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 <2, Z , ~ >

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

<~, E , ~ >. 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 <~, Z >

C.

In general, a random variable is a

to another measurable space

This concludes the elementary introduction I promised.

<~l' Z1 >" To see the enormously

rich applications of these elementary probability notions you should read the books by William Feller [2].

~.

STOCHASTIC PROCESSES

A stochastic process is a family of random variables bility space be into

(0,~) R

<~,Z,~ >

indexed by a set

T.

For the time being

so a stochastic process will just be a map so that for each

t ¢ (0,~), xt(')

{xt]te T

x(.)(.)

on a probaT

of

will just (0,~) X

is a measurable function on

~.

Notice that for each fixed

~ ~ ~, xt(~)

You should think of

as the value of some observable quantity at time

xt(~)

is just a real-valued function on

R° t;

for example the rate of arrival of particles at a counter, or the position of a particle undergoing Brownian motion on the x axis.. That is, each fixed sponds to a possible history of the value of some parameter for different

~l ¢ ~

functions

xt(~)

~

t E (0,~).

correA

will correspond to a different history since (in general) the and

xt(~l)

will be different.

t-axis

k/vk/ The measure

~

on

~

tells us essentially which (collections of) histories are

more likely and which are less likely.

~{~]

For example

G_< Xto(~) _< 63 : fG d~x to

is the probability that if we look at time be greater than or equal to probability that

tO

the value of the parameter will

and less than or equal to

~0 <- xt 0 <- ~0

and

G1 -< Xtl < 61 --

So ! Xto(~) ~ ~o; ~l ~ Xtl(~) ! BI] where

~to,t I

example, set

is the Joint distribution of MT(~) =

sup

t~(O,T)

(see below), ~{~I MT(~) ~_ 6] will not go above

~

xt(~).

~.

Similarly, the

is

=

Xto(')

f~lf ~0 d~to,t 1 G1 Go and

Then, assuming that

Xtl(').

As another

~(')

is measurable

gives us the probability that the path or history

before time

T.

The point here is that the underlying

dynamics of the situation is reflected in the measure

~

which allows us to

assign probabilities to certain subsets of the set of all paths in which we are especially interested.

For any given dynamics, of course, the measure

~

is not

given a priori but must somehow be constructed out of the physics of the problam being studied.

That is, the underlying dynamics should tell us how to construct

~, or better yet how to directly construct a measure on the set of paths for it is this measure which contains the answers to probabilistic questions about the

10

dynamics

We will now attempt such a construc-

such as the ones suggested above.

tion for the case of the Brownian motion of a particle on the real line. We make three assumptions: i.

The particle

starts at the origin at

2.

Given that the particle is at will be in a set

1

BC~

k

at time

t = O.

at time t > s

fB e-(~-k)2/2Jt-sld~.

s

the probability that it

is

For convenience we set

J2~It- sl P(k,~,t) ~

1 .~ e-(~-kj2/2t~ / 42~t

3.

(The Markov Condition) time

Given that the particle

is in a set

s, the future history of the particle will depend on

on the past history before

A A

at but not

s.

Condition one Just says that we are only interested in paths which start at the origin so for our space

~

= [all real-valued

we take

functions

f

on

[0,~]

Our stochastic process will just be evaluation at each

f

at time

is (a priori) t.

xt(. )

we want all sets of the form A

in

t, i.e.

considered as a possible history,

Since we want

each Borel set

so that

~.

to be a measurable

Ill xt(f) E A)

f(O) = 0}

.

xt(f) = f(t)

f(t)

where

being the position

function on

to be measurable

~

for each

for each

t

t

and

Any a-algebra which contains these sets also contains

all sets of the form

{fl

where

t I < t 2 < ... < t n

<Xtl(f),xt2(f),

and

A C ~ n.

. . . , X t n ( f ) > ~ A}

These sets are called the cylinder sets

and we will denote the collection of cylinder sets by

Z O.

tions 2 and 3 determine the measure we want to construct If we set

s = 0

on

Let us see how condiZ 0.

in condition 2 we see that the distribution of

xt

is

11 given by

d~t =

Let

~tlt2

and

B

e -k2/2t dk = P(O,k,t) dk

denote the joint distribution of Xtl

be Borel sets in ~.

terval ~kI xt2

1

is

The probability that

P(O,kl,tl)~AI

and given that

will lie in the small interval

probability that

Xtl

(2.1)

is in Ak I

~k2

is

and xt2

and xt2 , tI < t2, and let A Xtl will be in a small in-

Xtl = kl, the probability that P(kl,k2,1t 2 -tlI)2~k2.

is in Ak2

Thus, the

is the product

P(O,kl,tl)P(kl,k2,1t 2 -tll)AklAk2

Therefore,

(2.2)

~tlt2(a × B) = fkleA fk2eB P(O,kl,tl)P(kl,k2,1t2-tll ) dkI dk2

Thus condition 2 also determines the Joint distributions of any pairs

Xtl

and

xt2" To determine the joint distributions for three or more of the condition 3. Let

xt we need

tI < t2 < t3 be given. Then the probability that

a value in Akl, xt2

in Ak2, .and ....... xt3

conditional probability that

Xtl

has

is in hA 3 is Just (2.1) times the

xt3 ¢ Ak 3 given that

Xtl g AkI __and xt2 ¢ Ak2.

But, by condition 3 this conditional probability is Just the conditional probability that

xt3 e ~ 3

xt I e ~kl, xt2 ~ ~k2

(2.3)

given that

xt2 ¢ 2~2.

and xt 3 ~ Ak 3

Thus, the probability that

is

P(O,kl,tl)P(kl,k 2, It2 -tll )P(k2,k3, It3 -t2 1)Akl~2Ak 3

Using (2.2) we can now easily write down an integral expression analogous to for gtlt2t3(AXBXC), the joint distribution of Xtl , xt2

(2.2)

and xt3 . The

other Joint distributions work in exactly the same way so we see that conditions

i2

1), 2), and 3) determine a measure

~

on

ZO.

Such a measure is called a

cylinder measure for obvious reasons. This brings us to our first real mathematical problem, namely, E0 algebra of sets but not a ~-algebra. smallest ~-algebra

Z

containing

random variables (say

~0

r.0

in general.

[*I (~ <_ y(w) <_ 6]

whenever it can be.

~

will be in

xt r

and i but not

in such a way that the exten-

There is a general theorem (see Eoyden [ll]) which

says that this can be done in a unique way if E0

to the

because, one wants to consider various

The problem is to extend

sion is countably additive.

on

~

y(~)) which are limits of combinations of the

for such functions sets of the form in

It is important to extend

is an

That is, if

[A ]

~

is already countably additive is a sequence of disjoint sets

n

in

E0

..... A = U A n E ZO, then and

thing as showing that if then

~(E n) --> O.

~(A) = n_E1 ~(An).

E1 D E 2 D . . .

D E n D ...

This amounts to the same are in

Z0

and

O E n = ~,

Using a compactness argument this condition may be checked

in our case (for the proof see for example [7] or [15]).

Theorem.

(Kolmogorov)

A finite consistent cylinder measure on

x

[o,..,)

lq

can be

extended to a unique countably additive measure on the smallest e-algebra containing the cylinder sets.

The word "consistent" just means that cylinder mea-

sure has the property that

~tl.. .tn

the extra variables whenever

arises from

~Sl" "'Sm

by integrating out

[tl,...,t n} C [Sl,...,Sm].

The second mathematical problem that now arises is: Where does the measure have its support?

We have constructed

all real-valued functions on

[0,~)

~

so that it is a measure on the set of

such that

f(O) = O.

But surely if our

model is a realistic one for Brownian motion the support of the measure must be on a much smaller class of functions, for example on the set of continuous functions since we believe intuitively that Brownian paths are continuous.

So one

reason to investigate the support is to find the analytical properties of Brownian paths. we set

MT(~) =

The second reason is technical but very important. sup [xt(~)]. tE(O,T)

Suppose that

We pointed out before why we might be interested

13

in

MT(W)

MT(-)

as a randcm variable.

The trouble is that there is no reason for

to be measurable since it is the supremum over uncountably many tandem

variable and we are only guaranteed in elementary measure theory that the supremum of countably many measurable functions is measurable. let

%(~)

= sup Ixt (~)} t n

where

[t n]

On the other hand,

is a countable dense set in

(0,~).

Then

n

MT(')

is measurable and

fore if the set

%

and

C0(0,~) = [fl

f

can take our space of paths to be

MT

agree on the continuous functions.

continuous and C0(0,~)

f(0) = 0]

There-

has measure one we

in which case the sup function

MT

is measurable. Now, what sort of condition should guarantee that paths are continuous? condition should say roughly that if xt - x s

It - sl

The

is small then the distribution of

should be concentrated close to the origin.

cient condition is the requirement that for some

An example of such a suffi-

~, 8 > O,

(2.4)

E(Jxt - ~s j~) ~ MJt - sJ ~

For a nice proof see [15].

In our case the joint distribution of

xt

and

xs

is

z

e.~2/2s

~2~s 4 ~ ~anging

to new variables

~=

k1 + k 2 2

we find that the distribution of

~(fxt - ~s141

e

:

-(Z2-Zl)2/2Jt-

and

xt - x

~ = k2 - k I

is

1

sl

dk I dk 2

and integrating out e-~2/21t - sl d~.

Thus,

1 f~ ~ e-~2/21t-sJ d~ ~(t-s)

= 31t - sl 2

so the condition is satisfied and almost all paths are continuous. much more is known about Brownian paths.

In fact,

For example, almost all of them are

14

nowhere differentiable.

For a proof of this and other facts abot~t the modulus

of continuity see [6] or [l~]. The stochastic process which we have been using as an example is very special for several reasons.

First of all it is a Gaussian process which Just

means that all of the joint distributions of finitely many Gaussian.

Xtl ,...,Xtm

Secondly, it is a Markov process since condition 3 holds.

will reformulate condition 3 in more precise analytical terms. treated the case of a Brownian particle starting at zero. could have started with the particle at xt

would be

distribution

P(ko,k,t) dk d~o(k)

k0

are

In §3 we

Finally, we

More g e n e r a ~

we

in which case the distribution of

or we might only know that

x0

in which case the distribution of

xt

has some initial will be

(IR P(~,~,t) d~o(~)) d~ The construction of the cylinder measure is similar to what we have done and the extension is again by the Kolmogorov theorem.

Of course, the resulting measure

on the space of paths will be different for different initial distributions. What we have learned from this simple example is that two non-trivial mathematical problems immediately arise in dealing with stochastic processes, namely the construction of measures on spaces of paths and the support properties of these measures.

§3.

CONDITIONAL EXPECTATIONS

The Brownian motion which we constructed in the last section was a very special kind of stochastic process; a Markov process.

The Markov character re-

suits from the special assumption 3 which we used in calculating explicit expressions for the desired measure on cylinder sets.

In order to formulate pre-

cisely the notion of Markov process we need to introduce the concept of conditional expectation and to derive some of its properties.

15

Let Let

be a probability space and let

f ¢ Ll(~,Z,~)

Z0C Z

be a sub G-algebra.

and define

S(B) = / f(~)×B(~) d~(~) for each

B ~ Z0.

Then

S(B)

is a finite signed measure on

Z0

clearly absolutely continuous with respect to the restriction of (we denote this restriction again by

~).

and ~

S(B) to

is

Z0

Thus, by the Radon-Nikodyn theorem

there is a unique function (up to sets of measure zero)

E(fIZ0)

in

LI(~,Z0,~)

so that

S(B) = /B E(flZ0)(~) d~(~) for all

B ~ Z0, i.e.

The point is that may not be since

E(flk0)(~) Z0

is a subalgebra of

tional expectation of Example 1.

Let

Let

Z0

f

with respect to

G =~2, Z

things simple) that

is measurable with respect to E(fIZ0)(~ )

while

f(w)

is called the condi-

Z0.

equal the Borel sets in R 2, and suppose (to make

d~ = u(x,y) dx dy

where

consist of all sets of the form

Then the restriction of

Z.

Z0

u(x,y) dx dy

to

u(x,y) > 0

B x R E0

where

B

for all

<x,y>

is a Borel set in ~.

is Just the measure on ~

by

(fR u(x,y) dy) dx

and the conditional expectation of a function

E(flZ0)(x) = ~

~ E LI~2,Z,~)

f(x,y)u(x,y) dy fR u(x,y) dy

¢~2.

is given by

given

16

Example 2.

Let

~ = lq, let

E

probability measure on lqo val

[-t,t]

Ft

assigns

flxl> t d~(x)

be the Borel sets and suppose that

Let

Et

[-t,t]

to any Borel subset of

to the complement of

and lq. [-t,t]

[-t,t].

I E(flEt)(x ) =

If

The restriction of

E(flZt)(x)

Ixl~t

flxl~ f(x) ~(~)

is to be measurable with respect to

Ixl >t

Ixl > t

which it must be if

E(flZ t)

Et.

Conditional expectations have the following elementary properties. Proposition.

i.

If

f ~ O, then

2~

If

ZOO

E(fIEo) ~ O.

is a sub G-algebra of

~0'

then

E(E(fI%)IE00 ) = E(flZ00) •

3.

If

g E L (,EO,~)

then

E(fglzo) : gE(flzo) ~o

If

~'

is a sub G-algebra of

independent, then if

Z

so that

Z0

and

Z'

are

f e LI(~,Z',~),

E(flZO) = E(f) i.e. Proof:

E(fIZo)

to

f e Ll(lq,Z,d~) then

f(x)

is constant for

~

and assigns the number

fhxl~d~(x>

Notice that

is any

consist ef all Borel subsets of the inter-

plus the complement of ~(B)

~

is a constant funetion.

Property i is immediate and property 2 follows from the definition of

conditional expectation since

fB E(flEo) d~ = fB f d~ = #B E(flz00) d~

17 for all

B e ZOO.

Suppose

B g ZO, then by definition

f f~B d~ = f E(flZO)XB d~ so if

B i e ZO, i = I,...,N N

f f z ciXB. d~ : f E(flz o) z oi×B d~ . i

By the dominated convergence theorem this implies that

(B.~) for all

f fg d~ = ~ E(flEo)g d~

g ~ L~(fl,E0,d~).

In particular, if

B ¢ %,

fB fg d~ = fB E(fIEo)g d~

so by the uniqueness statement of the Radon-Nikodyn theorem This proves (3). Finally, suppose

E'

B ~ Z o.

so

Then, ~ ( A n B )

:~(A)~(B)

and

E0

Ai ~

~t

: g~(flzo).

are independent and

]B ×A d~ = ~(A N B) = ~(A)~(B)

Thus, if

E(fglZo)

.

, i = I,...,N,

fB Z CiXAi d~ = (Z ci~(Ai))~(B ) .

So, using the dominated convergence theorem again we have

for all

f ~ LI(~,E ',d~) which proves (~).

Given a sub a-algebra

E0 C E, the conditional expectation is a map

E% f

~->

E(flEo)

Z'

18

of Z-measurable functions into Zo-measurable functions.

EZO

is clearly linear

and has the following important additional properties: Proposition.

~o

EZn

is a contraction from LP(o,Z,d~)

to LP(~,Zo,d~)

for

each i ~ p ~ ° 6.

EZO

is the orthogonal projection of L2(~,Z,d~)

onto

L2(n,Zo,d~). Proof:

If f e Ll(~,Z,d~)

and f ~ 0 then

IIEz0fllI : ; Ez0f d~ : ; f ~ : llfllI

For a general f we write disjoint support.

f = f+ - f_ where

f+,f. _> 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

<2,Z >

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

=E^EX 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 <_ Ilfll If

Ile-tAflll < Ne'tAI f(x) lll 1

=

IIfll 1.

Thus, for all

f E L 2,

1.

f ~ L ~ C L 2, then

IIe-tAfll

=

sup

(e'tAf,g)

NgIll=1 g ~L 2 =

sup

tlgllS1

(f,e-tAg)

geL 2

_<JJffl so

e

-tA

is a contraction on

L

~

also.

Thus by the Riesz-Thorin Theorem, e

-tA

is a contraction on all the L p spaces. To prove that S (p)

e -zA

on

LP(~,d~), 1 < p < ~, is holomorphic in the sectors

and bounded in the sub-sectors one needs Stein's generalization of the

Riesz-Thorin Theorem.

The reader can find the details in [13].

clude by proving the strong continuity.

Jle-tAf

for

f ¢ L 2.

Since

L2

suppose some

2 < p < ~

~ E L q.

for all

adjoint of dense in

e -tA L p.

Let

e -tA

strong continuity follows since

P + ~q = 1.

e'S~ = 0

for all

s ~ tO

e -tA Lq

for each

on

Lq

e -tA

~ = e

is -t0A

Suppose

t > 0. on

$, $ ~ L p.

e "t0A~ = 0

and by analyticity

is strongly continuous on

on

L2°

is strongly continuous on

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]

is strongly continuous on

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

<2,Z,~ >

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

~ ~ B)

= ~F(B)

so

{~(fi)]~=l

and

[~(fi)]~=l

have the same joint distributions.

The only

34

step left to show that show that the measure

~(') ~*

can be realized as

$(')

on

<E~,~>

is to

which is presently only defined on the cylinder sets

can be extended to a probability measure on

E E.

This situation is analogous to

the extension problem which we faced in §2 and which was solved by the Kolmogorov theorem.

Theorem.

(Lenard)

Every cylinder measure on

countably additive measure on Proof:

Let

~* = { ~ ]

theory lemma [ll], ~*

E* alg

can be extended to a

<E~lg,EE>O

be the cylinder measure on

E*.

By a standard measure

can be uniquely extended if and only if for each de-

creasing sequence,

... A n ~ A n + 1 o.., of cylinder sets, ~(An) > 5

implies that

% ~.

0 A n

n

Each

A

for all

n

is of the form

n

-~(n), c~k(n)] A n = {T ~ E~!g I < T ( f n)),..o,T(Ik(n)) > ¢ B n

for some

f!n) e E, i = l,...,k(n).

We may assume that all the

f!n)

selected from a linearly independent set B

n

are

1

l

by expressing

vious ones.

a non-independent

f

[gi]i=l , because we can always change as a f i n i t e

linear

combination of pre-

Now, consider the map: ee

@:

E* --> alg

K lq i=l

given by

e(T) :

8

is onto since given

<w i>

linear combinations of the in general

T

cylinder set in

we set

T(gi) = wi, extend by linearity to finite

gi' and define

T

to be zero elsewhere.

is not 1-1 and is not onto when restricted to ±~i ~' we define

cylinder measure on additive extension to

i~l ~

~(S) = ~*(e-l(s)).

Thus

E*). ~

(Note that If

S

is a

is a consistent

and so by the Kolmogorov theorem has a countably

Z, the smallest u-algebra containing such

S.

Therefore,

$5

since

~(8(An))>5

for all

<Wi>

e n~l 0(An)"

Since

n 0

and

e(An)De(An+l)

is onto, there is a

there is a point T e n~l An"

has

Thus, ~*

an extension.

Corollary i.

Every generalized stochastic process

as the canonical process

~

on

<E~g,~,~*>

~

over

where

E

~*

can be realized

is the probability

measure constructed above.

From now on we will denote measures on

Corollary 2. such that

Let

C(-)

C(tf)

is a continuous function of

(7.2)

~

C(f) -- SE~g~

on

[fi}n=l

To say that

C

of vectors in

~*.

of positive type

E

for each

<E~g,Z E >

e i

is called the characteristic function of

Proof:

t

instead of

by

be a complex-valued function on

Then there is a probability measure

C

<E*lg,~>

f e E

and

C(O) = i.

so that

~(T)

~.

is of positive type means that for any finite collection E, (C(fi-fj)}

a finite-dimensional subspace in

E.

is a positive matrix on Then the restriction of

continuous positive definite function with

C n. C

to

Let F

F

be

is a

C(O) = l, so by the classical theorem

of Bochner, there is a unique probability measure

iF

on

F*

so that

c(f) = IF. ei gF(T )

for all ~F

on

f e F. Z F.

We now lift this measure exactly as we did before to a measure

[~F }

is a consistent (by the uniqueness in Bochner's theorem)

family and so defines a cylinder measure extends to a probability measure on

Since

E~g

when measures on lar on

E*.

~

on

E* alg"

By the theorem,

<E*ig,Z E >.

is difficult to handle analytically it is important to know <E*lg,~E>

have their support in smaller spaces, in particu-

A sufficient condition is provided by the following theorem of

$6

Minlos ([3], [4], [9]).

Theorem.

Let

~

be a probability measure on

<E~lg,~E>O

nuclear space and if the characteristic function of is supported by

E*.

Corollary.

~

Let

nuclear space that

E.

Suppose

¢

~

on

<2,Z,W >

is continuous in the sense that

in measure on

<~,Z,~ >.

<E*,~E,~*>

where

Then ~*

E

is a

is continuous on

be a generalized stochastic process on

~(fT) --> ~(f)

canonical process

~

Then if

~

f7 E >

E,

over a f

implies

can be realized as the

is a probability measure on

<E*,~E>. Proof:

We already know that

~

can be realized on

<E* v .i* ) alg,~E,~ •

continuity condition implies that the characteristic function of tinuous on

E

so the support of

We remark that if

E

~*

is on

is con-

E*o

is metrizable (like

can be replaced by the requirement that

~*

But the

fn E >

s~n)) f

the continuity condition

implies that

~(fn) --> ~(f)

pointwise almost everywhere.

Example (the CCR). product maps

(',').

Let

E

A representation of the Weyl relations over

f --> U(f), f --> V(g)

Hilbert space W

(1)

(~.3)

be a real nuclear space with a continuous scalar

from

E

E

is a pair of

to the unitary operators on a separable

so that

u ( f + g) = u ( f ) u ( g ) , v ( f + g) = v(f)V(g)

(2) V(g)U(f) = ei(f'g)u(f)V(g) (3) f7 E > f

implies that

U(fT) --> U(f)

and v(~ 7) - - > v(f)

strongly.

Gelfand and Vilenkin show in [3] how any representation which has a cyclic vector ~0

for

[U(f)}f~ E

can be reslized on

L2(E*,EE,d~)

for an appropriate

~.

will sketch the result to show how the Minlos theorem is used; for details see

We

37

[3].

Define

L(f) = (U(f)20,20) H Then

L

lying

is a continuous complex-valued function of positive type on

E

satis-

L(O) = l, so by the Minlos theorem

L(f) = rE* e i < f ' T >

for some probability measure

on

<E*,~>.

Define

n

S:

n

Z ~iU(fi)~ O .....> Z ~.e i 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 ]

<

(p2 + 1)n/2(pl2 + 1)~ l

for all ~ > 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,

can be majorized by

const, x ~

less rapidly growing function of if

p

x.

is any non-zero element of

T*p(x)

can be majorized by

rapidly growing function. like

for large

~

a typical

Ixl T

x

but not by any

Moreover, we will prove that,

SdR2), then, for almost every

const × l ~ o g

Ixl

but not by any less

Thus, we may say that a typical

at infinity.

T

grows

It should be understood, however, that

does not "behave like

v ~ "

instead, it is usually much smaller than ally takes values which are this large.

for large

~

x;

and only occasion-

Indeed, a typical

T

haves in some respects as if it were bounded; for example, if is any square-integrable

T,

function, the set of distributions

bef(x)

T

such

that

<

_d 2 )-~12 dX-~l + 1 T(x) f(x)

is square-integrable Since

is a set of

~o-measure one.

S'(~ n) is not a locally compact space, measure theory on

it is not quite standard.

We will not give a systematic investiga-

tion of the subject, but there are a few simple remarks which should make it seem less strange.

To begin with,

S ' ~ n)

comes equipped

with three c-algebras which might in principle be distinct: ~0:

the

c-algebra generated by the continuous linear

functionals Ew:

the

c-algebra generated by the weakly closed sets

47

Zs:

Evidently,

the

a - a l g e b r a g e n e r a t e d by the closed sets.

Z 0 C Zw C Z s.

Gaussian measures are defined on

Z O,

but it is frequently easier to verify that sets we are i n t e r e s t e d in belong to

Z s.

Fortunately, we have in fact

ZO : Zw : ~s"

There

are general theorems which imply this equality, but in the case at By using Hermite

hand it is easy to give a direct elementary proof: expansions, we can identify decreasing sequences and b o u n d e d sequences.

S(~ n) with the space

S'(~ n) with the space

s s'

of rapidly of p o l y n o m i a l l y

Let

!

K n = {(aj) e s : lajl < n(l+j n)

It is easy to check that topology on B

of

s'

ZO(Zs)

agree on

Z,

w

n.

K n. = E . s

ZO(Z s)

Kn,

Hence, E

that

s

if and only if

and since and Z

0

B A Kn

A subset

belongs to

is compact, the strong and w e a k Kn

is metrizable,

agree on

and w h e n we say that a subset of

measurability

.

s' = n~iKn •

K

n

We will f r o m now on denote this

that it belongs to E. the

and that

Since each K n

agree on

j}

is compact and m e t r i z a b l e in the strong

K n 6 ~0'

that

belongs to

for all

topologies

E0 = E

s',

Kn

for all

for all

ZO n

and

Zw

so

G - a l g e b r a simply by

S'aR n) is m e a s u r a b l e

, we mean

We will usually leave the p r o b l e m of v e r i f y i n g

of the sets we c o n s i d e r to the reader.

The fact

S'~R n) is a c o u n t a b l e union of compact m e t r i z a b l e subsets is

frequently useful; to a large extent,

it makes p o s s i b l e the r e d u c t i o n

!

of measure theory on

S (~n) to measure theory on compact metric

spaces. We will make one small change in the notation used in the Reed !

lectures.

If

U

is a G a u s s i a n measure of mean zero on

covariance of

~

was defined to be the p o s i t i v e s e m i - d e f i n i t e bi-

linear form (''')U

on

S aR n) given by

S aRn), the

48

(f,g)

By the nuclear theorem,

(f,g)B

where

: I~(dT)

we can write

: Idx dy f(x)g(y) X (x,y),

X (x,y) : X~(y,x)

is a tempered distribution

will also refer to the distribution ~.

Furthermore,

if

~

X (x,y)

on

~n

as the covariance

is translation-invariant,

We

× ~n. of

we can write

A

X (x,y) : X~(x-y),

where the

X ^

is a distribution and write,

of positive type on

~n.

We will drop

~.

In the reverse

for example,

X (x,y) : X (x-y),

again referring to direction,

X (~)

as the covariance

if we start with a distribution

there is a uniquely

determined

with mean zero and covariance

of

X

of positive type,

translation-invariant

Gaussian measure

X(x-y).

We make very limited claims to novelty for the results presented here.

Although we have not been able to find these results

literature,

in the

the ideas involved in proving them are quite standard

and, indeed, old-fashionedo processes has undergone good list of references,

The general theory of gaussian stochastic

vigorous

development

see the reviewer's

in recent years--for

a

remark in MR 42#4994

We are using "translation invariant" as a synonym for the probabilists' term "stationary", i.e., a measure ~ on S'OR n) is said to be translation invariant if it is invariant under the natural action on S ORn) of the group of translations in ~n.

49 (1973)--but recent general results do not seem to be directly applicable to the questions investigated here.

In any case, our principal

objective in writing this appendix was to provide a reasonably selfcontained discussion of the answers to these questions in a language accessible to mathematical physicists.

II.

Let

> 0.

REGULARITY

The mapping

T~

q--~+l dx I

T

is a continuous linear mapping of arily denote this mapping by

~.

S'(~ 2)

onto itself.

We tempor-

Schematically, what we want to

prove is that

B0(¢-I~ ) = 1

where

~

is some appropriate space of locally Holder

continuous

!

functions, regarded as a subspace of

S (IR2). This suggests that we

investigate the "support properties" of the measure.

= ~0 o #-i

It is easy to see that

B

is again Gaussian with mean zero but has

eovariance

X (x-y) : (2~) 1 2

I dp eip'x (p2+l)-i (pI2+I)-~

(2.1)

50 The Fourier t r a n s f o r m a t i o n

can be understood

in the sense of distributions, grable.

This also implies

since

rather than

(p2+l)-l(p~+l)-~

X

that

literally,

is a continuous

is inte-

function.

In

fact:

Proposition

2.1 (a).

For

0
1

IXp(0) - X (x) I < const x Ixl 2e

(b)

For

Ix

e

=

1

,c0)- x (x>l

oonst x Ixl

log( )

for

Ixl

We will return later to the proof of this proposition; the moment.

We are now in the following

Gaussian measure

p

on

situation:

assume

it for

We have a

S'CI~2) with mean zero and with a covariance

which is Holder continuous

at zero.

We want to conclude

that

assigns measure one to the set of locally Holder continuous

functions.

We can in fact prove quite a general and precise theorem in this direction; we do not need to assume that the measure invariant,

nor are we restricted

formulate our result be real numbers,

to a two-dimensional

efficiently,

with

~ > 0.

is t r a n s l a t i o n index set.

we need some notation.

For

0 < e < i,

Let

To ~,6

define

-6+{ ~ ,B(~) : E~Ilog (~)]

1 (Note the ~ convenience ~n,

define

(2.2)

in the exponent of the logarithm; later on.)

For any real-valued

it is put there for

function

T

defined on

5!

HTil

:

~,B

sup

I IT(x)

T(Y)I

i } ,

(2 3)

0
and let

-)(: {T: iJTii < =}. ~,S ~,~

Then

may be identified with a subspace of

-~a,8

S'OR n)

by

f e scRn).

= If(x)T(x)dx,

Our principal regularity result is the following:

!

Theorem 2.2.

Let

~

zero and covariance for some

a > O, 8,

be a Gaussian measure on

X(x,y). and

x,y

with

1

ix-Yi ~ ~- .

u()~

with mean

is continuous

and that

C

1 IX (x, x) +X (y,y)-2X (x,y) I~-

for all

X(x,y)

Assume

S ([Rn)

c

x

Then

,B) = i.

Note that, by combining Theorem 2.2 with Proposition 2.1 we obtain a sharper version of statement b) of Theorem i.i, i.e., we show that, 1 for 0 < e < [, ~0-almost every T has the property that

52

_d 2 )-e/2 -~l + i T

belongs

to

~

~,0

~,_~

(~

2

Belonging

to

continuity with

~e,0

is a slightly weaker

with exponent

all exponents In outline~

~

on

X

define

X

but is stronger

UTHe,8,

is the uniquely for any

in

for

~n

T E X,

to belong

to

determined

Xl,...,x m 6 D n,

are jointly

Gaussian,

T(x)~(dT)

2.3.

with dyadic rational

by the right-hand D n.

: 0;

Assuming

Let

Dn

co-ordinates,

be the Gaussian

X(x,y).

probability

side of (2.3) with measure

on

By this we mean that

measure

on

X

such that,

T ~ T(Xl),..o,T(x m)

and such that

IT(x)T(y)~(dT)

: X(x,y)

(~)

for all

x,y E D n

Then

: i.

the lemma, we can easily prove the theorem:

changing

we can restrict their Gaussian

there is a natural given by

~

X = {T e X: |TU~, 6 < ~}

Let

g(X) = !,

without

Let

the random variables

^

Since

than Holder continuity

the proof of the theorem goes as follows.

with mean zero and covariance

Lemma

than H61der

denote

restricted

I

local property

[I n ~, the space of all real-valued functions x ~-D We will again denote elements of X by T, and we will

D n.

x,y

2

e' < ~.

denote the set of points and let

1 e : [).

for

the rando~ variables

character

norm-preserving

T(x)

or their covariance.

bijection

i

of

~ ~,8

to Also, onto

53

i(T)

Let D

D

= Tin D

be the probability

under

i -I.

measure

The random variables

T ~ T(x)

are, by the definition covariance in

Dn

X(x,y).

converging

of

If to

T E ~a,~

implies

"

f E S(Rn),

~

,D

jointly Gaussian

= lim j ÷=

This,

given by

with mean zero and

and if we take a sequence

(xj)

T(x-) J

together with the continuity T ~ T(x)

are

D-measurable

of

X(x,y),

for all

Gaussian with mean zero and covariance

x,

X(x,y).

