Non-perturbative quantum field theory : mathematical aspects and applications

INON PERTURBATIVE QUANTUM FIELD THEORY Mathematical Aspects and Applications

ADVANCED SERIES IN MATHEMATICAL PHYSICS

Editors-in-Charge H Araki (RIMS, Kyoto) V G Kac (MIT) D H Phong (Columbia University) S-T Yau (Harvard University) Associate Editors L Alvarez-Gaume (CERN) J P Bourguignon (Ecole Polytechnique, Palaiseau) T Eguchi (University of Tokyo) B Julia (CNRS, Paris) F Wilczek (Institute for Advanced Study, Princeton)

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Mathematical Aspects of String Theory edited by S-T Yau

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Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras by V G Kac and A K Raina

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Kac-Moody and Virasoro Algebras: A Reprint Volume for Physicists edited by P Goddard and D Olive

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Harmonic Mappings, Twistors and o-Models edited by P Gauduchon Geometric Phases in Physics edited by A Shapere and F Wilczek

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Braid Group, Knot Theory and Statistical Mechanics edited by C N Yang and MLGe Vol. 10: Yang-Baxter Equations in Integrable Systems edited by M Jimbo Vol. 11: New Developments in the Theory of Knots edited by T Kohno Vol. 12: Soliton Equations and Hamiltonian Systems by L A Dickey

Advanced Series in Mathematical Physics

Vol. 15

NON PERTURBATIVE QUANTUM FIELD THEORY Mathematical Aspects and Applications

Selected Papers ofJurg Frohlich

fe World Scientific •

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We thank the following publishers for permission to reproduce the papers in this volume: Academic Press [III. 1]; American Mathematical Society [1.2], [V1.6]; Elsevier Science Publishers [II.3], [III.4], [IV.l], [VI.3], [VI.7]; Kluwer Academic Publishers [III.5], [VI.2]; Plenum Publishing Corp. [III.3], [VI.l]; Springer-Verlag [1.1], [1.3], [11.1], [II.2], [II.4], [V.2], [V.4]. While every effort has been made to contact the publishers of reprinted papers prior to publication, we have been unsuccessful in some cases. Where we could not reach the publishers, we have cited the source. Proper credit will be given in future editions of this work after permission is granted.

NON-PERTURBATIVE QUANTUM FIELD THEORY MATHEMATICAL ASPECTS AND APPLICATIONS. Selected Papers of Jiirg Frohlich. Copyright a 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyanymeans, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 981-02-0432-9 981-02-0433-7 (pbk)

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V

FOREWORD In J a n u a r y 1956, in an address to the All Union (USSR) Conference on Functional Analysis and Its Applications, I. M. Gelfand said: "Two areas, in my opinion, will exert the strongest influence of all on the future course of development of functional analysis. T h e first of these is hydrodynamics (the problem of flow of a viscous fluid, the theory of a compressible gas, the theory of turbulence). T h e second area is theoretical physics, more precisely, the q u a n t u m theory of fields and the theory of elementary particles. This region of physics, it is true, itself stands at a crossroads and it is not clear what the nature of its development will be, b u t , however it develops, one thing is clear: it and functional analysis are to be fellow-travellers." T h e three and a half decades which have since elapsed have seen m a t h e m a t i c a l bones p u t into the flesh of the second part of this pronouncement. First, several mathematically precise definitions of what should constitute a q u a n t u m theory of fields were given and a general theory of quantized fields developed. T h e initial stages of this development took roughly a decade (from the mid-1950s to the mid-1960s), but work on the general theory continues to this day. T h e creation of this theory almost immediately paid off in a strong and fruitful interaction with the theory of von Neumann algebras, a core area in non-commutative functional analysis. T h e next decade (from the mid-1960s to the mid-1970s) saw the initial stages of development in what has come to be called constructive q u a n t u m field theory: the m a t h e m a t i c a l construction of fields t h a t are solutions of specific Lagrangian quant u m field theories. T h e first examples treated {\, P()2, Y2, $3 in the shorthand of the subject) were deliberately chosen so as to eliminate all inessential complications. T h e y were therefore of somewhat limited physical interest. However, these examples did answer the question of whether non-trivial theories of interacting fields exist. T h e answer was yes, at least in spacetimes of dimension two and three. T h e next logical step might appear t o be the construction of theories in four dimensional spacetime, b u t t h a t is not in fact what happened; the history in the decade and a half leading up to the present have been more complicated. On the one hand, it has been shown t h a t the methods used so successfully in spacetime dimensions two and three to construct the theories \ and $\ of a self-interacting scalar field, <j>, fail for 4. On the other hand, q u a n t u m field theory in two and three dimensions has turned out to be applicable to a rich variety of physical phenomena. T h e papers of Jiirg Frohlich collected here contain many i m p o r t a n t original contributions to the developments of the last decade and a half, b u t what may make t h e m even more useful to the reader is the remarkable overview they provide. It would b e fruitless and r e d u n d a n t t o spell this out in detail here, since the author

vi himself has provided an admirable commentary in the following introduction. It remains only to wish the reader many happy hours reading this book.

Princeton University

Arthur S. Wightman

Vll

CONTENTS Foreword

v

Introduction

1

I Phase Transitions and Continuous S y m m e t r y Breaking [1.1] "Phase transitions, Goldstone bosons and topological superselection rules", in Schladming Lectures, 1976, P. Urban (ed.); Acta Phys. Austnaca Suppl. 15 (1976) 133-269.

27

[1.2] "The pure phases (harmonic functions) of generalized processes" or "Mathematical physics of phase transitions and symmetry breaking", Bull. Am.tr. Math. Soc. 8 4 (1978) 165-193.

164

[1.3] "Spontaneously broken and dynamically enhanced global and local symmetries", in Unified Theories of Elementary Particles. Critical Assessment and Prospects, P. Breitenlohner and H.P. Diirr (eds.), p p . 117-136, Springer Lecture Notes in Physics 1 6 0 .

193

II

N o n - P e r t u r b a t i v e Q u a n t i z a t i o n of T o p o l o g i c a l S o l i t o n s

[II.1] " Q u a n t u m theory of non-linear invariant wave (field) equations" or "Super selection sectors in constructive q u a n t u m field theory", in Invariant Wave Equations, G. Velo and A.S. W i g h t m a n (eds.), pp. 339-413, Springer Lecture Notes in Physics 7 3 .

215

[II.2] with P.-A. Marchetti, "Bosonization, topological solitons and fractional charges in two-dimensional q u a n t u m field theory", Commun. Math. Phys. 116 (1988) 127-173.

290

[II.3] with P.-A. Marchetti, " Q u a n t u m skyrmions", Nucl. B 3 3 5 (1990) 1-22.

337

Phys.

[II.4] with M. Struwe, "Variational problems on vector bundles", Commun. Math. Phys. 1 3 1 (1990) 431-464. III

359

G a u g e T h e o r i e s , i n c l u d i n g ( t h e Infrared P r o b l e m in) Quantum Electrodynamics

[III.l] with G. Morchio and F . Strocchi, "Charged sectors and scattering states in q u a n t u m electrodynamics", Ann. Phys. (N.Y.) 119 (1979) 241-284.

395

Vlll

[III.2] with P.-A. Marchetti, "Magnetic monopoles and charged states in four-dimensional, abelian lattice gauge theories", Europhysics Letters 2 (1986) 933-940.

439

[III.3] "Some results and comments on quantized gauge fields", in Recent Developments in Gauge Theories, G. 't Hooft et al (eds.), p p . 53-82.

447

[III.4] "Some comments on the crossover between strong and weak coupling in SU(2) pure Yang-Mills theory", Phys. Rep. 6 7 (1980) 137-149.

477

[III.5] with P.-A. Marchetti, " Q u a n t u m field theory of anyons", Lett. Math. Phys. 16 (1988) 347-358.

490

T r i v i a l i t y of \cp4

IV

[IV.1] "On the triviality of \ip\ theories and the approach to the critical point in dfaA dimensions", Nucl. Phys. B 2 0 0 [FS4] (1982) 281-296. V

R a n d o m G e o m e t r y (Quantum Gravity and Strings)

[V.l]

"Regge calculus and discretized gravitational functional integrals", I.H.E.S. Preprint (1981).

[V.2] "The statistical mechanics of surfaces", in Applications of Field Theory to Statistical Mechanics, L. Garrido (ed.), p p . 31-57, Springer Lecture Notes in Physics 2 1 6 . VI

505

523

546

Low-Dimensional Q F T : Two-Dimensional Conformal Field Theory, Three-Dimensional (Gauge) Theories

[VI.1] "Statistics of fields, the Yang-Baxter equation, and the theory of knots and links", in Nonperiurbative Quantum Field Theory, G. 't Hooft, A. Jaffe, G. Mack, P.K. Mitter, and R. Stora (eds.), pp. 71-100.

575

[VI.2] "Statistics and monodromy in two- and three-dimensional q u a n t u m field theory", in Differential Geometric Methods in Theoretical Phystcs, K. Bleuler and M. Werner (eds.), pp. 173-186.

605

[VI.3] "On the structure of (unitary) rational conformal field theory", in Proceedings of the LAPP Meeting on Conformal Field Theory, Nucl. Phys. B (Proc. Suppl.) 5 B (1988) 110-118.

620

[VI.4] "New developments in q u a n t u m field theory", in Proceedings of the XXIV International Conference on High Energy Physics, R. K o t t h a u s and J. K u h n (eds.), pp. 219-249.

629

IX

[VI.5] with C. King, "Two-dimensional conformal field theory and three-dimensional topology", Internal. J. Modern Phys. A 4 (1989) 5321-5399; also in Proceedings of the Erice Summer School, 1988, G. Velo and A.S. Wightman (eds.).

660

[VI.6] "Quantum statistics and locality", in Proceedings of The Gibbs Symposium, D.G. Caldi and G.D. Mostow (eds.), pp. 89-142.

739

[VI.7] with T. Kerler, "Universality in quantum Hall systems", Nucl. Phys. B354 (1991) 369-417.

793

1

INTRODUCTION 1 . This book is a collection of some of my articles on non-perturbative aspects of q u a n t u m field theory. More t h a n half of the papers reprinted here have a review character. Most of my review articles and many of my papers tend to be written under pressure of time and are therefore far from perfect. T h u s , my ambition is, in general, limited to describing some ideas t h a t I feel are beautiful or i m p o r t a n t , or b o t h , and the main results t h a t can be derived from them. It happens quite often t h a t , in the course of the writing, I get sidetracked on a detour which, it later t u r n s out, could have been avoided. Only rarely have I been able to do some rewriting or to pursue some scholarly ambitions going beyond the more obvious purposes of an article. T h e reader will notice t h a t , for all these reasons - and, most certainly, for reasons rooted more deeply in my personality - most of the papers collected in this book have the character of sketches, even though they are sometimes rather elaborate and lengthy. I hope and believe t h a t , in favourable cases, my sketches reproduce some of the main contours of what is being portrayed. However, the interested reader should take some time to rearrange the main lines of my drawing, or t u r n to the original sources, in order to understand more clearly what the main ideas and the main purpose of an article are.

2 . In order to preserve visual impressions and in order to try out new techniques or improve old ones, artists usually fill many sketch books with more or less rough sketches. W h a t is being published here is in many ways analogous to an artist's sketch book. One should ask whether such imperfect sketches are of interest to the public. I a m not sure what the right answer is. I suppose I leave the answer to the publisher - and to the reader. Artists usually try to convert those sketches which they feel are really good and interesting into more or less perfect paintings. I sometimes try to do the same with some of my sketches. But often I do not succeed, either because I find it too difficult, time-consuming or painful to a t t e m p t to transform a scientific sketch into a scholarly work of science, or because other people can do it better, in a shorter time, t h a n I could. Sometimes t h a t can b e a little awkward. T h e reader may permit me one last, more general comment on the relationship between art and basic science. I think t h a t artists and scientists engaged in basic research share closely related concerns, and t h a t the motivation for their efforts and their successes originate from inter-related sources. They b o t h contribute to the development of general h u m a n culture, albeit most often in very modest ways. But the cumulative effect is significant! T h e question of whether what they do is useful and applicable is and should remain secondary. At the root of their efforts, intuition, insights and visions of relatively little precision and of sometimes quite irrational n a t u r e play an important role. They are not only intellectually, b u t most

2 often also emotionally involved in their work. T h e y tend to be fairly ambitious, often rather egocentric people, and they are driven by forces other t h a n those of becoming wealthy rapidly, or of having a comfortable and leisurely life. In the course of trying to implement an idea they tend to have the experience t h a t their work develops its own dynamics, partially gets out of their control, forces t h e m around corners or guides t h e m through hidden doors to open the view to unexpected, new scenery - as if ideas had some pre-existence outside their minds, and the process of their incarnation followed rules which they had not set themselves. Taken together and averaged over sufficiently long periods in time, the contributions of artists and scientists to the evolution of culture and civilization are clearly i m p o r t a n t . Reducing art to purely functional or decorative purposes would eventually suffocate art - as experience during certain periods in the history of art would, I believe, tend to confirm. Reducing science to merely "useful" or applied science would eventually cut it from its life roots. It would surely turn out to be a disservice also to applied science. (I hope society is not in the process of testing this prediction by experiment.)

3 . Many papers reprinted in this book represent the fruit of a collaborative effort in which my collaborators and friends have played fundamental roles. W h e r e my review articles are based on such collaborations the original papers may well be more perfect t h a n my reviews, and the interested reader is advised to consult the original papers. Furthermore, some of my best work, often in collaboration with T . Spencer, does not concern q u a n t u m field theory, but areas like equilibrium statistical mechanics, disordered systems theory, dynamics of systems with many degrees of freedom and non-relativistic q u a n t u m theory. It is hardly represented in this volume. It tends to be rather more professional t h a n most of the articles on the following pages. Finally, I do not consider the articles reprinted in this book to represent anything like my "collected works" on non-perturbative q u a n t u m field theory - I still try to do work in t h a t area, from time to time. T h e reader should keep these remarks in mind! Also, my selection of papers is personal and does not cover some of the most important chapters in nonperturbative q u a n t u m field theory. I hope those chapters will appear in similar reprint collections by authors more competent to write t h e m t h a n I.

4 . I now t u r n to some brief comments on the different parts of this book and on some of the papers, and to some indications of relations between them. P a r t I, "Phase Transitions and Continuous Symmetry Breaking", describes work carried out in collaboration with R. Israel, E.H. Lieb and, primarily, with B. Simon and T . Spencer. Our main aim was to understand phase transitions accompanied by the spontaneous breaking of continuous symmetries in systems like multi-

3 component scalar field theories in three dimensions and (classical) Heisenberg ferromagnets on three and higher dimensional lattices. Thinking of phenomena like chiral symmetry breaking in Q C D , it becomes clear t h a t a mathematically precise study of such phase transitions is important for q u a n t u m field theory. We wanted to understand mechanisms driving phase transitions with spontaneous symmetry breaking and the associated emergence of Goldstone bosons in a mathematically precise way. More or less known to us were heuristic analyses of the spontaneous breaking of continuous symmetries in models of q u a n t u m field theory and statistical mechanics (droplet picture, spherical model, 1/N-expansion), some proofs of the Goldstone and the closely related Mermin-Wagner theorems [1,2], some forms of spin-wave theory and of Bose-Einstein condensation. After some futile a t t e m p t s , trying, for example, to p u t the droplet picture on a firm basis, it eventually turned out t h a t the right idea was t o try to establish an analogue of the Kallen-Lehmann spectral representation for the connected two-spin correlation function, with a t e m p e r a t u r e dependent b o u n d on the spectral measure t h a t , in relativistic q u a n t u m field theory, would correspond t o a b o u n d derived from canonical commutation relations, with Planck's constant, fi, and t e m p e r a t u r e , T, playing analogous roles. Such a result would yield bounds on the connected two-spin correlation function which, when combined with a sum rule (e.g., Sx • Sx = 1, in a classical Heisenberg ferromagnet), implies t h a t the full two-spin correlation function, < ST • Sy >T, exhibits long range order for sufficiently small values of the t e m p e r a t u r e T . Adaptations of the arguments of Mermin and Wagner [2] then turned out to prove the presence of a Goldstone boson in various more or less equivalent forms. It was found t h a t the simplest proof of suitable bounds, so-called infrared bounds, on the connected twospin correlation could be p a t t e r n e d on some results of Glimm and Jaffe [3] known by the n a m e "ir- and Vy>-bounds" in canonical q u a n t u m field theory. There is no doubt t h a t the concept of Osterwalder-Schrader, or reflection positivity - which had first been introduced and exploited in [4] - played an imp o r t a n t role in our thinking and in many subsequent applications. In fact, our proofs of infrared bounds are based on the hypothesis of reflection positivity or, equivalently, of existence of a positive transfer matrix. (This represents a limitation of our m e t h o d s which is also present in later generalizations. Reflection positivity is completely n a t u r a l in the context of Euclidean (lattice) field theory, but far less n a t u r a l in statistical mechanics.) Our first results (by B . Simon, T . Spencer and myself) appeared in [5]. They were subsequently developed into a fairly comprehensive theory of phase transitions with continuous symmetry breaking, with contributions by Dyson, Israel, Lieb, Simon, Spencer and myself. This may become clear from the reading of the second and third papers of P a r t I, and references there. Although the work described in P a r t I is conceptually not terribly original, it has the advantage of being quite elegant and technically transparent. Moreover, it t u r n e d out t o have many important implications. Besides offering one ascent - there are some others - to a mathematically rigorous understanding of phase transitions, continuous symmetry breaking and its relation to the presence of

4 massless excitations, the Goldstone bosons, in a class of models of physical interest, our work found applications in the study of the following problems: (1) Existence of a critical point in some classical lattice spin systems [6]. (2) Bounds on field strength renormalization, with implications for the triviality problem, in four-dimensional quantum field theories, such as A^-theory and QED4. This is discussed in Part IV, "Triviality of Xtpf', and references given there. Some remarks are also contained in the first paper of Part I (bounds on two-point function of field strength in abelian lattice gauge theory, etc.). (3) Study of the deconfining phase transition in lattice gauge theories at positive temperature and of chiral symmetry breaking in some strongly coupled lattice gauge theories by C. Borgs, M. Salmhofer and E. Seiler [7,8]. Although the work in [5] is mathematically complete, our arguments are simpler than many of the heuristic analyses of the same phenomena. One might have expected, therefore, that they will find their way into most new textbooks on phase transitions and critical phenomena, or on field-theoretic methods in statistical mechanics. Actually, that expectation does not appear to have been fulfilled, and, therefore, it makes sense to reprint some reviews of our work. After the work summarized in the first two papers of Part I had been completed, Tom Spencer and I went on to analyze, using mathematically rigorous methods, the Berezinski-Kosterlitz-Thouless transition [9] in two-dimensional models such as the two-component Coulomb plasma (transition from the plasma phase with Debye screening to the dipolar phase without screening) [10] and the classical XY-model, the transition in the one-dimensional Ising model with l/r 2 -interaction energy [11], and the deconfinement transition in the four-dimensional [/'(l)-lattice gauge theory [12]. This work is referred to in the third article of Part I, but see [13,14,15,16] for the original results. It is reviewed in another book [17] and has been elaborated upon, extended and simplified in numerous subsequent papers. Our results are the basis of the work described in the second paper of Part HI ("Magnetic monopoles and charged states in four-dimensional abelian lattice gauge theories", with P.A. Marchetti). On a more conceptual level, the work of Spencer and myself mentioned above led us to propose a rather precise form of the concept of "asymptotic enhancement of symmetries" which is discussed in some detail in the third paper of Part I. Clearly, this concept is of obvious importance in "non-perturbative quantum field theory". It also illuminates in a useful way the nature of the transition from the plasma phase to the massless Kosterlitz-Thouless phase in the two-dimensional, classical two-component Coulomb gas (enhancement of a spontaneously broken discrete symmetry Z, to a spontaneously broken continuous symmetry, R), of the deconfining transition in the four-dimensional, compact U(l) and Z„ lattice gauge theories, and of many related phenomena. The best known (and least understood) example of symmetry enhancement is the enhancement of lattice symmetries to full Euclidean symmetries in the continuum limit of lattice field theories. Symmetry enhancement at very high energy scales has, of course, been discussed in connection with grand unified theories of elementary particles.

5 The third article of Part I also discusses the "spontaneous breaking" of local gauge invariance and sketches a gauge-invariant formulation of the Higgs mechanism and the role of field configurations with topological defects in gauge theories, based on joint work with G. Morchio and F. Strocchi.

