WAVELETS AND RENORMALIZATION
APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
Vol. 1:
Wavelets: A...
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WAVELETS AND RENORMALIZATION
APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
Vol. 1:
Wavelets: An Elementary Treatment of Theory and Applications Tom H. Koornwinder, ed.
Vol. 2:
Approximate Kalman Filtering Guanrong Chen, ed.
Vol. 3:
Multivariate Approximation: From CAGD to Wavelets Kurt Jetter and Florencio I. Utreras, eds.
Vol. 4:
Advances in Computational Mathematics: New Delhi, India H. P. Dikshit and C. A. Micchelli, eds.
Vol. 5:
Computational Methods and Function Theory Proceedings of CMFT '94 Conference, Penang, Malaysia R. M. Ali, St. Ruscheweyh and E. B. Saff, eds.
Vol. 6:
Approximation Theory VIII Approximation and Interpolation - Vol. 1 Wavelets and Multi-level Approximation - Vol. 2 C. K. Chui and L. L. Schumaker, eds.
Vol. 7:
Introduction to the Theory of Weighted Polynomial Approximation H. N. Mhaskar
Vol. 8:
Advanced Topics in Multivariate Approximation F. Fontanella, K. Jetter and P. J. Laurent, eds.
Vol. 9:
Approximate Solutions of Operator Equations M. J. Chen, Z. Y. Chen and G. R. Chen
Vol. 10: Wavelets & Renormalization G.Batt/e
Series in Approximations and Decompositions - Vol. 10
WAVELETS AND RENORMALIZATION G. Battle Texas A&M University, USA
1II1»
World Scientific Singapore • New Jersey· London • Hong Kong
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World Scientific Publishing Co. Pte. Ltd. POBox 128. Farrer Road. Singapore 912805 USA office: Suite lB . 1060 Main Street. River Edge. NJ 07661 UK office: 57 Shelton Street. Covent Garden. London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Battle. G. (Guy) Wavelets and renormalization / G. Battle. p. cm. -- (Series in approximations and decompositions; vol. 10) Includes bibliographical references and index. ISBN 9810226241 (alk. paper) I. Renormalization group. 2. Wavelets (Mathematics) 3. Mathematical physics. I. Title. II. Series. QC20.7.R43B38 1998 530.14'3--dc21 98-28570 CIP
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Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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To My Parents Guy and Martha
To My Wife Jane
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Approximations and Decompositions
Mathematical physicists played a major role in the early development of the subject of wavelets in the decade of the eighties. The list of such pioneers includes A. Grossmann, I. Daubechies, A. Arneodo, M. Holschneider, T. Paul, and the author of this text , G. Battle, just to name a few . However, the motivation for Guy Battle to the study of wavelets was somewhat unusual: he needed a family of functions with certain special properties to solve the mathematical physics problems of interest to him and P. Federbush. This gave birth to the so-called Battle-Lemarie wavelets. The author 's motivation in writing this valuable book is also different from all the existing ones, in that he uses wavelets to introduce the renormalization group in condensed matter physics: the wavelet cutoff in the continuum, the canonical wavelet manifold, wavelet diagrams , wavelet refinement of the hierarchical model, etc. In addition , using wavelets, Guy Battle presents an elementary analysis of certain Euclidean field theory: wavelet estimates, combinatorics for summing over polymers, reducing from an infinite to a finite number of situations, assigning numeric factors , etc. What is valuable about this text is that it leads a reader with relatively minimal mathematical background to the frontier of research. I would like to congratulate the author for his outstanding contribution to this book series.
World Scientific Series in APPROXIMATIONS AND DECOMPOSITIONS EDITOR-IN-CHIEF :
CHARLES
K.
CHUI
Texas A&M University, College Station, Texas
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Preface Mathematical physics has always played a unique role in the history of mathematics . It is quite unique in the sense that it has been the origin of entire subjects in mathematics as well as an application of them . As a colleague once pointed out, it is often more appropriate to refer to mathematical physics as "physical mathematics." Examples of this regenerative impact on mathematics date back to ancient Greece, but at no time has the impact been greater than during the past one hundred years . The theory of Hilbert spaces grew out of an attempt to understand the integral equations arising from boundary-value problems in electrostatics, thermodynamics, and mechanical vibrations. The spectral theory of unbounded self-adjoint operators on a Hilbert space was stimulated by the development of quantum mechanics. To be sure, there are instances where a subject that was originally developed in a purely mathematical context has found application to physics . Riemannian geometry is an obvious case in point . Even in that case, however, the application to general relativity profoundly influenced the further development of geometry. Wavelet analysis provides another fascinating interface between physics and mathematics. It must be acknowledged that harmonic analysts, approximation theorists, and applied mathematicians - particularly signal analysts - were more instrumental in the explosive growth of the subject than were mathematical physicists. Indeed it was Y. Meyer who ignited that explosion. On the other hand, physicists had anticipated the existence of wavelets long before. When K. Wilson founded the subject of renormalization group analysis, he postulated the existence of well-localized, regular basis functions that were related to one another by dyadic scaling and discrete, scale-commensurate translation. Although he did not prove existence at the time, it became clear a number of years later that a mathematically rigorous implementation of the block-spin renormalization group transformation was actually related to a method for constructing such basis functions! In the meantime, P. Federbush had vigorously promoted the phase space analysis of problems in classical statistical mechanics and Euclidean field theory based on decomposing the field into a scale-coherent set of modes. His basis functions were mathematically concrete (albeit discontinuous) with an abundance of vanishing moments, and he introduced them several years before the mathematical community was interested in wavelets. This book is devoted to both the application of wavelet analysis to mathematical physics and the impact of mathematical physics on wavelet analysis. The major application is to a prototypal problem in constructive field theory that was originally solved ix
x
PREFACE
by J. Glimm and A. Jaffe. We describe an elementary cluster expansion of the weakly coupled Euclidean field theory based on a wavelet decomposition of the field. The convergence proof dates back to joint work of this author with P. Federbush, and an interesting by-product of this expansion is the replacement of Feynman diagrams by wavelet diagrams.
The twin goal is to describe the natural connection between wavelet analysis and the renormalization group formalism. The starting point is to itnroduce the renormalization group itself, and the point of departure is to show how the continuum picture of the lattice Gaussian fixed point leads to wavelets. A general method for constructing wavelets is contained in the variational calculation used to intenate - out the fluctuations in a renormalization group transformation. The type of wavelet produced by this machine depends on the type of averaging transformation on which the renormalization group transformation is based. This book is eclectic in its basic character, emphasizing as it does, other issues in addition to the renormalization group and phase cell analysis in Euclidean field theory. One significant theme is reflection positivity (alias Osterwalder-Schrader positivity). ~ l t h o u ~ this h condition has nothing to do with wavelets, it plays a vital role in an understanding of constructive quantum field theory, while in condensed matter physics it is the key to replacing the famous Peierls argument with the more general Frohich-Simon-Spencer argument. Another major theme is the combinatorial description of cluster expansions in both areas of physics. The literature on expansion methods is extensive, but the approach we choose appears in the joint work of this author with P. Federbush. The introduction of this approach is necessarily accompanied by a review of more standard material. Yet another important theme is the basic theory and construction of wavelets from a purely mathematical point of view. We review the multi-scale resolution analysis of S. Mallat and Y. Meyer as well as the renormalization-groupinspired variational method. We also discuss no-go theorems and other constraints on the existence and properties of wavelets. An additional theme is the accessibility of quantum field theory to mathematicians, facilitated by the Euclidean point of view. Introductions exploiting this pedagogical advantage have occasionally been written, and our own itnroduction to the subject is inspired in part by an expository article by P. Federbush entitled L'Quantum Field Theory in Ninety Minutes." It is a matter of mild interest that in none of our cluster expansions do we ever decouple covariances. In the context of a wavelet cluster expansion we choose the modes to be orthogonal with respect to the kinetic energy norm associated with the covariance, so the covariance is already diagonal with respect to the phase cell variables. Since the decoupling of covariances is second nature to an expert, our specific formalism is more a cultural choice than a technical advantage. We believe there should be at least one introduction to the phase space analysis of quantum fields which does not decouple covariances. In the context of classical equilibrium statistical mechanics for lattice spin systems, we consider high-temperature expansions, which control small perturbations of product states. For the Ginzburg-Landau spin model, we do not review the three dimensional lattice version of the celebrated Glimm-Jaffe-Spencer cluster expansion (which decouples covariances). Although the expansion converges in a regime different from the high-temperature regime, the model is just the lattice approximation of the
PREFACE
xi
Euclidean field theory, and the former regime is the weak-coupling regime from the latter point of view. We provide the bare bones of an introduction to the equilibrium statistical mechanics of classical spin systems and to the basic theory of phase transitions in that context. Our motivation is three-fold: (a) The renormalization group was originally conceived in an effort to analyze the critical behavior of such models. Moreover, the connection between wavelets and renormalization group analysis is a central theme for us, and the averaging of spin variables is very much involved in this connection. (b) An impressive application of reflection positivity is the Frohlich-Simon-Spencer argument. This argument is a proof that multiple phases exist at low temperature for a certain family of classical spin systems. This family is larger than that covered by the Peierls argument, in that systems with continuous internal symmetries are included. Within the scope of this book, our only interest in multiple phases is this application of reflection positivity. We do not review the Peierls argument. (c) The high-temperature analysis of classical spin systems with long-range interactions provides an excellent setting for the introduction of cluster expansions to the novice. Such expansions are an order of magnitude easier to control than any phase cell cluster expansion in Euclidean field theory, and yet they contain all of the important combinatorial ideas. All of the expansions are based on the widely accepted polymer formalism, which isolates the universal combinatorial notions. We give our own review of abstract polymer expansions in the interests of self-containment and also to adapt it to our needs. The Glirnm-Jaffe exposition of the abstract polymer theory is impossible to compete with, but in most of our expansions, the activity estimation and the entropy estimation cannot be separated so neatly. With regard to the combinatorial notions that are peculiar to a given expansion, the most important idea concerns ordered tree graphs, which arise from inductively defined cluster expansions. The only quantum fields considered in this book are scalar boson fields, as this case alone provides a full plate for any introduction to either Euclidean field theory or the renormalization group. We do not discuss either fermi fields or gauge fields. Chapter One is a fast-track introduction to constructive field theory which begins with a review of the most basic, abstract notions of puantum mechanics. Both the speed of light and Planck's constant appear as parameters in the early part of the exposition. Not until 51.8 do we set both constants equal to unity. The asterisk is our notation for complex conjugation, a s the more mathematical tradition of using the bar clashes with other notation we use. Also, our notation for the scalar boson fi$d is not quite standard. We denote the field operator in Minkowski space- time by Q( x ,t ) and the random field in Euclidean space-time by @(;,s), while the Euclidean lattice field is denoted by 4. Although this is a book for mathematicians, we have endeavored, as much as p o s sible, to follow the style of an old-fashioned theoretical physics book. While we derive formulas and prove many of the known results in the course of our discussions, there is no theorem-proof format. We also avoid arguments that would interrupt the flow of a discussion. Moreover, when the convergence proof for the wavelet cluster expansion of is reduced to checking a finite number of wavelet diagrams, we describe how the
xii
PREFACE
estimation can be done efficiently by examining a few diagrams and then we leave the remaining diagrams for the reader to check. Our referencing is done on two levels. At the end of each section we list the references most relevant to that section in alphabetical order, but we do not localize them in the discussion. Within each section an idea or method is often labeled by the name of the person or persons credited with it, but no reference numbers appear in the exposition. On the other hand, each chapter has an introductory exposition preceding the first section, and it contains reference numbers drawn from the total bibliography. The enumeration of references in the total bibliography involves a letter and a number, where for example, [G17] denotes the 17th reference among the G's, listed in alphabetical order. In every case of collaboration, we name the co-authors strictly alphabetically in the reference and assign to that reference the letter for the name of the first co-author. As large as our bibliography is, omissions are still inevitable, and we apologize for them. September 30, 1998
Acknowledgement I wish to thank Paul Federbush for the valuable experience of collaboration on research that is reviewed in this book. I am also grateful to Charles Chui for including this expository effort in his series of volumes on wavelet decomposition theory. A number of helpful suggestions by Brian DeFacio during the final weeks of preparation are also appreciated. I am greatly indebted to Robin Campbell for her technical skill and efficiency in the typing of this manuscript. Many thanks are also extended to the Texas A&M Mathematics Department for making its facilities available and to World Scientific for its remarkable patience. The encouragement of my parents during the writing of this book was very important as well. Finally, I thank my wife Jane for her endurance and understanding.
Contents .......................................
iu
1 Mathematical Sketches of Q u a n t u m Physics 1.1 Measurement and Dynamics . . . . . . . ..... .... 1.2 Imaginary Time Correlations . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Feynman-Kac Formula . . . . . . . . . . . . . . . . . . . . . . . .... 1.6 Quantum Mechanics of One Relativistic Particle . . . . . 1.7 Crisis in the Theory and Formalism . . . . . . . . . . . . . . . . . . . . 1.8 Canonical Formalism of Quantum Field Theory . . . . . . . . . . . . . . 1.9 Functional Integration for Quantum Field Theory . . . . . . ..... 1.10 Axiomatic Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . 1.11 The Constructive Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 A Tale of Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 The Need for Phase Cell Analysis . . . . . . . . . . . . . . . . . . . . .
1 6
Preface
11
15 23 30 36 40 47 56 64 73 82 97
2 Wavelets . Basic T h e o r y and Construction 105 2.1 The Balian-Low Theorem . . . . . . . . . . . . . . . . . . . . . . . 110 . . . . . . . . . . . . . . . . . . . 117 2.2 Lemari6 and Meyer Wavelets 2.3 Daubechies Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 . . . 131 2.4 Vanishing Moments . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Uncertainty Relations for (P)= 0 Wavelet States . . . . . . . . . . 135 2.6 &ther Constraints of Heisenberg Type . . . . . . . . . . . . . . . . . 140 2.7 Variational Construction . . . . . . . . . . . . . . . . . . . . . . . . . 150 2.8 Without Intrascale Orthogonality . . . . . . . . . . . . . . . . . . . . . 156 163 2.9 Chui-Wang Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Multi-Dimensional Wavelets . . . . . . . . . . . . . . . . . . . . . . . 169 3 Equilibrium S t a t e s of Classical Crystals 179 3.1 Classical Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.2 Phase Transitions and Ergodic Decomposition . . . . . . . . . . . . . . 194 3.3 Phase Transitions and Reflection Positivity . . . . . . . . . . . . . . . . 201 3.4 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.5 The Ginzburg-Landau Model . . . . . . . . . . . . . . . . . . . . . 220
CONTENTS
xiv
3.6 3.7 3.8 3.9 3. 10 3.11 3.12 3.13 4
5
Polymer Expansions . . . . . . . . . . . . .. Expansion for Nearest-Neighbor Interactions Inductive Interpolation. . . . . . . . . . . .. A Generalization of the Polymer Estimation. Unit-Vector Spins with Long-Range Coupling Combinatorial Properties of Interpolation Weights Scalar Spins with Long Range Coupling . . . . Extra liN! Factors in the Inductive Expansion ..
228 237 244 254 257 263 271
275
A Wavelet Introduction to the Renorrnalization Group 4.1 Averaging Transformations . . . . . . . . . . 4.2 The Renormalization Group Transformation. 4.3 Minimizers and Orthogonality. . . . . . . . . 4.4 Connection to the Continuum Wavelet Cutoff 4.5 Canonical Wavelet Manifold and ZF Transformation 4.6 Relevant, Marginal, and Irrelevant Parameters 4.7 Canonical Inverse-Limit Manifold 4.8 Wick Ordering and Linearized RG Analysis 4.9 The Second-Order RG Transformation. 4.10 Wavelet Diagrams . . . . . . .. . . . 4.11 The One-Loop-per-Scale Transformation 4.12 The Wilson Wavelet Recursion Formula 4.13 The Baker-Dyson-Wilson Hierarchical Model 4.14 The Hierarchical Model to Second Order . .. 4.15 Hierarchical One-Loop-per-Scale Transformation 4.16 Wavelet Correction of the Hierarchical Model
281
Wavelet Analysis of 'I>~ 5.1 The Cutoff - A Finite Set of Modes 5.2 Wavelet Estimates . . . . . 5.3 Inductive Operations . . . . . . . . . 5.4 Estimates on Numerical Factors. . . 5.5 Expansion Rules and Wavelet Diagrams 5.6 Organizing the Completed Expansion . 5.7 The Phase Cell Polymer . . . . . . . ' .' 5.8 Estimating the Activity I: Stability and Quartic Positivity . 5.9 Estimating the Activity II: Internal Combinatorics 5.10 Combinatorics for Summing Over Polymers . 5.11 Strategy for Number Divergence Cancellation 5.12 From Infinitely Many Cases to Finitely Many 5.13 Wavelet Diagrams for Cases .. . . 5.14 How to Assign Numerical Factors .
401
Bibliography . . . . . . . . . . . . . . .
290 297 303 310 318 325 332 340 348 356 360 369 376 382 389 394
407 412 417 423 431 443 451 456 464
471 478 483 492 504 513
WAVELETS AND RENORMALIZATION
Chapter 1
Mathematical Sketches of Quantum Physics One of the most interesting theoretical challenges of modern physics has been the combination of special relativity and quantum mechanics into a mathematically consistent theory. As revolutionary as the special theory of relativity was at the time of its development, it was also the crowning achievement of classical (deterministic) physics. On the other hand, quantum mechanics was much more radical, but its early (nonrelativistic) development was more ad hoc. Subsequent attempts to unify the theories eventually led to the formulation of quantum field theory. It has often been said that the most rapid advances in physics are made when there is an experimental contradiction to accepted theory. This has not always been the case for internal theoretical contradictions. However, quantum field theory was as much a response to experimental observation as to theoretical demands. Systems with infinitely many degrees of freedom were necessary to account for the creation and annihilation of particles in high-energy experiments. At the same time, the internal theoretical development of the subject has a fascinating history. Attempts to formulate the wave (quantum) mechanics of a single particle in relativistically covariant terms date back to the Schrodinger equation [Sl1-814J. 8chrodinger himself tried without success to develop a relativistic mechanics for de Broglie waves before deciding to study the non-relativistic case. The celebrated wave equation he subsequently derived has been immensely practical to the physical sciences. The first wave equation that successfully predicted a range of quantum-mechanical phenomena in the relativistic regime was the Dirac equation for the electron [D15J . The most serious problem with that equation, however, is that it also predicts the existence of states that have arbitrarily negative energies. Dirac proposed an ad hoc theory to recover the property of stability - a theory of the positron , which was identified as a hole in an "infinite sea" of otherwise-occupied negative-energy states [D16, D17J. Although this explanation ignored the interaction of an infinite number of negative-energy electrons with the positive-energy electrons, it was an ingenious application of the Pauli Exclusion Principle. Moreover, the mathematical structure for a quantum field theory 1
2
CHAPTER 1 . MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
of particles and anti-particles was suggested by the Dirac sea. It was recognized that a self-consistent relativistic quantum mechanics for a single particle was possible only under certain circumstances. Systems with infinitely many degrees of freedom were clearly more fundamental. Quantum field theory had already been developing from a different direction specifically as an attempt to explain the spontaneous emission of radiation. Since Maxwell's theory predicts that a classical electron orbiting a nucleus must emit radiation, spontaneous emission in quantum mechanics had a sound qualitative basis, but attempts to quantize the classical phenomenon were plagued with difficulties when one insisted on an interaction between a quantum-mechanical particle and a classical electromagnetic field. Even the more modest goal of realizing the radiation damping of the particle motion in quantum-mechanical terms had been problematical. Ultimately, the radiation field had to be treated as part of a coupled quantum-mechanical system, and therefore quantized as an infinite system. Dirac himself [Dl41 showed this approach to be powerful in describing emission and absorption processes a couple of years before his work on the relativistic electron. Heisenberg, Jordan, and Pauli were primary advocates of quantum field theory [H21, H22, J16-J181, but even they were surprised later by Dirac's "hole theory." From the beginning, formal calculations led to divergences. The basic approach was to perturb a free field theory of non-interacting particles with the quantum analog of a classical field interaction and expand basic physical quantities, such as elements of the scattering matrix, in powers of the interaction parameter. The terms in this perturbation theory reduce to multiple space-time integrals where the multi-argument functions are product combinations of two-point functions. These two-point functions are actually Green's functions corresponding to the types of fields involved - i.e., classical potentials between point sources, where the potentials are scalar, vector, etc., depending on the type of field interaction chosen. Products of local singularities in the integrands obviously result, and so there are divergent integrals t o be reckoned with. Without a physically acceptable way to avoid or subtract out these divergences, the perturbation theory is meaningless. From a technical standpoint, one regularizes the interaction first, so that all of the multiple integrals are convergent. Such a regularization is known as an ultraviolet cutoff, and the regularized integrals introduced in the interaction for cancelling the divergences at the perturbative level as the ultraviolet cutoff is removed are counter-terms. The requirement of relativistic invariance (in the ultraviolet limit) imposes restrictions on the structure of the counter-terms, but the most restrictive physical requirement is that these additional terms in the interaction must reflect only shifts in the parameters of the theory - albeit divergent shifts as the ultraviolet cutoff is removed. This scheme for removing infinities from the theory is known as renormalization, and as long as one obtains a well-defined quantum field theory with interacting particles in the absence of all regularization, the construction is perfectly valid. The ultraviolet-divergent parameter shifts only signify that perturbing a free quantum field theory with a given field interaction may not be a natural approach. However, even a renormalized perturbation series cannot converge for any non-zero value of the interaction parameter. The Taylor series was assumed to be asymptotic to the quantity expanded. During the early history of this program, it was Bethe [B32] who explained the celebrated Lamb shift with a mass renormalization of
the electron The next phase of development was initiated by Feynman [F36-F39] and Schwinger [S16-S22], who fully renormalized the quantum electrodynamics of electrons and positrons. The renormalized perturbation theory correctly calculated the elements of the scattering matrix within the margin of experimental error existing at the time. The Lamb shift and the anomalous magnetic moment of the electron were completely explained as well. This stunning scientific achievement gave considerable impetus to further investigations, especially of short-range interactions. Unfortunately, the success of perturbation theory could not be extended to nuclear interactions. A Taylor series is good enough for quantum electrodynamics because the interaction parameter (= fine structure constant) has a small experimental value roughly 11137. By contrast, the coupling constants (= interaction parameters) for nuclear forces are not small, so even with the standard assumption that the power series are asymptotic, such expansions are quantitatively useless in this regime. On the other hand, the severe restrictions imposed by quantum-field-theoretic properties alone on the scattering matrix provide a great deal of information about any interaction, whether it has been successfully modeled or not. The requirements of relativistic covariance, causality, and unitarity of the scattering matrix together with analyticity of the matrix elements as functions of energy and the internal symmetries of the interaction lead to dispersion relations. Gell-Mann, Goldberger, and Thirring first proposed deriving such relations and proving rigorous results [G34, G82]. This program produced the spectral representations of two-point functions for the various theories and firmly established the connection between spin and statistics already discovered by Pauli [Pg]. Moreover, it inspired the famous Lehmann-Symanzik-Zimmermann formalism, which expresses the scattering matrix in terms of time-ordered vacuum expectation values of the given . notion of regarding the scattering matrix as the most fundatheory [L20, ~ 2 1 1 The mental physical object - originally suggested by Heisenberg - was now superseded by the idea of using the vacuum expectation values as the starting point in extracting the physical information. In the meantime, there was a great deal of concern about the mathematical consistency of quantum field theories. It was proven by Gell-Mann and Low that the scattering matrix derived from quantum electrodynamics could not be unitary [G33]. This problem clearly indicated that the theory still had no solid foundation, so axiomatic quantum field theory became a major industry. Wightman and Haag each founded a school of thought introducing a mathematical structure with axioms reflecting the most essential properties of a quantum field - e.g., causality, relativistic covariance, positivity of the energy spectrum - and rigorously proving theorems based on those axioms. Stimulated by the Lehmann-Symanzik-Zimmermann formalism, the Wightman program was based on the properties of vacuum expectation values, which have come to be known as Wightman distributions. It is straightforward to verify that free quantum field theories (i.e., field theories for non-interacting particles) satisfy the axioms, so consistency of the axioms was established - at least from a "legal" standpoint. The goal of constructive quantum field theory was to construct interacting field theories that satisfy the axioms. Concurrent to these developments was the growing use of the functional integral approach, which dates back to the Feynman integral for non-relativistic quantum me-
4
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
chanics [F35]. Previous to his work in quantum field theory, Feynman had formally realized the kernel of the Schrodinger propagator in a potential as an integral over possible histories of classical particle motion. Originally motivated by an interest in formulating a least-action principle for quantum mechanics, he derived this infinitedimensional integral from products of phase factors, with each phase identified as a term in a Riemann sum approximation of the time-integral of the classical Lagrangian of the particle. The integral over trajectories is obtained in the refinement limit of the time partition used to approximate the propagator. At the formal level, this scheme can actually be extended to Lagrangians of systems with infinitely many degrees of freedom . It even applies to field Lagrangians, where the resulting functional integration is now over space-time field configurations instead of over particle trajectories. In particular, it provides a Feynman integral formula for the vacuum expectation values of a quantum field theory, and the non-perturbative nature of this formula is the most important advantage it has to offer. We have already explained the need to go beyond perturbation theory. Even a proof that the power series is asymptotic would have to involve non-perturbative estimation. From a measure-theoretic point of view, these functional integrals do not exist, and this limitation makes non-perturbative estimates hard to come by. Indeed, this problem was evident in the original non-relativistic Feynman integral over trajectories. It was Kac who suggested a study of this representation in imaginary time, where the measure theory is well-defined [KI] . As for the Schrodinger equation itself, one can interpret the imaginary-time version as the heat equation with a position-dependent heat source activated with an intensity proportional to the temperature itself. The resulting integral is rigorously controlled because the phase factors in Feynman's derivation are replaced by thermal decay factors. The representation of this perturbed heat kernel is known as the Feynman-Kac formula, and it actually reflects a Wiener process [W5]. It has led to many rigorous results in non-relativistic quantum mechanics. More generally, the replacement of the one-parameter unitary group representing the Schrodinger evolution of wave functions with the positivity-preserving contraction semi-group arising from analytic continuation to imaginary time is very useful because the latter quantity yields much more spectral information about the self-adjoint generator (= energy operator) . The idea of indirectly controlling a quantum field theory by analytic continuation to imaginary time first came up when Dyson showed that the renormalization prescription for an interacting field theory is unchanged if the space-time Green's functions generating the divergences in the perturbation series are all replaced by their imaginary-time analogs. There is interest in this result because the singularity of the Green's function of an elliptic operator is simpler than the singularity of the Green's function of a hyperbolic operator. Moreover, the perturbation series associated with imaginary time is Euclidean-covariant instead of Poincare-covariant. An early advocate of Euclidean methods was Schwinger [S25] , who studied the properties of the analytic continuation of an n-point Wightman distribution to imaginary time-differences, and indeed the resulting n-point distribution is called a Schwinger function . It was Symanzik who first proposed the formulation of Euclidean field theory as a mathematical theory in its own right [S94- S99] . The set of axioms needed for Schwinger functions to analytically continue in time-differences back to Wightman distributions were eventually
5 found by Osterwalder and Schrader [011, 012]. In particular, they isolated an important positivity property that a Euclidean field theory must have - now known as Osterwalder-Schrader positivity . The early work of constructive field theory was spear-headed by Glimm and Jaffe, and the focus was on quantum field models with space-time dimension less than four. The point is that ultraviolet singularities are milder in lower dimension, and so the renormalization is less involved for such models. The motivation was to isolate the constructive problems, as even the infinite-volume limit was not very well understood in the beginning. As far as technique was concerned, the use of the positivity-preserving contraction semi-group generated by the energy operator was important. Thus, even before the Euclidean approach was widely accepted, the operator-theoretic methods initially used in the constructive program were semi-Euclidean. As a constructive tool, Euclidean field theory came into its own with the functional integral approach of Nelson [N4- N7] . Extending the Feynman-Kac formula to the formalism of the scalar field, Nelson found that the Euclidean field associated with the free quantum field was actually a Markov field . The functional integral representation of the Schwinger functions perturbed by a regularized interaction is known as the Feynman- Kac-Nelson formula, and the Markov property of the free Euclidean field plays a role in controlling the infinite-volume limit, once the ultraviolet cutoff has been removed. In this chapter we give a sketchy introduction to quantum physics that eventually leads up to Euclidean field theory from a mathematician's point of view. In §1.1 we introduce quantum mechanics in its most general form, including an abstraction of Planck's Law, stability, incompatible observables, and the freezing effect that is manifest when the same observable is measured twice over an arbitrarily short period of time. In §1.2 we immediately describe the analytic continuation to imaginary timedifferences of the time-correlation functions of the ground state in this rarefied setting. This expository choice was made in order to advertise that such an analytic continuation depends on stability and is not peculiar to relativistic quantum field theory. The importance of reflection positivity is also promoted at this early stage. In §1.3 we describe the Correspondence Principle for a single particle in one dimension and realize the Uncertainty Principle in terms of the incompatibility of the momentum and position observables. We also introduce the Weyl correspondence formula as an extension of the Correspondence Principle and calculate the quantum-mechanical observables corresponding to powers of a couple of standard classical observables. In §1.4 we quickly review the harmonic oscillator of quantum mechanics, including the Hermite functions. We discuss the Hamiltonian, the annihilation operator, the Lagrangian operator, and the scaling generator in this context as well. In §1.5 we introduce both the Feynman integral and the Feynman-Kac formula in non-relativistic quantum mechanics. The description is a thumbnail sketch, but our focus is on deriving the path integral formula for the ground-state imaginary-time correlations of the perturbed harmonic oscillator. In §1.6 we discuss the relativistic quantum mechanics of a single, spin-zero, free particle, which is described by the Klein-Gordon equation and has a self-consistent interpretation. In §1. 7 we explain how this consistency is generally destroyed when interactions are introduced and why the phase space path integral cannot reduce in any case to a position space path integral for a single relativistic particle.
6
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
The remainder of the chapter is devoted to thumbnail sketches of quantum field theory. Canonical quantization - i.e., the extension of the Correspondence Principle to the Hamiltonian formalism of a free classical field - is given in §1.8 for the real scalar field. FUnctional integration of field amplitudes is introduced in §1.9, where the realization of a relativistic field as a system of non-relativistic mechanical vibrations is the key. We quickly review axiomatic field theory on §1.10 and give a general sketch of the constructive problem for self-interactions of the scalar field in §1.l1. In the former section Oster walder-Schrader positivity (reflection positivity for Euclidean field theory) is seen in terms of its physical importance, while in the latter section it is seen in terms of the obstacle it poses to the construction of non-trivial examples. In §1.12 we discuss the early success of the constructive program in two space-time dimensions, where the regularity properties neatly separate the short-distance problem from the long-distance problem. In this context OS positivity is brought into playas a constructive tool. Finally in §1.13 we explain the need for phase cell analysis in the construction of higher-dimensional models.
1.1
Measurement and Dynamics
A state of a physical system is represented by a unit vector in a complex Hilbert space, where scalar multiplication by a number with unit modulus does not change the physical state. An observable is a self-adjoint operator on the Hilbert space. When one measures an observable A for a system in a state 1jJ, the probability that the observed value of A will lie in the interval (ao , al) is given by (1.1.1) where II . II denotes the Hilbert space norm and FA (ao, ad is the spectral projection of A over (ao, al). If the information that the value of A indeed lies in (ao, ad is the result of the measurement, then the state has been altered. This alteration is what one means by "collapse of the wave function" in some contexts , and the new state is given by (1.1.2) In the case where A has only pure point spectrum and the result of the measurement is exact, the observed value can only be an eigenvalue of A . The probability of observing the eigenvalue a is given by (1.1.3) where F~a) denotes the orthogonal projection onto the eigenspace associated with a. If this value is actually observed, then the altered state is given by (1.1.4)
1.1. MEASUREMENT AND DYNAMICS
7
If the same measurement is performed many times on the same state 'IjJ, then the expected value of the observable A is given by (1.1.5)
where (., .) denotes the Hilbert space inner product. Now if a system is allowed to evolve in time undisturbed, the time-evolution of the state is purely deterministic. It is implemented by a strongly-continuous one-parameter unitary group U(t) on the Hilbert space. Thus, the state at time t is given by
'IjJ(t) = U(t)'ljJo
(1.1.6)
if 'ljJo is the state at time t = O. To study the time-evolution of a system, one clearly needs to prepare initial states. How can this be done? Let {A",} be a commutative set of self-adjoint operators - i.e., commuting observables. (In the case of unbounded operators, this simply means that their spectral projections commute.) For the sake of simplicity, suppose they have only pure point spectrum. If these observables are measured simultaneously for a system in state 'IjJ, then (1.1.7)
where the operator product is unambiguous because the spectral projections commute. Q, then the altered state is
If a", is indeed the observed value of A", for each
(1.1.8)
Even if 'IjJ is unknown, this altered state is a prepared state if every such product of spectral projections is a one-dimensional projection . In this case, there is only one state that is an eigenstate of every A", with specified eigenvalue an, so 'IjJ becomes irrelevant. If the set {A",} has this property, then it is a complete set of commuting observables. Obviously, one can use a single observable to prepare a state if its spectrum is nondegenerate. Two observables are said to be incompatible if they do not commute. The meaning of their simultaneous measurement is ambiguous because their spectral projections do not commute. One certainly cannot measure one observable without affecting the measurement of the other if they have only pure point spectrum and share no eigenstates, for example. More generally - since lack of commutativity need not be that extreme - one considers the result of measuring only A many times for the same state 'IjJ and compares it to the result of the same large number of experiments for the other observable B. One can underestimate the product of the variances as follows:
((A - a)2)",((B -
W)'"
~ ~
I((A - a)(B - b)'IjJ,'IjJ)1 2 IIm((A - a)(B - b)'IjJ, 'ljJW I([A, Bj'IjJ, 'ljJW,
(1.1.9)
8
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
where [., 'J denotes the operator commutator and a = (A)",. Obviously, the Schwarz inequality is the key to this estimation. The commutator is directly involved in the lower bound on the product of the variances, so it quantifies the uncertainty for the pair of observables. We return to this issue for specific observables in §1.3. Now consider the case of a single quantum-mechanical particle in one dimension with no forces acting on it . The Hilbert space is the space L2(JR) of wave functions and the time-evolution is given by (1.1.10) where ¢o denotes the Fourier transform of 1/Jo. The form of the frequency w(k) as a function of the wave number k depends on whether the particle is Newtonian or relativistic. The point is that - by Planck's Law - if the evolving wave associated with a particle is monochromatic with frequency w, then the particle has a definite energy given by (1.1.11) E: = fiw, where fi is Planck's constant. Inserting this law in our wave-packet expression, we see that the time evolution is represented by a spread in possible energies: 1/J(x, t)
= 2~
J
e-itE(k)/fteikx¢o(k)dk.
(1.1.12)
Obviously, the Fourier transform diagonalizes the self-adjoint operator (1.1.13) given by the functional calculus, and we may write
1/J(-' t)
= e- itH / ft 1/Jo .
(1.1.14)
Thus H generates the one-parameter unitary group for the time-evolution of this special system. The most general dynamical principle of quantum mechanics is just the abstraction of Planck's Law in this form. If a strongly continuous one-parameter unitary group U(t) is the time-evolution for an arbitrary system, the infinitesimal generator whose existence is guaranteed by Stone's Theorem is the self-adjoint operator H given by U(t)=e- itH / ft .
( 1.1.1 5)
The identification of H as the energy observable is the abstract content of Planck's Law . In keeping with the Correspondence Principle - described in §1.3 - H is also referred to as the Hamiltonian of the system. Now suppose the time-evolution is repeatedly interrupted by measurements. With 1/Jo the state at t = 0, consider the measurement of observables AI, ... , An at successive later times t l , . .. , tn· If 1/JI, . . . ,1/Jn are the altered states at those respective times, then the rules above clearly indicate the induction step (with to = 0)
(1.1.16)
1.1. MEASUREMENT AND DYNAMICS
9
where Fk is the orthogonal projection associated with the kth measurement . The final state is the normalization of the vector
v
= U(tn)Fn(tn)Fn-1 (tn-t} · · · FI (tt}1j;o, Fdt) = U(t)-l FkU(t).
(1.1.17) (1.1.18)
In general, the operators AI, .. . , An need not commute with one another or with the Hamiltonian H. An observable A is conserved if it commutes with H . The definition (1.1.18) for the undisturbed time-evolution of an orthogonal projection can obviously be extended to any self-adjoint operator. Thus (1.1.19)
(A),pCt) = (A(t)),po
for an arbitrary observable A, where 1j;(t) is given by (1.1.15) . The representation of the time-evolution of the expectation in terms of the solution of the vector-valued ordinary differential equation (1.1.20) fi~(t) = iH1j;(t), 1j;(0) = 1j;o, is the Schrodinger representation. The representation in terms of the solution of the operator-valued ordinary differential equation
fiA(t) = i[H, A(t)],
A(O)
= A,
(1.1.21)
is the Heisenberg representation. Since all physical information is embodied in the expectations of observables, these two representations are physically equivalent. A basic subtlety of quantum mechanics is that the 1j;o-expectation of A(t) cannot be realized as the limit of a difference quotient based on measuring A at successive times. To illustrate this limitation in the simplest possible way, assume that the spectrum of A consists only of nondegenerate eigenvalues. If A is measured at time t, the probability of obtaining the eigenvalue all - and thus altering the state 1j;(t) to the associated normalized eigenvector 1j;/l - is given by (1.1.22) In contrast to classical states, 1j;(t) cannot be recovered from the statistics. Instead, we get a mixed state given by the density matrix (1.1.23) with the expectation of an observable B given by (1.1.24) Clearly,
(A),pCt)
= (A)
p(t) ,
(1.1.25)
but on the other hand , (1.1.26)
10
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
The trouble is that once A has been measured, the expected value of A at any later time is affected, even if the result of the measurement is given by nothing more than unconstrained statistics. Thus we expect (1.1.27) because in general, (A(s))",(t)
(A)"'(HS) (A)p(r+.) "1= (A(s))p(t).
(1.1.28)
It is even more interesting to note that
tr pet) A(s)
L 1(1/!(t) , 1/!1') 12(A(s)1/!1' , 1/!1') L
(1/!(t) , 1/!I'W{al'
1
+ O(S2)}
(1.1.29)
I'
since the equation A1/!1'
= al'1/!1'
annihilates the commutator term. Therefore,
.
hm
6t-+0
1
A(A(~t)
ut
- A)p(t )
= 0,
(1.1.30)
and this stationary effect is interpreted as the freezing effect that measurements have on states.
References 1. P. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1958. 2. G. Mackey, The Mathematical Foundations of Quantum Mechanics, Benjamin, New York, 1976. 3. A. Messiah, Quantum Mechanics, Vol. I, North- Holland, Amsterdam, 1965. 4. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1972. 5. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
1.2. IMAGINARY TIME CORRELATIONS
1.2
11
Imaginary Time Correlations
A basic tenet of quantum physics is that the mathematical quantities with the physical meaning are the expected values of observables, but for a given model one would like to describe everything with a smaller and more detailed set of expectations. Let '!f;o be a state and A an observable for a system whose dynamics is given by a strongly continuous, one-parameter unitary group U(t) . Suppose that all observables can be derived from operator products of the form (1.2.1)
and that '!f;o is cyclic for the time-evolution of A. This means that linear combinations of vectors of the form (1.2.2) are dense in the Hilbert space. Then one expects to extract all physical information from the time-correlation functions (1.2 .3)
These functions are generally non-symmetric in the time variables, since one cannot expect the operators A(t,) to commute. One of the basic successes of quantum physics is the explanation of atomic stability. This means the Hamiltonian operator H of a given model is bounded below. Assume without loss of generality that H ~ 0 (adding a multiple of the identity modifies U(t) only by a phase factor, which has no effect on A(t)). A ground state is any state annihilated by H. Suppose the cyclic vector '!f;o is such a state. Then
U(t)'!f;o
= '!f;o,
(1.2.4)
and therefore (1.2.5)
Wn(t 1 +t, .. . ,tn+t) =Wn(tI, .. . ,tn ). This translation-invariance enables us to write
Wn(O, t2 - tl, t3 - h,· . . , tn - td Wn
(0,
t2 - tl, (t2 - t l ) + (t3 - t2)' . . . ,
I:
(tk+1 - tk))
k=l
Un-l(t2 - t l , t3 - t2, . .. ,tn - tn-I)·
(1.2 .6)
Clearly, the inner product formula is
Um(TI, . .. , Tm) = (AeiT1H/h A · · · eiT~H/ h A'!f;o, '!f;o) ,
(1.2 .7)
and the condition H ~ 0 allows analytic continuation of the unitary group to a holomorphic semi-group:
a
~
O.
(1.2 .8)
12
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
The analytic continuation
Um {T1
+ iO"l,' . . ,Tm + iO"m)
= (AeiTIH/lie-UlH/1i A· . . eiT=H/lie-u=H/1i A'l/Jo, 'l/Jo) (1.2.9)
is defined for 0"1 , .. . , O"m 2: 0, and the imaginary-time-ordered correlation is given by U
n-1 {iS2 - is 1, . .. ,is n - isn-d
(Ae-(S2- sd H / 1i A· . . e-(Sn -sn_dH/1i A'l/Jo, 'l/JO)
(1.2.1O)
for Sl ::; S2 ::; . . . ::; Sn . The collection of such correlation functions is expected to contain all physical information because in uniquely determines Wn by this analytic continuation in the time differences. Actually, one could just as well have expressed the analytic continuation in the time-differences (1.2.11) because the domain of Wn{O, t2 , t3 ," . , tn) (Aeii2H/1i Ae i (i 3 -i2)H/1i A · .. xei(in -in_dH/1i A'l/Jo , 'l/Jo)
clearly has the extension tv using
f-7
tv
+ is" with
Sl
=
(1.2.12)
°and Sv 2: sv-1 . We are simply (1.2.13)
after setting Sv = Sv + Sl . This choice of time-difference will be useful when we analytically continue the time correlations of the harmonic oscillator in §1.4. How do we extend in to the whole space IRn of arguments? Since in (Sl' . . . ,Sn) is defined only on the domain Sl ::; S2 ::; . . . ::; Sn thus far, we may remove this restriction by symmetrization. The physics of the system is now embodied in the symmetric functions {fn(Sl, . . . ,sn)} , but what properties characterize such a set offunctions? In addition to the translation-invariance property (1.2 .14)
in has a valuable property known as reflection positivity. Consider M complex numbers together with corresponding finite sequences
Q1, .. . ,QM
such that
°: ; sij) ::; s~j) ::; . .. ::; s};{~ . Then
1.2. IMAGINARY TIME CORRELATIONS
13
and this manifest positivity is the positivity referred to, as it involves reflection in imaginary time. This is the property that enables us to recover the Hamiltonian H and the space of states from the imaginary-time-ordered correlation functions . Accordingly, consider continuous, polynomially bounded, symmetric functions fn(Sl , . . . ,sn) such that for 0::; s\j) ::; s~j) ::; ... ::; s~;, 1::; j::; M,
fn(-Sl, . .. ,-Sn) = fn(Sl, . . . ,Sn)*, fn(Sl + s, . . . , Sn + s) = fn(Sl, . . . , sn), M ~
L...J
*f
(k)
QjQk rnj+mk -SmA:"
(k)
' " -51
(j)
,8 1
(1.2.16) (1.2.17) (j)
, ... , 8 m ;
> 0 -
(1.2.18)
j ,k=1
for arbitrary finite sequences (Ctl, ... , CtM) of complex numbers. These properties are sufficient for recovering quantum mechanics, and the construction is of an abstract universal type. Consider finite , increasing sequences (SI, . . . , Sn) of positive numbers together with the form (.,.) on these sequences defined by (1.2 .19)
and then extend this to a sesquilinear form on the linear space generated by formal linear combinations of the sequences. The reflection positivity yields a semi-norm, with respect to which one mods out those linear combinations that are null. The sesquilinear form is an inner product on the resulting linear space, which is then completed to obtain the desired Hilbert space 1i. The operator A is induced by the mapping A: (SI, .. . , sn)
1-7
(0, SI,· · ·, sn),
and its self-adjointness is an easy consequence of (1.2.16) . The ground state 1/Jo is just the null sequence in this construction. Finally, we induce the semi-group K(s), s ~ 0, by setting (1.2.20) K(S)(SI, . . . ,Sn) = (SI +s, . .. ,sn+ s). Since
fn+ti( -5;;" ... , -51 , SI +s, .. . , sn +s)
= fn+;;,( -5;;, -s, . . . , -51 -s, SI,· . . , sn),
(1.2 .21)
K(s) is obviously self-adjoint. Combining the semi-group property with the Schwarz inequality, one obtains
(K(2s)v,v)~
IIK(s)vll ::;
IIK(2s)vll~llvll~ (K(4s)v, v) ~ Ilvll ~
(1.2 .22)
Thus (1.2.23)
for N iterations. One can then use the assumption that every function fn(Sl, ... , Sn) is polynomially bounded to show that lim IIK(2N s)vIl 2 N-HXJ
N
=1
(1.2 .24)
14
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
for each abstract vector of the form M
=
V
L Cl'k(S~k),
...,
S~~).
(1.2.25)
k=1
Thus IIK(s)vll ~
IIvll
(1.2.26)
on a dense subspace of vectors, and so K(s) is a contraction. Given the regularity assumption, it is easy to verify that K( s) is strongly continuous as well. From these properties of K(s) we may now infer the existence of an infinitesimal generator - i.e., an operator H ~ 0 such that (1.2.27) This completes the reconstruction of the quantum mechanical Hamiltonian. The motivation for analyzing the imaginary-time-ordered correlations {fn} instead of the real-time correlations {w n } is to extract information about the spectrum of H by a more straightforward estimation. For example, suppose one expects 0 to be a nondegenerate eigenvalue of H and the rest of the spectrum to be contained in [c:,oo) with c: > O. To establish this, one only needs to prove that for s ~ 0, (1.2.28)
for an arbitrary vector v. This would be reflected in the estimate Ifm+n( -s~, . . . ,-s~, S1
+ s, . .. ,Sm + s)
- fn( -s~, . .. , -sD (1.2.29)
so the proof lies in the analysis of these correlations. The methods for analyzing fn( S1, . . . ,sn) are particularly powerful if the model has the additional structure f( S1, ... ,sn)
=
JIT
q(sk)dJ.L(q)
(1.2.30)
k=1
for some probability measure dJ.L on the cylinder sets of the space of real-valued continuous functions q( s) on lIt The properties (1.2.16-18) of the f n (S1' . . . ,sn) can be expressed as properties of the probability measure. If we define B as the reflection transformation (Bq)(s) = q( -S), (1.2.31) then the properties are: dJ.L(Bq) = dJ.L(q) ,
= dJ.L(q) , F(q)* F(Bq)dJ.L(q) ~ 0
dJ.L(q(- - a))
J
(1.2 .32)
a E IR,
(1.2.33) (1.2 .34)
1.3. CORRESPONDENCE PRINCIPLE
15
for all dtt-square-integrable random variables F(q) such that
F(q + h) = F(q), h(s) = 0, s ~ O.
(1.2.35) (1.2 .36)
References 1. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966. 2. J . Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View , Springer-Verlag, New York, 1987. 3. E . Nelson, "The Construction of Quantum Fields from Markov Fields," J. Funct. Anal. 12 (1973) , 97- 112. 4. K. Osterwalder, "Euclidean Green's Functions and Wightman Distributions," in Constructive Quantum Field Theory , G. Velo and A. Wightman, eds., SpringerVerlag, New York, 1973. 5. B. Simon, Functional Integration and Quantum Physics , Academic Press, New York,1979.
1.3
Correspondence Principle
There are a number of quantum-mechanical phenomena with no analogs in classical mechanics, but on the other hand, a great deal can be understood by studying the quantum-mechanical analog of a familiar classical system. The rule by which one derives this analog is known as the Correspondence Principle, and we give a thumbnail sketch of it here. Consider a single particle in one dimension with a time-independent potential as its environment. The classical dynamical variables of the particle are position x and momentum p, and an observable is a real-valued function of x and p. The Poisson bracket for any pair of observables is given by {a, b}
= 8a 8b 8p8x
_ 8a 8b . 8x8p
(1.3.1)
The classical Hamiltonian is a distinguished observable ~(x,p) =
£(p)
+ V(x),
(1.3.2)
where V(x) is the potential field and c(p) is the kinetic energy, whose expression in p depends on whether the particle is Newtonian or relativistic. A classical state of the system is an ordered pair (x,p) specifying the position and momentum of the particle. The set of all classical states is referred to as phase space, and the classical dynamics
16
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
is a continuous one-parameter group of homeomorphisms on phase space. An orbit of this group action is a solution of the system
= {1)(x,p),p}I(x,p)=(x(t),p(t»,
(1.3.3.0)
±(t) = {1)(x,p),x}I(x,p)=(x(t) ,p(t»,
(1.3.3.1)
]:itt)
which obviously reduces to the familiar classical equations of motion: ]:itt)
=-
. x(t)
dV dx (x(t)),
(1.3.4.0)
dE:
= dp (P(t)),
(1.3.4.1)
where (with Ma as the rest mass) dE: {piMa, Newtonian case, dp = pclJp2 + MJc2 , relativistic case.
(1.3.5)
The orbits are also the level curves of the Hamiltonian I) , and Liouville's Theorem states that each homeomorphism (x(O),p(O)) >-+ (x(t),p(t))
preserves Lebesgue measure on phase space. Perhaps the easiest way to understand this last observation is to regard the group of homeomorphisms as incompressible fluid flow on phase space: ±(t)i + ]:i(t)] =V'"(x ,p) I(x,p)=(x(t),p(t)),
(1.3.6)
where the "velocity field" is given by ~
v(x,p)
dE:
,
= dp(P)i -
dV dx j ·
(1.3.7)
The point is that V'"(x,p) is a divergence-free vector field on phase space. There are several one-parameter groups of measure-preserving homeomorphisms of interest to us. For example, if we replace the Poisson bracket action of the observable I) in Eqs. (1.3.3) with the bracket action of p as an observable, we obtain dp d>" = {p,p}I(x,p)=(x().) ,p().» == 0,
(1.3.8.0)
dx d>"
(1.3.8.1)
= {p, x }I(x,p)=(x().) ,p().»
== 1.
The solutions of this system all have the form p(>..) = p(O),
(1.3.9.0)
1.3. CORRESPONDENCE PRINCIPLE
17
= X(O) + A,
X(A)
(1.3.9.1)
which are the orbits of the group of translations in position space. Similarly, we generate the group of translations in momentum space by introducing the bracket action of the observable x. Every solution of the system dp dA = {x,p}l( x,p)=( x( >.),p(>.))
= -1,
dx dA = {x,x}b,p)=(x(>.)'p(>,))
=0
(1.3.10.0) (1.3.10.1)
obviously has the form p(A) = p(O) - A, X(A)
= x(O).
(1.3.11.0) (1.3.11.1)
Suppose we now consider the bracket action of xp: dp dA dx dA
= {xp,p}l (x,p) =(x(>.),p(>.)) = -p(A),
(1.3.12.0)
= {xp ,x }l (x ,p)=(x(>.),p(>.)) = X(A).
(1.3 .12.1)
The solutions all have the form p(A) = p(O)e->.,
(1.3.13.0)
= x(O)e>.,
(1.3.13.1)
X(A)
so the group of measure-preserving homeomorphisms generated here is a group of scaling transformations, where the direction of scaling in momentum is opposite to that of position. What is the Correspondence Principle for this one-dimensional system? The idea goes back to de Broglie, who associated a monochromatic wave to every particle with a definite momentum. The relation is p
= tik ,
(1.3.14)
where k is the wave number - i.e. , the wave amplitude is Ilt(x, t)
= ce-itw(k)+ixk
(1.3.15)
at a given time t . The Hilbert space for the quantum mechanics is the space L2(~) of square-integrable superpositions of the t = 0 monochromatic waves . Thus, an arbitrary state of the particle has a spread in possible momenta given by the Fourier transform: (1.3.16)
18
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
This yields the correspondence z
* x,
(x$)(~= ) x$(x),
(1.3.17.0)
for the position and momentum observables, respectively. The extension of this correspondence to all polynomials in p and x is generated by
which is actually an application of the celebrated Weyl transform. The study of the properties of this transform - such as its range of operators for a given space of functions of p and x - has been quite an industry. On the other hand, this extension of (1.3.17) is only part of the Correspondence Principle, which also makes the replacement {a, b) 4 ili-'[A, B] in the generation of transformation groups. The Weyl transform of the Poisson bracket is not the commutator of the Weyl transforms unless the observables are linear in position and momentum, so this bracket replacement is an additional part of the Correspondence Principle. In particular, the fundamental commutation relation
obviously corresponds to the Poisson bracket relation
More generally, the bracket action of a classical observable a(x,p) given by the system
is replaced by the operator action of the (self-adjoint) quantum observable A:
where A is obtained from a(x,p) by the Weyl transform. In the case of the Hamiltonian - or energy observable - we have the Schrijdinger equation
1.3. CORRESPONDENCE PRINCIPLE where H corresponds to
~.