For

the mapping

T ~

is again

which is the image of

we have

that the mappings

and are jointly

on

x ~n\Dn, x,

~, 8

(x e D n)

D,

T(x)

for all

on

D-measurable,

I~(dT)

(approximate if we regard

f

: ]f(x)T(x)dx

Gaussian,

the integral D

= ]dx dy fl(x)

defining

as a measure

and of mean Zero,

on

~(S'ORn)\}=,~)

S'(~)

= O,

and

f2(Y)X(x,y)

by Riemann by putting

sums).

Hence,

54

is a Gaussian

measure

with mean zero and covariance

Since a Gaussian

measure

is uniquely

determined

X(x,y).

by its mean and

covariance,

~ : ~,

and the theorem

is proved,

once we have proved Lemma

Now let A = [0,i] n ~ D n, right-hand reduce

Lemma

side of (2.3) with

Lemma

2.4.

ing only on

and define x,y

2.3 to the following

~TIIA,~, 8 on

restricted

~{T:

Lemma

by the A.

We next

statement:

There exist strictly positive constants such that, for all positive

n,c,a, 8

X

to lie in

local regularity

Cl, c 2

depend-

l

-c212

^

To prove

2.3.

IlTllA

> 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 ) ) ~<

~ j=l

8 x (2J+1)2.2. c3.2 -c4Y ( j + l )

side of (2.7)

-c~ = 0(4

2 )

as

y + ~.

57 ^

Since

~(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

<

xe~nIJl'og(Ix 1+2 )' To prove a), we combine Lemma 2.4 with the argument deriving Theorem 2.2 from Lemma 2.3. X(x~x)

To prove b), we note that a) and the fact that

is bounded imply the existence of

Cl0,Cll

such that

-Cllk2 p{T: sup {ITa(X) I} > 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~
~<

,~,(u2'fI~m(X'l'dx

191 denotes the total variation of

~.

By (3.1) and the

Monotone Convergence Theorem,

If~)(dy) ~m(X-y)l m

which implies

(3.2).

< =

a.e.

on

U1 ,

(3.2)

62

Now let with

be a

f(x)

ff(x)dx ~ 0 J

C~

function of compact support on

and, for each positive

fk(x) = knf(xx)

For each fixed

x,

X,

~n

define

.

the mapping

T m (T*fx)(x)

I

is a Gaussian random variable

on

S (/~n), with mean zero and variance

I X(dp)If(P/X)I2

(where form of

X

is the Fourier transform of f).

Since

X

X

and

f

the Fourier trans-

is a positive measure of infinite total mass,

this variance

goes to infinity with

tive numbers

a

k.

Choose a sequence of posi-

with m

am

<

=;

(3.3)

m

then a sequence

Xm

a m2

going to infinity so rapidly that

" IX(dp)l

f(P/%m) l2

+

~

;

(3.4)

and put

~m(X)

It follows

= amfx m (x).

from (3.3) and Lemma 3.2 that, if the restriction

some non-empty

open set is a signed measure,

tive Lebesgue measure

in

~n

on which

of

T

there is a set of posi-

to

63

lim

T*~m(X)

: 0.

m+~

We will complete

For almost

all

To prove this, continuous

T • S ' ~Rn),

lim sup

and hence Borel

is a Borel subset

e S'aR n)

'

of

going to infinity

S OR n)

× ~n:

S (I~n) x

T ÷ T~m(X)

'

from

l~n

.

lira sup

×

~n

lim sup

(3.5)

(T,x) ~ T*~n(X) to

~.

is

Hence,

I T e m ( X ) I < =}

Now for any fixed

(m:i,2,3,...)

(by (3.4)),

by showing

I T e m (X) I = = a . e .

we note first that the mapping

Y ~ {(T,x)

variables

the proof of the proposition

x,

the random

are Gaussian with variance

so

for almost

IT*~m(X) I : ~

all

T°

m-J-~

In other words,

for each

{T:

is a set of

b-measure

® dx measure that,

zero.

for almost all

gue measure

zero.

x,

(T,x) e Y}

zero.

By Fubini's

Applying

Fubini's

T 6 S'~Rn),

This

is exactly

{x:

Theorem*

, Y

is a set of

Theorem again, we conclude (T,x) 6 Y}

the desired

is a set of Lebes-

statement

(3.5).

There is a slight subtlety at this point. To apply Fubini's Theorem, we need to know that Y belongs to the product a-algebra, which may in principle by strictly smaller than the q-algebra of Borel sets in the product space. It is, however, not hard to show~ using the fact that S'(Iqn) is a countable union of compact metrizable spaces, that in this case the product q-algebra and the Borel a-algebra on the product space actually coincide.

64

IV.

Proposition (in some sense) set of every

2.5b implies that the set of distributions grow no more rapidly than

B0-measure T

BEHAVIOR AT INFINITY

one.

to say that a function

at infinity

at infinity

In this section, we prove that

does in fact grow as fast as

reasonable

Ixl

~log

~log

T(x)

Ixl .

which is a

B0-almost

It is certainly

Jiog Ixl

grows as fast as

if i lim sup T(x)[log(Ixl]

> 0.

Ix1÷ The meaning of the corresponding little more problematical

4.1.

for a distribution

is a

, but the following two propositions

seem to cover any reasonable

Proposition

statement

would

interpretation.

Let ~ > O.

Then

~ o - a l m o s t every

T

satisfies.

1

lim supfl + IxI- [\dxi Proposition

1

--

4.2. Let P e S QR2)

.

Then

Po-almost

1

lim sup

cp +l)

every

T

satisfies

1

[T* ~(x) [log(Ixl)] -~ = 2[((-A+ l)-ip,p)] ~

We will prove these propositions

by proving the following

more

general result. Theorem 4.3.

Let

~

be a t r a n s l a t i o n - i n v a r i a n t

Gaussian measure

!

mean zero on

S (I~n)

with covariance

X(x-y)

which is Ho'lder

continuous at zero and satisfies

Note that 2x cos(x 2) is the derivativ~ of the bounded sin(x2). Is the distribution 2x cos(x L) bounded?

function

of

65

a.

X(0)

b.

X(x-y)

= i

Ix-Yl n

is b o u n d e d

as

Ix-yl

+ ~

Then 1 ~{T:

l i m sup

T(x)

[log(Ixl)] -~ =v~}

To d e r i v e the p r o p o s i t i o n s ,

= i.

we a p p l y the t h e o r e m to m e a s u r e s

of

the f o r m

= ~0 o ¢-i

where

for P r o p o s i t i o n

4.1 we t a k e

_d 2 ~(T)

and f o r P r o p o s i t i o n

4.3.

L e m m a 4.4.

Let

There

exists

=

are c h o s e n

in T h e o r e m

A

~)-a/2

× \d--~l 2 +

T

4.2 we t a k e

#(T)

the c o n s t a n t s

: const

eonst

× T* o ;

to s a t i s f y the n o r m a l i z a t i o n

be any bounded

a constant

K 1 such

set in

An ,

4.5.

For any

6 > O,

and let

a.

1

K 2 < [.

that

~{T: sup IT(x) I ~> I} ~ K 1 exp['-K212] x EA

Lemma

condition

for all positive

1

66

1

IT(x) lElog(Ixl)]

p{T: lim sup

Lemma 4.6. in

~n

For any

6 > O,

> ~ }

there exists a sequence

O.

=

xi

going

to

such that

1 ~{T: !im sup T(xi)[log(Ixil)] i

< ~ }

=

o

(4.1)

The theorem follows at once by applying Lemmas 4.5 and 4.6 to a sequence of positive

Proof of Lemma 4.4. X(0) = 1

Let

converging to zero.

x 6 ~ n.

[he normalization condition

implies

~{T:

Hence,

6's

for any

IT(x) I > k}% K 3 exp[-12/2].

Xl,...,Xk,

~{T: sup IT(xi) I > k} ~ K3"k exp[-i2/2]. l~i~k By Proposition 2.5a,

~{T:

there exist

IT(x)-T(y)I < lIx-yl ~ for all

Define

e

e > 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
is within

we get

B{T: sup iT(x) I > (l+g)l} xEA or (replacing

A

: 1

(

sup{IT(x-a)l}[log(laI)]

')

> ~2n

(i+6)}

a 6~ n

lim ~ ~{T: sup{IT(x-a) I} > ~2n(l+6)log(laI~} A~ la ~A xEA a~ n

.

]

Now choose

K2 < ~

so that

(and translation invariance)

and apply Lemma 4.4,

2K2(I+~) > i, to get

~{T: sup {IT(x-a)l} >~2n(i+6)log(ialY} ~A where

n' : 2K2(l+~)n > n.

lim

~

laI>A proving the lemma.

~

Klla

-n

t

Hence

~{T: sup {IT(x-a)I} >

~2n(i+~)iog(lai)'}

: 0

68

Proof of Lemma 4.8. and let

(x i)

Let

R

be a large number

be a enumeration

lal~/2.a,

a e Z n,

(to be chosen

of the vectors

lal > R.

We choose the enumeration

in such a way that

Ixil

in

for log(Ixil)

T.

i,

and we write

variable

ii

T ~ T(xi).

later)

and

is increasing for the r a n d o m

l

By Lemma 4.5 1

lim sup i+~

for almost all

T,

0 = ~{T:-v~

i.e.,

for each

ITil-£

< v~

i

so to prove

(4.1)

i ~ T i ~ ~2n(i-~)£' i

it suffices

for all but finitely many i}

I

0 : ~{T:- ~3~-£i < Ti ~ ~2n(i-6)£'i

We will in fact only prove this equation result

for general

sufficiently For each

to prove

I

for all

for

I : 1

and obtain the

be observing that our argument works

large values

of

k : 1,2,3,...

What we want to show is that

~.

=

0.

for all

R. define

~k : ~{T:- 3 v / ~ i ~ T i ~ 2 h ( i ~ 6 ) £ ' i

lim

i > I}.

for

1 ~ i ~ k}

,

69 We estimate the

~s

recursively,

~k ~ ~k-i "ess sup I~{Tk ~

Here,

D{. I--}

denotes

~

I TI = tl,...,Tk_l

conditional

supremum is taken over all 1 ~< i < k-l.

using the inequality

probability

tl,...,tk_ 1

(We could, of course,

a smaller set of

t's.)

-

k-i ~ aki i=l akk

ti

Iti I~< v ~ k

take the essential

easy to check that the conditional T 1 = tl,...,Tk_ 1 = tk_ 1

and the essential

such that

Since the measure

B

= tk_l}l.(4.3 )

supremum over

is Gaussian,

distribution

of

Tk

for

it is

given

may be chosen to be Gaussian with mean -i akk

and variance

where A

is the k × k

=

(a..)

l]

matrix with inverse

A -I = B = (bij) = (X(xi-xj)). By condition

a) in Theorem 4.3,

bii = 1

for all

i;

also, by condition b) and the fact that the

x. l

are sparsely

dis-

n

tributed

in

~

,

we can make

k

;~

j:l

Ibijl

j~i small,

uniformly

Now if

in

i,k,

C = (cij)

by making

is any

k × k

R

large. matrix,

the operator norm of

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,

EUCLIDEAN GREEN'S FUNCTIONS AND WIGHTMAN DISTRIBUTIONS Konrad Osterwalder*) Harvard university Cambridge, Massachusetts I.

INTRODUCTION

The use of Euclidean methods has a long history.

in relativistic

quantum field theory

The idea to pass to imaginary

times or energies

in

order to replace the indefinite Minkowski metric by the Euclidean metric appeared

first in Dyson's work on renormalization

[Fr i], Nakano[Na

tinued to imaginary

dean field operators. purely Euclidean

functions

equations

or as the n-point

It was Symanzik

tions at imaginary

density the construction

He realized

of the Green's functions

man's famous history

a

that func-

or

- might be simpler than the direct construction

distributions.

rigorous meaning,

of Eucli-

[Sy i, 2] who first advocated

times - called Euclidean Green's

functions

con-

as solutions

functions

approach to quantum field theory.

given a formal Lagrangian

[Sc l,

of a field theory,

times, and to study their properties

of certain differential

Wightman

Fradkin

1], Wick [Wi i] and most forcefully Schwinger

2] proposed to consider the Green's

Schwinger

[Dy i].

of

It is only in the Euclidean context that Feyn-

integral

[Fe i, 27 can be given a mathematically

see e.g. ref.

[Ne 4] for the case of boson theories

and los i, 2], [Oz i] for theories

involving bosons and fermions.

In

his work, Symanzik developed many of the concepts and ideas which these days belong to the basic theory.

instruments

More recently Nelson's mathematically

Euclidean bose field theories reconstruction scheme

of the Euclidean

approach

rigorous

in terms of Markov fields

theorem and Guerra's

first application

to field

formulation

of

[Ne 1-47, his

of Nelson's

[Gu I] paved the way for all the various recent applications

of Euclidean methods in constructive field theory. Both in Nelson's and in Symanzik's work and also in most other work on Euclidean essential role.

field theory, Euclidean

field operators play an

But the existence of such Euclidean

is not automatically

field operators

guaranteed by what one usually knows or assumes

about the relativistic

theories.

It only follows

from the so-called

Nelson-Symanzik positivity condition on the Schwinger functions, which is expected to hold in theories describing scalar bosons only, but *)Supported

in part by National Science Foundation

Grant GP40354X.

72

not necessarily in cases where fermions are involved. It is therefore natural to try to work with the Schwinger functions only and to study their relations to the relativistic theory° Since the Schwinger functions are not necessarily the vacuum expectations of a

(Euclidean)

field~ one does not have a "Euclidean field

theory" anymore, but only a "Euclidean formulation of

(relativistic:)

field theory'~ It is the purpose of these lectures to define and to study the Schwinger functions within the framework of the Wightman axioms.

The

main task will be to determine under which conditions Schwinger functions are indeed the Euclidean Green's functions of a well defined Wightman theory. The plan of these lectures is as follows:

First we shall briefly

review some of the main definitions and results of axiomatic quantum field theory and give a precise definition of the Schwinger

functions.

Second we will inquire which properties of the Schwlnger functions can be derived as consequences of the Wightman axioms.

Finally we

will show how to reconstruct the Wightman distributions

from Schwinger

functions ~

satisfying the following conditions n (E0) A distribution property (El)

Euclidean covariance

(E2)

Positivity

(E3)

Symmetry

(E4)

Cluster property.

These conditions

(E0) - (E4) are n ecessar Y and sufficient for the

existence of Wightman distributions satisfying all the Wightman axioms.

In detail the connection is as follows. Euclidean

Relativistic

istribution

[iistributi°n~

I!

ovariance

ovariance

/

ositivity

ositivity

|

pectrum

J

[

] + Symmetry

[

] + Cluster

For applications

~

[

] + Locality

[

] + Cluster

in constructive field theory the distribution proper-

ty (E0) does not seem to be appropriate.

We therefore introduce a

73

second distribution property

(E0') and prove that

(E0'),

(El) -

(E4)

are still sufficient for the reconstruction of the Wightman theory. The above chart holds again with all the arrows pointing to the right only.

(E0') is a simple temperedness condition with a restric-

tion on the order of the distributions ~n tions), for large n.

(the Euclidean Green's

func-

It is this second result which I think will turn

out to be useful in constructive field theory; the equivalence result, though more general, seems to be of purely esthetical interest. Remark:

As discovered first by Schrader and Simon, there is a mistake

in the proof of a technical lemma for Euclidean Green's functions

in the original paper on the axioms

(ref. [OS 3], lemma

sequence, the temperedness condition of ref. weak

8.8).

As a con-

[OS 3]seems to be too

to allow for the reconstruction of the Wightman theory. In these lectures we restrict our attention to theories

involving

one neutral scalar field only; the generalization to finitely or countably many fields of general transformation properties requires only minor and mostly obvious modifications, [OS 3, 4].

see Osterwalder-Schrader

Most of the material covered in these lectures is taken

from references

[OS 3, 4].

It is a pleasure to thank Dr. J. Fr6hlich for a careful reading of this manuscript and for many critical remarks.

I would also like

to thank Prof. B. Simon for sending me prior to publication a copy of chapter II of his ZUrich lecture notes [Si i].

It was very useful

during the preparation of these notes. II.

WIGHTMAN DISTRIBUTIONS, WIGHTMAN FUNCTIONS, EUCLIDEAN GREEN'S FUNCTIONS

In this section we state the Wightman axioms in terms of the Wightman distributions and summarize some of the classical results of axiomatic field theory.

For details see the monographs of Jost [Jo 1]

and of Streater and Wightman

[SW I], and references given there.

Let ~(x) be a neutral scalar field obeying all the Wightman axioms. Then its vacuum expectation values or Wightman distributions

have the following properties: Distribution property: tion:

For each n, ~)n

is a tempered distribu-

74

Notation:

x-

( x , , . × n) 6_ ~4~

×~ ~

(x~ ~ ) ~ ~ •

Relativistic

for all where

co variance:

(a, A) A~ +o.

Positivity:

~ =

(Ax,+~, ...

fo 6 ~

vl)n9 ?~, ~ f~,

Ax,+~L)

, ~n &

sequences ~(R4")

is defined by

i,~,,."~s

(C~$,)

(K,~)=

.,>

if

f: f~) f~ (~_)

---

For any n and k = l, .... n - 1 "l,v~, ( x , , - - .

(W3)

of test

, D= I,... N ,

and

Locality:

invariant:

,

For all finite

functions,

where

~

/

For each n, ~)~ is Polncare

~k,X~., , . . .

X t)

~ -k)~(X,,...

X,.+, ' Y ' I , . ' ' ' '

X.. )

(Xk - ×~+, )z < O.

Clustex Property: x = Xl,

.... x k, Y = Yl,

Spectral

condition:

we conclude that

For any spacelike

is the Fourier

Remark:

the translation

where ~ = (~.....~..,)

transform

8~

=

Equations

we can reconstruct

×~

.

forward

(W5) have to be interpreted

Then

light

in distri-

sense.

the physical

This

~=x~÷,-

of ~V,_, , ~+ is the closed

(WI) -

the field operators .

and

q~ o _ ~q~

From a given set of Wightman

of ~: in ~

of the ~ n '~ ~4(n-i). ~/n~ & ~'(-), such

- (~v) -{("-') f e * p [-;Zff,~.] W._, ( ~ ) ~"~"I E

butional

_CL~,

invariance

that there exist distributions

;,., (~)

cone, and

.... Yn-k,

Using

~Q,(~_) = ~/~-i(~)

where

a and k = i, .... n - i,

distributions

Hilbert

~(x)

space~,

and a unitary

is the Wightman

satisfying

(W0) -

(W5)

the vacuum vector representation

reconstruction

theorem;

~C~,A)

see Wight-

man [Wg 1]. Let us now introduce by

~n

(xj,~)

where ~ . ~,,...~_,) tempered

-

vector

~(x,) .. and

distribution

~

valued distributions

~(~,)-0-

defined

,

. ~k., " ~

with values

formally

in ~-.

for k =l . . . . . n-1. ~ The scalar product

is a of two

75

vectors ~.(f)and ~.(~)for f e ~ ( ~ " )

, @~(~4m)

is

The Fourier transform of ~n is defined as usual by ~(~)=~.(f~ where is the Fourier transform of ~ e

~(~n).

Theorem i: For n = 1,2, "'~ the support of ~ ) i- s

contained in --n V+

For a proof we just take a test function $~ ~(R ~") with ~(~)= O

for

write down the norn% IIA.~)~ in terms of the distribution V~'~_ I

~e~",

and show that it is zero, using the spectrum condition. We can now study the Laplace transform ~.(~,~ ) of ~t~(q ) .

The

Laplace transforms of tempered distributions have been extensively studied, e.g. in references [Sz i, Li I, Vl i, SW i].

The particular

theorem we quote below is due to Bros, Epstein and Glaser [BEG]. Theorem 2:

Let ~(~) be a distribution in {~(~4.) with support i n ~ "

Then there exists an analytic function ~ 6 ~ ) ,

tube ~"~ {~1 ~ -

×+~ ~ ~"

_

,

, s o , h a t

a) the Fourier transform ~ of F

for all

~

~tO ~")

,

~ e

V2

regular in the forward

is the boundary value of ~

:

,

b) For some positive constants c, u, ~ and for all R e

~,

_

The function ~ ( ~ ) is called the Laplace transform of ~ and can be written

(heuristically) as

Theorem 2 can be immediately generalized to hold for distributions F with values in some Banach space.

Applied to our vector valued

distributions ~ . it shows that the Laplace transform

is an analytic vector valued function, regular in ~-; , with boundary value

~_~ (x,~). Applying Theorem 2 to the distributions

~A/n(q% we obtain the

Wightman functions ~/.C~), which are analytic functions for ~ e J ~ Obviously we have

76 translation

invariance

does not depend Lense

o f / L implies

on z.

The following

3 (see [Jo I], p. 74):

Using relativistic

lemma

, ( ~ _~,)~ ~

valued analytic

n °

extension

some

set of complex Lorentz

- into the extended ,

transformations

in ~ [ ~" - -

distributions

J ~_ ~ ~ +,ex£ }

~),(x)

where

with determinant

valued analytic

extension

of

~)~(~)

~ n (~-)"

group

It is invariant

~ } .

(of non-coinciding

Definition: called

_<~

values

the

a single

{--~ I

complex Zl,

, Points

again

Lorentz

.... z n.

o

in ~ n a r e

, called Euclidean

arguments).

The restriction

the n-point

, are

~ "ex%,peP~ --

of the arguments

~.

The func-

We denote this extension

contains

z k # z6 for all 1 < k < ~ points

tube

~÷,- ~

under the inhomogeneous

;h~{~) and under permutations The set ~t,p~r.~

I.

(W3), we obtain

into the set

( ~ B , - ~ c . ) ) ~ E ~ fOr some permutation by

locality

func-

denotes

and have as boundary

Finally using

theo-

of the W i g h t m a n

tions ~>~ (~__) , 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 ~ <

THEORY

I.

of sequences f - (~,f,~... )

82

where ~ 6 ~

, fn ~

some finite N.

For

By

is a positive semi-definite inner product.

(E2),

<, >

= {~

I _~ ~ ~ <

~(~2~ )

Hilbert space. we obtain

~ llf ~=<~_,f>=0~.

form

&-

~

, ~

We also define

Then the completion of

~

~

e

for

(×,,~) = ~

~,~ 6 ~< .

~(~<"), some n.

= 0 for k ~ n, we write ~

#< /v~

the natural injection of ~_<

,~(9)) = < ~ , ~ > Let

Set

, which will turn out to be the physical

Denoting by (~(~

~((i, 0, 0 ...)).

= 0 for n > N ,

~ , @ & ~
defines a Hilbert space ~

~,

, for i ~ n ~ N, a n d S ,

(x)

a vector valued distribution in

• where

into

We set ~ =

Then if ~

is of the

~(~) = ~ , ( ~ ) = ~ n ( x ~ ~ ( x > ~ . ~

~/(~4- )

= x~+ I - ×~ By

, and % ~

is

(E2) positivity, the

scalar product of two such vectors is again given by eq.

(4) in

Leman 5. In the following we will analytically extend the "functions" %~

(x,j~)

in the zero components

arguments, after smearing for

~ ~ ~(~BD),

(12)

~ ~

~°~°I~

Theorem Ii:

of their

them in the remaining variables.

We define

q~ (~B~),

) =

Let ~ + - {£ I~ee > O

x,°, ~,o, . . ~o~_i

}

j~;

(x,~)

and

~(~. ~

) ~,~

,

and

~ +k =(~+ ) ~

For fixed gh, the distributions

~n+~_z~}9

h)

are

restrictions to the product of positive real half axes of functions 5~+~_i(~°J~h) functions

, analytic in

lJ~(~°,~°I 9)

Furthermore

~+m~(~

o~ n + m -

I.

analytic in

I~h)

for some Schwartz norms

--+

i Is

There are vector ~

satisfies for

valued

, such that

_~°e

~+

and some constants a, b, c depending

83

Remarks:

(i) By standard arguments,

is the Laplace ~+

(15) implies that

transform of some distribution

, hence

~,+m_i(~ ) is of the form

is the Fourier transform of the difference bution.

in

of the

Positivity

I • J n/o -continuity

(W2) follows easily

(2) The proof of T h e o r e m Schrader

from

distri-

I this fact was a direct

assumption

for the

of the analytic c o n t i n u a t i o n

Constructing" the H a m i ! t o n i a n

( e ~ f_) , ,

<,_~ _

property of

f__>l

×.°

~,

t L

)

, for

f ,

By the Schwarz

9 e ~_4 , ~ - ~ < _ f , ~ >

is

,

c(,+t ~)

~

for t>/O ,s>/O

Furthermore

(17

f,,(,,,_° ~ +,,,,,

and for some c, m, depending on ~

I
(16)

of~,was

[GI 1,2].

q-t : ~--<--~ -~< by

By the distribution continuous

/s

is due to O s t e r w a l d e r -

Proof of T h e o r e m

For t >/ 0 we define

5~

(13).

and independently by Glaser

A.

in

~/n (q_)

also found s i m u l t a n e o u s l y ii:

with support

(Ii), and again

ii as presented below

lOS 4]. The construction

~'

variable W i g h t m a n

Note that in the proof of proposition

consequence

~m~m_,(~°lg~1)

, ~t~

" ~e÷s

, and

=<%f

inequality

I<_.!,t_f>l-=

for any ~ ~ ~_4

with

II ~ Jl -- I

Iterating

(17) and substituting

(16) we obtain

i
f>l As

~t

we may define a semigroup it by continuity and

~

one parameter

Tt = ~ % H

in ~

to all of ~@_ .

( T t ~ , ~') - ( ~ , % ~ ' )

continuous and

f

leaves the set JV ~ of norm zero vectors

, where H

by T ~ ( ~ )

= ~(~tf]

As for all

~- , ~'~-~£-

invariant,

and extend

,

llTt~-I{ <~ {I~

, we conclude that T ± , %>/o , is a weakly semigroup of selfadjoint

contractions

is a selfadjoint positive

operator.

on ~

is

the Hamiltonian. B.

The A n a l y t i q C o n t i n u a t i o n

We will use the h o l o m o r p h i c the analytic c o n t i n u a t i o n functions.

semigroup ~

, q~z~ > 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

<~Tz}, which continues

theorem

tions of a function

= ~w ( ~ )

if

e~~k

,

in w

and for I~nw-~0,

(see Epstein

[Ep i])

, analytic

in

~ _ l ( ~ c _ ,W , tt~ )

all the

-~~I

analytic

~=

functions

~(w,,...ww )

E q u i v a l e n t l y we may say that the 5 ~ (~,,...~)

,where

in the u-variables,

tells us now that there is a function {v_/l ~

More precisely,

are analytic

I~-~nwJ < m/p_ and distributions

con-

5 k and we are in the situation

of an edge of the wedge theorem for flat tubes. we define

if properly

C~ ;~,~')

are all restric-

, k = n + m - I, which

is

in

C(,,)

We c l a i m

that

the

functions

products

of two vectors

in~

£k(~') can be interpreted

Let us define

as t h e

scalar

~ (') C- ([~+

by

4----

(18~

Z) ~'~ =

Lemma 12: analytic

{ ~, ~_) I ~ > 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<'('~, the

85

(~, ~ )

point

,

with ~

the first variable

~

small enough• so that

P ~

let ~ ( ~ , ~

)

is convergent for be a ~

~¢~-~>

(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

~<~)

valued

functions

such that e q u a t i o n

~J ~ { u ) = ~ ÷ x ~ - I N

contains

as

we end up w i t h an a n a l y t i c

C (~)

ii follows

For all N = 3, 4,

~c~)

replacing

N times,

to a d o m a i n

The first p a r t of T h e o r e m

Lemma

~

this p r o c e d u r e

(19) holds.

if we can show that V We w i l l p r o v e

..., n = i~ 2,

CC~)k ~ ~

a stronger

result:

..o

the set

n

(B)

C 6 ~) --n contains

the sets

where

~

= max

~)

Proof:

By c o n s t r u c t i o n

the t r a n s f o r m a t i o n bases

, and

the regions

~-

w~

) =-_~(~) ~ and

We define

for

--k ~cN) b e c o m e c(~)

and

e '% )

6- ~ k

--

[ V I v~ = I m w ;

, I_~ ~_~ i~ ,

(e

,...

= c'*' k and

definitions

of

(23)

d"' -k

C (N) k and

are subsets

of I-w/2_ ', T/~

~(~) ~ we find

(see

(21),

after

to b e the

{ ~ ( v = (-~-' V, V"

c ~N, = c o n v e x h u l l of

(24)

}

Equations

(U)

~" I (-z,o,,,)

~(M)

}

in.

(22)

F r o m the

)

for

(O,v") ~ d ~I

(0,v') 6 ~(u)

vc(-~/~%)~

c~ w)

,

(24) t o g e t h e r w i t h c~(°) = (0, ... 0) ~ ) d k for all k and N inductively. Observe

c z(~)

d~u)

and

d ~~") are convex.

in

For a p r o o f of L e m m a r = 1,2,...,

- ~~

~(*.0)

--

-~--

(

..., such that

( 0 , O, . . .

0 for ~

m

if

with

define

that all 6 c (~ k

(v I, ... v k) corners

the

(zv~.

+_vk)

d ~¢") resp. ) .

13 we first c o n s t r u c t

N = i, 2,

~V(~,S,N)

W e choose

Moreover

), then the w h o l e h y p e r r e c t a n g l e

is also c o n t a i n e d

m~-~-k

}

(23),

the sets

(25)

tubes

-k

...e W'-')

" N o t e that

(or

N.

of these tubes: Ck

sets

~>

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

- ~<

,O,...O,k(I,L),H(Z,L)

it a l s o c o n t a i n s

....

hCs,t))

the p o i n t

(-IE~(~,L)-,- ~,(,-,,L)], .... ~[~(~,,I~-~(,,LI1, -~E~,<',l.~+~], %--" 0 , iE~(',') +'~3, ~ This means

\

that with

k(~,O)(26)

-i

O

b(I)L+I)

)

= ~[h(I,L

h(%bl)=

~'=

1,2.,..

)

~[H(%L)

+

H(~-I)L~ 1

,

~-Z,3,... (L+()

the .points w ( t , s , L + I), d e f i n e d b y take

(26) as i n d u c t i v e

shows

that the s o l u t i o n

w h e r e we d e f i n e Because

~

just use

C.

of

h(~,N)

v.k

~_,

, which proves

the d e t a i l e d

Sk ~

>

arguments

part 5

I~1

=

~/~E,-~-"~

~

(A) of L e m m a ( ~_ )

13.

(-~-)

For part

(~ /~ (B) we

~, ( ~ , g_~ ) .

~E ~

the s p a c e

variables

In p a r t i c u l a r

completely,

derivation

for

~

(15).

All

we s h a l l c o n -

w h i c h of c o u r s e w e

of i n e q u a l i t y

(E0') to c o n c l u d e

s u c h that

and define

lead to t h e e s t i m a t e

w i l l b e in lOS 4].