5 . The first article of Part I contains a rather long section on the quantum theory of topological solitons which is the main topic of Part II, "Non-Perturbative Quantization of Topological Solitons". I was working on the problem of soliton quantization in two dimensional quantum field models with kinks, in 1974/1975, just before and during my work with B. Simon and T. Spencer on phase transitions. The last part of the first paper in Part I, the first paper in Part II, the third paper in Part III, and the first paper in Part VI provide somewhat complementary introductions to the theory of superselection sectors and topological charges in quantum field theory, in particular to the quantum theory of topological solitons. In this area I have made what I believe are among my more original contributions to non-perturbative quantum field theory to-date. In the course of my work on solitons, in 1975, I found examples of "dual algebras", or "exchange algebras", of fractional charges, of "parafermions", in the sense of Zamolodchikov (but without conformal invariance), and, a few years later, of the connection between topological solitons and topological (line) defects in Euclidean functional integrals. I do not think that any of this work - with the exception of the first paper in Part VI - became well-known. So, on hindsight, one should ask: what went wrong? Presumably, I knew too little about the physics of quantum fields to draw clearly explicit physical consequences from my analytical findings, and I knew too little mathematics to understand that there were some interesting, general mathematical structures to be uncovered. But it was also a question of style. My results were embedded in tedious, lengthy mathematical analyses which many people find somewhat inaccessible and, perhaps, somewhat beside the point. Some of the most original ideas remained unpublished, because I could not implement them in the form of mathematically rigorous constructive field theory. I was working on problems concerning superselection sectors and topological charges in quantum field theory during three different periods, from the end of 1974 till the middle of 1977, in 1979, and, on and off, from 1984 till the present. In recent years, my efforts were supported or superceded by those of P.A. Marchetti, G. Felder, G. Keller, F. Gabbiani and T. Kerler. One goal during the first period was to study two-dimensional models exhibiting topological solitons as examples for the deep axiomatic results of Doplicher, Haag and Roberts [18] concerning superselection rules and quantum statistics; to better understand the connection between vacuum degeneracy and symmetry breaking, on one hand, and the existence of soliton sectors with non-trivial topological charge in such models, on the other hand; and to put the semi-classical analysis of quantum solitons then carried out by many distinguished physicists, see [19] and refs. given there, on a non-perturbative basis. All this led, in a natural way, to a primitive

6 notion of "exchange algebra" (see e.g. [20], formula (71); see also §6 of [20]) which I found before I learnt about the work of Kadanoff and Ceva [21], and Wegner [22]. Exchange algebras are algebras generated by unobservable, charged fields, ipa, (or by bounded functions of such fields) with quadratic relations, e.g. of the form rpa(x,t)ipp(y,s)

= R(±)lSpip1(y,s)ip6(x,t)

,

for (t — s)2 — (x — y) 2 < O and x ^ y. Here (x,t) and (y,s) are points in two-dimensional Minkowski space; the order relations x > y or x < y are then Poincare-invariant, for space-like separated points. The matrices R(±) turn out to be Yang-Baxter matrices. Such unobservable fields play an important role in analyzing the statistics and scattering of charged particles. The ansatz just described has been criticized for valid reasons, but is perfectly correct and general for field algebras with abelian fractional statistics. The models studied in the seventies gave rise to such field algebras - examples t h a t one now regards as somewhat trivial. Nevertheless, there was something a little new in the idea that, in two space-time dimensions, charged fields need have neither Bose nor Fermi statistics, but could have what is now called fractional statistics interpolating between Bose and Fermi statistics. Exchange algebras have a precursor - the algebra of Weyl operators, satisfying the quadratic Weyl relations W(a,b)W(a',b')

= ei(-ab'-a'^W(a',b')W(a,b)

W(a,b),

,

with W(a,b)

= expi[a

• p + b • q] ,

where q and p satisfy Heisenberg's canonical commutation relations. The Weyl relations for systems with infinitely many degrees of freedom and their representation theory (developed by Garding and Wightman, and by Segal) are at the basis of canonical q u a n t u m field theory. The free, massless field in two space-time dimensions is an example of a canonical field theory. This field theory describes left and right-moving waves created from the vacuum by what has become known as vertex operators, e.g., Vx{x,t) = Nexpi\[Tr(Q±,t) + 4>{x,t)], where ©^(jT) = 1, for j / ^ x,= 0, otherwise, and where N denotes normal ordering. Bounded versions of such vertex operators were first studied by Streater and Wilde [23] as examples for axiomatic results of Doplicher, Haag and Roberts. The analysis of Streater and Wilde was inspired by the seminal work of Skyrme on bosonisation in two dimensions [23]. Their results were extended in [20] (see also the second paper of P a r t II) to interacting field theories, including the sine-Gordon model. In 1979 and in the eighties, then in collaboration with P.A. Marchetti, I became interested in the analysis of quantized solitons within the framework of Euclidean

7 functional integrals. It became clear in 1979 that the existence of topological solitons in some q u a n t u m field model is intimately connected with the existence of topologically stable line defects in Euclidean field configurations, such as domain boundaries (contours) in the A^ 4 -theory in two dimensions, or vortices in threedimensional Higgs models. This was first discussed in the third paper reprinted in Part III. Soliton Green functions can be constructed by minimally coupling the Euclidean matter fields of the model to an external gauge field whose curvature has singularities in points of Euclidean space-time. In every field configuration there must be open line defects connecting those space-time points. This method has been further developed in the work of Schroer, Swieca and collaborators [25], and in the work of Marchetti and myself; see the second and third article of P a r t II, and the second and last article of Part III. The masses of solitons turned out to be given in terms of corresponding defect free energies. Such a connection had been envisaged by Glimm and Jaffe for the two-dimensional Ising model and was first made precise for the kinks of the Xtp^-model in [24], in 1977. This is discussed in detail in the first article of Part II. T h a n k s to M. Struwe, the analysis of soliton Green functions, expressed in terms of Euclidean functional integrals, with the help of semi-classical methods has recently led to a decent piece of mathematics reprinted as the last paper of P a r t II. Through studying the work of Kadanoff and Ceva [21] and of Jimbo, Miwa and Sato [26], it had become clear in 1979 t h a t dual (or exchange) algebras for order and disorder fields were in direct correspondence with non-trivial monodromy properties of their Euclidean Green functions. This was also noticed by Schroer, Swieca and collaborators. These findings (largely unpublished) turned out to have been a useful preparation when two-dimensional conformal field theory was reborn in 1984 [27]. The work of Belavin, Polyakov and Zamolodchikov [27] and of Knizhnik and Zamolodchikov [28] on conformal field theory and of Jones [29] on knot theory provided the motivation and justification to finally systematically develop the idea of exchange algebras and their relation with non-trivial monodromy properties of Euclidean Green functions and with the representation theory of the braid groups.1

6 . Apart from the examples of exchange algebras that had appeared in the q u a n t u m theory of topological solitons, as discussed above, exchange algebras in the context of two-dimensional conformal field theory first appeared in the work of Gervais and Neveu on the quantization of Liouville theory [30]. Their work did not receive the attention it deserved. Further important work was carried out later, independently, by Tsuchiya and Kanie [31], Rehren [32] and myself [33] (a summary of my results of t h a t period is contained in the first paper reprinted in Part VI). In retrospect, some of the developments around this theme, between 1987 and 1990, can perhaps best be characterized by recalling the following quotation from Blaise Cendrar's "L'or": 'Reverie. Calme. Repos. / C'est la paix. / Non. Non. Non. Non. Non. Non. Non. Non. Non. / C'est l'or. / Le rush.' (I believe that a few people did find some gold - but some found less precious metals.)

8 My own efforts were also motivated by an analysis, due to Marchetti and myself, of anyons in Chern-Simons-Higgs models, reprinted as the last paper of Part III, t h a t we had started in 1986, under the influence of talks by Y.S. Wu and G. Semenoff on related matters which we had attended, and motivated by a desire to understand some aspects of the fractional quantum Hall effect. While the work of Gervais and Neveu, Tsuchiya and Kanie, and Rehren specifically addressed two-dimensional conformal field theories, the work of Marchetti and myself on anyons and my study of exchange algebras in two-dimensional theories suggested t h a t these algebras represented a concept reaching beyond the realm of conformal invariance, t h a t similar mathematical structures would reappear in threedimensional gauge theories, and that the algebraic methods of Doplicher, Haag and Roberts might be adequate for a general, model-independent analysis. These ideas are developed systematically in [34] and, for three-dimensional theories, in [35], and in several more recent papers. (The general structure in three space-time dimensions is now understood, while the most general version of field algebras in two-dimensional theories is still unknown.) The role played by exchange algebras in two-dimensional conformal field theory, generated by chiral vertex operators, was investigated in detail by Rehren and Schroer [36]; Moore and Seiberg [37]; Felder, Keller and myself [38]; Babelon [39]; King and myself [40]; and many other people. All these developments are reviewed in the first six papers reprinted in P a r t VI, where many references to the original literature can be found.

7 . Fractional statistics was not only found in the analysis of field statistics in two and three dimensional quantum field theory, but turned up in a study of particle statistics in q u a n t u m mechanics in 2 + 1 space-time dimensions, due to Leinaas and Myrheim [41], in 1977. They considered point-like particles carrying electric charge and magnetic flux and noticed that, as a consequence of the Aharonov-Bohm effect, such particles exhibit fractional statistics under exchange of positions. This work went unnoticed, and the ideas of Leinaas and Myrheim were rediscovered, a few years later, by Goldin, Menikoff and Sharp [42] in their study of current algebra in non-relativistic physics, and by Wilczek [43]. It deserves to be mentioned that Goldin, Menikoff and Sharp made the connection between fractional statistics in two dimensions and representations of the braid groups, whence the currently used expression of "braid statistics". This was popularized in [44] which was the paper from which I originally learnt about the braid groups. In contrast to other people working on braid statistics, they also conceived the possibility of non-abelian braid statistics, a form of quantum statistics whose mathematical aspects have been understood only recently; see [34,35] and the sixth paper reprinted in P a r t VI. Unfortunately, their work did not receive much attention either. T h e situation only changed when Wilczek's work [43] appeared and its relevance for the theory of the fractional quantum Hall effect [45] (Laughlin vortices in incompressible quantum Hall fluids with filling factor ^, | are fractionally charged excitations with fractional statistics), and, perhaps, for the theory of layered high Tc superconductors [46] was

9 noticed. All this is described in a reprint collection introduced and edited by Wilczek [47]. The results in [43] through [46] triggered the interest of P.A. Marchetti and myself in braid statistics, in the context of (2+ l)-dimensional Chern-Simons gauge theories. Our early results are summarized in the last paper of Part III and are also mentioned in the second paper of Part VI. One of the puzzles we attempted to understand was how the exchange algebras of charged fields of the type of Mandelstam strings in three-dimensional Chern-Simons gauge theories would lead to the braid statistics of particles, in the sense of [41,42,43], in the asymptotic (scattering) states of such theories. A sketch of our insights appears in the last paper of Part III, a more precise theory was developed in [48]; but there is room for better understanding of these matters in theories with non-abelian braid statistics.

8 . Since 1986, the exploration of braid statistics has been one of my main scientific interests. With the appearance of the work of Reshetikhin et al. [49] it became plausible that quantum groups (more generally, "quasi-Hopf algebras") had a role to play as generalized symmetries of quantum field theories in two and three spacetime dimensions, whose representation theory would reproduce the super-selection rules and sectors and the structure constants of the exchange algebras - the braid statistics - of these field theories. In spite of early insights and partial results in this direction (see especially [50], and the fifth paper of Part VI, and refs. given there) this turned out to be a tricky and therefore fascinating problem. The best results in this area known to me - examples where the idea of quantum groups as generalized, global symmetries of low-dimensional quantum field theory, can actually be proven, mathematically, to work - are due to T. Kerler [51], in whose work I have had the luck to be somewhat involved, to G. Mack and V. Schomerus [52], and to G. Felder and C. Wieczerkowski [52], and refs. given there. (The work of Kerler and our joint efforts are briefly sketched in the sixth paper of Part VI.) On hindsight, it is clear that we should have followed the lead of S. Doplicher and J.E. Roberts and formulated braid statistics and the problem of constructing Hopf algebras as generalized symmetries whose representation theory reproduces the super-selection structure and braid statistics of some quantum field theory using the language of abstract tensor categories. See [53] for their fundamental results. Braided tensor categories offer a point of view unifying the theory of super-selection sectors with braid statistics in low-dimensional quantum field theory, the theory of quasi-triangular (quasi-) Hopf algebras [54], discrete versions of current algebras and WZNW models, and the theory of those invariants of knots and links, and of invariants for 3-manifolds [55], that generalize the Jones polynomial. (The connection between the braid statistics and operator product expansion (fusion) of chiral vertex operators in conformal field theory and the theory of invariants for links in 3-manifolds and other related objects (ribbon graphs), is described in the fifth paper of Part VI. We learnt from those efforts that all one needed for the construction of such invariants were certain combinatorial data, six-index symbols related to "braiding" and "fusing", which

10 rational conformal field theories - but quantum groups, too - provide. But the full structure of conformal field theory is irrelevant. In mathematical jargon, any braided monoidal category with unit and conjugates, subobjects and direct sums - we call this a "quantum category" - provides suitable combinatorial data for the construction of link and 3-manifold invariants. This can be extracted from [56]. In that paper we also suggested how quantum groups might appear as symmetries of conformal field theories. Our ideas were similar to independent ideas of Moore and Reshetikhin.)

9 . One might be worried that, over all that abstract mathematics, one would forget the physics. Clearly, two-dimensional conformal field theory has brought about a revolution in our understanding of the theory of critical phenomena in two dimensions. These developments are well-known. In 1989/90, it turned out that chiral current algebra (especially f/(l) and SU(2) current algebra) and the closely related topological Chern-Simons gauge theories of Witten [55] were perfectly natural tools in the analysis of the fractional quantum Hall effect - almost as if they had been invented for that purpose. This observation can be extracted from a paper of Halperin [57] on chiral edge currents in integer quantum Hall systems and was made, independently, by several people, including Wen, whose work came first, Stone, Kerler and myself and, most probably, other people as well. The beginnings of our work in this area which is still continuing are described in the last paper reprinted in this volume. In the form of chiral electric and spin currents flowing around the boundaries of incompressible quantum Hall fluids, nature manifests itself as an anomalous, (1 + l)-dimensional, chiral gauge theory; it cancels the gauge anomaly by a (2 + l)-dimensional, topological Chern-Simons term - as it should [58]. In other two-dimensional, incompressible quantum fluids, nature appears to have realized non-abelian braid statistics (related to the one t h a t can be reconstructed from the representation theory of the q u a n t u m group Ug(s£2), 4-theory which, with encouragement by J. Glimm and E. Nelson, we had adapted to lattice spin systems and field theories. We had shown that it can be used to prove, in a systematic way, many old and new correlation inequalities. The new inequalities in Aizenman's and in my paper made precise the intuition that two simple or (self) repelling random walks on Zd (not starting at the same point), miss each other with probability one, in the scaling limit, for d > 4 (and for d = 4, in the case of simple random walks). The form in which this intuition is cast is an inequality on the connected four-point Euclidean Green function somewhat motivated by earlier conjectures of A. Sokal. But what is the connection between two random walks on 7Ld and the connected four-point function of A^-lattice field theory? With the help of a partially resummed version of Symanzik's polymer gas representation of (lattice) Ay^-theory one can express the connected four-point function as a sum over weights indexed by two random walks connecting the arguments of the four-point function and then

13 estimate (using a correlation inequality) its absolute value in terms of intersection probabilities of the two walks. Another correlation inequality then bounds these intersection probabilities by tree diagrams with coupling-constant-independent coefficients. It is shown in my paper that one and two-component Xip^-theory, constructed as a continuum limit of ferromagnetic reflection-positive lattice \ 4 dimensions, and that in d = 4 dimensions, and under the same hypotheses, the only scale-invariant \% theory in the symmetric phase based on correlation inequalities and due to D. Brydges, A. Sokal and myself, besides many other topics in the theory of critical phenomena, are reviewed in much detail in a forthcoming book by R. Fernandez, A. Sokal and myself [63], where the reader will find lots of results and more references to the original literature than he may care for. I therefore do not wish to go into further details about Part IV, but see also the fourth article of Part VI. Nevertheless, one final comment should be added. The triviality results of Aizenman and myself have been applied to proving upper bounds on Higgs masses in the standard model. Actually, our results do not directly apply to exactly this problem; only a reasonable but unproven extrapolation thereof does. There would be a few interesting things to say about these matters which are not settled in a final way, but that would take us too far.

13 . I come now to some brief comments on Part V, "Random Geometry". The first paper, "Regge Calculus and Discretized Gravitational Functional Integrals", has not been published before, and, maybe it should have remained that way. It was written in the winter of 1980/81. I have made only very minor revisions since then. It is a piece of science fiction about the lattice approximation to

14 Euclidean gravity [64]. Of course, we do not really understand what "Euclidean gravity" means. My preprint encountered some interest, and quite a few people asked for copies. Now it suffices to buy this book to find out what is in this article. It reviews, in a somewhat crude way, some results in combinatorial topology and geometry which are due to other people and states some conjectures which were then proven by some bright colleagues, in particular by Cheeger, Miiller and Schrader and by Feinberg, Friedberg, Lee and Ren [65]. I suppose it poses some good problems, some of which are still open (e.g. the counting of the number of isomorphism classes of triangulations of, for example, the 3-sphere, or the 4-sphere, with N simplices, as N becomes large. Does this number grow like const.N, for a sphere?). Furthermore, I made some attempts to reconstruct a quantum-mechanical space of states with positive-definite scalar product from discretized gravitational functional integrals. The main idea is to assign to essentially combinatorial d a t a associated with a manifold with boundary a vector in a Hilbert space with a scalar product that can be expressed in terms of discretized gravitational functional integrals. This is somewhat related to the "wave function of the universe", a la Hartle and Hawking, and to ideas in topological quantum field theory, in the sense of Witten [55]. But my construction was plagued with certain combinatorial ambiguities that could not be explained away, except by appealing to a "deus ex machina": universality. (My problems were most likely a consequence of not having isolated quite the right concepts. Perhaps, ideas and results due to M. Gromov would clarify the situation.) In the same article, I also suggested t h a t the incorporation of spinor fields in my formalism imposes strong constraints on space (imaginary) time topology if one insists on a form of Osterwalder-Schrader (reflection) positivity. (These constraints say more than that the second Stiefel-Whitney class must vanish.) Maybe this is worth looking into in more depth. Nowadays, my paper is, at best, of some historical interest, apart from the circumstance that it draws attention to some problems that are still open and t h a t I find somewhat interesting, and to some useful literature. I have never become a professional in matters of quantum gravity, and this explains why my article and other dreams in this area (concerning Hawking radiation) remained unpublished. Furthermore, I did not pursue mathematical problems related to Regge calculus either. This was done by more competent people. I think that Regge calculus has not been a very useful tool in gravity theory, so far. However, ideas reminiscent of it (in particular, the work of Ponzano and Regge and of Hasslacher and Perry [64]) have recently proven, in the hands of Turaev and Viro [66], to lead to fascinating mathematics. Crudely speaking, what they have found is the following. Associate with each edge (1-cell) in a triangulation of a threedimensional topological manifold (compact and without boundary) an irreducible object of a "rational" braided quantum category (e.g., an irreducible representation of Ug(s£2), with q a root of unity, or of a chiral current algebra, such as s~u(2)k)Associate with the objects on the six edges of every tetrahedron a corresponding

15 six-index symbol. Take their product over all tetrahedra in the triangulation, multiply by some phase factor, and then, for each edge, sum over all possible irreducible objects of the rational tensor category ("rational" means there are only finitely many irreducible objects, so the sums converge). Turaev and Viro show [66] t h a t this produces a 3-manifold invariant related to (the modulus square of) W i t t e n ' s invariants [55]. This is certainly a very beautiful result. My attempts to understand Regge calculus turned out to have been a useful preparation for the study of random surface theory. My interest in random surface theory was triggered by the work of Tom Spencer and myself on the roughening transition in the SOS model, by the confinement problem in lattice gauge theory and by Polyakov's paper on string theory [67]. Moreover, motivated by remarks of Giorgio Parisi, M. Aizenman and I started to study plaquette percolation, a problem which in the competent hands of M. Aizenman, J. T . Chayes, L. Chayes and L. Russo turned out to have a number of fascinating features. T h e results of our efforts appear in [68]. The methods developed in this paper turned out to be quite useful and powerful in disordered-systems theory. Subsequently, Durhuus, Jonsson and I became interested in the theory of selfavoiding and of planar (genus 0) lattice random surfaces. Our main motivation came from lattice gauge and string theory. For the simplest models of planar random surfaces, we were able to show, modulo a bound on a critical exponent only "verified" numerically, t h a t the "mean-field" theory of Drouffe and Parisi becomes exact in dimension d ^ 2. 2 It was felt (and argued) t h a t this "collapse to branched polymers (or sea weed)" was related to the tachyon problem of bosonic string theory - which is likely to be correct. These developments are briefly reviewed in the second paper of P a r t V, "The Statistical Mechanics of Surfaces". This paper - which was written in the spring of 1984 - is not only a review paper. It introduced some new ideas on, and some new models of random surfaces, most importantly the triangulated random surface models - which I viewed as decent, discrete approximations to Polyakov's functional integral formulation of string theory - and the matrix-model formulation of triangulated random surface models. I also suggested interpreting models of this type with zero-dimensional target space as models of two-dimensional gravity. The triangulated random surface models were then analyzed by Ambj0rn, Durhuus, Orland and myself, and by Frangois David, and later by numerous other people. Giovanni Felder and I spent some time studying the matrix model formulation of triangulated random surface models. But we did not find results that would have appeared to extend, in an interesting way, those obtained by Brezin, Itzykson, Parisi and Zuber, and by Bessis, Itzykson and Zuber [69], who had had different applications in mind. Matrix models of random surfaces were then reintroduced and studied by V. Kazakov and A. Migdal, and by F. David, who eventually went considerably beyond what we had found. Our result is an example of a "theorem" only proven to be true with high probability. Our arguments are rigorous but require an inequality, whose truth has been demonstrated numerically with a certain (presumably high) probability, but not with certainty.

