W Greiner· J. Reinhardt

QUANTUM ELECTRODYNAMICS

Greiner Quantum Mechanics An Introduction 3rd Edition

Greiner Quantum Theory Special Chapters (in preparation)

Greiner· Milller Quantum Mechanics Symmetries 2nd Edition

Greiner Relativistic Quantum Mechanics Wave Equations

Greiner· Reinhardt Field Quantization (in preparation)

Greiner· Reinhardt Quantum Electrodynamics 2nd Edition

Greiner· Schafer Quantum Chromodynamics

Greiner· Maruhn Nuclear Models (in preparation)

Greiner· Milller Gauge Theory of Weak Interactions

Greiner Mechanics I (in preparation)

Greiner Mechanics II (in preparation)

Greiner Electrodynamics (in preparation)

Greiner· Neise . Stocker Thermodynamics and Statistical Mechanics

Walter Greiner· Joachim Reinhardt

QUANTUM ELECTRODYNAMICS

With a Foreword by D.A. Bromley

Second Corrected Edition With 155 Figures,

and 52 Worked Examples and Exercises

Springer

Professor Dr. Walter Greiner Dr. Joachim Reinhardt

Institut fUr Theoretische Physik der Johann Wolfgang Goethe-Universitlit Frankfurt Postfach 111932 0-60054 Frankfurt am Main Germany

Street address:

Robert-Mayer-Strasse 8-10 0-60325 Frankfurt am Main Germany

Title of the original German edition: Theoretische Physik, Band 7: Quantenelektrodynamik, 2., iiberarbeitete und erweiterte Auflage 1994 © Verlag Harri Deutsch, Thun 1984, 1994

This volume was originally published with the title: Theoretical Physics - Text and Exercise Books Vol. 4.

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This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1992, 1994

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: Camera ready copy from the authors using a Springer T EX macro package Production Editor: P. Treiber SPIN 10123143 56/3140- 5 43 21 - Printed on acid-free paper

ISBN 978-3-540-78049-6 ISBN 978-3-642-88022-3 (eBook) DOI 10.1007/978-3-642-88022-3

Foreword to Earlier Series Editions

More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modem theoretical physics - with mathematics, the most fundamental of sciences - using Walter Greiner's textbooks as their guide.

The idea of developing a coherent, complete presentation of an entire field of science in a series of closely related textbooks is not a new one. Many older physicists remember with real pleasure their sense of adventure and discovery as they worked their ways through the classic series by Sommerfeld, by Planck and by Landau and Lifshitz. From the students' viewpoint, there are a great many obvious advantages to be gained through use of consistent notation, logical ordering of topics and coherence of presentation; beyond this, the complete coverage of the science provides a unique opportunity for the author to convey his personal enthusiasm and love for his subject.

The present five-volume set, Theoretical Physics, is in fact only that part of the complete set of textbooks developed by Greiner and his students that presents the quantum theory. I have long urged him to make the remaining volumes on classical mechanics and dynamics, on electromagnetism, on nuclear and particle physics, and on special topics available to an English-speaking audience as well, and we can hope for these companion volumes covering all of theoretical physics some time in the future.

What makes Greiner's volumes of particular value to the student and professor alike is their completeness. Greiner avoids the all too common "it follows that ... " which conceals severnl pages of mathematical manipulation and confounds the student. He does not hesitate to include experimental data to illuminate or illustrate a theoretical point and these data, like the theoretical content, have been kept up to date and topical through frequent revision and expansion of the lecture notes upon which these volumes are based.

Moreover, Greiner greatly increases the value of his presentation by including something like one hundred completely worked examples in each volume. Nothing is of greater importance to the student than seeing, in detail, how the theoretical concepts and tools under study are applied to actual problems of interest to a working physicist. And, finally, Greiner adds brief biographical sketches to each chapter covering the people responsible for the development of the theoretical ideas and/or the experimental data presented. It was Auguste Comte (1798-1857) in his Positive Philosophy who noted, "To understand a science it is necessary to know its history". This is all too often forgotten in modem physics teaching and the

VI Foreword to Earlier Series Editions

bridges that Greiner builds to the pioneering figures of our science upon whose work we build are welcome ones.

Greiner's lectures, which underlie these volumes, are internationally noted for their clarity, their completeness and for the effort that he has devoted to making physics an integral whole; his enthusiasm for his science is contagious and shines through almost every page.

These volumes represent only a part of a unique and Herculean effort to make all of theoretical physics accessible to the interested student. Beyond that, they are of enormous value to the professional physicist and to all others working with quantum phenomena. Again and again the reader will find that, after dipping into a particular volume to review a specific topic, he will end up browsing, caught up by often fascinating new insights and developments with which he had not previously been familiar.