In a f i r s t s t e p we u s e

6>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
We only sketch

Now l e t

of

can b e c h o s e n a r b i t r a r i l y

~°(,..-,..) are in

{.~J~)=

definition

(25), are a g a i n

(15).

t h a t t h e r e are

> 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-

(~
and

(32)

I S.,c=. ( ~ ) {

-~

C.

for m a x

I'~"I~ ~'~-I - ~'cJ('l~÷;t'2_. ) >/ T/+

finally obtain

for s o m e c n

<33)

c_ )l

I

N o w we c o m b i n e

(33) w i t h

P--

=

(.Z {-' ) ~"

c.

(31) and

Combining

(28) a n d o b t a i n b y c h o o s i n g

~:

:

neglect

the s p a c e

By s t a n d a r d

butions

w~(~)

(32) w e

~ ~,',,, 1~.~,. &.

I

~for some c o n s t a n t s

,

~.1~ [~,+ ~-.)(,+.-"~]~'I .L-(,÷,~.,~]:-L[)(~+~-')] ~° ~ ('+ I~-I) = (' +("~z" ~'4') +

a, b, c, d e p e n d i n g

on n.

This p r o v e s

(15), if we

variables. arguments,

in

theorem

ii implies

~,(~4.) with supp W.

W e d e f i n e the W i g h t m a n

and we h a v e to v e r i f y

distributions

(W0) -

(W5).

by.

~

that there

~÷

are d i s t r i -

, such that

90 Verification of the Wi~htman Axioms (W0) The distribution property

(E0) follows from (34) and

(Wl) Invariance of ~), under translations and space rotations follows immediately from (34) and the corresponding properties of S n .

To

prove invariance under Lorentz rotations we need only show that X0j ~ ( q ) - 0

, where ~ o j = ~ ( q l ~ i

2, 3

are

the infinitesimal generators of Lorentz rotations, see [Ne 2].

But

~-I

~

~k ~k

+ ~

)

)

, j = 1

' the infinitesmal generators of

rotations in Euclidean space, we conclude from (El) that ~ [~exp[-i~-(~:

÷;~k~

Xoj ~ n ( _ ~ ) ~

Fourier and Laplace transforms is zero,

"

0 = ~/oj~,(_~)=

Because the kernel of Xoj ~/n(~)=O, j = i, 2, 3.

(W2) Positivity follows simply from Lemma 12 and the fact that the distribution

~,(~) can be obtained as boundary value of the analytic

continuation of the Schwinger functions (W5) Lorentz covariance of A~

~.

~/. (@) and supp~/n c_ ~ 4+n _

imply that

~ n

(W4) The cluster property <~,n~(~-,~')

(E4) implies that lim (~, U~,~) ~ l )

for any a = (0, ~) and arbitrary vectors ~ , ~ ' ~

This immediately gives the cluster property

= ~

.

(W4) for the Wightman

distributions. (W3) Locality follows from the symmetry of the Schwinger functions and a theorem in Jost [Jo i], p. 83~ This ends the proof of proposition II. Remark:

I) It has become clear from our proof, that condition

(E0')

was picked somewhat at random; other, similar conditions would do the same job.

Work in progress even suggests, that (E0') can be replaced

by a similar condition on the Schwinger functions ~ n Sn, see [OS 4].

rather than on

This would probably be the most convenient form for

applications. 2) For the Schwinger functions of P(~)2 models with small coupling constant, all the axioms

(EO') , (El) - (E4) follow easily from the cluster

expansion estimates of Glimm, Jaffe and Spencer [GJS 1 ~. In particular (EO') is a consequence of corollary 1.1.9 in [GJS l~, which implies that

91

for #

having support in the product of unit lattice squares

~, x. x&~ ~ ~i

(We use the notation of [GJS I].) For, let ~&- ~( ~Z(,-LL ~ )

and let 7 i ( ~ }

be the characteristic

function of

~j, ×-. x Zij,_,

then

I s~_,(~) Furthermore

for any

I ~-

Z_ /a~_,Ch~(, i) I g

9 e

-f(Rz), s u p p g 6_ Zi0 , ;g(×)d~x = I , f =xu÷-x~,

(36)

-

Observe, that g(x.)~i(~) ~£(~_ ~

x~ ~ ~ ,

is different from zero only if

This implies that for fixed j, not more than ~

of i give non-vanishing obtain using

-

contributions

to ~-

in (36)

values

Hence we

(35)

In the last step we have used the fact that the substitution has Jacobean i.

Hence for

and this gives immediately

x - ~ (x,,~)

~ = B. × ~z x... h,_,

(E0').

References [BEG]

BROS, J., EPSTEIN, H., GLASER, V., Commun. Math. Phys. 6, 77

[Dy i] DYSON, F J., Phys. Rev. 75, 1736

(1967).

(1949).

[Ep i] EPSTEIN, H., in: Axiomatic Field Theory, Brandeis Lectures 1965, Volume I; ed. M. Chretien" and S. Deser, Gordon and Breach, New York, 1966. [Fe i] FEYNMAN, R. P., Rev. Mod. Phys. 20, 367 [Fe 2] FEYNMAN,

R. P., Phys. Rev. 76, 749

(1948).

(1949).

[Fr i] FRADKIN, E. S., Dokl. Akad. Nauk. SSSR, 125, 311 transl, in Sov. Phys. Dokl. 4, 347 (1959).

(1959); engl.

92 [GIi] [GI 2~ [GJS]

GLASER, V., On the equivalence of the Euclidean and Wightman Formulation of Field Theory, CERN Preprint. GLASER, V., Problems of Theoretical Physics, 1969, p. 69.

"Nauka", Moscow,

GLIMM, J., JAFFE, A., SPENCER, T., The Wightman Axioms and particle Structure in the P(~)2 Quantum Field Model. Preprint.

[Gu i] GUERRA, F., Phys. Rev. Letto 28, 1213

(1972).

[JO i~ JOST, R., The General Theory of Quantized Fields, Amer. Math. Soc. Publ., Providence, R.I., 1965. [Li i~ LIONS, J. L., J. Analyse Math., ~, 369 (1952). [Na i] NAKANO, T., Progr. Theor. Phys. 21, 241 (1959). [Ne 17 NELSON, E., Quantum Fields and Markoff Fields, Amer. Math. Soc. Summer Institute on PDE, held at Berkeley, 1971. [Ne 2~ NELSON, E., J. Funct. Anal. 12, 97 (1973). [Ne 3] NELSON, E., J. Funct. Anal. 12, 211 [Ne 4] NELSON, E., Erice Lecture Notes [0S I~ OSTERWALDER, (1972).

K., and SCHRADER,

(1973).

(1973). R., Phys. Rev. Lett. 29, 1423,

lOS 2~ OSTERWALDER, K., and SCHRADER, R., Euclidean Fermi Fields and a Feynman-Kac Formula for Boson-Fermion Models, to appear in Helv. Phys. Acta. lOS 37 OSTERWALDER, (1973).

K., and SCHRADER, R., Commun. Math. Phys. 31, 83

lOS 4].OSTERWALDER, K., and SCHRADER, R., Axioms for Euclidean Green's Functions II, in preparation. [Oz 11 oZKAYNAK, H., Euclidean Fields for Arbitrary Spin, to appear. [Sc i~ SCHWINGER, J., Proc. Natl. Acad. Sci. U.S. 44, 956

(1958).

[Sc 2~ SCHWINGER, J., Phys. Rev. 115, 721 (1959). [Sz 11 SCHWARTZ, L., Medd. Lunds. Univ. Math. Semin. 196 (1952).

(Supplementband),

[Si 17 SIMON, B., Euclidean Formulation of (Relativistic) Quantum Field Theory, ZUrich Lectures 1973, to appear in: Princeton Series in Physics. [SW l~ STREATER, R. F., WIGHTMAN, A. S., PCT, Spin and Statistics and All That, New York: Benjamin, 1964. [Sy I] SYMANZIK, K., J. Math. Phys. !, 510 (1966). [Sy 2] SYMANZIK, K., Euclidean Quantum Field Theory, in: Proc. of the International School of Physics "ENRICO FERMI", varenna Course XLV, ed. R. Jost, New York: Academic Press, 1969. ~VI i] VLADIMIROV, V. S., Methods of the Theory of Functions of Several Complex Variables, Cambridge and London: MIT Press 1966.

93 [Wi i] WICK, G. C., Phys. Rev. 96, 1124 (1954). [Wg 17 WIGHTMAN, A. S., Phys. Rev. !0 l, 860 (1956).

PROBABILITY TKEORY AND EUCLIDEAN FIELD THEORY Edward Nelson Department of Mathematics, Princeton University

I.

Let on

H.

H

be a real Hilbert space, and let

That is, $

dexed by

SECOND QUANTIZATION

H

with mean

is a random variable on f

in

be the unit Gaussian process

is the unique (up to equivalence) real Gaussian process in0

and covariance given by the inner product on

Thus t~ere is a probability space

with

~

H, and if

Baire function on ~n

(~,S,~)

such that for each

(~,~,~), the ~-algebra fl,...,fn

~

are orthonormal in

H

and

F

H, $(f) $(f)

is a bounded

then

is immaterial.

tion is to choose an orthonormalbasis Cartesian product, indexed by

2

~ n F(x)e 2 dx .

ff2 F((P(fl)"'''(P(fn )) d~ = ~

(Z,~,W)

in

is generated by the

x

The choice of

f

~.

Perhaps the simplest concrete construc-. [en}

n, of copies of

of

and let (~,~,~) be the ± x2/2 (~, B, (27D-2e - / ~x), where H

is the c-algebra of Borel sets° If G

is a positive or integrab!e random variable on

its expectation by

For

(G,~,~), we denote

E G , so that

the unit Gaussian process

$

we

have the formulas

(1)

E $(fl)-.-~(f2n+l ) = 0 ,

(2)

E $(fl)..-$(f2n) = Z ''"

,

95

where the sum is over all pairings of

il < "'" < in;

and

(il,Jl,...,in,Jn)

l,...,2n

il < Jl' "''' in < Jn '

is a permutation of

from the fact that the symmetry

-- that is,

1,...,2n.

The equation (i) follows

~(f) --> ~(-f) = -@(f)

We need only prove (2) for the case

fl . . . . .

preserves expectations.

f2n = f' by virtue of the general

polarization identity

m

m

1 Z (-1)m-r (Zil + ... + Zir )m ~ zi : mT.' Z i=l r=l iI <... < i r

(3)

which is valid for commuting indeterminates unit vector.

z..

l

We

may assume that

f

is

a

Then (2) reduces to

(4)

E ~(f)2n = ( 2 n - l ) ( 2 n - 3 ) ' " 3 " l

,

since the right hand side of (4) is the number of pairings of

1,...,2n.

But

(4) is Just the one dimensional integral

co

2

X

i x2n; -~ dx = (2n- l)(2n- 3)'''3"i • f ~2Tf -o~

(5)

--=

We denote notation

r(H)

LP(~,S,~), for for

~L2(H)-~ Let

1 _< p _< ~, simply by

LP(H), and we also use the

I~(H)
all elements of the form

$(fl)"-$(fm)

orthogonal complement of

P(H)<(n_l)

with in

m < n, and let

P(~)
For

r(H)n

fl,...,fn

I~(H)~ of

be the in

H, we

define

to be the orthogonal projection of

$(fl)"'$(fn) i

dimensional, so that

into

r(H) n.

(If

is one

2

p(H) = ~2(~, B, (21r)-ge-X /2 d.x), then P(~)n

dimensional space spanned by the n'th Hermite polynomial, and

n

:x :

is the one is the n'th

96

Hermite polynomial normalized so that the leading coefficient is i.)

We have

the formula

(6)

<:~(gl)--'@(gn):,:~(fl)"'~(fn): > = Z '°" ,

where the sum is over all permutations

v

of

l,...,n.

Since the left hand

side of (6) is symmetric, it suffices to prove (6) for

gl = fl' " " ' gn = fn'

and by (3) it suffices to consider the case

fn = f"

that

f

is a unit vector.

fl . . . . .

We may assume

Then (6) becomes

<:$(f)n:,:$(f)n:> = n~ ,

(7) or

2 X ---

co

(8)

,,,~L

j.

(:xn:)2e 2 &x = n:

which is the well-known normalization of Hermite polynomials. Let

HI ~

be the complexification of

space symmetric tensor product of

~i

H, and let

with itself.

H

be the n-fold Hilbert

We give

Hn

the inner

product such that

(9)

<sym gl

where

Sym

® "'" ® gn' Sym fl ® °'" ® fn > = 7. ''' 7r

is the symmetrization operator

Sym fl ® . . .

It is c l e a r ,

,

® fn = 5

Z fw'(1) ® " ' " ® f ~ ( n )

'

b y (6) and (9), that

:~(fl)" "~(fn):~--> 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 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

on a probability space

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).

~

on a probability space

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)

on a probability space

(~,~,~)

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

<~.

The mapping

given by

Sn(fl,...,fn) = E $(fl).-.~(fn) is continuous.

By The Schwartz kernel theorem, there is a tempered distribution

S n

on Edn

such that

Sn(fl,...,fn) = S n ( f l ® ... ® f n ) •

Theorem 2.

Let

q~ be a Euclidean field satisfying assumption (B). Then

the sequence of distributions

Sn

satisfies the axioms

E0, El, E2, and E3 of

104

0sterwalder and Schrader [ii]. Proof.

We use the notation of

[l1].

Axiom E0 asserts that

SO = i

is true since the empty product is by convention l, and

E1 = l) and that

is in

~,(~dn).

~(~dn)

is in

~,(Edn)

(which S n

Since

~(~dn)

is a closed subspace of

it defines, by restriction, an element of

and since

Sn

~,(~dn).

Axiom E1 asserts Euclidean invariance:

Sn(f ) = Sn(f(a,R)) , R e S0(d) ,

We need only show this for

f

agE d , f E ~dn)

o

a tensor product, f = fl ® "'" ® fn' in which case

it reduces to

E ~(fl).'.~(f) = E T(~)(~(fl)'"~(fn))

for

q = (a,R), which is true by Euclidean covariance. Axiom E2 we prefer to call positive definiteness (to distinguish it from

the positivity condition of Symanzik [15] which results from the fact that the expectation of a positive random variable is positive).

(5)

Z

Sn+m(E~n × fm ) _> 0 ,

It asserts that

f e S_+ .

II , m

It is enough to prove (5) for the ease that each

f

fn = fnl ® "'" ® f n n "

means that

TO say that

f

is in

- -

S --+

is a tensor product,

fn(Xl .... ,xn) = fnl(Xl)'"fnn(Xn) = 0 unless

o <x I

d

<

. . .

<x n

d

Let

-- z ~(fnl) " o ' ~ ( ~ ) n

this is a finite sum.

In our notation, (5) becomes

105

(6) Let

~[(w(p)G)~] > 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

(-A + m 2 ) - l h >

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.

be a Markoff field over

(£,~,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 Markoff field over

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
Hence

fill <~

dk i K2 + m 2 k 2 + m 2 = ~-~ log m2 = 0(log K) .

is well-defined.

For

g

in

~I D ~L~(E2), let

:~Kn :(g) : f :~K (x)n:g(x) ~x .

If

K>K

(7) where

then we claim that

ll:@kn:(g) - :$Kn:(g)ll~ =

-

I13

GK(x ) =

i

1

(2Tr) 2 This follows from (6,1). as

K - > ~.

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
by

: n (g) = f g ( x ) : ~ ( x ) n :

is open in ~2

and

g

~2(fl,~,~)

d.x.

is in

~i A ~ 2 )

with

then

(8)

: n:(g) e O(A) •

To see this, let

MK(A)

be the closed linear subspace of

the

x

A

~k(x)

with

in

subspace spanned by the k ~ K.

Let

OK(A ). Let

OK(A) ~(A)

and

sk(f)

k ~ K. with

f

~2(fl,~,~)

This is the same as the closed linear in

L1 D ~L~(~2), supp f C A ,

be the a-algebra generated by

MK(A).

Then

be the closed linear subspace spanned by the

in ~l A ~=(~2), supp f C A .

spanned by

and

:@Kn:(g) @(f)

is in

with

f

Then

Q ~K(A)

= ~(A)

and consequently

OK(A) = O(A) Since

.

:~Kn:(g) __> : 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 <~

such

p,

n (161

II:p(¢ : (g) - :P(~KI : (g)11p_< (p - l) 2C~-~

Now suppose that is even and :P($):

a

n

> 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

<

llIhlll (~£ + I)

and a suitable norm,

Ill

(i)

III. Here

~£

is defined by:

H~ = H 0 + /~/2:P(~(x)):dx

(2)

-4/2 E(A) = inf spec (A) = A-

(3a)

E(A)

(3b)

Shortly after the appearance of GJ, Guerra, Rosen, and Simon (1972) [henceforth GRS] provided an abbreviated proof of bounds of the form (i).

The GRS

bounds were weaker than the GJ bounds in the types of functions, h , allowed and in the norm,

II]

GRS do not.

llI, used.

In particular, GJ allow

HI

III to be the

eI

norm and

For the original applications, this distinction did not matter but

recently Fr6~lich (1973) exploited the Ll-bound to prove the existence of equal time VEV's in the infinite volume limit.

One of our goals is the extension of the

GRS proof to cover these Ll-bounds. It is possible to merely modify one step in the GRS proof.

However, we wish

to rephrase the GRS proof in a way that we think makes the mechanism of proof more transparent.

To explain our point, we recall the GRS proof: one rewrites the bound

(i) as a set of bounds on matrix elements of the semigroup then uses Nelson's symmetry.

exp[-t(H£±~(h)]

and

In this new form, one bounds the matrix element as

tlle norm of an operator times the product of the norms of two vectors. symmetry is then applied to each of the vector norms.

Nelson's

Our improved proof can be

phrased as applying Nelson's symmetry also to the operator norm.

But then we have

exploited Nelson's symmetry twice which suggests that the two uses of the symmetry "cancel" and that somehow the symmetry is not needed. In a narrow sense, this is the case: what we wish to demonstrate is that what is really critical is the Markov propertf for constant space planes which

* A. Sloan Foundation Fellow t Research partially supported by USAFOSR under Contract F44620-71-C-0108 and U S N S F u n d e r Grant GP 39048

126

provides a sort of decoupling of spatial regions.

GRS ~or at least a subset of

theml) did not really understand the Markov property and so used Nelson's symmetry to reduce to the semigroup property in time-llke directions. emphasize the GRS.

#

While we will

bound, our remark applies equally well to the other material in

Of course in a deeper sense, "Nelson's symmetry" is involved as the critical

element in a Euclidean invariant path

integral.

What we will prove (in the third section) is the following result which is a large part of Spencer's Theorem i

Let

Then for any

F

(19731

N ,loc-bounds

[which generalize

(i)]:

be any function of the (time zero) fields smeared in

Z. with

[a,a+l] c

[a,a+l].

[-£/2,£/2]: -i

- F ~ H£

for suitable constants Corollary 2 c

and any

Let f

Cl,C 2

+

c I - c2 E(R 0 +

independent of

£

c 2 F)

and

F .

llfll_l = 4/If(k) 12(k2+m2)-Idk) ½ .

with

supp f c

(4)

Then for a suitable constant

[a,a+l] C [-Z/2,~/2] ~(f) ~ H£ + c( IlfII~ 1 + i)

451

~(f) ~ c'IlfIILl(H £ + i)

(6)

In particular

We prove (5) for Theorem I in the next section. in GJ.

(6) follows from (5)

as

We also have:

Corollary 3 constant

Let

ilfii_ ½

: (]if(k)i2(k2+m2)-½dk)½,,t, :

Then for a suitable

c ~(f)2 $ cllfll ~

(~£+ i)

(5'1

In particular 2 ~(f)2 $ c, IlfllL I (~£+ i)

§2.

(6')

Nelson's Bound Our notation for the free Euclidean field follows Simon (19741; see also

Guerra et el. 41973). is

A-measurable if

{~(f) If~N; {(x,s) Ix e A} and

A= ~ ,

generated by

If F

supp f ~ A }

A ~ R 2 , we say

F', a function of the Euclidean fields,

is measurable with respect to the •

If

AC~

, we use

(resp. {(x,s)Is ~ A}) •

A × R

(resp.

~-field generated by R × A)to denote

Later when we deal with the time zero fields

A-measurable will denote measurable with respect to the {#F(f) If ~ F , supp f C A }

•

Finally

Ja

4resp. ~a )

@-fleld

will denote the

127

isometry of ~7 ~ into 7Z induced by the map

Ja:F + N (resp. ~a )

given by

jaf(X,S) = f(x)~(s-a) (resp. (3af)(x,s) = ~(x-a)f(s)). A basic role is played by: Theorem 4

(Nelson's Bound)

p = 2/1 - exp[-m(b-a)] .

V

is

Proof

be the mass of the free Euclidean field.

Let

to

(resp. II~ W~bl { ~ IIVl[p)

~ IlVllp

~ x [a,b]-measurable

inequality.

m

Then as a map from ~

IIJ~ v Jhll if

Let

(7)

(resp. [a,b] x ~-measurable).

This is just an expression of hypercontractivity and H~ider's The basic idea is Nelson's (1973a) although we have used a result

from the later Nelson (1973b).

For details see Guerra et al. (1973) or Simon

(1974). Proof of Corol!ary 2

By Nelson's bound and the FKN formula:

-E(~0+¢(f))

/~ $ Ilexp(-

e

ds ~(f,s))ll p

= [/ mY 0 exp(-P$(f~x(0,1))] I/p exp(cllf ~ X(0,1)II ~) ~ exp(c'llfll21 )_ From this and (4) we im=nediately conclude (5). Thus for all and supp f c

f with

JJflJ_ I

=

[a,a+l]

± ¢(f) $ K£+ d < d(H Z + i) for some fixed

d 5 1 .

By homogenity,

± +(f) ~ dllfII_l(H £ + i) (6) follows from this. Remark. By simply modifying the above one shows that for any d(e)

e > 0 , there is a

with + ¢(f) ~ I IfI[el(e H£ + d(e))

Proof of Corollary 3

This is similar to that of Corollary 2.

that e

-E (H0-¢ (f) 2)

i .< l lexp(+ /£ ds(J ,(f))2ds)IIp ~< llexp(+ *(f)2)ilp ~<

SO long as

cons t

fll F .< d for d sufficiently small.

We use that fact

I

128

§3. The Proof of Theorem i Theorem i depends on the following result of some independent interest: Theorem 5 tively

Let

VI,V2,V 3

(-~,a],[a,a+l]

reflection of

be functions of the time zero fields which are respecand

[a+l,~)

measurable.

V I (resp. V3) in the point

I0 = 2/1 - exp(-m) .

Let

V l(resp. V 3) be the

x = a (resp. x = a+l).

Let

Then

(8)

-E(H0+VI+V2+V3) $ -1/2 E(H0+VI+~I)- i/2 E(H0+V3+~3) - i/I 0 E(H0+IoV 2) Proof

We need only show that <~0,e-t(H0+Vl+Vo+V3)~ n0> ~ <~0 e-t(H0+vl+~l)~0 >½ ...

By the FKN formula

(9)

and the Markov property:

<e0,e-t(H0+Vl+V2+V3)n0 = / FIF2F 3 d~ 0

/ (JaFl)(JaF2Ja+l)(Ja+iF3)d~0

t with

Fi = exp(- ~ (JsVi)ds) .

On account of Nelson's bound, this last quantity

is bounded by: ) a~ O)

-, 2 ½ (/ F2%0)II10(/ (Ja+iF3) d~ 0)

and by the Markov property again :

-

0

SO (8) follows. Remark that

Among other things, (8) implies the linear lower bound of Glimm-Jaffe (1970) -E. < d£

for

£

large.

For (8) implies that (with V 2 = ~_½ P dx

and

VI+V2+V 3 = /~_½ P dx) - E(2£+I) .< - 1/2 E(2Z) - 1/2 E(2Z) + c = - E(2Z) + const. Proof of Theorem I

Let

V I = /a Z 2:P(#(x)):dx ; V 3 = / ~ I

V 2 = F + /aa+l :P(#(x)):dx .

:P(~(x)):dx

and let

Then, by (8):

-E(R£+F) $ - 1/2 E(H%+2a) - 1/2 E(HZ_2a_I) - I/B 0 E(H0+BoV 2) Now, by bounds of GRS:

so that : -

1 ~ g(H£+2a ) - 1/2 E(H£_2a_I ) .< - E(H£) - S== -

(i0)

129

Moreover, by the convexity of the

-E(A)

in

A :

-E(~0+~0F+~0 -a~a+l :p:) $ _ 1/2 E(K0+2~0F~ - 1/2 E(E0+2~0 ~l:p)

so that (i0) becomes: -E(~+F) which implies

$

- E(H~) - i/2~ 0 E(H0+2u0F) + C

(4).

§4. Fr6~lich's Bounds Fr6~lich (1973) remarked a very convenient form of the ~-bound which provides the neatest form of the bounds needed to complete the proof of Nelso~s convergence theorem for the half-Dirichlet

Schwinger functions

(see Nelson's and Rosen's

lectures).

Since Rosen uses these bounds in his lecture, we sketch their proof:

Theorem 6

(FrO~lich(1973))

for a fixed

P(~)2 model.

k-independent

constants

Let

d~ A

denote the spatially cutoff Markov measure

Then for any bounded region D , there exists cI

and

c2

so that for any

lim f e~(f)d~[_~/2,~/2]x[_t/2,t/2] t-~ so long as

f

in

L2(D), real-valued:

< c I exp(c211flI~)

(ill

D ~ [-~/2,~/2] x ~ . ^

Proof

If

~

is the vacuum for

H£ , the right side of

(ii) is (12)

Without loss of generality,

take

D = [-1/2,1/2] × [-1/2,1/2]

.

Then (12) is

bounded by

½([[ftJ[L2+l)) f_½ =

~< exp(c which is (Ii). Theorem 7

(Fr~lich

(1973))

Let

d~ D

deonote the spatially cutoff half-

Dirichlet theory for a fixed

P(~)2

Then for each bounded region

D , there are constants

all

A ~ D

with

A

bounded and I/ e~(f)d~Dl

model with

f~L2(D)

P(X) = Q(X) - ~X cI

and

c2

with

Q even.

so that for

, complex valued:

2 $ ci exp(c211fll2)

(13)

130 Proof

Clearly

If e #Of)"aulID, A [ ~< f e# CRef~ d ~ y

l e~CRe:~)d,.,~~ / e~(l~fl~d'.,~ ~

/e'~(lfl)d~

. By the first GKS ~ inequality.

.

Let

A~:

y

Then by Nelson's Monotonicity theorem and ~he bound

S

[-£/2,~/2]

~

[-t/2,t/2].

_F.ree :

"< ~A

f e~(Ifl)duhD ~< lim f e~(If[)duHD t-~ [-4/2,£/2]×[-t/2,t/2] -5 lim t-w=

55.

/e~(Ifl)dg[_i/2,Z/2]×[_t/2,t/2] .

(ii) now implies (13) .

The Lower Bound on the Wave Function Renormalization We have reported on one simplification of the paper GRS.

There is a new

bound which helps illuminate another of their results, namely that in(~£,~ 0) - c~

.

Theorem 8

For any vector

~

in

L2(QF,d~ 0)

[Jell2=

with

1 (14)

[l~[ll ~ exp[- (~,N~)] Remarks i.

This bound in a disguised form was first shown me by C. Newman who proved

it by the method GRS use to prove 2.

Since

in(~£,~0) ~ -cZ .

(~,N~) $ ~^(~,HoD) = ~0(~,2(H0+V£)~)

~0(~,(H0+2V£) ~)

~012~£(I) - ~(2)]£ (14) implies the Proof

in(~£,~0) ~ -c~

By the infinitesimal form of hypercontractivity of Gross (1973): - (~,N~)

Since

bound.

l~12d~0

4 - f

I~I 2 i n l ~ l

d~ 0.

(15)

is a probability measure, exp(- / I~I21nl~Id~o ) $ / exp(-inI~l)l~12d~o (16) =

(14) follows from (15) and (16) .

f [fl[ d~ 0

131

Re fe ren ces FROHLICH, J. (1973) : Schwinger Functions and Their Generating Functionals, Univ. of Geneva Preprint, in preparation. GLIMM, J., JAFFE, A. (1970): Acta. Math. 125 203-261. GLIMM, J., JAFFE, A. (1972): J. Math. Phys. 13 1568-1584. GROSS, L. (1973) : Logarithmic Sobolev Inequalities ~ Cornell Preprint. GUERRA, F., ROSEN, L., SIMON, B. (1972): Commun. Math. Phys. 27 10-22. GUERRA, F., ROSEN, L., SIMON, B. (1973): The P(~)2 Euclidean Quantum Field Theory as Classical Statistical Mechanics, Ann. Math., to appear. NELSON, E. (1973a): in Proc. summer Institute of F.D.E., Berkeley ~ 1971. NELSON, E. (1973b): J. Func. Anal. 12, 211-227. SIMON, B. (1974): The P(~)2 Euclidean (Quantum) Field Theory, to appear Princeton Series in Physics. SPENCER, T. (1973: J. Math. Phys., to appear.

THE

PARTICLE

P(~)2 OF

HIGH

STRUCTURE

MODEL

AND

OTHER

TEMPERATURE

PHYSICS

OF

THE

APPLICATIONS

FIELD

A n Overview

l.Z. Survey of Results 1.3.

The Ooldstone Picture of

P (~0)2

1.4.

Field Theory and Statistical Mechanics

1.5. S o m e P r o b l e m s 2.

F r o m Estimates to Physics 2. i. Functional Analysis 2.2. Relevance to Physics

3.

Bound States and Resonances 3. I. Introduction 3.2. F o r m a l Perturbation Theory 3. 3. O n the Absence of Bound States 3.4.

4.

O n the Presence of B o u n d States

Phase Space Localization and Renormalizatlon 4. I. Results for

4 ~3

4.2.

Elementary Expansion Steps

4.3.

Synthesis of the Elementary Steps

References

PART

I:

MODELS

Arthur Jaffe t Harvard University Cambridge, Mass.

Five Years of Models I.I.

WEAKLY C O U P L E D

EXPANSIONS,

QUANTUM

James Glimm C ourant Institute N e w York, N.Y.

I.

OF

T h o m a s Spencer $ Courant Institute N e w York, N.Y.

133

i. F I V E

YEARS

1.1.

OF MODELS

An Overview

C o n s t r u c t i v e q u a n t u m field t h e o r y has m o v e d r a p i d l y in the p a s t few y e a r s . c o n t i n u a t i o n of this p r o g r e s s

A

c l e a r l y r e q u i r e s t h e i n t r o d u c t i o n of n e w f o r m a l i d e a s ,

methods and/or points of view.

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

we w i l l r e v i e w p a s t p r o g r e s s ,

p r e s e n t s o m e n e w r e s u l t s a n d p o i n t out p o s s i b l e d i r e c t i o n s f o r f u t u r e w o r k .

In t h i s

P a r t I. w e e m p h a s i z e t h e b a s i c i d e a s a n d c o n c e p t s , w h i l e i n P a r t II we p r e s e n t i n full d e t a i l s o m e c o r e r e s u l t s f r o m our p r o g r a m . v a c u u m c l u s t e r e x p a n s i o n and e s t i m a t e s F o r t h e two d i m e n s i o n a l this structure

P(~)

we p r e s e n t the

to e s t a b l i s h i t s c o n v e r g e n c e .

model,

is in q u a l i t a t i v e a g r e e m e n t

a fairly detailed structure is known, and with b a s i c i d e a s of p h y s l c s .

Haag-Ruelle axioms (verified, for instance, fields describe discrete mass

In p a r t i c u l a r ,

particles.

f o r s m a l l c o u p l i n g ) , we k n o w t h a t

An i s o m e t r i c

es the interactions between these particles.