16 My own interest in this subject had a payoff in mathematics. I told W. Thurston and R. Penner about the work of the Saclay group on matrix integration theory and that it had something to do with summing over Feynman diagrams and, in particular, triangulations of surfaces. R. Penner apparently found these remarks helpful in his work on the virtual Euler characteristic of moduli space [70]. Through their work, V. Kazakov and I. Rostov kept alife the physicists' interest in matrix models of two-dimensional gravity. In 1989/90, this led to beautiful work on non-perturbative two-dimensional gravity with important contributions by Brezin and Kazakov, Gross and Migdal, Douglas and Shenker, and many others. These developments are reviewed in [63], where many references t o the original articles can be found (there are too many to list all of them here). While I have some diffculties in understanding the relevance of the present results on two-dimensional gravity to physics, primarily because of the famous d = 1 barrier, they again triggered activities whose main payoff is in pure mathematics, at least so far: Witten has proposed a formulation of two-dimensional gravity in terms of intersection theory on the moduli space of curves and conjectured that his formulation was connected to matrix models [71]. This conjecture has recently been proven in beautiful work of Kontsewich [72]. These developments are likely to leave some traces in the mathematics of the moduli space of curves.

1 4 . It may be worthwhile to ask what purpose a reprint collection like this may serve, with all the many gaps it leaves open. My hope is that it illustrates a few basic ideas and concepts in the subject of non-perturbative q u a n t u m field theory - of course the perspective is personal - but, primarily, that it might arouse the reader's interest in this important and still active branch of theoretical physics and guide him to some of the relevant literature (not to all of it, of course). The bibliographies of this introduction and of several of the papers reprinted in this volume should be helpful in finding one's way to some good literature on axiomatic quantum field theory, constructive quantum field theory, conformal field theory, gauge theory, quantum solitons, equilibrium statistical mechanics, renormalization group methods, and combinatorial-geometrical methods. The second and third papers of Part I, the first and second papers of Part II, the fourth paper of Part VI and the last two papers of Part VI may be particularly useful in this respect (besides the literature quoted at the end of this introduction). Of course, many important topics in non-perturbative q u a n t u m field theory are not surveyed at all, and not adequately referred to. I believe that the four most important areas of application for field-theoretic methods are: gauge theories of elementary particles, the theory of critical phenomena in statistical mechanics, (many-body theory in) condensed matter physics, and astrophysics and cosmology. It is regrettable that none of these areas is discussed in sufficient detail, although glimpses are provided. Some topics are not touched upon at all. For example, one does not find anything about the fascinating area of exactly solved models in

17

(l-rT)-dimensional q u a n t u m field theory (I have never worked in i t ) . But, of course, this is not meant to be a textbook! Although many of the papers reprinted in this volume are not very technical, they describe results which, by and large, have grown out of technical work, work carried out in the spirit and with the methods of mathematical physics. Mathematical physicists tend to be confronted with a discrepancy between results they would like to establish, because they would be physically relevant, and results t h a t are accessible if one insists on mathematical precision. The reader is likely to encounter this discrepancy, but this is, perhaps, quite educational. To be somewhat immodest, I think the reader may also find out t h a t working in the spirit and with the methods of mathematical physics does not prevent one from sometimes being a little ahead of the crowd, idea and conceptwise. Some of the papers in Parts V and VI may illustrate this point. Physics, including theoretical physics, and, in particular, q u a n t u m field theory have had considerably more brilliant periods than the present one. Many areas in theoretical physics which, in former times, were exclusive to people engaged in fundamental research are now in the hands of mathematicians, applied physicists or engineers. One may think of much of classical physics, quantum mechanics, nuclear physics, parts of condensed matter physics, optics, etc. Often our colleagues in other departments are more successful in improving our understanding of these areas of physics and making them applicable. The present situation appears to diminish the glamour of the job of a theoretical physicist proper and induces some people to conclude that support of fundamental research in physics should be reduced. I am not pessimistic! I believe that physics, including classical physics, will remain a vital source of genuinely deep and important problems in basic research for a long time. We may not live through revolutions comparable to the ones in the first quarter of our century (relativity theory and quantum mechanics), or during the late sixties and seventies (gauge theories of fundamental interactions in particle physics, renormalization group approach to critical phenomena), every few years. But we can do important work even in the absence of such revolutions! And we may keep in mind that further revolutions will be necessary before we understand how events in space-time and the structure and dynamics of space-time itself and of interactions mediated by gauge fields emerge from a more fundamental q u a n t u m theory of nature whose basic formulation does not anticipate a model of space-time and of gauge fields, but will be able, at least in principle, to predict it. There is always reason to think that behind some steep mountains new horizons will open up.

In conclusion I wish to apologize to all those people whose important work is underrepresented, or even grossly underrepresented, in this book, even though it may be concerned with non-perturbative q u a n t u m field theory. Once again: this volume is neither a textbook, nor does it have scholarly intentions comparable to

18 those of a textbook! It only reprints some of my (review) papers on q u a n t u m field theory. I hope it will be somewhat useful. I do not think that anybody will read a long book like this from cover to cover. The idea would be that, by turning the pages of one or another section or paper, some readers will develop a little enthusiasm for the beauties of non-perturbative quantum field theory and some of the important problems left open. I wish to thank all those colleagues, whose influence on me, guidance, inspiration, or collaboration with me has made this book possible! There are too many of t h e m to thank them individually. Colleagues, who have been directly involved in work described in this volume, and to whom I am particularly indebted, are: J. Ambj0rn, M. Aizenman, D. Brydges, B. Durhuus, G. Felder, T. Jonsson, G. Keller, T. Kerler, C. King, E.H. Lieb, P.A. Marchetti, G. Morchio, E. Seiler, B. Simon, T. Spencer, F. Strocchi and M. Struwe. Special thanks also to my family, my teachers and my friends within and outside the scientific community! May they take this book as a token of gratitude.

Jiirg Frohlich

19

References 1. J. Goldstone, Nuovo Cimento 19, 154-164 (1961). J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127, 965-970 (1962). 2. D. Mermm and H. Wagner, Phys. Rev. Letters 17, 1133-1136 (1966). D. Mermin, J. Math. Phys. 8, 1061-1064 (1967). 0. Mc Bryan and T. Spencer, Commun. Math. Phys. 53, 299-302 (1977). See also: R.L. Dobrushin and S.B. Shlosman, Commun. Math. Phys. 42, 31-40 (1975). J. Frohlich and C.E. Pfister, Commun. Math. Phys. 81, 277-298 (1981); 104. 697-700 (1986). 3. J. Glimm and A. Jaffe, J. Math. Phys. U , 3335-3338 (1970). For a general review of constructive quantum field theory see: / . Glimm and A. Jaffe, "Quantum Physics (a Functional Integral Point of View)", New York, Berlin, Heidelberg: Springer-Verlag 1981, 1987 ( 2 n d edition). See also: B. Simon, "The P()2 Euclidean Princeton University Press, 1974.

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20

J.M. Kosierlitz and D.V. Thouless, J. Phys, C 6, 1181-1203 (1973). J.M. Kosierlitz, J. Phys. C 7, 1046-1060 (1974). 10. J. Frohlich, Commun. Math. Phys. 47, 233-268 (1976). / . Frohlich and T. Spencer, J. Stat. Phys. 24, 617-701 (1981). 11. P.W. Anderson, G. Yuval and D.R. Hamann, Phys. Rev. B 1, 4464 (1970). P.W. Anderson and G. Yuval, Phys, Rev. B 1, 1522 (1970). J.L. Cardy, J. Phys. A 14, 1407 (1981). 12. A. Guth, Phys. Rev. D 21, 2291-2307 (1980). 13. /. Frohlich and T. Spencer, Phys. Rev. Lett. 46, 1006-1009 (1981); Commun. Math. Phys. 81, 527-602 (1981). 14. J. Frohlich and T. Spencer, Commun. Math. Phys. 83, 411-454 (1982). 15. /. Frohlich and T. Spencer, Commun. Math. Phys. 84, 87-101 (1982). 16. J.Z. Imbrie, Commun. Math. Phys. 85, 491-515 (1982). M. Aizenman and C. Newman, Commun. Math. Phys. 1QZ, 611-647 (1986). J. Frohlich and B. Zegarlinski, J. Stat. Phys. 63, 455-485 (1991). M. Aizenman, J.T. Chayes, L. Chayes and CM. Newman, J. Stat, Phys. 5J), 1-40 (1988). 17. "Scaling and Self-Similarity in Physics", J. Frohlich (ed.), Progress in Physics, Basel, Boston: Birkhauser-Verlag 1983. 18. S. Doplicher, R. Haag and J.E. Roberts, Commun. Math. Phys. 13, 1-23 (1969); ibid. 15, 173-200 (1969). Commun. Math. Phys. 23, 199-230 (1971); ibid. 35, 49-85 (1974). 19. S. Colemam, "Classical Lumps and Their Quantum Descendants", in: "New Phenomena in Subnuclear Physics", A. Zichichi (ed.), New York: Plenum Press, 1977; and refs. given there. See also: S. Coleman, "Aspects of Symmetry", Cambridge: Cambridge University Press, 1985. C. Rebbiland G. Soliani, "Solitons and Particles", Singapore: World Scientific, 1984. 20. J. Frohlich, Commun. Math. Phys. 41, 269-310 (1976). See also: "Poetic Phenomena in Two-Dimensional Quantum Field Theory..." in: "Les Meihodes Mathematiques de la Theorie Quantique des Champs", F. Guerra, D. W. Robinson and R. Stora (eds.), Paris: Ed. du C.N.R.S., 1976; unpublished (lecture) notes 1975/76. See also: Phys. Rev. Lett. 34, 833-836 (1975). 21. L. Kadanoff&nd H. Ceva, Phys. Rev. B U, 3918 (1971).

21 22. F. Wegner, J. Math. Phys. 12, 2259-2272 (1971). 23. T.H.R. Skyrme, Proc. Roy. Soc. A 262, 237 (1961). R.F. Streater and I.F. Wilde, Nuclear Physics B 24, 561-575 (1970). 24. J. Bellissard, J. Frdhlich and B. Gidas, Phys. Rev. Lett. 38, 619 (1977); Commun. Math. Phys. 60, 37-72 (1978). 25. E.C. Marino and J.A. Swieca, Nucl. Phys. B 17Q [FS 1], 175 (1980); E.C. Marino, B. Schroer and J.A. Swieca, Nucl. Phys. B 200 [FS 4], 473 (1982); R. Koberle and E.C. Marino, Phys. Lett. 126B, 475 (1983). 26. M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto University, 15, 871 (1979). 27. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys., B 241. 333 (1984). 28. V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247, 83 (1984). 29. V.F.R. Jones, Inv. Math. 72, 1-25 (1983); "Braid Groups, Hecke Algebras and Type II\ Factors",in: "Geometric Methods in Operator Algebras", Pitman Res. Notes in Math. 123, 242-273 (1986). Bull. Amer. Math. Soc. 12, 103-112 (1985); Ann. Math. 126, 335-388 (1987); Notes on a talk in Atiyah's seminar (1986). F.M. Goodman, P. de la Harpe and V.F.R. Jones, "Coxeter Graphs and Towers of Algebras", MSRI Publ., S.S. Chern et al. (eds.), New York, Berlin, Heidelberg: Springer-Verlag, 1989. 30. J.-L. Gervais and A. Neveu, Nucl. Phys. B 238, 125 (1984). 31. A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13, 303 (1987). 32. K.-H. Rehren, Commun. Math. Phys. U6 , 675 (1988). 33. J. Frohlich, "Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links", in: "Non-Perturbative Quantum Field Theory", G. 't Hooft et al. (eds.), New York: Plenum 1988. 34. K. Fredenhagen, K.H. Rehren and B. Schroer, Commun. Math. Phys. 125. 201 (1989). 35. J. Frohlich, F. Gabbiani and P.-A. Marchetii, "Superselection Structure and Statistics in Three-Dimensional Local Quantum Field Theory", in: "Current Problems in High Energy Particle Theory", G. Lusanna (ed.) Singapore: World Scientific Publ. 1989. /. Frohlich, F. Gabbianiand P.A. Marchetii, "Braid Statistics in ThreeDimensional Local Quantum Theory", in: "the Algebraic Theory of Superselection Sectors", D. Kastler (ed.), Singapore: World Scientific Publ. 1990. 36. K.-H. Rehren and B. Schroer, Phys. Lett. B 198, 84 (1987); Nucl. Phys. B 312, 715-750 (1989).

22 37. G. Moore and N. Seiberg, Phys. Lett. B 212, 451 (1982); Nucl. Phys. B313. 16 (1989); Commun. Math. Phys. 123, 177 (1989). 38. G. Felder, J. Frohlich and G. Keller, Commun. Math. Phys. 124, 417-463 (1989); 124, 647-664 (1989); 130, 1-49 (1990). 39. 0. Babelon, Phys. Letts. 215B. 523-529 (1988). 40. J. Frdhlich and G King, Intl. J. Mod. Phys. A 4, 5321-5399 (1989); Commun. Math. Phys. 126, 167-199 (1989). 41. M. Leinaas and / . Myrheim,

Nuovo Cimento B 37, 1 (1977).

42. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 22, 1664 (1981); G.A. Goldin and D.H. Sharp, Phys. Rev. D 28, 830 (1983). G.A. Goldin, R. Menikoff and D.H. Sharp, Phys. Rev. Letts. 54, 603 (1985). 43. F. Wilczek, Phys. Rev. Letts. 48, 1144 (1982); 49, 957 (1982). F. Wilczek and A. Zee, Phys. Rev. Letts. 51, 2250 (1983). 44. Y.S. Wu, Phys. Rev. Letts. 52, 2103-2106 (1984). 45. R.B. Laughlin, Phys. Rev. Letts. 50, 1395 (1983). B.I. Halperin, Phys. Rev. Letts. §2, 1583-1586 (1984). D. Arovas, J.R. Schrieffer and F. Wilczek, Phys. Rev. Letts. 53, 722-723 (1984). 46. P.W. Anderson, Science 235, 1196 (1987); R.B. Laughlin, Science 242, 525 (1988). 47. "Fractional Statistics and Anyon Superconductivity", pore: World Scientific Publ., 1990. 48. / . Frdhlich and P.-A.

Marchelti,

F. Wilczek (ed.), Singa-

Nucl. Phys. B 356, 533-573 (1991).

49. N.Yu. Reshetikhin, Algebra i Analis 1, 169-188 (1989). L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, 178-207 (1989).

Algebra i Analis 1,

50. L. Alvarez-Gaume, C. Gomez and G. Sierra, Nucl. Phys. B 319. 155-186 (1989); see also the second and third paper in ref. 38. P. Bouwknegt, J. Mc Carthy and K. Pilch, Phys. Letts. 234B, 297-303 (1990). C. Gomez and G. Sierra Phys. Letts. 240B, 149-157 (1990). A. Alekseev, L. Faddeev and M. Semenov-Tian-Shansky, "The Unravelling of the Quantum Group Structure in the WZNW Theory", and "Hidden Quantum Group Inside Kac-Moody Algebras", preprints, Leningrad 1991. 51. T. Kerler, "Darstellungen der Quantengruppen und Anwendungen", ETHdiploma thesis 1989; paper in preparation. T. Kerler, "Fusion-Rule Algebras: A Classification for small Indices", in preparation.

23 J. Frohlich and T. Kerler, "On the Role of Quantum Groups in Low-Dimensional, Local Quantum Field Theories", Lecture Notes in Mathematics, to appear; preliminary draft 1990. 52. G. Mack and V. Schomerus, "Quasi Quantum Group Symmetry and Local Braid Relations in the Conformal Ising Model", and "Quasi Hopf Quantum Symmetry in Quantum Theory", preprints, Hamburg, July 1991. G. Felder and C. Wieczerkowski, "Topological Representations of the Quantum Group Uq(st2)", Commun. Math. Phys., in press (1991). 53. S. Doplicher and J.E. Roberts, Bull. Amer. Math. Soc. 11, 333-338 (1984); J. Funct. Anal. 74, 96-120 (1987); Invent, m a t h . 98, 157-218 (1989); preprint Rome 1989. 54. M. Jimbo, Lett. Math. Phys. 10, 63-69 (1985); Lett. Math. Phys. 11, 247-252 (1986). V.G. Drinfeld, "Quantum Groups", in: Proc. of ICM Berkeley 1986, A.M. Gleason (ed.), Providence R.I.: Amer. Math. Pupl., (1987). S.L. Woronowicz, Commun. Math. Phys. H i , 613-665 (1987), Publ. RIMS, Kyoto University 23, 117-181 (1987). 55. E. Witten, Commun. Math. Phys. 121, 351 (1989). 56. N.Yu.

Reshetikhin

and V.G. Turaev, Commun. Math. Phys, 127, 1-26(1990).

See also: J. Frohlich and C. King, Intl. J. Mod. Phys. A 4, 5321-5399 (1989). N.Yu. Reshetikhin and V.G. Turaev, "Invariants of S-Manifolds via Link nomials and Quantum Groups", Invent, math., to appear.

Poly-

57. B.I. Halperin, Phys. Rev. B 25, 2185 (1982). 58. X.G. Wen, "Gapless Boundary Excitations in the Quantum the Chiral Spin States", Phys. Rev. B, t o appear (1991). J. Frohlich and U. Studer, preprint, to appear. 59. J. Frohlich, T. Kerler and P.A. Marchetti, Dimensional Condensed Matter Physics",

Hall States and in

"Non-Abelian Bosonization preprint, to appear.

in Two-

60. "The Algebraic Theory of Superselection Sectors, Introduction and Recent sults", D. Kastler (ed.), Singapore: World Scientific Publ., 1990.

Re-

61. M. Aizenman, Commun. Math. Phys, 86, 1-48 (1982). M. Aizenman and R. Graham, Nucl. Phys. B 225 [FS 9], 261-288 (1983). 62. / . Glimm, A. Jaffe and T. Spencer, Commun. Math. Phys. 45, 203-216 (1975). J. Glimm and A. Jaffe, "Quantum Physics", see ref. 3, and refs. given there. 63. R. Fernandez, J. Frohlich and A. Sokal, "Random Walks, Critical Phenomena and Triviality in Quantum Field Theory", Texts and Monographs in Physics, New York, Berlin, Heidelberg: Springer-Verlag, to appear. 64. T. Regge, Nuovo Cimento 19, 558 (1961).

