Lecture Notes in

Edited by J. Ehlers, MCinchen,K. Hepp, ZiJrich, and H. A. WeidenmCiller, Heidelberg Managing Editor: W. BeiglbSck, Heidelberg

39

International Symposium on Mathematical Problems in Theoretical Physics January 23-29, 1975, Kyoto University, Kyoto/Japan

Edited by H. Araki

Springer-Verlag Berlin.Heidelberg • New York 1975 Editor Prof. Huzihiro Araki Research Institute for Mathematical Sciences Kyoto University Kyoto 606/Japan

Library of Congress Cataloging in Publication Data

International Symposium on Mathematical Problems in Theoretical Physies, Kyoto University, 1975. Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics (M~) January 23-29, 1975, Kyoto University, Kyoto, Japan. (Lecture notes in physics ; 59) Bibliography: p. Includes index. i. Mathematieal physics--Congresses. I. Araki, Humihir o, 1952- II. Series. QCI9.2. I57 1975 550.1'5 75-17580

ISBN 3-540-07174-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-07174-1 Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo- copying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1975 Printed in Germany Offsetprinting and bookbinding: Julius Beltz, Hemsbach/Bergstr. FOREWORD

This volume is the Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics which was held at Research Institute for Mathematical Sciences (RIMS) and Research Institute for Fundamental Physics (RIFP) of Kyoto University, Kyoto, Japan on January 23-29, 1975. The subject of the Symposium is mainly mathematical problems in and along with related mathematics. The Symposium was attended by both mathematicians and theoretical physicists working in different fields such as theory of elementary particles, quantum field theory, statistical mechanics, theory of probability, theory of operator algebras, theory of hyper- functions, mathematical scattering theory, linear and nonlinear differential equations. The aim of the Symposium is to provide an opportunity to find out recent results and outstanding problems in boundary areas of mathematics and physics and to promote an interaction between mathematicians and physicists. On the side of physics, attention is focussed on methods in quantum field theory and statistical mechanics and as a consequence phenomenology is not included. On the side of mathematics, attention is focussed on those areas which have lively interactions with quantum field theory and statistical mechanics. Obviously there are other areas of physics, such as fluid mechanics and general relativity, which have lively interaction with mathematics but which could not be included. This kind of international meetings of mathematical physicists was started by the International Conference on Mathematical Problems in Quantum Field Theory and Quantum Statistics held at Steklov Mathematical Institute in Moscow, USSR, on December 12-18, 1972. This Symposium is the second in the series. The symbol mark Mn ¢ adopted in both meetings denotes the intersection of mathematics and physics and well indicates the interdisciplinary character of these meetings. It is hoped that this series of international meetings of mathematical physicists continue to be held biannually or triannually. In writing contributions in this Proceedings, each speaker of the Symposium has been asked to make it suitable for Lecture Notes in Physics. A full paper with technical details is supposed to be iV published elsewhere. Review of a subject, report on recent new results, description of outstanding problems and well-defined proposal or con- jecture, along with appropriate references, are supposed to be included. Thus this volume is hopedto be useful for graduate students in finding out an appropriate subject for his research and for mathematicians and physicists in finding out what is happening in their neighbouring fields as well as in his own field. Discussions on each contribution in the SympoSium are written mostly on the spot by relevant persons and are given to the speaker. Each speaker is then given a complete freedom in deciding whether he includes a certain question or comment into "Discussion" part of his manuscripts for this Proceedings. Contributions are grouped into 30 Sessions, and arranged in the order of presentation in each Session. Each morning Session (numbered 1 to 6) includes two one-hour review talks. Afternoon Sessions, which were done in parallel 2 at a time and 4 altogether in each afternoon, are arranged with odd numbered Sessions (held at RIMS) first and even numbered Sessions (held at RIFP) afterwards in each afternoon. Other- wise Sessions are arranged in chronological order. Authors of the last two articles were unable to attend the Symposium but kindly sent in their manuscripts which they had already prepared for presentation at the Symposium. Grouping of the contributions into Sessions were originally planned according to the subject of contributions. However, due to various reasons, last minutes rearrangement of the schedule caused some mixing. In addition, some contributions are related with more than one subjects and some others do not fit well into the original classification. Therefore we delete the subject title of each Session in this Proceedings. Instead we provide a Subject Index in which a contribution may appear more than once under different subjects.

