Mathematical Theory of Quantum Fields

Total Page：16

File Type：pdf, Size：1020Kb

Mathematical Theory of Quantum Fields HUZIHIRO ARAKI Department of Mathematics Faculty of Science and Technology The Science University of Tokyo Japan Translated by Ursula Carow-Watamura Department of Physics Tohoku University Japan OXPORD UNIVERSITY PRESS CONTENTS 1 States and observables 1 1.1 Probabilistic description l 1.2 States and observables 2 1.3 Functions of observables 3 1.4 The expectation value of an observable 7 1.5 Mixture and pure states 8 1.6 The physical topology of the states 13 1.7 Equivalent theories 14 1.8 Symmetries 16 2 Quantum theory 19 2.1 The quantum mechanical description 19 2.2 Algebraic viewpoint 27 2.3 The representation associated with a state: GNS construction 33 2.4 The symmetries of quantum mechanics 42 2.5 Symmetries from the algebraic point of view 53 3 The relativistic symmetry 58 3.1 Minkowski space 58 3.2 The inhomogeneous Lorentz group 59 3.3 The relativistic symmetry in quantum mechanics 61 3.4 Irreducible representations and one-particle states 66 3.5 Free particle system and Fock space 70 3.6 Energy momentum 75 4 Local observables 78 4.1 General properties of local observables 78 4.2 The vacuum state 80 4.3 Irreducibility 84 4.4 Mass gap and exponential clustering property 89 4.5 The JLD representation 92 4.6 Truncated expectation values and multiple clustering property 97 4.7 Additivity assumption and Reeh-Schlieder theorem 100 4.8 Quantum field theory 102 4.9 From quantum fields to local observables 106 5 Scattering theory 108 5.1 The concept of scattering states and the S-matrix 108 5.2 Description of asymptotic states 111 Xll CONTENTS 5.3 Construction of asymptotic states 113 5.4 The counter interpretation of asymptotic states—verification of an appropriate asymptotic behaviour 128 5.5 Asymptotic conditions and reduction formula for the S-matrix 133 5.6 Analyticity and TCP symmetry 151 6 Sector theory 160 6.1 Superselection rules and localized excitations 160 6.2 Localized endomorphisms and sectors 166 6.3 Permutation of excitations and statistics of a sector 171 6.4 Charge conjugate sector 181 6.5 Operator algebra system of fields and gauge groups 184 6.6 Theory of local charges 187 Appendix A: Hilbert space and operators 193 A.l Hilbert space 193 A.2 Pre-Hilbert space and completion 194 A.3 Orthonormal basis 195 A.4 Unitary and antiunitary maps 196 A.5 Subspaces and projection operators 198 A.6 Direct sum and direct integral 199 A.7 Spectral decomposition 201 A.8 Trace 203 A.9 Unbounded operators 204 A. 10 One-parameter group of unitary operators 205 A. 11 Tensor product 206 Appendix B: Operator algebras 208 B.l C* algebras 208 B.2 Von Neumann algebras 210 Appendix C: Free fields 216 C.l Charged scalar field 216 C.2 One particle system with positive mass and arbitrary spin 217 C.3 Examples of free fields with positive mass and integer spin 218 C.4 Examples of free fields with positive mass and half odd integer spin 220 General literature and references 222 References for Chapter 1 224 References for Chapter 2 224 References for Chapter 3 225 References for Chapter 4 226 References for Chapter 5 227 References for Chapter 6 230 Literature for appendices 232 Index 233 .

