**Unformatted text preview: ** Quantum Electrodynamics
Fourth Edition Greiner
Quantum Mechanics
An Introduction 4th Edition
Greiner
Quantum Mechanics
Special Chapters
Greiner Müller
Quantum Mechanics
Symmetries 2nd Edition
Greiner
Relativistic Quantum Mechanics
Wave Equations 3rd Edition
Greiner Reinhardt
Field Quantization
Greiner Reinhardt
Quantum Electrodynamics
4th Edition
Greiner Schramm Stein
Quantum Chromodynamics
3rd Edition
Greiner Maruhn
Nuclear Models
Greiner Müller
Gauge Theory of Weak Interactions
3rd Edition Greiner
Classical Mechanics
Systems of Particles
and Hamiltonian Dynamics
Greiner
Classical Mechanics
Point Particles and Relativity
Greiner
Classical Electrodynamics
Greiner Neise Stocker
Thermodynamics
and Statistical Mechanics Walter Greiner Joachim Reinhardt Quantum Electrodynamics
With a Foreword by
D.A. Bromley Fourth Edition
With 169 Figures,
and 58 Worked Examples and Exercises Prof. Dr. Walter Greiner
Frankfurt Institute for Advanced
Studies (FIAS)
Johann Wolfgang Goethe-Universität
Ruth-Moufang-Str. 1
60438 Frankfurt am Main
Germany
[email protected] Dr. Joachim Reinhardt
Institut für Theoretische Physik
Johann Wolfgang Goethe-Universität
Max-von-Laue-Str. 1
60438 Frankfurt am Main
Germany
[email protected] Title of the original German edition: Theoretische Physik, Band 7: Quantenelektrodynamik,
2., überarbeitete und erweiterte Auflage 1994 © Verlag Harri Deutsch, Thun, 1984, 1994 ISBN: 978-3-540-87560-4 eISBN: 978-3-540-87561-1 Library of Congress Control Number: 2008937903
© 2009, 2003, 1994, 1992 Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
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Cover illustration: eStudio Calamar S.L., Spain
Printed on acid-free paper
987654321
springer.com 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 modern
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 several 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 modern physics teaching and the 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
V VI Foreword to Earlier Series Editions 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 Fourth Edition We are pleased by the positive resonance of our book which now necessitates a fourth
edition. We have used this opportunity to implement corrections of misprints and
amendments at several places, and to extend and improve the discussion of many of
the exercises and examples. We hope that our presentation of the method of equivalent
photons (Example 3.17), the form factor of the electron (Example 5.7), the infrared
catastrophe (Example 5.8) and the energy shift of atomic levels (Example 5.9) are now
even better to understand. The new Exercise 5.10 shows in detail how to arrive at the
non-relativistic limit for the calculation of form factors. Moreover, we have brought
up-to-date the Biographical Notes about physicists who have contributed to the development of quantum electrodynamics, and references to experimental tests of the theory. For example, there has been recent progress in the determination of the electric
and magnetic form factors of the proton (discussed in Exercise 3.5 on the Rosenbluth
formula) and the Lamb shift of high-Z atoms (discussed in Example 5.9 on the energy
shift of atomic levels), while the experimental verification of the birefringence of the
QED vacuum in a strong magnetic field (Example 7.8) remains unsettled and is a topic
of active ongoing research.
Again, we thank all colleagues and readers for their comments and information
about misprints in the book, and are grateful to the team at Springer-Verlag and especially to Dr. Stefan Scherer for smoothly handling the preparation of this fourth
edition.
Frankfurt am Main,
October 2008 Walter Greiner
Joachim Reinhardt VII Preface to the Third Edition Since the need for a third edition of this book has arisen, we have endeavoured to
improve and extend it in several ways. At many places small changes were made,
misprints have been corrected, and references have been added. In Chap. 5 new theoretical and experimental results on the Lamb shift in heavy atoms and on the anomalous magnetic moment of the muon are reported. We have also added a number of
new topics in Chaps. 3, 5, and 7 in the form of examples and exercises. Example 3.19
contains a detailed treatment of electron–positron pair production in the collision of
a high-energy photon with a laser beam. This is supplemented by Exercise 3.20 where
a closed solution of the Dirac equation in the field of a plane wave is derived. Furthermore, Example 5.3 on the running coupling constant in QED and Example 7.5 on
the supercritial point charge problem have been added. Finally, Example 7.8 treats the
birefringence of the QED vacuum in a strong magnetic field.
