Theory of Interaction of Elementary Particles at High Energies [1st ed.] 978-0-306-10899-0;978-1-4757-6824-4

Table of contents :
Front Matter ....Pages i-ix
On the Use of a Curved Momentum Space to Construct Nonlocal Quantum Field Theory (I. E. Tamm, V. B. Vologodskii)....Pages 1-20
Bilocal Formalism in Quantum Field Theory (N. A. Batalin, E. S. Fradkin)....Pages 21-46
Aspects of Axiomatic Quantum Field Theory and Current Algebra (R. E. Kallosh)....Pages 47-98
Theory of Peripheral Interactions at High Energies (I. M. Dremin, I. I. Roizen, D. S. Chernavskii)....Pages 99-143
Statistical Theory of Hadron Interaction at High Energies (I. N. Sisakyan, E. L. Feinberg, D. S. Chernavskii)....Pages 145-221
Spontaneous Breaking of Translational Invariance in Quantum Electrodynamics (E. S. Fradkin, A. E. Shabad)....Pages 223-244
Gases Consisting of Ultracold Neutrons and Possibilities of Investigating Fundamental Properties of the Neutron (A. V. Antonov, B. V. Granatkin, A. I. Isakov, M. V. Kazarnovskii, Yu. A. Merkul’ev, V. E. Solodilov)....Pages 245-258

Citation preview

The Lebedev Physics Institute Series

D. V. Skobel'syn Hrs.

Theory of Interaction of Elementary Particles at High Energies

THEORY OF INTERACTION OF ELEMENTARY PARTICLES AT HIGH ENERGIES

TEORIYA VZAIMODEISTVIYA ELEMENTARNYKH CHASTITS PRI VYSOKIKH ENERGIYAKH

TEOPIUI B3Al1MO,[I,EHCTBHH 3JIEMEHTAPHbiX qACTHU: IIPH BbiCOKHX 3HEPr:HHX

The Lebedev Physics Institute Series Editor: Academician D. V. Skobel'tsyn Director, P. N. Lebedev Physics Institute, Academy of Seiences of the USSR Volume 25 Volume 26 Volume 27 Volume 28 Volume 29 Volume 30 Volume 31 Volume 32 Volume 33 Volume 34 Volume 35 Volume 36 Volume 37 Volume 38 Volume 39 Volume 40 Volume 41 Volume 42 Volume 43 Volume 44 Volume 45 Volume 46 Volume 47 Volume 48 Volume 49 Volume 50 Volume 51 Volume 52 Volume 53 Volume 54 Volume 55 Volume 56 Volume 57 Volume 58 Volume 59 Volume 60 Volume 61 Volume 62 Volume 63 Volume 65 Volume fi7

Optical Methods of Investigating Solid Bodies Cosmic Rays Research in Molecular Spectroscopy Radio Telescopes Quantum Field Theory and Hydrodynamics Physical Optics Quantum Electronics in Lasersand Masers, Part 1 Plasma Physics Studies of Nuclear Reactions Photomesic and Photonuclear Processes Electronic and Vibrational Spectra of Molecules Photodisintegration of Nuclei in the Giant Resonance Region Electrical and Optical Properties of Semiconductors Wideband Cruciform Radio Telescope Research Optical Studies in Liquidsand Solids Experimental Physics: Methods and Apparatus The Nucleon Compton Effect at Low and Medium Energies Electronics in Experimental Physics Nonlinear Optics Nuclear Physics and Interaction of Particles with Matter Programming and Computer Techniques in Experimental Physics Cosmic Rays and Nuclear Interactions at High Energies Radio Astronomy: Instrumentsand Observations Surface Properties of Semiconductors and Dynamics of Ionic Crystals Quantum Electronics and Paramagnetic Resonance Electroluminescence Physics of Atomic Collisions Quantum Electronics in Lasersand Masers, Part 2 Studies in Nuclear Physics Photomesic and Photonuclear Reactions and Investigation Methods with Synchrotrons Optical Properties of Metals,and Intermolecular Interactions Physical Processes in Lasers Theory of Interaction of Elementary Particles at High Energies Investigations on Nonlinear Optics and Hyperacoustics Luminescence and Nonlinear Optics Spectroscopy of Laser Crystals with Ionic Structure Theory of Plasmas Stellar Atmospheres and Interplanetary Plasma: Techniques for Radioastronomkai Devices Nuclear Reactions and Interactions of Neutrons with Matter Stellarators Physical Investigations in Strong Magnetic Fields

In preparation Volume 64 Volume 66 Valurne 68 Valurne 69

Prirnary Cosmic Radiation Theory of Coherent Aceeieration of Particles and Emission of Relativistic Bunches Radiative Recombination in Semiconducting Crystals Nuclear Reactions and Accelerators of Charged Particles

Proceedings ( Trudy) of the P. N. Lebedev Physics Institute

Volume 57

THEORY OF INTERACTION OF ELEMENTARY PARTICLES AT HIGH ENERGIES Edited by Academician D. V. Skobel'tsyn

Director, P. N. Lebedev Physics Institute Academy of Seiences of the USSR, Moscow

Translated from Russian by Julian B. Barbour

SPRINGER SCIENCE+BUSINESS MEDIA. LLC

Library of Congress Cataloging in Publication Data Main entry under title: Theory of interaction of elementary particles a t high energies. (Proceedings (Trudy) of the P.N. Lebedev Physics Institute, v. 57) Translation of TeoriÎa vzaimode'îstviiâ elementarnykh chastits pri vysokikh energiiâkh. Includes bibliographical references. 1. Particles (Nuclear physics)-Addresses, essays, lectures. 2. Quantum field theory-Addresses, essays, lectures. I.' Skobel'fysn, Dmitru Vladimirovich, ed. II. Series: AkademiÎa nauk SSR. Fizicheskii institut. Proceedings, v. 57. QC1.A4114 voi. 57 [QC793.28] 530'.08s [539.7'21] 73-83900 ISBN 978-1-4757-6826-8 ISBN 978-1-4757-6824-4 (eBook) DOI 10.1007/978-1-4757-6824-4

The original Russian text was published by Nauka Press in Moscow in 1972 for the Academy of Sciences of the USSR as Volume 57 of the Proceedings of the P. N. Lebedev Physics Institute. This translation is published under an agreement with the Copyright Agency of the USSR (V AAP).

© 1974 Springer Science+Business Media New York Originally published by Plenum Publishing Corporation New York in 1974

A1l righ ts reserve d No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

PREFACE The investigations into the theory of particle interaction published in this volume have been completed in the Theoretical Department of the P. N. Lebedev Physics Institute in recent years. The first paper, by I. E. Tamm (deceased) and V. B. Vologodskii, is the first full exposition of the results achieved in a program proposed by Tamm several years ago- the elimination of the fundamental theoretical problems encm.mtered at high energies and very small space-time separations by the introduction of an elementary length into a model in which the momentum space has a variable curvature. This theory, which is a further development of attempts to quantize space and time, poses exceptionally difficult problems. The authors have succeeded in showing that many divergences encountered in the ordinary theory can be eliminated. However, the program is far from completed, and many questions are nowhere near resolution. It is therefore particu1arly desirable to have a comprehensive exposition of the fundamental ideas and the results already achieved in this demanding undertaking. The second paper, by I. A. Batalin and E. S. Fradkin, is a further development of the functional method in conventional field theory. Jnvestigations in past years have shown that the functional method can be very effective for attacking certain problems that are beyond the scope of perturbation theory. Batalin and Fradkin propose a new version of the functional formulation in which a closed solution for the generating functional (S matrix) is represented as a functional integral with respect to variables that depend on two space-time points (bilocal variables); this contrasts with the usual approach, in which the integration is with respect to "classical fields" that depend on a single four-dimensional point. The new formulation yields a system of functional equations for the Green' s functions and an eigentime formalism for Bose particles similar to the eigentime method for Fermi particles in the usual approach. The new approach is evidently better adapted to solving the most important problern of the dynamical approach - how to allow for polarization effects correctly. In the third paper, R. E. Kallosh investigates some fundamental aspects of one of the currently most important and productive directions of particle interaction theory- current algebra. It is by no means clear whether certain elements of the postulates on which current algebra is based can be reconciled with the general requirements of the theory; for example, is the usual practice of omitting the so-called Schwinger and quasilocal terms in the basic relations permissible? Qle of Kallosh' s results is to establish the conditions under which these compensate each other and to verify that these conditions are fulfilled in the special models generally employed. The next two papers are ccncerned with various aspects- multiple production, elastic scattering, etc.- of the theory of hadron interactions at high energies. They proceed from a common assumption: To a good approximation these interactions can be split into peripheral and central processes, with markedly different properties. Butthisdivision is by no means V

vi

PREFACE

entirely consistent, since - and this is one of the important results - processes that are peripheral as a whole contain central interactions as individual parts. In the first of these two papers, I. M. Dremin, I. I. Roizen, and D. S. Chernavskii give a systematic exposition of a series of investigations they have made during the last five years. They have developed a theory which is based on a rigorous equation of quantum field theorythe Bethe-Salpeter equation - and two specializing postulates: the cross section is asymptotically constant at high energies and there is no completely destructive interference between the two-meson and the multi-meson diagrams. This theory provides a natural means for distinguishing peripheral processes, guarantees the correct analytic properties of the elastic scattering amplitude, relates the properties of elastic and inelastic interactions, predicts the existence of the so-called fireballs with properlies in broad agreement with experimental data, etc. Several results invite experimental verüication with accelerators or cosmic rays. In particular, the authors concluded some years ago that true asymptotic behavior is established only at very high energies, ~1o 13 -1o 14 ev.

I. N. Sisakyan, E. L. Feinberg, and D. S. Chernavskii also review their investigations. In this case, we have a cruder interaction model- the statistical theory of central interactions, fireball decay, etc. They show that if the theory (or rather model) is used with circumspection it is capable of describing correctly a large nurober of effects and properties; for example, it gives the mean nurober of particles produced at high energies, the momentum distribution, the composition of the created particles, the production cross section as a function of the mass of the particles, large-angle elastic scattering, etc. Many of these properties were correctly predicted while others await experimental verification. But the approach also raises fundamental questions. In the sixth paper, E. S. Fradkin and A. E. Shabad find exact solutions of the nonlinear Maxwell equations with allowance for all radiation corrections when external sources are absent.

The ever-increasing flow of experimental data on high-energy interactions, the completion of the 76-GeV Serpukhov accelerator, and the expected commencement of investigations on accelerators of even higher energies in the near future - all these stimulate interest in theory. A consistent theory of high-energy interactions is long overdue. While special models do exist (even though inconsistent) for processes in which very few particles (between two and four) participate and are created, the main high-energy process -multiple production- has long remained clouded in mystery. Clearly, success can only be achieved by approaching the problern from two sides; that is, by developing rigorous fundamentals of the theory and simultaneously constructing semiphenomenological theories that can be compared with experimental data and cast light on the physical processes in specific mechanisms. The papers presented in this volume are tobe understood in this spirit. E. L. Feinberg

CONTENTS On the Use of a Curved Momentum Space to Construct Nonlocal Quantum Field Theory I. E. Tamm and V. B. Vologodskii . . . . . . . . . •. . . . . . . .

1

. . . . • •

4 6 8 11 11 20

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Modified Representation for the Generating Functional . . . . . . . . . • . . . . . . . • 2. System of Functional Equations for the Green's Functions in Terms of the Bilocal Variables Qij(X,y) . . . . . . . . . . . . . . . . . . . . . . . . 3. Eigentime Method for the Bosons (Mesons) . . . . . . . . . . . . . . . . . . • . . . . . Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . .

21 23

Introduction . • . . . . . . . . . . . . . . . . . . . . . . • . . . . . 1. Generalization of the Perturbation Theory Series for Matrix Elementstoa Curved Riemannian Space • • 2. Convergence and Unitarity ••..••. • . • • . . . . . . . 3. Alternative Method of Constructing the S Matrix . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices .· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . .

• . . • . .

• . . . . .

. . . . . .

. . . . . •

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . •

. . . . . .

. . . . . .

. . . . . . •. . . . .

. . . . . .

Bilocal Formalism in Quantum Field Theory N. A. Batalin and E. S. Fradkin

32 34 46

Aspects of Axiomatic Quantum Field Theory and Current Algebra R. E. Kallosh Introduction . . • . . . . . . . . . . . Chapter 1. Fundamental Axioms; Covariance of the T Product . . . . • • . . . . . . . • 1. Fundamental Axioms . . . . . . . . . . . . . . . . . • . . . . . . . . . . . • . . . . . . . • 2. Covariance of the T Product and the Principle of Minimal Singularity . . . . . . . • . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 3. Invariant Variables . . . . . . • . . . . . . . . . . . . . . • . . • . . • • • • . . • • . • . . Chapter 2. Equations of Axiomatic Field Theory . . . . . . . . . . . . . . . . . . . . . . . 1. Equations for the Lowest Matrix Elements . . . . . . . . . . . . . . • • . • . . . . . . 2. Choice of Independent Invariants in n-Point Functions • . . . . . . • • . . . . . . . 3. Equations for n-Point Functions in Difference and Integredifferential Forms . . . . . . . . • . . . . • . . . • . . . • . . . . . . . . • . . . • Chapter 3. Current Algebra and Dispersion Sum Rules . . . . . . . . . . . . • • . . . . . 1. Difficulties in the Derivation of Sum Rules in Current Algebra . . . . . • . . . . . . . . • . • . . . . . . . . . . . . . . . . . • . . . . . .

vii

47 51 51 54 56

58 58 63

64 67

67

viii

CONTENTS

2. Sum Rules in Current Algebra: Superconvergent Sum Rules 3. Axiomatic Relations in the Infinite-Momenturn Frame

. . . • . . . . . . . .

71

•••••••••••••••••••••••••••••••••••••••••••••••

. • . • .

75 77 78 80 83 85

. . . . ••. • . . . •

87 89 92

. ••. •. . . . . . .

92 93 93

. . . . . . . •

94 95

. . . . . . . . . • . • . . . . •. . . . . . . . . •. . •

99 101 105

. . . . . . . . . •

108

. . . . . . . •. • . •. . . . . •. . . . . . ••. . ••

110 114 117

. . . . •• . . . • . . . . . • . • . . . . . • . • . . . •

120 . 121 122

. ••. . . . . . • . . • •

128 134 136 139 140 141

Introduction • . . . . . . • . . • . • . . . . . . . . . • • • . • . . . . . . • . . . • . . • • • • • • • • . 1. History . . . . • . • . . . • . • . • • . • .. . . . • • . . • . . • . • . . . . . . . . • • • . • . • • • 2. Mean Multiplicity and Energy of the Particles • . . . . . . • • • . . • . • • . • • • . • • 3. Distribution over the Total and the Transverse Momenta . • • • • • • • • • . • • • • • 4. Composition of the Created Particles • . • • • • . • • • • • . . • • . • • • • • • • • . . • •

145 147 156 161 166

(p-co)

Chapter 4. Current Algebra in Axiomatic Field Theory . . . . . . . . . • . • . . . 1. The Axiomatic Amplitude and Its Divergence . . . . . • . . . . . . . . • . . . . 2. Condition for Curr€nt Algebra to Hold in Axiomatic Theory • . . . • • • • • 3. Current Algebra in Lehmann-Symanzik-Zimmermann Axiomatics . • • • 4. Partially Conserved Currents • . . . . . . . • . . • . . . . • . . . . • • • • . • . . 5. Conditions under Which the Algebra of Vector Currents Is Valid in Axiomatic Field Theory . . • • . . . . . . . . . . . . . • . . . • . . . . Conclusions . . . . . • . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . • . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . • . . • . • • . . . . . • • • . . • • . . • 1. Derivation of Equations of Motion for Current- Like (Quasilocal) Operators . . . . . . . . • . . . . . • . . . . . . . • . . . . ·. . . • . 2. Derivation of the Expression sii = f (s 12 s 13 s 23 s1i s 2 i s 3 i s1j s 2 i s 3i ) • • 3. Perturbation Theory for Six-Point Function . . . • . . . . . • . . . . . . . . . 4. Transformation of the Quasilocal Terms in LehmannSymanzik-Zimmermann Asymptotics . . • . • . . . . . . . • . . . . . . . . . . Literature C ited . . . . . . . . . . . . . . . . . . . . . . . • . . • . . . . . . . . . . • . . .

. . . . .

. . . . .

• . . . .

Theory of Peripheral Interactions at High Energies I. M. Dremin, I. I. Raizen, and D. S. Chernavskii Introduction . . . . . . . .. . . . . . . . . . . • . . . . . . . . . • . • . . • . . . 1. Bethe-Salpeter Equation • . . . . . . . . . . . . • • . . . • . . . . . . . . 2. Physical Meaning of the Equations and Fundamental Assumptions 3. General Properties of the Equation; Conditions Imposed on the Salutions and the Expression for the Total Cross Section . . . • 4. Structure of the Equation's Kerneland Estimate of the IVIain Parameters . . . . . . . . . . . . . . . . . . • . . . • . . . • . • . 5. Models . . . . . . . • . . . ; . . . . . . . . . . . . . . . . • . . . . . . . . • . 6. Pre-Asymptotic Behavior of the' Cross Section • . . . . • . . . . • . 7. Mean Multiplicity in Peripheral Collisions and the Nurober of Fireballs . . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . . . 8. Slope of the Vacuum Trajectory . . . . . . . . • . . . . . . . . . . . . . 9. Treatment of Real Processes and Criteria for Selecting Events • 10. Relationship between the Düferent Methods of Describing Inelastic Processes at High Energies . . . . • • . • . • . . . • . • . 11. Irreducihle Interaction Mechanism for s < s 0 • • • • • • • • • • • • • 12. Does a Pomeranchuk Pole Exist? . . . • . . . . . . • . • . . . . . . . . 13. Discussion of the Results • . . . . . . . . . . . . . • . . . • • . . . . . . . C onc Jusions . • . . . . . . . . . . • . . . . . . • . . • . . . . . • . • • . . . . . . . Literature Cited • . . . . . . . . • . . . . . . • . . . . • • • . . • . . . . • • . . •

• • • • • • • • • •

. . . .

• • . .

• . • .

. . . •

. . . •

. • . .

• . . .

. • . .

. . . •

Statistical Theory of Hadron Interaction at High Energies I. N. Sisakyan, E. L. Feinberg, and D. S. Chernavskii

CONTENTS

ix

5. 6.

Pair Production of Heavy Particles. The Problem of Quarks . Electromagnetic Production of Hadron Pairs and Universality of the Results of Section 5 . . • . . . • . . . . . . . • . . . • • . 7. Statistical Scattering Through large Angles. General Introduction . . . . . . . . . . . . . . . . . . . . . . . . • . . . . • . 8. Large-Angle Scattering. Statistical and Statistical Diffraction Scattering . . . . . . . . . . • . . • . . . • . . . . . . 9. Large-Angle Scattering. Conclusions and Comparison with Experiments . . . . . . . . . . . . . . . . . • . . . . . . • . . . . . . 10. Fundamental Questions in the Statistical Theory of Multiple Production . . . . . . . • • . . • . . . . . . • . . . . . . . . . . . • . 11. Time Irreversibility and Energy Dissipation . . . . . . . . . . . . 12. Ericson Fluctuations . . . . . . . . • . . . . . . . . . . . . . . . . . . • Conclusions . . . . . . . . . . . . . . . . • . . . . . . . . • . . . . . . • . • . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . Literature Cited . . . . • . • . . . . . . • • . • . • . . . • • • . . • . • . • . .

. . . •. . . •. .

170

. . . . . . •. . •

177

. . . . . . . . . .

179

. . . •. . . . . .

185

. . . . •. . . . .

19 5

. . . . . .

. . . • . •

2 04 205 213 217 218 219

. . . . . . ••. . . . . . . . . . . . . . . . . . . . . . . . •. . .

223 223 227

. . . • . .

. . . . • .

228 232 2 36 239 241 243

lntroduction . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . . 1. On the Behavior of a Neutron Gas in Traps, • . • . • . . . . . . . • • . . . • . . • • • 2. Reflection of Cold Neutrons from Mirrors . . . . . . . . . . . . . . . . . • • • . • . • 3. Possible Kinematic Arrangements of Systems for Slowing Down Neutrons • . . . . . . . . . . . . • . • . • . . . . . . . . . . . . . . . . . . • . . 4. Accumulation of Ultracold Neutrons in Traps ...••..••.•..••••..••• 5. Prospects for the Investigation of Some Fundamental Properties of the Neutron • . • • • • • . • • . • . . . • . . . . • . • . . . . . . . . . Literature Cited . . . . . . . . . • • . . . . . • • • • . • . . . . . . • • • • • • . . • . . • • • • .

245 246 249

. . . . . .

• • • • . .

. • . . . •

. . . . . •

. . . . . •

. • . • . .

• . . • . .

• . . • . •

Spontaneaus Breaking of Translational Invariance in Quantum Electrodynamics E. S. Fradkin and A. E. Shabad Introduction . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . 1. Statement of the Problem . . . . . . . . . • . . . . . . . . . . . . . 2. On a Class of Elementary Excitations . . . . . . . . . . . . . . . 3. Vacuum Field as an Electromagnetic Wave Polarized along the Light Cone . . . . . . . . . . . . . . . • . . • . . . . . . . . 4. Spectrum of Boson Excitations in the Vacuum Field . . . . . 5. Spectrum of an Electron in the Vacuum Fie ld . . . . . . . . . . 6. Vacuum Field in the Form of a Free Electromagnetic Wave Conclusions . . . . . . . . . . . • . . . • . . . . • . • • . . • . . . . . . . . Literature Cited . . . . . . . . . . . . . • • . • . • . • • . . . • . . • . • .

. . . . . •

. . . . • .

. . . . . • . . . . ••

. . . . •. •. . . . •

• . . . • •

. . • • • •

. . . . •. •. •• •.

Gases Consisting of Ultracold Neutrons and Possibilities of Investigating Fundamental Properties of the Neutron A. V. Antonov, B. V. Granatkin, A. I. lsakov, M. V. Kazarnovskii, Yu. A. Merkul'ev, and V. E. Solodilov

250 253 256 257

ON THE USE OF A CURVED MOMENTUMSPACE TO CONSTRUCf NONLOCAL QUANTUM FIELD THEORY I. E. Tamm and V. B. Vologodskii In troduction 1. Many would now agree that the development of physics has led to a situation in which certain of the fundamental physical principles on which our notions about the structure of matter are based must be changed. And it is probable that the change will be as radical as that which gave birth to the theory of relativity or quantum mechanics. One of the main reasons for thinking such a change is necessary is the difficulties posed by the divergences of the existing field theory. It is true that in many cases the difficulties can be avoided by applying the well-lmown renormalization procedure; but- even leaving aside the existence of unrenormalizable interactions - can one be satisfied by a theory in which divergences are endemic and can be eliminated only by means of an artificial device? A certain modification of the existing theory is also needed to answer one of the most important questions of elementary particle physics: how do the multitudinous species of elementary particles arise and what is the explanation of their fundamental properties? This question cannot even be posed in the majority of the existing dynamical theories. The most promising candidate at present in sight- Heisenberg's theory of the fundamental spinor field- is still very far from completion, largely because of divergence difficulties. At the present time there is hardly anyone who can say with conviction which principles of the existing theory must be modified and in which direction. The present paper is based on our ccnviction that the concepts relating to a particle's coordinate must be reviewed. In the spirit of modern physics, this concept must be directly related to the procedure for measuring the corresponding quantity. But the notion, taken from classical physics, and still employed today, that a particle's .coordinate can be measured with unlimited accuracy is clearly Contradieted by the fundamental facts• of high-energy physics. For suppose one attempts to measure a particle' s coordinate by letting it scatter test particles like photons or mesons; increasing the accuracy of such a measurement forces one to employ shorter wavelengths, i,e ., higher energies; but the scattering process is then necessarily accompanied by the creation of new particles, many of which are also unstable and give rise to further new particles when they decay. The obvious impossibility of distinguishing the particles involved in the primary scattering event and the numerous secondary particles separated from the original particle by some finite distance cannot but set a limit to the accuracy of the coordinate measurement.