Copy Link

Recommended publications

April 2010 Contents
International Association of Mathematical Physics News Bulletin April 2010 Contents International Association of Mathematical Physics News Bulletin, April 2010 Contents Reflections on the IAMP Geography 3 Aharonov-Bohm & Berry Phase Anniversaries 50/25 5 The 25th anniversary of the founding of HARL 8 An interview with Huzihiro Araki 10 Shing-Tung Yau the Wolf Prize laureate 2010 in Mathematics 14 News from the IAMP Executive Committee 22 A new associated member: PIMS 26 Bulletin editor Valentin Zagrebnov Editorial board Evans Harrell, Masao Hirokawa, David Krejˇciˇr´ık, Jan Philip Solovej Contacts [email protected] http://www.iamp.org Cover photo (courtesy of Professor A.Tonomura): From double-slit experiment to the Aharonov-Bohm effect. See a comment at the end of the page 7. The views expressed in this IAMP News Bulletin are those of the authors and do not necessary represent those of the IAMP Executive Committee, Editor or Editorial board. Any complete or partial performance or reproduction made without the consent of the author or of his successors in title or assigns shall be unlawful. All reproduction rights are henceforth reserved, mention of the IAMP News Bulletin is obligatory in the reference. (Art.L.122-4 of the Code of Intellectual Property). 2 IAMP News Bulletin, April 2010 Editorial Reflections on the IAMP Geography by Pavel Exner (IAMP President) The topic of today’s meditation was inspired by complaints of American colleagues about the shaky position our discipline enjoys in the U.S. True, such woes are ubiquitous since com- petition for resources in science was and will always be tough.
[Show full text]
Arthur Strong Wightman (1922–2013)
Obituary Arthur Strong Wightman (1922–2013) Arthur Wightman, a founding father of modern mathematical physics, passed away on January 13, 2013 at the age of 90. His own scientific work had an enormous impact in clar- ifying the compatibility of relativity with quantum theory in the framework of quantum field theory. But his stature and influence was linked with an enormous cadre of students, scientific collaborators, and friends whose careers shaped fields both in mathematics and theoretical physics. Princeton has a long tradition in mathematical physics, with university faculty from Sir James Jeans through H.P. Robertson, Hermann Weyl, John von Neumann, Eugene Wigner, and Valentine Bargmann, as well as a long history of close collaborations with colleagues at the Institute for Advanced Study. Princeton became a mecca for quantum field theorists as well as other mathematical physicists during the Wightman era. Ever since the advent of “axiomatic quantum field theory”, many researchers flocked to cross the threshold of his open office door—both in Palmer and later in Jadwin—for Arthur was renowned for his generosity in sharing ideas and research directions. In fact, some students wondered whether Arthur might be too generous with his time helping others, to the extent that it took time away from his own research. Arthur had voracious intellectual appetites and breadth of interests. Through his interactions with others and his guidance of students and postdocs, he had profound impact not only on axiomatic and constructive quantum field theory but on the de- velopment of the mathematical approaches to statistical mechanics, classical mechanics, dynamical systems, transport theory, non-relativistic quantum mechanics, scattering the- ory, perturbation of eigenvalues, perturbative renormalization theory, algebraic quantum field theory, representations of C⇤-algebras, classification of von Neumann algebras, and higher spin equations.
[Show full text]
Solutions in Constructive Field Theory
Solutions in Constructive Field Theory Leif Hancox-Li ∗ [email protected] June 7, 2016 Abstract Constructive field theory aims to rigorously construct concrete, non-trivial solutions to Lagrangians used in particle physics, where the solutions satisfy some relevant set of axioms. I examine the relationship of solutions in constructive field theory to both axiomatic and Lagrangian quantum field theory (QFT). I argue that Lagrangian QFT provides conditions for what counts as a successful constructive solution, and that it provides other information that guides constructive field theorists to solutions. Solutions matter because they describe the behavior of QFT systems, and thus what QFT says the world is like. Constructive field theory, in incorporating ingredients from both axiomatic and Lagrangian QFT, clarifies existing disputes about which parts of QFT are philosophically relevant and how rigor relates to these disputes. ∗I thank Bob Batterman and Michael Miller for comments and encouragement. Several anonymous referees also contribued valuable comments. Copyright Philosophy of Science 2016 Preprint (not copyedited or formatted) Please use DOI when citing or quoting 1 Introduction To date, philosophers of quantum field theory (QFT) have paid much attention to roughly two kinds of QFT: axiomatic approaches to algebraic QFT (Halvorson and Muger,¨ 2006), and Lagrangian-based QFT as used by particle physicists (Wallace, 2006). Comparatively less attention, however, has been paid to constructive QFT, an approach that aims to rigorously construct non-trivial solutions of QFT for Lagrangians and Hamiltonians that are important in particle physics, ensuring that such solutions satisfy certain axioms. In doing so, constructive QFT mediates between axiomatic approaches to QFT and physicists’ Lagrangian-based QFT.