Having used a number of Greiner's volumes in their original German in my teaching and research at Yale, I welcome these new and revised English translations and would recommend them enthusiastically to anyone searching for a coherent overview of physics.

Yale University New Haven, CT, USA 1989

D. Allan Bromley Henry Ford II Professor of Physics

Preface to the Second Edition

The need for a second edition of our text on Quantum Electrodynamics has given us the opportunity to implement some corrections and amendments. We have corrected a number of misprints and minor errors and have supplied additional explanatory remarks at various places. Furthermore some new material has been included on the magnetic moment of the muon (in Example 5.6) and on the Lamb shift (in Example 5.8). Finally, we have added the new Example 3.17 which explains the equivalent photon method.

We thank several colleagues for helpful comments and also are grateful to Dr. R. Mattiello who has supervised the preparation of the second edition of the book. Furthermore we acknowledge the agreeable collaboration with Dr. H. J. K6lsch and his team at Springer-Verlag, Heidelberg.

Frankfurt am Main, July 1994

Walter Greiner Joachim Reinhardt

Preface to the First Edition

Theoretical physics has become a many-faceted science. For the young student it is difficult enough to cope with the overwhelming amount of new scientific material that has to be learned, let alone obtain an overview of the entire field, which ranges from mechanics through electrodynamics, quantum mechanics, field theory, nuclear and heavy-ion science, statistical mechanics, thermodynamics, and solid­state theory to elementary-particle physics. And this knowledge should be acquired in just 8-10 semesters, during which, in addition, a Diploma or Master's thesis has to be worked on or examinations prepared for. All this can be achieved only if the university teachers help to introduce the student to the new disciplines as early on as possible, in order to create interest and excitement that in tum set free essential new energy. Naturally, all inessential material must simply be eliminated.

At the Johann Wolfgang Goethe University in Frankfurt we therefore confront the student with theoretical physics immediately, in the first semester. Theoretical Mechanics I and II, Electrodynamics, and Quantum Mechanics I - An Introduction are the basic courses during the first two years. These lectures are supplemented with many mathematical explanations and much support material. After the fourth semester of studies, graduate work begins, and Quantum Mechanics II - Symme­tries, Statistical Mechanics and Thermodynamics, Relativistic Quantum Mechanics, Quantum Electrodynamics, the Gauge Theory of Weak Interactions, and Quantum Chromodynamics are obligatory. Apart from these a number of supplementary courses on special topics are offered, such as Hydrodynamics, Classical Field The­ory, Special and General Relativity, Many-Body Theories, Nuclear Models, Mod­els of Elementary Particles, and Solid-State Theory. Some of them, for example the two-semester courses Theoretical Nuclear Physics or Theoretical Solid-State Physics, are also obligatory.

This volume of lectures deals with the subject of Quantum Electrodynamics. We have tried to present the subject in a manner which is both interesting to the student and easily accessible. The main text is therefore accompanied by many exercises and examples which have been worked out in great detail. This should make the book useful also for students wishing to study the subject on their own.

When lecturing on the topic of quantum electrodynamics, one has to choose between two approaches which are quite distinct. The first is based on the general methods of quantum field theory. Using classical Lagrangian field theory as a starting point one introduces noncommuting field operators, builds up the Fock space to describe systems of particles, and introduces techniques to construct and evaluate the scattering matrix and other physical observables. This program can be realized either by the method of canonical quantization or by the use of path

x Preface to the First Edition

integrals. The theory of quantum electrodynamics in this context emerges just as a particular example of the general formalism. In the present volume, however, we do not follow this general but lengthy path; rather we use a "short cut" which arrives at the same results with less effort, and which has the advantage of great intuitive appeal. This is the propagator formalism, which was introduced by R.P. Feynman (and, less well known, by E.C.G. Stiickelberg) and makes heavy use of Green's functions to describe the propagation of electrons and photons in space-time.

It is clear that the student of physics has to be familiar with both approaches to quantum electrodynamics. (In the German edition of these lectures a special volume is dedicated to the subject of field quantization.) However, to gain quick access to the fascinating properties and processes of quantum electrodynamics and to its calculational techniques the use of the propagator formalism is ideal.

