Solitons and Particles [only, Hardcover ed.] 9971966425, 9789971966423

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SOLITONS AND PARTICLES ClAUDIO REBBI GIULIO SOLIANI

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SOLITONS AND PARTICLES

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SOLITONS AND PARTICLES Claudio Rebbi

Physics Department Brookhaven National Laboratory Upton, NewYork 11973

Giulio Soliani

Dipartimento di Fisica dell' Universita Lecce, Italy and lstituto Nazionale di Fisica Nucleare Sezione di Bari, Italy

World Scientific

Published by

World Scientific Publishing Co Pte Ltd

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PO Box 128, Farrer Road, Singapore 9128

The authors and publisher are indebted to the original authors and the following publishers of the various journals and books for their assistance and permission to reproduce the selected articles found in this volume: Academic Press Inc. (Ann. Phys.); American Institute of Physics (JETP Lett., Sov. J. Nucl. Phys. and J. Math. Phys.); American Physical Society (Phys. Rev. and Phys. Rev. Lett.); D. Reidel Publishing Co. (Lett. in Math. Phys.); John Wiley & Sons (Commun. Pure and Appl. Math.); North­ Holland Publishing Co. (Nucl. Phys., Phys. Lett. and Physica); Socie-ta Italiana di Fisica (Lett. al Nuovo Cimento ); Springer-Verlag ( Commun. Math. Phys. and Leet. Notes in Physics). Copyright © 1984 by World Scientific Publishing Co Pte Ltd.

ISBN: 9971-966-42-5 9971-966-43-3 pbk

Printed in Singapore by Singapore National Printers (Pte) Ltd.

V

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PREFACE

For mathematical physicists studying non-linear evolution equations solitons are isolated waves which preserve their shape even after a collision. Solitons have been discovered in the nineteenth century and for a few decades have formed the object of numerous investigations. It has been established that many non-linear wave equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. The discovery that solitons may be relevant also in the domain of particle physics is more recent, dating to approximately ten years ago, although some pioneering investigations appeared much earlier. It was then that particle physicists realized that many field theoretical models for particle interactions possessed soliton solutions and that the solitons ought to be interpreted as additional particle­ like structures in the theory. During the last decade many investigations have been devoted to the study of solitons in the context of particle physics. In the course of these investigations it was established that, at the quantum level, solitons are associated with a variety of novel and remarkable phenomena. Also, particle physicists found it necessary to depart in some aspects from the more traditional theory of solitons, relaxing certain of the criteria used in the definition of solitons and concentrating on properties, such as the topology of the field configuration, which had not formerly received so much attention. The purpose of this book is to present the reader, through a collection of reprints, with a broad panorama of the results established in the application of solitons to particle physics. The emphasis is therefore on the particle physics aspects of the soliton phenomenon. However, although, as we have already mentioned, the lines followed by particle physicists and mathematical physicists in their analyses of solitons have somehow departed, we believe that there are still enough fundamental interconnections between the two approaches, that neither should be followed to the complete exclusion of the other. Thus, together

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vi with the main body of papers dealing with studies of solitons in the context of particle physics, we decided to include in the collection several reprints, where the most important concepts and methodologies in the more traditional line of investigations can be found. The book is ideally addressed to particle physicists and non-particle physicists as well. It is our hope that the former, beyond the topics which are more closely connected with their interest may find in the reprints on the mathematical theories of solitons useful ideas and inspirations, while the latter may find in this volume interesting and challenging applications of the concept of solitons in the domain of particle physics. We would like to express our gratitude to the many colleagues, in particular to Sidney Coleman, Neil Craigie, Roman Jackiw and Ed Witten, who have given us advice in the selection of the reprints. We are also thankful to Dr. Phua and World Scientific Publishing Co. for giving us the opportunity of editing this book and to Mrs. Isabell Harrity for her prompt and excellent typing of our introductory notes.

vii

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CONTENTS

Preface

V

Introductory Chapters I. 2. 3.

Introduction Mathematical Theory of Solitons: An Outline Solitons in Particle Physics: A Guide Through the Literature

7 31

REPRINTED PAPERS Interactions of "solitons" in a collisionless plasma and the recurrence of the initial states N.J.Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15 (I965) 240-243

57

Method for solving the Korteweg-de Vries equation C. S. Gardner, J.M.Greene, M.D. Kruskal and R. M.Miura, Phys. Rev. Lett. 19 (1967) 1095-1097

61

R3

Integral of nonlinear equations of evolution and solitary waves P.D. Lax, Comm. Pure Appl. Math. 21 (1968) 467-490

64

R4

Method for solving the sine-Gordon equation M. J. Ablowitz, D. J. Kaup, A. C.Newell and H. Segur, Phys. Rev. Lett. 30 (I973) 1262-1264

88

General derivation of Backlund transformations from inverse scattering problems H. H.Chen, Phys. Rev. Lett. 33 (I974) 925-928

91

On relativistic-invariant formulation of the inverse scattering transform method B. G. Konopelchenko, Lett. in Math. Phys. 3 (I 979) 197-205

95

RI

R2

RS

R6

viii R7

Canonical structure of soliton equations via isospectral eigen­ value problems M. Boiti, F. Pempinelli and G. Z. Tu, Nuovo Cimento 79B 231-265

104

Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron F. Calogero and A. Degasperis, Lettere al Nuovo Cimento 16 (I976) 425-433

139

Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation F. Calogero and A. Degasperis, Lettere al Nuovo Cimento 23 (1978) 150-154

148

RIO Exact theory of two-dimensional self-focusing and one­ dimensional self-modulation of waves in nonlinear media V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34 (1972) 62-69

153

RII Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method V. E. Zakharov and A. V. Mikhailov, Sov. Phys. JETP 47 (1978) 1017-1027

161

RI 2 Backlund transformation for solutions of the Korteweg-de Vries equation H. Wahlquist and F. B. Estabrook, Phys. Rev. Lett. 31 (1973) 1386-1390

172

RI3 Prolongation structures of nonlinear evolution equations H. Wahlquist and F. B. Estabrook,]. Math. Phys. 16 (1975) 1-7

177

R14 Prolongation structures of nonlinear evolution equations. II F. B. Estabrook and H. Wahlquist, J. Math. Phys. 17 {1976) 1293-1297

184

RI 5 Prolongation analysis of the cylindrical Korteweg-de Vries equation M. Leo, R. A. Leo, L. Martina and G. Soliani, Phys. Rev. D26 (1982) 809-818

189

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R8

R9

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ix RI6 Nonlinear evolution equations and nonabelian prolongations M. Leo, R. A. Leo, L. Martina, G. Soliani and L. Solombrino, J. Math. Phys. 24 (1983) 1720-1730

199

Rl7 Backlund transformations and the symmetries of the Yang equations H. C. Morris, J. Math. Phys. 21 (1980) 256-260

210

Rl8 Inverse scattering problem in higher dimensions: Yang-Mills fields and the supersymmetric sine-Gordon equation H. C. Morris,]. Math. Phys. 21 (1980) 327-333

215

RI9 "IST-solvable" nonlinear evolution equations and existence An extension of Lax's method I. Miodek,J. Math. Phys. 19 (1978) 19-31

222

R20 Evolution equations possessing infinitely many symmetries P. J. Olver,J. Math. Phys. 18 (1977) 1212-1215

235

R2l Symplectic structures, their Backlund transformations and hereditary symmetries B. Fuchssteiner and A. S. Fokas, Physica 4D (1981) 47 -66

239

R22 A simple model of the integrable Hamiltonian equation F. Magri, J. Math. Phys. 19 (1978) 1156-1162

259

R23 Some new conservation laws D. Finkelstein and C. Misner, Ann. Phys. 6 (1959) 230-243

266

R24 Kinks D. Finkelstein,J. Math. Phys. 7 (1966) 1218-1225

280

R25 Sine-Gordon equation J. Rubinstein, J. Math. Phys. 11 (1970) 258-266

288

R26 Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons R. F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D10 (1974) 4130-4138

297

R27 Quantization of nonlinear waves J. Goldstone and R. Jackiw, Phys. Rev. D1 I (1975) 1486-1498

306

R28 Soliton quantization V. E. Korepin, P. P. Kulish and L. D. Faddeev, JETP Lett 21 (1975) 138-139

319

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X

R29 Quantum sine-Gordon equation as the massive Thirring model S. Coleman, Phys. Rev. D11 (1 975) 2088-2097

321

R30 Solitons with fermion number -} R. Jackiw and C. Rebbi, Phys. Rev. D13 (1 976) 3398-3409

331

R31 Soliton excitations_ in polyacetylene W. P Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B22 (1 980)2099-21 1 1

343

R32 On the magnetic properties of superconductors of the second group A. A. Abrikosov, Sov. Phys. JETP 5 (1 957)1 1 74-1 1 82

356

R33 Vortex-line models for dual strings H. B. Nielsen and P. Olesen,Nucl. Phys. B61 (1 973)45-61

365

R34 Classical vortex solution of the Abelian-Higgs model H. J. de Vega and F. A. Schaposnik, Phys. Rev. D14 (1976) 1 100-1 1 06

382

R35 The stability of classical solutions E. B. Bogomol'nyi, Sov. J. Nucl. Phys. 24 (1 976)449-454

389

R36 Interaction energy of superconducting vortices L. Jacobs and C. Rebbi, Phys. Rev. B19 (1 979)4486-4494

395

R37 Arbitrary N-vortex solutions to the first order GinzburgLandau equations C. H. Taubes, Commun. Math. Phys. 72 (1 980)277-292

404

R38 Fractional charges and zero modes for planar systems in a magnetic field R. Jackiw, Phys. Rev. D29 (1 984) 2375-2377

420

R39 A 1 /n expandable series of non-linear a-models with instantons A. D'Adda, M. Lilscher and P. Di Vecchia, Nucl. Phys. B146 (1 978)63-76

423

R40 Scattering of massless lumps and non-local charges in the twodimensional classical non-linear a-model M. Lilscher and K. Pohlmeyer, Nucl. Phys. B137 (1 978) 46-54

437

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xi R4l Factorized S -matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models A. B. Zamolodchikov and A. B. Zamolodchikov, Ann. Phys. 120 (1979)253-291

446

R42 The theory of magnetic poles P. A. M. Dirac, Phys. Rev. 74 (1948)817-830

485

R43 A unified field theory of mesons and baryons T. H. R. Skyrme, Nucl. Phys. 31 (1962)556-569

499

R44 Magnetic monopoles in unified gauge theories G. 't Hooft, Nucl. Phys. B79 (I 974)276-284

513

R45 Particle spectrum in quantum field theory A. M. Polyakov,JETPLett. 20 (1974)194-1 95

522

R46 Poles with both magnetic and electric charges in non-Abelian gauge theory B. Julia and A. Zee, Phys. Rev. D11 (1975)2227-2232

524

R47 Exact classical solution for the 't Hooft moi:iopole and the Julia-Zee dyon M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. 35 (1975)760-762

530

R48 Class of scalar -field soliton solutions in three space dimensions R. Friedberg, T. D. Lee and A. Sirlin, Phys. Rev. D13 (1976) 2739-2761

533

R49 Quantum expansion of soliton solutions N. H. Christ and T. D. Lee, Phys. Rev. D12 (1975)1 606-1627

556

R50 Canonical quantization of nonlinear waves E. Tomboulis, Phys. Rev. D12 (1 975)1678-1683

578

RS l Soliton quantization in gauge theories E. Tomboulis and G. Woo, Nucl. Phys. B107 (1976) 221-237

584

R52 Fractional quantum numbers on solitons J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 986 -989

601

R53 Spin from isospin in a gauge theory R. Jackiw and C. Rebbi, Phys. Rev. Lett. 36 (1 976)11 1 6-111 9

605

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xii R54 Fermion-boson puzzle in a gauge theory P. Hasenfratz and G. 't Hooft, Phys. Rev. Lett. 36 (1976) 1119-1122

609

R55 Connection of spin and statistics for charge-monopole composites A. S. Goldhaber, Phys. Rev. Lett. 36 (1976) 1122-1125

613

R56 Current algebra, baryons, and quark confinement E. Witten, Nucl. Phys. B223 (1983) 433-444

617

R57 Stability analysis for singular non-abelian magnetic monopoles R. A. Brandt and F. Neri, Nucl. Phys. B161 (I 979) 253-282

629

R58 Exact multimonopole solutions in the Bogomolny-Prasad­ Sommerfield limit P. Forgacs, Z. Horvath and L. Palla, Phys. Lett. 99B (1981) 232-236

659

R59 Generating monopoles of arbitrary charge by Backlund trans­ formations P. Forgacs, Z. Horvath and L. Palla, Phys. Lett. 102B (I98I) 131-135

664

R60 Monopoles and dyons in the SU(5) model C. P. Dokos and T. N. Tomaras, Phys. Rev. D21 (1980) 2940-2952

669

R6 l Cosmological production of superheavy magnetic monopoles J. P. Preskill, Phys. Rev. Lett. 43 (I979) l 365 -1368

682

R62 Cosmological density fluctuations produced by vacuum strings A. Vilenkin, Phys. Rev. Lett. 46 (1981) 1169-1172

686

R63 Adler-Bell-Jackiw anomaly and fermion-number breaking in the presence of a magnetic monopole V. A. Rubakov, Nucl. Phys. B203 (1982) 311-348

690

R64 Dyan-fermion dynamics C. G. Callan Jr., Phys. Rev. D26 (1982) 2058-2068

728

R65 Pseudoparticle solutions of the Yang- Mills equations A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin, Phys.Lett. 59B (l975) 85-87

739

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xiii R66 Symmetry breaking through Bell-Jackiw anomalies G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8-11

742

R67 Computation of the quantum effects due to a four-dimensional pseudoparticle G. 't Hooft, Phys. Rev. D14 (1976) 3432-3450

746

R68 Vacuum periodicity in a Yang-Mills quantum theory R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172-175

766

R69 The structure of the gauge theory vacuum C. G. Callan Jr., R. F. Dashen and D. J. Gross, Phys. Lett. 63B (1976) 334-340

770

R70 Dyons of charge e0 I 2 rr E. Witten, Phys. Lett. 86B (1979) 283-287

777

R71 Condition of self-duality for SU(2) gauge fields on Euclidean four-dimensional space C. N. Yang, Phys. Rev. Lett. 38 (1977) 1377-1379

782

R72 Some aspects of the linear system for self-dual Yang-Mills fields L. L. Chau, M. K. Prasad and A. Sinha, Phys. Rev. D24 (1981) 1574-1580

785

R73 Self-dual Yang-Mills as a totally integrable system L. L. Chau, in Lecture Notes in Physics, vol. 180, "Group Theoretical Methods in Physics", Proc. ofthe Xlth Int. Colloq., Istanbul, Turkey, M. Serdaroglu and E. Inonu, eds., SpringerVerlag, 1983

792

R74 Instantons and algebraic geometry M. F. Atiyah and R. S. Ward, Comm un. Math. Phys. SS (1977) 117-125

800

R75 Construction of instantons M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Yu. I. Manin, Phys. Lett. 6SA(l978) 185-187

808

R76 A Yang-Mills-Higgs monopole of charge 2 R. S. Ward, Comm un. Math. Phys. 19 (1981) 317-325

811

1

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1 . INTRODUCTION

Solitons made their first appearance in the world o f science with the beautiful report on waves, presented by J . Scott Russell in 1 842 and 1 843 at the British Association for the Advancement of Science [ l ] a : "I was observing the motion of a boat which was rapidly drawn along a narro w channel by a pair of horses, when the boat suddenly stoppe d - not so the mass of water in the channel which it had put in motion ; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind , rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded , smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horse back , and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished , and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1 8 3 4 , was my first chance interview with that singular and beautiful phenomenon . . . " Scott Ru ssell conjectured that the propagation of an isolated wave , such as the one he ob served , was a consequence of the properties of the medium rather than of the circumstances of the wave's generation . This was not universally accepted and a long time had to elapse before it became established that some special nonlinear wave equations admit solutions consisting of isolated waves that can propagate and even undergo collisions without losing their identity. Land­ marks in the evolution of the subj ect were the proposal by Korteweg and de Vries a Simple numerals denote general references; references to the reprinted papers are indicated by a number preceded by R.

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2 in 1895 of an equation, incorporating both nonlinear and dispersive effects, f or the propagation of waves in shallow water [2] , and, at a much later date, numerical experiments on wave propagation, which became possible with the advent of modern computers. Pioneering among such experiments were those of Fermi, Pasta and Ulam [3] , of Perring and Skyrme [4] and of Zabusky and Kruskal [ R l ] . Very much as in Scott Russell' s observation, the computer experiments revealed the existence of wave-like excitations which, rather than disperse their energy, maintained a stable shape in the course of their propagation and emerged from collisions unaltered. It is remarkable that an analytic expression describing one such collision had been derived some ten years before the computer experiments by Seeger, Donth and Kochendorfer [5] . Zabusky and Kruskal introduced the word "soliton" to characterize waves that do not disperse and preserve their form during propagation and after a collision. Because of these defining features, the soliton might appear the ideal mathematical structure for the description of a particle. Yet particle physicists did not need, nor had to wait for the emergence of the concept of soliton to find analytical instruments for the study of particle phenomena. In the course of the evolution of field theory and quantum mechanics the notion of a particle became associated with the elementary excitation of a quantized field. The propagation of a free particle is described by a quantized mode of a linear system, rather than by the solution of a nonlinear wave equation . Non­ linear interactions among the quantized fields can be described in perturbation theory and lead to scattering phenomena : the individuality of particles emerging from the collision is guaranteed by their being quanta of some field ; apart from this the collision can change their state of motion and their quantum numbers. The formulation of particle physics in terms of quantized fields had been established for many years when the theory of solitons began to assert itself; it had proven extremely successful in describing some particle interactions, less capable of accounting for others, but altogether irrenounceable in its basic premises. It is therefore not surprising that little attention was generally paid by particle physicists to the newly developing concept of soliton. The situation began to change when the difficulties encountered in providing a perturbative description of some particle phenomena, especially in the domain of strong interactions, induced many theorists to reconsider from a new perspective the consequences of the nonlinearities in the equations of motion. Thus it was that in the early and mid-seventies the existence of soliton-type solutions was basically rediscovered by several particle physicists. These solitons, however, were not to be interpreted as the "elementary" particles of the theory, for which the standard

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3 association with the quantized modes of linearized small oscillations was to be maintained, but as new, additional particle-like excitations that the system possessed by virtue of its nonlinear nature. The solitons of particle physics had to be quantized, and frameworks to carry out the quantization were soon devised. It was then realized that some special features of the soliton solutions, in particular some non-trivial topological properties of the field configuration surrounding the soliton, gave origin to novel, rather unexpected quantum phenomena. This further aroused interest on the subject, many more theoretical investigations were performed, further remarkable properties of the solitons were discovered or clarified. By now the literature on solitons in particle physics is very rich : the major purpose of this book is to present the reader with a selection of some of the most important contributions to the development of this field. The selection is not exhaustive, not even of the papers which under any reasonable criterion might be considered the most important. Reasons of space forced us to omit many articles which otherwise we would have liked to include and no judgment of merit is implicit in our choices, which have been determined mainly by the attempt to provide a broad and yet coherent panorama of concepts, methods and results. The very b rief outline of how the notion of a soliton has recently become important for particle physics left out some pioneering investigations, where the relevance of soiiton-type solutions and of their topologies for particle phenomena was recognized way before the subject became of widespread interest. Skyrme and Perring, Enz, Finkelstein, Misner and Rubinstein are prominent among the early investigators. Their intuition preceded by several years the main body of research on solitons in particle physics : some of their works are reprinted in this book, others will be quoted. As particle theorists began to investigate solitons and their implications, they extended and somehow modified the original notion of soliton. To mathematical physicists studying the propagation of waves the most salient, indeed the defining feature of the soliton had been the capability of preserving its shape after a collision. Some nonlinear wave equations have solutions with this property, hence admit solitons. A small perturbation of the equation may destroy this feature. For the particle physicistb the most important charac-

b

Forced by the need of defining a distinction of interests, we shall contrapose the "particle physicists" , as those mainly concerned with the application of solitons to particle theory , with the "mathematical phy sicists", as thos;; studying solitons in the more traditional context of nonlinear wave propagation, although ours is an abuse of terms: one can hardly conceive of a particle theorist who is not also a mathematical physicist, nor does the study of mathematical physics rule out interest in particle phenomena.

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4 teristic of the soliton is its being stably confined into a small, definite volume of space. The fact that a collision might alter the shape of the classical wave is not so relevant in a quantum mechanical context, because, whereas a classical confined wave may exist in an infinite variety of forms, the quantized soliton can �mly exist in a discrete set of levels. Thus, quantum-mechanically, the distinction whether the two classical confined waves emerge unaltered from the collision or not translates into the distinction whether the two particles will preserve their original quantum states, after a collision, or may be found in a state of excitation. This second possibility does not destroy the individuality of the quantized soliton, and, indeed, may be considered phenomenologically more interesting. Thus particle physicists began to call freely solitons what mathematial physicists would refer to at most as solitary waves. The distinguishing features of the soliton for the mathematical physicists were not properties that the particle physicist would consider essential (although still important). At some point one could raise the question whether particle physicists and mathematical physicists were talking about the same object at all. Indeed, some of our colleagues are of the opinion that the use of the term soIiton by particle physicists is a misnomer, no less unfortunate than if one had discovered a new particle and decided to call it silicon. But we do not share such a drastic view. It is definitely true that the most fundamental property of the soliton for the mathematical physicist is the property of emerging from collisions unaltered, which can be variously related to conservation laws or symmetries, and that this property is not shared by many of the confined waves that the particle physicists would call solitons. However, some of the models for particle physics which can be solved owe their solvability to the same set of conservation laws which underlie the absence of scattering of solitons. Also, mathematical methods, which are proper of soliton theory and have been used to find multi-soliton solutions, have been successfully applied to find similar multi-particle (be they monopoles or instantons) solutions in field theories. Thus we feel that the fundamental unity of science, which has so often fruitfully related phenomena in different fields, is at work also in the context of the solitons proper, and of that extension of the concept which particle physicists have made. The spirit of this book will there­ fore be of bringing together a few articles, where some of the most basic notions and methods of the mathematical theory of solitons are explained, with a selection of papers, where solitons are considered from the point of view of particle physics and where some of the most remarkable consequences of the existence of soliton solutions and of their quantization are illustrated. Our hope is that the collection may thus be appealing both to the "particle physicist" and

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5 to the "mathematical physicist" . The former will find in it a description of soliton phenomena but also at least some fundamentals of those mathematical methods which can lead him to new analytical insights and discoveries. The latter may find in it applications of solitons to particle physics, which, although sometimes permissive in terms of the original concept, frequently embody beautiful mathematical notions and may suggest new directions to explore. The plan of the volume is as fo llows. The next section gives a general intro­ duction to the mathematical theory of solitons. The main techniques, which are used to establish the existence and properties of soliton solutions, and which form the subject of the first 22 reprinted articles are briefly illustrated. Sec. 3 is meant to provide the reader with a guide through the applications of soliton ideas to particle physics. No attempt will be made there to produce an even concise set of lecture notes: the subject is too vast to try to condense all the fundamental points in a few pages. Rather, Sec. 3 aims at giving a motivation for the selection and a logical path through the interconnections among mathematical solitons and the solitons of particle physics. Moreover, both in Sec. 2 and Sec. 3 additional references will be quoted : just as f or the reprints, however, no judgment of value is implicit in the presentation or omission of references. The number of titles in the literature on solitons runs in the thousands and it would be impossible for us to give an exhaustive list of the relevant contributions. List of additional references follow all sections : references to research articles are denoted by simple numerals, references to books and review articles, or to the reprinted paper are denoted by B and R, respectively, followed by a numeral. The reprints themselves are organized in the following manner. The collection is opened by works on the mathematical aspects of the more traditional theory of solitons, then come the papers on the applications of the concept to particle physics. These are grouped according to the dimen­ sionality of the model or full theory where the phenomenon occurs, in a progression ranging from the one-dimensional kinks of the sine-Gordon equation, to the four-dimensional instantons of gauge field systems. References for Section 1 1. 2.

J. Scott Russell, "Report on waves", Proc. of the Br itish Assoc iation for the Advancement of Sc ience, London, 1845, p. 311. D . J. Korteweg and G . de Vries, "On the change o f form o f long waves advancing in a rectangular canal, and on a new type of long stationary waves" , Ph il. Mag. 39(1895), 422 .

6

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3 . E. Fermi, J. R. Pasta and S. M. Ulam, "Studies of nonlinear problems", Los Alamos Sci. Lb. Rep . LA-1 940, 1 95 5 ; reprinted in Collected Works of Enrico Fermi, vol. II, p . 978, Univ. of Chicago Press, Chicago, 1 965. 4. J. K. Perring and T. H. R. Skyrme, "!',. model unified field equation", Nucl. Phys. (1 962) 5 50. 5. A . Seeger, H. Donth and A. Kochendofer, "Theorie der Versetzungen in eindimensionalen Atomzeihen. III. Versetzungen, Eigenbewegungen und ihre Wechselwirkung" , Z. Phys. 134 ( 1 953), 1 73 .

7

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2. MATHEMATICAL THEORY OF SOLITONS: AN OUTLINE

2-1 Introduction Among the nonlinear partial differential equations of physical interest, an important role is played by the class of evolution equations which possess a series of remarkable features, such as, for instance, solitary wave solutions which preserve their shape through nonlinear interaction. Because of their particle-like character, these solutions were named s olitons by Zabusky and Kruskal, in their pioneering work on the Korteweg-de Vries (KdV) equation ( 1 965) [ R l ] . Nonlinear evolution equations (NLEE) having soliton solutions share many special properties, i.e. an infinite sequence of conservation laws and Lie-Backlund symmetries [ B3] , multsoliton solutions, Backlund transformations, and reduction to ordinary differential equations of Painleve type [BIS, B9] . Furthermore, these equations may be obtained via the compatibility of two associated linear operators ; in other words, they can be put in Lax's form [ R3] . This says then that the equations are exactly solvable. When one has to deal with a certain exactly solvable NLEE, two fundamental problems arise: i) the development of a method of solution, and ii) the investi­ gation of the algebraic properties of the equation. The latter concerns the existence of an infinite sequence of conserved quantities in involution (in the sense that their Poisson brackets are vanishing), the possession of infinitely many commuting symmetries [ R20, R21] , the Hamiltonian, bi-Hamiltonian and action-angle-formulations, and others (see, for example, (36] ). After the introduction of the concept of soliton, in 1967 Gardner, Green, Kruskal and Miura (GGKM) [ R2] developed a method of solution of the initial value problem for the KdV equation resorting to the ideas of direct and inverse scattering. This method, known as the method of inverse scattering transform (IST), is one of the basic tools for the study of NLEE's. Other important

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8 procedures are: i) the Estabrook-Walquist (EW) prolongation scheme, and ii) the so-called symmetry approach (SA). For a description of further significant methods, which are mostly convenient for finding particular solutions of a given NLEE, such as the bilinear approach of Hirota [49, B23 ] , the dressing method of Zakharov-Shabat [99] , and others, we refer the reader to [36] . The 1ST method has received such attention in the literature that we shall limit ourselves, in Sec. 2-2, to outline a guide through the vast bibliography, pointing out only a few outstanding results achieved in the last years. On the contrary, the procedures i) and ii) will be illustrated in a more detailed manner. This standpoint is motivated by the fact that many review articles and text books on the 1ST technique and its applications are available to the uninitiated reader, unlike what happens for the methods i) and ii). Within the EW prolongation scheme one can relate to NLEE' s Lie algebras which can be used, in principle, to determine constants of motions, isospectral eigenvalue problems, Backlund transformations and nonlinear superposition formulae. A discussion on this point aimed at the beginners is made in Sec. 2-3. The SA is quite suitable for obtaining infinitely many commuting symmetries [ B3 ] of an exactly solvable NLEE, which can be generated from each other by a recursion operator [ R20] , and the Hamiltonian and bi-Hamiltonian structure of the equation under consideration. (For a deep discussion on this point, see for example [ 100] .) The SA is presented in Sec. 2-4. To facilitate the reader interested in getting an insight into soliton theory, beyond the list of reprinted articles, we have inserted additional references concerning both papers and books. Most of the reprinted articles and the additional references on papers will be fairly mentioned in Secs. 2-2 - 2-4. Conversly, we shall spend here a word about some additional references on the books. The readers who wish to learn about the basic ideas of soliton theory may consult profitably the texts [B l , B7, B i l , B l 2, B l 3, B l 9] . Many subjects are treated extensively and from a pedagogical point of view. The classical and quantum aspects of the inverse scattering theory are widely dealt with in [ B2] and [ B8] . The books [ B3 , B 1 4, BI S , B22, B23] concern the theory of Backlund transformations and the application of Lie group techniques to the study of nonlinear differential equations. References [B4, BS, B6, B18, B20, B21, B25] are conference proceedings or collections of papers devoted to several topics relative to the soliton theory . Finally, in [ B10] the role of the soliton is discussed

9 in connection with the mechanism of transmission of energy in certain biological system.

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2-2 Introductory Bibliography Notes on the 1ST Method One of the important goals achieved by GGKM [ R2 ] , is the fact that solutions of the KdV equation with suitable decay behaviors for l x l ➔ 00 are isospectral potentials for the Schrodinger equation. This result implies a series of questions, which are carefully examined in [ BI2 ] , Chapter 3. In this connection, here we recall only that there are other equations, besides the KdV equation, whose solutions are isospectral potentials for the Schrodinger equation. A rigorous mathematical approach suitable for dealing with such problems was set up by Lax in 1 968 [ R3] . Using Lax's technique, Zakharov and Shabat (ZS) in 1 972 [ RIO] introduced a linear scattering problem to solve the so-called nonlinear Schrodinger equation, which is of special importance in many branches of physics (see, for example, [ B l ] and [ 1 91 ). Later on (1 973), Ablowitz, Kaup, Newell and Segur (AKNS) tried successfully the same ideas on the sine-Gordon equation [R4] . These authors considered also a generalization of the ZS problem, permitting us to solve the initial value problem for a broad class of NLEE's of physical significance (3, 99] . An extension of Lax's method, which allows one to recover the more general NLEE's of AKNS, was carried out by Miodek in 1 978 [R 1 9] . A significant topic related to NLEE's solvable by 1ST is the search of Backlund transformations and conservation laws. Here we limit ourselves to cite very few contributions ; an exhaustive literature on this subject can be found in the texts devoted to soliton theory listed in the References - Books (see, for example, [ B1 3] , (B7] , [ B l 9] and [ B l l ] ). Backlund transformations for the NLEE's covered by the AKNS scheme were derived by Chen in 1 974 [RS ] . A link among the 1ST method, Backlund transformations and the existence of an infinite set of conservation laws is discussed in (90] . Finally we mention [45] , where an infinite sequence of conservation law is determined for coupled NLEE's. Beyond NLEE's whose eigenvalue problems are of the second order, there are certain NLEE's of physical interest which are related to higher scattering problems. One may see, for example, Refs. [98) and [5 I] where a 3 X 3 matrix eigenvalue problem is associated with the three- waves resonant interaction (3WRI) equations. An extensive account on this subject is expounded in [Bl] , while an analysis of the full three-dimensional 3WRI within the 1ST framework is given in [52] . Until recently, only inverse spectral problems of the second order

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10 [5] and their matrix generalizations [ 1 3, 91] were solved. Inverse scattering problems of the third order are dealt with successfully in (53] and [92] . An approach to a general n-th order problem is described in [ 1 5] , where an application to the Boussinesq equation is carried out. For a discussion on the inverse scattering analysis associated with eigenvalue problems higher than third order, see for instance [ B l ] , Chapter 2. See also Chapter 3 of [ B5] . Detailed treatments on the inverse scattering on the line are given in [29] and [21] . Several questions pertinent to this argument are examined in [B7J , Chapters I and 2. We notice that an important program of study on IST in multidimensions has been recently undertaken by Ablowitz and Fokas [35, 36, 2] . The readers who wish to be acquainted with the problem of relativistic formulation of IST method, may consult for example [ R6 ] and [ R11] . Related to IST there are some outstanding topics which have been widely considered in the literature, as for instance i) the problem of finding solutions of NLEE's with periodic boundary conditions, and ii) the problem of establishing whether a given NLEE exhibits the so-called Painleve property. Problem i) was tackled first by Novikov [76 , 77] , who worked on the KdV equation . His ideas were based essentially on methods of algebraic geometry. Afterwards other special NLEE's were studied, within this scheme, by several authors. Accounts of developments on this interesting area of research can be found in [57] . Problem ii) is connected with the possibility that a certain NLEE may be solvable by I ST (see [4, 92] , [ Bl] , Chapter 3 and [10] for an application to nonlinear Schrodinger equation). Another topic of great interest and related to IST concerns the possibility of building up quantum field models based on NLEE's having soliton solutions (see, for instance, [88] and references therein). This question leads to the concept of quantized soliton. To conclude these short bibliographic notes on the 1ST method, we would like to advise the beginners to consult first, for example , the review article [84] , where many subjects of soliton theory and applications to some remarkable NLEE's of physical interest are considered in a very understanding way. Sub­ sequently, one can, say, focus on the KdV equation and try to become familiar with the basic concepts of 1ST and related topics working on this "laboratory". To this aim one can benefit by the survey by Miura [ 67] and by Refs, (97, 75, 56, 89, 30, 55, 50] . In the last two references, perturbation methods are applied to some NLEE's which differ slightly from exactly solvable NLEE's (see also ( B19] , Chapter 9). Finally, terse examples of detailed calculations concerning certain NLEE'S

11 solvable by IST can be found in the works by Calogero and Degasperis [ R8] and [ R9] . In [ R8] the main references are also reported on a powerful technique set up by these authors, by means of which one can provide classes of NLEE's solvable by IST and investigate the properties of their solutions.

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2-3 The Estabrook-Wahlquist Prolongation Method The prolongation scheme was devised by Estabrook and Wahlquist in 1 975 [ Rl3, Rl4] . It consists essentially in introducing, through certain differential equations, new dependent variables (which can be regarded as components of a vector function named pseudopotential), in such a way that the integrability condition for the existence of these variables is just the original NLEE. The equations which define the new variables, called prolongation equations, can be interpreted as a generalization of Backlund transformations. The prolongation method can be developed resorting to the theory of the external differential forms or using the "jet bundles" formulation [82] , [B23] . However, for practical purposes, compatible with the nature of the equations under consideration, it is convenient, as far as it is possible, to exploit the technical procedures without the formalism of the differential geometry. To favour non-expert readers, we shall adopt later this standpoint in dealing with the onset of the prolongation technique. The EW method has been applied to many NLEE' s of physical interest. A few of these are : ut

+ ux x x + u ux = 0

u t + Ux x x + u 2 ux u tx

=

0

+ sin u = 0

i u t + ux x

+ X l u 12 u = 0

u t + uux - vu x x

=

0

3 u t t + uxxxx + 6( u ux )x

=

0

3( u t t + Uyy ) + ( ux x x + 6 u ux )x

=

0

(Korteweg-de Vries)

(2-I )

(modified Korteweg-de Vries)

(2-2)

(sine-Gordon)

(2-3)

(nonlinear Schrodinger)

(2-4)

(Burgers)

(2-5 )

(Boussinesq)

(2-6)

(Kadomtsev-Petviashvili),

(2-7)

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12 where u = u (x , t), x , y and t are respectively space and time variables and A and v are constants. Subscripts denote partial derivatives. One has found that with any exactly solvable NLEE a non-Abelian Lie algebra can be associated, which generally is (presumably) infinite-dimensional. Concerning this, we notice that for the KdV equation (2-1) a rigorous proof exists that such algebra is infinite-dimensional [86] . For other exactly solvable NLEE's this property is reasonably conjectured, on the basis of indications coming from the so-called closure operation (see later). In fact, this gives rise to a finite dimensional non-Abelian quotient algebra which contains a free parameter" whose role is leading in finding Backlund transformations and in deriving the isospectral eigenvalue problem typical of the 1ST method [24] . The above considerations make likely the existence of a correspondence between exactly solvable NLEE's and non-Abelian prolongations . Anyway, the lack of a general theory suggests that one should examine the largest number of cases which con firm such feature, having in mind the goal of acquiring useful information for building up, if possible, the theory. In this context, one may benefit from investigating a class of NLEE's rather than a single equation . Reprint [ R16] goes this way. This is devoted to a prolongation analysis of the class of equations : (2-8) where ( u ) (38, 40] . The SA is especially indicated for analyzing the Hamiltonian and the bi­ Hamiltonian structure of NLEE's [9, 65, R7, R21, R22] . In this context, it has been proved that if Eq. (2-43) is a bi-Hamiltonian system, i.e., if K(u ) can be written as (2-46) where 0 1 , 0 2 are implectic operators ( R21] and [1 , !2 are gradient functions

21 [ R2 l] , assuming that 0 2 is invertible, then

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(2-47) is a strong symmetry for Eq. (2 - 43) [ R2 1] . Moreover, if 0 1 and 0 2 are compatible (which means that their sum is again an implectic operator [ R2 1] ), the operator Eq. (2 -47) turns out to be hereditary [ R2 1] . This important statement establishes a connection between symplectic structures and hereditary symmetries of NLEEs. Another notable feature of the SA is the possibility of finding conservation laws for Eq. (2-43), under the hypothesis that the hereditary operator admits the implectic-symplectic factorization as in Eq. (2-47), and that (2-48) is a gradient function [ R2 1 ] . In this case the quantities Gn = 0 '2- i 'I',1, n uX

(n

=

1, 2 , . . . )

(2-49)

are gradient functions of Eq. (2-44), whose constants of motion are expressed by (2-50) where ( , ) is the scalar product (see [ R2 1] and [40] ). Hereditary symmetries can also be related to Backlund transformations of NLEE's [39] . For this purpose, let B (u, s ) be a function of two variables u € S 1 and s € S2 , where S 1 and S2 are suitable vector spaces, with values in some vector space S3 • Following Fokas and Fuchssteiner [39] , we shall call this function admissible if, for B ( u , s) = 0, the partial derivatives Bu and B are invertible linear maps S 1 ➔ S3 and S2 ➔ S3 , respectively. An admissible function B (u , s ) is said to be a Backlund transformation between the evolution equations

ut

=

K(u ) ,

u ( t) e S 1

(2 - 5 1 )

st

=

G (s ) ,

s(t) e S2

(2 -5 2 )

22 if for any t :

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B(u ( t), s (t)) = 0

(2-53)

whenever B ( u (O ), s ( O)) = 0. Since Bu and Bs are invertible, Eq. (2-51) ((2-52)) is uniquely determined by Eq. (2-52) ((2-51)) and the admissible function B(s, u ). Consequently, a Bllcklund transformation may be interpreted as a transformation of variables which puts in mutual relation Eqs. (2-51) and (2-52) [39] . An important role is played by the operator T : S 1 ➔ S 2 given by :

T

=

s-s 1 Bu

(2-54)

where B(u , s ) is assumed to be admissible. In fact, one has that (u ) is a hereditary symmetry if and only if the operator : (2-55) is hereditary (see Theorem 2 of [39] ). As a corollary, all the equations of a hierarchy generated by a hereditary symmetry possess the same Backlund transformation. Notable contributions to the SA are due to Anderson ( [843 ] and references therein), Fokas [ R21, 34, 37] , Fuchssteiner [ R21, 41, 42] , Gel'fand and Dikii [43 ] , Ibragimov [ 83 ] , Magri [ R22, 65 ] , Olver [ R20] and, more recently, to Boiti, Pempinelli and Tu [ R7] . Many applications have been performed [ l , 6, 23, 34, 40, 61, 78, 79, 87] mainly for continuum NLEE ' s. An extension of the SA formalism to one-dimensional discrete systems, such as the Toda lattice [ 824] , is described in [63] and [12] . References for Section 2 1. 2. 3.

L. Abellanas and F. Gui!, "Intrinsic properties for conservation laws and Lie-Backlund invariances of evolution equations", Le tt. in Math. Phys. 4 (1980), 257-263. M. J. Ablowitz and A . S. Fokas, "Comments on the inverse scattering transform and related nonlinear evolution equations", in Lecture Notes in Physic s, n. 189, Springer, Berlin, 1983, p. 3 . M. J. Ablowitz, D. J. Kaup, A. C . Newell and H. Segur, "Nonlinear evolution equations of physical significance", Phys. Rev. Le tt. 31(1973), 125-127.

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23 4. M. J . Ablowitz, A . Ramani and H. Segur, "Nonlinear evolution equations and ordinary differential equations of Painleve type", Lett. Nuovo Cimento 23 ( 1 978), 333-338. 5. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, "The inverse scattering transform - Fourier analysis for nonlinear problems", Stud. Appl. Math. 53 ( 1 974), 249-3 1 5. 6. R. N. Aiyer, " Recursion operator for coupled KdV equation", Phys. Lett. 93A (1 983), 368-370. 7. J. M . Alberty, T. Koikawa and R. Sasaki, "Canonical structure of soliton equations" , I. Physica SD ( 1 982), 43- 65. 8. M. Boiti, C. Laddomada and F. Pempinelli, "The permutability theorem for the nonlinear Schrodinger equation", Phys. Lett. A83 ( 1 98 1 ), 1 88- 1 90. 9. M. Boiti, C. Laddomada, F. Pempinelli and G. Z. Tu, "A new hierarchy of Hamiltonian soliton equations", ]. Math. Phys. 24 ( 1 983), 2035 - 2041 . 1 0. M. Boiti and F. Pempinelli, "Nonlinear Schrodinger equation, Backlund transformations and Painleve transcendents", Nuovo Cimento 59B (I 980), 40-58. 1 1 . M. Boiti, F. Pempinelli and G. Z. Tu, "A general method for obtaining the double Backlund transformations", Phys. Lett. A93 ( 1 982), 1 07- 1 1 0. 1 2. M. Bruschi and 0. Ragnisco, "The Hamiltonian structure ofthe non-Abelian Toda hierarchy", J. Math. Phys. 24 ( 1 983), 1 4 1 4- 1 42 1 . 1 3 . F. Calogero and A. Degasperis, "Nonlinear evolution equations solvable by the inverse spectral transform associated with the matrix Schrodinger equation", in Solitons, eds R. K. Bullough and P. J. Caudrey, Springer, Berlin, 1 980, p. 301 . 1 4 . P. J . Caudrey, "The inverse problem for the third order equation uxxx + q (x )ux + r(x)u = i� 3 u " , Phys. Lett. 79A ( 1 980), 264-268. 1 5. P. J. Caudrey, "The inverse problem for a general NXN spectral equation", Physica 6D ( I 982), 5 1 -66. 1 6 . P. J. Caudrey, "The inverse spectral problem for a system of three coupled first order equations", Proc. R. Irish Acad. 83A ( 1 983), 23-3 1 . 17. A. R. Chowdhury and T. Roy, "Prolongation structure for Langmuir solitons", J. Math. Phys. 20 ( 1 979), 1 559- 1 56 1 . 1 8. A. R. Chowdhury and T. Roy, "Prolongation structure for a nonlinear equation with explicit space dependence", J. Math. Phys. 21 (1 980), 1 4 1 6- 1 4 1 7. 1 9. 1. Corones, "Solitons, pseudopotentials, and certain Lie algebra", /. Math. Phys. 18 ( 1 977), 1 63 - 1 64.

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24 20. M. Crampin, F. A. E. Pirani and D. C. Robinson, "The soliton connection", Lett. in Math. Phys. 2 (1 977), 1 5- 1 9. 21. P. Deift and E. Trubowitz, "Inverse scattering on the line", Comm. P ure Appl. Math. 32 ( l 979), 1 2 1 -25 1 . 22. P . Denes and J. D. Finley Ill, "Backlund transformations for general PDE's", Phys ica 4D (1 983), 236- 250. 23. R. Dodd and A. Fordy, "On the integrability of a system of coupled KdV equations", Phys. Lett. 89A (1 982), 1 68-1 70. 24. R. Dodd and A. Fordy, "The prolongation structures of quasi-polynomial flows", Proc. R. Soc. L ondon A385 (1 983), 389-429. 25. V. G. Drinfel'd and V. V. Sokolov , "Equations of the Korteweg-de Vries type and simple Lie algebras", Sov. Math. Dok/. 23 (1981), 457-462. 26. H. N. van Eck, "The explicit form of the Lie algebra of Wahlquist and Estabrook. A presentation problem", Proc. ofthe K oninkl ijke Neder landse Akademie van Wetenshappen Series A, vol. 86 (2) (1 983), 1 49-1 64. (Also published as lndagat iones Mathemat icae, vol. 45, Fasc. 2 ( 1 983).) 27. H. N. van Eck, P. K. H. Gragert and R. Martini, "The explicit structure of the nonlinear Schrodinger prolongation algebra", Pr oc. of the K oninkl ijke Neder landse Akademie van Wetenshappen, Series A, vol. 86 (2), ( 1 983), 165-172. (Also published as lndagat iones Mathemat icae, vol. 45, Fasc. 2 (1 983).) 28. F. B. Estabrook and H. D. Wahlquist, "Prolongation structures, connection theory and Backlund transformation", in Research Notes in Mathemat ics 26, ed. F. Calogero, Pitman, London, 1 978, 64-83. 29. L. D. Faddeev, "Properties of the S - matrix of the one-dimensional Schrodinger equation", Am. Ma th. Soc. trans. (2) 65 (1 967), 1 39- 1 66. 30. L. D. Faddeev, "Inverse problem of quantum scattering theory I I," J. Sov. Ma th. 5 (1 976), 334-396. 3 I . J. D. Finley III, "Criteria for the existence of Backlund transformations", in Lecture Notes in Phys ics, n. 1 89, Springer, Berlin, 1 983, p. 365. 32. H. Flaschka, A. C. Newell and T. Ratiu, "Kac-Moody Lie algebras an:d soliton equations. IL Lax eq11ations associated with A 1 (1), Phys ica 9D ( 1 983 ), 300-323. 33. H. Flaschka, A. C. Newell and T. Ratiu, "Kac-Moody Lie algebras and soliton equations. III. Stationary equations associated with A 1 < 1 ) , , , Phys ica 9D ( 1 983), 324-332. 34. A. S. Fokas, "A symmetry approach to exactly solvable evolution equations", J. Math. Phys. 21 ( 1 980), I 3 I 8-1 325.

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25 3 5 . A. S. Fokas, " Inverse scattering of first order systems in the plane related to nonlinear multidimensional equations", Phys. Rev. Lett. 51 ( 1983 ), 3-6 . 36. A. S. Fokas and M. Ablowitz, "The inverse scattering transform for multi­ dimensional (2 + I ) problems", in Lecture Notes in Physic s, n. I 89, Springer, Berlin, 1983, p . 137. 37. A. S. Fokas and R. L. Anderson, "On the use of isospectral eigenvalue p roblems for obtaining hereditary symmetries for Hamiltonian systems", J. Math. Phy s. 23 ( 1982), 1 066-1073. 38. A. S. Fokas and B. Fuchssteiner, "On the structure of sympletic operators and hereditary symmetries", Lett. Nuovo Cimento 28 ( 1 980), 299-303. 39. A. S. Fokas and B. Fuchssteiner, "Backlund transformations for hereditary symmetries, Nonlinear Analy sis, TMA S ( 1 980), 423-432. 40. B. Fuchssteiner, "Application of hereditary symmetries to nonlinear evolution equations", Nonl inear Analysis TMA 3 ( 1 979), 849-862 . 4 1 . B. Fuchssteiner, "The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional Abelian symmetry groups", Prag. Theor. Phy s. 65 ( 1 98 1 ), 86 1-876. 42. B. Fuchssteiner, "The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems", Prag. Theor. Phys. 68 ( I 982), 1 082-11 04. 43. I. M. Gel'fand and L . A. Dikii, " Fractional p owers of operators and Hamiltonian systems", Funct. Anal. Appl. 10 ( 1 976), 259-273. 44. P. K . H. Gragert and R. Martini, "The explicit structure of the prolongation algebra of the Hirota-Satsuma system", Phys. Let t. 101A ( 1 984), 469-472. 45 . R. Haberman, "An infinite number of conservation laws for coupled nonlinear evolution equations", J. Math. Phy s. 18 ( I 977), 1 137-1 1 39. 46. J. Hamad and P . Winternitz, "Pseudopotentials and Lie symmetries for the generalized nonlinear Schrodinger equation", J. Math. Phys. 23 ( 1 982), 5 1 7-525 . 47. B. K . Harrison and F. B. Estabrook, "Geometric approach to invariance groups and solutions of partial differential systems", J. Ma th. Phys. 12 ( I 97 1 ), 653-666. 48 . B. K. Harrison, "Backlund t ransformation for the Ernst equati on of general relativity" , Phys. Rev. Le tt. 41 ( 1978), 1197-1200. 49. R. Hirota, "Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons", Phys. Rev. Le tt. 27 ( 197 1), 1 192-1 1 94. 50. V. I . Karpman and E. M. Maslov, "Structure of tails produced under the action of perturbations on solitons", Sov. Phys. JETP 48 ( 1 978 ), 2 5 2-259.

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26 51 . D. J. Kaup, "The three-wave interaction -- a nondispersive phenomenon" , Stud. Appl. Math. 5 5 (1 976), 9-44. 52. D. J. Kaup, "The inverse scattering solution fo r the full three dimensional three wave resonant interaction", Physica 1D (I 980), 45-67 . 53. D. J. Kaup, "On the inverse scattering problem for cubic eigenvalue problems of the class l/lxxx + 6 Q l/lx + 6R i/1 = "X I/I " , Stud. Appl. Math. 62 (1 980), 1 89-21 6. 54. D. J. Kaup, "The Estabrook-Wahlquist method with examples and applica­ tions" , Physica 4D (1 980), 391-41 1 . 55. D. J. Kaup and A. C. Newell, "Solitons as particles, oscillators, and in slowly changing media : a singular perturbation theory" , Proc. R. Soc. London A361 (1 978), 41 3-446. 56. T. Kotera and K. Sawada, "Inverse problem and classes of nonlinear partial differential equations", J. Phys. Soc. Japan 39 (1 975), 501-508. 57. I. M. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations" , Russian Math. Surveys, vol. 32, n. 6 (1 977), 1 85-21 4. 58. M. Lakshmanan, "Rigid body motions, space curves, prolongation structure, fiber bundles, and solitons", J. Math. Phys. 20 (1 979), 1 667-1 672. 59. M. Leo, R. A. Leo, L. Martina, F. A. E. Pirani and G. Soliani, "Non-Abelian prolongations and complete integrability", Physica 4D (1 981 ) , 1 05-1 1 2. 60. M. Leo, R. A. Leo, G. Soliani and L . Martina, "Backlund transformations of nonlinear evolution equations without using representations of related closed non-Abelian prolongation algebras" , Preprint Dept. of Physics, Univ. of Lecce, Italy (Dec. 1 983). 61 . M. Leo, R. A. Leo, G. Soliani, L. Solombrino and L. Martina, "Lie-Backlund symmetries for the Harry Dym equation", Phys. Rev. D27 (1 983), 1 4061 408. 62. M. Leo, R. A. Leo, G. Soliani and L. Solombrino, "On the isospectral eigenvalue problem and recursion operator of the Harry Dym equation", Lett. Nuovo Cimento 38 (1 983), 45-51 . 63 . M. Leo , R. A . Leo, G . Soliani, L. Solombrino and G. Mancarella, "Symmetry properties and bi-Hamiltonian structure of the Toda lattice" , to be published in Lett. in Math. Phys. (1984). 64. G. Liebbrandt, R. Morf and S . Wang, " Solutions of the sine-Gordon equation in higher dimensions" , J. Math. Phys. 21 (1 980), 1 61 3-1 624. 65. F. Magri, "A geometrical approach to the nonlinear solvable equations" , in Nonlinear Evolu tion Equations and Dynamical Systems, eds. M. Boiti F. Pempinelli and G. Soliani, Lecture Notes in Physics n. 1 20, Springer, Berlin, 1 980, p. 233.

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27 66. R. Martini, "Prolongation structure and Lax representation of the boomeron equation", Proc. of the Koninklijke Nederlandse A kademie van Wetenschap­ pen, Series A, vol. 86 (2) (1983), 221-227. (Also published as lndagationes Mathematicae, vol. 45, Fasc. 2 (1983).) 67. R. M. Miura, "The Korteweg-de Vries equation : a survey of results", SIAM Review 18 (1976), 412-459. 68. H. C. Morris, "Prolongation structures and a generalized inverse scattering problem", J. Math. Phys. 17 (1976), 1867-1869. 69. H. C. Morris, "Prolongation structures and nonlinear evolution equations in two spatial dimensions", J. Math. Phys. 17 (1976), 1870-1872 . 70. H. C. Morris, "Prolongation structures and nonlinear evolution e quations in two spatial dimensions. II. A generalized nonlinear Schrodinger equation", J. Math. Phys. 18 (1977), 285-288. 71. H. C. Morris, "Soliton solutions and the higher Korteweg-de Vries e quations", J. Math. Phys. 18 (1977), 530-532 . 72. H. C. Morris, "A prolongation structure for the AKNS system and its generalization", J. Math. Phys. 18 (1977), 533-536. 73. H. C. Morris, "A generalized prolongation structure and the Backlund transformation and the Backlund transformation of the anticommuting massive Thirring model", J. Math. Phys. 19 (1978), 85-87. 74. H. C. Morris, "Prolongation structure and nonlinear evolution equations in two spatial dimensions: a general class of equations", J. Phys. Al 2 (1979), 261-267. 75. A. C. Newell, "The inverse scattering transform, nonlinear waves, singular perturbations and synchronized solitons", Rocky Mountain J. of Math. vol. 8, Nos. 1 and 2, Winter and Spring 1978, pp. 25-52. 76. S. P. Novikov, "A periodic problem for the KdV equation", Funct. Anal. 8 (1974), 236-246. 77. S. P. Novikov, "New applications of algebraic geometry to nonlinear equations and inverse problems", in Nonlinear Evolu tion Equations Solvab le by the Spectral Transform, ed. F. Calogero, Pitman, London, 1978, pp. 8496. 78. W. Oevel, "On the integrability of the Hirota-Satsuma system", Phys. Lett. 94A (1983), 404-407. 79. W. Oevel and A. S. Fokas, "Infinitely many commuting symmetries and constants of motion in involution for explicitly time-dependent evolution equations", Preprint Univ. of Paderborn, W. Germany (1983). 80. M. Ornate and M. Wadati, "Backlund transformations for the Ernst equation", J. Math. Phys. 22 (1981), 961-964.

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28 81 . M. Omote and M. Wadati, "The Backlund transformations and the inverse scattering method of the Ernst equation", Prag. Theor. Phys. 65 (1981), 1 621-1 631. 82. F. A. E. Pirani, D. C . Robinson and W. F. Shadwick, "Local jet bundle formulation of Backlund transformations" , Mathematical Physics Studies, vol. 1 , Reidel, Dordrecht, 1979 , pp. 1-1 32. 83. R. Sasaki, "Canonical structure of soliton equations. II. The Kaup-Newell systems" , Physica SD (1 982), 66-74. 84. A. C. Scott, F. Y . F . Chu and D. W . McLaughlin, "The soliton : a new concept in applied science" , Proc. IEEE 6 1 (1973), 1443-1 483. 85. W. F. Shadwick, "The Backlund problem for the equation ( 3 2 z )/ ( o x 1 ox 2 ) = f ( z ), J. Math. Phys. 19 (1 978), 231 2-2317. 86. W. F. Shadwich, "The KdV prolongation algebra", J. Math. Phys. 21 (1 980), 454-461 . 87. K. M. Tamizhmani and M. Lakshmanan, "Infinitely many Lie-Backlund symmetries for a quasi-linear evolution equation" , Phys. Lett. 90A (I 982), 1 59-1 61 . 88. H. B. Thacker and D . Wilkinson, "Inverse scattering transform as an operator method in quantum field theory" , Phys. Rev. D19 (1979), 3660-3665. 89 . M. Wadati, "The modified Korteweg-de Vries equation" , J. Phys. Soc. Japan 34 (1 973), 1 289-1 296. 90. M. Wadati, H. Sanuki and K . Konno, "Relationships among inverse method, Backlund transformation and an infinite number of conservation laws," Prag. Theor. Phys. 53 (1975), 41 9-436. 91 . M. Wada ti, "A generalized matrix form of the inverse scattering method" , in Solitons, eds. R. K. Bullough and P. J. Caudrey, Springer, Berlin, 1 980, p. 287. 92. J. Weiss, M. Tabor and G. Carnevale, "The Painleve property for partial differential equations" , J. Math. Phys. 24 (1983), 522-526. 93 . G. Wilson, "The affine Lie algebra C2 ( 1 ) and an equation of Hirota and Satsuma" , Phys. Lett. 89A (1 982), 332-334. 94. P. Winternitz, "Lie groups and solutions of nonlinear differential equations" , in Lecture Notes in Physics n . 189 , Springer, Berlin, 1 983, p . 265. 95. C. N. Yang, "Condition of self-duality for SU(2) gauge fields on Euclidean four dimensional space" , Phys. Rev. Lett. 38 (1977), 1 377-1 379. 96. V. Zakharov and A. Belavin , "Yang-Mills equations as inverse scattering problem" , Phys. Lett. B73 (1 978), 53-57. 97. V. E. Zakharov and L. D. Faddeev, "KdV equation : a completely integrable Hamiltonian system" , Funct. A nal. Appl. S (1 9 72), 280-287.

29

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98. V. E. Zakharov and S. V. Manakov, "The theory of resonant interaction of wave packets in nonlinear media", Sov. Phys. JETP 42 (1976), 842-850. 99. V. E. Zakharov and A. B. Shabat, "A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, I." Funct. Anal. Appl . 8 {1974), 226-235. 100. V. E. Zakharov and B. G. Konopelchenko, "On the theory of recursion operator" , to be published in Comm. in Math. Phys. (1984). References for Section 2 - Books B1 B2 B3 B4 85 B6 87 88 89 8 JO Bl1 Bl2 B 13 B14

M. J. Ablowitz and H. Segur, Sol itons andthe Inverse Scatter ing Transform, SIAM, Philadelphia, 1981. Z. S. Agranovich and V. A. Marchenko, The Inverse Scatter ing Theory, Gordon and Breach , New York, 1963. R. L. Anderson and N. H. Ibragimov, L ie-Back lund Transformations in Appl ications, SIAM Studies in Applied Math. , Philadelphia, 1979. M. Boiti, F. Pempinelli and G. Soliani (eds.), "Nonlinear Evolution Equations and Dynamical Systems", Lecture Notes in Physics n. 120, Springer, Berlin, 1980. R. Bullough and P. Caudrey (eds.), Sol itons , Springer, Berlin, 1977. F. Calogero (ed.), Nonl inear Evol ut ion Equat ions Solvable by the Spectral Transform, Pitman, London, 1978. F. Calogero and A. Degasperis, Spectral Transform and Sol itons. I, North­ Holland , Amsterdam, 1982. K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1977. H. T. Davis, Introduction to Nonl inear Differential and Integral Equations, Dover, New York , 1962. A. S. Davydov, Biology and Quantum Mechanics, Pergamon Press, Oxford, 1 982. R: K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Sol itons and Nonl inear Wave Equations, Academic Press, New York, 1982. W. Eckhaus and A. van Harten, The Inverse Scattering Transformation and the Theory ofSol itons, North-Holland, Amsterdam, N. Y. , 1981. G. Eilenberger, "Solitons" , Ser ies in Sol id-State Science n. 19, Springer , Berlin, 1981. A. Forsyth , Theory of Differential Equations, vol. 6, Cambridge Univ. Press, 1906.

30 B15 R. Hermann, The Geometry of Nonlinear Differential Equations, Backlund

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816 817 818 819 820 821 B22 B2 3 B24 B25

Transformation, and Solitons. Part A. Interdisciplinary Mathematics, vol. 12, Math. Sci. Press, Brookline, 1976. E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956. N. Jacobson, Lie Algebras, Dover, New York, 1979. M. Jimbo and T. Miwa, "Nonlinear Integrable Systems-classical Theory and Quantum Theory", Proc. RIMS Symposium-Kyoto, Japan, May 1981, World Scientific Pub!. Co., Singapore, 1983. G. L. Lamb, Jr., Elements of soliton theory, J. Wiley and Sons, New Yrok, 1980. K. Lonngren and A. Scott, Solitons in action, Academic Press, New York, 1978. R. M. Miura (ed.), "Backlund Transformations", Lecture Notes in Mathe­ matics, n. 515, Springer, Berlin, 1976. P. J. Olver, Applications of Lie Groups to Differential Equations, Math­ ematical Institute, Oxford, June l 980. C. Rogers and W. F. Shadwick, Backlund Transformations and Their Applications, Academic Press, New York, 1982. M. Toda, Theory of Nonlinear Lattices, Springer, New York, 1981. K. 8. Wolf (ed.), "Nonlinear Phenomena", Lecture Notes in Physics n. 189, Springer, Berlin, 1983.

31

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3. SOLITONS IN PARTICLE PHYSICS: A GUIDE THROUGH THE LITERATURE

3 - 1 One-dimensional Systems Most of the results in the mathematical theory of solitons apply to the one-dimensional propagation of waves. Particle phenomena are essentially three-dimensional, yet frequently particle physicists have studied one-dimensional systems as useful, simplified models where to test ideas, which may find applica­ tions in the more realistic three-dimensional setting. Thus it is in the context of one-dimensional solitons that the largest overlap between the mathematical physicist's and the particle physicist's concepts of soliton is bound to occur. The system described by the sine-Gordon equation 3 2

3 2

3t

3x

-2 - -2 =

- sin t/)

(3-1 )

is probably the one where the above overlap can be better identified. The various mathematical properties which are at the root of the existence of soliton and multisoliton solutions have been considered already. Yet, the existence of the basic soliton t/)

=

4tn - 1 exp (x)

(3-2)

can also be inferred from simple topological arguments. Equation (3 -1) follows from a Lagrangian where the kinetic energy and potential energy terms are given by EK

=

J +G:)2 dx

( 3 -3)

(3 - 4)

32 One sees that the configuration of minimum energy, or vacuum configuration, is not unique, but is obtained in correspondence to any of the solutions

= const. =

2 nn

(3-5)

with integer n . Thus the sine-Gordon system possesses multiple vacua, labelled by an integer parameter n. For a configuration to be of finite energy , the field (x ) must approach one of its vacuum values, say 2 nn -, for x ➔ and another vacuum value, 2 nn + , for x ➔ + Adding a common integer to n ­ and n + is irrelevant, since the system i s invariant under global shifts ➔ + 2 nn ; but the difference !:-.n = n + - n - carries important physical meaning. No continuous deformation of the field configuration preserving the finiteness of the energy can alter !:-.n. Therefore, all finite energy field configurations are separated into topologically inequivalent sectors, characterized by the topological quantum number t:-.n . The soliton solution of Eq . (3 - 2) is the field configuration of minimal energy in the sector with t:-.n = 1. This line of argument relates the existence and stability of the soliton solution to the topological properties of the field configuration at the boundaries of space. It is dutiful to observe that such topological considerations do not account for all the properties of the soliton solutions, most notably for the conservation of shape after a collision. Indeed , a modification of the potential energy by some higher harmonic of the field would destroy this latter property, but would preserve the existence of multiple vacua with its topological implications. We begin to encounter here the distinction between the solitons of mathematical physics and the solitons of particle physics mentioned in the Introduction. Yet, topology is an exquisitely mathematical subject, and recent research has shown that the topological properties of soliton configurations have a wealth of physical implications. Pioneering ideas about the consequences of nontrivial field topologies were expressed in 1959 by Finkelstein and Misner, and, although their considerations are not restricted to one-dimensional systems, because of the interrelations discussed above and of the historical prominence of their work , their article [ R23 ] opens the series of reprints on the solitons of particle physics. More about topology is found in the subsequent article by Finkelstein [R24] . The article by Rubinstein [R25 ] concentrates specifically on the sine-Gordon equation and its particle interpretation. It is, incidentally, in this work that the name sine-Gordon equation was introduced. Use of the sine-Gordon as a model particle system appears also in earlier works of Skyrme [58-60] and Enz [22] .

