P(0)2 Euclidean (Quantum) Field Theory 9781400868759

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The P( c P) 2 Euclidean (Quantum) Field Theory

Princeton Series in Physics edited by Arthur S. Wightman and John J. Hopfleld Quantum Mechanics for Hamiltonians Defined as Quadratic Forms by Barry Simon Lectures on Current Algebra and Its Applications by Sam B. Treiman, Roman Jackiw, and David J. Gross Physical Cosmology by P. J. E. Peebles The Many-Worlds Interpretation of Quantum Mechanics edited by B. S. DeWitt and N. Graham The Ρ(Φ)2 Euclidean (Quantum) Field Theory by Barry Simon

The Ρ(ψ)2 Euclidean (Quantum) Field Theory by Barry Simon

Princeton Series in Physics

Princeton University Press Princeton, New Jersey · 1974

Copyright (C) 1974 by Princeton University Press Published by Princeton University Press, Princeton and London All Rights Reserved Library of Congress Cataloging in Publication data will be found on the last printed page of this book

Printed in the United States of America by Princeton University Press Princeton, New Jersey

To Ed Nelson who taught me how unnatural it is to view probability theory as unnatural

PREFACE These lecture notes are mainly based on a series of lectures given at the Seminar fiir Theoretische Physik of the ETH/EPF — Zurich in the Spring of 1973. It is a great pleasure to thank the many people who helped in my efforts: Klaus Hepp for inviting me to lecture at the ΕΤΗ, Jean Lascoux (Ecole Polytechnique-Paris), Paul Urban (Schladming), Daniel Kastler (CNRS-Marseille), Andre Lichnierowicz (College de France), R. Gerard (Strassbourg) and John Lewis (Institute for Advanced StudiesDublin) for the opportunity to present lecture series on this material; these "dress rehearsals" allowed me to experiment in many ways with presenta­ tion of the material, David Ruelle and Walter Thirring for convincing me the time was right for such written up lecture notes, James Glimm, Arthur Jaffe, Ed Nelson, and Arthur Wightman for all they have taught me, Francesco Guerra and Lon Rosen for the joy of collaboration and for permission to use material we are still in the process of writing up, Miss R. Hintermann for typing the bulk of the first draft (7½ chapters). This was done during a ten-week period which was exceptionally grueling for both of us! Mrs. G. Anderson and Mrs. C. Jones for the rest of the typing of the first draft and Mrs. H. Morris for the final typed copy, Arthur Wightman for his enthusiasm at publication of the notes (and for his enthusiasm in general!),

viii

PREFACE

Gail Filion and John Hannon for their editorial advice and their patience in the face of delays in promised manuscript, Lon Rosen for a careful proofreading of the final copy, The ΕΤΗ, the Sloan Foundation, the UASFOSR (Contract F44620-71C-0108) and the USNSF (Grant GP 39048) for financial support during the preparation of the manuscript, And Martha Simon; those ten weeks were pretty grueling for her too.

BARRY SIMON

CONTENTS PREFACE

vii

INTRODUCTION

xiii

CHAPTER I. GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

Gaussian Random Variables Models of Q-space Fock Space Second Quantized Operators Hypercontractive Bounds Relative Absolute Continuity of Gaussian Processes

4 19 23 25 34 40

CHAPTER II: AXIOMS, I II. 1. The Garding-Wightman Axioms and the Wightman Reconstruction Theorem 11.2. The Theory of Laplace Transforms: A Technical Interlude.... 11.3. Schwinger Functions and Their Properties (The Osterwalder-Schrader Axioms) 11.4. The Osterwalder-Schrader Reconstruction Theorem 11.5. Free and Generalized Free Fields

48 53 60 68 73

CHAPTER III: THE FREE EUCLIDEAN FIELD III. 1. 111.2. 111.3. 111.4. 111.5.

Beyond the Osterwalder-Schrader Axioms The Free Euclidean Field as a Path Integral Conditional Expectations and the Markov Property Products of Projections in Sobolev Spaces LP Estimates and Asymptotic Independence of Distant Regions

82 85 91 97 102

CHAPTER IV: AXIOMS, II IV.1. Nelson's Axioms IV.2. The Nelson Reconstruction Theorem IV.3. When Does a Garding-Wightman Theory have an Associated Euclidean Field Theory? IV.4. A Counterexample ix

107 116 120 127

X

CONTENTS

CHAPTER V: INTERACTIONS AND TRANSFER MATRICES V.l. V.2. V.3. V.4. V.5. V.6.

The Basic Strategies L p Properties of the Exponential of the Interaction Construction and Identification of the Transfer Matrix Vacuums for the Transfer Matrix, H(g) Some Miscellaneous Results The Hoegh-Krohn model

133 148 155 165 170 172

CHAPTER VI: NELSON'S SYMMETRY AND ITS APPLICATION VI. 1. The Glimm-Jaffe Linear Lower Bound and Guerra's Theorem VL2. /Soo and T7oo VI.3. The Glimm-Jaffe φ and π Bounds VI.4. Nelson's. Commutator Theorem VI.5. Frohlich's Bounds

180 185 190 203 204

CHAPTER VII: DIRICHLET BOUNDARY CONDITIONS VII.1. VII.2. VII.3. VII.4. VII.5. VII.6.

The Non-interacting Dirichlet Field Conditioning, Dirichlet States and Half-Dirichlet States and the HD Transfer Matrix D αOO Μ = α®= a" OO OO φ-Bounds for Half-Diriehlet States Half-Dirichlet States for the Hoegh-Krohn Model

215 223 232 239 247 252

CHAPTER VIII: THE LATTICE APPROXIMATION AND ITS CONSEQUENCES VIII. 1. Definition and Convergence of the Lattice Approximation .... 257 VIII.2. The Lattice Approximation with Dirichlet Boundary Conditions 264 VIII.3. The GKS Inequalities 271 VIII.4. The FKG Inequalities 279 VIII.5. Nelson's Convergence Theorem 285 VIII.6. Properties of the Infinite Volume Theory 293 VIII.7. Coupling to the First Excited State 302 VIII.8. The Hoegh-Krohn Model 307 CHAPTER IX: THE CLASSICAL ISING APPROXIMATION AND ITS APPLICATIONS IX. 1. The Basic Strategy and an Improved DeMoivre-Laplace Theorem

316

CONTENTS

IX.2. GHS Inequalities and Lebowitz' Inequalities and Their Applications IX.3. The Lee-Yang Theorem IX.4. The Last Wightman Axiom IX.5. Broken Symmetry — Some Generalities

xi

326 335 344 348

CHAPTER X: ADDITIONAL RESULTS AND TECHNIQUES: A BRIEF INTRODUCTION X.l. X.2. X.3. X.4. X.5.

High Temperature Expansions Fugacity Expansions Other Boundary Conditions Equilibrium States and Variational Principles The Work of Dobrushin-Minlos

REFERENCES

361 367 369 372 376 .379

INTRODUCTION These lecture notes are intended to introduce the reader to Euclidean ideas in quantum field theory and then to develop one approach, the "correlation inequality" method, to the simplest model of an interacting quantum field theory, the P(^) 2 model of a self-coupled Bose field in two dimensional space-time. We have tried hard to make them accessible to non-trivial subsets of both the mathematics community and the physics community. We have emphasized the probabilistic Euclidean strategy toward P(^) 2 over the Hamiltonian strategy, which in the hands of Glimm and Jaffe dominated the period from 1964 to 1971 and which has played such an important role in shaping the Euclidean strategy. The reader may consult [70,71] for lucid discussions of the Hamiltonian strategy. At the outset, we should emphasize that these notes are, in a sense, one-sided. The correlation inequality method in P() 2 is not all of the statistical mechanical approach to Euclidean fields, far from it — there are many other powerful techniques such as the expansion techniques of Glimm, Jaffe and Spencer (discussed in barest detail in Chapter X). And the statistical mechanical analog is not all the Euclidean field theory for P(0) 2 . And the Euclidean approach does not embrace all the results for P( χ

< o l T exp ^i J* H(x)d 4 x^ | o >

and the formula for correlation functions in a lattice system

σ

σ.=

denote the integral with respect to n (expectation).

Given a formal power

series in f, i.e., formal series where a) we don't worry about convergence, b) we don't identify two series which are identical by virtue of substituting in f (e.g., f and

are distinct as formal power series even if f = 1),

we define DEFINITION.

Let

f be a random variable with finite moments.

Then

is defined recursively by:

(1.14a) (1.14b) (1.14c) is called the nth Wick power of f. Notice that Wick powers depend on both f and the underlying measure. Thus,

e.g.,

10

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

Properties of Wick powers are found most easily using the formal generating function (1.15) Clearly, by (1.14b)

and by (1.14a, c)

Thus

(1.16) (1.16) holds in the sense of formal power series in a. If f is a Gaussian random variable, (1.16) is especially useful because the formal power series converge (for example in

and (1.17)

(1.17) can be obtained by direct computation from (1.8) or by noting it holds if a = it (t real) on account of (1.6) and then analytically continuing or by using (1.9). Thus, for a g.r.v. of variance (1.18a) By multiplying the series for exp (af) and

together,

we find that (1.18b) Conversely,

(1.19a) so that (1.19b)

§1.1. GAUSSIAN RANDOM VARIABLES

11

Remarks: 1. We emphasize that (1.17), (1.18) and (1.19) are for the special case of g.r.v. 2. If

where

is the nth Hermite poly-

nomial. This follows from (1.18a) and the fact that is the generating function for the Hermite polynomials. 3. If (M, 2 ) supports two measures /z and v so that f is a g.r.v. w.r.t. both ii and v, we can form

and

and ask for

transformation laws from one to the other. From (1.18a) we find (1.20a) so that

n

4. (1.9), (1.13), (1.18) and (1.20) all generally go under the name of "Wick's theorem." One can use (1.18a) to compute expectations of products of Wick powers. We will compute for the product of two powers, but a similar method works for more than two factors. In particular, in Section 1.5, we will quote the result for the product of four Wick powers without proof.

THEOREM 1.3. Let

f and g be g.r.v.

Then

(1.21) Proof.

Thus

(1.21) follows by expanding the exponentials. •

12

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

Remark: (1.21) is special to g.r.v.

In fact, if f is a random variable with

and

then f is a g.r.v.

note that

allows us to compute

and thus

To see this

in terms of

inductively.

Now consider several random variables

. The Wick product

is defined recursively in and

With this definition one has a binomial theorem

(1.22) There is also a multinomial theorem. WARNING! Not all algebraic relations are preserved by : : . For example

but if

and

then

COROLLARY 1.4

(a) If

and

;

are g.r.v. and

then (1.23a)

(b) If

are g.r.v. with

then (1.23b)

Proof.

Follows from the multinomial theorem and Theorem 1.3.

§1.1. GAUSSIAN RANDOM VARIABLES

DEFINITION.

13

Let (Μ, Σ, μ) be a probability measure space. Let V be

a (real) vector space. A random process indexed by V is a map φ from V to the random variables on M, so that (almost everywhere): φ(ν+ w) = φ(ν) + 0(w) φ(αν) = αφ(ν)

all v.wiV

all α e R, ν e V .

Remarks: 1. Often V is a topological vector space and φ is required to be continuous when the random variables are given the topology of convergence in measure or (with restriction on the range of φ) an LP-topology. 2. In many applications, V is a vector space of functions on R n such as C^(R n ) or

S (R

n

) [145; Chapter V] in which case one

thinks of φ as a "random-variable-valued distribution" and writes φ(f) = f φ(χ) f(x) d n x (formally). 3. Much of the probability literature uses the term "random process" in a more restricted context, namely as a random variable valued function on R, t κ q(t). Byletting φ(f) = Jf(t)q(t)dt, we see that this restricted notion is indeed a special case of the definition. What we call a random process is then called a "random field" (or a "generalized random process"), but it seemed unwise to use this terminology (at this stage) in a set of notes on quantum field theory. Later we will use the term Euclidean field in a situation where our definition would suggest "Euclidean process." 4. "Random processes" are often called "stochastic processes." DEFINITION.

Asetofrandomvariables

space (Μ,Σ,μ) is called full if Σ/3

on a probability measure

is the smallest measure algebra,

H-

with respect to which each f a is measurable (i.e., if the equivalence classes of {f~ 1[Ω] |a e I, Ω

C

R, Boreli in Σ/ίΙ

in any proper σ-subring of Σ/ί ).

are not all contained

14

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

LEMMA 1.5. Let

be a family of random variables on

Let

The following are (1)

equivalent:

^ is fuU as a subset of

(2)

is dense

(3) The von Neumann algebra

generated by

family of all multiplication

We concentrate on

Proof.

operators,

is

which is what we will use below.

The rest is fairly standard (but somewhat tedious). and

viewed as a

. For suppose

family of projections in

We will show

is full and let S be the

Each of these projections is also a multi-

plication operator so S is a subset of submeasure algebra and that each Thus S is all of

It is easy to see it is a is measurable with respect to S.

' is all multiplication operators.

that

we note that

closure of

Thus

contains

contains

> prove

since „ ' is the strong

which is dense in

i

Remark:

Here and throughout our _ valued functions.

Similarly

spaces

are always spaces

of

complex-

denotes complex-valued functions. We

use A preliminary and to denote the second spaces main of real-valued functions. version of the object of this section is: DEFINITION. A random process indexed by V is called a Gaussian random process indexed by V if and only if: (a) Remark:(b) Each

is full is a g.r.v.

Since V is a vector space, (b) implies that any

are

jointly Gaussian. Given a Gaussian random process, the map fines a (semi-definite) inner product on V. In the usual way, we can

de-

15

§1.1. GAUSSIAN RANDOM VARIABLES

quotient out by iv \ = OS and complete to a Hilbert space, K. It is not hard to see (after developing a part of the theory below!) that nothing is lost by supposing V to be a Hilbert space to begin with: DEFINITION. Let K be a real Hilbert space. The Gaussian random

process indexed by K is a random process indexed by H so that (a) \φ(ν) I ν e H! is full. (b) Each φ(ν) is a g.r.v. (c) = ί , the inner product on K. Remarks: 1. The

in = ^ is somewhat unnatural. We

add the \r so that when we consider the connection with Fock 1 (A *(v) + A(v)). If space in Section 1.3, we can write φ(ν) = = V2 one suppressed the \/2 in this last formula, one could eliminate the y above, but we prefer to conform to the usual physicist's ^ -J convention. Many authors (e.g., [90, 135]) do not include the . 2. Notice that the phrase "Gaussian random process indexed by K" means more if K is a Hilbert space than if K is merely a vector space. In Section 1.6, we will have occasion to deal with two Gaussian processes indexed by the vector space, K. We will then use the phrase "Gaussian random process with general covariance.' Of course, we will have to justify the use of the article "the" by proving a uniqueness theorem. In fact: THEOREM

1.6. Let K be a real Hilbert space. Let φ and φ' be two

Gaussian random processes indexed by K on probability measure spaces (Μ,Σ,μ) and (M', Σ', μ') respectively. Then there exists an isomorphism between the two probability measure spaces so that for every corresponds to φ'(ν) under the isomorphism.

VfK,

φ(ν)

16

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

Proof.

This proof will not be the shortest possible and it will have a

functional analytic flavor rather than a probabilistic one, but it has the advantage of introducing useful additional structure. Let the space

denote

be the closed subspaces generated by

On account of Corollary 1.4(a), for any

. Moreover,

converges in

, This

follows by direct computation on the single random variable contains each

any

and so each n

~

Thus

In particular, for

__

(1.24)

is in

. But such random variables are dense in

by Lemma

1.5. Thus (1.25)

Let

be the von Neumann algebra generated by the multiplication

operators (1.24). By the proof of Lemma 1.5,

, is isomorphic to the

ring of projections in Now let

by

By Corollary 1.4, U is unitary and well defined and by (1.25) and its primed analogue, it is defined from all of Since i

so

is given by a convergent series,

to all

§1.1. GAUSSIAN RANDOM VARIABLES

for all

Thus

of

and

so

13

sets up an isomorphism

. Under this isomorphism

and

> clearly

correspond. Remark: This theorem is a special case of a general theorem of Kolmogorov which asserts that measure spaces are completely determined by consistent joint probability distributions; see [15, 144], At this point, we have not yet proven the existence of the Gaussian random process indexed by section.

. This we will do at the start of the next

For the time being, we assume the existence of such a process.

We introduce some notation: denotes the underlying measure space denotes the underlying measure denotes the process denotes denotes the von Neumann algebra denotes the subspace of

spanned by the

Remarks: 1. Of course,

is not canonical — we consider different " m o d e l s "

for Q in the next section. 2. When a fixed

is involved, we will often drop the subscript

We c l o s e this section by considering some relations between operations on

and on

PROPOSITION 1.7. Let

Then:

(1.26)

18

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

Remark:

What a) means is that given any models spaces associated to

and

and

for the Q

when given the measure

b) and process c ) is a model for the Q-space over Proof.

Define

by (1.26) on the product space

with the

product measure. Then the joint probability distribution for

( f j ) and

is a product of Gaussians, so these two r.v. are jointly Gaussian and thus their sum is a g.r.v. since and similarly PROPOSITION 1.8. Let

space.

K

Moreover

be a (closed)

Then there is a natural imbedding,

that (1)

defines a homomorphic imbedding of

Proof.

of K ,

H, of

a real Hilbert

) into

x

so

for all defined by

(2)

positive

subspace

(bounded) random variables

Write

i n t o i n on ~

and let of Proposition 1.7.

particular a

into positive

takes

r.v. on

under the identification

§1.2. MODELS OF Q-SPACE

References

19

for Section 1.1

For general probability theory: Breiman [15], Feller [42], For Gaussian process: Gel'fand-Vilenkin [54], Hida [98], For Wick Products (from partially different points of view): Caianiello [17], Dimock-Glimm [27], Segal [166, 167], Wick [201], §1.2. Models of Q-space In this section, we fix a real separable Hilbert space H and consider explicit choices for a measure space

on which we can construct the

Gaussian random process indexed by K . Model 1 (Infinite Product Space).

Let

be a basis for H.

Let

be the product of an infinite number of copies of the one point compactification of R. Q is a compact Hausdorff space. 1 be the measure on Q which is an infinite product of explicitly given a function in

Let

exp

only dependent on

finitely many coordinates, let

Then

is well defined, linear and

are dense in

Since such functions

(by Stone-Weierstrass),

defines a measure

on

Let

which is a function from Q to

extends to C(Q) and so be multiplication by

which a.e. takes values in R,

so it can be viewed as a random variable.

• and oo

is clearly full. If pansion. Then

be its Fourier ex-

) converges in

putation; call the a.e. defined limit

by explicit comIt is easy to show that

is a g.r.v. (e.g., by computing its moments) and that . Thus process indexed by K.

provides a model for the Gaussian random

20

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

Remarks: 1.

is a measurable set with can take Q to be adding the

— measure 1. Thus we

. This is in many ways more natural. By

above, we get a " c h e a p " proof of the countable

additivity of djnQ (only " c h e a p " by relying on functional analytic machinery).

There is, of course, a direct proof of the countable

additivity, see e.g., Gel'fand-Vilenkin [54], Glimm-Jaffe [71]. 2. The above construction works if H is non-separable; thus:

THEOREM 1.9. For any (real) Hilbert space, indexed by H

K,

the Gaussian

process

exists.

3. The above model for Q is, of course, basis dependent. This is both a virtue (see e.g., Section 1.6) and a sin (it's "unnatural"). Model 2 (Spectrum Model). Once we know some choice of can form cal,

, as in the proof of Theorem 1.6. While i

• exists, we is not canoni-

is, in the sense that Theorem 1.6 sets up a natural isomorphism

of the

associated to two different choices of Q. Let Q s P e c be the

Gel'fand spectrum of the commutative Banach algebra then Exp (A) defines a positive linear functional on so a measure

and

, on

Remarks: 1. As we constructed it above, to form struct a preliminary model of Q to obtain

, one had to first con. By using Fock

space (see Section 1.3), one can avoid this; see e.g., [184], 2. The only advantage of

is that it is " c a n o n i c a l , " only de-

pending on H and not on some additional structure. Model

(Suggested by W. Faris).

There is another " c a n o n i c a l " model

for Q about as silly as model 2! Namely take for Q an uncountable product of copies of

one for each

) be

§1.2. MODELS OF Q-SPACE

multiplication by

21

As measure, n, take the measure whose restric-

tion to functions o

f

i

s

the Gaussian measure with covariance

. Or one could take one copy for each unit vector in K . Model 3

There has developed a general theory of

"cylinder set measures" on

the set of tempered distributions.

The same theory works on

or on the dual of any nuclear space.

Let us consider the theory associated with

A cylinder set in

the set of distributions T so that are n fixed elements in

and

is

where is a fixed Borel set in

which

indexes the cylinder set. A cylinder set measure is a measure, \i, on the a-algebra generated by the cylinder sets with tion, each

defines a measurable function

and

is full. If

By construcon

by

weakly, then

pointwise.

Thus, by the dominated convergence theorem f exp (i 2.

Proof.

Let

Let

-

Then Then

is a c o n t r a c t i o n from

§1.5. HYPERCONTRACTIVE BOUNDS

39

Remarks: 1.

T h u s the constant

2 in Lemma 1.18 can b e r e p l a c e d with 1

T h i s ( a n d in g e n e r a l

i s b e s t p o s s i b l e , f o r if

for a l l t h e n

by f o l l o w i n g the path from L e m m a 1.18 1

to 1.21,

, t h i s would v i o l a t e

the b e s t p o s s i b l e nature of T h e o r e m 1.17. 2.

S i n c e the c o n s t a n t

a r i s i n g in the proof of L e m m a 1.18 o b e y s ,

w e s e e that

In f a c t , by u s i n g the " b e s t p o s s i b l e n a t u r e " of T h e o r e m 1.17, w e s e e that

So F o c k s p a c e i s c a p a b l e of d o i n g c o m b i n a t o r i c s ! Added

Note:

L . G r o s s [ 8 5 ] has r e c e n t l y g i v e n a n e w and e l e g a n t proof of the b e s t p o s s i b l e bounds (1.42). hypercontractivity.

G r o s s a l s o d i s c u s s e s the i n f i n i t e s i m a l form of

E x p l i c i t l y , (1.42/3) are e q u i v a l e n t to t h e bounds:

where

In particular, for

which s h o w s that w h i l e in the O r l i c z s p a c e

_ In L .

may not be in

to S e c t i o n 1.4, 1.5:

Gross [85], N e l s o n [135].

one f i n d s :

for any

, it i s

F e i s s n e r [ 4 0 ] h a s e s t i m a t e s i n v o l v i n g higher

d e r i v a t i v e s of h y p e r c o n t r a c t i v i t y . References

p = 2,

40

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

§1.6. Relative Absolute Continuity of Gaussian Processes In this section, we wish to consider three related questions: (1) Given a real Hilbert space, K, and a linear transformation A on K, when is there a unitary map U on Γ(Η) so that U φ(f)U _1 =

CjS(Af)

for all f f H?

(2) Given A, as in (1), when is there an F in Γ(Κ) so that each φ{f) is also a g.r.v. with respect to |F| 2 d/n 0 but with variance ί IlAfII 2 instead of |!|f|| 2 ? (3) Given two Gaussian random processes with general covariance, indexed by the same Hilbert space, K, when can they be realized as processes on a single measure space but with two mutually absolutely continuous measures? (1) and (2) are essentially active and passive versions of the same question. Suppose K is complete in the norm defined by each of the covariances in (3); it is not hard to see that then (2) is equivalent (3), for we need only take one of the covariances as new norm. With this in mind, we will consider problem 2, with the additional hypothesis that A is bounded with a bounded inverse. These problems are really special cases of a general problem in the theory of the Canonical Commutation Relations concerning the implementability of Bogoliubov or sympletic transformations. This problem was solved by Shale [171], after partial results by Friedrichs [50], GardingWightman [52] and Segal [159, 160], Actually, Shale quite easily reduces the more general problem to problem (1) and tackles that. In our discus­ sion below, we basically follow Shale's proof with some simplifications of Klein [115], Since B = A*A is all that enters in problem 2, we may as well suppose that A is positive and self-adjoint. The basic result is:

§1.6. RELATIVE ABSOLUTE CONTINUITY OF GAUSSIAN PROCESSES

THEOREM 1.23 (Shale [171]). bounded inverse cient

on a real Hilbert

condition

continuous

Moreover,

A—1

if

on

A

be a bounded positive

space,

for the Gaussian

to be realizable absolutely

Let

process

H.

by K,

• with a measure

relative

to t

h

e

A—1 be n

w

h — r-

for some

p>l

and

and

with

dp which

is that

IS Hilbert-Schmidt,

operator

Then a necessary

indexed

41

with suffi-

covariance is

mutually

Hilbert-Schmidt. e

r

e

'

for some

Remarks: 1.

Shale considers general Hilbert-Schmidt.

A

and then asks that

Since

be and

i n v e r t i b l e , this i s equivalent to demanding

is

be Hilbert-

Schmidt. 2.

See [145; Section V I . 6 ] for a discussion of Hilbert-Schmidt operators.

Proof: 1.

Suppose f i r s t that basis

A — 1 i s Hilbert-Schmidt.

for K

and

s o that

hypothesis, there i s a for all

Then w e can find a By with

n and (1.48)

2.

T a k e the model of Q - s p a c e where with measure .

Formally,

is an i n f i n i t e product of and

v

where

R's

i s multiplication by

i s just the product of measures Thus w e l e t

(1.49)

We w i l l then take in 3.

A s a general result w e note:

which we must show c o n v e r g e s

42

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

L E M M A 1.24 ( S e g a l [ 1 5 9 ] ) . subset

is independent

product).

(i.e.,

Let

be random

so that any joint

variables,

probability

Suppose

so that

distribution Finally

for some with

Then,

there

any is a

suppose

is an

I as

Proof

of lemma.

Let

Then

(1.50)

w h e r e w e h a v e used p o s i t i v i t y of the

, n o r m a l i z a t i o n and i n d e p e n d e n c e .