From the

scattering operator express-

We k n o w tha t t h e v a c u u m s t a t e i s

u n i q u e f o r c e r t a i n c o u p l i n g c o n s t a n t s (e. g. , s m a l l c o u p l i n g , o r a l a r g e - o r i n s o m e cases, there

a nonzero - external field). are

multiple

w i t h k 2> m 0 interactions,

.

phase

D o b r u s h i n and Minlos have a n n o u n c e d that

solutions

for

even

p(@) + m 02 ~0Z

S y m m e t r y b r e a k i n g p l a y s a k e y r o l e in c u r r e n t t h e o r i e s

of w e a k

h e n c e t h e i n t e r e s t i n t h i s p h e n o m e n o n . T h e r e i s no d i r e c t e x p e r i m e n t a l

e v i d e n c e for or a g a i n s t o c c u r e n c e of b r o k e n s y m m e t r i e s physics,

models

since the i n t e r p a r t i c l e

in e l e m e n t a r y p a r t i c l e

coupling constants cannot be varied experimentally

(in d i s t i n c t i o n to t h e c a s e o f s t a t i s t i c a l m e c h a n i c s w h e r e we c a n , f o r e x a m p l e , off a m a g n e t i c f i e l d ) . symmetries

turn

C o n s e q u e n t l y t h e d e f i n i t i v e a r g u m e n t in f a v o r o f b r o k e n

may come from constructive quantum field theory.

T h e Y u k a w a 2 (Yz)

and 4

formal ideas developed for m o d e l s in g e n e r a l .

models are less highly developed.

~(~)2

Yet m a n y of t h e

m o d e l s a p p e a r to a p p l y t o s u p e r r e n o r m a l i z a b l e

C l e a r l y t h e n , o n e s e t of p r o b l e m s i s to d e v e l o p s t r o n g e r

134

4 techniques, to m a k e these ideas applicable to Y2' 03 and

Y3

W e propose, in

fact, four groups of problems. I. Physical Properties o O n e important direction for future w o r k is to develop further the physics of existing q u a n t u m field models.

T h e particle structure pro-

g r a m , bound states, resonances and scattering present interesting problems. Likewise, the long distance and infrared behavior of our models contains m u c h physics.

The general particle structure p r o g r a m is: W h i c h interaction polynomials

and which coupling constants give rise £o which particles, bound states and resonances?

H o w do the m a s s e s and half lives depend on the coupling constants?

do cross sections behave asymptotically? Section

How

W e discuss these p r o b l e m s further in

I. 5 and Chapter 3.

T h e long distance behavior of our models pertains to the existence of multiple phases, to the existence of a critical point and to the scaling behavior of the m o d e l s 4 W e ask: Does the ~Z m o d e l have a critical point? Does it

at a critical point.

admit scaling properties with a n o m a l o u s dimens{ons? the critical point?

W h a t p a r a m e t e r s describe

W e discuss these questions further below and in Section 1.5.

II. Four D i m e n s i o n s (Renormalizable Models).

A second important direction

is the question of four space-time dimensions, or in other words h o w to deal with renormalizable interactions, since there are no super-renormalizable interactions in four dimensions.

Clearly this is our m o s t challenging goal, to prove the axis-

tence of, for example,

4 ~4

" O u r present m e t h o d s have been tied to superrenormal-

izability (4 - ~ dimensions) and for ~ = 0 n e w ideas are required.

W e ask:

C a n an

understanding of the renormallzation group be an aid to r e m o v i n g the ~ = 0 ultraviolet cutoff?

D o the ideas in the lectures of Syrnanzik y~eld insight into charge

renormalization?

W e discuss these questions further in Section 1.5.

III. Simplification.

Aside f r o m these two m a j o r directions, there is the ques-

tion of simplifying the present methods.

Clearly the m a j o r need for simplification

concerns p r o b l e m s with ultraviolet divergences, and a m a j o r goal of such a p r o g r a m would be to i m p r o v e the techniques and isolate their essential elements in order to m a k e tractable m o r e complicated superrenormalizable models ,such a s Y 3 , or e v e n ~ .

135

IV.

Esthetic Questions.

questions.

F o r example,

~(~)Z m o d e l s ; what

Furthermore, there are esthetic or foundational the

If

.

Schrodlnger representation ~ = ~z(dq)

exists for

is the fermlon representation corresponding to this non-

Gausslanboson measure

on ~' ?

sures in m o d e l s with interaction?

W h a t are the properties of the path space m e a Related are interesting, but purely mathematical

questions motivated by field theory, which w e do not pursue here. In this connection, w e r e m a r k that the drive toward simplicity and elegance is important and also has been quite successful in the P(~)Z model.

However, we

e m p h a s i z e here those m e t h o d s that admit (or appear to admit) generalization to other m o r e singular interactions. First,

T h e reason for this emphasis is two-fold.

w e believe that, in the long run, our ability to handle m o r e singular prob-

l e m s will determine the extent to which the m o d e l p r o g r a m has succeeded. Second,

w e believe that a p r e m a t u r e emphasis on the simplicity and elegance of

the details can divert energy a w a y f r o m central issues, and thereby delay or obstruct progress.

136

I. 2.

To begin,

we r e v i e w t h e s t a t u s of t h e ~04 , Y2 a n d

chronological versus

summary

results

these results

complexity.

In t h i s c h a r t ,

X / m 0Z << 1 , a l s o h o l d f o r

domments:

We give a

complexity

we e n t e r t h e y e a r s

namely the existence

X•((P)2

of a s i n g l e v e c t o r ,

Alternatively,

s t a t e of a f i n i t e v o l u m e t h e o r y , state yields a representation, as required

models with

in which

of t h e b o u n d a r y

conditions,

the

C"" a l g e b r a v a c u u m

refers

Each infinite volume Uniqueness

to vectors

a s a l i m i t of f i n i t e v o l u m e s t a t e s . ~(x)

The infinite volume The latter are deter-

and the boundary conditions.

t h e n t h e r e i s s a i d to b e a u n i q u e p h a s e , Convergent

cluster

expansions

of

in this Hilbert

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

there are multiple phases,

We

a unique ground state

of t h e r e p r e s e n t a t i o n .

in the energy density

i n ~(x)

~ , / m Z << 1 .

and a Hilbert space vacuum vector.

vacuum state is determined

The

invariant under inhomogeneous

we c o n s i d e r

for the Wightman axioms,

the parameters

4 ~2 models.

and its infinite volume limits.

a n d is e q u i v a l e n t to i r r e d u c i b i l i t y

mined by parameters

for various

The Wightman axioms require

Lorentz transformations.

space,

models.

l , p l o t t i n g m o d e l s of i n c r e a s i n g

2 we give details and references

make several

the vacuum,

4

were proved.

quoted for

(vacuum),

in Figure

of i n c r e a s i n g

In F i g u r e results

S u r v e y of Results

If

independent

and otherwise

[G1 5a Sp 1, Z] y i e l d f o r

certain couplings both a unique vacuum and a single phase. In a P(~)2 the vacuum,

theory satisfying the Wightman axioms,

the decomposition

the observables

and recover

theorem

of B r a t t e l i

a unique vacuum.

except for the uniqueness

[ B r 1] a l l o w s u s to d e c o m p o s e

The local perturbation

estimate

[Ol Ia IV] and a result of Streater [St] ensure the spectral condition for the d e c o m posed theory.

In this m a n n e r

. 1 Z 2 I 4 +-Zm0~ - D~

interaction.

that the decomposition In Figure

w e arrive at a W i g h t m a n In the case

is unnecessary

theory for an arbitrary

D ¢ 0, , the L e e - Y a n g t h e o r e m

shows

[ G r Si, Si II].

3, we h a v e d e t a i l s a n d r e f e r e n c e s

4 n e e d s to b e d o n e to b r i n g i t to t h e l e v e l of ~o2 .

for

Yg ' a n d c l e a r l y m u c h w o r k

of

137

Critical Point

3 = 1973

Asymptotic Completeness

0 = 1970

Resonances

9 = 1969

B o u n d States

3

Broken Symmetries

3

A n a l y t i c i t y i n Coupling

3

Perturbation Theory Asymptotic

3

Single P a r t i c l e States

3

M a s s Gap

2

Wightrnan A x i o m s , V -~ C o n v e r g e n c e

2

Euclidean Formulation

1

Wightman Functions

1

Physical Representation

9

Haag-Kastler Axioms

9

E q u a t i o n s of Motion

9

1

Space T i m e C o v a r i a n c e

8

0

H V = H V

8

9

O~H V

5

8

2

Local Observables

4

7

1

t-IV

4

7

8

0

Y2

4 ¢P3

3 ¢P4

4 ~ 1964

L

P(m) 2

Figure

I.

M a i n Results and Y e a r Established

Y3

138

Renormalized Charge, Anomalous Dimensions Renormalizatton Group Critical Points S-Unitary (Asymptotic Completeness) Chapter

Bound States Resonances Multiple Phases (Broken Symmetries)

2 Y e s : ~.>'> m 0 (Dob Min)

No: GI J a S p I

No: G r Si, Sill

R e 1.>0, 0<]l[
A n a l K t i c i t y of the Schwinger Fns.

R e ).> 0 Sp 2

G1 J a Sp 2

O n e Particle States and S -Matrix

M e a s u r e dg_ SchrSdinger Repn. Perturbation A s ymptoti c

3

GI Ja Sp i

New 1.Fr 2 Fr 2

Theory is

Di 3 m>0 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 << 1, -1t~2 <<1 • m0 m0 Also ke (~0)z .

m>0 Sp 2

~=0

Details for @~0 4 + ~1m 0 ~2 2 - ~ 0 ) Z

~0

[ ~1 l a r g ,

139

Bound States Particle I!

Structure .

Schrodmger

representation

Wightman axioms Euclidean Properties

Fields and Feynman-Kac

Os Sch 4

formula

of C u r r e n t s

McB Di 1

Equations of Motion Physical

1

Representation

Exists

Sch Z

Space - Time Covariance

G1 :[a I0

0 < HV = HV

GIZ,3,GI Ia 9, II

Figure 3. Details for YZ

W e conjecture that the Euclidean fields given by Osterwalder and Schrader [Os Sch 4], and by O z k a y n a k [Oz] for higher spin, can be used as a starting point for a cluster expansion, yielding for small coupling: the V -. co limit, the W i g h t m a n axioms, a unique vacuum,

and single particle states.

W e also hope that Euclidean

gauge fields will lead to an understanding of function space integrals for higher spin.

In particular, w e mention as problems the factorlzation of the integration

over the s y m m e t r y

(gauge) group, and the renormalization s c h e m e of F a d d e e v and

Popov [Fa Po, Sa St]. The ~43 interaction is at an even m o r e primitive stage.

The 4

renormallzed

Hamiltonian is bounded f r o m below [GI Ja 8]. Similar estimates [F'e Z] s h o w that in a finite space time volume, approximate Schwlnger functions have see Chapter 4.

x ~ co limits,

4 W e hope that the W i g h t m a n axioms will soon be verified for ~3

There are several recent developments which do not fit into the above description and w e mention in particular: Federbush's study of the generalized Y u k a w a z model,

P(~) + ~@O(~)

, see [Fed]

the w o r k of Albeverio and H o e g h - K r o h n on

bounded nonpolynomial interactions [ A H K 1- 4], and recent w o r k on the polaron m o d e l by Gross[Grol] and by Fr~hlich [Fr I]. F u r t h e r m o r e Fr~hlich's recent

140 r e s u l t s on ~(~)Z m o d e l s [ F r Z] yield the M a r k o v p r o p e r t y ,

DLR e q u a t i o n s and

cyclicity of the v a c u u m (for time zero fields)when ~. << m 0z or I~tl>> m z , ).. T h e r e are three m a i n unifyingtechn[ques, yielding the results charted above:

I. Euclidean Formulation.

T h e use of imaginary time in constructive q u a n t u m

field theory dates to Syrnanzlk's Euclidean field formulation and to Nelson's posi4 tivity proof for the (~2,V

Hamiltonian.

This approach was used in m o s t of the key

4 technical estimates of the ~(~0)z and 7)3 boson models, and in proving finite propagation speed for the

YZ

model.

T h e covariant formulation in t e r m s of Euclidean

fields or Green's functions is important as a conceptual advance yielding Euclidean axioms, the relativistic F e y n m a n - K a c

formula and the M a r k o v property [Nel 4, Os

Sch 3, Sy I, Nel 3, Fel I]. T h e covariant formulation is important as a technical advance in proving estimates [Gu I,GI Ja 8], in simplifying estimates [Gu R o Si i, Si], and in reducing the n u m b e r of properties which m u s t be verified in order to construct a Wightrnan-Haag-Ruelle theory [Nel 3, Os Sch 2]. F u r t h e r m o r e , the Euclidean f r a m e w o r k m a k e s clear the connection of field theory with statistical mechanics, a central tool in the heuristic approach of Fisher and Wilson and in the w o r k of Guerra, R o s e n and Simon.

The lattice approximation [Ko Wi, G u Ro Si 3]

then b e c o m e s natural, and it relates boson interactions to an lsing m o d e l with continuous spin and nearest neighbor interaction.

2.

Correlation Inequalities [Gu R o Si 3, Nel 5, G r Si, Sill]. T h e convergence

of the lattice approximation and the existence of the t h e r m o d y n a m i c limit leads to the generalization of correlation inequalities and the L e e - Y a n g t h e o r e m f r o m statistical mechanics to q u a n t u m field theory.

T h e s e methods appear especially suited to

boson interactions and are discussed in the lectures of Guerra, Rosen, Simon, Nelson and in Chapter 3 of these lectures.

3.

C l u s t e r E x p a n s i o n s [G1 Ja Sp 1, Z],

T h e s e high t e m p e r a t u r e e x p a n s i o n s

p r o v i d e (within t h e i r r e g i o n of c o n v e r g e n c e - see F i g u r e 6) the m o s t d e t a i l e d i n f o r m a t i o n about ~(~)2 models.

In combination with low temperature expansions

(which w e believe to exist), the expansions should converge for all interactions

141

sufficiently far from a critical point. y i e l d a n a n a l y s i s of a n a r b i t r a r y nian 8].

H

.

interval

[0, a]

R e l a t e d a r e the i n d u c t i v e b o u n d s f o r

The bounds for

are known for restriction

In f a c t t h e i n d u c t i v e e x p a n s i o n s f o r

P(~)2

4

9(~)Z

of t h e s p e c t r u m of the H a m i l t o P(¢)Z

[G1 J a IV] a n d f o r

¢4

[G1 3a

h a v e not y e t b e e n r e f i n e d to y i e l d s u c h d e t a i l e d r e s u l t s a s "

In c o n t r a s t to the e x p a n s i o n s ,

on the m a g n i t u d e of the c o u p l i n g c o n s t a n t s .

the b o u n d s h o l d w i t h o u t

142

l. 3.

The GoIdstone Structure

We n o w r e t u r n to t h e There

P(t~)Z m o d e l ,

is a very simple picture,

ics folklore,

o f 9(~) z

and give a more detailed discussion.

p a r t l y d u e to O o l d s t o n e ,

and now part of the

which accounts for the fact that some interaction

duce degenerate

vacuums,

recent discussions

w h i l e o t h e r s do n o t .

s e e [Co 1, W i ] .

polynomials

physP pro-

S e e [Go, Go Sa W e , J o L ] , a n d f o r

Classically,

t h e ( E u c l i d e a n ) g r o u n d s t a t e of t h e

P(~) model is the function ~ 6 S' (iKz) which minimizes the classical (Euclidean) energy y[v@Cx) 2 + e(¢)]dx

This function ¢

is a constant,

Quantum mechanically, $ ' ( R 2)

, centered

~ which minimizes

the (Euclidean) ground state is a probability

(in s o m e s e n s e ) a b o u t t h e c l a s s i c a l

mechanical

picture

order

counterterms,

mass

e q u a l to the v a l u e

requires

corrections,

minimum.

p(~) distribution

on

The quantum

including Wick ordering

s e e C o l e m a n a n d W e i n b e r g [Co W e ] .

and the higher In P(~)2

models,

H e p p obtained the classical limit ~-~0 explicitly, s e e [He 3]. The Ooldstone picture also states that (i) the physical m a s s

m

is the curvature of P at its m i n i m u m

(ii) in the case of a double (or multiple) m i n i m u m ,

multiple phases exist.

discussion, we include in p the m a s s in the free Hamiltonlan

and

In our

H 0 , but because we

ignore quantum corrections beyond (bare) Wick order, our discussion is approximate.

(a)

1 Z Let ~ m 0 be the coefficient of ~Z in P

. W e consider the polynomials

P(~) = ~ Z ~ cj~ j ; CZn > 0 , ~Im 0Z = c 2 >> cj

, j¢ 2

=

2~z

1

(b)

Pig) = l ~ 4 + ~ m 0

(c)

P(~) = P0 (¢) - gg

(d)

1 Z Z ; ), > 0 , m ~ < 0 , ~0 P(~) = ),pO(~) + ~ m 0

(e)

p(~) = ) ,

4

;

;

- ~

P0

),> 0

, X> 0

,

bounded below,

tt ~ 0

~ >> 1

even.

143

See F i g u r e 4. In c a s e (b) a c u b i c t e r m m a y b e a d d e d t h r o u g h the t r i v i a l t r a n s f o r m a t i o n - [ + const.

In (c),

In (d), a p o l y n o m i a l

~ m a y be r e p l a c e d b y any odd p o w e r 6 +...

~ZJ+I, Zj+l < d e g p 0

of h i g h e r d e g r e e m a y h a v e m o r e t h a n two m i n i m a .

(a)(c)

(b)

(d) Figure 4. Various interaction polynomials (9 (~').

(e)

144

In cases (a) - (c) the polynomial curvature.

C a s e (d) has a double m i n i m u m ,

corresponding to the s y m m e -

, with positive curvature at each m i n i m u m .

phases, with positive m a s s particles in each pure phase.

Thus w e expect two pure The two v a c u u m

for the two pure phases are localized approximately at m i n i m a expect in the v a c u u m field,

(~>

.~ = -+ a

states Thus we

.

states for these two phases that the expectation value of the

, approximately equals

+a or

-a

. Case (e) is the limiting case

between Case (d) and a particular C a s e (a). H e r e = 0

v~th positive

This suggests the existence of a single phase, and the existence of

m a s s i v e particles. try ~ -, - ~

p(~) has a unique m i n i m u m

, with curvature zero.

P(£) has a unique m i n i m u m

at

The Goldstone picture suggests the presence of zero

m a s s particles, and is the Ooldstone picture of a critical point. yield the m e a n field approximation, rather than quantitative, picture.

These figures

which w e believe gives the correct qualitative, Thus the existence of the limiting case (e) sug-

gests the existence of critical values of the coupling constants for which

m

= 0

In this connection, Figure 5 represents the Goldstone conjecture for even

l

Z ~Z models.

l£0(~) + -~m 0

R. B a u m e l

[Ba] r e m a r k e d that changing the definition of

W i c k ordering (e.g., W i c k ordering [n a bare v a c u u m with a different bare m a s s ) permits the s a m e

H

to be written with different bare p a r a m e t e r s

location of the critical point, w h e n w r [ t t e n [n these parameters,

rn 0, ~.

T h u s the

can be changed,

so

the sign of m 0 at the critical point [s not significant.

mphys

Two Pure Phases

~'Critical Point

m~ z ~2

Figure 5. Conjectured symmetry breaking in even ,kPo(95)+~mo

145

Results.

We g i v e t h e r i g o r o u s r e s u l t s p r o v e d in c a s e s {a) - (e).

convergent cluster expansions show that the mass operator isolated eigenvalues at

0 and

m > 0

g e n e r a l i z e d to c a s e (c) [Sp 2]. mass

M = m

irreducibly ( r n . 2ml

, [G1 J a Sp 1].

In c a s e (a),

M = (H Z - p 2 ) l / Z

These results have been

A single p h a s e e x i s t s , the v a c u u m is u n i q u e and the

e i g e n v a l u e is n o n d e g e n e r a t e in the s e n s e that the L o r e n t z group a c t s

on the one particle space.

Any additional mass

spectrum in the interval

should be d i s c r e t e and would d e s c r i b e bound s t a t e s .

In c a s e (b), G r i f f i t h s a n d S i m o n [Gr Si] e x t e n d t h e L e e - Y a n g t h e o r e m a n d u s e t h i s r e s u l t to s h o w [ S i l l ] t h a t t h e g r o u n d s t a t e i s u n i q u e . small,

has

the structure

of t h e m a s s

For

4 to ~2

)~ l a r g e a n d

s p e c t r u m a b o v e z e r o is n o t k n o w n .

In c a s e (d), D o b r u s h i n a n d M i n l o s [Dob IV[in] h a v e a n n o u n c e d t h e e x i s t e n c e o f a t l e a s t two p h a s e s .

Continuous Symmetry

Breaking.

We] c o n c e r n s a c o n t i n u o u s s y m m e t r y g r o u p ~ -. - ~ P(~)

that we d i s c u s s e d

A n o t h e r f o r m o f t h e c l a s s i c a l p i c t u r e [Go Sa group, rather than the discrete

above.

In t h e c o n t i n u o u s c a s e , t h e m i n i m u m

o c c u r s on a m a n i f o l d of d i m e n s i o n g r e a t e r

manifold leaves

p{~)

of broken syrnnaetry these particles

constant.

than zero,

of

and t r a n s l a t i o n along this

The Goldstone picture now states that in the case

(6 ~ 0 a s d e f i n e d b e l o w ) p a r t i c l e s

have a mass

reflection

given by the minimum

of mass

zero occur.

c u r v a t u r e o f P(~)

Thus

at its mini-

mum.

In t h e p h y s i c s l i t e r a t u r e , conserved current, field theories,

b~j/~ = 0

this broken symmetry

i s d e f i n e d in t e r m s

, associated with the symmetry

a standard variational argument establishes

conserved current.

The generator

of the s y m m e t r y

group.

of the

For classical

the existence of the

group is

Q = ~J0dx

, and the

Goldstune picture concerns the vacuum expectation value

8-- <[io. ~]> If 6 ¢ 0

, discrete

zero mass particles

In c a s e t h a t t h e s y m m e t r y

( G o l d s t o n e b o s o n s ) a r e b e l i e v e d to e x i s t .

g r o u p of a u t o m o r p h i s m s

can not he u n i t a r i l y i m p l e m e n t e d ,

146

Kastler, Robinson and Swieca have s h o w n that the m a s s s p e c t r u m extends d o w n to zero [Ka R o Sw], and Swieca [Sw] has s h o w n that zero m a s s particles exist. T w o simple e x a m p l e s of such conserved currents are (I) a zero m a s s free field, with energy density ~rZ + (V~)2 , invariant under translations ~ - ~ + const. , and (Z) a multicomponent field invariant under orthogonal transformations in the space of components. In case (I), J~ = ~A~

, the p a r a m e t e r

the zero m a s s free particles.

0 = I and the Goldstone bosons are just

(We note this a r g u m e n t is not applicable in two

dimensions w h e r e zero m a s s free scalar fields do not exist. ) C a s e (Z) corresponds to Figure 4(d) with a rotational s y m m e t r y axis.

If our field has two components

tation value of the field components. f o r m e d of the c o m p o n e n t s the broken s y m m e t r y

, then 8 i = + <~j) is the v a c u u m expec-

T h e Goldstone bosons occur w h e n the vector,

6i , does not vanish i. e. , ~ ~ 0

In t e r m s of ~

,

condition requires that the v a c u u m expectation <~) ~ 0 . Let

us suppose that (~> ~ 0 decompose

~i

about the ~ = 0

, and let ~ be the unit vector in the

<~> direction.

~ into longitudinal and transverse parts, ~ = ~ L + ~ T

~-" L

=

(~

•

n n)-"

,

~T

We

' where

. n-" =0

T h e conventional picture is that along its trough,

P(.) has zero curvature and

displays zero m a s s excitations, while in the orthogonal direction the curvature of ~(.) is positive and one expects m a s s i v e particles. clustering for ~ L

Thus one expects exponential

'

{~L{XI~L(~}}

but polynomial clustering for ~ T

<_ O(e -rod)

,

m "- 0

,

o

A s w e r e m a r k e d above, two dimensions displays especially singular infra red behavior.

In fact, C o l e m a n [Co Z] has proved that for d = Z , a continuous s y m -

m e t r y as above always yields 8 = 0 . Thus the usual Goldstone picture never ensures the existence of zero m a s s particles.

T h e basic ingredient of Coleman's

147

proof is the distribution character of the two point function.

This precludes zero

m a s s particles appearing in a scalar two point function and forces a r g u m e n t does not, however, prevent the m a s s Therefore w e as]<: Does the conjecture that the m a s s icalpoint

~ = 0

s p e c t r u m f r o m going d o w n to zero.

)~(~2)2 m o d e l with )~ >> 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 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-X, e'tH(8- X)> <

Proof.

For

8 6 ~

lie- xllZe "(a+()t

,

II Ea(e - ×)II = II ~aetHe-tH(0

- ×)11

<-- e a t ( 8 - X , e " 2tH(8 _ X)> I / z

.1/2 -¢t m8 e

Thus

Ea~ 0 = Ea~

, which is dense in

Eag

.

0

By applying the Schwarz inequality

n times,

<e - x,

e-tH(e - X)> < lie -xll

11e'tH(e

-x)ll

< II e - ×11Z'z'n.e'(a+~)t~z'n

110 "Xll Ze" (a+~)t

159

W e n o w let }6 be a Hilbert space carrying a unitary representation U(a ,A) of the inhomogeneous Lorentz group. momentum,

PO < a,

IPI ~ b .

Proposition Z.I.Z. tion subgroup U(a)

, then the s p e c t r u m of IV[~

contains exactly one point and

.

{g(P) , P}

on }~

, so any c o m m u t i n g oper-

. In particular the energy m o m e n t u m

, and by Lorentz invariance

(Here w e a s s u m e the nontriviality of }~0 " ) Thus M = m

(UU(a,A)}~O)-=E •

T h e f~mily U(~) is m a x i m a l abelian on ~0

ator is a function of 5 the f o r m

U(a)}COC~ 0 and

If ~0 ~ {0} contains a cyclic vector for the space transla-

U(a ,Jl) is irreducible on ~

Proof.

Let

Let )~0 c )~ be a subspace of bounded energy and

s p e c t r u m is a set of

H = (~Z + i~n2)l/Z for s o m e M = m

on K0

and by Lorentz

Since reducibility would be a c c o m p a n i e d by multiplicity in the

m a s s spectrum, the representation U(a,A)

is irreducible.

m

.

160

2.2.

Relevance to Physics

The Schwinger functions with a space time cutoff h are given b y

(2. z.1)

Sh(X I,''., x n) = ~ ( X l ) ' ' ' ~ ( X n ) d q h

where

dqh is the m e a s u r e

e-V(h)d~ dqh - fe_V(h)dff~

(2.2° 2)

,

d~} is the Gaussian m e a s u r e with m e a n zero and covariance

2-i (-~ + m O) : C

, and

V(h) = X [ : e(,~,(x)) : h(x)dx

is the P(@)2 ]Euclidean action. A C

R2

with area

IAI

If h(x) is the characteristic function for a set

, then V(h) is the action for A • W e denote the corre-

sponding Schwinger functions S A . W e state the v a c u u m cluster expansion, which bounds the rate of asymptotic factorization of the v a c u u m state.

Let A

n

A =

(2.2.3)

where

fi(x) is either an

17 ~[ : ~(x) i=l

be a function of Euclidean fields,

n.

i : fi(x)d x

LZ(I<2) function, or else 8s (t)fi(~) , w h e r e

LZ(K)

Let (~} be a cover of R Z by unit lattice squares

suppt. A

as the smallest union of ~ 's containing suppt, fl U "'" U suppt, fn

the following,

Theorem with ¢ > 0

A , B

~

fi(~) is

, and define In

have the f o r m (Z. 2.3).

2.2.1 ( V a c u u m Cluster Expansion [GI J[a Sp i, 2]. ) Let ~,= m 0 - ~ Consider

)~P(~)2 models with )~ < )~(~ , P, m 0 )

A , suppto B} ° T h e n there exists a constant M A , B

such that

,

Let d = dist. {suppt.

161

(z. z.4)

uniformly

in

h

.

[The constant pose each localized

M A , B c a n b e b o u n d e d e x p l i c i t l y in t e r m s

fi is s u p p o r t e d in a s i n g l e A's).

We l e t

N(~)

Ai

(an a r b i t r a r y

b e t h e s u m of n i ' s

of t h e

f'l

We s u p -

A is a s u m of s u c h

, for

s u p p t , fi E ~

•

Let

K1

b e g i v e n and l e t

=

We d e f i n e f o r a l o c a l i z e d

n (E,N(m:) ,L

A

n

(z. z.5)

Here

IIAll = ~ n Ilfill i=l

[(f[[ = [[f[[Lz(R2 ) f o r

L e t us a s s u m e

ni < n

.

f E LZ(R 2)

Then for

(Z. 2 . 6 )

MA,

If it is not t h e c a s e t h a t

n.1 N< n

'

, or if

f ( x ) = 6s(t)f(x) , IIfff=

flf(l)IILz(R)

K1 s u f f i c i e n t l y l a r g e ,

B <

llAI[ ][B[I

we o b t a i n ( 2 . 2 . 6 ) w i t h 7 3 r e p l a c i n g

in (2. 2.5).

Also

(2.2.7)

uniformly

in

h

Theorem as

A -* R 2

Proof. to p r o v e

<__ [[AI[

~Adqh

.]

Z. 2. Z.

The

, to limits

As

S(x I , " • " , Xn)

explained

convergence

Schwlnger

as

functions ' S A { X 1 , • • • , Xn)

o b e y i n g the O s t e r w a l d e r - S c h r a d e r

in the lectures h - 1

converge

of O s t e r w a l d e r

, and a simple

~

and Nelson,

in ~'(R Zn) axioms.

it is sufficient

b o u n d that follows f r o m

the

162

vacuum

cluster

expansion.

Here

two space-time

cutoffs,

with

A=

)

suppt.

¢(fl )'''¢(fn

, let

be the set of lattice

we establish

hI - h 0

supported

A c suppt,

squares

convergence.

h0

on a bounded

and let

intersecting

F

Let

0
set

F

be

Let

d = dist (F, suppt.

A)

.

Let

o

Define the function

F(~) = ~ A d q ~

where

and

dq( x = d q h

h

= {2h1 + (1 - tv)h 0

.

T h e n by differentiating (2. Z. 2) w e

tv

obtain

By Theorem

2.2. I and (2 2. 6), the s u m above is

t

sup &

for y'= m 0- 2~ Then

d -* oo as The

~

MA, V(A)e -yd

dependence of the cluster expansion shows immediately that S(fl, • • • ,fn)

Theorem

Proof.

2

A0 , A1 -* i~2 and w e obtain the desired convergence.

establish analyticityin

~

O(1)e "'l'd

• W e let h 0 , h I be characteristic functions of sets A 0, / h c R

are continuous functions of i

vacuum

~

In fact, in Part II w e use the cluster expansion to

I in the half circle

2o 2.3. (Mass Gap).

For

0 < Ill < ~0

' l~el > 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> <

o r t h o g o n a l to 0 ,

O(MAe'Yd)

The theorem now follows as planned. We h a v e e s t a b l i s h e d s t r o n g e r arbitrary

cluster properties,

intervals of the energy spectrum

w h i c h p r o v i d e a n a n a l y s i s of

[G1 J a Sp 1].

These expansions are

d e f i n e d i n d u c t i v e l y , r a t h e r t h a n in c l o s e d f o r m or in the f o r m of K i r k w o o d - S a l s b u r g equatlons. Let

We n o w s t a t e t h e

y = Z(m 0 - ¢ ) Theorem

with

Z.Z.4.

k~(~0)g m o d e l s w i t h L2(l~ ) f u n c t i o n h

n = 1 e x p a n s i o n , or one p a r t i c l e c l u s t e r e x p a n s i o n .