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00 ,

00 •

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33 A complete treatment of any particle phenomenon must be quantum mechanical. The soliton emerges as a particle already at the classical level. When theorists realized that many nonlinear field theories used to describe elementary particles also had soliton solutions and that these would correspond to novel particle-like excitations, the development of methods for soliton quantization became of fundamental importance. Various techniques of quanti­ zation were derived. Mostly these are based on the very fact that, as opposed to the elementary quanta of a field, the soliton remains a particle also at the classical level, i.e., in the limit h ➔ 0. One can then proceed to the soliton quantization by performing an expansion into powers of h, or of some other parameter which occurs in conjunction with h in the Lagrangian, whereby the classical soliton solution appears as the term of leading order in the expansion, whereas terms of higher order incorporate quantum mechanical effects. The reprints [ R26] and [ R27] reproduce two of the earliest and most relevant works in this direction. A few more articles dealing with methods of soliton quantization will be presented later, in connection with three-dimensional models. We would also like to mention at this point, as excellent references where the topics of soliton quantization is treated extensively, the review articles by Jackiw [87] Faddeev and Korepin [ BS ] and the book of Rajaraman [ B1 1] . Reprint [ R28] also deals with the quantization of a soliton system, with particular emphasis on the sine-Gordon equation. The existence of a non­ linear canonical transf o rmation which b rings the dynamical variables to variables of the action angle type is crucial to the analysis in [ R28) , and so this paper brings us back in contact with the solitons of mathematical physics. For related works see [42, 73) . If there could have been any doubt that the solitons of the sine-Gordon equations are quantum mechanical particles of the same status as the fundamental quanta of a field, this was dispelled by the remarkable equivalence between the quantum sine-Gordon equation and the massive Thirring model, discovered by Coleman [ R29 ] and Luther [45] . By a set of mathematical equalities between Green's functions in the two systems, it can be shown that the kinks of the sine­ Gordon equation propagate and interact exactly like the fe rmions in the massive Thirring model. Quantized fermionic fields, in the latter system, may be represented by suitable nonlinear functions of the quantized bosonic fields underlying the sine-Gordon system. Such methods of bosonization have proven very useful, especially in the analysis of one-dimensional systems. Mandelstam [46] did pioneering work in this direction; for a recent contribution on the subject and more references see, e.g., a paper by Ha [8] .

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34 If in connection with a soliton solution one may say that its most relevant particle-like features are exhibited already at the classical level, it is when solitons are set into interaction with other fields that very remarkable, novel quantum mechanical phenomena are found to occur. These phenomena are related to the nontrivial topological properties of the boundary configuration and do not manifest themselves in the theory of quantized fields in absence of soliton-like excitations. The first instance of such phenomena was discovered by Jackiw and Rebbi [ R30] , who considered the situation where solitons interact with fermionic fields. In the framework of the quantization methods mentioned above, the fermionic fields do not enter into the description of the system at leading order (O( h0 )), which is given by the classical soliton solution. At the next order (O(h)) one encounters the first quantum mechanical e ffects, which are described by the propagation of the fermions in the background given by the soliton configuration . For most of the fe rmionic modes the situation is not much different from what would happen in the presence of an ordinary perturbing potential: they occur in charge conjugate pairs and the corr:_Sponding coefficients in the expansion of the quantized Dirac fields, i/1 and i/1 , are interpreted as creation and annihilation operators for fermions and antifermions. However, with definite type of Yukawa couplings between the soliton field and the Dirac field, the nontrivial boundary conditions imply the existence of a further mode of excitation for the fermionic field. Such a mode occurs isolated, without charge conjugate counterpart ; it is self-conjugate and has very remarkable implications. The corresponding pair of expansion coefficients, a in i/1, a t in 'Ji satisfy the algebra

{a, a } {a t, a f

=

=

{a t , a t }

=

0

1

(3- 6)

The soliton must provide a representation for such algebra ; it cannot there­ fore be described by a single state, but must be capable of existing in two degenerate states. Arguments of charge conjugation show then that the two states must carry fermion number +½ and this implies that in the presence of a suitable quantized fermionic field, the soliton has become half a fermion ! The investigation in [ R30] was made in the context of field theoretical models for particular physics. It is extremely remarkable that quite independently the same conclusions about fermions fractionalization in the presence of solitons

7'"½ ;

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35 were derived by Heeger, Schrieffer and Su, studying a system of interest for condensed matter physics, namely dimerized polyacetylene linear chains (R3 I ] . The solitons of particle physics are very interesting theoretical constructs. Their existence follows necessarily from the underlying field theoretical models. However, at the present stage, there is not enough experimental evidence to single out a definite model, nor have the corresponding solitons been detected. But, in condensed matter physics, solitons and the ·consequences following from the topology of the field configu ration ( B9] can be experimentally revealed. Thus condensed matter systems indirectly provide a verification of ideas on solitons and topology, which play such an important role in the physics of particles. To conclude this subsection we would like to mention that the notions of fermion fractionalization extend also to three-dimensional models, as can be seen from ( R30] , and that more recent investigations have shown that the fractionalization is not restricted to half unit of fermionic charge. We shall return to this when dealing with three-dimensional models. Also, that fractiona­ lization is not realized in an average sense, by having expectation values of the fermionic charge equal to ± -½- , while the charge operator itself maintains integer eigenvalues. Rather, a soliton far separated from other soliton or antisoliton excitations can be shown to be in an eigenstate of charge with half-integer eigenvalue. This conclusion is reached by studying the fluctuations of the charge operator : for the theoretical argument, the reader may consult, f or instance, Ref. [ 41] . 3 - 2 Vortices and Two-dimensional Systems Just as in the case of the sine-Gordon system, topological considerations can be used to argue for the existence of stable soliton-like excitations in two­ dimensional models. The simplest situation occurs if the vacuum state of the system is characterized by some complex field taking a non -zero value with definite modulus o . The equation l I = o

(3-7)

obviously admits an infinite variety of solutions, (3- 8 ) which can be labelled by an angle 0 , 0 ,;; 0 < 2 1r , or, equivalently, which can

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36 be put into one-to-one correspondence with the points of the unit circle . The vacuum state is therefore not unique ; rather we have a continuum of degenerate vacua. The boundary of the two-dimensional space is also homeomorphic to the unit circle ; thus the requirement I - - oo. We turn now to the functions v ; since these satisfy a linear equation, it can be shown, by an argument similar to the one which led to inequality ( 1 .4) , that any v increases at most exponentially. In fact, since exponentially increasing solutions

76

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INTEGRALS OF NONLINEAR EQ.UATIONS OF EVOLUTION

479

of the variational equation usually indicate an instability, and since on the other hand numerical evidence indicates that double waves are stable, it is reasonable to expect that all solutions v of the variational equation grow at a rate slower than exponential. We return now to the functional (2.�) ; we have shown that as t ----->- - oo the first factor G(d) tends to zero exponentially, and that the second factor v tends to infinity slower than exponentially. It follows then that (G(d) , v) tends to zero as t ----->- - oo ; on the other hand, according to (2.2) , this functional is independent of t. Therefore it follows that (G(d) , v) is zero for all t. At any particular time t0 the initial values of v may be prescribed arbitrarily ; therefore it follows that

(2 .6)

G(d)

=0

at time t 0 , that is for any time. There remains the task of constructing an integral / whose gradient G is local and satisfies (2.5) , i.e., annihilates both solitary waves s 1 and s 2 • Here we rely on Lemma 1 .2 , relation ( 1 . 30) , according to which solitary waves are eigenfunc­ tions of the gradient of every integral :

I t follows that, given three independent integrals, an appropriate linear combination of them will have a gradient which annihilates any two given solitary waves. We turn now to the task of finding three independent integrals whose gradients are local operators. We have already noted in Section l that the energy (2 . 7) i is an integral for the KdV equation. Another integral was found by Whitham [7 ] :

(2 . 7) 2 A third one was discovered by Kruskal and Zabusky :

(2 . 7) 3

la (u)

=

f ( iu

4

-

3uu!

+ fu!.,)

dx .

Subsequently, Kruskal and Zabusky found two more explicit integrals and Miura four more ; an infinite sequence 3 of such integrals was constructed in [5 ] . 1 Denote by I,. the n-th integral of this sequence and by G,. its gradient. The relation discovered by Gardner between I,. and the n-th generalized KdV operator Kn described in Section 1 is Kn

=

DGn .

77 480

PETER D. LAX

For the discussion of the double wave we need only the first three integrals. The gradients of these are (2.8) 1 (2. 8) 2

G1 (u) G 2 (u)

(2.8)a

G3 (u)

=u, = u 2 + 2u.,., , = u 3 + 3u! +

6uu.,.,

+ 1lu.,.,.,., .

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Note that these are indeed local operators . The proof that /1 , /2 , /3 are integrals of KdV consists in verifying that the product of Gn (u) with K(u) = uu., + u.,.,., is a perfect x derivative. Indeed an explicit calculation gives (2 .9) 1 (2 .9) a (2.9) 3

G1 (u)K(u) G 2 (u)K(u ) G3 (u)K(u)

= = =

(uu.,., - ½ u! + ¼ u 3 ) ., = H1 (u) ., , (u!., + u 2u.,., + !u4 ) ., = H2 (u) ., , (fu!., ., + 1r}!u.,.,.,u.,u + !u!.,u - iu.,.,u!

+ fu!u + u.,.,u 3 + ¼u6 ) = H3 (u) ., . .,

We know that solitary waves are eigenfunctions of G11 eigenvalues K ( G11 ) as follows : G1 (s)

(2 . I 0) i (2. 1 0) 2 (2. I O) a

G 2 (s) Ch)

;

a calculation gives the

=s, = 2cs , = \,'flc 2s .

We form now the linear combination (2 . 1 1 ) whose gradient is (2 . 1 2 ) I n view o f (2. 1 0) ,

G(s)

=

( \.!!.c 2

+ 2Ac + B)s .

Thus G annihilates s 1 and s 2 if c 1 and c 2 satisfy the equation S(c)

( 2 . 1 3)

= 1lc 2 + 2Ac + B = o .

In view of the relation between coefficients and roots of a quadratic equation this means that

A

(2. 1 4) .f

=

- ! (c1

+ c 2)

,

(2. 1 4) B In view of (2.6 ) we conclude : Let d be a double wave wi th speeds c1 and c 2 and define the constants A, B by (2. 1 4) ; then for each t, d satisfies (2 . 1 5)

G(d)

= G3 (d) + AG 2 (d) + BG1 (d) = 0 .

,

78 INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION

48 1

This is a nonlinear ordinary differential equation of fourth order 4 ; therefore its solutions form a 4-parameter family of functions. On the other hand, the double waves form a 2-parameter family. We shall accordingly deduce from (2. 1 5) a second order equation satisfied by all double waves. To do this we make use of the earlier observation (see (2.9) ) that the invariance of I(u) for solutions of KdV is equivalent with the fact that G(u)K(u) is a perfect x derivative. Multiplying (2. 1 5) by K (u) and using (2.9) we see that

(H3 + AH2 + BH1 ) = 0 .

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.,

Integrating this and using the fact that d and all its derivatives are zero at x ± oo, we deduce

=

(2. 1 6) Next we make use of the translation invariance of the integrals I under considera­ tion , i.e., of the fact that I(u,) is independent of e, where u, is the translate of u by e in the x direction. Differentiating with respect to e, we get

J

0 = � l(u,) 1,- o = G(u) u., dx . This implies that G(u) u., is a perfect x derivative. I ndeed, an explicit calculation gives

=

(2. 1 7) i

G 1 (u) u.,

(2. 1 7) 2

Gh)u., = (u ;

(2. 1 7) 3

G3 (u) u.,

=

( ½u 2 ) .,

= J1 (u) ., ,

+ ½u 3)

.,

= J2 (u) ., ,

(1-l-u.,u.,"., - Ju ;.,

+ 3uu; + fu4 )

.,

•

Thus multiplying (2. 1 5) by u., we obtain, after integration and use of the fact that d and its derivatives vanish at oo, (2. 1 8) Both (2. 1 6) and (2. 1 8) are third order differential equations. Expressing d.,.,., from (2. 1 8) , substituting into (2. 1 6) and multiplying by d; , we get an equation of second order and fourth degree in d.,., which we write symbolically as (2. 1 9)

Q (d.,., , d., , d)

=

0 .

From this e9.uation d.,., can be expressed as a 4-valued function of d and d., ; since x does not appear explicitly in these equations, this second order equation is equivalent to a first order autonomous system of equations for d and d., . By studying carefully the geometry of all four branches of the corresponding vector­ field one can show that (2. 1 9) indeed has solutions which tend to zero as x -+ ± oo and has the shape of a double wave, i.e., has two maxima and one minimum. • This equation appears in [8] .

79

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482

PETER D. LAX

I shall not present the details because (i) I did not carry them out completely, (ii) the formulas are horribly complicated, (iii) an explicit formula for double waves (indeed N-tuple waves) was derived recently in [ l ] . We show now how to use equation (2. 1 9) to study the time history of double waves. For this purpose we consider the maximum value of d(x, t ) with respect to x, or rather the relative maxima of d as functions of time. Denote by m = m ( t ) the value of a relative maximum of d(x, t ) ; at the point y where the relative maximum occurs

=0.

d.,

(2.20)

I t follows from (2.20) by the implicit function theorem that if d < 0 at y, then y is .,., a differentiable function of I. Hence m = d(y, t) also is differentiable and satisfies mt

=

d.,yt

+

dt

= dt .

Since d satisfies the KdV equation, we have, in view of (2.20) , m 1 = -K(d) = - d= .

(2.2 1 )

We proceed now to determine dxxx at a local maximum. At a point where d., = 0, equation (2 . 1 6) simplifies considerably ; using formulas (2.9) and denoting the value of d by m, we get (2.22)

fd;xx

+ fd;xm + d,,,,m 3 + ¼ms + A (d; ,, + m 2d,,,, +

¼ m4 )

+ B (md,,.,

+ ½ m 3 ) = 0.

Similarly, at a point where d., = 0, equation (2. 1 8) becomes d ; ., = f ( ¼m4 + A ½ m 3 + B½ m 2 ) = P( m ) .

(2.23)

From (2.23) we deduce that d,,,, = _ p1l2 ( m) ,

( 2.24)

the negative sign being chosen since the second derivative is nonpositive at a local maximum point. Substituting (2.24) into (2.22) we get (2.25) where

d;xx

R( m) = - a ( m )

(2.26) with (2 . 2 7) 0 and (2.2 7h

= R (m)

a(m)

=

JmP(m)

+

,

b ( m) P1i 2 ( m) ,

+ ½ms + l!lP(m) + 3\Am4 + {'7Bm 3

b ( m) = Jm 3 + jA m 2 + fBm .

80 INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION

483

Combining (2.2 1 ) and (2.25) we have (2.28)

mt

=

± R (m) 1f2 ;

the sign to be taken as positive when m increases, negative when m decreases. Thus m as function of t is governed by equation (2.28).. To study the behaviour of m we have to know something about the function R (m) . First of all we investigate the sign of P(m) ; since P(m) is Jm 2 times the quadratic polynomial Solitons and Particles Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 01/28/16. For personal use only.

+ ¼Am + ½B ,

¼m 2

(2.29)

it will be positive if the discriminant of (2.29) is : discr

=

½B

- iA 2 •

Using formulas ( 2 . 1 4) for A and B we have discr

=

l5(3cif 2 - c:

- c;)

This quadratic form is positive if and only if � � 3

( 2 . 30) Thus we have proved

C2 -

+ vs = 2 .62 . 2

LEMMA 2 . 1 . If c 1 and c 2 satisfy (2 . 30) , P(m) is positive for all real values of m. (2. 30) is violated, P(m) is negative in the interval (n 1 , n 2 ) ,

If

( 2 .3 1 ) I t i s easy to verify that the interval (n1 , n 2 ) lies inside (m 1 , m 2 ) . We turn now to R(m) ; a lengthy calculation, the gist of which i s described in [ I O] , yields the following LEMMA

2 .2 .

R(m) has a double zero at m 1 and at m 2 and d 2R/dm 2 is positive at

2.3.

(a) In the range

these points. LEMMA

the function R(m) is positive in ( m 1 , m 2) . (b) In the range

3

+ VS

Ct

0 , 0 ( ; ) = 0 a t = < 0 ) for space-like i n t e rvals < 0. Choosin g t he hypersurface x 0 - y 0 = I = const as a , one arrives

=5 =i

at t he t rian g ular represe n t a t ions o f Re fs . 5 , 6.

200

99 MJ t n x kernels

1-\ a n J R may he 1 c p rcsc n t c d t h 1 ,1uf'.h s.:alar kernels as follows

f. µ =

(

Kµ

f. = ( O K ) .

0 ).

K'

o l- oo. To each point q in the configuration space M one associates a tangent Rpace P0 of smooth vector fields on the real line

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3

13

(contra.variant fields) satisfying the same boundary conditions as q. The space of field functions {J(x, t) = ({J1 (x, t), {J2 (x, t ) , • . . , {J8 (x, t ) ) , obeying the same boundary conditions, which is put in duality with P. via the sym­ metric (nondegenerate) bilinear form +..

f

(2.1)

({J, IX) = /JIX dX ,

is called cotangent space T; and its clements covariant fields. We deal with functionals G(q) """ G(q, q. , q•• , . . . ) defined in M and with values in M, T. , T; or in a space of operators which are assumed to be dif­ ferentiable according to the definition of Gateaux - C exists such that F'(q)[1X]

(2.3 )

=

(y, IX) .

It is well known that y is a gradient operator if and only if y' = (y') + '

(2.4) ( 13 )

G. Z. Tu : J.

Phys. A,

15, 27 7 ( 1 982 ) .

(") J. M. GEL'FAND and I . Y A D ORFMAN :

Funct. Anal. Appl. ,

13, 248 ( 1 979) .

109 236

M. BOITI, F. PEMPINELLI and G. z. TU

where + means adjoint with respect to the bilinear form is then given by

(2.5)

1

< , ).

Its potential If'

+ ..

= J = [K"> , U''] , Tr Km = 1 .

3·2. The reduced cases. - For studying the so-called reduced cases in which one imposes algebraic constraints on the spectral operator U, it is convenient to expand U in termii of a basis {R 1 , i = 1, 2, . . . , N 2} in the linear space of the N x N matrices : (3 .28)

N'

U = _L Z;R, , £-1

and KCO in terms of the dual basis {S 1 , i (3. 29)

= 1 , 2 , . . . , N•} :

N'

K' 0 = 2: K\0 S, . i-1

The basis {S.} is dual to the basis {R.} with respect to the inner product of N X N matrices A and B : (3.30) and is normalized as follows : (3.31)

(A, B)

= Tr (A r B)

116 243

CANONICAL STRUCTURE O F SOLITON E QUATIONS ETC.

From the chain rule (2.15) we get (i = 1, 2 , . . . , N 2 ) .

(3.32)

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In general, a reduced case is obtained by choosing some z;'s to be constants and some other z;'s to be independent functions of a vector field q(x, t) . Specifically we choose (3.33)

z , == C , 01 ) ,

i = I, 2, .. . , M ,

(3.34)

z" == C" (q ; ,1. ) ,

IX = M + I, M + 2 , . . . , N 2 •

In the following, for the sake of notational simplicity, small Latin indices and Greek indices will be used to denote, respectively , the first M-components and the last N• - M components of any operator expanded in terms of the basis {R,} or of its dual {S,} . The repeated Latin (or Greek ) index , when one index is raised, implies a summation over that index (raising or lowering indices is considered to have no effect on the value of the quantity considered) . For instance, eqs. (3.28) and (3.29) can be rewritten in the reduced case as follows : (3.35) and (3.36) We assume in the following that, for q decaying sufficiently fast for !x! -+ oo , all the C" (q ; ,1. ) vanish (rapidly) at !x i = oo. Notwithstanding, since the C.'s are x-independent, the integral defining the conserved quantity H diverges, in general, and Hm must be renormalized. A convenient renormalization, which works in rather general cases, is as follows : (3.37)

H�\C)

=

Jdx [ff('l(C) - 0 (x) (R Y)

+.,

_.,

0

1,(

+ oo ) - 0 (- x) (R0 Y) 1 1

(-

oo)] ,

where (3.38) and 0 (x) is the Heaviside step function. The K,. i n expansion (3.36) which is evaluated at z, = C, and at z,. = C. can be computed as a variational derivative of H�> with respect to C. : (3.39)

117 244

M. BOITI, F. PEMPINELLI and G. z. TU

Let G( U) be the subspace generated by the matrices R0 (II) and Rtx 's , G1 the subspace generated by the s e"'s and G2 that generated by the S(X"''s and let us assume that (3.40)

or, equivalently, that (3.41 )

[Ro , S'"'] = c�Q ( }.) S0"' '

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[Rtx , S'"'] = C! e S0"' ,

[Rtx , SP"']

(3.43)

=

C!, S'"' + C�0 S0 "' ,

[Ro , sPT] = c:i (l) S1" + c:p) S0"

(3.44)

with some convenient constants c: . . By inserting expansions (3.35) and (3.36) into (3.2 6) and by using eqs. (3.41)-(3.44) , we get (we drop for convenience the index (Z)) (3.45) (3.46)

Ka,c = K,,z =

C0 c:" Kp + c:a ( }.) Kp + C0 c;" x, + C�" (}.) K, ,

C0 cg, Kp + c:,( }. ) Kp .

The last equation can be once integrated to (3.47)

where I is the integral operator (3.48)

I[ ] = ½ (

j-T )

dx[ ]

and

(3.49)

If we introduce the matrices {3.50) (3.51) (3.52)

( CB ) all = C!" ' ( C: ) " = C�" , ' ( C; ) ,tx = c;,

with B = o, I, . . . , N• and the vectors (3.53) (3.54)

oHR oHR oHR)T K=( 0CM+1 ' oCM+. ' ... , oc;. ' " = ( ie1 , "" • • . ' ie,i" '

118 CANONICAL STRUCTURE O F SOLITON EQUATIONS ETC.

245

from (3.45) and (3.47 ) we obtain the integrodifferential equation for the gra­ dient K that we are looking for :

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It is worthwile stressing that the conserved quantity HR and the integro­ differential equation (3.55) satisfied by K"" = 'SHR/'S C"" are determined only by the principal spectral problem without any reference to the specific form of the auxiliary spectral problem. Moreover, up to this point, the dependence on q of the C"''s has not been specified. 3"3. Isospectral eigenvalue problems implying a canonical geometric structure. The integrodifferential equation found for K (3.55) or, directly, eqs. (3.26), (3.27 ) , if condition (3.40) is not satisfied, can be used to derive a complete characterization of the geometric properties of the hierarchy of soliton equations (1.6 ) . The details of this derivation depend upon the specific principal spectral problem considered and they are described in the sect. 5 for various hierar­ chies of soliton equations. Nevertheless we can outline the main points of the method proposed which is common to the various cases considered. Let us consider the gradient of H� > with respect to q : (3.56) and let us suppose that we are able to solve the Lax representation (1.5) and to find the integrodifferential operators L and J defined in (1.6). We consider an equation (3.57) in the hierarchy (1.6) and assume that, by using the integrodifferential equa­ tion (3.55) and eq. (3.47 ) , we can compute LI'Cll and find that I'CZ>(q ; ).) sati;;fies the linear eigenvalue problem (3.58) where µ and v are functions of the isospectral parameter ). and yc 1>(q) is a con­ served covariant for eq. (3.57). Then L satisfies the equation (3.59)

(L'[s]

+ [s'+, L]) I', s) = 0. In order to achieve the strong symmetry property of L+, we need to com­ plete the set {I'� 1 (A), I'� \ J.)} with some clements satisfying the operator equation (3.6+) which are conserved covariants but not conserved gradients or renormalized conserved gradients . Let us consider the derivative of 1 with respect to }.

ri:

nil) _ a TTIJ l R = CA 1 R .

(3.65)

Since s and L are }.-independent operators and their Gateaux derivatives are linear continuous operators, I'�' is a conserved covariant and it satisfies the equation (3.66)

(L '[s]

+ [s ' +, L]) !'�1 = 0 .

120 CANONICAL STRUCTURE OF SOLITON EQUATIONS ETC.

247

If the principal spectral problem considered admits both discrete and con­ tinuum eigenvalues and direct and inverse spectral transform can be defined, we conjecture that the set of conserved covariants {I',!> (.J. ) , r (1.), f'li> (l. )}, where 1 and 1. belong, respectively, to the continuum and to the discrete spec­ trum of J.., is a complete set. If this conjecture is satisfied, it follows directly from (3.64) and (3.66) that L+ is a strong symmetry for the soliton equation

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q, = s ( q) .

The completeness conjecture that we make i s supported by the connection of the conserved gradient I'( !) with the (J... ) , f'li!> (J... )} is complete. ( 18 )

M . D.

ARTHUR

and K. M.

J. Math. Phys. ( N. Y.) , 23, 1 17 1 ( 1982) . J. Math. Phys. ( N. Y. ) , 22, 1 1 70 ( 1 98 1 ) .

CA.SE :

( 17 ) A. P. FORDY and J. GIBBONS :

121

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248

M. BOITI, F. PEMPINELLI and G. z. TU

In the following section we consider the 2 X 2 Zakharov-Shabat spectral problem and we relate explicitly r�> to sm showing that the completeness of the set {I'� \}.) , r�>(},k ) , f'i�'(,1.k )} follows from the proof given by KAUP of the closure of the squared eigenstates . Since the proof o f the closure requires the solution o f the direct and inverse problem of the spectral equation considered, the generalization of Kaup's result and, consequently, of our conjecture to other eases is highly nontrivial and it depends strictly on the specifities of the spectral equation . .According to a well-known general procedure, the conserved density ff' (we drop for convenience the index l) can be expanded into a formal power (or, possibly, fractional power) series in the spectral parameter ,1. (3.73) whose coefficients .Yt'n are recursively determined by the Riccati equation (3.1 0 ) . B y inserting expansion (3. 7 3 ) into the eigenvalue equation (3 . 58), one gets, in general, a recursion relation of the form (3. 74) with en a convenient constant and m(n) a convenient integer index depend­ ing on n. Equation (3.74) together with the first coupling condition (2.23) imply that the equations in the hierarchy (2.22) are Hamiltonian systems with Hamiltonian in involution and commuting flows. Because the equations in the considered hierarchy are Hamiltonian systems, the cosymplectic operator J maps the conserved gradient r1 1> into a symmetry (3.75)

c,lll (q ; }, ) = J(q) I'( l) (q ; A )

of the same equation q, = s(q) . This follows directly from definition (2.17 ) and from e q . (2.20) , since H�' is a conserved quantity. As for the conserved gradient I'l ll, it is convenient to renormalize urn by defining (3.76) By using eq. (3.61) and the linearity of the Gateaux derivative, it is easy to verify that u�' is a symmetry for the equation q, = s(q) . Moreover, (3. 7 7 ) i s also a symmetry.

122 249

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CANONICAL STRUCTURE OF SOLITON EQUATIONS ETC.

If J is invertible, from the conjecture that the set of conserved covariants {I'� > (q ; }.), r�> ( q ; J..) , f'� > (q ; }.k )} is complete, it follows that the set of sym­ metries {a�\ q ; A) , a�\q ; J.. ) , &�\ q ; Jk )} is also complete. By taking into account that a strong symmetry transforms a symmetry into a symmetry, that the symmetries au> and L+ a co commute if L and J satisfy the first coupling condition and , finally, that the definition of a hereditary symmetry can be cast into the equivaJent form ( 2 .26) , one can easily prove that the operator L + is a hereditary symmetry. Therefore, we conclude that J and L define a symplectic Kahler structure on the manifold Jlf. In sect. 5 we consider various spectral problems and we show that the general integrodifferential equation (3.55) for the gradient K or, directly, eqs. (3.26 ) , (3.27) allow us to derive easily the linear eigenvalue.problem (3.58) . This equation is used to prove that the soliton equations i n the hierarchy ( 2.6) are Hamiltonian systems with commuting flows and, moreover, it furnishes the explicit form of the Hamiltonians which result to be related to the infinitely many polynomial conserved densities generated by J{W. 4. - Proof o f the completeness conjecture i n the 2 x 2 Zakharov-Shahat spectral problem.

The Zakharov-Shabat spectral equation is given by

'P� = U'P '

(4.1) wher(l (4.2 )

U = - i,la3 + q1

n0

coincides with

+ J.•0c1 - .t)J a!\&}1i .

and the soliton does not move at all it aoes not change with time, n(t) = (i.e. , E(t) = �o) if J. < l, while if J. > l it moves with the constant speed ± v (i.e. , E(t) = �o ± v(t - t0 ) if 0 = n,,), where (11) Solitons and Particles Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 01/28/16. For personal use only.

n

(3 1 ) I f instead the initial polarization n0 coincides neither with n + nor with n_. the boomeron polarization n(t) does change with time : it never coincides with n+ nor with 11_ but, if J. > l, it tends asymptotically to the values n± (i.e. , n( ± oo) = n±), while if J. < 1 it precedes periodically (with period (2n/a)( l - J.• J-l ). As for the behaviour of the boomeron co-ordinate E(t), it is best understood notin g ( 1 8) that it coincides with that of a particle of unit mass, with initial position �(t0 ) = Eo Md initial speed E,( t0 ) = - pb, moving in the external potential dl(� - �0 ), with (32)

i = ½ (bfJ.>• exp [- 2px]{(( [( cx• + r•> J. + rJ• + cx• ) /( cx• + r'>] •

- exp [- 2px] - 2( 1 + J.y)} .

This (« Morse •) potential vanishes as :i; ..... + oo ; it has a single (negative) minimum at X = x = (2p)-1 log { ( [ ( cx• + r') J. + r]• + cx1 ) /[( cx• + ;,1)( l + J.y)l} if J.y> - 1 ; a.n d it diverges to positive infinity as x ..... - oo (unless ex = l + J.y = 0, when it vanishes identically ; this happens if and only if 0 = n + or 0 = ,i_). Note that the total energy of this particle has the simple expression

n

(33)

n

E = 41 (0) + ½/J' b• = ½v• sign (J. - 1 )

with v defined by eq. (31). Thus it is positive if J.> 1 , in which case the boomeron escapes, or recedes, to the right as t-+ ± oo, moving asymptotically with the con­ stant speed v of eq. (31 ). The total energy E is instead negative if J. < 1, so that in this case the boomeron behaves as a particle trapped in the potential, oscillating in­ definitely around the equilibrium position x (see above). This oscillatory behaviour is of course also given, in this case with ). < 1, by the explicit eq. (27a) ; it is periodic, with period (2n/a) ( l - ).2 J-½. The marginal case i. = 1 corresponds to the motion of a zero-energy particle in the potential (32 ) that always escapes (or recedes) to positive infinity, but with a vanishing asymptotic velocity, as evidenced by the explicit formula (34)

�(t) = p-1 log ! all + �0 + (2p )-1 log [½ ( l + y) ] + 0( ! t l-1 log !t i )

implied, for t -->- ± oo , by (27a) and (28) (with J. = l) ( 11) . (") 6(:z:> - 1 if :z: > o, 6(:z:) = o If x < o, 6(0) � l . (19) This can be proved as above; see footnote cu). (") We Msume here y # - 1 ; otherwise n, - ;.. - a. (see (30)), implying n(l) = n., �(ll = ,;••

147 433

COUPLED NONLINEAR EVOLUTION EQUATIONS SOLVABLE ETC.

Note that the analogy of the motion of the boomeron to that of a particle in the exte rnal potential (32) covers also the cases when the initial polarization coincides with ii + or i_ , since the potential then vanishes identically ( and moreover the initial speed of the boomeron - pb vanishes if A < l ; sec eqs. (29) and (30) ).

Ji'inaZ considerations. It should be emphasized that, for a given NEE of type ( l ) with a • {; = 0, in a solution containing several solitons all the types of soliton behaviour described above may be simultaneously present-the oscillatory one, the boomeranging one ( Ht) -+ + oo as t -+ ± oo), the static one ( W) = �0 ) , the uniform one ( W) = �0 ± ± r(t - t0 ) )-depending on the Yalues of the shape parameter p ( in particular, whether or not it exceeds the quantity a / 2b ) and of the initial polarization 0 of each soliton ; the latter two behaviours listed correspond however to a set of initial conditions having a lower dimensionality. Note moreover that, in order that the phenomenon of soliton trapping (oscillatory behaviour) may occur, it is required that ·b = 0, since this phen­ omenon is characteristic of this special case only ( this indicates a remarkable form of structural instability of the NEE ( l ), concerning the possible existence of localized solutions that never escape asymptotically ). lu view of the richness of the behaviour displayed by the solutions of the NEE ( 1 ) it will be most interesting t o invt:stigate natural phenomena that are modelled b y it ; but this exceeds our present scope ( 1 8 ) . We finally remark that the fact that the solitons move with variable speed, with a corresponding variation of their polarization (or, equi'valently, of the relative magnitude of the different components of the solution ), occurs generally for the systems of NE Es solvable by the IST associated to the matrix Schriidinger equation {3 ) ; indeed it is easy to construct examples in which the solitons (or boomerons) are accelerated even asymptotically ( 1 •) .

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n

a

(") Additional results for tho NEE ( 1 ) , including the explicit display of Backlund transformations and conserved quantities, are presented in a separate paper : F. CAWOERO and .! . DEGASPERIS : Lett. }.tum,o Cim.�nto Un press).

148 23 Settembre 1 97 8

VOL. 23, N. 4

LETTERE AL NUOVO CIMENTO

Solution by the Spectral-Transform Method of a Nonlinear Evolution Equation Including as a Special Case the Cylindrical KdV Equation.

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F. CALOGERO and A. DEGASPERIS

Istituto di Fisica dell' Universita - Roma, 00 1 85 Istituto Nazionale di Fisica Nucleare - Sezione di Roma {ricevuto il 22 Giugno 1978)

The preceding paper (1 ) opens the possibility to investigate a novel class of non­ linear evolution equations solvable by the spectral- transform method. In the present paper we introduce and discuss only the simplest {nontrivial) equation of this class, postponing to a subsequent paper the treatment of the whole class (the attentive reader will notice that the notation introduced below is indeed aimed towards such an exten­ tion ; even at the cost of some clumsiness) . The starting point of our analysis are the following two formulae (2 ) :

+co

f

(1)

f'(z) - f(z) = n dx F' (x, z) (x, z) [u'(x) - u(x)] ,

(2)

- f(z) Jax [u' (x) - u(x) ]Jdy g(y) -

+co

_.,

CD

"

+ca

+co

f

f

- 4zn dx F' (x, z) (x, z) g (x) = n dx F' (x, z) (x, z) A1 g (x) , the integrodifferential operator A 1 being defined by the equation (3)

A1 G(x) = G,,(x) - 2 [ 2x + u' (x )

+ u(x)] G(x) +

"'

"'

"'

+ [ 2 + u;(x) + u,(x)]Jay G(y) + [u' (x) - u(x)]Jdy[u'(y) - u(y)]fdz G(z) , V

that displays its action on the generic function G(x) . ( ' ) F. CALOGERO and A. DEGASPERIS : Lett. NUQVO Oimento, 23, 1 4 3 ( 1 9 7 8 ) . This paper is hereafter referred to as I and its notation is used here without defining it anew. ( ') Clearly these formulae remain valid if the role of primed and unprimed quantities Is exchanged.

150

149 151

SOLUTION BY THE SPECTRAL-TRANSFORM METHOD ETC.

Formula ( 1) is a straightfo1·ward consequence of the ·wronskian theorem applied to I ( l) (using 1 ( 7)-1 ( 1 1) ) . The proof of (2) is also easy ( 3) ; but for its validity it is required that the generic function g( x) be integrable, that it vanish asymptotically with its first derivative and that i ts integral jr01n - oo to + oo vanish :

+"'

fdx g(x) = 0 .

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(4)

Assume now that the potential u (and therefore everything else as well) depends on a second variable t (> ( 9 ) (2 1 )

q, + q,yy - 6 q, q + (2t)-' q = 0 ,

q = q (y, t) .

The transformation of this equation into the canonical form ( 1 5) is achieved by ( 1 6) with (22)

(') ( ') (a)

y = ( 1 2t)i ,

/J = y-' ,

(Xo = 0 '

F. CALOGERO and A. DEGASPERIS : Lett. Nuovo Cimento, 22, 263 ( 1 9 7 8 ) . F . CALOGERO a n d A. DEGASPERIS : Lett. Nuo�o Cimcnto, 2 2 , 2 7 0 ( 19 7 8 ) . T h e irtroduction of a n e w t i m e variable , -,; = T(l) , or a t-dependent shift of the V varia ble,

y ~ y(l) x + c(t), amount merely to redefinitions of the functions cx,(I) and cc , (t) in (1 5 ). ( 8 ) Of course the numerical constants in front of quw and quq can be changed rescaling q nn ar rJt Ox 2

(2 )

als o holds for the c omplex amplitude 1/J of a quas imono ­ c h romatic one -dimensional wave in a m edium with dis ­ persion and inertialess nonlinearity . In (2 ) , w k is the w av e dispersion law and q i 1/J 1 2 is the nonlinear c orrec ­ tion to the frequency of the wav e with amplitude 1/J . Equations ( I ) and ( 2 ) c an be reduc ed to a standard dimensionless form

i

au

7,

+

iu P

az1

+ "' l u l i u = O.

(4 )

el I a, = t[l, A).

1 '

W ithout loss of generality, we can assume that K > 2 2 0. and p The inverse-problem method was discov ered by Kruskal, Green, Gardner, and Miura '' l and was applied by them to the w ell-known Korteweg-de Vries (KDV) equation. In tqe c as e c onsidered by them, the role of the operator L was played by the one-dimensional Sc hrodinger operator. They rev ealed the fundam enW role played by the particular s olutions of th e KDV equa­ tion-solitons , whic h are direc tly c qnnec ted with the disc rete spec trum of the operator L, namely, it was established that the asymptotic state as t - ± � of any initial c ondition is a finite set of solitons . In our prob­ lem, an analogous role i s play e d b y the particular s olu­ t i ons of Eq . ( 3) :

>

( 3)

It is convenient to assign the variable t the meaning of time. In the present paper w e shall inves tigate Eq. (3) with K > 0 . As applied to Eq . ( 1), this m eans 6n n1 > 0 . Un­ der this condition, Eq . (1) desc ribes s tationary two­ dimensional s elf-focusing and the assoc iated transverse instabil ity of a plane monoc hromatic wav e . C' l For Eq. (2), the c ondition K > 0 is equivalent to the require­ ment q w k < O , whic h, when satisfied, produc es in the nonlinear medium longitudinal instability of the mono­ c h romatic wave -self-modulation . ' 3 ' "' s ] Equation (3) c an b e solved exa.c tly b y the inverse­ problem m ethod. This method is applicable to equations of the type

S,

A

i.

and are linear differential operators c on ­ Here taining the sought function u(x, t) in the form of a coef­ fic ient. If the c ondition (4) is s atisfied, then the spec ­ trum of the operator L does not depend on the time, and the asymptotic charac teristic s of its ei genfunctions c an eas ily be c alc ulated at any instant of time from their initial values . The rec onstruction of the function u(x, t) at an arbitrary Instant of time is realized by �olving the inverse sc attering problem for the operator L. As c an easily b e verified, Eq . )3) c an,be w ritten i n th e form (4), with the operators L and A taking U1e form • . I+p o 2 L=l [ -a + [uo Ou· ] ' X = -(5) 0 1 -- p ] Ox 1 - p1 iu." A = -p [ I O ] � + [ l • l 'i(l + p) 0 I &x' -iu, - 1 ,.l '/ ( l - p)

Ou I cit = §[11J

where generally speaking, is a nonlinear operator diffe rential in x), whic h c an be represented irl. the form (see C ' l ) 62

( - �') t - 2ii. + lq,) , u ( x, t) � yT,;"� exp {-4i !;'

ch (2� ( x - x, ) + 811st)

as

(6)

w ill b e whic h we shall also c all s olitons. A soliton, shown by us, is a stable formation . In the s elf- foc us ing problem, the soliton has Ui e meaning of a homogeneous waveguide c hannel inc lined 1 to the z axis at an angle 0 = - tan- 4�. In the s elf-

154 EXAC T T H E ORY

OF

T W O - D I M E N S I O N A L S E L F - F O C U S IN G

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modulation p roblem, the soliton is a single wave packet propagating without distortion of its envelope and with a veloc ity v = - 4(. The soliton is c haracterized by four crostants: 7J , { , x., and ,p. Unlike the KDV soliton, the constant 71 characterizing the amplitude and the c onstant ( which determines the veloc ity of the soliton are inde ­ pendent and can be chosen arbitrarily. The soliton (6) is the simplest representative of an extensive fan1ily of exact solutions of Eq. (3), which c an be expressed i n explicit form. I n the general case such a solution-we shall call this an N-soliton solution - de­ pends on 4N arbitrary c onstants : 'lj , (j , "oj , and 'l'j ; for non-c oinc iding (j this solution breaks up into indi­ vidual solitons if t - ± �. The N-soliton solution thus describes the process of scattering of N solitons by one another. In this sc attering, the amplitudes and veloci­ ties of the solitons remain unchanged, and only their center c oordinates x,, and phases ,p are altered. Just as in the case of KDV solitons, C • 1 only paired c olli­ sions of solitons contribute to this change. What is principally new c ompared with the KDV case is the possibility of formation of a bound state of a finite num­ ber of liolltons having identical velocities. In the sim­ plest case of two solitons, the bound state is a periodic ­ in-time solution of Eq. (3), and in the c ase of N solitons it is an arbitrarily-periodic solution charac terized by N periods. As applied to the self-focusing p roblem, the N -soliton solution desc ribes the intersection of N homo ­ geneous channels, and the bound state describes an os­ cillating "complic ated" c hannel. 1. THE DIRECT SCATTERING P ROBLEM

Let us assume that u(x, t) decreases sufficiently rapidlJ as I x I - �, and le! us examine the scattering problem for the operator L. To this end, we c onsider the system of equations

i,i,

=

•'i>, ,j,

=

{!: }

and make the change of variables -, z } u4 ,i,, - fl - p exp { -i -• 1-p

(7)

,i,, - l't + p exp { -1 -).z } v,. f - pl

Equation (7) can be rewritten in the form of the follow­ ing system: 9 = tu / yl - p',

t = Ap / (1 - p') .

(8 )

fn spite of the fact that this system is not self-adjoint, the problem of sc attering for this system is analogous in many respects to the problem of sc attering for the one-dimensional Sc hrodinger equation. Let v and w be solutions of the system (8) at t = t , and t., respectively. Then d ";j; (v, .,, - w,v,) •H!s, - t,) (v,w,

+ v,w, ) - 0.

In addition, if v i s a solution of the system (8) at t, = E 1 + iTJi , then v= -v.· -

{ ··· }

&atisfies the system (8) at t 2 =

t:

= t 1 - 117 1 •

(9)

63

We define, for real t = �. the Jost func tions ,p and ,p as solutions of Eq . (8) with asymptotic values q,- { �} ,-•• ,i,- { � } ,..

as :r - - oo,

as z -+- + oo.

The pair of solutions ,i, and ,/, forms a c omplete system of solutions, and therefore q, = a{s)iji + b(, - q,,¢, ) (x, ,l ,

>

and therefore a(( ) also admits of an analytic continua­ tion. It Is c lear that a ( t) - 1 as ! ; I -· 00 , Im t ;;, 0.

0 , t = tj , The points of the upper half-plane Im t j = I , . . • , N, at which a(t) = O, correspond to the ei­ genvalues of the problem (8). Here

>

••) = � In det IIA II ,

where IIAII is the mat rix of the system ( I 7'). For t he one - d imensional Schrodinger equation a similar for­ mula was obtained b y Kay and Moses. I 12 l To prove formula (22), we note that d - ln dct IIA II = dr

•

I d /;k, and the slowe r one x ln l(t m shifts backwards by an amount 11 )/ ( /;m - /;k) 1 - The total sollton shift is x In I (t m equal to the algebraic sum of Its shifts during the paired c ollisions, so that there Is no effect of multiparticle col­ lisions at all. The situation is the same with the phases.

"k

Here

(27a)

and a total change of phase

'10:

5. BOUND STATES AND MULTIPLE EIGENVALUES

The rate of divergence of a pair of solltons is pro ­ portional to the difference between the values o f the parameters (; at equal values of � . the solitons do not diverge, but form a bound state . Let us consider the bound state of N solitons, putting for simplicity � = 0. I Then c1 (t) = c j (0) exp ( - 4i11 } t). It i s seen directly from the general formula (22) that the bound state Is an arbi­ trarily periodic solution of Eq. (3), characterized in the general c ase by N frequenc ies Wj = 4 >J j . Actually the answer contains all the possible frequency differences, and therefore the bound state of two solitons is charac ­ terized only b y one frequency w = 4( 11 f - 11 !), an d con­ s titutes a periodic solution of Eq. (3). As 11z - 0 w e have w- 4 r,!, i.e., the period of the osc illations tends to a constant limit. A more detailed analysis shows, however, that the depth of the oscilla­ tions tends to zero, and the bound state goes over into a soliton with amplitude 7/ 1 • As r, 2 - 7/ 1 the zeroes of a(!;) c oalesce; the bound state then becomes aperiodic . The limiting state resulting from the coalescenc e of the zeroes and formation of a multiple zero of a(t), Is best investigated by starting directly from the Mar­ chenko equations (20). Let us c onsider the kernel F(x, t) of the Marchenko e quation for a system of two solitons with close values of the parameters : F ( z, t) = c, exp {isz + 4i1;'1) + c, exp (i (s + At).r + 4i (t + �) 't} a. exp (i;r + 4ic'I} [c, + c, + ic.As(.r + Bst) + . • .].

Going to the limit as A!; - 0, c2 A/; - y , c 1 + c 2 - a,., we obtain where

F ( r, t)

= [a. ( 1) + a, (t) z]e•",

a,(1) = a, ( O) ( I + By�t)e"'", a,(1) = a,(O) e'''"·

158 E X AC T

T H E O R Y O F T W O - D I M E N S I O N A L S E L F - F OC U S I N G

Solving the Marchenko equations, w e obtain after trans­ formations q ( x) =

µ

1 4T1A• a1•

I + l•• l ' l '- I ' I + 1 µ 1 ' 1 1, 1 ' • 2� (•, + 2 (z + 1 /2�) •,l_ ,.,. A= 2� 1 + 1 •, 1 ' 1 1• 1 ' - 2 + 1 i, l 'µal•

(28)

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An analysis of this expression shows that as I t i - 00 , it c onstitutes a superposition of two solitons wiU1 ampli­ tude 'I , the distance between which Increases with time like ln (4 q 2t). The solutions resulting from the c oales­ cenc e of a large number of simple zeroes are of simi­ lar form.

67

the solution tends asymptotically to this perturbed s oli­ ton. 7. QUASICLASSICAL APPROXIMATION

Let us c onsider for Eq. (3 ) initial conditions such that the quasiclassical approximation c an b e applied to the operator L. To this end, we eliminate the function v2 from the system (8). We have for v 1

(32) u, " + ( l q l ' + t') v, -- (v,' + i(u, ) q' / q = O. In the quasiclassical approximation, q'/q O, __ __ -f:: t,· - t. t. - t.· I/ii,•, / = 0.

1 V . I. Talanov , ZhETF Pis. Red. 2 , 223 (1965) (JETP Lett. 2 , 141 ( 1965)] . 2 P . L. Kelley, Phys. Rev. Lett, 15 , 1005 ( 1965). I . Bespalov, A. G, Litvak, and V. I. Talanov, II Vsesoyuznyi s!mpoz!um po nelinelnol optike (Second All-union Symposium on Nonlinear Optics), 1966, Col­ lection of Papers, N auka, 1968. • V . E . Zakharov , Dissertation, Institute of Nuclear Physics , Siberian Division, USSR Academy of Sci enc es, 1966. ' V. I. Bespalov and V. I. Talanov, ZhETF Pis . Red. 3, 47 1 ( 1966) (JETP Lett. 3, 307 ( 1966)] . ' A. G . Litvak and V. I. Talanov, Izv. Vuzov, Radlo­ fizika, 10, N o . 4, 539 ( 1967). ' P . D . Lax, Comm. on Pure and Applied Math. 2 1, 467 ( 1968). ' C . S, Gardner, J. Green, M. Kruskal, and R. Miura, Phys. Rev. Lett. 19, 109 5 ( 1967). ' V. E . Zakharov, Zh. Eksp, Teor. Flz . 60, 993 ( 1 9 7 1 ) [Sov. Phys • ..J ETP 33, 538 (19'1 1)] . "' V. A. Marchenko, Dokl. Akad. Nauk SSSR 104, 695 ( 1955), u L. D . Faddeev, Ibid. 12 1, 63 ( 1958) [3, 747 ( 1959)] . 12 J, Kay and H. E. Moses, Nuovo Clm. 3, 2 2 7 ( 1956). " C . S. Gardner, M. Kruskal, R. Miura, and N . Zabusky. J, Of Math. Phys, 9, 120 4 ( 1968); 11, 952 ( 1970),

' v.

Translated by J . G, Adashko 12

161 Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method V. E. Zakharov and A. V. Mikhallov

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L. D. Landau Institute for Tlwomlcal Physla. USSR Academy of S.), analytic in­ side the contour, such that on the contour

(4.2)

'1', (l.) '1',(l.) -G ('!.).

If the problem is posed this way the solution of the Riemann problem Is manifestly nonunique , since it al.­ lows the transformation w 1 - w ,g , w,-g,,i,, with arbi­ trary nonsingular matrix g. In order to remove the nonuniqueness we normalize the Riemann problem by fixing the value of one of the functions ,i, 1 or ,i, 2 at an arbitrary point. We call the normalization canonical if w,{"') = 1. We call the Riemann problem regular if det (w,, ,) " 0 in the domains of analyticity of the functions '1' 1 and 'i'2 , If a regular Riemann problem has a solution then this solution is unique. If the determinants of the functions '1!1 and +- 2 have finite numbers of zeroes i n their domains o f analyticity w e say that w e have a Rie ­ mann problem with zeroes. We show how to multiply the solutions of the system ( 1.2 ) by means of the solution of the Riemann problem. We first consider the case of a regular Riemann pro­ blem. Consider an arbitrarily given contour and on it a function G0(.\). We form the new function

r

(4.3 )

where 'i' 0 is a known particular solution o f th e system ( � .l ) , e . g . , the solution (4. 1).

w e differentiate the relation (4. 3) with respect to (. From ( 1 . 1 ) , (4 .2), and ( 4 . 3 ) it follows that

We now define the function U as

(4.4)

(4.5)

Eq. (4.5) shows that the function U can be analytically continued from the contour r onto the whole complex plane of >. . Since in their domains of analytic ity the functions w 1 and w, are nonsingular (the Riemann pro­ blem is regular), the singularities of the function U co­ incide with the singularities of Thus , U is a rat­ ional function with poles at the points ,\ = >.,.

u•.

1023

Sov. Phys. JETP 47(6), June 1978

Similarly one can define the function V : V - 'I' ,-' (- i'I',.+ V''I' , ) ~ ( i'I' ,,+ 'I' ,V') 'I',-•.

The poles of the function V coincide with these of The function '1t 1 is subject to the equations Setting 'I\ = w0

x, w e obtain

(4.6)

v•.

(4.7) (4.8)

Thus we have obtained a new solution of the system ( 1 . 1 ); consequently U and V satisfy the system ( 1.2).

Assuming that u• and v• are given by the expansions ( 1 . 3 ) , we obtain the following expressions for the com­ ponents of the expansions of the matrices U , V(n = 1 , 2 , 3 , . . . ):

v.-,.-•v:, . .

,.-'1', 1 ,-• .

u.--iq,- 1qoi+q,-'U.'q..

q, = '¥ 2 - 1 ( 00 ) ,

(4.9)

(4. 10)

The equations (4.9) show that a change in normaliza­ tion of the Riemann problem (a transition from w 1 to ,i, 1 g, where g is an arbitrary matrix function of � and �) leads to a gauge transformation of the system (1.2) with the matrix g. T o different gauges of the system ( 1.2) correspond different normalizations of the Rie ­ mann prob1em. Thus , the gauge leading to the problem of the principal chiral field ( 1 .9) (U 0 = V 0 = 0) corres ­ ponds to the canonical. normalization q 0 = w,("') = 1. The gauge leading to the U -V system ( 1 . 12) corresponds to the normalization We now go over to the consideration of our concrete system ( 1.9). As the contour r we pick the real axis -o> < X < ., and set G(l ) = l . We show that the above­ mentioned procedure of proliferation of solutions is equivalent to the traditional inverse scattering problem method.

The solution of the Riemann problem reduces to sol­ ving a singular integral equation. We set ( 'I' 1 - 1 + ...!... J p ;. �. l.' ) dl.' nt c J..-J.'+iO

'

(4. 11) (t l.' 'l' ,· • - 1 + ...!... J p •1. ) d/,.' • ni J.-).'-iO r Substituting into (4.4) we see that the quantity fl satis­ fies the equation (4.12)

(here f denotes the principal value of the integral), where the matrix T is a Rayleigh transform of the ma­ trix G: T- ( 1 -G) (l+G) - ' - '1',T,'1',·', T,-(1-G,) ( t+G,J - • .

V. E. Zakharov and A. V. Mikha'flov

1023

168 As can be seen from the equations (4 .9) , the quanti­ ties A and B do not change if, when the matrix '1- 1 is multiplied from the left and the matrix '1< 2 is multiplied from the right by arbitrary matrices , the matrix G 0 which commutes with A0 and B 0 is multiplied from the left and the rigjlt by the same matrices , respectively. Thus , there is an indeterminacy in the definition of the matrix T. In order to lift it we require that the diagonal elements of the matrix T vanish. The remainder of the discussion will be given for the special case when A0 and B0 are Hermitian numerical (diagonal) matrices . In this case

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We order the real numbers a, , putting As � - ± ,. the matrix T oscillates rapidly: T.(t, �. ·).) -TN().) exp [-l(a,-a,) V (H!)+J( b,-b,) �/().-1) ], therefore the asymptotic values of the matrix p as f - ,. are also rapidly oscillating: p-p' ( I,

�- •>.

p.,•-p.�, (1.) exp[-1(•,-•,) t/(H!) +i(b,-b,) �/ (1.- 1) ] .

(4.13 )

The matrices � can be found by explicitly solving the equation (4.12). For this one must make use of the formula

...

Um P� - - buign H (1.) ,

111--

).

(4.14)

which transforms (4. 1 1 ) into an easily integrable sys ­ tem of singular integral equations.

We denote by '1-* the limiting values of the function '1- 1 as � - ± "' . The matrices ,i,• are defined by the equa­ tions (4. 1 1 ) where p* from (4.13) are substituted for p. It follows from (4. 1 1 ) that '1-* are triangular matrices : for the matrix '1- • all elements below the diagonal van­ ish , and for the matrix ,i,- all elements above the diag­ onal vanish. It is obvious that 'Y•-exp [ - l (A,l;+B,�) ]S' (l.) exp [ i(A,t+B,�) ], where p•(;>.) are triangular matrices . The matrix S = ,i, • -•,i,· is the usual scattering matrix. Representing it in the form of a product of triangular matrices one can determine W" , and reconstruction of T and G in terms of these shows the equivalence of our approach to the usual inverse scattering problem method (cf. , e . g. , C"l ). From the condition G ( ± l ) = 1 it follows that T( l ) • T( -1 ) = 0 , therefore '1< 1(>- = ± 1 ) - 1 as � - ± "' and con­ sequently A - A0 , B - B0 • Thus , the matrices A00 and B 0 are indeed the asymptotic values of the matrices A and B . We also note that the procedure desc ribed above al ­ lows one to obtain solutions not only for the system ( 1 .9), but also directly for the chiral field g. Indeed, 1024

Sov. Phys. JETP 4716) . June 1 978

(4 . 1 5) Therefore the determination of the function >It automat­ ically solves the problem of calculating g. § 5. SOL ITON SOLUTIONS The procedure of proliferation of solutions by means of the regular Riemann problem does not make it pos ­ sible to determine all the solutions of the system ( 1 .2). For a complete description even of this class of solu ­ tions , which asymptotically, as x - ± ,. go over into A 0 , B 0 , it is necessary to resort to a Riemann problem with z.eroes . Of particular interest is a special Rie ­ mann problem with zeroes , for which G • 1 . The solu ­ tions of the system ( 1 .2) obtained by means of this Riemann problem will be called soliton solutions. In the case G = 1 the function '1l. = µ0) by

M

*.- = >-.).

In the general case of a Riemann problem with zer ­ oes, in addition to the function G , one must prescribe the positions of the zeroes of the function + 1 (>.1 , :1. 2 , • • . , >-.), the positions of the zeroes of the function '112( µ1 , 1-'2 , • • • , µ 0) , and also the two collections of sub­ spaces

This problem can be reduced to the regular Riemann problem by means of successive annihilation of the zer ­ oes . We represent the solution in the form

where "'l'' , ls a solution of the type (5.1) with the projector P 1 constructed according to the subspaces M1 , h1 , at the points >. = >- 1 , >-= µ 1 • It Is obvious that the functions i' 1 , w 2 no longer have zeroes at the points :!. = :1. 1 , :1. = i,1 , respectively. At the points >-r (i = 2 , . . • , n )

w�"

ii.-Im. i' 1-'¥! 1 l-.i. 1 .i'"

k1-Ker ij, 1-'1'! 1,- �,,

1

0

r:-w: , cv:•' . 1

For G = l we obtain G= 1, therefore the soliton pro1025

Sov. Phys. JETP 47(61, June 1978

Going over to the c:ase under consideration of a prin­ cipal chiral field , we choose the matrix w in the form O '!', (),) - exp

.

.

{ >+ �. t J B, (�') d�'} .

( 5.8)

In terms of g the one-soliton solution has the form g- ( I +

A, - X,

--,;;--

P, ) 'l',(O) .

(5.9)

We limit ourselves to a consideration of the groups SU(2) and SU(3). In these cases we can confine our ­ selves to projectors with one-dimensional range. Such a projector is characterized by a single vector c 1 :

(5. 1 0)

(c;•• is an arbitrary constant complex vector), and has the form

(5. 1 1 )

I t i s obvious that the vector c I s defined only accurate to multiplication by a complex number.

In the case of the group SU(2) the soliton solution de­ pends on two complex parameters (>. 1 = >./ -1- ix : , c,' " I c \ " ) and represents a soliton proper , i.e . , a solitary wave whkh has constant velocity at constant A.( �) and Bo( ij) v-

(b.-ba) l t+A.1 1 1- (0:1 -a:1 ) 1 1-A, f • (b,- b,) 1 1 +i..1 l•+ (a.-a.) l t -,.,1 1

One can calculate the energy and momentum of the soiiton from ( 2 . 24) and ( 2 . 25) and obtain an expression for its mass For the soliton the matrix P II has the form ch y P - [ 0 the soliton is an ordinary par ­ ticle, and for (a1 - a2 ) (b 1 - b2) < 0 i t i s a tachyon.

In the case of the group SU(3) the soliton solution is characterized in addition to the number >. 1 by the inte ­ ger vector (c! 1 > , c!1 > , c�1 >). If one of the components of this vector vanishes we obtain the simple soliton solu­ tion; in this case there are evidently three types of sol­ !ton. Two of these solitons (for cl 1 1 = 0 1111d c�" = 0) will be called simple solitons and the third (c\" = 0) will be called composite . It follows from the mass formula ( 5 . 1 2) that for the case of normal solitons the mass of V. E. Zakharov and A. V. M ikhailov

1025

170 From the relations ( 1 . 1 ) and the fundamental formulas ( 1 . 1 0) it follows that one can select a fundamental ma­ trix of solutions '11, satisfying the conditions

the composite soliton is larger than the sum of the masses of the simple ones: (o,-o,) ( b,-b,) > (a,-a,) (b,-b,) ;I- (a,-a,) ( b,-b,)

(6.3)

for a 1 > a2 > a 3 and b 1 > b 1 > b 3 • Thus , decay of the composite soliton Into simple sol ­ itons Is possible. In the case of the general formula­ tion (all three cl'' "' O) the soliton solution desc ribes in­ deed such a decay. From an analysis of the general solution (5.10), ( 5 . 1 1 ) one can easily deduce that as

t-

- GO

'l',(J.-') -1,'l', (J.),

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As t - + "'

P.,-o, I Pu J - t/2 ch ()./'cxu'z-)./'a.111 t+lo J c,lc:I ) , 1 Pu l - t/2 cb (Ai''ci,,'z-)./'au 1t+ln ! c,/c-, J ) .

Thus , the numbers c.fc, , c,!c, characterize the final coordinates of the decay products. The equations (2.2) are time-reversible, therefore there exists the inverse process of fusion of simple solitons into a composite one. The processes are contained in the soliton solu­ tion In which the image (range) of the complementary projector is a one-dimensional subspace. The solutions of the Riemann problem with G = l with two zeroes of the functions w 1 , w 2 describe collisions of soliton solutions, in particular, collisions of simple solitons as well as processes of "induced" decay of composite solitons in collisions with simple ones . The corresponding formulas can be easily derived but we do not list them here since they are rather clumsy. An analysis of soliton solutions for the groups 51,'(N) is not trivial. Already for N = 4 three exist composite solitons consisting of several (two, three) simple ones, and the processes of nontrivial interaction of solitons are quite varied. The corresponding analysis will be published elsewhere.

§ 6. I NTEGRATION OF CHIRAL FIELDS ON GRASSMANN MANI FOLDS

M.,

W e first note that Eqs . ( 2 . 3) and t h e condition g' = 1 imply the relations Bg+gB-0.

( 6. 1 )

and also , taking into account g = 1 - 2 P , the relations A --2i[PP,J , 1 026

B--2i[PP,J .

Sov. Phy� JETP 47{6) , June 1 978

r,-'l' (O),

(6.4)

is also valid for the original prescribed solution of the system ( 1 . 1) , the functions '1> 1 and w 2 satisfy the invo­ lution 'I', ( ).-' ) -g 'I', ( - r, 'I' ,(J.) r.

(6. 5)

which implies (6.6) it follows that Eq. (6.6) represents a condition on the "dressing" matrix G 0(J.). From Eqs . (6 .2) follow the relations PA+AP -A,

P.A.,+A.P,-A,,

PB+BP-B,

PJJ,+B.P,-B�

(6. 7)

Here P 0 ls the "bare" projector , g0 = l - 2P0 , and from (6.5) follow the formulas for "dressing'' the bare pro­ jector: P-f,P,t,-•.

/, - '1', ( I ) .

(6.6)

Therefore for the complete solution it ls necessary to establish the form of P O and A 0 • Remembering that A0 and B 0 are diagonal, it follows from (6.7) that

(6.9)

We now consider the procedure of integration of fields which appear as a result of reduction. We restrict our ­ selves to the reduction g• = 1 , considered in S e c . 3 for the principal chiral field on the group SU (N), and lead ­ ing to chiral fields on complex Grassmann manifolds. In order to carry out the integration it is necessary to clarify how the reduction reflects the data of the Rie ­ mann problem, i . e . , what restrictions does it impose on the matrices A0 , B0 , G (,I,), on the position of the zer­ oes of the functions + 1 and W" 2 , and on the structure of the subspaces 1 and

Ag+gA -0.

Ii.I =

Since the equation

Pn-0. Pu-0, I Pu l - t/2ch ('-.''a1/:r-'-/'e1u1t+lo lc.fc, I ) .

N

In 1>ther words , the zeroes of the function '1> 1 are dis­ posed symmetrically with respect to inversion in the unit circle and can be divided into simple zeroes for 1 and double zeroes . In the theory or the SG equa­ tion this fact has been known for a long time.hi Accor­ dingly there appear simple and double soliton solutions.

(6.2)

We order the squares of the numbers a, in decreasing order. The diagonal of the matrix A' splits into seg­ ments with equal squares of the eigenvalues. In each of these segments we let the negative eigenvalues fol­ low the positive ones. It follows from (6.9) that the matrix P O has block-diagonal form , and in each block the matrix P • reduces to a matrix P: or the form P ' - � [ I R, ' 2 R,• I

1

'

(6. 1 0)

where R, is a unitary matrix. From the equations ( 6.7) it is easy to establish that the matrix B 0 , which commutes with A0 must also de­ compose , for the above ordering of the numbers a0 , into diagonal matrices in accordance with the structure of the matrix A 0 • Finally , the k - th block of the ma­ trices A 0 and B 0 is characterized by the functions a,W and b.( �). Substifuting (6.10) into (6.2) we obtain V. E. Zakharov and A. V. Mikhailov

1026

171 the ..v, = 0

( v = y,),

Y , + Y ..,. - 6y 'y, + 6>..y, = 0 (y = Y.l,

(47)

The second of these Is essentially the modified KdV equation. Miura' s discovery that this equation is trans­ formable to KdV was one of the earliest results used in the analytical treatment of solitons. " We shall now briefly review the relationship of the prolongation structure to the known solution methods for the KdV equation. The most significant Pfaffian of the set in Eq . (45) is w 8 , defining the pseudopotential Y a for which we henceforth use the symbol y . On a solution manifold of the prolonged ideal, we will have from w. = o Y, = - ( 2u + y' - >.. ) , Y , = 4 [(u H )( 2u + y' - >.. ) + ½P - zy ) . ( 48) The first of these is a Rlccatl equation linearizable by the substitution J. Math. Phys.• Vol. 16, No. 1, January 1 975

These are linear in ,p and � and constitute the Plaffian differential form representation of the first-order In ­ verse scattering equations. Both Eqs . (50) and (55) have been used to develop linear techniques for solving the initial value problem for the KdV equation. '- • Another technique fo r generating analytic solutions can be deduced from the prolongation structure. Suppose that one particular solution of the prolonged ideal {a 1 , wJ Is known . We may inquire whether another solution , say u ' , of the KdV equation can be written as an alge ­ braic function of all the variables In the space of the prolonged Ideal; i . e . , u ' = u'(u, z , P, y 1 ) , The answer can be found by substituting this ansatz inio the set of fo rms a i ' = du ' A dt - z ' dx /\ di , a ,' = dz 'A dt - P' dx A dl, a , ' = - du' I\ dx + dp'A di + 12u'z' dxA dl,

(56)

and, as usual, demanding that these b e in the ring of the prolonged Ideal. After a tedious but straightforward calculation, the re­ sult Is that (57)

u ' = - u - y2 + A is always another solution.