24 G. Ponzano and T. Regge, in: "Spectroscopic and Group Theoretical Methods in Physics", F. Bloch (ed.), Amsterdam: North-Holland, 1968. B. Hasslacher and M. Perry, Phys. Letts. 103B, 21-24 (1981). 65. J. Cheeger, W. Muller and R. Schrader, Commun. Math. Phys. 92, 405 (1984). G. Feinberg, R. Friedberg, T.D. Lee and B.C. Ren, Nucl. Phys. B 245, 343 (1984). 66. V.G. Turaev and O.Y. Viro, "State Sum Invariants of 3-Manifolds and Quantum 6j-Symbols", preprint, 1990. V.G. Turaev, "State Sum Models in Low Dimensional Topology", preprint, 1991. V.G. Turaev, "Quantum Invariants of 3-Manifolds and a Glimpse of Shadow Topology", preprint, 1991. 67. A.M. Polyakov, Phys. Lett. B 1£3B, 207 (1981). 68. J. Frohlich, C.E. Pfisier and T. Spencer, "On the Statistical Mechanics of Surfaces", Lecture Notes in Physics 173. 169-199 Berlin, Heidelberg, New York: Springer-Verlag 1982. M. Aizenman, J.T. Chayes, L. Chayes, J. Frohlich and L. Russo, Commun. Math. Phys. 92, 19-69 (1983). M. Aizenman and J. Frohlich,,Nucl. Phys. B 235 [FS 11], 1-18 (1984). 69. E. Brezin, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59, 35-51 (1978). D. Bessis, C. Itzykson and J.B. Zuber, Adv. Appl. Math. 1, 109-157 (1980). 70. R. Penner, J. Diff. Geom. 27, 35-53 (1988). 71. E. Witten, Nucl. Phys. B 340, 281-332 (1980). 72. M. Konisewich, "Intersection Theory on the Moduli Space of Curves, and the Matrix Airy Function", preprint, to appear.

I Phase Transitions and Continuous Symmetry Breaking

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27 Acta Physica Austriaca, Suppl. XV, 133-269 (1976) © by Springer-Verlag 1976

PHASE TRANSITIONS, GOLDSTONE BOSONS AND TOPOLOGICAL SUPERSELECTION RULES* by J. PROHLICH +,X Department of Mathematics Princeton University Princeton, N.J. 08540

Instead of an abstract: Table of Contents 1. Introduction and Program Part 1: 2. Ferromagnetic Models in Classical Statistical Mechanics and Relativistic Bose Quantum Field Theory; Main Results 3. Two General Methods in the Theory of Phase Transitions 4. Infrared (Gaussian) Domination and Thermodynamic Limit 5. Phase Transitions and Spontaneous Symmetry Breaking for the Classical N-Vector Models present address: ZiF, Universitat Bielefeld, D-4800 Bielefeld, Wellenberg 1, F.R.Germany supported in part by U.S. NSF under grant GP-39048 and by ZiF, Universitat Bielefeld Lecture given at XV. Internationale Universitatswochen fur Kernphysik, Schladming, Austria, February 16-27, 1976.

28 134

6. The

(-)2 - Quantum Field Model

7. Phase Transitions in Two Dimensional Quantum Field Models 8. Quantum Crystals and More About the Peierls Argument Part 2: 9 . Phase Transitions and the Spontaneous Occurence of Soliton-Sectors. 10. Remarks on the Proof of Poincare - Covariance of the Soliton-Sectors.

1. INTRODUCTION AND PROGRAM A warning and a reflection: The material I propose to cover in these four lectures is quite large, and ideas from different fields in mathematical physics must be combined. Therefore not all the details will be explained. I have tried to select proofs for presentation according to their technical simplicity and elegance. This should not mislead you to believe that mathematical physics is a simple thing. Some of the most outstanding and admirable recent results of, say, constructive quantum field theory (e.g. [GJ1] [GRS] [GJSl] [OS]; see also [CQFT]) require an enormous amount of sophisticated and hard analysis. These results concern the existence of relativistic quantum fields and their detailed properties, e.g. their non-triviality, (in the sense that the scattering matrix is different from the identity [EEF, O S e ] \ The fact that the proofs of many of these results are very hard and intricate may seem or be unpleasant. Yet it tells us something that I feel is important: The foundations of relativistic quantum field theory and statistical mechanics may neither be wrong nor do they necessarily require major modifications, but it cculd be that the

135

mathematical problems occuring in the construction of quantum fields and models for systems with an infinite number of degrees of freedom are just very difficult and complicated, and that with many problems one has not yet been successful on a mathematical level. If this should indeed be true then the fact that one of two weeks of a school on "Kernphysik" was devoted to mathematical physics requires only minor defense, and it may then also seem plausible that progress in physics does not only come from the very important efforts of experimentalists and theoreticians but even a little bit from the attempts of mathematical physicists. Rather than presenting some more reflections I should like to refer you to some nice thoughts in the literature: [K] (some danger of producing "dehydrated elephants" in mathematical physics),[st E ] (if what was intended to represent" a boa that has swallowed an elephant" appears to you to represent "a hat" (or worse an "old hat") it may be that I am a bad writer and I wish to apologize myself for that). Next I describe the program of these lectures: The first part is centered around the phenomenon of phase transitions which is sometimes accompanied by the spontaneous breaking of a (discrete or continuous, internal) symmetry (of the "dynamics"). A simple (or very difficult) example (depending on one's point of view) for a phase transition is a ferromagnet: This is a macroscopic system of matter which at high temperatures (i.e. above some critical temperature T ) does (or may) not have any particularly exciting or extraordinary features; but, at temperatures below T , it has the remarkable property that it remains magnetized after an external magnetic field has been turned off. Here "macroscopic" means that the system consists of -10 23 elementary magnets - atoms

136

or molecules - (mathematically: infinitely many degrees of freedom). It turns out that such a material may have as many pure phases as there are directions in space. What is a pure phase? In the case of H_0 one distinguishes three pure phases: ice, water and vapor. These are especially pure states (or manifestations) of H ? 0. Mathematically, the pure phases of a physical system correspond to time translation invariant) states of the system with the property that the algebra of (time translation invariant) observables at infinity is trivial; see Sect. 3 The Hamilton function (the Hamilton operator, respectively) of a ferromagnet is generally assumed to be invariant under an arbitrary, simultaneous rotation of all the elementary magnets of the ferromagnet. Yet, below the critical temperature, in a pure phase, there exists a preferred direction; the direction of spontaneous magnetization. We say that the state of the ferromagnet in a pure phase breaks the symmetry of the Hamiltonian, or: the symmetry is spontaneously broken in the pure phases. The main issue of the first part of my lectures is the construction and analysis of simple models for phase transitions and spontaneous symmetry breaking. Some of these models are carricatures of (classical) ferromagnets, one class of them are relativistic quantum field models at temperature 0, the so called (-cj>) ^ models. Mathematically ,such models are related to models of classical ferromagnets, e.g. the Ising model; see [GRS], [ S G ] , [ N ] , [ D N ] . The role of the temperature is played by some coupling constant (related to the field strength), and phase transitions occur, as this coupling constant is varied. - Some other models are merely of conceptual

137

interest: One serves to exemplify the concept that there must exist phase transitions which are not accompanied by the breaking of any symmetry, another one shows the possibility of "triple points". Field theorists are presently very much interested in phase transitions and the spontaneous (or even more the "dynamical") breakdown of continuous, internal (or even nicer: "gauge") symmetries, because most current theories of the fundamental interactions involve as a central theoretical element the possibility that continuous symmetries may be broken by the physical vacuum. (It could be that mathematical consistency of such theories even requires symmetry breaking).

In the second part of my lectures I will illustrate, in the context of Bose quantum field models in two spacetime dimensions, how at temperature 0 phase transitions (as some coupling constant is varied) may be accompanied by the occurrence of new superselection sectors, [St w ] , [DHR], and non-trivial, dynamical (or "topological") charges. In order to illustrate what I try to describe I first sketch an example: Consider an infinite quantum mechanical chain of equally spaced, elementary dipoles (e.g. electric dipoles that are anharmonically bound and ferroelectrically coupled). Such a chain may have a degenerate groundstate, namely two pure, spontaneously, polarized groundstates with opposite polarizations (± p ) : Pictorially ffff+t+t ....

or

.... ++++++++

+ Fig. 1

138

We may then ask the natural question whether there exist states with a well defined, continuous time evolution which very far to the left look like groundstate - and very far to the right like groundstate +, or vice versa. Pictorially • • • • I / y ^ f ft ••••

°r

. . . . f f S —.> \ \ ir

s

s Fig. 2

(The pictures represent e.g. the expectation of the "dipole field" p(i), i £ Z , in the states +, -, s and s) What we will show in Part 2 is that the chain can indeed be twisted over some bounded space region by an angle of 180° to be in a state s (or s) that interpolates between the groundstates - and +. The state s (s) so obtained is not a groundstate (not even a discrete eigenstate) of the Hamiltonian. The states +, -, s and s are vectors in mutually orthogonal Hilbert spaces (super selection sectors, denoted X ,, X_ / H i H~z) • There exists a conserved charge Q = {3>; Qi|i = 2pi/j, for all * £ # s ;

such that Q "%

Q = - 2p^, for all ty£ %-.

If the groundstate of the chain were unique there would not exist any superselection sectors, and

Q s 0,

on all physical states. On the other hand for a chain with n-fold degenerate groundstate one can construct n(n-1) charged sectors. The proper mathematical framework for the construction and analysis of these sectors appears to be the framework of local observables and local morphisms axiomatically developped by Doplicher, Haag and Roberts, [DHR].

33 139

We will construct charged states by composing a groundstate with a charged local morphism (= a "generalized transformation") of the observables. The analysis of the spectrum of the energy-momentum operator on the new sectors

X , Z~/ however, must

apparently be done in analogy to the analysis of the surface tension in classical ferromagnets; [GMJ. Unfortunately the phenomenon described here is typically one (space) dimensional. In more than one space dimension more complicated constructions of the kind described may be possible in much more complicated models (possibly only in models involving gauge fields, i.e. gauge theories). In one space dimension, however, the phenomenon of spontaneous occurence of charged super selection sectors seems to be an observed fact (e.g. in linear (long, thin) systems of non-linear optics, where the twist regions in Fig. 2 are associated with pulses of the electromagnetic field [La]). Thus it is not merely a mathematical curiosity. Even in two space dimensions it can be observed experimentally: It describes the occurence of vortices in superconducters (described by a gauge theory!), and the charge Q is then related to flux quantization. What I have said here is a rough picture of the subject material of my lectures. In the following sections we want to make that precise: I will describe the framework, formulate the results in a mathematically precise manner and present some of the proofs. For brief overall information consult Section 2, 5-7 and 9

140

Remark: We will often add in between brackets some comments of mathematical or technical character directed towards the more mathematically inclined reader. If some reader finds such a comment confusing he should simply ignore it. Acknowledgements: The reason why I can present some new results on phase transitions that I find rather exciting is that I had the luck of collaborating with two clever colleagues: Barry Simon and Tom Spencer. I wish to thank them for the joy of collaboration and for permission to present results that are not yet published} (Sections 3.1, 4-6). I am much indebted to Erhard Seiler and Sidney Coleman who have taught me many things about the material in Part 2. I am grateful to James Glimm, Elliott Lieb and Charles Pfister for useful discussions.

Part 1: Section_22 FERROMAGNETIC MODELS IN CLASSICAL STATISTICAL MECHANICS AND RELATIVISTIC BOSE QUANTUM FIELD THEORY: MAIN RESULTS 2.1 The framework and the class of models: All the models discussed in my lectures may be interpreted as models of classical statistical mechanics. A classical, physical system is specified by its phase space r, and a state of such a system is represented - mathematically - by a probability measure du on r.

35 141

(Technically, one can always choose a topology on r such that r is a compact Hausdorff space and then choose as a o -algebra Z

the Borel sets; dy is then assumed to be a

regular Borel probability measure). The dynamics of the system is given in terms of a (Z

-measurable, often once

continuously differentiable) Hamilton function H on r. Let {Q},{P} be canonical coordinates for some local (e.g. some bounded, open) region in r. We will always deal with a Hamilton function of the form H({Q},{P}) = HQ({P}) + V({Q})

(2.1)

(with H e.g. a quadratic form in {P} the canonical momenta, V some potential only depending on the coordinates {Q}). The existence of solutions of the Hamilton equations of motion is not discussed at all. We limit our attention to the construction and analysis of the Gibbs equilibrium states which are formally given by z -1 e - e H( { Q},{P}) d { Q } d { p }

=

(2.2) =

Z;

1

e-

eH

o(tP}) d {P} • Z"1 e " B V ( { Q } ) d(Q},

where d{P} d{Q} is some factorizing a priori measure on r. We note that this state factorizes with respect to {P}, {Q}, (and that, for H Q a quadratic form in {P}, the first factor is simply a Gaussian measure, denoted d f ({P}) . «o If F is some observable, i.e. F = F ({Q},{P}), a E measurable function on r, we define ({Q}) =

F({Q},{P}) dj> o ({P})

142

The expectation of F in the Gibbs state is then formally given by

Z-1 [F({Q}>

e"

eV({Q})

d{Q}

(2.3)

These observations permit us to eliminate the discussion of the momenta {P} completely, and hence forth we limit our attention to the construction and analysis of Z -1 e-ev({Q}>

d{Q}

The observables of the system are functions F({Q}) of the coordinates {Q} alone. We also change our notations: {a} H {Q} (2.4) H s H({a}) = V({Q}) For details about the foundations of classical equilibrium statistical mechanics see [ R ] .

The class of model systems the Gibbs states of which we are going to analyze consists of classical lattice systems; These are systems on a cubic lattice 2 V with lattice constant 6 > 0; (unless otherwise stated 6 = 1 ) . The observables of these systems are functions of classical "spin" random variables {a } _v : with each site a£Z a a€Z there is associated a random variable a

with values in

1R ; N = 1,2,... is the number of components, and a be

may

interpreted as a classical spin (for N=3) or as the

37 143

position coordinates of some family of oscillators attached to site a; (i.e. we are dealing with classical spin systems, or, interpreted differently, with anharmonic crystals) . With each finite set B C 1 algebra £ D of Borel sets in B and with B = Z v of

X

we associate the aN N N 1R, .,(where TR. . - TR ) , (a) (a)

X _ a£Bn the a-algebra I of Borel cylinder sets

J)®, . .

ae* a.e.2

(2.4)

a.

In order to be able to specify the Hamilton function we must introduce periodic boundary conditions: Let A be a finite rectangle in 2 ; A = {a£?V:a = a

+ 1,6, + O

1

1

+ 1 6 } , V

(2.5)

V

where a o is some fixed lattice vector,' 6.I is the unit vector with components 6. ., j = 1,....,v, and the 1.'s are integers with O < 1. < L., for some positive integers L^, i=1,.... ,v . A point a of the form a = a + 1..6.. + .. . .+ (L. + 1) 6 . + . v..v .+1 6 o 11 J j with

is indentified

144

a = a + 1.6. +....+ 06. +....+ 1 6 £ A , o 1 1 D v v^ in the sense that a s a - . a a Given a, we set a, s a + 6.. The j

(2.6)

component of a is denoted a , a a

and 3 X a j = - (aj - aj.) , a a 1 "+ Fij

E

(2.7)

31 aj - 3j a 1

With A we associate a cutoff Hamilton function

H*Ua}) = §

I I Oia ) a ct£A i=1

2

+ h- ( I a a£A

)

(2.8)

(In the analysis of lattice field theories involving vector fields one also encounters Hamilton functions of the form (2 9) Z F a j F ij,a ]a£A The constant J > 0 is the nearest neighbor-ferromagnetic

"I

({

°}) = 1

coupling (related to the field strength in the case of lattice field thee theories [GRSJ) anc^ h £ 7R (magnetic) field",

is the "external

The finite volume equilibrium state of the system defined by (2.4) and (2.8) with periodic b.c. at 3A (the boundary of A) at inverse temperature = 1 is given in terms of a probability measure dy

on

l.z

39 145

du^

A

({a}) = Z~ 1 e x p [ - H ^ { a } J] A

A

~[J dX (a ) ''

a £ A

a

(2.10) = Z" 1 e x p [ j I I a >a J Ji 2, d\(o)

= ^ {6 (0 + D + 6 (a-1) }da

(or, more generally, dx(a) some o+-a measure on JR, ^ S(a)da).

(2.13)

invariant probability

2i_Classical_rotator2. N = 2, v > 3, dX(a) = <S ( | a |-1 )d2a

(2.14)

3i_Classical_Heisenberg_model2 N = 3, v > 3, dX(a) = 6(|a|-1)d3a

(2.15)

4. (0 is variable. For conventional reasons we change our notation: a = = ( ,...., ) . We redefine 3 , namely 3 1 ». The limiting measure du is a probability measure on E.

15

The expectation (state) determined by dy noted by =

J,h

.

is de(2.22)

Theorem B; (Gaussian domination, [FSS]) Under the assumptions of Theorem A

(2.23) independently of dX and h. _p

Theorem C: (0(k

) bound, [FSS])

Under the assumptions of Theorem A, the Fourier transform dw (k) of the two point correlation function has the form doi(k) = [a 6

(k) + F

(k) ] d 3 k,

I

(2.24)

where 0 < F 6 (k) < ^ f ^ " ,

J independently of dX and h; (a is the long range order) Theorem C is a direct corollory of Theorem B. Although the proofs of Theorem B and C are elementary and based on a well known tool, the transfer matrix formalism, these results seem to appear for the first time in [FSS]. They form the technical core of method 1.

In Section 5 we combine method 1 with the bounds of Theorem C to prove occurence of phase transitions in

152

Models 2,3 and 6 and others. In particular we prove: Theorem D: (Phase transition in the classical N-vector models, [FSS]) Let v > 3, N = 1,2,3,... and dX(a)

e

= g ' 6(|a| -1) d N a

(2.25)

Then there exists some J < » such that, for all J > J , there is long range order (i.e. a > 0 ) . The state < - > ' is a mixture. There are at least S - many pure phases, and the internal symmetry group 0(N) is N-1 broken in at least S - many pure phases (i.e. there is spontaneous magnetization). Remarks: N-1 "S - many pure phases" is to be read as "as many pure phases as there are points on the unit sphere N-1 S in N dimensions". The group 0(1) consists of the two elements a •+ a and a -*• -a. Theorem D can be extended to all single spin distributions dX € K

invariant under 0(N) and ^ 6 (a)d a,

and, for N = 1, to a class of single spin distributions dx€ U K, without any symmetry, at all, and to ferro|h| a , the long range order a is positive and the physical vacuum of the theory is degenerate;

(2) for N = 1,2,3, all h = h«e, where e is an arbitrary unit vector in m , h ^ 0, all Wightman axioms [stW,Jo,0S]are satisfied, including uniqueness of the vacuum , and

154

f,

h. .-*•,

-*•

,

.

+-+•

l i m d u (<j>) (o) = M e , M h+o >

*

> 0;

for N = 2,3 and h = 0 there exist N - 1 Goldstone bosons; (zero mass, scalar one particle states).-1

In Section 7 we give a simplified proof for the 4 occurence of phase transitions in the _ quantum field model (Model 4 for v = 2 and N = 1 ) , [GJS2]. The proof is based on the Peierls argument in the form of [GJS2]; see Section 3. Our techniques can be extended without difficulties to general P()? models [E,Si], (where P is a positive, "almost even" but not necessarily even polynomial) the pseudoscalar Yukawa- TY„] and the sineGordon model (Model 5 ) ; see [F3] and Section 7. What we shall show in Sections 6 and 7 is, in essence, that the following principle "governs" the occurence of phase transitions in super renormalizable field theories in two or three space-time dimensions:

l

this result is due to [F4] based on results of [Fe 0, Ma Se].

155

Principle: [F3, FSS] Let V be some superrenormalizable interaction term which depends on (possibly among others) a (pseudo-) scalar Bose field £ and is "almost" invariant under a substitution transforming into -. Assume that, associated with the formal Lagrangean V U ) - ja ,-*.

-*•

-*•

: O, there exists a theory satisfying all Wightman axioms with the possible exceptions of Lorentz covariance and uniqueness of the vacuum, (put differently, V(J) - -| : t ' t • is stable, or V(£) stabilizes - -~ : • :, for all a > 0) . Then (a) for N=1 (i.e. ) and a large enough and (b) for N=1,2,3..., large enough and in three space-time dimensions, the physical vacuum is degenerate, i.e. there is a phase transition. We will see in Part 2 that in two space-time dimensions this principle has an interesting consequence: the existence of a phase transition implies the existence of new, charged superselection ('soliton') sectors, [F7J.

In Section 8 we mention some results for classical ferromagnetic lattice systems that follow from the Peierls argument, and we draw some general conclusions.

156

Section 3: TWO GENERAL METHODS IN THE THEORY OF PHASE TRANSITIONS The two methods of proving the existence of phase transitions we explain in this section, Infrared Domination and the Peierls Argument, are of a general nature (as opposed to methods based on deriving exact solutions, etc.; see [PTJ). In principle they are applicable to general systems with infinitely many degrees of freedom in a space (-time) represented by ill i or by some v dimensional, regular lattice, typically £ , and v > 3, v > 2, respectively. In practice, however, successful applications have thus far been limited to (classical), ferromagnetic systems, (a class which is larger than it may perhaps seem). Our systems are described in terms of some C -algebra 01 of local observables and a state < - > on 01 ; ( Ql may be an algebra of bounded functions of some fields, an abelian algebra of random variables, etc. In statistical mechanics the state < - > is typically a Gibbs equilibrium state, in quantum field theory e.g. a Euclidean vacuum expectation value). In the following we may imagine that the underlying space is j/lv,

(since a v-dimensional lattice can

be embedded in 7/1 ) . We assume that OL has a local structure:

The following general remarks are not to be read with too much emphasis on precision.