The members of the Organizing Committee were as follows:

Huzihiro ARAKI, Hiroshi FUJITA, Masakuni IDA, Teruo IKEBE, Kyozi KAWASAKI, Ziro MAKI, Sigeru MIZOHATA, Noboru MAKANISHI, Kazuhiko NISHIJIMA, , Miklo SATO, Vasilie Sergeevfch VLADIMIROV, Hisaaki YOSHIZAWA(Chairman).

The Symposium was sponsored by International Mathematical Union, Japan Society for the Promotion of Science and Science Council Of Japan. We wish to thank these organizations for their financial and moral V

support of the Symposium. We wish to express our appreciations and thanks to all authors of contributions in this Proceedings for their splendid works and for their co-operation with assigned page limitation and early deadline, to all participants of the Symposium and to all secretarial and administrative staffs of host Institutes for their effort in bringing this Symposium to success.

March, 1975 Huzihiro ARAKI Executive Secretary of the Symposium Table of Contents

Session ~ (Chairman: H. J. Borchers) D. Iagolnitzer: Analytlcity properties of scattering amplitudes: a review of some recent developments ...... l~ M. Sato: Recent development in hyperfunction theory and its application to physics. (Microlocal analysis of S-matrices and related quantities.) ...... 13\i~ Session ll (Chairman: H. Komatsu) M. Kashiwara: Micro-local calculus ...... 30~ T. Kawai and H. P. Stapp: Microlocal study of ! S-matrix singularity structure ...... 38~ M. Morimoto: Convolutors for ultrahyperfunctions ...... 4~ Session l~ (Chairman: M. Sato) \ T. Kinoshita and A. Ukawa: Mass singularities of q Feynman amplitudes ...... 55 K. Pohlmeyer: Large momentum behaviour of Feynman amplitudes ...... 59J G. Sommer: Spectral dependence of the analyticlty domain of local vertex functions ...... 66 Session 12 (Chairman: K. NishiJima) E. Brining and P. Stiche!: Asymptotics and light-cone singularities in quantum field theory ...... 72 S. Schlieder: Structure of singularities of Wightman- ~V distributions and the Wilson-Zimmermann-expansion respectively lightcone-expansion ...... 85\~, B. Schroer: Conformal invariance in Minkowskian quantum ~\I field theory and global operator expansions ...... 92 K. Symanzik: On some massless superrenormalizable and non-renormalizable theories ...... 102 Session ~4 (Chairman: Y. Ohnuki) Y. 0hnuki and S. Kamefuchi: The locality condition in F parafermi field theory ...... 107~ M. Ninomiya and K. Watanabe: Bound state nature and deep inelastic structure functions ...... lll V V. P. Shelest: The physical content of the statistical bootstrap and high energy hadron interaction ...... ll4 VIII