19
The correspondence is obviously H
= c(P) + V(X),
(1.3 .24)
and every solution has the form 1/;(t)
= exp ( -~t(E(P) + V(X))) 1/;(0).
(1.3.25)
Since P and X do not commute, the free propagation cannot factor out as an operator. (Quantum-mechanical evolution would be a triviality if that were the case!) Determining the spectral properties of H - even establishing self-adjointness - can be a difficult problem, depending on the growth and regularity of the potential function V(x) . In the case of the momentum observable, we have the simple equation (1.3.26)
where the solutions have the form (1.3.27)
Thus P generates the unitary group of translations on the Hilbert space L2(JR) in the sense of Stone's Theorem. On the other hand, if we measure the position of the particle when it is in state 1/;, the probability of finding it in the interval (xo, Xl) is given by prob",{xo
< X < xI} (1.3.28)
so the amplitude 11/;(x)12 is a probability density in position space. Thus , the classical generation of translations in position space by the momentum observable has a quantum version as well. In the case of the position observable , the quantum analog of the bracket action is just (1.3.29)
where the solutions have the form (1.3.30)
On the other hand, if we measure the momentum when the state of the system is given by 1/;, the probability that the value will lie in the interval (Po, PI) is given by (1.3.31)
Since the Fourier transform diagonalizes P , this means (1.3.32)
20
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
so the amplitude 1~(Ii-lpWIi-l is a probability density in momentum space. On the other hand, the Fourier transform of Eq. (1.3.30) is obviously
1/J>.(k) = 1/Jo(k
+ ).),
(1.3.33)
so the Correspondence Principle yields a quantum analog of the classical generation of translations in momentum space by the position observable. As far as the measurements of position and momentum are concerned, it is important to remember that the observables are incompatible. For a given state 1/J, set (P)", = p,p,
(1.3.34.0)
= x",.
(1.3.34.1)
(X)",
If we apply (1.1.9) from §1.1, we actually obtain a lower bound on the product of
variances:
(1.3.35) This inequality is the Heisenberg Uncertainty Principle . Let us now consider the classical observable xp, whose bracket action was found to generate scaling transformations on phase space. The Weyl transform yields xp
liS
r-t
1
= 2(PX + XP),
(1.3.36)
where we have chosen S to be dimensionless. Alternatively, one may write .1
liS
PX+z 1i XP-
2 i~1i 2 .
(1.3 .37)
The solutions of the equation d
-1/J>. = -iS1/J>. d)'
(1.3.38)
have the form
1/J>.(x)
= e->./21/Jo(e->'x),
(1.3.39)
so S generates the unitary group of scalings on L2(IR) . Moreover (1.3.40) so the direction of the scaling in momentum space is opposite to the direction of the scaling in position space - just as in the classical context. As a self-adjoint operator, S has a simple continuous spectrum with generalized eigenfunction
IPs(x)
= Ixl-~-is
(1.3.41)
1.3. CORRESPONDENCE PRINCIPLE
21
corresponding to generalized eigenvalue s. Its commutators with Poisson bracket relations (1.3.12). We have
X and P mirror the
[S,PI = iP,
(1.3.42.0)
[S, XI = -iX.
(1.3.42.1)
To gain a greater familiarity with the Correspondence Principle, one finds it helpful to compute the Weyl transforms of various other classical observables. For example, consider the Newtonian Hamiltonian
which describes a simple harmonic oscillator with mass Mo and natural frequency po. Obviously, the corresponding quantum-mechanical Hamiltonian is just
In the study of quantization, one is often interested in deriving the classical quantities that transform into operator products of Weyl transforms, but here we shall consider the more straightforward problem of finding the Weyl transform of a multiplicative power of a classical observable such as 5 . The Weyl transform of fjn is assuredly not H n Indeed, one can calculate: n
(n!)2 W ( b n ) = 2-" (hpO)n 2m m=o ( m ! l 2( n - m ) !
C
The key to generating this formula is to write
and describe the Weyl transform of
in terms of the complex differential operator
and the Cauchy-Riemann differential operator d* As for the formula, the n = 2 , 3 , 4 cases are ri"& (1.3.49.0) W ( b 2 )= H 2 4 '
+
22
CHAPTER 1 . MATHEMATICAL SKETCXES OF QUANTUM PHYSICS
As another example, consider the nth power xnpn of the classical scaling generator. Applying (1.3.18) directly, we have
a2n ( e - j u x e - i u ~ - lP ei i u v ) l u = u = ~ . aunavn
w ( x r L p n )= ( - l ) n f i n -
(1.3.50)
First, notice that
Next, observe that
x n ( n - 1 ) ... ( m + 1 )
( -
(
-
P-
)
(1.3.52)
Combining these formulas, we obtain
(-ifil)m m=O On the other hand, there is the identity
( -
n-m
XmPm.
(1.3.53)
which we shall have occasion to verify and apply in Chap. 2 when we discuss higher order uncertainty relations for wavelet states. Inserting this formula in (1.3.53), we obtain for the scaling generator the same kind of formula obtained for the harmonic oscillator Hamiltonian - a polynomial in the quantum observable for the Weyl transform of each power of the classical observable. We have n
n
(n!)2 ( m ! l 2 ( n- m ) ! m=O and for powers n = 2 , 3 , 4 this reduces to w ( x q L p n= ) ( - i f li)
a)
w ( ~= a2~(s2~- ~ , )
(1.3.56.0)
1.4. THE HARMONIC OSCILLATOR
23
References 1. G. Baym, Lectures on Quantum Mechanics Benjamin, New York, 1969. 2. L. Hormander, "The Weyl Calculus of Pseudodifferential Operators," Commun. Pure & Appl. Math. 32 (1979),359-443. 3. E . Merzbacher , Quantum Mechanics , John Wiley & Sons, New York, 1961. 4. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover , New York, 1931.
5. E. Wigner, Group Theory and Quantum Mechanics, Academic Press, New York, 1959.
1.4
The Harmonic Oscillator
The application of the Correspondence Principle to the classical harmonic oscillator yields a quantum-mechanical system that has a special relationship to the Fourier transform and to the scaling generator. The classical Newtonian Hamiltonian is given by (1.3.43) , which we write again here: ~(X,p)
1
1
2
2 2
= 2M/ + "2MO/1-o X
,
(1.4 .1)
where Mo is the mass and /1-0 is the natural frequency. The Weyl transform is given by
H
1
2
1
2
= 2Mo P + "2Mo/1-oX
2
.
(1.4.2)
This system is well understood. The spectrum of H consists of the non-degenerate eigenvalues En = fi/1-O
(n + D'
n=O,1,2, ...
(1.4.3)
where the corresponding normalized eigenfunctions are just the Hermite functions. Concisely speaking, '{in
1
-qA yn!
A
tn
~(JMO/1-0/fiX+ ~p) 1 (
..,fi /3oX '{iO(X)
(1.4.4)
flO or,
I
1
d)
+ /30 dx '
$a exp (_ /35 X2) ~
2
(1.4.5) (1.4.6)
24
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
The Hermite functions are generated by the operator At from the ground state t.po· A is the "annihilation" operator, while
At
1 = J2
(
d)
/301 dx
(1.4.7)
= hn (v'2/3ox)t.po(x),
(1.4.8)
/3oX -
is the "creation" operator. More explicitly,
((At)nt.po)(x) At.po = 0,
(1.4.9)
where h n is the nth Hermite polynomial:
[t n ]
,
h (Z ) _ ~ ( ) j n. n-2j n L -1 ( _ 2·)'2 j ·,z j=O n J . J.
(1.4.10)
[.J denotes the greatest-integer function in this context. The Hamiltonian has an interesting relationship to the Fourier transform. If we apply the time-evolution to the observables P and X - i.e., if we set
= eitH/ 1i Pe- itH/"', X(t) = eit H/IiXe - it H/1i
P(t)
(1.4.11.0) (1.4.11.1)
- then the commutation relations
[H, PJ = ifiMoJ-L6X,
(1.4.12.0)
= -i fiP/Mo
(1.4.12.1)
[H, XJ
yield the Heisenberg equations of motion for the harmonic oscillator:
= -MoJ-L6X (t) , X(t) = MOl P(t) . Since the initial conditions are P(O) = P and X (0) = X, ?(t)
X(t) = X cos J-Lot P(t)
(1.4.13.0) (1.4.13.1)
the solution is:
+ (MOJ-LO)-l PsinJ-Lot ,
= P cos J-Lot -
MOJ-LoXsinJ-Lot.
(1.4.14.0) (1.4.14.1)
In particular,
X(-Tr/2J-Lo)
= (MOJ-LO)-l P,
P(7r/2J-Lo) = MOJ-LOX,
(1.4.15.0) (1.4.15.1)
so for an arbitrary square-integrable function 1j;, [exp( -i7r H/2fiJ-Lo) 1j;J
(;6) J =
e- ixk 1j;(x)dx.
(1.4.16)
25
1.4. THE HARMONIC OSCILLATOR
This identification of the Fourier transform as the t = I unitary operator in the time evolution implies that the eigenfunctions of H are also eigenfunctions of the Fourier transform - albeit , highly non-unique eigenfunctions , as the Fourier transform has only four distinct eigenvalues. Now consider the scaling generator S together with the Lagrangian operator L defined by L = _1_p2 _ ~M 2X2 . 2Mo 2 ofJo
(1.4.17)
It is straightforward to derive the evolution eitH / fi Le- itH / fi = L cos 2fJot
+ fJofiS sin 2fJot,
(1.4.18.0) (1.4.18 .1)
from (1.4.14) , so in particular, Land fiS are unitarily equivalent via the half-Fourier transform exp( -i7r H / 4fifJo), Since S has a simple, continuous spectrum, it follows that L does also. As far as commutation relations are concerned , we have
~[S, Hl
(1.4.19)
-iL,
~[H,Ll
(1.4 .20)
~[L, Sl
(1.4 .21 )
iH.
The Lie algebra generates the transformations ei AS H e- iAS = H cosh 2A e' AS Le- i AS = L cosh 2A eirL / fi H e- irL / fi = H cosh 2fJoT eirL/fiSe- i rL / fi = S cosh 2fJoT
+ L sinh 2A ,
+H
sinh 2A,
+ fJofiS sinh 2fJo T ,
+ fi- 1 fJo l H
sinh 2fJOT
(1.4 .22.0) (1.4.22.1) (1.4.23.0) (1.4.23.1)
in addition to (1.4.18) , so the Lie group SO(l, 2) is generated - the Lorentz group for three space-time dimensions. The generators H, Land S can also be written in terms of the "creation" and "annihilation" operators: 1
~
L
-"2 fifJo (At + A2),
(1.4.24)
S
i~(At2
(1.4.25)
H
_ A2) ,
fifJo (At A
+
D'
(1.4.26)
26
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
while the latter operators satisfy the commutation relations [A , At)
1
(1.4.27)
[A, H) [A,S) [A,L)
li/-Lo A , iAt, -1i/-Lo At
(1.4.28) (1.4.29) (1.4.30)
As far as an understanding of the harmonic oscillator is concerned, A and At are the operators that playa central role. For example, straightforward induction yields
IT (1i- /-Lo 1
1
H
+k -
k=l
tl (1i-
1
/-Lo
1
H - k
+
~) ,
(1.4.31)
D.
(1.4.32)
In particular (1.4.33)
More significantly, the operator h n (V2 !3oX) has a canonical expression in these operators. We define the Wick ordering of an arbitrary operator product in At and A only. Consider
#, is either the dagger or the absence of it. If r # is the number of ~ such that #, = t, then the Wick ordering of this product is given by
where
(1.4.34) If we extend this definition to linear combinations of the products by linearity, an
induction argument shows that
: (At +A)n:
~ (~)(Att A n- v [~nl , " ( l)j n. (A t j (n - 2J·)'2 L j=O . J. .,
+ A)n-2 j
hn (V2!3oX) .
(1.4.35)
We also have
[H !3on xn = " L j=O
,
n. !3n-2 j xn-2j (n-2j)!22jj! 0 :
:
(1.4.36)
1.4. THE HARMONIC OSCILLATOR
27
as a result of the standard inversion
[~nl , zn = L ( _ j n- 2j (z). j=o n J. J .
;:)'2 .,h
(1.4.37)
Wick ordering plays a basic role in the ultraviolet renormalization of quantum field theories. The operators A and At yield the most natural way to compute time correlations for the harmonic oscillator because their time-evolutions are trivial. The commutation (1.4.28) implies A(t) == eitH / 1i Ae- itH/ 1i = e-iJ.Lot A. (1.4.38) On the other hand, all physical information about the harmonic oscillator can be recovered from time correlations of the form
In particular, the ground state CPo is cyclic with respect to operator products of this form . Now one may certainly write
X(t)
= ,6o~(e-iJ.LotA+eiJ.LotAt),
(1.4.39)
so it is straightforward to verify inductively that (1.4.40.0)
pairing partitions l' of{l, ... ,n}
II
{j,j' }E1'
n even,
(1.4.40.1)
where the notation {j, j'} of each pair is chosen such that j < jf The basic ingredients of this calculation are (1.4.9), (1.4.27), and (1.4.39). Now the imaginary-time-ordered correlations are quite explicit in this case. Recall that in §1.2 that the analytic continuation Tj >-+ Tj + iO"j, O"j ~ 0, was done in the time differences Tj = tj+l - tj, but since v
tv == tv -
tl
= L(tj+l -
(1.4.41)
tj),
j=l
tv
we could just as easily have done the analytic continuation >-+ tv monotonicity conditions 8v ~ 8v - l and 81 = O. Thus for n even,
(X(t1)X(t2) .. . X(tn)cpo, cpo) (X(0)X(t2) " . X (in) CPo , CPo) Tn/2,6on L
II
pairing partitions {lI,v'}E'P l' of {1, ... ,n}
eiJ.Lo(iv'-iu)
+ i8 v
with the
(1.4.42)
28
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
has the analytic continuation
+ is 2) ... X(tn + isn)'Po, 'Po)
(X(0)X(t2 rn/2f3C;n
L
II
eil'o(iv,-i v )
pairing partitions {v,v'}EP P of {1, ... ,n}
x
II
(1.4.43)
{ v,v' }EP
so the imaginary-time-ordered correlations are given by
(X(0)X(iS2 - isd · ·· X(is n - isd'Po,'Po) rn/2f3C;n e-I'o(sj'-s;) (1.4.44)
L
II
pairing partitions {j,j' }EP P of {1, ... ,n}
in this case. As we have defined this function only for S1 :::; S2 :::; ... :::; Sn, there are a number of ways to extend the definition to an arbitrary ordering of imaginary times, one of which is to use the same formula. However, that choice allows exponential growth in these more general correlations and may not even make sense for less explicit models. On the other hand, the extension by symmetrization can always be defined, and the resulting correlation functions often have a classical probabilistic structure. In this case we get
II
pairing partitions {V,K }EP P of {1, ... ,n}
n even,
(1.4.45.0) (1.4.45.1)
If we let q(s) be the Gaussian random process with mean zero and covariance
C(s s')
,
= _1_ e -I'ols-s'l 2133
'
(1.4.46)
then our correlation functions are identified as (1.4.47)
where Oc denotes the expectation functional for this Gaussian process. Note that C(s, s') is the Green's function of the differential operator
2
Ito-2130
(dds2 + Ito2) -
2
1.4. THE HARMONIC OSCILLATOR
29
The characteristic function of the random process is the mapping f ds))c, where f ranges over class CO' functions. Explicitly, \ exp
(i Jf(s)q(S)dS) ) c
I--t
(exp(i J f(s)q(s)
= e-!(Cf,f)
(1.4.48)
because our random process is Gaussian. What does Wick ordering mean in this context? In the real-time context,
L II ei/Joo t; II e-i/Joot;(At)card AC{l, ... ,n}jEA
A An-card
A,
jrf-A
(1.4.49) and therefore n
: X(td··· X(tn) : y perturbing the relativistic Hamiltonian of a free particle with the static potential V (x). As long as we ignore the spontaneous loss of energy - an effect of the particle's own field - this propagator is perfectly valid in this static-potential reference frame. With
CHAPTER 1 . MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
42
this reservation we have the Hamiltonian
-.
and it is certainly natural to ask whether the generated time evolution has a path integral representation. If one generalizes path integrals to phase space trajectories, the answer is affirmative. Since the kinetic energy is no longer quadratic in the momentum, the free propagation no longer has the special structure of the Schrodinger propagation. However, exp ( - i c A t J
82
+
-)
M:c2
J d $ exp ( i E 1
= (2sD-'
(
.(; -
~
1
G) - i f i - ' e ~ t
+ , 7) p
M;c2
2
(1.7.7)
(-cijm) (-gv)]
and if one inserts this in the operator product approximation
N
[exp
L = ~ ~ x L
exp
to the interacting propagation, the result is a multiple phase-space integral - namely,
(
N
(
J
)
(Vnl
/ d
j=l
2
2
A
;(
G ~ ~ )X P
N $j
( ; j - ;,-I)
'
2
with x o = x and X N = Y . In the N = ca limit one obtains the formal representation
where the functional integration is now over the paths
1.7. CRISIS IN THE THEORY AND FORMALISM
43
in phase space. The phase of this more general path integral is still the classical action, but instead of the Lagrangian, it is the phase space action. This is the formalism which is independent of the momentum expression chosen for the kinetic energy. The difference between the Schrodinger case and all other cases boils to the surface when one tries to integrate out the momenta. In the relativistic case, one assuredly does not obtain a path integral in position space with the relativistic Lagrangian (1.7.10) as the phase. Does one obtain a reasonable functional integral over position space trajectories at all? To pursue this issue, return to (1.7.7) and note that we can apply the Taylor formula -1
Ii
2
Moc C
-41i
Ii-I ~ P
+ 2Mo
11 1
2
~
8
~
0
0
~
,
(1.7.11)
+ p(p), ~
(s' P
(p 2)2 2
+ MJc 2)3/2
,
(1.7.12)
and complete the square in the exponent to obtain
~p (-,co.,
(l?,y)
(1.7 .13)
where the last step is just the change (1.7.14)
44
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
of integation variable. If we now insert this formula in the operator-product approximation N [ex
( m (-gv)] ) 8%
exp
to the perturbed propagation
[ ;(..\lTgZ) 1
U ( t ) = exp - -
+
0.7.15)
,
we obtain an integral over the polygonal structure in Fig. 1.7.2 with the potential
Figure 1.7.2:
v( o. If (u~, v~) denotes the conjugate pair of dynamical variables at lattice site;; E €Zd, then the Hamiltonian formalism becomes: ~(u, v)
1
-2€
d
""' L~EEZ.l
{a, b}
-d
€
""'
L
~ EEZ.l
d ) - - u-) L..-J (v_ + mou- + ""'(u2
2
x
2
x
X+ee
l
x
2 -2 €
'
(1.8.8)
,=1
(8a 8b 8a 8b) 8v- 8u- - 8u- 8v- ' x
x
x
x
(1.8.9)
1.B. CANONICAL FORMALISM OF QUANTUM FIELD THEORY
49
where -;;, denotes the Lth unit coordinate vector. With regard to physical units, the dimension of u:;; is (length)1/2-d/2, while that of v:;; is (length)-1/2-d/2 . This describes a system of harmonic oscillators that are harmonically coupled to their nearest neighbors as well. The equations of motion are
V:;; (t)
{~(U, v), v:;; }I(u,v)=(u(t),v(t)) d
-m~u:;;(t)
+ .0- 2 Z)u:;;+e, (t) -
2u:;;(t)
,=1
+ U:;;_ e, (t)),
{~(u, v) , U:;; }I(u,v)=(u(t)v(t)) v:;; (t) .
(1.8 .10)
(1.8.11)
u:;;(t) should not be confused with any spatial direction, as it is actually an amplitude with a spatial location. One can certainly imagine a non-relativistic particle bound by these forces at each location, but it moves in a one-dimensional space that is independent of physical space. Such a particle certainly has nothing to do with the relativistic particles whose quantum mechanics is ultimately derived from the quantized field theory. Nevertheless, the canonical formalism is based on regarding these amplitudes as abstract mechanical displacements and applying the correspondence principle accordingly. Even in the lattice approximation, however, there are still an infinite number of variables to reckon with. In order to define the quantization unambiguously, we impose a finite-volume cutoff as well. Let
A; =
.0 { -
r, . .. , -1, 0, 1, . .. , r - 1} d,
(1.8.12)
and associate to each -; E A~ a copy 11.:;; of the Hilbert space L2 (lR) . The Hilbert space of this doubly truncated system is the tensor product
11.~
=
®
(1.8.13)
11.:;; .
X"EA~
Therefore , the states of the system are normalized square-integrable functions 1j;(u) of the variables u:;;, -; E A; . Now to apply the correspondence principle, one must first note that = c- d8_XiX' (1.8.14) {u-x" v-} x so the conjugate pairs are given the assignments (based on dimensional analysis) u:;; t--+ Q:;;,
v:;; t--+
P
(Q:;;1j;)(u) . -1 / 2-d/ 2
:;; = -Zc
= c1 / 2 - d/ 2 u:;;1j;(u)
a. au-
(1.8.15) (1.8.16)
x
We choose periodic boundary conditions for the Hamiltonian. This means the Hamiltonian is given by
i)
HE ,T
o
d = !c 2
""
L
:;;E~
(P'!. + m Q:' + ~(Q_ L 2
x
0
x
_ _ Q_x )2C 2)
X+Ee,
=1
(1.8.17)
50
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS A
+
2
with the understanding that x E, ; is identified as - ~ ( 2 -r 1) e , when x, = E(T - 1). The spectral analysis can be carried out quite explicitly. If we introduce the discrete Fourier transforms
we can write the Hamiltonian in the form
where r: is a dual set of lattice points - i.e., we have the discrete resolution of the identity:
k a-5
We choose the phase such that
is symmetric about the origin:
The annihilation and creation operators are given by
respectively, where
Since
[Q;, P;,] we have
= i~-~d;,;,
1.B. CANONICAL FORMALISM OF QUANTUM FIELD THEORY
51
from which it follows that (1.8.26) Obviously (1.8.22) can be inverted: ~
Q~(k) = W"(j;1/2 (A~(- k) + A~(k)t),
(1.8.27.0)
(1.8.27.1) Now consider ~
~
w" (k )A~ (k ) t A~ ( k )
1
~
~
1
~
~
2w,,(k)2Q~(k)Q~(- k) + 2P:(k)P:(-k) 1
~~
~~
~
1
~~
~~
~
+ i2w,,(k)Q~(k)P:(-
k)
- i2w,,(k)P:(k)Q~( - k) .
Summing over k and making the index change k I-t one obtains
-
k in the 3rd contribution only,
_2: w,,(k)A~(k)tA~(k) = (2rc)d (HQ - ~ _2: W"(k)) k Er~
(1.8 .28)
(1.8.29)
k Er~
from the commutation relation (1.8 .25) . Thus (1.8.30)
and this alternative expression for H~ , r is often referred to as the decomposition into normal modes. Also note that the constant term diverges as r --+ 00, even for fixed c > O. The ground state ofthe system is obviously the state !1~,r such that for every normal ~
mode k E
r~,
(1.8.31) This implies (1.8.32)
52
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
and this equation can be written as
8 E r( U, ) ---00'
(1.8.33)
(:)d _L
(1.8.34)
8u-x
wik)ei(;-;')k .
kEr
-t variables 81 ~ 82 ~ ... ~ 8 n , where
Tk
tk+l -
tk .
+ iak is defined for
17k ~
O. Now introduce the (1.10.17)
and define
Sn((i" 1,81),' . . , (i" n, 8 n )) (Q(i" 1, O)e-O"lH Q(i"2, 0) ... e-O"n-l HQ(i" n, 0)0, 0).
(1.10.18)
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
68
These are the Schwinger fun ctions . As in §l.2, we can remove the restriction Sl ~ S2 ~ . .. ~ Sn by symmetrization, but there is an extra subtlety in this case. This ~mmetric extension cannot be well-defined unless Sn is symmetric in any succession of x-arguments for which the corresponding succession of s-arguments happen to coincide. However, this condition is satisfied because
[QC;;,O),Q(x ',0)] = O.
(1.10.19)
The commutation property is a special case of the locality property. In the mainstream of constructive quantum field theory, the approach is to construct the Schwinger functions of a given model and verify those properties which insure the recovery of Wightman distributions. These properties are the Osterwalder-Schrader Axioms. OS Axiom 1. For each positive integer n, there is a tempered distribution Sn(X1, .. . , xn) on IRnd, where d = d + l. In this context, the x, are the Euclidean points . OS Axiom 2. For every transformation (R, a) in the Euclidean group,
Sn(RXl
+ a, .. . , RX n + a) = Sn(X1, . . . , xn),
(1.10.20)
where R ranges over all rotations in IRd and a ranges over all translations . OS Axiom 3. Sn(X1, .. . , xn) is symmetric in the Euclidean arguments. OS Axiom 4. Let 0 be the reflection operator
(Of)(x, s)
= I(x, - s).
(1.10.21)
For every sequence (It, ... , f n) of test functions, (1.10.22)
OS Axiom. 5. Consider any collection of sequences (ff, · ·· , I~l) "'" (fl, " ., f::'m) of test functlOns supported in the half-space d
IR+
~
= {(x ,s) E IRd :
s > O} .
(1.10.23)
Then m
L
Snj+nk(fr, . .. , f~;,Oft,· .. ,O/~k)a;ak ~ 0,
a1 ,··· ,am E C.
(1.10.24)
j,k=l
This is the field-theoretic version of reflection positivity, which we have already discussed in §1.2. In this context, the condition is often referred to as OS positivity . OS Axiom 6. For every pair of sequences (It, ... , I n), (gl , . .. , gm) of test functions, ;~~ Sm+n( It ,···, In , gl(- - a), .. . ,gm(- - a))
= Sn(It , · ·· , I n)Sm(gl, ... ,gm). (1.10.25)
1.10. AXIOMATIC QUANTUM FIELD THEORY
69
This is the cluster property for Schwinger functions . The last axiom is a technical condition that does not necessarily hold for the Schwinger functions of a given Wightman theory, but is sufficient for the recovery of Wightman distributions from a given set {Sn(X1, . .. , xn)} satisfying the above axioms.
as AxiOIn 7.
There is a J > 0 and a positive integer N such that for every sequence (II, ... , In) of test functions n
ISn(II, . .. ,ln)l~cn(n!)1-6rr sup
II(l+I'I)Na'l'I,lloo, ,=ll'l'I:5 N
(1.10.26)
where "I ranges over multi-indices. This has proven to be an estimate one can realistically expect to prove in the construction of various models. Notice that 8 = in the free field case. The Osterwalder- Schrader Reconstruction Theorem states that if {Sn(X1, . .. , xn)} satisfies these axioms , then it is the set of Schwinger functions for some Wightman theory. Actually, the proof directly reconstructs the Hilbert space and the Hamiltonian in much the same way that it was done for the simple imaginary-time-ordered correlations discussed in §1.2. After obtaining this structure, one then applies the theory of Laplace transforms to analytically continue the Schwinger functions back to real time differences . Although this analytic cqntinuation fixes the spatial directions, the Lorentz covariance of the Wightman distributions is obtained by showing that the generators of the group annihilate them. These differential conditions are easily derived from the parallel conditions on the Schwinger functions involving the Euclidean generators. For the class of field theories of interest to us, the Schwinger functions are realized as moments of a probability measure on an infinite-dimensional space. The probability measure dJ-L is a cylinder measure on the space S' (JRd) of real-valued tempered distributions, and the smeared Schwinger functions are given by
!
=
S(II ,· ··, In)
JIT
4!(f,)dJ-L(if?).
(1.10.27)
,=1
In this case the axioms reduce to a set of conditions on the measure. E Axiom 1. For every transformation (R, Q) in the Euclidean group, (1.10.28)
where 4!R,o(X)
= 4!(Rx + Q) .
E Axiom 2. With () defined as before, dJ-L(()4!)
= dJ-L(4!) .
(1.10.29)
E Axiom 3. Let FE L2(dJ-L) such that F(4!
+ h)
= F(4!),
(1.10.30)
70
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
for all tempered distributions h with supp he lRd\lRi
= lRd x
(-00,0). Then (1.10.31)
This is the measure-theoretic version of OS positivity. E Axiom 4. For F, G E L 2 (dJ.1) satisfying (1.10.30) for all tempered distributions h supported outside some bounded region, (1.10.32) E Axiom 5. There is a Ii > 0 and a positive integer N such that (1.10.33) for every test function
f.
It is straightforward to verify these axioms for the free imaginary-time field 4'>(X",s) and its Gaussian measure constructed in §1.9. The problem is to construct a non-Gaussian measure satisfying these axioms, as it will then yield a Wightman theory that is non-trivial (Le., interacting) . We have already begun to discuss this program by introducing a perturbation of the Gaussian measure. Generally speaking, the Euclidean invariance and the OS positivity are the most important axioms to worry about. It is easy to construct a probability measure with either property, but the two conditions together are very restrictive. Any perturbation of a Gaussian measure that meets both requirements will involve a very singular interaction, and this - together with the issue of stability - is what makes constructive quantum field theory so difficult.
References 1. H. Araki, "On the Algebra of All Local Observables ," Prog. Theor. Phys. 32 (1964), 844-854. 2. N. Bogoliubov, A. Logunov, and R. Todorov, Introduction to Axiomatic Quantum Field Theory, translation of 1969 publication by S. Fulling, Benjamin, Reading, 1975. 3. H. Borchers, "On Structure of the Algebra of Field Operators," Nuovo Cimento 24 (1962), 214-236. 4. H. Borchers, "Energy and Momentum as Observables in Quantum Field Theory," Commun. Math. Phys. 2 (1966) ,49-54 5. H. Borchers and J. Yngvason, "Necessary and Sufficient Conditions for Integral Representations of Wightman Functionals at Schwinger Points," Commun. Math. Phys. 47 (1976), 197- 213.
1.10. AXIOMATIC QUANTUM FIELD THEORY
71
6. J . Bros, H. Epstein, and V. Glaser, "On the Connection Between Analyticity and Lorentz Covariance of Wightman Functions," Commun. Math . Phys. 6 (1967), 77-100. 7. S. Doplicher , R . Haag, and J . Roberts, "Local Observables and Particle Statistics. I," Commun. Math. Phys. 23 (1971), 199-230. 8. S. Doplicher, R . Haag, and J. Roberts, "Local Observables and Particle Statistics. II," Commun. Math. Phys . 35 (1974),49-85. 9. F. Dyson, "Connection Between Local Commutativity and Regularity of Wightman Functions ," Phys . Rev. 110 (1958), 579-581. 10. F. Dyson, "Integral Representations of Causal Commutators," Phys. Rev. 110 (1958), 1460-1464. 11. J.-P. Eckmann and H. Epstein , "Time-Ordered Products and Schwinger Functions ," Commun. Math. Phys. 64 (1979),95-130. 12. J. Frohlich, "Schwinger Functions and Their Generating Functionals. I," Helv. Phys. Act a 47 (1974), 264-306. 13. J. Frohlich, "Schwinger Functions and Their Generating Functionals. II," Adv. Math. 23 (1977), 119- 180. 14. L. Garding and A.S . Wightman "Representation of the Commutation Relations ," Proc. Nat. Acad. Sci. 40 (1954),622-626. 15. L. Garding and A. Wightman, "Fields as Operator-Valued Distributions in Relativistic Quantum Theory," Arkiv for Fys. 28 (1965) , 129-184. 16. M. Gell-Mann, M. Goldberger, and W . Thirring, "Use of Causality Conditions in Quantum Theory," Phys. Rev. 95 (1954), 1612-1627. 17. V. Glaser, "On the Equivalence of the Euclidean and Wightman Formulation of Field Theory," Commun. Math. Phys. 37 (1974), 257-272. 18. J. Glimm and A. Jaffe, "Energy-Momentum Spectrum and Vacuum Expectation Values in Quantum Field Theory I," J . Math. Phys. 11 (1970), 3335-3338. 19. J . Glimm and A. Jaffe, "Energy-Momentum Spectrum and Vacuum Expectation Values in Quantum Field Theory II," Commun. Math . Phys. 22 (1971), 1- 22. 20. J . Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987. 21. R . Haag, "On Quantum Field Theories ," Mat.-Fys. Medd . Kong. Danske Videns. Selskab 29 (1955), No. 12. 22. R. Haag, "Observables and Fields ," in Lectures on Elementary Particles and Quantum Field Theory, S. Deser et al., eds., MIT Press, Cambridge, 1970.
72
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
23. R. Haag and D. Kastler, "An Algebraic Approach to Quantum Field Theory," J . Math. Phys. 5 (1964), 858-861. 24. K. Hepp , "On the Connection Between the LSZ and Wightman Quantum Field Theory," Commun. Math. Phys. 1 (1965) , 95-111. 25. R. Jost, The General Theory of Quantized Fields, AMS, Providence, 1965. 26. A. Klein and L. Landau, "From the Euclidean Group to the Poincare Group Via Osterwalder-Schrader Positivity," Commun. Math. Phys. 87 (1983),469-484. 27. H. Lehmann, K. Symanzik, and W . Zimmermann, "On the Formulation of Quantized Field Theories," Nuovo Cimento 1 (1955), 205-225. 28. V. Malyshev, Introduction to Euclidean Quantum Field Theory, Moscow University Press, Moscow, 1985. 29. E. Nelson, "The Construction of Quantum Fields From Markov Fields," J . Funct. Anal. 12 (1973) , 97-112. 30. K. Osterwalder and R. Schrader, "Axioms for Euclidean Green's Functions I," Commun. Math. Phys. 31 (1973),83-112. 31. K. Osterwalder and R. Schrader, "Axioms for Euclidean Green's Functions II," Commun. Math. Phys. 42 (1975) , 281-305. 32. W. Pauli, "The Connection Between Spin and Statistics," Phys. Rev. 58 (1940), 716-722. 33. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis and Self-Adjointness , Academic Press, New York, 1975. 34. D. Ruelle, "Connection Between Wightman Functions and Green Functions in p-Space," Nuovo Cimento 19 (1961),356-376. 35. J. Schwinger, "On the Euclidean Structure of Relativistic Field Theory," Proc. N.A.S. 44 (1958), 956-965. 36. B. Simon, The P( O. (1.11.16), (1.11.19), and (1.11 .20) together imply
J(iJ»
=
r
JR,d+
dxV(<J>(x)) -
~!dll(w)Kw(<J>)*Kw(<J>) . 2
(1.11.21)
Does this mean that we have an interaction that is non-local? Not really. Minlos ' Theorem implies there is a probability measure djj,o such that for every f E S(lRd ), (1.11.22) Thus
! (! dx~(X)G(<J>(x))) djj,o(~) G! J exp
exp
dx
dyC(x,Y)G(<J>(X))G(<J>(Y))) ,
(1.11.23)
76
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
and so we have the formula
e- I (4))
=
J
exp ( -
J
dx(V((x)) - (x)G((x))l) dP-o( 1, the support of dll o requires distributions whose pointwise powers are meaningless . The d = 1 case corresponds to zero-dimensional space, where the "Euclidean field theory" is given by the imaginary time correlations for the ground state of the perturbed harmonic oscillator discussed in §1.5. From a physical point of view, one is most interested in the d = 4 case, but the intermediate cases d = 2,3 have proven to be at the same time amenable to rigorous treatment and interesting to physics . If we impose the cutoffs introduced in §1.9, the resulting Euclidean measure is given by (1.11.27)
zreg A,s
J J
exp (-
ISsda 1d -; V(reg(-;,a))) dllo( is not regularized in the imaginary time direction, it is regularized in all space directions, so the random field <J>reg shares the regularity properties of the d = 1 random field . Thus d/-!o is supported by the space of <J> such that <J>reg is continuous, and so the functional integration of the exponential makes sense. This is the measure-theoretic understanding of why the only ultraviolet cutoff needed to define the mathematical objects comprising the vacuum expectations in the Hamiltonian picture is a regularization of the time-zero field QCi!). The coefficients A// in (1.11.26) are physical parameters of the theory. The most formidable obstacle in constructive quantum field theory is that for fixed parameter values, the limiting measure d/-!A ,S as <J>reg approaches <J> simply cannot exist for dimension d > 1, even as a singular measure relative to d/-!o. One can obtain an ultraviolet limit only by renormalizing some of these parameters - i.e., introducing a dependence A//(g) of the parameters on the ultraviolet cutoff function 9 - indeed, introducing a dependence A~ (X'; g) approaching A// (g) as A -t IRd in the sense of set inclusion - such that a measure d/-!~e'~ exists in the limit as g(X') -t 8(X') . These g-dependent parameters are called bare parameters, and some of them may approach infinity in this limit, but the original interaction is actually the wrong focus of attention. The point is that if the measure d/-!~e'~ converges - in the sense of expectation values - to a measure d/-!ren as A -t IRd (in the sense of set inclusion) and s -t 00, and if d/-!ren satisfies the Euclidean Axioms, then we can recover a quantum field theory. The unique vacuum is the renormalized vacuum in this case, the Hamiltonian one reconstructs is the renormalized Hamiltonian, etc. A major problem, of course, is to find the dependence A~(X';g) for which a limiting Euclidean measure indeed exists. One is certainly not interested in the trivial solution to this problem - i.e., to drive the parameters to zero sufficiently fast as g(X') -t 8(X') so that one merely recovers the Gaussian measure d/-!o! More generally, one does not want the g-dependence to drive the measure to a Gaussian, which describes a theory of free particles only. On the other hand, the key property that the A~ (X'; g) must have to insure existence of the measure d/-!ren in the limit as all cutoffs are removed is ultraviolet stability: (1.11.30) In Euclidean language, the property becomes
J
exp (-
~l dX'
ISs
dlTA~(X';g)<J>reg(X"lTt) d/-!o(<J» ~ eeslAI,
(1.11.31)
but the point is that the exponential bound with respect to the Euclidean volume is independent of the ultraviolet cutoff. We explain later how ultraviolet stability plays a central role in the convergence of the entire set of vacuum expectation values. The v = 0 term in the interaction is just a number, and therefore has the effect of shifting the energy. The dependence A~ (X'; g) is the vacuum energy renormalization. Actually, energy shifts cannot change the physics of the field theory, and this is reflected in the obvious fact that a numerical shift in the exponent cancels out in the normalized expressions for the vacuum expectations. The ultraviolet stability is another matter,
78
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
however; it holds only if the renormalization maintains the ground state energy of the Fock space Hamlitonian
at a fixed value, known as the effective vacuum energy. To describe the renormalization of the other parameters, we next consider the mass operator
(1.11.32) for an arbitrary quantum field theory. M is an operator instead of a number because it need not be the rest mass of a single relativistic particle. It can be the rest mass of any closed relativistic system, which includes the internal energy due to interactions among the constituent particles. If m is the rest mass of a single, free particle for the given theory, then the spectrum of M includes all real values 2: 2m for this reason. On the other hand, it includes no values between m and 2m because it takes two particles to contribute internal energy. The spectrum of the Hamiltonian H includes all real values in that interval because a single particle can have any amount of kinetic energy. Neither M nor H have spectra between 0 and m , so if dEM is the spectral measure of M, then the time-evolution is
e-
itH
=
exp ( -it
V;'
+
2
M2)
l )O exp ( -itV-P
2
+ a2) dEM(a)
EM({O}) +exp (-itV-P2 +m2) EM({m})
+
1:
exp ( -itV-P 2 + a 2) dEM(a)
(1.11.33)
and P commutes with the spectral measure. Formally, (Q(-;, t)Q(-;
I,
t')O, 0)
= (Q(-;)E( {O} )Q(-; ')0,0)
+ ( Q(-;) exp (i(t
+
1: (
l -
t)
Q(-;) exp (i(t
V;'
l -
2
+ m 2) E( {m} )Q(-; ')0,0)
t)V-P 2 + a2)
dE(a)Q(-; 1)0,0)
= (Q(O)O, 0)2 + (Q(-;, t)E( {m} )Q(-; " t')O, 0) +
roo (Q(-;, t)dE(a)Q(-; " t')O , 0) . 12m
(1.11.34)
1.11. THE CONSTRUCTIVE PROBLEM
79
Since the spectral measure c 2 are realized as formulas based on truncated vacuum expectations, which are given in general by
(ip(h)· ·· ip(f,,))T
=
L
Cp
PEII.
IT / IT ip(fk)) ,
(1.11.40)
AEP \kEA
where II" is the set of all partitions of {1, ... ,I/} and the coefficients Cp are inductively defined. A more transparent definition of truncated expectations can be given by a characteristic function as follows:
(ip(fd · · · ip(f,,))T = fIO:hk In (ex
p
~ hkip(fk)) L="'=h.=O
(1.11.41)
k=! Thus,
(ip(f))T (ip(fI)ip(h))T (ip(h )ip(h)ip(h)) T
=
(ip(f)), (ip(fI)ip(h)) - (ip(fI))(ip(h)) ,
(g
(1.11.42) (1.11.43)
ip(fk)) - t(ip(fe))
(U (Q x
ip(fk))
+2
g
(ip(fd),
(1.11.44)
ip(fk)) - t(ip(fl))
x /
IT ip(fk)) - t(ip(ft)ip(f4)) / IT
\k,U +2
e=!
ip(fk))
\k¥l,4
t; (ip(fe)) (ip(fi' )) (!t ip(ik)) (1.11.45)
k=! If the Euclidean field theory is Gaussian, then all truncated expectations of order 1/ > 2 are zero. The formulas for the effective coupling constants are ultimately derived from
1.11. THE CONSTRUCTIVE PROBLEM
81
a basic analysis of the scattering processes of the quantum field theory in real time, but the Euclidean versions involve these truncated vacuum expectations. Such derivations are beyond the scope of this introduction to quantum field theory, but the effective value of the v = 4 parameter is
(1.11.46) with m given by (1.11.37). Obviously, this constant is zero if the Euclidean field theory is Gaussian, so the renormalization prescription described above yields a non-trivial (non-Gaussian) field theory if the effective parameter >"4 is fixed at a non-zero value as cutoffs are removed. Actually, not all the types of renormalization have been mentioned yet. The constant w in the Kallen-Lehmann representation is the effective field strength, and this parameter must be held fixed as well . This constraint induces a renormalization of the Gaussian measure not to be confused with the mass renormalization, since it scales the kinetic part J(\7/,)) is ",-dependent. How, then, does one control the", = 00 limit? The fundamental strategy is to show that this exponential converges in the space U(d/-Lo) for 1 ~ q < 00 - i.e., that it fails to converge in U"'(d/-Lo) only marginally. The key ingredient in the argument is a property of the free measure d/-LO known as hypercontractivity. The most natural formulation of hypercontractivity is given in terms of the Fock space structure of d/-LO, but in this context we are concerned with the following special formulation . Let 11, ... , 1M be test functions in S(JR2 ) and PI, . .. , PM be vth-degree polynomials. Then for 2 ~ r < 00,
L J:Pj(<J>Kj(X)) : /j(x)dx M
3=1
Lr(dJ.'o)
L J: Pj(<J>Kj(X)) : /j(x)dx M
< (r _1)"/2
j=1
(1.12.19) £2 (dJ.'o)
for any values "'1, ... , "'M of the ultraviolet cutoff parameter. The application of this estimate to proving (1.12.20) for 1 ~ q < 00 is part of the famous Glimm-Jaffe-Nelson argument. To convey the ingenuity of their reasoning, we find it most convenient to simply establish the ",-uniform bound (1.12.21) for 1 ~ q < 00. If this is to be proven for all Iren,A(<J» of the form (1.12.10), then by scaling the parameters Av we may assume without loss of generality that q = 1. One tries to control the Duhamel expansion
j
x II(Iren,A(<J>K) - Iren,A(<J>KJ),
,=1
(1.12.22)
86
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
where so = 1, 0 = KO < ~1
< ... < KN
= K , and one has iterated the formula
Now, if we combine (1.12.16) and (1.12.18) with (1.12.10), we have
and therefore Holder estimation implies
On the other hand, the hypercontractive bound (1.12.19) implies
so if we estimate out the J'dsL-integrations, we obtain
from the convexity of the linear combination in the exponent and the factorial decay of the iterated integration. Moreover,
1.12. A TALE OF TWO DIMENSIONS
87
and standard convolution estimation in momentum space yields the bound CAI',;:o for some 6 > O. Hence
II exp( c::::
X
i
( X
X Fig. 1.13.3:
Iii A
Iii ~ (i iii £)
X
r,
Fig. 1.13.4:
The graphs in Fig. 1.13.4 converge in the K, = 00 limit, while the graphs in Fig. 1.13.3 are divergent. Obviously, the number of topologically different Feynman graphs escalates with the order, so the ultraviolet renormalization problem may appear formidable
1.13. T H E NEED FOR PHASE CELL ANALYSIS
101
just from a combinatorial and algebraic point of view. However, the problem can be simplified by the notion of primitive graph. A primitive graph is a connected divergent graph which becomes convergent if any one of its internal legs is replaced by two external legs. The second graph in Fig. 1.13.2 and the first graph in Fig. 1.13.3 are both primitive; there are no others in the entire perturbation series. Moreover, every connected divergent graph other than the third-order primitive graph is divergent only because it contains at least one second-order primitive graph as a subgraph. For convenience, we denote the basic quantities associated with these two primitive graphs by:
Note that for an arbitrary set of points XI,. . . ,X J ,
+ 4a2 j' , j l ' distinct
(
@ x j ) o j+j1.jtr
+
7
(1.134
while
It is a fundamental algebraic fact that the power series in X of the free expectation
is precisely the series obtained from the power series in X of
(fi j=1
m(ji) exp (-A
L
: @ , ( x ) ~: dx
by discarding all graphs containing primitive graphs as subgraphs. The algebra is a variation on the Linked Cluster Theorem, which we shall not pursue here, but will discuss in Chap. 3 while introducing polymer expansions. The point here is that a simple, direct modification of the interaction generates a perturbation series all of whose terms are finite because the divergent Feynman graphs are automatically omitted.
102
CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
On the other hand, this alteration is no renormalization - specifically, the new term NII.,,,(iP) in the interaction is not a mass renormalization. Actually, £A~~ poses no problem because it has no iP-dependence; it is just an energy renormalization. To implement a mass renormalization, define the parameter shift by
8m~,,,(x)
dx'
= 48>.2 [
(iP,,(x)iP,,(x'))~
(1.13.8)
and consider the problem of replacing Nil., " (iP) with the local term MII.,,,(iP)
= [ dx8m~,,,(x)
: iP,,(X)2 : .
(1.13.9)
On the other hand, (1.13.6) indicates that : NII.,,,(iP) : +£,e, 48>.2 [
dx [ dx'
(1.13 .10)
(iP,,(x)iP,,(x'))~ .
(1.13.11)
Moreover, it is the difference 48>.2 [
dx [
dx'
(iP,,(x)iP,,(x'))~
x : iP,,(x)(iP,,(x) - iP,,(x')) :
(1.13.12)
that contains no divergences as K, -t 00, so the additional energy renormalization £A~~ is required in this formal, perturbative scheme. Thus (1.13.13) is the actual renormalization of the interaction, while the renormalization cancellations are implemented term-by-term in the formal perturbation series by writing
The random variable (1.13.12) controls the difference between the algebraic cancellation of divergences and the renormalization cancellation. Note that
[<XX>
-2
+
], (1.13 .15)
which can be shown to be finite in the K, = 00 limit - i.e., the divergences in the diagrams cancel - so this random variable is square-integrable with respect to the free measure dJ.Lo .
1.13. THE NEED FOR PHASE CELL ANALYSIS
103
It has already been emphasized that the power series in the coupling constant ).. cannot converge, so term-by-term control of the series does not establish rigorous control of the model. In §1.11 the key importance of an ultraviolet stability bound (1.11.29) to a renormalized interaction was emphasized. In our case, the required bound is
Unfortunately, the underlying K, = 00 measure is actually singular with respect to the free measure dflO, so the exponential cannot be dflo-integrable. Thus, the Glimm-JaffeNelson ultraviolet expansion, which involves U(dflo)-estimation, cannot be applied here. The Guerra-Rosen-Simon results, while independent of dimension, are useless for the same reason. On the other hand, the Seiler-Simon estimation used to derive A-uniform bounds on normalized expectations can be adapted to this model because it depends on ultraviolet stability as an input bound and exploits OS positivity. The greatest difficulty lies in establishing (1.13.16), and what is significantly different about this problem in dimension d > 2 is that the ultraviolet limit and infinite-volume limit are not so neatly separated. Phase space must be decomposed into phase cells, subject to the constraints of the uncertainty principle.
References 1. T . Balaban , "Ultraviolet Stability in Field Theory: The c/>~ Model ," in Scaling and Self-Similarity in Physics, J . Frohlich, ed., Birkhiiuser, Boston, 1983. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical c/>~ Model," Commun. Math. Phys. 88 (1983),263-293. 3. G. Benfatto, M. Cassandro, G. Gallavotti, F. Nicolo, E. Oliveri, E. Presutti, and E. Scacciatelli, "On the Ultraviolet Stability in the Euclidean Scalar Field Theories," Commun. Math. Phys. 11 (1980),95-130 . 4. A. Bovier and G . Felder, "Skeleton Inequalities and the Asymptotic Nature of Perturbation Theory for <J>4- t heories in Two and Three Dimensions," Commun. Math. Phys . 93 (1984), 259-275. 5. D. Brydges, J . Frohlich, and A. Sokal, "A New Proof of the Existence and NonTriviality of the Continuum c/>~ and c/>~ and Quantum Field Theories," Commun. Math. Phys. 91 (1983), 141-186. 6. P. Federbush, "Unitary Renormalization of (c/> 4 h+1 ," Commun. Math. Phys . 21 (1971), 261-268. 7. J . Feldman, "The )..c/>~ Field Theory in a Finite Volume," Commun. Math. Phys. 37 (1974) , 93-120. 8. J . Feldman and K. Osterwalder, "The Wightman Axioms and the Mass Gap for Weakly Coupled (c/> 4 h Quantum Field Theories," Ann. Phys. 97 (1976) , 80-135.
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CHAPTER 1. MATHEMATICAL SKETCHES OF QUANTUM PHYSICS
9. G. Gallavotti, "Renormalization Theory and Ultraviolet Stability for Scalar Fields via Renormalization Group Methods," Rev. Mod. Phys. 57 (1985),471- 561. 10. J. Glimm, "Boson Fields with the ¢4 Interaction in Three Dimensions," Commun. Math. Phys. 10 (1968) , 1-47. 11 . J. Glimm and A. Jaffe, "Positivity of the ¢~ Hamiltonian," Fortschr. Phys. 21 (1973) , 327- 376. 12. K. Hepp, TMorie de la Renormalisation, Springer-Verlag, New York, 1970. 13. A. Jaffe, "Divergence of Perturbation Theory for Bosons," Commun. Math. Phys. 1 (1965) , 127-149. 14. J . Kogut and K. Wilson, "The Renormalization Group and the c:-Expansion," Phys. Rep. 12 (1974), 75-200. 15. J . Magnen and R. Seneor, "The Infinite Volume Limit of the ¢~ Model," Ann. Inst. H. Poincare 24 (1976) , 95-159. 16. Y. Park, "Convergence of Lattice Approximations and Infinite Volume Limit in the p.¢4 - U¢2 - l.ufJ)S Field Theory," J . Math. Phys. 18 (1977),354-366. 17. V. Rivasseau, From Perlurbative to Constructive Renormalization, Princeton University Press, Princeton, 1991. 18. E. Seiler and B. Simon, "Nelson's Symmetry and All That in the YUkawa2 and ¢~ Field Theories," Ann. Phys. 97 (1976), 470-518. 19. A. Sokal, "An Alternate Constructive Approach to the ¢~ Quantum Field Theory, and a Possible Destructive Approach to ¢:," Ann. Inst . H. Poincare A37 (1982) , 317-398. 20. E. Speer, "Analytic Renormalization ," J . Math. Phys. 9 (1968), 1404-1410. 21. K. Symanzik, "Small Distance Behavior in Field Theory and Power Counting," Commun. Math. Phys. 18 (1970),227- 246.