By H o l d e r ' s inequality

s o that (1.51)

S i n c e the product

c a n b e made arbitrarily

c l o s e to

1.

T h u s by (1.50) and (1.51),

c l o s e to

0,

so

can b e made arbitrarily

c o n v e r g e s to s o m e

. B y a s i m p l e argument,

( o f lemma). 4.

N o w w e claim that for s u i t a b l e

77, p > 2

and a l l

n: (1.52)

§1.6. RELATIVE ABSOLUTE CONTINUITY OF GAUSSIAN PROCESSES

where

rj and

p depend on

c,

In (1.52),

43

the constant with (

i s an

) norm.

To

prove (1.52), w e note that the integral

c o n v e r g e s only i f

and that in that c a s e :

(1.53a) where (1.53b)

Pick

so that so the integral c o n v e r g e s .

for

and

Moreover, (1.53) i s c l e a r l y . Thus by T a y l o r ' s

theorem: (1.54) where

(1.52) f o l l o w s from (1.53) and (1.54). 5.

On account of (1.48), and

(1.55)

44

GAUSSIAN RANDOM PROCESSES, Q-SPACE AND FOCK SPACE

6.

B y L e m m a 1.24, and t h e bounds (1.52), (1.55), the i n f i n i t e product c o n v e r g e s in .

Let

) to a function and

measure and e a c h

is a

g.r.v.

. w.r.t.

Then v

v

F

with

is a probability

with variance

for

w e can compute

7.

B y L e m m a 1.24,

'

for s o m e

so

for s o m e

B y turning around t h e a n a l y s i s and l e t t i n g s e e that

I for s o m e

we

. But c l e a r l y

T h i s c o m p l e t e s the proof o f s u f f i c i e n c y . 8.

S i n c e w e w i l l not u s e n e c e s s i t y , w e d o n ' t g i v e the f u l l proof which can b e found in Shale [ 1 7 1 ] or other r e f e r e n c e s .

9.

H o w e v e r , to i l l u s t r a t e t h e phenomena c o n s i d e r the c a s e T h e n on

w e can d e f i n e t h e i n f i n i t e product, and t h e i n f i n i t e product,

i

-

Similarly

Choose

s o that

Then

v,

of

A = 2. of

§1.6. R E L A T I V E ABSOLUTE CONTINUITY OF GAUSSIAN PROCESSES

while

so

n

i s not a b s o l u t e l y continuous with r e s p e c t to

References for Section 1.6: B e r e z i n [9], K l e i n [115], S e g a l [158, 159], S h a l e [171].

45

CHAPTER II AXIOMS, I In this chapter, we will discuss three axiom schemes for relativistic scalar quantum fields: the Garding-Wightman axioms for fields [207], their translation to vacuum expectation values [203] and the OsterwalderSchrader axioms for Euclidean region "Green's functions" (or, as we shall call them, Schwinger functions) [143], Unfortunately, the situation for the Osterwalder-Schrader axioms has been complicated by an error in their original paper [143]. The equivalence of the first two and some of the resulting theory has worked its way into the monographs of Jost [110] and Streater-Wightman [189] and for this reason, we intend to be especially sketchy on those aspects of the theory (Section II. 1, part of Section II.3). Our treatment of the recent results of Osterwalder-Schrader (Section II. 2-4) follows their papers [140, 143] for the basic ideas but includes some notational and technical simplifications. Considering how basic are the Garding-Wightman axioms (unless one rejects fields completely, the only axiom which one can reasonably imagine not holding in a relativistic quantum mechanics is the temperedness), it is surprising how rich is the resulting structure and how difficult it is to construct models (these two facts are not unrelated!). In Section II.5, we will describe the "trivial" but basic free field models. A final word about the role of axioms in quantum field theory seem:5 in order, especially since the motives of "axiomatists" have been occa­ sionally (mistakenly!) questioned. Axioms present nothing other than a framework for quantum field theory and, in particular, for constructive field theory. The first goal for constructive field theorists should be the

AXIOMS, I

47

verification of the axioms — but this is just a first goal and, at that point, one must begin to ask the really interesting questions about the models. We suppose the reader familiar with the basic notions in distribution theory and with the theory of their Fourier transforms — on the level of [189; Chapter 2] or [145; Section V.3; Section IX.1], We discuss only the case of a single, neutral (hermitean) Bose, scalar field. The results of this chapter extend to arbitrary spin and statistics (with the usual correlation between the two!). We also work in two dimen­ sional space-time but avoid using any special features of two-dimensions which preclude generalization to higher dimensions. In addition, much of the theory of this chapter has been extended to test function spaces other than

S. Jaffe [108] developed the "classical" theory for test functions

whose distributions can have growth in p-space roughly as exp(|p|^) for some β < 1 rather than just polynomial growth. Continescu and Thalheimer [23, 24], have synthesized the work of Jaffe and Osterwalder-Schrader to extend the OS Axiom scheme to Jaffe fields. In the long run, these ex­ tensions are probably needed if we are ever to control non-renormalizable interactions and, if exactly solvable models are to be trusted, perhaps even some renormalizable interactions [195], The use of Euclidean methods in field theory has a long history. It appeared first, like so much else in field theory, on the level of perturba­ tion theory — see Dyson [33], Wick [201], Weinberg [200], Analytic con­ tinuation of vacuum expectation values to a region including the Euclidean region was first considered by Wightman [203] and Hall-Wightman [93] but the Euclidean region itself was not emphasized by them. Euclidean field theory was first considered by Schwinger [157] and Nakano [211]; it was Symanzik [192, 193, 194] who first emphasized a purely Euclidean approach to field theory, especially model field theories. Symanzik also developed many of the ideas special to these theories. It was Nelson [132,134,135] who, by combining these ideas with constructive field theory techniques and introducing new ideas and techniques, demonstrated the power of the approach. We return to these ideas in Chapters III-V. Finally, by

48

AXIOMS, I

s u p p r e s s i n g a part o f the S y m a n z i k - N e l s o n i d e a s , O s t e r w a l d e r - S c h r a d e r [ 1 4 3 ] d i s c o v e r e d theorems r e l a t i n g M i n k o w s k i f i e l d t h e o r i e s and E u c l i d e a n r e g i o n expectation

values.

F o r p e d a g o g i c a l reasons, w e have presented the Osterwalder-Schrader a x i o m s b e f o r e N e l s o n ' s a x i o m s ( s e e Chapter I V ) e v e n though t h e h i s t o r i c a l order i s o p p o s i t e .

We e m p h a s i z e that s o m e of the arguments in S e c t i o n II.4

a r e borrowed from N e l s o n [ 1 3 4 ] and that the w h o l e e m p h a s i s on the E u c l i d e a n region which l e d O s t e r w a l d e r - S c h r a d e r to their a x i o m s i s due to N e l s o n and Symanzik. §11.1.

The Garding-Wightman Theorem

Axioms

and the Wightman

Reconstruction

T h e b a s i c p r o p e r t i e s demanded of any quantum f i e l d theory b a s e d on a s i n g l e Hermitean s c a l a r f i e l d are: (GW 1 ) ( H i l b e r t Space). unit v e c t o r ,

There is a Hilbert space

c a l l e d the

(GW 2 ) ( F i e l d s and Temperedness). and for e a c h For each

(b)

Each

with a d i s t i n g u i s h e d

vacuum. There exists a dense subspace

, an operator

(a)

K

> with domain

s o that

i s a tempered distribution. for

f

real v a l u e d i s symmetric, i . e . ,

all (c)

leaves

(d)

and arbitrary,

n

(GW 3 ) ( C o v a r i a n c e ) .

invariant, i . e . ,

is ( a l g e b r a i c a l l y ) spanned by arbitrary!. T h e r e i s a unitary r e p r e s e n t a t i o n of the proper

P o i n c a r e group (a) (b) (c)

I s o that

leaves . .

i m p l i e s that

invariant, i . e . ,

implies

for a l l i . e . , for any

and

§11.1. THE GARDING-WIGHTMAN AXIOMS

49

(II. l a ) where (II. l b )

T h e j o i n t spectrum of t h e i n f i n i t e s i m a l generators of

) (Spectrum). U ( a , 1)

l i e s in the forward l i g h t c o n e .

(Locality

or Microscopic

If

Causality).

f

and

g have space-like

supports, then (II. 2) for all (GW 6 ) ( U n i q u e n e s s of Vacuum). a l l the

T h e only v e c t o r s in

K

l e f t invariant by

are the m u l t i p l e s of

Remarks: 1.

(GW 1 - 6 ) are known a s the Garding-Wightman

2.

We h a v e s p l i t and rearranged the a x i o m s as s t a t e d in [110, 189]

axioms.

with a v i e w t o w a r d s the natural partition of the Wightman a x i o m s for vacuum e x p e c t a t i o n v a l u e s .

In particular, w e h a v e hidden

c y c l i c i t y of the vacuum in (GW 2). 3.

S e e [189, Chapter 1] for a r e v i e w of our r e l a t i v i s t i c notation.

In

particular, w e w i l l employ a M i n k o w s k i inner product in the F o u r i e r transform and d e f i n e the e n e r g y ,

H,

and momentum,

by (II. 3)

if

(Of c o u r s e , in two d i m e n s i o n s , a l l our s p a c e " v e c t o r s '

are o n e d i m e n s i o n a l . ) and

We s y s t e m a t i c a l l y w r i t e

x

f o r i t s ( o n e d i m e n s i o n a l ) s p a t i a l part.

for a v e c t o r in We usually w r i t e

( I I . 3 ) in the form: (II. 4) 4.

(GW 3 a ) f o l l o w s from (GW 2 d ) and (GW 3 b , c ) .

50

AXIOMS, I

Now fix

T h e n for

w e can d e f i n e (II. 5)

on account of

i s c l e a r l y multilinear in

s e p a r a t e l y continuous.

and by

It f o l l o w s that it i s j o i n t l y continuous and that

there e x i s t s a distribution, denoted by (II. 6 ) in

is c a l l e d the Wightman

or Vacuum

distributions

Expectation

Values.

T h e i r properties are summarized b y :

T H E O R E M II. 1.

The Wightman

ing the Garding-Wightman (W 1) (Temperedness). which

distributions

axioms For each

associated

to a theory

obey-

obey: n,

is real in the sense

is an element

of

that

(II. 7)

for any (W 2) ( C o v a r i a n c e ) .

Moreover Each

for all

is Poincare

invariant,

i.e.,

where

(W 3 ) ( P o s i t i v e Definiteness). by

Given

and

define

§11.1. THE GARDING-WIGHTMAN AXIOMS

51

by (II. 7). Then given

and

(W 4 ) ( S p e c t r u m Condition). in

For each

supported

there exists

a

distribution

in ,

the closed

forward light

conel

so that

(W 6) ( C l u s t e r Property).

If

a

is space

like and (11.10)

Remarks: 1.

( I I . 8 ) , ( I I . 9 ) and (11.10) must b e interpreted in distributional s e n s e . ( I I . 8 ) can, of course, b e expressed in terms of a change in variable plus Fourier transform statement.

2.

Roughly speaking In a precise sense, Other subset equivalences exist!

3.

We w i l l not g i v e a d e t a i l e d proof, which can b e found in [110,189]. We note that just s a y s that

are immediate transcriptions. (W 3)

AXIOMS, I

52

has a nonnegative norm where

is defined as f o l l o w s :

By using the Nuclear theorem, one e a s i l y shows that is a separately continuous multilinear vector-valued function, s o by the nuclear theorem ( a g a i n ! ) , w e can d e f i n e a v e c t o r valued distribution the formal

T h e intuition behind (W 4) is

expression:

on account of the

formal

T h e actual proof i s more technical ( s e e a l s o [145, Section I X . 8 ] ) . F i n a l l y (W 6 ) can b e proven as f o l l o w s : multiplication by

k on

L e b e s g u e lemma,

exp

Suppose first that

weakly as

onto the family of vectors invariant under

4.

Thus, if

is

Then, by the Reimann-

e a s i l y f o l l o w s that for a s p a c e - l i k e ,

topology.

p

. From this it the projection

U(a, 1),

in the weak

then

B y f o l l o w i n g the argument in 3, one can a l s o prove time-like cluster properties.

T h e s e seem to h a v e r e c e i v e d much l e s s attention,

partially b e c a u s e the rate of f a l l o f f in time-like directions is typic a l l y much s l o w e r than in s p a c e - l i k e directions. T h e point of (W 1 ) - ( W 6) is:

§11.2. THE THEORY OF LAPLACE TRANSFORMS

53

THEOREM 1.2 (The Wightman Reconstruction Theorem). Given a family

ί® η ^η=0 obeying (W 1)-(W 6), there exists an essentially unique theory obeying (GW 1)-(GW 6) for which the ® n are the Wightman distributions.

Remarks: 1. Again we don't give the proof but only note the main ideas, which parallel the GNS construction in the theory of C*-algebras. Let

S

be the family of all sequences f Q , ..., f n ,... with f^eSiR 2 ^)

and with f^ = 0 for all large k. By (W 3), the vector space

S

possess a natural inner product, so in the usual way we can quo­ tient and complete to get a Hilbert space, H. D 0 is then taken to be the image in K of those elements of

S

with ^(X j ,...,x n ) =

g 1 (x 1 )... g n (x n ) and φ(ί) is defined on D q in the obvious way. 2. Except for the equivalence of (W 6) and (GW 6), this theorem is due to Wightman [203]. The connection of clustering and uniqueness of the vacuum was first emphasized by Hepp et al., [97]. 3. Certain subsets of (W 1-6) are equivalent to corresponding subsets of (GW 1-6). See remark 2 after Theorem II.1. References for Section II.1: Wightman Axioms and Theory: Jost [110], Streater-Wightman [189], Wightman-Garding [207]. ' Relativistic Invariance: Barut-Wightman [7], Mackey [127], Wigner [208], Distributions and Their Fourier Transforms: Gel'fand-Shilov [53], ReedSimon [145, Chapter V, IX], Schwartz [156], Yosida [210].

§11.2. The Theory of Laplace Transforms: A Technical Interlude Unfortunately the details of the Osterwalder-Schrader theory depend on some rather technical results in the theory of Laplace transforms. The basic results of this theory are discussed in various texts [110, 145, 156, 189] but rather specialized results will be needed in Section II.4. On a first reading, it is probably best for the reader to skip the proofs, checking statements in order to settle the notation. We will first discuss the method of taking Laplace transforms in a general setting.

54

AXIOMS, I

DEFINITION.

Let

C

b e a c o n v e x c o n e in

by

T h e dual c o n e i s d e f i n e d

for a l l

T H E O R E M II.3.

Let

form has support

in a cone

Let

T

be a distribution C

in

whose Fourier

whose dual cone

C'

has non-empty

be the tube in

there exists

an analytic

transinterior

Then, F

function

in

so

that: (II. 11a)

C,k,m.

for suitable C,

k

and

(b)

T

dist(z, w )

Here

i s in a Euclidean

m only depend

on the norm and constant

is the boundary

value

of

F

norm on

bounding

in the sense

T.

that for any

and

(11.12)

T h i s theorem i s stronger than the usual one stated in the axiomatic t e x t s [110, 189],

In this form, it i s due to B r o s , E p s t e i n and Glaser [16]

( s e e a l s o [145, Section I X . 3 ] ) . T h e result i s o b v i o u s if in C ,

T h e i r proof is very s i m p l e and e l e g a n t :

i s a polynomial bounded function with support

for in that c a s e o n e can d e f i n e

F

by the L a p l a c e transform formula:

(II. 13a)

where the integral is absolutely continuous s i n c e T h e a n a l y t i c i t y of tive exists.

for

i s e s t a b l i s h e d by showing a complex deriva-

T h e bound (11.11) i s a l s o e a s y to e s t a b l i s h as i s (11.12).

Suppose now that ator and that

F

Im

P(D)

i s a constant c o e f f i c i e n t partial d i f f e r e n t i a l operwhere

function with support in the cone

^ is a p o l y n o m i a l l y bounded continuous C.

T h e n , a g a i n , the result f o l l o w s by

writing (II. 13b)

§11.2. THE THEORY OF L A P L A C E TRANSFORMS

55

T h u s the theorem f o l l o w s from:

L E M M A II.4 ( B r o s - E p s t e i n - G l a s e r L e m m a ) . support

in a convex

continuous coefficient)

Sketch

G

function

where

If

S has order

in

exists

k,

bounded (constant

s o that

with the " o c t a n t "

I

and i s C'.

with

a polynomially

C ' and a differential

P i c k coordinates inside

it h a s support in

then there

with support

P(D)

operator

of proof.

C",

cone,

S i s a distribution

If

Let

1 if

t > 0.

Clearly

is

and

M o r e o v e r , in d i s t r i b u t i o n a l s e n s e :

it i s not hard to s h o w that

and

o b e y s a l l the c o n d i t i o n s of the lemma. Remark-. F o r later u s e w e note that L e m m a I I . 4 (and thus a l s o T h e o r e m I I . 3 ) h a s an e x t e n s i o n t o distributions with v a l u e s in a f i x e d Banach s p a c e .

We w i s h to c o n s i d e r t h e c o n v e r s e problem of when a f u n c t i o n , more g e n e r a l l y a d i s t r i b u t i o n ) on

J

C

is

in

(or

or on o n e of i t s s u b s e t s i s a L a p l a c e

transform, i . e . , h a s a r e p r e s e n t a t i o n of t y p e (11.13). special c a s e where

F,

R.

We w i l l c o n s i d e r the

In our a p p l i c a t i o n s , w e w i l l

smear in s p a c e v a r i a b l e s and thereby reduce to t h i s s p e c i a l c a s e or a m u l t i v a r i a b l e v e r s i o n of t h i s s p e c i a l c a s e .

We t h e r e f o r e d e f i n e :

56

AXIOMS, I

and l e t

d e n o t e the s e t o f f u n c t i o n s in

inside

g i v e n the r e l a t i v e t o p o l o g y .

in

on

with support s t r i c t l y i s the s e t o f a l l f u n c t i o n s

, w h o s e d e r i v a t i v e s a l l extend continuously to

and for which

for a l l m u l t i - i n d i c e s

a

and

T h e s e norms d e f i n e a t o p o l o g y on

B y the Whitney e x t e n s i o n theorem ( s e e e . g . , [103, 1 8 8 ] ) any t h e restriction to

of s o m e

f

in

so

is

can b e v i e w e d a s

a quotient s p a c e

i

~r

and, b y t h e open mapping theorem, the t o p o l o g y i s j u s t the quotient t o p o l o gy.

A s a result, d i s t r i b u t i o n s

l i f t naturally to

systematically use the same symbol for

T

and i t s l i f t i n g .

i s naturally i d e n t i f i e d with t h o s e Finally, we will use objects like DEFINITION.

A function

and only if there is a

.

. We w i l l

with

In this w a y , supp

without comment. on

i s called a Laplace

transform

if

with (II. 1 3 c )

Remark: It i s e a s i l y s e e n that f o r e a c h f i x e d T h e f o l l o w i n g i s t h e standard c h a r a c t e r i z a t i o n of L a p l a c e transforms:

T H E O R E M II. 5. If

F

is a Laplace

to tube

transform,

then

of a function and

F

F(z)

is the

restriction

analytic

in the

obeying: (II. l i b )

§11.2. THE THEORY OF L A P L A C E TRANSFORMS

C , k, m

for some

T.

the distribution Fourier sense

which

only depend T

Moreover,

transform, of (II. 12).

a continuation

to

„

obeying

of some

for all

on the norm and constant

can be recovered

is the boundary Conversely,

if

F,

57

a function

(II.lib),

then

value

from

F

of

F(z)

on

F

bounding because in the

is given

i s the Laplace

and has

transform

Moreover

f

in

its

(11.14)

where

is a norm only

depending

on the

integers

k, m in ( I I . l i b ) .

T h e f i r s t half of t h i s theorem i s j u s t a restatement of T h e o r e m II.3. P r o o f s of t h e s e c o n d h a l f can b e found in [156], [189] or [145; S e c t i o n I X . 3 ] , T h e bound (11.14) d o e s not appear e x p l i c i t l y but i s i m p l i c i t in the p r o o f s . We next want to e x a m i n e s o m e g e n e r a l f e a t u r e s of L a p l a c e transforms a s distributions: P R O P O S I T I O N II.6.

Let

F

d e f i n e s an e l e m e n t o f

b e the L a p l a c e transform of

T.

Then

F

M o r e o v e r , f o r any ( I I . 15a)

where ( I I . 15b)

Proof.

If

then g i v e n

with remainder about

for any

m.

T h u s , g i v e n any

and making a T a y l o r e x p a n s i o n w e s e e that

m, k,

w e can f i n d a norm

so that (11.16)

58

AXIOMS, I

for all h t S(R"). By (II. 11) and (II. 16), for any Laplace transform F and any h e S(R"), the integral f F(y)h(y)d n y converges and defines a dis­ tribution. To prove (11.15), we note that it certainly holds if T is a func­ tion in S(R") (by Fubini's theorem) and that S(R") is sequentially dense in the weak topology in S(R")'. Moreover, if T n -» T weakly, it is easy to show that their Laplace transforms converge pointwise with a uniform bound of type (ILll) and so weakly. • PROPOSITION II. 7.

h h > h, given by (II.15b) is a continuous map of

S(R") to S(R") with a dense range and zero kernel. Proof. Continuity is easy and the fact that the kernel is trivial follows

from the fact that the Fourier transform of h can be obtained from h by analytic continuation and the taking of boundary values. If the range were not dense, there would be a non-zero T e S(R")' with T(h) = 0 for all h. Let F be the Laplace transform of T. Then F is zero as a distribution (and so as a function) by (II. 15a). But then T is zero since its Fourier transform is a boundary value of the analytic continuation of F. • THEOREM II.8.

°n

Let T be a distribution on S(R") so that for some norm

S(R+)'

ITOOI < Ilhll

(II-17)

for all h f S(R"). Then T is a Laplace transform.

Proof. For any g e S(R") of the form g = h let S(g) = T(h). Then S is

continuous by (11.17) and densely defined by Proposition II.7. Thus S extends to a map in S(R1J)'. Let F be the Laplace transform of S. By Proposition II.6 for all h

F(h) = S(h) = T(h)

C S(R").

Thus T is a Laplace transform. •

Remark:

An estimate of type (11.17) is, of course, very hard to verify. For example, ||h||

can be small without HhHoo being small.

§11.2. THE THEORY OF L A P L A C E TRANSFORMS

59

T h e r e i s a natural q u e s t i o n i n v o l v i n g L a p l a c e transforms in s e v e r a l v a r i a b l e s which e n t e r s in a c r i t i c a l p l a c e in the p r o c e s s of g o i n g b a c k w a r d s from the E u c l i d e a n region to the M i n k o w s k i r e g i o n . paraphrased:

T h i s q u e s t i o n can be

I s a distribution w h i c h i s a L a p l a c e transform s e p a r a t e l y in

e a c h v a r i a b l e a j o i n t L a p l a c e transform? E x p l i c i t l y , s u p p o s e and f o r e a c h

i

and each f i x e d

i s a L a p l a c e transform.

Is

F

a L a p l a c e transform?

In the o r i g i n a l

O s t e r w a l d e r - S c h r a d e r paper [ 1 4 3 ] , an i n c o r r e c t proof that t h e a n s w e r i s y e s appeared.

T h a t the answer is no i s s e e n by the f o l l o w i n g e x a m p l e of

Schrader: Example: Let

If w e smear in

i s a L a p l a c e transform. if it w e r e ,

F

But

F

c a n n o t b e j o i n t l y a L a p l a c e transform f o r

w o u l d h a v e an a n a l y t i c continuation to

p o l y n o m i a l l y bounded at i n f i n i t y ( T h e o r e m I I . 5 ) . tion of

F,

, clearly

expi

i with

which w a s

But the a n a l y t i c continua-

i s not p o l y n o m i a l l y bounded if

say A c t u a l l y there are t w o o b s t r u c t i o n s to p r o v i n g transform.

O n e i n v o l v i n g the f a c t that

a n a l y t i c continuation to

F

a joint L a p l a c e

may not h a v e the n e c e s s a r y can b e s o l v e d in the f i e l d

theory c a s e b y a p p e a l i n g to a d d i t i o n a l structure of c u s s i o n b e l o w and [60, 1 4 0 ] . )

F

It appears

F.

( S e e the b r i e f d i s -

that the s e c o n d problem of

b o u n d e d n e s s i l l u s t r a t e d by Schrader's e x a m p l e cannot b e o v e r c o m e without c h a n g i n g a x i o m s from t h o s e in [143], References

for Section

"Classical Theory":

II. 2: Schwartz [ 1 5 6 ] , Streater-Wightman [189],

60

AXIOMS, I

§11.3. Schwinger Schrader

Functions Axioms)

and Their

Properties

(The

Osterwalder-

Our f i r s t g o a l in this s e c t i o n w i l l b e the " a n a l y t i c c o n t i n u a t i o n " of the distributions

to a l a r g e region o f

. In this, w e f o l l o w

" c l a s s i c a l a x i o m a t i c f i e l d t h e o r y " [110, 189],

We f i r s t use the theory of

L a p l a c e transformations and (W 4 ) to f i n d a n a l y t i c f u n c t i o n s in

the

forward tube, then L o r e n t z c o v a r i a n c e (W 2 ) to e x t e n d t h e s e f u n c t i o n s to , the e x t e n d e d forward tube and f i n a l l y , l o c a l i t y (W 5 ) to extend them to

',

the permuted e x t e n d e d forward tube. w i t h the property that each

If o n e a s k s which p o i n t s

h a s a purely real s p a c e com-

ponent and a purely i m a g i n a r y time component l i e in that a l l such points with

, one finds

d i s t i n c t l i e in

. T o assure this

o n e must g o through the e l a b o r a t e three s t e p e x t e n s i o n to s t r i c t i o n of the a n a l y t i c f u n c t i o n s

to this s e t o f

.