[Show full text]
Operator Algebras and Applications, Volume 38-Part 1
http://dx.doi.org/10.1090/pspum/038.1 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 38, Part 1 OPERATOR ALGEBRAS AND APPLICATIONS AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1982 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT QUEENS UNIVERSITY KINGSTON, ONTARIO JULY 14-AUGUST 2, 1980 EDITED BY RICHARD V. KADISON Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 79-27061 Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics (1980: Queens University, Kingston, Ont.) Operator algebras and applications. (Proceedings of symposia in pure mathematics; v. 38) Includes bibliographies and index. 1. Operator algebras-Congresses. I. Kadison, Richard V., 1925— II. Title. III. Series. QA326.S95 1982 512'.55 82-11561 ISBN 0-8218-1441-9 (v. 1) ISBN 0-8218-1444-3 (v. 2) ISBN 0-8218-1445-1 (set) 1980 Mathematics Subject Classification. Primary 46L05, 46L10; Secondary 43A80, 81E05, 82A15. Copyright © 1982 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. TABLE OF CONTENTS PARTI Author index xi Preface RICHARD V. KADISON xvn* Operator algebras—the first forty years 1 RICHARD V. KADISON On the structure theory of C*-algebras: some old and new problems 19 EDWARD G. EFFROS Homological invariants of extensions of C*-algebras 35 JONATHAN ROSENBERG Geometric resolutions and the Kunneth formula for C*-algebras 77 CLAUDE SCHOCHET The J£-groups for free products of C*-algebras 81 JOACHIM CUNTZ The internal structure of simple C*-algebras 85 JOACHIM CUNTZ K homology and index theory 117 PAUL BAUM AND RONALD G.
[Show full text]
Is Our Mathematics Natural? the Case of Equilibrium Statistical Mechanics
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 1, July 1988 IS OUR MATHEMATICS NATURAL? THE CASE OF EQUILIBRIUM STATISTICAL MECHANICS DAVID RUELLE ABSTRACT. IS human 20th-century mathematics a natural or an arbi trary logical construct? Some insight into this ill-posed but fascinating question is gained from the study of mathematical physics. A number of ideas introduced by physical rather than mathematical necessity turn out later to be mathematically useful. (Equilibrium statistical mechan ics has been a particularly rich source of such ideas.) In view of this the author argues that our mathematics may be much more arbitrary than we usually like to think. 1. Is our mathematics natural? The story goes that, when he reached heaven, Wolfgang Pauli requested to see his Creator, and asked Him to explain why the fine structure constant of electrodynamics has the value a « 1/137. The Almighty went to the blackboard, and was busy writing formulae for a couple of hours. During that time Pauli sat, shaking his head and not saying a word. When finally the answer came: a~l — 137.0359..., Pauli stood up, still shaking his head, took the chalk and pointed to an essential flaw in the calculation. I heard the story from Res Jost, and I wouldn't bet that it is completely authentic. Anyway, I think that many of us would like to ask some questions about physics and mathematics of Him who knows—when the opportunity arises. There are a number of obvious questions. For instance, about the consistency of mathematics: has He
[Show full text]
Rudolf Haag's Legacy of Local Quantum Physics And
Rudolf Haag’s legacy of Local Quantum Physics and reminiscences about a cherished teacher and friend In memory of Rudolf Haag (1922-2016) submitted to the Eur. Phys. J. H Bert Schroer permanent address: Institut für Theoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin, Germany November 2016 Abstract After some personal recollectioms about Rudolf Haag and his thoughts which led him to ”Local Quantum Physics”, the present work recalls his ideas about scattering theory, the relation between local observables and localized fields and his contributions to the physical aspects of modu- lar operator theory which paved the way for an intrisic understanding of quantum causal localization in which fields ”coordinatize” the local algebras. The paper ends with the presentation of string-local fields whose con- struction and use in a new renormalization theory for higher spin fields is part of