Chapter 2
Wavelets - Basic Theory and Construction Phase space analysis is any useful decomposition of a function into modes that are well-localized and - at the same time - have a small spread in momentum. The properties of the Fourier transform impose limitations on such a decomposition at the very outset, but the need for phase space analysis in quantum field theory has already been made clear in Chap. 1. It is important in other areas as well - e.g., partial differential equations and signal analysis. One limitation is that a mode cannot be sharply localized in both momentum and position - i.e., no compactly supported function has a compactly supported Fourier transform. Another limitation is that - while exponential localization in both momentum, and position is possible - a small standard deviation in momentum (resp. position) creates a lower bound on the standard deviation in position (resp. momentum) by the uncertainty principle. Further limitations are inevitable if one desires the set of modes (expansion functions) to be coherent in some sense - e.g., related to one another through group operations. Perhaps the oldest type of phase space analysis is the discrete windowed Fourier transform - an analysis based on expansion functions of the form
with 6 and ranging over cubic lattice sites in d-dimensional space with integer coordinates. These functions are coherent in the sense that they are generated by discrete translates in 2d-dimensional phase space with f as the generating function. Naturally, the regularity and decay properties of f are an important issue. In particular, if f is the characteristic function of the unit cube [0, lId, then each expansion function is sharply localized, but discontinuous. Moreover, {f;;) is an orthonormal basis for the Hilbert space L ~ ( R in ~ )this case; indeed, the decomposition is just a cubic partition of @ with Fourier series analysis applied on each cube. The jump discontinuities are the obvious drawback, as they correspond to poor localization in momentum sp_acz This implies that singularities in the function analyzed are not reflected by the (m, n)-dependence of the expansion coefficients. Can a more regular f be found such
106
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
that {f1ii;} is still an orthonormal basis? The answer is yes, but only by sacrificing localization; the trade-off is very strict, as it turns out. It has been shown by Balian [B9] and also by Low [L47] that if {f1ii;} is an orthonormal basis, then the Heisenberg uncertainty of f must be infinite! This means that no orthonormal basis of this type can be much better than the one generated by the characteristic function of the unit cube. The common practice in phase space localization of this type is to sacrifice orthogonality to obtain more regularity for f . The obvious constraint is that the calculation of expansion coefficients for a given function must still be realistic. One of the earliest proposals - independently examined by Gabor [G1] and von Neumann [unpublished] - was to choose f as a Gaussian function peaked at the center of the unit cube - a function with the best possible phase space localization. The price of this analysis was immediately evident in the over-completeness of the set {f1ii;} generated. Expansion coefficients were not unique for an arbitrary function decomposed in this way. Indeed, such an expansion was known to be numerically unstable in the sense that the sum of squares of inner products of these functions with the function to be analyzed could be arbitrarily small relative to the L 2-norm of the given function. On the other hand - since this set is over-complete anyway - one can enhance this overcompleteness by generating a set with phase space translates smaller than unity in the spatial directions or smaller than 211" in the momentum directions. Expansions of this type are numerically stable, and they have been exploited in signal analysis despite the computational inconvenience due to the redundancy (see, e.g., Baastians [B1] and Janssen [J8-J10J) . This decomposition scheme is an example of a frame - a certain generalization of orthonormal basis where the elements are not even necessarily linearly independent. Frames were first studied in the context of nonharmonic analysis by Duffin and Schaeffer [D28] and can be defined for an abstract Hilbert space. The frame property is a generalization of Parseval's identity to a double inequality for the sum of squares of inner products, where the upper and lower bounds are proportional to the square of the norm of the test vector. From a computational point of view, frame expansions are "painless non-orthogonal expansions," and their usefulness to phase space localization in signal analysis was widely advertised by Daubechies, Grossmann, and Meyer [D4]. They even constructed frames generated by a-translates in position space and translates < 211"/ a in momentum space where the generating function f is class Coo with compact support. Even as they established impressive results for the old-fashioned windowed Fourier transform, Daubechies, Grossmann, and Meyer were already engaged in the development of a rather different type of phase space analysis based on scaling - namely, wavelet analysis . The meaning of the word "wavelet" has varying degrees of generality in the literature, but the coherence of a wavelet basis or wavelet frame usually involves scaling operations. Indeed, the orthogonality of discrete scalings is a property that shall be included in our own definition of the word, so there are results on wavelet frames that lie outside the scope of our concerns. For our purposes, a dyadic wavelet is a square-integrable function llJ such that (2.2)
for all non-zero integers r. Note that this definition does not require orthogonality of lattice translates on the same scale or completeness of the basis generated. Negative values of r correspond to dyadic scalings of 9 to large length scales, while positive values of r yield small length scales. Throughout most of this chapter we concentrate on dimension d = 1 (the dimension appropriate to signal analysis in any case) and extend the wavelet constructions to arbitrary dimension in the very last section. This decision is due, in part, to the ease with which the L2-orthogonal constructions extend to the multi-dimensional case. On the other hand, we shall also construct Sobolevorthogonal wavelets in that section. Obtaining well-behaved wavelets of this type is more involved because the coherent structure is rectangular while the Sobolev norm is elliptic. The earliest example of an orthonormal wavelet basis was constructed by Haar [H6] decades before the subject was born. It is a dyadic scale hierarchy - including arbitrarily large and arbitrarily small length scales - of piecewise constant functions whose integrals vanish. On the same scale, the discrete translates are orthogonal because they have disjoint supports, while functions on differing scales are orthogonal because the smaller-scale function is supported on some rectangle over which the larger-scale function is constant. This type of decomposition has the same regularity and decay properties as the windowed Fourier decomposition with the characteristic function the functions are discontinuous, but their localization is sharp. The Haar basis was originally introduced in the context of a general study of orthogonal systems, and at the time, no one seemed to have any curiosity about the existence of similar bases of more regular functions. The special features of the construction may have led some mathematicians to suspect that nothing better than the Haar basis was possible. Since that time, ideas involving scaling transformations have slowly emerged in other areas. In statistical mechanics, for example, Wilson introduced the renormalization group [W16] as a means of isolating the long-distance behavior of the generalized Ising model. The basic transformation averages spin values over a block of sites with the cubic lattice partitioned into such blocks and the new configuration defined by scaling the coarser lattice back to the unit lattice. The Gibbs measure itself is transformed by integrating out the fluctuations and applying the additional transformation induced by the scaling. Iteration of this transformation has the effect of reducing the long-distance decay of correlation functions to behavior at large length scales. Moreover, in his effort to justify a rather drastic approximation aimed at obtaining qualitative results, Wilson assumed the existence of a wavelet basis [W17]. At roughly the same time, Glimm and Jaffe were developing phase cell methods in their effort to construct the pure scalar quantum field theory with quartic selfinteraction in three space-time dimensions [G45, G57]. We have already discussed this model in Chap. 1, where we advertised the need for phase space analysis. While their decomposition of phase space into phase cells was primitive and their expansion method complex, the convergence proof contained several ingenious arguments that involved scaling. In signal analysis, the use of scaling as an analytic tool was first proposed by Grossmann and Morlet [G96], inspired in part by the work of Morlet et al. [M48]. Specifically, they applied the wavelet transform to the analysis of seismic data in geophysics, using a basic signal function f (t). The one-dimensional wavelet transform \EE of an arbitrary
108
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
function E(t) is given by QjE
1 (a, s) = -
m
dt f (a-l(t - s))*E(t),
-,
a
# 0,
(2.3)
with a fixed choice of f (t). Subsequently, Grossmann, Morlet, and Paul studied this integral transform as a square-integrable representation of the affine group [G97, G981. The resolution of the identity (i.e., the inversion formula) is given by
((t) =
1,
1 da lal-512 , c(f) -
1, w
ds (a-'(t
- s ) ) 4 ( a ,s),
(2.4)
The finiteness of the constant c(f) is a condition on the basic signal f ( t ) ,which is regarded as a "wavelet" in this context. Note that if f ( w ) is continuous, then f(0) = 0.-This weak vanishing-moment condition is satisfied by wavelets in virtually every context, independently of how they are defined. Meanwhile, Stromberg constructed a one-dimensional orthonormal basis of wavelets more regular than the Haar wavelets [S85]. Inspired by the construction of an orthonorma1 basis for L2([0,11) due to Franklin [F53] - which is more regular than the Haar basis but has little wavelet coherence - Stromberg derived a piecewise linear, continuous function with exponential decay that generates an orthonormal basis for L2(R) through dyadic scaling and discrete scale-commensurate translation. This achievement attracted little attention at the time, as it occurred in the context of harmonic analysis, and interdisciplinary communication failed in this instance. In mathematical physics at about the same time, Gawedzki and Kupiainen were engaged in establishing a sound mathematical basis for the renormalization group ideas of Wilson. For the transformation they rigorously determined the flow of the iteration in the neighborhood of the Gaussian fixed point for the dipole gas [G16, G251. The type of phase space localization used for the definition of the renormalization group transformation varies with the context, but in the case where the transformation is based on block spin averaging, the constrained minimization technique for integrating out the fluctuations can be adapted as a method for constructing wavelets (see, e.g., [B12, B131). Concurrent to the work of Gawedzki and Kupiainen was the work of Federbush on a class of infrared problems [FlO]. Instead of using the renormalization group formalism, however, he developed a phase cell cluster expansion in much the same spirit as Glimm and Jaffe, except that his phase cells were expansion functions. Not only was his orthonormal basis scale-coherent, but it was a polynomial generalization of the Haar basis! His construction combined the Haar scheme with the homogeneity of monomials to yield a coherent basis of piecewise-polynomial functions with as many vanishing moments as one wished. The vanishing moments directly implied the interscale-orthogonality but were most important to the multi-scale decomposition of long-distance singularities. On the other hand, short-distance number divergences in the cluster expansion of a Euclidean field theory could not be cancelled in this phase cell formalism for a technical reason related to the jump discontinuities in these wavelets, so
109
the rigorous treatment of ultraviolet problems in this formalism had to wait on the discovery of smoother wavelets [BI2. BI3). The Stromberg construction was unknown to the mathematical physics community, but the Stromberg wavelet would have required modification, as it does not have an abundance of vanishing moments. Interest in wavelets finally quickened when Meyer constructed an orthonormal basis of wavelets that were smooth and yet of rapid decrease [M40). Indeed, the Fourier transform of a Meyer wavelet is class e oo with compact support! On the other hand, if a function has compact support in momentum space, it cannot have exponential decay in position space - only the rapid decrease, at best. Any basis of wavelets that are class eN for conveniently chosen N and have exponential decay would be more useful to mathematical physics in general. Subsequent to the Meyer construction, Lemarie found such orthonormal bases - one for each degree N of smoothness [L22) . Mallat and Meyer almost immediately realized that the construction of a variety of wavelets could be organized into a scheme which they dubbed multi-scale resolution analysis [M41 , M42). Any input function for this constructive machine is called a scaling junction, and its regularity and decay properties are shared by the wavelet produced. This point of view was central to the decomposition and reconstruction algorithm of Mallat [M8MIO], and that algorithm, in turn, eventually led to Daubechies' discovery of compactly supported wavelets with a finite but arbitrary degree of smoothness [DI). As in the case of windowed Fourier analysis, there was immediate interest in the advantages gained by sacrificing orthogonality in favor of some weaker property. One idea - already suggested by the constrained minimization technique [BI2) - was to sacrifice intrascale orthogonality while retaining interscale orthogonality. In fact, the variational approach differs from the multi-scale resolution analysis approach, since wavelets that are only interscale-orthogonal are derived first . In the Lemarie case, these wavelets are more explicit than those comprising the orthonormal basis. At the same time, the realization of the dual basis - necessary for the calculation of expansion coefficients - is equally explicit because the interscale orthogonality reduces it to a single-scale derivation (see, e.g., [BI7) and [CIO)) . Chui and Wang subsequently constructed a basis of this type where the wavelets are compactly supported splines [Cll, CI2). This is a computational advantage over Daubechies wavelets, which are not splines. On the other hand, the wavelets dual to the Chui-Wang wavelets do not have compact support. Another industry was the construction of wavelet frames, which we shall not pursue. However, they have had great impact on signal analysis, and it is worth remarking that Daubechies, Grossmann, and Meyer were the founders [D4) . Yet another generalization of an orthonormal basis is a bi-orthonormal basis, and the multi-scale resolution analysis for constructing bi-orthonormal wavelet bases was introduced by Cohen, Daubechies, and Feauveau [CI7) . In particular, they constructed a bi-orthonormal basis of compactly supported class eN wavelets that are more amenable to decomposition and reconstruction algorithms than are the Daubechies wavelets. We shall not pursue this construction either. The aim of this chapter is an introduction to wavelets suitable for Euclidean field theory and to rigorous results on the properties of wavelets. In the first section we discuss the limitations of the discrete windowed Fourier transform, including the BalianLow Theorem. In the following section we introduce the multi-scale resolution anal-
110
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
ysis of Mallat and Meyer, and we apply it to the construction of Meyer wavelets and Lemarie wavelets. We devote the third section to a sketch of the Daubechies construction of compactly supported wavelets. Actually, in mathematical physics there never seems to be any need for compactly supported wavelets; exponentially localized wavelets are always good enough for our purposes. We describe the construction of Daubechies wavelets because it is the crown jewel in wavelet analysis. The succeeding section is devoted to the important issue of vanishing moments. We show how the condition of interscale orthogonality implies that the number of vanishing moments for a wavelet is comparable to its degree of smoothness. This relationship, in turn, implies that interscale-orthogonal wavelets - functions that meet our own definition of "wavelet" - cannot be exponentially localized in both position space and momentum space. In §2.5 and §2.6 we investigate further theoretical restrictions on wavelets in the form of Heisenberg inequalities. In particular, the position-momentum uncertainty of a real-valued wavelet in one dimension is 3/2 instead of the universal 1/2. In §2.7 we describe the constrained minimization method for constructing wavelets. This variational approach can be used to construct a variety of wavelets - including Meyer wavelets and Lemarie wavelets - but it cannot produce Daubechies wavelets. Following this point of view, §2.8 is devoted to a description of the Lemarie-type basis that is not intrascale-orthogonal. For the mother wavelet there is an explicit expression depending on the zeros of Euler-Frobenious polynomials in the unit disk. We devote §2.9 to a description of the Chui-Wang wavelets. Finally, we introduce the extension of the wavelet constructions to the multi-dimensional case in §2.10. At the same time, we construct exponentially localized Sobolev-orthonormal wavelet bases, using the variational method. The multiscale resolution analysis, which is suited for the construction of Daubechies wavelets, does not appear to be suited for the construction of Sobolevorthogonal wavelets with good localization. On the other hand, Lemarie has shown that it can be generalized to construct a Sobolev-bi-orthonormal system of compactly supported wavelets [L26] .
2.1
The Balian-Low Theorem
In the search for orthonormal bases of L2(~) that localize phase space, perhaps the most obvious type to try is a basis of the form
B={ei21TmXf(x_n): m,nEZ},
(2.1.1)
where f is a square-integrable function . The example that comes to mind most readily is f = X, where X is the characteristic function of the unit interval [0,1]. In this case, one is simply applying the orthogonal decomposition 00
L2(~) =
ffi
L2([n,n + 1])
(2.1.2)
n=-oo
to the Hilbert space and then using Fourier series analysis on each interval. Unfortunately, X is a discontinuous function, which means its Fourier transform has peor
2.1. THE BALIAN-LOW THEOREM localization. The latter is regular but not integrable. Indeed,
where p = 0 is a removable singularity. It is natural to wonder if an orthonormal basis of the above form is possible for a function f that has better localization in phase space. Roughly speaking, one would be partitioning phase space into "phase cells" with a rectangular lattice structure given by Fig. 2.1.1. The basis decomposition of an arbitrary function would yield its spread over those phase cells, where the area of each phase cell is 27r. Obviously, sharp localization of a basis function in its associated phase cell is impossible, as no function can have compact support in both position space and momentum space. However, a Gaussian function has excellent decay in both position and momentum. Can such a function generate any kind of basis with these phase space translates? Obviously, a set of the form
cannot be orthonormal, but it does span L2(R); indeed it is over-complete. This set has been called the uon N e u m a n n basis in the physics community and the Gabor basis in the mathematics community. In spite of its drawbacks, this "basis" has actually played a role in the understanding of phase space localization. In particular, the study of this set led to some intuition that resulted in the Balian-Low Theorem. This result states that if a square-integrable function f generates an orthonormal basis of the form (2.1.1), then either fl(x) or x f (x) fails to be square-integrable. The original proof was actually quite ingenious: it was a topological argument involving the winding number of the phase of an expression. There is now a more elementary proof available, which we present here.
Figure 2.1.1:
112
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
Suppose that P f and X f both lie in L2 (JR). translation operator defined by
(TmnCP)(x)
Let Tmn denote the phase space
= ei27rmxcp(x -
n).
(2.1.5)
By hypothesis, {Tmnf: m, n E Z} is an orthonormal basis, so by Parseval's identity,
(Pf,Xf)
= "I)Pf, Tmnf) (Tmnf, Xj).
(2.1.6)
m,n On the other hand,
(f, [P, Tmnlf) (f[Tmn, Xlf)
= 0, -n(f, Tmnf) = 0,
27rm(f, Tmnf)
(2.1. 7) (2.1.8)
so the expansion may also be written as
(Pf,Xf) = "I)T- m,-nf,Pf)(Xf,T-m,-nf), m,n
(2.1.9)
which - again by Parseval's identity - implies
(Pf,Xf)
= (Xf,Pf).
(2.1.10)
Since [X, Pl = i, we have the desired contradiction. Actually, one can get around this difficulty if some of the discrete translational symmetry in phase space is sacrificed. Orthonormal bases of the form
B
=
{sin(27rmx)f(x-n): nEZ,mEZ+}U {cos(27rmx)f(x - n): nEZ, mE Z+ U {O}}
(2 .1.11)
have been constructed where f and j both have exponential decay. Such a basis is a Daubechies-Jaffard-Journe basis. Obviously, a given basis element is roughly localized on a phase cell defined by the intersection of a set of the form
{(x,p) : n::;x::;n+l} with a set of the form
{(x,p): 27rm - 71'::; p::; 27rm + 71' or - 27rm - 71' ::; p::; -27l'm + 7r}.
i- 0, the cell consists of two disjoint rectangles, each with area 271', and there are two basis elements associated to this disconnected cell . If m = 0, the cell is a single rectangle, and it has only one basis element. The two separated peaks in momentum space are what make such a tremendous difference in the phase space localization. It has been shown that if one insists on an orthonormal basis of the form
If m
B = {fm(x - n): m,n
E
Z}
(2.1.12)
2.1. THE BALIAN-LOW THEOREM
113
im
localized roughly at 27rm (but not necessarily a momentum-space translate with of by 27rm units), then one can beat the conclusion of the Balian-Low Theorem, but only marginally. X f and P f can both be square-integrable, but either IX l1+e f or IPI1+e f will fail to be. There has long been an industry in phase space localization where one works with a set of the form B = {ei27ramx f(x - /3n) : m,n E Z} (2.1.13)
io
with a/3 < 1. The inner products of an arbitrary square-integrable function with these functions constitute the discrete version of a windowed Fourier transform of that function, and there is a trade-off implicit in the choice a/3 < 1. The expansion functions can have better phase space localization, but they cannot be orthonormal. The latter point can be established with an elementary calculation. Indeed, suppose a set of this form is an orthonormal basis. Then for an arbitrary square-integrable function 'P,
II'PII~
~ a-I
11.: ~ 1.:
2
'P(X)e-i2rrmax f(x - /3n)* dx I 'P(X)'P(X - a-lf)* f(x - /3n)*
f(x - /3n - a-lf)dx
(2.1.14)
by Parseval's identity and Poisson summation. Notice that if the support of'P lies in an interval of length a-I, then only the f = 0 term remains:
II'PII~ =
a-I
L n
1.:
1'P(xWlf(x - /3n)1 2dx,
diam supp
'P:S a-I
(2.1.15)
Since 'P is arbitrary otherwise, it follows that
L Applying the integration
I:
If(x - /3nW
= a.
(2.1.16)
n
dx to this equation, we obtain
1.:
2 If(x)1 dx
= a/3.
(2.1.17)
But IIfll2 = 1, so we have established the claim that the inequality a/3 < 1 rules out an orthonormal basis. On the other hand, the lack of orthogonality does not necessarily pose computational problems in the phase space analysis of an arbitrary function. The windowed Fourier analysis is done with sets that happen to be frames - expansion functions for which the orthogonality property is replaced by a similarity property. We give the definition in the abstract setting. A given set {gd of vectors in a Hilbert space is said to be a frame if Ilgk II = 1 and there are constants ao, al > 0 such that
aoll'P112 :S
L 1('P,gk)1 2 :S alll'P1I 2 k
(2.1.18)
114
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
for all vectors eN" :::;
Cw (
r
Ipk+1~(P)12dP)
1/2 [ (
i1pl <eN"
+ N- rk (
r
i1pl >eN"
r
i1pl>eN"
IPI- 2dP) 1/2]
IPI-2k-2 dP) 1/2
134
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
< _
CI\I
Ilw (k+ 1)11
2
[J
2k
2
+ 1 c -k-!N-rk-!r
+ V2c-!N- rk -!r] O(N-rk- !r).
(2.4.17)
To estimate the integral over the former region, we apply Schwarz estimation to the q-integration in (2.4.10) to obtain for the remainder a bound complementary to (2.4.14) - namely, (2.4.18) Thus
r
Ii1pl S,eW
:S
eiXOP~(p)Rk(N-rp)*dPI
cll~(k+1)1I2N-rk-!r
r
1~(P)llplk+!dp
i1pl S,eW
:S
cll~(k+1) IbN -rk-!r ( r
i1 plS,eW
x (
r
i1pl S,eW
(1
(1
+ IPI)-l dP)
1/2
+ IPI)2k+21~(P)12dP) 1/2
O(N-rk-~r vln(Nr)),
(2.4.19)
so we have the desired type of bound on the large-r behavior of the integral. This completes the induction step in the proof (2.4.6). An interesting consequence of this fundamental theorem is that an N -adic wavelet cannot have exponential localization in both position space and momentum space. Indeed, suppose that w(x) and ~(P) both have exponential decay. Then xl-
IIP\]!1121IX\]!1I2 I(P\]!,X\]!)I 1((S+iD\]!,\]!)1 1
(2 .5.1)
2
(Although Ii, appeared as a parameter in §1.3, recall that we set Ii, = 1 in §1.8.) For states that are wavelets, the scaling generator will playa more quantitative role than it seems to play in this most basic estimation. First, suppose the state happens to be the derivative of some square-integrable function, with no other assumption made. Thus
\]!
\]! =P(,
(2.5.2)
so we have
IIXP(112 I (s + iD (112 II(S-iD(L Hence
"P\]!"211 (s -iD (II
(p2)Ij2(X2)Ij2
>
I( (s - D()I P\]!,
i
(2 .5.3)
136
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
(2 .5.4) because [P,S]
= -iP.
(2.5.5)
In this case, S plays a useful role before it is discarded. Next, suppose 1J1 is the nth derivative of a square-integrable ( . How is the Heisenberg inequality generalized? With (2.5.6) we propose to obtain a lower bound on (2.5.7) We appeal to the identity (2.5.8) which is easy enough to prove by induction. Indeed, if this identity holds for a given n, then
x
(g (S + (k - D))P i
XPg(S+i(k+D)
(2.5.9)
because (2.5.10) Combining (1.3.37) with (2.5.9) completes the induction step. Now apply this identity as follows: IIxnpn( lI ~
Ilg (S+i (k- D) (II: (g (S2 (k - D2) (, () +
2.5. UNCERTAINTY RELATIONS FOR (P) ~
II(II~
= 0 WAVELET STATES
tl (k - ~) IIS(II~ t II (k _~) tl (k - ~) ) ([ , 2
+
2
I (1nS ± i
( r n
l:II
1n
1=1 ki-I (
Thus,
(p2n)Ij2 (xzn)IjZ
1
k--2)
137
(2.5.11)
2
(2.5.12)
tl (k - D)(liz ±itl (k- D)) (I tl (k - ~) ) tl (k - D)
~
IlpnwllZl1 (1nS ± i
~
I(pn w, (1nS
I(w, (1nS - in1n ± i
pn() I
I(w, (1nS - in1n ± i
w) I'
(2.5 .13)
where we have used
lpn ,S]
=
-inP.
(2.5.14)
Clearly, the better choice in sign is -i, so we may conclude that
(pZn)IjZ (xzn)IjZ ~ n1n +
IT (k - ~) .
(2.5 .15)
k=1
This is the generalization of (2.5.4) that we infer from (2.5.8) . Since (2.5.6) is a translation-invariant condition, it is trivial to extend this result to the observation that (2.5.6) implies a E JR.
(2 .5.16)
Now for an arbitrary observable A, define the nth-order deviation as (2.5.17) Then (2.5.16) amounts to the nth-order uncertainty principle (2.5.18)
138
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
provided that
(P)q,
= o.
(2.5.19)
This additional condition is satisfied by all real-valued W as well as by W that are either symmetric or anti-symmetric about their barycenters (X)q,. For n = 1,2,3 we have: (2.5.20.1) (2.5.20.2) a( 3) (P)a( 3) (X) 'II
'II
> -
3
~v'259 + 15 ~ 77 . 4
8
32
(2.5.20.3)
For large n , the lower bound (2.5.18) is essentially linear in n. What does any of this have to do with wavelets? The point is that if W is a wavelet state and if (p2n)q, and (x2n)q, are finite - otherwise, any lower bound on the product would hold vacuously - then w satisfies the hypothesis of the vanishing moment theorem, with the conclusion that ~(k)(O)
= 0,
k~n-l.
(2.5 .21)
Clearly, the condition (2 .5.6) is a close relative of this vanishing moment condition. Indeed, suppose the wavelet state W satisfies the slightly stronger condition that (2 .5.22)
for some c5 > O. Obviously, this includes all of the wavelets we have constructed thus far: the Meyer wavelets are Schwartz functions , the Lemarie wavelets are exponentially localized, and the Daubechies wavelets are compactly supported. Now (2.5.21) implies the Taylor formula (2.5.23)
and we wish to infer (2.5 .6) - i.e., to show that p-n~(p) is square-integrable. The region Ipi ~ 1 is obviously the issue, so we need only show that
Ipi
~ 1,
(2.5.24)
for some E:: > O. To this end, we apply the Holder inequality - with exponent r > 2 to obtain
(2.5 .25) (2 .5.26)
2.5. UNCERTAINTY RELATIONS FOR ( P ) = 0 WAVELET STATES
139
Since s < 2, we have the desired bound with
provided
I ~ $ ( ~ ) I I 0. In summary, if an N-adic wavelet state Q satisfies the decay condition (2.5.22), then
+
(P~~)!$'~((x - a)2n)i'2 2 n-y,,
-+
(2.5.31) k=l
and this inequality is an uncertainty inequality if Q is also a (P)= 0 state (real-valued, for example).
References 1. R. Balan, "An Uncertanity Inequality for Wavelet Sets," to appear in Appl. and Comput. Harmonic Anal. 2. G. Battle, "Phase Space Localization Theorem for Ondelettes," J. Math. Phys. 30, NO. 10 (1989), 2195-2197.
3. G. Battle, "Heisenberg Inequalities for Wavelet States,'' Appl. and Comput. Harmonic Anal. 4 (1997), 119-146.
4. A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam, 1965. 5. M. Reed and B. Simon, Methods for Modern Mathematical Physics, Vol. 11, Fourier Analysis and Self-Adiointness, Academic Press, New York, 1975.
140
2.6
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
Further Constraints of Heisenberg Type
In the previous section, the inequalities we derived essentially depended on vanishingmoment conditions only. We now derive phase space localization inequalities that are more peculiar to the wavelet state condition defined in that section. For an arbitrary N-adic wavelet state IJI, it is interesting to measure the phase space deviations about the points mE Z, (x,p) = (m,O), (2.6.1) in phase space - i.e ., to obtain lower bounds on (2.6.2)
Since every integer translate of an N -adic wavelet state is still an N -adic wavelet state, we may set m = 0 without loss. Consider the n = 1 case and notice the effect of the orthogonality property on the expectation of the scaling group. We have -ie>./2 \ (
S + ~i) e-i>.S )
~
-ie>./2 (X Pe-i>'S)~.
Combining this with ei>'S Pe-i>.s
= e->' P
(2.6.3) (2.6.4)
and the fundamental theorem of calculus, one obtains
VN (e-i(ln N)Sh
- 1
['nN e- i >./2(X e-i>.s Phd>'
= -i J
(2.6.5)
o
for an arbitrary state. For an N-adic wavelet state, (e-ir(ln N)S)~
= 0,
r E Z\{O},
(2.6.6)
so set r = 1. By the Schwarz inequality, 1
~
IIPIJI 11 2II X
['nN
11% J
e->' / 2d>'
o
2(1 - 1jVN)IIP1JI11211X1JI1I2
(2.6.7)
for such a state IJI . Thus we have the lower bound (2.6.8)
In contrast to the last section, we have not assumed the decay condition that IXl 1+01J1 be square-integrable for some c5 > O. The generalization of this estimation to arbitrary n is an application of Rolle's Theorem in elementary calculus together with the property (2.6.9)
2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE
141
for all non-zero integers r. First, observe that
(_i)ne(n-~).\(xn pne-i.\S )w.
(2.6.10)
On the other hand, (2.6.4) obviously generalizes to (2.6.11) so we have (2.6.12) Next, observe that the operator
acts separately on the real and imaginary parts of any function of A. Thus
(_i)ne-A/2(xne-iAS pn)w = Un(A) if we define
+ iVn(A)
(2.6 .13)
e-.\ (e Ad~) k U(A),
(2.6.14.0)
(e d~) k V(A),
(2 .6.14.1)
Uk (A)
=
Vk(A)
= e- A
e.\/2 (e-iAS)w
A
= U(A) + iV(A) .
(2.6.15)
The property (2.6 .6) implies
u(rln N)
= v(rln N) = 0
(2 .6.16)
for all non-zero integers r. In particular, (2.6.17.0)
(2.6.17.1) Now observe that by Rolle ,s Theorem, there are sequences CBP) , ,B~l), ,B~l), ... ) such that
r In N < Ct~ll, ,B~l) < (r + 1) In N,
(1) (1) (1) ) Ct l ,Ct 2 ,Ct 3 , . ..
an d
(2.6.18)
142
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
Ul(a~I»)
= 0,
(2.6.19.0)
(,6~1»)
= o.
(2.6.19.1)
VI
Thus (2.6.20.0)
(2 .6.20.1)
Now apply Rolle's Theorem again - this time, to these zeros of e Aul (.>.) and . (2) (2) ) to obtam sequences (a 1(2) ,a2(2) ,a 3(2) ,.. . ) and ((2) ,61 ,,62 ,,63 , . .. where a(1) T
< a(2) < a(l) r r+l'
u2(a~2») ,6(1) r
l l'
cx \2)
(.>.) -
(2.6.21) (2.6.22)
< ,6(2) < ,6(1) r r+l'
V2 (,6~2»)
Clearly,
= 0,
e Avl
= o.
(2.6.23) (2.6.24)
d'>'3U3('>'3)
= - eA2u 2('>'2),
(2.6.25.0)
d'>'3V3('>'3)
= - e A2V2 ('>'2).
(2 .6.25.1)
A2
(3(2)
A2
Obviously, we can iterate this procedure for as long as derivatives exist. In general, we ) (k) (k) (k) . . .. have sequences ( a 1(k) , a 2(k) , a 3( k, » . . . and (,61 ,,62 ,,63 , .. . ) satlsfymg the conditIOns a(k - I) r
< a(k) < a(k-I) r
r+l'
uk(a~k») = 0, f./(k-I)
Pr
'k+JUk+J('>'k+I)
lA,
= -eAkuk('>'k),
(2.6.30.0)
= -eAkvd'>'d·
(2 .6.30.1)
(3~k)
{
lAk
d'>'k+J Vk+d'>'k+d
Pulling this together, one obtains (2.6.31.0)
2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE
143
(2.6.31.1) The second iterated integral equation is useless; the imaginary part of (2 .6.13) contributes nothing to our lower bound. We focus on the first equation and notice that the smallest numbers o~k) in the sequences have the ordering In N
< 0 (2) < . .. < 0 (n-l) . < 0 (I) 1 1 1
(2.6.32)
Moreover,
oik ) < O~k-l) < 0~k-2) < . .. < 0~1) < (k
+ 1) InN,
(2 .6.33)
so we have the estimate
Finally, we apply (2.6.13): (2.6.35) We obtain 1
< (2.6.36)
and so we have the lower bound (2.6.37) We have derived this inequality without the assumption that IXl n +c51J1 be squareintegrable for some /j > O. Notice also that this lower bound can be estimated independently of the basic scale factor N:
::; r
1
10 n
dTI
( ' dT2 ...
10 1
II k_l'
k=1
(n-,
10
dTn
Fn (2.6.38)
2
so we also have the weaker bound (2 .6.39)
144
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
On the other hand, for the dyadic (N = 2) case, the bounds for n = 1,2,3 are:
= ~(2 +../2)
(P2)1/2(X2)1/2> _1_ IJI IJI - 2-../2
-V~
IJI
(p6);f6(X6);f6?:
3
(2.6.40.1)
'
f 6 = V17 /~(5 + 2../2) '
(p4)1/4(X4)1/4 > IJI
2
120 = 54 - 23V2
3
~(54+23../2) . 929
(2.6.40.2)
(2 .6.40.3)
This case is of interest because the wavelets that are routinely constructed are dyadic unless one desires a special property that requires N to have some other value. How can the estimate (2.6.37) enhance the previous lower bound derived from the assumption that IXI~+8 is square-integrable? In the n = 1 case, one certainly has the estimate (2.6.41) from the previous section. To date, no N -dependent enhancement of this lower bound has been found - only the N-dependent enhancement (2.6.8) of the universal lower bound. In this lowest-order case, we can only compare the lower bounds. Clearly, (2.6.40 .1) is stronger than (2.6.41) but for N ?: 3, (2.6.41) is stronger than (2.6.8) . For n > 1 we have more interesting estimates. In these higher-order cases, we can derive an N-dependent enhancement of the corresponding lower bounds derived from this decay in the last section. Consider the n = 2 case first: with IXI 2+°'l1 assumed to be square-integrable, we may infer from the previous section that (2.6.42) for some square-integrable ( . Thus IIX 2P2( 11 ~
II(S+iD (S+iD (II: ( ( S2
+
(S4(, ()
D + D(, () (S2
+ ~(S2(, () +
9 16 ((, ()
1 5 9
?: 4 >,4 ((1 - COS>'S)2(,() + 2(S2(,() + 16(('() 4 :411(1-
cos>'S)(II ~ + II Gv'IOS -
iD (Ii:,
(2.6.43)
where we have used the operator inequality (2.6.44)
2.6. FURTHER CONSTRAINTS OF HEISENBERG T Y P E Therefore,
1
= 4-I(*,
(1- cos X(S - i2))p25)I2
A4
Now set X = In N and apply (2.6.6) to obtain
so we have 4 1/4 x 4 1/4
( P ) (~ )
-
>
(
10 + 16 + (1nN)
Notice that without the N-dependent term, this inequality would be given by t h e n = 2 case of (5.2.15), so we have enhanced the latter inequality with a dependence on the basic scale factor N . How does this new inequality compare to the n = 2 case of (2.6.37)? It is stronger regardless of the value of N . Indeed
so actually, the n = 2 case of (5.2.15) is itself stronger for all N. The advantage of (2.6.37) lies in the absence of the decay assumption. We now derive the N-dependent lower bound for n > 2 with the decay assumption that IX)n+6KPis square-integrable for some 6 > 0. Let be the square-integrable function such that !I = PnC, (2.6.49)
0 defined by (2.6.51) Now for
e=
2m (even
e)
we apply (2.6.44) with A = In N to obtain
(s4m(, () 2: 4m (In ~)4m ((1 - cos (In N)s?m(, () , while for
e= 2m + 1 (odd e)
(2.6.52)
we apply (2.6.44) as follows:
(s4m+2(, () 2: 4m
A~m (S2(1 -
(2.6.53)
cos Ams?m(, (),
m
where the value of Am will be chosen below. Consider the case where n itself is odd, and pair consecutive terms in the following way:
IIxnwll ~ 2:
~(n-l)
[
(n)
+ a~~,!l (S2(1 ~(n-l)
fo
_ cos Am s)2m(, ()
]
mI Ctm Ja~'J.+l (1 - cos Ams)m S
4
- i (In
Hence
(n)
~ 4m (Jna~)4m ((1 - cos(Jn N)s)2m(, ()
~)2m Ja~'J. (1 -
cos (In N)s)m)
(II: .
(2.6.54)
2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE
- i (In
~)2m Va~~ (1 -
~(n-l)
~O
4
m
147
cos (In N)8)m() 12
(it, >.tm va~~+l(l- cos>'m(8 - in))m
1
x (8 - in)pn( - i
(ln~)2m Va~~(l _cos (In N)(8 _ in))m pn() 12
~(n-l)
'" 4 (it ' _1_ g;;J (1 _~ei>·~(S-in) L >.2m V 2 m 1
m=O
m
a2~+1
(2.6.55) Now notice that the expansion
(1 _~eiA(S-in)
_
~CiA(S-in)) m
f (m) (-~)~t
~=O x
J-L
2
ei(~-2q)A(S-in)
q=O
(J-L) q
(2.6.56)
together with (2.6.6) implies
(2 .6.57) Define (2.6.58)
and observe that our inequality may be written in the form
148
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
The significance of wm(A) is the wavelet property w,(lnN)
= wm(21nN) = 0.
Applying Rolle's Theorem to Im wm(X), we may choose A,
such that
For such a choice of A, we obtain an explicit lower bound by discarding the real part of the modulus, which includes the contribution by the real part of wA(Xm). Thus
In the case where n is even, the nth term is not paired with anything, and our estimation yields:
In both cases, we recover the lower bound (5.2.15) in the N = ca limit. In the n = 3 case, (2.6.63) is applicable and the m-summation consists of an m = 0 term and an m = 1 term. We have
2.6. FURTHER CONSTRAINTS OF HEISENBERG TYPE
149
as the q-summation consists of only the q = 0 term. It is easily seen that 225 o - 04'
a(3) _
a(3) _ 1
(3) _ -
a2
a 3(3)
259
-10'
-
(2 .6.66)
35
4' 1,
and therefore
(P6)1/6(X6)1/6> [45 v'259 9549 III III 16 + 64
_1_ (3V35
+ (In N)4
+
149)] 1/6 4
(2.6.67)
Since In 2 < V2, the dyadic case enhances the n = 3 case of (5 .2.15) considerably. In the n = 4 case, (2 .6.64) is applicable, but the m-summation still consists of only an m = 0 term and an m = 1 term. We have 2
1 ao + (InN)4 (4j:Jfi + Pf) 16 x (2 M + 2Pf) a + (InN)8 (4)
(4)
(4)
(4)
2
2
1 2
+4
( 1 ( 1) )
,(2.6.68)
where the q-summation in the last contribution consists of both a q = 0 term and a = 1 term. The coefficients a~4) are:
q
(4) _
ao
11025
--zso'
_ 3229 a 1(4) -"10'
_ 987 a 2(4) 8'
(Z
+ 41)
(z
9) +4
(z
25) + "4
(z
49) = t:o ~ at(4) z t , +"4
(2.6.69)
a4(4) -1 , and so we have:
Notice that in the dyadic case, this lower bound is drastically larger than the lower bound provided by (5.2.15).
150
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
References 1. G. Battle, "Heisenberg Inequalities for Wavelet States," Appl. and Comput. Harmonic Anal. 4 (1997), 119-146.
2. A. Messiah, Quantum Mechanics, Vol. I, North-Holland, Amsterdam, 1965.
2.7
Variational Construction
We have already described the Meyer-Mallat construction of various wavelets, and their approach is the most widely accepted one. We now describe a variational method, which is actually more natural to the renormalization group formalism of Chap. 4. It is also the natural way to construct multi-dimensional Sobolev wavelets in $2.10. On the other hand, this method of construction cannot produce Daubechies wavelets. Our starting point is a scaling function q whose integer translates are not necessarily orthogonal. The first and major step in the variational approach is to construct a function $ E L2(JR) such that
in that Hilbert space - i.e., a dyadic wavelet as defined in the introductory discussion of this chapter. If the set generated spans the Hilbert space, then a mother wavelet P for an orthonormal wavelet basis is given by
i.e., the standard Z-translation-covariant orthogonalization of the set of unit-scale basis functions. In position space, this is nothing more than the application of the inverse square root of the overlap matrix to the integer translates of $, and the same operation was applied directly to the integer translates of the scaling function q as a first step in the Meyer-Mallat construction. The function $ to be constructed is first defined as the solution of a constrained minimization problem: $ minimizes the L2-norm llClla with respect to the constraints
where {c,) is the sequence of coefficients in the scaling relation (2.2.1). The driving principle is quite simple. The linear constraints define a closed afine subspace of L2(R) - closed, because the linear functionals are bounded - so $ exists and is unique; it is the point in the affine subspace nearest to the origin. Therefore $ is orthogonal to every vector in the linear subspace associated with the affine subspace - i.e., if
2.7. VARIATIONAL CONSTRUCTION
151
for all m, then $ I C in the Hilbert space. How is this related to the scales? The point is that for r > 0,
..-m, and the transformation ml t+ ml, m2 I-, m2,. . .,m, I+ 1-2rm+2n-2r-1ml of summation indices has the effect of reversing the sign of this multiple sum. Therefore, the sum vanishes, and so -a
This establishes the interscale orthogonality property by the argument above. Now since $ minimizes a quadratic form - namely 1ICII; = (C, C) - with respect to linear constraints of the form (2.7.7) (C, gm) = dm, $ has the formal representation
On the other hand, K can be explicitly realized in momentum space if we set gm(x) = 2q (22 - m) . Indeed, by the Poisson summation formula, we get
(2.7.10)
152
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
Thus, an arbitrary power of the operator yields (2 .7.12)
and so it is trivial to verify that (2.7.13)
Since K 9n(P)
Gp)~ g,;(p +
17
e-i~np17 Gp) L
=
e
Gp + r Gp + 12
41ft)17 117
2m:
21ft)
(2 .7.14)
it follows that
e-i~np17 Gp) X
(1 + ~ Gp + e-i~np17 Gp) ~ Gp + 12 + ~ Gp + e-i~np17 Gp) ,
(1
Q
Q
117
Q
117
21ft)
117
21ft) 12) -1
21ft)
n
-1
(2.7.15)
and therefore (2.7.16)
In this case, the linear constraints are (2.7.17)
and so we have the solution explicitly given in momentum space: (2.7.18)
with h given by (2 .2.6) . Property (c) in §2.2 and the summability of {cm } guarantee that this momentum expression is well-defined and continuous, so '1j; is certainly squareintegrable if T/ is. It is worthwhile to remark that (2.7.19)
2.7. VARIATIONAL CONSTRUCTION
153
with llr defined via Property (a) in §2.2. To see this, we set r = 0 without loss of generality (by the dyadic scale covariance of the construction) and note that it suffices to show that 1)(2x) and 1)(2x - 1) can both be expanded in the 'IjJ(x - n) and 1)(x - n). In momentum space this would mean
fJ Gp) = f(p)fJ(P)
+ g(p);P(P) ,
(2.7.20)
e-i~fJ Gp) = j(p)fJ(P) + g(P);p(P)
(2.7.20')
for 27r-periodic functions f, g,j, g with square-summable Fourier coefficients. Inserting (2.2.7) and (2.7.18) and dividing out the resulting common factor fJ(!p), these conditions reduce to 1=
~h(P)f(P) - 2e-i~p ( ~ IfJ Gp + 27r£) r)
1=
~ei~P h(p)](p) - 2 ( ~ IfJ Gp + 27r£)
-1
n
-1
h(p + 27r)* g(p),
h(p + 27r)* g(p) .
(2.7.21)
(2.7.21')
On the other hand, the 27r-periodicity of the unknown functions imposes the additional conditions
n
1=
~h(P + 27r)f(P) + 2e-i~p (~lfJ Gp + 27r£ + 7r)
1=
-~ei~Ph(p + 27r)j(p) - 2 (~Ii) Gp + 27r£ + 7r) r)
-1
h(P)*g(P) ,
(2.7.22)
-1
h(p)*g(p) .
(2.7.22')
(2.7.21) and (2 .7.22) constitute a 2 x 2 linear system for f(p) and g(p), as do (2.7.21') and (2.7.22') for ](p) and g(p). Solving for f(p) and g(p), we obtain
f(P)
g(p)
~
(~I" Gp + ,., + • ) IT'
,, 1 and it is the only case where there is a residue at Z = O. We now derive the position space representation for the dual basis, which can be calculated on each scale because the basis is interscale-orthogonal. As always, we concentrate on the unit-scale functions - i.e., the unit translates of the mother wavelet - without loss of generality. The overlap matrix is Smn
J ~ 1 e-i(m-n)pl~(p)12dp ~ 1" L I~{p +
=
1(J{x - m)1(J{x - n)dx 00
27r
- 00
e-i(m-n)p
27r -"
so the dual basis {u m
}
27r£Wdp,
(2.8.26)
l
is given by
n
~ 27r
L
1(J{x - n)
n
1" -"
e-i(m-n)p
(L I~{p +
27r£)12) -1 dp . (2.8.27)
l
The Poisson summation formula
(2.8.28) n
160
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
implies
~ 27r
2: 17T-7T
ei(x-m)(p+27T j );j(p + 27rj)
~1
00
27r
(2: 1;j(P +
27rfW) -1 dp
l
j
ei(x-m)p;j(p)
(2: 1;j(P +
27rf) 12)
-1
dp.
(2 .8.29)
l
-00
Thus
Um(X)
= uo(x -
m),
(2.8.30)
and in momentum space, (2 .8.31) On the other hand, (2.8.11) implies
(2.8.32) Hence
uo(p)
-~e-i!P(1 -
ei !p)N+1 RN
(-COS Gp) ) fJ Gp) N+1
X[RN (COS Gp) ) 11 + ei !PI2N+2 + RN
(-COS Gp) ) 11 - ei!PI2N+21-1
(2 .8.33)
For the complex integration variable (2.8.16), we obtain Uo(x)
(2.8.34) m
1
1 z - -z 1 -1) -1. z m-2( 1- Z )N+1 RN (-27rt Izl=l 2 2
2.8. WITHOUT INTRASCALE ORTHOGONALITY
161
1 - 2Z-1 1 ) (1- z)N+l(l_ Z-I)N+l + RN ( -2Z
~
1
]-1
dz
zN+m-l(l_ z)N+IKN(-z)(-I)N
27ft 1%1=1
X
[KN(Z)(1
+ z)2N+2 -
(1 - Z)2N+2 KN( -z)tldz,
(2.8 .35)
while the alternate substitution (2.8.18) yields
(2.8.36) As involved as these contour integrals appear to be, the residue calculations are very closely related to (2.8.23)-(2.8.25), as we now show. First note that the variable relation z = eipl obviously yields the relations
4z (z-I)2 '
. dfl
tZ-
dz
(2.8.37) (2.8.38)
,
z=eip'
from which it follows that 1
RN(COSp)
=
4 N . 2N+2 (2N + I)! sm ~N X
2
dpI2N
CSC
(12 PI)
(12 PI) 4N = (2N
(z _1)2N+2
+ I)! (_4z)N+I
d )2N [ -4z J x ( iz dz (z _ 1)2 1
(2N
+ I)!
(z - 1)2N+2 ( d) 2N [ z 1 zN+I z dz (z - 1)2 .
(2.8.39)
Second, we insert (2 .8.20) to get 1
KN(Z)
= (2N + I)!
(z - 1)2N+2 ( d) 2N [ Z ] z z dz (z - 1)2 ,
(2.8.40)
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
162
which immediately implies KN(z)(l
+ Z)2N+2
- KN( -z)(l - Z)2N+2
(z2 _ 1)2N+2 ( d) 2N [ z z ] z(2N + I)! z dz (z - 1) 2 - (z + 1) 2 .
(2.8.41)
Next, we observe that Z
z
(z - 1)2 and the variable relation w
= Z2
(z
+ 1)2
4z2 - (z2 - 1)2'
(2.8.42)
yields
d f( z2) z -d
z
df I = 2wdw w = z2
(2.8.43)
Finally, (2.8.44)
and we may now insert this in (2.8 .35) and (2.8.36) to obtain bm
4-N~1 27l'l
1
4- N - 1 ---.27l'l
1
Izl=1
zm+N-2(1 - z)N+l KN( - z) (-1): dz, KN(Z )
(2.8.45)
(_l)N z -m-l( z - l)N+l KN( - z) - - 2 - dz , KN(Z )
(2.8.46)
Izl=1
respectively. Both 1j; and Uo are exponentially localized, and we are now in a position to determine the rates of exponential decay. Clearly, it suffices to determine the exponential decay rates of the sequences {am} and {b m }, since T] has compact support. Without loss of generality, we consider m ~ 2 - N or m ::; min{ -1,1 - N}; in the former case, am is given by (2.8.23), and therefore am = O(pm) for large positive m, where p =
max , la,1< 1.
(2.8.47)
In the latter case, am is given by (2.8.24), and so am = O(p-m) for large negative m. This means the rate of exponential decay in either direction is the positive number -Inp. Now consider the same two cases for the sequence {b m }; in the former case, we apply (2.8.45) to obtain N 1 bm = _4- - ~ Res (zm+N-2(z - l)N+l i:v~~:1'
Thus bm
= O(pm/2) for large positive m.
±yQ;) .
(2.8.48)
In the latter case, we apply (2.8.46) to obtain
N 1 bm = 4- - ~ Res (z-m-l(l - z)N+l i:v~~:1'
±yQ;) ,
(2 .8.49)
which implies bm = O(p-m/2) for large negative m. Therefore, the rate of exponential decay in either direction is - ~ In p. This incidentally proves that the exponential decay rate of uo(x) is half that of 1j;(x), regardless of the value of N.
2.9. CHUI-WANG WAVELETS
163
References 1. G. Battle, "Cardinal Spline Interpolation and the Block-Spin Construction of Wavelets," in Wavelets - A Tutorial in Theory and Applications, C. Chui, ed., Academic Press, San Diego, California, 1992, pp. 73-93. 2. C. Chui and J. Wang, "A Cardinal Spline Approach to Wavelets," Proc. Amer. Math. Soc. 113 (1991), 785-793. 3. P. Lemarie-Rieusset, "Une Remarque sur les Proprietes Multi-Resolutions des Fonctions Splines," C.R. Acad. Sci. Paris 317 (1993), 1115-1117. 4. P. Lemarie-Rieusset and J . Kahane, Fourier Series and Wavelets, Gordon and Breach, Luxembourg, 1995.
2.9
Chui-Wang Wavelets
There is a class of interscale-orthogonal wavelet bases whose mother wavelets are compactly supported splines. For such a basis, the mother wavelet of the dual basis is not compactly supported, but the interscale orthogonality of the compactly supported splines in the former basis is remarkable. The mother wavelet is a Chui-Wang wavelet, and the point of view of the construction lies somewhere between the Meyer-Mallat approach and the variational approach. We begin with a brief discussion of the wavelets constructed in the previous section to describe this intermediate point of view. Let T)M be the scaling function given by (2 .9.1) and consider the L2-subspace H(M) spanned by the integer translates of T)M . In approximation theory the fundamental M th-order cardinal interpolating spline is the solution to minimizing 11(112 in H(M) with respect to the constraints mEZ .
(2.9.2)
Standard notation for the solution of this problem is LM(X), and we denote the expansion of this fundamental spline by (2 .9.3) n
Obviously, this is not an orthogonal expansion, but the coefficients are uniquely determined. Indeed, they can be explicitly realized in terms of the zeros of the 2Nth-degree Euler-Frobenius polynomial in the unit disk of the complex plane. The crucial observation here is that the derivative function
164
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
is actually the mother wavelet of an interscale-orthogonal basis. Indeed, if we integrate by parts we have
J
Lg)(2x - 1)1)J(2 r+1 x - m)dx
(_I)J2 rJ
J
L2J(2x
2rJ / L2J(2x -1) J
2rJ - r- 1 ~)-I)j
_1)1)~J)(2r+lx -
m)dx
~(-I)j G)8(2 + r
1
X -
m - j)dx
(~)L2J(Tr(m + j) -1).
