From Perturbative to Constructive Renormalization [Course Book ed.] 9781400862085

Table of contents :
Contents
Acknowledgements
Part I. Introduction to Euclidean Field Theory
Chapter I.1 The Ultraviolet Problem
Chapter I.2 Euclidean Field Theory. The O.S. Axioms
Chapter I.3 The Φ4 Model
Chapter I.4 Feynman Graphs and Amplitudes
Chapter I.5 Borel Summability
Part II. Perturbative Renormalization
Chapter II.1 The Multiscale Representation and a Bound on Convergent Graphs
Chapter II.2 Renormalization Theory for Φ44
Chapter II.3 Proof of the Uniform BPH Theorem
Chapter II.4 The Effective Expansion
Chapter II.5 Construction of “Wrong Sign” Planar Φ44
Chapter II.6 The Large Order Behavior of Perturbation Theory
Part III. Constructive Renormalization
Chapter III.1 Single Scale Cluster and Mayer Expansions
Chapter III.2 The Phase Space Expansion: The Convergent Case
Chapter III.3 The Effective Expansion and Infrared Φ44
Chapter III.4 The Gross-Neveu Model
Chapter III.5 The Ultraviolet Problem in Non-Abelian Gauge Theories
References and Bibliography
Index

Citation preview

From Perturbative to Constructive Renormalization

Princeton Series in Physics edited by Philip W. Anderson, Arthur S. Wightman, and Sam B. Treiman Quantum Mechanics for Hamiltonians Denned as Quadratic Forms by Barry Simon The Many-Worlds Interpretation of Quantum Mechanics edited by B. S. DeWitt andN. Graham Homogeneous Relativistic Cosmologies by Michael P. Ryan, Jr., and Lawrence C.Shepley The Ρ(Φ)2 Euclidean (Quantum) Field Theory by Barry Simon Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann edited by Elliott H. Lieb, B. Simon, and A. S. Wightman Convexity in the Theory of Lattice Gases by Robert B. Israel Works on the Foundations of Statistical Physics by N. S. Krylov Surprises in Theoretical Physics by Rudolf Peierls The Large-Scale Structure of the Universe by P. J. E. Peebles Statistical Physics and the Atomic Theory of Matter, From Boyle and Newton to Landau and Onsager by Stephen G. Brush Quantum Theory and Measurement edited by John Archibald Wheeler and Wojciech Hubert Zurek Current Algebra and Anomalies by Sam B. Treiman, Roman Jackiw, Bruno Zumino, and Edward Witten Quantum Fluctuations by E. Nelson Spin Glasses and Other Frustrated Systems by Debashish Chowdhury (Spin Glasses and Other Frustrated Systems is published in co-operation with World Scientific Publishing Co. Pte. Ltd., Singapore.) Weak Interactions in Nuclei by Barry R. Holstein Large-Scale Motions in the Universe: A Vatican Study Week edited by Vera C. Rubin and George V. Coyne, S.J. Instabilities and Fronts in Extended SystemsfryPierre Collet and Jean-Pierre Eckmann More Surprises in Theoretical Physics by Rudolf Peierls From Perturbative to Constructive Renormalization by Vincent Rivasseau

From Perturbative to Constructive Renormalization

Vincent Rivasseau

Princeton Series in Physics

PRINCETON

UNIVERSITY

PRINCETON,

NEW

PRESS

JERSEY

Copyright © 1991 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford All Rights Reserved

Library of Congress Cataloging-in-Publication Data Rivasseau, Vincent, 1955— From perturbative to constructive renormalization / Vincent Rivasseau. p. cm. — (Princeton series in physics) Includes bibliographical references and index. ISBN 0-691-08530-7 1. Renormalization (Physics) 2. Quantum field theory. 3. Perturbation (Mathematics) 4. Constructive mathematics. I. Title. II. Series. QC174.17.R46R58 1991 530.1'43—dc20 90-49514

This book has been composed in Times Roman Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton University Press, Princeton, New Jersey 10 9 8 7 6 5 4 3 2 1