We thank all colleagues and readers who have informed us about misprints in the
book and are grateful to the team at Springer-Verlag for expertly handling the preparation of this new edition.
Frankfurt am Main,
August 2002 Walter Greiner
Joachim Reinhardt VIII 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 5.7) and on the Lamb shift (in Example 5.9). 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. Kölsch and
his team at Springer-Verlag, Heidelberg.
Frankfurt am Main,
July 1994 Walter Greiner
Joachim Reinhardt IX 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 learnt, 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
solid-state 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 turn 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 integrals. The theory of quantum electrodynamics in this context emerges just as a particular example of the general
X Preface to the First Edition 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. Stückelberg)
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 Stückelberg–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 (Møller) 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 modern 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 Jürgen Augustin, Volker Blum, Christian Borchert, Snježana Butorac, Christian Derreth, Bruno
Ehrnsperger, Klaus Geiger, Mathias Grabiak, Oliver Graf, Carsten Greiner, Kordt
Griepenkerl, Christoph Hartnack, Cesar Ionescu, André Jahns, Jens Konopka, Georg
Peilert, Jochen Rau, Wolfgang Renner, Dirk-Hermann Rischke, Jürgen Schaffner,
Alexander Scherdin, Dietmar Schnabel, Thomas Schönfeld, Stefan Schramm, Eckart
Stein, Mario Vidovic, and Luke Winckelmann. XI XII Preface to the First Edition We are also grateful to Prof. A. Schäfer for his advice. The preparation of the
manuscript was supervised by Dr. Béla 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 Contents 1 Propagators and Scattering Theory . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . .
1.2 The Nonrelativistic Propagator . . . . . . . . . . .
1.3 Green’s Function and Propagator . . . . . . . . . .
1.4 An Integral Equation for ψ . . . . . . . . . . . . .
1.5 Application to Scattering Problems . . . . . . . . .
1.6 The Unitarity of the S Matrix . . . . . . . . . . . .
1.7 Symmetry Properties of the S Matrix . . . . . . . .
1.8 The Green’s Function in Momentum Representation
1.9 Another Look at the Green’s Function . . . . . . . .
1.10 Biographical Notes . . . . . . . . . . . . . . . . . . .
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38 2 The Propagators for Electrons and Positrons . . . . . . . . . . . . . . 39 3 Quantum-Electrodynamical Processes . . . . . . . . . . . . . . .
3.1 Coulomb Scattering of Electrons . . . . . . . . . . . . . . . .
3.2 Scattering of an Electron off a Free Proton: The Effect of Recoil
3.3 Scattering of Identical Fermions . . . . . . . . . . . . . . . . .
3.4 Electron–Positron Scattering . . . . . . . . . . . . . . . . . .
3.5 Scattering of Polarized Dirac Particles . . . . . . . . . . . . .
3.6 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Compton Scattering – The Klein–Nishina Formula . . . . . . .
3.8 Annihilation of Particle and Antiparticle . . . . . . . . . . . .
3.9 Biographical Notes . . . . . . . . . . . . . . . . . . . . . . . . .
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239 4 Summary: The Feynman Rules of QED . . . . . . .
4.1 The Feynman Rules of QED in Momentum Space
4.2 The Photon Propagator in Different Gauges . . . .
4.3 Biographical Notes . . . . . . . . . . . . . . . . . .
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253 5 The Scattering Matrix in Higher Orders . . . .
5.1 Electron–Positron Scattering in Fourth Order
5.2 Vacuum Polarization . . . . . . . . . . . . .
5.3 Self-Energy of the Electron . . . . . . . . .
5.4 The Vertex Correction . . . . . . . . . . . .
5.5 Biographical Notes . . . . . . . . . . . . . . .
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