K. Kinoshita: Fleld theoretical approach to composite particle reactions ...... ll6 Session 2 (Chairman: J. R. Klauder) J. Glimm and A. Jaffe: Particles and bound states and / progress toward unitarity and scaling ...... ll8 K. Kikkawa: Field theory of relativistic strings ...... 128 S~ssion 21 (Chairman: N. Mugibayashi) / K. Hepp: Results and problems in irreversible statistical // mechanics of open systems ...... 13~ J. S. Feldman and K. Osterwalder: The Wightman axioms ~/ and the mass gap for weakly coupled (¢4) 3 quantum / field theories ...... 151 Session 2~ (Chairman: H. Ezawa) J. R. Klauder: Soluble models and the meaning of nonrenormalizability ...... 160f Y. Kato: Bound states and asymptotic fields in model field theories ...... 170/'~ Session 22 (Chairman: M. Ida) M. Minami: Feynman propagators associated with the Veneziano model ...... 175 ! T. Yoneya: Dual string models and quantum gravity ...... 180 G. Konisi and T. Saito: Physical states satisfying ,~ supergauge conditions in dual resonance models ...... 184/ Y. Chikashige and K. Kamimura: The canonical quantization f of a relativistic string ...... 187 Session 24 (Chairman: K. Kawarabayashi) F. G~rsey: Algebraic methods and quark structure ...... 189 K. Sekine: Unitarity and asymptotic condition in a model with dipole ghost ...... 196 i K. Yokoyama and ReiJiro Kubo: Invariant gauge families ~ inherent in Abelian-gauge field theory ...... 199/ N. Nakazawa and M. Yamada: Space-tlme approach to anomalies in the Ward-Takahashi identities and implications in physical processes ...... 202 Session ~ (Chairman: V. S. V!adimirov) L K. NishiJima: Dispersion approach to Ward-Takahashi identities ...... 205 A. A. Slavnov: Renormalizatlon of supersymmetric gauge theories ...... 214 Session 31 (Chairman: T. Hida) K. ItS: Stochastic calculus ...... 218 Masatoshi Fukushima: Asymptotic properties off the spectra2 distributions of disordered systems ...... 224 M. Miyamoto: A remark to Harris's theorem on percolation .... 228 Session 33 (Chairman: K. ItS) ~ I. Kubo: The ergodicity of the motion of a particle in a potential field ...... 230 ~ T. Niwa: Ergodicity of some simple model systems of infinitely many particles ...... 236 Session 32 (Chairman: Z. Maki) A. Hosoya and J. Ishida: New exact solutions of the classical Yang-Mills field equation ...... 238 ~kj j. Arafune, P. G. O. Freund, and C. J. Goebel: Topology of Higgs fields ...... 240 T. Maskawa and H. NakaJima: Spontaneous breaking of chiral symmetry in a vector-gluon model ...... 242 ~! N. Nakanishi: Quantum field-theoretlcal approach to ~, spontaneously broken gauge invariance ...... 245 \J H. Reeh: Reviews on axiomatic study of symmetry breaking. 2~9 Session 34 (Chairman: N. Nakanlshi) \~ 0, I. Zavialov: Structure of renormalized Feynman amplitudes. .256 \] Y. FuJii and Y. Takahashi: On the regularization in the Callan-Symanzik equation ...... 261 Session ~ (Chairman: N. M. Hugenholtz) ~,,J S. Doplicher: The statistics of particles in local quantum theories ...... 264 \'J Ryogo Kubo: Relaxation and fluctuation of macrovariables. . .274 Session 41 (Chairman: K. Sekine) \2 V. S. Vladimirov: Holomorphic functions with a nonnegative imaginary part in the future tube ...... 281 \~ H. J. Borchers: Algebraic aspects of Wightman quantum field theory ...... 283 [] G. Reents: The infrared problem and non-Fock representations of the canonical commutation relations ...... 293 G. Lassner: Continuous representations of the test function algebra and the existence problem for quantum fields ...... 297 Session 4~ (Chairman: T. Nakano) D. W. Robinson: Unbounded derivations of C* algebras ...... 303 R. Arens: A method for making relativistic non-quantum systems by constructing the Hamiltonian and the other nine generating functions ...... 312 !/ G. G. Emch: An algebraic approach to the theory of K-flows and K-entropy ...... 315 Sessi2~_~ (Chairman: R. Abe) \j R. Jackiw: Symmetry restoration at finite temperature ..... 319 / G. Jona-Lasinlo: Critical behaviour of stationary random fields ...... 326