A l' unique étoile

Contents

Acknowledgements Part I Introduction to Euclidean Field Theory 1.1 The Ultraviolet Problem 1.2 Euclidean Field Theory. The O.S. Axioms 1.3 The Φ 4 Model A Introduction B The Gaussian Measure C Cutoffs D The Lattice Regularization E Connected Functions, Vertex Functions 1.4 Feynman Graphs and Amplitudes A Graphs B Amplitudes C Trees D Parametric Representation 1.5 Borel Summability Part II Perturbative Renormalization II.l The Multiscale Representation and a Bound on Convergent Graphs II.2 Renormalization Theory for Φ*

ix 1 3 15 23 23 24 27 33 34 37 37 42 48 50 54 57 59 74

viii

Contents

II. 3 Proof of the Uniform BPH Theorem A The biped-free case B The general case, with bipeds II.4 The Effective Expansion II. 5 Construction of "Wrong Sign" Planar Φ\ II.6 The Large Order Behavior of Perturbation Theory Part III Constructive Renormalization III. 1 Single Scale Cluster and Mayer Expansions A The cluster expansion B The Mayer expansion C Further topics D The φ\ theory 111.2 The Phase Space Expansion: The Convergent Case A The vertical expansion and convergent polymers B Overview of the domination problem C Proof of Theorem III.2.1 111.3 The Effective Expansion and Infrared Φ\ A Model and results B Overview C The cluster and vertical expansion D The Mayer expansion and the definition of counterterms E The functional bounds F The behavior of the effective constants G The inductive choice of mr = 0 H Analyticity and Taylor remainders I Weak triviality 111.4 The Gross-Neveu Model A Two dimensions B Three dimensions III. 5 The Ultraviolet Problem in Non-Abelian Gauge Theories A Introduction B The model C Computation of the counterterms due to the ultraviolet cutoff D Perturbative results E The positivity and domination problems F The Gribov problem References and Bibliography Index

90 90 100 111 123 144 169 171 173 186 194 199 210 210 221 229 241 241 244 252 253 262 264 266 267 268 272 272 282 289 289 291 294 301 305 309 321 333

Acknowledgements

The material presented in this book grew out of common work with C. de Calan on uniform bounds for renormalized Feynman amplitudes and with J. Feldman, J. Magnen and R. Seneor on phase space analysis applied to perturbative and constructive renormalization theory. They are in this sense coauthors of these notes. I thank them deeply, together with all my other collaborators and colleagues, for the joy, stimulation and sharings of day to day work. I am particularly indebted to J. Lascoux, A. Wightman and E. Speer for guiding my first steps in this domain. For many years I have benefited from the friendly encouragement of Professors A. Wightman and F. Dyson. I hope to thank them by this small testimony to the inspiring beauty of the subject that, in different ways, they pioneered. I thank D. Ruelle for the first suggestion to write such a book, and I apologize to him and many others for taking such a long time to complete it. I thank the Universities of Princeton and of Belo-Horizonte for support and for allowing me to experiment with the teaching of the material of this book (in this respect I thank more particularly Profs. A. Wightman, M. O'Carroll and R. Schorr, and all the students which attended the corresponding courses for their help). Support of

X

Acknowledgements

the Institute for Advanced Study and of my home institutions, the CNRS and the Ecole Polytechnique, is also gratefully acknowledged. I thank C. Kopper and A. Wightman for their careful critical reading of the manuscript. Finally I thank Marie-France, Jean-Noel, Christian, and Marie, Annie and Jacques, Francois and Therese; their affection was also essential for this book to exist.

PART I

INTRODUCTION TO EUCLIDEAN FIELD THEORY

Chapter 1.1

The Ultraviolet Problem

Quantum field theory is an attempt to describe the properties of ele­ mentary "point-like" particles in terms of relativistic quantum fields. It is now widely believed to offer a coherent mathematical framework for relativistic models (like the "standard U(I) χ SU(T) χ SU(3) model"). These models include all the particles and interactions observed up to now except gravity. Therefore, together with general relativity, field theory is the backbone of our current understanding of the physical world. In the future a new, more unifying framework may be adopted, like the currently promising superstring theory, which is a relativistic and quantum description of extended one dimensional objects instead of point-like particles; nevertheless even in this case it is extremely likely that field theory will remain important in many situations, just as classical mechanics is still today. This situation is relatively recent. Until the 70's the very state­ ment that quantumfieldtheory might provide a coherent mathematical framework at all was not widely accepted. The main doubts on the mathematical consistency of quantumfieldtheory were due to the per­ sistence of ultraviolet problems (and to the lack of successful models for strong interactions: QCD, the present field theory of strong inter­ actions, did not exist). Let us sketch what these ultraviolet problems are and why they are important.

