Table Of ContentQuantum and Non-Commutative Analysis
MATHEMATICAL PHYSICS STUDIES
Series Editor:
M. FLATO, Universite de Bourgogne, Dijon, France
VOLUME 16
Quantum and
Non-Commutative Analysis
Past, Present and Future Perspectives
edited by
Huzihiro Araki
Research Institute for Mathematical Sciences,
Kyoto University,
Kyoto, Japan
Keiichi R. Ito
Department ofM athematics and Information,
College ofL iberal Arts,
Kyoto Unversity,
Kyoto, Japan
Akitaka Kishimoto
Department ofM athematics,
Hokkaido University,
Sapporo, Japan
and
Izumi Ojima
Research Institute for Mathematical Sciences,
Kyoto University,
Kyoto, Japan
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Quantum and non-commutative analysis : past, presant, and future
perspectives 1 edited by Huzihiro Araki ... [et al.].
p. cm. -- <Mathematical phys1cs studies : v. 161
ISBN 978-90-481-4334-4 ISBN 978-94-017-2823-2 (eBook)
DOI 10.1007/978-94-017-2823-2
1. Mathematical physics--Congresses. 2. Quantum field theory
-Congresses. 3. Solid state physics--Congresses. 4. Statistical
physics--Congresses. 5. Quantum groups--Congresses. I. Araki,
Huzihiro, 1932- II. Series.
QC19.2.Q36 1993
530.1'43--dc20 93-31537
ISBN 978-90-481-4334-4
Printed on acid-free paper
Ali Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Foreword ix
Preface xi
Partl Quantum Field Theory
R. Haag Local Quantum Physics and Beyond 3
H. J. Borchers A Non-commuting Realization of Minkowski Space 11
J. E. Roberts Goldstone's Theorem Revisited 31
K. Fredenhagen Global Observables in Local Quantum Physics 41
D. Kastler State of the Art ofA lain Cannes' Version of the Stan-
dardModel 53
A. Arai Supersymmetric Extension of Quantum Scalar Field
Theories 73
Part2 Statistical and Solid State Physics
E. Lieband Optimal Two-Uniform Convexity and Fermion Hyper-
E. Carlen contractivity 93
J. Bricmont and Renormalization Group and Non-linear Media 113
A. Kupiainen
K. R. Ito Random Walk Representations and Mayer Expansion 119
R. F. Streater The Free-energy Theorem 137
Y.M.Park Dynamical Entropy of Quasi-local Algebras 149
L. J. Landau The Weak Coupling Limit fora Fermi Gas in a Random
Potential 167
J. Bellisard The Gap Labelling Theorem: the Case ofA utomatic
Sequences 179
vi
H. Araki Operator Algebra Approach to Soluble Models of
Quantum Spin Lattice Systems 183
T. Matsui On Conservation Laws of the XY Model 197
Part3 Quantum Groups
P. P. Kulish Reflection Equation Algebras and Quantum Groups 207
M. Plato and Quantum Groups, Star Products and Cyclic 221
D. Sternheimer Cohomology
K. Gawedzki Quantum Group Symmetry in Conformal Field Theory 239
T. Kakehi, Spectrum of an Operator Appears in the Quantum
K. Ueno and SU(l,l) Group 253
T.Masuda
Y. Nakagami Takesaki Duality for the Crossed Product by Quantum
Groups 263
Part4 Subfactors and Index Theory
T. Natsume A New Role of Graph Projections in Index Theory 285
M. Choda Endomorphisms and Automorphisms for Factor
Inclusions 291
H. Kosaki Automorphisms in the Irreducible Decomposition of
Sectors 305
M. Izumi Fusion Rules and Classification of Subfactors 317
S. Yamagami Vector Bundles and Bimodules 321
Y. Watatani Lattice Structure of Intermediate Subfactors 331
R. Longo Minimal Index Unimodular Sectors 335
J. Evans and Subfactors and Conformal Field Theory 341
Y. Kawahigashi
vii
PartS Operator Algebras and Related Topics
G. Elliott A Classification of Certain Simple C*-Algebra 373
H.Su K-Theoretic Classifications for Certain Real Rank Zero
C*algebras 387
K. Thomsen Inductive Limits ofI nterval Algebras
The Simple Case 399
S. Doplicher Operator Algebras and Abstract Duals: Progress and
Problems 405
C. Pinzari Regular Actions of Compact Groups on Cuntz Algebras 419
0. Bratteli, Almost Shift Invariant Projections in Irifinite Tensor
D. Evans and Products 427
A. Kishimoto
D. W. Robinson Strongly Elliptic and Subelliptic Operators on Lie
Groups 435
Former Students Postscripts 455
Editors List of Speakers 461
Foreword
In the past decade, there has been a sudden and vigorous development in
a number of research areas in mathematics and mathematical physics, such
as theory of operator algebras, knot theory, theory of manifolds, infinite
dimensional Lie algebras and quantum groups (as a new topics), etc. on the
side of mathematics, quantum field theory and statistical mechanics on the
side of mathematical physics. The new development is characterized by very
strong relations and interactions between different research areas which were
hitherto considered as remotely related.