(2.9.4)
J
j=O
< 0, so
By (2.9.2) these evaluations of L2J all vanish if T
L~~)(2x - 1) ..L span{1)J(2 r+1 x - n): n E Z},
T
< 0.
On the other hand,
n
(2.9.5) Therefore, T
< 0,
and so the interscale orthogonality is established. This wavelet basis is precisely the interscale-orthogonal Lemarie basis constructed in the previous section. To see this, observe first that
JL~~)(2x JL2J(2x-l)1)~J)(2x-m)dx J I) (J) - 1)1)J(2x - m)dx
(_I)J
L2J(2x - 1)
-1)j
)=0
8(2x - m - j)dx
J
"(J)
J ~(-1)1 j L2J(m + j - 1)
_(_l)m
t (~)81-m,j , j=O
J
(2.9.6)
2.9. CHUI-WANG WAVELETS
165
Second, observe that this sum is just -( -1)mci~m' where c(J) m
= ~ (J)8 . . mJ
(2 .9.7)
~
j=O
J
are the coefficients for the scaling relation (2.9.8) n
so the constraints (2.7.3) are satisfied by -L~~~~(2x -1). On the other hand, for any L2-function f satisfying (2.7.3), the orthogonal projection E(N+l) f of f onto 1i(N+l) satisfies (2 .7.3) as well- and it is the only element of 1i(N+l) satisfying (2.7.3). Thus E(N+l) f is the solution to minimizing the L 2-norm with respect to the constraints (2.7.3) . This implies -L~~~;)(2x -1) is that unique element of1i(N+l) and therefore the function 1j;(x) that minimizes the L2-norm with respect to (2 .7.3) . This function is the wavelet given in the previous section, so the claim is established. We could have made just this identification _
(N+l)(
1j; (x ) - -L2N+2 2x - 1
)
(2.9.9)
by simply calculating both quantities in momentum space, but the reasoning we have just given is more to the point. We now consider the Chui-Wang wavelet, which can actually be written as (2.9.10) v
The verification of interscale orthogonality is the integration by parts
J
r 1j; (x) r/J (2 +lx - m)dx
L ( -W'T}2J(1I v
+
1)/ 'T}~~)(2x
x 'T}J(2 r +lx - m)dx (_I)J2 rJ L ( -W'T}2J(1I
+ 1)
- II)
J
'T}2J(2x - II)
v
x 'T}:,J) (2T+lX - m)dx 2rJ L(-lt'T}2J(1I v
+ 1)
J
'T}2J(2x -II)
166
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
This numerical expression is identically zero for r tion exploits the familiar identity
< O! Why? This ingenious construc(2 .9.12)
v
which follows from the observation that the sum is reversed in sign by the index change v t-+ 2n - 1 - v. We have already used it in other wavelet constructions, but this application is more subtle. In this case, clL = 1J2J(/-l) and n = 2- T (m + j), r < O. We may now argue as we did in the case of the spline Lg) (2x - 1) above. Since we have just shown that r
< 0,
(2.9.13)
we conclude that r
< 0,
from the observation that
where the point of this last equation is that '1j;(2TX - m) lies in the span of the 2- r - 1 Z_ translates of 1JJ(2 T + 1 x - 2m). How do we know that '1j; has compact support? Since 1J2J is obtained by convolution of X with itself 2J times, we know that each half-integer translate of 1J~~) (2x) is compactly supported. On the other hand, the linear combination (2 .9.10) of these translates is a finite sum because 1J2J(v + 1)
= 0,
v
< 0 or v
~
2J - 1.
(2.9.15)
Specifically, '1j;(x) is supported in the interval 0 :S x :S 2J - 1. As far as regularity is concerned, '1j; has the same degree of smoothness as 1JJ has - namely, class C J - 1 - , We now examine the dual basis, which is itself interscale-orthogonal. To compute the unit-scale elements, we apply (2.8.30) because general formulas from the previous section apply here. The functions are unit-scale translates of u, where u(p) =
(~I~(p + 211"1!12) ~(p).
(2.9.16)
-1
On the other hand, the Fourier transform of (2 .9.10) is just
~(p)
=
r
(i~P
TJ2J
Gp) 2) Gp) 2)
-W1J2J(V
+ 1)e- i !vp
v
(_1)J (e-i!p - 1)JTJJ
-lt1J2J(v
+ 1)e- i !vp
v
(2.9.17)
2.9. CHUI-WANG WAVELETS
167
where the Fourier sum (2.9.18) v
is 47r-periodic. Hence
le-i~p +
11 2JI>'(p + 27r)12 ~
IfiJ Gp + 27rt) 12
le-i~p + 11 2J I>,(p)12 ~ 117J Gp + 7r + 27rt) 12
le-i~p -11 2J I>'(p+ 27rWR J- 1 (cos Gp)) + and so
2~
u(x)
le-i~p + 11 2J I>'(PW R J- 1 (-cos (~p) )
i:
eipx{i;(p) (
lt
I{i;(p + 27rf) 12 )
11 e,pxilJ (1) 2 >.(p + 00
=
27r
-00
.
P
x I>'(p + 27r)12 R J -
1
(2.9.19)
dp
.
1 1 27r)(e-' 2P - 1) J [ le-'.2P - 11 2J
(cos Gp) )
XRJ-1 (-cos Gp))
-1
,
r
+ le-i~p + 112JI>,(p)12
1dP
-2 I:b m1)J (2x - m),
(2.9.20)
m
where - following the general formulation in the previous section - we have
_~ 47r
tTf ei~pm >.(p + 27r)(e-i~P _ I)J [le-i~P _ 112J
10
x I>'(p + 27rW RJ-1 (cos
(~p ) ) + le-i~p + 112J
r 1
x 1>'(pW RJ-1 (-cos Gp) )
~
1
27r% Izl=1
z-m-1
I: 1)2J(V + 1)( -zt(z -
I)J [( z - I)J (Z-1 _1)J
v
xI: 1)2J(V + 1)1)2J(v' + V1V'
dp
1)( -zt- v ' RJ-1
Gz+ ~ z-1)
168
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
if we set z = e - i a ~ . On the other hand,
[
+
~ - 1 ) expression Same V ~ J ( V l ) ( - ~ ) -(2-l
1-'
LIZ (2.9.22)
if we set z = e e p . The coefficients are computed by residues; to avoid high-order poles at z = 0, it is most convenient to use (2.9.21) for m 5 0 and to use (2.9.22) for m 2 1. By (2.8.20) we may write (2.9.21) as
and (2.9.22) as
x
[polynomial same I-'
dz.
The residues are determined by the zeros of the bracketed polynomial. Since almost none of the coefficients b, prove to be zero, the dual wavelet u(x) cannot have compact support. On the other hand,
so u(x) is exponentially localized.
References 1. G. Battle, "Cardinal Spline Interpolation and the Block Spin Construction of Wavelets," in Wavelets - A Tutorial in Theory and Applications, C.K. Chui, ed., Academic Press, San Diego, 1992, pp. 73-93.
2. C. Chui and J. Wang, "On Compactly Supported Spline Wavelets and a Duality Principle," Trans. Amer. Math. Soc. 330 (1992), 903-915. 3. C. Chui and J. Wang, "A Cardinal Spline Approach to Wavelets," Proc. Amer. Math. Soc. 113 (1991), 785-793.
2.10. MULTI-DIMENSIONAL WAVELETS
169
4. C. Chui and J. Wang, "A General Framework of Compactly Supported Splines
and Wavelets," J. Approx. Theory 71 (1992),263-·304. 5. P. Lemarh~-Rieusset, "Une Remarque sur les Proprietes Multi-Resolutions des Fonctions Splines," C.R. Acad. Sci. Paris 317 (1993), 1115·-1117.
2.10
Multi-Dimensional Wavelets
The extension of the one-dimensional constructions to an arbitrary number of dimensions is quite easy, but it is based on tensor product combinations of the mother wavelet with the unit-scale-orthogonalized scaling function cp given by -1 / 2
ip(p) ==
(
~ Ii)(p + 27reW )
i)(P) .
(2.10.1)
Too often, the remark is made that a one-dimensional wavelet basis can be extended to a multi-dimensional basis by a tensor product. Such a remark is misleading because a straight tensor product of copies of a wavelet basis yields basis functions that can have length scale 2100 in the north-south direction and length scale 2- 100 in the east-west direction! The point is that a multi-dimensional wavelet basis with only one scale parameter is more desirable - actually, a necessity from a practical point of view. For d dimensions, consider the special vectors -; E {O, 1 }d\ {O} and define
WCe-)(T·) ==
II cp(x,) II w(x,). Et=O
(2.10.2)
El.=l
The functions (2 .10.3) with r E Z, r;;E Zd, constitute an orthonormal basis of L 2(JRd) . The functions wCe-) are all mother wavelets in this construction, and there are 2d - 1 of them; the function cp(x,) is not included among the mother wavelets.
n
The moments of a function ~
if the functions j( x)
n x" d
,=1
d
E s, :S M.
,=1
Notice that if
je:;)
are said to vanish up to order M in d dimensions
are integrable and have vanishing integrals for s, 2: 0 and
je:;)
happens to have the product form d
je:;) ==
II j,(x,),
(2.10.4)
,=1
then the moments of je:;) vanish up to order M if and only if the moments of j, vanish up to order M for some £. Since there is at least one factor of W in (2 .10.2) , it follows that every multi-dimensional mother wavelet w(e) inherits the vanishing moments
170
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
property that the one-dimensional mother wavelet I)i happens to have. (Recall that the scaling function 'P - in contrast to I)i - does not even have a vanishing Oth-order moment.) A function f(-;) is said to be class C N in d dimensions if the functions
g d
(
8 ') ~ 8x:' f(x) S
d
are continuous for s, ~ 0 and
L
,=1
_
s, ~ N. Clearly, each mother wavelet I)i(e) inherits
the degree of smoothness possessed by 'P and I)i . Each I)i(-;) obviously has compact support if'P and I)i do, so the existence of multidimensional Daubechies wavelets is no problem. In general, this multi-dimensional extension preserves long-distance decay properties. For example, if 1'P(x)1 ~ eoe-molxl ,
(2 .10.5.0)
II)i(x)1 ~ eoe-molxl,
(2.10.5.1)
then
11)i(-;) (-;)1
~
cgexp(-motlx,l)
~ ci~+~~t.X1)'
(2.10.6)
so multi-dimensional Lemarie wavelets can be built from one-dimensional Lemarie wavelets, preserving even the rate of exponential decay. As another example, consider the decay property 1'P(x)1 ~ eo(l + x 2)-N, (2.10.7.0) 2 II)i(x)1 ~ eo(l + x )-N. (2.10.7.1)
In this case, d
11)i(-;) (-;)1
~
cg II (1 + x~)-N ,=1
~
d )-N cg ( 1 + ~x~ ,
(2.10.8)
so power laws are preserved as well . Finally, consider the smoothness and decay of a Meyer wavelet - i.e., the property rp, ~ E Ccf(lR). Since
---
I)i(-;) (p)
= II rp(p,) c,=O
II ~(p,), e,=1
(2.10.9)
2.10. MULTI-DIMENSIONAL WAVELETS
171
\[Ies is Sobolev-orthogonal to all ( such that (2.10.22) holds. On the other hand, (2.10.22) applies to all functions $".'(2' - ); for which r > 0. This is a consequence of iterating the scale relation $"gs
-
174
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
and applying the algebraic identity ""' ~ cm,, ""' ~ cm," . .. ""' ~ m~
m~'
Cm,( r)
(_l)m~r) C1_ 2 r m,+2n,_2 r -
1
, " - .. . --m,(r) r-. m,_2 :lm,
=0
m~r)
(2.10.25) for an ~ such that £, = 1. To derive from this a Sobolev-orthonormal basis of wavelets, one simply applies the inverse square root of the overlap matrix to the sequence of integer translates of the 'Ij; -; ,S. In momentum space, the mother wavelets are given by
(2.10.26)
The question of how much smoothness and long-distance decay these wavelets have must now be addressed, as ~sts the value of this construction. It is resolved by analyzing the expression for 'Ij; -; ,8 (p) . The momentum expression is derived in exactly the same way for 'Ij; E ,s as it was derived for the one-dimensional 'Ij; in §2.7, except the factor I p I-s is involved in the Poisson summation. The formula parallel to (2 .7.16) is just
(2.10.27) which can also be written in the form
~(p)
=
llo }I (2e-i~p, h(P,)
x
gIry Op,
h(p, + 27r)*) (
~ 1~ P+27r 71-
2S
e
+ 27rf,)
12) -11 P1-
25
gry Op,) ,
(2.10.28)
where the choice of '" for the construction of exponentially localized wavelets is given by d
II ry(p,) ,=1
=
,=1 (2.10.29)
2.10. MULTI-DIMENSIONAL WAVELETS Now write
so that ( 2 . 1 0 . 2 8 ) assumes the form
Recall that for this choice of q,
and therefore
h(p,
+ 2 ~ ) =* ( 1 - e ' + ~ ~ ) N + l .
(2.10.34)
The replacement of each real variable p L with a complex variable zL analytically continues all of these functions of momentum. We have the analytic continuations
-
rj(pL) ++ 4(pL)*
iN+l(e-izc
- l)N+lz;N-1
( - i ) N + 1 ( ei z . - 1 )N+l h ( p L ) ++ ( 1 + e-'+zL)J"+l h(p, + 2 r ) *
9
-N-l
2,
9
(1-eiiz~)N+l,
~ 6 ) rSG),
which yield the analytic continuation d~y.s(;) c--t ff IS(;)
with
176
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
Obviously,
and since r,(?;) is manifestly positive, it is easy to bound Re J?,(T) away from -1 on the region where: (a) each zL is bounded away from 47rl for all e E Z \{O), (b) Im zL is sufficiently small for each z,. These observations imply that f is analytic everywhere in the region
for some small 6 > 0. The crucial point about (2.10.36) is that analyticity at = ; 0 is manifest. To establish exponential decay of $;ts(;), we need to show that f ;IS(;) is analytic in the Cartesian product region
a ~ has d some mild polynomial decay in Re ; on that region. Pick an arbitrary non-zero m and observe that
so we may rewrite (2.10.36) as
2.10. MULTI-DIMENSIONAL WAVELETS
177
Since
(2 .10.40) (2 .10.41)
the same reasoning as above shows that {e,s(-;) is analytic everywhere in the region
The factor
(z;= (t z,) 2) -s poses no problem, because this region is bounded away from
-; = O. In particular, (2.10.40) and (2 .10.41) establish analyticity at -; = -47r r;;,. Now our choice of non-zero r;;, is arbitrary, so if we take the union of all the regions of analyticity generated, we finally have the desired Cartesian product of strips centered on the real axis. Why does {e,s (-;) have polynomial decay in Re -; for sufficiently small 1m-;? To answer this, consider the analytic continuation of the formula (2 .10.28):
(2 .10.42)
Every factor in this expression is periodic in the real directions except the last tw...2 factors. Moreover, the periodic contribution to this product is bounded for small Imz because
is bounded away from zero in such a region. As for the last two factors,
clearly decays in Re -; for small 1m -;, while
e-i!z, 1
I
2 Z,
11 ~ eiRe z,l- , 1
(2 .10.43)
178
CHAPTER 2. WAVELETS - BASIC THEORY AND CONSTRUCTION
This decay serves two purposes. On
o~and,
it allows the contour-shifting argument
Ch
for translating the real analyticity of 'IjJ -;,8 (p) into the exponential decay of 'IjJ -;,8 On the other hand , it establishes that the smoothness of 'IjJ -;,8 (-;) is class C N + 2 s- 1 , so the wavelet we have constructed shares the properties of a Lemarie wavelet. We return to (2.10.26) to consider the wavelets III c,. which generate an orthonor~ basis instead of just an interscale-orthogonal basis. The analytic continuation 'IjJ -;,s (p)
I----t
F -;,s (-;) is given by
(2.10.44)
n*
which follows from inspecting the analytic continuation of 'ljJc,s(p +2lT The proof that III -;,8 (-;) shares the regularity and decay properties of 'IjJ -;,s (-;) now follows from arguments very similar to those given above.
References 1. G. Battle, "A Block Spin Construction of Ondelettes, Part I: Lemarie Functions," Commun. Math. Phys. 110 (1987), 601-615. 2. G. Battle, "A Block Spin Construction of Ondelettes, Part II: The QFT Connection," Commun. Math. Phys. 114 (1988), 93-102. 3. J. Kahane and P. Lemarie, Fourier Series and Wavelets, Gordon and Breach, Amsterdam, 1995.
Chapter 3
Equilibrium States of Classical Crystals One of the basic industries of mathematical physics is to exploit the mathematical similarities between phenomena that are physically unrelated. The cross-fertilization between Euclidean field theory and classical equilibrium statistical mechanics is a case in point . Even before the evolution of quantum field theory into Euclidean field theory, there were parallel developments in these two different areas of physics. After all, one of the fundamental problems in both subjects is to control an infinite-volume limit, and the issue of stability is important to each area, both technically and conceptually. Another similarity lies in the basic quantities to be analyzed. Wightman distributions were not unlike the correlation functions of classical spin systems in thermal equilibrium, although the space-time cluster properties of the former were necessarily different from the spatial cluster properties of the latter. Only when the Wightman distributions were analytically continued to Schwinger functions did the similarities become striking. Classical equilibrium statistical mechanics, while far less fundamental than quantum field theory, is also far more ambitious in scope. The aim is to derive the bulk properties of matter from the microscopic interactions of its constituent particles. For a magnetic crystal, the constituent particles are atoms fixed in space, with internal degrees of freedom, and the bulk properties involve specific heat, magnetic susceptibility, regime of spontaneous magnetization, etc. For a gas, the constituent particles are molecules moving in space independently, with potentials defining the interaction, and the bulk properties involve specific heat, compressibility, condensation point, etc. Such diversity means that "the task of rigorous formulation cannot be unified, so a great variety of essentially different mathematical models is called for. The earliest work approaching mathematical rigor was on various expansion methods in the analysis of thermodynamic functions for dilute gases. The grand canonical ensemble is a power series in the activity, and Mayer wrote the density and pressure in powers of the activity with rules for computing the coefficients systematically [M22] . The viTial expansion is the expansion of the pressure in powers of the density, and it is older than the Mayer series. It was first introduced by Ursell [U4], and Mayer him179
180
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
self later simplified it by using graph techniques [M21] . Indeed, it was Mayer's heavy reliance on graphs for both virial and activity expansions that later inspired Feynman to use graphs so extensively in quantum field theory [F36-F39]. Convergence of these expansions (uniform in volume) for small values of the expansion parameters was not immediately established at the time - in part, because it certainly depended on the intermolecular forces assumed for the gas. As an alternate approach, systems of integral equations were formulated to characterize the correlation functions . The Mayer-Montroll system was the first of this type to be introduced [M23], and it is useful for microscopic interactions that are essentially repulsive. The Kirkwood- Salzburg equations were introduced later, but have a greater range of application with regard to the potentials considered [KI3]. Like the Mayer-Montroll equations, they comprise an infinite, linear, inhomogeneous system, and finite truncations have been accurate in low-activity calculations. It was Ruelle who employed the Kirkwood- Salzburg equations, with rigorous estimation, to prove volume-uniform convergence of the Mayer series for small activity and a large class of pair potentials characterized only by regularity and decay conditions [R17- R20]. Indeed, Ruelle and Fisher independently gave the first complete proofs that the infinite-volume limit of the pressure exists for arbitrary activity [F46] . For a special class of interactions, a proof had been given by Lee and Yang years before [LI6], but there had been limited interest in mathematically rigorous statistical mechanics until it was vigorously promoted by Ruelle and others. Subsequently, Lebowitz and Penrose proved uniform convergence (in volume) of the virial expansion for low density [LI3] and this marked the beginning of a renewed interest in investigating a variety of models. Our focus is on classical crystals - especially magnetic crystals, which have a distinctive history of their own. Since the crystalline state of matter is far from understood, the magnetic interactions between atoms at different lattice sites are not derived from first principles. The modeling is empirical in the sense that one investigates various interactions in a search for the model from which one derives the correct bulk properties of the magnetic material, but the derivation itself is mathematical. For a ferromagnet, one may believe the magnetic interactions to be the fundamental long-range coupling for magnetic dipoles fixed at the lattice sites by an unspecified force, but such an assumption only provides yet another input model from the empirical point of view. The earliest model proposed for ferromagnets was the Weiss theory [W2], which is essentially a mean field approximation. The calculations are simple and explicit, and some of the predictions are in qualitative agreement with experiment, but it was immediately clear that a more complicated model was needed. The Ising model - originally proposed and developed by Ising [19] - proved to be such a model, although its basic description is deceptively simple. Peierls gave an ingenious argument (which was not quite a proof) that the two-dimensional Ising model undergoes a phase transition as the temperature is lowered [PI3] . A few years later, Onsager [05] solved the same model explicitly! His solution is difficult, but it yields a wealth of information about the model, including the occurrence of a phase transition at a critical temperature and zero external magnetic field . For the Ising model in arbitrary dimension, it was suspected that a phase transition cannot occur unless the external magnetic field is zero. Lee and Yang extended this wisdom to all ferromagnetic pair interactions with
integrable range, provided the space of spin configurations was still the space of Ising configurations [L17]. Meanwhile, a rigorous study of the thermodynamic limit had been initiated by van Hove [H44]. Both the Onsager solution and the Lee-Yang Theorem were mathematically rigorous as well - a rare virtue at the time. Not until the time of Ruelle7s basic work did rigorous theorems on the classical spin models of crystals began to appear rapidly. Griffiths closed the loop-holes in the Peierls argument [G83] at about that time. In spite of the Onsager solution, this was important because the argument could be adapted to higher dimensions. A couple of years later, Griffiths also proved his celebrated Eorrelation inequalities assuming arbitrary ferromagnetic interactions for Ising configurations [G84-G86]. At the same time, Gallavotti and Miracle-Sole established existence and convexity of the pressure in the thermodynamic limit for a very general class of interactions and spin configurations [G8, G9]. There was also a great deal of interest in the mathematical structures associated with classical spin systems. As long as the space of a single-spin distribution is compact, the entire space of spin configurations for the infinite lattice of spin sites is also compact. The C*-algebra of continuous functions on this space is a complete set of classical observables for the system, while the states of the system are the normalized positive linear functionals on the C*-algebra. By the Riesz-Markov Theorem, this means that a state can also be given by a probability measure on the compact space of configurations. To determine whether an equilibrium state is unique for a given interaction and temperature, one must be able to characterize equilibrium states in the first place. The starting point is to characterize it as the Gibbs state for a finite system (so it is unique in that case), and it is reasonable to define an infinite-volume equilibrium state as the limit of the sequence of Gibbs states associated with some sequence of sets of lattice sites approaching the whole lattice in the sense of set inclusion or perhaps in a stronger sense. However, the basic theory of pure and mixed phases in a regime of multiple phases dictates that the set of equilibrium states be convex, and it is not clear how to obtain the geometry of a set of limit points, except in very simple cases. Dobrushin, Lanford, and Ruelle characterized equilibrium states with equations that link the finite-volume Gibbs states to conditionings of the infinite-volume state, and the set of solutions is automatically convex. Ruelle himself gave another characterization of equilibrium as a solution of a variational problem involving pressure and entropy [R23]. Equilibrium states with lattice-translational symmetry have an elegant formulation in terms of this functional analysis. This convex subset is a simplex whose extreme points are ergodic states, which are the pure phases into which a lattice-translation-invariant mixed phase is uniquely decomposed. Lanford and Ruelle applied the Choquet theory of abstract integral decompositions on convex sets to establish this theoretical result [L2]. This picture also provides a framework for the phenomenon of internal symmetrybreaking. If the single-spin distribution has a group symmetry of its own and if the interaction is invariant with respect to this group as well as the lattice-translation group, then a mixed phase may have both symmetries, but its integral decomposition includes ergodic states which are not invariant with respect to this internal group. The occurrence of spontaneous magnetization in the Ising model is the simplest example of this. The Griffiths inequalities inspired the derivation of other correlation inequalities
182
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
for spin systems in general and the Ising model in particular. Lebowitz proved strong correlation inequalities of Griffiths type for the Ising model, using properties more peculiar to that model [L6). These more powerful inequalities provided bounds on arbitrary correlations in terms of bounds on two-point correlations. On the other hand , Ginibre generalized the Griffiths inequalities considerably - especially to scalar spins where the single-spin distribution is even [G39). For another large class of scalar spin systems, Fortuin, Kastelyn, and Ginibre showed that any two random variables that are monotone increasing in the spin variables are also positively correlated by the Gibbs state [F52). At the same time the rigorous results in equilibrium statistical mechanics began to appear so rapidly, Symanzik was promoting Euclidean field theory [S94-S99). He was also the first to emphasize the formal resemblance between the thermodynamic limit of the expectations associated with the statistical mechanics of a classical field and the infinite-volume limit in space-time of vacuum expectation values associated with the perturbation of the free Euclidean field by an interaction. His idea was to use the methods of classical statistical mechanics to obtain control over the Euclidean field theory, and specifically, he used equations of Kirkwood-Salzburg type [S95) . Unfortunately, no one at that time knew how to analytically continue a Euclidean field theory to a relativistic field theory in the time differences. When Nelson isolated and exploited the Markov property of the free Euclidean field [N4-N7J, Symanzik's program attracted much more serious attention. Guerra, Rosen, and Simon obtained some control over interacting Euclidean scalar fields through correlation inequalities such as the Griffiths inequalities [G103, G104). They proved the Griffiths inequalities in the continuum limit of a lattice approximation in which the free part of the Euclidean field action is actually the Ising interaction and the interacting part defines an even single-spin distribution for a scalar spin system. They also proved the FKG inequalities with this lattice approximation, where the field interaction is not necessarily even. At the same time, Glimm, Jaffe, and Spencer developed a cluster expansion for controlling the infinite-volume limit of two-dimensional Euclidean scalar field theories with weak interaction [G76) . Although it was the first expansion of its kind, this expansion was very much in the spirit of statistical mechanics. Actually, it involved no lattice approximation, since the expansion was designed for the infinitevolume limit of the ultraviolet limit, but the systematic decoupling of random variables in distant regions was still the idea. Glimm, Jaffe, and Spencer were also able to combine a phase boundary expansion with the Peierls argument to show that the quartic self-interacting scalar field has two pure phases for strong coupling in the continuum i.e., with no lattice approximation [G78-G80). The connection between the two subjects attracted even more interest when ideas in quantum field theory found application to problems in statistical mechanics as well. An elegant example of this was the question of phase transitions for the classical Heisenberg ferromagnet, which was more difficult than the question had been for the Ising model. In two dimensions it was believed to have a unique phase for all temperatures because Mermin had proven that it had no spontaneous magnetization in that case [M35) . (The classical Heisenberg model does not satisfy an abundance of correlation inequalities, so one could not infer a unique phase from Mermin's result.) In higher dimensions it was believed to have a continuum of phases at low temperature - one for each orientation
183 of a single spin - but the Peierls argument could not be adapted to such a model. The input from Euclidean field theory was Osterwalder-Schrader positivity, or rather the lattice version of this reflection positivity property. Although the classical Heisenberg ferromagnet does not involve scalar spins, it shares the reflection-positivity property of the Ising ferromagnet . Frohlich, Simon, and Spencer actually derived from this property an infrared bound on the Fourier series whose coefficients are the two-point spin correlations [F77] . This implies existence of a phase transition at sufficiently low temperature in dimension greater than two. A short time later, Bricmont, Fontaine, and Landau proved that in two dimensions the equilibrium state is indeed unique at all temperatures [B54, B55] . An outstanding example of the impact of Euclidean field theory on classical statistical mechanics is the rigorous proof of Debeye screening. It had been shown by Edwards and Lenard that the grand canonical partition function for the lattice approximation of the classical Coulomb gas can be written as the partition function associated with the lattice Euclidean sine-Gordon field [E5] . This identification is called the sine-Gordon transformation. Brydges and Federbush combined this transformation with the phase boundary expansion methods of Glimm, Jaffe, and Spencer to rigorously prove that the high-temperature, dilute, Coulomb gas effectively screens the interaction of test charges [B65, B70] . This phenomenon was an experimental fact for which the only previous explanation had been the self-consistent field intuition of Debeye and Hiickel [D9]. The long-distance analysis of Brydges and Federbush is very difficult, but the intuition of the sine-Gordon representation is that the quadratic contribution to the cosine interaction is an effective mass term. This chapter is devoted to methods of analysis for classical spin models of crystals, with particular emphasis on the analysis of reflection-positive interactions on one hand , and high-temperature expansions for general lattice systems on the other. We confine our attention to three dimensions, not because we have to, but because in this chapter we are not concerned with the dimensional parameter, and dimension d = 3 is certainly the realistic value. In §3.1 we describe a ferromagnetic model where each spin is fixed in both magnitude and position on a cubic lattice, and the interaction is the fundamental one among magnetic dipoles - a pair interaction whose long-distance decay marginally fails to be integrable. We use the most naive high-temperature expansion (powers of the inverse temperature) to illustrate how a long-distance problem develops when one tries to control correlations in the infinite-volume limit. This is an example of an infrared problem, whose solution lies beyond our discussion. Then we shift our focus to lattice models with pair interactions that are integrable over distance and review the basic mathematical structure of equilibrium statistical mechanics in this context. In §3.2 we continue this description from a functional-analytic point of view, where states are realized as positive linear functionals on commutative C'-algebras. The Choquet simplex theory applied to the convex space of states provides an excellent framework for the ergodic decomposition of a lattice-translation-invariant equilibrium state into pure phases. In §3.3 we introduce the property of reflection positivity for lattices and show how it implies the infrared bound of Frohlich, Simon, and Spencer. Actually, there are two slightly different types of reflection positivity for lattices: one involves reflection through a plane of lattice sites, while the other involves reflection through a plane lying between two consecutive planes of lattice sites. The latter property is
184
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
the more important one , but one needs both properties to establish monotonicity in the long-distance approach of the two-point correlation to a (possibly non-zero) limit. This rules out pathologically singular behavior in the infrared estimation. In §3.4 we shift our attention to the Ising model and review its low-temperature phase structure and its critical properties. In §3.5 we introduce the Ginzburg-Landau model and review its similarity to the Ising model. In §3.6 we review a standard version of the abstract theory of polymer expansions in the interests of self-containment. The work of Glimm, Jaffe, and Spencer triggered a whole industry of rigorous cluster expansions for a host of models, each one different in some non-trivial way from every other. It was gradually recognized, however, that all of these expansions - including the MayerMontroll expansion and its contemporaries in early statistical mechanics - have basic combinatorial features in common. A unified framework for automating control over a given expansion by the convenient insertion of estimates peculiar to that expansion was developed and promoted through the efforts of Glimm, Jaffe, Malyshev, Seiler, and others (see [G7, G73, G99, M12 , S35] and references given therein). In §3.7 we review the most basic example of a high temperature expansion in the polymer formalism . The classical Heisenberg model and the Ginzburg-Landau model are the cases considered there. In §3.8 we introduce inductive interpolation as another expansion method cast in the polymer formalism. In §3.9 we consider long-range pair interactions (defined by a decay condition) for fixed-magnitude spins because it calls for a generalization of the polymer estimation given in §3.6. We examine the wholesale expansion of the exponential in all couplings and prove convergence at high temperature. Then in §3.10 we show how an expansion based on inductive interpolation yields a better polymer expansion in the sense that the decay condition on the interaction is weakened as a sufficient condition for high-temperature convergence. We study the combinatorics associated with inductive interpolation as well. For example, an Nth-degree ordered tree graph is a mapping TJ: {2, . .. , N} -t {I, ... ,N I} such that TJ (i) < i . For interpolation parameters h , . .. , t N there is a monomial associated with TJ given by (3.1) The oldest combinatorial inequality involving such monomials is an exponential bound on the sum over ordered tree graphs of the integrals:
L
II N
Nth-degree 71
(
i=1
1
10
dti
)
f 7l (t):S eN-I.
(3.2)
A standard application is to prove convergence of the Mayer series for a sufficiently dilute gas. It was later discovered by Robinson that this inequality can actually be replaced by an identity [B21] which shows how sharp this inequality is. We have
L
Nth-degree 71
NN-l N rl) (II In dti f7l(t) = N t· i= 1
0
.
(3.3)
3.1. CLASSICAL SPIN SYSTEMS This combinatorial estimation has also been replaced by
where d q ( j ) denotes the cardinality of 7 7 - 1 { j } . This refinement has no effect on the convergence of some expansions, such as the Mayer series, but it makes all the difference in the world for other expansions, such as the phase cell cluster expansion in Chap. 5. Shortly after (3.4) was established, it was shown by Speer that this inequality can be replaced by an identity as well [S66]. We have
Again, the identity adds nothing to the application of the corresponding estimate, but it lends a great deal of elegance to the combinatorics. This combinatorial interlude is the content of 53.11. In 53.12 we consider unbounded scalar spins with long-range pair interactions and a large-spin damping condition on the single-spin distribution. We apply exactly the same wholesale expansion used in 53.9 and show how the large-spin estimation burns up some of the long-distance decay of the interaction. In 53.13 we apply the inductive interpolation and show how the combinatoric estimate (3.4) controls the large-spin estimation without burning up any of the long-distance decay. This means convergence of the latter expansion requires no more long-distance decay than it required in the bounded-spin case.
3.1
Classical Spin Systems
From the standpoint of modeling the critical behavior of macroscopic systems in terms of the equilibrium statistical mechanics of their microscopic structure, perhaps no examples have been more heavily studied than the various classical models for the ferromagnet. The atoms are arranged in a cubic lattice of fixed positions, so the only dynamical variables are internal. Each iron atom is elegrically neutral, but has a magnetic moment proportional to its angular momentum s , whose magnitude is assumed to be unity, for example. Such a model is based on the assumption that each atom is a constantly spinning gyros2ope whose inertial axis is affected by the mqnetic field. Accordingly, every element n of a finite set A c Z3has the spin variable s ;t assigned to it, and one possible assumption for this crystal is that the Hamiltonian has the form
186
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
where h is a uniform external magnetic field and J is a constant. This expression for the interaction between atoms at different lattice sites is based on the elementary field of a magnetic dipole. We omit the kinetic terms because they will cancel out in the normalization of a classical equilibrium state, as the momentum variables are independent. For inverse temperature fJ, the equilibrium state of this system is the probability measure on the space of spin configurations defined as the normalization of the measure. e- f3HAC;)
II do-(~;:;),
(3.1.2)
;:; where do- denotes normalized Lebesgue measure on the sphere §2 of possible spin values. In general, do- will denote the single-spin distribution for the systems we study. The exponential factor is the Gibbs factor, and it is the solution of the elementary variational problem that is involved in deriving one probability measure from the constrained statistics of another. In this case, the original measure is just the product of singlespin distributions, the constraint is that the statistical average of the energy is fixed, and the inverse temperature fJ is the Lagrange multiplier. The partition function for a finite set A is the mass of the unnormalized measure (3.1.2) given by the Gibbs factor - Le., (3.1.3)
The probability measure dJ.L~' h for the Gibbs state is just
dJ.L~,h (~) = ZA(fJ, h)-le- f3HA (s) II do-(~ ;:;),
(3.1.4)
;:; M
and so the expectation of an arbitrary product
IT sj=l ni'i
of spin components is given
by
(3.1.5) A standard problem is to control the A = Z3 limit of such expectations for high temperature (small fJ) · From the limiting expectations one can recover a probability measure by applying some functional analysis - the Riesz- Markov Theorem in this case, where the space of spin configurations over Z3 is compact, or Minlos ' Theorem in the case where the space of spin configurations is linear instead of compact. Such a measure is an infinite-volume equilibrium state . To control the A = Z3 limit for small fJ - Le., the thermodynamic limit - one might try to expand a given expectation in powers of fJ. This is the most naive example of
3.1. CLASSICAL SPIN SYSTEMS
187
a high-temperature expansion, but in the case of this long-range interaction, it is only asymptotic - not convergent. Let (')0 denote the expectation functional of the product
n
measure (on the compact space
§2)
of the single-spin distribution, and set h= 0
~EZ3
for simplicity. To first order in the inverse temperature we have the expansion
/IT S~j")
\J=l
=/
IT S~j'j)
\J=l
A
- (3 0
/ (J~_~, --; ~,). --; ~ IT S~j'j)
L
~,~'E{~l""'~M} \ where
J~
j=l
+ 0((32),
(3.1.6)
0
denotes the 3 x 3 matrix defined by
~=O, (3.1.7)
Of the spin variables that are initially differentiated down from the exponent, only those --; ~ for which ~ occurs in {~1"" , ~ M} can appear, because the Oo-expectation is zero if any spin component at any lattice site has an odd power. This also implies (3.1.8) so the (3-derivative of the partition function in the denominator has contributed nothing here . It is significant that in (3.1.6) the first-order contribution does not depend on A at all . The second-order contribution is a little more involved:
d~2 / IT S~j'j) \J=l
1_ = / HX A
~-o
\
IT S~j'j)
J=l
0
- (HX)o / IT S~j'j) \ J=l
0
L /((J~_~" --;~,,) --;~)((J~,_~" --;~,,). --;~,) IT --;~j'j)
L
~,~'E{nl, ... ,nM} ~"EA \ +
J=l
0
/((J~n-rn~ --;~). --;~)((J~ ~ --;~). --;~) rrM --;~nitj ) n n'-m' n' n'
'" ~
\
m
r;,;' ,i1i.,ni.'E{n1! ... ,nM} ni.' :;en;,
j=l
0
(((J~ ~ --;~). --;~)((J~, -m~, --;~,). --;~,))o /rrM S~ m m
'~ "
n'-m
~,n'""m,ni'E{nb" " nM}
n
n
n
\
. .)
nJl- J
j=l
0
~¥~'
L
L
(((J~_r;" --;r;,,)' --;r;)((Jr;,-r;" --;r;,,)' --;r;,))o
n:,Ti"E{nI, ... ,nM} ;IIEA
(3.1.9)
188
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
because, for example,
Now the second and third double sums are clearly independent of A, while the first and fourth are not. On the other hand, the sum over A converges absolutely as A -+ ;[,3 because c
IJ~_~"I
and therefore
~ L,.;
_
(3.1.11)
::; -I~"""--~:--1-'1-3' n-n
IJ-n-n" - IIJ-n'-n" - 1< c -
In(2 + 1 r;; 1~ /I
,_ r;; I) 13 .
n - ~ n
(3.1.12)
n"
Thus, the thermodynamic limit is controlled to second order in /3. Similar estimation controls this limit to all orders in /3, but this control is only perturbative. Does the power series in /3 have a nonzero radius of convergence independent of A? Actually, the number of products of expectations of powers of H" contributing to the Nth-order /3-derivative at /3 = 0 is bounded by cN N!, so the Taylor coefficient has a bound of the form cN , provided each such product of expectations has such a bound. On the other hand, an example of the multiple sums to be estimated is (3.1.13)
Although we have a bound of the form (In(2 + 1 r;;
for every positive exponent
Ct,
'_ r;;
I))N ::; cN,,,, (1
+ 1 r;; '_ r;;
I)'"
(3.1.14)
we also have the dependence CN ,,,,
= O(N!)
(3.1.15)
for any choice of Ct . This is where convergence of the power series fails, and this type of divergence is known as an infrared divergence . The point is that this problem would not arise if J~_~, had just a little more long-distance decay - i.e. , if J~ were summable. The infinite-volume expectations can be controlled nonperturbatively for high temperature by the application of renormalization group methods, but such technology did not exist at the time that rigorous work on ferromagnetic spin systems was initiated. Moreover, magnetic crystals represent a very complicated state of matter from the most fundamental point of view, so it is not at all clear that the most basic interaction between magnetic dipoles is the correct one here. For these reasons, the traditional effort has been a search for shorter-range spin interactions that yield the correct bulk
3.1. CLASSICAL SPIN SYSTEMS
189
properties of ferromagnets. From this point on, we make the convenient assumption that J;. _;., is a scalar and that (3.1.16)
The Hamiltonian now has the form
-;-+h."sn' L n
(3.1.17)
;'EA
for an external magnetic field h . Boundary conditions may be modifications of the coupling coefficient J;. _;., or constraints on the possible spin values at the boundary. As far as high-temperature expansions are concerned, it will be proven later that the power series in (3 does indeed have a nonzero radius of convergence independent of A and its boundary conditions for this class of spin models. The study of the spin product expectations is closely related to the analysis of thermodynamic functions, which are generated by the free energy, defined by 1
~
fA((3 , h) =
~
!AI In ZA((3, h) .
(3.1.18)
First, it is a trivial consequence of (3.1.16) that
ZA((3, h) :S ei3 (C+ 1hI)IAI,
(3.1.19)
so the logarithm has a bound linear in 111.1 . Such a bound is a stability bound, and it plays a central role in the context of Euclidean field theory, as we have already seen in Chap. 1. Its proof was difficult in the field context, but is a triviality in this context because the spin variables are bounded. It obviously provides a A-independent bound on the free energy. In addition, the infinite-volume free energy
f((3, h)
= A-+Z3 lim fA ((3, h)
(3.1.20)
is actually unique for arbitrary values of (3 if the limit is taken in the sense of van Hove. For every positive integer r , define N;!(A) (resp. Nr-(A)) as the number of ~ for which {I, 2, ... ,r P + r ~ intersects with (resp. is contained in) the set A. A sequence {Aj} converging to Z3 in the sense of set inclusion is said to converge in the sense of van Hove if and only if (3.1.21) for every r. The uniqueness of the van Hove limit of fA ((3, h) is a general type of result. The elementary inequality
ZAUN((3, h)
:S ZA((3, h)ZN((3, h)exp (2(3
~ ;= IJ;_;,I) , nEA n'EN
A n A' = 0,
(3.1.22)
190
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
lies at the heart of the proof, which we do not pursue. However, this convergence was one of the early important theorems proven in the original program to develop mathematically rigorous statistical mechanics. Notice also that the free energy has the convexity property on the space of interactions - i.e., with 1
~
/(/3, h, {J.,;}) = lim(van
Hove)
A-+Z3
-IAl ln Z(/3, HA),
(3.1.23)
we have the inequality /(/3, t h +(1 - t) h ',{ tJ.,; + (1 - t)J~)) ~
:::;
t/(/3, h; {J.,;})
~
+ (1- t)/(/3, h
'; {J~}),
0:::; t:::; 1,
(3.1.24)
which follows from the Holder inequality
0:::; t :::;
1.
(3.1.25)
The thermodynamic functions are de~ned to be the partial derivatives of the free energy with respect to the components of h. The A = :£3 limit of either the free energy or anyone of these partial derivatives is the thermodynamic limit of the given function. For example, consider the first-order partial derivatives of the free energy:
(3.1.26) The thermodynamic limit of such a function is an infinite-volume average of single-spin expectations and is therefore a bulk property of the lattice spin system. Its r~gularity properties are important to the study of critical behavior with respect to h at low temperature (large (3). Closely related examples of thermodynamic functions are the second-order partial derivatives:
(3.1.27) The existence of the thermodynamic limit of such a derivative obviously depends on the behavior of the truncated expectation
(3.1.28) If the long-distance decay is summable uniformly in 01'., then this second partial derivative exists in the A = :£3 limit. How does one characterize an infinite-volume equilibrium state? In particular, the limiting state for any convergent sequence of Gibbs states based on a sequence {Aj} of finite subsets converging to :£3 in the sense of van Hove should qualify as an equilibrium state, and the existence of at least one such sequence is guaranteed by a compactness argument. Moreover, there should be only one equilibrium state in the
3.1. CLASSICAL SPIN SYSTEMS
191
case where these sequences of Gibbs states all converge to the same state. This case occurs for high temperature (small P ) , where one may regard the Gibbs factor as a small multiplicative perturbation of the product of single-spin distributions. In the case where uniqueness fails, infinite-volume characterizations are very i m ~ o r t a n t . ~ S2 of the form p( s ) do( s ;), Consider an arbitrary probability measure on
n
2
n
n€Z3
where the density
has the property that p~ l n p ~is integrable with respect to
n da(;;).
The entropy
2
n €A
of the state p on A is defined by
while the internal energy density of this state for a given interaction is defined by
with HA given by (3.1.17). It is yet another basic result that the infinite-volume limits S(p) = lim(van Hove) SA(p),
(3.1.32)
A+23 2
E,(h,{J;})
3
1
= lim(van~ove)~~(h,{~;})
A+23
(3.1.33)
exist. Moreover, the entropy S(p) is actually an affine function on the convex set of states - i.e., S(tp
+ (1 - t)pf) = tS(p) + (1 - t)S(pf),
< t < 1.
0
(3.1.34)
This property is an easy consequence of the concavity and subadditivity of the function E(x) = -xlnx. Indeed,
E(tx + (1 - t)xf) 2 t[(x)
because Eff(x)< 0, while
+ (1 - t)E(xi),
(3.1.35) 0
< t < 1,
(3.1.36)
E(x + xf) 5 E(x) + o. The uniqueness of this phase in the ferromagnetic case is assumed here, and the spontaneous magnetization M (13) ~ 0 is given by M(f3) = lim(-; 0)(13, h -:;;) . -:;; .
(3.2.1)
h.j.O
By convexity of the free energy, this limit is monotone decreasing, and the idea is that if the model has a phase transition, then M(f3) > 0 for sufficiently large 13. Notice that the lattice site at which the single-spin expectation is evaluated is irrelevant because the equilibrium state is translation-invariant - i.e., (3.2.2)
for all ~ E Z 3. The physical meaning of the case M (13) > 0 is that at sufficiently low temperature the ferromagnet remains magnetized in the direction of the external magnetic field after it is switched off. This also means that ~e model has a continuum of special equilibrium states parametrized by -:;;E S2 when h= O. Let S ~ be the set of translation-invariant equilibrium states for inverse tempera~
f3,h
ture 13 and external magnetic field h. By the characterization (3.1.45), S f3,h~ is a convex set because Ep is linear in p and S(p) is affine in p. S ~ is obviously compact in the f3 , h
topology of pointwise convergence of linear functionals on the commutative C' -algebra of continuous functions on II §2. An extreme point of a given convex set cannot be ~EZ3
written as a non-trivial convex combination of other elements of the set. The extreme points of S ~ are said to be the pure phases or the ergodic phases, and we first consider f3 , h
the problem of decomposing an arbitrary element of S ~ into these pure phases. Let dJ-t denote the probability measure on
II
f3 , h
§2
associated with a given translation-invariant
~EZ 3
equilibrium state p and consider the Hilbert space
1{
= L2(d/J,). Let
{U(~): ~E Z3}
196
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
be the unitary group induced by the group of dj.L-preserving homeomorphisms defined by translation on the lattice - i.e.,
(3.2.3) Let P denote the orthogonal projection onto the subspace of vectors invariant with respect to this unitary group. Moreover, let A denote the commutative CO-algebra §2. of operators on 11. defined as multiplication by the continuous functions on
n
;;EZ3
Then the C' -algebra B generated by PAP is also commutative, so the positive linear functional B EB, (3.2.4) can be represented by a measure dv on the compact space X of characters (multiplicative 'linear functionals) on B. Thus
(Bl, 1h-i
=
L
BEB.
X(B)dv(X),
n
Since df.L is a probability measure on
§2
and PI
(3.2.5)
= 1, it follows that dv is a proba-
;;EZ3
bility measure on X. In the special case B = P A( P, where A( denotes multiplication by some continuous function (on §2, we have
n
;;EZ 3
p(()
/ (df.L (A( 1, 1)1i (P A(Pl, 1)1i
L
X(PA(P)dv(X) ·
(3.2.6)
On the other hand, we may define the state
(3.2.7) so we have a convex decomposition of the arbitrary translation-invariant phase: p(()
=
L
Px(()dv(X)·
(3.2.8)
X is realized as the index set for the states into which P has been decomposed. How do we know this decomposition essentially includes only the states in S - - i.e., how do we show that Px is a translation-invariant phase for dv-almost all X ~'}(? First, the translation-invariance is obvious. Indeed, if ( is a continuous function on §2 and if we introduce the notation
n
;;EZ3
(3.2.9)
3.2. PHASE TRANSITIONS AND ERGODIC DECOMPOSITION
197
then X(PA(-;;P) X(PU(r;;)-l A( U(r;;)p) X(PA(P)
(3.2 .10)
Px(() ·
Next, we verify that Px is an equilibrium state for dv-almost all X, using the variational principle. Since P satisfies Eq. (3.1.45), we have (3.2 .11) because S(p) is an affine function of p and Ep is linear in p. Thus (3 .2.12) while in any case (3.2.13) for all X by the Gibbs variational inequality. Hence, the integrand vanishes for dvalmost all X, but this vanishing condition is again the characterization adopted for equilibrium states. Notice that this last argument does not use the special nature of dv - i.e., it appli~s to any probability measure that decomposes p into other states. This means that if S is the convex set of all states, then the subset of equilibrium states is not "interior" to S, but rather a "face" of S . As far as the measure dv is concerned, we have yet to show that it decomposes p into pure translation-invariant phases - i.e., extreme points of S - - i.e., ergodic phases. (3,h
Consider an arbitrary convex decomposition p = a.Pl + (1 - a.)po of p into two translation-invariant phases Po and Pl. Obviously, the corresponding probability measures dll O and dillon IT §2 are absolutely continuous with respect to dll, so there ~EZ3
exist positive dll-integrable functions
",0
and
",1
such that (3.2.14) (3.2.15)
where we have used the notation (3.2.9). If A, = A1)" property implies
the translation-invariance (3.2.16)
198
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
so both A, commute with P as well . Thus
J
(T/'d/1-
(A(A,I,I) (PA,PA,Pl,lhl
L L
X(P A, P A'I, P)dv(X) X(PA(P)X(PA'I,P)dv(X)
L
Px(()Px(T/,)dv(X)
(3.2.17)
because X ranges over the characters of B. Now there is a partial ordering of probability measures on convex compact sets known as Bishop- de Leeuw ordering, denoted by -Cs\ii
(3.3.22)
First consider an arbitrary continuous function F(-;) that depends on only those spin variables -;;ii for which r;;E A and m, > o. In terms of Gibbs expectations, we have: (3 .3.23) (3.3.24)
3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY
205
The reflection-invariance is a direct consequence of the J!:. _ translation-invariance n-m combined with (3.3.16), while the positivity condition follows from applying (3.3.17) to a wholesale power series expansion of the exponential factor
exp
~
[-2f3
L-
J!:.n-m - --; m- . --;-) n
~,~EA ml. > O nL~O
(3.3.25)
that contributes to the Gibbs measure. The residual exponential factor is
(3 .3.26)
11.-;
= {~E A: m, ~ OJ,
(3.3.27- )
11.;-
= {~E A: m,
(3.3.27+)
> OJ.
This type of reflection positivity plays the major role in establishing the existence of a phase transition . Now consider the slightly different reflection transformation 8~) With the (slightly different) condition that F(--;) depend on only those spin variables --; ~ for which ~E A and m, 2: O. We have:
J1~ (8~) (£))
=
J1~ (£),
J(F08~»)F*dJ1~
(3.3.28) (3.3.29)
2: O.
This positivity condition does not follow from (3.3.17) at all; instead, we observe that the symmetry (3.3.16) implies
+2
L
J!:.n-m_
8 -
8-
n
m
~EA? ~EA;-\A?
L
A
J-n-m -
~
8-
n
8m
206
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
(3.3.30)
where
~
A,O = {mE A: m,
= 0 or m, = 2N - 1 }.
(3.3.31)
The point is that (3.3.32) so the
s~ in (3.3.30) may be treated as coefficients with respect to the action of the s~-dependence . Thus m
transformation on the
n
(3.3.33)
s
Note that F(s) may depend on the ~ for which r;;,E A? without violating this positivity. This type of reflection positivity plays a relatively minor role in the analysis, but we need it to rule out pathological long-distance behavior. Let Pf3 denote our choice of limiting state for the sequence - indexed by the volume parameter N - of Gibbs states described above. Our task is to show that it has the
3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY
207
property (3.3.1), and the next step in this direction is to establish that the function ) actually has monotonic behavior. In an infinite volume, the reflection
p{3(s~ s~ nit
OK
transformations (}(L), jj(L) are obviously defined by (}(L)(--;)~
(3.3.34)
jj( L) (--;) ~
(3.3.35)
m
m
and P{3 has the positivity properties
2 0, 2 0,
p{3((F 0 (}(L»)F*) p{3((F 0 jj(L»)F*)
(3.3.36) (3.3 .37)
where F(--;) depends on only those spin variables --; ~ for which m L > O. Now we may analyze p{3(s ~ s~) for positive integers q most easily. By translation-invariance and qe" ,K OK by (3.3.37) we have
P{3((S2re",,- - S(2r_1)e",)(Se,,,, - so)) P{3((Sre"" - S(r-lJe,)(S(l-r)e"", - S_re",,)) (3.3.38)
< 0 for all positive integers r, because
S
~
-re",n.