The first chapter of this volume contains an introduction to nonrelativistic prop­agator theory and the use of Green's functions in physics. In the second chapter this is generalized to the relativistic case, introducing the Stiickelberg-Feynman propagator for electrons and positrons. This is the basic tool used to develop per­turbative QED. The third chapter, which constitutes the largest part of the book, contains applications of the relativistic propagator formalism. These range from simple Coulomb scattering of electrons, scattering off extended nuclei (Rosen­bluth's formula) to electron--electron (M0ller) and electron-positron (Bhabha) scat­tering. Also, processes involving the emission or absorption of photons are treated, for instance, Compton scattering, bremsstrahlung, and electron-positron pair anni­hilation. The brief fourth chapter gives a summary of the Feynman rules, together with some notes on units of measurement in electrodynamics and the choice of gauges.

Chapter 5 contains an elementary discussion of renormalization, exemplified by the calculation of the lowest-order loop graphs of vacuum polarization, self-energy, and the vertex correction. This leads to a calculation of the anomalous magnetic moment of the electron and of the Lamb shift. In Chap. 6 the Bethe-Salpeter equation is introduced, which describes the relativistic two-particle system.

Chapter 7 should make the reader familiar with the subject of quantum electro­dynamics of strong fields, which has received much interest in the last two decades. The subject of supercritical electron states and the decay of the neutral vacuum is treated in some detail, addressing both the mathematical description and the phys­ical implications. Finally, in the last chapter, the theory of perturbative quantum electrodynamics is extended to the treatment of spinless charged bosons.

An appendix contains some guides to the literature, giving references both to books which contain more details on quantum electrodynamics and to modem treatises on quantum field theory which supplement our presentation. We should mention that in preparing the first chapters of our lectures we have relied heavily on the textbook Relativistic Quantum Mechanics by J.D. Bjorken and S.D. Drell (McGraw-Hill, New York 1964).

We enjoyed the help of several students and collaborators, in particular Jiirgen Augustin, Volker Blum, Christian Borchert, Snjezana Butorac, Christian Der­reth, Bruno Ehmsperger, Klaus Geiger, Mathias Grabiak, Oliver Graf, Carsten Greiner, Kordt Griepenkerl, Christoph Hartnack, Cesar Ionescu, Andre Jahns, Jens Konopka, Georg Peilert, Jochen Rau, Wolfgang Renner, Dirk-Hermann Rischke,

Preface to the First Edition

Jiirgen Schaffner, Alexander Scherdin, Dietmar Schnabel, Thomas Schonfeld, Ste­fan Schramm, Eckart Stein, Mario Vidovic, and Luke Winckelmann.

We are also grateful to Prof. A. Schafer for his advice. The preparation of the manuscript was supervised by Dr. Bela Waldhauser and Dipl. Phys. Raffaele Mattiello, to whom we owe special thanks. The figures were drawn by Mrs. A. Steidl.

The English manuscript was copy-edited by Mark Seymour of Springer-Verlag.

Frankfurt am Main, March 1992

Walter Greiner Joachim Reinhardt

XI

Contents

1. Propagators and Scattering Theory ........................... . 1.1 Introduction ............................................ 1 1.2 The Nonrelativistic Propagator ............................. 2 1.3 Green's Function and Propagator ........................... 3 1.4 An Integral Equation for 1/J ......•......................... 6 1.5 Application to Scattering Problems ......................... 12 1.6 The Unitarity of the S-Matrix .............................. 20 1. 7 Symmetry Properties of the S-Matrix ........................ 21 1.8 The Green's Function in Momentum Representation

and Its Properties ........................................ 23 1.9 Another Look at the Green's Function for Interacting Particles ... 28 1.10 Biographical Notes ...................................... 37

2. The Propagators for Electrons and Positrons ................... 39

3. Quantum-Electrodynamical Processes .......................... 77 3.1 Coulomb Scattering of Electrons ........................... 77 3.2 Scattering of an Electron off a Free Proton: The Effect of Recoil . 96 3.3 Scattering ofIdentical Fermions ............................ 130 3.4 Electron-Positron Scattering: Bhabha Scattering

and Muon Pair Creation .................................. 137 3.5 Scattering of Polarized Dirac Particles ....................... 148 3.6 Bremsstrahlung ......................................... 155 3.7 Compton Scattering - The Klein-Nishina Formula ............ 174 3.8 Annihilation of Particle and Antiparticle ..................... 187 3.9 Biographical Notes ...................................... 219

4. Summary: The Feynman Rules of QED ........................ 221 4.1 The Feynman Rules of QED in Momentum Space ............. 222 4.2 The Photon Propagator in Different Gauges .................. 226 4.3 Biographical Notes ...................................... 230

5. The Scattering Matrix in Higher Orders ....................... 231 5.1 Electron-Positron Scattering in Fourth Order ................. 231 5.2 Vacuum Polarization ..................................... 234 5.3 Self-Energy of the Electron ............................... 263 5.4 The Vertex Correction .................................... 269 5.5 Biographical Notes ...................................... 294