Quantum theory is, of course, based on the uncertainty relations for Coordinates and momenta, which restriet the accuracy of a simultaneaus measurement of these quantities. In 1

I. E. TAMM A.i'ID V. B. VOLOGODSKII

2

view of what we have said, it seems reasonable to base the new field theory on uncertainty relations for the coordinate components, these relations imposing a restriction on the accuracy with which the coordinate alone can be measured. Accordingly, the components must now be regarded as operators that do not commute with one another. 2. In existing field theory, elementary particle interactions can be described equally well in the coordinate or the momentum space; the transition from one to the other is effected simply by a Fourier transformation of the corresponding quantities. If the coordinate components do not commute, it is clearly impossible, or at least extremely difficult, to use a Coordinate space. Now Snyder [1) showed as early as 1947 that noncommuting Coordinates correspond to momenta which can still be regarded as ordinary numbers* but which now form not a pseudo-Euclidean space, but a curved Riemannian space; they are defined as the operators of an infinitesimally small displacement with respect to the momenta, have the form xa =in Faß (B/8 pß) (here Faß is a function of the momentum), and obviously do not commute 'Nith one another. We may therefore base our modification of field theory on a purely geometrical principle, postulating a certain geometry of the momentum space and introducing all the parameters in the expressions for observables in accordance with this geometry. Putting it more precisely, we must construct a theory that is generally covariant under arbitrary transformations in momentum space. By inspecting the expression for the matrix element of a particular process in the momentum representation, one can form an idea of which are the quantities that must be modified. Such an expression contains the following structural factors: integrals over a momentum-spacevolume; o functions from the sum of the momenta in the given vertex; the Green's functions of the free particles; and, finally, functions corresponding to the free ends of the diagram. In introducing the curved momentum space, weshall modify only the first two of these quantities. In accordance with the requirement of general covariance, the volume element in the momentum space must undergo the transformation

where gaß is the metric tensor of the momentum space and d4p is the pseudo-Euclidean volume element. In addition, the usual definition of a vector sum based on the parallelogram law obviously ceases to hold in a curved momentum space. In Sec. 1 below we shall formulate a generaliied principle for adding momentum vectors that is compatible with general covariance and is needed to calculate the arguments of the 6 functions that occur in the conservation laws. Of course, our going over to covariant integration and generalizing the rule for adding momenta does not yet lead to generally covariant matrix elements; we must also generalize the propagator D F covariantly. In a rigorous treatment we should quantize the free field in a curved momentum space and find the propagator from the resulting scheme. In the present paper this will not be done. we therefore leave the propagator in its form in the usual theory and investigate the consequences of going over to covariant integration and generalizing the addition of momenta. 3. Some very simple arguments connected with the modification of the volume element give reason to hope that the transition to a curved momentum space will make it possible to eliminate the ultraviolet divergences inherent in ordinary field theory. Indeed, by choosing the metric in the momentum space tobe so strongly nonpseudo-Euclidean that det Jgcxa I is a

* The

physical reason for this is that nothing as yet indicates the existenc~ of fundamental restrictions on the accuracy with which a particle•s momentum can be measured.

I. E T AMM AND V B VO LOGODSKII o

0

3

0

rapidly decreasing function of p2, one can always ensure that integrals with respect to p2 converge Thus, the new field theory Ieads naturally to a cutoff factor that has a purely geometric origin 0

0

In reality, however, this is by no means sufficient to guarantee convergence of complete four-dimensional integrals with respect to the momentum The pseudo-Euclidean volume element d 4p can be represented (if we omit an unimportant coefficient) as the product p 2dp2d4Q, where d4Q = d3S1 sh2 ~ d~ is the surface element of a unit hyperboloid and ~ is a hyperbolic angle The surface of this hyperboloid is itself infinite, and the convergence of integrals with respect to ~ requires special verification In the existing field theory, the appearance of such integrals can be avoided, for one can always go over to integration with respect to the fourth momentum component p0 to the imaginary axis, transforming the pseudo-Euclidean space into a Euclidean space and the hyperboloid into a sphere However, in a nonlocal theory with cutoff factor, of which our theory is a special case, one cannot, as a rule, make the transition to the imaginary axis because of the unusual analytic properties of the integrand in the complex p0 plane; ultimately, this is because of the violation of microcausality cne is therefore confronted with the problern of the ~ divergences- the so-called angle divergences In fact, these divergences are one of the principal shortcomings of Snyder's original theory 0

0

0

0

0

0

0

In the earlier papers on field theory in a curved momentum space (1-3] a space of constant curvature was usedo Apart from considerations of simplicity, this choice of the metric was evidently dictated by the desire for the momentum space to be homogeneaus But unlike the similar requirement for the coordinate space, this requirement is clearly superfluousthere is no translational invariance in the momentum space Jndeed, a particle' s mass changes under translation is a very complicated manner, which depends moreover on the direction and magnitude of the velocity In the present paper, we do not consider a momentum space of constant curvature because det lg cxal has then singularities for real values of the momentum, which leads to further difficulties 0

0

0

0

In Sec 1 the ordinary perturbation theory series is generalized to a curved momentum space In particular, we postulate a way of adding momenta in the argument of the ö functions that correspond to vertices of Feynman diagrams In Sec 2 we show that the direct generalization to a curved momentum space in Sec 1 does not lead to success because of divergences (angle divergences) in the matrix elements and the violation of unitarity In Sec 3 we give another method of constructing the S matrix in a curved momentum space; in this case the S matrix is expressed as an exponential function ordered antichronologically with respect to the charge [4,5]o Thls ensures automatically that the S matrix is unitary and, evidently, divergence free. However, there is a difficulty here too: to preserve relativistic invariance the construction of the S matrix uses the identity e (x) [.:l (x),.:l (0) L = e (x)e [(x + -r) 2 l [.:l (x), .:e (O)L [where 5e (x) is the Lagrangian and T is an arbitrary vector in the forward light cone] and this leads to Violation of the correspondence principle according to which one can make the passage to the limit ofa local theory This difficulty is also discussed in Appendix m. The main formulas for the volume element and addition of momenta in a curved momentum space are obtained in Appendix I. In Appendix II we find some restrictions on the form of the metric tensor; these are derived from the following requirements: absence of singularities in the volume element, existence of a sum for any pair of momenta, finite value of this sum if the added momenta are finite, and the absence of divergences in the si mplest matrix elements * 0

0

0

0

0

0

0

0

0

In this paper we use a system of units in wb.ich c

= ti

= 1.

* The

last condition is formulated for the usual perturbation theory series considered in Sees 3 and 4 for time-like external momenta. However, the results of Appendix II can, by and I arge, also be applied to the form of the theory considered in Sec 3 0

0

0

4 1.

I. E. TAMM AND V. B. VOLOGODSKII

Generalization of the Perturbation Theory Series

for Matrix Elements to a Curved Riemannian Space In a local theory the S matrix can be expressed in a variety of forms: as a perturbation series in the coupling constants, the terms of the series corresponding to different Feynman diagrams; as a functional integral (see, for example, [6]); in operatorform [71; as a so-called exponential function [81 in the form proposed in [21. etc. Allthese expressions for the S matrix are identical. However, when we come to generaUze the ordinary local theory to a curved momentum space, the form of the local theory with which we begin is not immaterial; for the different forms give different, inequivalent theories in the curved momentum space.

Tg

In this section we shall proceed from the perturbation series in momentum space. The generalization to a curved momentum space can be achieved naturally by introducing the following postulates [9]. First, the momentum space' s metric tensor must, as is clear from the requirement of relativistic invariance, have the form ga.ß (x) =

go.:r.ßJ (x) + p"pßh (x),

(1)

in which the nonvanishing components of the tensor gO • o: B are gO ,oo = -g0 •11 = -gO ,22 = -gO .~3 = -1; 0 0: B N ~ x = g o:a p p ; and f and h are certain functions. It follows from dimensional considerations that

J (x) =I ( A~' ) , where M, which has the dimensions of a mass, characterizes the departure from a pseudoEuclidean metric. The value of M must be appreciably greater than the elementary-particle masses, for otherwise the theory would contradict experimental facts. To preserve the correspondence with the ordinary theory it is necessary that

It is shown in Appendix II that the requirements listed in the introduction mean that f and h must also satisfy the conditions

, the function that defines a sum of momenta, or on the modified propagator D F * (or on the form factor F which can be associated with the vertices of diagrams). 2.

Convergence and Unitarity

Although we have been guided by simple geometrical considerations in generalizing the expression for a matrix element to a curved momentum space, success does not come so easily. Difficulties arise from the existence of angle divergences and violation of the S matrix's unitarity. Let us first consider the problern of the angle divergences. t In the introduction we have already pointed out the origin of this difficulty; in this section, we want to discuss the question in more detail. Take the expression for a scalar particle's self-energy (Fig. 2); we integrate with respect to all the virtual momenta except one. Relativistic in variance enables us to write the result in the form

where f decreases sufficiently rapidly as its arguments increase. In this integral we separate out the region of the forwardlight cone, q2 > 0, q0 > 0, and introduce the parametrization q0 = (q2)t/2ch1f!, lql = (q2)t,h shl/1. Then 00

!Vfdp) =

1

00

Sq2dq 2 Sd'ljJ sh 'ljJ S d eos 8/ (p 2, q2, rqa(p 2

0

0

0

eh 'ljJ -I p Ish 'ljJ eos 8)),

-1

where Mt is the contribution to M from the integration over the forward light cone. If p is a time-like vector, we can verify that there are no angle divergences by going over to a frame of reference in which p = 0; for the function f decreases sufficiently rapidly with increasing hyperbolic angle l/1. The situation is different for a space-like vector p: the domain of integration always contains subdomains of infinite volume within which cos e - 0 as 1f - oo, and the invariants q2 and pq vary little, so that f is almost constant. The integral over these domains in momentum space is therefore infinite. If the integral is to converge for integration with respect to the angular variables over any domain, it is necessary that the matrix element depend on at least one time-like momentum or three noncoplanar space-like vectors, p, r, s. It is clear that divergences can be eliminated in the second case: in a coordinate system in which the components p0 , Pt• p2 , r 2 of the external momenta vanish, the invariants qp, qr, qs can be written (for q2 > 0, q0 > O) in the form pq = - v'?"sh 'ljJ cos 8p3 ,

rq =Reh 'ljJro- Rsh 'ljJ(sin 8eo.s(j). r1 sq

= ffch 'ljJso - R

sh 'ljJ (sin 8 cos (j)S1

+ eos 8. r 3 ),

+ sin esin (j)S2 + eos 8sa),

and since lf! - oo , eh lf! ~ sh lj! - oo as lf! - oo , it is clear that at least one of these in variants tends to infinity in the same Iimit; for it is impossible to make the three quantities -cos ep0 , ro- Sin 8 COS 0; the geodesics have the form shown in Fig. 8 and do not intersec~. There is another reason why a metric tensor with C (1- a) > 2 must be excluded; for if it does, the sum of certain momenta may be infinitely large, for the same reasons as f (x) cannot decrease. We can see this as follows. If p and q in this case lie in the forward light cone and ~ » 1, then

:J (oo, p 2 , s)

+ :J (oc, q s) < 2 In (s + rs 2,

2

t),

from which it is clear that Eq. (I.7) has a solution (for ~) when x = oo. It is readily seen that this last argument can be applied unmodified to any metric tensor for which C (1 - a) > 2 [and not only the metric tensor defined by (II.3) and (ll.4)). Examining the signs of zp - zq_ and zp' - z'q and also the positions of the points at which dz/ dx = oo , we can show that when 1 < C (1- a) < 2 a sum is defined for any pair of momenta [provided, as we have assumed throughout, they satisfy (pq) 2 - p 2q2 > 0). Suppose, for example, p and q lie in the forwardlight cone, p2 = q2 and ~ » max{p 2 , 1/p2}. A sum's existence in this case is obvious because Eq. (I. 7) can be satisfied; for if x = p 2

while as x -

oo

If (pq) 2 - p 2q 2 < 0 the treatment is simpler and the results are unaffected. True, if

f

(x)

behaves asymptoticall y as lxl, the situation is complicated by the geodesics' having the form of spirals (Fig. 9), i.e ., they intersect innumerably many tim es. To find which of the points

Fig. 9

I. E. TAMM AND V. B. VOLOGODSKII

18

of interseetion should be regarded as the sum, we must make an adiabatie transition to flat spaee and in voke the eorrespondenee prineiple. This poses teehnieal (but not fundamental) problems. Finally, the polarization operator's not having angle divergenees in the case of a timelike external momentum imposes some restrietions on the metrie tensor (this faet has already been used to a certain extent). In general, the restrietions depend on the ehoice of the model and the order in which the momenta are summed in the 6 function's argument. As an example, let us consider a sealar-meson field's interaetion with spinless nueleons, ehoosing adefinite order to sum the momenta:

+ •e) y det I g ~ß (p) I y det I g ~ß (q) I d4pd4q-

rr (k) ~ i (' ö[(pEBkl eqJ ö f(pEBk~l eqJ ,, (p 2

_

i ('

~

ö [(pe:;>k)

(p'- m2

-

m'

+ •e) (q' -

e (pEBk'll

+ ie) [(pEflk)2- m2 + iej

m'

V det I g,ß (PlI d•p

yd

( ) I d 4 ~ .ö (k _ k') (' et I go:ß P P l J (p 2

m'

-

+ ie)[(p€:)k)2-

m'

+ iej

'

i.e., the polarization operator's real part is (II. 5 )

(' Ö(p2 -m2 )fdetlgo:ß(PJid4 p

(pEBk)2- m'

ReiT (k) ~ ~

In a hyperbolie eoordinate system, d 4p ..., p 2dp 2sh2l/l dl/1, where eh l/1

= (pk) / (p 2k 2) 1h, eon vergenee

with respeet to l/1 entails (p EB growing faster than (eh for large lj! [in flat spaee (p + k)2..., eh l/1 ]. To see what restrietions this imposes on the metric tensor, let us find x = (p ~ k) 2 as a function of ~=eh 1/J forlarge ~. Ifthe above conditions are satisfied, f(p 2)x/p 2 ( ~ 2 -1 )f (x) can be negleeted ceropared with unity in the radicand of the expression for :J ; that is, approximately k) 2

r J

X

1/! ) 2

V( +/Ti)) 1

c

dx

xh (x) · (

1

I

(p 2) x

+ p' (f.'- 1) I (x)

)

~ .)

x

V

dx

1

+

xh (x)

I

(x)

For suppose p and k lie in the forward light c one; this approximation in (I. 7) for large therefore large x) then yields Ct In

X

P'

+ C1

In f(f X

-

-

2 J n ~," ?t

~

(and

(II.6)

i.e.,

At the sametime lxl /(~ 2 -1)f (x)..., lxl 1-o:/ ~ 2 "'~c (t-o:)- 2 « 1, since C (1- a) < 2 by virtue of theconditionobtained earlier. Thus, our above simplifieation is justified. Obviously this is also true when a = 1. It can be seen from (II .6) that (p EB k) 2 grows. faster than (eh 1/!) 2 for large l/1 only if C > 2. Ill. The Correspondence Principle for the Simplest Matrix Element We obtain the expression (10) for the matrix element eorresponding to the diagram in Fig. 4. we shall not use the general formula (3) but a slightly different method that is equivalent for this matrix element. We write down the identity* T;t (x) :i (y)

=

8 (x- Y) [:i (x), :i (y)L

+ :i (y) :i (x)

=

8 [(x- y

+ llJ

8 (x- y) [:f, (x), ;t (y)]_

+ ;t (y) ;t (x),

(lll.1) where T denotes ehronologieal ordering; [ ... ] is the eommutator; and T, an arbitrary timelike vector, will later be allowed to tend to zero. As the term :i (y):i (x) makes no eontribu* Instead of (Ill.1) we may use the identity T 2 (x) 2 (y) = 1/2\) [(x- y

+ T)2] sgn (xo- !JO) [2 (x), 2

(y)j_

+ 11z {2 (x), 2

(y)l+.

I. E. TAMM AND V. B. VOLOGODSKII

19

tion to the matrix element, we omit it and go over to the momentum space, obtaining the S matrix in the second perturbation order: S

2

I'

J exp

=

( .

- ~ p-r:)

2(p\.Z'(-p\-Z(-p\ (p' iepo)'

+

d4

where :f (p) is the Fourier transform of the Lagrangian, and p 2

p,

= (p0) 2 - p 2•

This expression's matrix elements are generalized to a curved momentum space in the same way as in Sec. 1. However, to investigate the passage to the limit of a local theory, we shall consider a simplified model, replacing certain of the generalized sums of momenta by simple sums. Summing the momenta in adefinite order and taking, for example, a threeboson interaction, we obtain the desired matrix element:

For simplicity, let us consider the case P2 =Pt. Jntegrating with respect to ,P. we can take the remaining ö function in front of the integral. N oting also that (det Jg a 8 (q) J)112 = const ""' 1 [because of ö (q2- m 2) in the integrand] and neglecting small terms (T - 0) in the exponential flmction, we obtain formula (10). To establish whether a passage can be made to the limit of a local theory in the expression (10), it is important to know whether (10) has additional poles on the real axis. Using the explicit expression (1.13) for the sum of the momenta, when the metric tensor has the very simple form (1.10), and ignoring small terms of the form af (pi, q2) (the function f depends on q only through q2) we obtain !(q

where k

= p1

e P1l- P2F=q

2 -

2qk + P -

2 (a.lplql)b(qk + qpl) + 2 (a.lplq ll 2 b P1q,

+ p 2 • We introduce a coordinate system in which

k= {k,O,O,O},

p 1 ={p~,pLO,O}, q={q 0 ,

Noting that the integrand in (10) contains ö (q2 ( (q8pl)- Pil 2 = m2

+ qO p~- p~ V (q0)2 -

m2 cos 8)

-

2q 0 k

+ k2 -

+ 2 (a I qOp~ -

-

qcos8,

qsin8cos; (x) öQu~x. 1

4

y)

Q ii (x, y) q>i (y) } lo=o =

l

q> 1 (y) F ( Q) lo=o,

(56)

which holds for an arbitrary functional F(Q), we obtain from (54)* CZ lTJ, 11, f] = exp {i

+ i ~ d'xd

4

i- ~ (- i ÖQjj~X,

XJ

d4X

+ fl ( - i ÖÖQ) +

ylj (x) G+(x, y 1- i 0 ~) T] (y)) ~ (Dq;) exp {i ~ [-} q; 1 (x) (0-

r..2 -T- ie) x

66Q)q; 1 (x)]d'x}exp{i~d 4 xd 4 yq;;(x)Q;j(x,y)q; 1 (y)}IQ=o'

x q;1 (x)+Ji(xl-i

(57)

We represent the integral with respect to the cp field in (57) in the form exp

Jl

i \

62

.

,

J öQu (x, y) öQi X q> 1 (y)

1

(x, y)

d4 xd 4 y

d 4 xd'yjl. \ (Dq;) exp ! --; i ( q;i (x) Di/'(x, y I Q) X

J

J

l

+ i ~ J;(x [ Q') q>i (x) d x} lo·=o'

(58)

4

where

n-;J (x, y I Q) = (i; X I.D-

1

(Q) I j; y) = (- 0-+- X2 -

ie) ö(x- y)Ö;; - Q;j (x, y)- Qii (y, x).

(59)

Integrating with respect to cp, we obtain the following expression for the generating functional: CZ [TJ, 11• l]

=

exp

{i {-~ (- i ÖQiiö(x, x/ d x +TI (- t 660 ) +

+ i ~ d xd ylj (.r) G+ ( .r, y 1 4

+

+ ~ d xd yJ ( x 4

4

I

4

4

i

6 ~ ) 11 (y)j· exp {-

Q - i 0 ~,) D ( x, y 1 Q' - i

i Sp

ln n- 1 ( Q - i

6 ~,) +

0~ )1 (Y\ Q - i 6 ~,)} ex p {i {- ~ Q~1 (x, x) d x + n (Q) + 4

+ i ~ l1 (x) G+ (x, y I Q) 11 (y) d xd'y} lo=o· 4

(60)

Q'=O

*Here andin what follows, II (Q), G+(Q), etc., aretobe understood as functionals of the solutions of Eqs. (45)-(47) for arbitrary Qii (x,y). Accordingly, II (-io /oQ), G.Ji-io/ oQ), .6(-io/ oQ), etc ., are to be understood as the values of these quantities after the formal substitution Q -io/ oQ.

29

N. A. BATALIN AND E. S. FRADKIN

Hitherto, our arguments have consisted entirely of identical transformations. We now a make generalization by lifting the conditions Q = 0 and Q• = 0 in the expression (60). We then obtain a pair of generalized generating functionals :* CZ

(ri, Tj,J, Q, Q'] =

exp

+ i) d4u;ilyTj (x) G+ ( x, y + T~d4 xd4 yJ CZ(Yj, TJ,

+

( x I Q'- i

xexp

4

2

0~)

D

4X

(x, y I Q- i 0 ~,) J

ÖÖQ)

+

i 6 ~,)

+

Q)},

(y I Q'- i 66

(61)

Q,Q'J=exp{--}spinD- 1 (Q'-i 6 ~)+

J,

+ ~ d xd yJ ( x 4

1 -

li { ~ (- i ÖQjj~X, x)) d + n (- i i ööQ) TJ (y)} exp l- f Sp In n- 1 ( Q -

1

Q- i 6 ~,) D ( x, y I Q' - i ö~) J ( y \ Q- i 6 ~,)} X

{i i ~ Qjj(X, x)dtx + n (Q) + i ~ Tj"(x)G+ (x, y I Q)TJ (y) d4xd4y},

(62)

suchthat

cz ril. TJ, J, o, 01 = cz rfi",

l],

J,

o,

OJ

=

cz rfi",

l],

JJ,

(63)

i.e., the limiting values of the functionals (61) and (62) are equal to the generating functional of the ordinary formulation. This makes it possible to regard the modified representations (61)(62) as the basis of a new field-theory formulation in which the "dynamical variables" arenot the local fields l[i, 1/!, and cp associated with each point of four-space but certain new "dynamical variables" Qii (x,y) and Qb (x:,y) associated with a pair of four-dimensional points x and y (bilocal variables). An important property of the representations (61) is this: the local sources 7j (x), J (x), and 1J (x) occur :in them parametrically or, in other words, without the participation of operations of differentiation with respect to these sources. This makes possible an explicit transition to the limit of switched-off sources of the. nucleon or meson fields directly in the representations (61)-(62) or their derivatives with respect to the sources (Green•s functions). The greatest simplifications in this sense arise in the limiting case 1) (x) = 1J (x) = 0, i.e., when the nucleon sources are switched off. In this case the modified current J (x I Q) is equal to the external current J (x), and the variables Q• bec ome parametric; going over directly to the Iimit Q' = 0, we find that (61)- (62) yield

cz [0, 0, J, Q, Oj =

exp {i {

xexp {-+SpIn n- 1 (Q)

cz [0, 0, J, Q, Oj =

~ (- i ÖQjj~x,xY d x + n (- i ö~)}x 4

+ T~

exp {- fsp In n-1

x(x, y 1- i 6~) J (y)}exp

(64)

d4 xd4 yJ (x) D(x, y 1 Q)J (y)},

(-

i ö~)

)i {~ QJi(x,

+ T~d4xd4 yJ (x). Dx

x)d4 x

+ IT(Q)}.

(65)

*Note that from the formal point of view the expressions (61) and (62) for the generating ftmctional would be almost tmchanged if the theory were not CT 2-invariant and II (cp) were not an even function of the field. In this case, the procedure (52), which is directly applicable to (22), would lead to the appearance in the expression (55) for the modified current of an additional term due to the cp-odd part of II(cp), while the even part of II~) would replace II(Q) or II (-i6/6Q).

N. A. BATALIN AND E. S. FRADKIN

30

Comparing (61), (62), and (64), we see that when Ti ,TJ ~ 0 the representations (61) and (62) include the variables Q and Q', which come with, respectively, the parts of the Green' s function in the external field that are even and odd with respect to the meson field. If Ti = TJ = 0, the nucleon degrees of freedom enter only through ll (Q), which, as we have shown, is an even functional of the meson field and, therefore, the variables Q do not occur in (64) and (65). Further, we see directly from (64) and (65) that all the odd derivatives of these expressions with respect to J vanish. At the same time, any even derivative of, for example, (64), with respect to J is proportional to the derivative of two times lower order with respect to Q. This follows naturally from the construction of the representations (61) and (64); for using the identity (56), which induces the substitution (48), we in fact introduced the variables Qii (x,y) as the source of the product of the fields cp i (x) and cp i (y). This is manifested in the fact that, for example, the operation Ö' öJi(x)ÖJj(y)'

applied to (57) is equivalent (before the passage to the limit Q -

.

