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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