T h e re-

"non-coincident

Euclidean p o i n t s " w i l l be c a l l e d Schwinger functions.

In this s e c t i o n w e

w i l l p r o v e that the S c h w i n g e r f u n c t i o n s h a v e a certain s e t of p r o p e r t i e s (OS 1 - 5 ) and in the n e x t w e w i l l s e e that t h e s e p r o p e r t i e s s u f f i c e to b e a b l e to reconstruct a Garding-Wightman theory. We f i r s t ( f o l l o w i n g Wightman [ 2 0 3 ] ) , apply the method of L a p l a c e transforms to the

. By (W 4 ) and the f a c t that the product of l i g h t c o n e s i s

i t s own dual c o n e , it f o l l o w s that a function

i s t h e boundary v a l u e of

in the r e g i o n

for

the forward

function of

On a c c o u n t of

tube.

i s only a

and m o r e o v e r for any

the proper

L o r e n t z group, (11.18)

Let mations,

d e n o t e the f a m i l y of determinant 1 complex i . e . , c o m p l e x transformations on Let

Lorentz

transfor-

so that

the extended

forward

tube ( w h e r e

+ • •

a c t s on

by

(n

times)).

T h e n a fundamental result a s s e r t s :

§11.3. SCHWINGER FUNCTIONS AND THEIR PROPERTIES

LEMMA II.9 (Bargmann-Hall-Wightman Lemma). It and obeys (11.18) for jext

then

61

is analytic in

has an analytic continuation

to

Remarks: 1. For a proof, see [189, Chapter 2]. 2. By definition, any

is of the form

for some

We can obviously try to define

and

The key problem is then to prove consistency of this attempted definition (i.e., if

is also

then

This is the bulk of the proof. Consistency depends critically on analyticity of While, by definition,

contains no real points,

does.

is called the set of Jost points and a fundamental lemma of Jost asserts, that

is a Jost point if and only if for any

non-negative with

(IL19)

in the Minkowski inner product. The set of Jost points is open. On this open set, the Wightman distribution,

is given as a distri-

butional limit of analytic functions which have a perfectly good (locally uniform) pointwise limit.

Thus

points is a real analytic function equal to

restricted to the Jost Moreover, at any

Jost point, each difference is space-like, so that for any permutation

,

(11.20) on account of (W 5). Finally, we define the permuted extended forward tube, for each

Namely

be the extended forward tube in the variables

AXIOMS, I

6 2

(11.21) The analytic continuation of

presents a continuation of

Moreover, by and

(and its analytic continuation)

agree on on

s o we can define a single function,

(equivalently we can continue

to all of

boundary values in different ways, we can recover the

Taking distributions

from the function There is an especially interesting subset of DEFINITION.

is called Euclidean

A point

if each

has a purely imaginary time component and purely real spatial component. Henceforth we parametrize Euclidean points in s o that

by a vector

and we write

The set of Euclidean points with the additional property that no is zero will be called the non-coincident

Euclidean

region,

denoted

LEMMA II. 10.

Proof.

The non-coincident

Let

Euclidean

region,

is contained

The set of vectors orthogonal to some

is a family of at most

codimension 1 planes, s o we can

find some unit vector e with

for all

We can now reorder

i s o that We claim that the corresponding

lies in

simultaneous rotations on each y j lie in takes

clearly takes

For the

and the rotation which into

Below, we will need a slightly stronger version of this last proof: For

in

the

"Edge of the Wedge" theorem.

are disjoint so the "continuation" requires the

§11.3. SCHWINGER FUNCTIONS AND THEIR PROPERTIES

LEMMA 11.11. For each

n, there exists

so that lor each

there is a unit vector

with

In particular, there is a permutation

Proof.

63

with

Suppose we can show that given any

unit vectors

we can find

Then clearly we

are done if we take

The sphere

is compact.

Define

is clear-

ly continuous and strictly positive, s i n c e m i n lies in one of

hyperplanes.

DEFINITION. The restriction of

takes a minimum value,

to the non-coincident Euclidean

region is called the n-point Schwinger function. the

implies that

Viewed as a function of

we denote it by

Remarks: 1.

Occasionally, we will later have a natural definition on coincident points in which c a s e we refer to the above as

Schwinger

non-coincident

functions.

2. It is common to think of the S n as continuations of the Green's functions (time-ordered Vacuum expectation values) because of the ordering conditions of the Im

They are, of course,

also analytic continuations of the (unordered) Wightman functions. In order to describe the properties of the S n which we wish to single out, we need to introduce two special spaces of test functions: will denote the family of all test functions in

which vanish

64

AXIOMS, I

(together with their derivatives) on each hyperplane is a closed subspace of

we put the relative topology on

elements of the dual space are called distributions will denote the set of those functions in

on

with support in the region

Remark: By the Hahn-Banach theorem, every distribution on to a distribution on

but functions in

lar than those in

extends can be more singu-

For example,

defines a distribu-

tion on

for by Taylor's theorem for any

While this distri-

bution has extensions to all of

, none of them are given by functions.

Basically, the functions in

are those which have at worst poly-

nomial singularities at coincident points. Below, when we say a function F lies in

we mean that the integral

is absolutely convergent for all

and that it defines a continu-

ous functional. THEOREM 11.12. The (non-coincident) any scalar Wightman theory

to

obey.

(OS 1) (Temperedness). For each of

Schwinger functions associated

with

defines an element obeys the following reality

property: where (II. 22b) and where

is given by

§11.3. SCHWINGER FUNCTIONS AND THEIR PROPERTIES

(OS 2) (Covariance).

Each

is Euclidean invariant, i.e.,

the proper

for all

takes

(OS 3) (Positive Definiteness).

INote:

65

Euclidean group where

into i t s e l f . ]

Let

For all sufficiently large

Remarks: 1.

Osterwalder-Schrader take a slightly different cluster property.

2. Most of these properties have been known for some time. (OS 4) has been emphasized by Jost [110]. Osterwalder-Schrader first emphasized the role of (OS 3) and of the totality (OS 1-5). 3. (OS 1-5) are called Osterwalder-Schrader

axioms.

66

AXIOMS, I

Proof:

(OS 1). On account of Theorem

obeys:

in the forward tube for suitable D, k, m. By Lemma 11.11, there is a c that for each

we can find

Using the invariance of

and

so

with

under rotations and permutations (see the

proofs of (OS 2) and (OS 4)) and (II.3a) we find that: (11.25)

Given

we have by Taylor's theorem that

so that for any where

(11.26) we conclude that

is an

dy converges for each

; and defines a continuous func-

tional. (OS 2). Since the integral defining S(f) converges, this follows from a pointwise invariance for S. This is a direct result of Lemma II.9 (the Bargmann-Hall-Wightman Lemma). (OS 3). By the vector-valued nuclear theorem (see remark 3 after Theorem II. 1) we can define tion. Its Fourier transform in

as a vector valued distribuhas support in a

product of light cones, so by the vector-valued version of Theorem II.3, it is the boundary value of a vector-valued function analytic in a suitable forward tube. (Note:

As (11.18) doesn't hold, we can't extend this function

§11.3. SCHWINGER FUNCTIONS AND THEIR PROPERTIES

to the extended forward tube.) In particular, letting spatial vectors and

1

67

be fixed

we can naturally define:

as the value of this vector-valued analytic function at a suitable point. We first claim that

follows by simple analytic continuation of the vectors back to the Minkowski region (!). On account of (11.28) and of the existence of vectors of the form (11.27), continuous in x and

is easily seen to assert positivity

of the norm of a suitable vector. (OS 4). As with (OS 2) we need only a pointwise property. This follows from (11.28). (OS 5). On account of (11.28), this is a direct consequence of the fact that

for any

Remarks: 1. On account of Proposition II.6 and the definition of 11.12 remains true if For eacl so that for any

is replaced by the stronger: defines an element of

Theorem

68

AXIOMS, I

for a suitable norm,

where

Laplace transform. Moreover,

and

is the

obeys

2. We denote

§11.4. The Osterwalder-Schrader

Reconstruction

Theorem

After the appearance of the original Osterwalder-Schrader paper [143] in'preprint form, it seemed that one had the happy state of affairs Unhappily, after about six months, a technical error was found in their last lemma and in [140] Osterwalder discussed various ways of improving the situation. From a purely axiomatic point of view, the best thing is to note that

is equivalent to

THEOREM 11.13 (O.-S. Reconstruction Theorem). A set of Schwinger "functions"

obeying

some (essentially

is the Schwinger functions associated

to

unique) Wightman theory.

Remarks: 1. A priori, (OS' 1-5) only supposes that the

are distributions

on

A posteriori, they are shown to be functions, in fact, real analytic functions, on 2. There are also relations between certain subsets of

and

We do note that 3. The real miracle is that the spectral condition " c o m e s for f r e e . " We begin our proof with the basic idea behind this. consist of finite sequences,

Proof: r

forms a vector space by thinking of

infinite sequence f,

let

and adding components.

as the Given

THE OSTERWALDER-SCHRADER RECONSTRUCTION THEOREM

69

is a positive semi-definite inner product. In the usual way, we can form a Hilbert space,

(which turns out to be the physical

Hilbert space) by quotienting out the elements of zero norm and completing. 0

will denote the equivalence c l a s s defined by We now turn to the critical idea. The spectral condition will basically

follow from positive definiteness, translation covariance and temperedness. Temperedness enters in the following way:

being a contraction for all t. If

_

is equivalent to

weren't positive,

exponentially in t and this would violate temperedness.

would grow To make this

precise, we define

For each, distribution.

On the one hand, since

is clearly continuous since

is a

and let

is tempered, (11.29)

for suitable On the other hand, by the Schwartz inequality where we have used the symmetry property (a consequence of Iterating (11.30) and using (11.29) we see that

(11.30) (11.31)

70

AXIOMS, I

Taking

we conclude that (11.32)

From (11.32), we first conclude that themselves and thereby Then by continuity

takes vectors of zero

lifts to the quotient of extends to all of K

Clearly for

into

by these vectors.

and by

. Since by the above Tt is strongly

continuous and self-adjoint, (11.33) where H is self-adjoint and Moreover,

(11.34)

will be the physical Hamiltonian and, once we

have Lorentz covariance, (11.34) will yield the spectral condition. Fix By

Consider the distributions

in

and Theorem II.8, T is a Laplace transform of some Using the Nuclear theorem to unsmear in the f ' s ,

we

define distributions

where

the space component integrations are

short hand for Fourier transforms while the time component integrations are in terms of legitimate

functions.

§11.4. THE OSTERWALDER-SCHRADER RECONSTRUCTION THEOREM

71

Now we can define the Wightman distributions and check

where

and where all integrations are to indicate Fourier

transforms. (W 1) is evident by construction.

The reality condition follows from

the reality condition (11.22). Translation invariance is evident from (11.36).

Following

Nelson [134], one proceeds as follows: Let Y be the infinitesimal generator of rotations in Euclidean space, i.e.,

Since

is rotation invariant,

But writing

and using (11.36, we see that

where

By the fact that the kernel of the Fourier and Laplace transforms is zero, (W 3). By construction,

so that

is Lorentz invariant.

has support in the region This plus Lorentz invariance

implies the support is in a product of light cones. (W 4). Let us show that

The general proof is similar.

Suppose first that f is of the special form where each

has a Fourier

72

AXIOMS, I

transform whose restriction to

is a Laplace transform;

Then,

where Thus

for those special

similar argument

works for the sum of such f ' s . But by Proposition II.7, such sums are dense in that

It follows from the continuity of for any

(W 5). That given

locality is implied by the symmetry of the

Wfl is a theorem of Jost [110], metry of the

By analytic continuation, sym-

implies symmetry of the

(W 6). We can now identify

and so locality.

as the physical Hilbert space and H

as the Hamiltonian. By (OS 5),

for a dense set of

so

as opposed to

is the only vector with

was used critically in concluding that is a Laplace transform in the u's.

portant to see how far one can go with

It is im-

or some strengthened version

of it since it is almost inconceivable to constructively prove (OS 1') without actually constructing the

a priori (as we do in Chapter VIII), so

to use the Osterwalder-Schrader axioms in a constructive program (which is not of major importance in two dimensions but may be useful in higher dimensions) we must look at the totality From the construction of exp (—sH), it is clear that if we smear in T has an analytic continuation in Uj (continue exp(—UjH)!) to the region that show that when smeared

One can even prove bounds T is a Laplace

§11.5. FREE AND GENERALIZED FREE FIELDS

73

transform in Uj. But, as we have discussed in Section II.2, such a separate Laplace transform need not be a joint Laplace transform. In fact, a joint Laplace transform will be analytic in the region

while a

separate Laplace transform will (by using analytic completion!) be analytic in

and perhaps no more. Now T is more than merely

a separate Laplace transform for

is related to Using this, it is possible to increase the

analyticity domain of

as has been noted by Glaser [60] and Osterwalder

[140]. Proceeding inductively and bringing in

one can prove

that T n actually has a continuation to Ju|arg Thus to prove that

imply

one only needs suitable

bounds on the behavior of this analytic continuation as arg bounds appear unlikely just on the basis ol plemented by a bound on the growth of While the new scheme,

. Such can be sup-

as

is not equivalent to

to yield it implies

(W 1-6) and is constructively "natural" to prove. Reference

for Section II.3, 4:

Osterwalder-Schrader [143].

§11.5. Free and Generalized

Free

Fields

We now want to construct a class of "physically trivial" models obeying all the GW axioms. The fundamental idea of "Lagrangian field theory" and of constructive field theory, in particular, is to build up more interesting models by suitably perturbing these trivial models. Let us begin by analyzing the structure of On account of translation covariance, of

is a " f u n c t i o n , "

We can summarize the properties that

must have:

only

74

AXIOMS, I

(II. 37a) has support in

(II. 37b) (II.37c)

if

is space-like .

(II.37d)

If, in addition (11.38) then is space-like .

(II.37e)

(II.45a-e) are respectively consequences of the Lorentz covariance, spectral, positive definiteness, locality and cluster properties. Remarks: 1. By translation covariance,

is a constant so (11.38)

can always be arranged by transforming 2. A priori, (II.37a, d , e ) are distributional statements but as d, e deal with space-like vectors and

is analytic there (on account

of (II.37a, b) and the analysis in Section

hold

point wise. It is easy to parametrize all distributions obeying (11.37):

THEOREM 11.10 (The Kallen-Lehmann representation). Any

distribution

obeying (II.37a, b, c , e) is of the form: (11.39) where (II.40a)

§11.5. F R E E AND GENERALIZED FREE FIELDS

75

with (II. 40b) and where p is a polynomially

bounded (positive)

measure.

Remarks: 1. (II.37d) follows automatically from the rest of (11.37). 2. The normalization (II.40a) is chosen so that (II.40c)

i.e., so that

in case 3. This result was discovered by several authors about the same time

4. For a detailed proof, see Reed-Simon [145; Section

The

basic idea is that by an extension of Bochner's theorem (Theorem 1.1) to distributions, .

. implies that

is a measure s o by

(II.37a, b) one need only analyse Lorentz invariant measures supported in 5. In two dimensions there is an

anomaly which requires that

p give no weight to 0 and that

be bounded as a -lO.

This anomaly is not present in more dimensions. sider cases where p is supported in

with

There is a special class of field theories in which by

will only con-

is "determined"

in a certain way. We introduce a useful shorthand terminology due

to Caianiello then the symbol

. Suppose a distribution

in two variables is given;

is defined by:if n is odd

76

AXIOMS, I

(11.41) where the sum is over all

ways of writing

with

T H E O R E M 11.15.

Let

as Then:

obey

(11.37) and let

Suppose

that

(11.42) obeys all the Wightman axioms (W 1 - W 6).

Then

Remark: We will not give a detailed proof. Only (W 3) (positive definiteness) is not immediately evident. One way of proving (W 3) is to construct in terms of Gaussian processes as we will do below. The field theory associated with the a generalized

given in

is called

free field (they were introduced by Greenberg [75]). In case we call the theory the free field of mass

Hence-

forth, we restrict ourselves to the free fields.

It is natural to ask for the Schwinger functions associated to the free field of mass

We claim that: (11.43)

Remarks: 1. The inner product in the exponential in (11.43) is a Euclidean inner product. 2. Explicitly,

where

is an associated

Bessel function related to the Hankel functions by

§11.5. FREE AND GENERALIZED FREE FIELDS

To see that

77

is the Schwinger function for the free field of mass

m 0 , we first note the following basic relation:

if

and s is real.

will occur again and again in relating

Euclidean field theory and Minkowski field theory (see Section

It

is already famous in physics as the link between " o l d fashioned perturbation theory" and the manifestly covariant perturbation theory of Feynman [10]. Of course (11.44) follows by an elementary contour integral. In particular, (11.44) and (11.43) imply

Thus

and

agree at the points

(11.43) is real analytic,

Since

as given by

agrees with the analytic continuation of

in the extended forward tube. For Having analytically continued

this is the entire Euclidean region. it is easy to continue

given by

(11.42):

T H E O R E M 11.16.

Let

given

by (11.43).

Let

(11.45)

Then

Proof.

is the family of Schwinger functions for the free field of mass.

Since (11.42) clearly extends to the forward tube and S 2 as given

by (11.43) agrees with

there,

in the forward tube. Since the

as given by of

agrees with

is invariant under rotations

and permutations, it agrees with the Schwinger function in all of

78

AXIOMS, I

Remark: The function

defined by (11.43) is integrable at

Schwinger functions distributions

Thus the

(given by (11.45)) have a natural continuation, as

to the coincident points, i.e., from

The formula (11.41) is, of course, formally identical to (1.13) and thus, it is natural to try to interpret the

given by (11.40/42) and the

given by (11.43/45) in terms of Gaussian random processes.

For the

we return to this idea (which goes beyond the O.S. axioms) in Section Let us consider the situation with regard to the represent the quantum fields

We cannot hope to

as random variables, since

they do not commute with one another. Locality and positivity (GW 3, 5) suggest that we might succeed in doing this for time zero fields. In fact: THEOREM 11.17. Let

be the Hilbert space of all real

distributions

on R whose Fourier transforms are functions with the property that the norm:

is finite.

(11.46)

Let

fi be the pseudo-differential

operator on (II.47a)

i.e., for f

let (II. 47b)

where (II.47c) (n is then the closure of the operator defined on and let Q0

be the Gaussian random process indexed

be the function

1 in

and let

b y L e t

§11.5. FREE AND GENERALIZED FREE FIELDS

79

(11.48) and for (II.49a) where (II.49b) Then 3" can be identified with the physical Hilbert space of the free field of mass the vacuum and

in such a way that

is the Wightman field,

is

is the Hamiltonian.

Remark: The normalization in (11.46) is chosen so that (11.46') for all Proof.

On account of (11.44), (11.50)

so that (11.51) By analytic continuation and the taking of boundary values, (11.52) From (11.50), we conclude that

for all f,

and (1.13), we get agreement of

and

80

AXIOMS, I

Remark: The proof is mote natural in terms of the complex version of Fock space described in Section 1.3 (see e.g., [207], [145, Section X. 7]). References for Section II.5: Bongaarts [11], Glimm-Jaffe [70,71], Reed-Simon [145, Section X.7], Garding-Wightman [207].

CHAPTER III THE FREE EUCLIDEAN FIELD In this chapter, we will consider the field theory associated to the free field Schwinger functions (11.43/45). The possibility of finding a "field theory" whose expectation values are the Schwinger functions is not an automatic consequence of the axiom schemes of the last chapter. We will return to what properties of general Wightman fields imply the ex­ istence of Euclidean field theories in the next chapter. Here we will rely on a positivity property first emphasized by Symanzik [192, 193, 194] who used it to develop the free Euclidean field. As we will see in Section IV.3, this positivity property is a consequence of Theorem 1.12 and (11.47) — but in this chapter, we will derive it directly. Symanzik, in particular, emphasized that Euclidean fields at distinct points should commute. Formally, this is easy to understand — for [(x, 0),

φ( y, 0)] = 0

by locality because the Euclidean fields at imaginary

time zero should agree with the time zero Minkowski fields. But any two Euclidean region points can be both brought to zero (imaginary) time by a Euclidean motion. It was Nelson [132, 134, 135] who first emphasized the power of Symanzik's framework and showed how to use the techniques of construc­ tive field theory to advance Symanzik's program. In particular, he empha­ sized the importance of the Markov property of the free field (see Section III. 3), a property which had been noted by Symanzik, but only in a unpub­ lished report [192]. Nelson also emphasized the role of Euclidean field theories as path integrals, an idea we discuss in Section III.2. Implicit, in addition, in Nelson's sketch [132] were certain L^ estimates which

82

THE FREE EUCLroEAN FIELD

will play a role in Chapter V and later. We discuss these estimates in Section III.4, 5 following the treatment of Guerra, Rosen and Simon [90]. §111.1. Beyond the Osterwalder-Schrader Axioms In order to understand exactly what is needed to form a Euclidean free field theory, let us consider the properties of a free field S 2 in one spacetime dimensions. In general, in d dimensions, one has (III.l) Thus, when d = 1 S2(X) = ± e"mlx

(III. 2)

This follows from (III.l) and (11.44). S 2 has three distinct positivity properties: (A) S 2 is the Laplace transform of a positive measure on [0,oc). (B) S 2 is pointwise positive. (C) S 2 is positive definite, i.e., S 2 has a positive measure for its Fourier transform (this follows from the realization (III.l)). (A) is the direct translation of (OS) positivity, i.e., it says that S2

is the analytic continuation of a positive definite function O 2 for which ffi 2 has certain support properties. (B) is related to certain ideas we re­ turn to in Chapter VIII. (C), as we will see, is the critical property for a field theory, for if S 2 (x,y) Ξ S2(X—y) is to be an expectation value Exp (φ(χ)φ(γ)) then surely Exp (φ(ί) 2 ) = / f(x) S 2 (x—y)f(y)dxdy > 0 for all real-valued f. Remarks:

1. For S 2 , OS positivity actually implies (C) (by use of Theorem 11.14 and (11.44)) but, in general, i.e., for S fl , the OS positivity condition is distinct from the one needed for fields. However, since S n for the free field obeys (11.45), the field positivity condition needed

§111.1. BEYOND THE OSTERWALDER-SCHRADER AXIOMS

83

follows from that for S 2 - Thus, for free fields, (OS) positivity does imply the possibility of fields. 2. But we emphasize (see the example in Section IV.4) that in general the (OS) axioms do not imply a field structure, so in terms of general theory, the structures of this chapter go beyond the (OS) axiom scheme. 3. Real progress in constructing quantum fields by Euclidean methods has depended on the extra structure of Euclidean fields and their Markov property and not just on the (OS) axiom scheme. The im­ portance of an (OS") axiom scheme in constructive field theory lies in its providing a convenient route to the Wightman axioms, especially in cases where one can control Schwinger functions rather than Markov field measures. In the two dimensional case, there are alternate routes which tell one more ([72] and Chapter VIII below). The positive definiteness property (C) and (11.45) suggest we try to form a Gaussian random process: DEFINITION. Let Nm be the Hilbert space of all real distributions, f, in

S(R2)'

whose Fourier transforms are functions and for which the norm

(III·3) is finite. DEFINITION. The free Euclidean field is the Gaussian random process

indexed by N ffl . We will use the symbol Tl occasionally for T(N m ), and Q n for the associated Q -space.

When we wish to avoid confusion between a Euclidean field φ( · ) and the corresponding time zero Minkowski field, we will denote the latter by Φ¥(·).

Of course, the point of constructing φ is:

84

THE FREE EUCLIDEAN FIELD

THEOREM III. 1. Let

denote the natural extension

free field Schwinger functions to coincident

points.

Follows from the equality of

a

Proof.

n

of the

Then for any

d

a

n

d

the

formulae (11.45) and (1.13). For later reference, we note that the inner product in N can be written:

(III. 4)

Thus while the kernel

defining the inner product is non-local

can be non-zero even if f and g have disjoint supports) it is the inverse of a local object.

This will have profound consequences

(see Theorem III.9). (III.4) also tells us that N is essentially a classical Sobolev space [145, 188], There is a natural action of the full Euclidean group, (i.e., the group generated by translations, rotations and reflections of given a Euclidean transformation

let

on

For

be defined on N m by (III. 5a)

u()3) is orthogonal so that (III. 5b) is a unitary operator on

i induces an automorphism of

so, in model 2 of Q-space, a pointwise map

on

f

and

so that

The realization of T^g as a pointwise map is model dependent (although such a realization is also possible in model 3), but in any model, Moreover, since

is induced by an automorphism of the ring is unitary

§111.2. THE FREE EUCLIDEAN FIELD AS A PATH INTEGRAL

for any measurable set A. equivalently

85

Finally (III.6a) (III.6b)

In addition, we see that if

is the group of translations

and we denote

t

h

e

n

o

n

account of

the Riemann-Lebesgue lemma. Thus

so

PROPOSITION III.2.

The family of measure prescribing

acts ergodically, such

i.e.,

transformations

the only events left invariant by each

are the trivial events

and

Added Note: According to Minlos' theorem, the in such a way that free field measure (i.e., of sets

space for

can be taken as

Studies of the support of the with

or with

have recently been made by Cannon [19], Colella-Lanford [22] and Reed-Rosen [212], These authors partially rely on abstract Gaussian process results of Gross [83], Minlos [129] and Umemura [198]. References

for Section III. 1:

Nelson [135], Symanzik [194],

§111.2. The Free Euclidean Field as a Path Integral We now want to examine the connections between the physical Hilbert space,

for the free Minkowski space field and the Hilbert space,

for the free Euclidean field. Throughout this section we fix a bare mass, m.