-
S(l -r )~eL l'" is the jj(LLreflection of S re~
L ,/\'
S(r-1)e"" On the other hand, (3.3.36) implies P{3((s(2r+1)e"" - s2re"")(se,,, - so)) P{3((s(r+l)e"" - sre,)(s(l_r)e"" - s_re,)) :S because
S
-T
~e
to ,/';;
-
(3.3.39)
0
S(l -r )~e /. ,,,, is the (}(LLreflection of s( r +1)~e /. ,1'\.
-
sr~e " " Thus
(3.3.40) for all positive integers q, and this inequality may be written as
1 P{3(Sqe""so) :S '2P{3(s(q-1)e""so) Therefore p{3(s ~
q e 1..,'"
1
+ '2 P{3(s(q+l)e""so)
s~) is a convex function of q for q OK.
has the bound
(3.3.41)
> O. Since this function also (3.3.42)
Ip{3(s q ~e ",I'\, s~)I:S 1, 0K
it can only be monotone decreasing for q > O. Since p{3 is isotropic, translationinvariant, and jj(LLinvariant, it follows that p{3(S~ s~ nK
OK
) :S
p{3(s~ s~ mK
OK
)
(3.3.43)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
208
for In,1 ~ Im,l, t = 1,2,3. Moreover, the monotonicity in each coordinate direction together with these symmetries imply (3.3.44) exists, and this long-range condition implies lim
J--+OO
IAI . I ~ L...J
nEAj
P(J(s-n~ s-OK, )
= c(J
(3.3.45)
for every nested sequence {Aj} of finite rectangular sets of lattice sites converging to Z3 in the sense of set inclusion. We have reduced the problem to showing that the more straightforward limit (3.3.44) is non-zero, in which case c(J is known as a long-range order parameter. Aside from the symmetry and reflection positivity already assumed for the interaction, there is yet another property it must have, if we are to show that c(J -I 0 for sufficiently large (3. The strategy is to consider the distribution (3.3.46)
-n
on the torus [-7l',7l'P Since p(J(s-nK. s-OK ) - c(J is a non-negative function on the lattice Z3 that decreases ~ the limit zero in every direction, it follows from elementary Fourier analysis that T(J( k) is actually a measurable function; indeed, one can show that 3
II (e ik ,
,=1
~
-
I)T(J(k) is a continuous function. Thus p(J(k)
= c(J6(k) + T(J(k),
T(J measurable,
(3.3.47)
is the structure of the Fourier series with coefficients p(J(s- s- ). What is the adnK OK ditional property the interaction must have? We shall presently use the reflectionpositivity to derive the bound
J-m
= J- L~ Jm rnO ~ n'
(3.3.49)
~
and this leads to a bound on p(J( k) as follows. If we scale {C;,;} with the replacement C;,; --+ A C;,; and expand the exponential in powers of A, we have 1 - (3A
22:-J-
-
m-n
m,n
~ Q- . ~ Qm n
222:-J-
+2(3 A
-
rn-n
2:-J - -
m'-n'
m ,n
(3.3.50)
3.3. PHASE TRANSITIONS AND REFLECTION POSITIVITY where
as a consequence of translation-invariance and the normalization
Now subtract 1 from the inequality, divide by X2, and then take the X = 0 limit:
If we make the choice a ; = e ,a;, then 2
2
Since this inequality may be written as
2
As h ( k ) is arbitrary, this implies
and so from (3.3.47) an infrared bound is derived:
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
210
The additional condition on the interaction is that (J( k) - J(0))-1 be integrable. Indeed, since
(3.3.61) we have
c~
>
(g i:
dk,)
(2'llip~(s~
p~(k) - ~/3-1
)-
0"
~(27r)3 - ~/3-1
~r1 2
(g i:
(IT,=1-11" 111"
(gi:
dk,) (J(k) - J«}))-1
dk,) (J(k) - J(0))-1
dk,) (J(k) - J(0))-1
(3.3.62)
because
(3.3.63) and p~ is invariant with respect to the internal symmetry group 50(3) . Thus c~ > 0 for sufficiently large /3 if the integral is finite . At last we have a class of interactions for which a phase transition occurs. It remains to verify the bound (3.3.48), and to this end, we return to the finite box A of lattice sites with periodic boundary conditions. For a technical reason we also approximate da-(; ;;;) with lim (s-s-) m 0
°
m-+oo
for sufficiently large (3. Obviously, the magnetic susceptibility is infinite in this parameter range. Correlation inequalities have played a major role in establishing properties of the Ising model, since the special structure admits a wealth of useful monotonicity relations. We briefly mention some of these inequalities with no attempt at completeness. 1) Griffiths Inequalities. Let A and B be arbitrary subsets of an arbitrary finite A C Z3 Then for h ~ 0,
(SA)A 2: 0, (SASB)A 2: (SA)A(SB)A, where
SA
=
II Sr; .
(3.4.11) (3.4.12)
(3.4.13)
r;EA Moreover, for arbitrary finite A' C Z3 containing A, (3.4.14)
provided (.) A is defined with no boundary conditions. This last inequality implies existence of the A = Z 3 limit for such a sequence of Gibbs states, since the uniform bound (3.4.15) obviously holds. The Griffiths inequalities are also useful in addressing the issue of phase transitions. If a given ferromagnetic interaction yields spontaneous magnetization at a given temperature, so does any interaction that is "more ferromagnetic" . 2) Lebowitz Inequalities. Introduce duplicate Ising spin variables tand set m
Then for h
~
qm
1 -(s..j2 m
+ t-) m'
(3.4.16)
rm
-(s..j2 m - t-). m
(3.4.17)
1
0,
((qAqB))A 2: ((qA)h((qB))A, ((r ArB)h 2: ((r A))A ((rB))A, ((qArB))A 2: ((qA))A((rB))A,
(3.4.18) (3.4.19) (3.4.20)
3.4. THE ISING MODEL
217
where the expectation « .)) A is defined by (3.4.21)
«SAtB))A = (sA)A(tB)A
with 01. the Gibbs expectation in the duplicate spin variables. These inequalities have significant consequences. The last inequality, for example, enables one to bound the expectation of a product of spin variables with a sum of products of expectations of products of fewer spin variables. Such bounds can be used to derive the long-distance decay of general correlations from the decay of the two-point function . 3) Fortuin-Kastelyn-Ginibre Inequality. If F(s) and G(s) are monotone increasing functions with respect to each spin variable s~, then
(F(S)G(S))A
~
(F(S))A(G(S))A.
(3.4.22)
One application of this correlation inequality is that the expectation of such an observable is monotone increasing in the external magnetic field . Since uniqueness of the equilibrium state can be established for large values of the external field by an expansion, this implies uniqueness of the equilibrium state for values of h where the free energy is analytic - i.e., all h 0, by the Lee-Yang Theorem.
t=
h
~c
Figure 3.4.1: The phase diagram in (3 and h for the three-dimensional Ising model is simple. The model has a unique phase for all parameter values other than «(3,0), (3 ~ (3e, where (3e is the critical value of the inverse temperature (3. On this half-line, the model has equilibrium states only of the form 0:':;0:':;1.
For the
0
(3.4.23)
= 1/2 state, the spontaneous magnetization is obviously zero, but (3.4.24)
because
,O) = (s-s-)+«(3,O) (S-S-)-«(3 mom 0
(3.4.25)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
218
by the sr;;
t-7
-sr;; change of variables. Hence
c(3
=
lim (S~S~h/2({3) mo
~ m~oo
=
lim (s~s~)+({3,O). mo
(3.4.26)
~ m~oo
Since O+({3, 0) is a pure phase, lim
~
m-+oo
(s~s~)+({3,O) m
((SO)+({3,0))2
0
(3.4.27)
M({3?, so we have the relation
(3.4.28)
For {3 > {3c, the spontaneous magnetization is non-zero, but it is also continuous in (3, so M({3c) = O. This means that there is a unique phase at {3 = {3c and that the long-range order parameter c(3 vanishes there as well. The properties of the critical point are governed by critical exponents . Consider the expectation (s~s~ ) ({3c, 0) for example. The large-n;, decay implicit in the equation m 0 c(3, = 0 is actually very weak, and indeed, the decay is not exponential in this special case. Instead, it is a power law decay - namely, (3.4.29)
m~oo,
where", is an example of a critical exponent. In the next section we shall see the similarity of the Ising ferromagnet to scalar Euclidean field theory on a lattice. By comparison to (3.4.29), the two-point correlation of the free massless field on a lattice is given by
(rr 1" ) 3
1 (21lV
_" dp,
,=1
.~ ~P e-,m.
3
2
1
L (1 -
,=1
~
~ 1-1, cl m
(3.4.30)
m~oo,
cosp,)
where this lattice function is the Green's function of the lattice Laplacian. For this reason, 3 - ", is called the anomalous dimension of the critical state. The value of ", cannot be computed in closed form, nor is there any known method that determines the value to an arbitrary degree of accuracy, but it is believed to lie between 0.04 and 0.06. In any case, the magnetic susceptibility
x({3, 0)
= {32 "(s~s~)({3,O) , ~ m 0
(3.4.31)
is clearly infinite at {3 = {3c. Another exponent 'Y governs the divergence of the magnetic susceptibility at {3 = {3c: {3 /' {3c. Yet another exponent v controls the divergence of the correlation length by m~oo, (S~SO )({3, 0) ~ cexp( -I n;, 1/~({3)),
(3.4.32) ~({3)
defined (3.4.33)
3.4. THE ISING MODEL for (3
219
< (3c· Its divergence has the form (3.4.34)
and the relation ,,(=(2-1))11
(3.4.35)
is a consequence of the celebrated Widom scaling hypothesis. In contrast to the magnetic susceptibility, the spontaneous magnetization has the property M((3) = 0, and is continuous for all (3 . For (3
(3.4.36)
> (3c, the dependence has the behavior (3.4.37)
where 8 is another critical exponent. The
28 = 311 - "(
(3.4.38)
is another consequence of the scaling hypothesis. The values of these exponents are known to be universal . This means that each exponent has the same value for large families of spin systems that include the Ising model. The point is that each critical exponent is independent of the short-distance details of a model. They reflect long-distance properties shared by many systems. The renormalization group sheds a great deal of light on this universality, as the values of the critical exponents are actually associated with a fixed point of the RG transformation . One of the earliest achievements in the study of the Ising model was the proof that spontaneous magnetization indeed occurs for sufficiently low temperature. (Originally applied to two dimensions, it is adaptable to three dimensions.) This proof did not depend on reflection-positivity methods (which were unknown) and the strategy was to construct each of the two pure phases for h = 0 by taking the thermodynamic limit with boundary conditions that favor the chosen phase. Outside the finite region A of lattice sites, one may specify all spin values to be +1, for example. This was the Peierls argument, and the representation is the earliest example of a low temperature expansion , which we shall not pursue.
See Bibliography for Papers by These Authors 1. D. Abraham
2. 3. 4. 5. 6. 7. 8.
M. Aizenman M. Aizenman and R. Graham M. Aizenman and B. Simon H. Araki and D. Evans T . Asano G. Baker W. Barbosa and M. O'Carroll
9. H. van Beijeren and G. Sylvester 10. J . Bricmont, J . Lebowitz, and G. Pfister 11. R. Dobrushin 12. F . Dyson 13. C. Fortuin , P. Kastelyn, and J. Ginibre 14. G. Gallavotti and S. Miracle-Sole
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
220 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
R. Griffiths R. Griffiths, C. Hurst, and S. Sherman R. Griffiths and B. Simon P. Hohenberg R. Holley W. Holsztynski and J . Slawny C. Hurst and H. Green J . Imbrie D. Kelley and S. Sherman J . Kogut and K. Wilson J. Lebowitz J. Lebowitz and A. Martin-USf J . Lebowitz and J. Monroe T . Lee and C. Yang E. Lieb, D. Mattis, and T . Schultz B. McCoy, C. Tracy, and T . Wu B. McCoy and T . Wu H. McKean A. Messager and S. Miracle-Sole M. Moore, D. Jasnov, and M. Wort is C. Nappi
3.5
36. 37. 38. 39. 40. 41. 42. 43. 44. 45 . 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.
C. Newman M. O'Carroll M. O'Carroll and W . Barbosa M. O'Carroll and F . Sa Barreto L. Onsager R. Peierls J. Percus S. Pirogov and Y. Sinai J. Rosen R. Schor S. Shlosman B. Simon J . Slawny A. Sokal M. Suzuki G. Sylvester D. Thouless F . Wegner B. Widom K. Wilson
The Ginzburg-Landau Model
One of the fundamental connections in mathematical physics is the close resemblance of the Ising model to the lattice approximation of the Euclidean scalar field theory with quartic interaction - more commonly known as the Ginzburg-Landau model. The latter is given by a probability measure on the space of all real-valued configurations on :£;3, but the nearest-neighbor coupling is the same. From the point of view of classical statistical mechanics, there is no longer any constraint on the scalar spin values - i.e., SO = { -1, I} is replaced by ~ at each lattice site. On the other hand, the form of the field interaction assigns a strong probability to "spin-up or spin-down" behavior in a certain parameter range. The lattice approximation of the ~ Euclidean field theory is similar in spirit to the standard ultraviolet cutoff, but the starting point is a lattice field ¢ with a discrete Gaussian covariance: (3.5.1)
As an infinite matrix,
clat
is the following inverse matrix: (3.5.2)
3.5. THE GINZBURG-LANDAU MODEL
221
-6, ~'=~, 1, ~,~' nearest neighbors , { 0, otherwise.
(3 .5.3)
,:llat is the lattice Laplacian, since the quadratic form is given by
L
(_Ll lat ¢, ¢) =
(¢(~ ') _ ¢(~))2.
(3.5.4)
(n,';' )
We define d/-Lbat as the Gaussian measure on IRz S with mean zero and covariance C lat Formally, d/-Lbat is the normalization of the measure exp
(_~((_,:llat + m6)cp,cp))
1J dcp(~) . n
For an arbitrary finite set A C Z3 define XA as the characteristic function of A of Z 3 The interacting probability measure on IRA is given by d/-L~' T (cp)
ZA(A, T)-1 exp( -Ilat(XACP))d/-L&at(XA¢)
(3.5.5)
Ilat(XAcp))d/-L~at(XA¢)
(3.5.6)
J
ZA (A, T)
exp( -
L V(¢(r;;)),
Ilat(¢)
(3.5.7)
~
V(z)
=
Az4
+ TZ2,
A > O.
(3.5.8)
Naturally, the continuum limit involves the lattice spacing parameter, which we set equal to unity here. Also the restriction of the Gaussian measure to XA¢ reflects a choice of boundary conditions. How do we recognize d/-LA as the Gibbs state of a spin Hamiltonian for inverse temperature f3? First we identify the spin variable: (3.5 .9)
Thus
~((_Lllat +m6)XA CP,XA¢) - 13
_~ s,;s~+f3f= (3+~m6)s~ , (n ,m)CA
Af32S~
V(¢(r;;)) Second, we give A and
m
T
(3 .5.10)
mEA
+ f3TS~m .
(3 .5.11)
the f3-dependence that makes the single-spin distribution
dO'( z ) = exp
[-13
(3 + ~m6 + T(f3)) Z2 - A(13) 132Z4 ] dz
(3.5.12)
222
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
independent of f3. Accordingly, we set
T(f3) = )..(f3)
~m6 -
f3-1f -
(3.5.13)
3,
= f3- 2~,
(3.5.14)
with ~ and f the independent parameters, so
ZA(f3)-l exp (f3
~~
sr;sr;;.)]I du(sr;),
(n ,m)CA
(3 .5.15)
nEA
f3-(card A)/2 ZA ()..(f3) , T(f3)),
(3 .5.16)
How do we discern anything resembling the behavior of the Ising model in this Gibbs state? In terms of the independent parameters, we write the single-spin distribution as follows : exp( -fz 2 - ~z4)dz du(z) exp( _~(z2
+ f /2~)2 + f2 /4~)dz .
If f is negative, the completed square has a local maximum at z at
z
(3.5.17)
= 0 and local minima
= ±j-f/2i
(3.5.18)
In this case the graph of the completed square is a double well. The point is that at a given lattice site the single-spin distribution has its maximum probability density at the
±j
spin values -f/2i This is a major reason why the lattice field theory has the same critical behavior as the Ising model for sufficiently negative f. Another, rather different reason is that this theory is approximated by the Griffiths-Simon superposition of Ising models, which we do not describe here. The Lee-Yang Theorem applies to the Ginzburg-Landau model as a consequence. For fixed ~ and f, one expects to have uniqueness of the 11.= Z 3 limit of dp,~,T in the case where f3 is small. This uniqueness breaks down for negative f if f3 is sufficiently large relative to ~/f2, but every possible measure is still a convex combination of two pure phases characterized by the spontaneous magnetization, which is defined exactly as it was for the Ising model. Indeed, if ZA().., T, h) and dp,~'T,h are defined by replacing V(z) with V(z) + hz, then
N
lim
A->Z 3
(rr Jrr ¢J(r:"j)dp,~'T,h(¢J)
¢J(r:"j)) ()..,T,h)
j=l
A
N
lim
A->Z3
(3.5.19)
j=l
exists for h i- 0 - i.e., the infinite-volume state is unique for h i- 0 by the same reasoning that was applied to the Ising model. The spontaneous magnetization is given by
3.5. THE GINZBURG-LANDAU MODEL
223
z
Figure 3.5.1: lim (3-1/2 (¢(O)) ()..((3), 7((3) , h) ,
(3.5.20)
h '-,. O
where this expectation is Z3-translation-invariant. Also, (3-1/2 (¢( 0)) _ ()..((3), 7((3))
=
lim r1/2(¢(O))()..((3), 7((3) , h) h /' O
= -M((3)
(3.5.21)
by the z f-t - z symmetry of V(z). If dJ.L ranges over convex combinations of probability measures realized as limits of the net {dJ.L~'T } , then dJ.L is parametrized by the unit interval as the convex combinations dJ.L
= exdJ.L+ + (1 -
ex)dJ.L_ .
(3.5.22)
If ex = ~, then the ¢C:;;') f-t -¢C:;;, ) symmetry is unbroken for dJ.L and the long-range o~er parameter is given by c((3) = Jim (3-1
/¢(~)¢(O)dJ.L(¢) .
(3.5 .23)
m-+oo
On the other hand, the ¢(:;;,) /
>-t
-¢C-;') change of variables yields
¢(~)¢(O)dJ.L-(¢) =
J
¢(~)¢(O)dJ.L+(¢),
(3.5.24)
so we have the same conclusion as in the case of the Ising model: c((3)
=
Jim (3-1
J ¢(~)¢(O)dJ.L+(¢)
m-+oo
(3-1 ( / ¢(O)dJ.L+(¢)) 2 = M((3)2.
(3.5.25)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
224
There is long-range order for those parameter values that yield spontaneous magnetization. Now consider the phase diagram of the Ginzburg-Landau model in the parameters A and T for fixed inverse temperature (J and variable X, r . In the h = 0 plane, this system has been proven to have the regimes for a unique phase given by Fig. 3.5.2. Actually, there is only one critical curve. We have drawn the boundaries of parameter regions for convergent expansions, where E. > 0 is understood to be sufficiently small. The region {A ::; E.JT} is the regime of the Glimm-Ja£fe-Spencer cluster expansion, while in the region there is a convergent cluster expansion that decouples the Hamiltonian directly - an expansion that will be introduced in §3. 7. Both expansions are often referred to as high-temperature expansions because they establish the properties of a unique phase. In the two-phase regime there is a low-temperature expansion based on the Peierls argument, but we shall not pursue it. The actual phase diagram is given by Fig. 3.5.3, where the boundary is called the critical manifold .
unique phase
---- ---- -- -- .-
umque
/
EArl
I
1=0
I
,,
)
,
--- --- --
A
phase
1=-EJ"l
Figure 3.5.2: The Ginzburg-Landau model has the same critical exponents as the Ising model. The spontaneous magnetization is a continuous function of the parameters, and (3.5.26.0)
3.5. THE GINZBURG-LANDAU MODEL
225
unique phase
)
spontaneous magnetization
Figure 3.5.3: (3.5.26.1) with 8 :::::: 0.31. The magnetic susceptibility has the divergent behavior
X(A,r) ~ e(r - rc(A))-T', X(A,r)
~
(3.5.27.0)
e(A - Ac(r))-T',
(3.5.27.1)
with f :::::: 1.24, where
X(A,r,h) f(A,r,h)
82
(3.5.28)
8h2f(A,r,h), .
1
lim -IAllnZA(A,r,h).
A--+Z 3
(3.5.29)
Clearly, (3.5.30) is well-defined even for h = 0 in the unique phase region arbitrarily close to the critical manifold. The summand is an example of a truncated correlation (already defined in §1.12) and such correlations are known to decay exponentially when this particular one does, and with the same correlation length etA, r, h) . This is a result of the Lebowitz inequalities for the Euclidean lattice 14 field theory. In the case h = 0 and A > Ac(r), the picture is: (3.5.31)
(O
x n
{
5 CgCpeclmcard (ml , . . . ,m,):
mu 5 rn,mu 2 0 v=l xcard{(al, . . . , a n ) E S n : all distinct).
(3.6.28)
The first cardinality counts the number of points with integer coordinates in the positive oct5nt of n-di_mensional Cartesian space enclosed by the simplicia1 face with vertices m e 1, . .. ,m e ,. This number is roughly the volume mn/n! of that region of space. Allowing for boundary effects, one has
The second cardinality is obviously the number
m! -(:In! Perm(m, n) = (m - n)!
234
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
of permutations of m objects taken n at a time. Thus n
II Iz((v)1
L
(l, ... ,(n mutually v=l disjoint , supp (v cS
(3.6.31)
< so if COC1 < 1, then 00
1
Ln!
n=O
n
L
II Iz((v)1
1) of the polymer expansion
3.6. POLYMER EXPANSIONS
tree graphs T on {I,...,n}
m l , ...,m,=O
235
P = ( c l r . . . , < n ) ~ K n ,=I T is a subgraph of G p
ICY I=mY
where we have further decomposed the sum with respect to volumes of polymers. Controlling the inner-most sum is the crucial part of the estimation. If one tries to count the connected graphs of which T is a connected subgraph, the resulting oversummation is disastrous. It is important to note that one is summing over all connected n-polymers - not n-vertex graphs - on which T lives. As one uses the structure of T to sum over all such n-polymers, the polymers P for which Gp is not simply connected are automatically included in the summation. To control the sum with T , one finds it convenient to use the standard notion of coordination number. For a tree graph T on (1,. . . ,n ) , the coordination number d,(T) of a vertex v is just the number of lines in T that meet v. Every tree graph can be reduced to a single vertex by the following iterative procedure. Let A be the subset of vertices v for which d,(T) > 1 and let T' be the tree graph on A obtained by removing the single lines meeting the other vertices. Obviously, there will now be vertices v E A such that d,(T1) = 1, so now remove those single lines to obtain the tree graph T" on the subset A' obtained by removing those vertices from A. This procedure can be iterated for as long as there are distinct vertices. The summation for fixed T is carried out as follows. First apply the energy estimate to obtain
and pull this bound outside the inner-most sum, which now becomes (with SF to be defined)
Here we have applied the entropy estimate to each sum over polymers C, such that a, E supp C,,. This replaces the sum over polymers with a supremum over polymers, but one still has to sum over the possible a, determined by the T-connectivity property of a given n-polymer. This vital control is implicit in the definition of SF, which reflects the tree reduction just described. First, for every vertex v such that d,(T) = 1, a, E supp CC, where {v, D) is that single line. Then for every v' such that d,~(TI) = 1, a,, E supp &,, where {v', D') is that single line, and so on. It is easy to see that this definition of S; is the correct one for the bound (3.6.37) and that, since the multiplicity o f p = t# is d,(T) - 1,
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
236
where the last polymer in the tree reduction is not connected to anything. This means that Q" E S is the only restriction for the last remaining vertex v , so card(S) appears as a factor . Combining (3.6.36-3.3.38) with (3.6.35) , one obtains interesting tree graph estimation of the nth term (with n > 1) of the polymer expansion:
I:
00
In(Gp)llz(P)I:S
PEK n
tree graphs T
I:
ffil , .. . ,mn=O
on {I •... . n}
co-cre
(Cl-CO) Em_ card(S)
IT m~p(T)-1 n
(3.6.39)
1'=1
The next step is to decompose the sum over all tree graphs with respect to coordination numbers, and then apply the celebrated Cayley Theorem, which states that the number of tree graphs on {1 , ... , n} with coordination numbers d 1 , . . . , d n is just the ratio n
(n - 2)!/
I1 (dl'
- 1)! . Hence
1'=1
I:
In(Gp)ll z(P)1
PEK. n
(n - 2) !Co-Cr lSI
00
< (n - 2)!Co-Grcard(S)
~
~
e
(Cl-CO) Em_ -
IT e n
mp
~=l
ml, . ·· ,mn=O
(n - 2)!Co-Crcard(S)(1 _ eCl-CO+I) - n,
(3.6.40)
provided CI < Co - 1. This yields the bound 00
1
l:,n . I:
n=O
In(Gp)ll z(P)1
PEK. n 00
1+
1
I: Iz (OI + card(S) I: n(n _ 1) Co-Cr(1- e (EP
Cl
-
co
+1))-n
(3.6.41)
n=2
on the polymer expansion of In Z(S) . Clearly, the n given by
= 1 contribution has the
bound
(3.6.42) where we have summed over volumes and over Q E S , applying the entropy estimate in summing over ( E P with fixed volume and supports containing fixed Q . In other words,
3.7. EXPANSION FOR NEA REST-NEIGHBOR INTERACTIONS the basic estimation is the same as in the n estimation to do. In summary,
>
237
1 case, but there is no combinatoric
and the dominating series converges if CoCl 5 1- ecl-co+l
References 1. G. Gallavotti, A. Martin-Lof, and S. Miracle-SolB, "Some Problems Connected With the Description of Coexisting Phases at Low Temperatures in the Ising Model," in Statistical Mechanics and Mathematical Problems, A. Lenard, ed., Lecture Notes in Physics 20, Springer-Verlag, New York, 1973. 2. J. Glimm and A. Jaffe, Quantum Physics: A hnctional Integral Point of View,
Springer-Verlag, New York, 1987. 3. C. Gruber and H. Kunz, "General Properties of Polymer Systems,'' Commun. Math. Phys. 22 (1971), 133-161.
4. V. Malyshev, "Uniform Cluster Estimates for Lattice Models," Commun. Math. Phys. 64 (1979), 131-157. 5. V. Malyshev, Introduction to Euclidean Quantum Field Theory, Moscow University Press, Moscow, 1985. 6. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969. 7. E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics 159, Springer-Verlag, New York, 1982.
3.7
Expansion for Nearest-Neighbor Interactions
As applications of the theory of polymer expansions, the high-temperature expansions of the classical Heisenberg model and of the lattice $J; model are probably the most elementary. This section is devoted t o describing these two specific examples as part of our progression toward more difficult expansion methods. We consider the hightemperature classical Heisenberg ferromagnet first. S2 and the For a finite set A of lattice sites, the space of configurations is 2
nEA
single-spin distribution is surface measure du on S2 For zero external magnetic field,
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
238
the Hamiltonian is just 3
HA=-J
s;;; (G,~)cA
2
s;,
(3.7.1)
and our goal is to control the A = Z3limit of the Gibbs expectations
for small p (high temperature). These quantities are generated by the generalized partition function
and our task is to develop a disconnected polymer expansion of
where we set
2
2
~ ( he - ) =Jexp(hi
2
2
si)do(si).
(3.7.6)
Once the expansion is developed, we establish the energy estimate and the entropy estimate for A-uniform convergence of the corresponding polymer expansion of
-
1 lnZ(h-: C E A) = card A e card A 2
2
(3.7.7)
e e EA
in connected n-polymers in the high-temperature regime. In this example, the most obvious expansion method works. Simply expand Z(h;: C E A) in powers of p: 2
2
3.7. EXPANSION FOR NEAREST-NEIGHBOR INTERACTIONS
239
However, a power series in {3 is not the most natural point of view, even in this case. We first write this in terms of multi-indices on nearest-neighbor bonds. Let BII. be the set of such bonds contained in A and A~ the set of all multi-indices Ie BII. -t {O, 1,2, ... } such that 111;1 = 11;( (T:t, 1{)) = N . (3.7.9)
L
(;;;',~)Cll. A
.
---lo...-..l...-..l.
---10.
The element of AN associated with a sequence ((ml, nl)' ... ' (mN, nN)) is defined by (3 .7.10) and there are
N!
n
11;( (T:t, 1{))!
(;;;',n)
such sequences with which a given multi-index II; is associated. Since the terms in the expansion depend only on the associated multi-index, we have Z(h~ : l
fE A)
( II Z(h 7 )) fEll. \supp
x
.I (~L f E supp
X
A
~ ~ (m,n)
~
00
A~ and supp
N=O
II;
7)
~
fE supp
K
II (-;;;;.. -;~)K«;;;"~)) II = U
da(-;
II Z(h7)!exp ( L h7·-;7)
fEII.\supp
l E supp
where All.
II f E supp ~
(m,n)
K
L II ((3J):«:,n;) 1I;((m, n)).
KE A
IT (-;; ;. .-; ~)"'«;;;., n))
h7 · -;7)
exp
K
da(-;7)'
K
(3 .7.11)
K
is the set of lattice sites joined by those bonds to
which II; assigns non-zero weights. We define a polymer to be any connected set of nearest-neighbor bonds (T:t,1{). The support of a polymer is just the set of lattice sites joined by the bonds, while the activity of a polymer is to be inferred from a reorganization of this power series expansion. For a given multi-index 11;, there is a uniquely determined set {(I, ... , (n} of polymers with mutually disjoint supports which partitions supp 11;. There are n! permutations of this set of polymers, and each permutation is an element of V n . Thus Z(h~: l
f E A)
240
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
(3.7.12) _
n
lEU supp k=l
0, consider the regime 7
i:
(3.7.28)
:S c..J)..,
We have the reductions
O'o().., r)
= )..-1/4
exp ( -
W '4 -
)..-1/2
(~m6 + 3 + r)w '2 )dw l ~ c).. -1/4,
(3.7.29)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
244
so the activity estimate becomes
A
for the parameter restrictions. Since L A ( C )
4
> 1 for C E
therefore have
in this regime. It is now clear that we have the desired activity bound for suficiently large A, relative to the arbitrary constant in T 5 c 6 . To see how this regime can be interpreted as a high-temperature regime, we recall that the scaling (3.5.9) of integration variables induces the parameter transformations
with .i and
ifixed independently
of
P. Note that (3.7.28) is equivalent to
while large X is equivalent to small P for fixed choose c = ?/A.
A.
The size of .idoes not affect P if we
References 1. J. Glimm and A. Jaffe, Quantum Physics: A finctional Integral Point of View, Springer-Verlag, New York, 1987. 2. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.
3.8
Inductive Interpolation
Our next example of a polymer expansion is another analysis of the lattice C#J~ model with mass mo and coupling constant X that is similar in spirit to the expansion presented in the last section in the sense that it attempts to decouple the quadratic form
3.8. INDUCTIVE INTERPOLATION (-alat4, 4). Recall that the partition function is given by the integral
for every finite A C Z3. The generalized partition function is given by
The polymer expansion we develop in this section is based on inductive interpolation, which shall be the basis of the phase cell cluster expansion introduced in Chap. 5. In the statistical-mechanical context of this chapter, however, it is just a refinement. Indeed, for this particular model, it is superfluous. Its introduction at this level is useful because of its sophistication relative to the wholesale expansion discussed in the last section. Accordingly, we introduce a linear ordering of A and base an inductive expansion on it - an expansion where eachJerm reflects a history of interpolaticns. The first step is an attempt to decouple 4(n l ) from the other variables, where n is the first element of A. By the fundamental theorem of calculus
'
so this first interpolation yields A
z(hb:
e E A)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
246
This leading term is a decoupled term, while the other terms are remainder terms. The second step in the inductive expansion is applied to each term, but the interpolation now varies with the term. For the decoupled term we apply the same formula to Z (h-:
e
e
E A\ {r; 1}) , except ¢(r; 2) is now the variable we try to decouple from the
other variables, where r; 2 is the successor of r; 1 in A. For a given remainder term labeled by (r; , r; 1), we decompose the d¢>-integral with the interpolation that attempts to decouple both ¢(r;) and ¢(r; 1) from the other variables. In this way, a sum over possible sequences of interpolations is induced by an algorithm that dictates the next step for a given term based on the sequence of past interpolations that developed it. Consider an arbitrary term that has developed at an arbitrary stage in the expansion. Since each previous step was an interpolation followed by a choice of new term, a product ¢(T:t )¢(r;) associated with a nearest-neighbor bond appeared in an integral with each expansion step selecting a remainder term, so sets of variables have also developed . If the term under consideration happens to be a decoupled term, it has the form
where K-y is some complicated
n
d¢( e)-integral developed by the ')'th segment of
e EA, successive expansion steps. The sets A-y of lattice sites are mutually disjoint, and each A-y labels the variables differentiated down by the ~th succession of expansion steps. Given this
d~couPled term, the next step is to decompose Z(he :eE A\ YA-y)
exactly as Z (h-:
eE A) was, except that if r; , is the first lattice site in A\ U A-y with -y
l
respect to the linear ordering on A, then ¢(r; ') is now the random variable one tries to decouple from the others. Now suppose the term under consideration is a remainder term instead. Let {til denote the set of parameters for past interpolations. The term has the form
K.
II K-y(h e: eE A-y) J_II -y
¢(r;)N-;:
(II 11
nEA\UA,'
-y
0
dti) f(t)
3.8. INDUCTIVE INTERPOLATION
247
exp(~ L h-;¢(i)-Ilat(XA\yA~¢)+WA\yA~(¢jt)
x
l EA\UA~ ~
~~ L l
(m6
+
~A(i))¢(i)2) ~ II
EA \ U A~
d¢( £),
(3.8.6)
lEA \ U A~
~
~
where W A\ U A~ (¢j t) denotes the interpolation of ~
L
¢(~)¢C;;)
(3.8.7)
(~ , n:}CA\UA~ ~
that has developed and f(t) is the product of powers of the ti having grown as old t-dependencies have been differentiated down by new interpolations. In this case, the next step is to base the interpolation on the attempt to decouple the variables in A' from the rest of the variables in 11.\ Uk.p where A' includes those ¢(:;, ) that have been "(
differentiated down - i.e.,
(3.8.8) The interpolation formula yields the form
K
= Z(h-;: fEA\(A'UyA"())K'(h-; : fEA')I]K"((h-; : fEA"() +
II
Ky(h-; :
f E A"()
11
J¢(~ ')¢(~)
~ ~L
dt'
"(
(m',m}CA\UA~ ~
~'EA',~I/;A'
x
II
¢(:;')N-;:
n:EA'
xexp
11 (II 11 dt'
0
r~ L l
dti)
f(t)f~,~ (t)
,0
h-;¢(f)_I
1at
(XA\yA~¢) +t'WA\yA~(¢,t)
EA\ UA~ ~
+
(1 - t')(WA' (¢j t)
+ WA\(A'UUA~)(¢j t) - ~
L
(m6
~
l EA\UA~ ~
+ ~A( £))¢( £)2
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
248 x
IT
(3.8.9)
d¢( f).
,
iEA\UA,
Jr;., r;.(t) is just the product of the old interpolation parameters ti associated with
dec~upling those complementary pairs of sets that separated n;, , from
n;,.
This means (3.8.10)
Wi\' (¢; t) denotes the restriction of the sum to those nearest-neighbor pairs both of whose lattice sites lie in 11.', and
K'(h-; : eEli.')
=
J.F!. (TJ.l1 (_2: ¢(:;{)N-;:
h-;¢(i) -
x exp
dti) f(t)
[lat (Xi\' ¢)
+ Wi\'(¢;t)
iEi\'
(3 .8.11)
The next step in the development of a history is to pick either the decoupled term or a remainder term in (3.8.9). This completes our inductive definition of the expansion. Since A is a finite set, it is clear that every possible history of steps eventually terminates with a decoupled term of the form
K=
IT K')'(h-;: eE A,),),
(3.8.12)
')'
Such a decoupled term is a completed term. The A')' are disjoint, and the factorization over this partition of A is due to the history of choices. In the ,th segment of successive expansion steps, every step except the last one chooses a remainder term. The last step chooses the decoupled term, and so one starts all over again with the residual set of variables, initiating the next sequence of expansion steps. We have now expanded the original partition function in these completed terms, and our task is to control the expansion by organizing the terms and writing the K')' more explicitly for estimation. Let Xo be a distinguished element of an N-element set r. An ordered connectivity graph on (r, xo) is a mapping QJ from {2, . .. , N} into r x r such that with QJ(i) = (go (i), gJ( i)),
gl (i) =J: gl (j), i =J: j, go(i) E {xo} U {gl(j): j < i}.
(3.8.13) (3.8.14)
3.8. INDUCTIVE INTERPOLATION
249
The point here is that the ')'th sequence of expansion steps is naturally described by an ordered connectivity graph (1)1' on (AI" r; 1') where go(i) and g1 (i) are nearest-neighbors and r; I' is the lattice site that this succession of steps initially tries to decouple from other lattice sites. Every time a remainder term is chosen for an interpolation step, an additional product ¢,(gJ (i) )¢'(gi (i)) appears in the dcf>-integral, where ¢,(gJ (i)) is a variable that has already been "differentiated down" from the exponent and is among the variables the interpolation is attempting to decouple from the variables that appear only in the exponent thus far - among them, the variable ¢,(gi (i)), which is now "differentiated down" for the first time. This variable cannot occur as a ¢,(gi (i)) for any other i-only as a ¢,(gJ (i)) for later i. This ')'th history of expansion steps has the tree structure described by (3.8.13) and (3.8.14) above. The KI'-factor corresponding to this history in the completed term has the form
KI'(h-;: f
E
xexp
AI')
=
(If 11
dti) fC!37 (t) J
(~L h7¢,(f)-I
1at
If
(¢'(gJ (i))¢'(gi (i)))
(XA,¢,)+WA,(¢,;t)
tEA,
~ ~L
(m6
+ LA(f))¢'(f)2)
tEA,
)1
d¢,Cy)'
(3 .8.15)
tEA,
where the monomial fC!37 (t) is yet to be determined. It is important to note that a factor KI' can be an integral in just one variable. After all, the initial interpolation in a new sequence of expansion steps attempts to decouple the first of the remaining lattice variables from all the other variables. If that first step chooses the decoupled term, then the sequence is terminated already, and the corresponding o.c.g. (ordered connectivity graph) is empty. In this case, N = 1 and
KI'
Z(h~,) n]
=
Jexp(h~,¢'(r; 1') n
V(¢'(r; 1'))
1
-~(m6 + LA(r; I'))¢'(r; 1')2)d¢'(r; 1')
(3.8.16)
occurs with no interpolation parameters at all. The point is that the polymers we define below do not index such factors, and indeed, the single term in the whole expansion consisting of only those factors - i.e., the term developed by the series of decouplings that never "differentiates down" anything - must correspond to unity in the polymer expansion of an abstract partition function . As in the previous section, the abstract partition function in this case is actually the normalization
ZA
= Z(h 7 : f
E
A)
II Z(h7 )-1 tEA
of the generalized partition function .
(3.8.17)
250
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
4
•
6
•
•
10
8
• •
•
11
•
•
13
•
• Figure 3.8.1:
Now from the standpoint ofrealizing f(1j,(t) explicitly, it is clear from the inductive step described above that (if we drop the superscript ')') (3 .8.18) so it is a matter of examining the f m'm (t) introduced there. By way of illustration, suppose the ordered connectivity graph
f(1j(t)
=
fdt)f13 (t)f17(t)!I,12 (t)h4 (t)hs(t) x 129 (t)!s6 (t) !s,1O (t)f4S (t)h,u (t) f11 ,13 (t)f11,14 (t) 1 X tl x (tlt2t3t4tS) X (tlt2t3t4tSt6t7tst9tlO) x t2 X(t2t3t4tSt6) X (t2t3t4tSt6t7) x (tst4) x (t3t4tSt6t7tS)
3.B. INDUCTIVE INTERPOLATION
251
xl X (t7tst9) x tu x (tut12) t1 3 t2 5 t3 6 t4 6 t 55 t6 4 t7 4 ts 3 tg 2tlO tu 2 t12 .
(3.8.20)
In general,
g1(i') where
7)(5
= go(i)
=> i'
= 7)(5 (i),
(3.8.21)
is defined by (3.8.22)
with the convention that g11(go(2)) = 1. We combine (3.8.18) with (3.8.19) to obtain the formula (3.8.23) and this completes our description of an interpolated factor in a completed term. The product of lattice field amplitudes that has developed in the dcf>-integral associated with a given ordered connectivity graph (!5 is just
II(¢(go (i) )¢(g1 (i))). Since every (gO(i),g1(i)) must appear as a nearest-neighbor pair for this model, and since a lattice site can occur no more than once as some g1 (i), notice that a given lattice site ~ can occur no more than seven times as some go(i) . With 0 ~ ti ~ 1, we have the trivial estimation (3.8.24) in the formula (3.8.10), so
L
1¢(i7i ')¢(i7i)1
(T;;! ,T;;)cA'
1
~
2 ~L
~
~A' ( e)¢( 1!)2
(3.8.25)
lEA'
Since
~A'
(I!)
~ ~A (I!),
JII
it follows that
(¢(go(i))¢(g1 (i))) exp
,
lEA'
L
1
+WA'(¢;t)--
~ ~II lEA'
(~L h7¢(e) - I1at(xA'¢)
i:
2~
l 'EA'
Iwl(card
~ ~
(m~+~A(e))¢(1! ')2 )
9,1(7))+1
II d¢(I!)
~ lEA'
exp (h7w - V(w) -
~m~w2) dw.
(3.8.26)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
252
The single-spin estimation is now done in the same manner as in the previous section. The combinatorial obstacle in the expansion is to sum over the ordered connectivity graphs. Once again, consider an N -element set r with a distinguished element Xo. An unordered connectivity graph on (r,xo) is simply the range of an ordered connectivity graph on (r, xo). In the combinatorics literature, the synonym for this term is "tree" not to be confused with "tree graph," already defined. An ordered connectivity graph may be regarded as a history of tree growth. Now to apply the abstract theory of polymer expansions, we specify our polymers to be unordered connectivity graphs on sets of lattice sites, where only nearest-neighbor bonds belong to it. The support of a polymer is just the set of lattice sites involved, so the volume is the number of these lattice sites. The entropy estimate is easy in this case. Indeed, if M is the given volume for all polymers whose supports contain a fixed lattice site r; 0, then such a polymer consists of M - 1 bonds. The number of these polymers is therefore bounded by the number of paths (consisting of nearest-neighbor bonds) of length ~ 2(M - 1) originating at r; o. It is obvious that a path can cover a tree of bonds without using any bond more than twice. On the other hand, the number of all paths of r bonds originating at r; 0 is bounded by 8T , so the bound (3.6.20) is established in this case. How do we realize ZII. as an expansion in activities of these polymers? The expansion we have developed is actually a sum over all finite sequences ((!)1, . .. , ~n), n = 1,2,3, . . ., of ordered connectivity graphs on disjoint pointed sets (A-y, r; -Y) such that: (a) (b)
r; -y+1 succeeds r; -y with respect to the linear ordering of A, r; -y
is the first element in A-y with respect to that ordering. n
K-y(h-: eE A-y), where -y=1 e each K-y-factor is entirely determined by the corresponding history of interpolation steps described by ~-y . In particular, there is a mapping
IT
With each such sequence is associated a completed term
~>-+K~(he: eEA~)
from ordered connectivity graphs to the interpolated factors that occur in completed terms. Now for an unorr!:!Ired conn~ctivity graph T consisting of nearest-neighbor bonds on the pointed set (A', n ') with n ' the first site occurring in A', specify the activity of this polymer T to be z(T)
K~(h-I : eE A')
= ordered connectivity graphs ~ on (11.,';') T",=T
where
T~
is the range
of~.
II
Z(h_)-l,
e EA'
e
(3.8.27)
Thus, we have the expansion n
II z(T
i ),
n=O (T, ,... ,Tn)EV~ i=l
(3.8.28)
3.8. INDUCTIVE INTERPOLATION
253
where V~ denotes the set of all n-tuples (T1' .. . ,Tn) of mutually disjoint polymers such that the first element in supp Ti+1 succeeds the first element in supp T i . If Vn denotes the set of disjoint n-tuples of polymers without the order condition, then ~
ZA
00
=L n=O
1
,
n.
n
L
II z(T
(3.8.29)
i ),
(T, •... •Tn}E"Dn i=l
so control of the induced expansion of
IAI- 1 In ZA = IAI- 1 (In Z(h~: t
e A) - '" E
~
In Z(h~)) t
(3.8.30)
tEA
in connected n-tuples of polymers is guaranteed by the general theory in the range of parameters for which we have an activity estimate (3 .8.31) with Co sufficiently large. We infer the desired bound from (3 .7.26), (3.7.29), (3.8.26), and a fundamental combinatorial result . For every unordered connectivity graph T on a pointed set (A', ~ ') , (3.8.32) ordered connectivity graphs I!l on (A'.;')
We prove this identity in §3.11.
References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field
Theories," Ann. Phys. 142 (1982),95-139. 2. D. Brydges and P. Federbush, "A New Form of the Mayer Expansion in Classical Statistical Mechanics," J. Math. Phys. 19 (1978),2064- 2067. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360. 4. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer-Verlag, New York, 1987. 5. R. Seneor, "Critical Phenomena in Three Dimensions," in Scaling and SelfSimilarity in Physics , J. Frohlich, ed., Birkhiiuser, Boston, 1983. 6. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.
254
3.9
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
A Generalization of the Polymer Estimation
We now recall the infinite-volume problem brought up in §3.1 (for high temperature). Returning to the lattice system whose single-spin distribution is Lebesgue measure da on §2, we consider the long-range Hamiltonian in a zero external magnetic field given by (3.9.1) 'L" J~n-n' ~ s ~n . s ~n t ) where in this section we assume only the decay condition (3.9.2) which is just summability. This spin model is treated differently from the classical Heisenberg model, as the theory of polymer expansions presented in §3.6 must be generalized in this context, and the Konigsberg bridge-island estimation is too crude as well. The algebraic structure of the expansion is the same, however. One expands Z(h~: iEA)
(3.9.3)
e
IT
Z(h~)
~
e EA
Z(h~ :
e
iEA)
e
J (.?= he· se exp
eEA
x
" -(3 '~
J~n-n' ~ s~n
n,n'EA
II da(se)
(3.9.4)
e EA in all couplings. We have already learned from the nearest-neighbor case that a straight power series in may converge for small but from the point of view of polymers it is more natural to recombine terms into an expansion of the form (3.7.15). Accordingly,
(3
exp
(-(3 '" L.. n,n'EA
(3,
J~ ~, s~ · s~,) = n-n
n
n
'" L...J
II [exp(-(3J~ ~ s~· s~) -1] n-n'
n
n'
rcAxA (n,n')Ef
(3.9.5) is the identity we use in this case. Clearly, our polymers will be subsets of Ax A that connect the lattice sites in their own supports, but we already have a problem with the entropy estimate. If we define the volume of a polymer to be the cardinality of the set of lattice sites supporting it, then - since arbitrarily distant lattice sites in A are directly coupled by a long-range interaction - the bound on the number of polymers with fixed volume and having a fixed lattice site in their supports depends on A and diverges as A /' Z3. An alternative that might seem tempting is to define the volume of a polymer to be the sum of iI-lengths of the line segments associated with the ordered pairs of lattice
255
3.9. A GENERALIZATION OF THE POLYMER ESTIMATION sites. The iI-length of a line segment {n;.,~} is defined to be ~ 1m,
,
- n,l, where the
motivation here is to give integer values to the p0!rmer volumes. With this understanding, we fix an integer L > 0 and a lattice site n E ;;Z3 and try to count all polymers with volume L and ~ in their supports - i.e., all connected sets of line segments having these constraints. By the Konigsberg bridge-island solution, every connected set of line segments has a path consisting of precisely those line segments, originating at any chosen point, and traversing each line segment no more than twice. Therefore, it is enough to bound the number of all possible paths of line segments originating at ~ and having total length L with L :S L :S 2L. Now classify such paths according to the number N of line segments they consist of and the sequence (Ll" ' " L N) of lengths according to the sequence of line segments. With Nand (L 1 , . • . , LN) fixed, it is clear that the number of such paths originating at ~ is bounded by (3.9.6) On the other hand, we know that the number of sequences (Ll" ' " LN) of positive N
A
•
integers with ~ Lk = L is bounded by e L -
1
Summing over all such sequences and
k=1
over all N, we bound the number of paths originating at ~ with length
L00 eL-lc~ ( ~ A
N=1
rN ,L-, (~ ':1' un' Nr
L by the sum
:S
:S :S
eL - 1
(00 ~ (L';;) A
exp(2LJCO + L - 1).
(3.9.7)
In this way, we obtain the desired entropy estimate. Can we balance this with an energy estimate in terms of the polymer volume defined here? Clearly, the activity of a polymer ( is given by
z(()
=
1
IT l Esupp (
x
Z(h 7
(7 II
J II )
[exp(-,lJJr;_r;,
s~
n
8'r;,)-1]
{r;,r;'}E(
~. ~) II
h -;
ell
7E
E supp (
dU
(~87 ) '
(3.9.8)
supp (
so by (3.7.16) we have the bound
Iz(() I:S
II {r;,n' }E(
(,BIJr;_r;, leiJlJn" -n"' I) .
(3.9.9)
256
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
This is inadequate because the number of factors can be much smaller than the volume of ( - i.e., the number of line segments has nothing to do with the sum of their [1_ lengths because the line segments can be arbitrarily long. One can still extract a bound that has exponential decay in the volume of (if the pair interaction J ri _ ri , has
sufficiently fast exponential decay in I ~ - ~ 'I, but such an interaction is essentially "nearest-neighbor" in character - a property much too restrictive. These two failed alternatives hardly represent a crisis, however. After all, hightemperature expansions for long-range interactions were developed long before the abstract polymer formalism was around. Moreover, the latter has a trivial generalization that is ideal for this high-temperature problem. Accordingly, we return to defining the volume of a polymer as the cardinality of its support and re-examine the abstract theory. Instead of insisting on both an energy estimate and an entropy estimate, we note that it is sufficient to require only the estimate (3.9.10) (: E supp (
I(I=M
for
C2
> 0 large enough . Recall that the estimation of the inner-most sum n
II Iz((v)1
2:
P=( l •. ·· .(n )EiC n 1'=1 T is a subgraph of G p
I(vl=mv
in (3.6.35) was done by combining the energy estimate with the entropy estimate. The resulting bound was CoC~e
(cl-co)L;m v v
II m~~(T)-l , n
(card S)
1'=1
where Co and Co (resp. C1 and C1) were given in the energy estimate (3 .6.19) (resp. the entropy estimate (3.6.20)) . On the other hand, the estimate (3.6.37) was necessary only because (3 .6.36) estimated the product of activities right out of the summation. If we apply (3.9 .10) to each sum of activities over polymers (v such that (tv E supp (v, the bound on the multiple sum simply becomes
C;e
- C2
Em" v
(card S)
II m~~(T)-l, n
1'=1
which is all that is needed for the convergence theory. In the present context, one has to establish (3.9.10) by using (3.9.9), but the Konigsberg estimation applied in the nearest-neighbor context is no longer sharp enough. Indeed, consider a fixed lattice site ~ 0 and a polymer ( containing ~ 0 with 1(1 = card supp ( = M. Once again, by the Konigsberg bridge-island solution, there is a path o!jine segments consisting of precisely those line segments that are in ( , originating at n 0, and traversing no element of ~ more than twice. Let 7r( denote such a path
3.10. UNIT-VECTOR SPINS WITH LONG-RANGE COUPLING
257
for each (. Then Iz(() 1:S 13 M (: ;oE supp (
(: ;oE supp (
I(I=M
I(I=M
(3.9.11) where D,,< is the number of line segments that have been traversed twice and we have applied 2 (3.9.12) IJ;;_;;, I :S cIJ;;_;;, 11 / to those line segments f~, ~
'}
that have not been traversed twice. Thus M'
L II (IJ;;i_;;i_lI1/2e~IJ;;;i-;;;i-ll)
(3.9.13)
M'=M m ---"I , ... ,m -->OM'i=l
(: ;oE supp (
I( I=M
because the lengths of the sequences 1l'( range from M to 2M. The idea is to estimate the multiple sum from the inside on out, applying a summability condition when summing over for fixed but this iteration depends on the summability condition.
ni
ni-1,
(3.9.14) which is stronger than condition (3.9.2). One can still establish convergence of this polymer expansion without this restriction on the long-range interaction. Indeed, the required combinatorics is standard and predates the polymer formalism. We can label these line-segment polymers with connected graphs and control the multiple summation with those subgraphs which are tree graphs. This control is quite similar to the key estimation for convergence of the abstract polymer expansion of the abstract free energy in §3.6. We shall not pursue this, however. Instead, we devote the next section to the polymer expansion based on applying inductive interpolation to this long-range model. The point of this alternative is that the tree structure is already there in an individual polymer.