¢ > 0

.

(One P a r t i c l e

Cluster Expansion).

k < k(e , m 0 , P) such that for

(0 A - X ,

X = (~,

Then given 8A 8 A ) ~ ] + ~0{h)~

e'tH(0A-X))

60

the span of

[f~,cp(h)f~}, where

e > 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,

<~(Xl)~(xz)~(x3)~(x4) > - <~(Xl)@(x2) >(~(x3)#(x4)) (3.3. l)

<~(Xl)~(x3)><~(Xz)~(x4)> + <~(xl)~(x4)> <~(xz)~(x3)) • Remark. (truncated)

Since

< ~(x)> = 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

z = ~ £ Lebowitz proves

e-H(£ )

[Leb 2]

<(Yic;j(;k(Y£> - <(Yi(Tj> <(Yk{Y£> ~

<(Y[(yk> <(Yj(7~> + <(yj(~k>

The inequality (3.3, 4) follows immediately, have proved that the Euclidean form, where

4 ~02

sinc~ Grifflths and S i m o n

[Gr Si]

model is a lixnit of Ising models of the above

~(x) can be expressed as a l[mlt of a s u m of spin variable

(y[.

173 W e recall that the relativistic time zero field Euclidean field

(f,£=0).

(3.3. Z)

= ~0(fl)~(fz)n

where

O(fl,f z)

W e let

f. 6 g(R) L

equals the time zero

and define

- (e,~0(fl)~(fz)e)

e

,

f2 is the v a c u u m vector. Corollary 3. 3.4. The vectors Proof.

O(fl,f2)

have energy

It is no loss of generality to choose

f.

~ 2m.

real and positive. By the

Feynrnan-Kac formula,
= <~(gl)~(gz)~(g3)~(g4)> where

x = (Xl,X O) gl(x)

<#(gl)~(g2)) <~(g3)~(g4))

and

-- fl(Xl)~(Xo)

g3(x) =

-

fl(Xl)St(x O)

,

gZ(x) = fz(xt)8(x0)

,

g4 (x) = f2(xl)bt(x0)

,

By (3.3. 1) and the F e y n m a n - K a c f o r m u l a , 2
~

i~__l <~°(fi)~2'e'tH~0(fi)f2)

+ l<~0(fl)f2,e'tH~0(fz) f2>12

Since

< f2,~0~) = 0 , the s p e c t r u m condition yields

l(~o(fi)Q, e-tH~o(f~)~) I~ O(1)e"tin

and therefore ( e, e-tH O> completing the proof. ary to this point.

<

O(1)e -2tm

W e r e m a r k that only vacuum cluster expansions are necess-

174

Next we state a result expansion. interval

We let

E> 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>

<

II O ] l >

(i)

states;

and

<0,M0)

1 ;

(iii)

(H)

O

is

<8,M8> < 2m.

Since

by

2m

In a theory with w e a k coupling, the cluster expansion shows that the low m o m e n t u m part of the m a s s

interval

(3.4. Z)

0

see [GI J a S p in addition,

(m, gin)

is spanned by vectors

= at~ + a*(f)f~ + e tH a*(f[) a

I] and Section 3.3. •(q0) is even, w e m a y

Alternatively, w e can replace

a

Here

a

choose

f = 0

If

X = a (f)~,

:

and

a*(~f) + [HI, a*(f)]

,

H

If,

at = - < f~, a (fl)a (f2)f~> .

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

<×,H×> = (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<~0,~(b,K)>[ 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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Small distance behaviour, analysis and Wilson expansions, Commun.

[Sy 7]

_ _ ,

Math. Phys. 188, 227-246 (1970).

Math. Phys. Z__~3,49-86 (1971).

Small distance behaviour in field theory, in: Stron~ interaction physics, Springer tracts in m o d e r n physics, Vol. 57, Springer-Verlag,

[Wi]

N e w York, 1971.

A. Wightrnan, Lecture at the 197Z Coral Gables Confe/ence, Gordon and Breach Sci. Pub. , to appear.

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
< c ,

-~/2 < arg k < ~/2

Let d be the set of convex combinations C c~,

are

in I, for I in the bounded sector

(2.1)

operators

P(¢)2

and in this

theory the Wightman analytic

in

(They are also

the infinite

f r o m the Schwinger

are satisfied

converge

volume S c h w l n g e r

cluster property.

reconstruction

in a

From these

functions

volume limit, and that the infinite

satisfy an exponential

independent

cluster proper-

(1,6).

The important

are summarized

properties

(2.2)

V(A)

of Dirichlet

covariance

of the measures

in §9 (as p r e p a r a t i o n

In order to make elementary definitions, a bounded measurable

.

dec,

for the estimate

we mention merely

of §i0).

that for A

set and for C c ~ ,

=

[ : P(¢(x)):dx

e Lp(~',d¢ C)

A and

e

(2.3)

for all

p E [i,=)

(2.2) the polynomial

-XV(A)

and

s L p ( ~ ' , d ¢ C)

Re k ~ 0.

(See Prop.

P is arbitrary,

The Wick ordering in (2.2) is taken relative of (1.4), even when C # C 0. to be a union of lattice restrict

A to be a square

(2.4)

Z(A)

=

To simplify

squares, centered

Z(A,C)

=

and in the limits at the origin. 5 I

dq A

=

dqA, C

=

In

to the fixed measure

the discussion,

d@c~

we take A

A ÷ =,

we further

Let

e-IV(A)d¢c"

In Theorem 6.1 below we show that Z(A) ~ 0. (2.5)

9.3, Th. 9.5.)

while in (2.3), P is semibounded.

Z(A,C)-Ie-/V(A)d¢c

Let .

bMore general cutoffs, as required for the proof of Euclidean covariance of the limit A ÷ ~, can be introduced by inserting a nonnegative L= function h in the integral (2.2). Such an h does not affect the subsequent discussion.

206

The finite volume Schwinger functions are by definition the moments of the measure dqA: (2.6)

SA(Xl'''''Xn)

=

I ~(Xl)'''$(Xn) dqA

S A is defined as a tempered distribution in

~'(R2n).

To see this,

take a test function of the form w(x) = Wl(Xl)...Wn(X n) integrate against (2.6). The n+l factors ¢(w ) = I ¢(Xi)wi(xi) dxi

and

and

e -kV(A)

in the integrand are bounded by HSlder's inequality,

(2.3), and the

following consequence of Prop. 9.3: U~(wi)ULp(J, )

~

MpUWiUL2(R2 )

In addition to the monomials in (2.4), it is useful to integrate products of Wick ordered polynomials, namely (2.7)

A

We assume

=

i

:¢(x l)

w E~(R2J),

nI

:...:¢(xj)

nj

:w(x I .... ,xj)dx .

although weaker bounds would suffice.

We

define suppt A to be the intersection of all closed subsets C c R 2 with (2.8)

suppt w c C x...x C

Theorem 2.1. and let

Let

~

belong to the closure of the half circle (2.1)

e/m~ be sufficiently small.

of the form ~(2.7). Let suppt A and suppt B. constant

(J factors)

Let A and B be functions on

d be the width of a strip in R 2 separating

There is a constant

m independent of A and B

M = MA, B

and a positive

such that

_

'fABdqA,C~ uniformly in

A,

as A ~ ~

IAdqA,c~fBdqA,C~I~MA,B

e

-md

,

Furthermore, M is independent of transla-

tions in either A or B. Theorem 2.2. Let k belong to the closure of the half circle (2.1), andr let ~/m~ be sufficiently small. For A of the form (2.7), I J A dqA,C I

is bounded uniformly in ~ and in A, as A

From analyticity of the finite volume Schwinger functions, from

207

V i t a l i ' s t h e o r e m and from the convergence,

as

A ~ ~,

for k real and

small, we have

Corollary 2.3. for s / m 2 s m a l l . o

The S c h w i n g e r functions are analytic in k in (2.1)

208

§3.

THE CLUSTER E X P A N S I O N

The p r o o f of the main theorems are based on a cluster expansion, w h i c h we now derive.

Let~

be a set of line segments in R 2.

i n t e r e s t e d in two examples: lines

= (Z2) * ~ F

The labeled

with F a finite subset of (Z2) *.

subsets by

r is

We identify a subset

formed

r c.

Thus

by

decoupled

(1.1).

lines

our

expansion.

The

term

r

a Gausslan

b in

r c are

measure called

differences

between

These

differences

are

theorem

in

across

are

fundamental

derivative

terms

=~9 ~

choosing

the

there

b c R2

L_J bEF label

rc

and

the

:

Y c~

(3.1)

in

We are

the set of all lattice

w i t h the subset r

e F

= (Z2) *

(bonds) Joining nearest neighbor lattice sites in Z 2 or

F c ~

b

Either ~

of

with

Dirichlet

Dirichlet coupled

expressed

calculus.

lines.

and in

Thus

covariance

terms

the

For

decoupled

b

of e F

the

on lines

measures

as

derivatives are

by

called

lines.

The

covariance

the operators to measure

operators

C r.

we c o n s i d e r

are

convex

combinations

of

For each b e ~ , we i n t r o d u c e a p a r a m e t e r s b ~ [0,i]

the strength of the coupling across b.

The p a r a m e t e r value

s b = 0 corresponds to zero Dirichlet data on b, hence to zero coupling across b, while sb = 1 corresponds to full coupling across b. (3.2)

s

=

(Sb)be ~

With

,

we define the m u l t i p a r a m e t e r family of covariance operators

(3.3) Since

C(s) the

1

= ~ r

bBT~r s b I I

~

coefficients

=

in

~ = ~3c1

(3.3)

are

the

terms

bsT~ [Sb + (l-Sb)]

(3.3) is a convex sum as asserted.

C~

(1- Sb)C

b~F c

=

C(I,I,...)

'

r e in

the

c(o,o,...)

of

,

The free covariance is now denoted

=

(-A + m2)-lo-

and the completely d e c o u p l e d covarlance is

c~5 =

expansion

:

( - ~ + ma)-lo

209

The Schwinger natural way,

functions

functions

I

Z(S)Ss(X

(3.~)

= Z(A,S)SA,s(X)

Z(s) = Z(A,s)

where

and the partition

in a

=

dec(s)

The goal of the cluster expansion (s b e l) in terms of decoupled The decoupled quantities

is to express

=

coupled quantities

(and hence finite volume)

have s b = 0 for many

them, it is convenient

s(r)

= ]H$(xi)e-kV(A)d¢ s

= /e-kV(A)dCs

de s

formulate

function are,

of s, and we use the notation

b e~,

quantities.

and in order to

to define

[s (r)b)beT~

by the formula Sb,

s(r) b O,

For F finite, large distances boundary

s(F) specifies

(on Fc), while

conditions conditions

Definition

b~r.

Dirichlet boundary

s can be thought

conditions

Thus the

is a way of saying that F is independent

of the

at ~.

A function F(s) is called regular at infinity

3.1.

at

of as giving general

on r c, and agreeing with s(r) on F.

following d e f i n i t i o n boundary

b g F

if for

each s, (3.6)

F(s)

=

lim{F~:

F finite}

Proposition

3.1.

The functions

S converges

in~',

and~

(3.4) are regular

We omit the proof,

since it is a

in Chapters

the limit

Here

consequence

(3.6) is elementary. of

techniques

7,9.

The first step in the cluster expansion tal theorem of calculus

at infinity.

c (Z2) *

Because A is fixed and bounded, to be developed

F(s(F))

is to apply the fundamen-

to the finite number of nonzero parameters

in

210

F(s(F)).

Define

c3.7)

brlJ

and for two coordinate

values

s and ~

define order coordinatewise,

so that

5 Proposition

(3.8)

3.2.

F(s)

Let

F(s)

=

Let

F(s(B))

= G(s(B)),

the sum in (3.8)

sb

,

qb 6

~FF(~(F))

f 0
G(s) denote the right side of (3.8).

convergence

to sets

F c Bo

f(s b)

in the assertion,

7

Then

Q

that is just

F(s) is regular from the assertion.

as B ~ .

of a single variable,

(~bf)(s b) = f(s b) - f(0) =

d~

Now G(s(B))

Since

of the sum in (3.8) follows

follows by limits

For a function

~.

We assert

for B any finite subset of ~ .

(3.8), restricted

at infinity, Also

!

be smooth and regular at infinity.

[ is finite}

{FC~:F Proof.

S * ~b

let

~bf(a b) d~ b

0 (E~f)(s b) Then

b 6b I = E0 +

(fundamental

f(O)

theorem of calculus)

! = I I (E~ + ~ b ) :

(3.9)

=

beB

B~F ~

Eo

and so

6r

FCB

where

6 r = I'l

~b ,

E~ ~r =

I I

bey It is easy to see that

E b0

beB~Y (3.9) yields

the desired identity

F(s(B)) = a(s(B)). The next step

in the cluster expansion

and partial resummation (3.10) so that each X i

of (3.8).

is a factorlzation

Write

R 2 % F c = X 1 u X 2 u ..,

u

Xr

is a union of connected components

xln xj--;~ for i~J.

such that

211

DefiNition

3.2.

A function F(A,s)

decouples

r

(3.11)

F[A,s(r))

1'"'1 F(Anx i , i=l

whenever

at s=0 provided

s(rnxi))

(3.10) holds.

Proposition

3.3.

Given

(3.10),

each factor into a product is defined o n ~ ' ( X i ) .

the measures

of r measures,

des(F) and e-kV(A)dCs(r)

and the ith factor measure

To avoid regularity

questions,

we assume

~ c (z2)*. Sketch of Proof. is the direct

C(s(r)) has

Since

zero Dirichlet

The factorization

C(s(F))

~ L2(X i)

of des(F)

follows

from this fact,

from the standard formula for evaluation polynomial

(Theorem 9.1).

exp[-kV(A))

also factors,

Corollary 3.4.

Because

interest

rC, c(s(r))

ZS and Z of (3.4) decouple

from connected graphs.

(e.g.

of a

is a local function,

and so does exp(-kV)dCs(F).

The functions

at s=0.

F as a sum of products

of con-

We start with some set X o of

X o = {Xl,...,x n} in (3.4)).

meet X o are resummed,

and can be seen

of a Gaussian integral

:P[¢(x)]:

The cluster expansion represents tributions

data on

sum of the operators

giving a single

The graphs which do not

factor of F in an exterior

region. We now carry out this resummation (3.11) in the expansion

(3.8).

in more detail.

Substitute

This yields

s(ri ) (3.12)

F(A,s) = ~

~

ariF(An Xi,c(ri))do

i=l where

ri = P n X i.

products

If the X i are connected,

of connected graphs.

union of all components remaining

components.

fixed, while

this is the sum over

Now we c~ooseX 1 in (3.10) to be the

meeting X ° and let X 2 be the union of the The r e s u m m a t i o n

summing over all choices

consists of r 2.

of holding X 1 and F 1

Explicitly,

212

F(A,s)

=

S(il)

~ Xl'F1

~FIF [An Xl,~(rl)) d~

0

s(i2)

x ~2

~F2F[AnX2'°(F2))dq

"

0 The F 2 sum runs over all finite sets F 2 of bonds i n q ~ X f .

For

this reason, the F 2 sum can be evaluated by (3.8) as F ( A n X 2 , s ( ~ X i ) ) . Setting X = Xf and writing F for F1, the expansion has the form

F(A,s)

~

=

X

(3.13) K(Xo,X ) =

~

r

K(Xo,X)

F(A~X, s (~*X))

slr)

~rF(A n X, ~ ( r ) ) d~

0

In these sums, X ranges over finite unions of closed lattice squares while F ranges over finite subsets o f ~ such that (i)

Each component of X ~

(ii)

F~Int

meets X °

X.

If no such F exists, for a given X, then K(Xo,X) ~ 0o Theorem 3.5.

Let X ° be bounded and let F be smooth, regular at

infinity and decouple at s=0.

Then the cluster expansion (3.13)

holds. Example I.

Let~=

Xo= {Xl,...,Xn}. Observe that s=l (3.14)

(Z2) * be the set of all lattice lines and let

We substitute ZS = F.

F(A*X,s~X))

=

See (3.4)°

I e-~V(A~X)dCs(1~X)

= Z(A~X,s(~X)) =

Z(A~X,Csx)

~

Z~x(A~X)

since a change in the data within X does not affect the integral. Thus if we divide

by Z, (3.13) yields

213

(3.15)

SA(X) =

~ S dr S Ni ¢ ( x i ) e-kV(AOX) d e s ( F ) ds(F) X,F × Z2x(A~X)/Z(A)

I f we w r i t e X ~ r c as a union o f c o n n e c t e d components, the i n t e g r a l ( 3 . 1 5 ) can be f a c t o r e d , as i n ( 3 . 1 2 ) . Example

2.

F 1 c (Z2) * and F 2 = r 1 ~ b I be finite

Let

lattice bonds, where b I is the first element graphic

order of

(Z2) *?

pair

take

F(A,s)

Also F is smooth,

for

= ZF1 - ZF2 ,

regular at infinity

depends

is independent

Dirichlet

on the exclusion

of the variables b e F2

From

for

blC A

,

bI n A =

and decouples of

sb

o F2 = @ .

(3.13), we have

on

~F

:

at s = O.

b I from ~ . )

for

lines in the cluster expansion

F

,

b c A . (The

We note that

b e F 2 , and so

Thus in all nonzero

impose the further restriction (iii)

of the

is arranged so that

F(s=l)

last property

- Z(A,s(rC),i)

= ~[Z(A,s(FC), 0)

Ths definition

~bJF = 0 ,

Z is now a function

We define rz(A,s(rC),o)

F

d¢Cr

/3 = (Z2) * ~ b I , X 0 = b I .

(S,Sbl).

(in all bonds

ZF1 - ZF2 where

Z F = ] e -kV(h) and so we

sets of lattice

of F 1 in some lexico-

We use the cluster expansion

b ~ b l) to study the difference

in

terms,

F 2 consists

of

for F, and in (3.13), we may

214

F(A,s=I)

=

ZFI (A)

-

z r 2 (A)

X K(bl,rl,X)F(A~X, s(~X)) X

=

X~K(bI'~'X)ZFI UX*(A~X)

Here X* is the set of lattice lines in X and the last line is justified by the fact that the second factor above is independent b ~ Int X. We multiply and divide by

of Sb,

Z~a(A)tA nXl where A is a single lattice square. Z~A (~)IAnxl

Zr I U X • (A~X)

Since

=

zrl U x,(A)

,

we obtain (with a new K)

(3.16)

Zrl

=

Zrl~bl + X~ K(bI'FI'X)ZFI U X *(A)

and now (3.17)

K(DI,FI,X)

: z~a(a)-IAnXJ

I

~ F U b l z [ A n X , s ( F U b!))ds(FU b!). F

These equations have a structure intermediate between the Kirkwood-Salsburg and the Mayer-Montroll equations. They will be studied in §6 in order to obtain bounds on ZF/Z , i.eo the second factor in (3.15).

215

§4.

C L U S T E R I N G AND ANALYTICITY:

PROOF OF THE MAIN RESULTS

We prove the cluster property and analyticity -- the m a i n results of these lectures

-- in this section, using as hypothesis

of the cluster expansion,

(3.15).

Let T(x,A,X,F)

convergence

denote the X-F term

in (3.15), so that

SA(X) =

~

T(x,A,X,r) .

(4.1)

X,F For each test function

w e ~(R2n),

SA(x) w(x)

the series

x =

~

<w,T>

(4.2)

X,F converges

absolutely.

area of X.

The rate of convergence is governed by

With K > 0,

IXI, the

we prove that

l<w,T,l --< Iwi e where

lwl

T h e o r e m 4.1.

is some Let

(n-dependent) ~ - n o r m K > 0

be given.

Let

(4.3)

on w. m 0 be large and ~ be small

(depending on K) and let k b e l o n g to the closure of (2.1). ~-norm

w

such that

There is an

(4.3) holds uniformly in k, m 0 , A and D, and

is invariant under translations

lwl

in any of its variables.

The p r o o f allows integrands A of the form (2.7) to be s u b s t i t u t e d in (3.15).

Thus T h e o r e m 2.2 and

Corollary 2.3 follow from the case

K = i, D = i of (4.3). T h e o r e m 2.1 also follows from the convergence of the cluster expansion.

The p r o o f for the two point function with an even inter-

action is conceptually easier, but still illustrates we consider that

the main idea, so

case first.

Proof of T h e o r e m 2.1, for n = 2, P even, assuming T h e o r e m 4.1. Because P, the i n t e r a c t i o n p o l y n o m i a l in

(1.2), is even,

is a symmetry of the theory.

For Gaussian integrals

factorizing measure

d¢c(s(F) )

exp(-~V)

¢(x) ÷ -#(x)

defined by the

(cf. P r o p o s i t i o n

3.3),

more

is true:

~(x) ~ c(x) ~(x)

(4.4)

is a symmetry, where a(x) = ±i; Because of the symmetry

~(x) = const, on each X i •

¢ ÷ -¢, S A ( X l , . . . , x n) = 0 for n odd. Similarly,

I x ¢(x i)

e-~V(ANX)

d¢c(s(r) ) = 0

216

unless each of the connected number of xi's.

components

XI,...,X r contains

Since ~F0 = 0, the same restriction

applies unless each Xj has an even number of xi's. at least one x i by

Let d be defined by Theorem

In other words X~F c is

2.1 and let w = W l • W

nected and suppt w i ~ X ~ ~, we have d ~

IX I + 1.

IfSA(Xl,X2)Wl(Xl)W2(x2)dx} Since the one-point

functions

SA(Xl)

~

2.

Because

connected. X is con-

Thus by (4.2-3),

lwle -K(D-2)

vanish for P even,

~ M w e-Kd this bound

the proof of Theorem 2.1.

Proof of T h e o r e m as before, components

2.1 (general case) assuming Th. 4.1.

is to reduce the cluster

one connected case

= 0

Since each Xj has

(i) of §3, each Xj must have at least two xi's.

Thus for n = 2, there is only one Xj.

completes

an even

T(x,A,X,F)

component

X 1 = X.

are to vanish,

(P not even)

and introduce

The terms with two or more connected

because

of some symmetry.

does not have such a symmetry,

Since the general

we follow Ginibre

a new theory containing an artificial

To construct

the new theory,

of the measure d@c,

C~)l and normalized

let d#~, denote an isomorphic

physical

de c x d¢~,,

[2]

symmetry.

(C*~C) defined on an isomorphic

The new theory has free measure

The idea,

expansion to terms involving only

copy

copy ~'* of ~'.

covariance

+ I ~)C* = C ~ measure

z"-le -V(A)e-V(A)*d~ C x d~*c, = dq ~

and a field

¢~" = ¢ O 1 This theory is invarlant changes

+ I@¢*

.

(even) under the symmetry

~-~@*, which inter-

the two factors.

We now apply the cluster expansion

The covariance

to the expression

Z ~ J(A-A*)(B-B*)dq ~ operators which arise in this expansion have the form c(s/

So that the symmetry

= c(s) •

I + I O

¢<--~ ¢* is preserved

c(s)*

in the Gaussian measure

of

each.term of the expansion.

(However the expansion

is modified by

chooslng ~3=

F is the set of lattice

lines in two

(Z)*

~F, where

connected sets, one containing

suppt A = suppt A-A*, and the other

217

containing

suppt B.

X i ~ suppt A = @,

Thus for each component Xi, X i m suppt A or

and the same for suppt B.

there are at most two components. the number of components,

as one can check, n = 2 in (4.3).)

sider a term in (3.15) with components suppt A ~ X i , For this term, X i.

With this restriction,

Since the n in (4.3) is a bound on

the symmetry

¢~-~*

XI,X2,...

suppt B ~ X j

must vanish. component

The n o n v a n i s h i n g

,

i ~ J

can be applied separately

However A-A* is odd for the X i symmetry,

Now con-

satisfying (4.5) in each

so terms satisfying

terms, which violate

(4.5)

(4.5) contain a

X 1 = X with d ~ 01X[, and so for d ~ 4,

If(A-A*)(B-B*)dq~I Expanding term the dq

~ MABe

-md

the integrand on the left yields

integral

factors.

and Theorem 2.1 follows.

We evaluate

four terms,

the result

as

and in each

218

§5.

CONVERGENCE:

THE MAIN IDEAS

There are two main ideas in part II of these lectures. is the formula functions.

(3.17) giving the cluster expansion

The second is the estimates

of this expansion, estimates

uniformly

as a series

proof of Theorem

as A ÷~.

and using them,

The simpler estimates

the harder ones are postponed of these estimates

which lead to the convergence In this section we state these

of three propositions,

4.1.

The first

for the Schwinger

for later sections.

is contained in Prop.

sense the core of the paper, we discuss

give the

are then proved,

5.3.

while

The most difficult

Since it is in some

the ideas involved in its

proof at the end of this section. The proof of convergence

depends

on estimates

of the following

types: (a)

Combinatoric

pansion, special (b)

estimates

and especially

to count the number of terms in the ex-

to count or bound the number of terms of some

type.

Single particle

estimates,

tors, and their derivatives, (c)

Estimates

type

(a) and (b) will be components Returning

on function

to bound kernels

to specifics,

term is a product

space integrals.

Typically

estimates

of an estimate of type a ratio of partition

The first proposition

toric - type

(a)- and it counts the number of terms with The second and third propositions

value of

~.l. 5.2.

e~IXI

are hybrids,

(b) and (c). with a fixed

K2, independent

IZ~x(A~X)/Z(A)I 5.3.

Ixl = area x

~ 0(1)elgI~ .

There is a constant

sure of (2.1), and of A and m 0, such that

Proposition

(a),

The number of terms in (3.15-3.16)

IXI is bounded by

Proposition

of types

times

combina-

bound the two factors

These latter two propositions

since their proof useg estimates Proposition

of

(c).

functions

is purely

held fixed.

making up a term in (3.15).

opera-

(4.3) or (3.15) is a sum in which each

of two factors:

a function space integral.

of covariance

~rC(s).

There is a constant

of k in the clo-

for E small and m 0 large,

~ e K2]XI K 3 and a norm

lwl on test func-

tions such that for any K > 0 , for any A and for ~ in the closure (2.1) with m 0 large I < I ~F f

n i=l

(the m 0 bound

~(xi)

(The m 0 bound depends any of the variables

depends

e_XV(A)d~c(s (r)) on K, and of w.)

of

on K)

ds(r),w~l ~ e -KIFI+K31AI

lwl is invariant

lwl

under translation

in

219

Remark. integrand

Wick polynomials,

as in (2.7), are also allowed in the

above.

Proof of Theorem 4.1. in (3.15),

We replace A by A D X in Prop.

5.3.

For the X

we have X =~ri=l X[ with r _< n and X i connected. r

F C ~i=l

Moreover

Int

and so "many" of the lattice lines in X[ belong to F.

In fact since

- F C : x i is connected, Xi~ (5.1)

IXi[ - 1 ~ 2IF ~ I n t

Thus we replace

X;I and

IXI - n ~ 21F I .

the upper bound in Prop. e -K(IXI-n)

wlth a new choice of K and

lwl.

this bound combined with Prop. 2roof of Proposition

5.1.

[w[ , Theorem 4.1 follows

We identify

located at its center,

directly

from

5.1 and 5.2.

We consider

(3.15).

number of ways of choosing the component xj, 1 i J ~ n.

5.3 by

First we bound the

X i containing a particular

each lattice square A with a lattice point

and we identify

each lattice line b g ~

lying

between two squares A and A' with the line Joining the center of ~ to the center of A'

Hence we must count the number of ways of drawing

a connected graph with nearest neighbor bonds. such graph may be constructed by starting

We assert that each

at xj and drawing an oriented

path formed of unit line segments which traverses twice. segments

as bridges,

this simply follows

Konigsberg bridge problem 6. Since £ ~

81Xil

for pairs X,F

from the solution to the

for X~ lis at most 0(i)2161xi[

for F is at most 4 IXI, the number of

is at most

o<1)2 f41xfv7 since 21xl bo=ds the n= er

216Jxil = o
This completes

The case (3:16) is similar.

Proposition

length

and starting at xj is at most 4 ~.

, the number of choices

Since the number of choices

at most

and the line

The number of connected paths.of

formed from lattice llne segments

choices

each segment

In fact if we regard the lattice sites as islands

the proof.

5.2 is proved in §6 using equations

related to the

6We thank O. Bratteli for this observation. For a solution of the Konigsberg bridge problem see, D.W. Blackett, Elementary Topology, Academic Press, New York, 1967, Page 159.

220 Kirkwood-Salsburg equations.

We only remark here that ~he change in

partition function involves both a change in the interaction region and a change in covariance (A~X Discussion of Prop. 5.~.

+ A and C~X ÷ C ). F

Each derivative d/ds b in S

either the measure or the integrand.

differentiates

Derivatives of the measure are

evaluated by (1.7), and produce s-dependent kernels C'(s) in the integrand. Repeated differentiation yields (a) A sum of terms coming from the iterated functional derivatives 32/9¢ 2 in (i.?). (b) A sum over terms arising from possible repeated derivatives ~FLc(s) of each C'(s) introduced by (1.7). Each derivative d/ds b also produces convergence in one of two ways. Differentiation of the measure introduces a kernel (c) C' and C are small in the sense that

(c I) tIC'(x,y)IIL

C', and

_< 0(moe) P -m 0(dist (x ,b)+dlst (y,b))

(c 2) 0 _< C'(x,y) = 8bc(x,y) < e

-m 0 Ix-yl (c 3) 0 _< C(x,y) _< e

, Ix-yl >i .

Repeated differentiation of C(s) produces the further convergence (d)

SFC is small in the sense that

(dl)II~rC(x,Y)l~

Z 0(m0 -~I~I ) P

(d 2) 0 ~ ~Fc(x,y) ! e-mOd where d is the length of the shortest path connecting x and y and passing through each b 6 7. In fact from the Welner integral representation for (-A+m~) -I, 8Fc(s) is the integral over Wiener paths from x to y passing through each b ~ F , We use (c 2) to control (a).

see §7-8° In fact by (c 2) , only x and y near b

contribute to the Laplaclan C'(S).A¢ of (1.7)o

By the nature of the

expansion, there is at most one Laplaclan A¢ per bond b 6 ~ , hence a bounded power, A n , locally. Evaluation of.A¢n leads to K(n) <~ terms, and this evaluation, repeated over disjoint local regions leads to e 0(VOl')' terms. We use (d2) to control (b). The basic cQnvergence,

-~trl

m0

, comes from (dl). After performing these steps, there remains a function space

integral of the form fRe-kVd¢. J

Then

221

I I Re-~V(A) d~l <-- ( I R2 d~)I/2 ( ~j e-2Re~V(A) d~)1/2 and each factor on the right is e 0(V°l). Now (c 3) produces the decoupling of distant local regions in R 2. With some practice, estimates such as Proposition 5.3 can be done by inspection. In Sections 7-9 we develop some general machinery, the goal of which is to make these estimates more routine.

222

§6. Equations

AN EQUATION

(3.16-3.17)

OF KIRKWOOD-SALSBURG

can be rewritten = Z(A)l

TYPE

as a Banach space equation

+ :~

(6.1)

with a unique solution p =

(I-~) -I

Z(A)I

satisfying the bound

Ipl <_ I ( Z - ~ ) - I i I z ( A ) I This bound is essentially Let ~ subsets of

f

Proposition

<5_4iZ(A)J 5.2, as we shall see°

be the Banach space of functions

F c (Z2) * to n

For

f e ~

element subsets.

let

fi defined on finite

(fn)n>0

The norm on

=

|fl

(6.2)

denote the restriction

~

is defined to be

Irn(r) I

sup 2 -n n I"

Ir[--n We define

to be the function

PA - (PA,n)n>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

<_

IZ~A(A)I-IAnXI

21x'fa(l_~)-llIZ~A(~)S-IAnXl

< e Proof of Theorem 6.1° the equations

_< 2

K21xl

Let

1 = (1,0,0,...)