15

With each bounded, open subset B c | v there is associated a local algebra Oi (B) , and if B.. and B„ are disjoint - (in local, relativistic field theory, if B. and B_ are space-like separated; a case not considered in the following) - all the elements of 01 (B1) commute with all the elements of 01 (B ? ); Ql is assumed to be the norm closure of

U

Ot(B) •

Bcnv

From 01 and < - > we obtain a Hilbert space JC, a cyclic vector U, and a representation i of Ct on X , by the G,N.S. construction; see e.g.

[R,H].

Let Ot (~B) denote the weak closure (on X ) of D

y

D

TT(&(B..)).

Furthermore, let

<X„ = B c f v ( ^ ( ~ B ) ' and

%

(3

= {Afi:AG p/

so

(3.12)

that

C > T Thus, if

(3.13)

c*A = - ||2 - C > o,

(3.14)

then dim

%_ > 1 , dim It I

> dim oo —

T

> 1 , and a„ > 0.

J.

A

See [RjH] for more information and references.

3.1 Infared domination, [FSS]: This is the first of the announced two methods for proving the existence of phase transitions. Without loss of generality we may choose an observable A that can be approximated in norm by strictly local observables

"arbitrarily rapidly and such that the Fourier transit form doo (k) of has support in some compact set B. (In the case of a theory on a lattice this is automatically satisfied; B is the first Brillouin zone. In the o continuum case, let A be a strictly local observable, h a Schwartz test function, the Fourier transform of which has support in B, and define A

=

h(x) A ) . x

J Obviously C = + .

By the definition of P , is independent

Therefore *

=

(3.15)

(where, for a lattice theory, "grad" is the finite difference gradient). In the case of a relativistic field theory the results of [AHR} prove that < C ||h||2

2

where A (h)

(3.16)

2 h(x) A•x' x , ||h|\\ I N H == J |h(x)| , and, in the

case of a canonical Bose field theory, the constant C is finite and independent of the specific model. The analogy between canonical Bose field theories in the Euclidean description and classical ferromagnets [GRS, SG, DN] suggests that, for a suitable choice of A, inequality (3.16) holds in classical spin systems with

Acta Physica Austriaca, Suppl. XV

162

nearest neighbor-ferromagnetic couplings, with a constant C only depending on these couplings, but independent of the single spin distribution dX. This is proven in Section 4; see [FSSj for the original results. We now assume inequality (3.16) and discuss its implications. It is asserted that it immediately gives an upper bound on , provided the dimension v of space is at least 3: From (3.16) we obtain doi(k) = [y«0(k) + F c (k)]d v k, where y = |[2 + aA' A , and 0 < F c (k) < C k

(3.17)

2

The proof is immediate, But from (3.17) it follows that .* c „ f dvk < C _

(3.18)

which is finite for v > 31 grable at k = 0 ) . In order to prove that a - II2 > C

(For v < 2, k

-2

is not inte-

> 0 it now suffices to show that

^ ^k 2

(3.19)

This estimate always depends on the characteristic features of a specific model, in contrast to (3.16) so that no general methods are available.

163

In a pure phase *

1 1 9

*

C

||2 = - > - C ( —

(3.20)

This inequality is often very useful for proving discontinuity of , e.g. "spontaneous magnetization", as some coupling constants of a given model are varied, which also proves the existence of a phase transition. We note that (for A = A ) inequality (3.16) follows from |<eZ ^ a d A ( h ) H ^ e C'|z|* ||h||2

(3>igi)

This is a consequence of analyticity in Z and the Cauchy estimate; see Section 4 and [FSS]. Finally we remark that the method for proving the existence of phase transitions consisting of inequalities (3.16) or (3.16') and (3.19) does not explicitly depend on the internal symmetries of the system described by 0i>

< ~ >•

As phase transitions are often accompanied by the spontaneous breaking of such symmetries it is now clear if not already obvious from (3.18) - that this method does in general not apply to two dimensional systems , as in two dimensions continuous symmetries cannot be broken; [M,E S W ] .

164

3 .2 The Peierls argument, [Pe, GJ52]: This is the second general method for proving occurence of phase transitions we propose to review. Let the algebra 01 and the state < - > be as discussed above. We assume that the G.N.S. Hilbert space X reconstructed from 01 and < - > is separable. n,

We cover

with a grid of mesh 1 . Let 0 denote

the family of all unit cubes of this grid, and let $ be the collection of all faces of all unit cubes in £. As described above, with each D € C (or each finite union of cubes in C ) there is associated a local algebra

Ol ( D ) , and t ((!(•)) = f f ( D J , f o r a l l x f / , X

(3.21)

X

where Q is the translate of 0 by the vector x. Let P.. ,P_ , . . . . ,P be m > 2 commuting, self adjoint projections in 01 ( O ) with P.+P-+...+P 1 2 m

= I

(3.22)

and > e > 0, (e < i

-

-

- ) ,

(3.23)

m

for all i = 1,....,m, and all x in Z"v. Here P ± (x) E T X ( P . ) £

OL (D

) • (If < - > is translation in-

variant it suffices to assume (3.23) for one x ) . Next,suppose we are able to prove that, for all (sufficiently large ) x and y and all i ^ j

< 6 e 2

(3.24)

for some constant 6 < 1. Then, for any sequence {x }°°_ of points diverging to » for which w - lim P.(x ) exists, for some i, n

•+ «»

w - lim P.(xn) f xx ->• oo

{lim n - *" 00

} I,

(3.25)

an immediate consequence of (3.23) and (3.24) . Since It

is separable, we can always choose a sequence

{x } „ diverging to such that n n=o IT. s w-lim l

P. (x ) i

(3.26)

n

exists, for all i = 1,...., m. Clearly ifj_ . e Qi , so that \l>. =iir.ft e °° If . We therefore -* ^uoo i conclude from (3.25) that the state < - > is not a pure phase in the sense that dim

3?

> 1 .

From (3.23) and (3.26) we have that (fi,i)j.) = > E , for all i, so that (i|».,ij)i) > e 2 , in particular \p. ^ 3, for all i. Furthermore for all i ^ j ,

(3.27)

166

(*.,*.) = lim [lim ] 1

•,

J

J-

11

n

(3.28)

Js.

< 6 e 2 , by (3.24) and,since

= < P i ( x n ) * P.(xk) P i ( x n )> > O, (*i,^ c(z')jt±

,


n (Dz,

(• ^

K Y i e" I I

(3.34)

Dz,)eN(Y)

for some constant K > K (v,m,e,6), where K(v,m,e,6) is a fixed constant only depending on the dimension v of space, the number m of projections and two positive numbers e and 6 (see (3.23) and (3.24)). Then

(P^X)

P.(y)^

For K > K(v,m,e,6

< 6 e 2 , for all j?£i.

(3.35)

) the state < - > is a mixture of

170

at least m pure phases; (see Proposition 3.1).

Remarks: For the purposes of these lectures it suffices to prove Theorem 3.1 for the case v = m = 2 considered in [GJS2]. We remark however that the analysis of the general case is perhaps a little more than an academic exercise: It is important in the analysis of multidimensional systems which are expected to have tripleor m-tuple points (i.e., for certain choices of the parameters of the system,' it is expected to have at least three, or m, pure phases). We also note that method 3.1

(infrared domination)

generally only proves existence of phase transitions without

proving more than obvious information (derived

from the structure of the internal symmetries) about the manifold of pure phases. This is not so in method 3.2 (the Peierls argument), as one learns from Proposition 3.1 and Theorem 3.2. Sometimes it is necessary to combine (among others) both methods to get the desired information on the structure of the pure phases of a system. Proof of Theorem 3.2 (for \>=m=2) ; For the purpose of good intuition we write P

=

P + , P 2 = P_ ("spin up-down"). 2 We now pick two arbitrary points x and y in % . It is then to be proved that (3.34) implies (3.35). The family C

is now a family of unit squares, the set 1Q

171

of their faces a family of bonds. Let R , , be the (x,y) smallest rectangle of unit squares containing the squares • x and • y', and let 3R (x,y) be the length of its boundary, (i.e. the number of bonds in 3R, , ) . (x,y)' Let now A vx,y; . . be an arbitrary rectangle of unit squares with the property that all contours Y G r , , , (x,y)' where r.vx ,y;. has been defined in (3.31), which have distance 0 from 3A, . (i.e. touch 3A, .) have at least length

3R (x,y)

There are two adjacent faces f.,,f9 of R. (of i ^ ix ,y) total length -^ |3R,„ „,|) such that each contour y£V , , c ^ ix,y) (x,y) contains at least one bond b.(y) (a "bond of entrance into R, ,") contained in the interior of R, . that ix,y) (x,y) touches f u f but none of the other two faces of 3 R , l 1 (x,y) Figure 3 below shows two sites x and y, the corresponding rectangles R, . (the shaded region) and A, 3 (x,y) (x,y) the faces f11 ,f and three contours in r, . with some ^ ix,y) bonds of entrance f«

2 A

JL. •i

1

l

--

-, r -

—j r

""

--

:;

W£?&Y*3 P r

Rr.

*>y)

(x,y)

ti

3

W— -~MW? s z £ ^ T^=: i'

i — . —i

ii

—J| 1

'i i1

*ir-

, 7 ^ 11

: [ — n_ i!

2

f?

3

Fig.3

172

There are TT |3R, ^

> | different bonds of entrance

ix tY)

for the contours in r, ,. This follows from the de(x,y) finition (3.31) of r. . and the definition of the class ix ,y) of bonds of entrance. For the class of contours in r, > \X r Y /

of length n there are no more than min {n, ^ |3R (x y ) I>

(3.36)

different bonds of entrance, a consequence of (3.31), the definitions of R, , and of the bonds of entrance. (x,y) Using now the definitions of 1, . and r , . we con^ (x,y) (x,y) elude from (3.36) that, for fixed n, there are at most n different bonds of entrance into R, ., for the class (x,y) of all contours in r, . of length n. (x,y) ^ If we now choose the bond of entrance as the first bond of a contour in r, . and apply a standard (x,y) ^ x argument we conclude that, for fixed n, there are at most n 3

n-1

(3.37)

contours y€ T.vx ,y; . with |y| = n. Finally we note that the length of a closed contour Y € r,

. is even. For closed contours (3.37)

may obviously be replaced by n 3

.

(3.38) (3.37')

173

Lemma 3.3 Let K be the constant introduced in Theorem 3.2, (3.34) , and assume that K > log 3. Then (p (x) P (y))

I ( TT N( "^ (xfy) °z'nz'> r

P+(z) P (z')\ " '

oo

V i < 2 2 n n=2

,2n-2 3

-K-2n e

'

where E(A, . ) represents the sum over all terms (x,y)' ^ labelled by contours Y£T, , that are not closed. But, (x,y) by assumption (3.34) and (3.37), E(/\. . )*0, ix, y) 2 v / as A, fL (x,y)^/ "

174

The proof now follows immediately from (3.37), (3.38) and the assumed inequality (3.34) . Q.E.D. Remark: For v>2 and m arbitrary (3.39) is replaced by (PI(X)P

(y))


2) follows from oo

—

3.1 inequalities (3.16) and (3.19), o£ 3.2 inequalities (3.23) and (3.34), and that information on the number of pure phases can be obtained from combining 3.1 and 3.2 with the structure of the internal symmetries.

175

In Sections 4-8 we prove these inequalities for various model systems (Models 1-6), thus establishing the results announced in Section 2.

Section 4: INFRARED (GAUSSIAN) DOMINATION AND THERMODYNAMIC LIMIT In this section we prove inequalities (3.16) and (3.16') in the form of Theorem B, (2.23), for the correlation functions of the classical, ferromagnetic systems introduced in Section 2, (2 .8)- (2 .11) , and for a class of Euclidean field theory models. These estimates, in particular Theorem B are basic for (1) taking the thermodynamic limit A / 2 , (2) applying method 3.1 to proving the occurence of phase transitions, see [FSS]; (3) the proof of the basic inequality (3.34), Theorem 3.2, in method 3.2 (the Peierls argument). The inequality we are going to prove asserts that the correlation functions of such systems are dominated by Gaussian correlations with covariance (= two point _2 function) in momentum space ~0(k ) . It is related to and motivated by the grad *- bounds of canonical quantum field theory, [GJ2, He].

176

4 .0 The 9 1 a- (grad (ft-) bounds: Theorem B Let H ({a}) (Hamilton function) and dp ({a}) (Gibbs state) be as defined in (2 .8) , (2.10) . We pick some direction % , hence forth called Tdirection (the direction of transfer). Without loss of generality this may be the 1-direction. The hyper plane in Z v perpendicular to T and passing through the origin is denoted T x . Let the space cutoff A be a rectangle of the form A.= [-L,L] x A. (4.1) W l th

Kx = [-L2,L2] x....x [-Lv,Lv]c

J T

The point a = (Jlj,...., L .+1 ,...., X. ) is identified with a = (l 1 , . . . . , - Z , . , . . . . , 2. ) ,

(4.2)

(corresponding to periodic boundary conditions). Let , = .' A

denote expectation in dp,. c

A

A

Theorem 4.1: (H Theorem B, [FSS]) For v=1,2,...., N=1,2,.... and all single spin

dis-

tributions dX in the class K, defined in (2.20) r

u

/

\

">%•

T

^ „ hl „ ( a ) - 3 aa) ^ , AJ (e a.'*

1


4 a , _ —

• ^ »

z

/

T J

-

,

- j ' ' " " * '

> 8 a, , ,

i

(f)-3a

T, -

(f) > i * rlj2

v

' ' ' ' ' '

\i

(f) - 4 a , , ,

T V

f

»

(f) r

f

y

- 3 a

> 8 a, , , £t

I £t j & I m m •

(f)

(f) - 7 a , f ±J

and we h a v e u s e d

\ f \ f \ f m t m f

(4.34)

again,

L V

(f)

> 2Va-

(f)-(2 V -1) ai

.(f)

This completes the proof of (2). Q.E.D. The following inequality is sometimes more convenient (see Section 6 ) : Suppose that, for all L.. , . . . . ,L aT

(f) < 5(f)

T

1

V

Then aT

T

(f) > 2Va„

Estimating a

(f) - (2V-1)o(f)

(4.36)

_ (f) for our lattice systems is a

straightforward task which we leave to the reader. An immediate consequence of Lemma 4.6 is Corollary 4.7; If

f(0) e h '°dx(o) > 0

lim a.(f) = in f a.(f) = a A A7>ZV A A

(f) exists.

We are now prepared to extend Theorem 4.1 and Lemma 4.5 to arbitrary single spin distributions in the class: ,~ . , K, = {dX: £j e > o such that e eI'a 'I e h'tr dx(a) < (A)

X

and, for dy -almost all x £ X ,

is a pure phase state. A

The measures corresponding to different pure phase states are mutually singular. If dim

X

= m there exist m different pure phase

states, (a consequence of the definition of Ot

and (3.2))

Remark: For the interested reader we remark that, for dy OO

almost all x £ X , the states

and

satisfy the

199

same "Dobrushin-Lanford-Ruelle equations"; [R,GRSj. Theorem 4.10 has been extended to the continuum limit (6=0; ffv -v flV) , i.e. the case of (Nelson-Symanzik positive) Euclidean field theory. More precisely, let du be some probability measure defined on the a-algebra generated by the Borel cylinder sets of $

'

, ( )H )

real which satisfies Osterwalder-Schrader positivity

[OS] in

the form of [F5] and the moments of which are Euclidean Green's functions (Wightman functions restricted to the Euclidean points) of some relativistic quantum field theory. Such a measure is called a quantum measure, [F5]. Let Z

be the smallest a-algebra such that all Tt

functions in du T = du|

(see (3.11) — (3.12)) are Z -measurable,

. Let X be a convenient model for the support

of du .

Theorem 4 . 1Q' :

[F5]

For all A££ u(A) =

du T (x) U V (A) , , A

1

X

where, for du-almost all xeX, u

is a Euclidean in-

variant quantum measure ergodic under the action of the translation group. Different "pure phase" quantum measures are mutually singular. Theorems 4.10 and 4.10' will be used freely in the subsequent sections without explicit reference. The statement "there are n - f

Let dai (k) be the Fourier transform of

/ a a > . From \ o a' Theorem 4.9 we obtain, (see also Theorem C, Section 2 and (3.17)) doi(k) = [y60(k) + F C (k)]d V k, where y = I (o ) I + a , and 'No'1 a0 C 0 < F (k) < ^T [2v - 2 { I cos k * } ] - 1 Jl=1

> I

(5.3)

202

Integration of (5.3) and use of (5.2) gives ,1-1 V 6 2 < y +. I 7f J R d.v,.^ k [ 2 v - 2„ r{ ^E ___ c o s k, . 1 }] J B £=1 1 = Y + const. —

(5.4)

Here B is the first Brillouin zone and the constant on the r.h.s. of (5.4) is finite, provided v >_ 3; (for v=2 the integral diverges logarithmically at k=0). Next we note that Theorem 4.9, since

^

CT

(a ) = 0 . This follows fr om J = 0, for all finite A. 0)

Thus, using (5.3) and (5.4) 2 > 6

a o

1 - const. —

—

(5.5)

j

0

This is obviously positive for large enough J. We may now apply Theorem 4.101 In a pure phase there is n£ long range order. Thus / \ J \ o o /

/ \ o

a

\ J,T o'

o du

i.e.

( ){ (a -a ) J - | (a > J | 2 } 6o(k) + F C (k)]d 4 k, where y = | ->

-X (A -A a a e

. 2

(5.9) -+

-*•

+aA -A ua

. d A

01,4,

a

From Jensen's inequality and the chess board estimate (Lemma 4.5 and Theorem 4.8) we get -2a (A -A )

° °

-2aA -A

i /e

a

° °\

a (1 ,X,-a)-a (1 ,X,a) (5.10) From (5.9) and (2.9) we get (1) a

(J,A,-1) < a (0,0,-1)

(2) a

(J,X,1)

> Const. X

, as X ^ 0,

uniformly in J £ [0,1] . ,1 ^r,1) A n ,+ 1 „ (3) a„ (1,A,a) = ara(^,

205

Combining

a

(1,X,-o)

(1)

-

-

a

CO

(3)

we g e t

(for

a>1)

(1,X,a) 0°

< aoo(0,0,-1) -a„

(1, -^,1)

< - Const, a 2 , as a /

(5.11)

Taking the logarithm of (5.10) and using (5.11) we conclude : (A-A)>0(a),asa;"°° x o o' Theorem 5.2 now follows from (5.7), (5.8) and

(5.12) (5.12). Q.E.D.

Remark: Of course, (5.12) agrees with the prediction of the naive Goldstone picture, [Co]. A physically more interesting result of the type of Theorem 5.2 is discussed

in [F3] . Theorem 5.3: ( [FSS]) Let v>3, N=1 and dX some Borel probability, measure on % with the properties that 1 . dX has no symmetries 2. dX£ /O K, . h

h

3. There exist positive numbers e and 6 such that X ( (—,-6]) > e, X ([6,«) ) > e.

206

Let HB ({a}) =

v 7 I { I a aeA 1=1

a . + h a }; a

(see (2.10)) .

Then there exists some finite J„ such that, for all J > J ,

(a

\

'

is discontinuous in h at some h=h

(J), for at least one h

.,

(J), and there is a phase

., (J)) which is not accompanied c crit. by the spontaneous breaking of any symmetry. transition at (J, h

Proof: We prove Theorem 5.3 for the special case, where X([-6,5]) =-- 0

(5.13)

For the general case see [FSS] . Cleary estimates (5.3) and (5.4) apply to this case, thus using (5.13) and (5.4),

e', for all J > J_(e) - C

(5.15)

Using now hypotheses 2. and 3. we see by a very simple argument that lim

h/>-

(a X

\

°

h

' J > o, lim

h^ —

(a X

\h'J < 0

(5.16)

o/

h J Combination of (5.15) and (5.16) proves that \/a )\ ' has at least one discontinuity in h. Thus , for some h = h . , (J) , there exist at least two differentr pure crxt. phase states.