ö

L

ÖQii (x, y)

= 0)

to the operation

etc. This makes it possible to go to the limit J = 0 in (64) and (65) in order to describe the closed system of fields and regard (64), for example, as a functional of only the variables Qii (x,y). The Green's function of a system of 2n mesons is equal (to within a factor) to the Z [O,O,O,Q,O] with respect to Qii (x,y). n-th derivative of Z [Q)

=

Thus, setting J

=0

in (64), we obtain

CZ [ Q] = exp {i

i ~ (-

i ÖQii ~x, x)

r

d 4x

+ fi (- i ö~)} exp {-

+

Sp ln D-1 ( Q)

t.

(66)

We now introduce (Dij(x,y)), which when Qii(x,y) = 0 is equal to the single-meson Green's function of the usual formulation; we find 'D·(x "-

'1

)'=-iö'lnZ[O,o.J,Q,Oil

'

Y' -

öJi (x) ÖJj (Y)

_ölnZ[QJ ÖQii (x, y) '

J=O -

(67)

An important property of the modified generating functional (61) is that not only it, but also its = TJ = 0, J = 0, can be expressed as a and TJ taken at second derivative with respect to

Ti

Ti

normal operator form in the variables Qii (x,y) alone. The reason for this is that, in accordance with (55), the modified current J (x I Q) depends linearly on Ti and TJ, so that the expression J

(x I Q'- i 6~) D ( x, y I Q- i 6 ~,) J (y I Q'- i 6~) ,

in (61) is a functional of fourth degree in Ti and TJ in the absence of the external current J (x) and makes no contribution to the second derivative with respect to Ti and TJ when Tj = TJ = 0. Thus, for (G(x,y)) , which goes over into the single-nucleon Green•s function of the usual formulation when Qii (x,y) = 0, we obtain i (y) d4 xd~y

(80)

where

y)=(-0+x. 2 -ie)Ö(x-y)Ö;i-

D01 (x,yiQ, R)=(i, xiD- 1 (Q, R)lj, - (Q (x, y)

+ Q (y, x))Ö;j-

f;jlc

(Rk (x, y)- Rk (y, x)).

( 81)

Integrating with respect to cp in (80) and arguing as above, we obtain the modified pair of functionals: 1 . t. \ ( . ö ·)o 4 . CZ[O,O,J,Q]=expll"4.) -löQ(x,x) dx

xexp

1

( 1 . " l2 sp In D- (Q, R) + :T ~J (x) D (x, 1

CZ [0, 0, J, Q, Rl

+ T~ J (x) D ( x, y

1-

= exp {-

+

,

(

.

ö

. ö ) 1. ,

IT -löQ'-löRJX

y I Q·,

R)J (y) d4 xd4 y},

Sp ln D- 1 ( - i ööQ, - i 6~)

l ~~Q

i 6~, - i ö~) J (y) d4 :id~y ]- exp i

2

(x, x)

(82)

+

d~x + rr (Q,

R)} .

( 83)

Like (64) and (65), the functionals (82) and (83) can be used as a basis for formulating the theory; the only difference is that in ( 64) and ( 65) the quantities ~i (x,y) form a three-dimensional isotensor, whereas we introduce the sc alar Q(x ,y) and isovector Ri (x,y) in ( 82) and (83); this will have certain advantages when we come to formulate the eigentime method. 2.

System of Functional Equations for the Green's

Functions in Terms of the Bilocal Variables Qii(x,y) For the purposes of this section we shall find it convenient to use the expression (66) of Sec. 1 for the generating functional with I = 0; we represent it in the form CZ [ Q] = exp

tTI (- i ö~ ) ]· exp {-

; Sp ln

i; ~

Sp ln G~ 1 ( Q),

n-1 (Q)},

(1)

where

TI (Q) =

Qji (x, x) d1x

+;

(2)

N. A. BATALIN AND E. S. FRADKIN

33

and D- 1 (Q) is defined by (59) and G+(Q) is the solution of Eq. (45) of Sec. 1 for arbitrary Qij(x,y). Using(l) for (Dii(x,y)) defined by (67) of Sec. 1, we obtain* =

where we have the transformation

(4) by virtue of which ( 5)

in accordance with the symmetry relation Ö(D;j(x,y))

_

öQ ro (x', y')

-

_ Ö(Dr 8 (x',y')) öQ rs (x', y') öQ ii (x, y)

ö~lnZ[QJ

öQ ii (x,

y)

(6)

Further,

exp[TI(-i ö~ =

Q ii {x, y)

-i)]Qii(x,y)exp[-TI(-t

+ -} ÖuÖ (x -

Y) (- i) ( öQ ss ~x.

x)

ö~

-i(D>)]=

+ ( D ss (x, x))) (7)

In addition, an arbitrary functional F(Q) satisfies an identity that follows from (5): F

( 6~

+) m

•=1 0

r

as

~ + ~ Jr + (k: (;); x~ (~). x: (s)) r~ (~) dS] , •=1 0

(32)

where (33) (34)

(35)

The values of r +(k,x,y) and G+(x,y) on the virtual trajectories of the bosons (mesons) are obtained from the generating system of integral equations obtained by substituting (27) and (28) into Eq. (3). We can close the'generating system of equations because the kernels (27) and (28) are concentrated on the meson virtual trajectories; as a result, we obtain a system of integral equations on the set of trajectories of an integral equation infour-dimensional space: n

G+ (x, y)

+~

S=l

.:ts

g 2'A

JT~> (S) G (x- x, (~}) r 0

0

+ (k,

(~); x, m.y) dS

+

38

N. A. BATALIN AND E. S. FRADKIN n

Jr (~) r~> (k., (;);X- Xs (~)) G+ (x. (s). y) dS +

"•

+~

g 2A.

S=l

0

r

~.

+~

j' r~ (~)Go (x- x: (s)) r+ (k: (s); x: m. y) ds +

g A. 2

s=l

JT~

r

~.

+~

g 2 A.

•=1

r + (k;

0

m

(S}

r~> (k: (s); X -

0

x, Yl

n

g 2 A.

g 2 A.

S=l

X-

x. (s)) G+ (x. m. Y) ds

JT~(s) m r~> (k;

X-

0

+

x; (s)) r + (k; (s); x: (S), y) ds +

"s

r

Here

m no2 (k; k. (S),

"•

g 2 A.

•=1

S=l

sr~>

(~); x. m. Yl ds +

0

r

+~

+ (k.

0

(36)

Go (x- y),

"s

n

+~

Jr~> (s) r~> (k, x- x. (s)l r

"•

+~ -1

+~

x: (s)) G+ (x; (s), y) ds =

g 2 A.

Sr~ (s) r~2 (k; k; m, x- x: (s)) G+ (x; m. y) ds =

0

r~>(k;x-y)=~ (~f.

r~> (k; x- y).

(37)

( 3 S)

exp(i.p(x-y))i1,G 0 (p+k)rr,Go(P),

(O) k· . r_ ( ,x-y)= ('j (2d'p n)• exp(lp(x-y))G 0 (p)y,G 0 (p-k)y,,

no2 (k, k';

X-

y) =

~ (::~.

exp (ip (x- y)) Y;Go (p

( 39)

+ k) Y;Go (p) i;Go (p- k') Y;·

(40)

To obtain closed integral equations on the trajectories we must make the following Substitutions in the generating equations (36) and (37), respectively:

(41)

(42)

The simplest special case of this system of integral equations on trajectories is obtained when one considers the single-meson Green's function in the approximation obtained by neglecting the term -% SplnD- 1 (-i = =
0 and p Pi. > 0, we obtain lim qiqi'

•-+o

=

P;Pi' (1- ie) 3

=

PiP;· - iO,

i

> i' =

1, ... , k;

(1.48)

+ ie) (1- ie) =

lim q;qj = -PiPi (1

•-o

- P;Pi•

i =/= j.

Therefore, the on-shell r-function depends only on the scalar products PiPi', PjPj', pipi, is regular at the points PiPi• and is a generalized function of PiPi' and PjPj', Let us elucidate this result for a four-point function: r (p 1 , Pa! p 3 ) (P 1 P2! j (0) I P3) = r (mi- iO, m~- iO, m; + iO, P1P2- iO, - P1P3, - P2Pa),

=

r(p

1,

p 2, p 3 !)

= (Pl• P2• Pa! j (0) I 0) = r(m~- iO, m;- iO, m;- iO,

(1.49)

P1P2 - iO, P1Pa - iO, P2Pa - iü).

As long as we arenot concerned with the analytic properties of the r-functions, we can simply write

i.e ., assume that they depend on invariant scalar products. Let us now find the range of independent variation of the invariants for real values of the momenta. It is readily shown that the three four-momenta p1 , p.z, and p3 are related by the inequality (1.50)

For equal masses (m 1

= m 2 = m 3 = m) s3

+u

2

1

'

t2

2sut " _... 0 . ----;n2-m""""

(1 .51)

CHAPTERII

EQUATIONS OF AXIOMATIC FIELD THEORY §

1.

Equations for the Lowest Matrix Elements

We consider a self-interacting neutral scalar field with mass m and assume that there are no bound states. A formal solution is obtained for the r-functions in [22, 27] by proceeding from the equal-time commutation relations. It consists of a system of m equations for each function r(p 1 ,

...

Here

1 ... ,

Pm)= r(1, 2, ...

1 ... ,

m)

=rm,

r(1, ... , k! k

+ 1, ... , m) =

R; (1, ...

! ... , m) + ~\..;(1, ... 0; ...

1 ... ,

m).

R. E. KALLOSH

59

where 0.l denotes absence of the particle with momentum i if i = 1, ... ,kor RJ1, ... , k I k

+ 1' ... , m) =

i (2::t)-'"

sd~xe-iP;X

.

( 2 .2b)

if i = k + 1, ... , m; A stands for arbitrary Pi-independent functions. This is a formal solution, containing, as it does, undetermined and, in general, divergent Ai terms (or quasilocal terms) .*

In the Bogolyubov-Medvedev-Polivanov metbad [see (1.32)] a similar expression is obtained if the quasilocal Operators contain no derivatives of the ö function. From what follows we assume that the Splitting of (2.1) is covariant for the scalar field, i .e ., the equal-time commutator of the scalar currents does not contain gradient terms. In more realistic cases (electrodynamics [32, 34] and mesodynamics [30-31]) thisis not so, but the covariant method of eliminating quasilocal terms developed for a scalar field [1, 22, 29] has proved helpful in both electro- and mesodynamics . By virtue of the m different equations (2.1) existing for each r-function and these functions' covariant properties, the undetermined Ai terms can be eliminated from the equations. This was done for the firsttime in [22, 27] by formal differentiation of the system (21) with respect to the invariant variables for the example of simple r-functions (including a six-point function). Taking a five-point function as example, we shall show how the integredifferential equations and additional conditions are obtained in [22, 27]. The ftmction (2 .3)

depends on six independent invariants sii =PiPi =Ei Ei- PiPi. It corresponds to a transition of the particles p1 and p2 into 1>3, p 4, and p 5 , where p 5 = p1 + p 2 - P:! - p 4, and p~ ;: m 2 • Here, the four-momen turn p 5 is off-shell. This is necessary in order to formulate microcausality in the axiomatic approach. Using (2 .1), we can express r(12l34) in terms of four different functions Ri (considering which momentum is taken outside the brackets):

r(lZ/ 34)=

l

R 1 (12j34) + A 1 (2j34), R 2 (12J34) + A 2 (1/34), R 3 (12J34)+A 3 (12j4),

R, (12j34)-!-

(2 .1a)

A~ (12j3).

Differentiating each of the equations formally with respect to the invariants on which the corresponding A terms depend, we obtain six equations: or

oR;

.

-~- = -:;---- ' l = 1' vsii vsii

j

= 2, 3, 4;

i

= 2,

i

=!= j = 1' 2, 3, 4.

j = 3, 4;

i

= 3,

j = 4

(2.4)

and six additional conditions (of solvability): oR 1 OS;j

=

aR i OS;j

I

(2.5)

The arbitrary A functions can be eliminated not only by differentiating the original relations (2 .1) but also by successively subtracting with certain fixed invariants. Thus, one can obtain

* Indeed, it is

the quasilocal terms' being independent of a certain momentum (as a result of minimal singularity) that is the basis for eliminating them from the equations.

R.E.KALLOSH

60

a difference (or integral) form of expression of the equation. These equations are, of course, the solution of the corresponding integrodifferential equations in [22, 27} with allowance for the additional conditions and the range of variation (1,50) of the independent invariants. The equations for the lowest r-functions (for three-, four-, and five-point functions) have the most practical interest. Let us therefore consider their derivation in more detail; for a four-point function we shall write down the right-hand side explicitly in terms of other r-functions. For simplicity, we shall here consider identical particles. Three-Point Function. Ingeneral, there are three different r-functions: r(12[ 0)- r(P1P2I 0) r (0 [12)

_r (1[2) Since r+( 112)

= r(12 I),

= r (I P1P2) =

j (0) I 0),

(Ü I j (0) I P1Pz),

(2.6)

=r (Pli P2) = (Pli i (0) I Pz),

it is sufficient to analyze only the two functions r(12 r+(s)

=r(121),

r_(s)

=r(112)

I)

and r(1[2). We write

= r+(- s).

R7(s), Ri(1 I 2) = Ri(s) = Rt(-s), where s p 1p 2 is the only invariant on which the three-point function depends.

It can be seen* from (1.48) that Ri(12)

E1 E 2 -

= h

1

~ 1iT

s

n=l

dk!. .. dk,.. {

2w 1 ... Zw

n

r (p2! k1 · · · k..) r (kl · · · kn J Pa) X

+ Pl - pa) ) k + E1- Ea- ie + r (J Pa 1 ' .. k.,) X

Ö (2: k;

ö (2: k, - p1 - p1) -2: w. + E1 + E2- ie -

!; w.

'

'

( 2 .12) w1

=(k~ +

n

m 2 )'iz,

~ w 1 = ~ w 1, i=-1

n

L: k = ~ 1

k,.

i=l

Five-Point Diagram. We restriet ourselves to considering r (12J34) = (pJ.P 2 l j (0) I P3P 4 ) = r (sii),

i

3): * This conclusion was drawn independently of [65]; a real three-dimensional space was considered. An explicit expressionwas obtained for f (see Appendix II). But since it has a rather cumbersome form, we here use the determinant derivation of [65-67].

R. E. KALLOSH

64

1) k=1, 2, 3,

l=4, ... ,n-2,

2) l = 1' 2, 3,

k

= 1' 2, 3,

or

l = j;

l ,' k.

Let us compare this choice of the independent invariants with Asribekov's [67]. Asribekov considers only on-shell matrix elements, so there is one independent invariant less. He chooses not only double scalar products but also more complicated combinations of the fourmomenta as independent invariants. It is then possible to choose kinematically independent invariants symmetrically with respect to all the momenta. But, since they satisfy a geometric relationship, the absolutely independent invariants are still chosen asymmetrically. For example, in the case of a six-point function, one geometrical condition (s 12 , s 23 , s 34 , s 45 , s 56 , s 61 , s 123 = s 456 , s 234 = s 561 , s 345 = s 612 ) is imposed on the nine kinematically independent invariants, i.e., any one of these invariants can be expressed in terms of the remaining eight, and the symmetry is lost. If the independent invariants are chosen as in [22, 27], the kinematically independent invariants are symmetric with respect to the n- 1 momenta. This is simply all the double scalar products, but the geometric restriction reduces this symmetry. For example, in the case of a six-point function one can also write the geometric condition as a symmetric function of all nine invariants, but if one of them is expressed in terms . of the others, the symmetry is lost. This is a simpler way of choosing the independent invariants. Now there is a universal function that determines how the remaining invariants depend on the independent invariants for an n-point function. In Asribekov's method the geometrical condition is symmetric with respect to the invariants that are kinematically independent. If the equations could be written down for a wider range of variation of the invariants (where they are independent), a symmetric condition of this kind would, of course, be better; but then the nurober of invariants would increase with increasing n quadratically, and not linearly. Moreover, as n increases, the geometric conditions become ever more complicated. In Asribekov's paper they are written down only for a six-point diagram. But if, as is the case in the construction of these equations, one uses explicitly geometric relations, the symmetry is violated by both Asribekov's choice and the choice of double scalar products . Clearly, independent invariants (if they are double scalar products) cannot be chosen symmetrically. For example, in the case of a six-point function one must choose nine invariants from ten; perhaps one should not choose double scalar products, but Asribekov' s choice is fairly complicated, and it is extremely difficult to write down an analytic dependence in the case of an n-point function. Thus, at the cost of losing symmetry, we reduce the nurober of invariants on which the amplitude depends (we choose only independent invariants and, in addition, using the universal dependence of the invariance, we can write down the equation for an n-point function with arbitrary n). §

3.

Equations for n-Point Functions in Difference and

Integrodifferential Forms Dependent invariants first appear among the scalar products Sij =PiPi with a six-point function. Weshall explain the method for eliminating the Ai terms for a six-point function and then generalize this method to an arbitrary n-point function.

65

R. E. KALLOSH We write down the system (2.1) for one of the six-point functions: R 1 (123\45)+~d23j45),

(2.18a)

~ 2 (13145),

(2 .18b)

r(123145)= R 3 (123l45)+r.1 3 (12145),

(2.18c)

R 4 (123145) + •.1 4 (123\5),

(2 .18d)

+ 1 5 (12314);

(2 .18e)

R 2 (123145) +

R 5 (123\45)

The number of invariant products formed from the kinematically independent momenta Pi (i = 1, ... ,5) is ten. They are related by one relation of the type (2 .15). Let us take it, for example, in the form (2.15a) i.e., we assume s 45 tobe dependent. Then At, A 2 , and A3 depend implicitly on all nine invariants, so that a simple subtractional procedure for three-, four-, and five-point functions fails. The sequence in which the invariants are fixed and the elimination of the A terms is not arbitrary. As before, we choose the boundary value tobe r(123l45)Jsii =t = A. 6 . We firstfix s 45 = 1 and eliminate the A 4 term from (2 .18). Then, because of (1.50), s 14 = s 15 and s 24 = s 25 , s 34 = s 35 • We then set s 35 = 1, s 13 = s 15 , s 23 = s 25 , etc. We then obtain the equation

23 24 25 34 35 45

r(

\

.J.. I

3

= t.,

..J__ B

- R., .

(! 15

1:

R ('

23 24 25 34 35 45

4

'

12 13 15 15 23 25 25) _ 35 35 \ 1/ \

R(

12 13 15 15,

/12 13 14 15)

/12 13 14 15)

R

(12 15 15 15) 25 25 25 1 1

3

1: 1: .) + 11 1

23 25 25 ) 35 35 + 1)

_ R ( 4

+R

(12 15 15 15, 25 25 1

2

25) _ 1

1

1

R(15 1: 1: 1:) - R(1 : ~ :) 1

1

11 1

1

11. 1,

( 2 .19)

Let us consider the derivation of an equation for a six-point function in integrediffere ntial form. In cantrast to the case of a five-point function, the A terms cannot be eliminated by simply differentiating equations (2.18). We differentiate (2 .18a) with respect to s 14 and note that At depends on s 14 through the function f: ( 2 .20) Differentiatin g Eq. (2 .18d) with respect to s 14 , we obtain ( 2 .21)

R. E. KALLOSH

66

f. We can therefore find an expression

since A 4 depends on s 14 neither explicitly nor through for BA/ 'Os 45 : oA1

I

o (R4- R1) osu

=

as."

(2.22a)

fjf

OS14 .

Similarly, aAa as.s

o (R•- Rz) asu

1as24 at

'

8(R•-R3) as34

Ia~~

•

=

aAz as.s

(2.22b) (2.22c)

We can now write down nine equations for the six-point function in integrodifferential form and six additional conditions: ar aR. ---'+ as .. - as.. tJ

i = 1,

j =

tJ

at I at '-· -, as.}. as .•

8(R.-R.) as.. t...

2, 3, 4, 5;

i = 2,

i=1,

j=2,3;

\'A

1

j = 3, 4, 5;

i = 3,

j = 4, 5;

(2 .23)

aR.t asij i

= 2,

j

= 3.

(2 .24)

These equations and additional conditions are identical with those obtained in [22, 27] by variation of the equations. We generalize the derivation of the equations to an arbitrary n-point function. The system of equations has the form r (1. .. n -1) = Rd1 ... n- 1) r (1. .. n- 1)

+ :\d2 ... n -1),

R,._l(!. .. n- 1)

=

+ A,._t(l. .. n- 2).

1. Equations for an n-Point Function in Düference Form. We derive the equations (that is, eliminate the quasilocal terms) in the same way as for a six-point function. If the independent invariants are taken tobe (2 .16), one of the difference equations with boundary condition at the threshold has the form

12 13 ... (1,n-1) r

(

12 13 ... (1, n-1)

23 ... (2,n-1))

.·

.

= Ä.n

+ Rn-2 (

(n- 2, n- 1)

23 ... (2,n-1)l ..

.

(n- 2, n- 1)

(2.25) 1213 ... (1n-1)(~n-1)

- R n-• ( ·

.

)

· -+- ..• (n-3,n-l)(n-3,n-11'

/(1,n-1) ... (1,n-1)

+R

1

~

1... 1 •

: 1

)

11 ... 1,

-

R1

( 1... 1) .

: 1

·

R. E. KALLOSH

67

Here An = r for all Sij = 1. For an example of a difference equation for an n-point function with boundary value at an infinitely distant point of the independent invariants, see [28]. 2. Equations in Integrodifferential Form. As before, the independent invariants are chosen as in (2 .15). Consider A 1 , A 2 , and A 3 • All the invariants that are dependent in the r function occur in these three A terms. But since a A term depends on n- 2 momenta, there are several more relations between the invariants in the A terms. The number of dependent invariants in the A terms is [ (n-1)2(n -2) -

(3n- 9)]- [(n-2)t-3)- (3(n -1)- 9)] = n- 5.

Their number increases linearly with n. We choose them as follows: s4ö• Ssa• ••• , S(n-2) (n-1) (n-5 in all).

To obtain equations without A terms, we must eliminate quantities of the type BAi/ ds a (i = 1,2,3; a = 45, ... ,(n- 2)(n- 1) in the same way as for a six-point function. We then obtain 3n- 9 equations and six additional conditions:

where sii are all the independent invariants; Sa

{s45•

Sse• · · . , S(n-2) (n-1)} •

0 0 e:"i

a

osn-3. n-2

Os;,

n-2

a....

° °

9n-2.n-l

= asi4 •.•

08 n-2, n-1 •

09 ;, n-2

'

9 ;, n-2

is e:"i, but with the a column replaced by

a(Rn-2-Ri) asi.

t:.~ =Oifi~

n-2

1,2,3.

The additional conditions are o(Ri-Ri) asii

=

"" (~i6i - -;;-~j

'-.J

a

a

a )

as'a as ii '

j = n -1,

i = 1,

j = 2,3;

i = 2,

j

= 3;

i = 1, 2, 3.'

CHAPTER III

CURRENT ALGEBRA AND DISPERSION SUM RULES §

1.

Difficulties in the Derivation of Sum

Rules in Current Algebra Despite the great successes achieved in current algebra, there remain a number of obscurities in dispersion sum rules [30-35]; these concern the convergence of integrals in the

R. E. KALLOSH

68

dispersion sum rules, difficulties with the Schwingerterms in the equal-time commutators of the current, the problern of current algebra for higher spins and the choice of invariant amplitudes, the düference between current algebra and sum rules for strong interactions, the question of the subtractional functions in dispersion sum rules, etc. Let us consider the sum ru1e that was first obtained by Adler [48] in connection with the high-energy neutrino reactions. He used equal-time current commutators and infinite-momentum frame (p - oo). This relation was subsequently obtained by the covariant technique by Fubini, Furlan, and Rossetti (68] and Gell-Mann and Dasben [36]. This sum rule has been derived in a number of papers (49, 69, 70] for electroproduction. To demonstrate the difficulties posed by the derivation of these sum rules, let us reproduce the usual method by which they are obtained [53, 71, 72]. One postulates unsubtracted dispersion relations for a certain amplitude, and current algebra in fact gives information about this amplitude's realpartat the point v = 0. The upshot is dispersion sum rules. In the lowest order in the weak interaction, one considers the amplitude of a reaction of the type V + p- p + e + + V , in Which two lepton pairs partic ipate:

e

(3.1) where Xa = eya (1 + y 5)v, G = 10- 5/M 2 ; M, L, L', and N differ by the form of the interaction in the vertex parts. We are interested in the matrix element (2:t) I 04 (pl

+ ql- P2 -

q2) ü (p2) :li;:; (p2q2plql) u (pl)

=

i ~ d 4 xd'y exp (i ('

~

Im M 1 (v 1 , q2) d v'

1

V·

(3 .12)

Separating the contribution from the pole terms and invoking the optical theory, we obtain the electroproduction sum rule* * This transitionwill be discussed in more detail in the next section.

