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From Microphysics to Macrophysics I + II Methods and Applications of Statistical Physics By R. Balian

Variational Methods in Mathematical Physics A Unified Approach By P. Blanchard and E. Briining

Quantum Mechanics: Foundations and Applications 3rd enlarged edition By A. Btlhm

The Early Universe Facts and Fiction 3rd corrected and enlarged edition By G. Btlrner

Geometry of the Standard Model of Elementary Particles By A. Derdzinski

Random Walks, Critical Phenomena, and Triviality In Quantum Field Theory By R. Fernandez, J. Frohlich and A. D. Sokal

Quantum Relativity A Synthesis of the Ideas of Einstein and Heisenberg By D. Finkelstein

Quantum Mechanics I + II By A. Galindo and P. Pascual

The Elements of Mechanics By G. Gallavotti

Local Quantum Physics Fields, Particles, Algebras 2nd revised and enlarged edition By R. Haag

Elementary Particle Physics Concepts and Phenomena By O. Nachtmann

Inverse Schriidinger Scattering in Three Dimensions By R. G. Newton

Scattering Theory of Waves and Particles 2nd edition By R. G. Newton

Quantum Entropy and Its Use By M. Ohya and D. Petz

Generalized Coherent States and Their Applications By A. Perelomov

Essential Relativity Special, General, and Cosmological Revised 2nd edition By W. Rindler

Path Integral Approach to Quantum Physics An Introduction 2nd printing By G. Roepstorff

Advanced Quantum Theory and Its Applications Through Feynman Diagrams 2nd edition By M. D. Scadron

Finite Quantum Electrodynamics The Causal Approach 2nd edition ByG. Scharf

From Electrostatics to Optics A Concise Electrodynamics Course By G. Scharf

Large Scale Dynamics of Interacting Particles By H. Spohn

General Relativity and Relativistic Astrophysics By N. Straumann

The Dirac Equation By B. Thaller

The Theory of Quark and Gluon Interactions 2nd completely revised and enlarged edition By F. 1. Yndurain

Rudolf Haag

Local Quantum Physics Fields, Particles, Algebras

Second Revised and Enlarged Edition With 16 Figures

t Springer

Professor Dr. Rudolf Haag Waldschmidtstra,se 4b. 0- 83727 Schliersee-Neuhaus, Germany

Editors

Roger Balian CEA Service de Physique Theorique de Saclay F-91191 Gif-sur-Yvette, France

Wolf Beiglbock Institut fUr Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 0-69120 Heidelberg, Germany

Harald Grosse Institut fur Theoretische Physik Universitat Wien Boltzmanngasse 5 A-1090 Wien, Austria

Elliott H. Lieb ladwin Hall Princeton University, P. O. Box 708 Princeton, Nl 08544-0708, USA

Nicolai Reshetikhin Department of Mathematics University of California Berkeley, CA 94720-3840, USA

Herbert Spohn Theoretische Physik Ludwig-Maximilians-Universitiit Munchen TheresienstraBe 37 0-80333 Munchen, Germany

Walter Thirring Institut fur Theoretische Physik Universitat Wien Boltzmanngasse 5 A-1090 Wien, Austria

Library of Congress Cataloging-in-Publication Data. Haag, Rudolf, 1922- Local quantum physics: fields, particles. algebras I Rudolf Haag. - 2nd rev. and enl ed. p. em. - (Texts and monographs in physics) Includes bibliographical references and index. ISBN 3-540-61049-9 (Berlin: alk. paper) I. Quantum theory. 2. Quantum field theory. I. Title. II. Series. QCI74.12.H32 1996 530.1 '2-dc20 96-18937

ISBN·13:978·3·540·61049·6 e·ISBN·13:978·3·642·61458·3 001: 10.1007/978·3·642·61458·3

ISSN 0172·5998

ISBN 3·540-61049-9 2nd Edition (Softeover) Springer-Verlag Berlin Heidelberg New York

ISBN 3-540-61451-6 2nd Edition (Hardcover) Springer-Verlag Berlin Heidelberg New York

ISBN 3·540·53610·8 1st Edition (Hardcover) Springer-Verlag Berlin Heidelberg New York

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Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH (!'J Springer-Verlag Berlin Heidelberg 1992, 1996. Printed in Germany

The use of general descriptive names, registered names, trademarks. etc. in (his publication does not imply, even in the absence of a specific statement. that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