Since u � o satisfies KdV, u0 ' = - y• +A must also be a H.D. Wahlquist and F.B. Estabrook

182 solution. From E q . (48) for this case Y, = - (y 2 - X ) ,

1 12 2 u ' = u + [ (x , - x 1 ) (u, - u, ) ]/[ (x , - u, - u) ' 12 - (x 1 - u1 - u) ] ,

(58)

y 1 = 4X (y 2 - X ) = - 4X y,,

the regular Integral of these being 1 2 2 Y = X 1 1 tanh [ x 1 (x - x, - 4x t) ] .

(59)

The result for u,,• i s the regular 1 -soliton solution. n equation like Eq. (57) is c learly equivalent In gen ­ eral to a B"/tcklund transformation. Simply solving for y and substituting into Eq. (48 ) will produce the usual form of the Backlund transformation. In fact, for the KdV equation, 1 2 it is simpler to use the potential function w . To see this , we use the Pfaffian to get A

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w,

(60)

so that Eq. (57) can be written (61) where we have used Eq. (48) to write the second equal­ ity . Integrating and absorbing the Integration constant in the potentials , we have (62)

y = w - w ', so that Eq. (57) can finally be written as - w t ' - wx = ll' + u = X - (w 1 - w) 2 .

(63)

With x = k', this is precisely the space part of the Back­ lund transformation presented In Ref . 12 . By using Eq. (57) and Eq. (62 ) , the y , equation in Eq. (48) can also be r.ewritten to give (64) w , ' + w 1 = 4 (1t ' 2 + u 'u + 112 ) + 2 (w ' - w) (z ' - z), which expresses the other half of the Backlund tran s ­ formation of Ref. 12 i n a symmetric fo r m . An extension of the solution gene rating technique lead­ ing to Eq. (57) c an be used to derive directly the hie r ­ archy o f solutions which are known to result from r e ­ cursive application of the Backlund transformation. So far we have treated the Pfaffian w, as a single 1 -form, but we can also consider it to represent a 1 -parameter infinity of independent Pfaffians , parametrized by X . That is , for any given solution {11 , z , p} of the KdV equa­ tion , "'• defines a 1 -parameter family of pseudopoten­ tials , y (x , t , x ) , which are linearly independent functions. This suggests that we attempt to find more general solu ­ tions than Eq. (57) by entering the fo rms of Eq. (56) with the ansatz 11 ' = 11 ' (11 , _v (x , ) , y (x 2 )),

(65)

fo r exampl e . Another tedious calculation (we certainly suspect there mus.t be a neater way to obtain these r e ­ sults) shows that II

,

= II + (x . - x , )[y'(x .) - x . - .v'(,,) "- x ,] [y (X , ) - y (X 1 ) J'

(66)

is indeed always another solution. Since we know from Eq. (57) that 11, = - 11 - y'(x . ) + x 1 , 11, = - 11 - .v'(x ,) + x ,

(67 )

a r e solutions , we can eliminate the .v ' s from E q . (66) to obtain the superposition princ iple , or recursion relation, J. Math. Phys., Vol. 16, No. 1, January 1 9 7 5

(68) which can then b e used to generate the Backlund hierar­ chy . , A gain, one has much simpler expressions if this analysis is carried out for the potential w as in Ref. 1 2 , and we shall not pursue i t further her e ,

VI I . CONCLUSION

The formulation of nonlinear evolution equations in terms of ideals of differential forms leads in a very clear fashion to the derivation of potential functions and conservation theorems. The natural and Important gen­ eralization to pseudopotential functions results in discovery of new conservation theorems; and even more Importantly, pseudopotentials appear to be the unifying concept for understanding the relations between diverse known solution techniques (Backlund transformation, as ­ sociated Inverse scattering problems ) . The discussion here has been made concrete by reference throughout to the treatment of the Korteweg--de Vries equation. The systematic search for the pseudopotentials of a closed set of forms leads to consideration of an associ­ ated overdetermined set of first-o rder nonlinear partial differential equations wtilch we denote a prolongation structure. The prolongation structure is integrable pre ­ cisely b ecause It has the form of (a subset of) the com ­ mutation relations o f a Lie group. W e have not in the present paper been able to exploit some of the deeper known results fo r Lie groups systematically, but it seems c lear that extremely powerful mathematical tech ­ niques are now at hand. For example, the search fo r linear , or matrix, representations of the group (or structure) can be undertaken In a completely algorith ­ mic way . This results in representations of the vector generators of the form X, = a 1 1.y 1 a/ay•, where the a 1 1, are constants; and th e pseudopotentials y• thus found will enter the prolongation forms w linearly . The vari­ ables ,P and !/J and forms w, and w1 0 of Eq. (55) thus are seen to belong to a two-dimensional matrix repres.enta­ tlon of the prolongation structure for the KdV equation, Eq. (37 ) . T h e inverse scattering technique h a s been shown to provide a linear method of solving the Initial value prob ­ lem for many nonlinear evolution equations . One of the primary obstacles to extending the method Is the discov ­ ery of the appropriate linear equations or operator s . T h e search f o r linear representations of t h e prolongation structure would appear to provide a straightforward ap ­ proach to this problem which does not require ad ho c restrictions . It also suggests generalizations; for In ­ stance , an intriguing possible generalization of the known method of inverse scattering may result from higher -dimensional matrix representations . At present we c an only speculate that for l inear representations of sufficiently high dimension, the pseudopotentials gene r ­ ated will provide s o m e ultimate linearization of the o r i ­ ginal problem similar to that achieved for KdV. A final comment is to remark on the close connection of Lie groups and generalized Fourier analysis; the natural ex­ pression of the superposition rules intuitively felt to underlie the soliton phenomenon may well be found in the use of the Invariant functions dual to the vector genera­ tors of the prolongation structure. H . 0 . Wahlquist and F.8. Estabrook

6

183 A . R . Forsythe , Theory of Differential Equations a:>ove r , N e w York, 1 959) , V o L IV , C hap, XXI, pp• . 425-55. 8M . J. Ablowitz , D.J. Kaup, A , C . Newe l l , and H . Segur , Phys . Rev. Lett. 31 , 1 2 5 (1 9 73) . 9H . F landers, Differential Form s (Academic , New York, 1 963) , pp. 1 - 7 3 ; Y. Cboquet-Bruhat, Geometrie differentielle et system es exteYieurs (Dunod, Parts , 1 968) , pp. 54 - 84 . 1 °E . Carta n , Les syst€!'tn es differentielles exterieurs et leurs applications geometrique (Hermann, Par i s , 1 94 5) ; W . Slebodz inski, Exterior Fonn s and Applications (Polish Sci­ entific Publishe r s , Warsaw, 1 9 7 0) ; B . K. Harrison a nd F . B . E s tabrook, J, M a t h . Phy s . 1 2 , 6 5 3 (1 971). 1 1R . M . M iura, J . Math. Phys. 9, 1 202 (1 968) . 1Z H , D. Wahlquist and F . B. Estabrook, Phys, Rev. Le tt . 31 , 1 386 (1 9 73).

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"'This paper presents the results of one phase of research car­ ried out at the Jet Propulsion LabQratory, California Institute of Technology, under Contract No. NAS7- 1 0 0 , sponsored by the National Aeronautics a nd Space Administration. 1 R . M . M iura , C , S. Gardner, and M . D . Kruska l, J. Math. Phys . 9, 1 2 04 (1 968) . 2 P. Lax, Comm. on Pure and Appl. Math. 21 , 467 (1 968). 3 A . C . Scott, F . Y . F . Chu, and D.W. Mclaughlin, Proc. IEEE 61 , 1443 (1 973). 4 c . s. Gardner, J. M . Green, M , D. Kruskal, and R , M . M iura , Phys. Rev. Lett. 19, 1 095 (1 967) ; N , J. Zabusky, Phys . Rev. 1 68, 124 (1 968) . 5G . L. Lamb, Jr. , Rev. Mod. Phy s . 43, 99 (1 9 7 1 ) . 6F . B. Estabrook a n d H . D. Wahlqu ist, t o be published. 7 L. P. E isenhart, A Treatise on the Differential Geometry of Curves a nd Suifaces (Dover, New York, 1 960) , pp. 280-90;

J. Math. Phys., Vol. 16, No. 1, January 1 975

H.O. Wahlquist and F.B. Estabrook

184 Prolongation structures of nonlinear evolution equations. II F. B. Estabrook and H. D. Wahlquist

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103

(Received 21 October 1 975)

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The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schrodinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is opcn-i.e., does not terminate as a set of structure relations of a finite-dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Backlund transformation. and derive the double Bii.cklund transformation, This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.

I. I NTRODUCTION

In the first of these papers1 we introduced a geometric method for finding a hie rarc hy of potentials and pseudo­ potentials (denoted ,_ ; ) for sets of nonlinear partial dif­ ferential equations with two independent variables . By " geometric" is meant the systematic use of the formal­ ism of differential geometry, in particulac the represen­ tation of the partial differential equations as a closed ideal of differential forms . To find the potentials and pseudopotentials, it turns out that one must solvefind representation of- a Lie structure of auxiliary vec­ tor fields . We have denoted this the "prolongation struc­ ture, " or PS . The vector fields, say X1 , X2 , • • • , and their commutation relations are defined in the space of " prolongation" variables .'\' f , while the defining equations for the potentials, or pseudopotentials, appear as 1 forms i n the space o f all variables- the original, o r primitive variables (independent variables x and t, say, and dependent variables z•) plus the .,· ' , w i = rly i + p i ( z- A ' y i ) dx + G i (z A ' y i ) dt .

The F' and G' depend on the ,, ' by being linear in the components X i =X - d_,, i of the X ' s . These 1 - forms each lead to a conse rvation law, since dw 1 is required to be in the p rolon1;ed ideal: the ideal representing the original partial differential set, to which is also adjoined the w i . Consequently, for any solution manifold of the ideal, the w i are exact and Stokes · theorem says that ef> w 1 , con­ fined to that manifold, vanishes. This hierarchy of high­ er conservation laws is essentially different from those previously recognized [for equations such as the Korteweg-de Vries (KdV) and nonlinear Schrodinge r ], which involved repeated partial derivatives .

boundary value problem as the method of inverse scat­ tering. More general linear representations surely can be found, and their application remains to be investigated .

The PS is further invariant under less trivial trans­ formations in the sp�ce of the y 1 : Linear superpositions of the X's can be found which keep their commutation relations invariant. This is a group of automorphisms which, in the KdV case, at least, turns out to be a 2parameter group isomorphic to an invariance group of the initial KdV equation, before prolongation. These automorphis ms imply that some of the pseudopotentials )' i can in fact constitute a continuous family. For the case of a two-dimensional representation of the KdV, this degenerates to a 1 - parameter family of pseudo­ potentials, the automorphism paramete r appearing as the so- called eigenvalue in the associated linear problem. The existence of prolongation structureg also seems to be c losely related to the possibility of solution meth­ ods by Backlund transfor mation. We have found these to be derivable as discrete invariance transformations of the prolonged differential ideal, when a true pseudo­ potential exists . They again involve the automorphism parameters explicitly .

Coordinate transfor mation i n the space o f the ,v' leaves the abstract algebraic re lations- the PS- among the X's unchanged: It is equivalent to changing the X's in­ dividually by local similarity transformation, while forming trivial superpositions of the pseudopotentials and associated conservation laws. It is important, how­ ever, that coordinates must exist for which the X ' s are homogeneous first degree in the l - the finding of these is equivalent to finding matrix (or linear) representa­ tions of the X ' s , a standard procedure in the discussion of Lie groups . It is in such coordinate frames that one finds the linear auxiliary partial differential equations for pseudopotentials already known in application to the

We speculate that the existence of a nontrivial PS may be a useful defining algebraic characteristic for the en­ tire class of nonlinear equations now under intensive study in many contexts, roughly, those equations having solutions with nonlinear superposition properties, such as "solitons . " By nontrivial we mean that the PS is non­ Abelian, for then true pseudopotentials exist-viz . , those y i whose p i and G i cannot by coordinate trans­ formation be made independent of y 1 • Three now- classic s oliton equations are the Korteweg-de Vries, sine­ Gordon and nonlinear Schrodinger . We have given a PS for the first, 1 involving seven vectors X 1 and an ap­ parently open set derived from their commutators . A nontrivial PS also is readily derivable for the sine­ Gordon equation. In the present paper we consider the PS of the third, which again is nontrivial and seems almost certainly to be open . As with the first two equa­ tions, this PS has useful low dimensional representa­ tions, and allows Blicklund transfor mations. We intro­ duce the use of multiple pseudopotentials, belonging to

1 293

Copyright © 1 976 American I nstitute of Physics

Journal of Mathematical Physics, Vol. 1 7, No. 7, July 1 976

1 293

185 different values of the automorphis m parameter , for the derivation of the multiple B:icklund transformation. The existence of simple pseudopotentials has been carefully discussed by Corones 2 ; he also derives a pro­ longation structure for the Hirota equation. Prolonga­

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tion structures for the Boussinesq equation and the non­ linear wave-envelope equations have been obtained by

Morris . 3 Most recently Morris has devised an algorithm for extending an equation of evolution in one spatial di­ mension, which has a PS, to derive a related evolution equation in two spatial dimensions also having a PS . • In this way h e has systematically discussed both the ex­ tension of the B oussinesq equation to the Kadomtsev­ Petviashvili-Dryuma equation, and the extension of an interesting new nonlinear system having a PS to the gen­ eralization of the nonlinear Schrodinger equation of Ablowitz and Haberman. T he nonexistence of a non-Abelian PS does not of c ourse mean that the systematic methods of differential geometry cannot then be useful for treating other classes of nonlinear differential syste ms . Many properties such as invariance operations, variational formulations, cha­ racteristics, and special solution sets such as simi­ larity solutions, can in our opinion best be unde rstood in this way . But the search for a PS, and the conse­ quent conservation laws and potentials, does seem to be a first operation to try on a " well-formulated" set of partial differential equations-i. e , , a set belonging to the regular integral manifolds (of maximum dimension) of a closed differential ideal. In the Appendix it is shown how such a search differentiates the KdV and modified KdV equations from those with higher order nonlineari­ ties- the latter having only an Abelian PS .

11. THE PS OF THE NON L I N EAR SCH RODI NG E R EQUATION

T he cubically nonlinear Schrodinger equation has been treated extensively in the recent literature of non­ linear wave equations, cf. Whitham5 and Scott et al . 6 With one sign (< = - 1 in the following) of the nonlinear term, the equation desc ribes stationary two-dimension­ al self-focusing of plane wave trains in nonlinear media (cf. Zakharov and Shabat, 7 Hirota, 8 and references therein), or the time dependent phenomenon of s elf­ modulation (leading to the so- called " envelope" s olitons) . With the other sign (< = + 1) the solutions have much greater stability; nevertheless, s o- called envelopehole solutions, etc . , have been studied. 9 • 1 0 The local analytical method expounded in the present paper is ap­ plicable in either case. We present the prolongation structure, and construct some of the resulting new po­ tential and pseudopotential conservation laws, and the inverse scattering equations. We use the pseudopotential to find the single Backlund transformation (independent­ ly given by Lamb1 1 ) and are able also to find a double Backlund transformation, or theorem of permutabi1-ity. The equation (1) ilf1: + ¢u - 1 E1f;lj!2 = 0,

where E = ± 1 and the bar denotes complex conjugate, can be expressed as the set of differential forms c,1 = d� A d/ + s dx ., dt c,2 = id,j; A dx + d U, dt + ½ ,�,j;2 dx I\ di, 1 294

J. Math. Phys. , Vol. 1 7 , No. 7, July 1 976

(2)

together with the complex conjugates 01 and 02 • These four 2-forms live in a six-dimensional space of primi­ tive variables iii, If;, t 1, x, f , and have two-dimension­ al integral manifolds which are the solutions of ( 1 ) . In the prolonged space of variables Ip, "'Ji, t 1, x, t, y\ we search for Pfaffians of the form w' = dy' + F" ( ,ti, �'

s, 1, y ) dx - iG'( I/J, ¢, s, 1, y ) dt 1

1

(3 )

(the factor - i in the dt term proves convenient later ) , which a r e such that dwk a r e in t h e prolonged ideal a1 , a2 , £11 , 02 , w" . Following the same procedures as in Sec . I, we find overdetermined partial differential equa­ tions requiring a decomposition into polynomials in the primitive variables F' = ½ [xt + �"Jix) - 2 1/!Zl - 2�.zl ] , c• = ½ [ ( s� - 1,pJx; - 2 sz: + 21.z: + Y'; + 1/1,PY'; +

(4)

,pz; - �� ],

where the vectors Z ,,. (v i ) are complex, while X,,,. (v i ) are real and Ym(/ ) pure imaginary,

(5)

The remaining partial differential equations involve only dependency on the y 1 and are all of commutator form, and so they define the prolongation structure : [x1 , x, ] = [x, , Y2 ] = [x, , Y1 ] = [x, , z, ] = [z 1 , z, ] = o , (6)

½[x,, z,] + [ Y,, z,J - EZ 1 = o , [x1 , z,J + 2[ Y,, z, ] = o,

[x1 , Y1 ] + [x2 , Y2 ] + 2[zi , .z, J - 2 [.z1 , z,] = o, together with complex c onjugates.

A number of further relations are derivable using the Jacobi identity. W e find that X2 must commute with all these vectors, and also that (X1 , Y1 ] = [ Y1 , Y2 ] = 0,

(7)

from which follow [ Yi , Z 1 ] = EZ i ,

[ Yi , Z, ) = EZ 2 •

(8)

At this point we have all the noncomplex vectors {Xm , Ym} constituting an Abelian subalgebra. Also, the set { Y1 , Zi , Z1} satisfies [ Y1 , Z, l = EZ 1 ,

[ Zi , Z1 ] = ½ Y1 ,

(9)

which is the algebra of the 3- parameter rotation group in complex notation. If we define a new complex generator

z, • [x1 , z 2 ] ,

(10)

we can split up one of the re maining relations in Eq. (6) to get [ Y, , z, J = - ½z, ,

(11)

as well as [Z 1 , Z3 ] = [Z 2 , Z3 ] = 0, [ Yi , Z3 ] = J '(M,N ). Such a map Xautomatically preserves as and a4 as x•a, = (Yi - Z1 f, )a; , (4.8) X*a4 = ( Y, Z2 - Z, Y, )a; . (4.9) The invariance of a 1 , a, , and a3 requires only that + z, Y, Y, + Y, Y, = z, (4. 10)

z,

Y,

Z;

z,

+ Y, Z; = O = Z,

z,'

Y1 + z, Y2 •

(4. 1 1) From these equations one easily shows that the functions Y, Y,Z, Zare linear functions of their respective arguments. We write (4. 1 2) and (4. 1 3) where A and B are constant matrices and C and D constant vectors. The C and D correspond to translation invariance, and we set them both to zero. Equations (4. 1 0), (4. I I) can be expressed neatly in terms of A and B as

AB T = µI, where µ = Yi Y, + Y, Y, is a constant. The action on the generations a, is given by x•a, = µa; ,

(4. 1 4)

i = 1 ,2,3, (4. 1 5) = detAa; , X*a, = detBa; . (4. 1 6) We denote the coordinate symmetry expressed by (4. 1 2)-(4. 14) by X (A ,B ). The mapping X(A ,B )-.J 0(M,N ) --,J 0(M,N ) defined by

X*a•

X(A,B ):

260

[C). G) (�). (::)]

J. Math. Phys., Vol. 2 1 , No. 2, February 1 980

can be lifted to a unique transformation

X(A ,B )-.J '(M,N)-->J '(M,N ). As both iiand X(A ,B ) act on

the same space J 0(M,N), we can stay at the J °(M,N ) level and construct a family of"Biicklund maps" B(A,B ) from B by defining 1 (A ,B ) BX(A,B ) = (id M X B (A,B )) . B(A,B ) = (4. 1 8) One readily determines that B (A,B ):N-->N is given by

x-

B (A,B ):(,f,',,f?,,f,',,t,•,,f,')-->((,t,') - ' , 4

detB ,f,

µ (,f,')"

__it;,,

detB µ (,f, )

L)

detA _j!_ _ detA . (4. 1 9) µ (,f,')" µ (,f,') ' 2 From (4. 1 9) we have detA detB = µ , and we see that B (A,B ) simply results from B by a scaling symmetry of the type in Eq. (2. 1 8). As we determined all possible B, this had to happen, but it is interesting to see how it has come about. ACKNOWLEDGMENTS

It should like to thank my colleague Roger Dodd for stimulating discussions and also acknowledge the support of the Department of Mathematics, Oregon State University, where the author was visiting Professor for the academic year 1 977-78.

'R.M. Miura, Ed. "B8cklund Transformations, Lecture Notes in Math­ ematics S I S (Springer-Verlag Berlin, 1 976). 'H.D. Wahlquist and FB Estabrook, Phys. Rev. Lett. 31, 1 386-90 ( 1 973). 'H.D. Wahlquist and F.B. Estabrook. J. Math. Phys. 16. 1 -7 ( 1 975). 'J.P. Corones, J . Math. Phys. 18, 1 63-{)4 ( 1977). sH.C. Morris, "B8cklund transformations and the sine-Gordon e.quation," in The 1 976 Ames Research Center (NASA ) Conference on the Geometric Theory o[Non•Linear Waves (Math. Sci. Press., Brookline. Mass. 1 977). 6H. Chen. Phys. Rev. Lett 33, 925-28 ( 1 974). 7M . Crampin, F.A.E. Pirani, and D.C. Robinson, Lett. Math. Phys. 2, I 51 9 ( 1977). 1F.A.E. Pirani, D.C. Robinson, and W.F. Shadwick, "Local jet bundle fonnulation ofBiicklund transfonnations," preprint, Kings College, Lon• don, 1 978. 9P.F.J. Dhooghe, "Jet bundles and Backlund transformations" (in press, 1 978). 10R. Hennann, Vector Bundles in Mathematical Physics Benjamin, New York. 1970), Vol. I. 1 1 R. Hennann, Geometry Physics and Systems Dekker, New York, 1 973). 1 2R. Hermann, The Geometry of Nonlinear Equations, Biicklun.d Trans/or• motions and Solitons, Par/ A, lnterdi�iplinary Math., Vol. XU Math Sci. Press, Brookline, Mass., 1 976). i.1 R.K. Dodd, preprint, Trinity College, September 1 978 "C.N. Yang, Phys Rev. Lett. 38. 1 377-79 ( 1 977). "G. 't Hooft, Phys. Rev Lett. 37, 8-1 1 ( 1 976). H•L.v. Ovsjannikov, "Properties of Differential Equations," translated by G.W. Bluman (unpublished). 17E.F. Corrigan, D.B. Fairlie, R.G. Yales, and P. Goddard, Commun Math. Phys 58, 223-40 ( 1 978).

Hedley C. Morris

260

215

Inverse scattering problems in higher dimensions: Yang-Mills fields and the supersymmetric sine-Gordon equation H. C. Morris Department of Mathematics. •> Oregon State University, Corvallis, Oregon 97331 and School of Mathematics.>> Trinity College, Dublin, Republic of Ireland

(Received 1 1 October 1 978)

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It is shown that the notion of a prolongation structure can be extended to higher dimensions and used to determine inverse scattering problems . The relationship to generalized Lax representations is also considered. The method is illustrated on the self­ dual Yang-Mills equations. A generalization to include Grassman algebra valued variables is shown to provide a scattering problem for the supersymmetric sine-Gordon equation.

1. INTRODUCTION In previous work we have attempted to extend the pro­ longation structure method'·' in two distinct directions. We have tried to generalize and develop the technique in higher dimensions..., and also to extend it by the inclusion of Grass­ man algebra valued variables. " In this paper we further ex­ plore the development of this approach for the determina­ tion of inverse scattering problems in three and four dimensions and illustrate its utility by the construction of an inverse scattering problem for the self dual Yang-Mills equations. 11 · 1 2 We then go on to combine the two approaches in order to determine an inverse scattering problem for the supersymmetric sine-Gordon equation. """ In Sec. 2 we examine the equations that may be related to a particular type of linear prolongation form in three di­ mensions, derive in a slightly different way some of the re­ sults of Ref. 6, and show the connection to the Lax represen­ tation approach of Zakharov and Shabat. 16 In Sec. 3 we develop the analogous equations for four dimensions and emphasize in particular the relationship to a generalized Lax representation . " We wish to stress the point that the prolongation structure method can allow the direct determination of such a representation from the equations if it exists. The process is not trivially implementable however, and considerable experience and insight is required to imple­ ment the method in its current state of development. To illustrate the direct approach we tackle in Sec. 4 equations of the self-dual Yang-Mills equations in their sim­ plest form due to Yang" and Zakharov and Belavin. " It is not necessary to use this particular complex form, but it does reduce the algebra to a more compact and manageable form which is easier to follow. In Sec. 5 we link this work to that of Ref. l 0 by consider­ ing a theory involving superfields. This example is the first of its kind in that it involves not only four dimensions, two of which are fermionic, but also leads to a coordinate depen­ dent prolongation form related to a generalized prolonga­ tion structure which is an incomplete graded Lie algebra. We do not attempt to determine solutions using the sc attering problems defined. The purpose of this paper is to 327

J. Math. Phys. 21 (2), February 1 980

show the all-embracing nature of the prolongation structure idea, as applied to partial differential equations, to determine inverse scattering problems. However, a detailed analysis of the new results of this paper will be presented elsewhere in the context of specific equations.

2. LINEAR PROLONGATION FORMS IN THREE DIMENSIONS

Consider linear prolongation forms n of the following

structure,

(2. 1 ) where o' = ½E"bcdx" /\ dx• and the "'• are a set o f constant matrices which commute with one another, (2.2) [ "'••"' • J = o, \fa, b . In general some coordinate.dependence is possible in the "'• but we will not consider that possibility in this paper in any generality. We will present an example of coordinate depen­ dent "'• in our construction of an inverse scattering problem for the supersymmetric sine-Gordon equation in Sec. 5. Taking the exterior derivative of (2. 1 ) gives dn = (dF, /\ o') s + (F/1') lf 7/ is a I -form of the type then

/\ds

7/ = 7/,PX",

(2.3) (2.4)

(2.5) where u = dx' /\ dx' /\ dx'. We see from (2.5) that fl is a prolongation form for the ideal spanned by the elements aij of the matrix valued 3form a defined by

a = dF, /\ u, - '1/'P

where 'la and F, are related by F, = '7,p1,,E"1>c.

© 1 980 American Institute of Physics

(2.6)

(2.7) 327

216 As [w.,wb ] = 0 it follows that we have a constraint on the F. given by Fµ, = 0.

(2. 8)

If we now suppose that w, is invertible, then by redefining 5 it may be taken to be unity. To determine the content of Eq. (2.6), which we would like to express in terms of the F. , we must simplify the quantity ri,F, which occurs. We note that

ri,F, = ri,F, + ri,F, + rJ,( - F,w, - F,w, ) = (1), - ri,w,)F, + ( 1), - ri,w, )F,

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- ri,( [ F.,w, ] + [F,,w,J ) ,

and s o b y virtue of Eq. ( 2 .7) w e have

1'/J, = [ F,,F, J - ri,( [F,,w , J + [F,,w,] ) ,

(2.9)

(2. 10) (2. I I )

Therefore, if we choose w, and w, so that [ F, ,w, ] + [F,,w,J = 0,

(2. 1 2)

our basic 3-form takes the form a = dF, f\ u, - [F,,F, J u,

(2. 1 3)

which is closed and consequently equivalent to a set of par­ tial differential equations. We should also recall that we have the cot1dition imposed upon w.,w, that, [w.,w,J = 0. (2. 14)

The existence of the nontrivial solution for w, and w, to (2. 12) and (2. 14) means that the three-dimensional equa­ tions (2. I 3) have an inverse scattering problem obtained by restricting n to a solution manifold of that system. In that case we obtain the equations (2. 1 5) E0 t,,uJ.,S, b = - FJ,

matrices and/or parameter dependent. For example, if we take (2.24)

w, = ,1 and w, = ,1 ' and

F1 = A2 - A 2AJ,

(2.25) (2.26) (2.27)

F, = A,,1 - A , , F, = A ,,1 ' - A,,1,

which conforms to the constraint (2.8) which in this case is expressed in the form F, + ,!F, + ,1 'F, = 0.

Equations (2. 19) then become F., = A.,, - A,., + [A.,A, J = 0

(2.28)

(a,b = 1 ,2,3), (2.29)

the vanishing of the Yang-Mills fields in three dimensions. It should be recalled that this gives only special solutions because only in the case of self dual fields in four dimensions is F., = 0 equivalent to the full field equations. However, we have obtained an interesting inverse scattering problem for the SU(2) Yang-Mills equation F., = 0 in threee dimen­ sions which is given by (2. 30) 5. 2 - ,l 's,, = (A , - ,1 'A ,)5, (2. 3 1 ) s, , - As,, = (A , - ,lA,)s,

3 . LINEAR PROLONGATION FORMS I N FOUR DIMENSIONS Consider a linear prolongation form il of the type

which for c = I, 2 are s.2 - w,s,, = F,s

(2. 1 6)

(3. 1 ) il = ! (w•.dx" /\ dx' ) /\ d5 + F,., - (r,F, + r.F, >.• = [F, .F, J ,

J. Math. Phys .• Vol. 2 1 . No. 2, Feb•uary 1 980

(3.22)

Again the equations have a generalized Lax representation

If we take the first two rows as independent, then there exist r, such that

r1(0,w1.,Wmlt}21 ) + ,.( - tu1,. ,0,er, .,,lt1n ) = ( - W21 , - Cd31, - Wu ,O);

l 14Ct1 23 ).

(3.32)

These equations greatly generalize some of those presented in Ref. 9. We could go on to determine that the analog of the free Yang-Mills fields that arise in our study of three-dimension­ al examples in Sec. 2 are the self-dual Yang-Mills equations. Rather than adopt the same approach we attempt in the following section to show how one may, by using the prolon­ gation method ideas, derive an inverse scattering problem for the self-dual Yang-Mills equations directly from the equations themselves. 4. THE SELF-DUAL YANG-M I LLS EQUATIONS The self-dual Yang-Mills fields expressed in the com­ plex coordinates introduced by Yang" and Zakharov and Belavin 1 2 take the form a, , B, - a,,B, + [B , ,B, ] = 0,

(4. 1 )

a,, B ,+ - a,, B ,+ + [B ,+ ,B ,+ ] = 0,

(4.2)

a,,B ,+ + a,,B ,+ + a,, B , + a,,B, + [ B, ,B ,+ I + [ B, ,B ,+ J (4.3)

= 0,

and may be expressed by the closed ideal of 4-forms spanned by the forms a, defined by H.C. Morris

218 a,

=

a,

=

(dB, /\ dz, + dB, /\ dz,

=

[B, .B,]dz, /\ dz,")di, /\ di,,

(4.4)

( dB ,+ /\ di, + dB ,+ /\ di, [ B ,+ ,B ,+ J di, /\ di, J dz, /\ dz,,

+ a,

+

(4. 5)

( dB ,+ /\ dz, - dB ,+ /\ dz, ) /\ di, /\ di, + (dB, /\ di, - dB, /\ di,) /\ dz, /\ dz, + ( [B,,B ,+ ]

+ [B,,B ,+ ) ) a,

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where a = dz, /\ dz, /\ di, /\ di,.

(4.6)

Consider a linear prolongation form n having the structure

n = ds /\ (w , ,dz, /\ dz, + w 1dz, /\ di, + w 2dz, /\ di, 1

+ (F,dz, /\ di, /\ di, + F,dz, /\ di, /\ di, + F,dz, /\ dz, /\ di,

(4.7)

+ F2dz, /\ dz, /\ di,)s,

where the w•• are scalars. The requirement that n prolong the ideal (a , ,a,,a,> spanned by a, . a, and a , is

dll = Pa, + ( 17 ,dz, + 17,dz, + 11,az1 + 17,di, ) /\ fl , (4. 8)

and this shows us that we m ust have F, = F, (B,,B ,+ ) ,

(4.9)

F, = F, ( B, ,B ,+ ) ,

(4. 1 0)

F; = F; (B,,B ,+ ) ,

(4. 1 2)

Fl'B/ = Fl'B / '

(4. 1 4)

F, = F, ( B , .B ,+ ) , Fi ·B,

(4. 1 1 )

= F2·s,,

Fl 'B t = F1,,s,

the relationships

=

(4. 1 3)

- F2-s1

=

- F2' B t '

Fi = - Wn_1/l + W221/i - W2T1/i , F1

(4. 1 6)

= - wu 11 1 + w 1 21J1 - (LJ 1 11/2 ,

(4. 1 7)

F1 = - (1)221/ 1 + W 1 i1/2 - W r 21/i ,

F;

=

[ B ,,B, J F, ,B,

+

(4. 1 8)

- w 2 1 17 , + w 1 111, - w 1 2 17 1 ,

and the central equation

+

[ B ,+ ,B ,+ J F1 •B;

[ B, ,B ,+

+ F, .B ,-

(4. 1 5)

(4. 1 9)

! [ B , ,B ,+ ]

l l = ( 1/ ,F1 - 11,F, + 11,F, - 11 ,F, ) .

(4.20)

(4. 2 1 )

=

( x 1 B,

+ x,B ,+ ) ,

F1

=

(x,B 1

+ x,B ,+

F, = (x ,B, - x,B ,+ ) ,

330

),

J. Math Phys., Vol. 2 1 , No. 2 , February i 9BO

Fi

=

which is equivalent to x, =

(4. 26)

- J..F, .

+ J..x ,

and

(4.22) (4.23)

(4.27)

x, = - ,lx,.

This in not necessary but greatly reduces the required alge­ bra and leads to useful, if not most general, results.

The term (11,F, - 11,F, + 11,F, - 17 ,F;) must be con­ verted into commutation relations between the Fa if the nor­ mal prolongation ideas are to be used. Using our ansatz (4.25)-(4.26) this term becomes

+ J.. 17, J F, +

( J.. 17 1 - 11, ) F,,

and we need to express the combinations ( 1/ , + 117 2 ) and (A 1/1 - 1/ , ) i n terms of the F0 i f we are to obtain a Lie alge­ bra-like structure. Equations (4. 1 6)--{4. 1 9) give 1/ , ( w2 ; - -lw12 ) - 17,u1 1 2

+ 1/ ; ( w, , - -lw 1 j ) = 0,

1/ , ,,, , , + 11, ( ..lwu - w 1 1 ) + 1/, ( w , , - ..lw 22 )

= 0.

These are solved by the relations w 2 2 = Awn ,

(4.30)

I. Miodek

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Laboraroire de Physique Marhl!motique, b) Uniw!rsite des Sciences et Techniques du La11guedoc, 34060 Montpellier Cedex. France (Received 17 January 1 977; revised manuscript rec.civcd 12 May 1 977)

We present here a new and easy method, a natural extension of Lax's method, for obtaining gener.al "IST­ solvablcH nonlinear evolution equations. These arc evolution equations for the potential function(s), v. of a Hamilconian, H. when the logarithmic t derivatives of H's inverse scattering data are given by a r ­ dcpendcnt ratio o f entire functions o f E, fl(t ,£). Here E is the energy variable and n i s the . .dispersion relation.. of Abowitz, Kaup, Ncwe11, and Segur (AKNS). We pose the question of existence of the evolution equation's solution. This qucsti�n is answered completely in the one-dimensional Schrodinger case (first example). In a second ex.ample we derive the evolution equation for an n X n matrix generalization of the Zakharov-Shabat-AKNS equation. Our method displays the central role of analyticity in E in the 1ST method as a whole.

INTRODUCTI ON Inverse methods were introduced into nonlinear evolution equations by Gardner, Green, Kruskal, and Miura (GGKM) 1 in 1967 ; they showed that the Korteweg­ de Vries (KdV) equation and its higher order genera l­ izations could be interrelated b y means of the one ­ dimensional S chrodinger equation and that this family of nonlinear equations could be solved exactly by letting the inverse scattering data of the Schrodinger equation evolve in a manageable way which is determined by the nonlinear equation. Much of the motivation for the study of the KdV equation came from experimental and numer ­ ical studies which showed the presence of stable "particlelike" solutions, named solitons. GGKM were able to show that soliton solutions were associated exactly with the presence of proper eigenvalues of the Schrodingcr Hamiltonian, and thus their stability was essentially due to the invariance of this Hamiltonian's spectrum. In 1968 Lax' introduced an elegant unitary operator formalism whi ch recovered the GGKM r esults for the one-dimensional Schrodinger Hamiltonian: Lax showed clearly, in an abstract setting, that what was involved was the "isospectral" evolution of a Hamiltonian H =H(t) = U(/)H(O)U*(I) , induced by the unitary operator U = U(t) {U(O) = l] and that the evolution equation for H was then given by the commutator equation H , = [B . , H J, where B, = U,U• = - B : . (Notation : /, = of/os for s = x or t, )

Here B11 , a linear antisymmetric operator, is u·s generator. Now if H ' s operator structure is restricted so that only its potential " can evolv e , a s is the case When an inverse method is applied, then the commutator equa tion becomes an evolution equation for v . The great inter est is that the evolution equation is in general non ­ lin ear, is exactly solvable, and has soliton type solu­ ti ons whe never H has proper (discrete and separated) aJ Th\s work was completed under Contract No. 264 of the n . c. P. ,, Phys i�'!_e Mathematique et Theorique, Equipe de Rechcrche Assoc tce au C. N. R. S. 19

J. Math. Phys. 19(1), January 1 978

eigenvalues . Given the above, Lax 's method consists of somehow finding or constructing antisymmetric linear differential operators, BL = - BL •, such that [B L , HJ has the correct form to be identified with " • · He explicitly constructed such operators in the Schrodinger case to recover the GGKM results , Lax also looked at s,,veral other Hamiltonians.

Zakharov and Shabat' applied these methods to a 2 x 2 matrix eq,uation, whose Hamiltonian wasn ' t symmetric, but which had proper eigenvalues and a solvable inverse scattering problem: They obtair.ed solvable nonlinear evolution equations which had soIi ton solutions, some of which they explicitly displayed. The added interest here i s that their nonlinear equation which is a kind of nonlinear time -dependent Schrodinger equation, has applications in plasma physics and, more i,nportantly, the IST method worked again.

In 1974 Ablowitz, Kaup, Newell, and Segur• (AKNS) presented a comprehensive analysis of the above meth ­ ods, including a new and more systematic method 0f deriving even more general evolution equations which could be written in a beautifully concise form. They showed that the proces s was a nonlinear generalization of the Fourier transform and so they called it the in ­ verse scattering transform (1ST ) . Their method bypa sses B, and derives the evolution equations by cs:ng convenient integral representations for the inverse scattering data, Applied to their 2 x 2 generalization of the Zakharov -Shabat equation they prove that if their two component potential U satisfies the evolution equa ­ equation U , = fl(L *)U,

Then the inverse scattering data evolv e a ccording to their "dispersion" relation fl(E). Here !1(E) is poly ­ nomial (or entire or a ratio of entire functions) in E, th e energy variable , and L i s a linear matrix differen­ tia l operator whose eigenfunctions are e ssentially the squared eigenfunctions of the Hamiltonian , They i;av e similar results for the Schr0dinger equation.

Our method , in its present formulation, is a natural extension of Lax 's method and is motivated by a general theorems which guarantees the existence of a unitary © 1978 American Institute of Physics

19

223

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U (and thus of B. = - B: ), given that the evolution of H = H* has been induced "isospectrally , " e . g. , by caus ­ ing H 's inverse scattering data to evolve without altering H's spectral structure which includes the proper eigen ­ values, continuous spectrum, and their degeneracie s . Given H = H(I) (iind consequently having H, " an/at) w e systematically solve the two following commutator equa ­ tions , [B, H]l/! = H ,,P = [B•, H ],J, (where H,j, = Er/1 and B.1/1 = B,J,), first for the linear E -dependent operator B, which in turn gives us the linear E -independent operator B., all modulo the addition of operators which commute with H .

[ U H* = H, then HT = H , and B. = ½(B-- B-•) must satisfy the same commutator equation as B· since B· + a-• com ­ mutes with H automati cally . I The operator equation for B, applied to ,i,, takes on a more convenient form which we call the "reduced commr/I;:-. I/!• ; It ls that subset which can be freely specified and from which the rest of the scattering data can be derived. Finding a conv e ­ nient subset Is not a trivial pa r t o f the inverse problem.

A given inverse method constructs the unique Hamiltonian, with a given operator structure, from the data. We remark that two Hamiltonians defined on the same space but with different structures (say one with a local potential and the other with a particular kind of nonlocal potential) can both reproduce exactly the same data. The inverse scattering data thus gives us a unique . Hamiltonian within the class assumed by the given in­ verse method-we will use this remark when discussing existence for solutions of evolution equations generated by Lax's method. The potential is l/1at part of H's sln,cture r.olzicl1 can uary . B. The inverse scattering transform ( 1ST)

The inverse method can be used to generate an evolu­ tion of the potential ,, by causing the data to evolve with t; ,, must change if the data changes because the direct scattering problem determines the data uniquely from v. U we can Identify the evolution equation satisfied by ., when the data evolves in a given way, then 11 's evolution equation can be solved by' 11(t = o)!!! data(/ = o)W data(n'£J 11 ( 1) .

Here:

(a) indicates solving the direct s cattering problem at 1 = 0;

(b) indicates integration of th e data 's evolution equation;

(cl indicates solving the inverse problem for 11(/ ) given data(/). I. Miodek

20

224 'l'his is what AKNS' later entitled 1ST .

The evolution equation for the data i n step (b) must, of course, be solvable if the !ST i s to work, and in the cases considered this is trivially true. The nontrivial part is the identification of the evolution equation for v associated with a given (trivially integrable) evolution of the data. The data 's evolution will be characterized entirely by what AKNS called the "dispersion relation" fl; schematically

D. The reduced £-dependent forms and analyticity in E

In the examples to follow, O ' s analyticity in E is the only restriction imposed on the data ' s evolution.

We are able to solve (1 . 5) for B in the examples con­ sidered thanks to analyticity in E and to completeness of the ,P 's . Completeness permits us to "reduce" E­ independent linear differential operators A" to more convenient £ -dependent forms A. If A = ,A(n; x, • • )E" i s a polynomial in E, then, when applied to �. it is " equivalent to A " A(n; x, . . )H ; A i s the reduced form of A". In the reverse sense , for example in the case of the v-dimensional Schrodinger equation

Lax ' s method' for finding the evolution equation for v is based on the existence' of unitary operators U=' U(I), = l, which generate the isospectral evolution of the Hamiltonian JI ='H(t) [and of the generalized eigenfunc • lions 1"" 1J;(I) ] via

a �" 4 is equivalent to a !"(u - E - A ' ) , where t:,.' = A - a �, and so forth till we obtain the "reduced" form cr(E, • • · ) + fl(E, • • ·)il,, where the linear operators er, /J no longer When v= 1, er and /J are no longer differen ­ contain tial operators.

il ll(l , E l = at (ln data(/, £)) .

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which commutes with H . This remark has relevance when H i s not symmetric , as, for example, in the Zakharov ..:.shabat3 equation and its AKNS generalization.

C. Lax's method and the present generalization

uu•

H= UH(O)u• [and ,P = U;/1(0) ]

o r equivalently via

Jf,= [B., ll] (and ,P, = B,,P),

(1. 2)

where BIJ = u, u- 1 = - B: is U's generator. The evolution of the data is then given by the asymptotic form of B D, -ii, via the equation

(1. 3)

Rem,zrk: This, and everything else that we do , ob ­ viously extends to any number of / -like parameters (cl. ,·esults in Ref. 6 ) . Lax n o w seeks (somehow) t o construct linear anti ­ synuuetdc operators B L such that [n• , JI ) " n ; Z n,

( 1 . 4)

has the correct form to be identified with H, (a multi­ plication operator for the Schrodinger case ) .

We, o n U1e other hand, seek to solve the "commutator equation" [B, If ] = H ,

( 1 . 5)

for B, modulo operators which commute with H. Once oolveJ, ( 1 . 5) yields the evolution equation for the poten­ tial, In comparison, the AKNS method seek'!; to find (it nearly solves for) the evolution equation directly from the dispersion relation , bypassing B.

Our B need not be antisymmetri c . The symmetry of II, constructed by the !ST, implie s the symmetry of H, ar.d [B, HJ and thus B, = B + B• automatically commutes With H; s o the antisymmetric part of B , B, = ½ (B - B* ) , ilso satis fies ( I . 5 ) .

The explicit form o f t h e antisymmetric part o f B 1s Only nee ded asymptotically , to obtain th e data ·s evolution 1·ia Eq. ( 1 . 3 ) . In fact the anti symmetry of the generator , •·•· , the unitarity of U, should not be overstressed. In lhe c ases cons idered it suffices to conserve the n.} •0i-ms of the proper eigenfunctions (i. e . , bound states ) 01 Ii and this can be arranged b y adding t o B a n operator J. Math. Phys.. Vol. 1 9, No. 1, January 1 978

r

¥=r

- ' H = - L> + v , l> � u1.,P.} follows easily from the linear independence of ,P. and ,Ji.; e . g. , uae Eq. (4 . 3b) to consider (h/1/!!)., then put h = O. {li) For E E {E,},

(A - E. Dlh, = o,

lr, e n}(R; dx) n C'(R) = h, a: ,i,,,p.,

(4 . 4a ) {4 . 4b)

where (H - E,),P, = 0, 1/J. E 1L 2 (R; dx) n C'{IR ) . Thus l/>,,P, E IL 1 {1R ; dx) n C'{R) c lL'(JR; dx) . (If v , v * , then (A• + E:D)h : = 0 = h, a: ,P.,P• • ] {iii) Orthogonality:

J,:: ax ,P,,P, v , = O for s = x or t.

.\!(E)f"{E) = f{u; E ) ,

(4 . 8)

which is obtained in Lemma 2 . For our immediate pur 11ose we state that M(E ) , its determinant det}!(E), and f(,r ; E) are analytic in k = E ' 1 ', Imk > 0; that f(u; E) is linear in " and that b(E) ls a linear functional of !°{E) , so for example if 11 , = 0, then b(E) = O.

This matrix equation, (4 . 8 ) , can be solved for the initial value f"(E) for all E E o:'\ [E�I detM(E�) = o} . This initial value when used in (4 . 6) yields an / which satis­ fies all the conditions on b ., except possibly when E ,:{£�}. To finish, we now need to show that f"(E ) , and the f it generates , are regular when E - E,. Since M(E) and its determinant are regular in E , f"(E) can only have polar singularities on the point set {E�}. We show in Lamma 3 that {E�} c {E.}. Now we can make use of the orthogonality property (4. 5 ) with which we showed that when E = E., E q . (4. 2) has a solution (b -b)(x, E,) in IL'(JR; dx) . We deduce that when E = E�, Eq. (4. 8) must also have at least one solution. This solution is the init1al value vector of b (x, E.) at x= x0 , noted as b0 {E,) . So f(,,, ; E,) is in the range of M(E.). This, together with analyticity in E, implies the existence of (4 . 9 )

( s e e Lemma 4 for details) . Thus f0(E) is regular o n {�} as well.

( 4 . 5)

These three properties show that the null space of (A :.. ED) is spanned by the squares of the Schrodinger eigenfunctions and that (4 . 2 ) has a solution in 1L2 (R; dx) when E E {E .} • [Our proof does not extend to v ; v • be­ cause v , is orthogonal to ,P,,P, but not to (,P,,P,) * . ]

Now, ln order to construct the solution of (4. 1) with the appropriate analytic properties in E, we consider the related initial value problem for "regular" solutions /E ('(R), (A - ED)f = u(E) = ½ b_(E)v, ,

(4 . 6)

lim(x - x0 ) ( f, f., f.,) = (fi , /2°, f3°) = f"{E) ,

(4. 7 )

and we look for conditions on the initial value f"(E) so that /= b . of ( 4 . 1).

If v , v x , and v , are "smooth enough" locally and if they also dec rease "fast enough" as Ix l - oo, standard theory tells us that f is well defined and entire in E if J. Math. Phys., Vol. 19. No. 1, January 197B

Three linear equations must b e satisfied b y the three components of l°(E) so that J(x, k') {with k 2 = E, lmk > 0) remains bounded as x - ± "" and so that Iim, __ J(x, k') exists uniformly in k as Imk - 0 + . Then lim,_.f(x, E) = b (E) gives us the "dispersion relation. " These three linear constraints on !°(E) are analytic in E , and they can be conveniently put into the 3 X 3 matrix for m

lim E •E� M(E) "' !(v ,; E ) = b0 (E ,)

[If v ; v •, then ( 4 . 5) still holds for v ,; but this is not enough -it needs to hold for both (v * ± v),. ]

25

f'(E) and u(E) are entire in E . We can also show that the asympt.>tic behavior as I x I - "" can be controlled by imposing analytic conditions on the initial value !0 (E). More details of this construction are given in proving Lemma 2 .

We have obtained what we wanted. The solution of Eq. (4. 8 ) gives us the appropriate initial value for (4. 6) which then generates both the appropriate solution of (4 . 6 ) and its asymptotic limit. By construction this solution also satisfies (4. 1 ) and all the analyticity re ­ quirements in E. Schematically f"(E) = M{E)" 1 f(v , ; E), and then via (4. 6 )

f"tE) - f(x, E) = b (x , E), b(E) = lim.-.b (x, E)

gives us the pathway of this construction.

a:,

{4 • o ) t Q. E. D

(If ,- • ,.. , then {E�} c {E.}c but we do not have the appropriate orthogonality property . I

Sum mary of the proof: The study of v, = iJv/iJs where s = x or I permits us to prove that the squares of the generalized Schrodinger eigenfunctions span the null space of (A - ED) and that 11, is orthogonal to the squares of the Schrodinger bound states . This orthogonality i mplies existence of an lL " ( JR ; d.r) solution b of (A - ED)b = ½r I at £ = £ 11 This in turn perm its us to show that, when we study the associated initial value problem for a ureg,Jlar" solution f, we can solve for the initial value •

I . M iodek

25

229 vector f'(E) = (J, f,, fn> � .. which is analytic in E E O: \{E.}U {E.} = a: . By " regular" we mean (i) regular in E for fixed xe R, (ii) bounded asymptotically as l x l - ± ., for fixed E e a:, and (iii) having an x-indepen ­ dent limit as x - + "' uniformly in k = E ' 1 ' , lmk 1' 0 . L e m m a 1: The squares ,/1' of the generalized Schrodlnger eigenfunction s , (H - E)I/J = 0 span the space of homogeneous solutions h of Eq. (4 . 3a) , i . e. , h satisfies

(A - ED)h = O with �'!' D'h = O for j = 0, 1 , 2 .

(4. 1 1 )

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Furthermore, the squares 1/1! of proper Schrodinger eigenfunctions are orthogonal to v , ,. av /as , with s = x or t, in the sense of Eq. (4 . 5 ) ,

J.:

(4 . 12 )

dx ,P! v, = 0,

Proof:

(4. 1 3 ) =-

J.: dx ,J,',,,

(4 . 13) =

(,J,1/J., - 1/J, 1/1,) I :�---

(4. 14)

For E = E., ,P. ec ,i,. cc ,i,_ [defined by Eqs. (3. 12a) and (3 . 1 2 c ) I and so ,J,.

1 , j• • exp( - l x l I Imk. l )

(k = E ' 1 ' · for v = •• • E .; O) and (,P) - 0 as l x l _ .,_ Thus (4 . i4) Implies' (4."12) when E e'\E.}.

[For E E a: \ {E.}, (4. 14) with 1/J = 1/1. generates the integrai representations for R. (when s = x) and (R.) , (when s = I) . These are basic to the AKNS method. ]

To prove that ,Ji' satisfies (A - ED),i,' = 0 in a way that clearly relates this property to Eq. (4. 1 ) we first note that the ansatz [cl. Eqs. ( 3 . 4 ) and (3. 7) ] l/>, = (n ° - ½b ,),P + b,P., a� = O

(4. 1 5a)

when substituted into Eq. (4 . 1 3 ) with s = I yields Eq. (4 . 1). Similarly the ansatz

(4. 1 5b)

when substituted into Eq. (4. 1 3 ) with s = x yields ½!' , = (A - ED}g.

(4. 1 6a)

(4 . 16a) e:e> ( A - ED)�' = O.

(4. 1 6b)

But recalling that ½ !! , = A l and noting the equivalence in (4 . 1 5b ) , we see that

If {!/,., ,i,_} were any pair of linearly independent Schrodinger eigenfunctions of energy E, {$! , 1/J� , $. �•.} would also be linearly independent. Thus , noting that 2 1/!.l/J. = (,J,. + ,i,.)' - f. - ,p\ we see that for E E a: \{E.}, an arbitrary homogeneous solution of (4. 1 1 ) can be ex ­ panded as a linear combination of the squares of the Schrodinger (generalized) eigenfwictions defined by ( 3 . 12a) and (3 . 1 2 c ) together with ( � . + 1/J J ' . However when E = E ,,, then IJ,11 cc 1/J. a:: 1/J�, but even so any other linearly independent solution of the Schrodinger equa ­ tion , �- �, must diverge as x - oc in such a manner that lj,i ,, /J,� .:. 0 as r - ce. Thus i;i! spans the solutign space of (4 . 1 1 ) , boundary condition inc luded , when E = E •• This completes the proof of Lemma 1 .

26

J. Math. Phys., Vol. 1 9, No. 1, January 1 978

[Note : If " * " * , Eq. (4. 14) still Implies Eq. (4. 12) because the linear dependence I/J. "" 1/1. cr O. ]

L e m m a 2 : The solution of E q . (4, 6) will be bounded as x- ± 00 and will have a constant limit as x- + .. wilformly in k = E ' 1 2 , lmk " O, if and only if the Initial value vector at x = x0 e R, f'(E) = (f, f., f.,)� ...0 , satisfies three linear constrains which are analytic in E; these constraints can be put in the matrix form of E q . (4 . 8).

Proof: We start by sketching a standard method for solving Eq. (4. 6 ) . Equation (4 . 6) can be put into the integral form

(1 - A(E)(vD + ½v ) ]f = C(E) - f0(E) - A(E)11(E) (4. 17a) where A(E) is a right inverse of (¼ D' + E)D = ¼ exp(± 2ikx)D exp(ac 2ikx)D exp(± 2ikx)D. Defi ning

r

[so DD"' g(x) = g(x) ;< D" ' Dg{x) = g(x) - g{xol l , ' A (E) = 4D" 1 exp(D + ½ , , ,) ]"' A (E )u(E) [recall that 11(£) = ½ii.(E),, , ] be well defined and so that it has standard asymptotic behavior with the coefficients that are entire i n E . The factor b.(E) multiplying ,, , indi cates that if 11(E) - ½1·. , then the term in ,, , will generate poles at E E {E� l b.(E�) = O }. Given that c· satisfies these conditions we solv e

(4 . 1 7a) b y inverting [ 1 - A (E)(,,D + } r , ) ] and obta i n a sum of a homogeneous solution f,(1° ) (linear in f") and a particular solution f,( - 1•,) (linear in ,, , ) . What concerns us are the asymptotic behaviors respectively a s x- • ., from which we c a n obtain boundedness and limit conditions. Defining

I. Miodek

26

230 and

f,(v,; E) ce ½ [l -A(E)(vD + ½v,) J- 1 A(E)b.(E)v ,

(4. 18)

f.(r'; E) = [l -A(E) (11 D + ½v .)}" ' C(E) , f'(E) ,

J=l,(t'; E) -f,(v ,; E)

f"'(E) = (detM(E)) · 1 MiE)f(v , ; E).

,..... exp( - 2 ikx)( t:.• -1;; ) + exp(2 ikx)(t:: - 1;.•)

(4 . 19 )

+ O, only the dominant term counts after the terms with null coefficients have been eliminated.

l

From (4. 19) we see that, for Imk > O, f is bounded as l :d - .. if and only if the two following equations hold:

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t:,;lf'; E)

= 1;;(v ,; E).

( 4 . 20a±)

The first term ls linear in !', the second is linear in v,. and both are analy�c in E.

Furthermore from (4 . 19 ) we see, that we have eliminated the term in exp(- 2ikx) a s x- "', that f will take a constant limit as x - «> uniformly as Imk - 0+ If and only if a third linear equation holds, J;:(f'; E) = f;.'(v,; E).

(4. 20b)

(4. 20)

Left multiplying b y M,(E.) now gives us the sought after result,

M(E)t"(E) = l(v ,; E),

From equations ( 4 . 19) and (4. 20) we now have J bounded as I .r I - ,. and the existence of

(4 . 2 1 )

uniformly ln k = E ' 1 ', Imk ;, O. [Whenever ( 4 . 2 0 ) ls satisfies we can Identify I with b •• ]

Lemma 3: {E'! I detM(E�) = O} = {E . I W(,P., ,PJ(E.) = 0}.

•

Proof: At E= E'! U1e homogeneous form of (4. 20) , which is obtained when v , = 0, has a nontrivial solution f';(E). This implies that at E = E'!, t',,(E) Is the initial value of a homogeneous solution /, of (4. 6) satisfying all the conditions of Lemma 2. This f, is therefore in the null space of (A - ED) and so, by Lemma 1, f, can he expanded as a linear combination of {,p!, ,P.,P.}. But If £'! ,;t {E.} It ls straightforward to verify that no linear combination of {if., ,P�, ,P .,P.} satisfies all the conditions (boundedness at both x- ± "' and the existence of a con ­ stant limlt as .r- + oo uniformly in k = E ' 1 2 , Imk ;, O) of Lemma 2. Thus {E"J c {E J. [This much holds whether or not v = v•. ]

,t-,

ff E = E., then f.(a: f. a: f.) Is always a homogeneous solution of (A - ED) which satisfies the conditions of Lemma 2 . So, If E = E., then (,f., (if.)., (if.) u ) �,_ ,.. = �(E.,) " 0

(4 . 22 )

•

[I! v " v*, then it is possible for E. > 0 and then if. does not have an x-independent limit as x- + "'. ] Lemma 4: The limit E - E� of M(E)" 1 f(v 1 ; E) = f"(E) exists. 27

J. Math. Phys., Vol. 19, No. 1, January 1 978

Considering (4 . 23) under these conditions, and re ­ membering that ((v ,; E) too is analytic in E, we see that ub.ether or not the limit exists depends on whether or not M,(E)f(v , ; E) has a zero at E = E, to compensate the zero of detM(E): It thus suffices for

(4 . 24 )

where M(E) and f(v ,; E) are analytic in E and the rhs is linear 1n v ,.

must be In the n ull space o f M(E) and {E.}c {E"J.

(4. 23)

M,(£ ) and detM(E) are analytic in E because M(E) is analytic in E. Let us s;mplify matters a little by assum ­ ing that detM(E) has only simple zeros. [Multiple zeros ean then be treated as the limit where simple zeros coincide or by using the following observation whose proof is quite standard: If a matrix M (E) is analytic In £ and if M(E,) has a null space of dimension j , then detM(E) must have a zero of order at least j at E = E,. J

But since we know from Lemma 1 that (4. 1 ) h a s a solu ­ tion at E = E., b.(x, E,) , its initial value at x = x0 , de­ noted by b�(E.), must satisfy (4. 20) at E = E,,

We rewrite equations (4. 20u) and (4. 20b) in the 3 x 3 matrix form

� .(E) = ��'!'j(x, E ) = J;,;(v ,; E) -f;"/f'; E)

Proof: We know that {E� c {E,} so it suffices to prove that the limit exists when E - E •• Now if M /E) denotes the cofactor matrix transpose of MtE) we have , from (4 . 2 0),

M(E.)b0 (E.) = f(v ,; E,).

(4. 2 5 )

M,(E.) f(v,; E,) = detM(E,)b0(E,) = 0 .

(4 . 26}

The existence of a solution of (4. 1) at E = E, has p.rmitted us to show that C(v ,; E) was in the range of M(E.l [Eq. (4 . 26) ] W!rlch in turn implied that the analytic fUAction of E, M ,(E)f(v ,; E) had a zero at E = E. to com ­ pensate that of detM(E) (assuming simple zeros) and so the limit of (4. 2 3 ) exists as E - E, E {E.} ::, {E�}. Exten • sion to multiple zeros proceeds as indicated above.

5. NONLIN EAR EVOLUTION EQUATIONS FOR LINEAR F I RST ORDER n X n MATRIX EQUATIONS O F THE ZAKHAROV-SHABAT TYPE A. The linear matrix equation

We will follow the model set up in Sec . 3 for the one ­ dimensional Schrodinger equation to show that the same method also yields evolution equations for the off­ di.agonal (i. e . , diagonal part is zero) n Xn matrix "potential" Q = Q0 " + Q "' • = Q'" of the following first order differential matrix equation for /: UD - iQ - kCJ)f= O,

I. e . , l, = Zf = (Q - ikCJ)f.

The Hamiltonian form is

(5 . la) (5. lb)

Here D = ii/ox; k Ea: is the energy vartable in this case; CJ i s a constant (I. e . , Independent of t, x, k) diagonal matrix (so CJ= CJ4""); trCJ = trace 11= O; the matrix ele • ments of o satisfy = (Ai ., ) (L (b ,., ( L - - kl "' - -= 1,,. ( 1 ; I ) + � t ,. ( j; / ) B l, 1 " 1 k 1 ' (5. 16b) J •2

tel

given that the matrix entries of [(k) and i., (k), are polynomial (or entire) In k , with expansions

�1., (l , k) = � § 1.,(j; Ilk' (showing the I dependence). , (5. 16c)

[We apologize for the flood of indices in Eqs. (5. 16). ] The same method as employed in the Schrlidinger case also works here to generalize the r esult to ratios of polynomial (or entire) functions of k -we will not repeat it here. Assuming that [(k) is regular in k at all points where

f has one of its (assumed) finite number of proper

eigenvalues, and that i, s i,(1, x, k) is regular in k every ­ where where [(k) Is regular, we conclude that both sides of (5. 15a) must be zero for the same reasons as in the Schrodinger case. Namely, (5. 1 5a) is equivalent to (5, 1 5b)

2 : A·" - [Q, JT ' [Q, A 0 " ]"'"1 + [Q, A 0 " ] , i : A 0 " - (;) , A 0 " ] ,

(5 . 1 4d)

where e(l , m ) ., = 6 , .6,.1 Is the natural basis for n xn matrices . (The complexities due to giving up the re­ striction Q = Q'" are similar to those due to permitting say a., = a.., because In the latter case the off-diagonal matrix Q 12e ( l , 2) + Q 2 1e(2, 1) commutes with a; so a has a null space in both these cases . ]

Both sides of (5. 1 5b) must be zero because one side or the other is regular in k at all points k E a: , while the right -hand side tends to zero as l k l - 00 • The results are the solution giving i, in terms of [i. e . , terms of y = y(l , k) = li m , •• B•"•] and the evolution equation

r

Q , = r_(t , f )Q.

(5. 17)

"!,!e note that, since trB'h ll' = O, our dispersion relation, l', contains n - 1 scalar functions of k and I. H. Relation to the scattering data

We have obtained the evol11/io11 eq11atio11 (5 . 1 6 ) for Q I. M iodek

29

233 on the basis of analyticity assumptions in the k plane , and we have done so 1vifhout solving the inverse problem for ( 5 . 1 ) . Since the inverse problem has not been solved we cannot relate y(I, k) to the inverse scattering data, because the latter is unknown. (Most recent s tudies• which obtain evolution equations for linear operators share this difficulty. ) However it is straightforward to relate Y = lim r . B to the asymptotic behavior of F as x- "'· Equation ( 5 . le) becomes

d1

•

( 5 . 18a)

F - F = exp( - ikxa) F0 , F0 , = 0 .

