN(t) -- - i £ S N ( t )

-

i~N-~2SN(t)

and

a~(t)

:

+ i [ H R, SN(t)]j

aN(t)SN(t)

(2.6)

) + i [H R, SN3(t)] .

The very successful semiclassical theory of the laser ILl] R that H N should be chosen in such a way that

i [ H N,R aN(t ~

: _ ~ aN(t ) +

gN(t) )

i [ H R, SN(t) ]

= - ~ SN(t) +

FN(t) )

Here ~ > 0

and ~ > 0

while

~> 0

for the photon amplitude

and

-~2~

~

~2

describe the pumping of the atoms into a mean inversion Of course, the purely dissipative

of

that in a suitable topology the additional F~(t)

= 0(i)

and

should ~

terms alone on the r.h.s,

are inconsistent with the selfadjointness and

suggests

(2.7)

are damping constants

and the atomic polarisation,

SU 2 ,

+ i [H R, aN(t)] ~

+ 2i~N-~2S3(t)aN(t)

SN3(t) = ikN-~2( a N ( t ) S N ( t ) -

of

equations of motion for the

S Nk exp(_iHNt)

S (t) = exp(iHNt)

understanding

give a representation

~

S~/N .

of

(2.7)

H N . However, one hopes

fluctuation forces

gN(t)

become negligible in the limit N--~ ~ . Assume that gN(t) F~(t) = O(N ~2 ) , by some law of large numbers. Then it is

plausible that the intensive observables ~(t)

which at limit

=

~ )N -~2 ) aN(t

~Nk(t) =

sk(t)N -I )

t = 0

N--~ ~

have

O(N -I)

commutators,

(2.8)

become c-numbers

, and satisfy the ordinary differential

in the

equations

301

~-

= -

(iY

+~)o
be the states

O_nl-> where

: C+nl->

T + = c+c_ *

for the fermion pair

= 0 ,

b± )where

T+I -') = I + ~

. Then in obvious

(3.5)

notation

I _>N Here

(3.6)

N = M(2p+q)

and

M = 1,2~. .. Let

TN =

~

bN . Then

(3.7) defines

an irreducible

representation

of

SU 2 . The fluctuation

ope-

rators t N+ =

N-[2T

satisfy [ W ~ lim

and

t~, s ~=

t N3 =

,

= 0 R _t~ "

and

=I ( ~ t

[

all our results and their

introducing

(3.8)

t .m ]~_~.] if all "" ]

2~F . Let

~k(m))~>

= ~

k

'"

.m : + --

.

(3.9)

t ~ = t ~1 L, t ~ 2

t ~3 = 0

and

~

Let .i~ N = I N ~ IN : ~ - - >

N : "-~T k(m) t~L~,,s~

can be expressed

~>N ~N

I " *.r..+-s N~a~) (t~) ~

Then 11 I N II = 1 , and

GNS Hilbert and

~N

(3.1o) (3.11)

in terms of correlation

limits, the mathematics

, the

XI.],

Kr k(m)

a sequence of approximating

verges against

Define

~ NY2

Then

lira N ~ - I T [ ~ N (m) I £>N

functions

k.

0 , otherwise

ItS, t ~ ] :

!ira N < ~ I]TN ]/2 (~N (m)

Although

N-~2_3 iN +

:

m I -~N ~q~--It k N " " "tN

N-~

where

(tN)*

=

simplifies

Hilbert

considerably

spaces -~t N

space for

which con-

(3.1) ,and

be the span of all

*

~2~

by

(a -N ~ )

(3.10) r

+ s

.(tN) ~'~N'

by linear extension of

=

(a

*

-

N~2~)r

+ s (tN) - [ ~ N

lim U T N ~.~1 -- II ~..11 for N.-~p.

(3.12)

,

all

~&~,by

.

(3.9)

304

On the finite particle

lim

subspace,~,

IIXN,..YNI N ~ ,

-

one has

INX~...y . ~,II=

0

)

(3.13)

N-~

if

XN,..y N @ ~ (a~ - N Y ~ ~ ) ,

sense than in [H4] fluctuations

NY2(_~N -_~ ) ~

. In a somehow different

, the intensive observables

around

O~~, _~F in the sequence

OfN, KFN

~N

We shall say that a sequence of selfadjoint converges ~

,

in the Weyl-Trotter

A N --~--~ A

operators

sense to a selfadjoint

~ if for all

s E~

and all

lim )lexp(iANS) I N ? ~ - I N e x p ( i A ~ s ) ~ I I =

have normal

of coherent

~6

states [W~.