By hypothesis 1. this phase transition is not accompanied by spontaneous symmetry breaking. Q.E.D. Remark: Theorem 5.3 also holds in v=2 dimensions. The proof is then based on the Peierls argument. See Section 8. Remarks: The results proven in Theorems 5.1 and 5.3 are both very satisfactory and insufficient: They are very satisfactory, because they show that in v > 3 dimensions spontaneous breaking of continuous internal symmetries does occur and that phase transitions are not always accompanied by spontaneous symmetry breaking, (i.e. the concept of phase transitions is more fundamental than and independent of the one of symmetry breaking. This can also be shown in 2 dimensions; see Section 8 ) .

102 208

They are insufficient, because no more than trivial information, derived from the structure of the internal symmetries, about the manifold of pure phases has been achieved. By choosing suitable single spin distributions dX essentially any manifold of pure phases compatible with the internal symmetries can be achieved. This weakness of method 3.1 can often be cured by combining it with the Peierls argument, (method 3.2); see Section 8. The relevance of results of the form of Theorem 5.2 must be discussed elsewhere.

Section 6: THE (£•!') | - QUANTUM FIELD MODEL

In this section we prove the existence of a phase transition and the spontaneous breaking of the internal 0(N)-symmetry for the (•)? - quantum field model (4> = ($ *,. . . . , ) is an N-tuple of real, scalar Bose fields). For N = 2,3 we show that there are N - 1 Goldstone bosons, (zero mass one particle states). This establishes Theorem E, Section 2. 6.1 Existence and Wightman axioms We briefly summarize the basic results concerning

103 209

existence and Wightman axioms for the {§'$)':

- models.

These results are very deep and difficult to prove, so that all proofs will be omitted. First we recall the definition of these models, in the case of a positive lattice constant , where

(6.1) 2

2

L U ) = \(-cf>) - (M (A,6)+a) <j> • o, a > o and all h ^ o . \.a.Z,i, ,. dy (4) = lim

e

-|A|a (X.a) a:*-*:(xA) Xp l t , e dy ' (4)

A^fft3

A

exists, in the sense specified in Theorem 6.1. There exists a sequence {h } converging to 0 such that ^ n n=o ^ ^ for all unit vectors e e %~ , X,a,e,->. ,. , X,a,h e ,-*-. dy (

X

>°'*

= (M*)2 + — ! — (2TT)

3

f d \ e i k x F C (k) >

a consequence of Theorem 4.9 (see also (3.17)) and Theorem 6.2, ( 2 ) ; (in particular, the cluster property , a',)e .. of.c X ' Theorem 4.9 (in the limit 6=o, see (4.41)) gives

214

O < ( k X ) 2 F C (k) < N, for all i=1,2,3. From this one can conclude that 0 < F C (k) < ^ k

, (see [FSS] )

(6.9)

(This can also be concluded from (6.8) and the KallenLehmann representation) Thus , \\ (o)X),a ,e aJl.oJ-aJHrO) ,

(6.11)

and it suffices now to prove positivity of aoo(A,a) - a 12)

for all bounded cubes B cz y/[ . By scaling lengths, X and a it is easily seen that it now suffices to show that, for fixed a>o and some bounded cube B,

a:$•£:(x R ). X,o (e

\

> 1

(6.13)

for sufficiently small X. By the chess board estimate -a : • : (XTJI

^/O

[B| a (X,-a)

< e

(6.14)

and the r.h.s. is uniformly bounded in X € [o,X ] , for each finite X ; by

[GJ1J,

From [MS,Fe 01] we know that < [_:<j> - : (x B ) J

)

converges to

110 216

([:£-$: (x B )J m } 0 , as X 'H o, for all m < «. Here

is the Euclidean (Gaussian) expectation of a

free massless field; see (6.7). Finally

M I m=o

2m ^

„ < [:?•?: (xB> ] 2 m >

(6.15)

Q

diverges to +°° , as M->-°°, for large enough | B | , as o n e varifies by a n e x p l i c i t calculation. Combining these facts w e conclude that / a :3 the occurence of phase transitions can still be proven (by almost identical arguments), but Lorentz covariance of the theory is unknown. (2) For N=2,3 and o>o (.\),

the physical mass of the

217

theory tends to 0, as h ^ 0, linearly in h; see [F6 , LP] . (3) It is shown in [DN] by means of correlation inequalities that

<J(e4) (o) (e-f) (x))

X

' ° ' e = 0(( ((e/\+) (o) • /"*" t\

i \ \ X ,a (eA*) (x)^

as

,e.2.

) ) ,

I x I -*-» .

Section 7: PHASE TRANSITIONS IN TWO DIMENSIONAL QUANTUM FIELD MODELS In this section we consider the P() -, [E,Si], X cos(e<j>)2-f

[F1 , FSe] , and the pseudoscalar Yukawa.,

quantum field models, [ Y ? ] . We prove that, for certain values of the bare couplings, these models have phase transitions which are sometimes, but not necessarily, accompanied by the breaking of an internal symmetry transforming <j> into -. We consider the P() = lxm

Ml

2

, P+h n x ,,. dp ° () A

exists, in the sense of Section 6, (and the limiting measure is independent of "classical boundary conditions"; see [FS] ) . (2) &VP+ ()„ and pseudo-scalar Yukawa. are presented in [F3] . Proof: Cover %

with a grid of mesh 1. Let •

be some

unit square of this grid. According to method 3.2 we must define m

mutually orthogonal projections on

T = L2(J real ( ^ 2 ) ' ^ t * w h i c h a r e "supported" on D , i.e. Zn - measurable functions on / ' . (% ) , and the Lj real sum of which is the identity:

222

We let x + ( D ) be the characteristic functions of the measurable sets U e supp dy 0 : * ( D > 7

0 } c

^'real

(

^2)"

(7

'7)

In the notation of 3.2 we have m = 2, P = x , ( Q ) / P 2 = X _ ( D ) ; obviously X + ( D ) + x_(D) = 1. 2 Suppose now that, for all x e 2? , <X+(Oo>

X_(Dx))

±

< 1-

6,

(7.8)

for some 6>o. Since

+ is a pure phase state, by Theorem 7.1, (3),

<x + (D 0 ) x _ ( D x ) > ±

- <x+(Do)}±

<x_(D0))±.

as ]x| ->- oo.

But < x _ ( n o ) > We set

(7.9)

±

=1-

<x+(Do))±

( x + ( D o ) } ± = M*

(7.10)

By construction of the state ,, see Theorem 7.1, (2), and the assumed evenness of P M* > 1/2

(7.11)

Combining (7.8)-(7.11) we conclude M* - (M*)2 1 + /s T

-t~4

+

—

+ ~ Z

(7.12)

22

Furthermore M_ = 1-M

< -j - /«"

(7.13)

which proves the phase transition, and, since, for a state invariant under 4>-»— <J», /x

(•)/

= 1/2, this

also shows that the states + break the -»— $

symmetry

of the dynamics. (The chain of arguments (7.8) - (7.13) is taken from [GJS2J). We are now left with proving (7.8): By Theorem 3.2, inequalities (3.34), (3.35), (7.8) follows from the inequality / TT X . ( O ) X (D')\ + < e " K l Y | \ ( D , D')e N( Y ) / _ for all contours y g r ,

(7.14)

, (see Section 3) and some

(x,y) sufficiently large constant K. (By Lemma 3.3, K^log 3.5 yields (7.8)). Let J be some positive number to be chosen later. We let x + ( D ) be the characteristic functions of > J {*:•(•) < _j>, 2 and x + ( D ) the ones of {*:* (D ) £ [0,±J]}, Then

x±

= X+ + xj

(7.15)

We insert (7.15) into the l.h.s. of (7.14) and expand. This yields

118 224

/

TT

x.(D) X _(D") ) /

< ( • , D')€N(Y)

I

(7.16)

(

e (D )=1,2

(D) ,,_,. elU; ,-. x:E (D) (D)xI (D')(D') \+

TT

MD,D')£N(Y)

/ "

E(D')=1,2

Let N.(y) be some maximal subset of N(y) with the property that if (•.,,•.]) and ( D 2 , D ^ )

are two dif-

ferent elements of N.. (y) then • . f D'. , for i=1,2. Obviously Y

1 . |N1 (y) I > j |N(Y) I = '-j-' ,

(7.17)

where |N.(y)| is the number of pairs (•,•') in N (y) . We now label the pairs (•,•') in N 1 (y) by some index i e {1 , 2 ,...., |N1 (y) |} = I . We then define Xl(i)

= xl(D) xl(D'), x,(i) = x+(D) z

(7.18)

x3(i) = x?(D')i x4(D = x+(d) xf(O') S i n c e 0 min (y2+(1-y)2) - 2 (1 - 6) y€[o,i] = 26,

(7.33)

where

+'

are the clustering states corresponding to

the polynomial P defined in (7.32) which have been constructed in Theorem 7.1, (2). Next, Theorem 5.2 of ref. [FSj tells us that, for all but countably many h ^ o, lim

( X + ( D ) - x-

(D))^'h=

( x + ( D ) - x_ (•))

°' h (7.34)

and the r.h.s. of (7.34) is strictly positive, for h>o, and strictly negative, for h!'h . for h = h c r i f c >

Q.E.D.

Conjecture: For all positive polynomials R and fixed ae(1,2ir2) there exists some X

= \

(R,a) > 0 with the property that

for O < X < X , there is an h_ such that, for P(x) = XR(x)- | x 2 + h c x, P +

,

P

r

Our present estimates on a

and a

are not sharp enough

to provide a proof of this conjecture. Remark.: The results of Section 7 are the basic input for our construction of soliton sectors in Part 2.

Section 8; QUANTUM CRYSTALS AND MORE ABOUT THE PEIERLS ARGUMENT 8.1 Quantum Crystals (a straightforward application of [FSS]

)

234

Here we briefly sketch an extension of our results of Sections 4 and 5 to systems which may be interpreted as simple models of

anharmonic (quantum) crystals.

We consider a cubic lattice 2. = 2 . x....x 2. , where o o -i o — 1 v 1. = {x:x = 6-m; m £ Z } ,

(8.1)

o

i.e. 6. is the lattice constant in the direction i. i

We set „i

i

1

%

1

a

+

with a^, 3 1 as in (2.6), (2.7). v We abbrviate JJ 6 . by TT_6; i=1 1

(8.2)

Z,. o

A denotes a rectangle in

The Hamilton function associated with A is of the form H, ({a}) = 4r

I a£A

TT6

{ J J. (31a ) +ha } i=1

(8.3)

with periodic boundary conditions at 3A. Let dX be some single spin distribution in the class K introduced in (2.20), and let . denote the expectation associated with H. and dX. Then, for all Ac?^, A

—

0

the following slightly more general version of Theorem 4.1 holds:

h

129 235

I u(S 3g. (a) •3 1 a '. - l a\ (e 0 " 1 )

1

-j A

< e

7 - 1 i ,. 2 ir6 — |g. (a) i ' "- J i ' ^ (8.4)

/ •

2 a

The proof is almost as in Section 4 (Steps 4.1-4.3), details are left to the reader. An interesting example for this somewhat more general situation is an anharmonic quantum crystal: Here v=4, the number N of components of o is arbitrary; (a is now interpreted as an N tuple of oscillator coordinates). We let J 1 , J ? , J be arbitrary positive numbers; (for simplicity we may suppose that J =J = J = J > 0 ) , and e.g.

6

1

=

6

2

=

6

3

=

K

Furthermore

^4

=

M' ^4

=

2L+1' ^

=

1'2'3''"*"'

(8.5)

for some positive constants M and g. The rectangle A has the form A = A1 x [o,g] .3 with A1 a rectangle c. 2' The single spin distribution is given by

(8.6)

236

d\(a)

= e

- & *

Via) dNa

(8.7)

where V is some r e a l , measurable function on W f e - B V(a)

d

Na


const. > 0,

(8.11)

o —

uniformly in B, (which must be derived from the properties of V; see e.g. Section 5, Theorem 5.2) yields existence of phase transitions for sufficiently large B, for many (physically interesting) choices of V; (note that the truncated second moment is bounded above by OCB~ 1 ) by (8.4)) . We leave it to the reader to make a list of potentials V for which (8.11) is valid and to exploit consequences of the Lee-Yang theorems of [SG,DN], for the case where V is a quartic polynomial.

8.2 Some more consequences of the Peierls argument (a) Theorem 5.3 remains true in v=2 dimensions: This is an extension of a recent result of Sylvester and van Bejeren [SvB] to single spin distributions that are not necessarily invariant under a-+-a.

132 238

The (somewhat complicated) proof follows from the Peierls argument and is very similar in spirit to the proofs of Theorems 7.2 and 7.6. Rather than proving discontinuity of ' some h = hcrit. .. (as in Theorem tinuity x j of <x,(o)A+ A _ (°) >

at

5.3) one proves discon^

at some h = h c r i..t . (see

Theorem 7.6). Details are contained in some unpublished work of the author. (b) Anisotropic N-vector models; [Ma,KuJ Let dX (a) ^ S (a) d a be some 0(N)-invariant Borel probability measure on f/l • We define dX_(a) = dx (2-) , J /J and v > 2.

(8.12)

Let a finite volume Gibbs measure be given by dvl'J({a})

=

Z A ( E ,J)" 1 exp[-l

I aeA |a'-a|=1

W_ (a..) x ]J e e a dXj(a a ), a£A

{(a

_0

)2

+

e(a1

1 )2}-[

(8.13)

12 E J where W (a) = 2e(a ) - za • a, and let dy ' be some infinite volume limit of (dy^' }.

239

Theorem: For arbitrarily small, fixed e>0 (or for e=e J 1 log J, with e > -j(N-1)) there exists some J < ~ such F

7

that, for all J > J , there is a phase transition: dy ' is a superposition of at least two pure phase (ergodic) measures some of which are spontaneously magnetized. Remark: This theorem extends work of Malishew [Ma] and Kunz [Ku]. The proof is again based on the Peierls argument and is contained in (unpublished) work of E. Lieb and the author.

(c) Triple points Let v = 2,3,..., N = 1, and dX(o)e=g'2w1+1 J

(Wj 6(a-J)+6(a) + W j 6(o+J)}da,

(8.14)

where W j is e.g. a monotone increasing function with the property that W_ = 1, for J = J

.

> J , (for a

certain J^ < °°) . Let H A be the usual Hamilton function and du

some infinite volume Gibbs measure corresponding

to H A and dX

(as in

(8.14)).

Then

( d ) for sufficiently small J>0 du

is ergodic and

unique (in the sense that it is independent of boundary conditions used in the construction of the thermodynamic limit). There is no spontaneous magnetization. The proof

240

is by means of a "high temperature expansion". (c2) for J=J . dy is the superposition of at least three different pure phase (ergodic) measures, at least two of which have spontaneous magnetization. Proof: This follows by combining the results of Section 3.2 (Proposition 3.1 and Theorem 3.2, with m=3 orthogonal projections at each site: onto a=±J, a=o),

the

lattice version of the first inequality in (7.20) (see also (7.24)) and Gaussian domination (Theorem 4.1). Q.E.D. (c3) Suppose e.g. that W T / , « , as J / » . Then we conjecture that, for J>>J . there are precisely two different pure phase equilibrium states with nonvanishing, opposite spontaneous magnetization. (Presumably this follows from low temperature methods. It is known that under these conditions there are at least two different pure phases.) Remark: Examples of the type of (c) with an arbitrary degree of sophistication may be invented; (also for the case where N>1; e.g. dX (0) a convex combination of {6(|a[-j JO J)d 0} Jo — i , 0 < j.i )„ quantum field models, (with deg P > 6) .

Part 2: Section 9: PHASE TRANSITIONS AND THE SPONTANEOUS OCCURRENCE OF CHARGED SUPER-SELECTION (SOLITON-) SECTORS In this section we explain a basic connection between phase transitions and the existence of non-trivial super-selection rules ("topological charges") in two dimensional quantum field models. We show that (under suitable hypotheses specified in [F7]) the following holds. Theorem 9.1: [ F 7 ] In two space-time dimensions, any quantum field theory with an n-fold degenerate physical vacuum has n(n-1) charged super-selection (soliton) sectors (disjoint from the vacuum sectors). These sectors are labelled by the elements of a finite group, (the so called soliton-group).

Acta Physica Austriaca, Suppl. XV

242

We shall not present a general proof of this theorem at this place. This has been done in [F7,§6]; see also [Ro]. We rather illustrate the general theory of solitonsectors in two dimensions by studiing one specific model, (the anisotropic (*•) -model) . But we remark 2 that Theorem 9.1 applies to all the models analyzed in Section 7 and proves the existence of non-trivial topological charges and soliton states for those models; [F7J. Particularly clear interpretations of the physical mechanism responsible for the existence of soliton states can be given in the context of the sine-Gordon theory, with or without mass term; see [Co 2,3, F1, F Se, F7j; Perhaps the most interesting model is the pseudo scalar Yukawa model. Mathematically, the P (<j>) 2 -models (with non-even P) are most fascinating from the point of view of solitons: For these models the existence of solitonstates is still unknown. Here we propose to study the anisotropic (• 0

(9.4)

The Osterwalder-Schrader reconstruction gives two pure Wightman states u and m_ with the property that

(f) = 1 (f ) + „(f ) is the smeared relativistic quantum field, f^£^(t 2 )r for all i,j and W ( ^

are the

(pure phase) n-point Wightman distributions(obtained as a boundary value of the analytic continuation in the time arguments of the n point Euclidean Green's functions, i.e. the n

moments of ) .

244

In the following we propose to construct two states a) and u- interpolating between u

and OJ_ (the soliton-,

anti-soliton state, resp.). Our construction is best explained in the framework of the theory of algebras of local observables [HKJ adapted to the two dimensional models; see [GJ 2,3] Let $ (x) = (x,o) denote the time O-quantum field, and £(x) = (|^ 1)

(x,o)

(9.6)

As the ($•) - model is a canonical quantum field theory, TT1 ,TT are the momenta canonically conjugate to $..,*„, respectively. Let O be some finite union of compact intervals on space. We define a real, local Sobolev space of distributions supported in O by ,0V) , ( a (x) =

IT,

(9.12)

for x < -L, a (x) = 0, for x > L;

^-«(x)£L2(|). dx (9.12) is called the "soliton-condition". The automorphism a

is given by the following Bogoliubov

transformation _ ^-j)

=

$

> c

°'

Then (1) There exists a continuous unitary U„ of the Poincare group on automorphism group { T ,

H,

representation

implementing the

. } of

256

The advantage of this representation of the cocycle r -•(£;) is that the operators U and detailed estimates

. are known explicitly,

(in particular Trotter product

formulas) are available; see [CJ, McB] , and § 3 , Lemma 3 of

[F7] for an application.

These estimates combined with (1) duality for the free field, first proven by Araki [A],

(see Lemma 1, §3 of [F7] ) ,

(2) the energy - momentum - and boost densities of the anisotropic

( •)_-theory are invariant under j>

(i.e. under the substitution ^ - ^ j a r e used in [F7] , (see Lemma 3, §3 of [F7] and [F7 1 ]) to prove

Theorem 10.1:

[F7, F7']

Given 0 and N, t h e r e are compact diamonds B(0,N)

2

(J 5 £ £ N 5

a n d B(0->-,N) x

= B(0,N)+ X

2

?

£

U N

0+ x ' ?

such that if |x| is so large that B(0,N) and B(0->-,N) are space-like separated

r

a,x(?) =

for all

r a

U ) T _ ^ < x

then

€

) ,

(10.8)

C C N , and

(1) for all 5 e N, r ( ? ) £ 0£(B(O,N)) is a local cocycle that is strongly continuous in E, , (2) r -aj

( e ) £ 01 (B(0+,N) ) is a local cocycle that is •»

x

strongly continuous in E, .

25

Remark. (1) Theorem 10.1 is the technical core in the proof of Theorem

9.4.

(2) In § 3 , Lemmas 3 and 4 of [F7j Theorem 10.1 has been proven for the special case where the group elements £

are space-time translations. The proof of the

general result is very similar to the one of this special case is however technically a little more complicated. It is too long to be reproduced here. (3) We emphasize again that the existence of locally correct generators for the Poincare automorphisms, i.e. the results of

[CJ, M c B ] , and duality

(for the

free field only) are basic ingedients of the proof; technically the Trotter product formula and finite propagation speed, i.e.

(9.10),play a basic role.

(4) The main point of Theorem 10.1

(in the realization

of the announced program) is that, by approximating the

automorphism a

by inner

automorphisms

implemented by the local operators V i.e. by approximating u

-+, (as x->— °° ;

by vector states in X . ) ,

we achieved to construct strictly local r

1-cocycles

( £ ) , a direct construction of which appears to

be difficult. For the expert it is now clear that Theorem 10.1 immediately yields a proof of Theorem 9.4. The last step is the following.

