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Francisco 1. Yndurain

Relativistic Quantum Mechanics and Introduction to Field Theory

, Springer

Professor Francisco J. Yndurain Universidad Aut6noma de Madrid Departamento de Ffsica Te6rica, C-XI Canto Blanco, E-28049 Madrid, Spain

Editors

Roger Balian Nicolai Reshetikhin CEA Department of Mathematics Service de Physique Theorique de Saclay University of California F-91191 Gif-sur-Yvette, France Berkeley, CA 94720-3840, USA

Wolf Beiglbock Institut fUr Angewandte Mathematik Universitlit Heidelberg 1m Neuenheimer Feld 294 0-69120 Heidelberg, Germany

Harald Grosse Institut fUr Theoretische Physik Universitlit Wien Boltzmanngasse 5 A-1090 Wien, Austria

Elliott H. Lieb Jadwin Hall Princeton University, P. O. Box 708 Princeton, NJ 08544-0708, USA

Herbert Spohn Theoretische Physik Ludwig-Maximilians-Universitlit Miinchen TheresienstraBe 37 0-80333 Miinchen, Germany

Walter Thirring Institut fUr Theoretische Physik Universitlit Wien Boltzmanngasse 5 A-1090 Wien, Austria

This book is a completely revised translation of the Spanish original edition: F. J. Ynduniin, Mecanica Cmintica Relativista, Alianza editorial s/a, Madrid 1990

Library of Congress Cataloging-in-Publication Data applied for.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Yndurain, Francisco J.: Relativistic quantum mechanics and introduction to field theory Fancisco J. Ynduniin. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Texts and monographs in physics)

ISSN 0172-5998 ISBN-13: 978-3-642-64674-4 DOl: 10.1007/978-3-642-61057-8

e-ISBN-13: 978-3-642-61057-8

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© Springer-Verlag Berlin Heidelberg 1996 Softcover reprint of the hardcover I st edition 1996

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Preface

A fully relativistic treatment of the quantum mechanics of particles requires the introduction of quantum field theory, that is to say, the quantum mechan­ics of systems with an infinite number of degrees of freedom. This is because the relativistic equivalence of mass and energy plus the quantum possibility of fluctuations imply the existence of (real or virtual) creation and annihilation of particles in unlimited numbers.

In spite of this, there exist processes, and energy ranges, where a treat­ment in terms of ordinary quantum mechanical tools is appropriate, and the approximation of neglecting the full field-theoretic description is justified. Thus, one may use concepts such as potentials, and wave equations, clas­sical fields and classical currents, etc. The present text is devoted precisely to the systematic discussion of these topics, to which we have added a gen­eral description of one- and two-particle relativistic states, in particular for scattering processes.

A field-theoretic approach may not be entirely avoided, and in fact an introduction to quantum field theory is presented in this text. However, field theory is not the object per se of this book; apart from a few examples, field theory is mainly employed to establish the connection with equivalent potentials, to study the classical limit of the emission of radiation or to discuss the propagation of a fermion in classical electromagnetic fields.

The bulk of applications of our tools is for electromagnetic interections, as is only natural. Nevertheless, some applications to nuclear physics (Yukawa interactions), weak interactions (parity violation in atoms) and strong inter­actions (the Bogoliubov bag model of quark bound states) are also presented.

This book is the English version of the author's text Mecanica Cuantica Relativista (Alianza Editorial, Madrid 1990). It is not merely a translation. Enough new material has been added that the original 260 pages have become well over three hundred, and a few chapters are entirely new.

To end this introduction, a few words on notation, and units, are in order. Carets over operators, AIL' iI etc. and tildes under matrix objects, IlL' f! etc. are used whenever there is danger of confusion. When the meaning -of an ob­ject is clear from the context we dispense with these aids. Three-dimensional vector objects are denoted by boldface characters, e.g., r. The convention

VI Preface

r = Irl will be used when no confusion (particularly with Minkowski vectors) may arise.

A peculiarity in which our text diverges from most presentations (a no­table exception is the treatise of Bogoliubov and Shirkov, 1959, which agrees with us on this) is our almost entire avoidance of contravariant components for Minkowski tensors, so that we also seldom employ Einstein's convention of summation over repeated indices. This makes some expressions a bit clumsy, a price that we consider worth paying in an introductory textbook which, moreover, continuously oscillates between relativistic calculations and non­relativistic (or semirelativistic) limits and concepts: the use of only one kind of component makes it easier to avoid a large number of ambiguities, notably when identifying nonrelativistic component notations with the corresponding relativistic ones. This ease of connection with nonrelativistic quantum me­chanical conventions is also the reason why we have kept Ii and c explicit in many formulas. As for units, we use, unless explicitly stated otherwise, the SI system with Gauss units (so that the potential between charges el, e2 is eledr). In a number of instances, however, the Heaviside system (in which the Coulomb potential is ele2/41fr) and/or natural units, with Ii = c = 1, is employed. Explicit warning is given in such cases. Other matters of notation, which are not very standardized in our subject, may be found throughout the text.