70

R. E. KALLOSH

(3 .13) where F 1 and F 2 are, respectively, the nucleon isovector electric and magnetic form factors, (.ll

a=

+ J.l)2- Jf2- q2 ,vJ

(3.14)

'

and aL+T is the total cross section of virtual photoproduction. We differentiate (3 .13) with respect to q 2 and set q2 = 0: = ( llp -lln ) 2 2m

where aF(q ) (r02) = o -~-"2

uq

I

q'=O

,

+ _2a•a: i - C~ (2o'i• ~ v vo

o'i•)

(3 .15)

'

FI(O) = 1,

This is the well-known C abibbo-Radicati sum rule [42]. Thus, the important assumptions in this derivation of the sum rules are: 1) the amplitude's divergence (3 .4) contains no terms of a polynomial in powers of Q except the zeroth; the electroproduction sum rules would be modified by any polynomial term, and the CabibboRadicati sum rule would be modified by the right-hand side of (3.4) taking on a term proportional to q2 ; 2) the assumption that Iim M 1 (vq 2 ) -l!v', where r- > 0, i.e ., unsubtracted dispersion relations hold for the amplitude M1 (v

,

q 2) in the energy for fixed q2 •

Note that violation of either of the conditions 1 or 2 could change the sum rule. For example, if (3 .4) is changed, i.e., the current algebra ceases to hold: (3

while the condition (3 .10) - the unsubtracted dispersion relations for M1(v, q2) we obtain the electroproduction sum rule

-

.4a)

is retained,

( 3 .16)

and similarly the Cabibbo-Radicati sum rule (3 .17)

Conversely, one could have (3.4) in the sameform but instead of (3.10) write down a oncesubtracted dispersion relation of the type (3 .10a)

where j(q2) is, in general, an arbitrary function of q2 • All that one can say about this function (see the proof of the dispersion relations in [12,20]) is that it is analytic in q2 for q2 < 0. The relation (3.10) is an example of a so-called subtraction at i:rifinity, i.e., the subtraction is not intended to improve the convergence of the imaginary part's integral, but it is as-

R. E. KALLOSH sociated with the behavior of Re M 1 (v, q 2) as persion sum rule is

11 -

oo •

71

In this case the electroproduction dis-

( 3 .18)

and the C abibbo-Radicati sum rule is ( 3 .19) If the conditions (3.4) and (3.10) do not hold but instead (3.4a) and (3.10a), the sum rules be-

come (3 .20)

_!.. (..k

k=O

i

=

( 4 .25)

0.

a sufficient condition for (4.24) to holdisthat (k-1)

ß

A ~.

+ -~2'

(k)

ß

I>

A~. i

( 4 .26)

R. E. KALLOSH

83

should be proportional to ~ ..i, where 1 ~ i ~ k. The condition for current algebra to hold in the axiomatics has a particularly simple form for forward scattering in the limit A !J - 0

=

o

(4 .27)

and entails the vanishing of the time component of the quasilocal term of each order. For nonforward scattering, the condition (4.24) relates the nonvanishing time component of the quasilocal term of (k - 1)-th order to the quasilocal term of k-th order. §

3.

Current Algebra in Lehmann-Symanzik- Zimmermann

Axiomatics The amplitude has the form

iVJ~f

=

5exp (iq x - iq y) d xd yKxK -,nv.s~~> {(A, a".A) +

j)}

+ rzv.S~,t {(alLA, j)- (j, a11A)} [ i).

(IV .4)

Besides (IV .2), we need new recursion relations:

iSi~i,{(a 11 A, j)- (j, a 11 A)} = ~11 S~t {(A, j) + (j, .4)} + { (g11 l., + d:>.. rzv.) xS~/{(.4, j) + (j, A)}

iS~,t {(A,

j)

+ (j,

A)}

= - -1 2 d_!!__dd

=

+(

nA, nA,

g11l., + d~,

X

rzv.n.s {(a".A, A) + (A, a

(IV .5)

11 A)}.

Since [see (IV .2)] s(~{(BJ.t A, A) + (A, 8J.L A)} = 0, so also s(~ >.. {(A, j)- (j, A)} l 2 nonvanishing terms in (IV .3) can be reduced to

= 0.

All the

(IV .6) 121

where A~f is given in (4.32). Let us consider the third-order quasilocal term: Ql.,Q)..,Q).., .,)..,l i) n>.,- 2S(O) (A, A.)

- n>.,Si~,t{(j, .4) + (.4, j)}

+ g)..,).,rzv.S~1; {(.4,

aw4)- (av.A, A)}-

+ n~'-S~!>.,>., {(a A, j)- (j, a~'-A)}

(IV. 7)

11

and use the recursion relation

iS~3!>.,>., {(a~'-A,

j)- (j, a 11 A)}

= -} (2g11>., + d:>.,

n 11)

S~~t {(.4,

j)

+ (j,

A)}.

(IV .8)

As before, using the recursion relations (IV .8), (IV .5), and (IV .2), we can express all the Schwingerterms of each commutator (S~,L•• S~2,L S~~) in terms of the coefficient of the ö function (S 0) of certain other commutators. The third-order quasilocal term becomes (IV .9) (3)

ß

where A~.

is given in (4.32).

Proceeding from (IV .3), (IV .6), and (IV .9) in the case of the Bogolyubov-MedvedevPolivanov axiomatic amplitude, we have written the quasilocal term in the general form

LITERATURE CITED 1. 2.

A. Wightman, Phys. Rev ., 101:860 (1956). R. Jost, Helv. Phys. Acta, 30:409 (1957).

96 3. 4. 5.

R. E. KALLOSH

G. Ludersand B. Zumino, Phys. Rev., 110:1450 (1958). R. Haag, Dan. Mat. Fys. Medd., 29:12 (1955). R. F. Streater and A. S. Wightman, PCT, Spin and Statistics andAll That, Benjamin, New York (1964). 6. R. Jost, The General Theory of Quantized Fields, AMS, Providence, Rhode Island (1965). 7. I. T. Todorov, Lectures at the International School at Dubna, Vol. 1 [in Russian], Izd-vo OIYai (Dubna) (1964), p. 5. 8. H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento, 1:205 (1955). 9. H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento, 6:319 (1957). 10. V. Glaser, H. Lehmann, and W. Zimmermann, Nuovo Cimento, 6:1122 (1957). 11. N. N. Bogolyubov, Izv. Akad. Nauk SSSR, Seriya Fiz., 19:137 (1955). 12. N. N. Bogolyubov, B. B. Medvedev, and M. K. Polivanov, Aspects of the Theory of Dispersion Relations [in Russian], Fizmatgiz, Moscow (1958). 13. N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields, Interscience (1959). 14. H. Lehmann, Nuovo Cimento, 11:343 (1954). 15. H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento, 3:425 (1955). 16. R. Jost and H. Lehmann, Nuovo Cimento, 5:1598 (1957). 17. F. Dyson, Phys. Rev ., 110:579 (1958). 18,19. F. Dyson, Phys. Rev., 110:1460 (1958). 20. H. Lehmann, Nuovo Cimento, 10:579 (1958); Suppl., N 1, 153 (1959). 21. F. Rohrlich, Phys. Rev ., 80:666 (1950). 22. V. Ya. Fainberg, Lectures at the International School at Dubna, Vol. 1 [in Russian], Izd.-vo OIYai (Dubna) (1964). 23. L. S. Brown, Phys.Rev., 150:1338 (1966). 24. B. S. Schroer and P. Stichel, Commun. Math. Phys ., 3:2158 (1966). 25. H. Tonin, Nuovo Cimento, 47, N 4, 919 (1967). 26. R.E. Kallosh, Preprint [in Russian], P. N. Lebedev Physics Institute, Moscow, No.111 (1967). 27. V. Ya. Fainberg, Zh. Eksp. Teor. Fiz., 47:2285 (1964). 28. R. E. Kallosh, Diplom. Thesis [in Russian], Moscow State University (1964). 29. R. E. Kallosh and V. Ya. Fainberg, Zh. Eksp. Teor. Fiz ., 47: 1611 (1965). 30. B. L. Voronov, Zh. Eksp. Teor. Fiz., 49:1802 (1965). 31. B. L. Voronov, Preprint No. 11 [in Russian], P. N. Lebedev Physics Institute (1966). 32. V. D. Skarzhinskii, Preprint No. 57 [in Russian], P. N. Lebedev Physics Institute (1966). 33. V. D. Skarzhinskii, Zh. Eksp. Teor. Fiz., 52, No. 4, 910 (1967). 34. V. D. Skarzhinskii, Candidate's Dissertation [in Russian], P. N. Lebedev Physics Institute (1967). 35. G. Furlan and C. Rossetti, Lectures given at the 1966 Balaton Meeting. 36. R. Dashen and H. Gell-Mann, 1966 Coral Gables Conference on Symmetry Principles at High Energy. 37. V. de Alfaro, S. Fubini, G. Furlan, and C. Rossetti, Torino Preprint, December, 1966. 38. L. D. Solov'ev, Talkat the International Conference on Low and Intermediate Energy Electromagnetic Interactions, Dubna, February 7-15, 1967. 39. V. A. Matveev, L. D. Solov'ev, A. N. Tavkhelidze, and V. P. Shelest, Preprint JINR R2-3118 [in Russian], Dubna (1967). 40. L. D. Solov'ev, Talkat the International Theoretical Physics Conference, Rochester, Preprint, Dubna 2-3511 (1967). 41. J. Schwinger, Phys. Rev. Lett ., 3 : 296 (1959). 42. K. Johnson, Nucl. Phys ., 25:431 (1961). 43. S. Okubo, Nuovo Cimento, 41A:586 (1966); 44A: 1015 (1966).

R. E. KALLOSH 44. 45.

97

J. D. Bjorken, Phys. Rev ., 148 : 1467 (1966).

B. V. Medvedev and M. K. Polivanov, Lectures at the International School at Dubna, Vol. 1 [in Russian], Izd-Vo OIYai (Dubna) (1964). 46. H. Gell-Mann, Phys. Rev., 125:1067 (1962); Phys. Lett., 1:63 (1964). 47. H. Cabibbo and L. A. Radicati, Phys. Lett., 19:697 (1966). 48. S. L. Adler, Phys. Rev ., 143: 1144 (1966). 49. F. Buccella, G. Veneziano, and R. Gatto, Nuovo Cimento, 42A: 1019 (1966). 50. A. A. Logunov and L. D. Solov'ev (Soloviev), Nucl. Phys ., 10:60 (1959). 51. L. D. Solov' ev, Preprint JINR E-2343, Dubna (1966). 52,53. V. de Alfaro, S. Fubini, G. Furlan, and C. Rossetti, Phys. Lett., 21:576 (1966). 54. R. E. Kallosh, Phys. Lett., 22:519 (1966). 55. R.E. Kallosh, V. G. Pisarenko, and R. N. Faustov, Yad. Fiz., 5:1270 (1967). 56. J. D. Bjorken, Phys. Rev. Lett ., 16: 408 (1966). 57. B. L. Ioffe, ZhETF Pis. Red., 4, No. 9, 376 (1966). 58. B. L. Ioffe and E. P. Shabalin, Preprint ITEF (1967). 59. V. Ya. Fainberg, Zh. Eksp. Teor. Fiz., 40:1758 (1961). 60. V. Ya. Fainberg, Doctoral Dissertation [in Russian], Inst. Theor. Eksp. Fiz. (1961). 61. B. V. Medvedev, Doctoral Dissertation [in Russian], V. A. Steklov Mathematics Institute, Moscow (1964). 62. A. R. Sukhanov, Zh. Eksp. Teor. Fiz., 51:1195 (1966). 63. B. Schroer and P. Stichel, Preprint NY0-3829-5 (1967). 64. D. Hall and A. S. Wightman, Kgl. Danske; idensl. Selsk, Mat. Fys. Medd., 31:5 (1957). 65. A. Frenkel', Zh. Eksp. Teor. Fiz., 47:221 (1964). 66. Chan Hong-mo, Nuovo Cimento, 23:181 (1962). 67. V. E. Asribekov, Zh. Eksp. Teor. Fiz ., 42 : 565 (1962). 68. S. Fubini, G. Furlan, and C. Rossetti, Nuovo Cimento, 40 :A1161 (1965). 69. M. Gourdin, Preprint Orsay TH/127 (1966). 70. G. Framerand K. Meetz, Preprint DESY 66/8 (1966). 71. V. N. Gribov, B. L. Ioffe, and V. M. Shekhter, Phys. Lett., 21:457 (1966). 72. V. N. Gribov, B. L. Ioffe, and V. I. Shekhter, Yad. Fiz., 5:387 (1967). 73. M. Gourdin, Preprint Orsay TH/146 (1966). 74. I. B. Bronz an, I. S. Gerstein, B. W. Lee, and F. E. Low, Phys. Rev. Lett.,18: 32 (1966). 75. V. Singh, Phys. Rev. Lett., 18:32 (1967). 76. D. Amati and R. Jengo, Preprint CERN Th, 737 (1966). F. I. Gilman and H. I. Schnitzer, Phys. Rev ., 150: 1362 (1966). 77. 78. S. B. Gerasimov, Preprint JINR R4-3222 [in Russian], Dubna (1967). 79. H. Pagels, Phys. Rev. Lett., 18:316 (1967). 80. H. Harari, Phys. Rev. Lett ., 18 : 319 (1967). 81. 0. W. Greenberg and F. E. Low, Phys. Rev., 124:2047 (1961). 82. A. Martin, Nuovo Cimento, 42:930 (1966). 83. F. Low, Phys.Rev., 97:1392 (1955). 84. D. Amati, R. Jengo, and E. Remiddi, Preprint CERN Th, 759 (1967). 85. M. K. Polivanov, Lectures at the International School at Yalta, Naukova Dumka (1967). 86. V. P. Pavlov and A. D. Sukhanov, Zh. Eksp. Teor. Fiz ., 54:138 (1968). 87. S. Adler and Y. Dothan, Phys. Rev ., 151 :1267 (1966). 88. B. V.Medvedev and M. K. Polivanov, Dokl. Akad. Nauk SSSR, 143:1071 (1962). 89. M. V. Terent'ev, Preprint ITEF No. 463 [in Russian], Moscow (1966). 90. H. Lehmann, Nuovo Cimento, 11:342 (1954). 91. H. Weltman, Phys. Rev. Lett., 17:553 (1966). 92. H. Gell-Mann and M. Uvy, Nuovo Cimento, 16:705 (1960). 93. R. Haag, Phys. Rev., 112:669 (1958). 94. R. Nishijima, Phys. Rev., 111:995 (1958).

98 95. 96. 97. 98. 99. 100.

R. E. KALLOSH W. Zimmermann, Nuovo C imento, 10 : 597 (1958). W. W. Wada, Preprint Th. (1967). G. F. Chew and F. E. Low, Nuovo Cimento, 8:132 (1968); Phys. Rev ., 101:1571 (1956). G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev., 106:1337 (1957). B. L. Voronov and V. Ya. Fainberg, Preprint No.17 [inRussian], P. N. Lebedev Physics Institute (1969). K. Bardakci and E. C. G. Sudarshan, Nuovo Cimento, 21: 722 (1961).

THEORY OF PERIPHERAL INTERACfiONS AT HIGH ENERGIES I.M. Dremin, I. I. Roizen, ai1d D. S. Chernavskü Introduction This review is concerned with theoretical aspects of inelastic high-energy hadron (nucleon and pion) interactions and their effect on elastic collisions. The authors' papers on peripheral collisions and their relationship to nonperipheral collisions form the basis of the review. It is not the intention to confront theoretical and experimental data in detail; indeed, there is reason to believe that this would be meaningful only after a number of parameters have been made more precise, a question to which we shall return at the end. We shall merely describe briefly experiments that played an important role in forming the theoretical conceptions. The notion of peripheral interactions of nuclear-active particles was put forward as long as twenty years ago by, among others, Zatsepin, Feinberg, and Chernavskii [1-3]. Initially, the notions were elementary, but even these brought out fairly clearly several basic features of the process which have remained virtually unmodified in the modern theory. In peripheral collisions, the interaction is due to the exchange of a single quantum, the 7T-mesonthe lightest nuclear-active particle - playing the most important role. Cosmic-ray experiments have profoundly influenced the theory of inelastic processes; the following results were especially important. 1. The interaction cross section is constant to within 50% in the entire investigated energy range from 10 10 to "'1015 eV. 2. The secondary particles are strongly collimated along the collision line in the centerof-mass system at high energies. The degree of collimation increases with the energy, so that the transverse momentum remains constant on the average in almost the whole of the energy range ( !i1 "' 2p. , where p. is the 1T -meson mass; here and below (!i 1= c = 1). 3. The mean multiplicity increases rather weakly with the energy. 4. Interacting high-energy nucleons retain the major part of their energy: the inelasticity coefficient K - the ratio of E1 , the energy lost on forming new particles, to E0 , the primary particle's cms energy- is about 0.5 on the average. These results, which have been known for a comparatively long time, arenot in doubt. (The value given for K is averaged over different possible classes of processes .) The subsequent and more detailed information obtained on the elementary process is the subject of contention, primarily because of the düficulties of obtaining and interpreting cosmic-

99

100

I.M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

ray data. This is not the place for a detailed discussion; the fact remains that, although controversial, these results have played a very important role in forming theoretical notions. The results are as follows. 1. Two centers from which secondary pions are emitted are formed when nucleons of energy Er_ '5 10 12 eV interact. In the center-of-mass system the centers move along the collision line with relativistic velocity, the cms Lorentz factor being Yc = (1- v~)-1/2 ~ 1.5. 2. About six or seven pions are emitted from each of the centers, the mass ~ of a center being ~3 GeV. These results were mainly obtained by Miesowicz, Gierula, and collaborators [4] (Poland) and independently by Niu [5] (Japan). The erniss ion centers were dubbed fireballs by Cocconi [6]. 3. Data obtained subsequently by Dobrotin, Slavatinskii, and collaborators [7] suggested that a single emission center is formed when nucleons of slightly lower energies collide (EL ~ 10 11 -10 12 eV). The main characteristics (mass, number of emitted particles, etc.) of the single "fireball" were the same. This prompted a simple idea: when the energy increases, so does the number of fireballs, but their characteristics- essentially, their mass- remain unchanged. Evidence has been found by Hasegawa [8] in Japan and by the Cracow group [9] that a greater number of fireballs are formed at higher energies, ~ '5 10 13 e V, but it is not so convincing as that for the formation of one and two fireballs. These experiments are all reviewed by Miesowicz [10]. The theory of inelastic interactions cannot but attempt to explain this striking phenomenon offireball formation. Precisely how theory matches experiment we shall discuss below; it suffices now to say that the theoretical notions developed in step with, and under the influence of, the experimental data. First came the von Weizsäcker-Williams method, which contained many parameters. Reformulation of the model in terms of Feynman diagrams made it possible to calculate quantities previously postulated ad hac; for example, the total cross section of peripheral interactions, which was calculated for the firsttime in the pole approximation for proton-proton collisions at 9 GeV by Dremin and Chernavskii [11]. However, it was found that the cross section calculated in this approach increases rapidly with the energy [12]. Using the diagram technique, Berestetskii and Pomeranchuk [13] tried to explain the cross section' s high-energy behavior. They encountered a problern that was to play an important role in the subsequent development of the theory: under what seemed natural assumptions, the cross section was found to increase asymptotically with the energy. The root of the problern was later discovered, and it was shown that the cross section could be asymptotically constant. We still have to contend with the problern in discussing the pre-asymptotic behavior, as we shall see below. An important new step in the theory of peripheral interactions was the multiperipheral model proposed by Amati, Fubini, Stanghellini, and Tonin [14] (the AFST model).

The model was reformulated in the language of Feynman diagrams, but, unlike earlier models, a closed expression describing a complete set of diagrams simultaneously was proposed. Much work was done on the model, though comparison with experimental data did reveal practical shortcomings: it could not guarantee asymptotic constancy of the cross section nor describe the formation of fireballs. All the same, the main features of the model are very good; its defects are due primarily to an unfortunate choice of the main parameters and not allowing for various conditions. Many characteristic features of the model are retained in the more rigorous theory expounded below.

I. M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

101

The next important factor in the development of the theory was the rapid progress made in investigating the properties of the elastic scattering amplitude. We are referring to the use of dispersion relations, formulation of the crossing symmetry principle, the Mandelstam representation, and, finally, the method of complex orbital angular momenta. Unitarity relates the forward elastic scattering amplitude (or rather its imaginary part) to the cross section of inelastic processes; restrictions on the former are restrictions on the latter. Here, the Bethe-Salpeter equation, which takes account of this connection and will play an important role in this review, was found tobe a very convenient tool. What is more, an effect similar to fireball formation occurs automatically in the theory if one imposes apparently abstract conditions such as the existence of a solution and a leading vacuum singularity (or, which is the same thing, asymptotic constancy of the cross section). In other words, the "abstract" conditions determine parameters (at least their order) that were arbitrary in the earlier models; they impose serious restrictions on the models of peripheral interactions. All models of inelastic processes must: 1) describe the characteristics of the inelastic processes in a wide energy range, in particular, at very high energies;

2) explain the structure in the amplitude of the elastic scattering due to a given inelastic process; 3) satisfy the conditions imposed on the elastic scattering amplitude: the amplitude must have the correct analytic properties. with respect to not only the Mandelstam variables s, t, and u, but also in the complex orbital angular momentum representation [15]; it must be unitary and give the correct asymptotic behavior of the cross section. The theory expounded below was proposed in [16], developed and refined in [17-20], and briefly discussed from the point of view of its conclusions in [21] as weil. We attempt to show that it satisfies the above requirements. However, one cannot get by entirely without model representations, for these t:P,eoretical restrictions do not lead to a complete quantitative description of the process. C ertain dependences must be assumed, and a number of parameters remain arbitrary prior to the comparison with experiment. Thus, our main aim is to expound the theory of peripheral interactions as rigorously as possible. Webegin by considering the exact Bethe-Salpeter equation in all the representations tobe used subsequently. We then formulate and justify an additional assumption needed to give a physical meaning to the quantities in the equation. We then investigate the solutions and discuss the various results. We then compare ours with other models of inelastic processes and discuss some problems related to the superasymptotic energy range. The exposition emphasizes the consequences - above all, the very existence of processes with fireball formation- that follow from the general assumptions. 1.

