Lecture Notes in Physics 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 quantum field theory and statistical mechanics 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, David RUELLE, 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