AN

on

operator

A~

on

~(~

0 ,

(3.14)

N-)~

Let ~FN(t) = exp(iH~t) _N

are ~-correlated and

~ m ( S)~m(t)>N

=

~ + ( s )~+ (t)>N

~m(S)~+(t)>N

=

~(s-t)~ (~2 - ~ ) ( 1 - 2 ~t)

(S)fm(t)> N

=

[(s-t) ~(~4 + ~ t )

~+ ~m3(S)fm(t)>N

=

~ ~3. m<S)~m( t)~N :

~+(S)fm~3(t)>N

=

~ m (s)~m3(t)>N = - ~(s-t) ~(~2 + ~ )
N . is defined by linear

mapping

extension of

I N a~*iA~'~'.<s l)..A~(sj)(t+)kfn+(tl )..fn+(t r) ~'~. =

=

(4.27)

~+ ~-n+. M-(k+r)/2%-l, ~+ tmk f~k*,(tl] • "~n~+(tmk, ~ r)~ z,_ 0

+ 2i~(ts

and initial condition

Also the correlation

= I~'~N@J~ equations

converge

in the WT sense

of the linearized equations

(4.29)

-ik Am(t) ] (t) '+ 2 ~ F

s• 3 (t) = - ~ s i (t) + i ~ (%(ts~(t) * with

~N

~t, _~/t of the classical

~o = ~ , ~ e : q" . The fluctuation operators

a~(t), ~ ( t )

= -(i¥+~)a

:

the intensive obser-

~'N(t), _~'N(t) converge in the WT sense along

(with (3~)) (2.9)

: In the DHL model

N"

a.(t) + f2(t) ]

+~j Finally,

it is clear that the really hard problems of irreversible statistical mechanics are not the construction of quantum mechanical cuckoo clocks but the understanding of continuous Coulomb forces.

systems with short range and

316

References [A~ F.T. Arecchi, Schulz-Dubois, E.0., "Laser Handbook", North-Holland Publ. Co., Amsterdam (1972) [G~

P. Glanssdorff,

I. Prigogine, "Thermodynamic Theory of Structure,

Stability and Fluctuatlons " " ~

Wiley, London (1971)

R.J. Glauber, Phys. Rev. 131, 2766 (1963) J.M. Radcliffe, J.Physo A 4, 313 (1971)

~

F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys.Rev. A6~ 2211 (1972). R.J. Glauber "Quantum Optics", Varenna Lectures, Academic Press~ N.Y.

(1969)

~

R. Graham, in "Springer Tracts in Modern Physics", Vol. 66,

~q

H. Haken, Handbuch der Physik~ Vol. 25, 2c, Springer, Berlin (1970)

~

K° Hepp and E.H. Lieb, Annals of Physics 76, 360 (1973)

[H~

K. Hepp and E.H. Lieb, Phys. Rev. A, to appear

~

K. Hepp and E.H. Lieb~ Helv. Phys.Acta, to appear

~5]

K. Hepp~ "The Classical Limit for Quantum Mechanical Correlation

~

E. Hopf, Ber.d.math.phys. Kl.d. Sachs.Akad.d. Wiss., Leipzig 94,

Berlin (1973)

Functions", in preparation 3 (1942) ~q

S.M. Kay', A. Maitland,

"Quantum Optics", Academic Press, London(1970)

~

W.E. Lamb jr., Phys. Rev. 134, A 1429 (1964)

~

V.P. Maslov, YMN 15, 213 (1960)

~

W. Thirring and A. Wehrl, Comm. Math. Phys. ~, 303 (1967)

~

H.F. Trotter, Pacific J. Math. ~, 887 (1958)

~

A. Wehrl, Comm. Math. Phys. 23, 319 (1971)

~

W.F. Wreszinski, Thesis, ETH (1973)

PERTURBATION EXPZ{SIO~ FOR THE P(÷). n ~ SCHWINGER FUNCTIONS Jonathan Dimock Department of Mathematics SUNY at Buffalo The Schwinger functions (imaginary time Green's functions) for a ~ ) a

theory are formally given by

SC ; where

-

~ = S~R ~) is the space of real-valued tempered distributions

and the measure

#~

is given by

Here A~,O is the coupling constant, V(I)= I : e ( I C , ~ : J x

, and d~o

is the Gaussian measure with mean zero and covariance C - ~ 4 ~ ) "I. Expressed in this form the Schwinger functions have a particularly accessible perturbation series. tials

e "~V .

