Table Of Content•
Frontiers in Physics
A Lecture Note and Reprint Series
DAVID PINES, Editor
P. W. Anderson CONCEPTS IN SOLIDS: Lectures on the Theory of Solids
N. Bloembergen NUCLEAR MAGNETIC RELAXATION: A Reprint
Volume
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Geoffrey F. Chew S-MATRIX THEORY OF STRONG INTERACTIONS: ·
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A Lecture Note and Reprint Volume .
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R. P. Feynman QUANTUM ELECTRODYNAMICS: A Lecture Note
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and Reprint Volume · · .
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R. P. Feynman THE THEORY OF FUNDAMENTAL PROCESSES: · . .
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A Lecture Note Volume · • .
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Hans Frauenfelder THE MOSSBAUER EFFECT: A Collection of Reprints .
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with an Introduction
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Robert Hofstadter NUCLEAR AND NUCLEON STRUCTURE: A Reprint
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Leo P. Kadanoff and Gordon Baym •
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QUANTUM STATISTICAL MECHANICS: A Lecture •
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A. M. Lane RECENT DEVELOPMENTS IN NUCLEAR THEORY: •
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A Lecture Note and Reprint Volume • •
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R. Omnes and M. Froissart • • • •
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MANDELSTAM THEORY AND REGGE POLES: • •
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An Introduction for Experimentalists
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George E. Pake PARAMAGNETIC RESONANCE: An Introductory •
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Monograph · .
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David Pines THE MANY-BODY PROBLEM; A Lecture Note and · .
Reprint Volume
J.
E. SqUires COMPLEX ANGULAR MOMENTA AND PARTICLE
PHYSICS: A Lecture Note and Reprint Volume
L. VanHove, N. M. Hugenholtz, and L. P. Howland
PROBLEMS IN THE QUANTUM THEORY OF MANY
PARTICLE SYSTEMS: A Reprint Volume
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MANDELSTAM THEORY
AND REGGE POLES
AN INTRODUCTION
FOR EXPERIMENTALISTS
*
,
R.OMNES
,
Institut des Huutes Etudes Scientiflques
M. FROISSART
Centre d'Etudes NucIeaires de Saclay
W. A. BENJAMIN, INC.
1963 New York Amsterdam
MANDELSTAM THEORY AND REGGE POLES
An Introduction for Experimentalists
Copyright © 1963 by W. A. Benjamin, Inc.
All rights reserved
Library of Congress Catalog Card Number: 63- 22795
Manufactured in the United States of America
The final manuscript was received on July 1, 1963; this volume was published
December 15,1963
The publisher is pleased to acknowledge the assistance of Zeb Delaire, who
composed the volume; Cecilia Duray-Bito, who produced the illustrations;
and William Prokos, who designed the Idust iacket
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W. A. BENJAMIN, INC.
New York, New York
EDITOR'S FOREWORD
The problem of c:ommunicating in a coherent fashion the recent devel
opments in the most exciting and active fields of physics seems particularly
pressing today. The enormous growth in the number of physicists has
tended to make the familiar channels of communication considerably less
effective. It has become increasingly difficult for experts in a glven field
to keep up with the curn:nt literature; the novice can only be confused.
What is needed is both a consistent account of a field and the presentntion
of a definite "point of view" concerning it. Formal monographs cannot.
meet such a need in a rapidly developing 6eJd, and, perhaps more im
portant, the review article seems to have falIen into disfavor. Indeed, it
would seem that the people most actively engaged in developing a given
field ace the people least likely to writc at length about it.
"'Frontiers in Physics~ has been conceived in an effort to improve the
sihlation in several ways. Fir~t, to take advantage of the faet that the
leading physici~t:s today frequently give a series of lectures, a graduate
seminar, or a graduate course in their special fields of intere5t. Such lec
tures serve to summarize the present status of a rapidly developing field
and may well constitute the only coherent account available at the time.
Often, notes on lectures exist (prepared by the lecturer himseH, by gradu
ate students, or by postdoctoral fellows) and have becn distributed in
mirneogr.l.phed fOITn on a limited basis. One of the principal purposes of
the "Frontiers in Physics" series is to make sllch notes available to a wider
audience of physicists.
It should he emphasized that lecture note.~ are necessarily rough and
informal, both in style and content, and those in the series will prove DO
exception. This is as it should he. The point of the series is to oHer new,
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EDITOR'S FOREWORD
VI
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rapid, more informal, and, it is hoped, mOre effective ways for physicists
to teach one another. The point is lost if only elegant notes qualify.
A second way to improve communication in very active fields of physics
is by the publication of collections of reprints of recent articles. Such
collections are themselves useful to people working in the field. The value'
of the reprints would, however, seem much enhanced if the collection
would be accompanied by an introduction of moderate length, which
would serve to tie the collection together and, necessarily, constitute a
brief survey of the present status of the field. Again, it is appropriate'that
such an introduction be informal, in keeping with the active character
of the field.
A third possibility for the series might be called an informal monograph,
to connote the fact that it represents an intermediate step between lecture
notes and fOllnal monographs. It would offer the author an opportunity to
present his views of a field that has developed to the point at which a
summation might prove extraordinarily fruitful, but for which a formal
monograph !night not be feasible or desirable.
Fourth, there are the contemporary classics....,.papers or lectures which
constitute a particularly valuable approach to the teaching and learning
of physics today. Here one thinks of fields that lie at the heart of much of
present-day research, but whose essentials are by now well understood,
such as quantum electrodynamics or magnetic resonance. In such fields
some of the best pedagogical material is not readily available, either be
cause it consists of papers long out of print or lectures that have never
been published.