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Dedicated to Eugene P. Wigner in deep gratitude

and to the memory of

Valja Bargmann and Res Jost

Preface to the Second Edition

The new edition provided the opportunity of adding a new chapter entitled "Principles and Lessons of Quantum Physics". It was a tempting challenge to try to sharpen the points at issue in the long lasting debate on the Copenhagen Spirit, to assess the significance of various arguments from our present vantage point, seventy years after the advent of quantum theory, where, after ali, some problems appear in a different light. It includes a section on the assumptions leading to the specific mathematical formalism of quantum theory and a section entitled "The evolutionary picture" describing my personal conclusions. Alto­gether the discussion suggests that the conventional language is too narrow and that neither the mathematical nor the conceptual structure are built for eter­nity. Future theories will demand radical changes though not in the direction of a return to determinism. Essential lessons taught by Bohr will persist. This chapter is essentially self-contained.

Some new material has been added in the last chapter. It concerns the char­acterization of specific theories within the general frame and recent progress in quantum field theory on curved space-time manifolds. A few pages on renor­malization have been added in Chapter II and some effort has been invested in the search for mistakes and unclear passages in the first edition.

The central objective of the book, expressed in the title "Local Quantum Physics", is the synthesis between special relativity and quantum theory to­gether with a few other principles of general nature. The algebraic approach, that is the characterization of the theory by a net of algebras of local observ­ables, provides a concise language for this and is an efficient tool for the study of the anatomy of the theory and of the relation of various parts to qualitative physical consequences. It is introduced in Chapter III. The first two chapters serve to place this into context and make the book reasonably self-contained. There is a rough temporal order. Thus Chapter I briefly describes the pillars of the theory existing before 1950. Chapter II deals with progress in understanding and techniques in quantum field theory, achieved for the most part in the 1950s and early 1960s. Most of the material in these chapters is probably standard knowledge for many readers. So I have limited the exposition of those parts which are extensively treated in easily available books to a minimum: in partic­ular I did not include a discussion of the path integral and functional integration techniques. Instead I tried to picture the panorama, supply continuous lines of argument, and discuss concepts, questions, and conclusions so that, hopefully,

VIII Preface

Chapters I-Ill by themselves may serve as a useful supplementary text for stan­dard courses in quantum field theory. Chapters IV-VI address more advanced topics and describe the main results of the algebraic approach.

The remarks on style and intentions of the book are quoted below from the preface to the first edition.

April 1996 Rudolf Haag

From the Preface to the First Edition

. .. Physical theory has aspects of a jigsaw puzzle with pieces whose exact shape is not known. One rejoices if one sees that a large part of the pieces fits together naturally and beautifully into one coherent picture. But there remain pieces outside. For some we can believe that they will fit in, once their proper shape and place has been recognized. Others definitely cannot fit at all. The recognition of such gaps and misfits may be a ferment for progress. The spirit of tentative search pervades the whole book.

The book is not addressed to experts. I hope that at least the essential lines of argument and main conclusions are understandable to graduate students interested in the conceptual status of presently existing theory and sufficiently motivated to invest some time, thinking on their own and filling in gaps by consulting other literature.

Local Quantum Physics is a personal book, mirroring the perspective of the author and the questions he has seriously thought about. This, together with the wish to keep the volume within reasonable bounds implied that many topics could only be alluded to and that I could not attempt to produce a well balanced list of references.

Thanks are due to many friends for valuable advice and correction of errors, especially to Detlev Buchholz, Klaus Fredenhagen, Hans Joos, Daniel Kastler, Heide Narnhofer, Henning Rehren, John Roberts, Siegfried Schlieder, and Wal­ter Thirring. I am grateful to Wolf Beiglbiick for his unwavering support of the project at Springer-Verlag ... , to Michael Stiller, Marcus Speh, and Hermann Heflling for their perseverance in preparing files of the manuscript.

Last but not least I must thank Barbara, without whose care and constant encouragement this book could not have been written.

November 1991 Rudolf Haag

Organization of the Book

The book is divided into chapters and sections. Some sections, but not all, are subdivided. Thus III.2.1 denotes Subsection 1 of Section 2 in Chapter III. Equations are consecutively numbered in each section. For example (III.2.25) is equation number 25 in Section 2 of Chapter III. All types of statements (theo­rems, definitions, assumptions ... ) are consecutively numbered in each subsection and the chapter number is suppressed; thus lemma 3.2.1 is the first statement in Subsection 3.2 of the running chapter. The index is divided into three parts:

1. Bibliography (listing books). 2. Author Index and References; it combines the listing of authors with the

quotation of articles in journals and the indication of the place in the text (if any) where the respective article is referred to.