86

THE FREE EUCLIDEAN FIELD

At first sight one might think that J =

since the Hilbert space

constructed in the Osterwalder-Schrader reconstruction theorem is just 3\ This is not true as is most easily seen by the fact that the "time translations,"

of their reconstruction theorem are a self-adjoint semigroup on

, while the time translations U^ of the last

section are a unitary group on realized as a subspace of

. What we will see is that ?

is naturally

so that (III. 7 )

where E is the projection onto

as imbedded in

To establish the connection between connections between the indexing spaces Gaussian processes.

and and

we first study the and then pass to the

We first note that:

PROPOSITION III.2. Let

f e F and let a be fixed.

Then the distribution

given by

is in ,

(III. 8 )

and

Thus, we can use (11.52)

Proof. to do the p integral in

and thereby obtain that The properties of the various DEFINITION. Let PROPOSITION

's are given by:

be the projection in N onto Ran (III.9) (III. 1 0 ) (III. 1 1 )

87

§111.2. THE FREE EUCLIDEAN FIELD AS A PATH INTEGRAL

where

fx is the pseudo-differential

operator (11.47)

(III. 12)

Proof-, (a) states that j a

is an isometry and (b) follows from the defini-

tions. (c) is a consequence of (11.52). Finally (d) follows from (c) since

and (III. 11) implies

Multiplying b y o n

the left a n d o n

the right and using (III.9)

yields For technical purposes, it is useful to state some properties of the and their ranges. We will use e[-a ^

to denote the projection onto

all elements of N with support in the region PROPOSITION III.4. (a) Ran eff consists

precisely

of those elements

of

N with support in the hyperplane (b) Let

r denote reflection

leaves (c) Ran e [ a

Ran

in the hyperplane

pointwise

is the smallest closed

(d) Let

then

invariant. subspace

of N

containing

Then (III. 13)

(e)

is strongly

continuous.

88

THE FREE EUCLIDEAN FIELD

Proof,

(a) If

has support in the hyperplane

then

for suitable

1

can the integral defining

and n. Only if

converge in the p-direction.

(b) follows from (c) By shrinking, convoluting and multiplying, we can see that is dense in Ran e [ a j j s o we need only show that any such f is a limit element in + Ran

Using the dominated convergence theorem, it is not

hard to show that for any such converges in N as

to

(d) follows from ( c ) and Proposition : (e) By Proposition III. 3 (a, b), Henceforth, we will usually surpress j 0 subspace

Thus

and associate

F with the

becomes identified with e Q .

We now drag this structure over to the Gaussian Random process. DEFINITION .

T H E O R E M III.5.

(III.9')

(III. 10') (III. 11') (d) Let

denote the o-algebra

generated

by

Then the set of measurable the smallest

functions

is

X-algebra

Moreover, for which

is is

measurable

for all (e) (Primitive Markov Property).

Then

§111.2. THE FREE EUCLIDEAN FIELD AS A PATH INTEGRAL

89

(III. 13') (f)

Proof.

is strongly

continuous.

Except for (d), which requires Theorem III.8 below, a direct trans-

lation of Propositions We are now ready for a main theorem which exhibits the Euclidean free field as a kind of path integral over the free Minkowski field — this idea will be further developed in Chapters IV, V and the precise result below will be extended in Section III.5:

THEOREM III.6 (The Free Field FKN Formula). ments of F, and let be given.

be bounded measurable

be functions

ele-

on

Then: (III. 14a)

where

is arbitrary and (III. 14b)

(III. 14c)

(III. 14d)

Proof.

where

Let

denote the vector

90

THE F R E E EUCLIDEAN F I E L D

On account of the basic imbedding relations of Proposition 1.8,

so (III. 15)

Consider the r.h.s. of (11.15). in Ran

To the tight of any

appears a vector

and to the left a vector in Ran

Thus by the

primitive Markov property (III. 13'), we may remove all t

h

e

!

(III. 14)

results. • Remarks: 1. FKN stands for Feynman-Kac-Nelson and (III. 14) is called the FKN formula. 2. In one space time dimension, the analogue of (III. 14) is .

(III. 16)

This is just the Feynman-Kac formula. Feyman [44, 45, 46] first used such formulas "formally" for e l t H .

Kac [111, 112] realized

that using the Ornstein-Uhlenbeck velocity process, one could prove (III. 16) rigorously.

(Kac's proof is quite different, making

l e s s use of the general theory of Gaussian processes and more of the explicit Mehler kernel (1.38)). The theory of the OrnsteinUhlenbeck velocity process is in turn a development of the fundamental work of Wiener [202] on path integrals. 3. The key discovery of Nelson is not only that one can develop a path integral formalism over the free Minkowski field but that it is a manifestly Euclidean invariant path integral. This will be useful throughout the developments from Chapter V onwards, but most especially in Chapter VI. 4. Letting i.e., maps from R to

we can think of

as a space of paths,

§111.3. CONDITIONAL EXPECTATIONS AND THE MARKOV PROPERTY

References

for Section

91

III. 2:

" C l a s s i c a l " Theory of Path Integrals:

Babbitt [6], Cameron [18],

Gel'fand-Yaglom [55], K a c [112], Nelson [130], F r e e Euclidean Field Theory:

§111.3. Conditional

Guerra-Rosen-Simon [90], Nelson [132].

Expectations

and the Markov

Property

Following Symanzik [192] and Nelson [135], we wish to provide a new proof of (III. 1 3 ' ) and more importantly a vast generalization of considerable use later.

While it is not necessary, it is useful to express this generali-

zation in terms of conditional probabilities.

We thus begin by discussing

this probabilistic notion.

THEOREM III.7.

Let

be a sub-o-algebra there exists

be a probability of

Let

f be an element

a unique function

(i)

is

measure

space

of

and let .

Then

I so that

measurable

(ii)

g which are

for all

'-measurable

and

in

Suppose both

Proof of Uniqueness. . Then, by (ii),

and 1

are candidates for

for any

g which is

able and, in particular for any bounded function F

of

Then, letting

measur. Let

1

for any n.

By the monotone convergence theorem (Doob [31]).

First Proof of Existence of jj. to

Let

For any

vergence theorem,

By the dominated condefines a finite signed measure and clearly

absolutely continuous with respect to for some

is

. Thus

by the Radon-Nikodym theorem.

definition, this some

denote the restriction

By

obeys (i) and it obeys (ii) whenever

Since such

g's

are total in

(ii) holds in general.

for

92

THE F R E E EUCLIDEAN F I E L D

Second Proof of Existence.

Let

those functions which are

-measurable and let

tion onto

denote the subspace of

since

for

is in

so

Thus, by Theorem 1.13,

any

. denote the projec-

. Moreover if

so its negative part,

Since

g, is in

,, and thus

. extends to a contraction on

is dense in

is dense in

is a pointwise limit of elements of

•measurable.

As a result the candidate

Moreover, if

of

Thus , and so is

for

obeys (i).

then (ii) holds and so by a limiting argument

(ii) always holds. Remarks: 1. Doob's proof has the advantage of also defining way if

even if f isn't integrable.

2. By the second proof, we s e e that

3. 4.

in a natural

is called the conditional

i is a contraction on each

expectation

of f given

As with so many probabilistic notions, conditional expectations have a straight-forward functional analytic definition, but one that does not seem especially promising from a functional analytic viewpoint. After all, why single out the projections onto the special subspaces

?

5. To understand the name and the intuition behind conditional expectation, suppose that is M and let

are disjoint sets whose union

be the algebra (with

elements) generated by

Every measurable function is set. of the formit is easy Let B be some other measurable Then j. —' to see that (III. 17a)

§111.3. CONDITIONAL E X P E C T A T I O N S AND THE MARKOV P R O P E R T Y

93

with (III. 17b)

As a result,

at a point

is the intuitive "conditional

expectation" that the event B occur given that we know that occur.

This means that we can think of

for general

as a sort of general label of the probability that B occur knowing which events in

are occurring.

6. Henceforth, in line with the second proof we write place of or if

in

. We will occasionally use the shorthand is labelled as e.g.,

, then

For Gaussian random process, it is easy to describe many of the conditional expectations.

THEOREM III.8. Let subspace the

H be a real Hilbert space,

and let p be the orthogonal projection

-algebra of measurable

subsets

of

let

a

on

Let

generated

closed '

by the

Then

Proof.

denote

(III. 18)

Let

be an O.N. basis for H .

Then any

a limit of finite sums of the

is

and thus of polynomials in

It follows that any : we see that it lies in Ran

tains each

Ran

for

the proof of Lemma 1.4, it follows that

> is in 1

if

, By expanding so Ran

_ is in Ran

> con>. As in

Thus

Ran

We are now able to state the fundamental Markov property of the free field:

94

THE F R E E EUCLIDEAN F I E L D

DEFINITION.

Let

be an open or closed set and let

closure of set of distributions in N with support in orthogonal projection onto

^ and let

denote the Let

, Let

be the be the

ff-algebra generated by Remarks: 1.

By Theorem III.8,

is the conditional expectation with respect

to 2.

For many open s e t s , e . g . , a disc or square,

so that

most results can be expressed in terms of closed s e t s . 3.

Let

be closed.

Then clearly, on account of the definition of

support,

open! so

- lim

' implies

- lim

> (for

• implies

Thus our definition above agrees with one that could be made more in the spirit of the

-algebra theory of quantum fields [3, 92],

THEOREM III.9 (The Markov Property). of __

with

Let

A and

B be closed

Then:

(i) (ii)

subsets

(HI. 19) If F

is any

expectation (iii) If F

is any

-measurable is -measurable

function,

then the

conditional

-measurable. function,

then (III. 20)

Remarks: 1. (III. 19) is a strict extension of (III. 13'), for clearly since

so that by (III. 19)

§111.3. CONDITIONAL EXPECTATIONS AND THE MARKOV PROPERTY 2. The reason for the name is the following:

Consider the analogous

result for the one space-time dimensional theory. s a y s that if F -measurable. taking F

to be

is

95

One special c a s e

measurable, then

is

Thinking of the underlying real line a s time, for

and using the interpretation

of conditional expectation, this s a y s that the probability of an event taking place in the future knowing the entire past depends on the past only through the present

i

This is the

usual Markov property for " p r o c e s s e s " and so Theorem III.9 is a generalization of the Markov idea from one dimensional p r o c e s s e s to multidimensional objects. 3.

As with so much of the free field theory, the core of the proof is a result about the index space,

4.

N.

Generalizations of this result appear in [90, Section II. 1].

5. Merely for convenience do we take A and B closed.

There is a

result for arbitrary measurable s e t s . 6.

As the proof will show, this theorem depends critically on the fact that while N has a non-local inner product, the kernel is the inverse of a local operator.

LEMMA III.10 (The Pre-Markov Property). sets with

Let

A and

B be closed

sub-

Then (III. 21)

Proof.

Since

, (III.21) is equivalent to (III. 2 1 ' )

which is clearly the same as (ii). has support in

To prove (ii) we must show that

a s a distribution.

Since A is closed,

support in A so we need only prove that

has

96

THE F R E E EUCLIDEAN F I E L D

(III. 22) if g has support in

and is in

The integral in (III. 22) is

formal standing for distributional action.

proving (III.22).

On account of (III.4):

The second equality depends on the fact that

is an

N-orthogonal projection and the next on the fact t h a t a l s o support in

since

is local.

Proof of Theorem III.9. (III. 19) holds.

has

(Lemma)

Second quantizing (III.21), we conclude that

Since

, (ii) and (iii) follow.

Occasionally, we will use the Markov property in a slightly different form:

COROLLARY III.11. that C separates with

Let

A and

A and B, and

B be disjoint closed sets.

in the sense

that there exists

(see Figure

III. 1).

Suppose D, E

closed

Then (III. 23) (III. 24)

Fig. III. 1. The Situation in Corollary III. 11.

§111.4. PRODUCTS OF PROJECTIONS IN SOBOLEV SPACES Proof.

Since

97

. Similarly

(by Theorem III.9)

References

for Section

III.3:

" C l a s s i c a l " Theory of Markov P r o c e s s e s : Markov Property of the F r e e Field:

Dynkin [32], Ito-McKean [104],

Guerra-Rosen-Simon [90], Nelson [135],

Symanzik [192].

§111.4. Products

of Projections

in Sobolev

Spaces

For technical purposes, we will need information in Section III.5 and in Section VII. 1 about the product case

and

are disjoint.

theory of the Sobolev space,

of two projections in N in This is essentially a problem in the . For the time zero-free field theory

the analogous space is

and the product

by field theorists in this c a s e , s e e [141, 175].

was first studied Our discussion here follows

Section III. 1 of Guerra, Rosen and Simon [90] — one of their arguments is based on an argument for

found in Osterwalder-Schrader [141].

For many purposes, one only needs the following simple result implicit in Nelson [132]:

THEOREM III. 12. the projections

Let

and

in N onto

be convex

sets in

.

L e t b e

. Then

(III. 25)

where

m is the mass used to define

the distance

between

and

the norm in N and

is

98

THE F R E E EUCLIDEAN F I E L D

Proof.

Suppose first that

and

ness, we can find

are bounded. Then, by compact-

with i

Let

and

be the planes through x and y orthogonal to the line segment By simple geometry

separates

and

xy.

so by the Markov property

(in the form of Corollary III. 11)

so (III. 26)

By rotation covariance, we may as well suppose that xy is the time direction in which c a s e

by (III.9/11) so that (III. 27)

(III.25) clearly follows from (III.26) and (III.27). If

are arbitrary, let

ball of radius r. Since

be their intersections with the are bounded and

the result follows from the bounded c a s e . Remarks: 1. (III.25) is best possible for take

parallel planes.

2. For certain regions one can do better than (III.25), for example, if is bounded and

is large (see Theorem III.15 below).

For general regions we can recover the main features of (III.25):

§111.4. PRODUCTS OF PROJECTIONS IN SOBOLEV SPACES

DEFINITION.

99

closed with

THEOREM III. 13.

(a) For small

d (III. 2 8 )

(b) For large

d (III. 2 9 )

Remarks: 1. (a) is due to E . Stein (unpublished); (b) follows ideas of OsterwalderSchrader [141] used in a slightly different situation.

Both results

appear explicitly in [90], 2.

On the b a s i s of Theorem III. 12, one might guess that This is definitely false.

Guerra, Rosen and Simon [90] present an

example showing that for large d, ture that Proof of Theorem if

_

and they conjec-

for d large. III. 13 (a).

Let x

with

be a function in

Let

f(x) be the function which is

which is i. Given

, let

1 if

and zero otherwise.

0

if Let

"

. Then where

the constants ate independent of closure of by

in the norm

g is bounded on

duality

(N is the dual of

, and j

Let i

j

with bound const

denote the . Then multiplication By

we have: (III. 3 0 )

Let

and suppose

.

Then

100

THE F R E E EUCLIDEAN F I E L D

by (III.30) SO that

from which the bound

follows.

(Part (a)).

For (b), we need:

LEMMA III. 14. If (M,dx) is a measure space and

a(x, y) is a

"kernel"

on M x M with (III. 31)

)dy defines

Then

Proof. so

a bounded map on

By Schwarz inequality I

Proof of Theorem III. 13 (b). Suppose

As in the

proof of (a) we can use convolution to find and

and

if dist

with

on

and with bounds ion

independent of d. Suppose so that

with

and let Let

with . Then (III. 32)

§111.4. PRODUCTS OF PROJECTIONS IN SOBOLEV SPACES

101

so we need only prove that a s an operator on

(III. 3 3 ) By simple manipulations (and the fact that (III. 34) so to prove (III. 33) we need only\>btain a bound of the form

(III. 3 5 ) if

are multiplication operators with support a distance d > 1

apart.

But

has a kernel

k(x, y) with

(III. 36) (III. 35) and thus (III.33) follow from (III.36) and the lemma. For bounded regions one can do better than (III.29) for large d:

THEOREM III. 15.

Fix

a.

Then there exist

c, R

with (III. 37)

if d = dist

Proof.

and diameter

i

Under the geometric hypotheses, we can find concentric c i r c l e s with radii a and d—2a respectively w i t h i n s i d e

outside

and

. Thus, by the pre-Markov property

By a partial wave expansion,

can be computed explicitly

(s-waves produce the maximum) and a (const.) log bound results from which (III.37) follows.

102

THE F R E E EUCLIDEAN F I E L D

Finally we will need the following:

THEOREM III.16. Let

and .

Suppose (a)

be closed disjoint regions with

is bounded.

is

Then:

Hilbert-Schmidt

(b)

is trace-class

and if

with

then

Proof,

(a) As in the proof of Theorem III. 13 (b), we need only show that

i s a Hilbert-Schmidt operator on with dist (supp rj 1 , supp

if

and

, are ~

and supp rj^ bounded. This follows

from the bound (III.36). (b) By (a), a

is Trace class.

Moreover, by the pre-Markov

so Ran a C Ran

property, §111.5. L^ Estimates

and Asymptotic Independence

of Distant

Regions

In this section, we will combine the estimates of the last section with the hypercontractive bounds of Section 1.5 to obtain certain bounds involving

These bounds will play an important role in our

construction of the spatially cutoff interacting Hamiltonian in Section V.3. These estimates also play an important role in some aspects of the infinite volume limit. In a quantum field theory, fields at distinct points are coupled. This is most easily seen in the commutative Euclidean framework. Two random variables are called independent

if their joint

probability distribution is the product of their individual distributions. In particular, if f and g are independent, then For the free Euclidean field so the fields at distinct points are not independent. The fact that goes to zero exponentially as and

suggests that in some sense

might become exponentially independent as

_

i

§111.5. L p

ESTIMATES

103

The first results of the type (of asserting exponentially decoupling of distant regions) were found in the time zero theory by Glimm-Jaffe in their study of the infinite volume limit [64], of the kernel relating

They stated their results in terms

to a set of "Newton-Wigner f i e l d s , "

which were independent at distinct points.

,

Later, Simon [175] and

Osterwalder-Schrader [141] stated results, still in the time zero theory, in terms of norms of

but for projections in

. Guerra, Rosen

and Simon [90] proposed the following idea as a notion of asymptotic independence:

If F

and G are independent random variables, then

, while for general random variables,

, is

the best one can hope for. Thus a statement like with p, q near 1 is a statement of almost independence.

As a result,

the estimates below are an expression of asymptotic independence of distant regions. It is useful to restate Theorem 1.17 using Holder's inequality and the fact that

is equivalent to

THEOREM 1.17'. Let contraction.

H be a real Hilbert space and let

be a

Suppose that p, q are given with

Then (III. 38)

This result and Theorem III. 12 imply:

THEOREM III. 17. Let be-measurable 1

and .

and-measurable. i

be convex subsets Suppose

of p, q

. Let

F

obey

Then (III. 39a)

104

THE F R E E EUCLIDEAN F I E L D

More generally,

if

then

(III. 39b)

Remarks: 1. Since

-measurable implies that

is

(III.39b) follows from (III.39a) upon taking F

-measurable, to

and G to

in (III.39a). 2. In particular, as

and q in (III.39a) may be

taken exponentially close to 1. Finally Theorem III. 13, 15 can be used in conjunction with

THEOREM III. 18. Let

be disjoint closed regions

in

and let

Then (III. 40a) so long as

: and (III. 40b)

As an application of

THEOREM III. 19. is only assumed

Proof.

bounds (of a simple sort) we note that:

The Feynman-Kac

formula, (III. 14), holds when each

to be polynomially

bounded.

On the one hand, by Holder's inequality, the right hand side of c.

(III. 14a) is bounded by a product of other hand, since

bounded so that

On the

is a contraction on each _ , the left hand side

is bounded by a product of and

norms of

norms

if G is polynomially bounded, we can find in all

§111.5. L p

I and

ESTIMATES

105

in all

Thus (III. 14) in the bounded case, implies (III. 14) in the polynomially bounded case. Reference for Sections

III.4, 5:

Guerra, Rosen, Simon [90],

CHAPTER IV AXIOMS, II The considerations of the last chapter suggest that first it should be useful to consider quantum field theories associated with Euclidean region fields rather than just with Euclidean Green's functions and secondly that we might expect a connection between such theories and the theory of Markov processes. In fact, our development of the connection between Euclidean-Markov field theories and Wightman field theories with additional properties closely parallels the discussion of the connection between Markov processes and positivity preserving semigroups [32]. In Sections IV.1,2 we present Nelson's germinal work [132, 134] on con­ structing Wightman fields from Euclidean field theories and in Section IV.3 Simon's partial results [180] in the opposite direction. Finally in Section IV.4, we present an example, due to Simon [180] of a theory in one spacetime dimension obeying all the GW axioms but not possessing Euclidean region fields. The version of Nelson's theory that we discuss in Sections IV.1, 2 is restricted to theories for which there are time zero-smeared fields as selfadjoint operators and thus to interacting fields in two and three spacetime dimension, at least according to the current folklore (which is based on perturbation theory considerations). Nelson [134] has presented an alternative theory which may be applicable in four dimensions. It is an interesting but difficult question to extend the considerations of Section IV.3 to this more general setting of Nelson. Unlike the situation for the Axiom schemes of Chapter II, we do not know how to extend the Axiom schemes of this chapter to higher spin. In

§IV.l. NELSON'S AXIOMS

107

particular, it is a major open question to accomodate relativistic local Fermion field theories into a Markov Euclidean framework.

§IV.l. Nelson's

Axioms

In this section, we will present Nelson's axioms for what he calls a Euclidean

We first recall two technical definitions:

field over

I denotes the Sobolev space of all real distributions,

DEFINITION. f, for which

Thus DEFINITION. space.

in our previous notation. Let

f be random variables on a probability measure

We say that

converges

to f in measure

and write

if

and only if for every

Remarks: 1. Since

convergence implies

convergence in measure (a simple but separate argument is needed for 2. By passing to a subsequence, we can be certain that It follows that

except

on a set of measure zero. Thus convergence in measure lies intermediate between

convergence and (sufisequential) pointwise

a.e. convergence. The properties we will demand of Euclidean fields are (Nl-5): (Nl) (Fields).

There is a probability measure space

random field

indexed by

(a) (b) If

is full ^

in

then 0 with a bound Thus by employing the theory of Laplace transforms and the KallenLehmann representation (IV. 12)

where

is the free field Schwinger function and dp is poly-

nomially bounded. 4 By definition (IV. 13)

where Z is called the field strength renormalization the integral diverges). '(Finite

(we set

Z = 0 if

Then

field strength renormalization).

Remarks: 1. We emphasize that (N6) will play no role in the basic reconstruction theorem but will have some use in Section IV.3. 2. On the other hand folklore suggests that if one can make sense out of zero-time Wightman fields with (IV.7) holding, then Z will be non-zero. 3. We note that there is a version [134] of Nelson's axioms which could hold in c a s e s with Z = 0.

3

n

For

"

n

" is bounded and

For is then the L a p l a c e transform of a distribution to which one can apply the standard Kalle"N-Lehmann analysis (see e.g., [ 1 4 5 B ] ) . 4

§IV.l. NELSON'S AXIOMS

115

The use of (N'6) is illustrated by:

PROPOSITION IV.4.

If (Nl-5) are supplemented

by (N'6), then tor each

) and the map of

Proof.

is

continuous.

On account of (IV. 12) and (III. 1):

by (N'6).

The motivating example for Nelson's axioms is, of course, the free field: THEOREM IV.5.

The Free

Euclidean

Field

obeys (NL-6) with

in

Axiom (N4). Proof.

We have already verified ( N l , 2, 3, 5 , 6 ) .

We wish to prove (N4) in

the form: To proveIV.6.(Segal's (IV. 14) we use the following: LEMMA If A, A B + are and and (IV. 14) is Lemma). bounded, then B self-adjoint is bounded below

Proof.

Let

so by the

operator monotonicity of log ([145, Problem VIII.51)] or Lowner [ 1 2 6 ] ) (Lemma).

116

AXIOMS, II

Returning to the proof of (IV. 14), we consider ~ By the hypercontractive properties of

is in ~

_

_

.

and the fact that e

(as a Gaussian random variable), we see that5 : exp

, Thus

by Segal's lemma.

By homogenity, one concludes that (IV. 14) holds. Remarks: 1. (IV. 14) follows also from number estimates [70]. In many ways this proof is more "elementary" but as we have not introduced the number estimates, we use the "hypercontractive" proof above. 2. Segal's lemma appears in [165].

It is not unrelated to the Golden-

Thompson inequality (see [124] and references therein). for Section IV. 1:

Reference

Nelson [134],

§IV.2.

The Nelson Reconstruction

Theorem

Suppose now that (Nl-5) hold. Let

Then f defines a con-

tinuous map with polynomial decrease of R into

where

As a result, by (N4),

is con-

tinuous with polynomial decrease for any the quadratic form,

We thus define,

on K with form domain

by (IV. 15)

We can now state the main result of this section (and of this chapter):

® H m

e

~

is bounded from

is bounded from

to

to

for suitable

p > 2 and then

by Holder's inequality.

§IV.2. THE NELSON RECONSTRUCTION THEOREM

THEOREM IV.7 (The N.-Reconstruction Theorem [134]). obeying

(Nl-5) is associated

plicitly,

K is the physical

117

Every

theory

unique (GW) theory.

with an essentially

Ex-

defined by (IV. 15) is an

Hilbert space,

1 is the vacuum, and H is the

operator and is the field operator, Hamiltonian.

Remark: Actually only ( N ' l - 5 ) will be used.

Our proof begins with some elegant abstract machinery of Nelson [133]; similar ideas (although slightly weaker and not systematized) appear in Glimm-Jaffe [63,65], We define the scale

Suppose that A is a positive self-adjoint operator. of s p a c e s

with norm

as follows.

If

and if k < 0, we take

to be the com-

pletion of K in the norm (IV. 16) It is easy to see that

and that

and

are

naturally dual s p a c e s of one another by the action

if

. Here we use the fact that

isometry of

o n t o W e

l

e

t

a

' defines an n

d

put the Frechet

k topology generated by the seminorms same as from

on it.

is, of course, the

will denote the family of all bounded maps to

and

> its norm. If

i s naturally

associated to the family of all quadratic forms, b, on K with a

n

d

,

„_ . (IV. 17a)

AXIOMS, II

118

or equivalently if b is symmetric (IV. 17b) Since ,

for any k, whenever (IV. 18)

is an element of

We say (by a simple abuse of notation)

that

if and only if

for every

, and

We now have:

LEMMA I V . 8 .

