Springer-Verlag Berlin Heidelberg GmbH

W. Greiner 1. Reinhardt FIELD QUANTIZATION Springer-Verlag Berlin Heidelberg GmbH

Greiner Quantum Mechanics An Introduction 3rd Edition Greiner Quantum Mechanics Special Chapters Greiner. MUller Quantum Mechanics Symmetries 2nd Edition Greiner Relativistic Quantum Mechanics Wave Equations Greiner Mechanics I (in preparation) Greiner Mechanics II (in preparation) Greiner Electrodynamics (in preparation) Greiner Neise. Stocker Thermodynamics and Statistical Mechanics Greiner Reinhardt Field Quantization Greiner Reinhardt Quantum Electrodynamics 2nd Edition Greiner. Schafer Quantum Chromodynamics Greiner Maruhn Nuclear Models Greiner MUller Gauge Theory of Weak Interactions 2nd Edition

Walter Greiner Joachim Reinhardt FIELD QUANTIZATION With a Foreword by D. A. Bromley With 46 Figures, and 52 Worked Examples and Problems Springer

Professor Dr. Walter Greiner Dr. Joachim Reinhardt Institut fiir Theoretische Physik der Johann Wolfgang Goethe-Universitiit Frankfurt Postfach 1119 32 D-60054 Frankfurt am Main Germany Street address: Robert-Mayer-Strasse 8-1 O D-60325 Frankfurt am Main Germany email: greiner@th.physik.uni-frankfurt.de (W. Greiner) jr@th.physik.uni-frankfurt.de (J. Reinhardt) Title of the original German edition: Theoretische Physik, Ein Lehr- und Obungsbuch, Band 7a: Feldquantisierung Verlag Harri Deutsch, Thun 1986, 1993 Library of Congress Cataloging-in-Publication Data. Greiner, Walter, 1935- [Feldquantisierung. English] Field quantization 1 Walter Grei ner, Joachim Reinhardt: with a foreword by D. A. Bromley. p.cm. lncludes bibliographical references and index. ISBN 3-540-59179-6 (pbk.: alk. paper). 1. Quantum field theory. 2. Path integrals. 1. Reinhardt, J. (Joachim), 1952-. Il. Ti tie. QC174.45.G72 1996 530.1 '43-dc20 95-45449 ISBN 978-3-540-78048-9 ISBN 978-3-642-61485-9 (ebook) DOI 10.1007/978-3-642-61485-9 This work is subject to copyright. AII 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 thereofis permitted only under the provisions ofthe German Copyright Law ofseptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are Iiable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1996 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.

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

VI Foreword to Earlier Series Editions 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 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 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 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. 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 Chromo dynamics 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. The present volume of lectures covers the subject of field quantization which lies at the heart of many developments in modern theoretical physics. The observation by Planck and Einstein that a classical field theory - electrodynamics - had to be augmented by corpuscular and nondeterministic aspects stood at the cradle of quantum theory. At around 1930 it was recognized that not only the radiation field with its photons but also matter fields, e.g. the electrons, can be described by the same procedure of "second quantization". Within this formalism, matter is represented by operator-valued fields which are subject to certain (anti-) commutation relations. In this way one arrives at a theory describing systems of several particles (field quanta) which in particular provides a very natural way to formulate the creation and annihilation of particles. Quantum field theory has become the language of modern theoretical physics. It is used in particle and high-energy physics, but also

VIII Preface the description of many-body systems encountered in solid-state, nuclear and atomic physics make use of the methods of quantum field theory. The aim of this volume is to present an introduction to the techniques of field quantization on their own, putting less emphasis on their application to specific physical models. The book was conceived as a companion to the volume Quantum Electrodynamics in this series, which concentrates on the evaluation of results and applications without working out the field-theoretical foundations. The two books, however, do not rely on each other and can be studied independently. As a prerequisite for the present volume the reader only needs some familiarity with quantum mechanics, both in the nonrelativistic and in the relativistic form. In the first two chapters we review the classical and quantum mechanical description of oscillating systems with a finite number of degrees of freedom, followed by an exposition of the classical theory of fields. Here the emphasis is laid on the important topic of symmetries and conservation laws. The main part of the book deals with the method of "canonical quantization" which is applied, step by step, to various types of fields: the nonrelativistic Schrodinger field, and the relativistic fields with spin 0, 1/2, and 1. The subsequent chapters treat the topic of interacting quantum fields and show how perturbation theory can be employed to systematically derive observable quantities, in particular the scattering matrix. This is complemented by a chapter on discrete symmetry transformations, which play an important role for the development of models of elementary particles. The last part of the book contains the description of a formalism which in a way stands in competition with the "seasoned" method of canonical quantization. The method of quantization using path integrals, which essentially is equivalent to the canonical formalism, has gained increasing popularity over the years. Apart from their elegance and formal appeal path-integral quantization and the related functional techniques are particulary well suited to implement conditions of constraint, which is necessary for the treatment of gauge fields. Nowadays any student of theoretical physics must be familiar with both the canonical and the path-integral formalism. We repeat that this book is not intended to provide an exhaustive introduction to all aspects of quantum field theory. In particular, the important topic of renormalization is covered only in passing. Our main goal has been to present a detailed and comprehensible introduction to the methods of field quantization themselves, leaving aside most applications. We hope to attain this goal by presenting the subject in considerable detail, explaining the mathematical tools in a rather informal way, and by including a large number of examples and worked exercises. We would like to express our gratitude to Dr. Ch. Hofmann for his help in proofreading the German edition of the text. For the typesetting of the English edition we enjoyed the help of Mr. M. Bleicher. Once again we are pleased to acknowledge the agreeable collaboration with Dr. H.J. KOlsch and his team at Springer-Verlag, Heidelberg. The English manuscript was copy edited by Dr. Victoria Wicks. Frankfurt am Main, December 1995 Walter Greiner Joachim Reinhardt