an ongoing reformulation of gauge theory in the conceptual setting of Haag’s LQP. 1 First encounter with Rudolf Haag arXiv:1612.00003v1 [math-ph] 30 Nov 2016 On his return from the Niels Bohr Institute in Copenhagen to the University of Munich Rudolf Haag passed through Hamburg to meet his colleague Harry Lehmann, at that time the newly appointed successor of Wilhelm Lenz who held the chair of theoretical physics since the 1920 foundation of the University of Hamburg. It was the year 1958 shortly after the decision to construct the DESY particle accelerator in Hamburg which created a lot of excitement. I had nearly completed my diploma thesis under Lehmann and begun to worry about my career.
[Show full text]
Current Groups and the Hamiltonian Anomaly
CURRENTGROUPSANDTHEHAMILTONIANANOMALY ossi niemimäki Doctoral dissertation to be presented for public examination with the permission of the Faculty of Science of the University of Helsinki, in Auditorium 107, Athena Building, on the 27th of May 2020 at 14 o’clock. Department of Mathematics and Statistics Faculty of Science University of Helsinki Helsinki 2020 isbn 978-951-51-5976-2 (paperback) isbn 978-951-51-5977-9 (pdf) http://ethesis.helsinki.fi Unigraa Oy Helsinki 2020 if you’re going to try, go all the way. there is no other feeling like it. you will be alone with the gods and the nights will ame with re. — Charles Bukowski ABSTRACT Gauge symmetry invariance is an indispensable aspect of the eld-theoretic models in classical and quantum physics. Geometrically this symmetry is often modelled with current groups and current algebras, which are used to capture both the idea of gauge invariance and the algebraic structure of gauge currents related to the symmetry. The Hamiltonian anomaly is a well-known problem in the quantisation of massless fermion elds, originally manifesting as additional terms in current algebra commutators. The appearance of these anomalous terms is a signal of two things: that the gauge invariance of quantised Hamiltonian operators is broken, and that consequently it is not possible to coherently dene a vacuum state over the physical conguration space of equivalent gauge connections. In this thesis we explore the geometric and topological origins of the Hamiltonian anomaly, emphasising the usefulness of higher geometric structures in the sense of category theory. Given this context we also discuss higher versions of the gauge-theoretic current groups.
[Show full text]
Nine Lessons of My Teacher, Arthur Strong Wightman a Talk at Princeton University, 10 March 2013 by Arthur Jaffe (Cambridge (MA), Basel, and ZÜrich)
Nine Lessons of my Teacher, Arthur Strong Wightman Nine Lessons of my Teacher, Arthur Strong Wightman A Talk at Princeton University, 10 March 2013 by Arthur Jaffe (Cambridge (MA), Basel, and Zürich) At Harvard University, Arthur Ja↵e is the Landon T. Clay Pro- fessor of Mathematics and Theoretical Science. He is currently on sabbatical leave at the University of Basel and at the ETH, Zürich. He has worked in several fields of mathematical physics. Arthur Ja↵e received his doctorate in physics from Princeton as a student of Arthur Wightman, after studying mathematics as a Cambridge undergraduate and chemistry as a Princeton un- dergraduate. He has served as president of IAMP, and also as president of the American Mathematical Society. Preamble. All of us here today have been students of Arthur Wightman. But his thirty-six formal doctoral students form a very special club, which I represent on this part of today’s program. A few members appear in this 1965 snapshot of Arthur Wightman’s seminar in Palmer 301, with Arthur smiling benevolently in the center. IAMP News Bulletin, April 2013 3 Arthur Ja↵e Postdoctoral fellows and other visitors to Princeton surround the students during those exciting times some 48 years ago. To the best of my knowledge, here is the complete, wide-ranging list of graduate students: Arthur Wightman’s student club Silvan Samuel Schweber 1952 Gerald Goldin 1969 Richard Ferrell 1952 Eugene Speer 1969 Douglas Hall 1957 George Svetlichny 1969 Oscar W. Greenberg 1957 Barry Simon 1970 Huzihiro Araki 1960 Charles Newman 1971 John S.