One has only to expand the exponen-

This is the imaginary time version of Feynman's

original approach to field theory. In this note we report the result that the perturbation series for a Schwinger function is asymptotic to all orders as 9%-~0 4 • details will be published elsewhere [0q.

The

Such a result establishes

the extent to which one can use perturbatlon theory to extract information about a field theory.

By extending these results one

might reasonably expect to prove that the S-matrlx for the model h~s an asymptotic series.

Since the series is non-trlvlal it would

follow that scattering is non-trivlal. of asymptotic series in P(#)Z are

Other possible applications

sketched in [B].

Let /k be a bounded region in R.~ , define %/AE~-JA:(PC~(~X, and define ~IA,A as in (2) with V well-defined measure on ~

: f [ " ' q ~@ {,~,,x .

replaced by %/A" Then d~,A is a

and we denote integrals by

~ "' ~a,A

The main input we require is the following result of

318

Glimm, Jaffe, and Spencer [a],[3] . Let ~ be a product of Wlck monomials ~:~c~)~ : w ( , ~ support.

with each W bounded and having compact

Then there exists an interval ~ = [o>Ao] such that

z.

~,

I~ - (Z(~))-lsz.5 "'" ~z.~ Z(~) 31 3m for any choice of mean that

Jl . . . . . Jm 6 [i ..... N].

Re z. > 0 3

for all

j.

We write

Re ~ > 0

to

The extension of the Lee-Yang

Theorem is then:

Theorem I. particular,

when

Z (~) ~ 0

when

Re ~ > 0

~ = (z ..... z), Z(~)

(or when

Re z < 0) .

can only vanish when

z

In is

pure imaginary. The measures corresponding distributed continuous

to s p i n - ~

n

and to uniformly

spins are easily shown to belong to

yielding a new proof of the Lee-Yang theorem for these cases

~, thus

323

[Griffiths,

1969].

even polynomial,

The measures,

d u / d s = exp(-Q(S))

which arise in q u a n t u m field models,

be shown to b e l o n g to

~

when

Q

is fourth degree,

producing

the Lee-Yang results of [Simon,

not known

(at least to the author)

polynomials

satisfy condition

involves the w e a k e n i n g all the zeros of

a measure

in

b

concerning

to show that

2.

a measure

For any fixed m

)

Z(~+

~ >0

For any

in

and

> 0

J'Jl . . . . .

when

when

Re z > 0.

• Sjm>~0

and

In addition,

yields

for

~.

m I . . . . . m N = 0,i,2, .... x >0.

Jm E {i ..... N},

(<s s ....

(Sjl

of

for exponential-inter-

together yield:

3.

that

1 is:

~ E RN

)

with

Theorem 2 and the fact that

Theorem

[Ruelle,

the v a n i s h i n g

Q(S) = kcosh(S) + b S 2

A strong version of Theorem

Theorem m

it is

anv higher degree

Q(S) = k cosh (S)

also yields

1973];

occur in a strip with a corre-

~, and it would be relevant

arbitrary real

thus re-

(B) to the r e q u i r e m e n t

of the conclusions

action field theories

an

can d i r e c t l y

A new result of Ruelle

of condition

It can be shown that

Q

for these higher degree cases

~exp(zS)du(S)

sponding w e a k e n i n g

Griffiths,

whether

(B).

1973] which may prove important

Z(~).

with

L

3 31

i

31

s

>41

3m z | R e l ~ ~ . - ~ 4 -~ > 0 3m z

3

324

<SjSk) ~ - <Sj)~<Sk) ~ ~ 0 -~

when

4

z= iy

4

and

Z(iy)

~ 0 .

The above result generalizes equalities

[Griffiths,

1967; Kelly

certain of the GKS in-

, Sherman,

1968] to complex

z°

The final theorem we mention can be used to show that

(certain of)

the truncated Schwinger

~

theory alternate

functions

in sign as the order increases;

case of a stronger conjecture (Ursell functions) proven

m+i/ d 2m

(-i)

for

If

[Lebowitz,

Z(~) ~ 0

log = 0, then

325

REFERENCES

Griffiths,

R.B.:

J. Math. Phys. 8, 484-489

Griffiths,

R.B.:

J. Math.

Guerra,

Kelly,

Phys.

i0, 1559-1565

F., Rosen, L., Simon, B.: as Classical Statistical (1973) . D. G., Sherman,

S.:

(1969).