Bethe-Salpeter Equation

Our theory of peripheral collisions, whose foremost feature is fireball formation, is based on the Bethe-Salpeter equation. Let us make it clear immediately that we are not using the "ladder approximation"; we use the exact Bethe-Salpeter equation to study the general questions. Of course, it is not an equation in the ordinary sense of the word (unlike the latter approximation), relating, as it does, two unknowns; it were better known as a relation. However, certain general properties of the scattering amplitude that do not follow from the BetheSalpeter equation, above all analyticity and unitarity, impose far-reaching additional restrictions on the quantities in this equation. This is why this very general equation can yet yield

102

I.M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKIT

s-

~ Ät Pz

P3

_

I{

)=(1 +~)(PJ /{ /{

P,

f

2

Pq

pz

P2

P,

Fig. 1. Diagrammatic form of Eq. (1.1). The circle stands for the total amplitude A; the square, for its irreducible part A. definite information about interactions at high energy. More precisely, it relates properties of the interaction at high energies to characteristics of low-energy processes. One has a_ situation similar in certain respects to that which leads to sum rules. For simplicity we shall first assume that the interaction involves only identical neutral pseudoscalar particles of mass J.L (for example, pions) * and see what general consequences flow from our adoption of the Bethe-Salpeter equation in conjunction with analyticity and unitarity. In this special case the Bethe-Salpeter equation can be written in the form

A (s, t, pi_ 3 ) = .:f (s, t, Pi)- (2.~) 4 ~ d4 k1.4 (s 1 , t, Pi. 3 , k~.s) A (s2 , t, ki. 2) D(k~) D (ki)o

(1.1)

Here A is the scattering amplitude, Ais its irreducible (in the t channel) part,t s and t are the ordinary Mandelstam variables: t

st is obtained by replacing the momentum p2 by Pt by -kt:

= -

kt

(pl - Pa) 2,

(102)

in the expression for s; and s 2 , by replacing

(1.3) Pf,3 means Pr and p~ and, finally, D(k:l) is the propagator; ~ =Pt + P3- kt. The metric is chosen such that ~ = k2 - kJ 0 Equation (1.1) is shown diagrammatically in Fig 0 1 0 We expand both A and

Ä in partial wavest in the t channel: A (s2 , t, k~,2) =

~ (2l 1-o

Ä (s1, t, Pi_ 3 , k~,2) =

+ 1) / 1(t, ki. 2 ) Pc(z 2 ),

L. (2l + 1) f1(t, Pi,a•

(1.4)

k 1 , 2 ) Pc (z 1),

l=O

(1.5)

where zt and z 2 are the cosines of the scattering angles of particles whose squared masses are (-pf, -p5, -ki, -Iq) and (-ki, -Iq, J.L 2 , J.L 2), respectively, i.e., (1.6)

*In Sec. 9 we generalize to real 1r N and NN collisions. t That is, the part that does not contain two-particle intermediate states in. this channel. t The latter is expandible within the Martin-Lehmann ellipse, its analytic properties differing from those of A only in the absence of the branch point at t = w2 0

I.M. DRE:MIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

103

and z 2 is obtained from z 1 by replacing s 1 by s 2 and setting p{ = p§ = -~J, 2 • Using (1.4), (1.5), and (1.1) and the orthogonality of Legendre polynomials, we obtain the Bethe-Salpeter equation for the partial waves: (1.7) where q=lkll 2 ,

_ pro + pso k _ Vt k _ kzo - kro _ k~ - ki q o - - - 2 - - lo-----z-- l o - - - 2 - - 2 y7·

(1.8)

Equations (1.7) and (1.1) arevalidfort > 41-t 2 and fort< 41-t 2 , but in the latter case all the singularities of the propagators in the integrand of (1.7) lie in the second and fourth quadrants of the ~ plane. If all the integrand•s other singularities do so too (and this, at least, is the case in perturbation theory), and if the integrand decreases rapidly enough as ICio 1- oo in the first and third quadrants, the contour of integration can be Wiek-rotated through rr/ 2 in the ~ planefort > 4~J. 2 • This means we go over to a Euclidean metric; it is shown in [22, 23) that this can be done if the scattering amplitude satisfies some very weak conditions. In the physical region of the s channel (t : 4!J, 2 to integration in the ~ plane along the contour shown in Fig. 2a. In the case t < 4j.L 2 , the contour of integration has the form shown in Fig. 2b. At the same time, considering Eq. (1.1) directly in the region t < 0, we see immediately that the integration with respect to ki and k~ does not extend beyond real values, i.e., the contour of integration for (1.7) in the ~ plane coincides with the imaginary axis, as shown in Fig. 2c. C omparing the contours in Figs. 2b and 2c we see that for t : O, k2 > 0 the function ft (p2, k 2) cannot have a pole at l = 1, for otherwise the integral term in (1.12) would have a second-order pole, which is not present in the total partial amplitude. Thus, either the position or the nature of the singularity (or both) must depend on the external masses, andin the most general case lz (p 2 , k 2) must have the form

fz (pz, k2)

Rz (p2,

= [l -

l (p''

k2) k 2) j ä (p 2 ' k 2 )

(5.1)

'

where Z(- JJ. 2 , -JJ. 2) = a; (-JJ. 2 , -JJ. 2) = 1. In [18J. two Iimiting cases are considered: ä(p 2, k 2) = 1 and T (p 2 , k 2) = 1. In accordance with this behavior, the nonperipheral cross section is taken (for example, in the second model) in the form a - const, ; (s, p 2, k2) = / 0° Öa 0

* This

4f.L 2

for

So

so+ k2

+ p'

a 0 in all the cases considered; we shall consider this question separately below. Figure 7 shows the eigen-

* The corresponding term in ( 5 .1)

illustrates one way in which a noncontracting düfraction peak (on the mass shell) can exist without violating unitarity in the t channel. It can be shown that the Bethe-Salpeter equation itself gives rise to the necessary branch-point in the l plane [19]. This is clear; for the solution of the Bethe-Salpeter equation is unitary, this being true, in particular, in the two-particle range of values (this is discussed in more detail tagether with a similar possibility in [16, 19]). In this case, oa 0 has the same order as the nonperipheral cross section of real particles (an the mass shell) at high energies.

116

I.M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII R, (p')

Fig. 7. Numerical solution of Eq. (5 .5); s 0 = 12 GeV 2 • 0

{

5

IN (x') (

Fig. 8. The distribution W(k2); s 0 = 12 GeV 2 •

n function R 1(p 2), which depends fairly smoothly on its argument until k 2 = s 0 • Figure 8 shows W (k2) = k~ D2 (P)ln----;:.-- R 1 (k2).

(5.6)

... min

This is of interest because it reflects the distribution over the squared momentum transfers between the blocks in the asymptotic region, i.e ., it is observable in principle. The figure shows that the k 2 values that contribute effectively to the integral are much less than kJ ;: : s 0 , and that the distribution W(k 2) is "pressed" into the region of small k; that is, it is peaked at a k 2 equal to a few J..L 2 , which is much less than the mean value 0. Finally, some comments on models in which kJ and s 0 are independent parameters, s 0 • Table 2 illustrates one possibility. This "freeing" of the parameters, which can now vary freely, has the following consequences (cf. Table 1, for which k5 = s 0 , and Table 2 for which k5 < s 0): if s 0 is fixed and k5 decreases, u 0 and u~ of course increase (cf. Tables 1 arid 2), so that kJ is bounded below in this case.

k5
s 0 ; then Eq. (6.5) is a homogeneous Volterra equation with positive kernel. Its minimal eigenvalue - the one of interest- therefore corresponds to an eigenfunction of fixed sign. To determine the sign, we must first determine the value of s at which the previously omitted terms begin to contribute little. Since (s 0 I s) 2 :(

I.M. DREl\tllN, I. I. ROIZEN, AND D. S. CHERNAVSKII

119

(10 GeV 2 /s) 2 , they cannot contribute more than a few percent (at the most several tens) when s > 100 GeV, and certainly cannot affect the sign of the solution to Eq. (6.5). We solved the exact inhomogeneaus equation (6 .1) in the most "unfavorable" case s 0 = 12 GeV 2 • The corresponding curves are shown in Fig. 9. The only important point is that for s > 150 GeV 2 we always have* d(J (s, p2) d.>

> 0.

(6.6)

The omitted terms being small in the part of the analyzed energy range where s > 150 GeV 2, the solution of Eq. (6.5) is also positive in this region. But by what we have already said, it is then positive in the whole region s > 150 GeV 2 , and so is the solution of the exact equation for the derivative da (s, p2)/ ds. We therefore expect the peripheral cross section to increase when s :5 150 GeV 2 (which corresponds to E1ab :5 75 GeV for pp scattering). But how does the total cross section behave? To answer this question we need to know how der (s) I ds behaves for s » s 0 (hitherto we have assumed ä == const for s > s 0 ). But about this we know virtually nothing. All the same, the effective interaction range being roughly inversely proportional to the mass of the exchanged particles, simple arguments suggest that the nonperipheral cross section is much smaller than the peripheral for s » s 0 • If this is so, the dependence of 0' (s} on s can hardly affect the above estimate. Recently, Gribov, and Migdal have found another indication that the cross section may increase in the pre-asymptotic region. They deduced this from their proposed version of Regge theory with weak coupling [32]. Physically, the cross section's pre-asymptotic growth can be understood as follows. Berestetski and Pomeranchuk [13] have shown that if lim ö (k, s2) = 0 (P) 0 in any finite range CO

>

of values k 2 > 0, Eq. (1.18) is incompatible with the total cross section O'(s) being asymptotically constant, since the integral term on the right-hand side of (1.18) increases unrestrictedly with s. Of course, this last property is manifested the more strongly, the greater the effective region of k 2 • On the other hand, we have seen above that if the peripheral cross section is to be asymptotically constant the values of s 0 and k 2, which determine the effective region of integration of Ü(s, k 2) with respect tos and k2 , must be fairly large. Therefore, the growth of the integral term in (1.18) must have an effect up to very large values of s and may raise the total cross section in the pre-asymptotic region. t We shall discuss below how this can affect the cross section of peripheral nucleon collisions, causing the cross section to grow in the pre-asymptotic region. It is also clear that the existence of a growth region is intimately related to the asymp-

totic constancy of the cross section; for if it tended asymptotically to zero, it could reduce the effective values of s 0 , and 0' 0 in such a way that a growth region would be entirely absent. And, conversely, a growth region•s being absent would indicate that the peripheral cross section "dies out" asymptotically.

kJ,

*Other properties of the numerical solution shown in Fig. 9, for example the minimum at s :::J 100 GeV 2, are of no particular interest, depending strongly on the model chosen for Ü(s, p 2, k 2). In particular, the assumption ö = 0 is responsible for the minimum, which disappears if ä is a smoother function. But the inequality ( 6 .6) always•holds. t When real pions interact, both neutral and charged particles can be exchanged. Then ärn < 0' 0 and pre-asymptotic growth is favored even more.

120

I.M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

Fig. 10. Comparison of the theoretical predictions for N with experimental data. The straight line 1 is drawn in accordance with the experimental data; the calculated curves 2, 3, and 4 correspond to s 0 = 5, 8, and 12 GeV 2 •

7.

0

Mean Multiplicity in Peripheral Collisions and

the Number of Fireballs Having solved Eq. (5 .5) numerically, we can calculate the mean multiplicity N of a peripheral collision to within logarithmic terms in the energy [20]. It is convenient to use the method developed in [14], which expresses N in terms of (dalda 0 )a=i• where a is the position of the vacuum pole, and in terms of the mean number ii of particles in a fireball:

-N = -(da.') na0 d-:::-

' vo 'cx=l

s In --=:;,

2k ..

( 7 .1)

where the omitted term does not depend on s. Now (da I da 0 ) a= 1 can be readily determined by generalizing Eq. (5.5) to the case when 'Pz (p 2) has a pole at the point l = a = 1 + e: ( 7 .2)

Then da I da 0 is found in exactly the same way as the correct normalization of the solution to Eq. (5.5). The upshot is (7 .3)

where Rr( -J.L 2) is the on-shell solution of Eq. ( 5 .5) normalized to unity. Substituting ( 7 .3) into (7 .1), we obtain ( 7 .4)

and then, substituting the numerical value,* 6.0 lg (s;iP)

for

N = \ 6.8 lg (s/2:2 )

for

7.7lg(s/2k 2 )

for

m=

N;n.

( 7 .5)

and the mean number of fireballs ( 7 .6)

The dependences ( 7 .5) and ( 7 .6) are compared with the experimental data in Fig. 10 under the condition 2k 2 = 4 GeV 2 • As the experimental points allow for only charged pions, the right*The statistical theory of [33] [see Eq, (11.1)] was used to calculate ii.

I.M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

121

hand side of ( 7 .5) was multiplied by 73 for the comparison. The theoretical estimate may be too low because some of the pions are formed by an isobar' s decaying. 8.

Slope of the Vacuum Trajectory

Finally, let us consider the slope of the vacuum trajectory [20}, i.e., y = [da(t)/dt]t=O. Of course, the special case t = 0 no Ionger suffices for this, and we must consider the BetheSalpeter equation near and not only at the point t = 0. In other words, we must deal directly with Eq. (1.7b). = 1 near t = 0. We represent cp 1(t, r, v) [see (l.lla)

Let us consider Eq. (1.7b) with and (1.11b)] in the form

(t

lfll

'

r,

V

) _ !l.I(t, r, v) 1 -CL (t) ' -

(8.1)

where the function R (t, r, v) is regular at the point l = a(t) and we expand* all the functions in (1. 7b) with respect to t at t = 0. Then

the prime denoting derivatives with respect tot at t = 0. Taking the terms of zeroth and first order in t in ( 8 .2), and noting Eq. (3 .2), we obtain rlfl1

3 • , 2 ~· dk"'• (k") + -1ön.

00

_

CO

R1. = -

2

r.'1 D 2 (k") :P1.n - [-:-

' , I - 41-1 2 , compensating each other entirely at t = 4J,L 2 , and for t < 41-1 2 compensating in some manner the pole due to the integral term in the BetheSalpeter equation. But, first, this would only be possible if interference is very important at high energies which, as we have attempted to show, is implausible and, secondly, at the present phenomenological Ievel of the theory it is hardly sensible to complicate the models until the simplest variants lead to contradiction. t Strictly speaking, one ought also to allow for the presence of absorptive parts in the total energy of each pair of "j ets." *In principle, one could conceive of models in which the function

130

I. M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKII

Fig. 17

Fig. 18. Scheme of multiRegge model.

In the case of elastic scattering, z"' s/2M2 » 1, ü the energy is sufficiently high. It was assumed (without there being sufficient justüication for this) that z is always large in inelastic processes as weil, and it was concluded that processes in which a region with the largest value of ai (0) is exchanged make the main contribution. A problern with the model is that it is neither well defined nor closed, and this has meant that it has been possible to use it in two essentially different variants. This comes

about because to calculate a quantity one must represent the diagram (or part of it) as a fourline diagram ( Fig. 1 7). This can be done in s everal different ways. For example, one can distinguish two blocks in the diagram and regard them as "particles" with masses ~ 1 = .fs 1 and ~ 2 = .fs 2 , res pectively (here s 1 and s 2 are the squares of the total energies of all the particles of the first and the second group in their center-of-mass systems). Then in the s channel the process can be regarded as the formation of two heavy particles from two primary particles with masses J1. and m. One can then use the crossed channel (t channel), in which the particles with masses J1. and ~ 2 go over into particles with masses m and ID? 1 • But can one really identify such unwieldy formations with particles? One can adopt a different approach, distinguishing a number of internal four-line diagrams in the diagram of Fig. 18, as is shown in this figure; for each of these four-line diagrams one then calculates the value of z and introduces them into the total amplitude of the process. It is true that some of the exterior ends of the four-line diagrams are then virtual. This approach was employed in [41] and then in [42J, but the values of z were not, in fact, calculated; rather expressions of the type s/s0 , where s 0 is regarded as a variable parameter, were simply substituted. * In general, the two approaches lead to düferent results. All the problems disappear if one is concerned with the simplest diagrams of specialtype in which relatively few particles are formed (for example, in binary processes, the formation of isobars, etc .) .

We shall show below that it is correct to use the model in which a vacuum region is exchanged to describe low-productive events which are suchthat the masses ~ 1 and ~ 2 are small and, most importantly, do not increase with the energy. Earlier, these processes were assumed to accompany elastic scattering and were called diffraction production processes [37, 44]. As regards complicated processes in which \1J? 1 and ~ 2 increase with the energy, we shall attempt to show that the vacuum region model can hardly be justified in this case. We may also mention that the application of Reggeism in traditional form to even binary processes with different masses Ieads to difficulties. It has been shown [37] that, in cantrast to elastic scattering, the values of z at the boundary of the kinematically allowed region are small (I z I = 1) even at an arbitrarily high energy. *In addition, the idea of "clusterization" (already discussed in Sec. 4) is used in [42]. This Ieads to good agreement when the results of the model are compared with the experimental data from the C ERN accelerator.

I. M. DREMIN, I. I. ROIZEN, AND D. S. CHERNAVSKTI

131

This problern gave rise to a series of papers terminating in Domokos' s proposing [45] a new four-dimensional Reggeism, which weshall discuss in detail. Followmg [45], we decompose the amplitude of the process with respect to irreducible representations of the group of fourdimensional rotations, i.e ., with respect to the four-dimensional spherical functions (we operate for the time being in the Euclidean region of the variables s, t, u > 0):

= Pnz (ß) Y?' (tt, cp)

Z~ (ß, tt, cp)

where '" m 2 , where m is the mass of the created particles) .

I • N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

147

5. The collisions considered under 4 may be collisions between a virtual particle in a peripheral process with one of the incident particles, so that one is dealing with the form factor of strong interactions in certain cases. 6. The theory of the cöllisions of particles with fairly heavy nuclei and nuclei with nuclei at an energy which is sufficient for many created particles tobe present as well as the nucleons involved in the collision. -~ It is probable that this list does not exhaust all the possibilities. For all these cases we have no other theoretical scheme capable of serving as an effective tool in calculations; at the same time, the basis of the statistical model allows for the presence of very many degrees of freedom. This is why it is capable of explaining many facts qualitatively and quantitatively.

Let us now turn to a generat description of the various forms of statistical model that have been proposed and, in particular, the formthat at present appears tobe the mostsensible and best confirmed by experiment. 1.

History

The first suggestion that in a collision of strongly interacting high-energy particles could give rise to a system with many degrees of freedom that reaches thermodynamic equilibrium is tobe found in Beisenberg's paper on the collision of high-energy weakly interacting particles [1]. In the version of the theory popular at the time (proposed by Knopinskii and Uhlenbeck) the interaction Lagrangian contains derivatives of the field Operators with respect to the time, i.e., it is essentially proportional to gE 11 , where g is the constant of the weak interactions, Eis the characteristic energy of the created particles, and v is a number, v :=:: 1 (cases with v "' 5 and higher were considered). Of course, in such a theory (and, we may mention, in the modern nonrenormalizable theory of weak interactions) at a sufficiently high energy, perturbation theory breaks down and multiple production is possible (in the modern theory, only at a cms energy Ec 5> 300 GeV). Beisenberg noted that in such a case a bunch must be formed with a high temperature which, expanding and cooling, must ultimately decay into individual particles with a Planck spectrum. Beisenberg was attempting to explain wide atmospheric showers, but the appearance of the Bethe-Beitler electromagnetic cascade theory removed that problern. When mesons were subsequently discovered, a comparison of their number (and relatively weak absorption) in the depth of the atmosphere with the number of particles of the primary flux at the boundary of the atmosphere showed that they must be created in groups of particles, about five in one collision event between primary protons and the nuclei of the atoms in the air [13]. Since the primary protons have a mean energy "'10-15 GeV, it is readily verified that this estimate was correct. The confirmation of the mesonnature of the interaction processes between primary protons of high energy and matter and the multiple nature of the production processes led Wataghin [2] to a statistical model of this effect, whose bases were the same as the more developed model proposed in 1950 by Fermi [3] and named after him. The statistical model proceeds from the fact that, on the average, many particles are produced in a single collision event at high energies. It can then be assumed that the process cannot be well described by approximate quasiclassical methods; for many particles means !arge quantum population numbers. On the other band, the energy of the process and of an individual particle is much greater than the pion mass iJ., and therefore the wavelength of the particles is A. "' 1/ q (where ti is the energy of the i-th particle) and much less than the range of the strong interactions, which is about 1/ p. • These two conditions prech.ide a classical treatment, and we are forced to a paradoxical but, in principle, inescapable conclusion: the

148

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

interaction and multiple production of particles at superhigh energies - if they take place in a small volume V 0 with dimensions of the order of the interaction range, "'1/1-' - must exhibit classical rather than quantum-mechanical features and, moreover, be described by thermodynamics and statistics. Fermi assumed that in a collision in a small volume V 0 the entire energy of the process is liberated and that the final number of particles and also their distribution over the particle species, energies, etc ., are determined by the thermodynamic formulas of blackbody radiation. Fermi' s model immediately attracted a great deal of interest. Rowever, Pomeranchuk (4] noted that Fermi' s treatment contains an inconsistency: no allowance is made for the interaction of the created particles with one another after they have left a volume ..... v 0 • Even approximate allowance for this interaction appreciably modifies the theory' s results and. as we shall see, improves it in the framework of the statistical approach when the multiplicity is not too high (n ~ 10-20). But if the multiplicity is very high, consistent allowance for the interaction leads to the idea of hydrodynamic expansion of the bunch of nuclear matter, which decays into individual particles as it cools (as Reisenberg also pointed out [1]). This was immediately noted by Landau, and Pomeranchuk's paper remained neglected in all the subsequent years. Nobody realized that Pomeranchuk's model has its own field of applicability, and that it is one limiting case (for not very large n) of a single theory whose other limiting case (for very large n) is the hydrodynamic theory. In the exposition that follows we shall attempt to justify this point of view, and we shall make wide use of this very model. The hydrodynamic theory of multiple production was developed by Reisenberg [14] and in a more consistent form by Landau (15]. It greatly improved its predecessors. Let us dwell in more detail on the basis of the statistical theory. The probability of a collision process between two particles of four-momenta P 1 and P 2 leading to a final state containing n particles with momenta Pi, i = 1, ... , n, is described in the general case of a quantum-mechanical formula that can be expressed in the form (1.1) Here Mn is the matrix element of the process and Pn is the statistical weight or phase volume of the final state. Neglecting spin factors (for example, by assuming that they have already been summed over) we can assume that Pn

=

n

II

(dp V

(2n,)" ) •

(1.2)

t=l

Here V is the volume of the system. The main idea of the theory is that for large n the statistical weight Pn is a function that depends strongly on the momenta and n itself (since it depends on n essentially exponentially); for, integrated over all p i• this factor is, by dimensional considerations (it is assumed that relativistic particles for which the mean energy pO and mean momentum IPI are large, po . . . Ipl » ~, where mi is the particle mass, are effective), of the order n

n

5fi

i=l

dptö 4 (

Pl

+ Pz- ~Pt)~ (J P 1) 3,.._~.

(1.3)

Obviously, I p I is of order E/n if E is the total energy of the system. Thus, the integral is of the order exp [-nln(n/n0)], where no is a characteristic parameter that determines the maximum with respect to n; for n < llo the integral increases with increasing n; for n > n 0, it decreases. There is no justification for assuming that Mn depends so strongly ( exponentially) on n and the final momenta. This then is the basic hypothesis: 1Mn1 2 in (1.1) can be replaced by

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

149

a constant with dimensionality that ensures the correct dimensionality of the complete probability. We now note that the assumption that IJ.Vfu I is constant, i.e., that the process's probability is determined by the phase volume of the final state (1.2), corresponds exactly to the assumption that there is a microcanonical distribution in a classical statistical system in equilibrium; for according to this assumption the probability of the system' s being in a given volume element in the multidimensional phase space (with 6n dimensions if there are n particles) is determined solely by the size of this element - the phase volume between the surfaces of constant energy E and E + dE. Such an assumption automatically leads to a canonical distribution for any small part of the system and, ultimately, to the statistical and thermodynamic formulas for the most probable (for large n, this is equal asymptotically to the mean value) distribution over n and the momenta. We are therefore fully justified in assuming that the system has a temperature. One can also use the statistical thermodynamic formulas that relate the energy density e = E/V (E is the total energy of the system) to the temperature T, the entropy density s, the total entropy S = sV, and the number of particles n. Generally speaking, these relations depend on the actual physical properties of the system. In our case it is a relativistic system whose temperature in the Fermimodel is usually much greater than the mass of the particles, while the nurober of particles is not fixed. A well-known investigated object of this kind is blackbody radiation- a photon gas (see, for example, [16]'). Using this for the pions and noting that they have three internal degrees of freedom, and not two like a photon, we have E

e = V = const T',

nz

const = 10 = 0.99

(1.4)

which is the Stefan-Boltzmann law (T is measured in energy units, i.e., k, Boltzmann's constant is equal to unity). In addition, we assume that n = c = 1 and, therefore, the ordinary coefficient const = 4a /c, a = tr2k 4 /60n 3c 3 takes the form s

= const T a,

4n• = . 1.3?· const = 30 ~-

(1.5)

The total entropy S = sV is proportional to the nurober of particles: S = sV = constn,

const

=

2,""(4

45 1;. (3)

= 3.61

(1.6)

[~ (3) = 1.202 ... ] . As we see~ the mean nurober of particles is proportional to certain powers of the energy and the volume: E 'I• n- TlV- ( v) V= EI•V'I•.