4

I: Introduction to Euclidean Field Theory

An ultraviolet problem is one which is due to the existence of arbitrarily small length scales, or equivalently of arbitrarily large frequencies in the Fourier analysis of a theory. Such problems are inherent to the formalism of quantumfieldtheory, because it is a crucial assumption that thefieldslive on a continuous space time. One might wonder whether this continuity condition has anything to do with physics and whether the whole problem is not a mathematical artifact. After all it is reasonable to expect that space time will conserve its smoothness only until the Planck scale, where quantum aspects of gravitation might distort it significantly. This Planck scale might provide a physical ultraviolet cutoff; this is what seems to occur in superstring theories where it is conjectured that at least in perturbation theory, there are no ultraviolet divergences. However the Planck scale is much higher than the typical scales that field theory tries to describe, and is completely inaccessible to direct experiments. In fact the most compelling reason for which we are interested in the continuous formulation of field theory is the same for which we are interested in the thermodynamic limit of statistical systems. In statistical mechanics this limit corresponds to systems of infinite volume. We know that in nature macroscopic systems are in fact finite, not infinite, but they are huge with respect to the atomic scale. The thermodynamic limit is an adequate simplification in this case, since it allows one to give a precise mathematical content to the physically relevant questions (like dependence of the limit on boundary conditions, existence of phase transitions etc ). Since a limit has been taken, the power of classical analysis may be applied to these questions. It would be much harder and less natural to try to define the analogous notions for a largefinitesystem, just as it is difficult and often inappropriate to make discrete approximations to some typically continuous mathematics like topology. From this point of view the ultraviolet problem appears central and inescapable infieldtheory. A limit has to be performed; if this limit were not to exist, the corresponding mathematical formalism would be of little interest. Historically quantum field theory was plagued by two successive ultraviolet "diseases" which raised doubts on the existence or consistency of the ultraviolet limit. In both cases the situation looked bad for many years until a way out of the crisis was found. The first and most famous ultraviolet disease has been recognized almost since the birth of quantum field theory. It is the presence of divergences due to

5

1.1: The Ultraviolet Problem

the integration over high momenta in the loops of Feynman integrals. In the φ\ theory which will be discussed soon, one of the simplest of these divergences is the divergence of the second order graph which we call the "bubble" (see Fig. 1.1.1). By momentum conservation the amplitude for this graph is only a function of k = k\ + kj. In Euclidean space, this amplitude (apart from combinatoric coefficients discussed later) is given by the integral: μ 2 2 2 2 / (p + m )[(p + fc) + m ] '

(1.1.1)

which diverges logarithmically for large values of p. (Similar di­ vergences occur of course in the more physical theory of quantum electrodynamics). Around 1950, this disease was cured by the invention of perturbative renormalization by Feynman, Schwinger, Tomonaga, Dyson and others (see e.g., [DyI]). Basically it amounts to a redefinition of the physically observable parameters of the theory which pushes the in­ finities into unobservable "bare" parameters. It took more than a decade to put this perturbative theory of renormalization on a com­ pletely firm mathematical basis. Roughly speaking, the main result states that theories which are renormalizable from the naive power counting point of view can indeed be renormalized without changing the formal structure of the Lagrangian. More precisely one can replace the bare parameters of the Lagrangian by formal power series in the renormalized parameters (usually the coupling constant), so that the resulting perturbative expansion in the renormalized coupling is finite to all orders, as a formal power series. We call this important theorem the BPH theorem (Bogoliubov-Parasiuk-Hepp [BP][HeI]) although as usual it incorporates a lot of former work and was followed by impor­ tant extensions and refinements. It was somehow a surprise to discover that this theorem, developed for quantum electrodynamics or the φ* model, remained also true for non-Abelian gauge theories [HHl][LZ], in which case it is highly non trivial to check that the counterterms

R, and large external momenta. Let m be the mass of the particles. One finds (in Euclidean space): S4n'R(k) ^

0

O eSDfc log(k/m)]"-1,

(1.1.2)

where gR is the renormalized coupling constant and βι is a numerical coefficient (for single component m

Figure 1.1.2 (Left) The triangular graph. (Right) The triangular graph with a chain of bubbles attached.