[Show full text]
Algebraic Quantum Field Theory--An Introduction
Algebraic Quantum Field Theory – an introduction Christopher J Fewster∗1 and Kasia Rejzner†1 1Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom. November 19, 2019 Abstract We give a pedagogical introduction to algebraic quantum field theory (AQFT), with the aim of explaining its key structures and features. Topics covered include: algebraic formulations of quantum theory and the GNS representation theorem, the appearance of unitarily inequivalent representations in QFT (exemplified by the van Hove model), the main assumptions of AQFT and simple models thereof, the spectrum condition, Reeh–Schlieder theorem, split property, the universal type of local algebras, and the theory of superselection sectors. The abstract discussion is illustrated by concrete examples. One of our concerns is to explain various ways in which quantum field theory differs from quantum mechanics, not just in terms of technical detail, but in terms of physical content. The text is supplemented by exercises and appendices that enlarge on some of the relevant mathematical background. These notes are based on lectures given by CJF for the International Max Planck Research School at the Albert Einstein Institute, Golm (October, 2018) and by KR at the Raman Research Institute, Bangalore (January, 2019). Contents 1 Introduction 2 2 Algebraic quantum mechanics 4 2.1 Postulatesofquantummechanics. ....... 4 2.2 Algebraicapproach ............................... ... 5 2.3 Statesandrepresentations. ....... 7 arXiv:1904.04051v2 [hep-th] 18 Nov 2019 3 Case study: the van Hove model 10 4 Algebraic QFT 13 4.1 Basicrequirements ............................... ... 13 4.2 Examples ....................................... 15 4.3 Quasifreestatesforthefreescalarfield . ........... 20 5 The spectrum condition and Reeh–Schlieder theorem 24 5.1 Thespectrumcondition.
[Show full text]
Arthur Jaffe's Publications
Arthur Jaffe’s Publications Revised August 2020 [1] Complex Angular Momentum in Two-Channel Problems, with Y.S. Kim, Phys. Rev., 127 (1962), 2261{2266. [2] Asymptotic Behavior of the S Matrix for High Angular Momentum, with Y.S. Kim, Phys. Rev., 129 (1963), 2818{2823. [3] Divergence of Perturbation Theory for Bosons, Commun. Math. Phys., 1 (1965), 127{149. [4] Entire Functions of the Free Field, Ann. Phys. (N.Y.), 32 (1965), 127{156. [5] Nonpositivity of the Energy Density in Quantized Field Theories, with H. Epstein and V. Glaser, Nuovo Cimento, 36 (1965), 1016{1022. [6] On the Approximation of Quantum Field Theories, J. Math. Phys., 6 (1965), 1172{1178. [7] Dynamics of a Cut-off λφ4 Field Theory, Princeton University Thesis (1965). [8] Existence Theorems for a Cut-off λφ4 Field Theory, in Mathematical Theory of Elementary Particles, R. Goodman and I. Segal (eds.), MIT Press (l966). [9] Local Quantum Fields at Operator-Valued Generalized Functions, in Conference on Dispersion Theory at MIT, I.E. Segal (ed.), MIT (1966). [10] A Preliminary Study of Infinite Volume Limits in a Cut-off λφ4 Field Theory, Conference on Functional Integration and Constructive Quantum Field Theory, Editor, I.E. Segal, MIT (1966). [11] Wick Polynomials at a Fixed Time, J. Math. Phys., 7 (1966), 1250{1255. [12] Form Factors at Large Momentum Transfer, Phys. Rev. Lett., 17 (1966), 661{663. [13] High Energy Behavior of Local Quantum Fields, SLAC PUB 250 (1966), Stanford Linear Accelerator Center. [14] A Generalization of the Paley-Wiener-Schwartz Theorem, SIAM Review, 15 (1967), 1046{ 1047.