3. Subject Index. Only those page numbers in the text are given where the item is defined or first mentioned or where some significant further aspect of it appears.

References from the text to the bibliography are marked in italics with the full name of the author written out; references to journal articles are indicated by an abbreviated form of an author's name and the year of publication such as [Ara 61J.

For the benefit of the reader who likes to start in the middle, here are a few remarks on notation and specific symbols. Vectors and matrix elements in Hilbert space are usually written in the Dirac notation: 11Ji), (1JiIAlcP). In the case of vectors the bracket is sometimes omitted. Generically, the symbol A is used to denote a general "-algebra, whereas the symbol 21 is used in the case of a C'-algebra generated by observables and R is used for a von Neumann ring or W" -algebra.

Contents

I. Background 1

1. Quantum Mechanics Basic concepts, mathematical structure, physical interpretation.

2. The Principle of Locality in Classical Physics and the Relativity Theories 7 Faraday's vision. Fields. 2.1 Special relativity. Poincare group. Lorentz group. Spinors. Conformal group. 2.2 Maxwell theory. 2.3 General relativity.

3. Poincare Invariant Quantum Theory 25 3.1 Geometric symmetries in quantum physics. Projective representations and the covering group. 3.2 Wigner's analysis of irreducible, unitary representa­tions of the Poincare group. 3.3 Single particle states. Spin. 3.4 Many particle states: Bose-Fermi alternative, Fock space, creation operators. Separation of CM-motion.

4. Action Principle 39 Lagrangean. Double role of physical quantities. Peierls' direct definition of Pois­son brackets. Relation between local conservation laws and symmetries.

5. Basic Quantum Field Theory 42 5.1 Canonical quantization. 5.2 Fields and particles. 5.3 Free fields. 5.4 The Maxwell-Dirac system. Gauge invariance. 5.5 Proc~sses.

II. General Quantum Field Theory 53

1. Mathematical Considerations and General Postulates 53 1.1 The representation problem. 1.2 Wightman axioms.

2. Hierarchies of FUnctions 58 2.1 Wightman functions, reconstruction theorem, analyticity in x-space. 2.2 Truncated functions, clustering. Generating functionals and linked cluster

XII Contents

theorem. 2.3 Time ordered functions. 2.4 Covariant perturbation theory, Feyn­man diagrams. Renormalization. 2.5 Vertex functions and structure analysis. 2.6 Retarded functions and analyticity in p-space. 2.7 Schwinger functions and Osterwalder-Schrader theorem.

3. Physical Interpretation in Terms of Particles 75 3.1 The particle picture: Asymptotic particle configurations and collision the­ory. 3.2 Asymptotic fields. S-matrix. 3.3 LSZ-formalism.

4. General Collision Theory 84 4.1 Polynomial algebras of fields. Almost local operators. 4.2 Construction of asymptotic particle states. 4.3. Coincidence arrangements of detectors. 4.4 Generalized LSZ-formalism.

5. Some Consequences of the Postulates 96 5.1 CPT-operator. Spin-statistics theorem. CPT-theorem. 5.2 Analyticity of the S-matrix. 5.3 Reeh-Schlieder theorem. 5.4 Additivity of the energy-momentum­spectrum. 5.5 Borchers classes.

III. Algebras of Local Observables and Fields 105

1. Review of the Perspective 105 Characterization of the theory by a net of local algebras. Bounded operators. Unobservable fields, superselection rules and the net of abstract algebras of ob­servables. Transcription of the basic postulates.

2. Von Neumann Algebras. C'-Algebras. W'-Algebras 112 2.1 Algebras of bounded operators. Concrete CO-algebras and von Neumann algebras. Isomorphisms. Reduction. Factors. Classification of factors. 2.2 Ab­stract algebras and their representations. Abstract CO-algebras. Relation be­tween the C'-norm and the spectrum. Positive linear forms and states. The GNS-construction. Folia of states. Intertwiners. Primary states and cluster prop­erty. Purification. W' -algebras.