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Now use Eq. ( 5 . 1 0 ) , taking heed of the remark which follows it concerning the right-hand normalization of F, B "' F , F" 1 - FA 0 F"' = B = F,F· 1

-F A0 i· 1 ,

(5 . 18b)

O, and A0 0 if and only if E is appropriately where A 0 right-normalized. The condition trii = 0 is equivalent to

=

,=

trF 1 F" 1 = trF0 1 01 = trA0 = O ,

( 5 . 18c)

F'

a condition on the right -normalization of F . We can also adjust the right-normalization so that ..B B dias = i. e. , so that lim, • • B0 1 1 = 0 . This last condition is equivalent to

r,

=

(5. 1 8d)

again a condition on the right-normalization of F. Given these conditions Y = lim, • • F 1 F· 1 •

(5. 1 8e)

Recalling (5. ld), its analog with res pect to F, = BF rtrB) is (= T

,=

( 5 . 1 8!)

This gives us another interp retation, in terms of T = detF, of trB = 0.

Further than this we cannot go without solving, or at least starting to solv e , the inverse problem for (5 . 1 ) . This is not within the scop e of this article . However, it may turn out that finding the evolution equation may help in solving the inverse p roblem -at least in identify ­ ing the inverse scattering data. This need not be sur ­ prising, if true, because the approach is based on analyticity in the energy variable as are the solutions of inverse p roblem and the related completeness relations. I . The AKNS data as the 2 X 2 case

When n 2 and a= a, the following choice of the right ­ hand normalization o f F ,

=

1 /a O ] • F o - LR"/a r a '

J

a

(5 . 19a)

_ R•)''• ( '

immediately yields B = lim x 4 • F t F· � F st F; = B•HM = y(I, k ) , 1

i. e.

1

y(I , k) = ex p( - i k x a{R; � •{; . ,�a] exp( + i kx a) a , !a a = - ½ o,(lnR "( I , k)) 1 = - a, fl ( I, k) . ,

( 5 . 19b)

Here R · is the coefficient of refle ction to the right, T is 30

J. Math. Phys., Vol. 1 9, No. 1, January 1 978

the transmission coefficient. {The existence of a solu ­ tion with "normalization" F0 can be p roven b y separately Imposing appropriate asymptotic conditions on the first and second columns of F = (F ,; F , ] , e . g . , for Imk " O,

J)

{ - rexp( - ikx) T·' rtl ·l . exp ( - ,kx) aT F , \" . .. Lexp( + ikx) ,-· 1 R•

LoJ , ..•

exp ( + ikxr] ,.::. ( a T) "1 F • (-"...

li

f

xp ( -

ikx) r·'R

bxp ( + ikx)T·

'

]) •

which are linearly independent if and only If T-, ;(u ),

are just the higher -order analogs of the KdV equation . 4 Example 4: Consider Burger s ' equation U t = B(u )= u x x --1- Wt x .

(12)

D = D + ½u + ½u ,D·' .

(13 )

W e show that B possesses the recurs ion operator Here

A (B)= A= D2 + 11D + 11, . Hence

and

A • /J = D3 + �1D2 + ( iz, x + ½u2 }D + i(uxx + uux ) + ½(u XlX + Ul/ XX + u! )D- 1 D • A = D3 + �uD2 + (izi :c + ½z/)D + (uxx + uu) .

Therefore ,

[A , D ] = ½{ux:c + uux ) + ½(unx + uu n + u! )D- 1 •

Furthermor e

[D i ,D ] = ½ut + ½uxtn- 1 ,

which proves condition ( 8 ) formally . Therefore, we have an infinite sequence of flows u 1 = B'J > (u)� D'B(u) all of which preserve the flow given by Burger s ' equaPeter J. Olver

1213

237 tion. The first few of these flows are u 1 --,-. B 10 1 (11 ) = u n --1- u11 :c , 11 : B' 1 1 (1 l ) = l l nr + liw r;.- + �fl ! + �/1 2 11 :,: ,

H ence

7:

l/ t ::-- B 12 l (u ) = l l nn + 21m nr + 5U,/ l n + ½u 211 n + 3u11 ! + ½u 3 1(,: ,

(14)

Therefore ,

lA,D] = Ju ( u:ux + u 2u) + 1 (u :o:xx + u 2u xx

U t :;:;; B l3 l (u ) = lfnxn + �ltlt nn + Jfu x1� ,:n + 5 u !x + �1t 2ll xn + ,¥UH/I n + !_rll! + �U 3/l n + !Ju 2u! + h11 4 t1 :,: .

+ 2uu!)D- 1 • u + } u xD- 1 · (u ur + lt2 u ) .

E-:r:ample 5 : F inally , w e consider the modified KdV equat ion u t = K(u ) = u xxx + u u x ,

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2

(15)

which i s known t o also possess infinitely many conserva ­ tions laws . In fact, the or iginal proof of the existence of infinitely many conservation laws of the KdV equation u t = K(u ) stemmed from the remar kable transformation of Miura 8 relating the two equations . Explic itl y , if then

[A , � up- 1 • u] = 2 (u11 n + u! }D + 2(rm ur + U/-l,) + H u rxn + rt 2 11: n --+ 2uu!)D- 1 • u + � ,, xn- 1 - (11 ,: n --t 1/ 211 ) .

z1 = u 2 + µu r , where µ = ✓-=-6 , ( µD + 211 )(11, - K(11 ) J = ,,, - K(v ) .

Let us assu me_for the moment that K possesses a recu r ­ sion operator D and that furthermore the higher -order analogs u 1 = !{en =D/K are related to the higher-order analogs of the KdV equation by the same formula ,

On t h e other hand, D t • u rD- 1 • 11 = 1t rt D- 1 • u + up- 1 • u t + u rD- 1 • u D t . Hence

[D t , D ] = !1m t + iu rt n- 1 • u + � u rn- 1 • 11 t .

Compar ing the expressions for [ A , OJ and [D, , OJ shows that condition (8) holds formally under the plausible assumption that 11 0 - 1 • " t and u n- 1 • (u x + 1/u ) define the same operato; modulo the icteal {,� � R(u )}. We con­ c lude that the flows "• = K' 1 '(u) = D'K(u)

all preserve the modified KdV equation . The first few of these flows are 2 ut = K ( O l (u) = u:r:rx + 1t u r ,

2 lt t = K O l (u) = ll rr:r:rr + jn u x:r.i: + ?j uu i' rx + iu! + �-u u ,: , 4

u t :::: K

'( u ) [ v J = t ,J> ( u + €v) I uE

e -0

has to exist and the map ,J> '( u )[ · ] is assumed to be linear. In addition we always assume that the chain rule holds (in a topological context that means that we deal with Hadamard derivatives). If no confusion can arise we write or '[ · ] instead o f ,J> ( u ) and ,J>'(u)[·]. An operator II : S* ➔ S is called symmetric if

241 B. Fuchssteiner and A. S. Fokas/Symplectic structures

(a, Ob) = (b, 8a) for all a, b E S*, and skew­ symmetric if (a, 8b) = -(b, 8a) always holds. For operators J: S ➔ s• these notions are

defined analogously. A function ,f,: S ➔ S • is said to be a gradient function if it has a poten­ tial, i.e. a map p : S ➔ R such that

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(,f,(u),v) = p '(u)[v ], for all u, v E S .

A necessary and sufficient condition for the existence of a potential is that ,f,'(v) is always symmetric. Then a potential p is given by

J I

p(v) =

(,f,(,\v), v ) d,\ .

(2)

Basic notions

This paper is concerned with evolution equa­ tions ( I ) . An important role with respect to symmetries is then played by the derivative K ' ,;,, K '(u)[·J and its transposed K '*. (i) A function o-: M ➔ S i s called a symmetry of ( I ) if we have always

Definition I .

o-'[KJ - K '[o-J = 0.

49

operators s• ➔ S is said to be a Noether opera­ tor for ( I) if we have always 8'(K J - 8K '* - K'8 = 0 .

(v) A function J from M into the space of operators S ➔ s• is called an inverse Noether operator if we have always:

J'[KJ + JK' + K '*J = 0 .

Note that all the functions considered above are functions of the variable u E M, and by 'always' we mean that the relations above hold identic­ ally in this variable. Remark I .

All the notions we have considered above are intimately connected with the evolution given by ( I ). To see this, take an arbitrary solution u(t) of ( I ), put K'(t) = K'(u(t)), K '*(t) = K'*(u(t)) and consider the corresponding per­ turbation equation v, = K '(t)v,

(3)

v(t) E S

and its adjoint w,

= - K '*(t)w,

w(t) E

S* .

(4)

O"(u(t)), -y(t) = -y(u (t)), etc., we

(ii) y: M ➔ s• is said to be a conserved covariant of ( I ) if we have always

Then for have:

y'(K J + K '*(y] = 0 .

(i) O"(t) is a solution of (3), (ii) y(t) is a solution of (4), (iii) 4>(1) maps solutions of (3) into solutions of

The reason for this name is that if y also satisfies y' = ( y')*, then y is the gradient of a conserved quantity (see section 4). (iii) A function from M into the space of operators S ➔ S is called a strong symmetry of ( I ) if we have always '[KJ - (K', J = 0 . (iv) A function 8 from ,M into the space of

17( 1) =

(3),

(iv) (r )* maps solutions of (4) into solutions of (4), (v) 1/(t) maps solutions of (4) into solutions of (3),

(vi) J(t) maps solutions of (3) into solutions of (4).

Furthermore-in case that the initial value prob-

242

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50

B. Fuchssteiner and A. S. Fokas/Symplectic structures

lem for (3)-(4) is properly posed-the validity of (i) to (v) (for any solution u ( t ) of ( 1 )) is equivalent to the properties required in definition I .

Jacobi identity is fulfilled for the bracket { } defined on S*' by

This remark suggests that there are very many relations between these notions. Let us list the basic ones:

However, the bracket { } makes also sense (and plays an important role) even for operators 8 which are not invertible.

Remark

Definition 2.

2.

(i) CT and 8 -y are symmetries, JCT is a con­ served covariant; (ii) 81 is a strong symmetry ; (iii) 8 is a Noether operator and J is an inverse Noether operator; (iv) taking the inverse (if it exists) goes from Noether operators to inverse Noether operators (and vice versa); (v) the set of strong symmetries is an algebra. All these statements can be checked by ele­ mentary differential calculus.

2. Symplectic-implectic operators, Hamiltonian systems and Magri's results An operator-valued function J ( u ) : S ➔ S * , u E M, which is skew-symmetric is said to be a symplectic operator if the bracket defined on S' by [a, b, e ll = [a, b, c ll( u )

* (J '(u )[a ]b, c)

(5a)

satisfies the Jacobi identity

[a, b, c D + lb, c, a D + [c, a, b l = 0 for all a, b, c E s. 0� (The above means that the e xterior derivative of the corresponding two-form vanishes, i.e. the two-form is closed.) If J has an inverse 8 = r' there is a cor­ responding identity for 8. In that case the

{a, b, c) = {a, b, c ) ( u ) = ( b, 8 '( u )[8(u ) a ] c) .

An operator-valued function 8(u ):

s•

(6)

➔

S,

u E M, which is skew-symmetric is called i m ­ plectic if the bracket defined by (6) satisfies the

Jacobi identity, i.e.,

(b, 8 '[8a ]c ) + (c, 8'[8b ] a ) + (a, 8'[ 8c ] b ) = 0, for (7) all a, b, C E s• . Implectic is-by abuse of language-a short form for inverse symplectic ; in the literature there are also other names for this structure, c.f. [2] or [ 1 7]. We do not call 8 co-symplectic because this would imply that e- 1 exists ; however, the e xistence of e - 1 is not necessary here. It should be observed that every constant skew-symmetric operator is implectic. Obviously the inverse of symplectic operators (if they exist) are implectic. We shall now give a formula playing an im­ portant role in the characterization of implectic operators : Let f be a function S ➔ s• and let 8 ( u ): s• ➔ S, u E M, be a skew-symmetric operator. Then if G (u ) = 8 ( u )f(u ) it follows that for all a, b E S * (b,(8'[G] - BG '* - G'8)a ) = ( b. 8'[8f]a) + (f, 8'[8a ] b ) + (a, 8 '[ 8b ]f) - (b, 8(f' - f'*)8a) .

(8a)

(Sb)

The above formula is a consequence of the skew-symmetry of 8 and of the obvious iden-

243 B. Fuchssteiner and A. S. Fokas/ Symplectic structures

Further in the set of functions a-; S ➔ S one considers the Lie-algebra commutator

tities: G '[ · l = 0 '[ lf + of'[ l . (b, 0G ' * [ a ]) = - (lib, G '*[a ]) = - (a, G '[ 0b ]) .

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Proposition 1 . Let the 0(u), u E M, be skew­ symmetric operators S* ➔ S. Then the following are equivalent : (i) /J is implectic ; (ii) /J is a Noether operator for every evolu ­ tion equation of the form u, = 0( u )f(u), f gradient function ;

(9)

(iii) for all gradient functions f and g we have ( /Jf) '[ /Jg ] - ( IJg) '[IJf ] = 0(f, /Jg) '[ ·] . Proof.

⇒

(i) (ii): Put G ( u ) = 0( u )f(u). Then since f is a gradient function we have f' = f'*. Hence the right side of (8b) vanishes by virtue of (i). So, the left side must be equal to zero, and /J is a Noether operator for the evolution equation u, = G( u ). (ii) (i): One has to keep in mind that con­ stant functions are gradient functions. So, from (ii) it follows that the right side of (8b) must vanish for arbitrary constant f. This implies (i). (i)�(iii): Performing the differentiations in (iii) for constant functions f,g one sees that (ii) is a special case of (iii). For nonconstant gradient functions f and g the contributions due to the f' and g ' of the right and left side of (iii) cancel out.

⇒

Let us list some corollaries of eq. (Sb) and of the above proposition.

Remark 3 .

(i) (Gelfand and Dikii or Magri) If f and g are gradients and 0 is implectic one usually defines a Poisson bracket by If, g H = (f, t!g) ·

51

Then (iii) of prop. ! says that the operation li ( u ) x gradient maps the Poisson bracket onto the commutator of the Lie algebra. This is well known (cf. [ 1 3] or [ 1 8]). (ii) (Noether's law). Usually the equivalence of (i)�(iii) of prop. 1 is used to show that /J maps the gradients of conservation laws of the Hamiltonian system (9) onto symmetries of that system (cf. [ 18]). But (i)�(ii) actually tells us more. N amely, that /J maps conserved covariants of (9) onto sym­ metries ( whether the conserved covariants are gradients or not). (iii) Assume that /J is invertible or that /J(u)S* is weak-star dense in S. Then if /J is a Noether operator for the equation u, = O ( u )f(u) and O is implectic, then f must be a gradient function. This follows directly from (8b): (a, O(f' - f'*)Ob) = 0, for all a, b E S * . Hence, f' must be symmetric. Let us recall that a Hamiltonian system is an evolution equation of the form u, = /J(u)f(u),

u(t) E M ,

where /J is implectic and f is a gradient function. We call a system bi-Hamiltonian if it can be written in two different ways as a Hamiltonian system, i.e.

u, = K ( u ) where

( 10a) (!Ob)

with implectic 9,, 9 2 and gradient functions f ,, f,.

244 52

B. Fuc:hssteiner and A. S. Fokas/Symplectic structures

Now, using (iii), (iv) of remark 2 and prop. I (ii) we immediately obtain,

Magri's construction of a strong symmetry : Consider the bi-Hamiltonian system ( 10) and assume that 8 1 is invertible. Then

terion for compatibility is provided by con­ sidering the following brackets which are mixed, that is involve both 11 , and 82

*

la, b, c j . (b, 8 i [ ll2 a)c) , la, b, c l, = (b, lli[8 1 a ] c ) . A simple calculation yields :

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is a strong symmetry for u, = K(u). In his beautiful paper [ 1 8] Magri has shown that many of the popular soliton equations are bi-Hamiltonian systems (in fact they are, more or less, N -Hamiltonian systems for arbitrary N in a sense to be explained later.)

3. Hereditary symmetries admitting a symplec• tlc-lmplectlc factorization. Fir�t let us recall the notion of hereditary function operator-valued An symmetry. 4>(u ) : S ➔ S, u E M, i s called a hereditary sym­ metry [ IO] if ['(u), (u)] is a symmetric bil­ inear operator for all u E M. By this we mean that ( v, w) ➔ '[ v ] w - '( v ] w

is symmetric i n v, w E S . The importance of these operators lies in the fact [ I O] that if a hereditary symmetry is a strong symmetry for 11,

= K(u)

then. it is also a strong symmetry for the equa­ tion

Lemma I (see footnote I O in [ 1 8]): 11 1 a n d 11 , are compatible if and only if the bracket la, b, c l = la, b, c l , + la, b, cl, satisfies the Jacobi identity. An immediate consequence of the above is

Remark 4. If 11 , . 11, are compatible, then 11 1 + all, are, for all a E R implectic. Theorem I. Let 8,. 82 be compatible implectic operators and assume that 82 is invertible. Then = 11 , ll , ' is hereditary. Proof. B y { }, and { }, we denote the brackets with respect to 11, and 11,, respectively. Consider arbitrary b E S* and a, c E S and let b = 1 'b, cl = 11 , ' a, c = 8 2 c. We have to prove that A = (b, [', ](a, c )} - (b, [', J(c, a )} is equal to zero. So, let us calculate this quan­ tity : 1 A = (b, ll i [ 8 , 11 , ' a J B , ' c ) - (b, ll i [ ll , 8 , 1 c ) ll 2 a ) 1 1 - (b, 8 , 8 , 1 11 i [ a J 8 , ' c ) + (b, 8 1 8 2 9 i [ c ] 8 2 a ) - (b, 8 , 8 , ' 8i[8 , 8 , ' a l8 , ' c ) + ( b , 9 1 9 2 1 8i[ 8 , 8 2 1 c J 8 , ' a ) 1 1 + ( b , 8 , 8 , ' 8 , 82 8i[ a J 8 2 c ) - (b, 9 , 8 2 ' 9 , 8 , ' 8i(c J 8 , ' a ).

u, = ( u ) K ( u ) .

The first line of this expression is equal to

N o w , consider t w o implectic operators 8,(u) and 8 2(u). We call them compatible if 8 1 ( u ) + 9,(u ) is again an implectic operator. A cri-

{cl ,

b, c }, + /c, a, b } , .

By the Jacobi identity and by the definition of

245 B.

Fuchssteiner and A.

I I, this then equals . -{b, c, ii ) = l b, ii, c l ,

( I l a)

( I l b)

Theorem 2. Under the assumptions of theorem I all the operators "112, n = I , 2 , . . . are implectic.

and the third line equals

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- Iii, 6, cl, + l e, b,ii l, .

( I l e)

Finally , for the fourth line we get {ii, *6, c l, - /c, *b, a l, . Again , by the Jacobi identity and the definition of I I, this is shown to be equal to {*6, a, cl, = (a, 11,r11, 6Jc) = 16, a, c l,.

( I I d)

Hence , we obtain A = l b, ii, c l, + la, c, b l , + I t, 6, a l , + 1 6, a, c l, + la, t, 61; + le, 6, a l, . Then lemma I implies A = 0.

Remark 5.

53

Foka1/Symplectic strucrures

Backlund transformations (see section 5). Another method to find new implectic operators is given by:

The second line is equal to Iii, c, bl , + l e, 6, ii i , ,

S.

( I l e)

■

(i) Actually the proof of the theorem shows a little bit m ore. If (*t' exists or, if * s• is (weak-star) dense in s• then by virtue of (I l e) we can conclude that if is hereditary, then 11 1 and 112 are compatible. Hence if •s• is always then 8 , , 82 are compatible if and dense in only if 11, 8 , ' i s hereditary. (ii) If in addition to the assumptions of theorem I 11 1 is also invertible, then and - 1 are hereditary.

s•

The above theorem makes it desirable to find as many implectic operators as possible, in order to construct hereditary symmetries from the compatible pairs. A systematic method to construct new implectic operators is to use

Proof. To avoid the cumbersome computing of the derivatives involved we give a short proof for a restricted case , namely for the case that 8 1 - /\82 are invertible for all /\ in a neighbour­ hood of zero. Since 11, - /\1/2 are again implectic , the opera­ tors ( 8 1 - /\ 82t ' are symplectic. Since the sym­ plectic operators form a vector space the n th derivative df. ( 11 , - /\ 11,t ' ,

n = 0, I, . . .

at /\ equal to zero must again be symplectic. But this n th derivative is equal to 8 1 1 (82 8 1 1 )". Hence, the inverse of that operator must be implectic. But this inverse is equal to =

'8 1

" 1 112,

n = 0, 1 , 2, . . .

■

4. Motivation and Illustrations of the basic notions In this section we illustrate the notions of a gradient function , a symmetry , a conserved covariant, a strong symmetry , a Noether operator and an implectic operator by using the KdV as an example. We shall consider the KdV in the form U1

=

U x:rx + 6UUx .

Then K ( u ) = u,,, + 6uu.,

K ' = D' + 6uD + 6u, .

246 B. Fuchssteiner and A. S. Fokas/Symplectic structures

54

4. 1 The gradient of a functional Recall that the gradient of a functional I is defined by the equation

I'[v] = (f'[v ) , g) + (f, g'[v]) = ((f ')*[g), V) + ((g')* [f), V ) .

I '[ v ] = (grad I, v) ,

Hence

Example I . Let

It is well known that 1' is a gradient function, i.e. there exists a potential I such that 1' = grad I, iff y' = (y')*.

where I '[v] "' a/ad(u + ev)l,-o and ( ,) denotes the relevant scalar product.

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where f, g are functions of x, u, u,, . . . . Then

I=

Example 3. Let

f ( - ¥ + u') dx.

Then I'[v) =

J

'Y = ll:u + 3u 2 ,

(-u,v, + 3u 2 v ) dx =

= (u,, + 3u 2 , v) .

J

then ( u,, + 3 u 2) v dx

4.2 Symmetries

grad I = u,, + 3 u 2 • Note that in general if

J

p ( u ) dx, p (u) = p (x, u, u,, . . . ) ,

then

grad I = (� - D __!_ + D 2 _a_. - . . . ) p ( u) . Ou

au:(

Example 2. Let I=

f f( u )g(u) dx,

y ' = D 2 + 6u = ( y')*

Hence, 1' is a gradient function as expecteJ since its potential is the functional I of example I.

Hence

I=

grad I = (f')*[g] + (g')*[f] .

Ouu

For a motivation of the definition of sym­ metries given here the reader is referred to [6). It is obvious that the KdV is invariant under translation in x. To this invariance there cor­ responds (see [6)) the symmetry a = u,. Let us verify this: u

= Ux, u' = D ,

then

K'[a) - Franco Magri

lstituto di Marematica del Politecnic-o, Piazza Lt•onardo da Vinci 32. 21 033 Milano. /tail'

(Received 18 April 1 977)

A method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators, It leads to a simple and sufficiently genera) model of integrable Hamiltonian equation. of which the Korteweg�e Vries equation. the modified Korteweg--de Vries equation, the nonlinear Schrodinger equation and the so-caJled Harry Dym equation turn out to be particular examples.

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INTRODUCTION The aim of this paper i s to suggest a constructive approach to the infinite- dimensional integrable Hamil­

tonian equations, i. e. , to the evolution equations pos­

sessing an infinite sequence of independent integrals which are in involution. The present analysis is based on the study of the connection between the symmetries and the conservation laws of the evolution equations. The main result is in showing a simple model of inte­ grable Hamiltonian equation, of which the Korteweg­ de Vries equation, the modified Korteweg- de Vries equation, the nonlinear Schrodinger equation, and the so- c alled Harry Dym equation turn out to be particular examples.

The analysis proceeds as follows. In Sec . 1 it is shown that any conservation law of an infinite- dimen­ sional Hamiltonian equation is connected with a sym­ metry transformation. The study of the connection be­ tween the symmetries and the conservation laws of a given evolution equation is thus reduced to the study of its Hamiltonian structures. In Sec . 2 it is shown by an example ihat a given evolution equation may be endowed with different Hamiltonian structures , Each of them provides a way of connecting the conservation laws with the symmetries. Let us then consider an equation endowed with two of such connections, and let us use

the former to associate the conservation laws with the symmetries and the latter to conversely associate the symmetries with the c onservation laws. One is thus able to obtain a new conservation law from a given one. In Sec. 3 it is shown that highly ordered chains of inte­ grals which are in involution can be constructed in this way for special twofold H amiltonian equations. Suc h equations provide a simple model of integrable H amil­ tonian equation. The examples of Sec . 5 seem to sug­ gest that this model is not only conceptually simple but also effective in the applications. 1. SYMMETR I ES AND CONSE RVATION LAWS OF HAM I LTON IAN EVOLUTION EQUATIONS In thi s section an operator approach to the symme­ tries and to the conservation laws of any system of a> This work has been sponsored by the Consiglio Nazlonale delle Ricerche , Gruppo per la Fisica-Matematica. 1 1 56

J. Math. Phys. 19(5), May 1978

evolution equations (1 . 1 ) o ,1t A 0E , t ) = k A (u s , u7, u 7. , • • · ) , is suggested. The field functions u A (x , t) are supposed to be defined, at any instant of time,-in a fixed region

SJ of R 3 and to vanish on the boundary of this region; the subscripts denote the partial derivatives of these functions with respec t to the spac e coordinates x1 •

We set up the study of Eq. ( 1 . 1) into the linear space U of the field functions regarded as functions of the spac e coordinates only . Consequently , any n- tuple u A (x , t 0 ) will be simply denoted by u (t 0 ) and will be re­ ferred to as a point of th is spac e. The given evolution equations will be synthetized into the single operator equation

( 1 . 2)

a,u = K(u) ,

where K is the formal differential operator defined by the functions k A (u s , uf, • • • ). The space U will be c alled the configuration space associated with the abstract evolution equation (1. 2). The purpose of this operator approach is to suggest a simple way of extending to in­ finite- dimensional systems the geometric analysis de­ veloped for the c lassical Hamiltonian mechanics in the phase space, 1

A. Symmetries The object of the theory of the symmetries is the study of the manifold of the solutions of Eq. (1 . 2 ) in the configuration space U. We s hall limit ourselves to a local study of such a manifold and so we only consider the infinitesimal symmetry transformations. They are th e infinitesimal point transformations

u = u + ,S(u) ,

(1 . 3)

o f t h e configuration space into i tself which map every solution again into a solution. 2 The operator S is called the generator of the symmetry mapping and is regarded as defining a "contravariant" vector field on the space

U. The lines of this vector field are the orbits of the symmetry mapping.

The symmetry condition is readily obtained if one ob­ serves that for any solution u(t) it is a,ii - K(ii) 1 �} > a,,u + f o ,S(u) - K,(a) - EK�(u) © 1978 American Institute of Physics

1 1 56

260 '�' ,[ s ; a ,11 - K:S (u ) ]

(I . 4)

' �A[S{,K (ll ) - K{,8(11 ) ] ,

where s: is the Gateaux derivative of the operator S, which is supposed to not expliciUy depend on the lime (see Appendix A). Hence, the S) m metry condition is 1

(1. 5)

s;K(u) - K:S(u) ,g, 0,

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where t h e symbol !e means that the equality is required to hold only !or the solutions. For simplicity , however, in this paper we shall only consider the symmetry gen­ erators !or which condition ( I . 5) is identically verilied (the condition being a fortiori verified on the manifold of the solutions).

Equation (I. 5) expresses the structural relation which connects the given equation to its symmetries, inde­ pendently of the specific form either of the equation or of the symmetry mappings. It is a commutation rela­ tion, the left side being the commutator of the two non­ linear operators S and K. 3 Therefore , the set of the generators of the symmetry mappings constitutes a Lie algebra. This m eans that if two of such generators S 1 and s. are composed according to the formula [s , , s.j (u) = s;,s,(u) - s;,s,(u) ,

(1. 6 )

a third generator is obtained again. B. Conservation laws

The study of the manifold of the solutions is the ob­ ject also of the theory of the conservation laws , but the standpoint is different and, so to speak, dual to that of the theory ol the symmetries.

Besides the configuration space U, one considers a second space V, put in duality w ith U.by a convenient bilinear foi-m (v , u), 4 and then one considers the opera­ tors Q : u - V (see Fig. 1 ) . Such operators may be re­ garded as defining the "covariant" vector fields on U, For such lields it is possible to introduce the concept of elementary circulation

oc = (Q(u) , oo),

(I . 7)

and so it is possible to consider the conservative co­ variant vector fields, for which the circulation does not depend on the line but only on the endpoints. As i s known, ' in order that t h e field be conservative it is

C O N T R O VA R J A N T

F I ELDS

C OV A R I A N T

FI E LDS

u

M A N I FO L D OF THE

SOL U T I O N S

FIG, 2.

necessary and sufficient that (Q� du , M = (Q: 6u , du),

(1 . 8)

oF[u] = (Q(u) , oo).

( 1 . 9)

!or any pair ol variations du and Ou ol the field !unc­ tions. The operators Q verifying this condition are called potential operators . • For such operators the circulation lrom a fixed point u 0 to any point u defines a functional F[u], so that the elementary circulation is given by For this reason, the operator Q is also called the gradient of the functional F,

The theory of the conservation laws associates a special set of conservative covariant vector fields with the given equation by the requirement that the corre­ sponding functionals keep their value F[u(t)] independent of t for any solution u(t), These functionals are c alled mtegrals 5 of the given equation and the c orresponding potential operators may be called "mtegratmg" opera­ to rs. ' Therefore, the theory of the symmetries studies the manifold of the solutions by using contravariant vector fields while the theory of the conservation laws studies the same manifold by using covariant vector fields. In this sense the two formalisms are dual. The following condition (Q(u), K(u)) !e 0,

(1. 10)

on the integrating operators Q is readily obtained if one observes that it is a ,F[u(t)] ' 1='' (Q(u), a ,u) \

I

' I I/

!

/'

< v,u >

(Q(u), K(u)), (1 . 1 1 ) fo r any solution u(t) . A s i n the c as e o f the symmetry generators, however, we shall only c onsider the inte­ grating operators for which the c ondition (1 . 10) i s identically verified. IR

C. Connecting the conservation laws with the symmetries The problem of c onnecting the conservation laws with

FIG. I, 1 1 57

'1="

.>

J. Math. Phys., Vol. 19, No. 5, May 1978

Franco Magri

\ 1 57

261 the symmetries requires the introduction of a metric

operator, which associates the covariant with the con­ travariant vector fields as is usual in Riemannian

geometry . By a metric operator it is meant a linear operator L, : V - U (which may nonlinearly depend on the point u) mapping the covariant vector fields Q1 into the c ontravariant vector fields S 1 according to the relation S,(u) = L.Qiu)

(1. 12)

(examples will be give n in Secs. 2 and 5 , see Fig. 2 ).

It is the purpose of this subsec tion to study the spe­ cial class of the metric operators verifying the follow­

ing two conditions:

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(dv , L,Ov) = - (Ov , L, dv),

(flv , L;(ov ; L,av )) + (6v , L;(av ; L, dv)) + (.i.11 , L;(dv ; L,ov)) = 0 ,

To this end, consider any operator S; associated with

(1. 15)

( s e e Appendix B). This condition i mplies that t h e com­ mutator [S J o S,] of any pair of H amiltonian operators verifies the relation

< ,g (flv , L,Q,,.S,(u) + L;(Q,(u) ; S,(u)) - S,,L,Q;(u )) >

F

( �. : 1i -� I

a

0

a,

1 · -------- i' · � \

t

s

s,

I ----------- [ S; S . ] � C O M M U T A T OR

FIG. :J 1 1 58

J. Moth.

s;,

Phys .. Vol. 1 9 . No. 5, Moy 1 978

( 1 . 17)

where is the adjoint of s;, with respect to the pre­ fixed bilinear form (,, , u) (see Appendix A). Sinc e the

operator

(1 . 1 8 )

Q,.(u) a Q;,s,(u) + s;.Q,(u) ,

F 1,[u ] = (Q1 (u) , S,(u)),

(1 . 1 9 )

6F 1,[u )

= (Qj,6" , S,(u)) + (Q,(u ) , s;,ou)

«� (Q,,S, (u), liu) + (s;,Q1 (u), ou) >

= (Qj,S,(u) + s,.Qiu) , 6u) ,

(1. 20)

relation (1 . 1 7) shows that [ S ; , S , ] is again a Hami ltoni an operator relative to L u . Therefore, the operators S i make a Lie algebra, and this Lie algebra structure in­ duces a corresponding structure on the operators Q i and on the functionals F; acc ording to the scheh1e of Fig. 3 . The func tional Fi k is the Poisson brae/wt of the functionals F1 and F, associated with the Hamiltonian operators S1 and S, (see Ref. 3 , Sec. 5). A simple property of this algebraic structure is that the c ondition F 1 ,[n ] = (Q1 (u ) , S,(u)) = 0

( 1 . 21 )

[S 1 , S,] = 0.

( 1 . 22)

?," = S,(u) • L,Q,(u ) ,

(1 . 2 3 )

implies

a (dv , S,,.S,(u) - s;,.s,(u))

F

[S;, S,](u) = L,(Q1,.S,(u) + s;,.Q/u)),

( 1 . 14)

a potential operator Q, by means of a symplectic opera­ tor L., . It is c alled a Hamiltonian operator and it veri­ fies the following condition,

(flv , [ s, , s,j (u))

which s hows that

as is proved by

spec t to the prefixed bilinear form (v , u).

(1. 16)

":,'," (dv , L,(Q 1,.S,(u) + S,,Q1 (u))),

i s the gradient o f the func tional

generators of the evolution equations. Such metric operators will be called sy mplectic operators with re­

= (flv , L;(ot' ; S1 (u))),

- (Q1 (u) , S,,L. dv) - (dv , L;(Q1 (u); S,(u)))

(1. 13)

where L; is the Gateaux derivative of Lu with respect to u (see Appendix B), and to show that they allow to connect the integrating operators with the symmetry

(dv , S 1 ,L,ov) - (6v , Sj,L, dv)

" :,'," (dv , L.Q,,.S,(u)) + (dv , L; (Q, (u) ; S,(u)))

From the point of view of the theory of the symmetries and of the conservation laws, c onditions (L 2 1 ) and ( 1 . 22) mean that Q1 and S 1 are respectively an integrat­ ing operator and a symmetry generator of the evolution equation

Therefore, t h e symplec tic operator L,. , associating Q; with S; according to ( 1 . 1 2 ) , connects the integrating operators of Eq. ( 1 . 23) with its symmetry generators . This property explains t h e importance of t h e symplectic operators in the study of the evolution equation s ,

T h e problem of connecting: t h e conservation laws with the sym metries o[ the �iven equation {I. 2 ) is thus re­ duced to thal of recasting this equation into the H amil­ tonian form (1 . 23), hv decomposing the operator K as follows,

(1 . 2 4 )

K(1t ) = L,Q(1t ) ,

where L 11 is a suilable symplec t i c operator and Q is a potential operator. To fin d such operators, if any , it is useful to observe that, ac cording: to ( 1 . 1 5) , L 11 must be F ranco Magri

1 1 58

262 (d,, , K'.L,6,,) - (6v , K'.L, dv) = (dv, L'.( 6v ; K(u))),

(I . 25)

operator Q2 • By iterating this process, first obtain from Q, s, (u) a M,,Q,(u)

(Q(u) , K(u)) = (Q (u) , L,Q(u)) "�" 0

(1 . 26)

and then

coupled to K by the condition

and that Q must be an integrating operator of the given equation , as is proved by

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(examples will be given in Sec . 5). When condition ( I . 25) is fulfilled, one says that the symplectic opera­ tor L u makes the given equation Hamiltonian. It c an thus be stated that every symplectic operator L v making the given equation Hamiltonian maps its integrating operators Q1 into its symmetry generators S i according to relation (1 . 1 2 ). This is the main result on which the following analysis of the integrable Hamiltonian equa­ tions rests upon. 2, AN EXAMPLE: THE KORTEWEG-de V R I ES EQUATION It is the purpose of this section to show by an example how the integrable H amiltonian equations may be analy­ zed by using only the connection between the symmetries and the integrating operators previously pointed out. Consider the KdV equation u 1 + aUU:c + uxxx = O

(2. 1 )

and observe that it admits the following two Hamilton­ ian decompositions: U.t + ax(f u2 + ux..) = 0, U: + .(a,:x,: + }auax

+f U,: l)u = O,

where the operators

L ui:p = ,.) such that q,(x,O) = i/,o(x) , cf,(x,1) = i/,1(x) , q,( � ,>..) = 0. It is readily seen that the relation of homotopy is an equivalence relation, or that it is possible to divide the col­ lection of all continuous fields ct,(x) up into classes of mutually homotopic fields. The classes resulting from this partition are called the homotopy classes of the theory. For us their fundamental property is the fact that the passage of time, being a homotopy, cannot make the field j ump from one homotopy class to an­ other ; and the main property of I for us is that it labels the homotopy class in a 1-1 manner : two fields are homotopic if and only if they have the same index. Finally we point out that the homotopy classes are themselves the elements of an additive group in a natural way. Namely, if C and C' are such classes then C © C' is formed by selecting representatives , ct,' of each class, respectively, and taking the class containing cf>(x) + cf>'(x) . Thus we have a group whose elements are constructed in the following way, to summarize : We start with the circle C : 0 :::; :::; 21r on which 4> _ranges ; we consider functions defined on the infinite I-dimensional x interval with values on C, assuming a fixed pre-assigned value on the boundary of the infinite interval. Then we . class to­ gether or identify functions of this type which are homotopic to each other. The resulting classes are the elements of this group ; and the group operation has been described in a special way above. The group constructed for the circle in this manner is called the Poincare group or the fundamental group of the circle, 1r1 (C) . For later use, we first point out the generalization of this construction which consists in replacing the ex­ pression " I-dimensional interval" in the above paragraph by "n-dimensional interval", or Euclidean n-space. We are then dealing with cp-fields in n-space. The resulting group is called the nth homotopy group of the circle, 1rn ( C) . (For n > 1, 1r n (C) consists of one element, zero, so cf>-fields cannot have twists in higher dimensional spaces.) Secondly in the next section we will have to con­ sider taking the nth homotopy group of other spaces than the circle, and there the definition of the group operation must be recast in more general form. The index I = /[ct,(x) ] establishes an isomorphism between this group and the additive group of the integers : I[C © C'] = I[C] + I[C']. The homotopy class belonging to I = I is a generator of the entire group, and thus represents the basic structure of "'hich all others are composites. li'or this one-dimensional scalar field the nature of this basic twist is easily visualized. In the next section we consider a field with 10 components in 3 dimensions. There the intuition boggles and the professional topologist must be invoked.

273 N EW COXSERVATIO N LAWS

237

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I I I . ILL USTRAT I V E EXAMPLE : G E � E R A L R E LATIV ITY

The underlying continuum has one t ime and three space dimeusions. A hist ory of t he system is a metric field g a /l(x0 , :r 1 , :r2 , x3 ) , a,/3 = 0, I , 2, :1 , of const ant sig­ nature ( + - - - ) . We assume t he (x " ) coordinate system covers t he continuum neatly, explicitly excluding such t opological st ruct ur �

'.v

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N EW CO NSERVATION LAWS

239

pre:-ent t ime, we will simply call the structure counted by M and schematized in Fig. 2 a metric twist for now. It is readily seen that a metric t wist necessarily contains regions in which t he direct ion of time and causality are anomalous. Suppose it is decided that a vect or at a large distance from the twist is pointing in the direction of increasing x0 \\·hen it is directed t owards the future. Then such a vector when transported in ;.:ide t he t wist according to any continuous rule that keeps it time-like ( of posit ive norm) ,,·ill in some regions point toward decreasing x0 , and on some 2-surface in 8 will be tangent to 8. Moreover each twist is surrounded by a one­ way :-urfacc t hrough which causes can propagate in only one sense ( in or out) . ( 3) 0 If t he t ransformation x0 ---" - x is represented by T then TMT-I = - M ;

M i s changed i n sign b y "time"-reversal (but not b y reversals of the other coordinates) . It will be observed that we do not present a dynamical discussion analogous to t hat of Section I I ; we cannot even m;sert that the free-spa�e gravitational equations possess any regular solutions with M � 0. In any case there are no static twists. In the free-space case, it is possible to assert that there are no cylindrically symmetric stationary twists. If stationary free-space twists do exist at all , there i:,; no constant of the dimensions of a length available in the classical theory to fix their scale ( as the scale of the twists was fixed in Section II) . In­ stead, there would merely be a relation between their active mass, as represented by their asymptotic gravitational field, and their scale of size, the scale of this relation being set presumably by the gravitational constant K ,..._, 10-2 7 cm/g. Of course in a theory with nonvanishing cosmological constant there does exist a natural scale of length, of cosmic magnitude. IV. THE GENERAL CASE

Let w(x ) be a field whose values w at each point x a of space-time form the a same continuum n, and suppose the value (say) of w(x ) at spatial infinity is fixed by boundary conditions. Then we proceed as before to introduce the con­ 2 3 cept of homotopy for ;;tates of the field : wo = wo(x1 x2x3) and w1 = w1(x 1x x ) are homotopic, w0 ,..._, w 1 , means that there exists a continuous function JCX) = 2 J(x 1 x x3 , :\) satisfying the same boundary condition as w(O ::; :\ ::; 1 ) such that f(O) = w0 , f( l ) = w 1 • By uniting mutually homotopic fields into a single class we obtain the elements of the third homotopy group 1r3(Q) . We define the group operation (t) on homotopy classes in the way indicated for the theory of gravity : if w and w' are representatives of two homotopy classes C and C', we iron out the twists in w and w' on opposite sides 8, S' of some plane (that is, we take w = a

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240

FINKELSTEIN AND MISNER

const on S', w' const on S) and then sew them together along the plane ( that is, we define w" = w on S, = w' on S') to form a representative of C EB C'. Then since the passage of time x0 is a homotopy, the dynamical variable whose values correspond to the group elements of 7r3(Q) , the third homotopy group of Q, is constant in time. The assertions of this note are simple consequences of this elementary topological remark. We term the field theory intrinsically nonlinear if Q is not homeomorphic to some vector space. If Q is simply connected in 3 dimensions [7r3 (Q) = O] then the above dynamical variables are the same for all states anyway and the homotopic conservation law is trivial . In particular this is the case if Q is any vector space. For Section 1 I I , Q is the circle S1 and it is the first homotopy group 7r1 ( S ) (the "fundamental group") that appeared because a one-dimensional field theory was treated. For Section III, Q is the collection S4 ,3 of symmetric 4 X 4 matrices of signature + - - - . It happens that both homotopy groups are the same : 7r1 ( S 1 ) = 7r3 ( S4 , 3) = oo , the infinite cyclic group ; indeed the example of Section II was chosen to this end. This is not the general situation, however, and a special calculation of 7ra(Q) is necessary in each case. Nor is it the case that time-reversal T always changes the value of this dynamical variable into its negative with respect to the group operation EB (the " anti-particle") ; for the strip this is ac­ complished by space-reflection instead. The additivity property of this variable is evidently general, in the sense that its values combine according to an additive group under the operation of juxtaposition of field structures. To summarize : to find whether an intrinsically nonlinear field theory possesses new conservation laws of the kind presented here, we form the range Q of the values assumed by the field and ask the topologist for 7r3 (Q) . If 7r3 (Q) = 0, there are no twists ; otherwise there are. If 7r3 (Q) = 2, they possess a conserved parity (oddness or evenness) only. If 7r3 (Q) = oo they possess a conserved number. If 7r3 (Q) = oo + oo , there are two kinds, each with a conserved number. Simple examples of all these species of field theory may be constructed. In the next section , we shall see that we should also inquire about 7r4{Q) . V. ON THE SPACE THEORY OF MATTER

By the space theory of matter we mean the program of the geometrization of physics founded by Riemann and Clifford and given its principle advance hy Einstein . ( 1 ) A peculiarity of the space theory of matter is that the theoretical formulations are so simple that it is at first difficult to make out in them a rich enough complex of qualitatively different structures to image the real world. Then after a little study the picture suddenly reverses itself. To bring this out , we will dwell on the theoretical potentialities of general relativity in this section . First a matter of terminology. The vocabulary of the space theory of matter already contains in addition to the conventional concepts of field theory such special expressions as "geon" , "wormhole" and "metric twist " . \Ve will see below

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NEW CONSERVATION LAWS

24 1

that there is in fact ample opportunity for an infinity of barbarous neologisms. To forestall them we propose an open-ended terminology into which the struc­ tures we describe can be fitted. As root of the system of names let us take Wheel­ er's word "geon" , (4) to be regarded here as pidgin-Greek for "geometrical entity" . It stands for any of the semilocalized structures encountered in the space theory of matter. Conceptually the space theory of matter begins with a blank differentiable manifold, devoid of metric, affine connection , or any other physical fields. How­ ever, even in this void can he found distinguishable structures of interest. We will call these 0-geons. They are of a purely topological nature. If we think of a two-dimensional plane for illustration the simplest structure of the kind in­ tended is gotten by drawing two nonoverlapping disks, removing their interiors, and identifying their edges, or what is much the same thing, j oining them by a tube. This structure is essentially a two-dimensional torus, if we close it off with a point at infinity. In the two dimensional case there is only one other elementary structure which is gotten by identifying the edges of the disks the wrong way around ; this gives the cross-cap or Klein-bottle. The number of possibilities in the three-dimensional case is embarrassing. The essential feature of the disc is that it has an inside and outside which are distinct. In three dimen­ sions, there are many surfaces having this property : the sphere, the torus, the sphere with n-handles, the over-hand knot, the three-ring symbol, · · · . Two replicas of any one of these structures can be used t o construct an 0-geon, and that in more than one way. The simplest possible 0-geon , two spheres with matching orientation , gives the "wormhole" of Wheeler. Any of these structures, if provided with a suitable metric to begin with, possesses an evolution which can be governed by, say, the gravitational equations for empty space, and is thus an obj ect of the space theory of matter. A complete enumeration of all possible 0-geons is not possible at present. Now a metric has to be added to the differentiable manifold. I -x in (l)] defines a field with one antikink. 3. A rotator field :

4> = P•

The three-dimensional projective space P3 is de­ fined as the space of rays in R', i.e., a point of P' is the set of all multiples of some nonzero vector in R'. P3 occurs as the range of the field variable in some field theories. A "rotator" field, whose field variable is an element of SO (3, R) (the rotation group in three dimensions) is an example, since SO(3, R) P'. Because• 1r,(P') = "' , there is a single nontrivial homotopic number. The kink in this field is best described after the next example.

~

4. A wlit 4-vector field : of> = s•

The three-sphere S' is the covering space of the projective sphere P' of the previous example, and 3 ,r3 (S ) = "' (Table 1). S' is the range of the field variable in a theory of strong interactions of Skyrme' which suggests that the fundamental variable is not an isovector .,, = (1r', 1r2, 1r') but an element II of SU,, to which an isovector .,, can be related

according to

n = exp (i.,, • ½-o) (3) for IT sufficiently close to 1. Here ,. is the usual isovector operator (T , T', T') generating the algebra of infinitesimal spin-½ rotation operators. Thus, Skyrme's theory, in addition to implying multipion interactions because of the dynamical nonlinearity, implies the existence of a strictly conserved homo­ topic quantity because of the topological non­ linearity. The class of unit homotopic number is represented in the stereographic coordinates (I) by the identity mapping 1

II = II(u) = u.

With this result we return to Example 3. A rotator field cp,(v) of unit homotopic number can be con­ structed from Il, (v). Let D(II) be the spin l rep­ resentation of the group SU,, defined by Then

D(II)-. = II-.W'.

cp,(v) = D [II,(v)] .

5. General Relativity•

Let us postpone the consideration of topologically nontrivial space-times and continue to describe space-time by four coordinates x• ranging over R', with the boundary conditions g., -> -r• ., (x')'

+ (x')' + (x')' ->

"'

imposed on a distant cylinder. The coordinate x0 is supposed to be timelike on this distant cylinder ; indeed we take -y., = diag (1 , - 1 , - 1 , - 1) .

The previous discussion of conservation of homo­ topic quantities in classical field theories can now be applied, provided we replace the concept of time by the coordinate x0 throughout. The conserved quantity is associated with a cross section x0 = const of space-time ; this cross section is not in general spacelike throughout, and in general there is no cross section which is spacelike throughout. The field variable cp is now a symmetric tensor g., of signature + 1 -3, and the space of these tensors has known homotopy groups. In particular• ,r,() = co . Therefore, there is again just one conserved homo­ topic number, called metricity in an earlier work.' The heuristic argument concerning the definition and the conservation of these homotopic quantities

285 KINKS

1223

=

in quantum field theory is somewhat less than con­ conversely, suggesting that the fundamental field vincing for this case, because the quantization have a built-in double-valuedness with W - 1. process is not a.t all understood when the direction Some physicists have followed the "geometric" path, of time is as tangled as it is in most of the metrical others the "algebraic". universes introduced in this computation. For ex­ 2 . Here, we show that certain appropriately non­ ample, the field of time-directions in a. gravitational linear but single-valued fields ,p(x) admit double­ kink is shown in Ref. 2. valued but continuous wave-functionals iJ>'[,p (x)) that have transformation properties belonging to 6. Displacement field W - 1. We shall inspect each of the kinks we Suppose a. continuous medium undergoes a. con­ have mentioned to see whether it admits odd-spin tinuous differentiable deformation x ----> x' = f(x), parity in this way. We shall also see examples of which can be regarded as an idealized crystal lattice fields which do not admit kinks but still admit deformation. Then, at each point, x is defined a. structures of odd-spin parity. The connection be­ Jacobian matrix l"(x) = ax'/ax belonging to GL(3, R) . tween spin partity and statistics are discussed in We suppose l"(x) ----> 1 as x ----> "' • Are there kinks a later paper. in the field I"? We are led to seek ra[GL(3, R)]. 3. The tool for the study of multivaluedness is the The polar factorization homotopy theory, since a multivalued function on a. = I" p = R", .-,(, ,p0 ) = 1 . No spin here. possess odd-spin parity.' 6. Displacement field 2. Unit 3-vector field The reduction of this problem to the rotator = S' . .,,., = 2 according to Table I, so that double-valued wavefunctionals are admitted by this problem was already performed in the analysis of theory. For the second test we need the action of the the kinks in the displacement field, and enables us rotation group G on . First (cf. Sec. V. 13) the to limit our inspection to a field ,p(x) with ,p in SO,. boundary value 0, and the being a rigid translation ; against it ,f,0 is in neutral .X: real constants. In particular, ,f, = 2116(x - t) can be equilibrium, because w• = 0. Note, again, the analogy interpreted as the limit as v _,. I of ,f,•: with particles. The remaining (unbounded , scattering) solutions of X - Vt ,f," = 4 tan-• exp � - 21tll{x - r). (4. 1 3) (S.2) are obtained by Morse and Feshbach, 1 also as a (1 - V 1' •-• combination of hypcrgeometric functions ; after some Since elements like 2116(x ± t - x:) do not interact manipulations they can be reduced to with each other nor with ,f,, where ,f, satisfies the !f,(x) = (2,r)!we'... (k + im tanh mx), (5.5) sine-Gordon equation, we can have an indeterminate number of "nothings" (for them ,f, = 0 mod 2,r), with where k• = co1 - m1 and - oo < k < co. an arbitrary total particle number and an energy The fact that ( S.2) has this type of solutions means that the potential - (m'/2) cosh-1 mx, for which (5.2) I lim � = co. is the Schrodinger equation (particle of mass = I), ••-• (1 - v, ) is completely transparent at any energy of the incoming Although this strongly suggests a nonquantum particle. This is a remarkable property indeed , and approach to the divergence problem in field theory, we shall elaborate on it later on. we have not pursued this line further. It is easy to check that (5.5) is a solution of (S.2) and 5. SMALL OSCILLATIONS We next analyze small departures from the single kink state, i.e., we consider ,f, = tf,0 + tp, tp « I . The

• L . 0 Landau and E. M. Lifshitz, Quantum M«honlrs, Non• R,latitisli,; Theory (Pergamon Press. London. 1 958). • P. Mrse and H. Feshbach, Methods of Theoretical Physics (Mc],p.(x y,;'(x') ) [e _.., 2w + t - t' ,p.(x),p.(x'). (6.4)

Sm

Comparing (6. 3) and (6.4), we get

Sm[q, tj] = -i, [at , 0:,) = t3(k - k'), For completeness we give here Je in terms of 'I' and (6.5) [at , a•. ] = [a. , q] = [at , 41 = 0. = 'fl ,r : Je i( �:• + 4m 1 s in 1 ! ,f,") The relatiqns (6.5) imply that (6.2) contains an infinite zero-point energy (as i n the ,f,0 = 0 case) and + l(,j,1 + v,! + m 1(1 - 2 cosh-• m x).,2) that Smq can be interpreted, in this approximation, as (5.7) the bona fide momentum of the kink. and the closure relation One point of caution should be made about the quantization process: Although 'I is assumed small in y,.{x)v,.(x') = "(x - x'). dk y,:(x)',p,.(x') + (5.1). it is easy to see that ( fJ,+ 1 , we have n

n-1

/J, = 2 L k, . fJ, = 2 2 k, ' 1 ft-2

1

{33 = 2 L k, + 2 k. , etc., 1

(7. 1 8)

R )'e(fr'P,l• ( R _ 1'2

1 2 1' 1

a.�e

2/J1z

� e'fl2-/J1) ::t = e - 2kn::t, X _., oo , � -2/J.,ua.i.re/JJl ;c t"" 2 , E v < 0 and E,[¢, J is a maximum for varia­ tions in a at a = 1, which implies that the fie ld is not s table agains t oscillations i n a. 4 Consequently in three dime nsions , s ometh i ng more complicated than a set of spinles s fields must be used i f one wants to consider c l as s ical s olutions which lead to a finite and minimized field e nergy -for ex­ ample. fields with spi n . ' To investigate stability in m o re detail, we look for s olutions of the fiel d equations (returning to one dimension) with ,P (x. t) = ¢c(x) + IJ,(x)e•wt. The perturbation IJ, must satisfy

308 1488

J . G O L DSTONE AND R . J A C K IW

[- d� 2 + U"(¢,)] I/' = w2 1p

(2 6 )

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2

and the stability condition i s that all e igenvalues w 2 of th is Schri:idinger-like equation should be non-negative. This is, of course, precisely the requirement that 02 E,(¢ )/04>(x)li ¢ ( y ) I • =• , h ave a non-negative spectrum, which is nec essarily true if E,[ ¢] is to have ·a local minimum at ¢ = ¢ , . The assumed symmetry of U(¢ ) implies that U"(¢ 1 ) = U"( ¢ 2 ) = µ.2 > 0, where µ. is the lowest ap­ proximation to the meson mass in the sector built on either of the vacuum states w ith ¢ = ¢ 1 , 2, U" ( ¢ ) must ce rtainly g o negative betwee n ¢ 1 a n d c/) 2 • In ge neral (2 . 6 ) has a continuous spectru m of eigen ­ values w ' = µ.2 + k ', with ,i, - e'" at large l xl and possibly some isolated bound s tates w ith w2 < µ.2 • But, by differentiating (2 . 2 ) we see that

[- d�•, + U"(cp ,)]

between " in" states, the other between "out" state s . We shall not exhibit the " in, " "out" label explicitly; it will matter only in selecti ng boundary conditions for the differential equations which are encountered below. (Of course the single -baryon -no -meson state is both an "in" state and an "out" state , since the baryon is stable. ) The two sets of matrix el ements are related by the S matrix, which can therefore be calculated onc e both are known. The matrix element ( p ; {k} l 4> l q ; { 1}) is the sum of all possible terms of the form (2,r)6 (kl - 1 1 )(2 ,r)6 (k, - 1, ) ( p; k, . 1 4- lq ; l, · . · ) C in which any numbe r of mes ons are "dis connected" (but not the baryon). 1 3 The essential assumption is that the connected matrix elements between m and n mesons, denoted by the subscript c, have an expansion i n powers of ;. with leading order A < .,+ • -1 )! 2 _ Thus (p 14> jq) is of o rder .>. - 1/ , and (p 14' l q; k) of order .>. 0, while

·

(p ; k j jq; {) = (2ff)0(k - l) (p j 4> jq) + ( p ; kj4> )q; l ) c where the connected part is of order .>. 11 '. Finally we set to zero matrix eleme nts of any product of 4' " s between no-baryon and one -baryon state s, since the baryon is stable (see Sec . V ). We write down the equation of motion for ( x, I ), at first for the ¢ 4 theory a' (x, t) a 2 ( x, t) , + 2m , ( x, 1 ) _- 2>-4> ( x,. t ) , 81 , ax '

and take matrix elements . The left -hand side give s

(4. 1 )

+ [ E (p ) + L w (k, ) - E(q ) -L wU, )] ' + 2m ' }

x ( p ; {1,) 1 4> jq; {1} )

To evaluate the matrix element of 4> ', we i nsert complete sets of multimeson state s . It is then easy to see that we obtain a set of equations for the connec ted matrix elements of the form [ - (t::.p )2 + (AE)2 + 2m')(p; { k}/ 4' /q; {!}) c

� 2.>.( p; { k) 14>' lq; { l)) . , (4 . 2a )

312 J . G O L D S T ON E A N D R . J A C KI W

1492 ( p ; {k} W Jq ; { 1}>.

= }: < P ; { kJ l + J r; { 1 1 }> .< r; {k,} l + l s;

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x ( s ; {kJ l + Jq ; { 1,}> . .

{IJ> .

(4 . 2b )

T h e sum i s ove r baryon momenta r, s and s ets o f me ­ son momenta { k,} • • • { !J consisting of (i ) the external mom enta {k} divided among {k,. } , {k2} , { k,} i n any way and {!} divided among { 1 1 } , {!.} , {IJ i n any way ; (ii) i nternal m om enta each occurring twic e , once i n a set {I} and once in a set {k} to its right. i.e . • i n {1 1 } and {k,} . o r {IJ and {k,} , o r {IJ and {k,} . T he c rucial point is that the leading term on the right -hand side of (4.2a) is now the te rm with no internal meson mom enta and is of order A ]jq)I , ..

= i(211)6( p - q) 6(x - y) .

(4. Ba )

-(p j ♦ ( y , O)l r; {k})( r ; {k}l ♦ (x, O) Jq) [ E (p ) - E ( r) -I: w ( k1 >] = O(x - y )(2 11 ) 6 (p - q ) .

(4 . 8b )

= a sum of terms involving only fewer mesons than J({k},

{!}; x)

(4 .7 )

Insertion of a complete set of intermediate states gives

J. ( J> l + (x, O) I r, {k}) ( r; {k} l + ( y , O ) l q) [E ( r ) - E(q ) + }:w ( k ) ]

,".r.}

1

The no- meson terms give a c ontribution of order >. 0, since each matr ix element is o f order A - 1/• while the

313 1493

Q U A N T I Z A T I O N OF N O N L I N E A R W A V E S

11

e nergy differenc e s are of order >. . The one -meson terms are of the same order, since each matrix ele­ m ent is of order >. 0 and so is w (k). All other contributions are of higher order in >. . Thus we obtain two types of contribution to this sum rule. (i) From the one-meson states

L w ( k)[(p l 4> (x, 0)l r; k) (r; ki ol> ( y , 0 )lq) + ( x - y )] = L w( k ) [ jdxof*(k; x - x )J( k; y - x )e • ->lxo + (x - y )lJ . 0

0

II

r , Jt

iC

If we take for f( k; x) the orthonormal solutions of (4. 6 ) ¢,(x) divided by the u sual boson factor of

[2w(k)] 11 2, then

(4 . 9a )

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f(k; x) = {1 /[2 w (k)]112},µ,(x)

and

L 2 w ( k)J* (k; x)f(k; y ) + N ¢ /(x)¢ / (y ) = o (x - y ) , 2

where the second term is the contribution from the w = 0 state . [Any bound-state solutions of (4. 6) are included in the sum, with f i nterpreted as the matrix element of 4> between the baryon and a baryon ex ­ c ited state . ] The c ontribution of the one - meson terms to the sum rule is the refore b(x - y )(2,r)o(p - q ) -N 2

j dx e• ( , -P)•o¢/(x - x )¢ / ( y - x } 0

0

(ii ) From the kinetic energy of the no-meson states

4--

" (

\P l 4> (x, 0 } j 17 ( rj 4> (y , 0} i q)

2;

y2

q'

(4 . 9b )

0

+ ( P lol> (y , 0 ) j 17 ( ,j ol> (x, 0 ) jq)

2; 2 - q2 y2

"' , (x - xo ) .+. Jdxody o-,, = f .'!!:_ -,,, ( ;• - y o ) (r 2M (2 ,r )

=

=

f

e H r -J>)xo + i( 9 -rho +

2 2 y _ p e H o-T)xo + i ( r -J)yo) 2M

- h H , -,),o ( r -p )( r - q ) "' , (x - xo ) "' E_ fdxod" .,,, ( y - y o ) e ' < • • o • , o-,, M (Z ,r )

M ¢/ (x)¢/ (y ) (2 n }o (p - q ) 1

[the second equality follows from the first by changing r to p + q - r in the second ter m ] . Thus prov ided M = 1 /N' = dx(¢ / )2 = E J ¢ , l , we find that the translation-mode state is not needed to saturate the sum rul e to leading order in >. . This identifies M and shows that it is indeed of o rd e r >. -, . We ought also to determ ine the rest energy of the baryon. The ene rgy is give n by the matrix element of the Hamiltonian density JC:

J

E (p ) = (P IJC (0) I P ) •

p2 )

>. m' 1 1 a 4> ' JC (x) = - l1 2 + - ( - ) - 111 2 4> 2 + - 4> 4 + 2 2 ii x 2 2>. '

(4. 1 1 )

a,i, Il = at

The matrix e lements of products of fields are evaluated by expanding in intermed iate state s. When th e computation is carried to terms of orde r >. -, and >. 0, the relevant i ntermediate states are the following: ( i ) The no-meson s tate gives an

(4 . 10 )

O(>. - , ) contribution dx[½ (¢/ )2 + (1 /2>.)(m 2 - >. ¢ f f ] . (The matrix e l em e nt (p I Il lq) = i[E(p ) - E(q ) ](Pl ol> lq) is order >. 11 2 and is dropped. ) (ii) The O(>. 0 ) te rms come from one -meson state s . IT' gives

J

f (� ) i (p I II lq ; k) ( q ; k l II Ip) =

=

f (� ) �

[w(k)]'(p i,t,i q; k) (q ; k i ol> I P)

J dx L• ½w ( k)1/i:(x),i,, (x) .