1.1: The Ultraviolet Problem

7

These contributions are not summable over η (since they add up with the same sign they are also not Borel summable). Therefore the renormalon problem appears as a difficulty in summing perturbation theory. However one might also consider the asymptotic behavior in k of the bare (unrenormalized) 4-point function: 4

5„ f (0) - ^ 0 O (-gBr\fii log(k/m)]"-\

(UA)

where gB is the bare coupling constant. Since in the theory of perturbative renormalization this function should be the counterterm of the theory, i.e., the difference between the renormalized coupling gR and the bare coupling ge, one gets the formula: OO

Therefore if ge > 0, H m ^ 0 0 gR = 0 (no matter how gB is chosen as a function of k). This is the phenomenon of "charge screening" called also the triviality problem for inserted: .

(1.2.12)

This formula is difficult to justify because the usual argument, based on the so called "interaction picture" is wrong: by a theorem of Haag, there is no way to relate the free field

W. For a discussion of this and a full mathematical presentation of these axioms we refer to [OS] [SiI]. The important result is: Osterwalder Schrader reconstruction theorem Any set of functions satisfying OS1-OS5 determine a unique Wightman theory whose Schwinger functions form precisely that set. From now on we forget the Minkowski space and all the background briefly reviewed above. We always assume that we are in a d dimensional Euclidean space Ud. Our starting point is the Euclidean Feynman-Kac formula; our goal is to makerigoroussense out of it, and to check the validity of Osterwalder-Schrader axioms for the corresponding Schwinger functions.

Chapter 1.3

The 4 Model

A

Introduction

The simplest interacting field theory is the theory of a one-component scalar bosonic field

2 we cannot define directly the fourth power of φ in (1.3.1) and we have to build the in­ teracting measure (1.3.1) by a limiting process. In one way or another this means that we have to introduce cutoffs. Then the exact regularity properties of the sample fields for the Gaussian or the final interacting measure (they may not be the same) will not be very important for the construction of the theory. This is because typically in this book we are going to use very smooth cutoffs which correspond to Gaussian mea­ sures with very regular sample fields. Then we study the convergence of the corresponding moments (the Schwinger functions) as the cutoffs are removed and eventually we use Osterwalder-Schrader axioms to check that a sensible theory has been obtained, but for that we do not need support properties of the underlying interacting measure.

C

Cutoffs

Let us define a smooth ultraviolet cutoff first. One of the most con­ venient is the "α-space cutoff." It may also be called a heat kernel regularization of the propagator, or a regularization of the proper time of the path in the Wiener representation of the propagator as an integral over random paths. It suppresses in a smooth way the high frequencies in (1.3.2). To define it we write the α or parametric representation of

28

I:

Introduction to Euclidean Field Theory

the propagator: (1.3.5)

(1.3.6) (remark that (1.3.6) is well defined except at coinciding points x = y, where for d > 2 it is a divergent integral). We suppress the contributions of parameters a less than K and get: (1.3.7)

(1.3.8) When K —• 0, one recovers the full propagator. In contrast with (1.3.6), (1.3.7) is well defined everywhere. The corresponding bilinear form on S(Rd) is BK(f,g) = ( / , [l/(27r) r f ][e-^ + " l 2 V(p 2 + m2)]g), where / and g are the Fourier transform o f / and g. It satisfies obviously the hypothesis for Minlos theorem, but the corresponding Gaussian measure d/xK has much better support properties than dp. It is in fact supported with probability one by smooth C°° functions. We can get an intuitive understanding of why this is true by remarking that in the naive formulation (1.3.1) the inverse of the covariance appears, so that for functions whose Fourier transform does not decay exponentially fast (at least approximately as e~Kp n) the corresponding exponential quadratic factor in (1.3.1) blows up and the measure is 0 for such fields. But fields whose Fourier transform decays that fast have infinitely many derivatives. Of course this is not completely correct because (1.3.1) is only a formal guide. To make this idea more precise one can follow the strategy of [CL], and prove: Theorem 1.3.2 The support of d\iK is made of distributions which are locally smooth C°° functions and which grow at infinity as y^ogjxf. More precisely

1.3: The4Model

29

and if we call also n the restriction of we have fiK(SFCd) = 1, where SFCd, the set of Sample Fields with Cutoff in dimension d, is defined as