If

B

and

(ADA)B

lie

in

) then

and

Proof.

Let

. By hypothesis so

and or

and

moreover:

By induction on Lemma IV.8 one clearly has:

LEMMA IV.9. If B , ( A d A ) B , then B takes

into

. Explicitly

for any a,

for some fixed

k,1,

there is a

and

C (only depending on k, (.) so that (IV. 19)

§IV.2. THE NELSON RECONSTRUCTION THEOREM

Proof of Theorem IV.7. Taken into

119

the bound (IV.7) now translates

and moreover (IV. 20)

Since

is unitary on each

(IV.20) implies (IV. 21)

Moreover, by (IV. 20),

so


and

bounded measurable functions on (IV. 24a)

where

is arbitrary and

and where (IV. 24b)

and (IV. 24c)

Remark: This theorem (and the proof we will give) is due to Simon [180] but it is essentially an expression of the fact that a positivity preserving semigroup generates a Markov process, see Dynkin [32, Section III.2], We use a trick of Nelson [130] to make the countable additivity proof for (i easy. We first need two general lemmas:

LEMMA IV. 12. Let

X and Y be compact Hausdorff spaces and let

i be a positivity

preserving

linear map with

Let v

§IV.3. AN ASSOCIATED EUCLIDEAN FIELD THEORY

be a probability measure on Y. measure

Then there exists

a unique

123

probability

77 on X x Y with (IV. 25)

for all

Proof.

Uniqueness is evident since

is total in

the Stone-Weierstrass theorem. Let

by

denote the probability mea-

sures on Y. Then

since for for f positive because A is positivity preserving

and each

since Then

for

is continuous in the weak-* topology since Then h defines a uni-

formly continuous map of

with

Let

by (IV. 26a) and define

by (IV. 26b)

Then £ defines a measure rj on X x Y. Moreover, if then

so

Thus

so rj obeys (IV. 25).

LEMMA IV. 13. Let for preserving with

compact Haussdorff spaces

suppose we are given . Let

.

be given and I

be a probability measure on

there is a unique probability measure

ft on

with

positivity Then

124

AXIOMS, II

(IV. 27a) where (IV. 27b)

Proof.

By induction. The case k = 2 is Lemma IV. 12, so suppose the

result true for

Then we can find v on Let

with and let

) by

Then by Lemma IV. 12, there is a measure on X x Y with

This proves existence of /x.

Uniqueness follows as in the last lemma. • Proof of Theorem IV.11. Let fine a measure

on Q b

the Gel'fand spectrum of y v

T

h

e

n

Hilbert space, is naturally isomorphic to

, De-

K, the physical

) and

to

(since ... is maximal abelian). [Note: !.. is also isomorphic to C(Q) by construction.

Thus every bounded measurable f on Q is equal a.e.

to

a continuous function — which shows how crazy the topology on Q is]. By (S4),

is positivity preserving and since

Thus by Theorem 1.13,

takes

to

a positivity preserving map Now take a copy,

measure -

each

_ for each

obtained by taking

to

by using "

l

e

t

defines

_.

, of

map

and so e ~ ®

b __e

and for each t, the

. For each

and the identifications. For

the measure on

by using Lemma IV. 13 with and for a function only of

define a

and

let

b—a, the Markov property allows us to write (F,U t F) = (G 1 , exp[(—|t|+b —a)H] G 2 ) for suitable Gj with (Ω, Gj) = 0. Butsince such F's are dense in the set of all F's orthogonal to 1, we see that Iim (F,U t F) = 0 if (F, 1) = 0. From this ergodicity follows. • t -» OO

Reference for Section IV.3:

Simon [180], §IV.4. A Counterexample We can summarize our axiomatic considerations by the diagram (GW) (W)
(N') or equivalently (GW) => (S) is false. Of course, it is false because (S), (N') have strong regularity conditions. For example, there are generalized free fields with Z = O s o that (N'6) fails and in four or more dimensions, there are strong indications that K_ 1 is not suitable for interacting fields. But we are really asking if (OS) —> (N') fails for non-technical reasons. It fails for two reasons. First there are Wightman theories in which (S3) fails, e.g., generalized free fields, see [139], Secondly we can ask if (W) ==$» (S4) can fail. -The answer is surely yes in general dimension but since we do not have many examples of interacting fields in two (or more) dimensions we will have to settle for a one-dimensional counterexample. In particular, we will show that (OS) positivity does not imply Nelson-Symanzik positivity. Our example is explicitly due to Simon [180] but once again, it is basically an expression of a fact from the theory of Markov processes, namely that the only differential operators suitable as the generators of Markov process are second order operators.

128

AXIOMS, II

Let K be

let q be multiplication by q and p be

, Define (IV.34a)

H is unitarily equivalent to the operator

(under Fourier

transform) so we can determine a great deal about H by studying ... This can be done by using the methods of Chapter V or by other methods (an exhaustive study of

can be found in [107, 172]). In particular

is essentially self-adjoint on degenerate lowest eigenvalue, function in Then

has purely discrete spectrum, a nonE, with associated strictly positive eigen-

Let

(IV.34b)

is essentially self-adjoint on

vector

with

Moreover,

nite. In particular,

, positive and there is a unique is in

and

, is positive defi-

is strictly positive for

If we take

for some

as our "Wightman f i e l d , "

as vacuum

and H as Hamiltonian, then we have a theory obeying all the one spacetime dimensional Wightman axioms and so the OS axioms.

But Nelson-

Symanzik positivity fails for: THEOREM IV. 15. Let eigenvector.

Proof. on

Let

be given by (IV.34) and let

Then there exists

t > 0 and positive

and let with continuous boundary value

obeys the differential equation

be its

functions

lowest

F, G with

. Then and (IV.35)

§IV.4. A COUNTEREXAMPLE

129

In particular

Thus by continuity,

for t small and so for some small

and all

Let

positive with support in

= min

and let F be

Then

Letting

we see that

Added notes: 1. It is probably a good idea to remark on the connection of OsterwalderSchrader positivity to Nelson's axioms. Since Nelson's axioms imply the Wightman axioms, they clearly imply (OS) positivity. More directly, (OS) positivity) follows from (NS) positivity, the Markov property and the reflection axiom. (formal) positivity

For example, the (OS)

follows by writing (by the Markov property)

(by the reflection axiom)

(by

NS positivity) 0. 2. The last remark illustrates that given a Euclidean Markov field, (OS) positivity (and the related connection to Minkowski fields) is destroyed by a non-locality in the time direction (but not by a nonlocality in the space direction).

Thus, for example, an ultraviolet

cutoff involving smearing in the space direction alone will not destroy (OS) positivity but a smearing in time will. 3. The above remarks also answer the following:

A gas in classical

statistical mechanics clearly defines a Euclidean covariant field with (NS) positivity.

Does it perhaps lead to some kind of "non-

l o c a l " Minkowski region field? The answer is no because non-

130

AXIOMS, II

localities in time built into classical gases destroy (OS) positivity which is essential for the existence of Minkowski fields. Reference for Section IV. 4: Simon [180],

CHAPTER V INTERACTIONS AND TRANSFER MATRICES Thus far we have described general frameworks and trivial models. We now begin the serious business of constructing nontrivial models. We will try to construct models by local perturbations of the free field model — following thereby the time-honored practice of Lagrangian field theory [10]. It is at this point that the famous infinities of quantum field theory enter. There is a natural hierarchy of formal models going under codenames: P( 2,

the integral (V.4) diverges. In that case

cannot be

defined as a random field in Euclidean space time and more serious renormalizations non-linear in g must be used.

Since

similarly considerations apply to Wick products of the time-zero Minkowski space field. We summarize this result for the free field as follows:

THEOREM V.2. Fix free field of mass field.

m > 0. Let

m and

denote the two-dimension

the corresponding

Euclidean

time zero Minkowski

Then: (a)

exists each

for any n as a random process , Moreover, for any such

for

for

and any (V.6a)

2

We use the word " e s s e n t i a l l y " s i n c e the integral might make s e n s e formally without converging absolutely.

138

INTERACTIONS AND TRANSFER MATRICES

(b) :

: exists

for any n as a random process

each

Moreover, for any such

for

for

£ and any (V.6b)

Remarks: 1. The constant c in (V.6) is only dependent on

it is indepen-

dent of p and g. 2. The hypothesis on g is certainly not minimal, especially in case (a) where g of the form

with

is possible

(by case (b) and the imbedding theory of Section III.2!). Proof.

We consider case (a). Case (b) is similar. On account of

Theorem 1.22 and the fact that

by definition, we need

only prove (V.6a) in c a s e p = 2. By the proof of Proposition V.2, this follows if we can show that (V.7)

Rather than prove (V.7) on the basis of the x-space behavior of

(see

e.g. Rosen [147] and Guerra, Rosen, Simon [89]), we follow Segal [168] and use L p -inequalities in p-space. (In their original proofs, Glimm and Jaffe also relied on p-space methods but for

For

(V.8)

Since some lution with

(by Hausdorff-Young ([145, Section IX.4])) for Since is in

Thus precisely of this form.

its multiple convoby Young's inequality ([145, Section IX.4]). using Holder's inequality. But (V.8) is

§ V . l . THE BASIC STRATEGIES

139

Finally we want to link this up with the idea of defining limit of

as a

. We state it in very general terms:

DEFINITION. We say that a Gaussian random process indexed by is of type

C if and only if (V.9a)

with (V.9b) Remark: Both

and

THEOREM V.3. Let of type

C. Let

(as in Theorem V.2) are of type C for suitable C.

_ be a Gaussian random process

indexed by

be a family of functions in

with

(a) (b) Then for each

m and

some

dependent on m,p, d and C (otherwise

there exists independent of

a D

only

and

so that (V.lOa) where (V.lOb)

Proof.

By direct computation

140

INTERACTIONS AND TRANSFER MATRICES

for some r > 1 by mimicking the proof of Theorem V.2. • Remark: By more detailed analysis in x-space, Rosen [147] shows that in (V.10) can be replaced by

when

Now we are ready to describe the Euclidean strategy for constructing interacting field theories. Fix a polynomial P(X). We will call P bounded if and only if

i.e., if and only if with

call P

semi-

If

we

normalized.

Formally, the interacting field theory with interaction P lives on the same measure space but has measure:

(V.ll)

f V . l l ) is based on a variety of intuitions: (1) The theory of multiplicative functionals of Markov orocesses (see [32, Section 9.1] for the classical theory; [90,132,136,137] for the field theory intuition). The basic idea is that the field with

on

§V.l. THE BASIC STRATEGIES

141

φ0 exp (—U(g)) άμ 0 / I exp (-U(g)) (

U(g) =

(V.12a)

I g(x) :Ρ(φ(χ)): d 2 x

(V.12b)

has the Markov property (see Proposition V.4 below). For ν to be formally translation invariant, we try to take g = 1. (2) The FKN formula as built up from the Hamiltonian theory (see Section V.5). (3) The formal continuation of the Gell Mann-Low formula to Euclidean space-time (see the Introduction). (V.ll) cannot hope to be anything but formal if dv Φ d

0

for the only

probability measure absolutely continuous with respect to μ 0 and in­ variant under Euclidean translations is μ 0 (on account of the ergodicity of Euclidean translations). This suggests that one try to form di^ for suitable g and then take some kind of limit as g -» 1; a limit which does not require the output to be absolutely continuous. To normalize dv we o need the finiteness of f exp(—U(g))d/i 0 . Formal considerations suggest that this will not be possible unless P is of degree 1 or 2 (in which case the Gaussian piece of d

0

can cause convergence) or if P is

semibounded and g is non-negative. Henceforth the symbol P will denote a semibounded polynomial. Since : proof that f

P

i

snot a random variable bounded from below, the

βχρ(—ϋ^))άμ 0 < °° for a large class of g (and in particu­

lar for g = X^ the characteristic function of a bounded region, Λ) is non-trivial and will be the main topic of the next section. Taking the results of the next section for granted, we define: DEFINITION. The P{φ) 2 (Euclidean) field theory with space cutoff

is the free random process but with measure (V.12). If g = Xp i , the characteristic function of Λ, we call this theory the P(φ) 2 theory in region Λ and write dv

g

142

INTERACTIONS AND TRANSFER MATRICES

Later, in Chapter VII, we will discuss the possibility of replacing άμ 0 with a different Gaussian process which has different "boundary

conditions." We will then occasionally call the above theory the free boundary condition P(φ) 2 theory with cutoff

g.

In summary then, the construction of an infinite volume P (φ) 2 theory in the Euclidean strategy consists of two pieces: (A) Construct the measure di^ (B) Take g -» 1 to obtain a new theory. (A) is solved for Ρ(ςδ)2 in Section V.2. The analogous problem for 4 ο the (φ ) 3 , etc. theories is still unsolved. There are strong formal indi­ cations that once one solves (A) for these more complex theories, the methods of (VIII, IX) should allow one to solve (B) for at least large sub­ classes of these more complex theories. There are several distinct meanings one can give to (B): (1) Local LP convergence for some ρ > 1. Write dν^ = ί Λ Φ 0 · fixed bounded Λ', let f^ ^ =

For

(f^) the free field conditional

expectation. While f^ cannot converge as A-»» (by the above considerations), f

could converge in LP(Q, d 0 ) for each

fixed bounded Λ'. Such convergence is to be expected on the basis of the local Fock property, a Hamiltonian strategy result of Glimm-Jaffe [64]. This sort of convergence has only been proven for the exactly solvable linear and quadratic models (which lead to "trivial" field theories). (2) Local L -ConvergenCe. The same as (1) but with' ρ = 1. This 1

has been established by Newman [137] for small coupling constant but only for Λ a sequence of rectangles (see Chapter X).

Λ

/4

Recently announced results of J. Feldman change this situation, for (φ )g.

§ V . l . THE BASIC STRATEGIES

1 4 3

(3) Convergence of Minlos-type characteristic functions. Define r

exp I

vergence of

for real-valued to some

and that

Prove conobeys the hypothesis

of Minlos' theorem. This has been accomplished by Frohlich [51] for small coupling and (with the boundary conditions of Chapter VII) for

with Q even (see Chapter X and Chapter

VIII). (4) Convergence of the Schwinger functions (or of the non-coincident Schwinger functions),

for

. This has been accomplished by Glimm-JaffeSpencer [72] for small coupling (see Chapter X) by Nelson [90, 136] (with the boundary conditions of Chapter VII) for with Q even (see Section VIII), and by Spencer [187] for P(X) with Q even, R odd and

large (see Chapter X).

Remarks: 1. The logical relation of (l)-(4) is

2. (3) and (4) are closely related but (4) does not imply (3) because of the uniqueness of moment problem questions and (3) does not imply (4) because

is unbounded.

This concludes our discussion of the general features of the Euclidean strategy. We note, however, one result of a precise nature (supposing the integrability of e x p ( - U ( g ) ) for the time being):

PROPOSITION V.4.

The measure

is the conditional expectation

has the Markov property, i.e., if with respect

to

then

INTERACTIONS AND TRANSFER MATRICES

1 4 4

(V.13) for any

Proof.

-measurable

Write

A.

with

. We claim that for any such A

(V.14) By the Markov property for

the right hand side of (V.14) is

measurable so that (V.13) holds. To prove (V.14) we let B be bounded and

-measurable. Then, by definition of

From this (V.14) follows.

The formulae for the spatially cutoff Schwinger "functions" are thus

This looks exactly like the formula for correlation functions in classical statistical mechanics on account of the occurrence of a Gibbs' factor The analogy between statistical mechanics and quantum field

§V.l. THE BASIC STRATEGIES

145

theory is quite old being associated with the work of Fradkin [49] and Jona-Lasinio [109]. It was Symanzik [194] who first emphasized that by going to the Euclidean region one obtained something akin to classical statistical mechanics and might try to employ the infinite volume tech­ niques of statistical mechanics to control the g -» 1 limit for Sg. This statistical mechanics analogy was developed in constructive Euclidean field theory especially by Guerra, Rosen and Simon [90]. Subsequent to the GRS preprint, the statistical mechanical analogy was further elaborated by Glimm, Jaffe-Spencer [73]. In any event, the remainder of these notes will have a marked statisti­ cal mechanical bias in terms of topics studied, methods used and termi­ nology employed. This begins in Section V.3 with the interpretation of the basic objects of the Hamiltonian strategy as a "transfer matrix." We emphasize now (and will frequently return to) the fact that there is an important difference between field theoretic statistical mechanics and the usual kind of statistical mechanics, for say spin systems. In the latter case,

f • άμ 0

is replaced by a sum over spins. With respect to

this spin system free theory, random variables associated with disjoint regions are independent and the coupling between disjoint regions in spin systems comes entirely from the interaction. In field theory, disjoint regions are not

0 -independent

as we have seen; on the other hand, the

interaction in field theory does not by itself couple disjoint regions but only mediates the basic coupling of the free theory. This distinction between spin systems and field theory will be especially clear in the lattice approximation of Chapter VIII. *

*

*

Our description of the Hamiltonian strategy will be much briefer since we will be following the Euclidean strategy. More details can be found in Glimm-Jaffe [70]. However, the reader should read this subsection care­ fully for the notation defined herein.

146

INTERACTIONS AND TRANSFER MATRICES

Let HQ be the Hamiltonian for the free field Let

•"o

Define

Then, by

Theorem V.2,

for each

show that any dense subset of some multiplication operator on core for both

and

I.

It is not hard to is a core for

as a

and in particular, that

is a

The first two steps in the Hamiltonian

strategy are now easy to describe: (A) Let (V.15) Prove that

is semibounded and essentially self-adjoint on at least for suitable g including

the character-

istic function of (B) Let (V.16) Prove that H(g) has an eigenfunction

with eigenvalue E(g). One

generally defines (V.17a) so that (V. 17b) and normalizes

so that (V.18a) (V.18b)

(It is a significant result that

is never orthogonal to

so that

(V.18b) can be arranged by choice of phase.) The semiboundedness part of problem (A) was solved by Glimm [61] partly on the basis of a sketch of Nelson Ll31] who dealt with a different sort of cutoff. For

self-adjointness was proven by Glimm-

§ V . l . THE BASIC STRATEGIES

Jaffe [62, 66] at least for

147

and positive. The general P result is

due to Rosen [147]. Independently of Rosen, and at the same time, Segal [165] constructed a self-adjoint operator H(g) which was formally — only shortly later [168] did he prove it essentially selfadjoint on

Problem (B) was solved by Glimm-Jaffe [63].

(More complete references to alternative proofs appear in Section V.3; Section V.4.) In Section 3 below, we will present solutions to (A) and (B) because the objects H(g) and

are useful technical constructions even in the

Euclidean strategy. We will require that g be in

and positive

although it is possible to use the bounds of Section VI. 1 to handle the case

whenever P is normalized and

We are mainly concerned with the case

(see [89]).

in which case we write

in place of The infinite volume limit question for the Hamiltonian strategy will not concern us at all so we give only the barest outlines. One constructs a certain _ -algebra into itself. (C) Let

i so that

takes

Then 1

denote the automorphism

Prove that for fixed (D) Let

on

Prove that

exists. be the state

has a weak

build a new representation of A in which ^

_

__ is then used to is unitarily implementable

in accordance with the GNS construction (see e.g. [204]). (C) is actually solved by solving (A). This is a theorem of Segal [164] based partly on earlier intuition of Guenin [86] (see also Glimm-Jaffe [62, 70]). (D) has

148

INTERACTIONS AND TRANSFER MATRICES

not yet been solved purely within the Hamiltonian strategy but it has been solved in some cases by making translations from Euclidean strategy results (see Chapter VIII below). References

for Section V.L:

Wick powers of free fields: Dimock-Glimm [27], Glimm-Jaffe [70], Jaffe [105], Segal [166,167], Simon-Hoegh-Krohn [184], Wightman-Glrding [207]. properties of U(g) and

: Glimm [61], Nelson [131], Rosen [147].

Euclidean strategy: Nelson [132]. Hamiltonian strategy: Glimm-Jaffe [70].

§V.2.

Properties

of the Exponential

of the

Interaction

In this section, we wish to show that large class of g ' s (and all

for a

We first consider the case where

and then allow for the possibility of x-dependent lower order terms. -estimates on the exponential of the interaction were first obtained within the Hamiltonian framework by Nelson [131] (extended by Glimm [61]). In early 1971 Nelson [132] and Guerra (unpublished) realized that Nelson's methods easily controlled the Euclidean strategy exponential also. We follow Nelson's original ideas below. Reference to other methods of controlling

can be found at the end of this section

(we use the fact that on account of the FKN formula, a proof of the semiboundedness of

automatically implies bounds on

The basic technical result for

LEMMA V.5.

Let

, is:

for some

and let

n and let

Then for some

is independent of

g; b is

^-dependent) (V.19)

for all large K (how large is g-dependent).

§V.2. L p PROPERTIES OF THE EXPONENTIAL

149

Remarks: 1. The bound (V.19) says that V can only get very negative on a very small set. Such a result is obviously connected with a statement that

In fact (V. 19) is much stronger than

implying, for example, that

for any odd m.

2. Detailed estimates (Rosen [147]) show that a in (V.19) may be taken arbitrarily c l o s e to Proof.

Choose some and

Let

l/2n. with

Let

if so that

be given by (V.lOb) and let

Then by Theorem V.3 and Theorem 1.22: (V.20) for suitable

and g-dependent C 1 .

By (1.18) (V.21)

where

and a m are constants with Then Q is bounded below so

Let

inf

By (V.21)

as a function in Q-space. Since c K is independent of

and L 1 : (V.22)

150

INTERACTIONS AND TRANSFER MATRICES

Now

for all large K.

Thus (V.23)

for all large K, all

q and suitable g-dependent

Now we claim that (V.20) and (V.23) imply (V.19) for if then

by (V.23) so

(V.24) by (V.20).

We now choose p in a K-dependent way, namely

Thus

Clearly for K large, Thus for K large, (V.24) is certainly bounded by Letting

and

we conclude (V.19). •

For later technical purposes, we need to generalize lemma V.5 in several ways: first we want to deal with the more general process of type C; secondly we want to allow lower order terms and to know how the bounds diverge as the lower order terms become very negative.

§V.2. L P PROPERTIES OF THE EXPONENTIAL

LEMMA V.6. Let

be a Gaussian random process

be given together with

and let

of type

151

C.

Let

Let

and

Let Then there exists

m so that for all

an integer

(V.25)

K with

(V.26) (V.27) where

m, a and a are constants

independent o

f

a

n

d

a

only dependent on the constant n

d

of which process

of type

is chosen. Proof. We just follow the proof of Lemma V.5. (V.20) becomes where

is now only dependent on the type of

we claim if

C but C

(V.20')

To replace (V.22) (V.22')

For

tion. low byOne can write

by an elementary calculaeach as a sum of which of terms is thus of the bounded form be-

152

INTERACTIONS AND TRANSFER MATRICES

since

This proves (V.22'). Thus

is

allowed, s o (V.27) follows for large enough K. We must thus only examine how large k must be in the various approximations made in the proof of Lemma V.5. In order for

log K we

need

Since

(say) and for (V.23) we need that const, this follows if

const. We also need K to be such that

and

All these conditions are obeyed if

K > some constant. Finally we need

Thus the

requirements on K are obeyed if (V.26) holds for m suitable. • These last two lemmas have three important consequences:

THEOREM V.7. Let

P be a fixed semibounded polynomial.

(a) If

Then

then (V.28a)

(b) If

then (V.28b)

Remark: More generally, if U is of the form (V.25), then Proof. let

Let f be a function on a probability measure space (M,X,fi) and Let F be a bounded positive, C 1

non-decreasing function on R.

Then F(x)dm£ (Stieltjes integral)

monotone

§V.2. L p PROPERTIES OF THE EXPONENTIAL

153

In particular, by the monotone convergence theorem: (V.29)

where both sides are infinite simultaneously.

for

If —f obeys:

then

This proves (V.28) if

THEOREM V.8. Fix

But

P a semibounded polynomial of degree

Let

2n.

for for

.

Then for A large

Proof.

(V.30)

.

On account of the argument in the proof of Theorem V.8 and

Lemma V.6, for some c > 0: (V.31)

Thus, by scaling argument it is certainly enough to show that

(V.32) u

1

Let to bound is clearly bounded (V.32) follows from

Then the integral whose logarithm we wish b

y

s

o

154

i n t e r a c t i o n s and t r a n s f e r matrices

all large and this in turn clearly follows from large where x^ is themaximizing

x. But for

(V.33)

large, it is easy to s e e that

s o that (V.33) holds. This proves (V.32) and s o (V.30). •

T h e o r e m V.9.

Fix

g, n.

Let

or the same quantity with

Then for all (V.34a)

where (V.34b)

Proof.

By Lemma V.6, there are Cj, c 2

so long as

By letting exp (const

so that

Thus, as in the last proof:

this last integral is bounded by on account of (V.32). •

§V.3. c o n s t r u c t i o n o f t h e t r a n s f e r m a t r i x

155

There is one final result on exponentials we wish to prove — among other things this result demonstrates that many of the Q N bounds follow directly from Q F

bounds:

PROPOSITION V. 10. Let

Proof.

and suppose

that

By a simple limiting argument, we need only prove (V.35) in case By Holder's inequality:

since

and

is an

-isometry on positive

functions. Now we take References

for Section V.2:

Original Exponential Bounds: Glimm [61], Nelson [131], Other Proofs: Dimock-Glimm [27], Federbush [39], Glimm-Jaffe [68]. Coupling Constant Behavior: Guerra-Rosen-Simon [88, 89, 90],

§V.3. Construction Fix

and Identification

of the Transfer Matrix

. For each a,

denote the function dt

let

For each polynomial, P, we can form and by Theorem V.7 the measure

156

i n t e r a c t i o n s and t r a n s f e r matrices

In this section and the next, we wish to study the a, b dependence of The analogous problem in statistical mechanics of considering the dependence of quantities on varying the size in one direction is solved by the transfer matrix (see e.g. [125]). In this section and the next, we too will construct a kind of transfer matrix and obtain formulae very similar to those of the transfer matrix formalism for spin systems. We will then see that the transfer matrix is essentially

exp(—(b—a) H(g))

thereby obtaining a link between the Hamiltonian theory and the Euclidean theory. Our construction of the transfer matrix following Guerra, Rosen and Simon [90] borrows heavily from Nelson's construction of Theorem IV. 1:

THEOREM V.10. For fixed

(a, b), let

Let (V. 36) as a map a priori from

Then:

(i) (ii) P(t) is bounded from

in fact (Nelson [132]): (V.37)

where (V.38) with m the mass of the free

field.