The remaining terms are similarly evaluated and the total energy becomes E (P ) = fdx[ ½ { ¢/ l' + U(¢ c l] + ½ +½

j dx G -' (x, x)

f dx dy G(x, y ) [- ::, + U" (!l' ' at

(5 . la )

'f x

(5 . lb )

where

H = -,

o2 ¢] d 6 ¢ (x)6¢ (x) + E J

Thus the c lass ic al field e nergy Ej ¢ ] plays the role of a potential e ne rgy in the Schrodinger e quation. E.[ ¢ ] is infinite unles s ¢ (x) - ¢ 1 or ¢ 2 as x - ±«>, and so the c onfigurati on space is divided into four regions s eparated by infinite pote ntial barriers. The s olutions of Eq. (5. 1 ) may be divided into four sectors, in each of which >1'[¢] is nonzer o in only one of the regions. The whole space of s tates ;s divided into four orthog­ onal subspaces, listed below, with no trans itions between subspac e s . Sector I. >1'[ ¢ ] = 0 unless ¢ (x) - ¢ 1 as x - ± "". The lowest energy eigens tate s are multimeson s tate s built on a vacuu m 11 1 • Howeve r, the clas­ sical multibaryon scattering states with an even number of transition regions alternately from ¢ 1 to ¢ 2 and f rom ¢ 2 to ¢ 1 must also c orrespond to quantum states in th is s ector. Sector II. This is re lated to sector I by the sym­ m et ry which takes ¢ 1 - ¢ 2 and c onsists of s tates built on a vacuum 112 • Sector III. >1' [¢ ] = 0 unle s s ¢ (x) - ¢ 1 as x- -00

1495

and ¢ (x) - ¢ 2 as x- + 00 • The lowes t e nergy eigen­ states are the baryon-multimeson states de ­ s c ribed in this pape r. But there must also be scattering states corre sponding to the clas s ical s olutions w ith an odd number of transition regions alternately from ¢ 1 to ¢ 2 and ¢ 2 to ¢ 1 • Sector JV. This is related to s ector III by th e symmet ry ¢ 1 - ¢ 2 • To make a sensible theo ry of a real (one -di ­ m e nsional ) world, we should retain only two s e c ­ tors , o n e chose n arbitrarily from sectors I and II, and one chosen arbitrarily from s e ctors III and rv. The discarded sectors are identical double s o f those r e tained. Since n o transitions take place between sector I (even number of baryons ) and s ector Ill (odd number of baryons), the baryons c arry an additive quantum number + 1 , which is conserv ed modulo 2. Thus the o ne -baryon s tate i s stable , a s we have assumed. It should be noted that whe th e r a baryon localized at x0 is as­ s ociated with a field expectation value ¢ ,(x - x0 ) or ¢ ,(x0 - x) depe nds on how many baryons are to its left; however, this does not obviously contradict local causality since insertion of an extra baryon far to the left does not affect the physical behavior far to its right, it merely changes the description in effect from sector I to II or III to IV. If this is a c orrect desc ription of the quantized theory, there is no baryon-antibaryon c onjugation. The symmetry ¢ 1 - ¢ 2 is broken, since it send s a retained sector into a discarded sector. How ­ ever, within sector I (x, t) - ( -x, I) is a parity transformation; within sector III this must be com­ bined with the ¢ 1 - ¢ 2 transformation. Thus for the ¢ ' theory, the field 4> behaves as a scalar i n s e ctor I, but as a pseudoscalar in sector Ill . This s tructure is specific to the particular type of model [with a ¢, - ¢ 2 symmetry ] . but should be capable of generalization. For example, still in one s pa c e dimension, in the theory with U(rp ) = 1 - cos ¢ we would retain one s ec tor for each integer N, with rp (oo ) - ¢ (-., ) = 2Nn; there are bary ons and a ntibaryons with a conse rved additive quantum nu mber ± 1 ; the symmetry (x, I ) - - ( x, t) survives as a baryon-antibaryon con­ jugation. We have no syste matic method of verifying th ese c onj e ctures or of calculating scattering of baryons . We should draw attention to the rather puzzling analytic properties of the matrix ele­ ments w e have c alculated . For the ¢' theory, we found (P ' l I P) = A

-,;,f dx e

iC

• -,•,x tanh m x

!!:, A -1/2 1 m sinh(n/2m )(p -p ' ) ·

( 5 2a ) ·

316

(The singularity at P -P ' = 0 is a princ ipal part and is due to the lar ge -x behavior of tanhx. ) We may write this in a Lorentz -covariant form as _ iE.µ P µP , u , ( p [ \ p) - [2 E(p 'u)2E(p )] 112 G (I ) ' t = ( P - P ' ) • (p - p ' )" ,

2 u,\ -v, 1 G(t \ = - IT sin(u/2m )IT · m

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11

J . G O L D S T O N E A N D R . J A C K IW

1496

(5 . 2b )

(5.2c)

T h e pseudoscalar form is expec te d from the argu ­ ment about parity above. There are poles at I = (2 111>1 )'; wheth e r these can be associated with meson thresholds in the c rossed two-baryon to vacuum matrix element is not c lear. Indeed any attempt to interpret Eq. ( 5 . 2 b ) in the c rossed channel immediately runs into the difficulty that i t is antisymmetric under interchange of P and ' . p Is thi s a pecu l iarity of the two-dimensional theory o r a hint that the baryons are fermions ? (The c lassical picture of the multibaryon stat e s does not obviou s ly force a particular statisti c s o n the quantized the ory . ) The s tability of the baryons may als o be under­ s tood in te rms of a conse rvation law. A l l models i n one di mension possess a c onserved current J" = E ""a u (x, I). The c harge is ( 00 , 1 ) - ( -oo, I). In sector I (or II) , matrix elements of this charge are zero, s ince the matrix e lements of the field tend to the same value as x - ±00 , while in sec ­ tors III (or IV ) the matrix eleme nt is nonze ro, since the field matrix elements have d iffe ring asymptotic values . It i s the conse rvation of this charge that rend e rs the baryons s tabl e . 1 6

In the one -baryon sector which we have ex­ amined , the que stion of highe r-order calculations remains open. The corrections in ,\ come from two distinc t plac e s . First there a re c ontributions of the multimeson states to the right -ha nd side of (4 . 4 ) whi ch are of higher order in ,\ than the left­ hand side. Secondly, the use of correct relativis tic kinematics for the baryon will produce correc tions in A to the static approximation which we have employed. As an example we derive a formula for the bar ­ yon form factor which includes fir st-order cor ­ rections to (4 . 5 ) . In this particular case, the kinematic correc tions only occur at second orde r . Returni ng to (4. 4 ), w e find that f o r the no-meson matrix element the order ,\ 11 ' te rm s o n the right ­ hand side involve only one -mes.on terms . The e quation s atis fied by / therefore becomes /" - U' ( /) - ½G( x, x)U"' ( /) = 0 .

(5.3 )

T his is similar to (3 . 2 ), w ith the c rucial diffe r ­ e nce that the propagator G(x, x), given by (4 . 1 2b). is infrared finite . E quation ( 5. 3 ) may be solved

iteratively . Setting / = ,P,(x) + o,p (x) and construct ­ ing G(x, x) from the so lutions of (4 . 6), we find" (5.4)

For consistency , the' right -hand side mu st be orthogonal to the trans lation mode :

fdx G( x, x)U"'( q, ,),P ,' (x) = 0 .

(5.5)

T o s e e tha t this vanishes w e proceed a s fol lows. The integral in ( 5 . 5 ) can be expressed as � 2 :(k )

fdx 1j1: (x)1j,, ( x) U" ( ,P , )c/lc' (x) .

( 5 . 6a )

But by d iffe rentiating (4. 6 ) with respect to x, we s e e that the quantity 1/1,( x) U"' (c/J c lc/> c ' (x) can be r e ­ p l a c e d by ,J,,'" {x) - [ U" (c/J,) - w ' ( k )],i,;(x). Thus using (4 . 6 ) once more to elim inate f: ( x)U" ( ,P, ), ( 5 . 6a ) may be rewritten as � Z �k w ) =

fdx[I/J: (x)l/;. '" (x) - 1/1:" (x) �.' (x) ] � Z :( k )

[,Pt ( x),P, '' (x) - 1/Jt '(x),p,' (x) ]: :.'.' e •

(5 . 6b )

F o r discre te s tates th i s c learly vanishe s ; for con­ tinuum states (5 . 6b ) can be evaluated from (4 . 1 6a ) . W e find wi th the help o f (4 . 1 6b) 2

f

dk �;!,) I B( k ) [ 2 -2

L

dk w�:) [A (k) l 2 •

(5 . 6c )

Changing k to - k i n th e second integral a nd using (4 . 1 6d ) shows that this quantity indeed vanishe s . [Equatio n ( 5 . 4 ) does not determine contributions to o,p pr oportional to c/J c' . This is as it should be, s i nce such te rms can be c ompe nsated by adj us ting the phase of the form factor . ] Consiste ncy condi tions, like ( 5 . 5 ), may be ex­ pected to arise at each new ord e r of the calcula­ tion. It would be most inte resting to for mu late them in some ge neral way, s o that the consistenc y of ou r theory i s evident to all o rde rs .

VI. CONCLUSION We have paid rather close atte ntion to the quan­ tum inte rp retation of c lassical s olutions to fie ld theory and to the trans lation mode . Though it was

317 11

Q U AN T I Z A T I ON O F N ON L I N E A R W A V E S

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c lear that such a n inte rpretation had to be in terms of new particles, it was not so obvious how to carry it out in a way that would extend beyond the crudest approximation. Once the properties of the quantum theory have been established, one can proceed w ith more confidence to phy sically interesting calculations on field theories in three dimensions . Also one must develop approxima ­ tion techniques suitable to larger c oupling con ­ stants. I t may be , a s has been long speculated, 1 8 that the ba ryons occurring in nature will be found to coinc ide with the mathe matical bar y ons which we have discussed. 1 9

* This work is supported in part through funds provided by the Atomic Energy Commission under Contract No. AT (ll-1)-3096. t Permanent address: Department of Applied Mathemat­ ics and Th eoretical Physics, Silver Street, Cambridge, England . 1 There exists a vast literature about solutions of non­ linear field equations . We are concerned with quantized solitary waves; much of the classical work concerns itself with "soliton s , " which are solita ry waves that emerge from collisions unchanged , We do not use the name soliton since we do not know whether our baryons have, or possibly must have, the soliton property . For a review of solitons, see G. Whitham, Linear' and Non -linear Wares (Wiley, New York, 1974); A. Scott, F. Chu, and D. McLaughlin, Proc . IEEE .§!, 1443 (1973) . 2 R . Dashen, B. Hasslacher, and A. Neveu [ Phys . Rev . D 1 0 , 4 114 (1974) ; 10 , 4130 (1974/; 1 0 , 4138 (1974)) ha� also examinedthe quantum-me�anical interpre­ tation of classical solutions. Their method is entirely different from ours, though some of their conclusions are similar. 3 The notation U'(¢J , u•(rpJ, etc. will always denote d erivative s of U \i:.b) with respect to ¢, while ¢ ' , ¢'\ etc. will mean derivatives of q,(x) with respect to x . 4 Thi s stability theorem is generally known t o workers in this field. 5 H . B. Nielsen and P. Olesen, Nucl . Phy s . B61, 4 5 rl973 ) ; G , ' t Hooft, Nuc l . Phy s . B79, 276 (1974/ ; L.D. Faddeev , M ax-Planck Institut report(unpublished) ; A. M. Polyakov, Landau Institute report (unpublished) ; R. Dashen et al . , Ref . 2 ; T . Eguchi and H. Sugawara, Phy s . Rev . D 1 0 , 4257 (1974). 6 See for exampi;-P . Morse and H . Feshbach, Methods of Theoretical Physics (McG raw- Hill, New York, 1953 ) , p. 1650 . 7 Additional field-theoretic model s which possess static , stable s olutions in the clasS'ical approximation can be constructed in the foll owing \vay. Let j \x ! be any in­ tegrable function without zero s . Define ¼ (xJ = f x dy f (J' I + ¢ 1 and express V (9c ) = 1 1 2 i n terms of %, , I t f;ll ow s that/ i s a zero-energy eigenstate of - d2/d x 2 + V ' ' (¢c ) ; s ince/ has no zeros, it is the lowest state. The class of field potentials that lead to (2 . 1 1) correspond to setting! (x ) = 1/ coshLx . For L = 1 the sine-Gordon

1497

ACKNOWLEDGMENT

We be gan this research indepe ndently. One of us (J . G . ) is grateful for the hospitality at the Laboratoire de Physique Theorique e t Hautes E nergies of the University of Paris at Orsay, where he benefited from discussions with Curtis Callan, Sidne y Coleman, David Gross , Andre Neveu, and Tony Zee. The other (R. J. ) was aided by conversations with Roger Dashen, Ju lius Kuti , E lliot Lieb, and Francis Low . Both of us thank Arthur Kerman for his explanation of the method used in Sec . IV .

theory i s obtained . The sine-Gordon equation in class­ ical mechanics has been recently analyzed by L . Faddeev and L. Takhtajan, JINR report (Unpublished ] , while S . Coleman [ Phys . Rev. D (t o be published) ) has discussed the quantum theory. 8 J. M . Cornwall, R. Jackiw, and E. Tombouli s , Phy s . Rev . D 10, 2428 (1974) . 9 1n other -;-ords, we are as suming that the exact equation 6 I' [¢]/o,,(x , t ) = 0 only has solutions with constant ¢, and the l owest- order nontranslationally invariant solution (k)} , - "' < k < ., , 0 � p(k) < GO , O � tf>(k) � 211' , { p. ; q11} , a = 1 , . . . , A , - 00 � p0 ; qo � oo ; { t,, ; 71,,} , { a ., ; ,Sb} ; b = l , . . . , B , - ao < t., ; 71,, < oo ; 0 ,; a, ,; 2n ; 0 ,; ,8 ,; 8,r/y, where the positive integers A 6 and B c an assume arbitrary values . The total energy and the momentum are expressed in terms of the fore­ going variables in the following m anner:

-

P. ::. f (i ' t ,. 2p 1 2p (lt,JtlA +

: < •• l

11

-•

.

I ,,: + _JI. IJ 1 /t

•

I

••l

already noted in 1 1 1 , is confirmed by an investigation of

the quas iclassical express ion for the amplitudes of the sc attering of the particles of the s econd so rt-solitons . A soliton corresponds to the solution of the equ ation of motion al p = O , A = l , B = O , u 1 (x , t/p , q) = 4 t an·• x (exp{, n,(r - vl - q)/,ff::,jf} ) , v = p ( p2 + M 2 )·1 1 • , and • is the charge of the soliton (see 1 1 • 1 ) . The equation of motion h as a two -soliton solution 1', (r , t) that breaks up as t - ± oo into a sum of s ingle-soliton solutions with 2

ch aracteristics l 4 I

II

•

p. + I.

6• 1

411 ')

I

f1+ • 9 1 - + (; : + M 2 } 1 t ,: ln\ l - -.- ; 1ti ;

II • 8• / y .

When quantizing in the quasiclassical approximation

we can use arbitrary canonical variables, and in part­ ticular the ones just described. In this approximation,

we can state that the system consists of three sorts of 1 38

representations U I shows that these representations can

be qu antized only if its total area (number of stales ) , equal in our c as e t o 16,r2/y, I s a multiple o f 2 ,r . In other words , the coupling constant is qu antized -y = Bn/N, with N an integer , the eigenvalues v, take the form n (n + 1 /2 )/ 2N , and the m as s es of the particles of the third sort are equal to 2M s in[n(n + l/2)/2N} . This paradoxical result ,

•• l

\ + (2M■i■ v ) 1 ) \fi ;

P 1 .:. f l p (lc ) JI, + v • y ,8/ 1 6 ;

+

particles w ith masses m, M , and 2Msinv. The canoni­ c al v ariables a and fl run through the compact phase spac e . Experience with the theory of the Lie group

JETP Lett., Vol. 2 1 , No. 5, March 5, 1975

f'i + . 91 __ _" ___ 1 ''

P1+

,, : ♦ •

'"

'J

2

In� - � ) ;

P i - ; P 2+ • P2- ;

•

1

• = fp u + P uJ - fp l

+

p z l 1.

These formulas constitute a canonical t ransformation from the in t o the out variables, i.e., they specify the classical S matrix. The corresponding generating func­

tion is

Copyright © 1 9 7 5 American I nstitute of Physics

1 38

320 /up l i .( Ji fl1'- ■, - HI •; all I D II P,.l a; al - pJl(X\ a; •l - ff• J l ,c

14 f .,., ,. + 1 g .,, ( . - i ll + l H(p l : D ., l = .. - f --- ln--- r. l - r ,IR ln ----...J J • )' I s s-1 £ + • tJ

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,

_______ • - 2M 1 + V� 2M 2

The quantum S m at rix in the quasiclassical approxi­ m ation is given by ( p1 ; p2 1 s 1 p; ; p;.} = 6 ( p1 - p;l6 ( p2 - p;) x exp[ - iE 1, 2H} and we obtain , depending on the particle charges , two S functions , S, = exp['l' iH(p 1 ; p,)]. T h e c rossing- invarian c e condition S, (4M' - S ) = (- l l'S,(s ) w i t h integer c i s s at isfied only if y is quant ized in t h e m anner indic ated above . The express ion fo r s, (s ) in not real in the unphysi c al region O ,: s ,: 4111" , In contradic­ tion to the general p rinciples of analyticity . We hope that the quantum corrections will improve the s itu ation . Indeed , by approxim ating the integral with integral sums and by assuming y to b e quantiz ed , we obtain for s. the expression s .(• J = u

f.. , ... I s [ 46 1n -�

••

l .. �-' 111

s• • .'.'· (• • ½) . N

which has the correct analytic properties in the c ase when the pole is att racted to the points S = (2m sinv.l"­ in other wo rds , the particles of the third s o rt (double solitons ) are bound states of solitons . 2 . Quantum corrections . To investigate them it is convenient to use a continual integ r al , Let us illu strate this using as an example the corrections to the soliton m ass . The Green's fun ction G ( p1 ; p2 ; t1 - 12 ) , which de­ s c ribes the trans ition from the stat e "soliton with mom entum p 1 at t = 11 " into the state "soliton with m o ­ m entum p2 at I= t2 , " is given as T = 11 - 12 - 00 by t h e integral

.. ,

'•

f n Ja J'• n p l i / Ji

-j

U ( n , - H h; s:gl ;

H( u ; .,] • .!.. .,, s + 2_ • •, + � • 2

2y

y

where u = u, (x , t l p 1 q 1 ) at l = t , . with the limit independent of q 1 and q2 • To calculate the coefficient F at 6 ( p 1 - p,) it is convenient to integrate over a subm anifold with fixed total momentu m . The natu ral additional condition associat ed with the const raint f ""• dx + p = 0 takes the fo rm x � f dxxH/f dxH =f(tl . The general presc ription 1 5 1 s hows that the sought coefficient F[(p2 + M') 1 1 2 MT] is given by the integr al

1 39

..,

r

'•

w here

JETP Lett., Vol. 2 1 , No. 5, March 5, 1 975

D

,1.,, ,1••

It suffices to c alculate It at p :f(J) - 'J . We assume " = uJ< , t • 1 0 . 0) + y' 1 'w�< , I) and 11 = ,- ·• l •v (x , t), and con­ fine ourselves at first to the orders ,-·• and --,P (trees and one loop) . The answer is

f

F(TJ = exp l - i M T l np • 1- T, "' I• - + D = F 1 F , ) - • 2 JJ!i I

where D = - (d 2/ dr) + m 2 - 2m 2/ cosh'm.r. The symbol Tr' m e ans that when the trace is taken, one leaves out the contribution of the eigenfunction �0 (x) of the opera­ tor D with zero eigenvalu e . This limit ation is brought about by the condition with the additional conditio n . As T - ., , u s ing the known formulas for the traces'"' of the Schrodinger operato r , we have

l

I • ,,, T • y J-F. = up l - 1 -tr D 1 1 1 TI • ..� " - .:iT 2 f f2.r) 41 + • I where t h e t r a c e I r' i s taken only i n .r space . The re­ sultant divergence is eliminated by the same renormal­ i z ation of m which elim inates the divergenc e in o rdinar y perturbation theory in this approxim ation. 111 (The equal ­ ity of these renorm alizations was pointed out to us by A . M . Polyako v . ) We see that the quantum correction In the single-loop approximation has been reduced only to a renormaliz atio n . The multiloop correc tions m ay m ake a nontrivi al contribution to the soliton mass . The cal­ culation of the latt e r becomes s implified if one uses t' Hooft• s s tratagem , replacing 6 (x - /(1)) in the continual integral by exp[- iµ.2M f dtX"]. Perturbation theory is now constructed with a propagator A (x1t/.r2 12 ) such that (d 2 dl � + D + µ.2P )A = 6 (x 1 - .r,)6 (t1 - t2 ) , where P is the proj ector on �al and a spinor field it, with a Lagrangian density of the form

.c = ½ a . a " - 7 U (g ) + i�y"a µ ,i, - G+V(g )>I-.

(2. 1 )

G i s a pos itive constant with mass dimensionality. We fu rther suppose that, in the absence of fer ­ mions , the static field equation for possesses a classically stable, finite -energy solution 1P ± ; p + > = (2 11 )o(p' - Pl ± ) = o(x - y)(211) o (P' - P) .

(2 .20)

Two term s are to be evaluated : (P' ± ) 'l-'(x)'l-( y)I,> ± ) and (P' ± ) 'l-( y)'l- t(x)jp ± ) . In the intermediate states, which saturate the product of the two op­ erators, we retain only the no -fermion and one­ fermion states . T he no-fermion states contribute to the first term only for the + sign (soliton ex­ ternal state, antisoliton intermediate states), while to the second term the no-fermion states are p resent only for the - sign (antisoliton exter­ nal state, soliton intermediate states) . In either case, the one- soliton intermediate states contrib­ ute to the sum of the two terms

f dz e

HP

• -P > •ut (x + z) 11o ( y + z ) .

(2 .2 1 a)

The one-antifermion intermediate states contrib­ ute to the first term

f dz e " ' -P > , f (:!) vt (x+ z)v,(y + z ), P

n, = ½ { ± = ±½ .

(2.2 1 b)

f dz e t < P• -P ), f (2dp) u,* (x + z )u, ( y + z ) . ,r

(2 . 2 1 c )

Adding the three, w e s e e from (2 .9) that i f Uo is identified with the normalized, zero-energy solu­ tion I/J0 of (2 . 6) , "• with the positive-ene rgy solu­ tions 1/J,• of (2.7), and v, with the fermion-number conjugate of the negative -energy solution 1/J,- of (2 . 7), then the commutator is correctly repro­ duced. Thus we establish that our Ansatze lead to a complete, normalized set of states . (The an­ ticommutator {+, +} vanishes trivially since there are no intermediate states that c an contribute.) The wave functions u, and v, desc ribe the scatter­ ing of fermions off solitons . (Bound solutions of the Dirac equation, other than the one with zero energy, describe soliton-fermion bound states.) Next we consider the meson wave function (2 . 1 3d) . The equation is of the form (2 . 10); the interaction with the ferm ions can be neglected s ince its is of O ( g) . Therefore J.(x) satisfies the same equation as in the absence of fermions• : -J. ' + U"( g c)f. = w2 (k)J. '

{2 .22)

The zero-frequency mode c' is associated with translations of the soliton, while the other modes describe meson- soliton bound and scattering states . Since the norm off• is fixed by the Bose field commutator to be unity,2 it is established that all the one-particle wave functions are indeed of O( g° ) . Finally, we calculate the fermion number of our soliton . The conserved Fermi-number current is ify "+; the charge must be properly ordered so that it transforms correctly under fermion-number conjugation. T he ordering is determined in the vacuum sector to be and the fermion number is given by

(2 . 2 3)

(2 .24)

The calculation of this matrix element is analogous to the one performed for the anticommutator, ex­ cep t that the two terms enter with opposite sign. Hence we find

f d z u.;' ( z ) uo(z ) + f dz f (�!) [ v; ( z ) v,( z ) - u; (z ) u,( z ) ] }

(2. 2 5)

335 R . JAC KIW AND C . RE B B I

3402

The soliton has fermion number + ½ ; the antisoli­ ton has - ½ . These Kerman-Klein calculations can be ex­ tended to higher orders; alternatively they may be summarized by introducing c ollective coordinates through a c anonical transformation. 8 The field transformation is ( x) =lf> ( x - X) + ¢c ( x -X) ,

-v ( x) = iir ( x -X) +a,P 0( x -X) ,

(2.26a)

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with subsidiary conditions

(2 . 2 6b) X is the collective position operator; a is an op­ erator associated with the degenerate soliton and antisoliton states . Its anticommutation relations ( 1 .2 ) are realized by a ! P + ) = I P -) ,

(2 . 2 7)

We do not give further details , since they are a s traightforward generalization of previous re.­ search. 8 The quantum theory possesses two discrete sym ­ metries: fermion conjugation 3' = 3' -1 = [f t , if,f,3' = - i y 1 ,i, t ,

3'4> 3' - 1 =4>

(2.28)

[compare (2 .8) ] and discrete chirality ­ (A3)

(A4 )

and using -r2f 1' = - 'r-r , one obtains for 31t' the ma­ trix equation

;j • p 31t ' - ½ A (;j x r)•m 'a• ,f'½ i Gtlffi ' a" r° = E31l ' ,

(AS)

where it is no longer necessary to distinguish be­ tween a and f.

339 R . JAC KIW AND C . R E B BI

3406

We now expand 31! ' in terms of two scalar and two vector functions: mt.. ( r) = g ' ( r)o , ., + g

: ma-; ., .

K� = P a + ½ i A 'Ya �½ i GcpYa .

(A6)

which Implies

(A7a)

g ' ( r) = c ' exp {½

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(A7b)

1 ./

3

+ C ; (r)e� J , (n ) ] .

(A8a)

u,

J

(A1 6a)

(ABb)

This means that g: c an be expre ssed in terms of "potential" functions J • ( r) as g: = (a. 'F ½G .pi'. )J ' ,

which acc ording to (A 7b ) satisfy

(A9b)

,

(A9c )

or

Before displaying the final form o f the equa­ t ion s , let us show how the existence of zero-energy solutions can be investigated directly from (A 7). We tal x · The order purameter -,,, is ploned radially. �

351 ll

SOLITON EXCITATIONS IN POLY ACETYLENE

half is created or destroyed at a chain end, ensuring that K rn mers theorem is satisfied. 24 A t each site 11, the c\ertron density missing from the valence band is exactly compensated by the den­ sity l u( 11 ) I ' of the unpaired electron. This can be proved as follows. For any configuration of the chain, the local density of states µ 11" ( E ) satisfies the sum rule

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L: p.. ( £ ) ii£ = I

w h ich follows from the completeness o f the eigen• slates of II ( [ ,i,, I ) !Nole Eq. (4. 1 S) follows from Eq. (4. 1 7 ) by summing on 11. J Therefore the change of p ,,,, ( E ) due to the presence of the soliton in­ tegrates to zero. Furthermore. A. p1111 ( E) = !1 µ1111 ( - E ) . s o that w e find t h e local compensation s u m rule

• f-

2 __ fl p •• ( E) d£ + l u ( 11 ) i ' = O

(4. 1 8 )

which proves that the m i ssing electron density i n the v,liencc band i s exactly compensated at each site by the 1/, 0 electron density. Thus. a neutral soliton is both globally and locally charge neutral. A n analo­ gous relation of local charge compensation has been derived by B razovskii 4 i n the continu u m model . Si nce the e nergy of lhe system is lhe s a m e if the occupancy of 11 ( 11 ) for even and odd 11 are uncoupled. there are two linearly in­ dependent solutions. one of which decreases ex­ ponentially as 11 - ± oo , while the other d i verges and is not normalizable. I f the soliton is l'entered on 11 - o. ± 2 , . . . the normalizable stale only in volves even 11. while i t only i n volves odd n for a soliton cen­ tered at ± 1. ± J . . . The state is a linear combina­ tion of even and odd ,, if the soliton ·s center is between sites. I n the presen t case, the soliton is cen­ tered on 11 = 0 and 11 ( II + 2 ) =

=

-211

2107

-1� l ln +2,n + l

1 11 ( II )

1 1

-. -. . . + 1 fn +2,w +I 1 � , l.• , - ,.• - ,

� 1 11( 0 ) l2, 1

(4 2 1 )

while u ( 11 ) - 0 fo r odd 11. 0 ( 0 ) i s determined so lhal o is normalized lo unity. l u< 11 ) I ' is plolled in Fig. 1 3 . for / = 7 and / = 5, showing little variation with /. Roughly speaking, 0 ( 11 ) ?:£

1

11 ! sec h 7 cos2 1r u

I n passing we n o t e t h a t Coulomb interactions /f'

(4.22)

352 W. P. SU, J. R. SCH RIEFFER, AND A. J. HEEGER

2108

0. 20

-- L. = 7

- -- -' ' 5 -

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0. 1 6

FIG. 1 3 Gap•center state probability density 1 4>0 ( 11 ) \ 2 plotted for a soliton centered on 11 = 0. Two soliton widths I = 5 and ? ure considered. By symmetry. ct,0 ( 11 ) - 0 for odd II if the soliton is centered on even 11.

between e lectrons split the ,t, 0 level for different oc­ cupancies in a nonintuitive fashion. 25 The mass M, of the soliton can be determined by calculating the energy of a slowly moving domain wall, t/l, ( 1 ) - 11 0 tan h [ ( na - v, 1 )//a ]

(4. 2 3 )

From time-reversal symmetry, a n y change in wa11 shape, e.g . . /, must be of order v': and does not con­ tribute to M, for small v,. Continuing to work within the adiabatic approximation, we find

f M, v} = f M I I/I;

•I" I

Mu J v ; - --sech I 2 1 2a 2 I ,,

(4.24)

Therefore, using the parameters for £0 - 1 .4 eV one obtains. M, - 3/ 4

I1 Uo --;;

M

= 6111,

where m� is the free electron mass. The small value of M, is a consequence of the smallness o f the di• merization length u 0 compared to the lattice spacing a. One would expect that the soliton would have h igh mobility because of its small mass, and it must be treated as a quantum particle. 2

V. DOPING EFFECTS

In the traditional semiconductor picture of doping. an impurity donates an electron (or hole) to the con­ duction (or valence) band of the solid and no struc­ tural change of the solid occurs. I n ( C H ) , . one must consider whether the state of lower energy is a free electron (or hole) as in the semiconductor picture, or

a charged soliton ( Q - ± ,. ). Choosing the center of the gap as the origin of energy. the minimum energy to inject an electron (or hole) is A, while the energy to make a charged soliton is £, . Thus. if £, < A

soliton doping occurs through the formation of charged solitons, while if £, > A

(5.l l (5.2)

semiconductor band doping occurs. F o r a range o f gap sizes w e found £, - 0.6A. Therefore, soliton doping is favored i n (CH ),-like systems. This result implies that for each donor ( K , Na, etc. ) or acceptor (Cl, AsF5 , etc.) which transfers an electron or a hole to the chain, 26 one charged soliton is formed. Since charged solitons have zero spin. no spin resonance or Curie-law susceptibility would be associated with the charge carriers. as is experimentally observed . One might ask, how is it possible to transport a single charge + e without having spin transport. I n essence, the charged soliton carries one m issing electron, half of which is i n the up-spin valence band and half in the down-spin valence band. This is accomplished by slightly deforming all of the states in the valence band so as to reduce locally the up- and down-spin electron density each by a total of half an electron in the vicinity of the soliton. Far from the soliton, the electron density returns precisely to its value without the soliton. For Q - - e, the soliton has two elec­ trons spin paired i n o and the missing electron den­ sity from the valence band is doubly compensated for by the ,t,0 electrons. Next we calculate the interaction energy of � E, of a charged soliton Q = ± ,, interacting with an impurity of opposite sign of charge Q ' - + e. For simplicity we assume the impurity to be a point charge located a distance d from the chain and centered at 11 = 0. When the soliton is centered on site ns - the Born-

353 ll

Oppe n h e i m e r e n e rgy has t h e additionctl t e r m l,( 11 - 11, ) I ' A E, = - .C E ,, [ ( 11a ) 2 + d 2 ) 1 1 2

(5 J )

& E1 ( x, ) = - E,, + f A6 x/ + 0 ( x/ ) .

(5.4)

!

w h e re w e assume the i n teraction i s screened b y t h e macroscopic d i e l ectric consta n t E . For E,, = 1 .4 e V a n d , = 1 0 . o n e fin d s fo r l x, l /,t < I ,

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2 1 09

SOLITON EXCITATIONS IN POL Y ACETY L E N E

w h e re Xs = 11:i "· One fin d s Ek = 0.33 e V a n d k ,, = 0 0029 eV I A' for d = 2 . 0 A a n d 0.30 eV a n d 0.0028 c V / A 2 , respecti vely, for d = 2.4 A . I f t h e soliton i s treated classica l l y , the equilibrium b i n d i n g energy is £6 • com pared to a measu red activation e nergy for conductivity in t h e d i l u te a l l o y � E rnnd := 0 .30 e V . F o r s m a l l a m plitude m o t i o n of the s o l i t o n about t h e position o f t h e c h arged i m purity, on e obtains t h e q u a n t u m or v i b r 1/,/2 i s negative . Thus superconductors of the s e c ond group should h ave properti e s very different from tho s e of th e first

Ginzburg and Landau considered only th e case i n wh i ch x « 1 /,/2. It has been shown by Zavari tsk i i , 3 h owever, that the propert i e s of pure metal thin fi lms condensed at liquid-he lium temperatures are not described by such a theory. Zavaritskii and th e pre s e n t auth or have therefore suggested that such films correspond to x > 1/,/2, and th at superconductors can thu s be divided into two groups. Th e critical fie ld for su­ perconductors of the second group h as already been I calculated as a function of the fi lm th ickness. Th e agreement obtained with Zavarits k i i ' s e xperimental data was not bad. In the pre s e n t work, a more detai led in vestiga·

gro up.

tion of the magn e t i c properties of bulk superconduc·

For pure metal, x i s found to b e small . For in­ stanc e for mercury , x = 0 . 16 . In view o f this.

tors of the second group (a cylinder in a l ongitudi• nal fie ld) i s undertak e n . The results obtained show

357 M A G N E T I C P R O P E R T I E S OF S U P E R C O N D U C T O R S

that >< > 1/-.,/2 for a large number of superconduct­ ing alloys whose magn e t i c propert i e s had not pre­ viously been well understood.• I . T RAN SITION T O TI I E NORMAL STAT E

It was shown by G i n z burg and Landau that i f >< > 1 /y'Z, the superconductivity i s maintained a t fie lds greater t h a n fl c m at wh i c h e qu i librium c ould exist between the n orma l and superconducting state s . At fields h igher than H e m • a state with 'I' = 0 is unstable, an d superconducting sections with 'I' -/, 0 may ari s e . It was shown 1 that this in­ stab i l ity continues to some value fl c , , wh ich for a bulk superconductor is >< v'Zfl c m . At this value of the fi e ld th e s uperc ondu ctor undergoes transition to the n ormal state by means of a s e c on d-order phase transition . In th i s s ection we shall investigate the properties of a superconductor i n the n e ighborh ood of the transition point, that i s

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2

( 1)

The Ginzburg-Landau e quations (in units pre vi­ ously used by the author, 2 and to be used h e n c e­ forth) , t can be written - curl curl A

= I 'Y 1 2 A + !x (o/'v'o/-'f'v'o/*) .

( 2)

( 3)

We shall assume that the superconductor fills all space, and that th e external field f10 i s directed along the z axis . Let the potential A b e dire cted along the y axis . Close to t h e transition point 'I' « 1 , s o that in the first approximation we may neglect the influ­ en c e of 'I' on the field. We then obtain

I I'

H = canst = H0 ,

A = H0x.

( 4)

We n ote that the point x = 0 may be lo cated any­ where in spac e . Let us now go on to Eq. (2) . In­ serting ( 4) into the equation for 'I!, neglecting· the term 'I' 'I' and c onsidering 'I' to b e a fun ction of x only in the first approximation , we obtain the o s c i l­ lator type of e quation

I I'

d"'F / dx 0 - x 2 ( I - H�') 'f" (x)

= 0,

(5)

�suggestion that x. may be greater than 1//2 for alloys was first made by L. D. Landau. t We note that in these units He m :;:; I/y2.

1 175

as found previ ously . 2 Th i s e quation h a s solutions when fl 0 = >< c orresponds to 'f" = exp (- x 2x2 / 2}.

(6)

k, )'} exp { iky - x (x - --;;2

( 7)

In addition to such solution s , Eq. ( 2) is satisfi ed b y the function >

Since the point x = 0 has no special properties, the conditions will be exactly the same at all points of spa c e , and it i s natural to pick 'I' in the form 1f = , (x) 'In

� Cn e ikny 't n (x) ,

= exp

X- 7 [ - - x• 2( kn

)'J

( 8)

with arbitrary coeffic i ents k and C• . Thi s form of 'I' is a solution of the lin ear equa­ tion , and refers in facts to H0 = "· In order to find a s o lution for HO < >< , and 'I' from Eq. (8) into Eq . (3). Thi s leads t o � C.C:e•< n -m) ky [xk - {n -i;_;> k ] n (x) •� m (x) , n ,m ( 9) PA . i i � • k LJ CnC me•< n- m)ky (n - m) ,;>. (x) 'f m (x) . - a;a- = z;;!I n,m

It can he shown easi ly that

H = aA ; ax = Ho - l 'Y l' / 2>1

• / 41t (l" K• ( V

+ m • + Im )'

).

V3 xB

x (H 0 - H 0 . 0394 .

(45)

We note that (aB/al/0) r i s always positive, which indicates that th e s ituation d i s cussed i s stab l e . 7

For very large values of >< Eq. (35) can be used

(43)

to find He , • T o d o th i s one must solve ( 26) an d

(27) numeri cally, alter wh i ch & is calc ulated from

(3 1 ) . On the othe r h and, finding B (H0 ) in th i s c ase 'I • /� • ;-V ; \ ;! ) • ex p ( - y ,iif} , re quires solving a separate and very c omplex prob-

one s e e s c l e arly that for suffi c i e ntly small B th ere should be a tran si t i on to th e triangular modification . Th e v a l u e o f H0 at the tran s i tion point i s determined by setting th e free e n ergy for a given field strength equal to F, _ 28 08 , and i s H�

[ 2Ko (X1,m) + x1, mK1 (x1 , m) ] , xi,m+o

Xt , m = V 2:-: ( l 2

By comparing the asymptotic form of th i s expression

for small B with asymptoti c form o f -(41),

+�

= H,1 + 0 . 0394/>< .

(44)

Th i s tran s i t i on is c l e arly a first order phase tran• sition . Th e induction undergoes a j ump from

B,

= 0 . 286 / >< t o 8 2 = 0 . 294/ >< , A t fie lds greater than H:, th e s qu are latt i c e i s more favora b le , wh i ch j us­ tifies to some degree the assumption made in th e

previous section. We note that th e transition tak e s p la c e i n th e im­

mediate n e i ghborhood of

11 .,. ,

and that the d;scon­

tinuity i n the induction i s s o small (about 3%) th at i t would be extremely difficult t o observe i t on a

B (H 0) curve . It is n ot di ffi c u l t to s e e th at wh e n

11

= fl , . , th e

B (IJ 0) curve has a verti ca l tange n t . Indeed, wh en H = H, . , th e H 0 -depen dence of B i s de termi ned pri­ marily by the term H 0 "- exp ( -(4,r /./3>
(x)) ,

giving the equation of motion

a µ a µ 1,c>(x) + cd sin (d1,c>(x)) = 0 , i.e,, the Sine-Gordon-equation. It is readily seen that this theory classically allows for a static solution 4 1,c>(x) = d arctan exp (ycdx 2 ) describing a vortex line along the x 1 -axis. The field 1,c>(x) changes by 21r/d across the vortex line, which has a wid th of the order of magnitude 1 /../ed. Analogous to the case of the Ginzburg-I.andau theory, this width is equal to the Compton wavelength for the particle of the theory, as is seen by considering the Klein-Gordon equation with mass square m 2 = cd 2 obtained as the weak field limit of the Sine-Gordon­ equation. The energy density along the vortex line is 21T£t'

=

£+oo [{j a:

arctan exp (y'cdx)

}2

+ 2c sin 2 (2 arctan exp (✓cdx)] dx �.Jcid ,

s o that strings that are narrow (vortex lines) compared t o the hadronic length ,Jc?',,., d½ c-¼ are obtained in the strong coupling (and super quantum mechanical) r limit m✓a ,,., c- i are obtained in the strong coupling (and super quantum mechanical) vortex lines move as Nambu dual strings. We have shown the equivalence of string models with a certain .set of solutions of some field theories in a classical approximation, namely the vortex lines. We be­ lieve, but we have not proven that in the strong coupling limit at low energies all states of the Ginzburg-I.andau field theory are states that can be described as states of some system of strings. This hope cannot be taken to be true for all theories having vortex lines, since theories can be cooked up, which have e.g. zero-dimensional structures in excess of the one dimensional one. That is to say one could make a field-theory model which also had kink like type of singularity similar to the solu­ tion in the I + I dimensional Sine-Gordon theory. Such theories could be built in higher dimensions too. That we have to take a super quantum mechanical limit necessitates that, (i) the

378

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58

H.B. Nielsen, P. Olesen, Dual strings

field theories to be used should be quantized and (ii) the equivalence of the field theory and the corresponding string theory be proven on a quantum mechanical level (that can only be done, if at all, in a low energy and strong coupling limit). Also we have to understand how it can happen that tachyons appear in a model having at first a positive definite Hamiltonian density, as is easily seen to be the case for the Higgs-Ginzburg-Landau model. Further, we should like to know what the sifnificance of the critical dimension (d = 26 in the conventional model) is in terms of field-theory models like the ones discussed, but we have not even made the 26-dimensional classical Ginzburg-Landau model yet . If some day one understands the quantum properties better, the possibility of constructing field-theory models for strings like the ones we discussed, might be an easier way to come across a good (possibly unitary) Veneziano-model than to make a string model directly. First o f all we want to thank Don Weingarten for pointing out how to make vortex-line Sine-Gordon models in any dimensions, and C.H. Tze for finding litera­ ture. We also thank B. Zumino for calling our attention to the fact that the equation we discussed is well known as the Ginzburg-Landau equation. Secondly we want to th:mk our colleagues at the Niels Bohr Institute and CERN for helpfur discussions . Appendix. Discussion of the Yang-Mills type of model It is natural to ask whether it is possible to produce strings from Lagrangian having internal degrees of freedom, e .g. isospin or SU(3). If we can manage to keep the vortex solutions for such Lagrangians, this would indeed be very nice from the point of view of the dual string, since the string would then carry internal degrees of freedom , and would therefore perhaps lead to a more realistic spectrum of hadrons (and perhaps to a more realistic dual amplitude, provided we could solve the problem of colliding strings). This, unfortunately, does not seem to b e the case . An example of a Lagrangian with internal degrees of freedom which immediately comes to the mind, is the Yang-Mills type of Lagrangian . Here we introduce a field BIJV ' Bµ, v = o µ, Bv - o vB - g B X Bv (A . I ) µ, Defining the dual field

Bµ,* v = !.2 e/HXll.{3 Jf'-/3 , it is easily seen that

(o" + g B"X) B:v = 0,

µ,

(A.2) (A.3)

379 H. B. Nielse,1, P. Olesen, Dual strings

59

which is the analogue of the Maxwell equation a µ F* = 0 ,

(A.4)

µv

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which states that magnetic monopoles do not exist. The form of eq. (A.3) poses a problem with respect to the interpretation of (magnetic) flux lines. From eq. (A .4) one can derive that O = J aµ Fµ*v dVV = f Fµ v da µ v _ f Fµ v da µ v ' (A .5) s 1 s2 vµ where Vµ is a volume bounded by the two surfaces S 1 and S 2 . Eq. (A.5) tells us that no magnetic flux lines can start inside the volume. As far as eq. (A.3) is concerned, a similar procedure leads to -g f Bµ X B*µ v d V v = f Bµ v da µ v - J Bµ v da µ v . Vµ

�

�

(A .6 )

Thus, at least in general flux lines can originate inside the volume. If, however, we concentrate on the static situation, we can take B0 = 0, and B0k = 0, and hence we have the interpretation that the flux is given by Cl> = JBµ v da µ v (static case) .

(A.7)

Next let us consider a specific Yang-Mills Lagrangian , namely

2 £ = - ¼Bµ v Bµ v + ½ [( a µ + g Bµ X ) ♦] 2 + ½ ((a µ + g Bµ X ) lj, ) 2 2 2 2 2 + c2 cl> 2 - c/cl> 2 ) + d2 1j, - d/lj, ) + e 2 cl> ;!, - e4 (q, 1j, ) '

(A.8)

where the fields cl> and lj, are isovector fields. The reader may wonder why we intro­ duce two fields cl> and lj, and not just one field . The reason for this will turn out later, where we shall see that in order to have a vortex solution at least two isovector fields are needed . Now we are looking for a solution which quantizes the flux (A.7) in a way similar to the Higgs {Ginzburg-Landau) Lagrangian discussed in sect . 2. Defining the cnrrent to be -- aµ Bµ v + g Bµ v X Bµ

= jv '

{A.9)

i.e . , jv = 0 for a free Yang-Mills field, we obtain from the Lagrangian (A.8) by_ varying Bµ jV = g( cl> X a V cjl) + g( lj, X

a

V

2 2 lj, ) + g (BV X cf,) X cf, + g (BV X lj, ) X lj, .

(A. 1 0)

Considering now the static solution with cylindrical symmetry we see that the term Bµ X B11 in eq. (A. I ) is smaller than the term aµ Bv - a v Bµ in Bµ v for large distances

380 60

H.B. Nielsen, P. Olesen, Dual strings

from the axis of symmetry. Thus we can write the flux (A.7) as Cl> = cj5 B,,. (x) dx µ. (static case, large distance) ,

(A. I ! )

provided we integrate over a circle with large radius. For such large distances ;., vanishes. and eq . (A. I O) leads to (Bv X cj, ) X cj, + (Bv X lj, ) X lj, = (B., cj,) cj, + (Bv 1j,) lj, - (cj, 2 + lj, 2 ) Bv

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= 1- (ct> g

x a cl> + + x a +) . V

V

(A. 1 2)

Now the ground state {the vacuum) of the Lagrangian ( A.8) corresponds to (A. 1 3) where we assume that cj, and lj, are not in the same (or opposite) direction, i.e. (A. 14) Thus, in the vacuum the lenghts of the isovectors cl> and lj, are fixed, and in addition the projection of one vector on the other is fixed. This ensures that in any frame of reference in isospace at least three components of cl> and lj, are non-vanishing (e.g., 1/> 1 , 1/1 1 and 1/1 2 ) Now let us go to a frame of reference where 3 = 1/1 3 = 0. This corresponds to selecting the flux lines in the 3-direction. It is easily seen that the condition (A. 1 2) for the current to vanish at large distances imposes the condition on B., that B V1 = BV2 = O

'

I g

BV = - - 3 V X ' 3

(A. I S)

where X is the phase of 1/> 1 + i2 . Notice that due to the last condition (A. 1 3 ) the phase of 1/1 1 + i i/1 2 is the same as x + constant. Inserting eq. (A. I S) in eq. ( A. I I ) and using the fact that the phase is only unique modulus 21rn we get that the flux is given by I Cl> I = 21rn/g ,

(A. I 6)

i.e., the flux is quantized* . This result only follows if we have at least two fields lj, and Having obtained the flux quantization we then go to the strong coupling limit, in order that the width of the vortex line is made sufficiently small. The width is given

+.

* We have not convinced ourselves that there are no other quantas than 211/g.

381 61

H.B. Nielsen, P. Olesen, Dual strings

b y the order o f magnitude o f the Compton wave lengths of the Higgs particles. As before , we have scalar particles. with masses

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(A. 1 7) The Higgs vector particle is obtained from the seagull terms in the Lagrangian (A.8), i.e. from 2 2 ½g (Bµ X ) + h 2 (Bµ X lj, )2 = ½K2 [BZ ( 2 + lj, 2 ) - (Bµ ) 2 - (Bµ lj, ) 2 ] . (A. 1 8) Inserting the vacuum values (A. 1 3) it is easily seen that all vector mesons Bµ acquire a mass, my = g

(A. 1 9)

Proceeding as in sect. 4 we can now go to the strong coupling limit g ➔ 00, c4 ➔ 00 , d4 ➔ 00 (or m s , m s, my � 1 /vf/), which then gives us the string solution. From the very general argument in sect. 3 , we know that this solution corresponds to the classical Nambu Lagrangian. However, one can easily see that due to gauge invariance no new degrees of freedom are introduced in the string Lagrangian. Note added in proof In addition to the term ( 4.9) the magnetic energy-density also contains a term coming from the seagull term in the Lagrangian. The latter term gives rise to an energy-density of the order of magnitude As long as log (t,/ t) is not too large, this term does not change any of the conclu­ sions in sect. 4. See ref. [8] for example. A preprint of L.J. Fassie [9] with a similar idea has appeared. References [ l ] Y. Nambu , Proc. Int. Co nf. on symmetries and quark models, (Wayne State University, 1 96 9 ) ; H . B . Nielsen, 1 5 th I n t . Conf. on h igh energy physics, Kiev, 1 97 0 ; L. Susskind , N uovo Cimento 6 9 A ( 1 9 70 ) 45 7 . [ 2 ] Y . Namb u , Lectures a t the Co penhagen Sum mer Sympo sium, 1 97 0 . [ 3 ] L.N. Chang a n d J. Mansouri, Phys. R e v . D5 ( 1 97 2 ) 2 5 3 5 . [4 ] P . Goddard, J. Goldstone, C. Rebbi a n d C.B . Thorn, Nucl. Phys. B 56 (1973) 1 09. [ 5 ] H.B. Nielsen and P. Olesen, Nucl. PHys. B 5 7 ( 1 973) 3 6 7 ; P. Olesen , Niels Bohr I nstitute preprint NBI-HE-7 3-9, Nuovo Cimento, to be published. [6 ] D. Saint-James, G. Sarma and E.J. 'I110mas, Type ll su per-co nductivity (Pergamon Press, 1 969). [ 7 ] G. 't Hooft , CERN preprint TH 1 66 6 ( 1 9 7 3 ) . [ 8 J A. De Gennes, Superconductivity of metals a n d alloys ( 1 966). [9] L.J. Fassie, Lines of quantized magnetic flux and the relativistic string of the dual resonan­ ce m odel of hadrons, Canberra preprint ( 1 9 7 3).

382 I' ll \ S l 2,

(4.6)

and the recursion relation for s ;,, n + 2

(4 . 7 )

l • -• r/J' D' -' V.• • • · - s (s - 11 - l ) r� ,. n l

(4 .8)

f.W = � [ (u + l )D:• 1 ] 1 ' t[ 1 + 0 ( (2 ) ] .

(4 . 9 )

Relation (4 . 6 ) ensures that the Higgs field is zero for p = O . M oreover. Eq. (4 . 7 ) shows that this zero is of order 11 . Indeed, from Eqs. (4 .3) and (4 .5), for E, - 0, 1

The recursion relation (4 .8) fixes all the D! In terms of D! and n:• •. We know that D! = - 1 , but do not yet know the value of v:• 1 • This coeffic ient will be fixed by the boundary condition at infinity. We have thus obtained an expression for :IC( (), good for small v alues of � - We now seek an ex­ pression of the same solution valid at large dis ­ tances . In order to do that, we will write an integral equation equivalent to Eq. (4 . 2 ) with b oundary con­ tions (4.4 ) . Taking into account the fact that the left - hand side of Eq. (4 . 2 ) can be related to the modiiied Bessel equation of order one , one can obtain a nonlinear integral equation by using the well -known Green's function method 3C,,W = n 2 E, K 1 (2 ()

- 2 £, K I (2 E, )

J\

I

( d (2!J) :JC , !J ) JC. d!J IJ d!j

( d - 2 E,J I (2 E,) [ :JC,, d!J . ' K I (2 1J ) 3C,, 1Jl d!J IJ

(4 .10)

It is easy to check that Eq. (4. 5 ) gives the same solution as E q. (4. 1 0) for s mall E,. We will use this fact in order to find the numerical value of v:••. The first ter m on the right - hand s ide of Eq. (4 . 1 0 ) corresponds to the approximate solution given by Nielsen and Olesen for the vector poten­ tial . It is not necessary to determine v:• 1 to ob­ tain s ome general properties of the solution. For example , we can find the asymptotic behavior of 3C for large (. Integrating Eq. (4. 10), one obtains

= h zJ f e -21 ( 1 + 0 ( ( - 1 ) ] - ½ (h z.)2 e - 3C.( �) > O ,

- h _l_ d JC,, < O 2 � d�

(4 .1 4 ) (4 . 1 5 )

In the preceding relation, the equality holds only for � = 0. Thus we see that the vector poten­ tial A(p) is a monotonically decreasing function of p . Because the magnetic field H, is given by the expression m 2 1 d JC. 4e H, ,if '

f

(4.16)

m• O < H, "' 2e ·

(4. 1 7 )

=

we see from relation (4 . 1 5 ) that

Moreover, one c a n show from Eqs. (4 .3), (4 .5), an d (4 . 1 6 ) that the derivative of H, is negative. Hence, the magneti c field decreases monotonical­ ly from its value at the or igin to zero at infinity . The preceding relations allow us to make a qualita­ tive picture of the field behavior : the magnetic field decreasing monotonically with a characteris ­ tic length 1/m and the scalar field inc reasing with the same length from zero at the origin to its vac ­ uum value at infinity. One can obtain more restrictive upper and lower bounds for the functions 3{,,( { ) , the constants z., and the coefficients O::' 1 • We show in the Appendix that the following inequalities hold: (4 . 1 8 )

( w e s e e that the approximate solution o f Ref. l i s i n fac t a lower bound o f the exact solution), z. > ¾ n + ( JT3 - !) n2 = 1 . 511 + 0 . 1046112 , 1 . 6046 < Z 1 < 3 .230 ,

e"' T D"' • < II + l 1

(4 . 1 9)

(4.20)

(4 .2 1 )

(here r Is the Euler -Mascheronl constant),

l !n (,a + l ) D; ♦ 1 1 < l + 211Z. K0 (2) ,

and, for the case

,i

= l,

0 . 128 < D� < 1 . 586 .

(4 .22)

(4 . 2 3 )

V . EXPLICIT ONE-QUANTUM VORTEX SOLUTION

We have obtained in Sec . IV the exact vortex solution as a powe r -series expansion [ Eq. (4 . 5 ) ] . A l l the coefficients in this expression can be ob­ tained frcm the recursion relation (4 . 8 ) provided one knows the value of D�" . As we have said be­ fore, this c oefficient must be fixed by the bound­ ary c ondition at infinity . The aim of this section is to find the numerical value of n:" for the low ­ est-energy vortex, namely n = 1 . (It is also possi ­ ble to get the corresponding solution for n > l in the same way . ) From Eq. (4 . 1 3 ) a n d the boundary condition (4 .4 ) it is easy to derive the following integral equation for 3C.( �): JC,,(�) = n - t2 + 2 X

£

{

exp [2 £• 1n11 X.: (1J ) d1J ] }

d!J 1J 1 + 2-"n ( • > exp [ - 2

f

£"

ln!; 3C.'( !; ) d/;]

(5.1 )

Expanding the r ight-hand side in Eq. ( 5 . 1 ) for small values of � and comparing with the expan­ sion (4 . 4 ), it follows that D:" =

n !l

exp [ 2

£•

d � !n � Je. •W] .

(5.2)

For the n = 1 case, Eqs . (5 . 2 ) and (4. 1 2) can be written as ln� = 2

i. 0

d ( ), d � ln � � 3C '

Z = t - £·1 (2 ' ) l

O

1

'

,

3C W �

� d' . d� '

( 5. 3 )

( 5 .4)

Splitting the integration in these equations into two i ntervals (0 , b) and (b , oo) and replacing the adequate expression in each one -the power series (4 . 5 ) i n the first interval and the asymp­ totic expansion (4. 1 1 ) i n the second one - one arrives at a system of two coupled nonlinear equations for Z, and D;. In this way, one has re­ duced the problem of solving the system of non­ linear differential equations (2 . 5 ) to a syste m of nonlinear transcendental numerical equations . We have solved these equations numerically. In doing so, we have used not only the power series (4 . 5 ) in the inte rval (0 , b) but also its analytic continuations . In th is way, we have checked that the solution was independent of the value chosen for b. We have also verified the stabi lity of the results . The values obtained for Z 1 and D; are Z 1 1 .707 9 . . . , =

� � 0 . 727 91 . . . .

(5.5) ( 5 .6)

386 I 1 04

H

J . DE VEGA AND F

W i th these values for Z 1 and D i and the recur ­ sion relation (4 .8) 1 D'• = ---

t-

s(s - 2 ) • "

one knows all the coeffici ents of the power ex­ pansion of the solution

i

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A (r) = 2 1 - Z 1 mK1 (m p ) ee p

-

e

I:-

J,

q, ( r) = e · • • - � s D/{c, p' )' " ' e •.a

"' [ 'e'"'

]'''

( 5 .8) ( 5.9)

and also its asymptotic expansion from Eq. (4 . 1 1 )

- (Z 1 )2 2m p [r1 (m p)

J.�

o da K, (a'i'K0 {c,) - K, (m p)

J.�

d d-, K, (a)f,(,., )K0 (a)] + o((Z 1 )'e "'••) } ,

(5.10)

+ 2 (Z1 ) 2 [ro< m p)

1•

a da K.'b)K0(a) + K. ( mp)

1•

a d a /1 (a)K1 (o)K0(o)l + O ((Z,)'e·•••) } ,

(5.11)

,P(r ) = e ·• • -; {1 - Z 1 K0(m p) -

(�.>• K0( m p)2 MP

In Fig. 1 we plot the exact solution for the modu­ lus of the scalar fi�ld and the magnetic field. In Fig. 2 we present the plot of the vector potential. We have investigated the convergence of the series expansion ( 5 . 1 1 ) and its analytic continua­ tions from the numerical values of its coefficients . We found that the radius of convergence of series (5 . 1 1 ) is R = 1 .2 5 5 .

( 5.12 )

This , together with the convergence radius of the analytic prolongations of series ( 5 . 1 1 ) , shows that the solution has singularities at E = ± iR . VI. FORM FACTORS

We have thus obtained the exact classical solu­ tion of a vortex of one quantum of flux, provided 1.

l

A (rl = � c,p' v:(c,p' ) • p • .a

( 5 . 7)

r v•• v•• - •

14

A . S C II A P O S N I K

dltt

j

relation (3 .7) between the coupling constants holds. It is clear that one can also get the corresponding solution for n > 1 by the same method as the one exposed in Sec. V. We also obtained bounds for the 11-quanta solution and its parameters. The mass spectrum of the theory at the class ical level is found to be composed of a scalar and a vector particle w ith the same mass m and the vor­ tices , witb. mass 11m • M• = n 7 .

(6 . 1 )

This expression can b e regarded a s the vortex mass per unit length. It can be pointed out that the approximate solution of Ref. 1 disagrees with this result because it gives an estimate of the energy

a

2 4 6 8 mp FIG. 1. The curves a and � refer to (e/m)l4' (r)i and 2 (2e Im ) H, (P), respectively, for the case n = 1.

2

10

FIG. 2. A plot of (e/m)I A (P)I for the case ll = 1.

mp

387 1 1 0S

CLASSICAL VO RTEX SOLUTION OF THE ABELIAN HIGGS MODEL

with inc reas e s w ith n 2 instead of n . We see from Eq . ( 6 . 1 ) that the de cay of an n- quanta vortex (n :;, 2 ) in n 1 - quanta vortic e s (n = 61 n 1 ) is energetic ­ ally allowed. However , we do not know if there is any conse rvation law that makes them stable , as is the case in the sine- Go :-don model.• A s a first application of the solution , we have c alculated the scalar and vector field form factors in the c la s sical limit.' In this limit, from Eq. ( 2 . 3 )

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=

f

J �;

J

e · li• (f - j•> ,i,.(x)

d • e · 1 1- < = 1/v'2. 171

"'• = filF Is the mass of the scalar field, m v = eF is the mass of the vector field, and

4. STABI LITY OF THE VORTEX LINE (STRING)

p-2 � - � . r ,..._

o.-c'ttu,, f•• -v.r.-v_._ D. Q.-V .Q.+•,...Q..

We write the expression (3. 4) in the form

Q,-cot nw,S(r, 'I') , Q,-1in ,.,,S(r, 'l') ­

J { 1 (J--•- 0-Q.Q.) ) •+ I c••.D.Q.+o..ll.Q,)'

l lf- j;' l'g

p- 1

4

2

f

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+ 7 , sin 6 sin mt.p, �os 8).

We rewrite the expression (5. 1) in the following form:

J {

,··•G· D Q' I • + (Q! - F) ' } · E - a'z 4 (G •• -, •• ,D,Q-) '+ [ ;• ' A . 8 (5 . 3) As for the string, the expression in the square brackets is equal to the divergence of a vector

l

I

(5. 4)

z e•"•G:...D,.Q"= V ...S,.,

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l • S, - 2 e.,..._.,G.,..Q".

The integral of this vector over the area of a large sphere is proportional to the c harge of the monopole. This can be checked by substituting the asymptotic val­ ues for the fields (5. 2) or by making use of the formula for the gauge-invariant definition of the electromagnetic 131 field-strength tensor F�Finally £. -

l I • Q"a�·•kQ"D.Q'DIJ•. w IQI

J [

(5. 5)

fa.If� I • A • m + d'z (G•• -, •• ,Q ' ( ---,,4 ,D ') + 8 Q; - F') •

l

(5. 6)

where M1 = gF. For /3 = 71./g1 - 0 the minimal value E = •bMw/g1 is attained on functions which satisfy the equation G:.-,.,,D ,Q".

(5. 7)

For m = 1 we shall s earch for the solution of this equa­ tion in the form'u Q'-n't(r)F,

(5. 8 )

where the vector n• coincides with the unit vector of the vector ,.-•. Going over to the dimensionless variable r = gFY we obtain a system of equations for v and s dv -;i; - • (1-v),

dt v(2-v) -;;: - --:::'-

(5. 9)

with the boundary conditions v - 1 , s - 1 for ,.. ., "'· The,e equations are easily solved" and we obtain final­ ly Io,· m = 1 and /J- O the monopole mass 4n,V..

,- ---,,-

E

(S.. 10)

and the field distributions i ( r) - � - � . !b r- r

r v(r) -1 - - . sil r

(5. 11)

I n Ref. 8 the mass of the monopole was determined by nume rical int egration for m = I and all � . For � - 0 the

equations (5 . 7). If this assumption is valid, then the energy of the monopole with charge m > 1 is in the limit /3 - 0 4:"l:A/w

(5. 12)

E. - -, - m.

i. e. , the mass of a monopole with charge m is equal to the sum of the masses of m monopoles of unit mass. In this case the point /J = 0 is analogous to the point /J = 1 for the string, and one may assume that monopoles with m ;, 2 and {3> 0 are classically unstable.

6. THE DYON

In the preceding section we have considered the clas­ sical equations of the 0 (3 ) - gauge theory with sponta­ neous symmetry breaking describing a state with mag­ netic charge-a monopole. There are solutions of these equations describing states with magnetic and electric charges, called dyons. "81 The classical equations of the theory under consider­ ation are obtained from the usual Lagrangian (6.1)

where

Q• is a triplet of scalar fields, W� is a triplet of vector fields; these equations have the form (6. 2) A solution of the magnetic-monopole type is obtained Ji one considers that t he fields Q' and the spatial compo­ nents of the vector fieldS differ from zero, are time -­ independent, and satisfy boundary conditions of the type (5. 2).

A solution of the dyon type can be obtained ii the time components of the vector fields are also diiferent from zero. We separate in Eqs. (6. 2) t he spatial and temporal components, taking into account the time-independence of the fields: D.G•"·•--p.-�Q"D•Q�+,r.�•w:D� w:, D.D•w: - g•d'Q"D,Q', k D,D'Q-+D.D•Q•-- (Q'Q,-F')Q". 2

It is easy to see that for /J = 71./g• - 0 one can search for a solution of these equations in the form

result of the numerical integration coincides with ( 5 . 1 0).

It is natural to expect that the solution which de­ scribes a monopole with charge m > I satisfies the 45 3

Sov. J. Nud. Phys.• Vot 24, No. 4, October 1976

(6. 4) where C is a constant to be determined below. With E. B. Bogomol'nyi

453

394 w;

this substitution D,,Q• = 0 and the equations for and Q" coincide in the limit fl- 0, so that we obtain for the functions w:, and Q" the system of equations: D.G••. •-- g( I-C')•"''Q'D'Q.. < I and repu lsive for >.. > I. They also show that for A = I the lower bound o n the e nergy which can be t h e n derived is actually reached at all separa tions a n d , therefore. that in this case vortices do not i nteract.