(Remark that SFCj, our best estimate for the support of dfiK, is not a vector subspace of C°°). Proof In what follows, the value of K is fixed (we may suppress the corresponding subscript to simplify the notations). In [CL] it is shown that if the propagator C(x, y) is Holder continuous with some exponent 2a then the sample paths are in particular Holder continuous for all exponents a' < a (in fact logarithmic local behaviors are also studied in [CL]). It is also shown that for a Holder continuous translation invariant propagator C(x - y) with C(0) = 1 and C(x - y\d bounded, then the behavior at infinity of the sample paths is in more precisely it is exactly as specified in the definition of the set SFCj. Hence the main difference between our Theorem 1.3.2 and the results of [CL] is in the local smoothness; we start with a cutoff propagator CK which is not only Holder continuous but in fact smooth, and we want to prove that the resulting sample fields are smooth. We give now an outline of how this can be done using the following theorem of [Ree], which is in fact a corollary of the proof of Minlos theorem [Hid]. Theorem 1.3.3 Let £ be a nuclear space, djx a measure on E*, c its characteristic function. Let (, )„ be a continuous inner product on E and H-„ be the completion of E with respect to this scalar product. Suppose that c is continuous on H-„. Let T be a Hilbert Schmidt operator on H-„ which is one to one, and such that E c Ran T, T~l(E) is dense in 77_„, and the map T~l : E —> //_„ is continuous. Then the support of dfi is in fact contained in i w h e r e is the dual of H-n. We apply this theorem to the scalar product where (,) is the regular L2 product, and P = -A+m2. Pm is a unitary map from Ho = L2 into Hn, the completion of 5 with respect to (,)„. Our covariance is continuous on H-„ for any n since we have CK(f, g) = which is convergent on H-„ and bounded by . We apply the theorem with

30

I:

Introduction to Euclidean Field Theory

where and Then T a a is Hilbert Schmidt on H-n because T0 = p~ Q~ is Hilbert-Schmidt on Ho, since it is an integral operator with kernel in L2(U2d). (This is because is integrable in p and Q~2a is integrable in jc). Furthermore T satisfies the hypotheses of the theorem. The dual of H-n is the space Hn of functions which are sent to L? by the differential operator P"n. These functions are n times weakly derivable, which means that they areat least of class Cn~l. Therefore dfiK has support on the set Since this is true for any n, we conclude that dfiK has support on smooth C°° functions. This together with the results of [CL] completes the proof of Theorem 1.3.2. There is an other point of view which is even more enlightening. Since the covariance with cutoff is just the multiplication in p space of the standard covariance by the function one can consider that the corresponding Gaussian process is the same as the standard one without cutoff but realized on a space which is obtained by convoluting the standard sample distributions with a fixed deterministic kernel, namely the Fourier transform of . In this point of view this convolution automatically smears out the rough distributions on which the measure with no cutoff was supported and make them C°°. By Theorem 1.3.2 multiplications are well defined on the support of dfiK and derivatives can be taken in the ordinary sense. Every polynomial in the field, like q>4 has a meaning. However the fields are not bounded and they are not typically in LA, so that in order to make sense out of (1.3.1) we need still to introduce an infrared or volume cutoff. The simplest thing is to integrate the interaction only in a finite volume box A, usually some ^-dimensional cube. In this way we obtain the functional integral (1.3.9) This is the theory with ultraviolet cutoff and finite volume interaction, and (1.3.9) is now a well defined formula for complex g such that Reg > 0. Indeed by Theorem 1.3.2 above the exponential is a well defined function on the space of smooth functions on which dnx.K is supported; this exponential is bounded by 1, hence it is in provided that we can check that it is measurable. For that the essential step is to check that on the support of d/iK the evaluation at a point

