Download CAUSALITY AND DISPERSION RELATIONS

CAUSALITY AND
DISPERSION RELATIONS
H. M. Nussenzveig
INSTITUTE FOR FUNDAMENTAL STUDIES
DEPARTMENT OF PHYSICS AND ASTRONOMY
THE UNIVERSITY OF ROCHESTER
ROCHESTER, NEW YORK
ACADEMIC
PRESS
New York and London
1972
CONTENTS
Preface
Part I
CAUSALITY AND ANALYTICITY
Chapter 1 Causality and Dispersion Relations
1.1. Introduction
1.2.
The Damped Harmonic Oscillator
1.3.
Causality and Analyticity
1.4.
Light Propagation in a Dielectric Medium
1.5.
Physical Origin of Dispersion Relations
1.6.
Titchmarsh's Theorem
1.7.
Subtractions
1.8.
Dispersion Relations and Distributions
1.9.
The Kramers-Kronig Relation
1.10. The Optical Theorem
References
3
10
15
17
20
21
28
33
43
47
52
Chapter 2 Partial-Wave Dispersion Relations
2.1.
2.2.
2.3.
Introduction
Classical Field: j-Wave Scattering
The Causality Condition
54
55
59
vii
viii
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
Contents
Analytic Continuation to /_
Product Expansion
Extension to Higher Angular Momenta
Nonrelativistic Quantum Scattering
The Schiitzer-Tiomno Causality Condition
Van Kampen's Causality Condition
The ^-Function
Wigner's Causal Inequality
Completeness
References
61
63
70
72
75
82
96
108
116
122
Chapter 3 Dispersion Relations for the Total Scattering Amplitude
3.1.
3.2.
3.3.
3.4.
Introduction
Dispersion Relations for Fixed Scattering Angle
Dispersion Relations for Fixed Momentum Transfer
Extension to Nonrelativistic Quantum Scattering
References
124
127
131
147
154
Chapter 4 Physical Interpretation of S-Matrix Singularities
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
Introduction
Effects on the Cross Section
Complex Poles and Unstable States
Vibrating String and Oscillator
.
The Transient-Mode Propagator for the Schrodinger Equation
Application to an-Explicit Model
References
155
156
159
162
169
175
189
Part II
POTENTIAL SCATTERING
Chapter 5 Analytic Properties of Partial-Wave Amplitudes
5.1. Introduction
5.2. The Jost Function
5.3. Analytic Properties of the Jost Function
5.4. The Singularities of the 5-Function
5.5. Cutoff Potentials
5.6. An Example: Square Well or Barrier
5.7. Mittag-Leffler and Transient-Mode Expansions
5.8. Extension to Higher"Angular Momenta
References
193
194
197
203
214
219
223
234
241
Contents
ix
Chapter 6 Analytic Properties of the Total Amplitude
6.1.
Introduction
243
6.2.
6.3.
6.4.
6.5.
6.6.
The Resolvent Operator in Banach Space
Analytic Properties of the Total Scattering Amplitude
High-Energy Behavior of the Scattering Amplitude
Dispersion Relations for Fixed Momentum Transfer
Analyticity in Momentum Transfer and Finite Range of the Interaction
References
248
255
262
268
273
281
Chapter 7 Regge Poles
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
Introduction
Regular and Irregular Solutions
The Jost Function and the S-Function
Properties of the Pole Distribution
Asymptotic Behavior of S{\,k) as \X\->• co
Watson Transformation and Analytic Continuation in cos 0
Regge Poles
References
282
287
292
300
306
314
317
323
Chapter 8 The Mandelstam Representation
8.1.
8.2.
8.3.
8.4.
Derivation of the Mandelstam Representation
•
The Unitarity Condition
Determination of the Scattering Amplitude from Mandelstam's
Representation and Unitarity
Cutoff Potentials
References
326
333
336
347
360
Appendix A Distribution Theory
Al.
A2.
A3.
A4.
A5.
A6.
A7.
A8.
A9.
A10.
All.
Introduction
The Space 3 and Schwartz Distributions
Operations with Distributions
Differentiation of Distributions
Product of Distributions
Support of a Distribution
Direct Product
Convolution
Fourier Transforms and the Space Sf
Temperate Distributions and Their Fourier Transforms
Fourier Transform of ^(l/O'and Related Distributions
References
362
363
367
368
373
374
375
377
381
384
389
390
x
Contents
Appendix B
Passivity and Causality
391
Appendix C
Properties of Herglotz Functions
393
Appendix D
Properties of /^-Functions
396
Appendix E
Asymptotic Time Behavior of Free Schrodinger
Wave Packets
402
Appendix F
Compact Operators in Banach Space
404
Appendix G
Asymptotic Behavior of Green's Function
411
Appendix H
Appendix I
The Path T(v)
414
Dispersion Relation for the Basic Mandelstam Integral
Author Index
Subject Index
418
421
•
426