The typesetting of this text has been a rather tough enterprise for a Iffi.TEX beginner like the author. That the job has been completed was due to the invaluable help of Gabriel Martinez, Stephan Titard, Elena Ynduniin and, above all, Lourdes Rey and Kassa Adel (whose program "Kdraw" was used for the drawings), help which is here gratefully recorded.

Madrid, April 1996 F. J. Yndurtiin

Table of Contents

1. Relativistic Transformations. The Lorentz Group . . . . . . . . . 1 1.1 Rotations, and Space and Time Reversal

for Particles with Spin .................................. 1 1.2 Galilean Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Lorentz Transformations. Normal Parameters. . . . . . . . . . . . . . 6 1.4 Minkowski Space. The Full Lorentz Group. . . . . . . . . . . . . . . . . 8 1.5 The Lorentz Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 1.6 Geometry of Minkowski Space ........................... 14 1.7 Transformation Properties of Physical Quantities

Under the Lorentz Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.8 Covariant Form of the Maxwell Equations. . . . . . . . . . . . . . . .. 19 1.9 Minkowski Space: Metric, Conventions. . . . . . . . . . . . . . . . . . .. 20

2. The Klein-Gordon Equation. Relativistic Equation for Spinless Particles ..................................... 23 2.1 The Klein-Gordon Equation. Generalities ................. 23 2.2 Wave Equation for Free Spinless Particles ................. 24 2.3 Plane Waves. Current. Scalar Product. . . . . . . . . . . . . . . . . . . .. 27 2.4 Interaction with the Classical Electromagnetic Field.

Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 2.5 Particle in a Coulomb Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

3. Spin 1/2 Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 3.1 The Dirac Equation ..................................... 35 3.2 Invariance Properties of the Dirac Equation. . . . . . . . . . . . . . .. 37

3.2.1 Rotations....................................... 37 3.2.2 Boosts.......................................... 38 3.2.3 Parity.......................................... 41 3.2.4 Time Reversal ............................. . . . . .. 42

3.3 Density of Particles. Current. Scalar Product .............. 42 3.4 Minimal Replacement. Gauge Invariance. Large and Small

Components: Nonrelativistic Limit of the Dirac Equation. . .. 45 3.5 Plane Waves. States with Well-Defined Spin ............... 47 3.6 Radial Form of the Dirac Equation. Free-Particle Solutions.. 51

VIII Table of Contents

3.6.1 Radial Form of H ......... . . . . . . . . . . . . . . . . . . . . . .. 52 3.6.2 Free Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55

3.7 The Problem of Negative Energies in the Dirac Equation. The Dirac Sea. Hole Theory. Charge Conjugation. . . . . . . . . .. 57 3.7.1 Negative Energies. The Dirac Sea. Holes. . . . . . . . . . . .. 57 3.7.2 Charge Conjugation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58

3.8 Covariants and Projectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60 3.8.1 Covariants....................................... 60 3.8.2 Projectors .................. ·..................... 61

3.9 Massless Spin 1/2 Particles .............................. 62

4. Dirac Particle in a Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67 4.1 Dirac Particle in a Spherical Well. . . . . . . . . . . . . . . . . . . . . . . .. 67 4.2 Particle in a Coulomb Potential: Continuum States. . . . . . . .. 69 4.3 Scattering States. Phase Shifts. Cross-sections.

Wave Function at the Origin ........................... " 73 4.3.1 Scattering States. Phase Shifts. . . . . . . . . . . . . . . . . . . .. 73 4.3.2 Cross-sections.................................... 74 4.3.3 Wave Function at the Origin. . . . . . . . . . . . . . . . . . . . . .. 75

4.4 Bound States in a Coulomb Potential ................... " 76 4.5 Semirelativistic Approximation:

the Foldy-Wouthuysen Transformation. . . . . . . . . . . . . . . . . . .. 80 4.5.1 General Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80 4.5.2 Electromagnetic Interactions ..................... " 84 4.5.3 Free Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85

5. Massive Particles with Spin 1. Massless Spin 1 Particle: Photon Wave Functions. Particles with Higher Spins (3/2, 2, ... ) . . . . . . . . . . . . . . . . .. 89 5.1 Particle with Spin 1 and Mass m -# 0 ..................... 89 5.2 Particle with Spin 1 and Zero Mass: The Photon.

Plane Waves. Photon Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92 5.2.1 Photon Wave Function. Gauge Fixing.