I. INTRODUCTION

It has been k nown for a long time that the G i n z b u rg-Landau equations' admit localized solutions of the vortex type. 2 M ore rece ntly it has been ob­ served that vortices of essentially the same structure a ppear as solu tions to model field theories for strong­ ly i n teracting particles.' A typical model, known as the A belian H iggs model, describes the i n teraction of a matter ( H iggs) field (analogous to a n order para me­ ter) with a n A belian gauge field, the e lectromagnetic potential i n the G i nzburg-Landau theory, i n a m i n i m a l gauge-invariant fashion. The energy func­ tional, and the field equations which follow from its m i n i m ization , are mathematically identical to those in the G inzburg-Landau theory. The purpose of this paper is to present a highly ac­ c u ra te variational comp u tation of the interaction o f vortices fo r arbitrary separation. O u r starting point is a trial configuration where the matter field q:, van­ ishes a t two points, the locations of the vortices, an d which reduces, for large separation, to two single­ vorte x configurations. The trial field is then modi­ fied i n the i n teraction region , constra i n i ng the zeros of q:,, until the energy i s m i n i m ized. The m i n i m u m can be interpreted as t h e energy of t w o vortices kept at a fixed separatio n . The main releva nce of o u r resu lts l ies i n a complete description of the behav ior of the i n teraction energy as a fu nction of the separa­ tion and of the coupling constant which measures the relative strength of the m a tter self-cou p ling and the e lectromagnetic coupling. The e n e rgy functional contains th ree coupling constants, but two inessential ones may be eli m i nated by straightforward rl)sca ling of the fields . The remaining coupling constant /\ (re­ lated to the G i nzburg-Landau para meter) is physica lly significant. With o u r normalization conven tion , in

12

the sector w here >< < I the range of the matte r self­ i n teraction exceeds that of the electromagnetic one, whereas for 1' > I the opposite is true. Materials with /\ < I ( >< > 1 ) exhibit Type I (Type II) superconduc­ tivity. The in teraction between widely separated vor­ tices has been stud ied and i t has been found that, asy m ptotically, vortices attract (repel) each other for A < l {>, > l l . 4 A lso, the stability of a rotationally sy m m e tric configu ration of many superimposed vor­ tices has been i nvestigated: it is found that for 1' > I the system is unsta ble against decay into separated vortices. 5 In the case where A = l , it is possible to derive a lower bound on the energy . 6 The bo u n d is satura ted if the fields satisfy a set of first-order (non­ li near) partial differe ntial equations and the energy is then proportional to the n u m ber of vortices. These first-order equations have been solved for two vor­ tices a t zero separation 7 and, of course, are solved by a configuration o f infinitely separated vortices. S i nce the bound is additive i n the n u mber of vortices, the e xistence of solutions to the first-order equations for arbitrary separations i m plies that A = I vortices do not i nteract. However, solutions with finite separation between vortices have not been constructed nor ex­ istence proofs been given, and therefore one can not infe r the absence of i n teraction from the bo u n d . O u r resu lts s h o w t h a t t w o vortices attract e a c h o t h ­ er for A < I a n d repel for A > l at a l l separatio ns. M oreover, for /\ = I we find that the e nergy of a two-vortex field config u ration is constant as a func­ tion of the separation within a n error of less than two parts in 1 04 , attributable to the method of appro xima­ tion. We thus obtain a very strong indication that /\ = l vortices do not i nteract, i mplying a high dege n­ eracy of the solutions of the G inzburg-Landau equa­ tion s for this v alue of the coupling constant. The paper is organized as follows: I n Sec. I I we

4486

396 4487

INTERACTION ENERGY OF SUPERCONDUCTING VORTICES discuss the asymptotic behavior of the fields, derive the bound o n the energy, and summarize our results. In Sec. Ill we construct the variational A nsatz and explain the procedure followed to minimize the ener­ gy functional. Section IV contains a discussion of the results. 11. THE ENERGY FUNCTIONA L AND ITS M I N I M U M VALUE

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The expression for the free energy i n the G inzburg-Landau theory (or equivalently the poten­ tial energy in the A belian H iggs model) is £=

J d3x [ ½ I Ul; - ieA,) l ' + + F;1Fij + c,( 1 ¢1 2 - cJ ) ']

(2, 1 )

i s a complex scalar field, A , i s the gauge potential, and Fij = cl;A1 - 1l1 A; is the field strength. 8 The minimum of the energy density is reached when l I = c0 ac 0. R escaling lengths and fields as follows: (2 2 ) the energy functional is written

I n this last equation il,

a stand for

and the factor I / ,r has been introduced for later con­ venience. The tilde over will be om itted hence­ forth. For S to be fin ite, lI must tend to l as l z l goes to infinity. H ence must approach the value e ; >fHl as z - oo with fixed argu ment 0. Continuity r demands that we have e i x (e + i 1 > = e i x( 9> and therefore X ( 0 + 2 1r ) = X ( O) + 2 7r n

(2.9)

with i nteger n. The local behavior of the phase X as a function of fl is immaterial because it can be changed at will through a gauge transformation. But the integer n has physical significance. I ndeed, finite­ ness of S also requires

or

lim ( - iA ) c/> = 0 1,1 - - a

( 2 1 0)

A = -i oln + lvi J � a x + o lvi l o

(2 . 1 I )

a s lz I - oo . The total magnetic flu x through the plane then follows from the Gauss theorem: (2.3) where 1.. 1 = 8 c 4 /e 1

(2.4)

We shall be interested i n field configurations in­ variant under translations along a definite axis. Tak­ ing this one to be the third axis, the fields depend only o n the coordinates x 1 1 x 1 , and A 3 =0. It is then convenient to i ntroduce complex coordinates (2.5) and a com plex potential A = ½ < A , - iA 1 ) , A = ½ < A , + ;A ,)

(2.6)

In terms of these variables the energy per unit length along the third axis is

with

E=

co 1r S e

(2.7)

s = _!__ J dz di [ I ( il - iA ) ;j, I' + I ( a - iA ) ;j, I' 2 1r

( B ) = - i e

J dz di (clA - aA )

2 n = .!_ J im j (A dz + Adi) = l § d x = 1r e e \z l - oo e

( 2 . 1 2)

Thus, finite-energy field configurations are divided into classes, labelled by n. Each class contains all field con figurations (with finite 8) which can be con­ tinuously distorted into each other (for this reason n is called a topological invariant) and within each class the total magnetic flu x is 2 1r n /e. Continuity also imposes a relation between n and the n u m ber of zeros of . The integral 2 1r

}'

n =-i f d in e/> , y

along a n y closed contour y, takes an integer value which can change u nder continuous deformations of the contour only when it goes through a zero of . Assuming that there are n + points z} +l, where van­ ishes as z - z/ +l and n _ points z, H where it vanishes as z - 'i; H , taking for y, first, a circle which encloses all the points z; and letting then the radius of y de­ crease to zero, one readily verifies that n = n+ - n-

(2.8)

(2 1 3 )

(2 . 1 4)

We shall say that the field configuration exhibits a

397 + ( li - iA ) ( il - iA ) i/> - + -� ',f,(,f,;j; - 1 ) - 0

4 /liiA - 4 il A - ; ;j;i),f, + i ,t, il;j; - 2 A ,f,;j; = O 2

(2. 1 5) ( 2 . 1 6)

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One can i nsert i n t o these equations a rotationally symmetric A nsatz ,f, - e 1• •.f ( r ) ,

A - - ( ni /2 z ) a ( r ) , r

• lz l ,

( 2 . 1 7)

with .f ( oo ) = a ( oo ) = I . The symmetry is compatible with the E u ler-Lagrange equations, w h ich reduce to the following ordi nary nonlinear d i fferential equa­ tions for .f( r ) a n d a ( r ) :

!!:.f.. + l. !!L _ r dr

dr 2

n ' ( a - 1 ) ' J l. 2 - 2 A 'f (.f' - l ) - O r2

( 2 . 1 8)

d 2 a _ l_ da l - (a - 1 ) / = O r dr dr 1

From the asym ptotic behavior of the fields one recognizes that the configuration has vorticity n. Correspondingly ,f, m u s t have n zeros, which, be­ cause o f the sy m m e try, m u s t be degenerate at z = 0. Regu larity of H ence we have / { r ) = 0 (,•) for r A (z, z ) requires a (r) = 0 ('r 2 ) for r -0.

-o.

8 = 1- f dz di i l ( a - ;A ) l ' + [ - i ( ilA

"

< li' - ;A ) ,t, = 0

(2.22)

(2.23)

a r e satisfie d . The second term is e /2 " t i m e s the to­ tal magnetic flu x , i . e . , n. Thus for >. = I , 6 is bou n d ­ ed below by n, the bound b e i n g saturated when Eqs. ( 2 . 2 2 ) and ( 2 . 2 3 ) are satisfied. For negative n an a n a logous procedure shows that 6 i s bounded by - n a n d t h a t we have 8 = - n i f ( il - iA ) ,, for vorticity two, and other expansion coefficients are the variation a l parameters. Expressions 0 . 1 ) and ( 3 . 2 ) are chosen so as to reproduce the correct behavior of the fields a t the origin and the correct ap­ proach to their asy mptotic values. Otherwise, the choice of the e x pa n sion i s gu ided by the convenience o f the n u merical a nalysis.• I nsertion of the A nsatz i n to the for m u la for the energy produces a poly n o m i ­ a l of the fourth o r d e r i n the variational parameters with coefficients w h ich can be all evaluated ana lytical­ ly. In the actual c o m p u tation we have set n = I 0. The coefficients of the poly nomial ( taking i n to ac8 ( V,) = & 0 + l; s / n v, + l; s,J1 1 V, V1 + i ?, J

s , -- � � vj I vI - vI V

I

i �J � k

s ,jp v, v1 v, +

(3.4)

( v, - v,< ))

v/m

cou n t the sym metry of the i n dices) then for m a set of less than I O' n u m bers, which can be handled very easily by a large computer. The search for a m i n i m u m has been carried out appro x i mating the quartic 8 by a quadric tange n t to it a t the current values of the variational param e ters. The m i n i m u m o f t h e quadric is t h e n chosen as t h e n e w value o f the variational parameters. The process converges e x ­ tre m e ly rapidly (typically i n fou r or five iterations, i n spite o f t h e large dime nsionality of para meter space) to the m i n i m u m of 8 . C learly, it is the physical na­ ture o f the problem w h ich makes the surface 8 (f, , a,) concave and well behaved, and a llows for a very effi­ cient variational procedure. The steps for the m i n i m ­ ization a r e ex pressed b y t h e following equations, w here V, stand for the collection of variational parameters:

0 .6) 1 (x T d ) fro m the orig i n , the argu­

ment of i u ndergoes a rota tion by 41r. For the two­ vortex config uration we choose then the phase factor of ,f, to be the transform of e x pression 0 . 8 ) a n d set

,f, (z,n = { [, 2

- [1 1']/[ z' - [1

n rf( z . n

(3 1 0)

with real I f must vanish at z = ± ½ d, be having

there as

(3 1 1 )

This guarantees that ,f, has two zeros of the appropri­ ate type a t the location of the vortices. Further specializing the A nsatz we demand that / consists of an asym ptotic term capable of reprodu ci n g a con figuration of separated vortices, plus a correc ­ tio n , co ntaining the variational parameters. A lso, we want to take advantage of the i n formation a lready ob-

400 INTER A CTION ENERGY OF SU PERCONDUCTING VORTICES TABLE I I . T h e e n e rgy o f t w o vortices a t a separation for A ~ 0. 7 , 1 .0 and 1 . 3 . ( U nits a r e e xplained m t h e te xt.) A ~ 0. 7

A ~ 1 .0

A ~ 1 .3

0 I

1 .653 1 .653 1 .656 1 .665 1 .680 1 .696 l .7 I O 1 .7 1 8 1 . 7 23 1 . 7 28

2 .000 2.000 2 .000 2.000 2.000 2.000 2.000 2 .000 2.000 2 .000

2.309 2 .308 2.299 2 . 27 6 2.254 2 , 242 2.236 2.234 2.233 2 .232

6

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d

d

4

tained for vanishing separatio n . We set therefore f (z, i l - wf' " l l z - { I Jr ( l ) l l z + { I I

2.28 2 .24

- - - - - - - -- - - - - - --==-"'-�----

2 04 2 0 1----------------

1 .96 I 76 1.72

1 .68

� •0.7

4

2 + ( 1 - w ) 1 , _ � 2) ' 1

x r2> ( 1 z l )

1; (

Hf (z.n

449 1

(3 . 1 2 )

where w is a weight factor and / 1 1 l , f' ' l are the func­ tions given by Eq. ( 3 . 1 ) with the variational parame­ ters previously determined for a s ingle vorte x and two superim posed vortices. The Fu nction f' " ap­ proaches one exponen tially when its argu ment be­ comes larger than 1 / 'A. . For d >> 2/'A., the product

then reduces to the fu nction

i n the right- (lert-) halF plane and the first term on the right- hand side or Eq. ( 3 . 1 2 ) , with w - I, repro­ duces a n asym ptotic configuration or separated vor­ 1 tices. In the second term the factor l z ' - ( d /2) 2 / l z ' I has the effect or replacing the double zero or {' 2 > at the origin with two zeros at z - ± T d ; for d - o and w -o this second term reproduces the field or two su­ perimposed vortices. 6/ conta ins the variational

6

8

IO

F I G . 2. E nergy of a :wo-vorte x field configu ration for A = 0. 7 , 1 .0 a n d 1 . 3 as a function of the i n ter-vortex separa• tion. T h e dashed lines correspo nd to asy m ptotic values.

parameters, and is e x pa nded as follows: 6/( z, i ) - l z ' x

- 11

I±

, -01-0

r l (cos h 'A. l z l )

fv

(z

z 2Y

·I

1 J

11� 1 II. I ] z

+ z

(3 . 1 3 )

The first factor on the right-hand side or Eq. (3 . 1 3 ) ensures that vanishes at z - ± ½ d; the secon d Factor i ntroduces an exponential cut-off, consistent with the analytic behavior or ) have the following behavior as I x/ approaches infinity : l c/.> 1 -> I DA ¢ = d cf.> - i A cp ->O,

as

(2.2)

/x/ -> oo .

The usual topological arguments imply that the vortex n umber, I n= 21t

J d zxF 1 2

(2.3)

R'

is an integer and is unchanged by finite action perturbations of the fields. The integer n is a topological invariant ; the first Chern number of the complex line bundle in which A is a connection. As Bogomol' nyi [ 6] pointed out, a lower bound on the action results from rewriting the action via an integration by parts as

+ ½ [ ( 8 2 ¢ 1 + A 2 cf.> 2 l ± ( 8 1 cf.> 2 - A I cf.> 1 lJ 2 + ½ [ F 1 z ± ½ ( c/.>f + c/.>� - l ) J 2 } ± ½

J d2 xF 1 2 •

R'

(2.4)

The upper sign corresponds to positive vortex number and the lower sign to negative vortex number. In Eq. (2.4) c/.> 1 = Re cf.> and c/.> 2 = Im cf.>. We shall treat the case of positive vortex number only, the case of n < 0 being completely analogous.

406 279

Vortex Solutions to the Landau-Ginzburg Equations

The lower bound of Bogomol'nyi is

(2.5)

d � nn .

This lower bound is realized if and only if ( 0 1

O. The second term is an infinite sum which converges for t < ( J/2ell v l l 1 , 2 ,11O. For the second term in (4. 1 6) we use Holder's inequality to write

I

lim J e"0

t -+ O

IR 2

(eth - 1 (

-

th)

I

(e " - 1 ) ;£ [ J (e " - 1 ) 2 ] 1 1 2 i im 2 r-o IR

. ! J le' h - l - thl 2 1 1 2 . ] t [ JR'

(4.22)

The function (e0 - 1 ) for V E H is square integrable on L2 (IR 2 , dx). This follows from the bound

J (e' - 1 ) 2 � J le 2 '' - 2v - 1 1 + 2 f le0 - v - l l

IR.2

IR.2

IR 2

which is finite due to Lemma 4.6. We remark that

(4.23)

J le'h - 1 - thl 2 is order t 4 •

IR '

413

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286

C. H. Taubes

For, given any UE H we have

The last step follows from the Monotone Convergence Theorem and the fact that the sum has strictly positive terms. Using Lemma 4.8 to bound ll u ll �.• : JR' and choosing an integer N so that inequality (4.20) holds we have N II nn 2 12 J, l e" - l - u/ ;"; 2 [ J0 (n - k ) ! k ! 2" 1 / u ll � . 2 :JR' " +2

I

n -;;;;;:_ N + 1

(2 0 e ll u l1 1 , 2 ,1R2 )" .

(4.25)

Setting u = th, we see that for sufficiently small t, the right hand side of (4.25) is finite and order t 4• This implies that as t->O the right hand side of (4.22) vanishes, proving Lemma 4.7. To prove the remaining assertions of Proposition 4.2 notice that l a '(v ; h)I ;";

J 1 _ --�- ( 1 + k ll v ll 1 . 2 ) -

(5. 1 )

The proof of Proposition 5. 1 rests on the following properties of the functions u 0 and g 0 :

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Lemma 5.2. Let u 0 ,;nd g 0 be defined by Eq. (3.2) and (3.4) respectively. Let A > 4n. Then a) There exists a constant c 1 > 0 such that for all xE IR.2 1 - g 0 (x) � c 1 .

b) For all x E IR.

(5.2)

2

(5.3)

1 - g 0(x) - e•0 > O .

Proof of Lemma 5.2. For part (a) we note that for l > 4n 9o �

4n

-y < 1 .

(5.4)

For part (b) we have g o(xJ + e•o(x) = 4

k

I=" t ((x - a l) 2 + A.)• 2

Define y = 4n/A < 1 and

zk = ((x - a.) 2 /l + 1 ) -

k

n=" t ((x(x- -a a)1)+ l) . 2

+•

k

1

(5.5) (5.6)

With this notation we rewrite (5.5) as

I

(5.7)

g 0 + e"0 = l1 ( 1 - zk) + l'. zf . n k= l k= l

Each z1 is less than 1 so that

nk = l ( 1 - z ) < 1 - -n1 Lk z n

k

(5.8)

k

Using this inequality in (5. 7) gives finally Y n . ( 1 - y) n 1 n z +zf < l - -z 4n. Any two values, .l. 1 , .l. 2 > 0 of A. give the same solution. For let V; satisfy (3.6a) and (3.6 b) with u 0 (i) = -

I In (1 + �)2)

k= I

(x - ak

i= l, 2 .

We note that u 0 ( 1 ) - u o (2) is in H. By uniqueness then we have v 2 = u 0 ( 1 ) - u o (2) + v 1 .

(5.25)

(5.26)

VI. Properties of the Solution

We now prove the regularity of the weak solution obtained in Sect. V. The Sobolev space W'"·P(JR2 ) is defined as the completion of Cg'(IR 2 ) in the norm (6. 1 ) with et 1 , et 2 nonnegative integers. We restate the final assertion of Theorem I ' in a proposition :

417 290

C. H. Taubes

Proposition 6. 1. The unique weak solution, v 0 to Eqs. (3.6a) and (3.6b) in H is real analytic in IR. 2 • 0

Proof of Proposition 6. 1. L1 v 0 E L2 ( 1R.2 ).

Since v 0 is a weak solution of (3.6a) and (3.6b),

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Thus v 0 is in the Sobolev space W2 • 2 (1R2 ). By the Sobolev Imbedding Theorem ( [ 9], p. 97), v 0 E C0 (JR2 ). Repeating this argument for derivatives of v 0 we obtain v E Ck(JR2 ) for k = 0, 1 , . . . . By a standard theorem ( [ 1 0], pp. 1 70- 1 80) we have v 0 is real analytic in IR.2 • Appendix

The purpose of this appendix is to prove the following proposition.

Proposition A1.1 Let A and be respectively a C 00 connection and Higgs field on IR. 2 which satisfy the first order Ginzburg-Landau equations. Then {xE IR2 1 (x) = 0} is disc rete.

□

Proof of Proposition A l. I. Our plan is to assume the converse and show that a contradiction results. Therefore assume that there exists A and satisfying the conditions of Proposition A l . 1 with {xE IR 2 1 (x) = 0} not discrete. Denote by Z the zero set of . If Z is not discrete then there exists a Jordan arc, y (for definitions, see e.g. Ahlfors [ 1 1 ], p. 69), in Z. Given any open set U intersecting y there exists an open set V C U such that y divides V into two nonempty sets V+ and V_ such that V+ n V_ = y and y n V+ is open in y. Because Z is the zero set of the C''' function l 1 2 on JR 2 , it is possible to choose a Jordan arc y C Z and an open set V as above such that l 1 2 1, = 0

(Al . 1 )

l 1 2 1v. > 0 .

By taking a smaller open set i f necessary we may assume that V i s simply connected. Select an open set WC V such that y divides W into two nonempty sets W+ and compact ; its boundary a w+ a W_ with W+ simply connected ; its closure Jordan curve and y n W+ open in y. In the interior of V+ , ln l l 2 = u + is a C 00 function which satisfies

»'+

(A l .2)

By assumption , l1 is C'" in V. Standard arguments (see e.g., Lions and Magenes [ 1 2] ) imply that there exists a unique C 00 function h on W+ satisfying 2

- Ll h + 11 2 - 1 = 0 in with h = 0 on

int W+ a w+ .

( A l .3)

In the interior of W+ we have - Ll(h - u + ) = 0 .

( A l .4)

418 291

Vortex Solutions to the Landau-Ginzburg Equations

Hence, in the interior of U+

=h

W.+

( A l .5)

+u

with u a harmonic function in int W+ . Since lef,1 = 0 on y and is continuous in V and hl 1 = 0 it follows that for fixed ye y 2

Jim u(x)--+ - oo . lx - yl - O

(A l .6)

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xeint if+

Thus we are led to consider the possible behavior on the boundary, aw+ , of a function u harmonic in int W+ . We note that it is sufficient to study this question for W+ the unit disc in IR 2 and y a subset of the unit circle. This follows from the following two fundamental theorems on conformal mappings. Theorem At.2 (Riemann Mapping Theorem). The interior of any simply-connected domain R with more than one frontier-point can be represented on the interior of the unit circle by means of a one to one co,iformal transformation.

□

Theorem A1 .3. If one Jordan domain is transformed conformally into another, then

the transformation is one-one and continuous in the closed domain, and the two frontiers are described in the same sense by moving a point on one a11d the corresponding point on the other.

□

For proofs and discussions of these two theorems, see, e.g., Caratheodory [ 1 3 ], pp. 70-86. Without loss of generality we take for 0 0 > 0 y = {ei9 E S 1 10 ;;i; 0 ;;i; 0o } .

Here we are representing the unit disc, D, in JR2 as the set of complex numbers with modulus less than or equal to one. Lemma At.4. Given n > 0, u(x) as above and W+ = D, there exists 0 < e. < 1 such that

L) The set {0 e [0, 2n] l u(Q.ei8) < - n} has Lebesgue measure 0. > 0 0 •

Proof of Lemma A J.4. Statement L) follows from ( A l .6) and the continuity of u in

int D.

Let fo.} :'� 1 be a sequence of positive numbers which satisfy the conditions and statement L) of Lemma A l .4, with the added proviso that e. < e. + 1 for n = 1, . . . co . W e have J�� e. = l . Since u is harmonic in int D,

1 2,r ( Al .7) u(0) = - f u(e.eio"Jd0 . 21t 0 (For a proof of this statement, see e.g., Ahlfors [ 1 1], p. 1 64.) u(0) is a finite constant independent of n. Equation ( A l .7) implies that n0

sup u(e.e ill) ;;;; -o + u(0) . 21t 9e[O. 2,t)

( Al.8)

419 292

C. H. Taubes

Let z. E { lz / = r. } be such that sup u ( e.e;e) = u ( z.) .

( A l .9)

0 € [ 0 , 2n]

The compactness of the interval [O, 2rr ] insures that z. exists. The set { z. } ;;'= 1 form a sequence in D and hence have a limit point z 00 e D. Since u is harmonic, z 00 e iJD and u ( z 00 ) = + oo. Applying the inverse mapping of D into »'+ we see that this contradicts that l 0, u -,_ � ,-,4' is normalizable for e > I . Also f may be any integer power less than e- 1 . Thus there are [e - 1 ] normalized zero-energy states:

Here [v] denotes the largest integer less than v. When flux is quantized e=N, the charge is integral or half­ integral and there are N - l states. 10 There is a mismatch between the value of quantized flux and the number of zero modes. The reason is that the N = I mode is asymptotic to r - 1 , which cannot be normalized on the plane with measure r dr d8; rather the norm is logarithmically divergent. This discrepancy may be removed when the R 2 manifold is compactified to S2 by stereographic projection. The eigenvalue equation for nonzero eigenvalues acquires a weight, but the zero eigen­ values are unchanged. The measure becomes r dr d82R 2 /(R 2 + r 2), where R is the radius of the sphere whose surface is S2 • With this measure the N = I mode is normalizable, giving N zero-eigenvalue states when e =N. What is being done here is to take cognizance of the circumstance that mathematical index theorems, which relate the number of zero eigenvalues to topological properties of the background field, take their simplest form on closed compact manifolds like S • 1 1

Another way to understand our zero-energy eigenstates is to consider the square of H which coincides with the Pauli Hamiltonian: 2

422

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H 2 = 2m

!-2

FRACTIONAL CHARGE AND ZERO MODES FOR PLANAR . . .

1 ( p - e A J2 - .E:.. B 2m m

j.

( 1 6)

The latter is an example of supersymmetric quantum mechanics. 1 2 When supersymmetry is not broken, neces­ sarily zero eigenvalues exist. Thus we have closed the circle between zero modes, unexpected quantum numbers, and vacuum currents in three dimensions. The situation is quite analogous to two and four dimensions. The signal for topologically in­ teresting effects is nonvanishing flux, and its magnitude measures the degeneracy of the zero modes. The present results may be relevant to condensed­ matter situations where electrons move in a magnetic field which is constant in one direction. However, before ap­ plying our theory to actual phenomena, the relevance of the three-dimensional massless Dirac equation must be established. Certainly electrons are not massless three­ dimensional particles. Rather one may expect that the tw�by-two matrix equation emerges in a well-defined ap­ proximation to a one-component nonrelativistic theory, where the energy dispersion law is linearized and a tw� component structure emerges ldnematically. Examples. of such constructions have been found in other dimensionali­ ties. The continuum limit of the Su-Schrieffer-Hecger � lyacetylene Hamiltonian 1 3 and its generalizations" yields a Dirac equation in tw�dimcnsional space-time. 1 5 Simi-

IA. N. Redlich, Phys. Rev. Lett. �. 18 ( 1984); preceding paper, Phys. Rev. D 22, 2366 ( 1984); L. Avarez-Oaume and E. Wit­ ten, Nucl. Phys. B2M, 269 ( 1984). Earlier perturbative calcu­ lalions on this are in S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N.Y.) .Hll, 372 (1982); as well as in l. Affleck, J. Harvey, and E. Witten, Nucl. Phys. � 413 ( 1982). 2A. Niemi and G. Semenoff, Phys. Rev. Lett. ll, 2077 (1983). lR. Jackiw and S. Templeton, Phys. Rev. D ll, 229 1 (1981); J. Schonfeld, Nucl. Phys. �. I 57 ( 1 98 I ). 4R. Jackiw and C. Rebbi, Phys. Rev. Lett. ll. 172 ( 1976). 5R. Jackiw and C. Rebbi, Phys. Rev. D ll, 3398 ( 1976). 6R. Jackiw and J. R. Schrieffer, Nucl. Phys. B.1.2!! [FS3J, 253 (1981). 7V. Rubakov, Pis'ma Zh. Eksp. Teor. FIZ. ,ll, 658 (1981) [JETP Leu. ,ll, 644 0981)]; Nucl. Pbys. ll2lU, 3 1 1 (1982); C. Cal­ lao, Phys. Rev. D ll, 2141 ( 1982); � 2058 (1982). IJ. Goldstone and F . Wilczck, Phys. Rev. Lett. 41, 968 (1981) 9L Landau and E. Lifshitz, Quantum Mechanics, 3rd ed. (Per• gamon, Oxford, 1977).

2377

larly, an approximate description of electrons near degen­ eracy points (points where two tlectronic energy bands are in contact) in a hypothetical gapless, parity-nonvariant semiconductor gives rise to a two-component Dirac (Wey!) equation 16 in four-dimensional space-time. Thus one may hope that with planar systems, a physical role for Eq. (4) will also be found. In this connection, it is in­ teresting to note that (2) implies that an external, constant electric field, produces a current perpendicular to it, and the conductivity is e 2 /4,r. Physical electrons possess two states, spin up and spin down. For these one should mul­ tiply by 2, yielding a conductivity of e 2 /2,r. Evidently there exists a quantum Hall effect in our system. 1 7 Note added in proof: The quantized Hall effect has also been analyzed from the point of view advocated in this paper by K. Ishikawa, Hokkaido University Reports Nos. EPHOU 83 Dec 005 and EPHOV 84 Feb 002 (unpublish­ ed); Y. Srivastava and A. Widom, Lett. Nuovo Cimento (to be published); M. Friedman, J. Sokoloff, A. Widom and Y. Srivastava, Phys. Rev. Lett. (to be published). Conversations with F. Wilczck, who also has investigat­ ed similar questions, 1 8 arc gratefully acknowledged. This research is supported in part through funds provided by the U.S. Department of Energy under contract No. DE­ AC02-76ER03069. 10Massive fermion modes in the field of the vortex have been ex• amioed by C. Nohl, Phys. Rev. D .U, 1 840 0975); H. de Vega, ibid. ll, 2932 ( 1 978). Zero modes for massless fer­ mions interacting with a vortex and a scalar field are dis� cussed by R. Jackiw and P. Rossi, Nucl. Phys. IU.2Q [FS3J, 681 (198 1 ). I I M. Ansourian, Phys. Lett. ll!B, 301 ( 1 977). llA. Barducci, R. Casalbuoni, and L. Lusanna, Nuovo Cimcnto -"A, 377 ( 1976). uw.-P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. il, 1698 ( 1 979). l♦M. J. Rice and E. J. Mele, Phys. Rev. Lett. � 1455 ( 1982). ISH. Takayama, Y. Lin-Liu, and K. Maki, Phys. Rev. B ll. 2388 (1980); R. Jackiw and G. Semenoff, Phys. Rev. Lett. � 439 (1983). 16H. Nielsen and M. Ninomiya, Phys. Lett. 13.QB, 389 (1983). 17K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. il. 494 ( 1980). IBF. Wilczek (unpublished).

423 Nuclear Physics B 1 46 ( I 978) 6 3 - 7 6 © North-Holland Publishing Company

A 1 /n EXPANDABLE SERIES OF NON-LINEAR a MODELS WITH INSTANTONS

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A. D'ADDA * and M. LUSCHER

**

The Niels Bohr Institute, University of Copenhagen, DK-21 00 Copenhagen (p, Denmark

P. DI VECCHIA

NORD/TA, DK-21 00 Copenhagen (), Denmark

Received 28 August 1978 We formulate and discuss in detail the recently discovered (Cpn - l non-linear a models in two dimensions. We find that the fundamental particles in these theories are confined by a topological Coulomb force.

1 . Introduction

The most attractive feature of two-dimensional non-linear a models is their simi­ larity with Yang-Mills theories in four space-time dimensions, a parallelism, which has already inspired some exciting conjectures on the latter. For example, the Pohl­ meyer ( or "dual") symmetry [ 1 ] of the a models feeds the hope that such a sym­ metry might exist in four dimensions [2] , too. A somewhat disappointing aspect of the analogy between Yang-Mills and a models is the absence of stable instantons in the 0(n) a models (n > 4). In particluar, effects in the 0(3) a model due to instan­ tons [3] *** were not accessible to the powerful 1 /n expansion and could therefore be explored only by the infrared-divergent dilute-gas approximation [4] . In this paper we define and analyze a new series of SU(n ) non-linear a models, which are 1 /n expandable and whose members are all topologically non-trivial. These new models have first been proposed by Eichenherr [5] , who also showed that they have the dual symmetry and that the n = 2 case is equivalent to the familiar 0(3) model. In sect. 2 we outline the construction of a general non-linear a model and then specialize to the crpn - 1 models, the new series announced above. The instanton

*:

On leave of absence from the Istituto Nazionale di Fisica Nucleare, Sezione di Torino. Address after September l 'st: DESY (Theory), Notkestrasse 85, D-2000 Hamburg 5 2, Germany. *** Refs. [ 3a,b ] are recent reviews of instanton physics.

63

424 64

A. d 'A dda et al. / Non-linear a models

structure of the (Cp n - I models is made explicit in sect . 3 and the quantization via the 1 /n expansion is undertaken in sect. 4. The physical interpretation of the results including a discussion of 0 -vacua is contained in sect. 5 . In sect. 6, we draw conclu­ sions and indicate some interesting possibilities to extend our work.

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2. Definition of the cp n - I non-linear a models * The definition of a general non-linear a model proceeds along the following steps. Let G be a compact Lie group and H some closed subgroup of G. The G/H non-linear a model is a theory of fields .. (p) does not imply tha t there is a zero-mass particle: the two-point function of the gauge-invariant field fµv O µ Av is analytic in momentum space for Re p 2 > - 4m 2 and consequently falls off expo­ nentially in position space. Also, there is no other particle associated with Aµ , * We thank K. Symanzik for a clarifying letter concerning this question.

434 74

A. d 'A dda et al. /Non-linear a models

because (any p 2 =t- 0) .

(56)

The relation ( I 5) between the G:P model and the 0(3) a model now becomes physically meaningful. From the l /n expansion of the O(n) a model we know that the spin field q a is the interpolating field for a triplet of massive mesons. By eq. ( I 5 ) , they can be thought of as two-particle b ound states of the G:P 1 partons. For the general Q: p n - t model the meson spectrum is not known , although for large n one should be able to calculate it within a non-relativistic or perhaps a semiclassical approximation . So far we have been discussing the a:p n - l models in the 0 = 0 vacuum . The 0 =t- 0 vacuum is defined by the modified action [ 1 2]

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1

s8 = S - i0Q ,

S being the ordinary action (30) and Q the topological charge. By eq. (53) this amounts to a change of the effective action Seff according to

(5 7)

(5 8) � fd 2XEµv aµ Av(x) . 2rrv1n Correspondingly, the Feynman rules for the 1 /n expansion are supplemented by a new vertex where a single Aµ •line ends in the vacuum picking up a factor -(0 /2rryn) EµvPv , p ➔ 0. For example , the topological density q (x) = (2rr) - 1 Eµ va µA v (x) n ow has a n on -zero vacuum expectation value 3m 2 (59) ( q (x)>e = iF(0) , F(O) = - 0 + 0 (7 n nrr Stff = Se ff - i

I)

Thus, in a 0 vacuum, there is a constant background topological density . That the addition of the "boundary term" i0Q to the action has an in fluence on the correlation functions, is due to the pole ( 5 5) of the propagator of the topolog­ ical field. If that pole was absent, the zeros EµvP v , p ➔ 0, from the 0 vertices would not be cancelled and there would be no 0 dependence at all . Thus, we see that the confining Coulomb force is linked to the contribution of topologically n on-trivial field configurations to the functional integral. It has been argued [ 1 3 ) that this con­ nection is a very general one, but it is not clear at present, what the implications of the analogous Coulomb force in higher dimensions are. Because the topological charge Q of smooth fields takes on integral values only, one naively expects that the physics of Q:p n - t models is periodic in 0 with period 2rr. At first sight, the result (59) seems to contradict this conclusion. The discrep­ ancy can be explained as follows. The phenomenology of 0 vacua in our model is very much the same as· in the massive Schwinger model [ 1 4] . We thus expect that if 1 0 1 > rr the vacuum breaks down by pair production of partons until the strong topological background fields has decreased to a value corresponding to 1 0 1 < rr. With respect to the 1 /n expansion , this is a non-perturbative effect so that one

435 A. d 'Adda et al. /Non-linear a models

75

should trust eq. (59) only for 1 0 1 < rr. Alth ough the argumen t given here is perfectly plausible , we do not consider it rigorous enough to settle the importan t question of whether or not field con figu rations with fractional topological charge [ 1 5 ) con­ tribute to the fun ctional in tegral .

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6. Conclusions and outlook

Classical of q b > . Up to an integration constant (which can be fixed by specifying q '( -=) 1 q b > (-=)), qM ' is determined by [2 ] a

a�

, , (q < 'f > + q < 'f > ) I I (q b > - q b > ) ,

a 1) (q (-y ) ' - q < 'f )) I (/'f l ' + q b > ) ' a q b )' can then be constructed from u('t ) and u( ± i'f) : _q < 'I J ' = wh ).,;( 't l(-=) J.1,h ) - 1 , with

< > p 'f ( -=) =

C1'/ ) 4h >

( 1 6)

' ( --=) ifo't) - 1 '

w('t) = (u('t ) 0i't ) ) - I v b ) ,

vM = ½ (u( h )

( 1 5)

+ u(-i't)) p( 't)( --=) - 1 (u( i't ) - u( -i't ) ) f/ (-=) . 2i

( I 7)

Thus we see that the eigenvalue problems (9) provide a linearization of the Backlund t ransformation ( 1 5) * . Let us now turn to the general case n ),, 3 . As before , we start from the set of com­ patible equations � 0 ( 1' ) == ( 1 - -y - l ) q a � q bf1b 0 ('{)

a�

a�

'

* For the expert, we note that eq. (9) could have been obtained from the Backlu nd transforma­ tion ( 1 7) via a pair of matrix Riccati equations.

442 M. Uischcr, K. Pohlmcyer / Non-linear a-model

Oh)T o b ) = o a) and (oµ r a )(t0 , x) = 0 (x locality of Qn we then find that, in general , Qn [q ] 'F Qn [l] + Qn [r] .

< a) . Due to the non­ a

(32)

Here , Qn [s] denotes the nth charge evaluated for the spin string s (t, x). Nevertheless, we still have an addition law ; for, as is easily seen from eq. ( I Ob) , the respective generating functionals Q [q ] (w) , Q [l] (w) and Q [r ] (w) satisfy : Q [q ] (w) = Q [r] (w) · Q [l] (w)

Expanding in powers of w we get , for example , Q'; [q ] = Q'; [r] + Q'; [l]

(33)

!f2 [q ] = � [r] + {f2 [l] - e°bc Qf [r] {ti [!] .

(34)

w a U(t, x) = - X W 1 a

(3 5 )

In view of the non-Abelian composition law (33) it is sensible to look more closely at the charges carried by special spin strings q a (t, x). Let us for example consider a massless lump moving to the right (sect. 2). Eq. ( 10b) then reduces to (q x X q)° (t, x ) ia° U(t, x) .

For lumps, Q(w) has a simple geometric interpretation. Namely, eq. (3 5 ) tells us that Q(w) is the product of all the infinitesimal rotations (t is fixed) R (x) = 1 - � wa �) ia° ,

1 -w

445

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54

M. Liischer, K. Pohlmeyer / Non-linear a-model

From this picture it is obvious that there exist non-trivial lumps with Q(w) = l for all w. Indeed, this happens always if q 4 (t, x) runs back the same curve when x goes from x O to -too, as it moved along when x increased from --00 to x0 • We the refore conclude that we cannot construct a complete set o f integrals of motion in involu­ tion (i.e. the invariant manifold in the phase space for q 4 ) from the charges � and the energy-momentum tensor E>µv alone . The statement above suggests that there are more constants of motion. One might speculate that these can be obtained by forming Poisson brackets among the old charges. Unfortunately , due to non-vanishing boundary terms, the Poisson bracket between say � and Q!!i is not unambiguously defined through the funda­ mental b rackets. In order to get a safe definition of a Poisson bracket { � , Q!!i } let us introduce volume cutoff charges Q;;' 4 • These are the same as � where , however, the multiple integrals involved range only from -L to +L . Then the limit (36) is well-defined. Had we first taken the L 2 ➔ 00 limit, the outcome would differ from the above by a polynomial in (tj , z ..;; n + m - i, and q(±oo ) . By a rather Ienghty argument (it will be omitted here) one can show that {�. Q� } is a polynomial in Qf , 1 ..;; n + m - I and q ( ±00) . For example , we found that {(ti , Q!!i } = _e4bc {fm+ 1 + P';.;/

'

{3 7)

where P'fn is a combination of Qf , 1 ..;; m , and q (±00) . Thus, we cannot p roduce new constants of motion this way . We remark finally that a simple interpretation (such a s particle-number conserva­ tion etc.) of our charges is lacking so far. Of course, this is due to the fact that clas­ sical spinwaves do not decay into a superposition of Abelian waves for large times. b

One of us (K.P.) would like to thank the staff of the 11 . Institut fiir Theoretische Physik der Universitiit Hamburg for their kind hospitality. M .L. thanks H. Lehmann for discussions. References [ I ) A. Polyakov, Phys. Lett. 59B ( 1 9 75) 79 ; A. Belavin and A. Polyakov, JETP Lett. 22 ( 1 9 75) 245 ; A. Jevicki, Princeton preprint C00-2220-108 ( 1 9 7 7) ; D. Forster, Nucl. Phys. B l 30 ( 1 9 77) 3 8 . [ 2 ) K. Pohlmeyer, Comm . Math. Phys. 46 ( 1 9 76) 207. [3) S. Coleman, Classical lumps and their quantum descendents, Erice Lectures, 1 9 75 . [ 4 ) H. Boemer, Darstellungen von Gruppen, (Springer Verlag, Berlin.Cottingen-Heidelberg, 1 95 5 ).

446 ANNALS OF PHYSICS 120, 253-29 1 ( 1 979) Copyright Q 1979 by Academic Press, Inc.

Factorized $-Matrices i n Two Dimensions as the Exact Sol utions of Certain Relativistic Quantum Field Theory Models ALEXANDER

B.

2AMOLODCHIKOV

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Institute of Theoretical and Experimental Physics, Moscow, USSR AND ALEXEY

B.

2AMOLODCHIKOV

Joint Institute for Nuclear Research, Dubna, USSR Received August J, 1978

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation Jaws of the underlying field theory is discussed. The fac­ torization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different propert­ ies for the cases N = 2 and N > 3. For N = 2 the general solution depends on one para­ meter (of coupling constant type), whereas the solution for N > 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliion S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N > 3 there are two "minimum" solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear a-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.

I . INTRODUCTION The general two-dimensional relativistic S-matrix (not to mention higher space-time d i mensionalities) i s a very compl icated object. In two space-time dimensions, how­ ever, a situation is possible in which the total S-matrix being nontrivial is simplified drastically. This i s the case of factorized scattering. Generally, the factorization of a two- d i mensional S-matrix means a special structure of the multiparticle S-matrix elemen t : it is factorized into the product of a number of two-particle ones as if an arb itrary process of multiparticle scattering would be a succession of space-time separated elastic two-particle collisions, the movement of the particles in between being free.

253

447

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254

ZAMOLODCHIKOV AND ZAMOLODCH IKOV

The factorized S-matrix has been first discovered in the nonrelativistic problem of one-dimensional scattering of particles i nteracting through the o-function pair potential [J -3). Furthermore, the factorization is typical for the scattering of solitons of the nonlinear classical field equations completely integrable by the i nverse scattering method (4-6). Note, that all the dynamical systems leading to the factorized S-matrix possess, as a common feature, an infinite set of "close to free" conservation laws. 1 This set of conservation laws is considered to be a necessary and sufficient condition for the S-matrix factorization [7- 1 I ]. Some speculations about this point are presented in Section 2. The expressibility of the multiparticle S-matrix in terms of two-particle ones pro­ vides an essential simplifi cation and enables one to construct in many cases the total S-matrix up to the explicit calculation of the two-particle matrix elements themselves. In the present paper we construct a certain class of the relativistic factorized S-matrices being invariant under O(N) isotopic transformations. We use the method first suggested by Karowski, Thun, Truong and Weisz ( 1 2] (in the sine-Gordon context). The selfconsistency of the factorized structure of the total S-matrix turns out to impose special cubic equations (the factorization equations in what follows) on the two-particle S-matrix elements (see Section 2). Therefore, the factorization, unitarity and crossing symmetry provide a nontrivial system of equations which is basic for the method mentioned above. The general solution of these equations has an am­ biguity of CDD type: there is a "minimum solution" (i.e., the solution having mini­ mum set of singularities) ; one obtains the general one addi ng an arbitrary number of auxiliary CDD poles. Are there any two-dimensional quantum field theory (QFT) models that lead to these S-matrices ? Most of the nonlinear classical field equations have evident QFT versions. The problem of factorizing quantum S-matrices of these models (which is closely connected with that of " surviving" classical conservation laws under quanti­ zation) is nontrivial and requires special investigation in each case. In this paper we consider three models, the aim is to show that they lead to O(N)�symmetric factorized S-matrices. ( 1 ) The quantum sine-Gordon model, i.e., the model of a single scalar field

ef,(x), which is defined by the Lagrangian density:

!t'sG = ½ (8� c/>) 2 +

1

2

;�

cos(/3cf,),

(I. I)

where m 0 i s a mass - like parameter and {j is a coupling constant. It is well -known that the classical sine - Gordon equation is completely integrable [6 ] . The structure of the quantum theory has been also studied in detail . The mass 1 The meaning of this term is as follows. I n the asymptotic states where all particles are far enough from each other these conserva tion laws tend to those of the theory of free particles. The latter laws lead to conservation of the individual momentum of each particle and can be formulated, e.g., as the conservation of sums of the entire powers of all particle momenta La Pa•; n = I , 2, . . . [7].

448 FACTORIZED S-MATRICES IN TWO DIMENSIONS

255

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spectrum of this model has been found by a quasiclassical method [ 1 3-1 5). I t contains particles carrying the so-called "topological charge" 2 - quantum soJitons and corre­ spondi n g antisoliton s - and a number of neutral particles (quantum doublets) which can be thought of as the soliton-antisoliton bound states ; the "elementary particle" correspond ing to field ef, turns out to be one of these bound states. Some of the quasi­ classical results (the mass formula for the doublets) proved to be exact ( 1 5). The other exact result has been obtained by Coleman [ 1 6) (see also Refs. [ 1 7, 1 91 ) . The quantum sine-Gordon model is equivalent to the massive Thirring model, i.e., the model of cha rged fermion field, defined by the Lagrangian density ( 1 .2) provided the coupling constants are connected by g/71 = 471/P2 - 1 .

( 1 .3)

Funda mental fermions of ( 1 .2) are identical to quantum solitons of (1 . 1 ). There is a considerable amount of results in support of the factorization of the quantum si ne-G ordon S-matrix; these results are mentioned in Section 4. (2) The quantum chiral field on the sphere s-v- 1 (N = 3, 4, ... ) (O(N) symmetric nonlinear er-model) defined by the Lagrangian density and the constraint ( 1 .4) wh ere g 0 is a (bare) coupling constant. This model is O(N) symmetric, renormalizable and asymptotically free (20, 1 2]. The infrared charge singularity of this model seems to cause the disintegration of the Goldstone vacuum [22]. True vacuum i s O(N) symmetric and nondegenerate; all particles of the model are massive and form O(N)-multiplets. This situation is surely the case when N is large enough [23, 24] and we suppose it is valid for N � 3. In Section 5 some arguments i n favour of S-matrix factorization in the model ( 1 .4) are presented. The first evidence of this phenomenon is based on the properties of the 1 /N expansion of the model (23, 24] . Namely, the absence of 2 -+ 4 production ampl itude and the factorization of 3 -+ 3 amplitude can be shown to the leading order in I / N [25). A more rigorous proof of the er-model S-matrix factorization follows from the recently d iscovered infinite set of quantum conservation laws [26, 27]. In Section 5 we review briefly the results of Ref. [26). ' In model ( I . I ) the topologic charge q is connected with the asymptotic behaviour of the field 4,(x, t) as x -- ± oo ; q = P.'2rr

J

.,

d,f,

- clx = P/2rr(,f,( oo ) - ,f,( oo )] . _ ., dx

449 256

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

(3) The Gross-Neveu model, i.e., the model of N-component self-conjugated Fermi-field tf,;(x); i = I , 2, ... , N (N � 3) with four-fermion interaction j

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!i'aN = 2

N

L {,;-y,. o,.tf,, +

,-1

g N ]2, o[ S •-1 Y,; t/,;

L

( 1 . 5)

where {,; = tf,,y0 • Like the chiral field this model is renormalizable, asymptotically free and explicitly O(N) symmetric. Model ( 1 . 5) has been studied by Gross and Neveu in the limit of N -+ oo [28]. They have found a spontaneous breakdown of discrete y 5-symmetry {the fie ld 1 VJ ;t/, ; acquires a nonzero vacuum expectation value) leading to the dynamica l mass transmutation. Using quasiclassical method Dashen, Hasslacher and Neveu [29] have studied the model in the same N -+ oo l i mit. These authors have found a rich spectrum of bound states of the fundamental fermions of this model and deter­ minded their masses. We support the factorization of the Gross-Neveu S-matrix by arguments which are quite analogous to those for model (I .4). The paper is arranged as follows. I n Section 2 general properties of factorized scattering are considered, factorization equations are introduced and their meaning is cleared up. Furthermore, a convenient algebraic representation of the factorized S-matrix is suggested. Sect. 3 contains the general solution of analyticity, unitarity and factorization equations for the S-matrix having O(N) i sotopic symmetry. The "minimum" solutions of these equations turn out to be essentially different for N = 2 and N � 3. The solution for N = 2 depends on one parameter of coupling constant type. As i t is shown in Section 4 this solution turns out to be the exact S-matrix of quantum si ne-Gordon solitons. In this Section we construct the total sine-Gordon S-matrix too, wh ich i ncludes all bound states (doublets). For the case N � 3 "mini­ mum" solutions of Section 3 do not depend on any free parameters. They correspond to asymptotically free field theories with the dynamical mass transmutation. In Sections 5 and 6 one of these solutions is shown to be the S-matrix of model ( I . 4) and the other-to be an exact one of elementary fermions of ( 1 . 5) .

I;r_

2. FACTORIZED SCATTERING, GENERAL PROPERTIES, fACTORIZATIOS EQUATIO'.' p 2 > . . . > p,., , the wave function in each domain 3 I t may appear that (i) and (ii) mean that the S-matrix is diagonal in the momentum representa­ tion . It is not true if the theory contains different particles (having different internal quantum numbers, e.g., particle and antiparticle) of the same mass. In this case the exchanges of momenta between these par t icles and other nondiagonal processes are possible (see Sect. 3).

451 258

ZAMOLODCHI KOV AND ZAMOLODCHIKOV

should be a superposition of waves, the set of wave vectors being selected by these rules:

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'Pp(x1 , . . . , xN )

= L C(P, P ') exp{ipP1Xp• 1 + · · · + P'

ipPNXP' N} ;

{x} E Xp .

(2. 5)

H ere summation is carried out over all permutation P' of Pt , . . . , PN , perm itted by (i) and (ii). Symmetrization (antisymmetrization) in the coordinates of identical particles is i mplied in (2.5). The coefficients C(P, P') are functions of the domain XP and of the permutation P'. In particular, the coefficient C(P, I) describes the inci­ incident wave in the domain Xp ; to obtai n the scattering wave function one puts C(P, I) = 0 if P ,f, I and C(/, /) = 1 (here / is identical transposition). The coefficients C(P, l) (I is the inverse transposition /(I , 2, . . . , N) = (N, N - I , . . . , I )) describe outgoing waves in these domains and thus they a re elements of the N-particle S-matrix. For example, in the case of two particles of the same mass the wave function becomes

(2.6) I n Eq. (2.6) SR and Sr are two-particle S-matrix elements corresponding to backward scattering (reflection) and forward scattering (transition). It i s convenient to picture the situation as the scattering of the N-di mensi onal plane wave in the system of semipenetrable hypersurfaces X; = X; (for any i and j). Far enough from these hypersurfaces the wave is described by (2. 5); near them the m otion is more complicated because of the interaction between the particles. More­ over, if the relativistic problem is taken into consideration the motion in the inter­ action region cannot be treated in terms of the wave fu nction of a finite n umber of variables (because the virtual pair creation is possible). The determi nation of the coeffi­ cients C(P, P') in (2. 5) requires the extrapolation of the wave function from one domain of free motion to another through the boundary between them, where the particles are in interaction. The solution of the problem of i n teracting particles is, in general, a very complicated task. Note, however, that the extrapolation of the wave function can pass through the region of the boundary, where two particles are cl ose and others are arbitrary far from them and each other (e.g., I x 1 - x2 I ;:;; R, I X; - x1 l ► R, I x2 - X; I ► R and I X; - X; I ► R; i, j = 3, 4, . . . ). These r egions describe two-particle collisions and there the extrapolation conditions are the same as in the two-particle problem. Therefore, in this case the knowledge of the two­ particle S-matrix elements provides one with a sufficient i nformation to dete rmin e all the coefficients C(P, P') and, therefore, to obtai n the multiparticle S- matri x. N-particle S-matrix element turns out to be a product of N(N - 1 )/2 two parti cle ones. Such a structure is spoken about as the factorized S-matrix. Note t hat the possibility of this structure is d u e to the fact that the wave function in each d o ma in Xp is a superposition of a finite number of waves, the latter being a conse q ue nce of the infinite set of conservation laws.

452

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FACTORIZED S-MATRICES IN TWO DIMENSIONS

259

Of course, this consideration connecting the factorization and the existance of infinite number of conservation laws is not a rigorous proof; a complete evidence can be found in a recent paper [ 1 1 ]. All the considerations presented in the above para­ graph are of exact sense in the case of the one dimensional problem of nonrelativistic particles i nteracting via the S-function potential [ I -3]. The factorized S-matrix corresponds to the following simple scattering picture. In the infinite past particles of momenta P i > p2 > · · · > P N were spatially arranged in the opposite order: x1 < x2 < ·· · < x N . In the i nteraction region the particles successively collide in pairs; they move as free real (not virtual) particles in between. The set of momenta of particles is conserved in each pair collision ; if the particles are of different mass the transition is possible only, the collision of particles of the same m:iss may result in the reflection too. After N(N - 1 )/2 pair collisions the par­ ticles are arranged along the x axis in the order of momenta i ncreasing. This corre­ sponds to the final state of scattering-outgoing particles.

F10. 1.

Toe space-time picture illustrating the multiparticle factorized scattering.

The space-time picture of the multi particle factorized scattering can be represented by a spacial diagram ; an example is drawn in Fig. I . Each straight linein the diagram corresponds to any value of momentum, obviously connected with the slope of the line (in this diagram time is assumed to flow up). Two-particle collisions are repre­ sented by the vertices where the lines cross each other; the corresponding two particle amplitude should be attached to each cross. The total multiparticle S-matrix element of the process drawn in the diagram is given by a sum of products of all the N(N - l )/2 two-particle amplitudes corresponding to each vertex. The summation mentioned above should be carried out over all possible kinds of particles flowing through the internal lines of the diagram and resulting in a given final state. The following is to be mentioned. The same scattering process can be represented by a n umber of different diagrams in which some of the lines are translated in parallel (e .g. , sec Figs. 2a and 2b). The amplitudes of these diagrams should not be added in the m ultiparticlc S-matrix element. In terms of the wave function in sectors X,. ampl itudes drawn in Figs. 2a and 2b correspond to different semifronts of the same

453

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260

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

P, a F1G. 2.

Two possible ways of the three-particle scattering.

outgoing wave. Both should have the same amplitudes and phases (because of (i) and (ii)), i.e. , be coherent. This requirement makes two-particle matrix elements satisfy special cibic equations, the latter being necessary conditions of the factorization. In what follows these equations play an essential role and we shall call them the factorization equations. 4 In the present paper the relativistic scattering is mainly considered. The following notations are convenient in this case. We shall u se rapidities 0a instead of momenta Pa of particles (of mass m 0) (2.7)

Two-particle amplitudes S( Pa , pb) become functions of the rapidity difference of colliding particles 00b = 00 - 0b , the latter being simply connected with the s-channel invariant Sab = ( P a" + P b") 2 (2.8)

(m a and m b are masses of the particles).

® rig ht cut le ft cut FIG. 3 .

The analytical structure o f two-particle amplitudes in the physical sheet of the s-plane.

' Factorization equations and their physical sense in the problem of nonrelativistic particles interacting via the a-function potential have been considered in Ref. [2]; in the case of the sine­ G ordon problem they were obtained in Refs. [30, 3 1 ] and used in Ref. [1 2].

454 FACTORIZED S-MATRICES I N TWO DIMENSIONS

26 1

Two-particle amplitudes S(s) are the analytical fu nctions in the complex s-plane with two cuts along the real axis s ¾ (m a - m b) 2 and s ), (m a + m b) 2 (see Fig. 3). The points s = (ma - m b) 2 and s = (m a + m b) 2 , being the two-particle thresholds, are square root branching point of S(s). In the case of the factorized scattering there is only the two-particle unitarity and it is natural to suppose functions S(s) not to exibit other branching points. If it is the case, the functions S(B) should be meromorphic. Mapping (2. 8) transforms physical sheet of the s-plane into the strip O < Im 0 < -rr (if it cannot lead to a misunderstanding we shall drop subindices 0 0ab) in the 0-plane, the edges of the right and the left cuts of the s-plane physical sheet being mapped on the axes Im 0 = 0 and Im 0 = -rr, respectively (see Fig. 4). The axes Im 0 = /-rr; I = - 1 , ±2, . . . correspond to the edges of cuts of the other complex s-plane sheets.

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=

"5

i I

ul

.211

:C l

'- I

® 1I

Ph � sica[ strip

I

I I I I

bound

;states I I I -1

JmB

�. GI

...., 1 -1

FIG. 4.

The structure of the 8-plane.

The functions S(0) are real at the imaginary axis of the 0-plane (real analyticity). In particular, at Im 0 = 0 the relation S(- 0) = S*(0) is valid. Crossing symmetry transformation s -,. 2m/ + 2m b2 - S corresponds in terms of the variable 0 to substitution 0 -,. i-rr - 0. ln the nonrelativitic limit p/ � m a rapidities can be replaced by the nonrelativistic velocities 0a -,. Va = Pa lma . All the following expressions (except those connected with the crossing relations) can be applied to the case of nonrelativistic S-matrices after replacement ea -,. Va , eb -,. vb , 0ab -,. Va - vb . I t is convenient to describe a general structure of the factorized S-matrix by means of a special algebraic construction [30, 25]. Consider a factorized scattering theory containing several kinds of particles (A , B, C and so o n ; particles of the same kind are supposed to be identical ; statistics is not i mportant for our consideration). These particles are represented in our constructio n by the special noncommutative symbols A (0), B(0), C(0), ... , the variable 8 being the rapidity of the correponding particle. These symbols are frequently called the particles. The scattering theory is stated as follows. Identify asymptotical states of the scattering theory with the products of all the particles in the state. The arrangement of the symbols in the product corresponds to that of particles along the spatial axis x: in-states should be identified with the products arranged in the order of decreasing

455 262

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

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rapidities of particles while out-states with those arranged in the order of i ncrea sing rapidities. For example, in-state of three particles A, A and B having rapidities 81 , 82 and 83 , respectively (81 > 82 > 83), acquires the form A(81 ) A(82) B(83) . Any product can be rearranged by means of a number of subsequent commutati on s of neighbour particles (the associativity of the symbol multiplication is supposed) . Each commutation corresponds to the certain two-particle collision ; this leads to commutation rules for the symbols A (8) , B(O), . . . . For example, if particles A and B are of different mass, one writes (2.9)

where s; 8(81 2) is the transition amplitude for the reaction AB -+ AB (remind, that in the case of different masses, reflection is forbidden by (ii)). If particles of differerent kinds (say A and C) but of the same mass are under consideration the reflection is permitted and we should write (2. 1 0) Reflection and transition are indistinguishable i n the case of identical particles, therefore (2. 1 1)

As it was mentioned above (see footnote 3) if there are different particles of the same mass one of them is permitted to turn into the other in the process of two-particle scattering. It means that additional channels in the two-particle scattering are open and, hence, corresponding terms should be added into the right hand sides of Eqs. (2.9), (2. 10) and (2. 1 1 ). We shall not discuss this point here, there are some examples of such situation in the next section. The consistency of the commutation relations of type (2.9), (2. 1 0), (2. 1 1 ) in the calculation of symbols A(O), B(8) and so on and their associativity requires certain equations for the two-particle amplitudes to be satisfied. The latter are of two kinds. The identities of the first kind arise when one performs the opposite transposition of symbols after the direct one, and requires the result to be equal to the initial com­ bination; these identities coincide with the two-particle unitarity relations. The multi­ particle in-states may be rearranged i nto out-states in many possible successions of pair commutations but the result should be the same. This leads to the i dentities of the second kind. Clearly, it is sufficient to consider three particle states only and require the same result of permutations i n two possible successions. One obtains all the required identities which coincide, of course, with the conditions ensuring the equality of triangle diagrams (see Figs. 2a, 2b), and so they are the factorization equations. If identities of both kinds are satisfied the commutation relations permit one to rearrange unambiguously any in-state into a superposition of out-states and then this construction represents the total factorized S-matrix. Its unitarity is trivial. One obtains the matrix s-1 after the rearrangement of out-states into in-states : it d i ffers

456 FACTORIZED S-MATRICES IN TWO DIMENSIONS

263

from the S-matrix in the signs of the arguments of all two-particle amplitudes 80 b -+ - 8.b . This change of signs leads to the complex conjugation of the two­ particle matrix elements. Taking into account the symmetry of the S-matrix one obtains s- = s-1 •

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3. RELATIVISTIC S-M ATRIX WITH O (N)-JS0SYMMETRY. G ENERAL SOLUTION Followin g the general consideration of the previous section we treat now the class of relativistic factorized S-matrices characterized by the isotopic O(N) symmetry. To introd uce the O(N) symmetry we assume the existance of isovector N-plet of particles A , ; i = l , 2, . . . , N with equal masses m and require the O(N) symmetry of the two-particle scattering (this ensures O(N) symmetry of the total S-matrix due to the factorization) . Namely, we assume for two-particle S-matrix the form: ik S;1 = (A ; ( p{) A , ( p;), o u t I A ;( P i) A k( P 2), i n ) = S ( Pi - PD S( p2 - p�)[S;kS;,Si(s)

+ S;;Sk1Sh) + SilS;kSa(s)] (3 . 1 )

where s = ( P i" + P2 u) 2 and the + (-) refers to bosons (fermions). The functions Sis) and Sh) are the transition and reflection. amplitudes, respectively, while Si(s) describes the "annihilation" type processes : A; + A; -+- A1 + A 1 (i -¥= j). The S-rnatrix (3. 1 ) will be cross-symmetric provided the amplitudes S(s) satisfy equations Sh) = Sl4m2 - s) and Si (s) = Sa(4m2 - s). After introducing the rapidity variables (2.7), (2.8) we deal with meromorphic functions S1 (8), Sz(8) and Sa(0), where s = 4nr ch'!(B/2), and the cross-symmetry relations become Sz( 0) = Sz(i-rr - 0),

(3.2a)

S1 (0) = Sa(i,r - 0).

(3.2b)

To describe now the factorized total S-matrix let us introduce, following the general method of Section 2, symbols A;(0) ; i = 1 , 2, ... , N. The commutation rules corre­ sponding to (3 . 1 ) are A ;(81 ) A;(02) = S,;S1 (012)

N

I A i8 ) A i8 ) k-1 2

1

lt is straightforward to obtain the u n itarity conditions for two-particle S-matrix (3 . 1 )

+ Sa(B) Sa(- 0) = 1 , + Sz( - 0) S3(0) = 0, NS1 (8) S1( - 8) + S1 (8) Sz(- 0) + S1(8)Sa(- 8) + S2(0) Si( - 0) + Sa(0) Si( - 0) = 0. Sill) Sz( - 0)

(3.4a)

Sill) S3(- 0)

(3.4b) (3.4c)

457 ZAMOLOD C H I KOV AND ZAMOLODCHIKOV

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264

Obviously, Eqs. (3.2) and (3.4) a re not sufficient to determine the functions 5(0). Further restrictions arise from the factorization equations (see Section 2). One obtains the factorization equations considering all possible three-particle in-products A;(B1) A;(B2) At(63) , reordering them to get out-products by means of (3.3) and requiring the results obtained in two possible successions of two-particle commu­ tations to be equal. The equations arising are evidently different for the cases N = 2 and N � 3 (fewer different three-particle products are possible at N = 2). Therefore it is convenient to make a notational distinction between these two cases. Dealing with the case N � 3 we redenote the amplitudes S1 , S2 and S3 by a1 , a2 and a3 , respectively, reserving the original notations for the case N = 2. The factorization equations have the form (the derivation is straightforward but somewhat cumbersome) S2 S1S3

for N

+ S2S3S3 + S3S3S2 = S3S2S3 + S1 S2S3 + S1 S1S2 ,

SaS1Sa + S3S2S3 = S3 S3S1 + S3 S3S2 + S2 S3 S1 + SiS3S3 + 2 S1 S3S1 + S1 S3S2 + S1 S3S3 + S1S2 S1

= 2 and

+ S1S1S1

(3.5a) (3. 5b) (3. 6a)

Na1a3U1 + a1 a3 a2 + 0"10"3 0"3 + a1 a2 a1 + 0"20"30"1 + 0"30"30"1 + U1U10"1 = 0"30"10"3

(3.6b) (3.6c)

for N � 3. For each term in (3 .5) and (3.6) the argument of the first, the second and the third S (a in (3.6)) is implied to be e, e + 0' and 6', respectively. The factorization equations turn out to be rather restrictive. They allow one to express explicitly all the amplitudes in terms of one function. General solutions for both systems (3. 5) and (3.6) satisfying the real-analyticity condition (all the amplitudes are real if 8 is purely imaginary) are derived in Appendix A. For system (3.5) (i.e., for N = 2) this solution is 41r6 41r8 . Sa(0) = 1 ctg ( --) cth ( --) S,(0) y y 41T(i8 - O) 4 8 S1 (6) = i ctg ( ; ) cth ( ) SiB) y

(3. 7a) (3. 7b)

with arbitrary real y and 8. The general solution for {3.6) contains only one free parameter ,\ and have the form : aa (0) = -

i.\

B al0)

;;,,.

ai(0) = - i N [( - 2)/2 ] ,\ -

(3.8a)

e aiO) .