E~,

and define ~

by

223 (~f)n(r) for n h 2 connected

=

fn_l(F~bl)

+

[ ~ K(bl,r,X) m=l X, IXI=n

flx,url(X*ur)

where the sum over X (as in (3.13) and (3.17)) ranges over unions

of lattice

squares

such that b I e X.

For n = i, we

omit the first term on the right of (6.4) and for n = 0, ( ~ f ) 0 We now show that to obtain

(6.4)

(6.2).

1 + sup 2-1rl 2 rc~

~

is a contraction,

It suffices

~ m=l

xX

x,l I=m

I~I < 3/4,

e 0.

in order

to show that

IK(bl,r,x)l

2 IrUx[

< ~

(6.5)

The proof of (6.5) is similar to the proof of Theorem 4.1 given in §5. In particular it depends on Propositions 5.1 and 5.3. By Proposition 5.1 there are at most e K1]XI terms for each fixed value of Ixl.

From

(3.17)

each term has at least one derivative

Using this fact and (5.1) the bound from Propositions

IK(bl,Y,X) t ~ e -K(Ixl+l) eK3 Ixl For

K sufficiently

showing Thus

PAW

Z(A) W 0. 0

large

(6.5) follows.

By our choice of

and by (6.2),

Z(A) ~ 0.

c,

We complete

abl.

5.1 and 5.3 is

the proof by

ZA, = IZaA(A)I[A] ~ 0.

224 §7.

COVARIANCE OPERATORS

The basic facts for the kernel

C@(x,y) of Cg = (-A+m~) -1 are

(a)

C~(x,y) is a function of the difference

(b) (c) (d)

0 < C@(x,y) for Ix-yl h l, as Ix-yl * 0,

(e)

for

Ix-yl.

_m0(l_6)Ix_y I Cg A e C@ % log Ix-yl loc ~4, r > 2/(2-6), 6 > o, c@ e ~ r , 6 ( n ) loc _4. ~r,6(~ )

By definition, the space e ~ ' which satisfy

is the space of distributions

D~D~r,6 = q(l+k2)6/2 (~)~aLr < whenever ~ e CO . For r and r' dual indices and for i ~ r' ~ 2, ~loc r,6 is a space of functions with fractional Lr, derivatives, by the Hausdorff-Young inequality.

See [8, Chapter 2]

for a general

theory of such spaces. (e) follows from the fact that as a function of x-y, Cg has the Fourier transform (k2+m2) -I. Property (a), which is based on the translation invariance of the Laplaclan, does not extend to the operators C e ~. Recall that ~ i s the set of all convex combinations of the Dirichlet covarlance operators

(1.6), or in other words, operators of the form

ties (b) and (c) are valid for all C e ~ ,

C(s).

Proper-

while (d) is valid as an

upper bound for all C e ~ . Proposition 7.1.

The kernel

C(x,y)

of

C e~

satisfies

0 i C(x,y) i C~(x,y) Proof.

z(~)

Let dz T be the conditional Wiener density on paths x,y T starting at x at T = 0 and ending at y at T ~ T. Let Jb be

the function

jT(z) = I 0

(7.1)

b

defined on Wiene; paths.

(T.2) cr(x,y) =

] e-m~T I

[1 Then

if

z(~) ~ b,

0 < T < T

otherwise C F and

C(s)

have the representations

ber J (z) dz ,ydT

0 (7.3) C(s)(x,y) =

7 2f

e -m0T Kbe~(Sb 0 See [0, §5.3, 6.1 and 6.3].

+ (l-Sb)JbT) dzTx,y dT .

225 The inequality of Proposition 7.1 comes from substlhuting jT 0 ! (sb + (1-s b) b ) ~ 1 in the formula above for C(s). The next estimate combines the exponential decay (c) with a local bound coming from (d). We label each lattice square A = Aj c R 2 by the lattice point

J e z2

in the lower right corner of A.

Any lattice square (or any set X c R 2), will be identified with the multiplication on L2(R2) by its characteristic function. j = (Jl,J2) 6 Z 4 be a pair of lattice points. ized covariance operator

(7.4)

Now let

We define the local-

C(J) = AJl C Aj2

and the distance

(7.5)

d(J) = Dist (AJI'AJ2)

'

which measures the nonlocality of C(J). Proposition K4(q,6)

7.2.

For

independent of

1 < q < =

m0 > 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

<

the proposition

(and slightly more). since it is not needed in

follows. Proposition. 7.4.

with

to

Tr B A*A B A*A

The reader may skip the next result, what

Schmidt

2 ~ C ~(-A+m~.~ - ~ + I / 4 B-HS

< Tr (B A'A) B A*A =

proves

is Hilbert

Schmidt norm

= Tr A B A*A B A*

This inequality

,

C e ~ l°c ,Jq,~ Proof.

For any

q ~ (4/3,~,

and the bounds

With

there is a ~ = ~(q)

are independent

of C 6

k = (kl,k 2) e R 4 , define

f(k)

= l+k 2

h(k)

,

g(k)

=

(l+k~)(l+k~)

= (~ C ~) (k) .

,

>

0

227 Then g By Proposition

-e-i

e LI ,

7.1 and the fact that

f/g • L

.

~Cz~ e L I ,

h6L and by

(7.6), g-e+i/4

We apply these

facts

h 6 L2

and

to the integral

(7.7) I ff /2 hlq= I 1g- +l/%l First

take

6 > 0

The factors For

e

Since

and suppose

in (7.7) belong

small

and

h e L

q

near

I q/21gl( q/2)+ -l/4thlq-1

that

h 6 Lql ,

to L2, L4+ e ,

h 6q Lq

for small

~.

Pr0position

completes

7.5.

For

h e

h e L

q > 4/3. n

L

Lq also.

for

(7.7) shows thatq

, q > 4/3,

This

ql >

and Lql/q_l

ql ' this implies

for q in an interval,

Now with

h 6 L2

q e (4/3,o].

f~/2h ~ Lq

also,

the proof. 1 ! q < ~ ,

c(x) = limy+x

C(x,y)

- C~(x,y)

There is a constant

K5(q),

independent

that for any lattice

square

A,

of

e L l°c q

m 0 h 1 and

C e ~

such

~CULq(A ) ! K 5 m0 I/q Proof. Let

F =~

By scaling,

we may take

be the set of all lattice

m 0 = l, as in Proposition lines.

7.2.

For x ~ F,

0 ! - c(x) i C~(x,x) - Cr(x,x) ! This inequality For

x ~ F,

o(I +

completes

Jlog dlst(x,r)l)

the proof.

It is proved in [7] as follows.

y ~ F , (-Ay+m~)[Cg(x,y)-CF(X,y)]

Hence by the

maximum p r i n c i p l e ,

0 ~ C~(x,y)

- Cr(x,y)

and the

~ SUpyer

fact

C~(x,y)

= 0 . that

CF(x,y)

~ O(l+llog

= 0 for

dist(x,r)I).

yet,

228

§8.

DERIVATIVES OF COVARIANCE OPERATORS

For the differentiated covariance operator a strong decay, path in R 2 segment

%e-m0 d,

where

d

~Yc,

there should be

is the length of the shortest

Joining x to y and passing through each lattice line

b e y.

This can be seen by inspection from the Wiener inte-

gral representation (sYC(s))(x,y) = 7 e-m~T I Kbex(l-J ~) 0

(8.1)

x ~bE~y[Sb

+ (l-Sb)J~] dZx,y T dT ~

We need the improved bounds on ~Yc for two reasons. is to localize x and y, with y given. For this purpose (8.2)

The first

d(J,y) = sup {Dist(Ajl,b) + Dist(Aj ,b)} bey 2

is sufficient, as a crude lower bound on d. We now explain the second use of bounds on

~Yc.

Let ~ ( r )

be

the set of all partitions

~ of the set of lattice llne segments F. f In Proposition 5.3, we are called on to bound ~r # F de s , which by

Leibnitz' rule and by (1.7) is Just

(8.3) The second use of the bounds on ~YC is to control [ ~ e ~ ( F ) As in §7, we also find a factor m00IYI, which yields the overall convergence of the expansion. Proposition 8.1. large.

Let

1 3 q < =

There are constants

K6(q,y)

and let and

m0

KT(q) ,

be sufficiently independent of m 0 ,

such that

(8.4)

usYcu Lq(AjI×Aj2 ) ~ K6(q,y) mo IYI/2q exp(-m 0 d(J,y)/2)

(8.5) Proof.

We use the Wiener integral representation

(8.1) for ~Yc.

The proof consists of estimates on the Wiener measure of paths Z(T) which cross the lattice lines

b e y

with combinatoric arguments to count

in some definite order together the number of ways the lines

229 b e T

can be so ordered. Let

lines

L(y)

be the set of all possible

b e y,

paths which ing is £. (8.6)

and for

cross

£ e L(y),

all

lines

let

b ~ y,

linear

~(~)

orderings

of the

be the set of Wiener

and whose

order of first

cross-

Then

0 < ~YC(s)

<

e

[[ bey

0 and (8.7)

~YC@(x,y)

= e~L(y )

e

Let

,

bl,b2,..,

b 2 be the first

be the elements

of

y,

of the b's not touching

first of the b's after

b2

dZx,y

~/~)

0

as ordered by £.

and not touching t

Let

,

bI = bI , b~

let

b

, etc.

be the

Set

!

aj = Dist(bj+l,b j)

,

I < J < m

and de fine I£ I

m = ~i=l ai -

T

If there

is no such

With these for

b 2 , we set

definitions,

I~I = o, by convention.

we bound the

£ e L(y) term in (8.7),

I*I >_ l, by

/

2

e -m2T Hi(2~ti )-I exp

(- ½

m ~i=l qai)

dt dT

ti=T - (m~-2)T <

K~

--

sup

-

e

~

m 2 [i=lai/t i

e

T,[ti= T

since dt _< e T

,

I e -T dT

 l, (2wti)-l thereby

defining

K 8.

Using the method we bound the

£ E L(y)

of Lagrange

I£1 >_ 1.

For

multipliers

to evaluate

the maxima,

term by

~ for

_2 6 e ai/ ti ,< K 8 < ~ ,

exp

ICm~-2~1/2 I~t),

I£1 = 0 , we use t h e r e m a r k f o l l o w i n g

Prop.

7.2.

230

There is an entirely similar estimate, based on the distance d(J,y) of (8.2), and taking geometric means of these two bounds yields (8.8)

|sYcm L

for m 0 large.

i _m0 i~i/(2+~) -mod(J ,y)/(2+~) _< ~ K YI e e q (AIXA2) £ L(y) If I£1 ~ 1

for all

~ e L(y),

then we can include a

factor mo IYI on the right side of (8.8), by increasing 6. If 141 < 1 for some £, then 141 = 0, and in this case ITI_7 A 4 With IYI ! 4 and d(J,y) 5 i, we can still include the factor m~ IY| in (8.8) by increasing ~. Finally for IYI < 4 and d(J,y) = 0, the factor mo IYI/Rq ! mo 2/q in (8.4) comes from scaling, as in Proposition 7.2. We define (8.9)

K6(q,y) : K 4

~

E~ ¥I e -m0[~I/(2+6)

~eL(y)

With this definition,

(8.4) follows; in the case d(J,y) = 0 and

I~I = 0 for some ~, K 6 ~ K 4 , and we use the bound of Proposition to establish (8.4). We complete the proof by establishing (8.5) as a separate proposition. Proposition

(8.10)

8.2.

For

[

E

m0

sufficiently

[

e

-m0[~l/3

7.2

large,

< e

K9tFI

~E~Q(F) ye~ ~ L ( y )

Proof.

Let

~(F)

be the set of linear orderings defined on

subsets of F. Thus L(F) ~ ~ ( F ) . As before, we define £ ~ $~(F). We assert that the number of ~ ~ ~ ( F ) with bounded by (8.11)

141 for l~I ~ r

is

IF[ e KlO(r+l)

Using (8.11), we complete the proof. Let A£ = exp(-mol£1/3). Expanding ~ E [ A£ in (8.10), we get a sum of terms of the form A~I A£2 ... A~j where the ~j are distinct elements of ~ ( F i . this form, we bound (8.10) by <

<

E

-- ~ e ~ ( r )

•

exp A~ = exp

=

~

~ ~e ~(r)

Adding all terms of

(I+A~)

A~

< -

exp

(O(1)IF I) .

231

Here in the last expression, we used the bound (8.11.) to estimate X~(F

) A~

and we choose m 0 sufficiently

Next we establish

(8.11).

large.

Suppose the integer part

[a i]

of the

V

distances a i. are given. We choose b I = b I in Irl ways, and we choose the b's between b~ and b~ in 0(1) ways, since they all must !

overlap b 1. Next b 2 is chosen in O(1) lattice line segments b with [a l] ~Dist(b,b~)

[a 1] ways, namely from the

< [al]+l .

Continuing in this fashion, we choose all the b's in E i 0(i)

Irl ways. the 2 r.

Finally

number

of

we c o u n t ways

In

fact

ri

as

follows:

goes

to

a 1 or

the J+ist

[a i] ~ I r l

of

suppose The a2

(one

the

first binary

0(i) [ [a i]

number

choosing ~ ri

e

of

integers

= r,

~ Irl

choices ri ~

of

the

1 with

a n d we d i s t r i b u t e

1 goes

into

choice).

a 1 (no If

the

eO(1)r [ai]. X ri the

r

choice). Jth

goes to a i or a$+ I (one binary choice).

1 goes

This ~ r,

is

ne~mely

units

in

The second to

ai

1

,

Thus there are

r-1 binary choices, or 2 r-± ways to choose r i with ~ r i = r ~ 1. Summing J = ~ r i gives ~ = 1 2 J - l = 2 J - 1 . Finall~ we get one more choice from

l~I = 0

(no ai's).

232

§9. The integral can be e v a l u a t e d

GAUSSIAN INTEGRALS

of a polynomial with respect in closed form.

to a Gaussian measure

The closed form expression

and each term in the sum is labelled by a graph. complicated polynomials

of high degree,

will also be complicated. estimates

We will e n c o u n t e r

and the resulting

However we present

for such polynomials;

the structure

is a sum,

graphs

some very simple of these estimates

can

be seen easily from the associated graphs. We define a

localized monomial

(9.1)

r ~ i=l

R =

where

w(x)

to be a polynomial ni

:¢(x i)

is supported in a product

: w(x) dx , Aj ×...×Aj

We also require w e LI+ e and it is convenient assume

of the form

of lattice

butrnot

essential

to

a bound

(9.2)

ni ! ~ ,

the bound

(9.2) does not restrict

Polynomials

which arise naturally

the kernels

w are not localized.

written

i ~ i ! r . r, nor the total degree of R. are not ushally

of this

However any polynomial

form because can be

as a sum of localized monomials.

Associated with R of (9.1) is a graph G(R) consisting vertices

and at the ith

vertex, we draw

X Fig.

I.

G[f

This formula measure

d¢C

n i legs.

:¢(xi)4:

:¢(x2)4: w(x) dx ). ] R de C , we integrate by parts:

can be proved by passing to the Fock space 7 of the , expanding

¢

as a sum of a creation

operator and using the canonical

commutation

Theorem 9.1 below.

by parts

monomial

of r

See Figure i.

X

In order to evaluate

tions,

squares.

We integrate

R we want to integrate.

the monomial

After

and an annihilation

relations.

See also

to reduce the degree of the r (~i=l ni)/2 partial integra-

is replaced by a sum of constants,

and since

7See for example Theorem 3.5 of J. Glimm and A. Jaffe, Boson quantum field models, in Mathematics of Contemporary P ~ s i c s . Ed. by R. Streater, Academic Press, New York, 1972.

233 f

i, the integral is evaluated explicitly. In applying this procedure, we encounter ¢'s in a Wick ordered factor :¢(xi)ni: in R. For such ¢'s, we use the formula de C

(9.3)

f

:¢(x)n: R de C = (n-l) c(x)

f

:¢(x)n-2: R de C

+ f :¢(x)n-l: C ( x , y ) ~ - ~~RV - de C dy

,

with c(x)

= C(x,x)

-

C¢(x,x)

defined by Proposition 7.5. The first term arises from the differf ence between the covariance C@ in : : and the covariance in J...d¢ C. The second term is exactly as before. The integration by parts formula (9.3) has a simple expression in terms of graphs. In case the ¢(x) is a factor in :¢(xi)ni:, we label the terms on the right side of (9.3) by drawing a line connecting one leg of the xi-vertex to a distinct leg at the same or a distinct vertex. The graph with a llne from the x i to the xj vertex labels each of the nj terms (9.4)

f

C(xi,x j) :¢(x i)

ni-i

: :¢(xj

)nj-l. .

H £#i,J

:¢(x£)

coming from a single integration by parts in (9.1).

I c××l

f

--

+

Fig. 2.

Integration by Parts.

n~

: w(x) dx

See Figure 2.

,Oc.

As an example, we evaluate the integral of Figure 2. integrations by parts, we have

After four

If :~(Xl)4::~(x2 )4: w dx d~c -- 4, f C(Xl'X2)4w dx

The absolute value of the first term is bounded by 4! I

C(Xl,X2 )4 W(Xl,X 2) dx I <_ 4! ICIL8(AJlXAj2 ) < 4! K4mo1 e-m0 (l-~)4d(j)

if suppt w c AjlX Aj2.

|W|L2

In the last line, we used Proposition 7.2 to

234

bound |Cm

See Figure 3

Lq(AjlXAj2 ) "

Fig. 3. To evaluate

Evaluation of a Gaussian Integral.

[ R de C

~//(R) of all vacuum graphs.

in the general case, we form the set A vacuum graph is a graph obtained from

G(R) by Joining pairwise distinct legs until all legs are so Joined. If the total number of legs is odd, ~ R ) For each G e ~ ( R ) , define

(9.5)

= @

(and ~ R de C = 0). J

I(G,w,C) = I w(x) El' c(xi(£'))E£ C(Xil(£)'xi2(£)) dx .

The first product,

H~,

, runs over all lines £' of G which connect

two legs of a single vertex, labeled

i(£').

The second product runs

over all lines £ Joining pairs (i1(£),i2(£)) of distinct vertices. We now state the formula for evaluation of Gaussian integrals. The above discussion of integration by parts and its graphical interpretation gives the formal derivation. Theorem 9.1. Then

Let

R e L r ( ~ ' , dec) ,

(9.6)

w be localized and let r e [1,~),

w e Lq

for some q > 1.

and

I R d~ C = [ G ~ R )

I(G,w,C)

.

The proof has a combinatoric aspect (sketched above) and an analytic aspect.

The latter is an approximation argument, which uses

a modified (momentum cutoff) form of (9.6) to prove that the approximants converge.

For this reason, we bound the integrals I(G,w,C)

before sketching the proof of Theorem 9.1. We consider a function of the form F(Xl .... 'Xn) = ~Z F£(Xil(£),xi2(£)) where

Ai(X)

Ei Ai(xl)

is the characteristic functi,on of a set of area I, and

suppose that for each index £, 1 < il(Z) < i2(£) < n • Lemma 9.2.

With the above notation, BFJ L q -< H£ ]F£~Lq£(AiI(,% ) ,Ai2(~ ) )

235

provided that for each q

case

Proof: q = i.

i,

~ ~

q~ ; il(~) = i or i2(~) = i

.

With the substitution F ~ F q , we are reduced to the We apply HSlder's inequality successively in each of

the n variables (vertices) of F, and we use the language of graphs, in order to visualize this process. Each factor F£ corresponds to a line Joining the

il(~) vertex to the ~i

i2(£)

vertex.

Let

= {£: i!(~) < i = i2(£)} !

~i The hypothesis

= {£: ii(£) = i < i2(£)}

concerning

q~ is -i

~ Ziu fi (~i

= @ = ~i

n

IAi £e ~i

--

is also allowed, trivially.)

]F~(Xil(~)'xl)l~ £ i

Thus

UF~(xi")mLq£(Ai2(Z)) dxl

Lq£

q (Ail(£)×Ai2(£))

by H~Ider's inequality. A finite induction on i (decreasing f r o m i = n to i = I) now completes the proof. Remark.

An obvious modification

depend on a single variable, i.e.

allows some of the factors F£ to

F£ = F£(xi(£)).

Sketch of proof of Theorem 9.1. t

We introduce momentum cutoffs, This ~(x i) + J ¢(y) Jk(Y-X i) dy, and omit the x-integration. replaces R by a polynomial cylinder function Rk(X) , based on a finite dimensional subspace o f ~ . Rk(X) is integrable by the definition of de C , and for Rk(X) the formulas (9.3) and (9.6) are valid by thef above combinatorial arguments. Thus (9.6) holds for R k = ] Rk(X) w(x) dx , and can be used to evaluate the right side of the inequality I IRk-Rk' I dec ~ ( I

(Rk-Rk')2 d¢C)I/2

Removing the momentum cutoff in one linear factor ¢ at a time, the estimates of Propositions 7.2 and 7.3 are sufficient. For details see [I].

236

~loc C e ~q,~(q)

We remark that the stronger hypothesis of [1], Vq > 1

the present lectures, since only used.

,

(used in [1] to control C-Wick ordering) is not required for CZ = (-A+m~) -1

Wick ordering is

See also Proposition 7.5. Proposition 9.3.

I [ P- d¢cI

(9.7)

Let

w be localized.

Then

<_ |WNLp ~Ge~'(R) fIR, lc| L (Aj

q

(~')

)

x H£ |CN L (Ajl ) q (£),AJ2(£) where

q >__ p'n, n is defined by (9.2) and J(£) = Ji(£) is the localization of the ith vertex. Proof.

I(G,w,C). factors°

By Theorem 9.1, it is sufficient to bound each integral We use the H~lder inequality to separate w from the c and C

The integral of the c and C factors is bounded by Lemma 9.2

and the remark following it. There are

O( ~ ni/2)!

This completes the graphs in

~(R).

proof. Because of the expon-

ential decay in C as Ix-yl + ®, in general most of these graphs are very small. Efficient estimates must take advantage of this fact. For each lattice square

A,

we define

N(A) = [ {ni: AJi = A } as the number of legs Theorem 9.4.

Let

(linear factors

~(xi)) of R localized in A .

w be localized.

There is a constant

Kll such

that

for q =

I I R d¢cl i Nw| L [~A N(A)I P p'n, n defined by (9.2)°

(Kll mo 1/2 qlN(A)

]

Proof. and Z.5.

We bound the L q norm in (9.Y) by Propositions Having done this, it is sufficient to show that -m 0 ( 1 - ~ ) d ( J

[Ge~/(R) ~£ e

(£) ) < KA'(const. N(A) N(A)I)

Let ~ denote a leg of R. ized and given G ~ ~ ( R ) , Joined to

~ by G.

can be written as a sum over the choices

Also let N(A~)

.

Let A9 be the square in which 9 is local! and let A~ be the square of the leg

G e ~/(R)

and a sum over the

7.2

d(9) = dist (A ,A~).

possible contractions in

Then the sum over A'

for each

A~, for each

9.

237

The summand is independent within each square

of the choice of possible

A (v * A' held fixed).

we have only to count the number of terms. can be obtained permutation HAN(A)!

(nonuniquely,

in general)

of the legs in each square

terms in this sum.

contractions

To estimate

this

sum

Since an arbitrary

term

from a single given term by

A,

there are at most

Hence it remains

to show that

-m 0 (1-6)d(v)/2 H

{A~}

since each

d(J(£))

summation lattice

index

< ~ (const.)

-

occurs

{A~}

squares.

summation

e

v

as a

d(v)

for exactly two

is simply a set of functions

to include

change the sum and product,

all such functions.

We note

< --

• that

the set of

Then we can inter-

-cd(v)

e v

The

obtaining

-cd(v) H

v's.

from legs to

We increase the left side by enlarging

indices

~{A~}

,

v

~

~ v

e

=

H

{A~: v fixed}

Kll is independent

const. v

of s

and

m0 ,

for

m 0 bounded

away from zero. Theorem 9.5. Re k ~ 0.

Then

Let

A

be a union of lattice squares

e -V(A) e L p ( ~ ', de C )

a constant K12 independent

With

Re k

bounded

chosen independent Remark. contained.

of

~121AI

Corollary

of [I, §II.3, p. 22-27]

is self

Theorem 9.4

7.4), the proof of (9.8) is nearly

proof that

e -V(A) e L I ( ~ , , d~c).

identical

This proof is

to that of [3].

9.6.

IR e-V(A)

KI0 can also be

k and m 0.

Proposition

close in structure

Proof.

e

In fact, using a slight generalization

(standard)

I

A

m 0 bounded away from zero,

The simple proof

(incorporating to the

e -xv d~cl

and

There is

of C such that

Izl

(9.8)

and let

for all p ~ [1,~).

d¢cI

For

 1 and

e KI2[AI

q _> p'n,

|wl L

P By the Schwartz inequality,

[RA N(A)!

(2KII mol/2q)N(A)

].

238

The factors

on the right

Theore_~m 9.7. A

Let

are estimate

w be a localized kernel in

be a union of lattice

Then

by Theoresm 9.4 a n d 9.5°

squares

and let

Lp, p > i, let

F = Re -V(A)

in (1.7).

(1.7) is valid. Sketch of Proof:

suppose

In order to present

first that F is a polynomial.

explicitly respect

in terms of graphs by

to

sb

in these

formulas

the formal ideas, we

Then

J F dec(s)

(9.5) - (9.6).

df ; d~c(s) = [a~9'(F) f [4 (ds-~c4) ~4'~

C£

denotes

C(Xil(%),xi2(£)).

E4'~%

in effect removes

The product G ~ ~(F).

Equivalently,

by the line %. linear factors

from F, or the same as differentiating

removed

is equivalent

sum is Just

G E ~/(F), of F

with respect

} A¢ F ,

so

The proof in the general starting with

approximations

F

of linear factors.

F = R e

a polynomial.

is given by Corollary

Such a

the right side of (9.9) as

• A¢] F dec(s)

case,

F with respect

~4 • the sum over lines

to a sum over mixed second

to pairs

we identify

IC I) [(ds~ 2

mations,

from F the two legs Joined

H o w e v e r removing legs from F is the same as removing Thus we see that

from

c4' w dx

one line from the v a c u u m graph

one could remove

to these linear factors. derivatives

with

yields

(9.9) ~ where

is given

Differentiating

9.6.

-V

,

.

is based on approxi-

The control For details

over these see [I].

239

§i0.

CONVERGENCE:

THE PROOF COMPLETED

Proof of Proposition 5.3. Without loss of generality, the kernel w is localized, and in this case we take ~w| = gwJ 2 . The expression we want to estimate is (i0 i) "

< f] ~F f]

n e -kv(A) ds(r), w> Ei=l ¢(xi) des(F)

Let # ( F ) be the set of all partitions and (1.7), (i0.I) equals (10.2) where

w

F.

By Leibnitz'

rule

[lIyew ~i 8Yc "A¢]~ ¢(xi)e -kV(A) de s (r) ds(r),w>

C = C(s(r)). As in (7.3), we define ~YO(Jy) = Aj

~Yc Aj l,y

where

of

J

~Yc(Jy).A¢

= (Jl,y,J2,y) 6 Z 4, are localized in

2 'Y

so that the two derivatives in Ajl and

Aj2

respectively, and

~YC = ~jy ~Yc(Jy) . We substitute this identity into (10.2) and expand. The resulting sum is now indexed by localizations {iT} and partitions w e ~(F). For a given term let M = M(~,{Jy}) be the number of terms resulting from the differentiations A¢ in (10.2). By Corollary 9.6 each of the resulting terms can be estimated by |w'g

Lp

eEl21Al ~A N(A)!

(2Kll)N(A)

for m 0 ~ i. Here w' is the w of (i0.I) multiplied by the kernels ~Yc arising in (10.2). From Proposition 8.1 and Lemma 9.2 we have (for p < 2, and q large), HW'0Lp < gW|L2 l~ye w ~YC(Jy)|Lq -Irl/2q -m0d(Jy,y)/2 flwnL2 m 0 Kye w K6(q,y) e Now using (8.5) to control the sum over w e ~(F)

we can bound

240

(10.2)

by

RWIL2

eKFIrl m0 -Irl/2q

The proof control

M and

of

[

max

Proposition

the

sum over

number of elements Lemma 10.1°

M

{jy} ~e f ( r ) 5.3

K

follows

{Jy}

e- m 0 d ( j Y ' Y ) / 2

year

a

from

two

respectively.

in the set

{Ji,y: AJ i

There exists

a constant

~ N(A)!

lemmas

Let

which

M(A) b e

the

= A, i = i or 2, y e ~}. KI3

, independent

of

m0 ,

such that P

M < e KI3[I~]

n (M(A) !) A

and II N(A)] A where

<

e KI31FI

p is the degree of the interaction Lemma 10.2.

constant

KI4

Given

~ e ~(r)

, independent

of

{Jy} ye~

differentiations

in

in

Let

A.

A

N0(A) ! [A N0(A) from all

there exists

a

m 0 , such that M( A)!r

i

e

Kl41rl

N0(A)

The number

be the number

of

of terms resulting

x i , l~i<_n, from M(A)

is bounded by

(N0(A)+l)(N0(A)+p+l)

resulting

P.

A

Proof of Lemma i0.I: which are localized

polynomial

and r > 0,

~ e-m0d(JY 'Y)/2

Since

H (M(A)I)P A

= n,

~/~¢(y)

...

(No(A)+p(M(A)-I)+I)

we have M, the total number of terms differentiations,

M ! KAPPM(A)(N0(A)+I+pM(A))

bounded by

No(a)+l

M(A)!P ;

using the inequalities (a+b)! Furthermore

with

< (a+b)a(b!) N(A),

after differentiation,

and

(ab): < a a b ( b ! ) a .

as defined in §9, the n u m b e r of legs we have

N(a) _< NO(a) + (p-Z)M(a)

in A,

241 and so

~N(A)!

is bounded as above.

Proof of Lemma 10.2: exponentially

The sum

decreasing distance

[{jy} is controlled by the

factor, so it is sufficient

~A M(g)!r <--]IV eC°nst']Y]

eC°nst'[Y

to show

d(J,y)

with constants independent of m 0 , X, {J~} and w. Recall that d(J,y), defined by (8.2), contains the distance

from

Jl,y ~nd J2,y to some b e y . Thus for fixed A, there are at most O(1)r ~ values of y within a fixed partition w such that (10.4) and

Ajv,y = A ,

d(J,y) ~ r.

(10.4).

By definition there are

The most distant half

(= M(A)/2)

~ = I or 2, M(A)

y's which satisfy

of these y's must also

satisfy M(A) I/2 £ const, d(J,y) + const. because the

y's are nonoverlapping.

M(A) 3/2 _< const.

Hence

[¥ {d(J,y): Ajv, Y = A} + const.

and so the proof is completed by the inequality H A M(A)Z r _< exp

(r [A M(A) £n M(A)]

! exp (0 [ {M(A)I+~: M(A) > 0})

~_ e~p (o [~ d(J,~-)] exp ( o l r l )

242

REFERENCES 0.

Z. Ciesielski, potential

I.

Lectures

theory,

J. Dimock

on Brownian motion, heat conduCtlol~ and

Aarhus

Space and Applications 2.

Universitet,

and J. Glimm.

J. Ginibre.

General

Comm. Math.