"Frontiers in Physics" is designed to be flexible in editorial format.
Authors are encouraged to use as many of the foregoing approaches as
seem desirable for the project at hand. The publishing format for the series
is in keeping with its intentions. Photo-offset printing is used throughout,
and the books are paperbound, in order to speed publication and reduce
costs. It is hoped that the books will thereby be within the financial reach
of graduate students in this country and abroad.
Finally, because the series represents something of an experiment on
the part of the editor and the publisher, suggestions from. interested
readers as to format, contributors, and contributions will be most welcome.
DAVID PINES
Urbana, Illinois
August 1961
PREFACE
The present book stems from separate lecture courses given by the
authors at Professor Leprioce-Ringuet's laboratory in the Ecole Poly"
technique and at Saclay. As the audience was essentially made up of
experimentalists, these courses were of a descriptive character.
The book is addressed to students who arc not acquainted with the
most recent work in the theory of strongly interacting elementary particles.
Our aim has been to extract the most important physical ideas, with
proofs frequently omitted or replaced by heuristic arguments. The level
has been kept as elementary as possible; the reader is assumed to know
only elementary qu,.'mtum mechanics. We have made no attempts at
mathematical rigor or completeness, since further developments may be
found in the book by G. F. Chew, S-Matrir Tlzeory of Strong Interactions,
published in the same series.
As is customary in elementary textbooks, we have not explicitly referred
to the original authors of the work under description. We feel ju.sti.£ied
in not doing so because our heuristic presentation would hardly give them
due credit. In addition, we have given only a short bibliography, intended
to provide a reading program for those who may want to acquaint them
selves further with the subject.
t
One of tlle authors (R. O. thanks Mr. L. Motchane for his hospitality
at the Institut des Hautes Etudes Scientifiques. We both thank Mile
Helene Noir for her invaluable aid in preparing the manuscript.
R. DMN",
M. FROISSAnT
Pam, France
September 1963
vii
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CONTENTS
Editor's Foreword v
Preface vii
Chapter 1 Nonrelativistic Scattering Theory 1
1-1 Formulation of the Problem 1
1-2 Integral Form of the Schrodinger Equation 4
1-3 The Born Series Expansion 10
1-4 Calculation of the Scattering Amplitude 11
Chapter 2 Properties of Partial Waves 15
2-1 Separation of the Schrodinger Equation
in Partial Waves 15
2-2 Study of the Radial Equation 20
2-3 Analytic Properties of the Partial-Wave
Amplitude as a Function of k2 21
2-4 The Left-Hand Cut 22
2-5 The Right-Hand Cut 23
2-6 Analytic Properties of the Partial-Wave Amplitude 25
2-7 Physical Aspects of the Partial-Wave Formalism 26
Chapter 3 Analytic Properties in Energy and Momentum Transfer 29
3·1 Kinematics 29
3-2 Analytic Properties in Energy 30
3·3 Analytic Properties in Momentum Transfer 33
3-4 Treatment of Exchange Forces 36
3-5 Bound States 37
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CONTENTS
Chapter 4 Introduction of Complex Angular Momenta 41
4-1 Interpolation between Physical Values
41
of Angular Momentum
43
4-2 The Sommerfeld-Watson Fonnula
4-3 Physical Interpretation of Regge Poles 46
4-4 Behavior of Regge Trajectories 48
4-5 Exchange Forces-Signature 50
52
4-6 Asymptotic Behavior in Momentum Transfer
Chapter 5 Kinematics of Relativistic Scattering, Crossing 54
5·1 Kinematics 54
5-2 Crossing 58
5-3 Crossed Reactions 62
5-4 Kinematics of Crossed Reactions 64
5-5 First Example: p-n Scattering 66
5-6 Second Example: Pion-Nucleon Scattering 66·
5-7 Third Example: Decay and Dalitz Plot 66
K/L~
5·8 A General Case: Associated Production 70
"a
Chapter 6 Relativistic Scattering Theory la Mandelstam" 72
6·1 FomlUlation of the Problem 72
6·2 The Invariant Scattering Amplitude 74
6·3 75
The Nonrelativistic Region
6·4 Another Energy Cut 77
6-5 An Example: Proton·Antineutron Scattering 78
6-6 The Mandelstam Hypothesis 80
6-7 Application to the Properties of
Nucleon- Nucleon Potential 80
6-8 Exchange of Other Particles 82
6·9 Application to Pion-Nucleon Scattering , 86
6-10 The High-Energy Domain 88
6·11 Asymptotic Properties of A (8, t, u) 92
94
6-12 Dispersion Relations
Chapter 7 Relativistic Regge Poles 100
7-1 100
The Paradoxes
7·2
Comparison with Nonrelativistic Theory 102
7·3 Chew's Hypothesis 104
7·4 Exchange Forces 105
7·5
Trajectories 106
7-6 Generalizations 107
Chapter 8 Comparison with Experiment 110
8-1 Experimental Tests in the Direct Channel 110
CONTENTS xi
8-2 Experimental Tests in the Crossed Channel U2
8-3 Total Cross Sections. The Pomeranchuk Pole U3
84 Total Cross Sections. The Satellite Poles U5
8·5 The Diffraction Peak U7
8·6 The Inelastic Experiments 120
Bibliography 122
Index 123