~J J. Zittartz: Phase transitlons~of continuous order ...... 330 S¢~slon 44 (Chairman: S. Katsura) H. Nakano: Order of the phase transition related to the degeneracy and interaction with proposal for an exact problem ...... 336 T. Ogawa: The Heisenberg model on the Cayley tree ...... 339 C. Di Castro: Generalized Gell-Mann and Low group trans- formations: scaling variables and cross-over effects. . 342 ! Y. Yamazaki: Generalizations and applications of Callan- Symanzlk equations to statistical mechanics in critical phenomena ...... 349 ~ T. Izuyama: On the rigorous definition of superfluidity and superconductivity ...... 353 Session 5 (Chairman: D. Ruelle) J.-L. Lions: Variational problems and free boundary problems ...... 356 J. L. Lebowitz: Uniqueness, analyticity and decay properties of correlations in equilibrium systems ...... 370 Session 51 (Chairman: H. Fujita) ~ S. Matsuura: On the propagation of support of solutions to general systems of partial differential equations .... 380 ~ M. Toda: Wave propagation in a non-linear lattice ...... 387 ~/ R. Jackiw: Quantization of non-linear waves ...... 394 Session 53 (Chairman: M. Yamaguchi) Y. Kametaka: On the non-linear diffusion equation of Kolmogorov-Petrovskii-Piskunov type ...... 401 ~j T. Nishida and M. Mimura: Global solutions to the Broadwell's model of Boltzmann equation for a simple discrete velocity gas ...... 408 Session 52 (Chairman: N. Saito) H. Mori: ~ Scaling method for asymptotic evaluation in nonequilibrium statistical mechanics~ ...... 413 Y. Kuramoto: Self-entralnment of a population of coupled non-llnear oscillators ...... 420 ×I

\j M. Suzuki: Thermodynamic limit of non-equilibrium systems extensive property, fluctuation and nonlinear relaxation ...... 423 Session ~4 (Chairman: H. Nakano) \ J. L. Lebowltz: Equilibrium states and ergodic properties of infinite systems ...... 432

~J H. Hasegawa: Variational principles for Markov processes and Onsager principle ...... 433 B. M~hlschlegel: Functional averages in statistical physics ...... 437 M. Takahashi: On the validity of collective variable description of Bose systems ...... 446 Session 6 (Chairman: G. G. Emch) D. Ruelle: Equilibrium statistical mechanics of one- dimensional classical lattice systems ...... 449 k~ T. Ikebe: A look at the development of spectral and scattering theory in Japan ...... 458 Session 61 (Chairman: T. Ikebe) ~.~ J. M. Combes: On the Born-Oppenhelmer approximation ...... 467

j S. T. Kuroda: Abstract approaches to spectral and scattering theory, contributions from Japan ...... 472 V~ Y. Sait~: Eigenfunction expansions for differential operators with operator-valued coefficients and their applications to the Schr6dinger operators with long- range potentials ...... 476 Session 63 (Chairman: S. T. Kuroda) \ L. E. Thomas: Asymptotic completeness in three-partlcle quantum mechanical scattering ...... 483 ~ K. Mochizuki: Decay and asymptotlcs for wave equations with dissipative term ...... 486 t~ T. Sasakawa: Treatment of three-body problems in coordinate space ...... 491 Session 6~ (Chairman: H. Matsuda) B. Souillard: Links between decay properties of correla- tions and analyticity of the pressure and correlation functions ...... 497 A. Verbeure: Linear response, stability, cluster properties. . 504 ~ G. L. Sewell: Equilibrium and metastable states of classical systems ...... 510 J. T. Lewis and J. V. Pul~: Dynamical theories of Brownian motion ...... 516 XII

Session ~,4 (Chairman: K. Kawasaki) \K~ Ikeda: Distribution of zeros and the equation of state ...... 520 Y. Karaki: New theorems on algebraic equation and its application to statistical physics ...... 524 \ Masahisa Fukushlma: 0perator-valued-measure approach to spectra of two-dimensional classical harmonic lattices ...... 528 M. E. Mayer: Models for relativistic statistical mechanics ...... 532 Contributions by mail C. DeWitt-Morette: Path integrals in Riemannian manifolds 535 B. Simon: Bose quantum field theory as an Is!rig ferromagnet: recent developments ...... 543

Author Index ...... 554 Subject Index ...... 555