3. The Net of Algebras of Local Observables 128 3.1 Smoothness and integration. Local definiteness and local normality. 3.2 Symmetries and symmetry breaking. Vacuum states. The spectral ideals. 3.3 Summary of the structure.

4. The Vacuum Sector 143 4.1 The orthocomplemented lattice of causally complete regions. 4.2 The net of von Neumann algebras in the vacuum representation.

IV. Charges, Global Gauge Groups and Exchange Symmetry 149

1. Charge Superselection Sectors 149

Contents XIII

"Strange statistics". Charges. Selection criteria for relevant sectors. The pro­gram and survey of results.

2. The DHR-Analysis 156 2.1 Localized morphisms. 2.2 Intertwiners and exchange symmetry ("Statis­tics"). 2.3 Charge conjugation, statistics parameter. 2.4 Covariant sectors and energy-momentum spectrum. 2.5 Fields and collision theory.

3. The Buchholz-Fredenhagen-Analysis 174 3.1 Localized I-particle states. 3.2 BF-topological charges. 3.3 Composition of sectors and exchange symmetry. 3.4 Charge conjugation and the absence of "in­finite statistics" .

4. Global Gauge Group and Charge Carrying Fields 184 Implementation of endomorphisms. Charges with d = 1. Endomorphisms and non Abelian gauge group. DR categories and the embedding theorem.

5. Low Dimensional Space-Time and Braid Group Statistics 192 Statistics operator and braid group representations. The 2+ I-dimensional case with BF-charges. Statistics parameter and Jones index.

V. Thermal States and Modular Automorphisms 199

1. Gibbs Ensembles, Thermodynamic Limit, KMS-Condition 199 1.1 Introduction. 1.2 Equivalence of KMS-condition to Gibbs ensembles for fi­nite volume. 1.3 The arguments for Gibbs ensembles. 1.4 The representation induced by a KMS-state. 1.5 Phases, symmetry breaking and the decomposi­tion of KMS-states. 1.6 Variational principles and autocorrelation inequalities.

2. Modular Automorphisms and Modular Conjugation 216 2.1 The Tomita-Takesaki theorem. 2.2 Vector representatives of states. Convex cones in 1i. 2.3 Relative modular operators and Radon-Nikodym derivatives. 2.4 Classification of factors.

3. Direct Characterization of Equilibrium States 227 3.1 Introduction. 3.2 Stability. 3.3 Passivity. 3.4 Chemical potential.

4. Modular Automorphisms of Local Algebras 245 4.1 The Bisognano-Wichmann theorem. 4.2 Conformal invariance and the theorem of Hislop and Longo.

XIV Contents

5. Phase Space, Nuclearity, Split Property, Local Equilibrium 254 5.1 Introduction. 5.2 Nuclearity and split property. 5.3 Open subsystems. 5.4 Modular nuclearity.

6. The Universal Type of Local Algebras 267

VI. Particles. Completeness of the Particle Picture 271

1. Detectors, Coincidence Arrangements, Cross Sections 271 1.1 Generalities. 1.2 Asymptotic particle configurations. Buchholz's strategy.

2. The Particle Content 279 2.1 Particles and infraparticles. 2.2 Single particle weights and their decompo­sition. 2.3 Remarks on the particle picture and its completeness.

3. The Physical State Space of Quantum Electrodynamics 289

VII. Principles and Lessons of Quantum Physics. A Review of Interpretations, Mathematical Formalism and Perspectives 293

1. The Copenhagen Spirit. Criticisms, Elaborations 294 Niels Bohr's epistemological considerations. Realism. Physical systems and the division problem. Persistent non-classical correlations. Collective coordinates, decoherence and the classical approximation. Measurements. Correspondence and quantization. Time reflection asymmetry of statistical conclusions.

2. The Mathematical Formalism 304 Operational assumptions. "Quantum Logic" . Convex cones.

3. The Evolutionary Picture 309 Events, causal links and their attributes. Irreversibility. The EPR-effect. En­sembles vs. individuals. Decisions. Comparison with standard procedure.

VIII. Retrospective and Outlook 323

1. Algebraic Approach VB. Euclidean Quantum Field Theory 323

2. Supersymmetry 329

Contents XV

3. The Challenge from General Relativity 332 3.1 Introduction. 3.2 Quantum field theory in curved space-time. 3.3 Haw­king temperature and Hawking radiation. 3.4 A few remarks on quantum gra­vity.

Bibliography 349

Author Index and References 355

Subject Index 387