Symmetries in Particle Physics PDF
Preview Symmetries in Particle Physics
SYMMETRIES IN PARTICLE PHYSICS Professor Feza Cursey SYMMETRIES IN PARTICLE PHYSICS EDITED BY Itzhak Bars University of Southern California Los Angeles, California Alan Chodos and Chia-Hsiung Tze Yale University New Haven, Connecticut Springer Science+Business Media, LLC Library of Congress Cataloging in Publication Data Main entry under title: Symmetries in particle physics. "Proceedings of a symposium celebrating Feza Gürsey's sixtieth birthday, held April 11, 1981, at Yale University, New Haven, Connecticut"—T.p. verso. "List of publications of Feza Gürsey": p. Bibliography: p. Includes index. 1. Symmetry (Physics)—Congresses. 2. Particles (Nuclear physics)—Congresses. 3. Gursey, Feza. I. Bars, Itzhak. IL Chodos, Alan. III. Tze, Chia. IV. Gürsey, Feza. QC793.3.S9S93 1984 539.7 84-13418 ISBN 978-1-4899-5315-5 ISBN 978-1-4899-5315-5 ISBN 978-1-4899-5313-1 (eBook) DOI 10.1007/978-1-4899-5313-1 Proceedings of a symposium celebrating Feza Gürsey's sixtieth birthday, held April 11, 1981, at Yale University, New Haven, Connecticut © 1984 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1984 Softcover reprint of the hardcover 1st edition 1984 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Editors' Foreword On April II, 1981, many friends of Feza and Suha Giirsey descended on the Yale Campus from near and far for a day-long celebration of Feza's sixtieth birthday. The present collection of articles is, in part, a memento of that joyful gathering. Besides the talks delivered at the Symposium that was part of the festivities, this volume includes additional invited papers from a representative sample of Feza's long list of close friends and colleagues. We hope to have captured between the covers of this book some small part of Feza's remarkable breadth of knowledge and interests, as well as his unique style of doing physics. We dedicate it to him as a sincere expression of our affection and admiration. The production of this book benefited greatly from the efforts of several members of the Yale Theory Group. We wish to thank Thomas Appelquist who was instrumental in making the Yale Symposium an immense success. We gratefully acknowledge the skills and labor of John Gipson, Mary LaRue, Sait Vmar, Jeff Hersh and Roger Ove who spent interminable hours typesetting the text and equations. We warmly thank the Yale Printing Service without whose essential collaboration this book would not have been possible. Finally, we deeply appreciate the cooperation of Ellis Rosenberg of Plenum Press who shepherded this enterprise through to its completion. Itzhak Bars Alan Chodos Hsiung Chia Tze New Haven, April, 1984 Contents Some Perspectives on the Nonlinear Sigma Model 1 T. Appelquist The Turbulent Aether 9 Yoichiro Nambu Does Quantum Chromodynamics Imply Confinement? 19 Gerard 't Hooft Chaos and Cosmos 33 Luigi A. Radicati di Brozolo Dynamic Symmetries in Nuclei, Atoms and Molecules 47 F. Iachello The Symmetry and Renormalization Group Fixed Points of Quartic 63 Hamiltonians Louis Michel Relativistic Heavy Ion Collisions and Future Physics 93 T. D. Lee The Fermion Determinant in Massless Two-Dimensional QCD 105 Ralph Roskies Dynamic Mass Generation for Fermions 115 Abdus Salam and J. Strathdee Tomographic Representation of Quantized Fields 127 Charles M. Sommerfield On Spontaneously Broken Supersymmetry 141 G. Domokos and S. Kovesi-Domokos An Action in Superspace for SO(N)-Supergravity 159 S. W. MacDowell vii viii Contents Intrinsic Geometry of Supergravity 177 v. I. Ogievetsky On the Physics of Dimensional Reduction 191 Peter G. O. Freund Backlund Transformations and Deformations of Linear Differential 201 Equations with Applications to Diophantine Approximations G. Chudnovsky Backlund Transformations and Geometric and Complex-Analytic Back- 221 ground for Construction of Completely Integrable Lattice Systems D. V. Chudnovsky Unfashionable