XIV Contents

6. Two-Particle Systems ........................................ 297

6.1 The Bethe-Salpeter Equation .............................. 297 6.2 Biographical Note ....................................... 327

7. Quantum Electrodynamics of Strong Fields ..................... 329

7.1 Strong Fields in Atoms ................................... 333 7.2 Strong Fields in Heavy Ion Collisions ....................... 356 7.3 The Effective Lagrangian of the Electromagnetic Field ......... 365 7.4 Biographical Notes ...................................... 381

8. Quantum Electrodynamics of Spinless Bosons ................... 383

8.1 The Klein-Gordon Equation ............................... 384 8.2 The Feynman Propagator for Scalar Particles ................. 385 8.3 The Scattering of Spin-O Bosons ........................... 387 8.4 The Feynman Rules of Scalar Electrodynamics ............... 392

Appendix ..................................................... 399

Subject Index .................................................. 401

Contents of Examples and Exercises

1.1 Properties of C ............................................ 5 1.2 Relation Between C+ and c- ................................ 15 1.3 The Free Green's Function and Its Properties .................... 16 1.4 An Integral Representation for the Step Function ................. 24 1.5 Green's Function for Diffusion ................................ 30 1.6 Kirchhoff's Integral as an Example of Huygens' Principle

in Electrodynamics ......................................... 32 2.1 Plane-Wave Decomposition of the Feynman Propagator ............ 48 2.2 Feynman Propagator for a Fermi Gas .......................... 54 2.3 Nonrelativistic Limit of the Feynman Propagator ................. 61 2.4 Time-Evolution of Dirac Wavefunctions ........................ 63 2.5 The Explicit Form of SF(X) in Coordinate Space ................. 67 3.1 Calculation of a Useful Integral ............................... 86 3.2 Lorentz Transformation of Plane Waves ........................ 87 3.3 Mathematical Supplement: Traces and Identities Involving

,),-Matrices ................................................ 89 3.4 Coulomb Scattering of Positrons .............................. 93 3.5 Rosenbluth's Formula ....................................... 112 3.6 Higher-Order Electron-Proton Scattering ........................ 120 3.7 Static Limit of the Two-Photon Exchange ....................... 127 3.8 Mandelstam Variables Applied to M011er and Bhabha Scattering 142 3.9 Degree of Polarization ....................................... 152 3.10 Static Limit of Bremsstrahlung ................................ 167 3.11 The Bethe-Heitler Formula for Bremsstrahlung .................. 169 3.12 Relations of Energy and Momentum ........................... 186 3.13 The Total Cross Section of Pair Annihilation .................... 192 3.14 Electron-Positron Annihilation in the Centre-of-Mass Frame 196 3.15 Pair Creation by Two Photons ................................ 200 3.16 Pair Creation in the Field of an Atomic Nucleus ................. 203 3.17 The Method of Equivalent Photons ............................ 207 3.18 Angular Integration of the Hadron Tensor ....................... 217 4.1 Furry's Theorem ........................................... 224 4.2 Supplement: Systems of Units in Electrodynamics ................ 228 5.1 Evaluation of an Integral ..................................... 247 5.2 The Photon Polarization Function ............... . . . . . . . . . . . . .. 249 5.3 The Uehling Potential ....................................... 252 5.4 Muonic Atoms ............................................. 260

XVI Contents of Examples and Exercises

5.5 Mathematical Supplement: Gaussian Integrals .................... 262 5.6 The Form Factor of the Electron .............................. 272 5.7 The Infrared Catastrophe ..................................... 281 5.8 The Energy Shift of Atomic Levels ............................ 283 6.1 Solution of the Initial-Value Problem for Fermions ............... 306 6.2 The Bethe-Salpeter Equation for Positronium .................... 308 6.3 The Nonretarded Limit of the Bethe-Salpeter Equation ............ 312 6.4 The Breit Interaction ........................................ 318 6.5 Nonrelativistic Reduction of the Two-Body Equation

Applied to Positronium ...................................... 322 7.1 The Wave Function at the Diving Point ......................... 335 7.2 Fano's Formalism for the Description of Resonances .............. 342 7.3 The Product of Two Principal-Value Poles ...................... 349 7.4 Time-Dependent Decay of the Vacuum ......................... 350 7.5 An Alternative Derivation of the Effective Lagrangian ............ 376 7.6 The Solution of the Dirac Equation

in a Homogeneous Magnetic Field ............................. 379 8.1 Compton Scattering at Bosons ................................ 393 8.2 The Electro-Production of Pion Pairs ........................... 396