P(~)~ Euclidean Field Theory Mechanics: ~ Princeton preprint

J. Math.

Phys. 9, 466-484

Lebowitz,

J.: GHS and Other Inequalities: (1973).

Lee,T.D.,

Yang,

C.N.:

(1967).

(1968).

Yeshiva Univ. preprint

Phys. Rev. 87, 410-419

(1952).

Newman,

C.M.: Zeros of the Partition Function for Generalized Isinq Systems: Courant Inst. preprint (1973) (to appear in Comm. Pure Appl. Math.).

Ruelle,

D.:

private communication

Simon, B., Griffiths,

R.B.:

(1973).

Phys. Rev. Lett.

30, 931-934

(1973).

E U C L I D E A N FERMI FIELDS Konrad O s t e r w a l d e r * Harvard University Cambridge, M a s s a c h u s e t t s Introduction C o n s i d e r i n g the great success of E u c l i d e a n methods

in the c o n s t r u c -

tion of t w o - d i m e n s i o n a l m a s s i v e b o s o n q u a n t u m field theory models, is natural to ask w h e t h e r these m e t h o d s and ideas would also w o r k models

i n v o l v i n g b o t h b o s o n s and fermions.

it in

In p a r t i c u l a r we w o u l d

like to k n o w the answers to the following questions: --Are there free E u c l i d e a n fermi fields such that the S c h w i n g e r functions of a free fermi theory are the vacuum expectation values of these E u c l i d e a n fields? --Is there an integration theory for the E u c l i d e a n fermi fields?

Are

there L -space methods a v a i l a b l e ? P --IS there s o m e t h i n g like a Markov p r o p e r t y for E u c l i d e a n fermi fields? --Can we introduce interaction through a which

is related to a

(cut off) E u c l i d e a n a c t i o n

(cut off) H a m i l t o n i a n via a F e y n m a n - K a c

formula? --What are the properties of cut off S c h w i n g e r functions of a b o s o n fermion m o d e l ?

Can we use the G l i m m - J a f f e - S p e n c e r cluster e x p a n s i o n

methods to obtain estimates and to pass to the no cutoff limit?

Do

the S c h w i n g e r functions satisfy c o r r e l a t i o n inequalities? In the f o l l o w i n g I try to summarize w h a t is k n o w n about these questions. Free E u c l i d e a n Fermi fields Free E u c l i d e a n spin ~ fields have b e e n constructed by O s t e r w a l d e r and S c h r a d e r ozkaynak

lOS i], the case of a r b i t r a r y spin has b e e n c o n s i d e r e d by

[Oz i]; here we discuss the spin ~ case only.

i)

We want to express the S c h w i n g e r functions of a free fermion theory as vacuum e x p e c t a t i o n values of a free E u c l i d e a n fermi field. theory is free,

it suffices to study the 2Lpoint S c h w i n g e r

i) E u c l i d e a n fermi fields w e r e already studied by S c h w i n g e r

As the

function

[Sc 1,2].

*Supported in part by N a t i o n a l Science F o u n d a t i o n Grant GP40354X.

~27

J'-0 and

pz =

>-p~

)

n~f

is the bare fermion mass.

We can not find a Euclidean fermi field

~.

~

(×)

such that for

>,,

= (V~f

because this would imply that II~

tl~ ( n ) -C~II~- = ~@ ( ~ , ~(~,)~ % ( ~

is not necessarily positive.

The way out of this problem is to intro-

duce two distinct fields ~',

and ~

~,~ The f i e l d s

'

qi~

such that

.~

a c t in t h e E u c l i d e a n F e r m i Fock s p a c e 6 f and t h e y

satisfy the anticommutation relations

The vacuum On ~

-~

~

Ef is

cyclic for the smeared fields

there is a unitary representation

~(~)

, ~e~-I~(~¢)°

~,(~) of the translation

group and a unitary representation ~z(A) of the universal covering group SU(2) x SU(2) of the four dimensional rotation group SO 4 such

t h a t with

~(A,~)

=

L,(CA,a) _0_~

~z(,~ ) ~ , ( ~ ) =

,

~_=

and similarly for ~z(×) . Here A ~ R(A) is a four dimensional representation of S U ( 2 ) ~ S U ( 2 )

and A ~ r(A) is the homomorphism of SU(2)× SU(2)

onto SO 4 . F eynman Kac formula lOS I] In the boson case the relativistic Fock space

~

can be naturally

embedded in the Euclidean boSCh Fock space

~ b [Ne 1,2].

hold for fermions and the relation between

Ef and ~f is more compli-

cated.