(1. 7)

It is possible to obtain more detailed formulas for the distribution of the particles over the momenta, etc .; this has been done an many occasions [1 7]. In particular, the pions must have

a Planck energy distribution. However, before we consider these questions, let us dwell on the two fundamental hypotheses employed above. The first is th~ assumption that all the particles in the final state, including those newly created, form a single system, during whose existence an approximate equilibrium can be established. In other words, multiple production resembles this classical process: a hard photon is absorbed by 2. body, which is heated as a result. and then many soft photans are emitted in the established equilibrium state. The second hypothesis concerns the undetermined parameter V, the volume of the system. This is a very important, although more special circumstance. In the quantum-mechanical formula (1.1), V in the usual use is essentially an auxiliary parameter, the normalization

150

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNA VSKII

volume, which disappears from the final result. But here V is the real volume of the system in which the thermodynamic equilibrium is established, and it must be chosen on the basis of certain independent considerations. Fermi arbitrarily took it equal in order of magnitude to the volume of the colliding particles. At rest, a particle (nucleon or pion) has a radius -1/.u and volume v 0 = 41T /3,u 3 • In the center-of-mass system of two colliding nucleons the volume is reduced in the longitudinal direction by a fa.ctor E c/M (Ec is the energy and M is the nucleon mass). Therefore (1.8) ( 1.9a) The thickness of the region in which the nucleons overlap is (1.9b) In such a case v1/ 4

.....

E-1/~, and (1.7) yields c

(1.10) Here it is assumed that all the final particles are of one species, for example, pions. Of course, when nucleons collide, there must be at least two nucleons in the final state, but if n » 1, their fraction is small; this will be discussed below. If one uses the numerical coefficients valid for black body radiation, (1.4)-(1.6), and takes the volume V from (1 .8), then n= 3.8

(E ,!-; )'/. = 3.2 (E )'!• = 1.5 (ET )'/• = 2.0 (Ell" L

L )'/•

iH

.

(1.10a)

This formula, which gives the mean multiplicity as a function of the energy, does not contradict the experiments (to be sure, they are still rather crude) in the energy range 10 ;( EL ;( 105 GeV (see [18], Fig. 27). Since there are reasons to believe that at superhigh energies the process has a peripheral nature, this agreement is largely fortuitous (see below). But, as we have already mentioned, the second hypothesis has an inconsistency: if the particles interact strongly, they will, while flying apart, continue to interact and undergo mutual transformations (in particular, multiplying) until the mean distances between the particles exceed the interaction range 1/,u . It is therefore more consistent to assume that the volume in which equilibrium is established is not VF (1.8) but, for example, the final volume of all the n final particles at the moment when the interaction ceases: V=

Vp~nV Tcr /iJ. > 0.90

and it is moreproba ble that (for Tcr/ iJ. = 0.93) gqq 1.0 · 10-15 • If gqq = 36, this means that there are . . . 4 · 10-14 antiquarks (and as many quarks) per collision (i.e., the effective quark production cross section is . . . 10-39 cm2). This lies at the limit of present-da y experiment al feasibility. Formula (4.2), which holds for nq » 1, also has a region of applicabilit y. For example, there has been recorded a case of a Ca nucleus with an energy of 1012 GeV per nucleon in a oosmic-ray flux colliding with a photoemuls ion nucleus, more than 500 particles being formed in the collision [43]. In this case, n 11 is solarge that n Ncan also be appreciable . Although the condition for the theory of (1.20) (which is derived for a nucleon-nu cleon collision) to apply is violated, the energy of an individual particle in the final state is not high, the Lorentz contraction of its interaction sphere is insignifican t, and there is no transition to hydrodynam ics. For antinucleon s (IDq/M = 1) formula (4.2) gives nq »1. However, even for deuterons (and, a fortiori, for mq/M = 3) one must use formula (4.6a). Under the conditions of such a collision, (4.2) gives ~ /M = 1. and 2 and nq: = 12 and 0.6 · 10-3 , respectivel y. Let us briefly consider other papers in which the nurober of created heavy particles has been calculated. Hagedorn [41] has calculated the production of d and light nuclei (d, H3 , He 4) and obtained a nurober of important results. However, he based his work on the Fermi model and accordingly did not use experiment al data to determine the decay temperatur e. For d production our results depart quantitative ly from Hagedorn's . To predict the quark production cross section, Domokos and Fulton [44] used formula (4.2), which is also derived from the statistical model. They did not use experiment al data but specified arbitrarily V = V0 in accordanc e with the Fermi model. They thus obtained TCJ: ~ 2 iJ.. Further, formula (4.2) cannot be used when nq « 1- for pair production in an experiment one must have 2mq and not mq. As a result, the formula used in [44] contains an exponential dependence cc exp(-amq) with an argument that is reduced by a factor . . . 4. Accordingly, the conclusions drawn in [44] concerning the possible detection of very heavy quarks are too optimistic .

ucr

In [45] Hagedorn obtains a formula that is almost identical with (4.6). Essentially his treatment is along lines correspond ing to hydrodynam ics except that, first, the distribution of the hydrodynam ic velocities is not found by solving the hydrodynam ic equations, as in the Landau theory, but is described by two functions chosen by comparison with experimen ts. Secondly, Hagedorn uses a specialfor m of thermodyn amics developed by hirnself in which there is a maximum possible temperatur e, T 0 = 158 MeV ~ 1.16 iJ. • It is not surprising that the particle mass distribution is found to be similar to ( 4 .6), since hydrodynam ic effects do not influence the compositio n of particles produced at a fixed temperatur e.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

175

Hagedorn's model differs from our treatment significantly in that he assumes a temperature that does not exc eed T 0 ,..., f.L for all stages of the expans ion and flying apart of the bunch. But in our treatment the temperature may be high in the initial stages, so that an admixture of large-mass particles can initially be significant. It is only during the subsequent expansion and cooling that they are annihilated and transformed into pions. There remains, therefore, the possibility of heavy particles' "escaping" during the early stages because of the small dimensions of the system. We here come up against the important question of possible departures from thermodynamic equilibrium and the effect this has on our conclusions. It is undoubtedly a crude approximation to assume that such a small and short-lived sys-

tem as an expanding bunch of meson-nucleon plasma of mass ,..., (3-8) M is in quasiequilibrium at each instant of timeandin its entire volume. The only justification is subsequent successful comparison with expeFimental data. The departure from equilibrium will be especially important for subtle effects and improbable events. The success of the crude model in describing p and d production is remarkable, but it does enable one to rely on the model. In reality, however, a collision may not be statistical but direct, in particular, peripheral. On the other hand, even in the compound state, created heavy pairs can escape in the early expansion stages, when T ,.. mq, and the equilibrium number of heavy pairs is accordingly large. Even if the departures from the equilibrium scheme are relatively small, they can, on the transition to large masses- for mq ;;; 3mN, when nqq is very small- be important and lead to llqq having a value I arger than that indicated by formula (4 .6). We do not have at our disposal experimental data for production of pairs with mq > 2mN. At the same time, we consider below (see Sec. 7) statistical elastic scattering, for example pp- pp, when additional particles arenot formed; this is just such a case of escaping during the initial stage of expansion, when V) and is therefore a rare fluctuation. It was quite probable that the statistical model would be too crude to describe such subtleties, butthiswas not the case. In the course of subsequent investigations new interesting features of the phenomenon and the fruitfulness of the statistical model were revealed. But details were also found that were interpreted as contradicting the model. It is necessary to distinguish two such kinds of contradiction. On the one hand there are facts like these: there is no symmetry with respect to () c = -rr I 2 in pp scattering at eriergies 1 Ge V, there is a diffraction peak that decreases rapidly with increasing -t, in good agreement with peripheral collisions due to the exchange of a vacuum pole or some particle (at angles Be near 1r, i.e., at a four-momentum transfer in the crossed channel (-u). 112 0. If there were no statistical scattering, then Im fd > 0, whereas now Im (JJ + ff )> 0. Defining, as usual, elastic scattering O"el and inelastic scat..:. tering O"inel in terms of S 1 (see, for example, [621), we obtain

a., = ainel

~ ~(2l + 1) I!~+ /!1 2 , I

= ~ ~(2l + I

Otot

1)

[21m(/~+/~) -Jt~ +

= :. ~(2l + 1) [2Imf~ + 2Im/!J. I

/!1 2 ], (8.12)

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

189

Fig. 17 The unitarity condition is shown graphically in Fig. 17. The first four diagrams express the contributions of the various forms of scattering: a) peripheral, b) diffraction peripheral, c) statistical diffraction, d) elastic statistical. Finally, the diagram e represents the interference terms between the statistical and the peripheral processes. We now allow for the incident wave's not having a well-defined energy in practice. If the energy spread .6E for a given width r of the compound system's level is sufficiently large, so that ( 8 .8) holds, it is expedient to average over the energy in all the relations. In doing so, we shall assume that the entire "fast" dependence on E is contained in the phase shifts o~ , all the moduli of the amplitudes depending less strongly on the energy. Averaging over the energy in Eq. (8.3), we obtain, since the term prol?ortional to the rapidly varying function cos (oJ - of) disappears, and the term with cos (o~ - o~ ) remains only when l = l', d:;

t

"Q (z

d"Q=kfLJ .

l ,I' '

=

1 , (t' 1\ z z· z z· . z z· +T) +T)Pz(cos8)Pl'(cos8){j/d///d 1 cos(öd-öd) -t- 1 /,Jj/,JÖu·}= \

~,t[~(l+ ;)t~P 1

(cos8)/ 2

-r~(l+

;r/f!J 2 P((cos8)}=(;~)d+(:~),

( 8 .13)

The first term in the curly brackets is the differential cross section of statistical diffraction scattering (d o/ cill) d, and the second term is the statistical scattering cross section (du/ dQ) 5 • Averaging in ( 8 .12) yields the expressions ~ -,.;el

;t LJ "Q(?l..L ""J(! - i 1) [ l

6inel

=

;2

/1 1d 1: ..L I'/•1 /2 ] • 1

~(2/ + 1) [21m/~ -J/~/ 2 -j/!/ 2 ],

( 8 .14)

l

Cirot

= :, ~(2l + 1)2Im/~. l

In Fig. 18 we represent the unitarity condition after averaging over the energy. It differs from the unitarity condition in Fig. 17 in the disappearance of the term Im f i and the interference terms between the statistical and the peripheral processes (the diagram e).

We now note that the partial amplitude f ~ of dle statistical diffraction scattering is related to the partial cross section of the inelastic interaction that leads to the compound state in the s ame way as the total and the elastic cross sections are always related (see, for example, [62]): Os(I)

~tot

'.Js

= k2nl ( l _

-

"Q

+ 2t

) (21 m f dz -

Itzd /")- ,

( 8 .15)

( 8 .16)

..,(1)

LJvs · l

Fig. 18

190

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKI I

Since we assume that all the impact parameters satisfy p :s Prnax, all the inelastic processes proceed through the formation of a compound system (this assertion is equivalent to our neglecting all processes intermediate between peripheral and statistical), tot

acomp = a, •

( 8.1 7)

Substituting (8.17) into (7.1) and remembering (8.15) and (8.16), we obtain

a~ 1 = !~ ~(z ++)(2Im/~-I/~\ 2 )W2.

( 8 .18)

I

On the other hand, by definition, ( 8 .19)

Therefore, * ( 8 .20)

This important relationwill be used later, especially to analyze specific models. Since W is 2 exponentially small at high energies ~see Sec. 7), we can say that the amplitude f f is small in absolute magnitude compared with f d· This means that if the unitarity condition is written down for the partial amplitudes the diagram d of Fig. 17 will make a contribution that is exponentially small compared with the contribution of the diagram c. If only one partial wave (or few waves) were to contribute to the sum in the expression (8.13), the second term could be ignored compared with the first. Then statistical scattering would always be exponentially small compared with statistical diffraction scattering. It is this case that essentially is considered in [63], in which the correct conclusion is drawn that the experimental data cannot be explained in such an approach.

However, there is in reality another case, omitted from the treatment in [63]. If many waves (although, possibly, a bounded number) occur in the sum in (8.13), interference, which is moreover destructive, plays an important role in the first term, as can be readily demonstrated for the optical model or Van Hove's model of uncorrelated jets [64]. As a result, the two terms make similar contributions, although f~, which is due to all possible decay channels, is much greater than f ~, which corresponds to a single, exponentially weak channel. In Sec. 9 we show that for the optical model with f ~ = fo expf-a/k 2 ( l + 2] (k are the cms momenta) in such a case and for large angles

X)

(

;~

)a

~ exp ( -

k: )

and (

;~

). - exp ( - A

yS).

Then at sufficiently high energies and for angles greater than some angle Ocr defined by the condition (da(Ocr) /ili2)d = (da(OcrV drl)s, statistical scattering is mo:i:e important that statistical düfraction scattering. As a function of the energy for fixed angle, the diffraction cross section has a point of inflection, corresponding to the transition from the diffraction to the statistical regime. Of course, the regime change must also. be observed in the angular distribution at a fixed energy. These questions are considered in Sec. 9. Having obtained general expressions, we can calculate the angular distribution of the scattering at large angles . Let us first consider the angular distribution for statistical scattering (da I dQ) s . For spinless particles, it follows from ( 8 .13) that (the entire treatment is in the center-of-ma ss *Strictly speaking, in deriving (8.20) we have assumed that

w 1 does not depend on l.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

191

system) (8.21) Assuming that at high energies many partial waves are significant, and using the asymptotic expression for Pz (cos e) that holds when l e » 1: (8 .22)

we obtain d:l ) ( dQ • =

We a.re interested in angles 1r I 2 - e ; then

"? + 21 ) I ls 1- [1 + sm(2l + 1) 9].

1 ~( k':tsin8 l

1 ,

.

(8.23)

e near 1r I 2, and it is therefore convenient to use the variable x = (8 .24)

It can be seen that the second term in the square brackets gives a small contribution after

summation. Ignoring it and replacing the sum by an integral, we obtain [cf. (8 .19)] z ( ~) dll s

k'n

i.

SIU

8

I 112 = "?~(z + ...!..) 2 /s -

a!l

2...1:'

Sill I:J

=

e er • * It is determined by the condition ( dr;(ecr(E);E)) =(dG(ecr(E);E)). dQ

d

dQ

(9 .2)

s

The transition from one regime to the other at the point E = Ecr(8) or e = e cr(E) must be very abrupt; for under these regimes the angular dependences are very different [exponential or Gaussian for (da/dQ)d; cosec e for (da/dQ) 5 ] and so are the energy dependences [exponential and Gaussian, and it would be remarkable if the constant factors in the arguments of the exponential functions were equal]. Thus, the transition from one regime to the other must be marked by a point of inflection in both cases. This result, which is obtained in [54-56], corresponds well, as we shall see below (in Figs. 19 and 20), to the pp scattering experiments. Accordingly, the symmetry of the angular distribution about e = 11" /2, which is characteristic of the statistical scattering but not the diffraction scattering, can be manifested only at a sufficiently high energy, E > E cr(1T /2), which is, in general, different for collisions of different particles. The experimental dependence of da/ dQ on the energy in the statistical regime (i.e., after the inflection) agrees weil, as we have already seen (Sec. 7), with the predictions of the statistical theory. A sensible effective temperature is also obtained. The most important thing is that the statistical theory predicts (in agreement with the experiments) a very strong- exponential, or almost exponential- energy dependence. Thus, neither the departure from symmetry about e = 11"/2 nor the more rapidly increasing angular dependence than "'Cosec e in the neighborhood of this point (which are observed at insufficiently high energy) are, in general, arguments against the statistical theory .t Rather the opposite: the change of the regime and the concomitant appearance of symmetry and approximate isotropy, which are observed when the energy increases, are arguments in favor ot:. the statistical theory. The next important r~sult of the theory is the pronounced difference between pp and pp (and, in general, NN and NN) scattering (8.33). This prediction of the theory has not yet been verified (see below). Finally, the relationship between pp and np scattering at measured sufficiently accurately.

e = 11" /2

has not yet been

We now turn to a more detailed quantitative comparison of theory and certain experiments. A . E n er g y D i s tri b u t i o n in p p Sc a t t er in g . In Fig. 19 the points are the experimental values of da/dQ in pp scattering in accordance with the data of [70-72]. The continuous curve corresponds to formula (8.13)

(9.3) *In reality, statistical scattering may also be manifested if Eis somewhat less than E cr(11" /2), but this occurs at angles e > 11" /2, and the characteristic symmetry about e = 11"/2 is absent. t But such a criticism was put forward in [69].

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

I

-Jif

\

\

197

Fig. 19. Dependence of da I dn on k 2 (Ge V I c) 2 • The experimental points correspond to the following papers: 0, [72]; •, [71]; x, [70]; 1) (daldn)d; 2) (daldn) 5 ;3) daldn.

in a model in which f ~ is taken in accordance with (8.37), so that the expressions (8.38) and (8.39) arevalid for definite values of a and fo and also Tel in the formula w~.

PP=

exp (- .4

Vs + B),

A--1-

(9 .4)

- T",

v's = E = 2E c• where A and B are constants. Namely, (9 .5)

This corresponds to the following values of the parameters in (8.37) and (9.4): Cl.=

0.297,

fo = 2.6,

A

= 3.24::::::::

1

2 _211 ,

B = 5.5

(9.6)

( all the energies are in GeV). Thus, A and B correspond almost exactly to formulas ( 7 .2) and ( 8.3 7) with T = Tel = 2 .2J.L. The same continuous curve in the region 5 ~ EL ~ 30 Ge V (to within deviations that cannot be noted in the figure) can be obtained by taking

(9. 7)

with the parameters a

= 0.267,

fo = 1,

A

= 3.24,

B

= 5.5.

(9 .8)

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

198

Strictly speaking, the value fo = 2.6 contradicts unitarity; for the purely imaginary f~ ::S fo ::S 2. Butthis is hardly significant, since the real part of the amplitude, which we ignore, is known tobe not small.

in (8.37) one must have 0

Figure 19 shows that the statistical model can well describe the observed dependence, and that the critical energy of the transition from the diffraction to the statistical regime corresponds to the momentum PL.cr ::::::8 GeV/c.

kcr = 2.4 GeV/c 2,

(9.9)

B. p p Sc a t t er in g. One of the most interesting predictions of the statistical theory is tlie dllerence between !arge-angle pp and pp scattering: a;~ must be appreciably smaller than a el • PP

Experimental data for pp scattering at Be = 1T /2 are hitherto available only up to E L = 6 GeV. They are plotted in Fig. 20 (squares); for comparision we also give the experimental points for pp (points and the upper curve) and np (circles) scattering. The continuous curves are drawn through the experimental points. In the region EL ~ 6 Ge V, the pp and pp curves have appreciably different slopes; the values of the cross sections themselves are also different (app is smaller than app by a factor of approximately 25). However, this difference is due to another factor and not the difference in statistical scattering. 6 Ge V, düfraction scattering is predominant and the slope of the curve is determined primarily by the interaction radius, as we have already discussed. Note that it is sufficient for there to be a small difference between a PP and O!pp for it to be possible to explain the observed difference between the diffraction ends of pp and pp scattering in accordance with ( 8 .37). In the region E L

~

Statistical pp scattering can be manifested only in the region EL > 7 GeV. The expected curve is shown in the dot-dash form. It is constructed on the basis of the known data for pp scattering and formulas (8.33) and (8.35) for the cross-section ratio. If the experimental data for pp scattering are extrapolated to the right of the point E L ::::; 6 GeV, the intersection with the expected curve occurs in the region E L ::::; 7 GeV, where the point of inflection must be found. Above this energy statistical scattering must predominate,

["g_

Q

'

1 - - -1

PP

'\1

i\

{

5

I

'J ' ·

{

0.00{

p

:

'.

I

5

8

Fig. 20

"

,,

2 10 12 {I( t=s/2,(GeV!c)

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

and there must be symmetry of the pp scattering about tistical theory requires experimental verification.

ac

= 1f12.

199

This prediction of the sta-

We should however emphasize the approximate nature of the predicted curve, since it is very sensitive to the value ofT eff• and, though to a lesser extent, to that of CJ ~P. It is clear from what we have said that for EL < 7 GeV, for which diffraction is predominant, we cannot expect symmetry of pp scattering about a = 1rl2. The absence of symmetry has also been confirmed experimentally. Unfortunately, it was falsely interpreted as an argument against the statistical theory of scattering [69].

C . A ng u 1 a r Distribution. In the statistical model, the angular distribution at large angles is almost isotropic' 00 I dU~ 11 sin a . Unfortunately' there are few data that could be used to verify the angular dependence of statistical scattering. The problern is aggravated by the need to know the parameters of the diffraction scattering, so as to subtract its contribution from the experimental angular distribution. At the present time this problern can hardly be resolved. An attempt to describe the experimental data [73] at large angles as the sum of multiple peripheral diffraction (in accordance with the classüication of Sec. 8) and a c ertain constant quantity S has been mad~ as we have already mentioned, in [52] with the formula* 1030

dcr

dQ =

S

+ exp (c- b(k)k8].

( 9.1 0)

The experimental data were considered for the following values of the momentum p: 7.1, 8.1, 9.2, 10.1,11.1, 12.1 GeVIc. Thex 2 testwasusedtodeterminec andb,whichdepend on k but not on the angles. When c and b are substituted into (9 .10), the value of S can be found. It was indeed independent of the angles. It is readily seen that this agrees with the statistical theory; indeed, it was found that the energy dependence of S is well described by (8.23). Thus, S can be interpreted as the contribution of statistical scattering. There is, however, a more complicated problem: what role is played by the statistical düfraction scattering in the a-dependent term in (9 .1 0) ? It is possible that at relatively low energies (EL ~ 7-12 GeV) the statistical düfraction is still insignificant on the background of the peripheral scattering and does not appear in (9 .10) at all. Then, at high energies, when the peripheral diffraction cone contracts appreciably, the statistical diffraction may be manifested at angles between the peripheral diffraction and the statistical scattering. An argument in favor of this possibility is the following fact. The expression found in [52] and interpreted as multiple peripheral diffraction (see Sec. 7) is valid only for kfl 50°, as well (after the multiple peripheral diffraction has ceased to make an appreciable contribution to the cross section), while in accordance with (9.10) dCJ/dU should alreadyhave reached the statistical regime. It is possible that this decrease is a manifestation of statistical diffraction. D. Comparison of pp and np Scattering at Large Angles. Wehave shown in Sec. 8 that the differential cross sections of statistical elastic pp- pp and np- np scattering behave at large angles as (dCJ/dU) ~ cosec 8 and are equal at 8 = 1r/2.

*In this formula [formula (3) of [52]] there is also a term of the form A exp (c- (d + b/2)k8] · cos (dk8 - q;), which gives an excellent description of the oscillations found in [73] at intermediate angles. This term also belongs to the düfraction scattering "tail" and is small for 8 ~ 1fl2.

200

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

Fig.21. Dependence -30 of da I ein on 8 . The points correspond to the experimental data -32 of [71, 73]; the continuous curve to multiple peripheral diffraction [the second term in ( 9 .1 0)] . 0

30

50

/t/,

fOO.l

?.0

~

3.0

lf.U

0

2.0

'-1.0

0

3.6

ceev;d

2.0 1(.0 6,0 0 2.0 '-1.0 6.0

lf.5 GeV/c

'-l.f

~

[t

fi

t

1:-

t -~ N~ u O,OffO .

.......

> ~

->:::,100,0

o

.0

E

:;;'

0

-f,O

f.O

0

11.0

8.0

o

1(.0

I

~~

- f, 0 {. 0

8.0

fi,f

~6

mor lt ~

::\ 401"

O.OOf f.O

y 0

-to f.O

0

-1,0

-f,O 1.0

1(,0 8.0 .;:.,orr-r,.:;...,rr=..;..,

~

~

0

f.O

Fig. 22

0 - f,O cosft (GeV/c) 2 '-1.0 S.!J

/t/,

o

6,8 Ge V I

201

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

ltl, to-zsO

lf

(GeV/c)

2

8

Fig. 23. Angular distribution of np (circles) and pp (continuous curve) scattering for 5 .9-7 .2 and 7 .0 and 7.06 GeV/ c, respectively.

Figure 22, which is taken from [74], shows the angular distributions for np scattering at different energies. It can be seen that at low energies, when PL is in the range between 3 and 5 GeV/ c, there is no symmetry about e = 1r/ 2. We conclude that diffraction scattering still predominates over statistical scattering. As the energy increases, symmetry begins to appear. This can be attributed to the statistical scattering' s becoming comparable with the diffraction, and then exceeding it. It can be seen from Figs. 23 and 24 [75] that the pp and np angular distributions do not differ strongly, and that even for EL"' 5-7 GeV the np and pp cross sections at ec = 1T /2 are comparable and, at least, do not contradict the value R = 1 (8.29). Thus, both the equality of the np and pp cross sections and the symmetry of the np scattering about ec = 1r/2, which appears as the energy is increased, agree with the statistical theory. We emphasize that absence of symmetry in the angular distribution at high energies could only mean, strictly speaking, that (do/dQ~ at e = 1T /2 decreases with the energy faster than (do/dQ)d. This is evidently not the case; there is symmetry, and we can observe the statistical scattering, which is not a priori necessary.