(iii) P(t) is bounded from For any such

so long as s and t there is an r so that

for all interaction polynomials and all

g's.

157

§v.3. c o n s t r u c t i o n o f t h e t r a n s f e r m a t r i x

Proof, (ii)

(i) follows from translation covariance of the free Euclidean field. Let

Then

so Moreover:

(V.39) where q(t) is given by (V.38) and

The

first inequality above is Holder's inequality. The second inequality follows from Theorem III. 17 for

so that (III.39) holds. (V.37) follows from (V.39). (iii) This is more or less a mimic of the proof of Theorem IV. 1. ting U f be reflection in the time zero hyperplane, and

so P(t) is self-adjoint.

Thus

Moreover

where we have used the Markov property to remove

Let-

158

i n t e r a c t i o n s and t r a n s f e r matrices

On account of the strong continuity of it is easy to prove that

(free time translation),

is continuous if f,

Since P(t) is clearly uniformly bounded on each

by (V.37),

P(t) is clearly strongly continuous on (0, ). T o obtain strong continuity at 0, we note that by the semigroup property

so that

for any a and thus for (V.40) As a result, P(t) is uniformly bounded on [ 0 , 1 ] so strong continuity at 0 results. obeys

where

so

implies there is an

with Thus by Theorem III. 17:

for all

Thus:

so that P(t) is bounded from

to

•

On account of this last theorem

for some H. We can

identify:

Theorem V.12. Let (V.15).

Let

H(g) be given by

Then (a) H(g) is essentially

self-adjoint

on

(b) H(g) is bounded from below, explicitly (Nelson's

we have the bounds:

Bound) (V.41)

where

q(t) is given by (V.38).

159

§V.3. CONSTRUCTION OF THE TRANSFER MATRIX

(NGS Bound) - E(g) < ^ log ||exp(—2Hj(g)/m)|| , 2

(V.42)

L 1 (Q p )

(c) The transfer matrix, P(t), of Theorem V.10 obeys P(t) = exp (—tH(g)) .

(V.43)

Remarks:

1. The semiboundedness of H(g) is a result of Gliram [61] partly based on a sketch of Nelson [131], The proof was considerably simplified by Segal [165,168]; hence the name NGS bound. See the references at the end of the last section for alternate proofs or Simon [182] for an "annotated bibliography." 2. Nelson's bound is implicit in [132] although we have used his re­ sults in [135] to improve the constants. 3. A proof of Nelson's bound without using Euclidean methods appears in [88]. Using the same idea, Guerra, Rosen and Simon [89] proved a bound of the same type but with q(t) replaced by q'(t) = 1 + (mtr 1 (4 log 3)

(V.38')

This bound has the advantage that it allows q'(t) to become arbi­ trarily close to 1 while (V.38) requires q(t) to be larger than 2. 4. The essential self-adjointness of H(g) on D(Hq ) Π D(H1) (for

more restrictive g) is a result of Glimm-Jaffe [62] for P(X) = X 4 and of Rosen [147] for general P. That C00(Hq) is a core is also a result of Rosen [148]. 5. There are now a wide variety of proofs of self-adjointness (see the references in this section or [182]). The particular proof we give is that of Guerra, Rosen and Simon [90] but it is essentially a translation of the Rosen [147] and Segal [168] proofs to Euclidean Q-space. We employ a trick of Semenov [170]. The proof that COC(Hq) is a core is taken from Simon [176].

160

i n t e r a c t i o n s and t r a n s f e r matrices

6. The NGS bound is clearly a consequence of the best hypercontractive estimates. Conversely, the NGS bound with N replacing H q ,

i.e.,

implies the best hypercontractive bounds. For taking V = —log | f| the NGS bound implies that

which is Gross' infinitesimal form of the best hypercontractive estimate [85], (See the remark at the end of Section 1.5.) Proof,

(a), (c). Let H be the generator of the semigroup P(t),

i.e.,

. By the spectral theorem, ® is a core for H. Our first goal will be to prove that (V.44) for any

where

, This is known as Du Hamel's

formula. We will then use (V.44), to prove that

and

that Let

so that

Then (V.45)

pointwise in Q N . One way of proving (V.45) is to note it holds if V is (by series expansion of the exponential) and then approximate V with bounded

. By (V.45)

where we have used the Markov property to insert an extra

161

§v.3. c o n s t r u c t i o n o f t h e t r a n s f e r m a t r i x

Now

as operators, so (V.44) is proven.

Now let

by (iv) of Theorem V . l l .

(since

and by the strong continuity of

As a result u on each

and of

As a result, by (V.44),

as

This shows that so Finally let

all

Then and

as

Thus

(b) On account of (V.37):

so Nelson's bound holds by taking logs. By Proposition V.10:

and is a core

162

i n t e r a c t i o n s and t r a n s f e r matrices

Since

(V.43) follows.

Remark: By a limiting argument using the hypercontractivity of

(or of

exp(—tH) - see Theorem V . l l (iv)), Du Hamel's formula extends to any

c o r o l l a r y V.13

(Coupling Constant Behavior of E(g)). and P semibounded of degree

2n.

Fix Then

(a) ([89]) For large A:

(b) ([184]) If P is normalized (i.e.,

then

for A small.

Proof,

(a) follows from (V.42) and Theorem V.8. To prove (b) we first

note that, if P is normalized, then

so that

the relation:

implies that

From this and (V.42), (b) follows. • Remarks: 1. In [184], it is proven that Rayleigh-Schrodinger perturbation theory is asymptotic as

§V.3. c o n s t r u c t i o n o f t h e t r a n s f e r m a t r i x

163

2. See [151,173,174,184] for complex coupling constant results. 3. It is known [184], that

is not bounded by (const) A as

and there are indications [8] that

is the actual

behavior. By the construction of the transfer matrix (V.36), (V.43), one has the following Feynman-Kac type formula:

(V.46)

One can write the right hand side more suggestively as

where we think of functions on write

as functions of the

and we

In particular (we have used this in the proof of

Nelson's bound):

COROLLARY V . 1 4 .

One also has the more general:

T h e o r e m V.15

(FKN formula-interacting case). Let

be measurable functions on suitable

C,N and all j .

Let

Let

with

for with

and

Then:

(V.47)

164

i n t e r a c t i o n s and t r a n s f e r matrices

where

Proof.

is arbitrary and

Suppose first that

are bounded. Writing

and using

we see that the left hand side of (V.47) is

I Using the Markov property, the E ' s can be removed and the right side of (V.47) results. This proves (V.47) in case are bounded. For arbitrary polynomially bounded Gj we approximate with bounded Then in each

by the dominated convergence

theorem, so the right hand side of (V.47) converges as

Similarly

using properties Remark: (Theorem Moreand generally, V not .in l l each (iv)) only the of if above isso holds athe function left if the hand ofhypercontractive only side finitely converges. many and • fields.

§V .4. VACUUMS FOR THE TRANSFER MATRIX H(g)

165

References to Section V.3: Transfer Matrix Philosophy: Guerra-Rosen-Simon [90], Nelson [132], Self-Adjointness Proof, Original: Glimm-Jaffe [62], Rosen [147]. Self-Adjointness Proof, Hypercontractive: Faris [36,37], Segal [168], Semenov [170], Simon-Hoegh-Krohn [184], Self-Adjointness Proofs, other: Gidas [56], Glimm-Jaffe [68,71], Konrady [116], Masson-McClary [128], Rosen [148],

§V.4. Vacuums ίοτ the Transfer Matrix, H(g) In statistical mechanics, the transfer matrix is useful because it allows one to control the infinite volume limit in one direction. In this section we wish to follow the same idea in controlling the Euclidean measure dv

as a,bThekeytoolfordoingthisinaddition

to the machinery of the last section is the existence of a unique lowest eigenvector Og, for H(g), equivalently largest eigenvector for exp(—tH(g)). This is in direct analogy with the situation in statistical mechanics. This spatially cutoff vacuum is also a natural object in the Hamiltonian strategy and it is in this framework that Glimm-Jaffe [63] first demonstrated its existence and uniqueness. We emphasize that while we have presented E(g) and fig

as "derived" quantities, we feel they are of considerable

interest in their own right. As a preliminary to proving existence and uniqueness of O g we note:

THEOREM

V.16. exp(—tH(g)) is positivity improving for each t > 0.

Remark·. This theorem and its proof is due to Simon [177] but it is only a mild improvement of a result of Glimm-Jaffe [63] (see also [84,169]). -tH Proof. Let f , h f T(F) with f, h positive. Then, since e Γ(β

.

^) is positivity improving (Theorem 1.16), (f, e

tH

h) > O so that

166

i n t e r a c t i o n s and t r a n s f e r matrices

We conclude that

is a.e. non-negative

and is not identically zero. Now strictly positive since

_

is a.e.

.

and so a.e. finite. Thus

T h e o r e m V.17 (Glimm-Jaffe [63]). vector

There exists

a strictly

positive

with (V. 48)

(equivalently

, Moreover

is the unique (up to

constant multiple) vector with

Remarks: 1. Existence of

follows from a stronger result of Glimm-Jaffe

[63], namely that H(g) has purely discrete spectrum in [E(g),E(g)+m).

Our proof of the weaker existence result follows

Gross [84], 2. The uniqueness proof of Glimm-Jaffe [63] depended on the fact that exp(—tH(g)) is ergodic. The stronger positivity improving result (Theorem V.16) simplifies the uniqueness proof. Proof.

1. Since H(g) is reality preserving, we can suppose that any

eigenvector is real valued. Suppose that that

Then we claim

for:

ill since e

is positivity preserving. Thus

2. Next we claim that

and

implies that

a.e. strictly positive or a.e. strictly negative. For either or

is not identically zero and so positive. Thus, by

is

§v.4. v a c u u m s f o r t h e t r a n s f e r m a t r i x

h(g)

Theorem V.18, either

167

is strictly

positive. But

(by 1.) so either is strictly positive, i.e., either

or

a.e.

a.e.

3. On account of 2., if

for

then

so the eigenvalue cannot be degenerate. This proves uniqueness and the strict positivity of any eigenvector.

Thus we

need only prove existence. 4. Let A = exp (—H(g)). For each finite Q f , let

p a r t i t i o n o f

onto

(V.49)

Clearly

is positivity preserving,

(if a

is

ordered by refinement) and because F^ is positivity preserving and

is a contraction on each L^ (Theorem 1.13).

5. Let

. Then clearly is positive, positivity improving and for some

and some

on account of 4., Theorem V.16 and Theorem V . l l (iv). 6. A a

leaves the finite dimensional space ran Ran

Thus

[ is an eigenvalue of

1 . - 3 . the corresponding eigenvector so that

then

invariant and and by

If we normalize

const for all a (on account

of 5.). 7. By the weak compactness of the unit ball in subnet

of

w-lim

8. Since and thus, by 6.,

with

we can find a

weakly. Clearly so we need only show that

for suitable But then

168

i n t e r a c t i o n s and t r a n s f e r matrices

SO

As immediate corollaries of this last theorem, we gain control of as a, b -» oo . T h e o r e m V.18. Fix Equivalently:

Then:

(V.50a) (V.50b)

Remark: (V.50b) was first emphasized by Glimm-Jaffe [65], Proof.

By Schwarz' inequality: e~ t ]

Since

we can take log s, divide by

and obtain

(V.50b). • Theorem V.19 (Cutoff Gell'Mann Low Formula)

be as in Theorem V.15.

Then: (V. >1)

§V.4.

VACUUMS F O R T H E T R A N S F E R MATRIX

H(g)

169

In particular:

(V.52)

Proof.

By the FKN formula, Theorem V.15, the left hand side of (V.51)

is equal to N/D where: (V.53a) and (V.53b) Multiplying N and D by

we may replace H(g) by

in

both (V.53a) and (V.53b). Now, by the functional calculus (see e.g. [145], Theorem VIII.5(d)) (V.54) so that

On account of the hypercontractive bound, Theorem V . l l ( i v ) , the convergence in (V.54) is in each

so

Remark: As for Theorem V.15, this extends to arbitrarily

170

i n t e r a c t i o n s and t r a n s f e r m a t r i c e s

References

for Section

V.4:

Existence and Uniqueness o f O r i g i n a l Proofs: Glimm-Jaffe [63], Rosen [147]. Existence and Uniqueness, Additional Discussion; Faris [35], Gross [84], Segal [169], Simon-HOegh-Krohn [184],

§V.5. Some Miscellaneous

Results

In this section, we collect a number of facts about H(g) some without proof, which will not be used again in these notes but which we feel may be of some use. In any event, the reader may skip this section. First, we wish to demonstrate how one can derive the FKN formula from a Hamiltonian point of view. This follows Nelson's derivation [130] of the FK formula for non-relativistic systems (see also [ l , 41,142,186]). One first establishes the free Euclidean theory as a path integral following Section III.2. Now suppose that one establishes that is essentially self-adjoint by non-Euclidean means. Then (if

by the Trotter product formula. Thus, e.g.,

where

and we have used (III. 14). In a similar way, one can

derive the full FKN formula. *

*

*

Next we mention some technical estimates of Rosen [148] which are sometimes useful:

171

§V. 5. SOME MISCELLANEOUS RESULTS

THEOREM V.20 (Higher order estimates). Fix g e L 1 Π L 1+£ (R). Then for any j, there exists a constant, c, ( depending on g and j) so that

N j < c(H(g)+l)J' .

(V.55)

Moreover, for any ε > 0, there is a j ( depending only on ε) and a c (depending on g and ε) so that

H Q-ε < c(H(g)+l)j .

(V.56)

Remarks:

1. For proofs, see Rosen [148]. A sketch of the main techniques appears in Simon [182]. 2. For P(X) = X 4 , and ε = 1, one may take j = 2 in (V.56). This result of Glimm-Jaffe [62] is interesting since it implies that D(H) = D(H q ) Π D(H i ), i.e., H is self-adjoint rather than merely

essentially self-adjoint on D(H q ) Π D(H 1 ). It is an open question about whether such a bound holds for general P; that such a bound might hold is suggested by the fact that such bounds do hold for one space-time dimensional theories [107,172], 3. For applications of these estimates and similar estimates, see Hoegh-Krohn [100], Rosen [148,149], and Theorem V.22 below. *

*

*

Finally, we will summarize what is known about the spectral proper­ ties of H(g). First we have a strengthened version of the fact that E(g) is an eigenvalue: THEOREM V.21 (Glimm-Jaffe [63]). H(g) has purely discrete spectrum in [E(g), E(g) + m Q ). In particular, E(g) is an isolated point of σ (H(g)).

Proof. See Glimm-Jaffe [63] and Rosen [147].

172

I N T E R A C T I O N S AND T R A N S F E R M A T R I C E S

We also have:

THEOREM

V.22 ([100], [113]).

if g is

with

Among the interesting open questions are proving concerning eigenvalues in References

and

(see however Simon [213]).

for Section V.5:

Feynman-Kac-Nelson from the Hamiltonian viewpoint: Albeverio-HOeghKrohn[l], Feldman [41], Nelson [132], Osterwalder-Schrader [142], Spencer [186], Higher Order Estimates: Rosen [148,149], Simon [182], Spectral Properties: Glimm-Jaffe [63,70], Hoegh-Krohn [100], Kato-Miguboyashi [113], Rosen [147,150], Segal [169], Simon [182], Simon-Hoegh-Krohn [184].

§V.6.

The Hoegh-Krohn

Model

In this section, we discuss an interaction not of the form

but

rather of the form :exp a \ following Hoegh-Krohn [101], This model turns out to be much simpler than the reason:

since

:exp a 0 :

model for the following

is formally

(1.18a)),

(see

should be positive so that we don't

need to work hard as we did for

in Section V.2 to prove that

it will automatically be in

Since it is in

we won't need the hypercontractivity theorem. In addition, formally a power series (if

while every

:exp a cf>\ is

with is formally

with some b

< 0.

This will have important consequences when we come to applying GKS inequalities to the model (see Section VIII.8) making them more powerful and simpler to use than for

models.

Thus the Hoegh-Krohn model is of especial pedagogic interest since it is so simple. There are strong formal indications that the Hoegh-Krohn

§v.6. t h e h o e g h - k r o h n m o d e l

173

interactions are purely repulsive and in particular, there is no spontaneously broken symmetry phenomena in the model (and probably no bound states).

Thus

is a "laboratory" for certain physically interesting

field theoretic phenomena not present in the Hoegh-Krohn model. The fact that one can define the

cutoff interaction without

ultraviolet renormalizations is a consequence of the fact that Thus, for like

purposes a singularity in

would be just as good as

On

the other hand, formally (V.57)

on account of the proof of Theorem 1.3. Thus, the fact that only has a logarithmic singularity is critical and the constant in ffl2(x, 0)

In |x| is important in determining which a are allowed. We

thus begin by analyzing this small behavior.

We also note that as

if space-time has dimension 3 or more, the ultraviolet divergence in the

:exp a (f>: model are very severe in 3 or more

dimensions.

PROPOSITION V . 2 3 .

Let

F(x)

be defined

equivalently

or

, Then-.

(a) F is real analytic

on (0,

positive

for

large.

(b) (c)

by

and monotone

decreasing.

is bounded as

Remark: (b) is not ideal.

The actual behavior is

We have

used this better behavior in Section III.4. Proof.

Analyticity follows from the fact that

non-coincident Euclidean region.

is real analytic on the

On account of the Markov property, if f

174

i n t e r a c t i o n s and t r a n s f e r m a t r i c e s

real-valued and spherically symmetric has support in a sphere of radius £ and

and if

is the translate of

then (V.58)

with

orthogonal to

Thus, the left side of (V.58) is mono-

tone decreasing and positive and so letting

has this property.

(b) If f is as in the proof of (a), then:

by (V.58). (c) S 2 defines a tempered distribution.

Since

and

C 1 -function. In particular since

we see that

from which

follows. • We are now able to prove:

Thus:

are both in

is a

175

§V.6. THE HOEGH-KROHN MODEL

THEOREM V.24. (a) Let / g(x) :exp a

and let

Then

is defined as an

positive

vector in the

following two ways-. (i)

converges

to an

vector

(ii) If hK is a family as in Theorem V.3 a n d i s (V.10),

converges

given by

then-.

in

:exp a

as

to the same vector.

Here

is given by (1.18a).

Moreover the norm

is uniformly bounded on each

(b) Let

and let

Jg(x):expa

. .. _ _

dx is defined as an

Then, positive

function by methods analogous to (i), (ii).

Proof.

We give a proof of (a). The proof of (b) is analogous. We first

claim that if

then: (V.59)

with a bound uniform for

while

on account of the last proposition.

For

176

i n t e r a c t i o n s and t r a n s f e r matrices

Moreover, letting (V.60) (V.60) follows from

Thus, using the

analog of (V.57) which is valid s i n c e i s

a

random variable, we see that

where

Now, if P n is the projection onto

with

and by Theorem V.3,

orthogonality of the Since

it follows that

converges as

. Using

converges to some V in

and clearly

so that (V.62) converging in Remarks: 1. VK -» V in

Since

and

§V.6. THE HOEGH-KROHN MODEL

177

(V.63) (V.64a) we have that (V.64b) so that in

But since

is bounded

Holder's inequality implies that

> in

each LP space with 2. We regard the fact that the ultraviolet divergences are less severe in Euclidean space

as opposed to

as a very

hopeful sign for handling ultraviolet divergences by Euclidean methods. We are now ready to define: DEFINITION. Fix

and v a finite measure on

We

define the cutoff Hoegh-Krohn model Markov field with cutoff and weight v to be the free field Q-space and field but with measure (V.65a) where (V.65b)

As with the

case, we can construct a transfer matrix and show

that it has a unique ground state. Actually, the proof is easier than in the

case since

needed. If

and thus no hypercontractivity is so that the time zero potential is in

we

can mimic the proof of Theorem V.12 (with some modification since but maybe not contraction on each

for summarizing

on the other hand,

is a

178

INTERACTIONS AND TRANSFER MATRICES

THEOREM V.25. For any

so that exp(—tH(g)) is a contraction

is an operator each

and Hoegh-Krohn model, there

and so that

then H(g) is essentially

If, moreover,

and equal there to

In all cases,

self-adjoint

an FKN formula of form (V.47) holds and

for Section V.6:

Hoegh-Krohn [101].

on

where

and transfer matrix formula of type (V.50) -(V.52) hold.

Reference

on

Gell'Mann-Low

CHAPTER VI NELSON'S SYMMETRY AND ITS APPLICATION The FKN formula tells us that the Euclidean field is a path integral over the time zero Minkowski field. This path integral is manifestly Euclidean covariant. In particular, invariance under rotations by yields:

THEOREM VI.1 (Nelson's Symmetry). (VI.l)

Proof.

By the FKN formula, we need only prove that:

and this follows from the invariance of

under Euclidean motions and

the covariance of At first sight (VI. 1) is striking looking although once one understands that it is an expression of Lorentz invariance, it is not quite so mysterious. What is perhaps more surprising is how powerful it turns out to be. One reason for the power is the following: the problem of controlling the behavior of behavior of

is a priori difficult. On the other hand, the is relatively simple to control. But (VI. 11 relates the

two! 179

180

NELSON'S SYMMETRY AND ITS APPLICATION

Nelson's symmetry was first stated (implicitly!) in Nelson [132] who used it to sketch a proof of the Glimm-Jaffe linear lower bound. Its great power was appreciated first by Guerra [87]. Following Guerra's discoveries, the applications were further developed by Guerra, Rosen and Simon [88,89],

The Glimm-Jaffe Linear Lower Bound and Guerra's

Theorem

Fix P a non-zero semibounded polynomial which we also suppose to be normalized

will denote the lower bound on

and we define (VI.2) Occasionally, we will want to consider

the ground state energy of

Our first goal will be to give Nelson's proof [132] of the linear lower (const)

bound,

of Glimm-Jaffe [64] and to establish (following [88])

Guerra's theorem [87] that

exists and is strictly positive:

THEOREM VI.2. (a) ag is strictly positive f

a

c

for all t where (b)

and bounded from above, in

t

,

„ (VI.3)

q(t) is given by (V.38).

is a monotone increasing function of exists

In particular

and we define (VI. 4)

Proof,

(a) On account of Nelson's bound (V.41):

§ V I . l . THE GLIMM-JAFFE LINEAR BOUND

181

In the second step we use Nelson's symmetry and in the last the equality

Finally to prove that

we need only show that

since P is normalized,

But,

so by the variational inequality,

with equality only if

But the latter is easily seen to

be false so (b) Let 11 be a probability measure on

Then for any

Holder's inequality implies that

Thus for any self-adjoint positive operator A and any unit vector

and

any

(by the spectral theorem).

Thus:

By Nelson's symmetry:

Taking logarithms, dividing by t, we can take

and use Theorem

V.18 to conclude:

Since

is arbitrary and

creasing. Remark: One also has [89]

is arbitrary,

is monotone in(VI.3')

182

NELSON'S SYMMETRY AND ITS APPLICATION

with

given by

This has been used by Guerra, Rosen and

Simon [89] to prove t h a t e x i s t s

and is given by the relevant

Feynman diagram. On account of the bound ( V I . 3 ) and Corollary V . 1 3 , we have:

COROLLARY VI.3

Proof,

(a) for small

follows for all concave.

and (b) follow from Corollary V.13 and then (a)

To prove (c), we need only prove that each

is

This follows from the fact that it is an infinimum of linear

functions:

(d) follows from (c). One can ask about control of E(g) for more general g. First we have:

THEOREM VI.4 (Improved linear lower bound [89]). For any (VI. 5) In particular.

§ V I . l . THE GLIMM-JAFFE LINEAR BOUND

Proof.

183

Suppose first that g is of the special form

with

Then one has the extended

Nelson's symmetry:

(VI.6)

(VI.6) is proven by writing both sides in Euclidean Q-space by employing the FKN formula. Thus:

where we have used

in the last step. On account of Theorem

V.18, this proves (VI.5) in case g is of the special form. A simple approximation argument [89] now proves (VI.5) for arbitrary g. (a), (b) and (c) now follow from (a), (b) and (d) of Corollary VI.3. Remark: was only needed to assure that

Thus the bound

(VI.5) suggests that one should be able to prove essential self-adjointness for every

This is done in [89].

As for generalization of Guerra's theorem, we state without proof the following result from [89]:

184

NELSON'S SYMMETRY AND ITS APPLICATION

THEOREM VI.5. of intervals,

be a sequence

of functions and

a

sequence

so that

In the next section, we will need to know that

is not constant, so

we note [88]:

PROPOSITION VI.6. i4s

In particular,

is not a

constant.

Proof.

By Nelson's bound and Nelson's symmetry:

Since P is normalized,

so by Taylor's theorem

for all

Thus References

holds for to Section VI. 1:

The Glimm-Jaffe Linear Lower Bound: Glimm-Jaffe [64,68], Nelson [132], Simon [175]. Convergence of Guerra [87], Guerra-Rosen-Simon [88,89], 04terwalder-Schrader [141].