Transformation Properties ....................... " 92 5.2.2 Plane Waves. Helicity States. . . . . . . . . . . . . . . . . . . . . .. 95 5.2.3 Field Variables as Wave Functions for the Photon.

The Schwinger Gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96 5.3 Angular Momentum Eigenstates for the Photon.

Vector Spherical Harmonics. Multipoles .......... . . . . . . . .. 97 5.3.1 General Useful Formulas. . . . . . . . . . . . . . . . . . . . . . . . .. 97 5.3.2 Multipoles ....................................... 100 5.3.3 Photon Wave Functions with Well-Defined

Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 Particles with Higher Spins. Rarita-Schwinger

and Bargmann-Wigner Equations. The Graviton ........... 104

Table of Contents IX

5.4.1 Rarita-Schwinger Equations ....................... 104 5.4.2 Bargmann-Wigner Equations ...................... 105 5.4.3 The Graviton .................................... 106

6. General Description of Relativistic States ................. 109 6.1 Preliminaries ........................................... 109 6.2 Relativistic One-Particle States: General Description ........ 112 6.3 Relativistic States of Massive (m -:f- 0) Particles ............ 116 6.4 Massless Particles ................ .' ..................... 119 6.5 Many-Particle States. Creation-Annihilation Operators.

Fock Space ............................................ 123 6.6 Connection with the Wave Function Formalism ............. 125

7. General Description of Relativistic Collisions: S Matrix, Cross-sections and Decay Rates. Partial Wave Analyses .. 129 7.1 Two-Particle States. Separation of the Centre of Mass

Motion. States with Well-Defined Angular Momentum ...... 129 7.2 Kinematics of Two-Particle Collisions ..................... 131 7.3 The S Matrix. Scattering Amplitude. Nonrelativistic Limit .. 134 7.4 Cross-sections and Decay Rates. The Optical Theorem ...... 136 7.5 Partial Wave Analysis and Phase Shifts.

1. Spinless Elastic Scattering. Effective Range Expansion .... 140 7.5.1 Partial Wave Analysis ............................. 140 7.5.2 Effective Range Formalism ........................ 143

7.6 Partial Wave Analysis. II. Several Two-Body Channels ...... 143 7.6.1 Multichannel Analysis ............................ 143 7.6.2 Effective Range Approximation .................... 146

7.7 Partial Wave Analysis. III. Particles with Spin ............. 146 7.7.1 Spin Analysis .................................... 146 7.7.2 Scattering of Spin 0 - Spin 1/2 Particles ............ 148

7.8 Evaluation of the S Matrix .............................. 150 7.8.1 The S Matrix and the Interaction Picture ........... 150 7.8.2 The S Matrix in the Lippmann-Schwinger Formalism. 154 7.8.3 Scattering by Two Interactions ..................... 157

8. Quantization of the Electromagnetic Field. Interaction of Radiation with Matter ..................... 159 8.1 Normal, or Wick, Products .............................. 159 8.2 Quantization of the Electromagnetic Field (Coulomb Gauge).

The Casimir Effect ..................................... 160 8.2.1 Quantization of the Electromagnetic Field ........... 160 8.2.2 Multipole Expansion .............................. 165 8.2.3 The Casimir Effect ............................... 166

8.3 Interaction of the Radiation with Slowly Moving Particles ... 169 8.3.1 Radiative Decays, and Absorption of Radiation ...... 171

X Table of Contents

8.3.2 Low-Energy Compton Scattering ................... 175 8.4 Bremsstrahlung ........................................ 180 8.5 The Classical Limit. Coherent States ...................... 184 8.6 Uncertainty Relations for Field Variables .................. 186

9. Quantum Fields: Spin 0, 1/2, 1. Covariant Quantization of the Electromagnetic Field .............................. 189 9.1 Generalities ............................................ 189 9.2 Localization of Particles in Relativistic Quantum Mechanics . 189 9.3 Retardation and Consistency ............................. 191 9.4 Quantization of Scalar Fields and of Massive Vector Fields ... 193

9.4.1 Second Quantization for Spinless Particles ........... 193 9.4.2 Massive Vector Particles .......................... 198

9.5 Quantization of the Dirac Field. Weyl and Majorana Particles 198 9.6 Covariant Quantization of the Electromagnetic Field ........ 204

9.6.1 The Gupta-Bleuler Space ......................... 205 9.6.2 Covariant 'fransformation ......................... 210 9.6.3 The Metric Operator Method ...................... 212

9.7 Time-Ordered Product. Propagators ...................... 213 9.8 Interactions in Quantum Field Theory.

Lagrangian Formulation ................................. 217 9.8.1 Lagrangian Formalism for Fields ................... 217 9.8.2 Interactions ...................................... 219

9.9 Gauge Invariance in Quantum Electrodynamics ............ 222

10. Interactions in Quantum Field Theory. Nonrelativistic Limit. Reduction to Equivalent Potential .. 229 10.1 Potentials Equivalent to Field-Theoretic Interactions.

General Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 10.2 Equivalent Potential for Two Particles

in Electromagnetic Interaction ................ . . . . . . . . . . . 232 10.2.1 Elastic Collision of Two Charged Particles

in the Born Approximation ........................ 232 10.2.2 Nonrelativistic Limit .............................. 236 10.2.3 Relativistic Corrections. The Breit Term ............ 240

10.3 Hydrogenlike Atoms: Hyperfine Structure. System with Two Electrons: the Helium Atom ............. 243 10.3.1 Hydrogenlike Atoms .............................. 243 10.3.2 System With Two Electrons. The Helium Atom ...... 244

10.4 Electron-Positron Collisions: Effective Potential. Positronium 247 10.4.1 Scattering Amplitude in the Born Approximation .... 247 10.4.2 Annihilation Channel ............................. 252 10.4.3 Positronium ..................................... 256

10.5 Scalar and Pseudoscalar Interactions. The Yukawa Potential . 258 10.6 Weak Neutral Currents. Parity Violation in Atoms .......... 262

Table of Contents XI

11. Relativistic Collisions in Field Theory. Feynman Rules. Decays ................................... 267 11.1 Electron-Positron Annihilation into Two Photons,

and Pair Creation by Two-Photon Collisions ............... 267 11.1.1 e+e- Annihilation into 2, ......................... 267 11.1.2 Creation of an e+e- Pair by Two Photons ........... 270

11.2 Feynman Rules. Gauge Invariance ........................ 271 11.2.1 Feynman Rules for the Evaluation

of Transition Amplitudes .......................... 271 11.2.2 Gauge Invariance ................................. 276

11.3 Polarized and Unpolarized Cross-sections. Sums Over Polarizations ....................... . ....... 279

11.4 Compton Scattering (Relativistic) ........................ 281 11.5 Decay of Bound States .................................. 284

11.5.1 General Theory .................................. 284 11.5.2 Decays of Positronium ............................ 286 11.5.3 Decay of Muonium into e+e-. Decays of Quarkonium . 288

12. Relativistic Interactions with Classical Sources ........... 291 12.1 Interaction with a Fixed (Classical) Potential .............. 291

12.1.1 Scattering by an External Field .................... 291 12.1.2 Bremsstrahlung .................................. 293

12.2 Photon Emission by a Classical Source. The Bloch-Nordsieck Theorem. Classical Limit ............. 294 12.2.1 Classical Radiation ............................... 294 12.2.2 Photon Emission by a Classical Current ............. 296 12.2.3 Radiation of Coherent States ...................... 300 12.2.4 The Bloch-Nordsieck Theorem ..................... 300

12.3 Propagation of an Electron in a Classical Potential. The Proper-Time Method ............................... 301 12.3.1 Electron in a Coulomb Potential ................... 302 12.3.2 The Proper-Time Method ......................... 304 12.3.3 Dirac Particle in a Constant Field, or in a Plane Wave 307

Appendices ............................. '.' .................... 309 A.1 Spherical Harmonics, Clebsch-Gordan Coefficients,

Matrix Representations of the Rotation Group ............. 309 A.1.1 Spherical Harmonics .............................. 309 A.1.2 Some Specific Values .............................. 310 A.1.3 Multiplication Formulas ........................... 310 A.1.4 Gegenbauer-like Formulas ......................... 310 A.1.5 Spinor and Vector Spherical Harmonics ............. 310 A.1.6 Clebsch-Gordan Coefficients ....................... 311 A.1.7 Rotation Matrices ................................ 313

A.2 Special Functions ....................................... 313

XII Table of Contents

A.2.1 Kummer, or Confluent Hypergeometric Functions .... 313 A.2.2 Bessel Functions ................................. 314 A.2.3 Spherical Bessel Functions ......................... 315 A.2.4 Bessel Functions of the Second Kind ................ 315 A.2.5 Laguerre Polynomials ............................. 315

A.3 Relation Between the Lorentz Group and the Group SL(2, C) 316 A.4 'Y Matrices ............................................. 319

A.4.1 The Pauli Realization ............................. 321 A.4.2 The Weyl Realization ....... .' ..................... 321 A.4.3 The Majorana Realization ......................... 321

A.5 Three Lemmas on T and Wick Exponentials ............... 322 A.6 Physical Quantities ..................................... 324

A.6.1 SI (Gauss) Units ................................. 324 A.6.2 Natural Units: c = Ii = 1 .......................... 324 A.6.3 Other Relations .................................. 325

References . ................................................... 327

Index ......................................................... 329