Pursuits 265 Freeman J. Dyson Epilogue 287 Maurice Goldhaber List of Publications of Feza Giirsey 291 Curriculum Vitae of Feza Giirsey 299 List of Contributors 301 Index 303 SOME PERSPECTIVES ON THE NONLINEAR SIGMA MODEL T. Appelquist J. W. Gibbs Physics Laboratory Yale University New Haven, CT 06511 Abstract When the nonlinear sigma model was first written down in 1959 by Feza Gursey [1], it was modestly offered as "an illustration of the possibility that the symmetries of the weak interactions may be essentially contained in those of the strong interactions, and certainly not as a definite proposal for a theory of elementary particles." In the more than twenty years since that paper appeared, the nonlinear sigma model has been extended in several ways, applied to a variety of low energy phenomena, and analyzed as both a classical and quantum field theory. While it has never gained the status of a fundamental theory of elementary particles, it has certainly played a key role in the development of particle physics. Its beautiful and realistic symmetry properties and its deep dynamical structure have made it much more than an illustration of possibilities. Most of the important features of the nonlinear sigma model were already empha sized in Gursey's original paper. He introduced a proton-neutron doublet and con structed the pion-nucleon interaction to be invariant under the chiral symmetry group SU(2)L xSU(2)R. He pointed out that a nonlinear realization of this symmetry requires the presence of pion self-interactions of a definite form, accompanying the usual pion kinetic energy term in the Lagrangian. He noted that the nonlinear constraint would lead to a nucleon mass along with pion-nucleon interactions and, most importantly, that it would not allow the existence of a pion mass. The symmetry group SU(2)L xSU(2)R is perhaps the simplest, and yet still the most physically relevant, on which to base a nonlinear theory. Nonlinear realizations of 1 2 T. Appelquist larger groups, such as SU(3)L xSU(3)R are of less obvious physical importance and, furthermore, they lead to very little that is new in terms of field theoretic structure. In the SU(2)L xSU(2)R I)1odel, the fields are represented by the two-by-two matrix M(x) == CT(X) + iT· iT (1) which transforms from the left and right according to the (112, 112) representation of SU(2)L xSU(2)R. The nonlinear constraint (2) can be enforced by requiring (3) The Lagrangian density for the purely pionic part of the theory is (iT . a iT)2 5f = ~ Tr a MtafLM = ~(a iT)2 + ~ u. (4) 4 fL 2 fL 2/-iT2 and, as with any Lagrangian field theory, it can be cast into a variety of different forms by making changes of variables [2]. With the Lagrangian written in terms of the 7r fields, invariance under the isomorphic group 0(4) ==SU(2)L xSU(2)R is nearly manifest. The set <l> = (CT,iT) transforms as an 0(4) vector with iT transforming line arly under the 0(3) subgroup. In addition, there is a set of three "boosts" under which 7r transforms nonlinearly: (5) The work of Nambu and lona-Lasinio [3] and Goldstone and others [4] showed that many of the properties of the nonlinear sigma model are common to a larger class of models. The unifying feature is the spontaneous breakdown of continuous symmetries, leading necessarily to the presence of massless Goldstone bosons. The nonlinear reali zation of SU(2)L xSU(2)R is an example of spontaneous breakdown, with the massless 7r fields creating the Goldstone bosons. A simple way to enter the larger class of models is to remove the nonlinear constniint (Eq. 2). Then the CT field becomes an independent coordinate and invariant interactions of the form f-L2MtM, A( MtM)2, etc., can be added. This is the linear sigma model, and with an appropriate choice of f-L2 and A the symmetry will again break spontaneously and the 7r field will be massless. The mass of the CT field will be f-L.
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