185

In this section, we consider two quantities, the correction to the behavior of

(VI.6)

and vacuum overlap (VI. 7) is a surface energy. One expects

to be bounded from below by a

strictly positive constant on the basis of the following intuition [184]: If the regions

were not coupled by

would be a product of

then

for

integral

functions, each looking like a vacuum

in orthogonal coordinates.

but

As the coupling goes exponentially to zero,

should look " a l m o s t " like a product as

at least as far as

distant q ' s are concerned. On this basis, one expects

constant.

We set

is hounded for for

Proof.

large;

suitable.

On account of the proof of Theorem V . l l (iv), for T is bounded from

equivalenth:

to

(for all

suitable,

and (VI.8)

for some r. By the FKN formula (Corollary V.14)

186

NELSON'S SYMMETRY AND ITS APPLICATION

so by (VI.8): (VI.9) for suitable c. Thus, since

But, by Holder's inequality:

so

i.e., Remark: The bound on

also follows from the inequality

and the linear lower bound. This inequality on

due to Simon [181]

follows from Gross' infinitesimal hypercontractivity [85]. As a corollary of the proof and the bound

we have:

C O R O L L A R Y VI.8.

Proof.

Tracing through the above proof,

for suit-

able T and r. We also have:

THEOREM VI.9 (Guerra's equation [87]). For any (VI. 10)

187

Taking logarithms

which reduces to (VI. 10) upon dividing by

and using the definitions of

a and We are now ready to prove:

THEOREM VI. 10 ([88]). (a)

is strictly negative and is bounded from

below, in tact (VI. 11) (b) jSt is monotone decreasing

in t, so that (VI. 12)

exists

Proof,

(a) Picking a subsequence of

from Guerra's equation (VI.10). Since Proposition VI.6, from (b). (b) By Theorem V.18:

for t small, so

with

(VI. 11) follows is bounded below. By

strictly negative follows

188

NELSON'S SYMMETRY AND ITS APPLICATION

Thus

or

Since

are arbitrary positive numbers,

is monotone decreasing,

(c) Since this follows from Corollary VI.8. COROLLARY VI.11 (The Van Hove Phenomena).

(VI. 13)

and since

is strictly negative and decreasing

Thus

189

Remark: That w-lim

is a conjecture of Van Hove [199] in general field

theories. Guerra [87] proved that

and thus that

The stronger exponential falloff is a result of Guerra, Rosen and Simon [88],

It is a general conjecture of Guerra, Rosen and Simon (unpublished) that

exists and equals

equivalent to

On account of (VI.13), this is

In general, this conjecture remains open, but

for small coupling constant, it is a result of Newman [137] based on a result of Glimm-Jaffe-Spencer [72] which we discuss in Chapter X. For general

define: (VI. 14)

i.e.,

is the " m a s s - g a p " for

Glimm-Spencer result says that

Then the for all small

and

THEOREM VI.12 (Newman's Equation [137]). (VI,15)

Proof.

Dividing both sides by

and taking logarithms (VI. 15) results.

NELSON'S SYMMETRY AND ITS APPLICATION

190

THEOREM VI. 13 ([137]). If

In particular, for

small coupling constant,

Proof.

Choose

exists

and equals

so that

Let

in (VI. 15).

Then as (VI. 16) Choose a subsequence, so that

Then by Newman's equation

and (VI. 16):

the equality is proven. References

to Section VI.2:

Guerra, Rosen, Simon [88]; Newman [137],

The Glimm-Jaffe

and

Bounds

Our main goal is to prove some basic bounds of Glimm-Jaffe [65,67] of the form

(VI. 17)

for an I independent constant c and for A of the form {f) or The point of (VI. 17) is that c is expect (VI.17) to remain true in the

independent so one can

limit (see Bratelli [14] and

Glimm-Jaffe [65] in the Hamiltonian Framework and Sections VI.5, VII.5 and expecially VIII.6 below in the Euclidean framework for a discussion of the passage to the

The main technical

estimate for the bounds is: LEMMA VI. 14. Let

P be a fixed polynomial.

nomial so that either

Q is semibounded or

Let

Q be a second

poly-

and tor any f with supp f contained in an integral of

THE GLIMM-JAFFE

length

BOUNDS

191

1 and in

(where

is required if

where

AND

Finally,

let

is a translate of f with support in

be the corresponding

ground state energies.

Let Then-. (VI. 18a)

where

and

(b) If moreover,

are those associated

to P and

Q is either linear or semibounded (and

the non-linear case),

in

then: (VI. 18b)

Proof. Let

Suppose supp f be the embedding of

i.e., the image of

where into

and at constant space coordinate x,

under a rotation by

Then by the Feynman-Kac-

Nelson formula and the Markov property:

(VI. 1 9 )

where

192

Now, let

NELSON'S SYMMETRY AND ITS APPLICATION

be the image of Fj under reflection in the plane x = a.

Then, again by the Markov property

so that, by translation covariance (VI.20a) Similarly (VI. 20b)

Moreover by using hypercontractivity and mimicing the proof of Nelson's bound (see the proof of Theorem V . l l (ii) and Equation (V.37))

(VI. 20c) By (VI. 19), (VI.20) and Theorem V.18 (suitably extended to ( V I . 13)

Now, on account of

THE GLIMM-JAFFE

AND

BOUNDS

193

(VI.22) (VI. 18a) follows from (VI.21). Finally to prove (VI.18b), we note that

so that (VI.23) (VI. 18b) follows from (VI.18a) and (VI.23). Remarks: 1. (VI. 18) is a slight improvement of the original Glimm-Jaffe bounds [65], In this form, it is due to Simon [181] but the basic idea is borrowed from [88]. 2. While we have not explicitly used Nelson's symmetry, we have used the

rotations of the more usual

We could just as

well have followed [88] and used Nelson's symmetry twice, and the FKN formula in place of the explicit

Markov property [in that

case (VI.20a) would just be Nelson's bound and not merely its analogue]. However, the point of this proof is that what is really essential for the

bound is the Markov property and not Nelson's

symmetry. In [88], Nelson's symmetry was needed only because no explicit use of the Markov property was made. 3. The basic idea of the above proof yields the following abstract version [181], Let

be multiplication operators by

functions of the time zero fields supported respectively in Then

194

NELSON'S SYMMETRY AND ITS APPLICATION

where

and where

sum of

(resp.

is the

and its reflection in the point

THEOREM VI. 15 ([65]). Let be the renormalized

Q be a polynomial with deg associated

for the interaction

with P and let

Then for any

integers: (VI. 24) where

Remarks 1. Since

(VI.24) says that

for an

independent constant 2. For example, if we require that that

(VI.24) easily implies where

is M-dependent and

|supp f| is the size of the smallest interval containing supp f. This is the original formulation of the bound by Glimm-Jaffe [65], Proof.

By an analog of the proof of Theorem The theorem now follows from (VI. 18a) by breaking unit intervals, and using the estimate

THEOREM VI. 16 (Glimm-Jaffe bounded polynomial and let for some suitable

Bounds [65]). Let be the associated

(independent of

P be a fixed

Hamiltonian.

and all h on R with

semi-

Then

THE GLIMM-JAFFE

(i)

AND

BOUNDS

195

supp h C some unit interval we have (VI.26a) with supp

In particular, for all

supported on an interval of length

(but not

necessarily

1) (VI. 26b)

Proof.

Let h obey (i), (ii). Then, by Nelson's bound:

But it is easy to see that and all h with

so for suitable obeying (i):

by (VI. 18b). Thus, by (VI.25) for all such h (VI.27) for

Putting

for arbitrary

in (VI.27), we

obtain (VI.26a). To prove (VI.26b), we first note that as its Fourier transform,

is in

Thus by

This proves (VI. 26b) when h is supported in a unit interval. If h has support in have support in

we write and note that since

where the

196

NELSON'S SYMMETRY AND ITS APPLICATION

we have (VI.26b). Remarks: 1. The bounds in Theorems VI.15 and VI.16 are due to Glimm-Jaffe [65] (except that the

in (VI.26a) is replaced by

for suitable

Most of their results were

recovered by Guerra-Rosen-Simon [88] using Nelson's symmetry but their methods were not quite strong enough to prove (VI.26b) [instead, they got

for a suitable

By basically following their proof but using Nelson's bound and Nelson's symmetry an extra time, we have recovered the stronger Glimm-Jaffe result. The one application of the 2. Spencer [186] has proven " l o c a l

— result is needed in (Section VI.5 below). -estimates" which are stronger

than (VI.26a). 3. By a small modification of the above proof one can show that for any

there is a d depending only on

so that under the

hypotheses of (VI.26b):

(VI. 26b')

4. Frohlich [51] has noted another version of the if f has support in

-bounds. Namely,

then (VI. 28)

where

For (VI. 18b), the argument

above and the bound

imply that

(VI.29)

THE GLIMM-JAFFE

AND

BOUNDS

for any h with support in a unit interval inside

197

Given

f, write

where

has support in

Using (VI.29) on the

and the bound

(VI.28) follows. As an application of the above and the method of Lemma IV.9, one obtains:

THEOREM VI.17. For any

there exists

a norm

that

where supp

We now turn to the

We must first develop a simple set of

"N-bounds" in order to define Let

Let

We use Gaussian Process language.

be the " f i e l d " on

and define (VI. 30)

on

On account of (1.31):

so that

and thus

(VI.31a)

198

NELSON'S SYMMETRY AND ITS APPLICATION

which follows from (VI.31a). From (VI.32) and (VI.34) also follows: (VI.35) Finally we want to establish:

LEMMA V I . 1 8 .

F o r any

and

(VI.36)

THE GLIMM-JAFFE

Proof.

Using (VI.32), that

AND

BOUNDS

199

is a direct computation. Let

Then (again applied to vectors

Thus

Now we are ready to extend the If

is a real Hilbert space and

realizing

so that If

to an arbitrary Gaussian process. we define

on

by

and taking

is a densely defined self-adjoint operator, for

we define PROPOSITION Then: VI.19. With the notation of the last paragraph: (VI. 37) (VI. 38) Proof.

If

has a complete set of eigenvectors so that

(VI. 40) 39)

200

NELSON'S SYMMETRY AND ITS APPLICATION

the results follow from (VI.28) and (VI.34)-(VI.36). limit in norm resolvent sense of such

Since any

is a

the results hold in general.

In particular, for the free field of mass m, we define

so that (VI.37)-(VI.40) become:

(VI.37') (VI.38') (VI.39') (VI.40')

It is now easy to prove: THEOREM VI.20 (The Glimm-Jaffe

Bounds [67]). For any

interaction

and any (VI.41)

(VI. 42) (VI.43)

Proof.

Since

commutes with

(VI.43) follows from (VI.37')

and by (VI.40'):

(VI.41) follows immediately by taking taking

and (VI.42) follows by

THE GLIMM-JAFFE

AND

BOUNDS

201

Remarks: 1. Of course,

is just the usual annihilation operator, p is the

momentum and (VI.28)-(VI.33) are standard harmonic oscillator results, here expressed in terms of Gaussian processes. 2. To the reader used to the harmonic oscillator on point out that p is not

we

because we are on as

in (VI.31).

On account of Lemma IV.8:

LEMMA VI.21. Let

be the scale of spaces

as in Lemma IV.8. If

associated

to some

is symmetric and

then:

Proof.

By Lemma IV.8,

Section IX.4])

so by interpolation (see [145, which yields the

symmetry

bound since by The

bound

comes from the monotonicity of the square root (see [145, Section IX.4]) and the

bound.

Combining Lemma VI.21 and the

THEOREM VI.22. Let supp

and

P be fixed and let

Then for some constant

bounds:

with C independent of

and h: (VI.44) (VI.45)

202

NELSON'S SYMMETRY AND ITS APPLICATION

Remark: The bounds (VI.44) and (VI.45) hold with placed by

and

and

re-

by Spencer's bounds [186].

There is an alternate proof of the bound (VI.44) in a stronger form. Namely: THEOREM VI.22 B. Let of length

1 contained

In particular,

Proof.

h be in

with

in

supp h in a unit

interval

Then

with supp h

for any

One just mimics the proof of Theorem VI. 16 using the following

bound for

By the NGS bound:

and moreover:

Remark: In fact since

for

(or more general h's) one

can also prove by this method

if supp

For details, see Simon [181].

The major applications of the

-bounds made thus far are:

§VI.4. NELSON'S COMMUTATOR THEOREM

203

(1) (Glimm-Jaffe [65]; see also Glimm-Jaffe-Spencer [72].) Temperedness of the Wightman functions uniform in 2; see Theorem VI. 17. A similar idea is used to obtain bounds on the Schwinger functions useful for control of the infinite volume limit (see [90] and Sections VIII.5, 6). (2) (Glimm-Jaffe [65]; see also Nelson [133].) Self-Adjointness of infinite volume Wightman fields smeared in space and (Minkowski) time. We describe the abstract theorem in the next section and the application to the infinite volume limit in Section VIII.6. (3) (Nelson [133].) Existence of infinite volume Green's functions. We will not have occasion to discuss this further. (4) (Frohlich [51].) Certain Euclidean region bounds which will be very useful to us. We discuss them in Section VI.5 and apply them extensively in Sections VIII.5,6. References for Section VI.3: Glimm-Jaffe [65,67]; Guerra, Rosen, Simon [88], Simon [181].

§VI.4. Nelson's Commutator Theorem In their original paper on the 0-bound [65], Glimm-Jaffe showed how the bounds plus commutator technology implied self-adjointness of infinite volume smeared fields. We give here an abstract theorem of Nelson [133] which strengthens the Glimm-Jaffe theorem. It will allow us to use the infinite volume φ-bounds to prove essential self-adjointness of the Wightman fields on C 00 (H) (see Section VIII.6). THEOREM VI.23 ([133]). Let

he the scale of spaces associated to

some A>0 (see Lemmas IV.8,9). Suppose Bf £(H +1 ,K_ 1 ) with [A 1 BltS (H 11 ^ 1 ) and with B symmetric. Then B is essentially

self-adjoint on any core for A.

204

NELSON'S SYMMETRY AND ITS APPLICATION

Proof:

(Faris-Lavine [38]). On account of the arguments in Lemma VI.21, i.e.,

I and (VI.46)

On account of this, for any core,

so we need

only prove that B is essentially self-adjoint on D(A). Let we will prove that

has zero kernel, thus proving

essentially self-adjoint ([145; Theorem VIII.3]). For let and let

Then:

Thus

i so that

implies

implies Remark: Faris-Lavine [38] have a beautiful application of this theorem to Schrodinger operators (see [145, Section X.5]) and also an interesting intuition. §VI.5. Frohlich's

Bounds

Frohlich [51] has a very convenient transcription of the bounds to a Euclidean-statement. The basic idea behind Frohlich's transcription is that in a theory obeying the basic axiom schemes of Chapter IV, the bound of axiom (N4) (with (VI. 47) is formally equivalent to: (VI.48)

§VI.5. FROHLICH'S BOUNDS

205

for if one can establish a Feynman-Kac-Nelson formula for (which may not be easy since

is unbounded, but it is certainly

formally true), then (VI.47) implies (VI.48). Conversely, if one knows that

has a positive ground state (which again is formally true),

then (VI.48) implies that One result we can prove is that: THEOREM VI.24. Let that

obey all the Nelson axioms and

suppose

and (VI. 49a) (VI.49b)

Then

(VI. 50a)

(VI. 50b) Conversely,

if

obeys all the Nelson axioms except

, (VI.50) holds, Proof.

(N4) and for some

then (VI.49) holds.

Suppose first that F is a bounded function. Then by following

either of our general proofs of the FKN formula (in Section V.3 via the Markov property and Du Hamel's formula, or in Section V.5 via the Trotter product formula): (VI.51) In particular, by Schwarz' inequality: (VI.52)

206

NELSON'S SYMMETRY AND ITS APPLICATION

Conversely, we have that (VI.53) by the following argument: Since

is doubly Markovian on the

physical Hilbert space

it is bounded on _

since F is bounded is

Thus since

is a core for H —F, there is a set of H —F. Thus given e, we can find

Since

and so

functions which is a core for with

pointwise:

so that

Since

is arbitrary and we have (VI.52) to control

(VI.53) holds.

Now suppose (VI.49) holds. Let F fi be the function: Then so by (VI.52) Letting (VI.50b).

and using the monotone convergence theorem, we obtain

§VI. 5. FROHLICH'S BOUNDS

207

On account of (VI.49b), it is easy to show that Thus using (VI.52), (VI.50b) and the dominated convergence theorem we obtain (VI.50a). Conversely by using (VI.53) in place of (VI.52) and similar arguments, one shows that (VI.50) implies (VI.49). Remark: This last theorem has important consequences from an axiomatic point of view. In the first place, it means that Nelson's axiom (N4) can be replaced with a (stronger) purely "Euclidean" statement (VI.50b), i.e., with something which does not refer to the construction of Theorem IV.1. More significantly, it implies that if (S2) is replaced with

then the measure dv constructed in Section IV.3 is Euclidean covariant. For (VI.50a) (in a slightly extended form) implies a uniform bound on the (coincident or non-coincident) Schwinger functions so that the coincident Schwinger functions are a limit of non-coincident Schwinger functions. Thus covariance of the non-coincident Schwinger functions implies covariance of the coincident Schwinger-functions. Since converges to is analytic near

for

, small and

Thus invariance of the coincident

Schwinger functions implies invariance of

and so of dv.

The above only depends on the time translation invariance of the Markov field and not on the rotation invariance and so it holds in a theory associated to a (VI.49) and so:

model. In such a case, we have

208

NELSON'S SYMMETRY AND ITS APPLICATION

THEOREM VI.25 (Frohlich's Exponential Bounds [51]). Fix a semibounded interaction

polynomial

(a) For each

P.

Then, for all there exists

I:

a c(f) with

(VI. 54a)

(VI. 54b) where

is the

measure: Normalization .

(b) For each

there exists

d(f) with (VI.55)

(c) For each

e, there exists

a constant,

so that (VI.56)

so long as (ii) Proof.

(i) supp

obeys: for all

t.

As in the proof of Theorem VI.4,

Thus (a), (b) follow from Theorem VI.15 and (c) from Theorem VI.16 in form (VI.26b') (see remark 3 following Theorem VI. 16). For purposes of Minlos' theorem, we want bounds on

§VI. 5. FROHLICH'S BOUNDS

209

Since:

so that we shall want bounds on THEOREM VI.26 (Frohlich's action polynomial.

uniform in I: -Bounds [51]). Let

P be a fixed

inter-

Then, for all C:

(a) There exists

an

-norm (independent

of £) so that for all

(VI.57) where (VI. 58a) i.e., (VI. 58b) (b) There exists

an with

•norm (independent supp

of t), so that for all

and all

t,s: (VI.59)

and (VI.60)

Proof : VI.25(c),

j, implies that

Then by Theorem

210

NELSON'S SYMMETRY AND ITS APPLICATION

Since

so that by homogenity

Since

by (NS) positivity, (VI.57) follow by "renormalization" of (b) Returning to the proof of Lemma VI.21, we see that (VI.61a) for a suitable Schwartz space norm, where the norm of

as a map from

scale associated to

is with J

1

the

Thus: (VI.61b)

so that

(VI. 62) Letting

(VI.62) says that

from which (VI.59)/(VI.60) immediately follow. There is one final abstract result of Frohlich [51] that we will require; since it does depend on Euclidean invariance, it will require carrying over the bounds of the last two theorems to easy!).

(which will be

§VI. 5. FROHLICH'S BOUNDS

THEOREM VI.27 (Frohlich [51]). Let 2

S R ( R ) which is Euclidean

covariant.

(a) For each

there is

211

be a random process Suppose

indexed

by

that:

) so that (VI.63)

so long as

for all

t.

(b)

(VI. 64) for a suitable

(c) (VI.65) where Then: (1) For a suitable

exists

sequence

for all

(2)

is integrable

for all

(3)

" > all

(4) If Proof.

so that

l

is the translate

of

then (VI.59), (VI.60) hold.

Let

I and

Thus, by (VI.65):

212

NELSON'S SYMMETRY AND ITS APPLICATION

so (1) and (4) follow immediately. To prove (2) and (3), we need only prove that

is

uniformly bounded for a real, for (VI.66) for

so that

proving (3) for a real and the boundedness (2) for a real. But since , (2) holds for all a and then (3) follows by the Vitali convergence theorem. By the Euclidean covariance, L

e

t

T

h

e

n

t independently of n, so that (VI.63) implies Reference

for Section

Frohlich [51].

VI.5:

CHAPTER VII DIRICHLET BOUNDARY CONDITIONS In the theory of statistical mechanical systems such as the Ising model, the clever use of one or more kinds of boundary conditions plays a major role [80,154, 47], It is therefore not surprizing that a similar situa­ tion occurs in the P (φ) 2 model, especially when statistical mechanical methods are employed. In this chapter we discuss in detail one type of boundary condition, the Dirichlet boundary conditions. An analysis of more general kinds of boundary conditions is possible (see [90] for the one-dimensional case, and [91] for P(^) 2 ) and it is our expectation that other kinds of boundary conditions will eventually play a role in construc­ tive quantum field theory. Since the applications we discuss in the later chapters only rely on free or Dirichlet boundary conditions, we restrict ourselves to their study. However, see Section X.3. One of the most natural questions involving boundary conditions is making precise the sense in which two theories are really the same theory with "different boundary conditions." For the non-interacting free boundary condition and Dirichlet boundary condition theories, this question is answered by Theorem VII.2. For the interacting theories, this question is most naturally answered by reference to the analog of the equilibrium equations of Dobrushin [28] and Lanford-Ruelle [117]. The P(φ) 2 version of these equations is discussed in Section VII of [90] and we urge the reader to consult that reference for further details; see also Section X.4. Given an open region Λ C R 2 , the free boundary condition (noninteracting) field is the Gaussian random process indexed by N^, i.e., (f)0(g)> f r e e = L 2

(VII.1)

214

DIRICHLET BOUNDARY CONDITIONS

where

_

is the infinite volume operator and

The Dirichlet

boundary condition (non-interacting) field is also Gaussian but with covariance

(VII. 2)

where -

is the Dirichlet boundary condition operator of

) (de-

fined in Section VII. 1). Dirichlet boundary conditions are especially natural for the following reasons: (1) In a sense we make precise in Section VII.2, they are minimal among all boundary conditions. (2) By (VII. 1), free boundary conditions are obtained by restricting the measure

to

; thus the variables outside are "integrated

out." Dirichlet B.C. correspond to setting the variables outside to 0. This is seen most clearly in the lattice approximation (see Sections VIII. 1, 2). (3) The free B.C. field has a covariance matrix obtained from that for the free field by restricting to

In a sense made precise in

the lattice approximation (see Section VIII.2), the Dirichlet field has an inverse

covariance matrix obtained from that for the free

field by restricting to

. Of course, for general random varia-

bles, the inverse covariance matrix is of limited interest but on account of (1.12), it is extremely important for Gaussian random variables. By using model 3 for Q-space, we think of free B.C. or Dirichlet B.C. non-interacting field. r.v. but there are two measures

and

^

for either the is then a fixed

and thus two notions

of Wick ordering related by (1.20). This means that in adding an interaction i

we must decide on which choice of

: : to use. Thus, even if we decide on

as basic unperturbed

measure we can obtain two interacting states, the Dirichlet : : is w.r.t.

L

or the Half-Dirichlet

state where

state where : : is w.r.t.

215

§VII.l. THE NON-INTERACTING DIRICHLET FIELD

Both are important in applications and so we discuss both: The Dirichlet states in Sections VII.2, 4 and the Half-Dirichlet states in Sections VII.3,4,5,6. The importance of Dirichlet boundary conditions in connection with GKS inequalities was first understood by Nelson [136] who emphasized the importance of such theories. An extensive study of the Dirichlet theories was then undertaken by Guerra, Rosen and Simon [90, 91]. §VII.l.

The Non-interacting

Dirichlet

Field

We begin by studying the Dirichlet Green's function: DEFINITION. Let

be an open set. Let

_

be the Friedrichs

extension ([145; Section X.3]) of the operator in i and action

The Dirichlet

is the kernel,

I with domain

Green's function for region

of the distribution (VII. 3)

We will let

i denote the kernel of

THEOREM VII. 1. Let

be the projection

(in N) onto

and let (VII.4)

i.e.,

is the projection

onto the orthogonal

complement

of

Then: (VII. 5) for any

. In particular,

the completion

of

norm contains Proof.

Suppose first g is of the special form (VII. 16)

216

DIRICHLET BOUNDARY CONDITIONS

Then since

has support in

Thus

This proves (VII.5) if g is of the special form (VII.6). Let the completion of

in

-norm. By the definition of Fried-

richs extension,

is dense in

, so proving (VII.5)

for g of the form (VII.6) proves it for all of

On account of (VII.5), in

and in particular for

f

on

so the completion

clearly contains the completion of

given

, since

denote

is open,

. But

has support in

if j has suffi-

ciently small support and if f has compact support. It is therefore easy to prove that

is the completion of

DEFINITION. Markov

field

Let

in

be open.

T h e free

B.C.

(non-interacting)

is the G a u s s i a n random p r o c e s s with indexing s p a c e

and covariance

. The Dirichlet

B.C. (non-interacting)

Markov field is the Gaussian process with indexing space

and co-

variance There are two distinct ways of thinking of the connection between the two theories. First there is the active picture in which we (following Model 3 of Q-space) think of two distinct Gaussian measures and

on

functions

. In this case both fields are just the coordinate on

Then there is the passive picture in

which we define (VII. 7) and think of a fixed measure,

is a model for the free

§VII.l. THE NON-INTERACTING DIRICHLET F I E L D

B.C. field and

217

a model for the Dirichlet B.C. field. We will

generally use the active picture but (especially in Section VII.2) will have occasion to employ the passive picture (which we will then specifically point out). There is a direct connection between the measures, THEOREM VII.2 ([90]). Let \J

J *

a

n

d

/

be a compact

are relatively

subset

absolutely

and

of

Then

continuous,

and, in

fact (VII.8)

where

F is a Gaussian,

is

'—measurable

and lies in some

Remark: (VII.8) is clearly an expression of the fact that

differs from

only in a "boundary term." Proof.