Contents Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems..... 3 1.1 Introduction... 3 1.2 Classical Mechanics of Mass Points........................ 4 1.3 Quantum Mechanics: The Harmonic Oscillator.............. 6 1.3.1 The Harmonic Oscillator........................... 8 1.4 The Linear Chain (Classical Treatment).................... 10 1.5 The Linear Chain (Quantum Treatment)... 18 2. Classical Field Theory... 31 2.1 Introduction... 31 2.2 The Hamilton Formalism... 34 2.3 Functional Derivatives... 36 2.4 Conservation Laws in Classical Field Theories............... 39 2.5 The Generators of the Poincare Group... 49 Part II. Canonical Quantization 3. Nonrelativistic Quantum Field Theory... 57 3.1 Introduction... 57 3.2 Quantization Rules for Bose Particles...................... 58 3.3 Quantization Rules for Fermi Particles..................... 65 4. Spin-O Fields: The Klein-Gordon Equation......... 75 4.1 The Neutral Klein-Gordon Field... 75 4.2 The Charged Klein-Gordon Field... 91 4.3 Symmetry Transformations............................... 95 4.4 The Invariant Commutation Relations... 100 4.5 The Scalar Feynman Propagator... 106 4.6 Supplement: The.1 Functions... 109 5. Spin-~ Fields: The Dirac Equation... 117 5.1 Introduction... 117 5.2 Canonical Quantization of the Dirac Field... 123

x Contents 5.3 Plane-Wave Expansion of the Field Operator... 124 5.4 The Feynman Propagator for Dirac Fields... 132 6. Spin-I Fields: The Maxwell and Proca Equations... 141 6.1 Introduction... 141 6.2 The Maxwell Equations... 141 6.2.1 The Lorentz Gauge... 144 6.2.2 The Coulomb Gauge... 144 6.2.3 Lagrange Density and Conserved Quantities... 145 6.2.4 The Angular-Momentum Tensor... 148 6.3 The Proca Equation... 152 6.4 Plane-Wave Expansion of the Vector Field... 154 6.4.1 The Massive Vector Field... 154 6.4.2 The Massless Vector Field... 156 6.5 Canonical Quantization of the Massive Vector Field... 158 7. Quantization of the Photon Field... 171 7.1 Introduction... 171 7.2 The Electromagnetic Field in Lorentz Gauge... 172 7.3 Canonical Quantization in the Lorentz Gauge... 176 7.3.1 Fourier Decomposition of the Field Operator... 177 7.4 The Gupta-Bleuler Method... 180 7.5 The Feynman Propagator for Photons... 185 7.6 Supplement: Simple Rule for Deriving Feynman Propagators.. 188 7.7 Canonical Quantization in the Coulomb Gauge... 196 7.7.1 The Coulomb Interaction... 200 8. Interacting,!uantunn Fields... 211 8.1 Introduction... 211 8.2 The Interaction Picture... 211 8.3 The Time-Evolution Operator... 215 8.4 The Scattering Matrix... 219 8.5 Wick's Theorem... 225 8.6 The Feynman Rules of Quantum Electrodynamics... 233 8.7 Appendix: The Scattering Cross Section... 267 9. The Reduction Fornnalisnn... 269 9.1 Introduction... 269 9.2 In and Out Fields... 270 9.3 The Lehmann-Kallen Spectral Representation... 278 9.4 The LSZ Reduction Formula... 282 9.5 Perturbation Theory for the n-point Function... 290 10. Discrete Symmetry Transformations... "... 301 10.1 Introduction... 301 10.2 Scalar Fields... 301 10.2.1 Space Inversion... 301 10.2.2 Charge Conjugation... 305 10.2.3 Time Reversal... 306

