Table Of ContentSYMMETRIES
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.
