Walter Greiner, Joachim Reinhardt-Quantum Electrodynamics (2009).pdf

Walter Greiner, Joachim Reinhardt-Quantum Electrodynamics (2009).pdf

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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 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. Violations are liable to prosecution under the German Copyright Law. 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. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 6 12 20 21 23 29 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 96 131 139 150 157 177 190 239 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 244 248 253 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 255 257 291 298 327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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