Corollary 10.2:

[F7,F7']

(1) The unitary operators {U + ( £)r

(? ) : ? £ p | } form

a continuous, unitary representation U

Acta Physica Austriaca, Suppl. XV

of the

258

Poincare group na , on

£+.

(2) For all A d 01 OS(TC

(A)) = U ± U

)ra(C )as(A) (U ± (5)r a (C))*,

^ (10.9) as an operator equation on ) that commute with the energy-momentum - and boost densities. For a general analysis we refer the reader to §6 of [F7]; (see also [RoJ).

10.2 Conclusions (1) Presently we are checking whether directly

that the spectrum of the

operator

(H. ,P ) on

M s , (for sufficiently

3?

one can

prove

energy-momentum

has a positive mass

small \ € ( 0 , \

gap

) ) . Heuristic

c arguments suggest that M is bounded below by some analogue of what one knows as surface tension x in the statistical mechanics of classical spin systems; (see e.g. [ G M ] ) . This surface tension can be esti-

mated directly by means of the Peierls argument; (see Section 7 ) . The author has shown that it diverges to +» (i.e.x* +») as X*Or (with X the coupling constant of the (J-|")2 - term, and a>1 fixed). (2) There seems to be an interesting the existence

connection

of soliton states and the

between

soectrum

154 260

of bound states in such two dimensional quantum field theories. (Example: For fixed CT>1, ^sA„ and 0< [ y | 2 + ji^j model in two dimensions should have a rich spectrum of bound states

(of "would-be" solitons and

-anti-solitons)

which decay into soliton-anti-soliton pairs, as ij->-0) . This has been briefly discussed in remark 5) §6 of ref.

[F7] (and, independently, by Glimm, Jaffe

and Spencer in a forthcoming p a p e r ) . (3) The next urgent task in the theory of quantum solitons might be the study of quantum vortices in scalar QED in three space-time dimensions;

(abelian

Higgs m o d e l ) . On a formal level this looks rather promising. In the case of non-abelian Yang-Mills theories in four space-time dimensions

(perhaps the

only candidates for soliton-behaviour in four d i mensions) the situation does not seem to be well understood, y e t , even on a purely formal level, although the existence of solutions of the classical field equations like the t'Hooft monopole is very intriguing. (4) For refs. to many original papers on quantum see

solitons

[Co4, F 7 j . Since the number of such papers is

rapidly diverging a proper list of refs. could not be included here.

155 261

References: [A]

Araki,H.: J. Math. Phys.4_, 1343, (1963), J.

Math. Phys.5_, 1

, (1964),

see also K. Osterwalder, Commun. Math. Phys. 29_, (1973) [AHR]

Araki, H. K. Hepp and D. Ruelle, Helv. Phys. Acta J35_, 164, (1962).

[CJ]

Cannon, J.T. and A. Jaffe, Commun. Math. Phys., V7_, (1970).

[Co1]

Coleman, S., "Secret Symmetry" 1973 Erice Lectures (Int. School of Subnuclear Physics, "Ettore Majorana").

[Co2]

Coleman, S. Phys. Rev. D11 , 2088, (1975).

[Co3]

Coleman, S., R. Jackiv and L. Susskind, Annals of Physics 9_3, 267, (1975).

[Co4]

Coleman,S., "Classical Lumps and Their Quantum Descendants" 197 5 Erice Lectures (Int. School of Subnuclear Physics "Ettore Majorana").

[CQFT]

Symanzik, K. , J. Math. Phys. 1_, 510, (1966) Symanzik, K., "A Modified Model of Euclidean Quantum Field Theory", N.Y.U. Preprint, 1964. Nelson, E., "Quantum Fields and Markoff Fields' in: Proceedings of the Summer Inst, on Partial Diff. Equ., D. Spencer (ed), Berkeley 1971, A. Math. S o c , Providence 1973. Nelson, E., J. Funct. Anal. J_2_, 97, (1973). Guerra, F., Phys. Rev. Letters 28, 1213,(1972).

Glimm, J. and A. Jaffe, refs. [GJ1,3] Glimm, J., A. Jaffe and T. Spencer, refs. [GJS1,2] Guerra, F., L. Rosen and B. Simon, ref. [GRS] Osterwalder, K. and R. Schrader, ref. [OSj. See also refs. [E] [Si] [Y ] . [DHR]

Doplicher, S., R. Haag and J. Roberts , Commun. math. Phys. _2_3, 199, (1971) and _3_5 49, (1974).

[DS]

Dunford, N. and J. Schwartz, "Linear Operators Interscience, New York, 1963.

[DN]

Dunlop, F. and C. Newman, Commun. math. Phys. 44, 223, (1975) .

[EEF]

Eckmann, J.-P., H. Epstein and J. Frohlich, "Asymptotic Perturbation Expansion for the S-Matrix and ...." to appear in Ann. Inst. H. Poincare, 1976.

[E]

"Constructive Quantum Field Theory", G.Velo and A.S. Wightman (eds.), Springer Lecture Notes in Physics, vol. 25., (1973) .

[ESw]

Ezawa, H. and J. Swieca, Commun. math. Phys. 5,

330, (1967) .

[Fe]

Feldman, J., Commun. math. Phys. 3_7, 93, (1974).

[Fed]

Feldman, J., and K. Osterwalder, "The Wightman Axioms and the Mass Gap for Weakly Coupled (^L Quantum Field Theories", to appear in Annals of Physics.

263

[Fe02]

Feldman, J. and K. Osterwalder, "The Construction of X C^) o Quantum Field Models" Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille 1975.

[FO]

Follmer, H., "Phase Transition and Martin Boundary", in: Seminaire de Probabilites IX, Universite de Strasbourg, p. 305, Springer Lecture Notes in Math..vol. 465, (1975).

[F1]

Frohlich, J., "Quantum Sine-Gordon Equation and Quantum Solitons in Two Space-Time Dimensions", in: "Renormalization Theory" Int. School of Math. Physics" Ettore Majorana". 1975, Reidel, Dordrecht - Boston, 1976.

[F2]

Frohlich, J., "Classical and Quantum Statistical Mechanics in One and Two Dimensions: ", to appear in Commun. math. Phys. 1976.

[F3]

Frohlich, J., "Phase Transitions in Two Dimensional Quantum Field Models", ZiF, University of Bielefeld, Preprint 1976; (extended version in preparation).

[F4]

Frohlich,J., "Poetic Phenomena in Two Dimensional Quantum Field Theory: ", Proceedings of the Int. Colloqu. on Math.Methods of QFT, Marseille 1975.

[F5]

Frohlich, J., "The Pure Phases, the Irreducible Quantum Field ", to appear in Annals of Physics, 1976.

[F6]

Frohlich, J., "Existence and Analyticity in the Bare Parameters of the [x (<j> -<j>) 2-a<j> ^-ut,] Quantum Field Models" to appear.

158

[F7]

Frohlich, J., "New Super-Selection Sectors ("Soliton-States") in Two Dimensional Bose Quantum Field Models", to appear in Commun. math. Phys., 1976, and paper in preparation (referred to as [F7'J).

[FSeJ

Frohlich, J. and E. Seiler, "The Massive Thirring-Schwinger Model (QED_): Convergence of Perturbation Theory and Particle Structure", to appear in Helv. Phys. Acta.

[FS]

Frohlich, J. and B. Simon, "Pure States for General P () _ Theories: Construction, Regularity and Variational Equality", submitted to Ann. Math.

[FSS]

Frohlich, J., B.Simon and T. Spencer, "Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking", Princeton University, Preprint 1976; results announced in Phys. Rev. Letters 3_6, 804, (1976).

[GM]

Gallavotti, G. and A. Martin-Lof, Commun. math. Phys. 2S.i

87

' (1972) .

[GJ1]

Glimm, J. and A. Jaffe, Fortschritte d. Physik 2A_, 327, (1973) .

[GJ2]

Glimm, J. and A. Jaffe, Commun. math. Phys. ££ 1, (1971)

[GJ3]

Glimm, J. and A. Jaffe, (<j>^ II,III, I V ) , Ann. Math. 9_1_, 362, (1970), Acta Math. 125, 203, (1970), J. Math. Phys. J_3, 1568, (1972).

[GJS1]

Glimm, J., A. Jaffe and T. Spencer, "The Particle Structure of The Weakly Coupled P()2 Model and Other Applications of High

265

Temperature Expansions, Part II: The Cluster Expansion", contribution in ref. [E] and refs. given there. Glimm, J., A. Jaffe and T. Spencer, Commun. math. Phys. 4_5, 203, (1975), and paper in preparation. Guerra, F., "Exponential Bounds in Lattice Field Theory", Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille 1975. Guerra, F. , L. Rosen and B. Simon, Ann. Math. 101 , 111, (1975). Haag, R. and D. Kastler, J. Math. Phys.5_, 848, (1964) . Herbst, I. "Remarks on Canonical Quantum Field Theory", Princeton University, Preprint 1975. "Statistical Mechanics and Quantum Field Theory", Les Houches 1970 C. De Witt and R. Stora (eds.), Gordon and Breach, New York, 1971. Jost, R., "The General Theory of Quantized Fields", A. Math. Soc., Providence, R.I., 1965. Kac, M., "On Applying Mathematics: Reflections and Examples", Quarterly of Appl. Math. 2P_/ 17 » (1972) • Kunz, H., private communication• Lamb Jr. G., Rev. Modern Physics £3, 99, (1971) • Lebowitz, J,, and 0. Penrose, Phys. Rev. Letters 35, 549, (1975) •

160

[MaSeJ

Magnen, J. and R. Seneor, "The Infinite Volume Limit of the (ty1*)^ Model", to appear in Ann. Inst. H. Poincare.

[Ma]

Malishev, V.A. Commun. math. Phys. £0, 75, (1975) .

[McB]

McBryan, 0., Nuovo Cimento, 18A, 654, (1973).

[M]

Mermin, N.D., J. Math. Phys. 8_, 1061, (1967).

[N]

Nelson, E., "Probability Theory and Euclidean Field Theory", contribution in ref. [EJ.

[OSJ

Osterwalder, K. and R. Schrader, Commun. math. Phys. 2 1 '

[OSe]

83

' (1973) and £2, 281, (1975).

Osteralder, K. and R. Seneor, "The S-Matrix is Non-Trivial in ($**)....",

to appear in

Helv. Phys. Acta. [Pa]

Park, Y.M., "Convergence of Lattice Approximations and Infinite Volume Limit in the (A.<J>^-atji 2 -u<J>) 3 Field Theory", Schladming Lectures, 1976, and J. Math. Phys. 16,1065, (1975), and to appear in J. Math. Phys.

[Pe]

Peierls, R. , 477, (1936).

[PT]

general refs. on phase transitions: See refs. [H],

Poc. Cambridge Phil. Soc. , 32,

[R], [FSS], [GM] and refs. given there.

Results that are somewhat complementary to what is discussed in these lectures may be found in L. Onsager, Phys. Rev. 6_5, 117, (1944), (Ising model), H. Araki and E.J. Woods, J. Math. Phys. _4, 637, (1963), (free Bose gas) T.H. Berlin and M. Kac Phys. Rev. 8£, 821, (1952),

267

E. Lieb and C.J. Thompson, J. Math. Phys. 10, 1403, (1969), and refs given there; (spherical model). N.N. Bogolinbov, Soviet Physics JETP, 7_, 41, (1958), W. Thirring and A. Wehrl, Commun. math. Phys. £, 301, (1966); (BCS model). K. Hepp and E. Lieb, Ann. Phys. 7_6, 3 60, (1972), and contribution to ref. [ E ] ; (Laser models). R. Griffiths, Phys. Rev. 136A, 437, (1967), R. Dobrushin, Funct. Anal. Appl. 2_, 44, (1968), (Peierls argument) R. Israel, Princeton University Series in Physics, Monograph, to appear; (general results on phase transitions), and Commun. math. Phys. 43_, 59, (1975) . F. Dyson, Commun. math. Phys. Jl_2_, 91, (1969). Roberts, J., "Local Cohomology and Superselection Structure", Centre de Physique Theorique, C.N.R.S. - Marseille, Preprint 1976. For results concerning the solitons of the free massless, scalar fields in two dimensions, see R.F. Streater and I.F. Wilde Nuclear Physics B24, 561, (1970); and P. Bonnard and R.F. Streater, ZiF, University of Bielefeld, Preprint 1975. Ruelle, P. "Statistical Mechanics - Rigorous Results", Benjamin, New York, 1969. Seller, E. and B. Simon, "Nelson's Symmetry and All That in the (Yukawa), and (4),

268

Field Theories", to appear in Annals of Phys. [Si]

Simon B., "The P () Euclidean (Quantum) Field Theory, Princeton University Press, Princeton 1974.

[SG]

Simon, B. and R. Griffiths, Commun. math. Phys. 32/ (1973) .

[St E]

St. Exupery, A. "Le Petit Prince".

[St W]

Streater, R.F. and A.S. Wightman, "PCT, Spin and Statistics and All That", Benjamin, New York, 1964.

[SvB]

Sylvester G. and H. van Beijeren, "Phase Transitions for Continuous Spin Ising Ferromagnets", Yeshiva University, Preprint 1975.

[T]

See ref s. [H] [RJ ; also K. Huang, "Statistical Mechanics", Wiley and Sons, New York-London 1963.

[Y2]

Refs. on the Euclidean description of the (Yukawa)- quantum field model are: E. Seller, Commun. math. Phys. £2, 163, (1975) . E. Seller and B. Simon, J. Math. Phys. to appear ("finite mass renormalizations") and ref. [SeSi] . E. Seller and B. Simon, Commun. math. Phys. .45, 99, (1975) . O. McBryan, Commun. math. Phys. £4_, 237, (1975) . 0. McBryan, Commun. math. Phys. £5, 279, (1975) .

2

O. McBryan, Contribution to the Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille, 1975. J. Magnen and R. Seneor, "The Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions", ZiF, University of Bielefeld, Preprint, 1975. A. Cooper and L. Rosen, "The Weakly Coupled (Yukawa)

Field Theory: Cluster Expansion

and Wightman Axioms", Princeton University, Princeton, 1976.

THE PURE PHASES (HARMONIC FUNCTIONS) OF GENERALIZED PROCESSES1 OR: MATHEMATICAL PHYSICS OF PHASE TRANSITIONS AND SYMMETRY BREAKING BY J. FROHLICH2

Contents I. Introduction. (Description of the problem, symmetries and broken symmetries). II. Lattice Systems and Generalized Processes. (The mathematical structure, interactions, finite systems, thermodynamic functions, equilibrium states, uniqueness theorems). III. The General Notion of Phase Transitions. (Definition of phase transitions, strategy for proving the existence of a phase transition, connection with symmetry breaking). IV. Reflection Positivity. (The cone of "reflection positive interactions", a generalization of the Holder inequality for traces and the "chessboard estimates", examples of reflection positive interactions, infrared bounds). V. Application to Classical Lattice Systems: Phase Transitions for Gibbs Random Fields. (Application of results of §§III and IV to the proof of existence of phase transitions for a large class of systems, conclusions). I. Introduction. This paper is written for mathematicians and mathematical physicists with some knowledge of stochastic processes and of the basic notions of statistical mechanics, but I have tried to explain what I believe are all major concepts, notions and definitions required for the understanding of the main results, i.e. I have tried to write these notes for the nonexpert at the risk of boring the expert and, perhaps, being a little imprecise here and there. (The expert may find some new results in §§IV and V.) All major recent or new results I am describing in this paper were obtained in collaboration with B. Simon, T. Spencer, E. H. Lieb and R. Israel. The reader is advised to consult references [l]-[4] for statements of the original results and complete proofs. Reviews of some of the material contained in these references and applications to relativistic quantum field theory may be found in [5], [6]. The reader may consult [7], [8], [1] for the original results on phase transitions in This is an expanded version of an invited address presented at the 83rd Annual Meeting of the American Mathematical Society under the title, Mathematical physics of phase transitions and symmetry breaking: New rigorous results in St. Louis, Missouri on January 28, 1977; received by the editors May 16, 1977. AMS (MOS) subject classifications (1970). Primary 60G20, 81A18. 'Work supported in part by the National Science Foundation under Grant MPS 75-11864. 2 A. P. Sloan Foundation Fellow. © American Mathematical Society 1978 165

Reprinted from "The Pure Phases (Harmonic Functions) of Generalized Processes" or "Mathematical Physics of Phase Transitions and Symmetry Breaking", by Jiirg Frohlich, Bulletin of the American Mathematical Society, Volume 84, Number 1-3 (1978), pp. 165 193, by permission of the American Mathematical Society.

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relativistic quantum field theory and their proofs. The general point of view adopted in this paper is developed in [9]-[ll] and references given there (see also [12]-[14]). Some of the weighting of different concepts and a few results have grown out of a course I have taught at Princeton University in the fall semester 1976. ACKNOWLEDGEMENTS. I am very much indebted to E. H. Lieb, B. Simon and T. Spencer for all they have taught me and for the joy of collaboration and to these colleagues and R. Israel for permission to present results that are in part not yet published. 1.1. Description of the problem. In these notes I try to outline a new mathematically rigorous theory of phase transitions and symmetry breaking which is rather general. It applies to Gibbs random fields and noncommutative generalizations of these, namely some class of quantum lattice systems and Fermion (Grassmann) lattice systems; the general concepts and methods involved may however equally well be applied to other physical theories, in particular relativistic quantum field theory. Many of the results I am going to describe were actually first obtained in the context of relativistic quantum field theory or at least motivated by it. This illustrates once again that mathematics can sometimes profit a lot from theoretical physics. The few proofs contained in these notes also show that, to use some words of Mark Kac, "in the right hands, Schwarz's inequality and integration by parts are still among the most powerful tools of analysis".3 Mathematically speaking, we shall be concerned, in this talk, with certain aspects of the theory of stochastic processes and their noncommutative versions; aspects that are somewhat related to probabilistic potential theory. In particular I want to discuss an analogy between phase transitions and the existence of nonconstant harmonic functions of a generalized process. The simplest example of a generalized process is a (multi-time) Markov process (or a multi-dimensional Markov chain), but the concept of generalized processes such as has emerged from the work of the past few years [9], [10], [15], [16], [12], [13], [11], [17] is more general and includes "Gibbs lattice fields" which are some sort of noncommutative random fields. The general problem I shall discuss may be posed as follows: Suppose we are given the local characteristics of a generalized process, in the commutative case e.g. a system of conditional probabilities or some equilibrium equations of the Dobrushin-Lanford-Ruelle (DLR) type [9HH], m the noncommutative case e.g. a "Gibbs condition" [13] or a "Gibbs variational equality" (all cases), can we prove general theorems giving a complete description of all harmonic functions of the generalized process (i.e. all Gibbs lattice fields with given local characteristics) or, in a physicist's language, the pure phases of the process"} Our results are two-fold: 1. Uniqueness theorems: Dobrushin's theorem [15], [11], [18] and its noncommutative versions [19], [20]. 2. A general method for proving the existence of "nonconstant" harmonic 3

M. Kac, On applying mathematics: Reflections and examples, Quart. Appl. Math. 30 (1972).

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167

functions, or, in other words, of several distinct pure phases with identical local characteristics. For aesthetical and educational reasons I shall emphasize the discussion of symmetries and symmetry breaking, by which I mean that the local characteristics of a generalized process (and in particular its "Gibbs potential"* resp. its Hamiltonian) may be invariant under a symmetry group which does not leave invariant some or all its nonconstant harmonic functions, i.e. which permutes the pure phases of the process among themselves. I briefly want to motivate this emphasis on symmetries and symmetry breaking. 1.2. The role of symmetry in mathematics and physics. For a beautiful discussion of the significance and the history of symmetry as a concept in mathematics, the natural sciences and the arts I refer the reader to Hermann Weyl's book entitled Symmetry [21]. (A new, somewhat more modern treatise on this subject would be desirable.) There are two aspects of "symmetry" of direct relevance to this paper: 1. A purely geometric aspect. 2. A dynamical aspect. The symmetries of geometric objects lead to the mathematical concept of symmetry and are one of several major roots for the development of group theory. (One might recall, here, Felix Klein's program of characterizing geometries by their invariance groups.) Geometric symmetries played and still play an important role in chemistry, crystallography, biology and other natural sciences. Historically they also played a role in dynamics, especially celestial mechanics. The Greeks believed that the motions of the planets and the moon would necessarily have to be circular or a superposition of circular motions, as the circle is a geometric object of maximal symmetry. The Platonic solids, namely the tetrahedron, the cube, the octahedron, the pentagon dodecahedron and the icosahedron found their way into celestial mechanics: Kepler tried to reduce the distances in the planetary system to the shapes of these solids which he alternatingly inscribed and circumscribed to spheres. The six spheres correspond to the six planets Saturn, Jupiter, Mars, Earth, Venus and Mercurius, known at that time, separated in this order by cube, tetrahedron, dodecahedron, octahedron and icosahedron. Kepler's attempt to understand laws of nature in terms of static, geometric symmetries is typical for natural philosophy in pre-Galilean and preNewtonian times. One of the most significant steps in the history of human thinking may well have occurred when the static, geometric concept of symmetry and its rather successless applications to dynamics were abandoned in favour of a dynamical concept of symmetry. With Newton physicists started to conceive the idea that it is the laws of physics describing the motion of particles, e.g. the planets, which are invariant under certain symmetry groups rather than the orbits of the particles themselves. This dynamical concept of "symmetry" is at the basis of some

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of the major revolutions in the physics of this century and is the one of main importance for the following. 1.3. Spontaneously broken symmetries. The idea that the symmetry group leaving invariant the laws that describe a physical system may be broken in the space of states of the system is in some sense the main theme of these notes. This idea will take a mathematically precise shape in the following discussion. The statement that some symmetry of the laws describing a physical system is broken means that the states of the system fall into equivalence classes invariant under the time evolution and under all possible measurements one can do at such systems which are however not invariant under a symmetry operation. Rather, symmetry operations permute these equivalence classes among themselves. A very striking example of a broken symmetry is found in biology: Living organisms contain the dextro-rotatory form of glucose and the laevo-rotatory form of fructose. There seems to be no a priori reason why it should not be the opposite or why living organisms of both kinds should not coexist (though perhaps coexistence might necessarily result in the extinction of one kind). We all know that this striking asymmetry in the chemical constitution of living organisms has been preserved over centuries and has so far not been destroyed by any changes of the environmental conditions. Thus it really presents an example of a broken symmetry. A trivial example of a broken symmetry in every day physics is a dumbbell shaped balloon with two distinct, asymmetric equilibrium shapes.