_,

i z

"'u

> toJ

s

4)

Fig. 24. Dependence of da I dt on t for pp- pp, 5.0 Ge VI c (1) and np- np, 5.1 GeV /c (2).

.0

::1.

....,' to 2

~ ~

0

z

S

t,..(GeV/c)

2

202

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII fOYr------------. J(--p

Fig. 25. Dependence of da I dt on t for 1T -p scattering. The points are the experimental data for 3 .63 (1), 8 (2), 8.9 (3), and 12 (4) GeV lc.

Our theoretical conclusion that eer < 1T I 2 is based, first, on the statistical scattering' s energy dependence being given by (8.32). It can be obtained either by choosing adefinite equation of state for the compound system or by assuming that the compound system decays from a volume V ""' V0n. Secondly, to calculate the statistical diffraction scattering, we chose a particular form of the partial amplitude in analogy with the optical model. However, at e c "' 1T 12, such an approach may give an inaccurate result. The optical model (Kirchhoff approximation) is only valid for angles e that arenot too large. Therefore, even if symmetry does not appear in a particular elastic scattering, this will only mean that in the given process the statistical scattering is suppressed compared with the diffraction, while in other processes this may not be so. In any case, it does not prevent one•s applying the statistical concept to particle production processes.

JJ

E. Elast ic 1T ± p Sc a t t er i ng. The experimental data are very sparse for this process at large angles and high energies and are less accurate than for pp scattering [76, 77J. We can therefore consider only crude aspects of the situation. Experiments were carried out at p 1 = 3-12 GeV I c for 1T -P and at p 1 = 4-12 GeVI c for 1T +p scattering. The results [761 are shown in Figs. 25 and 26. At large momentum transfers

Fig. 26. Dependence of da ldt on t for 1T+p scattering. The experimental data for 4, 8, 12, and 12.8 GeV I c correspond to the curves 1, 2, 3, and 4.

203

I. N. SISAKYAN, E. L. FEINBERG, Ai'ID D. S. CHERNAVSKI I

• I

oZ •3

t:>i{

Fig. 2 7. Angular distributions in 1r+p (1) and 1r-p (2) scattering for 3 GeV /c; in 1r+p (3) and 1r-p (4) scattering for 4 GeV I c. 2

3

lt

5

-t,

5

2

(GeV/c)

(-t) 112theresultso f acalculation made in accordance with (7.3) in the statistical model [49] without allowance for diffraction:*

(:~

t= :l~~mpexp[-3.17(E-1.4)], "P

( 9 .11)

agree to within the experimental errors with the experimental data [76]. The free parameter was O'comp• which was determined from the data for p L = 8 GeV. It was found that O'comp.rr p = = 3.3 • 10-27 cm 2 ;::::: 0.1atot,rrp. The differential cross sections for PL = 3.6 and 4.0 GeV /c were then calculated. The values obtained are in good agreement with the experimental data. From themadiffere ntial cross section of about 5 ·1o-33 cm2/sr was predictedfor PL = 12.0 GeV/ c. This quantity is almost an order of magnitude lower than what can be reached by the experiments that have been made. Indeed, the experiment did not reveal scattering in this region. To within the experimental errors (which are still very large), the angular distributions at large angles were practically isotropic at the larger investigated energies and symmetric about e = 1r /2, while at a lower energy, PL = 3.0 GeV /c, neither isotropy nor symmetry was observed [77] (Fig .. _27). This again agrees with the statistical theory which allows for diffraction. However, there have recently been published [78] preliminary results on elastic scattering experiments at large momentum transfers with PL = 5.9, 7.9, and 9.8 GeV /c. In these experiments, which had a somewhat higher accuracy than those in [76, 77]. symmetry about e = 71"/2 and isotropy We:"'e clearly manifested at large angles only when PL reached 9.8 GeV /c (Fig. 28, which is taken from [78]). If we now take the point

e = 1T /2 for PL = 9.8 GeV /c and use the formula dcr

~

dQ

=

crcomp,

2..'t2

np

VV.,..,

-, "P •

( 9 .12)

where W 2,rrp "'"exp [-3.17(E -1.40)], then O'comp.1Tj is found tobe much smaller than the experimental value [73, 74], namely acomp,!!p"'" 3 · lo- 2 "'"lo-2atot,1Tp . It should, however, be noted that our value of Wz,rrp was obtained by interpolating computer results. As we have already

* Formula (9 .11)

is obtained under the assumption that statistical scattering is isotropic. At the same time, formula (8 ,26) contains a dependence 1/ sin e; accordingly one should introduce afactor 2/71" in (9.11), which leads to O'comp,1Tp;::::: 5.2 ·10- 27 cm 2 or O'comp,rrp"'" 0.16atot,1Tp.

204

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNA VSKII

Fig. 28. Angular distribution for -P scattering at 5.9 (1), 7.9 (2), and 9.8 (3) GeV /c.

1r

70

90

iiO

130

150° 9

pointed out, these results are sensitive to the assumptions made in the calculations. For example, in [49] a somewhat different value, 2.8 instead of 3.17, is given for the coefficient of E in the exponential function. We recall that, physically, this coefficient is equal to the reciprocal temperature at which the compound system decays into 1r and p. At the present time it is not possible to determine the decay temperature experimentally for 1r p scattering in the way this was done for pp scattering because of the sparsity of the experimental data. Therefore, the value of O"comp,rrp is also determined unreliably.* Thus, the 1r ± p scattering experiments can be explained qualitatively by the statistical concept, but a detailed quantitative comparison is not yet possible. Summarizing our confrontation with experiment, we are inclined to say that in pp, pn, and 1r p scattering at angles near e = 1r /2 there are exhibited all the main features of statisticcal scattering predicted by the model (provided, or course, statistical diffraction is allowed for): at a sufficiently high energy the scattering becomes symmetric about e = 1r /2, is approximately isotropic, decreases exponentially with increasing Ec, and the experimentally determined parameters of statistical scattering- the temperature of the system and the cross section O"comp of a central collision- have reasonable values: T ~ 2/-L, and Ucomp is less than u tot and (depending on the particle species) O"comp"' ( 0.05-0 .20)Urot . For pp collisions the statistical model makes the very interesting prediction (8 .32), which awaits experimental verification. The model is not Contradieted at all. However, in this entire analysis we have ignored an important problern- the absence of Ericson fluctuations in the cross section with varying angle and energy; as we pointed out at the beginning of Sec. 7, these would be expected by analogy with the compound states in lowenergy nuclear physics. This questionwill be discussed in Sec.12. 10 .

Fundament a 1 Questions in t h e Statist i c a l

Theory of Multiple Production From the inception of the statistical theory of multiple production it has generally been assumed implicitly that this theory is only an approximate method of calculating the characteristics of the process by which two particles interact and that the future rigorous theory of strong interactions- of the type of a dynamical S-matrix theory- will be able to give exact results, which we are at present forced to approximate by the formulas of the statistical theory. Accordingly, it is generally assumed that there cannot be any fundamental problems in this theory, and that everything reduces to the problern of constructing anS matrix. We should like to counter this opinion by emphasizing that such an approach is not only not forced upon us, but may even be false [79]. For we have here the same problems as arise when classical statistics and mechanics are considered. The mostfundamental is the problern of time irreversibility of statistical processes and the possibility of describing a process dynamically. *Note that ucomp,rrp ~ 2.4 ·

lo-27

or O"comp,ITp ~ 0 .06utot,pp.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

205

In questions of multiple production this is not purely abstract; for its solution will affect the

prediction of observable effects such as the Ericson fluctuations, which we shall discuss below. The role of fundamental problems in the statistical theory should be neither overestimated nor underestimated. For example, it cannot be concluded that because the statistical theory contains unresolved problems it is not sufficiently well founded and cannot yet be used. The statistical theory of multiple production is no less secure than classical or quantum statistics. The latter are widely and successfully used, although the problern of their justification has not yet been completely solved. But the fundamental problems of the statistical theory should not be underestimated. Their presence brings the statistical theory into a direct relation with certain fundamental and acute problems of the modern t:Q.eory of elementary particles. We have already discussed the different formulations of the statistical theory (see Sec .1). Fora long time it seemed that Fermi's formulation, which is based on the expression (1.1), was not subject to doubt and was true irrespective of the process's being statistical or dynamical. Subsequently, however, data were obtained that forced one to consider the problern more closely. For example, attempts to calculate the cross section on the basis of Fermi's original formulation with very varied forms of the matrix element M(p1 , . . • , Pn) generally employed in the statistical theory [2 7] gave an exponentially increasing cross s ection (80 ]. This problern and its relation to entropy increase will be considered below, in Sec. 11. Another problern arose in the analysis of large-angle scattering. The formula used to describe this process, formula (1.1) - again under fairly natural assumptions about the form of the matrix element- predicts Ericson fluctuations, which were not observed in the case of pp scattering (see Sees. 7-9). In Sec. 12 we shall show that this question is also related to fundamental problems and that, if the statistical theory is correctly formulated, there is a possibility that it ca.n be successfully resolved.

11 .

T im e Ir r e ver s i b il it y an d E n er g y D i s s i.p a t i o n

This problern is well known in classical physics, in which it can be stated as follows: the equations of mechanics, which must describe all systems, including statistical systems, arereversible in time. The solutions of the equations of mechanics arealso reversible; however, in statistical systems processes take place that are irreversible in time and are well described by kinetic equations (in particular, the equations of hydrodynamics and thermodynamics). The latter, in cantrast to the equations of mechanics, contain terms that are explicitly noninvariant under time reversal. But is this not a contradiction? Can one "deduce" kinetic equations from mechanical equations? To solve this question, one must revise and make more precise a number of concepts, for example, that of an isolated closed system. The situationwas clarified after the investigation of the stability of mechanical solutions. It was shown that mechanical solutions frequently lose their stability, which corresponds to the cases when the system becomes statistical. Let us consider this in more detail. Suppose that in a system in which there are multiple interactions of particles there is a random per-

turbation at the time t 0 of the traj ectory of a particle by an amount ßXo; then this deviation will develop in accordance with the law ßx (t) = Llx0 exp A. (t).

(11.1)

206

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

Here A (t), which is known as the characteristic number, is a function of t that vanishes when t = t0 • If ReA (t) ~ 0, the motion can be regarded as stable, and the solutions also; with the course of time, any perturbation is damped, and the system returns to the original state. Deviations from it are small, of the same order as the perturbation. A similar situation obtains even if ReA (t) > 0 but in the whole time range A (t) remains bounded, for example, ReA(t) ~ 1.

But if ReA (t) > 0 and after a finite interval of time one can have ReA (t) » 1, the solutions cannot be assumed stable. Because of the exponential dependence on A (t), the deviations ~x(t) become large, and the original solution is destroyed with the passage of time. Strang instability of this kind arises if A is a strong function oft, i.e., power-type or stronger, but not logarithmic. Clearly, if there is strong instability, the process cannot be reversible in time in reality because the system departs strongly from the original traj ectory. Under time reversal, the addition of a random external perturbation similar to the first gives rise to a new exponential departure of the trajectory, but it does not lead to the original trajectory (except in the completely improbable case when the magnitudes of the first and second perturbations compensate each other). Investigation of the classical systems that are generally regarded as statistical revealed they always exhibit strong dynamical instability. The characteristic number has the form A (t) = t/ T 0, where T 0 is the collision time of the particles in the system, and with the passage of time it may therefore become arbitrarily large. This result can be regarded as decisive in the question of the correspondence between statistical mechanics and dynamics. In an intuitive form, arguments of this kind were put forwardlang ago- by Borel [81}, Poincare [82], and Born [83]. In a more rigorous mathematical form they were developed by Krylov [84], and then significant progresswas achieved by Kolmogorov, Sinai [85], and others. Let us consider the fundamental aspects of the problem.

First, if the dynamical solutions are unstable, it is meaningless to consider an absolutely isolated system. A treatment of this kind will certainly not describe a real process. The result obtained when external influences are ignored differs radically from that obtained when they are allowed for. In other words, the dynamical solutions are here unpredictable. It is impossible to predict, for example, the positions or the velocities of individual particles in a statistical s ystem. However, using the equations of mechanics, one can formulate equations - kinetic equations- for averaged quantities: the pressure, temperature, etc. In "deriving" these equations, one must carry out an averaging procedure that has not hitherto played a role in the postulates of mechanics. The averaging is the fundamental factor, made necessary by the instability of the original solutions, and it must augment the postulates of mechanics. The kinetic equations for the mean values do have stable solutions and they describe irreversible processes; the t-noninvariant terms in these equations arise from the averaging procedure. Secondly, it is very important that the deviation ~x(t) depends linearly on the external perturbation ~x 0 but exponentially on A (t) . This means that the fundamental factor in the stability problern is A (t) while the magnitude ~x0 of the initial perturbation is of almost no significance; it can be very small. The characteristic number A (t) does not itself depend on the external perturbations and is an internal characteristic of the system itself. It enables one to determine whether the system is statistical or dynamical irrespective of the value of ~Xo.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

207

Systems that are stable or weakly stable [for them Re.\ (t) t:: 1] can be assumed tobe dynamical. Strongly unstable systems, for which Re.\(t) may take values Re.\(t) » 1, aretobe classed as statistical. We should emphasize that this criterion (the sign and magnitude of the real part of the characteristic number) is not, in general, directly determined by the number of particles in the system. One can give examples of systems with many particles, tut in which, nevertheless, .\ (t) t:: 1, and the behavior is dynamic. On the other hand, there exist systems of only two or three objects in which, however, Re.\ (t) » 1 and the behavior is statistical. Therefore, a criterion for statistical behavior that is based on the number of particles in the system is, in general, false. We have already mentioned the problern of the relationship between a dynamical description and the statistical nature of systems in classical physics. It is readily seen that in the statistical theory of multiple production the problern is just as acute; for in this theory we have spoken of an entropy increase of the system (accompanying multiple production), but at the sametime we have spoken of an elementary event, the collision of two "elementary" particles. Is irreversibility possible in such an event? In quantum mechanics the relationship between a dynamical theory and statistics h~s not yet been fully elucidated. Here, too, there is a paradox similar to the classical one: it can be shown that the entropy, defined as the expectation value of a certain operator during the dynamic development of the system, remains constant. We wish to emphasize that once again we come up against the problern of the stability of dynamical characteristics such as the amplitude, phase, etc. However, it is readily seen that these quantities are stable in systems with a nondegenerate discrete spectrum; instability can arise only in systems with a continuous or strongly degenerate spectrum. The problern has been considered in detail only for individual examples of model systems. Nevertheless, by analogy with the classical case, we shall attempt to establish the consequences of instability and the cases in which manifestation of this instability is to be expected. We consider a system with multiply degenerate (multiplicity N0) energy level E. We take basis functions 'IJ.! 1 , 'IJ.! 2 , ••• , 'IJ.!N 0 • Suppose that at the initial instant t 0 the system is in a state '1J.! a that can be expanded with respect to a system containing N0 functions: No

(11.2)

c:

11

where the coefficients are normalized in such a way that the probability of finding the system in one of the N0 states is equal to unity: No

(11.3) n

We now assume that the state -wo: is unstable. A definite, but not the only,formofperturbation under which the system can exhibit instability is a random perturbation oV of the potential. It is well known that in the degenerate case the state can change (i.e., the coefficients C ~ can change) by a finite amount under the influence of an arbitrarily small perturbation;* the set *Note that this kind of quantum-mechanical instability differs from the classical. The difference is above all this: the time in which a new regime is established when there is aperturbation oV is of the order of ßt ~ ti/ß V, i.e., it is large because oV is small. Wehave seen above that in the classical case the characteristic time does not depend on the perturbation ßXo but is determined by the properties of the system itself. But the important thing for the subsequent treatment is this fundamental property of the system: a small difference be-

208

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKTI

of coefficients C~ after the perturbation has been applied depends on the form of the perturbation, but not on its magnitude. Weshall assume that the state \{Ia corresponds to a given (a-th) realization of the random perturbation öVa.. No

No

rt. = ~ c~·c~onm =~I C~[ 2 0nm + ~ c~·c~onm· n,m

n

~m

(11.4)

A

The value ( 0) a depends on the form of the perturbation öV a • Note that none of the states that arise as a result of a specüic realization öva. is yet statistical. Statistical behavior arises when we come to consider an ensemble of similar systems that düfer only in the form of the perturbation. Since the amplitude of the perturbations is arbitrarily small, one can observe only expectation values with respect to this ensemble. The main statistical hypothesis, which marks the transition from a dynamical to a statistical system, is this: as a result of averaging with respect to a sufficiently large nurober v » 1 of representatives in the ensemble, the second term in (11.4) becomes arbitrarily small, and we obtain (11.5)

V

where W n

=

__!__"" 1 Ca.[ 2 • the bar denotes averaging over the ensemble (in contrast to the quan'V ..:::::J n ,

rt.

turn mechanical averaging ( ) ) . The quantities Wn satisfy the normalization condition

(11.6) n

We see that for the transition from (11.4) to (11.5) it is not necessary to assume that N0 is large; it is only necessary that v be large. This corresponds to a classical system's exhibiting statistical properties even if it contains only a few particles. In the simplest variant, when the external perturbations are random, all the indices n are on an equal footing, and, with allowance for the normalization, we obtain i

Wn=-v • 1 0

(11.7)

The expression (11.5) can be rewritten in the usual form: (11.8) The index N0 gives the number of components in the basis in which the treatment is made. It can be seen from (11. 7) and (11.8) that the system is not, in fact, a pure state, but a mixture of N0 components. If the entropy is defined as the operator S = -ln W, then from (11. 7) and (11.8) (11.9)

tween two small perturbations ova. and öVC. gives rise to a large difference in the final states.

I. N. SISAKYAN, E. L. FEIN"BERG, Al'ID D. S. CHERNAVSKII

This corresponds completely to the physical meaning of entropy, since W0 case the probability of finding the system in a given state.

= 1/N0 is

209

in our

Note that the expectation value of the operator depends not only on the form of the functions that describe the system but also on their number, or rather on the number of pure states that make a contribution to the given mixed state. This number may be called the "dimensionality of the mixture basis"; in particular, the entropy depends on it alone (for a pure state, N0 = L and ( S) = O). Hitherto we have for simplicity considered a degenerate system with a well-defined energy. However, it is not difficult to extend the conclusions to systems characterized by a continuous spectrum and an energy spread of order AE. In this case, the "dimensionality of the mixture basis" is N = öEp(E),

(11.10)

where p(E) is the density of levels, and N is equal to the phase volume of the system. Systems of just this kind are of interest when we come, as we now do, to discuss the temporal development of a system. For dynamical systems (which are those in which either there are no random perturbations at all or the system is irrsensitive to them) d

~

Tt (0)

i

=T

~ ~

(11.11)

[HO].

Herewe have assumed that the operator 0 does not itself depend on the time explicitly. If, in addition, 0 commutes with the Hamiltonian, its observable value is also independent of the time.

" These properties are shared by the entropy operator S not increase in a dynamical evolution.

=

"

-ln W, so that the entropy does

Note that these conclusions arevalid only if the development is dynamical and the "mixture basis" does not change with the time (in other words, if a pure state remains pure, andin a mixed state the number of contributing pure states does not increase), but if in its evolution a system is sensitive to small random perturbations (i.e., it develops in an unstable manner) the situation is changed. In this case the "dimensionality of the mixture basis" increases and formula (11.11) no Ionger describes the variation in time of the expectation value. It is this situation that corresponds to real statistical systems. To elucidate what we have said, let us consider a very simple example- the expansion of a system. Suppose that up to the moment t = 0 a particle is in a volume V 0 and is described not by a pure state but by a mixture of N0 components. Suppose that at t = 0 the walls confining the volume become penetrable and the system can expand to a volume V1 > v 0 • Then the basis functions vary instantaneously, since we go over from the Hamiltonian H0 (with a system of eigenfunctions H0\.ll1 = Ei\.lli0 )to a different Hamiltonian H1 (with some system of functions '.{l ! : : : : exp [ii (ln fi- 1)]; for as the number of particles increases, so does the number of states of the complete system, which also means an extension of the basis. The matrix element, if it really is a moderately increasing function, cannot essentially change such a strong dependence. If the statistical theory is applied consistently, the increase in the phase volume associated with the extension of the mixture basis is compensated by the normalization (since the condition 'LW n = 1 must still be satisfied when the basis is extended). This is usually allowed for by replacing dPn by the relative phase volume: dp 0 /Ptot(E), where Ptot(E) is the total

212

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

phase volume at the energy E; then the cross section does not increase. However, such a procedure is not contained in the expression (1.1) and cannot arise in any dynamical theory. In addition, in a dynamical theory of the S matrix it is, in principle, impossible to nor-

malize an intermediate state. This is due to the absence of the so-called half S matrix S(i:t, ~) a unitary operator realizing a transition of a state defined at time 11 into a state at the final time t 2 • As yet, wehavenot considered the role of the various parameters that determine the stability or instability of the system (we have merely said that continuity or degeneracy of the system's spectrum are necessary conditions for one or other possibility tobe realized). Nevertheless, it is extremely important to solve this problem, even for practical needs, as we shall see in Sec. 12. To this end, an attempt was made in [79] to treat a nonrelativistic model problern in which a particle of momentum p is scattered by a random potential V (r). The potential is defined in a region of radius R as a random function of the Coordinates (in the simplest form, a spherically symmetric potential) with correlation interval r 0 and variation amplitude V 0 • The investigation starts from the Schrödinger equation reduced to the usual form of a nonlinear differential equation for the partial phase function k 1 , which is frequently called the K matrix andin the spherically symmetric case* is equal to K 1 - tgö 1 , where ö1 is the phase shift of the partial wave of angular momentum l : d

-d

z

K 1 (r)

2m

= -

p

•

B FB

(Zer) (Zer! .

It is obvious that approximately ( e Ir) ,... ~ ,... a - 1/ 3 • The multiplicity is then determined in accordance with (n 0 = W / ( t1r) . The mean transverse momentum (P .L) of the pions is cal-

culated in accordance with (3.2). The mean transverse momentum of the heavy particles of mass mq » p. can be calculated from the approximate formula (3 .11) and is simply proportional to T 1er12 ,.... a- 116• It is probable that, given the present accuracy of the experiments and the not very good (for such large (n)) accuracy of the statistical approach, the a-dependence of the quantities given in Table 1 is of no significance and the value a = 1.0 adopted for estimates is perfectly reasonable. However, for physical quantities in whose expressions Ter enters in an exponential function, the dependence on a may be more important. For example, for the probability of production of heavy particles, we obtain the curves of Fig. 10 for different a, and comparing these with the experimental values we choose Ter :::::J 0 .93p. , which corresponds to a :::::J 1 .22. The experimental dependence of the p .L distributions for different 11s (Fig. 6) also indicate a > 1. The distribution over the momenta at E L ,... 300 Ge V (Fig. 5) is a Planck distribution with T ~ 0.80 GeV, i.e ., a :::::J 2. One must however bear in mind the errors in the experimental data for cosmic-ray experiments.

On the other hand, in accordance with the remarks made in the text after formula (2 .7), one could imagine that the theoretical value for (n) should be lowered, i.e., one should take a < 1. However, in this case comparison of the theory with experiments requires the knowledge of an additional, only approximately determined quantity: (K), the inelasticity coefficient.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

219

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30. 31.

32. 33.