Let

~ ~ ~m

~

This does not

be the Euclidean boson-fermion Fock space,

(X) the free Euclidean bose field of mass mb, and define ~ + t o be the

328

c l o s e d linear hull of vectors of the form

where

f

and h k are test functions w h i c h vanish at "negative times"

i.e. for

x e L_

WX

O . T h e n there is a m a p W:

:TT

~ (×) =

and

^

and

~

;

(2)

~ -f~E

07i'c~@-'

where

= -¢>-a

~/ ~, ( % , S ) X ~,(%,~)

on

~

such that for all vectors

j

0% =-

and for all

6 " t H ° k/~
/ O ,

,

is the u n i t a r y t i m e translation group in ~ .

the F e y n m a n - K a c

turns out

(Ox,Y)e

--

°" *

~

(3)

~

It

(I)

(w×,wy)~

Furthermore

H 0 b e i n g the free

the free relativistic fields.

that there is a u n i t a r y involution X, Y of the form

such that

""

e.-xPl'4° 50 (o,:2 ~ e x~H~

H a m i l t o n i a n , %0, , ~

~+-~ ~

Eq.

(3) is

formula for a free theory.

N o w let us c o n s i d e r fermions

interacting with a b o s o n

field.

The

cutoff E u c l i d e a n action is given by

A where ~

denotes a p r o p e r l y chosen u l t r a v i o l e t cutoff, A a finite vol-

ume in ~3.

P is a real polynomial, w h i c h is b o u n d e d below.

is not a symmetric o p e r a t o r on ~

holds, with the O For XE~+and

(5) where

t > O

, but

introduced above. we now h a v e

V e_-V~ (t,A~

d,(~,~)×

-

e-tHe,^

WX

V~(£,A)

329

A is the cutoff Hamiltonian. Equation

(5) is the Feynman Kac formula.

Remarks:

I)

becomes

If we leave out the Yukawa

the well-known

[Ne l],[Fe I ] , [ G ~ 2)

Feynman-Kac

term in V and in H, then

formula for boson theories,

fermi fields,

formula is a good substitute

With

V.

a Markov property

it appears that in many situations

where one would like to use the Markov property,

Cutoff Schwinger

see e.g.

1].

Though there is no obvious way of formulating

for the Euclidean

(5)

the Feynman-Kac

for it.

functions

Va(%,A)

as above, cutoff Schwinger

functions

can be de-

fined by

In the case of

~(~)z

models,

the volume cutoff Schwinger

satisfy all the Euclidean axioms exception of Euclidean

(see [GJS I],[OS i],[0 i]) with the

convariance.

(This property holds

cutoff limit only and follows easily

(6). Notice that the denominator

(2) and

(5) it is equal to

Because of the ~,t,A

are

(anti)symmetric

in [0S 2] or [O 1]. from equation

(2).

estimates on

~,%,A

constant,

IIe -~H~,A

(anti)commutation

of the Euclidean

axiom

fields the

this is axiom

(E3)

(E2) again follows easily

To prove the distribution

the GJS-cluster

defi-

in (6) is not zero. because by

in their arguments:

are needed.

functions

_c~ll1

relations

The positivity

in the no

if this limit is unique). We con-

jecture that the same is true for the cutoff Schwinger ned by

functions

and the cluster axioms,

For a Y2 model with small coupling

expansion me£hod should lead to the neces-

sary bounds, were it not for the non hermiticity

of the action V,

which might cause problems. At present there seems to be no reason to believe that the Schwinger functions of, for example, inequalities.

a Yukawa model satisfy correlation

330

Functional inte@ration The problem of functional integration for fermions has been extensively discussed in the literature. (a)

There are two main approaches:

The "physical" approach, which grew out of Feynman's formulation

of quantum mechanics in terms of "sums over histories"

[Fe 1,23, inter-

prets fermi fields as anticommuting c-number functions.

The mathemat-

ical objects resulting from a "functional integration" over fermions are Fredholm determinants depending on the bose field ~(×) !

References

for this approach are Matthews-Salam [MS 1,2,3], Edwards [E 1], v. Novozllov-Tulub [NT i], Berezin [Be 1] and further references given in these publications. (B)

The "mathematical" approach of non-commutative integration,

intro-

duced by Segal [Se 1,2]; see also Gross [Gr i] and Nelson [Ne 3]. The Euclidean fermi fields introduced above naturally lead to the "functional integrals" of (A).