Focussing on these new developments in mathematical physics and theory
of operator algebras, the International Oji Seminar on Quantum Analysis
was held at the Kansai Seminar House, Kyoto, JAPAN during June 25-29,
1992 by a generous sponsorship of the Japan Society for the Promotion of
Science and the Fujihara Foundation of Science, as a workshop of relatively
small number of (about 50) invited participants. This was followed by an
open Symposium at RIMS, described below by its organizer, A. Kishimoto.
The Oji Seminar began with two key-note addresses, one by V.F.R. Jones
on Spin Models in Knot Theory and von Neumann Algebras and by A. Jaffe
on Where Quantum Field Theory Has Led. Subsequently topics such as
Subfactors and Sector Theory, Solvable Models of Statistical Mechanics,
Quantum Field Theory, Quantum Groups, and Renormalization Group Ap
proach, are discussed. Towards the end, a panel discussion on Where Should
Quantum Analysis Go? was held.
A half of this volume consists of written version of these lectures. The
other half is from the subsequent RIMS Symposium described below. We
hope that these lecture notes provide readers a glimpse at the present status
of the fast developing subject of quantum analysis.
As the organizer, I would like to express my gratitude to many people whd
worked for the organization of this Oji Seminar for their devoted help, to
all participants for their enthusiastic participation, to all speakers for their
excellent talk and for writing up of their lectures for this volume and to the
Japan Society for the Promotion of Science and the Fujihara Foundation for
their generous support.
January, 1993 Huzihiro ARAKI
ix
Preface
THE EDITORS
January, 1993
The symposium on Non-commutative Analysis held in Kyoto, June 29
- July 2, 1992, was supported for overseas participants by Commemora
tive Association for the Japan World Exposition (1970), The Great Britain
Sasakawa Foundation, Japan Society for the Promotion of Science, Inoue
Foundation for Science, and The Kajima Foundation. I would like to ex
press my sincere thanks to them on behalf of the organizers.
I was advised and assisted by Professor Y. Nakagami among others during
the course of the financial arrangements of the symposium and was helped
very much by Ms. M. Ohmura, Hokkaido University, and Reiko for secretarial
works. I am grateful to them for their generous help.
Akitaka Kishimoto
Department of Mathematics, Hokkaido University
Sapporo, Japan
Two international symposia, Oji International Seminar on Quantum Anal
ysis and International Symposium on Non-commutative Analysis were held
in Kyoto in early summer of 1992, with many distinguished participants
from overseas. The Oji seminar was held at the Kansai Seminar House lo
cated 2.5 km north of Kyoto University. The International Symposium on
Non-commutative Analysis was subsequently held at the Research Institute
for Mathematical Sciences (RIMS).
The symposia were held on the occasion of the 60th birthday of Professor
Huzihiro Araki, and the lectures were dedicated to him. There were more
than 40 lectures, and most of them are described in these proceedings. We
would like to thank the speakers for presenting their recent work.
Professor Araki had many students who are now working in many fields of
mathematical sciences, and the two symposia were planned by these former
students and materialized with many advices from Professor Araki himself
and with enthusiatic help from secretaries in the RIMS, to whom we would
like to express our sincere acknowledgements.
We list the names, addresses and e-mail addresses of the speakers in the
final part of the proceedings. We here again would like to say "Arigatou
xi