[Show full text]
Notices of the American Mathematical Society
ISSN 0002-9920 of the American Mathematical Society February 2004 Volume 51, Number 2 Geometrization of 3-Manifolds via the Ricci Flow page 184 RIMS, an Institute for Japan and the World page 194 Los Angeles Meeting page 290 Lawrenceville Meeting page 292 Houston Meeting page 295 Heawood's demonstration of Kempe's error (see page 207) Document conversion between Microsoft™ WORD and LaTeX is easier than watching PING-PONG! Paintwork © 2003 Alex Chikrii. a, rd \i Word2TeX and TeX2Word are either registered trademarks or trademarks of Chikrii Softlab in the United States and/ or other countries. Microsoft, Microsoft Word are either registered trademarks or trademarks of Microsoft Corporation in the United States and/ or other countries. Other product and company names mentioned herein may be trademarks of their respective owners. ~ Chikrii Soft lab ~ http:/jwww.chikrii.com International Mathematics Research Notices Editors Website: http://imrn.hindawi.com Morris Weisfeld IMRN provides very fast publication of research articles of high current interest in all Managing Editor areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics. Issues are published as frequently as nec Dan Abramovich essary. IMRN is expected to publish 80± issues in volume 2004. The articles of the IMRN are Enrico Arbarello reviewed/indexed in COMPUMATH Citation Index, Current Contents, lSI Alerting Services, Joseph Bernstein Mathematical Reviews, Science Citation Index, SciSearch, and Zentralblatt fUr Mathematik. Enrico Bombieri There are no page charges. Submissions are made by email to [email protected]. Richard E. Borcherds New print subscribers shall receive a free copy of all back volumes, i.e., volumes 1991-2003.
[Show full text]
50 Years of Communications in Mathematical Physics !
50 Years of Communications in Mathematical Physics ! 50 Years of Communications in Mathematical Physics ! by Arthur Jaffe (Harvard) Arthur Jaffe, the Landon T. Clay Professor of Mathemat- ics and Theoretical Science at Harvard University, studied at Princeton and Cambridge Universities, starting in experimen- tal chemistry, transitioning through mathematics, and ending in mathematical physics. He has been on the faculty of Harvard University since 1967, and has served as President of the In- ternational Association of Mathematical Physics, President of the American Mathematical Society, and Chair of the Council of Scientific Society Presidents. He was a founding Member, Director, and first President of the Clay Mathematics Insti- tute. He hears the mathematical physics community speaking with one voice. This summer marks a milestone in the history of our subject: the first issue of Com- munications in Mathematical Physics appeared just fifty years ago! This remarkable initiative originated with a small group of researchers who sorely felt the need for a jour- nal that specialized in mathematical physics. Thinking back it is hard to imagine how our subject would have evolved so coherently without this top-quality, high-profile journal. A key person in the founding of CMP was Res Jost in Zürich. He believed in the need for a journal in which physics and mathematics were both central. Jost was a friend of Konrad Springer, heir to the family dynasty that became famous for publishing outstanding scientific books and journals. In those times, personal connections and a close camaraderie led to many good developments. Res Jost, Jr. described to me how Konrad Springer met his father in the Josts' new home in Unterengstringen (a Zürich suburb) to discuss the possibilities for a new journal.
[Show full text]