W. Heisenberg, Z. Phys., 101:533 (1936); 113:61 (1939). G. Wataghin, Symposium on Cosmic Rays, Acad. Bras. Ciencias, 129 (1941); Compt. Rend., 207:358 (1938); Phys. Rev., 63:137 (1943). E. Fermi, Progr. Theor. Phys ., 5:570 (1950); Usp. Fiz. Nauk, 46: 71 (1952). l. Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR, 78:889 (1951). P. Amati, S. Fubini, and A. Stanghellini, Nuovo Cimento, 26:896 (1962). I.M. Dremin, I. I. Roizen, R. B. White, and D. S. Chernavskii, Zh. Eksp. Teor. Fiz., 48 : 952 (1965). I. M. Dremin, I. I. Roizen, and D. S. Chernavskii, this volume, p. 99. T. Regge, Nuovo Cimento, 14:952 (1959); 18:947 (1960). C. F. Chew and S. C. Frautschi, Phys. Rev. Lett., 7:334 (1961); 8:41 (1962). V. N. Gribov, Zh. Eksp. Te'or. Fiz ., 41:1962 (1961); 42:1260 (1962). Chan Hoang-Mo and I. Loskiewicz, Nuovo Cimento, Ser. X, 57A: 93 (1968). K. A. Ter-Martirosyan, Zh. Eksp. Teor. Fiz., 44:341 (1963). L. W. Nordheim and G. Nordheim, Phys. Rev ., 54:254 (1938). W. Heisenberg, Z. Phys.,164:65 (1949). L. D. Landau, Izv. Akad. Nauk SSSR, Seriya Fiz., 17:51 (1953). L. D. Landau and E. M. Lifshitz, Statistical Physics, London (1958). A. M. Baldin, V. I. Gol'danskii, V. I. Maksimenko, and I. L. Rozental', Kinematics of Nuclear Reactions [in RussianJ, Atomizdat, Moscow (1968). V. S. Murzin and L. I. Sarycheva, Cosmic Rays and Their Interaction [in RussianJ, Atomizdat, Moscow (1966). S. Z. Belen'kii and L. D. Landau, Usp. Fiz. Nauk, 56:309 (1955); S. Z. Belen'kii and A. I. Nikishov, Zh. Eksp. Teor. Fiz., 28:744 (1955); S. Z. Belen'kii, Zh. Eksp. Teor. Fiz ., 32:1171 (1957). G. A. Milekhin, Trudy FlAN, 16:50 (1961). E. L. Feinberg, Trudy FlAN, 29:155 (1965). I. Gierula, R. Holynski, A. Jurak, M. Miesowich, T. Saniewiska, P. C iok, 0. Stanisz, and I. Pernegr, Nuovo Cimento, 8: 166; 10: 741 (1958). G. Cocconi, Phys. Rev., 111:1699 (1958). S. Z. Belen'kii, V. M. Maksimenko, A. I. Nikishov, and I. L. Rozental', Usp. Fiz. Nauk, 62:1 (1967). A. N. Diddens, W. Galbraith, E. Lillethun, et al., Nuovo Cimento, 31:961 (1964); D. Dekkers, J. A. Geibel, R. Mermod, G. Weber, T. R. Willitts, and K. Winter, Phys. Rev ., 137: B962 (1965). R. Hagedorn, Nuovo Cimento, 15:434 (1960). I. L. Rozental' and V. M. Maksimenko, Zh. Eksp. Teor. Fiz., 39:754 (1960). 0. Czyzewski, Proc. Top. Conf. on High-Energy Collisions of Hadrons, Vol.1, Geneva, 15-18 January, 1968, p. 345. F. Turkot, Proc. Top. Conf. on High-Energy Collisions of Hadrons, Vol. 1, Geneva, 15-18 January 1968, p. 316. G. A. Milekhin, Proc. Intern. Conf. on Cosmic Rays (Moscow 1959), Vol. 1 [in Russian], Izv. Akad. Nauk SSSR, Moscow (1960). V. V. Guseva, E. V. Denisov, N. A. Dobrotin, N. G. Zelevinskaya, K. A. Kotel'nikov, A. M. Lebedev, V. M. Maksimenko, A. Ch. Morozov, and S~ A. Slavatinskii, Izv. Akad. Nauk SSSR, Ser. Fiz ., 33:1408 (1969). G. A. Milekhin and I. L. Rozental', Zh. Eksp. Teor. Fiz., 33:197 (1957). I. W. Elbert, A. R. Erwin, S. Mikamo, D. Reeder, Y. Y. Chen, W. D. Walker, and A. Weinberg, Phys. Rev. Lett., 20:124 (1968).

220 34.

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII

A. D. Krisch, Lectures on High Energy Proton-Proton Interactions, Preprint, Univ. of Michigan, 1968; D. G. Grabb, I. L. Day, A. D. Krisch, et al., Phys. Rev. Lett., 21:830 (1968). 35,36. M. Koshiba, Proc. Tenth Intern. Conf. on Cosmic Rays, Canada, 1967, Part A, p. 525. 37. B. Peters, Proc. Conf. on Cosmic Rays, Jaipur, Vol. 5, 1963, p. 423; B. Peters and Yash Pal, Mat. Phys. Medd. Danske Videnskabernes Selskab (1964). 38. W. F. Baker, R. L. Cool, G. W. Jenkins, et al., Phys. Rev. Lett., 7:101 (1961). 39. I. N. Sisakyan, E. L. Feinberg, and D. S. Chernavskii, Zh. Eksp. Teor. Fiz., 52:545 (1967). 40. V. M. Maksimenko, I. N. Sisakyan, E. L. Feinberg, and D. S. Chernavskii, Zh. Eksp. Teor. Fiz ., Pis. Red., 3:340 (1966). 41. S. Hagedorn, Nuovo Cimento, 25:1017 (1962). 42. T. Massam, Th. Muller, P. Richini, H. Schnagans, and A. Zichichi, Nuovo Cimento, 39:10 (1965). 43. K.Rybicki, Nuovo Cimento, 28:1434 (1963). 44. G. Domokos and T. Fulton, Phys. Lett., 20:546 (1966). 45. R. Hagedorn, Preprint CERN 67/269/5- Th 751, April, 1967. 46. H. Conversi, T. Massam, Th. Muller, and A. Zichichi, Nuovo Cimento, 40:690 (1965). 47. A. Jaffe, Phys. Rev. Lett., 17:461 (1968). 48. G. Fast and R. Hagedorn, Nuovo Cimento, 27:208 (1963); G. Fast, R. Hagedorn, and L. W. Jones, Nuovo Cimento, 27:856 (1963); L. W. Jones, Phys. Lett., 8:287 (1964). 49. R. Hagedorn, Nuovo Cimento, 35 : 216 (1965). 50. T. Ericson, Phys. Rev. Lett ., 5: 430 (1960); T. Ericson, Advan. Phys ., 9: 425 (1960). 51. J. V. Allaby, G. Bellettini, G. Cocconi, et al., Phys. Lett ., 23: 389 (1966). 52. I. V. Andreev and I.M. Dremin, Yad. Fiz ., 8:814 (1968). 53. I. V. Andreev, I. M. Gramenitskii, and I. M. Dremin, Preprint No. 97 [in Russian], P. N. Lebedev Physics Institute, Moscow (1968); Nucl. Phys ., B10: 137 (1969). 54. I. N. Sisakyan and D. S. Chernavskii, Izv. Akad. Nauk SSSR, Ser. Fiz ., 30:1587 (1966). 55. I. N. Sisakyan and D. S. Chernavskii, Izv. Akad. Nauk SSSR, Ser. Fiz., 32:346 (1968). 56. I. N. Sisakyan, E. L. Feinberg, and D. S. Chernavskii, ZhETF Pis. Red., 4:432 (1966). 57. A. Bia.fas and V. F. Weisskopf, Nuovo Cimento, 35:1211 (1964). 58. G.Auberson and B. Escoubes, Nuovo Cimento, 36:628 (1965). 59. G. Cocconi, Nuovo Cimento, 33:643 (1964). 60. D.S.Chernavsky, Postepy Fizyki, 18:247 (1967). 61. J. V. Allaby, F. Binon, A. N. Diddens, et al., Fourteenth Intern. C onf. on High-Energy Physics, Vienna, 1968. 62. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergarnon Press, Oxford (1965). 63. C. H. Woo, Phys. Rev ., 137: B449 (1965). 64. L. V an Hove, Rev. Mod. Phys ., 36:655 (1964). 65. E. L. Feinberg and D. S. Chernavskii, Preprint No. 127 [in Russian], P. N. Lebedev Physics Institute, Moscow (1968). 66. I. N. Sisakyan and D. S. Chernavskii, Preprint No. 24 [in Russian], P. N. Lebedev Physics Institute, Moscow (1968). 67. V. I. Savrin and 0. A. Krustalev, Yad. Fiz ., 8:1016 (1968). 68. W. N. Cottingham and R. F. Peierls, Phys. Rev ., 137: B147 (1965). 69. A. Biaias and 0. Czyzewski, Phys. Lett., 21:574 (1966); Nuovo Cimento, 49:273 (1967). 70. G. Cocconi, V. T. Cocconi, A. D. Krisch, J. Orear, et al., Phys. Rev. Lett., 11:499 (1963); Phys.Rev., 138:B165 (1964). 71. J. V. Allaby, G. Cocconi, A. W. Diddens, A. Klovning, G. Mathiae, E. I. Sacharidis, and A. M Wetherell, Phys. Lett., 25B :156 (1967); Preprint CERN, ps/5985/k1 (1967). 72. C. Akerlof, R. H. Hieber, A. D. Krisch, K. W. Edwards, L. G. Ratner, and K. Ruddick, Phys. Rev. Lett., 17:1105 (1966); Phys. Rev ., 159:1138 (1967).

I. N. SISAKYAN, E. L. FEINBERG, AND D. S. CHERNAVSKII 73.

74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91.

221

J. V. Allaby, A. M. Diddens, A. Klovning, E. Lillethum, E. I. Sacharidis, K. Schliipmann,

and A. M. Wetherell, Proc. Top. Conf. on High-Energy Collisions of Hadrons, Vol. 1, Geneva, 15-18 January, 1968. I. Cox, H. L. Perl, N. M. Kreisler, M. I. Lougo, and S. T. Powell, III, Preprint SLACPUB-407 (1968). M. L. Perl, Proc. Top .Conf. on High-Energy Collisions of Hadrons, Vol. 1, Geneva, 15-18 January, 1968, p. 252. I. Orear, R. Rubinstein, D. S. Scarl, et al., Phys. Rev., 152:1162 (1966). M. L. Perl, Y. Y. Lee, and E. Marguit, Phys. Rev ., 138: B707 (1965). I. Orear, D. P. Owen, F. C. Peterson, et al., Fourteenth Intern. C onf. on High-Energy Physics, Vienna (1968). I.M. Pukhov and D. S. Chernavskii, Preprint No. 29 [in Russian], P. N. Lebedev Physics Institute, Moscow (1968); Teor. Mat. Fiz., 7:219 (1971). T. I. Efremidze, V. Ya. Fainberg, and D. S. Chernavskii, Preprint A-53 [in Russian}, P. N. Lebedev Physics Institute, Moscow (1965). E. Borel, Probability and ReHability [Russian translation], Fizmatgiz, Moscow (1961). H. Poincare, J. Phys ., 4:369 (1906). M. Born, Phys. Blätter, 11:49 (1955); Z. Phys., 153:372 (1958). N. S. Krylov, Papers on the Bases of Statistical Physics [in Russian], Izd. Akad. Nauk SSSR, Moscow (1950). A.I. Kolmogorov, Dokl. Akad. Nauk SSSR, 119:861 (1958); Ya. G. Sinai, Dokl. Akad. Nauk SSSR, 158:1261 (1963); D. V. Anosov and Ya. G. Sinai, Usp. Mat. Nauk, 22:107 (1967). T. Ericson and T. Maeyr-Kuckuc, Annual Rev. Nucl. Science, 16:183 (1966). E. L. Feinberg, Proc. Intern. Conf. on l'.'uclear Physics [in Russian], Tbilisi (1966). U. Facchini, E. Sactta-Menichelli, and F. Tonolini, Phys. Lett., 1:209 (1962). L. Colli, 0. Facchini, I. Iori, G. M. Marcazzan, M. Milazzo, and F. Tonolini, Phys. Lett., 1:120 (1962). A. S. Wightman, Fourteenth Intern. Conf. on High-Energy Physics, Vienna (1968), p. 431. V. Ya. Fainberg, Preprint No.137 [inRussian], P. N. Lebedev Physics Institute, Moscow (1967); Lectures at the School on Elementary Particle Physics [in Russian], Otepya (1967).

SPONTANEOUS BREAKING OF TRANSLATIONAL INVARIANCE IN QUANTUM ELECTRODYNAMICS E. S. Fradkin and A. E. Shabad Introduction In this paper we seek solutions that break translational invariance spontaneously. The

usual procedure of quantum field theory is adapted to finding solutions with the same symmetry as the original Hamiltonian. But, physically, it is perfectly possible for solutions with broken symmetry to be realized. In recent years there have therefore been many investigations (following Nambu' s pioneering work) devoted to spontaneaus symmetry breaking in quantum field theory. One usually proceeds from the unrenormalized system of equations. It seems to us more consistent (and, at the least, more convenient) to use the renormalized system of equations, which enables one to avoid infinities and the need to introduce cutoff parameters and all that this entails. One can formulate a completely renormalized system of equations adapted to all possible spontaneaus symmetry breaking. In the present paper we shall take as our point of departure a special form of this system which is especially convenient for studying the spontaneaus symmetry breaking associated with the appearance in the theory of a nonvanishing (and nonconstant) vacuum expectation value of the operator of the electromagnetic field. The results are reviewed and discussed in detail at the end of the paper. The reader can find some additional details in [1, 2}. 1.

Statement of the Problem

It is well known that quantum field theory can be formulated in terms of Green's functions, for which there is an infinite system of coupled equations. In 1954, Fradkin [3} (see also [4}) renormalized this system; it then took a form such that, solved by an expansion in the interaction constant, it reproduces the renormalized perturbation series of the ordinary Lagrangian approach augmented by the subtractional procedure. No divergences (not even the so-called overlapping divergences) arise (in (5} the Schwinger-Dyson system of equations is also renormalized; this excludes all the divergences except the overlapping ones). We write down the system of renormalized equations for the Green's functions in quantum electrodynamics in the presence of an external source J Jl of the electromagnetic field All (x) [4,6}, which will soon be allowed to tend to zero:

(ip + m + ~R(p))G(p,

+~

I(l)

p')-

ie~ ft.J.(p,

(p, s) G (s, p') ds

223

p- s, s) (At.J.(s)) G(p- s, p')d11

= Ö (p- p'),

,

(1)

224

E. S. FRADIGN AND A. E. SHABAD (ö".N-k".q(1-:t(kZ))(A.(k))=J".(k)-

r

eZt

(2n)•Sp~y".

-

[c (q-r' k ,q)-J['

öG (q

+ k,

q)

ö.

k".~:J)

_

I=o'/A• ( ) d ~1d

(2)

q,

s)

(3)

Here G(p, p'), Du p(k, k'), and r Jl (k1, k.z, s) are, respectively, the exact electron, photon, and vertex Green's functions in the presence of the classical source J !l (k): D.P (k, k')

(Z~)• ~ {(T (A". (x),

=

A. (x'))) -

( A". (x)) ( Av (x'))} exp (- ikx

+ ik' x') dx dx',

(4)

(Z~)• ~ (T ( 'Y (x), o/ (x'))) exp ( -ipx +ip' x') dx dx',

G (p, p') =

(5) (6)

roperty [certainly the case if 7T(k 2) is calculated [8] to e 2]: there exist space-like vectors k = k with fixed K' 2 = const = K' 2 - 1{6 > 0 suchthat (17) (as is well known, .

.

.

k 2 ......

m 2 exp {127T 2e-2 } » m 2 in the second order of perturbation theory). In

this sltuatwn, the photon propagator D ll

v -

öl'-"- k.".k)k 2 k 2 (t -n (k 2))

has a false pole in a space-like region

whose appearance contradicts the Kallen-Lebmann representation. This is the so-called "zero-charge" problern [9, 10]; it has been discussed often (see, for example, [11-16]). In a translationally invariant theory, the property (17) leads to the appearance of free "particles" described by a wave function Av(k) = auö(k- k'>, the solution of Eq. (16) (without right-hand side). However, these "particles" are forbidden by causality, as they move faster than light. Many authors assume that the false pole does not exist in real ~ty, and that its occurrence is due to the incorrect use of perturbation theory in the high-energy region, where the effective expansion parameter (e 2 1n k 2 /m 2) is not small. Nevertheless, the false pole has not yet been eliminated from the photon propagator. Redmond•s well-known procedure [11] cannot be regarded as satisfactory in view of its essential ambiguity [12] and basically prescriptive nature. In our approach we take the false pole seriously. Even if it does not exist, it is meaningful to do this in an attempt to construct a formalism different from perturbation theory that would not contain the difficulties of the false pole even in simple ap-proximations. In Sec. 3 we show that the false pole leads to the appearance of a transverse (in the fourdimensional sense) mean field ( A ll(x)) with constant polarization vector lying in the light cone and space-like wave vector. From the vacuum field obtained one can, in particular, construct a periodic spatial lattice.

In Sec. 4 we investigate the spectrum of boson excitations in the presence of the selfconsistent field we have found. To this end we solve the homogeneaus equation (16). Its solutions are regarded as "elementary excitations," i.e., as free quanta present in the theory. All the elementary excitations found tobe generated by the false pole have a remarkable property: Any wave packet formed from their wave functions propagates at exactly the velocity of light, despite their having a space-like wave vector. Their existence does not contradict the principle of macroscopic causality. Certain of these excitations have much in common with the photon. In Sec. 3 we introduce the concepts of the electrodynamics of continuous media and define an electric displacement and magnetic induction in addition to the field strengths. These quantities are calculated for a wave of excitations and the density of the macroscopic energymomentum is expressed in terms of them. This last is a quantity that lies on the light cone, which means that the elementary excitations can be ascribed a vanishing macroscopic mass, they themselves being massless vector Goldstone bosons corresponding to the spontaneaus symmetry breaking.

E. S. FRADKIN AND A. E. SHABAD

227

In Sec. 5 we obtain the Green' s function and wave function of a Dirac electron in the vacuum field. The electron spectrum is investigated in the diagonal representation of the macroscopic energy-momentum (the quasienergy-quasimomentum representation). It is shown that for attainable values of the quasienergy-quasimomentum the electron spectrum does not differ from the free spectrum, but that there are gaps in the spectrum at large quasimomenta. In Sec. 6 a further exact solution of the infinite system (1)-(14) is obtained for (AIJ(x)) with J IJ = 0. It represents a free plane electromagnetic wave. The corresponding Goldstone bosons are photons. This solution is discussed in connection with the problern of infrared divergence.

2.

On a Class of Elementary Excitations One can show [1,2] that Eq. (2) with JIJ = 0 reduces to (ÖI'-,k 2

-

kl'-k,-

II~, (k)) . (k, s) ( A, (s)) ds, 0

(18)

where rr(Uv;~. (k,s) is obtained by replacing (A) by ;\. (A) everywhere in rr(t). Comparing (18) with (16), we see that the kernels of these two equations differ only by integration with respect to ;\. on the right-hand side. This suggests that if Eq. (18) has a solution, Eq. (16) does not have solutions in the general case, since their Fredholm determinants are somewhat different, and it is therefore necessary that special conditions be satisfied if they are to vanish simultaneously. One can attempt to exploit this circumstance to eliminate the false pole (see Sec. 1) of the photon propagator. The property (17) of the polarization operator may be the cause of some of the solutions of the nonlinear integral equation (18). At the same time, there is hope, as we have said, that Eq. (16) does not have solutions, i.e., the appearance of the false pole is a sign that in reality the physical system has a translationally noninvariant state in which there is no false solution. For just such a Situation was successfully demonstrated in manybody theory [17], in which a false pole also arises at low temperatures in the translationally invariant solution for the temperature Green' s functions. However, there are serious difficulties bindering one from implementing such a program in quantum field theory; for one needs to find a solution for (A ix)) that would modify the asymptotic behavior of the electron Green's function G(p, p'), since this is responsible for the false poles' appearance. It is however clear that it would hardly be possible for any of the simple approximate methods of solution of the system (1)-(14) to lead to essential modifications in the asymptotic behavior. At the present time we dispose of a solution of the integral equation (18) generated by the false pole. Butthis solution is suchthat there are fulfilled those special conditions that render Eq. (16) soluble s imultaneously with (18): the solution satisfies the conditions of the theorem formulated and proved in this section. Therefore, the pole in the space-like region in the D function (with allowance for the spontaneously arising mean field) remains in its place, but now the structure of the D function is such that th.e class of elementary excitations (found by us) generated by the pole is much smaller than the class of elementary excitations to which the pole leads in the translationally invariant theory. Moreover, the properties of the elementary excitations are such that their presence does not contradict causality. Of course, it remains highly desirable to implement the above program in its original form. This is especially so for quantum mesodynamics, in which the solutions that we find for quantum electrodynamics in the following sections do not exist. However, this requires at the least approximate solutions of Eq. (18) that differ from those presently at our disposal.

E. S. FRADKIN AND A. E. SHABAD

228

We now formulate and prove a general assertion relating to the very special class of solutions of Eq. (18) and the corresponding elementary excitations. The two exact solutions that we obtain in Sees. 4 and 6 satisfy the conditions of the following theorem. THEOREM: If (Ail(k)) is a solution of the system of equations (1), (18), (3)-(14) such that a(Ail(k)) is also a solution of this system, then Eq. (16) has the solution All(k) = ß(A 11 (k)), where ß is any number. Proof. In the system of equations (1), (18), (3)-(14) we take a (Ail(k)) instead of (All). By hypothesis, this does not violate the system of equations. At the same time, rr ds = 1

a:

0

s sII~~>..(k, "

df..

0

s) (A.(s)) ds.

(20)

Differentiating with respect to a twice, we see that the integrand does not depend on A and, therefore, the integral with respect to A in (18) can be allowed with A replaced by unity. Then, substituting ß ( A v(k)) as a solution for A v(k) into (16), we see that (16) is satisfied, being identical with (18) after cancellation of ß. This proves the theorem. Wehave also proved that on the class of solutions described by the theorem's conditions the right-hand side of (18) vanishes sincefii (c11)>-.(k, s) (A V(s)) ds does if it does not depend on A, something which must ,V be clear from the efinition (12). Note that in the proof we proceeded from the exact system of renormalized equations of quantum electrodynamics without recourse to any approximations. The essence of the theorem is this: if the amplitude of the mean (vacuum) field is not fixed in the theory, it must contain a particle (elementary excitation) that is a quantum of this field. To the vacuum field one can add or remove a certain number of quantahomogeneaus with this field without violating the system (1)-(14) (at this juncture we do not discuss the energy aspect of such modifications). In other words, the vacuum contains an indeterminate nurober of quanta, or elementary excitations, that exist in the theory as free quanta- the vacuum is degenerate with respect to the particle number. Such a situation arises whenever there are excitations or bound states in the system that differ from the original in- and out-stat es. The former vacuum now becomes degenerate with respect to the new quantum numbers, the vacuum field amplitude's indeterminacy making it poss ible to quantize with respect to the new excitations (the case described by the above theorem is not the only possible way of achieving indeterminacy of ( A ~ ) . However, the quantization of this field requires the separation out of a certain part of the "classical" field, only the remaining part of the amplitude being quantized. 3.

Vacuum Fieldas an Electromagnetic Wave

Polarized along the Light Cone Suppose that Eq. (2) with right-hand side omitted has a solution corresponding to the false pole; in other words, suppose that in the exact translationally invariant D function there is a pole in the space-like region (in our approach this assumption was discussed in Sec. 1).

E. S. FRADKJN AND A. E. SHABAD

229

Let us consider the transverse four-vector quantity (21) where k is a space-like vector chosen suchthat the left-hand side of Eq. (2) vanishes on the substitution of (21) (this condition fixes only the invariant k' 2 , so that k has arbitrary direction). In the second perturbation order k' 2 ~ m 2 exp {127re- 2}, k 2 » m 2 • In the coordinate representation, (21) has the form (22) The four-dimensional polarization vector n 11 lies on the light cone, and the expressions (21) and (22) are transverse in the four-dimensional sense, n 2 = nk = 0. These two conditions are compatible only ü the vector k is space-like. Weshall discuss the specific properties of the electromagnetic field generated by the vector potential (21)-(22) a little later. We mention only that the expressions (21)-(22) are just as simple as the electromagnetic field of a free plane wave. We now show that the right-hand side of Eq. (2) (J = O) also vanishes when (21) is substituted, i.e., (21) is a solution of the system (1)-(14). Tothis end we expand G(q + k, q) in a functional series in (A) . Having (15) in view, we prove that the right-hand side of (2) has the structure (23) Since the mean field is set equal to zero after the variational derivative has been taken, the variational derivative contains a 6 function: '

::,py~-'6