View

as a Hilbert space. The free boundary condition field

is clearly the natural G.R.P. indexed by

• while the measure

defines a second G.R.P. with covariance

where

. Thus 1 — A as an operator on

' equals

. By Theorem III. 13, by Theorem III. 16, 1 - A

is trace c l a s s . Thus A is invertible and

moreover, by Theorem 1.23, absolutely continuous with an proof of Theorem 1.23, with i

and

and

_

are relatively

Radon-Nikodym derivative. By the where the

and

are

218

DIRICHLET BOUNDARY CONDITIONS

determined by

Only those

F . But by

with

will enter into

' and Theorem III.16,

that

i.e., F is

implies

'-measurable.

We also note that: THEOREM VII.3. The Dirichlet field,

i.e.,

i is closed,

if

Boundary Condition and F is

field is a Markov —measurable,

then (VII.9)

where Proof.

is the conditional

expectation

with respect

to

As in the case of the free field, it follows from the fact that is a local operator.

Consider the region

Let

denote the

Dirichlet field measure for this region. Clearly, this Dirichlet theory is time translation invariant, so there should be an operator

on a

suitable "time-zero" Hilbert space which acts as a transfer matrix for this theory. It is not very hard to identify LEMMA V I I . 4 .

Let

explicitly. We first note:

and

(VII. 10a) (VII. 10b) with (VII. 10c)

§VII.l. THE NON-INTERACTING DIRICHLET F I E L D

219

(VII. 11)

Proof.

Let

and let . (VII. 12)

One easy shows that

is

on the lines

vanishes exponentially as

and

. Moreover:

Thus,

I and so

, (VII.11) now follows

from (VII. 12). THEOREM VII.5 ([90]). Let

H be the Hilbert space

and let (VII. 13)

(VII. 14a) where

is defined

by: (VII. 14b)

(Here

is an orthonormal

H into the Dirichlet

basis for

field Hilbert space

for

a n d i s

Map

by: (VII.15)

Then

is an isometry with

Ran

and

(VII. 16)

220

DIRICHLET BOUNDARY CONDITIONS

Proof.

On account of Lemma VII.4,

so this result follows by mimicking our construction in Section III.2.

Dirichlet theories in boxes

| will later

play an important role. The Dirichlet conditions on trolled by using the Dirichlet transfer matrix tions at to

" : will be con-

. To control the condi-

we use the following intuition: Passing from corresponds to setting the fields in

to zero. By

the Markov property, it is sufficient to set the fields if we only consider the fields in .

to zero

. [That is

Thus intuitively we should have that (Normalization factor) where

is a product of S-functions. The key to making this intuition

rigorous is to note that while

is not a legitimate vector,

is! One first sees this in the one dimensional case: 1 THEOREM VII.6 ([90]). View

as

and let (VII.17)

f o r L e t "Euclidean"

o be the measure field,

for the corresponding and every

q(t), of mass Dirichlet

u which is

for the one dimensional m = 1. Let

field in

denote

free the

Then for every -measurable'.

measure

§VII.l. THE NON-INTERACTING DIRICHLET F I E L D

221

(VII. 18)

Remarks: 1. (VII. 17) comes from (1.38) by taking Thus (VII. 18) intuitively comes from:

by the Markov property and the relation:

l 2. Since

^

in (VII. 18) and since

for the Gaussian measure is

(

-norm preserving on

implicit positive

functions, we expect that as can be checked by explicit computation. Proof.

Let

Then, by either explicit computation or the semi-

group property for the kernel

of (1.38) (i.e.,

(VII. 19)

To prove (VII. 18), we need only consider the case since both sides define Gaussian measures. Thus, using

we need only prove that for

I

(VII.20)

222

DIRICHLET BOUNDARY CONDITIONS

agrees with the Dirichlet Green's function on the interval (VII.19), (VII.20) is independent of

and so the

By made symmetric

in s , t can be pieced together to form a single function . Since as s or

as

or

, for

one can show that

Moreover, using (VII.20) and the commutation

relation

it is easy to show that (see [90]; Section II. 6)

Thus F is the Dirichlet Green's function and so (VII.18) follows. THEOREM VII.7 ([91]).

Let

be given by (VII.17).

Let

be as

in Theorem

VII.5.

Then:

(1) For any (VII.21) converges (2) For any

in each -measurable

u:

(VII.22a)

(3) For any (VII. 22b)

Proof.

(1) By direct computation

_

. Moreover Thus, by

§VII.2. CONDITIONING, DIRICHLET STATES AND

Lemma 1.24, in

converges in boundedness plus

and so

223

i

converges

convergence implies convergence in

each (2) We need only consider

or equivalently

For this case, (VII.22) follows by combining Theorem VII.6 and Theorem VII.5. (3) Follows from (VII. 19). • §VII.2. Conditioning,

Dirichlet

States and

We begin by considering in detail an abstract theory developed by Guerra, Rosen and Simon [90] dubbed by them "the theory of conditioning" The reader may consult their paper for the motivation behind the name. The basic definition is: DEFINITION. Let

and

, be two Gaussian random processes in-

dexed by the same vector space V. Let respective covariances. We say that

i and is obtained

) be their from

by

condi-

tioning if and only if (VII. 23) for all When a relation of the form (VII.23) holds, there is a natural realization of

on the Q-space for

with the same measure. We caution

the reader that this natural realization is thus in the passive picture so he should shift gears from the active picture of the last section. Given a 1 relation of the form (VII.22), we can complete V in the norm and realize

as the "natural" Gaussian random process over this new

Hilbert space H. Since

(VII. 27)

224

DIRICHLET BOUNDARY CONDITIONS

we can find a unique positive symmetric operator, A, with

Of course

by (VII.24). Now consider the Gaussian random process . Clearly its covariance is

so that

presents a model for the process c . Henceforth

we write

)

for

and

for DEFINITION. If

is called an interaction, then

called the conditioned

interaction.

We'll write

The reason for this name is simple. If „

is

for

I

then

: (by the definition (1.32)). Thus, if are processes indexed by

and

and then (VII.25)

A word is in order about : : in (VII.25). In the passive interpretation of

as a variable on

Wick ordering (of

the Wick ordering in (VII.25) is the

I with respect to

active picture and thinks of

. But when one passes to an

as the field

with a new measure

dlx o A , then one should write (VII.25')

that is the Wick ordering is w.r.t. to the measure then

in the passive picture and so in the active picture.

. For example if

225

§VII.2. CONDITIONING, DIRICHLET STATES AND i

There is an extremely useful way of rewriting a conditioned theory. Let

so that

. Let

be the com-

pletion of H in the norm

(after quotienting out the vectors of zero A-norm) and let analogous object built on

,. Then K

realized as a subspace of K (by tions 1.7 and 1.8, we can realize

and we can realize

(

jection of ^v onto Thus for any

be the is naturally

On account of Proposias

and

simultaneously on

. If

, then

is the pro-

I for any

:

and more generally (VII. 26)

To see this note that

i (if all are realized in

not in their realizations of

in

i and

;

is independent

j so that:

so that using

(VII.26) follows.

To use (VII.26) we note that: LEMMA V I I . 8 .

( a ) For

any

(VII. 27)

226

DIRICHLET BOUNDARY CONDITIONS

(b) ( J e n s e n ' s inequality) (VII.28) f with

for any

Proof,

/ exp

(a)

by Holder's inequality so that (VII.27) holds,

(b) Suppose first that f > 0

and bounded. Then

for any n by (a) so that summing, (VII.28) holds. By the monotone convergence theorem, we can allow any if it is not bounded. Next suppose

1

even

for some c.

Then

so (VII.28) holds for f + c and so for f. Finally by employing the monotone convergence theorem, we can remove the

restriction.

Combining (VII.26) and Lemma VII.8, we see that: THEOREM VII.9 (Conditioning Comparison Theorem). In the picture

where

_

passive

' is given by (VII.25)

(VII. 29)

(VII.30) In the active

picture

where

is given by (VII.25'):

(VII.29')

(VII. 3 0 ' )

227

§VII.2. CONDITIONING, DIRICHLET STATES AND

Proof.

For example:

As a final result in the abstract theory:

THEOREM VII.10 (Conditioning Convergence Theorem). If obey

and if

,

all

then

(VII.31)

(VII.32) For any U with Proof. Since

in

in

and so in each

Thus we need only show that i . But from (VII.33a) (VII. 33b) if Theorem VII.9,

Now

in and

since

and by have uniform

228

DIRICHLET BOUNDARY CONDITIONS

_ -bounds

. From the boundedness of

_ -convergence we obtain Thus in each

in 1

convergence

) and so in

Now we can define the Dirichlet state in region active point of view so that

and

We return to the

is the new measure, but

same function as for the free field. :

I is the

will denote Wick ordering with

respect to DEFINITION.

normalized

for any

(VII. 34a)

(VII.34b)

(VII. 3 4c) Remark: In (VII.34a) we do not require that supp _ We set

a.e. w.r.t.

if

Thus for general

but allow arbitrary g. so that only

matters.

. For example (VII.34d)

Since the Dirichlet theory is a conditioned theory (VII.29'), (VII.30') are applicable and we see that for

(and in particular, for

Thus we can define:

if

is bounded or 1

if

is bounded),

CONDITIONING, DIRICHLET STATES AND

DEFINITION. The Dirichlet B.C. region,

229

field theory in a bounded

is given by

where

(VII.35) and the Dirichlet Schwinger functions

by: (VII.36)

More generally we can c o n s i d e r a n d b y

replacing

by Remark: We can now make clear what we mean by the statement that Dirichlet B.C. are "minimal", so long as

be any extension of

Then

a general theorem (see e.g., Kato [214; pp. 330-

333]) assures us that

Thus conditioning relations hold which assure, e.g., that

DEFINITION. The Dirichlet pressure, open

is defined for any bounded

by: (VII.37)

DEFINITION.

The Free pressure,

is defined for any bounded open

by

(VII.38)

230

DI RICH L E T BOUNDARY CONDITIONS

We also use

(VII.39)

(VII. 40) The real power of the theory of conditioning is shown by

THEOREM VII.11

(a)

for any

(b) If P is normalized, (c) If P is normalized, for any (d) For any disjoint open (e)

are disjoint,

then

Proof: (a) Since

this follows from the general conditioning com-

parison theorem. (b) By Jenson's inequality (Lemma VII.8b)

exp

= 1 if P is normalized. (c) By the theory of conditioning and the fact

by (VII.34d). (d) We can realize

and Thus so that

231

CONDITIONING, DIRICHLET STATES AND

Thus

and

so that

(d) follows,

(e) follows by ( c ) and (d). • and similarly for already seen in Section VI. 1 that monotone in namely

exists (since

with the more general limit). We also have:

P be a normalized polynomial. in I (resp t) for each fixed

is superadditive exists

of

is

we may replace the limit we studied in Section VI. 1,

THEOREM VII. 12 ([90]). Let

Proof,

We have

t (resp £).

and

(a) Since we can write and

Then:

as a disjoint union of translates

(plus a little extra border)

from which superadditivity follows.

232

(b) Since and

DIRICH L E T BOUNDARY CONDITIONS

is bounded above and below (by Theorem VII. 11(a), (b) this follows from (a) and the standard theory of subaddi-

tive functions (see e.g., Kato [214, p. 27]). We turn to the natural questions of proving vergence of

( r e s p . f o r

and of the con-

more general

in

Section VII.4 below. References

for Sections

VII.1,2:

Guerra, Rosen, Simm [90,91],

§VII.3. Half-Dirichlet Let

States and the HD Transfer Matrix

be an open bounded set. For any

with

compact,

we can define not only

where

Wick-ordering (and we are in an active picture). is absolutely continuous w.r.t.

Since and the

Radon-Nikodym derivatives in both directions are

so that

Our first goal will be

to prove that this remains true as the measure

which will then allow us to form We follow the treatment of

GRS [90]. The first thing we need to do is to rewrite of

using (1.20). The difference of

and

in terms thus enters

naturally and we begin with a preliminary study of this object. Since (VII.41)

for

obeys the distributional equation

233

§VII.3. HALF-DIRICHLET STATES

so that

away from

(by the elliptic regularity

theorem). Formally,

vanishes on

so formally

We thus single out: DEFINITION. We call an open region, as

or infinity (if

normal if for every is unbounded).

It is known that the interior of any Jordan curve is normal. DEFINITION.

THEOREM VII.13. Let (a)

be a normal region. exists

Then:

for all

(b) (c)

(d) (VII.42) In particular, const for a

independent constant and for d

(VII.43) large: (VII.44)

is also normal, then

Proof.

We first note the general fact that if f is continuous on a closed

bounded set S with

on S l n t and if

(resp

on S then f is subharmonic (resp. superharmonic) so that f takes its maximum value (resp. minimum value) on d S. (a) Since

for all

is

in • for

all x by the elliptic regularity theorem. Thus, in particular, the limit in question exists.

234

DI RICH LET BOUNDARY CONDITIONS

(b) Fix y. By (a),

s o that

= 0 on the open set

Thus the minimum of

is taken on implies that either e s i s ) or

But

vanishes on

(so that

since

by the normality hypoth-

is both negative and non-negative in a neighborhood of x.

(c) holds by the same argument as (b) if we note that

(d) Since

is positive and

its maximum when

takes

at which point

Thus

Since

is monotone in

The explicit bounds follow from Proposition V.23. (e) Fix y.

as a function

on

obeys

Moreover, when following the proof of (b),

By for all x and, in particular

for By (1.20b), we may write (VII.45)

where (VII.45) is intended in the sense of holding apriori when smeared with

On account of the bounds (VII.43), we conclude that if

we define: DEFINITION. We call a set bounded and for all

then:

log-normal if it is normal, open,

HALF-DIRICHLET STATES

THEOREM VII.14 ([90]).

For any log-normal

set

235

and any

P

where

Proof.

Using (VH.45), we can write

where

Then

by the log-normality of

and

Theorems V.2 and V.7. On account of the conditioning comparison theorem and We similarly fordefine exp can now DEFINITION. The Half-Dirichlet normal region,

B.C.

A is given by

Field Theory in a logwhere (VII.46)

and the Half-Dirichlet

Schwinger functions by: (VII.47)

236

DI RICH LET BOUNDARY CONDITIONS

Remarks: 1. The possibility of defining a Half-Dirichlet state depends heavily on the fact that

has only a logarythmic singularity for the free

field. Analogs of half-Dirichlet states probably do not exist in three or more dimensions. 2. At first sight, the Half-Dirichlet states seem unnatural but they are useful for the following reason. As due to

changes.

increases the part of This is not true for

This has important consequences in the application of GKS inequalities (see Section VII.5) and also considerably simplifies the theory of the Half-Dirichlet transfer matrix as opposed to the Dirichlet transfer matrix.

Fix

and let

be the Hilbert space of Theorem VII.5.

Define

(VII.48)

where

: : is

ordering, i . e . ,

for the region

is defined by (VII.45)

[or by following the general procedure of

Section V . l with

By mimicking the proof

of Theorem VII. 14, we see that

and exp

in the realization of

Thus follow-

ing the construction of Section V.3:

THEOREM VII.15.

is essentially

The Half-Dirichlet

self-adjoint

lie in

transfer matrix

and bounded below on

HALF-DIRICHLET STATES

We will let

denote its ground state energy, and

ground state. As one might guess,

DEFINITION.

and

237

its

is a transfer matrix where

Let

the objects when

THEOREM VII.16 (GRS [90]).

Then:

(i) (ii)

for all

where

This result clearly follows by mimicking the methods of Section V.4 once we have: PROPOSITION VII.17. For each in some

there exists

a positive

vector

so that: (VII. 4 9 )

where

is given by (VII.21).

238

DIRICHLET BOUNDARY CONDITIONS

where Proof.

Let

be given by (VII.21). We will first prove that

exists. For we fix

and let

(VII.51)

Then, by the

Feynman-Kac formula, and (VII.22a): where By (VII.22b),

independently of

and by Theorem VII.14, exists as holds. By definition: and so

for some

exists. independently is Cauchy in

since

Thus

so that so that (VII.51) (VII. 52) has "hypercon-

tractive" properties by mimicking the proof of Theorem V.10.

239

(VII.49) and (VII.50) now follow easily from (VII.22), (VII.51) and the fact that

Reference

in

as

for Section VII.3:

Guerra, Rosen, Simon [90], §VII.4.

In this section we prove two results of GRS [91], that that

and

We prove the latter result first as it is technically

somewhat simpler. The biggest technical complication involves different Wick orderings. We will use the phrase fixed interaction with ... Wick ordering to indicate (VII. 53)

(or its Hamiltonian analogue) where

obey the hypotheses of

Lemma V.6. Thus V will change as our notion of Wick ordering changes. The phrase

ordering is self-explanatory and we use L-space,

T-time and L,T-space-time-ordering to denote respectively Wick ordering relative to

LEMMA VII.18. Let

and

g 2 n be a function with support in

V L denote a fixed interaction with L-space ordering note the interaction with

ordering.

and let Let

V

de-

Then: (VII. 54)

Proof.

Let

ditioning comparison theorem (Theorem VII.9),

Then by the conis monotone in-

creasing in L and by the conditioning convergence theorem (Theorem

240

DIRICH LET BOUNDARY CONDITIONS

VII.10), Now, by the FKN formula, and the proof of Theorem VI.2 (b),

Thus

is monotone increasing and its limit is

LEMMA VII.19. Let

V be a fixed interaction of the form ordered.

Proof.

Then (see Fig, VII. 1)

Write

(VII. 55)

with L-space ordering and

the object whose limit appears in (VII.55). By Theorem

VII.16, slightly modified, its proof,

and by Lemma VII. 18 and By the conditioning comparison and conver-

gence theorems,

We claim that (VII. 56)

for given if

find L with Then for

so (VII.56) holds.

and

with

241

For each t, T find

so that

Thus

T-time Wick-ordered is where P(. T

is a poly-

nomial with the same leading term as P and lower order terms bounded by

for

for

and by C exp

(on account of Theorem VII. 13). Then:

where the first inequality uses the conditioning convergence theorem and the second an argument similar to that used in proving Theorem VI.4. By the bound Theorem V.9, and the

bound on the

lower order coefficients:

const.

independent of T. By the convexity of the coefficients of P and the

for all t, T with

in bound:

Thus

const.

242

DIRICH LET BOUNDARY CONDITIONS

so that

THEOREM VII.20 (GRS [91]). Fix

P.

Then

the free B.C. energy per unit volume.

Proof.

By Theorem VII. 16, we need only prove that (VII. 57)

Now, by Nelson's symmetry (see Figure VII. 1)

so that, by Lemma VII. 19: (VII. 58) By (VII.58) and the bound

we see

that for any

Fig. VII. 1.

243

so that

Next we note the existence of a linear upper bound This follows by rewriting

in terms of £-space ordered objects and

using the conditioning comparison theorem. Thus by mimicking the proof of Theorem VI.7 (which only depended on the linear upper bound and hypercontractivity)

for all

Thus, for each

T

s o that by (VII.58): for any T. Taking

(VII.57) results.

We now turn towards showing

The effect of Wick ordering

is more severe than in the last proof but on account of the conditioning comparison theorem, certain inequalities are easier. By the construction of Section V.3: THEOREM VII.21. where

The Dirichlet

Hamiltonian

(VII.59)

244

DI RICH L E T BOUNDARY CONDITIONS

(VII.60)

is essentially

self-adjoint

We will let

on

denote its ground state energy, a n d i t s

ground

state. We write

THEOREM VII.22.

Proof.

By the conditioning comparison theorem:

so that

Moreover, by mimicking the proof

of Theorem VII. 16 (a): (VII.61)

The whole problem then is that the interaction in

is space and time

Wick-ordered and not just space ordered. Write this space time Wickordered object as

and define

by

(VII.62)

Then by the Schwarz inequality for the measure

245

Thus by (VII.61),

so that the theorem follows if we can prove that

(VII.63)

To prove (VII.63), we first use the conditioning comparison theorem to replace

and then an argument identical to that in the

second half of the proof of Lemma VII. 19. Remark: The above argument is related to an argument of Fisher-Lebowitz [47] in their study of the independence of classical gas pressures on boundary conditions.

LEMMA V I I . 2 3 .

Fix

where the Wick ordering is in the time direction.

Then (VII. 64)

where

ordered.

248

DI RICH L E T BOUNDARY CONDITIONS

Reference

for Section VII.4:

Guerra, Rosen, Simon [91],

§VII.5.

Bounds for Half-Dirichlet

States

In this section we wish to discuss

Bounds for Half-Dirichlet states.

The key will be to treat them in Frohlich's formulation (see Section VI.5) and to employ certain consequences of the GKS inequalities of the next chapter. For the case of

(or more generally

one can also prove Frohlich's bounds for the Dirichlet states but we will not give details.

bounds for Dirichlet states await the development of

a transfer matrix treatment of Dirichlet states. One can also give a Markov proof of the

bounds for HD states by mimicking [181].

Given a bounded region

and g with support in

we define the

free B.C. and Half-Dirichlet B.C. Schwinger generating functions by

For application of Minlos' theorem, we are especially interested in for

(real valued) but for Frohlich's bounds

for

such g enters most naturally. In Sections VII.3, 5 we will prove:

PROPOSITION. For any P of the form

even and any

and in (a) (b) (c)

all bounded all bounded bounded.

§VII.S.

249

BOUNDS FOR HALF-DIRICHLET STATES

We thus have:

THEOREM VII.27 (Frohlich's Exponential Bounds; Half Dirichlet States [51]). Fix

P a semibounded polynomial.

(a) For each

Then, for all bounded

A:

c(f) with

there exists

(VII.67a)

(VII.67b) with supp

for all (b) For each

is independent of there exists

a d(f) with (VII.68)

with supp

for all (c) For each

there exists

a constant

with (VII.69)

with supp

for all for all

Proof.

and

t.

We first note that Frohlich's bounds (VI.54, 55, 56) readily extend

to functions which are absolutely values of

functions. Moreover, if

f is real-valued,

by (a) of the Proposition.

Thus (VII.70)

250

DIRICH LET BOUNDARY CONDITIONS

Since, for complex-valued

f, (VII.71)

(VII.67a), (VII.68) and (VII.69) follow respectively from (VI.54a), (VI.55) and (VII.56), and the following consequence of (b), (c) of the Proposition; if

then: (by (c)) (by (c)) (by (b))

To prove (VII.67b) we use the remark following Theorem VI.22B and first prove that

Next we note as above that for f real-valued

so that (VII.67b) follows from Thus by Theorem VII.27 and Theorem VI.24: THEOREM VII.28. Fix supp

P. Then for any

and all I with

§VII.5.

BOUNDS F O R H A L F - D I R I C H L E T STATES

THEOREM VII.29 (Frohlich's Let

P be a fixed polynomial. (a) There exists

an

251

Bounds for Half-Dirichlet States [51]). Then for all bounded

A:

norm so that for all

with

support in A: (VII.72) (b) There exists

an

norm so that for all

f, g with

supp (VII.73) Moreover, if

then

(c) For a suitable

with support in

(VII. 74)

Proof.

By the proposition (as in the last theorem): (by (a)) for I, t suitable (by (c))

(by (b)) So (a), (b) fo'low from their free B.C. analogs (VI.57) and (VI.59). To prove (c), we first note that there are Half-Dirichlet 77-bounds, for

and

252

DIRICH LET BOUNDARY CONDITIONS

because the functions

of (VII. 10) obey

O.N. basis for

are an

Thus, as in the proof of Theorem VI.20:

so that (VII.14) follows by mimicking the proof of (VI.60). References

for Section VII.5:

Frohlich [51], Simon [181].

§VII.6. Half-Dirichlet

States for the Hoegh-Krohn

Model

We now wish to briefly develop the necessary estimates to assure us that we can define Half-Dirichlet States in the Hoegh-Krohn model and that the corresponding Schwinger functions are not identically zero. By (1.20a), if

: : is

ordering: (VII.75)

Thus:

THEOREM VII.30. Let with

be a finite measure with support in

For bounded

let (VII.76)

Then

Proof.

Since

(by Theorem VII. 13), (VII.75) implies that

253

HALF-DIRICHLET STATES-HOEGH-KROHN MODEL

with

Thus by the conditioning comparison theorem and

Theorem

Since

(b) follows.

Remark: This result appears (in a different form) in Albeverio-Hoegh-Krohn [2]. DEFINITION. The Hoegh-Krohn model Half Dirichlet measure for region A and weight v is given by:

The Hoegh-Krohn model H.D. Schwinger functions are given by

PROPOSITION VII.31. Let supp

Proof.

Then, in the Hoegh-Krohn

be non-zero functions in

with

model:

By direct computation

so Since Reference

is a positive (not identically zero a.e.) function. a.e. so that for Section VII.6:

Albeverio-Hoegh-Krohn [2].

is strictly positive.

CHAPTER VIII THE LATTICE APPROXIMATION AND ITS CONSEQUENCES We now turn towards controlling the infinite volume limit of the cutoff Schwinger functions. Since there is an analogy between statistical mechanics and Euclidean field theory, it is natural to ask what methods are available for controlling the infinite volume limit of the correlation functions there. There are three [154]: (a) The transfer matrix method. This is restricted to onedimensional systems or at least to going to infinity in only one direction. We have already seen in Section V.4 how to extend the method to Ρ(φ) 2 -models; in particular, this method completely solves the problems for one-dimensional theories (anharmonic oscillators). (b) High temperature and low density expansions. The analog of high temperature is small coupling constant and of low density a large linear term in P (i.e., Q(X) — μΧ for μ large). Thus the ideas of GIimm-Spencer, Glimm-Jaffe and Spencer to which we turn in Chapter X can be viewed as the translation of the ex­ pansion methods of statistical mechanics to the P(^>) 2 -model. (c) Correlation Inequalities. For a very specialized class of systems which in some sense are "ferromagnetic", there is a powerful method for controlling the infinite volume limit with­ out restriction on temperature (coupling constant). The P(