References 1. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View,
Springer-Verlag, New York, 1987. 2. D. Ruelle, Statistical Mechanics, Benjamin, Reading, 1969.
3.10
Unit-Vector Spins with Long-Range Coupling
We continue our analysis of the lattice system whose single-spin distribution is Lebesgue measure da- on §2 and whose Hamiltonian in a finite region A C Z3 is given by
HA =
L
~
~
J-n-m - s -n . sm
(3.10.1)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
258
We now describe an expansion to control the A = Z 3 limit of the expectations for high temperature, assuming only that
J o = o.
(3.10.2)
This condition is the weaker of the two conditions mentioned in the previous section. The stronger condition seemed to be called for, but there are various ways to remedy this. Here we generate the products of exponential differences in a more controlled way - using inductive interpolation, as we did in §3.8 - only now we use the more general polymer estimation. Introduce a linear ordering of A and attempt to decouple -; -, from all other spin ---lo.
n
.....:..I.
variables in the integral expression for Z ( h - : eE A) by interpolation, where ri: f the first lattice site in that ordering. We insert the identity exp
(-(3'" J-
n-m
~
-; -n
1
is
-;-) m
n,m
exp
(-(3 '"
J-n-m - -; -n
~
-;-) m
; ,n;:#;; 1
+11o dt1(-(3 x exp
(-t
'"
J-n-m - -;_. -;-) L-, n m ;=;1 or ;;=;1
" J-n-m - -; -n . -; m - -(1 1(3 'L-,
tt}(3
n,m
X
J-n-m - -; -n . -;-) m
' " L...J
(3.10.3)
n,J;#nl
in the integral to obtain ~
Z(h
- (3
~
X
1
or
J-n-m -
T1i=nl
11
---lo.-lo.
= Z(h n,)Z(h
'" ;=n
X
.....),.
7 : eE A)
7 : eE A\{ri: 1})
J-;-.-;n
m
" J-n-m - -; -n . -; m - -(1 - t1)(3 '~ " J-n-m _ -; _ . -;_) o dt1 exp (-t 1(3 '~ n m
II du(-; 7)'
n,m
n,ni#nl
(3.10.4)
f EA
We use the same terminology as before: the factorized term is called a decoupled term, while the other terms are called remainder terms . The second stage in the development
3.10. UNIT-VECTOR SPINS WITH LONG-RANGE COUPLING
259
of the expansion is applied to each term, with the variables we attempt to decouple dictated ~ th:."term. In the decoupled term, the same interpolation is applied to the
eE A\{ri: I}), but -;n is the variable to be decoupled from the other variables, where ri: 2 is the first element in A\ {ri: I}. For a given remainder term labeled by (ri:, n;,) with either ri: =ri: 1 or n;,= ri: 1 - the interpolation is based on the
factor Z(h-;=
2
attempt to decouple both -; ~n and -; m~ from the other variables. The iteration of this procedure develops a sum over histories of interpolations, with each interpolation branching out that development in many possible directions for every partial history previously developed. We still refer to a step as an interpolation followed by a choice of term, so every term that has developed at a given stage is labeled by a history of previous steps. The nature of the next interpolation depends on the term. If a given term is a decoupled term, it has the form
where K'Y is an integral in only those spin variables -; ~ for which eE A'Y' As in §3.8, l K'Y is called a completed factor, and it is developed by the 'Yth succession of steps, where "succession" has the same meaning here as it did there. For such a term, the next interpolation is applied to the
fact~ Z ('h e:
the same manner as it is applied to Z ( h ~ : l
eE A\ YA'Y) only, and in exactly
eE A)
- as an attempt to decouple the
spin -;~, in the integral from the other spins -; ~, where n n
ri: ' is the first
lattice site in
the given set of sites ri: with respect to the linear ordering. On the other hand, if the given term happens to be a remainder term, then it has the form
fft>(t)j~!1
(-f3 J :;;_r;.-;:;;'-;r;.)
(n ,m)EU.,
h~
x
l
s~
l
Let A' denote that set our previous interpolation was trying to decouple from the other lattice sites in A \ UAT This remainder term represents a failure to decouple,
'Y
and it contains a factor J~n-n ~, -; ~n
-;~, in the integrand where n
ri: '
~ A'. i-I and 7](i') < i}
(3.11.33)
candidates for p(i) . The latter set has the same cardinality as the set of candidates because {2, . .. ,i - I} C {i' : 7](i') < i} . (3.11.34) On the other hand , {i': 7](i') < i < i' + I} {i' : 7](i') :S i -1 :S i' -I},
{i' : i' > i-I and 7](i') < i}
(3 .11.35)
so it follows from (3.11.28) that the number of candidates for p(i) is w1/(i - 1) Therefore, the number of possible p is just N
II (w1/(i i=2
1)
+ 1),
+ 1.
3.11. COMBINATORIAL PROPERTIES OF INTERPOLATION WEIGHTS
269
and this completes the proof of (3.11.27). The Speer identity now develops as follows. For any two ordered tree graphs TJ and TJ', we can find a permutation p of {2, . . . , N} with TJ' = TJOp if and only if d7)' (j) = d7)(j) for all j. Pick a representative TJ from each of these equivalence classes and index these ordered tree graphs: TJI, . .. , TJM. Then
N,"~'~' OI d'(il') (fJ l' dt.) I,(ti ~ OI .,,(il) "E,.. (fJ l' "t.) I,(ti ,
(3.11.36)
On the other hand, the permutations p of {2, .. . , N} for which TJ 0 p = TJ are precisely those p that permute the sets TJ- I ({j}) independently, so the number of such N-I
permutations is
IT
d7)(j)! Hence
j=l
.D
N_I
1l'(TJ)
(
)
card {TJ': TJ'
d7) (j)!
= TJ 0 p for some permutation p} (3.11.37)
Combining this with the interpolation weight identity (3.11 .27), we reduce (3.11.36) to:
1
M
L L
k=I7): d,=d'k
card{TJ' : d7)'
= d7)k}
M
L1 k=l
(3.11.38)
M,
so the dependence of M on N is the answer. M enumerates the equivalence classes defined by the equations d7) (j) = d7)' (j) , so M is the cardinality of the set WN
= {(nl,. ' " nN-I):
nj
= d7)(j)
for some ordered tree graph TJ} ·
(3.11.39)
On the other hand, integer sequences (nl, . . ' ,nN-I) arising from ordered tree graphs are characterized by the constraints nj ~ 0 and k
k ~ Lnj ~ N-1. j=l
(3.11.40)
270
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
Given this alternative definition of WN, it is a standard combinatoric fact that M
1(2NN _-12) '
(3 .11.41)
= cardWN = N
and so (3.11.25) is proven. We conclude our discussion of pure combinatorics by stating the Anghel-Tataru identity without proof. For complex variables WI, ... , W N,
p
permU~{I,
... 'N}
Nth-~ree ( 1/
N
}] W
p
1) !! 10
)(N (1/(i))
dti
f1/(t)
(N )N-2 = ~ Wi
p(I)=I
(3.11.42) This result is a master identity for interpolation weights. It reduces to the Robinson identity if we set Wi = 1 for all i, while we can obtain the Speer identity by applying the integration
to this equation.
References 1. V. Anghel and L. Tataru, "New Identities for Cluster Expansions," Lett. Math. Phys., to appear. 2. G. Battle, "A New Combinatoric Identity for Cluster Expansions," Commun. Math. Phys. 94 (1984), 133-139. 3. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 4. G. Battle and P. Federbush, "A Note on Cluster Expansions, Tree Graph Identities, Extra liN! Factors!!!" Lett. Math. Phys. 8 (1984), 55-57. 5. D. Brydges and P. Federbush, "A New Form of the Mayer Expansion in Classical Statistical Mechanics," J. Math. Phys. 19 (1978), 2064-2067. 6. F. Harary and E. Palmer, Graphical Enumeration, Academic Press, New York, 1973. 7. E. Speer, "Combinatoric Identities for Cluster Expansions," Lett. Math. Phys., to appear.
3.12. SCALAR SPINS WITH LONG RANGE COUPLING
3.12
271
Scalar Spins with Long Range Coupling
We return to the case of scalar spins with assumptions defining a rather large class of models. Recall that the generalized Ising model associated with the Euclidean lattice r/lj theory (i.e., the Ginzburg-Landau model) had only nearest-neighbor couplings. We now complicate this unbounded spin case with the introduction of long-range couplings as we have already done in the §2- valued spin case - and consider once again the wholesale expansion in couplings. Let
HA
=
~ ~
J~ s~s~ m-n m n'
(3 .12.1 )
r;;.r;EA
where Jr;, -n satisfies the decay condition (3 .12.2)
This condition is stronger than condition (3.9.14), let alone condition (3.9.2), but we reserve refinements for the next section. The non-inductive expansion together with the Konigsberg estimation is the most convenient context in which to describe how the extra long-distance decay can be used to deal with the large spin values whose uncoupled probabilities are quenched by a single-spin distribution da(z)
= e-U( z)dz
(3.12.3)
with U(z) a continuous function such that
U(z) 2: f z 2
-
c,
f
> O.
(3.12.4)
The Ginzburg-Landau model meets this single-spin requirement for arbitrary,..,. The transcription of the non-inductive expansion formula (3.9.5) to the present model is obvious. The polymers themselves are exactly the same - subsets of Ax A whose supports are connected by their elements. We have
exp (-fJ
~~
Jr;;_r;sr;;sr;)
n ,mEA
L II
{exp(-fJJr;;_r;sr;;sr;) -I}
rcAxA (r; .r;;)H 00
1
Ln!
n=O
l
L
Z - exp (-h-Z-':; {e} e - {e} e {e}
ze-{3Joz
2
dU(Z))
-00
(3.12.14) '
we obtain the following bound on the activity:
Iz(()1
S cl(l{3card (r card (f({3)-!I(1
II
(NJe)!)1/2
II
IJ;;_;I
eEsupp (
x exp _
eE
L (C({3)h e + ~f({3)-1 h7) .
(3.12.15)
supp (
Before we try to sum over polymers, we must find something in this product bound against which to cancel the factorial growths. Since powers of small numbers cannot accomplish this, the product of varying coupling strengths is the only candidate. Accordingly, one writes
(3.12.16)
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
274
and observes that for aI, . .. , aN > 0, (3 .12.17)
by log concavity. This means
(3.12.18)
and in our context we get
II
(3.12.19)
The point is that (3 .12.2) bounds the sum with a universal constant. Thus
II
IJ-m-n_12/3
(;:; ,;;)E(
x exp _
L
(C(fJ)h 7 + ~f(fJ)-lh7) ,
(3 .12.20)
f Esupp (
and we are ready to sum over polymers. The estimation is now done exactly as in §3.9, because the bound no longer differs in any essential way from that which occurred in the §2- valued spin case. To be sure, a fractional power of each IJ;:;,_;:; 1 has been burned up, and the Konigsberg bridge-island estimation will cut the remaining power by half. This is the reason why the assumption (3.12 .2) is convenient. Once again, there are ways to refine the expansion technique. Although estimation such as (3.12.19) is important to a phase cell cluster expansion, we show in the next section that this consumption of small factors is unnecessary for the class of unbounded scalar spin systems considered here.
References 1. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981) , 327- 360.
2. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer- Verlag, New York, 1987.
3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION
3.13
Extra
1/N!
275
Factors in the Inductive Expansion
We now replace the wholesale expansion of the previous section with the inductive expansion already applied to the §2- valued spin case in §3.1O. Compared with the inductive expansion in that case, the expansion rules have not changed, but the estimation on the polymer activity has changed, since there are factorial growth bounds on single-spin expectations in this case. The point to be made in this section is that the combinatorial properties of the interpolation weights enables us to cancel the factorial growths without using any of the long-distance decay of the interaction! Thus we can replace the long-range requirement (3.13.1) of the less refined expansion and estimation with the weaker requirement (3.13.2) The challenge in this section is to find that alternative cancellation of the factorial growths. The inductive interpolation described in §3.1O generates the desired polymer expansion of a generalized partition function: ~
1
00
Z(h-: fE A) l
= L, n. n=O
n
II Z(Ui)
L
(3.13.3)
(U" ... ,Un)E'Vn i=l
where the polymers are unordered connectivity graphs. The activity is given by
z(U)
II
= eE
Z _ (h- )-1 {f}
(II 11 dti) fQ)(t) ,
X
l
o.c .g. Q) : U"'= U 90(2) = first site in supp U
supp U
exp (_
L
_II
(-(3J-;;,_-;)
J _II
(n,m)EU
his
i
-
(Sn S-;;')
(n,m)E U
(3H~pp U(t)) _ II eE
l E supp U
du(s-;),
(3.13.4)
supp U
where "first site" refers to the fixed linear ordering chosen for A, and H~pp u(t) is the interpolated Hamiltonian for the history of attempted decouplings associated with ~. It is easy to establish by induction that it has the convex form:
H~pp u(t)
L L
I:. partitions supp U
I:. partitions supp U
w~(t)
L
(3.13.5)
HA"
NEI:.
w~(t) = 1,
w~(t) ~
o.
(3.13 .6)
276
CHAPTER 3. EQUILillRlUM STATES OF CLASSICAL CRYSTALS
On the other hand, (3.13.2) implies
L
HI\' -> -J _
(3.13.7)
s:'t
tEl\'
for an arbitrary restriction of the Hamiltonian. Thus
S:.,
(3.13.8)
l
t Esupp
U
and so we have the activity bound: Iz(U)1
:s
II
Z - (h_)-I(3card {l}
1:
l E supp U
II
X _
II
U
l
IJ;,_~I
(~.;')EU
IzINu(7)l7Z+2~JZ2 du(z) ,
(3.13.9)
l E supp U
where we have also used the u.c.g. identity (3.8.22). Nu (f) denotes the number of (~ , iii) E U meeting f. - as in the previous section - and (3.12.4) implies
1:
IzINu(l\h7z+~cz2 du(z)
:s
(1:
:s
(f - 2(3J)-1/4cr~Nu(7) exp(~U'-2~J)-lh7) (Nu( f)!)1/2
z2NU(t)dU(Z)) 1/2
(1: e2h7Z+2~Jz2
dU(Z)) 1/2 (3 .13.10)
for small (3. Combining this with (3.12.14), we reduce (3.13.9) to: Iz(U)1
:s
c IUI(3 card Ur card
II
U
t E supp
x exp
L
(C((3)h t
(Nu (f)!)1 /2
II
IJ;,_~I
U
+ ~(f - 2(3J)-lh7) ,
(3.13.11)
l E supp U
which is certainly more refined than (3.12.15) . Following the general polymer estimation scheme , we seek to reduce the obvious bound (with card U ~ lUI - 1)
L
Iz(U)1
u.c.g . U : ~oE supp U
IUI=M u .c .g . U : ~oE supp
IUI=M
u
(t:;,ni.)e U
3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION
II
x
277
((Nu(i )!)1/2 X((3, h- )),
(3.13.12)
+ ~(f -
(3.13.13)
i
i E supp U
X((3,h)
exp (C((3)h
2(3J)-1 h2 )
,
to a bound of the form (3.9.10). We cannot throwaway the combinatorial advantage of summing over unordered connectivity graphs if we wish to control these factorial growths without burning up fractional powers of the factors IJ~_,:;I. The point is that we were indeed throwing away that advantage in §3.10. To see this, we write the sum in terms of tree graphs on {I, . .. , M} with line segments directed by choosing 1 as the root. First, we have the summation identity
=M u.c.g. U: ':;°E supp U !U!=M
because the summand is symmetric in pairs. Second,
(3.13.14) u.c.g. U: !U!=M ':;0= root of U
en:, r;;,) and the root only orders the individual 1
L
u.c .g. U: !U!=M ':;0= root of U
tree graphs T on {l, oO .,M} rooted at 1
(M -I)!
(3.13.15) Q: {l, oO .,M}-+A injective, Q(1)=-;:-O
where the summand is altered as follows:
II
IJ~_,:;I
II
IJQ(j)-Q( i) I,
(i,j)ET M
X((3, hi)
f----t
iEsupp U
eE
II
f----t
(,:; ,~)EU
II X((3, hQ(i)), i=l
II
i=l
supp U
The (M -I)! in the denominator compensates for the over-counting of subsets of A due to the permutation of {2, . .. , M} corresponding to different injections with the same range. Hence (with Ihil ::; c)
L
Iz(U) 1
u.c.g. U: ':;°E supp U !U!=M
cM (11:- 1 (3)M-1 M (M - I)! x
L
tree graphs T on {l,oO.,M} i=l rooted at 1
II
Q: {l,oO .,M}-+A (i,j)ET Q(l)=':;o
IJQ(j)-Q(i) I,
(3.13.16)
278
CHAPTER 3. EQUILIBRIUM STATES OF CLASSICAL CRYSTALS
where we have also over-estimated by dropping the requirement that Q be injective. For fixed T, the sum over Q is just a multi-index summation, with the iterative application of (3.13.17) IJQW-Q(i) I ~ J
L
Q(;)EA
indicated by the structure of T . At each stage, only an index that occurs in only one of the remaining factors is summed over. We described this in the context of abstract polymer expansions in §3.6. Thus
M
II (d;(T)!)I/2,
x
(3 .13.18)
tree graphs T on {I •. .• M} ;=1 rooted at 1
so we may now apply Cayley's Theorem, which states that the number of tree graphs on {I, ... , M} with specified root and coordination numbers d I , . ·., dM is just M
(M - 2)!/
II (d; -
I)!
;=1
The sum over T is bounded by M
II(d !)-I/2d; ~ 2M/2(M -
(M - 2)!
i
d, •. ..• dM>O
2)!
i=I
L:i di=2M-I
x card
{(d
I , . ..
,dM):
d
I , . . . ,
dM
> 0 and
~ d; = 2M -
I} ,(3.13.19)
but the cardinality of the set is known to be bounded by 4M, so we finally have the desired bound for convergence of the polymer expansion at high temperature (small (J) . The reciprocal factorials were essentially hidden in the summation over unordered connectivity graphs, which, in turn, arose from the u.c.g. identity (3.8.32) . That combinatorial identity exploits the smallness of the interpolation weights to a degree comparable to the estimate (3.11.26) rather than (3.11.23) . One can derive the required polymer estimate by using ordered tree graphs instead of the polymers (u .c.g.) themselves. To this end, we write u. c .g. U: IUI=M ;:;0= root of U
o.c.g. (1; : deg (1;=M 90(2)=;:;°
3.13. EXTRA liN! FACTORS IN THE INDUCTIVE EXPANSION
279
M-I X
II ((d1)'" (j) + 1)!)1/2 ,
(3.13.20)
j=1 where the previously defined coordination numbers are related by (3.13.21) Thus u. c .g .
~UI=M Iz(U)1 ::; (7-1 eM
f3)M-I
Mth-~ree 1) (;g 11 dti) f1)(t)
~o= root of U M-I
M
II ((d1)(j) + 1)!)1/2
X
j=1
II 119, (i)-90(i) I,
o.c.g . 15 :
(3.13 .22)
1)", =1) i=1
90(2)=~0
and we estimate the inner most sum by the iterative applicatio~ of (3.13 .2) dictated by ~e ordered tree graph TI, which specifies which site variable f. j a given site variable
ei is "connected back to." u .c. g .
This reduces the bound to:
~UI=M Iz(U)I::; e
M
I
(r- f3)M-I
Mth-~ree 1) (;fill dti) f1)(t)
~o= root of U
M-I
X
II ((d1)(j) + 1)!)1/2,
(3.13.23)
j=1 and this is the point where (3.11.26) may be directly applied. The more traditional but weaker estimate (3.11.23) would require the use of some of the long-distance decay of the interaction to cancel the factorial growths in the manner described in the previous section. This, in turn, would require the condition (3 .9.2) on the interaction , in spite of our use of inductive interpolation.
References 1. G. Battle, "A New Combinatoric Estimate for Cluster Expansions," Commun. Math. Phys. 94 (1984), 133-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 3. G. Battle and P. Federbush, "A Note on Cluster Expansions, Tree Graph Identities, Extra liN! Factors!!!" Lett. Math. Phys. 8 (1984),55-57. 4. F. Harary and E. Palmer, Graphical Enumeration, Academic Press , New York, 1973.
Chapter 4
A Wavelet Introduction to the Renormalization Group In classical statistical mechanics and quantum field theory, no tool has had greater impact - either analytically or computationally - than the renormalization group. In the analysis of systems with infinitely many degrees of freedom, the iteration of a renormalization group transformation systematically eliminates irrelevant behavior and reduces the set of degrees of freedom. This reduction is most efficient because the idea is to realize the entire set of degrees of freedom as a scale hierarchy, with the functional integration over the smallest-scale variables implementing the transformation at a given stage. In the lattice formalism of the renormalization group, these variables are actually wavelet amplitudes. The natural connection between wavelets and the renormalization group is the central theme of this chapter. In an attempt to derive the non-analytic behavior of thermodynamic functions for scalar spin systems from microscopic interactions, Kadanoff and his co-workers introduced the idea of considering a block of strongly correlated spin variables on the lattice as a single spin variable [K3]. For parameter values close to critical values - i.e., close to singularities in thermodynamic functions - the exponential decay of correlations is very slow, so a block of spins can be treated in this way. The difficulty lies in writing down the effective interactions between blocks and then regarding super-blocks as single spin sites. The iteration of this procedure necessarily weakens correlations, so the block-spin description loses validity for blocks of blocks of .. . blocks of spins. Nevertheless, Kadanoff considered the effective interaction between super-blocks as a function of the effective interaction between super-blocks corresponding to the next level down the hierarchy. With no information about this function, he and his team assumed that it was analytic and derived non-analytic behavior for the thermodynamic functions! Naturally, the result was not quantitative at all, but it inspired the work of Wilson [W16, W17]. From the standpoint of developing Kadanoff's idea rigorously, Wilson recognized that there was no help for it but to successively integrate out fluctuations in successively larger blocks of spins, rescaling the coarser lattice to the unit-scale lattice with 281
282
CHAPTER 4. THE RENORMALIZATION GROUP
each step. The partial functional integration and rescaling for a given partition of the lattice into blocks came to be known as the renormalization group transformation, and the primary aim was simply to calculate Kadanoff's effective action at each stage by this functional integral transform. The iteration was complicated at the very outset, but the success of this approach was unparalleled in the study of critical behavior. In the case of three dimensions, for example, a non-Gaussian state was discovered which is actually fixed with respect to the renormalization group transformation. Moreover, every critical state is driven toward this fixed point by iterations of the transformation. There is a Gaussian fixed point as well, but the flow of the iteration on the manifold of critical states is away from that state. Since the RG (renormalization group) transformation changes the correlation length by a constant factor and a critical state has infinite correlation length, the critical manifold is invariant with respect to the RG iteration. On the other hand, the manifold has unstable directions in the space of possible interactions. Indeed, if a state is not critical, it has finite correlation length and is therefore driven away from the critical manifold by the RG iteration because the transformation scales down the correlation length. This implies the non-Gaussian fixed point is semi-stable, and such a property enables one to calculate critical exponents by analyzing the linearization of the RG flow in a neighborhood of that fixed point. Moreover, this picture of critical behavior explains the important principle of universality i.e., the principle that very large classes of models share the same values of critical exponents. The point is that any scalar spin interaction in a neighborhood of the critical manifold is drawn by the RG flow into a neighborhood of the non-Gaussian fixed point before it is driven away from the manifold. Since the correlations after many iterations are supposed to reflect the long-distance behavior of the original correlations, the fixed point governs that long-distance behavior. Early and very qualitative calculations were done in second-order perturbation theory with reasonable results. In an attempt to analyze the RG iteration beyond perturbation theory, Wilson introduced an approximation in the functional integration that reduced the RG transformation formula to an integral transform in one integration variable [W17] . There is no more control over the error in this approximation than over the error in the crudely modified second-order perturbative calculations, but this simple integral transform promotes RG calculations to the non-perturbative level. Wilson gave a qualitative justification of his reduction by describing the fluctuations at all scales in terms of an orthonormal basis of expansion functions for the Euclidean scalar field. The basis had a multi-scale coherence and its existence was not proven. Wilson simply assumed that the functions were well-localized in phase space [W17], but he anticipated wavelet analysis fifteen years before the subject really began to develop . We shall refer to his integral transform as the Wilson wavelet recursion formula . Shortly after Wilson's derivation, Baker pointed out that this integral transform gives the exact RG transformation for a toy model where the interaction between nearestneighbor blocks is indirectly realized through their interaction with a larger-scale block containing both of them [B4] . This model is a variation on a scheme originally introduced by Dyson [D45-D47] and is known as a hierarchical model. The study of such models proved to be very fruitful [B36- B38, C22, K22-K24] , so naturally the attention of the mathematical physics community was drawn away from the wavelets postulated by Wilson.
The program to obtain rigorous results for realistic models through renormalization group techniques was enthusiastically promoted by Gawedzki and Kupiainen [G17G28]. For example, the approximate RG calculations referred to above predict that in four dimensions the Gaussian fixed point is the only fixed point for scalar selfinteractions and that it is attractive on the critical manifold. Gawedzki and Kupiainen proved that this fixed point is attractive for the iteration of the full RG transformation with no approximations [G26]. An immediate application was to control the infinitevolume limit of the massless @: model, whose long-distance behavior poses problems for any straightforward expansion method. A convergent phase space expansion in the infrared regime was developed by Feldman, Magnen, Seneor, and Rivasseau [F30]. Their ideas were similar to the ideas in the RG analysis of Gawedzki and Kupiainen. While the Ising model was the original focus in the study of critical exponents for scalar spin systems, Kadanoff and Wilson worked with the Ginzburg-Landau model, which can be regarded as a generalization of the Ising model. Let 4 be the free lattice field in d dimensions with mean zero and covariance matrix H;', where Ho is minus the lattice Laplacian, already given in 53.5 for d = 3. For arbitrary d, 2
2d, -1, 0,
2
nl=n, n and n ' nearest neighbors, otherwise. 2
2
(4.1)
Equivalently, 2
(H04, 4) =
(4(;
'1 - 4(n))27
(4.2)
where (., .) denotes the inner product on e2(Zd). Let dPHo denote the Gaussian measure associated with the covariance of this free lattice field - i.e.,
Consider the Euclidean lattice action
The non-Gaussian perturbation corresponding to this quartic interaction is given by the A = Zd limit of
CHAPTER 4. THE RENORMALIZATION GROUP
284
where X A denotes the characteristic function of the finite set A c Z d We discussed this model in $3.5, and the advantage in working with this variation on the Ising model is that the spin configuration 4 can be averaged over blocks of lattice sites with no restrictions on spin values. This block-spin averaging is realized as the KadanoffWilson transformation, given by
where the reason for the averaging weight 2-4-I is to preserve the order of magnitude of the quadratic form (4.2). The renormalization group transformation 'RTassociated with the averaging transformation T acts on probability measures d p on the space of spin configurations 4, and it is given by
for arbitrary random variables X. This innocent-looking definition is computationally convenient in cases where the expectations for dp are explicitly known - if d p is a Gaussian measure, for example. Otherwise, one must deal with what this transformation means in terms of functional integration. The fluctuations to be integrated out must be defined, and the residual functional integral must be expressed in terms of the blockspin variables. This complicated transformation drastically affects the form of dpH. Not only does the interaction I($) lose its locality, but the quadratic form (Ho4,d) loses its nearest-neighbor structure. The iteration of the transformation escalates these complications. Recognizing the essential irrelevance of some of this complexity, Wilson replaced the Euclidean lattice field with its continuum analog, where the short-distance regularization is realized as the standard ultraviolet cutoff. In this setting, the averaging transformation is replaced by a truncation in momentum space. Let be thz gee continuum field in d dimensions with mean zero and covariance kernel -A-'(x, y). Consider the regularization
of that kernel. The action (4.4) is now given by
but the non-Gaussian perturbation of the Gaussian measure dpTegassociated with is given by the 0 = IRd limit of ~1(;,
G) p
=
(z,H)-~ exp (-u2
a,g(32d
;
z,H
=
/
exp ( - u 2
Jn area
(;)'d
; - ~
4
where R is an arbitrary bounded region in Rd. Let dfiregbe the Gaussian measure with mean zero and covariance kernel
The RG transformation is defined in this formalism by
Since -~
4 , =~/ () v~* : ~ ~ ) ~ ,
(4.20)
the free part of the action is already a fixed point - clearly an advantage in calculating the effect of R on a perturbation of this free action. The early calculations done by Wilson were based on this modification of the generalized Ising model [K25, W16], except that the mass was included in the covariance. Nevertheless, this scheme poses a serious technical problem as it stands. Since the ultraviolet cutoffs are sharp, the correlations among fluctuations have slow longdistance falloff. On the other hand, the entire renormalization group program owes its success to the fact that fluctuations are weakly correlated - specifically, with exponential falloff in the correlations. This conflict can be resolved by the following scheme. For r E Z+ define the covariance kernel
CHAPTER 4. THE RENORMALIZATION GROUP
286 Clearly,
~ c(r)(~, y) = (2~)d Jd P
eip,(;-y)
~
1 12
Cp I~ + 1) ,
(4.22)
which is a Pauli-Villars regularization [G28, P12] of the covariance kernel (4.23) For each r E Z+, the covariance kernel c(r)(~, y) has exponential decay with correlation length 2- r + 1 , while on the other hand, the decomposition (4 .22) can be realized by a direct sum 00
cp(~)
= EB cp(r) (~)
(4.24)
r=l
of Gaussian random fields . Let covariance c(r)(~, y):
djl(r)
be the Gaussian measure with mean zero and (4.25)
If dfJP) is the Gaussian measure with mean zero and covariance
( ~) (~))
(cp x cP Y
_ dl'(l ) -
1 (27l')d
Jd~
ip.(;_y)_l_ (
pel
P12
1)
1P 12 + 1 '
(426) .
then 00
dJ.L(l)
= EB djl(r) ,
(4.27)
r=l
(4.28) where
cp(r)
(~) is not to be confused with
This rigorous improvement over the sharp ultraviolet cutoff is used in much of the work of Feldman, Magnen, Seneor, and Rivasseau as well as in some of the work of Gawedzki and Kupiainen. We do not use this "momentum-slicing" formalism, but return to the block-spin transformation of lattice variables. Gawedzki and Kupiainen [G28] introduced a formalism based on the lattice Gaussian fixed point. Perturbing the lattice field action associated with that Gaussian, they carried out an RG analysis of interacting lattice field theories based on the Kadanoff- Wilson transformation which also had the advantage offered by the standard ultraviolet cutoff on the continuum field . We shall display
287 the wavelets that are hidden in this lattice formalism. Indeed, there is a continuum picture canonically associated with the lattice picture. If Hoo denotes the matrix for the free lattice field action one has in this formalism, then the covariance matrix is given by
= _1_ ( H-1)~~ n n' (211")d 00
(rr 1" d
d )
,=1 -"
p,
e
ip.(-,:;--,:;')
~ I£(p +211" ~
~e EZ d 1
l:')12
P+211" l:' 12
(4.29)
where X denotes the characteristic function of the standard cube [o,l]d. On the other hand, (4.30)
where X-,:; (;I;) X( x n) and 00 denotes the expectation functional of the free (massless) continuum field oo
(4.38)
We shall introduce the zero-fluctuation transformation R'f', which throws away the fluctuations instead of integrating them out. If we define the interaction
f J~l Uv
(;;)" d
;;,
(4.39)
v=l 00
LQ~)~W~~)(;; -2~), r=1
--+ iIt(( by (4.1.8) The averaging transformation of ¢>( reduces to
T~-s
L c~¢>d2 ~ + ni) ~
.I ~ 2-~-s .I G -; -~) 2~-s J ~)((2 T
~-s 2; c~
'f/(-; -2
- ni)((-;)d -;
m
((-;)d-;
'f/
'f/(-; -
¢>u,d~),
-;)d-;
Us = Us (1n2),
(4.1.9)
so if T denotes the mapping ( t-+ ¢>( from the Sobolev space to the space of lattice configurations, we have the answer to our question:
ToT=ToUs.
(4.1.10)
How does the averaging procedure extend to vector-valued configurations in a geometrically natural way? First of all, a vector-valued function on the lattice may be regarded as a scalar-valued function on the set of neare~-~igh~or bonds. ~n oriented nearest-neighbor bond is any ordered pair of the form (n, n ± e It), ~here e It denotes the j.tth-coordinate unit vector. An arbitrary vector-valued function A on Zd assigns a number A(b) to each oriented bond b such that (4.1.11.+)
292
CHAPTER 4. THE RENORMALIZATION GROUP
(4.1.11.-) Secondly, one would like to define the vector averaging such that it is induced by the scalar averaging in the case where the vector configuration is given by the lattice gradient of a scalar configuration. If A(b) is given by
(\71at--+ A -;; such that if v = \7 0
1
p
= c # O.
(4.2.28)
2s
1
In the case where H o(' minimizes (Ho¢>, ¢» with respect to the constraint T r ¢> = ¢>' . With Mr also a right inverse of Tr, we can infer that MrTr is an (Ho¢>, ¢»orthogonal projection. Moreover, for r ~ q, MrM(r+l) . .. M(q)Tq
(MqTq) (Mrrr)
=
Mq(TqMr)rr Mq(Tq-r)T r = MqTq,
= MqTq,
(4.3.29) (4.3.30)
4.3. MINIMIZERS AND ORTHOGONALITY
307
so the sequence of projections MrTr is decreasing in the operator-theoretic sense. Now for an arbitrary configuration ¢', we can easily derive the formula for Mr¢' in momentum space:
p).
(4 .3.31)
where hr is defined by (4.2 .27) . This follows from a lattice version of the derivation given for the variational problem in §2.1D. This formula will be useful, but the decreasing sequence of projections enables us to introduce a scale hierarchy of fluctuation configurations for an arbitrary configuration ¢ - namely, (4.3.32) Clearly, N
1. However, M r _ I (1/J(Kl(. - 2 n:i,)) = (Mr _ I 1/J(K))(. - 2r n:i,), (4.3.44) and M r _ I 1/J(K) is given in momentum space by (4.3.31). In this case we have _ ~ M r _ I 1/J(K)(p)
Ihr _ I (2 r -
=
2(3~+s)(r-l)
~ Ho(p)
(
l
1~p +21T ~£)1 2
L E{O,I, ... ,2
~
1
r -
1
-I}d
~
)-1
Ho(p +2- r + 2 1T e) xh r _ I (2 r - 1
p) L1/J(I 1,
(4.4.15)
and this conclusion suffices. Thus the basis generated by this scaling is a wavelet basis with respect to a Sobolev norm. We may also write (4.4.11) as ((M oo )-r'+l1/J(K'»)(. - 2r '
n; ') .l(H~ with covariance 1V'1- 25, we recall from the previous section the alternative formulas
(MT<J»(-;)
(4.5.27) r; 00
(MT<J»(-;)
2: 2: Q~k 1lt~1'),
(4.5.29)
4.5. CANONICAL WAVELET MANIFOLD AND ZF TRANSFORMATION
323
which implies (4.5.30) Hence K('\; q>')
= N-+oo lim n(O) (H(uO,N))(q/) r,N
(4.5.31)
is the effective action on the unit-scale lattice for the formal continuum limit with a fixed wavelet cutoff at the unit scale. Alternate expressions are:
K('\; T4!)
~J(IVISMT4!)(-;)2d-; + 'f'\v J(MT4!)(-;td-;, (4.5.32) v=l
K(,\;T4!)
(4.5.33) Note that the limiting behavior of UZ,N is given by (4.5.24) . We have lim uO,N = 0
N-too
lim ue,N
N -+00
v
2d v
,
< d _ 28' 2d
= oo(sgn ,\),
v> d _ 28'
(4.5.34-) (4.5.34+ )
and the latter case would be of serious concern if a genuine continuum limit of the lattice N measure dJ-Lr (¢2-N) were involved. To control the effective action as N -+ 00, one has to reckon with the effect of the smaller-scale fluctuations that have been integrated out, and the infinite uZ,N -limits pose a problem for the 2- N -dependence of the '\V needed for this purpose. This issue does not come up in the sequence of ZF transformations, which throwaway those fluctuations. The manifold of these lattice actions K (¢) parametrized by s, and '\V with v = 1,2,3, ... is the canonical wavelet manifold, and it resembles the ultraviolet cutoff on a Euclidean field theory that is yet to be renormalized. The property shared by the wavelet cutoff and the ultraviolet cutoff is that the free part of the action is a fixed point with respect to integrating out the fluctuations and rescaling. However, in the case of the wavelet cutoff, this functional integration is implemented by the averaging transformation, while in the case of the standard ultraviolet cutoff, the connection is only qualitative. If we now apply the iteration of the exact renormalization group transformation Rr to K(u) = K(u; ¢), we have e-'R.¥(K(u))(')
= e-~(H=',')
lim [(Z(N))-l A-+Zd
A
CHAPTER 4. THE RENORMALIZATION GROUP
324
x
(OXp ( - ~ u.
+
~
ZY')
f
tE ·;k \
d? {
"f"(? -2';;;)
;;)((Moo)N ¢')(;;)
ntJ
(4.5.35)
~uv Jdx { ~~2r~A a~k
exp ( -
XW~K)(X _2r ni) where the partial expectation
~ g(? -
Or~N
f) )r~N'
(4.5.36)
is defined by (4.5.37)
with the relation (4.5.38) We have used
L9(X -
~)'l/!::"(K)(~)
= W~")(x),
(4.5.39)
n and the leading exponential is unaffected because (4.5.40) Notice that the integration variables are no longer the a-variables that appeared in (4.3.52), but the a-variables introduced in (4.4.33) - i.e., the hierarchy of fluctuation amplitudes on the lattice have been replaced by wavelet amplitudes in the continuum. This formula is an immediate consequence of the connection between Hoc> and the continuum wavelet cutoff discussed in §4.4. It is natural to base the zero-fluctuation approximation on the wavelet amplitudes in this context, because the form of the interaction does not change under the transformation, and the sequence is now an iteration of (4.5.41) Thus
r
n~oo' (K(u))(¢') = ~(Hoo¢" ¢') + ~ u. f d? ( ~ g(? - ;;)(MOO¢')(;;)
(4.5.42)
4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS
325
Recall that (4.5.43)
-n It follows from (4.4.8) that
(4.5.44)
(U;l M1/)(:;)
2'-~(M1/) (~ :;) ~ ~) ¢'(n). ~ 2-'L: g (1"2x-n •
d '"
(4.5.45)
n
Thus
~)(K(u))(¢')
or more concisely, the iteration has the effect n~OO)N (K(u))
= K(>..O,N (u)).
(4.5.47)
with >.~,N given by (4.5.13).
References 1. G. Battle, "Wavelets: A Renormalization Group Point of View," in Wavelets
and Their Applications, B. Ruskai et al., eds., Bartlett and Jones, Boston, 1992, pp. 323-349.
2. G. Battle, "Wavelet Refinement of the Wilson Recursion Formula," in Recent Advances in Wavelet Analysis, L. Schumaker and G. Webb, eds., Academic Press, Orlando, 1994, pp. 87-118. 3. K. Gawedzki and A. Kupiainen, "Rigorous Renormalization Group and Asymptotic Freedom," in Scaling and Self-Similarity in Physics, J. Frohlich, ed., Birkhauser, Boston, 1983, pp. 227-261.
4.6
Relevant, Marginal, and Irrelevant Parameters
As non-interactive as the zero-fluctuation transformation is - at best, it is a zeroth-order approximation of the renormalization group transformation - it maps the canonical
CHAPTER 4. THE RENORMALIZATION GROUP
326
wavelet manifold into itself. Its only effect on any action belonging to this manifold is a scaling of each coefficient of V(z), so it is actually a linear transformation on this linear manifold of the V(z) with the functions z" as eigenvectors. This is the transformation used to describe the notions of relevant parameter, marginal parameter, and irrelevant parameter. Henceforth, we set s = 1, as this is the Sobolev index of physical interest. Thus
(4.6.1) with . The associated wavelets are well-localized in this case. They have exponential falloff because their momentum expressions are real-analytic in the case s = 1 as we have already shown in §2.1O. This exponential localization improves with the localization of the averaging transformation upon which one chooses to base the wavelet construction. After N iterations of the ZF transformation R'f', the output parameter corresponding to the input parameter u" is given by
(4.6.2) as 2v tion
vd
/2+d
is the eigenvalue for the eigenvector z". For those v satisfying the condiv - vd/2 + d
< 0,
(4.6.3)
the output parameter is driven to zero - i.e.,
(4.6.4) because the corresponding eigenvalue is less than unity. In this case, the parameter is said to be irrelevant. If there is an integer solution Vd of the equation v - vd/2 + d
= 0,
(4.6.5)
then the corresponding output parameter remains fixed:
(4.6.6) because the eigenvalue is unity. There is no V2, but V3 = 6 and V4 = 4. This parameter is dimensionless from the standpoint of dimensional analysis, and it is said to be a marginal parameter. Corrections to the ZF transformation would introduce logarithmic behavior of the iteration with respect to length scale for such a parameter (Le., algebraic behavior in N). Finally, for those 1/ such that 1/ -
vd/2
+ d > 0,
(4.6.7)
= (sgn uv)oo
(4.6.8)
the output parameter is driven to infinity: lim >.~,N (u)
N-t oo
4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS
327
because the eigenvalue is greater than unity. In this case, the parameter is said to be relevant. Note that the v = 2 parameter is always relevant and that its dimension is the square of a mass. Corrections to the ZF transformation may drive relevant parameters to finite limiting values. Now if all of the parameters of an initial interaction are irrelevant, then iteration of the ZF transformation damps it out: lim R~OO)N (K(u)) = Hoo .
(4.6 .9)
N-+oo
Indeed, this property extends - for such an interaction - to iterations of the full RG transformation: lim R!j. (K(u)) = Hoo. (4.6.10) N-+oo
Moreover, one can prove existence and uniqueness of the infinite-volume limit of the expectations arising from the initial action K - namely, existence and uniqueness of
}~~}Zr)-l
zr
JX(.bare,N ; I/>':i!-N) =
f f f
2J1Nd/2-vN-Nd>.~are,N j
d
Y (MTU1 N ~)(y)"
1'=1
>.~are,N j d Y [2N ~ -N (MTu1- N~) (2N yW
1'=1
>.~are,N
1'=1
I
.
d Y (U(l MTU1 N~ )(y)"
(4.6.33)
The composition MT is actually the (-~ . " )-orthogonal projection onto the continuum configuration subspace spanned by the wavelets with length-scale 2: 1. By the nature of a wavelet basis, this means the scaling Ui'MTU1 N is the (-~ " ·)-orthogonal projection onto the subspace spanned by the wavelets with length-scale 2: 2- N . This is the type of short-distance cutoff that this fine lattice field really imposes on the bare interaction in the continuum. In this wavelet context, the short-distance problem is to find an N -dependence of the bare parameters that controls a non-trivial N = 00 limit of the expectations M
=
jrrl/>':i!-N(-;j)dP,hN\I/>':i!-N),
(4 .6.34)
j=l
(Zt))-l exp( -I~N (>.bare,N j I/>':i!-N Xn(N»))
dp,!/-N (I/>':i!-N),
(4.6.35)
4.6. RELEVANT, MARGINAL, AND IRRELEVANT PARAMETERS Z(N)
n
331
J
exp( -I~N (Abare,N j ¢>~NXrHN»))
dp,~~ (¢>~N),
(4.6.36)
!1 n 2- N Z d ,
(4.6.37)
where the region !1 C IRd is fixed and dp,r~ denotes the Gaussian measure on .))) = K(>.) ue,N (>.) = 2(v~-sv-d)N >'v.
(4.7.10) (4.7.11)
This transformation property of K extends to iterations:
RTN (K(uO,N (>.))) = K(>.).
(4.7.12)
Once we have defined the lattice field 8"" we shall see that H(>.; rjJ) also has this transformation property. Moreover, there is a sequence {h(>.;rjJ): -00 < k < oo} of interactions such that
RT(Hoo + h(UO,l(>.))) lim I k (>.) = 100 (>') , k ..... oo lim h (>.) = Loo(>') · k ..... -oo
= Hoo + h+l(>'),
(4.7.13) (4.7.14) (4.7.15)
This is a sequence of canonical manifolds driven toward the canonical wavelet manifold and away from the new canonical manifold by iterations of the transformation
Hoo
+ ~ >'vJv t - t RT (Hoo + ~ Ue,l(>')Jv)
CHAPTER 4. THE RENORMALIZATION CROUP
334
The latter manifold is the limit of extrapolating back, so we call it the canonical inverse-limit manifold. Let X denote the characteristic function of the standard unit cube. The idea of defining the lattice field s¢ is to write 00
Loo(u;4»
= LUv Ls¢(it v=1
(4.7.16)
and realize
l: xC:Z;- - -; )s¢ (-;)
as a step-function approximation of l: g(x - n)4>(n).
~
~
n
Now (4.7.17) since 9 is determined by our choice of averaging transformation - namely the block-spin transformation (4.7.6). It may seem obvious to set stf> (n) = 4>(n) and take X as the stepfunction approximation of g, but then Loo would be precisely J, and jj(u; 4» would not have the transformation property (4.7.11) in that case. The piecewise constant approximation of 9 must have a multi-scale structure. We introduce the step-function replacement
~(It)(x) -+ Lw(It)(f)x(x - e),
(4.7.18)
l
w(I g,
(4 .7.36)
from which it follows that the r > g contribution collapses to
00
L 2(1-~)(T-q-1)LQ~kLw(It)(C) T=q+1 It,m d
X
II b[2-
r
= s~q)(h.
+ O+ 1 j,J,e,+2m,
,=1
What about the r
= g contribution?
T~-1 L Q~~
(4.7.37)
We have
L
w(lt) (2 -; -2
r;;, + 1) =
0,
(4.7.38)
;;" E{O,1}d
It,m
and so we have verified the desired compatibility:
T s¢(q-1) -- s¢(q)
(4.7.39)
s~q) = Tqs¢ .
(4.7.40)
This implies It is important to notice that in (4.7.38) we have used the wavelet property
L
r;; + 1) = 0,
w(lt) (2
(4.7.41)
;;" E{O,1}d
which we shall verify later.
This is not to be confused with the vanishing moment
property
L
e
w(Itl(C) =
J
w(lt) (X")d
X" = 0
(4.7.42)
CHAPTER 4. THE RENORMALIZATION GROUP
338
that all wavelets have. (4.7.41) is a stronger property that reflects the block-spin construction of our wavelets, which are certainly not supported on cubes. Given that H(u;') = e-!(H~4>',4>') }~~d [(.ZlN))-l ( exp ( +
- 100 (u; (Moo)N 1//
L t L a~k1/J;:"(Itl(. - 2 ~)))) J, r
It
(4.7.43)
r5,N
r=12rniEA
where we have inserted (4.5.39) in the fluctuations. Obviously, the effect on obtained by replacing 100 with Loo :
e-R~ (H(u))(4>') = e-!(H~4>' ,4>') A~~d [(Zt'))-l ( exp (
+Lt L a~k1/J;:"(It)(--2r~)))) It
Zt')
r=12rniEA
= ~XpCLoo(Lt \
\
It
H is
- Loo (u; (Moo)N 1//
],
(4.7.44)
r5,N
~ a~k1/J;:"(Itl(.-2r~0\) J) r5,N,
(4.7.45)
r- 1 2 r mEA
with the partial expectation Or5,N defined by (4.5.37). On the other hand, (4.7.46)
so if we iterate (4.7.30), we also have 1_ 00 (u; (Moo)N '
+L It
t
L a~k 1/J;:"(Itl(. - 2r ~))
r=12 r niEA
~ ,i"-"'i'H)N"" Jd; { ~ b(;, n)ql' (n) + 2(~-1)N L It
f
L a~k L b(2N ~, ~)1/J;:"(It)(~ _2r ~)r, (4.7.47)
r=l 2rniEA
-;;
J
where we have made the change of variable ~ I-t 2N ~ . Hence
e-R~(H(u))(4>') = e-!(H~4>',4>') Al~~d [(ZlN))-l( exp ( _ ~2(V-Vd/2+d)Nuv
4.7. CANONICAL INVERSE-LIMIT MANIFOLD
XL
L
2- Nd
-; 7 E{O,1, ... ,2 N
339
{sq,.c'}) + 2C~-1)N L f)(1-~)(r-1)
_l}d
K
r=l
~ Q~kwCK)(2N-r+l -; + €,Cr) -2 n;)}"))
J,
(4.7.48)
r5,N
2"mEA
(4.7.49) where we have used the identity (4.7.50) This completes our description of the action Loo (u; ')(n) 2('-''')"u JdO:. (p G 0:, n) .'(0) oo
u
2" -"'f a,u 2v -
vd
JdO:'. (~b(O:, n) v', and then introduce the shift i/ = v' + ij. Thus
X
tq!(VI) (VI +ij)3S~~v'+'v
J ~,
~, d x 0
C¥l,C¥2'
JII w, J 4
W1,W2,W3 W4,
,=1
CHAPTER 4. THE RENORMALIZATION GROUP
360 4
L jII'li, ,=1 3,4
2',3' ,4'
T31r4~O
T2"T3"T 4 , ~O
respectively. From the standpoint of analyzing ultraviolet divergences, wavelet diagrams create a mind-set that is somewhat different from the more familiar Feynman diagram point of view. As we have seen above in the example given by Fig. 4.10.3, the analysis reduces to counting sub-cubes in a given cube!
References 1. G. Battle, "Ondelettes: The QED3 Connection," Ann. Phys. 201, No.1 (1990), 117- 151. 2. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360. 3. J. Kogut and K. Wilson, "The Renormalization Group and the c-Expansion," Phys. Rep. 12 (1974), 75-200.