Contents XI 10.3 Dirac Fields... 311 10.3.1 Space Inversion... 312 10.3.2 Charge Conjugation... 313 10.3.3 Time Reversal... 315 10.4 The Electromagnetic Field... 318 10.5 Invariance of the S Matrix... 324 10.6 The CPT Theorem... 326 Part III. Quantization with Path Integrals 11. The Path-Integral Method.... 337 11.1 Introduction... 337 11.2 Path Integrals in Nonrelativistic Quantum Mechanics... 337 11.3 Feynman's Path Integral... 343 11.4 The Multi-Dimensional Path Integral... 350 11.5 The Time-Ordered Product and n-point Functions... 356 11.6 The Vacuum Persistence Amplitude W[J)... 360 12. Path Integrals in Field Theory.... 365 12.1 The Path Integral for Scalar Quantum Fields... 365 12.2 Euclidian Field Theory... 371 12.3 The Feynman Propagator... 375 12.4 Generating Functional and Green's Function... 380 12.5 Generating Functional for Interacting Fields... 384 12.6 Green's Functions in Momentum Space... 391 12.7 One-Particle Irreducible Graphs and the Effective Action... 400 12.8 Path Integrals for Fermion Fields... 408 12.9 Generating Functional and Green's Function for Fermion Fields...................................... 419 12.10 Generating Functional and Feynman Propagator for the Free Dirac Field... 421 Index... :... 433

Contents of Examples and Exercises 1.1 Normal Coordinates... 15 1.2 The Linear Chain Subject to External Forces.................... 20 1.3 The Baker-Campbell-Hausdorff Relation....................... 27 2.1 The Symmetrized Energy-Momentum Tensor... 47 2.2 The Poincare Algebra for Classical Fields....................... 51 3.1 The Normalization of Fock States... 68 3.2 Interacting Particle Systems: The Hartree-Fock Approximation.... 69 4.1 Commutation Relations for Creation and Annihilation Operators.. 83 4.2 Commutation Relations of the Angular-Momentum Operator...... 85 4.3 The Field Operator in the Spherical Representation.............. 86 4.4 The Charge of a State... 94 4.5 Commutation Relations Between Field Operators and Generators.. 99 4.6 The FUnction Lh(x - y) for Equal Time Arguments... 105 5.1 The Symmetrized Dirac Lagrange Density... 121 5.2 The Symmetrized Current Operator... 134 5.3 The Momentum Operator...,... 135 5.4 Helicity States... 136 5.5 General Commutation Relations and Microcausality... 138 6.1 The Lagrangian of the Maxwell Field... 149 6.2 Coupled Maxwell and Dirac Fields... 150 6.3 Fourier Decomposition of the Proca Field Operator... 160 6.4 Invariant Commutation Relations and the Feynman Propagator of the Proca Field... 167 7.1 The Energy Density of the Photon Field in the Lorentz Gauge... 175 7.2 Gauge Transformations and Pseudo-photon States... 183 7.3 The Feynman Propagator for Arbitrary Values of the Ga.uge Parameter (... 189 7.4' The Transverse Delta FUnction... 202 7.5 General Commutation Rules for the Electromagnetic Field... 205 8.1 The Gell-Mann-Low Theorem... 220 8.2 Proof of Wick's Theorem... 231 8.3 Disconnected Vacuum Graphs... 243 8.4 M!illler Scattering and Compton Scattering... 245 8.5 The Feynman Graphs of Photon-Photon Scattering... 249 8.6 Scalar Electrodynamics... 251 8.7 4 Theory... 261 9.1 Derivation of the Yang-Feldman Equation... 276 9.2 The Reduction Formula for Spin-! Particles... 288

XIV Contents of Examples and Exercises 9.3 The Equation of Motion for the Operator U(t)... 294 9.4 Green's Functions and the S Matrix of 4 Theory... 295 10.1 The Operators P and C for Scalar Fields... 309 10.2 The Classification of Positronium States... 320 10.3 Transformation Rules for the Bilinear Covariants... 330 10.4 The Relation Between Particles and Antiparticles... 331 11.1 The Path Integral for the Propagation of a Free Particle... 345 11.2 Weyl Ordering of Operators... 347 11.3 Gaussian Integrals in D Dimensions... 353 12.1 Construction of the Field-Theoretical Path Integral... 368 12.2 Series Expansion of the Generating Functional... 386 12.3 A Differential Equation for W[J]... 387 12.4 The Perturbation Series for the 4 Theory... 392 12.5 Connected Green's Functions... 398 12.6 Grassmann Integration... 415 12.7 Yukawa Coupling... 424