4

1.3: The Φ Model

31

φ -» φ(χ) is measurable for all x; then any power of φ evaluated at a point will be measurable, and integrals of such powers on a compact domain will be also measurable because they are limits of correspond­ ing Riemann sums and the limit of a sequence of measurable functions is measurable. To check that the evaluation at a point is a measurable function, d we can for instance introduce some countable approximation to U , like the η dimensional dyadic numbers Ifl considered in [CL], and 0 let X be the space of all real valued functions φ on D . We consider the Gaussian measure djx on X with covariance C defined by (1.3.7) restricted to dyadic values. This is a Gaussian measure on the natural dyadic cylinder σ-algebra (the smallest one for which, for all dyadic x, the evaluation functions φ —• φ(χ) are measurable). Because there is a natural identification between continuous functions on M.d and their restriction to dyadic values we can find a corresponding isomorphism between the Gaussian measure on X and the true one, and the evalu­ ation at a point in the true problem inherits the measurability of the evaluation at a point for the dyadic problem [CL]. Still an other sim­ pler way of reasoning (already mentioned above) is to consider the measure with cutoff as the measure without cutoff but on the space of distributions convoluted with a smoothing kernel. Then the eval­ uation at a point, which is the same as applying the field to a Dirac test function, which is normally illegal, becomes the same as applying the unsmearedfieldto a smeared Dirac function, and this is directly a measurable function from the definition of the cylinder σ-algebra on S'. The well defined measure (1.3.9) is our usual starting point in this book, but similar measures with different cutoffs such as Pauli-Villars ultraviolet cutoffs of sufficient degree can be defined rigorously in the same way, and may be convenient in some particular situations. Typ­ ical Pauli-Villars cutoffs suppress the high momenta in a polynomial way, hence they correspond to samplefieldsof a given class Ck. Also there are situations where one may want to restrict not only the interaction but also the Gaussian measure to a finite volume. For this purpose one can still use afinitebox Λ, with some set of prescribed boundary conditions X for the Laplace operator. Periodic (X = p) or Dirichlet (X — D) boundary conditions are the most usual. The corre­ sponding Gaussian measure and normalization factors with ultraviolet and infrared cutoffs are noted Cx^x, άμχ^κ, Zx^x or simply C, άμ, Z, depending on context; we try to forget subscripts or superscripts when it seems harmless. We have to be careful however that these

32

I: Introduction to Euclidean Field Theory

volume cutoffs may spoil theorem 1.3.2; for instance if we define a finite covariance CA, (.X,}') = XA(X)C (X, y)xdy) with XA(X) the charac­ teristic function of Δ it is no longer continuous in χ or y, and theorem 1.3.2 does not hold. However again the best point of view is to consider the corresponding measure άμ^κ as applied to functions ΧΑΨ where φ is a sample field for άμκ, hence is smooth. Such products are no longer continuous functions because the characteristic function is not con­ tinuous; nevertheless multiplication can still be defined and functions like / Δ φ* are still measurable, so that the corresponding theory is still well defined. It is this theory which is the natural starting point in the Brydges-Battle Federbush cluster expansion scheme considered in section III. 1. Still an other possibility is to use a smooth function with compact support for the volume cutoff instead of a sharp characteristic function; this may be technically useful in some cases. However we must remem­ ber that compact support in Λ: space is not compatible with fast decay in ρ space and conversely, so that ultimately some compromise on the infrared and ultraviolet properties of the sample fields is unavoidable. Theories with cutoffs such as (1.3.9) violate some of the axioms; for instance the volume cutoff breaks Euclidean invariance and an ul­ tra violet cutoff such as (1.3.7-8) violates reflection positivity (axiom O.S.3 in section 1.2). To construct a satisfying theory we have to per­ form both the ultraviolet limit κ -» 0 and the infinite volume limit, also called thermodynamic limit. The thermodynamic limit consists in defining in fact a sequence volume cutoffs, typically a sequence of boxes An such that υ„Λ„ = IR'', and studying the limit of the theory in Λ„ as η -» co. In this book we do not pay to much attention to the particular shape of the boxes and the particular boundary conditions used because the models that we are going consider are in their "hightemperature phase," in which case the thermodynamic limit (when it can be constructed) turns out to be independent of the particular sequence of boxes and of boundary conditions chosen. Let us recall that although (after appropriate renormalization) the normalization Z and the unnormalized Schwinger functions Z · SN in a fixed box can have separately an ultraviolet limit, the thermody­ namic limit makes sense only for so called intensive quantities such as the pressure (1/|Λ|) log Z(A) or the normalized Schwinger functions SN (defined in (1.3.4)). The reader might conclude from this lengthy discussion that even functional integration with cutoffs is something quite complicated. There is however a conceptually simpler regularization scheme, which K

X

1.3: The 4 Model

33

provides a cheaper road to a well defined functional integral formula for the cutoff