4.11
The One-Loop-per-Scale Transformation
Having extensively discussed the canonical structures associated with the zero-fluctuation transformation and the linearized RG transformation, we now examine a more interesting modification of the RG transformation that is quadratic in a certain sense, but quite different from the second-order approximation. There can be problems with the stability of this approximation, but it is actually a scale-by-scale version of what is known in quantum field theory as the one-loop approximation. The traditional version is defined at the perturbative level for all scales, with all Feynman diagrams discarded except the one-loop diagrams. What is introduced here will not be that simple, but each application of the transformation is just Gaussian integration with the (highly nonlinear) contribution of the background field changing with each step. On the other hand, it is widely believed that the only fixed point for this modified transformation is the trivial (Gaussian) fixed point. We shall discuss this issue in a later section. First consider an action K(u) from the canonical wavelet manifold together with its RG transformation R(K(u)) given by e-'R(K(u »)(¢')
e-~(H~¢' ,¢') AEZ. lim [z-11 ex p { A \
-JdX
-
V
(~g(; ~)(Moo.,)(;;)
+
L ~ a,,~'li(")(~ -2~))}) "
2mEA
(Q.~)
],
(4.11.1)
4.11 . THE ONE-LOOP-PER-SCALE TRANSFORMATION
361
where we suppress the T-dependence because the choice ofT has been fixed by (4.1.3). The modification discards all terms in the exponent that are of order three or higher in the fluctuation amplitudes O:"m' If we denote this transformation by R 2 , we have e-R.2(K(u))(¢')
L ~ "
= e-R.~(K(u))(¢')
lim A ..... Zd
O:"m'l1(")(--; -2 n;)V
[Z-1 / exp { - Jd--; 2,A \
(~g(--; - r;)(Moo¢,)(~))
2mEA
n
(4.11.2)
Z2,A
\ exp { -
~V(O) Jd--;
L L
O:"m'l1(")(--; -2 n;)O:"'m' i[J("')(--; -2 n;
",,,, 2m ,2m'EA
,)l) f
.(4.11.3)
(ar.;;::)
The first-order term in the expression for Z2 ,A is zero because the wavelets have vanishing moments, while the zeroth-order term has been absorbed into the zero-fluctuation transformation. The calculation of R2(K(u)) is just an application of the Gaussian formula
(II 1 dti 00
i
)
e
-! Et~-! L;A.;t.t;+L:.B. t •
-.J
-
-
-00
det(l
+ A)-1/2 exp
(-~ L:[(1 + A)-lli /J;/3 j
j )
(4.11.4)
',J Thus Roo(K(u))
+ ~ ~ ~ [(1 + A)-lt""mm,/J"m/J"'m' K.,K
m,mi
+ ~ In det(l + A) - ~ In det(l + A(¢' = 0)), 2
A KK' ,mm' --
2
J
(4.11.5)
d --; 'l1(")(--; -2 n;)'l1(1' ,¢>') + ~ R(V(u))(s¢, (j )), j
(4.12.7)
4.12. THE WILSON WAVELET RECURSION FORMULA
371
e-R(V)(z)
(4.12.9)
This is one version of the celebrated Wilson recursion formula. Note that the application of this transformation still requires an integration in 2d-l variables, e.g., 15 variables in four dimensions. There is a way to reduce the formula further with the same kind of brutal approximation, however. Recall that our wavelets were never completely specified; the mother wavelets were derived from an arbitrary basis of a (2d - I)-dimensional space. By (4.4.2), (4.4.24), and (4.4.62), we have
(4.12.10)
where the set {woo(l 4, the continuum limit of the hierarchical ¢~ model is nonrenormalizable because the parameter U4 is now irrelevant. The quantitative difference is that U4 is now driven to zero exponentially fast by the flow of the RG iteration. In contrast to the d = 4 case, there is a non-trivial fixed point with U4 < 0, but it is repulsive. Long-distance behavior is governed by the trivial fixed point. Perhaps the most remarkable achievement in the study of the Baker-Dyson- Wilson models has been the proof of existence and analysis of the non-trivial fixed point in the case d = 3. The single-spin distribution of this fixed-point hierarchical model analytically continues to an entire function! It is also bounded by exp( -cz 6 ) on the
380
CHAPTER 4. THE RENORMALIZATION GROUP
U
2
~! /1\ u4
trivial ,,~
fixed point
critical manifold
\
non-trivial fixed point
Figure 4.13 .1:
U2
trivial fixed point critical manifold Figure 4.13.2:
4.13. THE BAKER- DYSON-WILSON HIERARCHICAL MODEL
381
real axis - i.e., the potential V(z) has z6-positivity. This is consistent with the fact that the parameter U6 is no longer irrelevant, as it becomes marginal in the d = 3 case. In order to replace the Baker-Dyson-Wilson model with a hierarchical model that reflects the wavelet refinement, we need only make the replacement ~I< f-t 2(~-~)«0 v+2ij>0 V+II>O
q,ij=O
2~-~UII+2q
x
U V+ 2ijZ II+V .
Thus 00
RII(V)(Z)
=
L~£(u)z£,
(4.14.9)
£=1
00
L
- 2£-~+1
{(T
2q 2ij + )(T) -
(T
2q
)(T) (T
2ij
)(T»)
q,ij=O l
X
L
V (
+v 2 q )
(f -
v
f _
+V 2ij) Ul - II +2ij U II +2q .
(4.14.10)
11=0
1I+2q>0 II --+ 2d4-(,·-l)/d+,./28/t/t"
4. Now to verify (4.16.25)
p'(v) = vf'(v) ; f(v) ::; 0 v or rather
v!'(v) ::; f(v)
(4.16.26)
= lov !,(u)du
(4.16.27)
because f(O) = O. Clearly, this inequality holds if f' is a decreasing function as well, so the problem now is to verify that
!" ::; O.
(4.16.28)
On the other hand, the chain rule implies (4.16.29) If we differentiate again, the chain rule and product rule yield
n
!" = (f~
0 fd-l 0 •. . 0 h 0 fr)(f~-l 0 . . . 0 h 0 Jr)2 ... (f~ 0 fr)2 + (f~ u fd-l 0 ..• 0 h 0 fr)(f~-l 0 •.• 0 h 0 fr) ... (f~ 0 fr)2!'i + ... + (f~ 0 fd-l 0 . . . 0 h 0 fr)(f~-l 0 ••. 0 h 0 fr) . . . (f~' 0 fr)!'i + (f~ u fd-l 0 ' " 0 h 0 fr)(f~-l 0 '" 0 h 0 Jr) ... (f~ 0 fr)f{'. (4.16.30)
Since
1'/ == -2~72w1,
(4.16.31)
4.16. WAVELET CORRECTION OF THE HIERARCHICAL MODEL
399
we need only show that those first-derivative factors
fj 0 Ii-I 0 ... 0 h not appearing as squares are manifestly positive - although
fj(v)
= 2~-1
- 2~
X
72WjV4
(4.16.32)
certainly is not. The key observation here is that j > 1 for this kind of first-derivative factor and that
fj
0
fj-I ~
°
(4.16.33)
is the final reduction of the claim. On the other hand,
fj(v) ~ 0,
1 - 144wJ'
vj model. Actually, it was natural to expand cf> in the Bessel potential (-~ + m5)-! of these basis functions, so there was enough regularity to carry out the perturbative analysis in terms of these phase cells. The point is that this formalism created a technical problem for the non-perturbative analysis of the cluster expansion developed in [B2l, B22]. Nevertheless, important combinatorial ideas were introduced in the wavelet (phase cell) formalism [Bll, B2l-B23] which made it clear that phase cell cluster expansions should apply to a broad range of Euclidean field theories. The authors simply had the wrong kind of wavelet at the time, and even the resulting technical problem could be avoided by complicating the new expansion in substantial but familiar ways. In the meantime, Brydges, Frohlich, and Sokal found an entirely different and elegant approach to the construction of the cf>j model. Instead of using expansion methods, they controlled the continuum limit with a class of correlation inequalities based on a random walk representation [B72, B73]. Not long afterward, a phase space analysis based on a multiscale momentum-slicing of the free covariance ((;;)(y))o was developed by Feldman, Magnen, Seneor, and Rivasseau [F27- F30]. It was shown to be applicable to asymptotically free models that are strictly renormalizable - not just j, which is super-renormalizable. Even in the case of the cf>! model (which is not asymptotically free) these authors controlled the long-distance behavior for a fixed ultraviolet cutoff and zero mass. They proved existence of the infinite-volume limit for weak coupling - a difficult infrared problem. Gawedzki and Kupiainen had proven the same result by showing the Gaussian fixed point to be attractive on the critical manifold of models with respect to the flow of the renormalization group [G26]. A couple of years later, the subject of wavelets began to develop rapidly as a result of widespread interest in signal analysis. The type of wavelet needed for short-distance analysis in a phase cell cluster expansion was found at this time. The Lemarie class of wavelets has already been discussed in Chap. 2. It has an integer-valued constructibnparameter that determines both the class of smoothness and the number of vanishing moments. A Lemarie basis of wavelets is ideal for a phase cell expansion of the ~ model, or any other scalar model that is asymptotically free . Such expansion functions are not sharply localized like the piecewise-polynomial expansion functions originally tried, but they have exponential decay, which is the long-distance decay that is actually required of a function associated with a cube of a given scale [B12 , B13]. The wavelet cluster expansion developed in [B2l, B22] was finally vindicated [B24]. In this chapter we describe and control the wavelet cluster expansion of the cf>~ Euclidean field theory with the advantage of hindsight . The starting point is the decomposition cf>{-;;) = L:>:tkUk{-;;) (5.l) k
of the free Euclidean field with mass mo, where
Uk = {-~+m6)-h}lk
(5.2)
403 and {\II kl is an orthonormal basis for L2 (I~.3). This basis is actually a modification of a Lemarie wavelet basis inspired by the presence of the mass rna in the theory. There is no infrared problem for weak coupling because the correlations of the free Euclidean field have exponential decay. With no multi-scale decomposition required for the longdistance analysis, the idea is to include all elements of the Lemarie basis with length scales :::; 1 and then complete the orthonormal set with special unit-scale functions . The most natural set of such auxiliary functions is the set of all unit-scale translates of a function which is not quite a wavelet but plays a role in the construction of the wavelet basis. In the end, one must obtain an interacting continuum field theory satisfying the Euclidean Axioms . As we discussed in Chap. 1, the most important properties to be verified are Euclidean invariance and reflection positivity - both of which are destroyed by the wavelet cutoff. However , once the wavelet cutoff is removed by a phase cell cluster expansion, the proof that the limiting correlations have these properties is not as difficult as one might believe at first glance. To obtain rotational invariance, for example, one superimposes a spherically symmetric ultraviolet cutoff on the wavelet cutoff and applies a double-limit argument, although one has to be careful with the type of ultraviolet cutoff. While the spherical symmetry insures rotational invariance of the easily-controlled infinite-volume limit of a fixed ultraviolet cutoff with the wavelet cutoff already removed, the double-limit argument cannot work unless the convergence of the phase cell cluster expansion is uniform with respect to the removal of the ultraviolet cutoff. This can be guaranteed by uniformity of the input estimates , so the point is that the dependence of these bounds on length scales must be uniform in the ultraviolet cutoff. Such a condition cannot hold for the standard (sharp) ultraviolet cutoff, which we used in Chap. 1. However, the Pauli-Villars regularization [P12] adapted to Euclidean fields works very well . If we apply the cutoff
with M = 00 as the ultraviolet limit, it is easy to show that the essential estimates are uniform in M ~ 1 for wavelets regularized in this way, provided that arbitrarily large length scales are ruled out . On the other hand, our choice of orthonormal basis has done just that. Another advantage offered by the choice of expansion functions is that Z3-translational invariance of the correlations is guaranteed by the Z 3-translational invariance of the basis, provided the infinite-volume limit is unique. On the other hand, it is a nice grouptheoretic exercise that rotational invariance together with Z3-translational invariance imply full Euclidean invariance, as continuous translations are easily generated by the combination. Symmetry arguments of the type we have given above are the reason why the expert is so often satisfied with the convergence of a phase cell cluster expansion, provided that the superposition of ultraviolet cutoff and phase cell cutoff poses no problem in itself. As long as the interaction Lagrangian is formally invariant, there is no cause for concern beyond model-specific details of a routine argument. Nor does the verification of Osterwalder-Schrader positivity pose any problem again, provided that the Euclidean probability measure is formally OS-positive. Sup-
CHAPTER 5. WAVELET ANALYSIS OF §
404
pose one wishes to verify this property in the x3-coordinate direction. If the interaction Lagrangian is formally local, then the ultraviolet cutoff , ~
(p)o--+
2
M2 2
, ~
M2(P)
PI +P2 + preserves locality in the x3-coordinate and that locality property can be rigorously exploited to establish the refiection positivity because - although this regularization is not smooth for sample fields - it is still continuous. The double limit proof of the x3-refiection positivity in the absence of both this cutoff and the wavelet cutoff applies here because the wavelet estimates are uniform in M ~ 1 for this ultraviolet cutoff as well. The wavelet cluster expansion we develop is based on the polymer formalism described in Chap. 3, so the expansion rules themselves determine the definition of an arbitrary polymer, which we shall refer to as a phase cell polymer. In [B21, B22], where the polymer formalism was not fully exploited, the same type of polymer was developed, but it was called a Representation 2 graph . Naturally, such polymers have a complicated structure that contain renormalization cancellations, but they also involve sums over "histories" of the type explained in Chap. 3. The expansion will be inductively defined, but the algorithm is nontrivial because the inductive interpolation with which we are already familiar must be interrupted by integration by parts whenever a short-distance divergence begins to develop. One great advantage of a phase cell cluster expansion is that no energy counter-terms are really needed for the renormalization because infinite-energy diagrams never develop if our inductive expansion is clever enough. Now in §1.13 we saw that the counter-terms required in formal perturbation theory for the § model are a second-order mass term, a second-order energy term (which compensates for the Wick-ordering of the mass counter-term), and a third-order energy term. This means that all we need for our expansion is the mass counter-term, and we do not even Wick-order it. It is important to remember that there is no heavy-duty functional integration involved in this expansion. At any given time, only an elementary integral for a finite number of the variables Ctk is ever considered. They are accompanied by products of numerical factors given by f d ?-integrals of products of the expansion functions Uk(?) . We have seen in Chap. 4 how such quantities are associated with wavelet diagrams, which we shall use here as well. In this chapter, however, we are not using the renormalization group formalism at all, but rather a grand cluster expansion that encompasses arbitrarily small length scales. The identification lines in our wavelet diagrams here are generated by integration by parts, which are expansion steps developing the diagrams. It is also important to remember that the free part of the Euclidean action is diagonal in these phase cell variables, and so the expansion does not decouple covariances. The convergence proof consists of two intricately interwoven but very different parts - the perturbative part and the large-amplitude part. The former contains the cancellation of short-distance divergences and is the part of the phase space problem that guides the development of the expansion. Experts have a more routine attitude toward this problem than toward the large-amplitude problem. The latter is very critical because it can only be solved by both stability and phase cell positivity of the interaction.
405
To be sure, the Gaussian (free) part of the measure quenches the probability of large amplitudes too, but such estimation only yields the bound (5 .3)
for high powers of the amplitude Ok of each mode k . This is too much factorial growth to admit convergence of the expansion, and we shall refer to this phase-celllocalized growth as the number divergence . A major task is to show that each factorial growth is really much weaker and then cancel that weaker growth with a product of numerical factors that decrease in size at a certain geometric rate. The extraction of these factors from the small factors differentiated down by the expansion steps will be referred to as the assignment of numerical factors . The phase cell positivity of the interaction is applied to weakening the factorial growths, as that positivity is quartic in the amplitudes Ok in contrast to the quadratic positivity of the kinetic (free) part of the Lagrangian. The extra liN! factors arising from the combinatorics in Chap. 3 plays an indispensable role in the assignment of numerical factors. This chapter is the most technical and, at the same time, the most elementary. Only the most basic concepts are invented to deal with the convergence problem. The expansion rules never involve any operations more sophisticated than interpolation and integration by parts, so the generated terms are represented by Representation 1 graphs [B2l, B22J, which consist of simple wavelet diagrams and chains thereof. Our terminology differs in a couple of respects from the terminology used in the original papers. For example, we refer to the simplest wavelet diagrams as links, and any link u associated with a space integral of the form (5.4)
is a 4-link. We still use the term composite [B22) for a link (u, u ' ) associated with w{u)w{u ' ), where at least one mode in u is a mode in u ' and the case requires that u and u ' be considered together. On the other hand, we use the term combination link for those special cases where a composite link has to be combined with an additionaI4-link. Moreover, since Representation 2 graphs are the polymers in this phase cell expansion, we speak of a "Representation 2 graph" and "phase cell polymer" interchangeably. The notion of Representation :1 graphs [B2l, B22) is avoided by applying combinatoric estimation that is more widely used (see §5.1O) , but of course one can argue that the concept is still there. In the first section we describe the <J>~ Euclidean field theory in the wavelet formalism, where the regularization is just the restriction of the field configurations to a finite but arbitrary set of wavelet amplitudes. We refer to the wavelet indices as modes, but actually, the discussion in that section is independent of the wavelet analysis . The Federbush stability bound derived there applies to any orthonormal basis. Related results have been proven by Lieb [L37). In the second section the analysis becomes wavelet-specific. We exhibit the most basic estimates on Wk(a\ uda;), ~k(p), and partial derivatives of these functions.
CHAPTER 5. WAVELET ANALYSIS OF ~
406
The primary objective here is to establish the basic quartic positivity bound (5.5) where Lk and
= 2-
rk
is the length scale of
wd:;;), A is a finite but arbitrary set of modes, (5.6) kEA
The scale parameter r ranges over the non-negative integers, but - in contrast to the notation and circumstances of Chap. 4 - the length scales are arbitrarily small instead of arbitrarily large . Since we are solving a short-distance problem with a phase cell cluster expansion, it is most convenient to parametrize shrinking length scales with increasing r - a convention we followed in Chap. 2. In the third section we introduce the two inductive operations for the expansion interpolation and integration by parts. The former is based on attempts to decouple variables, but it is a little more elaborate than the interpolation used in Chap. 3. As the coupling agent for the phase cell variables, the quartic part of the interaction is interpolated in almost the same straightforward way as the nearest-neighbor interaction for a spin system - actually, its Wick-ordering is interpolated in exactly the same way but the mass counter-term is interpolated in a different way. Indeed, the latter interpolation is more convoluted than that chosen for the mass counter-term in [B22]. The aim is to avoid the development of any infinite-energy diagrams - an option made available by the nature of phase cell cluster expansions. The price we pay is that the proof of stability in the number of modes for the total interpolated interaction is a little more involved. The proof is a major part of §5.8 and depends on a combination of Federbush stability with estimates on numerical factors for bounding the negative contributions to the interpolated mass counter-term. In §5.4 we establish those estimates - indeed, all estimates on numerical factors that will be needed for proving convergence of the expansion. This includes the estimate on the type of numerical factor containing the mass renormalization cancellation. In §5.5 we introduce the expansion rules. With the interpolation of the interaction already defined, the rules specify exactly when to interrupt interpolation with integration by parts along a given branch of the inductively defined expansion. Naturally, the algorithm chooses the a-variable with respect to which the integration by parts is done, and wavelet diagrams are very useful visual aids in understanding the numerical factors that are generated by a sequence of integration by parts steps. The algorithm we choose is by no means unique - and it is not the same one chosen in [B22] - but there is a guiding principle for all of these choices. The set of expansion rules is motivated by the need to avoid ultraviolet divergences in the perturbation theory, and nothing more. Cancellation of the number divergence - the nonperturbative part of the problem - is done by clever estimation, and none of the difficulti es therein influence the expansion rules . As far as the expansion rules in this chapter are concerned, we choose to integrate by parts as little as possible . In §5.6 the completed expansion is realized as a sum over histories, which we proceed to organize with the notions of links and chains. The renormalization cancellations are
5.1. THE CUTOFF - A FINITE SET OF MODES
407
automated here by the appropriate combination of histories. This means that every such cancellation is realized as a special numerical factor, on which the necessary estimate has already been established in §5.4. Some important notation is fixed in this section as well. The actual terms in the completed expansion are labeled by Representation 1 graphs. In §5.7 we relate this expansion to the polymer expansion formalism and identify the polymer as a Representation 2 graph, which is basically a Representation I graph without its history of development. This means that the activity of the polymer is the sum of all expansion terms whose Representation I graphs all correspond to the given polymer. Therefore, the preliminary estimation of the expansion terms in §5.8 is not an activity estimate, but this is the stage where the stability of the interpolated interaction is proven and the quartic positivity is extracted. The preliminary estimation of the polymer activity itself is the content of §5.9, and this is very important, as it depends on a combinatorial identity summing the weight factors associated with the Representation I graphs and realized as integrals over products of interpolation parameters. It is a generalization of the Federbush identity, which has already been discussed in Chap. 3. In §5.10 we consider the problem of summing activities within the framework of the polymer formalism, and this is where the important notion of attachment is introduced. From the combinatorics of the summation we extract the extra liN! factors associated with the attachments, and in §5.11 these factors are used to reduce the problem of cancelling the number divergence to a problem concerning the assignment of numerical factors. §5.12 is devoted to reducing the whole convergence problem to the consideration of just a finite number of cases. Chains are a main source of concern here, as the necessary smallness their development extracts must be distributed over the links before the chains are broken up. The notion of attachment is extended to the notion of pinning for each link in a chain, which depends, in part, on whether a link precedes or succeeds the attachment link of the chain. The case structure of a chain is further complicated by the possibility of a chain being internally connected in a way that affected the expansion steps. As in [B22J, this case is dealt with by introducing a hook and a hook link - important topological notions that affect the distribution of small factors over the links of the chain. In the end, we leave the case-by-case assignment of numerical factors to the reader as one great exercise. However, in §5.13 we bring back the wavelet diagrams already used to describe the original expansion rules. The diagrams provide the best way to think about the assignment of numerical factors. It is similar in spirit to the power counting for Feynman diagrams, and in §5.14 we illustrate this game by successfully assigning numerical factors in a number of cases.
5.1
The Cutoff - A Finite Set of Modes
Let {wd denote an orthonormal basis in L2(1l~.3) which we specify in the next section as a certain wavelet basis. Our initial remarks are independent of the multi-scale structure so vital to a phase cell analysis. As we have already indicated above, we expand the
CHAPTER 5. WAVELET ANALYSIS OF ~~
408
Euclidean field ~(X") in random amplitudes ak as follows :
L akuk(X"),
(5.1.1)
k
(-~
+ m6)-1/2 wk .
(5.1.2)
Equivalently, we define the random variables (5.1.3)
with the motivation that since (5.1.4)
these variables have the property (5.1.5)
which makes them independent with respect to the Gaussian measure dJ1.o . The rules for Wick ordering are simple:
II .
. II U.k' ~Nk .
.
~Nk
.
. Uk
(5.1.6)
"'
k
k
~N e -!"'~ .. U.k
dN
1
2
(-1) N --e- ,"'k
..
daf:"
(5.1.7)
Let A be a finite subset of the set of indices and define
~A(X")
=L
(5.1.8)
akuk(X").
kEA
Let the expectation functional (-) A be given by ZA'I (F(a)e-I(A))o,
(5.1.9)
(e-I( A))O,
J:~A(~f + J ~A(X")2 J
(5.1.10)
: d 'X"
A
48A
2
d'X"
dy
(~A('X")~A(Y))~.
(5.1.11)
In this a priori renormalization of the interaction, the mass counter-term is not Wickordered, nor are the second-order and third-order energy counter-terms included. This omission is possible because at each stage in the removal of the cutoff, only a finite number of variables are involved. Naturally, the phase cell expansion must still be inductively defined such that the infinities cancelled by the omitted counter-terms can never develop.
5.1. THE CUTOFF - A FINITE SET OF MODES
409
No functional analysis is needed to make sense of this regularized functional integral (5.1.9) because it is an ordinary integral in a finite number of variables. More explicitly,
=
(F(Q))A
(II 1
00
(27r)-!
cardA ZA"1
kEA
e
-!
a~-I(A)
L kEA
,\
dQk)
-00
F(Q),
(5.1.12) 4
L
w(kl,k2,k3,k4):IIQk,:
k"k2,k3,k.EA
,=1
k, , ... ,k5EA
(5 .1.14) Our goal is to control the limit of the expectations
\Jl ~(fj))
A
k'~kN \Jl
Qk; ) A
Jl J
/j(-;)ud-;)d-;
(5 .1.15)
as A approaches the set of all modes in the sense of set inclusion. The interpretation of the A-expectation when not all k1 , . . . , kN lie in A is understood to be:
II
N Qk; ) \ j=1 A
=
\
II
Qk; )
j: k;EA
II
\
A
j: k;rlcA
(5.1.16)
Qk; ) 0
It suffices to control the limit of such an expectation for a fixed product, provided the bounds obtained imply the condition
(5.1.17) The point is that if
L 1(( -~ + m6)-no
Uk )(-;)1
~e
(5.1.18)
k
for some positive integer no, then (5.1.17) implies
N
~
(N!)1/2e N
II II( -~ + m6tofjlll j=1
~
N
(N!)! eN
II sup 1(1 + 1-; 12)no (-~X" + m6)no Ij (-;) 1 j=1
x
(5.1.19)
CHAPTER 5. WAVELET ANALYSIS OF ~
410
for the test functions h, ... , f N if we also choose no > 1. This scheme already imposes a significant condition on the orthonormal basis {'l1 d, but (5.1.18) is a property that is typical of a wavelet basis. The generating functional to study for the expectations (5.1.16) is obviously given by (5.1.20)
and the partition function for the polymer formalism will be (5.1.21)
The polymers for the phase cell expansion will be rather complicated, but the lower bounds on I(A) required for estimates on polymer activities will be relatively simple. Indeed, the most basic lower bound holds for any orthonormal basis - not just a wavelet basis. First, one completes the square to obtain: (5.1.22) Second, one expands this last integral: (5.1.23) Observe that the lower bound (5.1.24)
follows from the Federbush stability bound :
L k,lEA
JUk(~)2Ul(~)2
d ~~ c card A,
(5.1.25)
which holds for an arbitrary orthonormal basis {'l1 k}. Although this quantity is a double sum in the modes, we have a bound that is linear in the number of modes. To see this, set
p - PIW
~'k2k3(:;)2 + (1 - td(iI>k'k2 k3 (:;)2 + iI>k,k2k3 (:;)2)] (5 .3.11) B\{k'} , {k' }nB
(5.3.12)
Then
I(tIiiI>A) x
L
= I'(tIiiI>A) +48>.2
.I d:;
w(k l ,k2,k3)
k, ,k2,k3EA Uk,
(:;)Uk2(:;)Uk3(:;){E~,k2k3(tli iI>(:;)) 2
- E~'k2k3(tli iI>k,k2 k3(:;))2}
+ QA(tli iI>A),
(5.3 .13)
and the new interpolation is given by
I(tii t2i iI>A)
= t2I'(tl i iI>A) + (1- t2)[I'(t l i iI>B) + I'(iI>A\B)]
+ t2QA(tl i iI>B) + w(k l ,k2,k3)
.I
(1 - t2)QA(t l i iI>A\B) + 48>.2
L
k, ,k2,k3EA ~ ~ ~ ~ kkk ~ d x Uk,(X)Uk2(X)Uk3(X){EA' 23(tl , t2iiI>(X))2 (5 .3.14)
E~,kok3(tl,t2;iI>(:;))
=
t2E~,k2k3(tIiiI>(:;))
+ (1-t2) x[E~'/;lk3(tli iI>(:;)) + E~\kt3(tli iI>(:;))] . (5 .3.15)
The remainder terms now are contributed by the a-products generated by the t2derivative of the modified interaction I(tl, t2i iI> A)' If the decoupled term is chosen by this expansion step, then the next step is to interpolate Z(Tk: k E A\B) by trying to decouple the first variable in A\B from the other variables in A\B. The iteration of this procedure generates a sum over histories of expansion steps, with each step a choice among many possibilities provided by the partial history already developed. For a given partial history, the term is either a decoupled term or a remainder term. In the former case, it has the form
IT dak-integral developed by the £th sequence of previous expansion kEA, steps , which is composed of successive interpolations occasionally interrupted by a where K, is the
CHAPTER 5. WAVELET ANALYSIS OF ~
422
succession of integration by parts operations and finally terminated by choosing the decoupled term. The sets A, are mutually disjoint sets of modes, and the next expansion step is to interpolate Z A\
(Tk:
k
E
A\
YA,)
by trying to decouple the first mode in
UA, with respect to our fixed linear ordering of A from the other modes in A \ UA, . , '
Suppose the term associated with the partial history is a remainder term instead. It has the form
(II 11 ,
L
dti) J(t) / exp (
0
Teae - J(t;
A\YA,)) G(t; a))
eEA\ UA,
\
,
0
where {til is the set of parameters for past interpolations and J(t) is some product of powers of the ti having developed as old t-dependences have been differentiated down by new interpolations. The quantity G(t; a) depends on only those ak for which k E A\ UA" and the modes which indeed occur in G(t; a) comprise the interior set for
,
the next interpolation of the
J IT
dak-integral, provided the next step is indeed
kEA\UA.
interpolation. An integration by parts may be in order - with AI
= A \ UA, in
(5 .3.2)
- but only if
G(t; a)
= F(t ;a)
: a~a :
(5.3.16)
for the variable aka with respect to which the integration by parts is to be done, where F(t; a) does not depend on aka . Since A is a finite set, every possible history of expansion steps eventually terminates with a decoupled term of the form
and such a decoupled term is a completed term (as it was §3.1O, where we were decoupiing lattice variables). The generating functional is now expanded in these completed terms, but of course the estimates for controlling this expansion must be independent of A. Bear in mind that we have only described the inductive operations themselves and not the rules governing their application, so the expansion we propose to generate in this way is given only schematically at this point. As we shall see later, we have brutally swept some messy but important issues about G(t; a) under the rug as wellissues concerning what type of expression constitutes a term when a formula is applied. However, our next step is to derive the input estimates that will actually motivate the rules for the integration by parts.
5.4. ESTIMATES ON NUMERICAL FACTORS
423
References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 2. G . Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical j Model," Commun. Math . Phys. 88 (1983),263- 293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math . Phys. 81 (1981), 327-360.
5.4
Estimates on Numerical Factors
The basic estimation of terms in the expansion obviously falls into two categories - estimation of a-integrals and estimation of space integrals involving products of functions corresponding to modes. We refer to these space integrals as numerical factors, and the quantity w(k 1, ... , k n ) defined by (5.1.14) is the simplest type of numerical factor. There is more than one way that we will have occasion to estimate this quantity, but the most straightforward way is to insert the bound (5.4.1) which is a special case of (5.2.11) . Since w(k1, ... , k n ) is symmetric in the modes, we may assume without loss that (5.4.2) Clearly, n
Iw(k 1, ... , kn)1 :::; c
IT L~,1/2 sup((1 + L;21-; - -;k n I)-N+1+' x
t=l
x IT(1+L/;,11-; -
-;k,
I)-N-2)
,=1 cL~~2
Jd-;
(1 +L/;n11 -; -
-;k n 1)-3-,
n-1
IT L;;,1/2s~~'p((1+L/;:I-; - -;k n I)-N+1+' t.=1
x
n-1 X
IT (1 + L/;,11-; - -;k, I)-N-2),
(5.4.3)
,=1 and the desired estimate involves the use of one mode k, as a fixed point of reference. By the triangle inequality,
(1+L/;,11-; -
2: 2:
1
---lo.
-;k, 1)(1+L/;,11-; - -;k, I) -..lo.
---lo.---lo.
1+L/;, (I x - Xk, 1+1 x - Xk, I) --" --" L')
(>')u. C;oJ(
a,-' u,) (>')d >',
(5.4.39)
we sup out every factor in each integrand except the function O;-1 Ut . Integrating IO;-1Utl, we obtain
x (l + L;;11 ;;k -1
~
;;t I)-NO (1 + L;;311;;k3 - ;;e I)-No ~
N
X(l+Lkll Xkl - Xk 1)- 0 x (l + L;;21 ;;k 2 - ;;k3 I)-No 1
(5.4.40)
for a positive integer No which can be made large by adjusting the wavelet construction parameter N.
5.5. EXPANSION RULES AND WAVELET DIAGRAMS
431
References 1. G . Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No. 1 (1982),95-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical 4>~ Model," Commun. Math. Phys. 88 (1983), 263-293.
5.5
Expansion Rules and Wavelet Diagrams
As we have already described in §5.3, there are two types of expansion steps - the type arising from interpolation and the type arising from integration by parts. The latter type of step is aimed at avoiding ultraviolet divergences, so naturally the cases in which it is called for involve certain combinations of small length scales. In this section we describe those cases, and this description will complete the expansion algorithm. Recall the terminology we have already introduced. A completed term has the form
where each factor has been developed by a history of expansion steps finally terminating with an interpolation step that chooses the decoupled term. A completed term is a decoupled term where the complementary set of variables is empty. No further expansion steps are called for in the case of a completed term . More generally, a decoupled term has the form
Such a term calls for an expansion step, but we have previously specified that in this case, the next expansion step is a choice of term arising from the interpolation of
that attempts to decouple the first mode in A \
U , A, from the other modes in that residual
set. By "first mode" we mean first with respect to a fixed linear ordering of A . A remainder term has the form
x
II K , (Te : eE A,),
and the expansion step that such a term calls for is the issue here. The modes occurring in the expression for G(t; 0:) constitute the interior set.
CHAPTER 5. WAVELET ANALYSIS OF ~
432
We require a little more terminology. The interpolated interaction may be written in the form I (tjA\YA,) =>.
L
fk, ...k,(t)w(k 1 , ... ,k4 ):ak, ... ak, :+48>.2
k, , .. . ,k, X
L
ak,akJk, .. ks(t)w(kl,k2,k3)W(kl, .. . ,k5),
(5.5.1)
k" .. ,ksEA\UA, , where fk, ...k,(t) and lk, ... ks(t) are products of powers of all previous interpolation parameters tl, .. . , tn determined by the history regarding which subsets of A \ U A, we
,
have attempted to decouple from their complements relative to A \
U A,. ,
Note that in
contrast to fk, ... k,(t), the case structure of lk, ... k5(t) will be a little more complicated because - as we have already described in §5.3 - the interpolation of the mass counterterm is a little involved. Each of these products of variable expressions with parametric monomials and numerical factors in the exponent will be called a unit. The units Teae are form 0 units, while the Wick-ordered a-product units in the O(>.)-contribution are form 1 units. The units that are second-order in >. are form 2 units . Now every remainder term is the result of an expansion step that has either differentiated some unit down from the exponent or simply a-differentiated an already-differentiated unit very previously brought down from the exponent. In the former case, that previous expansion step could arise from either interpolation or integration by parts. In the latter sub-case, that step could have been called for by a situation where the preceding step a-differentiated an already-differentiated unit already down from the exponent. This means the induction hypothesis for the expansion rules must involve several cases, some of which reach back two or three expansion steps into the history of what has developed thus far. On the other hand, there are a couple of general rules, which we state immediately in the interests of case reduction. Form 0 Rule. Whenever a form 0 unit has been differentiated down from the exponent (which can happen only as a result of integration by parts), the next step is generated by interpolation based on the current interior set (which is also the previous interior set). Form 2 Rule. Whenever a form 2 unit has been differentiated down from the exponent (whether as a result of interpolation or of integration by parts) , the next step is generated by interpolation based on the current interior set.
These two rules imply that a chain of integration by parts steps is automatically terminated if a form 0 or form 2 is differentiated down from the exponent. Therefore, the only cases left to be covered by our expansion rules involve form 1 units only. We first consider the form 1 case where the previous step was induced by interpolation - designated henceforth as Case I. Let G(t j a)
= F(tj a)U(tj a),
(5.5.2)
5.5. EXPANSION RULES AND WAVELET DIAGRAMS
433
where U(t; Q) is the differentiated unit - differentiated with respect to the most recent interpolation parameter tn. Thus (5.5.3) The modes occurring in the expression for F(t; Q) obviously form the interior set on which the interpolation is based, and F(t;Q) does not depend on tn. Let B c A\UA,
,
denote that interior set. The occurrences of modes in U(t; Q) that do not lie in Bare new occurrences, while those that lie in B are old occurrences. Recall that we fixed a linear ordering of A for the expansion. We now stipulate that the linear ordering must be chosen such that a latter mode has equal or smaller length scale, and we shall often refer to it as the scale lexicon or the scale-lexicographic ordering. One may now state the Case I rules concerning how to determine the next step. Consider (5.5.3) and assume without loss of generality that (k 1 , ... , k 4 ) is the linear ordering of the occurrences. Thus (5.5.4) by our scale-lexicographic convention. The case structure is based on which of these occurrences are new. Case la. If either k3 or k4 is old, then the next step is induced by interpolation, where B U {new occurrences} is the interior set. Case lb. If both k3 and k4 are new, then the next step is induced by integration by parts with respect to Qk,. Now consider the case where the previous step was integration by parts but the step before that was interpolation. We classify this scenario as Case II, and in this event,
F(t;Q)
= F'(t;Q)U'(t;Q),
(5.5.5)
where the circumstances of U'(t; Q) are Case Ib, because Case la did not call for integration by parts. Let B' be the interior set for the interpolation that differentiated down U'(t; Q) - i.e., the set of modes that occur in the expression for F'(t; Q) . Then B=B'U{k~,
. .. ,k~},
(5.5.6)
where k~ , .. . ,k~ are the modes occurring in this tn-differentiated form 1 unit. We again assume without loss of generality that (k~, . .. , k~) is the linear ordering of those modes, so by Case Ib, the integration by parts is done in the variable Qk~, and therefore
U'(t;Q)
= -)..8~/k~ ... k~(t)W(k~, . .. ,k~): Qk~Qk;Qk; :,
(5.5.7)
while the Case II unit is differentiated with respect to Qk~' The possibilities for the latter unit are
U(t; Q)
(5.5.8)
CHAPTER 5. WAVELET ANALYSIS OF j
434 U(t; a) U(t; a)
=
but we already know that (5.5.8) and (5 .5.10) call for interpolation, so (5.5.9) is the expression whose expansion directive is not yet given. As far as Case II is concerned, we shall say that an occurrence is new if it occurs in either unit but does not lie in B'; the modes in B' are old occurrences . Thus every new occurrence in the preceding Case Ib scenario is still regarded as new. In particular, ka and k~ are new. Case IIa. If either k3 or k4 is old, then the next step is induced by interpolation with interior set B U {k 1 , •.• , k4 }. The same step is called for if k2 is old and either k3 = k~ or k4 = k~ - also if the latter occurrence in {k 1 , k;} is old and either k3 = k~ or k4 = k~. Case lIb. If both k3 and k4 are new and k~ # k 3, k4, then the next step is to integrate by parts with respect to ak•. Case IIc. If none of the above cases hold - i.e., if both k3 and k4 are new, if either k~ = k3 or k~ = k 4 , and if the latter occurrence in {k 1 , k;} is new - then the next step is to integrate by parts with respect to ak;'
This third possibility is the case where a divergence may develop that has to be cancelled by the mass renormalization. Next we consider Case III, in which the last two steps were induced by integration by parts and the third step in the past was induced by interpolation, but the integration by parts that has just taken place only a-differentiated one of the two units differentiated down by those two preceding steps. We have the same basic form G(t; a) = F'(t; a)U'(t; a)U(t; a)
(5.5.11)
that we had in Case II, only this time U and U' have been altered by this most recent integration by parts step, which could have been called for only in Case lIe with ka = k3, with ka = k 2, or with ka = k 1 • The rules for Case lIa and Case lIb can only differentiate down a new unit . Therefore, the differentiated units are given by
U(t; a)
{)2
->"fk, ... k.(t)w(k 1 ,oo.,k4){)
{) Qka
U'(t;a)
>..
Qk~
{)~n fk; . k~ (t)w(k;, . .. , k~) : ak; ak;
:ak,oo.ak.:, :.
(5.5 .12) (5.5.13)
In this case, the next step is to integrate by parts in the variable ai, where:
(i) If ka = k 2, then e is the latter occurrence in {k 1 , k 2} with respect to the scale lexicon. (ii) If ka = k 1 , then e is the latter occurrence in {k 2, k 3, k 4 } \ {kU. Case IV is the designation we give to the case where the last three steps were generated by integration by parts, the fourth step in the past was due to interpolation,
5.5. EXPANSION RULES AND WAVELET DIAGRAMS
435
and neither of the two most recent steps have differentiated down a new unit, but have only a-differentiated the units that are already down from the exponent. This means that previous to the integration by parts that has just taken place, the situation was clearly Case III, and since this integration by parts did not differentiate down a new unit, we have f = k~ . The differentiated units are now given by
U(t; a)
-)..jk'oo.k4 (t)w(k 1 , • . . , k 4) ()3
x ()
U'(t;a)
ak~
()
"
ak;uak~
: ak, ... ak4 :,
-A {)~n fk~ oo.k~ (t)w(k~, . . . k~)ak~ .
(5.5.14) (5.5.15)
In this case, the next step is interpolation based on the interior set B U {k 1 , ... , k4 }. We still have to consider the same hierarchy of cases where the first unit was differentiated down by an integration by parts instead of by interpolation. We designate the corresponding cases by Case I, Case fi, Case ill, and Case iV, but we need to allow for overlap with the cases already considered. With this constraint in mind, we state the expansion rule for each case. Case Ia. If either k3 or k4 is old, then the next step is induced by interpolation with B U {k 1 , ... , k4 } as the interior set, where B is understood to be the interior set prior to the integration by parts that has just taken place. Case Ih. If both k3 and k4 are new, then we have a situation that may be Case lIb with the given unit as the second unit. We extend the Case lIb rule to this generalization - i.e., we integrate by parts with respect to ak.. This is also the Case Ib rule. Case fia. If either k3 or k4 is old, then the next step is induced by interpolation with interior set B U {k 1 , ... , k 4 } where B is now given by (5.5.6) and B' is understood to be the interior set prior to the initial integration by parts. The same step is called for if k2 is old and either k3 = k~ or k4 = k~ - also if the latter occurrence in {k 1 , k~} is old and either k3 = k~ or k4 = k~. Case fib. If both k3 and k4 are new and k~ i- k 3, k 4, then the second unit is now Case lb. Therefore we integrate by parts with respect to ak • . Case fie. For the remaining Case fi situations - i.e., if both k3 and k4 are new, if either k~ = k3 or k~ = k4' and if the latter occurrence in {k 1 , k~} is new - the next step is to integrate by parts with respect to ak;. This is also the Case IIc rule.
m.
Case This case does not overlap with any case previously considered. We apply the Case III rule. Case IV. This case does not overlap with any previous case. Using the same interior set as in Case fia, we interpolate in this situation.
This completes the description of our expansion algorithm. Wavelet diagrams are useful visual aids in understanding these cases, but here they are a little different from those introduced in the renormalization group context . The mode identifications correspond to selective integration by parts for a term rather than to integrating out all variables for a given length scale and then looking at a term.
CHAPTER 5. WAVELET ANALYSIS OF ~
436
However, the correspondence between markings and quantities is the same. Following the terminology already introduced, we refer to indices k, that appear in a unit as occurrences of modes, since the k, need not be distinct. We use vertical line segments to represent spatial integration variables, and any occurrence that labels a function in a given integrand will be on the corresponding line segment, either as a dash or as an xmark. We also use dotted lines to connect occurrences of the same mode. The position of an occurrence as a vertical coordinate will represent the length scale of the mode, where the smaller-scale modes occur below the larger-scale modes. A real occurrence is any index k, where the variable Ci!k, actually appears in the unit ; otherwise, k, is a ghost occurrence. We use the dash for a real occurrence and the x-mark for a ghost occurrence. Thus, the three basic types of units have the graphical representations exemplified by Fig. 5.5.1. For the actual cases , we enhance the graphs as shown in Fig. 5.5.2, where Fig. 5.5.2.1a gives examples of Case la, Fig. 5.5.2.1b gives examples of Case Ib, and so on. -f.
Form 0
k J k2
f k4
k3
Form 1
k4
~~! --
ks
kJ k2 k3 Fonn 2
~ ~]
--
kJ
k4
k2 k3
ks
Figure 5.5.1:
k I old
k I new
I I k 2 new
k 2 old
k 3 old
k 3 new
k 4 new
k 4 old
Figure 5.5.2.1a:
5.5. EXPANSION RULES AND WAVELET DIAGRAMS
f
I
k I new
k 2 new
k 2 old
k 3 new k 4 new
!
k , I old k 2 new
k4 new
k ' I Old
j
k I new
k ' 2 old
k 3 old - - -
f
k 3 new
Figure 5.5.2.1h:
k I old k 2 new
k 3 new
k' - n~w-
k I old
437
k 2 new
k ' 3 new
-------
k '4 new
k 4 = k '4
4
k 4 new
k I old k I old
k 2 old k 2 new
k 3 new
k 4 new
Figure 5.5.2.I1a: k I old
:: : : f~ -t: ~~e:~ ;_n:w_____
k 4 new
j
k 2 old
k 3 new
- ------ -- -
k 1= k,j
k 4 new
k3 0ew
k 4 new
k 2 new k3
new
k 4 new
Figure 5.5.2.I1b:
CHAPTER 5. WAVELET ANALYSIS OF ~~
438
: :: !
k' l old
k 'J new
k 3 new
----- --k'4 new
l
, k 4 = k4
k'l old
:~
old
.-L.
new
kj
new
k 2 new
- -- -- - - - -
k 4 new
k 4 new
Figure 5.5.2.lIc: k ' l old
k I new
k'2 new
k 2 new
k I old k I new
k 2 new k '3 new
- - ----- - -
k 3= k 3
k ')
new
- -- -- - - -
---- - ----
k 2= k 3
k'4 new
k'4 new
k 4= k 4
--- -- - -
k 3= k4 k 4 new
k I old
:~~;~j
k I old k 1 new
k 2= k 3
k'2 old
-~;-";: --]
k 1= k 3 k 2 new
k 3 new
k 3 new
k '4 new k 4 = k 4
-- -
--- -
-
k 4= k4
k 4 new
Figure 5.5.2.111:
The Case IV graphs are cancelled by the respective graphs in Fig. 5.3 to implement the ultraviolet renormalization, as we shall see later. For future convenience we define a composite as the pair of differentiated form 1 units that occur in a Case lIa, Case IIc, Case III, Case fia, Case fic, or Case ill
5.5. EXPANSION RULES AND WAVELET DIAGRAMS
!-T~:-:-:-]
k
439
3 new
k 4 ne w
k I o ld
F~~----] _____
k I new
k '2 ne w
k 2= k 2
---------
k '3 new
=k 2
K
1
K
2= k 3
_k ~ ~e:
k ) = k 3
k 3= k 4
k 4 new
k 4=
k~
,
Ie.
4 new
k 4 ne w
Figure 5.5.2.1V:
k I old
I
---f:::~:
k lold k I new
--jk neW --Tk )neW 2
- - -X
--
"2 ncw
--
k3new k 4 new
k 4 new ,
Figure 5.5.3:
k I new
k I new
k 2 old
k 2 new
k 3 new
k 4 new - --- ---
k I new
~
-
- --
k I new
k 2 old
k 2 new
k 3 new
k 3 new
k 3 new
k 4 new
k 4= k 4
Figure 5.5.4.IIa:
----- -Figure 5.5.4.lla:
k 4= k4
CHAPTER 5. WAVELET ANALYSIS OF ~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981),327-360.
5.6
Organizing the Completed Expansion
For the arbitrary finite set A of modes, every sequence of steps allowed by the expansion rules eventually terminates with a completed term because the interior set grows monotonically for every possible history. It is possible that an integration by parts step does not introduce new modes, but the rules allow at most three successive steps of this type. Even when the growth of the interior set is arrested by choosing the decoupled term in an interpolation, the next step gives birth to a new interior set in the residual set of modes. Thus, the completed expansion of Z(T/ : P. E A) consists of terms of the form A, = A, A, n A" = 0,
U
and our next task is to describe an arbitrary factor K,(T/: P. E A,) in a manner suitable for estimation. Unfortunately, these completed terms are not indexed by the possible partitions {A,} of A, but rather by the histories (of expansion steps), many of which induce the same partition. Our notation for a factor in a term has been somewhat tentative.
CHAPTER 5. WAVELET ANALYSIS OF j
444
We define an n-link as an n-tuple of occurrences, ordered by the fixed scalelexicographic order of modes, together with an indication of which occurrences are real. Thus to a form 1 unit that has been differentiated down - and possibly altered further by integration by parts - we associate a 4-link. We associate a pair of links to a form 2 unit that has been differentiated down - a 3-link consisting of ghost occurrences only and a 2-link having at least one real occurrence. We call such a pair of links a mass link . Obviously, in the case of a form 0 unit we have a I-link due to integration by parts - i.e., a ghost occurrence of a single mode. We associate to a composite the pair of 4-links corresponding to the pair of modified form 1 units that constitute the composite. We call this pair a composite link. Moreover, it will be important to the case structure of the convergence proof to consider combination links . If a sequence of integration by parts steps terminates with a 4-link preceded by a composite link, then we regard the pair as a single link - a combination link. For any history of expansion steps, consider a sequence of integration by parts steps such that the step preceding the first step and the step succeeding the last step are both induced by interpolation - i.e., a complete, uninterrupted sequence of steps induced by integration by parts. These steps generate links because they differentiate down units, possibly differentiating them again, and possibly forming composites. In this sequence of links the expansion rules imply that the last link can be anyone of the four types, but that the others are either 4-links or composite links. We call such a sequence of links a chain, but this does not include the case where the sequence of integration by parts steps is so short that it develops a single composite link or a single combination link and then calls for interpolation. Such isolated links are not regarded as chains, even though several integration by parts steps may be involved. After all, our convention does not regard a mass link due to interpolation as a chain, and in the renormalization scheme it may be paired with Case IV composite link. Now a link is only recording the unit or composite that has been differentiated down and/or developed by the expansion steps, and our notation for those quantities has been tentative up to now. Notice that in the case of, say, a 4-link a, the quantity U(a; t) depends on the past history of expansion steps - not just on a. It consists of the factors:
U(a;t) = -Av(a)w(a)q" (t)F,,(a),
(5.6.1)
where q,,(t) is a product of previous interpolation parameters, v(a) is the number of permutations of distinct modes in a, w(a) is the numerical factor w(k 1 , .. . , k4 ) associated with the occurrences k 1 , •. . , k4 in a, and F" (a) is the modification of : 4
IT ak, : appropriate to the situation.
,=1
In contrast to q,,(t), w(a) is independent of the
previous history, with F,,(a) almost independent because a also designates the ghost occurrences among kl' . .. , k 4 . It is worthwhile to inspect this more closely. If a is not in a chain, then 4
F,,(a) = :
II ak, :, ,=1
(5.6.2)
5.6. ORGANIZING THE COMPLETED EXPANSION
445
while if a is the first link in a chain, then 3
Fu(O:)
= : II O:k,
:.