Phys.

Measures

to P(¢)2

field theories.

formulation

16 (1970)

1965.

on the Schwartz distribution

of Griffiths

inequalities.

310-328. h

3.

J. Glimm and A. Jaffe. cutoffs.

III

The 1(¢)~ quantum field theory without

The physical

vacuum.

Acta Math.

125 (1970)

203-261. 4.

J. Glimm and A. Jaffe. cutoffs,

IV.

The 1(¢)~ quantum field theory without

Perturbations

of the Hamiltonian.

J. Math.

Phys.

13 (1972) 1568-1584. 5.

J. Glimm and A. Jaffe. Fort°

6.

der Physik.

of the ¢~ Hamiltonian.

J. Glimm, A. Jaffe and T. Spencer. particle

structure

7.

F. Guerra.

8.

L.

9.

J. Klauder.

L. Rosen and B. Simon.

HSrmander.

statistical

Berlin,

Ultralocal Phys.

J. Lebowltz

and

To appear.

The P(¢)2 quantum field

mechanics.

Linear Partial Differential

Springer-Verlag, Comm. Math.

The Wightman axioms

in the P(¢)2 quantum field model.

theory as classical

I0.

Positivity

To appear.

Operators.

1964. scalar field models.

18 (1970)

307-318.

and O. Penrose.

Decay of correlations.

Preprint. ii.

12.

R. Minlos and

Ja. Sinai.

The phenomenon

at low temperatures

in some lattice models

Trans.

Soc. Vol.

Moscow Math.

C. Newman.

Ultralocal

Comm. Math.

Phys.

13.

D. Ruelle.

Statistical

14.

T. Spencer.

of phase separation of a gas II.

19 (1968), 121-196.

quantum field theory in terms of currents.

26 (1972) 169-204. Mechanics.

Benjamin,

New York,

1969. The mass gap for the P(¢)2 quantum field model with

a strong external 15.

field.

K. Wilson and J. Kogut. c-expansion.

Preprint. The renormalization

Phys. Reports,

to appear.

group and the

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 , ~ )

,

~ -<~ < ~

,

(smoothnessand l o c a l i t y ) .

b) U^ = Umi+ Um~ , if A = A~ u A~ with zero Lebesgue measure ( a d d i t i v i t y ) .

and A i

and / i ,

have intersection

c) d)

e -u^ E L~'(Q,Z^, p )

,

~. -~-P ~ ':~'.

The Gibbsian factor plays a very important role in two respects. First of a l l i t allows to write the cut off interacting Hamiltonian Hz in ~ o ~

space

tlO,41J , defined by

in the following particularly simple form UA (4,~) where xz and ~ are Q space functions and 0 .~c2 - ~

A(-#,y#)

is the rectangle-2/2.<~ci~<~/2 ,

. We call this expression the Feynman-Kac-Nelson formula [24,18] . In the particular case Xz:zr=-fZo

, taking into account the Euclidean

covariance of the free Markov f i e l d , we obtain the following celebrated Nelson's symmetry

< ~ o , e_~,,e rz > = < ~ 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,~)

<

ZA(.e,~t)

and

On the other hand by the spectral theorem we have and therefore

..lZo )

,

252

~>~ ~(~,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<~< oo

to oo i f

Even though ~

~

cannot be absolutely continuous with respect to ~

and tends

, there is no

general principle to prevent the occurrence of such absolute continuity locally, i.e. when the measures ~

and F

are restricted to every sub-C-algebra >-A ,

associated to a compact region A , at least for two dimensional theories. In fact there are the following indications that such absolute continuity does occur a) i t holds for spatially cut off theories, b) i t can be explicitely verified in the exactly solvable linear and quadratic models, c) i t is true in perturbation theory d) i t is suggested by the locally C ~ o K

property of the Hamiltonian theory proved

by Glimm and Jaffe [9]. e) i t follows form recent results of Newman ~9] for the PC~ in the small coupling region. Thus we are led to the following definition. Definition 12. A smooth state ~

on~Q,Z) is a family I ~ I

~^ , labelled by compact regions A a) each ~^

of Q space functions

of Rz , so that

is Zm rmasurable, almost everywhere non negative, in

LI(Q,>-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 ~^

~^I

=

, 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<~ &(^)

as A-~ oo

A /p-smooth state

, where ]AI

"is the area of

A

F is called weakly tempered i f there are cons-

and ct>o such that for all regions A with s u f f i c i e n t l y large diameter

we have

~

II ~ Ilt~ ~ -~e [ ~ ~ ~/'~ ]

•

255 Weak temperedness is a l l we need for the control of the i n f i n i t e

volume

l i m i t of the entropy density, but we expect that the i n f i n i t e volume P(@)astates s a t i s f y a stronger condition: D e f i n i t i o n 16. I f f o r some constant A we have

for all large &(A) then ~

is called tempered.

Now we can state the main r e s u l t of t h i s section. Theorem 17.

Let F be a t r a n s l a t i o n i n v a r i a n t , weakly tempered,~-smooth state.

Then the f o l l o w i n g l i m i t exists

s~) = ~,~

Iz

1-~ ~^

(~-)

as k - > ~

(Fisher). The functional S(~) is a f f i n e in

A i + A z =#-

, we have

~

, i.e. for

O _cAt _~ i

We r e f e r to [18] f o r a complete proof of t h i s theorem. Here we give a l l d e t a i l s of the proof in the case of A square, in order to show how hypercontractiv i t y , in p a r t i c u l a r the checkerboard estimate (Theorem 4) , allows us to decouple d i s t a n t regions and get a thermodynamic behaviour.

Proof.

Fix

Given ~'

~Z~ so that .~<~<~[ . For each fixed ~

, let ~(=~?'and ~-=~+Z~(

, write /'=~Z÷ ~ , with~ integer and os } < ~ . The square A' of side

can be partially f i l l e d by ~

squares A~] , ~,j"

square A;] contains a concentric square A;]

&, .,.~.

, I f side ~

Each

of side ~ surrounded by a border of

width ~ . Now we use the decrease and weak subadditivity of the entropy to write

S(p

-

Then by t r a n s l a t i o n invariance of

C

and the checkerboard estimate we have

where A is a square of side ~ and -- ~ -

F6z - !

<- ~'~p(-C ~ Z ' )

)

, so that

for large

Now we let Z'-~o~ with ~ fixed. Since I,~ I =

~Z

,

IAI

= .~z

, we have

Therefore, with the d e f i n i t i o n ~(~) = Y2(---~ I A ' I "~ ~

(A')

, we have

IA'J= (,. Z-,-

)"

,

256

SC1C) ~< I/~1 -~ ~ ( A ) ~ )

tAII~I-~-_~

Since

rL(1)-> 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

~<~

, on the other

, therefore -zM).->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~<;

Here L)~ is the D i r i c h l e t i n t e r a c t i o n in the region

/l; )

and

Z i = JQ~ e " L ) I c { F ~

is the D i r i c h l e t p a r t i t i o n function f o r a square of

side Proposition 21. The state ~ introduced above is

)p-smooth for any ~ : , a n d

pered. Proof. In fact consider

A = U i AZ

, so that

IAI=~ 2 ~

, then

tem-

259

Uz

-Z_ Since the f i e l d s

qP~ are independent, by translation invariance we have

.~# ]lfA 114,

c# I AI

=

with

c~ = 7 -z z-, ~ IJ ~ u~ i]4, Proposition 22. For the state ~ introduced above the entropy density Sf~) is expressed as

£-z~P'~Z~

+2-zZ2

~

S ( ~c ) =

fo

Ut e

c~

and the mean interaction is given by

Proof. The expression of s(~J follows from straightforward calculation. For (~ we must take into account that Elm : ~^' p((8(.~,): ~x :

]~ U~ ~

£

U^~ :

U;

and i

=

~,~,..

,

as explained in Section 4. The state F defined above is i n v a r i a n t only for l a t t i c e translations of amplitude ~

, but a translation i n v a r i a n t state ~ can be easily obtained by

averaging over translations, according to a well known procedure, in such a way that £

where A is a Q space function and T, the translation

R2_> ~2+ ~

, z c R z , is the operator implementing

. As in s t a t i s t i c a l mechanics ~3] this averaging

procedure does not change the entropy density and the mean interaction. Theorem 23. (Gibbs variational p r i n c i p l e ) . For any semibounded polynomial

where the supremum is taken over a l l smooth, translation i n v a r i a n t , tempered states. Proof. For the state

~

constructed before we have

by Proposition 22. But we know from Theorem 10 that, for ~

large enough, the r i g h t

hand side can be taken as near to dm as we l i k e and the theorem is established. Consider now the anharmonic o s c i l l a t o r with Hamiltonian where ~

~= Ho÷ F(R)

is a polynomial bounded below, and l e t _o_ be the ground state and E the

ground state energy. The free Markov f i e l d in the Euclidean (imaginary time) region

260 for the harmonic oscillator is a Gaussian stochastic process 9(4) , ~ , defined by the expectations E (q(~J)=O and E ( q ( ~ q ( ~ g ) = ~ x p ( - I ~ - 4 ' l ) Using the Feynman-Kac-Nelson formula i t is very simple [18] to write the Euclidean infinite volume state associated to the giveninteraction ~ . For each f i n i t e interval [~,6] the system of compatible densities is given by ~[,,b] = ( J ~ ' p - ) ( I b ' O ' )

e-I~P{q(~))~t e ( ~ ' ~ ) E

,

where ~ are the embedding operators introduced in Section 2. It is simple to verify that F~a,~] is a positive, normalized, L~(~,>-/a,&],~) function on the space of the free process q ( t ] . Here ZE~,&] is the ~--algebra generated by ~(~] for 0. ~ . Moreover for large 6 - ~ we have

therefore the state

F

is tempered.

Now by e x p l i c i t simple computations we can prove the f o l l o w i n g theorem. Theorem 24. For the one-dimensional Markov theory (the anharmonic o s c i l l a t o r in the Euclidean region) we have

cx~C~)

:-

E

, 9(f,P)

: <-cl, P - n - >

,

s(~)=-<-rz, Horz>,

therefore 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 is v e r i f i e d , because = E-F2.

(~÷P)JT_=

and moreover i t is equivalent to the usual Rayleigh-Ritz v a r i a t i o n a l

p r i n c i p l e for the ground state energy. This example shows that the entropy density is the Euclidean counterpart of the free Hamilton expectation value. In the case of n o n - r e l a t i v i s t i c quantum mechanics general results of t h i s type have also been obtained by W. C r u t c h f i e l d ~2]. The one dimensional example suggests the f o l l o w i n g conjecture for the two-dimensional Euclidean theory. Conjecture 25. The v a r i a t i o n a l p r i n c i p l e for the entropy density in the two-dimensional Euclidean theory is equivalent to a Rayleigh-Ritz v a r i a t i o n a l p r i n c i p l e f o r the energy density in the Hamiltonian theory. From t h i s point of view the i n f i n i t e

volume states of the Hamiltonian

theory associated to a given i n t e r a c t i o n would be characterized as those states f o r

which the total energy density takes the minimum value - o ~ ( P ) . 7. THE EQUILIBRIUM EQUATIONS In classical s t a t i s t i c a l

mechanics Dobrushin [3,4,5]

introduced a set of r e l a t i o n s f o r the i n f i n i t e

and Lanford and Ruelle [21]

volume states, which express in a

compact way the fact that the state is an e q u i l i b r i u m state f o r a system with some given i n t e r a c t i o n at some f i x e d temperature. Following our i n t e r p r e t a t i o n of E u c l i dean f i e l d theory as classical s t a t i s t i c a l

mechanics we w i l l introduce equations of

261

the same type, according to the basic i n t u i t i v e idea, following from Markov property, that a change of interaction outside a bounded region

A

can affect the state

restricted to A only through a multiplicative factor concentrated at the boundary. A

Definition 26. A measure ~

on the measurable space (Q,Z) of the free Markov f i e l d

is called Gibbsian for the interaction P

( e - G i b b s i a n ) i f and only i f , for every

compact region A in Rz , we have i ) the r e s t r i c t i o n of ~ to ~A is absolutely continuous with respect to the free measure i~

, i.e. d ~ : f ^ ~

o n Z ^ , with

F~

in

L}(~,~A,~)

i i ) for any bounded, Z m measurable Q space function A

(A) = where E)^

(A uA) /

,

we have

(e -u")

and E~A denote conditional expectations with respect to Z-~m for

the measures ~

and ~

respectively.

As a motivation for the assumption that ~-Gibbsian states are the relevant physical states associated to a given interaction, we consider the case in which the measure ~

is obtained through a l i m i t i n g procedure from the approximate

measures

corresponding to volume cut o f f interacting theories, taking

A ~-> 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

(1.6)

we make the following definitions:

The lattice cutoff field

f~,n (x)

p(k) -~

to have covariance matrix

; i.e.,

Cnn'

[F°r

~(n)

~8(n) = %(f6,n ) =

~(n)

where

(2~)-dI d eik'(x-n6) JT 6

' f~,n' >~

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

^ext -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, q<-->p

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
2.2, the integral

off-diagonal

(2.8)

part of the Gaussian.

(f (q)-f (P)) (g (q) -g (P) l

is nonnegative

by Lemma

in Theorem

2.1 that

for both

~

(2.1)

(2.8)

by expanding

the

But each term in this expansion

i' (-Aii') (qiqi' + PiPi'

d~° (qldg° (p)

2.3.

It is of some interest

examples

can be evaluated

> 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

<#(Xl)'''~(Xr):~2(x): >A >

AS for the satisfies c o n d i t i o n

FKG

inequalities,

(A)

o b t a i n T h e o r e m 2.8 b e l o w

<#(Xl)'''~(Xr)>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 ~+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[<#(Xl)

a

as an example,

It is easy to see that

and that

2

.#(Xr):#a(x):> A

..

A

• <:~2 (x) :>~] 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, <~(fi)~(fj) > T icA j~B

to the increasing

~ ~ T icA j~B

<- 1 i~AZ

FKG

inequality

~B "

from adding the pair of

FKG

inequalities,

in

0

and

<(~B - K B ) ~ A > T ~

function. H_I( ~ 2) Axiom A

applied

0 .

Similarly

for the

(3.3).

"the field couples

in crude terms,

(3.3)

The second inequality

Simon has exploited Lemma 3.6 to prove that in theories

=

7 <#(fi)~(fj)> T . j~B

is just the

field

Then we have

and

T ~

functions

hold.

in a Euclidean

HA

last inequality

(see

are finite index sets).

The first inequality

follows

[ii].

in terms of the variables

theory for which the 0 < ~A --

the

P(~)

2

the vacuum to the first excited state",

that the mass gap must show up in the two point

In general,

consider a Euclidean Markov field theory over

satisfying Nelson's [12, 10] and the

FKG

axioms including the sharp time inequalities.

The relativistic

or

285

Hilbert zero

space

subspace

~

is o b t a i n e d

in Nelson's

of

L z(Q,d~)

~ = E L 2(Q,dp)

;

construction ,

E

being

0

orthogonal

projection

the c o r r e s p o n d i n g vacuum, t

< ... < t 2

--

Hamiltonian

--

--

(~'¢0 (gl)e

"time"-

the

0

supported

operator

then the F e y n m a n - K a c - N e l s o n

< t i

onto v e c t o r s

as the

on

at

~

t = 0

If

~

its u n i q u e

and

formula

asserts

that

H

is

for

, n

- ( t - t )H - (tn-tn_,) H 2 I ¢0 (g2) ... e ¢0 ( g n ) ~ ) ~ (3.4) t =(¢(gi

where gt(x

1

~0

is the time

,x 2) = g(x

l -I o(Ig) Lemma

to

3.7.

i

satisfying

Using

as

Consider

axioms

< t i

O

and the c o n v e r g e n c e

Markov

field theory

< ... < t }

is total

for s i m p l i c i t y

is an e i g e n v a l u e

in

(3.4)

and

t.+t f~3 = gJ 3

<~A

the

t >

hA

spectral

T

=

over

~_I(]R2)

Q

E

that

> 0 .

If

the

next

~

'

n ~A = 3~Ip(fj).=

theorem,

(~, e-t~

point

in

a(H)

is a c o r r e s p o n d i n g i

e i g e n v e c t o r , then by L e m m a 3.7 there is some t tn = E 0 p ( g * i) "'" P(gn ) ~ S such that (~'~i) ,

of

n

i

t. fj = gj3

and

including

2

Now s u p p o s e above

(3.4)

gj E S (IR I)

A x i o m A for sharp time fields. t t S = {E0 P(gl I) "'" P(gn n) Igj E S ( ~ I) ,

T h e n the set of v e c t o r s 0 < t

~,

I ÷ 0 , Simon has shown:

a Euclidean

Nelson's

g. > 0 , 3 --

i) "'" ¢(gn n)>,

zero field on

)~(x -t) 2

¢(g)

t

we h a v e

~) - I (~,~)

and

for

fl 0 .

Let

~At = K p(f~) . t

By

> 0

12

-tE

>_ l(*, a~)l ~ e Now L e m m a (3.3)

that

3.6 e x t e n d s

to t h e s e

i sharp time

f. 3

and we d e d u c e

from

286

-tE (@,n)I 2 e

* < 1 E E < 9(fi)9(f t) >

•

--

i

j

T

-

(t+tj-ti) H

4j li7 7 [ (90 (gi)~' e

(3.5)

90 (gj)~)

- (~,90(gi)~)(~,90 (gj)n)]

assumlng

t ~ t i - tj

that for some E,

,

gi

, for all

i, j .

We conclude from

and some eigenvector

(90(gi)~ , ~i) ~ 0 .

If

El

~i

of

H

with e i g e n v a l u e

is in the continuous

a similar argument using spectral measures

(3.5)

spectrum,

carries through,

and so

we have proved: Theorem

3.8.

~_I(IR2)

Consider

a Euclidean Markov field theory over

satisfying Nelson's

Then the vectors

{90 (g)~

axioms

and the

FKG

I g ~ S ( ~ i) , g ~ 0}

first excited state of the Hamiltonian

inequalities.

are coupled to the

H .

Remarks ' i.

A related result has been obtained by Glimm,

Spencer

[6] and Jaffe's

(see

2.

An almost identical

P(9)

theory (see Simon 2 In the case where H

3. (say

P(9)

2

with

P

lectures).

argument works

~_I(IR2)

is invariant under the symmetry

~

corresponding

i

<9(f) > = 0

by ordinary

9 + -9

,

to

E

1

which is odd

so that we can replace

expectations:

Consider a E u c l i d e a n M a r k e v field theory over

which satisfies Nelson's

and for which ~ 9 ( f ) > = 0 . lowest points

cutoff

9 + -9 -

truncated expectations 3.9.

for the spatially

[15]).

In this latter case

Corollary

and

even) -it is a corollary of Theorem 3.8 that

there is an eigenvector under the s!nT~etry

Jaffe,

axioms

Then the gap

in the spectrum of

H

and the bE

FKG

between

inequalities the two

is given in terms of the two

287

point

Schwinger

AE

=

function

sup f>0

-

by

f

lira ~ log t~

S(x,t;y,0)

f(x) f ( y ) d x d y

.

(3.6)

f~S (IR 1)

For

a spatially

cutoff

P(~)

(g)

theory

with

P

even,

2

formula h(x,s)

(3.6)

holds

= g(s)

tells

Below

us n o t h i n g

(bE) -I

i.

Confining

immediately Theorem

is the S c h w i n g e r

the c r i t i c a l

turn

length

gap

S

because

We n e x t correlation

where

AE

point,

for I s i n g

(3.6)

for~ula

with

(3.6)

of

cutoff course

H 0

to a p p l i c a t i o n

formula

function

(iii)

ferromagnets

with

: the a n a l o g u e

is the i n v e r s e

the m o n o t o n i c i t y

of

of

the

the m a s s

of L e m ~ a

3.1 w e

with

even,

obtain:

3.10.

Consider

any of the

P(~)

theory

P

1

P(x)

n x2 i = Z a2i

; a spatially

cutoff

or infinite

volume

P(~)2

1 theory

with

P

even,

P(x)

= Pe(X)

+ a x2 o

Then

the m a s s

gap

bE

0

is an i n c r e a s i n g 2.

There

are

function

a number

on the c o r r e l a t i o n that

bE

result

is c o m p l i c a t e d of W i c k

can be

controlled

3.

When

the f a c t

3.8 s t i l l

function

Theorem

3.11.

permit

[16]

of

term

m in

the

defined

3.10

a change

special 3.4)

[i0] P the

a monotonicity

Consider

P = ax ~ + b x 2 - ~x

that

In the

results

,

or

a0

for

AE

based

suggests,

of the b a r e m a s s

(as in T h e o r e m

is a l i n e a r

Theorem

Theorem

function

ordering.

is an i n c r e a s i n g there

by

a2,..,a2n

monotomicity

inequalities.

bE

to

of o t h e r

is an i n c r e a s i n g

definition change

of the p a r a m e t e r s

in

case

m m

of

for .

Such

changes

GHS

out

the c r i t i c a l inequality

a the

a~ ~ + b~ ~

a n d it turns

above

instance,

this

that point.

and

result:

infinite

by Theorem

volume 3.4.

theory Then

corresponding

the m a s s

gap

288

AE

is an increasing

function of

This result expresses phase

transition

phase

transitions

be elaborated

as

i~i ÷ 0 .

the idea that we get closer to a There is a preliminary

(application

on in Simon's

i~i

(iv))

in

GRS

discussion

of

[i0] and this topic will

lectures.

References i. S. Albeverio

and R. Hoegh-Krohn,

Gap for Strong Interactions Space-Time,

University

2. C. Fortuin,

of Exponential

Type in T w o - D i m e n s i o n a l

of Oslo preprint.

P. Kasteleyn

on Some Partially

The Wightman Axioms and the Mass

and J. Ginibre,

Ordered

Correlation

Sets, Comm. Math.

Inequalities

Phys.

22

(1971)

89-103. 3. J. Frohlich,

Schwinger Functions

University 4. J. Ginibre, Math.

16

(1970)

5. J. Glimm and A. Jaffe, Physo

(1972)

Particle 7. R. Griffiths, Math. (1967) 8.

and T. Spencer, in the P(~)

Correlation 8 (1967)

The Wightman Axioms 2

Quantum Field Theory,

in Ising Ferromagnets, 478-483,

and the

484-489,

preprint

I, II, III, J.

Comm. Math°

Phys.

6

121-127. , Phase Transitions,

Quantum Field Theory, Editors, 9. R. Griffiths,

Gordon and Breach, C. Hurst,

Phys.

ii

in Statistical

Les Houches

(1970)

Mechanics

1970, C. De Witt,

New Y~rk,

and S. Sherman,

of an Ising Ferromagnet Math.

Comm.

1568-1584.

Structure

Phys.

Inequalities,

The ~ ( ~ ) z Quantum Field Theory Without of the Hamiltonian, J. Math.

IV. Perturbations

6. J. Glimm, A. Jaffe,

of Griffiths'

310-328.

Cutoffs. 13

Functionals,

of Geneva preprint.

General Formulation Phys.

and their Generating

1971.

Concavity of M a g n e t i z a t i o n

in a Positive External

790-795.

and R. Stora,

Field,

J.

289

10. F. Guerra, L. Rosen,

and B. Simon, The P(~) Euclidean Quantum z Field Theory as C l a s s i c a l Statistical Mechanics, Ann. Math., to appear.

ii. J. Lebowitz,

Bounds on the C o r r e l a t i o n s

of F e r r o m a g n e t i c (1972) 12. E. Nelson,

Ising Spin Systems, Comm. Math.

Phys.

28

313-321. C o n s t r u c t i o n of Q u a n t u m Fields from Markoff Fields,

Funct. Anal. 13.

and A n a l y t i c i t y Properties

12

(1973)

J.

97-112.

, The Free Markoff Field, J. Funct. Anal.

12

(1973)

211-

227. 14. D. Ruelle,

S t a t i s t i c a l Mechanics,

15. B. Simon, C o r r e l a t i o n

Benjamin, New York,

1969.

Inequalities and the Mass Gap in P($)

I. D o m i n a t i o n by the Two P o i n t Function,

Comm. Math.

2 Phys.,

to appear. 16. B. Simon and R. Griffiths, Ising Model, 17. K. Symanzik, Theory,

The

Comm. Math.

(~4) Field Theory as a Classical 2 Phys., to appear.

E u c l i d e a n Q u a n t u m Field Theory,

in Local Quantum

Proceedings o f the I n t e r n a t i o n a l School of Physics

"Enrico Fermi", Course 45, R. Jost, Editor, A c a d e m i c Press, New Y~rk,

1969.

BOSE FIELD THEORY AS CLASSICAL STATISTICAL MECHANICS, III. THE CLASSICAL ISING APPROXI~IATION Barry Simon *'t Departments of Mathematics and Physics Princeton University

§i. Introduction and General Strategy The techniques which have been developed initially or solely for Ising ferromagnets fall generally into two broad categories.

One group, which includes

correlation inequalities of GKS and FKG type, holds for general kinds of ferromagnets with more or less arbitrary (even) single spin distributions and with many body (ferromagnetic) forces allowed.

The other group, which includes the correlation

inequalities of GHS and th~ zero theorem of Lee-Yang, has been proven directly only for spin 1/2 ferromagnets (each spin takes the values ±I with equal probability in the non-interacting systems) with pair interactions.

In fact, counter examples

exist with four-body interactions and spin 1/2 or with pair interactions and spins taking the values ±2,0 (but with 0,±2 having different weightings). The lattice approximation of Guerra, Rosen, and Simon (1973) discussed already in these lectures by Nelson and by Rosen, approximates P(¢)2 by general Ising models and thus obtains GKS and FKG inequalities.

Here, we wish to discuss a

further approximation of Simon and Griffiths (1973) [henceforth SG] which approximates (¢4)2 theory by "classical Ising models", i.e. systems with spin-i/2 spins and pair interactions.

SG thereby obtain GHS and Lee-Yang theorems for certain

P(¢)2 theories with deg P = 4.

In the interests of emphasizing the main ideas we

propose only to discuss the Lee-Yang theorem and one of its main applications. will also not give certain technical details.

We

The reader interested in further

details and applications and in the GHS inequalities should consult the original papers of SG and of Simon (1973b) or the lectures of Simon (1974). In the remainder of this introduction we want to state the Lee-Yang (1952) circle theorem and explain the general strategy of Griffiths (1970) for extending this theorem fromspin-i/2 spins to more complicated situations. Theorem i z l,...,z n

Let

a.. ~ 0 ij of degree i

for

1 ~ i < j ~ n .

in each

z.l with

P(zl'''''Zn) = Ol=±l,...,On=±l

Then if each

zi6

D~

2n

Let

P

be the polynomial in

terms given by:

exp( ~ aijoioj)zl ½(O1+I) ..z ½(°n+l) \i<j " n

{zlJzl < i ) o { i )

, then

P # 0

* A. Sloan Foundation Fellow f Research partially supported by USAFOSR under Contract F44620-71-C-0108 and USNSF under Grant GP39048

291

Remarks I.

The connection with Ising ferromagnets is the following e-8(hl+'''+hn)p(e 2~hl ,...,e2~hn)

and represents the partition function of an Ising ferromagnet if spin magnetic field

hi .

Of course, it is not a priori clear why zeros

partition function are important.

This idea of Yang-Lee

~. is in a l of the

(1952) is further discussed

in §3 below. 2.

Since

(ai + -oi ) , = 0

P(z I .... ,zn) = (Zl...Zn)P(Zl I ..... Zn I) P

zi~ D-I

is also non-zero if each

can only happen if

Izl = 1 , i.e.

by spin-flip syn~metry and in particular

P(z,...z)

P(z,z, ... ,z)

has its roots on the unit

circle, hence the name "circle theorem". 3.

There are various proofs of this theorem: the original Lee-Yang (1952) proof

found also in Ruelle (1969); an unpublished proof of S. Sherman found in Simon (1974) ; a proof of Asano (1970) described also in

Simon (1974); and a proof of

Newman (1973). By combining Remarks 1 and 2, Theorem i is easily seen to be equivalent to: Theorem i'

Let

aij >i 0

be fixed for

Z(h I ..... hn) =

Then

Z # 0

if each

Griffiths

1 . 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

<2)
looks like a spin--i

2n/a~(hl ..... hn) =

I=±1 Oil

~i2 =±I

spin.

ox,
=

/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

<'>

denote a spatially cutoff

expectation value with free, Dirichlet or half-Dirichlet boundary conditions (see Guerra etal. (1973)) and P(x) = ax4+bx 2 ; a > 0 .

Let

h >i 0

be in

L~/) LI(~ 2) •

Then F(z) = <exp(z~(h))> is an entire analytic function whose zeros For

lie on the axis Rez = 0 .

deg Q .> 6 , we have the following negative situation (SG):

definitely sixth degree

ferromagnets and for which the Lee-Yang theorem fails. approximation, the Lee-Yang theorem fails for certain

Thus, in the lattice Q's •

This suggests, but

certainly does not prove, that the Lee-Yang theorem is false for some theories with

There are

Q's which are not the limit of spin-i/2 pair-interactlng (*)

P(~)2

deg P = 6 .

(*) Of course if four-body interacting is allowed, there is no problem in approximating sixth degree re-scaling leads to

Q's as a

exp(-i/30 s6).

1/12(Esi)4/N3

term in the energy plus

294 §3.

Clustering of the Schwinger Functions

of

P = ax 4 + bx 2 - Bx~(B~O)

Theorem 2 is a striking looking but, at first sight, apparently not very powerful

theorem.

That it is intimately

is a discovery of Yang-Lee

connected with analyticity

(1952) translated

to

(~4) 2

by SG.

strong bounds and falloff is a discovery of Lebowitz-Penrose (¢~)2

by Simon (1973b).

Fix

(1968), developed in

We wish to indicate these ideas in this section.

discuss the case of Dirichlet boundary half-Dirichlet

of the pressure

That it implies

conditions

We

although similar results hold for

B.C.

a,b

and for

B

real and

A C ~ 2 , hounded,

let

i ~A(~) = ~

~n / [exp(- fAla¢4 + be 2 - ~ ¢ : D ) ] d ~ , A

Then, by a result of Guerra et al. (1973), for any real A ÷ ~

(Fisher).

aA(~) ÷ ~ ( ~ )

B

as

The main point of the Lee-Yang theorem (Theorem 2) is that

has an analytic continuation

to the right half-plane,

Re ~ > 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[ [<¢(XA)¢(XA)>A - <¢(XA)>A<¢(XA)>A]

s~p d2ah/d~2 < ~ ,

there is a

D with

[<¢(XA)¢(XA)>A- <¢(XA)>A<¢(XA)>t] ~< 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 ) > - <~(~)>®<¢(y)> is positive and monotone decreasing

as

[x-y] ÷ ~ •

If the limit is some

C > 0 ,

then

[<¢(XA)¢(XA)> (6) and (7)

- <¢(XA)t,, <¢(XA)>,= ] >. 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 <~(0)> 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

<~(0)>

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

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)> @ 0 .

But by

<~(0)> = 0 o

~ ~ -4

Thus the

fails so do (A),(B).

symmetry in the Dirichlet B.C. theories the value of

<'>~=0

theory should not have a unique vacuum.

Thus away from the critical point:

Since (D)

dlfferentiality of the

pressure should be a sensitive test of dynamical instability.

At the critical point

one expects (B)-(D) to hold on the basis of most star. mech. models although we emphaslze that there are star. mech. models where (B),(C) fall at the critical point. 3.

By general arguments

~(~)

Lee-Yang it is analytic away from

is convex in

~

and

so continuous.

By

~ = 0 °

Sketch of proof (A)=>(B)° We need only a bound on de~A d~2 uniform in

A

and

]~] .(C).