As a matter of fact, just in terms of

these Euclidean fields one can understand why the c-number functions and ~

--see [NT 1]--corresponding to the fields ~(x} and ~ ~xl have

to be independent functions; not each other's complex conjugates: they correspond to the uncorrelated Euclidean fields lJ-' and ~ z .

It

should also be remarked that the way renormalization can be dealt with in this formalism seems quite attractive

(in a Y2 model, for example).

It is an interesting but still open question whether there is a natural way to formulate a Euclidean fermi theory within Segal's framework of non commutative integration. References [Be i]

F. A. BEREZIN, The method of second quantization, Academic Press, New York, 1966.

[Bo I~

N. N. BOGOLUIBOV, Dokl. AKad. Nauk SSSR 99, 225 (1954).

lEd i]

S. F. EDWARDS, Phil. Mag. 47, 758

[F i]

J. FELDMAN, Nuclear Physics B52, 608

[Fe i]

R. P. FEYNMAN, Revo Mod. Phys. 20, 367 (1948).

(1954). (1973).

[Fe 2]

R. Po FEYNMAN, Phys. Rev. 76, 749 (1949).

[Fr i]

J. S. FRADKIN, Dokl. Akado Nauk SSSR 98, 47

(1954).

[GJS i] J. GLIMM, A. JAFFE, T. SPENCER, The Wightman axioms and particle structure in the P(~)2 quantum field model, preprint.

331

[Gr i]

L. GROSS, J. Functional Anal. i0, 52,

(1972).

[GRS i~ F. GUERRA, L. ROSEN, B. SIMON, The P(9)2 Euclidean quantum field theory as classical statistical mechanics, preprint. [MS i]

P. T. MATTHEWS, A. SALAM, Proc. Roy. Soc. A221, 128 (1953).

LMS 2]

P. T. MATTHEWS, A. SALAM, Nuovo Cimento 12, 563 (1954).

[MS 3~

P. T. MATTHEWS, A. SALAM, Nuovo Cimento, ~, 120 (1955).

[Ne i]

E. NELSON, Quantum fields and Markoff fields, Amer. Math. Soc. Summer Institute on PDE, held at Berkeley,

[Ne 2]

E. NELSON, J. Functional Anal.

1971.

12, 97 (1973).

[Ne 3]

E. NELSON, J. Functional Anal. 12, 211 (1973).

[Ne 4]

E. NELSON, Notes on non commutative integration,

[NT i]

J. V. NOVOZILOV, A. V. TULUB, Uspechi fiz. Nauk 61, 53 (1957) German translation in Fortschr.

lOS i]

K. OSTERWALDER,

preprint.

d. Physik ~, 50 (1958).

R. SCHRADER, Euclidean Fermi Fields and a

Feynman-Kac Formula for Boson-Fermion Models, to appear in Helv. Phys. Acta. lOS 2]

K. OSTERWALDER,

R. SCHRADER, Commun. Math. Phys. 31, 83

(1973)

and Axioms for Euclidean Green's functions If, to appear. [0 i]

K. OSTERWALDER, distributions,

Euclidean Green's functions and Wightman Erice lectures 1973+

Lsc lJ

J. SCHWINGER,

Proc. Natl. Acad. Sci. U.S. 44, 956 (1958).

[Sc 2]

J. SCHWINGER,

Phys. Rev. 115, 721 (1959).

[Se i]

I. E. SEGAL, Ann. of Math. 57, 401 (1953).

[Se 2]

I. E. SEGAL, Ann. of Math.

58, 595 (1953).

Constructive Quantum Field Theory. The 1973 ''Ettore Majorana'' International School of Mathematical Physics

Mathematical Aspects of Quantum Field Theory

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Ettore Majorana: Unpublished Research Notes on Theoretical Physics

Nonequilibrium Quantum Field Theory (Cambridge Monographs on Mathematical Physics)

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Constructive Quantum Field Theory. The 1973 ''Ettore Majorana'' International School of Mathematical Physics

Mathematical Aspects of Quantum Field Theory

Mathematical notions of quantum field theory

Ettore Majorana: Unpublished Research Notes on Theoretical Physics

Nonequilibrium Quantum Field Theory (Cambridge Monographs on Mathematical Physics)

Quantum field theory in condensed matter physics

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Quantum field theory in condensed matter physics

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