(3.8b)

458 265

FACTORIZED S-MATRICES IN TWO DIMENSIONS

The restrictions on the amplitudes Sl0) and u2(0) come from the unitarity condi­ tions (3.4). The equations (3.4b) and (3.4c) are satisfied by (3.7) and (3.8) identically, while equation (3.4a) gives 41r0 . Sill2 ( --) S h 2 ( --) 41r8

Sl0) S2( - 0) =

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for N = 2 and

{ 4778 41r ) • 2 ( -sm ) s h 2 ( -Y

y

ui0) ui- 0) = 2 8

y 8 41r8 + cos2 (-41r ) ch 2 (-) y

92

+ ,\2

(3.9)

y

(3. 1 0)

for N � 3. Until now we have deliberately avoided the use of the cross-symmetry relations (3.2). Although the above consideration concerns the relativistic case, the unitarity conditions (3.4) and factorization equations (3.5), (3.6) are valid for any non­ relativistic O(N) symmetric factorized S-matrix as well, under the substitution: (3. 1 1 ) where k1 and k2 are momenta o f the colliding particles. Therefore, the general solutions (3.7), (3.9) and (3.8), (3. 10) are still valid (after the substitution (3. 1 1 )) in a nonrelativistic case. This will be used at the end of Section 4. Equations (3.2) are especially relativistic. They turn out to give. restrictions on free parameters in (3.7) and (3.8). It is easy to see that (3.2) is satisfied only if (3 . 1 2) in (3.7), (3.9) and

A=� N-2

(3. 1 3)

L sh 8 + i sin cx k k-1 sh 8 - i sin cx k '

(3.14)

in (3. 8), (3. 1 0). Thus, the formulas for N � 3 do not actually contain any free para­ meter. This circumstance will be important in Section 5. Equation (3.2a) (which is certainly valid for ui(0) as well as for SJO')) together with (3.9) and (3. 1 0) will be used to determine S2(0) and u2(8). In both cases N = 2 and N � 3 the solution admits the CDD-ambiguity only (32]: an arbitrary solution can be obtained multiplying some "minimum" solution by a meromorphic function of the type f(B) =

TI

459 ZAMOLODCffl KOV A N D ZAMOLODCHIKOV

266

where Ot'.1 , Ot'.2 , ••• , Ot'.L are arbitrary real 5 numbers. It is the " minimum" solutions, i .e., the solutions having a minimum set of singularities in the 0 plane, that will be of mo st interest below. For N = 2 such a solution can be represented in the form Sl0) where

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U(0)

R"(0)

si n (

4

;2 ) sh ( 4;8

)

sh [

41r(i1r - 8) U( , 0) ] y

(3. 1 5)

11 r (1 + i � ) I' (1 - � - i �) y y y •-i

Rn (0) R. ( i,r .- 0) ' R.(O) R.( m) (3 . 1 6) r (2n � + i �) r ( 1 + 2n � + i �) Y Y Y Y 8 r ( (2n + l) g; + i � ) r ( t + (2n - l ) ; + i � ) ·

= r (�) y

=

= :

In the case N ;;;, 3 there are, in general, two different "minimum" solutions (the exceptional cases are N = 3 and N = 4, when these two solutions coincide). We denote these solutions a-!+ 1 (0) and at1 (0) ; they can be written in the form (3 . 1 7) where Q

(B) and at 1 (0) for N ;;;, 3 give the exact S-matrice s for the quantum chiral field ( 1 .4) and for the "fundamental" fermions of Gross­ Neveu model ( 1 .5), respectively. • We co nsider the solutions having singularities on the imaginary () axis only, i.e., the sol uti ons exhibiting bound and virtual states only.

460 FACTORIZED S-MATRICES IN TWO DIMENSIONS

267

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4. EXACT S-M ATRIX OF THE QUANTU M S INE-GORDON MODEL

Quantum sine-Gordon model { 1 . 1 ) is the most known example of relativistic quantum field theory leading to the factorized scattering. There are various results ensuring the factorization of the sine-Gordon S-matrix. Complete integrability of the classical sine-Gordon equation [6] means the existance of an infinite number of conservation laws in the classical theory which are "deformation of free ones". The analogous set of conservation laws is also present in the "classical" massive Thirring model (which corresponds to the "tree" approximation for the Lagrangian ( 1 . 2) [33, 34]). The important problem of the conservation laws in quantum theory has .been treated in Ref. [34] where such conservation laws were shown to survive after quantization by the perturbation theory approach (in all perturbational orders). The absence of particle production and factorization of multiparticle quantum S-matrix which are the consequenses of conservation laws has been previously demonstrated applying the direct sine-Gordon perturbative calculations by Arefyeva and Korepin [8]. 6 The same result can be obtained i n perturbation theory of massive Thirring model, i.e., for the soliton scattering [35, 36]. The semiclassical arguments for the soliton S-matrix factorization are also possible [ 1 4]. We use here the results mentioned above and treat the total sine-Gordon S-matrix as a factorized one. The bound states of quantum solitons {the quantum doublets) and the soliton scattering have been i nvestigated by a semiclassical approach in Refs. [ 1 3-1 5, 37, 38]. We represent here some semiclassical formulas which will be necessary below. The two-particle scattering amplitude S(8) for the solitons of the same sign and the transition amplitude Sr (8) for soiton-antisoliton scattering calculated in the main semiclassical approximation have the form ( 1 3 , 1 4, 37]

!i r

s � em ) (8) = s exp s 81r, provi ded the standard renormalization technique is used; the phenomenon is of the ultraviolet nature. This scarcely means the failure of the theory with ff' > 81r, but rather indicates a Jack of superrenormalizability property and suggests that another renormalization prescription is necessary at fl' > 811. Throughout this paper we restrict our consideration to the case fl• < 3,,, 5 9 5 / 1 20/2- 2

461 268

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

The semiclassical soliton-antisoliton reflection amplitude (which takes into acco unt a n imaginary time classical trajectory, see Ref. [38]) is (4.2)

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The derivation of the semiclassical mass spectrum of the quantum doublets was carried out in papers [ 1 3-1 5). In the first two semiclassical approxi mations it i s m�em)

= 2 m sin

( �� ) ;

n

=

8 1T 1 , 2, . . . < , ,

(4.3)

where m is a soliton mass. The authors of [ 1 5] have presented some arguments for formula (4.3) to be not only semiclassical but exact. Independent supports for this hypothesis have been given in [ 1 0, 39, 40). The exact solution for the S-matrix which is derived in this section also confirms the exactness of spectrum (4.3). We begin constructing the quantum sine-Gordon S-matrix stressing that the model exhibits an 0(2) isotopic symmetry. In terms of the massive Thirring fields If this symmetry is quite obvious; it corresponds to the phase invariance If ........ e'"'f of (1 .2). From the view-point of the sine-Gordon Lagrangian 0(2) symmetry is of a more delicate nature; it is the rotational symmetry of the disorder parameter (see Ref. [41 ) fo r the concept o f the disorder para"meter). The detailed discussion of the last point i s beyond the scope of this paper. For our purpose it i s sufficient to note only that the soliton and antisoliton of model ( I . I) can be incorporated i nto an i sovector 0 (2) doublet. Following the convention of Section 3 we denote real components of this doublet by symbols A ;(8) ; i = I, 2. Then the soliton and antisoliton themselves will be the combinations (4 . 4) In terms of the particles A (0) and A(8) comm utation rules (3. 3 ) take the form A(81) A(82) = Sr(81 J .if(8J A(8J A(01) A(82) = S(812)A(82) A(01), A(8J A(82) = S(81 2) A(82) A(81) .

+ Si81 2) A(82) A(81) ,

(4 . 5 )

In (4. 5) Sr(8) and SR (8) are transition and reflection amplitudes for the soli ton-an ti­ soliton scattering while S(8) i s the scattering amplitude for identical solitons. They are connected in a simple way with amplitudes Si (8), Si8) and Sa(8) from (3. 3) S(8) = Sa(8) Sr (0) = S/8) SR (0) = S1 (8)

+ Sz(8),

+ Sz(0),

+ Sa(8).

(4. 6)

462 269

FACTORIZED S-MATRICES IN TWO DIMENSIONS

I t is seen from (3.2) that S(8) = Shrr - 8) ;

(4.7)

The factorization and 0(2) symmetry of the sine-Gordon soliton S-matrix allows one to apply immediately the results of the previous section. It follows from (3.7 1 , b), (3. 1 2) and (3. 1 5) that

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Sr(8) = - i

S(8) = - i

sh ( � 0) � • SR (8), sin ( ;- )

(4.8a)

sh (� (i7T - 8)) y 8 2 SR( f!), sin

(4.8b)

(-f-)

where (with arbitrariness of the CDD-type (3. 1 4) only) (4.9)

® -i l[ :

-2tl[ : I

II

{l[ I

0

I

I

I

I

I • I x• 1 x

-2 i 1!' :

I

I

I

' ": x+ " I

I

-t:11 I

"

0

I

2i:J!I

I

I

I I

•

.

I I I

@

I I •

i1l I'

I

I

I

.

I

• JJmf:l

wqI

F1G. 5. The soliton-antisoliton scattering amplitudes. Location of poles (dots) and zeroes (crosses) in the 8-plane. (a) Transition amplitude SR(8). (b) Reflection ampl itude Sr(B). Some of the dots and crosses are displaced from imaginary axis for the sake of transparancy ; actually all the singularities are at R,8 = 0. I

and U(8) is given by (3. 1 6). The location of zeroes and poles of functions Sr (8) a nd SR (8) (4.8), (4.9) is shown in Fig. 5. Note the equidistant (with separation y/8) positions of Sr (8) poles in the physical strip O < Im 8 < 7T. Such positions are i n accord with the semiclassical mass spectrum (4.3). The correspondence is exact i f 'Y

= y'.

(4. 1 0)

463 270

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

Therefore, the whole bound state spectrum (4.3) is already contained in the "minimum" solution (4.9), (3. 1 6) and CDD poles need not be added. This solution automatically stisfies also another necessary requirement for the exact sine-Gordon S-matrix. If y = 81r(f32 = 41r) the massive Thirring-model coupl ing vanished and the S-matrix should become unity. In fact when y = 81r one has from (4.8), (4.9) and (3. 1 6)

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S7 (0)

= S(0) = 1 ;

These two remarkable properties of the "minimum" solution (together with its obvious aesthetic appeal) may serve as initial arguments to choose i t as the exact S-matrix of quantum sin-Gordon solitons. We present below a number of checks which confirm such a choice. If y --+ 8 1r the formulas (4.8), (4.9) and (3. 1 6) can be expanded in powers of 2g/-rr = 8 1r/y - 1 and the expansion coefficients can be compared with the results of diagram­ matic calculus in massive Thirring model ( 1 .2). Such a comparison has been carried out in [42] up to g3 and the coincidence has been found . Another check is a comparison with semiclassical formulas (4. 1 ), (4.2). The semi­ classical limit for Lagrangian ( 1 . 1) corresponds to {32 --+ 0. At 0 fixed and f32 --+ 0 exact relation (4.8a) converts into semiclassical one (4.2). Furthermore, it can be easily verified that a symptotics of the exact amplitudes S(B) and Sr(B) as y --+ 0 coincide with (4. 1 ). To do this one represents the exact Sr(8) from (4. 8a), (4.9) in the form

., S r(B) = fl

i -1

8 ) r (!_=:-__!._ r - i 8 r (_!L - i2 -rr ) 2 -rr l 6 1r l 61r - ; _0_) r ( - ! !_=:-___!._ r - ; _0_ ) r (!2 + _!L_ 2-rr 2-rr 2 ' l 6 1r l 6 1r _.L

I' (

3

+

/y

+.

8

)

I' I (

+

I- I

y

, .

0

)

I 2 7T 2 � T I 2 1r 2 1 6 7T x ----�-----,�---��----�-

1 r (1 + 1 t + ; L ) r (1 + \;7T r + i

2� )

(4. 1 1 )

Changing i n (4.4) the infinite product by a sum i n the exponent and then replacing at y --+ 0 the summation by the integration one reprod uced (4. 1 ) exactly. The larger the coupling parameter, the larger the mass of each bound state (4.3) (in units of the soliton mass). The n-th bound state acquires the soliton-antisoliton threshold when y = 81r/n, and when y ?c 81r/n i t disappears from the spectrum converting into the virtual state. At y ?c 81r all bound states (4.3) including the "elementary" particle of sine-Gordon Lagrangian ( 1 . I) become unbound (re mind that "elementary" particle is one of states (4. 3), corresponding to 11 = I [ 1 5, 1 4]) . Thus, at y ?c 81r the spectrum contains soliton and antisoliton only. The valu es y � 81r correspond to g ,,;; 0 in (1 .2), i .e., to the repulsion between soliton and anti­ soliton. Note that at y = 81r/n the reflection amplitude (4.9) vanishes identically (t his

464 FACTORIZED S-MATRICES IN TWO DIMENSIONS

27 1

property appears already in semi-classical formula (4.2)) while transition apmplitude ST (0) acquires, as a result of special cancellation of poles and zeroes in Fig. 5, a simple form :

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S T(0)

=

e i n rr

n- 1 e8 - i ( rr k / 11 )

Il

k-l

e9

' I T

+ e-i(812) B. (02) A(81) ,

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where

2/ -rr . (n n . sh 8 + I cos y n-l sm 2 --- y + I. 28) 32 16 4 s(O) = ----- Il -----,,-,c-------,,8 n y ' i . 2 n - 2/ s h 8 - 1 cos -s m (--- y - -rr - 1• ) 2 4 32 16 -

.

is the amplitude of two-particle scattering A deration leads to commutation rules

+ Bn -+ A + B• .

(4. 1 7)

Analogous consi­ (4. 1 8)

where S < ti . m >(O) is the two-particle amplitude for B,. Its explicit form i s n

m

+ B,,. -+ B,. + Bm

)

scattering.

)

n . ( -+ m y . (sh e + 1. sm -- y s h O + 1. sm 1 6 16 s< n, m >(O) = --------- --------n + m y ) s h B - 1. sm n m y) . (. (s h 0 - 1. sm --16 16 X

m -i

• sm 2

( m - n - 2/ y + z· 8) cos·• m + n - 2/ y + · 8) ( 1 2 32 32 2

[l sm. 2 ( m - n - 2/ y - . 8) cos2 ( m + n - 2/ y - . 8) ; 32

1

2

32

1

2

n � 111.

(4. 1 9)

Amplitudes (4. 1 7) and ( 4. I 8) turn out to be 21ri-periodic functions of 8 (in fact, this property is dictated by the cross-symmetry and the two-particle unitarity of s(0) and s< n , m l (0)). The location of poles and zeroes of these amplitudes is shown in Fig. 6. Note the set of double poles 8 1 = i(-rr/2) + [(2/ - n)/ 1 6] y; I = I , 2, . . . , n - I of s(O) for n � 2 ; these "redundant" poles do not correspond to any bound states. Single poles 8 = i(-rr/2) + in(y/ 1 6) and 8 = i(1r/2) - in(y/ 1 6) are the s-channel and u-channel sol iton poles, respectively: in the s-plane these poles are at s = m2 and u = m 2 (m is the soliton mass). In amplitude s< n, m >(O) only the poles 0 = i[(n + m)/1 6] y and 8 = i1r i[(n + 111)/ I 6] y correspond to the real particle Bn +m , all other poles are redun dan t. The appearance of poles Bn +m in the amplitude s(O) allows one to interpret

466 FACTORIZED S-M ATRICES IN TWO DIMENSIONS

...

273

®

-rn I I I I •

I

" " JmB

i1f t I

0

I

I

@1

I I

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I

•

0

- i1I : I I

•

•

,rr : I

Jm8

I I

FIG. 6. Poles and zeroes of the soliton-bound state and the bound state - bound state scattering amplitudes. (a) 5 1 " 1 (0) amplitude for n = 5. (b) 51 " • 1 (8) amplitude for n = 4, m = 2. '"

any particle Bi for I ;;;;, 2 as a bound state B n + B,. with n + m = I' and, conse­ quently, to interpret Bi as a bound state of / "elementary" particles B1 • A possibility of such interpretation was mentioned in Ref. TJ 5]. In the case m = n = I Eq. (4. 1 9) gives the two-particle amplitude of "elementary" particles sn .i> ( 0)

=

sh 0 + i sin(y/ 8) sh 0 - i sin(y/8) ·

(4.20)

This expression can be expanded in powers of fr' and compared with fr'-pertur­ bation theory results for Lagrangian ( 1 . 1). Formula (4.20) together with its pertur­ bation verification was presented i n [8-1 OJ as a solution of analyticity and unitarity for particles B1 Formulas (4. 1 6-4. 1 9) solve the bound state problem of the sine-Gordon model. Together with (4. 5), (4. 8-4. 1 0) and (3. 1 6) they represent the total quantum sine­ Gordon S-matrix. To conclude this section let us consider the nonrelativistic version of O(2)-symmetric S-matrix. A fter substitution (3. 1 1 ) the general solution of the factorization equations becomes, instead of (4. 8), (see Appendix A) ' One can verify that definition

is consistent with (4. 14) and completely self-consistent [3 1 ].

467 274

ZAMOLODCHIKOV AND ZAMOLODCHI KOV

(4. 2 1 a)

S(K)

=

-i

S

h (.

8 '7Tk )

/ '7T K - --

m

.

sm ( '7TK /

SR(k),

(4.2 1 b)

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where K = &S/y. Since we do not require any crossing-symmetry now, y and K are independent parameters. The unitarity condition, then, gives the following equation (4.22)

The "minimum" solution of (4.22) is

r ( -i ¾ - r ( -i ¾ + + 1 ) r ( -i �) r (1 - ; � ) ym ym K

K)

(4.23)

Formulas (4.2 1-4.23) clearly give the nonrelativistic l i mit of the sine-Gordon soliton S-matrix. Furthermore, amplitudes (4.23) and (4.2 1 a) are just reflection and transition ones for the scattering on the potential VA,;(x) = - m 64

y2 G

my

ch2 (-- x ) 8

(4. 24)

while amplitude (4. 2 1 b) describes the scattering on the potential

VAA(x) =

m

64

y2 G my sh 2 (-- x ) 8

(4. 25)

where G = K2 - K + !- It is known that a system of N + M nonrelativistic particles of two different kinds described by the Hamiltonian

M

+ I

j (0) from (3. 1 7), appears to be the most natural. Below we pre se nt some arguments in support of this choice. At first let us note that CDD-poles (3 . 1 4), if added, in general, result in add itional poles in all three channels of two-particle scattering: isoscalar, antisymmetric-tensor

474 FACTORIZED S-M ATRICES IN TWO DIMENSIONS

28 1

and symmetric tensor12 • Such a strong isospin degenerasy of states seems to be un­ natural. The "minimum" solution a2(8) = ai+> (()) possess no poles in the physical strip O < Im () < TT and therefore "elementary" isovector particles A; of ( 1 .4) produce no bound states. Furthermore, a calculation of two-particle amplitudes for model ( 1 .4) by 1 /N­ expansion technique (see Appendix B) in the order of 1 /N leads to the result:

P,

P,

Rz +

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62 (0) = Pz

R

03 (6) = P2 01 (6)

=

P,

R

Pz

Pz P,

= -

P. � Pi = Pi

1

2 1f i

Pz = - NShe

P2

t�

i

(5. 1 2)

21ii

N ( [1f-B)

The signs i n (5. 1 2) mean that the interaction between A; is of repulsive type (at least for large N). Hence, A , is unli kely to form bound states. It is easy to verify that expressions (5. 1 2) really coincide with the first terms of 1 /N-expansion of exact solution (3.8), (3. 1 3), (3 . 1 7) with a2{8) = a�+>(fJ). Thus, the latter choice is in accordance with 1 /N-expansion of ( l .4}. It is i nteresting to compare the solution of Section 3 with the results of the ordinary g-perturbation theory of model (l .4). Adopting the S-matrix (3.8), (3. 1 7) to correspond to some renormalizable asymptotically free field theory, one can expand the scattering amplitudes, which are the functions of variable s s ln -2 = In . + m

,-,.-

J u i .. i

dg a

,., ( g)

{ 5. 1 3)

+ O ( g3)

(5. 14)

in the asymptotic series in powers of g(µ,). Using the first term of u-model Gell­ Mann-Low function fJ( g) [20)

N-2

one obtains up to g2 ( g

fJ(g) = - -- g2 4TT

= g(µ,))

2

+

ai{s)

=

1 - i�

aa{s)

=

-i J

ai {s)

=

2 2 i � - i N - g2 I n .!._2 - N - g2 + O( g3). 2 871' 8 ,.,.

O ( g3),

2 + i N S� g2 J n :z + O ( g3),

(5. I 5)

12 The only exceptional case is that of single CDD-pole a 1 = ,\ added to a�+ 1(8), where bound states appear in isoscalar and antisymmetric tensor channels only. This case corresponds to o.(6) = a�- 1 ( 8) (see (3. 1 9)) and is under consideration in the further Section.

475

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282

ZAMOLODCHIKOV AND ZAMOLODCHIKOV

In (5. 1 5) the asymptotics s --+ ro is written down and power terms in s are dropped. The usual perturbation theory (expansion in g) is based on Goldstone vacuum and deals with N - l plet of Goldstone particles i nstead of N-plet of massive particles A, . l n this perturbation theory the loop diagram calculation leads to infrared divergencies. H owever, all the infrared divergencies are cancelled among the diagrams contri­ buting to the scattering amplitude of Goldstone particles in the order of g 2 • Hence, one can believe that calculation of these diagrams results in the correct ultraviolet asymptotics of the reeal A. particle scattering amplitude . This calculation is straight­ forward and does lead to (5. 1 5). 6. S-MATRIX

OF

THE GROSS-NEVEU "ELEMENTARY" FERMIONS

Another example of an asymptotically free field theory exhibiting the properties (a) and (b) of Section 5 and therefore leading to the factorized S-matrix of Section 3 with N � 3 is Gross-Neveu model ( 1 . 5). An i nfinite set of nontrivial conservation laws for a classical version of model ( 1 .5) has been found i n a recent paper [49]. These classical conservation laws are quite analogous in their structure to those of the nonlinear a-model found by Pohlmeyer. Again the conformal invariance of the classical theory ( 1 . 5) (which is broken i n a quantum case) plays the crucial role i n the derivation of these conserved currents. H owever, it is natural to expect that higher conservation laws are present in quantum theory (1 .5) as well. Dashen, Hasslacher and Neveu [29] have investigated the classical field equatio n which determine the stationary phase points of the effective action (B.5') (see A ppendix B). They have been able to find out explicitly a series of time­ dependent solutions . It means almost surely the complete integrability of the system determi ned by these equations. To make sure that quantum theory (1 . 5) really possesses higher conservation laws let us derive the fi rst nontrivial law following Polyakov's method [26]. All the con­ siderations will be quite parallel to those applied in Section 5 to the case of the a-model. It is convenient to use the motion equations of (I . 5) explicitly in terms of right and left-handed components of Majorana "bispinors" ,plx) = (,p/(x), ,p;' (x))

I:;..

i = l , 2, . . . , N,

(6. I )

where w = g0 1 ,p/,p/. The momentum-energy conservation and conform al invariance of equations in classical case imply analogously to (5.3) (6.2)

476 FACTORIZED S-M ATRICES IN TWO DIMENSIONS

283

which should be, of course, replaced in the quantum theory

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(6.3)

Using these quantum equations one can easily verify that the following equations can be satisfied by an appropriate choice of the parameter C

(6.4)

These equations are just the first nontrivial conservation law of the Gross-Neveu model. The existence of higher conservation laws implies that the Gross-Neveu S-matrix satisfies property (b) of Section 5 [26] . Alternatively o n e could discover this property of the Gross-Neveu S-matrix in I/ N-expansion [50]. The I / N-expansion technique for this model has been developed in Ref. [28] (it is described briefly i n Appendix B). It is similar in the main to that used in the case of the nonlinear a-model. In particular, the diagrammatic consideration of previous section can be repeated word for word in the Gross-Neveu case. There is an important difference, however, between the a-model 1 /N technique and that of the G ross-Neveu model. Note the additional minus sign in (B.9') against (B.9) which is connected with the fermion nature of - ay• aJ(x-y) A,(x).,-ge,>-,, ff------dridT1 (21) ax. aro OT1 integrated over all the sheet, where e,�,. is the antis}mmetrical tensor of the fourth rank with eom = 1. To verify (21), we note that it leads tc aA.,.. e"'0,-- = ge"'0�E,Ap• ax�

iJyA ay• a•J(x-y)

- ---�d.-rtl-T1 ffOTO cir, ay•ay.

489 THEORY OF MAGNETIC POLES

Using

12 is the action integral for the field alone,

where -(a/3) means that we must subtract all the preceding terms with a and {3 interchanged, we get

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aA.,, ,i,•all--=g ox•

=g

JJ{o2

oy"

J(x-y) aya a•J(x-y) ____ --+---or,iJya iJro iJrooY11 iJr1

[ J

iJy«

ay"

+-- □ J(x-y) oro or1

I

dr(lir,-(afJ)

dz" aJ(x-y) --ds-(a{3)+4irG«II, --iJy11 ],-,ds

with the help of Stokes' theorem (16), and (19) and (15). According to (13), this gives the re­ tarded field (Ff),. all= eP•«lliJA .,,/iJx• -4,.Call,

with

821

= -g

J

iJJ(x-z) dz" -----ds-(a{3), ds iJx11

= -aB,."/ox11+iJB,l/iJxa, B,,"=g

J

J(x-z)(dz"/ds)ds.

(22)

(23)

This is seen to be the correct value for the re­ tarded field produced by a pole, from the analogy of (23) to the Lienard-Wiechert potential (20). In the usual electrodynamics the potentials are restricted by the condition aA,/iJ;c, = 0

or

aA,*/ax, = O.

(24)

This condition can be retained in the present th,cory, as it is satisfied by the retarded poten­ tials (20), (21). The two forms of (24) are equiva­ lent because of the linear connection between the starred and unstarred field (see Eq. (29)). IV. THE ACTION PRINCIPLE

The action integral of ordinary electrody­ namics may be expressed as a sum of the three terms, J1+J2+Ia, where 11 is the action integral for the particles alone, l1 = I:,, m

f

ds,

and Ia is the contribution of the interaction of the charges with the field, la =

:E, efA•(z)(dz,/ds)ds.

(25)

The F,, in I, are to be regarded as functions of the potentials. The same action integral will do in the present theory, provided the sum in I, is extended to include the particles with poles as well as those with charges,

f

I, = L,+.m ds.

(26)

No further term is needed to give the interaction between the poles and the field, this interaction being taken into account in 11, in which F., is now to be regarded as a function of the potentials and the string variables y,(r0, r,) given by (13) and (15). In order to avoid infinities in the equations of motion arising from the infinite fields produced by point charges and poles, we shall make a small modification in the field equations, by replacing 11 by ft'= (l61r)-1

JJ

F,,(x)P.•(x')-y(x-x')d'xd•x',

where -y(x) is a function which approximates to the function o,(x), and is made to tend to o,(x) in the limit. We shall assume that -y(-x) =-y(x),

(27)

and shall assume other properties for -y(x) as they are needed, but the precise form of ,,(x) will be left arbitrary. We may write Is' as U = (l61r)-1

f

F,., *(x)F�•(x)d'x,

(28)

using the notation that for any field quantity U(x), U*(x) =

f U(x')-y(x-x')d'x'.

(29)

490 822

P.A. M. DIRAC

It will now be verified that the variation of

a((Ft)•• *oy•) oy'

=l:.gff { -----

I = I1+U+J3

OTo

leads to the correct equations of motion. The variation of 11 is well known and gives

f

U1 =-Lo+am (d'z./ds2)oz•ds.

(30)

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The variation of Ia may be carried out the same as in ordinary electrodynamics and gives

dz• = E. g (Ft)••*(z)oz"--ds ds

f

ola = L• ef l[(ilA./ox•)-(ilA./ox•)J-,oz•

+(M.)-,} (dz• /ds)ds. (31)

-I:. g

The variation of 12' gives, using (27) oU =(8r)-1

= (8r)-1

ff f

Substituting for F•• its value given by (13), we get oI,'=-(4r)-1

f

F.,*(ooA•/ox,)d•x +½La

= (4r)-l

f F,.*o(Gt)••d•x

f(oF••*/ox,)M• d•x +n::a f (Ft) ••*oG-•d'x.

(32)

(33)

Using (15), the second term here becomes

= Lag

J

(Ft),.*d4x

JJ { oc:: ::)

o,(x-y)

oy• oy• oo,(x-y) ----oy' } drodr1 oro on oy•

+=Lag

Jf {

(ooy• oy• ooy• oy' (Ft)•.*(y) ----- )

o(

oTo OTJ

OT1 OTo

) .*(y) oY" oY -oy• } drodr1 oy• oTo 0T1

Ft • +----

f( I

a(Ft)• .* a(Ft).. • --+-ay" a y•

o( t),. ay• ay•

F +--)--oy•dTodT1

F•• (x')oF••(x)-y(x-x')d4xd•x'

F•• *(x)oF••(x)d•x.

0T1

o((Ft)•.*oy•) ay• o(Ft)•• * ay• ay• --oy• OT1 oy• ( aTo aT1 aTo

oy•

OTO OT1

(34)

by a further application of Stokes' theorem (16). The total variation oI is given by the sum of (30), (31), (34) and the first term in (33). By equating to zero the coefficient of oA•(x) in ol, we get precisely Eq. (8).By equating to zero the coefficient of oz• for a charged particle we get m(d'z,/ds')

= e[(ilA./ax.)-(oA,/ox•)],_,(dz• /ds).

This agrees with the equation of motion (10) provided the charged particle does not lie on any of the strings, so that G•• (z) =0. By equating to zero the coefficient of oz• for a pole, we get precisely (11).Eq\lation (9) is a consequence of Eqs. (13) and (15), which express F,. in terms of the potentials and string variables. Thus all the equations of motion (8), (9), (10), and (11) follow from the action principle of =0, provided we impose the condition that a string must never pass through a charged particle.

By equating to zero the coefficient of the variation oy• in a string variable, we get o(Ft) ,.*/ ay•+o(Ft)., •/ ay•+a(Ft),.*/ oy• = o

or

oF••*/ay. = O, holding at all points on the sheet. From (S) this is automatically satisfied, provided the string never passes through a charged particle. Thus the action principle leads to no equations

491 T H E O R Y OF M A G N E T I C P O L ES

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of motion for the string variables, in conformity with the unphysical nature of these variables. The action integral I is a correct one and may be used as basis for a theory of electrodynamics, but it leads to some inconvenience in the Hamil• tonian formulation of the equations of motion, since it makes the momentum conjugate to A o vanish identically. This inconvenience may be avoided by a method due to Fermi, which con• sists in adding on a further term to the action in tegral

I, = (8,,.)-J (aA,*/ax,.) (iJA ,./ax,)d'x. (35)

This gives

f

H, = (4r)-1

(aA ,•jax,.) (a 6A ,./ax,) d�

= - (4r)-1

I

A .•/ax.ax-) BA•a•x,

(a2

(36) (37)

and leads to a further term -a2A,•jax,ax• on the left-hand side of Eq. (8) . This further term in (8) docs not affect the equations of motion, because it vanishes when one uses the supple­ mrntary condition (24), but in the Hamiltonian formulation it is necessary to distinguish be­ t wrrn those equations that hold only in virtue of supplementary conditions and those that are i ndependent of supplementary conditions. There­ fore, we must leave this term in (8) to have an equation of the latter kind. Equation (8) may now be written, with the help of (13) and (1 5) , DA.•(x) = 4,r I:. e

J

(dz,. /ds) 8 4 (x-z) ds

+4 ... I:, a (Gt) ,. .*/ax,.

V. THE METHOD OF PASSING TO THE HAMILTONIAN FORMULATION

(38)

When on e has the equations of motion of a dyn am ica l system in the form of an action pri ncipl e, one must put them into the Hamil­ ton ian form as the next step in the process of 'f "_lffl tization. The general procedure for doing th 11 1_ 5 t take the action integral previous to a _ � arta, n tim e I and to form its variation allowing : to �ary . This variation U appears as a linear imctio� of �t and of the variations 8q i n the d!)'na m rc:a l coordinates at time t the other terms . Ill &f can celli· ng wh en one uses ' the equations of

823

motion. One introduces the total variation in the final g_'s Aq = Bq+ q&t,

and expresses 81 in terms of the Aq's and One puts this equal to

at.

U = I: p.Aq,- WBt,

(3 9)

W-H(pq) =O.

(40)

(or the corresponding expression with an in­ tegral instead of a sum) and so defines the mo­ menta P, and the energy W. The Pr and W ap­ pear as functions of the coordinates ,fA .,*e'("lk,- d'k, 1

(44)

(kx) = ko:co - k1x1 - k,x,- k&Xs, d'k = dk,dk2dk,, ko = ± (k1'+k2'+ ki")I,

and Lko means the sum over both values of k:0 for given k1, k2, ka. The factor k 0-1 is introduced because ko-1 O of (49), ( 51) and (53). and u se (45} and (46), With this expression for H, we cannot directly introduce the momenta in accordance with for­ 2 .-'i 'Yk(A ,.oA.•-A. oA.• )ko-1 d 3k ,. mula (39), since the A•, A• * whose variations occur in (53) are not independent, and since we have not varied S,. A convenient way of pro­ = 2-rio 'Y.A i.,Ai.• ko-1d'k ceeding is to pass to the Fourier components of the potentials, for which we may use the Fourier -4n -y..J .,M .• ko-1d 3k. (54) resolutions given by (43) and (44), as we are concerned in expression (53) with the potentials on the surface S,. Let us take a varied motion The first term in (54) is a perfect differential which satisfies the equation of motion, so that and may be discarded. We thus get the final the Fourier resolutions (43) , (44) are valid on result that lil is equal to, apart from a perfect Sr also for the varied motion.Then expression differential, the sum of (49), (51), and the second term of (54) . (53) becomes, with the help of (47) , We take as dynamical coordinates the co­ ordinates z, of the particles when they go out 1 (8ir) - i Lkoko' k, (-y.+-y,,) of existence, the coordinates y,.(.·1) of points on the strings when they go out of existence (pro­ viding a one-dimensional continuum of co• ordinates for each pole and each value of µ), If we take the surface S, to be x0 constant for and the Fourier components A .,, with ko >O, of simplicity (any space-like surface must give the the potentials after the particles and strings have same final result) , this becomes, on integrating gone out of existenc;e. The coetlicients of the with respect to x i, x 2 and xa, variations of these coordinates in the expression for oI given by the sum of (49), (51), and the second term of (54) will be the conjugate mo­ ('Y1+-y,,)A 1,M i.•• .-'i L•oko' menta. Thus the momenta of a charged particle Xexp[i(ko+ko')xo]oa(k + k')d'kko'-1d3k', are p, = mdzp/ds+ eA p(z), (55) where oa(k) means o(k,)o(k 2) o(k,). The factor those of a particle with a pole are li,(k+ k') here shows that the integrand vanishes P, = mdz,/ds, (56) except when k.' = - k,(r = 1, 2, 3), which implies ko' ±k 0• Thus the expression reduces, with the the momenta conjugate to the string variables

f

JJf

=

JJ

=

I

J

495 827

THEORY OF MAGNETIC POLES

y•(ri)-let us call them fJ• ( r 1)-are Phi ) = g (Ft) .,*dy•/dr ,, and the momentum conjugate to A .,• is - 4,r2i-y.,A••k.-1 •

(57) (58)

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The string momenta P. ( r ,) form a one-dimen­ sional continuum of variables, corresponding to the one-dimensional continuum of coordinates y• ( r 1) , and the field momenta (58) form a three­ dimensional continuum, corresponding to the three dimensional continuum of field coordinates. We may introduce Poisson brackets in the usual way. For the coordinates and momenta of each particle we have [p. , z,] = g.,.

(59)

For the coordinates and momenta of a string we have (60) [P. ( r 1) , y, ( ri')] = g., o(ri - ri') , and for the field variables we have, according to (58) , [A • • , A •. ,] = i (4,r2) -1g.,-y.,-1kooa{k - k') .

(61)

The other P.B . 's all vanish. In the limit when -y(x)-+o,(x) , we have -y.,-+1 and Eq. (61) gives the usual P.B. relation for the Fourier amplitudes of the elctromagnetic poten­ tials. If we take 'Y•-1 = cos(H,) , where X is a small four-vector satisfying X2 > 0, and make X->O, we get a limiting procedure which has already been used in electrodynamics, classical and quantum , and which gets over some of the difficulties connected with the infinite fields caused by point particles. This value for -Y• might be suitable in the present theory, but I have not investigated whether i t would be compat i ble with all the requirements of the function -y(x) . From Eqs. (55) and (56) we can eliminate the veloci ties dz.Ids and get { p. - eA . (z) ) ( p• - eA • (z) l - m2 = 0

(62)

for each charged particle and

so the Fourier resolution of (48) in this region gives, with ko > O, k •-y,Ab - (4,r2)-1

E, ee-• = 0,

k'-yk A., - (h2) -1 E , ee'(, ,> = 0.

(66) · (67)

These equations involve only dynamical co­ ordinates and momenta, so they are of the right type to form Ham i l ton-] acobi equations. One can easiiy verify that they and the previous Hamilton-Jacobi equations (62), (63), and (64) form a consistent set of differential equations for I, by verifying that the P.B.'s of their left­ hand sides all vanish. VII. QUANTIZATION

{ 63) for each particle with a pole. These equations should be joined with (57) or fJ.(ri) - g(Ft).,*(y)dy•jdr , = 0.

With the A . (z) in (62) and the (Ft),,*(y) in (64) expressed in terms of the Fourier components A ,. , A •• (the validity of this was discussed near the end of the preceding section) , Eqs. (62), (63) , and (64) are equations involving only dynamical coordinates and momenta. They are differential equations satisfied by the action integral I, when the momenta are looked upon as derivatives of I, and they are the Hamiiton­ J acobi equations of the present theory. Since they are known to have a solution, namely I itself, we can infer from the theory of differential equations that the P.B.'s of their left-hand sides all vanish, as may also be verified directly from (59) , (60) , and (61). The supplementary conditions (48) should be brought in at this stage and treated as further Hamilton-Jacobi equations. The various equa­ tions (48) obtained by taking different field points x are not independent of the equations of motion or of one another, and we get a complete independent set of equations from them by making a Fourier resolution in the region be­ tween Sp and S,,. In this region we may, from (18), replace J(x -z) by A(x - z) , whose Fourier components are given by

(64)

From the foregoing Hamiltonian formulation of classical electrodynamics one can pass over to quantum electrodynamics by applying the usual rules. One replaces the dynamical · coordinates and momenta of the classical theory by opera-

496 P . A . M . D I RAC

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828

tors satisfying commutation relations corre­ sponding to the P.B. relations (59), (60), and (61), and one replaces the Hamilton-Jacobi equations by the wave equations which one gets by equating to zero the left-hand sides of the Hamilton-Jacobi equations (now involving oper­ ators for the dynamical variables) applied to the wave function y,. The wave equations obtained in this way will be consistent with one another, since the operators on ,f, in their left-hand sides commute, as may be inferred from the vanishing of the P.B.'s of the left-hand sides of the Hamil­ ton-Jacobi equations. This straightforward quantization leads to wave equations of the Klein-Gordon type for all particles, corresponding to their having no spins. For dealing with electrons one should replace these wave equations by the wave equations corresponding to spin ½h. We have no informa­ tion concerning the spins of the poles, and may assume provisionally that they also have the spin ½h, as this gives the simplest relativistic theory. The change from zero spin to spin ½h does not affect the mutual consistency of the wave equations. We now have the following scheme of wave equations, expressed in terms of a set of the usual spin matrices a 1 , a 2 , a3, a,.. for each par­ ticle : {Po-tA o(z)-a ,[p,-eA ,(z) ]- a,.m } ,f, =0 ;

for each charged particle,

(68)

{Po- a,p,- a,.m ) ,f = O ;

(69)

I.Bh 1) -g(Ft) • .*(y)dy •/dr, },p=O ;

(70)

for each particle with a pole,

for each string, and { 4 ,r2k•-y� k>- L • ee-•Ck J ,f, = 0, } { 4 rk'-y1A b- r;. eei (b) ),t, = O,

(7 l )

for the field variables.The wave function ,p may be taken to be a function of the particle variables z., suitable spin variables for each particle, the string variables y µ (r1) with 0 (x), the particle number is additive, N

(26)

= Nm +N.. (1 ) + .Al"A (2 ) + ( i/4n2 )e>.. µ v p o[B / (2 )gp" B p" ( 1 ) ] / ox, . Since the last term is a divergence it contributes nothing to the integrated particle number N, establishing (27) . In particular for the separation ( 5) we have fl = 0 by definition and for the 'source' S, the particle number may be calculated by formula (1 6). This gives N

=

- ( I / 2:,r,2 ) [2e/ (e2 + ( B," (x-x0 ) ;) 2 ) ] det

f

B ," d3x,

508 A U N I FIED FIELD THEORY

which tends in the limit e

--+

565

0 to the value

N

=

-sgn det B ;" (x0 ) .

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This j ustifies the definition o f particle/antiparticle character. Two particles far apart might be described approximately by the product field (26) with U (i) equal to the separate free-particle fields. We substitute (28) into the energy density C = 9"'44 , which gives (3e/4n 2 ) (l.1 J ! + K2 J .1 J i)d3x. We can establish two results from this inequality. First the integrand is greater than 2K J .1 1 , so that

f

E > (3e K/2n 2 ) J.1 Jd3x > (3eK)N.

(30)

Thus the energy per particle must be greater than (3e K), the same limit as we found by a different, more special, argument in ref. 3 ) . The present argu­ ment shows therefore that the average binding energy per particle could not exceed E K approximately. Secondly we can apply Holder's inequality to the first term only: which gives

E > (3e/4n2 ) (2n2N ) W-l.

(3 1 )

510 A U N I FIED FI ELD THEORY

567

Thus the energy must rise with r0½ for small mean separation r0 • The numerical limit is weak as the inequalities used have been very crude. In the discussion of mesons it is natural to use the description (1 8) in terms of fields Ila. that have a linear domain . Then we we get

.

B/B/ = Q2 [�a. �a.] I [Q2 +¼(IIa.) 2J 2 .

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,.

(32 )

To lowest order the system is described by the usual free field Lagrangian (20 ) . The next terms, quartic in the fields Ila. , give the interaction Hamiltonian 2 2 2 �1nt = - ( 1 / 4Q ) ( Ila. ) ( olla. / ox ,. ) 2 2 (aII, ) _ ( oIIa. ) ( oII, ) ( oIIa. ) ( oII, ) } . + (1 /4Q2 K2 ) { ( ox,. ox, ox, ox,. ox,. ox, In an eventual quanta! treatment these terms would describe a meson­ meson interaction, and it is interesting therefore to analyse them in terms of the usual representation of the quantized fields Ila. . The Hamiltonian �Int describes the annihilation of four mesons with 4-momenta k,. and isobaric spin directions t, for which the matrix element is equal to

0IIa.)

c5 (t1 , t2 ) c5 (t3 , t4 ) 4 Q 2 (w1 WzW3 W4 )½

(33)

{ (k1 • kz) (ka · k,) + (1/« ) [2 (k1 • k2 ) (k3 • k4 ) - (k1 • k3 ) (k2 • k4 ) - (k1 • k4 ) (kz · ks)]}

+

2

two permuted terms.

In the centre-of-mass system for the scattering of two mesons we have k3 , k 4

=

ff k ', -iw ) .

Then in a state of isobaric spin T = 0, the matrix element equals (1/ 2 Q 2w2 ) [- (w 2 + 3k2 ) + ( 4w2k 2 + 2k4 - 2 (k' • k ) 2 } / K2] ,

which is negative, corresponding t o an attraction, for low momenta changing to repulsion at energies of the order of K . In a T = l state the matrix element equals - (k' • k)/ (Q 2w 2 ) ( 1 + w 2/ K2 ) ,

representing an atraction in the J = T = 1 state at all energies. Meson-particle interactions can be studied by perturbation theory applied to the free-particle state. We write, as in equation (26 ) , U (x)

=

Umes (x ) Up (x ) ,

where UP is the static free-particle solution, and expand Umes in powers of the field; as in equation (28), we get

511 568

T. H. R, SKYRME

B/ = b / +gpa. B/ , where b and g refer to the meson fields, and B is the particle field. We substitute this into the Lagrangian, and retain terms up to the second order in the meson fields. The negative of the interaction terms gives the effective interaction Hamiltonian

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.n"1nt

=

- P/ grzp b ,,_ + (e/4n2 ) ( b / b/ ) { (B>. r B>. r ) tJ ,,. A./J - B/ B v r tJa.p+ 2B/ B / - B>-" B/ tJ ,,. v - B v" B /} ,

{34)

where P / , defined by eq. {10), is proportional to the isobaric spin current of the source. To second order we have ga.pb/ = ( olla.fox ,,_ ) + ea.py (oflpfox ,,. ) Il1 .

The linear term gives no contribution on account of the equations of motion oP,,_"/ox ,,. = 0; the second term contributes 2..F/ erz/Jr Ilp (oll1 /ox,,. ) , a n interaction between the isospin current o f source and meson field which is familiar in the usual theories of nucleon-pion coupling derived from an assumed symmetrical pseudoscalar interaction. The second term in the interaction ( 34) is a P-wave meson-particle inter­ action, repulsive on the average. There is no indication of the strong attraction observed in the pion-nucleon resonant state, but this would hardly be expected in a static classical treatment where the rotational splitting of the particle states has been ignored. In so far as this analysis can be compared with ex­ periment it should represent an average effect of pion-baryon interactions at energies above the resonance regions. 4. Prospects

In this paper we have developed a classical field theory somewhat further from its statement in ref. 3 ) and have shown that there are a number of en­ couraging resemblances between its consequences and the phenomena of the strongly interacting particles; quantitative comparison is hardly significant however, until the quantized theory is understood. The challenging problem which is unsolved is to understand the nature of the states in the quantized theory which corresponds to the classical solutions with N = I , etc. The usual apparatus of canonical quantization can be applied to the original Lagrangian, but the only obvious starting point for a perturbation theory is the meson Lagrangian ( eq. { 20) ) ; in this way, since all the operators of the theory commute with N, only states with N = 0 can be built. A different starting-point is needed to describe the particle states. We conjecture that this will involve the definition of singular operators that in-

512

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A UNIFIED FIELD THEORY

569

troduce branch-points, such as described by S ( eq. (6) ) . We believe also that these operators may have many of the properties of Fermion field operators in conventional theories. This was demonstrated for the model theory in ref. 2 ) , but the proper extension t o the theory o f this paper has not been found. The chief difficulty arises, as might be expected, in the introduction of spins. The simplest operators that change the particle number are characterised by an orthogonal matrix et , whereas the proper operators that we seek must be states of definite angular momentum. Isobaric spin is contained implicitly in the structure of the basic field quantity U; ordinary spin apparently arises from a strong-coupling between the spin and isospin directions of the source, but its mathematical formulation is elusive. We are indebted to Mr. A. J. Leggatt for carrying out the calculations re­ ported in sect. 3, while a vacation student at A.E.R.E.

References 1) T. H. R. Skyrme, Nuclear Physics 31 (1962) 550 2) T. H. R. Skyrme, Proc. Roy. Soc. 262 (196 1 ) 237 3) T. H. R. Skyrme, Proc. Roy. Soc. 260 (1961) 127

513 Nuclear Physics B79 (1 974) 276-284. North-Holland Publishing Company

MAGN ETIC MONOPOLES IN UNIFIED GAUGE THEORIES G. 't HOOFT

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CERN, Geneva

Received 3 1 May 1 9 74 Abstract: I t is shown that in all those gauge theories in which the elec tromagnetic group U ( I ) i s taken to b e a subgroup o f a larger group with a compact covering group, like SU(2) or SU (3), genuine magnetic monopoles can be created as regular solu tions of the field equa­ tions. Their mass is calculable and of order 1 3 7 Mw , where Mw is a typical vector boson mass.

1. Introduction The present investigation is inspired by the work of Nielsen et al. [ I ] , who found that quantized magnetic flux lines, in a superconductor, behave very much like the Nam bu string [ 2 ] . Their solution consists of a kernel in the form of a thin tube which contains most of the flux lines and the energy ; all physical fields decrease ex­ ponentially outside this kernel. Ou tside the kernel we do have a transverse vector potential A, but there i t is rotation-free: i f we put the kernel along the z axis, then A (x) a: (y , -x, O)/(x 2 + y 2 ) .

(I .I)

A (x) can be obtained by means o f a gauge t ransformation Q (sp) from the vacuum . Here sp is the angle about the z axis: Q (O) = Q (211) = I It is obvious that such a string cannot b reak since we cannot have an end point : it is impossible to replace a rotation over 211 continuously by Q (sp) ➔ I . O r : magnetic monopoles do not occu r in the system. Also it is easy t o see that these strings a re oriented: two strings with opposite direction can annihilate ; if they have the same direction they may only join to form an even tighte r string. Now, let us suppose that the electromagnetism in the_ superconductor is in fact described by a unified gauge theory , in which the electromagnetic group U ( J ) is a subgroup o f, say , S0(3). In such a non-Abelian theory one can only imagine non­ oriented st rings, because a rotation over 411 can be continuously shifted towards a fixed fl. Wha t happened with our original strings? The answer is simple : in an S0 (3) gauge theory magnetic m onopoles with twice the flux quantum (i.e . , the

514 G. 't Hooft, Magnetic monopoles

277

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Schwinger [3 ,4 ) value), occur. Two of the original strings, oriented in the same direction, can now annihilate by formation of a m onopole pair [ 5 ] . From now on we shall dispose of the original superconductor with its quantized flux lines. We conside r free monopoles in the physical vacuum. That these mono­ poles are p ossible, as regular solutions of the field e quations, can be understood in the following way. Imagine a sphere, with a magnetic flux entering at one spot (see fig. I ). Immediately around that spot, on the contou r c0 in fig. I , we must have a magnetic potential field A , with f(A · dx) = . It can be obtained from the vacuum by applying a gauge transformation /\. : A = V I\..

( 1 .2 )

This /\. is multivalued. Now we require that all fields, which transform according to ( 1 .3 ) to remain single valued, so must be an integer times 211 : we then have a com­ plete gauge rotation along the contour in fig. 1 . In an Abelian gauge theory we must necessarily have some other spot on the sphe re where the flux lines come out, because the rotation over 2k11 cannot con­ tinuously change into a constant while we lower the contour C over the sphere. In a non-Abelian theory with compact covering group , however, for instance the group 0 (3 ) , a rotation over 411 may be shifted towards a constant , without singularity : we may have a vacuum all around the sphere. In other theories, even rotations over 211

Fig. ! . .The contour C on the sphere around the m onopole. We deplace it from Co to C 1 , etc., un til it shrinks at the bottom of the sphere. We require that there be n o singularity a t tha t point.

515

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278

G. 't Hooft, Magnetic monopoles

may be shifted t owards a constant. This is why a magnetic monopole with twice or sometimes once the flux quantum is allowed in a non-Abelian theory, if the elec­ tromagnetic group U( J ) is a subgroup of a gauge group with compact covering group. There is no singularity anywhere in the sphere, nor is there the need for a Dirac string. This is how we were led to consider solutions of the following type to the clas­ sical field equations in a non-Abelian Higgs-Kibble system : a small kernel occurs in the o rigin of three dimensional space. Outside that kernel a non-vanishing vector potential exists (and other non-physical fields) which can be obtained from the vacuum* by means of a gauge transformation Q (8 , I/'), At one side of the sphere (cos 8 ➔ I ) we have � rotation over 471', which goes to unity at the other side of the sphere ( cos 8 ➔ - I ). For such a rotation one can, for instance, take the following SU(2) matrix: 0

0

. ) + sin � e ( . i e- 1,p

1 . )· 0

( 1.4)

Now consider one rotation of the angle IP over 271'. At 8 = 0, this n rotates over 471' (the spinor rotates over 271'). At 8 = 'II', this n is a constant. One easily checks that

nnt = 1 .

( 1.S)

In the usual gauge theories one normally chooses the gauge in which the Higgs field is a vector in a fixed direction, say , along the positive z axis, in isospin space. Now, however, we take as a gauge condition that the Higgs field is Q (8 , 1P) times this vector. As we shall see in sect. 2 , this leads to a new b oundary condition at in­ finity, to which corresponds a non-trivial solution of the field equations : a stable particle is sitting at the origin. It will be shown to be a magnetic monopole. If we want to be conservative and only permit the normal boundary condition at infinity, with Higgs fields pointing in the z direction, then still monopole-antimonopole pairs, arbitrarily far apart, are legitimate solutions of the field quations. 2. The model We must have a model with a compact covering group. That, unfortunately, ex­ cludes the p opular SU(2) X U ( l ) model of Weinberg and Salam (6]. There are two classes of possibilities. (i) In m odels of the type described by Georgi and Glashow (7) , based on SO (3), we can construct m onopoles with a mass of the order of 1 37 Mw , where Mw is the * As we shall see this vacuum will still contain a radial magnetic field. This is because the in­ coming field in fig. 1 will be spread over the whole sphere.

516 G . ' t Hooft, Magnetic monopoles

279

mass of the familiar intermediate vector boson. In the Georgi-Glashow model, < 53 GeV/c2 . (ii) The Weinberg- Salam model can still be a good phenomenological description of p rocesses with energies around hundreds of GeV, but may need extension to a larger gauge group at sti!Lhigher energies. Weinberg [8] proposed SU(3) X SU(3) which would then be c ompact. Then the monopole mass would be 1 3 7 times the mass of one o f the superheavy vector bosons. We choose the first possibility for our sample calculations, because it is the sim­ plest one. We take as our Lagrangian:

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Mw

(2. 1 ) where

wi and Q 0 are a triplet o f vector fields and scalar fields, respectively.

(2.2)

We choose the parameter µ 2 to be negative so that the field Q gets a n on-zero vacuum expectation value [6,7,9 ] : (2.3) Two components of the vector field will acquire a mass: Mw 1 ,2

=

eF '

(2.4)

whereas the third component describes the surviving Abelian electromagnetic inter­ actions. The Higgs particle has a mass: (2 .5) We are interested in a solution where the Higgs field is not rotated everywhere towards the p ositive z direction. If we apply the transformation n of eq. ( 1 .5 ) to the isospin-one vector F(O , 0, I) we get F(sin 0 cos ip, sin 0 sin ip, cos 0).

(2.6)

We shall take this isovector as our boundary condition for the Higgs field at space­ like infinity. As one can easily verify, it implies that the Higgs field must have at least one zero. This zero we take as the origin of our coordinate system. We now ask for a solution of the field equations that is time-independent and spherically symmetric, apart from the obvious angle dependence . Introducing the vector

r0

=

(x , y , z ) ,

(2 . 7)

517 G. 't Hooft, Magnetic monopoles

280

we can write Q a (x,

t) = ra Q (r) ,

(2.8)

where eµ ab is the usual e symbol if µ = 1 , 2 , 3 , and e4 ab = 0. In terms of these variables the Lagrangian becomes

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L

=

J E d 3x = 4rr 0j r2 d r [ - r

2

( d W ) - 4 rW d W - 6 W 2 - 2 e r2 W 3 2

dr

dr

d 2 d - !2 e 2 r4 w4 - 2! r 2 ( Q ) - r Q Q - �2 Q 2 - 2 e r2 WQ 2 - e 2 r4 W 2 Q 2 dr dr + ¼ X F 2 r2 Q 2 - ! Xr4Q4 - i XF 4 ] ,

(2.9)

where the constant has been added to give the vacuum a vanishing action integral. The field equations are obtained by requiring L to be stationary under small varia­ tions of the functions W (r) and Q (r). The energy of the system is then given by

E = -L ,

(2 . 1 0)

Q (r) ➔ F/r .

(2. 1 1 )

since our system is stationary. Before calcu"iating this energy , let us concentrate on the boundary condition at r ➔ 00• From the preceding arguments we already know that we must insist on The field W must behave smoothly, as some negative power of r : W(r) ➔ a , - n .

(2. 1 2)

From (2.9) we find the Lagrange equation

.!.(2,4 dd

W

dr

r

= r2 [4r

+ 4 r 3 w)

dW

dr

+ 1 2 W + 6 e r 2 W 2 + 2 e 2 r 4 W 3 + 2 e r 2 Q 2 + 2 e 2 r 4 WQ 2 ] .

(2. 1 3)

So, substituting (2. 1 1 ) and (2 . 1 2), (3 - n) (4 - 2 n)ar 2 -n - -4nar2 -n + 1 2ar 2 - n + 6 ea 2 , 4 - 2 n + 2 e 2 a 3 ,6 - 3 n , - 00

(2 . 1 4 ) The only solution is n=2,

a

=

- 1 /e .

So, far from the origin, the fields are

(2 . 1 5)

518 G. ' t Hooft, Magnetic monopoles

Q/x, t) ➔ F ra ir .

281

(2 . 1 6)

Now most of these fields are not physical . To find the physically observable fields, in particular the electromagnetic ones, Fµ v ' we must first give a gauge invariant de­ finition, which will yield the usual definition in the gauge whe re the Higgs field lies along the z direction everywhere. We propose:

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(2 . 1 7) because , if after a gauge rotation, Q a = I Q 1 (0, 0, 1 ) everywhere within some region, then we have there Fµ v = o µ WJ - a v wJ

,

as one can easily check. (Observe that the definition (2. 1 7) satisfies the usual Maxwell equations, except where Q a = O ; this is one other way of understanding the possibility of monopoles in this theory .) From (2 . 1 6), we get (see the definitions (2.2)): F Q a G µa v -- - 3 E va ra ' er µ

(2 . 1 8) (2. 1 9 )

Hence (2.20) Again, the € symbol has been defined to be zero as soon as one of its indices has the value 4 . So, there is a radial magnetic field (2.2 1 ) with a total flux 4 rr/e . Hence, our solution is a magnetic monopole , as we expected. It satisfies Schwinger's condition eg = I

(2.22)

(in units where h = I ). In sect . 4 , however, we show that in certain cases only Dirac's condition eg = ½ n , is satisfied.

n integer ,

519 G. 't Hooft, Magnetic monopoles

282

3. The mass of the monopole

Let us introduce dimensionless parameters : w = W/F 2 e ,

x

q = Q/F 2 e ,

= eFr ,

/3

= A/e 2 = M�/Mi .

(3 . 1 )

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From (2.9) and (2 . 1 0) we find that the energy E of the system is the minimal value of

[

4 11Mw = dw 2 dw dq 2 -x 2 dx x 2 ( - ) + 4xw - + 6 w 2 + 2x 2 w 3 + 2! x 4 w 4 + !2 x 2 ( -) 2 dx dx dx e o

1

+ xq

!;

+ � q 2 + 2x 2 w q + x 4 w 2

2

q

2

- ! f3x 2 q 2 + ! f3x4 q 4 +

l/3] .

(3 .2)

The quantity between the brackets is dimensionless and the extremum can be found by inserting trial functions and adjusting their parameters. We fo und that the mass of the monopole (which is equal to the energy E� since the monopole is at rest) is 4 11 Mm = Mw C( /3) , e2

(3 .3)

where C(/3) is nearly independent of the parameter /3. It varies from 1 . 1 for /3 = 0 . 1 t o 1 .44 fo r /3 = I O *. Only in the Georgi-Glashow model (for which we did this calculation) is the parameter Mw in eq. (3.3) really the mass of the conventional intermediate vector boson. In other models it will in general be the mass of that boson which corre­ sponds to the gauge transformations of the compact covering group : some of the superheavies in Weinberg's S U (3 ) X SU(3) fo r instance. 4. Conclusions

The relation between charge quantization and the possible existence of magnetic monopoles has been speculated on for a long time ( 1 0] and it has been obse rved that the gauge theories with compact gauge groups p rovide for the necessary charge quantization ( 1 1 ] . On the other hand, solutions of the field equations with ab­ normally rotated boundary conditions for the Higgs fields have also been considere d before [ I , I 2 ] . Nevertheless, it h a d escaped to our notion until now that magnetic monopoles occu r among the solutions in those theories, and that their p roperties are predictable and calculable. * These values may be slightly too high, as a consequence of our approximation procedure.

520

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G. 't Hooft, Magnetic -monopoles

283

Our way of formulating the theory of magnetic monopoles avoids the introduc­ tion of Dirac's string [3 ] , We expect no fundamental p roblems in calculating quantum corrections to the solution although they might be complicated to carry out. The p rediction is the most striking for the Georgi-Glashow model, although even in that model the mass is so high that that might explain the negative experimental evidence so far. If Weinberg's SU(2) X U ( l ) model wins the race for the p resently observed weak interactions, then we shall have to wait for its extension to a com­ pact gauge model, and the p redicted monopole mass will be again much higher. Finally , one important observation. In the Georgi-Glashow model , one may intro­ duce isospin ½ representations of the group SU(2) describing particles with charges ± ½ e. In that case our monopoles do not obey Schwinger's condition, but only Dirac's condition qg =

½'

where q is the charge quantum and g the magnetic p ole quantum, in spite of the fact that we have a completely quantized theory . Evidently , Schwinger's arguments do not hold for this theory [ I 3 ] . We do have , in our model !1 qg

=

I '

whe re !1q is the charge-difference between members of a multiplet, but this is certainly not a general phenomenon. In Weinberg's SU(3 ) X SU(3) the monop ole quantum is the Dirac one and in models where the leptons form an SU(3) X SU(3 ) octet [ 1 4 ] the monopole quantum is three times the Dirac value (note the possibili­ ty of fractionally charged quarks in that case). We thank H. Strubbe for help with a computer calculation of the coefficient C (�), and B. Zumino and D. Gross for interesting discussions. References ( I ] H.B. Nielsen and P. Olesen, Niels Bohr Institu te preprint, Copenhagen (May 1 9 7 3 ) ; B. Zumino, Lectures given at t h e 1 97 3 N a t o Summer Institu te in Capri, CERN preprint TH. 1 7 79 ( 1 973). [ 2 ] Y. Nambu, Proc. Int. Conf. on symmetries and quark models, Detroit, 1 96 9 (Gordon and B reach, New Y ork, 1 970) p. 269 ; L. Susskind, Nuovo Cimento 6 9 A ( 1 970) 4 5 7 ; J . L . Gervais and B . Sakita, Phys. Rev. Letters 3 0 ( 1 9 7 3 ) 7 1 6 . [ 3 ) P.A.M. Dirac, Proc. Roy. Soc. A l 3 3 ( I 934) 6 0 ; Phys. Rev. 7 4 ( I 948) 8 1 7. [ 4 ] J. Schwinger, Phys. Rev. 1 44 ( 1 966) 1 087. [ 5 ] G. Parisi, Columbia University preprint CO-2 2 7 1 -29. (6] S. Weinberg, Phys. Rev. Letters 1 9 ( 1 96 7) 1 264. [ 7 ] M. Georgi and S.L. Glashow, Phys. Rev. Le tters 28 ( 1 972) 1494. [ 8 ] S. Weinberg, Phys. Rev. D5 ( 1 972) 1 96 2 .

521 2 84

G. 't Hooft, Magnetic monopoles

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[ 9 ] F. Englert and R. Brout, Phys. Rev. Letters 1 3 ( 1 964) 3 2 1 ; P.W. Higgs, Phys. Letters 1 2 ( 1 964) 1 3 2 ; Phys. Rev. Letters 1 3 ( 1 964) 508 ; Phys. Rev. 1 4 5 ( 1 966) I 1 56 ; G.S. G uralnick, C . R . Hagen and T.W. B . Kibble, Phys. Rev. Letters 1 3 ( 1 964) 5 8 5 . [ ! O J D. M. S tevens, Magnetic monopoles : a n updated bibliography, Virginia Poly . Inst. and S t;te University preprint VPI-EPP-7 3-5 (October 1 97 3 ) . [ 1 1 ] C.N. Yang, Phys. Rev. D I ( 1 970) 2 360. [ 1 2 ] A. Neveu and R . Dashen, Private communication. [ 1 3 ] B . Zumino, S trong and weak interactions, 1 966 I n t. School of Physics, Erice, ed. A. Zichichi ( A cad. Press, New Y ork and London) p . 7 1 1 . [ 1 4 ] A . Salam and J .C. Pati, University o f M aryland preprin t (N ovember I 9 7 2 ) .

522 Particle spectrum in quantum field theory A. M. Polyakov L D. Landau ln.stitute of 'llteonticol Physics (Submitted July 5, 1974) ZhETF Pis. Red. 20, No. 6, 430-433 (September 20, 1974)

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We wish to call attention in this article to a hitherto uninvestigated part of the spectrum of o rdinary Hamil­ tonians used in quantum field theory. As the first example we consider the model of a self-interacting scalar field in two-dimensional space-tim e , with Hamiltonian II = - f Is •' +(- - •' ♦• + 1 ♦• 2 -• ,ls l I •- I • l•J = - i �B ♦(s/

'•J

I

(1 )

In this model , the vacuum is filled with the Bose con­ densate ;;;• = µ2/). . Over the vacuum there is a single­ particle state with mass -IIµ , which represents small oscillations of the constant condensate . However, con­ stant 4i is not the only stable equilibrium state . There is another extremal of the potential energy in (1) , de­ termined from the equation

,i1;• • •• ♦, - A ,il!