The physical laws describing the balloon do not distinguish between left and right. Without going into any detail I want to recall the fundamental role played by dynamical symmetries (spatial or internal) and their breaking in elementary particles physics. The idea that symmetries of physical laws may be broken spontaneously (or dynamically) in the state space of a system is a fundamental ingredient in all recent theories of elementary particle physics [22]. To conclude this introduction let me mention some examples of symmetry breaking in solid state physics that have a certain bearing on the subject of my talk: The first example is a ferromagnet, i.e. a system of bulk matter (e.g. iron) that has the property that when an external magnetic field in a fixed direction is turned off and the temperature is low enough it remains magnetized in the direction of the turned off field. Quantum mechanically, this phenomenon is not yet well understood. Another theoretically closely related phenomenon is Bose-Einstein condensation. In a quantum gas of particles satisfying Bose-Einstein statistics the ground state of the gas may have, at low temperatures, a macroscopic occupation. This is accompanied by the spontaneous breaking of a gauge

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169

group of the first kind isomorphic to SO (2) which leaves the physical laws describing the gas invariant. We shall also meet examples where a discrete symmetry is spontaneously broken. One of the most striking and fundamental phenomena is however, no doubt, the existence of crystals in nature, that is to say of states of matter which break the translational invariance of all physical laws. In the past two years mathematically rigorous theoretical understanding of the phenomenon of phase transitions and symmetry breaking in the framework of admittedly somewhat too simple models has made great progress. What I intend to do is to describe some of the mathematical and analytical aspects of this progress. I hope this introduction has convinced the reader that the problems I am going to discuss are important and that it has indicated what kind of mathematics is involved (generalized processes, Gibbs random fields, probabilistic potential theory). II. Lattice systems and generalized processes. II. 1. Description of the mathematical structure. Let £ denote some p-dimensional lattice. For simplicity I shall in general assume in these notes that £ = Z"; the simple cubic lattice. Many of the results I am going to indicate in the following depend however only on a certain reflection invariance property of the lattice £, i.e. a geometric symmetry property of £. (Some of the results, e.g. the uniqueness theorems, do not depend on any special properties of the lattice, at all.) Since there are only finitely many crystallographic groups in v dimensions (17 for v = 2, 230 for v = 3), it is a matter of consulting a table of these groups in order to give a complete list of all lattices having the required reflection invariance. At each site / G £ we are given an algebra 31, of operators. We must distinguish two cases (if we included Fermions it would be three): (C) classical case

% * c(a,-), where S2, is a copy of some fixed, compact Hausdorff space J20. In these notes Q0 c R^4, N = 1, 2, 3 , . . . , but in interesting cases (lattice gauge theories) Q0 may be a nonabelian compact group. We equip 21, with the sup norm and complex conjugation as an involution*. (QM) quantum mechanical case here 3C, is an isomorphic copy of some fixed, finite dimensional Hilbert space %j. The norm and the * operation on 2t, are defined in the usual way. For X G 9f(t) (the algebra of bounded subsets of the lattice £), we define STjr = ® 21,. H X c A" we consider %x to be the subalgebra of %x. defined by 4

Here RN is the one-point compactification of Rw.

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J. FROHLICH

( ® «,-) ® ( ®

0

where 1, is the identity element in 91,. Technically speaking, {21*: X & ^ ( E ^ i s a family of C* algebras (in case (QM) von Neumann algebras), and they are called "local algebras". If a is a translation in the lattice £ and * is a finite subset of £ then X + a denotes the translate of X by a. The natural identification of 21* with 21*+,, is denoted ra. The * algebra 21 = U x ^ x i s called the algebra of all local observables; 21 is normed in the obvious way with the norm denoted ]| • ||. The group ( v a e £ } acts as a * automorphism group on 21. The completion of 21 in the norm || • || is denoted 2t and is called the algebra of all quasi-local observables. This algebra is a C* algebra. In the classical case, 21 is isomorphic to C(fl), with

a- x a,. zee

(Stone-Weierstrass theorem). A state p on 21 is a positive linear functional on 21 normalized such that P(l) = 1. with 1 = ®,-6eli ( t n e identity in 2t). The space of states on 21 is denoted 21*. The structure of 21* is analyzed in [14]. In the classical case, where 21 = C(Q), 21* is simply the class of all regular Borel probability measures on U. For all A e 21*, define

(C)

tr04)=f

II 4i«MK)>

a

x iex where fl* = X ,af(A) =

ei,H*Ae-"H*.

In case (C) a,A is trivial, but in case (QM) it is in general not. The equilibrium state ("state of interest" in these notes) of a finite, dynamical system specified by the region A and the interaction $ is then defined by

pg;*(A) = tvie-^yhrie-^U),

(A e 3TA).

It is unique and it is invariant under all * automorphism groups of 9tA commuting with H® and leaving tr invariant. This means that "finite systems do not have phase transitions or symmetry breaking". Here some examples of interactions that are interesting for physics: (CI) fi0 = { - 1 , 1}; S0 is the function on fi0 defined by 5 0 (±1) = ± 1; 5, = T,(5 0 ); S, is called the (Jsing) spin at site /. $({/}) = - hSif h real; $({',./'}) = - J,-JS,SJ, with J,_j > 0 and /,_, > 0, for |« - j \ = 1; $(X) = 0, for |jr| > 3;

M{±1})~5-

J. FROHLICH

172

This is the so-called ferromagnetic Ising model. We note that, for h = 0, //* has a discrete symmetry H

A(SA)

=

#A(-5A)>

i.e. the dynamics is invariant under flipping all spins in A. This symmetry is shared by d\i and tr. (C2) Here S20 = SN~', the unit sphere in R*; SQ is the function assigning to a unit vector in S20 its /th component, and S 0 = (S0\ . . . , SQ), S, = T,(S 0 ). $({/}) = - h S„ h e R"; *({/J}) = - / , _ , S , - S,, with e.g. /,_, > 0, and Jt_j > 0, for |/ -j\ = 1; $ ( * ) = 0, for |A"| > 3; rfM(So) = fi(|So| -

l ^ X

This is the classical, ferromagnetic TV-vector model (when N = 3 it is frequently called the classical Heisenberg model). For h = 0, its symmetry group is obviously 0 ( A ) . REMARK. If in the definition of the interaction 2, we obtain examples of multidimensional Markov chains. (The equilibrium expectations of these models have the local Markov property, [16], [17], [23].) (QM1)

%> = C 2 S + 1 ;

S = 1/2, 1, 3 / 2 , . . . ;

{SQ: i — x,y, z) a 2S + 1 dimensional, irreducible representation of the Lie algebra of SU(2); S 0 = (S$, S&, 50z); S(- = T,(S 0 ); $ as in (C2). This is the spin-S Heisenberg ferromagnet. For h = 0 it has 0(3) as its symmetry group. This is a difficult model which is not completely understood yet. (QM2) The same as (QM1), but in the definition of $ we require./,- • < 0, for |/ — j \ = 1, Jj_j = 0, otherwise. This is the spin-S Heisenberg antiferromagnet. Again the symmetry group is O (3) (for h = 0). For results see [2]. The main part of these notes is devoted to the discussion of new results concerning the equilibrium statistical mechanics of a dynamical system specified by the observable algebra 3lA and the dynamics //*. We are mainly interested in the systems obtained by taking the thermodynamic limit A—» Z>\/2- This limit must be taken before one can start to discuss phase transitions and symmetry breaking. (Finite systems never exhibit symmetry breaking!) I shall now introduce some basic objects of thermodynamics and statistical mechanics, in particular the so called thermodynamic functions. They are needed to define the "states of interest" for the infinite systems (A = Z\/2)11.4. Thermodynamic functions. We define the canonical partition function for a system in the region A with interaction <J> by Z A (/?,0) = t r ( e - ^ ) , and the free energy / A (/S, $) per unit volume by #A(J8,*)=

"(1/|A|) In Z A (/?,$).

Here /? is the inverse temperature. Let p be a translation invariant state of the infinite system, i.e.

PURE PHASES OF GENERALIZED PROCESSES P{T,{A))

=

173

p(A),

for all / e Z", all A e 21. We set pA = p/2I A

(the restriction of p to 21A).

We define the internal energy per site by

«A(P. *) = (l/|A|)p(//*), and the specific entropy by 5A(p)=-(l/|A|)tr(pAlnpA). (In case (QM) we consider arbitrary translation-invariant states on 21, in case (C) only those states p with the property that pA is absolutely continuous with respect to 11,,=A dn{u,). The class of all these states is denoted E'.) The following results (see [9] and references given there) summarize some rigorous thermodynamics for lattice systems with interactions 4> €E $ . THEOREM 1. For all real ft $ £ S , p £ E', the following limits exist {and are "independent of boundary conditions'" and of the sequence {A} —* Z"; A —> X" "in the sense of van Hove").

/ ( £ < ! > ) = Iiir. / A (/?,4>), A—*U

u(p, ) - 0 / ( 0 , * ) {Gibbs variational equality [9]). Any cluster point of the sequence of states in E' ( Z A ( A 4»)-'tr A (e-*"J - ) ® M - ) } { A K Z , 2 satisfies the Gibbs variational equality, i.e. any limiting state of the equilibrium states of the finite systems, as A ^ Z ' / 2 satisfies this equality. REMARK 3. Existence of at least one such limiting state follows from a standard compactness argument. Moreover, under suitable assumptions on the interaction $, one can prove that a.(A) = n-lim a*(A) exists, for all A €E 2t; A^-Zi/2

see [9]. A different construction of the time-translations for infinite systems is discussed in [24].

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We are now prepared to define the "states of interest" for the infinite systems. II.5. Equilibrium states, uniqueness theorems. Any state p G E1 satisfying the Gibbs variational equality for an interaction 4> G $ and inverse temperature B is called an equilibrium state of the infinite system (specified by , at inverse temperature fi. If A'5'* happens to be the family of all stationary states of a multidimensional Markov chain (see the Remark in §11.3, Example (C2))-or some more general stochastic process-then the probability measures supported on the extreme points of A"-* are in 1-1 correspondence with the harmonic functions of the Markov chain. The next result asserts that, for small B, A'3-* contains typically only one state.

THEOREM 5. Let 3> be of finite range. Then, for sufficiently small fi ("high temperature"), A^* contains precisely one state p^*. If A and B are arbitrary operators in some local algebra then \p^(ArJ(B))-p^(A)p^(B)\ decays exponentially in \j\ ("exponential clustering"). REMARKS. 1. A more general result has been groven in the classical case (C) by Dobrushin [15]: For any interaction $ G ® (||$||_ < °°) A'3'* contains precisely one state, for B. small enough (but generally no exponential clustering); see also [11], [18]. 2. In the quantum case (QM) Theorem 5 is due to the author [19]. In one dimension (v = 1) Araki has proven a more general result of this type, for all B [20]. Preliminary results of this genre were obtained in [26], [27]. 3. Let G be a connected Lie group acting as a nontrivial, local, continuous * automorphism group on 21. Let $ be an interaction of finite range that is G-invariant. For v = 1 and 2 Dobrushin and Shlosman [28] have proven that

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175

in the classical case (C) all states in A'3'* are G-invariant. Hence there is no spontaneous symmetry breaking. We shall show that if $ has very long range then this conclusion is in general false. 4. The proofs of Theorem 5 and the results mentioned in Remarks 1-3 involve a great deal of concrete, hard analysis, in particular expansion methods, fixed point theorems, trace-inequalities, etc. HI. The general notion of phase transition. We consider an infinite lattice system characterized by an interaction 4> and the family A'3-* of all its translation-invariant equilibrium states at inverse temperature /?. DEFINITION OF PHASE TRANSITIONS. We say that a system with interaction has a phase transition if the number of extremal equilibrium states in A^'* is not constant as a function of /?. From Theorem 5 we already know that, under suitable assumptions on $ and for \fi\ small enough, A^'* contains precisely one state. In this situation we speak of a phase transition if, for sufficiently large | /? |, Aft* contains more than one extremal state. Thus we will have proven the existence of a phase transition if we can find some /? and a state p^'* £ AA* which is not an extremal state. We must therefore formulate a criterion which permits us to decide whether some state p^'* is extremal or not. Since in the following p'5'* is some fixed element of AA*, we do not need the labels ft and 4> and write is denoted du>{T)(k). It is a positive measure on the first Brillouin zone B = {k: k' G [ — IT, IT], i = 1 , . . . , v}. Clearly (ASA,) = (ASA^

+ (ir(A0)Q, (1 -

P')*(A$l).

PURE PHASES OF GENERALIZED PROCESSES

177

The second term is independent of /, since (1 — P') projects onto the space of translation-invariant vectors. Therefore doi{k) = c8{k)dpk + doiT(k), for some c > 0; if c > ||2 then (PT) and (MF) hold. Our strategy for the proof of the existence of a phase transition can now be formulated as follows: (0) Choose a suitable local observable A. (1) Derive an upper bound for (A*A}r: (A*A}T

= f duT(k)

< c,.

(II) Derive a lower bound on (A*A} — \(A)\2: (A*A) - \(A)\2 > c2. If c, < c2 then (A*A)-\(A)]2>(A*A)T, therefore dim %' > 2, i.e. < —> = p ^ * is not extremal (that means that < - > has macroscopic fluctuations in the sense of inequality (MF)). Next we want to show why a phase transition may be accompanied by the spontaneous breaking of a symmetry of the system. Let G be some compact topological group acting as a nontrivial, local * automorphism group {rg: g E G) on the algebra 3( of all quasi-local observables, and rg(%x) = 31*, for all X G 9f{£). Suppose now that the interaction 4> of the system is (7-invariant, i.e. Tg(-in the subsequent sections. IV. Reflection positivity. In this section we consider a certain cone of interactions which satisfy a positivity property called reflection positivity [4]. This property can only be formulated for lattices which have a certain reflection in variance (alluded to in §11.1). For simplicity we only consider simple, cubic lattices; but see [4] for more general results in the classical case (Q. In the language of a mathematical physicist reflection positivity expresses the existence of a selfadjoint transfer matrix. IV. 1. Reflection positivity infinite systems. Unless otherwise stated all the following results are proven in [4]. We let A c Z\/2 = Z" + ( 1 / 2 , . . . , 1/2) be the rectangle X [ - / , + 1 / 2 , / , - 1/2], i=

/,. = 1 , 2 , 3 , . . . ,

I

and we identify lt + 1/2 with - /, + 1/2, i.e. we wrap A on a torus. We then define A4 = X [-/,. + 1/2, ( - 1/2] x [ ± l / 2 , ±lt+

1/2].

The following figure represents a cross section of A (viewed as a torus): 5/2

0'

3/2

(,-1/2

/

-//+1/2 s

/

/ %

/

y

/

J K112

1-1/2 -3/2

180

J. FROHLICH

For / = ( / ' „ . . . , /„) G Z^ /2 we define OH = (/„ . . . , - / } , . . . ; , / „ ) . Thus if X is some subset of Z\/2, 9jX will denote the reflection of X at the plane /} = 0. We now define & to be the * automorphism of 21A which when restricted to 21,. is the identification map: 21,-* 21^, and @J(WX)= ® , e * &(%)• Obviously e>siA4 = 2t M . In the following we generally suppress the superscript^'. Any statement that does not contain explicit reference to a distinct j is true for ally = 1 , . . . , v. LEMMA 7. For all A G 2tA+, tr(AQ(A*)) > 0. PROOF.

t r ( / J 6 ^ * ) ) = tr A+ (/()tr A (©(/(*)) = tr A ^(^)tr A+ (^*) = tvA+(A)T^jA)

> 0. Q.E.D.

Note that because of translation invariance of A and tr the choice of an origin on the torus A is arbitrary, as indicated in the figure! DEFINITION. An interaction $ satisfies reflection positivity iff, for all finite X, 0()= \d2x \^{V2 =

W

W = WR = C-

Then (3.7) reads

)>=—(Jc-yT 1 , (3.7) i

(c(x)b(y)}=~(x-y)- , and <M>> = = = = 0 ,

(3.9)

which follows from (rpaxpa> = =1

/

pairings, p \ ^ 7 t /

A

1=1 \-*-p(2i-l)

p(2i)/

Similar identities hold if N(bc) is replaced by N(bc). Moreover, by (3.9),

n mbcHxd n N(bc)(y)j = (.n

MMC*^

( n w o ^ , (3.20)

at non-coinciding arguments. At coinciding arguments, there is a contribution from the anomaly. For example, (N(bc)(x)N(5c)(y)y=~5(x-y).

(3.21)

It is not hard to extend (3.19)—(3.21) to find the expressions for the Euclidean Green functions of arbitrary products of N{bc)- and N(5c)-currents. [This yields a proof of Schwinger's formula <exp(yMM)> = e x p [ i < ^ ^ " l ^ > ] . ] (C) One may also derive, somewhat less easily, from (3.10) and (3.11) that if F is a monomial in bb and cc then i . d2y I^F(bb,cc) oF(bb,cc) (N(bc) (x)F(bb, cc)> = ^ f2]- ^ ( ^ ) J ((bb)>W(y) - ^ J (cc) (y)) , 4n x-y \S(bb)(y) 5(cc)(y) (3.22) and similarly for (N(bc)F(bb,cc)y. A proof of (3.22) will emerge from our discussion of bosonization. The identities derived in (A)-(C) permit us to calculate the Euclidean Green functions corresponding to arbitrary products of currents, axial-vector currents and chiral densities. [Identities (B) are the most fundamental ones. They extend to general Riemann surfaces.] Next, we review the bosonization of the b-c system described above. This is a well known and much exploited procedure. Since it will turn out to be very useful for the material in Sects. 4 and 5, it is worthwhile to briefly present our views of these matters. We define the zero-mass Gaussian measure, d^i° with mean 0 and covariance "47i( — A)~ln (see [12]) in terms of its characteristic functional: <e'>o=

J d j U 0( z ) e '

=

g2«)

( 3 2 3 )

.9"

if / is a test function whose Fourier transform, /, vanishes at the origin, and J dfi°(x)ei<x'f> = 0, sr

(3.24)

Bosonization, Topological Sohtons and Fractional Charges

149

if / i s real and /(0) + 0. In (3.23), (3.24), the sample space !~f" is chosen to be the space of real, tempered distributions on R 2 . We shall simplify our notations by writing simply \, instead of J . -9"

Heuristically dl*°{x)= ^

e x

P

(3.25)

Dlo,

an

where Dx is the formal Lebesgue measure and the subscript "0" indicates that % has zero-Dirichlet b.c. [i.e., symbolically, x(x)->0, as bc|-»oo]. We define the Wick-ordered exponential, :expi:, by :e>.=eig2ir

(3.26)

for some m 0 > 0 . Then a straightforward calculation shows that, with :e'CK:(x) s

.eie.

I e ; *0

0, if Sdfi°U)

n

(3.27)

••