<~(×A)~(XA)>T,~, A

]A] <~(x)~(y)>T,B, A. as

Ix-y] ÷ -

See SG. ~(~)

- ~(0)

= A-~"lim Of ~

<~(XA)> B d~

and this yields

297

By using the GI,II inequalities

one shows that

%(~) with

m(~) = < # ( 0 ) >

~(0) = I v

-

m(v) dv

m(0+)

m(0+) = lira m(B) exists. One then proves that U+0 is the right derivative of ~®(~) and m(0-) , so dlfferentiability of

em(~)

at

~ = 0

and that

is equivalent to continuity of (C)=>(D). ' Suppose (D) fails.

x - y -> ~

so

But b y s y m m e t r y

$2,~> 0 + d 2 >~ c 2

<$(0)> > 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)> > 0 ~< c

so

<$(0)>

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<

-

= -

(ig

+ ~)~-

+ 2i~

These equations

i~-

have remarkable

to the qualitative

picture

~ 30~ )

(2.9)

properties

of laser action.

[H4] that they have global

solutions

in

which correspond

quite well

It can be r i g o r o u s l y

(2.10)

shown

for all physical

ini-

tial conditions

(2.lo)

_~ There exists a unique

~3~

~

)

and a 1-parameter

(t)

= ~ exp-i~t

At ~ = ~ ¢ ,

stationary

~ =

~-- = 6

=

a Hopf bifurcation

damping dominated,

%~%2 = g ( ~

)

3,

if

-~c)/2~

discussed

solutions,

and o n l y i f

EH6] occurs

~r-(t) :
~2 ~

is attained

for

These equations

: below threshold

by Prigogine

which this heuristic

the laser is

I~]2, goes to

and coworkers H NR

value

t--')~ , starting from almost any suggest that a n o n e q u i l i b r i u m and that the attracting

state is one of the dissipative

give a Hamiltonian model

,

> ~¢ }where

state, while above threshold the stationary

transition occurs at laser threshold oscillatory

(2.11)

and the mean photon number per atom,

zero from any initial

initial condition.

solution

family of periodic

, ~3(t)

~4 .

structures

picture holds rigorously

in

expansion O(N)

self-

which have been

[G~ . In the following

and an asymptotic

phase

.

in

we shall N ~2

in

302

§3.

The C l a s s i c a l Limit for Atoms and the R a d i a t i o n Field

There are several topologies

in which the infinite volume limit

can be studied. A very c o n c r e t e and r a t h e r e l e m e n t a r y d i s c u s s i o n is possible in the S c h r 6 d i n g e r picture,

by passing with

N-~

along a

sequence of p h o t o n and atomic coherent

states

theory of a p p r o x i m a t i n g Hilbert spaces

[T2]. These coherent

haust all p h y s i c a l

initial conditions

[G2] using the T r o t t e r

system.

However,

states of

if one measures finer properties

of the

system by looking at the f l u c t u a t i o n observables, then one finds that the p h y s i c a l

can be d e s c r i b e d in the Hilbert space generated from a coherent in f l u c t u a t i o n operators.

we shall c o n s i d e r the classical The i n t e r a c t i o n

first c o n s t r u c t k a~ and SN .

Let ~ 6 ~

[H~

s t a t i o n a r y states are not coherent, but that they

by applying all p o l y n o m i a l s

and radiation.

ex-

for the intensive o b s e r v a b ! e s ~

and hence they can be c o n s i d e r e d as the true pure classical the infinite

states

initial states

, U(~)

= exp( ~ a

state

In this s e c t i o n

limit for the elosed system of atoms S H N has the form (2.4) , and we shall I~>

N = ~I N

- ~ a)

and

a a . Then the sequence of photon coherent

which are c o h e r e n t

I0> be the ground states

IN~>

in

state of

= U(N~)IO>

satisfies

-- <0 iqTa~10>

lim
(3.1)

and

= UT~ ~ (3.2)

using the r e p r e s e n t a t i o n t h e o r y of ~J~ 1 !

= q" ~ ~2

has the form

(3.1) (3.2)

for fermions can be c o n s t r u c t e d SU 2 [ G ~ .

~ = R _~0

, where

Any

__~6~ 3

with

R ~ S0(3)

and - - ~ k = - ~

k ~ 3 . For simplicity, we r e s t r i c t ~ to a dense set of p h y s i c a l

initial c o n d i t i o n s

cr =

:

q/(2q + 4p)

,

0 ~ p,q e ~[

,

q t, 0 ,

(3.3) e

Take

A @ SU 2 acting on the Pauli matrices as

J

A* "C kA : ~ - ~ i R l k

and

define new fermion a n n i h i l a t o r s by (b+n' b -n ) =

(e+n , c -n ) A

(3.4)

303 Let I~> 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) _
and

= 2-~2(~N(t) + iTTN~~

~N(t)

~N(t) -~'~ ~t ' "TrN(t) --~-)ql-t ' ~FN(t)----)WT --~-t ' N~2(%N(t)

- ~ t ) ---~--~ q,(t)

,

(3.15)

N~2(qq-N(t) - T t) ~ p ~ ( t ) ,

NF2(~N(t) - ~Ft) - ~ ( t ) Here = ~

~t = ~

classical and

and = 0

satisfy the classical

and initial conditions

are solutions

~r+]2

at

with

~

=

t = O . Clearly the since

|~12 + and

~3

~(t)

equations around the classical orbit

:

-- - i Y a~(t) - i ~ sL(t) )

'3 s (t) = i % ( ~

s2t) + 2ix * t s[(t)

with initial conditions

Let

(~,~)

(2.9)

are invariant under the flow. a~(t)

of the linearized

2t) = -

classical

equations

equations have always global solutions,

(~3)2 +

a~(t)

~F t

(3.16)

(~ ,~ )

t s.3(t) ÷

+ v~

a~(0)

a~(t) *

= a~ ,

a.(t) h.co

s~(0)

(3.17)

)

= s~ .

be any initial state. Around the corresponding

solution we develop

H NS

in powers of

N ~2 into a mean field

305

and a fluctuation part,

H M(t N ) =

HNS =

- N+ + I0( tS

~ S N3

N~/2(~'~ + k
HM(t) + HF(t)

+ "/~t S N *

+ N~2(Y~ t + k ~ )a"" + , _

is well known from the thermodynamics model

[H~)with

The action of

of the atomic observables.

leads t o when

H NM(t)

- N~ t)

is purely group theoretical,

=

desoribes cor-

~t+ )

, which linearize for uM(t)

H~(t)

T exp-i~%s

~ ( a '~- N ~ 2 ¢ (*t ) N ~ 2 ( ~

+

N--)~°

-

CF~) (3.19)

nonlinear Heisenberg equations

N (~

or of the Dicke

if the initial condition

We shall see that

A(a - N~2~t )NF2( ~ N+ -

+

of the BCS [T~

of the intensive observables when

Y(a * - N ~ 2 ~ ) ( a

=

(3.18)

of the boson operators by the photon part and a rotation

rectly the time-evolution

HF(t)

+

an explicit time dependence,

is not stationary. a translation

:

for the fluctuation operators,

N--)~.

Consider

(3.20)

HM(s)

O

Clearly this propagator Dyson series on ~ M x UN(t) satisfies ~NCt) Similarly,

t

using a strongly convergent

~ daN(t)/dt

= 0

and

~N(0)

= U~(t)*(a-N~2~t)x

= a - N ~2&

= a - NY2~

~N(t)

dgN(t)/dt

exists for all

and a unitary extension. ~N(t)

(3.21)

= uM(t)*N~2(-<~N - ~--t)UM(t)

= -i£

SN(t)

, hence

+ 2iXC~t ~

(t)

satisfies

, (3.22)

d~3(t)/dt

~ -i~

t

sN(t) + i ~

t ~(t)

,

and has a global solution in terms of a 1-parameter

~N(t)

=

M(t)

family of rotations

N~2( ~ N - ~ )"

Let us consider Weyl operators

WN(t)

= WN(~,~,t,x,y, ~)

(3.23) of the form

306

WN(~,~,t,x,y,z)

: exp i[x(q-N~2~)+y(p-N~)+Z(~FN-~)N~2].

(3.24)

Then S

exp(iH t) WN(t) exp(-iHNSt) A

M

~. (t) a~N~

VN(t,0)

(3.25)

,

*

WN(t) : UN(t) WN(t) U (t) ~N(t,0)

A

: ~N(t,0)*~N(t) ,

(3.26)

S = U~(t )* exp(-iHNt)

is obtained from

by replacing in

M(t)(_~ N- _~) . CN (t,O)

and _W~-~4 by

dVN(t,0)/dt

WN(t)

(3.27)

=-i

~F(t )

(t,0)

(3.24)

a-N~20< t

by

is the solution of

, ~F(t)--UN(t)H

(t)U

(t) (3.28)

The propagator ~N(t

s) = T exp-i ~dr ~F HN(r)

(3.29)

s

can again De defined by a Dyson series on tained from

HF(r)

~

, and

HN(r)

is ob-

by going over to fluctuation observables

at time

zero° Hence (3.29) can converge, when we pass to

N-->~o along _fiN .

Theorem 3.1 : For every physical initial condition ~ ~ ~

(with(3.3)),

the intensive observables and the fluctuations along the classical orbit converge to Proof : For

(3.15)

and

It-s i sufficiently

"~ V~(t,s)

£

: T exp-i~dr

~F

H~(r)

(3.16)

along

IW,~>N

for

N--~

small, the Dyson series for ,

(3.30)

5

AH~(r) F

= Y a ~*a .

s (r) =

+

M(r) R t

k( a * s -(r) + s+(r)a~)

,

(3.31)

(3.32)

,

converges in the strong topology on ~ o

Using

(3.25)

and the

Schwar z i n e q u a l i t y III N ~ ( t

,0) ,~A W(t)

AL ( t , 0 ) ~

~VN(t,0) ,~ WN(t) /~ VN(t,O)

I N ~ ~ II "I

307

~ II~N ~ (t,0)~ (t)~ (t,0)T~ +

II I N

W (t) V ( t , O ) ~ .

+ II IN V ~ ( t , 0 ) ~ For small

~N(t,0)*TN~(t)<(

-

- WN(t) I N V (t,0) ~

II +

II,

- VN(t,0) IN ~ .

(3.33)

~t-s I , the Duhamel formula can be used on

A llIN V b ( t , s ) ~ .

II 4-

t,0)T~

~

:

/k - VN(t,s) IN U/.. ~

(3.34)

% =l)i~dr ~N(t,r) [~F(r) r

)I ( H N ( r )

IN

IN - IN ~:(r)] ~ - IN H~(r)

(r,s)~

) V (r,s) ~

II II ~

$ since the Dyson series for (3.34)

A Vm(r,s)

converges to zero for small

on all ~ ,

- INv~(s,O)hu..ll

+ II

e.g. ~VN(t,0)

(t,s)INL,(s,O)

the exponential on any ~ ,

)11NW~(t) ~U~

and therefore

t) V (t,0)

It-sl can

uf~ - INV ( t , O ) ~,~.11 .By e x p a n d i n g

one proves that 0

N --" ~

for

by uniform boundedness.

(3.33)

Hence

I N ~ ~- IN V~.(t,0)~LLIIVN(S,0)IN~ ~

W N ( t ) I N ~U~ II - - >

and therefore on ~ the r.h.s, of (3.24)

~t-sl and ~ - ~ 6 ~ 2 ,

since ~I N I~ = llVN(t,s)l[ =I~ V~(t,s) ~= i . Large

be estimated stepwise,

V~(t,0)

solves all domain problems.

go to zero for all

t

(3.35)

)

Hence all three terms on and all ~ . e ~ .

has an exponential representation

Now,

of the form

, where a (t) = < ( t , 0 ) * < ( t )

¢~(t,0)

,

(3.36) s (t) = V (t,0)~s-~ ~ ( t ) t ( t One checks that = ~

(3.36)

satisfies

,0) (3.i7)

, and by uniqueness the WT-convergence

with

a (0) = a

(3.18)

, ~(0)

follows.

(3.15)

can be derived similarly. Corollary 3.2 : For all

tm,Xm,Ym,Zm,

lim (_[IN, ~ exp(iHSt) N--~ m=l

i { m L n ,

%(0(,~,tm,Xm,Ym,Zm ) exp(_iHNStm ) ~ ]

_--

=

308

=

n (_O.,,q-[ W~(~,~,tm,Xm,Ym,Z m) .O.~. ) m:l

and analogously

Proof

for the intensive

: By repeated

application

gence for a product (3.37),

since

It is typical

functions

[H~

3.1

one obtains

Weyl operators.

= (~,~

products

of motion.

observables

of solutions

N--~

~

fixed

There

This

implies

limit of quantum mechanics

that the quantum mechanical in coherent

states

orbits

(withamplitudes

equations,

and that

follow linearized

boson

is a close connection between the limit

in mean field models and the limit ~ --) 0

nary quantum mechanics

in

correlation

values converge)

of the classical

around the classical

equations

WT-conver-

by polarization.

so that the time zero expectation

the fluctuations

and

IN~)

states

of intensive

tend towards

of Theorem

for the classical

minimum uncertaincy

which are scaled

observables.

of time translated

lim -.(IN~'

(3.37)

of a fixed number of degrees

in ordi-

of freedom.

~4. The Lossy Laser

After having studied the classical

limit for the closed

atoms and radiation we shall now construct

system of

a model of a quantum mecha-

nical reservoir with simple damping and pumping properties. reservoir

for photon losses

in the laser is a quantized r a d i a t i o n

field of infinitely many boson degrees all the non-lasing Hamiltonian

HP =

photon modes

is linear

of freedom,

of the world.

a ~w ' which describes

The simplest

interaction

:

~ dw ~wa*aw

where for notational

A natural

~ dw k w (awa * + a *a w )

+

simplicity we take

w e~

,

(4.1)

and a c o n t i n u u m

normali-

zation

[aw

, a w,

:

(w-w')

H NS

Under the influence of

aN(t)=-i~aN(t)

-

i~

,

and

aw

HP

SN(t)N-v~ -

, a w,

:

0

,

the Heisenberg

i]

(4.2)

equations

k w awN(t) )

become

(4.3)

309 $

awN(t)

= - i$wawN(t)

In the initial

(4.4)

- ik w a N ( t ) .

state we shall assume no correlations

b e t w e e n the cavity

and the reservoir, -(lN : - ~ ~ ' ~ , and take the photon reservoir to R be empty, aw~)- N = 0 for all w . A dissipation of the form (4.1) will in general (2.7)

not lead to phenomenological

with a f l u c t u a t i o n

the small system, ciently for 0 ~

"white"

awN(t)

unless

force

gN(t)

in terms of the initial condition

~w -~> w . This leads for

~N(t) where

: - (iY+~)~N(t)

~=~Fk z

,

~N(t)

some care,

abusive notation, as a derivation

and

and

aN(s)

for

limit

t ~ 0 , to

XN(t)

and where

;

(4.5)

A(t)

= ~ dw a w e -iwt

Singular equations of motion of the type including R HN

(4.5) some

justified by treating

as an a u t o m o r p h i s m

of the

which on a dense domain is defined by a converseries.

degrees of freedom it is more complicated

a phenomenologically

acceptable

and mathematically

We shall couple the electrons

of infinitely

aw

(4.4)

and pass to t h e ,

+

exp(iH~t)(.)exp(-iH~t)

For the atomic

is suffi-

We can solve

but all the steps in the following,

gent m u l t i c o m m u t a t o r

coupling

- i~N(t)

can be rigorously

Weyl and CAR algebra,

process.

(4.3) ~N(t)

= -ikN-~2A(t)

is "white boson noise". require

of the type

of the state of

the interaction with the reservoir

to destroy all memory effects.

s ! t , insert the solution'into

kw'-~ k '

equations

independent

many degrees

of freedom,

to find

simple d i s s i p a t i o n

of each atom to an electron again by a singular

sea

linear

: N n=l

~n using

±n

~

(4.6)

'

÷~

w(b±nwb±nw

Fermi

normalization the atomic

fields (4.2)

with

+ e±nwC±nw)

normal

. Under

Fermi operators

+

dw

ant±commutation

n ( f + b + n _w _ relations

S + HNA + HP ! t h e HN = HN

time

) + h.e].

g÷C+n w _" in

continuum

evolution

of

becomes (4.7)

~.n(t)

= -(i~ + ~)b+n(t)/2

- i ~(t)b_n(t)

- if+B+n(t)

- ig+C+n(t) j

310

b_n(t) = -(~-i~)b_n(t)/2

- i~(t)b+n(t)

- if_B_n(t)'- ig_C_n(t). (4.8)

Here B~n(t) = ~dw b±n w exP(-iwt) and C,n(t) are "white fermion noise" operators and ~ : 2~(~g+i 2+Igj2). We have assumed ~f~2 + ig~2: = ~fj2+Ig32 and ~f~2 = [gj2 in order to ensure macroscopic charge neutrality for the dissipation process, so that stays at N up to N y2 fluctuations.

~(b+nb+n

+ b_nb_n)

The intensive observables obey the following differential equations &N(t)

= -(i£+~) ~N(t)

+ 2iX~N(t)~FN3(t)

+ ~N(t)

)

(4.9)

where the fluctuation force ~N(t) = ~N(t)* unfortunately depends on the observables of the small system : (t) =

~a

(t) +

(t)

(4.10)

N

~ N +(t) = i N - l ~ [ f ~ B * n ( t ) b = -iN- ~-'

~N-(t)

(4.11)

n(t) + g_C n(t)b+n(t) ] j

- b n(t)B -n (t) + g+b-n (t)C +n (t)

.

Similarly

"3 ~N(t)

3 (t) _ ~ ) : _ ~ (CFN

~3(t)

=

~3N(t) + ~3N(t)

N(t) = ( 2 N ) - I ~

In

(4.12)

+ i~N(t) ~N(t)

~3N(t)

=

,

- ig C n(t)b n(t)

+

+ if;B;n(t)b+n(t)

- if_B_n(t)b n(t

.

the "unsaturated inversion"

=

~3N(t)*

[ ig+C+n(t)b+n(t)

off, in order to obtain for

i (t)

,

+ ~N3(t) , (4.12) (4.13)

- iX~N(t)~[N(t)

i (t )

~ = TF~-l(~g~2-1g32)-

(4.14) was

split

i = ±,3 _0. N

:

o

,

(4.15)

if the B-reservoir is empty, B+n(t) "f)-NR = 0 , and the C-reservoir full, C n(t)-f~N = 0 . By a suitable choice of [g~ 2_ , we can reach any -Y2 ~ ~ y2 .

311

By generalizing

the approach

of Section

3 we shall

study the WT

convergence of the intensive and fluctuation observables along i'i N = S~ R S =~ ~N ' where ~ is the vacuum of the reservoir fields and _f-ZN is coherent we split

~ ~'-~ = I ~ , ~

HN

along the classical

a fluctuation

part,

H (t) = H P +

HNF(t) : Under

~

M

*

+

:3

+ H~(t)

initial ~ondition

~t,qt

into a mean

,

field and

(4.16)

+

+

and the different

is a product

~, ~

:

+ * ~" ~(~tSN + ~ t S N ) - N(~Ct~

N~2~t)N~2(~N-

the photons state

orbit

- Y2 * + ~N (a vrt +
a +

SN

~(a*-

HN(t)

the initial ties,

HN = H~(t)

Ya

A + HN

+

. Given any physical

+ q-t+ ~t ) ) (4.17)

h.c.

Since

atoms are uncoupled.

state with strong

this will lead to "boson white noise"

factorization

fluctuation

forces

properfor the

fluctuation observables, when N--9~. In the corresponding interaction ~F picture, HN(t) will again become quadratic in boson operators. We will study the evolution in the form

(3.25,26,27)

uM(t)*(. )UNM(t) S~N(t)

of Weyl operators with

considered

replaced

by

as an automorphism.

from their differential

d~N(t)/dt

H NS

(3.24) which can be represented

equations

= - (iY + ~ ) ~ N ( t )

HN

and with

We compute

~aN(t)

:

- ik A(t).) £

~N(t)

= e x p - ( i Y +~Ot

and

(a-N[2~)

- ik]ds

(4.18) A(s)

exp-(iY+~)(t-s)

)

O

dSN(t)/d t

_ (

~3

fN(t ) J (4.19)

~3 dSN(t)/dt

= _ ~s3(t)

+ i~t~N(t)-

In terms of the propagator

~N(t)

In

(4.19)

£~(t)

P(t,s)

: P(t,0)N~2(_~N-_~)

, the fluctuation

= M -~2

(t) ,

of

i ~ S N^+ (t) (4.19)

+ f3(t ) one has

÷ S ds P(t,s)IN(S)

forces

fN(t)

= fN(t) +

(4.20)

.

+ fN(t ) _

fm(t ) = (2p+q)-~2 ~__ Fn(t )4

,

satisfy (4.21)

312 where ~ (m) = (4.10,11,13,14) under

~(t).

((2p+q)m, (2p+q)(m+l) ] . The F!(t) h~ve the form n without N -I and ~ , and with the b±n(t) evolved

The

h±n(t)

replaced by the o-number

therefore obey ~t

(4.7,8)

with

product structure of ]'L N , one has for all

l{ m £ M

*

+

(2p+q)-i ~--- (J~N' b+n(t)b-n(t) /~LN) ~ ) (2p+q) -I ~ where the

~

: ~t (4.22)

(~,(b+n(t)b+n(t)-b'~n(t)b_n(t))]'i N) = 2 ~ t

satisfy the classical equations

correlation functions in independent of m :

(lqN,

(2.9) . Hence the

~ N ) = < ' ">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 + ~ ) <

3

^3

~N(t)

and, as a consequence of the direct

-- 0

~(s-t ) ~

(4.23) (~2 - ~ )~t +

Furthermore, sin~e ~ m ( t ) ~ N = -~4(t)*-6~ N = 0 and m for m~n , one has independent of N = M(2p+q)

f •(s)f ~N

:

~,"

E~ (s), '~ (t)]

= 0

~

z,'

(4.24)

The 2-point function of ~:'f~(s) with A~(t) or any fluctuation operator at t = 0 vanishes. However, the higher correlation funtions of such quantities are far from Gaussian for finite Let ~

be the Fock space of two boson modes

boson fields L2([0,~))

a~

and

N . t~

and four

i+

(t) , f ~ (t) , i=1,2,3, with test function space and cyclic vacuum ~)-~, satisfying

f~i - (t)]l~ = a~._O-~= A~(t)9-~= tL _9_~= o

(4.2s)

313

fi+(t)~ = f~(t)i- *, t~+ = t ~ ", f

and with c-number commutators, , [am, a,]= I,

(t) = f~-(t) _+ if

which all vanish except for

+ . ItS, t+] = 2~- , [ f ~ ( s ) , f ~ ( t ) ]

The identification

(t) ,

j (4.26) = ZfN(s)fN(t)>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,_
where

~I

extends over all

: M-}'2 tN

~ 9 ; £ j and

~+m

of all vectors of the type product struture of ~'~N

II IN1; = i

as in

(4.21)

(4.27)

with

with

m k ~ mL

h ~ ( ~

. Let

and

for

be the span

k + r : h . Because of the

there exists an isometry

I Nh

such that

h-i = T ~ (i - g/M) ~2 I Nh g=l

IN ~ ~ 2 Hence

m I ,...m~+~-- I,..M

(4.28)

lim III~T~.II= II~,;;

for all ~ e ~ .

Nowjwe are

ready to prove the main-theorem on the time evolution of the laser Theorem 4.1 vables

aN(t)

(2.4),(4.1,6)

towards the solutions

with initial conditions = N~2(~N(t)-~t ) , ~N(t)

towards solutions a~(t)

~ N~2(~N(t)-~t) in ~

(t) - i~s~(t)

s.(t) : -(i£+~)s~(t)

t>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) ]

+
a~(0)

functions converge

= a~,

s

(0) -- s ~

in the sense of

in~.

(3.37).

314

Proof : As in theorem 3.1swe consider the Weyl operators

exp(iHNt) WN(W,~,t,x,y,z) Here by

WN(t) : exp(i~N(t)) aN(t)

and

N

to the l.h.s, of

: ~N(t ,0) *~ WN(t)VN(t,0) .

is obtained from

(gN-~t) (4.30)

exp(-iHNt)

by

SN(t)

(3.24) by replacing

a-N~ t

, which gives a rigorous meaning

. VN(t,s) -- : T e x p - i ~

dr ~FHN(r)

a Dyson series with the same substitution as in (3.33) proves convergence,

(4.30)

if we can show for

WN(t) 0!u~l

is defined by . The estimate and

0~ si r~t

with It-sb0sufficiently small and for every ~ . e ~ t h a t

-

)IN)V~(r,s)

~

II

,

(4.31) II(INK.(t) - KN(t)IN)exp(iuK.(t)) ~

II

are uniformly bounded in N and converge to zero for every r,u, if ~F A N-~ . Here H~(r), K~(t) and V~(r,s) are analogously defined using

~a~(t) and

~(t)

, the IN-preimages of

The uniform boundedness and convergence of by expanding The limits

~(r,s)~ am(t)

and

§5.

(4.29)

and

(4.20)

(4.31) can be easily proved

exp(iu~(t))~

= V~(t,0)*a~(t )V~(t,0) ~

by the differential equation

(4.18)

and by using

and

s~(t)

(4.28) .

are identified

which they satisfy.

Conclusion

We have discussed in considerable detail a microscopic quantum mechanical model for a laser cavity coupled to different reservoirs and have given a precise sense in which the laser shows irreversible classical and stochastic behavior in the thermodynamic limit. We have, however, only barely prodded the sleeping giant of nonequilibrium statistical mechanics.

There are many open.problems in this field,

both of a technical and conceptual nature. There is the immediate question, whether the choice of the reservoirs,

in particular the use

of regular Hamiltonians and nonlinear couplings, can qualitatively change the phase transition in this class of mean field models. A nontrivial generalization of our approach is necessary in order to treat the finite mode laser. Our results on the intensive and flue-

315

tuation observables are the first two terms in an asymptotic expansion for large finite lasers. if one is interested linear system for

It is not clear how good this expansion is,

in the equilibrium properties of the finite non-

t-@~.

Hopefully,

the two limits, N--)~ and

can be exchanged for certain observables away from threshold.

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
for

I < A h , ^_
o

all

A

and

as ,~,.*c^~^',o')-,.~o

Ifor some: m (independent of A)A') and all A j ' / , . Here A~ is ~

translated by X~'RZ, and < A , B ~ ~ = ~ A B ~

is the truncated two-point function.

~;s# (..A)o) --~ oo

By I, A) A

-<~>

converges as

to a Euclidean invariant limit functional
Let ~ E C~(~)and set

F~}~= <~>

... < ~ >

. The Schwinger

functions may be defined as distributions by S(l~t)...,~,) = < F ~ ) ~ This provides a well-defined realization of (I)~(2). • heorem:

The Schwinger functions

(including endpoints).

(3)

~.

at"

~ F ~

Defining ~

I
C °o on ~ -

.: D ; < F T ~ ) ~ I A = O

N

- 7

are

~.x~/.!

I

[O,]Lo ~

we have

- o

Equation (3) is the statement that the perturbation series is asymptotic and it follows immediately from the C~property and Taylor's theorem.

The

.4,~ can be evaluated explicitly with the

result being a sum of terms labeled by n th order connected Feynman diagrams. A more general theorem also holds which states that < F ~ ) ~ C ~ o n any interval J

where I, II hold.

is

In the case at hand if

we omit the crucial endpoint it is known that ~P~A,.0 is real analytic on (O,~o)

[3] .

The proof of the theorem is given in L~]. that

<~>~)

is C I.

Here we only show

It is straightforward to show that < F ~ , A i s

319

C I and that centered on ~

D>~,^- -
Let ~

be a unit square

and let /k be a square centered on the origin

which is a union of the Z&~ •

Then we may write

O

By I) II we have

From

It

(%)) (6) and the dominated convergence theorem we conclude that

follows that )c~is c i o n J

.

The higher derivatives are expressed in terms of truncated n-polnt functions.

Vith a space cutoff we have T - c-,,

(

,

To prove differentiability in the limit the crucial estimate turns out to be

where ~(.')is

the diameter of the set.

from I,II directly.

zI'.

This estimate does not follo~

Instead one uses:

,-r I
x.J

I~.
whenever the localizations of A , A ' c a n be separated by a strip of width 4

•

Then I) II' together with some elementary geometry give (7).

The

improved cluster property II' is obtained by showing that II implies that the Hamiltonian in the associated Hilbert space theory has a uniform mass gap) and then showing thab this in turn implies II'. This argument is carried out by ~p,proximatlng the infinite volume theory first by a cutoff theory in which and then by a theory in which ~ details.

~

is an infinite strip,

is a rectangle.

See C I J for

320

References ~S,

J. Dimock, Asymptotic Perturbation Expansion in the Pg~)Z Quantum Field Theory, preprint.

EaS. J. Gllmm, A. Ja~ffe, and T. Spencer, The Wightman Axioms ~nd Particle Structure in the P C ~ Quantum F£eld Model, to appear in Annals of M~thematlcs.

~3S, J. G~imm, A. Jaffe, and T. Spmncer, Erice lectures.

NONDISCRETE SPINS AND THE LEE-YANG THEOREM 1 Charles M. Newman

Courant Inst. of Math. Sciences; Dept. of Mathematics;

New York, N.Y.,U.S.A.

Indiana Univ.; Bloomington,

The Lee-Yang Circle Theorem [Lee, Yang,

Ind.,U.S.A.

1952~ is an

important tool for the rigorous study of phase transitions ferromagnetic spin systems.

The technique of approximating

Euclidean quantum fields by "general Ising models" Simon,

in

[Guerra, Rosen,

1973] indicates that the Lee-Yang Theorem is also applicable

to the study of symmetry breaking in quantum field theory providing that the theorem is extended from its original context of spin - ½ to

the

continuous

context.

"spin" variables arising in the field theory

This extension has been accomplished for

[Simon, Griffiths, [Griffiths,

~

4

models

1973~ by the method of "analog s p i n - ½

systems"

1969~. In this note we report a result which essentially

determines the entire class of spin variables to which the LeeYang Theorem is applicable. Griffiths' theorem;

The proof of this result combines

method of analog systems with the Hadamard factorization

it appears in complete form together wi~h an analysis of

the other results presented here and a new elementary proof of the iThe research for this paper was supported in part by the National Science Foundation, Grant NSF-GP-24003, while the author was at the Courant Institute.

322

spin - ½ Lee-Yang Theorem in [Newman,

Definition. positive)

~

measures

1973].

is the set of signed ~

(real but not necessarily

on the real line which are either even or

odd and which satisfy

(A)

~ e x p ( b S 2) dI~(S) I < ~

(B)

rexp(z S)d~(S) ~ 0

for all

when

b ~ 0, and

Re z / 0 .

We consider for a fixed choice of Jkj 2 0

(for

and

k, j = 1 ..... N), the partition function, N

N

Z(~) ~ ~ e x p ( ~

zjS 3 . + ~

j=l where

Ul . . . . . u N E h

J j k S j S k ) d U l ( S I) ...dUN(S N)

j ,k=l

z = (z I . . . . . ZN), and the correlations

<Sjl "'" Sjm>~ - (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 =<)(01

m = 1,2, . . . .

concerning

4

field

it is a special

the truncated c o r r e l a t i o n s

of arbitrary order which has recently been

for fourth order

Theorem 4.

of even order in a

0

and

1973].

Z~(z) ~ Z ( z ~ )

with

~ > 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/~<

X 6_ ~_~+ , Jc>/ 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).

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