• 0

(2)

with boundary conditions �!C: .,) = µ2/)., which m ean that the vacuum is perturbed only within a finite volume . The solution o f (2) is ,J, · (s)

=

.l:.. ,11 � • ,12

v•

(3 )

To calculate the spectrum of the vibrational energy

levels near the considered equilibrium point, we write

( s l • , (s) + ♦ /•J

and neglect the terms �• and �• in the Hamiltonian (the latter is valid if ). « µ•) . Diagonalizing the obtained quadratic Ham iltonian , We obtain the mass spectrum ' 2 ,12,• ,/3+ 2 • · '. (-. - • J p. + M,. :: -Y- """"i° 1v• t'Jf � n • O. 1, 2 .

(4 )

In fo rmula (4), the first term is the potential energy at the equ ilibrium point , the second is the zero -point oscillation energy , and the third is the excitat ion

energy . Thus , in this model there are three types of particles w ith anomalously large mas s es . We call these obj ects uextrernons . "

In the generalized formalizatio n , the results consists

in the fact that each stationary regular solution of the

class ical equations of motion corresponds in quantum

field theory with weak coupling lo its own set of ex­ tremons . the masses of which can in princ ipl e be calculated . We shall show that extremons exist in three ­ dimensional models. We consider the theory of the Higgs isovector fie ld ,f>,(x), a = l , 2, 3 , 1 94

JETP Lett., Vol. 20, No. 6, September 20, 1 974

H = Jl22 ••• + 22 r v,ii• > ' - -�2 • ' Fp , ' ' •>< c/>,.D µ c/> ,D v c/>c , elc/>l

( 2 .9 )

whe re le/J I = ( 'J, · 'J,) 1 12 • II is easily verified that if = (0, 0, 1 ), say, then � "" = aµA ! - a.At - In the ,resent case we find that the electric field is

p

• d

� 0 , = - x, dr [J(r)/er] .

fhe electric charge can be written as Q=

f dS , � ,

=

-�1•

(2 . 1 0)

o

e

O

dr

'

JK

r

'

(2. 1 1 )

JI

ZEE

where we have used the constraint equati on (2. 3 ) and Eq. (2 . 6). Thus, the charge is given by an expres­ sion of the form Q=

(?) et(� '

f) ,

(2. 1 2 )

where M i s the scale o f A � and has the dimension of a mass. (This scale will be introduced below . ) The function s can b e determined only numerically . In contrast, the magneti c charge g is quantized exactly as- in ' t Hooft ' s solution since K (r ) - 0 at infinity and �I J - E ;,a x,( - e�)

as r - oo .

By Gauss ' s theore m , g = llc /e .

Ill. BOUNDARY CON DITIONS

As r - 0 we impose J- 0, H - 0, K - 1 . Using E qs . ( 2 . 6 ), ( 2 . 7 ) , and (2 . 8 ) w e find J - const x r2 , H - cons! xr', K - 1 + const x r'; the fields are d if­ ferent iable at the origin ensuring a finite - energy solution. As r - «> the Higgs field approaches its "vacuum expectation value , " that is, H (r), c:. .{3µr + • • • . F rom Eq. (2 . 6) we see that if K , c: . O ex­ ponent i ally we may have J(r), c:. .Mr + b + • • · . M is a parameter with the dimension of a mass and sets the s cale for J. The parameter b determines the charge. As can be seen from Eq. ( 2 . 1 1 ) the asymptot i c form of Eq. ( 2 . 8 ) i s then r2K " = [ (/3µ)' - M ' } r'K + · • ·

and ad mits one particular solution dec reasing l ike e�' , where a = ! (13 µ )' - M ' } 'l2 _ Thus we. require /3µ > M . In particular, the hope t hat the Higgs field could h ave been omitted cannot be realized for this type of solution. The A: c omponents of the gauge field a ct l ike an isolriplet Higgs held with negative metric, and by themselves would cause the othe r components of the gauge field A I t o os­ cillate rather than decrease exponentially as r - co. We have studied the nature of the solutions to Eqs . ( 2 . 6), (2 . 7 ), and (2 . 8 ). we· have also integrat­ ed them numerically . The result is discussed in Appendix A, and the general form of the solutions i s displayed in Fig . I . We will just note here that the electric charge can assume a nonzero value . From Eq. ( 2 . 6), we see that starting with J"(0) > 0, J stays positive, and the elect ric charge g iven by Eq. (2 . 1 1 ) is nonze ro. In the l imit of small J on the effect of J on the equations is small, and thanks to the Higgs field, the solution should exist as in ' t Hooft's example with J = 0. The Higgs fields ensure that our s olution has a finite ene rgy but int roduc e two parameters A and

526 P OL E S WIT H BOT H M A G N E T I C A N D E L E CT R IC C H A R G E S. . .

11

µ which .enter in the electric c harge t(Ve' , M /µ ) Eq. (2 . 1 2 ). The electric charge is then apparently continuous; we will return to this problem.

the covariant energy- momentum tensor coupled to the gravitational field. The "gravitational" Ham­ iltonian density in this case has the form

The mas s of the dyon is given by ft xT00 accord­ ing to the principle of equivalence, where T • • is

This expression is gauge-invariant, of course. Explicitly, the mass o f the dyon is

I V . MASS OF TH E OYON

P

E=

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2229

J

d'JtJC ,

= (411M ,,/e2 )

1• 0

JC, =

F0 , • F01

+ D 04> · D 0

i - J! .

dx{(K ' )' + (K 2 - 1 )' /2x2 + ½x'W )' + ½x' (,i, ' )2 + .K2 W + ,i,2 ) + (il/4/l" )(t - ot 2 )']

where x = M .,r, M ., = {3µ , -t = H/M ,,r, 'f[ = J/M .,r, and /' = df/dx. Some numerical examples are given in Appendix A . Dyons are not much more massive than magnetic monopoles. A dyon with an electric c har§e Q - 1 3 7e is only about 40% heavie r than the magnetic monopole (with g = 1 /e). In Appendix B we give some clarifying re mark s on the relationship between the "gravitational" Hamiltonian and the " canonical" Hamiltonian. V. STABILITY CONSIDERATIONS ANO CONCLUDING REMARKS

't Hooft' s monopole is guaranteed to be stable by topological arguments. 10 There is no such argu­ ment for our solution. We find apparently a solu­ tion for each M in a range of values O ' Qs , the lowest soliton mass is < Q m , where m is the mass of the free charged meson field; therefore, there are solitons that are stable quantum mechanically as well as classically. When Q is between Qc and Qs, the soliton mass is > Qm; nevertheless, the lowest-energy soliton solution can be shown to be always classically stable, t hough quantum-mechanically metastable. The canonical quantization procedures are carried out. General theorems on stability are established, and specific numerical result� of the soliton solutions are given.

I. INTRODUCTION In t h i s paper , we shall pre sent a class of sol iton solutions in three spa c e d i m e n s ions; these soli­ tons are made of scalar fields and are of a non­ topological natu r e ' (to be d i st inguished from the monopole - type solutions given by 't Hooft' and by Nie l s e n and Ole sen3 ) . 4 A brief des c ription of such nontopological solitons has been given in Ref. I. As w e shall see, they s e rve a s prototype s of a rat he r general c l a s s of soliton solutions, whose realization hinges on the existence of some int e ­ g r a l const ra int s on the f i e l d s , which are, in turn, the consequenc e s of the appropriate phy sical con­ se rvation law s in the theory, suc h a s charge, iso­ spin, etc . Gene ral ization of suc h sol iton solutions t o include fields of nonzero spins will be g iven in a subsequent paper. Throughout ou r d i scu s sion, w e consider only relativistic local fiel d s w ith nonlinear coupl ings t hat are renormal izable in the u sual sense ( i . e . , in terms of the u sual perturba­ tion s e r i e s expan sion around the plane-wave solu ­ tions of the free-field equat ion s ) . To begin w ith, it m a y be useful to g i v e the de­ finition of a sol iton solution that i s appropr iate to part i c l e phy s i c s . Following Ref. 1, w e define a c l a s s ical sol iton solution to be one that ( i) has a finite and nonzero r e st mass and ( ii) is confined i n a finite region in space at all t i m e s ( i . e . , non­ d i spersive ) . It can then be shown5 ' that for every suc h c l a s s ical sol iton solution there exist s a cor­ re sponding quantum sol iton solut ion. The quan­ tu m sol iton solution (i) also has a fin ite and non13

zero mass, expre ssed in terms o f t h e u sual r e ­ nor mal ized quantities which a r e defined b y the usual pe rtu rbation series around the free plane­ wave solut ions, and · ( ii) has a spat ial e xtension which gives rise t o "soft " form factors' that go to zero at large momentum t ransfe r . Becau s e of the unce rta inty principle, it i s clearly not poss ibl e to construct a nondi spersive wave packet of the quantum soliton solution. However, it can be readily shown that w hen the appropriately defined nonl inear coupling con stant g becomes suffic iently small, both the mass and the form factor redu c e asy mptot ically to their respective classical e x ­ pre s s ions; the mass i s O ( ,,-') a n d the f o r m factor O(g 0 ) . When g dC'c rease s , the spat ial e xtension of a quantum sol iton, a s determ ined by it s form factor , remains finile , in accordance w ith its classical l im it . This remarkabl e feature d i s ­ t inb,u ishes a quantu m sol iton from either an atom or a molecule, whose size approac h e s infinity as the fine - st ructure constant a - 0 . T h i s difference under score s , once again, that in thf:' context of a relat ivistic field theory the sol iton solution already C'Xist s on a classical level, while atoms and mo­ lecul e s exist only in the quantum theory. (Our definition of solit on differs from a more narrow one, used in some mathe mat ical and engine e r ing l iteratu r e . " In th i s narrow definition, the t e r m sol iton i s c onfined only to s o m e e xtremely spe c ial­ ized nonl inear equat ions that have solitary \vave solut ions whose shape and vel o c ity re main un­ changed even after a head - on col l i s ion. Su c h a h ighly restrict ive definit ion would automatically 2739

534 R . FRIEDBERG, T . D. L E E , AND A . SIRLIN

2740

exclude all t h e four-dimensional local field the­ ories that are of interest to particle phy sicist s . ) F or clarity, we consider first t h e simplest example of such a soliton solution . We assume the system to cons ist of only two spin-0 field s : a complex field c/> and a Hermit ian field X- The Lagrangian density £ is assumed to be (general­ ization will be given later in Sec . VI)

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£,

a c/> t a c/> 1 ( a x ' ) a x" a xµ 2 ax" t 2 - J 2 x c/> c/> - ½ g'( x ' - x ,./) 2 ,

=-

( 1 . 1)

where x " = (r, i i ) , c/> t is t h e Hermit ian conjugate of c/>, and f, g, and X "' are constant s . The theory possesses a disc rete symmetry x- -x ' besides the U ( l ) symmetry cp - e• e c/> ;

(1.2)

( 1 . 3)

because of this U ( l ) symmetry, there is the cur ­ rent conservation aj µJ a Xµ = 0 where j" = - i[ c/> \ a c/> / a xµ ) - cp(a cp t/ax µ) ] . Consequently, the charge

Q = - i fi, d'r

( 1.4)

i s a constant of motion. We shall consider first the classical solution, and leave the discu s s ion of the quantum solution to a later section, Sec. IV. Since Q depends lin­ early on �. the classical solution for Q ,e Q must be time d ependent; for the lowest energy state, c/> oc exp(-i w /). It is convenient to scale away b oth the physical d imension and the nonlinear coupling g. We introduce

x(r, 1 ) = ( µ/g) A (Pl ,

c/>(r, I ) = 2 - ' 12 ( µ/g) B(p) e - i wt ,

( 1 . 5)

where A and B are both dimensionless and real, and

P = µ "r , µ

=

gx vac

( 1 . 6)

( 1 . 7)

is the mass of the neutral x me son as can be readily seen from the Lagrangian ( 1 . 1 ) . T he cor­ responding mass of the c harged c/> meson is m == fX uc .

( 1 . 8)

v 'A - K' B 2A - ½ (A 2 - l ) A = O

( 1 . 9)

F rom ( 1 . 1 ) and ( 1 . 5) it follow s that the functions A (p) and B(p) satisfy and

2 2 V 2 B - K2A B + v B = O ,

( 1 . 1 0)

where V is the grad ient operator with respect to

the dimensionless parameter p, v = w/ µ., and K =m/ µ. .

13 ( 1 . 1 1)

The charge Q is related to the frequency w, or v, by The energy of the system is g iven by E = ( µ./g') f 8 d 'p ,

where

( 1 . 12)

( 1 . 1 3)

8 = ½ ( VA )2 + ½ ( V B) 2 + ½ ( V 2 + K 'A 2 ) B 2 + ½ (A 2 - 1 ) 2

( 1 . 14) T hrough ( 1 . 1 2) , v may be regarded as a function of Q and a functional of B(p) . Upon substituting v = v(Q, B) into ( 1 . 1 4 ) , we may expre s s E as a func ­ tion of Q and a functional of A ( p} and B(p) . Equa­ tions ( 1 . 9) and ( 1 . 1 0) can also be derived by keep­ ing Q fixed and setting the functional derivatives M/M (p) = /J E//JB(p) = 0

( 1 . 1 5)

As we shall see, there exist two c rit ical value s of charge : Q s and Qc with Q s > Q c . Sol iton solu­ tions exist when t he total charge Q is greater than the lower c r it ical value , Q > Qc . In general, at any given Q > Qc , there is more than one sol iton solution. A stability theorem will be e stablished, which states that among these sol iton solutions the one with the lowest energy is classically always stable against arbitrary small functional variat ions, while the others are not . When Q is greater than the upper critical value Q s, the rest mass of the soliton solution (w ith the lowest energy) is less than Q m, the corresponding energy of the free - meson solution. Thus, when Q > Q s, the sol iton solution (with the lowest energy) is absolutely stable against complete dissociation into free me­ sons . (In- fact, it is absolutely stable against any decay . ) When Q s > Q > Qc , the soliton mass is > Q m. Nevertheless, classically one still has stable soliton solutions. The corre sponding quan­ tum soliton solution is, of course, metastable; however, its l ifet ime can be quite long in the weak-coupling l imit, the lifetime - 00 when the nonlinear coupling - o. The existence of the sol iton togethe r with some general properties of the solution are discussed in Sec. II. The question of stabil ity is examined in Sec . III, and also partly in Sec . IV. The quan­ tization is carried out in Sec . IV. The result of a nume r ical calculation of the soliton solution is given in Sec. V. Most of the method s developed for the simple Lagrangian ( 1 . 1 ) are appl icable to a much wider class of problems. In particular,

535

if ( 1 1 ) is generalized to

a .C = - a x� q/ � 2 a x JJ ax JJ

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.! (�,;__)'

- f ' x' ].

(2. 1 0)

This inequality holds for arbitrary l engths R and /. As Q inc rease s, the optimal value of R in­ c reases with Q, while that of / remains 0 ( µ- 1 ) . Thus, for Q large, the right - hand side o f ( 2 . 10) becomes R - ' nQ + ¼ ( n µ4 R ' /g 2 ) + O(R 2 µ'/g 2 ) . Tak­ ing its minimum, which occurs at we find

E m;, ,;

t nµ(2g ) - 1 14 Q3 2

14

+ O( Q 1 12 µ/g) .

( 2 . 1 1)

(2.12)

By comparing this value with Q m, the ener gy of the plane -wave solution, we see that

(2. 13)

536 R . FRIEDBERG, T . D . L E E , AND A . SIRLIN

2742

where

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Q . -(

4 ,r µ ' s-;;;- )

1 2g '

(2. 14)

Therefore, when Q > Q s the soliton solution exists and is absolutely stable . I t c a n be readily seen that t h e trial function ( 2 . 8) actually sat isfies the differential equations ( 1 . 9) and ( 1 . 10) when r ,; R, and it approaches the correct boundary condition at "' · Because R in­ creases with Q, for Q sufficiently large the upper bound ( 2 . 1 2) is the correct asymptotic expression for Emin • Thus, (2. 14) also gives the correct limiting value of Q s when K =mlµ - 0. As will be shown in Appendix A , when K - ., on upper bound on Q s can be obtained: Q s < 7 5 1T 4 ( 1T 4 - 36) 1 /2 /(5 1 2 g 2 K 3 ) 2' 1 1 1 . 8 (g 2 K 3 ) - 1 • ( 2 . 1 5)

Th erefore, when K - "', Q 5 - 0. A s we shall see, in the quantum theory Q must be an integer. Ac­ cording to ( 2 . 1 5), for K sufficiently large, it is possible to stay in the weak-coupl ing region (g and f = gK both s mall) and yet have Q 5 < 1; the quan­ tum solitons would then be stable for all Q I 0. C. Variational principles and virial theorem

As already noted in the Introduction, the dif­ ferential equat ions ( 1 . 9) and ( 1 . 10) can be derived from the variational princ iple ( 1 . 1 5) , keeping Q fixed, i . e . , (6E) 0 = 0 .

( 2 . 16)

d E( Q) = w . dQ

( 2 . 17)

The resulting stationary value of E is a function of Q. Its derivative is given by The Legendre transformation F = E - wQ

( 2 . 1 8)

F = .J!:.. ffJ d 'p ' g'

( 2 . 19)

5' = 8 - v' B 2 •

( 2 . 20)

(6 F\ w = 0 .

(2.21)

define s a functional F of A and B. We may write where the function fJ is related to 8 of ( 1 . 14) by

The variational princ iple ( 1 . 1 5), or ( 2 . 16), is equivalent to requiring F stationary against arbi­ trary functional variations in A and B but keeping w fixed; i . e . ,

W e may regard t h e re sulting F as a function o f w ; i t follow s then

13

(2.22)

T here exist s still another variational formal ism that is particularly useful in our later study of the stability problem. We define a functional G of A and B: G = .J!:.. f11 d'p '

where

g'

(2.23)

S = ½ ( VA )' + ½ ( V B)' + ½ K 'A'B ' + ¼ (A' - 1 )2 •

(2.24)

The stationary condition (2 . 1 6), or ( 2 . 2 1 ) , can also be expressed in terms of G. We require the functional G to be stationary, keeping / = ( µg ') - 1

fB ' d'p

fixed. The condition (liG)1 = 0

implies

and

6G =O M(p)

� = ( µg 2 ) - 1 w'B 6B(p}

( 2 . 2 5) ( 2 . 26) (2. 27) ( 2 . 2 8)

which are identical to ( 1 . 9) and ( 1 . 10), w ith w' now appearing as the Lagrange multiplier. The resulting stationary value of G is a function of the constraint /, From ( 2 . 28), it follow s that d

di G( / ) = 2l w'

The functional G is related to E and F by G = E - ½ wQ

= F + ½ wQ ,

and / is related to Q and w by Q =Iw .

(2. 29) ( 2 . 30) (2.31)

T h e above three variational formulations, (2. 16), (2 . 2 1) , and ( 2 . 26), are applicable to both the soli­ ton solution and the plane-wave solution, pro­ vided that we assume a finite (but large) volume and impose the periodic boundary condition. Of course, for the soliton solution, we may directly assume an infinite volume w ith the boundary con­ dition : At "', A - 1 , and B - 0; in addition, B is square - integrable . We note that both 8 and S are positive, but fJ is not. The virial theorem for the soliton solution can be most easily derived by using the variational

537 C L A S S O F S C A L A R - F IE L D S O L I T O N S O L U T I O N S I N T H R E E . . .

13

formalism ( 2 . 2 1 ) . We c onsider the variation A (r) - A ( Ar) and B(r) - B(Ar)

for a sol iton solution in an oo volume, where A = 1 + E and € = O + . By setting ( a F ) = o at A = l ' aA w

we find

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fu a'p = ½f r a'p ,

where and

T = ½ ( VA ) ' + ½ (v B)'

U = ½ ( v' - « 2 A 2 ) B ' - ½ (A ' - 1) 2 •

(2 . 3 2 ) ( 2 . 3 3)

D. Soliton solution when w is near m

From ( 2 . 9) and ( 2 . 1 1 ) we see that as w increase s from O, t he rad ius R of the sol\ton varies as rr/ w and its c harge Q decreases from 00 in proportion to w-•. On the other hand, for the plane-wave solut ion, as the volume - 00 , the lowest-energy state at a given Q is one w ith w = m. Now, both the plane -wave solution and the sol iton solution satisfy the same set of differential equations ( 1 . 9) and ( 1 . 10). As we shall see, assuming that the volume of the sy stem is sufficiently large, when w inc reases from O to m, the soliton solution would evolve c ontinuously and f inally join onto the plane-wave solut ion. To find the connection be­ tween these two types of solutions, we shall in­ vestigate the soliton solution when w is near m ( i . e . , v near «). We define i; = ( « ' - v ') ' I' .

(2.34)

and

B = z - , 1, 2 . The plane-wave solution has an /" - 0 as r.l - oo , while that of the soliton solution remains nonzero.

J

C. Classical stability

( 3 . 13)

where w i s , by choi c e , positive. Proof. Cond it ion ( i) is nec essary because othe r ­ w ise, i f t he re are two e igenfunc lions of H that have negat ive e i genval\.Jes , by a suitable l inear combinat ion of these two e igenfunc t ions one can easily cons truc t a I/! o rthogonal to b ; because of (3 . 1 ) , the cor responding ( O 'E )Q is < 0 . To understand cond i t ion ( i i) , we first define

-

a (P )

"ava \n(ACP� (p)) ,

(3 . 14)

in wh ich A (p) and B ( p) , be ing solut ions of ( 1 .9) and ( 1 . 10) , are regarded as func t ions of the param eter v that appears in ( 1 . 10) . By different iating ( 1 . 9) and ( 1 . 10) , we find Ha � 2 vb ,

La, i/!

1

(P)

(3 . 1 6)

From (3 . 1 5) , it follows that

thus ,

(3 . 1 5)

where b is given by ( 3 . 2) . The e igenfunc t ions of H can be normal ized to fo rm a complete ortho ­ normal set of real functions { O , ,9(z ) = O has n o negative root. There­ fore, cond itions ( i) and ( i i) are suff ic ient to insure that (o'E) 0 >' 0 . From Theorem 1 , H i s known to have a t least one negative e igenvalue . Hence , by exam ining the plot JJ(z ) vs z , one can easily verify that condit ion ( i i) is also necessary. This completes the p roof of Theorem 3 . As show n in Fig. 2(a) , the sol iton solution exists only if Q >' Q c . Theorem 4 . At any g ive n Q :;,, Q0 the lowest-ener­ gy sol iton solution is always class ically stable. Proof. In F ig. 2(d) , the point C l ies to the r ight of the point C ' . By using Theorem 2, one sees that in Fig. 2 ( a) , along the entire lower branch CS'S H has o nly one negativ e e igenvalue . Since Q - '(dQ /dw) is negative, it follows from Theorem 3 that the soliton solutions on this entire lower b ranch CS'S are all class ically stable . Theorem 4 is then proved. We note that the soliton solutions along the upper b ranch CC' oo in Fig. 2(a) are all class ically un­ stable because Q - '(dQ/dw) is positive. In Sec . II D , ii was shown that at a g iven Q there can be other exc ited sol iton solutions which have radial nodes. As we shall see, those solutions are also class ically unstable. Theorem 5. If in the spher ically symmet r ic soliton solution the function B has one or more node s , then ( 62 £) 0 can be negative. Proof. Let us cons ider Eq . ( 1 . 10) , which is l in ­ e a r - i n B . T h e assumption that B h a s nodes im­ plies that there exists a B w ith no node, or fewer nodes than B , which satisfies ( 3 . 2 7)

(v' - • 'A 'li'i + v'B = o ,

with °i,2 < v 2 . The funct ion Ii is, of course, ortho ­ gonal to B. Thus by choos ing we find, th rough (3 . 1) and ( 3 . 3 ) , (0 2E lo = ½ ( µ/g2)

f(v' - v')B d P < 0 , 2 3

(3 .2 8) ( 3 . 29)

which establ ishes the theorem . We note that if B has n nodes, then there are at least n such I inearly independent B func t ions. ( There may be more, s ince B does not have to be spher ically symmet ric .) Thus, the higher the number of nodes a soliton solut ion cont a ins, the more unstabl e it is. Rem arks . The class ical solut ions A and B are real functions . So far , for clar ity of presentation, the ir variations 6A and 6B are also assumed to be real . Since in a class ical theory, x is a real field, the variation 6A must be real . On the other hand ,

13

is a complex field; the variation 6B can be com ­ plex . The general case of a complex 6B will be exam ined in Sec . IV. As we shall prove , the clas ­ s ical stability theorem (Theorem 4 g iven above) remains val id even when 6B is complex. [ See ( 4 . 78) and the remarks made at the end of Sec . IV D .] IV. QUANTIZATION

To der ive the quantum sol iton solution, we shall follow the general canonical method developed by Christ and Lee. 5 For clarity of presentation, in this section the quantization procedure is carried out only for the c ente r -of -mass system in which the total momentum P is zero. ( The details for a moving system , P;a O, are d iscussed in Appendix B.) A. Collective coordinates

We introduce four collective coordinates , the three components R• (f) of the center -of-mass position vector (k = 1, 2, 3) and an over -all phase variable t (I) for the c harged field fr , I ) . Follow­ ing Ref. 5, the quantum expans ion of the operator s t x fr , l ) , fr , I) and i t s Herm it ian conjugate fr, I) is g iven by x fr , t) =J A Qm . Furthermore, as K- 00 , the inequality ( 2 . 1 5) hold s for the general case as well.

551 C L A S S O F S C A L A R - F I E L D S O L I T O N S O L U T I O N S IN T H R E E . . .

13

Except for th e replacement of ½(3A 2 - 1 ) by d 2w ( A )/dA 2 in (3 . 3 ) , t h e entire discuss ions given in Secs . III and IV on the stability and quantiza ­ tion are valid in the general cas e. The numer ical integration of the differential equation ( 2 . 3 7 ) dis ­ cuss ed in Sec . V is also applicable to the general case; of course, the spec ific s o lutions of ( 1 . 9 ) and ( 1 . 10) given there h o ld only for w (A ) = ½ ( 1 - A 2 )2 • The generalization to include spin - ½ fermion fields and spin - 1 boson fields will be discus sed in a subsequent pape r . ACKNOWLEDGMENTS

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We would like to express our deep apprec iation to Dr. Gerald Dorman for his advice and assistance in the c omputer operations, and to thank Professor R . Jackiw for a helpful discussion. Two of us (T.D. L. and A . S . ) would also like to thank the CERN T heo­ ry Divis ion for the warm hospital ity shown to us du ring the summer of 1975, when s ome aspects of this problem were investigated. We gratefully acknowledge partial support of the c omputer operation by ERDA under Contract No. E ( l l - 1 )-3077 at New York Univers ity . APPENDIX A

In this appendix , we derive the upper bound (2 .15) for Q 5 when K = m /µ - oo , where µ and m are , respectively , the masses of the neutral x and the charged field s . ( 1 ) W e first show that when K - 00 , Q 5 - 0 faster than K -z . It is convenient to define and

z = pK = mr

(Al )

n = II/K = w/m .

F or a spherically symmetric solution , (1 .9) and ( 1 .10) may be written as and

.!_2 _!!_ z dz

(z

2

d.4. ) - B 2A = _!_ (A 2 - l )A dz 2 K'

],_ i!.. (z' dB ) - A'B + n 2B = O . z• dz dz

Correspondingly , (1 . 1 2 ) and ( 1 . 1 3 ) become and

Q = 4 11n (gK ) _,

1.•

E = 4 1r111 (gK)"2

where

E=

z 2B 2dz

1• 0

z 2Edz ,

_! dA 2 .!fdB 2 _! • • • ) + (n +A )B ( ) + 2 dz 2 \"dz 2 + ( S K2 ) °1 (A 2 - 1 )2 •

(A2 ) (A3)

(A4) (A 5 )

(A6)

275 7

When K - 00 and for Q ~ O (K"2 ) , the right-hand side of (A2 ) and the last term in (A 6) may be ne­ glected , except at large z . This suggests a trial function of the form and

A (z ) =

for z :,; z { 1 - (l - a)(;/z) e•U•Z > I • for 2 e,c z

(b /z) sin (,rz/Z) for z :,; z , B (z) = { for z e,c z ' 0

(A7)

where a , b, and Z are parameters to be deter­ mined by minimizing the energy E . The charge Q of this trial function is given by Q = 2 ,rn (gK )-,b 2 Z .

(AS)

,rm(gK) ""b 2 Z [n2 + a2 + ( ,r /Z)2 ] ,

(A9)

By substituting (A 7 ) into (A5 ) , we find E is equal t o a b - independent term plus

where n' is , accord ing to (A S ) , proportional to (Q/b')' . The minim ization of E with respect to b can be carried out by d ifferentiating (A9) with respect to b , but keeping a, Z , and Q fixed ; this leads to n2 = a2 + (,r/Z)2·•

(A lO)

q = (4,r) "' (gK) 2 Q .

(A l l )

E :5 4,rm(gK) ... e ,

(A 1 2 )

The s ame result can also be obtained by requiring the trial function (A 7) to satisfy the field equation (A3 ) for z < Z . It is convenient to define

B y using the t rial function (A 7 ) , we derive a n up­ per bound on the energy

where

e = qn + ½ ( 1 - a)'Z'(Z"' + K 01 ) + (24K2 )"1 (1 - a')' Z3 + ( ,

and

€ = (2K')"1

1• z

z' ( M )' [l + ¼ ( M ) yiz ,

(A 1 3 )

(A 1 4 )

6 A = - ( 1 - a)(Z /z ) exp[- (z - Z)/K J.

L e t us c onsider the m inimum o f e when K - 00 , but assuming Q - O(K "" ) , i .e . , q ~ O( l ) .

(A 1 5 )

A s w e shall see from (A 1 7 ) and (A 1 8 ) below , in this case the optimal values of the param eters a and Z are both 0 ( 1 ) . F rom (A 1 3 ) , it follows that when K - 00 e = qn + ½ ( 1 - a )2Z + O(K"1 ) (A 1 6)

552 R . F R I E D B E R G , T . D . L E E , A ND A . S I R L I N

2 7 58

where n is given by (Al0 ) . Neglecting O(K-1 ) , the m inimum of (A1 6 ) occurs at (Al 7) a = [1 + (q/n)2 J-1 / 2 and

Z-1 = [½a ( 1 - a ) ]1 1 2 /n;

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the corresponding minimum value of e is 2 2 2 1r-f2 { [1 + (q/,r) ]l / _ l }l / 0

(Al8)

(A19)

Since (A1 9 ) is always u'

a V(o) _

ax,/ - � - 0 '

where V(o) = V(o ( r, t, z?, . . . , zi)). Since the above equation is independent of g, so is o'. We will al­ ways introduce a sufficient number of parameters z�, . . . , z� so that translation of our solution in space or time can be accomplished by changing the parameters z�, . . . , z�. In particular, there will exist a set of values z.(t, z �, . . . , z�) w hich allow us to write 1

o'fr, 1, z1, . . . , 4)

= a' (r'1 0, z 1( t, z�, . . . , z1 ), . . . , zK (t, z�, . . . , zi) )

( 1 . 2)

For the single- soliton solution in a D-dimensional space , we w ill use at least D integration constants, say z�, . . . , z'1,, representing the center-of-mass position of the soliton at a particular time . In the case of an l- soliton solution in a D-dimensional space, there should be at lea st ID integration con­ stants ·corresponding to the initial positions of the l solitons. It will often b e useful to choose the pa­ rametrization z., . . . , z K of our family of classical solutions so that the z.( t, z?, . . . , 4) defined by Eq. ( 1.2) have the s imple form z. = u.t +z� ,

where the u.'s are constants (k = 1, 2, . . . , K), so that

has no explicit dependence on t, and it satisfies - , v •a -

....

L u.

u.,

a 'o-' a V(o) az az . , - � = 0 . .

(1.3)

1 607

For a time- independent solution, one has u• = 0 and therefore z, = z�. That suc h a s imple choice, z. = u,t + z�, is always possible for any time-de­ pendent solution follow s from the time-transla­ tional invariance of the original field equation. We can always choose one of the integration constants, say zi, to be related to the time translation, t- / + constant, and this leads to the special choice that only uK f' 0, but all other "• = 0. In general, for an l- soliton solution in a D-dimensional space, it is most convenient to choose u 1 , • • • , u 10 to be simply the velocity component s of these l solitons as t - - 00 , and therefore z,, . . . , z 10 become their positions at least in the asymptotic region. For the quantum theory, we expand the field operator q, 1 (r, I ) about the classical solution g- 1(/ (r, Zu . . . 1 z .{.): +

..

L

n=K+1

q.( t ) ,p! ( r, z . , • . . , z K ) ,

( 1 .4)

where z 1 , • • . , z K , qx + u qK +2 , . • • are treated as coordinates and the N-component c-number func­ tions ,p!(r, z 1 , . • • , z K ) form a complete set of real functions, subject to the constraints

ao' dr L'Ill f,p! 8z t

=0 ' and the orthonormality relation

N

t '= 1

J ,j,� ,p! , dr = O••' ,

( 1. 5 ) ( 1 . 6)

where dr = d0r. We now propose to apply the usual methods of canonical quantization to the Lagran­ gian density ( 1 . 1), written in terms of these new coordinates z 1, • • • , z K , qK+ i , qK+ 2 , • . . . This approach will be developed in the next sec­ tion. The quantization procedure can be carried out in a completely standard and straightforward way . The Hamiltonian of the system, as a function of the coordinates zu . . . , z K , qK + u . . . and their conjugate momenta (- ;a /az 1 ), • • • , ( - ;a/az K ), (- i a /aqK + il , . . . can then be explic itly given. By using this Hamiltonian, we can attempt to express the solution of the resulting quantum-mechanical problem a s a power series in g. The leading term for small g is O (g -2 ) , and it can be derived by solving a K - dimensional quantum-mechanical prob­ l em, depending only on the coordinates z 1 , • • • , zK and their conjugate momenta . It turns out that to O(g- 2 ) the solution of this K-dimensioc.al Schro­ d inger equation is precisely that given by the WKB approximation; it leads to a direct connection be­ tv.•een the quantum-mechanical solution and the classical solution of interest. The next-order

558 N . H . CHRIST AND T . D. LEE

1608

solution is O(g 0), and it involves also q. and 'It• = - i8 /aq.. Since terms in the Hamiltonian that depend on cubic or higher powers of q. and rr. carry additional powers in g, to O(g 0) the Hamiltonian is reduced to one that depends, besides on z. and P. = - ia /a z., only quadratically on q. and rr•• This then reduces to a system of harmonic oscillators whose frequenc ies may depend on z.. Its quantum­ mechanical solution Is closely tied to the corre­ sponding classical problem, as shall be explicitly exhibited in Sec. In Sec . m, our general method is Illustrated by application to field theories in one space dimension and one time dimension. Independently of the de­ tailed form of the Lagrangian, we derive the quan­ tum-mechanical solution of a single soliton, either at rest or in motion. The general form of the characteristic frequencies of the harmonic oscil­ lator s in q. and "• is discussed, which allows us to examine the inverse problem of starting from the energy spectrum and then deriving the corre­ sponding field theory . The question of statistics of solitons in one space dimension is fundamentally different from that in two or more spac e d imensions. In space dimen­ sions other than one, there is a continuous set of points at infinity. Except for some special the­ ories involving gauge vector-meson fields, be­ cause of continuity, we can assume ,p' ("') to be of a constant value equal to that in the vacuum state. Thus, w e may at l east formally expand the soliton state in terms of the usual free-meson creation operator s acting on the vacuum state. A soliton can then be viewed as a bound state of an indefinite

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n.

-½ i: l;1

k • c va' ) +

t,

norK+ l

q. ( vl/i! f - g->

in which the derivatives of a' and I/!! are taken by regarding z" . . . , z K and as independent vari­ ables. In the following, for convenienc e we shall adopt the convention that all repeated indices are to be summed over. The superscript i varies from 1 to N, while the subscripts k (or k ' ) and n (or n') are always treated differently; they vary as fol­ low s :

number of mesons; its statistics are like those of any bound state, determined by its constituents. In one space dimension, the above argument fails since the value of ' at x = ., usually d iffers from that at x = - ., in a soliton state . On the other hand, as is well known and shall also be analyzed in Sec. m, the statistics of any one-dimensional system of interacting identical particles is entirely a matter of convention, not of principle. Section IV contains a brief discussion of the quantization of the time-dependent two-soliton and soliton-antlsoliton solutions to the sine-Gordon equation. This section serves primarily as an Illustration of the use of the coordinates z., q. of Eq . ( 1 .4) in a nontrivial case . For the soliton­ soliton or soliton-antisoliton scattering solutions only the terms of order g ..,, are discussed. The quantization of the periodic breather mode (soliton­ antisoliton scattering with imaginary velocity) is carried out through order g 0 and results similar to those of Dashen, Hasslacher, and Neveu• are obtained. Finally, In Sec. V, w e introduc e and quantize a new one-dimensional scalar field theory whose classical solutions include soliton pairs permanently confined in a "bag. " II. GENERAL METHOD

L et us now use the usual canonical methods and the coordinates defined 1n ( 1 .4 ) to quantize the dy­ namical system described by the Lagrangian den­ sity ( 1 . 1 ) . In terms of the coordinates z ., • • • , zc , qK + , , qK + , , • • • , the Lagrangian becomes

v(a' +g

but

n ( or n') ?- K + 1

The momenta conjugate to z, and . - , ) ( VI/>.•) - , + I/>,, 8CTa• V(er) 1 1 8 1 1/!n• ] d-r

F•• • -

er

c_, = - C,,,n == - u

J

(2.31)

8 1/>nl !(r, z 11 • • • , z K ) to be the re­ sult of the same symmetry operation applied to 1/>!(r, 0, . . . , 0) . In this way the momenta conjugate to the variable s z. become generator s of the K­ parameter symmetry and commute w ith H. In particular, the m atrix elements F•• , and eigen­ values w.' are independent of the variables z•. Thus the products ( 2 . 1 5) are eigenstates of H ac cu­ rate through order g 0 • If a sufficient number of variables z• has been introduced so that the K functions ac, /az. comprise all the eigenvectors of (2 . 33) w ith zero eigenvalue, then all the w. will be different from zero. Consequently, if w e consider states with definite values of the conserved mo­ menta P., • . • , P,r: , our ground state of H w ill be nondegenerate to order g 0 • 2. Time�dependent cose

Next, we consider the general case u #a O. The eigenstates x a can be written down formally as ( 2 . 3 6)

Here OpK is a constant, X a(z 1 , . • . , Z,r - ,, 0, q0 ) is any function of z 1 1 • • • , Z ,r -1 and the q0 , while U(z,r ) is a "time" develop ment operator which satisfies w here

(2.37 )

562

J'2 = ½( fl',. 11,. + q,. Fmi ' q,. , + fl'" GM , q,. , - q,. G,.,. , 1r,. ,)

(2.38)

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and

12

N . H . CHRIST AND T . D. LEE

1612

(2.39)

U( O) = 1 .

In H 2 , the subscript 2 indicates its quadratic de­ pendence on q. and ir•. The operator U(zK ) is uni­ tary. U the variable zK /u is interpreted as the time, Eq. (2.37) can be viewed as a Schrildinger­ type equation for an Infinite-dimensional harmonic oscillator with time-dependent coefficients. Al­ though not shown explicitly, the operator U(ZK) will, in general, also depend on the variables z., . . . , z K -•• The energy eigenvalue 8 ,. for the state x ., specified by (2.36) is (2.40) The somewhat unusual Hamiltonian H2 appearing

in (2.37) has a very direct and simple Interpreta­ tion. Consider our original problem expressed in terms of the variables z., q0 • A solution to the classical equations lying close to the solution z." (t ) = u.t +z0., q. = O,

1T. = O

2 p" • ( t ) -- K- � (z "'1 , - .. - • znd )

az,.

,

(2.41)

may be written z.( t ) = zI' ( t ) + liz.( t ), 85 P.(t ) =g-• 8z. (z� + Oz 1 ,

•••

, z� + liz.) + lip.(t ) , (2.42)

U the classical Hamiltonian is expanded In powers of the fluctuations Oz., lip., liq., liir., the terms quadratic in these quantities and of leading order in g are

H�' = ½((6110 )2 + F•• ,(liq0) (/iq0 ,) + 2G•• ,(liir.) (liq0 , ) ] aa a,p + -., g 2 0P. (M0 - 1 )•• • •uJ>,,, - 2g u• - -!!.. (MO -• ) J.11, d r lip.ti' 6qn

j

az ,. 8z I

(2.43)

As expected, 6z• does not appear in H ;' , and the choice liP,, = O is thus consistent with Hamilton's equa­ tions and eliminates the variables /iz., lip• from H�1 altogether, leaving simply the Hamiltonian ( 2 .3 8) . Therefore, the operator U(zK ) is precisely the time development operator for the system obtained b y quan­ tizing those small oscillations about the classical trajectory z • = u.t + z� for which all the lip• are zero. Thus the problem of finding eigenstates of the complete Hamiltonian (2.13) accurate to order g 0 has been reduced to that of solving the Schr!Sdinger equation for a system of harmonic oscillators with time-depen­ dent coefficients and an infinite number of degrees of freedom. Since such an equation does not have an explicit general solution, we will limit the remainder of this general discussion to expansions about two specific types of classical solutions: (a) slowing varying and (b) periodic. Examples of other siblations can be found in Sec. IV. (aJ Slowly varying. U we assume that u Is sufficiently small, so that compared to any characteristic frequency "'• of H2 we have

I I

aa1 u Bz « l w. a' I , K

(2.44)

Eq. (2.37) can then be solved by the usual adiabatic approximation. The resulting eigenfunctions x..,(z., q.) are X a(z., q.) =

.

¾ e1 < 6•K •K rr { exp [- j; (N. + ½) >

I

•.r w.( zf) dzf ] (2.4 5)

where hN (x) is a Hermite polynomial of order N and the q. and "'• are the coordinates and corre­ sponding eigenfrequencies which diagonalize the quadratic form F••• obtained from �2 . 3 1 ), by re­ placing the explicit factor u' by zero. (b) Periodic. If we assume that our classical

solution a 1 (r, t; z., . . . , z K ) is periodic with period T, then

(2.46}

563 QUA N TU M E X P A N S I O N O F S O L I T O N S O L U T I O N S

The coordinate po ints Z i, • • • , zl!, q/! + i, • • • and z , . . . . , Z,r + u T, q"• " • • • determine through Eq. (1.4) the same configuration of our physical sys­ tem. Thus in our quantum - mechanical description we will use zx In the range (2. 47)

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and require that our wave function have the same value at the physically identical points z K = 0 and zK = u T. This condition will give the familiar quan­ tization of energy. For example, If we make the adiabatic approximation of Eq. (2.45), then for the ground state N. = 0 the entire wave function it of Eq. (2. 1 5) will be periodic If

! .r

�x (zx) -

i;. L.

w.(zx)] dzx + (0Px) uT = 2 lfll , (2 . 48)

where 6pK ls a constant and Px " (8S/8zx) = (M 0)xx", on account of (2.29). By using (2.40) , we have u = B 8.,/8(0Px), so that (2.48) may be written as

!.r

8 6 ) {Px] } t1zx = hn • (2 .49)

Thus the elgenenergies g 8 + &., are shifted from those given by the usual WKB values by a sum of the zero-point energies of the harmonic oscillators averaged over one period of the classical motion, ½W11, where ->

( 2 . 50)

H Eq. (2 .49) ls to be satisfied for all values of z,. • • • , Zx -, wlth a constant value of &.,, it is suf­ ficient t6 require that (M0) n and w.(zx) be inde pendent of z ,. • • • , Z,r -,• This can be achieved if we assume lt ls possible to choose variables z., • • • , ZK -u as described in the paragraph fol­ lowing Eq. (2.3 5), so that their conjugate momenta - i(8/8 .a 1) , • • • , - i(8 /8.t l! -,) commute wlth H . Again, In most cases of interest, a family of degenerate periodic classical solutions a' (x, z,. . . . , Zx - u z� +uT) exists because the original Hamiltonian po s­ sesses a continuous (K - 1)-parameter symmetry. In order to quantize fluctuations about a general periodic orbit, we must know some properties of the operator U(uT) determined by the Schrlldinger equation ( 2 . 37) and Eq. ( 2 .3 9 ) . In fact, this oper­ ator Is completely determined by the solution to the clas sical problem of small oscillations. It can be easily shown from the classical theory of small osclllations 14 that through canonical transforma­ tion our variables Oq.(t ), 0 7r0 (t ) can be chosen

so that

1 613

18 r a. Oq,( T) + b. t'l7r.( T) = e- . [a, t'lq.(O) + b. t'1 7r.(O) ) ,

(2.51)

8 a: oq.( T) + b: 0 11.( T) = e •" [a: o q.(O) + b: 0 11.(0)] , 1

where P. is a characteristic exponent (or stability angle) and we have required that our classical orbit be stable so that P. is real . We will choose a. and b0 so that and define the quantum - mechanical operator A.

'"

a. q. + b. (- i8/aq.) ,

which obeys

(2. 52)

(2 . 53)

[A., A!, ] = ll.. , •

Because the Hamiltonian H2 in ( 2 . 37) is quadratic in the q0 and 110, the Heisenberg equations of mo­ tion for the operators q. and 110 are identical to the classical equations . Therefore, ( 2 . 5 1 ) can be written 1 8 :r 1 ( 2 . 54) U( Tu)- A. U( Tu) = e- • A. which implies that

U( Tu) = exp [- i

.

L (A!A. + ½) P.T] .

( 2 . 55)

The constant ½ in the exponent can be determined from the formula

r

( O j U(t ) I O ) = {detT [expf- t

JC ( t') dt')i....

r/•i ,

(2 .58) where the matrix JC can be written in four blocks, JC

=

0

(� MfiAr. a•n. ) aA. aA., • H.

a - --=-=8A 8A

e •H.

- i'i-fu 8A 0 8A 0

and only the determinant of the corresponding lower right - hand block is to be taken in evaluating Eq. ( 2 . 56) . Using Eq . (2. 55), we can impose the condition of periodicity in z" for the wave function 0

0,

ti' ., (z• , q.) = e11

.

-·

,

,

1 U • )" - V(I/>) ,

572 1 62 2

N. H . C H R I S T A N D T . D . L E E

where V(,P) is proportional to (g 2 q,

2

H

2

) ( 1 - g 2 q,2 )

2

,

( 5 . 1)

with a proportionality constant = ¼ (µ /g)'( l H 2 ) - 1 . The constant E is assumed to be real and positive. One sees that the potential function V( = ±(g- • + 0 ¢) , then

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3 +e 2 '(II l' . . . . V ( cp ) = , µ '( o l' + t , ¢ 2 "f+€Tgµ c/l +

Therefore, µ denotes the meson mass and g Is the dimensionless coupling constant. As e varies, the height of the lo cal minimum at ,j, = 0 changes. When e - "' , the local minimum disappears, and the potential V reduces to the special quartic func ­ tion considered before . By following the general discussions given in Sec. III, one finds that the field equation a•cp dV() _ 0 axµ* d¢ -

admits a time -independent classical solution c/l = g- 1a(x) where

state. The interesting feature is that (for arb i ­ trary e finite and nonzero) there is 110 single-soli­ ton state. The state � = 1 has an infinite energy. In accordance with our general discussion, IJ!.' s are C -number functions which satisfy the " Schro ­ dlnge r" equation ( 3 . 12) d2 d2 V(a) ] ,i,, (x) = w, ',P,(x) . [ - dx' +

(5.5)

tia'

As will be shown later, this equation can be trans­ formed Into the Heun' s equation . 19 The lowest frequency of ( 5 . 5 ) Is and its eigenfunction is d ,p, (x) = ; .

(5.6)

Because of the orthogonality condition (3. 1 1 ) , the sum ( 5 . 3) extends over all eigenfunctions of (5.5) except n = 1 . The energy spectrum of the system i s given by (assuming the total momentum = O) ( 5 . 7)

a(x) = [1 +E _, + sinh2 (½µ x)J - ' 1 ' sinh(½ µ x) .

(5 . 2)

,P (x, t) = g- 'a ( x - Z ) + q. ( t ) ,P,(x - Z ) ,

( 5 . 3)

A typical shape of o Is given In Fig. 1. The cor ­ responding quantum -mechanical operator cp(x, I) is then given by

where, as In (3.9), the repeated index n Is summed over from n = 2 to "' . The ,p/ s again satisfy both the orthonormal condition ( 3 . 10) and the orthogo ­ nality relation ( 3 . 1 1 ) . It is convenient to define the soliton -number operator !J! e g( q,(«> ) - ( - «> ) ] .

12

(5 . 4 )

The solution ( 5 . 2) corresponds to m = 2. As can be seen from Fig . 1 , It consists of two k inks , and therefore may be regarded as a two -soliton bound

where the constant m denotes the mass of the two­ soliton ground state and the quantum number N. = 0 , 1, 2 , . . . . If we negl e ct the radiative correc ­ tion, then m = !. -!;- [1 - ½ e ' 4 g

l 2 + ( l H ')l/ 2 ( 1 + ,€ ) ln 2£

2

1 + (1

+ € 2) 1 / 2 ]

E

(5.8)

A s e - 0, m - ¼ . ' vanishes by symmetry. However, we do not us e this fact. Lorentz invariance is shown by proving that H, P, and L form the Poincare algebra,

J

{H, P } = O,

{ L , p } = H,

( 17)

(18a)

(18b)

{ L , H } = P.

The relations ( 1 8a) and ( 1 8b) are trivial. Equation (18c) follows after a somewhat lengthy calculation.

(1 8c )

ill. QUANTIZATION

We now proceed t o quantize the theory by postulating the usual correspondence between c lassical brack­ ets and commutators . The only nonvanishing ones are i [ rr(x, 1), x (y , l)] = � (x - y) - � . ' (x)q,, ' (y ) ,

i [ p, X] = 1 .

(19)

0

We must also order the noncommuting factors in t h e canonical transformation (2) a nd (3). W e take (x, I ) = 4> .(x - X) + x(x - X, I),

Il 0 (x, t) = n(x - X, t) - � [ 4>. ' (x - X) l �� ( + Mo P + 2 0

J

f 11x') l + :/Mo 4>, ' (x - x)] . xi 0

x ' •)+(p +

(20) (21) (2 2 )

581 CANO N I C A L Q U A N T I Z AT IO N · O F N O N L I N E A R W A V E S

12

1681

A stra ightforward calculation shows that, just as i n the c lassical case, Eqs . { 19 ) and {20) ensure that + and ll 0 satisfy conventional commutation relations . We now proceed to evaluate U." as a preliminary step in obtaining the quantum expression for T p • • De­ fining a a [p +

J

we have

a t ., l + � [P + M,,

IIX'L + V ' M,,

f

x ' 11] .

{23)

A e ½{a + a ' ) ,

1 , ] 1 l [ ,. 2 1 " i Do• = 1r • - M,, ll'c/>. , A + + 2A#l [ c/>c • A J+ - 2 M,, [11, 1 + (/ ] 4'o M,,

9?;7k

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+ �• { ½ [a", ,t,.'"] + ½ [ c/)0 '", a '" ] - 2 ia 1

i

+ 2 i /:;/( a ' +

t [ ,t,. ' , a J 1 +� M,,

i !b,v,, 1

[a, c/>.' ] } '

where [ , l+ denotes an anticommutator, and we reorderd all factors in the expression that results from squaring {22) so that a and a t are the rightmost or leftmost factor in each term. Evaluating the commuta­ tors we finally obtain

n0•c

x

, t ) = 1r " '..,. -X, t ) - 1 [ lf(x - X, t ) ,t,0 ' (x - X) , A l + -

Afo

x) d '" 1 ( �•(t + (/M,,'r' (tk ,t,, " - )

f

,•

+

X 'Po +

1 2A#l [ ,t,• '" (x - X) , A·1 1

4Ml

1 • - 4Ml

t, ""{x - X) ( l + l/Afoi2

1 d1 '" 4'o {" - X) • {l + (/Afo'f'

u•

(24)

In evaluating integrated expressions in the classical theory we shifted the variable of integration x - x +X. In the quantum case we can do this only If every factor under the integral sign commutes with X. It ls for this purpose that we brought n 0• in the form {24), the last three terms of which give quantum contributions absent in the classical expression. Using {24) and proceeding as before we obtain H=

J

T.. = M,, +

P = - f T,,. =P ,

1 A1 + 2 M,,

f3C, - 8M/ 1 J,t,. •,t,. " � ,

L = fxT00

• ½ [x, H]. + ( 1 + +

l

BM!

l

2

!.•

( 1 + (/M,,)'

A" )

J,

J

X c/>o • 11

Xc/>. 'c/>.' - 2 �

[A,

(25) (26)

fx c1>, 1 ] + fx:JC, - ski &"!&{, 11

•

(27)

Comparing with the classical expressions we note that apart from symmetrizations the last term in (25) and the last two terms in (27) are the new quantum contributions. The equations of motion now are x (x, t) = i [ H, x (x , t)]

(x ' {x, t ) - � tc/>.'{xl ) , A ] , = 11 (% , t ) + 2 � [i + � M,, .

i(x, t) c i [H , 11(%, t)]

(28a)

582 1 682

E . TOMBOULIS

•

1 [ 1 l+t/M

X = i[ll, X]=

2�

o

,A

]+

•

(28c)

By a slightly tedious calculation one can verify that E qs. (28), together with (21 ) and (22), are equivalent to the quantum equations of motion for the original variables + and n 0 • The presence of the last term in '28b), which is absent in the clas­ sical expression, is important in this. connection. Again it follows trivially that

12

L 1/1. (x) I/J:(y ) = O (x - y) - !, r/>.'(x) cp.'(y) . 0

•

Furthermore , there are two possible sets, {1/1:} ui and {ij.,; J , corresponding to "in" and "out" states as determined by the appropriate boundary con­ ditions for (3 1 ). We can work In terms of either of these two and in the following we shall not ex­ hibit the "in" or "out" label explicitly. The c om ­ mutation relations ( 1 9) are then satisfied by

i[p,H] = 0

( 33 )

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and i [ p , L] = H . The proof o f the relation i[H, L] =P , which establishes Lorentz invariance , is very lengthy and will not be reproduced here. The ex­ istence of the new quantum terms in H and L is necessary for obtaining the result. IV. PERTURBATION THEORY

The Hamiltonian (25) can be separated into a free and an interacting part (29a)

H = ll0 +H 1 ,

H0 = �+

f[ ½ir (x, t) + ½ x 2

12

(x, t ) + ½ x 2 (x , t ) U*(q,0 ) ] . (29b)

H0 is the Hamiltonian of a static particle of mass � plus that of the free meson field X· The quan­ tization of the free x Is performed in the standard fashion by expanding x (x , t ) = � Th � ''' [b• 1/!.(x)e -•w•• w ) + b: I/Jl(x )e 1w•• ] and 11 (x, t ) =

(3 2 )

(30a)

1

- bll/Jl (x )e 'w•'] ,

(34 ) which breaks the symmetry + - - + in the vacuum states +,, 2 ± m/>..112 • The classical solution Is =

m ,P0 (x - X) = i1Tz tanh m(x - X) ,

( 3 5)

with (x) between m and n ( "in " or "out") mesons . The no-meson to one-meson matrix element to lowest order is clearly given by (P' / w ( x)/P , k} = i 11

f e• . M?

(2 .7a) (2.7b)

(2.7c)

The field equations imply the constraint equation o 1 fi l + geab '11ilb = 0 .

(2.8)

If we now fix. the gauge by setting A 1 == O ,

(2.9)

we can solve the constraint (2.8) for A 0 : (2. 1 0) where G(x - y) is a Green function. In term of independent canonical variables the

587 E. Tomboulis. G. Woo / Scliton quantization

224 Hamiltonian then is

f dx (½ n0n0 + ! n 2 + } a 1 '\Jl03 1 '\Jl0 + ½o 1 o 1 cf> + U('\JI, )] - ½g2 f dx dy e0bn0(x) '\Jlix)G(x - y) ecdITc(y) '\J!Jy) .

H=

(2.1 1 )

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We now look for solitpn solutions to Hamilton's equations. The static equations will have a solution of the form cf>(x, f) :,: (x - q(t)) + x(x, t)'

(2.1 Sa) (2.15b)

where a(t) and q(t) are the collective coordinates corresponding to the charge and translation symmetries respectively. Introducing (2.15) into the Lagrangian (2.4), we find that the various momenta defined by - 3L

P=--

3q(t)'

_ aE

rr = -

ax'

(2.16)

satisfy the constraint equation

J dx7i €abDbc(o:),i,/x - q) = 0, -J _ a • J _ a P - dxrraDab(o:) aq '¥ ix - q)- dx1T aq (x - q) = 0. Q-

a

(2.17)

By making use of these constraints we find that

H =Pq +Qa +

J dx (1r/ia + n:x)- L[D

ab,fr b

+ 11a ,Dab,i, b +

Tia, 4> + X. + X] (2.18)

where the last line in (2.18) denotes the same functional form as in (2.11 ), but with 110 and fl replaced by rr0 and rr respectively. At this point it is convenient to intro­ duce the row-notation M; = (o:(t), q(t)) ,

1'; = (Q, P) ,

(i = 1, 2) ,

589 226

E. Tomboulis, G. Woo/ Soliton quantization

Then (2.17) can be written compactly as (2.19) Assuming naive canonical commutation relations

i[fis(x, t), K/Y, t)] = 5sf>(x - y),

(2.20)

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eqs. (2.17) are first class constraints. We now impose the subsidiary conditions (2.21) We then have (2.22)

i["i• A;l = JJ.i;- 'Z.;;* 0, where

(2.23) The constraints " i = 0 become second class and the associated zero-frequency modes get eliminated. The canonical brackets (2.20) must now be modified according to Dirac's prescription {14] in order to have a consistent operator formalism. However, it is more convenient to quantize by means of Faddeev's functional transcription [14) of Dirac's procedure, although there are possible ordering ambiguities. The functional integral representation of the transition amplitude is then given by j{DqDPDoDQDf/i/)fri/)xDrr} Detl[Ki> A;JI,

IJ I

r, [K;]

n I

5 [A;] exp{i

fdt(Pq + Qa +f dx(1fa11a + nx)-H)}.

(2.24)

We now make the change of variables

-1- -1 n = � [J i' - µ .=. )i'i , .:I';

(2.25)

and go to the soliton-fixed coordinate system, (2.26a)

590 227

E. Tomboulis. G. Woo/ Soliton quantization 1r0

(x, t) = v;l(a)":;ix + q(t), t),

(2.26b)

x(x. t) = x(x + q(t), t) '

(2.26c)

rr(x, t) = 1r(x + q(t), t) .

(2.26d)

We then shift x ➔ x + q in all integrations over x and make the further change

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(2.27) We finally obtain

f

j{DqDPDaDQDT10D1rdDxD1r} DetlµI X 5[e ef 1re'j_,fl5 [e0 • b ,J 11a '�b '] x0

cf1rb01 �b +J1ra1Jo rf11d•01�d' +J x01�l

J

J

(2.28)

X exp {i dt(Qa + Pq + (1ra11a + 1rx) - Herr)} ,

with the effective Hamiltonian

J 1Ta'TlbJ2Ni} + ½ [P+J 1T T/ +f 1T31xl2N2l + [Q - €ab J 1T 'Tlb]NiJ [P+J 1Td'l11ld+ f 1T01Xl+ J dxg( •

Herr = Eo + ½ [Q - €ab Here

a0 1 a

a

Eo = ½

f [a1q,aal,i,a

+

0131cti]+ ju(,i,,ci>),

(2.29) (2.30)

-l -l -1- -1 Nij-1 = {J - µ -1.::.];;• µi'j' [J - µ. .::.]if'i •

(2.31)

(2.32)

with

na

= [Q- €cb J11' T/b] [I- µ-lzJ1/fµii e 'l1d- µii ° '11 ] + [P +J 11' 111c + f 1ra1xHJ - µ-I:::]i; [µ.ij ead,i,d _ µiia1,i,a], c

c0

1

1

ad

1

1

a

1

au . .

au . .

(2.33)

. .

• • X, Tl)= U('1! • + T/, + • , Vi(\fr, x) - 77 � ("1, ) - X aci> ("1, ) - U(w, ).

(2.34)

591 E. Tomboulis, G. Woo I Solito11 quantization

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22 8

Here, £0 is 0(1/g2) and is the classical energy of the static neutral solution. There is a term quadratic in the meson fields which is 0(1) and defines the free meson field Hamiltonian. This can be shown to have a positive-definite spectrum. All other terms are of higher order ing and are treated as interactions. In particular, note that these include terms linear in the quantum fields 1Taand 'Tia which are O(g). Since we decided to expand around a static neutral solution, in the lowest ap• proximation the Hamiltonian gives the classical rest mass of a neutral soliton. If however we had chosen to expand around a non-static charged solution, then in the lowest approximation the Hamiltonian would have given the corresponding c)assfoal energy which would have exhibited the usual relativistic dependence on momentum as well as a certain dependence on charge. Now it is well-known (3] that by starting from the effective Hamiltonian obtained by quantizing around a static solution, it is possible to obtain the relativistic momentum dependence by summing the tree graphs. Here we provide the corresponding demonstration for the charge dependence. In doing this we shall ignore for convenience all terms coming from tl1e transla• tion mode since they are kinematical. The sum of all tree diagrams is equal to the value of the action evaluated wifh the fields 1Ta , 1T, 110 , x being the solution of the equations of motion that follow from our effective Hamiltonian (2.29). Therefore, we have to show that these equations ac­ tually reproduce the equation satisfied by the time-dependent solutions (2.13). The equations of motion are (Q -

1Ta + Ea b1lb

ecdf 1fc4,d1

T(l -1 1Z)2

f.

-geac(1lc + '¥c)A

.

. .

geac1lc +---'¥j_17d+'¥d)A + + 1r�ak. wct + i'ak ..J,,)J Ii

X exp {i

...iill

cxak .. ?, + T1dJOk"Wd1 + a,ak ..

J

dt [Qa:

J

+ P'flik +

(1rq,X + 1r!11a1 + 1r!B1) - HeffJ} ,

where, if we define

1 -1 11.r-l = {/ - µ -1" "' [/ - µ- l':'J-1 ""ll .:. J.LYrr' - r'"r' ' r r

{3.23)

599 2 36

E. Tombou!is. G. Woo / So/iton quantization

we can write Heff as Heff = ½ [Q - Eab f-rr!z11b;]

J

-Vil

2

+ ½ [Pk + (1T3kX + -rrfzak'11a; + 11';3,tB;)]Ni°A,k'+l X [Pk• +J (1T,ik•x + �Ok''lcq + rlak,Bj)]

J

f

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+ [Q - Eab 1r!flb;J.Ni.".):..-1 [Pk + (1T4i3kX+ 1T�3k'11dj + trfil,IJ1)J

f ½01{� + x) o,{$ + x) +f U(� + x) f + f ¼ (ai1 and cp, are slowly varying in space and

(c )

FIG. 1. (a) The two degenerate ground states for l'lectronic structure of poiyacetylene. (bl An Imperfec­ tion interpolatlng between the two ground states. (cl A chain w Ith two imperfections.

FIG. 2. A form of polymer with single-single-double bond pattern in the ground state.

987

603 VOLU M E

P H Y S I C A L R E V I E \'(' L E T T E R S

47, NUMBER 1 4

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time, i.e. , their gradients are « g ( q; / + cp, 2) 1h , we may conveniently calculate the change in the expected value of j µ = ij"y µ !J, in the no-partic le state by considering the Feynman graph of Fig. 3. Since the interaction ( 1) is chirally invariant, we may first suppose that only cp 1 c# 0 at a given point, and then express the result in a chirally symmetric form. We then need only do a very simple calculation for an effectively massive fermion to find

= ..!.. . µ • a • tan-• � . 211 'P,

( 2)

If the scalar fields do not propagate (they repre­ sent very massive particles) more complicated graphs need not be considered. If in the end we reach the soliton state by slow changes, we need only to evaluate ( 2) to find the fermion number charge on the soliton. It is im­ portant to remark that the resulting state will be a true eigenstate of the charge, not a superposi ­ tion o f states o f different charge (even though we only derived an expectation value). For this it is only necessary to note that there are no de­ generate states of different charge. In this the localized char ge on a soliton differs from, for instance, the "localized charges ." of ! on the top and bottom of an ammonia ion. Two general features of the result deserve com­ ment. First, the divergence 8 µi P vanishes identi­ cally, reflecting the conservation of fermion number. Second, the charge Q= fl' dx' = (2 rr) - • x �(tan- 1 2 -