(5 .6.3)
L=l
If a is the last link in a chain , then (5.6.4)
for some i > 2, while if a is any 4-link in a chain that is neither first nor last, then (5 .6.5)
for some i < 3. Except for the multiplicity factor, we may think in terms of ghost occurrences without regard to the place of a in a history. If a has no ghost occurrences, then a is not in a chain and Fu(O:) is given by (5.6.2). If only k4 is a ghost occurrence, then a is either the first link or the last link in a chain, but notice that the integer factor is the only dependence of Fu(O:) on these cases, because ~o = 4 in the latter case. If only k3 is a ghost occurrence, then a is the last link in a chain and (5.6.6)
If k4 and k i are the only ghost occurrences with i < 3, then Fu(O:) is given by (5 .6.5). There are no other possibilities for a with regard to ghost occurrences. In the case of a composite link (a',a), the general observation is the same: the only dependence on previous expansion steps (that is not already implied by ghost occurrences) lies, for the most part, in the product of previous interpolation parameters. The associated quantity is given by
U(o: ; t)U'(o:; t)
= >.?v(a)v(a')w(a)w(a')qu(t)q", (t)F" (o:)Fu' (0:)
(5.6.7)
with w (a#) and Fu# (0:) virtually determined by a# alone. We mention only a couple of cases here, where (kr, .. . , kt) denotes the quadruple associated with a#. For example, suppose k4' k~ , and k~ are the only ghost occurrences. Then k~ = k4 and the composite is Case IIc with the next step (integration by parts in O:k;) differentiating down a unit from the exponent. The composite link is the first link in either a chain or a combination link, but these remarks are superfluous to the formula 3
II O:k, : . L=l
(5.6.8)
CHAPTER 5. WAVELET ANALYSIS OF ~~
446
As another example, suppose k1 , k 2, and k~ are the only real occurrences. Then k~ = k3 and k~ = k 4 , where the composite is Case III with the next step (integration by parts in ak ) differentiating down a unit from the exponent. Moreover, kl precedes k~ in the 2 scale lexicon - a constraint on (a', a) itself. (5.6.9) for this composite. As yet another example, suppose k~, k~, and k2 are the only real occurrences. Then, again, the composite is Case III with k~ = k 3 , k~ = k 4 , and the next step differentiates down a unit from the exponent, but in this case, the integration by parts is in ak,. Thus kl succeeds k~ in the lexicon, and (a' , O") itself obviously determines the sub-case. (5.6.10) for this composite. There is a very special type of composite link that can reflect two different developments with the same composite expression. Suppose k, = k;, £ = 2,3,4, and that kl and k~ are the only real occurrences. The composite must be Case IV, and the a-expression is: (5.6.11) but suppose kl as well as k~ is old with respect to the previous expansion steps. In this case, the development that differentiates down the primed unit after the unprimed unit is also a possibility, and the a-expression is still given by (5.6.12). Thus (a, a') and (a' , a) have identical composite expressions as well as identical previous histories in this situation. This case is combinatorially relevant to the renormalization cancellation. Case IV and Case iV composites in a given history are important because they contribute to the ultraviolet divergence. In Case IV, the composite link (0"', a) has the property that (k~, k~, k~) is a subsequence of (k 1 , ... , k 4 ) and the only real occurrences are k~ and the occurrence in (k 1 , ... , k 4 ) not in the subsequence. In Case iV, the composite link has the same properties, except that k~ is a ghost occurrence - i.e. , the only real occurrence is the occurrence in (kl' ... ' k 4 ) not in the subsequence. An exact link is either type of composite link. The possible composite expressions are: (5.6.12) for Case IV with k;; not in
(k~, k~ , k~),
and (5.6.12)
for Case iV because qui (t) = 1 in that case. The mass link we need for the renormalization cancellation against the exact link is given by the pair ((k~, k;;) , (k~, k~, k~)),
5.6. ORGANIZING THE COMPLETED EXPANSION
447
and such a mass link we call a counter-link. There is a minor abuse of notation here: (k~, k,) is in scale-lexicographic order except possibly when l = 1. Clearly, the possible form 2 expressions are:
U (a ; t)
= -48), 2v(k~, k,)3!w(k;;, k~, ... , k~)w(k~, k~, k~)ak; ak,qk; k"k~k;k~ (t)
(5.6 .13)
in the case where the form 2 unit has been differentiated down by interpolation, and (5.6 .13) in the case where the form 2 unit has been differentiated down by integration by parts in the variable ak; . Now suppose l = 1 and kl' k~ are both real occurrences with kl preceding k~ in the scale lexicography. Further assume that kl is a new occurrence. In the cancellation of (5 .6.13) against (5.6 .14), the graphs to be compared are given by Fig. 5.6.1. Clearly, the cancellation cannot be implemented unless k I new
k I old
k I new
k I old
k 2= k 2 k 3= k 3 k 4= k 4
newk 4
----1 ----l
k 2 new k 3 new k 4 new
Figure 5.6.1:
U(a ;t)U'(a;t)
+ U(a;t)
DC
w(a-)w(a-') - W(kl,k~, ... ,k~)w(k~ , k~ , k~),
(5.6.14)
so we need the equation (5.6.15) Since kl i- k~, we have V(kl' kD = 2, and so the integer products match. Also, qu (t) because kl is a new occurrence. The desired equation reduces to
=1
(5.6.16) Now by inspection of the interpolation defined for form 2 units described in §5.3, qk,k' ,k' k' k' (t) is the product of all interpolation parameters associated with those previods i~te;ior sets containing k~. But that is exactly what qu' (t) is. Having dealt with this particular case, we need to organize this matching. Consider an arbitrary sequence of chains and isolated links corresponding to a term in the expansion and pick any exact link. Can we replace that exact link with the corresponding counter-link such that a history of allowed expansion steps is still reflected
CHAPTER 5. WAVELET ANALYSIS OF ~
448
by the sequence? If so, do the products of integers and interpolation parameters match for that replacement so that a proportionality like (5.6.15) holds? The first question is easily answered in the affirmative because: (i) The interpolation that generates the step initiating the development of a Case IV composite also generates the step differentiating down the corresponding form 2 unit. (ii) The integration by parts that generates the step initiating the development of a Case IV composite - i.e., the step differentiating down the first form 1 unit in the composite - also generates the step differentiating down the corresponding form 2 unit. There is an extra wrinkle here. If the exact link reflects Case IV with k; = k" t = 2,3,4, then kl' k~ are both real occurrences, and in the case where kl is an old occurrence, ((7, (7') and ((7', (7) are both given by (5.6.12), as we have already mentioned above. The point to be emphasized here is that replacing one with the other does not violate any rules. We state this as a third observation: (iii) Consider the interpolation that generates the step initiating the development of the Case IV composite whose exact link is ((7', (7) with kl an old occurrence as well as k~ . The same interpolation generates the step initiating the development of the Case IV composite whose exact link is ((7, (7') . To answer the second question, we begin by observing that in any case, k~ is old and k~, kg, k~ are new. Therefore, qu' (t) is the product of all interpolation parameters associated with those previous interior sets containing k~. Beyond that observation, we must inspect each case. In the example given by Fig. 5.6.1, we have already established the match. Consider the case where the occurrence k~ is real and t: = 4. Then k4 is real and new, and the corresponding counter-link is ((k~, k4)(k~, kg, k~)) with k~ the only old occurrence. The graphs to be compared are given by Fig. 5.6.2, while k I old
k I old
----l
k 1= k2 k 3 new
k 4 new
- - - - - - - - - - -
----l
k 2= k 3
k I new
k 2 new k 3 new
k 3= k 4
k 4 new
k 4 new
Figure 5.6.2:
U(aj t)U' (aj t)
+ U(aj t)
ex: W((7)W((7') -
w(k~, ... ,k~ , k4)W(k~, k~, k~)
(5.6.17)
is the desired proportionality in this case. We need the identity 48v(k~, k4)3!qk;k4,k2k3k~ (t)
= (4!)2qu(t)qu' (t),
(5.6.18)
5.6. ORGANIZING THE COMPLETED EXPANSION
449
but this holds for the same reason that (5.6.16) held : kr is a new occurrence, so qu(t) = 1, v(k~, kr) = 2, and (5.6.19) Now consider the Case IV situation, where k~ is a ghost occurrence. For an example, pick "[ = 3, in which case k3 is real and new. The corresponding counter-link is ((k~,k3),(k2,k~,k~)), and the graphs to be compared are given by Fig. 5.6.3. The required matching in this case is the identity k I old
Figure 5.6.3:
96(3!)qk'1 ka 'k'2 k'3 k'4 (t)
= (4!?qu(t)qu' (t),
(5.6.20)
which is even more trivial than (5.6.19) because there are no products of previous interpolation parameters - i.e., (5.6.21) in this pure integration by parts scenario. The initial integration by parts in the variable o.k'1 means that k~ has never been in an interior set. k I old
k I old
k 2 new
-------
k 3 new k'4 new
-------
k I old
k I ?Id
k 3= k 3
---] ., . ----
k3 new
k 4= k 4
----
k new 4
k 2= k 2
Figure 5.6.4:
"
CHAPTER 5. WAVELET ANALYSIS OF ~
450
The scenario relevant to the observation (iii) above is the most interesting one. Suppose "[ = 1 but kl is an old occurrence strictly preceding k~, and k~ is a real occurrence. The graphs to be compared are given by Fig. 5.6.4. If we inspect the interpolation defined in §5.3 for this case, we find that qklk~,k;k;k~ (t) is twice the interpolation parameter just introduced times the product of all interpolation parameters associated with those previous interior sets containing either k~ or kl' where the interior sets containing both modes have the squares of their interpolation parameters appearing in the product. The interior set for the interpolation parameter just introduced contains both modes, but this parameter has just been designated as special - it is doubled instead of squared, by differentiation. Now clearly, (5.6.22)
from which we obviously get
48v(kl' k~)3!qklk;,k~k;k~ (t) where V(kl' kD
= 2 because kl i- k~.
2U(0:; t)U' (0:; t)
+ U(o:; t)
= 2(4!)2 q".(t)q"., (t),
(5.6.23)
This match yields the proportionality
ex: w(O")w(O"') - W(kl' k~, . .. ,k~)w(k;, k~, k~),
(5.6.24)
so the renormalization cancellation requires two copies of the composite expression U(o:; t)U' (0:; t). But that is exactly what the two exact links (0"',0") and (0",0"') provide. To automate the renormalization cancellation, we now collect the equivalence classes of completed terms into single terms, where the equivalence relation is that two terms are equivalent if one can be obtained from the other by some set of link replacements allowed by the observations (i-iii) above. By way of illustration, suppose a completed term is given by the sequence [0"1,0"2, (0"3,0"4, (0"~,0"5)), 0"6, (K;7,K7),0"8,0"9], where (0"3, 0"4, (O"~, 0"5)) is a chain terminated by an exact link (O"~, 0"5) and (K;7, K7) is a counter-link whose two real occurrences are both old. Denote the corresponding expression by
There are five other terms equivalent to this one in the sense we have just defined, because the replacements (O"~, 0"5) (K;7, K7)
f---t
(K;5, K5),
f---t (O"~, 0"7)
or (O"~, 0"7 )
are allowed. The idea is to collect these six terms into the product
U1 ... U4(U~U5
+ U5)U6(U7 + 2U;U7)U8U9
and denote the corresponding sequence by
[0"1,0"2, (0"3,0"4, {(O"~, 0"5), (K;5, K5)}), 0"6, {(O"~, 0"7), (0"7, O"~), (K;7, K7)}, 0"8, 0"9] . Each grouping represents a cancellation of graphs. We call each grouping a renormalized link.
5.7. THE PHASE CELL POLYMER
451
References 1. G. Battle, "Ondelettes: The Spinor QED3 Connection" Ann. Phys . 201, No. 1 (1990),117- 151. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical 4>~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981) , 327-360.
5.7
The Phase Cell Polymer
As involved as our inductively defined phase cell cluster expansion appears to be - and a certain degree of complexity is inescapable for the 4>~ Euclidean field theory - we still realize it as a polymer expansion to which the abstract theory applies. It is the polymer that is complicated and the input estimates that are hard to establish. The terms of the completed expansion are labeled by histories for which we now have a concrete description, and we have just combined those terms in such a way that renormalization cancellations are all contained in the consolidated terms. The quantity expanded is the generating functional Z(re : f. E A), and the sequence of interior sets in a given history consist of segments where the sets are increasing (in the sense of inclusion) in each segment and the maximum interior sets in these monotone segments actually partition A. Any sequence of links and chains giving rise to such a segment of the sequence shall be called a Representation 1 graph. Many can have the same monotone sequence of interior sets, while on the other hand, there is one kind of monotone segment with no Rl (Representation 1) graph - namely, a segment consisting of only one interior set. Such an interior set can only be a singleton, so a history is reflected by a sequence of isolated modes and Rl graphs, where the growth of each Rl graph is terminated by the selection of the decoupled term in the interpolation, whose interior set is therefore the maximum interior set in the monotone segment. Now, actually, the partition function for the polymer formalism is given by the normalization Zh : f. E A) = Z(rl: f. E A) (5 .7.1)
n Z{l}(re)
eEA
and the denominator is the leading term in the expansion of Z(re: f. E A) . It is the result of choosing the decoupled term in every interpolation; no situation calling for integration by parts ever arises and the interpolations never differentiate down anything, so there are no Rl graphs in the leading term. More to the point, those factors in an arbitrary term that correspond to isolated modes are cancelled out in the normalization, so the terms in the expansion of Z(re: f. E A) are labeled by sequences of Rl graphs only. Let (g1, .. . , gn) be a sequence of Rl graphs consistent with the development of a term and define the support of an Rl graph as the set of all modes occurring in the links and chains of links - as both real and ghost occurrences. This is actually the maximum interior set associated with the development of the Rl graph.
CHAPTER 5. WAVELET ANALYSIS OF j
452
Then the sets supp gi are mutually disjoint, but the union is not necessarily A, as the isolated modes are not included. However, there is another restriction: if k i is the leading mode in the development of gi - i.e., the singleton for the initial interior set then (k 1 , . . . , kn) must be in lexicographic order. Let En be the set of all such n-tuples of Rl graphs . Schematically,
~( ) ~ Z Te: £ E A = 1 + L
'" L
lIn Kgi(Te : £ E supp g')
n=1 (g', ... ,gn)Ecn i=1
n
eE supp gi
Z
( ) ,
(5.7.2)
{e} Te
and our task here is to write the factors Kgi more explicitly. To this end, we choose the notation 9 for the sequence of links obtained from an Rl graph 9 by ignoring the chain structure. Enhancing previous notation, we denote by UJ(o.; t) the differentiated unit or units associated with the jth link in g. On the basis of our organization of the expansion, we observe that UJ (0.; t) can be a composite, so we adopt the notation (5 .7.3) in that case, where UJo (0.; t) is the development of the form 1 unit initially differentiated down from the exponent and UJI (0.; t) is the development of the subsequent form 1 unit. It is important to remember that UJ(o., t) can also be the expression associated with a combination link, in which case we write
(5.7.4) where UJ+ (0.; t) is the form 1 unit following the development of the composite. If UJ(o.; t) is a renormalized expression, we let UJx (0.; t) be the counter-term, in which case we write
UJ(o.; t)
= UJo(o.; t)UJl (0.; t) + UJx (0.; t)
(5.7.5)
unless gj has two real occurrences that occurred previously in 9 (which also means cannot belong to a chain in g). Under that condition ,
gj
(5.7.6) as we have already discussed in the previous section. As far as the links themselves are concerned, it seems appropriate in the case of a composite link to use the notation
gj = (aJo,aJl)'
(5.7.7)
and in the case of a mass link we choose the notation
(5.7.8) For a combination link we write
(5 .7.9)
5.7. THE PHASE CELL POLYMER
453
In the case of a renormalized link,
gj
= {(aJo,aJl)' (II:J, ~J)}
(5 .7.10)
unless the above condition holds, and · (5.7.11) under that exceptional condition. The other types of links are the I-links, with the single occurrence automatically a ghost occurrence, and the 4-link, which has at least two real occurrences. If gj is a I-link with its single mode occurrence denoted by f gj , then the corresponding differentiated unit is just
UJ(cv.; t)
= Te,
f = f gj ,
(5 .7.12)
because a I-link can appear only at the end of a chain - i.e., a form 0 unit can be differentiated down from the exponent only by integration by parts, and the subsequent expansion step is induced by interpolation. If gj is a 4-link with ~th occurrence denoted by k" then we write (5 .7.13) This notation is appropriate because the q-factor and F-factor depend on the previous development of g, in contrast to the numerical factor, which depends only on the link. Now in the previous section we saw how every UJ(cv. ; t) can be written as this type of product - in particular, those expressions arising from renormalized links - and we now adopt some master notation accordingly. Let w(gj) be defined by (5 .7.14) if gj is a 4-link, (5 .7.15) if gj is a I-link,
w(gj)
= w(aJO)w(aJl)
(5.7.16)
if gj is a composite link given by (5.7.7) , (5 .7.17) if gj is a mass link given by (5 .7.8) with k4' ks the occurrences in II:J and kl' k2 ,k3 the occurrences in ~J, (5.7.18) if gj is a combination link given by (5.7.9), and
w(gj)
= w(a'JO)w(aJl) -
w(k 1 , ... , ks)w(kl' k2' k3)
(5.7.19)
CHAPTER 5. WAVELET ANALYSIS OF j
454
if 9j is given by either (5.7.10) or (5.7.11) with k1,k2,k3,k4 the occurrences in oJo and k1,k2,k3,k5 the occurrences in OJ1 ' Let qJ(t) denote the product of interpolation parameters in UJ(a; t) for all cases. In the case of a composite, for example, (5.7.20) where qJo(t) (resp. qJ1 (t)) is the product of interpolation parameters appearing in the differentiated form 1 unit UJo(a; t) (resp. UJ1 (a; t)). In the case of a differentiated form 2 unit, we introduced the notation (5 .7.21)
"'1
in the previous section, where, again, k4' ks and k1' k2, k3 are the occurrences in and ;;.,1, respectively. In the case of a renormalization cancellation, qJ(t) is still given by this formula. Let the notation Fl(a) be extended in the same way. In the composite case, (5 .7.22) 4
where Flo(a) (resp. Fl1 (a)) is the modification of:
IT ak:
,=1
4
: (resp. :
IT ak.
,=1
:) that
appears in the differentiated form 1 unit UJo(a; t) (resp. UJ1 (a; t)), where k~, . .. , k~ (resp . k1, .. · , k4) are the occurrences of I7Jo (resp. I7J1)' In the case of a differentiated form 2 unit, then - with k 1 , . .. , ks defined as in (5.7.17) -
Fl (a) = ak4 aks if the mass link
9j
(5.7.23)
is not a link of any chain in g, and (5.7.24)
for ~ = 4 or ~ = 5 if 9j is the last link of a chain (which is the only other possibility for a mass link). Fl(a) is given by these formulas in the case of a renormalization cancellation as well. Finally, we extend the notation for the combinatorial factor in UJ(a; t). Let (5.7.25) if 9j is a 4-link, (5.7.26) if 9j is a I-link, V(9j)
= v(k 1, . .. ,k4)V(k~, ... ,k~)
(5.7.27)
if 9j is a composite link, and V(9j) = v(k 1, ... , k4)V(k~, ... , k~)v(k~/, ... ,k~)
if
9j
(5.7.28)
is a combination link. (5.7.29)
~,
5.7. THE PHASE CELL POLYMER
455
is the combinatorial factor - again, with k 1 , .. . , k5 defined as in (5 .7.17) - for both mass links and renormalized links. We can write the Kg; factors more explicitly with this master notation. Let [. denote the set of all I-links in a representation one graph g. Consider the cardinalitie~ ng n~
= card{j: 9 is a 4-link},
= card{j: gj
(5 .7.30.0)
is a mass link},
(5.7.30.1)
= card {j: gj is a composite link}, n~1 = card{j: gj is a renormalized link},
(5 .7.30.3)
ngIV = card{'J: gj
(5.7.30.4)
n~
A
IS •
(5.7.30.2)
a comb"matlOn I'mk} .
With Ig(t; ~k2k3 (X")) 2 P BEP L 'Y~(t) L O~'k2k34>~k2k3(X")2 P BEP L'Y~(t) L 4>~k2k3(X")2 (5.8 .10) P
BEP
because BnB'
= 0.
(5.8.11)
This means that the total quantity in braces in (5 .8.7) is positive, and so from the standpoint of obtaining a lower bound we may throwaway the Wt,k2 k3-contribution. As far as the w k'k2 k3-contribution is concerned, we may obviously discard the second combination in the braces because it is manifestly negative. Thus
I~(t;4>suppg)
2
L
_,).2
jdX"W k'k 2k 3(X")
k"k2,k3Esupp 9
x{ (L'Y~(t) L o~'k2k34>B(X"))2 P
+ L'~ (t) P
2
_,).2
BEP
L 4>~ k2 k3(X")2 }
BEP
L 'Y~(t) L P
L
BEP k, ,k2,k3Esupp
x{O~,k2k34>B(X")2
j d X" Wk,k2 k3(X") 9
+ 4>~k2k3(X")2},
(5 .8.12)
where we have now applied convexity and (5.8.11) to the first combination in the braces. We must use the Wick-ordered quartic part I~ (t; 4>supp g) to dominate this negative quadratic quantity. Combining (5.8.12) with (5 .8.1) and (5.8.2) eliminates the interpolation from the problem once again. We need only show that
1'(4)B)
L j d X" Wklk,k3(X"){O~'k2k34>B(X")2 + B+A
1'(<J» B
2:
/(14
22 22) 3"<J>B-6(<J>B)O<J>B+3(<J>B)O
~A/ <J>~+CA2)1+lnL;;1)-2LkQ:k -
cAcard(B),
(5.8.14)
kEB where we are estimating as in (5.8.3) above, except that we have set aside part of AJ<J>~. With the negative part of WkJk2k3(~) dominated by its absolute value, the desired bound (5.8.13) will follow from:
f
~A <J>B(~)4d ~ _A 2 .
L / d ~ IWklk,k3(~)I<J>B(~?
kJ,k2,k3EB
2: - CA card(B),
(5.8.15)
L
cAL(1+lnL;;1)- 2L kQ:k- A2 kEB ~,h , ~ 2: -C(c)A card(B),
!d~lwkJk,k3(~)I<J>~k2k3(~)2 (5.8.16)
for c > O. We prove the inequality (5.8.15) first . The estimation of W(k1' k2, k3) is our initial concern. We have
Iw(k 1,k2,k3)1 :scLtL~2~L~J~(1+L;;J11 ~kJ - ~k31)-No 1 x (I+L;;J 1~kJ - ~k21)-N°(I+L;;211 ~k2 - ~k31)-No by virtue of (5.4.3). It follows from (5.4.14) that
(5.8.17)
(5.8.18) On the other hand, if we complete the square, the integrand in (5.8 .15) is bounded below as follows:
(5.8.19) with
L kJ ,k2,k3EB
=
288
IWkJk2k3 (~)I
:s 6 kJ,k2,k3EB: LkJ ?:Lk2?:L.3
CHAPTER 5. WAVELET ANALYSIS OF ~
460
By the Schwarz inequality together with (5 .8.18) , we also have
~ C.
, " "B,
E,
h,
,L., Iw(k" k" k, )I') ,
(5.8.21)
At the same time, (5.4.1) implies (5 .8.22) from which we may infer that
(5.8 .23)
Finally, combining this with (5.8.19) and integrating, we obtain (5 .8.15) from the Federbush stability bound (5 .1.25). We now turn to the verification of (5.8 .16). Ordering the scales of kl ' k2' k3 with the estimate (5 .8.20), we have
~k,k3CX')
= ~CX' ) =
L
Uk(a;)Ok ,
(5.8.24)
k EB Lk < L . ,
and therefore
L ~
288
J a; IWk,k,k3 (a;)I~k2k. (a;)2 d
k, ,k2,k3
L kl ,k2 ,k.
Iw(kl' k2' k3) I
J a; d
IUk, (a;)Uk2 (a;)Uk3 (a;) I~' (a;) 2
L kl ;;: Lk2 ;;: L • •
288
(5 .8.25)
5.8. STABILITY AND QUARTIC POSITIVITY
461
(5.8.26) we see that c>.
L (1 + In L;1)-2 Lkak -
>.2
kEB
L k"k2,k3
Jd;
IWk1k2 k3(;)I~1kok3 (;)2
L
2: c>'L(1+lnL;1)-2Lk(at-288>.c-1L;1(1+lnL;1)2 kEB
eEB: k k2,k3 L. , ?L'"2 ?L' 3 L.,L,L.
(5.8 .29)
L' 1 ?L' 2 ?L' 3
so the desired inequality will follow from :
Divide the sum as follows:
B~
Br
Bk == {(k 1,k2,k3 ,P) : Lk1 2: Lk2 2: Lk3 and Lk1 > Le > L k},
(5.8.31.0)
== {(k 1,k2,k3 ,P): Lk1 2: Lk2 2: Lk3 and Lk1 > Lk 2: Le > Lk3}' == {(k 1,k2 ,k3 , P): Lk1 2: Lk2 2: Lk3 2: Le and Lk1 > Lk 2: Le}.
(5.8.31.1)
For the set Bk we do not have to be particularly subtle. We estimate
(5.8.31.2)
CHAPTER 5. WAVELET ANALYSIS OF j
462
~
Iw(k 1,k2,k3;k,£)1
Jd~IIIUd~)IIUk(~)llueCX")1 3
(5.8.32)
,=1 and we apply (5.8.22) to the indices k1 and k2 in the following way:
IUk, (~)I L k,: Lk, >Lk
~
IUk2(~)1 L k2 : Lk;?Lk3
~
Thus
L~~-6LLt.Hluk,(~)1 ~ cL~~-o,
(5.8.33)
k,
L~}-6LLt6Iuk2(~)1 ~ CL;3~-6.
(5.8.34)
k2
3 Lf3Iw(k1' k2, k3; k, £)1 L (k"k2 ,k3,e)EBk
J
~ L~~-6
(5 .8.35) L L~~6 d ~ IUk3(~)llud~)llue(~)I· k3,e L,>Lk We have just taken the crucial step, as the restriction Lk, > Lk was essential. If we now apply (5.8.22) to the indices k3 and £, this inequality further reduces to: 3
Lf3Iw(k1' k2, k3; k, £)1
~ cL~~-6 l:
~
CL;1-20
L L,>L k
J ~ IUk(~)llue(~)1 d
Jd~ IUk(~)ILLtH lul(~)1 l
~
CL;1 - 20 /
dxlud~)1
CL~-20
(5 .8.36)
k
If we choose 8 < ~, we have the bound (5.8.30) on the Bk-sum. For the set B~ we apply the same crude estimate (5.8.32), and the crucial step is still to sum over k 1 :
(5 .8.37) where we have also summed over both k2 and £ by using (5.8.34). Since
J~ d
I
Uk 3 (~)lluk(~)1
L i': L t l=2- r
(1
+ L;11
~ cLtL~~ (1 + L;11 ~ k3
~f' - ~k I)-No ~ C (2~kr ) 3,
-
~k 2- r
I)-No,
(5.8.38)
~ Lk ,
(5.8.39)
5.S. STABILITY AND QUARTIC POSITIVITY
463
= k3)
it follows from (5.8.37) that (with f'
(5.8.40)
!,
Again, if we choose 8 < the bound (5.8.30) holds for the B~-sum with room to spare. For the set B~ we apply the more delicate estimate (5.4.40), but the crucial step is still to sum over k1 with respect to k and over k2 with respect to k3. We obtain:
L
L%3 min{L;!, L;3! }lw(k 1, k2, k3; k, f)1
(kl ,k2,k3,l)EB~
~ ~
l,k3: L,L.
L. 3' C
card(supp g),
(5.9.1)
where we have applied (5.8.46) to (5.7.31). The only dependence on the interpolation parameters that remains in this bound lies in the qg-factors. The preliminary step in controlling the activity (5.7.35) of the whole phase cell polymer (representation two graph) is to bound the other factors in (5.9.1) by quantities depending only on Ug . The goal in this section is to use the J dt-integrals of the qg-products to control the combinatorics of the sum over Rl-graphs in (5.7.35) . The combinatorial factors v(gj) are irrelevant to the combinatorial problems of the expansion, and we have worried about them up to this point simply because they are there. The value for a composite link can be as large as (4!)2, while the value for a mass link can be as large as 96(3!), but since there are only finitely many possible values, we can bound every combinatorial factor associated with some link by the same universal constant: v(gj) ::; c. (5 .9.2)
5.9. ESTIMATING THE ACTIVITY II: INTERNAL COMBINATORICS Note that the set of links in
465
9 of a given type depends only on Ug , so we may also write n# 9
supp 9
#
nu. '
(5.9.3)
supp Ug .
(5.9.4)
Moreover, the numerical factor w(gj) associated with a given link on how gj appears in g, but only on gj. Thus
II Iw(gj)1 = II
g. does not depend J
Iw(J)I,
(5.9.5)
JEU.
where t1 is the set of all links in U and of the chains in U . Finally, in estimating the Fg-product, we find it convenient to define the number NJ(f.) as follows: (i) If J is a 1-link, then NJ(f.) == O. (ii) If J is a 4-link, then NJ(f.) is the number of times that the mode f. occurs in that 4-tuple. (iii) If J is a composite link (a',a), then (5.9.6) where k~ is the last occurrence in a' and Nu#(f.) is defined as in (ii). (iv) If J is a mass link (11:, ii:), then NJ(f.) is the number of times that the mode f. occurs in the pair II: of occurrences. (v) If J is a renormalized link whose mass link is (11:, ii:), then NJ(f.) is defined as in (iv). (vi) If J is a combination link ((a" , a'),a), then (5.9.7) where
k~, k~
are the last two occurrences in a"
Actually, it would be more natural to define NJ(f.) simply as the number of real occurrences of f. in the link J, but this choice would also make it slightly dependent on how .J arises . Our definition of NJ(f.) over-counts the real occurrences in a manner dependent on J alone. The occurrence in a 1-link is always a ghost occurrence. For a mass link (11:, ii:), the occurrences in ii: are always ghost occurrences. A composite link (a', a) cannot even develop without integration by parts in the variable ak;'. This means k~ is a ghost occurrence in a' and that the same mode occurs at least once in a as a ghost occurrence, so one may safely subtract 2ok ;,e in (5 .9.6) for all cases without throwing away any real occurrences. Similarly, a combination link ((a", a'), a) can develop only through integration by parts in ak~ and ak~ . There may also be integration by parts in ae, where f. is the latter occurrence in {k~, k~ , .. . , kD \ {k~, kn, but (5.9 .7) does not subtract possible ghost occurrences - only those that necessarily occur. What does this have to do with estimating Fg-factors? If we introduce the obvious estimate
N':S:N:S:4 ,
(5 .9.8)
CHAPTER 5. WAVELET ANALYSIS OF iI>~
466 we may write
(5.9.9)
rlj.
The step from (5 .9.8) to (5.9.9) may treat a ghost occurrence as a for every link real occurrence, but the resulting bound depends on the link Qj itself and not on how it arises in g. Pulling all of these preliminary observations together, we may now bound the activity of a representation two graph U as follows :
Iz(U)1 ~ oX
+2
n U
t
'
card(supp U)
1 II (clw(J)I)-=----:--:II Z{l} (rl)
JEU
x /
II(1 + lall)Jfa NJ(l) exp L
\
X
+ In L/1)-2 Lla:))
lEsupp U
l
g:
(real - coX(l
lEsupp U
f=u (I] 11 dt I} qJ(t). m)
0
(5 .9.10)
Since 00 is a product expectation, the a-integration now splits into a product of one-variable integrals. For convenience, we introduce (5.9.11) so that we may write
Iz(U)1
(5 .9.12) It is the remaining sum of interpolation weights over R1 graphs that we now need to control, and this is the internal combinatorial problem for the phase cell polymer (R2 graph). To this end, we define an Mth-degree generalized ordered connectivity graph on a set A of modes to be a mapping A : {2, . .. , M} ~ P(A) with the following properties:
(i) A(2) contains the first mode in A with respect to the scale lexicon, (ii) For 2 < m ~ M, m-1
A(m) n
U A(i) i= 0,
i=2
(5.9.13)
5.9. ESTIMATING THE ACTNITY II: INTERNAL COMBINATORICS
467
m-1
A(m)\
U A(i) # 0,
(5.9.14)
;=2 M
(iii) A
= U A(i). ;=2
Obviously, an Rl graph induces a generalized ordered connectivity graph on its own support with the sets in the sequence identified as the supports of the links and chains of links in the sequence defining the R1 graph, deleting the I-links. We return to this point below. For an Mth-degree generalized o.c.g. A on A, we define the mapping TJA : {2, ... ,M}-+{I, .. . ,M-l}by
TJA
(
M)
={
1, if A(m) contains the first mode in A, min{i: A(m) n A(i) # 4>} otherwise.
(5 .9.15)
By the property (i) we have TJA(2) = 1, while (5.9.13) insures TJA(m) < m for A(m) not containing the first mode in A, so TJA is an ordered tree graph. Let (5 .9.16) with f'1(t) defined by (3.11.8), and define on Mth-degree generalized unordered connectivity graph on A as the range TA of some Mth-degree generalized ordered connectivity graph A on A. We claim (5.9.17)
for an arbitrary generalized u.c.g. T on A. The argument is a little different from the proof of the o.c.g. identity given in §3.11, but the basic strategy is the same. To each set B in T we assign a complex variable ZB, consider the function exp 2:= ZB, and our first step is the interpolation: BET
exp
L
sup
ZB
L
ZB
+
BET iortB
BET
W( Z; t)
h
L
L
ZB'
B'ET ioEB'
ZB+
BET ioEB
L
11
dt 1e W (z ;t),
(5 .9.18)
0
(5 .9.19)
ZB ,
BET iortB
where fo denotes the first mode in A with respect to the scale lexicon. The next step in the expansion of our multi-variable function is to interpolate the exponential appearing in each B'-term as follows :
exp (
L
BET iortB
ZB
+ t1
L
BET ioEB
ZB)
= exp
L
BET BnB'=0
ZB
CHAPTER 5. WAVELET ANALYSIS OF 4i~
468
+ B"ET loEB" ,B'nB"#0
(5.9.20)
+ B"ET lo rtB" ,B' nB" #0
L
WB'( Z ; t)
ZB
+
L
t2tl
ZB
BET loEB,BnB'#0
BET BnB'=0
L
+ t2
ZB·
(5.9.21)
BET lortB ,BnB'#0
The iteration of this expansion is inductively defined by pursuing an arbitrary branch. Suppose we have chosen remainder terms m - 2 times, thereby introducing the sets B', B", . .. , B(m-2): our (m -1)st step is to interpolate between the current remainder term and the expression obtained from it by keeping only those variables ZB for which m-l
U B(i-l) = 0,
Bn
(5.9.22)
i=2
and then choose either that terminal expression or one of the remainder terms generated by that interpolation. Since T has only M - 1 elements and a new element is chosen every time a remainder term is chosen in the inductive expansion, every branch must terminate in less than M steps. Thus, every term is completed and is labeled by a history - specifically, a sequence (B', B", . .. , B(m-l)) of elements of T. The point is that the term depends only on the complex variables ZB',ZB", .. . ,zB(~-l) together with those ZB for which m
Bn
UB(i-l) = 0.
(5.9 .23) i=2 If there are such B, then the connectivity property of T implies that the term cannot depend on all of the complex variables. If there is no such B, then the term depends only on Z8', ZB", .. . , ZB(~-l), in which case the term cannot depend on all the complex variables unless m = M. Hence
exp
L BET
aM - 1
ZB
IT
a ZB
exp
BET
aM - 1
IT
L
zB
BET
aZB
(
sum of all terms whose ) histories have M - 1 steps .
(5.9 .24)
BET
On the other hand, it is clear from our expansion procedure that the sequences (B', B", . . . ,B(M-I)) we have built are exactly the generalized o.c.g. A on A such that TA =
5.9. ESTIMATING THE ACTIVITY II: INTERNAL COMBINATORICS
469
T Moreover, the differentiation with respect to t m - 1 associated with the (m - l)st interpolation step in the development of A brings down the product m-2
z A(m)
IT
ti
i=1JA(m)
from the exponent for the choice of remainder term labeled by A(m) . Therefore, we may write sum of all terms whose ) ( histories have M - 1 steps =
IT ZB BET
generalized o.c.g. A TA=T
(5.9.25) where W A (z j t) denotes the final form of the exponent for the sequence of interpolations determined by A. Since this function is linear in the complex variables, we have eWA(z;t)
IT (1 + O(ZB)),
=
(5 .9.26)
BET
and so if we combine (5.9.24) with (5.9.25), we obtain 1 =exp
L ZB/
BET
z =O
(5.9.27)
We have proven the identity (5.9.16) . What application does this have to (5.9.1l)? For a given representation one graph g, we let Ag denote the induced generalized o.c.g. Clearly, TA. depends only on Ug , as it is only the set of supports of the links and chains of links in g. Also note that while Ag does not determine 9 uniquely, Ag together with Ug does. Therefore, summing over all 9 such that Ug = U corresponds to summing over all A such that TA = T For convenience we set
= fA. (t), ng(m) = nA.(m) . fg(t)
(5.9.28) (5.9.29)
Now for an Rl graph 9 consisting of M - 1 links and chains of links - again, deleting the I-links - there are M - 1 interpolation parameters tl, .. . , tM-l and M - 1 factors
CHAPTER 5. WAVELET ANALYSIS OF ~~
470
b~(t), . .. , bL_l (t), each a product of old interpolation parameters brought down from the exponent by differentiation with respect to a given interpolation parameter. Thus M
II qJ(t) = II b~_l (t)
(5.9.30)
m=2
j
because b~_l(t) is a single q9-factor in the case of a single link and a product of q9-factors in the case of a chain. On the other hand, M
f9(t)
#
II b~_l(t)
(5.9.31)
m=2
because the factor b~_l (t) is certainly not as simple as the product m-2
II
ti
i=1)g(m)
we had at the more abstract level. Recall that the interpolation of the mass counterterm was not straightforward. If the single link is a mass link or a renormalized link, then b~_l = qJ can include the square of an interpolation parameter or twice that parameter. Nor is the additional complexity confined to this issue. The square of an interpolation parameter can appear in the case of a composite link as well. In the case of a chain, b~_l can definitely be a complicated monomial, because a large number of q9-factors may comprise b~_l. An arbitrarily high power of a given parameter can appear in such a monomial, which we have never explicitly described; each q9-factor was implicitly given by the expansion procedure. On the other hand, the factor 2 can appear at most once in b~_l (t) - even in the case of a chain - and (5 .9.32)
Therefore, j-2
II
bJ-l (t) :::; 2
ti,
(5.9.33)
f9(t).
(5.9.34)
i=1)g(j)
which, in turn, implies M
II bJ-l (t) :::; 2
M
-
1
j=2
Combining this estimate with the generalized o.c.g. identity (5.9.17), we obviously obtain the bound (5.9.35)
5.10. COMBINATORICS FOR SUMMING OVER POLYMERS
471
and so we have reduced the activity estimate (5 .9.12) to Iz(U)1
(5.9.36)
References 1. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for Euclidean Field Theories," Ann. Phys. 142, No.1 (1982),95-139. 2. G. Battle and P. Federbush, "A Phase Cell Cluster Expansion for a Hierarchical ¢~ Model," Commun. Math. Phys. 88 (1983),263-293. 3. P. Federbush, "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360.
5.10
Combinatorics for Summing Over Polymers
Having eliminated representation one graphs from the bound on the activity z(U) of a given representation two graph U, we now have the freedom to pick a representation one graph gU such that (5.10.1) To fix a choice for each U is essential because we still need the notion of "old" and "new" occurrences of modes in a given link. We also need the notion of attachment, which we now define. As important as it is, this notion is simply described. Whether a given element g~, m > 1, is an individual link or a chain of links, we say the attachment of g~ is the last mode in the set m
supp g~ n
U supp g~-l i=2
with respect to the scale lexicon. In the special case m = 1, the attachment is just the first mode in supp U with respect to that order. We shall denote the attachment of g~ by e~ . Our foremost concern at this point is the combinatorial scheme for summing over the R2 graphs (= phase cell polymers). For an arbitrary R2 graph U, we can certainly pick a tree structure for this connected set of links and chains of links based on selecting some of the support intersections and ignoring others. Since the chosen tree structure still connects all elements of U, distinct representation two graphs lead to distinct tree structures because the sets of elements are not equal. (The structures may still be isomorphic.) Therefore, we can replace the sum over U by the sum over these representatives. However, we have to be careful in our selection of a representative
CHAPTER 5. WAVELET ANALYSIS OF ~
472
for each U because, at some point, the gU-dependent estimation implied above must be inserted in the combinatorial scheme. Accordingly, we define the representative in terms of gU as follows . For a given link or chain r E U we have r = g~ for some m. Let i(U ,m) = min{i : f~ E supp g~}, (5.10.2) If indeed f~ -I f~ , we regard r = g~ as connected to g~u .m) and to all g~, such that i(U, m') = m . If f~ = f~, we say r = g~ is connected to f~ and to all g~, such that i(U,m') = m . Thus we have defined a tree on the set U U {fn. Ultimately, we must establish the bound demanded by the polymer expansion formalism - namely,
Iz(U)1 ~ c
(5.10.3)
R2 graphs U fEsupp U
for an arbitrary mode
e. We focus on the more restrictive sum f SML Iz(U)1
(5.10.4)
R2 graphs U card U=M-l fEsupp U
with the intent of proving (5.10.5)
Since f. must be the fixed mode for the tree summation, we cannot choose f~ as the root of the representative tree in general, and this is a problem. Our first concern is to estimate with a sum over R2 graphs U such that f~ is the fixed mode. To this end, consider the sequence (m2, " " mv) of integers such that
st
i(U,m,+d = m" i(U,m2) = 1, mv
(5.10.6) (5.10.7)
= min{i : f. E supp g~},
(5.10.8)
and define (5 .10.9)
with pr(k) defined for r E U and k E supp U according to a few cases. If individual link other than a combination link, then
r
is an
(5 .10.10) If r is an individual combination link ((a", a'), a), then
+ (max{Le,Ld)-ll ~k E supp(a", a'),
P(u" .u,)(f)(1
f¥
Pc(k)
={ Pu(f)(1
f¥
+ (max{Le,Ld)-ll
E supp a,
-
~e 1)-3-E,
~e - ~k 1)-3-",
(5.10.11)
5.lD. COMBINATORICS FOR SUMMING OVER POLYMERS
473
where e is the last occurrence in supp an supp(a" , a'). If r is a chain, there is a finite sequence (k 1 , • • • , kN) of modes such that kl = k and kN = and
e¥
N-l
prCk)
=
II (1 + (max{Lkj,Lkj+,))-l\-;kj -
-;kHl 1)-3-, .
(5.10.12)
j=l
We postpone the description of this sequence to §5.12; all we need here is the estimate prCk)
~
(1 +
(max Lk 15,j5,N
)
)-l\-;k _ -;eur
\)-3-"
(5 .1O.13)
which follows from the triangle inequality. Extending this simple estimation for the case of a chain to the entire product in (5.10.9), we obtain (5.1O .14) The point is that Le~ ~ every length scale that arises in the journey from f back to e~ , because e~ is the first mode in supp U with respect to the scale lexicon. This enables us to estimate:
L
R2 graphs U card U=M-l iEsuPP U
(5.10.15)
x R2 gra phs U card U=M-l =e' ,iESUPP U
e':.
On the other hand,
(I
+ L i'-1 \ Xi
~
--10.
-
X i'
I)
-3-E
(5.1O.16)
~ c,
so by geometric series estimation, (5 .10.17)
Hence
L
R2 graphs U card U=M-l iESUPP U
\z {U)\ ~
C
L
SUp
e' E A
L,,?L ,
R2 graphs U card U=M-l =e' ,iESUPP U
e':.
Pu{f)-l\ Z{U)\,
(5.10.18)
CHAPTER 5. WAVELET ANALYSIS OF ~~
474
and therefore we have the desired reduction - to prove (5.10.19) R2 graphs U card U=M-l e~ =e' ,iESUPP U
for Ll' 2: Le' Now the hardest part of the estimation - done in the next section - is to control the number divergence. That control will yield an activity estimate which can be written in the form
Iz(U)1
:s du(it)! II du(r)! II rEu
w(g~, g«u,m))
II
w(g~),
(5.10.20)
m : e';!,i-e~
where the du-integers are the coordination numbers. For the combinatorics, we use this as our input estimate. Let [. denote the set of all pointed links and chains of links supported by the original set A of modes. The case-by-case determination of w(g~,g«U,m)) when e~ -j. e~ and of w(g~) when e~ = e~ is postponed to the next section, but in the meantime we extend this function w to the domain (£ x £) U £ in the following trivial way. If there is no R2 graph U such that (r, r') = (g~, g«u ,m)) and e~ (resp . e«u,m)) is the distinguished point of the link or chain g~ (resp. g«u ,m)) for some m, then w(r, r') = O. If there is no R2 graph U such that r = g~ and e~ is the distinguished point of the link or chain g~ for some m with eZ;:. = e~ , then w(r) = O. The idea is to replace the sum over R2 graphs with the sum over all rooted trees connecting M - 1 elements of £ and one mode £' E A, where that fixed £' is the root and precedes every mode in the given tree with respect to the scale lexicon. Now label the trees - i.e., introduce one-to-one mappings !1 from the set {2, .. . , M} into £ - so that we are now summing over both rooted tree graphs T and maps !1 while dividing by the number of permutations. We still need to bound Pu(i)-l with a quantity depending on!1 and T only - not on U. We can make the obvious over-estimation (5.10.21) where m = mv in the sequence defined by i and we have simply included extrapr(k)-lfactors. The factor Pg~ (e~2) in (5.10.9) has not been dropped, since i(U , m2) = 1. The loose cannon here is m defined by i, which can lie anywhere in supp U. Therefore, we must sum over m: M
PU(i)-ll z(U)I:S R2 graphs U card U=M-l e~=e' eEsupp U
I:
(5 .10.22)
m=2 R2 graphs U card U=M-l
ft.;=t' lEsupp g';:,
5.10. COMBINATORICS FOR SUMMING OVER POLYMERS
475
The bound (5.10.19) is certainly implied by the same type of bound on the inner sum, so we fix m with this constraint on i. If we let fr denote the distinguished point of a pointed link or chain r, then by (5.10.20) and (5.10.21),
L
Pu(i)-llz(U)1
R2 graphs U card U=M-l e~=e' tEsupp g~
~
(M
~ I)! (~e:
e
r
tree graphs T on
{! •...• M} rooted at 1
M
II dT(m)! m=l
one-to-one mappings 11 : {2 •...• M}-+C (l.m)ET~en(~)=f'
iEsuPP l1{m)
X
II
(
PI1{m') (fl1{m»)_l)
(m'.m)ETm
II
X
w(O(m),O(m'))
II
(5.10.23)
w(O(m)),
{l.m)ET
{m'.m)ET m'#l
where Tm is the unique path along T from 1 to m. Here we are permuting M -1 labels and the coordination numbers of the tree graph are the coordination numbers of the tree. By Cayley's Theorem, there are exactly (M - 2)! I1(d m - I)! m
tree graphs T on {I, ... , M} rooted at 1 such that dT(m) Therefore,
= dm
gd (Le')
.,e x
s~p (exp (-~e
ce Tl sup(e-te(1
-
:S ( [ : de e-te+T'~)
cALe(1
+ InL(1)-2e)
(1
+ IWJ~U NJ(l))
+ Iwnewu(l))
~
x sup(exp( -cALe(1
L:
+ In L( 1)-2e4)(1 + IWJEU
old~(e)
),
~
(5.10.30)
where the integer newu (f) counts the first 3 occurrences of f in gU if the total number of real occurrences is > 3, and all occurrences of f otherwise, while old~ (f) is just NJ(f) minus the number of these special occurrences that happen to belong to the link J as well. Since newu(f) :S 3, the first supremum is bounded by a universal constant. Combining this estimation with (5 .10.29) we reduce the activity bound (5 .9.36) to nu+2
Iz(U)1 :S A
t n~)+3nLv II (clw(J)I)ccard(supp
,=1
L:
TF
U)e'E'uPp u
JEU
L:
x
II sup(exp(-cALe(1 + InL(1)-2e)(1 + IWJEU e
old~ (l)
).
(5.10.31)
~
It is the remaining supremum that contains the number divergence. This number divergence is much weaker with the quartic exponent than it would have been with the quadratic exponent. Nevertheless, the latter exponent has just played an important role by separating out certain occurrences that will not have to be reckoned with when we extract small factors from the w(J) to implement cancellation of the number divergence. A more basic necessity met by this estimation is to save powers of the small parameter A. The point is that we pay a certain price when we use the quartic exponent - namely in negative powers of A and Le. We have
L:
sup (exp ( -cALe(1 ~
+ In Lei )-2e4)(1 + IWJEU
old~ (e)
)
CHAPTER 5. WAVELET ANALYSIS OF . in the activity bound becomes (5.10.33) as we shall see in the next section. By contrast,
(5.10.34)
5.11
Strategy for Number Divergence Cancellation
In order to reduce the sum over polymers to the combinatoric estimates (5.10.27), we used an estimate of the form (5.10.20). We have yet to realize the w-factors, and our next concern is to apply (5.10.31) and (5.10.32) to this end. Logically, the proof of (5.10.20) precedes the estimation carried out in the previous section, since we are now concerned with a delicate estimation of the polymer activity. The basic issue here is the number divergence
If ((~ Old~(e)) ,) ,/. For a given R2 graph U, define the sequence (J';;.) of attachment links as follows : (i) If g~ is an individual link, then
J';;.
= g~ .
(ii) If g~ is a chain, then J';;. is the latest link in g~ such that the attachment £~ of g~ lies in the support of the link. Let [1' be the set of all links in f] that are not attachment links , and decompose the number divergence accordingly:
(
~ old~ (e)) ! (2: old~~ (e))! JEU'
(5.11.1)
m
Small factors will have to be burned up to cancel these factorial growths . For each J E f] and e E supp U we assign a factor 'Y~ (e, L) to the Lth old occurrence of in the link J. This is the assignment of numerical factors, for which the rules must be determined. Obviously, these factors cannot be the same small size, as an exponential decay in the number of old occurrences cannot beat a factorial growth in that number. For each cancellation, the relative sizes of the factors are governed by
e
5.11. STRATEGY FOR NUMBER DIVERGENCE CANCELLATION
479
a summability condition, which is exploited by some variation of the geometric mean inequality. For the fl' -assignments the summability conditions are old~ (t)
L L
,~(e,~)4 S c
(5.11.2)
suggested by the inequality old~ (e)
II II JEU'
,~(e, ~)4
,C
(l,m)ET""ln(=>=l'
where we have applied the m-bound to every O(m)-factor - not just to the O(m)-factor. This choice eliminates the i-dependence as well as the Tm-dependence. We now reduce the problem to an estimation of single sums. The standard tree graph summation is essential here - first summing out the factors with coordination numbers equal to 1, then summing out those factors whose subsequent coordination numbers are equal to 1, etc. However, there is an extra wrinkle in the issue of "connecting back" because we have to sum w(O(m), O(m')) over all possible fn(m) is supp O(m') for fixed O(m') as well as over all possible O(m) for fixed fn(m) ' This is easily dealt with by counting the possibilities in the obvious way:
L
w(O(m),O(m')):~ Model," Commun. Math. Phys. 88 (1983), 263-293. 3. P. Federbush , "A Mass Zero Cluster Expansion," Commun. Math. Phys. 81 (1981), 327-360.
5.14
How to Assign Numerical Factors
For every enhanced link J and every mode f. E supp J, the assignment of the numerical factor to the ~th old occurrence of f. in J is yet to be made. This is highly casedependent, but it is clear from (5.13.12) that we can make the universal assignment (5.14.1) without loss of generality. The challenge to the reader is to find a minimal set of rules that will cover all of the many cases to be reckoned with. In this section we illustrate the nature of the game by assigning numerical factors in a few of the cases. Consider the species given by Fig . 5.13.1. There are only two old occurrences counted by XJ, and one of them is k J . Thus (5.14.2)
(5.14.3) Combining this with (5.13 .9), we must establish (5.14.4) Fig. 5.13.1 links J kJ=k'
with the choice of , (k 1 ) constrained by the requirement (5 .14.5) lEA Fig . 5 13.1 links J kJ=l.kl=f
In this case we choose (5.14.6)
5.14. HOW TO ASSIGN NUMERICAL FACTORS
505
because on the set of enhanced links we sup over, we want
-'{j(kd
= (1 + (max{Le,Ll})-II-;e
-
-;ll)-~+ (~!) ~ +
(5.14.7)
(5.14.5) now follows from the basic estimates
L
(1
+ (max{Ll,Ll})-ll-;e
- -;[1)-3+
l: L i =2- r
(5.14.8)
L
(2-rLil)"~C
(5.14.9)
r: 2- r $L ,
already discussed in §5.4. Does the extraction (5.14 .6) from w(J) leave enough smallness for (5.14.4) to hold? To answer this question, we insert (5.14.3) and (5.14.6) in (5.13.9) to obtain f"J
x
[g(l +
(m=(L,,,L',,W'1 C;" - C;",
-~ +
1+
x lw(J)ILkJ4 Lkl (1
+ (max{Lk"LkJ})-
1
1)'H1'
-I.
I
Xk , -
-~
XkJ
In the summation, we sum over all of the occurrences except kJ = k' over kl and k2' exploiting the estimates
3+
1)4 .(5. 14.10) First we sum
k, : L k ,=2- r
2- r 2': L kJ ,
L
(2r LkJ)" ~ c.
(5.14 .11) (5.14.12)
r: 2 - r~LkJ
Then we sum over k4 ' exploiting the estimates (5.14 .8) and (5.14.9) with the replacements f. H kJ, I H k 4 . Now, in this game, there are always enough long-distance decay factors in w(J) for every purpose; we need only to adjust the construction parameter of the wavelet to obtain enough vanishing moments for the Bessel potentials of the wavelets to have the degree of polynomial decay required. For this reason, we can focus on length scales only , with the understanding that when we sum over smaller-scale modes k relative to a fixed mode k', we "use up" the small factor
CHAPTER 5. WAVELET ANALYSIS OF .~
(
II
Tl ) L;} - ' Lk3'
e
tEsupp J
x [_
~ II
(1
+ (max{LJ.:L;;;})-11 :l" - :l"k
1)3+,]2 IW(J)I .(5.14.23)
k . kEsupp J
By (5.13 .13.2) it follows that
c = 0(L 5- , L -1 L -~-, L -1 L -}) k3 k2 kJ k. k '
.~ ( II
Lt-' Lt;-' L~}-' L-;;: L-;;~'
eT ; )
lEsupp J
X [_=
II
(1
+ (max{L k ,L;))-II-;k - -;;;; 1)3+,]2
k , kEsupp J2
X [_
=
II
(1
+ (max{LkJ,Lk,})-II-;kJ - -;k,
1)3+,]2 IW (J)1
k,kEsupp J 1-'-
-.l.
3+
+ (max{Lk" L kJ })- Xk, - XkJ 1)4 1-'--" 3+ x(l + (max{Lk;,LkJ})- Xk; - XkJ 1)4 ,
x(l
1
(5 .14.43)
1
where J 1 and J2 are the 4-link and composite link comprising 1. By (5.13.13.5), it follows that (5.14.44)
Now, in general , we handle a combination link by summing first over the possibilities for the constituent link other than the one whose support contains k J . In this example it means summing over the possible J 1 = (k 1 , . .. , k 4 ) with k3 fixed. If we sum over kl and k2 relative to k 3 , we obtain
and subsequent summation over k4 yields
Since kJ succeeds k~ and precedes k~ in this case, our next step is to sum over k~ relative to kJ to get 0(L 1 - ' L 5 ; , L -,I L -} L -~- ' ). k3
k4
k3
Now since k3 = k 2 , we now sum over k~ , k3 obtain the order of magnitude
This is all we need in this case.
k2
= k3'
kJ
and k2 relative to k~
= k~
to finally
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